ci<o2>/.3/35tik. Q'l'2 & 
 Trinity Coileg^ Library 
 Durham, N. C. 
 
Digitized by the Internet Archive 
 in 2016 with funding from 
 Duke University Libraries 
 
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Engineering Science Series 
 
 ALTERNATING CURRENTS AND ALTERNATING 
 CURRENT MACHINERY 
 
ENGINEERING SCIENCE SERIES 
 
 EDITED BY 
 
 DUGALD C. JACKSON, C.E. 
 
 Professor of Electrical Engineering 
 Massachusetts Institute of Technology 
 Fellow and Past President A.I.E.E. 
 
 EARLE R. HEDRICK, Ph.D. 
 
 PK0FES80B OF MATHEMATICS, UNIVERSITY OF M I8SOUP.I 
 Member A.S.M.E. 
 
ALTERNATING CURRENTS AND 
 ALTERNATING CURRENT 
 MACHINERY 
 
 NEW EDITION, REWRITTEN AND ENLARGED 
 
 BY 
 
 UUGALD C. JACKSON, C.E. 
 
 PROFESSOR OF ELECTRICAL ENGINEERING, MASSACHUSETTS INSTITUTE OF TECHNOLOGY 
 PAST-PRESIDENT AND FELLOW OF THE AMERICAN INSTITUTE OF ELECTRICAL 
 ENGINEERS, PAST-PRESIDENT OF THE SOCIETY FOR THE PRO- 
 MOTION OF ENGINEERING EDUCATION, ETC. 
 
 AND 
 
 JOHN PRICE JACKSON, M.E., Sc.D. 
 
 COMMISSIONER OF LABOR AND INDUSTRY, COMMONWEALTH OF PENNSYLVANIA, DEAN 
 OF THE SCHOOL OF ENGINEERING, PENNSYLVANIA STATE COLLEGE, FELLOW 
 OF THE AMERICAN INSTITUTE OF ELECTRICAL ENGINEERS, MEMBER OF 
 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS, ETC. 
 
 
 Neto Iforfe 
 
 THE MACMILLAN COMPANY 
 1922 
 
 All rights reserved 
 
Copyright, 1896, 1913, 
 
 By THE MACMILLAN COMPANY. 
 
 Set up and electrotyped. New edition. Published September, 191 
 Reprinted July, 1914; October, 1917. 
 
 NortoootJ J9rfss 
 
 J. S. Cushing Co. — Berwick & Smith Co. 
 Norwood, Mass., U.S.A. 
 
2-A 3/3 3 
 
 
 ; 3 (X^ 
 
 PREFACE 
 
 This is a new edition of the authors’ book on Alternating 
 Currents and Alternating Current Machinery which was first 
 published in 1896, and which has now been rewritten and 
 greatly extended. The former editions met so favorable a re- 
 sponse that the authors cannot help but believe that it served 
 an important place in connection with the development of the 
 methods of teaching the subject of alternating currents and 
 their applications ; and they earnestly hope that the new edi- 
 tion will be equally useful. 
 
 In this edition are maintained the well-known features of 
 the earlier book in which were worked out the characteristics 
 of electric circuits, their self-induction, electrostatic capacity, 
 reactance and impedance, and the solutions of alternating cur- 
 rent flow in electric circuits in series and parallel. More atten- 
 tion is paid to the transient state in electric circuits than was 
 the case in the original edition. A considerable amount of 
 related matter has been introduced in respect to vectors, com- 
 plex quantities, and Fourier’s series which the authors believe 
 will be useful to students and engineers. The treatment of 
 power and power factor has been given great attention, and a 
 full chapter is now allotted to the hysteresis and eddy current 
 losses which are developed in the iron cores of electrical ma- 
 chinery. More space and more complete treatment have been 
 assigned to synchronous machines and to asynchronous motors 
 and generators. The treatment of the self-inductance and mutual 
 inductance of line circuits and skin effect in electric conductors 
 which was found in the old book has been extended, and it has 
 been supplemented by a treatment of the electrostatic capacity 
 of lines and the influences of distributed resistance, inductance, 
 and capacity. 
 
 In all these features as well as in others the book has been 
 
Y1 
 
 PREFACE 
 
 brought up to the requirements of present day teaching. The 
 book covers the ground that is needed to give a fairly complete 
 course in the essential elements of alternating currents and 
 their applications to machinery. It is longer than will be 
 needed in the courses in many of the engineering schools, but 
 chapters may be selected so as to meet the requirements of 
 each school. Thus, if an abbreviated course is required, Chap- 
 ters V and XIII may be omitted, and if the course does not go 
 into electrical machinery, the first eight chapters only need be 
 considered. In those colleges where the students have a course 
 in alternating current measurements previous to their entering 
 upon the subject of alternating current machinery the second 
 chapter may be omitted during the study of this text, but the 
 chapter will be useful in connection with the course in alter- 
 nating current measurements. The book is also serviceable in 
 connection with instruction in the electrical transmission of 
 power. For this purpose the authors suggest more particularly 
 the use of the portions of Chapter I which deal with complex 
 quantities and Fourier’s series and the whole of Chapters IV, 
 V, VI, VII, VIII, XIII. These are all particularly related to 
 matters which are fundamental not only to electrical machinery 
 but also to the transmission of power and the other branches 
 in which alternating currents are usefully applied. 
 
 The authors have endeavored to make the phraseology simple 
 and to illustrate the applications of the principles by examples 
 drawn from the best practice in the art. As in the first edi- 
 tion, original methods have been introduced in various instances 
 to gain simple paths to results, every effort being made to pre- 
 sent a full physical conception of phenomena to the reader's 
 mind. The mathematics used are merely logical means for 
 accomplishing the end, and are by no means to be considered 
 from any other standpoint. It has been sought by the authors 
 to avoid either the error of presenting unnecessary formulas or, 
 on the other hand, of giving results without supporting them 
 on reasons, both of which are fatal to a student’s effective prog- 
 ress, since they leave him without a true physical conception of 
 the phenomena studied. It has been the authors’ aim to pro- 
 duce a text which avoids the fault of being cursorily descriptive 
 and at the same time avoids the reasonable objections to a 
 treatise which is unrelieved mathematics. 
 
PREFACE 
 
 vii 
 
 The original edition of the book was noticeable for utilizing 
 the word “ active ” for representing the true power component 
 of voltage or current, and that practice, which has now been 
 approved by the Standards Committee of the American Insti- 
 tute of Electrical Engineers, is followed in this edition, while 
 the component at right angles is in this edition called either 
 the quadrature component or the reactive component, the latter 
 phraseology being also now in accordance with the recommen- 
 dations of the Standards Committee of the American Institute 
 of Electrical Engineers. In various respects the book contains 
 original treatments which will be obvious, but the authors have 
 made an effort to set forth the best treatment of each of the 
 problems of alternating currents and therefore have brought 
 under contribution most of the literature and practice in the 
 art in the course of their selections of methods of presentation. 
 
 The examples which are introduced at intervals throughout 
 the text for the purpose of illustrating the way in which the 
 principles may be utilized will be useful to teachers and stu- 
 dents. Footnotes referring mostly to correlated articles in the 
 book itself are numerous. The many historical footnotes which 
 were in the former editions have been mostty omitted from this 
 edition, since the early literature has been now far outstripped 
 and is no longer an essential source of knowledge for the under- 
 graduate student. 
 
 The book is placed in the hands of electrical engineers in the 
 belief that it will prove valuable as a textbook in the engineer- 
 ing schools, and also prove of service as a reference book with 
 respect to the principles used in the numerous applications of 
 alternating currents to practical purposes which now make so 
 large a part of electrical engineering. 
 
 The authors wish to express their sincere thanks to Mr. H. 
 P. Wood, Professor of Electrical Engineering at the Georgia 
 School of Technology, Mr. Charles L. Kinsloe, Professor of 
 Electrical Engineering at the Pennsylvania State College, and 
 Mr. Charles W. Green, Instructor in Electrical Engineering at 
 the Massachusetts Institute of Technology, who have read the 
 proof and made many suggestions. It is not to be expected 
 that a book of the extent and importance of this, the manu- 
 script of which is written and the proof read in the midst of 
 the exacting employments of men in important duty in engi- 
 
PREFACE 
 
 viii 
 
 neering schools, can be entirely free from ambiguities and 
 errors ; but the authors venture the hope that any errors or 
 ambiguities which have escaped their vigilance are few. They 
 will greatly appreciate having brought to their attention any 
 which readers may find. 
 
CONTENTS 
 
 CHAPTER PAGE 
 
 I. The Voltage developed by Alternators. The Use of 
 
 Vectors and the Complex Quantity. Fourier’s Series 1 
 
 II. Elementary Statements concerning Transformers and 
 
 Measuring Instruments 61 
 
 III. Armature and Field Windings for Alternators. Ma- 
 
 terials of Construction 77 
 
 IV. Self-induction, Electrostatic Capacity, Reactance, 
 
 and Impedance ......... 128 
 
 V. The Use of Complex Quantities Extended . . . 241 
 
 VI. Solution of Circuits. Application of Graphical and 
 
 Analytical Methods 257 
 
 VII. Power. Power Factor 312 
 
 VIII. Polyphase Circuits and the Measurement of Power 
 
 therein 359 
 
 IX. Hysteresis and Eddy Current Losses .... 400 
 
 X. Mutual Induction. Transformers 443 
 
 XI. Synchronous Machines. Alternators, Motors, Rotary 
 
 Converters, Frequency Changers 597 
 
 XTI. Asynchronous Motors and Generators .... 784 
 XIII. Self-inductance, Mutual Inductance, and Electro- 
 static Capacity of Parallel Wires. Skin Effect. 
 Effects of Distributed Resistance, Inductance, and 
 Capacity. Corona 897 
 
 INDEX 955 
 
 IX 
 
ALTERNATING CURRENTS 
 
 CHAPTER I 
 
 THE VOLTAGE DEVELOPED BY ALTERNATORS. THE USE OP 
 VECTORS AND THE COMPLEX QUANTITY. FOURIER’S 
 SERIES 
 
 1. Direct and Alternating Currents obey the Same Laws. — A 
 
 deeply rooted belief seems to have been cultivated in the 
 minds of many that phenomena connected with the flow of 
 direct electric currents and of alternating electric currents are 
 almost entirely unrelated. This popular idea, however, is 
 erroneous ; the principles which relate to the flow of electric 
 currents, whether direct or alternating, and which are applied 
 to the design and construction of machines and circuits, are 
 one and the same. 
 
 When Oersted, in 1820, made known his signal discovery 
 that an electric current exerts a magnetic influence in the 
 space around it, the foundation was begun for our knowledge 
 of the laws of the flow of alternating currents. Within a 
 dozen or fifteen years thereafter much knowledge of the electric 
 current had been thrashed out experimentally by men like 
 Ampere, Arago, Faraday, Henry, and others. The last two 
 named laid the finishing stone on the foundation by searching 
 out and making known the laws of electro-magnetic induction. 
 These basic laws, developed early in the nineteenth century, 
 apply with equal force to continuous, pulsating, and alternat- 
 ing currents. 
 
 In dealing with alternating currents, several variables enter 
 into the problem which make it impracticable to use, without 
 modification, the results gained in the study of direct currents. 
 But, as already said, the same fundamental laws control the 
 phenomena of both, and if care is taken to apply these with 
 
 1 
 
 B 
 
o 
 
 ALTERNATING CURRENTS 
 
 due regard to tlie limiting conditions, it will be found that 
 the subject of alternating currents may be clearly grasped and 
 be reduced almost to the simplicity of direct-current work. 
 
 The design and operation of alternating-current machinery 
 may differ considerably from those of direct-current apparatus 
 in certain particulars because the commercial requirements are 
 materially different, but the same principles fundamentally 
 apply in both, and in the matter of good workmanship and sub- 
 stantial construction there should be no difference in the two 
 classes of machines. 
 
 2. Alternating Voltage and Current. — An alternating voltage 
 or electric pressure, as the term indicates, is an uninterrupted, 
 rapid succession of electric pressure impulses which are alter- 
 nately in opposite di- 
 rections, These voltage 
 impulses produced by or- 
 dinary commercial alter- 
 nating-current machines 
 reverse direction many 
 times per second * and 
 may be very irregular in 
 form, though the typical 
 and simplest form is that 
 
 Fig. 1. — Two Loops of a Sinusoidal Alternating 0 £ a s p luso i ( ] - Figure 1 
 Voltage or Current Curve. 3 
 
 shows the plot of an alter- 
 nating voltage curve of sinusoidal form. The ordinates of the 
 curve represent voltages, while the abscissas represent time. 
 The abscissas of such a curve are ordinarily marked in degrees, 
 or in fractions of 7r; the base of the two loops which constitute 
 a complete cycle of a sine wave being given the assigned value 
 of 360°, or 2 7r radians. The ordinates may be scaled in volts. 
 The alternate loops are drawn above and below the zero or 
 datum line to indicate reversal of voltage. 
 
 An alternating electric current flows whenever an alternating 
 voltage is applied in a closed electric circuit of fixed constants. 
 This current is not necessarily proportional at each instant to 
 the voltage ordinate at the same instant, as will be shown later, f 
 A curve representing the form of an alternating current wave 
 may be drawn exactly as was indicated for an alternating volt- 
 * Art. 10. t Chap. IV. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 3 
 
 age, but the ordinates represent amperes instead of volts. The 
 simplest typical form of current curve is also sinusoidal, but this 
 is very seldom attained in commercial practice; while on the 
 other hand highly irregular current curves are quite common. 
 
 The successive loops in either current or voltage curves are 
 usually assumed to be of identical form and dimensions as long 
 as the conditions of the circuit are constant.* 
 
 3. Period and Frequency of an Alternating Current. — The 
 time in seconds, T \ required to pass through a complete cycle 
 (that is, two loops of a curve) is called the Period of an alter- 
 nating electric current or voltage. The number of periods in a 
 second is called the Frequency of the current or voltage (this term 
 was adopted by the Paris Electrical Congress). Usually an al- 
 ternating current or voltage is designated by its effective f value 
 and frequency . It is common, however, to use the number of 
 half -periods, or the number of Alternations, in a minute, instead 
 of the frequency. In this case, the number of alternations is 
 equal to 2 x 60 x the frequency. Example : a current with a 
 frequency of 60 periods per second makes 7200 alternations per 
 minute. The term Periodicity is sometimes used for frequency. 
 
 4. Effective Alternating Voltages and Currents. — The heating 
 effect or power activity of a current flowing through electrical 
 resistance varies directly as the square of the current. This 
 was proved by Joule in 1841, and the statement is often called 
 Joule's law. Putting the statement of the law in symbols gives 
 
 where H is heat measured in calories ; /, current measured in 
 amperes ; T, the time measured in seconds ; R , resistance 
 measured in ohms ; and P is the power expended in the circuit, 
 measured in watts. 
 
 Alternating currents and voltages are constantly varying in 
 intensity, and the heating effect of an alternating current is 
 found by summing up instantaneous values through a com- 
 plete cycle, as follows : 
 
 in which i represents instantaneous values of current ; and 
 the power is 
 
 . 24 I 2 R T , and the power is P = I 2 R , 
 
 * Art. 12. 
 
 t Art. 4. 
 
4 
 
 ALTERNATING CURRENTS 
 
 The expression v t X Pdt obviously represents the average of 
 
 the squared instantaneous currents , and is equal to the average 
 ordinate of a curve plotted on the same base as the original 
 alternating-current curve, but with each ordinate of a length 
 equal to the square of the corresponding ordinate of the origi- 
 nal curve. 
 
 If the average value or ordinate of an alternating current is 
 
 C T 
 
 squared (that is, [1/Ti idt ] 2 ), the result is not the same as 
 
 taking the average of squared ordinates or instantaneous 
 values ; and the amount of the difference depends on the 
 
 form of the current curve. 
 It is evident that this must 
 be true, since squaring the 
 larger instantaneous values 
 produces more than a pro- 
 portional effect upon the 
 resulting average. It is 
 well known that the average 
 of the squares of a series 
 of numerical values is 
 always larger than the 
 square of the average of the 
 values, unless the numeri- 
 cal values are all equal to 
 each other. 
 
 Figure 2 shows a sine 
 curve and its curve of squared instantaneous ordinates desig- 
 nated respectively A, B , (7, J), E, and A , _P, (7, Q , E. The 
 height of the line EG represents the average value of the 
 squared current ordinates, while the height of the line A' T 
 represents the square of the average of the current ordinates. 
 
 When a sinusoidal current flows through electrical resistance, 
 the average power expended in producing heat is proportional to 
 the average of the squares of instantaneous values of current, 
 that is, to the height of the line EG ; and the square root of this 
 ordinate is equal in amperes to a direct current which (flowing 
 
 P Q 
 
 Fig. 2. — Curve of Squared Instantaneous 
 Ordinates. 
 
T3E VOLTAGE DEVELOPED BY ALTERNATORS 
 
 b 
 
 through the same resistance) will have the same heating effect 
 as the alternating current under consideration. This square 
 root of the average of the squared instantaneous values, that is, 
 v/Av. i l or VAv. e 2 , is called the Effective value of the current 
 or voltage. 
 
 Proh. 1. The resistance of a circuit is 20 ohms and 500 watts 
 are expended in heating it. What is the effective current in the 
 circuit? 
 
 Proh. 2. Four circuits in parallel and having 1, 2, 4, and 5 
 ohms resistance respectively, absorb 400, 200, 100, and 80 watts. 
 What is the effective current in each branch ? 
 
 Prob. 3. What are the effective voltages across the terminals 
 of the circuits in problems 1 and 2, supposing that there is no 
 live counter- voltage in the circuits? 
 
 Prob. 4. A circuit of 10 ohms resistance is in series with the 
 four parallel circuits of problem 2, and 2000 watts are expended 
 in the sj^stem. What are the effective currents and voltages 
 in the series portion and in the parallel portion of the whole 
 circuit, supposing no live counter-voltages are present ? 
 
 5. Value of Effective Current and Voltage in Terms of the 
 Maximum when the Curve is Sinusoidal. — If the current curve 
 is of the sine form, it may be expressed as follows : 
 
 i = i m sin a, 
 
 where i is the instantaneous value of the current for any angle 
 a and i m is the maximum instantaneous value. To obtain the 
 average heating effect of this current when flowing through 
 a resistance, square both sides of the equation, multiply through 
 by i?, and find by integration the area of one loop or one-half 
 period of the squared curve. The area thus found divided by 
 the base nr gives the average of the squared ordinates. The 
 expression may conveniently take the form, 
 
 (Av. « 2 )i? = — i} n P sin 2 ad a, 
 
 7 T *A ) 
 
 7?; 2 
 
 from which, ( Av. i 2 ) R = — -• 
 
G 
 
 ALTERNATING CURRENTS 
 
 Dividing by R, Av. 
 
 9*2 
 
 or 
 
 V Av. i 2 — ~~= — A07*„ 
 
 V2 
 
 The value of VAv. i 2 is called the Effective value of the 
 alternating current, as has already been pointed out. 
 
 The effective value of an alternating current is the square root 
 of the average of the squared instantaneous values of the current. 
 This is correct whether or not the current is of sinusoidal form. 
 The foregoing equations show that, when the current is sinu- 
 soidal in form, there is a fixed relation between the effective 
 and maximum values of the current and that the effective 
 
 value is .707 ( = — ] times the maximum value. 
 
 V V2 J 
 
 In the case of a sinusoidal pressure (voltage) curve the for- 
 mula may be written, 
 
 e - e m sin «, 
 
 where e is the instantaneous value of the voltage for any angle a 
 and e m is the maximum instantaneous value. By writing the 
 integral for the heating effect as before, a similar result is ob- 
 tained, or 
 
 Av. e 2 _ Mj 
 2 R 
 
 and 
 
 R 
 
 v/Av. e 2 = 
 
 V2 
 
 = -707 e m . 
 
 Rffective voltage is the square root of the average of the squared 
 instantaneous values of the voltage. When the form of the 
 voltage wave is sinusoidal, the effective voltage is equal to 
 
 .707 f = times the maximum value of the voltage. 
 
 V V2 / 
 
 The average value of the ordinates of a sine curve may be 
 found from the expression 
 
 Av. y = j sin ada, 
 7 r 
 
 where y is the instantaneous ordinate and y m the maximum 
 ordinate of the curve ; and therefore 
 
 Av. y = — .037 
 
 y mi 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 7 
 
 2 
 
 or the average value of the instantaneous ordinates is — ( = .637) 
 
 7T 
 
 times the maximum ordinate. From this it is seen that 
 VAv. y 1 — (Av. «/), 
 
 or VAv. ?/ 2 = 1.11 (Av. y'). 
 
 Therefore the ratio of effective current or voltage to the average 
 ordinate of the curve, in the case of sinusoidal curves, has the 
 fixed value of 1.11. 
 
 The same considerations follow for alternating current and 
 voltage waves of any form, but the ratios of the average and 
 the effective values to the maximum are different, and depend 
 upon the form of each curve considered. Alternating current 
 and voltage waves are always single-valued ; that is, a single 
 value of the ordinates corresponds to each single value of the 
 abscissas. They may be represented by the expression 
 
 x=af(u~). 
 
 Manifestly, the ratio which the average bears to the maximum 
 ordinate takes a value nearer unity for a curve that is flatter 
 than a sinusoid, and recedes farther from unity for a curve more 
 peaked than a sinusoid. 
 
 For a curve made up of alternating rectangular loops, such 
 as is shown in Fig. 17, the maximum, average, and effective 
 values are equal to each other ; but for a curve made up of 
 alternating triangular (isosceles) loops, the average value is 
 one half the maximum value, and the effective value exceeds 
 the average value in a ratio of 1.155 to 1. The ratio of the 
 effective to the average value may be quite large in the case of 
 a curve which is very peaked. This ratio is sometimes called 
 the Form factor of the curve. 
 
 Prob. 1. What are the maximum ordinates of the sinusoidal 
 voltages having effective values of 100 and 200 volts, and of 
 sinusoidal currents having the effective values of 50 and 75 
 amperes ? 
 
 Prob. 2. What are the average ordinates of voltages and 
 currents in problem 1? 
 
 Prob. 3. What are the effective and average values of a 
 
8 
 
 ALTERNATING CUR 1 i ENTS 
 
 voltage wave which is of the form of an equilateral triangle, 
 and lias an altitude of 110 volts? 
 
 Prob. 4. What are the effective and average values of a 
 voltage wave in the form of a right triangle in which the alti- 
 tude is 10 and the base 10? 
 
 Prob. 5. What are the average and effective values of a cur- 
 rent wave having the form of a semicircle with the base equal 
 to 100? That is, compute the average ordinate and the square 
 root of the average of the squared ordinates, in terms of the 
 radius. 
 
 6. Vectors representing Sinusoidal or Harmonic Voltages and 
 Currents ; Lead and Lag- — A physical quantity which may be 
 represented by the length, position, and direction of a line is 
 called a Vector quantity and the line so representing the 
 quantity is called a Vector.* Thus, two forces may be repre- 
 sented (1) in magnitude, sometimes termed Scalar value, by two 
 lines having lengths proportional to the intensities of the 
 forces ; (2) in relative angular position, or Phase, by the angle at 
 which the lines, extended if necessary, intersect ; and (3) in di- 
 rection by arrow heads placed upon the lines. In the following 
 discussions, if one or more vectors radiate from or turn upon a 
 common center, and no arrow heads are shown, it is assumed 
 that their directions are from the center outward. Vectors 
 may be combined or resolved into components by the well-known 
 laws of mechanics relating to the composition or resolution of 
 forces or velocities. Thus, the centrifugal force acting upon a 
 uniformly rotating crank may be represented by a vector rotat- 
 ing uniformly around one end as a center of revolution , and its 
 component value at any instant in the direction of some fixed 
 line (assume for convenience a vertical line passing through 
 the axis of rotation) is found by projecting the vector from 
 its position at the instant, upon that line. If the center is 
 moved (for convenience, horizontally to the right) a given dis- 
 tance for each degree of angular rotation of the vector, and 
 the instantaneous vertical projections for each degree of ad- 
 vance are plotted at the proper points along the path of the 
 center, the points thus obtained when joined form a sine curve. 
 In Fig. 3, OA is a rotating vector. OX is the initial line or 
 
 -* For a discussion of the use of vectors in graphical methods of solving elec- 
 trical problems, see Chapter VI. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 9 
 
 horizontal axis from which the angle a is measured, OY is the 
 vertical axis upon which the point A is projected as the vector- 
 rotates, and SS is a 
 sine curve produced 
 as explained above. 
 
 In the figure, OA has 
 advanced 30°, and 
 the ordinate a' of the 
 sine curve, which is 
 
 placed over the 30 Produced Thereby, 
 
 division of its base, 
 
 is made equal to OA sin 30° or Oy. Likewise, when the vector 
 has advanced to 60° the ordinate a" of the curve is equal to 
 OA sin 60°. When the vector is at 90°, the ordinate a’" is 
 equal to OA sin 90°, which equals OA\ hence the length of a 
 rotating vector required to produce a given sine curve must be 
 equal to the maximum ordinate of the curve. 
 
 By geometry it is seen that in the figure 
 
 A = Vy 2 + x 2 = A (sin 2 « + cos 2 a)^, 
 where A = OA, x = Ox — A cos a, y = Oy = A sin a ; and 
 
 tan a = ^ and tan 
 x 
 
 x 
 
 These relations express the angular position and value of a 
 Rotating vector in terms of its instantaneous rectangular com- 
 ponents with great simplicity, and are much used in the compu- 
 tations of electrical phenomena. 
 
 Rotating vectors may be combined, by the ordinary methods 
 of mechanics, with other vectors having the same speed of 
 rotation, since their relative angular positions will remain 
 always the same. Thus, in Fig. 4, OA and OB are two rotat- 
 ing vectors having the same speed or frequency, and therefore 
 at a fixed Phase difference or angular position with respect to 
 each other. Their resultant is OO, the diagonal of the paral- 
 lelogram OACB. It may be seen by inspection that the ver- 
 tical projection of OC, which is Oc , is equal at any instant to 
 A sin a + B sin (u-0)=Oa- f Ob, since BO is equal and par- 
 allel to OA and ac is therefore equal to Ob. These vectors 
 A, B, and 0 are the generators of the sine curves or harmonics 
 
10 
 
 ALTERNATING CURRENTS 
 
 A', B ' , and C . If the ordinates of A' and B' corresponding 
 to any abscissa are added together, they give the corresponding 
 
 Fig. 4. — Composition of Rotating Vectors OA and OB by Parallelogram of Forces. 
 
 ordinate of the curve C' . This follows from the construction 
 of the curves. If the vector OC is given, it can be resolved 
 
 into the component vectors OA 
 and OB with their particular phase 
 relations to OC , or it may be re- 
 solved into components at any 
 other fixed angular relations, by 
 reversing the process of construc- 
 tion. Or, if OC and one compo- 
 nent, OA, are given, the other 
 component may be found by com- 
 pleting the parallelogram of which 
 OC is the diagonal and OA one 
 side. Three or more vectors may 
 be combined by similar processes. 
 
 In the case of two vectors 90° apart the relations are very 
 simple. Thus in Fig. 5, if A and B are two such vectors, rotat- 
 ing about Z, which form the resultant C, 
 
 B 
 
 Fig. 5. — Illustration of the Relations 
 of Two Vectors 90° apart. 
 
 C= V^L 2 + B' 2 , tan 6 = —, tan 6' = 
 
 A -B 
 
 a = A sin a, and b = B sin (« -f- 90°) = B cos «. 
 
 where A, B , and C are the values of the vectors, a and b are 
 the vertical components of A and B. a is the angle of advance 
 of A, and 6 and 6' are the fixed angles AZC and CZB. Further, 
 
 c—C sin (« + 6), 
 
 and c = a + b. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 11 
 
 where a, b , and c are the instantaneous vertical components of 
 A, B, and C; from which, remembering that cos « = sin (a + 90°), 
 
 c = A sin u + B cos « ; 
 
 hence, A sin « + B cos a = (7 sin (« + 0). 
 
 Also A sin a + B cos a — C cos (« — 6 
 
 In using these expressions it is desirable to always measure a 
 in the positive (counter-clockwise) direction in order to pre- 
 vent confusion in the algebraic signs of the angles. 
 
 Since B cos a = B sin (« + 90°), it is evident that a sine 
 curve generated by a rotating vector 90° in advance of the 
 vector OA of Fig. 3 would be a curve with its positive maxi- 
 mum at a = 0°, its negative maximum at a = 180°, and its zero 
 values at a = 90°, 270°, etc. That is, this cosine curve would 
 be ahead of or in the Lead of the curve drawn in Fig. 3 by 90°, 
 and vice versa , the sine function Lags 90° behind the cosine 
 function. Also, the phase or relative angular position of a re- 
 sultant vector is between the phases of its components, and the 
 angles between a resultant and its components depend upon 
 the phase difference and the ratio of scalar values of the com- 
 ponents. 
 
 Vectors having different speeds of rotation cannot be com- 
 bined by the parallelogram of forces, as the resultant vector 
 varies in length with the time.* 
 
 Harmonic (sinusoidal) voltages and currents can evidently 
 be represented by rotating vectors and can be combined with 
 as much facility as mechanical forces or velocities. In doing 
 this'the maximum ordinate, or Amplitude, must be taken for the 
 scalar value of the vector if it is desired to determine the in- 
 stantaneous components ; but if it is desired to find the effec- 
 tive resultant of two or more voltages or currents, the effective 
 values of the components may be used. This is because 
 
 E = : and therefore the parallelograms using maximum and 
 
 effective values are similar, the former being convertible into 
 the latter by dividing its sides and diagonals by V2. Effective 
 voltages and currents may possibly not be represented as vec- 
 tors in the broadest mathematical sense, but for our purposes 
 such representation is eminently satisfactory. 
 
 * See Art. 71. 
 
12 
 
 ALTERNATING CURRENTS 
 
 K"/ 
 
 If a number of vectors are to be combined, the resultant of a 
 pair may be found and this then combined with another vector, 
 
 and so on until the final result- 
 ant is obtained. In Fig. 6 the 
 resultant of the vectors A, A', 
 A", and A'" is found by the 
 parallelograms OAa'A', 
 Oa'a"A", and Oa"a"'A'". 
 This is cumbersome and can 
 be simplified by drawing OA , 
 Aa' , a' a", and a" a'", respec- 
 tively, parallel and equal in 
 x length to the vectors A , A', 
 
 Fig. 6. Vector Addition by Parallelo- i A" 1 Bv ioillino- the 
 
 grams of Forces and Vector Polygon. . ‘ ' 
 
 free points O and a" the re- 
 sultant OR is obtained. The figure OAa'a" a"' 0 is called a 
 Vector polygon. If only two vectors are combined, giving, say, 
 OAa' , the result is called a Vector triangle. When the figure 
 shows all the vectors radiating from a center as do OA, OA', 
 OA", and OA'", it is called a Phase diagram, as the relative 
 angular positions of the vectors are directly indicated. 
 
 Prob. 1. Two current vectors differing 30° in phase have 
 values respectively of 20 and 40 amperes. What is their com- 
 bined value ? 
 
 Prob. 2. Four voltages of 25, 50, 75, and 100 are represented 
 by vectors having 0°, 30°, 60°, and 90° absolute phase positions. 
 What is the combined value of all the voltages in series, and 
 what is its angle with reference to the component vector hav- 
 ing 0° angle ? 
 
 Prob. 3. A current of 100 amperes divides into two branches 
 in such a manner that one branch current leads the main cur- 
 rent by 30°, and the second branch current lags behind the 
 main current by an angle of 60°. What are the values of the 
 two currents ? 
 
 Prob. 4. A current vector of 50 amperes is divided into two 
 rectangular components, one of which lags behind it by an 
 angle of 45°. What is the value of the second component and 
 its angular position ? 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 13 
 
 Prob. 5. A voltage vector of 100 volts is composed of two 
 components, one of which lags behind it by an angle of 10°, 
 and the other leads by an angle of 20°. What are the scalar 
 values of the components ? 
 
 Prob. 6. There are four voltages, A, B , (7, i), in series in a 
 circuit. Assuming the first to have an angle of 0° and the others 
 to be measured in angular positions with reference to it, the 
 scalar and angular values of the vectors are 10, A 0° ; 20, Z 15° ; 
 50, Z — 15° ; 30, Z — 45°. What is the resultant voltage and 
 its angular position with reference to the component having 
 Z0°? 
 
 Prob. 7. The resultant in problem 6 is to be divided into 
 two rectangular components, one on the horizontal axis (Z 0°), 
 and the other on the vertical axis. What are the scalar values 
 of the components? 
 
 7. Polar Coordinates and the Method of obtaining Effective 
 Voltages from the Polar Curve. — It is sometimes convenient to 
 plot alternating voltage and current curves to polar coordi- 
 nates ; i.e. the instantaneous values of the voltage or current 
 are laid off on radius vectors occupying the corresponding in- 
 stantaneous positions of the rotating vector thought of as gen- 
 erating the alternating function. 
 
 The effective value may be directly derived from the primary 
 polar curve, as originally shown 
 by Steinmetz,* if it is plotted on 
 polar coordinates taking 360° to 
 a complete period. This gives a 
 symmetrical curve which crosses 
 the origin at 0°, 180°, 360°, etc. 
 
 For an exact sinusoid the curve is 
 of the form shown in Fig. 7, and 
 has its maximum values, positive 
 and negative, at 90° and 270°; 
 i.e. each loop is a circle with the 
 pole on its circumference and the 
 initial line, C*A, tangent to the Fig. 7. — Polar Diagram of a Harmonic 
 circumference, the maximum Function. 
 
 * Trans. Amer. Inst. E. E., Yol. 10, p. 527 ; Elektrotechnische Zeitschrift , 
 June 20, 1890. 
 
14 
 
 ALTERNATING CURRENTS 
 
 ordinate, a , being equal to the diameter. The area of the curve 
 in this form may be shown to be directly proportional to the 
 square of the effective value of the ordinates as follows : In 
 the case of a sinusoidal curve, the polar curve has the equation 
 e = a sin a, where e is the instantaneous voltage corresponding 
 to an angular advance a. In plotting the curve, values of e are 
 laid off on the radius vectors having vectorial angles equal to 
 the corresponding values of a, and a line is drawn through 
 the points thus located (Fig. 7). Each loop of this curve, that 
 is, the part of the curve taken between « = 0° and « = 180°, or 
 a — 180° and a = 360°, is a circle, and its area is A = ^ ird 2 , 
 where d is the diameter of the circle. By the construction, d 
 is equal to a of the formula e = a sin «, and the area of a loop 
 of the curve is therefore A = \ira 2 . The effective ordinate of 
 a sinusoid has already been shown to be * 
 
 a 
 
 V2 
 
 Consequently 
 
 = .798 VA. 
 
 ' 7 r 
 
 This may be taken for most purposes as E = .8VA. 
 
 The same expression applies to any single-valued periodic 
 curve of equal positive and negative loops. The area of one 
 loop of any such polar curve is 
 
 A = r e 2 da ; 
 
 for if the curve is divided up into elementary triangles having 
 their apices at 0 and bases equal to eA «, each triangle has an 
 area approximately equal to A « x e 2 , where e is the average 
 altitude of the triangle; and the total area is 
 
 A = e 2A a. 
 
 ^0- 
 
 This, when Aa is reduced to the limit, becomes the integral 
 given above. The mean of the squared ordinates of the func- 
 tion is Av. e 2 — — f e 2 da. The effective ordinate is therefore 
 
 TT'-'O 
 
 E — VAv.« 2 = \/— f e 2 du — \/ : ^ = .798 Vi. 
 
 ' 7r*A> ' 7 r 
 
 As before, this may be taken as E = .8 VA. 
 
 * Art. 5. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 15 
 
 Figure 8 shows the voltage curve plotted to rectangular 
 coordinates, the curve of squared ordinates, and the polar curve 
 for the voltage wave of an 
 alternator which bj special 
 design gave a very distorted 
 wave. The mean ordinates 
 of the rectangular curves 
 are readily determined by 
 measuring the area by plani- 
 meter and dividing the area 
 by the length of the base. 
 
 Prob. 1. Construct the 
 polar loops of a wave of 
 voltage which, laid out to 
 rectangular coordinates, has 
 rectangular loops with a 
 base of 10 and an altitude 
 of 5. 
 
 Prob. 2. Make the same 
 construction as in problem 
 1, when the wave plotted on 
 rectangular axes is a semi- 
 circle having a diameter of 8. 
 
 Prob. 3. Find the effec- 
 
 Fig. 8. — Irregular Alternator Voltage Loop 
 plotted to Rectangular and Polar Coordi- 
 nates, and the Curve of Squared Ordinates. 
 
 tive values of the voltages in problems 1 and 2 by the method 
 explained in this article. 
 
 8. A Fundamental Conception of the Vector Quantity. — In 
 Art. 6 it has been pointed out that currents and voltages 
 may be shown in magnitude and relative phase by means of a 
 phase diagram. Thus, in Fig. 9 suppose OX to be the initial 
 line and OA', OA ", and OA !" to be voltages in series, or cur- 
 rents entering or leaving a junction, which are represented in 
 relative phase by the angular positions and in magnitude by 
 the lengths of the lines. It has been pointed out that the 
 resultant of two or more similar electrical quantities may be 
 found by treating their representative lines as vectors ; such 
 vectors may be combined by algebraically adding the vertical 
 and horizontal components of the individual lines, by which 
 
1G 
 
 ALTERNATING CURRENTS 
 
 means the vertical and horizontal components of the resultant 
 are determined. If a', V ; a", b " ; and a b'" are the hori- 
 zontal and vertical components of OA', OA", and OA'", and 
 A, B, the components of the resultant, then 
 
 or 
 
 A + B = (V + a" + a'"') + (V + b" + 5'"), 
 
 A + B = O' + 5') + ( a " + b") + (ct!" + V"). 
 
 W 
 
 o 
 
 1'lvi 
 
 
 
 X/! ' 
 
 
 
 \ 
 
 \ 
 
 \ 
 
 \\ 
 
 A 
 
 rtt s' 
 
 s' 
 
 / / ! 
 
 / 1 
 
 
 r / 
 
 ! 
 
 / " 
 J A 
 
 : 7^P 
 
 /,X''Z 
 
 O' / 
 
 i i 
 
 'r 1 1 
 
 
 
 i 
 
 9. — Illustration of Relation of Vectors 
 and tlieir Components. 
 
 In this expression there is 
 nothing to distinguish the hor- 
 izontal from the vertical com- 
 ponents except the difference 
 between the letters A or a and 
 the letters B or b ; or, in gen- 
 eral, there is nothing to indi- 
 cate the angular positions of 
 the components, or of the lines 
 represented by them, with 
 reference to the initial line ; 
 and the ordinary algebraic 
 processes afford no convenient 
 means for giving these indica- 
 tions. To fully indicate the magnitude and position of a line 
 by its rectangular components, we must abandon the methods 
 of algebra for geometric processes. Therefore we may con- 
 sider, for the moment, that tire compo- 
 nents t and u of the vector A (Fig. 10) 
 both lie on the initial line OX , but in 
 order that t and u may determine the 
 vector A, u must be vertical, that is, 
 rotated 90°.* To indicate such a ro- 
 tation, a prefix such as j may be used. 
 
 Then A will be represented in mag- 
 nitude and angular position by the Fig 
 expression 
 
 t x 
 
 10. — Single Vector and 
 Components. 
 
 A= t +ju, 
 
 where the sole function of the letter j is to indicate that the 
 component u stands 90° from the initial line and the addition is 
 
 * The letter denoting a vector will be written with a vinculum, thus, A, to dis- 
 tinguish it from its scalar value. When its components are on the horizontal or 
 vertical they will have this designation omitted. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 17 
 
 geometric; that is, the joint effect of two vectors in the direction of 
 their resultant is equal to the effect of their resultant. The effect 
 of t and ju is equal to the effect of A. The expression t A ju 
 does not indicate 
 arithmetical addition, 
 hut represents the 
 combination of the ef- 
 fects of t and u. It 
 is sometimes written 
 t , ju. As ju is posi- 
 tive, it is said to have 
 rotated u ahead 90°; 
 
 — ju would indicate 
 that u had been ro- 
 tated 90° in a nega- 
 tive direction, or that 
 u is measured down- 
 wards (the negative 
 
 direction) from the 
 
 . . 1 . . Fig. 11. — Illustration of the Effect of the Operator j. 
 
 origin. It t and ju 
 
 are both negative, they are both measured in the negative 
 direction ; hence, if t +ju be multiplied by — 1, there results 
 
 — t — ju , t and u are both reversed in direction, and the vector 
 line OA is rotated 180° (Fig. 11); jt — u means that the line 
 has been rotated forward 90°, since t is positive but stands at 
 90° from the initial line, and u is negative ; — jt + u means that 
 the line has been rotated back 90°. Finally, multiplying by/ 
 means advancing the vector line 90°; for, j(t + ju) =jt -f (+/) 
 (/m), and as j 2 indicates rotation twice forward, j 2 u becomes 
 
 — u, and therefore j (t + jii) is geometrically equal to jt — u ; j 
 is, therefore, seen to be similar to the imaginary term V — 1 
 since (V— 1) 2 = — 1. Also multiplying by —j means turning 
 the vector back 90°, for — j(t + ju)= —jt + (—/)(/«), and as 
 
 — j 2 (namely, +/ x — /) indicates rotation forward 90° and 
 back 90°, — j 2 u = u , and there results — jt + u. 
 
 Quantities comprising associations of real and imaginary 
 quantities, such as the quantities described in the preceding 
 paragraph, are called Complex quantities. 
 
 The vector expressing a sine wave may now be represented 
 in magnitude or scalar value, as heretofore shown, by 
 
18 
 
 ALTERNATING CURRENTS 
 
 A = Vt 2 + u 2 , 
 
 where A is the length or scalar value of the vector A and is a 
 purely arithmetical quantity; in phase by 
 
 tan 6 = - ; 
 t 
 
 in phase and magnitude by the complex quantity 
 aL(cos 6 +j sin 6), 
 
 since A cos 8 — t and A sin 6— u ; and also, as just indicated, 
 by the equivalent complex quantity 
 
 t +ju. 
 
 The addition of the vectors given in the first illustration 
 of this article now becomes 
 
 A +jB = ( a ' + a" + a'"') + j(b' + b" + b'"). 
 
 The assumptions made with respect to the symbol j leave it 
 subject to the fundamental laws of algebra, even though it 
 does not represent an ordinary numerical quantity and may be 
 considered purely as a sign of operation. 
 
 For a further discussion of the complex quantity and its 
 application to various types of electric circuits, refer to 
 Chapter VI. 
 
 Prob. 1. The complex quantity expressing a vector of vol- 
 tage is 10 + /8. What is the angular position of this vector 
 with reference to the horizontal axis, Z 0°, and what is its 
 scalar value in volts? 
 
 Prob. 2. A vector of current is 1= 15 — j 6. What is its 
 angular position and scalar value? 
 
 Prob. 3. Three vectors of voltage E x = 5 +j 6, = 8 + 
 
 j 10, E z = 20 +/ 2 are to be added together. What is the angu- 
 lar position and scalar value of the resultant voltage ? 
 
 Prob. 4. Complex quantities expressing three voltages which 
 are in series are E x = 80 — j 10, E 2 = 100 +j 0 and E z = —j 50. 
 What is the scalar value and angular position of the resultant 
 voltage? 
 
 Prob. 5. A current having a scalar value of 100 amperes 
 and an angular position of Z 30° may be expressed as a com- 
 plex quantity in what terms ? 
 
TIIE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 19 
 
 Prob. 6. Two vectors of voltage having respectively scalar 
 values of 20 and 50 and angles of Z 30° and Z — 30° are to be 
 combined. Form their complex expressions and from these 
 obtain the complex expression of the resultant, after which 
 find the scalar value and angular position of the resultant. 
 
 Prob. 7. Find the scalar value and angular position of the 
 resultant of the four following voltages : 
 
 -#i = 8 +j 4, J? 2 = 10 -j 6, Jj 3 = 15, E i =j 20. 
 
 Fig. 12. — Simple Alternator Diagram. 
 
 9. The Principle of an Alternating-current Generator. — The 
 simplest form of Alternating-current generator, or Alternator, is 
 
 a single coil revolving in a constant magnetic field and having 
 its terminals connected to two rings fastened upon the shaft. 
 Such a simple ar- 
 rangement is shown 
 diagrammatically in 
 Fig. 12. The rings 
 to which the ends 
 of the coil are at- 
 tached are called 
 Collector rings or 
 Slip rings, and upon 
 these bear copper or 
 carbon brushes for 
 the purpose of connecting the coil with the external circuit. 
 Revolving the coil in the magnetic field causes a voltage to be 
 generated in one direction during the time it is under the influ- 
 ence of one pole piece, and the direction of the voltage reverses 
 as soon as the coil begins cutting lines in the opposite direction 
 under the influence of the other pole. If the instantaneous 
 voltages set up in the revolving coil are plotted as explained 
 in Art. 2, a curve of two loops, or one complete period, will 
 represent the results in each complete revolution (Fig. 1). 
 As an alternating current will not serve to magnetize the fields 
 of alternating-current generators of this type, which 'are usually 
 called Synchronous generators, some special arrangement for 
 obtaining a direct current for the excitation must be made. 
 This may be done either by commutating or rectifying all or 
 a part of the alternating current produced by the machine, or 
 
20 
 
 ALTERNATING CURRENTS 
 
 a small auxiliary direct-current dynamo called an Exciter may 
 be supplied for the purpose. Sometimes the exciter is 
 mounted on the bed plate of the alternator. 
 
 10. Commercial Frequencies and Multipolar Generators. — . 
 The frequencies of alternating currents used for the general 
 commercial purposes of the present day vary widely, but in 
 nearly all cases fall within the limits of 15 and 135 periods or 
 Cycles per second (1800 and 16,200 alternations per minute). 
 The majority of American alternating-current dynamos, or 
 Alternators, give a frequency of 25, 40, and 60 periods per 
 second. 
 
 The rotation of a coil in a two-pole field gives one complete 
 period, or two alternations, for each revolution ; and the fre- 
 quency is therefore equal to the number of revolutions per 
 second. Since the armatures of two-pole machines, unless 
 driven by direct-connected steam turbines or the like, would 
 be required to run at impracticable speeds in order to give 
 the ordinary commercial frequencies, alternators are usuallv 
 made with a considerable number of poles. The number of 
 poles depends upon the size of the alternator and other condi- 
 tions which may control the speed of the armature, but, in 
 general, it may be said to vary from eight upwards. An ex- 
 ception to this statement must be made in the case of generators 
 which are designed for direct connection with steam turbines. 
 The unusually high speed of these turbines renders a small 
 number of field poles, even as few as two or four, requisite in 
 the direct-connected alternator. 
 
 Figure 13 is a diagram of a six-pole generator with revolving 
 armature. In the figure A 7 ", S, JY, S, JY, S, are the pole pieces, 
 Y is the yoke, A the armature, B the brushes, and TF TF the 
 field windings. Two of the magnetic circuits of the machine 
 are shown by the dotted lines marked n , n. Instead of using 
 one coil on the armature as was shown in Fig. 12, it is usual in 
 this type of machine to use as many as there are poles for the 
 purpose of utilizing the working space to the best advantage. 
 These coil's may be conveniently connected in series and their 
 free ends connected to the collector rings, as is shown at B, 
 Fig. 13, but they may be connected in series parallel or parallel 
 relations if desired. In the series connection, the alternate 
 coils, which move under pole pieces of opposite polarities, must 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 21 
 
 be connected in circuit in relatively opposite directions so that 
 their voltages may add together. 
 
 The frequency which is produced by an alternator of this 
 type is equal to one sixtieth of 
 the product of the number of 
 its pairs of poles by the number 
 of revolutions per minute made 
 by the armature. It is evident 
 that this must be the case, since 
 an armature coil will have com- 
 pleted one cycle or period when 
 its center has passed from a 
 point under one magnetic pole 
 to a similar position with re- 
 spect to the next pole of the 
 same sign. The number of al- 
 ternations per minute is equal to the frequency, as shown 
 above, multiplied by 120. 
 
 In any alternator, one complete cycle of the alternating vol- 
 tage is generated while an armature conductor moves from a 
 given point under a pole of one polarity to a like point under the 
 next pole of the same polarity ; and the frequency is equal to the 
 number of cycles per second. This definition applies to all 
 alternators, while the commoner definition involving the num- 
 ber of poles, given in the preceding paragraph, only applies to 
 alternators possessing field magnets with adjacent poles (in the 
 direction of rotation) of opposite signs. 
 
 The angular distance of movement of the conductor required 
 to generate one complete cycle is called 360 Electrical degrees. 
 The electrical degrees correspond with the mechanical degrees 
 of armature rotation only in a bipolar machine or its equiva- 
 lent. Three hundred and sixty electrical degrees are em- 
 braced in the span from a given point in the magnetic field 
 under one pole to the corresponding point in the magnetic 
 field under the next pole of the same sign ; so that there are as 
 many times 360 electrical degrees measured around an arma- 
 ture as there are poles of like sign in the related field magnet. 
 In speaking of the movement of the armature conductor in the 
 magnetic field, the relative movement of the armature with 
 respect to the field is here referred to, and it may be obtained 
 
 Fig. 13. — Diagram of Multipolar 
 Generator. 
 
22 
 
 ALTERNATING CURRENTS 
 
 by the actual rotation of either the armature or the field mag- 
 net. In modern machines the latter is most frequently rotated. 
 
 Prob. 1. What is the frequency of an alternator having 10 
 pairs of poles and a speed of 600 revolutions per minute? How 
 many alternations does it give per minute? 
 
 Prob. 2. An alternator gives a frequency of 25 periods per 
 second and lias 5 pairs of poles. At what speed (in revolutions 
 per minute) does it run? 
 
 Prob. 3. An alternator is to give a frequency of 60 periods 
 per second at a speed of 1200 revolutions per minute. How 
 many poles must it have? 
 
 Prob. 4. Can an alternator be built to furnish a frequency 
 of 47 periods per second when run at a speed of 600 revolu- 
 tions per minute ? Why ? 
 
 Prob. 5. Can an alternator be built to furnish a frequency 
 of 60 periods per second when running at 700 revolutions per 
 minute? Why? 
 
 11 . Form of the Voltage Curve of an Alternator. — The volt- 
 age set up by a conductor or coil moving in a magnetic field 
 is equal to the rate of cutting lines of force, or to the rate of 
 change of the number of linkages of lines of force with the coil, 
 divided by 10 8 ; or (for a single conductor), 
 
 1 deb 
 
 io 8 x dt ’ 
 
 where e is the instantaneous voltage and — is the time rate of 
 
 dt 
 
 cutting lines of force. In the case of an armature revolving in 
 a magnetic field as shown in Fig. 12 the voltage will be at any 
 instant, 
 
 1 S dcf> 
 
 e — - ^ X — X — £ 
 
 10 8 
 
 P 
 
 dt 
 
 where S is the number of active conductors on the armature, 
 p' is the number of paths for the current through the armature, 
 
 S 
 
 and — is the number of active conductors in series cutting the 
 
 P 
 
 lines of force. If the armature revolves at a constant speed of 
 V revolutions per minute in a magnetic field of p poles, with 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 23 
 
 useful lines of force emanating from each north pole, the aver- 
 age voltage set up may be deduced from this formula to be, 
 
 E„ 
 
 pS®V 
 p' x 10 8 x 60’ 
 
 since is the average rate of cutting lines of force by each 
 conductor. 
 
 This is the formula used to state the continuous voltage of 
 an ordinary direct-current dynamo. Since the voltage produced 
 by direct-current dynamos is essentially constant, the effective 
 and average values are equal to each other in this case.* 
 
 In the alternating-current dynamos the same averaging does 
 not occur as in direct-current commutating machines with arma- 
 
 ture conductors uniformly distributed over the surface of the 
 armature, and the voltage between alternator armature ter- 
 minals varies from instant to instant. 
 
 Thus, suppose the coil of Fig. 12 has negligible width but 
 comprises S conductors arranged to rotate in the uniform mag- 
 netic field as illustrated ; then the voltage in the conductors is, 
 at any instant, as before stated, 
 
 1 S dd> 
 e io sX p /X dt' 
 
 If is the number of useful lines of force emanating from the 
 
 north pole, then under the conditions described, becomes, 
 dt 
 
 for the bipolar machine, 
 
 d(f> _ 2 7i -r <t> 
 
 27 
 
 x sin « = —<!> sin a, 
 
 pv 
 
 where a is the angular displacement of the conductor from a 
 diameter perpendicular to the lines of force, r is the radius of 
 the armature, and T' is the time in seconds of one revolution. 
 
 If there are more than two poles, each having <3? lines of force 
 uniformly distributed over its face so as to make a multipolar 
 magnetic field analogous to the bipolar magnetic field illus- 
 trated in Fig. 12, the voltage induced is obviously increased in 
 proportion to the number of poles, and the last formula be- 
 comes, in general, 
 
 * Jackson’s Electromagnetism and the Construction of Dynamos , Chap. 4. 
 
24 
 
 ALTERNATING CURRENTS 
 
 dd) p 2 Trr <b 
 dt 2 T 2 r 
 
 sm a. 
 
 It must be remembered that « is an angle measured with re- 
 spect to a normal to the lines of force, and it changes by the 
 extent of 360° in passing from a given point in the magnetic 
 field to the corresponding point under the next pole of like sign. 
 
 The voltage at any instant now becomes 
 
 7T S p 
 
 e = — X V X 7 2 X rpi 8111 «• 
 
 10 8 
 V 
 
 p 
 
 But is equal to where V stands for the speed of the 
 
 armature in revolutions per minute. The formula, therefore, 
 reduces to 
 
 irpSQV 
 
 e ~ 2 p' 10 8 x 60 Sm 
 
 which is the equation for a sinusoidal or harmonic curve * such 
 as is shown in Fig. 1. The equation may be written 
 
 e = e m sin a, 
 
 in which the maximum ordinate, e m , is equal to 
 
 ^ _ irpS^ V 
 
 2 x p' x 10 8 x 60’ 
 
 That is, under the conditions assumed for the magnetic field, 
 and the arrangement of the armature conductors, the alternat- 
 ing voltage of the armature partakes of the form of a sine curve. 
 
 The symbol p in this formula has been defined as the number 
 of poles comprised within the magnetic field. Since one com- 
 plete cycle of the voltage is generated while an armature con- 
 ductor moves from a given point under a pole of one polarity 
 to a like point under the next pole of the same polarity, — that 
 is, while the conductor moves through twice the polar pitch, — 
 it is plain that p in the last formulas may be replaced by 
 
 2 — /, where/ represents the frequency, and the formulas become 
 
 7T/SW . 
 
 « = — — TbS sin 
 
 p' x 10 8 
 
 and 
 
 7 rS&f 
 e ™~ p' x 10 8 ’ 
 
 * Art. 5. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 25 
 
 The effective voltage for this case is 
 E- rS *- f - 
 
 2 22 
 
 s*f 
 
 V2 p' x 10 8 p' x 10 8 
 
 Now, if it is assumed that the field strength under the pole 
 faces illustrated in Fig. 12 is altered so that it is weakest near 
 the center, then the 
 rate of cutting lines 
 no longer varies as a 
 sine curve, but the 
 curve rises more rap- 
 idly in its lower por- 
 tions, and does not 
 reach so great a maxi- 
 mum. On the other 
 hand, if the field is 
 concentrated towards 
 
 AVERAGE E 
 EFFECTIVE E 
 
 Fig. 14. — Flat-topped Voltage Curve. 
 
 the center, the curve becomes higher at the center, but does 
 not rise so rapidly in its lower portions. Figure 14 shows the 
 
 curve of voltage de- 
 veloped in a coil 
 which revolves in a 
 field of the same total 
 number of lines of 
 force as that of Fig. 
 12, but in which the 
 magnetic density at 
 the centers of the 
 poles is about 35 per 
 cent less than the 
 average. Figure 15 
 shows a similar curve 
 when the magnetic 
 density at the center 
 is 50 per cent greater 
 than the average. In 
 each case, the polar 
 density is assumed to change gradually and uniformly from the 
 center to the edges. In all cases where a certain total number 
 of lines are cut per revolution by a coil revolving at constant 
 
 Fig. 15. — Peaked Voltage Curve. 
 
26 
 
 ALTERNATING CURRENTS 
 
 1ig. 1(1. — Imaginary Magnetic Field for obtain- 
 ing Rectangular Voltage Curve. 
 
 speed, the average voltage remains constant regardless of the 
 magnetic distribution, but the effective voltage (VAv.f 2 ) is by 
 
 no means independent of 
 the distribution. Taking 
 the maximum voltage of 
 the sine curve shown in 
 Fig. 1 as the unit, the 
 average voltage* in each 
 2 
 
 figure is ~ = -637. The 
 
 effective voltage of the 
 sine curve shown in Fig. 1 
 
 is — - = .707. The maxi- 
 
 mum voltages shown in 
 the curves of Figs. 14 and 15 are .84 and 1.5, and the effec- 
 tive voltages are .58 and .84, respectively. If the field were 
 distributed as in Fig. 16 
 (the total magnetization 
 remaining constant, and 
 the coil of inappreciable 
 width), the maximum, av- 
 erage, and effective volt- 
 ages would be equal to 
 each other (Fig. 17), and 
 of a numerical value of 
 .637. This form of curve 
 gives the least possible 
 relative effective voltage. 
 
 
 
 
 
 
 
 — / / / 
 
 
 
 — 1 / / 
 
 |( w) I 
 
 \ \ / / — > 
 
 \ N. s' /- — > 
 
 X. ^ — > 
 
 
 Fig. 18. 
 
 -Magnetic Field for obtaining a Peaked 
 Voltage Curve. 
 
 Fig. 17. — Ideal Rectangular Voltage Curve. 
 
 On the other hand, if the field is 
 greatly concentrated 
 towards the center, 
 as in Fig. 18, the 
 maximum voltage is 
 very great, and the 
 effective voltage is 
 considerably greater 
 than the average, 
 though it by no 
 means approaches in 
 value the maximum 
 
 * Art. 5. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 27 
 
 voltage. In the case of modem alternators the windings are 
 usually Distributed * over a large portion of the polar area, so 
 that the resulting curves of voltage tend to approximate the 
 sine form. 
 
 12. Equations of Curves by Fourier's Series. — It has already 
 been explained that the curves of current and voltage in com- 
 mercial alternating-current circuits may deviate widely from 
 the sine form, hut are always single- valued. A general formula 
 is needed, therefore, that will express such curves whatever 
 may be their forms. The most usual and satisfactory method 
 of obtaining such a formula is by means of Fourier’s Series. 
 This series is based upon the proposition that any single- 
 valued recurrent function may be represented by a constant 
 term, plus a sine term and a cosine term having the same period 
 as the function in question, plus a sine term and a cosine term 
 of one half the period, plus a sine term and a cosine term 
 of one third the period, and so on up to a sine term and a 
 cosine term of one nth the period f ; where n is dependent upon 
 the irregularity of the curve in question and the accuracy of 
 representation desired. 
 
 If/(£) is a single-valued recurrent function of time having 
 a period T , it may be expressed in accord with the above state- 
 ment as follows : 
 
 r a , a -27 rt . t> 2 irt , | . 4 7rt . t» 4 irt 
 
 f (t) = A + A 1 sin — + B x cos — + A 2 Bin — + B 2 cos — 
 
 . i • 6 irt . D 6 irt 
 
 + ^8 sm ~jT + ^3 cos -yT 
 
 , . ■ 2 m rt . r, 2 nirt 
 
 • + A a sin — — b B n cos 
 
 T 
 
 T 
 
 where A, A v B v A v B v etc., are constants. 
 
 2 irt 
 
 Since the expression represents a proportion of 2 7r (or 
 
 360° in angular measure), the equation may be written as 
 follows : 
 
 /(«) = A + A l sin a + B 1 cos « + A 2 sin 2 a +B 2 cos 2 a • • • 
 
 + A n sin na + B n cos na. 
 
 The measure of « at the end of a cycle is 2 7 r. 
 
 Evidently /(«) is represented by a summation comprising a 
 constant term plus a series of sine and cosine terms having 
 
 * Art. 20. 
 
 t Byerly, Fourier's Series and Spherical Harmonics , pp. 30 etseq. 
 
28 
 
 ALTERNATING CURRENTS 
 
 maximum instantaneous values of A v B v A 2 , B v etc., with 
 frequencies of one, two, three, etc., times that of the original 
 function, and with the zero ordinate of each cosine term oc- 
 curring simultaneously with the maximum ordinate of the sine 
 term of equal frequency. The value of the ordinate of the 
 periodic curve in question at any time, £, or angle, «, is equal 
 to the algebraic sum of the ordinates of all the terms at the 
 particular instant. Any single-valued recurrent curve, no 
 matter how irregular, may be thus resolved into component 
 sinusoids. 
 
 a. Examples of Irregular Waves . — Figure 19 shows a curve, 
 S, of alternating voltage, which is composed of two sine waves 
 
 having respective maxi- 
 mum values A x and A 3 . 
 The sinusoidal component 
 of the same frequency as 
 the wave is called the 
 Primary or Fundamental 
 harmonic ; remaining 
 component waves may in 
 general be called Minor 
 harmonics, or specifically, 
 the Second, Third. Fourth, 
 etc., harmonics, depending upon their frequencies when com- 
 pared with that of the primary wave. In Fig. 19 the primary 
 and third harmonics only are present, so that the Fourier's 
 Series for the curve is 
 
 Fig. 19. ■ 
 
 -Periodic Curve composed of Funda- 
 mental and Third Harmonics. 
 
 /(«) = y = A x sin a + (— A 3 ) sin 3 a, 
 
 where y is the ordinate of the curve S for any time, t , or 
 angle, a. 
 
 The constant A is zero in this as in all periodic curves having 
 equivalent loops above and below the x axis.* All the other 
 terms except those given are also zero as the sum of the first 
 and third sine harmonics exactly satisfy the curve S. The 
 constant term A 3 is negative because the negative loop of the 
 third harmonic starts at zero of time. Figure 20 shows the curve 
 that is formed by the same harmonics, but with the third har- 
 monic moved forward 180°, so that A 3 is positive. Figure 21 
 * For proof see Art. 13. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 29 
 
 shows a curve composed of a primary and a fifth harmonic, with 
 A l and A h both positive; and Fig. 22 shows a curve derived 
 from the following series : 
 
 x = A x sin a + A 3 sin 3 a -f A 5 sin 5 a. 
 
 It is to be observed that the loops of the curves shown are 
 symmetrical on each side of a vertical axis drawn through the 
 
 Fig. 20. — Periodic Curve containing Fundamental and Third Harmonics. 
 
 axis of abscissas at 90° or 270°. If, now, a cosine term is added 
 to the components of Fig. 19, for instance, as in the expression 
 
 y = A l sin a — A 3 sin 3 a — B 3 cos 3 «, 
 
 which is shown by the curves in Fig. 23, this symmetry of the 
 loops with respect to the vertical axis is destroyed. The loops 
 
 Fig. 21. — Periodic Curve containing Fundamental and Fifth Harmonics. 
 
 of this type, as of the preceding curves, are, however, sym- 
 metrical one with the other. That is, if the negative loops are 
 revolved 180° on the axis of abscissas so as to stand between 
 the positive loops, all of the loops look just alike, and if one 
 
80 
 
 ALTERNATING CURRENTS 
 
 of the negative loops is then moved along the axis one half a 
 period so as to lie upon one of the positive loops, it will exactly 
 coincide with a positive loop. From the brief description 
 already given of the arrangement of armature windings and uf 
 the magnetic field of the ordinary commercial alternator,* it is 
 
 Fig. 22. — Periodic Curve containing Fundamental, Third, and f ifth Harmonics. 
 
 seen that the same magnetic lines are cut in the same order by 
 exactly the same arrangement of conductors during the gener- 
 ation of both the positive and negative loops of voltage which are 
 as a result alike in area and form, provided the machine is 
 
 Fig. 23. — Periodic Curve containing Sine and Cosine Terms in the Third 
 Harmonic. 
 
 mechanically and magnetically rigid. It will likewise be 
 shown later that the negative and positive current loops also 
 ordinarily have the same characteristic likeness. It is there- 
 fore evident that such waves can be expressed by such formulas 
 as have preceded and which include only the odd harmonics. 
 
 * Arts. 10 and 11. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 31 
 
 If the even harmonics appear, this characteristic symmetry 
 no longer exists. Figure 24 shows a curve answering to the 
 formula : 
 
 y = sin a + A 2 sin 2 a. 
 
 Fig. 24. — Periodic Curve Containing Fundamental and Second Harmonics. 
 
 Figure 25 shows a curve compounded of the first, secondhand 
 third sine harmonics ; and Fig. 26 has first and second sine 
 and second cosine harmonics — the latter two combined into 
 
 Fig. 25. — Periodic Curve Containing Fundamental, Second, and Third Harmonics. 
 
 one curve, which may be done by adding their ordinates 
 algebraically. The resultant irregular curves in these figures 
 do not have positive and negative loops which will superpose 
 by revolution around and sliding along the axis of abscissas as 
 
32 
 
 ALTERNATING CURRENTS 
 
 above described, because for each half period of the primary 
 curve the even harmonics complete one or more entire cycles 
 and have relatively opposite effects upon the positive and nega- 
 
 Fig. 26. — Periodic Curve having Second Harmonic which is composed of Sine 
 and Cosine Terms. 
 
 tive loops of the primary curve. On the other hand, the odd 
 harmonics keep at all times the same relation to the primary, 
 i.e. when the primary reverses, the odd harmonics also reverse. 
 
 Prob. 1. Construct the voltage curve expressed by the 
 formula e = 100 sin « + 50 cos a + 30 sin 3 a + 20 cos 3 a. 
 
 Prob. 2. Construct the curve of current expressed by the 
 formula i = 200 sin a — 25 sin 5 « + 15 cos 5 «. 
 
 Prob. 3. Construct the voltage curve expressed by the 
 formula e = 200 sin a + 200 cos a + 50 sin 3 a + 50 cos 3 « 
 
 --- 25 sin 7 « — 25 cos 7 a. 
 
 Prob. 4. Construct the curve represented by the formula 
 y = 50 + 50 sin a + 40 cos « + 30 sin 2 a + 20 cos 2 a. 
 
 h. Fourier's Series with Sine and Cosine Terms Combined . — 
 Sine and cosine harmonics of equal frequency may be combined 
 into one curve, as explained with reference to Fig. 26, and a 
 sine term is formed thereby having a maximum which may be 
 designated C and which is out of phase with the original sine 
 component harmonic by some angle /3„. As a result Fourier's 
 Series may be expressed in a more general form, thus: 
 
 f = A. -f- C± sin + /3\) ~b sin (2 a. + fdf) ■ • • 
 
 + C n sin Qua + /3„) ; 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 33 
 
 or where /3,[ is the angle between the new harmonic and the 
 original cosine component, 
 
 /(«) = A + <7j cos (« — ySj) + C 2 cos (2 « — /Sg) ••• 
 
 + O n cos (na - /3') . 
 
 In these expressions O n = A\ + B\, tan /3 re = and 
 
 tan /3'„= —5, since (7„ is the resultant of A n and B n , which differ 
 B» 
 
 from each other in phase by 90°. It is usual in drawing the 
 harmonics of a curve to plot the total harmonics as given by 
 either of these formulas rather than to use their sine and cosine 
 components, which add to the complication.* 
 
 13. Determination of the Value of the Constants in Fourier’s 
 Series. — The value of Fourier’s Series to electrical engineer- 
 ing consists in the possibility of resolving irregular waves of 
 current and voltage into their harmonics and by that means 
 obtaining' mathematical expressions to which the elementary 
 laws of electricity can be applied. To do this it is evidently 
 necessary to obtain the numerical values of the constants con- 
 tained in the formulas. 
 
 Having given the formula, 
 
 /(a) = A + A x sin a + B x cos a + A 2 sin 2a + 5 2 cos 2 a ••• 
 
 + A n sin na + B n cos na , 
 an expression for the value of A may be obtained by multiply- 
 ing both sides of the equation by da , and integrating between 
 0 and 2 7 r. Thus : 
 
 X %r r /~2rr 
 
 f(a)da = Aj da-\-A x J sin ada -f B l cos ada--- 
 
 + A 
 
 0 si 
 
 sin tiada + B 
 
 l ( 
 
 cos nada. 
 
 * A theoretical discussion of Fourier’s Series will be found in the first three 
 chapters of Byerly, Fourier's Series and Spherical Harmonics ; J. W. Mellor, 
 Higher Mathematics , Chap. VIII ; Merriman and Woodward, Higher Mathe- 
 matics ; and similar works. The following articles are of considerable interest 
 in this connection: Periodic Functions Developed in Fourier Series; The 
 Graphical Method, by Professor John Perry, London Electrician , Vol. 35, p. 285 ; 
 Wave Form Synthesis, London Electrician, Vol. 35, p. 257 ; Trans. Amer. Inst. 
 Elect. Eng., Vol. 12, p. 476 ; Fleming’s Alternate Current Transformer in 
 Theory and Practice, Vol. II, p. 454 ; Graphical Analysis of Harmonic Curves, 
 by Wedmore, London Electrician, Vol. 35, p. 512 ; Approximate Method, 
 Houston and Kennedy, Electrical World, Vol. 16, p. 580 ; Graphical Compu- 
 tations in Alternating Current Circuits, Eclairage Electrique, Vol. 16, p. 397. 
 
34 
 
 ALTERNATING CURRENTS 
 
 All the terms on the right, except the first, reduce to zero, as 
 may be shown by integrating the general sine and cosine terms 
 as follows : 
 
 A 
 
 Jo 81 
 
 sin nada — A r , 
 
 1 V* 
 
 - cos na = 0. 
 n Jo 
 
 Therefore 
 
 J --277 f \ \2n 
 
 cos nudu — A n [ + - sin na) =0. 
 u V n Jo 
 
 A= jzr^ d ' 1 - 
 
 en- 
 
 J r* 2n 
 
 /(«) da is the net area of the curve for an 
 
 0 
 
 tire period, and 2 it is its base ; so that A is the average ordi- 
 nate for an entire period. Since, as has already been explained, 
 the positive and negative loops in the ordinary alternating elec- 
 
 i n* 
 
 trie voltage and current curves are closely alike, - — I /(«) da 
 
 2 IT 
 
 reduces to zero because the area of the positive loop in a period 
 is equal to the area of the negative loop. The constant A , 
 therefore, need not be included in writing the series for compu- 
 tations of such quantities ; that is, the axis of abscissas is 
 placed symmetrically with respect to the positive and negative 
 areas. 
 
 To obtain A x multiply the series by sin ada and integrate 
 
 between the same limits as before. Thus 
 
 sin ada 
 
 J ^2n 
 
 , /(«)si 
 
 X 2n f2n _ ^ /*2ir 
 
 s\n ada + A x I sin 2 ada + B X J cos a sin ada 
 
 J ' 2 " C*2tt 
 
 # sin na sin ada + B n J cos na sin ada. 
 
 Each of the terms in the right-hand member of the equation 
 except the second term is of the form of one of the follow- 
 ing : 
 
 J r2n _ P*2tt 
 
 I sin ada , I sin a cos ada, 
 o 
 
 X 2 7T _ _ /' 2 ^- 
 
 sin pa sin qada, J sin pa cos qada, 
 
 /*2 7T .^2ir 
 
 I sill pa cos pada, or | cos pa cos qada. 
 
 where p and q are unequal integers. As is well known, these 
 expressions reduce to zero. This is equivalent to saying that 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 35 
 
 any curve which may be represented by one of these expres- 
 sions incloses as much negative area below the axis of abscissas 
 as positive area above the axis, in any length equal to one 
 period measured along that axis. The rule also applies for full 
 periods, when, of p and q , one is an integer and the other dif- 
 fers from an integral number by one half. 
 
 The second term in this equation is of the form 
 
 X 2 tt 
 
 sin 2 ^«d«, 
 
 where p is any integer. This reduces to i r, as also does the 
 analogous expression 
 
 J ^2n 
 0 ( 
 
 J "2ir 
 0 * 
 
 cos 2 pada. 
 
 The original equation therefore reduces to 
 f (a) sin ada = ttA v 
 
 1 r 2n 
 
 A x = - I f (a) sin ada. 
 
 TT 
 
 A like process in which the original equation is multiplied by 
 cos ada , and reduced to find the expression for B l by sin 2 ada, 
 and reduced to find the expression for A v etc., shows that the 
 expressions representing the various constants are as follows : 
 
 and 
 
 1 r 2*- 
 
 B, — — I /(«) cos ada, 
 
 7W0 
 
 1 C 2k 
 
 A 2 = — /(a) sin 2 ada, 
 
 Bn = — f f (a) cos 2 ada, 
 
 7T»/o 
 
 A n = - I /(a) sin nada, 
 
 IT i/O 
 
 1 r2w 
 
 I /(«) cos nada. 
 
 7T»/0 
 
 14. Method of Finding the Harmonics of a Periodic Curve by 
 Fourier’s Series. — If f («) (that is, each ordinate of the re- 
 current curve) is dependent upon a by some known law, its 
 value can usually be incorporated into the formulas for the 
 constants given above and the integration performed. For 
 
36 
 
 ALTERNATING CURRENTS 
 
 instance, such would be the case if the loops of the curve were 
 rectangular. In the case of current or voltage loops such a 
 relation between the ordinates and abscissas of the curve is not 
 apt to exist, so that it is necessary to change the integral sign 
 into a summation sign and obtain the result as accurately as 
 time will permit or necessity require. 
 
 Tims erect m equally spaced ordinates on the axis of abscissas 
 of an experimentally determined curve, beginning at the incep- 
 tion of a period. The ordinates divide the base line into m — 1 
 equal divisions in the period. The lengths of the ordinates, 
 from the axis of the abscissas to the curve, may be called e 0 , e v 
 ••• e m . The base of one complete period being 2 v radians in 
 
 Fig. 27. — Showing Periodic Wave divided into Elementary Areas for obtaining 
 
 Harmonics. 
 
 length, the distance between any adjacent two of the equally 
 
 2 t r . 
 
 spaced ordinates is -radians (see Fig. 27). Then the value 
 
 m — 1 
 
 1 C 2,t 
 
 A, = — I /(«) sin ada can be written approximately 
 
 1 7 rJn' 
 
 2 
 
 1 m — 1 Z— ■ -i 
 
 e,.sin [ k 
 
 vi 
 
 
 where e k represents the ordinate of the curve or ,/(«) at any 
 particular division point, k the designating number of the cor- 
 
 2 7T 
 
 responding ordinate, and — distance between adjacent ordi- 
 
 , vi — 1 
 
 nates. 
 
 Hence, 
 
 9 
 
 A = -Ar 
 
 m — 1 
 
 e , sin 
 
 Vi 
 
 — 1 / 
 
 ■ + sin 2 
 
 vi — 1 
 
 + e m sin 
 
 f 2 ’ 
 
 m — 
 
 V vi — 1 /_ 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 where e v e 2 , etc., are the ordinates of the curve at the division 
 points, 1, 2, 3, ••• m, and in general 
 
 B = 
 
 on — 1 
 o 
 
 m- 
 
 ■ 1 
 
 • / 2tt V . f 0 2tt ) , . ( 
 
 e.sin n- + e 9 sin In — He m sin nm 
 
 1 V m — lj 2 V m-lj \ 
 
 f 2 IT \ f 0 2 7T \ ( 
 
 e.cos n +e 9 cos 2 n — be»cos : 
 
 1 V m-lj 2 V m-lj 
 
 \ on — 1 
 
 V m — 1 
 
 The positive and negative loops of ordinary alternating cur- 
 rents and voltages are similar, and for them it is only necessary 
 to divide up one loop ; and for the same reason even harmon- 
 ics of the series are commonly unimportant. The larger m 
 is made, the more accurate will be the result ; likewise the ac- 
 curacy is increased by making n large, though it is seldom nec- 
 essary to obtain more than the first, third, and fifth harmonic. 
 
 Having found A v A 3 , etc., and B v B 3 , etc., the values of C v 
 C 3 , and /3 V fi 3 may be at once obtained for substitution into the 
 general formula.* Having obtained these values, the harmonics 
 represented by each term of the series can, if desired, be plotted 
 with the periodic curve to which it belongs. It must be re- 
 membered in doing this that a = 0 when the irregular curve 
 crosses the X axis in an upward or positive direction. 
 
 The values of the constants may be obtained approximately 
 by the use of a planimeter instead of by the laborious com- 
 putation involved in solving the foregoing equations. Meas- 
 ure the ordinates e v e 2 . e 3 , •■■e m as before and multiply each 
 by the sine of the proper angle as shown in the formula of 
 the desired constant or A v or B x or B 2 , etc.). Plot a 
 curve over the points 1, 2, 3, 4, ■■■m with ordinates equal to 
 
 ( 2 
 
 these products, that is, place the ordinate e x sin [ n 
 
 m — 1 
 
 m — 1 
 
 , over the point 1, e 2 sin (2 n 
 
 m — 1 
 
 or 
 
 e 9 cos Z n 
 
 on — 1 
 
 , over the point 2, and so on (see curve z in 
 
 Fig. 28). Measure the areas of the resultant loops by means 
 of a planimeter and add them algebraically, considering areas 
 above the axis of abscissas as positive and those below the axis 
 as negative. If m is sufficiently large, this area, l T , will be 
 with sufficient accuracy 
 
 * Art. 12 6. 
 
38 
 
 ALTE R NATING C UR RENTS 
 
 /'2n 
 
 U = I /(«) sin nudu or ( f (a) cos nudu. 
 
 «/0 %) 0 
 
 i r n . . u 
 
 Hence, A n =— I /(a) sin nudu. = — . 
 
 7T *Z° 7T 
 
 In Fig. 28 is shown a recurrent curve, X , and the process 
 of obtaining H 3 is indicated. To do this, 37 values of e and 
 the corresponding 37 values of sin 3 u were used, u being re- 
 
 2 77" 2 7T i • 
 
 spectively 0, — r , 2 — , etc., at the successive points. These 
 36 36 
 
 values multiplied into values of e, e v e v etc., and plotted on the 
 corresponding ordinates give the points on the curve Z. If 
 the areas of the positive and negative loops of curve Z are meas- 
 
 Fig. 28. — Product of an Alternating-current Curve with Sine Third Harmonic 
 Analyzer, x is the Irregular Curve, and z the Curve of Products. 
 
 ured by a planimeter, it will be found that there is a small posi- 
 tive area in excess. The constant A 3 is therefore positive : i.e. 
 the sine component of the third harmonic starts at zero with a 
 positive loop, and has a maximum height equal to twice the 
 length obtained by dividing the algebraic sum of the areas of the 
 loops of curve Z by the length of the base for the period. This 
 is readily converted into volts by applying the scale used in 
 plotting the values of e. The dotted line in Fig. 28 shows the 
 harmonic plotted in position and to scale. 
 
 The other constants may be obtained by following a similar 
 process. While the foregoing example is applied to a com- 
 plete period of the curve, it is sufficient to carry out the analy- 
 sis on a single loop or half period of ordinary alternating 
 current or voltage waves which have half periods alike, as 
 already pointed out. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 39 
 
 14 a. Components of Alternating Curves do not always fall 
 within the Fourier Series. — The foregoing articles deal with 
 the harmonic analysis of alternating voltage and current curves 
 on the hypothesis that the components of such curves are all of 
 frequencies represented by integral numbers of times the pri- 
 mary frequency, and this is an accurate hypothesis with respect 
 to curves produced under perfectly stable recurrent conditions, 
 such as voltage curves induced by the uniform rotation of the 
 armature of a synchronous generator constructed with perfect 
 mechanical rigidity and possessing a perfectly rigid magnetic 
 field. But these conditions do not always exist and sometimes 
 are departed from widely. The rotation velocity may lack 
 uniformity, the shaft of the armature may spring and alter the 
 magnetic conditions, and other occurrences may likewise intro- 
 duce influences into the voltage curve that are independent of 
 the primary frequency. Also, even though the voltage wave 
 should be free from such effects, variations in the circuit 
 through which the current flows may introduce distortions into 
 the current wave that are independent of the primary frequency. 
 
 Variations of the kind here referred to are generally not 
 strictly recurrent, but this may not be indicated by the analysis 
 of a single period of the curve and is not likely to be indicated 
 by the analysis of a half period. As such variations are likely 
 to cause the successive loops to differ in form and area, the 
 analysis of several periods is likely to indicate their existence 
 and magnitude. They are generally too small a factor in most 
 of the existing commercial conditions to make them of serious 
 import, but in case of necessity they may be treated like inter- 
 ference phenomena which introduce recurrent effects embracing 
 several or many of the periods of the alternating curve in ques- 
 tion and have a frequency which is some fractional multiple of 
 the primary frequency. 
 
 15. Concrete Examples of the Resolution of an Alternating- 
 current Curve into its Harmonic Components. — The positive 
 loop of a voltage wave of an alternator is represented by the 
 heavy line of Fig. 29, and the constants of Fourier’s Series for 
 this curve have been determined up to the seventh harmonic, 
 giving the values, 
 
 A 1 = + 98.6, 
 
 A a =- 13.3, 
 
 A = -14.7, 
 B z = 8-18.2, 
 
40 
 
 ALTERNATING CURRENTS 
 
 A = ~ !- 6 . 
 
 A 7 = + 0.56. 
 
 B 5 = - 4.8, 
 B 7 = + 1.2. 
 
 The equation for the curve as determined by this means is 
 
 e =98.6 sin « — 13.3 sin 3 « — 1.6 sin 5 a + 0.56 sin 7 « 
 
 — 14.7 cos a + 18.2 cos 3 « — 4.8 cos 5 a + 1.2 cos 7 a. 
 
 Substituting various values of a in this equation, the corre- 
 sponding values of e are given, and the corresponding curve, 
 which is dotted, has been plotted in the figure. It will be 
 noticed that the calculated curve very closely approximates to 
 
 0° q e 2 q q e q c 7 lso' 
 
 Fig. 29. Irregular Curve of Voltage, and Calculated Curve composed of tlie First, Third 
 
 Fifth, and Seventh Harmonics. 
 
 the exact form of the original curve. If a larger number of 
 constants had been determined, the calculated curve would 
 have crossed the original curve a larger number of times, and 
 the approximation would have been still closer. The cal- 
 culated curve must coincide with the original curve at each 
 ordinate which was used in the computation. The calculated 
 curve cannot, except in special cases, exactly coincide with the 
 original curve unless m = ac. The series used is rapidly con- 
 vergent, and in this particular curve the effect of the fifth and 
 seventh harmonics is quite small, so that the curve is sufficiently 
 well represented for practical purposes by the fundamental and 
 third harmonics, in which case the equation is 
 
THE VOLTAGE DEVELOPED BY ALTERNATOKS 
 
 41 
 
 e = 98.6 sin « — 13.3 sin 3 a 
 — 14. T cos « + 18.2 cos 3 «. 
 
 The corresponding sine and cosine terms of the series, 
 
 A, sin a ) ( A„ sin 3«] f A, sin 5 « 
 
 + + 
 
 B 1 cos « J [ + B 3 cos 3 a J [ + B 5 cos 5 « 
 
 + 
 
 \ + etc., 
 i + etc., 
 
 may be conceived as representing the rectangular sine compo- 
 nents of the terms of a single sine or cosine curve.* The equa- 
 tion when reduced to the more general form (using /3) has the 
 following constants : 
 
 C\ = 99.7, /3 X = - 8° 29', 
 
 C 3 = 22.5, /S 3 = - 53° 50', 
 
 C 5 = 5.1, B h = + 71° 34', 
 
 G 1 = 1.3, /3 7 = + 66°, 
 
 and the equation is 
 
 e = 99.7 sin (« - 8° 29') - 22.5 sin (3 a - 53° 50') 
 
 — 5.1 sin (5 a + 71° 34') + 1.3 sin (7 a + 66°), 
 
 and its value to a considerable degree of approximation is 
 e = 99.7 sin (« - 8° 29') - 22.5 sin (3 « - 53° 50'). 
 
 The example which has been taken fairly represents the 
 complexity of the average distorted waves. Some alternating- 
 current curves are so greatly distorted that larger numbers of 
 terms of the series are required to closely represent them, but for 
 practical purposes three or four terms are generally sufficient. 
 In a large number of waves the forms are so simple that two 
 terms of the series give sufficient approximation for practical 
 purposes. 
 
 Following are examples of the calculation of the constants of 
 these equations : 
 
 m = 9, (-^r) = 221°. 
 
 \m — 1 / 
 
 Values of e from curve by measurement : 
 
 e x = 18, e 3 = 70, e 5 — 125, e 7 - 30. 
 e 2 = 42, = 110, e 6 — 80, e 0 = e s = 0. 
 
 a _ i j 18 sin 22|° + 42 sin 45° + 70sin67|° + •••! _ qn 
 1 ~ 1 1 + 30 sin 1571° “ J “ ’ 
 
42 
 
 ALTERNATING CURRENTS 
 
 , . f 18 sin 674° + 42 sin 135° + 70 sin 2021° "I 100 
 
 A ‘ = i { + .. + 80 8 inll2i» = j = - 13.3, 
 
 Q 1 f 18 cos 674° + 42 cos 135° + 70 cos 2024° + ••• ] 1QO 
 ^“‘i '+30co S 112>° ' |= 18 ' 2 ’ 
 
 = v'A;?TB7- = 22.5, /3,= tan 1 ^* = - 53° 50.' 
 
 A 
 
 As C is a second root, it is algebraically either positive or 
 negative, but the conditions of the problem require it to enter 
 the equation with the algebraic sign pertaining to the corre- 
 sponding A. 
 
 Prob. 1. Find the primary and third harmonics of an equi- 
 lateral triangular voltage wave having an altitude of 10 volts. 
 
 Prob. 2. Find the primary and third harmonics of a voltage 
 wave which is in the form of a semicircle with a diameter of 10. 
 
 16. Effective Value of an Irregular Curve of Voltage or Cur- 
 rent. — The effective ordinate of an alternating-current curve 
 may be determined by integrating directly from its equation. 
 The effective value squared is 
 
 ^ 2 =Av.e 2 =- CeHa 
 7 r*4o 
 
 _ 1 C ,r (A 1 sin « + A z sin 3 « + etc. + _Bj cos « + B 3 cos 3 « 
 
 — 77 - J 0 4- etc.) 2 da; 
 
 X 7T > S*1 7 " /»7T 
 
 sin pa sin qada, sin^a cos qada, | cosy>« cos qada, 
 
 J f sin px cos pxda are zero when p and q are unequal integers, 
 there results 
 
 1 r n A 2 a 2 /**■ 
 
 E 2 = — ( e 2 da = — L ( sin 2 ad a + — 3 - ( sin 2 3 ada 
 
 r jp*J 0 77 0 77 0 
 
 A . 2 C n B 2 C n 
 
 sin 2 5 ada + etc. 4 L I cos 2 ada 
 
 7T *'0 7T •/> 
 
 B 2 B 2 C 17 
 
 4 3. I cos 2 3 udu 4 5 - I cos 2 5 ada 4- etc. 
 
 7 r c/0 77 c/0 
 
 4 2 , ,6 2 , i?„ 2 , 5 2 , , 
 
 -y- 4- etc. 4- — 4- — y- 4- 4- etc., 
 
 But X 
 
 7T ( %7r 
 
 sin 2 pud a and 1 cos 2 
 
 hence, 
 
 A 2 1 2 
 
 pj2 = Ai. + th_ + 
 
 and 
 
 (S ^K=n A 2 
 
 E ={L K ,^- 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 43 
 
 The effective value of the curve is therefore independent of /S. 
 The area and the mean ordinate of the curve, on the other hand, 
 are not independent of /3, which, of course, depends on the 
 relative magnitude and algebraic signs of the sine and cosine 
 functions. That is, the effective value of the current depends 
 only on the amplitudes of the harmonics, and is not influenced 
 by their relative positions ; but the average value depends 
 jointly upon the amplitudes and relative positions of the har- 
 monics. 
 
 The following table gives various ratios relating to periodic 
 curves of various forms with known equations or which are 
 illustrated in various figures in this book : 
 
 TABLE 
 
 Characteristic Features of Different Forms of Alternating- 
 current and Voltage Curves 
 
 Name of Curve 
 
 Ratio of 
 Average to 
 Maximum 
 Value 
 
 Ratio of 
 Effective to 
 Maximum 
 Value 
 
 Ratio of 
 Maximum to 
 Effective 
 Value 
 
 Ratio of 
 Effective to 
 Average 
 Value 
 
 Area of 
 Squared 
 Curve corre- 
 sponding to 
 Unit Maxi- 
 mum Value 
 
 Triangle 
 
 .500 
 
 .577 
 
 1.732 
 
 1.155 
 
 .333 
 
 Sinusoid 
 
 .637 
 
 .707 
 
 1.414 
 
 1.112 
 
 .500 
 
 Parabola 
 
 .666 
 
 .730 
 
 1.369 
 
 1.096 
 
 .533 
 
 Semicircle 
 
 .785 
 
 .816 
 
 1.225 
 
 1.040 
 
 .667 
 
 Rectangle 
 
 1.000 
 
 1.000 
 
 1.000 
 
 1.000 
 
 1.000 
 
 Prob. 1. Find the effective value of a voltage having a wave 
 expressed by the formula e = 50 sin (a + 30°) -f 30 sin (3 a + 
 60°)+ 20 sin (5 a + 45°). 
 
 Prob. 2. Find the effective value of a current curve ex- 
 pressed by the formula i = 200 sin (a + 90°) — 50 sin (3 a-f 20°). 
 
 17. Alternating Current Voltmeters and Amperemeters measure 
 Effective Volts and Amperes. — Alternating voltages and cur- 
 rents cannot be measured by instruments depending upon per- 
 manent magnets, so that instruments for such purposes usually 
 fall within three general classes : * 
 
 1. Electromagnetic instruments, depending on the magnetic 
 attractions of two coils, one fixed and the other movable, 
 * Jackson’s Elementary Electricity and Magnetism , Chaps. VIII and XIV. 
 
44 
 
 ALTERNATING CURRENTS 
 
 carrying the current to be measured ; or of a fixed coil or coils 
 and a small movable vane-like bit of iron or disk of conducting 
 metal. The coils may or may not be provided with soft iron 
 cores, with the iron carefully subdivided to prevent excessive 
 eddy currents. 
 
 2. Hot ivire instruments, in which the expansion of a wire 
 due to the heating effect of a current is used to obtain a 
 deflection. 
 
 3. Electrostatic instruments, depending upon electrostatic 
 forces for obtaining a deflection, as in an electrometer. 
 
 In the first two classes the force tending to deflect a needle, 
 suitably attached, is proportional to the square of the current 
 flowing in the coil or wire, and hence is also proportional to the 
 square of the voltage at the terminals of the instrument when 
 it is arranged for use as a voltmeter, except in the cases of 
 instruments with iron cores or vanes, in which the forms of 
 pole pieces or the saturation of the iron may be caused to 
 modify the force. In the third class the tendency of the needle 
 to deflect is proportional to the square of the voltage- im- 
 pressed upon the terminals ; and it is proportional to the square 
 of the current flowing' through a shunt across the terminals 
 of the instrument, if it is arranged with a shunt for the pur- 
 pose of using it as an amperemeter. 
 
 When an instrument of either of these types is inserted in 
 an alternating-current circuit, it tends at each instant to deflect 
 its needle by a force proportional to the square of the instanta- 
 neous current flowing through it or instantaneous voltage im- 
 pressed upon its terminals. The deflection of the needle is 
 resisted by a spring or like device affording a resisting force 
 proportional to the extent of movement. On account of the 
 great frequency of commercial alternating-current circuits the 
 resultant deflection of the needle is ordinarily constant and is 
 proportional to the average of the squared instantaneous values 
 of the current or voltage. The square root of this deflection is 
 proportional to the effective amperes or volts impressed upon 
 the instrument. 
 
 Such an instrument can therefore be calibrated to read 
 effective amperes or volts. When the instrument is so cali- 
 brated, it will read direct or alternating currents or voltages 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 45 
 
 on the same scale, since the force causing the needle’s deflection 
 is in every case proportional to the average of the squares of 
 the instantaneous values of the current or voltage. It is 
 further evident, for the same reason, that instruments of these 
 types will correctly measure alternating currents or voltages 
 whatever may be the forms of their waves, provided that modi- 
 fying influences from iron cores or similar extraneous influ- 
 ences are not allowed to creep in. 
 
 Direct currents and voltages and effective alternating cur- 
 rents and voltages are uniformly indicated throughout this 
 hook by the same symbols, I and U, since they are equivalent 
 in their electro-dynamic or heating effects. 
 
 18. The Effective Voltage developed by an Alternator. — In 
 Art. 11 it is shown that the instantaneous voltage of a multi- 
 polar armature generating a sine voltage curve is 
 
 irpSQ V 
 
 2 xp' x 10 8 x 60 
 
 sin a = — sin cu 
 
 p' x 10 8 
 
 and that then the maximum voltage is 
 
 7rpS<b V _ 
 
 2 x p' x 10 8 x 60 p' x 10 8 
 
 To obtain the effective voltage of a machine which fulfills 
 the conditions of this formula it is only necessary to divide by 
 the square root of 2, since that is the ratio between the maxi- 
 mum and effective values of the sine voltage curve. 
 
 Therefore, 
 
 e m ^ 1.11 p#I> V ^ 2.22 +SW 
 V2 p' x 10 8 x 60 p' x 10 8 
 
 The numerical coefficient in this formula is accurate only for 
 a machine with very narrow coils and with such a distribution 
 of magnetism that the voltage curve is of the sine form. It is, 
 however, correct within a few per cent for a large proportion 
 of the alternators of the present day, as will be seen later.* 
 
 Prob. 1. What is the voltage generated by an alternator 
 having 500 active conductors in series, a magnetic flux from 
 each pole of 2,000,000 lines of force, a speed of 1200 revolu- 
 tions per minute, and five pairs of poles, supposing the formula 
 above to be applicable ? 
 
 * Art. 20. 
 
46 
 
 ALTERNATING CURRENTS 
 
 Prob. 2. The magnetic field of an alternator has been de- 
 signed with six pairs of poles so that 2,000,000 lines of force 
 emanate from each pole. The armature is to run at 500 revo- 
 lutions per minute and generates a sinusoidal voltage. What 
 must be the number of conductors in series to give 1000 volts? 
 
 19. Comparison of the Voltages developed by an Alternator 
 and a Direct-current Dynamo. — In multipolar direct current 
 dynamos 
 
 E _ P S<$>V 
 p' x 10 8 x 60 
 
 Assuming two machines in which /S', p , p ' , <F, and V are alike, 
 one of which has the armature conductors uniformly spaced 
 over the armature surface and produces direct currents, and the 
 other of which has the conductors arranged in a narrow coil 
 and produces alternating currents, the formulas show that the 
 average voltage developed in the alternator armature is equal 
 to that developed in the direct-current armature, but the effec- 
 tive voltage of the alternator should be greater than the aver- 
 age voltage. Calling the ratio of effective to average voltage k, 
 it is seen that the effective voltage of the alternator is k times 
 that of the direct-current machine under the conditions repre- 
 sented by the formulas. The ratio of effective to average volt- 
 age when the curve is sinusoidal is, as already shown, 
 
 v/2 7r 
 
 Hence, the value of k is 1.11. 
 
 The foregoing value of k is computed on the hypothesis of a 
 uniform magnetic field for the alternator, corresponding to that 
 shown in Fig. 12, but the value of k in commercial alternators 
 depends upon the ratio which the width of the poles bears to 
 the polar pitch (distance between poles, center to center), the 
 form of the Polar surfaces and the distribution of the magnet- 
 ism thereover. It has been shown * to have a minimum limit of 
 unity and a maximum limit which may be large, depending 
 upon the form of the voltage wave set up in the individual 
 armature conductors. 
 
 * Art. 11. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 47 
 
 20. The Effect of Distributed Coils on Armature Voltage. — 
 
 It is now well to examine the relative sizes of the two arma- 
 tures compared above, and the effect of the arrangements of 
 the windings upon their voltages. 
 
 Thus far the alternator windings have 
 been assumed to be in narrow coils, or 
 arranged so that at each instant all of 
 the conductors are equally effective. 
 
 This is not the case in actual ma- 
 chines, as the coils must have apprecia- 
 ble width. If two collector rings, C and 
 C v are placed upon the shaft of a direct- 
 current armature, such as a Gramme 
 ring mounted to rotate in a bipolar 
 field, and are connected to armature 
 conductors which are in opposite coils 
 (Fig. 30), an alternating current may 
 be taken from the rings. Such an 
 arrangement is called a Double-current 
 machine. The voltage between the 
 rings has its maximum value when the 
 conductors connected to the rings are 
 under the points of commutation, and 
 the voltage is zero when the armature 
 has revolved 90°.* 
 
 The maximum of the alternating 
 voltage is manifestly equal to the 
 direct voltage for which the armature was designed, and the 
 effective alternating voltage is .707 times the direct voltage 
 if the field is uniform and the voltage wave sinusoidal. In 
 commercial machines the field is not uniform, but it is not 
 likely to be sufficiently irregular to materially disturb the ratio 
 of the direct and effective alternating voltages when such an 
 armature is carrying little current. When the armature carries 
 considerable current, armature reactions may disturb the rela- 
 tions to a greater or less degree. An armature thus arranged, 
 with the conductors uniformly distributed over its surface, there- 
 fore gives substantially a sine alternating voltage, and the value 
 of k would be 1.11 except for the distribution of the conductors 
 * Jackson’s Electromagnetism and the Construction of Dynamos , p. 90. 
 
 Two-pole Field, connected to 
 Collector Rings. 
 
48 
 
 ALTERNATING CURRENTS 
 
 over the armature surface ; but the effective alternating voltage 
 is only .707 times that of the direct voltage developed by the 
 same conductors. The question at once arises as to the cause 
 of this loss of 40 per cent or more. A little consideration shows 
 that the Gramme ring acts like two broad coils in parallel, which 
 cover the whole armature and unite at the points where the col- 
 lecting rings are connected. When in the position of zero vol- 
 tage, the two halves of each coil are so located in the field as to 
 cut lines of force in opposite directions, and hence the voltage 
 is zero. Similar opposition but to less degree is found in the 
 case of windings which cover only a portion of the armature 
 core, the extent of the effect depending upon the ratio of the 
 width and pitch of the poles to the width of the coils. It is 
 largely avoided if the coils are not distributed over a greater 
 width than the distance between the pole tips. Economy of 
 material and the form of the voltage wave are also factors in 
 determining the width the coils should have. 
 
 On account of the differential action of distributed coils 
 which is here described, it is necessary to include another con- 
 stant in the formula for the voltage developed by alternating- 
 current armatures. Calling this constant k \ and replacing the 
 product k'k , by AT, the formula for the voltage of an alternator 
 armature becomes 
 
 KpS&V _2KS<S>f 
 ~ p' x 10 s x 60 ~ p' x 10 s ’ 
 
 Kapp states that K varies from .29 to 1.15,* but in the greater 
 number of commercial cases it is between 1.00 and 1.11. 
 
 Instead of having the windings laid together upon the surface 
 of the armature, it is usual to have them laid in slots as shown 
 in Fig. 31, where the winding is distributed in four slots per 
 pole. If the winding is progressive, as in the Gramme ring, the 
 wires in different slots will set up voltage waves with their max- 
 ima separated by the angular distance between the centers of 
 the slots, as illustrated in Fig. 32, where each slot is supposed 
 to contain the same number of conductors, and the whole width 
 covered by the winding is equal to one half the pitch. The 
 resultant curve, due to connecting the conductors all in series, 
 may be found by adding the instantaneous ordinates together 
 
 * Kapp’s Dynamos, Alternators, and Transformers, p. 374. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 49 
 
 at each abscissa. If the individual curves are sinusoids, the 
 summation is of this form : 
 
 where e is the instantaneous voltage of the resultant curve, 
 e sm is the maximum ordinate of any one of the individual 
 curves, a is the angular distance from the point where the first 
 
 Fig. 31. — Multipolar Armature with Winding distributed in Four Slots. Dotted 
 Lines show Cross Connections at the Rear. 
 
 curve cuts the x axis, /3 is the angular distance between slots, 
 center to center, measured in electrical degrees, and n is the 
 number of individual curves composing the resultant curve. 
 By trigonometry this summation becomes* 
 
 For the purpose of finding the value of a. at which the resultant 
 curve has its maximum height, the first derivative of this 
 function with respect to « may be equated to zero and reduced 
 to 
 
 e = e sm \ sin« + sin (a + /3) + sin(« + 2 /3) + ••• 
 + sin (a + O — l)/3)b 
 
 cos a cos 
 
 n — 1 
 
 o 
 
 or, 
 
 * Hobson's Plane Trigonometry, 2d ed., p. 89. 
 
50 
 
 ALTERNATING CURRENTS 
 
 From this the value of « required to give the maximum ordi 
 nate of the resultant curve is found to be 
 
 « 
 
 m 
 
 7 T 
 
 2 
 
 / 3 , 
 
 Fig. 32. — Curve of Voltage set up in the Separate Slots of Progressive Distributed 
 Windings, and the Resultant Total Voltage. 
 
 and substituting this value of « in the formula for e gives 
 
 n — 1 a , 71 — 1 
 x P + — — - 
 
 a \ ■ 
 
 p ) sin — y cosec 
 
 = e. 
 
 sin (71 3/-) 
 sin (fi/2) ' 
 
 i 
 
 7 T 
 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 51 
 
 The effective voltage is therefore 
 
 sin Q/ 3/ 2). 
 - yo sin (/3/2) ' 
 
 If E' = — is the voltage that would have been set up if all 
 the wires had been in a single narrow slot, the formula becomes 
 
 E' sin (n/3/2) 
 n sin (/3/2) ’ 
 
 but E' is the voltage given by the formula in Art. 18 ; hence, 
 
 2.22 ;SW sin Q/3/2) 
 
 p' x 10 8 n sin (/3/2) 
 
 Evidently K of Art. 20, in this case, is equal to , 
 
 J > n w sin (/3/2) 
 
 k and k' being respectively equal to 1.11 and s ^ n . i n 
 
 61 J ^ n sm (/S/2) 
 
 any instance in which the conductor voltages are of sinusoidal 
 
 form. 
 
 Suppose there are four slots equally distributed through 90 
 electrical degrees in an alternator armature with the conductors 
 of one coil equally distributed in those slots ; then n = 4 and 
 /3 = 221°, whence 
 
 k' = \ sin 45° cosec 11° 15' = .906 (approx.) 
 
 and AT= 1.11 x .906 = 1.006 (approx.), provided the con- 
 ductor voltages are of sinusoidal form. Suppose now that 
 there are a total of 500 active conductors in series, and equally 
 distributed in the slots, 2,000,000 lines of force per pole, and 
 that the frequency is 60 periods per second, then 
 
 F _ 2 KS$>f _ 2 x 1.006 x 500 x 2,000,000 x 60 _ 12n7 j 
 p' x 10 8 1 x 10 8 
 
 Figure 31 shows the slots for a winding such as contemplated 
 in the problem above, and Fig. 32 shows the voltage curves. 
 
 The following table gives the value of K for various propor- 
 tions of the armature core covered with winding and various 
 numbers of slots per pole up to four, assuming sinusoidal con- 
 ductor voltages. 
 
52 
 
 ALTERNATING CURRENTS 
 
 TABLE I 
 
 Values of K for Progressively Distributed Windings 
 
 Group of Slots 
 
 LOCATED IN 
 
 Number of Slots in a Group 
 
 
 i 
 
 2 
 
 3 
 
 4 
 
 45° 
 
 l.n 
 
 1.09 
 
 1.085 
 
 1.08 
 
 60° 
 
 l.n 
 
 1.075 
 
 1.07 
 
 1.065 
 
 90° 
 
 l.n 
 
 1.025 
 
 1.015 
 
 1.01 
 
 135° 
 
 l.n 
 
 .925 
 
 .895 
 
 .885 
 
 180° 
 
 l.n 
 
 .78 
 
 .735 
 
 .725 
 
 In a single phase winding, i.e. an armature having a single 
 winding, any angular width for the group of slots may be used, 
 though about 90° is common. Where there are two independ- 
 ent circuits on the armature, 90° is also used ; while in three- 
 phase circuits, where there are three independent circuits, 60° 
 is common.* 
 
 The values given for K in the table can be used with suffi- 
 cient accuracy for most commercial forms of alternators, as the 
 deviations of the curves from sinusoidal form are usually not 
 great enough to change the constant very much ; but this con- 
 dition cannot be relied on unchallenged.! 
 
 When desired, the curves of conductor voltage may be plotted, 
 and the ordinates of the voltage curve of the machine may be 
 obtained by adding corresponding ordinates of the conductor 
 voltage curves. 
 
 The vector relations of sinusoidal conductor voltages and the 
 machine voltage are illustrated in Fig. 33, where OA , AA ' , A' A", 
 and A"S are the voltages developed by the conductors in the 
 several slots of the foregoing example. These are laid off in a 
 vector polygon. The effective voltage produced in the con- 
 ductor (or conductors) of each slot is 333 volts, and the resultant 
 of the four voltages is 1207 volts as already computed by the 
 formula. These vectors are plotted in their proper relative 
 positions. Their lengths are laid down equal to their respect- 
 
 * Art. 28. 
 
 t De la Tour’s Moteurs Asynchronous Polyphases, Chap. 2. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 53 
 
 ive effective values. The corresponding instantaneous values for 
 the instant corresponding to Fig. 33 are proportional to the pro- 
 jections of the various vectors on the vertical axis. The reason 
 that the coefficient k' is less than unity is plainly shown by 
 this figure, in which it is seen that the component or conductor 
 
 Fig. 33. — Graphical Representation of Voltage generated in an Armature having 
 
 Four Slots. 
 
 voltages are in different phases and their resultant is therefore 
 less in magnitude than their arithmetical sum. 
 
 The complex expression for these vectors is 
 
 OS = COS") +j(OS') = [(06) + (bb 1 ) + (b'b") + (b"S")] 
 
 + j [( Oa) + (aa') + (a' a") + (a" S')], 
 or, ( OS") = (Ob) + (bb') + (b'b") + (b"S"), 
 
 and ( OS') — ( Oa) + (aa') -f (a' a") + (a" S'). 
 
 Having thus obtained the numerical values of the terms of 
 the complex expression for the resultant, the numerical or scalar 
 value of the resultant is obtained from the relations 
 
 ( os) = V( os'y + ( osy. 
 
54 
 
 ALTERNATING CURRENTS 
 
 The value of Oa, aa\ Ob , bb', etc., may readily be found by use 
 of the trigonometrical table ; for instance 
 
 AA' = ( bb ') +j (aa r ) = AA' [cos (« + 221°) + j sin (a + 221°), 
 
 and therefore, as a, in this case, at the instant indicated in 
 Fig. 33, is Ilf, 
 
 (bb') = 333 cos 33|° and (W) = 333 sin 33|°. 
 
 A winding which creates several equal elementary voltage 
 curves following each other at equal angles, but added to obtain 
 a resultant at the collector rings, is called a Distributed winding. 
 Such windings are shown in Figs. 67-70, Chapter III. 
 
 Prob. 1. In an alternator armature there are five slots in 
 a group and the angle covered by them is 120°. What is the 
 value of the constant K for the group ? 
 
 Prob. 2. The effective voltage set up in the conductors in 
 each slot of problem 1 is 50 volts. Determine graphically the 
 total voltage of the machine. 
 
 Prob. 3. Obtain the answer to problem 2 by the method of 
 complex quantities. 
 
 Prob. 4. An armature has four slots in a group covering 
 60°. The conductors in each slot develop 400 volts. Determine 
 the total voltage developed by each group, when the conductors 
 are all connected in series, both graphically and by the method 
 of complex quantities. 
 
 21 . Most Economical Width of Pole Face. — The output of an 
 alternator is proportional to the product £<!> = swbw where s 
 is the number of conductors per unit width of coil, b the number 
 of lines of force per unit width of pole face, and w and w' are 
 respectively the widths of coil and pole face. In order to 
 economize material, the distance between pole tips may be taken 
 as equal to the width of the coil, for the reason explained in the 
 previous article, and this makes w + tv' equal to the pitch of 
 the poles, which is constant ; this makes the product ww\ and 
 hence the output of the machine, a maximum when w is equal 
 to w\ or when the width of coil and pole face are each equal to 
 half the pitch. The result thus derived must be modified to 
 suit practical conditions, since fringing tends to increase the 
 width of field, and armature reactions tend to crowd the field 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 55 
 
 towards the trailing pole tip, thus narrowing the field. Experi- 
 ment has shown that it is best to have the coils somewhat wider 
 than half the pitch, and it is often advantageous to have the 
 poles slightly less than half the pitch in width. 
 
 22. Polyphase Alternators. — As has been shown, only 
 slightly over one half the armature surface is utilized in the 
 
 machines of the type which has been dealt with, and which are 
 called Single-phase machines or Single-phasers from the fact 
 that they give out a single current. 
 
 A second set of windings may be placed between the first, as 
 in Fig. 34, which will give out from a second pair of collector 
 rings or terminals another alternating current displaced 90° from 
 the first. This is indicated in Fig. 35, where A, A\ and B , B', 
 represent the two currents. Such a machine is called a Two- 
 phaser. Figures 36 and 37 show by diagram two methods, and c, c' 
 
56 
 
 ALTERNATING CURRENTS 
 
 of connecting up the coils of two-pkasers. In the former the 
 phases are joined together at a central point called a Neutral 
 
 ARMATURE 
 
 TERMINALS 
 
 Fig. 3(3. — Two-phase Armature. 
 Four Terminals. 
 
 ARMATURE 
 
 TERMINALS 
 
 Fig. 37. — Two-phase Armature. 
 Four Terminals. 
 
 ARMATURE 
 
 TERMINALS 
 
 Fig. 
 
 38. — Two-phase Armature. 
 Three Terminals. 
 
 point ; while in the latter the two phases are entirely inde- 
 pendent. The figures indicate a simple bipolar ring, to avoid 
 
 the complexity that arises in 
 diagraming the windings for 
 multipolar machines, and rep- 
 resent either a stationary or 
 rotating armature. Figure 38 
 shows the coil terminals passing 
 to only three armature termi- 
 nals, which is made possible by 
 utilizing one wire as a common 
 conductor for both circuits. 
 The current in the common 
 wire, as will be shown later,* is V2 times the current in either 
 of the independent wires when the currents are equal in the 
 two circuits and are of 90° difference of phase, which is the 
 condition of a balanced two-phase system; and the voltage be- 
 tween the two independent wires is then V2 times the voltage 
 between either of those and the common wire. 
 
 Instead of two independent circuits being placed on the arma- 
 ture, the winding space may carry three circuits, as illustrated 
 in Fig. 39, and the machine is then a Three-phaser. The three- 
 pkaser gives three voltage curves ordinarily differing from each 
 other in phase by 120° as indicated in Fig. 40, where A. A', B . 
 B\ and C, O ’ , are the three curves. Figures 41 and 42 illustrate 
 such a winding for a simple Gramme ring in which a , a', b , b\ c, c\ 
 
 * Art. 100. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 57 
 
 represent the coils of each phase or winding. In both figures 
 the windings are so connected together that only three arma- 
 
 ture terminals are used instead 
 of three independent pairs. A 
 simple diagram of the connec- 
 tion of the coils of Fig. 41 is 
 shown in Fig. 43. This wind- 
 ing is called a Mesh or Delta(A) 
 winding. The currents which 
 combine from two coils at the 
 armature terminals (A, B , (7) 
 are V3 times greater than the 
 current in a single phase,* 
 
 Fig. 41. — Three-phase Winding. Mesh 
 Connection. 
 
 * Art. 100. 
 
58 
 
 ALTERNATING CURRENTS 
 
 provided the currents are equal in the three windings and are 
 120° apart in their phases, which is the condition of a balanced 
 three-phase system. 
 
 The arrangement shown in Fig. 42 is called a Star or Wye (Y) 
 winding, and the connection of the coils is diagrammed in Fig. 
 
 44. The voltage measured 
 between any two armature 
 terminals connected to the 
 Y winding is V3 times the 
 voltage in the winding of a 
 single phase,* provided the 
 voltages are equal in the three 
 windings and are 120° apart 
 in their phases. 
 
 Machines having more than 
 
 Fig. 42.— Three-phase Winding. one phase or winding are 
 
 St.u Connection. called in general Polyphase 
 
 and sometimes Multiphase machines, or Polyphasers or Multi- 
 phasers. 
 
 The methods of connecting up two-phase armature windings 
 are very simple. If the armature is wound with independent 
 concentrated coils, each coil may be con- 
 nected to its individual armature termi- 
 nals, in which case four terminals are re- 
 quired and the two circuits are entirely 
 independent ; or, one end of each coil may 
 be connected to a common armature ter- 
 minal and the other ends to independent 
 terminals, in which case 
 but three terminals are 
 required and the circuits have a common 
 point. If the armature has a direct-current 
 or closed-circuit distributed winding, such 
 as is treated in Art. 20, it may be made 
 into a two-phaser by connecting collector 
 Fig. 44. - Diagram of Y pings to the windings at the ends of two 
 
 Connection. .. 
 
 diameters which are 90 apart, it the ma- 
 chine is bipolar ; if the machine is multipolar, the connections 
 must be made as described in Art. 28. Four rings are nec- 
 
 Fig. 43. — Diagram of A 
 Connection. 
 
 * Art. 100. 
 
THE VOLTAGE DEVELOPED BY ALTERNATORS 
 
 59 
 
 essary in this case, as the use of a common ling would cause 
 the permanent short-circuiting of one quarter of the armature. 
 A direct-current armature converted into a polyphaser (of any 
 number of phases) in this manner has a capacity when used 
 as a polyphase alternator which is nearly equal to its direct- 
 current capacity, though it has already been shown that its 
 capacity as a single-phaser is only seven tenths as great as its 
 direct-current capacity.* 
 
 The manner of connecting three-phase armatures is not so 
 immediately evident, but is perfectly simple. It is illustrated 
 in Figs. 64 and 65. Three armature terminals are universally 
 used, and if the armature is wound with three independent 
 coils, these may be connected to the terminals in either of two 
 ways : (1) one end of each of the coils may go to a common 
 point, and the other ends go to independent terminals ; or (2) 
 the coil ends may be connected together two and two, form- 
 ing a sort of triangle, and connections be carried to the arma- 
 ture terminals from these points. The latter arrangement makes 
 each coil terminate at each end in an armature terminal, and, 
 since there are six coil ends and three terminals, each terminal 
 is connected with two coil ends. These two arrangements are 
 illustrated respectively in Figs. 65 and 64, each of which shows 
 the winding diagrammatically as developed and as projected. 
 The following considerations make it perfectly easy to connect 
 the coils in the proper order: Fig. 40 shows that when the cur- 
 rent in one coil is at its maximum point, the currents in the 
 other two are equal to each other and opposite to the direction 
 of the first ; then, considering the instant at which the conduct- 
 ors of one coil (such as C in the figure) are directly under the 
 poles, if we connect its positive end to the common or neutral 
 point the negative ends of the other two coils must be con- 
 nected to the same point. Each of the free ends may then be 
 connected to one of the armature terminals and the connection 
 is completed according to the first or Y arrangement (Fig. 65). 
 To make the second or A arrangement, the coils must be con- 
 nected so that, at the instant considered, the' current flows by 
 two paths through the armature from the negative to the posi- 
 tive end of the first coil (coil C of the figure). Consequently 
 the negative ends of the first and second coils go to one arma- 
 
 * Art. 20. 
 
60 
 
 ALTERNATING CURRENTS 
 
 ture terminal, the positive ends of the first and third coils to 
 another terminal, and the free ends of the second and third coils 
 to the third terminal (Fig. 64). 
 
 If the armature has a closed-circuit or distributed winding, 
 the connection is very simple, as the rings (terminals) are con- 
 nected to points in the windings 120 electrical degrees apart. 
 If the machine is multipolar, it must be remembered that one 
 cycle, or 360 electrical degrees, is comprised within the space 
 of twice the polar pitch. 
 
 In a three-phase machine, if the armature is A connected, 
 the voltage between any two armature terminals is equal to the 
 voltage developed in one coil, while the current leaving a ter- 
 minal is composed of the vector currents in two coils; and, if 
 the armature is Y connected, the voltage between the terminals 
 is composed of the vector voltages developed in two coils, while 
 the current leaving a terminal is equal to the current in a coil. 
 The line currents with a balanced load are V3 times the current 
 in a single coil in the A connection, and the voltages between 
 lines or terminals are V3 times the voltage of a single coil in 
 the Y connection, so that the capacity of a machine is independ- 
 ent of the way in which its armature is connected, but, for a 
 given voltage and output, the windings will differ (though the 
 weight of copper will be the same) for the two arrangements.* 
 
 Prob. 1. 100 volts are set up in each winding of a balanced 
 Y connected three-phase alternator armature. What are the 
 voltages between the terminals of the machine ? 
 
 Prob. 2. 200 volts are generated in each coil of a balanced 
 A connected three-phase alternator armature. What are the 
 voltages between the terminals of the machine ? And when 
 50 amperes flows in each coil, what is each line current? 
 
 Prob. 3. The two windings of a balanced two-phase armature 
 are connected to three line wires. The voltage developed in 
 each winding is 100 volts; the line wires are designated A, B. C , 
 where B is the common return wire. What are the voltages 
 between A and 74, B and (7, and A and C ? 
 
 Prob. 4. In problem 4 the current flowing in each phase is 
 50 amperes. What is the current in the common return wire ? 
 
 * Chap. VIII. 
 
CHAPTER II 
 
 ELEMENTARY STATEMENTS CONCERNING TRANSFORMERS 
 AND MEASURING INSTRUMENTS 
 
 23. The Fundamental Principle of the Transformer. — If a coil 
 of wire is wound upon a closed iron core of high magnetic con- 
 ductivity, i.e. low reluctance, as illustrated in Fig. 45, a small 
 current passed through 
 the coil will produce a 
 powerful magnetic flux. 
 
 If an alternating current 
 is passed through the coil, 
 the magnetic flux changes 
 with the cycles of the cur- 
 rent and the chaneino - ^ig. 45. — Coil Wound upon an Iron Core of Low 
 ’ , .. ‘ ° ° Reluctance, 
 
 number ot lines ot force 
 
 linked with the coil create a Counter voltage in the coil which 
 opposes the tendency of the voltage impressed at the terminals 
 to send current through the coil. This action is in accord with 
 a natural law which may he stated as follows : When a current 
 
 in a conductor is caused to change in value , or a conductor is 
 moved in a magnetic field so as to cut lines of force , an induced 
 
 electro-motive force is set up in the 
 conductor which tends to oppose the 
 change in current value or the move- 
 ment of the conductor. This state- 
 ment is a corollary of the Laiv of 
 the Conservation of Energy. 
 
 If another coil is wound upon 
 the same core, as illustrated in 
 Fig. 46, the magnetic flux set up 
 by current in the first coil passes 
 through the second coil also, and 
 the changes in the strength of the 
 field set up an induced voltage in this second coil similar to 
 that in the first. An Induction coil of this character, used with 
 
 61 
 
 Fig. 46. — Low Reluctance Iron 
 Core carrying Primary and Sec- 
 ondary Coils. 
 
62 
 
 ALTERNATING CURRENTS 
 
 alternating currents, is called a Transformer. The coil which 
 receives the current from an outside source is called the 
 Primary coil and that in which voltages are induced by the 
 magnetic flux set up by current in the primary coil is called 
 
 the Secondary coil. 
 
 The phenomenon is called Self-induction when electro-motive 
 force is induced in a coil as the result of changes of the current 
 in its own conductors ; and the phenomenon is called Mutual- 
 induction when an electro-motive force is induced in a coil as 
 the result of changes of the current flowing in a neighboring 
 coil. When an alternating current flows through a primary 
 coil, it is obvious that the magnetic flux set up in the iron core, 
 such as is illustrated in the transformer of Fig. 46, must also 
 be alternating and of the same period ; and that the induced 
 voltages in both the coils are also alternating and of the fre- 
 quency of the primary current. 
 
 The maximum voltage induced per turn of wire in either coil 
 is proportional to the maximum value of the alternating mag- 
 netic flux which links the coil and to the rate of change of the 
 flux. The latter is proportional to the frequency of the current. 
 Therefore, if all the turns of each coil are in series and the 
 same magnetic flux links the two coils, the voltages induced in 
 the primary and secondary coils are respectively proportional 
 to their number of turns ; or 
 
 where R[ and U' 2 are the induced voltages and n v n 2 are the 
 turns in the primary and secondary coils, respectively. 
 
 In the case of commercial transformers, when no current is 
 flowing in the secondai-y coil, the self-induced voltage in the 
 primary coil almost equals the voltage impressed at its termi- 
 nals. This is because only a very small Exciting current is 
 required for the core magnetization necessary to set up an 
 induced voltage equal to the impressed voltage, and the voltage 
 used in sending the current through the resistance of the con- 
 ductors of the primary coil is practically negligible. The in- 
 stantaneous value of counter voltage at each instant must be 
 smaller than the corresponding instantaneous value of the 
 impressed voltage by an amount equal to the IR drop at that 
 
ELEMENTARY STATEMENTS 
 
 63 
 
 instant. The IR drop in a commercial transformer is usually 
 very small when the secondary circuit is open. 
 
 It may therefore be said that, in an ordinary unloaded com- 
 mercial transformer, the ratio of the primary impressed voltage 
 to the secondary induced voltage is substantially equal to the 
 ratio of the numbers of turns in series respectively in the 
 primary and secondary coils. That is, substantially, 
 
 where R 1 is the primary impressed voltage. 
 
 The induced voltage set up in the winding of the secondary 
 coil is obviously in phase with the counter voltage of the pri- 
 mary coil, assuming them to be set up by the same magnetic flux, 
 and this secondary voltage is practically opposite to the voltage 
 impressed on the primary coil. Closing the circuit of the 
 secondary coil through resistance (such as a number of incandes- 
 cent lamps) allows a cui’rent to flow under the impulse of the 
 secondary induced voltage, which current is opposed in phase 
 to the current caused by the impressed voltage to flow through 
 the primary coil. The result is that the magnetizing effect of 
 the primary current is opposed by the magnetizing effect of the 
 secondary current, and the primary current must increase suf- 
 ficiently to overcome the magnetic effect of the secondary cur- 
 rent and continue to magnetize the core as before. The ampere 
 turns of the primary coil must therefore increase by an amount 
 equal and opposite to the ampere turns of the secondary coil. 
 That is, ... 
 
 where n 2 i 2 are the instantaneous ampere turns of the secondary 
 coil, n 1 i 1 are the ampere turns in the primary coil, and n^, u rep- 
 resents the part of the primary ampere turns required to set up 
 the magnetic flux in the core at the corresponding instant. 
 
 The current in the primary coil, then, must be slightly 
 
 greater than 2 i 2 in order to supply the core magnetization, 
 
 n i 
 
 but in commercial transformers this extra magnetizing current 
 is small, and may usually be neglected ; and the statement may 
 therefore be made that the currents in the primary and secondary 
 coils of the usual commercial transformers designed for constant 
 
64 
 
 ALTERNATING CURRENTS 
 
 voltage service are almost inversely proportional to the number 
 of turns in the two coils. That is, approximately, 
 
 When currents flow in the secondary coil, the voltage lost in 
 the resistance of the coils can no longer be neglected. This 
 drop at full load amounts to from one to five per cent of the 
 voltage at no load in the transformers of ordinary practice, and 
 the ratio of the voltages at the primary and secondary terminals 
 is no longer correctly represented by the ratio of the numbers 
 of primary and secondary turns. Magnetic leakage, whereby 
 the magnetic flux linking one coil becomes different from the 
 flux linking the other coil, also affects the ratio of voltages. 
 These deviations from ideal action are fully discussed in a 
 later chapter. 
 
 A transformer may be defined as a combination of coils and 
 magnetic core for transforming alternating currents of a given 
 
 voltage into propor- 
 tional alternating 
 currents of another 
 fixed voltage with- 
 out the intervention 
 of physical motion. 
 The core of a trans- 
 former must be 
 Laminated, that is, 
 made up of very 
 thin sheets of soft 
 iron or steel laid 
 parallel to the lines 
 Figure 47 shows 
 
 TERMINAL OF 
 PRIMARY 
 W'NDING 
 
 TERMINALS OF 
 SECONDARY 
 WINDINGS 
 
 TERMINAL OF 
 PRIMARY 
 WINDING 
 
 TERMINALS OF 
 SECONDARY 
 WINDINGS 
 
 Fig. 47. — Skeleton View of a Transformer. 
 
 of force to prevent the flow of eddy currents 
 a skeleton view of a commercial transformer. 
 
 If there were no losses of power in the operation of a trans- 
 former, the power given out by the secondary coil would 
 obviously be equal to the power received by the primary coil ; 
 that is, P 2 would be equal to P r But an actual transformer 
 cannot be operated without I 2 R losses in the conductors, 
 hysteresis losses in the core, and eddy current losses in the 
 core and perhaps other parts, so P 2 is always smaller than P y 
 
ELEMENTARY STATEMENTS 
 
 65 
 
 The difference is only a few per cent of the input in the case 
 of a well-designed modern constant voltage transformer oper- 
 ated at normal full load. 
 
 A modified transformer having the secondary mounted upon 
 a rotating core and so arranged as to deliver mechanical instead 
 of electrical power is called an Induction motor. A discussion of 
 transformers and induction motors is to be found in Chapter XI. 
 
 Prob. 1. A certain transformer delivers to the external 
 secondary circuit a power of 50 kilowatts. If the efficiency 
 of the transformer is 96 per cent at this load, what power must 
 be supplied to it ? 
 
 Prob. 2. In a fully loaded transformer the voltage supplied 
 is 1000 volts, the current 50 amperes, and the secondary voltage 
 is 100 volts. What is the approximate value of the secondary 
 current ? 
 
 24. Alternating-current Amperemeters, Voltmeters, and Watt- 
 meters.* — The three general classes of direct-reading instru- 
 ments most suitable for making measurements in alternating- 
 current circuits have already been given. f Working gear for 
 each of the three most important of the instruments for com- 
 mercial electrical measurements (namely, amperemeters, volt- 
 meters, and wattmeters) may be made on the principles of 
 either of the three types named in Art. 17. Paragraphs a, d , 
 and e. following, deal with the first class. 
 
 a. Electro-dynamometers . — Large numbers of the best known 
 alternating-current instruments depend upon the electro-dyna- 
 mometer principle, i.e. the magnetic action of the current in 
 one coil upon the magnetic field of a current in another coil. 
 Such instruments must be constructed without masses of solid 
 conducting material about them, or eddy currents may be set 
 up and the readings of the instruments become affected by the 
 frequency of the current to be measured. This is due to the 
 dynamic effect which eddy currents (induced short-circuit 
 currents) circulating in the metallic masses produce upon the 
 
 * Jackson’s Elementary Electricity and Magnetism, Chaps. XIII, XYI, and 
 §231. 
 
 t Art. 17. 
 
 F 
 
66 
 
 ALTERNATING CURRENTS 
 
 currents in the moving parts of the instruments.* Likewise, 
 iron cores can be used only with the utmost caution in the 
 construction of the working gear, even though laminated (built 
 up of thin wires or sheets) to lessen eddy currents, as the 
 hysteresis of the iron is likely to impair the accuracy of the 
 readings.) If the instrument is to be used as a voltmeter, 
 the self-induction and electrostatic capacity of the wdn dings 
 must be reduced to negligible values, or the readings may be 
 affected by the circuit frequency. The elimination of self- 
 inductive or capacity effects from the voltage coils of watt- 
 meters is also very important, as is fully demonstrated later. ) 
 
 If any of these harmful elements are present, the instrument 
 must be calibrated under the exact conditions of frequency, 
 current, or power, with which it is to be used, or the readings 
 may prove to be quite erroneous. On the other hand, such 
 
 instruments, property constructed, 
 can be relied on to read with equal 
 accuracy in alternating-current cir- 
 cuits having any of the commercial 
 frequencies ; and they may ordinarily 
 be calibrated in a direct-current cir- 
 cuit (unless iron is used in the work- 
 ing gear). Figure 48 illustrates a 
 long-used form of electro-dynamom- 
 eter arranged for use as an ampere- 
 meter. This is often called the 
 Siemens Electro-dynamometer . This 
 form gave excellent service, though 
 it is somewhat clumsy. One coil. F, 
 in this instrument is fastened to the 
 frame of the instrument, and the 
 
 Fig. 48. — Siemens Electro-dyna- 
 mometer and Diagram of Con- 
 nections to Circuit. 
 
 other coil, M, which stands at right angles to the first, is 
 suspended by a heavy silk fiber, so that it is free to move. 
 The ends of the wire composing the movable coil dip into 
 little cups, (7(7, containing mercury, which are connected with 
 the main circuit so that the current can enter and leave the 
 coil. The movable coil is attached to a spring 6r, the other 
 end of which is connected to a thumbscrew T, called a Torsion 
 
 * Arts. 90 and 111. 
 
 t Art. 106. 
 
 I Art. 90. 
 
ELEMENTARY STATEMENTS 
 
 67 
 
 head , by means of which the spring may be twisted. When a 
 current flows in the coils, the magnetic force tends to turn the 
 movable coil around so as to place it parallel with the fixed 
 coil. This force is balanced by twisting the spring by means 
 of the thumbscrew. The amount of twist, as shown by a 
 pointer B attached to the screw, is proportional to the force 
 exerted by the coils on each other. This force is proportional 
 to the square of the current flowing in the coils, since the coils 
 are connected in series and the magnetism set up by each coil 
 is proportional to the current, and they act upon each other 
 mutually. The pointer N indicates whether the movable coil 
 is at its zero position. The “ binding posts ” for connecting 
 the instrument into circuit are shown at AA. 
 
 According to the known laws of electro-dynamics, the torque 
 acting at each instant to rotate the movable coil is proportional 
 to the product of the simultaneous instantaneous values of the 
 currents flowing in the coils, and is equal to this product multi- 
 plied by a constant of the instrument which is fixed by the rela- 
 tive dimensions, the numbers of turns, and the relative positions 
 of the two coils. The average torque acting on the movable 
 coil through each period of an alternating current is therefore 
 proportional to the average (taken through the period) of the 
 products of the corresponding instantaneous currents in the 
 two coils ; and when the two coils are in series relation, as in 
 the electro-dynamometer type of amperemeter or voltmeter, 
 the current at each instant is the same in the two coils, and the 
 average torque is proportional to the average of the squares of 
 the instantaneous values of the current. That is, the average 
 torque is equal to iT(av. * 2 ), where K is the constant of the 
 instrument fixed by the construction. This average torque is 
 balanced by the spring attached to the torsion head in the 
 Siemens electro-dynamometer; and if the spring is uniform, 
 the movement of the torsion head required to hold the mova- 
 ble coil in its initial position is proportional to (av. « 2 ) = I 2 . 
 
 Instruments of the electro-dynamometer type are more con- 
 venient if the movable coil carries a pointer which moves over 
 a scale. This arrangement has largely displaced the Siemen’s 
 type. Such an instrument, of Weston make, is illustrated in 
 Figs. 49 and 50. This instrument is like that of Fig. 48 in 
 principle, but it has lighter coils connected in series with a large 
 
68 
 
 ALTERNATING CURRENTS 
 
 non-inductive resistance JK to reduce the time constant ot the 
 instrument circuit, and make it practicable for use as a volt- 
 meter.* The resistance of the commercial alternating-current 
 
 voltmeters of this 
 type is from about 
 20 to 40 ohms per 
 volt of the maximum 
 reading of the scale. 
 The construction of 
 the voltmeter shown 
 in Figs. 49 and 50, 
 which are lettered 
 alike, is as follows : 
 B is the movable 
 coil, and AA is the 
 stationary or fixed 
 coil ; AT, iV T , 0 are 
 
 Fig. 49. — Perspective View of Interior of Weston Alter- ]ji n( ^p n( -r posts • J 
 nating-current Voltmeter from below. e I ‘ ’ ’ 
 
 K are extra resist- 
 ance coils, wound non-inductively \ L is a special variable 
 resistance used to correct the readings for variations of tem- 
 perature ; D is a push-button switch ; C , , C 1 are springs through 
 which the current en- 
 ters and leaves the 
 
 movable coil ; P is the 
 needle or pointer ; H is 
 the scale, which is en- 
 graved with two sets of 
 figures; and 6r is a ther- 
 mometer with its bulb 
 near the coils of the in- 
 strument, and its stem 
 in view near the scale 
 of the instrument. The 
 instrument 
 
 boxed up so that only 
 the scale H , IT, over which the pointer moves, the end of the 
 pointer, the dial of L , and the stem of the thermometer are 
 
 is usually Fig. 50. — Diagram of Weston Alternating-current 
 
 Voltmeter. 
 
 * Jackson's Electricity and Magnetism, pp. 250, 251. 
 
ELEMENTARY STATEMENTS 
 
 G9 
 
 visible. The thermometer is not always essential when the best 
 modern resistance material of low temperature coefficient is 
 used. 
 
 When the instrument is in service, the voltmeter is connected 
 to the circuit by means of the binding post 0 , and either the 
 binding post M or the binding post N. When the button D is 
 depressed, current flows through the fixed and movable coils, 
 and through the resistance J -f K or J alone (depending upon 
 which binding post, M or W, is used), and the pointer is made 
 to move over the scale by the movement of the’ movable coil, 
 which action is caused by the electro-magnetic attractions be- 
 tween the current in its windings and the current in the wind- 
 ings of the fixed coil. 
 
 A pointer on the dial L is set at a mark which corresponds 
 to the temperature indicated by the thermometer, and more or 
 less of the resistance coils connected to the dial are thus in- 
 cluded in the voltmeter circuit. In this way the resistance of 
 the voltmeter, measured from binding post to binding post, may 
 be kept uniform, regardless of the temperature of the instru- 
 ment, and the readings are thus corrected for the variations of 
 temperature. The resistance of the windings AA and B , added 
 to the resistance of J", is just equal to one half of the resistance of 
 the same windings plus the resistance of J -\- K. Consequently 
 only half the voltage between the instrument terminals is re- 
 quired to cause a given movement of the needle when the bind- 
 ing posts 0 and N are used, as when the binding posts 0 and 
 M are used. The scale which reads up to 7.5 volts, in the 
 instrument illustrated, is therefore used in connection witli 
 binding posts 0 and W, and the scale which reads up to 15 volts, 
 with the binding posts 0 and M. The movement of the coil 
 and pointer is opposed by the springs C, C v and the scale is 
 engraved so that the instrument is direct reading. As the 
 springs offer an opposing force which is practically proportional 
 to the extent of movement of the movable coil, and the 
 torque acting to move the movable coil is proportional to the 
 square of the current, it is obvious that the scale of the instru- 
 ment must widen out from left to right. That is, the scale is 
 a “ square-root scale,” but by proper design of the coils of such 
 instruments it is possible to make the widest divisions at the 
 most important part of the scale. 
 
70 
 
 ALTERNATING CURRENTS 
 
 Fig. 51. — Diagram of Wattmeter Connections. 
 
 If an electro-dynamometer is to be used as a wattmeter, the 
 movable coil is usually placed in series with a large non-induc- 
 tive resistance and connected across the circuit, and the station- 
 ary coil is of low 
 resistance and is con- 
 nected in series with 
 the circuit. This is 
 illustrated in Fig. 51. 
 Under these circum- 
 stances the instru- 
 ment can evidently 
 be calibrated to 
 measure watts, since 
 the torque is proportional to the average of the products of 
 the instantaneous voltages with the corresponding instanta- 
 neous currents. Figure 52 shows a portable form of wattmeter 
 with its cover removed so as 
 to show the working parts, 
 which are indicated by let- 
 ters. MM and 0 0 are bind- 
 ing posts, the former of 
 which are terminals for the 
 “ voltage coil,” and the latter 
 terminals for the “ current 
 coil ” (one of the latter is at 
 
 the far corner of the instru- Fig. 52. — Partial Perspective View of Hoyt 
 ment and is hidden); A is 
 
 the stationary or fixed coil; BB is the movable coil: P is the 
 pointer or “needle”; H is the scale; K is a non-inductive 
 extra resistance coil which is placed in series with the voltage 
 coil; B is a torsion head by means of which the spring E may 
 be turned so as to bring the movable coil, which is attached to 
 it, into zero position. This position is indicated by the pointer 
 P ' . When the pointer P' points to zero, the pointer _P, which 
 is attached to the torsion head, points to the reading of the 
 instrument. R and Q are the wooden base and supports of the 
 instrument. CC are flexible conductors for affording the cur- 
 rent ingress and egress to and from the movable coil. 
 
 Wattmeters are usually made so that the needle P is directly 
 attached to the movable coil in the manner illustrated in FJcr. 
 
ELEMENTARY STATEMENTS 
 
 71 
 
 50; and in that case the flexible conductors CC are replaced 
 by conducting springs, also as illustrated in Fig. 50. 
 
 Electro-dynamometers are made with an endless variety of 
 details. One type of special accuracy is the Kelvin balance, 
 which is frequently used as a standard for calibrating less per- 
 manent instruments. Such a balance, illustrated in Fig. 53, 
 
 consists essentially of two electro-dynamometers which tend to 
 turn an arm pivoted at the center between the movable coils. 
 The fixed and movable coils in these instruments are parallel 
 to each other and horizontal. The two movable coils are con- 
 nected in circuit in series with each other so that the current 
 in one circulates in the same direction as the current in the 
 other. This balances the effect of any reasonably uniform 
 external magnetic field (such as the earth’s field) on the bal- 
 ance arm and eliminates such effects from the readings, which 
 is very desirable when the instrument is used in connection 
 with direct currents. When the instrument is used with alter- 
 nating currents, the external fields would not, in any event, 
 affect the readings unless these fields also were alternat- 
 ing, and of the same frequency. The force with which the 
 movable coils tend to move when a current flows in the sets of 
 instrument coils is directly balanced and weighed by means of 
 a slider A moving on a scale beam B. Some of the balances 
 having a large number of turns are not suitable for alternat- 
 ing currents on account of the coils being of too high self- 
 inductance. 
 
 b. Hot-wire Instruments. — If a heated wire is carefully 
 inclosed so that its temperature is not affected by air currents, 
 it will rise a definite number of degrees in temperature for 
 every current that is passed through it, and the rise is propor- 
 
ALTERNATING CURRENTS 
 
 VI 
 
 tional to the energy expended in the wire and therefore to the 
 square of the current. The length of wire increases practically 
 in direct proportion to its rise in temperature when it is heated, 
 and the length again decreases when the wire is cooled. Con- 
 sequently, when currents of different strengths flow through a 
 wire, it will take up a corresponding length with each current, and 
 measuring its length therefore measures the square of the current. 
 
 Hot-wire instruments evidently average up the instantaneous 
 currents squared, since the heat developed in the wire is 1 2 B and 
 (the wire being protected so that the coefficients of radiation 
 
 and convection are constant) 
 the temperature is proportional 
 to the rate at which heat is lib- 
 erated in the wire. Figrure 54 
 shows a hot-wire switchboard 
 instrument. The current passes 
 through the platinum silver 
 wire A and this fine wire is so ar- 
 ranged, by the use of special 
 springs, that its elongation oper- 
 ates the pointer B , which is 
 mounted on jewels. The wire is 
 of too low resistance to adapt the instrument for direct use as 
 a voltmeter on ordinary voltages, and a resistance coil, non-in- 
 ductively wound, is put inside the case and connected in series 
 relation with the wire to make a voltmeter. The wire is of 
 small current-carrying capacity and must be shunted to adapt 
 the instrument for use as an amperemeter unless the current is 
 only a few milliamperes. Hot-wire instruments have not been 
 used commercially as wattmeters. 
 
 A shunt, except for very 
 small currents, is ordi- a wire ot 
 
 Fig. 54. — Hot-wire Instrument. 
 
 MAIN CIRCUIT 
 
 MILLI AMMETER 
 
 narily placed within the 
 case and permanently at- 
 tached in most types of 
 portable amperemeters ; 
 unless the instruments are 
 to be used for large currents or on switch-boards, when the 
 shunt is separated from the instrument and they are connected 
 in circuit as illustrated in Fig. 55, or, as is frequently the case, a 
 
 Fig. 55. — Ammeter with Shunt. 
 
ELEMENTARY STATEMENTS 
 
 73 
 
 current transformer takes the place of the shunt. The shunt 
 should be made of material having a low temperature coefficient 
 and large surface and should be substantially non-inductive. 
 
 c. Electrostatic Instruments. — Electrostatic instruments are 
 built on the principle of the electrometer. Figure 56 shows the 
 plan of a quadrant electrometer. If the 
 needle (which is indicated by the dotted line 
 in the illustration) and a pair of opposite 
 quadrants are connected to a point in an elec- 
 tric circuit and the other pair of quadrants to 
 another point in the circuit, the needle will 
 tend to deflect with a force proportional to 
 
 x A’ IKj. tIU. A 1 iX 11 UA 
 
 the square of the difference of potential be- Quadrants and 
 
 tween the points. This must be so, since Needle for a Quad- 
 
 , , ... , • rant Electrometer. 
 
 the force tending to move the needle is pro- 
 portional to qq', where q and q' are the electric charges on the 
 two elements of the instrument respectively, and since both 
 these quantities vary with the difference of potential between 
 the points of connection with the circuit. Evidently qq' may 
 
 be written Aq 2 , where A is a constant 
 depending upon the shapes and dimen- 
 sions of the parts of the instrument. If 
 the movement of the needle is opposed 
 by a uniform spring, as usual in elec- 
 trical instruments, the deflections will 
 be proportional to the torque and there- 
 fore the readings will be: 
 
 D = .5( av. j 2 ) = _ST( av. e 2 ), 
 
 where B and K are constants. Such 
 an instrument can therefore be cali- 
 brated to read effective volts, hut (like 
 the electro-dynamometer voltmeter) it 
 has a “square-root scale.” Figure 57 
 shows a Kelvin electrostatic voltmeter 
 arranged to be used commercially. It 
 is seen from the figure that there are 
 a large number of sets of quadrants 
 and needles, one above the other. The needles receive their 
 charges through a fine phosphor bronze wire which acts as a 
 
 Fig. 57. — Kelvin Electro- 
 static Voltmeter. 
 
74 
 
 ALTERNATING CURRENTS 
 
 supporting filament for the needles and also furnishes the nec- 
 essary restraining force. 
 
 If an instrument of this class is to he used as an ammeter, it 
 must be in connection with a shunt, though it is not well suited 
 for this purpose. In this case the needle deflects proportionally 
 to the square of the voltage drop in the shunt, and hence to the 
 square of the current flowing. Special arrangements are made 
 when electrostatic instruments are applied to power measure- 
 ments, as will be explained later.* 
 
 d. Magnetic Vane Instruments . — Magnetic vane voltmeters 
 and ammeters are instruments in which a very light vane or 
 needle of soft iron is caused to turn under the influence of a mag- 
 netic field. The magnetic vane will theoretically have a torque 
 exerted upon it proportional to the instantaneous current 
 squared, since the torque is proportional to the product of the 
 magnetic field due to the current in the coil and the field in- 
 duced in the iron vane, which is in turn proportional to the 
 current if the intensity of magnetization is low. Hysteresis 
 will not materially affect the readings in connection with 
 alternating currents if the vane is of very small volume. Such 
 instruments cannot be used readily as wattmeters, and they are 
 generally looked upon as less reliable than electro-dynamometer 
 
 instruments for 
 standard am- 
 peremeters and 
 voltmeters. 
 They are much 
 used for switch- 
 board instru- 
 ments which 
 may be expected 
 to be subjected 
 to currents of a 
 single fixed fre- 
 quenc)'. An in- 
 strument of this 
 type is illus- 
 
 Fig. 58. — Plan of Thomson Alternating-current Amperemeter. ^ p;) led with its 
 
 cover taken off so as to expose the working parts, in Fig. 58. 
 
 * Art. 99. 
 
ELEMENTARY STATEMENTS 
 
 75 
 
 The parts of this instrument are indicated by the letters, where 
 j D is the current coil, C the thin movable iron vane, B the 
 needle, S the scale, and AA the binding posts which are con- 
 nected to the coil I) by the wires WW. 
 
 e. Induction Instruments. — Induction instruments are made 
 on the same principle as the integrating or recording instru- 
 ments fully described later. In the usual form, they include 
 coils which induce currents in other coils by transformer action. 
 The magnetic fluxes due to the currents of the various coils 
 
 WATTMETER 
 
 CURRENT COIL- 5, 
 
 combine to form what is called a rotating field. This in turn 
 sets up eddy currents in a movable disk, or shell of metal, to 
 which the pointer is attached, which cause a torque between 
 the disk and coils. 
 
 25. Arrangement of Instruments for Measuring High Voltages 
 or Large Currents. — In using a wattmeter where very high 
 voltages are met, considerable difficulty is found in arranging 
 a satisfactory non-inductive resistance for the voltage coil. 
 This difficulty may be over- 
 come by the use of a trans- 
 former. Instead of connect- 
 ing the voltage coil of the 
 wattmeter across the termi- 
 nals of the test circuit, the 
 primary of a transformer is 
 so connected, and the vol- 
 tage coil of the wattmeter 
 is connected to the sec- 
 ondary of the transformer (Fig. 59). The constant of the 
 wattmeter is then dependent upon the ratio of transformation 
 of the transformer, which may be readily measured. This 
 method gives fairly reliable results, since the phases of the 
 primary and secondary voltages of a very slightly loaded trans- 
 former are almost exactly 180° apart. If the wattmeter con- 
 stant is determined without the transformer, its constant when 
 in use with the transformer must be multiplied by the ratio of 
 transformation; but wherever practicable, the calibration ought 
 to be made of the wattmeter and transformer as a unit. In 
 like manner if the current is too great to conveniently pass 
 through the instrument, a series transformer may be used, as 
 shown in Fig. 60. In such a case if the wattmeter has a scale 
 
 Fig. 59. — Wattmeter connected to High-vol- 
 tage Circuit through a Voltage Transformer. 
 
7G 
 
 ALTERNATING CURRENTS 
 
 ■WATTMETER 
 
 Fig. 60. - 
 
 "“ITSTr* - 
 
 SERIES TRANSFORMER 
 
 -Wattmeter connected to a Circuit of Large Cur- 
 rents through a Series Transformer. 
 
 for use without the transformer, the readings must be multi- 
 plied by the ratio of transformation (the ratio of the main cur- 
 rent to that in 
 the transformer 
 secondary). 
 Changes of fre- 
 quency change 
 the reading 
 slightly where a 
 series transform- 
 er is used, so that 
 it is necessary to 
 calibrate the 
 wattmeter with 
 the transformer 
 for the frequency of the circuit to be measured. Evidently 
 amperemeters and voltmeters can have the current or voltage 
 reduced in the same manner. For high-voltage switchboard 
 instruments it is very desirable to use both series current and 
 voltage transformers with the switchboard wattmeters, and 
 indeed with all such switchboard instruments, as in this way 
 the danger of accident from the high voltage can he greatly 
 reduced. 
 
 Non-inductive resistance coils can be used in series with the 
 voltage coils of any instrument, as already described, hut it is 
 quite difficult to secure a satisfactory resistance device without 
 appreciable self-inductance or capacity for voltages larger than 
 a few hundreds of volts. Shunts may be provided for the cur- 
 rent coils, but in the case of instruments of the electro-dynamic 
 or magnetic vane type the shunts are apt to cause frequency 
 errors, and in any event the dangers due to direct connec- 
 tion with the high-voltage circuits are not eliminated as when 
 transformers are used. 
 
CHAPTER III 
 
 ARMATURE AND FIELD WINDINGS FOR ALTERNATORS. 
 
 MATERIALS OF CONSTRUCTION. 
 
 26. Classification of Armatures. — Dynamo armatures may 
 be classified under three divisions : 
 
 1. Where a wire cuts lines of force by moving across them, 
 
 as in the case of a slider or of a wire moving around a 
 
 magnet pole. 
 
 2. Where a coil, or set of coils, is moved parallel to itself, or 
 
 nearly so, between points of different field strength. 
 
 3. Where a coil, or set of coils, is wound on a ring or drum 
 
 and given a rotary motion in a fixed magnetic field. 
 
 The first division includes only the so-called unipolar arma- 
 tures, and requires no treatment here. Alternator armatures 
 are in general classed in the second division. 
 
 It is most usual for the field to move instead of the armature, 
 and in some cases neither the armature nor field coils move, but 
 the magnetism linking the former is varied by the revolution of 
 iron Inductors. Where the field rotates, or the variation of the 
 magnetism is effected by moving inductors, the voltage gen- 
 erated in the armature windings is produced in exactly the 
 same manner as if they moved, and such armatures therefore 
 belong to the second division. That is, the generation of the 
 voltage is dependent upon the relative motion of armature with 
 respect to field, and this relative motion may be effected by the 
 mechanical rotation of either. 
 
 The construction of the machines thus enumerated requires 
 an additional classification into : 
 
 1. Alternators with moving fields. 
 
 2. Alternators with moving armatures. 
 
 3. Inductor alternators. 
 
 The armatures of alternators belonging to the first and sec- 
 ond of these classes are almost always, in the United States, 
 
 ’ 77 
 
78 
 
 ALTERNATING CURRENTS 
 
 Chord wound, i.e. the winding of a coil passes across the face of 
 the armature under one pole face, and returns under the next 
 pole of opposite sign without spanning an entire diameter of 
 the armature. The arrangement is the same whether the arma- 
 ture revolves within the field or the field revolves within the 
 armature. Such a disposition is termed Drum winding, in con- 
 tradistinction to Ring winding, where the wires pass through 
 and around a ring-shaped core, as in the armature invented by 
 Gramme. 
 
 The windings may further be distinguished as Bar and Coil 
 windings. In the former, rectangular bars are laid in slots 
 across the face of the armature, and these are properly con- 
 nected together at their ends by suitable welded, brazed, or 
 bolted connections ; and in the latter, coils of either small rec- 
 tangular or circular wire are wound upon a former, insulated, 
 and then placed in the armature slots. 
 
 The wires of a coil are sometimes bunched together in a 
 
 single slot, as shown in Fig. 34, but are more commonly divided 
 into several small coils, each occupying its own slot. The lat- 
 ter arrangement is called Distributed winding, because the active 
 conductors are distributed more or less evenly over the face of 
 the armature. (See Figs. 31, 67, 68, 69, 70.) 
 
 27. Examples of Armature Windings. — Essentially the same 
 
 forms of windings are used 
 for machines with revolving 
 armatures and machines with 
 revolving fields, and those 
 described in the following 
 paragraphs are applicable to 
 botli classes. In some of the 
 illustrations, collector rings 
 and an external crown of 
 poles are shown, for conven- 
 ience, for revolving arma- 
 tures ; but the same type of 
 winding may as well be used 
 for a stationary armature, in 
 which case the crown of 
 poles is internal and the armature terminals are stationary. The 
 original type of armature winding used in America is like that 
 
 Fig. Cl. — Coil Winding having as many 
 Coils as Poles. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 79 
 
 shown in Fig. 61. In this case the field magnet comprises a 
 ring of poles of alternate polarity, and the armature consists of 
 as many coils A laid on the face of the armature core as there 
 are poles in the field magnet. The ends of the set of coils are 
 connected to two collector rings It , on which bear brushes B, 
 by means of which the current reaches the external circuit. 
 In this figure the coils are shown for clearness as though lying 
 in the plane of the page, but in reality they lie upon the face of 
 the cylindrical armature. If another set of coils is placed 90 
 electrical degrees from those shown, and connected to addi- 
 tional collector rings, the winding is of the two-phase variety * ; 
 while if three sets of coils are placed so that the centers of the 
 coils are 60 electrical degrees apart, the winding is three- 
 phase.* In the latter case, the coils are almost invariably con- 
 nected to three collector rings, and the connections must be 
 made in the manner described in Art. 22. Referring again to 
 Fig. 61, it is seen that adjacent coils move at any instant under 
 magnet poles of opposite 
 signs, and that the voltages 
 developed in them are in op- 
 posite directions. They must 
 therefore be connected in 
 circuit with each other so 
 that they are alternately 
 right and left handed. Like- 
 wise, when an armature 
 winding is composed of a 
 number of coils equal to one 
 half the number of poles, as 
 illustrated in Fig. 62, the 
 coils must all be connected 
 in the same direction in the 
 circuit. 
 
 To provide a winding similar to Fig. 62 for two or three 
 phases involves adding the additional one or two independent 
 sets of windings properly spaced and suitably connected to 
 appropriate collector rings, as already explained for the winding 
 of Fig. 61. 
 
 When the coil connections cross each other in their progress 
 
 * Art. 22. 
 
 Fig. 62. — Coil Winding having half as 
 many Coils as Poles. 
 
80 
 
 ALTERNATING CURRENTS 
 
 around the armature, as in Fig. 61, or one half of Fig. 63 a, the 
 windings may be called, after E. Arnold* and S. P. Thompson,! 
 
 Lap windings.^ 
 
 Alternator armatures are frequently connected with the two 
 halves of the windings in parallel, or, where the machine is to 
 
 work at low voltage and de- 
 liver heavy currents, there 
 may be several circuits in par- 
 allel in each phase. In this 
 case, instead of being ar- 
 ranged as illustrated in Fig. 
 61, the coils must be con- 
 nected in each half so that 
 they are alternately right- 
 handed and left-handed ; and 
 where the coils join the col- 
 lector rings, both must have 
 the same polarity. This is 
 indicated in Figs. 63 and 
 63 a, which are diagrams for 
 Such a winding, having two circuits 
 
 Fig. 63. — Coil Windings like Fig. 61, but 
 with Halves connected in Parallel. 
 
 single-phase machines, 
 in each phase, is called a 
 Two-circuit winding. When 
 all the coils are connected in 
 series, the first and last coils 
 lie side by side; consequently, 
 in armatures built for high 
 voltages, a severe strain is 
 thrown upon the insulation 
 separating them, and the 
 strain is especially trouble- 
 some if parts which differ 
 largely in potential occupy 
 the same slot. When the 
 halves of the armature are 
 connected in parallel, the 
 
 Fig. 63 a. — Coil Winding like Fig. 62, but 
 with Halves connected in Parallel. 
 
 * Die Ankerioickelungen der Gleichstrom-Dynamomachinen . p. 13. 
 t Dynamo Electric Machinery, 7th ed., vol. 2, p. 168. 
 
 | For numerous diagrams of alternator windings, see Parshall and Hobart, 
 Armature Windings , Chap. XII. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 81 
 
 4 - 
 
 
 n 
 
 
 first and last coils in the armature circuit lie on the opposite 
 sides of the armature, and effective insulation is therefore less 
 difficult to maintain. In the latter case the voltage between 
 the adjacent coils cannot be greater than the total machine 
 voltage divided by half the number of coils. When the coils 
 are all in series, the voltage between adjacent coils, except the 
 two end coils, is 
 equal to the total 
 voltage divided by 
 the total number of 
 coils. When a coil 
 winding of this type 
 is connected with its 
 halves in parallel, 
 precautions must be 
 taken to make the 
 two halves of the 
 winding alike, and 
 have the field poles 
 of equal strength ; 
 as, otherwise, the in- 
 stantaneous voltage 
 of one half of the 
 winding is likely to 
 differ from that of 
 the other half, and 
 injurious cross cur- 
 rents may occur in 
 the winding. This 
 precaution, of 
 course, applies 
 wherever there are 
 several parallel cir- 
 cuits per phase. 
 
 Those forms of windings in which the end connections of any 
 one phase need not cross each other at the ends of the armatures 
 are called by S. P. Thompson* Wave windings, or according to 
 more recent and acceptable nomenclature, Progressive windings. 
 Figures 64 and 65 illustrate three-phase progressive windings. 
 
 * Thompson’s Dynamo Electric Machinery , 7th ed., vol. 2, p. 168. 
 
 Fig. 64. 
 
 -Three-phase Progressive Windings with 
 A Connection. 
 
82 
 
 ALTERNATING CURRENTS 
 
 
 The first is connected in A and the second in Y. Figure 66 
 shows a winding of the same type for two phases. In these 
 figures, with the exception of the developed views at the top of 
 Figs. 64 and 65, the active conductors lying in the face of the 
 armature are shown as radial lines, while the end connections 
 
 are indicated by the 
 diagonal lines con- 
 necting the active 
 conductors together 
 in such a way that 
 the electro-motive 
 forces of each circuit 
 add together. Each 
 line may represent 
 a number of wires in 
 the same slot, which 
 may be arranged as 
 coils in each phase 
 in the manner indi- 
 cated in Fig. 62. 
 
 The principles of 
 armature windings 
 used in direct-cur- 
 rent machines are, in 
 general, applicable 
 to alternators, but 
 practical considera- 
 tions make it often 
 advisable to modify 
 the windings for the 
 
 Fig. 65. — Three-phase Progressive Windings with 
 Y Connection. 
 
 latter machines. 
 Synchronous alter- 
 nator armatures often have One-circuit windings ; that is, the 
 conductors of each phase are connected in series, making one 
 circuit per phase, inasmuch as high voltage is usuall) desit ed. 
 Direct-current armatures, on the other hand, have necessarily 
 two-circuit or multiple-circuit windings; hence, while diiect- 
 current windings may be adapted to alternating-current work, 
 the reverse is not always true. 
 
 The possibility of converting a direct-current dynamo into a 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 83 
 
 double-current machine has already been referred to in Art. 20. 
 A machine so constructed, with a direct-current commutator 
 and alternating-current collector rings, may he used to convert 
 a direct-current which 
 is fed into its commu- 
 tator end, and by which 
 it is driven, into an al- 
 ternating-current which 
 is taken from the collec- 
 tor rings. Or, the con- 
 version may he from 
 alternating to direct 
 currents, if the arma- 
 ture is properly syn- 
 chronized so that it 
 runs as a synchronous 
 motor. Or, alternating- 
 current and direct-cur- 
 rent may he simulta- 
 neously produced or 
 utilized in such machines, or either one of them alone. When 
 these double-current machines are used for converting one 
 kind of current into the other, they are called Rotary convert- 
 ers. The simple diagram, Fig. 66 a, shows the principle 
 
 clearly, though such ma- 
 chines are ordinarily mul- 
 tipolar. (For a discussion 
 of such machines see 
 Chapter XII.) 
 
 It is possible in this 
 manner to make a two- 
 phaser to be used with 
 separate circuits out of any 
 direct-current machine 
 with Gramme or Siemens 
 armature, by arranging 
 four collector rings on the 
 shaft and connecting them 
 to the armature windings 
 at points which are 90 electrical degrees apart. It is also possi- 
 
 Fig. 66. — Two-phase Progressive Winding con- 
 nected to Four Collector Rings. 
 
 Fig. 66 a. — Double-current Dynamo. 
 
84 
 
 ALTERNATING CURRENTS 
 
 ble to make a three -pliaser out of a direct-current machine by 
 arranging three collector rings on the shaft, and connecting them 
 to the armature winding, at 120 electrical degrees apart ; and, 
 in general, a polyphaser of any number of phases may be made 
 out of a direct-current machine by providing a proper number of 
 collector rings and connecting the rings to appropriate points on 
 the armature winding. The taps for the different rings should 
 
 360 
 
 connect with the armature windings at points electrical 
 
 n 
 
 degrees apart, where n is tire number of phases, except that, for 
 single and two-phases, n must be taken equal to two and four 
 respectively. Suck machines may be used to transform direct 
 currents into polyphase currents or vice versa. (See Fig. 66 a.) 
 
 In connecting the collector rings of such machines to the 
 armature windings the relative angles corresponding to the cur- 
 rent phases must be carefully distinguished. One complete 
 revolution of an armature in a two-pole field corresponds to 
 one complete period of the alternating current, and therefore 
 360 mechanical degrees correspond to 360 electrical degrees in 
 that instance, but in multipolar machines a rotation of the 
 armature equal to twice the angular pitch of the poles corre- 
 sponds to one complete period, so that, in general, the relation 
 
 V 
 
 of electrical degrees to mechanical degrees is ~ : 1, where p is 
 
 the number of poles. Two-pole alternators of the form here 
 described evidently utilize the whole of the armature winding 
 with each collector ring connected to a single point, as the 
 winding has only two paths for the current from commutator 
 brush to commutator brush, and the same is true of multipolar 
 machines with two path windings. If single connections to 
 the collector rings are used in multipolar machines with multi- 
 ple path windings, a proportion of the armature equivalent to 
 360 electrical degrees only is occupied in the delivery of alter- 
 nating currents, and the armature capacity is therefore not 
 fully utilized. To fully utilize the armature in this case, each 
 collector ring must be connected to the winding at as manv 
 points as there are pairs of paths, the points being 360 electrical 
 degrees apart ; but the different collector rings must connect 
 
 Q0| ! 
 
 successively into points on the armature winding which are - — 1 
 electrical degrees apart, as already stated. 
 
ARMATURE AND EIELD WINDINGS FOR ALTERNATORS 85 
 
 In general, any form of winding used for direct-current ma- 
 chines, in which alternating voltages are induced in the coils, 
 may be used for alternating-current machines by properly pro- 
 viding collector rings. The only forms of windings excluded 
 from this are the so-called acyclic or homopolar machines, 
 wherein the induced electro-motive forces are unidirectional. 
 
 28. Distributed Windings. — At the present day distributed 
 windings are used almost universally, as they have a tendency to 
 reduce the armature reactions, and also may be more readily 
 arranged to create a curve of voltage which closely approxi- 
 mates the sinusoidal form.* The simplest form of distributed 
 winding is of the coil type, distributed as in Fig. 67, which 
 
 Fig. 67. — Distributed Single-phase Progressive Coil Winding. 
 
 shows three partial coils per pole, or three slots per pole. 
 Each coil division may he wound complete on a former. The 
 formed divisions may then be placed in the slots and correctly 
 connected together to make up the armature circuit. Wind- 
 ings of this type can be made for two or three phases by super- 
 imposing one or two more circuits at 90 or 120 electrical 
 degrees apart respectively. For single-phase windings the 
 
86 
 
 ALTERNATING CURRENTS 
 
 distances AB and CD in the figure are each usually made 
 nearly equal to BC to reduce differential action.* 
 
 A common winding for polyphasers is a distributed winding 
 of the progressive type and is similar to the two-circuit chord wind- 
 ings such as are used on multipolar continuous-current machines. 
 This arrangement is sometimes called “ barrel winding.” Figure 
 68 shows such a winding for six poles having the Y connection. 
 In this particular armature there should be either three or six 
 slots per pole per phase, depending upon whether one or two 
 
 Fig. 68. — Three-phase Winding with Y Connection and Three Slots per Phase per 
 Pole in a Six-pole Machine. 
 
 sets of conductors are placed in a slot ; and all conductors in a 
 phase are connected in series, thus making it a one-circuit wind- 
 ing. It is noticed that, in Fig. 68, after the winding of one 
 phase has passed around the armature three times it doubles 
 back upon itself and passes around the armature three times 
 
 * Art. 20. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 87 
 
 again. This is to permit connecting the armature with its 
 halves in parallel if desired. If the terminals of the coils, in- 
 stead of passing to collector rings at BI, BII , and Bill , re- 
 spectively, are properly connected into the windings to make 
 them reentrant, and the junctions of the interior connection 
 wires are connected to commutator bars, the winding is suitable 
 for a direct-current machine. In fact the alternator windings 
 of the present day are frequently the same as those used for 
 multipolar direct-current machinery. In tracing the winding 
 
 Fig. 69. -Three-phase, Two-circuit Winding with Y Connection for Six-pole 
 
 Machine. 
 
 of one phase through its length it is necessary to follow around 
 the armature six times for the winding illustrated in Fig. 68, 
 which is for a six-pole machine, and the conductors in each 
 complete path traced in going once around the armature gen- 
 erate a voltage wave which differs in phase from the voltage 
 wave of the next set of conductors in the same circuit by an 
 angle equal to the electrical angular distance between the 
 
88 
 
 ALTERNATING CURRENTS 
 
 centers of adjacent slots. As the six slots in each group of 
 Fig. 68 span 60 electrical degrees, the value of K will he 1.055 * 
 approximately. This same winding can be used for a two-phase 
 machine, when, instead of having three sets of windings using 
 60° of width each, the connections are so made that there are 
 two sets using 90° of width each. In like manner a single- 
 phase machine can be constructed by removing one of the two- 
 phase windings. 
 
 CONDUCTOR IN TOP OF SLOT 
 CONDUCTOR IN BOTTOM OF SLOT 
 
 Fig. 70. — Distributed Winding with Four Slots per Pole per Phase connected in 
 either Three-phase A or Three-phase Y Connection, for a Four-pole Machine. 
 
 Figure 69 illustrates the same armature as Fig. 68, with the 
 halves of the winding of each of the three phases connected in 
 parallel. In this figure, starting from any one of the three 
 terminals BI, BII , or Bill , there are two paths to follow. 
 Each of these paths passes entirely around the armature three 
 times and occupies three slots. In the case of a three-phase 
 winding, as shown, the six slots of a phase should occupy one 
 third of the polar pitch on the armature circumference. 
 
 Figure 70 shows a distributed winding to occupy four slots 
 per pole per phase and to be connected in either three-phase A 
 or three-phase Y style, intended for a four-pole machine. In 
 this figure the winding is shown in the developed form. i.e. the 
 armature is supposed to be rolled out upon a flat surface. 
 
 * Art. 20. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 89 
 
 The connections at the points a , b , c are brought together in 
 the finished winding. The terminals A, B, (7, and A', B\ C 
 are the terminals which pass to collector rings, as shown by the 
 A and Y diagrams. 
 
 Figure 70 a shows another form of distributed three-phase 
 
 Fig. TO a. — Distributed Winding with Four Slots per Pole per Phase, connected in 
 either Three-phase A or Three-phase Y Connection, for a Four-pole Machine. 
 
 winding with four slots per phase per pole, which may be con- 
 nected in either A or Y style. This is a distributed winding of 
 the type shown for one phase in Fig. 62. 
 
 The advantage of distributing the winding in several slots, 
 as pointed out in Art. 20, is the reduction of armature reaction 
 and more particularly the closer approximation toward a 
 sinusoidal wave form; but the larger number of conductors 
 used to produce a given effective voltage results in increased 
 self-induction in the armature. The question of the number 
 of slots per phase is settled after balancing the influences of 
 these several effects and the conditions of manufacture. Prac- 
 tice indicates that three or four slots per pole per phase form a 
 satisfactory compromise, though cost of insulation and manu- 
 facture often make it desirable to reduce this number at the 
 expense of the wave form. When a sine wave form is of prime 
 
90 
 
 ALTERNATING CURRENTS 
 
 importance, a larger number than four slots per pole per phase 
 is sometimes used. 
 
 29. Disk Armatures. — The disk form of armature for alter- 
 nators is one of the earliest that came into service. The first 
 commercial alternator was a magneto machine (i.e. with per- 
 manent field magnets) known as the Alliance Dynamo. This 
 was originally devised as early as 1849, but was not developed 
 into commercial form until after 1860. It had a ring armature. 
 
 In 1867 Wilde built an alternator with electro-magnets and a 
 
 Fig. 71. — Disk Armature with Coil 
 Winding. 
 
 disk armature. From that time on, the disk armature received 
 
 much attention in Europe and 
 was an element of many success- 
 ful machines designed by such 
 eminent designers as Siemens, 
 Ferranti, and Mordey. It has 
 received less attention in Amer- 
 ica, and no machines of impor- 
 tance are at the present day 
 being actually manufactured 
 with disk armatures on either 
 side of the Atlantic. This may 
 be due to the'essential peculiarity 
 of the disk which admits of no 
 substantial iron core and is there- 
 fore difficult to build in a solid and workmanlike manner. Also, 
 it is probably not possible to build 
 machines with disk armatures as 
 economically as those in which the 
 armatures are built upon substan- 
 tial iron cores; at least when the 
 two types are made of equal me- 
 chanical stability. Disk armatures 
 may be wound either with lap or 
 progressive windings. Figure 71 
 illustrates three coils of a coil- 
 winding arranged to rotate be- 
 tween poles of opposite polarities, Fig. 72. — Disk Armature with 
 
 i ,, , r Progressive Winding, 
 
 and a section through one pair ot ° 
 
 poles. Figure 72 shows a progressive winding. Either the 
 
 armature or the field may revolve. (Examples: Ferranti 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 91 
 
 Alternator; Mordey Alternators; Brush Alternator.) The 
 absence of iron in disk armatures reduces the losses due to 
 hysteresis and eddy currents to a minimum, but it increases 
 the depth of the air gap. Hence, a greater exciting current 
 is apt to be required for the field magnets, and many turns 
 must be placed upon the armature. That this objection may 
 be overcome is shown by the small amount of energy required 
 to magnetize the old Mordey alternators. In a 75-kilowatt 
 machine of this type the I 2 R loss in the armature is 2.3 per 
 cent and in the field is 1.5 per cent, which compare favorably 
 with the losses in other machines. The curve of voltage in 
 disk armatures is in general quite near to that of the sine func- 
 tion. The first experimental determination of the form of the 
 curve was made in 1880 by Joubert, who experimented upon a 
 Siemens machine having a disk armature. The curve proved 
 to be practically a sinusoid. This is also true of the curve of 
 voltage developed by the Mordey alternator. 
 
 A diagram of a Mordey alternator is shown in Fig. 73. It 
 is seen that in the construction of this machine only a single 
 exciting coil F is required, and thus the 
 expense of construction is in some de- 
 gree reduced. In this machine all poles 
 of the same sign lie upon the same side 
 of the armature. The armature coils are 
 arranged in a disk, which is stationary 
 between the faces of the revolving poles. 
 
 The Ferranti alternator has two crowns 
 of pole pieces and magnet coils at either 
 side of the disk, as shown in Fig. 71. 
 
 The poles in this machine which lie on 
 one side of the armature alternate in po- 
 larity. The Ferranti alternator is of 
 especial interest on account of the Fer- 
 ranti machines installed at Deptford 
 Station near London, which was the first 
 high-voltage, long-distance transmission plant of any impor- 
 tance. The Ferranti and Mordey machines are now substan- 
 tially obsolete. 
 
 In iron-cored machines, armature reactions have a more dis- 
 torting effect and therefore tend to modify the form of the vol* 
 
 Fig. 73. — Diagram of a 
 Mordey Alternator. 
 
92 
 
 ALTERNATING CURRENTS 
 
 Fig. 74. — Diagram of a Revolving 
 Ring Armature. 
 
 tage curve. The variation is usually not great in machines hav- 
 ing distributed windings, but where single-coil windings are 
 used on single-phasers, the curves of voltage may deviate wide- 
 ly from the sinusoid, and sometimes become quite irregular. 
 
 30. Other Types of Armatures. — Ring-wound alternator ar- 
 matures were early used with commercial success, and some of 
 the older magneto machines of the De Meritens type with per- 
 manent field magnets and ring armatures are still in existence. 
 The invention of the ring armature for alternating-current 
 machines is usually ascribed to Gramme or Wilde, who inde- 
 pendently patented the form in 
 France and England in 1878. 
 In America ring armatures have 
 never been viewed with as great 
 favor as have drum armatures, 
 and they are not used, partially 
 on account of the small mechani- 
 cal stability of the ring and the greater self-induction of its 
 windings, hut more especially on account of the fact that the 
 windings must be placed by hand and are of greater length, 
 which conditions result in excessive expense. Figure 74 is a 
 diagram of a segment of a revolving ring armature intended to 
 revolve inside of 
 an inwardly point- 
 ing crown of poles 
 of alternate polar- 
 i t i e s . Such 
 armatures can, of 
 course, be mechan- 
 ically arranged so 
 that field magnets 
 with outwardly 
 pointing poles 
 may be located 
 inside the ring 
 instead of the re- 
 verse, and either armature or fields may be arranged to revolve 
 in any particular instance. 
 
 In some of the older types of foreign alternators the arma- 
 ture coils were wound upon long projections similar to the field 
 
 Fig. 75. — Alternator with Pole Armature. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 93 
 
 cores, and these were called Pole armatures. On account of 
 the great variation in reluctance of the magnetic circuit as the 
 armature revolves, such armatures cause large losses in the 
 pole pieces by eddy currents, unless the iron field is entirely 
 laminated. One form of pole armature is shown in Fig. 75. 
 In this machine the armature is stationary while the field re- 
 volves. (Example: early Ganz alternators and others.) In the 
 form illustrated in the figure, the reluctance of the magnetic 
 circuit varies less as the poles move past the armature coils, on 
 account of the fact that the armature coils are really wound in 
 deep slots, thus reducing the type to a close approximation of 
 the drum windings on slotted cores, as already described. 
 
 31. Methods of Applying the Wires to Chord-wound Arma- 
 tures. — The coils for alternator armatures are ordinarily 
 wound on formers and then fastened upon the armatures after 
 having been well insulated. At present the armature wind- 
 ings are nearly always placed in slots, which are properly 
 formed in the surfaces of the cores, but most of the older 
 American machines had their windings fastened upon the sur- 
 faces of smooth armature cores. 
 
 In the old machines where 
 surface windings were used, 
 the coils were frequently ar- 
 ranged to bend down over the 
 ends of the cores, as illustrated 
 in Fig. 76, where they could 
 be securely fastened by end 
 plates or blocks of wood or 
 fiber. It was usual to fill the 
 
 . , „ Fig. 76. — Surface-wound Alternator 
 
 spaces in the centers of the Armature. 
 
 coils with wood blocks screwed 
 
 to the cores or held by binding wires. (Examples: early 
 Westinghouse and National alternators.) In some machines 
 the coils were flat or pancake-like, and of the same length as 
 the armature cores. In this case they were laid upon the 
 cylindrical surfaces of the armature cores and securely bound 
 with wire bands. (Examples: early Thomson-Houston and 
 similar alternators.) The high peripheral velocities of alter- 
 nator armatures make heavy bands essential to the preserva- 
 tion of surface windings, and all blocking must be securely 
 
94 
 
 ALTERNATING CURRENTS 
 
 fastened. The wood blocks which fill the center of the coils 
 make excellent driving teeth, and therefore serve a good pur- 
 pose if they are fastened securely to the core. 
 
 When imbedded coils are used, they may be made upon 
 formers (lathe-wound), and then applied to the core, or the 
 conductors may be threaded through the grooves which are pro- 
 vided with insulating linings. When the core teeth are 
 T-shaped, the coils are sometimes wound of sufficient width to 
 slip over the head, and when in place, they may be narrowed by 
 squeezing. The coils for this purpose must be wound with the 
 ends of such shape that they will bend without injury, as is 
 shown, for example, in Fig. 77. The methods used in manufac- 
 turing the coils 
 and applying 
 them to slotted 
 armature cores 
 of direct-cur- 
 rent machines 
 may also be 
 used in the con- 
 
 Fig. 77. — A Method of Inserting Coils over T-shaped Teeth. . „ 
 
 struction of al- 
 ternator armatures with distributed windings ; but the alternator 
 armatures are usually of much higher voltage, and the insulation 
 must be carefully and specially designed. Construction of this 
 character (namely, using small distributed coils) has largely 
 superseded construction of the character illustrated in Fig. 77. 
 
 Lathe-wound coils are of decided advantage for armatures 
 designed for high voltages, as their insulation may be made 
 particularly safe. Such coils are usually served with layers of 
 japanned canvas, special fuller or press board made for insulat- 
 ing purposes, vulcanized fiber, and mica. The slots between 
 the core teeth may be made of sufficient area to permit the use 
 of any desirable thickness of insulation, and the teeth afford 
 very complete mechanical protection for the coils. Toothed 
 armatures with lathe-wound coils should therefore be thoroughly 
 reliable. If the magnetic surface of the armature is fairly 
 complete, the wires are protected from magnetic drag, which 
 decreases the chances of the conductors chafing and injuring 
 the insulation. The winding shown in Fig. 77 is not distrib- 
 uted, but grouped in one slot per pole, as shown also in the 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 95 
 
 Fig. 78. — Two-phase Armature Construction. 
 
96 
 
 ALTERNATING CURRENTS 
 
 diagrams of Figs. 61 to 63. Distributed windings have very 
 largely superseded this type, as already explained. 
 
 The construction of a two-phase group-wound armature, hav- 
 ing open rectangular winding slots in the core, is shown in 
 Fig. 78. Each phase in this case forms a complete single-phase 
 winding, as in Figs. 61 and 63. It is to be noticed that the 
 forms upon which the coils of one phase are wound are of such 
 a shape that the ends of the coils bend down over the ends of the 
 armature, while the coils of the other phase are rectangular. 
 This construction keeps the coils well apart at points of cross- 
 ing. After the coils are properly insulated they are slipped 
 into the rectangular slots, and are held in place by wedge-shaped 
 strips of hard wood. Grooves are arranged near the upper 
 edges of the iron teeth into which the wooden strips fit. In 
 this manner the windings may be held very firmly without the 
 aid of band wires. The four peripheral slots to be observed 
 in the core are ventilating openings, maintained by fingered 
 grids inserted between the adjacent core stampings. 
 
 The way in which coils are arranged for insertion into the 
 partially closed slots of the modern distributed armature wind- 
 ings is shown in Fig. 79. One end of each coil, which in this 
 case is composed of several turns of round wire, is left open. 
 The coil is then slipped through the proper slots, and the several 
 wires are joined to complete the coil. In this figure, each coil 
 occupies four slots and is therefore a distributed winding. 
 The coil terminals whereby the coils are connected with each 
 other are shown as spirally insulated wires at the top of the 
 figure. Another and similar winding for partially closed slots 
 is shown in Fig. 80. This winding, as seen, is placed upon a 
 stationary armature intended to surround a rotating field, and 
 copper straps or bars are used for making the coils instead of 
 round wire. To complete the winding, the open ends of the 
 coils must be brazed together and the coils themselves be inter- 
 connected. Figure 81 shows how the straps are brought 
 around ready for brazing, and also shows the completed joints 
 after the final insulating wrapping has been applied. In the 
 left-hand side of the figure may be seen the connectors be- 
 tween coils. The method of insulating the slots is shown in 
 the lower right-hand corner of the figure ; the two slots, which 
 have not as yet received their coils, show the tubes of insulat- 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 97 
 
 Fig. 79. — Grouping of Wire Coils for Partly Closed Slots. 
 
98 
 
 ALTERNATING CURRENTS 
 
 J?ig. 80. — Strap or Bar Coils in Partly Closed Slots. Incomplete Winding. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 99 
 
 ing material — micanite, fuller board, or other insulator — 
 extending out a short distance from the core. The method of 
 taping the coils to complete the insulation can also be readily 
 seen. The conductors are held firmly in place by wooden 
 wedges driven in the tops of the slots in the manner more 
 plainly seen in Fig. 78. 
 
 In Fig. 82 is shown a three-phase, coil-wound, stationary 
 armature having a distributed winding of four slots to the coil. 
 
 Fig. 81. — Method of Connecting Open Ends of Strap Coils. 
 
 The particular machine from which the photograph was taken 
 is wound to generate a voltage of 13,000 volts, and is therefore 
 very heavily insulated. 
 
 Figure 83 shows a model of a three-phase partially distrib- 
 uted winding applied to one of the 5000-kilowatt generators 
 
100 
 
 ALTERNATING CURRENTS 
 
 Fig. 82. — Three-phase Winding for 13,000 Volts. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 101 
 
 Fig. 83. — Model of Three-phase Partially Distributed AViuding. 
 
102 
 
 ALTERNATING CURRENTS 
 
 of the Manhattan Station in New York City. It is especially 
 interesting as showing the manner in which the windings are 
 arranged for crossing at the ends and the methods of construc- 
 tion and insulation. 
 
 The lap, progressive, and barrel windings given heretofore 
 are suitable for any type of generator, including those used 
 with steam turbines. Figure 83 a shows the stationary arma- 
 ture of a horizontal steam turbine generator of 1000-kilowatt 
 capacity of Westinghouse type. It is evident from the con- 
 
 Fig. 83 a. — Turbine Generator Armature. 
 
 struction shown in the figure that this type of generator is of 
 the same character as those driven in a different manner. The 
 high speed of such machines makes it necessary to wind them 
 for a comparatively small number of poles. The winding in 
 the figure is two-phase, four-pole, of the distributed coil type. 
 
 32. Collectors. — In all the alternators heretofore described, 
 either the field or the armature windings are arranged to re- 
 volve. In the case of inductor alternators to be described 
 later it is possible to arrange the construction so that both the 
 magnetizing and armature coils may remain stationary, but 
 mechanical and other considerations ordinarily render this inad- 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 103 
 
 visable. It therefore may be said generally that either the arma- 
 ture or the field windings must revolve with the iron cores on 
 which they are wound. This makes essential the use of some 
 means for conveying the current to and from the revolving coils. 
 
 In either case no modification of the current is required and 
 therefore plain insulated Collector rings serve the purpose when 
 they are properly mounted on the shaft so that the brushes may 
 be arranged to bear against them. Figure 8-1 is an illustration 
 of collector rings for conveying the current to a multipolar 
 revolving field. These rings, often called Collectors, are some- 
 times made of copper or bi'onze, though they are now com- 
 monly made of cast iron. In the older construction the rings 
 were imbedded in cylinders of vulcanized fiber or the like 
 keyed to the shaft of the rotating part. In modern construc- 
 tion the rings are usually placed upon a metal spider and hub. 
 This makes it possible to do away with such a large bulk of 
 insulation, and thus adds materially to the mechanical rigidity 
 of the construction. The insulating materials used are ordi- 
 nary pressed fiber, vulcanite, or some form of mica board. 
 Figure 92 shows four rings mounted between discs of solid 
 insulation, for a four-phase rotating armature ; and Fig. 84 
 shows two rings on metal spiders affording air insulation. 
 
 Collection is obtained by means of brushes fastened to the 
 brush holders and hearing upon the rings. These brushes are 
 usually of carbon, electroplated with copper to increase their 
 conductivity. The copper coating is cut away from the end 
 of the brush which bears on the ring. As carbon has a high 
 resistance it is not suitable for machines delivering a very 
 large amount of current. In such cases it is common to use 
 brushes made of fine woven copper wire gauze. Former prac- 
 tice employed brushes made of copper strips, but as it is diffi- 
 cult to make strip brushes which will not cut the rings, the 
 substitution of carbon and woven wire has become almost 
 universal. 
 
 A carbon brush can collect without undue heating from 
 about 40 to 80 amperes per square inch of contact surface, 
 while a woven-wire brush can collect over two or three times 
 that amount. A number of brushes may make contact with 
 each ring. 
 
 The brushes are mounted in brush holders, which in turn are 
 
104 
 
 ALTERNATING CURRENTS 
 
 Fig. 84. — Rotating Field Magnet, showing Collector Rings, and also showing Exciter Armature mounted on Same Shaft. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 105 
 
 fastened to the frame of the machine some convenient arms 
 or brackets; the circuit connections are often made to the 
 brushes by means of flexible copper cords or strips soldered on, 
 in addition to the contact of brush with holder. The brush 
 holders are so arranged through the medium of springs that 
 the brushes may be made to bear upon the collector rings with 
 any desired pressure. Ordinary practice calls for a pressure of 
 from 1 to 1^ pounds per square inch. A greater pressure is apt 
 to cause cutting of the rings and a less pressure may permit 
 the brushes to vibrate sufficiently to cause sparking. 
 
 The safe limit for the amount of current that can be col- 
 lected per square inch is fixed by the heating alone and there- 
 fore may be quite large without danger. If mechanical 
 considerations required it, doubtless as much as 500 amperes 
 per square inch of contact might be satisfactorily collected 
 by means of copper gauze brushes and as much as 200 amperes 
 by carbon brushes, but such extreme cases do not often arise 
 and it is not generally advisable to exceed one fourth these 
 amounts. 
 
 Instead of brushes the collection may be effected by means of 
 flexible weighted copper bands hung from the rings. Figure 85 
 indicates such an arrangement. An 
 arrangement somewhat similar to 
 that of Fig. 85 was used on the 
 Ferranti alternators of the famous 
 Deptford Station already referred 
 to.* By this construction a large 
 collecting area may be gained with- 
 out unduly increasing the size of 
 the rings, but the arrangement of 
 carbon brushes rubbing on flat rings 
 is mechanically preferable. As 
 modern alternators are usually of 
 the revolving field type and the collectors are used for trans- 
 mitting exciting current to the revolving field coils, the collec- 
 tors need not be so highly insulated as would be necessary for 
 a high alternating voltage. The field voltage seldom exceeds 
 220 volts. It is good practice to mount the two rings, for this 
 
 Fig. 85. — Copper-band Contactors. 
 
 * Art. 29. 
 
106 
 
 ALTERNATING CURRENTS 
 
 purpose, upon separate spiders attached to one hub, so that there 
 is an air space between the rings. 
 
 33. Field Excitation of Alternators. — The exciting current 
 for synchronous alternators must be unidirectional, and it is 
 ordinarily obtained from auxiliary direct-current shunt or com- 
 pound dynamos called Exciters, as illustrated in Fig. 86. If 
 the alternator field magnets are stationary, this exciting current 
 
 is fed directly into the coils, 
 
 A RM ATI IDP 
 
 but if the machine is of the 
 revolving field type, the ex- 
 citing current must be led 
 into the coils through col- 
 lector rings ; in this case 
 the armature current is led 
 directly from the terminals 
 of the stationary armature. 
 
 Fig. 86. — Diagram of a Separately Excited The eXciter is fluently 
 Alternator, with Bus Bars for Parallel Connec- driven from the same prime 
 tion of Exciters and Alternators. Switches, moygr as is its alterna- 
 etc., not Shown. . . 
 
 tor, and is sometimes even 
 
 mounted upon the same shaft. In generating plants having 
 large units, however, it is better practice to have the exciters 
 driven by separate prime movers, in which case two or three 
 exciters may be provided to furnish the exciting current for all 
 the main generators. The regulation of the alternator voltage 
 is secured, by varying the resistance in the field circuit of 
 the alternator or by varying the field strength of the exciter. 
 These are the usual methods in 
 America and are done both by 
 
 Fig. 87. — Self-excited Alter- 
 nator. 
 
 Fig. 88. — Compositely Excited Alter 
 nator. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 107 
 
 hand or by an automatic controller, as is explained in detail in 
 the chapter dealing with alternator operation. Self-excitation 
 and composite ex- 
 citation, as shown 
 below, were much 
 used earlier and 
 have sufficient pos- 
 sibilities to war- 
 rent study. 
 
 By suitable com- 
 mutation the 
 alternator may 
 evidently be made 
 to furnish its own 
 exciting current 
 either wholly or in 
 part, and the windings of the field magnets of alternators may 
 be classified according to their arrangement in circuit. The 
 principal divisions are three : separately excited, self-excited, 
 compositely excited; so-called, respectively, when the exciting 
 current is supplied from an external source (Fig. 86), when it 
 is supplied through a rectifying commutator from the armature 
 of the machine under consideration (Fig. 87), or when these 
 two arrangements are combined (Fig. 88).* Self-excited al- 
 ternators may again 
 be d*i v i d e d into 
 series-wound and 
 shunt-wound, de- 
 pending upon, first, 
 whether the whole 
 current is rectified 
 and led through a 
 comparatively small 
 number of turns 
 around the field 
 magnets (Fig. 89), 
 or, second, whether 
 only a portion of the current is rectified and led through a 
 
 * Compare Jackson’s Electromagnetism and the Construction of Dynamos, 
 
 p. 136. 
 
 Fig. 90. — Diagram of a Shunt-wound Alternator. 
 
 Fig. 89. — Diagram of a Series-wound Alternator. 
 
108 
 
 ALTERNATING CURRENTS 
 
 shunt circuit many times around the magnets (Fig. 90). In 
 the latter, either the whole voltage of the armature, or that of 
 
 Fig. 91.— Diagram of a Shunt-wound common, as the inconveniences 
 
 than outweigh any advantages derived from making the machines 
 self-contained. 
 
 Composite Excitation. — Evidently another division might be 
 added to those named in the preceding paragraph, which would 
 comprise a compound winding in which both the shunt and series 
 field currents are supplied by rectification. This, however, 
 would require two rectifying commutators, which at the best 
 are unsatisfactory, and for other reasons would not prove prac- 
 tical. To gain the result for which compounding is used in 
 direct-current dynamos, the composite winding is used. That 
 is, the alternator is externally excited to its normal voltage on open 
 circuit and the internal losses are compensated by series ampere- 
 turns from self -excitation. This method of excitation is common 
 with machines having rotating armatures and is very desirable in 
 small or medium-sized electric light plants or where the nature 
 of the service calls for constant regulation with meager switch- 
 board attendance. The series turns may be so arranged that 
 they will add more than the voltage lost in the machine itself, 
 thus providing some compensation for the voltage lost in the 
 feeder wires and tending to maintain the voltage at the center 
 of distribution approximately constant. Evidently, in the latter 
 case the voltage at the machine terminals will rise when the load 
 increases. Under such circumstances the machine is said to be 
 Over-compounded, and the term Compound-wound has of recent 
 
 one or more coils, may be im- 
 pressed upon the rectifying 
 commutator either directly, as 
 illustrated in Fig. 90, or by 
 means of a transformer at- 
 tached to the armature. Fig-- 
 ure 91 illustrates the trans- 
 former arrangement when the 
 fields rotate. Self-excited al- 
 ternators are now quite un- 
 
 Re volving Field Alternator, with the 
 Voltage reduced for Commutation by a 
 Transformer. 
 
 accompanying the rectification 
 of the current under usual con- 
 conditions of operation more 
 
ARMATURE AND FIELD WINDINGS EOR ALTERNATORS 109 
 
 Fig. 92. — Revolving Armature arranged with Series Transformers for Composite Excitation. A A, Series Transformers. 
 
110 
 
 ALTERNATING CURRENTS 
 
 years come into general use to designate these machines instead 
 of the more distinctive term composite-wound. 
 
 The self-exciting circuit of the composite winding may be 
 arranged in various ways. Thus the armature current may all 
 be rectified for use in excitation (Fig. 93) or it may pass 
 
 Fig. 93. — Diagram showing All of the Armature Current rectified for 
 Field Excitation. 
 
 through special series transformers attached to the armature, 
 and the secondary of these may then supply the current for rec- 
 tification and self-excitation (Figs. 92 and 112). The cores 
 of these transformers may be either independent of the arma- 
 
 Fig. 94. — Diagram of Composite Excitation with Series Turns on All Poles. 
 
 ture core, as at A in Fig. 112, or may consist of the lami- 
 nated spider or other portions of the armature core. Again, 
 the rectified current may be passed through a few turns of wire 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 111 
 
 on each pole (Fig. 91), or all the necessary series turns may be 
 concentrated upon one or two poles (Figs. 95 and 96). In the 
 latter case, the series turns must always be equally divided 
 between two poles with symmetrical positions, as in Fig-. 96, 
 
 Fig. 95. — Diagram of Composite Winding with Series Turns on One Pole. 
 
 when the armature winding is connected with its halves in par- 
 allel. Composite windings may be arranged with the self-exci- 
 
 Fig. 96. — Diagram of Composite Winding with Series Turns on Two Poles. 
 
 tation in a shunt circuit, but no advantage is gained by this 
 arrangement over complete self-excitation in shunt or over sepa- 
 rate excitation, and the arrangement is never used. In some 
 
112 
 
 ALTERNATING CURRENTS 
 
 old-type self-exciting alternators, a separate set of exciting 
 coils was wound on the armature and connected to a rectifying 
 
 commutator. These were 
 wound either directly with 
 the main armature coils, or 
 across a chord of the arma- 
 ture core, as illustrated in 
 Fig. 9' 7, in which the coils A 
 for generating exciting cur- 
 rent are wound across a long 
 chord of the armature, as 
 
 Fig. 97. — Method of Winding Special shown in the upper diagram 
 
 Alternator Exciting Coil. „ , . „ . .. 
 
 ot the figure, while the mam 
 armature coils B are wound pancake fashion. The compound- 
 ing may be effected in self-excitecl alternators by means of 
 shunt and series transformers combined as in Fig. 98, which 
 shows a machine with stationary armature. This arrangement 
 pertains to some early European examples put out by Ganz and 
 Co. The arrangement 
 permits a self-excited 
 alternator to be com- 
 pounded by the use of 
 only one commutator. 
 
 The shunt transformer 
 furnishes what is equiva- 
 lent to the shunt current 
 of an ordinary com- 
 pound dynamo, and the 
 series transformer adds 
 additional voltage to increase the current as required for com- 
 pounding. This latter voltage increases with the load on the 
 alternator as the primary winding of the transformer is in 
 series with one of the alternator leads. In the illustration, S 
 represents the series transformer, T the shunt transformer, and 
 li a variable rheostat to control the degree of excitation. 
 
 For the purpose of varying the magnetizing effect of the 
 series turns, a variable shunt is often connected across their 
 terminals in a manner similar to the usage with direct-current 
 machines, as at A in Fig. 99, and a shunt is sometimes placed 
 across the rectifier terminals in such a way that only a fixed 
 
 Fig. 98. — Diagram of Compounding by Means 
 of Shunt and Series Transformers. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 113 
 
 proportion of the total current is rectified and passes through 
 the series field winding, as at B in Fig. 99. This latter shunt 
 reduces the difficul- 
 ties caused by spark- 
 ing by reducing the 
 current to be rec- 
 tified. 
 
 Polyphase alter- 
 nators may be com- 
 pounded in the man- 
 ner described above 
 by rectifying the 
 current of one phase. 
 
 Another method 
 of compounding is 
 based upon the mag- 
 netic reactive effect of current in an armature upon the field. 
 The revolving field magnet of such a machine, called a “ compen- 
 sated alternator,” is shown in Fig. 100. Upon the same shaft 
 as the revolving field magnet is placed the direct-current ex- 
 citer armature and its commutator. Surrounding the revolv- 
 ing field magnet is a three-phase stationary distributed armature 
 winding of the ordinary type, and supported by the same frame 
 
 ALTERNATOR FIELD RINGS 
 
 Fig. 100. — Revolving FieldMagnet and Exciter Armature of a Compensated Alternator. 
 
 Fig. 99. Diagram of Composite Excitation having 
 the Series Field and the Rectifier Shunted. 
 
 is the exciter field, which has the same number of poles as the 
 field of the alternator. Direct current is drawn from the com- 
 mutator of the exciter, and part is taken through the alter- 
 nator field rings to the revolving field magnet, and part goes 
 through a rheostat to the stationary exciter fields. So far 
 as described the arrangement corresponds with an ordinary 
 
 i 
 
114 
 
 ALTERNATING CURRENTS 
 
 separately excited rotating field alternator having the exciter 
 mounted upon its shaft. In order to procure the compounding 
 effect, however, the alternating currents from the secondary 
 windings of three series transformers placed in series with the 
 three-phase alternator leads, are led by means of the special 
 exciter armature rings into the exciter armature winding. As 
 explained above, the exciter and alternator field magnets have 
 the same number of poles, and hence the frequency of the electro- 
 motive force is the same in the exciter and alternator armatures, 
 and therefore the alternating current will pulsate through the ex- 
 citer armature with the peaks of its waves in a definite position 
 with reference to the exciter pole pieces. By a proper organi- 
 zation, the reactions of the three-phase currents thus fed to the 
 exciter armature can be caused to increase the strength of the 
 exciter fields in a degree dependent on both the amount and 
 phase position of the currents. The field magnet of the exciter 
 is mounted so that it can be angularly shifted, slightly, so as 
 to bring the impulses of the alternating current into proper 
 relation with the polar positions. Such machines can be made 
 for any number of phases. 
 
 34. Rectifying Commutators. — The rectifying commutator 
 has as many segments as there are poles on the alternator, and 
 alternate segments are joined in electrical connection, making 
 
 two sets composed of al- 
 ternate segments, as illus- 
 trated in Fig. 101. When 
 series excitation is to be 
 provided by the rectified 
 current, one terminal of the 
 armature winding and one 
 terminal of the external 
 circuit are attached respec- 
 tively to these two sets of segments, as shown in the figure, 
 or the secondary terminals of a series transformer are respec- 
 tively attached to the two sets if the excitation is to be 
 obtained through the medium of such a transformer, as shown 
 
 Fig. 101. — Rectifying Commutator. 
 
 in Fig. 92. Brushes bearing upon the commutator at non- 
 sparking points (that is, under the circumstances considered, 
 as nearly as practicable at points of zero current) then collect 
 a rectified current. The brushes are shown in Fig. 101 bear- 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 115 
 
 mg upon adjacent commutator segments, but they may ob- 
 viously be placed on the commutator so as to be any odd num- 
 ber of segments apart. It is evident that direct current will 
 flow in the series field between the brushes B, B when the 
 circuit connections are those indicated in Fig. 101, and the ma- 
 chine is in operation. The commutator possesses as many 
 segments as there are poles in the field magnet, and if the 
 brushes are properly placed, each brush will pass from one set 
 of commutator bars to the other at each alternation of the cur- 
 rent. That is, the polarities of the segments reverse, but the 
 brushes simultaneously slip from one set of segments to the 
 next, and therefore remain of fixed polarity. The current in 
 the circuit of the series field is unidirectional, although it is 
 electrically in series relation with the armature and external 
 circuit, and the current is alternating in them. 
 
 To obtain the maximum effect, the brushes must pass across 
 the insulation between the commutator segments at about the 
 time when the current in the armature reverses. Minimum 
 sparking will occur if the brushes are shifted to the point where 
 the current is zero in the circuit at the instant each brush 
 breaks contact with one segment as it slips on to the next. If 
 the brushes are shifted in the forward direction, commutation 
 will take place on a rising current, which is likely to cause 
 quite serious sparking. The character of the current in the 
 series field winding manifestly could be modified by shifting 
 the position of the brushes, if sparking were not prohibitive. 
 For instance, if the brushes could be maintained in the position 
 where commutation takes place when the armature current is 
 maximum, the current in the series field circuit would in each 
 period change gradually from a maximum in one direction to a 
 maximum in the other and be then reversed to a maximum in 
 the first direction, and the resultant effect on the strength of 
 the field magnet would be zero. On the other hand, if the 
 brushes are upon the neutral points, that is, commutate at the 
 instant the armature current is zero, the current is unidirec- 
 tional in the field circuit. 
 
 Various devices have been employed to avoid sparking at the 
 rectifying commutator, but in American machines no special 
 precautions are taken. In a Zipernowsky alternator, built 
 by Ganz and Co. of Budapest, the following arrangement of 
 
lie 
 
 ALTERNATING CURRENTS 
 
 the commutator was employed. Between the commutator 
 divisions are inserted narrow metallic sectors which are con- 
 nected together. Four brushes are used, two on each side of 
 the commutator. One brush of each pair is set a little in the 
 lead of the other, and the pair is connected together through a 
 small resistance. The leading brushes are connected directlv 
 to the circuits. When the commutating point is reached dur- 
 
 
 Fig. 102. — Special Rectifier for Minimum 
 Sparking. 
 
 self-inductance tends to uphold 
 and this, therefore, does not f; 
 tion, as shown in Fig. 103, hut 
 wavy line more like that of Fig 
 
 mg the rotation of the com- 
 mutator, the trailing brushes 
 move on to intermediate seg- 
 ments, while the forward 
 brushes are still on main seg- 
 ments. Hence both the field 
 circuit and supply circuit 
 are short-circuited for an in- 
 stant through the resistances 
 connecting the brushes (Fig. 
 102). Upon short-circuit- 
 ing the fields with this or an 
 ordinary commutator, their 
 the current in the windings, 
 ill to zero at each commuta- 
 the current curve becomes a 
 . 101. Picou says * that it is 
 
 Fig. 103. — Current Impulses that would 
 be produced by a Rectifier having no 
 Self-inductance in Circuit. 
 
 Fig. 101. — Curve showing the Wavy Cur- 
 rent which passes from a Rectifier to 
 the Alternator Fields. 
 
 preferable to place the brushes so that commutation occurs 
 slightly earlier than the point of least sparking. In this case 
 the spark is due to a decreasing current, and is thin and weak. 
 With the commutation occurring later than the point of least 
 sparking, the spark is due to a rising current, and it is of great 
 
 * Machines Dynamo-lSlectriques, p. 99. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 117 
 
 magnitude. The advantage of a wavy current in the field, in 
 stead of a discontinuous one, is sufficiently well gained by the use 
 of copper brushes of considerable thickness, on an ordinary recti- 
 fying commutator, which short-circuit the supply circuit and field 
 circuit at the instant when they bridge over the insulation be- 
 tween two segments. Figure 101 very closely represents the 
 form of current curve under these circumstances. 
 
 Figure 101 shows the circuit connections at the rectifying 
 commutator of a compound-wound machine with rotating arma- 
 ture. For a machine with a rotating field magnet, the series 
 field connections and the connections to armature and external 
 circuit are interchanged so that the former go directly to the 
 commutator segments and the latter to the brushes. 
 
 35. Inductor Alternators. — The windings of inductor alter- 
 nator’s may be made entirely stationary, thus avoiding collect- 
 ing devices. These devices, however, are 
 of so little expense in construction and 
 material that there is no marked advan- 
 tage in suppressing them, and the mechani- 
 cal and magnetic difficulties encountered 
 in the design and construction of inductor 
 alternators have not permitted them to 
 come into general use, though there are 
 several theoretical points of advantage 
 presented in their design. In order to 
 avoid excessive losses due to eddy cur- 
 rents and hysteresis it is important that 
 the magnetic density in the field magnets 
 be kept as uniform as possible. Since this 
 cannot be fully accomplished, it is neces- 
 sary to thoroughly laminate the iron in 
 which the density varies. The inductor 
 must be moved in such a way as to periodi- 
 cally short-circuit or break the lines of 
 force which naturally pass through the ar- 
 mature coils. This may be accomplished 
 as shown in Fig. 105, where the effective reluctance of the 
 total magnetic circuit is fairly constant for all positions of the 
 inductor. In this example it is seen that as an inductor travels 
 from one pole piece to the next of opposite sign, the mag- 
 
 Fig. 105. — Diagram of 
 an Inductor Alternator. 
 A, A are the Armature 
 Windings, NS the Field 
 Windings, and BB In- 
 ductors. 
 
118 
 
 ALTERNATING CURRENTS 
 
 netism through the armature coil changes from a maximum in 
 one direction to a maximum in the other. In spite of the fact 
 that the field poles are maintained at a fairly constant magnetic 
 density it is evidently necessary, in order to sufficiently reduce 
 the iron losses, to laminate all the iron of the magnetic circuit. 
 The figure shows that the reluctance cannot remain entirely 
 constant, and that the effective ampere turns in the magnetic 
 circuits also vary with the position of the inductor. In Figs. 
 106 and 107 are shown two types of inductor machines in 
 
 which no attempt is 
 made to keep the 
 magnetic circuit of 
 
 constant reluctance. 
 Each, of these forms 
 gains some economy 
 in construction by 
 uniting the coils. 
 
 In the Stanley al- 
 ternators which were 
 at one time consid- 
 erably used in this 
 country, lines of 
 force are caused by 
 the motion of the inductor to sweep across the armature coils, 
 while the total magnetism in the inductor remains fairly con- 
 stant. The field 
 windings, which are 
 arranged in the form 
 of a single coil, em- 
 brace the inductor 
 core, and though 
 this coil is stationary 
 the machines are 
 not true inductor 
 alternators. Figure 
 108 shows the so- 
 called inductor and 
 field coil of a 2000- 
 kilowatt alternator of the Stanley type. The field coil rests in 
 the frame, as seen in the figure, and does not rotate with the in- 
 
 f-ARMATURE COIL 
 
 Fig. 107. — Diagram of an Inductor Alternator having a 
 Single Magnetizing Coil and Single Armature Coil. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 119 
 
 Fig. 108. — Inductor and Field Coil of a 2000-kilowatt Stanley Alternator. 
 
 Fig. 109. — Illustration show- 
 ing the Armature Windings 
 of a 2000-kilowatt Inductor 
 Alternator. Two Coils are 
 partly Removed. 
 
 ductor. This coil magnetizes the inductor so that the crown of 
 inductor projections on one side of the coil are of one sign and 
 those on the other are of the opposite 
 sign. The armature coils are usually of 
 the distributed type, as shown in Fig. 
 
 109, and in modern machines are always 
 chord wound. The armature windings 
 are also in two crowns ; one surround- 
 ing each crown of inductor projections. 
 
 There are in each phase twice as 
 many coils in each crown as inductor 
 projections, and in order that they 
 shall add their voltages the coils must 
 be connected into circuit alternately right and left handed. 
 
 The Warren alternator was very similar to the Stanley ma- 
 chine. It had, however, only one 
 set of inductors and armature coils. 
 Figure 110 shows the inductor. 
 The armature windings were placed 
 on the inner surface of a laminated 
 stationary iron frame surrounding 
 the projections at the left hand 
 
 gle Crown of Projections. ° f the fi S U1 ' e ’ and the magnetiz- 
 ing coil surrounded the cylindrical 
 portion at the right. The magnetic circuit was completed 
 through the frame. 
 
120 
 
 ALTERNATING CURRENTS 
 
 36. Armature Insulating and Core Materials and Construction. 
 
 — Before the conductors are placed on an armature core it is 
 usual to insulate it, very much as in the case of a direct-current 
 armature,* but more thoroughly on account of the high voltages 
 usually produced in alternator armatures. For this purpose 
 mica, micanite, mica paper and cloth, shellacked canvas, fuller 
 board, oiled paper and cloth, sheets of vulcanite, vulcabeston, vul- 
 canized fiber, and similar insulating materials, are used. Mica, 
 micanite, vulcanite, vulcabeston, bonsilate, vulcanized fiber, 
 asbestos paper, and similar materials are also used to insulate 
 collecting rings and brush holders, and for insulation between 
 the armature coils. The wire used for high-voltage alternator 
 armatures is often triple-cotton covered, and is thoroughly 
 japanned during the process of winding. Vulcanized fiber is 
 made from paper fiber by a chemical process and is furnished 
 in sheets and tubes. Its convenient form, cheapness, and ease 
 of working have brought it into extensive use. It unfortu- 
 nately absorbs moisture when exposed to the air, which causes 
 it to expand and contract to a remarkable degree. The mois- 
 ture also reduces its insulating qualities to a large extent. It 
 is therefore unsafe to place entire reliance upon it where con- 
 tinuously high insulation is required. Of the various available 
 insulating materials mica is the only thoroughly reliable one, 
 but it is unduly expensive and of poor mechanical qualities. 
 On the latter account it is generally used in combination with 
 other materials. When made up in the form of micanite by 
 treating with varnish, laying in overlapping scales between 
 paper or cambric, and subjecting to high pressure and heat, its 
 mechanical qualities are somewhat improved, and it may be 
 formed into sheets and tubes or molded as desired. Materials 
 such as vulcabeston and bonsilate are advantageous for insulat- 
 ing collector rings and similar details, since they can be molded 
 into any desired form. Vulcabeston, which is a compound con- 
 taining rubber and asbestos, is also manufactured in sheets and 
 has quite satisfactory mechanical and electrical properties. 
 Bonsilate is not sufficiently tough for very general service. 
 Boxwood and paraffined maple or hickory are frequently 
 used for insulation where considerable bulk is required and 
 their mechanical properties will serve, though, as they are liable 
 * Jackson’s Electromagnetism and the Construction of Dynamos, p. 104. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 121 
 
 to be more or less affected by the surrounding condition of 
 humidity, they must be used with care. A special material 
 called asbestos wood is now manufactured in sheets and plates 
 adapted for use as insulating material in some of the situations 
 now unsatisfactorily filled by other materials. 
 
 Tubes or troughs of micanite, varnished fiber, or some other 
 insulator that will meet the conditions, are ordinarily slipped 
 into the armature slots before the wires are inserted. The coils, 
 after careful taping, are then placed within the tubes. The 
 method of insulating armature conductors is illustrated in Figs. 
 79 to 83. When a coil is made up of many turns of wire, it is 
 sometimes necessary to also insulate between layers, in addition 
 to the insulation afforded by the cotton covering of the wire 
 itself. The field coils of alternators are insulated with the 
 same materials as the armature windings, though the lower 
 voltages usually employed in the field circuits call for a less 
 thickness of insulation. 
 
 The field coils are generally wound upon formers, and insu- 
 lating sheets are placed between layers. The wire used is often 
 double cotton covered ; but a flat strip (sometimes bare) is not 
 uncommonly used. This latter may be wound on edge with 
 insulating strips between the conducting strips. Winding a 
 single layer of strips on edge is thought to improve the 
 mechanical and electrical stability of the field coil and improve 
 its heat-dissipating qualities. The taped coil may be placed 
 upon an insulating spool and then slipped upon the pole piece. 
 Formed conductors are often given a coating of enamel for 
 insulation; and coils are frequently heated to very high tempera- 
 ture during construction, while under the high pressure of a 
 former, to drive off volatile matter and leave the conductors 
 embedded in a compact and stable mass of insulation. 
 
 In insulating alternators care must be taken to use such 
 materials as will not deteriorate either electrically or mechani- 
 cally under the rather high temperature which the machines 
 attain when running. If the insulating material softens per- 
 ceptibly, it may permit points of high difference of potential to 
 come close together with a resultant short circuit ; or the same 
 failure may result in the permanent lowering of the specific 
 resistance of the insulating materials. 
 
 The Dielectric strength, or the property of an insulator to resist 
 
122 
 
 ALTERNATING CURRENTS 
 
 electric puncture, is the basis upon which to calculate insula 
 tion, rather than the electrical resistance, though the latter often 
 serves as an indication of the former. The following table 
 gives the specific resistance, or the resistance in megohms per 
 centimeter of length and square centimeter of cross section, of 
 familiar insulating materials at ordinary temperatures (namely, 
 about 70° F.). The values are only approximate, as different 
 samples of the same material are apt to have quite different 
 characteristics. 
 
 TABLE 
 
 Specific Resistance of Insulators (in megohms) 
 
 Mica 
 
 Paper 
 
 Paraffine oil 
 
 Rubber 
 
 Shellac 
 
 Yulcanized fiber 
 
 10« to 10 8 
 10 4 to 10 5 
 10 6 to 10 8 
 
 10 8 to 10 9 
 
 10 9 to 10 10 
 10 6 to 10 8 
 
 The dielectric strength of any insulating material is quite de- 
 pendent upon the character of the particular sample under test, 
 the character of the electrodes, the duration of the stress, and 
 other conditions surrounding the tests; and therefore the data 
 from tests should only be used for rough estimates or checks. 
 Dr. Steinmetz * has worked out a number of special formulas 
 from the results of tests which may be used in this way. In the 
 following table of these formulas, d is the thickness of dielec- 
 tric in milli-centimeters through which a disruptive discharge 
 will occur under the impulsion of E kilovolts. The given 
 voltage is the maximum value of a sinusoidal voltage wave 
 between flat electrodes. 
 
 dielectric materials 
 Air .... 
 Mica .... 
 Yulcanized Fiber 
 Dry Wood Fiber 
 Paraffined Paper . 
 
 Melted Paraffine . 
 
 Copal Varnish 
 Crude Lubricating Oil 
 Vulcabeston 
 Asbestos Paper 
 
 TABLE 
 
 FORMULAS 
 
 . d = 36 (e~ 1SE — 1) + 54 E + 1.2 E 2 
 . d = .24 E + .0115 E 2 
 . d = 7.66 E + 2.3 E 2 
 . d = 7.66 E 
 . d = 3 E 
 . d = 12.4 E 
 . d = 30 E 
 . d = 60 E 
 . d = 28E 
 . d = 23 E 
 
 * Steinmetz on Dielectrics, Trans. Amer. Inst. Elect. Eng., vol. 10, p. 85. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 123 
 
 In obtaining data upon which these very rough formulas are 
 based, alternating currents of about 100 cycles per second were 
 used, and the voltage was varied from two to thirty thousand 
 volts. In general, thin layers of dry air withstand about six 
 thousand volts per millimeter when the spark occurs between 
 needle points; oils from twelve hundred to a hundred thousand 
 volts, and specially selected oils from one quarter to a million 
 volts per centimeter between needle points, depending upon the 
 quality ; and mica from several hundred thousand to several 
 million volts per centimeter between flat electrodes pressing 
 the material between them. 
 
 The wire used for winding the fields and armatures of alter- 
 nators is almost always double or triple cotton-covered copper 
 wire or strip of high conductivit}'. The standard of conduc- 
 tivity adopted by the American Institute of Electrical Engi- 
 neers in 1893 is 9.59 ohms per mil-foot at 0° C.* The 
 average rate of change of resistance per centigrade degree of 
 change of temperature within ordinary ranges may be taken 
 as .42 per cent of the resistance; at 0° C. This makes the 
 resistance at any temperature, in comparison with the resist- 
 ance at 0° C., 
 
 R t = R 0 (l + .0042 0, 
 
 where R t is the resistance at the required centigrade tempera- 
 ture, R 0 at zero temperature, and t is the required temperature. 
 The copper conductors used in alternator windings should fall 
 little below this standard. For winding armatures and fields 
 the copper is made up in the form of bars (Fig. 83), strap 
 (Figs. 80 and 81), or the ordinary round wire (Fig. 79), as the 
 conditions make desirable. In large machines bar or strap 
 copper is used almost exclusively. 
 
 The armature cores themselves may be made of punched iron 
 disks, iron punchings of special shapes, iron wire, or iron tape. 
 In American machines the punchings are almost universal. 
 The old Ivapp machines had armature cores made of iron' tape.f 
 In the larger sizes of American machines, which are built to 
 connect directly to steam engines or large turbines, and there- 
 
 * Trnns. Amer. Inst. Elect. Eng., October, 1893. See also Standardization 
 Rules of 1907, Par. 260. 
 
 t Kapp’s Dynamos, Alternators, and Transformers, p. 467. 
 
124 
 
 ALTERNATING CURRENTS 
 
 fore have armatures of large diameters, the armature cores 
 are usually built up of segmental punchings put together in 
 such a way that the segments of alternate layers break joints 
 
 Fig. 111. — Punching for Rotating Alternator Armature. 
 
 (Fig. 111). In stationary armatures the teeth are, of course, 
 inside and the keys outside. 
 
 Figure 112 shows an old-style single-phase machine with 
 rotating armature constructed with one armature slot per field 
 pole. At A' may be seen one of the punchings which go to 
 make up the laminations of the armature. The field pole 
 pieces are solid in this machine, which is common, though they 
 
 Fig. Ill a. — Core Stamping with Distance Pieces for Procuring Ventilation. 
 
 are also sometimes made up of punchings secured in the frame 
 in numerous ways. 
 
 It is usual to provide ventilating ducts between core lamina- 
 tions, and these ducts are provided for in the assembling by the 
 insertion of distance pieces. The distance jueces c , c , illustrated 
 in Fig. Ill a are usually ribs of brass or T angle iron which may 
 be one half inch or more in depth in large size machines and 
 are riveted to core stampings. These are placed at the desired 
 distance apart in the core and make annular ventilating ducts 
 at intervals of a few inches in the length of the core. The 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 125 
 
 ventilating ducts are to be plainly seen where they come to the 
 face of the armature core in various illustrations of the text, 
 such as Figs. 80 to 84. 
 
 The sectional view in Fig. 
 
 113 also shows the ventilat- 
 ing ducts. 
 
 Figure 113 is a partial 
 side elevation and section 
 of a rotating-field alterna- 
 tor having a three-phase 
 armature winding with 
 one slot per pole per phase, 
 in which the general 
 method of mechanical con- 
 struction is shown. Much 
 care must be taken in the 
 selection of iron for the 
 magnetic circuit of alter- 
 nators, in order to keep 
 the iron losses at a mini- 
 mum. This subject will be discussed at some length in 
 a later chapter.* 
 
 Fig. 112. — Side Elevation of Alternator, 
 showing Armature Punchings. 
 
 Fig. 113. — Side Elevation and Section of Rotating Field Alternator, showing 
 General Construction. 
 
 In high-speed turbine generators the rotating field magnet 
 must be made very substantial to withstand the strains due to 
 
 * Chap. IX. 
 
126 
 
 ALTERNATING CURRENTS 
 
 Fig. 113 a. — Two-pole Ro 
 tating Field for a Tur 
 bine Generator. 
 
 the high circumferential velocity. Figure 113 a shows a West- 
 inghouse two-pole rotating field magnet with the lower half of 
 the field partly wound. After the coils 
 have been wound in the slots and insu- 
 lated, as shown in the lower half of the 
 figure, they are fastened firmly in place 
 by heavy metal strips driven through the 
 keyways shown near the outer ends of 
 the lugs. Thus the field is made entirely 
 metal-clad. Ventilation is obtained by 
 means of the holes seen in the ends of 
 the core and by the circumferential slots. The core itself is 
 built up of steel disks about .015 of an inch thick, which are 
 machined for the windings and ventilation. 
 
 Figure 113 b shows an alternator of the General Electric Co. 
 on a vertical shaft, driven by a Curtis steam turbine. This 
 figure shows the machine in one half section. The nozzles and 
 guide vanes of the turbine are within the lower cylindrical por- 
 tion. The shaft, which carries the multipolar rotating field 
 magnet of the alternator at its upper part and the turbine 
 runners at its lower part, is supported on a step-bearing sup- 
 plied with water under high pressure as a lubricant. 
 
 The fields and armature of the Westinghouse alternator illus- 
 
 trated in Figs. 83 a and 113 a are intended to be driven by a 
 horizontal steam turbine of the Westinghouse-Parsons type 
 which runs at a high speed, and the field has fewer poles, — in 
 this case it is bipolar. 
 
 * See Arts. 106 and 113. 
 
ARMATURE AND FIELD WINDINGS FOR ALTERNATORS 12 
 
 Fig. 113 6. — Alternator on Vertical Shalt of a Steam Turbine. 
 
CHAPTER IV 
 
 SELF-INDUCTION, CAPACITY. REACTANCE. AND IMPEDANCE 
 
 37. Self-induction. — Before proceeding with the develop- 
 ment of the design and operation of alternating-current ma- 
 chines, it is essential to discuss the relations which exist 
 between voltages and currents in circuits which carry alternat- 
 ing currents. It is to be remembered that the current pro- 
 duced by cutting lines of force sets up in turn magnetic force, 
 which is in opposition to the original field. Again, when a 
 current is introduced into a circuit, it produces a magnetic 
 flux the rise of which causes a counter-voltage. These condi- 
 tions are necessary under the law of conservation of energy 
 and its corollary, Lenz’s Law.* This counter-voltage is called 
 the Voltage or Electro-motive force of Self-induction, and the 
 phenomenon as a whole is called Self-induction. 
 
 Self-induction , therefore , may he defined as the inherent quality 
 of an electric circuit whereby the electro-magnetic induction set up 
 by an electric current in the circuit tends to impede the current's 
 own introduction , variation , or extinction in the circuit. 
 
 The voltage in a circuit which at any instant is due to self- 
 induction is evidently proportional to the rate of change of the 
 magnetic flux set up by the current of the circuit ; therefore, 
 the voltage of self-induction is proportional to the rate of 
 change of current in the circuit, provided that the reluctance 
 of the magnetic circuit remains constant ; or 
 
 e z ce- 
 
 de}) _di 
 dt X ~Jf 
 
 where e t is the instantaneous self-induced voltage in the circuit, 
 ej) the magnetism, and i the current, respectively, at the same 
 instant of time t.. The minus sign is used because the voltage 
 induced at each instant is in a direction to impede the changes 
 of current and magnetism. 
 
 * Jackson’s Electro-magnetism and the Construction of Dynamos, p. 77. 
 
 128 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 129 
 
 If the circuit exhibits the phenomena of electrical resistance 
 and self-induction, and none other, as would be true of a sim- 
 ple coil of wire, the voltage impressed on its terminals must at 
 each instant be sufficient to overcome the counter- voltage e : and 
 supply the iR drop corresponding to the current at the par- 
 ticular instant. That is, the voltage impressed on the termi- 
 nals of the circuit, or Impressed voltage, is made up of two 
 components, one equal and opposite to the counter-voltage and 
 the other equal to the drop of voltage through the resistance 
 of the circuit. The latter component represents the portion of 
 the force driving the current through the circuit which results 
 in expending power therein. 
 
 When the instantaneous values of current and voltage are 
 considered, the iR drop, the counter-voltage e h and the im- 
 pressed voltage e are scalar values, and therefore iR — e — e , ; 
 but a little consideration will show that the Phase of the alter- 
 nating' voltage of self-induction is not in unison with that of 
 the current, although their periods are the same ; because the 
 counter- voltage is proportional to the rate of change of magnet- 
 ism caused by the current, and, supposing the magnetism to be 
 alternating, its rate of change is zero when it is at a maximum, 
 and the voltage of self-induction is therefore zero at the same 
 time. Manifestly, therefore, a vector relation must be recog- 
 nized when the effective values of current and voltage are 
 considered. 
 
 When the curve of current is a sinusoid, the rate of change 
 of its ordinates at any point is 
 
 i m being the maximum value of current ; and the counter-electro- 
 
 assuming that the reluctance of the magnetic circuit is uniform. 
 The curve sin (a + 90°) is manifestly a curve which occupies a 
 position 90° earlier than (or in the Lead of) the curve sin a, and 
 the curve — sin (a + 90°) is 180° later than sin (a + 90°) and 
 therefore 90° later than the curve sin a. Hence the curve of 
 the counter-voltage of self-induction is a sinusoid, the phase of 
 which Lags 90° behind (that is, is 90° later than) the phase 
 
 di _ i m d (sin a) 
 dt dt 
 
 motive force e t is therefore proportional to — 
 
 sin ( a -f 90 °)da 
 dt 
 
130 
 
 ALTERNATING CUR R ENTS 
 
 of the current setting it up. It is evident also that this must be 
 the case, since when sinusoidal current is passing through the 
 zero value, rising in the positive direction of flow through the 
 circuit, its rate of change is a maximum and the self-induced 
 voltage is at maximum value in a direction opposing the rise 
 of current ; and when the current has risen to a maximum (a 
 quarter period, or 90°, later), its rate of change is zero, and the 
 self-induced voltage is zero and about to change in direction 
 from negative to positive. Thus it is seen that the self-induced 
 voltage goes through its cycle in a phase which is 90° later 
 than the current. 
 
 38. Vector Diagrams representing the Voltage Relations in an 
 Inductive Alternating-current Circuit. — Suppose the line 00 
 
 C describes a circle and its projection on the vertical axis 
 describes a simple harmonic motion. (In this book, such vec- 
 tors are assumed to rotate in a counter-clockwise direction.) 
 At each instant the projection of the line on the vertical axis 
 proportionally represents the magnitude of the instantaneous 
 active voltage (i.e. iR drop) corresponding to the angular 
 advance of the line. The projection of the line OjD, which is 
 90° behind 00, likewise represents the instantaneous counter- 
 voltage at each instant. The algebraic sums of the instanta- 
 neous projections of OC and CA (the reverse of OR') are al 
 ways equal to the simultaneous projections of OA, where OA 
 is the impressed voltage. The impressed voltage OA has the 
 magnitude and phase relation of the resultant of the active 
 voltage (iR drop) and the opposite of the self-induced voltage. 
 This may be expressed in still another way, which may pos- 
 sibly give a more useful conception : The impressed voltage ap- 
 
 plied to an alternating-current circuit must have such magnitude 
 
 Fig. 114. — Phase Diagram of Voltages in a Self-inductive 
 Circuit. 
 
 (Fig. 114) repre- 
 sents the maxi- 
 mum value of the 
 iR drop in a cir- 
 cuit carrying a 
 sinusoidal cur- 
 rent. If the line 
 is uniformly ro- 
 tated around the 
 point 0, its end 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 131 
 
 and jjhase relation that it will neutralize all other live voltages 
 and at the same time furnish the active voltage which drives the 
 current through the resistance of the circuit. From the relations 
 given it is seen that 
 
 ( OAf = (OCf + ( GA)\ or OA = V(<96 7 ) 2 + ( CA f. 
 
 The lengths of the lines in the figure have been assumed 
 to be proportional to maximum values of the voltages, but the 
 effective values of the voltages hold the same relations since 
 each effective value is equal to the maximum multiplied by 
 .707. Consequently, the effective value of the impressed vol- 
 tage operating in a circuit with resistance and self-inductance in 
 series is equal to the square root of the sum of the squares of 
 effective values of the active voltage and the inductive voltage 
 reversed. Or when E, E r , and E x are respectively effective 
 values of impressed voltage, active voltage, and self-induced 
 voltage reversed, 
 
 E=fE 2 + Ef. 
 
 The triangle of voltages representing these conditions is shown 
 in Fig. 115. (Such triangles, in this book, usually have the 
 active voltage line laid out in a 
 horizontal direction.) 
 
 If the heights of the points A, 
 
 (7, and D of the uniformly rotat- 
 ing vectors OA, OC , and OD are 
 plotted as ordinates to the corre- 
 
 , . . - . Fig. 115. — Vector Polygon of Vol- 
 
 spondmg talues ol time 01 of tages for Series Self-inductive Cir- 
 ce laid out on the axis of abscissas cuit. 
 
 as illustrated in Figs. 3 and 4, three sine curves are traced 
 like those at the right hand of Ffig. 114. These are marked 
 respectively E , E n and — E x . They represent the waves of 
 impressed voltage, IR drop, and inductive voltage. The in- 
 stantaneous value of the impressed voltage at each instant is 
 equal to the corresponding instantaneous values of E r and E x 
 added together algebraically. 
 
 The instant of time in the sine wave plot which corresponds 
 to the time when the rotating vectors arrive at the position in- 
 dicated by the left-hand part of Fig. 114 is shown by the 
 dotted vertical axis cutting through the sine curves. If the 
 zero of time is assumed at the instant when the impressed voltage 
 
132 
 
 ALTERNATING CURRENTS 
 
 is passing through zero from the negative to the positive direc- 
 tion, this would be indicated by the phase diagram of rotating 
 vectors at the instant when OA lies in the positive direction 
 along the W-axis. The relations of the voltage waves at that 
 instant are indicated at the left-hand terminus of the plot of 
 sine curves. It is obvious that the inductive voltage must be 
 equal to the iR drop at that instant, to maintain the current i, 
 since the then instantaneous value of the impressed voltage is 
 zero. 
 
 Prob. 1. Construct a vector polygon of voltages for a circuit 
 containing a self-inductive sinusoidal voltage of 50 volts and 
 an IR drop of 50 volts. 
 
 Prob. 2. Construct a vector polygon of voltages for a circuit 
 containing an IR drop of 100 volts and a self-inductive sinu- 
 soidal voltage of 5 volts. 
 
 Prob. 3. Construct a vector polygon of voltages for a circuit 
 containing 20 volts IR drop and 500 sinusoidal self-inductive 
 volts. 
 
 Prob. 4. What are the values of the impressed voltages in 
 problems 1, 2, and 3 ? 
 
 Prob. 5. Draw the curves of voltage in a circuit having an 
 IR drop (effective value) of 100 volts and a self-inductive 
 voltage (effective value) of 50 volts, on the supposition that 
 the curves are sinusoids. 
 
 Prob. 6. In problem 5 find the instantaneous values of the 
 self-inductive and impressed voltages when the IR drop is 0 ; 
 find the impressed voltage and IR drop when the self-inductive 
 voltage is 0 ; find the IR drop and self-inductive voltage 
 when the impressed voltage is 0. 
 
 Prob. 7. Find the instantaneous values of the three voltages 
 in problem 5 when a — 0, 45°, 90°, 135°, 180°, 225°, 270°, 315°, 
 and 360° ; find these values graphically from the rotated vector 
 polygon and from the curves of voltage. 
 
 39. Angle of Lag. — The angle 6 between the lines OA and 
 OC in Figs. 114 and 115 shows the amount by which the phase 
 of the active voltage (which is in phase with the current) lags 
 behind that of the impressed voltage. The figure makes it evi- 
 
LF-INDUCTION, CAPACITY, PEACTANCE, AND IMPEDANCE 133 
 
 dent that this lag is caused by the position of the self-inductive 
 voltage, and that its magnitude depends upon the relative 
 lengths of the line 6hi( = — OB') and the line 00. .The tangent 
 of the angle is 
 
 „ CA Inductive voltage reversed 
 
 tan 0 = — — — 
 
 00 Active voltage 
 
 The angle 0 is called the Angle of lag. The relations are 
 plainly shown in Fig. 115. 
 
 Prob. 1. What are the angles of lag in problem 1, problem 2, 
 and problem 3 of Article 38 ? 
 
 Prob. 2. Find the angle of lag in a circuit which has an 
 impressed sinusoidal voltage of 100 volts and an active sinu- 
 soidal voltage (IR drop) of 30 volts. 
 
 Prob. 3. Find the angle of lag in a circuit which has an 
 impressed sinusoidal voltage of 300 volts and a self-inductive 
 sinusoidal voltage of 150 volts. 
 
 40. Sinusoidal Voltage in a Self-inductive Circuit expressed 
 as a Complex Quantity. — The complex quantity which ex- 
 presses the vector representing the impressed voltage is 
 
 E — E r + 3 E x , 
 
 or E = Z'fTos 6 +j sin d), 
 
 where E is the numerical or scalar value of E , E r the scalar value 
 of IR drop and E x the scalar value of inductive voltage reversed. 
 
 Also 
 
 E = VE* + E} and tan 6 = ^. 
 
 E,. 
 
 If there are a number of circuits in series having impressed 
 voltages E v E 2 , ZJ 3 , etc., the combined voltage across the out- 
 side terminals is represented by the vector sum 
 
 E — E r +jE x — (E ri + E r 2 +E ra + etc . ~) + j (E Xi + E Xi + E Xi + etc.) 
 in which 
 
 E r — E ri -f- E rt -f- E ri -(- etc. and E x — E Xi + E Xt + E Xa -f- etc. 
 The angle of lag is equal to 
 
 g = tan -1 ^• ri ^ + ^ etc -) 
 
 (Zy 1 + E Vi -(- Zy 3 -f- etc.) E r 
 
 The angle 6 is the angle of lag measured between the phase of 
 
134 
 
 ALTERNATING CURRENTS 
 
 the current flowing in the circuit and the voltage impressed 
 upon the entire circuit. 
 
 Prob. 1. The IR drop in a circuit is 50 volts and the. self- 
 inductive voltage 25 volts. Find the impressed voltage and 
 the angle of lag of the current. (In this and the following 
 problems all voltages are assumed to be sinusoidal.) 
 
 Prob. 2. The IR drop in a circuit is 50 volts and the self- 
 inductive voltage 500 volts. Find the value of the impressed 
 voltage and the current lag. 
 
 Prob. 3. The angle of lag in a circuit is 30° and the im- 
 pressed voltage 200 volts. What are the values of IR drop 
 and self-inductive voltage ? 
 
 Prob. 4. The IR drop and self-inductive voltage of two cir- 
 cuits are respectively 50 and 30 volts, and 40 and 20 volts. 
 What will be the resultant voltage if the two circuits are placed 
 in series, and what will be the angle of lag between the current- 
 flow and the voltage impressed between the main terminals ? 
 
 Prob. 5. Two circuits in series have voltages impressed 
 upon them of 100 and 200 volts and angles of lag of 30° and 
 45° respectively. Calculate the total voltage across the two 
 circuits and the angle of lag of the current with respect thereto. 
 
 Prob. 6. Three voltages in a circuit of 100, 200, and 300 
 volts are in series, the angles of lag in the three parts of the 
 circuit being respectively 20°, 40°, and 60°. What are the 
 resultant voltage and angle of lag ? Also what are the IR 
 and self-inductive components of the total voltage ? 
 
 41. Self-inductance. — The magnitude of the voltage of self- 
 induction is dependent upon the number of lines of force 
 inclosed by the circuit per unit current flowing in it, the num- 
 ber of turns composing the circuit, and the current flowing 
 therein; which follows from the fundamental equation for vol- 
 tage* ; the formula is 
 
 ndd> 
 
 e ‘ ~ ~ l¥Jt' 
 
 where e t is the voltage of self-induction for the given rate of 
 change of magnetization, n is the number of turns in the coil. 
 
 * Art. ll. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 135 
 
 and, as before, — is the rate of change of magnetism. But in 
 dt 
 
 a long or closed solenoid, the magnetizing force is * 
 
 7i/T _ 4 tmi 
 10 ’ 
 
 where i is the value of the current at the instant under con- 
 sideration ; and the reluctance of the magnetic circuit isf 
 
 where l and A are respectively the average length and cross 
 section of the solenoid and y. is the permeability, assumed in 
 this paragraph to be uniform. Therefore, the number of lines 
 set up by a current i is f 
 
 and 
 
 4 nrniAy 
 10 1 ’ 
 
 nd<f> _ n ^ 4 irnAy 
 10 8 dt ~ ~ 10 8 X 10 z " 
 
 j, 4:7rnAu j. 
 
 d + = -wT x *> 
 
 , di _ 4 irn 2 Ay w di 
 
 ( jt~ io^ x it 
 
 The numerical value of 
 
 n ^ 4 irnA/i 
 
 To 8 x 10 l 
 
 is called the Self-inductance or the Coefficient of self-induction 
 
 of the coil, and is usually represented by the capital letter L. 
 The number of lines of force passing through the solenoid when 
 the current is one ampere is equal to 
 
 4 TrnAfj- 
 10 l 
 
 Hence the value of the self -inductance of a long solenoid of uni- 
 form ’permeability may be defined as times the product of the 
 
 number of turns in the circuit by the number of lines of force 
 inclosed by the circuit when carrying one ampere of current. That 
 is, in the case of a long solenoid in a medium of uniform per- 
 meability, n 0 
 
 L 10 8 x i ' 
 
 * Jackson’s Electro-magnetism and the Construction of Dynamos , p. 15. 
 t Jackson's Electro-magnetism and the Construction of Dynamos , pp. 6 
 and 7. 
 
136 
 
 ALTERNATING CURRENTS 
 
 Since the number of lines of force developed by a coil is pro- 
 portional to the number of turns composing it, its self-inductance 
 is proportional to the square of its turns. Also, it will be ob- 
 served that the formula for self-induced electro-motive force 
 reduces to 7 . 
 
 t -di 
 
 “ = ~ L it 
 
 42. The Henry. — The Chicago Electrical Congress formally 
 assigned the name Henry to the unit in which self-inductance 
 is measured, after Professor Joseph Henry, the notable Ameri- 
 can discoverer of electro-magnetic phenomena. Since the volt 
 is 10 8 times as large as a C. G. S. unit of electro-motive force and 
 the ampere is as large as the C. G. S. unit of current, it is 
 apparent from the formula that the Henry is 10 9 times as large 
 as the corresponding C. G. S. unit of self-inductance. 
 
 43. Self-inductance of a Short Coil ; Circuit containing Variable 
 Permeability ; Examples. — The definition of the henry is above 
 
 developed for a long solenoid 
 in which all the lines of force 
 pass through all the turns. 
 If the circuit is not so con- 
 structed, the definition still 
 holds, but the summation of 
 the number of lines of force 
 passing through each turn 
 individually must be taken, 
 since the number of lines 
 passing through any turn is 
 a variable which depends 
 upon the position of the turn 
 in the coil. Thus, suppose Fig. 116 represents a short solenoid 
 of eight turns in which are developed ten lines of force when 
 one ampere flows through the coil. Assuming the distribution 
 of the lines shown in the figure, the self-inductance is calcu- 
 lated as follows : 
 
 Fig. 116. — Solenoid and Magnetic Field. 
 
 10x2 + 8x2+6x2 + 4x2 
 10 s 
 
 56 
 
 io 8 
 
 henrys. 
 
 If an iron core is now placed in the coil, the number of lines 
 of force is increased directly as the reluctance of the magnetic 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 137 
 
 circuit is decreased. 
 
 Hence, assuming the distribution of the 
 
 56 P' 
 
 lines to remain unchanged, the self-inductance becomes 
 P' 
 
 henrys, where — 
 
 10 8 P 
 
 is the ratio of the reluctance before and after 
 
 the iron core is inserted. 
 
 In the case of a long solenoid 
 
 T n 4 7 rnA 
 L = — x 
 
 10 8 
 
 10 Z 
 
 when the permeability is unity, but when the permeability of 
 the magnetic circuit taken as a whole is g, the number of lines 
 of force per ampere, as shown before, is 
 
 4 7 ruA/i 
 
 101 
 
 and therefore 
 
 T 11 
 
 i = foS x 
 
 4 TrnAfi 
 
 10 l 
 
 In general, where the magnetic circuit is composed wholly of 
 non- magnetic material, the self-inductance is 
 
 t _ n <f> 
 
 10 8 i 
 
 (where </> represents the number of lines of force for current if 
 and the inductance is a constant for all values of i. However, 
 when iron or other magnetic material is included in the mag- 
 netic circuit, the value of the self-inductance varies with the 
 value of i because g varies with fi. As before, 
 
 10 H 
 
 hut this may have a different value for each value of i since 
 
 4 7 rnA/jii 
 
 4 > = 
 
 10 l 
 
 and /r varies with 0. In this case the inductance for any value 
 of i is n times as great as when no magnetic material is in- 
 cluded in the magnetic circuit, the value of fi taken being that 
 corresponding to the particular value of i. 
 
 Therefore, the self -inductance of a long solenoid ivliich contains 
 an iron core , when carrying a certain current , may be defined as 
 
138 
 
 A LT E RNATING C U li R ENTS 
 
 - times the number of turns in the solenoid multiplied by the 
 
 number of lines of force set up by the current and divided by the 
 number of amperes of the current. It is shown later that the 
 self-inductance found by this relation is not necessarily equal 
 to the apparent self-inductance found by the use of measuring 
 instruments because the disturbing effects of eddy currents and 
 hysteresis in iron cores sometimes mask the results. 
 
 As an example of the calculation of the value of A, consider 
 a uniform ring of wrought iron 100 centimeters in mean cir- 
 cumference and 20 square centimeters in cross section. Sup- 
 pose a coil of 2500 turns is uniformly wound on the ring, and a 
 current of two amperes is passed through the magnetizing coil. 
 Taking y as equal to 250, which is a fair value, 
 
 _ 4 7 mAyl _ 
 10 1 ~ 
 
 4 7r x 2500 x 20 x 250 x 2 
 10 x 100 
 
 314,200 ; 
 
 hence, 
 
 L = 
 
 n<f> _ 2500 x 314,200 
 10 8 / 10 8 x 2 
 
 = 3.93 henrys. 
 
 If the current in the magnetizing coil is taken as 1|, and it is 
 supposed that the value of y becomes roughly 300, then 
 
 T 2500 x 282,744 , 71 , 
 
 10 8 x 1.5 J 
 
 That is, the permeability having increased with the reduction 
 of the magnetic density in the iron core, the magnetism per 
 ampere is greater with the smaller current and the self-induc- 
 tance is therefore larger, although the total magnetism is smaller. 
 
 If a similar winding is put on a ring of the same dimensions 
 but of brass or other non-magnetic material, the magnetism set 
 up per ampere is: 
 
 4 > = 
 
 4 7T x 2500 x 20 x 1 
 10 x 100 
 
 628.3, 
 
 and the self-inductance is : 
 
 L = 
 
 2500x628.3 
 10 8 x 1 
 
 = .0157 henry. 
 
 In this case the self-inductance is not affected by the strength 
 of the current, because the permeability of the medium is fixed 
 and independent of the current flow. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 139 
 
 According to the foregoing definitions, the number of henrys 
 of self-inductance possessed by any circuit, whatever may be 
 
 the current flow, is equal to -i- times the change of the sum- 
 
 mation of the linkages of lines of flux through or around the con- 
 ductors of the circuit which would he caused by a change of the 
 current by one ampere, assuming that the magnetic reluctance 
 of the surrounding medium does not change at the same time. 
 In the case of the long solenoid or the ring solenoid, all of the 
 lines of flux link through the circuit as many times as there 
 are turns in the coil, and the summation of linkages may be 
 found by multiplying the total number of lines of flux by all 
 of the turns of the coil ; hut in other instances some of the 
 lines of flux make a different number of linkages with the cir- 
 cuit from others, and then the summation of the linkages can 
 be determined only by tracing the linkages of the lines indi- 
 vidually and summing them up as in the example on page 136. 
 The self-inductance ordinarily required in dealing with alter- 
 nating-current circuits, is that calculated upon the basis of the 
 circuital magnetic permeability obtaining for the maximum in- 
 stantaneous ordinate of the current under consideration. 
 
 In the usual practical problems that are met by the engineer, 
 the conformation and numerical constants of the magnetic cir- 
 cuits and their windings are often unknown or are so irregular 
 that the self-inductance cannot be determined by calculation, 
 and experimental determination must then be resorted to ; but 
 the same definitions apply to the units and the same physical 
 conceptions may be cultivated. 
 
 The formula e t = — L 
 
 di 
 
 dt 
 
 obviously only applies to conditions 
 
 in which L is constant, which is not the case when the magnetic 
 circuit includes an iron core. In the latter case, the formula 
 becomes : 
 
 d(Li) 
 
 e t =- 
 
 dt 
 
 _ a dJjn) = _a 
 dt 
 
 
 di , -da 
 a 1 - l — 
 
 dt dt 
 
 If the winding under consideration is a long solenoid on an 
 iron core, the constant a in this equation is expressed by 
 
 n 4 7 rnA 
 
 a = — x . 
 
 10 8 10 1 
 
140 
 
 ALTERNATING CURRENTS 
 
 It will be observed that the first term of the last member of the 
 
 foregoing equation for e t is equal to Z . The second term of 
 
 the same member shows the effect on the induced voltage which 
 is produced by the change of the permeability of the iron in the 
 magnetic circuit as the current and magnetic density change. 
 The algebraic sign of the first term is fixed by the direction 
 
 of change of the current, ^ ; but the second term may be 
 
 at 
 
 either positive or negative with respect to the algebraic sign of 
 depending on which limb of the permeability curve comes into 
 
 the conditions concerned, — that is, depending upon whether 
 the permeability increases or decreases as the current rises. 
 
 The induced voltage e t is therefore numerically equal to 
 
 L y only when L is constant, and it may be numerically either 
 larger or smaller than L -T when the permeability of the mag- 
 
 netic circuit concerned varies as the magnetic density and cur- 
 rent change. It is numerically larger than L ^ when the per- 
 
 meability varies similarly as the magnetic density and current 
 (i.e. increases when they increase and decreases when they 
 
 decrease); and it is numerically smaller than Z~ when the 
 
 permeability varies inversely to the magnetic density and cur- 
 rent, as it does when the magnetic density is larger than that 
 producing the maximum permeability in the iron. 
 
 Prob. 1. A closed ring of iron contains a flux of 2,000,000 
 lines when a current of 5 amperes flows through its magnetizing 
 coil of 1500 turns. What is the self-inductance of the coil ? 
 In this and the following problems the effect of magnetic leak- 
 age is assumed to be of negligible magnitude. 
 
 Prob. 2. A closed ring having a constant permeability of 
 1000 units and a cross section of 200 square centimeters and 
 a length of 150 centimeters, has wound upon it a coil con- 
 taining 2000 turns of wire. What is the self-inductance of 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 141 
 
 M 
 
 the coil? (Aid: remember that the flux <E> = — , where M is 
 
 4 7 nil 
 
 10 
 
 magneto-motive force and P reluctance ; that M = 
 
 and P = where l is the length of the magnetic circuit and 
 A/x 
 
 A is its cross section.) 
 
 Prob. 3. In problem 2 the ring is cut and its ends pulled 
 apart until there is an air space included in the magnetic circuit 
 having a length of a half centimeter and an effective cross 
 section of 280 square centimeters. What is the self-inductance 
 of the coil, supposing that the permeability of the iron has not 
 changed ? 
 
 Prob. 4. A ring of iron wire 100 centimeters long and 400 
 square centimeters in cross section is surrounded by a magnet- 
 izing coil of 200 turns. When the magnetic density in the 
 iron is 2000 lines of force per square centimeter, /x = 1000 ; 
 when the magnetic density is 5000, /x = 1200; when the mag- 
 netic density is 10,000, /x = 800 ; when the magnetic density is 
 20,000, /x = 300 ; when the magnetic density is 40,000, /x = 50. 
 What is the self-inductance of the coil for each value of the 
 five magnetizations ? 
 
 Prob. 5. The field windings of a bi-polar direct-current dy- 
 namo contain 2000 turns carrying a current of 2| amperes, the 
 field magnet has an average length of 150 centimeters, an aver- 
 age cross section of 500 square centimeters, and an average per- 
 meability of 800; the armature has the same average magnetic 
 cross section and a length of 30 centimeters with a permeability 
 of 1500; and the two air spaces each have a length of 1 centi- 
 meter and a cross section of 600 square centimeters. What is 
 the self-inductance of the exciting winding ? 
 
 44. Examples of Self-inductances. — Ordinary practical experi- 
 ence in electrical measurements and in handling wires soon gives 
 a capacity for estimating the values of resistances ; in the same 
 way facility is soon gained in roughly estimating electrostatic 
 capacities, or the current which maybe safely carried by a wire, 
 or even the ampere turns required to produce a given magnet- 
 ization in a magnetic circuit. Ordinary practice, however, 
 gives little clue to estimating the self-inductance in a circuit. 
 
142 
 
 ALTERNATING CURRENTS 
 
 It is true that, as already shown, the self-inductance is depend- 
 ent upon the magnetism inclosed in the circuit and the number 
 of turns thereof, but experience in dealing with coils and mag- 
 netic circuits is not usually regarded in such a way as to aid 
 in estimating self-inductances. The following values of self- 
 inductance are therefore presented here to give a foundation for 
 judgment. 
 
 The range of self-inductances met in practice is very w r ide. 
 The smallest which are practically met are in the doubly 
 wound resistance coils used for Wheatstone bridges and similar 
 devices. Since the wire in these is doubled back upon itself, 
 the magnetic effect of the current is almost neutral and the 
 inductance is often less than a microhenry (one millionth of a 
 henry). The inductance of a certain electric call bell of 2.5 
 ohms resistance has been found to be 12 microhenrys ; a tele- 
 phone call bell of 80 ohms resistance, 1.4 henrys; a modern 
 short-core telephone bell of 1000 ohms resistance, 5.5 henrys: 
 the armature of a magneto calling generator of 550 ohms resist- 
 ance, from 2.7 henrys when the plane of the coil lay in the plane 
 of the pole pieces, to 7.3 henrys when the plane of the coil was 
 perpendicular to the plane of the pole pieces ; more modern 
 magneto generator armatures, one measuring 540 ohms resist- 
 ance, 1.6 henrys with plane of coil in plane of pole pieces, 2.4 
 henrys with plane of coil perpendicular to plane of pole pieces: 
 one measuring 125 ohms resistance, .51 and .74 henry; and 
 one measuring 110 ohms resistance, .89 and 1.17 henrys ; a Bell 
 telephone receiver measuring 75 ohms resistance, with dia- 
 phragm, 75 to 100 millihenrys (thousandths of henrys), without 
 the diaphragm about 35 per cent less; a modern bipolar telephone 
 receiver with 70 ohms resistance, 35 millihenrys; mirror gal- 
 vanometers vary with their resistance from a few millihenrys 
 to 10 or 12 henrys ; a mirror galvanometer for submarine sig- 
 naling of 2250 ohms resistance, 3.6 henrys; astatic min'or 
 galvanometers of 5000 ohms resistance average about 2 henrys. 
 The single coil of a Thomson galvanometer of 2700 ohms 
 resistance measured 2.56 henrys; the coil of another Thomson 
 galvanometer having 100,000 ohms resistance measured 70 
 henrys ; the coil of an Ayrton and Perry spring voltmeter, with- 
 out iron core, measured 1.462 henrys. This coil had a length 
 of 2.88 inches, an external diameter of 3 inches, was wound on 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 143 
 
 a brass tube .58 inch in external diameter, and had a resistance 
 of 338.5 ohms. Each of the above measurements was made 
 with a current of a few milliamperes. The following are meas- 
 urements of telegraphic apparatus: 
 
 TABLE 
 
 Polarized Relays of Various Types 
 
 Type 
 
 1 
 
 2 
 
 3 
 
 4 
 
 P.esistance in Ohms 
 
 419 
 
 423 
 
 413 
 
 413 
 
 Self-inductance in 
 Henrys 
 
 1.99 
 
 1.89 
 
 1.69 
 
 1.31 
 
 Testing Current in 
 Milliamperes 
 
 6.3 
 
 6.3 
 
 6.3 
 
 6.3 
 
 All armatures were 4 mils (thousandths of an inch) from the 
 magnet poles. 
 
 A common Morse relay of 148 ohms resistance measured 
 10.47 henrys with the armature against the poles, and 3.71 
 henrys witli the armature 20 mils from the poles, the measur- 
 ing current being 6.3 milliamperes. In ordinary working ad- 
 justment the inductance of a Morse relay is about 5 henrys. 
 Telegraph sounders with bobbins, respectively, 1| by 1, and 1| 
 by 1^ inches, each wound to 20 ohms resistance, measured 191 
 and 150 millihenrys, the armatures being 4 mils from the poles 
 and the measuring current being 125 milliamperes. A single 
 coil of a Morse sounder with a resistance of 32 ohms, and hav- 
 ing an iron core .31 inch in diameter and 3 inches long, the 
 bobbin being .94 inch in diameter, was found to have a self- 
 inductance of 94 millihenrys. A complete sounder with a core 
 like that of the preceding coil, but with bobbins of 50 ohms 
 resistance having a diameter of 1.25 inches, was found to have 
 a self-inductance of 444 millihenrys. The self-inductance of a 
 complete sounder of 14 ohms resistance measured 265 milli- 
 henrys ; a modern telephone relay with 62 ohms resistance, 300 
 millihenrys ; another with 2000 ohms resistance, 840 milli- 
 henrys ; a supervisory relay for telephone service measuring 
 12.5 ohms, approximately 1 millihenry; a modern telephone 
 induction coil, primary coil resistance 17 ohms, self-inductance 
 117 millihenrys, secondary coil resistance 27 ohms, self-induc- 
 
144 
 
 ALTERNATING CURRENTS 
 
 tance 74 millihenrys, mutual inductance approximately 91 
 millihenrys ; a modern telephone repeating coil with 46 ohms 
 resistance in each coil, 1.10 henrys self-inductance in each coil 
 and 1.09 henrys mutual inductance. These measurements of 
 telephone apparatus were made with alternating current of a 
 frequency of 800 periods per second. 
 
 A single phase transmission line of No. 2 B. and S. gauge 
 copper wires spaced 36 inches apart is calculated to have a 
 resistance of about 1.62 ohms and a self-inductance of about 
 3.78 millihenrys per mile; No. 6 copper wire under similar 
 conditions to have a resistance of about 4.09 ohms and a self- 
 inductance of about 4.08 millihenrys per mile. 
 
 Bare No. 12 B. and S. gauge copper wire erected on a pole 
 line about 23 feet from the ground is calculated by Kennedy 
 to measure about 8.5 ohms and 3.15 millihenrys per mile; 
 No. 6 copper wire under similar conditions is calculated to 
 measure about 2.1 ohms and 2.95 millihenrys. A quadruplex 
 telegraph line, with all instruments in circuit, measures approxi- 
 mately 10 henrys. 
 
 The largest self-inductances met in practice are usually in 
 the windings of induction coils or of electrical machinery. 
 The secondary winding of an induction coil capable of giving 
 a 2-inch spark and having a resistance of 5700 ohms, meas- 
 ured 51.2 henrys. The primary winding of an induction coil 
 which is 19 inches long and 8 inches in diameter, measured .145 
 ohm and 13 millihenrys, while its secondary measured 30,600 
 ohms and 2000 henrys. The inductance of dynamo fields is 
 likely to vary from 1 to 1000 henrys ; direct-current dynamo 
 armatures measure between the brushes from .02 to 50 henrys ; 
 the fields of a shunt-wound Mather and Platt direct-current 
 dynamo built for an output of 100 volts and 35 amperes meas- 
 ured 44 ohms and 13.6 henrys at a small excitation ; the arma- 
 ture of the same machine measured .215 ohm and .005 henry; 
 a Mordey alternator armature of the disk type, with a capacity 
 for 18 amperes at 2000 volts, measured 2 ohms and .035 henry ; 
 a Kapp alternator armature of the ring type, with a capacity 
 of 60 kilowatts at 2000 volts measured 1.94 ohms and .069 
 henry ; another Kapp machine, 30 kilowatts, 2000 volts, meas- 
 ured 7 ohms and .0977 henry; the fields of a Ferranti alterna- 
 tor measured 3 ohms and .61 henry, while the armature of the 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 145 
 
 same machine built for an output of 200 volts and 40 amperes 
 measured .0011 to .0013 henry, with no current in fields ; the 
 armature of a British Thomson-Houston alternator, three-phase, 
 850 kilowatts, 5000 volts, measured .332 ohm in resistance 
 and .061 henry in self-inductance per phase ; the armature of a 
 General Electric alternator, three-phase, 2500 kilowatts, 6500 
 volts, measured .41 ohm in resistance and .035 henry in self- 
 inductance per phase ; while another British Thomson-Hous- 
 ton machine, a turbo-alternator, 1500 kilowatts, 1000 volts, gave 
 an inductance per phase of .0403 henry ; the primary and 
 secondary windings of transformers measure roughly from .001 
 of a henry up to 50 henrys, depending upon their output and 
 the voltage for which they are designed. 
 
 An electro-magnet designed by H. DuBois * had, when 
 carrying a current of 45 amperes, a self-inductance of 180 
 henrys. This electro-magnet had a core of iron made of two 
 half rings butted together, which had a radius of 25 centi- 
 meters and a cross section of 78.5 square centimeters. The 
 core was wound with 12 coils of 200 turns each. The self- 
 inductance given is for all coils in series. 
 
 The effect of the field magnets upon the self-inductance of 
 a disk-alternator armature is shown by some measurements 
 taken by Dr. Duncan f on a small Siemens eight-pole alterna- 
 tor, the results of which are given in the following table. 
 
 TABLE 
 
 Self-inductance of Armature in Place 
 
 
 Position of Armatuke 
 
 Current in Field 
 
 
 
 
 
 0° 
 
 111° 
 
 22j° 
 
 0 ampere 
 
 .120 
 
 .112 
 
 .100 
 
 2.5 amperes 
 
 
 .115 
 
 .108 
 
 4.5 amperes 
 
 .128 
 
 .115 
 
 .106 
 
 Self-inductance of armature removed from field, .082 henry; 
 resistance of armature, 7 ohms ; pitch of the poles, 45°. 
 
 Professor Ayrton found that the self-inductance of an unex- 
 
 * The Magnetic Circuit, H. DuBois, p. 264. 
 t Electrical World, Vol. 11, p. 212. 
 
 L 
 
146 
 
 ALTERNATING CURRENTS 
 
 cited Mordey alternator armature varied between .033 and 
 . 038 henry, and that this decreased about 10 per cent when the 
 fields were excited.* 
 
 45. Conditions of Establishment and Termination of Current 
 in a Circuit containing Resistance and Self-inductance in Series. — 
 
 a. Rise of Current under Constant Impressed Voltage. — If a 
 circuit containing constant self-inductance and resistance is 
 suddenly connected to a source of constant voltage, the current 
 is retarded so that its rise is along a logarithmic curve. The 
 voltage E of the source is absorbed at each instant in sup- 
 plying the iR drop and overcoming the counter-voltage 
 
 which represents the instantaneous value of the current flowing 
 at any moment while the voltage E , constant for the time 
 under consideration, is applied to a circuit of constant induc- 
 tance L. To find the value of the instantaneous current at 
 any particular time t, we have from the same equation, by 
 transposition, 
 
 — L — • Hence the equation 
 dt 
 
 and from it is given 
 
 Ldi 
 
 dt 
 
 di dt 
 
 E - iR L ; 
 
 whence 
 
 which gives 
 
 oi- 
 
 and finally 
 
 * Jour. Inst. Elect. Eng., Yol. 18, p. 662; also ibid., p. 654. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 147 
 
 Fig. 117. — Logarithmic Curve of Rising Current. 
 
 where e is the base 
 of the Naperian sys- 
 tem of logarithms. 
 Figure 117 shows 
 the logarithmic 
 curve of rise of cur- 
 rent in a particular 
 circuit and Fig. 
 117 a shows the 
 
 di E 
 
 rate of change — = — e r of the same current. 
 dt L 
 
 It will be observed that, since in this case the counter-voltage 
 Ldi 
 
 of self-inductance, — , is equal to — QE — iK), its instan- 
 
 ce 
 
 taneous value is represented by the exponential expression 
 
 _Rt 
 
 — Ee^, where t is the numerical value of the time for the 
 
 di 
 
 instant at which the ratio — is taken. 
 
 dt 
 
 b. Fall of Current on Withdrawal of Impressed Voltage. — 
 Likewise, when the voltage is withdrawn, E = 0, and if the 
 
 or 
 
 - (log i- log 7) 
 
 t_ 
 
 r 
 
 i i z Rt 
 
 and logj=-— , 
 
 R 
 
 in which I is the value of the current equal to -— . 
 
 9 
 
 . R 
 
 l = R e ~ L ' 
 
 Hence, 
 
148 
 
 ALTERNATING CURRENTS 
 
 which gives the instantaneous value of the current at any 
 instant during its fall, after the voltage is withdrawn. In this 
 Ldi 
 
 is the voltage of self-induction causing the current 
 
 case — 
 
 dt 
 
 to flow, and it is obviously equal at each instant to iR — EV~l. 
 
 Figure 118 shows 
 the logarithmic curve 
 of fall of current for 
 the circuit already 
 referred to by Fig. 
 117, and Fig. 118 a 
 shows the rate of 
 change of the falling 
 current. 
 
 c. Current Value 
 when the Impressed 
 — Under these condi- 
 
 Fig. 118. — Logarithmic Curve of Falling Current. 
 
 Voltage is a Sine Function of the Time 
 tions the voltage equation of the circuit becomes 
 
 Ldi 
 dt 
 
 e =f(t) = e m sin cot= Ri + 
 
 In this expression a> = 2 irf and cot — a. This is a linear differ- 
 ential equation of the first order, the solution of which is * 
 
 i — 
 
 V7U + 4 
 
 sin (« — d) + 
 
 in which 
 
 6 = tan -1 
 
 2 irfL 
 R 
 
 After a short time the 
 exponential term in the 
 right-hand member of the 
 equation becomes inap- 
 preciable since the expo- 
 nent is negative and t is 
 an increasing quantity. 
 It may be neglected for 
 the present. Its effect 
 will be discussed later, f 
 The current now becomes 
 a sine function of the time 
 
 Fig. 118 a . — Logarithmic Curve, showing Rate of 
 Change of Falling Current. 
 
 * Murray’s Differential Equations, p. 26. 
 
 t Art. 59. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 14S 
 
 and lags behind the impressed voltage by an angle 6 whose tarn 
 
 , . 2 7 rfL 
 gent is 
 
 R 
 
 It has the same frequency as the voltage. 
 
 It will be observed that the sinusoidal current has a maximum 
 value of 
 
 2 _ ® m 
 
 VW+ 4 7T 2 / 2 X 2 ’ 
 
 and since the effective values bear a fixed ratio to the maximum 
 values, we have 
 
 ~ V^ + 4tt 2 / 2 L 2 ’ 
 
 in which I is the effective value of the current and R the 
 effective value of the sinusoidal impressed voltage. The de- 
 nominator of this expression is measured in ohms and is called 
 the Impedance of the alternating-current circuit. This im- 
 pedance is composed of the square root of the sum of two 
 terms. The first of these terms is the square of the electrical 
 resistance, and the second of the terms is the square of the 
 expression 2 irfL. This latter expression is called the Reactance 
 of the circuit. The vector relations of the voltages of a circuit 
 containing resistance and self-inductance are shown in Fig. 
 119, which corresponds to Fig. 115. 
 
 46. Transference of Electricity during the Transient State. — 
 When the current in an inductive circuit has reached its full 
 value, a smaller quantity of 
 electricity has passed 
 through the circuit during 
 the interval since the start 
 of the current than would 
 have passed if the retarda- 
 tion, or momentum effect, 
 had not been present. This 
 decrease in the quantity of 
 
 electricity is proportional to the area OYQ between the curve 
 of the current and the horizontal line YQ (Fig. 117). The 
 ordinates of the curve representing instantaneous current 
 strengths in amperes and the abscissas representing time in 
 seconds, the area referred to obviously represents quantity 
 of electricity measured in coulombs. This area may be found 
 in terms of Z, Z, and R as follows : 
 
 
 Er =IR 
 
 Fig. 119. — Voltage Relations in Circuit con- 
 taining Resistance and Inductance. 
 
150 
 
 ALTERNATING CURRENTS 
 
 The area is equal to J' — i) dt, where T is the duration 
 
 of time from the instant of the introduction of the source of 
 voltage, E, into the circuit to the instant at which the current 
 
 E 
 
 reaches its full or ultimate value /= — • Also, since iR = 
 
 R 
 
 E — L— and * = — — it is manifest that 
 dt R Rdt 
 
 .r(i->=x 
 
 ’Ldi 
 
 ~R 
 
 E , 
 
 LI 
 R ‘ 
 
 The area OYQ A, which is equal to — T, represents the number 
 
 R 
 
 of coulombs which would have been transferred in the circuit 
 during time T had the current instantly come to its full or 
 E 
 
 ultimate value of 1= A; and the area OQA , which is equal to 
 ET LI 
 
 — — — , represents the number of coulombs that are actually 
 
 transferred through the inductive circuit while the current is 
 rising over the logarithmic curve to its full value. The quan- 
 tity coulombs is equal to the difference between the quan- 
 tity of electricity which actually flows through the circuit in 
 the period during which the current is rising to its permanent 
 E 
 
 value I— — and the quantity that would flow in the same time 
 R 
 
 if the current immediately rose to its full value. 
 
 If the voltage is suddenly reduced from E to zero without 
 breaking the circuit, the current does not stop immediatelj*, 
 but falls off along a logarithmic curve, and the quantity of 
 electricity passing through the circuit is increased on this 
 account. The increased quantity is proportional to the area 
 OYQ in Fig. 118, which shows a curve of falling current in an 
 inductive circuit corresponding with the curve of rising cur- 
 rent shown in Fig. 117. That this quantity, proportional to 
 area OYQ of Fig. 118, which is transferred through the cir- 
 cuit after the voltage E is removed, is equal to the deficit of 
 electricity during the starting of the current in the same cir- 
 
 di 
 
 cuit is shown thus: the counter-voltage is, as before, — Z — 
 
 dt 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 151 
 
 and = 
 
 R 
 
 Ldi 
 
 Rdt 
 
 But the area OYQ of Fig. 118 is equal to 
 
 dt 
 
 -Jf 
 
 0 Ldi 
 R 
 
 LI 
 R ’ 
 
 where T is the time in seconds measured from the instant of 
 removing the source of voltage E to the instant at which the 
 
 current reaches zero value, and I is the value ^ of the current 
 
 R 
 
 at the instant of removing the source of voltage. 
 
 The current, which is thus maintained by the disappearing 
 magnetic field after the external source of current has been 
 
 removed and which conveys the quantity of electricity 
 
 LI 
 
 R 
 
 coulombs, was formerly called the extra current of self-induc- 
 tion. It was then thought that the self-induction caused an 
 increase of the flow of electricity through the circuit ; but the 
 foregoing demonstration of the manner in which the current 
 rises and falls in a self-inductive circuit shows that the extra 
 current , which occurs when the external source of current has 
 been removed from the circuit and the current falls from I to 
 
 0, conveys a quantity of electricity, 
 
 which is exactly 
 
 equal to the deficit of coulombs which was caused at the time 
 that current was introduced into the circuit, — the aforesaid 
 deficit being due to the fact that the self-induction causes the 
 current to rise gradually instead of instantly when the source 
 of voltage is switched into the circuit, and the extra current 
 being due to the fact that the self-induction maintains the cur- 
 rent while the magnetic field is dying away after the external 
 source of voltage is removed. If the value I' is substituted 
 for the value 0 in the integrations, it will be observed that the 
 deficit coulombs upon the current being increased from I' to 1 
 and the extra-current coulombs upon the restoration of the 
 current to I\ assuming these operations accomplished without 
 
 changing the circuit constants, are each equal to 
 
 LCL-T) 
 
 and the principle may be stated as follows : 
 
 If a current is started or changed by changing the voltage 
 in a circuit having only resistance and self-inductance, and 
 these of fixed values, and the voltage is then restored to its 
 
152 
 
 ALTERNATING CURRENTS 
 
 original value and the current allowed to return to its original 
 or initial value without changing the resistance or self-induc- 
 tance of the circuit, the total number of coulombs transferred 
 through the circuit during the cycle are equal to the number 
 that would be transferred if the circuit were without self- 
 inductance. 
 
 The foregoing demonstration is founded on the proposition 
 that the self-inductance of the circuit is fixed in value, and that 
 the magnetic linkages existing at the respective ends of a cycle 
 of changes are proportional to the initial and final values of the 
 current ; but if the electric circuit has an iron core, the density 
 of the magnetism may not fall to its initial value upon restoring 
 the current, and the energy restored is not then equal to that 
 absorbed in building up the magnetic flux. The difference in 
 the energy remains stored in the magnetic flux in the form of mag- 
 netic linkages maintained by residual magnetism. Under the 
 circumstances here referred to, the permeability of the mag- 
 netic circuit usually varies as some complex function of the cur- 
 rent, and the current therefore does not rise and fall in plain 
 logarithmic curves because the value of L in the equation 
 
 i x dt — is a function of i of greater or less complexity. In 
 
 R 
 
 this case the integration along the curves of rise and fall may 
 not be readily accomplished, but it always remains a fact (pro- 
 vided the circuit is affected only by resistance and self-induc- 
 tance and the resistance is of fixed value) that the coulombs 
 conveyed through the circuit by the extra current upon restor- 
 ing a current to its initial value are equal to the deficit occur- 
 ring during the rise of the current, provided the magnetism 
 and the current both return to their initial values. The effect 
 of hysteresis must be negligible in order that this condition 
 may occur. 
 
 In case a current is started by switching a source of voltage 
 into a circuit surrounding an iron core which is already magnet- 
 ized, the current will rise more rapidly than if the core were not 
 previously magnetized, if the magnetic force of the current is 
 in the direction of the already existing magnetism; and the 
 rise of the current will be retarded if its magnetizing effect is 
 opposite to the already existing magnetism. This is an obvious 
 corollary from the foregoing demonstrations. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 153 
 
 Prob. 1. A steady current of 10 amperes is flowing through 
 a coil having a self-inductance of 100 henrys and a resistance 
 of 10 ohms. How many coulombs of electricity will be trans- 
 ferred through the circuit after the voltage is withdrawn with- 
 out breaking the circuit or changing its resistance? Assume 
 magnetic leakage and hysteresis to be negligible in this and 
 the following problems. 
 
 Prob. 2. A magnetic circuit in the form of a ring of 200 
 centimeters mean length and 100 square centimeters in cross 
 section, with a fixed permeability of 1000 units, has a coil 
 wound upon it of 1000 turns and 20 ohms resistance. If a steady 
 current of 10 amperes is flowing in the coil, what will be the 
 number of coulombs of electricity which will be transferred 
 through the circuit after the voltage is withdrawn without 
 breaking the circuit or changing its resistance ? 
 
 Prob. 3. Draw the curves of quantity and current when the 
 voltage is introduced and withdrawn in the circuits in problems 
 1 and 2, assuming no change of resistance or self-inductance in 
 either instance. 
 
 Prob. 4. What is the value in amperes of the current flow- 
 ing in the circuit of problem 2, at an instant ^ seconds after 
 the removal of the voltage? 
 
 47. Energy stored in a Magnetic Field associated with an 
 Electric Circuit. — The effect of self-inductance is manifested, 
 as already explained, by a counter-voltage which tends to 
 retard a rising current and to sustain or continue a falling 
 current. That is, when a current is introduced into a circuit, 
 the inductive effect of the rising magnetic flux created by the 
 rising current prevents the current from immediately coming 
 to a value equal to the impressed voltage divided by the 
 resistance of the circuit. Likewise, when the voltage is with- 
 drawn from the circuit, the dying out of the magnetic flux 
 prevents, by its inductive effect, the current from immediately 
 disappearing. The inductive effect of the magnetic flux is 
 directly proportional to the rate of change of the summation of 
 the number of linkages around the conductors of the circuit 
 that are made by the lines of force of the magnetic flux. 
 Each such magnetic linkage set up in connection with a cur- 
 
154 
 
 ALTERNATING CURRENTS 
 
 rent in a circuit represents a definite amount of stored energy ; 
 and the phenomena relating to the setting up of the magnetic 
 field (that is, these linkages) about a circuit, thereby storing 
 energy by introducing a current in the circuit, and the phenom- 
 ena relating to collapsing the linkages and recovering the 
 energy by withdrawing the current, have close analogies to the 
 phenomena relating to the energy of moving bodies. 
 
 If a mechanical force is applied to a movable mass or body, 
 the velocity of the body does not immediately rise to its full or 
 ultimate value, but is prevented from doing so by the inertia 
 of the mass, until an amount of energy proportional to the 
 product of the mass and the square of the ultimate velocity 
 has been stored in the body. Likewise, when the impelling 
 force is removed from a moving body, it will continue to move 
 until all the energy that was stored in bringing it up to speed 
 is dissipated in overcoming resistance to its movement. In 
 
 civ 
 
 case of tangible matter — — , Mv, and M — are respectively 
 
 the energy, momentum, and rate of change of momentum (that 
 is, counter-force or pressure caused by inertia) of the mass M 
 when moving at a velocity v ; while in the case of the electric 
 Li 2 di 
 
 circuit, — — , Li, and L — , as will be seen later, may be called 
 
 Jmi GLT/ 
 
 the energy, momentum, and rate of change of momentum 
 (counter-electric voltage) of its magnetic flux. In such anal- 
 ogies, electric current has its counterpart in velocity or rate 
 of motion, voltage in mechanical force or pressure, electric 
 resistance in mechanical frictional resistance, and self-inductance 
 in mass or inertia. 
 
 When a source of constant voltage is switched into a circuit 
 with self-inductance and resistance, the current gradually rises 
 
 LJ 
 
 to its final value of I = — . Also, a counter-electric voltage 
 
 R 
 
 of self-induction is produced in the circuit by the effect of 
 the rising magnetic flux set up by the current. This counter- 
 voltage at any instant may be represented by the expression 
 di 
 
 — L~, assuming the permeability of the magnetic circuit to 
 
 be constant. 
 
 The impressed voltage contains a component equal and 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 155 
 
 opposite to this counter-voltage at each instant and also sup- 
 plies the iR drop, or, as has already been pointed out, 
 
 E=iR + L-. 
 
 at 
 
 This equation manifestly is equally accurate when E is of con- 
 stant value or when it represents the instantaneous value of a 
 variable impressed voltage, provided the circuit contains only 
 resistance and self-inductance (that is, the circuit is not subject 
 to the effects of mutual induction or electrostatic capacity). 
 The rate at which work is being delivered from the external 
 source of voltage and energy and expended in the circuit is, 
 in watts, 
 
 Ei = i*R + Li 
 
 dt 
 
 and the work delivered and expended during a time interval 
 dt is, in joules, 
 
 dw = Eidt = PRdt + Lidi ; 
 
 and the joules delivered by the source and expended in the 
 circuit during a period of T seconds in which the current rises 
 from 0 to I amperes is 
 
 The first term of the right-hand member of this equation repre- 
 sents the aggregate heat produced by the current flowing 
 through the resistance of the wire. The second term of the 
 
 right-hand member is equal to 
 
 LI 2 
 
 2 
 
 and represents the work 
 
 done by the external source of voltage and energy in over- 
 coming the counter-electro-motive force, caused by the magnetic 
 flux while the current is changing. This energy is stored in 
 the magnetic linkages of the flux with the turns of the electric 
 circuit and is maintained as long as the current continues at 
 its value I. 
 
 As we know the value of current at every instant during 
 its rise,* it is practicable to compute the value of the energy, 
 
 * Art. 45, a. 
 
156 
 
 ALTERNATING CURRENTS 
 
 Fig. 120. — Energy Stored in Magnetic Linkages at Each 
 Instant during Period of Current Rise. 
 
 W= | Li 2 , stored in 
 magnetic linkages 
 at each instant dur- 
 ing the rise of cur- 
 rent. The curve 
 exhibited in Fig. 
 
 120 shows the value 
 of the energy at each 
 instant correspond- 
 ing to the conditions 
 illustrated in Fig. 
 
 117. It is also prac- 
 ticable to compute the rate of the rise of energy, — = Li — . in 
 
 dt dt 
 
 the magnetic linkages; and the curve exhibited in Fig. 120a 
 shows the value of this rate at each instant, corresponding to 
 the conditions illustrated in Fig. 117. 
 
 The equation shows that the energy delivered to the circuit by 
 a battery, dynamo, or similar source, switched into a circuit having 
 resistance and self-inductance is equal to the sum of the energy 
 converted into heat by the current flowing through the resist- 
 ance of the conductors and the energy stored in the linkages of 
 
 the magnetic flux with 
 the conductors of the 
 circuit. If the source 
 supplies a constant vol- 
 tage, the latter portion 
 of energy is fixed by 
 the character of the 
 circuit and the magni- 
 tude of the voltage, 
 and is supplied by 
 the source while the 
 current is rising 1 to its 
 
 E = ICO VOLTS 
 R = 5 OHM8 
 
 L = 0.1 HENRY 
 
 0 .01 ,02 .03 .04 05 
 
 Fig. 120 a. — Rate of Change of Stored Energy at 
 Each Instant during Period of Current Rise. 
 
 E 
 
 steady value of — . 
 
 As long as the current continues of that 
 
 value the magnetic flux is maintained without further expen- 
 diture of energy, and the source then furnishes work equal to 
 that converted into heat only. It will be observed that these 
 deductions are founded on the original premise that the phe* 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 157 
 
 nomena of hysteresis, eddy currents, and other effects of mutual 
 induction and the like are absent, and the deductions are lim- 
 ited to conditions in which the phenomena of steady resistance 
 and self-inductance alone occur. 
 
 di 
 
 From the formula E = iR + L — it is also to be observed that 
 
 dt 
 
 idt = — dt — — di. 
 R R 
 
 The expression idt obviously represents the quantity of electricity 
 (coulombs) transferred through the circuit by the current flow 
 during the period of dt seconds at the instant of time when the 
 current is equal to i amperes. If the current came instantly 
 
 Ij 
 
 to its full value of / = — upon introducing voltage R into the 
 R 
 
 circuit, the coulombs transferred through the circuit in any 
 period of T seconds would be IT ; but the current does not rise 
 instantly to its full value in a circuit containing resistance and 
 self-inductance, on account of the electro-magnetic inertia of 
 the magnetic linkages. During the time of T seconds which is 
 taken by the current in rising to its full value of I amperes, 
 the number of coulombs transferred is less than IT because the 
 current has had a smaller average value than I amperes. The 
 actual number of coulombs transferred during this transient 
 period of the current rise may be determined by integrating 
 the foregoing formula from zero of time and current through 
 the elapsed time of T seconds during which the current rises to 
 its full value of I amperes, thus, 
 
 fa-! f'a-Z fan 
 
 J o RJq R 
 
 or 
 
 R 
 
 R 1 R' 
 
 E 
 
 Since Q =— T = IT is the number of coulombs that would 
 R 
 
 be transferred through the circuit in ^seconds of time if the 
 current had its full value of I amperes all of the time 7, the 
 
 P 7 7 
 
 formula Q T = — T shows that during the rise of current to 
 
 R R 
 
 E 
 
 its full value of I = — amperes, there is a deficit of coulombs 
 R 
 
158 
 
 ALTERNATING CURRENTS 
 
 transferred through the circuit, compared with an equal time 
 of full current flow; and that this deficit has a numerical 
 
 value of Q' = coulombs, as already pointed out.* 
 
 The total energy expended in the circuit during the time of 
 T seconds while the current is rising to its full value oi I — 
 
 amperes is 
 
 Q t E = EIT-LP 
 
 R 
 
 LI 2 . 
 
 and, of this, — — joules goes into storage in the magnetic field. 
 
 The energy expended during the transient period of the rise of 
 current is therefore less than the energy {LIT) expended in the 
 circuit during an equal length of time with the current at its 
 full value. This is analogous to the energy expended on a 
 moving body. When a given force is externally applied to 
 bring the body in T seconds of time from rest to full speed at 
 which the force is just balanced by frictional opposition, the 
 distance moved by the body and the total work (joules or foot- 
 pounds) done in the time of T seconds are less than would have 
 been the case if the body had moved at its maximum velocity 
 throughout the period of T seconds. 
 
 If the external source of energy is switched out of the circuit 
 without changing the resistance and self-inductance when the 
 current is of value I amperes, the energy equation becomes 
 
 0 = + L f^idi. 
 
 Li 2 X-Z 2 C T • 
 
 The last term may be written — ; and— —is equal to I i 2 Rdt , 
 
 2 2 o 
 
 which shows that the magnetic field (under the circumstances 
 considered) discharges energy equal in quantity to that stored 
 at the time the current was started, and that this energy is now 
 all converted into heat during the flow of the “extra current.’’ 
 The energy stored in the magnetic field falls off as the current 
 falls, and the values of the stored energ} 7 corresponding to the 
 conditions illustrated in Fig. 118 are shown by the curve in 
 Fig. 121. The rate with which the stored work is given out 
 by the magnetic field at each instant for corresponding condi- 
 tions is shown by the curve in Fig. 121a. 
 
 * Art. 46. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 159 
 
 The voltage formula also becomes 
 0 = iR -f- L 
 
 di 
 Jt ’ 
 
 Fig. 121. — Energy Stored in Magnetic Linkages at 
 Each Instant during Period of Falling Current. 
 
 or, idt = ^ di. 
 
 R 
 
 The expression idt represents the coulombs of electricity trans- 
 ferred through the cir- 
 cuit during dt seconds 
 by the discharge of the 
 energy from the mag- 
 netic field, after the ex- 
 ternal impressed volt- 
 age has been removed. 
 
 That is, the current 
 does not fall instantly 
 to zero under the cir- 
 cumstances when the 
 external voltage is 
 switched out of circuit 
 without changing the resistance or self-inductance, but the 
 discharge of the energy from the magnetic field tends to main- 
 tain it briefly. 
 
 The total coulombs 
 conveyed through the 
 circuit by this “ extra 
 current ” may be ob- 
 tained by integrating 
 the foregoing formula 
 from the instant when 
 the external voltage is 
 switched out of the cir- 
 cuit (that is, from zero 
 Fig. 121 a. -Rate °f Change of Stored Energy at of time and j amperes 
 Each Instant during Period of Falling Current. r 
 
 of current) through the 
 period of T seconds during which the current falls to zero, thus 
 
 E = 0 VOLTS 
 R = 5 OHMS 
 [_ = 0.1 HENRY 
 
 r«ft— #r 
 
 RJi 
 
 or. 
 
 Q" = 
 
 R ■ 
 LI 
 R ' 
 
 di, 
 
160 
 
 ALTERNATING CURRENTS 
 
 It will be observed from this that Q" = Q', which is to say that 
 the number of coulombs conveyed through the circuit by the 
 “ extra current ” is exactly equal to the deficit in the coulombs 
 conveyed through the circuit during the time of the current 
 rise, as previously pointed out,* so that the number of coulombs 
 conveyed through the circuit as a result of the cycle of rise and 
 fall of current does not differ from the number that would be 
 conveyed through the circuit if L were zero or had any other 
 value, provided R remained unchanged in value. The time 
 occupied by the cycle consisting of the rise of the current to 
 
 E 
 
 its full fixed value of — and its succeeding fall to zero upon 
 
 switching out the external voltage R is affected by changing 
 the value of _L, but the number of coulombs transferred through 
 the circuit during the cycle is not changed. 
 
 The gradual decay of the electric current under the condi- 
 tions here described, accompanied by the consumption of the 
 energy stored in the magnetic field, is analogous to the decay of 
 the velocity of the moving body referred to above. Upon the 
 removal of the external force after the body has come to its full 
 speed, the body would continue in motion with gradually fall- 
 ing speed and finally come to rest when the energy stored in its 
 moving mass at the maximum speed had all been consumed in 
 overcoming the frictional resistance and been converted into 
 
 heat. 
 
 The work expended in a self-inductive circuit is therefore 
 manifested by (1) the iR drop relating to the conversion of 
 energy into heat and (2) the counter- voltage of self-induction 
 relating to the storage of energy in the magnetic flux by a 
 tendency to retard a rising current and accelerate or continue 
 a falling current. The effects are in many respects analogous 
 to the inertia of tangible matter, as already pointed out, in 
 
 which Mv, and are respectively the energy, the mo- 
 
 mentum, and the rate of change of momentum (force) acting 
 in the mass M when moving with the velocity v ; while in the 
 LP . di 
 
 electric circuit, , I/i, and L - — may be called the energy, 
 
 2 dt 
 
 momentum, and rate of change of momentum (counter- voltage) 
 
 * Art. 46. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 1G1 
 
 of its magnetic field. The manifestations of the electro-mag- 
 netic inertia are exerted through the linkages of the magnetic 
 lines of flux with the conductors of the electric circuit. 
 
 When the value of L varies with the current, as when iron 
 is magnetized by the electric current under consideration, the 
 energy stored in the magnetic linkages is measured by the value 
 of L corresponding to the current flowing at the instant under 
 
 TT 
 
 consideration; and the value of L corresponding to 1 = — is 
 
 Li 2 ^ 
 
 the correct value to apply in the expression when it is 
 
 desired to compute the energy stored during the rise of 
 
 U 
 
 current from 0 to — amperes in a circuit affected by a con- 
 
 stant impressed voltage, since the total number of linkages 
 only is required to obtain the stored energy and not the 
 instantaneous rates at which they are created or disappear. 
 The phenomena of hysteresis prevent the magnetic field from 
 fully discharging its energy as the current falls. For instance, 
 if the current is caused to increase from I x to I and the self- 
 inductance changes from L l to L , the energy that is delivered 
 to the magnetic field by virtue of the increase of current is 
 Lp l 1 2 
 
 equal to — LJ- • If the current now returns to the value 
 
 I x amperes, hysteresis may prevent the magnetic field from 
 assuming its first value, so that the value of the self-inductance 
 has the different value of L v In this case, the cycle of the 
 rise of current from I x to I and its restoration to I x has caused 
 an increase of the energy stored in the magnetic field which 
 
 is equal to ^2 — Lllh-. If the current changes in recurrent 
 
 cycles between — I and /, the conditions in the magnetic field 
 also become cyclic, and a certain amount of energy is converted 
 into heat in each cycle as the result of the hysteresis phenomena. 
 
 If a coil is wound on a closed ring of soft iron, which form 
 exhibits great magnetic retentiveness, the value of L is very 
 great if the ring is magnetized by an alternating current. 
 But if the ring is magnetized by a rectified periodic cur- 
 rent, that is, a periodic current which is unidirectional, the 
 apparent value of L may be practically the same as though the 
 iron core were not present. This behavior is due to the ring 
 
162 
 
 ALTERNATING CURRENTS 
 
 continuously retaining the magnetization caused by the maxi- 
 mum current, and since the magnetism in the core therefore 
 remains constant it does not set up a counter-voltage. By 
 making a transverse cut in the ring, its coercive force may be 
 reduced so much that the effects are practically the same for 
 rectified and alternating currents. 
 
 Prob. 1. A coil with a self-inductance of .02 of a henry has 
 a steady current of 50 amperes flowing through it. What is the 
 energy of the magnetic field? 
 
 Prob. 2. A magnetic circuit 100 centimeters long and 200 
 square centimeters in cross section, the material of which has a 
 constant permeability of 1000 units, is excited by a coil of 
 500 turns which has a resistance of 10 ohms. What energ}^ 
 has the magnetic field when 100 volts are steadily impressed at 
 the terminals of the coil, there being no magnetic leakage ? 
 
 Prob. 3. A steady current of two amperes flowing in a coil of 
 1000 turns creates a field which sets up 1,000,000 lines of force 
 through each turn of the coil. What is the energy of the field? 
 
 Prob. 4. A magnetic circuit is excited by a coil of 2000 turns 
 in which a steady current of 20 amperes flows and sets up a 
 total of 2,000,000 lines of force. What is the energy of the 
 field, if all the lines of force link with all the turns of the coil? 
 
 Prob. 5. In what degree does changing the reluctance of a 
 magnetic circuit affect the energy stored therein by an exciting 
 coil, other things remaining equal? 
 
 Prob. 6. The energy of a certain magnetic field is 20 joules. 
 This is created by a magnetizing coil having 100 turns carrying 
 a steady current of 10 amperes. What is the self-inductance of 
 the circuit if all the lines of force of the field link with all the 
 turns of the coil? 
 
 48. The Transient Transfer of Electricity in Divided Circuits. 
 Application to a Shunted Ballistic Galvanometer. — The fact that 
 the total quantity of electricity which passes through a wire 
 when subjected to a transient voltage is independent of the 
 self-inductance of the circuit when hysteresis effects are absent, 
 as is shown above, has a bearing upon the distribution of cur- 
 rents in divided circuits. With no external disturbing factors, 
 it is apparent that where a transient voltage is impressed upon 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 163 
 
 parallel circuits of different inductances, the number of cou. 
 lombs which flow through each circuit would also flow were the 
 circuits without self-inductance, but the phase of the flow in 
 each circuit is retarded so as to lag behind that of the voltage 
 by an amount which is proportional to the self-inductance of 
 the circuit. 
 
 This reasoning would make it appear that shunting a ballistic 
 galvanometer must change the constant in the ratio of the re- 
 sistances of galvanometer and shunt without regard to their 
 self-inductances, as is true when steady currents are in ques- 
 tion. This, however, is not correct, because one of the factors 
 in this problem has not been taken into account, i.e. the move- 
 ment of the needle which occurs before the end of the discharge 
 generates a counter-voltage in the galvanometer winding. This 
 reduces the proportion of the discharge which passes through 
 the galvanometer, but the effect is due to the needle and not to 
 the relation of the self-inductances of the galvanometer wind- 
 ing and the shunt. Assuming that the number of lines of force 
 due to the needle which link with the turns of the winding are 
 proportional to the sine of the deflection of the needle; calling 
 r g and L the resistance and self-inductance of the galvanometer 
 winding ; r s the resistance of the shunt (the inductance of the 
 latter being assumed negligible on account of its being wound 
 with doubled wire) ; and i g and i s being the respective instan- 
 taneous currents : then the instantaneous impressed voltage is 
 e = i s r s . The corresponding instantaneous value of the IR drop 
 in the galvanometer winding is i g r 0 and this is equal to the 
 impressed voltage less the corresponding instantaneous counter- 
 voltages caused by self-induction and the swing of the needle. 
 
 TI,eiefore: . . (Ldi kd( siiw.)' 
 
 \ dt + dt 
 
 where & is a constant representing the summation of linkages 
 of magnetic lines of force from the needle with turns of the gal- 
 vanometer winding when the needle stands perpendicular to the 
 plane of the winding. Whence i s r s dt — i g r g dt = Ldi g +Jcd ( sin a) 
 
 s*t nt /»o /* a 
 
 and r s I i s dt—r„ ) i g dt = L I di g + k I c?(sin «). 
 do d () 
 
 This is r s q s — r g q g = k sin «. If the swing of the needle is small, 
 
164 
 
 ALTERNATING CURRENTS 
 
 then sin a is sensibly equal to 2 sin^, but 2 2Tsin- = q T where 
 
 2 2 
 
 K is the ordinary constant of the ballistic galvanometer. Hence, 
 Calling Q the total discharge, which is equal 
 to q s + q g , this becomes 
 
 kq g 
 
 7 s r ls r g ( lg 
 
 % 
 
 r s(Q f dg) r g1g ’ an d q g 
 
 Q'\s 
 
 r a + r s + 
 
 K 
 
 This discussion shows that the coulombs of a discharge winch 
 pass through a shunted ballistic galvanometer are less than 
 might be predicted from the ratio of the resistances; that is, 
 
 do ^ 
 
 Qr, 
 
 r g + r s 
 
 This deficit is caused solely by the lines of force 
 
 from the needle cutting the galvanometer coils while the dis- 
 charge is passing, and its value is 
 
 Qr, 
 
 (?g + r s) 
 
 1 + j( r g + 
 
 The shunted ballistic galvanometer therefore gives readings 
 which are too small, unless the duration of the discharge is 
 very small compared with the time of vibration of the needle. 
 But it must also be remembered that, if the needle is removed 
 from the galvanometer or clamped in a fixed position, the divi- 
 sion of the discharge between the galvanometer winding and 
 the shunt is then independent of the self-inductance of the 
 winding and shunt, and may be predicted from the ratio of 
 
 resistances, or q a = - • 
 
 r 4 — x 
 ' g \ ' s 
 
 Only under special conditions can a single coil with self- 
 inductance be substituted for the coils in parallel so as to 
 produce the same effect as the latter upon transient cui’rents 
 of every duration. These conditions are fulfilled when the 
 
 ratio of — is constant for all the coils, and an equivalent coil 
 
 R 
 
 may then be substituted for the parallel circuits. In this case 
 the resistance of the equivalent coil must be 
 
 - — = — -| -\ -(- etc., 
 
 R R x R 2 i? 3 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 1G5 
 
 and its self-inductance must be 
 
 i = T + A- 
 
 L i, L % 
 
 H — — — (- etc. 
 
 These values make — equal to the constant value of the ratio 
 K 
 
 for the individual coils.* 
 
 49. Effect of Eddy Currents and of Hysteresis upon the Rise 
 and Fall of Current in a Self-inductive Circuit. — If a circuit 
 surrounds or lies near conducting material, the rise or fall of 
 current when a voltage is applied or removed is modified by 
 the magnetic effect of eddy currents induced in such conduc- 
 tors.! The eddy currents tend to quickly rise to a maximum 
 while the main current is changing most rapidly, and then to 
 fall gradually as the main current approaches its full value. 
 Eddy currents are similar to the secondary currents of a trans- 
 former | and are induced in the same way. As in the trans- 
 former, an additional current just sufficient to neutralize the 
 magnetic effect of the secondary currents at the instant flows 
 in the main or primary circuit. This extra neutralizing ele- 
 ment of the main current causes the curve of current to rise 
 more rapidly when the voltage is applied and fall more quickly 
 when the voltage is withdrawn ; that is, energy is expended 
 not only in building up the magnetic field and in i 2 R loss in 
 the main circuit, but also in the eddy current circuits, as the 
 main current rises, and while the main current is falling, the 
 energy yielded from the magnetic field is absorbed by i' 2 R losses 
 in the main circuit and also in the eddy current circuits. This 
 transfer of energy from the main circuit to the eddy current 
 circuits has the same effect on the rise and fall of the current in 
 the main circuit as would be produced by increasing to an equal 
 degree the energy changed into heat by the i 2 R loss in the 
 main circuit and excluding eddy current effects. Any trans- 
 fer of energy by induction from the main circuit while the 
 main current is changing has the same effect of hastening the 
 change of the main current, and thus reducing the apparent 
 effect of the self-induction. It is therefore possible to surround 
 a self-inductive circuit with conducting material in such a 
 manner as to largely mask the presence of self-inductance. 
 
 * The subject of circuits in parallel is fully treated in later pages, 
 t Art. 111. J Art. 23 and Chap. X. 
 
166 
 
 ALTERNATING CURRENTS 
 
 On account of the counter-magnetizing effect of eddy cur- 
 rents the lines of force set up by the main current tend to 
 crowd outside of the eddy current circuits, which decreases the 
 average cross section of their path and hence increases the 
 reluctance. This effect of eddy currents in apparently screen- 
 ing the interior of an iron core from magnetic influences de- 
 creases the number of lines of force per ampere in the circuit 
 and consequently also decreases the apparent self-inductance. 
 It consequently, also, causes an acceleration in the rise or fall 
 of the current. If the circuit surrounds an iron core, this 
 effect may be quite large unless the core is finely laminated. 
 
 The phenomena of hysteresis and residual magnetism tend 
 to produce a smaller average rate of change of the magnetic 
 field set up in an iron core when the exciting current is with- 
 drawn than occurs when the exciting current is introduced in 
 the circuit ; and a current therefore dies away more quickly 
 than it builds up in a circuit with an iron core, even though 
 the source of voltage is switched into the circuit and later 
 removed without altering the resistance of the circuit. The 
 change of permeability of iron for different magnetizations, 
 which results in a change in number of magnetic lines set up 
 per ampere in the electric circuit, causes a distortion of the 
 curves of rising or falling current, as already pointed out in 
 the latter part of Art. 46. This effect becomes particularly 
 noticeable when the magnetic circuit is completed through 
 iron, as in the case of a closed ring core or the transformer 
 core illustrated in Fig. 45, as any air gap in the path of the 
 magnetic lines of force introduces a space of relatively large 
 reluctance which is independent of the current. 
 
 50. High Voltage generated on Breaking a Self-inductive 
 Circuit. — The condition under which the curves of rising and 
 falling current are logarithmic and of exactly the same dimen- 
 sions when voltage is applied to and withdrawn from a circuit 
 requires that the resistance as well as the self-inductance of the 
 circuit shall remain constant. If the circuit is quickly broken, 
 by opening a switch or otherwise, it is a well-known fact that 
 the counter-voltage rises much higher than the original impressed 
 voltage , frequently rising to many times its value. The extreme 
 severity of the shock which may be received upon breaking a 
 circuit of large inductance attests the fact. This is due to 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 167 
 
 the exceedingly large increase of resistance in the circuit which 
 is introduced by the break. This increase in resistance causes 
 the current to fall off more quickly, and hence produces a 
 greater rate of change of magnetism. However, as before, the 
 energy given up by the field must be 
 
 where q is the current after the break, since the same number 
 of magnetic linkages disappear from the circuit, and the energy 
 given up must therefore be equal to the energy stored in the 
 circuit when the full current I was flowing ; and, also 
 
 where is the new resistance of the circuit, including the 
 break, assumed to be constant, and cp is the coulombs of the 
 discharge. The current is 
 
 for small values of t. The total number of coulombs trans- 
 ferred is, therefore, less, and the induced voltage greater, upon 
 breaking a circuit than upon making it.* If hysteresis is present, 
 the field may not be totally discharged, and the amount of 
 energy and quantity of electricity will then not be so great 
 as indicated by the formula, as is explained in the previous 
 article. 
 
 Prob. 1. A certain electric circuit supports a magnetic field 
 which is charged with an energy of 5 joules. This circuit is 
 opened by introducing resistance increasing to infinity in .01 
 of a second. What is the average power exerted by the mag- 
 netic field during the time of the break, assuming that the 
 magnetism falls to zero ? Give the answer to this and the 
 following problems in watts and neglect the effect of hysteresis 
 and eddy currents. 
 
 * Formulas for further expressing the conditions when the resistance of a 
 circuit is changed are given in Art. 55. 
 
 q = Ie i < 7e z, 
 
 for finite values of t ; and the induced voltage is 
 
168 
 
 ALTERNATING CURRENTS 
 
 Prob. 2. If 6,000,000 lines of force linking the turns of a 
 coil of 1000 turns are set up by a steady current of 5 amperes 
 in the coil, what is the average power exerted by the discharge 
 of the magnetic field if the exciting circuit is broken in .01 
 second by introducing resistance up to infinity, and the mag- 
 netism falls to zero ? 
 
 Prob. 3. A magnetic circuit having an average permeability 
 of 1000 units, a length of 200 centimeters, and a cross section 
 of 50 square centimeters is excited without magnetic leakage 
 by a coil of 1000 turns having a resistance of 25 ohms. A 
 steady voltage of 1000 volts is impressed at the terminals of 
 the coil. What is the average power exerted by the discharge 
 of the magnetic field when the exciting circuit is broken 
 through an arc maintained .01 of a second? 
 
 51. Capacity. — All insulated conductors have the property 
 of being able to hold a charge of electricity. When an insu- 
 lated conductor is connected to a source of a different potential, 
 electricity will flow to it or from it, until its potential is the 
 same as that of the source. The measure of the charge or 
 quantity of electricity which is held by the conductor when at 
 unit potential is its electrostatic Capacity, and the unit of ca- 
 pacity may be defined as the capacity of a conductor which con- 
 tains a unit charge of electricity when at unit potential. The 
 practical unit of capacity is the capacity of an isolated conductor 
 which ivould contain a charge of one coulomb when at a potential 
 
 of one volt. This is called a Farad, after Faraday. It is — jj 9 
 
 times as large as the unit of capacity of the C. G. S. set of 
 units. The farad is too large a unit of capacity to be conven- 
 ient in practice, and the Microfarad, or millionth of a farad, is 
 commonly used as the unit of measurement. The capacity of 
 a conductor depends upon its conformation and surroundings. 
 
 Two adjacent conducting bodies brought to different poten- 
 tials and isolated from external influences assume equal and 
 opposite charges. In this case, the capacity is measured in 
 terms of the difference of potential between the conductors and 
 the charge on each. The capacity is one farad when a differ- 
 ence of potential between the conductors of one volt produces 
 a charge of one coulomb on each, — the charges being (relative 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 169 
 
 to each other) positive on the conductor of higher potential and 
 negative on the conductor of lower potential. 
 
 Capacity effects caused by differences of potential between 
 neighboring conductors or parts of the same conductor, are of 
 constant occurrence and much importance in electrical engineer- 
 ing structures. In most of such instances the influences are 
 complex because many conductors of different potentials and of 
 different shapes and space relations relative to each other may 
 be involved ; but most of these instances may be reduced to 
 approximate analytical processes by considering the conductors 
 pair by pair and thus learning their influence on each other. 
 
 The term Condenser is applied to any pair of insulated con- 
 ductors having an appreciable capacity, although it is more 
 strictly used to designate a combination of sheets of conducting 
 material, insulated, and laid together with the alternate layers 
 connected in parallel. In the following discussion the term 
 condenser will be used in its broader sense. 
 
 The Dielectric of a condenser is the insulating medium which 
 surrounds the conductors ; and its dimensions and quality deter- 
 mine the condenser capacity. The capacity is directly propor- 
 tional to the area and inversely as the thickness of the dielectric 
 and depends upon the Specific inductive capacity, or Dielectric 
 constant of the dielectric material. The Charge of a condenser 
 is the quantity of electricity comprised on either conductor or 
 electrode of the condenser. 
 
 It should be kept clearly in mind that the capacity of a con- 
 ductor or circuit is a quality of the circuit and is not depend- 
 ent upon the current flowing or the voltage applied. In the 
 same way the capacity of a vessel to hold a liquid is a quality 
 of the vessel, dependent upon the conformation of the vessel, 
 and irrespective of the amount of liquid that may be in the 
 vessel or the rate at which it is running in or out. 
 
 From the foregoing definitions of capacity is at once derived 
 the fundamental relation, 
 
 Q = CE, 
 
 where Q represents the quantity of electricity (that is, the 
 coulombs) in the charge of a condenser, C the capacity in 
 farads, and E the voltage between the conductors or electrodes 
 of the condenser. 
 
170 
 
 ALTERNATING CURRENTS 
 
 Prob. 1. A condenser is charged by bringing its electrodes 
 to a difference of potential of 100 volts. What is its capacity 
 if, under that voltage, its charge is .002 coulomb? 
 
 52. Conditions of Establishment and Termination of Current in 
 a Circuit containing Capacity and Resistance in Series. — a. Cur- 
 rent and Rise of Charge upon, Impressing Constant Voltage. — The 
 work done during time dt on a circuit containing resistance 
 and capacity at instant t after a constant voltage E is im- 
 pressed thereon is 
 
 Edq = Eidt = edq + Ri 2 dt , 
 
 where e, i, and q are the instantaneous values at instant t of the 
 voltage of the charge, current flowing into the condenser, and 
 quantity of the charge. The last member contains two terms, 
 of which the first is the work stored in the increment dq coulombs 
 of the charge which is put in the condenser during the time dt 
 seconds, and the second is the woi’k converted into heat by dq 
 coulombs flowing through the resistance R of the circuit. If 
 this equation is divided by idt = dq, there results an equation 
 
 E = e -(- Ri , 
 
 of voltage 
 
 or 
 
 From these equations the charge at any instant may be 
 determined when the applied voltage E is constant during the 
 process of charging. The equation may be put in the form 
 
 dq dt 
 
 q-CE = “ EC 1 
 
 whence 
 
 Integrating gives 
 
 log (q- CE ) 
 
 V dt 
 o RC' 
 
 and q — CE = — CEe RC . 
 
 Therefore since CE = Q, where Q is the final value of the charge, 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 171 
 
 q = Q(l — e RC ), 
 
 and 
 
 i _dq_Q J rC ' 
 dt RC 
 
 Figure 122 shows the logarithmic curves of the rise of charge 
 in a condenser in a particular circuit and the current flowing 
 
 Q t_ 
 
 v e so during the same process. It is 
 
 R 2 C 2 
 
 into the condenser (that is, the rate of change of the charge) 
 while the charge is reaching its full value Q after 100 volts are 
 impressed on the circuit. Figure 122 a shows the rate of change 
 
 of the current — = 
 dt 
 
 to be noted that 
 the current at the 
 instant of intro- 
 ducing the voltage 
 E into the circuit 
 (i.e. t = 0) is i = 
 
 Q E 
 
 RC = R ampereS ’ 
 and it is therefore 
 instantaneously 
 equal to the cur- 
 rent that the vol- 
 tage E would cause 
 to flow through the resistance R of the circuit if the condenser 
 were absent. 
 
 b. Current and Fall of Charge on Withdrawal of Impressed 
 Voltage. — During the discharge, assuming that the impressed 
 
 Fig. 
 
 122a.— Curve showing Rate of Change of Current 
 during the Period of Charging. 
 
172 
 
 ALTERNATING CURRENTS 
 
 voltage is withdrawn from the circuit without changing R or 
 (7, the impressed voltage E is zero, and therefore 
 
 ,dq 
 
 0 = — - 
 
 c 
 
 R 
 
 dt ’ 
 
 
 dq 
 
 dt 
 
 or 
 
 9 ~ 
 
 = ~TlC ’ 
 
 and 
 
 4 — > 
 
 II 
 
 so 1^ 
 1 
 
 II 
 
 Integrating gives 
 
 
 
 
 log q f = 
 
 7- 
 
 RCjq 
 
 or 
 
 log «“ 
 
 t 
 
 RO ’ 
 
 whence 
 
 9 = 
 
 Qe~*a, 
 
 and 
 
 
 Q- €~ 
 
 dt 
 
 RC 
 
 Figure 123 shows the logarithmic curves of falling charge after 
 the impressed voltage has been removed and of current flowing 
 
 during the period of dis- 
 charging. Figure 123 a 
 shows the rate of change 
 
 of current, — , during 1 
 dt ° 
 
 the same period. 
 
 From the equations 
 showing the charging 
 and discharging cur- 
 rents, which are of 
 forms similar to those 
 showing the rise and 
 fall of currents in cir- 
 cuits containing resist- 
 ance and self-induct- 
 ance, it is seen that 
 the curves of charge 
 and discharge in cir- 
 cuits containing resist- 
 
 -.10 
 
 - 20 
 
 6 QBMS 
 60 MlCROftj 
 
 Fig. 123. — Curves of Discharge of a Condenser. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 173 
 
 Fig. 123a. — Curve showing Rate of Change of Cur- 
 rent during the Period of Discharging. 
 
 ance and capacity in series are logarithmic when an unvarying 
 voltage is impressed on the system. (See Figs. 122 and 123.) 
 In many cases of practice RC is so small that the charge and 
 discharge of a condenser are practically instantaneous, but in 
 other cases the dura- 
 tion of the rise or fall 
 of the charge may be 
 quite appreciable. 
 
 Some condensers have 
 a quality which has 
 some analogy to mag- 
 netic hysteresis and 
 which is often termed 
 dielectric hysteresis 
 from the fact of its 
 being due to a charac- 
 teristic of the dielectric. When hysteresis is present, the con- 
 denser does not at once proceed to a complete discharge when 
 the impressed voltage is removed from its terminals, but it 
 retains a residual charge. The amount of energy retained in 
 this residual charge is a measure of the amount of energy 
 absorbed by the condenser on account of hysteresis, and this 
 energy may be converted in each cycle into heat if the con- 
 denser is subjected to alternating charges and discharges. It 
 is probable that this effect is due to a small part of the electric 
 charge soaking into the dielectric (especially into those layers 
 near the condenser electrodes) and then soaking out again, 
 because of the imperfect insulating qualities of the dielectric, 
 rather than to molecular phenomena strictly analogous to mag- 
 netic hysteresis ; and the heating of the dielectric which is 
 sometimes observed as the result of repeated charging and dis- 
 charging may be ascribed to the I 2 R loss due to the current 
 flow in the dielectric which results from the aforesaid soaking 
 in and out of the charge. The effect is not ordinarily very im- 
 portant in engineering practice. 
 
 Prob. 1. A condenser of 50 microfarads capacity and negli- 
 gible internal resistance is connected in series with a wire 
 having a resistance of 10 ohms. Draw the curve of charge in 
 this circuit when 100 volts are impressed ; and of discharge 
 
174 
 
 ALTERNATING CURRENTS 
 
 when the impressed voltage is removed without breaking the 
 circuit. 
 
 Prob. 2. What quantity of charge has the condenser of 
 problem 1 received at the instant when the impressed voltage 
 has been applied for R C seconds? 
 
 Prob. 3. What quantity of charge remains in the condenser 
 of problem 1, RC seconds after the removal of the impressed 
 voltage? 
 
 c. Current and Charge when the Impressed Voltage is a Sine 
 Function of the Time. — Under these conditions the voltage 
 equation of the circuit becomes 
 
 e=f(f) = e m sin cot = Ri + & 
 
 0 
 
 =R i+ fiM 
 
 c 
 
 This is a linear differential equation of the first order, the solu- 
 tion of which is * 
 
 i =- 
 
 \ R2 -I 1 
 
 ' ' 4 7T 2 f 2 C 2 
 
 sin (« — 0) + B x e nc, 
 
 where a = c of, oo = 2 7 rf and 6 — — tan -1 ( — - R )• 
 
 \coC 
 
 In a similar manner the value of q is found to be 
 
 r =J idt = — 
 
 — 
 
 ' 4 t rf 2 C‘ 
 
 cos (« — d) + B 2 e ac- 
 
 After a short time the exponential term of the equations be- 
 comes inappreciable and may be neglected. The effect of this 
 exponential term will be discussed later, f The current then 
 becomes a sine function of the time and leads the impressed 
 
 voltage by an angle 0 whose tangent is 
 
 1 
 
 2 irfC 
 
 h- R , instead of 
 
 * Murray’s Differential Equations, p. 26. 
 t Art. 69. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 175 
 
 lagging behind it, as in the case of a circuit containing induct- 
 ance and resistance. The current has a maximum value 
 
 = 
 
 or an effective value of 
 
 ^ 2+ 47 r 2 f 2 C 2 
 
 /= 
 
 E 
 
 + 4 7T 2 f 2 C 2 
 
 in which E is the effective value of the sinusoidal impressed 
 voltage. The denominator of this expression is, as in the case 
 of the inductive circuit, called the Impedance of the circuit. 
 This impedance is composed of the square root of the sum of 
 two terms, one of which is the square of the electrical resistance 
 
 and the other the square of the expression 
 
 This latter 
 
 2 TrfC 
 
 term is called the Reactance of the circuit, and corresponds in 
 this case to the term similarly designated in the circuit contain- 
 ing inductance and resistance. 
 
 53. The Energy of a Charged Condenser. — As a condenser is 
 charged, a certain amount of work is done in raising the 
 potential of the charge. During the time dt this is equal to 
 
 dw = (eidt = edq ) : Cede , 
 
 in which e is the voltage of the charge, i the current flowing 
 into the condenser, and q the charge in the condenser at the 
 instant t ; from which, by integration. 
 
 w __CE 2 EQ 
 
 2 2 ’ 
 
 where E and Q are respectively the final values of the voltage 
 and the quantity composing the charge. 
 
 This represents a certain amount of work which is stored in 
 the condenser when its charge is increased from zero to Q 
 coulombs. When the condenser is discharged, an equal amount 
 
 CE 2 
 
 of work is returned to the circuit. The expression — — is 
 
 similar to that giving the work stored in a compressed spring, 
 NF 2 
 
 W = -- -, where N is the compressibility of the spring meas- 
 
176 
 
 ALTERNATING CURRENTS 
 
 ured by the distance compressed per unit of force, and F is 
 the force applied to perform the compression. These expres- 
 sions are proportional to the square of impressed electrical or 
 physical pressure and the stored energy is truly potential. 
 
 LI 2 
 
 These equations thus differ from the expressions — — - and 
 — —4, which are dependent upon momentum instead of pressure 
 and represent kinetic energy. 
 
 As we know the current and the value of the charge at 
 each instant during the rise of the charge, it is practicable to 
 
 d 2 
 
 compute the value of the energy, w — Ce 2 = | , stored in 
 
 v 
 
 the condenser at each instant during the rise of charge, and 
 the rate at which the energy is changing at each instant. 
 These values are exhibited in the curves of Figs. 124 and 124 a 
 for the conditions of the circuit corresponding to Fig. 122. 
 
 Fig. 124. — Energy stored in Condenser during Period of Charging. 
 
 The energy stored in the condenser at each instant during 
 the fall of the charge, and the rate at which the condenser gives 
 out energy (the rate of change of the stored energ} ) during the 
 fall of the charge, for circuit conditions corresponding to those 
 indicated in Figs. 123 and 123a are shown in Figs. 125 and 125a. 
 
 54. Vector Diagrams showing the Voltage and Current Rela- 
 tions in a Circuit containing R and C- — Expression in Form of 
 Complex Quantities. — When a condenser is connected to a 
 source of alternating voltage, as indicated in Fig. 126, a cur- 
 rent will flow into and out of the condenser. The value of the 
 current at any instant is proportional to the rate of change of 
 the voltage impressed on the capacity, because the charge in 
 the condenser at any instant is proportional to the charging 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 177 
 
 E = 1 00 VOLTS 
 R = 5 OHMS 
 
 C= 50 MICROF. 
 
 .0002 
 
 .0004 
 
 .0006 
 
 .0008 
 
 voltage then acting on the capacity, and the rate at which the 
 charge changes must be proportional to the rate at which the 
 voltage changes. The rate 
 of change of the charge is 
 equal to the number of 
 coulombs flowing per se- 
 cond into or out of the 
 condenser, and is there- 
 fore equal to the current o 
 flowing into or out of the Fig. 124 a . — Rate of Change of Energy stored in 
 , ' mi i Condenser during Period of Charging. 
 
 condenser. 1 he condenser 
 
 de 
 
 current (i c ) at any instant is represented by i c = c'~, and since, 
 
 when the alternating voltage is sinusoidal, 
 e — e m sin a = e m sin cot, 
 
 where e m is the maximum voltage acting upon the condenser 
 during a cycle, co is the angular velocity or advance of the 
 cycle, and t the time measured from the beginning of a cycle, 
 
 clc 
 
 — = e m co cos cot. 
 dt 
 
 But co = 2 rrf, 
 
 where /is the frequency of the cycles. 
 
 Therefore, 
 
 and 
 
 = e m cos a -4- 
 
 1 
 
 TTfC 
 
 = 2 7j -fCe m cos a 
 
 e m cos a = i. x 
 
 2 7 rfO 
 
 The voltage e c of the condenser charge, which is, of course, 
 equal and opposite to the voltage e impressed on the plates, 
 
 may be called the 
 Capacity voltage or 
 Condenser voltage. 
 This voltage is 
 purely reactive, due 
 to the charge in the 
 condenser. That it 
 must be 90° in ad- 
 vance of the current 
 when the impressed 
 
 E — o 
 
 R = 5 OHMS 
 C = 50 MICROF. 
 
 .0002 
 
 .0008 
 
 Fig. 125.- 
 
 .0004 1 .0000 
 
 SECONDS ’ 
 
 -Energy stored in Condenser at Each Instant Voltage is SlllUSOl a 
 during Period of Discharging. may be readily seen 
 
178 
 
 ALTERNATING CURRENTS 
 
 SECONDS 
 
 .0004 .0006 
 
 .0008 
 
 from the formula above and the reactions that occur in the cir- 
 cuit. When a sinusoidal voltage impressed at the terminals of a 
 
 resistanceless condens- 
 er is rising, a current 
 flows into the condenser. 
 This current is a posi- 
 tive maximum at the 
 instant the impressed 
 voltage passes through 
 zero in changing from 
 negative to positive 
 values, for the rate of 
 Fig. 125 a. — Rate of Change of Energy stored in change of voltage is 
 Condenser during Period of Discharging. then a positive maxi- 
 
 mum. When the impressed voltage passes through its maxi- 
 mum point, its rate of change is zero, and the current at that 
 instant is therefore zero. When the voltage is falling, a cur- 
 rent flows out of the condenser ; that is, in the direction which 
 discharges the condenser. There- condenser 
 
 fore, the current flowing is 90° in 
 advance of the voltage impressed on 
 the capacity. The capacity voltage 
 is a counter-voltage caused by the 
 potential of the charge, and is equal 
 and opposite to the voltage impressed 
 on the capacity, and is 90° in advance of the current. 
 
 From the formula for instantaneous current in the condenser 
 the maximum current is seen to be 
 
 ALTERNATOR 
 
 Fig. 12(5. — Diagram showing an 
 Alternator connected to a Ca- 
 pacity Circuit. 
 
 im = 2 7rfCe m , 
 and the effective current, 
 
 I c = 2 irfCE, 
 
 where E is the effective value of the voltage impressed on 
 the capacity. This current. I c being 90° ahead of the phase of 
 the voltage impressed on the capacity (which is 180° behind the 
 capacity voltage), the phase of the active voltage which drives 
 the current through the resistance of the circuit is also 90° 
 ahead of the voltage impressed on the capacity. The voltage 
 impressed upon the series circuit containing capacity and re- 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 179 
 
 sistance must therefore comprise two components, one equal 
 to the IR drop in the circuit and the other equal and opposite 
 to the capacity vol- 
 tage. A phase dia- 
 gram of these, ac- 
 companied by the D 
 corresponding 
 sinusoidal curves, 
 is shown in Fig. 
 
 127, where 00, 
 
 CA. and OA are ®' IG- 127. — Phase Diagram of Voltages in a Circuit con- 
 . taming Resistance and Capacity in Series, 
 
 respectively the 
 
 IR drop, the component equal but opposite to the capacity 
 voltage, and the voltage impressed on the circuit. The triangle 
 of voltages E r , E x , E, as shown in Fig. 128, is derived from this 
 diagram, and it may be observed that 
 
 E=-JE? + E?-, 
 
 and also that the angle of lead, i.e. the angle by which the 
 current and IR drop lead the voltage impressed on the circuit, 
 is the angle 0 , with 
 
 tan 0 — 
 
 --ffr 
 
 E r 
 
 : = E 
 
 The angle 0 is here called negative because it relates to a 
 
 leading current, and the 
 corresponding angle for a 
 lagging current has al- 
 ready been called positive. 
 
 The triangle of voltage 
 (Fig. 128), taken from the 
 phase diagram in Fig. 127, 
 is similar to the triangle 
 for a self-inductive circuit 
 illustrated in Fig. 119, except that the current in the circuit 
 and IR drop lead the impressed voltage instead of lagging 
 behind. 
 
 It must be carefully borne in mind that in self-inductive and 
 capacitj r circuits the self-inductive voltage and the capacity 
 voltage act under the same laws as any other electric voltages. 
 The impressed voltage , therefore, must comprise ttvo rectangular 
 
 Fig. 128. — Triangle of Voltages in a Circuit 
 containing Resistance and Capacity in Series. 
 
180 
 
 ALTERNATING CURRENTS 
 
 components , one equal and opposite to the reactive , and the other 
 equal to the active voltage ( i.e . the IR drop). The triangles 
 of Figs. 119 and 128 indicate this. The vertical sides E x 
 of the triangles are vector components of the hypothenuses 
 which represent the voltages impressed on the circuits, and are 
 respectively equal and opposite to the self-inductive or capacity 
 voltages in the circuit, while the active voltages (J72 drop) 
 represented by the horizontal sides of the triangles complete 
 the respective right-angled vector triangles. 
 
 The vector representing the impressed voltage in a circuit 
 containing resistance and capacity may be indicated by the 
 complex quantity 
 
 E=E r -jE x , 
 
 E—E (cos 6 — j sin O'). 
 
 E is the scalar value of E. The scalar value of E and the 
 angle 6 are determined in the same manner as explained in 
 Art. 40, as is also the sum of several voltages in series. 
 
 Since 
 
 6 = tan 1 
 
 the angle of lag 6 is negative, which follows from the fact that 
 the current leads the impressed voltage. 
 
 The expression 
 
 E = E (cos 0 —j sin 6) 
 obviously may be written 
 
 E = E (cos ( — 6) +j sin ( — 0)). 
 
 55. The Effect of Introducing Resistance in a Continuous Cur- 
 rent Circuit ; Opening the Circuit. — It has already been pointed 
 out in Art. 50 that the voltage must rise at the breaking of a 
 direct current circuit possessing self-inductance. Breaking 
 such a circuit has the effect of very rapidly introducing resist- 
 ance until the resistance becomes infinite, with a correspond- 
 ingly rapid forced decrease of the current. In case of a closed 
 circuit of resistance R and self-inductance L upon which a 
 steady voltage E is impressed, the equation representing the 
 conditions, while the circuit is in process of being broken, is 
 
 J!=z|+[i8+/(0]j, 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 181 
 
 in which /(£) is a function of time measured from the instant 
 of starting the process and represents the rate of introducing 
 extra resistance to open the circuit. 
 
 This may be transformed into 
 
 ' di [ig +/(Q] i _ E 
 dt L L 
 
 The rate at which extra resistance is to be introduced at 
 breaking the circuit — namely, the form of f(t) — must be known 
 before this equation can be solved. Assuming that the resist- 
 ance is added as a simple function of time, then f(f) = at , in 
 which a is a constant, and the equation becomes 
 
 di (R + at ) i _ E 
 
 It L ~ L 
 
 . / — - ■ di du di 
 
 Tutting R + at — — 2 aL and — 
 
 ( 2 ) 
 
 Va di 
 
 this becomes 
 
 di 
 
 — 2 ui= — 
 
 du V — 2 aL 
 
 dt dt du V — 2 L du 
 2 E 
 
 (3) 
 
 and the solution is of the form* 
 
 2 E 
 
 — J*2 udu 
 
 ie 
 
 = --±2= C £~P udu du + K. 
 
 I-taLd 
 
 Hence, i = 
 
 2 E 
 
 — t 
 
 V — 2 aL 
 
 I * 1 
 
 "X 
 
 e u2 du = — ■ 
 
 2 E 
 
 V — 2 aL 
 
 Ce~*du- Ce~ u2 du\. (4) 
 To Ju J 
 
 When this is integrated by parts and approximated by neglect- 
 ing evanescent terms and terms containing the reciprocal of 
 (u?) n , the equation reduces to 
 
 E 
 
 i = 
 
 R -|- at 
 
 The induced voltage caused by the process of opening the 
 circuit is _ di 
 
 e ’=- L Jt' 
 
 (5) 
 
 the 
 
 ( 6 ) 
 
 From equation (5) is obtained the value, 
 di _ _ aE 
 dt ( R + at ) 2 
 
 aLE aLIR 
 
 Whence, 
 
 «i = 
 
 (A + «£) 2 (R + aty 
 
 * Murray’s Differential Equations , p. 26. 
 
 (7) 
 
 ( 8 ) 
 
182 
 
 ALTERNATING CURRENTS 
 
 E 
 
 in which 1= — , that is, it is the current flowing in the circuit 
 
 R 
 
 before the process of breaking commenced. 
 
 When the process begins, t = 0, and at that instant, 
 
 (X T ~r LI 
 
 e, = — LI = — , 
 R R 
 
 (9) 
 
 a 
 
 in which LI represents the number of linkages of lines of mag- 
 netic force around conductors of the circuit before the process 
 
 of breaking the circuit was begun, and — represents the ratio 
 
 JAj 
 
 of the rate of introducing extra resistance into the circuit com- 
 pared with the initial or steady resistance of the circuit. If 
 the circuit is broken instantly, that is, a = ao, the induced vol- 
 tage obviously must be infinite, as the formula shows ; but this 
 fortunately cannot occur in practice, because the arc which 
 occurs at a break prevents the resistance thereat from going 
 instantly to infinity. 
 
 If the line voltage is not removed from the circuit upon the 
 incipiency of the break in the circuit (as in the case of open- 
 ing a switch), it must be added to the self-induced voltage e e , 
 and the resultant becomes 
 
 e — ei+E. 
 
 The induced voltage, e h is numerically equal at time t = 0 to 
 
 as many volts as LI is a multiple of — . The formula (8) shows 
 
 a 
 
 that e t falls off rapidly with the time, especially when a is large. 
 
 If the rate of change of resistance increases with t instead of 
 being constant, the maximum value of e t comes when t has some 
 finite value instead of when t= 0. 
 
 If a current of ten amperes is flowing through a circuit of 10 
 ohms resistance and .2 henry self-inductance, and the opening 
 of a switch gives an effect at the initial instant of introducing 
 resistance into the circuit at the rate of 10,000 ohms per second, 
 the induced voltage at that instant is approximately 2000 volts, 
 which is the maximum value if the rate of change of resistance 
 does not increase. If the drawing out of the arc following the 
 interruption of metallic contact causes the apparent rate of in- 
 troduction of resistance to increase, the induced voltage may 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 183 
 
 continue to rise to a higher value, after which it drops to zero 
 on the breaking of the arc. 
 
 By applying the same reasoning to the formula when resist- 
 ance and capacity are in circuit together, it may be observed 
 that e c = — E when t = 0 and there is no rise of voltage, which 
 is to be expected from the physical characteristics of capacity. 
 The stored energy remains stored in the condenser. Such 
 storage unaccompanied by How of current is manifestly impos- 
 sible in the case of electromagnetic energy except through some 
 exhibition of the phenomenon of coercive force in an iron core 
 or the like. 
 
 56. Rate of Expenditure of Work in Circuit Containing Self- 
 inductance and Capacity ; Analogies. — The effect of self- 
 inductance has been compared with the effect of inertia in 
 a moving physical mass. The inertia effects of water flowing 
 in a pipe, as suggested by Faraday, afford instructive analogies. 
 Thus, on impressing voltage upon a circuit containing resist- 
 ance and self-inductance, the current does not rise to its full 
 value instantly, but increases as a logarithmic function, the con- 
 stant of which depends upon the relation of the self-inductance 
 to the resistance of the circuit. In the same way, if pressure is 
 exerted upon water which fills a pipe, the water cannot begin its 
 full flow instantly, on account of inertia. If a gate is suddenly 
 closed in the pipe after the flow is fully under way, the momen- 
 tum of the liquid tends to continue the flow, and the gate suf- 
 fers a severe blow. In the same way, upon opening an electric 
 circuit a bright spark passes on account of the so-called extra 
 current caused by the tendency of self-inductance to uphold the 
 flow. It must always be remembered that the analogies be- 
 tween the flow of electric current and moving solids or liquids 
 are by no means exact. For instance, there is a marked differ- 
 ence between the effect of bends on the inertia effect in the 
 pipe containing water and in the electric circuit. Thus, in the 
 electric circuit, a solenoid has much more self-inductance than 
 has the same wire straightened out. On the other hand, bends 
 in a water pipe cause the inertia effect to be absorbed by 
 friction. Notwithstanding the differences, the analogies are 
 quite useful in fixing the meaning of the phenomena and 
 worthy of further consideration. 
 
 When water in a pipe is set in motion, a part of the force 
 
184 
 
 ALTERNATING CURRENTS 
 
 exerted upon it at any moment is utilized in overcoming the re- 
 Mdv 
 
 action, , caused by the inertia of the accelerating mass, and 
 
 the remainder in overcoming frictional resistances (J.v). 
 That is. 
 
 F = Av + 
 
 Mdv 
 
 where F is the pressure exerted, v the instantaneous rate of flow 
 (velocity) of the water, M is its mass, and A is a constant. It 
 is here assumed that the frictional resistance is proportional to 
 the velocity, which is true only when v is small. When the 
 velocity of the water has become so great that Av x = F. where 
 v x is the final velocity, the acceleration ceases, and the water 
 continues to flow at a uniform velocity v x as long as the force 
 is applied. 
 
 In the case of the electric circuit, tlie impressed voltage is 
 expended in overcoming the counter- voltage due to self-induct- 
 
 Ldi 
 
 ance and in causing the current to flow through the resist- 
 
 ance R of the circuit, or 
 
 E=iR + 
 
 Ldi 
 
 dt 
 
 This is similar to the expression for the flow of a liquid as 
 given above. The voltage exerted in overcoming the electric 
 resistance or electric friction of the conductor is represented by 
 
 iR, and represents the voltage exerted in stor- 
 
 ing energy in the magnetic field ; that is, in changing the 
 electro-magnetic momentum of the magnetic field.* When the 
 current reaches such a value that iR = E , the last term dis- 
 appears and the current becomes constant. The expression 
 
 Mdv^ f __ d(Mv) \ the f ormu i a relating to the flow of water 
 dt \ dt J 
 
 represents, of course, the pressure or force exerted in storing 
 energy in the water by increasing its momentum. 
 
 The power expended in the electric circuit at any instant is 
 
 Ei = i 2 R + 
 
 Lidi 
 
 ~1T' 
 
 * Art. 47. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 185 
 
 in which PR is the power expended in heating the conductor 
 di 
 
 and Li — is the power expended in storing energy in the mag* 
 dt 
 
 netic field. In this discussion, it is assumed that the electrical 
 resistance of a conductor is a constant, and is the same for a 
 variable current as for a constant one. This is correct within 
 practical limits, provided the rate of variation of the current is 
 not too great and the conductor is not too thick. 
 
 The capacity effect in a circuit has analogies with the effect 
 of the physical capacity of a system of water pipes upon the 
 flow of water ; or, to make the comparison more simple, suppose 
 that the pipes are so small as to have an inappreciable capacity 
 but lead to the bottom of a large tank. The force acting upon 
 the water must then accomplish two purposes ; first, it must 
 force the water through the resistance of the connecting pipes, 
 and second, it must raise the level or potential of the water to 
 the tank level. A formula may be made to express this as follows, 
 
 F = Av +JI— Av + 
 
 where H is the head of the water in the tank, Gr is the volume 
 of the water in the tank, and Y is the capacity of the tank in 
 units of volume per unit of height or head. In case of an elec- 
 tric circuit the analogous equation is, 
 
 F=Ri + ±. 
 
 The power which is used can immediately be obtained by 
 multiplying the hydraulic equation by the rate of flow of the 
 water and the electric equation by the current. The equation 
 of power in the electric circuit is then 
 
 Fi = PR + i l, 
 
 but as i = — 
 
 dt 
 
 iF=RP + %&- 
 
 Cdt 
 
 In case there are both capacity and self-inductance in series 
 in the circuit, the two equations must be combined and 
 
 F = Ri + 
 
 Ldi q 
 
 * Art. 52 c. 
 
186 
 
 ALTERNATING CURRENTS 
 
 and the power used is 
 
 iR=Ri 2 + ^ + 
 
 dt Cdt 
 
 It must be remembered in considering this equation that energy 
 is absorbed by the circuit on account of the second term on the 
 right only while the current is rising ; when the current falls 
 the magnetic field gives up energy which is returned to the 
 source. Likewise, energy is absorbed by the circuit on account 
 of the third term of the right-hand member of the equation 
 only while the voltage impressed on the capacity is rising and 
 the charge in the condenser is therefore increasing; when the 
 charge falls the condenser gives up energy which is returned to 
 the source. These reactions are analogous to the effects in 
 water pipes where the inertia absorbs energy while the velocity 
 is rising, to return it as the velocity falls ; while the tank 
 stores energy while being charged and returns it on discharge. 
 The resistance and iron and dielectric losses are analogous to 
 the frictional resistances in the pipes and tank. 
 
 57. Effect, on the Transient State in a Circuit, of Self-induct- 
 ance and Capacity Combined. — When an inductive coil of con- 
 , stant inductance L is included in a 
 
 steady-current circuit of resistance R. 
 and a condenser of capacity C is shunted 
 across a portion of the circuit of resist- 
 ance r, as in Fig. 129, the following 
 conditions obtain : 
 
 The condenser is charged with a 
 quantity of electricit)*, Q = CIr , where 
 I is the steady value of the current. 
 Fig. 129. -Circuit containing Now if the impressed voltage is sud- 
 Resistance, Self-inductance, denly removed by throwing the switch 
 and Capacity. shown in the figure to the horizontal 
 
 position, the condenser will discharge and the quantity of elec- 
 tricity which will pass from the condenser through the part of 
 the circuit beyond its terminals is 
 
 H 
 
 CIr 2 
 R 
 
 At the same time the self-inductance will cause a quantity of 
 electricity to be transferred through the circuit in the opposite 
 direction, which is equal to 
 
 1 = 
 
 LI 
 R ‘ 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 187 
 
 Hence, the total quantity of electricity transferred through the 
 circuit is r 
 
 qi ~ <1c = r^ l ~ 
 
 and the effect of the condenser is to apparently reduce the self- 
 inductance by an amount equal to the capacity of the condenser 
 multiplied by the square of the resistance around which it is 
 shunted. With iron in the circuit, L varies, but the general 
 relations remain the same. 
 
 The foregoing relates to the transient state upon removing 
 the impressed voltage which sets up the steady current I in the 
 circuit, but it is obvious that similar conditions are produced 
 during the transient state accompanying the establishment of 
 the current upon introducing the impressed voltage. If Cr 2 is 
 equal to L , it is obvious that the charging current neutralizes 
 the extra current of self-induction and the total circuit acts 
 like a circuit containing resistance R but lacking self-induct- 
 ance or capacity. The neutralization of the effect of self- 
 inductance cannot ordinarily be complete on account of 
 hysteresis in the iron core of the coil (if an iron core is used) 
 and the dielectric hysteresis of the condenser, which cause the 
 curves of discharge from the magnetic field and condenser to 
 vary from the logarithmic form. 
 
 58. Conditions of Establishment and Termination of Current in 
 a Circuit containing Resistance, Inductance, and Capacity in 
 Series. — a. Current and Charge under Constant Impressed Vol- 
 tage . — The equation of voltage when current is established by 
 introducing constant voltage E into the circuit is 
 
 E = Ri + ^l+Z 
 
 dt C 
 
 = Bi+L ft + hf idt 
 
 From (1) we obtain 
 
 0 — ^ , 1 dq 
 
 ~dC + Ld^ + LCdt' 
 
 Putting q = 
 
 the equation takes the characteristic form 
 
 R x 2 , a; 
 
188 
 
 ALTERNATING CURRENTS 
 
 the roots of which are 
 
 x-, = — 
 
 A 
 
 2 L 
 
 ^ VI T% TO 
 
 I R 2 
 
 i 
 
 k 4 L 2 
 
 LC ’ 
 
 
 1 
 
 x s = °- 
 
 The solution of such an equation as (2) is * 
 
 q = cqe* 1 * + a 2 e r2 * + a 3 , (4) 
 
 in which x 1 and x 2 are the aforesaid roots of equation (3) and 
 and a 3 are constants introduced by integration. Also, since 
 
 tty, &2P 
 
 ;=A, 
 
 dt 
 
 i = + a 2 x 2 e r - t . (5) 
 
 These equations (4) and (5) for charge and current assume three 
 forms according to the values of the constants of the circuit, 
 
 Z?2 1 7?2 1 
 
 (3) when 
 
 namely: (l)whenA>A ; (2) when A ; 
 
 R 2 1 
 
 = — under which conditions the roots x, and x n are 
 
 4 A 2 LG 12 
 
 respectively real, imaginary, and equal. 
 
 R 2 1 
 
 Case (1). When — — > — — - ; 1 Yon-oscillatory Effect. — Under 
 4 _£r L (J 
 
 these conditions the values of x x and x 2 in equations (4) and (5) 
 are real. The values of the constants a v a 2 , and a 3 may be 
 determined by solving when t is given the special values of 0 
 and oo. 
 
 For the conditions here named, q = 0 and i= 0 when f=0; 
 q = EC — Q and i = 0 when t = oo. Substituting these two sets 
 of values successively in (4) and (5) gives the following four 
 equations : ^ + a 2 + a 3 = 0, 
 
 a 1 x 1 + a 2 :r 2 = 0, 
 
 Q — a 3 = aff 1 " 11 + a.ff-" 0 , 
 
 4- a 2 x. 2 e nr = 0, 
 
 Qx 2 
 
 from which, 
 
 a. = 
 
 x i ~ X 2 
 
 - Qx\ 
 
 a 3 = Q. 
 
 * Murray’s Differential Equations, p. 64. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 189 
 
 Substituting these values of the constants in (4) and (5), we 
 obtain 
 
 Qx 2 ^ Qx ^ 
 
 Q = U 
 
 l nr nr nr nr 
 
 1 x 2 1 •* 2 
 
 = Q- 
 
 Q 
 
 and i = 
 
 i = 
 
 / R 2 
 
 1 
 
 ^-iL 2 
 
 LC 
 
 
 ( R 
 
 
 U L 
 
 Qx^c 2 C lt 
 
 Q 
 
 ~ x 1 -x 2 
 
 X 
 
 II 
 
 
 R 
 
 
 nJR 2 
 
 L 
 
 ” * 4 
 
 C 
 
 R + - 
 
 1C 
 
 IL + 
 
 '4 L 2 
 
 1 R 2 
 
 1 ' 
 
 '4 L 2 
 
 LC 
 
 V (6) 
 
 X, — x n 
 
 „ V2Z 4Z2 iW „ V2 L 4 L- LC) 
 
 • ( 7 ) 
 
 These equations (6) and (7) give the values of the charge 
 and current in a series circuit containing R , Z, and (7, when 
 R 2 1 
 
 ■ > , at any time t after the introduction of a constant 
 
 4 L 2 LG J 
 
 voltage R in the circuit, Q being the final charge corresponding 
 to the voltage R. 
 
 7? 2 1 
 
 Case (2). When ■ — — < — — ■; Oscillatory Rffect. — Under these 
 4 1 / L C 
 
 conditions the roots x x and x 2 are imaginary. Hence, using the 
 operator j to indicate the imaginary unit V — 1, we have 
 
 Substituting 
 where a = — 
 
 L. 
 
 2 L J ^ LC 
 
 R 2 
 
 4 Z 2 ’ 
 
 U i\\ 1 - 
 
 R 2 
 
 2 L J ' LC 
 
 4 L 2 ' 
 
 Xj = a +jb and 
 
 
 1 1 
 
 2 L V 
 
 LC 
 
 and are real quantities, 
 
190 
 
 ALTERNATING CURRENTS 
 
 equations (4) and (5) give * 
 
 q = a 1 e r,< + a 2 G 4 + a 3 
 = a l G a+jb)t + a 2 e^ b)t + a 3 
 = e at (A' cos bt + B' sin bt) + a 3 
 = A'e at cos bt + B' e at sin bt + a 3 , 
 in which A! — a x + a 2 and B' = j (a 1 — a 2 ). 
 
 Also i — (A! a + B'b)e at cos bt + (B 1 a — A'b)G t sin bt. 
 
 If p is an angle of which tan p = — , it follows that 
 
 n 
 
 m n 
 
 sin p — — — cos p = — . 
 
 's/m 1 + n 2 V m 2 + n 2 
 
 and m cos a + n sin u = Vm 2 + n 2 sin (a + p) . 
 
 Therefore, 
 
 q — e al a/A' 2 + B 12 sin (bt + 0) + a 3 , (8) 
 
 i = e a ' V (A'a + B'b ) 2 + ( B'a — A'b ) 2 sin (it + 0'), (9) 
 
 m 
 
 i • , a , _id' , /i, i (A/a + 15'i) 
 
 which 0 = tan 1 — , and 0' = tan 1 1 
 
 B 1 (B'a -A'b) 
 
 The values of the constants of the equations may he deter- 
 mined as in the previous case by solving when t is given the 
 limiting values of 0 and oo. As before, q — 0 and z = 0 when 
 t — 0 ; q = Q and i = 0 when t = go. Whence, 
 
 A'=-Q, 
 B'=-A'? = 
 
 ~QR 
 
 2 L 
 
 q = Q-Qe 2L 
 
 a 3 = Q. 
 
 \ L 0 
 
 I 1 R 2 
 LG 4 L 2 
 
 x Sill 
 
 LG\ 
 
 J 1 
 
 R 2 
 
 'LG 
 
 4 L 2 
 
 and (9), we 
 
 "Jl _ 
 
 R 2 
 
 ^ LG 
 
 4 Z 2 
 
 • <+ e 
 
 ( 10 ; 
 
 See Art. 71 and Murray’s Differential Equations, p. 67. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 191 
 
 and 
 
 i = e 2L - 
 
 E 
 
 ■ I 1 
 
 '^LC 4 
 
 R?_ 
 
 I? 
 
 . r i e 2 
 
 Sm 'X(7 4 L 2 
 
 ( 11 ) 
 
 in which 6 — tan 1 
 
 2 lJE-E 
 
 \ TO 1 T 
 
 LG 4 L 2 
 
 R 
 
 These equations (10) and (11) give the values of the charge 
 and current in a series circuit containing R , L , and C, when 
 R 2 1 
 
 — — < — — , at any time £ after the introduction of the constant 
 4 Ir LC 
 
 voltage E in the circuit. It will be noted that the expressions 
 for charge and current each contain the product of a logarith- 
 mic term and a sine term, and their values are recurrent 
 functions of time of an oscillatory nature. The value of the 
 charge at any instant is equal to the difference between a con- 
 stant term, Q = CE, and an oscillatory term which is in the lead 
 
 1 
 
 R 2 
 
 *LC 
 
 4 I? 
 
 R 
 
 of the current by an angle 6 of which tan 6 = 
 
 The current consists of a sinusoid modified by a vanishing 
 logarithmic curve, and its oscillations are of gradually vanish- 
 ing amplitude. 
 
 R 2 1 
 
 Case (3). When — — = — Under this critical condition, 
 
 4 L 2 LC 
 
 the roots x x and x 2 of the equation (3) become equal, or 
 
 R 1 
 
 X, = X„ = X = — — — 
 
 2 L 
 
 VLC 
 
 and the solution becomes : 
 
 q — a^C 1 + + a z , 
 
 i = a x xe ri -{- a z xte xt 4- 
 
 ( 12 ) 
 
 (13) 
 
 The values of the constants may be determined by solving 
 when t is given the special values of 0 and oo. As before, 
 q = 0 and * = 0 when t = 0 ; and q = Q and i = 0 when t = oo. 
 
 Then, a x = — Q, 
 
 *See Murray’s Differential Equations , p. 65. 
 
192 
 
 ALTERNATING CURRENTS 
 
 R 
 
 «2 = - Q = - — = 
 
 Q 
 
 2L 
 
 V LO 
 
 a 3 — Q‘ 
 
 Substituting these values in equations (12) and (13) we 
 obtain, 
 
 Rt\ " 
 
 q = Q~ (/ I + )e 2 £, 
 
 2 A 
 
 7?2 E 1 /« 
 
 i =QjL 2 te^ = -t^. 
 
 (14) 
 
 (15) 
 
 These equations (14) and (15) give the values of the charge 
 and current in a series circuit containing R, L , and (7, when 
 
 R2 1 
 
 = — at any time t after the introduction of a constant 
 
 4 A 2 LC 
 
 voltage E in the circuit. This condition affords the most rapid 
 rate of charging for the circuit. A decrease of R would put 
 the circuit in an oscillatory condition, and an increase of R 
 would put the circuit in a condition represented by equations 
 (6) and (7). 
 
 b. Current and Charge on Withdrawal of Impressed Voltage . — 
 On removal of the impressed voltage from a circuit containing 
 R, L , and C in series, the equation of the circuit is 
 
 = m+i § + c. 
 
 ir* 
 
 idt 
 
 - V d( i | | ? 
 
 ~ dt + dt 2 + c' 
 
 Putting q = €**, 
 
 the equation takes the characteristic form 
 
 the roots of which are 
 
 ( 16 ) 
 
 + L X + 
 
 1 
 
 LC 
 
 0, 
 
 
 lR 2 
 
 1 
 
 2 A + ^ 
 
 4 A 2 
 
 LC' 
 
 R 
 
 I A' 2 
 
 1 
 
 2 L ^ 
 
 “4 A 2 
 
 LC 
 
 ( 17 ) 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 193 
 
 The solution takes three forms according as these roots are real, 
 imaginary, or equal, or when 
 
 ... 1 , JR* 1 /Q N 1 R 2 1 
 
 (1) Tl^lo’ (2) when (3) when 4I? = LC' 
 
 R 2 1 
 
 Case (1). When— — >— — ; Non - oh dilatory Effe ct. — Under 
 T J-J -Lj 0 
 
 these conditions the solution of equation (16) takes the form 
 
 q = aqe* 1 * + (18) 
 
 i — a x x 1 e r ' t + a 2 x 2 e Xit . (19) 
 
 To determine the constants we have, when t = 0, q = Q and 
 i = 0 ; and therefore 
 
 Qz* 
 
 _ Qx-i 
 
 ‘X'i OCn 
 
 Substituting these values in (18) and (19), we obtain 
 R 
 
 q=Q 
 
 4 Lxl* 
 
 1 
 
 
 LC 
 
 and 
 
 i = ■ 
 
 -Q 
 
 Q 
 
 4 L' 
 
 2 LC 
 
 r 
 
 + q 
 
 1 
 
 i 
 
 R 
 
 
 R 2 
 
 i 
 
 4 L 2 
 
 LC 
 
 — (IL+ -\/ 
 
 P \-2L v 4L* 
 
 -(A+VWIT\ 
 
 : \2L + V 4£2 LC) 
 
 'll 2 LC 
 
 ( 20 ) 
 
 ( 21 ) 
 
 These equations (20) and (21) give the values of the charge 
 and current in a series circuit containing R , _Zi, and U, when 
 R2 l 
 
 > , any time t after the withdrawal of the impressed 
 
 4 U LC 
 voltage E. 
 
 Case (2). When ; 
 
 conditions the roots x x and x 2 of equation (17) are imaginary 
 and proceeding as in case (a. 2), p. 189, the equations for charge 
 and current become 
 
 Oscillatory Effect. — Under these 
 
 q --- e at y/ fA 1 ') 2 + ( B 1 ') 2 sin ( bt + d), 
 
 o 
 
 ( 22 ) 
 
194 
 
 ALTERNATING CURRENTS 
 
 i=e al V(A r a + B'b) 2 + (B'a-A'by sin (bt + 0'), (23) 
 
 where 
 
 R , ^ 1 
 
 a = , o — \ 
 
 9 T V 
 
 i(7 
 
 it * 2 
 4 X 2 ’ 
 
 , n .d/ -i , /w -I - R b 
 
 tan 6 = — , and. tan 0 = — — - — — . 
 
 B ' B'a - A'b 
 
 To determine the values of the constants in (22) and (23) 
 we have, when t = 0, q = Q and i = 0. 
 
 Then, 
 
 A' = Q and B' = - -A' = - ® Q. 
 
 b b 
 
 Substituting these values in (22) and (23), we obtain, 
 
 " vzv 
 
 q = Qe 
 
 2 L 
 
 LG 
 
 W 
 
 R 2 
 
 LG 4 i 2 
 
 x 
 
 and 
 
 sm 
 
 i = — e 
 
 '•LO 
 
 Rt 
 
 1L 
 
 Q 
 
 t + tan 
 
 -l 
 
 2ijxi^r 
 
 4 L 2 
 
 R 
 
 LG 
 
 w 
 
 i 2 
 
 sin 
 
 ( Vi- 
 
 i? 2 
 
 LG 4i 2 
 
 (24) 
 
 (25) 
 
 1 LG 4 X 2 
 
 These equations (24) and (25) give the values of the charge 
 and current in a series circuit containing R, L , and (7, when 
 R 2 1 
 
 - — - < — — , at any time f after the withdrawal of the impressed 
 
 voltage E. It will be noted that the discharging process is of 
 an oscillatory nature, the expressions for q and i being each 
 made up of the product of a logarithmic term and a sine term, 
 and that the charge leads the current by an angle of which 
 
 tan 6 = 
 
 2 lJ— - 
 
 'LG 
 
 R 2 
 4 L 2 . 
 
 R 
 
 The period of each oscillation is 
 
 z TV 
 
 1 1 _ 
 
 R 2 
 
 J LG 
 
 4 L 2 
 
 which is the natural oscillatory period of the circuit. If R is 
 small compared with L , this is approximately equal to 2 7rx LG. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 195 
 
 R 2 l 
 
 When — Under these conditions the 
 
 4 U LC 
 
 Case (3). 
 
 roots of equation (17) become equal, or 
 
 X, = x 0 = x = — 
 
 B 
 
 -1 
 
 2 L y/LC 
 
 and the solution of equation (16) becomes 
 
 q — a x C l + a^e* 1 , 
 
 and i = a^x^ 1 + a 2 xte Tt + a 2 € Tt . 
 
 (26) 
 
 (27) 
 
 To determine the values of the constants, we have q — Q and 
 * = 0 when t = 0. 
 
 Hence, 
 
 a x = Q and a 2 = — Qx = 
 
 Substituting these values in (26) and (27), we obtain 
 
 
 1? 
 
 i = -~te 2L . 
 -Lj 
 
 Rt 
 
 2L 
 
 (28) 
 
 (29) 
 
 These equations (28) and (29) give the values of the charge 
 and current in a series circuit containing R , L , and C, when 
 B 2 1 ■ 
 
 = — - at any time t after the withdrawal of the impressed 
 
 4Z 2 LC J * 
 
 voltage j E from the circuit. This relation of B , X, and C affords 
 
 the most rapid discharge of the circuit. A decrease of R would 
 
 throw the discharging process into the oscillatory state, and an 
 
 increase of R would put the circuit in a condition represented 
 
 by equations (20) and (21). 
 
 c. Current and Charge when the Impressed Voltage is a Sine 
 Function of the Time. — In this case the instantaneous voltage 
 conditions are represented by 
 
 e = e m sin wt = iR 4- + 7 
 
 dt C 
 
 
 dt CC dt 
 
 e m . , cPq R dq q 
 
 L Sin = dp + L dt + LC 
 
 or 
 
 (30) 
 
196 
 
 ALTERNATING CURRENTS 
 
 o ^ 
 
 In this case of a sine function, u> is equal to ~jT = 2 the 
 
 angular velocity of a rotating vector generating the sinusoid. 
 
 The above is a linear differential equation of the second 
 order which is to be solved to find the values of q and i in 
 terms of e m , R , L, (7,/, and t. The solution of such an equation 
 consists of the sum of two parts, the complementary function 
 and the particular integral. The complementary function is 
 the integral obtained by equating the second member to zero. 
 In other words, it is the solution of case (5) ante , i.e. when e = 0, 
 and takes the form * 
 
 q = a 1 e T,t + a 2 e T2 \ 
 
 i = a^x-^e 1 ' 1 + a 2 x 2 e Xlt . 
 
 The particular integral contains no arbitrary constants and is 
 equal to f 
 
 9 = 
 
 R 
 
 D 1 * + — D + — — 
 L LG 
 
 — ••• ~ sin a>t = - — % sin cot 
 
 1 L I) — a R 
 
 ^ m ,a( i c —at 
 
 R 
 
 /* 
 
 — e„ 
 
 and 
 
 tan 8 = 
 
 + 
 
 > 
 
 3 
 
 1 
 
 e 
 
 bn 
 
 1 
 
 lb 
 
 tc 
 
 
 co*L 
 
 1 
 
 Cl ■ — 
 
 R 
 
 OR' 
 
 
 e 
 
 bi 
 
 1 
 
 e u 
 
 coR 
 
 R 
 
 -cos (cot — d). 
 
 The complete solution is therefore 
 
 9 = Pm — — cos (cot - g) + u 1 e ri/ + a 2 e* 1 . (31) 
 
 + (»£—) 
 
 In a similar way the complete solution for the current becomes 
 i — — e ' n sin (cot — 8) + a,x,e rit + u 0 rr o e r!/ . (3d) 
 
 V^ + («,z-L) 
 
 * Murray’s Differential Equations, Chap. VI. 
 f Murray’s Differential Equations, Arts. 58 and 62. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 197 
 
 These equations give the values of the current and charge at 
 any time t after impressing the voltage in a series circuit con- 
 taining constant it, i, and (7, when such voltage is sinusoidal. 
 The above equation for current may be written 
 
 i = , . € "’ ~ — - sin (a-0) + a 1 x 1 e c i t + a 2 x 2 e x * t . (33) 
 
 The angle 6 is the angular difference of phase between the cur- 
 rent in the circuit and the impressed voltage ; that is, it is 
 the angle of lag (or lead) of the current for frequency/. 
 
 Neglecting the exponential terms, which are rapidly evanes- 
 cent where x x and x 2 are real, the equation for instantaneous 
 current under a sinusoidal voltage is 
 
 where e m is the maximum voltage and the angle 6 is either posi- 
 tive or negative according as capacity or self-inductance pre- 
 dominates. The current is evidently a maximum when sin 
 (a— 0) = 1, and therefore 
 
 The relation of effective current and voltage can be obtained 
 b} 7 dividing both sides by V 2, and then 
 
 I — 
 
 E 
 
 \/« 2 
 
 2 + 2 7T.fi - 
 
 2 7 tfC, 
 
 The term yjR 2 + (ZirfL — - - is called the Impedance of 
 \ 2 ttj G y 
 
 the circuit and is composed of the square root of the sum of the 
 squares of two quantities, the first of which, R, is the resistance 
 
 of the circuit and the other yl^fL — ■ ) is called the React- 
 
 \ ‘ 2 irfCJ 
 
 ance of the circuit. 
 
198 
 
 ALTERNATING CURRENTS 
 
 If L is zero and C is infinite, the equation (38) reduces to the 
 
 value _ g 
 
 i = V sin «, 
 
 R 
 
 as already derived for the current flow when a sinusoidal vol- 
 tage is impressed on a circuit containing resistance alone. 
 
 If C is infinite (which is equivalent to saying that no capacity 
 voltage arises in the series circuit), but R and L have finite 
 values, the equation reduces to that on page 148, or 
 
 ri 
 
 i — 
 
 ■ sin (a — 6') + A x e z. 
 
 4 - (2 77 -fLy 
 
 The value of A x may be derived by giving a and t the values 
 and t v corresponding to the instant of introducing the voltage 
 in the circuit, at which instant i = 0 and the second term of 
 the right-hand member must be equal to but opposite the first 
 term ; 
 
 whence 
 
 A x = - 
 
 Rti 
 
 e L sin (oq- 
 
 ■n 
 
 V R 2 + (2 tt/L) 2 
 
 which is constant, because t x and « 1 and O' have particular values, 
 
 and A x e l = _ 
 
 -2lL=hl . 
 
 z sin («j — 6 '). 
 
 Vi? 2 + (2 tt/L) 2 
 The value of t is measured from the instant when the 
 sinusoidal voltage passes through the preceding zero from 
 negative toward positive values ; and t x is the value of t at the 
 instant of switching the voltage into the circuit. 
 
 If L is zero, but R and C have finite values, the equation 
 reduces to that on page 174, or 
 
 e ’" — sin (a + 6"') 4- B x e 
 
 i = 
 
 _t_ 
 RC . 
 
 
 1 Y 
 
 2 77-/CV 
 
 The value of B v derived from giving a and t the values a x 
 and t v corresponding to the instant of introducing the voltage 
 in the circuit, is 
 
 B x =- 
 
 VR 2 + 
 
 B x e nc — _ 
 
 l Y 
 
 2 77 fC) 
 
 n 
 
 V ! + 
 
 — T 
 
 2 irfCJ 
 
 e RC sin («j + 6 
 
 e R c sin (aj 4- 0"'). 
 
 and 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 199 
 
 Fig. 130. — Effect of switching-in on Circuit containing R and L. 
 
200 
 
 ALTERNATING CURRENTS 
 
 59. Effect of Exponential Terms. — The exponential members 
 of the equations of case ( c ) show the transient effect on the form 
 of the current wave which occurs upon first impressing sine 
 voltage on a circuit containing resistance and inductance, resist- 
 ance and capacity, or resistance, self-inductance, and capacity, 
 in series. In circuits with resistance and inductance or resist- 
 ance and capacity the exponential term quickly reduces to zero 
 and can be then neglected. The actual influence of the expo- 
 nential term in either of these two cases is to distort the current 
 wave for the earlier periods, and the extent of the effect 
 depends on the value of (rq — d) at the instant of introdu- 
 cing the voltage into the circuit. The value of a 1 expresses 
 the advance of the voltage at the instant of its introduction in 
 the circuit, and the value of (cq — d) expresses the advance 
 of the corresponding normal current. Consequently, when 
 eq = d, the exponential value is zero and the current starts from 
 zero, following a regular sine wave ; and when cq = 90° -f d, the 
 exponential obtains its largest value and the current wave is 
 given the maximum of distortion. Figures 130 and 131 show 
 the form of current wave for the first few cycles where a sinu- 
 soidal voltage of 100 volts is switched onto a circuit containing 
 5 ohms and .1 henry and on a circuit containing 100 ohms and 
 100 microfarads, the frequency being 60 periods per second in 
 each instance. The curves are shown for cq = d, cq =45° + d, 
 cq = 90° + d, <q = 135°+ d. 
 
 In Fig. 130 a the sinusoid marked E shows the position of 
 the voltage ; the sinusoid marked I shows the current. The 
 current grows up in the circuit without deviation from the true 
 sine form, since the switch is closed at the instant when « = d ; 
 namely, at the instant when the sine current would normally 
 pass through zero, and the exponential term of the equation is 
 therefore zero. In Figs. 130 b , c, and d, the voltage curve has 
 been omitted. The dotted curve represents the normal sine 
 form of the current, corresponding to the curve I of Fig. 130 a, 
 and the full line curves represent respectively the exponential 
 term (which is equal and opposite to the value of the normal 
 sine curve at the instant of switching in) and the actual current 
 flow. The latter is distorted for several periods from the nor- 
 mal sine form by the effect of the reactance in the circuit. 
 
 Figures 131 a,b,.c, and d similarly represent the relations 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 201 
 
 when the sinusoidal voltage is switched on a 
 
 circuit of fixed 
 
 / ~ \ 
 1 1 \ I 
 
 Jh< 
 
 T 
 
 capacity and resistance. 
 
 The effects of switch- 
 ing a sinusoidal voltage 
 onto a circuit of fixed 
 resistance and self-in- 
 ductance or a circuit of 
 fixed resistance and ca- 
 pacity are thus made 
 plain. In case the self- 
 inductance varies much 
 during the cycles, as in 
 the case of a coil with 
 an iron core that be- 
 comes highly saturated 
 in each half cycle, or 
 the capacity varies dur- 
 ing the cycles, as in the 
 case of certain liquid 
 condensers, the expo- 
 nential terms become 
 very complex on account 
 of the variation of L or 
 (7, and the magnitude of 
 the first rush of current 
 may become serious. 
 
 The distortion of the 
 form of the current 
 wave during the first 
 few cycles may then be 
 much greater than is 
 indicated in Figs. 130 
 and 131. 
 
 When the voltage is 
 withdrawn from the cir- 
 cuit without changing 
 the constants of the 
 
 circuit, it is obvious that the first terms of the equations reduce 
 to zero, and the current dies down along an exponential curve 
 from its value at the instant of withdrawal of the voltage. 
 
 = 90°+ 0° 
 
 
 . 
 
 
 
 / 
 
 o 
 
 © 
 
 
 
 — T" T d 
 
 
 
 a, = 135°+0° 
 
 Fig. 131. — Effect of switching-in on Circuit con- 
 taining Ii and C. 
 
202 
 
 ALTERNATING CURRENTS 
 
 On the other hand, if the voltage is withdrawn in the usual 
 manner of opening the circuit, the value of R is rapidly changed 
 to infinity. An inspection of the two exponential terms shows 
 that this may cause a great disturbance in an inductive circuit 
 during the discharge of the magnetic field ; but the current 
 merely ceases in the circuit with capacity, leaving the con- 
 denser charged. In the latter case, a serious momentary current 
 rush and distortion of the current wave may occur if the vol- 
 tage, when switched on the circuit again, chances to be reversed 
 in relation to the condenser voltage. 
 
 Turning again to the equations representing the conditions 
 when resistance, self-inductance, and capacity are all three in 
 series, (equations (31) and (32)), these apply to all instants of 
 time, including the instant t = t x when the voltage is introduced 
 in the circuit. If the capacity of the circuit is without a 
 previous charge, the values of q and i are, of course, both zero 
 at this instant ; and, solving for oq and a 2 under these circum- 
 stances gives, when x x and z 2 are real, 
 
 e m e ~ Xlh sin (coty — 6 4- p') 
 
 d . . } 
 
 Z x 2 — x 1 cos p 
 
 _ _ e m e~ x - tl sin (coty — 6 + p'~) 
 
 a 2 '/ ' i ’ 
 
 Z Xy Xy COS p 
 
 in which Xy and x 2 have the values heretofore given them, Z is 
 the impedance of the circuit (= yjR 2 + (2 irfL - — ^ 
 ty is the value of t at the instant of introducing the voltage 
 into the circuit, tan p = - 2 , and tan p' = Therefore, 
 
 0) (O 
 
 (lyXyG^ + dyXylf 2 * — 
 
 2 zj&_ 
 
 V 4 Z 2 
 
 1 
 
 LO 
 
 JL _ e - G h- V S r S) <* - « Sin (coty-d + p ) 
 
 9 T. » -t T. 2 T.C! ) cos p 
 
 A 
 
 2 L 
 
 + ' 
 
 1 R 2 
 
 1 N 
 
 *4 I? 
 
 LGj 
 
 1 R 2 
 
 1 ^ 
 
 4 L 2 
 
 LG) 
 
 V (2* + V 4 % -Ic )«- « s in (tot 1 — d + p _ ' V 
 
 cos p 
 
 When the circuit constants are fixed, the right-hand member 
 of the last equation reduces to zero rather rapidly, the precise 
 
SELF-INDUCTION, CAPACITY, KEACTANCE, AND IMPEDANCE 203 
 
 rate of decrease as time grows depending upon the relative 
 values of R, L, and C ; and the current will quickly come to a 
 sinusoidal value with relatively little disturbance. However, 
 if R , L, and 0 vary during the cycles of current, the disturb- 
 ance may be serious on account of the instantaneous flow of an 
 excessive current. 
 
 1 R 2 
 
 When — — = — — , equations (31) and (32) take the special 
 
 LC 4 L 2 
 
 forms, 
 
 q = — ~ cos ( cot — 0) + (a x + a 2 £)e r ( 
 
 i 
 
 Z 
 
 sin (cot — 0 ) + (ape + a 2 xt -(- « 2 ) e Tt . 
 
 Solving for a x and a 2 in these equations when q = 0 and i = 0, 
 which are the values of q and i at the instant t = t x when the 
 sinusoidal voltage is introduced into the circuit, gives 
 
 = sin (<»U - d + p)e 
 cos p 
 
 Sill 
 
 e m . (cot, — 6 + »') 
 a„ — . — sm - — 1 ' e xl \ 
 
 cos p 
 
 in which tan p = ^ ^ and tan p' = -, where x = — 
 
 '•* co 2 L 
 
 cot , 
 
 
 Therefore (apr + a 2 xt + u 2 ) e** = (A -(- i?t) e 2 z . 
 
 When < —5— and the values of 2 ;, and 2 -, are therefore 
 4 L 2 LC 12 
 
 imaginary, equations (31) and (32) take the forms 
 
 q = — ^jr cos (cat — 0 ) -f Me at sin (bt + v ), 
 
 i = ~ sin (ait — 0) + Ne at sin (bt -f v'~). 
 
 z 
 
 Solving these for the values of M, iV", v and v' when t = t v and 
 therefore q and i are both equal to zero, results in expressions 
 affording for i the equation 
 
 p „ mt-h) li in 
 
 * = J sin (eat -O') + Pe 2 1 S mt-\— - 4 ^ 2 » 
 
 in which P is a real function of e m , R, L , C , and £ r 
 
204 
 
 ALTERNATING CURRENTS 
 
 These equations show that the current will grow to its 
 normal sinusoidal value without great disturbance after the 
 voltage is switched into circuit, and will die away without 
 great disturbance, if the values of X, X, and C give real roots; 
 but if the roots are imaginary, oscillatory effects (instead of a 
 plain exponential) of a frequency fixed by 
 
 are superimposed on the normal sinusoid at both switching in 
 and switching off the voltage, and the disturbances may be 
 quite great at either switching in or switching off the voltage 
 if «j chances to have an appropriate value ; especially if f x is not 
 very different from f, as is not unusual. If these conditions 
 are accompanied by variable values of X, X, or C , the disturb- 
 ances may be still more magnified. 
 
 Prob. 1. A circuit has resistance of 20 ohms and self-induct- 
 ance of .01 henry in series. What steady voltage impressed 
 on the circuit will bring the current to a value of 20 amperes 
 at the instant that it is increasing at the rate of 1000 amperes 
 per second? 
 
 Prob. 2. How much power is being expended in the cir- 
 cuit of problem 1 at the instant in question ? What proportion 
 of this power is being converted into heat in the conductor 
 resistance and what proportion is being stored in the magnetic 
 
 Prob. 3. A circuit of 10 ohms resistance has a capacity of 
 20 microfarads in series with the resistance. What steady 
 voltage impressed on the circuit will bring the current to a 
 value of 20 amperes at the instant when the charge equals ^ 
 coulomb ? 
 
 Prob. 4. How much power is being expended in the circuit 
 of problem 3 at the instant in question ? What proportion of 
 the power is being converted into heat in the conductor resist- 
 ance and what proportion is being stored in the condenser ? 
 What value does the condenser charge finally reach, and how 
 much energy is then stored in the charge ? 
 
 field ? 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 205 
 
 Prob. 5. A circuit has a resistance of 10 ohms, a self-induct- 
 ance of .02 of a henry, and a capacity of 100 microfarads, all 
 in series relation. At a certain instant the current is 20 
 amperes and is increasing at the rate of 2000 amperes per 
 second. What steady voltage is impressed on the circuit ? 
 The charge in the condenser at the instant is .001 of a coulomb. 
 
 Prob. 6. How much power is being expended in the circuit 
 of problem 5 at the instant in question ? What proportion of 
 the power is being converted into heat in the conductor resist- 
 ance, what proportion is being stored in the magnetic field, 
 and what proportion is being stored in the condenser ? What 
 value does the condenser charge finally reach, how much 
 energy is then stored in it, and how much energy is then stored 
 in the magnetic field ? What is the maximum value of the 
 energy stored in the magnetic field and how long after the 
 introduction of the impressed voltage in circuit is it reached ? 
 
 Prob. 7. If the impressed voltage of problem 5 instantane- 
 ously falls to zero after a steady state of condenser charge has 
 been reached without altering the electrical constants of the 
 circuit, how long does it take for the condenser to fully dis- 
 charge, and what is the maximum value of the discharge 
 current ? 
 
 Prob. 8. A circuit has 200 ohms resistance, .02 henry self- 
 inductance and 5 microfarads capacity, in series. Draw the 
 curves of rising and falling current when 100 volts are steadily 
 impressed on the circuit until the charge rises to its full value 
 and the impressed voltage is then withdrawn from the circuit. 
 
 Prob. 9. Draw the curves as in problem 8 when the resist- 
 ance is 1 ohm, the self-inductance 10 henrys, and the capacity 
 10 microfarads. 
 
 Prob. 10. Draw the curves as in the preceding problems 
 when the resistance is 1 ohm, self-inductance .2 of a henry, and 
 the capacity 200 microfarads. 
 
 Prob. 11. Draw the curves as in the preceding problems 
 when the resistance is .1 of an ohm, the self-inductance 5 
 henrys, and the capacity 500 microfarads. 
 
 Prob. 12. If the resistance, inductance, and capacity of Prob. 
 5 are individually segregated in the circuit, how much voltage 
 
206 
 
 ALTERNATING CURRENTS 
 
 is impressed on the capacity, how much is impressed on the 
 inductance, and how much is absorbed in the iR drop, at the 
 instant considered ? 
 
 60. The Time Constant of a Circuit. — The formula 
 
 • E ( i 
 
 shows, as already stated, that the theoretical curve represent- 
 ing the rise or fall of the current in a circuit containing re- 
 sistance and self-inductance, is logarithmic. The formula shows 
 that when L is very small, the current almost immediately 
 JE 
 
 takes its full value I = — on closing the circuit or falls to zero 
 
 R -Rt 
 
 when the voltage is removed, since e L quickly becomes negli- 
 gible in comparison with unity. Theoretically, when induct- 
 ance is present, the current can rise to its full value only after 
 
 -Rt 
 
 an infinite time, yet e L becomes practically negligible after 
 a comparatively short interval. 
 
 Since resistance has the absolute dimensions of a velocity 
 (a length divided by a time) and inductance has the dimen- 
 sions of a length, the ratio ~ has the dimensions of a time ; 
 
 R 
 
 this ratio, in the case of any circuit, is therefore generally 
 called the Time constant of the circuit, and may be represented 
 
 by the Greek letter r. In the preceding equation, — repre- 
 
 R 
 
 sents the value which the current would instantly reach when 
 under the constant impressed voltage, were there no inductance 
 in the circuit. This is the same as the ultimate value when 
 there is inductance. The equation may therefore be written 
 
 * = 7 ( 1 - 6 "). 
 
 When t = t, this becomes 
 
 I 
 
 i=I- 
 
 2.718’ 
 
 or 
 
 * = .632 7. 
 
 The current therefore reaches .632 of its ultimate or full 
 value in a time equal to t seconds. The value of the time con- 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 207 
 
 stant is therefore a measure of the growth of the current in a 
 circuit after a fixed voltage is impressed, and it is obvious that 
 in a circuit of great inductance and also great resistance, the 
 current practically reaches its full value as quickly as in a 
 circuit of small inductance and proportionally small resistance. 
 
 When the circuit is one containing resistance and capacity 
 instead of resistance and self-inductance, the time constant t is 
 equal to R C, and in this case it is a measure of the growth of 
 the charge in the circuit. When the circuit contains resist- 
 ance and capacity and inductance, it has no real time constant, 
 but the apparent time constant is less than that of the resist- 
 ance and inductance or that of the resistance and capacity. 
 
 Eddy currents evidently reduce the value of the time con- 
 stant by their distortion of the curves of current flow.* A 
 variable self-inductance due to magnetic saturation and hys- 
 teresis also distorts the curves of current change, and hence 
 changes the time constant. Thus, substituting the value of 
 self-inductance, 
 
 t _ ^ rrnn' Ag 
 
 ~ 10 s 109 ’ 
 
 in the formula for the fall of current gives 
 
 ._E -(; 
 * R e 
 
 1 pout 
 
 4 irnn'AfA. 
 
 ) 
 
 i 
 
 and at any given time i is proportional to e A similar 
 condition exists for rise of current. As g and therefore L varies 
 during the entire time of rise or fall of current in a circuit, it is 
 evident that the curve cannot be logarithmic. Raising the value 
 of i by the power g gives 
 
 
 / -p\y- _(uw\ 
 
 j € ^4 7rW7i '*4 / ' • 
 
 i can therefore be obtained for any instant of time (tj) if the 
 constants of the energizing coil are known, provided there is 
 no magnetic leakage. 
 
 ^ \R 
 
 7tt\ r ( \ 
 
 Jh \ l , \4 nnn’ A/ _ 
 
 where A is a constant for the instant t v or 
 
 = A, and g log q = log S. 
 
 * Arts. 49 and 111. 
 
208 
 
 ALTERNATING CURRENTS 
 
 If the constants of the circuit and the magnetization curve for 
 the circuit are known, the value of /a for each value of i can be 
 obtained. A curve can therefore be drawn in which the ordi- 
 nates are equal to p. log i — where /a is the permeability of the 
 circuit when the current i is flowing — and the abscissas are 
 log i. Log i l is then the abscissa of a point on this curve which 
 has an ordinate n log i v and i l can be obtained from logarithmic 
 tables. In this manner any number of values of the rising or 
 falling current can be obtained with due account of changes in 
 permeability caused by the phenomena of saturation and hyster- 
 esis. The curve of current will contain harmonics caused by 
 the phenomena of saturation and hysteresis. 
 
 61. Examples of Time Constants. — The following are the 
 time constants of some of the circuits for which inductances 
 have previously been given.* A Wheatstone bridge resistance, 
 when properly wound, generally has a time constant of a 
 millionth of a second or less; electric bell, 4.8 millionths of a 
 second; telephone call bell, nearly .02 of a second; modern 
 short-core telephone bell, about .006 of a second; armature of 
 a small magneto generator, from .005 to .013 of a second; 
 telephone receiver, with diaphram, about .001 of a second; 
 modern bipolar telephone receiver, .0005 of a second; mirror 
 galvanometer for marine signaling, .0016 of a second; mirror 
 galvanometer of 5000 ohms resistance, .0004 of a second; 2700 
 ohm coil of a mirror galvanometer, .001 of a second; 100,000- 
 ohm coil of a mirror galvanometer, .0007 of a second; coil of 
 Ayrton and Perry spring voltmeter, .0044 of a second ; polar- 
 ized relays, types 1, 2, 3, and 4, respectively, .0048, .0045, .0041, 
 and .0032 of a second; Morse relays, from about .070 to .026 
 of a second, with about .034 of a second as an average for in- 
 struments in working adjustment; two telegraph sounders, 
 .0095 and .0065 of a second ; modern telephone induction coil 
 primary about .006 of a second, secondary of same nearly .003 
 of a second; a mile of bare No. 12 B. & S. gauge copper wire 
 on a pole line, with ground return, .00037 of a second; No. 6 
 wire in a similar position, with ground return, .0014 of a sec- 
 ond; primary of large induction coil, .09 of a second; sec- 
 ondary of same, .065 of a second; dynamo fields, from about 
 .01 to 10 seconds; direct-current dynamo armatures, from .005 
 
 * Art. 44. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 209 
 
 to 5 seconds ; Mordey 36-kilowatt 2000-volt alternator arma- 
 ture, .017 of a second; Ivapp 60-kilowatt 2000-volt alternator 
 armature, .035 of a second; primary and secondary windings 
 of transformers, from several thousandths of a second to a 
 number of seconds. The Du Bois electro-magnet has a time 
 constant of 75 seconds. Finally, suppose 6 ohms is the resist- 
 ance of the magnetizing coil figuring in the example of Art. 
 43. Then assuming the value of L to be constant, which is 
 not exact when the core is iron, the time constant becomes in 
 the three cases, respectively, .655, .785, and .0026 of a second. 
 
 62. Vector Relations of Current and Voltage in a Circuit con- 
 taining R, L , and C ; Complex Expressions. — Where resistance, 
 self-inductance, and capacity are all in series in a circuit, the 
 effects of the self-inductance and capacity, being opposite, tend 
 to neutralize each other, and the diagram takes the form shown 
 in Fig. 132. Here OO is the active voltage (Ji2 drop), OB 
 
 Fig. 132. — Phase Diagram of a Circuit containing Resistance, Self-inductance, 
 and Capacity in Series. 
 
 and OB' are voltages respectively equal to the self-inductive 
 and capacity voltages, and OA is the impressed voltage. 
 Since OB' is 90° ahead and OB 90° behind the active vol- 
 tage, they are exactly in opposition and tend to neutralize each 
 other. If they are of equal values, the external effect is as though 
 both were absent and the impressed and active voltages are 
 then identical. In Fig. 132 the self-inductive voltage is 
 greater than the capacity voltage and their difference OF is the 
 reactive effect in the circuit. The external effect is as though 
 a self-inductive voltage OF were present without capacity. On 
 the other hand, if OB' should be the greater, the difference 
 would show a capacity voltage equal to the difference between 
 
210 
 
 ALTERNATING CURRENTS 
 
 the two reactive voltages. Figure 133 shows the polygon of 
 the voltages shown in Fig. 132. 
 
 A 
 
 and Capacity in Series. 
 
 The complex expression representing the voltages in such a 
 
 circuit is 
 
 E — E r +jE t — jE c , 
 
 or 
 
 E=E r +j{E l -E c ') 
 
 and 
 
 E — .27 (cos 6 +j sin 0), 
 
 where 
 
 E = Vi ?,. 2 + (E, — E c y 
 
 and 
 
 a f - 1 Ei — E c 
 
 E r 
 
 If Ei > E c , the imaginary term of the complex expression is 
 positive and the equation is similar to one for a circuit contain- 
 ing resistance and self-inductance only, and the angle 6 is posi- 
 tive. If E t < E c , the imaginary term is negative, and the angle 
 6 is negative. 
 
 The voltage arising from the combination of the voltages of 
 several circuits in series is 
 
 E = (E n + E r „ + etc.) -\- j(^E ti -(- E ln + etc. E Ci E Ci etc.) 
 
 ■ = S^ r +/(2^-2Jr e ), 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 211 
 
 and E = E (cos 0 +j sin 6), where 
 
 , 'LE, -IE 
 
 E= V(^,) 2 + (2-Ei - 2^,) 2 , and 6 = tan" 1 ' c . 
 
 Either or both E, and E c may be zero in any of the partial cir- 
 cuits and E r may be so small as to be negligible.* The angle 
 6 is the angle of lag of the circuit. It is measured from the 
 initial axis or axis of reals (which gives the direction of the 
 current vector or IR drop) towards the vector of impressed 
 voltage. When 6 is positive, the circuit carries a current which 
 lags with respect to the voltage impressed at the circuit termi- 
 nals ; and when 6 is negative, the circuit carries a current 
 which leads with respect to the voltage impressed at the 
 terminals. 
 
 The current in a circuit may be represented by a vector 
 (complex) equation in the same manner as a voltage. In this 
 case the current must be resolved into rectangular components, 
 preferably with one component parallel to the vector of voltage 
 impressed on the circuit. It will be shown later that the initial 
 axis in this case should correspond with the direction of the 
 vector of voltage impressed on the circuit, and therefore the 
 angle measured from the initial axis to the current vector is 
 the angle of lag reversed ; that is, it is — (±6). 
 
 Knowing the rectangular components of the several currents 
 in a number of branches or parallel circuits, the components 
 of the main current are obtained by adding the corresponding 
 components of the branch currents together algebraically ; as 
 
 1= tl r +/( 2 / r ), 
 
 and 1= /( + cos 0 — j sin #). 
 
 In this case 
 and 
 
 J= V(2J,) ! + (S4) 2 
 
 V T 
 
 6 = tan -1 
 
 63. Impedance and Reactance and their Expression as Com- 
 plex Quantities. — The quantity 
 
 \/*“ 2 
 
 + 2 7 rfL- 
 
 J_Y 
 
 called the Impedance of the circuit and ( 2 irfL — 
 
 2 TrfCj 
 
 1 
 
 2 irfC 
 
 is 
 
 is 
 
 * Art. 58. 
 
212 
 
 ALTERNATING CURRENTS 
 
 called tlie Reactance, whatever may be the values of R, L , or C. 
 The square of the impedance of a circuit is therefore equal to 
 the sum of the squares of its resistance and reactance. Imped- 
 ance and reactance are both of the dimensions of resistance 
 and are therefore expressed in ohms. Impedance may be de- 
 fined for circuits in general, as the total opposition in a circuit 
 to the flow of an alternating electric current, or the ratio of the 
 voltage to the current; and Reactance may be defined as the 
 component of the impedance caused by the self-inductance and 
 capacity of the circuit. 
 
 The reciprocal of impedance is called Admittance. It is 
 equal to the current divided by the voltage. It therefore 
 measures the tendency of a voltage to force current through a 
 circuit. It is measured in Mhos or the reciprocal of ohms, as 
 the term indicates. 
 
 The Capacity Reactance of a circuit is inversely proportional 
 to the capacity and the frequency in the circuit, since it is 
 
 equal to — 
 
 rrfC' 
 
 The Inductive Reactance of a circuit is directly 
 
 proportional to the self-inductance and the frequency in the 
 circuit, since it is equal to 2 irfL. The total reactance of a cir- 
 cuit containing capacity and self-inductance in series bears a 
 complicated relation to the capacity, self-inductance, and fre- 
 
 quency in the circuit, since it is equal to 2 Itis 
 
 -irfl 
 
 zero when, for any reason, 2 n rfL and 
 
 are both equal to 
 
 2 irfC 
 
 zero, which is the condition of a circuit of resistance alone ; or 
 
 whenever 2 7 rfL = , 
 
 J 2 irfO 
 
 that is, when 2nf= — 
 
 Vic 7 
 
 Polygons of impedance may be directly obtained from the 
 polygons of voltages as are shown in Figs. 119, 128, and 133, 
 by dividing each side of the polygon by the current I. Figure 
 134 shows the impedance diagram for a self -inductive circuit; 
 Fig. 135 for a capacity circuit; and Fig. 136 for a series 
 circuit which includes both self-induction and capacity. The 
 capacity effect predominates over the self-inductance in the latter 
 figure, which in this differs from the circuit represented in 
 Fig. 133. Impedance is usually represented in formulas by the 
 letter Z, reactance by the letter X , and admittance by the letter Ti 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 213 
 
 Impedance may be treated as a vector operator ; that is, as a 
 line having definite length and direction. It may be expressed 
 in the complex form in this way, 
 
 Z=R+jX 
 
 and Z - Z( cos 0 +j sin ; 
 
 also Z = V R 2 + X 2 and 0 = tan -1 — • 
 
 R 
 
 Circuit. 
 
 2?r/C 
 
 Fig. 135. — Impedance Triangle for a Capacity Circuit. 
 
 ing both Self-induction and Capacity, Capacity Pre- 
 dominating. 
 
 Since the triangles of impedance are geometrically similar to 
 the corresponding triangles of voltage, the angle 0 is the same 
 in each, i.e., it is the angle of lag in the circuit. If X and 0 
 are negative, capacity predominates, and the imaginary terms 
 become — ■ jX and — j sin 0.* 
 
 * Art. 81. 
 
214 
 
 ALTERNATING CURRENTS 
 
 The combined impedance of a number of circuits in series is 
 Z = (R 1 4- -B 2 + etc. ) j(^X^ 4" X 2 4- etc. ) = ~R 4- j^LX, 
 
 in which the reactances are to be added algebraically ; 
 
 Z = Z(cos 0 +j sin 6 ) 
 
 and 
 
 tan 0 - 
 
 IT 
 
 17? 
 
 Admittance being the reciprocal of impedance, 
 
 1_ 1 _ R —jX R . X 
 
 Z R+jX R 2 + A " 2 R 2 4 X 2 ^ R? + X 2 ' 
 
 Putting g ani b for the horizontal and vertical components, 
 respectively, gives 
 
 Y = 
 
 R . X ., 
 
 t*> . = 
 
 Thus, g = 
 Also, b - 
 
 R 
 
 R 2 + X 2 
 
 R 2 + X 2 ' R 2 +X 2 
 and is called the Conductance of the circuit. 
 
 X 
 
 R 2 + X 2 
 
 and is called the Susceptance of the circuit. 
 g R 
 
 Admittances in parallel can be combined in the same manner 
 as impedances in series. 
 
 Then, 
 
 Assume sinusoidal voltages and currents in the following 
 problems : 
 
 Prob. 1. A circuit has a resistance of 10 ohms and a capacity 
 of 50 microfarads in series. What is the impedance of the cir- 
 cuit when the frequency is 60 periods per second? Solve by 
 the graphical method and by means of complex quantities. 
 
 Prob. 2. A circuit has a resistance of 50 ohms and a capacity 
 of 200 microfarads in series. What is the impedance when the 
 frequency is 25 periods per second? 
 
 Prob. 3. If the resistance of a capacity circuit is 20 ohms 
 and its impedance is 50 ohms, what is the capacity in micro- 
 farads, and what is the angle of lag, the frequency being 120 
 periods per second? Solve graphically and analytically. 
 
 Prob. 4. A circuit has an impedance of 100 ohms. The 
 frequency is 60 periods per second and the angle of lag is 45°. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 215 
 
 Find the resistance and inductance of the circuit by graphics 
 and by vector equations. 
 
 Prob. 5. A circuit has an impedance of 200 ohms. The 
 angle of lag of the circuit is —60° and the frequency is 25 
 periods per second. Find the resistance and capacity by 
 graphics and by vector equations. 
 
 Prob. 6. A circuit having an angle of lag of — 90° and a 
 frequency of 40 periods per second has an impedance of 50 
 ohms. What is the resistance and what is the capacity? Solve 
 by graphics and by vector equations. 
 
 Prob. 7. A circuit in which there is an angle of lag of 90° 
 and a frequency of 40 periods per second has an impedance of 
 100 ohms. Find the resistance and the reactance by graphics 
 and by vector equations. 
 
 Prob. 8. A circuit has a resistance of 10 ohms, a self-induct- 
 ance of .01 of a henry, and a capacity of 50 microfarads in series. 
 If the frequency is 60 periods per second, what is the value of 
 the impedance and of the angle of lag? 
 
 Prob. 9. A circuit has a resistance of 20 ohms, a self-induct- 
 ance of .01 of a henry, and a capacity of 200 microfarads in 
 series. Find the impedance and angle of lag by graphics and 
 vector equations when the frequency is 40 periods per second. 
 
 Prob. 10. A circuit has a resistance of 50 ohms, a self- 
 inductance of .02 of a henry, and a capacity of 20 microfarads 
 in series. Find the impedance and angle of lag by graphics and 
 vector equations when the frequency is 25 periods per second. 
 
 Prob. 11. A circuit has an inappreciable resistance, a self- 
 inductance of .01 of a henry, and a capacity of 100 microfarads 
 in series. Find the impedance and the angle of lag when the 
 frequency is 60 periods per second. 
 
 Prob. 12. Two circuits having self -inductances of .01 and .02 
 of a henry and resistances of 8 ohms and 10 ohms, respectively, 
 are in series. Find their combined impedance and the angle of 
 lag of the circuit when the frequency is 40 periods per second. 
 
 Prob. 13. Two circuits having capacities of 100 and 200 
 microfarads and resistances of 10 and 20 ohms, respectively, are 
 in series. Find the impedance and angle of lag when the fre- 
 quency is 40 periods per second. 
 
216 
 
 ALTERNATING CURRENTS 
 
 Prob. 14. Two circuits have resistances respectively of 10 
 and 15 ohms, the first has a capacity of 100 microfarads and 
 the second a self-inductance of .01 of a henry. If the circuits 
 are connected in series, what is the combined impedance and 
 what is the angle of lag when the frequency is 40 periods per 
 second? 
 
 Prob. 15. If a voltage of 100 volts is impressed upon the 
 joint circuit of problem 14, how much current will flow? 
 
 Prob. 16. If it is desired to cause a current of 10 amperes 
 to flow through the joint circuit of problem 14, what voltage 
 must be impressed? 
 
 Prob. 17. A circuit has a resistance of 10 ohms, a self-induct- 
 ance of .01 of a henry, and a capacity of 200 microfarads all 
 in series, and a voltage of 200 volts with a frequency of 60 
 periods per second is impressed on this circuit. What is the 
 active, and what is the reactive, voltage? Find graphically and 
 by vector equations the current flowing through the circuit. 
 
 Prob. 18. Three coils of respectively 1 ohm, 2 ohms, and 
 3 ohms resistance having self-inductances of .01, .02, and .03 
 of a henry are connected in series. If a current of 10 amperes 
 flows through the series at a frequency of 25 periods per second, 
 what is the voltage between the terminals of each coil, and what 
 is the voltage between the terminals of the series ? 
 
 Prob. 19. Three condensers having internal resistances of 
 8, 10, and 12 ohms, respectively, and capacities of 50, 100, and 
 180 microfarads are connected in series. What is the total 
 voltage impressed on the series and what is the voltage be- 
 tween the terminals of each condenser when the current is 20 
 amperes with a frequency of 60 periods per second ? 
 
 Prob. 20. A circuit of 10 ohms resistance and .01 of a henry 
 self-inductance is in series with a circuit of 8 ohms resistance 
 and 200 microfarads capacity and these in turn are in series 
 with another circuit of 15 ohms resistance, .02 of a henry self- 
 inductance, and 150 microfarads capacity. When a current of 
 5 amperes with a frequency of 40 periods per second is caused 
 to flow through this series, what are the voltage and angle of 
 lag observed at the main terminals, and what are the voltages 
 and angles of lag observed at the terminals of each of the three 
 above-named parts of the total circuit? 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 217 
 
 Prob. 21. The current in a circuit is 50 amperes with a 
 period of one one-hundred-and-twentieth of a second. The cir- 
 cuit contains a positive (inductive) reactance of 20 ohms and 
 a resistance of 20 ohms. What are the impressed voltage and 
 the angle of lag ? 
 
 Prob. 22. Twenty amperes flow through a circuit with an 
 angle of lag of 30°, when the impressed voltage is 100 volts 
 and the frequency 40 periods per second. What are the imped- 
 ance, reactance, and resistance ? 
 
 Prob. 23. A circuit having an impedance of 40 ohms is in 
 series with a circuit having an impedance of 30 ohms. When 
 a voltage of 100 volts with a frequency of 60 periods per second 
 is applied to the two circuits, the angle of lag in the first cir- 
 cuit is 30° and in the second circuit is — 60°. How much 
 current flows through the circuits and what is the angle of lag 
 observed at the main terminals ? 
 
 Prob. 24. Three circuits are in series : the first has a re- 
 sistance of 20 ohms, the second a resistance of 10 ohms and a 
 positive (inductive) reactance of 20 ohms, the third a resistance 
 of 5 ohms and a negative (capacity) reactance of 30 ohms, the 
 frequency being 25 periods per second. What is the angle of 
 lag observed at the main terminals and what voltage is im- 
 pressed between the main terminals when a current of 50 
 amperes flows through the series ? 
 
 Prob. 25. What is the angle of lag and what is the voltage 
 observed between the terminals of each of the three parts of 
 the main circuit of problem 24 ? 
 
 Prob. 26. If a voltage of 500 volts, frequency 25 periods 
 per second, is impressed on the main circuit of problem 24, what 
 current flows and what is the angle of lag ? 
 
 Prob. 27. What is the voltage across each of the three 
 parts of the main circuit of problem 24 under the conditions of 
 current flow fixed by problem 26 ? 
 
 Prob. 28. A circuit of negligible resistance has a positive 
 (inductive) reactance of 10 ohms when the frequency is 60 
 periods per second. What current flows when a voltage of 
 100 volts is impressed on the circuit, and what is the angle 
 of lag ? 
 
218 
 
 ALTERNATING CURRENTS 
 
 Prob. 29. A circuit of negligible resistance has a negative 
 (capacity) reactance of 10 ohms when the frequency is 60 
 periods per second. What is the current when the impressed 
 voltage is 100 volts, and what is the angle of lag ? 
 
 Prob. 30. A circuit of negligible resistance has a positive 
 reactance of 10 ohms and a negative reactance of 10 ohms when 
 the frequency is 60 periods per second. What current flows 
 when 100 volts are impressed on the circuit ? 
 
 Prob. 31. A circuit has a resistance of 10 ohms, a positive 
 reactance of 10 ohms, and a negative reactance of 10 ohms, 
 when the frequency is 60 periods per second. What current 
 flows when a voltage of 100 volts is impressed upon the circuit, 
 and what is the angle of lag ? 
 
 Prob. 32. A circuit of negligible reactance has a resistance 
 of 10 ohms. When a voltage of 100 volts at a frequency of 60 
 periods per second is impressed upon the circuit, what current 
 flows and what is the angle of lag? 
 
 Prob. 33. Find the self-inductance and capacity in each of 
 the circuits given in problems 28 to 32, inclusive. 
 
 64. Application. — The application to circuits in general when 
 the currents and voltages are sinusoidal, and to alternator arma- 
 tures in particular, of the preceding deductions is evident. 
 
 Thus, suppose it is desired to design an alternator which is 
 to generate 25 amperes at an effective voltage of 1000 volts at 
 its terminals when operating on a non-reactive load, the fre- 
 quency being 100 periods per second. Take first, for example, 
 a disk armature without iron in its core, with a resistance of 
 1 ohm and an average self-inductance of .01 henry. The effec- 
 tive value of the total voltage to be developed in this armature 
 at full load is then 
 
 E = E r +jE x , 
 
 E = VA, 2 + E} = _ZV A 2 + (2 t r/A) 2 , 
 
 E = ,V(1000 + 25 x 1) 2 + (2 tt x 100 x .01 x 25) 2 , 
 
 which is equal to 1037 volts. Consequently, the effect of self- 
 inductance is to demand an increase of the total voltage equal 
 to 12 volts. Suppose, however, the armature is of a type hav- 
 ing an iron core and has a resistance of 1 ohm and an average 
 
SELF-INDUCTION, CAPACITY, KEACTANCE, AND IMPEDANCE 219 
 
 working inductance of .05 henry, the total voltage then becomes 
 
 E = V(1000 + 25 x l) 2 + (2 7T x 100 x .05 x 25) 2 , 
 
 which is equal to 1291 volts. Hence, the total voltage must be 
 increased by 266 volts on account of self-inductance. If the 
 two machines were worked at full load upon resistances of 
 absolutely no inductance or capacity, the lags of the currents 
 with respect to the induced voltages in the circuits in the two 
 cases would be respectively 8° 43' and 37° 27' (Fig. 137). 
 
 JE 
 
 These values are obtained from the relation 6 — tan -1 -^- 
 
 On the other hand, 
 in case the load con- 
 tains reactance, the fig- 
 ures are modified. For 
 instance, in case the load 
 contains 40 ohms of im- 
 pedance as before, but 
 in this instance is com- 
 posed of 32 ohms resist- 
 ance and 24 ohms 
 positive (inductive) 
 reactance, the induced 
 voltage required to af- 
 ford 1000 volts at the E _ IR 
 
 terminals of the genera- Fig 137 ._ Voltage Triangle calculated for an 
 tor with the iron-cored Alternator Armature, 
 
 armature is 
 
 E = V[25(l + 32) ] 2 + [25(31.4 + 24)p, 
 
 which is equal to 1612 volts ; while if the reactance of the load 
 is negative instead of positive, 
 
 E= V[25(l + 32)] 2 + [25(31.4 - 24)] 2 , 
 
 which is equal to 845 volts. In each case the terminal voltage 
 of the generator, that is, the voltage impressed on the load, is 
 1000 volts ; but the voltage impressed on the entire circuit, that 
 is, the induced voltage, is affected by the relations of the im- 
 pedance of the load to the impedance of the armature winding. 
 In case the impedance of the load contains negative reactance, 
 
220 
 
 ALTERNATING CURRENTS 
 
 it is to be observed that the terminal voltage may be actually 
 higher than the induced voltage on account of the interaction 
 of the capacity of the load on the inductance of the armature 
 winding. 
 
 Prob. 1. An alternator with terminal voltage of 10,000 volts 
 and frequency of 60 periods per second furnishes 50 amperes to 
 an external circuit. The armature has a resistance of 4 ohms 
 and a self-inductance of .02 of a henry. When the external 
 load is non-reactive, what total voltage must be generated by 
 the machine ? 
 
 Note. — This and the following problems are to be solved by the graphi- 
 cal method and by the use of the complex (vector) equations, neglecting 
 any effects of armature reactions. 
 
 Prob. 2. An alternator generates a total voltage of 5000 
 volts at a frequency of 40 periods per second. The armature 
 has 2 ohms resistance and .03 of a henry inductance. How 
 much current will this alternator deliver to a non-reactive load 
 of 20 ohms, and what is the terminal voltage? Also what is 
 the value of the angle of lag between the current and the ter- 
 minal voltage and the angle of lag between the current and the 
 total generated voltage? 
 
 Prob. 3. The generator of problem 2 is connected to a load 
 containing 10 ohms resistance and 200 microfarads capacity in 
 series. What are the values of current, terminal voltage, and 
 angle of lag between current and terminal voltage? 
 
 Prob. 4. The generator of problem 2 is connected to a load 
 containing 20 ohms resistance and .02 of a henry self-induct- 
 ance in series. What are the values of current, terminal vol- 
 tage, and angle of lag between the current and terminal voltage ? 
 
 Prob. 5. An alternator armature has a resistance of 2 ohms 
 and a self-inductance of .02 of a henry. It produces a termi- 
 nal voltage of 2000 volts at a frequenc} r of 100 periods per 
 second. Twenty-five amperes flow in the circuit, the angle of 
 lag between the current and the induced voltage is 30°. What 
 are the impedance, resistance, reactance, and self-inductance of 
 the external circuit? 
 
 Prob. 6. In problem 5 the angle of lag changes from 30° to 
 — 60° and the current to 50 amperes without changing termi- 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 221 
 
 nal voltage or frequency. What then are the impedance, resist- 
 ance, reactance, and capacity of the external circuit ? 
 
 Prob. 7. The external circuit in problem 5 again changes so 
 that the tangent of the angle of lag is .4652 ; the current 80 
 amperes. What are the impedance, resistance, and reactance of 
 the external circuit? 
 
 Prob. 8. What are the values of induced voltage correspond- 
 ing respectively to the conditions in problems 5, 6, and 7 ; and 
 what are the respective angles of lag between the currents and 
 the induced voltages? 
 
 65. Equivalent Resistance and Reactance. — When there is no 
 iron near an alternating-current circuit, the self-inductance is 
 constant and the triangles of voltage 
 given previously are true for all values 
 of the current, but if iron is present, L 
 varies with the current. In addition to 
 this, if iron is present, hysteresis and 
 eddy currents absorb power. Under a 
 given impressed voltage this power 
 reduces the reactive voltage and in- 
 creases the active voltage, so that instead 
 of leaving the triangle of voltages OAB 
 (Fig. 138) such as would be obtained Fig 138 . — Triangles of Vol- 
 in a certain circuit when a current I is tages with and without Iron 
 flowing and there are no Iron losses Losses ' 
 present, the triangle actually becomes more like OCD. The 
 reactive voltage is reduced and the active voltage is greater in 
 OCD than in OAB , and the angle of lag decreased. The Equiva- 
 lent or Working reactance of the circuit is 
 -P7 E' x Esm 6' 
 
 x = T -— I -. 
 
 ■ The Equivalent or Working resistance is 
 
 E' r E'cos0' 
 
 ~ I I 
 
 These conditions are discussed further in Chapter IX. 
 
 Prob. 1. A circuit has a voltage applied to- it of 100 volts 
 and a current of 10 amperes flows with a lag of 60°. What 
 is the apparent resistance and what is the equivalent reactance 
 of the circuit? 
 
222 
 
 ALTERNATING CURRENTS 
 
 66. General Equation for Current in a Circuit. — The voltage 
 impressed upon any circuit has already been expressed * in the 
 form T ,. 
 
 iK+ “ + $— W 
 
 in which f(f) represents any voltage which may be impressed 
 on the circuit. This general solution therefore includes, when 
 properly interpreted, the various cases already discussed in 
 Arts. 45, 52, and 58. 
 
 Since ^ idt = q , 
 
 (1) may be written 
 
 III di 
 ~L + dt 
 
 e 
 
 L' 
 
 ( 2 ) 
 
 Differentiating this with respect to time to rid the equation of 
 the integral sign, there results : 
 
 d I) 2 i Rdi i _ 1 de _ 1 
 di 2 + LJtLC~lTt~L 
 
 The term — can only have one finite value in an electric cir- 
 dt J 
 
 cuit at any specific time, and is therefore a single-valued function 
 of time, which for convenience is called /'(t). Equation (3) 
 is a linear differential equation of the second order with its 
 second member a function of t.f 
 
 The object is to find the value of current i, in terms of R, L, 
 C, and the voltage. 
 
 The equation may be written 
 
 /'(*)• (3) 
 
 RD IV /'(Q 
 L + LG) L ’ 
 
 (■*) 
 
 where D is a symbol of operation and represents 
 
 *1 1 
 
 dt 
 
 and 
 
 I) 2 means differentiation twice with respect to t. I) may be 
 used as an ordinary algebraic term.J 
 
 j?D 1 
 
 The “auxiliary equation” D 2 + -j- + ^~^, = 0 is a quadratic 
 equation with the two roots, 
 
 * Art. 58. 
 
 t Price’s Calculus, Vol. II, p. 458; Forsyth’s Differential Equations, p. 86; 
 Perry’s Calculus for Engineers, pp. 213-241 ; John Graham, An Elementary 
 Treatise on Calculus, p. 221. 
 
 1 Perry’s Calculus for Engineers, p. 231; Forsyth’s Differential Equations, 
 p. 43 ; Murray’s Differential Equations, Chap. VI. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 223 
 
 and 
 
 A + , 
 
 2 L 
 
 IB? 
 
 1 
 
 \ R i 
 
 J 1R2 
 
 1 -1 
 
 ^4 U 
 
 LC 
 
 _2 L ^ 
 
 MX 2 
 
 
 R y 
 
 \IB? 
 
 1 
 
 \h+' 
 
 J 1 B? 
 
 1 "| 
 
 2 L ^ 
 
 Mi 2 
 
 LC~ 
 
 M X 2 
 
 
 (5) 
 
 ( 6 ) 
 
 Letting a x and a 2 equal the bracket expressions respectively 
 of the second members of equations (5) and (6), then the 
 “ complementary function ” is 
 
 Ae~ a J + Be~ a ?. (7) 
 
 The particular integral is 
 
 1 
 
 z f(t) 
 
 -/'( 0 
 
 7)2 I BI> i JL C-® + (D + « 2 ) 
 
 L LC 
 
 - z /,(f) 
 
 
 ;} 
 
 ( 8 ) 
 
 _I) T rtj D -f- <x 2 . 
 
 in which the third member is obtained by resolving the second 
 member into partial fractions. This solves into the form, 
 
 1 
 
 L 
 
 N x e + (t)dt 
 
 (9) 
 
 The values of N x and iV 2 may be determined from the identi- 
 cal equation,* 
 
 = ^_ + 
 
 2V„ 
 
 whence 
 
 and 
 
 (D -f- a 1 )(i> -)- (1%) B q- $q B -(- ct 2 
 
 + « 2 ) + N 2 (^D + «j) = 1, 
 
 B(N t + N 2 ) = 0, N x a 2 + N 2 a 1 = 1, 
 
 jV,= -V 2 = - 1 1 
 
 ■i 2JIA J _A 
 
 '4 i 2 LC 
 
 (10) 
 
 (11) 
 
 ( 12 ) 
 
 (13) 
 
 The complete solution of equation (3), being the sum of the 
 complementary function and the particular integral, is 
 
 1 
 
 * = 
 
 2 iJT®_ J_ 
 
 V 1 T 2 TO 
 
 — |^e — e e a ~ l f (t)dt 
 
 4 1? LC 
 
 + Ae~° lt + Be _ “ 2< . 
 
 * Wentworth’s Complete Algebra , p. 409. 
 
 ( 14 ) 
 
224 
 
 ALTERNATING CURRENTS 
 
 The coefficient of the first term of the second member of the 
 last equation may also be written 
 
 C 
 
 V_Z2 2 C' 2 — 4 Lc' 
 
 The terms a 1 and a 2 may also be written 
 
 R \Tr? T _RC- CICC 2 -4 L C 
 
 2 L '4 L 2 LC 2 LG 
 
 R ilR 2 T~ RC + VR?C 2 — 4 LC 
 
 2L + ^4 L 2 LC 2 LC 
 
 A similar equation can be obtained in the same way for the 
 charge at any instant in a condenser in the circuit. 
 
 67. Current in a Circuit when Any Periodic Voltage is Im- 
 pressed. — In the case of a circuit with resistance, self-induct- 
 ance, and capacity in series on which any periodic voltage is 
 impressed, the voltage formula is 
 
 e =/(f) = iR 4- L — + — C idt. 
 
 at C* 7 o 
 
 The general solution of this is equation (14), Art. 66. 
 
 This contains too many unknown terms to be of much use to 
 the electrical engineer at present; but if e=f(t') is construed 
 to mean an alternating voltage, it may be expressed in terms of 
 the harmonics. This formula becomes 
 
 e = /( 0 = e m x sin (« + ^ : ) + e„, 3 sin (3 a + # 3 ) + ••• 
 
 + e TOj sin (no. + 5> n ), 
 
 and 
 
 j~ t =/( 0 = "e, ni cos (« + 0j) + 3 cos (3 « + 0 g ) ••• 
 
 + ncoe m cos ( na + ^„). 
 
 Substituting this value of f(t) in the general equation for 
 current developed in Art. 66, there results an equation in which 
 there is a pair of similar terms for each harmonic. These may 
 evidently be integrated by parts and the result is as follows : 
 
 sin (a + tfj) 
 
 i = 
 
SELF-INDUCTION, CAPACITY, KEACTANCE, AND IMPEDANCE 225 
 
 e m 
 
 H ■■ ■■ 3 — sin (3 u + do) 
 
 + 
 
 H n - sin (na + 0 ) 
 
 + Ae~ aii + Be~ a *. 
 
 The angle 0 n is determined by the relation 
 
 tan 0 n 
 
 2 t rf nL 1 = X n 
 
 I t 2 irfnCR II ' 
 
 Each of the terms of the right-hand member, from the first to 
 the wth, has the form of a sine function. 
 
 The formula may be more conveniently written 
 
 i = — 1 sin (« + 6 X ) + — 'sin (3 a + 0 3 ) ... 
 
 z x z 3 
 
 + >sin (na + 0 n ), 
 
 where Z v Z 3 , ••• Z H each represents the impedance which the 
 circuit offers to the corresponding current harmonic. The 
 exponential terms are omitted, as they ordinarily disappear 
 quickly after the current is started ; and the formula repre- 
 sents the conditions in the circuit after a permanent state is- 
 established. It will be noted that the impedances for the several 
 harmonics may have very different values even though they are 
 all affected by the same resistance, self-inductance, and capacity. 
 This is due to the frequency of the harmonic entering into the 
 reactance component of the impedance, as shown by the formula. 
 Each term of the second member represents the instantaneous 
 
 value at the instant t,(^ = — of a sine wave current. A similar 
 
 formula may obviously be derived to represent the condition 
 when even-numbered harmonics as well as odd-numbered har- 
 monics appear in the voltage wave. 
 
 The effective value of current is found by taking the square 
 root of the sum of the squares of the effective values of the 
 
226 
 
 ALTERNATING CURRENTS 
 
 harmonic currents given in the last expression, or 
 
 i=y/r, + r,+ -p,* 
 
 ^vfVtc. 
 
 Prob. 1. A certain voltage wave is composed of two har- 
 monics, the fundamental having an effective value of 150 volts 
 and that of three times the frequency having an effective value 
 of 25 volts. What is the effective value of the resultant voltage? 
 
 Prob. 2. The impedance offered by a circuit to the primary 
 harmonic of a certain voltage is 20 ohms, the effective value of 
 the harmonic being 100 volts ; the remaining harmonic of the 
 voltage is of five times the primary frequency, has an effective 
 value of 10 volts, and overcomes an impedance of 30 ohms. 
 Find the effective values of the voltage and current. 
 
 Prob. 3. A current is composed of two harmonics, a primary 
 and one of three times the frequency. The first has an effective 
 value of 50 amperes and flows through an impedance of 50 ohms. 
 The other has an effective value of 20 amperes and flows through 
 an impedance of 40 ohms. What is the effective value of the 
 voltage impressed on the circuit ? 
 
 68. Irregular Current and Voltage Waves expressed as Com- 
 plex Quantities. — A Fourier’s series may be used to represent 
 irregular current and voltage by means of component sine waves, 
 as pointed out in Art. 67. The effective value of such a vol- 
 tage is found from the relation, 
 
 E 2 = E\ + E\ + etc. + E\ + E 2 X3 + etc.,* 
 
 and E = V. E\ + E\ + etc. + E\ + E 2 Xs + etc., 
 
 where E r and E x are active and reactive effective voltages for 
 the various harmonics. This may be written : 
 
 E = V(^ 2 ri + E\~) + (. E\ + E 2 X3 ) + etc. 
 
 and the term within each bracket is evidently the square of 
 the scalar value of the corresponding sine voltage. A non- 
 sinusoidal alternating curve may be empirically indicated by 
 the expression 
 
 (-£>,, E ra , etc.) +j(E Xi , E X2 , E Xs , etc.); 
 
 * Art. 16. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 227 
 
 which is intended to convey the idea that each harmonic of the 
 voltage curve acts individually to set up a current of the same 
 frequency in the circuit, as though the other harmonics were 
 not present. The square root of the sum of the squares of the 
 effective values of the harmonic voltages, as already pointed out, 
 gives the effective value of the total voltage impressed on the 
 circuit. In the same way, the square root of the sum of the 
 squares of the effective values of the individual current har- 
 monics caused to flow in the circuit by the harmonics of voltage 
 gives the effective value of the actual current flowing. 
 
 The irregular current relations are similar to those given for 
 voltage. 
 
 Triangles of voltages for each harmonic may be drawn, and 
 if the corresponding harmonic of current is divided into the 
 value of each side, an impedance triangle results. Thus, 
 
 represents the impedance offered to the flow of the harmonic 
 of primary frequency ; 
 
 = R + jX 3 
 
 represents the impedance offered to the flow of the harmonic 
 of three times the frequency, etc. 
 
 In obtaining the reactances for the different harmonics, care 
 must be taken to use the proper frequency in the formula, 
 
 X=2t rfL, or X = 
 
 1 
 
 2 7 rfO' 
 
 Thus, if the resistance in a circuit is 10 ohms and the reactance 
 for the primary harmonic is 5, there results 
 
 Zj = 10 +j 5 ; Z 3 = 10 +/15 ; Z 5 = 10 +j 25, etc., 
 
 when the voltage and current contain harmonics of odd fre- 
 quencies and the reactance is inductive. If the reactance 
 results from capacity, the expression is 
 
 Z 1 = 10— /5; Z 3 = 10— if; Z 5 = 10 -j f , etc. 
 
 It will be noticed that the resistance is the same for all 
 harmonics. 
 
228 
 
 ALTERNATING CURRENTS 
 
 If the empirical expression for current 
 
 C^i’ I g ^ etc.) + j ( 1 j, i . Jf , 3 , etc.), 
 
 where and _ZJ, are the rectangular components of current, has 
 its several harmonic values divided by the respective harmonic 
 values of the voltage, the admittances which the circuit offers 
 to each current harmonic may be derived. Thus 
 
 where g and b represent the rectangular components of the 
 admittances. r r 
 
 is the admittance of the circuit for the current harmonic of 
 primary frequency, 
 
 is the admittance for the current harmonic of three times the 
 primary frequency, etc. 
 
 Vector voltage may obviously be multiplied by admittance 
 or divided by impedance to give vector current; or vector cur- 
 rent may be divided by admittance or multiplied by impedance 
 to give vector voltage.* This process gives the relations of 
 the equivalent sinusoids representing irregular curves of voltage 
 and current. The total vector impedance and admittance may 
 be written empirically 
 
 Z i -)- Z<£ T Zq -f- etc. 
 
 = (Ri + R<i + Rg + etc.) 4 - j (A j + A 2 + Ag + etc.) 
 
 T) + Vij + 1 3 + etc. 
 
 = ( 9 i + 92 + 9 s + etc.) -3 i b i + h + b 3 + etc -) 
 
 [(i^ + R 2 + R 3 + etc.) 2 + (aq + x 2 + x 3 + etc.) 2 ]*, 
 and [(#! + 92+93 + etc.) 2 + ( b i + b 2 + b s + etc.) 2 ] -. 
 
 The process of solving problems involving irregular curves 
 where the harmonic components are known is similar to those 
 
 — (, 9 v 92' etc.) -f f'3, etc.), 
 
 * Art. 63. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 229 
 
 already disclosed for sinusoidal voltages or currents, but each 
 sinusoidal component of voltage or current must be dealt with 
 as if acting alone in the circuit, and then the results may be 
 combined to obtain the total values according to the theorem 
 that the effective value of any single-valued periodic curve is 
 equal to the square root of the sum of the squares of the effec- 
 tive values of its harmonics. It should be noted that the 
 impedance obtained by dividing the total voltage by the total 
 current is dependent upon the shapes of the curves of voltage and 
 current on account of the fact that fixed values of self-induct- 
 ance and capacity do not have the same effect upon harmonics 
 of different frequencies. 
 
 Prob. 1. The voltage impressed on a circuit is composed of 
 two harmonics. The fundamental has an active component of 
 190 volts and a reactive component of —50 volts, while the 
 other harmonic has three times the frequency, an active com- 
 ponent of 25 volts, and a reactive component of 20 volts. 
 What is the value of the voltage ? 
 
 Prob. 2. A circuit contains a resistance of 10 ohms and self- 
 inductance of .01 henry. A current with a frequency of 60 
 periods per second flows in this circuit. It is composed of a 
 fundamental harmonic of 100 amperes and a third harmonic of 20 
 amperes. What voltage is impressed upon the circuit ? (Aid : 
 Multiply each harmonic current by the corresponding complex 
 expression for impedance to obtain the voltage harmonics.) 
 
 Prob. 3. What impedance is offered by a circuit in which 
 voltage J5 r = (100 1 , 10 3 ) + y(50 1 , 15 3 ) is impressed and current 
 I — (50 x , 5 g ) +/(40 x , 10 3 ) flows? (Aid: First find effective 
 voltage and effective current.) 
 
 69. Variation in the Impedance offered to Current and Vol- 
 tage Harmonics of Different Frequencies. — Following out the 
 remarks of the last paragraph of the preceding article, it is to 
 be observed that, since the reactance of a reactive circuit is 
 dependent on the frequency of the current flowing, it is obvious 
 that the impedance interposed by a circuit to the different 
 harmonics of a current may differ for the several harmonics. 
 Inductive reactance (2irfL') varies directly with frequency; 
 
 capacity reactance 
 
 varies inversely with frequency ; and 
 
230 
 
 ALTERNATING CURRENTS 
 
 the reactance of a circuit containing both inductance and capac- 
 ity ( 2 7 rfL ~~-—— 
 
 J V 2 t rfC 
 
 and may increase or decrease as frequency increases, depending 
 upon the relations of the inductive and capacity parts of the 
 reactance. 
 
 Under these circumstances, when an irregular alternating 
 voltage is impressed on a circuit, the harmonics of the current 
 which flow will not bear a uniform ratio to the voltage har- 
 monies unless the circuit is non-reactive. Assuming the self- 
 inductance and capacity to be uniform, 
 
 I 1 — ^ + I* + ••• + 
 
 in which J i = § 2 ’ ^ = etc - 
 
 But in a series circuit 
 
 z*-* + (s^z- s £J)\ 
 
 Zl=B> + (2„ A L-^)\ 
 
 ^ is a more complex function of frequency 
 
 Zl = & + (2*f„L- 
 
 in which /p/g, etc ,,f n represent the frequencies of the several 
 harmonics, the numerical value of the subscript showing the 
 number of times to multiply the fundamental frequency to 
 arrive at the particular frequency under consideration. 
 
 Scrutinizing these equations brings out a number of impor- 
 tant facts. 
 
 1. Only when reactance is negligible or the voltage sinus- 
 oidal can the current flowing in a circuit have the same wave 
 form as the impressed voltage which sets it up. 
 
 2. If the capacity reactance is negligible, as it ordinarily is 
 in the instance of a closed metallic circuit of relatively short 
 length, inductive reactance only is to be considered. In this 
 instance it will be observed that the reactance opposed to each 
 current harmonic is directly proportional to the frequency of 
 the harmonic and the impedance is larger for the harmonics of 
 higher frequency than for those of lower frequency. Under 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 231 
 
 these circumstances, the current produced by an irregular im- 
 pressed voltage will be less irregular than the voltage produc- 
 ing it ; that is, the inductive reactance has the effect of reducing 
 the amplitude of the higher harmonics, and thus reducing the 
 ripples or irregularities on the current curve. The lag angles, 
 8 V 6 V etc., of the several current harmonics with respect to the 
 corresponding voltage harmonics also differ from each other 
 since R is unaffected by frequency, which additionally alters 
 the form of the current wave compared with the voltage wave. 
 The tangent of the angle of lag between each harmonic of 
 current and its corresponding harmonic of voltage is obviously 
 proportional to the frequency of the particular harmonic cur- 
 rent, in this instance. 
 
 The above conditions are illustrated in Fig. 139. The upper 
 curves show, in a full line the voltage wave E, and in broken 
 lines its component harmonic waves, which are the fundamental, 
 the third, and the fifth harmonics. This voltage wave is im- 
 pressed on a circuit containing resistance and inductive react- 
 ance ; namely, 10 ohms resistance and ,01 henry self-inductance, 
 the fundamental frequency being 60 cycles per second. The 
 lower curves in Fig. 139 indicate in broken lines the harmonics 
 of current corresponding to the harmonics of voltage, while the 
 resultant current is shown by a full line. The impedance in- 
 creases with the frequency, that encountered by the first har- 
 monic being 10.7 ohms, by the third 15.1 ohms, and by the 
 fifth 21.4 ohms. It will also be observed that the resultant 
 wave of current is less irregular than the voltage wave produc- 
 ing it, showing the effect of inductance in a circuit in smooth- 
 ing out the irregularities in the current waves and dampening 
 the effects of the higher harmonics, provided the inductance is 
 fixed in value. 
 
 3. If the inductive reactance is negligible, capacity reac- 
 tance only is to be considered. In this instance the reactance 
 encountered by each harmonic of current is inversely propor- 
 tional to the frequency of the harmonic, and the impedance 
 encountered by each harmonic is therefore smaller as the 
 frequency of the harmonic is larger. Consequently, the cur- 
 rent produced in a condenser by an irregular impressed voltage 
 is more irregular than the impressed voltage. Each harmonic 
 of the voltage is represented by an harmonic of current, and 
 
282 
 
 ALTERNATING CURRENTS 
 
 the higher harmonics are exaggerated in comparison with the 
 fundamental. In the instance of a condenser without resist- 
 
 ance, the relative amplitudes of the higher harmonics of current 
 increase over the relative amplitudes of the harmonics of im- 
 pressed voltage in direct proportion to their several frequencies. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 233 
 
 The lag angles, 6 V d 2 , etc., of the several current harmonics 
 with respect to the corresponding voltage harmonics, are obvi- 
 ously smaller as the frequencies of the harmonics are greater. 
 The tangent of each angle of lag is inversely proportional to 
 the frequency of the particular harmonic. 
 
 These conditions of a circuit containing capacity reactance 
 are illustrated in Fig. 140. In this case the same voltage as 
 
 that shown in Fig. 139 is impressed on a circuit containing a 
 resistance of 10 ohms and a capacity of 100 microfarads. The 
 harmonics of current corresponding to the harmonics of voltage 
 are indicated in Fig. 140 by broken lines, and the resultant cur- 
 rent is shown by a full line. The impedance encountered by 
 the first harmonic is 28.4 ohms, by the third 13.4 ohms, and 
 by the fifth 11.3 ohms, while the respective angles of lag for the 
 current harmonics are 9 X = 69° 22' ; d 3 = 41° 30' ; = 27° 57'. 
 
234 
 
 ALTERNATING CURRENTS 
 
 It is observed that the resultant current is more irregular than 
 the voltage producing it, showing that the effect of capacity in 
 a circuit is to exaggerate the higher harmonics and distort the 
 resultant wave. 
 
 4. When the circuit contains both capacity and self-induct- 
 ance, these two factors of the reactance vary differently with 
 frequency and the effects on the current harmonics are not so 
 
 represent the reactance, it will be observed : that very low fre- 
 quencies, in general, make the first term negligible compared with 
 the second; that very high frequencies make the second term neg- 
 ligible compared with the first ; and that the two terms are 
 equal to each other and the reactance reduces to zero at some 
 intermediate frequency. Whatever may be the particular nu- 
 merical values of L and C in any instance, some frequency will 
 
 make 2 7 rfL = - — — • This frequency is, plainly, f — ^ 
 
 2 TrfC 1 * >' J 2-kVLC 
 
 A higher frequency makes 2 irfL > 
 
 2-rrfC 
 
 , and a lower fre- 
 
 quency makes 2 tt/A<- — — — • The condition in which 2 irfL 
 
 1 TTJ C 
 
 = in which X — 0 and Z — R, is called the condition of 
 
 2 TTJ C 
 
 Resonance, in analogy with the tuning of a wire or a windpipe 
 by adjusting the inertia and elasticity of the vibrating medium 
 so as to afford a maximum natural amplitude of vibration at a 
 particular frequency of disturbance. 
 
 Applying the above considerations to the circuit conditions 
 wheti, for instance, the numerical values of L and C make 
 
 2 7 rfL < o y -y for the fundamental frequency of the impressed 
 
 voltage, it is to be readily seen that the impedance encountered 
 by higher harmonics of current will decrease up to a harmonic of 
 
 the frequency at which 2 irfL = anc ^ the impedance over- 
 
 come by the harmonic of that particular frequency is equal to 
 the resistance of the circuit. Current harmonics of still higher 
 frequency encounter impedance which is again greater than R 
 and increases with the frequency. The angles of lagf for the 
 
236 
 
 ALTERNATING CURRENTS 
 
 various harmonics, as the frequencies increase, change from 
 negative values through zero to positive values. In conse- 
 quence of these relations, the form of the current wave in a cir- 
 cuit of this character may be greatly distorted from the form of 
 the wave of impressed voltage, through the exaggeration of a 
 certain harmonic or harmonics and the shifting of the relative 
 positions of the higher harmonics of current with respect to 
 corresponding voltage harmonics. These conditions are illus- 
 trated in Fig. 141. The upper curves show a voltage wave 
 in a full line and its harmonics in broken lines. It is observed 
 that the harmonics are the first, third, fifth, seventh, and ninth. 
 This voltage wave is impressed on a circuit of .1 ohm resistance, 
 .3 henry self-inductance and 1 microfarad capacity in series. 
 The fundamental frequency is 60 periods per second. The 
 impedance encountered by the fundamental (first) harmonic of 
 current is 2539.4 ohms; the impedance encountered by the 
 third harmonic is 544.9 ohms ; by the fifth harmonic is 35 ohms, 
 as at this frequency the condition of resonance is almost at- 
 tained; the impedance encountered the seventh harmonic is 
 412.8 ohms; and the impedance encountered by the ninth har- 
 monic of current is 725 ohms. The respective angles of lag 
 are each nearly 90°, being negative for the first two harmonics 
 and positive for the others. The remarkable exaggeration of 
 the fifth current harmonic caused by resonance illustrates 
 clearly the circumstances here discussed. 
 
 Looking upon the foregoing considerations, it is to be 
 observed that the form and magnitude of the voltage wave 
 have been assumed, and the conditions of current flow have been 
 derived. Converse^, in case a current of particular form and 
 magnitude is imposed on a circuit, the form and amplitude of 
 the voltage observed at the terminals of the circuit or any 
 part thereof are dependent upon the current harmonics and 
 the constants of the circuit. In these cases it is entirely pos- 
 sible to have high, and possibly injurious, voltages set up in 
 parts of the circuit, although the effective value of the voltage 
 at the main circuit terminals is within ordinary bounds. 
 
 Prob. 1. A certain circuit has a self-inductance of .01 of a 
 henry and a resistance of 10 ohms, the voltage impressed upon 
 this circuit has a frequency of 60 periods per second. The 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 237 
 
 voltage is composed of the first four harmonics having the 
 respective effective values of 100, 75, 50, and 25 volts. What 
 current flows under influence of each one of these harmonics, and 
 what is the impedance opposed to the current of each harmonic 5 
 What is the effective value of the resultant current ? 
 
 Prob- 2. The voltage wave of problem 1 is impressed upon a 
 circuit of 10 ohms resistance and 200 ^microfarads capacity. 
 Answer the questions asked in problem 1. 
 
 Prob. 3. The voltage wave of problem 1 is impressed on a 
 circuit of .02 henry self-inductance and 50 microfarads capac- 
 ity. Answer the questions asked in problem 1. 
 
 Prob. 4. Answer problems 1, 2, and 3 when the frequency 
 is changed to 120 cycles per second without changing the shape 
 of the voltage wave. 
 
 Prob. 5. Answer problems 1, 2, and 3 when the frequency is 
 changed to 25 cycles per second without changing the shape of 
 the voltage wave. 
 
 Fig. 142. — Current Locus in Series Circuit with Constant 
 Reactance and Varied Resistance. 
 
 70. The Envel- 
 ope of the Current 
 Vector when the 
 Conditions in Circuit 
 Vary. — (a) A con- 
 sideration of the 
 effects of the varia- 
 tion of the con- 
 stants in a series 
 circuit gives rise to 
 the following theorems for the locus of the current vector 
 when the voltage and frequency are fixed. 
 
 Case (1). Reactance Constant and Resistance Varied. — In 
 Fig. 142 let OR represent the impressed voltage R and OC 
 
 JE 
 
 represent the current J 0 = — when R = 0, in which case it is 
 
 X 
 
 obvious that OC must lag (or lead) OR by 90°. Let OA rep- 
 resent the current I for some value of R between zero and 
 infinity. 
 
 Now r=|and J 0 = |- 
 
238 
 
 ALTERNATING CURRENTS 
 
 Hence, 
 
 But 
 
 IZ = I 0 X and I = I 0 
 
 X 
 
 Z 
 
 X 
 
 Z 
 
 OB 
 
 OA 
 
 OA 
 
 00 
 
 - cos 6. 
 
 Hence, 
 
 I = I 0 cos 6, 
 
 E 
 
 which is the equation of a circle whose diameter is I C) — — , and 
 
 X 
 
 the circle is the locus of the current vector when the reactance 
 is constant and the resistance varies. 
 
 A consideration of the limits in the variation of R shows, 
 
 as above stated, that when R = 0, I— 
 
 E_ 
 
 X' 
 
 and when R = oo , 
 
 1= 0; and thus when R is positive, the current vector is 
 limited to the semicircle OAC. Now suppose a negative resist- 
 ance is introduced in the circuit, — R varying from 0 to go . 
 Negative resistance may he physically considered as the ratio 
 of a voltage to the current flowing, where the voltage may be 
 that of a generator, transformer or other means by which me- 
 chanical, chemical or other energy is transformed into electrical 
 energy, instead of electrical energy being transformed into heat 
 or mechanical energy as in ordinary resistance. The current 
 must have a component in opposition to the circuit fall of 
 potential and thus lies in the lower semicircle ODC. 
 
 E 
 
 It will he observed that the diameter OC= I 0 = — will fall 
 
 A 
 
 c' 90° to the right or 
 
 C" 
 
 Fig. 143. — Current Locus in Series Circuit with Constant 
 Resistance and Varied Reactance. 
 
 left of the voltage 
 vector OE according 
 as inductive react- 
 ance or capacity re- 
 actance p r e d o m i- 
 nates. Thus in Fig. 
 142 the circle to the 
 right of the voltage 
 vector represents 
 conditions when in- 
 ductive reactance 
 
 predominates, while the circle to the left is the locus of the 
 current when capacity reactance prevails. 
 
SELF-INDUCTION, CAPACITY, REACTANCE, AND IMPEDANCE 239 
 
 Case (2). Resistance Constant and Reactance Varied. — As 
 before, let OR , Fig. 143, represent the impressed voltage and let 
 
 E 
 
 OC represent the current I 0 = — when the reactance X = 0, in 
 
 R 
 
 which case it will obviously be in phase with OR since there 
 is only pure resistance in the circuit. Let OA' represent the 
 current for some value of X between 0 and infinity. Evi- 
 dently when „ 
 
 Now 
 
 Hence, 
 
 But 
 
 Hence, 
 
 X=cx> ,1= 0. 
 
 T R ij R 
 
 I= Z* niI °=R 
 
 1Z = I 0 R and 1=1 A 
 Zj 
 
 R 
 
 Z 
 
 OB' OA' 
 
 OA' OC 
 1= I 0 cos 6', 
 
 = cos O'. 
 
 Following the reasoning of Case (1), when R is considered 
 as positive and X is positive, the semicircle OA' C is the limit 
 of the current vector, and when X is negative the semicircle 
 OA" C is the limit of the vector. When R is considered as 
 negative, the current becomes opposed to the voltage and com- 
 prises a component 180° from it, whence the circle OB' C" I)" 
 is the limit of the current vector. As in Case (1) the current 
 vector will fall to the right or left of the voltage OR by an 
 angle O' of lag or lead, according as inductive or capacity 
 reactance predominates. 
 
 ( b ) The converse of the above cases, namely, when the resist- 
 ance and reactance are constant but the voltage and frequency 
 vary in turn, indicates the following : 
 
 Case (3). Voltage fi-xed and Rrequency Varied. — Any varia- 
 tion in the frequenc} r must cause a variation in the value of 
 the reactance according to the value of 
 
240 
 
 ALTERNATING CURRENTS 
 
 The effect of variation of the reactance, however, has already 
 been shown in Case (2), and the deductions of Case (2 ) apply 
 to this case. 
 
 Case (4). Frequency fixed and Voltage Varied. — Under these 
 conditions it is obvious that the current vector is fixed in phase 
 according to the constants of the circuit, but its magnitude 
 increases or decreases as the voltage varies. 
 
 Case (5). Voltage and Frequency varied proportionally to- 
 gether. — In this case, if the reactance is purely inductive, the rise 
 of voltage matches the increase in reactance, and the proportion 
 of the total volts absorbed in the reactance per ampere of cur- 
 rent is unaffected, hut the proportion of the total volts absorbed 
 in the resistance per ampere of current by the IR drop varies 
 inversely with the voltage. The effect is therefore like Case 
 (1), in which the reactance is constant and the resistance varies. 
 When voltage is zero, frequency and reactance are zero, and 
 
 Z=-^- = 0, which gives the same result as I— — = 0 and corre- 
 
 sponds to the point 0 of Fig. 142. When voltage increases toward 
 the limit of infinity, frequency and reactance increase toward 
 the same limit and R becomes of negligible influence on the 
 impedance. This condition, therefore, gives the same result as 
 E 
 
 I 0 = - — — and corresponds to the point C in Fig. 142. The 
 
 two circles of Fig. 142 correspond to F positive and 2 7 t/X 
 positive for the upper half of the right-hand circle, F negative 
 and 2 irfL positive for the lower half of the right-hand circle, 
 E positive and 2 71 fiL negative for the upper half of the left- 
 hand circle, and E negative and 2 7 rfL negative for the lower 
 half of the left-hand circle. The negative values for 2 irfL 
 may be secured by substituting the effects of mutual induction 
 in the place of self-induction. 
 
 When capacity is associated with resistance either alone or 
 in company with inductance, the reactance varies in a different 
 relation with the frequency, and the relationships of the current 
 vector are less simple to illustrate. 
 
CHAPTER V 
 
 THE USE OF COMPLEX QUANTITIES EXTENDED 
 
 71. Vector Analysis and the Complex Quantity. — It must be 
 continually borne in mind that as far as mathematics are useful 
 to the engineer, or for developing the applications and extend- 
 ing our knowledge of discoveries in science, they must be con- 
 sidered as a system of logic, which may be conveniently applied 
 to the premises which we possess for the purpose of disclosing 
 or predetermining additional phenomena. When they are 
 used in this manner, it is necessary for us to obtain adequate 
 physical conceptions of the results of all mathematical trans- 
 formations. Our ordinary systems of mathematics are based 
 upon certain assumed premises called axioms, and it is clear 
 that if, by abandoning these axioms and accepting other and 
 different premises which we may or may not call axioms, we 
 can thereby obtain a more convenient and satisfactory system 
 of logic to apply to our ends, then we are justified in abandon- 
 ing the old and adopting the new or of adding the new to the 
 old. This is what we do to a certain extent when we add the 
 processes of vector analysis to the processes of Euclidian and 
 Cartesian geometry for the solution of alternating-current 
 problems. We do an analogous thing when we abandon the 
 ordinary geometry and adopt Hamilton’s quaternion methods 
 or Grasmann's powerful space analysis for treating physical 
 problems involving vectors in space. It is just as irrational 
 for us to confine our attention to the mathematics of a line, as 
 is done in ordinary algebra, when we can usefully extend our 
 consideration to an entire plane or to space, as it was absurd 
 for the early algebraists to neglect all but positive roots in 
 their solutions of equations. The old-time axioms might still 
 be conclusive if viewed within the narrow horizon of Euclid’s 
 time, but in our wider horizon they extend to but a small por- 
 tion of the range of our observations and we can properly 
 extend our methods of reasoning to correspond with the 
 borders of our knowledge. 
 
 o 
 
 k 241 
 
212 
 
 ALTERNATING CURRENTS 
 
 Until the engineer has learned that the place of mathematics 
 in his professional training is no other than that of a system 
 of logic and therefore a tool, it is unsafe for him to use anv 
 form of mathematics higher than arithmetic and the elements 
 of ordinary algebra ; but when it is recognized that mathe- 
 matics compose a system of logic and, indeed, a system of the 
 most powerful kind which may be used in improving engineer- 
 ing processes, the propriety of the use of various branches of 
 higher mathematics in engineering studies is manifest. 
 
 In Art. 6 the following forms and conditions have been given 
 for the vector or complex quantity : 
 
 A = a + jb, 
 
 A — ra( cos 0 + j sin 0) = me 30 ,* 
 
 where m is the Tensor and 0 the Argument of the vector 
 quantity and a and b are the lengths of rectangular com- 
 ponents of the vector. Also, 
 
 m 2 = a 2 + b 2 
 
 , , b sin 0 
 
 a cos 0 
 
 In these expressions the scalar values of A and m are 
 equal, hut the former is a vector having direction, while m 
 may be considered as a simple number. The expression 
 (cos 8 4 - j sin 6) = e j0 is sometimes called the Direction coeffi- 
 cient or Versor of the expression of which it is a part. It may 
 be abbreviated into (cjs) 0. The tensor (also sometimes called 
 the modulus) of the vector represents its actual numerical 
 value and it is therefore a “ scalar ” quantity ; that is, it is a 
 quantity which has numerical or scale value only, without fixed 
 direction. It is by reason of the latter fact that it is some- 
 times called the “ absolute value ” of the vector. The versor 
 of a vector has direction fixed by the argument 8 , but is of 
 unit length only. The product of versor by tensor therefore 
 fixes the vector in terms of both length and direction. 
 
 The operator j and the coplaner-vector quantities treated in 
 this hook are assumed to be subject to the usual algebraic laws, 
 
 * See Williamson’s Differential Calculus, pp. 60-69. 
 
THE USE OF COMPLEX QUANTITIES EXTENDED 243 
 
 like the law of signs, the law of indices, and, in general, the 
 commutative, associative, and distributive laws.* 
 
 Such characteristics have been given the operator / that the 
 imaginary quantity V — 1 may be substituted for it, and the 
 equation 
 
 a -\-jb = a + V — 1 b 
 
 holds true; for / 2 = — 1 by definition. Hence / = V— 1. 1 
 The symbol V— 1 may therefore be considered to have a physi- 
 cal meaning like that of the operator/, and if it is met with in 
 the roots of an equation or elsewhere, it may be replaced by /. 
 When interpreted in this way, it is to be observed that an 
 “ imaginary ” expression assumes its true significance of a vec- 
 tor quantity having a numerical magnitude and definite direc- 
 tion. The real roots of an equation or any other “real” 
 expression may similarly be considered as vector quantities, all 
 of which lie on the same line which is at right angles to the 
 axis of “imaginaries.” 
 
 Since / 2 =— 1, — / 2 =+l, from which follows the relation 
 ^ = — / ; or, in words, the effect of applying the reciprocal of 
 
 the operator to a vector quantity is the same as the effect of 
 applying the reversed operator. 
 
 In general it must be remembered that any direction is rep- 
 resented equally by d, 0 + 2 7r, 6 + 4 tt, etc., to an infinite 
 number of angles, so that the argument of the vector has many 
 values, except where the conditions specially fix it. We shall, 
 as a rule, consider it for convenience as equal to 0 even where 
 it is not fixed by the conditions; but where distinct generaliza- 
 tion of the argument is desirable, we shall write it 0 + 2 7 rn. 
 The complex quantity which has been developed lies in one 
 plane and its particularly marked simplicity is a result of its 
 uniplaner condition. 
 
 Coplaner (uniplaner) vectors, with their corresponding com- 
 plex representations, are quite useful in representing the rela- 
 tions arising in problems which are caused by current flow in 
 alternating-current circuits. These problems deal with periodic 
 quantities which may be treated as though they were the result 
 
 * For a discussion of these laws in relation to vector algebra, see Hayward’s 
 Vector Algebra and Trigonometry , pp. 2, 14, and 39. 
 
 t Art. 6. 
 
244 
 
 ALTERNATING CURRENTS 
 
 of uniform rotating vectors, or stationary vectors in a plane 
 which itself rotates about a perpendicular axis. The methods 
 used in vector algebra are analogous to those which are so 
 largely used in the graphical solutions relating to the composi- 
 tion and resolution of forces in graphical statics and to the 
 composition and resolution of velocities and moving forces in 
 graphical dynamics. The alternating-current vectors are mov- 
 ing vectors, and, consequently, the methods must be similar to 
 those used in the vector processes applied to dynamics. 
 
 72. Addition and Subtraction of Complex Expressions. — Ad- 
 dition and subtraction of these expressions, as has already been 
 indicated,* is accomplished as in ordinary algebraic operations; 
 as, for instance, 
 
 O +jb) + 0 —jd) = a + c +j (b - d), 
 
 (a +jb) - (c —jd) = a — c +j(b + d). 
 
 The rules for addition are, — reals must be added numerically 
 to reals, and imaginaries must be added numerically to imagi- 
 naries. Whence, when addition is performed, (1) the horizontal 
 component of the resulting vector is equal to the sum of the 
 horizontal components of the vectors added; (2) the vertical 
 component of the resulting vector is equal to the sum of the 
 vertical components of the vectors added. 
 
 Subtraction being the reverse process gives: (1) the hori- 
 zontal component of the resulting vector is equal to the differ- 
 ence between the horizontal components of the minuend and 
 subtrahend ; (2) the vertical component of the resulting vector 
 is equal to the difference between the vertical components of 
 the minuend and subtrahend. 
 
 In case an expression has only a horizontal or only a vertical 
 component, then, of course, the addition or subtraction is carried 
 on as though the lacking component were equal to zero. 
 
 73. Multiplication and Division of Complex Expressions. — 
 The process of multiplication or division of a complex by a 
 scalar quantity (that is, a quantity which has only a numerical 
 value) is similar to the corresponding ordinary arithmetical 
 process of multiplication or division. For instance, 
 
 (a +jb) x 2 is 2 (a +jb) = 2 a + j 25 
 and (a +jb) -r2is|(a +jb) = | a +j b. 
 
 * Art. 8. 
 
THE USE OF COMPLEX QUANTITIES EXTENDED 245 
 
 A.' 
 
 k/ 
 
 i 
 
 kV i 
 
 
 <— 1_ _J 
 
 
 This is illustrated in Fig. 144 where OA is a vector. If this 
 vector is multiplied by 2, its direction is not changed, but its 
 length is doubled and the vector becomes 
 OA'. The proportions of the vector show 
 plainly that the horizontal component of 
 OA' is equal to 2 a and the vertical com- 
 ponent is equal to/2 6. If OA is divided 
 by 2, it becomes a vector of | the length 
 and the same direction as before, and is 
 
 represented by the vector 0A 1 . The pro- q a X 
 
 portions of the vector again show clearly Fig. 144.— Graphical Rep- 
 that the horizontal component now is Quantity multiplied or 
 equal to \a and the vertical component divided by a Scalar Quan- 
 is equal to j\ b. tlty- 
 
 Algebraical processes also immediately apply in the multipli- 
 cation or division of complex quantities by complex quantities, 
 as will be more fully illustrated. W e will suppose that the two 
 complex quantities a + jb and c + jd are to be multiplied 
 together and that these quantities are equal, respectively, to 
 m x (cos 6 X A / sin d x ) = and m 2 (cos d 2 + j sin d 2 ) = m 2 e jB \ 
 
 Multiplying the exponential forms of the two expressions 
 together gives as a result 
 
 m 1 m 2 e j{e i + e * ) = m x m 2 [cos (6 X + d 2 ) +/ sin (6 X + d 2 )]. 
 
 If we directly multiply together the expressions 
 
 m^cos 6 X +j sin 0 X ) and w 2 (cos d 2 +/ sin d 2 ), 
 
 we immediately come to the expression 
 
 m 1 m 2 [cos (dj + d 2 ) +j sin (d 1 + d 2 )]. 
 
 We therefore have a satisfactory demonstration that the ordi- 
 nary process of multiplication in algebra applies to the multi- 
 plication of these coplaner-vector expressions. Consequently, 
 
 (a +/6)(c+/d) must be equal to ac A j ad A job Aj^bd, 
 
 which, again, is equal to 
 
 ac — bdA j(bc + ad'). 
 
 This shows that 
 
 ac — bd is equal to m x m 2 cos (d x -f d 2 ) 
 be + ad is equal to m x m 2 sin (0 X -}- d 2 ). 
 
 and 
 
246 
 
 ALTERNATING CURRENTS 
 
 These equalities may also be proved by expanding the expres- 
 sions mpn 2 cos(d 1 -f $ 2 ) and m 1 w 2 sin(d 1 + d 2 ), and assigning to 
 m x sin 0 V m 1 cos 6 V m 2 sin d 2 , m 2 cos d 2 their respective values in 
 terms of a , b , c, and d, indicated by Fig. 145. 
 
 By an inspection of the trigonometrical form of the foregoing 
 product, it is made evident that the product of two complex 
 quantities is another complex quantity in which the tensor or 
 
 modulus is equal to the product of the 
 tensors of the multiplier and multipli- 
 cand, , and in which the argument is 
 equal to the su?n of the arguments of 
 multiplier and multiplicand (see Fig. 
 145) ; also that the product vector 
 has a horizontal component which 
 is equal to the difference between 
 the products of the horizontal com- 
 ponents and the vertical compo- 
 nents respectively of multiplier and 
 multiplicand, and a vertical compo- 
 nent which is equal to the sum of 
 the product of the horizontal com- 
 ponent of the multiplier with the vertical component of the 
 multiplicand and the product of the vertical component of the 
 multiplier with the horizontal component of the multipli- 
 cand. 
 
 By Complementary vectors are meant vectors with equal hori- 
 zontal components, equal vertical components, and equal but 
 reversed arguments. Thus, if we desire to multiply a 4 -jb bt T 
 a—jb (i.e. a vector quantity by its Complement), we get the 
 following result, 
 
 (a -f/4)(a — jb') = a 2 — jab +jab — j 2 b 2 = a 2 + b 2 = m 2 \ 
 
 0 a 0 
 
 Fig. 145. — The Product of Two 
 Vectors OA 1 and OA 2 is OR, in 
 which OR has a length equal to 
 the length of OAi multplied by 
 the length of OA. 2 . 
 
 and again 
 
 (a +i^)(d r — jb) = m 2 [cos {6 — @) + j sin (d — d)] = m 2 . 
 
 The product of a vector and its complement therefore is a vector 
 with zero argument and may be considered as an algebraic value 
 equal to the sum of the squares of the two components of either of 
 the complementary vectors. 
 
 In division a similar process may be followed. We have 
 
THE USE OF COMPLEX QUANTITIES EXTENDED 247 
 
 (a +jh) -h (<? - \-jd ) = mff 01 -j- mff 02 = - 1 [V( 0l_9a) ] 
 
 m 2 
 
 = ^ [cos (0j — 0 2 ) 4- j sin (^0 1 — Of ] . 
 w 2 
 
 The demonstration of this follows in a manner identical with 
 the demonstration for multiplication given above. The ratio 
 of two vectors is therefore another vector in which the tensor 
 is equal to the ratio of the tensors of dividend and divisor and the 
 argument is equal to the difference of the arguments of dividend 
 and divisor. 
 
 Also multiplying both numerator and denominator by (<? — jd~) 
 and remembering that — j 2 — + 1, there results 
 
 a +jb _ (a + jh)(c —jd) 
 c +jd c 2 + d 2 
 
 The denominator in this case is a real or arithmetical quantity 
 and the numerator is equal to the product of two vectors. Ex- 
 panding shows that the above ratio is equal to 
 
 whence, 
 
 and 
 
 ac + hd +j(hc — ad ) _ac + hd , .he — ad m 
 c 2 + ( p “ c 2 + d 2 +J c 2 + d 2 ’ 
 
 m, .a a N ac + hd 
 
 — 1 sin (6 l — Of — 
 
 
 he — ad 
 c 2 + d 2 
 
 Since m 2 is numerically equal to the square root of e 2 + d?, it is 
 clear that 
 
 ac -f hd 
 
 and 
 
 m x cos(dj — Off - 
 m 1 sin(d 1 — 0f = 
 
 (c 2 -f d 2 ) - 
 he — ad 
 O 2 + d 2 ff 
 
 The foregoing theorems of multiplication and division are 
 equally applicable to stationary vectors and to rotating vectors 
 of equal frequency. The multiplication together of rotating 
 vectors of unequal frequencies gives a different result for the 
 reason that Q x and 0 2 do not increase at the same rate. Since 
 
 sin ma sin nada = 0 when m and n are unequal integers, it is 
 
 obvious that the product of a rotating vector by a rotating 
 
248 
 
 ALTERNATING CURRENTS 
 
 vector having a different frequency which is any integral num- 
 ber of times the frequency of the first vector, is equal to zero. 
 This also follows from the usual theorems of the products of 
 simple harmonic motions. Also, since 
 
 f 
 
 sin ma sin nada = - 
 
 9 
 
 sin (m — n) a sin (m + ri) a 
 m — n m -\-n 
 
 it is obvious that the product of the two rotating vectors does 
 not reduce to zero in one-half period in case n is a mixed 
 number instead of an integer, hut the product does reduce to 
 zero in the number of half periods equal to the reciprocal of the 
 fractional part of n. The product of two rotating vectors of 
 different frequencies, therefore, always reduces to zero in some 
 definite number of half periods, although it may have a large 
 finite value for some particular half period. 
 
 74. Reciprocal of a Vector Expression. — The reciprocal of a 
 vector may be written this way : 
 
 1 = 1 ^ 1 _ 1 _l c - rt 
 
 /[ a+jb ra(cos 6 +j sin 6) me je m 
 
 = — [cos ( — 0) +/ sin ( — 0)] . 
 m 
 
 Also since 
 
 a — jb = m (cos 6 —/sin 6) = m [cos(— 0)+j sin(— #)], 
 
 it is clear that -. 
 
 1 _a — jb 
 
 a 4 -jb m 2 
 
 This may also be proved by rationalization. Thus, multiplying 
 the numerator and denominator by a — jb gives 
 
 a — jb _ a — jb 
 (■ a 4 -jbj(a —jb) a 2 4- b 2 
 
 a — jb _ a 
 
 9 r 9 
 
 m* nv* 
 
 Hence, the reciprocal of a vector is a vector whose tensor is equal to 
 the reciprocal of the original tensor , and whose argument is equal to 
 the argument of the original vector with reversed sign. 
 
 The horizontal component of the reciprocal is equal to the hori- 
 zontal component of the original vector divided by the square of 
 the tensor , and the vertical component of the reciprocal is equal to 
 the vertical component of the original vector with reversed sign and 
 divided by the square of the tensor of the original vector. 
 
THE USE OF COMPLEX QUANTITIES EXTENDED 249 
 
 75. Involution and Evolution of Complex Quantities. — The 
 
 square of the vector A is 
 
 A 2 = (me ?0 ) 2 = m 2 e ,2e = m 2 ( cos 2 0 +j sin 2 0). 
 
 This comes from the rules of multiplication given above ; or, 
 the square of a vector has a tensor equal to the square of the origi- 
 nal tensor and an argument equal to twice the original argument. 
 
 Other powers of vectors follow the same laws, which come 
 directly from the rules of multiplication. For instance, 
 
 (a + jby= [m(cos 6 + j sin 0)]”= m”e ine — m"( cos n0 + j sin n0f. 
 
 If the exponent n is negative, its sign is reversed in the 
 above expression as compared with the expression when n is 
 positive. We then have the reciprocal of the tensor and the 
 reversed angle. 
 
 Also 
 
 1 i i .0. if 0 , . Q\ 
 
 ( a + jb )" = [m(cosd+y sin 0)] ”= m n ” =m n ( cos -4 -j sm - j- 
 
 From this it can be seen that evolution is clearly a process 
 of multiplication and that involution may be looked upon either 
 as a process of multiplication or a process of division. 
 
 It has been previously shown that/ -1 =- — — / ; that is, / -1 
 
 3 
 
 is an indicator of rotation backwards by the amount of one 
 quadrant. In the same way j n indicates a rotation forward by 
 
 n quadrants and /“” = •“=— /" indicates rotation backwards 
 
 by n quadrants. 
 
 In case the vector a +jb is squared, we have 
 (a +jbf = m 2 ( cos 2 0 + j sin 2 0), 
 and this is also equal to 
 
 a 2 + / 2 ab +j 2 b 2 = a 2 — b 2 +j 2 ah , 
 and therefore 
 
 to 2 cos 2 0 — a 2 — b 2 
 and m 2 sin 2 0 — 2 ab. 
 
 The square of a vector therefore has a horizontal component 
 equal to the difference of the squares of the horizontal and, vertical 
 components of the vector , and a vertical component which is equal 
 to twice the product of the horizontal and vertical components of 
 the vector. 
 
250 
 
 ALTERNATING CURRENTS 
 
 In taking the root (such as the nth root) of a vector, we get 
 
 the product of the given root of the tensor and cos -+; sin ", 
 
 n n 
 
 That is, the tensor of the root of a vector is equal to the real 
 algebraic root of the tensor of the vector. But the root of a 
 vector has direction of its own which depends on the argument 
 of the original vector. The directions of the root vectors are 
 
 6 
 
 given by -, which affords n different directions. For instance, 
 let 0 = 30° and n = 2 ; then | = 15°, ^+ - 360 ° = 195° 6 + 720 ° 
 
 2 
 
 2 
 
 9 
 
 = 375° = 15° etc., giving just two roots which have opposite 
 directions. Again, let 0 = 30° and n = 5 ; then, 
 
 0 _ ao o + 360° 
 5 ’ 5 
 
 0 4- 1440° 
 5 
 
 = 78 o 0 + 720° _ 15QO 0 + 1080° 
 
 ' 5 ’ 5 
 
 _ 294°, fl+ 1800 ° : 
 
 6°, etc.. 
 
 999 ° 
 
 giving five different directions for the five roots. 
 
 Again, let 0=0, and n — 2, then ^=0°, = 180°, 
 
 0 4 - 790° ~ 2 
 
 — — = 360° = 0°, etc., which shows that the square roots of 
 
 Z 
 
 a real quantity, that is, a quantity with a zero argument, are 
 both real quantities (that is, they lie in the horizontal axis of 
 reals) and are of opposite signs. Again, letting 0 = 0,but taking 
 
 n = 4, gives 
 
 0 = ^ 0 + 360° = 9QO 0 + 720° = 18()0 0 + 1080° 
 4 4 4 4 
 
 = 270°, ? + 144Q ° =360° = 0°, etc., 
 
 4 
 
 showing that the fourth roots of a real quantity are in opposite 
 pairs, of which one pair are reals and the other pair pure 
 imaginaries. 
 
 Following up in this manner our rule for evolution, it will be 
 observed that the »th root of any vector quantity has a tensor of 
 fixed value equal to the nth root of the original tensor, but that 
 there are in fact n root vectors, each with a direction of its own. 
 
 l 
 
 The n roots of (1)" are 
 
 vt(> 
 
 2 t rA . . ■ 2 irk 
 
 cos V j sm — 
 
 V n n 
 
 _ . ink 
 
THE USE OF COMPLEX QUANTITIES EXTENDED 
 
 251 
 
 d (OA) 
 
 where k is to be given the values 0, 1, 2, 3 ••• n — 1, taken in 
 order. In this case (1) is considered to be a vector lying along 
 the horizontal axis of reals, and therefore 6 = 0. Since the 
 
 nth root of (1) is a vector with argument it is clear that 
 
 1 . n 2 7T 
 
 multiplying a vector by (l) re causes rotation by an angle - — • 
 
 n 
 
 We have here a justification of our use of the imaginary which 
 enables us to determine all the roots of an equation. It would 
 be just as irrational to neglect the imaginary roots as it was in 
 the early algebraists to also neglect the negative roots and con- 
 sider only the positive roots as of any significance. 
 
 76. Differentiation and Integration of Complex Quantities. — 
 Differentiation of a vector quantity d(A ) ~ d(a + jb) becomes, 
 in conformity with the distributive 
 law of algebra, the same as da -+- 
 d(jb), and this becomes in conformity 
 with the commutative law the same 
 as da + jdb. The j here acts like 
 any constant quantity in an equation 
 which is differentiated and serves as 
 an operator or indicator to show 
 that db is measured vertically. We 
 can illustrate the condition by Fig. 
 
 146 in which OA represents the vec- 
 tor and AA' may be considered to be its increment. AA' 
 therefore is equal to d( OA'), and da and jdb are respectively 
 the horizontal and vertical components of the increment. 
 
 It is evident that an integration performed on the expression 
 da + jdb must result in a +jb , provided the above reasoning is 
 correct. 
 
 Suppose the process of differentiation is applied to the ex- 
 pression m( cos 6 +j sin 6). We then have 
 
 d[m(cos 6 +j sin #)] = dm( cos 6 +j sin 6) + md( cos 6 +j sin 6) 
 = dm (cos 6 +j sin 6) + m(j cos 6 — sin 6)dd. 
 
 Now writing (cjs)d for the direction coefficient cos 6 +j sin 6, 
 the foregoing becomes 
 
 (cjs )6dm +/m(cjs)ddd = (dm -\-jmdd) (cjs)0. 
 
 This, then, is the differential of the expression a +jb. If the 
 
 0 « 
 
 Fig. 116. — Illustration of a Vector 
 increased by an Increment. 
 
252 
 
 ALTERNATING CURRENTS 
 
 expression is in the form me je and we differentiate it, we have 
 this result, 
 
 (e' j6 ')dm -f jme J0 dO = (dm +jmdd)e 36 . 
 
 It will also be observed from this that the real part of the 
 differential is 
 
 da = cos Odm — m sin QdO ; 
 and the imaginary part is 
 
 jdb = j (sin Odm + m cos 0O0). 
 
 We have written (cjs)d for the expression cos 0 +j sin 0 in 
 the preceding paragraphs. The expression (cjs) may be pro- 
 nounced as though it were spelled “ siss,” and it can be readily 
 seen that it is derived by combining the initial of cos with the 
 operator j and the initial of sin in the expression cos 0+j sin 0. 
 This abbreviation is a convenient one and will be used here- 
 after. It should be remembered that (cjs)# = e j0 . 
 
 77. Logarithms of Complex Quantities. — In the preceding 
 portion relating to the differentiation of complex quantities, 
 we have shown that where A is a complex quantity equal to 
 ?rc(cjs)#, then dA = (dm +jmdO) (cjs ) 0 ; and from these we 
 cl cl/TCL * 
 
 get = \-jdO, the process performed having been to 
 
 A m 
 
 divide the vector into its differential. Since the natural loga- 
 rithm of any quantity is by definition equal to the integral of 
 the ratio of the differential of a quantity to that quantity, that 
 
 is, log , it, follows that log A = j"— + J'jdO. In- 
 
 tegrating these gives, log A = log m +j(0 + 2 7 to), where n is 
 0 or an integer. The term 2 7 to in the final expression may be 
 looked upon as a constant of integration, and it will be noticed 
 that its addition to 0 does not change the real direction of the 
 vector, but shows that the angle 0 is not iixed except in 
 special cases and that the angle 0 may have an infinite num- 
 ber of values differing from each other by 2 7 -j or 360/°. 
 These expressions show that the logarithm of a complex num- 
 ber depends upon the logarithm of the tensor of the number 
 and upon its argument, and that any function may have an 
 infinite number of logarithms which arise in an arithmetical 
 series with the difference equal to 2 7r/. 
 
THE USE OF COMPLEX QUANTITIES EXTENDED 
 
 253 
 
 It is easy to get from this the logarithm of the ratio A x to 
 A 9 ; thus, 
 
 log = log A x - log A 2 = log m x + jd i - log m 2 - j0 2 
 
 A 0 
 
 = log^l + jQ0 1 - 02). 
 
 m„ 
 
 In the same manner log (A.J A 2 ) may be shown to be equal to 
 log m x m 2 +j(0 x + 0 2 ). 
 
 78. Combination of Admittances. — In the previous chapters, 
 series circuits have been dealt with in most instances. The 
 following pages of this chapter will deal more particularly with 
 circuits in parallel and series-parallel. 
 
 If impedances are in parallel, their reciprocals must be com- 
 bined, in which case the resultant is the reciprocal of the 
 impedance of the divided circuit. It is evident that the com- 
 ponents of the admittance (reciprocal of impedance) will not be 
 equal to the reciprocals of the components of the impedance, 
 but they may be found in terms of the impedance components, 
 as pointed out in Arts. 63 and 74. 
 
 The component, g, of the admittance that lies along the hori- 
 zontal axis is the component termed the Conductance and the 
 vertical component, 5, the Susceptance. The latter is positive 
 in direction if caused by capacity, and negative if caused by 
 self-inductance. If, then, g , 5, are the admittance components 
 (conductance and susceptance) and R, X , the impedance com- 
 ponents (resistance and reactance) of a single circuit, we may 
 write by the principles of geometric multiplication above 
 stated, 1 
 
 Y= — = g Tjb = 
 
 R±jX 
 
 O) 
 
 The numerical values of the first and second terms of the right- 
 hand member of the expression Y = g T jb are respectively 
 proportional to the active current and quadrature current in a 
 circuit. When a circuit contains inductive reactance only, X 
 is essentially positive, but the quadrature current lags 90° behind 
 the active current, so that b is essentially negative. When a 
 circuit contains capacity reactance only, X is essentially nega- 
 tive, but the quadrature current leads the active current by 90°, 
 so that b is essentially positive. When a circuit contains both 
 
254 
 
 ALTERNATING CURRENTS 
 
 inductive and capacity reactance, the signs of X and b are depend- 
 ent upon the relative magnitudes of the inductance and capacity. 
 To reduce the equation 
 
 9 Tjb = 
 
 1 
 
 R±jX 
 
 (&) 
 
 to a more convenient form, the numerator and denominator of 
 the right-hand member may be multiplied by R T jX ; whence 
 
 R t jX RTjX 
 
 (. R T jX) ( R ± jX) R* + X 2 ‘ 
 
 since y 2 indicates the operation which is equivalent to multiply- 
 ing by — 1.* 
 
 Hence, 
 
 9^ jf> = 
 
 R 
 
 R 2 + X 2 T J R 2 + X 2 ’ 
 
 X 
 
 but the numerical value of the impedance is 
 
 VR Tx 2 . 
 
 Therefore g T jb = T 3 
 
 R ii , ^ 
 
 or 9 = y* and b = 
 
 If R , X, and Z are known or can be determined from the con- 
 ditions presented, problems relating to parallel circuits can now 
 be solved. 
 
 Fig. 147. — Admittances in Parallel. 
 
 If in Fig. 147 g\ b' and g", b" are the components of the 
 admittances of two parallel circuits having impedances Z' 
 and Z" , 
 
 *Art. 8. 
 
THE USE OF COMPLEX QUANTITIES EXTENDED 255 
 
 g'-jv 
 
 -EL 
 
 z ,2 3 Z' 2 
 
 and 
 
 „ , R" , -X " , 
 
 ^ ^ ^//2 ^ 7/2 1 
 
 and if g and £ are the components of the total admittance, 
 Y = g + jb = g' + g" + j (b" 
 
 R' , R" , .X" .X' 
 
 7 Z' 2 Z" 2 ^ Z" 2 ' Z' 2 
 
 The intrinsic sign of jb depends upon the relative signs and 
 X' X" 
 
 magnitudes of and The impedance of the circuit is the 
 
 reciprocal of the admittance thus found. 
 
 These processes which enable us to find the joint impedance 
 and admittance of parallel or series circuits when the elements 
 of the individual parts of the circuits are known, equally enable 
 us to find the impedance and admittance of any combination of 
 such circuits by computations which are almost as simple and 
 rapid as those which would be used in dealing with a direct- 
 current system. Also, when the impedances of any combina- 
 tion of circuits have been obtained, it is possible to find the 
 voltages in any portion when a sinusoidal current is flowing 
 and to find the current when a sinusoidal voltage is applied. 
 
 The meaning of the terms in the expression for impedance 
 and admittance may be explained by multiplying (i2 ± jXj by 1 
 (current in the circuit), when it is evident that rl is the active 
 voltage and XI the component of voltage acting against the 
 reactance; and by multiplying (g T jbj by E (voltage im- 
 pressed on the circuit), when gE is the active component of 
 the current and bE the quadrature current. 
 
 The following is a recapitulation of the formulas for the 
 analytical solution by geometric processes of many problems 
 relating to alternating-current circuits. The use of a vinculum 
 over a letter (as Z ) indicates its value as a complex quantity 
 and the letter without the vinculum indicates the tensor or 
 numerical value. In this recapitulation the small letters r, x, y, 
 and z represent the resistance, reactance, admittance, and im- 
 pedance of parts of the circuit while the capitals i?, X , IT, and 
 Z represent the same quantities for the whole circuit. In the 
 same manner the capitals Cr and B stand for circuit parts and 
 the small letters g and b for the whole : 
 
256 
 
 ALTERNATING CURRENTS 
 
 Geometric Equations 
 
 z = r ±jx. 
 
 Z — IjZ — ±j"Lx 
 
 = ^ ±y* 
 
 = Z (cos 0 +y sin 6). 
 
 y = l = aTjB 
 
 r .x 
 — ~2 “ 2 ‘ 
 
 Y=^y = H r -Tj^\ 
 z 2 z 2 
 
 _r t x 
 
 Z 2 Z 2 
 
 y E - 
 I= 2 = * r - 
 
 e = iz = £- 
 
 Y 
 
 Algebraic Equations 
 
 g = S i =RYM = ^ i =XYK 
 
 z= VW+X 2 = 
 
 Y = 
 
 V^ 2 + b 2 — 
 
 n 
 
 cos 0 
 9 _ 
 
 X 
 
 sin 6 
 b 
 
 cos 6 sin 6 
 
 tan 0 = — = -■ 
 R g 
 
 T =i= £Y - 
 
 The currents and voltages in the geometric equations are 
 true vector quantities with magnitudes and relative directions. 
 The impedances and admittances are of the character of vector 
 operators. 
 
 It will be observed that when E is computed by means of 
 the last equation in the column of geometric equations, its 
 phase is measured with respect to the current ; and when I is 
 computed by means of the next to the last equation in the 
 column of geometric equations, its phase is measured with 
 respect to the phase of the voltage. The two measurements 
 give the same angle but it is taken in opposite directions. 
 
CHAPTER VI 
 
 SOLUTION OF CIRCUITS — APPLICATION OF GRAPHICAL 
 AND ANALYTICAL METHODS 
 
 79. Graphical Methods. — As previously pointed out graph- 
 ical methods often lend themselves satisfactorily to the solution 
 of problems relating to circuits upon which sinusoidal voltages 
 are impressed. This application of graphical processes was first 
 brought to general attention by T. H. Blakesley.* This 
 chapter is a general review of portions of chapters I and V, 
 but it goes further, dealing extensively with circuits in parallel 
 and with complex combinations of series and parallel circuits. 
 It also develops a number of important general relations which 
 have heretofore not been dealt with. 
 
 To go more into detail concerning the graphical combination 
 of vectors than lias been done previously, suppose the line OA 
 in Fig. 148 is conceived as swinging at a uniform angular 
 velocity oo around the point 0, the angle « which it makes 
 with the horizontal axis OX at 
 any instant is a = cot, where t 
 is the interval of time during 
 which the line describes the 
 angle a. The instantaneous pro- 
 jection Oa upon the vertical axis 
 OY, of the line OA, has a value 
 Oa — OA sin a. If OA is propor- 
 tional to the maximum value of a 
 sinusoidal function, its instantane- 
 ous values are proportionally represented by the instantaneous 
 projections of OA ; and if OA is proportional to the effective 
 value of a sinusoidal function, the instantaneous values of the 
 function are proportionally represented by the product of V2 
 into the corresponding instantaneous projections. It is there- 
 
 '! . * Blakesley’s Alternating Currents of Electricity. 
 
 & 2ol 
 
 Fig. 148. — Rotating Vector. 
 
258 
 
 ALTERNATING CURRENTS 
 
 fore possible to represent all the elements of a sinusoidal func- 
 tion : (1) by a straight line which rotates at a uniform rate 
 around one end; and (2) by the instantaneous projections of 
 the line. It is evident that the motion which the projection of 
 the end A of the rotating line makes along the axis OY is a 
 simple harmonic motion, and that all the theorems relating to 
 simple harmonic motion may be applied to these solutions. 
 As is ordinarily done, the rotation of the line will always be 
 considered to be left-handed ; and angles measured from right 
 to left will be considered positive, while those measured from 
 left to right will be considered negative. 
 
 If two sinusoidal voltages of the same frequency, but having 
 a phase difference 6, act in a circuit, the corresponding instan- 
 taneous values are, 
 
 e = V2 E sin «, 
 e' — V2 E' sin (« + #). 
 
 The total instantaneous voltage acting in the circuit is e -f e' . 
 In Fig. 119, the voltage E is represented by the line OA , 
 
 and the voltage E' by 
 the line OA' . Oa and 
 Oa' are the instanta- 
 neous values of the 
 voltage for the angular 
 positions shown. The 
 total instantaneous 
 voltage in the circuit 
 which corresponds to 
 the angular position 
 shown is equal to Oa 
 + Oa', or Oa". It is 
 readily shown that Oa" is the projection of the diagonal of the 
 parallelogram constructed upon OA and OA ' . This is true for 
 all angular positions, since the sum of the projections of the 
 lines OA and OA' must be equal to the sum of the projections 
 of the lines OA and AA" , which in turn is equal by construc- 
 tion to the projection of the diagonal OA". 
 
 The length of the line OA" therefore proportionally repre- 
 sents the magnitude of the effective or maximum total voltage 
 in the circuit, and its position relative to that of OA and OA! 
 
SOLUTION OF CIRCUITS 
 
 259 
 
 represents the relative phase position of the total voltage. If, 
 instead of two voltages acting in a circuit, there are three or 
 more, as OA, OA', OA ", OA'", and OA "", in Fig. 150, the 
 same construction is used. Thus, completing the parallelo- 
 gram for OA and OA', their resultant OA x is found. Com- 
 pleting the parallelogram for OA x and OA", their resultant 
 OA 2 is found, and again with this and OA"' the resultant OA z 
 is obtained ; finally OA 4 , the final resultant, is obtained by 
 combining OA 3 with OA"" . The figure shows that it is un- 
 necessary to complete all the parallelograms. It is only neces- 
 
 Fig. 150. — Resultant of Three or more Vectors. 
 
 sary to draw the lines OA, AA V A X A 2 , A 2 A 3 , ^- 3^4 respectively 
 parallel and equal in length to the lines OA, OA', OA" OA'", 
 and OA"", and the line drawn from 0 to the end of the last 
 line laid off gives the phase-position and magnitude of the total 
 voltage in the circuit, regardless of the number of the compo- 
 nents from which it is derived (Fig. 151). The composition of 
 voltages is therefore exactly analogous to the composition of velo- 
 cities or of forces. ^4.s in the case of velocities or forces, the re- 
 sultant of any number of voltages may be determined by this method. 
 
 The resultant of two sinusoidal alternating currents which 
 
260 
 
 ALTERNATING CURRENTS 
 
 flow in a divided circuit may be graphically determined in the 
 same manner. In Fig. 149, let OA and OA! be the currents 
 in the two reactive branches of a divided circuit. The two 
 partial currents differ from each other in phase by an angle 6. 
 The instantaneous values of the currents are represented by 
 the instantaneous projections of the lines as they revolve 
 around the point 0. At each instant, the total current in the 
 main circuit is equal to the sum of the instantaneous partial 
 currents, or to i + i' . Consequently, the magnitude of the 
 effective or maximum value of the current in the main circuit 
 is proportionally represented by the length of the line OA", 
 and the angular direction of OA" gives the angular relation of 
 
 the phase of the total current to the phases of the partial cur- 
 rents. When the divided circuit contains more than two 
 branches, the same method may be extended as already ex- 
 plained for the composition of voltages. 
 
 For convenience in using the graphical methods for solving 
 alternating-current problems, it is well to distinguish between 
 two different diagrams. The first diagram represents the mag- 
 nitude and relative phase positions of the voltages or currents 
 by means of lines radiating from a point. This may be called 
 the Phase diagram. It is analogous to the force diagram of 
 graphical statics. The other diagram is the polygon formed by 
 laying off lines equal and parallel to the lines in the phase 
 diagram. This may be called the Vector diagram or polygon. 
 It is analogous to the funicular polygon of graphical statics. 
 
SOLUTION OF CIRCUITS 
 
 261 
 
 Figures 150 (full lines only) and 151 are respectively phase and 
 vector diagrams for representing five voltages or currents. 
 The resultant voltage is represented in magnitude and phase 
 by the line OA 4 . If the closing line OA i of the vector polygon 
 is inserted in the phase diagram by drawing from 0 a line in 
 the direction obtained by following round the vector diagram 
 against the direction in which the lines were drawn (that is, 
 from 0 to A 4 ), the line so inserted evidently represents the 
 resultant of the component voltages or currents. If the line 
 he drawn from 0 in the opposite direction, it represents a 
 balancing voltage or current. 
 
 These simple propositions, which so evidently come from the 
 ordinary graphical mechanics (statics and dynamics), give all 
 the foundation that is usually necessary for the solution of 
 problems relating to the flow of current in simple and compound 
 circuits containing definite resistances, inductances, and capac- 
 ities in their different parts. For solutions of complicated 
 problems the analytical method using complex quantities is 
 usually preferable to the graphical. The graphical method 
 has the advantage of showing directly to the eye the relative 
 phases of the voltages or currents in different parts of the 
 circuit, but a drawing board is not always so convenient a 
 medium as a tablet of computing paper and the sharpness of 
 the lag angle sometimes to be dealt with may cause indefinite 
 intersections and inaccuracy in graphical solutions. The graph- 
 ical solutions have the same limitations in regard to alternating 
 currents or voltages which are not sinusoidal as have the ana- 
 lytical methods ; and where the wave form is not sinusoidal, 
 only an approximation can be arrived at by judiciously cor- 
 recting the results shown by the diagrams based on equivalent 
 sine functions, unless the component sinusoids are used. 
 
 80. Classification of Circuits. — The problems relating to alter- 
 nating-current circuits which can be solved by graphical meth- 
 ods may be divided into three classes : (1) where the cur- 
 rent flows through all parts of the circuit in series ; (2) where 
 the same voltage is impressed upon all parts of the circuit 
 (parallel circuits) ; and (3) where the first and second classes 
 are combined. Solutions in the third class are effected by 
 combining partial solutions of the first and second classes. 
 Such problems can also readily be solved by the use of the com- 
 
262 
 
 ALTERNATING CURRENTS 
 
 plex quantity. The articles immediately following give a num- 
 ber of illustrative problems which are solved by both methods. 
 
 In more complicated systems, it is frequently desirable to 
 use the relations, — usually termed Kirclioff’s Laws, — that the 
 algebraic sum of all the currents at any junction point in a 
 circuit , or network of circuits , is zero; and that the algebraic 
 sum of the voltages in a closed circuit , or any closed loop of a 
 network of circuits , is zero. In the first case, the currents 
 entering the point under consideration are usually termed posi- 
 tive and those leaving it, negative. In the second case, it is 
 usual to follow around the circuit in a clockwise direction, 
 and call those voltages, in the several parts or sections of the 
 complete circuit, positive which are considered to increase from 
 one terminal of a section to the next, and those negative which 
 are considered to decrease from one terminal to the next. 
 These relations are in accordance with the simple laws of elec- 
 tricity and in the form above are scarcely worthy of statement 
 as separate laws, but are rather convenient expressions of Ohm's 
 law. The relations should be carefully borne in mind as here 
 stated, for the purpose of affording systematic direction to the 
 solution of complex problems. 
 
 81. Series Circuits. — First Class. — Suppose a circuit is given 
 which has a certain resistance, self-inductance, and capacity, and 
 it is desired to know what impressed voltage with a frequency 
 f is required to pass through it a certain current I. In this case 
 the impressed voltage is made up of two components: (1) the 
 voltage equal to the IR drop caused by the current flowing 
 through the resistance of the circuit, which we call the active 
 voltage ; (2) the voltage required to balance or overcome the 
 reactive counter-voltage. The reactive voltage is 90° in phase 
 in advance of or behind the active voltage, depending upon 
 whether the reactance lias the effect of capacity or of induct- 
 ance, and the phase diagram which shows the relative phases 
 of the voltages in the circuit is therefore like that shown in 
 Fig. 152. The active voltage OA' is equal to IR. and the 
 reactive voltage OA" is equal to 2 irfLI, this representing a 
 circuit with inductive reactance. The reactive component of 
 the impressed voltage is required to balance 2 irfLl \ and is 
 therefore equal to and opposite to OA". An arrowhead is 
 therefore placed on OA' to show that in the vector pol}'gon its 
 
SOLUTION OF CIRCUITS 
 
 263 
 
 direction must be taken from A" to 0, instead of from 0 out- 
 wards, as is done with the other lines. The vector polygon is 
 therefore given by drawing 0 1 A 1 
 equal and parallel to OA', A 1 A 2 
 equal and parallel to A"0 , and clos- 
 ing the polygon by the line 0 X A 2 
 (Fig. 152). The line O x A 2 taken in 
 the direction from 0 X to A 2 repre- 
 sents the magnitude and relative 
 phase of the impressed voltage. 
 
 When inserted in the phase dia- 
 gram, it is the line OA'" . The 
 angle d, by which the current lags 
 behind the impressed voltage, is the 
 angle A 1 0 1 A 2 . 
 
 If a number of inductive circuits 
 are connected in series, the line in 
 the phase diagram which represents 
 the reactive voltage has a length 
 equal to 2 i rfI(L 1 + L 2 + L s + etc.), 
 and the line which represents the active voltage has a length 
 equal to I(R 1 -4- R 2 + R 3 + etc.), where L v L v L 3 , R v R 2 , R 3 , 
 etc., are the self-inductances and resistances of the different 
 parts of the circuit. If the circuit is non-reactive, the phase 
 diagram and vector polygon each become a single horizontal 
 line equal in length to IR, while if the inductive circuit con- 
 tains no resistance, the diagrams each become a single vertical 
 line which is equal in length to 2 71 fLI. 
 
 Dividing the lengths of the sides of a vector polygon of vol- 
 tages by the value of I in the circuit gives the impedances of 
 the several parts of the series circuit. A vector polygon of the 
 voltages impressed on the parts of a series circuit and a polygon 
 of the impedances of the corresponding parts may evidently be 
 converted, one into the other, by a simple change of scale. It 
 is to be remembered that in a series circuit the current has the 
 same phase, and is equal at any given instant, in all parts of 
 the circuit ; but the phase, with reference to the current, of the 
 voltage impressed at the terminals of different parts of the cir- 
 cuit depends for each part wholly upon the relation between 
 the individual resistance and reactance of the part. 
 
 Diagrams of Self-inductive 
 Circuit. 
 
264 
 
 ALTERNATING CURRENTS 
 
 Examples. — In the following examples it is desired to find 
 for each of the given circuits : (1) the impedance of the cir- 
 cuit ; (2) the angle by which the current lags behind the 
 impressed voltage ; (3) the current which flows through the 
 circuit when the impressed voltage is 100 volts ; (4) the im- 
 pressed voltage which is required to pass 10 amperes through 
 the circuit. The frequency in each case is taken just under 
 127-| periods per second, whence 2 irf is equal to 800, and 
 current and voltage are assumed to be sinusoidal. 
 
 Circuits containing Resistance and Seif-inductance 
 
 a. The circuit is non-reactive and has a resistance of 10 ohms. 
 The phase and vector diagrams for the first solution each con- 
 sist of a horizontal line 10 units in length. 
 
 R=io 
 
 | the 
 
 The impedance of the circuit is 10 ohms, and 
 
 current which flows when 
 
 voltage 
 
 of 
 
 100 volts is impressed on the circuit is 10 
 O R = 1 ° A amperes. The diagrams for the fourth solu- 
 
 oi IR ioo tion eac p coi:is i s t of a horizontal line IR( = 
 
 Fig. 153. — Solutiono. ... . . . 
 
 100) units m length, and the impressed vol- 
 tage required to pass 10 amperes through the circuit is 100 volts. 
 The angle 6 is zero. The complex expression for the imped- 
 ance of the circuit is 
 
 Z = r +jx = 10 4-/0. 
 
 Therefore, 
 
 (1) Z = v/10 2 -|- 0 2 = 10 ohms, and (2) 6 = tan -1 ^ = 0°. 
 When the impressed voltage is 100, 
 
 and (3) I— = 10 amperes, in phase with the vol 
 When the current flowing through the circuit is 10 anq 
 
 E—JZ= Ir 4-/70, 
 or R = 1 0 R 
 
 and (4) E = IZ = 100 volts, in phase with the current. 
 
 b. The circuit consists of an inductive coil of 10 ohms re- 
 sistance and 0.01 henry self-inductance. The phase and vector 
 
SOLUTION OF CIRCUITS 
 
 265 
 
 diagrams for the solutions are shown in Fig. 154. The imped- 
 ance is shown by the closing line of the impedance diagram to 
 be 12.8 ohms, whence it is seen that 7.81 amperes will flow 
 through the circuit when the impressed voltage is 100 volts. 
 
 R = 10 L= .01 
 
 Fig. 154. — Solution b. 
 
 The fourth solution shows that the impressed voltage required 
 to pass 10 amperes through the circuit is 128 volts. The 
 angle 6 is 38° 40'. The complex expression for the impedance 
 in the circuit is 
 
 Z — r +jx — 10 ,+y 8 
 
 and (1) Z= VlO 2 + 8 2 = 12.8 ohms. 
 
 (2) 6 = tan -1 =f$8° 40'. 
 
 When the impressed voltage is 100 volts, 
 
 and 
 
 Y Er .Ex 1000 .800 
 
 J =-®y=22-^=i6r^i6i’ 
 
 (3) /= 100 x — — = 7.81 amperes, lagging behind the 
 
 12.8 
 
 voltage. 
 
 The two terms in the right-hand member of the complex 
 expression are respectively the two rectangular components, 
 I g and I b , of the current. 
 
266 
 
 ALTERNATING CURRENTS 
 
 When the current flowing through the circuit is 10 amperes, 
 E = IZ = Ir +jlx = 100 +/ 80, 
 
 and (4) E — IZ = 10 x 12.8 = 128 volts, leading the current. 
 
 The two terms in the right-hand member of the complex ex- 
 pression, namely Ir and lx , are respectively the active and 
 reactive components E r and E x of the impressed voltage. 
 
 c. The circuit consists of a non-inductive coil of 5 ohms in 
 series with an inductive coil of 5 ohms and 0.01 henry. The 
 
 total phase and vector 
 R = 5 01 diagrams for this are simi- 
 
 lar to those in example b. 
 The vector diagram may 
 be laid off as shown in 
 Fig. 155. This shows 
 that the voltage O x A 3 im- 
 pressed upon the circuit 
 as a whole is not equal 
 to the algebraic sum of 
 the voltages O x A 2 and 
 A 2 A 3 measured between 
 the terminals of the parts of the circuit, but it is still equal to 
 their vector sum. 
 
 The solution by means of complex expressions is as follows : 
 Z 1= 5 +/ 8 tan 6 = = . 8. 6 = 38° 40' 
 
 t a " 
 
 
 27T/L 
 
 
 = 8 
 
 
 . ///' 
 
 R=5 * 
 
 Z^A38°40 
 
 R = 5 A > 
 
 Fig. 155. — Solution c. 
 
 5 +/ 0 
 
 Z = Vr 2 +:r 2 =VT0 2 -t-8 2 = 12.8 ohms. 
 
 Z = 10+/ 8 
 
 When 100 volts are impressed on the circuit, 
 
 I=EY=^-j - lift 
 
 and (3) 1= 7.81 amperes, lagging behind the voltage. 
 
 When 10 amperes flow through the circuit, 
 
 E=IZ= 100+/ 80, 
 
 and (4) E = 10 x 12.8 = 128 volts, leading the current. 
 
 d. The circuit comprises only inductance of 0.01 henry. The 
 phase and vector diagrams for the first solution each consist of 
 a vertical line 2 7 rf.L(= 8 ) units in length. The impedance of 
 the circuit is 8 ohms, and the current which flows through the 
 
SOLUTION OF CIRCUITS 
 
 267 
 
 u 
 
 
 ■J 
 
 27T/L 
 
 8 
 
 'iTTj'i I 
 
 = 8 
 
 *-^0=90° 
 
 = 80 
 
 
 A„ 
 
 
 Ai 
 
 «o 
 
 circuit under an impressed voltage of 100 volts is 12.5 amperes. 
 The diagrams for the fourth solution L= oi 
 
 each consist of a vertical line 2 irfLI r^^TSinrirVi 
 
 (=80) units in length, and the im- 
 pressed voltage required to pass 10 
 amperes through the circuit is 80 
 volts. The angle 6 is 90°, and the 
 current therefore lags 90° behind the 
 
 impressed voltage. hxe=90° b\0=9o° 
 
 (1) Z=0 +y 8, Z= 8, a' 
 
 (2) 6 = tan- 1 1 = 9Q°. 
 
 When a voltage of 100 is impressed on the circuit, 
 
 Y 100 .100 
 
 Z J 8 ’ 
 
 and (3) -§-= 12.5 amperes, lagging behind the voltage. 
 
 When 10 amperes are flowing through the circuit, 
 
 ^=ioz=y so, 
 
 and (4) _Z?= 80 volts, leading the current. 
 
 O, 
 
 A' 
 
 O, 
 
 Fig. 156. — Solution d. 
 
 The effect of a condenser placed 
 in series in a circuit may be shown 
 by diagrams which are very similar 
 to those relating to inductive circuits. 
 The charging current of the con- 
 denser has such a phase position and 
 magnitude that its effect on the total 
 current flowing in the circuit is the 
 same as the effect of a voltage which 
 
 is equal to 
 
 I 
 
 and is 90° in ad- 
 
 2t jfO 
 
 vance of the circuit current. This 
 may be called the Condenser vol- 
 tage.* The phase diagram is like 
 A 2 that shown in Fig. 157. The vol- 
 Phase and Vector Dia- t a g e impressed on the circuit when 
 current I flows must consist of two 
 components : (1) the voltage required to pass the current 
 
 Fig. 157 
 
 grams of a Capacity Circuit. 
 
 * Art. 54. 
 
2(58 
 
 ALTERNATING CURRENTS 
 
 through the resistance of the circuit: (2). the voltage 
 
 required to balance the condenser voltage. The active 
 voltage OA in Fig. 157 is equal to IR, and the condenser 
 
 voltage OA" is equal to 
 active voltage 
 
 and is 90° in advance of the 
 
 2 77/(7 
 
 The capacity component of the impressed 
 
 voltage is required to balance 
 
 I 
 
 2 t rfC' 
 
 and is therefore equal 
 
 and opposite to OA". An arrowhead is therefore placed on 
 OA" to show that in the vector polygon its direction must be 
 taken from A" to 0, instead of from 0 outwards. The vector 
 polygon is then as shown in Fig. 157 (compare with Fig. 152). 
 
 Examples. — The following examples are to be solved for the 
 quantities as before, the value of 2 nf being taken as 800 and 
 voltage and current being assumed to be sinusoidal. 
 
 Circuits containing Resistance and Capacity 
 
 e. The circuit contains simply a condenser having a capacity 
 of 100 microfarads (= 0.000100 farad). The phase and vector 
 diagrams for the first solution each consist of a vertical line 
 
 - — 12.5) units in length. The impedance of the circuit 
 
 is 12.5 ohms, and the current which flows through the circuit, 
 
 when 100 volts is impressed on it, is 
 8 amperes. The diagrams for the 
 fourth solution each consist of a ver- 
 
 C=ioo 
 
 A' 
 
 O, A' 
 
 ‘ *“ T “ 
 
 0>-9O 
 
 o, 
 
 tical line 
 
 I 
 
 27 rfc 
 = 12.5 
 
 12.6 
 
 1 
 
 27 rfc 
 =126 
 
 A, 
 
 0/1-90° 
 
 irfC 
 
 ( = 125) units in 
 
 126 
 
 A, 
 
 Fig. 158. — Solution e. 
 
 length, and the impressed voltage re- 
 quired to pass 10 amperes through 
 the circuit is 125 volts. The angle 
 6 is — 90°; that is, the current is 90° 
 in advance of the impressed voltage. 
 The lines composing the diagrams 
 for this example are drawn in a direction which is exactly 
 opposite to that of the lines in the diagrams of the example d. 
 
 (1) Z = 0 —j 12.5, Z — 12.5 ohms. 
 
 (2) 6 = tan" 1 - = - 90°. 
 
SOLUTION OF CIRCUITS 
 
 269 
 
 When the impressed voltage is 100, 
 
 and 
 
 T 100 .100 . q 
 
 1 — = 7 = 78, 
 
 £ J 12.5 J 
 
 (3) I— ~~ = 8 amperes, leading the voltage. 
 
 ±A, u 
 
 When the current flowing through the circuit is 10 amperes, 
 E = 10 Z = - j 125, 
 
 and (4) E = 10 x 12.5 = 125 volts, lagging behind the current. 
 /. The circuit 
 
 R =io 
 
 C '= ioo 
 
 2tt/C 
 = 12. 5 
 
 R =10 
 
 A' 
 
 contains a resist- 
 ance of 10 ohms and 
 a capacity of 100 A 
 microfarads. The 
 phase and vector 
 diagrams are shown 
 in Fig. 159. The 
 impedance of the 
 circuit is 16 ohms 
 and the current ° 
 which flows under 
 an impressed vol- 
 tage of 100 volts is 6.25 amperes. The impressed voltage re- 
 quired to cause 10 amperes to flow through the circuit is 160 
 volts. The angle 6 is — 51° 20' 
 
 (1) Z = 10 — j 12.5, Z = 16 ohms, 
 
 (2) S = tan" 1 - ^ = - 51° 20'. 
 
 v 7 10 
 
 For 100 volts impressed, 
 
 Fig. 159. — Solution /. 
 
 7=100 Y = — + j 1250 
 
 256 
 
 256 ’ 
 
 and 
 
 100 
 
 (3) I = — = 6.25 amperes, leading the voltage. 
 Zj 
 
 For 10 amperes flowing, 
 
 E= 10 Z= 100 -j 125, 
 
 and (4) E = 10 x 16 = 160 volts, lagging behind the current. 
 
270 
 
 ALTERNATING CURRENTS 
 
 Circuits containing Self -inductance and Capacity 
 
 C=ioo L = .oi 
 
 =^ nmnri 
 
 o-- 
 
 ♦ A, 
 
 2jt/C 
 
 = 12.5 
 
 2tt/L 
 = 8 
 
 /- 90 ° 
 
 4.5 
 
 Fig. 160. — Solution g. 
 
 g. The circuit consists of a con- 
 denser of 100 microfarads capacity 
 in series with a 0.01 henry induct- 
 ance, the resistance being negligi- 
 ble. The diagrams are as shown 
 in Fig. 160. The impedance of the 
 circuit is 4.5 ohms. The current 
 which flows when the impressed 
 voltage is 100 volts is 22.2 amperes, 
 at which time the voltage measured 
 between the terminals of the con- 
 denser is 277.5 volts and that meas- 
 ured between the terminals of the 
 inductance is 177.6 volts. The im- 
 pressed voltage required to pass 10 
 amperes through the circuit is 45 
 
 90° 
 
 volts. The angle 0 is 
 
 The complex expression for impedance is 
 
 Z\ = 0 +j 8.0 
 Z, = 0 -j 12.5 
 
 and 
 
 Z = 0-j 4.5 
 (1) Z= 4.5 ohms. 
 
 (2) The angle of lag is 0 — tan" 
 
 T5 
 
 0 
 
 = -90° 
 
 (3) When the impressed voltage is 100, 
 
 T _100_ .100 
 
 Z J 4. 5’ 
 
 I — = 22.2 amperes, leading the voltage. 
 
 Z 
 
 (4) When 10 amperes flow, the impressed voltage is 
 
 E — IZ = — j 10 x 4.5 = — j 45, 
 
 E = 1Z = 45 volts, lagging behind the current. 
 
 The voltage measured between the terminals of the inductive 
 coil when 10 amperes flow is 
 
 E x = 1Z X = j 80. 
 
 E x = 80 volts, leading the current, 
 
SOLUTION OF CIRCUITS 
 
 271 
 
 L = .016 
 
 r^nnr- 
 
 C = 125 
 
 A, 
 
 and the voltage measured between the terminals of the con- 
 denser is ^7 
 
 = IZ 2 = -3 125, 
 
 = 125 volts, behind the current. 
 
 Under a current flow of 22.2 amperes, the voltage measured 
 between the terminals of the condenser is 22.2 x 12.5 = 277.5 
 volts, and the voltage measured between the terminals of 
 the inductive coil is 22.2 x 8 = 177.6 
 volts as pointed out in the first 
 paragraph. 
 
 h. The circuit comprises a capacity 
 of 125 microfarads in series with an 
 inductance of 0.015 henry, the resist- a' x 
 ance being negligible. The diagrams 
 of Fig. 161 show that impedance of 2 tt/c 
 the circuit is 2 ohms and the current =1 ° 
 which flows under 100 impressed volts 
 is therefore 50 amperes, at which time 
 the voltage measured between the ter- 
 minals of the condenser is 500 volts 
 and between the terminals of the in- 
 ductance coil is 600 volts. The im- 
 pressed voltage required to pass 10 
 amperes through the circuit is 20 volts. 
 
 With this current flowing, the voltage A'-L 
 measured between the terminals of the 
 condenser is 100 volts and between the terminals of the in- 
 ductance is 120 volts. The angle 6 is 90°. 
 
 Solving by the complex expressions gives, 
 
 Z 1 = 0 + j 12 Z = Vo 2 + 2 2 = 2 ohms 
 
 a .‘in o 
 
 0 = 90°. 
 
 o ■■ 
 
 1 
 
 1 
 
 k-~ 
 
 
 
 i 
 
 <M 
 
 tH 
 
 01 — 
 
 
 !*. 
 
 
 
 4'- 
 
 <4 
 
 * 
 
 .0=90° 
 
 X 
 
 O, 
 
 2tt/U 
 
 =12 
 
 Fig. 161. Solution h. 
 
 tan 0 = - = oo 
 0 
 
 5 = o m io 
 
 z = 0+3 2 
 
 When 100 volts are impressed upon the circuit, the current is 
 
 T U .100 . „ 
 
 Z 2 
 
 and I — 50 amperes, lagging behind the voltage. 
 
 The voltage measured between the terminals of the condenser 
 at this time is 
 
 and 
 
 E 2 = IZ 2 = 50 x — j 10 = — j 500, 
 E„ = 500 volts, behind the current. 
 
272 
 
 ALTERNATING CURRENTS 
 
 The voltage measured between the terminals of the inductance 
 coil is 
 
 E x = 1Z X = 50 x/ 12 = j 600 
 and E x — 600 volts, ahead of the current. 
 
 L =.0156 
 
 a" 
 
 1 
 
 27T/C 
 
 =12.5 
 
 •> f 
 
 o-- 
 
 , V 
 
 C = ioo 
 
 Ai 
 
 'FT*' 
 
 .i.l.t. 
 0, A, 
 
 27T/L 
 = 12.5 
 
 A'.. 
 
 Fig. 162. — Solution i. 
 
 It will be observed that the 500 
 volts measured between the con- 
 denser terminals and the 600 volts 
 measured between the coil terminals 
 are in exact opposition, and the vol- 
 tage measured between the termi- 
 nals of the circuit made up of the 
 condenser and coil in series is the 
 difference of the two or 100 volts. 
 This combination gives a large rise 
 of voltage at the condenser and at 
 the coil caused by the interaction 
 of the two through their mutual 
 transfer of energy. 
 
 When current flowing through 
 the circuit is 10 amperes, the vol- 
 tage measured between the circuit 
 terminals is 
 
 E=IZ = j 20 
 
 and E = 20 volts, ahead of the current. 
 
 Under these circumstances 
 
 E 2 = IZ. 2 = - j 100 
 
 and E 2 = 100 volts, behind the current. 
 
 E x = IZ X = +j 120 
 E x — 120 volts, ahead of the current. 
 i. The circuit comprises a capacity of 100 microfarads in 
 series with an inductance of 0.0156 henry, resistance being 
 negligible. The phase and vector diagrams are shown in Fig. 
 
 162. In this case 
 
 2 irfC 
 
 = 2 7i fL, O x A x is equal and opposite 
 
 to A x A 2 , and the impedance of the circuit is zero. 
 
 Z x = 0 +j 12.5 Z= 0 
 
 Z 2 = 0 — j 12.5 That is, this acts like a 
 
 Z = 0 short circuit. 
 
SOLUTION OF CIRCUITS 
 
 273 
 
 If a current of 10 amperes is caused to flow through this 
 arrangement of capacity and inductance in series, the voltage 
 measured between the terminals is zero ; but the voltage meas- 
 ured between the terminals of the condenser is 
 
 and 
 
 E 2 = IZ 2 = - j 125 
 
 E 2 = 125 volts, behind the current ; 
 
 and the voltage measured between the terminals of the induct- 
 ance coil is --tor 
 
 E x = IZi =j 125 
 
 and E x = 125 volts, ahead of the current. 
 
 Circuits containing Resistance , Self- Inductance, and Capacity 
 
 R — lo L=.oi 
 
 C = ioo 
 
 j. The circuit 
 consists of 10 ohms 
 plain resistance, 
 
 100 microfarads, 
 and .01 henry in 
 series relation. 
 
 The impedance of 
 the circuit is shown 
 by Fig. 163, to be 
 10.96 ohms. The 
 current which 
 flows through the 
 circuit when 100 
 volts are impressed 
 at its terminals is 
 9.12 amperes. The 
 voltage required to 
 pass 10 amperes 
 through the circuit 
 is 109.6 volts. This 
 is the vector sum 
 of 125 volts and 
 128 volts, which are 
 the voltages meas- 
 ured respectively 
 between the condenser terminals and the remainder of the cir- 
 cuit. The angle 9 is — 24° 14'. 
 
 A'"" 
 
 
 l 
 
 
 2jt/C 
 = 12.5 
 
 
 
 R=lo A' 
 
 27T/L 
 = 8 
 
 A" 
 
 R =10, L= 01 
 
 Fig. 163. — Solution j. 
 
274 
 
 ALTERNATING CURRENTS 
 
 Solved by complex quantities, there results 
 
 z x = 10 +j 0 
 
 Z 2 = 0 -\-j 8 
 
 3 a =0-/12.5 
 3 = 10-/ 4.5 
 
 Here the prefix of the reactance term in the expression for 
 3 is — /, hence the angle 6 is negative. 
 
 When 100 volts are impressed on the circuit, 
 
 I_E_ F Y- 1000 • 450 
 
 z J (10.96) 2+ ' ; (10.96) 2 
 
 and I— = 9.12 amperes, leading the voltage. 
 
 Under these conditions, the voltage measured across the ter- 
 minals of the condenser is 
 
 E z = IZ z = — / 9.12 x 12.5 
 and ^ 3 = 114 volts, behind the current. 
 
 The voltage measured across the remainder of the circuit is 
 E 1+2 = KZ X + Z 3 ) = 9.12 (10 +/ 8) 
 
 and E 1+2 = 4- 72.9B 2 = 116.8 volts, ahead of the current. 
 
 When 10 amperes is caused to flow through the circuit, 
 
 J= 73= 100-/45 
 
 and E = 109.6 volts, behind the current. 
 
 The values of E 3 and E x+2 under these circumstances may be 
 computed in the same manner. 
 
 k The circuit comprises 10 ohms plain resistance, 150 micro- 
 farads, and 01042 henry in series. The diagrams in Fig 164 
 show that the impedance of the circuit is 10 ohms One hun- 
 dred volts is therefore the impressed voltage that gives a cur- 
 rent of 10 amperes. When 10 amperes flow in the circuit, the 
 voltage measured between the terminals of the condenser is 
 83.8 volts, and that measured between the terminals of the 
 remainder of the circuit is 180 volts. The angle 6 is zero. 
 
 Z x = 10 +/ 0 3=10 ohms. 
 
 Z 2 = 0 + j 8 . 33 tan 0 _ q _ qo_ 
 
 3 3 = 0 — j 8.33 10 
 
 Z =10+/0 
 
 Z = VlO 2 + 4.5 2 = 10.96 ohms, 
 tan e = - 6 = - 24° 14'. 
 
SOLUTION OF CIRCUITS 
 
 275 
 
 When 100 volts are impressed on this circuit, 
 
 100 
 
 10 
 
 = 10 
 
 and I— 10 amperes, in phase with the voltage. 
 
 The voltage measured between the terminals of the condenser 
 (assuming it to be resistanceless) is 
 
 j? 3 = -yiO X 8.38 = -/ 83.3 
 and Eq = 83.3 volts, behind the current. 
 
 The voltage measured between the terminals of the inductance 
 coil (assuming it to be resistanceless) is 
 
 E 2 = +j 10 x 8.33 = +j 83.3 
 and _Z ?2 = 83.3 volts, ahead of the current. 
 
 i 
 
 27T/C 
 = 8.33 
 
 O 
 
 R =10 
 
 A' 
 
 2 tt/L 
 =8.33 
 
 These two reactive R = 10 L = .01042 
 components of vol- 
 tage are equal and 
 opposite to each 
 other, and the vol- 
 tage measured 
 across the circuit is 
 the same as that 
 measured across the 
 plain resistance coil, 
 namely, 100 volts in 
 phase with the cur- 
 rent. 
 
 82. Conclusions in 
 Regard to Series Cir- 
 cuits. — The eleven 
 examples thus given 
 cover every funda- 
 mental arrangement 
 of series circuits 
 which may occur. 
 
 An examination of 
 the diagrams makes 
 the following state- 
 
 C = 150 
 
 R =10, L = .0104 
 
 83.3 
 
 t 
 
 8.33 
 
 10 
 
 o, 
 
 Fig. 164. — Solution k. 
 
 ments evident for 
 circuits in which are sinusoidal voltages and currents : 
 
 8.33 
 
 R=10, C =150 
 
 83.3 
 
276 
 
 ALTERNATING CURRENTS 
 
 1. When non-inductive circuits are connected in series, the 
 total impressed voltage equals the sum of the voltages measured 
 between the terminals of the individual parts, and the total 
 resistance of the circuit is equal to the sum of the resistances 
 of the individual parts. 
 
 2. When inductive circuits of equal time constants are con- 
 nected in series, the total impressed voltage equals the sum of 
 the voltages measured between the terminals of the individual 
 parts, and the total impedance of the circuit is equal to the 
 sum of the individual impedances. 
 
 3. When inductive and non-inductive circuits are connected 
 in series with each other, or when inductive circuits of unequal 
 time constants are connected in series, the total impressed vol- 
 tage equals the vector sum, which is always less than the alge- 
 braic sum, of the voltages measured between the terminals of 
 the individual parts, and the individual voltages are each less 
 than the total impressed voltage. The total impedance of the 
 circuit is equal to the vector sum of the individual impedances, 
 each of which is less than the total. 
 
 4. When condensers are connected in series by conductors of 
 7iegligible resistance , the total impressed voltage equals the sum 
 of the voltages measured across the individual condensers, and 
 the total impedance of the circuit is equal to the sum of the 
 impedances of the individual condensers. 
 
 5. When condensers are connected in series with non-hiduct- 
 ive circuits , the total impressed voltage equals the vector sum, 
 which is always less than the algebraic sum, of the voltages 
 measured between the terminals of the individual parts of the 
 series, and the individual voltages are each less than the total 
 impressed voltage. The total impedance of the circuit is equal 
 to the vector sum of the individual impedances, each of which 
 is less than the total. 
 
 6. When condense?-s are connected in series with inductive 
 circuits , the total impressed voltage equals the vector sum, 
 which is always less than the algebraic sum, of the voltages 
 measured between the terminals of the individual parts of the 
 series. Since the effects of capacity and self-inductance respec- 
 tively cause the lag angle to become negative and positive, the 
 individual voltages may be either greater OR less than the total 
 impressed voltage , depending upon the relation between the 
 
SOLUTION OF CIRCUITS 
 
 277 
 
 various resistances, capacities, and inductances in the circuit. 
 The total impedance of the circuit is equal to the vector sum 
 of the individual impedances, each of which may be either 
 greater or less than the total impedance. 
 
 The third, fifth, and sixth paragraphs above make the follow- 
 ing proposition evident : When in series circuits the angles 
 measured between the phase of the current and the phases of 
 the individual voltages measured between terminals of parts of 
 the series, are all either positive or negative, the total impressed 
 voltage is always greater than any of the individual or partial 
 voltages. When the angles measured between the phase of 
 the current and the phases of the partial voltages are in part 
 positive and in part negative, some or all of the partial voltages 
 may be greater than the total impressed voltage. 
 
 83. Parallel Circuits. — Second Class. — The graphical and 
 analytical treatment of problems relating to parallel circuits is 
 entirely analogous to that given for series circuits. As the 
 simplest cases of parallel circuits are those in which the same 
 voltage is impressed upon all the parts of the circuit, these will 
 be treated first. In this class the same general operations are 
 used in solving problems as in the first class (series circuits), 
 but alternating currents and admittances are dealt with instead 
 of alternating voltages and impedances. Suppose a circuit is 
 made up of two branches in parallel, each with a known resist- 
 ance and reactance, and it is desired to know what impressed 
 voltage with a frequency f is required to cause a sinusoidal 
 current I to flow through the circuit. In this case the total 
 current is made up of two components, each of which flows 
 through one of the branches and is inversely proportional to 
 the impedance of the branch, and the phase of which has an 
 angular retardation with respect to the impressed voltage 
 which depends upon the time constant of the branch. The 
 total current, which is inversely proportional to the equivalent 
 impedance of the parallel circuit, is equal in magnitude and 
 position to the resultant of the branch currents. The con- 
 dition is represented by Fig. 165, in which OA! and OA" are 
 the currents in the two branches respectively, O' and 6" being 
 their respective lag angles. The relative phase position of the 
 impressed voltage is taken on the horizontal line. Then the 
 resultant or total current in the circuit is represented in mag- 
 
278 
 
 ALTERNATING CURRENTS 
 
 Fig. 165. — Currents in Parallel Branches. 
 
 E 
 
 nitucle and phase by OA. OA! is equal to — , OA" is equal to 
 
 F E ■ Zx 
 
 — , and OA is equal to — where E is the voltage impressed on 
 Z 9 Z 
 
 the branches and the de- 
 nominators of the frac- 
 tions are the respective 
 impedances of the branches 
 and of the total circuit. It 
 is therefore evident that 
 the reciprocal of the joint 
 impedance Z(= the joint 
 admittance) of a parallel 
 circuit may be at once derived from the admittances of the 
 branches, by taking their vector sum, as is shown in Fig. 166. 
 The joint admittance or the joint impedance of a branched circuit 
 being known for a particular frequency, the voltage of that fre- 
 quency required to pass a 
 given current through it, or 
 the current flowing under a 
 given impressed voltage of 
 the same frequency, may 
 be at once dei’ived. In 
 dealing with series cir- 
 cuits, the phase of the cur- 
 rent (or active voltage) 
 has been assumed to be 
 
 along the horizontal line, .Y. x 
 
 or line of reference. In 
 dealing with parallel cir- 
 cuits, it is more convenient 
 to assume the phase of the 
 impressed voltage (or conductance) for the reference phase. 
 It must always be remembered, however, that angles of lag are 
 measured from lines representing current to lines representing 
 voltage. Thus, in Figs. 165 and 166 the angle 6 is negative 
 because the current leads the voltage. 
 
 The resultant current is also obtained by adding the com- 
 plex expressions representing the branch currents. Thus the 
 resultant of two currents in parallel circuits is 
 
 Fig. — 166. Vector Diagram of Admittances. 
 
SOLUTION OF CIRCUITS 
 
 279 
 
 1= I x + 1 2 = El \ + E Y 2 — E(g x + gf) — + ^ 2 )* 
 
 In this equation Eg x + Eg 2 is the active component I g of the 
 particular current, and Eb 1 + Eb 2 the component I b in quad- 
 rature. In the same way the combined admittance of parallel 
 circuits is obtained. The addition is performed graphically in 
 Figs. 165 and 166. 
 
 Examples. — In the following examples it is desired to find 
 for each of the given circuits : (1) the joint impedance of 
 the circuit at the given frequency, (2) the angle by which 
 the total current lags behind the impressed voltage, (3) the 
 current which flows through the circuit when the impressed 
 voltage is 100 volts, (4) the impressed voltage which is re- 
 quired to pass 10 amperes through the circuit. The frequency 
 is taken as in examples of the series circuit to be just under 
 127^- periods per second, whence 2 7r/is equal to 800 ; and cur- 
 rent and voltage are supposed to be sinusoidal. 
 
 R, - 10 
 
 Circuits containing Resistance and Self-inductance 
 
 a. The circuit consists of two non-reactive * branches in 
 parallel, one having 20 ohms resistance and the other 10 ohms 
 resistance. The phase diagram for 
 the solutions, using the admittances 
 as a basis of work, is two horizontal 
 lines superposed, of lengths respec- 
 tively .05 and .10 unit. The vector 
 diagram is obtained by drawing 
 these consecutively, and the equiva- 
 lent admittance of the circuit is OA 2 
 in Fig. 167 and is equal to .15 mho. 
 
 The joint impedance of the cir- 
 cuit is therefore 6.67 ohm. The current flowing under 100 
 volts impressed is therefore 15 amperes, and the voltage re- 
 quired to pass 10 amperes through the circuit is 66.7 volts. 
 The complex expressions for these circuits reduce to the ordi- 
 
 Ot- 
 
 0,t 
 
 A" 
 
 .10 
 
 A, 
 
 -05_ 
 
 -.16 
 
 Fig. 167. — Solution a. 
 
 * 
 
 * The terms inductive , capacity, and reactive circuit are used in this book 
 with the following significations : an inductive circuit is one containing in- 
 ductance, but not capacity ; a capacity or condenser circuit is one containing 
 capacity, but not inductance ; a reactive circuit is one containing either in- 
 ductance or capacity, or both inductance and capacity. A non-reactive circuit 
 is, therefore, one which contains neither inductance nor capacity, that is, one 
 which contains plain resistance only. 
 
280 
 
 ALTERNATING CURRENTS 
 
 R, = 10 
 L = .01 
 
 A, 
 
 nary forms derived from Ohm’s Law and are, therefore, 
 omitted. 
 
 b. The circuit con- 
 sists of a non-reactive 
 branch of 10 ohms 
 and an inductive 
 branch of .01 henry 
 in parallel. The phase 
 diagram consists of 
 two lines at right an- 
 b= gles (one being hori- 
 zontal), since the cur- 
 rent in the non-reac- 
 tive branch is in phase 
 A with the impressed 
 voltage and that in 
 the inductive branch 
 lags 90° behind. The lengths of the lines are respectively 
 
 The vector poly- 
 
 — (= .10) units and—- ■(= .125) units 
 R 2 irfL 
 
 gon is as shown. The admittance of the circuit is .16 mho and 
 the impedance is 6.25 ohms. The current flowing under an 
 impressed voltage of 100 volts is therefore 16 amperes, and it 
 requires 62.5 volts to cause 10 amperes to flow. The angle 6 
 is 51° 20'. The small triangle of Fig. 168 is an impedance 
 triangle drawn on a different scale from the larger admittance 
 
 - 1 
 
 triangle. Its sides are derived from the relation Z = = and 
 
 z = whence r = and x= Y*' 
 
 In the complex expression we have g = l = 
 
 Z - Z“ 
 
 and 
 
 -g + jb, therefore 
 10 
 
 F i = 100 + 0 
 
 F 9 = 0 
 
 -JO 
 
 -j 
 
 64 + 0 
 
 Y= .1 - j .125 
 
 Y = V.l 2 + .125 2 = .16 mho. 
 tan e = — p e = 51° 20'. 
 
 Z = ^ r = 6. 25 ohms. 
 
 When 100 volts are impressed on this branched circuit, 
 
SOLUTION OF CIRCUITS 
 
 281 
 
 I = EY = 10 -j 12.5 
 
 and I = EY = 16 amperes, lagging behind the voltage. 
 
 When 10 amperes flow through the branched circuit, 
 
 W-iz- 10 - 1 |j L25 
 
 .1-/.125 .0256 . 0256 
 
 and E = IZ = 62.5 volts, leading the current. 
 
 The current in each branch under either set of conditions is 
 equal to the admittance of the branch multiplied by the im- 
 pressed voltage corresponding to the conditions. 
 
 c. The circuit consists of a non-reactive branch of 10 ohms 
 in parallel with an inductive branch having a resistance of 10 
 ohms and an inductance of .01 henry. The impedance and 
 the angle .of lag for the inductive branch are found by the 
 method given under Series Circuits: First Class, and the 
 admittance of the branch is laid off 
 line making with the horizontal axis 
 an angle equal to the angle of lag 
 taken backwards. This is line OA" 
 in the diagram, Fig. 169. The line 
 OA' represents the admittance of, 
 and the relative phase of current in, 
 the non-reactive branch. The 
 length and direction of the line 
 0 X A 2 in the vector polygon shows 
 the value of the equivalent or joint 
 admittance of the circuit and the 
 angle by which the phase of the 
 main current lags behind the phase 
 of the impressed voltage. The 
 joint admittance of the circuit is 
 .168 mho and the joint impedance 
 5.95 ohms. The current flowing 
 under an impressed voltage of 100 
 volts is 16.8 amperes, and the voltage required to pass 10 
 amperes through the circuit is 59.5 volts. The angle 6 is 
 16° 52k The triangle OA! A is equal to the impedance tri- 
 angle of the self-inductive branch rotated on its base until 
 its apex points downward. This reversed construction of the 
 
 in the phase diagram on a 
 R, =io 
 
 R 2 = io, L = .oi 
 
 — ' kfinnrird'- 
 
 A, -io 
 
 Fig. 169. — Solution c. 
 
282 
 
 ALTERNATING CURRENTS 
 
 impedance line (used also in following problems) is made foi 
 convenience in obtaining the correct position of the admittance 
 line, since admittance is the reciprocal of impedance and taking 
 the reciprocal of an operator reverses its angle. If the position 
 of the impedance and its components are desired, an impedance 
 triangle should be laid out as in the case of any series circuit. 
 
 The problem may be worked out by the use of complex 
 quantities as follows : 
 
 Y — JLQ A 0 
 
 1 i — i o o J u 
 
 Y = 10 • §_ 
 
 2 100 + 64 100 + 64 
 
 Y= .161 -j .049 
 
 Y= V.l61 2 + .049 2 = .168 mho. 
 Z — 5.95 ohms. 
 
 tan e = e = 16° 52'. 
 
 .161 
 
 When 100 volts are impressed on the circuit, 
 
 I— EY = 16.1 — j 4.9 
 
 and /= EV = 16.8 amperes, lagging behind the voltage. 
 
 When 10 amperes flow through the circuit, 
 
 e= iz= ~ = — — + /-MiL 
 
 Y .0282 . 0282 
 
 and E = 1Z = 59.5 volts, ahead of the current. 
 
 The complex expressions for currents in the branches when 
 
 100 volts are impressed are 
 
 J 1 = .2^ = 100 (.1-/0), 
 
 /j = 10 amperes, in phase 
 with the voltage, 
 
 Li = .oi 
 
 L 2 =.0126 
 — ^ (5 X s * - 
 
 "‘ = tan ' 1 B =0 ° ; 
 
 and 
 
 Y = EY 0 = 100 (.061 
 -i.049), 
 
 72 = 7.81 lagging behind 
 the voltage, 
 
 = tan- 1 ^? = 38° 40h 
 2 .061 
 
 It will be observed that the arith- 
 metical sum of the currents, 10 
 amperes and 7.81 amperes, in the 
 two branches is greater than the current, 16.8 amperes, in the 
 main line. 
 
 A 2 - l 
 
 Fig. 170. — Solution d. 
 
SOLUTION OF CIRCUITS 
 
 283 
 
 d. The circuit consists of two inductive branches of respec- 
 tively .01 and .0125 henry in parallel. The diagrams consist 
 of vertical lines, as shown in Fig. 170. The admittance is 
 .225 mho and the impedance is 4.44 ohms. The current flow- 
 ing when 100 volts is impressed on the circuit is 22.5 amperes, 
 and it requires 44.4 volts to cause 10 amperes to flow. The 
 angle of lag is 90°. 
 
 The complex expressions for admittance are 
 ¥ 1 = -j. 1 Y = .225 mho. 
 
 Y 2 = -j. 125 Z= 4.44 ohms. 
 
 f=-jm 0 = tan-^5 = 90 ". 
 
 0 
 
 The currents in the branches when 100 volts are impressed on 
 the circuit are 
 
 I l — EY l =.lx 100 = 10 amperes, lagging 90°, 
 and I 2 = EY 2 = .125 x 100 = 12.5 amperes, also lagging 90°. 
 The main current is 
 
 1= EY = —j 22.5 
 
 and 1 = 22.5 amperes, lagging 90°. 
 
 When 10 amperes are caused to flow through the circuit, 
 
 10 
 
 .225’ 
 
 E = IZ = 44.4 volts, 90° ahead of the current. 
 
 two reactive branches of respec- 
 
 Ri=io, L, — .005 
 r-OS - 5"15Tr"5"'b~'* — 
 
 R =8, L, = .0125 
 
 e. The circuit consists of 
 tively .005 henry 
 and 10 ohms, and 
 .0125 henry and 8 
 ohms in parallel. 
 
 The diagrams are 10 
 
 as shown in Fig. 
 
 171. The admit- 
 tance of the circuit 
 is shown to be .165 
 mho, and the im- 
 pedance 6.06 ohms. 
 
 The current flow- 
 ing under a voltage of 100 volts is therefore 16.5 amperes, and 
 the voltage required to pass 10 amperes through the circuit is 
 
 Fig. 171. — Solution e. 
 
284 
 
 ALTERNATING CURRENTS 
 
 60.6 volts. The angle 8 is 35° 16'. The problem may be 
 solved by the use of complex quantities as follows : 
 
 y 10 • £_ 
 
 1 100 + 16 100 +16 
 
 Y — § j 10 
 
 2 64 + 100 J 64 + 100 
 
 Y= .1349 -j. 0954 
 Y — ^ . 1349 2 + .0954 2 = .165 mho. 
 
 Z = = 6. 06 ohms. 
 
 .165 
 
 tan 6 = = .7072, 6 = 35° 16'. 
 
 .1349 
 
 The currents in the branches when 100 volts are impressed on 
 the circuit are 
 
 I x = EY X — 9.3 amperes, lagging behind the voltage, 
 and / 2 = EY 2 = 7.8 amperes, also lagging behind the voltage ; 
 
 d x = 21° 48' 
 
 and 6 2 = 51° 20'. 
 
 The total current is 
 
 1= EY = 13.5 -j 9.54 
 
 and I = EY — 16.5 amperes, lagging behind the voltage. 
 
 The five preceding examples cover all the fundamental com- 
 binations of resistance and inductance in parallel circuits. The 
 following four in like manner cover the combinations of resist- 
 ance and capacity. The solutions in the two cases are similar, 
 but the lag angles become negative in the latter on account of 
 the influence of the capacities. 
 
 Circuits containing Resistance and Capacity 
 
 f. When two or more condensers of negligible internal re- 
 sistance are connected in parallel by wires of negligible resist- 
 ance, they evidently act upon the circuit exactly as though 
 it contained one condenser with a capacity equal to the com- 
 bined capacity of those in parallel. The impedance of a con- 
 
 denser is equal to 
 
 2 irfC 
 
 , and its admittance to 2 i -fC. The 
 
 admittance of several condensers in parallel is therefore 
 evidently 
 
 2 Trf{C x + C 2 + etc.). 
 
SOLUTION OF CIRCUITS 
 
 285 
 
 R - 10 
 
 C = ioo 
 
 A" 
 
 g. The circuit consists of a non-reactive branch of 10 ohms 
 in parallel with a capacity branch of 100 microfarads and negli 
 gible resistance. The dia- 
 grams are as shown in 
 Fig. 172. The admit- 
 tance of the circuit is .128 
 mho and its impedance is 
 7.81 ohms. The current 
 flowing under a voltage 
 of 100 volts is 12.8 am- 
 peres, and the voltage re- 
 quired to pass 10 amperes 
 through the circuit is 78.1 
 volts. The angle 0 is 
 - 38° 40'. 
 
 The complex expressions for this case are of the same form 
 as for Solution b except that the direction of the vertical 
 component is reversed. 
 
 Fig. 172. — Solution g. 
 
 Y^.l-jO 
 
 F„ = o+y.Q8 
 
 Y = .!+/. 08 
 
 Y= ^.l 2 + .08 2 = .128 mho. 
 
 Z - —— 7.81 ohms. 
 
 6 = — tan -1 = -38° 40'. 
 
 When 100 volts are impressed on the circuit, 
 
 I=EY=10+j8 
 
 and /= EY = 12.8 amperes; 
 
 I\ = EY X = 10 —j 0 
 
 and I x = EY X — 10 amperes ; 
 
 I 2 = EY 2 =j 8 
 
 and I 2 = EY 2 — 8 amperes ; 
 
 Q x = 0°, 0 2 = - 90°. 
 
 When a current of 10 amperes is caused to flow through the 
 circuit, 
 
 1 . 0.8 
 
 r 
 
 E = — 
 
 .0164 .0164 
 
 E — IZ =78.1 volts. 
 
 and 
 
286 
 
 ALTERNATING CURRENTS 
 
 R = 10 
 
 R = 10 , C = ioo 
 
 h. The circuit consists of a non-reactive branch of 10 ohms 
 in parallel with a reactive branch of 10 ohms and 100 micro- 
 farads. The ad- 
 mittance of the cir- 
 cuit is shown by 
 the diagrams of 
 Fig. 173 to be .117 
 mho and the im- 
 pedance 6.8 ohms. 
 The current flow- 
 ing under 100 
 volts is 14.7 am- 
 peres, and the vol- 
 tage required to 
 pass 10 amperes 
 
 -19° 20'. 
 
 through the circuit 
 is 68 volts. The 
 angle of lag is 
 The solution by means of complex quantities is 
 
 Y _ 10 • 0 
 
 1 100 + 0 J 100 + 0 
 
 v 10 ■ 12-5 
 
 2 100 + 156 J 100 + 156 
 
 Y = .1390 +/. 0488 
 
 r = V.l390 2 + .04 88 2 = .147 mho. 
 
 Z = —5— = 6.8 ohms. 
 
 .147 
 
 tan ^ = _iM§8 = _. 3 509, 0 = -19° 20'. 
 .1390 
 
 When 100 volts are impressed on the circuit, the currents are 
 I=EY= 13.9 +i 4.88 
 
 and 
 
 I = EY = 14.7 amperes; 
 I x = EY 1 = 10 -j 0 
 I x = EY X = 10 amperes ; 
 
 I^EY 
 
 1000 .1250 
 
 256 +J 256 
 
 and 
 
SOLUTION OF CIRCUITS 
 
 287 
 
 and / 2 = E Y 2 = 6.25 amperes ; 
 
 6 X = 0°, 0 2 = — tan" 1 = 
 
 When 10 amperes are caused to flow through the circuit, 
 
 51° 20'. 
 
 E = -- = 
 
 1.39 . .488 
 
 -3 
 
 and E - IZ = 68 volts. 
 
 i. The circuit con- 
 sists of two reactive 
 branches in parallel, re- 
 spectively, of 10 ohms 
 and 100 microfarads, 
 and of 20 ohms and 250 
 microfarads. The ad- 
 mittance of the circuit 
 is shown by the diagram 
 in Fig. 174 to- be .105 
 mho and the impedance 
 9.5 ohms. The current 
 flowing under a voltage 
 of 100 volts is 10.5 am- 
 peres, and the voltage 
 required to pass 10 am- 
 peres through the cir- 
 cuit is 95 volts. The 
 angle 6 is — 35° 10'. 
 
 .0218 " o0218 
 
 R = io, C= loo 
 
 R =20, 
 
 C = 260 
 
 R = 7.78 
 
 Fig. 174. Solution i. 
 
 The solution with complex quantities is 
 
 Y= A// .0862 2 + .0605 2 = .105 mho. 
 
 F= J0 .12,5 
 
 1 256 J 256 
 
 Y=™-+j±- 
 
 2 425 J 425 
 
 Z = — r = 9.52 ohms. 
 
 Y =.0862+ j. 0605 0 = - tan 
 
 _! .0605 
 
 = -35° 10'. 
 
 .0862 
 
 When 100 volts are impressed on the circuit, the currents are 
 I=EY = 8.62 +j 6.05 
 1= EY = 10.5 amperes ; 
 f 1000 .1250 
 
 I 1 = EY 1 =mt + 3 
 
 256 
 
 256 
 
 and 
 
288 
 
 ALTERNATING CURRENTS 
 
 and 
 
 I x = EY X — 6.25 amperes ; 
 
 i 2 = ey 2 
 
 2000 .500 
 
 425 + 3 425 
 
 and I 2 = EY 2 = 4.85 amperes ; 
 
 0. = - tan" 1 — = - 51° 20', 0 2 = - tan^ 1 — = - 14° 5'. 
 
 1 10 2 20 
 
 When 10 amperes are caused to flow through the circuit, 
 
 I -862 . .605 
 
 J ~Y~ -0111 3 . 0111 ’ 
 
 E = IZ = 95. 2 volts. 
 
 The impedance components of the combined circuit may be 
 found as in Problem b or from the formula 
 
 or 
 
 Z= --p (cos 0 +j sin 0) 
 %= — b (.817 —J .576), 
 
 Z= 7.78 —j 5.49. 
 
 The impedance diagram is shown in the triangle 0 2 BC, Fig. 
 174. 
 
 The following examples cover the fundamental combinations of 
 circuits containing resistance , capacity , and self -inductance. 
 
 j. The circuit consists of two reactive branches in parallel, 
 respectively of 5 ohms and .005 henry, and of 10 ohms and 100 
 microfarads. The admittance of the circuit is shown by the 
 diagrams of Fig. 175 to be .168 mho and the impedance 5.95 
 ohms. The current flowing under a voltage of 100 volts is 
 16.8 amperes, and the voltage required to cause a current of 
 10 amperes to flow is 59.5 volts. The angle 6 is 16° 50'. 
 
 The solution by means of the complex quantities is as 
 follows : 
 
 Y _ 5 4 
 
 1 25 + 16 J 25 + 16 
 
 Y= ’ s/ . 161 2 + .0487 2 = .168 mho. 
 
 Y - if 12 ‘ 5 
 
 2 100 + 156 '100 + 156 
 
 Z = - ^ = 5.95 ohms. 
 
 .168 
 
 tan 6 = •° 487 = .3025, 0 = 16° 50'. 
 .1610 
 
 Y = .1610 — j .0487 
 
SOLUTION or CIRCUITS 
 
 289 
 
 When 100 volts are impressed on the circuit, 
 I = EY =16.1-/4.87 
 and I = EY = 16.8 amperes ; 
 
 and 
 
 r 500 .400 
 
 I x — EY 1 = 15.6 amperes ; 
 
 E = EY„ = 1000 
 
 .1250 
 
 and 
 
 I 2 = EY 2 
 
 400 
 
 0 = tan-i = 38° 40', 0 O = 
 
 1 O00 2 
 
 256 ' 256 
 
 6.25 amperes ; 
 
 ! 1250 
 
 1000 
 
 tan" 
 
 = - 51° 20'. 
 
 R = 5 , L= .005 
 R = io, C = loo 
 
 When 10 amperes are caused to flow through the circuit, 
 
 E=IZ = 
 
 1.61 . 
 .0284 
 
 .487 
 
 .0284’ 
 
 and 
 
 E — IZ =59.5 volts. 
 
 The impedance diagram for the combined circuit is shown in 
 the triangle O^BC, Fig. 175, and is obtained as in the preced- 
 ing cases, thus : 
 
290 
 
 ALTERNATING CURRENTS 
 
 or 
 
 or 
 
 Z=R + jX, 
 
 Z = y (cos 0 + j sin 0), 
 
 Z = 
 
 .168 
 
 or Z = Z (cos 6 + j sin 0), 
 
 and from either of the last two, 
 
 (cos 16° 50' + j sin 16° 50'), 
 
 Z= 5.70 + j 1.72. 
 
 k. The circuit con- 
 sists of two reactive 
 branches in parallel, 
 respectively, of 10 
 ohms and .0156 henry, 
 and of 5 ohms and 200 
 microfarads. The ad- 
 mittance is shown by 
 the diagrams of Fig. 
 176 to be .127 mho, 
 and the impedance of 
 the circuit is 7.87 
 ohms. The current 
 flowing under a vol- 
 tage of 100 volts is 
 12.7 amperes, and 78.7 
 volts are required to 
 pass 10 amperes 
 through the circuit. 
 The angle 6 is — 22° 
 37'. 
 
 The solution using complex quantities is as follows : 
 10 • 12.5 
 
 1 100 + 156.3 J 100 + 156.3 
 
 r= 5 . 6.25 
 
 2 25 + 39 25 + 39 
 
 Y = .1173 + j .0489 
 
 Y =V.ii73 2 + .0489 2 = .127 mho. 
 
 1 
 
 Z = 
 
 .127 
 
 = 7.87 ohms. 
 
 tan 6 = - Q = - 22° 37'. 
 
 .1173 
 
SOLUTION OF CIRCUITS 
 
 291 
 
 and 
 
 and 
 
 When 100 volts are impressed on the circuit, 
 
 I— 11.73 +y 4.89 
 I = EY = 12.7 amperes ; 
 
 r rrtr_ 1000 ..1250 
 
 11 256.3 ^ 256.3 
 
 7 1 = FJ\ = 6.25 amperes; 
 
 7 w 500, .625 
 i = -EF„ = — — 
 
 t 2 — ^ ^ 2 
 
 and I 2 = UY 2 = 12.5 amperes ; 
 
 6.25 
 5 
 
 = - 51° 20'. 
 
 When 10 amperes are caused to flow through the circuit. 
 
 6, = tan 1 ^ = 51° 20', 6 '„ = — tan" 
 
 l io 2 
 
 E=IZ = 
 
 1.173 . .489 
 
 ■J 
 
 and 
 
 .0162 " .0162 
 Fj — IZ= 78.7 volts. 
 
 1. The circuit consists of two reactive branches in parallel, 
 respectively, of 10 ohms and .01042 henry, and 10 ohms and 
 
 R=10, L=. 01042 
 
 R =io, C = iso 
 
 Fig. 177. — Solution l. 
 
 150 microfarads. The diagrams, Fig. 177, show that the 
 admittance of the circuit is .118 mho and the angle 6 is equal 
 to zero. 
 
292 
 
 ALTERNATING CURRENTS 
 
 The vector expression for admittance is as follows : 
 
 Y 
 
 10 
 
 8.33 
 
 Y— 
 
 1 1 
 
 100 + 69.4 3 
 
 100 + 69.4 
 
 
 Y 2 
 
 10 
 
 8.33 
 
 z — 
 
 100 + 69.4 J 
 
 100 + 69.4 
 
 
 Y 
 
 = .1181 
 
 3 0 
 
 tan 6 — 
 
 .1181 
 
 0 
 
 .1181 
 
 = 8.47 ohms. 
 
 ,6 = 0 . 
 
 The effect of the reactances in this circuit is noteworthy. 
 It will be observed that the self -inductance and capacity neu- 
 tralize each other’s effects so that the angle of lag is zero, but 
 the admittance is not much more than one half as great as 
 would be the case if the two branches each contained 10 ohms 
 of resistance and no reactance. 
 
 When 100 volts are impressed on this circuit, 
 
 and 
 
 and 
 
 1= EY = 11.81 -j 0 
 I = EY = 11.81 amperes; 
 
 j EY = ^ <>( ^ — j 833. 
 1 1 169.4 J 169.4 
 
 I x - EY X - 7.7 amperes; 
 
 833 
 
 ^ — h : v — _i /i 
 
 I 0 = EY„=l^r+j 
 
 and 
 
 I, = EY,= 
 
 169.4 ' 169.4 
 
 7.7 amperes; 
 
 e,= 
 
 tan -1 _833_ _ 39 o 48 / Q = tan -i 
 
 1000 2 
 
 R=10, L=.01 
 
 r^nnnnr^n 
 
 C = ioo 
 
 833 
 
 = -39° 48'. 
 
 1000 
 
 When 10 amperes are 
 caused to flow through 
 the circuit, 
 
 10 
 
 E=IZ = 
 
 +j 0 
 
 Fig. 178. — Solution m. 
 
 .1181 
 
 and 
 
 E = IZ = 84. 7 volts. 
 
 Under the latter con- 
 ditions the currents in 
 the two branches are, 
 respectively, 6.52 am- 
 peres and 6.52 amperes. 
 
 m. The circuit con- 
 sists of two reactive 
 branches, respectively, 
 
SOLUTION OF CIRCUITS 
 
 293 
 
 of 10 olims and .01 henry, and of 100 microfarads. The 
 diagrams in Fig. 178 show the joint admittance to be .0685 mho. 
 The joint impedance is 14.6 ohms the impedances of the 
 branches are, respectively, 12.8 and 12.5 ohms, so that when 
 the impressed voltage is 100 volts, 6.85 amperes flow in the 
 main circuit, while 7.8 and 8 amperes, respectively, flow in 
 the branches. The angle 6 is — 27° 10'. 
 
 It will be noticed in this case that not only is the arithmetical 
 sum of the branch currents greater than the main current, but 
 each branch current is itself greater than the main current. 
 
 The solution by complex quantities is as follows: 
 
 Y 10 8 
 
 1 100 + 64 3 100 + 64 
 
 Z = 0 +./.08 
 
 Y = .0610 +/.0312 
 
 Y= ^.OOIO 2 + .0312 2 = .0685 mho. 
 Z — 14.6 ohms. 
 
 tand = -^^= - .511, 
 
 .0610 
 
 6 = - 27° 10'. 
 
 When 100 volts are impressed on this circuit, 
 
 
 I = EY= 6.1 +j 3.12 
 
 and 
 
 I = EY = 6.85 amperes; 
 
 
 T 1000 ' 800 
 
 1 “ 1 ~ 164 1 164 
 
 and 
 
 I x = EY 1 = 7.8 amperes; 
 
 
 T 2 = EY 2 =j 8 
 
 and 
 
 I 2 = EY 2 = 8 amperes; 
 
 - - tan 1 .8 = 38° 40', 0 2 = — tan 1 oo = — 90°. 
 
 When 10 amperes are caused to flow through the circuit, 
 
 E=IZ = -• 61Q - j ~ 312 - 
 
 .00469 . 00469 
 
 and E = IZ = 146 volts. 
 
 n. The circuit consists of two reactive branches in parallel, 
 respectively, of .01 henry, and of 10 ohms and 100 microfarads. 
 The diagrams, Fig. 179, show the admittance to be .0856 mho. 
 and the impedance to be 11.7 ohms. The impedances of the 
 branches are, respectively, 8 and 16 ohms, so that when 100 
 volts are impressed upon the circuit, 8.56 amperes flow in the 
 
294 
 
 ALTERNATING CURRENTS 
 
 main leads, while 12.5 and 6.25 amperes flow, respectively, in 
 the two branches. The angle 6 is 62° 53'. 
 
 L= .oi 
 
 Here one branch current is much larger than the main 
 current. 
 
 The solution by complex quantities is, 
 
 10 12J5 
 
 100 + 156.25 +J 100 + 156.25 
 
 .0391 -j .0762. 
 
 Y = V.0391 2 + .0762 2 = .0856 mho. 
 Z = ,. 0 -„ = 11.7 ohms. 
 
 tan 6 = - 
 
 0856 
 
 0762 
 
 0391’ 
 
 6 = 62° 53'. 
 
 When 100 volts are impressed on the circuit, 
 1= EY= 3.91 - j 7.62 
 and 1= EY — 8.56 amperes; 
 
SOLUTION OF CIRCUITS 
 
 295 
 
 and 
 
 I x = EY,= -j 12.5 
 I x = EY X = 12.5 amperes; 
 
 z-^F.-^l+y 1250 
 
 tan" 1 ^ 3 = - 51° 20'. 
 
 1000 
 
 2 2 256.3 ‘ " 256.3 
 
 and / 2 = EY% = 6.25 amperes; 
 
 0 X - - tan -1 oo = 90°, $ 2 = 
 
 When 10 amperes are caused to pass through the circuit, 
 E = IZ = 
 
 and 
 
 .391 , . .762 
 
 +J 
 
 ....33 ’ " .00733 
 E — IZ - 117 volts. 
 
 The impedance diagram of the combined circuit 0 2 BC is 
 given in the figure (Fig. 179). The sides of the triangle are 
 shown from the expression 
 
 Z = Z (cos 6 + j sin d) 
 
 Z= 5.34 + j 10.4. 
 
 L = .01 
 S = ioo 
 
 TT 
 
 ii 
 
 I O 
 
 • f 
 
 or 
 
 o. The circuit consists of 
 two reactive branches in 
 parallel, respectively, of .01 08 
 
 henry and of 100 microfar- 
 ads. The diagrams, Fig. 180, 
 show that the admittance 
 of the circuit is .045 mho, 
 and the admittance of the 
 branches are, respectively, 
 
 .125 and .08 mho. When 
 
 the impressed voltage is 100 i => /0=9O° 
 
 volts, the main current is 4.5 A 
 amperes, and those in the 
 branches are 12.5 and 8 amperes, respectively. The angle 6 is 90°. 
 
 The corresponding complex quantities are: 
 
 - 0.8 
 
 Y x = - 
 
 .125 
 
 + - 
 
 ±± 
 
 0 + 64 
 
 0 
 
 = 0 + 156 
 
 +.? 
 
 0 + 64 
 12.5 
 0 + 156 
 
 Fig. 180. — Solution o. 
 
 Y = .045 mho 
 
 Y = 
 
 0 -y.045 
 
 Z = 22.2 ohms. 
 
 045 
 
 tan 6 = — 7 - — = oc , 0 = 90°. 
 
296 
 
 ALTERNATING CURRENTS 
 
 In this problem the active (horizontal) component of current 
 is zero, which is a condition that can only be approximated in 
 practice. The vertical component of current composes the total 
 current. The branch currents are in exactly opposite phases, 
 and the main current is equal to their arithmetical difference. 
 
 When 100 volts are impressed on this circuit, 
 
 
 J = 
 
 EY 
 
 = 
 
 -j 4.5 
 
 and 
 
 I = 
 
 EY 
 
 = 
 
 4.5 amperes; 
 
 
 
 EY i 
 
 = 
 
 -y 12.5 
 
 and 
 
 
 EY, 
 
 = 
 
 12.5 amperes 
 
 
 I,= 
 
 EY , , 
 
 = 
 
 +ys 
 
 and 
 
 I 2 = 
 
 EY, 
 
 = 
 
 8 amperes ; 
 
 
 6, = tan -1 oo = 
 
 90°, 
 
 *2 
 
 = — tan -1 oo 
 
 When 10 amperes are caused to pass through the circuit, 
 E = IZ = j 222, 
 
 E = IZ = 222 volts. 
 
 12 
 
 p. The circuit consists of two reactive branches in 
 parallel, respectively, of .01012 henry and 150 microfar- 
 ads. The diagrams, 
 Fig. 181, show that 
 the two branch cur- 
 
 S = 160 
 
 L= .01042 
 
 X 
 
 .12 
 
 A, 
 
 rents are in opposition 
 and of equal value, 
 and that the admit- 
 tance is zero, so that the main current is zero. 
 A comparison of this problem with problem l 
 shows that this condition can only arise when 
 resistance in the branches is negligible. When 
 the impressed voltage is 100 volts, the branch 
 currents are each 12 amperes, and when 10 
 amperes flow in each branch, the voltage is 
 83.3 volts. No current can be caused to flow 
 in the main circuit leading to these branches. The complex 
 quantities representing the admittances are: 
 
 F x = 0-y.l2 Y= 0. 
 
 T„ = 0+,M2 
 
 Fig. 181.— 
 Solution p. 
 
 Y = 0 Tj.O 
 
 Z = oo ohms. 
 
 6 = tan -1 . -J = indeterminate. 
 
SOLUTION OF CIRCUITS 
 
 297 
 
 If there is any appreciable resistance, however little, in either 
 of these branches, Y and Z have finite values and 6 is zero. 
 In practical cases all resistance cannot be eliminated, and hence 
 the angle may be considered zero. 
 
 If circuits approximating those given in this problem are 
 constructed with very small resistance, excessively large local 
 currents may flow when the generator current is almost neg- 
 ligible. Such cases are not unknown in practical operation. 
 
 84. Conclusions in Regard to Parallel Circuits. — Second 
 Class. — - The sixteen examples just presented cover every 
 fundamental arrangement of simple parallel circuits. An 
 examination of the diagrams and the principles involved in 
 their construction makes evident the following statements, 
 applicable to circuits in which the voltages and currents are 
 sinusoidal, which are in many respects analogous to those here- 
 tofore given as applying to series circuits : 
 
 1. When non-reactive circuits are connected in parallel, the 
 total current equals the arithmetical sum of the currents in 
 the branches, and the joint admittance of the circuit is equal 
 to the arithmetical sum of the branch admittances. 
 
 2. When inductive circuits of equal time constants are con- 
 nected in parallel, the total current equals the arithmetical 
 sum of the currents in the branches, and the joint admittance 
 of the circuit is equal to the arithmetical sum of the branch 
 admittances. 
 
 3. When inductive and non-reactive circuits are connected in 
 parallel with each other, or when inductive circuits of unequal 
 time constants are connected in parallel, the total current is 
 equal to the vector sum, which is always less than the arith- 
 metical sum, of the branch currents, and the individual branch 
 currents are each smaller than the total current. The joint 
 admittance of the circuit is equal to the vector sum of the 
 branch admittances, each of which is less than the joint total. 
 
 4. When condensers are connected in parallel by wires of neg- 
 ligible resistance , the total current equals the arithmetical sum 
 of the branch currents, and the joint admittance equals the arith- 
 metical sum of the branch admittances. 
 
 5. When condensers are connected in parallel with non- 
 reactive resistances , the total current equals the vector sum, 
 which is always less than the arithmetical sum, of the branch 
 
298 
 
 ALTERNATING CURRENTS 
 
 currents, and the individual branch currents are each smaller 
 than the total current. The joint admittance of the circuit 
 equals the vector sum of the branch admittances, each of which 
 is smaller than the joint total. 
 
 G. When condensers are connected in parallel with inductive 
 circuits , the total current equals the vector sum, which is always 
 less than the arithmetical sum, of the currents in the branches. 
 Since the effects of capacity and of self-induct.ance, respectively, 
 cause the angle 6 to become negative and positive, the individ- 
 ual branch currents may be either greater or less than the 
 main or total current , depending upon the relation between the 
 various capacities and inductances in the circuit. The joint 
 admittance of the circuit equals the vector sum of the branch 
 admittances, each of which may be either greater or less than 
 the joint admittance. 
 
 The third, fifth, and sixth paragraphs make evident this 
 proposition, which is similar to that given for series circuits : * 
 When in parallel circuits the currents in the branches are all 
 either lagging or leading with respect to the impressed voltage, 
 the total or main current is always greater than the current in 
 any one of the branches. When the currents in part of the 
 branches lead the impressed voltage and in other branches lag 
 behind the voltage, some or all of the branch currents may be 
 greater than the total or main current. It is even theoretically 
 possible for the angles to have such a relation that a large cur- 
 rent may circulate in the branches, while the main current is zero. 
 
 This is obviously the result of the opposing relations of 
 inductive susceptance and capacity susceptance. It may be 
 conceived that a local current is caused to circulate between 
 the condenser and the inductance, under the stimulus of the 
 alternating impressed voltage, acting to transfer energy back 
 and forth between the capacity of the condenser and the 
 magnetic field of the inductance. 
 
 When the current is not sinusoidal, the deductions given 
 above do not strictly apply, but equivalent sinusoids f may 
 frequently be substituted, when the approximation of the de- 
 ductions is usually satisfactory. The deductions are general 
 when applied to the harmonics of current and voltage when the 
 latter are not sinusoidal. 
 
 * Art. 82. 
 
 f Art. 90. 
 
SOLUTION OU CIRCUITS 
 
 299 
 
 In every case referred to in these problems, it is assumed 
 that the parts of the circuits have no appreciable mutual mag- 
 netic effect. If the parts are mutually inductive, the mutual 
 effects must be added to those thus far treated, as will be seen 
 in a later chapter. 
 
 85. Solution of Parallel Circuits by the Impedance Methods. — 
 
 The solutions of parallel circuits may be made by another 
 method in which .voltages 
 and impedances are princi- 
 pally involved instead of 
 currents and admittance. 
 
 This method may be readily 
 exemplified by illustrations. 
 
 Suppose, for instance, it is 
 desired to find the joint 
 impedance of the branched 
 circuit in example e (Art. 
 
 83). It may be assumed 
 that an impressed voltage of 100 volts acts on the circuit. 
 Upon a line, OX , representing this voltage (Fig. 182) is drawn 
 a semicircle. From 0 draw the line OA making a lag angle of 
 
 6, with OX, where tan 6, = = XXLLa = .4. Then OA is 
 
 1 1 R x R x 
 
 equal to and XA is equal to I x X v since the angle at A 
 is a right angle. The current in this branch, when the irn- 
 
 OA 
 
 pressed voltage is equal to OX, is — and this may be laid 
 
 R i 
 
 off from 0 to B. The current in the second branch is given 
 by laying off the direction of the line OA' so that it makes 
 
 a lag angle of 0 2 with OX, where tan 0 2 = ^ — 1.25. 
 
 R 2 r 2 
 
 The current in the second branch is equal to OA' divided by 
 R 2 , and when laid off from 0 gives OB' . The total current 
 in the circuit is the resultant of OB and OB' , or OB" . Its 
 value in amperes is 16.5. The impedance of the circuit is 
 
 Fig. 182. — Resultant Current in a Parallel 
 Circuit. 
 
 then — : 
 6 = tan -1 
 
 OX _ 100 
 OB" 16.5 
 
 = 6.06 ohms. The angle of lag is 
 16'. Figures 183, 184, 185, 186, 187, 
 
 and 188 give the solutions by the same method for examples 
 
-12.5- 
 
 300 
 
 ALTERNATING CURRENTS 
 
 7=rl®2. . 5.95 
 
 u 16.8 
 
 Fig. 1S8. 
 
 Fig. 185. 
 
SOLUTION OF CIRCUITS 
 
 301 
 
 5, c, d , g , j of Art. 83. These show fully the application 
 
 of the method.* 
 
 It is seen that the triangles OXA, OXA and OXA" are 
 voltage diagrams with the apeces turned down for conven- 
 ience ; and that they can be converted into impedance dia- 
 grams by dividing the sides by the respective currents in the 
 branches and combined circuit. Therefore, it is possible to use 
 the complex quantities expressing the impedances for solving 
 the problem. Thus, 
 
 Z x = 10 -\-j 4. 
 
 Z 2 = 8 +j 10. 
 
 Z 1 = VlO 2 + 4 2 = 10.77 ohms. 
 
 Z 2 = V8 2 + 10 2 = 12.81 ohms. 
 
 t E 100 o Q 
 
 1, — — = — = 9.3 amperes. 
 
 1 Z x 10.77 1 
 
 T E 100 _ q 
 
 J2= ^ = m8i =7 - 8amperes - 
 
 6 1 = tan -1 .4 = 21° 48'. 
 
 6 2 = tan" 1 1.25 = 51° 20'. 
 
 I l = ij( cos 6 1 — j sin dj). 
 
 J x = 8.62 — j 3.45. 
 
 Likewise, J 2 =. 4.87 — j 6.09. 
 
 The two current vectors added give 
 
 1= 13.49 — j 9.54. 
 
 /= ^13.49 2 4 - 9.54 2 = 16.5 amperes. 
 
 
 E 
 
 I 
 
 100 
 
 16.5 
 
 = 6.06 ohms. 
 
 The angle of lag of the main current is 
 
 9.54 
 
 6 — tan 1 pgTjT) = 35° 16'. 
 
 This is evidently more cumbersome than using the branch 
 admittances directly, as in the process described in the preced- 
 ing article. 
 
 * Compare Bedell and Crehore, Alternating Currents , p. 292 ; Loppe et Bou- 
 quet, Courants Alter natifs Industriels, p. 111. 
 
302 
 
 ALTERNATING CURRENTS 
 
 R,= lo, L, —.005 
 R 2 =8, L 2 =.0125 
 
 BKWTSTinPH 
 
 86 . Series and Parallel Circuits Combined. — Third Class. — 
 Where series and parallel circuits are combined, the funda- 
 mental solutions already given apply, directly, and it simply 
 requires experience to acquire facility in the solutions relating 
 to the most complicated circuits. Several examples are given 
 below to indicate the general procedure. In these examples it 
 
 is desired to determine as before : 
 
 ( 1 ) the total impedance of the circuit, 
 
 ( 2 ) the lag angle between the total 
 current and the impressed voltage, 
 
 (3) the total current flowing under a 
 voltage of 100 volts, and (4) the volt- 
 age required to cause 10 
 amperes to flow. The 
 frequency is taken nearly 
 127J- cycles per second 
 so that 2 7rf = 800. 
 
 a. The circuit consists 
 of an inductive coil 
 ' x of 10 ohms and .01 
 henry in series with 
 a branched circuit similar to e , 
 Sect. 83. We know that the paral- 
 lel part of the circuit has an im- 
 pedance of 6.06 ohms, and that the 
 lag angle is 35° 16b Therefore OA 
 (Fig. 189) is laid off in the phase 
 diagram 6.06 units in length, and making the proper angle 
 with the horizontal. The line OA' is then laid off horizon- 
 tally R 3 ( = 10) units long, and OA" is then laid off verti- 
 cally 2 7 t/X 3 (=: 8 ) units long. In the vector diagram 0 1 A 1 
 is equal and parallel to OA , A^ 2 to OA'. and A^A 3 to OA . 
 The length of the line 0 X A 3 gives the impedance of the cir- 
 cuit, which is equal to 18.8 ohms. The current which flows 
 under a voltage of 100 volts is 5.32 amperes, and it requires 
 188 volts to cause 10 amperes to flow through the circuit. 
 The angle of lag, d, is the angle A 3 0 1 X. and is equal to 
 37° 34b 
 
 By means of complex quantities we take first the admittance 
 of the parallel branches: 
 
 Z =18.8 
 
 Fig. 189. — Solution a. 
 
SOLUTION OF CIRCUITS 
 
 303 
 
 Y a = .086.2 -j .0345 
 Y b = -0487 -j .0609 
 Tab = • 1349 -j. 0954 
 
 tan 6 aK = 0 AB = 35° 16'. 
 
 Z ... = 
 
 .1349 
 1 ( 
 
 1 
 
 Y ab = .165 mho. 
 
 .165 
 
 = 6.06 ohms. 
 
 The impedance Z AB can be divided into its components, thus 
 (note the change of sign caused by changing from admittances 
 to impedances) : _ 
 
 Z AB = -J~: 
 y ab 
 
 .1349 ..0945 
 
 .0272 .0272’ 
 
 or Z AB = 6.06(cos 9 AB +j sin 0 AB ) 
 
 = 6.06(.816 +j .577) 
 
 = 4.95 +j 3.50. 
 
 This impedance is to be added to the impedance of the 
 remaining portion of the circuit, giving : 
 
 Z JB = 4.95 +j 3.50 
 Z c = 10 + j 8 
 
 Z = 14.95 +j 11.50 
 tan e = 11^2, e = 37° 34h 
 Z = 18.8 ohms. 
 
 1= ^^ - =5.32 ampei’es when 100 volts are impressed. 
 
 18.8 
 
 U = 10 x 18.8 = 188 volts when 10 amperes flow. 
 
 The components of current when 100 volts are impressed on 
 the circuit are 
 
 I=* = 
 
 100 
 
 £ 14.95 +j 11.50 
 
 = 4.2 -j 3.23; 
 
 that is, the active current is 4.2 amperes and the quadrature 
 current is 3.23 amperes. 
 
 When 100 volts are impressed on the circuit, the current 
 flowing in the main circuit is 5.32 amperes, and the currents in 
 
304 
 
 ALTERNATING CURRENTS 
 
 the branches A and B are found from the condition that the 
 voltage across the branched part of the circuit is 
 
 E ab = Z AB I= 6.06 x 5.32 = 32.3 volts. 
 
 and 
 
 or 
 
 The currents in the branches are, therefore, 
 
 4=32.3 4 = 2.78 —j 1.11 
 4 = 2.99 amperes; 
 
 4 = 32.3 Y b = 1.57-/1.97, 
 
 4 = 2.52 amperes. 
 
 When 10 amperes are caused to flow in the main circuit, the 
 currents in the branches are in the same ratio as 2.99 to 2.52. 
 
 b. The circuit consists of an inductance of .01 henry in series 
 with a branched circuit having two parallel branches contain- 
 ing, respectively, 40 ohms and 100 microfarads. The joint im- 
 pedance of the branched part of the circuit is first found in 
 
 the usual manner. 
 
 40 — R, 
 
 100 = C2 
 
 -72° 40' 
 
 0 -026 A" 
 
 ’ L 3 =.01 
 
 This is 11.9 ohms, 
 and the lag is 
 - 72° 40'. In the 
 phase diagram, Fig. 
 190, OA! is there- 
 fore laid off equal 
 to 11.9 and making 
 angle of 
 40', and 
 laid off 
 
 a lag 
 - 72° 
 
 is 
 
 OA' 
 
 2 7 rfL( = 8) units 
 in length and mak- 
 ing a lag angle 
 of 90°. Laying off 
 the vector diagram 
 gives 0 X A 2 equal to 
 4.9 and making a lag angle of — 43° 40'. The current flowing 
 under a voltage of 100 volts is 20.4 amperes, and the voltage 
 required to cause 10 amperes to flow is 49 volts. When 10 
 amperes flow in the main circuit, the voltage at the terminals 
 of the branched circuit is 119 volts, and the currents which flow 
 through the resistance and the condenser are, respectively, 
 
SOLUTION OF CIRCUITS 
 
 305 
 
 3 amperes and 9.5 amperes. The voltage across the inductance 
 i 3 is then 80 volts. 
 
 By complex quantities the joint admittance of the parallel 
 
 branches is: ^ A „ r . „ 
 
 Y a = .025 +j 0 
 
 Y b — 0 + j .08 
 
 Y ab = .025 +j .08 
 
 0 ab = “tan 1 
 
 • uo _ _ 70° 40'. 
 .025 
 
 The components of the impedance of the branched circuit 
 
 r ab = 
 
 9 _ 
 
 .025 
 
 3.55 ohms, 
 
 
 Tab 
 
 (.084) 2 
 
 
 X AB = 
 
 b 
 
 .08 
 
 11.38 ohms. 
 
 
 T AB 
 
 (.084)2 
 
 
 Therefore the impedance of the total circuit is: 
 
 Z AB = 3.55 -j 11.38 
 Zc=0 +/ 8-0 
 Z = 3.55 — j 3.38 
 
 Z = V05 2 + fW = 4.9 ohms. 
 
 6 = - tan- 1 = - 43° 40' . 
 
 The components of the 
 branched circuit may be 
 obtained from the trigo- 
 nometric expression as 
 indicated in Solution a , 
 or, if more convenient, 
 from the logarithmic 
 formula. 
 
 b v If the frequency 
 in the preceding example 
 is cut down to 80, the 
 relations are materially 
 changed. The impe- 
 dance of the branched circuit becomes 17.8 ohms, and the lag 
 angle in it is — 63° 32'. The phase diagram, therefore, is as 
 
 3.55 
 
306 
 
 ALTERNATING CURRENTS 
 
 shown in Fig. 191. From the vector diagram it is seen that 
 the joint impedance of the whole circuit is 13.5 ohms, and the 
 total current is 54° ahead of the impressed voltage. The total 
 current flowing when the impressed voltage is 100 volts is 7.4 
 amperes, and it requires 135 volts to cause 10 amperes to flow. 
 When 10 amperes are flowing, the voltage at the terminals of 
 the branched circuit is 178 volts, and the currents which flow 
 through the resistance and the condenser are 4.45 and 8.9 
 amperes, respectively, while the voltage across the inductance 
 X 3 is 50 volts. To maintain a voltage of 100 volts at the 
 terminals of the divided circuit requires a voltage of 76 volts 
 impressed on the total circuit. With this voltage, 5.6 amperes 
 flow through the circuit. 
 
 tan e AB = 0 AB = -68°32'. 
 
 .02o 
 
 Y . R = .0561 mho. 
 
 Y A = .025 -/0 
 Y ri = 0 +/.0503 
 
 Y ab = . 025 +j .0503 
 Z A b =17.8 (cos 6 AB -j sin 0 AB ) 
 = 17.8 (.4456 -j. 8960) 
 Z_ AB = 7.94 -j 15.95 
 Z r =0 +y 5.03 
 Z = 7. 94 -j 10.92 
 
 ^ = ^k =17 - 8ol,ms - 
 
 tan0= e= - .54° O'. 
 
 7.94 
 
 Z = 13.5 ohms. 
 
 /= ““““7 = 7.4 amperes when 100 volts are imoressed. 
 13.5 1 
 
 E = 10 x 13.5 = 135 volts when 10 amperes flow. 
 
 When 100 volts are maintained on the divided part of the 
 circuit, 
 
 I AB = EY ab = .025 X 100 + j .0503 x 100 = 2.5 + j 5.03 
 and I AB = ^2.5 2 + 5.03 2 = 5.61 amperes. 
 
 The voltage measured between the terminals of the part C 
 under these conditions is 
 
 E c — Z c x 5.61 = (0 + j 5.03) x 5.61, 
 E c = 28.2 volts, 
 
 and 
 
 0 C = 90°. 
 
 The voltage impressed on the total circuit under these con- 
 
 dltlOllS IS T, -- r> -t y j j r *4-10 
 
 = o.61 Z = 44.5 —j 61.3, 
 
 E — 5.61 Z — 76 volts. 
 
 and 
 
SOLUTION OF CIRCUITS 
 
 307 
 
 It will be observed that the current, voltage, and angle of 
 lag in any part of the circuit under any conditions may be 
 computed by these processes when the conditions are suffi- 
 ciently known. 
 
 c. The circuit consists of a combination as shown in Fig. 
 192. The resistance of the branches of the circuit are R 1 = 8 
 ohms, i2 2 =10 ohms, R s = 5 ohms ; the inductances are Xj = .0125 
 henry, Z 2 = .01 henry, L z = .005 henry ; and the capacities are 
 
 0^ = 125 microfarads, (? 2 = 100 microfarads, C 3 = 150 micro- 
 farads. The diagrams show the impedance of the branched 
 part of the circuit to be 4.74 ohms, and its lag angle is 
 — 10° 12'. From the complete diagrams it is seen that the 
 joint impedance of the whole circuit is 10.95 ohms, and 
 the total current is 28° 10' ahead of the impressed voltage. 
 The total current flowing when the impressed voltage is 100 
 volts is 9.12 amperes, and it requires 109.5 volts to cause 10 
 amperes to flow. When 10 amperes are flowing, the voltage 
 at the terminals of the branched circuit is 47.4 volts, and the 
 currents which flow through the branches are 5.9 and 4.3 am- 
 peres, respectively. The voltage across the first part of the 
 circuit is then 66 volts. To maintain a voltage of 100 volts on 
 the branched part of the circuit requires an impressed voltage of 
 
308 
 
 ALTERNATING CURRENTS 
 
 231.1 volts on the whole. With this voltage 21.1 amperes flow 
 through the circuit. 
 
 The calculation by complex quantities may be made as 
 follows : — 
 
 8 +/ 10-/ 10 = 8 +/ 0 . 
 
 Z B = 10 +/ 8 -/ 12.5 = 10-/ 4.5. 
 
 Hence, 
 
 Y a = .125-/0 
 Y b =.083+/. 0374 
 Y ab = . 208+/. 0374 
 
 = V & 2 + y 1 = . 211 mho. 
 Z ab = -^1 = ohms- 
 
 tan d AB = — -, 6 ab = — 10° 12'. 
 9 
 
 Z AB = 4r~= 4.66 — / 0.84 
 
 Y 
 
 
 Z, = 
 
 5-/ 4. 33 
 
 Z 
 
 Z 
 
 9.66-/5.17 
 
 = ViZ 2 + AT 2 = 10.95 ohms. 
 
 9 = tan -1 — = — 28° 10b 
 
 R 
 
 The main current with 100 volts impressed between the cir- 
 cuit terminals is ri C\an r,PT 
 
 Jh _ 966 . 517 
 
 “ J~120 + ‘ ? 120’ 
 
 /= ^ = 9.12 amperes. 
 
 While, with 10 amperes flowing in the main circuit, 
 E = _ZZ = 109.5 volts, 
 
 and the voltage at the terminals of the branched circuit is 
 E AB = 1Z AB = 47.4 volts. 
 
 The currents flowing in the parallel branches are then 
 I A = E AB Y a = 5.9 amperes 
 and I B = E ab Y b = 4.3 amperes. 
 
 Also, the voltage across C is 
 
 E c — IZ C = 66 volts. 
 
SOLUTION OF CIRCUITS 
 
 309 
 
 To obtain a voltage of 100 volts on A and B requires that 
 
 T 100 oi 1 
 1 = = 21.1 amperes. 
 
 ^ A Ii 
 
 Hence the voltage impressed on the terminals is 
 E= IZ= 231.1 volts. 
 
 87. Solution of Series-Parallel Problems by the Impedance or 
 Impressed Voltage Method. — The second method of solution 
 for parallel circuits may be applied to circuits like those included 
 in the above article. Figure 193 shows the solution for 
 example b x made by that 
 method. In this it is 
 assumed that 100 volts 
 are impressed upon the 
 branched part of the cir- 
 cuit. Then lay off a 
 length OX on the hori- 
 zontal axis representing 7g 
 
 100 volts and mark Z ~ eT6i = 
 
 Q(J __ 100 _ 9 5 
 
 1_ 40"" ’ 
 
 Fig. 193. - 
 
 ■ Solution of Problem &i by Voltage 
 Diagram. 
 
 which is the current in the first branch. The current in the 
 second branch is 90° in advance of the voltage, and is repre- 
 sented by OC v which is vertical and 2 7rfC 2 B ( = 5.03) units in 
 length. The resultant of these currents is OC, which is 5.61 
 units in length. The impressed voltage measured across the 
 terminals of the entire circuit is the resultant of the 100 volts 
 at the terminals of the branched part of the circuit, and the 
 voltage required to pass 5.61 amperes through the inductance 
 L x = .01 henry. The line representing the latter voltage is 
 perpendicular to the line representing the current in the circuit. 
 Drawing a semicircle on OX , and from the intersection of OC 
 with the semicircle drawing a line to X gives the direction of 
 this voltage. The magnitude of the voltage is 2 71 -fL x I— 28.2 
 volts. This must be laid off from X to B , and the total im- 
 pressed voltage is represented by OB. This shows that when 
 100 volts are maintained at the terminals of the branched cir- 
 cuit, 76 volts must be impressed on the total circuit. The im- 
 pedances of the circuit may be calculated from the data thus 
 
310 
 
 ALTERNATING CURRENTS 
 
 found, as also can the voltage required to maintain a certain 
 current through the circuit. 
 
 Figure 193a shows the solution of example c by this method. 
 As before, the voltage at the terminals of the divided circuit is 
 assumed to be 100 volts for the purpose of the solution. This 
 is laid down as OX , and a semicircle is drawn upon the line as 
 a diameter. Tan 0 A is equal to zero, so that the current in the 
 first branch is laid off on OX to B , a distance of 12.5 units. 
 
 
 Fig. 194. — Solution of Problem c by Voltage Diagram. 
 
 Tan 6 A = .15, as shown by calculation, and the line OA' is laid 
 off at that angle from OX. From 0 on this line, OB' is laid 
 
 off equal to the current in the second branch, or ^ -OX . The 
 
 resultant of the lines OB and OB' is OB’\ which represents 
 the total current in the circuit. The total voltage impressed on 
 the circuit is the resultant of the voltage impressed on the divided 
 circuit, the active voltage due to resistance R y and the reactive 
 voltage due to L 3 and (7 3 . The active voltage required to pass 
 current OB" through R z is represented by 0(7, which is equal 
 to IR y The reactive pressure is perpendicular to this and 
 
 is equal to 2 7 rfL 3 I — r - ; it is represented by the line OB. 
 
 - irfCz 
 
 The voltage impressed on the parallel circuit is represented by 
 the line BE , which is equal and parallel to OX. The closing 
 line, OE, represents the impressed voltage E on the circuit 
 when the current is R and therefore the impedance of the circuit 
 
 isy = 10.95. The angle of lag is the angle COE = — 28° 10'. 
 
SOLUTION OF CIRCUITS 
 
 311 
 
 87a. Caution. — The foregoing deductions of Arts. 81 to 88 
 relate to the impedances and admittances in circuits within 
 which the currents and voltages are sinusoidal, and to the 
 effective values of such voltages and currents. They also relate 
 to the impedances and admittances encountered by single har- 
 monics of distorted voltages and currents. That is, impedances, 
 admittances, effective voltages, and effective currents combine 
 vectorially. But the student must constantly remember that 
 instantaneous values of voltages or of currents at a junction 
 point must be combined algebraically since the instantaneous 
 values are simple algebraic values and the vector relations 
 heretofore considered do not extend to them. It is frequently 
 possible also, for approximate solutions, to substitute equivalent 
 sinusoids * for the irregular curves, though on account of the 
 extreme activity which is sometimes exhibited by the higher 
 harmonics f — especially where a condition of resonance is ap- 
 proached — each problem should be carefully studied before 
 such a substitution is made. 
 
 * Art. 90. t Art. 69. 
 
CHAPTER VII 
 
 POWER, POWER FACTOR 
 
 88. The Power expended in a Circuit on which a Sinusoidal 
 Voltage is Impressed. — a. Non-reactive Circuit. In a circuit 
 without inductance or capacity, the current agrees in phase 
 with the voltage which sets it up. The rate of expenditure of 
 energy in the circuit at any instant is equal to the product 
 of the corresponding instantaneous current and voltage. The 
 average rate of expenditure of energy, or the average value of 
 the power expended in the circuit, during a complete period is 
 equal to the average of all the instantaneous products. Or, 
 
 ie At , but with the voltage and current sinusoidal 
 
 e 
 
 e m sin a and i = 
 
 R' 
 
 where T is the time, of a period and e m is the maximum voltage ; 
 hence, P = J ^ ie At = sin 2 ada = AA, 
 
 but E = -* 0 - and I = ^ == ■ 
 
 V2 E V2 R 
 
 Hence, P = ^ = IE, 
 
 I and E being the effective values of the current and voltage. 
 
 b. Reactive Circuit. If the circuit under consideration is 
 reactive, the current is caused to lag behind or lead the voltage 
 by the angle 6. The rate of expenditure of energy in the cir- 
 cuit at any instant is evidently still equal to the product of the 
 corresponding instantaneous values of the current and voltage. 
 
 312 
 
POWER, POWER FACTOR 
 
 313 
 
 The expression for the average power expended in the circuit 
 is, therefore, as before, 
 
 P — 2 ie A t. 
 
 o 
 
 In this case, however, with the voltage and current sinusoidal 
 e = e m sin a, and i sin (a — 6 ). * 
 
 Hence, P =•%= f sin a sin (a — 0)da — - m c< ^ ^ f sin 2 ada 
 ttZJo nZ Jo 
 
 e j sin Of 71 . , ej cos 0 
 
 - — — — I sin a cos ada = — — — — , 
 
 7 tZ 2 Z 
 
 and since e m = _E f V2, and — = IV 2, there follows 
 
 jP = IE cos 
 
 Also 
 
 cos 6 = 
 
 P 
 
 IE 
 
 Since sinusoidal current and voltage curves have equal posi- 
 tive and negative loops, the expression thus derived for the 
 power expended in a circuit during one half period applies to 
 every half period, and therefore to continuous operation. In 
 the ordinary measurements of current and voltage the effective 
 values of the quantities are determined. Consequently, the 
 product of amperes and volts , thus determined, does not represent 
 the powqr expended in a reactive circuit , but the product must be 
 multiplied by the cosine of the angle of lag. On the other hand, 
 a Wattmeter, that is, an electrodynamometer with one coil of 
 low resistance connected in series with the circuit and another 
 coil of high resistance connected in shunt with the circuit, aver- 
 ages the instantaneous products, and therefore gives readings 
 that are directly proportional to the power absorbed. 
 
 The preceding expressions show that the power expended in 
 a reactive circuit is equal to the voltage E, multiplied by a 
 component of the current which is in phase with the voltage 
 and is equal to I cos 0. This component may be called the 
 Active or Energy current. The remaining rectangular compo- 
 nent of current, I sin 6 , is in quadrature (that is, at 90° differ- 
 ence of phase) with the voltage and does not contribute to the 
 
 * Arts. 58 c and 62. 
 
314 
 
 ALTERNATING CURRENTS 
 
 power expenditure when taken through a complete period. 
 This quadrature component which does no work during a full 
 period may be called the Wattless or Quadrature current. For 
 illustration, suppose in Fig. 195 that OE is the vector voltage 
 
 impressed on a cir- 
 cuit, 01 is the cur- 
 rent, and 9 the angle 
 of lag. Resolving 
 01 into its compo- 
 nents in consonance 
 and in quadrature 
 with the voltage, 
 gives 0I a = 01 cos 9 
 and 0I X = 01 sin 9. 
 Multiplying the 
 former by E gives 
 the power expended, 
 El cos 9 , and the 
 component OI x is therefore inactive when averaged over a 
 whole period. 
 
 Since instantaneous power is equal to ei, it is obvious that 
 quadrature current must represent work done during some part 
 of the period, and that the wattless character must therefore 
 be due to alternate absorption and return of power by the 
 circuit. Taking the summation of eiAt for successive quarters 
 of a period proves this. Thus, 
 
 Fig. 195. — Vector Diagram of Current Components 
 and Voltage. 
 
 2 P 2 C'l 
 7 tZ ‘ / 0 
 
 and 
 
 r sin a sin (« y 90 °)da = — f sin a sin (« T 90 °)da, 
 •sn 7 rZ 
 
 2 
 
 f sin a sin (a T 90°)d« = 0. 
 
 7T Z Oo 
 
 This shows that the quadrature current is inactive only when 
 considered for any consecutive length of time equal to a full 
 half period. It in fact is a vehicle of energy which swashes 
 back and forth in the circuit, in one quarter period chargiug 
 up the magnetic or electrostatic fields and in the next quarter 
 period discharging the fields and returning the energy to the 
 source, the next quarter period charging the fields in the 
 opposite direction, followed again by the return of the energy 
 
POWER, POWER FACTOR 
 
 315 
 
 to the source in the succeeding quarter period, thus balancing 
 off the ebb and flow of energy in each half period and calling 
 for no continuous expenditure of power. 
 
 If 0 were 90°, the total current would be in quadrature with 
 the voltage and therefore inactive. This would be possible only 
 in a circuit having no electrical resistance or other grounds 
 for conversion of electrical energy into heat, light, or mechani- 
 cal power ; otherwise some power would necessarily be ex- 
 pended in heating the conductors, etc., and the current would 
 have to include an energy component. It is possible, how- 
 ever, to make the ratio of inductive reactance 2 nfL so great 
 in comparison with the resistance R, by using special inductive 
 coils, that the angle 0 is very nearly 90°. It is also possible to 
 make the capacity of a circuit so great in comparison with the 
 resistance that the angle 6 is nearly — 90° ; and thereby these 
 deductions in regard to wattless current may be tested in the 
 laboratory. 
 
 Prob. 1. The current flowing in a circuit is 65 amperes 
 and the voltage is 200 volts ; the angle of lag between the cur- 
 rent and voltage is 60°. What is the power expended in the 
 circuit? Would a change of frequency change the power, the 
 voltage, current and angle of lag remaining the same? 
 
 Prob. 2. Three reactive coils respectively comprising resis- 
 tance and self- inductance as follows: 4 ohms and .01 henry, 
 8 ohms and .02 henry, and 10 ohms and .05 henry, are con- 
 nected in series. Fifty amperes flow in the circuit. If the 
 frequency is 60 periods per second, how much power is ex- 
 pended in the total circuit, and how much in each part ? 
 
 Prob. 3. An overhead transmission line has a resistance of 
 2 ohms and a self-inductance of .0002 of a henry. This circuit 
 supplies fully loaded transformers which in effect give a non- 
 inductive load which has an equivalent resistance of 10 ohms. 
 If the voltage at the receiving end where the transformers are 
 placed is 1000 volts when the frequency is 60 periods per second, 
 what is the voltage at the generator, how much power is lost in 
 the line, and how much is utilized by the transformers ? 
 
 Prob. 4. The three reactive coils of Prob. 2 are placed in 
 parallel, and 100 volts with a frequency of 60 periods per 
 second are impressed between their terminals. What is the 
 total power expended in the circuit? 
 
316 
 
 ALTERNATING CURRENTS 
 
 89. Power in a Circuit. An Exercise. — Blakesley has given 
 a graphical proof of the formula P = IE cos 9 when I and E 
 are sinusoidal, which is presented here for the purpose of an 
 exercise. Applying the graphical representation of alternating 
 voltages and currents by means of rotating lines, let AB and 
 AO (Fig. 196) represent respectively the maximum value of 
 
 the impressed voltage in a 
 circuit and the maximum 
 value of the resulting cur- 
 rent. The angle BAC is 
 the angle of lag. If the 
 lines rotate counter-clockwise 
 about the point A, the in- 
 stantaneous projections of 
 the lines AB and AC upon 
 the axis of Y represent the 
 instantaneous values of the 
 voltage and current, when « is measured from the X axis. It 
 is therefore desired to determine the average value of the prod- 
 ucts of these projections for the purpose of obtaining the average 
 power for a period. Draw AB' and AC respectively perpen- 
 dicular and equal to AB and AC. These lines represent the 
 positions of AB and AC after revolving through 90°. In the 
 figure, the angle BAX represents a, and CAX represents a— 9. 
 Also, the angle B' AY = BAX , and CAY — CAX. It is then 
 seen from the figure that 
 
 AE x AD = AC sin CAX x AB sin BAX, 
 or ie = i m sin (a — d) x e m sin « ; 
 
 and in the same way 
 
 AE' x AD' = A C' cos CA Y x A B' cos B'A Y, 
 i'e' = i m cos (a — 6) X e m cos a. 
 
 The mean of these expressions is 
 
 I FrhU - — hAjn a s j n _ Qy _|_ cog a cos Q a _ J 
 
 _ l _m^m cog (a — 0)] = l C m . cos 9 = IE cos 9. 
 
 Fig. 196. — Voltage and Current Vectors. 
 
 or 
 
POWER, POWER F ACTOR 
 
 317 
 
 This is the expression for tire mean of the products of e and * 
 for two values of « which are 90° apart. This mean value is 
 independent of the positions of the lines in the figure, and 
 is therefore the mean for all positions. 
 
 90. Power expended in a Circuit when the Voltage and Cur 
 rent are Any Single-valued Periodic Functions. — If the current 
 in a circuit is not sinusoidal and is derived from a periodic 
 voltage of irregular form, the instantaneous values of current 
 and voltage are * 
 
 * = V sin (« + /3j) + i m% sin (2 « + /3 2 ) + i m% sin (3 « + £„) — 
 
 and e = e m ^ sin (a + 7l ) + e„ h sin (2 « + 7 2 ) + e m% sin (3 « + 7 3 ) • • • 
 + e mn sin (na + 7,,) . 
 
 Also, ie is the instantaneous value of power, and - eidt is the 
 average power ; therefore, 
 
 + J 0 imfm, sin (3 a + # 3 ) sin (3 a + 7 3 ) da — 
 
 + f 0 im n e mn sin (na + Bn) sin (% « + 7n ) da . 
 
 All the other terms obtained by multiplying the values of i 
 and e together are omitted as they become zero when integrated 
 between 0 and 2 7r.f 
 
 Expanding and integrating the above expression gives, 
 
 + *m, sin (na + £„), 
 
 sin ( a + £1) sin (“ + 7i) da 
 
 p = l i m e mi cos (/3j - 71) + \ SS c °s (/3 2 - 7 2 ) 
 
 + I hnf rn, COS (B, ~ 7 a ) • ■ • • + i COS (/3„ - J n ). 
 
 * Art. G7. 
 
 t Art. 13. 
 
318 
 
 ALTERNATING CURRENTS 
 
 Since each /3 represents the angular displacement of a current 
 harmonic from the zero of time and each 7 represents the angu- 
 lar displacement of a voltage harmonic from the zero of time, it 
 is apparent that each (/3 — 7 ) represents the angular difference 
 of phase between the voltage and current for the particular 
 harmonic ; that is, it corresponds with the angle of lag 9 associated 
 with the frequency of the particular harmonic. Therefore, 
 since 1 i m e m = IE , 
 
 P=I 1 E 1 cos 9 X + I 2 E 2 cos 9., 4 - I Z E Z cos 0 3 + ••• I n E n cos 6 n . 
 
 Each term to the right in this formula is the power expended 
 in the circuit by the particular harmonic and is independent of 
 the other harmonics. The total power expended in the circuit 
 is therefore the sum of the power independently expended b} r 
 the several harmonics. This theorem also follows from the 
 fact that the product of rotating vectors differing in frequency 
 is equal to zero, and hence I m E n = 0, where m and n are unequal 
 integers. It will be observed that the power expended in a 
 circuit when the current and voltage are sinusoidal may be de- 
 rived from the foregoing formula, since, in that case, all of the 
 higher harmonics are zero. 
 
 When 9 = 90° for any harmonic, that harmonic is inactive. 
 Also if a particular harmonic of current is present but the 
 corresponding harmonic of voltage is zero, the current harmonic 
 is obviously inactive ; and likewise, if a voltage harmonic is not 
 accompanied by a current harmonic of the same frequency, the 
 voltage harmonic does not contribute to average power, though 
 it may contribute to the energy that swashes back and forth in 
 the circuit and balances out for each half period. 
 
 The effective current squared in a circuit is 
 
 " 2 *- = I\ + 1% + I3 + "• In-> 
 
 and the power expended in heating a conductor of resistance R 
 when current I flows is represented by 
 
 ER = I? R + J 2 2 R + I* R + - I* R. 
 
 * Art. 67. 
 
POWER, POWER FACTOR 
 
 319 
 
 The active voltage in the circuit may be written for convenience 
 E cos 0, which, substituted for its equivalent IR in the last 
 formula, gives as before, 
 
 P = IE cos 0=1^ cos 0 l + I 2 E 2 cos 0 2 + I 3 E 3 cos 0 3 + etc. 
 With irregular waves the phase difference 0 cannot be 
 
 p 
 
 measured directly from the plotted curves, but 0 = cos -1 — . 
 
 IE 
 
 It is sometimes convenient to use the formula P = IE cos 0 in 
 place of the more complicated formula P = I l E 1 cos 0 X + etc., 
 and in that case I and E may be thought of as like sinusoidal 
 functions differing in phase by the angle 0. When so con- 
 sidered I and E are called Equivalent sinusoids. Under some 
 circumstances equivalent sinusoids can be substituted for com- 
 pound curves without error, but as harmonics of different 
 frequencies differ in their effects on circuits containing self- 
 inductance and capacity, the substitution must be made with 
 discretion. 
 
 91. Alternating and Pulsating Functions. — It has heretofore 
 been pointed out that commercial alternating currents and 
 voltages ordinarily comprise harmonics of only odd numbers of 
 times the fundamental frequency, namely, 1, 3, 5, 7, etc., in 
 which case the successive half cycles are like in form; but there 
 are nevertheless many instances in which harmonics appear of 
 even numbers of times the fundamental frequency. The fore- 
 going demonstration has therefore been made general in respect 
 to the frequencies of the harmonics, but the following dis- 
 cussion will be confined mostly to the formulas with the 
 even-numbered harmonics omitted. These may easily be re- 
 introduced, however, for use in connection with any particular 
 problem requiring their presence. It is also to be noted that 
 the introduction of a constant term into the general formulas 
 causes a larger average current to flow in one direction than the 
 other or converts the alternating functions into pulsating func- 
 tions. Then, 
 
 I 2 = 7 0 2 + I* + / 2 2 + I 2 + - I n \ 
 
 E 2 = E 2 + E* + E 2 2 + E 2 + - I n 2 ; 
 
 the power expended in heating a conductor of resistance R, by 
 I 2 R = I 2 R + Iy-R + I 2 R + I 2 R + • InR \ 
 
320 
 
 ALTERNATING CURRENTS 
 
 and, P = IE cos 8 = I^E^ + I\P\ cos 8 l + J 2 ^7 2 cos 0 2 
 
 + I 3 E s cos 0 3 + ••• I n E n cos 0 n ; 
 
 in which J 0 and E 0 respectively represent the average values of 
 the current and voltage for a whole period. 
 
 92. Power Loops. — - Power loops or curves may be plotted as 
 in Figs. 197 to 201, in which the ordinates of the dotted curves 
 
 represent the prod- 
 ucts of the corre- 
 sponding ordinates of 
 the current and vol- 
 tage curves (assumed 
 sinusoidal) plotted in 
 full lines. Figure 197 
 is for a non-inductive 
 circuit in which the 
 voltage and current 
 reverse their direc- 
 tions at the same 
 time, and the power 
 ordinates are there- 
 
 Fig. 197. -Power Loops in Non-reactive Circuit. fore always positive , 
 
 but their numerical values vary in each half period from 0 to 
 i m e m and back to 0, so that the power absorbed by the cir- 
 cuit varies continu- + + 
 
 ally during each half 
 period. In this case 
 6 — 0, cos 0 = 1 , and 
 the average power is 
 P = IE. Figure 198 
 shows the power 
 loops for a reactive 
 circuit in which the 
 angle of lag is 45°. 
 
 This may be taken to 
 represent equally the 
 condition when the 
 current leads or lags. It will be seen in this case that during 
 a portion of each half period the current and voltage are in 
 
POWER, POWER FACTOR 
 
 321 
 
 Fig. 199. — Power Loops in a Circuit having a 90° Lag. 
 
 opposite directions, and some of the ordinates of the power 
 loops are therefore negative. This must always be the case 
 when the current and voltage do not coincide in phase. During 
 the portion of the half period in which the ordinates of the 
 power loops are 
 positive the circuit 
 absorbs energy, but 
 during the portion 
 in which the ordi- 
 nates are negative 
 the circuit gives out 
 energy which was 
 stored as magnetic 
 field or condenser 
 charge and returns 
 it to the source. 
 
 The total work given to the circuit during the half period is 
 equal to the difference of that represented by the area of the 
 positive loop and that represented by the area of the negative 
 loop, and the average power absorbed by the circuit is equal 
 
 to this difference 
 divided by the 
 length of the half 
 period. When 
 
 0 = 45°, 
 
 P=IE cos 45° 
 = .707 IE. 
 
 When the ordi- 
 nates of the loops 
 represent prod- 
 ucts of instanta- 
 neous volts and 
 amperes, the 
 areas represent joules, and the areas divided by their respective 
 lengths measured in seconds give average power in watts. 
 
 Figure 199 shows the power loops for a circuit in which the 
 current and voltage differ in phase by 90°. Tn this case the 
 negative loops are equal to the positive ones, or the circuit and 
 source alternately give and take equal amounts of energy, so 
 
 Fig. 
 
 200. — Power Loops showing Negative Work. 
 Current lags 135°. 
 
322 
 
 ALTERNATING CURRENTS 
 
 that, taking each half period as a whole, no energy is absorbed 
 by the circuit. In this case cos 6 = 0, and therefore P = 0. 
 Figure 200 shows the power loops when the current lags 135°. 
 
 Under these circum- 
 stances the power is 
 negative, i.e. the cur- 
 rent is working against 
 the induced voltage, as 
 in a motor. 
 
 In case the curves of 
 voltage and current 
 are not sinusoidal, the 
 power loops are not 
 symmetrical. Such 
 loops are shown by 
 the broken lines in 
 Fig. 201. 
 
 Power loops are 
 periodic curves of 
 double the frequency of the voltage and current curves, since 
 they are obtained by multiplying current and voltage together, 
 and the argument of the product is equal to «+(a — d) = 2 u — 6. 
 That is, a cycle of the power loops is completed for each half 
 period of current and voltage. The power loops may be 
 expressed by means of Fourier’s series in the same manner as 
 any other single-valued periodic curve; thus, 
 
 Fig. 201. • — Power Loops derived from Irregular 
 Voltage and Current Curves. 
 
 p = P + p' m sin(2 a + 8 a ) + p" m sin (4 a + S 2 ) + etc.* 
 
 Where p is the instantaneous power corresponding to the 
 angle a, P is the net average power taken over a period, and 
 p' mi p" mi etc., are the amplitudes of the harmonics. 
 
 Prob. 1. A circuit has a resistance of 8 ohms and a self- 
 inductance of .01 of a henry in series. The voltage impressed 
 is 100 volts, at a frequency of 120 cycles per second. Con- 
 struct the power loops, assuming the voltage to he sinusoidal. 
 
 Prob. 2. A circuit has a resistance of 20 ohms, a capacity of 
 400 microfarads, and a self-inductance of .01 henry in series. 
 Draw the power loops produced when a sinusoidal voltage of 200 
 volts at a frequency of 40 cycles per second is impressed thereon. 
 
 * Art. 12. 
 
POWER, POWER FACTOR 
 
 323 
 
 Prob. 3. A circuit of 100 microfarads capacity and 5 ohms 
 resistance, lias a sinusoidal voltage of 50 volts at a frequency 
 of 60 periods per second induced in it. A sinusoidal current 
 of 50 amperes is caused by an external impressed voltage to 
 flow through the circuit in exact opposition to the induced 
 voltage. What is the magnitude and relative phase of the 
 impressed voltage ? Is the power positive or negative ? Con- 
 struct the power loops. 
 
 93. Trigonometrical Proof that Power Curves produced by Sin- 
 usoidal Voltages and Currents are Double Frequency Sinusoids 
 with their Axes Displaced. — Sinusoidal current and voltage of 
 equal frequency in a circuit having any fixed phase relation 
 produce instantaneous power that may in general be expressed 
 
 P = «»*» sin « sin (a - 0), 
 
 where 0 is the angle of lag. This expanded gives 
 P = e rJm [sin 2 a cos 9 — sin a cos a sin 0] . 
 
 The trigonometrical part of the expression on the right may be 
 written (i _ 1 cos 2 «) cos 9 — (1 sin 2 a) sin 9 
 
 = 1 cos 9 — I (cos 2 a cos 9 + sin 2 « sin 01 
 = | cos 9 — \ cos (2 a — 9). 
 
 Hence, the instantaneous power is 
 
 P = e rJm (I COS 0 - | COS (2 rt - 0)), 
 p = El cos 0 — El cos (2 « — 0). 
 
 The second term in the right-hand member of this equation is 
 a sinusoid having its maximum ordinate equal to AT, the appar- 
 ent power. It is displaced with reference to the voltage curve. 
 Since the instantaneous values depend on 2 a, it is obvious 
 that the curve goes through two cycles in the period of time 
 required for the current and voltage in the circuit to go 
 through one cycle, and the power therefore has double the 
 frequency of the voltage and current curves. 
 
 The first term in the right-hand member of the last equation 
 depends upon the cosine of the lag angle, which is assumed to 
 be constant. This constant term is added algebraically to the 
 
324 
 
 ALTERNATING CURRENTS 
 
 ordinates of the cos(2 a — O') curve of the second term, and there- 
 fore displaces the horizontal axis of symmetry of the power loops 
 above or below the axis of abscissas of the current and voltage 
 curves. Figure 200 illustrates the arrangement where the vol- 
 tage curve E has an effective value of 100 volts, and the current 
 curve / has an effective value of 30 amperes and lags 135°. The 
 axis of the power curve P is YY r , and is below the axis X in the 
 scale of watts a distance 
 
 ///cos 0 = 100 x 30 cos 135° = - 2121, 
 which is the average power expended in the circuit. 
 
 When the angle TO < 90°, the value of El cos 0 is positive, 
 and the average power is positive, i.e. is absorbed by the 
 circuit, and when t 0>9O°, as in the example above, the power 
 is negative, i.e. is delivered by the circuit. When 6 = 0, the 
 equation becomes p = EI(1 — cos 2 a), in which case the dis- 
 placement of the axis is maximum and the power loops just 
 touch the axis of abscissas as illustrated in Fig. 197. When 
 6 =90°, p = — El sin 2 a and the axis of symmetry of the power 
 loops corresponds with the axis of abscissas, as shown in 
 Fig. 199. 
 
 An examination of the power curves shows that an alternator 
 which is connected into a highly reactive circuit, or one in 
 which the phases of current and voltage are far apart, may be 
 under a serious strain even though it may be furnishing very 
 little power to the circuit. This follows from the fact that for 
 one quarter cycle the machine may be delivering large power to 
 the circuit and during the next quarter cycle, when the power 
 is negative, the circuit may be returning nearly as much power 
 to the machine, tending to drive it as a motor. This alternate 
 generator and motor action due to the positive and negative 
 loops is of special importance when there are alternators in the 
 circuit running in parallel or as synchronous motors.* 
 
 Prob. 1. Using the formula given on page 323, draw the 
 power loops for a sine voltage of 200 volts and a sine current of 
 50 amperes when the current lags 90°. Obtain the power loops 
 for the same current and voltage when the lag is 45°, 90°. 135°, 
 and 180°, 0°, — 45°, — 90°, and — 135°, and compare the sets of 
 loops with each other. 
 
 Chap. XII. 
 
POWER, POWER FACTOR 
 
 Prob 2. Draw the power loops in which the current and 
 voltage are expressed by the formulas i = 100 sin (« + 30°) 
 -f 10 sin (3 a + 60°), and e = 200 sin a + 15 sin 3 a + 5 sin 5 a. 
 
 94. Apparent Power ; True Power ; Power Factor. — The 
 
 product of the effective values of the current and voltage , IE, in a 
 reactive circuit is called the Apparent Power or Volt- amperes in 
 the circuit. The term Kilo-volt-amperes is also frequently used 
 (1 kilo- volt-ampere = 1000 volt-amperes). 
 
 The reading of a wattmeter applied to the circuit , which gives the 
 value of IE cos 6 , gives the True Power or Watts expended in 
 the circuit. 
 
 The ratio of the tvatts to the volt- amperes in a circuit is generally 
 called the Power factor, as originally suggested by Fleming. 
 
 The power loops for a circuit are exactly sinusoidal, provided 
 the original current and voltage curves are sinusoids. (Figs. 
 197 to 200.) When the voltage and current coincide in phase, 
 i.e. their phases are Coincident, the average ordinate of the power 
 loops is equal to one half of the maximum ordinate, since the 
 maximum ordinate is equal to i m e m , and the average ordinate is 
 
 equal to IE = 
 
 2 
 
 When the original curves are sinusoids 
 
 but their phases are not coincident, the average power ordinate 
 is equal to one half of the difference between the maximum 
 value of the positive ordinate and the maximum value of the 
 negative ordinate. When the curves are not sinusoids, the 
 power loops are not sinusoidal and the average power ordinate 
 does not necessarily depend at all upon the maximum ordinate. 
 
 When either or both the current and voltage waves are not 
 sinusoidal, the power factor may be determined by dividing the 
 volt-amperes, obtained from amperemeter and voltmeter read- 
 ings made in the circuit, into watts obtained from a watt- 
 
 p 
 
 meter reading; from which -— = cos d, where cos 6 is a positive 
 
 El 
 
 fraction less than unity. Under these circumstances there is no 
 fixed angle of phase difference which may be measured from the 
 relations of the plotted current and voltage curves and called 
 
 p 
 
 the angle of lag. But the angle 6 = cos -1 — may be called the 
 
 IE 
 
 equivalent angle of lag. It is equal to the angular displacement 
 
326 
 
 ALTERNATING CURRENTS 
 
 between equivalent sinusoids* * of voltage and current which 
 produce the same expenditure of power in the circuit. 
 
 It must not be assumed that curves of voltage and current 
 which are not sinusoidal and which have no apparent angular 
 displacement with respect to each other, or that dissimilar curves 
 in which the zero ordinates occur simultaneously will neces- 
 sarily give power factors of unity. In fact, potver factor of 
 unity can be obtained only when the voltage and current curves are 
 exactly similar and have no displacement with reference to each 
 other. Curves of current and voltage that are not angularly 
 displaced with reference to each other, but are not similar, pro- 
 duce a power factor less than unity. Curves of current and 
 voltage of relative symmetry, such as a semicircle associated with 
 a parabola, yield their maximum power factors when the zeros of 
 current and voltage occur simultaneously. Their maxima then, 
 likewise, occur simultaneously. When the current and voltage 
 curves are not of such relative symmetry, the maxima will not be 
 coincident in time when the zeros are, and the maximum power 
 factor is yielded when neither zeros nor maxima are coincident. 
 
 These theorems are easily proved and illustrated. 
 
 Power factor 
 
 p 
 
 ei~ 
 
 But e = e' m sin a + e" m sin 2 a + e"' m sin 3 a + etc. ; 
 
 - C eHu = Ef + Ef + Ef + etc. ; 
 
 7T 
 
 i = i' m sin (« — 0f)+ i" m sin (2 a — Of) + if" sin (3 a — Of) + etc. : 
 
 - f* Pda = If + If + If + etc. : 
 
 and 
 
 1 
 7 r 
 
 eida = E^f cos 0 X + E 2 I 2 cos 0 2 + E 3 I 3 cos 0 3 + etc. 
 
 Hence 
 
 p.f.= 
 
 E X I X cos 0 j + E 2 I 2 cos 0 2 + E 3 I 3 cos 0 3 + etc. 
 
 (Ef + Ef + Ef + etc.)^(/j 2 + If + If + etc .) 4 
 
 * Art. 90. 
 
POWER, POWER FACTOR 
 
 327 
 
 Two conditions must be fulfilled to make this unity : 
 
 E I E T 
 
 (1) - 1 — 1 — = 1, etc., which is equivalent to saying that 
 
 the voltage curve and current curve must be alike in form ; and 
 (2) = 0, # 2 = 0, 0 3 = 0, etc., which is equivalent to saying 
 
 that phases of corresponding harmonics must be in coincidence 
 with each other. 
 
 As an example of the association of two curves not of relative 
 
 Fig. 202. — Conventional Curves of Voltage and Current with coincident Zeros. 
 
 symmetry, consider those illustrated in Fig. 202, which shows a 
 right-angled triangle and a sinusoid with their zeros coincident. 
 The formulas of the curves for one half period are 
 
 1 * . . 
 
 e - — and i — i m sin «. 
 
 7 r 
 
 The effective values are, 
 
 and 
 
 The volt-amperes therefore become 
 
 The power is 
 
 — sin a — « cos a 
 
 •1 
 
 7T 
 
328 
 
 ALTERNATING CURRENTS 
 
 The power factor is p.f. = -p- = — - = .78. The two curves 
 
 Jhl 7 T 
 
 shown in Fig. 202 therefore give a power factor of only 78 per 
 cent when their zero ordinates coincide, and instrument readings 
 
 7T ® 
 
 Fig. 203. — Conventional Curves displaced — . 
 
 in the circuit would show an apparent angle of lag of 
 6 = cos -1 .78. Now suppose the current curve is retarded a 
 quarter cycle as shown in Fig. 203. The equations become, 
 for a half period, 
 
 e — 
 
 ^ m a 
 
 7T 
 
 and i = i m sin ( « — ^ )• 
 
 The power now is 
 
 P = - f « sin (a - p) d a 
 
 IT V '£J 
 
 — — sin a -f cos « 
 
 Hence the power factor is 
 
 1 e m i r 
 
 o + if 
 
 = -203 e m i m . 
 
 p.f. = T = .203f„;,x ^ = .497. 
 
 -tjl ^rn^rn 
 
 The power for any displacement k of the zero points of these 
 curves with respect to each other may be expressed thus : 
 
 p = gin , _ K ^ da 
 7T“ * / 0 
 
 e i C n 
 
 = I « ( sin a cos k — cos a sin /c) da 
 
 TT* J 0 
 
 = j^cos k ^sin a — a cos aj — sin k ^cos « + a sin rcj 
 = e -^f [tt cos k + 2 sin *]. 
 
 7T- L 
 
POWER, POWER FACTOR 
 
 329 
 
 The maximum power factor must occur when the power 
 itself is a maximum. Solving the above equation for a maxi- 
 mum gives 
 
 ^ = cos K + 2 sin *) 
 
 die <XK 7T^ 
 
 = ^(-7rsin« + 2cos *). 
 
 Equating this to zero and simplifying gives 
 
 2 
 
 — cos k — sin k = 0. 
 
 7T 
 
 Hence, k = 32° 30' when the power to be derived from these 
 
 p 
 
 two curves is a maximum, and then — - = .377 e m i m x — — = .924 
 
 ^ m i m 
 
 and the angle of lag indicated by instrument readings would be 
 6 — cos -1 .924 = 22° 29'. The maximum power factor thus be- 
 comes 92.4 per cent for curves of these particular forms. 
 
 It is obvious that the power factor will be zero when the re- 
 sultant power is zero or when 
 
 P = e -^P [tt cos K + 2 sin *1 = 0, 
 
 under which conditions the angle k is 122° 30' or — 57° 30' and 
 the angle of lag indicated by instrument readings would be 
 6 = cos" 1 0 = ± 90°. 
 
 The latter values of k are 90° different from its value when 
 power and power factor are at their maximum values. This 
 obviously must be so in this instance because one of the curves 
 is a pure sinusoid, and therefore maximum power occurs when 
 its phase and the phase of the fundamental harmonic of the 
 other curve are coincident, and zero power factor must occur 
 when these two are in quadrature. 
 
 The above example is especially convenient for purposes of 
 illustration on account of the simplicity of the curve formulas ; 
 and the results are comparable to those found in actual electri- 
 cal circuits. However, for the sake of closer approximation to 
 practical conditions, consider now a peaked curve of voltage and 
 a flat-topped curve of current each composed of two harmonics 
 as shown in Fig. 204, and represented by the following formulas : 
 
 e = 9 sin a — 3 sin 3 «, 
 i — 9 sin a + 3 sin 3 a. 
 
330 
 
 ALTERNATING CURRENTS 
 
 The effective values of these are 
 
 ■®=Vf+f = 6 -7and7=Aj| 
 
 92 , y2 ft 7 
 
 5 +T= 6 - 7 ' 
 
 The volt-amperes are, therefore, El = 45. 
 
 Fig. 204. — Peaked Voltage and Flat-topped Current Curves. Two Harmonics. 
 
 Angle k = 0. 
 
 The power is 
 
 P = - Ceida = - f 
 
 77*^0 77*^0 
 
 (9 sin a — 3 sin 3 a) (9 sin a + 3 sin 3 a) da, 
 
 -iff 
 
 7T L7 0 
 
 81 sin 2 ada 
 
 - r 
 
 •A) 
 
 9 sin 2 3 ada 
 
 = 36. 
 
 Or, using the formulas of Art. 90, 
 
 P = i COS + 2 COS 6>g, 
 
 = 4 1 - cos 0° + a cos 180° = 40.5 - 4.5 = 36. 
 
 An inspection of Fig. 204 affords an interpretation of this for- 
 mula. The first term of the first numerical member is evidently 
 the power given by the primary harmonics which are of coinci- 
 dent phases, and the power loops for which are marked P v 
 The second term is the power given by the third harmonics, 
 which is shown as P 3 in the power loops, and is negative as the 
 curves are in opposition or 180° apart. The power factor is 
 
 p 
 
 p. f. = -— =.80, and the apparent angle of lag is 0 = cos -1 
 El 
 
 .80 = 36° 52'. 
 
 In this instance not only are the zeros of current and voltage 
 coincident, but the pairs of harmonics each have unity power 
 factor (positive or negative) as they are either in exact coin- 
 cidence or exact opposition. 
 
POWER, POWER FACTOR 
 
 331 
 
 If the current curve is shifted 90° to the right, the formula is 
 P = — cos 90° + | cos 27 0° = 0, and p. f . = = cos 6 = 0. 
 
 Shifting the fundamental of current and voltage from coin- 
 cidence to quadrature generally does not cause all the harmonics 
 to fall into quadrature as in this example. For instance, if the 
 third harmonic of the current is i,„ 3 sin (3 a + 90°) instead of 
 i sin 3 «, the last formulas above become 
 
 P = -M. cos 90° + cos 0° =- 4.5 and p.f. = F r = cos 6 = .10. 
 
 A x 7 7T 
 
 In general the power factor of a circuit, as already demon 
 strated, may be expressed thus : 
 
 cos 6 = 
 
 where T7 and Tare the effective voltage and current; or in the form 
 
 cos 6 = F ^ C0S + F * T * C0s e ? ‘ + ••• + cos 6 n 
 
 PI 
 
 The following relations are expressed in the formulas : 
 
 a. The power factor in the case of sinusoidal curves is unity 
 when the phases of voltage and current are in coincidence or 
 opposition, and is zero when they are 90° apart. 
 
 b. When the voltage and current curves are not sinusoids, 
 the power factor is unity when the curves are in coincidence or 
 opposition, provided the curves are of exactly like forms. In 
 this case, coincidence occurs when the voltage and current curves 
 are in such relative phases that each current harmonic crosses 
 the zero value at the same instant and in the same direction as 
 the corresponding voltage harmonic ; that is, each current har- 
 monic is in coincidence with the corresponding voltage har- 
 monic. The current and voltage are in opposition when each 
 current harmonic is in opposition to the corresponding voltage 
 harmonic. 
 
 c. If the current and voltage curves are not of exactly like 
 forms, the power factor never becomes as large as unity for any 
 phase relation of the curves; and it may pass through zero 
 when the power factors of some or all of the harmonics are finite, 
 
332 
 
 ALTERNATING CURRENTS 
 
 but the algebraic summation of power produced by the several 
 harmonics is zero. 
 
 Power factor is expressed as a matter of convenience and 
 habit in terms of the cosine of an angle between 0° and 180°. 
 The foregoing discussion shows that in the case of sinusoidal 
 current and voltage, this angle is the same as the true angle of 
 lag or difference of phase between the curves. It is therefore 
 usual to call the cos' 1 p. f. the angle of lag, whatever form may 
 he assumed by current and voltage curves. When the current 
 and voltage are sinusoidal, the angle of lag may he scaled off 
 from a plot of the curves, hut this cannot readily be done when 
 the curves are of other forms on account of the effects of the 
 various harmonics. 
 
 When the power factor is positive , the circuit is absorbing poiver 
 from an outside source ; when it is negative , the circuit is deliver- 
 ing power to the source of the voltage under consideration. 
 
 Proh. 1. A circuit has an effective sinusoidal voltage of 500 
 volts impressed upon it and a sinusoidal current of 75 amperes 
 flows at a lag of 30°. What are the volt-amperes of the circuit, 
 the power expended, and the power factor ? 
 
 Proh. 2. Two circuits are in series, one having a resistance 
 of 5 ohms and a self-inductance of .01 of a henry, the other a 
 resistance of 10 ohms and a capacity of 200 microfarads, and a 
 sinusoidal current of 50 amperes flows through these two circuits 
 at a frequency of 40 periods per second. What is the power fac- 
 tor of the combined circuits and of each circuit separately ? 
 
 Prob. 3. A circuit is composed of four circuits in parallel, 
 the first having a resistance of 20 ohms, the second a resistance 
 of 5 ohms and a self-inductance of .01 of a henry, the third a 
 resistance of 8 ohms and a capacity of 100 microfarads, the 
 fourth a resistance of 2 ohms, a self-inductance of .02 of a henry, 
 and a capacity of 200 microfarads. When 1000 volts (sinu- 
 soidal) are impressed upon this circuit at a frequency of 60 
 periods per second, what is the power factor in the entire cir- 
 cuit, and of each of the four parts? Give the angle of lag or 
 lead in each part of the circuit and in the total circuit. 
 
 Proh. 4. A voltage having two harmonics, the first and the 
 third, of which the effective values are respectively 200 volts 
 and 10 volts, is impressed upon a circuit and a current flows 
 having first and third harmonics with effective values of 50 
 
POWER, POWER FACTOR 
 
 B33 
 
 and 5 each in coincidence with the corresponding voltage har- 
 monic. What is the power factor? 
 
 Prob. 5. A voltage having two harmonics, the first and the 
 third, with effective values of 200 and 50, is impressed at a fre- 
 quency of 60 periods per second on a circuit having a capacity 
 of 200 microfarads and a resistance of 5 ohms in series. What 
 current flows in the circuit and what is the power factor? 
 
 Prob. 6. The voltage of problem 5 is impressed upon a cir- 
 cuit having a resistance of 10 ohms and a self-inductance of .01 
 of a henry in series. What is the power factor ? 
 
 Prob. 7. The voltage of problem 5 is impressed on a circuit 
 having 5 ohms resistance, 200 microfarads capacity, and .2 of a 
 henry self-inductance in series. What is the power factor? 
 
 Prob 8. The voltage of problem 5 is impressed upon a cir- 
 cuit having a capacity of 200 microfarads. What is the power 
 factor? What would be the power factor if a sinusoidal vol- 
 tage of the same effective value were impressed on this circuit ? 
 
 Prob. 9. The voltage of problem 5 is impressed upon a cir- 
 cuit having a self-inductance of .2 of a henry. What is the 
 power factor? What would be the power factor if a sinusoi- 
 dal voltage of the same effective value were impressed on the 
 circuit? 
 
 Prob. 10. The voltage of problem 5 is impressed upon a cir- 
 cuit having 5 ohms resistance. What is the power factor? 
 What would be the power factor if a sinusoidal voltage of the 
 same effective value were impressed on the circuit? 
 
 Prob. 11. What would be the power factor in problem 4 if 
 the third harmonic of current was in opposition to the third 
 harmonic of the voltage ? 
 
 Prob. 12. A voltage curve having loops in the form of a 
 semicircle and with a maximum value of 200 volts is in coinci- 
 dence with a sinusoidal current having a maximum value of 
 100 amperes. What is the power factor? 
 
 95. Expression of Power Relations by Means of Vectors. — 
 
 The product of the vectors of current and voltage may be called 
 Vector power. Apparent power, which is given by the product 
 of the readings of the amperemeter and voltmeter in the circuit, 
 
334 
 
 ALTERNATING CURRENTS 
 
 or the “ volt-amperes,” is numerically equal to the product of the 
 tensors of current and voltage, and is a coefficient in vector 
 power. Apparent power may be resolved into two components 
 in quadrature about the argument 8. One of these components 
 is equal to IE cos 6 , or the reading of a wattmeter in circuit, and 
 is called real power or true power. The component perpendicu- 
 lar thereto is equal to jlE sin d, and is the power which is exerted 
 alternately positively and negatively in the circuit, but which 
 balances itself in and out during any half period so that its 
 average is equal to zero. The existence of the second compo- 
 nent is dependent on a condition that during one portion of the 
 half period the circuit absorbs power and during the remaining 
 portion the circuit delivers power to an exactly equal average 
 amount, so that a wattmeter reading will be zero with respect 
 to this latter component. Hence comes the absurdly inappli- 
 cable term wattless energy or power, which can better be called 
 Quadrature power or Reactive volt-amperes. It may also be 
 called the quadrature volt-amperes. A mechanical analogy of 
 the reactions of the quadrature power during a half period is 
 found in a cycle consisting of the frictionless raising of a weight 
 and its frictionless return to its initial position. In this case 
 mechanical power is exerted by some source to raise the weight, 
 but the weight delivers back the full power to the source as it 
 returns to its first position. The weight has here had work 
 exerted on it by the source' and has then returned an equal 
 amount of work to the source, and a mechanical power meter 
 would afford a zero reading as the result of this operation. 
 
 The power vector has the form 
 
 P = Pcjs-«/r, 
 
 and it is equal to the product of the current and voltage vectors, 
 or P = El = E( cos a+j sin oc) • Z[cos(« — 8) + j sin (a — 0)] 
 = Ee ]a x Ie j(a ~ 8) = IEe 3<2a ~ 9) = IEcjs (2 a — 8) 
 
 = (IE cos 8 — jlE sin 0)cjs 2 a. 
 
 Cjs-v/r is therefore equal to cjs 2 a, and it is clearly seen that 
 P has twice the frequency of Zand Z7, as heretofore explained. 
 Moreover, the apparent power P is shown to be composed 
 of two rectangular components. One of these (IE cos 9 ) 
 represents the true average power in the period, and the other 
 (jlE sin d) is the quadrature power which disappears when 
 
POWER, POWER FACTOR 
 
 335 
 
 taken over any complete half period and stands for the work 
 that swashes back and forth in the circuit but always gives as 
 much as it takes in any complete half period. 
 
 By drawing a curve through the points found by giving 
 various values to « from 0° to 360° in the expression Pcjs 2 a, we 
 get the well-known sinusoidal “power loops ” which are of twice 
 the frequency of the corresponding vectors of voltage and current. 
 The true power is zero when 
 
 cos 6=0, 
 
 that is, when 6 = ± 90°, in which case E and I are in quadra- 
 ture; and the quadrature power is zero when 
 
 sin 6 = 0, 
 
 that is, when 6 = 0° or 180°, in which case E and I are either in 
 coincidence or in opposition. 
 
 The algebraic sign of the angle 6 indicates whether the 
 quadrature power is the result of equivalent inductance or 
 equivalent capacity; and the numerical value of 6 indicates 
 whether the true power is delivered or absorbed by the circuit, 
 power being absorbed by the circuit when ± 6 < 90°, and de- 
 livered by the circuit when ± 6 > 90°. The application of the 
 commutative, associative, and distributive laws of algebra to 
 the vector formulas is not affected by doubling the frequency 
 in the equation. 
 
 These same deductions can be made, using the ordinary com- 
 plex form of the vectors. In this case the product of current 
 and voltage gives : 
 
 El = (a +jb')(^c +jd') = ac — bd + j ( ad + bc~) 
 
 = El j [cos « cos (« — d) — sin « sin (a — d)] 
 
 + j [sin a cos (a — d) + cos « sin (« — d)] j 
 = -ET[cos(2 a — d) +j sin (2 « — d)]. 
 
 This expression is the complex expression in which are shown 
 the horizontal and vertical components of the complete power 
 vector; that is, 
 
 ac — bd = El cos (2 a — d) 
 and ad + be = El sin (2 a — d). 
 
 This is not what we desire, as the valuable information to be 
 derived from the equation is the relative magnitude of the rec- 
 
336 
 
 ALTERNATING CURRENTS 
 
 tangular components of the apparent power corresponding to the 
 argument 0, since the average of the variable angle « over a 
 period is zero and 0 is constant. The forms given earlier have 
 shown that apparent power is a coefficient in the vector power 
 and is multiplied by (cos 6 — j sin d)cjs 2 « to give the vector 
 power. We now desire to transform El (cos 0 —j sin 0) into 
 terms of the rectangular components of voltage and current a, b, 
 c , and d. But, 
 
 cos 0 = cos [« — (« — #)] (see Fig. 205) 
 
 = cos a cos (a — 0) + sin a sin (a — 0) 
 a c , b d _ac-\-bd 
 ~e 1 + E' I ~ El 
 
 and, therefore, 
 
 El cos 0 = ac 4- bd. 
 
 Also, sin 0 = sin [« — (a — #)] 
 
 = sin a cos (a — 0) — cos « sin (a — 0 s ) 
 
 _ b c a d __bc — ad 
 
 ~E ' I~E ' I~ El ’ 
 
 and, therefore, 
 
 jEI sin 0 =j(bc — ad). 
 
 The true power is therefore equal to ac + bd and not to ac—hd\ 
 and the quadrature power to be — ad and not to be -f ad. 
 
 It is to be remembered 
 that the true power, ac + 
 bd, is the average value 
 for any full period of a 
 periodic quantity of double 
 frequency, Pcjs 2 a, the 
 origin of which is displaced 
 from the W-axis by a dis- 
 tance equal to El cos 0 ; 
 and the quadrature power 
 be — ad is the quadrature 
 component of El obtained by subtracting (ac -p bd ) 2 from 
 (Eiy. 
 
 The conditions are illustrated in Fig. 205, in which OA and 
 OB, respectively, represent vector voltage and current, OC rep- 
 resents vector power, and OB and BO, respectively, represent 
 IE cos 0 = ac + bd and IE sin 0= be — ad ; OE and EC repre- 
 
 c 
 
 Fig. 205. — Vector Diagram of Power Relations. 
 
POWER, POWER FACTOR 
 
 337 
 
 sent the horizontal and vertical components of vector power 
 obtained by the multiplication of the complex quantities, i.e. 
 ac — bd and ad + be. 
 
 The expression -Z/i (cos 6 —j sin 0) is of the nature of a vector 
 
 operator, similar in character to 
 
 — 7^2 
 
 El (cos 0 —j sin 0) = E 2 Y = — • 
 
 z 
 
 r = 
 
 Jr (cos 0-j sin 0) ; 
 
 and 
 
 Similar deductions are reached when the expressions are still 
 further generalized by assigning an initial fixed angle to the vol- 
 tage vector, in which case E = Ecjs (a — </>)and I = 7ejs(a — <p — 0). 
 
 Since the power factor is equal to true power divided by volt- 
 amperes, it must be equal to cos 0. In the same way the in- 
 ductance factor which is equal to quadrature power divided by 
 volt-amperes must be equal to sin 0. 
 
 We see from what goes before that the volt-amperes of a com- 
 pound circuit may not be equal to the algebraic sum of the volt- 
 amperes in the parts of the circuit, but they must always be equal 
 to the vector sum of the volt-amperes in the parts of the circuit. 
 
 Inasmuch as true power and quadrature power in the several 
 parts of a circuit are the rectangular components of the respec- 
 tive volt-amperes taken with respect to the corresponding argu- 
 ments 0 V 0 V etc., it is clear that the true power in any circuit 
 is equal to the algebraic sum of the powers in the parts of the 
 circuit, and the quadrature power in the total circuit equals the 
 algebraic sum of the quadrature power in the parts. 
 
 The conditions that are here set forth hold regardless of 
 whether the parts of the circuit are in series or in parallel. 
 
 Power will be assumed to be positive in every case where 
 it is absorbed by a circuit and negative when it is delivered 
 from the same. 
 
 Considering the semicircle diagrams of Figs. 142 and 143, 
 the impressed voltage is supposed to be constant and drawn 
 vertically upwards, and the vertical component of current in 
 each instance is the active component. Then the power ex- 
 pended in the circuit is equal to the voltage multiplied by the 
 vertical component of current. Consequently, the vertical com- 
 ponent of the current vector for each position in the semicircle 
 diagram is proportional to true power, since voltage is assumed 
 to be constant ; and the power corresponding to each current 
 
338 
 
 ALTERNATING CURRENTS 
 
 Fig. 206. — Locus of the Power Vector in a Series 
 Circuit where Resistance varies and Reactance 
 is maintained Constant. 
 
 vector may be laid off on that vector of a length from 0 which is 
 proportional to the vertical component. Following this process 
 
 for the Figs. 142 and 
 143, gives Figs. 206 and 
 207, in which the power 
 vector locus is repre- 
 sented by an oval-like 
 loop within each semi- 
 circle. The power per- 
 taining to any current 
 vector is equal to the in- 
 tercept on that current 
 between 0 and the oval 
 cutting it, as, for in- 
 stance, the power corre- 
 sponding to the current 
 01 with the given voltage is represented by OP , which is 
 numerically equal to the vertical component of current 01 when 
 P is unity. It will be observed that the ovals do not give any 
 new information about the nu- 
 merical value of the power for 
 each current vector, other than 
 that to be obtained directly from 
 the active current components 
 in the semicircle diagrams, but 
 the ovals show in a graphic 
 manner the way in which the 
 power varies with the current 
 over the entire range of condi- 
 tions considered. 
 
 It will be observed that each 
 semicircle in Fig. 206 contains 
 its own oval-like loop. Of the 
 four ovals occurring in this fig- 
 ure, two are positive and two 
 are negative as indicated. The 
 positive ovals represent condi- 
 tions in which power is absorbed 
 
 by and expended in the circuit by conversion into heat, light, 
 or mechanical power ; while the negative ovals represent the 
 
 Fig. 207. — Locus of the Power Vector in 
 a Series Circuit where Reactance varies 
 and Resistance is Constant. 
 
POWER, POWER FACTOR 
 
 339 
 
 conditions when the circuit generates and gives up energy as 
 in the case of an electrical generator. In each of these figures 
 the semicircles to the right of the vertical axis represent condi- 
 tions when the current lags behind the voltage, and the other 
 semicircles represent conditions when the current leads the 
 voltage. In passing from one oval to another in either figure, 
 it will be observed that either the current or the power factor 
 passes through zero. 
 
 It will be observed that the ovals of Fig. 206 show the maxi- 
 mum value of power when the angle of lag in the circuit is either 
 ± 45°, or ± 135°. It is easy to prove that the condition of maxi- 
 mum possible power transferred through an alternating-current 
 circuit with a fixed voltage and with fixed reactance and variable 
 resistance, as illustrated by Fig. 206, comes when the angle of 
 lag is ± 45° or ± 135°. The demonstration is as follows : 
 
 R E 
 
 Since P = IE cos 6 and since cos 9 = — and 1= — , there re- 
 sults the equation: 
 
 P = 
 
 E 2 R 
 R 2 + X 2 ' 
 
 This is obviously a maximum when R = X, at which time 6 = 45° 
 and the power factor is 70.7 per cent. It is equally easy to 
 prove that the condition of maximum possible power transferred 
 through an alternating current circuit with a fixed voltage and 
 with fixed resistance and variable reactance, as illustrated in 
 Fig. 207, comes when the angle of lag is 0° or 180°. In this 
 case, as before, 
 
 E 2 R 
 
 P = 
 
 R 2 + X 2 ' 
 
 whicli is obviously a maximum when X= 0. Under these cir- 
 cumstances the current is in coincidence with the voltage 
 (0 = 0°) when the circuit absorbs power, and in opposition to 
 the voltage (6 = 180°) when the circuit delivers power. 
 
 It may be noted in this connection that the condition of maxi- 
 mum power is a condition of little practical importance, as the 
 highest practicable operating efficiency or plant efficiency is 
 usually desired in the operation of electrical circuits and 
 machinery. A high plant efficiency is ordinarily incompatible 
 with a low power factor. 
 
340 
 
 ALTERNATING CURRENTS 
 
 96. Quadrature Components of Power Loops. — If 6 , the angle 
 of lag in a circuit, were ± 90°, the total current would be in 
 quadrature with the voltage, and therefore its vector product 
 with the voltage would be zero, as already indicated. A quad- 
 rature current may do a considerable amount of work in one 
 
 J ,I»r 
 
 sin a sin («— 90°)rfa = J,but during the 
 o 
 
 next quarter period the circuit returns an equal amount, and 
 the total work for the period is zero. (Compare power loops, 
 Fig. 199.) It is possible to make the ratio of inductive reac- 
 tance, 2 7r/X, so great in comparison with the true resistance 
 R of a circuit that the lag is nearly 90°. It is also possible to 
 make the capacity of a circuit so great in comparison with its 
 resistance that there is a lead of nearly 90°. The latter condi 
 tion is one not ordinarily met in practice, but the former may 
 quite easily be brought about in circuits including underloaded 
 transformers of poor design. 
 
 The effect of the quadrature element of a current lagging 
 less than 90° may be shown by components as indicated in Fig. 
 
 208, where the full line 
 curves E, I, I A , and I g , 
 respectively, represent 
 the voltage, current, ac- 
 tive component of cur- 
 rent, and the quadrature 
 component of current, 
 while the dash-dot line 
 represents the power 
 loops and the dotted 
 lines the components of 
 the power loops pro- 
 vided respectively by the 
 active and quadrature 
 components of current. 
 It is here seen that during each half period equal amounts of 
 power are received and given out on account of the quadrature 
 component, while the active component causes energ}' to be 
 absorbed by the circuit throughout the period. The power 
 loops due to the quadrature component really represent the 
 work stored in the magnetic field as the current rises, or stored 
 
 Fig. 208. — Power Loops caused "by Active and 
 Quadrature Components of the Current I. Lag 
 45°. 
 
POWER, POWER FACTOR 
 
 341 
 
 in the electrostatic field as the voltage rises, and given out as 
 it falls. 
 
 Figure 209 represents the component power loops formed by 
 the product of the current respectively with the active and 
 quadrature compo- 
 nents of the voltage. 
 
 The total current, to- 
 tal voltage, and total 
 power curves are iden- 
 tical with those of 
 Fig. 208. 
 
 The power loops 
 representing total in- 
 stantaneous power are 
 sinusoidal when E and 
 I are sinusoids, and 
 they Vary in position ^ IG - — Power Loops caused by the Active and 
 
 . . . Quadrature Components of the Voltage E, Lag 45°. 
 
 with respect to the 
 
 X-axis from just touching that axis when 6 = 0° to a symmetri- 
 cal location thereon when 6 = 90°. The power loops due to 
 the active component of current or voltage represent the irre- 
 versible power (the average of which over each period we call 
 true power) and are always located so as to just touch the X- 
 axis ; while the loops due to the quadrature component of 
 current or voltage represent the self-compensating quadrature 
 or reactive power and are always located symmetrically on the 
 X-axis with equal areas above and below the axis. 
 
 Prob. 1. A circuit has a reactance of 10 ohms and a resist- 
 ance of 10 ohms. When a voltage of 2000 volts, at a particular 
 frequency required to give the reactance named, is impressed 
 upon it, what are the values of the active and quadrature com- 
 ponents of the current ? Work this and the following problems 
 by means of complex quantities. 
 
 Prob. 2. A total current of 50 amperes at the given fre- 
 quency flows through the circuit in problem 1. What is the 
 impressed voltage, and what are the values of the active and 
 quadrature components of the current ? 
 
 Prob. 3. A circuit composed of three parts in series, the first 
 having a resistance of 10 ohms, the second a self-inductance of 
 
342 
 
 ALTERNATING CURRENTS 
 
 .02 of a henry, and the third a capacity of 100 microfarads, has 
 a sinusoidal voltage of 500 volts with a frequency of 120 periods 
 per second impressed upon it. What are the values of the active 
 and quadrature currents in the total circuit and in each part 
 taken respectively with reference, (1) to the voltage on the 
 total circuit, and (2) to that on the particular part considered ? 
 
 97. Reactive Factor; Table of Reactive and Power Factors. 
 
 — The value of the power factor of a circuit is equal numeri- 
 cally to cos 0, and the total current in a circuit multiplied by 
 
 TABLE OF POWER FACTORS AND REACTIVE FACTORS 
 
 Lag 
 
 Power 
 
 Reactive 
 
 Factor 
 
 
 Lag 
 
 Power 
 
 Reactive 
 Factor 
 sin 6 
 ± 
 
 
 Angle 6 
 
 Factor 
 
 
 Angle 6 
 
 Factor 
 
 
 ± 
 
 cos 6 
 
 ± 
 
 
 ± 
 
 cos 8 
 
 
 Degrees 
 
 
 
 Degrees 
 
 Degrees 
 
 
 
 Degrees 
 
 0 
 
 1.0000 
 
 .0000 
 
 90 
 
 23 
 
 .9205 
 
 .3907 
 
 67 
 
 1 
 
 .9998 
 
 •0175 - 
 
 89 
 
 24 
 
 .9135 
 
 .4067 
 
 66 
 
 o 
 
 .9994 
 
 .0349 
 
 88 
 
 25 
 
 .9063 
 
 .4226 
 
 65 
 
 3 
 
 .9986 
 
 .0523 
 
 87 
 
 26 
 
 .89S8 
 
 .4384 
 
 64 
 
 4 
 
 .9976 
 
 .0698 
 
 86 
 
 27 
 
 .8910 
 
 .4540 
 
 63 
 
 5 
 
 .9962 
 
 .0872 
 
 85 
 
 28 
 
 .8829 
 
 .4695 
 
 62 
 
 6 
 
 .9945 
 
 .1045 
 
 84 
 
 29 
 
 .8746 
 
 .4848 
 
 61 
 
 7 
 
 .9925 
 
 .1219 
 
 83 
 
 30 
 
 .8660 
 
 .5000 
 
 60 
 
 8 
 
 .9903 
 
 .1392 
 
 82 
 
 31 
 
 .8572 
 
 .5150 
 
 59 
 
 9 
 
 .9877 
 
 .1564 
 
 81 
 
 32 
 
 .8480 
 
 .5299 
 
 58 
 
 10 
 
 .9848 
 
 .1736 
 
 80 
 
 33 
 
 .8387 
 
 .5446 
 
 57 
 
 11 
 
 .9816 
 
 .1908 
 
 79 
 
 34 
 
 .8290 
 
 .5592 
 
 56 
 
 12 
 
 .9781 
 
 .2079 
 
 78 
 
 35 
 
 .8191 
 
 .5736 
 
 55 
 
 13 
 
 .9744 
 
 .2249 
 
 77 
 
 36 
 
 .8090 
 
 .5878 
 
 54 
 
 14 
 
 .9703 
 
 .2419 
 
 76 
 
 37 
 
 .7986 
 
 .6018 
 
 53 
 
 15 
 
 .9659 
 
 .2588 
 
 75 
 
 38 
 
 .7880 
 
 .6156 
 
 52 
 
 16 
 
 .9613 
 
 .2756 
 
 74 
 
 39 
 
 .7771 
 
 .6293 
 
 51 
 
 17 
 
 .9563 
 
 .2924 
 
 73 
 
 40 
 
 .7660 
 
 .6428 
 
 50 
 
 18 
 
 .9511 
 
 .3090 
 
 72 
 
 41 
 
 .7547 
 
 .6561 
 
 49 
 
 19 
 
 .9455 
 
 .3256 
 
 71 
 
 42 
 
 .7431 
 
 .6691 
 
 4S 
 
 20 
 
 .9397 
 
 .3420 
 
 70 
 
 43 
 
 .7313 
 
 .6S20 
 
 47 
 
 21 
 
 .9336 
 
 .3584 
 
 69 
 
 44 
 
 .7193 
 
 .6946 
 
 46 
 
 22 
 
 .9272 
 
 .3746 
 
 68 
 
 45 
 
 .7071 
 
 .7071 
 
 45 
 
 
 i 
 
 sin 6 
 Reactive 
 Factor 
 
 cos 6 
 Power 
 Factor 
 
 ± 
 
 e 
 
 Lag 
 
 Angle 
 
 
 ± 
 
 sin 6 
 Reactive 
 Factor 
 
 cos 6 
 Power 
 Factor 
 
 ± 
 
 8 
 
 La» 
 
 Angle 
 
POWER, POWER FACTOR 
 
 S43 
 
 the power factor is, therefore, equal to the active component of 
 the current. A factor which in the same way is proportional 
 to the quadrature current, may be called the Reactive factor of 
 a circuit. It is evidently equal in numerical value to sin 6. 
 
 Any table which, like the foregoing, gives natural sines and 
 cosines of the circular angles also gives the numerical values 
 of the reactive factors and power factors of the angles. 
 
 This table is given for angles of lag; that is, the values of 6 
 are positive. When the current leads, that is, 6 is negative, the 
 numerical value of the power factor (cos O') corresponding with 
 the particular value of 6 is unchanged, but reactive factor 
 (sin 6) changes from positive to negative value. The reactive 
 factor lias sometimes been called the Induction Factor, especially 
 in inductive circuits. 
 
 Prob. 1. What is the reactive factor in a circuit having a 
 resistance of 10 ohms, a capacity reactance of 10 ohms, and a 
 self-inductive reactance of 10 ohms in series, when a voltage of 
 100 volts at a frequency of 120 periods per second is impressed? 
 
 Prob. 2. The volt-amperes in a circuit equal 10,000 and the 
 reactive factor is .9. What is the power? 
 
 98. Method of Measuring Angle of Lag and Power Factor. — 
 
 The power factor, or cos 9 , whether the curves of voltage and 
 current are sinusoidal or not, may be determined by using vol- 
 tage, current, and power readings taken simultaneously, as already 
 explained.* The method for determining the angle of lag is as 
 follows : Measure the current flowing in a circuit by a correct 
 amperemeter ; measure the voltage at its terminals by an elec- 
 trostatic voltmeter or some type of non-inductive voltmeter of 
 relatively very high resistance. Finally, measure the power 
 absorbed in the circuit by means of a wattmeter, the voltage coil 
 of which is non-reactive and of relatively high resistance. The 
 power in watts determined by the wattmeter, when divided by 
 the product of volts and amperes, gives the cosine of the angle 
 of lag. If the curves of current and voltage are of irregular 
 form, this measurement will give the power factor which is 
 equal to the angle of lag between the equivalent sine curves for 
 the particular condition of the circuit.f We will later take 
 
 *Art. 94. 
 
 tArt. 90. 
 
344 
 
 ALTERNATING CURRENTS 
 
 up the effect of reactance in the voltage coil of the watt- 
 meter.* 
 
 Prob. 1 . The no-load terminal voltage of a generator having 
 a frequency of 40 periods per second is 2200 volts, as shown by 
 a voltmeter. When the machine is fully loaded on a non- 
 inductive load, the voltmeter measures 2000 volts, the ampere- 
 meter measures 100 amperes, and the wattmeter measures 200 
 kilowatts. The resistance of the armature is 1 ohm. What 
 is the angle of lag between the induced voltage (2200 volts) 
 and the current when the machine is fully loaded? Is the 
 wattmeter reading needed in this instance ? Why ? 
 
 99. Methods for Measuring the Power in an Alternating-current 
 Circuit. — It can be readily understood that the power in an 
 alternating-current circuit may be measured most accurately 
 and expeditiously by means of a wattmeter. The other methods 
 here given may serve a purpose in special cases or when a watt- 
 meter of the proper range is not at hand. They are all well 
 worth studying as problems. 
 
 1. Wattmeter Methods. Any instrument which directl} r 
 measures the true power in a circuit is called a wattmeter. 
 The common form of a wattmeter is an electrodynamometer f 
 with one of its coils connected across the terminals of the circuit 
 under test and the other in series therewith. Such instruments 
 of modern form are usually encased in a suitable box to make 
 them portable, and are proportioned so as to give direct read- 
 ings. The electrodynamometer in its ordinary amperemeter 
 arrangement with the two coils in series measures the value of 
 
 - Pdt, since the torque exerted on the movable coil at each 
 
 instant is proportional to the product of the currents in the 
 movable arid fixed coils. The average torque is therefore pro- 
 
 portional to 1 2 , since ■ fidt, as has been earlier shown, is 
 
 equal to 1 2 ; and the deflection of the movable coil, if its motion 
 is opposed by a spring, is D = KT 2 . when iThas a value fixed by 
 the conformation and character of the parts of the instrument. 
 To make such an instrument direct reading, it must be provided 
 
 * Art. 99, 1 a. 
 
 t Art. 24. 
 
POWER, POWER FACTOR 
 
 345 
 
 with a square-root scale, and the scale consequently is quite 
 open at the higher portion, but contracts badly toward the lower 
 parts, because the units of graduation are proportional to the 
 square root of the torque. As such a scale is inconvenient for 
 use, these instruments, when arranged for use as voltmeters or 
 amperemeters, are sometimes constructed so that the conductors 
 of the movable coil are located in the densest portions of the 
 magnetic field of the fixed coil when the deflections are small. 
 This arrangement manifestly counteracts in some degree the 
 contraction in the lower part that is inherent in the pure 
 square-root scale. 
 
 The assumption that the torque on the movable coil of any 
 such instrument is directly proportional to the product of the 
 currents in the two coils may be vitiated by various disturbing 
 influences, — particularly if the movable coil in its deflection 
 moves so as to appreciably alter the relations to each other of 
 the magnetic fields of the two coils, and if eddy currents can be 
 set up in any metal parts of the instrument in such a manner 
 as to introduce an additional and extraneous magnetic field. 
 If no such disturbing influences enter the situation, the calibra- 
 tion of such an instrument by means of currents of one fre- 
 quency, or even by means of direct currents, is sufficient to 
 standardize it for use with currents of any ordinary frequency ; 
 provided, however, that if one or both of the instrument coils 
 are to be used across the line as voltage coils they must be 
 non-reactive, or the calibration will be affected by frequency. 
 
 An additional explanation of these conditions is contained 
 in Art. 24. 
 
 When an electrodynamometer is arranged for use as a watt- 
 meter, one coil (ordinarily the movable coil) is planned for 
 connection between the terminals of the circuit within which 
 the power is to be measured and may be called the Voltage coil, 
 and the other coil is planned to be connected in series with the 
 circuit and may be called the Current coil. The connections 
 are illustrated in Fig. 51. In this case the torque on the mov- 
 ing coil, at each instant, is still proportional to the product of 
 the corresponding instantaneous currents in the two coils, and 
 
 i r T .. 
 
 the average torque is therefore proportional to — n^t, where 
 
 i represents the instantaneous current in the current coil and q 
 
346 
 
 ALTERNATING CURRENTS 
 
 the corresponding instantaneous current in the voltage coil. If 
 the coil is non-reactive or the frequency is constant, the current 
 in the voltage coil is directly proportional to the voltage im- 
 pressed upon the coil, which is the same as the voltage impressed 
 on the main circuit, and therefore i l is proportional to e , where 
 e is the instantaneous impressed voltage. Hence the average 
 torque acting on the movable coil of the wattmeter is propor- 
 1 C T . 
 
 tional to— I iedt = P ; or, D = KP , if the motion of the coil is 
 2 7 «/o 
 
 opposed by a spring and its deflection is represented by D. In 
 this case, P is the power expended in the circuit to which the 
 wattmeter is connected and K is a constant of the instrument 
 with a value fixed by the conformation and character of the 
 parts of the device. It will be observed that the deflection is 
 here proportional to the first power of P, and a uniformly 
 graduated scale is therefore used on such an instrument except 
 as modifications are required to correct for errors of construc- 
 tion of the parts. The torque of a wattmeter of this type, 
 where the coils are non-reactive, is therefore directly propor- 
 tional to the power, while the torque of instruments of the 
 same type used with the coils in series is proportional to the 
 square of the effective current. In the usual arrangement, 
 shown in Fig. 51, wattmeters of this class have a current coil of 
 a few turns of thick wire, which is placed in series with the cir- 
 cuit to be measured. The voltage coil is composed of a few 
 turns of fine wire in series with non-reactive resistance, and 
 is connected across the terminals of the circuit. 
 
 a. Errors in Wattmeter Readings on Account of the. React- 
 ance of the Coils. It is essential that the voltage coil of the 
 wattmeter be of entirely negligible inductance and capacity, or, 
 where the voltage and current of the circuit are harmonic, that 
 these constants be so mutually adjusted that the time constant 
 is practically zero. If this is not the case, the current in the 
 
 voltage coil is equal to — instead of where E is the 
 
 i?i R x 
 
 voltage in the circuit, 0 X the angle of lag in the voltage coil 
 
 X 
 
 which is dependent on the relation tan 6 X = aud R x and X x 
 
 At-i 
 
 are respectively the resistance and reactance of the voltage 
 coil. The currents in voltage and current coils now have 
 
POWER, POWER FACTOR 
 
 347 
 
 a difference of phase which is equal to 0 — 0 X instead of #, where 
 0 is the angle of lag in the main circuit. This is illustrated by 
 Fig. 210, where the vectors E and I are 
 the impressed voltage and the current 
 of the circuit to which the wattmeter is 
 attached, and I x is the current flowing 
 in the voltage coil and lagging 0 X de- 
 grees behind the voltage. In case the 
 voltage coil is affected by capacity 
 reactance, the angle 0 X is negative, 
 and should be so used in the com- 
 putations ; that is, current I x of Fig. 
 
 210 would lead voltage E and the 
 angle between I x and I would be- 
 come 0 — ( — 0 X ) = 6 + 0y The read- 
 ing of a wattmeter in which the 
 voltage coil is reactive is therefore proportional to 
 
 0 X cos (0 — 0 X ), R 
 
 Fig. 210. — Phase Diagram 
 showing Relation of Currents 
 and Voltage in Wattmeter 
 with Reactive Voltage Coil. 
 Lagging Current in Main 
 Circuit. 
 
 IE cos 
 
 since 
 
 T _ E cos 0 A 
 1 \ ~ 
 
 instead of — ■ 
 
 Z x = „ 
 
 COS <7 X 
 
 and therefore 
 
 A correct reading is propor- 
 
 R x 
 
 tional to IE cos 6. The readings of such a wattmeter must 
 therefore be multiplied by a factor equal to 
 
 -p _ cos 0 cos 0 
 
 cos 0 X cos {0 — 0j) cos 0 X (cos 0 cos 0 X + sin 0 sin d x )’ 
 
 in order that they may give the true power. This multiplier 
 may be called the “ correcting factor ” of a reactive wattmeter, 
 and is given here with the following discussion for the purpose 
 of illustrating the relation of the currents in the coils rather 
 than for use in measurement. 
 
 Since 
 
 cos 0 - 
 
 R 
 
 VA 2 + X 2 
 
 COS 0 X - 
 
 R^ 
 
 Vi^ 2 + X* 
 the correcting factor reduces to 
 RR X 2 + X x 2 R 
 
 sin 0 = 
 
 sin 0 X — 
 
 X 
 
 Vi? 2 + X 2 ' 
 
 X 
 
 Vi ? x 2 + X* 
 
 1 + tan 2 0 X 
 RR X + XX x R x 1 + tan 0 tan 0 X 
 
348 
 
 ALTERNATING CURRENTS 
 
 The formulas show that when tan 2 6 1 is negligibly small (in 
 which case 9 X is practically equal to zero), or tan 9 1 is equal to 
 tan 9 (in which case 9 X = 9 ), the correcting factor reduces to 
 unity, and the readings of the wattmeter are directly proportional 
 to power. When tan 9 1 is algebraically smaller than tan 9, the 
 correcting factor is less than unity, and the wattmeter reads too 
 high , and when tan 9 1 is algebraically greater than tan 9 , the cor- 
 recting factor is greater than unity, and the wattmeter reads too 
 low. The indications of a wattmeter with reactive voltage coil 
 may, therefore, be either correct, too high, or too low, depend- 
 ing upon the algebraic value of the time constant of the circuit 
 upon which measurements are being made. When 9 = 90°, the 
 correcting factor is equal to zero and the true power is zero 
 even though the wattmeter gives a finite reading, as it will do 
 under these circumstances if tan 9 1 has any appreciable value, 
 though it may be very small. As a general rule, the tan 9 
 of the circuit is likely to be positive 
 and greater than that of the wattmeter 
 coil, so that the readings of such a 
 wattmeter are generally found in prac- 
 tice to be too high ; but in ordinary 
 measurements it is impracticable to 
 make an advance determination of the 
 value of tan 9 , so that the correcting 
 factor of the wattmeter is unknown. If 
 
 Fig. 211. — Phase Diagram the main circuit has a leading current, 
 showing Relations of Cur- • •, • •. i ., , 
 
 . . w — is a capacity circuit, — and the vol- 
 
 rents and Voltage in Watt- 1 J 
 
 meter with Reactive Vol- tage coil of the wattmeter is reactive, the 
 tage Coil. Leading Current ve ctor relations are as in Fig. 211. If 
 
 in Main Circuit. 
 
 the voltage coil were also artected by 
 capacity, 9 t would be negative and 7 1 would lie, in Fig. 211, 
 between Jd and I. 
 
 It is evident that the correcting factors for either leading or 
 lagging currents depart far from unity when 9 approaches 90° 
 and that the indications of the instrument are likely^ to be far 
 from accurate under those circumstances for even microscopic 
 values of 9 V Indeed, as a rule ordinary electrodynamometer 
 wattmeters are not to be depended upon for accurate measure- 
 ment of power in circuits in which the lead or lag of the main 
 current approaches 90°. 
 
POWER, POWER FACTOR 
 
 349 
 
 The only safety in wattmeter measurements of power in alternat- 
 ing-current circuits , therefore, lies in the use of a wattmeter with 
 such a very small time constant in the voltage coil that it may he 
 considered absolutely negligible. This construction can be ac- 
 complished by placing a relatively very large non-reactive re- 
 sistance in series with the voltage coil, due care being taken to 
 avoid appreciable capacity between conductors of the resistance 
 coil, and also by making proper use of the mutual induction 
 between the coils.* It is very seldom that there is appreciable 
 self -inductance in the current coil of a wattmeter; but if there 
 is, the effect merely increases the current lag in the main circuit 
 or decreases its lead by a small amount and is therefore taken 
 into account in the above discussion. In ordinary commercial 
 instruments the capacity of the current coil is insignificant. 
 
 If the voltage is not sinusoidal, the inductance of the watt- 
 meter will cause a reactance having a value for each harmonic 
 proportional to the frequency of the harmonic. Thus the har- 
 monics of current through the voltage coil will each have dif- 
 ferent phase angles when compared with the harmonics of the 
 impressed voltage. As the power is equal to the sum of the 
 products of the effective values of the current and voltage of 
 each harmonic with the cosine of their phase difference, or 
 ^E n I n cos 9 n , it is evident that a separate correcting factor is 
 necessary for each harmonic. The large effect of any reactance 
 on tire higher harmonics may cause extremely untrue indica- 
 tions of a wattmeter with reactive voltage coil where it is used 
 to measure power in a circuit of low power factor. This may 
 occur with values of the voltage coil reactance that might be 
 considered negligible when the voltage and current are sinu- 
 soidal. The relation cos 6 = F cos 6 1 cos (d — 9f) does not hold 
 if equivalent sinusoids are substituted for the irregular currents 
 in the current and voltage coils and for the line voltage, for it 
 will be found that (0 — Of) + 0 1 > 9. Under these circum- 
 stances the current vector f of Fig. 210 cannot be in the plane 
 of the vectors I and E, but the three vectors must form a tri- 
 angular pyramid in which the faces give the angles IOE = 9, 
 IOI 1 — 9—9 v and I 1 OE=9 v A like relation holds for 
 Fig. 211. 
 
 b. Correction of Wattmeter Readings on Account of the Poiver 
 
 * Art. 23. 
 
350 
 
 ALTERNATING CURRENTS 
 
 Fig. 212. — Watt- 
 meter connected 
 so that the 
 Power used in 
 Voltage Coil is 
 added to that of 
 the Circuit being 
 Tested. 
 
 Absorbed by the Instrument. Another correction due to the 
 power used by the wattmeter itself is also necessary. Thus, if 
 the voltage coil is connected to the circuit be- 
 tween the current coil and the test circuit (Fig. 
 212), it is evident that the power measured 
 includes that absorbed by the voltage coil. If 
 the current coil is included between the point 
 of connection of the voltage coil and the test 
 circuit (Fig. 213), the power measured includes 
 that absorbed by the current coil. In either 
 case this power should be small and usually 
 may be neglected ; but when this is not the 
 case, it is easily determined from the resistance 
 of the coil included, if the voltage or current is 
 known. In some wattmeters a special correct- 
 ing coil wound over the series coil is introduced 
 in series with the voltage coil, which corrects 
 for the current in the voltage coil, — the in- 
 strument being connected as in Fig. 212. (Example : Weston 
 wattmeter.) 
 
 c. Effect of Eddy Currents in Wattmeter 
 Frame. As in the case of any electrodyna- 
 mometer or other instrument operated by 
 electrodynamic action, it is necessary that a 
 wattmeter of the type here discussed shall 
 have no metal in its frame in which eddy 
 
 currents may be developed.* If this precau- . 
 
 tion is not carefully looked after, the constant 
 of the instrument will vary with the fre- 
 quency, and a calibration is necessary for every 
 frequency. 
 
 The magnetic effects of any eddy currents 
 which are set up in the metal supports of an 
 instrument may have quite a marked effect 
 upon the magnetic fields around the coils, and as the intensi- 
 ties of the eddy currents are dependent upon the frequency 
 of the inducing current, the above statement about calibration 
 is evidently correct. For a properly built wattmeter, as said 
 above, which is used at a point near which there are no masses 
 
 * Art. 111. 
 
 Fig. 213. — Watt- 
 meter connected 
 so that the Power 
 used in the Cur- 
 rent Coil is added 
 to that of Circuit 
 being Tested. 
 
POWER, POWER FACTOR 
 
 351 
 
 of metal, a single calibration with direct currents is suffi- 
 cient. 
 
 2. Electrometer Method. If the two pairs of quadrants of a 
 quadrant electrometer are connected with points of potential, 
 respectively, V x and V 2 , and the needle is connected with a 
 point of potential V 3 , then the deflection of the needle is theo- 
 retically 
 
 ri + r? 
 
 Fig. 214. — -Diagram of Electrom- 
 eter Connections. 
 
 d = k (T\ - r 2 ) [y s - 
 
 where k is the constant of the electrometer. An explanation 
 of the truth of this formula may be made as follows. In Fig. 
 214 the electrometer needle is well under the quadrants, and, if 
 the needle mov.es slightly, the capac- 
 ity afforded by the quadrants toward 
 which the needle moves and the 
 portion of the needle under them is 
 increased slightly, due to the greater 
 surface exposed to the inductive 
 effect, while the quadrants from 
 which the needle moves and the por- 
 tion of the needle influenced by them 
 decrease in capacity. If c is the 
 change in capacity of either pair of 
 quadrants and the accompanying 
 portion of the needle for each very small angular movement of 
 the needle, then the amount of charge gained by the quadrants 
 toward which the needle moves is ca (E s — V 1 ') and by the 
 needle is ca (F[ — V 3 ~), where a is the angular distance moved. 
 Likewise, the charge lost by the quadrants from which the 
 needle moves is ca ( T r s — V^), and that lost by the needle is 
 ca(V 2 —V 3 ). These expressions come from the fundamental 
 equation, Q= CE. Equal charges of opposite signs must flow 
 to quadrants and needle. 
 
 It has been proved earlier that the work stored in a charge 
 is Q V, where V is the absolute potential of the charge. 
 Therefore, the energy gained by the first pair of quadrants is 
 I ca (Fg — V x ) V v and by the needle ca (V x — T r 3 ) V 3 . Like- 
 wise, the energy lost by the second pair of quadrants is 
 \ ca (V 3 — V 2 ) V 2 , and by the needle \ ca ( V 2 — V 3 ) V 3 . Sub- 
 tracting the energy lost from the energy gained, there results 
 
352 
 
 ALTERNATING CURRENTS 
 
 OT = c«(F 1 -F 2 )(r 3 -SAiS). 
 
 but the couple tending to turn the needle is equal to the work 
 done divided by the angular displacement. Therefore, the 
 torque _F, tending to turn the needle, is 
 
 T')(r 8 -G±iSj. 
 
 If the angle « is small, so that the surface distribution of the 
 charges is not materially altered, this torque should be constant 
 throughout the range of the deflection ; and when the restrain- 
 ing force of the needle suspension is proportional to the angle 
 of deflection, the deflection is proportional to the applied torque. 
 As a result, 
 
 d = k{v 1 -v 2 )(r 3 -^±Hy 
 
 as stated above. These conditions, however, cannot be rigor- 
 ously obtained in ordinary electrometers because the distribution 
 of the charges will not remain unchanged during the operations. 
 
 If v v v v and v 3 represent the instantaneous values of the 
 potentials at the points when varying alternatingly, similar 
 reasoning shows that the deflection becomes 
 
 d f Oi - ” 2 ') ( v 3 - dt - 
 
 If it is desired to measure the power absorbed by a reactive 
 
 circuit, the electrometer may be 
 used in the following manner : 
 The reactive resistance BC is 
 connected in series with the non- 
 reactive resistance AB (Figs. 
 215 and 216). Let the potential 
 of the points A , B , and C at any 
 instant be represented respec- 
 tively by v v v v and v 3 , when B is 
 
 Fig. 215.— Electrometer in First Posi- the junction between the react- 
 tion for obtaining; a Power Reading. . , , . . . 
 
 ive and non-reactrve resistance. 
 
 Then, if a quadrant electrometer is connected with its quad- 
 rants to A and B , and its needle and case to C (Fig. 215), the 
 deflection is 
 
POWER, POWER FACTOR 
 
 353 
 
 d = ^rSo 
 
 If the connection of the needle is interchanged so that it is 
 connected to B while the connections of the quadrants remain 
 unchanged (Fig. 216), this becomes 
 
 d ' = -f X^’ 1 ~ ~ 
 
 By subtraction, this results in 
 
 k C T 
 
 d' -d= — J o Oq - v 2 )(v 2 - v^dt. 
 
 Dividing this by kR , where R is the resistance of AB , gives 
 
 d' — d _ 1 r T v x 
 tX r 
 
 kR 
 
 ^(*’2 
 
 v^)dt. 
 
 Now 12 is equal to the instantaneous value of the cur- 
 R 
 
 rent passing through the circuit, and v% — v 3 is the correspond- 
 ing instantaneous value of the voltage between the terminals 
 of the reactive resistance BO. 
 
 d' — d l C T • 
 
 Consequently, ^ = — J iedt = P, where P is the power 
 
 absorbed by the reactive part of A B c 
 
 the circuit. AAAAAAn — 
 
 On account of structural de- 
 fects, the deflections of electrom- 
 eter needles do not always 
 follow the theoretical law, as 
 aforesaid. Consequently, it is 
 necessary to determine how great 
 the deviation is before the instru- 
 ment may be relied upon. Or, Fig. 216. — Electrometer ill Second 
 the instrument may be calibrated Posi *! on for obtaining a Power 
 by the use of direct currents 
 
 passing through known resistances, which are so adjusted that 
 v v v v and v 3 are nearly the effective values of the tests. 
 
 3. Electrostatic Wattmeter. A modification of the quadrant 
 electrometer may be made which reads directly as a watt- 
 
354 
 
 ALTERNATING CURRENTS 
 
 meter.* In this case the needle box is divided diametrically 
 into two parts instead of into quadrants. The needle consists 
 
 of a disc divided diametrically 
 into two parts (Fig. 217). The 
 points A and B of the circuit are 
 connected to the two halves of 
 the needle, and B and C to the 
 two halves of the needle box. 
 
 Then the force which causes 
 the deflection of the needle is 
 theoretically proportional to 
 the difference which is found by 
 subtracting the total loss of 
 energies of the quadrants and needles from the total gain due 
 to a small movement, exactly as was done in the previous case 
 for the electrometer. 
 
 Hence, 
 
 75 d 1 B r (v 1— fo)/- n. 
 
 Fig. 217. — Diagram of an Electro- 
 static Wattmeter. 
 
 since 
 
 -- 1 -— — : ^ is the instantaneous current and (p 2 — 1 > 3 ) the 
 
 R 
 
 instantaneous voltage in the circuit. 
 
 This instrument may also be calibrated, as explained above, 
 by passing a known direct current through a known resistance. 
 Wattmeters of this type 
 have been designed and 
 constructed, but they are 
 not very satisfactory. 
 
 4. Three-voltmeter Meth- 
 od .f As in the previous 
 method, a non-reactive re- 
 sistance must be connected 
 in series with the reactive 
 circuit to be tested (Fig. 
 
 218). Non-reactive voltmeters are then respectively connected 
 between the points A and B , B and (7, and A and C. Letting 
 
 'r ~©~ 
 
 r— ^ 
 
 I 
 
 k — (^y — 1 — (a^) — < 
 
 Fig. 218. — Connections for Three-voltmeter 
 Measurement of Power. 
 
 * Gerard’s Lemons sur V Blectricite, 3d ed., vol. I, p. 611 ; Hospitalier’s 
 Traite de V iSnergie Blectrique, vol. I, pp. 205 and 507. 
 
 t Suggested by Ayrton and Sumpner, London Electrician, vol. 20, p. 736; 
 Electrical World , vol. 17, p. 329. 
 
POWER, POWER FACTOR 
 
 355 
 
 e v e 2 , and e represent instantaneous voltages at the three volt- 
 meters, then e = e 1 + e v whence 
 
 p* p & p & — V p p 
 
 O C-i on — 01 Z', 
 
 1 D 2* 
 
 But the instantaneous value of the power in the inductive 
 
 • £ 
 
 circuit is p = ie 2 = e 2 . Substituting the value of e 1 e 2 already 
 
 R ' 
 
 found gives p = — (e 2 — <q 2 — e 2 2 ), and the mean power ab- 
 2 R 
 
 sorbed during a period is 
 
 p = = dUC<^ ~ e ' 2 - ^ 
 
 
 where E, E v and E 2 are the respective readings of the volt- 
 meters. If R is not known and the value of the current I is 
 known, the formula may be written 
 
 The letters E representing the effective voltages replace the 
 letters e representing the instantaneous voltages in the formula 
 after the integration is performed on the formula, because 
 
 — f e 2 dt = av • e 2 = E 2 , 
 
 tJo 
 
 as has already been proved.* 
 
 In order that the results of measurements by this method 
 may be the most accurate possible, E x should equal E 2 , which 
 makes the method inconvenient for use in ordinary testing. 
 Neither is the method sufficiently accurate to compensate for 
 its disadvantages. The accuracy of any particular measure- 
 ment made by this method may be checked by inserting a 
 known non-reactive resistance in place of the reactive cir- 
 cuit. This method, and methods 5 to 8, inclusive, are not 
 desirable for ordinary measurements and only prove useful 
 under special conditions. 
 
 * Art. 5. 
 
356 
 
 ALTERNATING CURRENTS 
 
 5. Three-ammeter Method. Instead of putting a non-re- 
 active resistance in series with the reactive circuit, it may 
 
 be placed in parallel with 
 it. In this case ampere- 
 meters must replace the 
 voltmeters of the preced- 
 ing method (Fig. 219). 
 One amperemeter meas- 
 ures the whole current /, 
 another measures the cur- 
 rent Jj in the non-reactive resistance B, and another meas- 
 ures the current I 2 in the reactive circuit. It is evidently 
 essential that the amperemeters are of negligible reactance. 
 Supposing i, i v i 2 are the instantaneous values of the currents 
 at any moment, we have 
 
 Pig. 219. — Connections for Three-ammeter 
 Measurement of Power. 
 
 i = i l + i 2 and i 2 — i 2 — i 2 = 2 ip v 
 while p = ei 2 — Rip 2 = ^ (i 2 — — t 2 2 ). 
 
 Whence P = ^ pdt = ^{I 2 -I 2 - I 2 ^>. 
 
 This may also be written 
 
 - 1 J i 
 
 In this case the greatest accuracy is given when and I 2 are 
 about equal; but, at the best, the method is not very exact. 
 Its accuracy may be checked, as in the previous case, by replac- 
 ing the reactive circuit by a 
 suitable non-reactive resist- 
 ance. 
 
 6. Other Three-instrument 
 Methods. Various modifica- 
 tions of the last two methods 
 have been suggested by Ayr- p IG ooo. — Connections for measuring 
 ton, Sumpner, Blakesley, and Power using Two Ammeters and One 
 
 others. One of the obvious Aoltnuter - 
 
 arrangements is to omit the amperemeter in series with the 
 non-reactive resistance of the fifth method, and connect a volt- 
 
POWER, POWER FACTOR 
 
 357 
 
 meter across the circuit as in Fig. 220. In this case the power 
 
 becomes 
 
 
 7. Split Dynamometer Methods. If separate alternating 
 currents of the same frequency are passed through the two coils 
 of an electrodynamometer, its reading will be proportional to 
 
 1 C T • 
 
 — J ip 2 dt. This is equal to I X I 2 cos 6. For 
 
 = V2 /j sin a and i 2 = V2 I 2 sin (a — 0 ), 
 
 where 0 is the angle between the two current waves, 
 stituting gives 
 
 \ rr i) r*T 
 
 •y Jo i ih dt =JiJ J 2 sin a sin (a - 0) dt, 
 which is equal to 
 
 - § sin a sin (a — 0 ) da = I X I 2 cos 0. 
 
 Sub- 
 
 An electrodynamometer used in this manner is called a Split 
 dynamometer. Now suppose we determine the values of I x and 
 I 2 by means of a dynamometer used as an amperemeter or by 
 other instruments, then the value of cos 0 is at once found. If 
 the measurements are all made by the same electrodynamometer, 
 its constant does not need to be known. Suppose the readings 
 in the two circuits are J x 2 = Jc$ v and / 2 2 = &S 2 , and the reading as 
 
 a split dynamometer is I X I 2 cos 0 = 7cS 3 , then 
 This plan was first suggested by Blakesley.* 
 
 8. Blakesley s Split Dyna- 
 mometer Method. Blakesley 
 planned various methods for 
 using a split dynamometer in 
 measuring the power absorbed 
 by a reactive circuit, f In 
 one of the methods a non- 
 reactive resistance 
 
 cos 0 - — - 3 — . 
 
 vs,s 2 
 
 Fig. 
 
 221. — Measurement of Power by a 
 Split Dynamometer. 
 
 is con- 
 nected in parallel with the reactive circuit to be tested (Fig. 
 221), and a split dynamometer is connected so that one coil 
 
 * Alternating Currents of Electricity, 2d ed., p. 97. 
 t Phil. Mag., vol. 31, p. 346. 
 
358 
 
 ALTERNATING CURRENTS 
 
 carries the total current, and the other carries the current of 
 the reactive branch. An amperemeter is also placed in the 
 reactive branch. Calling i the instantaneous value of the 
 total current, and i v i 2 , respectively, the instantaneous currents 
 in the non-reactive and reactive circuits, the following re- 
 lations hold: the reading of the split dynamometer is propor- 
 tional to 1 I 2 cos 9, and that of the amperemeter gives I 2 \ but 
 
 i\h = 0 h)h = ~ *2 2 ’ anc ^ Rhh = — f 2 2 )- e d ua ^ 
 
 to the instantaneous voltage between the terminals of the non- 
 reactive resistance, and therefore Ri 1 i 2 is equal to the instan- 
 taneous value of the power absorbed by the reactive circuit. 
 Integrating gives 
 
 = R(1 I 2 cos 9) - Rif = kRD - Rif, 
 
 where D is the scale reading of the split dynamometer, and k is 
 its constant. Hence, the power absorbed by the reactive cir- 
 cuit is equal to R times the difference between the reduced 
 split dynamometer reading and the square of the current in the 
 reactive circuit. 
 
 A similar result may be gained by putting the amperemeter 
 in the non-reactive branch, provided the instrument is itself 
 non-reactive. 
 
CHAPTER VIII 
 
 POLYPHASE CIRCUITS AND THE MEASUREMENT OF 
 POWER THEREIN 
 
 100. Polyphase Systems. — A Polyphase system is a system 
 comprising a number of simple alternating-current circuits 
 carrying currents of different phases used conjointly to obtain 
 the advantage of combined vector relations. When the simple 
 circuits number more than two, the several line voltages are 
 capable of forming a closed vector polygon, and the several 
 line currents of likewise forming a closed vector polygon. The 
 circuits of such a system may be operated as separate single- 
 phase circuits, or jointly as a polyphase circuit. 
 
 A polyphase system is said to be Balanced when the voltages 
 of the principal single-phase circuits are numerically equal to 
 each other and alike in form, and the currents of the same cir- 
 cuits are also numerically equal to each other and alike in form; 
 with the additional proviso that when the simple circuits number 
 only two, the difference in phase between the two voltages is 
 90° and the difference in phase between the currents is also 90°. 
 When the number of single circuits composing the balanced poly- 
 phase system is greater than two, the phase difference between the 
 
 2 7 r 
 
 successive voltages or successive currents is — , where n is the 
 
 n 
 
 number of phases ( i.e . the number of simple circuits composing 
 the polyphase system). Polyphase systems are usually operated 
 with either two currents with approximately 90° difference of 
 phase, called quarter-phase or two-phase currents, or three cur- 
 rents with approximately 120° phase difference, called three- 
 phase currents. 
 
 The transmission circuits for quarter-phase currents may be 
 arranged to be entirely independent of each other, four wires 
 being then required (Fig. 37) ; or, three wires may be used, in 
 which case one of them is common to the two currents (Fig. 38) 
 and the current in the third or common wire, at any instant, is 
 equal to the algebraic sum of the currents in the other two. 
 
 359 
 
360 
 
 ALTERNATING CURRENTS 
 
 The algebraic sum of the instantaneous currents in the three wires 
 is always equal to zero , the total return current being of course 
 
 equal to the total 
 outgoing current 
 at every instant. 
 The effective 
 current in the 
 common return 
 wire is equal to 
 the vector sum of 
 the two circuit 
 currents ; and it 
 is, therefore, 
 a/2 Z, where I is 
 the effective cur- 
 rent in one cir- 
 cuit, provided the currents are equal in the two circuits and 
 have a phase difference of 90°, which is the condition when the 
 system is properly designed and sym- 
 metrically loaded or Balanced. The 
 voltage between the two outside wires of 
 the quarter-phase system with common 
 return is the vector sum of the two cir- 
 cuit voltages, and is, therefore, a/2 E in 
 a balanced system, where E is the voltage 
 between one side and the common return. 
 
 The common current in a balanced sys- 
 tem is 45° from the phase of the current 
 in either of the independent wires and 
 the joint voltage is 45° from the voltage 
 of either phase. Figure 222 shows the 
 graphical composition of the voltages. 
 
 A and B are the two line voltages, and 
 R is the resultant voltage measured 
 across the outside wires. 
 
 The coils of quarter-phase machines 
 may be entirely independent of each 
 other, in which case an armature requires 
 four circuit terminals, or the circuits 
 may be joined so as to require only three terminals. In a 
 
 Fig. 223. — Methods of Con- 
 nection — Two-phase Arma- 
 tures. 
 
 Fig. 222. — Voltage Curves of Three-wire Two-phase System. 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 3G1 
 
 quarter-phase machine with rotating armature the armature 
 may be wound with the equivalent of a series-path direct- 
 current winding, and four collector rings and independent 
 circuits are then required to avoid short-circuiting portions of 
 the armature. Figure 223 shows diagrammatically various 
 ways of connecting 1 the coils of quarter-phase machines. (See 
 also Art. 22.) 
 
 It is possible, in three-phase systems, to use three entirely 
 
 Fig. 224. — Three-phase Delta Connection. 
 
 independent circuits, each consisting of two wires, and carrying 
 currents of 120° difference of phase ; but in practice the circuits 
 are almost invariably combined so as to use three line wires. 
 
 The three armature circuits of three-phase machines may be 
 connected together so that they form the sides of a delta with 
 the transmission wires connected to the three corners of the 
 triangle (Fig. 224), or one end of each of the three circuits 
 
 Fig. 225. — Three-phase Wye Connection. 
 
 may be individually connected to the transmission wires, the 
 free ends of the three circuits being connected together (Fig. 
 225). In either case the number of transmission wires is three. 
 
362 
 
 ALTERNATING CURRENTS 
 
 Fig. 226. — Vector Relations in Star-connected System. 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 303 
 
 and the algebraic sum of their instantaneous currents is always 
 equal to zero. 
 
 In the latter, called the Wye or Star arrangement, which is 
 represented by the symbol Y, the voltage between anj T two line 
 wires in a balanced system is V3 E, where E is the voltage in 
 one circuit of the machine. Thus, in Fig. 226 a, the triangle 
 ABC represents the vector polygon of line voltages, and the 
 arrowheads the directions of the vectors which are of successively 
 120° difference in phase.* The phase diagram of these vector 
 voltages (denominated E AC , E CB , and E BA ) is laid out from the 
 center of rotation 0 , with the vectors of the same dimensions 
 and directions as the sides of the triangle ABC. The wye 
 armature, or load branch, voltages in a balanced system are OA, 
 OB , and 00 (denominated E 0A , E 0B , and E uc ). This must be 
 the case since the three branches are equal and meet at a central 
 or neutral point, and their free terminals have the same instan- 
 taneous potentials as the points A, B , and C. It is evident, then, 
 from Fig. 226 a that the branch voltages — E 0A and E 0B form a 
 closed vector triangle with line voltage E BA , and that the vector 
 difference of E 0A and E 0B (i.e. the vector sum of E 0A and 
 
 — E ob ) is therefore equal to E BA f; likewise, branch voltages 
 
 — E ob and E oc form a closed vector triangle with line voltage 
 E cb . and E cb is therefore equal to the vector sum of E 0B and 
 
 — J£oc> and, in the same manner, it will be observed that E AC is 
 equal to the vector sum of — E 0A and E uc . It is sometimes 
 more convenient to illustrate these relations by laying down 
 
 — E ob as at OH , and completing the parallelogram on the sides 
 E oa and — E ob , whence it is at once seen that the diagonal, OB, 
 is equal by construction to E BA . Consequently E BA must be the 
 vector sum of E 0A and — E 0B . The vector — E 0B is of course 
 the same as vector E B0 . The line OB shows the position of E BA 
 in the phase diagram of branch and line voltages for a balanced 
 three-phase Y-connected circuit. The lines OGr and OF 7 likewise 
 show the positions of E AC and E CB in the phase diagram. 
 
 Figure 226 b shows the vector polygon of the voltages E 0A , 
 E ob , and E oc , which, being numerically equal and 120° apart in 
 phase for a balanced system, make a closed equilateral triangle. 
 
 Returning to Fig. 226 a, since the angle BOA in the phase dia- 
 gram equals angle OAB and OB and OA are of equal length, it is 
 * Art. 103. f Art. 79. 
 
304 
 
 ALTERNATING CUR 1 1 ENTS 
 
 obvious that OD = V3 OA and, therefore, that E BA — V3 E 0 , 
 = VSE B0 when these are given in effective values. Similar 
 relations holding for the other two line voltages, it is seen that 
 E l = v/3 E, when E L is the effective value of a line voltage 
 and E is the effective value of a branch voltage. The figure 
 shows that line voltage E BA leads branch voltage E 0A by 30° 
 and branch voltage E UB by 150° ( i.e . lags behind E B0 by 30°); 
 line voltage E AG leads branch voltage E uc by 30° and branch 
 
 voltage E 0i bj 150°; 
 and line voltage E clt 
 leads branch voltage 
 E ub by 30° and E oc 
 by 150°. ' 
 
 In Fig. 227, the 
 curve R shows the 
 potential difference 
 between A and B. 
 
 The line current in 
 any star-connected 
 circuit must always 
 be the same as that 
 passing through the 
 branch in which the 
 line terminates. 
 
 In the Delta or Mesh winding, which is often represented by 
 the symbol A, the voltage between wires is evidently that gen- 
 erated by one coil, and the current in the line wire is the com- 
 bination of those in two adjacent coils, or V3 I in a balanced 
 system, where I is the current in a coil. Thus, in Fig. 226 c, 
 the triangle 3INP is the vector polygon of the line currents 
 from the delta corners A , B, and C , of Fig. 224, the arrow- 
 heads representing the vector directions.* The phase diagram 
 of these vector currents, called I Al , I BL , and I CL , respectively, is 
 laid out from the center of rotation 0 with the vectors of the 
 same directions and dimensions as those forming the sides of 
 the triangle MNP. The branch currents between BA, AC, 
 and CB , Fig. 224, called I BA , I AC , and I CB , respectively, are 
 numerically equal and of successively 120° difference in phase. 
 But I BA and I AC join with I AL at the delta corner A and the 
 
 * Art. 103. 
 
 Fig. 227. — Voltage Curves of a Three-phase System. 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 365 
 
 three must therefore form a closed vector triangle, since the alge- 
 braic sum of their instantaneous values must always be equal to 
 zero in accordance with Kirchoffs law of currents at junction 
 points. Similar relations exist between the branch and line cur- 
 rents at the other corners of the delta. To fulfill these conditions 
 OR , OS, and OT may represent the vector currents I BA , I CB , and 
 I AC .* Then, from the parallelogram OTRU , it is seen that I AL 
 is the vector difference of I BA and I AC (i.e. the vector sum of I BA 
 and — / 4C ) ; parallelogram ORSV shows I BL is equal to the 
 vector sum of — I BA and I CB ; and parallelogram OSTW shows 
 that I CL is equal to the vector sum of — I CB and I AC . These 
 relations are in accord with the principle enunciated by Kirch off 
 that the algebraic sum of the currents meeting in a point is zero, 
 when those flowing toward the point are considered of one sign 
 and those flowing away from the point of the opposite sign. At 
 any point of meeting such as A, Fig. 224, the sum of the instan- 
 taneous currents must always be zero, and the geometric sum of 
 their vectors must therefore be zero. But we have taken I BA 
 to be from B to A and I AC and I CB must therefore be taken 
 in the directions, respectively, from A to C and C to B. 
 Hence I BA has a direction toward the junction A, and I AC has 
 a direction away from A and must be reversed in order that it 
 may add to I BA to give I AL . This is expressed in the vector 
 triangle OUR , where the sum of I BA and — I AC is vectorially 
 equal to I AL , or I AL = I BA + (— I AC '). The same relations 
 exist for the other corners. From Fig. 226 c it is seen that 
 I l = v3I, where I L is the effective value of a line current 
 in amperes and I the effective value of a branch current. 
 It is also seen that I AL lags behind I BA by 30° and I AC by 150°, 
 I BL lags behind I CB by 30° and I BA by 150°, and I CL lags behind 
 I AC by 30° and I CB by 150°. Figure 226 d shows a vector 
 polygon of the branch currents. 
 
 Figure 228 shows ways in which the coils of three-phase 
 machines may be connected to the external circuit. The 
 arrangements are either of the wye or delta connection. The 
 point of common connection 0 in the wye arrangement is 
 called the Neutral point of the winding. A line wire may or 
 may not be led from this point, depending on the conditions of 
 use of the current. 
 
 * Art. 103. 
 
366 
 
 ALTERNATING CURRENTS 
 
 In the case of a two-phase system, it is obvious that the 
 algebraic sum of the instantaneous currents must always be zero 
 at any section taken across the line wires when the two phases 
 are kept separate by the use of four main line wires, because the 
 instantaneous incoming current is equal to the instantaneous out- 
 going current in each single-phase circuit. When three wires 
 are used for such a circuit, the joint wire is a common return 
 for the other two, and the algebraic sum of the instantaneous 
 currents in the three wires is, therefore, clearly equal to zero. 
 
 Fig. 228. — Methods of Connection of Three-phase Machines or Loads. 
 
 In other polyphase systems, the vectors of the several line 
 currents are capable of making a closed vector polygon, and it 
 therefore follows that at any section taken across the line wires, 
 
 ij sin a + / 2 sin (« — /S 2 ) + ••• + I n sin (a — /3„) = 0, 
 
 \il v I v -I n are the effective line currents and /3 2 ••• are the 
 angular differences between the several current vectors. This 
 must be equally true for each harmonic of current. But each 
 of the terms on the left-hand side of this equation multiplied 
 by V2 gives the instantaneous value of the current, and the 
 algebraic sum of the instantaneous currents is therefore always 
 equal to zero. 
 
 This is plainly to be seen in the case of star-connected wind- 
 ings, when it is remembered that the electric current acts like 
 the flow of an incompressible fluid and the aggregate current 
 flowing from the neutral point through one or more of the 
 windings must at each instant be equal to the aggregate current 
 entering the neutral point at that instant through the other 
 winding's. This is in accordance with what is called Ivirchoff s 
 law of current flow at a junction point. In case of the use of 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 367 
 
 a neutral wire, or other extra wire in a polyphase system, the 
 instantaneous current flowing therein, across the section taken, 
 must be included in the algebraic sum. 
 
 When the system is balanced and has n phases, 
 
 *3 — ^ m ^ ^ ! 
 
 in = i m sin (a - 2 ^- n ~p— 
 
 . . . . . i . f 2 
 
 Hence, i x + * 2 + « 3 + ••• + i n = i m j sin a sin la 
 
 . f 4 7T \ . f 0 ( w — 1 )*7 t'\ ) 
 
 but evidently, sin a + sin q. s in ^ a — + •• 
 
 + sin l^a — 2 
 
 and therefore i 1 + i 2 + i 3 + ••• +i n = 0. 
 
 (n — l)7r^ _ 
 
 ; 0 , 
 
 101. Uniform Power in Polyphase Systems. — In general, the 
 power transferred in a balanced polyphase circuit is uniform 
 throughout each period, and the torque exerted by balanced 
 polyphase machinery is uniform. This is different from the 
 conditions in single-phase circuits, where the power has been 
 shown to vary between a maximum and a minimum during every 
 quarter period.* In the case of a single-phase circuit the 
 power at any instant is i m e m sin (a — 0) sin a = i m e m sin 2 a cos 9 
 — i m e m sin a cos a sin 0, which varies with a. In a balanced two- 
 phase circuit the instantaneous power is i m e m sin 2 a cos 9 + 
 i m e m sin 2 (a — 90°) cos 0 = i m e m cos 0 (sin 2 a + cos 2 a) = i m e m cos 0, 
 which is constant. In the same way the power in a balanced 
 three-phase circuit is, when i m and e m represent maximum values 
 of branch currents and voltages, i m e m cos 0 {sin 2 « + sin 2 (a — 120°) 
 + sin 2 (a — 240°) ( = § i m e m cos 9 , which is constant ; and, in gen- 
 eral, the power in any balanced polyphase circuit in which the 
 
 * Art. 92. 
 
368 
 
 ALTERNATING CURRENTS 
 
 9 Jr 
 
 phase differences are equal to - — , where n is the number of 
 
 n 
 
 phases, is, when i m and e m are maximum values of branch cur- 
 rents and voltages, 
 
 i m e m cos 6 | sin 2 « -f- sin 2 ^« — ^ j + sin 2 ^« — + ••• 
 
 which is equal to cos 6 , and is constant, since 
 
 s. n ] 
 
 sin 2 tt + sin 2 
 
 n J 
 
 -f- sin 
 
 . 2 { 2(w— l)7r\ n 
 
 -f sin 2 a x — = - . 
 
 This being true for one harmonic it is true for all harmonics 
 where the currents and voltages in a balanced circuit are not 
 sinusoidal. 
 
 The uniformity of power in a balanced polyphase circuit may 
 also be directly deduced from the proposition that the resultant 
 of n equal harmonic motions acting in lines having successive 
 
 angular differences of is a uniform circular motion with an 
 
 n 
 
 amplitude equal to - times the amplitude of the components. 
 
 mi 
 
 It is to be observed that uniform transfer of energy is in- 
 herent in direct current and balanced polyphase alternating- 
 current circuits, but that this attribute is not possessed by 
 single-phase circuits nor in general by polyphase circuits which 
 are unbalanced. 
 
 102. Delta and Wye Connections for more than Three Phases. — 
 
 With more than two phases the number of line wires may be 
 equal to the number of phases, as illustrated in Figs. 228 and 
 229 a to 7j, which show corresponding delta and star arrange- 
 ments of operating circuits and the connection of line wires 
 thereto, for various numbers of phases. Turning to the illus- 
 trations in Fig. 229, a to li inclusive, which show simple mesh 
 and star connections for five, six, seven, and eight phases, it will 
 be observed that there are always as many corners in the mesh 
 connection as there are operative circuits or phases, and it is 
 obvious that an equal number of line wires individually con- 
 * Todhunter’s Plane Trigonometry , p. 243. 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 369 
 
 j- k. 
 
 Fig. 22<J. — Connection of Line Wires, for Various Numbers of Phases. 
 
370 
 
 ALTERNATING CURRENTS 
 
 nected to these corners as shown will convey all of the currents 
 of the operative circuits. Each line wire will convey the vec- 
 tor difference of the currents in two of the mesh circuits. That 
 is, the current in line wire B is the vector difference of currents 
 in a and b, and so on. Four wires may thus be used for a four- 
 phase system, five wires for a five-phase system, etc. 
 
 When the star arrangement is used, one end of each operative 
 circuit lias a line wire connected to it and the minimum prac- 
 ticable number of line wires is again equal to the number of 
 operative circuits. The free ends of the operative circuits are 
 connected with each other at the neutral point, and each 
 operative circuit utilizes all the other circuits and lines as its 
 return lines. In this case, the current in a line wire is equal to 
 the current in the operative circuit to Avhich it is connected, but 
 the voltage between any two line wires is the vector difference 
 of the voltages of the corresponding two operative circuits. 
 
 An additional line conductor may be led from the neutral 
 point, if desired, and this may then act as a partial return wire 
 for all the circuits in case the system is unbalanced. 
 
 Figure 229 i, j, and k show three combinations of mesh and 
 star loads, so that six, eight, and twelve phase loads may be fed 
 from three, two (using four wires), and three phase lines re- 
 spectively ; the value of such combinations is explained later. 
 
 103. Relations between Currents and Voltages. — In both the 
 graphical and analytical representation of polyphase currents or 
 voltages, it is particularly desirable to adopt a notation which 
 logically and conveniently indicates the vector relations in the 
 circuit. This may be accomplished by following the methods 
 described in Chapter V and exemplified in Art. 100. A diagram 
 showing the connection of circuits (whether mesh or star) may 
 be drawn, and phase diagrams of currents and voltages then 
 laid down. The connection diagram being carefully lettered, 
 each vector in a phase diagram may be distinguished by the two 
 corresponding terminal letters taken from the diagram of con- 
 nections, and the vectors in analytical expressions may be 
 identified by the same letters used as subscripts. Vector 
 voltages may be considered as falling in the direction from the 
 point corresponding to the first subscript toward the point 
 corresponding to the second. For instance, in a circuit from 
 A to B , where A is taken as the point of positive or higher 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 371 
 
 potential, the voltage is E AB , the positive direction being from 
 A to B. Likewise vector currents may be considered as flowing 
 from the point corresponding to the first subscript to the point 
 corresponding to the second subscript when caused to flow by 
 the difference of potential of those two points. For instance, 
 in a circuit having the voltage E AB the current would be under 
 these circumstances I AB . If, however, the current is to be 
 represented as flowing from the point of low to the point of 
 high potential, under the influence of an electromotive force 
 in the circuit itself or other cause, it may be considered as 
 flowing from the point corresponding to the second subscript 
 to the point corresponding to the first subscript ; and in such 
 a case, the circuit having a voltage E AB , the current would be 
 I BA ; if the subscripts are not changed, this may be written 
 — Iabi 1-ba = -^-ab* 
 
 To illustrate further, Fig. 230 shows a wye-connected three- 
 phase generator 
 armature and the 
 corresponding phase 
 diagram of voltages. 
 
 The voltages of the 
 coils a , b , c, are re- 
 spectively E A0 , E b0 , 
 
 E c0 , and successively 
 differ 120° in phase. 
 
 The terminal vol- 
 tages become, vec- 
 torially : 
 
 Eab — E_ a0 + Eqb — E ao + ( — Ego") > 
 
 EjiC — Ego + Eq c = EgU + ( — ^_CO ) j 
 Eca — E co -T E oa = E co + ( — FT 1(J ). 
 
 Fig. 
 
 230. — Wye Connected Armature and Phase 
 Diagram of Voltages. 
 
 The currents in the coils are I A oi I B oi and I c0 , and lag behind 
 their respective voltages by angles of lag fixed by their respec- 
 tive circuits. 
 
 The following are the relations between the currents and the 
 voltages in the lines and coils of a balanced three-phase system 
 developed from the earlier discussion and illustrated in Fig. 228 : 
 
 1. Star Connection. I lme = J eoil ; E AB =■■ E BC = E CA = V3 E coil . 
 Line voltage E AB is the vector sum of coil voltages E A0 and 
 E ob (note that E AO = — E OA ) and is 30° behind the phase of coil 
 
372 
 
 ALTERNATING CURRENTS 
 
 voltage E ub . Line voltage E AC is the vector sum of coil vol- 
 tages E a0 and E oc and is 30° behind the phase of coil voltage 
 E au . Similar relations hold for the other two corners. For 
 instance, line voltage E CA (which is equal to — E AC ~) is the 
 vector sum of coil voltages E co and E OA and is 30° behind the 
 phase of E OA and 30° ahead of E co ; and E CB (equal to — E BC ~) 
 is the vector sum of E co and E 0B , and is 30° behind the phase 
 of E c0 and 30° ahead of the phase of E OB . 
 
 _ 2. Delta Connections. E AB = E BC = E CA = E coiU I Une = Vs J coil . 
 Ila is the vector sum of branch currents I AC and I AB (= — I BA ) 
 and is 30° ahead of I AC . I LB is the vector sum of branch cur- 
 rents I BA and I BC ( = — I CB ) and is 30° ahead of I BA . is the 
 vector sum of branch currents I CB and I CA ( = — I AC ) and is 30° 
 ahead of I CB . 
 
 Figures 226 a and 226 c show the relations that exist if non- 
 'reactive wye and delta circuits are placed on the same line wires. 
 Thus, the line voltage between each pair of wires is in phase 
 with the delta current between the same wires, so that I BA , I CB , 
 and I AC of c are parallel, respectively, to E BA , E CB , and E AC 
 of a. Also, the voltage and current in each branch of the 
 wye are in the same phase, so that E 0A , E 0B , and E oc of a are 
 parallel, respectively, to I AL , I BL , and I CL of c. Also, the wye 
 branch voltages in a balanced three-phase circuit successively 
 differ in phase from each other by angles of 120°, and they 
 occupy certain specific relations to the delta or line voltages, 
 which are illustrated in Figs. 226 and 230. It will be observed 
 from Fig. 226 a that E BA , which is in phase with I BA , leads E OA by 
 30°. But it will also be observed from Fig. 226 c that I AL lags 
 30° behind I BA , and I AL is therefore in the same phase as E OA . 
 That is, the line current flowing from a corner of the delta has 
 the same phase as the voltage measured from the neutral point 
 of the circuit to the same corner, when the delta circuit is non- 
 reactive. Now, the branch voltage E OA of the wye circuit is 
 identical with the voltage E 0A measured from the neutral point 
 to the corner A of the delta circuit, and they are therefore in 
 the same phase. Hence, I 0A of the wye circuit, which is in 
 phase with E 0A , is also in phase with I AL of the delta circuit. 
 By the same reasoning, I OB of the wye circuit is in phase with 
 I BL of the delta circuit, and I oc of the wye circuit is in phase 
 with I CL of the delta circuit. These relations are always true 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 373 
 
 for balanced wye and delta circuits when the phase of each 
 branch current coincides with the phase of the corresponding 
 branch voltage. 
 
 If the delta and wye branches of the balanced circuits, on the 
 other hand, are reactive, but all of the same power factor, the 
 voltage and current in each branch differ in phase, but the dis- 
 placement of the line current from the phases of the branch cur- 
 rents combining at a delta corner is not altered, and the delta 
 line currents Z Une therefore are still in phase with the correspond- 
 ing wye-branch currents I cojl . If the power factor of the delta 
 branches differs from the power factor of the wye branches, 
 the delta-line currents ij ine are then out of phase with the wye- 
 branch currents _T coil , and the main-line currents feeding the 
 wye and delta jointly are vector resultants of the delta and wye 
 line currents. 
 
 If the circuits of utilization in a polyphase system are not 
 machines (for instance, are incandescent lamps), the devices 
 must be connected exactly as would be the coils of a machine ; 
 unless transformers intervene, in which case the secondary 
 circuits may be independent ; but the load should be uniformly 
 distributed to keep the system balanced. 
 
 The following examples deserve careful study, as they 
 exhibit the phase and quantity relations of currents and vol- 
 tages in polyphase circuits which are unbalanced. The voltages 
 and currents are assumed to be sinusoidal to avoid undue com- 
 plexity in exhibiting the underlying principles. If irregular 
 curves are dealt with, a rigorous solution requires that they 
 be analyzed and their sinusoidal components be used in the 
 solution ; but in many instances the substitution of equivalent 
 sinusoids for the irregular waves is sufficientl} 7 accurate. The 
 first two examples deal with three-phase unbalanced wye 
 loads, in which the position of the neutral point of the wye 
 with reference to the three-line voltages must be determined.* 
 
 Example 1 . — A three-phase line (. A L B L C L , Fig. a) is con- 
 nected to a non-reactive wye circuit containing resistance of 
 one ohm in the branch AO. two ohms in BO. and three ohms 
 in CO. The voltage is balanced, being 100 volts on each phase. 
 
 (a) What are the values of the current, voltage, and angle 
 of lag in each branch? 
 
 * H. P. Wood. Electrical World, vol. 55, p. 1597. 
 
374 
 
 ALTERNATING CURRENTS 
 
 (S) What are the current values in the line wires AA L , BB L , 
 and CC L and their phase relations respectively with the vol- 
 c tages E ab , E bc , and E CA ? 
 
 CO M 
 
 A 
 
 Eca: 1 ") 
 
 L 
 
 A Rao=1 A 
 
 1 e bc 
 
 =10 V 
 
 / 
 
 /Rbo= 2 
 
 t 
 
 E ab =ioo 
 
 1 
 
 B, 
 
 B 
 
 
 
 Example 1. 
 
 — Figure a. 
 
 
 (c) What is the total 
 power absorbed ? 
 
 Figure a diagrammati- 
 cally represents the load. 
 
 (a) The triangle ABC 
 (Fig. 5) is drawn to scale 
 representing, by the 
 length and positions of its sides, the line voltages E AB , E BC , and 
 E ca . For convenience, the side AB is made horizontal. The 
 point O' is taken at any convenient position in the plane ; and 
 the lines O' A, O'B , and 
 O' C are drawn, representing 
 the corresponding voltages 
 E aA , E aB , and E^ c from O' 
 to A, B, and C. The posi- 
 tion of the point O’ is 
 assumed for convenience in 
 the process of determining 
 the location of the neutral 
 point 0 of the circuit, with 
 reference to the points A, 
 
 B , and C. Rectangular axes 
 are erected with the origin 
 at O' and the axis of abscissas parallel to AB. 
 
 From Fig. b we obtain the following vector equations, in 
 which the true values of a and b are to be derived : 
 
 B 0 -a = a -ft. 
 
 Bo b = — (100 - «) — jb. 
 
 Eq. v = a — 50 + j (86. 6 — 5). 
 
 As the circuit is non-reactive, the directions of the currents, 
 I(y A , I ob' lav, may be represented by the lines 0'A V 0'B V and 
 0'C V coincident with the lines O' A, O'B. and O' C. Dividing 
 each vector equation for voltage given above by the resistance 
 of its respective branch, gives : 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 375 
 
 Since the algebraic sum of the instantaneous currents meeting 
 at the neutral point of the system (currents flowing towards 
 
 the neutral point being taken 
 of one algebraic sign and those 
 flowing away being taken of 
 the other sign) must always 
 reduce to zero, it is plain that 
 the algebraic sum of the scalar 
 values of the three vertical 
 components, and likewise of 
 the three hori- 
 zontal compo- 
 nents, in these 
 three vector equa- 
 tions must each 
 reduce to zero. This also fol- 
 lows from the fact that the 
 three currents of a three- 
 phase system make a closed 
 vector triangle, and therefore 
 I n , + I nR + I or = 0. To cor- 
 
 Ex ample 1 .— Figure c. 0A B oc 
 
 rectly locate the neutral point 
 it is therefore only required to solve for a and b in the following 
 equations : 50 
 
 *-hr-i +i-T = # > 
 
 and 
 
 whence 
 
 V 2 2/ 3 
 
 , b 86.6 b n 
 - 4 - 5+~-3 = °; 
 
 a =36.4 and 5 = 15.8. 
 
 The values of the wye currents and the location of the neutral 
 point 0 are determined by substituting these values in the fore- 
 going vector equations of current, and are plotted in Fig. c. 
 
376 
 
 ALTERNATING CURRENTS 
 
 In this figure the three line-voltages are also drawn from 0 , 
 making a phase diagram. The numerical values of the voltages 
 E 0A , E ob , and E oc are determined by substituting the values 
 of a and b , just found, in their vector equations and obtaining 
 the tensors in the usual manner, thus : 
 
 E oa = V (36.4 ) 2 + (15. 8) 2 = 39.7 volts. 
 E ob — ^ (63. 6) 2 4- (15. 8) 2 = 65.5 volts. 
 
 E oc = V(13.6) 2 + (70. 8) 2 = 72.1 volts. 
 
 The numerical values of the currents are found in the same 
 way from their equations, or by dividing the voltages just 
 determined by the accompanying branch impedances with the 
 result that 
 
 I OA — 39.7 amperes. 
 
 I 0B = 32.8 amperes. 
 
 I oc = 24.0 amperes. 
 
 The angles of lag of the currents with respect to the voltages 
 of the respective branches are zero, since the branches are non- 
 reactive. 
 
 (5) The currents in the line wires, I AL , I BL , and I CL are, of 
 course, equal to respectively and in phase with the branch cur- 
 rents Iq A , IoB ’ ^IRl I oc . 
 
 The phase angle between E BA and I AL (=/ 0A ) is, from the 
 
 vector equation of the current, 0 BA = — tan -1 
 
 15.8 
 36.4 
 
 -15.8 
 
 = 23° 28'. 
 
 The minus sign is applied to tan 1 ( — 
 
 36.4 
 
 , because by conven- 
 
 tion the phase angle between current and voltage is positive 
 when measured from the current to the voltage in counter-clock- 
 wise direction , i.e. for a lagging current. In this case we have 
 taken the voltage as the initial line, and measuring from it 
 gives a reversed angle. 
 
 Moreover, since E CB lags 120° behind E BA or what is the 
 same thing leads by 240°, the angle between I 0B and E CB is 
 
 @cb — 
 
 240° - tan" 1 
 
 -15.8 
 
 — (100 — 36.4)/_ 
 
 = 46° 3'. 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 377 
 
 In like manner the phase angle is 
 
 6 
 
 AC — 
 
 120° -tan- 1 
 
 f 86.6-15.8Y] 
 V 36.4-50 )_ 
 
 = - 19° 9'. 
 
 Here E AC lags 240° behind E BA , or, for simplicity, in the formula 
 may be counted as leading by 120°. 
 
 It must be understood that these angles are the angles between 
 the respective line currents and line voltages, and do not deter- 
 mine the power factor, which is fixed by the phase relations of 
 the branch currents with respect to the branch voltages and is 
 unity in this case. 
 
 (c) The power in the circuit is E OA I OA + E 0B I 0B + E oc I oc 
 = 5460 watts, since the branches are 
 non-reactive. 
 
 A check can be applied to the com- 
 putations made in finding the currents 
 by constructing’ a vector diagram as 
 in Fig. d. Here, A'B' is equal and 
 parallel to I OA , B'C is equal and 
 parallel to I 0B , and O' A' is equal and parallel to I oc . Had 
 the triangle not closed, it would have been an indication of 
 error, as in such a case the instantaneous sum of the branch 
 currents could not be zero at the neutral point 0, Fig. a. 
 
 C 
 
 Example 1 . — Figure d. 
 
 Example 2 . — A three-phase line (. A l B l O l , Fig. a ) is con- 
 nected to a wye circuit having the following characteristics : 
 
 In branch OA, R 0A = 
 1 ohm in parallel with 
 in 
 
 -A l 
 
 X 0A = + 1 ohm ; 
 branch OB , R OB = 2 
 ohms in parallel with 
 X OB = — 1 ohm ; and in 
 branch OC . , R oc = 1 ohm 
 (Fig. a). Equal vol- 
 tages of 100 volts are impressed on the three phases. Answer 
 the questions a, b , and c of Ex. 1. 
 
 The same preliminary or trial diagram as was previously 
 used (Ex. 1, Fig. 5) may be employed here for construct- 
 ing the vector equation of branch voltages. They are as 
 before : 
 
378 
 
 ALTERNATING CURRENTS 
 
 Eoa = « ~jb- 
 
 £ 0B = “ ( 10 ° - a ) ~ J b - 
 
 U 0 c = « — 50 4- j (86.6 — £). 
 
 Example 2. — Figure b. 
 
 Then 
 
 (a) To obtain the 
 vector expression for 
 branch currents, the 
 voltage equations must 
 be divided by the 
 respective branch im- 
 pedances or multiplied 
 by the respective ad- 
 mittances. As a matter 
 of convenience, multi- 
 ply by the admittances, 
 which are 
 
 Y OA =l-jt; 
 
 1 ob = T ,7 1 5 
 and Y oc = 1.* 
 
 Iqa = (« (1 ~j 1) = « - b ~j (a + J). 
 
 I 0B = (« —100 — jb)Q +j 1) = | a — 50 + b +j( — | b + a — 100). 
 I oc — (a — 50 4 j (86.6 — £>)) (1) = a — 50 +/ (86.6 — J). 
 
 By placing the vertical and horizontal components respectively 
 equal to zero, there results 
 
 a = 40 and b = — 5.3, 
 
 and Fig. b can now be plotted to scale as shown. 
 
 The scalar values of the branch voltages are : 
 
 E oa = V (10 ) 2 4 (5.3) 2 = 40.3 volts. 
 
 E ob = V (60) 2 4 (5.3 ) 2 = 60.2 volts. 
 
 E oc — V (10) 2 4 (92) 2 = 92.5 volts. 
 
 * Arts. 74 and 78. 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 379 
 
 The currents may be obtained in the same manner from the 
 vector equations of currents, and are : 
 
 I OA = V (40 + 5. 3) 2 + (40 — 5. 3) 2 = 57. 2 amperes. 
 
 Iob = ^ (20 — 50 — 5. 3) 2 + 4- 40 — loo) = 67.4 amperes. 
 
 I oc = V (40 — 50) 2 4- (86.6 -f- 5.3) 2 = 92.5 amperes. 
 
 The angles of lag between the currents and voltages of the 
 branches are found directly from the admittances and are : 
 
 d 0A = — tan -1 — — - = 45° ; 0 OB = — tan -1 j = — 63° 26' ; 
 
 1 2 
 
 0 OC — — tan -1 ^ = 0°. 
 
 h. The currents in the branches are the same as in the lines 
 to which they are connected. 
 c. The power in the circuit is 
 
 1-OA OA ^ OA 4" I()B -® OB C0S ^ OB 4“ ^ OC ~^OC C0S ^ OC 
 
 = 12,000 watts. 
 
 The way in which the branch voltages are influenced by the 
 power factors of the branches, as well as by the magnitudes of 
 the branch currents, is clearly seen in this example. 
 
 Example 3. — A 
 
 neutral wire is con- 
 nected to the point 0 
 in the wye of Ex. 2, 
 Fig. a, and is main- 
 tained at fixed equal 
 voltages, 120° apart, 
 relatively to the re- 
 spective potentials of 
 the points A, B, and 
 C. Answer questions 
 a and c of Ex. 1, and 
 calculate the current 
 in the neutral wire. 
 
 (a) In this case the 
 point 0 in the trial 
 
 -Y 
 
 Example 3. — Figure a. 
 
380 
 
 ALTERNATING CURRENTS 
 
 vector diagram of Ex. 2, Fig. a, is fixed at the center of the 
 triangle by the effect of the neutral wire ; therefore, the effect- 
 ive voltages on the branches are equal numerically to each 
 
 other and of value = 57.7 volts. Figure a shows the rela- 
 V3 
 
 tions of the voltages. The vector currents in the branches, 
 each referred to its branch voltage as the initial line, also shown 
 in Fig. a, are as follows : 
 
 i 0A = Ka Yoa = 57.7 (1 -j 1) = 57.7 -j 57.7. 
 
 i 0B = YJ ub Y ob = 57.7 (1 +j 1) = 28.9 +j 57.7. 
 
 loc — E 0C Y 0C = 57.7 (1 +j 0) = 57.7 +j 0. 
 
 The scalar values and the angles of lag are : 
 
 I 0A — V(57.7) 2 -t- (57. 7) 2 = 81.7 amperes. 0 OA = 45°. 
 
 l OB = V (28. 9) 2 -f- (57. 7) 2 = 64.6 amperes. d 0B = — 63° 26'. 
 
 I oc = 57.7 amperes. 6 oc = 0. 
 
 The neutral current I ON must have such a value that the alge- 
 braic sum of instantaneous currents meeting at 0 will be equal 
 to zero, but each of the vector equations of currents just given 
 has its components referred to its own branch voltage for the 
 horizontal axis. These are 120° apart; the coordinates of the 
 current vectors must, therefore, be changed so as to refer to a 
 single pair of axes before they can be added. We will take as 
 the axes for this OX and OY, where OX coincides with the 
 line voltage E BA . It conduces to simplicity to use one of the 
 line voltages as the initial line in this manner. 
 
 Then, b} r using the proper direction coefficients, 
 
 I'oa = (57.7 — j 57.7) cjs* (— 30) = (57.7 —j 57.7)(.866 —j .5) 
 = 21.1 — j 78.8. 
 
 I' OB = (28.9 + j 57.7) cjs ( — 150°) = 3.87 — j 64.4. 
 
 I'oc = (57.7 +j 0) cjs (- 270°) = 0 + j 57.7. 
 
 But I'oa. + 1' OB + Y oc + I'ox = 0 5 
 
 * Art. 76. 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 381 
 
 hence substituting the right-hand sides of the above equations 
 in this and transposing gives, 
 
 I' ON = — 25 + / 85.5, 
 
 with a scalar value of l ON = 89.1 amperes, and an angle with 
 reference to the direction axis OX of 0' ON = 106° 18'. 
 
 (c) The power may be found by taking the product of the 
 current, voltage, and cosine of angle of lag of each branch and 
 adding the three products together. 
 
 This and the previous problems show clearly the advantage 
 of using a neutral lead wire on a star load for the purpose of 
 keeping the voltages in the branches as nearly as possible equal, 
 and to prevent severe insulation strains and poor service. 
 
 The check may be obtained as in Ex. 1 by drawing the 
 vector polygons of voltages or currents. The vector polygon 
 of line voltages is partly dotted-in, in Fig. a. In making the 
 check polygon of current vectors, the currents should be drawn 
 parallel to and of equal magnitude to their vectors in the phase 
 diagram ; and should, to avoid confusion, be preferably taken in 
 the order in which they appear on that diagram when following 
 it around in a clockwise direction. Thus, to I OA add I OB , then 
 add I 0N , and then I oc . In order that the solution may be 
 correct, it is evidently necessary for the neutral current to 
 close an open polygon of branch currents. 
 
 Example 4. — A star circuit having the same impedances in its 
 branches and the same arrangement as in Ex. 2 has impressed 
 voltages as follows : E AB — 141.4 ; E BC = 100 ; E CA = 100. 
 
 Answer the questions of Ex. 2. 
 
 Suggestion : Lay out the vector polygon of voltages as in 
 Ex. 2, making E BA horizontal. Assume a point O' and solve 
 for the position of the neutral point as before. Lay out the 
 phase diagram and proceed as in Ex. 2, using care to plot 
 the current phase angles correctly. The problem is in essence 
 the same and as simple as that of Ex. 2, and should be solved 
 without difficulty. These same unbalanced voltages are used 
 in Ex. 5, which has to do with a delta system. 
 
 Example 5. — The branch-load impedances of Ex. 2 are con- 
 nected in the form of a delta as shown in Fig. a\ 141.4 volts 
 
382 
 
 ALTERNATING CURRENTS 
 
 are impressed on the phase AB and 100 volts are impressed 
 upon each of the other phases. 
 
 (a) What is the current and angle of lag in each branch? 
 
 ( b ) What is the current in each line and its lag angle with 
 
 reference to one of 
 the line voltages ? 
 
 (a) First the vec- 
 tor polygon of vol- 
 tages is made and then 
 the phase diagram by 
 drawing the voltage 
 vectors parallel to those of the vector polygon ; or as the angles 
 of the triangle and their supplements can readily be obtained 
 by the elementary 
 formulas of trigo- 
 nometry, the vector 
 polygon may be 
 omitted. Using the 
 latter method, we 
 find that E, 
 
 CB 
 
 lags 
 
 E, 
 
 BA1 
 
 asrs 
 
 E b 
 
 voltages are 
 
 135° behind 
 and that E A 
 225° behind 
 These 
 
 laid out in Fig. b 
 with E ba on the 
 initial line OX , as 
 heretofore. The 
 vector equation of 
 each branch current, 
 with components 
 taken with reference to its particular branch voltage, is 
 obtained by multiplying the branch voltage by the correspond- 
 ing branch admittance ; so that, 
 
 Example 5. — Figure b. 
 
 I BA = 141.4 -j 141.4. 
 
 I CB = 50 +j 100. 
 
 4 = 100 . 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 383 
 
 From this the scalar values of the currents in amperes and 
 the angles of lag are found to be : 
 
 I BA = 200 amperes. 6 BA = 45°. 
 
 I CB = 111.8 amperes. 0 CB = —63° 26'. 
 
 I AC = 100 amperes. d AC = 0°. 
 
 (5) The currents in the three line conductors may be found 
 by combining the branch currents graphically as shown in Fig. 
 b, or they may be determined more accurately by adding their 
 vectors geometrically. Thus, referring each of the branch cur- 
 rents to the axes OX, OY, there results, 
 
 * ba = Iba <ds (0°) = 141-4 -j 141.4. 
 
 1'cb = Tcb cjs ( - 135°) = 35.3 -j 106. 
 
 I' AC = Iac cjs (-225°)= -70.7 + ,? 70.7. 
 
 Remembering that, by the convention adopted in this book, 
 current at a junction point is considered positive when it flows 
 toward the point, the two currents meeting at a corner of the 
 mesh may be added, giving the following results : 
 
 I'AL = I' BA + I'CA = I' BA + i-I’Ac) = 212-1 -j 212.1. 
 Likewise, T BL = I' CB + T AB = - 106. 1 +j 35.4, 
 
 and T cl = I' AC + I’ BC = - 106 +/ 176.7. 
 
 The scalar values of these currents from these equations are : 
 I AL = 300 amperes ; I BL = 112 amperes ; I CL — 206 amperes. 
 
 These currents have the following phase relations with the 
 line voltages : I AL lags behind X BA by 45° ; I BL lags behind E CB 
 by 63° 26' ; and I CL lags behind E AC by 14° 5'. These angles 
 are obtained from the vector equations of current representing 
 V T' iiiid V 
 
 - L AL' - L BL ' CL* 
 
 Example 6. — The wye circuits of Ex. 1 and Ex. 2 are placed 
 in parallel on a set of 100-volt, three-phase mains of the fre- 
 quency assumed in those examples. What total current flows 
 in the mains? 
 
384 
 
 ALTERNATING CURRENTS 
 
 All that is necessary is to geometrically add the currents 
 already determined in those problems. (A line is drawn under 
 the subscripts of the symbols from Ex. 1 to distinguish them 
 from those of Ex. 2.) Thus, 
 
 IgA + Ioa = (36.4 - j 15.8) + (45.3- j 34.7) 
 
 = 81.7 — j 50.5, 
 
 from which I AL = 96 amperes. The total currents in the other 
 lead wires can be obtained in the same manner. 
 
 Example 7.— A load of 10,000 watts is absorbed in a delta 
 three-phase system in the following way : in the branch AB , 
 4000 watts at 90 per cent power factor, current leading; in the 
 branches BC and CA, 3000 watts each at 80 per cent power 
 
 factor, current lagging. What are the currents in the branches 
 and in the lines, the voltage between A and B being 141.4 
 volts, between B and (7, 100 volts, and between C and A . 100 
 volts ? 
 
 From the data given we have, 
 
 Iba = I4i°4°x 9 = 31,4 am P eres - 9 ba = - 25° 50'. 
 
 I CB = = 37.5 amperes. 0 CB = 36° 52'. 
 
 I ag = " ( ^° g = 37.5 amperes. 0 AC = 36° 52'. 
 
 The combined currents can be obtained graphically as shown 
 in Fig. b , but more accurately by adding their vectors referred 
 to a common axis, OX , as has been done heretofore, care being 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 385 
 
 used to give the dividing currents at a branch junction the 
 correct sign. The vector equations are : 
 
 I' BA = Iba = 28 - 3 +3 13 - 7 - 
 
 I' CB = I CB e -7(36°a2- + i35°; = _37.i _/ 5.30. 
 
 l' AC = /J € -| (36 ° 5 * +225P) = - 5.30 + j 37. 1. 
 
 The exponential form of the complex expression is used here 
 as a matter of convenience in computation, as might have beeu 
 done also in earlier examples. From these expressions of the 
 vector-branch currents we obtain 
 
 Ial = I'ba + I'ca = 33 - 3 — j 23.4. 
 I BL = I'cb + I'ab = - 65.4 -j 19.0, 
 I CL = I’ AC + I'bc = 31-8 +j 42.4. 
 
 The scalar values of the line currents then are : 
 
 7^ = 40.8; 7 Bi = 68.1; and I CL = 53 amperes. 
 
 It should be noted that where the voltages or impedances are 
 unequal, as in this case and in Ex. 5, the delta branch currents 
 do not necessarily form a closed vector polygon, though the 
 three line currents do. 
 
 2c 
 
380 
 
 ALTERNATING CURRENTS 
 
 Example 8. — A two-phase receiver having a common return 
 wire (Fig. a ) has the following characteristics : In OA, R = 2 
 
 B in parallel with 
 
 -X" = — 1 ; and 
 
 B 
 
 R=i;X = i 0 
 
 E bo = 100 
 
 1 
 
 X-i R=2 
 
 ■0 L 
 
 F =141.4 
 
 Q AO 
 
 i 
 
 in 
 
 OB , R = 1 in series 
 with X = 1. Be- 
 tween the line wires 
 A l and 0 L , and B L 
 and 0 L , respectively, 
 141.4 and 100 volts are impressed. What are the currents in 
 the three line wires and the voltage between the lines A L and B L 
 when E oa leads B ob by 90°. 
 
 Example 8. — Figure a. 
 
 ■Al 
 
 Y oa = 2 +i 1 and Y 0 
 
 X _ n X 
 2 «/ 2 ’ 
 
 and the vectors of current referred to their respective voltages 
 
 lot = Eoa Yoa = 70.7 +j 141.4, 
 
 I 0 b — Bob 5 ob — 50 j 50 ; 
 
 from which, I OA = 158.1 amperes and d OA = — 63° 26'; and 
 I 0B =70.7 amperes and 0 OB = 45°. 
 
 Referring the vectors I OA and I 0B to the axes OX ’, OY , 
 
 I' OA = IoAe j ^^ = 10.1+jUlA, 
 and I' OB = I 0B e~ j (45 ° + 9 ° 0) = - 50 - j 50. 
 
 Adding these vectors gives, 
 
 IlO — I'oA + I'oB — 20.7 + J 91.4. 
 
 The scalar value of I L0 is 93.7 amperes. 
 
 The voltage between A L and B L is 
 
 B ba — E bo + B oa — 141.4 + j 100, 
 or B ba = 173 volts at an angle leading B OA by 35° 16'. 
 
 Prob. 1. A three-phase wye circuit has in series in each 
 branch a resistance of two ohms and an inductive reactance of 
 one ohm When a line voltage of 100 volts is impressed on 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 387 
 
 each phase, what current flows in each branch, what are the 
 branch voltage and angle of lag, and what are the power ex- 
 pended in the circuit and the power factor ? 
 
 (In these problems lay out the results of computations in 
 phase diagrams.) 
 
 Prob. 2. If the power of 10,000 watts is expended in the 
 wye system of Prob. 1, what must be the voltage across the 
 mains, and the cur- 
 rents in the mains 
 and branches? 
 
 Prob. 3. A three- 
 phase delta circuit 
 has a resistance of 
 one ohm and capac- 
 ity reactance of one 
 ohm in series in each 
 branch. When 500 
 line volts are im- 
 pressed on each 
 phase, how much 
 current flows in each 
 branch and line ? 
 
 How much power is 
 expended in the cir- 
 cuit, and what is the 
 power factor ? 
 
 Prob. 4. When 25 
 Ivw. are expended in 
 the delta of Prob. 3, what currents flow in the branches and 
 line conductors and what are the line voltages ? 
 
 Prob. 5. If two wye circuits, having the characteristics and 
 being under the conditions named in Prob. 1, are connected in 
 parallel to the same line wires, will current flow through a con- 
 ductor joining their neutral points ? Prove the answer. 
 
 Prob. 6. A three-phase transmission line receives from a 
 generating plant 1000 Kw., the line voltage being 5000 volts 
 in each phase. How much current flows in each wii’e if the 
 circuit is balanced and of 90 per cent power factor? 
 
 Prob. 7. The generator of Prob. 6 is wye-connected. What 
 voltage must be produced by the coils of each phase? 
 
 Y 
 
 Example 8. — Figure b. 
 
388 
 
 ALTERNATING CURRENTS 
 
 Prob. 8. A balanced load of 500 K\v. is transmitted over a 
 two-phase line of three wires, at a voltage of 2000 volts per 
 phase, to induction motors having an average power factor of 
 85 per cent. What current flows in each of the three wires 
 and what is the voltage between the two independent wires? 
 
 Prob. 9. A three-phase wye circuit has resistance of one 
 ohm and capacity reactance of .25 ohms in series in each 
 branch. The line voltages E AB , E BC , and E CA , respectively, 
 are 200, 220, and 240 volts. What are the values of current 
 and voltage in each branch, and what are the angles of lag of 
 the currents with respect to the corresponding branch voltages? 
 
 Prob. 10. A three-phase wye circuit has the following vec- 
 tor impedances in its branches: Z 0A — l-\-jl\ Z OB =\-\-j\\ 
 and Z oc = \—j 1. What are the branch voltages, currents, and 
 angles of lag when a line voltage of 500 volts is maintained on 
 each phase? What power is absorbed by the circuit ? 
 
 Prob. 11. The impedances of Prob. 10 are connected in 
 delta to the same mains. What are the branch currents and 
 angles of lag, and what are the line currents? What power is 
 absorbed by the circuit ? 
 
 Prob. 12. A neutral wire is connected to the wye junction 
 of Prob. 10 without changing the other conditions, except that, 
 by proper regulators, the voltages between the neutral point 
 and the line conductors are maintained numerically equal. 
 What current flows in the neutral wire, and what power is 
 absorbed by the circuit ? 
 
 Prob. 13. A transmission line supplies a three-phase wye 
 circuit with 100 Kw. in the proportion of 50 Kw. at 85 per cent 
 capacity power factor in one branch and 25 Kw. at unity power 
 factor in each of the other branches. The line voltage of each 
 phase is 240 volts. What are the values of the branch currents, 
 voltages, angles of lag, kilowatts, and kilovolt-amperes, what 
 are the line currents, and what are the total kilovolt-amperes 
 of the circuit? 
 
 Prob. 14. A transmission line supplies the three branches 
 of a delta circuit with 1000 Kw. in the proportion of 250, 350, 
 and 400 Kw., the voltage and power factor for each branch 
 being 2000 volts and 90 per cent power factor. What are the 
 branch currents, the line currents, the circuit power, and the 
 circuit kilovolt-amperes ? 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 389 
 
 Prob. 15. A three-phase transmission line of 2000 line volts 
 on each phase supplies the following circuit : a three-phase 
 
 delta circuit taking 20 Kva.* in each branch at 80 per cent 
 lagging power factor ; a three-phase wye circuit taking 50 Kva. 
 in each branch at 90 per cent leading power factor ; and a 
 single-phase load from one phase, of 25 Kw. at unity power 
 factor. How much current flows in each line wire ? 
 
 Prob. 16. A balanced load of 100 Kw. at 100 per cent power 
 factor is absorbed by a six-phase mesh-connected receiver. The 
 voltage between line wires is 250 volts. What is the current 
 per line wire? 
 
 Prob. 17. What would the branch voltages and line currents 
 be if the load of Prob. 16, at the same line voltages, was absorbed 
 by a six-phase star circuit ? 
 
 104. Measurement of Power in Two- and Three-phase Cir- 
 cuits. — The principles underlying the methods of measuring 
 power in polyphase circuits differ in no respect from those 
 already deduced in relation to single-phase circuits,! but it is 
 desirable to apply them in such a way as to reduce the number 
 of necessary readings to a minimum. For satisfactory measure- 
 ments, non-reactive wattmeters are of essential importance. 
 
 A. Two-phase Systems 
 
 A 1. Independent Circuits. In a two-phase system with 
 separate circuits, independent wattmeter readings are taken in 
 each circuit and the total power is the sum of the readings. 
 One wattmeter placed in each circuit, as shown in Fig. 231, 
 from which simultaneous readings are taken, is the best arrange- 
 ment ; but if two wattmeters are not to be had, one may be 
 inserted successively in the two circuits, and the sum of the 
 readings is equal to the power in the system, provided the load 
 does not vary while the readings are being taken. If the circuit 
 is perfectly balanced, twice the reading of a wattmeter in one 
 circuit is equal to the power, but this is a condition which can- 
 not be relied upon. 
 
 A 2. Circuits with Common Return. Two wattmeters may 
 here be used, one for each circuit, connected in the way shown 
 
 * Kva. is an abbreviation for kilovolt-amperes. t Art. 99. 
 
390 
 
 ALTERNATING CURRENTS 
 
 W 
 
 Fig. 231. — Measurement of Power in Two-Pliase System with Separate Circuits. 
 
 in Fig. 232. The arrangement shown in Fig. 233 is equivalent 
 to a single wattmeter connected as in Fig. 234, and is only 
 correct for a system in exact balance. When the single watt- 
 
 w 
 
 Fig. 232. — Measurement of Power in Two-phase System with Common Return. 
 
 meter is used in a balanced system, the current coil is placed in 
 the common wire, and a reading is taken with the free end of 
 the voltage coil connected to one outside wire. The voltage 
 coil terminal is then quickly transferred to the other outside 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 391 
 
 Fig. 233. — Measurement of Power in Two-phase System in Exact Balance. 
 
 wire and a new reading taken. The condition of exact balance 
 is not to be relied upon, so that the arrangement of Fig. 232 
 
 Fig. 234. — Equivalent Connection for Fig. 233, using One Wattmeter. 
 
 must ordinarily be used. The sum of the readings of the two 
 wattmeters then gives the power in the system. 
 
 B. Three-phase Systems 
 
 B 1. Three Wattmeters. a. If the power delivered by a 
 generator, or absorbed by a motor or other device, which is 
 connected in wye fashion, is to be measured, three wattmeters 
 may be used, connected as shown in Fig. 235, provided the 
 common or neutral point is accessible. It is evident that each 
 
392 
 
 ALTERNATING CURRENTS 
 
 W 
 
 Fig. 235. — Measurement of Power in Three- 
 phase System, Wye Connected. Three 
 Wattmeters used. 
 
 wattmeter measures the power in one branch so that the sum 
 of the readings gives the power in the system. If the system 
 
 is exactly balanced, three 
 times the reading of one 
 wattmeter gives the power. 
 
 b. If the devices are con- 
 nected delta fashion, three 
 wattmeters may still be 
 used, provided the current 
 coils of the wattmeters can 
 be inserted directly in the 
 branch circuits as shown in 
 Fig. 236. The power in 
 the circuit is equal to the 
 sum of the three wattmeter 
 readings, and if the circuit 
 is exactly balanced, three times the reading of one wattmeter 
 gives the power. 
 
 c. When it is impossible to insert the wattmeters in the coil 
 circuits of a device with delta connection, the three-wattmeter 
 method may still be used A 
 by the creation of an 
 artificial neutral point as 
 shown in Fig. 237. For 
 this purpose, three equal 
 non-reactive resistances 
 are connected together 
 at one end of each, and the 
 other ends are connected 
 to the respective corners 
 of the mesh circuit. In a 
 balanced circuit, the vol- 
 tage between the neutral 
 point and either corner 
 E 
 
 Fig. 236. — Measurement of Power in Three- 
 phase System, Delta Connected. Three Watt- 
 meters used. 
 
 is equal to — where E 
 V3 
 
 is the voltage of one branch, and the phase of this voltage is 0 
 degrees in advance of the current entering the corner. A watt- 
 meter with its current coil inserted in the circuit wire leading to 
 the corner carries a current equal to V3 /, where I is the branch 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 393 
 
 current, and if the free end of the voltage coil is connected 
 to the neutral point, the power reading of the wattmeter is 
 
 w 
 
 V3 IE cos 9 
 
 vF - 
 
 = IE cos 9, 
 
 Fig. 237. — Measurement of Power in 
 Three-phase System, Delta Con- 
 nected, with Artificial Neutral. 
 
 which is the power in the 
 branch. Care must be taken 
 that the resistances of the watt- 
 meter voltage coils are so large 
 compared with the three auxil- 
 iary resistances that connecting 
 them in circuit does not disturb 
 the voltage of the neutral point. 
 
 d. Tf the free ends of the three voltage coils are connected 
 directly together, the sum of the instrument readings gives 
 the total power, although the readings may not be individually 
 significant unless the resistances of the voltage coils are exactly 
 equal. 
 
 These methods are independent of the condition of balance in 
 the system or the current lag. 
 
 B 2. Two Wattmeters. The algebraic sum of the readings 
 of two wattmeters inserted in a three-phase circuit as shown in 
 yj Fig. 238, gives the power in the 
 
 system with entire independence 
 of the balance of the system or 
 current lag. When the current 
 lag in the circuit is less than 60°, 
 i.e. the power factor is greater 
 than .50, the arithmetical sum 
 of the readings is equal to the 
 power in the circuit ; but if the 
 lag is greater than 60° (the power 
 factor is less than .50), the rela- 
 tion of the currents in the current and voltage coils of one of 
 the wattmeters causes it to have a negative reading, and the 
 arithmetical difference of the readings of the two instruments 
 gives the power. There is some difficulty in distinguishing 
 which condition exists in some cases, especially when the power 
 absorbed by partially loaded induction motors, in which the 
 power factor is low, is under measurement. As a general rule, 
 
 Fig. 238. — Measurement of Power in 
 Three-phase System. Two Watt- 
 meters used. 
 
394 
 
 ALTERNATING CURRENTS 
 
 if the conditions do not make the case evident, the truth mav 
 be discovered by interchanging the positions of the instruments 
 without altering the relative connections of their main and 
 voltage coils. If the deflections of both needles are reversed, 
 the difference of the original readings represents the power, 
 but if the deflections are in the same direction as before, the 
 sum of the readings is correct. The proof of this theorem is 
 given in Art. 105. 
 
 A double wattmeter, consisting of two movable coils on one 
 spindle and two fixed coils, can be used in measuring power 
 by the two-wattmeter method. Such an instrument of itself 
 sums up the double reading algebraically, and a single reading 
 gives the power. Integrating wattmeters based upon this prin- 
 ciple are very useful in the commercial sale of power from two- 
 phase and three-phase circuits. 
 
 B 3. One Wattmeter. In a balanced circuit one wattmeter 
 may be very conveniently used by connecting the current coil 
 
 in one wire and connecting the 
 free terminal of the voltage coil 
 alternately to the other two leads 
 (Fig. 239), when the sum of 
 the readings gives the power. 
 For, the power reading of the 
 wattmeter in its first position is 
 V3 IE cos (0 + 30°), and in its 
 second position, 
 
 V3 IE cos (0 — 30°), 
 
 and the sum of the readings, 
 
 V3 IE \ cos (0 + 30°) + cos (0 - 30°)} = 3 IE cos 0, 
 
 where I E. and 0 are the current, voltage, and lag in a branch ; 
 but IE cos 0 is the power in one branch and 3 IE cos 0 is the total 
 power of the three branches, hence the one wattmeter gives correct 
 indications provided 0 is the same for all the branches, and the load 
 is uniformly distributed. A wattmeter having two independent 
 voltage coils could be used as a direct reading instrument for 
 this purpose. A similar wattmeter could also be used in one- 
 wattmeter measurements of power in two-phase circuits. 
 
 Thus, when one wattmeter is used as explained above in a 
 balanced three-phase circuit, the sum of the two readings gives 
 
 Fig. 239. — Measurement of Power in 
 Three-phase Balanced System. One 
 Wattmeter used. 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 395 
 
 3 El cos 9. In like manner it may be shown that the difference 
 of the same two readings gives V3 El sin 6. From which it 
 
 follows that ,,, nn 
 
 tan0 = V3^-^, 
 u \ + %< 2 
 
 where w x and w 2 are the two readings of the wattmeter. This 
 relation is sometimes convenient for obtaining the value of 9 and 
 thus the power factor of a balanced three-phase circuit. It may 
 also be shown by a similar process of reasoning that, when the 
 circuit is balanced, two wattmeters connected in the manner 
 illustrated in Fig. 238 give readings which may also be used in 
 this formula. 
 
 Wye Box. The power in a balanced three-phase system may 
 also be measured by using one wattmeter and what is called a 
 Y-box. This consists of two equal non-reactive resistance coils 
 with one end of each connected to a common binding post and 
 the other end of each connected to an individual binding post. 
 If each coil is of resistance equal to the resistance of the voltage 
 coil of a wattmeter with which it is designed to be used, insert- 
 ing the wattmeter current coil in one leg of the circuit, connect- 
 ing the individual binding posts of the wye box respectively to 
 the other legs of the circuit, and connecting the free end of the 
 wattmeter voltage coil to the common binding post of the wye 
 box, affords an arrangement equivalent to one wattmeter of 
 Fig. 240. The power for the 
 balanced system is three times 
 the wattmeter reading. If the 
 arms of the wye box are not each 
 equal in resistance to the voltage 
 coil of the wattmeter, a correc- 
 tion must be made. 
 
 105. Measurement of Power in 
 Any Polyphase Circuit. — In the 
 
 case of any polyphase system of n phases and n conductors, it 
 was originally proved by Blondel that the power in the circuit 
 may be measured by n — 1 wattmeters. Supposing A , B, G , D, 
 etc., are points where the n conductors of a polyphase supply 
 circuit connect to the circuits under test, then, as has been 
 already proved,* 1i = 0, if i represents the instantaneous cur- 
 
 Fig. 240. — -Measurement of Power 
 in. Three-phase System. One Watt- 
 meter used with Non-reactive Wye- 
 box. 
 
 * Art. 100. 
 
396 
 
 ALTERNATING CURRENTS 
 
 rent in any conductor. The power being supplied through the 
 
 A conductor at any instant is equal to c ^h- v a = i a v a , where q a is 
 
 the quantity of electricity brought to A during a time dt, and 
 v a is the absolute electrical potential of A. 
 
 The average power transferred through conductor A during 
 
 1 
 
 a complete period is — j i a v a dt , and the total power in the 
 
 T Jo 
 
 circuit, P 
 
 = 2 i r 
 
 TJo 
 
 ivdt. 
 
 The absolute potentials of the points are inconvenient to 
 measure, and it is desirable to introduce into the formula the 
 difference between the potentials at the points and some fixed 
 point of potential v' . Since = 0, we also have = 0, and 
 may therefore be directly inserted in the formula without 
 destroying the equality, or 
 
 P =1 ]pf T ( iv ~ iv ') dt = 2 - v ') 
 
 dt. 
 
 Writings for v — v' (the instantaneous difference of potential 
 between the fixed point and any given point in the system) gives 
 
 P = 2 \ Ciedt. 
 
 TJ 0 
 
 The fixed point may be taken at one of the corners of the cir- 
 cuit, A for instance, since SaV = 0 holds equally for it, and the 
 power formula becomes 
 
 P = 2 ’^J' T ijacdt + etc. + t n e ar dt , 
 
 or P — I b E ab cos Q' + I c E ac cos 6" + etc. 
 
 where 0\ 6 ", etc., are the differences of phase between the 
 respective line currents and the corresponding voltages to 
 point A. The terms on the right of the equation are the 
 familiar forms representing the power readings of a wattmeter, 
 so that if n — 1 wattmeters are inserted in circuit with their 
 current coils respectively in the n — 1 conductors, B. C , Z), etc., 
 and the free ends of their voltage coils all connected to ^4. the 
 algebraic sum of their readings is equal to the power in the 
 system. 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 397 
 
 When the circuit includes three phases only, the formula is 
 P = I b E ob cos O' + I c E ac COS0", and only two wattmeters, con- 
 nected as in Fig. 238, are required to give the power in the 
 circuit, but due regard must be had to the relative signs of 
 cos O' and cos 0" . 
 
 From the relative phases of the currents I b and I c and vol- 
 tages E ab and E ae it is easy to see that in a balanced three-phase 
 system O' = 0 X — 30°, and also that 0" = d 2 + 30°, and therefore 
 
 P = I b E ab cos (0 X - 30)° + I c E ac cos (0 2 + 30°), 
 
 in which 0 l and d 2 are the angles of lag of the branch currents, 
 and the subscripts indicate the respective branches. The for- 
 mula shows that the first term at the right, which represents the 
 reading of one wattmeter, is positive within the limits 0 l = + 90° 
 and dj = — 60°, and that its value is negative between the limits 
 0 l = —60° and 6 l = — 90°. The second term, which represents 
 the reading of the second wattmeter, is positive between the 
 limits — 90° and + 60° and negative between + 60° and + 90°. 
 Consequently, if the current lags equally in the circuits , i.e. 
 dj = : Both wattmeters have a positive reading and the power 
 
 in the circuit is the sum of the readings, for angles of lag between 
 -f- 60° and — 60°. It' the angle of lag is + 60° (the current lags 
 behind the voltage), the second wattmeter reading is zero, and 
 the power in the circuit is 
 equal to the reading of the 
 first wattmeter. If the lag is 
 greater than + 60°, the read- 
 ing of the first wattmeter is 
 positive and the second is 
 negative, and the power in 
 the circuit is equal to the dif- 
 ference of the two readings. 
 
 Again, if the angle of lag is 
 — 60° (the current leads the 
 voltage), the first wattmeter 
 reading is zero, and the power 
 in the circuit is equal to the 
 reading of the second instru- 
 ment. If the lead is more than 
 60°, the reading of the first instrument is negative and of the 
 
 Fig. 241. — Curves showing Relation of 
 Wattmeter Readings to Lag Angle in 
 Balanced Three-phase Circuit. 
 
398 
 
 ALTERNATING CURRENTS 
 
 second positive, and the power in the circuit is equal to the 
 difference of the two readings. When the angle of lag is ± 90°, 
 the readings of the two instruments are equal, but one is posi- 
 
 to Power Factor in Balanced Three-phase Circuit. 
 
 tive and the other negative. The relations of the wattmeter 
 reading's to the angle of lag between + 90° and — 90° are shown 
 by the curves in Fig. 241. These are two equal sinusoids with 
 a phase difference equal to 60°. The readings of two watt- 
 
POLYPHASE CIRCUITS AND MEASUREMENT POWER 399 
 
 meters in a balanced three-phase circuit at any value of the lag 
 are in the proportion of the corresponding ordinates of the two 
 curves. Figure 242 shows similar relations plotted in a different 
 manner. In this case the ratios of the readings of the watt- 
 meters are plotted as ordinates and the power factors as 
 abscissas. This curve is derived directly from the curves of 
 Fig. 241. 
 
 The same conditions obtain in an unbalanced circuit, pro- 
 vided equivalent angles of lag are considered. 
 
CHAPTER IX 
 
 HYSTERESIS AND EDDY CURRENT LOSSES 
 
 106. The Energy Losses Caused by Magnetic Hysteresis. — 
 
 When the number of lines of force passing through a coil is 
 changed, a voltage is generated in the coil the value of which is 
 
 e = — in which i s the change per second of the 
 
 10 s dt dt b L 
 
 number of magnetic linkages through the turns of the coil. 
 
 If a current is flowing in the coil when the change occurs, the 
 
 work dw , done upon or by the lines of force, is equal to eidt, 
 
 and therefore, dw= — But in case the magnetic circuit 
 
 is of uniform material and its cross section is of uniform area 
 
 d> = AB and d<b = AdB, and n= , where <£ is the mag- 
 
 4l7TI 
 
 netic flux in lines of force, B is the magnetic density (lines of 
 force per square centimeter), A is the cross section of the mag- 
 netic circuit in square centimeters, M is magnetomotive force 
 
 in Gilberts (one Gilbert = ampere-turns), and n is the num- 
 
 ber of turns in the coil. From this we have dw = iedt = 
 
 — AMdB j$ u t i n an y space sufficiently limited so that H, 
 
 10 7 x 4 ir 
 
 the field strength or magnetizing force, is uniform, M = H L, 
 where L is the length of the magnetic circuit, and conse- 
 
 ciuentlv dw = ~ HdB. It will be observed that AL is 
 
 1 J 10 7 x 477- 
 
 equal to the volume of the magnetic circuit under considera- 
 tion, and the coefficient of HdB is therefore proportional to 
 that volume. By integrating this we get the expression 
 
 W= f iedt = - * L — f HdB. 
 
 J 10 7 X 4 7T*' 
 
 400 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 401 
 
 The power in watts which is expended during the change is 
 of course the average rate of doing the work shown above and 
 is equal to the work divided by t. This work is taken from the 
 electric cii’cuit as the magnetism rises and returned to the cir- 
 cuit as the magnetism falls, when the varying magnetism is 
 
 f 
 
 HdB 
 
 B 
 
 
 r 
 
 s 
 
 
 / 
 
 — -/<J 
 
 0 
 
 H 
 
 Fig. 243. — Curve of Rising and 
 Falling Magnetism. 
 
 caused by varying the exciting current. The value of 
 equals an area between curves of rising and falling magnetism, 
 such as Ogjs of Fig. 243, when the operation starts with both 
 H and B of zero value, proceeds to y 
 
 a value of H equal to rj and the 
 corresponding value of B equal to 
 Or , and returns to the condition of 
 H equal to zero and the state of 
 remanent magnetism (as might be 
 the case if the magnetic circuit 
 was composed of iron) when B 
 equals Os. This work is stored 
 in the magnetic linkages or ex- 
 pended in heating the metal. That some of the work remains 
 stored in the magnetic linkages under the conditions indicated 
 by Fig. 243 is manifest by the finite value Os of the magnetism 
 at the end of the procedure ; and part of this stored energy is 
 recoverable, but it is impossible to say from the given premises 
 
 what proportion of the whole it con- 
 stitutes. However, when the j HdB 
 
 is carried through a closed (cyclic) 
 curve, such as is illustrated in Fig. 
 244 for instance, the available energy 
 of the magnetic linkages at the end of 
 the cycle is the same as at the begin- 
 ning, and in that case all of the energy 
 
 represented by — -- J HdB is con- 
 
 verted into heat. In Fig. 244 the 
 cyclic curve is JSBjsbJ. After the 
 iron part of a magnetic circuit has 
 come into a cycle of this form through continuous reversals 
 of the magnetism between the fixed limits of J and j, the 
 
 curve repeats itself with each reversal, and j HdB has a fixed 
 
 2d J 
 
 Fig. 244. — Cyclic Curve of 
 Magnetism. 
 
402 
 
 ALTERNATING CURRENTS 
 
 value per cycle. Upon first starting the cyclic magnetization, 
 B = / (-ff) travels over a line such as Ooj of Fig. 243, but it 
 takes up the fixed cyclic curve when the iron is subjected to 
 complete cycles or reversals of magnetism. 
 
 A mental picture of these phenomena may be formed by con- 
 sidering that work is done upon the magnetic molecules in 
 order to swing them into line when a piece of iron is mag- 
 netized. Part of this work is expended in overcoming an oppo- 
 sition to the swinging of the molecules which is akin to friction, 
 and this work is converted into heat ; but another part is ex- 
 pended in overcoming the live forces of mutual attraction and 
 repulsion between the molecules or groups of molecules, and 
 this part is returned to the electric circuit by any molecules 
 which swing back upon the withdrawal of the magnetizing force. 
 As no energy is required to maintain magnetic flux when 
 once established, there would be no loss of energy from mag- 
 netizing and then demagnetizing a piece of iron if the curves of 
 magnetism ascending and descending were the same, for in this 
 
 case jHdB for the descending curve would be equal and 
 opposite to CffdB for the ascending curve, and the two 
 
 would cancel. But in fact there is a difference between the 
 ascending and descending branches of the curves of magneti- 
 zation of iron and other magnetic metals, which is caused by 
 Hysteresis and there must be a loss of energy due to the oper- 
 ation of magnetizing and demagnetizing, which is proportional 
 to the area between the ascending and descending curves. 
 
 In Fig. 243 the area Orj is equal to j'HdB going up the curve 
 and srj is equal to ^ lid B going down the curve. The work 
 
 which was expended in going up the curve to magnetizing 
 force H and magnetic density B is proportional to area Orj; 
 and the work which was recovered in going down the curve 
 to zero magnetizing force is proportional to area srj. The 
 net work which has been expended in the operation, part 
 of which may yet be recoverable, though a portion of it 
 has been expended in overcoming what may be called mag- 
 netic molecular friction and is not recoverable, is propor- 
 tional to the area Ojs. The work proportional to the area Ojs 
 
 is W = 
 
 — AL 
 
 10 7 X 4 7T 
 
 The recoverable part of this is 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 403 
 
 retained by tbe remanent (residual) magnetic flux which has 
 a density B = Os. Of this part, all may he recovered except 
 that which is absorbed in molecular friction when the coer- 
 cive force is overcome and the magnetism is forcibly brought 
 to zero. 
 
 If the piece of iron is carried through a cyclic curve of 
 magnetization so that it coincidently returns to the same values 
 of magnetization and magnetizing force as those from which it 
 started, all the work which has been expended must have been 
 done at the expense of the magnetising circuit and is irrecov- 
 erable, as already explained. Consequently, in a cyclic curve 
 like Fig. 244, the area of the cyclic curve is directly pro- 
 portional to the lost energy, and is equal to the lost energy in 
 ergs when II is measured in gilberts per centimeter, B is given 
 in lines of force per square centimeter, L is given in centi- 
 meters, and A is given in square centimeters. If the cycles 
 are produced by reversals of the magnetization between equal 
 positive and negative values by rotating a cylindrical piece of 
 iron between fixed magnet poles or subjecting the iron to the 
 magnetizing effect of a coil carrying alternating currents, the 
 irrecoverable energy has a fixed value per cycle as shown by 
 the formula, which is proportional to the area of the cyclic 
 curve of magnetization or “hysteresis loop,” and this area 
 bears a close relation to the maximum magnetic density between 
 the limits of which the cycle of magnetism is carried. 
 
 This energy loss due to hysteresis was shown by Steinmetz, 
 as the result of a very brilliant series of experiments,* to be 
 closely proportional to the 1.6 power of the maximum magnetic 
 density, within the ranges of density used in ordinary engineer- 
 ing practice.' Later experiments carried on by Ewing and 
 others, including Steinmetz, have shown that the exponent need 
 not be exactly 1.6, but commonly approximates thereto, and for 
 most practical purposes a figure between 1.55 and 1.6 is suf- 
 ficiently close. At magnetic densities exceeding 20,000 lines 
 of force per square centimeter, the exponent seems to become 
 considerably smaller ; and it varies for different steels. 
 
 Warburg called early attention to the fact that the actual 
 loss of energy, when a metal subject to hysteresis is carried 
 through a complete magnetic cycle, is proportional to the area 
 
 * Trans. Amer. Inst., Elect. Eng., 1892, Yol. IX. 
 
404 
 
 ALTERNATING CURRENTS 
 
 of the cyclic curve of magnetism. This was also demonstrated 
 by Ewing independently of Warburg.* 
 
 Combining the results of the theoretical considerations, which 
 show that the area of a cyclic curve of magnetization is pro- 
 portional to energy per unit volume of the iron expended on 
 something analogous to molecular friction, and the experi- 
 mental considerations which show that the area is proportional 
 to approximately the 1.6 power of the maximum density of the 
 magnetic cycles, gives the following relations for the hysteresis 
 loss per cycle in a piece of iron: 
 
 W — ClIdB = v VB x x 10-7, 
 
 10' X 47 rj +s 
 
 in which rj is a numerical quantity depending on the quality of 
 the particular sample of iron and is often called the “hysteresis 
 constant ” of the iron, and V is the volume of the iron. The 
 numerical value of g for any sample of iron also depends upon 
 the unit of volume adopted, and the values hereinafter given 
 assume the volumes to be measured in cubic centimeters. 
 
 The expression is ordinarily written 
 
 W= gVB 16 x 10-7, 
 
 since the exponent x has approximately the value of 1.6 ; and the 
 hysteresis loss in watts is equal to this loss per cycle multiplied 
 by the number of cycles per second, or 
 
 P = v VfB 16 x 10-7, 
 
 where / is the frequency of the magnetic cycles. 
 
 When put in words, this is sometimes referred to as Stein- 
 metz’s law; namely, the hysteresis loss in each unit volume of 
 iron is equal to the product of (1) a hysteresis constant fixed by 
 the quality of the iron under consideration , (2) the frequency of 
 the magnetic cycles . and (3) approximately the 1.6 power of the 
 maximum magnetic density occurring per cycle. 
 
 107. Curves of Hysteresis, and of Current and Voltage dis- 
 torted by Hysteresis. — Figure 245 illustrates cyclic hysteresis 
 curves for the same test piece with varying degrees of satura- 
 
 * Ewing, Magnetic Induction in Iron and Other Metals, p. 99. 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 405 
 
 tion, when produced by alternating magnetizing currents with 
 a wave form in which the ordinates increase continuously to 
 a maximum and then 
 similarly decrease to 
 zero, as in a sine or a 
 parabolic wave. The 
 three cyclic curves 
 correspond to the three 
 exciting currents af- 
 fording maximum field 
 strengths, respectively, 
 of two, five, and fifteen 
 gilberts. , 
 
 Figure 246 shows a 
 similar curve resulting 
 from a magnetizing 
 wave which contains 
 marked indentations, 
 such as the current 
 illustrated by curve I 
 in Fig. 247 ; that is, 
 the wave does not in- 
 crease continuously to 
 a maximum, but has 
 ripples superposed on its form. It will be noted here that the 
 cyclic hysteresis curve contains small auxiliary loops. The 
 area now is proportional to the area of the main cyclic curve 
 
 plus the sum of the areas of the 
 auxiliary loops. Figure 247 shows 
 the current wave used in the last 
 series resolved into its harmonics, 
 which consist of a fundamental and 
 a fifth harmonic of considerable 
 amplitude. 
 
 If an air space exists in series 
 with the iron part of a magnetic 
 circuit, its effect is to increase the 
 exciting current which is required 
 to produce a given maximum mag- 
 netic density ; but, except so far as 
 
 Fig. 240. — Hysteresis Curve 
 caused by a Wave of Mag- 
 netizing Current which has 
 more than One Maximum. 
 
 Fig. 245. — Hysteresis Curves corresponding to 
 Three Maximum Magnetic Densities, each pro- 
 duced by an Alternating Magnetizing Current 
 with a Single Maximum Value. 
 
406 
 
 ALTERNATING CURRENTS 
 
 there is a greater loss in the electric circuit due to the larger 
 current, it does not increase the active component of the excit- 
 
 ing current, inasmuch as the area 
 of the hysteresis cycle is not 
 altered, assuming that the iron is 
 not changed in volume or con- 
 formation and the same maximum 
 magnetic density is maintained. 
 This case may be illustrated by 
 the tilted magnetic curve shown 
 in Fig. 248. The effect of the air 
 space is here represented by the 
 position of the tilted axis OY t 
 with respect to the axis OY. 
 Curve bOb' is the hysteresis loop of the ir - on in the magnetic 
 circuit, and the abscissa corresponding to any ordinate repre- 
 sents the magnetic force required to set up in the iron the 
 magnetic density corresponding to the particular ordinate. 
 The abscissa of the line GY 1 
 corresponding to the same 
 ordinate represents the mag- 
 netic force required to set 
 up, the same magnetic den- 
 sity in the air space. 
 
 Consequently these two ab- 
 scissas added together give 
 the magnetic force required 
 to set up that magnetic den- 
 sity in iron and air space in 
 series. It is thus to be seen 
 that the hysteresis loop for 
 the magnetic circuit com- 
 posed of iron and air gap in 
 series consists of a curve 
 like b x 0 b' v which has abscissas equal to the sums of the abscissas 
 of the hysteresis loop of the iron in the magnetic circuit and 
 the straight line which corresponds to the curve of magnetiza- 
 tion of the air gap. Manifestly, the area of the hysteresis loop 
 and the corresponding hysteresis loss is unaltered by the intro- 
 duction of the air gap in the magnetic circuit, provided the 
 
 Fig. 248. — Hysteresis Cycle of a Magnetic 
 Circuit containing an Air Space. 
 
 Fig. 247. — Component Harmonics 
 of the Magnetizing Current Giv- 
 ing Hysteresis Loop of Fig. 246. 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 407 
 
 limits of magnetization are kept the same ; but the plotted 
 curve is skewed off to the right by the effect of the air gap, 
 and the exciting current required to produce the maximum 
 magnetic density is increased on account of the reluctance of 
 the air gap. 
 
 108. Similarity between Magnetic and Dielectric Hysteresis 
 and Friction. — All magnetic materials seem to be subject to the 
 phenomena attendant upon a seeming molecular magnetic fric- 
 tion which are collectively called hysteresis. As already ex- 
 plained, the area of a cyclic hysteresis curve is proportional to 
 the energy which is converted by molecular action into heat 
 when magnetic material is carried through cycles of magneti- 
 zation. In the course of extended experiments already referred 
 to, Steinmetz demonstrated that the area of the cyclic curve 
 for iron is approximately proportional to the 1.6 power of the 
 maximum magnetic density within ordinary ranges of density. 
 Similar experiments were repeated by Professor J. A. Ewing, 
 and he therein demonstrated that the exponent 1.6 was not 
 necessarily fixed, but that 1.6 approximately represents the cor- 
 rect value. In some of Ewing's experiments the results were 
 more nearly represented by the exponent 1.55. He also showed 
 that it may be much larger (perhaps more than 2.5) at very 
 low densities. All experiments, however, go to show that the 
 area of the cyclic hysteresis diagram for iron is very nearly pro- 
 portional to the maximum magnetic density affected by an ex- 
 ponent of 1.6 within the range of densities in usual engineering 
 practice. 
 
 Some also suppose that dielectric hysteresis is subject to a 
 similar exponential law. When a condenser is subjected to 
 alternating voltage, energy is absorbed in the dielectric, and 
 this energy is proportional (according to the results of certain 
 experimenters) to the maximum charge of the condenser, — that 
 is, to the maximum electric stress or density of electrostatic 
 induction in the dielectric, — raised to the 1.6 power. It is 
 probable that the rotation of a dielectric body in an electrostatic 
 field would result in conversion of energy due to dielectric hys- 
 teresis in a manner analogous to the conversion of energy due 
 to magnetic hysteresis when iron is rotated in a magnetic field. 
 It may also be expected that the losses for equal frequencies 
 and comparatively small electrical densities should be practi- 
 
408 
 
 ALTERNATING CURRENTS 
 
 cally similar if a given volume of dielectric was rotated in an 
 electrostatic field, and if it was subjected to an equal frequency 
 of equal maximum electric stress by means of an alternating 
 electrostatic field. In the case of the rotating dielectric the in- 
 crease of loss should tend to become less and less as the electric 
 stress is increased, until with a certain value of the electric 
 stress the hysteresis loss should become practically zero. This 
 is analogous to the conditions that exist with respect to magnetic 
 hysteresis, and of course can only be true in case the molecules 
 act toward each other in a similar manner. 
 
 The foregoing conditions also have analogies in mechanical 
 friction, which has been experimentally proved to be affected 
 by an exponential law. Thus, for instance, a fluid friction, as 
 that of mercury at low velocity, has been experimentally shown 
 to vary approximately in proportion to the 1.6 power of the pres- 
 sure. This points the way to an inference that the molecular 
 actions which cause absorption and conversion of energy by fric- 
 tion may prove to be of the same fundamental nature and of the 
 same order as the molecular actions which cause the absorption 
 and conversion of energy through magnetic, and perhaps dielec- 
 tric, hysteresis. Professor Ewing has suggested that experiments 
 of his substantiate this view, but no complete demonstration of 
 these relations has yet been disclosed. Professor Steinmetz, 
 whose accomplishments in advancing alternating current phi- 
 losophy have been varied and great, has cast doubt on the rela- 
 tionship of dielectric hysteresis to the other phenomena, and some 
 of the later experiments seem to show that what was thought by 
 some to be dielectric hysteresis is in reality only the effect of cur- 
 rent seepage in the dielectric and is not of the nature of hysteresis. 
 
 109. The Magnetic and Dielectric Hysteresis Constants. — It 
 is easy to compute the energy which is absorbed through the 
 effect of magnetic or dielectric h}'steresis and converted into 
 heat thereby, provided measurements have been performed on 
 materials similar to the samples under consideration, so that 
 the coefficient y giving the heat developed per cycle in unit 
 volume of the material at unit magnetic density is known. 
 This coefficient, as already pointed out, is often called the Hys- 
 teresis constant of the material. 
 
 The constants of dielectric hysteresis have not been deter- 
 mined satisfactorily. It may be assumed that the dielectric 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 409 
 
 constant varies for different dielectric materials, but the evi- 
 dence of such experience as has been had in the commercial 
 world with condensers constructed with the ordinary dielectric 
 substances indicates that they do not differ very widely among 
 themselves in respect to losses per cycle, unit volume, and unit 
 electrostatic stress that may be due to hysteresis effects. 
 
 Large numbers of experiments have been made and are being 
 made by manufacturers of electrical machinery to determine the 
 constants of magnetic hysteresis related to various qualities of 
 iron and steel. Experiments on other magnetic materials ex- 
 cept iron and steel have been few. Cobalt and nickel have 
 been tested by Professor Ewing and others, and certain other 
 materials have been tested; but none of these have any impor- 
 tant significance in theory or practice at present. The range 
 of hysteresis constants of iron is well represented in the accom- 
 panying table. The better qualities of iron are such as would 
 prove satisfactory for use in alternating current transformers, 
 alternating current motors, and the like. 
 
 The following table, compiled by Messrs. Swenson and 
 Erankenfield, indicates in round figures the values of the coeffi- 
 cient r) for various irons ; but it is to be remembered that 
 variation in the processes of manufacture or in the treatment, 
 as well as in the composition of the metal, often make compara- 
 tively large changes in rj. A small addition of nickel or silicon 
 to soft iron intended for rolling into sheets tends to reduce the 
 value of 7] and increase the electrical resistance of the iron, and 
 such silicon iron is now generally used for the cores of trans- 
 formers and the like. 
 
 f 
 
 TABLE OF HYSTERESIS CONSTANTS 
 
 Best Wrought Iron and Steel Sheets 001 
 
 Good Soft Iron and Steel Sheets 002 
 
 Ordinary Soft Iron 003 
 
 Annealed Cast Steel 008 
 
 Ordinary Cast Steel ......... .012 
 
 Ordinary Cast Iron ......... .016 
 
 Hard Cast Iron and Tempered Cast Steel .... .025 
 
 The constants in this table are the values of rj to be used in 
 the formula with cycles per second, volume in cubic centimeters, 
 and magnetic density in lines of force per square centimeter. 
 
410 
 
 ALTERNATING CURRENTS 
 
 110. Measurement of the Energy absorbed by Hysteresis. — 
 
 Methods of obtaining the approximate values of magnetic 
 hysteresis constants of samples of iron are now so simple that 
 every well-equipped electrical engineering laboratory is fitted 
 for such work, and a large proportion of the manufacturers 
 carry out systematic tests upon their iron. Some of the methods 
 of measurement follow. 
 
 a. Some of the manufacturers do not make tests with special 
 instruments, but punch a certain volume of iron sheets into trans- 
 former stampings and determine the hysteresis loss by a watt- 
 meter when the magnetic density is at a fixed value. For this 
 
 purpose the stampings are built 
 up into a closed magnetic circuit 
 of constant cross section. There 
 is some uncertainty in determin- 
 ing the value of the magnetic 
 
 density, since this depends upon 
 the indications of a voltmeter and 
 is affected by the form of the im- 
 pressed voltage, because the maxi- 
 mum magnetic density depends 
 upon the area of the voltage wave 
 instead of upon its effective value 
 which is indicated by the volt- 
 meter. If two samples of iron 
 are tested by this means, using the 
 same exciting voltage, the results 
 of the test are of course compara- 
 tive ; but exact results cannot be 
 obtained readily unless the voltage 
 impressed upon the exciting winding is a strict sinusoid. 
 Figure 249 illustrates the arrangement of apparatus for this 
 mode of testing. The method is much used iu commercial 
 testing by manufacturers of electrical apparatus who only care 
 to obtain comparative data relating to the iron which they 
 use. It has the advantage of testing the iron under conditions 
 quite similar to those under which it is used iu transformer 
 cores and other such apparatus. If care is taken to impress a 
 sinusoidal voltage wave on the coil used for exciting the test 
 sample, the method can be used with reasonable accuracy. 
 
 Fig. 249. — Diagram of Arrange- 
 ments for approximately meas- 
 uring Hysteresis Losses. 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 411 
 
 The hysteresis constant is 
 
 P X 107 
 71 ~ VfB 1 - 6 ’ 
 
 where P is the power shown by the wattmeter after the copper 
 loss in the exciting coil and any loss from eddy currents in 
 the stampings have been deducted, Pis the volume of the iron 
 in cubic centimeters, / is the frequency, and B is the maximum 
 magnetic density in lines of force per square centimeter. The 
 magnetic density is obtained from the formula 
 
 rr_ irnBAf # 
 
 ' 10 8 
 
 where n is the number of turns of the exciting coil, A is the 
 cross section of the core in square centimeters, and B is the 
 voltage measured by the voltmeter. This is the same formula 
 as that earlier developed in the discussion of alternator voltage. 
 The symbol B in the formula, stands for the induced voltage set 
 up in the exciting coil by the alternating magnetism excited by 
 alternating current with a frequency of f periods per second 
 flowing in the coil. This is substantially equal to the volt- 
 meter reading, if the IR drop in the resistance of the coil is 
 negligible, and it is usual to assign to B the voltage read on 
 the voltmeter when testing according to this method. Caution 
 must be used, however, to avoid appreciable IR drop in the 
 coil because B in the formula, strictly speaking, is equal to the 
 vector difference of the voltmeter reading and the IR drop. 
 
 The foregoing method of measurement includes the eddy 
 current losses, and the distribution of the magnetism through 
 the cross section of the test sample may be disturbed by the 
 magnetic effect of the eddy currents. The magnetism set up 
 at any point in the cross section is dependent on the resultant 
 magnetic force at that point, and the eddy-current ampere turns 
 may be in substantial opposition to the ampere turns of the 
 exciting coil. The eddy currents, if of appreciable magnitude, 
 therefore tend to reduce the magnetic density in the interior of 
 the stampings, f Eddy current losses vary as the product of the 
 square of the frequency, the square of the maximum magnetic 
 density, and approximately as the reciprocal of the electrical 
 
 * Art. 11. 
 
 t Art. 113. 
 
412 
 
 ALTERNATING CURRENTS 
 
 resistance of the stampings, so that they may be calculated 
 and subtracted from the wattmeter reading. However, as the 
 laminations used in electrical machinery should be sufficiently 
 thin to reduce the eddy current losses to a small fraction of the 
 total Iron loss, which comprises the hysteresis and eddy current 
 losses, the wattmeter reading is often substantially equal to the 
 hysteresis loss alone. If the frequency of the alternating cur- 
 rent is very low, the eddy currents are almost always negligible. 
 
 Plans have been proposed for separating the hysteresis and 
 eddy current losses by computations based on test data. In 
 one process, wattmeter measurements are made of the losses at 
 two different voltages. The magnetic density is substantially 
 proportional to the impressed voltage, since the induced voltage 
 is equal to the vector difference of impressed voltage and IR 
 drop in the coil, the latter being of small value by the con- 
 struction of the apparatus; and the other factors of hysteresis 
 and eddy current losses are independent of the voltage. The 
 following expressions therefore connect the two wattmeter read- 
 
 in ° s: P = aE l - 6 + bE\ 
 
 P x = aE™ + bE ! 2 , 
 
 where the first term in each right-hand member represents the 
 hysteresis loss and the second term the eddy current loss. That 
 E and E l vary as the magnetic density is shown by the voltage 
 formula given earlier in the article. Hysteresis loss varies as 
 the 1.6 power of the maximum magnetic density, and hence as 
 E IS ; and eddy current loss varies as the square of the maximum 
 magnetic density, and hence as E 2 . The symbols a and b are 
 quantities depending upon the characteristics of the magnetic 
 circuit and the frequency, and are assumed to be constant. 
 From these equations 
 
 PE 2 ~P,E 2 
 a E™Ef - E 2 E™' 
 
 P, pi-6 _ PE,™ 
 
 ■ E 2 E 2 6 — E' 2 E 1 1S ‘ 
 
 As P, P x and P, E x are known by the instrument readings, 
 the values of a and b may be computed. The hysteresis losses 
 for the two voltages, then, are 
 
 H = P - bE 2 and H x = P 1 - bE 2 ; 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 413 
 
 and the eddy current loss for each voltage is 
 
 F= P — aE 1 - 6 and F x = P 1 — aE p 1,6 . 
 
 Since hysteresis losses are proportional to the 1.6 power of 
 voltage and eddy current losses to the square of the voltage, 
 the computed values of H, H v F , and F l ought to yield the 
 results : 
 
 H = H } Is = F i 
 
 E i-« E™ and E 2 E 2 ' 
 
 hut the fact is that results of the measurements and computa- 
 tions will seldom bear this test. The errors of observation 
 inherent in the measurements are large compared with the 
 quantities computed, and the effects of the errors are magnified 
 by tbe fact that observed quantities come into the computed 
 results by differences. The consequence is that this process 
 is not reliable and is little used. 
 
 If the frequency can be varied without altering the sine 
 form or amplitude of the voltage wave, the iron loss for different 
 frequencies f and f x may be written, 
 
 P =c f-.G + d, 
 
 Pi = <?/r - 6 + & 
 
 where the first term in each right-hand member is hysteresis loss 
 and the second is eddy current loss. The hysteresis loss varies 
 proportionally to/ --6 because it is proportional to the product 
 of the number of cjmles per second and _B 1,6 ; but B varies 
 inversely as the frequency when the voltage is maintained 
 constant in the arrangement of apparatus under consideration, 
 as can be seen from the formula for voltage given earlier in 
 the Article. The part of the iron loss due to eddy currents 
 should remain constant under the conditions here considered, 
 because the eddy current loss is directly proportional to the 
 product of frequency squared with magnetic density squared, 
 and these are inversely proportional to each other in the 
 arrangement of apparatus under consideration. The constants 
 c and d of the equations depend upon the characteristics of the 
 magnetic circuit and the voltage. Having observed P , P x and 
 /, f v c and d may be computed and the values of the hysteresis 
 loss and eddy current loss separated from each other by a 
 
414 
 
 ALTERNATING CURRENTS 
 
 process similar to that indicated on the preceding page ; 
 but this process is also subject to the causes of inaccuracy 
 accompanying the companion process. 
 
 b. Ewing Hysteresis Tester . — In this instrument a perma- 
 nent horseshoe magnet A, Fig. 250, is balanced upon knife- 
 
 edges and has a pointer attached 
 to it which swings over the scale 
 B. The test piece, made up of 
 laminated iron, is rotated between 
 the pole pieces of the magnet. If 
 no losses were present, the ten- 
 dency for the pointer to swing in 
 one direction would be as great as 
 in the other ; but with hysteresis 
 present, the torque upon the mag- 
 net is greater when the test piece 
 is leaving the magnet than when 
 approaching, and the average 
 torque exerted on the magnet is 
 proportional to the work done on 
 the test piece per revolution. 
 Since the hysteresis loss, that is, 
 the poAver absorbed by the effect 
 of hysteresis, is directly propor- 
 tional to the speed, the torque is independent of the speed pro- 
 vided disturbing influences like eddy current losses or wind 
 friction are not encountered. If, then, the test piece is rotated 
 rapidly enough to give a steady deflection of the pointer on the 
 magnet, but not to set up appreciable eddy currents or cause 
 Avind friction, the deflection of the pointer is proportional to 
 the hysteresis loss. 
 
 Test pieces of known hysteresis constants are furnished Avith 
 the instrument. The constants of unknown samples of the 
 same length and approximately the same cross section as the 
 standard samples can be determined by comparison with them. 
 As the hysteresis constant is approximately the same for all 
 values of the magnetic density within the range of ordinary 
 commercial practice, the use of a permanent magnet in such an 
 instrument seems justified, but the results obtained by the in- 
 strument are not very satisfactory. This is doubtless partially 
 
HYSTERESIS AND EDDY CURRENT LOSSES 415 
 
 due to the small size of the samples that may be used for 
 testing. 
 
 c. The Holden Tester . — In the Holden hysteresis tester the 
 pole pieces of an electro-magnet A revolve about a test piece 
 B , built up from disc or ring-shaped stampings as shown in Fig. 
 251. The magnetic flux traversing the test piece is measured 
 by means of a coil which surrounds the core and is cut by the 
 flux as it revolves. This coil is attached through a commutator 
 to a voltmeter. The density of the magnetism traversing the 
 test piece may be deter- 
 mined from the observa- 
 tions by means of the 
 formula 
 
 B = ciTFH > 
 
 where E is the voltmeter 
 reading, S the number of 
 test coil conductors on 
 the surface of the test 
 piece, V the revolutions 
 of the field magnet per 
 minute, and A the mean 
 cross section of the test 
 piece in square centimeters. The test ring is so supported that 
 it can twist against a spring, the torsion of which can be meas- 
 ured. The torque acting between the test piece and the field 
 magnet is proportional to the loss due to hysteresis and eddy 
 currents, as in the case of the Ewing tester, and can be deter- 
 mined from the torsion of the spring. This instrument can 
 be used either for comparison of different samples, by observ- 
 ing the deflections caused by each, other things remaining 
 equal, or for the direct determination of hysteresis constants by 
 computing the power absorbed by the hysteresis per cycle and 
 unit of volume at any desired magnetic density. The apparatus 
 has the advantage of permitting frequency and magnetic density 
 to be varied at the will of the observer. The former is varied 
 by altering the speed at which the field magnet is revolved by an 
 outside motive power, and the latter is varied by changing the 
 
 REVOLVING 
 
 Fig. 251. — Diagram of Holden Hysteresis 
 Tester. 
 
 * Art. 11. 
 
416 
 
 ALTERNATING CURRENTS 
 
 exciting current fed to the field magnet. The hysteresis and 
 eddy current losses may be separated at constant speed by ob- 
 taining the aggregate loss at two magnetic densities, the former 
 varying as B 16 and the latter as B 2 ; or at constant magnetic 
 density by obtaining the aggregate loss at two speeds, in which 
 case the hysteresis loss varies as V and the eddy current loss as 
 V 2 . The separation of the two losses may thus be established 
 by computations similar to those already explained. 
 
 d. Step by Step Method. In this method a core A, Fig. 
 252, is inserted in a magnetizing coil B and a measuring coil 
 
 C. A ballistic galvanom- 
 eter B is attached to the 
 coil C. After the residual 
 magnetism of the sample 
 has been destroyed by 
 some process, such as send- 
 ing an alternating current 
 through the exciting coil 
 and gradually reducing it 
 to zero, direct current is 
 sent through the coil B, 
 and increased by conven- 
 ient increments until the 
 highest desired excitation 
 is reached. The exciting 
 current is then reduced by the same steps. Upon reaching 
 zero, circuit connections are reversed and the process repeated. 
 Each change of the exciting current causes a change in the 
 magnetism set up in the test piece, which in turn induces volt- 
 age in the test coil and causes a throw of the galvanometer 
 needle, which must be read and recorded with the correspond- 
 ing current value. The magnetic density set up in the test 
 piece by the first increment of current may be computed from 
 
 the relation B = where R is the resistance of the galva- 
 
 2 An l 
 
 nometer circuit, K the galvanometer constant, /3 its throw, A the 
 cross section of the test piece, and n x the number of turns on 
 coil C. The added magnetic density for each additional incre- 
 ment of current may be computed from the same relation, using 
 the corresponding galvanometer reading. Plotting the mag- 
 
 Fig. 252. Step by Step Method of Testing for 
 Hysteresis Loss. 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 417 
 
 netic densities as ordinates with currents or magnetic force 
 
 (/T = _ 7rn ^- , w p ere n i s ti ie number of turns on the coil B, I is 
 
 10 * 
 
 the current shown by the amperemeter, and l is the average 
 length of the magnetic circuit) as abscissas, gives the hysteresis 
 loop, and the hysteresis loss per cycle may be estimated from 
 the area of the loop. 
 
 It is desirable to repeat the step by step current cycle a half 
 dozen or more times before taking readings, as otherwise the 
 curve may not be either closed or symmetrical above and below 
 the axis on account of the fact that the first cycle of magnetiza- 
 tion starts at zero //and B and is therefore unsymmetrical. As 
 the reversals proceed, the curve gradually assumes the form of 
 one of the curves shown in Fig. 245. This method is quite 
 laborious and liable to inaccuracies from various sources. 
 
 111. Computations of Eddy Current Losses. — In all conduct- 
 ing matter placed in a variable magnetic field there are in- 
 duced, as already frequently referred to, certain Eddy currents, 
 or Foucault currents as they are often called. The exact values 
 of such currents cannot, as a rule, be computed, because the 
 magnetic densities at different parts of a conductor and the rates 
 of change of magnetism cannot be fully known, nor can the 
 exact resistances of the paths of flow be fully determined. 
 There are some conditions under which the computation of 
 these eddy current losses is of value, as, for instance, in a trans- 
 former core where the character of the metal used is well 
 known and the paths of the magnetic lines of force can be 
 approximately predicted. Suppose the condition of a thin plate 
 of iron in which magnetic lines of force are set up parallel to 
 the sides of the plate and the magnetic density is uniform over 
 the cross section. When this magnetic density passes through 
 an alteration in strength, currents tend to circulate in paths 
 which are parallel with the edges of the cross section and per- 
 pendicular to the lines of force. The average length of these 
 paths and their aggregate cross section may be estimated if the 
 dimensions of the plate are known. The voltage set up in 
 each portion of the paths may also be calculated if the maximum 
 magnetic density and its rate of change are known. If we are 
 acquainted with the specific electrical resistance of the plate, 
 we can also determine, with a considerable degree of approxi 
 
 2 E 
 
418 
 
 ALTERNATING CURRENTS 
 
 mation, the average resistances in the paths of the eddy cur- 
 rents; and we are therefore enabled to compute with reasonable 
 accuracy the amount of power expended on account of the flow 
 of these currents. We can do the same when the magnetic 
 core is composed of an aggregation of cylindrical wires. In 
 the two following discussions it is assumed that the magnetic 
 material is so finely subdivided that the magnetic flux may be 
 considered constant over the cross section of a lamination. 
 This is permissible in view of the practice in modern manufac- 
 ture of electrical machinery. With such fine subdivisions of 
 the magnetic circuit, it is ordinarily sufficiently accurate to 
 assume that B = yH, where H is the impressed field strength 
 and y is the permeability of the metal, or that the magnetic 
 effect of the eddy currents is negligible. 
 
 A. Eddy Current Loss in Cylindrical Conductors. — Eddy cur- 
 rent loss varies directly as the square of the frequency, directly 
 
 as the square of the maximum mag- 
 netic density, and, if the resistance of 
 the circuits in which the eddy currents 
 flow is large compared with their react- 
 ance, the loss varies nearly inversely 
 with the resistance. These relations 
 are manifest from the fact that 
 
 Fig. 253. — Illustration of 
 Eddy Currents in a 
 Cylindrical Conductor. 
 
 P = I 2 R = 
 
 E 2 R 
 
 E 2 
 
 B + 
 
 X 2 ’ 
 
 It 
 
 and E is proportional to the rate of 
 change of the magnetic linkages in 
 the circuits. This is equally true whether the magnetic core 
 is rotated in a fixed magnetic field or is influenced by the mag- 
 netism induced by a coil carrying alternating currents. The 
 resistance of the eddy current circuits is dependent upon the 
 specific resistance of the metal concerned and the volume and 
 conformation of the metal within which the eddy currents may 
 be induced. 
 
 Assume that the large circle, Fig. 253, represents a cross 
 section taken perpendicular to the axis of a cylindrical wire of 
 magnetic metal and that the magnetism passes along the wire 
 perpendicular to the plane of the paper and of uniform mag- 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 419 
 
 netic density over the cross section. Now as the magnetism 
 varies, cylindrical currents will be set up which are projected 
 in Fig. 253 as concentric circles in the cross section of the 
 wire. If we consider the path of one of these currents of 
 radius r, the instantaneous voltage acting in the circumference 
 having 1 this radius is 
 
 0 dB 1n _ a 
 e = 7rr 2 — - x 10 8 ; 
 dt 
 
 which, when the magnetism varies as a sine function, becomes 
 
 e = t rr 2 d ( B m s[nn ) 10-8 = o 7 r‘ l rJB m cos a X 10" 8 , 
 dt 
 
 since a = 2 7 rft; and the effective voltage is 
 AE= V2 7 T 2 r 2 fB m 10 -8 . 
 
 The resistance of the circular paths in a unit length (centi- 
 meter) of an elementary cylinder of radius r and thickness dr 
 
 is 0 
 
 R = 2 7 -rp 
 
 1 1 x dr 
 
 where p is the specific resistance of the metal in ohms per one 
 centimeter of length and one square centimeter of cross section. 
 The instantaneous power (watts) expended in a conductor of 
 this resistance due to a current which is set up by voltage 
 having the value given in the first of the foregoing formulas, 
 considering the resistance of the eddy-current paths to be large 
 compared with their reactance, is approximately 
 
 e 2 _ Trr 3 (dB ) 2 dr 10 -16 
 
 r ~Ri~ - h * j 2 
 
 When the magnetism is sinusoidal, the average power (watts) 
 expended during a period in the filamentary path or cylinder 
 of radius r, thickness d>\ and unit length (remembering the 
 assumption that the magnetic effect of the eddy currents is 
 negligible in comparison with the inducing magnetic field), is 
 
 A 7 > (AiT) 2 _ 7r 3 r 3 (r?r) f 2 BJ 1 0 -16 
 
 R x 
 
 P 
 
420 
 
 ALTERNATING CURRENTS 
 
 and the joules per period are 
 
 A W= AP x T = 7r3r3 ^” > 2 1 ° ~ 16 dr ' 
 
 P 
 
 The next to the last expression gives the power expended 
 and converted into heat by eddy currents in an elementary 
 cylinder of unit length, thickness dr , and radius r; and the total 
 power expended and converted into heat by eddy currents in the 
 mass of the solid cylinder of conducting material of radius r l and 
 embraced between planes one centimeter apart is equal to that 
 expression integrated between the limits of 0 and r v We 
 therefore find that the loss of power in watts which is caused 
 by eddy currents in the unit length of the wire is 
 
 p= P ^PBJPdr _ 7r 3 PB m W 
 Jo 10 K p 4 p 10 16 ’ 
 
 and the work in joules per cycle converted into heat by eddy 
 currents is 
 
 W= P T = 77-3 
 
 4 p 10 16 ’ 
 
 T— ~ being the length of one period. 
 
 The strength of the eddy currents may be found in a similar 
 manner, by integrating 
 
 j_ P' irfB m rdr _ i rfB m r* 
 
 A B 1 Jo V210 V 2 v/2 10 8 p 
 
 To find the watts lost per cubic centimeter of the material at 
 frequency f and the joules lost per cubic centimeter of the 
 material and per cycle of the magnetism, the last two expres- 
 sions but one may be divided by the area rrr\ of the end of the 
 cylinder; and the watts expended per pound of material at 
 frequency f, or the joules per pound of material and per mag- 
 netic cycle, may then he obtained by multiplying the results by 
 the number of cubic centimeters of the material required to 
 make a pound in weight. 
 
 The formulas show that eddy current losses in a cylindrical 
 wire under the conditions assumed are proportional to the 
 fourth power of the diameter of the wire. If a core of given 
 configuration is to be made up of equal cylindrical wires, it is 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 421 
 
 manifest that the number of wires composing the core is 
 substantially proportional to the reciprocal of their diameter 
 squared. The total loss in a wire core of given configuration 
 composed of equal wires is therefore proportional to the square 
 of the diameter of the wires. 
 
 It is to be noted that the premises used in the analysis include 
 the assumption that the magnetic lines of force are parallel to 
 the walls of the wires, that they are uniformly distributed 
 over the cross section of the wires, that they vary sinusoidally, 
 and that the magnetic effects of the eddy currents may be 
 neglected. These assumptions cannot be accepted as rigidly 
 holding in practice, but any deviation is likely to occur in such 
 a way as to reduce the actual losses below those computed. In 
 particular, the magnetic effects of the eddy currents may not be 
 negligible, but introduce a disturbing factor which reduces the 
 
 power expended from to ^ GGs _ . This effect is generally 
 
 treated as though the eddy currents shielded the interior of the 
 conductor from the full impressed magnetic force, since the 
 magnetic effects of the eddy currents increase 
 with the distance from the surface of the 
 conductor.* 
 
 B. Eddy Current Loss in Sheets of Rec- 
 tangular Cross Section. — - A similar integra- 
 tion will indicate the losses in the case of 
 thin sheets of rectangular cross sections in 
 which the lines of force are parallel to the 
 sides and are uniformly distributed over the 
 cross section. 
 
 In a sheet of thickness d let a slice be 
 taken a distance x from the middle and of 
 thickness dx as indicated in Fig. 254, which 
 represents the sheet with its edge presented 
 to the paper. This slice, taken in connection 
 with a similar one on the other side of the 
 center line, may be considered equivalent to a complete ele- 
 mentary electric circuit if the disk is thin enough so that the 
 length required to complete the circuit at the ends is negligible. 
 
 -d 
 \/vvy 
 
 '%d* 
 
 i 
 
 ■I K 
 
 Ax 
 
 Fig. 254. — Cross Sec- 
 tion of a Disk or 
 Sheet of Iron much 
 magnified, illustrat- 
 ing Eddy Current 
 Circuits. 
 
 * Art. 113. 
 
422 
 
 ALTERNATING CURRENTS 
 
 The instantaneous number of lines of force embraced by such a 
 circuit is 2 IxB , where l is the length of the sheet parallel to the 
 page. The eddy currents flow perpendicularly to the lines of 
 force and parallel to the sides of the sheet, up on one side and 
 down on the other as indicated by the arrows in Fig. 254. 
 
 The instantaneous voltage set up by varying magnetism in 
 one centimeter of the elementary slice dx is 
 
 d -B 1 A —8 
 
 e — x— x 10 8 - 
 dt 
 
 When the magnetism varies as a sine function, this is 
 
 e = x s '“ u * x 10 -8 = 2 7i fxB m cos « x 10 -8 , 
 dt 
 
 and the effective voltage is 
 
 AJ2= 
 
 10 8 
 
 The resistance in ohms of a portion of the circuit one centi- 
 meter long, one centimeter deep, and dx thick is lt 1 = where 
 I o is the specific resistance in ohms. 
 
 The power expended in the elementary slice which is dx 
 thick, one centimeter long, and one centimeter deep (again 
 assuming the resistance of the eddy current paths to be large 
 compared with their reactance) is 
 
 A P = 
 
 A E 2 2 ir^PBJEdx 
 
 * i 
 
 10 16 X P 
 
 and the power expended in a part of the sheet one centimeter 
 long, one centimeter deep, and the full thickness d is 
 
 2 t rlPBJx^dx = 2 7T 2 PBJd 3 
 
 J-id 10 m p 4 x 8 x 10 16 x p 
 
 The joules per cycle converted into heat by the eddy cur- 
 rents are 
 
 w p -2^fB,:-dz 
 f 4 x 3 X 10 16 x P 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 423 
 
 The strength of the eddy currents may be found in a similar 
 manner by integrating 
 
 y A E _ C* d V2 TrfB„,xdx _ V2 7 rfB m d? 
 
 . ~ ^ 7/ 10 8 x} ~ ~ 8 x 10 8 x p ' 
 
 To obtain the power expended per cubic centimeter of metal 
 at frequency/, and the work expended per cubic centimeter of 
 metal and per cycle of magnetism, it is only necessary to divide 
 the expressions for P and W by d. This gives : 
 
 where 
 
 and 
 
 7 T*PBJd 2 
 
 6 x 10 16 x p 
 
 S(fB m dy, 
 
 
 6 x 10 i6 x P ’ 
 
 w 
 
 Tf cm 
 
 fBJd 2 
 
 6 x 10 16 x p 
 
 The value of p for soft steel and iron sheets varies from 
 0.9 x 10~ 5 to 1.25 x 10~ 5 with an average near 1 x 10 -5 
 ohms.* 
 
 The foregoing computations, like those relating to the eddy 
 current losses in wires, are founded on certain assumed premises, 
 including the assumptions that the magnetic lines of force are 
 parallel with the sides of the plate and uniformly distributed 
 over its cross section, and that the magnetic effects of the eddy 
 currents are negligible in comparison with the impressed mag- 
 netic force. Deviations from these assumptions in commercial 
 magnetic cores are likely to cause a reduction of the actual 
 losses below the computed values rather than an increase, and 
 the formulas are therefore safe for use in most commercial 
 instances. The magnetic effects of the eddy currents are often 
 of considerable importance instead of being negligible and are 
 therefore discussed in some detail in Art. 113. 
 
 The difference between the coefficients of r 1 and d in the 
 foregoing formulas for the eddy current losses in wires and in 
 
 * Smithsonian Physical Tables, 3ded., p. 255; Steimnetz, Trans. Amer. Inst. 
 Elect. Eng. Vol. 11, p. 60d ; London Electrician, Vol. 28, p. 631 ; Loppe’ et 
 Bonquet, Courants Alternatifs Industriels, p. 273. 
 
424 
 
 ALTERNATING CURRENTS 
 
 rectangular cross sections which are very thin compared with 
 their length shows that it is necessary to subdivide a core more 
 finely when the metal is in plates than when it is in wires if 
 the losses are to be equally small, but the greater mechanical 
 convenience during processes of manufacture and the more 
 satisfactory mechanical stability which may be obtained by 
 means of rectangular plates when placed in actual machinery 
 has led to an almost universal adoption of plates for the mag- 
 netic cores of electrical machines. It is to be noted that the 
 eddy current loss in a magnetic core of fixed dimensions is 
 
 proportional to the square 
 of the thickness of the 
 stampings used to build 
 up the core. 
 
 Figure 255, following a 
 curve calculated by Ewing, 
 shows the relative hys- 
 teresis loss and eddy cur- 
 rent loss in lamince of 
 different thicknesses when 
 the maximum magnetic 
 density is 4000 lines of 
 force per square centimeter 
 and the frequency 100 
 periods per second. The 
 curve which starts at zero 
 represents the eddy current 
 loss, the flatter curve repre- 
 sents the hysteresis loss, and the upper curve is the sum of the 
 other two and therefore represents the total iron loss. 4 he 
 curves take into account the magnetic effects of the eddy 
 currents, which become important when the sheets exceed a 
 thickness of one half a millimeter. The hysteresis curve rises 
 slightly with the thickness of the plates on account of crowding 
 of the lines of force by the magnetic screening effect of the 
 eddy currents. The figure makes it plain that eddy currents 
 may be serious sources of loss in the magnetic cores of electrical 
 machinery, and that it is desirable that the iron used in such 
 cores should be of high electrical resistance as well as of low 
 hysteresis constant. 
 
 0 9 18 27 36 45 64 63 72 81 
 
 THICKNESS OF PLATE IN MILS 
 
 Fig. 255. — Losses due to Eddy Currents and 
 Hysteresis in Sheet-iron Plates, produced 
 by Sinusoidal Alternating Magnetism of 
 100 Cycles per Second and 4000 Lines of 
 Force, Maximum Density. 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 425 
 
 112. Cyclic Curves of Eddy Currents. — If a periodic mag- 
 netization acts upon a laminated iron core, the eddy current 
 loss produced thereby may be represented 
 by the area of a loop such as is shown in 
 Fig. 256. In this curve the ordinates f\ 
 
 represent instantaneous values of the mag- 
 netic density B or of the total alternating 
 flux cj) in lines of force inclosed by eddy 
 current circuits, and the abscissas repre- 
 sent the corresponding values of induced V/ 
 
 eddy currents i in amperes. 
 
 If e = ■ - is the voltage causing i to 
 
 10 8 dt & 45 
 
 flow, it is easy to show that the area of the 
 cyclic curve of Fig. 256 is proportional to 
 the power expended and transformed into 
 heat by the eddy currents : The instantaneous value of this 
 
 Fig. 256. — Loop having 
 an Area Proportional 
 to Eddy Current Loss 
 occasioned by Sinu- 
 soidal, Varying Mag- 
 netization. 
 
 power IS 
 
 . . dd> 
 
 v = ze = i — , 
 F 10 \it 
 
 and the work expended during a time dt is 
 
 dW=iedt = i4i- 
 10 8 
 
 The differential of the area of the cyclic loop of Fig. 256 is 
 
 dA = id(f>. 
 
 Further, the work expended and transformed into heat by the 
 eddy currents during one cycle of the magnetism is 
 
 A 
 
 but this is equal to . 
 
 10 8 
 
 If the abscissas are laid off on the scale of one ampere per cen- 
 timeter and the ordinates on the scale of 10 8 lines of force per 
 centimeter, the area of the curve in square centimeters is equal 
 to the joules of eddy current loss per cycle. If other scales 
 are used in plotting the curve, the area must be multiplied by 
 a constant in-order to arrive at the joules lost. 
 
426 
 
 ALTERNATING CURRENTS 
 
 The cyclic curve in Fig. 256 is of the form that results when 
 the magnetism alternates as a sine function. If the magnetic 
 wave is irregular, the curve may lose the symmetrical regularity 
 notable in the curve of Fig. 256, since the abscissas of the curve 
 are proportional to the rate of change of the magnetism. Thus 
 if the curve of magnetism has harmonics which cause it to 
 have two maximum points per half cycle, there are two points of 
 rate of change of magnetism and current i equal to zero, which 
 result in subsidiary loops at the top and bottom of the cyclic 
 curve. Or, if the maximum value of the curve of magnetism 
 is pushed over towards « = 0° or « = 180°, the time rate of 
 change of the magnetism is decreased in one part of the period 
 and increased in another part, so that the cyclic eddy current 
 curve is skewed with reference to its axis. 
 
 An exciting coil which provides the impressed magnetic 
 force in a magnetic circuit must carry an additional energy 
 component of current on account of the eddy current loss which 
 is approximately „ 
 
 where P is the measured or calculated value of the eddy cur- 
 rent loss in watts, and E x is the induced counter- voltage in the 
 coil (substantially equal and opposite to the impressed line volt- 
 age in a highly inductive circuit as in the case of the unloaded 
 primary coil of a transformer). The combined eddy current 
 circuit is similar to the secondary coil of a transformer with one 
 turn in the coil, so that approximately 
 
 I 2 =I x n* 
 
 where I 2 is the effective value of the current in the combined 
 eddy current circuit and n is the number of turns of wire 
 in the exciting coil on the magnetic core. We then have 
 
 approximately 
 
 T> Eo 
 
 R, = r 
 
 A. 
 
 n 2 I x 
 
 In order that the cyclic curve of eddy current loss may be 
 plotted, it is necessary that the instantaneous values of mag- 
 netism and their corresponding rates of change shall be known. 
 
 * Art. 23. 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 427 
 
 B 
 
 Fig. 257 . ■ 
 
 ■ Cyclic Curve of Iron 
 Loss. 
 
 The rates of change are directly proportional to the corre- 
 sponding instantaneous values of induced voltage. The actual 
 plotting of the curve is not usually practiced, but the relation- 
 ships pointed out by means of theo- 
 retical considerations are valuable. 
 
 If the abscissas of this last curve 
 converted into terms of exciting 
 current are added to the corre- 
 sponding abscissas of the cyclic 
 curve of hysteresis for any par- 
 ticular periodic wave of magnetism, 
 a cyclic curve is obtained like that 
 in Fig. 257, where BTB'T' is the 
 hysteresis curve and BFB'F' is the 
 combined curve. The result is a 
 loop having an area proportional to 
 the total iron loss in the circuit and 
 it may suitably be denominated a 
 Cyclic curve of iron loss. 
 
 113. Magnetic Screening due to 
 Eddy Currents. — The law of the magnetic circuit asserts that 
 the magnetism in any part of the circuit is equal to the net 
 magnetomotive force in that part of the circuit divided by the 
 magnetic reluctance thereof. The net magnetomotive force is 
 the vector sum of all magnetic forces acting on the circuit. 
 When a varying magnetic flux is set up in material which is 
 an electrical conductor, the filaments of varying magnetism 
 become surrounded by eddy currents, and the magnetic effects 
 of the eddy currents may be quite large and even comparable to 
 the impressed magnetic force when the material is in large 
 masses and of high electrical conductivity. It is experi- 
 mentally well known that a copper or other highly conducting 
 plate placed in an alternating magnetic field has eddy currents 
 set up in it to so high a degree that their magnetic effect may 
 practically neutralize at the back of the plate the magnetic effect 
 of the impressed magnetic field, and the space beyond the plate 
 is practically screened from the original magnetic field. A 
 non-magnetic copper plate has no screening effect in a magnetic 
 field that is constant, and its screening action in an alternating 
 magnetic field is caused by the magnetic force of the induced 
 
428 
 
 ALTERNATING CURRENTS 
 
 eddy currents. This is in accordance with the theorem that 
 the magnetic force of induced currents must tend to reduce the 
 impressed magnetic flux. The induced voltage causing the 
 flow of eddy currents is in lagging quadrature with respect to 
 the inducing magnetic field, assuming sinusoidal functions ; 
 and if the path of the eddy currents is of very low resistance, 
 its impedance is mostly composed of the reactance component, 
 and the eddy currents are lagging nearly 90° behind the in- 
 duced voltage. Under these circumstances, the eddy currents 
 are nearly 180° in phase from the impressed magnetic force, 
 and their magnetic force is therefore nearly in opposition to 
 the impressed force. On the other hand, when the resistance 
 of the eddy current circuit is high, as has been assumed in the 
 computations of Art. Ill, the currents are small in volume 
 and lag little behind the induced voltage, in which case their 
 magnetic effect is small and nearly in quadrature with the 
 impressed magnetic force. It then has little effect on the 
 magnetism set up. 
 
 Certain important results of magnetic screening arise in 
 instances where the magnetic flux is constrained within a con- 
 ductor in a particular manner, as in the case of the magnetism 
 in the laminated iron core of a transformer or a dynamo arma- 
 ture which threads through the stampings parallel to their flat 
 sides. A simple example is a homogeneous cylindrical wire 
 of magnetic metal which may be considered as a part of a 
 magnetic core of some machine, and in which the magnetism 
 is set up parallel to the cylindrical sides of the wire bv" an im- 
 pressed magnetic force that is uniform over the cross section 
 of the wire. In case no other influences came into play, the 
 magnetism set up in this wire would be uniform over its cross 
 section, and this is indeed the case if the magnetizing force is 
 constant ; but if the magnetizing force is alternating, as when 
 the exciting current is alternating, circular eddy currents are 
 set up in the conductor which themselves produce magnetic 
 force and which may be in sufficient volume to strongly influ- 
 ence the net magnetic field acting on the iron of the wire. 
 
 Figure 258 shows the cross section of such a wire in which 
 the lines of force are supposed to be perpendicular to the plane of 
 the cross section. The magnetism being induced by an alternat- 
 ing exciting current is itself alternating of the same frequency, 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 429 
 
 and induces eddy currents. These flow in circles concentric 
 with the surface of the wire as illustrated by the dotted line 
 in Fig. 258, and their volume and phase relation to the im- 
 pressed magnetic force depend on the impedance of the eddy 
 current circuits. The magnetic force which the eddy currents 
 produce within the mass of the metal is parallel to the axis of 
 the wire, and its magnitude varies along a 
 
 radius from a maximum equal to v 2 ^ 77 ^ 
 
 1 10 
 
 gilberts at the center of the wire (in which 
 I is the aggregate volume of eddy currents 
 in effective amperes per centimeter of length 
 of the wire) to a minimum of zero at the 
 outer surface of the wire. The net vector 
 magnetic force at any point in the cross 
 section is equal to the difference between 
 the vector impressed magnetic force and the vector counter 
 magnetic force caused by eddy currents. The latter is equal 
 to zero at the outer surface of the wire, and the net mag- 
 netic force is therefore there equal to the impressed magnetic 
 force ; but at the center of the wire the counter magnetic force 
 may have a value comparable with the impressed force and a 
 phase nearly in opposition to it, and the net magnetic force may 
 there nearly disappear. The value of the eddy currents depends 
 upon the density of the inducing magnetism, its frequency of 
 alternation, and inversely upon the resistance of the eddy cur- 
 rent circuits ; and the phase of the eddy currents with respect 
 to the induced voltage depends on the frequency of the alter- 
 nations of the magnetism and the resistance of the eddy cur- 
 rent circuit. This resistance depends upon the specific 
 resistance and the dimensions of the conducting body. It 
 therefore follows that the magnetic force of the eddy currents 
 may be relatively large and its phase nearly in opposition to 
 the impressed magnetic force at the center of a thick metal 
 wire of low specific resistance, and the net magnetic force in 
 that case may be very small along the axis of that wire al- 
 though the impressed magnetic force is large. But in the case 
 of a thin metal wire of high specific resistance the eddy cur- 
 rents must be more nearly in quadrature with the magnetic 
 flux and small in volume, and the net magnetic force along the 
 
 Fig. 258. — Eddy Cur- 
 rent in Cross Section 
 of Wire. 
 
430 
 
 ALTERNATING CURRENTS 
 
 axis of that wire may not differ substantially from that paral- 
 lel to the axis near the surface of the wire. Since the resist- 
 ance of any circular path of the eddy currents of elementary 
 width and concentric with the walls of the wire is propor- 
 tional to the circumference of the circular path and therefore 
 to the radius of the path, and the self-inductance is propor- 
 tional to the area of the circle and therefore to its radius 
 squared, it is obvious that the lag angles of the eddy currents 
 will differ along any radius of the wire, and this will addition- 
 ally cause the net magnetic force to differ in phase from the 
 center to the circumference of the wire. 
 
 In consequence of these reactions, alternating magnetism 
 tends to forsake the central parts of the stampings of the cores 
 of electrical apparatus unless the stampings are thin enough or 
 of high enough specific resistance, or both, to make the resist- 
 ance of the eddy current circuits relatively large. This results 
 in a smaller aggregate magnetic flux being set up per unit of 
 impressed magnetic force, and gives the appearance of an in- 
 crease in the magnetic reluctance of the core, unless it is rather 
 finely laminated. 
 
 The extent of this effect in the laminated iron cores of elec- 
 trical machines subjected to alternating magnetism was treated 
 by Ewing in 1891, following a brief mathematical presentation 
 by J. J. Thomson.* 
 
 The formulas developed by J. J. Thomson, showing the ex- 
 tent of magnetic screening in stampings, take into account the 
 effect on the eddy currents of the modified distribution of the 
 magnetism and are in vector form, and numerical computa- 
 tions are complicated and laborious. Professor Ewing has 
 computed a table of the ratio of net magnetic force ( 7/) at vari- 
 ous depths iu a plate to the magnetic force (77 0 ) at the sur- 
 face, for iron plates of various thicknesses when the frequency 
 of magnetic alternations is 100 cycles per second. The table is 
 
 * London Electrician , Vol 28, pp. 599 and 631. See also Russell’s Alternat- 
 ing Currents , Yol. I, p. 359. 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 431 
 
 TABLE 
 
 Thickness of Plate 
 in Millimeters 
 
 :7atio of II at Middle 
 to IIq (at Surfaec) 
 
 Ratio of ff x to 7/ 0 
 
 Ratio of 77 a to 77 0 
 
 2. 
 
 0.120 
 
 0.342 
 
 0.250 
 
 1.5 
 
 0.245 
 
 0.42 
 
 0.341 
 
 1 . 
 
 0.520 
 
 0.629 
 
 0.564 
 
 0.75 
 
 0.739 
 
 0.82 
 
 0.793 
 
 0.5 
 
 0.925 
 
 0.940 
 
 0.932 
 
 0.25 
 
 0.995 
 
 0.995 
 
 0.996 
 
 here reproduced. The plotted results for three thicknesses are 
 exhibited in Fig. 259. In this figure the ordinates represent 
 the ratio of H to J? 0 , and the abscissas, measured from zero either 
 way, represent proportional distances from the center through 
 the thickness of the plate. 
 
 It will be seen that the 
 magnetic screening com- 
 puted by the formula for a 
 plate as thin as one half 
 millimeter (.0197 of an 
 inch) amounts to approxi- 
 mately 7.5 per cent at the 
 center, which is not of great 
 commercial importance in 
 most electrical machinery. 
 
 But when the thickness of 
 the plate is 2 millimeters 
 (.079 of an inch) the com- 
 puted magnetic screening at 
 the center of the plate is 88 
 per cent and the net magnetic force at the center as com- 
 puted is only 12 per cent of the impressed magnetic force. 
 The figures are made on the basis of iron with an assumed 
 uniform magnetic permeability of 2000 units, a specific elec- 
 trical resistance of 10,000 C. G. S. units (= 10~ 5 ohms) and a 
 sinusoidal magnetizing force having a frequency of 100 periods 
 per second. 
 
 The symbol R 0 in Ewing’s table and charts stands for the 
 value of the impressed magnetic force at the surface of the 
 plate, H l stands for the average value of the magnetic force 
 
 Fig. 259. — Distribution of Net Magnetic 
 Force across Iron Plates of Various 
 Thicknesses, when the Frequency of 
 Magnetic Alternations is 100. 
 
432 
 
 ALTERNATING CURRENTS 
 
 over the cross section of the plate at the instant of maximum 
 flux, and H 2 stands for the value of H which would produce, 
 with eddy currents absent, the maximum total magnetic flux 
 
 now reached in each period. 
 Figure 260 is a chart taken 
 from Ewing's article, which 
 gives the ratios of H l and 
 to H 0 for various thick- 
 nesses of iron plates. 
 
 The formulas show that 
 the effect of magnetic screen- 
 ing is to reduce the useful 
 thickness of a plate to super- 
 ficial external layers which 
 have a thickness dependent 
 upon 
 
 ' P_ 
 
 Pf 
 
 .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 
 
 THICKNESS OF PLATE IN MILLIMETERS 
 
 Fig. 260. — Chart showing Ratios of H l and 
 to H 0 in Iron Plates of Various Thick- 
 nesses. 
 
 in which p is the specific electrical resistance and p. the mag- 
 netic permeability of the material composing the plate. The 
 formulas also show that, when the thickness of the plate is 
 large the eddy current loss is greatly modified by the magnetic 
 screening, and is proportional to TT 0 2 V ppf which is independent 
 of the thickness of the plate. When the thickness is small, as 
 for iron under 2 millimeters, the heating in each plate is pro- 
 
 portional to — <UzjL , where a is the thickness of the plate in 
 
 P 
 
 millimeters. Since a laminated core is made up of many plates 
 of thickness a, the total heating of a core of given size is pro- 
 
 portional to — . The last deduction agrees with the 
 
 deductions reached in Art. 111. 
 
 The formulas do not take into account the variation of per- 
 meability with the magnetic density in iron, and the actual 
 effect of the magnetic screening in cores of ordinary electrical 
 apparatus is much less than the formulas indicate, because the 
 permeability goes up with a reduction of the magnetic force 
 when the magnetic density is as high as the usual densities of 
 commercial practice, and the smaller net magnetic force at the 
 center of a stamping may not be accompanied by a proportional 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 433 
 
 reduction of the magnetism set up. As a consequence the 
 actual effects of magnetic screening are, doubtless, ordinarily 
 of little or even negligible effect in the cores of the usual com- 
 mercial electrical machinery, which, when subjected to alternat- 
 ing magnetism, are commonly built up of stampings not more 
 than 20 mils thick. At rather low magnetic densities, on the 
 other hand, the permeability decreases when the magnetic force 
 is reduced, and the effect of magnetic screening may on this 
 account be actually greater than the formulas indicate. 
 
 Figure 259 and the table are based on magnetic alternations 
 of 100 periods per second. The corresponding data for any 
 other frequency may be determined by the relation that the 
 effect of magnetic screening changes in approximately direct 
 proportion to the first power of the frequency. 
 
 114. Method for finding the Form of Exciting Current required 
 to produce a Given Alternating Flux when the Curve of Magneti- 
 zation is Known. — In order to indicate the form of the Exciting 
 Current required to set up a given core magnetization, suppose 
 first an electric circuit surrounding an iron core possessing a 
 
 Fig. 261. — Determination of Exciting Current for a Magnetic Circuit without 
 
 Hysteresis. 
 
 cyclic curve of magnetization like AOB , Fig. 2(31, which pre- 
 supposes no appreciable hysteresis or eddy current loss. r lhe 
 abscissas of this curve may be scaled as amperes of exciting 
 current, and the ordinates as magnetic density. Also suppose 
 that the form of the wave of magnetism in the core is to be 
 sinusoidal. The problem is to find the form of current wave 
 that must flow under these circumstances. The curve LL in 
 
434 
 
 ALTERNATING CURRENTS 
 
 Fig. 261 represents one period of the sinusoidal wave of mag 
 netism in the core, and the curve A OB represents the instan- 
 taneous relations between the magnetism and the exciting 
 current which sets it up. The ordinates of the two curves are 
 drawn for convenience to the same scale. 
 
 To find the current required to set up the magnetism at any 
 instant, such as corresponds to the point 0 on the magnetism 
 wave, draw a horizontal line from 0 to I) and a vertical line from 
 J) to I. The length 01 evidently represents the current in the 
 exciting coil which is required to set up the magnetism 1)1 — CM. 
 Next, from the point M, directly under <7, lay off a vertical 
 distance MN, equal to the current 01. The point N is one 
 point on the curve of the exciting current necessary to set up 
 the curve of magnetism LL. If a large number of points on 
 the current curve corresponding to successive points on the 
 magnetism wave are obtained by the same method, the result- 
 ant curve drawn through the points so determined is I^I . 
 The peaked form of the exciting current here indicated is 
 notable. The peak is manifestly caused by the increasing ratio 
 between the current and magnetism as the magnetic density 
 becomes greater and the permeability of the core is less. If 
 the curve of magnetization AOB were a straight line, the ratio 
 between exciting current and magnetism would be constant, 
 and the form of exciting current wave would then obviously 
 be the same as the form of the magnetism wave, the scale only 
 being: changed. It is the saturation of the core which is evi- 
 deuced by the crook in the curve of magnetization A OB that 
 causes the peak in the wave of exciting current, and the degree 
 of the peakedness depends on the degree of saturation to which 
 the core is forced by the magnetism set up. The maximum 
 exciting current plainly must come at the same instant as the 
 maximum magnetism, and, since hysteresis is assumed to be 
 negligible, the zero of current comes at the same instant as zero 
 magnetism. 
 
 In practical cases where iron forms part of the magnetic sur- 
 roundings of the electric circuit, hysteresis is not negligible, 
 and the magnetization curve, therefore, takes the general form 
 AHBJA , Fig. 262 ; and if the sinusoidal alternating magnetism 
 wave LL in the core is laid out on the plot in the same manner 
 as in Fig. 261, the exciting current 1 1^ is obtained by the same 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 435 
 
 process as before ; but the increasing limb of the curve of mag- 
 netization now differs from the decreasing limb, and the current 
 wave is not symmetrical about a vertical line drawn through its 
 maximum point. A scrutiny of the hysteresis cycle shows that 
 the current is not zero when the magnetism is zero, but has a 
 magnitude ST = OJ when the magnetism is passing through zero 
 in the positive direction and the corresponding value S' T = OH 
 when the magnetism is passing through zero in the reverse direc- 
 tion. The current comes to zero while the magnetism has a con- 
 siderable value in the decreasing quadrants between maximum 
 and zero points. For the hysteresis cycle shown in Fig. 262, the 
 
 falling positive current comes to zero while the magnetism still 
 has the value ZW = OV, and the falling negative current comes 
 to zero while the magnetism has the value Z'W = OV ' . Since 
 the point A on the hysteresis cycle represents simultaneous 
 maxima of magnetism and current and the same is true of the 
 point B , the maximum values of the current occur at the same 
 instants as the maximum values of magnetism. The current 
 wave is therefore deformed from the symmetry shown in the cur- 
 rent curve of Fig. 261. It has its maximum points at the same 
 instants as the cyclic curve of magnetism, as in the other instance, 
 but the zero values of current come earlier than the zero values 
 of magnetism, and the phase of the exciting current is advanced 
 with respect to the wave of magnetism. The equivalent angle 
 of this advance was called by Steinmetz the hysteretic angle of 
 advance. The fact is, that the exciting current now consists of 
 two components : an exciting component similar to the current 
 
436 
 
 ALTERNATING CURRENTS 
 
 of Fig. 261 and. in phase with the magnetism ; and an energy- 
 component required to provide the power absorbed by the hys- 
 teresis loss in the core and the I 2 B loss in the exciting coil. 
 The latter component is in phase with the impressed voltage, 
 which is in leading quadrature with the magnetism and is pro- 
 portional to the rate of change of the magnetism at each instant 
 in case the I 2 H loss in the exciting coil is negligible ; but if the 
 I 2 R loss is not negligible the phase of the impressed voltage is 
 slightly retarded. 
 
 In like manner, the exciting current required to set up sinu- 
 soidal magnetism in a core when the effect due to eddy eur- 
 
 Fig. 263. Exciting Current in a Circuit containing Hysteresis and Eddy Currents. 
 
 rents is included may be determined by' using the cyclic curve 
 of the iron loss.* The construction is shown in Fig. 263, where 
 AFB GrA is a cyclic curve including hysteresis and eddy 7 currents. 
 In this case the current is still more advanced. The component 
 of the advanced current in phase with the voltage furnishes the 
 power which is absorbed in the losses. 'Where the wave of 
 magnetism is sinusoidal, as in Fig. 263, the eddy 7 currents are 
 also sinusoidal, hence the component they 7 add to the exciting 
 current is sinusoidal and 90° in advance of the magnetism. 
 This element of the exciting current can be laid out to illus- 
 trate its position. In Fig. 263 the dotted curve corres- 
 ponds to the hysteresis cycle AIIBJA . and the curve I f I f is the 
 portion of the exciting current required to overcome the mag- 
 netic effect of the eddy currents. The ordinates of I 1 ' I' are 
 the algebraic sums of the ordinates of these two. It wall further 
 
 * Art. 112. 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 437 
 
 be shown in the chapter dealing with the transformer that the 
 eddy currents in the core act like currents in a secondary 
 winding of one turn. 
 
 115. Relationship between Form of Magnetism Wave and Form 
 of Induced Voltage. — In the foregoing discussion, a sinusoidal 
 wave of magnetism has been assumed. This may be readily 
 produced by impressing a sinusoidal voltage between the termi- 
 nals of the exciting coil even when the core is of iron, provided 
 the resistance of the winding of the coil is small enough so 
 that the IR drop is substantially negligible. The impressed 
 voltage, counter voltage induced by the magnetism in the core, 
 and the IR drop make a closed vector triangle ; and if the IR 
 drop is substantially negligible, the counter voltage must be 
 substantially equal and opposite to the impressed voltage. It 
 follows from the relations of the induced counter voltage and 
 the Avave of magnetism set up in the core that the latter is 
 sinusoidal when the former is. Under the circumstances, the 
 exciting current which is caused by the sinusoidal impressed 
 voltage to flow through the coil inevitably takes on a form and 
 volume which produces sinusoidal magnetism of just sufficient 
 amount to induce a counter voltage equal to the impressed 
 voltage. 
 
 Since the instantaneous voltage induced in a generator 
 armature or transformer coil or any reactive coil Avinding is at 
 each instant proportional to the time rate of change of the mag- 
 netism threaded through the windings (i.e. is proportional to 
 
 Avhere cf> n represents the magnetic linkages betAveen the 
 dt 
 
 conductors and the lines of force), it folloAvs that the magnetic 
 
 relations it may be seen that the form of the voltage curve may 
 be derived from the curve representing the change of magnetic 
 linkages during a period ; and vice versa , the form of the mag- 
 netism Avave may be derived if the form of the induced voltage 
 curve is knoAvn. For instance, if 
 
 cj),, = A 1 sin (a + /3j) + A 2 sin (2 a + /3 2 ) + A 3 sin (3 « + /S 3 ) 
 
 + ••• A n sin (ri« + /3„), 
 
 linkages at each instant are proportional 
 
 From these 
 
438 
 
 ALTERNATING CURRENTS 
 
 then, sines da = 2 nfdt , 
 
 e = — cos (« + /Sj) + 2 A 2 cos (2 a + yS 2 ) 
 
 + 3 A 3 cos (3 a + /S 3 ) + b nA.„ cos (/?« + /?„)]. 
 
 2 7r/* 
 
 In other words the ratio — r ~ is introduced into the ordinates 
 
 10 8 
 
 of the curve of voltage, all the harmonics are moved one 
 quarter period, and the curve of voltage is therefore in quadra- 
 ture with the curve of the magnetic linkages, and the ordinates 
 of each harmonic in the voltage curve are affected by a coeffi- 
 cient equal to the ratio of the frequency of the particular har- 
 monic to the fundamental frequency. Every harmonic in the 
 wave of magnetic linkages enters into the wave of induced volt- 
 age, and the ordinates of the harmonics are exaggerated in the 
 voltage wave in comparison with the ordinates of the harmonics 
 of the wave of magnetism, in an order proportional to the ratio 
 of their several frequencies compared with the fundamental fre- 
 quency. It is plain from this that induced voltage of sinu- 
 soidal form cannot be derived from magnetic linkages varying 
 according to any other function. Any deviation of the mag- 
 netic wave from the sine form produces a greater deviation 
 in the induced voltage. 
 
 In the case of a transformer or similar coil under conditions 
 of negligible magnetic leakage, the number of magnetic linkages 
 at each instant is equal to the number of turns in the coil multi- 
 plied by the number of lines of force in the aggregate magnetic 
 flux at that instant ; but, in the case of alternator armatures 
 or other windings which are distributed or are affected In- 
 appreciable magnetic leakage, the number of magnetic linkages 
 at any instant is not equal to the product of turns in the wind- 
 ing by instantaneous flux, but the instantaneous summations of 
 flux embraced by the individual turns must be used as the 
 ordinates of the magnetism wave when plotting that curve. 
 
 116. Alternating Flux caused by a Current of Predetermined 
 Form. — The wave of current in the main wire of a circuit is 
 the geometrical resultant of the current in all the branches of 
 the circuit. An inductive coil placed in series with the line 
 will have a current flow through it which is dependent for its 
 form upon the characteristics of all parts of the circuit. Sup- 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 439 
 
 pose for convenience that the voltage impressed between the 
 terminals of a circuit which contains an impedance coil, among 
 other things, is of such a form that a sinusoidal current is 
 caused to flow through the circuit. This fixes the form of the 
 current in the winding of the impedance coil. The form of 
 the magnetism wave set up in the core of the impedance coil bj 
 the current in its winding may be obtained by reversing the 
 procedure set forth in Art. 114, if eddy currents are negligible. 
 The construction is shown in Fig. 264 for a core containing 
 hysteresis but neglecting eddy currents. Here curve II repre- 
 sents a sinusoidal current passing through the impedance coil, 
 
 Y 
 
 Fig. 264. — Wave of Magnetic Flux caused by Sinusoidal Current. 
 
 and AHBJA is the cyclic hysteresis curve. Taking any 
 instantaneous value of the current, such as MN, laying it off on 
 the axis of abscissas from 0, as 01, the ordinate of the hysteresis 
 cycle erected from I to the hysteresis cycle at D shows the 
 magnetic density corresponding to current MN. The point C 
 at the intersection of the horizontal line drawn from D with 
 the line MN extended, is one point on the wave of magnetism. 
 The curve representing the wave of magnetism may be deter- 
 mined for an entire cycle by continuing this process. The 
 curve LL in Fig. 264 shows the wave of magnetism set up by 
 the current II in the core with cjmlic curve of magnetization 
 AHBJA, assuming the latter to be plotted in magnetic densities 
 for ordinates and amperes flowing in the exciting coil for 
 abscissas. 
 
 The induced counter voltage in the coil is a very irregular 
 curve under the circumstances here considered. It is propor- 
 
440 
 
 ALTERNATING CURRENTS 
 
 tional at each instant to the time rate of change of the magnetism. 
 Since the maximum of magnetism comes at the same point as the 
 maximum of exciting current, the induced counter voltage 
 passes through zero at the instant the exciting current passes 
 through its maximum. It will be observed that the eddy cur- 
 rents in the core depend upon the induced voltage in circuits 
 surrounding the magnetism, and the eddy current cycle cannot 
 be assumed to be the same as one caused by sinusoidal eddy 
 currents. 
 
 The considerations of this article are especially applicable to 
 the study of current transformers used with amperemeters and 
 wattmeters. 
 
 117. Effect of Iron Losses on the Apparent Resistance and Re- 
 actance of a Circuit. — The foregoing discussion of iron losses 
 leads to a generalization of great importance. 
 
 The true electrical Resistance of a conductor is a quantity 
 which depends solely on the dimensions, temperature, and char- 
 acter of the conductor. It is the electrical resistance which is 
 measured by means of a Wheatstone bridge, using continuous 
 currents in the process. It is the same whether an alternating 
 current or a direct current flows through the circuit, though it 
 may be masked by various effects of alternating currents. 
 
 In a perfectly insulated circuit carrying continuous currents, 
 the causes of losses of energy lie in the passage of the currents 
 through the resistances of the conductors composing the circuit 
 and the currents through devices which absorb energy by 
 presenting a counter voltage. In the case of alternating cur- 
 rent flow, additional sources of loss of energy become evident. 
 Energy may be absorbed by the effects of hysteresis and eddy 
 currents. The hysteresis may be either magnetic hysteresis or 
 dielectric hysteresis. An effect also arises from the self-induc- 
 tion of the conductor, which causes a concentration of the 
 current near the surface of the conductor, which is treated in 
 a later chapter under the name of skin effect. 
 
 Leakage effects due to imperfect insulation, and the effects of 
 convection accompanying brush discharges and the development 
 of a luminous corona between conductors of the circuit at very 
 high potentials, occur in either continuous current or alternating 
 current circuits. These effects also result in losses of energy. 
 
 The angle of lag in an alternating current circuit is obtained 
 
HYSTERESIS AND EDDY CURRENT LOSSES 
 
 441 
 
 from the expression 9 = cos -1 
 
 where P is the power meas- 
 
 ured by wattmeter and El is the volt-amperes obtained by 
 multiplying current by voltage. This angle 6 is the angle sub- 
 tended between Z and R in the right-angled triangle defining 
 the relations of Z, R , and X, namely : 
 
 operator Z = R - f jX, and 
 tensor Z— V R 2 + X 2 . 
 
 From this triangle also follows the right-angled triangle 
 IZ = X — IR + jlX == E r + jE x . 
 
 In this instance, the angle subtended between E and IR is the 
 angle 9. The value of X in these triangles is determined by 
 the frequency of the current and the characteristics of the cir- 
 cuit which fix its inductance and capacity, and its value is not 
 affected by the power transferred through the circuit. The 
 question at issue is: Does the value of R (in ohms) of the 
 horizontal sides of these triangles correspond with the circuit 
 resistance as measured by a Wheatstone bridge? A consid- 
 eration of the relations of the sides of the triangles results in a 
 
 negative answer. Since the horizontal side of the impedance 
 
 p 
 
 triangle must be equal to Z cos 6 = Z — — and E = Zf it follows 
 
 p m pp 
 
 that Z cos 6 = — , which will always differ from R = ^ if any 
 
 energy is expended in the circuit besides the I 2 R loss. This 
 points the way to the generalization of the term resistance in 
 which it is referred to as either positive or negative in connec- 
 tion with the semicircle diagrams of Art. 70. 
 
 Since power may be either positive or negative, that is, it 
 may either be absorbed by and expended in a circuit or be gen- 
 
 P 
 
 erated in and delivered by the circuit, it is obvious that -^like- 
 wise may be positive or negative, since its sign depends on P. 
 In general, then. Equivalent resistance of a circuit may be de- 
 fined as the ratio of power expended in or by a circuit to the 
 square of the current flow. When an alternating current flows 
 through a circuit, the equivalent resistance ordinarily differs 
 
442 
 
 ALTERNATING CURRENTS 
 
 from the electrical resistance of the circuit as measured by a 
 
 P — I 2 R 
 
 Wheatstone bridge by an amount equal to — — where Pis 
 
 the wattmeter reading and R is the electrical resistance of the 
 circuit. 
 
 When the word resistance is used in this text in relation to 
 alternating current circuits, it may be taken as meaning the 
 equivalent resistance unless it is qualified by the word true or 
 the word electrical. The equivalent resistance is the horizontal 
 component of the impedance triangle. 
 
 In any circuit carrying a steady current, the equivalent resist- 
 ance and the true resistance are always equal unless the effects 
 of some voltage-generating device are included within the circuit. 
 
 As a comparison of equivalent resistance with the true resist- 
 ance, their values for the primary coil of a transformer may be 
 taken when the secondary circuit of the transformer is open 
 and therefore does not absorb any power. In the case of a cer- 
 tain transformer, the voltage was 1100 volts, the exciting cur- 
 rent .2 amperes, and the wattmeter reading which gives the 
 power expended in hysteresis and eddy currents in the core and 
 I 2 R in the winding 150 watts. From these figures, the equiva- 
 lent resistance of the winding is shown to be 3750 ohms. The 
 impedance is 5500 ohms. The electrical resistance of the con- 
 ductor composing the winding was only one and a half ohms. 
 In another instance of similar kind the voltage was 1100, the 
 exciting current .043 amperes, and the wattmeter reading 37 
 watts. This gives an equivalent resistance of 20,000 ohms and 
 an impedance of 25,600 ohms. The electrical resistance was 7 
 ohms. In another instance the voltage was 60 volts, the cur- 
 rent .86 amperes, and the power 43.3 watts. This gives equiv- 
 alent resistance of 58 ohms and impedance of 70 ohms. The 
 electrical resistance of the conductor was .08 ohms. 
 
 It is obvious that the power consumption in the iron cores of 
 these transformers enlarges the power factor by advancing the 
 current to a phase nearer the voltage. This is partially caused 
 by hysteresis and partially by eddy currents. The part, as stated 
 before, caused by hysteresis was called the “ hysteretic angle of 
 advance” by Professor Steinmetz. This is an angle of advance 
 or negative angle of lag, and is not the whole of the effect pro- 
 duced by the iron core, though it is often the greater part of it. 
 
CHAPTER X 
 
 MUTUAL INDUCTION, TRANSFORMERS 
 
 118. Mutual Induction and Elementary Transformer Diagrams. 
 
 — The remarkable development in the use of alternating cur- 
 rents for transmitting and distributing electric power is 
 mainly due to the facility and economy with which they may 
 be transformed from one voltage to another. The induction 
 coils that are used for this purpose are called transformers, as 
 already explained, and their action is due to the inductive 
 effect which a varying current in one circuit exerts upon an 
 adjacent circuit as well as upon the circuit itself.* Since this 
 effect is a mutually interacting one, it is called Mutual induction. 
 Evidently, if the magnetism due to a flow of current in one 
 coil, which may be called the primary coil, see Fig. 46, passes 
 through the turns of another coil, which may be called the sec- 
 ondary coil, a variation of the current in the primary coil sets 
 up a voltage in the secondary coil. This voltage, when a sinu- 
 soidal alternating current passes through the first coil, is 90° 
 behind the phase of the current in the primary coil, when core 
 losses in the magnetic circuit are negligible. This is also true 
 of self-inductive voltage, and for the same reasons. f In case 
 the external electric circuit of the secondary coil is open and the 
 magnetism in the magnetic circuit is sinusoidal and all passes 
 through both coils, a phase diagram similar to Fig. 114 may he 
 constructed to represent the phase relations of the primary cur- 
 rent, magnetic flux, impressed voltage, and primary and second- 
 ary induced voltages. Such a diagram is exhibited in Fig. 265, 
 in which OF represents the impressed voltage F x ; 01 the current 
 in the primary coil I x ; OF , the self-induced voltage in the pri- 
 mary coil F lm ', and OK the induced voltage of the secondary 
 coil E v In this case the ratio of OF to OK is evidently equal 
 to the ratio of the numbers of turns in the two coils, since it is 
 
 t Art. 37, 
 
 * Art. 23. 
 
 442 
 
444 
 
 ALTERNATING CURRENTS 
 
 assumed that all of the magnetic flux links both coils, and 
 there is no secondary resistance drop, since no current flows. 
 OC represents the IR drop in the primary coil, and the primary 
 ampere-turns n 1 I l are proportional to 01. The magnetic flux 
 is assumed to be in phase with the current. Had iron losses 
 
 not been neglected the angle 6 X would 
 have been less, as the active com- 
 ponent of 01 would then have been 
 required to furnish these as well as 
 the I 1 2 R 1 losses. Also the angle be- 
 tween 01 and OF would have ex- 
 ceeded 90° by the hysteresis angle 
 of lead, but the magnetism itself 
 would still remain in quadrature re- 
 lation with E' lm . In this and the 
 discussion immediately following the 
 subscript m is used to denote in- 
 duced voltages caused by variable 
 magnetic fluxes which pass through 
 both primary and secondary coils. 
 When the letter E is used to repre- 
 sent primary voltages it is given a 
 prime when representing induced 
 voltages, and is without the prime 
 when representing drops of impressed 
 voltages. 
 
 Thus far the effect of mutual in- 
 duction is similar to that of self-in- 
 duction. But when a current flows 
 in the secondary coil under the in- 
 fluence of the mutually induced vol- 
 tage, another effect is produced, 
 which is not so closely related to 
 the conditions in a self-inductive 
 circuit. The current in the second- 
 
 Fig. 265. — Phase Diagram of 
 Voltage and Current in a 
 Transformer having Negli- 
 gible Iron Losses with the 
 Secondary Circuit Open. 
 
 ary coil tends to demagnetize the magnetic circuit apper- 
 taining jointly to the two coils. But the induced voltage in 
 the primary coil must at each instant be equal to the alge- 
 braic difference between the values of the active and impressed 
 voltages at that instant ; that is, the instantaneous impressed 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 445 
 
 voltage must be equal to the algebraic sum of the corre- 
 sponding instantaneous active and reactive voltages (the 
 components of impressed voltage drop in primary resistance 
 and reactance), the latter being equal to the self-induced 
 voltage in scalar value but reversed in direction ; or e x 
 — e lm + e 1R ^ where e lm is the primary reactive voltage and e 1R ^ 
 is the active voltage which drives the primary current through 
 the primary resistance. If the waves of voltage are sinusoidal, 
 we may write E lm = V E x 2 - E 1R 2 or E x = VE 1R 2 + E lm 2 , in 
 which expressions E x is the effective value of impressed vol- 
 tage, E 1R of active voltage, and E lm of reactive voltage. This 
 is a fixed relation determined by the law that the algebraic 
 sum of instantaneous voltages taken around a circuit must re- 
 duce to zero, and is therefore independent of the amount of 
 current flowing in either the primary or secondary circuit. 
 Then if the characteristics of the secondary circuit (load on 
 the secondary) are changed so that more secondary current 
 flows with a resultant decrease of induced voltage below the 
 value shown by the relation e lm = e x —e 1Ri , an increased primary 
 current flows in order to maintain the relation ; and if the 
 characteristics of the secondary circuit are changed so as to 
 decrease the secondary current with a resultant increase of the 
 induced voltage as indicated by the equation e lm = e x — e XRl , the 
 primary current falls, so as to reestablish the relation. In other 
 words, if the magnetic flux which induces E Xm is disturbed, 
 as by introducing a magnetizing effect extraneous to the pri- 
 mary coil, such as is caused by current flowing in the second- 
 ary coil, the primary current will again adjust itself so as to 
 maintain the inducing flux at such value as still to produce 
 the relation e Xm = e x — e 1R . Consequently, the magneto-motive 
 force of the primary and secondary currents of a transformer 
 must so combine at each instant that the resultant magneto- 
 motive force will create a wave of magnetic flux in the core, 
 the rate of change of whose linkages with the turns of the 
 primary coil is equal to 10 8 times the primary self-induced vol- 
 tage, which voltage must in turn be at each instant equal to 
 the difference between the impressed and active voltages at 
 that instant, witli algebraic sign reversed. In the special case 
 where the voltages are sinusoidal, the two magneto-motive 
 forces must combine in such a way as to create a sinusoidal 
 
446 
 
 ALTERNATING CURRENTS 
 
 wave of flux which is advanced 90° ahead of the wave of self- 
 induced voltage and is of a magnitude that produces a vector 
 of self-induced voltage equal to — — E 1R ^) = E 1Ri — E r 
 
 The relation of impressed, active, and reactive voltage then is 
 E x — E lRi — E ]jn — 0. The vector OE lm ! of Fig. 265 is equal 
 to - E lm . 
 
 This means that when the secondary circuit is closed and a 
 current flows under the impulsion of the secondary induced 
 voltage, the primary current must take such magnitude and 
 phase position as will in conjunction with the secondary am- 
 pere-turns produce the requisite magneto-motive force to act 
 in the joint magnetic circuit. This further means that the pri- 
 mary ampere-turns must, at each instant, neutralize the effect of 
 the secondary ampere-turns in the joint magnetic circuit, and 
 must in addition supply the ampere-turns which maintain the 
 inducing flux, i.e. n l i 1 — n 2 i 2 -\- , where the terms in the 
 
 equation represent respectively corresponding instantaneous 
 values of the total primary ampere-turns, the ampere-turns 
 of the secondary circuit, and the resultant ampere-turns of the 
 primary and secondary coils. It follows from this that i l = 
 1 n 
 
 - i 2 + i/, in which s = - 1 , n x and n 2 being respectively the num- 
 8 » 2 
 
 bers of turns of conductors comprised in the primary and sec- 
 ondary coils. Since flux equals magneto-motive force divided 
 by reluctance, and the reluctance of an iron core varies with 
 magnetic density, the exciting current ij varies quite irregu- 
 larly in a transformer with an iron core, even though the im- 
 pressed voltage and magnetic flux vary sinusoidally. The 
 total primary current which flows, when the secondary circuit 
 is open, multiplied by the primary coil turns may be called the 
 Exciting ampere turns. 
 
 If the magneto-motive force of either or both coils sets up 
 flux which does not pass through the other coil, the effect is 
 similar to the introduction of external self-inductance into the 
 circuits of the respective coil or coils in addition to the mutual 
 inductance, as such fluxes surround only the currents producing 
 them. Such flux is usually termed Magnetic leakage. In cases 
 where there is no magnetic leakage or other variable flux link- 
 ing only the secondary coil or its external circuit, the phase of 
 the secondary current is coincident with the phase of the induced 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 447 
 
 voltage in the secondary coil. Then if the primary current, 
 the exciting element of the primary current, and the secondary 
 current, are assumed to he sinusoidal, and there are no core 
 losses in the magnetic circuit, as may be the case if the mag- 
 netic circuit includes no iron, the relation becomes M x = 
 Vff/j 2 4- Tf 2 2 , where the letters M v and A f 2 represent the 
 scalar values of magneto-motive forces set up in the joint mag- 
 netic circuit by the currents. (In this early discussion of the 
 transformer it is assumed that effects of electrostatic capacity 
 are absent.) The vector relation is then = M x — pro- 
 vided no magneto-motive forces act in the magnetic circuit 
 except those set up by the currents mentioned. 
 
 A simple diagram, using equivalent sinusoids for currents and 
 voltages, such as is shown in Fig. 266, develops these relations 
 graphically. Vectors E x ( = OE) and OA) represent the 
 
 impressed voltage and exciting ampere-turns (the latter exag- 
 gerated in length for the sake of clearness) of a transformer 
 having negligible leakage and a non-reactive secondary circuit. 
 The vector n x 1^ and the vector OB representing total primary 
 ampere-turns may be vectors scaled to also represent either 
 primary currents measured in amperes or magneto-motive forces 
 measured in gilberts, while the secondary vector of ampere-turns 
 (9(7, by using the same scales, equals the scalar value and posi- 
 tion of secondary current divided by the ratio of transformation, 
 s, or the gilberts of secondary magneto-motive force. The 
 vector n x IJ( = Off) represents the equivalent magnetizing ele- 
 ment of the exciting magneto-motive force and lags behind the 
 latter by an angle equal to the angle of advance caused by the 
 iron losses in the core. It is 90° in the lead of, and determines 
 the direction of, the primary and secondary induced voltages, 
 
 tors is laid out the secondary vector of magneto-motive force 
 n 2 I 2 ( = (9(7), the secondary cii’cuit here being assumed non- 
 reactive. The vectors n 2 I 2 and n x I^ must then, to conform to 
 the results of the discussion above, form a closed vector triangle 
 with the total magneto-motive force of the primary current 
 ttj/p or n x l x and n 2 I 2 must combine to furnish the resultant 
 magneto-motive force acting in the joint magnetic circuit. 
 On the line OB , n x I v a distance (9(7 is laid off, which represents 
 
 On the line of the latter vec- 
 
448 
 
 ALTERNATING CURRENTS 
 
 Y 
 
 Fig. 266. — Phase Diagram of Voltages 
 and Currents in a Transformer when a 
 Current flows in the Secondary Circuit. 
 The various vector quantities used in the 
 construction are represented by lines as 
 follows: Impressed voltage E 1 by OE; 
 primary self-induced and secondary 
 mutually induced voltages E\ m and 
 sE'om by OF; voltage drop in resist- 
 ance of primary coil E 1R by OG; excit- 
 ing ampere-turns n i/ M and current I ^ by 
 0 A ; secondary ampere-turns ?ioX> and 
 
 currenti/j by OC ; quadrature ampere- 
 
 a 
 
 that portion of the voltage E^ 
 which, multiplied by the cur- 
 rent I v furnishes the necessary 
 power to drive the primary 
 current through the primary 
 resistance, or E 1R . The vol- 
 tage E 1 minus this voltage E lf ^ 
 must be numerically equal but 
 opposite to the primary self- 
 induced voltage E 1 '(= — OF) 
 and also numerically equal 
 but opposite to the mutually 
 induced secondary voltage 
 
 E 2m = — (^OF^j multiplied 
 
 the ratio of transformation, s. 
 
 The triangle ODE of Fig. 
 266 corresponds to the tri- 
 angle OCE in Fig. 265 and 
 OCA in Fig. 114, i.e. OD, 
 OC , and OC are the respec- 
 tive components of impressed 
 voltage projected on the cur- 
 rent, and DE, CE , and CA 
 are the respective components 
 in quadrature to the current ; 
 but in the case of Fig. 266, 
 unlike the other two, onl} T 
 part, E 1Ri , of the voltage is 
 lost by reason of power used 
 in the primary circuit. The 
 remaining power represented 
 by the vector product of E l I I 
 is transferred to the magnetic 
 circuit, and thence, deducting 
 core losses, to the secondary 
 electric circuit. Therefore, it 
 is evident that the effect of 
 
 turns nil/ by OH; primary current 
 and ampere-turns nj/j by OH. 
 
 the current in the secondary 
 circuit has been to decrease 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 449 
 
 E 
 
 — 1 of the primary circuit, 
 
 h 
 
 X' 
 
 and the angle of lag, 6 X = tan -1 ■ , where X x and R x are the 
 
 -“'l 
 
 apparent primary reactance and resistance. Starting with 
 n x I x = n x I^ as shown in Fig. 265, for a transformer with open 
 secondai’y circuit, then closing the secondary circuit and gradu- 
 ally reducing its resistance until full load is reached, will cause 
 the vectors and n x I v in a commercial, constant voltage 
 transformer, quickly to become so extended in length that 
 the parallelograms of magneto-motive forces and voltages 
 will have flattened out so much that, for most practical pur- 
 poses, the primary current and secondary current may be 
 considered to differ substantially 180° in phase. Thus, in a 
 certain transformer of 20 kw. capacity, the exciting current is 
 about one half of an ampere, while the full load primary 
 current is ten amperes, or I x is twenty times larger than 1 ^ a 
 difference which is still more marked in transformers of larger 
 sizes. Therefore, in a transformer without appreciable mag- 
 netic leakage, and fully loaded with a non-reactive load, the 
 apparent self-inductance almost disappears, and the primary 
 and secondary act like a single circuit made up almost entirely 
 of non-reactive resistance. The apparent primary impedance 
 reduces to nearly the sum of the resistances of the primary coil 
 and equivalent resistance of the secondary coil and the sec- 
 ondary external load circuit. There is very little variation in 
 either the phase position or scalar value of 1^ with change of 
 load, as will be seen later.* 
 
 The various conditions and problems that arise in the opera- 
 tion of transformers are entered into extensively in later articles, 
 hut before taking up the special cases of sinusoidal or other 
 periodic impressed alternating voltages, we will consider the 
 more general case of the characteristics of mutually reactive 
 coils as they are affected by any change of currents or voltages 
 whatever. 
 
 119. Mutual Inductance. — Consider two adjacent coils sur- 
 rounded by air and in which currents are flowing. Then at 
 any instant the total number of linkages of lines of force with 
 the turns of the conductors composing either one of the coils, is 
 
 both the apparent impedance Z x 
 
 2g 
 
 * Art. 128. 
 
450 
 
 ALTERNATING CURRENTS 
 
 the number of linkages due to the current in that coil alge- 
 braically added to the number of linkages due to the other coil 
 which are embraced by the first coil. If the current is changed 
 in either of the coils, an instantaneous voltage equal to ej 
 
 _ _ i s induced in the coil under consideration, which 
 
 10 8 dt 
 
 may be called the primary coil, where n x is the number of turns 
 and (p 1 the instantaneous average magnetic flux linked per 
 turn of the coil. The average number of magnetic linkages 
 per turn with the turns of the coil due to its own current is 
 
 -^i j^ i n which L x and I are respectively the self-inductance 
 
 n i 
 
 and the current in the coil.* 
 
 The number of lines of force due to the current in the first 
 coil which pass through the second coil evidently depends upon 
 the relative positions of the coils, but it cannot be greater than 
 the total number of lines set up by the current in the first coil. 
 For any two fixed coils in a medium of constant permeability, 
 this number of lines is proportional to the current flowing in the 
 primary coil. The voltage developed in the second coil, due 
 
 to a change in the current flowing in the first, is eJ = — 
 
 ° & 2 10 8 cft 
 
 where « 2 and <£ 2 are the turns in the second coil and the instan- 
 taneous average magnetic flux linked therewith per turn of that 
 
 coil. The equation e-J = — ma y p e written eJ = — 
 
 ^ 1 10 8 dt J 1 dt 
 
 when the permeability is constant.* The equation e 2 ' = 
 
 — may be similarly written eJ = — — — 1 where 10 8 J/ 
 
 !U 8 (ft J J 2 dt 
 
 is the number of linkages with the turns of the second coil of 
 lines of force which are due to the first coil when one ampere 
 is flowing in the first coil, supposing the coils to be in air or 
 other medium of unchanging permeability. The expression 
 represented by M is called, by analogy, the Mutual-inductance 
 or the Coefficient of mutual induction of the coils. If k is a 
 coefficient numerically equal to the ratio of the reluctance of 
 the path of the lines of force which interlink the two coils to 
 the aggregate reluctance of the path of all the lines of force set 
 up by the primary coil, then the value of cf) 2 k is equal to <f> v 
 
 * Art. 41. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 451 
 
 If the coils are long solenoids in air, with current only in the 
 first coil, the flux in the first coil is (f> l = where A is 
 
 the cross section and l the length of the coil ; and if the sole- 
 noids are wound one over the other so that their dimensions 
 are practically equal, the value of k is unity, because 0 2 evi- 
 dently becomes equal to <f> v and M becomes equal to . 
 If the same current is now switched into the second coil, we 
 
 have <A = ^ rTn ^^ aiK l M' == But in this special case 
 
 , 10 1 10 8 / 1 , 
 
 = and hence M=M '. Also L x = ^ and = Ml . 
 <£ 2 n 2 1 10 8 / 2 10 8 / 
 
 Consequently = M 2 , or M — V L X L V 
 
 If k is not equal to unity, it is manifest that some of the 
 magnetic flux set up by the ampere-turns of each coil do not 
 link with the turns of the other coil. Such lines of force are 
 called Magnetic leakage or Leakage flux. 
 
 If the foregoing long solenoids are now separated by draw- 
 ing one out of the other, the value of M continually decreases 
 as the separation continues, because the paths of the magnetic 
 flux set up by the two coils become differentiated. The self- 
 inductances of the coils remain constant, so that as the coils 
 separate, the mutual-inductance becomes less than V L x L r As 
 the separation of the coils becomes greater, the value of M de- 
 creases towards a minimum of zero. It reaches the latter value 
 when the coils are at an infinitely great distance apart or the 
 axis of one is placed symmetrically but at right angles with 
 reference to the axis of the other, so that the summation of the 
 linkages of the magnetic flux from each coil through the turns 
 of the other reduces to zero. 
 
 The maximum possible value of the mutual-inductance of the 
 two coils is therefore a mean proportional between the values 
 of their self-inductances, and the minimum value is zero. The 
 maximum value can only be attained when all the lines of force 
 due to one coil pass through all the turns of the other, and the 
 value of M may therefore be written M L X L V from which 
 it is at once seen that the mutual-inductance of the two coils must 
 always be small if the self-inductance of one or both of the coils 
 is very small, while the mutual-inductance may be large if both 
 the self-inductances are large. It is shown in a preceding par- 
 
452 
 
 ALTERNATING CURRENTS 
 
 agrapli that the value of M may be generally written Ms-n^Jt. 
 In this expression </q is dependent on n v whence it is shown 
 that My n x njc. This is also to be derived from the relation 
 M OC v L x L 2 i since L x and Z 2 are respectively proportional to n x 2 
 and n 2 2 .* 
 
 The summation product of the number of lines of force which 
 interlink two coils by the numbers of turns in the individual 
 coils will not change by changing the point of reference from 
 one coil to the other, provided the reluctance of the magnetic 
 circuit and its shape are unchanged, and, consequently, the 
 mutual-inductance of two coils is the same when measured from 
 either coil. When the magnetic circuit is composed wholly or 
 partly of iron, it is necessary to have the same distribution of 
 lines of force in the parts of the magnetic circuit when the two 
 measurements are made, in order that they may give equal 
 results. When the magnetic circuit is made up partly of iron 
 and partly of non-magnetic materials, and the coils are dissim- 
 ilar in their relations to the magnetic circuit, as in the case of 
 armature and field windings of dynamos, the condition of equal 
 distribution of lines of force is difficult to fulfill on account of 
 the effects caused by unequal saturation of the iron, and the 
 value of M may be quite different when measured from the field 
 winding and from the armature winding. This difference is due 
 to the difference in the permeability of the magnetic circuit and 
 the form of the magnetic paths, during the two measurements. 
 
 As shown above, the mutual-inductance is homogeneous with 
 and therefore of the same absolute dimensions as self-inductance. 
 Its unit is therefore the Henry. The Henry is 10 9 times as large 
 as the G.G.S. absolute unit of inductance. f 
 
 If the lines of force due to one coil which enter another do 
 not all pass through all the turns of the second coil, as may be 
 tacitly assumed in developing the foregoing relations, the defi- 
 nition of mutual-inductance still holds as already given, but the 
 summation of the linkages of lines of force with the individual 
 turns must be taken in a manner similar to the summations 
 required to obtain the self-inductance of a coil through all the 
 turns of which all of the self-induced lines of force do not pass.ij: 
 
 If a change occurs in the reluctance of the magnetic circuit 
 
 * Art. 41. 
 
 t Art. 42. 
 
 1 Art. 43. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 453 
 
 common to the coils, the relative positions of the coils remaining 
 unchanged, the mutual inductance is altered from M to M' = 
 
 p 
 
 ilf— , where P and P' are the reluctances of the path before 
 
 and after the reluctance is changed, such as by inserting an 
 
 p 
 
 iron core. The ratio — is dependent on the permeability of 
 
 the iron in the magnetic circuit, and the value of M must 
 therefore vary with the current in the coils in any case where 
 iron is in the magnetic circuit, while it is independent of the 
 value of the current when magnetic material is absent. 
 
 120. The Energy of Mutual Induction. — Assume two adjacent 
 coils with a constant mutual-inductance, in which all of the 
 flux passes through all the turns of both coils, such as the 
 superimposed long solenoids described in Art. 119. In one let 
 a current of I x an^eres flow, and in the other a current of I 2 
 amperes, the two magneto-motive forces being additive in the 
 mutual magnetic circuit. The number of linkages of lines of 
 force due to the first coil with the turns of the second is 
 10 8 JiJ r and the time rate of change in this number, when the 
 
 current in the first coil changes, is 
 
 10*Mdi 
 
 dt 
 
 h where i 1 is the in 
 
 stantaneous value of the current in the first coil while it is 
 changing. If the current I x is varied, the work due to 
 mutual induction which is done in the second coil in chang- 
 ing the magnetic field against the effect of the current I 2 is 
 obtained from the following relations, in which I 2 represents the 
 current in the secondary coil and e 2 represents the voltage in- 
 
 duced in the second coil by the time-rate of change 
 
 of 
 
 the current in the first coil, and is considered positive in sign 
 when it tends to increase Z 2 : 
 
 and div = I 2 e 2 'dt ; * 
 
 hence dw — — MI 2 di v 
 
 If the current in the first coil is changed from zero to ,Zj (J 2 
 being maintained by a battery of constant voltage), the work 
 
 * Art. 47. 
 
454 
 
 ALTERNATING CURRENTS 
 
 done on the second coil is W= — J MI 2 di x = — MI X I V which 
 is all abstracted from the electric circuit of the primary coil 
 and stored in the joint magnetic field. If the current I x falls 
 again to zero, this work is restored as electrical energy to the 
 primary circuit. When M varies with the current, the work is 
 still M1 X I V but M in the expression must be assigned its proper 
 value corresponding to the maximum magnetization of the core. 
 As the current in the first coil rises to I v the total work stored 
 in the magnetic field is evidently the sum of the work due to 
 
 A/ 2 
 
 self and mutual induction, or - - 1 - MI X I 2 . 
 
 If the current is caused to vary at the same time by a source 
 of energy in each of the coils, the following condition exists at 
 any instant. The instantaneous voltage, which must be im- 
 pressed on each coil to maintain the current flow therein, is the 
 
 sum of the active voltage (*i2), the reactive voltage, which 
 
 is equal and opposite to the voltage of self-induction, and the 
 reactive voltage, which is equal and opposite to the voltage 
 of mutual-induction, whence 
 
 e i — 0^2 + 
 
 ( 1 ) 
 
 e 2 — -®2*2 d" ^ 
 
 where e x and e 2 are the instantaneous impressed voltages in the 
 two coils, i x and i 2 are the corresponding instantaneous currents 
 in the two coils, R v R 2 , L v L v are the resistances and self-in- 
 ductances of the two coils, and M is their mutual inductance. 
 By transformation, we have 
 
 (e x — = Mdi 2 + L 1 di l , (2) 
 
 (e 2 — i 2 R 2 ~)dt = Mdi x + L 2 di v 
 
 M, L v and L 2 being assumed to be constant. Multiplying 
 these equations respectively by i x and i 2 and adding, gives 
 
 (i x e x 4- qe 2 ) dt — (i x R x + « 2 2 i? 2 ) dt 
 
 = L x i x di x + L 2 i 2 di 2 + M(i x di 2 + i 2 di{). 
 
 ( 3 ) 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 455 
 
 The first terra of the left-hand member of this equation rep- 
 resents the total work done by the impressed voltages during 
 the interval dt, and the second term of the same member rep- 
 resents the work expended in heat during the same interval on 
 account of the currents flowing through the resistances of the 
 coils. The difference between these two terms represents the 
 work done on the magnetic field, and this is represented by the 
 right-hand member of the equation. The total work done on 
 the magnetic field during any change of the currents, as from 
 zero to I x and I 2 amperes, is found by integrating the right- 
 hand member of the equation. Thus, 
 
 J /, . C r - ■ • C Iv ‘- ■ ■ L / 2 
 
 i\di\ + L<i iodi<i + ( iydi 2 + i^di-y ) = ^ 
 
 + Ljl + M I l I v (4) 
 
 This is all absorbed from the two electric circuits and stored 
 in the magnetic linkages, which are set up by the rise of the 
 currents. If the currents now fall to zero again, the same 
 amount of energy is restored to the electric circuits as the mag- 
 netic field is restored to its initial condition. The work MI X I 2 
 is equally divided between the two coils when I x = I T 
 
 The foregoing relates to two coils receiving their current 
 through separate electric circuits, but if the two coils are con- 
 nected in series relation in one electric circuit, so that they 
 carry the same current, the expression for the work stored in 
 the magnetic field as the current rises from zero to I amperes 
 becomes 
 
 M 2 + M 2 + i l fj 2. (5) 
 
 When the two coils in series are of equal self-inductances and 
 are located relatively to each other so that there is no magnetic 
 leakage, in which case L x — Z 2 = M, the expression represent- 
 ing stored work becomes 
 
 -I- 2 + ^ + MI 2 = 2 MJP = 2 LP. (6) 
 
 2 2 J 
 
 Equation (3) is written on the assumption that no eddy cur- 
 rent or hysteretic losses are involved, but in case such losses 
 come into the operation an additional energy term representing 
 
456 
 
 ALTERNATING CURRENTS 
 
 them may be subtracted from the first term in the left-hand 
 member of the equation. Equation (3) is also written on the 
 hypothesis of the magnetic fields of the two currents being 
 additive, and if they are subtractive, the algebraic sign of one 
 of the currents is changed. The result in equation (4) then 
 
 becomes 
 
 
 MI X 1 2 ; 
 
 and in (5) becomes 
 
 
 while (6) reduces to 
 
 L 1 P + L. i P 
 2 2 
 
 - MP = 0. 
 
 121. Transfer of Electricity by the Effect of Mutual Induction. 
 Coils in Series and in Parallel. — Continuing the assumption 
 of two coils in separate electric circuits and constant L 
 and M, suppose that no voltage is initially impressed on the 
 second coil though its circuit is closed ; then when the current 
 in the first coil is changed, the conditions in the second coil 
 are given from the equation 
 
 Whence 
 
 and 
 
 + -^2*2) — 0 - 
 
 i 2 R 2 dt — — Mcli 1 — L 2 di 2 , 
 
 di 2 . 
 
 The reason for the two zero limits of the third term of the 
 above equation is evident when it is remembered that the sec- 
 ondary current grows from zero and returns to zero during the 
 operation. Since the last term reduces to zero, the quantity 
 of electricity which is transferred in the second coil under the 
 inductive influence of the first when its current changes from 
 zero to 7j is 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 457 
 
 If the current of the first coil is now brought to its original 
 value, we have 
 
 The two quantities are equal and of opposite sign, so that the 
 transfer of electricity in the secondary coil during the rise and 
 fall of the primary current reduces to zero, provided the origi- 
 nal and final values of the magnetism of the magnetic circuit 
 are the same.* If the current in the first coil is a simple peri- 
 odic one, a periodic current of the same frequency is set up in 
 the second coil. 
 
 The instantaneous voltages in the two coils connected in 
 series in an electric circuit, where the current varies, are 
 
 r, . t (^2 -jt/rdx 
 
 e x = R x i + — ± M-, 
 
 = /fni + Ln—r i M' 
 2 2 2 dt 
 
 di 
 dt ’ 
 
 assuming the reluctance of the magnetic circuit to remain con- 
 stant ; and the instantaneous voltage across the two coils is 
 
 j; 
 
 e = ( R j + R 2 )i + ( L x + Z 2 ± 2 M') — • 
 
 dx 
 
 The positive or negative algebraic sign for M - in these equa- 
 tions must be selected according to whether the coils are con- 
 nected in the electric circuit so as to cause their magnetic effects 
 to aid or oppose each other in the joint magnetic circuit. Revers- 
 ing the connection in the electric circuit of the winding of one of 
 
 the coils reverses the algebraic sign of the expression related 
 to both coils. dt 
 
 Upon switching a constant voltage E into this circuit, the 
 deficit of electricity transferred through the circuit during the 
 
 ]E 
 
 rise of current to its final value I = — — is 
 
 R x + R 2 
 
 (Z, +L, ±2 31)1 
 
 R x + R 2 
 
 coulombs. 
 
 * Compare Art. 46. 
 
458 
 
 ALTERNATING CURRENTS 
 
 The “ extra current ” occurring at switching out the voltage 
 without changing the circuit resistance comprises an equal 
 number of coulombs. 
 
 Returning now to the equation 
 
 e = (-Bj + -B 2 )i 4" + -^2^ 2 T^)— ', 
 
 assuming e to be a sinusoidal function of time, so that 
 e — e m sin cot, the solution of the equation is, after a short lapse 
 of time when the current wave has become sinusoidal, 
 
 i = -- sin ( cot — 9), 
 
 z 
 
 in which Z = V-B 2 + [2 n rf(L 1 + X 2 ± 2 M )] 2 , 
 
 where R = R x 4 - B 2 , 
 
 and Z = R +j 2 + i 2 ± 2 3T), 
 
 and tan 9 = L ^~ Ml . 
 
 R 
 
 In case L x = Zl 2 and there is no magnetic leakage so that 
 M= VX 1 X 2 = X, the self-inductance of either coil, the value of 
 Z becomes either 
 
 Z = V R 2 4- (8 irfL ) 2 , or Z = B, 
 
 and tan 6 = MU , 0 r tan 6= 0°. 
 
 R 
 
 If the coils referred to in the two preceding paragraphs are 
 of equal self -inductances and are located relative to each other 
 so that there is no magnetic leakage, it is obvious from the 
 voltage equation that (since M then equals VX 1 X 2 = X) the 
 reactive voltage caused by the two coils in series is either four 
 times as great as the reactive voltage caused bj r one of the coils 
 alone, uninfluenced by the other, or is zero (in the first case 
 they act like a single coil of twice the turns of either); and, 
 also, if the two coils are of equal self-inductances, but are lo- 
 cated relative to each other so that M is zero, the reactive 
 voltage caused by the two coils is twice as great as the reactive 
 voltage caused b} r one coil alone. 
 
 When the number of coils in series is «, the equations of 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 459 
 
 voltage become, when the subscripts 1, 2; 1, 3; etc., represent 
 the mutual inductances between coils of those numbers, 
 
 e x — R x i + L x — + ( ± M x 2 ± M x 3 ± • ■ 
 
 
 rj'j 
 
 e 2 = ^2* + -^2 ^ + ( ± ^2, 1 ± ^2, 3 i " 
 
 
 e n = Rn % + + ( ± j ± 2 ± • 
 
 
 and the instantaneous voltage measured across the series of 
 coils is 
 
 e = i1*R + + 2 }'* ± M. 
 
 dt dt 
 
 The impedance of such a circuit is 
 
 Z = R y + [2 t rf(T[L + 22^ 1, ’"±ilf)] 2 , 
 
 and the reactance is 
 
 X = 2 irf(XL + 2 s 1)1 " ± i!f). 
 
 As far as the effects in the circuit of coils in series are con- 
 cerned, the mutual inductance between the coils in the circuit 
 merely adds to or deducts from the reactions caused by self- 
 inductance. Mutual inductance between coils in different 
 electric circuits, however, affords different results by causing a 
 transfer of energy from one electric circuit to the other when 
 the currents vary in one or both of the circuits. 
 
 The conditions are more complex when the mutually induc- 
 tive circuits are connected in parallel to the same main circuit, 
 instead of being connected in series as heretofore considered. 
 In case of two mutually inductive coils connected in parallel, 
 the voltage formulas are 
 
 and e = + (2) 
 
 The following equations may be derived from these by differ- 
 entiation, 
 
460 
 
 ALTERNATING CURRENTS 
 
 and 
 
 de r, di , , t d 2 i, 7,rd 2 i 0 
 
 dt = B 'ft + L 'le ±M w’ 
 de = R„ih + hfiz ± M St ' 
 
 dt 
 
 dt 
 
 dt 2 
 
 di 2 
 
 (3) 
 
 (4) 
 
 The positive sign pertains to when the coils are con- 
 
 nected to the electric circuits so that their magnetic effects are 
 
 di 
 
 cumulative, and the negative sign pertains to fff— when the 
 
 dt 
 
 magnetic effects of the coils are in opposition. 
 
 The following equations in i 1 and i 2 are obtained by multi- 
 plying (1), (3), and (4) respectively by R 2 , Z 2 , and T M, and 
 adding to get (5), and multiplying (2), (3), and (4) respec- 
 tively by R v t M, and L v and adding to get (6) : 
 
 R 2 e + (Z 2 T + (,R\L 2 + L 1 R 2 )-^ 
 
 
 d ^ 
 
 dt 2 ’ 
 
 R x e + (Z x T M) — — R x R 2 i 2 + (ZjL 2 + L X R^ 
 
 + (ZjZ 2 — M 2 ') 
 
 d\ 
 dt 2 
 
 (5) 
 
 ( 6 ) 
 
 Equations (5) and (6) afford exact solutions when R v R 2 , 
 L v L 2 , and M are of constant values and the voltage is a sine 
 
 function of time. In this case e = e m sin cot, ^ = coe m cos cut, 
 
 dt 
 
 and — e - = — (o 2 e m sin cot. Therefore ^ ^ ^ = — to 2 . Substituting 
 dt 2 dt 1 
 
 these values in equations (5) and (6), and putting R X R 2 — 
 
 o) 2 (Z 1 Z 2 - M 2 ) = a, R X L 2 + L X R 2 = b, = Z) and = Z 2 
 gives by integration,* 
 
 _ [ Z^ 2 S ^ n tut + &)(Z 2 T TZ) COS ^ q € - mi i 
 
 1 a + 6Z> 1 
 
 ■ _ r Ri Sin a t + <»(£, T M) cos + ^ + ( g) 
 
 a + hi) 
 
 * Murray’s Differential Equations, Chap. VI. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 461 
 
 The exponential terms quickly become negligible after the 
 current is started. Neglecting the exponential terms and 
 rationalizing the right-hand member of each of these equations 
 by multiplying numerator and denominator by a — bD , simpli- 
 fying the expression, and factoring the resulting coefficient 
 in the numerator gives 
 
 . ^ e m V[aR,+co 2 b(L, t M)V+ co*[a( L 2 t M)- bE 9 J 2 { . 
 
 1 a 2 -f oo 2 b 2 1 
 
 = e ” Vi?2 2 + a)2 (^ T .I I 2 B i„(©t - 0 X ), 
 
 Va 2 + co 2 b 2 
 
 ( 9 ) 
 
 1 a 2 + c o 2 b 2 
 
 ( 10 ) 
 
 = t - Vfi l 2 + a,2 ( J l sin («i - A.) ; 
 
 V <! 2 + •«* 
 
 in which 
 
 tan Q x = ~ l1 ^ and tan d 2 = *[< L \ ~ h ^ i \ 
 
 1 aR 2 + co 2 b (i 2 T df) 2 ai2 x -)- &> 2 5(f x T df) 
 
 It follows from equations (9) and (10) that 
 
 and 
 
 j _ V R 2 2 + 6)2 (A t df ) 2 £ 
 V a 2 + o» 2 /i 2 
 
 j = a) 2 (f, TdP) 2 ^, 
 
 Va 2 + co 2 b 2 
 
 I , I RS + co 2 (L,tM) 2 
 
 f 2 V A *! 2 + « 2 (f x T J /) 2 
 
 7 
 
 Va 2 + o) 2 ^ 2 
 
 A V A 2 2 + « 2 ( A 2 t df ) 2 
 
 z x = 
 
 +d 
 
 
 ^2 = - = 
 
 VA 2 2 + a) 2 (X 2 t df) 2 VA 2 2 + ft) 2 (X 2 T df) 2 ' 
 E 
 
 Va 2 + « 2 f> 2 
 
 h V^ + ^T#) 2 ’ 
 
 Z 2 = 
 
 :+d 
 
 
 Vi ? x 2 + « 2 (X x T df ) 2 V ^! 2 + ® 2 (£ x T df ) 2 ' 
 
462 
 
 ALTERNATING CURRENTS 
 
 When the coils are superposed so that there is no magnetic 
 leakage, are connected in the electric circuit so that their 
 magnetic effects are cumulative, and the resistances and self- 
 inductances are equal, the foregoing expressions reduce to 
 
 tan dj - tan d 2 — , 
 
 Ji = / 2 = 
 
 E 
 
 Vi£2 + (2 coLy ’ 
 
 Z 1 = Z 2 = ^E 2+ (2 <oL ) 2 ; 
 
 since the named conditions make R x = R 2 , L x — L 2 = M, and 
 
 di 
 
 satisfy the uppermost of the algebraic signs associated with M — 
 
 and It will be observed that the mutual inductance has, 
 under these circumstances, the effect of doubling the reactance 
 and increasing the angle of lag, compared with the values 
 
 (a> L and tan 1 for corresponding resistances and self-induct- 
 ances with M zero. 
 
 It is also to be observed from the equations that when the 
 coils are connected in the electric circuit so as to make their 
 magnetic effects cumulative, the effect of the mutual inductance 
 is always to increase the impedance of the coils. The impedance 
 is a maximum when M — V L X L 2 ; that is, when magnetic leakage 
 is negligible. 
 
 When the coils are superposed as before, but connected in 
 the electric circuit so that their magnetic effects are in opposi- 
 tion, R x being equal to R 2 and L v L v and M being equal to 
 each other, the equations reduce to 
 
 tan 6 X — tan 0 2 = 0, 
 
 I — I — — 
 i2 ~ir 
 
 Z x = Z 2 = R. 
 
 The last condition satisfies the lower one of the algebraic signs 
 di 
 
 associated with M — and M. The mutual inductance under 
 • dt 
 
 these circumstances neutralizes the effects of self-inductance 
 and reduces the reactance of the coils to zero. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 463 
 
 When the magnetic effects of the coils are in opposition, the 
 effect of their mutual inductance is always to reduce the im- 
 pedance and angle of lag. The maximum influence of the 
 mutual inductance occurs when M = V L X L 2 , that is, when there 
 is no magnetic leakage. 
 
 The formulas representing the reactions of coils in parallel 
 may be extended to any number of coils larger than two, by 
 the same processes as are exhibited in the foregoing. If either 
 branch circuit has electrostatic capacity in series with its resist- 
 ance and self-inductance, a term representing condenser voltage 
 must be added to the voltage equations corresponding to equa- 
 tions (1) and (2). When the capacity measured from one coil 
 to another is appreciable, it may be treated approximately by 
 considering it as a condenser composing a separate branch in 
 parallel with the coils ; but an exact solution of this case requires 
 the use of equations which represent the effects of distributed 
 resistance, inductance, and capacity, and have the character of 
 those developed in a later chapter. 
 
 122. Ratio of Transformation in a Transformer. — The formulas 
 of Art. 119 show that the voltage developed in a secondary 
 coil is at any instant 
 
 , _ d(Mi^) 
 
 2 “ dt • 
 
 If the current wave is a sinusoid, this becomes 
 
 eJ = 
 
 d ( Mi lm sin cot ) 
 dt 
 
 If the conditions require that M be treated as a variable 
 dependent upon the varying permeability of an iron core, this 
 equation is practically insolvable. For a close approximation 
 to practical conditions, however, it is sufficient to assume M as 
 having a constant value which depends upon the iron of the 
 core and the maximum magnetic density used in the trans- 
 former. The equation then becomes e 2 = — • The 
 
 maximum value of the voltage is then e 2m ' = 2 7rfMi lm , where / 
 is the frequency of the current wave, since da/dt = w. The 
 effective value of the secondary voltage is therefore evidently 
 E 2 = — 2 7i fMI v where I x is the effective primary current. If 
 
ALTERNATING CURRENTS 
 
 464 
 
 the secondary circuit is open, the following equations may be 
 written 
 
 j- 
 
 1 VE* + 4 7 T 2 f*Lf 
 
 while — E[ = 2 7 t/L x I x = ~ i where — E[ is numeri- 
 cally equal but opposite to the total induced primary counter- 
 voltage. (This includes the voltage induced by both the leak- 
 age and joint magnetic flux.) E x has been used to represent 
 the impressed voltage, and the maximum number of lines of 
 force in the cycle. If the resistance of the primary winding is 
 considered negligible, the former equation becomes 
 
 and 
 
 2 t rfL 1 
 
 E x = 2 t TfL x I x =-E[; 
 
 and, in this case, if the primary and secondary coils are so com- 
 pletely superposed that there is no magnetic leakage, the value of M 
 becomes M — Vi 1 Z 2 ; whence 
 
 E^ _ 2 irfLy I x __ L x or E x _ VXj~ _ 
 2 7 TfMI x ^L~L 2 Vi; 
 
 E n 
 
 But — 
 
 L , since the magnetic circuit of the two coils is 
 
 assumed to be identical, that is, M = V L X L V and consequently 
 there is no magnetic leakage. Therefore 
 
 o o 
 
 E x 
 
 = s. 
 
 n 
 
 2 
 
 In other words, if the active voltage in the primary circuit may 
 be considered negligible when compared with the impressed 
 voltage, and there is no leakage of magnetic lines, the ratio of 
 the impressed voltage to the voltage induced in the secondary 
 winding is equal to the ratio of the number of turns of wire in 
 the two coils. The ratio of the primary voltage and the secon- 
 dary voltage of a transformer is commonly called the Ratio of 
 transformation. The ratio of transformation of well-designed 
 transformers intended for use on constant voltage circuits is 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 465 
 
 practically equal to — when the secondary circuit is open, 
 n 2 
 
 showing that the assumption that the active voltage and 
 magnetic leakage are negligible in most commercial trans- 
 formers, when the secondary circuit is open, is entirely allow- 
 able. An example will show this in a striking manner. In a 
 certain transformer of 22.5 kilowatts capacity the electrical re- 
 sistance of the primary winding is practically 1 ohm and 
 the inductance is 9.1 henry s. At a frequency of 60 cycles 
 per second and a voltage of 2000 volts, the square of the value 
 of the reactance is nearly 12,000,000. In another transformer 
 of 11.25 kilowatts capacity, designed for 2400 volts primary 
 voltage, the value of the electrical resistance of the primary 
 winding is 6.45 ohms (when squared 41.6), and the square 
 of the value of the reactance is 10,000,000. In three other 
 transformers designed for a voltage of 1000 volts and respec- 
 tively of 7.5, 4.5, and 1.5 kilowatts capacity, the electrical re- 
 sistances of the primary windings are 1.16, 2.15, and 8.90 ohms, 
 while, at a frequency of 60 cycles per second, the squares 
 of the values of the reactances are respectively 25,000,000, 
 
 31.000. 000, and 100,000,000; and in a transformer of .5 kilo- 
 watt capacity, the electrical resistance of the primary winding 
 is 25 ohms and the square of the value of the reactance is 
 
 400.000. 000. In each of these cases, which represent common 
 practice in the construction of transformers, the value of R x 2 
 is entirely negligible when compared with 4 7 
 
 When the secondary circuit of a transformer without magnetic 
 leakage is closed, neither R x nor i? 2 can be neglected,* and then 
 the ratio of transformation evidently is decreased when the 
 secondary voltage is higher than the primary, and is increased 
 when the secondary voltage is lower than the primary on account 
 of voltage due to the current flowing through R x and R 2 (that 
 is, for a given impressed primary voltage the secondary terminal 
 voltage is decreased). 
 
 123. Magnetic Leakage in Transformers. — The primary and 
 secondary coils in the discussion of Art. 122 are supposed to be 
 so sandwiched together that magnetic leakage is negligible 
 when there is no current in the secondary coil. This is not 
 necessarily the case. A case when magnetic leakage is always 
 
 * Art. 127. 
 
 2 H 
 
466 
 
 ALTERNATING CURRENTS 
 
 present is shown in Fig. 267. From the figure it is evident 
 that even if no current flows in the secondary winding, the 
 counter-voltage in the primary winding will be greater per turn 
 of wire than the voltage induced per turn in the secondary 
 winding ; hence, even if the self-induced or counter-voltage in 
 the primary winding is practically equal and opposite to the 
 impressed voltage, the ratio of transformation will be changed 
 by this relative decrease of the secondary induced voltage. If 
 a current flows in the secondary winding, the self-induction 
 due to magnetic leakage in the secondary (lines of force link- 
 ing with the secondary coil, but not linking with the primary 
 coil) will still further reduce the active secondary voltage, and 
 
 Fig. 267. — Diagram for showing Transformer Leakage, a, a, leakage flux; 
 b, b, mutual flux. 
 
 the ratio of transformation will be further changed. The effect 
 of magnetic leakage in altering the ratio of transformation 
 (decreasing the proportional voltage induced in the secondary 
 winding by decreasing the magnetic flux passing through it) 
 was early shown by an experiment reported by Professor 
 Ryan.* In the experiment recorded by him, the primary and 
 secondary coils were wound on opposite sides of a laminated 
 iron ring with much the same arrangement as indicated in 
 Fig. 267. The number of turns in the primary and secondary 
 
 windings were respectively 500 and 155, or -A = 3.2. When a 
 
 n 2 
 
 voltage of 75.6 volts was impressed upon the primary winding 
 with the secondary circuit open, a voltage of only 16.4 volts 
 
 * Some experiments upon Alternating Current Apparatus, Trans. Amer. 
 Inst. E- E., Vol. 7, p. 324. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 467 
 
 was induced in the secondary winding, or — i = 4.6. The 
 
 -<8 
 
 whole difference in the two ratios was due to magnetic leakage, 
 and the magnitude of the difference shows that M was much less 
 than VXjig. The magnetic leakage was, in fact, nearly 30 
 per cent ; that is, the number of lines of force that linked with 
 the primary winding, but not with the secondary winding, was 
 30 per cent of the total magnetic flux set up in the magnetic 
 circuit. 
 
 The change in the ratio of transformation, just spoken of, 
 occurred when the secondary circuit was open. When current 
 is permitted to flow in the secondary circuit of such an arrange- 
 ment, the effect is much more striking, for under such condi- 
 tions the magneto-motive force of the secondary winding op- 
 poses that of the primary winding, and there is a strong tendency 
 to force magnetic flux across the air space surrounding the coils. 
 That is, because of the counter magneto-motive force of the sec- 
 ondary winding, the difference of magnetic potential between the 
 points A and B in Fig. 267 may be very great. When the sec- 
 ondary current is zero, the maximum magneto-motive force 
 tending to send flux from A to B through leakage paths is 
 measured by m = <&P, where <I> is the maximum flux in the 
 core and P is the reluctance in the iron circuit connecting those 
 points through the secondary coil. When a current flows in 
 the secondary winding, the maximum magneto-motive force be- 
 
 tween A and B is m — <t> P + 
 
 V2 x 4 7 t« 2 T 2 
 10 
 
 where the second term 
 
 on the right represents the part of the primary magneto-motive 
 force equal and opposite to the back or counter magneto- 
 motive force of the secondary winding, which in commercial 
 transformers, except for small secondary loads, is many times 
 larger than the first term. The values of <E> and P usually 
 change slightly when current flows in the secondary coil. 
 
 The effect of magnetic leakage on the action of a transformer is 
 analogous to the effect which would be produced on a transformer 
 without leakage by inserting coils having inductive reactance, 
 i.e. Reactance or Impedance coils, in the primary and secondary 
 circuits outside of the transformer, as is illustrated in Fig. 268. 
 These coils would have such self-inductances as to increase the 
 self-inductances of the primary and secondary circuits in the 
 
468 
 
 ALTERNATING CURRENTS 
 
 ratio of a : 100, where a is the magnetic leakage in per cent. 
 Since leakage causes a proportional increase in the apparent self- 
 inductance of the primary and secondary circuits, it causes an 
 equivalent lag of the currents in the two circuits. The react- 
 ance of the windings of a transformer which is caused by the 
 leakage flux may, for convenience, be called the Leakage 
 reactance, to distinguish it from the reactance caused by the 
 total of magnetic linkages. The part of the self-inductance, 
 which is due to the leakage flux may also, for convenience, be 
 called the Leakage inductance, to distinguish it from the total 
 self-inductance of the windings. 
 
 TRANSFORMER 
 
 The effect of magnetic leakage in transformers which trans- 
 form at constant primary voltage — Constant voltage trans- 
 formers — is evidently to interfere with the regulation of the 
 secondary voltage, so that in such transformers the leakage 
 paths are made of as high reluctance as possible. While Con- 
 stant current transformers, that is, those which transform from 
 constant voltage in the primary circuit to constant current in 
 the secondary circuit, depend upon leakage for regulation.* In- 
 duction motors are affected in much the same way as constant 
 potential transformers, with which, in principle, they have much 
 in common. f 
 
 124. Diagram of Transformer with Magnetic Leakage. 
 Equivalent Circuit. — The effect of magnetic leakage, as dis- 
 cussed in the preceding article, may be shown bj" a simple dia- 
 gram, such as Fig. 269. In this diagram, 0 C represents the vector 
 magneto-motive force caused by the current I v which flows in 
 the secondary circuit when voltage E 1 ( OE) is impressed on the 
 
 * Art. 143. 
 
 t Art. 192. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 469 
 
 primary winding, as in Fig. 266, and this magneto -motive 
 force is just balanced by the magneto-motive force produced 
 
 Fig. 269. — Diagram showing the Relations of Currents and Voltages in a Trans- 
 former with Large Magnetic Leakage. 
 
 by the component 1 ^ ( 01 of the diagram) of the primary cur- 
 rent. These magneto-motive forces are equal and opposite to 
 
470 
 
 ALTERNATING CURRENTS 
 
 each other as far as the joint magnetic path through the pri- 
 mary and secondary coils are concerned, the mutual flux being 
 produced by the magneto-motive force of primary component 
 OA ; but the magneto-motive forces of I 2 and 1\ act in parallel 
 on the paths of leakage flux, such as the leakage path from A to 
 B in Fig. 267, and set up leakage fluxes which create the re- 
 actance-induced voltages U 1Li and H 2I/2 (OH and OG of the dia- 
 gram). Except for the effect of the magnetizing current 1^ on 
 these reactance-induced voltages are opposite to each other. 
 They lag substantially 90° behind their respective inducing 
 magneto-motive forces, since the reluctance of each leakage path 
 is mostly outside of the magnetic material and there is therefore 
 no appreciable iron loss angle of advance of the current. 
 
 For conqflete accuracy, the voltage induced by the primary 
 leakage flux should be taken in quadrature with the total pri- 
 mary current, OB , and it is not exactly opposite to the voltage 
 induced by secondary leakage flux. It may be considered as 
 comprising two components, of which one is OH in lagging 
 quadrature with 01 and due to the leakage flux caused by the 
 01 component of primary current, and the other is in lagging 
 quadrature with OA and is due to the leakage flux caused by 
 the exciting component OA of the primary current. The first 
 of these components is opposite to OG. For simplicity, the 
 second of the components is neglected in the figure, since it is 
 very small in commercial transformers because the exciting 
 current is small. If the highest accuracy is desired, both com- 
 ponents may be introduced in the diagram and treated in a 
 manner corresponding to the treatment of OH in the following 
 discussion. The reactance-induced voltages OH and OG are 
 not necessarily of equal numerical values per turn in the wind- 
 ings, because the reluctance of the leakage paths about the 
 primary and secondary coils may not be the same. 
 
 The diagram, Fig. 269, is drawn with all vectors reduced to 
 primary equivalents by multiplying secondary voltages and 
 dividing secondary currents by the ratio of the turns in the 
 primary and secondary windings. The counter-induced volt- 
 age in the primary winding is the vector sum of E' lm , the voltage 
 induced by the mutual flux threading both the primary and 
 secondary windings, and represented by OF in the diagram, 
 and F Ul , the voltage induced by the primary leakage flux and 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 471 
 
 represented by OH in the diagram. The secondary induced 
 voltage E 2m is represented by OF in the diagram. The voltage 
 drops in the secondary circuit when current flows therein are: 
 OJ = NT. \ which is the drop of voltage E 2R2 in the resistance of 
 the secondary winding when current 00 flows; TF= — OGr , 
 which is the drop of voltage E 2Lt required to overcome the 
 secondary leakage reactance ; and ON which is the drop of 
 voltage through the load, i.e. is the terminal voltage E 2 . 
 
 The voltage drops in the primary circuit are: 0L= — OF , 
 which is the component —E lm of impressed voltage required to 
 overcome the counter voltage set up in the primary winding by 
 the mutual flux which threads botli windings ; LM 4- PE= 
 OK + OR = 0 V \ which is the drop in the resistance of the pri- 
 mary winding caused by the flow of primary current OB ; 
 MP = — OH , which is the component of impressed voltage 
 — E 1Ll , required to overcome the primary leakage reactance. 
 The impressed voltage is OE— OL + LM+MP + PE. The 
 primary angle of lag is d v The angle between the secondary 
 current and secondary induced voltage is angle TOF, but the 
 angle of lag of the secondary current with respect to the ter- 
 minal voltage is zero in the diagram; that is, the external sec- 
 ondary circuit is assumed to be non-reactive. The vector OB' 
 is the total voltage representing power expended in the primary 
 and secondary circuits, and HE is the total reactive voltage in 
 primary and secondary circuits. 
 
 The active secondary voltage 0 T is E 2Rn + N 2R — E 2(R .+«) 
 which is required to send the secondary current I 2 through the 
 given resistance of the secondary coil and external circuit. The 
 value of E 2(Ri+R) is determined from the vector diagram com- 
 prising itself and the induced voltages E^ and E 2L ^ the relation 
 being E 2(R 2+ R) = E 2m — E 2Ls . The voltage used in the secondary 
 coil resistance is E 2R , and E 2R is the drop in the load. The pri- 
 mary winding, being cut by the same flux that sets up E 2m , has 
 a counter-voltage E lm set up in it, in phase with and equal to 
 sE 2m , where s is the ratio of the number of primary turns to 
 secondary turns. The impressed primary voltage E 1 must have 
 a component equal and opposite to this E lm . The impressed vol- 
 tage E x must also furnish the voltage E 1R required to drive the 
 element of the primary current 1 which lias a magneto-motive 
 force equal and opposite to that of I 2 , through the primary coil 
 
472 
 
 ALTERNATING CURRENTS 
 
 resistance. It must also have a component — E lLi equal and op- 
 posite to the primary leakage voltage E lL . These three vol- 
 tages are closed, as indicated in Fig. 269, by the vector voltage 
 represented by OP , forming the vector polygon OL3IPO. But 
 i^,the exciting current, joins vectorially with I x , in making up 
 the total primary current I v and I is in advance of E lm by 
 an angle of 90° plus the iron loss angle of advance. Its com- 
 ponent OQ is in quadrature with OF and the angle AOQ is the 
 iron loss angle of advance. The voltage required in phase 
 with 7 m to drive it through the resistance of the primary circuit 
 is E c (= OR), and this must therefore be vectorially added to 
 the voltage represented by OP. The primary voltage drops 
 from the vector polygon OLMPEO , which has the resultant 
 OE equal to E v Instead of separating the voltage drop due 
 to the resistance of the primary coil into two components E xr ^ 
 and E c , its combined value in phase with I x , the total primary 
 current, might have been used as was done in Fig. 266. 
 The primary and secondary phase angles are 6 X and 0 2 , 
 the latter in this case being zero as the load assumed is non- 
 reactive. 
 
 The triangle ODP is especially worthy of notice. The side 
 OB represents the voltage that would be necessary to drive 
 7/ through the resistance of the primary coil R x plus the 
 voltage necessary to drive the secondary current 7 2 divided 
 by s through the resistances of the secondary coil R 2 and 
 external load resistance P multiplied by s 2 . Dividing 7 2 
 by s gives a quotient equal to 7/, and therefore 7 1 ' 2 
 
 (R 0 + R)s 2 = I 2 2 CR 2 -f- R). For this reason 2 . R 2 s 2 , Rs 2 , and 
 
 s 
 
 E 2m s are sometimes called the primary equivalents of the 
 secondary quantities. For like reasons secondary reactances 
 and impedances must be multiplied by s 2 to reduce to pri- 
 
 mary equivalents; thus, = V (_R 2 + R) 2 + (x 2 + x) 2 , 
 
 7 2 s 2 I x 
 
 where x 2 and x are the reactances of the secondary coil and load 
 respectively, or s 2 V(if 2 + R ) 2 - f (a -2 + x) 2 . The side OB 
 
 of the triangle OBP is thus an active component of the voltage 
 represented by OP, while the side BP, 90° therefrom, is reactive 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 473 
 
 and equal to the primary leakage voltage drop E lL plus the 
 equivalent secondary leakage voltage drop sE 2L ^. 
 
 Such a triangle of voltages can also be formed if the secondary 
 winding is replaced by a similar coil having s times as many 
 turns and a resistance equal to s 2 i? 2 , connected in series with 
 and magnetically opposed to the primary coil and in series with 
 a load circuit in which the resistance equals s 2 R. Such an ar- 
 rangement is diagramed in Fig. 270. Then OP (Fig. 269) is the 
 
 Fig. 270. — A Conventional Arrangement of Coils to absorb the Same Power and 
 produce the Same Reactances as a Transformer of Equivalent Impedance. 
 
 voltage measured across the three parts, i.e. the first coil, the 
 second coil, and the load; OP is the active voltage, and PP is 
 the leakage reactive voltage, as in the case of the transformer 
 discussed above. Under this arrangement there is no mutual 
 flux in the core, but only the leakage flux. To make the sub- 
 stitution complete, a current equal in scalar value and phase to 
 1^ must be shunted around the second coil and the external 
 load. The magneto-motive force of this current, when it flows 
 through the first coil to the dividing point, sends the same 
 magnetic flux through the core as 1^ sets up in the transformer, 
 and by which the voltages E lm and E 2m are induced (Fig. 269). 
 This current also combines with the impressed voltage to fur- 
 nish the power absorbed by the iron losses. The component of 
 the impressed voltage required to drive 1^ through the primary 
 resistance is then E c ; this combined vectorially with the voltage 
 
474 
 
 ALTERNATING CURRENTS 
 
 represented by OP equals the impressed voltage E v which is 
 
 required to make the currents I x and - I 2 flow respectively 
 
 through the first and second coils, analogously to the trans- 
 former. The counter-voltages set up in the two coils by the 
 mutual flux which threads both coils exactly counterbalance 
 each other, and have no effect on either the voltage or current 
 of the supply wires or the load, so for purposes of analogy 
 they may be considered to be absent. The' combination of 
 resistance and reactance necessary to shunt the current 1^ 
 around the second coil and load is indicated in Fig. 270 by 
 the part marked I Regulator. 
 
 In this manner two coils in series opposition have been 
 substituted for the transformer reduced to terms of unity 
 transformation, in which the equivalent voltages for the 
 transformer are impressed on the system, driving equivalent 
 currents at the same angles of lag, and expending the same 
 power in each coil and in the load. 
 
 If the secondary load of a transformer is reactive, an inspec- 
 tion of Fig. 269 will show that 0 2 and 9 X tend to increase, while 
 with a load containing electrostatic capacity these angles tend 
 to lessen, or to reverse in sign if the capacity is sufficiently large.* 
 
 The leakage fluxes and internal losses of voltage indicated by 
 Fig. 269 are, for the purpose of obtaining a clear diagram, 
 made much greater in proportion to the impressed and mutual 
 induced voltages than would be found in commercial constant 
 voltage transformers. However, the leakage fluxes used for 
 the diagram are not necessarily excessive for conditions in in- 
 duction motors and constant current transformers, to which the 
 diagram is also applicable. Likewise, the exciting current 1 4 
 has been given a large value in Fig. 269, for the same reasons. 
 It is usually so small in constant voltage transformers that the 
 voltage drop OP may be considered numerically equal to and of 
 the same phase as OE (Fig. 269) for the solution of such prob- 
 lems as usually arise in commercial circumstances. However, 
 when the load current considered is a small fraction of the full 
 load current or the transformer is special, so that in either case 
 the drop due to the exciting current is material, the exact tri- 
 angle representing apparent primary impedances and resultant 
 
 * Art. 131. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 475 
 
 active and reactive voltages is OD' E. In this triangle, when 
 scaled as a triangle of voltages, the side in quadrature to the 
 current I x = OB must be composed of the quadrature projection 
 of the drops of primary voltage that are due to those induced 
 in .the primary and secondary coils and load and is equal to 
 D' E. It will be here remembered that OH , when fully ex- 
 pressed, is at right angles to OB. The active voltage OD' in 
 phase with I x must be composed of the projection of the active 
 drops of voltage caused by the work done in driving the 
 current through the primary and secondary coils and the load. 
 When the triangle OD' E is scaled to represent impedances, 
 i.e. voltages divided by currents, the line OE is the total ap- 
 parent impedance, 0D\ the apparent resistance, and D'E, the 
 apparent reactance offered by the two windings and the load 
 of the transformer. The power input is of course E X I X 
 cos 0 X — OE x OB x cos Z.EOD' and the output E 2 I 2 cos 0 2 = 
 00 x ON. 
 
 125. Transformer Exciting Current. — In the case of an ideal 
 transformer, that is, one without losses or magnetic leakage, 
 the lag of the primary current, when the secondary circuit is 
 open, is 90° with respect to the impressed voltage ; for R x and 
 cos 9 X are assumed to be zero. Since the induced secondary 
 voltage lags behind the magnetism, which is in phase with the 
 primary current when the secondary circuit is open, by an 
 angle of 90°, the phases of the primary impressed voltage and 
 the secondary induced voltage are exactly 180° apart ; that is, 
 they are in exact opposition. The current in the primary circuit 
 of an ideal transformer, when the secondary is open, is all 
 reactive (i.e. wattless), and of a magnitude which depends 
 only upon the total inductance L ' of the primary coil. 
 
 On the other hand, the losses due to hysteresis and eddy 
 currents in the iron core and to resistance in the primary coil are 
 by no means negligible in commercial transformers, but are of 
 such a magnitude as to decrease the lag of the primary current 
 until the power factor of the primary circuit is ordinarily 
 between 50 per cent and 70 per cent when there is no current 
 in the secondary circuit ; but the magnetism in the core 
 remains nearly in phase with the reactive component of the 
 primary current and is almost 90° behind the phase of the 
 primary voltage, so that the primary impressed and secondary 
 
476 
 
 ALTERNATING CURRENTS 
 
 induced voltages are still almost in opposition. The current 
 which flows in the primary circuit when the secondary circuit 
 is open may therefore be considered as composed of two compo- 
 nents, one of which supplies the power required to make up 
 the transformer losses, and the other of which serves simply 
 for setting up the magnetism, and is therefore wattless. The 
 primary current which flows when the secondary circuit is 
 open is sometimes called the leakage current, open circuit cur- 
 rent, or magnetizing current. In this volume, however, the 
 more satisfactory term, Exciting current, is used, the term mag- 
 netizing current being reserved to apply only to the magnet- 
 izing (i.e. reactive) component of the exciting current. 
 
 It is desirable to look a little farther into the effect upon the 
 cyclic curve of iron loss which is produced by substituting a 
 sinusoidal exciting current for the irregular current which actu- 
 ally flows, as has been done in the transformer diagram of Figs. 
 265, 266, and 269, and others following. In Fig. 271, <M> is a 
 
 Fig. 271. — Conventional Cyclic Curve of Iron Loss, constructed from the Curve 
 of Flux in a Transformer Core and the Equivalent Sinusoid of Exciting Current. 
 
 sinusoidal curve of magnetic flux which is setup in a transformer 
 core, and is the sinusoidal curve of exciting current, lead- 
 ing the curve by an iron loss angle of 18°. From these 
 curves the cyclic curve is constructed as heretofore explained,* 
 
 * Art. 114. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 477 
 
 with its ordinates and abscissas equal respectively to magnetic 
 flux and exciting current. The shape of the cyclic curve 
 is not that of a true cyclic curve of iron loss, as may be seen by 
 the fact that the maximum value of the exciting current ST 
 does not occur at the point of maximum flux, a condition which 
 could only occur in fact if the eddy current part of the iron loss 
 was relatively greater than is found in commercial transformers. 
 But if curve is the equivalent sinusoid of the actual irreg- 
 ular exciting current, the area of RR must be equal to the area 
 inclosed by the true cyclic curve of iron loss. It is thus seen 
 that the substitution of an equivalent sinusoid for the actual 
 curve of exciting current can usually be made without involv- 
 ing serious error. 
 
 126. The Effects of Variable Reluctance, of Hysteresis, and of 
 Eddy Currents on the Form of the Primary Current Wave. — In 
 
 the preceding discussions it has 
 been assumed that the reluc- 
 tance of the magnetic circuit of 
 a transformer can be taken at 
 an average constant value which 
 is practically equal to the value 
 when the current is at its maxi- 
 mum point. The low maximum 
 value of the magnetic density 
 which is used in commercial 
 transformers as ordinarily con- 
 structed makes this assumption 
 allowable, though it is by no 
 means exact; and if the magnetic 
 density is pushed above the 
 bend in the curve of magnetiza- 
 tion, the influence of the low- 
 ered permeability of the iron 
 becomes marked. The curve 
 OM in Fig. 272 may be taken to 
 represent the curve of magneti- 
 zation of iron in a transformer 
 core, plotted with volts induced 
 
 . . . Fig. 272. — Diagram showing the Effect 
 
 in the primary Windings as or- 0 j Variable Permeability, with Iron 
 dinates and exciting ampere- Loss Negligible. 
 
478 
 
 ALTERNATING CURRENTS 
 
 turns as abscissas, supposing the effect of hysteresis, eddy cur- 
 rents, and magnetic leakage to he negligible, and line OP may 
 be taken to represent a corresponding hypothetical curve of mag- 
 netization, assuming the effect of saturation to be negligible, i.e. 
 the reluctance to be constant. Vectors OB and OC are the pri- 
 mary and secondary ampere-turns respectively for a given load. 
 Then when the magnetizing ampere-turns equal n x I^ represented 
 by OA , the induced voltages in the primary and secondary coils 
 are reduced by the effect of saturation from OF and OJ which 
 would be reached with a constant reluctance, to OF' and OJ' . 
 And, conversely, if the induced voltages are to be OF and 
 OJ', OA ampere-turns are required to bring the magnetism up 
 to the requisite value when the curve of magnetization is 031 , 
 while OA! ampere-turns would be sufficient if the magnetic 
 circuits were not influenced by saturation. 
 
 The construction shows that the saturation of the iron makes 
 necessary more turns of wire on the primary and secondary 
 coils in order that a given output may be obtained. Since, in 
 this case, the permeability varies through each period with the 
 magnetizing ampere-turns, there is a periodic variation of Q , and 
 the primary current wave is distorted from the form of the 
 primary impressed voltage wave. This is shown clearly in 
 Fig. 261 of Art. 114 and is there described. 
 
 When the transformer has its secondary circuit open, the 
 drop of voltage due to the exciting current flowing through 
 the primary winding is usually negligible, so that the pri- 
 mary impressed voltage at each instant is proportional to the 
 tangent of the curve of magnetism ; and is 90° in advance of 
 the magnetization when the latter is sinusoidal. The tangent 
 relations between the curves of flux and voltage * must exist 
 as long as I X B X is negligible. The effect on the output of 
 impressing excessive voltages upon the primary winding of a 
 transformer is indicated clearly in Fig. 272. Thus, there is 
 but slight rise of magnetic flux in the curve 031 bevond the 
 point M (near saturation) for increases of exciting current. If, 
 therefore, OF is much increased, the exciting current must in- 
 crease excessively. It is thus seen that if transformers, or other 
 apparatus depending upon counter-induced voltage, are sub- 
 jected to voltages that demand rise of core flux beyond the point 
 
 * Art. 115. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 479 
 
 of saturation, the exciting currents demanded may reach danger- 
 ous or destructive proportions. Another important reason for 
 having transformers designed for low flux densities in the core 
 is because an excessive exciting current such as spoken of above 
 will cause a material resistance drop in the primary, while high 
 core reluctance will cause relatively high leakage flux, thus 
 tending to interfere seriously with the regulation. 
 
 As a very simple example showing that the primary induced 
 voltage is always equal and opposite to the impressed voltage 
 when I 1 R l is negligible, and that the exciting current varies in 
 such a way as to furnish this induced voltage in a transformer 
 with the secondary circuit open, we may consider an inductance 
 coil of negligible electrical resistance with a sinusoidal voltage 
 E x of, say, 100 volts impressed upon it. Suppose the frequency 
 is 60 periods per second, and the self-inductance L x is .01 of a 
 henry, then 27rfL x = 3.71 ohms is the impedance. The current 
 E 
 
 I x flowing is- — tL_ — 26.5 amperes with a lag angle of 90°. 
 
 The induced voltage E lm is 2 irfL x T x = 100 volts, which is equal 
 and opposite to the impressed voltage. Suppose the self-in- 
 ductance L x changes to L x = .005; then the current becomes 
 i 7 =53 amperes; but the induced voltage E Xm is 2m -fL x 'I x = 
 100 volts, which is the same value as before. It is seen, in this 
 case of negligible resistance, that whatever value the self-in- 
 ductance may have, the exciting current takes such a magni- 
 tude that the counter-voltage of self-induction is numerically 
 equal to the impressed voltage. Since the self-inductance of 
 the electric circuit is inversely proportional to the reluctance 
 of the magnetic circuit, when the reluctance changes, the excit- 
 ing current changes, so that, as before, the counter-voltage 
 equals the impressed voltage. Figure 262, described in Art. 
 114, shows that when core losses and copper losses are present, 
 the exciting current is advanced in phase so that it is less 
 than 90° behind the impressed voltage, and that the summation 
 of the products of the instantaneous values of impressed vol- 
 tage and current is no longer equal to zero for the inductance 
 coil, but becomes equal to the sum of the core losses and the 
 I 2 R losses. 
 
 The effect of hysteresis in the core of a transformer is to dis- 
 tort the form of the primary current wave to a still more marked 
 
480 
 
 ALTERNATING CURRENTS 
 
 degree than would occur from the effects of magnetic saturation 
 without hysteresis, and the higher the maximum magnetic 
 density is carried, the greater the distortion becomes. The 
 ordinates of the primary current wave are at each instant pro- 
 portional to the difference between the corresponding ordinates 
 of the wave of primary impressed voltage and the wave of in- 
 duced counter-voltage. The latter is, in an ordinary transformer, 
 practically similar to the form of the secondary voltage wave. 
 With the primary impressed voltage sinusoidal and the reluc- 
 tance of the magnetic circuit uniform, the primary current 
 wave would be sinusoidal. With a variable reluctance, hut no 
 hysteresis, the current wave becomes peaked, but remains sym- 
 metrical; but when hysteresis is taken into account, the sym- 
 metrical form is lost. This is illustrated in Figs. 261 and 262.* 
 
 The effects which eddy currents in the core have upon the 
 exciting current are shown in Fig. 257, which is described in 
 Art. 112, and in Fig. 263, described in Art. 114. The instan- 
 taneous value of the portion of the exciting current which is 
 required to make up the losses due to eddy currents at any 
 instant is equal to the corresponding instantaneous value of 
 the eddy current loss divided by the instantaneous value of the 
 primary voltage. The total core loss maybe shown by plotting 
 a cycle of the core flux having abscissas equal to the arithmetical 
 sum of the corresponding abscissas of the hysteresis and eddy 
 cycles (Fig. 263). The total transformer exciting current 
 may be plotted from this as shown by I' in Fig. 263. 
 
 The eddy current loss is similar to that which would be pro- 
 duced by a closed secondary circuit of one turn of appropriate 
 electrical resistance, and is unlike hysteresis loss, which is de- 
 termined by the magnetic quality of the magnetic core. 
 However, eddy current loss causes an increase in the angle of 
 advance of the exciting current, as also does current in the 
 secondary coil.f Moreovei’, the loss caused by the exciting 
 current in the resistance of the primary winding — usually very 
 small — causes a still further advance of the exciting current. J 
 The phase of the exciting current is thus caused to be less than 
 90° behind the impressed voltage on account of (a) the hyster- 
 esis loss, (5) the eddy current loss, and (e) the resistance loss 
 in the primary coil caused by the flow of exciting current. 
 
 * Art. 114. t Art. 126. 1 Art. 123 a. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 481 
 
 The angle by which the exciting current is advanced from lagging 
 quadrature with respect to the voltage is called in this volume the 
 Excitation Angle. Some inaccurately use the term hysteretic 
 angle of advance to express this relation, instead of confining the 
 use of the latter term to the advance caused by hysteresis alone. 
 
 If the eddy current cycle has a large area, its effect may 
 cause an advancement of the instantaneous maximum point in 
 the exciting current. This would be the case, referring to Fig. 
 263, if the current curve I f had a maximum greater than the 
 magnetizing current curve 1^. 
 
 127. Forms of the Primary Current Waves as affected by 
 Current in the Secondary Winding. — When the secondary cir- 
 cuit is closed, the form of the primary current is changed by 
 the effect of the secondary current. In Fig. 273, curve e x rep- 
 resents the primary impressed voltage; represents the 
 
 exciting current times the primary turns. Now, if by closing 
 the secondary cir- 
 cuit through re- 
 sistance free of re- 
 actance, a current 
 1 , g is caused to flow 
 and the instanta- 
 neous values of its 
 ampere-turns are 
 represented by the 
 curve « 2 «2 — the 
 effects of mag- 
 netic leakage are 
 here supposed to 
 be negligible — 
 the effect of the 
 current flowing in 
 the secondary cir- 
 cuit is to cause a 
 corresponding increase in the current flowing in the primary 
 circuit.* The ampere-turns of this increase of the primary cur- 
 rent may be represented by curve n x i^ . The total primary 
 wave of ampere-turns is represented by curve n 1 i v and is the 
 sum of (the exciting ampere-turns) and npq'. The pri- 
 
 * Art. 118. 
 
 2 1 
 
 Fig. 273. — Curves showing Effect on Primary Current of a 
 Transformer caused by Current in a Non-reactive Second- 
 ary Circuit. 
 
482 
 
 ALTERNATING CURRENTS 
 
 mary current and its components may be directly shown from 
 
 this curve by a 
 single change of 
 scale. It is thus 
 shown by the fig- 
 ure that the sec- 
 ondary current, 
 when in phase with 
 the secondary 
 voltage, tends to 
 reduce the distor- 
 tion and lag of the 
 primary current. 
 
 If the secondary 
 circuit is induct- 
 ive, the effect is 
 altered so that the 
 lag of the primary 
 current is larger, 
 as shown in Fig. 
 274. In this case n 2 i 2 lags behind e 2 ; and n x i x also lags behind 
 e v The sum of n x i x ' and n^i , which gives n x i v is therefore in 
 a lagging position 
 with respect to e v 
 the angle of lag be- 
 ing largely deter- 
 mined by the angle 
 of lag of i 2 with re- 
 spect to e 2 where 
 is small in compari- 
 son with n 2 i 2 . 
 
 If the secondary 
 current leads the 
 secondary induced 
 voltage, the curve 
 corresponding to n 2 i 2 
 leads e 2 , and n 1 i 1 
 (which is the sum of Fig. 275 . 
 
 Fig. 274. — Curves showing Effect on Primary Current of 
 a Transformer caused by Current in an Inductive Second- 
 ary Circuit. 
 
 and nJ ~) also 
 
 Experimentally Determined Curves of Cur- 
 rent in a Transformer with Open Secondary Circuit. 
 
 leads e x when is small in comparison with n 2 i 2 . 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 483 
 
 Figures 275 and 
 27 6 show transformer 
 curves experimental- 
 ly observed by Pro- 
 fessor Ryan in 1889,* 
 using sinusoidal im- 
 pressed voltage. 
 These show a strik- 
 ing resemblance to 
 the hypothetical 
 curves built up from 
 the loss cycles. The 
 unmarked half sinus- 
 oid in Fig. 275 is a 
 half cycle of the mag- 
 netic wave, which is 
 
 Fig. 276. — Experimentally Determined Curves of Cur- 
 rent in a Transformer with slightly Inductive Full 
 Load in Secondary Circuit. 
 
 in lagging quadrature with the primary impressed voltage. 
 
 128. Core Magnetization. — The maximum magnetic density 
 during a period, in the iron core of a transformer, is dependent 
 upon the maximum value of the magnetizing component of the 
 exciting current in the primary circuit, and is equal to 
 
 T = 
 
 4 
 
 10 P 
 
 = 1.257 
 
 P 
 
 where L is the maximum value of the magnetizing current 
 and P is the reluctance of the magnetic circuit, in gilberts, at 
 the time that the current is maximum. 
 
 Assume, for a moment, that the transformer has a constant 
 reluctance, no magnetic leakage, and neither iron nor copper 
 loss, which is, of course, an ideal case. Then, 
 
 T = 
 
 4V27m 1 J 1 _ 1 777 n 1 I 1 
 
 10 P 1 P ’ 
 
 when the secondary circuit is open. Closing the secondary 
 circuit so that a current may flow in it under the impulse of 
 the induced secondary voltage materially changes the condi- 
 tions. We will neglect hysteresis and eddy current losses in 
 the iron core and resistance losses in the windings, and first 
 assume that the secondary circuit is without reactance in which 
 
 * Trans. Amer. Inst. E ■ E., Vol. 7, p. 1. 
 
484 
 
 ALTERNATING CURRENTS 
 
 case the secondary current I 2 will be in unison with the induced 
 or secondary voltage E 2 . This current has its own magnetiz- 
 ing effect on the magnetic circuit. If i 1 and i 2 are the primary 
 and secondary currents at any instant, the total magneto- 
 motive force in the circuit at that instant is 
 
 4 nT(n x i, + n 2 i 2 ) 
 
 10 * 
 
 where <£>,- is the instantaneous value of the magnetic flux, and P 
 is the assumed constant value of the reluctance of the magnetic 
 circuit. From this is found 
 
 /10 P4>A 
 h ~ V 4 t rnj ) 
 
 ( 2 ) 
 
 If a sinusoidal voltage is impressed upon the primary wind- 
 ing, which under the assumed conditions results in sinusoidal 
 waves of magnetism and current, the instantaneous primary 
 current is 
 
 . 10 P<t> . n 2 V 2 7, 
 
 i, — sin a -cos a ; 
 
 1 4 irn^ n x 
 
 but 
 
 10 
 
 4 7 rrq 
 
 = V2 7 fl , 
 
 whence i x = V2 (7^ sin a — n 2 7 2 cos a), (4) 
 
 n i 
 
 where a is measured from the zero instant of the cycle of flux, 
 which in this case is in unison with I IL since losses are assumed 
 to be absent. Whence 
 
 n 1 i 1 = sin a — n 2 I 2 cos «), 
 
 and ( nfl sin 2 a 
 
 — 2 n x n 2 I^I 2 sin « cos a 
 + n 2 I 2 cos 2 a) da, (4) 
 
 where 7, is the effective value of the primary current when the 
 secondary current is equal to 7 2 ; and as before, is the watt- 
 less primary current when the secondary circuit is open. Per- 
 forming the integration gives 
 
 n 2i2 =n *I l ? + n*I*. 
 
 ( 5 ) 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 485 
 
 Remembering that I and / 2 have, in this case, 90° difference of 
 phase, the three terms of this formula may be represented by 
 the three sides of a right-angled triangle, as in the triangle 
 formed by the points OAB in Fig. 272, in which excitation 
 losses are assumed absent. The current I x amperes, which 
 flows in the primary circuit when the secondary circuit is 
 loaded, is in advance of the current 1^ by an angle the tan- 
 
 1 " is here a wattless magnetizing component of the primary 
 current, losses being absent by assumption, it is directly de- 
 pendent on the reluctance of the magnetic circuit and the coun- 
 ter-voltage in the primary coil. As the reluctance is usually 
 small, I is small. 
 
 When the transformer has an iron core, the exciting current 
 is no longer sinusoidal or in exact leading quadrature with the 
 secondary induced voltage, but is distorted and is advanced by 
 the amount of the excitation angle.* Call this angle 77 , and con- 
 sider it the advance of the equivalent sinusoid substituted for 
 the irregular curve of exciting current. This substitution can 
 be made with sufficient accuracy for this discussion. The 
 magnetism remains, as before, 90° ahead of the induced vol- 
 tage although reluctance varies, so that the following relations 
 obtain : 
 
 This is the formula of a triangle where 90° — 77 is the angle 
 included between the sides representing and — n 2 I v such as 
 
 10P<3> . 
 
 iUjrcp . /or-/ , \ 
 
 sin « = V 2 1 sin (oc + 77 ), 
 
 4 7 r/q 
 
 ( 6 ) 
 
 being the primary exciting current. 
 Substituting, as before, in (3) gives 
 
 IT Jo 
 
 — 2 n^I I 2 sin (a + 77 ) cos a 
 -f- n 2 2 I 2 2 cos 2 a) da, 
 
 or, 
 
 «i 2 A 2 = n i % 2 + n ‘i l i - 2 sin V 
 
 = zq 2 // 2 + « 2 2 Z 2 2 - 2 cos (90° - 77 ) . (7) 
 
 * Art. 125. 
 
486 ALTERNATING CURRENTS 
 
 OA and AB of the triangle OAB of Fig. 266, in which the 
 effect of excitation losses is shown. 
 
 129. The Circle Diagram for Non-reactive Secondary Circuit 
 and Active Voltage Locus. — In a constant voltage transformer 
 the load is zero when the resistance of the secondary circuit is 
 infinite, and the load increases as the secondar} 7 resistance de- 
 creases through the range of normal currents for which the 
 transformer is designed. Then, if leakage reactance is con- 
 sidered constant for all loads, as may be done without appre- 
 ciable error when the frequency of the exciting current is con- 
 stant, its relative effect in causing voltage drop will increase with 
 increase of load. These relations may be neatly shown by a con- 
 struction similar to that discussed before.* Thus, in Fig. 277, 
 
 OC represents the impressed 
 voltage of a transformer in 
 which, for convenience, a 
 unity ratio of transforma- 
 tion is assumed. The right- 
 angled triangle OG-C is 
 then constructed on OC. 
 The side GO is made equal 
 to the drop of impressed 
 voltage due to the reactive 
 voltage CF of the primary 
 coil and FG of the second- 
 ary coil or — G C = E l = 
 E lLl + The side OG 
 
 is made equal to the sum of 
 J G , the voltage expended 
 in the resistance of the 
 primary coil R x on account 
 of the flow of the current 
 //, which is equal and 
 opposite to the secondary 
 
 current _£>, the voltage KJ \ 
 AD 2 ° ’ 
 
 Fig. 277. - Transformer Diagram for a Trans- which is e( l ual and Opposite 
 former containing Leakage Reactance and to the voltage expended in 
 Non-reactive Secondary External Circuit. the secondary coil resistance 
 
 R v and the voltage OK , which is equal and opposite to the 
 
 / 
 
 * Art. 85. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 487 
 
 voltage expended in the load resistance R. In obtaining these 
 voltage drops the exciting component of the primary current 
 is neglected for simplicity, as its effect is ordinarily small. 
 Therefore 
 
 OG = Ey R< = — / 2 (Ry + R 2 + R) = ly ( /('j + R 2 + R), 
 
 the ratio of transformation being taken as unity. The sides 
 OG- and GC thus form the right-angled triangle of active and 
 reactive voltages in the transformer. 
 
 Since GC increases with the current and angle OGC must re- 
 main aright angle for any current,* the locus of the point G , 
 which moves as the current changes, is the arc of a circle with 
 OC as its diameter. In a well-made commercial transformer 
 with its primary and secondary coils divided into parts and sand- 
 wiched together to avoid leakage, the line GC is relatively very 
 short even at full load, so that the point G travels only a short 
 distance from C as the secondary current is increased from zero 
 to the full load current. The point G would be at the point 
 C for all loads if the load and coil reactances were zero. 
 
 The secondary terminal voltage KO = RI V the voltage in the 
 resistance JK= E 2R ^ or the reactance voltage FG = E 2U of the 
 secondary coil may readily be obtained for any ratio of trans- 
 formation by merely using a scale equal to - times that used in 
 
 s 
 
 the original construction and reversing the secondary circuit 
 voltage or current arrow shown in the figure. The total in- 
 duced voltage due to the mutual magnetism of the primary 
 and secondary windings is HO = E 2m = E' lm = —E lm , the ratio 
 of transformation being unity. The voltage drop in the primary 
 coil resistance, due to the current component opposite to the sec- 
 ondary current, is JG =HF= E 1R , and the voltage drop by rea- 
 son of the primary coil reactance is EC = E lLi . 
 
 The component of primary current used for excitation is indi- 
 cated by the line AO , and is composed of an active component 
 DO, in phase with OC and a quadrature component AD in phase 
 with the core magnetism. The component D 0 is of such value 
 that when multiplied by OC ', the product equals the excitation 
 losses. The exciting current is assumed for the purpose of the 
 diagram to be constant in scalar value and phase position, which 
 
 * Art. 85. 
 
488 
 
 ALTERNATING CURRENTS 
 
 is sufficiently accurate for most practical cases. Its assumed 
 constant value is usually taken as the amperes flowing in the 
 primary circuit when the secondary circuit is open, i.e. the 
 Open circuit current. On the other hand, the current I 2 of the 
 secondary coil and its opposing element I\ of the primary 
 vary directly with the load or inversely with the impedance. 
 For the present we will continue to consider the load non-reac- 
 tive and the leakage reactances constant. Then we have a case 
 equivalent to a circuit having variable resistance and fixed in- 
 ductive reactance. In such a circuit, as was proved earlier,* 
 the locus of the current vector is a semicircle, located in the 
 first trigonometrical quadrant, with its diameter on the X-axis, 
 one end being at the origin. The diameter of this semicircle 
 was shown to be equal to the impressed voltage divided by the 
 reactance. In Fig. 277, OC may correctly be used as the vol- 
 tage absorbed in the primary, secondary, and load impedances, 
 by the flow of the secondary current, or its primary opposing 
 component, as heretofore explained. Then a semicircle with 
 its center on OV extended, as shown at OBT \ with a diameter 
 equal to OC divided by the combined leakage reactances 
 
 m 
 
 , that is, equal to 
 
 K 
 
 „ , is the locus of the end of 
 
 Xiu+Xu. 
 
 OB ( = J 7 1 ) or of the secondary current vector reversed. In 
 this case, X, the reactance of the external secondary load is 
 assumed to be zero, that is, the load is non-reactive. When 
 the primary triangle of voltages is OGC, the secondary current 
 is B 0, which is in phase with the active element of the voltage, 
 JO = X 2(/?2 + ^). The primary current is then the vector AB = 
 AO + OB. The secondary current can be read directly from 
 the diagram for any ratio of transformation by using a scale 
 with s times as many amperes to the inch as that used in the 
 construction. 
 
 If the impressed voltage, kilowatt capacity, exciting current 
 as a percentage of full load current, core losses in percentage 
 of kilowatt capacity, and the resistances and reactances of the 
 coils and load of a transformer are known, the construction, 
 such as shown in Fig. 277, can at once be made. From 
 this the primary and secondary currents are scaled off on 
 AB and BO in amperes. The angle of lag of the primary 
 
 *Art. 70, Case (1). 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 489 
 
 current is the angle B WO. The angle of lag of the sec- 
 ondary current is the angle between the terminal voltage 
 KO = E % and the current BO = I V which angle is zero under 
 the circumstances here assumed. The per cent regulation for the 
 given load is one hundred times the ratio of 00— OK to OK, as- 
 suming unity ratio of transformation. The various reactance and 
 resistance drops are measured along G-0 and 0G- respectively. 
 The power received from the mains is 00 times the projec- 
 tion of AB thereupon, or W { = E X I X cos 6 V The power absorbed 
 in the load is KO times BO or W„= E^I V which, though ob- 
 tained in the diagram for a transformer with ratio of transfor- 
 mation (s) equal to unity is also true without change when s has 
 any other value, since the primary equivalent voltage is divided 
 by, and the current multiplied by, s to reduce to secondary 
 values. The total power losses in the transformer are : exciting 
 current losses BO x 00; the primary coil copper loss due to 
 current OB, JO x OB ; and the secondary copper loss, JK x BO. 
 The sum of these losses subtracted from the power input equals 
 the power output. The commercial efficiency of the trans- 
 
 IT 
 
 former is the total output divided by the total input or 77 = — ° . 
 
 W i 
 
 Changing the position of B to various points on its locus 
 OBT and moving O along the semicircle OMO so that OO is 
 kept in the same direction as OB, permits the determination of 
 the entire transformer performance for any value of the secon- 
 dary current or fraction of the rated capacity of the trans- 
 former. 
 
 When the reactive voltage OO is very short, as in the case of a 
 well-made commercial transformer supplying, for example, an in- 
 candescent lamp circuit, the line OKJO almost coincides with 
 the line 00. In this case the locus OBT has such a large diame- 
 ter that it is almost a vertical straight line from no load to full 
 load ; then, if OA, for the particular character of the calcula- 
 tion in question, may be considered of negligible length, KO, 
 AB, and BO may be considered without serious error to vary 
 with the load up and down the line 00, and the calculation of 
 the transformer regulation resolves itself into the simple prob- 
 lem of three resistance drops in series. The three resistances 
 causing these drops are the fixed resistances of the primary and 
 secondary coils and the variable resistance of the load. 
 
490 
 
 ALTERNATING CURRENTS 
 
 In the foregoing discussion AO was considered constant. 
 Though approximately true this is not exactly the case, as the 
 magnetizing current must be proportional to the total mutually 
 induced voltage HO, and leading it by an angle of 90° plus the 
 excitation angle of advance. An approximately accurate cor- 
 rection to calculations can be made, when necessary, by giving 
 A 0, for each load, such a length that it will have the same 
 ratio to OH that the open circuit current has to 00, and also 
 turning it to such a position that it maintains the constant 
 angular relation to OH. Inasmuch as the construction of the 
 length OH is dependent upon the resistance drop HF, the 
 length and phase of HF must be corrected if great accuracy 
 is desired so as to include the drop of voltage due to exci- 
 tation losses. The diagram presupposes, also, that the reluctance 
 of the core is constant for all loads, and that the leakage paths 
 are free from iron losses. These assumptions for commercial 
 transformers introduce no great error. The further assumption 
 that the voltages and currents are sinusoidal may, under some 
 circumstances, require correction, as when resonance is present 
 for some of the harmonics. The effects of irregular waves of 
 voltage and current will be treated later. For ordinary trans- 
 former calculations it is usual to assume that the voltage and 
 currents are sinusoidal. 
 
 130. Circle Diagram where the Transformer Load contains 
 Constant Reactance and Variable Resistance. — Suppose now, 
 without changing the unity ratio of transformation or other as- 
 sumptions, that the load contains inductive reactance X, then 
 the construction shown in Fig. 277 may be conveniently modi- 
 fied as in Fig. 278. Here the component of the impressed 
 voltage OC, used in overcoming the total reactive voltage, is QC, 
 in which OF and F(J are the primary and secondary coil re- 
 active voltages respectively and GrQ the load reactive voltage 
 transferred to the primary circuit. The total active voltage 
 in phase with the component of the primary current which is 
 equal and opposite to the current in the secondary coil is OQ. 
 This is expended in the resistance of the coils and load. The 
 secondary terminal voltage is KO , and is composed of KK , 
 which is equal and opposite to the reactive voltage of the load, 
 and K'O, which is the voltage drop in the load resistance. 
 The secondary angle of lag 0 2 the an gl e X' OK. The dia- 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 491 
 
 gram is similar in 
 construction to that 
 of Fig. 277, and its 
 further description 
 and use may be ob- 
 tained by the pre- 
 ceding discussion 
 concerning the latter. 
 
 Electrostatic ca- 
 pacity in the load 
 tends to neutralize 
 the self-inductance 
 of the primary and 
 secondary coils.* 
 Thus, if capacity is 
 introduced into the 
 secondary circuit of 
 such value that the 
 capacity voltage 
 
 equals the inductive ^78. — Transformer Diagram showing the Effect of 
 1 _ . _. Inductive Reactance in the Secondary External Circuit. 
 
 voltage (xQ m big. 
 
 278, the reactive voltage remaining is (7(7, and the construction 
 becomes like Fig. 277. If the capacity increases, the point (7 
 moves toward (7, reaching it when the capacity voltage equals 
 the inductive voltage CQ. Because of this increase of capacity, 
 the diameter of the current locus OB T changes from the value 
 
 X, 
 
 X Ul + 
 
 to the value 
 
 A. 
 
 x lLi +x^+x L -x c 
 
 , where X c is 
 E 
 
 the capacity reactance, and the diameter becomes equal to — = 
 
 when the capacity reactance in the denominator is equal to the 
 sum of the inductive reactances. For each increase in capacity 
 a locus corresponding to OBT must be drawn with a diame- 
 ter larger than the last, until the capacity balances the 
 self-inductance, when the diameter becomes infinite so that the 
 arc of the circle coincides with 00. When the capacity vol- 
 tage exceeds CQ, the system acts as one having resistance and 
 capacity alone, and the current takes a leading angle determined 
 by the excess of capacity reactance; then the point Q has a 
 
 * Arts. 81 and 82. 
 
492 
 
 ALTERNATING CURRENTS 
 
 locus on the left of OC , as shown in Fig. 279. Thus the current 
 locus is a semicircle to the left of OC, such as OB^T', and has 
 a diameter equal to the impressed voltage divided by the net 
 excess of capacity reactance. This is in accord with the 
 theorem earlier proved * for a circuit having constant capacity- 
 reactance and variable resistance. Otherwise the diagram is 
 essentially as in the case of an inductively reactive circuit, ex- 
 cept that the primary current is somewhat smaller and the 
 angle of lead somewhat less than would have been the current 
 and lag angle for an equal net amount of inductive reactance 
 instead of capacity reactance. This is because the exciting 
 component of the current, which need move its vector position 
 only slightly under ordinary conditions and is here considered 
 stationary, is in a different phase position in the two instances 
 with relation to the remaining component of the primary 
 current. 
 
 Figure 279 shows the conditions assumed above when the 
 secondary current is of the same value as in Figs. 277 and 278. 
 The lettering is similar in the three figures. In Fig. 279 is 
 shown the impressed voltage 0(7; the active voltage locus for 
 inductive and capacity reactances, which is the circle OMCN ; 
 the current locus OBT where the net predominance of inductive 
 voltage is CQ, as in Fig. 278 ; the current locus OB^T where 
 the net predominance of capacity voltage is CQQ ; and the 
 current locus OCT" when inductive and capacity voltages 
 neutralize each other or are both negligible. 
 
 To determine the lengths and phase positions of the lines rep- 
 resenting the vectors of mutually induced primary voltage and 
 the secondary terminal voltage, take, for example, the internal 
 and load reactive voltages of Fig. 278, which are reproduced in 
 the line CQ of Fig. 279, and suppose a net excess of capacity 
 reactive voltage is represented by the line CQJ. To give this 
 excess, the total capacity voltage must then he equal to CQ / 
 plus a voltage equal and opposite to the inductive voltage. 
 The point QQ is fixed by the length QQ C and the locus of the 
 active voltage ONC ; therefore, the total capacity voltage may 
 he laid out from QJ to Q', the length of CQ' being equal to 
 CQ but its direction being in quadrature with the current OB x ' 
 and therefore in line with Q^ C. The part of CQ' represented 
 
 Art. 70, Case (1). 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 493 
 
 by G'Q' neutralizes the effect on the transformer of the load 
 inductance. The triangle of voltages at the secondary termi- 
 nals is then OK^ K' and the impressed voltage OC may be con- 
 sidered to be applied in the following components : OK^ , which 
 
 Fig. 279. — Transformer Locus Diagram showing the Effects of Capacity or Induc- 
 tance Predominant or Equal. 
 
 is equal and opposite to the secondary active voltage ; K' 
 which is equal and opposite to the secondary capacity voltage 
 in excess of the inductive voltage of the load ; K'J the drop 
 equal and opposite to the drop in the secondary coil resistance ; 
 J 1 H ' , the drop equal to the secondary leakage voltage ; H'F’, the 
 
494 
 
 ALTERNATING CURRENTS 
 
 drop in the primary coil resistance ; and F'O, the drop equal 
 and opposite to the primary leakage voltage. The total pri- 
 mary mutually induced voltage is H' 0 and the total secondary 
 terminal voltage is K' 0. The coil resistance drops may be 
 transferred from K J' Gr' to the active voltage line at 
 and the diagram then becomes similar to those previously dis- 
 cussed. The secondary angle of lead is the angle K' OK j' and 
 the primary angle of lead is the angle between the lines CO 
 and B^A extended. 
 
 When inductive and capacity reactances neutralize, as when 
 the current locus is OCT ", the line Q^Q' swings to a horizontal 
 position and Q^C=CQ' ; the coil impedance triangle CGr'K' 
 therefore swings down so that its side CG-' is horizontal. It is 
 thus seen from Fig. 279 that for a given angle between the 
 current and impressed voltage the value of H' 0 is greater when 
 the capacity reactance is in excess compared with its value 
 when inductive reactance is in excess, but for commercial con- 
 stant voltage transformers operating with ordinary power factors 
 and normal secondary currents, the change of H’ 0 , the mutually 
 induced primary voltage, in either length or position is not 
 great. When neither inductance nor capacity is present, H' 0 
 swings into the line 0(7, and in practice it usually remains 
 close thereto. 
 
 The circular arcs 0T 2 , 0T 2 ', and 0T 2 " are the circular loci 
 that would be made by the secondary currents had they been 
 drawn outward from 0 instead of inward as shown in Fig. 
 279. When these arcs are used, a phase instead of a vector dia- 
 gram of the currents results. 
 
 Prob. 1. Given a 60-cycle transformer of 100 kilowatts rated 
 capacity, in which the primary voltage is 2200 volts, the ratio 
 
 of transformation, ^l, is 20, the copper loss at full load 2 per 
 n 2 
 
 cent of the rated capacity and equally divided between the 
 primary and secondary coils, the iron loss 1 per cent of the 
 transformer rated capacity, the exciting current (assumed con- 
 stant) 2 amperes, and the drop of voltage at full load due to 
 magnetic leakage 1 J, per cent of the impressed voltage when 
 current is being delivered to a non-reactive secondary circuit : 
 Draw for this transformer a large transformer diagram similar 
 to Fig. 277. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 495 
 
 (Note. — For convenience use secondary quantities in terms of primary 
 equivalents in making the diagram. After its completion read off the correct 
 secondary quantities by changing the scale. The point G for full load may 
 be found by laying out GC so that OG is 1’ per cent less in length than OC. 
 Since OG is the active component of OC it is proportional to the terminal 
 voltage plus the 2 per cent allowed for copper losses, and KG is therefore 
 0 ° 
 
 per cent of OG, while KO is the primary equivalent of the secondary 
 
 terminal voltage E 2 at the full load of 100 kilowatts. Divide 100,000 (the load 
 in watts) by the number giving the volts represented by the scale length of 
 KO, and the value of amperes in the full load secondary current, — //, 
 is obtained, expressed in primary equivalents. Then the total primary 
 equivalent leakage reactance X L is approximately the volts CG divided by 
 
 and the diameter of the current locus is . Make 01) of a length 
 
 which represents in amperes that number which wdien multiplied by the 
 volts represented by OC gives one per cent of the transformer’s rated capac- 
 ity. This fixes the phase angle of OA under the assumption that the mag- 
 netizing element of the current is constant in A T alue and at right angles to 
 
 OC.) 
 
 Assuming the exciting current constant in length and phase 
 position, find the primary and secondary currents, the secondary 
 terminal voltage, the primary and secondary angles of lag and 
 the regulation (ratio of primary voltage minus secondary ter- 
 minal voltage in terms of the primary to primary voltage) for 
 full load and for 25 per cent, 50 per cent, 75 per cent, and 125 
 per cent of full load. 
 
 Prob. 2. Assume the same specifications as in Prob. 1 for 
 the transformer and conditions there specified, except that the 
 power factor of the load has become 0.9 on account of the in- 
 troduction of self-inductance. Draw the diagram and compute 
 the currents, voltages, angles of lag, and regulation as required 
 in Prob. 1. Also give the amount (in henrys) of self-inductance 
 introduced into the load. 
 
 Prob. 3. Assume the same specifications as in Prob. 1 except 
 that the secondary circuit contains a net capacity reactance, so 
 that the power factor of the load is .9, with current leading. 
 Draw the diagrams and compute the currents, voltages, angles 
 of lag, and regulation as required in Prob. 1. Find the amount 
 (in microfarads) of capacity in the secondary circuit. 
 
 131. Short Circuits. — As explained in the preceding articles, 
 the point 6r in a transformer diagram, such as Fig. 277, does 
 
496 
 
 ALTERNATING CURRENTS 
 
 not, for the ordinary working power factors and loads used in 
 commercial, constant voltage transformers, move far from the 
 point C, within the practicable ranges of continuous load. If, 
 however, for any reason, the terminals of the secondary winding 
 are connected together — short-circuited — through a negligible 
 resistance, the secondary terminal voltage KO becomes zero 
 and Gr swings around the semicircle until iTlies on 0. This 
 means that the primary and secondary currents may become 
 enormous compared with the maximum currents for which the 
 transformer was designed and will be approximately equal to 
 Hj 
 
 - l , where E x is the impressed voltage, and Z x is the combined 
 
 A 
 
 impedance of the primary and secondary coils — the latter re- 
 duced to primary equivalents. 
 
 The conditions are represented in Fig. 280, where the current 
 locus is OBB' T, the voltage locus OMCN , the rated full load 
 
 Fig. 280. — Transformer Diagram showing Conditions when the Secondary Termi- 
 nals are Short-circuited. 
 
 secondary current BO , the primary and secondary impedance 
 voltage triangles HFC and KJH respectively, the diagram 
 being lettered to correspond with Fig. 277. All secondary 
 quantities are shown, as heretofore, as primary equivalents : 
 and, as before, the coil impedances are exaggerated for the sake 
 of clearness in the diagram. 
 
 When the secondary terminals are short-circuited, Gr moves 
 to Gr 1 and the entire impressed voltage OC is expended in 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 497 
 
 driving current through the coil impedances as shown by the 
 triangles H'F' C and K'J'H'. The secondary current is B' 0, 
 which is several times larger than BO. The mutually induced 
 primary voltage here becomes H' 0 , which is out of phase with 
 HO and is much shorter. Since H' 0 must be proportional to 
 the magnetic flux which is mutual to the primary and secondary 
 windings, this means that the mutual flux in the core decreases 
 with the increase of magnetic leakage and resistance drop caused 
 by the extraordinarily large currents and the accompanying large 
 magneto-motive forces impressed on the leakage paths. Exciting 
 current AO can no longer be considered constant but must be 
 changed by reducing its vertical (active) component in a ratio 
 
 approximately equal to 6 •> and, supposing that the core 
 
 reluctance remains approximately constant, reducing its hori- 
 zontal (reactive) component in the ratio of • It is then 
 
 A'O and the primary current is A'B ' . 
 
 For the sake of making a clear diagram of small dimensions, 
 the leakage and copper drops have been greatly exaggerated in 
 Figs. 277, 278, 279, and 280, as compared with the values they 
 would have in large, well-designed, commercial transformers in- 
 tended for constant voltage working. In such transformers 
 the radius of the semicircle OBB' T would be many times 
 larger than is shown in Figs. 277, 278, 279, and 280, and the 
 point B would move farther along the semicircle towards T in 
 case of short circuit, so that OB' may become enormously larger 
 in proportion to the full load current than is shown in Fig. 280. 
 
 The coils of a transformer are usually wound close together, 
 as seen in Fig. 46, or are sandwiched together in sections, to 
 prevent undue magnetic leakage, and are bound and supported 
 only by insulating materials. When a short circuit occurs, the 
 forces tending to repel the primary and secondary coils one 
 from the other, by reason of th'e large currents and excessive 
 leakage fluxes (proportional to F'C and Gr' F' in Fig. 280), 
 may become very great. The primary and secondary currents 
 being substantially in opposition, the forces set up between them, 
 by virtue of their magnetic leakage fluxes, are of repulsion. 
 Also, the leakage fluxes being proportional to the currents in 
 primary and secondary coils, this repulsion is proportional to 
 2k 
 
498 
 
 ALTERNATING CURRENTS 
 
 the product of the currents, and as the secondary and primary 
 currents of a transformer approximately vary together, the 
 mechanical force exerted between the coils is closely propor- 
 tional to the square of the current flowing in either winding. 
 
 The mechanical strains thus set up in a small transformer 
 which becomes short-circuited may not exceed the strength of 
 the insulating supports of the coils. In the case of large trans- 
 formers the short-circuit currents become larger in proportion 
 to the rated full load currents, because of the proportionately 
 lower resistances of the windings of the larger machines, and the 
 fluxes are larger, hence, the electro-magnetic stresses may be- 
 come very great if short circuits occur. For transformers, of 
 similar physical form but different sizes, the stresses referred 
 to may increase in proportion to the squares of the rated 
 loads; so that of two transformers, one of which has a capacity 
 of 10 kilowatts and the other of 1000 kilowatts, the repulsion 
 strain on conductors of the second may be in the neighborhood 
 of 10,000 times as great as in the first. The result is that the 
 windings of very large transformers are sometimes torn to 
 pieces, as by an explosion, when the secondary circuits are acci- 
 dentally short-circuited. 
 
 The burning out of the coils, due to the excessive I 2 R losses, 
 when a short circuit is maintained, will occur as in any other 
 electrical machine or circuit, but injury from repulsion may oc- 
 cur before the circuit is broken by a burn-out or by the action 
 of an automatic circuit breaker. The computation of the 
 electro-magnetic stresses has been attempted and may be ac- 
 complished to some degree of approximation.* 
 
 132. Locus of Current in a Transformer when the Load Re- 
 actance is varied and the Resistance is kept Constant. Relation 
 of Power Factor to Capacity. — In the discussions just preceding, 
 the load and coil reactances of a constant voltage transformer 
 were considered to be constant while the resistance varied. 
 It is also possible to have a load in which the resistance is ap- 
 proximately constant while the power factor varies because of 
 changing inductive or capacity reactance, as when the load 
 comprises a synchronous motor operating at constant torque 
 but with various field excitations, f Consider a case of the 
 latter kind and assume the transformer to have negligible mag- 
 
 * Trans. Amer. Inst. E. E., Vol. 30. 
 
 t Art. 160. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 499 
 
 netic leakage. Then let 00 in Fig. 281 be the impressed vol- 
 tage. The voltage locus is OMON \ as before. Consider first 
 that the load and coil reactances are zero. Then the voltages 
 of the primary and secondary coils induced by the mutual flux 
 lie on the line 0(7, if the slight deviation therefrom due to 
 
 Resistance is maintained Constant. 
 
 the small element of impressed voltage in phase with the ex- 
 citing current is neglected. The line 00 represents the im- 
 pressed voltage on the primary coil ; HO is the primary voltage 
 drop in the primary coil resistance ; KH is the primary vol- 
 tage drop due to the secondary coil resistance (the ratio of trans- 
 formation is taken as unity for convenience) ; OK is the 
 primary voltage drop due to the resistance of the load ; HO 
 
500 
 
 ALTERNATING CURRENTS 
 
 is the mutually induced voltage. In this case, for ease in con- 
 struction, the secondary current is scaled so that it equals 
 OC in length. It will coincide with that line because there 
 is no appreciable reactance present. 
 
 Now if positive or negative reactance is introduced so that 
 the apex of the right-angled triangle of voltages takes any posi- 
 tion on the locus OMON as Q ' , Q" , or Q'", the ends of the lines 
 representing the current vectors for the various conditions will 
 also fall on the locus OMCN ; for it was proved in an earlier 
 chapter that when a circuit contains variable reactance and con- 
 stant resistance, the locus of the current is a circle having a di- 
 ameter lying in the line of the impressed voltage and equal to the 
 
 impressed voltage divided by the resistance, or — . This is 
 
 R 
 
 the maximum current and is represented by 00 in Fig. 281. 
 The diameter of the circular locus is equal to this current. 
 The secondary current for a certain positive reactance is then 
 Q' 0 and the primary current for the same reactance is AQ' , 
 where AO is the exciting current. The line CH'K' being 
 drawn perpendicular to CQ' and K' K\ being parallel thereto, 
 the mutually induced voltage, if there is no magnetic leakage 
 in the coils, but when the reactance voltage of the load is CQ', 
 is H' 0, and the primary and secondary coil resistance drops are 
 respectively J\Q' = H'C and K\J\ = K'H ' . The total ac- 
 tive primary voltage is OQ' = K' 0 + OK' v and the secondary 
 terminal voltage is K' -f). It will be noted from the construc- 
 tion that as Q moves along its locus by reason of a change of 
 reactance, K moves around a circle having a diameter OK, and 
 H describes a circle with a diameter OH. The smaller circle 
 is evidently the locus of the mutually induced voltage HO and 
 of the resistance voltage drop HO of the primary coil, and the 
 larger circle is the locus of the resistance drop KO of the pri- 
 mary and secondary coils combined. 
 
 When magnetic leakage is present, the diagram for a non- 
 inductive load is as in Fig. 277. In Fig. 281 the coil voltage 
 drops are’ similarly shown for a non-reactive load where mag- 
 netic leakage is present, by the lines F X C, H 1 F V J 1 H V and 
 K X J V and the secondary current is then on the line Gq 0. Then 
 when reactance is introduced into the load and Q moves on 
 
 * Art. 70, Case 2. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 501 
 
 the locus OMCN , the figure CGr 1 K l H l swings around and 
 remaining symmetrical diminishes in size when the current de- 
 creases, as did the line OK when leakage was absent. When 
 Q reaches 0, the secondary current becomes zero and only the 
 exciting current remains in the primary circuit, for then the 
 reactive voltage equals the impressed voltage minus the small 
 component required to drive the exciting current through the 
 primary coil. As this requires infinite reactance it is equiva- 
 lent to a transformer with the secondary circuit open. The right- 
 hand semicircle OQ' C is the current locus for varying net in- 
 ductance and OQ"' C for varying net capacity. 
 
 The broken arc of a circle M 2 0N 2 is of the circular locus that 
 would be formed if the secondary currents for varying induct- 
 ance were drawn outward from 0 instead of inward. 
 
 Prob. 1. Given a 50-kilowatt 60-cycle transformer transform- 
 ing from 2200 to 220 volts, in which the primary and secondary 
 coil resistances are .8 and .008 ohm respectively, the core 
 losses 100 watts (assumed constant), the exciting current .3 of 
 an ampere, and the magnetic leakage negligible. When the 
 load resistance is maintained such that the output is equal to 
 50 kilowatts, the primary voltage being 2200 volts, what are 
 the primary and secondary currents for power factors of 1, .8, .5, 
 and .2, for both positive and negative secondary angles of lag? 
 
 133. The Use of Equivalent Impedances in Solving Transformer 
 Problems. Vector Formulas. — It is possible to substitute im- 
 pedance coils for the transformer which will duplicate the trans- 
 former diagrams in Figs. 277 to 281. Thus, in Fig. 282, Z v Z v 
 
 M N P 
 
 Fig. 282. — Combination of Impedances Equivalent to the Impedances of a 
 Transformer. 
 
 and Z represent three sets of impedances connected in series 
 which are equal respectively to the impedances of the primary 
 coil, Z x = V + A u 2 , the secondary coil, Z 2 — V Ii 2 2 + X 2L 2 , 
 and the load, Z — Vi? 2 + A 2 , of a transformer after its secon- 
 
502 
 
 ALTERNATING CURRENTS 
 
 dary quantities have been reduced to primary equivalents. A 
 branch, which carries a current equal to the exciting current, 
 is connected across the main circuit at MT. The arm R c is of 
 such resistance that an active current flows which when multi' 
 plied by the impressed voltage, E v gives a product equal to the 
 no load losses (see DO , Fig. 277); while the arm X m is such a 
 reactance as will permit the flow of the proper magnetizing 
 element {AD, Fig. 277) of the exciting current; the two com- 
 bine to give the total exciting current I^{AO, Fig. 277). 
 Under this arrangement the impressed voltage and current 
 of the series of impedances are E v 1^, the load voltage and 
 current are E v I 2 ; and the angles of lag are 0 V 0 V These 
 and component voltages and currents are as shown in Figs. 
 277 to 281; the particular figure and the character of the loci 
 which apply depending upon the dimensions and character of 
 the various impedances. 
 
 In order that the combination may be a complete reproduc- 
 tion of conditions of a transformer the exciting current path 
 should shunt the second impedance and load at XS. The 
 current flowing in it would then be proportional to the voltage 
 induced in the primary coil by the mutual flux {SO, Fig. 277 
 and other figures) and its vector would make a constant angle 
 therewith, instead of being proportional to and at a fixed angle 
 with the impressed voltage. 
 
 The first arrangement is sufficiently accurate for ordinary 
 calculations and is a less complicated arrangement. It is 
 noticed that by the substitution of impedances the solution 
 of transformer problems becomes in principle equivalent to the 
 simple solutions of circuits dealt with quite extensively in an 
 earlier chapter.* 
 
 The vector equations necessary for determining the most im- 
 portant characteristics of the transformer may be readily writ- 
 ten from Fig. 282, thus: Considering the first arrangement of 
 impedances illustrated in Fig. 282, the total impedance offered 
 to the impressed voltage bv the main circuit J\INPQ is 
 
 Zq = Z 1 + Zy -(- Z , 
 
 Z 0 = {R t -(- R 2 + It ) ■+■ j {X 1L + A 2i ± X) ; 
 where Z v R v and X 1L are the impedance, resistance, and leak- 
 
 * Al t. 86. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 503 
 
 age reactance respectively of the first coil, Z v R v and X 2i of 
 the second coil, and Z, R, and X of the load; X is negative 
 when capacity reactance predominates in the load. Let the first 
 term on the light equal R t and the second equal ±jX t . The 
 admittance of the circuit is then 
 
 Y — ^ 
 
 0 R ? + X , 2 
 
 L 3 
 
 R? + X? 
 
 and the current flowing through it, when the voltage E 1 is 
 impressed, is 
 
 
 R,R t , E,X t , 
 /.’r+x; 2 J R? + X? 
 
 The two terms in the right-hand member of this equation rep- 
 resent respectively the active and reactive components of I±. 
 This current represents the secondary current in a transformer 
 having a unity ratio of transformation and the resistances and 
 reactances of the windings are as denoted in Fig. 282, the vector . 
 angle being measured from the impressed voltage E v 
 The total primary current flowing is 
 
 I\ J\ T I ix ! Ifx — Ic 3 Y m , 
 
 where _Z^, I c , and I m are the total exciting current and its 
 active and reactive components respectively when referred to 
 the impressed voltage E 1 as the initial vector direction. Then, 
 
 calling 
 
 X\Rt _ j 
 
 R? + X? R ‘ 
 
 and 
 
 x,x t r 
 
 R? + X 2 
 
 there 
 
 results 
 
 I\ — (Ir, + Ic) —j(Im ± I, t ). 
 
 The angle of lag between E v the impressed voltage, and I v the 
 total current that would flow in the primary winding of the 
 equivalent transformer, is 
 
 
 tan 1 
 
 Im ± I.X t 
 
 I Rt d” Ic 
 
 The total voltage across the second coil and load, which is 
 equal and opposite to the total mutually induced primary and 
 secondary coil voltages in the equivalent transformer of unity 
 ratio of transformation, is 
 
 Ej 1 = I 1 , Z 2 + 7/Z 
 
 — I\(H 2 + H) I J Ii'(X iL ± X). 
 
504 
 
 ALTERNATING CURRENTS 
 
 The voltage across the load is 
 
 E 2 = I 1 'Z=I 1 'R±jI 1 'X . , 
 and the angle of lag of 1^ with respect to E 2 is 
 
 # 2 = tan -1 -^p, 
 
 which is the secondary angle of lag in the equivalent trans- 
 former. 
 
 The power input is -Eqij cos 6 l ; the power output, E 2 I 2 cos 0 2 ; 
 the excitation losses are I^R C \ the coil resistance losses due 
 to // are T 1 ,2 K 1 and I 1 ' 2 E 2 . The regulation for any load, neg- 
 
 lecting the small drop due to the exciting current, is — — 2 , 
 
 -®i 
 
 where E 2 is the voltage across an impedance which is equivalent 
 to that load. The commercial efficiency, or the ratio of the 
 output to the input for any load, can be readily calculated from 
 the data obtained by the formulas above. 
 
 If any difficulty attaches to a clear understanding of the 
 formulas just given, it is well to lay out impedance and admit- 
 tance triangles for the equivalent impedances and admittances 
 (Fig. 283) exactly as is done in Chapter VI, though reference 
 
 to Figs. 277 and 281 may 
 serve as well, as they repre- 
 sent voltage and current 
 diagrams of the same general 
 character. After the data 
 required for a transformer 
 Fig. 283. — Impedance Diagram obtained from are obtained by the use of 
 Equivalent Transformer Impedances. equivalent impedances, 
 
 which requires the reduction of the secondary quantities to 
 primary equivalents, the secondary currents and voltages may 
 be stated in their correct values by multiplying and dividing 
 respectively by the ratio of transformation s. 
 
 Prob. 1. A 500-kilowatt, 60-cycle, 5500-volt (nominal pri- 
 mary voltage) transformer having a ratio of transformation of 
 20 to 1 develops its full capacity with a voltage of 230 volts at 
 the secondary terminals. At full load the copper losses in the 
 primary and secondary coils are for each ^ per cent of the trans- 
 former’s output. The drop in voltage at full load due to mag- 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 505 
 
 netic leakage is 1 per cent of the no load voltage. The iron 
 losses are 1 per cent of the full load output, and the exciting 
 current is 2 amperes. The load reactance causes the current to 
 lead the secondary terminal voltage by an angle of 30°. Find 
 by the method given in this article the primary voltage, pri- 
 mary current, and primary angle of lag, and the secondary cur- 
 rent. Find, also, the primary current, and the secondary vol- 
 tage and current when the rated voltage (5500 volts) is impressed 
 on the primary winding, the power factor of the load being as 
 before. Give the regulation and efficiency in each case. 
 Answer the same questions when the load is reduced to one 
 half rated capacity (250 kilowatts), assuming that the iron losses 
 and exciting current do not change. 
 
 134. The Effect of Harmonics in the Waves of Voltage and 
 Current upon the Operation of a Transformer. — If the primary 
 voltage impressed on a constant voltage transformer in a single 
 phase circuit contains harmonics, the harmonics are transmitted 
 into the voltage of the secondary circuit, since the form of the 
 wave of magnetism in the core is determined by the form of 
 the impressed voltage wave after the small part that is absorbed 
 in copper losses has been deducted, and the mutually induced 
 
 secondary voltage is at each instant proportional to — 
 
 Ctb 
 
 If the secondary circuit contains considerable capacity re- 
 actance and comparatively small resistance, the higher harmonics 
 of the secondary voltage cause exaggerated current harmonics.* 
 The magneto-motive forces of these secondary current harmonics 
 would affect the magnetic flux of the core and they must be off- 
 set by opposing magneto-motive forces in the primary winding, 
 which results in harmonics of current flowing in the primary 
 circuit that are equivalent to those in the secondary circuit. 
 
 Thus, irregular voltages and currents in the circuits to which 
 a transformer is attached act very much as would be the case if 
 the transformer was replaced by the equivalent network dis- 
 cussed in the previous article. 
 
 As the exciting current of a transformer is quite irregular 
 in form, it contains harmonics of relatively large amplitude. 
 Therefore, in a circuit containing a large number of unloaded 
 or lightly loaded transformers, conditions may arise under 
 
 * Art. 69. 
 
,506 
 
 ALTERNATING CURRENTS 
 
 which the resultant flow of irregular current may prove 
 troublesome. The principal harmonic in the exciting current 
 of higher frequency than the fundamental when the impressed 
 voltage is sinusoidal is the third, although the fifth also may 
 be observed. The conditions may be understood from Fig. 
 263, which shows a hysteresis and eddy current cycle and the 
 current wave required to produce a sinusoidal magnetic flux. 
 The quadrature component of current is peaked as a result of 
 the increased reluctance caused by partial saturation of the 
 iron toward the top of the flux wave, which occurs at even 
 the relatively low magnetic densities used in transformers. 
 This symmetrical peaking of the quadrature current compo- 
 nent indicates a prominent third harmonic therein. 
 
 No matter what the primary voltage wave may be like, the 
 secondary voltage wave must have practically the same form, 
 the exciting current and magnetization curves varying to meet 
 this requirement. When the voltage curve is peaked, the 
 magnetization curve is flattened and does not reach so high a 
 maximum as for an equivalent sine curve ; and conversely, 
 when the voltage curve is flat, the magnetization curve will be 
 peaked, as has been hereinafter proved in Art. 157. Since the 
 iron losses of a transformer depend upon the maximum value of 
 the magnetic flux in the core, it is to be expected that a peaked 
 voltage curve will cause minimum iron losses. Roessler has 
 shown that the iron losses of a transformer depend upon the ratio 
 of the average value of the voltage wave to its effective value, 
 i.e. upon the Form Factor. Steinmetz cites a case where the 
 losses in a 200-kilowatt transformer were 13 per cent less when 
 worked on a peaked curve than when a sine curve was im- 
 pressed. The change in full load efficiency is, however, very 
 slight in ordinary commercial practice, although the “all da}'” 
 efficiency may be considerably affected by the form of the vol- 
 tage wave impressed on a transformer which runs for any con- 
 siderable part of the day very lightly loaded. Generally speak- 
 ing, constant voltage transformers have the best efficiencies 
 when operated on circuits having peaked voltage waves. Ex- 
 perience shows that alternating current arc lights give the best 
 results when worked on a flat-topped curve, while a sine curve 
 is most convenient for the operation of synchronous motors and 
 gives the best efficiency in the operation of induction motors. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 507 
 
 The last statement is shown later to be theoretically correct.* 
 Since a transformer may be required to supply any one of these 
 kinds of loads, an approximate sine wave probably gives the 
 best general results. 
 
 135. Effects of Changes of Frequency and Voltage. — The for- 
 mulas giving the voltages developed in the coils of an unloaded 
 transformer, 
 
 ^ = 2 
 
 V2 7 T<&fn 
 
 1 0 8 
 
 1 and -E" 2 — -%J2 V f 
 
 show that if the maximum value of the magnetic flux in the 
 core of a transformer is fixed, then for a given impressed vol- 
 tage the number of turns in the coils must vary inversely with 
 the frequency. To design transformers for all frequencies with 
 a fixed value of the maximum magnetic flux is not an economi- 
 cal plan, however, since the core loss per cubic centimeter of 
 iron depends upon the frequency. If a certain core loss is de- 
 termined upon as being that which will give the most satis- 
 factory general results in the case of a transformer of given 
 size and for a given duty, then the magnetic density in the 
 core must be designed to be smaller as the frequency is made 
 larger. This may be seen from the fact that the eddy current 
 loss varies as the square of the frequency, and the hysteresis 
 loss as the first power of the frequency, other things being 
 equal. Since the former is not uncommonly between one 
 tenth and one fourth of the latter, this makes the total core 
 losses vary somewhere between the first and second powers of 
 the frequency, if the maximum magnetic density is designed to 
 be the same for all transformers. The eddy current loss also 
 varies as the square of the maximum magnetic density, and hys- 
 teresis loss varies with some degree of approximation within 
 the limits of transformer practice, as the 1.6th power of the 
 maximum density. Consequently, the total losses vary as 
 some power of the maximum density between the 1.6th and 
 the second. It is thus shown that the core losses will not be 
 greatly changed if the maximum magnetic density, within 
 reasonable limits, varies inversely with the frequency. Now 
 it is evident from the formula that, if the turns and voltage re- 
 main unchanged, when the frequency changes, the maximum flux 
 
 * Art. 198. 
 
 t Art. 122. 
 
508 
 
 ALTERNATING CURRENTS 
 
 will change inversely ; hence a transformer should give nearly 
 the same efficiency for currents of different frequencies reason- 
 ably near that for which it was designed, only a small rate of 
 increase of the iron losses occurring as the frequency decreases ; 
 and the number of turns in the coils may be nearly the same 
 in a series of transformers of equal capacities designed for the 
 same voltages but different frequencies. This statement is 
 made under the supposition that the lower frequencies do not 
 call for a maximum magnetic flux which exceeds the saturation 
 point of the magnetic core. 
 
 Changing the frequency alters the quadrature component of 
 the exciting current in a proportion which depends upon the 
 saturation of the core, and transformers built of poor material, 
 or which operate with their cores near the point of saturation 
 when used on normal frequency, may give unsatisfactory service 
 and overheat on lower frequencies than their normal, on account 
 of an excessive magnetic density and an excessive exciting 
 current. Transformers under these circumstances may have 
 poor regulation due to large copper drop and increased magnetic 
 leakage. This is seen from the voltage formula, where d>/ 
 must be constant while is constant. It will be seen also from 
 the formula that raising E x above the rated value tends to cause 
 a greater maximum flux <f>, so that even if the insulation will 
 stand an increased voltage, the increase cannot go beyond the 
 point where the core becomes saturated, without causing ex- 
 cessive core losses and an excessive flow of exciting current. 
 
 Since the rates of variation of the hysteresis and eddy cur- 
 rent losses with the frequency and with the reciprocal of the 
 maximum magnetic density are not exactly the same, it is to 
 be expected that the efficiency of a transformer, so far as it is 
 affected by the iron losses, will vary to some degree with the 
 frequency. 
 
 Since the flux density varies inversely as the frequency of 
 the impressed voltage in any particular transformer, as is shown, 
 by the foregoing formula, and the eddy current loss is propor- 
 tional on the one hand to the square of the frequency and on 
 the other hand to the square of the flux density, a variation of 
 the frequency leaves the eddy current loss substantially unaf- 
 fected, provided the paths for the flow of the eddy currents do 
 not change. In the case of the hysteresis loss, different consid- 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 509 
 
 erations arise, for this loss is proportional to /T 1 - 6 , while <I> in 
 any particular transformer is proportional to 1//. The conse- 
 quence is that the hysteresis loss in a particular transformer, 
 when the frequency is varied, changes proportionally with 
 l//- 6 . In first class modern transformers, the eddy current 
 loss is usually much smaller than the hysteresis loss, and the 
 efficiency of a particular transformer increases appreciably as 
 the frequency of the impressed voltage is increased. 
 
 Transformers of equal outputs but for different frequencies 
 may be built with windings having like numbers of turns but 
 with the cross sections of the cores inversely proportional to the 
 frequency. The magnetic density would then be the same in 
 all the transformers. This is unsatisfactory, however, since it 
 makes low frequency transformers large and expensive ; and 
 since the weight of iron in a core must vary more rapidly than 
 the cross section, this method of construction also causes com- 
 paratively large core losses in low frequency transformers and 
 gives them comparatively low efficiency. 
 
 The conclusion is therefore derived that the core and wind- 
 ings of a well-designed and well-constructed transformer will 
 be satisfactory in performance for small variations in frequency, 
 but that transformers designed for high and low frequencies 
 respectively should be made with such numbers of turns, 
 weights of iron, and magnetic densities as will give in each 
 case the efficiency and regulation sought. With the modern 
 high grade alloy steels the hysteresis is so much less than in 
 common steels and irons that it is possible to carry the magnetic 
 density well toward the point of saturation for all frequencies, 
 and this is now standard practice. 
 
 The modern standard frequencies in this country are 60 
 cycles per second for lighting and small power purposes, 25 
 cycles per second for alternating current railroad and large 
 power units, and sometimes 40 cycles per second for combined 
 lighting and power circuits. 
 
 136. Transformer Iron and Steel. — Early experience showed 
 that the core losses in some transformers increased to a very con- 
 siderable extent during the first few months of operation. The 
 increase in losses was found to be due to increased hysteresis loss 
 per cycle, and was originally ascribed by Ewing to magnetic 
 fatigue of the iron. Later it was conclusively proved by 
 
510 
 
 ALTERNATING CURRENTS 
 
 experiments and the records of transformer manufacturers to 
 be caused by the continuous condition of heightened tempera- 
 ture at which the iron is operated. This effect of Ageing seems 
 to have the greatest effect upon poor qualities of iron, hastily 
 and imperfectly annealed, and the least effect upon the best 
 grades of iron which have been annealed with great care. The 
 conditions under which the annealing of transformer iron is 
 performed, especially with reference to temperature and dura- 
 tion of the process, have much to do with the extent of the 
 ageing effect, and by proper annealing it can be rendered very 
 small in cores made of proper qualities of iron. The iron 
 formerly and now much used for transformer cores is a very 
 mild steel made by either the bessemer or open-hearth process, 
 though puddled iron sheets were formerly used to some extent. 
 But the magnetic material most approved at the present day is 
 an alloy of silicon with mild steel. This is steel containing 
 from 2 1 to 4 or more per cent of silicon, called silicon steel. 
 Such steels are harder and more difficult to make and work, so 
 that they are more expensive than ordinary soft steel. But 
 these steels are apparently little affected by ageing — which 
 sometimes increases the hysteresis losses in ordinary steels as 
 much as 100 and 150 per cent after the metal has been in 
 service in a transformer core for a few months. Further, 
 these new steels have a lower hysteretic constant, and trans- 
 former cores made of some of them waste from 35 to 45 per 
 cent less power in hysteresis than do the ordinary steels. They 
 also possess higher specific resistances, and this reduces eddy 
 current losses. Thus, for the same efficiency and duty a trans- 
 former can be built having a core of the best transformer silicon 
 steel at a much less weight per kilowatt capacity, than when any 
 one of most of the standard soft steels is used. This is be- 
 cause the magnetic density can be increased from 45 to 65 per 
 cent, without increasing the loss due to hysteresis, which re- 
 sults in a consequent reduction in the cross section of the core 
 and a resultant lessening of the length of wire per turn.* A 
 fair value for the hysteretic constant for good soft transformer 
 steel is .0021 and for silicon steel .00094 in terms of watts per 
 pound and per cycle.* In Fig. 284 the curves A and A' are 
 plotted from the hysteresis equation f and show the hysteresis 
 
 * Trans. A. I. E. E., Yol. 28, p. 439; also Vol. 30. t Art. 106. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 511 
 
 loss in watts per cubic centimeter and per pound of metal when 
 the frequency of magnetic cycles is 100 per second and the 
 
 hysteresis constant is rj = .0021. Curve A is plotted for max- 
 imum flux densities from 0 to 5500 lines of force per square 
 
512 
 
 ' ALTERNATING CURRENTS 
 
 centimeter, and curve A! is plotted for maximum flux densities 
 from 5000 to 10,500; while the curves B , B ', and B " are plotted 
 for 7) = .00094, the maximum flux densities being 0 to 5500 
 lines of force per square centimeter for curve B , 5000 to 10,500 
 lines of force per square centimeter for curve B\ and 10,000 to 
 15,500 for curve B " . 
 
 It must be remembered that the character of steels, especially 
 of silicon or other alloyed steels, is apt to be quite variable. 
 Therefore, in designing magnetic apparatus, the exact kind of 
 material to be used should be tested and its qualities as thus 
 determined used in making calculations. As seen from the 
 hysteresis equation just referred to, it is evident that the 
 curves of Fig. 284 may be used for any other hysteresis 
 constant and frequency by multiplying the vertical scale of 
 
 losses by the ratio where t)f is the product of the hysteresis 
 VJ 
 
 constant and frequency taken in calculating the curves referred 
 to, and 7]'f is the product of the values of those quantities for 
 which the data are sought. 
 
 The eddy current losses are dependent both upon the thick- 
 ness of the sheets and the specific electrical resistance of the 
 steel. The thickness of laminations quite commonly used for 
 60-cycle transformers is from 14 to 15 mils, or more exactly No. 
 29 sheet-iron gauge, which has a thickness of 14.1 mils, while 
 for 25 cycles No. 26 gauge sheets, having a thickness of 18.7 mils, 
 do not permit the eddy current loss to exceed economical limits. 
 The specific resistance of iron varies quite widely, and this is 
 especially true of the alloy steels, which may have a much 
 greater resistance than the soft steels and irons. In Fig. 285 
 curves A , B , and O are the calculated eddy current losses in 
 plates of 10, 15, and 20 mils thickness respectively for various 
 maximum magnetic densities, a frequency of 100 periods per 
 second, and a specific resistance of 1 x 10 -5 , which is a fair 
 value for good transformer steel.* The curve shown in three 
 sections, Z), D\ D'\ is for a plate 15 mils in thickness and having 
 a specific resistance three times as great as for the first curves. 
 It represents some of the harder alloys. The curve can readily 
 be used for other frequencies and thicknesses of laminations, as 
 will be seen by an inspection of the eddy current formula. f 
 
 * Art. 111. 
 
 t Ibid. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 513 
 
 The total core loss in transformers varies more or less, de- 
 pending upon the characteristics desired, being from about 1.1 to 
 
 001 = / "WO -no S3d S11VM 
 
 3 per cent in small transformers, such as those under 5 kilowatts 
 capacity, and decreasing to about | of a per cent in those of 25 kilo- 
 
 2 * 
 
514 
 
 ALTERNATING CURRENTS 
 
 1GOOO 
 
 14000 
 
 12000 
 
 £ 10000 
 
 8000 
 
 C5 6000 
 
 4000 
 
 2000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 rB 
 
 
 
 
 c 
 
 
 
 
 
 -A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 12 
 
 Fig. 286. - 
 
 2 4 6 8 10 
 
 MAGNETIZING FORCE, H 
 
 -Curves of Magnetization of Transformer 
 Steels. 
 
 14 
 
 watts capacity, to § 
 of a per cent in those 
 of 50 kilowatts ca- 
 pacity, and to a 
 smaller percentage as 
 the capacity still fur- 
 ther increases. The 
 eddy currents seldom 
 comprise over 10 per 
 cent of the total core 
 losses. 
 
 As the best grades 
 of low carbon steels 
 and wrought irons 
 are used for trans- 
 former cores, the 
 ratio between the 
 
 magnetic 
 
 maximum 
 
 density <£ and the 
 field strength H is high. Curve B* in Fig. 286 is from a good 
 grade of transformer steel, and curve A represents a soft steel of 
 poorer grade. The 
 silicon steels do not 
 ordinarily have 
 quite such high 
 curves as the best 
 transformer carbon 
 steels. The rela- 
 tion between 
 curves B and C 
 fairly indicates the 
 difference. The 
 permeability of B 
 in Fig. 286 is shown 
 by B in Fig. 287. 
 
 Curve A is a poorer 
 grade of soft iron 
 
 and C represents 0 '-’ 00 ° 4000 6000 8000 10000 
 
 t. . - MAGNETIC DENSITY = <#> 
 
 tll6 permeability of Fig. 287. — Curve of Permeability of Transformer Steels, 
 
 * Smithsonian Tables, pp. 274 and 275. 
 
 12000 14000 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 515 
 
 a good grade of silicon steel. The silicon steels take somewhat 
 more exciting current than the best grades of soft iron and soft 
 carbon steels, on account of their lower permeability. 
 
 The maximum magnetic densities to which transformer cores 
 are worked is dependent somewhat upon the frequencies for 
 which the transformers are designed. For 60 cycles the den- 
 sities vary from 6000 to 10,000 lines of force per square centi- 
 meter, or slightly higher, with a tendency toward 10,000 when 
 the highest grade transformer core metal is used. For 25 cycles 
 they vary from nine or ten to twelve or more thousand lines of 
 force per square centimeter. The usual silicon steels cannot 
 be run too high in magnetic density or the exciting current 
 will be larger than desired. In silicon steels a maximum mag- 
 netic density of 10,000 lines of force per square centimeter is 
 apt to be about as near the point of saturation as it is wise 
 to go, while economy of material dictates as high a density as 
 possible, so that in modern transformers, using the new metals, 
 the density will not ordinarily be far from that figure. With 
 ordinary steels the greater iron losses tend to make lower den- 
 sities more desirable. 
 
 In addition to silicon there are other alloying materials used 
 with iron for the purpose of reducing the hysteresis coefficient 
 below figures common in ordinary soft carbon steels. Among 
 these are nickel,* tin, and arsenic, f The new pure irons re- 
 cently produced have also unusually good magnetic qualities. 
 The state of the art in the manufacture of iron and steel is 
 rapidly advancing, and it is probable that present practice will 
 be materially modified as a result. 
 
 137. Efficiencies of Transformers. — The average working effi- 
 ciency of a transformer is by no means equal to its full load 
 efficiency. In the case of dynamos placed in a central station 
 it is usual to divide the generating units so that the plant op- 
 erating during any part of the day will be well loaded. In 
 the same way the capacities of stationary motors may ordinarily 
 he chosen so that the motors seldom operate at very small par- 
 tial loads. The way that transformers are usually operated, 
 however, makes it quite difficult to keep a uniform load on 
 them, and, in fact, for a considerable portion of the day they 
 may have their secondary circuits open. When this is the case, 
 
 t Electromet. Ind., Vol. VII. 
 
 * Art. 109. 
 
516 
 
 ALTERNATING CURRENTS 
 
 the iron losses of transformers are of much greater influence on 
 their All-day efficiency than the copper losses, and it is desir- 
 able to reduce the iron losses to a minimum. For instance, sup- 
 pose a transformer of 5000 watts output at normal full load 
 has iron losses of 125 watts, and copper losses at full load of 
 100 watts. Then the full load efficiency of the transformer 
 is 95.7 per cent, the half load efficiency is 94.3 per cent, and 
 the quarter load efficiency is 90.4 per cent. Supposing that 
 the daily output of the transformer is equivalent to 25,000 
 watts for one hour (25,000 watt-hours), obtained by full load 
 operation for five hours and open-circuit operation for the re- 
 maining 19 hours, then the losses in the iron core are equivalent 
 to 3000 watts for one hour, and in the copper to 500 watts for 
 one hour (neglecting no-load copper losses), or a total of 3500 
 watts for one hour. The all-day efficiency is then 87.7 per cent. 
 To increase this all-day efficiency, it is evidently necessary to 
 decrease the iron losses. To do this for a fixed frequency 
 requires a decrease in the amount of iron in the magnetic cir- 
 cuit or a decrease of the maximum magnetic density. Either 
 process calls for an increase in the wundings and consequently 
 in the copper losses. Suppose that decreasing the core losses to 
 75 watts makes it necessary to increase the full load copper 
 losses to 150 watts ; then, other things being equal, the efficien- 
 cies become, full load, 95.7 per cent; half load, 95.7 per cent; 
 quarter load, 93.7 per cent; and the all-day efficiency, on the 
 assumption made above, is 90.7 per cent. There is a saving by 
 the latter construction of 950 watt-hours in twenty-four hours, 
 and in one year of 365 days the saving is nearly 350 kilowatt- 
 hours. If one kilowatt-hour is worth 4 cents to the central 
 station, the difference in the amount of energy wasted each year 
 by the two transformers has a value of nearly fourteen dollars, 
 which is a substantial part of the difference in the original cost 
 of the two transformers. If the average load of the transformer 
 were less than that assumed, which is frequently the case for 
 small transformers in practice, the iron losses would have a still 
 greater influence on the all-day efficiency. 
 
 A still greater decrease in the iron losses with its attendant in- 
 crease of copper losses would evidently raise the all-day efficiency 
 to a higher point. Here, however, is met the question of regula- 
 tion, which will not satisfactorily admit of too 8'reat a copper loss 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 517 
 
 at full load on account of the attendant drop of secondary vol- 
 tage, but this difficulty maybe met by increasing the cross section 
 of the copper. This alternative causes an increase in the cost 
 of the transformer, but a transformer with small losses and good 
 regulation is worth more to the central station than one with 
 large losses or poor regulation. The advantage of decreasing 
 the iron losses, which is thus shown, led Swinburne some years 
 ago to advocate and build transformers with a cyclindrical iron 
 wire core under the windings, but without closed iron magnetic 
 circuits. A diagram of this transformer, called the Hedgehog, 
 which was somewhat extensively used in England at one time 
 is shown in Fig. 288.* Decreasing the amount of iron in the 
 magnetic circuit decreases the core losses but 
 at the same time increases the reluctance, and 
 therefore increases the exciting current. This 
 is a decided disadvantage if carried to excess. 
 
 While the true magnetizing current is in quad- 
 rature with the mutually induced voltage of the 
 transformer, yet it does result in a continuous 
 I 2 R loss in the conductors leading to the trans- 
 former and in the primary winding of the trans- 
 former itself. It also causes an extra demand 
 for current from the dynamos supplying the 
 circuit, so that extra generators may have to be 
 operated at periods of light load in order to sup- 
 ply the quadrature currents. In other words, 
 the power factor of the system as a whole is de- 
 creased, with an accompanying loss of plant efficiency. Finally, 
 a large magnetic reluctance causes a considerable magnetic 
 leakage and consequent increase in the secondary drop of a 
 transformer, and therefore impairs its regulation. 
 
 Without entering into a controversy regarding the exact point 
 at which a high reluctance in the magnetic circuit of a trans- 
 former causes more disadvantage than is counterbalanced by 
 the decreased iron losses brought about by decreasing the 
 amount of iron, the examples may serve to show the necessity 
 of carefully designing the magnetic circuit to fit the conditions 
 if they can be predetermined. 
 
 *For test on transformer of this type see Trans. A. I. E. E ., Yol. 10, p. 497 ; 
 Elec. World, Yol. 22, p. 357. 
 
 Fig. 288. — Hedge- 
 hog Transformer. 
 
518 
 
 ALTERNATING CURRENTS 
 
 The commercial efficiency of a transformer is equal to the 
 watts delivered to the external secondary circuit by the trans- 
 former divided by the watts absorbed by the primary coil. It 
 may be written 
 
 Po P 0 
 
 71 — i = « , 
 
 Pi P, + L 
 
 where P x and P 2 are the power absorbed and delivered respec- 
 tively by the primary and secondary coils, and L represents 
 the total losses of power in the transformer. The losses are 
 made up of the I 2 R losses in the primary and secondary coils, 
 eddy currents in the windings caused by leakage flux, and 
 the losses due to hysteresis and eddy currents in the core. 
 The I 2 R loss in the secondary winding is directly proportional 
 to the square of the load (secondary output), assuming the 
 secondary terminal voltage to be constant, while the PR loss 
 in the primary winding is nearly proportional to the square 
 of the load, though it contains a small approximately constant 
 term due to the exciting current. The rated capacities of very 
 large transformers are not uncommonly expressed in kilo- 
 volt-amperes instead of kilowatts, since the PR heating effect 
 is dependent upon the current, which depends not only upon 
 the load but also inversely upon the power factor when the vol- 
 tage is constant; but ordinarily the rating of small transformers 
 is in kilowatts capacity, the assumption being that the power 
 factor for which they are specified is approximately unity. 
 The efficiency will obviously have different values for different 
 loads even when power factor and primary impressed voltage 
 are kept constant; and also if the load (i.e. the output in kil- 
 owatts) is kept constant, the efficiency will have different 
 values for different power factors. 
 
 The hysteresis and eddy current losses vary from .5 to 1.5 
 per cent of the full load capacity in transformers of 500 or 
 more kilovolt-amperes full load rating, and the percentage in- 
 creases as the capacity decreases. In a transformer of 100 
 kilovolt-amperes capacity the losses may be only a little higher 
 than the minimum given above, or they maybe as much as one- 
 half or more above the maximum figure given above. Intrans- 
 formers of from 5 to 100 kilovolt-amperes capacity of a given 
 make and type, the iron losses usually increase gradually with 
 decreasing rated capacities until at the small size named they are 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 519 
 
 from 1 to 2.5 per cent or more of the rated capacity. In very 
 small transformers they rise to from 3 to 5 per cent of the rated 
 capacity. The loss in watts per pound of iron varies from less 
 than J a watt per pound for the best grades of alloy steels, 
 when a low magnetic density is used, to as high as 2 or 3 watts 
 per pound where poorer grades of metal are used and the 
 magnetic density used is higher. 
 
 The copper losses at full load are apt to be from 25 to 100 per 
 cent greater than the iron losses, though the tendency in 
 modern designing is to keep these losses down, for by increas- 
 ing the weight of the core and decreasing the number of the 
 coil turns, the IR drop and the IX drop are reduced, and the 
 regulation is improved. The range of variation in watts lost 
 per pound of copper for full load current in commercial trans- 
 formers is in the neighborhood of from slightly less than one to 
 six or more watts per pound. For transformers to go into 
 lighting service or for duties where there are short full load 
 peaks, however, the copper losses may to advantage be made 
 relatively still higher, with an accompanying reduction of the 
 iron losses ; for then the copper losses varying with the load 
 do not have so much influence on the all-day efficiency and 
 the reduction of the fixed iron losses results in better all-day 
 efficiency. This is not always desirable, however, as there is 
 then a higher load on the circuit during the peak when all 
 power available in the circuits may be most valuable. In any 
 event, the copper losses must not exceed reasonably low values, 
 or the regulation of the transformer becomes too poor to 
 afford satisfactory conditions for electric lighting where incan- 
 descent lamps are used. 
 
 The United States Government specifies the iron and cop- 
 per losses shown in the table on page 520 for small step-down 
 lighting transformers designed for a frequency of 60 cycles 
 per second, 2200 volts primary voltage, and ratios of transfor- 
 mation of 10 and 20 to one. 
 
 The losses shown are low, and the relatively low iron 
 losses compared with the copper losses result in the maximum 
 commercial efficiencies occurring at loads less than the full 
 loads, with resulting high all-day efficiencies compared with 
 those which would be obtained were the same total losses at full 
 load equally divided between the copper and the core. 
 
520 
 
 ALTERNATING CURRENTS 
 
 U. S. TABLE OF IRON AND COPPER LOSSES 
 
 Kva. Rated 
 Capacity 
 
 Ikon Losses in 
 Watts 
 
 Full Load Copper 
 Losses 
 
 Total Fcll Load 
 Losses in Pee Cent 
 of Capacity 
 
 1 
 
 21 
 
 26 
 
 4.70 
 
 2.5 
 
 33 
 
 55 
 
 3.52 
 
 5 
 
 45 
 
 93 
 
 2.76 
 
 10 
 
 82 
 
 160 
 
 2.42 
 
 20 
 
 135 
 
 280 
 
 2.08 
 
 30 
 
 175 
 
 374 
 
 1.83 
 
 40 
 
 210 
 
 470 
 
 1.70 
 
 50 
 
 255 
 
 580 
 
 1.67 
 
 75 
 
 400 
 
 800 
 
 1.60 
 
 100 
 
 560 
 
 1000 
 
 1.56 
 
 The commercial efficiency of transformers at full load ranges 
 from 95 to 98 per cent or better in machines of from 10 kilo- 
 watts to the large sizes of several thousands of kilovolt- 
 amperes capacity. In smaller sizes the full load efficiency 
 becomes materially smaller and at one or two kilowatts capa- 
 city it may not exceed 92 or 98 per cent. Commercial trans- 
 formers designed for the standard lighting frequency of 60 
 cycles per second are apt to have a slightly higher efficiency 
 than those designed for 25 cycles per second, and likewise 
 transformers designed for 2200 volts in the high voltage coil 
 (whether primary or secondary) usually have a slightly higher 
 efficiency than those designed for higher voltages. In Fig. 
 
 289 the curves A and B represent the full load efficiencies of a 
 standard line of 500 Kva. transformers designed for voltages in 
 the high voltage coil of 2200, 6600, 11,000, 16,500, 22,000, 
 33,000, and 44,000 volts respectively. Curve A is for trans- 
 formers designed to operate on 60 cycles per second ; and B 
 for those operating on 25 cycles per second. The curve in Fig. 
 
 290 is the commercial efficiency curve of one of these 500 
 Kva. transformers designed for 2200 volts in the high voltage 
 coil and a frequency of 60 cycles per second. The curve is 
 typical of good efficiency curves for transformers of this size. 
 It is noted in this figure that the efficiency at ^ load is only 
 about 2 per cent less than that at full load, and in this regard 
 it is not far from the performance of other well-designed trans- 
 formers, except those of very small capacities, — five kilowatts 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 521 
 
 or less capacities, — which show relatively lower efficiencies 
 at light loads. 
 
 100 
 
 99 
 
 98 
 
 96 
 
 95 
 
 94 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 __ A 
 
 
 
 
 
 
 
 
 
 B 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5000 
 
 10000 
 
 15000 20000 25000 30000 
 
 TRANSFORMER VOLTAGE 
 
 35000 40000 45000 
 
 Fig. 289. — Curves showing the Full Load Commercial Efficiencies of the Same Size 
 and Style of Transformers when designed for Various Voltages and for Frequencies 
 of 60 and 25 Cycles respectively. 
 
 Fig. 290. — Curve of Commercial Efficiency of a 500-Kva., 60-cycle, 2200-volt (High 
 .Voltage Coil ) Transformer. 
 
522 
 
 ALTERNATING CURRENTS 
 
 The weight efficiency, that is, the weight of transformers 
 without their containing cases or tanks per kilowatt of rated 
 capacity, is from 7 to 20 pounds for transformers of 500 Kva. 
 capacity and over. This increases by 50 to 100 per cent 
 as the size decreases to 50 Kva. or thereabouts, and from there 
 to 10 Kva. capacity the weight increases by 10 to 25 per 
 cent further. For smaller sizes, especially those less than 5 
 Kva., the complete weight per kilo-volt-ampere of rated capacity 
 including oil and cases, may be as much as eight or more times 
 that of transformers of 500 Kva. capacity. Consequently, 
 large transformers are very much less expensive to build than 
 small ones per unit of output. The weight of a transformer is 
 dependent not only on the design, but also upon the frequency 
 and voltage of the circuit with which it is intended to be used ; 
 for low frequencies require a larger total magnetic flux to in- 
 duce a given voltage in a given number of turns in the wind- 
 ings and hence they require larger cores or greater windings ; 
 very high voltages require more insulation space to be allowed 
 in association with the windings ; and very low voltages 
 require the use of inconveniently thick conductors or conductors 
 composed of multiples of parallel elements. The ratio of weight 
 of copper to weight of iron varies from about 50 to 80 per cent, 
 when the design follows the usual practice. 
 
 The regulation of transformers is of the highest importance, 
 especially where carbon filament incandescent lamps are at- 
 tached to the secondary circuit, as such lamps are liable to serious 
 injury when subjected to varying voltages. A closely constant 
 voltage for what is termed constant voltage apparatus is al- 
 ways desirable, though the percentage permissible variation is 
 dependent somewhat upon the character and duty of the load. 
 The accepted general definition of “ regulation ” of a machine or 
 apparatus in regard to some characteristic quantity (such as 
 terminal voltage, current, or speed) is “the ratio of the deviation 
 of that quantity from its normal value at rated load to the 
 normal rated load value,”* and for a constant voltage trans- 
 former it is specifically : “ the ratio of the rise of secondary 
 terminal voltage from rated non-inductive load to no load (at 
 constant primary impressed terminal voltage) to the secondary 
 
 *Amer. Inst, of Elect. Eng., Standardization Rules, No. 187, Trans.. 1907, 
 p. 1809. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 523 
 
 terminal voltage at rated load.” * The regulation is dependent 
 upon the drop of primary voltage due to the leakage reactance 
 and the resistance in the primary and secondary coils of a 
 transformer, as may be seen by a study of the internal reactions 
 of transformers given earlier, f The total rise of the secondary 
 terminal voltage when the load changes from non-reactive full 
 load to no load, and the primary impressed voltage is maintained 
 constant, is 
 
 u d = [(u+ i 2 ny + / 2 2 x 2 ] * - e, 
 
 where 1^ is the full load secondary current, H is the sum of the 
 primary and secondary coil resistances, the former reduced to 
 the secondary equivalent, and X is the sum of the leakage re- 
 actances of the primary and secondary coils, the former reduced 
 to the secondary equivalent, and E is the secondary terminal 
 voltage at rated full load. The drop due to the primary exciting 
 current is neglected here. This formula becomes evident from 
 a study of the loci of the earlier article to which reference has 
 just been made. 
 
 The regulation with a load of unity power factor (as called for 
 in the foregoing definition) varies from less than 1 per cent to 
 about 2 per cent in transformers of 10 Kva. capacity and greater, 
 and may increase to 3 or 4 per cent in the very small sizes. 
 The regulation obtained when the power factor of the load 
 is other than unity is impaired if the load is inductive, 
 but improves with a load exhibiting capacity reactance until 
 the effect of the leakage reactance of the coils is neutralized 
 and then with further increase of capacity reactance it becomes 
 poorer. With inductive reactance in the load, the regulation 
 is always poorer than when the load is of unity power factor. 
 As inductive loads must ordinarily be reckoned with in com- 
 mercial electric circuits, it is common for buyers of transformers 
 to specify the regulation, not only for loads having unity power- 
 factor, but also for loads in which the power factor is .9, .8, 
 and sometimes down to .6. If the constants of a transformer 
 are known, it is easy to compute the regulation for any power 
 factor of the load by constructing transformer diagrams or using 
 the transformer formulas. | 
 
 *Ibid., No. 197, p. 1810. 
 
 t Art. 123. 
 
 t Arts. 117-118. 
 
524 
 
 ALTERNATING CURRENTS 
 
 138. Polyphase Transformers. — A saving in the amount of 
 material used, and therefore a reduction of the cost of con- 
 struction and an improvement of the operating efficiency, may 
 be effected for polyphase transformation by combining the 
 magnetic circuits of the individual transformers in the several 
 phases in a manner which is analogous to that in which poly- 
 phase electric circuits are combined into common circuits. 
 
 Figure 291 represents a quarter-phase transformer with a 
 combined magnetic circuit. Since the phases of the magnetic 
 fluxes in the two halves of the transformer are 90° apart (the 
 phases of the impressed voltages being 90° apart and here 
 
 assumed sinusoidal), the re- 
 sultant magnetism in the 
 middle tongue is their vec- 
 tor sum and is V2 times as 
 great as that in either of 
 the cores under the wind- 
 ings. Therefore this cen- 
 tral tongue should have V2 
 times as great a cross sec- 
 tion as the remainder of the 
 magnetic circuit. There is 
 a saving of iron in the com- 
 bined transformer, as com- 
 pared with two independent 
 transformers, which is equal 
 to ( V2 — 1) times the weight 
 of the central tongue. This saving is further emphasized by the 
 advantage of the small space occupied and the convenient form 
 obtained by inclosing this equivalent of two transformer units 
 in one case. In this illustration, the high voltage external cir- 
 cuits are shown with a common return wire, and the low vol- 
 tage external circuits are shown with independent wires ; but it 
 is obvious that the use of independent circuits requiring four 
 wires for the two phases, or the use of three wire circuits 
 utilizing one wire for the common return, are free alternatives 
 for use with the primary and secondary circuits. 
 
 A similar combination may be effected in tri-phase trans- 
 formers, and the magnetic circuits may be coupled in either the 
 wye or the delta arrangement. Figure 292 shows a conventional 
 
 Fig. 291. — Diagram of Circuits and Core of a 
 Quarter-phase Transformer. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 525 
 
 diagram of a tri-phase transformer of a form early made by several 
 manufacturers, in which the magnetism in the yokes DD', which 
 join the cores A, B , and (7, is 1/V3 times as great as that in the 
 cores. These yokes make a delta coupling for the magnetic 
 
 Fig. 292. — Diagram of Circuits and Core of a Tri-phase Transformer with a Yokes. 
 
 circuits. Figure 293 shows a similar diagram of a tri-phase 
 transformer, in which the yokes are joined so as to make a wye 
 coupling of the magnetic circuits. These constructions allow 
 a considerable economy in use of materials in comparison with 
 
 Fig. 293. — Diagram of Circuits and Core of a Tri-phase Transformer with Y Yokes. 
 
 separate transformers in the three phases, on account of the 
 fact that the magnetic circuit through the core under each 
 winding is completed through the cores under the other two 
 windings. 
 
 In Fig. 294 is shown a diagram of circuits and core for a 
 common type of tri-pliase transformer. To obtain an expres- 
 sion for the instantaneous magnetic fluxes in the three cores 
 
526 
 
 ALTERNATING CURRENTS 
 
 W, X, and y, shown in Fig. 294, assume the magnetic reluc- 
 tance in the path from A to A' indicated by the U-shaped dotted 
 line through the core W and the path indicated by the corre- 
 sponding U-shaped dotted line through the core Y to be equal, 
 and of constant value P w ; also let the reluctance of the path 
 directly from A to A' through X he represented by P Con- 
 sider that sinusoidal alternating exciting currents, 120° apart 
 
 Fig. 294. — Diagram of Circuits and Core of a Tri-phase Transformer having 
 Straight Yokes. 
 
 in their phases, flow in the three exciting windings, and let <f> w , 
 (f) x , and (j) y represent values of the magnetic fluxes in the cores 
 W, X , and Y, respectively, at a given instant, and m w , m x , and 
 m y represent the corresponding instantaneous values of the 
 magneto-motive forces exerted in the aforenamed paths through 
 the three cores. Then, using the same conventions in regard to 
 the algebraic signs of the instantaneous values of magneto- 
 motive forces and fluxes in the several parts of the transformer 
 as are set forth in Art. 103 for three-phase electric circuits, 
 the following simultaneous equations represent the conditions : 
 
 m w — m x = cf) lc P, c - <j> x P x , 
 m w — m y = 4> ir P lc — 
 m x -m y = 4> U P X - <$> y P w , 
 
 flA + 4>x + = 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 527 
 
 Solving these equations gives the values of cf> w , (p x , 
 terms of m w , m x , m y , P w , and P x , as follows : 
 
 and <f) y in 
 
 , _ m w — m x P x (rn w - m y ) 
 
 P +2P P (P + 2P) 
 
 (1) 
 
 , _ m w — m y 2 (m w — »q) 
 
 P + 2P P +2 P 
 
 (2) 
 
 , m v -m x P x (m u -m w ) 
 
 P w + 2 P x PJP W + 2 P x ) 
 
 (3) 
 
 Putting m w = rt/sin a, 
 
 
 m x = al sin (« — 1 20°), 
 m y - al sin (« — 240°), 
 
 
 a being a constant, these equations become, 
 
 </>» = p a fsin a - sin (« - 120°) 
 
 w + 1 + PJP w \s\n a — sin (a — 240°)] } (4) 
 
 <f> x = p a \ p <2 sin (« — 120°) — sin (« — 240°) — sin (5) 
 
 4> v = Jsin(a- 240°)- sin(a - 120°) 
 
 w ^ r ~‘ x + P z /P u ,[sin(a — 240°) — sin a]|. (6) 
 
 The hysteretic angle of lead being zero under the assumption 
 made, these fluxes come to their respective maxima when 
 a = 90°, a= 210°, and a= 330°, and have the numerical values, 
 
 = O + A/A) (7) 
 
 w "4 x 
 
 cp 
 
 1 X 
 
 Sal 
 
 P. + 2P,' 
 
 ( 8 ) 
 
 These values are obtained on the assumption that sinusoidal 
 magneto-motive forces of equal amplitudes and differing suc- 
 cessively in phase by 120° are impressed by the coils in the 
 three branches of the magnetic circuit, and also on the assump- 
 tion that P x differs from P w and P y , but that the last two are 
 equal to each other. 
 
 It is obvious from these formulas, that with equal magneto- 
 motive forces, the value of dq is larger than the value of dq 
 and dq, in case P x is smaller than P w on account of the shorter 
 
528 
 
 ALTERNATING CURRENTS 
 
 length of the middle magnetic circuit. The maximum fluxes 
 <£>„,, d>„ are all equal to one another when P x = P w = P y ; 
 
 but if P x is negligible compared with P w , <t> z becomes twice as 
 large as and dq,. 
 
 Under the usual conditions of transformer operation, equal 
 voltages, 120° apart in their phases, would be impressed on the 
 three equal primary windings. Under these circumstances, 
 equal fluxes would be set up in the three branches of the 
 magnetic circuit, and equations (7) and (8), which give the 
 relations between the fluxes and exciting currents, show that 
 the exciting currents must differ under these circumstances 
 unless P x = P w = P v . Such a difference of currents causes an 
 unbalancing of the supply circuits. Also on account of the 
 difference of form of the magnetic leakage path around the 
 middle limb compared with the leakage paths around the two 
 outer limbs, unbalancing is felt in the secondary voltage. 
 
 To partially avoid this unbalancing, the middle leg may be 
 made of smaller cross section than the other two legs, so that 
 P x is made more nearly equal to P w and P y , but this is likely 
 to increase the core losses on account of the increased magnetic 
 density in that part of the core, and it also changes the power 
 factor of the exciting current of that leg compared with the 
 currents exciting the other legs. As the yokes which complete 
 the magnetic circuits at the ends of the limbs obviously carry 
 the same maximum flux as the limbs themselves, if magnetic 
 leakage is neglected, the yokes are usually made of the same 
 cross sections as the outer limbs. The same advantages in 
 respect to reduced weight and bulk and reduced exciting 
 current relate to this form of construction as to the construc- 
 tion with wye and delta magnetic circuits. 
 
 The magneto-motive forces of the three primary coils being 
 out of phase with each other, magneto-motive force pulsations 
 between the points 8 and T, and U and V are created which 
 tend to set up leakage fluxes between these points, which is in 
 addition to the ordinary magnetic leakage between the primary 
 and secondary coils. Pulsations of the third harmonic are most 
 prominent and are in the same phase and the same direction 
 through their respective cores.* Quite large additional third 
 harmonic currents may therefore flow around the mesh made by 
 
 * Art. 198. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 529 
 
 the primary coils if they are delta connected as shown in Fig. 294. 
 If, however, the coils are connected in wye without a neutral 
 return wire, these currents comprising the third harmonic can- 
 not flow and the resulting leakage magnetism is absent. If 
 the neutral point of the wye is connected to a neutral or fourth 
 wire of the supply system, the currents comprising the third 
 harmonic will flow through the three lead wires and coils in 
 parallel relation and complete their circuit through the neutral 
 wire. 
 
 When the three-phase transformer is of the shell type (in 
 which the windings are embedded in the iron), as in Fig. 
 295, and the primary windings are all connected in the same 
 relative direction, the result- 
 ant magneto-motive force act- 
 ing along the whole length of 
 the common core is 
 
 M a + M b + M c = 0, 
 
 where M A , M B , and M c are 
 the magneto-motive forces in 
 the coils A , B , and (7, in case 
 the magnetizing currents are 
 assumed to be sinusoidal, equal 
 in scalar value, and 120° apart 
 in phase. This means that 
 the vector sum of the mag- 
 neto-motive forces, as in the 
 
 previous case, equals zero, but Fig - 295. — Diagram of In-phase Shell 
 1 l . 1 . Type Transformer. 
 
 in this case any two are m 
 
 series instead of parallel with reference to the third ; also, 
 ® A + ® B + $c = °, 
 
 where d> A , d> 5 , and d> c respectively represent corresponding 
 vector values of the three magnetic fluxes. 
 
 The distribution of the magnetism in the core is as follows : 
 The flux in Y is the combination of the fluxes threading coils 
 C and B , and that in X the combination of the fluxes threading 
 coils B and A. As the vector fluxes threading coils C and B 
 are numerically equal, but differ in phase by 120°, the flux in Y 
 
530 
 
 ALTERNATING CURRENTS 
 
 is (neglecting magnetic leakage) equal to vector (7-flux plus 
 vector ,5-flux reversed. That is. the vector addition is made 
 in the same manner as for three-phase electric circuits.* There- 
 fore Y, and for the same reason X, must carry a flux equal 
 to V3 times the flux threading either of the coils. The maxi- 
 mum flux in the shell at W, Z, and H has the same value 
 (neglecting magnetic leakage) as the maximum flux threading 
 the coils. 
 
 When the middle coil B in Fig. 295 is reversed relatively in 
 respect to A and (7, the magneto-motive force formula becomes 
 
 M r = M A -M B + M c . 
 
 The vectors on the right-hand side of this formula are now 
 only 60° apart in phase and the scalar value of their resultant 
 M r — 2 M, where M is the scalar value of the vector magneto- 
 motive force in each coil. But the fluxes in all of the cores 
 must be equal in scalar value and 60° apart in phase in order 
 to produce the proper counter-voltages in the primary coils. 
 Therefore the flux due to JEf r , or 
 
 A> r = 
 
 where <3?^, <3> 5 , and <3> c are the fluxes threading the coils A, B. 
 and C. In order to maintain the proper counter-voltages in the 
 windings and vector sum of fluxes in the main core, flux d> A is 
 set up in the magnetic circuit AWITXA, <3> fi in BXHYB . 
 and <3> c in CYHZC. Fluxes <£> 4 and <t > B are 60° apart in phase 
 in the central core or tongue, so that they are 120° apart in 
 phase in X. The tensor of their sum is then equal to the scalar 
 value of either of the two. The same conditions exist in Y, so 
 that the combined cross section of iron at A, B, (7, W. X . Y, Z. 
 and H may be made uniform for the same induction in all parts 
 of the iron. There is thus some saving obtained in material, or 
 reduction in the reluctance of the core, by reversing the mid- 
 dle coil connections. The exciting current is the same in each 
 phase as would be required in three single-phase transformers 
 having equal core reluctances. 
 
 When the voltages or currents or both are unbalanced, the 
 conditions still hold that the magnetic flux for each phase must 
 
 * Art. 103. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 531 
 
 be such that it will produce the proper counter-voltage, and 
 hence the flux under each coil remains proportional to the 
 counter-voltage in that phase ; and the three fluxes, however 
 different in value, will combine vectorially according to the 
 same laws as where the circuits are balanced. 
 
 139. Some Constructive Features of Constant Voltage Trans- 
 formers. — The formula 
 
 ™ _ V2 Trn<t>f 
 1 10 8 
 
 contains the four quantities U v n , <E>, and /, which may be 
 varied, but in practice and f are usually fixed by the re- 
 quirements of the service, and only n and <1> may be varied. 
 Their relative variation is also limited by the fact that the 
 product n<& is fixed when JE X and f are fixed. In designing a 
 transformer, the relative values of n and <3> depend upon the 
 relative weights which it is considered desirable to assign to the 
 copper and iron in the transformer. When the magnetic density 
 which is safe to use in the core has been chosen, the cross section 
 of the core depends directly upon fl? ; while n, and therefore 
 the weight of copper, assuming that the size of wire to be used 
 has been decided upon, is inversely dependent upon the same 
 quantity. The weight of iron depends upon the cross section 
 and length of the magnetic circuit, and must be limited so 
 that the losses occurring in the iron of available quality shall 
 not reduce either the full load efficiency or the all-day efficiency 
 below a reasonable figure. 
 
 The number and length of the turns ( n ) of wire used in 
 each winding and the size of the conductors which are chosen 
 for the purpose of preventing injurious heating, objectionable 
 reduction of full load efficiency, or poor regulation, jointly 
 determine the required weight of copper. The density of 
 current in the windings varies widely, from 400 to over 2000 
 circular mils cross section per ampere. This density is largely 
 dependent upon the form of the coils. These are seldom 
 made so that the heat liberated by the 1 2 R loss must travel 
 more than one half an inch before reaching the radiating 
 surface of the coil. It is not unusual to make the density 
 somewhat smaller in the low voltage winding than in the 
 high voltage winding, and the values in the best transform- 
 
532 
 
 ALTERNATING CURRENTS 
 
 ers frequently fall between 1000 and 1500 circular mils per 
 ampere for the high voltage winding and between 1200 and 
 2000 for the low voltage winding. On the other hand, some 
 designers make the density of current greater in the low vol- 
 tage windings, while others make the density about the same in 
 both. As the high voltage conductors carry less current and 
 are therefore of smaller cross section than the low voltage con- 
 ductors, the insulation of the high voltage winding takes up 
 more space in proportion to the space occupied by the con- 
 ductors ; and this condition becomes emphasized when the vol- 
 tage exceeds a few thousands of volts, on account of the exag- 
 gerated thickness of insulation required for the high voltage. 
 This heavier insulation also makes it more difficult for the heat 
 generated to escape, and it is often necessary to divide the high 
 voltage winding into a number of thin coils separated by ducts 
 for the circulation of air or oil. 
 
 The I 2 R loss in transformers at full load is ordinarily equal 
 to from per cent to 3-|- per cent of the full load capacities. 
 This is divided with approximate equality between the primary 
 and secondary windings. Sometimes this loss is permitted to 
 reach 5 per cent, but in the better transformers it is more 
 often between 1 per cent and 3 per cent. The percentage of 
 loss which is allowable depends primarily upon the trans- 
 former efficiency and regulation which are desired, and secon- 
 darily upon the duty to be performed, the voltage, and the 
 frequency. 
 
 Transformers are of two general classifications, termed the 
 Core type and the Shell type, these designations depending upon 
 the manner in which the windings and the iron core are disposed 
 with respect to each other. Both forms seem to lend them- 
 selves satisfactorily to the design of transformers for ordinary 
 duty, and, in fact, there is no definite line of demarcation be- 
 tween them. In general terms, core type transformers may 
 be defined as those in which each of the principal branches of 
 the magnetic circuit is embraced by windings, as illustrated in 
 Figs. 47 and 296, while shell type transformers are those in 
 which the return branches of the magnetic circuit embrace the 
 windings, as illustrated in Figs. 300 and 301. 
 
 Laminations for the cores are made in various shapes. Fig- 
 ure 296 represents the core and coils of a core type transformer, 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 533 
 
 and Fig. 297 shows a pair of the lami- 
 nations or stampings of which the core 
 is made up. When the stampings are 
 of this form, they are built up into a 
 core within the iinished coils. In every 
 other layer of the core the stampings 
 are reversed end for end so that they 
 break joints. This is for the purpose 
 of reducing the magnetic reluctance 
 of the magnetic circuit to a minimum. 
 
 It is important that joints in the mag- 
 netic circuit shall be as few as possible, 
 and be so made as to be of low reluc- 
 tance, as otherwise the exciting current 
 may be caused to be of undesirable 
 magnitude. Rolled iron or steel sheets 
 as they come from the rolling mills are usually covered by a 
 tough and closely adherent coating or 
 thin scale of magnetic oxide of iron, 
 which is developed in the process of 
 manufacture of the sheets. This is a 
 poor electrical conductor, and since it 
 sticks to the stamping, it usually affords 
 sufficient insulation between the lamina- 
 tions to prevent eddy currents from becom- 
 ing excessive. However, to give double 
 assurance, it is usual to dip the stampings 
 into a liquid varnish or enamel and then 
 bake them to solidify the coat before build- 
 ing them into a core. 
 
 Fig. 296. — Core Type Trans- 
 former, Core and Coils. 
 
 Fig. 297. — Laminations In Fig. 298 is shown a leg of the core 
 
 for Transformer shown a i ar g e transformer which is composed 
 
 of rectangular laminations built into a 
 cross section having a cruciform shape. This is for a core 
 type transformer such as is shown 
 in Fig. 47. The complete core com- 
 prises two such legs, each covered 
 with windings, and connecting yokes 
 completing the magnetic circuit at former Core buiIt up of Rect . 
 their ends. The crossbars con- angular Laminations. 
 
534 
 
 ALTERNATING CURRENTS 
 
 necting the ends of the legs are also made of rectangular 
 laminations which alternately slip between and butt against 
 those on the leg shown, every other one on the leg being made 
 short to give space for this purpose, or the dovetailing may be 
 
 done by bunches 
 of laminations 
 instead of by al- 
 ternate sheets. 
 The type of coil 
 used for the wind- 
 ings of a trans- 
 
 Fig. 299. — Division of a Coil for the Transformer of which 
 Part of the Core is shown in Fig. 298. 
 
 former using the 
 
 cruciform core is shown in Fig. 299. It is composed of 
 a cotton-covered rectangular cojiper strip wound edgewise. 
 Where possible to insulate properly proportioned rectangular 
 wire, its use is often desirable, as it occupies less space for 
 a given effective area than round wire. It is also frequently 
 desirable to wind it edgewise, that is, with its greatest width 
 in the plane at right angles to the core, as this construction 
 often makes it practicable to make the complete coil with 
 only one layer of the conductor, thus affording opportunity 
 for efficient cooling, and the con- 
 struction may be made rigid and 
 strong. Bare strips are sometimes 
 wound in the edgewise coils, with 
 thin strips of vulcanized fiber, var- 
 nished paper, or varnished cambric 
 laid on edge between the turns of 
 conductor. 
 
 Figure 300 shows a shell type trans- 
 former with the laminations horizon- 
 tal. The wire in this construction is 
 wound with its flat faces parallel to 
 the axis of the coil, and the windings 
 are made up of a number of flat sec- 
 tions, each having a thickness equal 
 to the breadth of one conductor strip. 
 
 It is usual in the larger trans- 
 formers, — 50 Kva. capacity and over, — to split up the primary 
 winding into sections or divisions and sandwich between them 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 535 
 
 the secondary winding, which is also wound in sections. This 
 reduces magnetic leakage and, when free space is left between 
 the coil sections, facilitates the dissipation of heat by providing 
 opportunity for the circulation of air or oil. 
 
 Figure 301 shows a shell type transformer in which the 
 laminations and coils are vertical. In this case the coils are 
 made elongated as in Fig. 299, and therefore the sections of the 
 primary and secondary windings are sandwiched within one 
 another, while in Fig. 300 the sections are flat and wide and 
 lie side by side. 
 
 Fig. 301. — Shell Type Transformer with the Coils and Laminations Vertical. 
 
 From the illustrations it may be observed that core type 
 transformers usually consist of a simple rectangular magnetic 
 circuit with half of the primary and secondary turns wound 
 upon each of two legs, and the cross sections of the two legs 
 and the two end pieces are equal. Likewise it may be observed 
 that the core of a shell type transformer usually comprises a 
 central branch or leg upon which the windings are located and 
 
536 
 
 ALTERNATING CURRENTS 
 
 which is of double the cross section of the remainder of the 
 magnetic circuit if the latter is closed by two branches as illus- 
 trated in Figs. 300 and 301. 
 
 Figure 301 a shows the construction looking down on the core 
 
 and windings from the top, 
 
 Mica 
 Shields y 
 
 Secondary 
 
 Winding 
 
 OH Duel 
 
 'Oil Channels 
 
 Fig. 301 a. — Transformer 
 Type. 
 
 of Intermediate 
 
 of a transformer belonging 
 to a type now much used, in 
 which the windings surround 
 a central branch of the mag- 
 netic circuit, and the mag- 
 netic circuit is closed by 
 several limbs which join 
 radiating end pieces. This 
 arrangement lias proved ad- 
 vantageous for transformers 
 of small and medium capac- 
 ities, since it affords windings 
 in which the mean length of 
 turn is reasonably short and 
 at the same time affords a magnetic circuit of the requisite 
 cross section while permitting the use 
 of a moderate bulk and weight of iron. 
 
 This results in satisfactory regulation 
 associated with reasonably good all-day 
 efficiencies. The construction also 
 has some advantages in reducing the 
 total bulk of small transformers with- 
 out interfering objectionably with the 
 opportunities for getting rid of the 
 heat produced by the operation. It 
 presents a little more difficulty in the 
 mechanical construction of the core, 
 more particularly in respect to obtain- 
 ing a magnetic circuit of the lowest 
 practicable reluctance without undue 
 expense. 
 
 Figure 302 shows a tri-phase trans- 
 former of the shell type with three cores fig. 302. — Tri-phase Traus- 
 
 meeting at the center for carrying the former Wlth the Ma -' ienc 
 ^ J ° Circuits connected in V ye 
 
 coils of the three phases. It is com- Fashion. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 537 
 
 mon in this type of transformer to have the coils of the three 
 phases distributed on a single core, as in the diagram of Fig. 
 295. A core type tri-phase transformer may be arranged as 
 in the diagram of Fig. 294, and the reluctances of the three 
 magnetic paths can be made equal by arranging the limbs at the 
 corners of a triangle, with their magnetic circuits joined in 
 either delta or wye, as indicated in Figs. 292 and 293. 
 
 The quarter-phase transformer core may be built conven- 
 iently as shown in the diagram of Fig. 291, or it may be built 
 of the shell type, similiar to that shown in Fig. 295. 
 
 The various arrangements in which the iron of a transformer 
 may be disposed, which will give approximately the same results 
 in operation, are quite numerous, and the particular one to be 
 selected in any instance is apt to turn upon the question of the 
 cost of manufacture rather than the needs of service. The first 
 and most important element to be dealt with in the construc- 
 tion of a transformer, as in the case of most other electrical 
 machinery, is the disposition of the heat occasioned by the 
 iron and copper losses. Upon the solution of this problem 
 depends the rise of temperature of the transformer per kilovolt- 
 ampere of load when in operation, and its ultimate safe capac- 
 ity. The windings of transformers are usually embedded 
 more or less in the iron cores, as is illustrated in the preceding 
 figures ; and the whole transformer is inclosed in a waterproof 
 iron case, as illustrated in Figs. 47, 303, and 304. The 
 rise of temperature when in operation is due to the heating 
 of the structure by the core losses and the copper losses. If 
 transformers were not encased, but were placed naked in the 
 open air, the entire external surface could be assumed to be 
 effective in dissipating heat by radiation and convection ; but 
 on account of the inclosing case, convection of heat to the ex- 
 ternal atmosphere cannot take place directly, and all the heat 
 must be radiated to the wall of the case or carried thereto 
 through the poor heat conductors which are used to electrically 
 insulate the transformer from its case. The conditions there- 
 fore point to the conclusion that, without some special method 
 of cooling, for a given liberation of heat per square centimeter 
 of surface, the working temperature is likely to be higher in 
 transformers than in dynamo field windings. In fact, to pre- 
 vent excessive working temperatures and permit the highest 
 
538 
 
 ALTERNATING CURRENTS 
 
 safe output per pound of iron and copper, special means of cool- 
 ing are usually provided for transformers. These are described 
 later. 
 
 Transformer coils are usually designed to be lathe wound, 
 and these may be very effectually insulated by a liberal use of 
 mica, varnished cotton cloth, fiber, and wood. It is therefore 
 possible to safely run transformers with the windings at a con- 
 siderably higher temperature than dynamos. Sixty degrees 
 centigrade (108° F.) may be set as a maximum limit to the 
 safe rise in temperature caused by operation, though so high a 
 temperature is undesirable, both because of depreciation of the 
 insulating materials and the tendency of the core to age unless 
 a non-aging steel alloy is used. A high temperature limit has 
 also a marked disadvantage in causing an undue impairment 
 of the regulation when the transformer is hot, which results 
 from the increased resistance of the windings. Many trans- 
 formers still in operation exceed the temperature limit named, 
 but most of the best types do not exceed 40 ° centigrade rise. 
 As the rise in temperature also increases the electrical resist- 
 ance of the iron core, it decreases the eddy current loss, so that, 
 as suggested by Elihu Thomson, it was early considered 
 advantageous to have the core of a transformer operated at a 
 high temperature while the windings were kept cool. This 
 could not be conveniently arranged in small transformers, but 
 the cooling of the conductors of very large transformers has 
 been experimentally effected by making the conductors tubular 
 and passing a cool liquid through them. 
 
 It is practically impossible to fix any averages for the external 
 surface of transformers needed per watt lost in the core and wind- 
 ings, on account of the very varied arrangements of the coils with 
 reference to the core, and the effect of the containing case. In 
 good commercial designs of transformers the superficial area of 
 core and windings varies from 3 to 7 square inches per watt of 
 energy to be dissipated. For transformers of small and medium 
 capacities it is usual to make the design as compact as possible, 
 and no particular trouble from heating is experienced if the 
 losses are not excessive when judged from the criteria of 
 efficiency and regulation, provided the inclosing case is filled 
 with oil for the purpose of improving the facility with which 
 the heat can escape to the outer surface of the case. A highly 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 539 
 
 refined petroleum oil carefully filtered and dried is used for this 
 purpose, and is commonly called Transformer oil, on account of 
 the principal object of its use. Such oil also posesses high insu- 
 lating qualities, which is an advantage in the transformers. 
 Transformers with which oil is associated for cooling purposes 
 are called Oil-cooled transformers. Those without oil are some- 
 times called Dry-core transformers. As transformers increase 
 in size, the weight per unit of capacity decreases, and the 
 superficial area per unit available for dissipating heat decreases 
 at a still more rapid rate, since the bulk and weight are pro- 
 portional to the cube of the linear dimensions, and the super- 
 ficial area is proportional to only the square of the linear 
 dimensions ; and some device, such as circulating the oil within 
 the case and through ducts in the core and between the divisions 
 of the windings, or blowing air through ducts in the core, must 
 be provided for forcing the more rapid conveyance of heat from 
 the core and windings. Otherwise the heating would be ex- 
 cessive unless the transformer is of unreasonably great and ex- 
 pensive bulk. Dry core transformers, arranged for cooling by 
 a blast of air, are called Air-cooled transformers. 
 
 A small transformer will evidently cool more readily than 
 a large one of the same type and design. Thus, in a number of 
 transformers of different capacities having the same efficiencies, 
 losses per pound of copper and iron, and voltages, and of similar 
 design, if we let A stand for linear dimensions, A for superficial 
 area, W for weight, K L for full load losses, and Kva for rated 
 full load output, we have 
 
 TFxA*; ioci 2 ; 
 
 K l x W x A 3 x A 2 , approximately ; 
 
 4. 4 
 
 Kva oc A 4 oc IF 3 x x A 2 , approximately. 
 
 It is evident that Kva varies approximately as A 4 , since when 
 the dimensions are varied the cross section of the magnetic cir- 
 cuit and hence the magnetic flux is varied as A 2 . With fixed 
 voltage, the number of turns in the windings therefore changes 
 
 proportionately to ^ and the space in which the windings go 
 
 changes proportionately to A 2 . Hence with winding space 
 
540 
 
 ALTERNATING CURRENTS 
 
 changed in proportion to L 2 and number of turns in pro- 
 
 portion to — , the area of each conductor may be changed in 
 
 proportion to X 4 , and its current-carrying capacity correspond- 
 ingly changed, provided the necessary insulating space can be 
 reserved and the heat produced by the PR losses at full load 
 can be carried off. 
 
 It is seen, therefore, that transformers of large sizes should 
 have normally less weight per kilowatt capacity, full load losses 
 lower in proportion, and higher efficiency than those of smaller 
 size, — which deductions are, in fact, borne out in practice. 
 But the dissipation of heat in the larger transformers is more 
 difficult, since K L oc X 3 oc AK and therefore the area exposed for 
 cooling in similar transformers of different sizes varies only 
 with the f- power of the losses, when the losses are taken to be 
 proportional to the weights of the transformers, and it also 
 varies as the square root of the rated transformer capacity 
 given in kilovolt-amperes. Hence it is evident that if a small 
 transformer of a given form heats at full load to the maximum 
 temperature allowed, say 40° centigrade, a transformer of ten 
 times the size but equivalent in form and all other constants 
 of the design will heat to a much higher temperature ; for, 
 using primed symbols to represent the larger transformer, 
 
 A _ (Kvay* _ ( Kva )- _ _J_ 
 
 A ' ~ ( K'vaf- 10* (Kvaf- 3 - 2 ’ 
 
 nnr1 k l _ ( Kva)* __ (Am)’ _ 1 
 
 K l ( K'vcCy 10 *(Kvay 5 - b 
 
 whence ^ = 1 : If* 
 
 A A 
 
 That is, the full load losses of the larger transformer are about 
 three quarters greater per square inch of the superficial area of 
 the transformer, although the losses are only about half as 
 great per kilovolt-ampere of rated capacity. Had the size 
 been 100 times increased, the rate of liberation of heat would 
 have been over 200 per cent greater per square inch of radiating 
 surface. Therefore, transformers of large size require either 
 the application of special measures for carrying away the heat 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 541 
 
 generated, or they must be given special designs which would 
 prove bulky and expensive in sizes of 50 Kva. capacity or 
 over. 
 
 Several methods for cooling are in use : 
 
 For very small transformers — a fraction of a kilowatt capacity 
 — no special arrangements are needed. 
 
 For transformers from a kilowatt capacity to those of several 
 hundred kilowatts capacity the most common method is to 
 immerse the transformer in a tank of insulating oil. For the 
 smaller sizes this is not necessary, but it is desirable, as the oil 
 tends to maintain the insulation and 
 close up punctures. The tanks for 
 smaller sizes of transformers, say to 
 about 50 Kva. or less capacity, are 
 frequently made of cast iron with a 
 smooth outer surface, as shown in 
 Fig. 47 ; while those of larger capacity 
 are made of cast iron or riveted 
 boiler plate with a corrugated surface 
 in order to present more radiating 
 surface, as shown in Fig. 303. The 
 action of the oil is to circulate up- 
 ward through ducts in the warm core 
 and over the surface of the warm coil 
 sections and downward alongthe cooler 
 interior of the the tank or case. The 
 coils and cores are so arranged that 
 the oil can readily circulate through 
 them ; thus in Fig. 298 the cruciform 
 shaped core leaves triangular channels 
 through which the oil can flow between 
 the core and the cylindrical inner face 
 of the coil. Where necessary, channels or ducts are left between 
 each 1|- or 2^ inches of thickness of the laminations. In trans- 
 formers of large size space is left for the oil to circulate freely 
 between sections of the coils ; thus in Fig. 302 is shown a trans- 
 former intended for oil cooling in which the coils are so 
 separated. 
 
 The radiation of the heat from the case is commonly at the 
 rate of one watt (joule per second) to each 4 to 8 square inches 
 
 Fig. 303. — Transformer Case 
 having a Corrugated Surface 
 for increasing the Area for Heat 
 Dissipation. 
 
542 
 
 ALTERNATING CURRENTS 
 
 of surface when the case temperature is from 35° to 40° centi- 
 grade above the temperature of the surrounding air. 
 
 In the case of large transformers, from a couple of hundred 
 to several thousand kilowatts capacity, it is found necessary to 
 artificially cool the oil, as the case will not radiate heat to the 
 surrounding air fast enough to keep the temperature down to a 
 reasonable value. In this case it is common to circulate water 
 in a spiral of pipes located inside of the case above the height 
 of the transformer proper. Such a transformer is said to be 
 Water-cooled. The arrangement is illustrated in Fig. 304, which 
 represents the design of a water-cooled transformer, the cooling 
 coils being at AA. This figure shows one of the ways for mak- 
 ing an oil-proof case of riveted or welded boiler plate, which in 
 this instance does not need to be corrugated since the major part 
 of the cooling is accomplished by the circulating water which 
 flows in the pipes. The figure also shows means which may 
 be used for supporting the cores, coils, and terminals, and for 
 draining the oil from the tank. The latter is accomplished by the 
 pipe containing the valve B. which leads to a suitably protected 
 receiving tank. This is most important in order to prevent dis- 
 astrous results in case the oil in the transformer catches fire. 
 Other constructive details are also illustrated in the figure. The 
 water pipes in this case are covered with cotton tape above 
 the oil to prevent the collection and drip of moisture. 
 
 When a water-cooled transformer, such as is illustrated in 
 Fig. 304, is in operation, the oil in contact with the core and 
 windings becomes warm and tends to rise, while the oil cooled 
 by contact with the cooling pipes tends to fall. This sets up a 
 circulation of oil upwards through the ducts and over the sur- 
 face of the transformer into contact with the cooling pipes, then 
 the cooled oil falls to the bottom of the case through the ample 
 space left around the transformer. In this manner heat is 
 carried away from the transformer and delivered to the cooling 
 pipes. The cooling water may be caused to circulate by a 
 pump and some means utilized to cool it after its exit from the 
 cooling coil. The water may thus be used over and over. For 
 transformers of very large sizes it is sometimes desirable to 
 also give the oil a forced circulation. In this case, oil is 
 drawn from the transformer by means of a pump, passed 
 through a surface condenser or cooler from which the heat is 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 543 
 
 abstracted by water, and returned to the transformer under 
 pressure, where it is compelled by the design to circulate 
 through the core and windings before again reaching the outlet. 
 
 A good method, and one of the oldest, of cooling indoor trans- 
 formers of low or medium high voltage is by means of a blast 
 
544 
 
 ALTERNATING CURRENTS 
 
 of air. The air blast is produced by a fan, and directed to the 
 transformer by way of an air chest under the transformer case, 
 thence through the core and windings, and thence out of the 
 top cover of the case. The windings and core are designed with 
 ducts to give as free circulation of air as possible. A large 
 number of air-blast transformers can be located over one air 
 chamber and cooled effectively and economically in this way. 
 
 Figure 305 shows the rise of temperature of the conductors 
 of an old type air-blast transformer of 200 kilowatts capacity as 
 a function of the period of operation, operated with the blast 
 
 Fig. 305. — Curves showing the Effects of an Air Blast in Cooling Transformers. 
 
 and without the blast. Point D shows the temperature of the 
 stampings after the transformer has been operated seven hours 
 with blast on; curve A shows the temperature of the windings 
 when operated at full load without blast; curve B shows the 
 temperature of the windings when operated at full load with 
 blast of 1040 cubic feet per minute; and curve C shows the 
 temperature of the air issuing from the transformer. 
 
 The insulation in transformers is composed largely of the 
 materials described for generator insulation,* but materials 
 which deteriorate in either hot or cold oil cannot be used in 
 oil-cooled or water-cooled transformers. The usual substances 
 used for the insulation between the cores and coils are mica and 
 
 * Art. 36. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 545 
 
 varnished cotton cloth or paper, with the addition of blocks of 
 wood or press board where large volumes of insulating materials 
 are demanded. The wire is usually double cotton covered, and 
 varnished cloth is used between the layers. The voltage 
 between layers seldom exceeds 200 volts. In fact, 200 volts 
 per layer is high. Each winding is divided into thin sections, 
 giving a total of not over 3000 or 4000 volts, which figures are 
 also rather high for the best construction. These coil sections 
 are thoroughly impregnated, usually by using a vacuum process, 
 with linseed oil, asphaltum, or compounds derived from coal 
 tar. It is evident that in transformers for high voltages, such 
 as from 60,000 to 100,000 or more volts, the insulation between 
 the cores and coils must have very high dielectric strength so 
 that much mica is employed. The insulation between coil sec- 
 tions is largely accomplished by the cooling oil which surrounds 
 them, as the sections are kept apart by narrow separators of 
 wood or pressboard. 
 
 The terminals of a transformer are troublesome to insulate, 
 as they are not in oil, and especially since high disruptive forces 
 are apt to be set up at those points upon opening or closing 
 circuits. Fig. 306 shows at A a pair of 66,000 volt leads for 
 a transformer located under roof. The 
 wires themselves are insulated to many 
 times their diameter and are bushed in 
 porcelain where they pass through the 
 iron cover plate of the transformer. 
 
 At B is shown a lead, for a similar 
 voltage, to be used in a transformer 
 intended for use out of doors. The 
 multiple petticoated porcelain insulator 
 is for the purpose of effectively disposing 
 of rain water. 
 
 A desirable form of terminal is now 
 manufactured with alternate layers of 
 insulating material and tin foil, the 
 function of the latter being to distribute the electric stress uni- 
 formly through the dielectric.* 
 
 The oil in transformers should withstand a breakdown test 
 of about 30,000 volts when the voltage is applied between 
 
 * Trans. A. I. E. E., Vol. 28, p. 209. 
 
 2 N 
 
 Fig. 306. — Terminals for 
 06,000 Volt Transformer. 
 A, for Indoor Service. B, 
 for Outdoor Service. 
 
546 
 
 ALTERNATING CURRENTS 
 
 PRIMARY SIDb 
 
 SECONDARY SIDE 
 
 electrodes having spherical ends of .5 centimeter diameter 
 placed .4 centimeter apart. It should be free from moisture 
 and other foreign substances, such as materials containing 
 sulphur, alkali, or acids; should be a mineral oil manufactured 
 by the fractional distillation of petroleum; and should be very 
 fluid and neither solidify or form wax at 0° centigrade. Its 
 flash point should be at least 170° centigrade.* The removal 
 of moisture which may have got into otherwise satisfactory oil 
 may be accomplished by filtering the oil through common soft 
 blotting paper. It may also be accomplished by heating the 
 oil to a temperature above 100° centigrade and stirring it. 
 
 140. Methods of connecting Constant Voltage Transformers 
 and some Features of their Operation. — Single-phase trans- 
 formers can be connected to single phase circuits either to raise 
 or lower the voltage, as shown by the upper or lower trans- 
 formers in the dia- 
 gram of Fig. 307. 
 If the three-wire 
 system is used in 
 the secondary cir- 
 cuit, two trans- 
 formers in series 
 can be used, but 
 
 more commonly the neutral wire is tapped to the middle of 
 the secondary winding, as shown by the dotted line in Fig. 
 307. Lighting transformers generally have their secondary 
 windings divided into two coils with their terminals brought 
 out to terminal blocks so that the three-wire connection can 
 be readily made. 
 
 Two or more transformers are sometimes connected with 
 their secondary windings in parallel on the same secondary cir- 
 cuit ; in which case it is necessary that they shall have equal 
 ratios of transformation, and that their internal resistances and 
 reactances shall be inversely proportional to their respective rated 
 capacities, or they will not divide the total load proportionally 
 to their rated capacities. Thus, consider two transformers 
 designed for equal primary and no-load secondary voltages and 
 equal rated capacities, but having their total internal impedances, 
 resolved in terms of secondary circuit equivalents, in the ratio 
 * U. S. Government Specifications. 
 
 Fig. 30T. — Diagram showing Connections of Transformers. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 547 
 
 of the vectors Z t and AZ t , where A is a constant, the trans- 
 formers being connected to the same primary circuit. The 
 
 currents in the two are respectively proportional to ~ and — . 
 
 Z t AZ t 
 
 The load on one transformer is therefore less than that on the 
 other transformer in the ratio of Z t : AZ t = 1 : A. Moreover, 
 the secondary terminal voltages may not be exactly in the same 
 phase, and an undesirable cross current will flow, so that the 
 arithmetical sum of the currents carried by the two transformers 
 
 X 
 
 is greater than the current in the load, unless — ^ is equal for the 
 
 ti t 
 
 two transformers. 
 
 If the ratios of transformation on open circuit are different 
 in the ratio of E. 2 + B to E v where E 2 is the secondary ter- 
 minal voltage of one transformer, the transformer having the 
 lower secondary voltage absorbs power from the other until the 
 total load becomes great enough to cause the terminal voltage 
 of the latter to drop to E 2 volts. With further increase of 
 load the first continues to carry the load required to bring its 
 voltage to E 2 and divides further increases with the second 
 transformer in the proportion stated in the preceding paragraph. 
 If the internal impedances are both very low compared to the 
 load impedance, differences in the division of load are little 
 noticeable, but it is always wise to keep these conditions in 
 mind when designing transformers for manufacture or selecting 
 them for use. 
 
 When secondary circuits of transformers fed from one primary 
 circuit are to be connected in parallel it is, of course, essential 
 that they be connected in the proper relation so that they 
 will not short-circuit upon each other. It is thus necessary to 
 know or determine the Polarity of the windings. The relative 
 polarity of two transformers may be obtained by connecting up 
 one transformer, and then, after attaching the primary terminals 
 of the second to the primary line, connecting one of its second- 
 ary terminals directly to a secondary line conductor and the 
 other secondary terminal to the other secondary line conductor 
 through a voltmeter. If the transformers are connected in 
 series instead of parallel relation, the voltmeter will read 
 double the normal secondary voltage. A light fuse may be 
 used instead of the voltmeter, and the blowing of the fuse will 
 
548 
 
 ALTERNATING CURRENTS 
 
 denote wrong connections. Figure 807 a shows correct connec- 
 tions for two transformers in parallel. Figure 307 b (1 and 2) 
 
 show incorrect connec- 
 tions for parallel opera- 
 tion, as the transformers 
 in each diagram are 
 short-circuited on each 
 other. Figure 30 7 c 
 shows the connections 
 whereby a three-wire, 
 may be obtained by means 
 two-wire primary circuit 
 
 Fig. 307 a. — Correct Connections for Two 
 Transformers in Parallel. 
 
 circuit 
 from a 
 
 
 1 
 
 
 1 
 
 CD 
 
 single-phase, secondary 
 of two transformers 
 when a middle con- 
 nection to the second- 
 ary winding is not avail- 
 able. 
 
 The transformation 
 of voltage in polyphase 
 circuits may be com- 
 passed by using single- 
 phase transformers in 
 groups. A quarter- 
 phase circuit then re- 
 quires two transformers 
 at each point of trans- 
 formation, and a tri- 
 phase circuit requires 
 either two or three 
 transformers. The individual transformers must each have a 
 capacit)' equal to the power required to be transformed by 
 
 it divided by the 
 power factor of the 
 part which it sup- 
 plies. As the power 
 factor of an incan- 
 descent lamp load 
 by is practically unity, 
 and since circuits 
 supplying induction motors are likely^ to have full load power 
 factors not higher than 0.8, and further since when such motors 
 
 Fig. 307 b . — Incorrect Connections for Two Trans- 
 formers in Parallel. 
 
 Fig. 
 
 307 c. — Three-wire Secondary Circuit supplied 
 Two Transformers. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 549 
 
 are only partly loaded the power factor is much decreased, it is 
 evident that transformers which supply currents to motors must 
 be of greater kilovolt-ampere capacity than those which supply 
 equal power to incandescent lamps. This rule applies equally to 
 single-phase and polyphase circuits and it is important to bear in 
 mind when purchasing transformers that their capacities should 
 
 Fig. 308. — Diagram of Transformer Connections for a Three-wire 
 Quarter-phase System. 
 
 GROUP 1 
 
 be suitable for the kilovolt amperes of the load to which they are 
 to be connected. Figure 308 represents the connections for a 
 quarter-phase circuit when a common return wire is used in 
 both primary and secondary circuits. In the case of a four- 
 wire quarter-phase circuit, the transformers may be kept inde- 
 pendent, one single-phase transformer ABC — 
 being provided for each phase of the 
 circuit. A four-wire circuit may be 
 used for either the primary or second- 
 ary circuit and a circuit with common 
 return for the other by associating 
 the four-wire connection with one 
 side of two transformers and the 
 three-wire connection of Fig\ 308 
 with the other side of the trans- 
 formers. In Fig. 309 are shown the 
 usual connections utilized for groups 
 of three single-phase transformers 
 for three-phase transformation. The 
 figure shows three groups each of 
 which comprises three transformers. 
 
 In Group 1 the primary windings ABC' 
 and the secondary windings of the Fig. 309. — Diagram showing 
 
 transformers are both connected in Ways of Connecting ihree 
 
 Single-phase Transformers to 
 
 A arrangement. In Group 2 the Tri-phase Lines. 
 
 GROUP 2 
 
 GROUP 3 
 
550 
 
 ALTERNATING CURRENTS 
 
 primary windings and the secondary windings of the trans- 
 formers are both connected in Y arrangement. In Group 3 the 
 primary windings of the transformers are connected in A ar- 
 rangement and the secondary windings are connected in Y 
 arrangement. The third arrangement may be reversed so as to 
 connect the primary windings in Y and the secondary windings 
 in A. 
 
 When transformers are to be connected in any of these 
 arrangements, the terminals must be joined correctly according 
 to the polarities, or the voltage phases will be unbalanced in 
 the secondary circuit. Thus, when the secondary coils are 
 properly connected and the system is balanced as to the vol- 
 tages. the three coil-voltages may be laid out graphically as a sym- 
 metrical phase diagram, 
 as shown by the vectors 
 represented by OA, OB , 
 and 0(7, in Fig. 310. 
 
 By reversing the con- 
 nections of one of the 
 windings of one of the 
 transformers in a Y ar- 
 rangement, its secondary 
 voltage is reversed. For 
 instance, reversing the 
 transformer giving the 
 voltage OB of Fig. 310 
 gives the voltage vector 
 OJD of the diagram, which, 
 using the relations shown in the figure, lies 60° behind OA 
 and 60° ahead of 0(7 instead of in the proper position 120° 
 ahead of OA and behind 0(7. The voltages between the sec- 
 ondary line wires A and B , B and C, and (7 and A are there- 
 fore changed to the values and vector relations shown by the 
 lines AT), DC , and CA in the vector triangle ADC , instead 
 of their balanced relations for the proper connections shown 
 by A B, B (7, and CA. The diagram shows by inspection that 
 the voltages AD and DC are numerically equal to the wye 
 voltage CO of the circuit, but CA is equal to V3 times the 
 wye voltage. The angles between AD , DC , and CA are re- 
 spectively equal to 60°, 150° and 150°. 
 
 Fig. 310. — Diagram of y-connected Transformer 
 Voltages in a Tri-phase Circuit to show the Effect 
 of reversing the Connections of One Winding. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 551 
 
 C B' 
 
 Fig. 310 a. — Diagram of A-connected Transformer 
 Voltages in a Tri-phase Circuit to show the Effect 
 of reversing the Connection of One Winding. 
 
 When the secondary coils are connected in delta, the diagram 
 of Fig. 310 a can be applied, but now OA, OB , and OC repre- 
 sent, not only the coil voltages, but also the line voltages. 
 Hence, when the con- 
 nection of the wind- 
 ing of transformer 
 B is reversed, the 
 voltage between B 
 and C is reversed and 
 the vector diagram 
 of voltages AB , B' C 
 ( = — BO) and OA 
 does not close. This 
 leaves an uncompen- 
 sated voltage B' B to make a circulating current in the mesh, 
 which would burn out the transformers. 
 
 In a wrongly connected wye system, unbalanced voltages are 
 impressed on the secondary circuit which would ordinarily lead 
 to the flow of unbalanced currents in both primary and secondary 
 circuits, and if maintained would cause inconvenience to 
 users of apparatus of which the load might be composed. 
 In a wrongly connected delta system the transformers are jeop- 
 ardized by a circulating current caused by the resultant vol- 
 tage in the mesh. This is generally obviated by use of proper 
 fuses and other circuit breakers, but it is well to guard against 
 such dangers by making polarity tests before cutting the trans- 
 formers into circuit. 
 
 It must be observed that the phases of the voltages between 
 three-phase secondary line wires are opposite to the voltages 
 impressed between the primary line wires when both primary 
 and secondary windings of the transformers are connected both 
 either in delta or in wye arrangement. This is not true, how- 
 ever, when the primary windings are connected in one arrange- 
 ment and the secondary windings are connected in the other 
 arrangement, as this results in a displacement of the secondary 
 voltage phase 30° from the position of opposition, for the reasons 
 heretofore explained.* The line voltages are shifted forward 
 30° when going through the transformers from delta to wye 
 and are shifted backward 30° when going through the trans- 
 
 * Art. 100. 
 
552 
 
 ALTERNATING CU RR ENTS 
 
 formers from wye to delta. For this reason groups of three 
 transformers operating in three-phase circuits cannot be con- 
 nected in parallel on both the primary and secondary circuits 
 unless due consideration is given to the manner of their con- 
 nections. Transformers connected delta primary and delta 
 secondary and transformers connected wye primary and wye 
 secondary can be paralleled on the secondary side with like 
 groups and with each other, when operated from the same pri- 
 mary circuit; but they cannot be paralleled with transformers 
 connected delta primary and wye secondary nor with trans- 
 formers connected wye primary and delta secondary. Each 
 of the two last-named arrangements may be paralleled with 
 like groups, but neither may be paralleled with the other. 
 
 When the connection of the windings is delta primary and 
 delta secondary or wye primary and wye secondary, it is mani- 
 fest that the ratio of primary and secondary voltages at no load 
 is equal to the transformation ratio of the individual trans- 
 formers, but tins is not true when the connections of primary and 
 secondary windings differ. When the arrangement is delta 
 primary and wye secondary, the wye voltage of the secondary 
 circuit is equal to the primary impressed voltage divided by the 
 ratio of transformation, and in a balanced circuit the secondary 
 line voltage is V3 times as great. When the arrangement is 
 Avye primary and delta secondary, the secondary line voltage 
 is equal to the primary wye voltage divided by the ratio of 
 transformation. The primary line voltage is V3 times the pri- 
 mary wye voltage, and the secondary line voltage is therefore 
 1/V3 as great in a balanced circuit as would be given by divid- 
 ing the primary line voltage by the ratio of transformation of 
 the transformers. 
 
 In Fig. 311 is shown a diagram for connecting two trans- 
 formers on a tri-pliase circuit in what is sometimes called the 
 Open delta or Vee connection. The two arrangements shown 
 are alike in result, but the transformers are attached to different 
 phases. The secondary voltages of the respective transformers 
 are represented in Fig. 312 by the lines AB and CA< cor- 
 responding with the connection shown in the upper part of the 
 figure, while the distance BO represents the resulting secondary 
 line voltage that is maintained jointly by the two transformers 
 between the wires B and C. When three transformers are con- 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 553 
 
 nectecl in delta arrangement and one becomes disabled so as to 
 be disconnected from the circuit, a large part of the full load 
 can still be carried by the two transformers a b C 
 that remain in V connection. With three 
 transformers connected in the delta arrange- 
 ments the total power delivered to the cir- 
 cuit is equal to V3 cos # 2 , and each 
 transformer delivers one third of this power, 
 
 E„Ir, COS Or, 
 
 or 
 
 2 2 
 
 V3 
 
 After the circuit of one of 
 
 the three transformers has been opened, as- 
 suming the load to remain the same, each 
 of the remaining transformers must deliver 
 
 VBAJgig cos 0 2 ^ to the receiving circuit. 
 
 Since the current in each line wire remains 
 unchanged as does the voltage between line 
 wires, it is evident that in the transformer 
 coils there will be additional angular dis- 
 placement corresponding to an angle of 
 v/3 
 
 cos _1 - — , that is, ± 30°. The total phase dis- 
 
 Fig. 311. — Connections 
 for using Two Trans- 
 formers on a Tri- 
 phase Circuit. V Con- 
 nection. 
 
 SC' 
 
 placement in the two transformers is then re- 
 spectively # 2 + 30° and 0 2 — 30°. From this it will be seen that 
 the current in each transformer increases in a greater proportion 
 than does the power delivered by the trans- 
 former. Hence when two transformers are op- 
 erating in open delta (i.e. V connection) on a 
 tri-phase circuit the transformers will not 
 (without being overloaded) deliver their full 
 rated capacity in kilowatts even when supply- 
 ing power to a non-reactive load. Thus, three 
 transformers each of 200 Ivva. rated full load 
 capacity being delta connected to a tri-phase 
 showing Voltages circuit and delivering a total of 600 kilowatts 
 
 of Secondary to a balanced receiving' circuit having unity 
 Phases of a Tri- ° J 
 
 phase System when P ower factor, when one of these transformers 
 
 Two Transformers has its secondary circuit opened by accident 
 are used, and the eac p 0 f tq-, e two remaining transformers will 
 Wrong Connections, deliver 300 kilowatts to the receiving circuit, 
 
554 
 
 ALTERNATING CURRENTS 
 
 but the current in each transformer will be out of phase 
 with its secondary voltage by an angle of 30° ; hence, the kilo- 
 volt amperes delivered by each transformer must be equal to 
 
 V3 
 
 300 h — — = 346 Ivva. and the transformers are 73 per cent over- 
 loaded. This is but one of many possible transformer combi- 
 nations which result in a phase displacement between voltage 
 and current in the transformers ; and in every case when such 
 a combination is made the kilowatt output of the transformers 
 is less than the sum of the kilovolt amperes on account of the 
 internal phase displacement of the current. 
 
 If the connection of either the primary or the secondary 
 winding of the transformer giving voltage CA in Fig. 312 is 
 reversed, the three secondary line voltages become AB , BC, 
 and C'A. The last is equal to — CA. This is a workable com- 
 bination because the third side of the vector diagram BC is not 
 fixed in position or magnitude by its own transformer, as it is 
 in the case of delta connection of three transformers ; but it 
 gives unbalanced secondary voltages. The diagram shows by 
 inspection that the voltage BC is numerically V3 times vol- 
 tages AB and C'A, and the angles between the voltages AB , 
 BC , and CA are 150°, 150°, and 60°. This arrangement is 
 sometimes used to get a 60° phase difference. 
 
 A substitute for the V connection may be made with two 
 transformers, in which the windings of 
 
 Ci 
 
 0 
 
 B, 
 
 Ax 
 
 a/vvWa Wvw\ 
 
 B,. 
 
 one transformer possess ^ times as many 
 
 turns as the windings of the other. In 
 this case, one end of each winding of the 
 former is connected to the middle of the 
 corresponding winding of the latter, as 
 illustrated in Fig. 313. This is called 
 the Tee connection of transformers. The 
 windings of fewer turns which are pos- 
 sessed by one transformer are sometimes 
 called Teaser windings. The potential of 
 A v the line end of the primary teaser 
 winding, is fixed by the tri-phase relations 
 of the line voltages, while the potential of the other end is fixed 
 by being midway between the potentials of B x and C v by reason 
 
 ) \ 
 
 LOAD / ' 
 
 Fig. 313. — Tee Connection 
 of Transformers for Tri- 
 phase Circuit. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 555 
 
 of the connection of the teaser at the mid-point 0. The vector 
 diagram of voltages is given in Fig. 314, in which BO is the 
 voltage from B x to 0 of Fig. 313, 00 is the voltage from 0 to 
 O v and OA is the voltage from 0 to A v The line voltages are 
 represented by AB , BO, and CA. The tensor value of voltage 
 
 V3 
 
 OA is — - times that of either line voltage. The voltages of 
 2 
 
 the secondary circuit are proportional to those of the primary 
 circuit, by reason of the correspondence of the pri- a 
 
 mary and secondary windings and, therefore, are 
 also represented by the vectors of Fig. 314. That 
 is, the secondary line voltages must be proportional 
 in magnitude and angular relations to the vectors 
 represented by AB, BO, and OA of the figure. 
 
 Assuming the system to be balanced, the line Fl g G ran fo{ v7itages 
 currents flowing in the two halves of the coil produced by 
 B 1 0 1 or B 2 C 2 are numerically equal to each other arranged 0 in* Tee 
 and are 120° apart in phase. At 0 they add Connection, 
 together vectorially to form the line current which flows 
 through the teaser, which is numerically equal to each of the 
 others and 120° in advance of one and 120° behind the other, 
 in accordance with the law that the vector sum of the currents 
 at a junction must be zero. The kilovolt-auqjeres capacity of 
 the transformers manifestly must be greater than the kilowatt 
 output even when the load is non-reactive. 
 
 The T connection of two transformers for tri-phase circuits 
 possesses the disadvantage, compared with the V connection, that 
 the two transformers cannot be alike unless part of the winding 
 of one is allowed to be idle and its capacity is therefore sacri- 
 ficed ; but the T connection has a countervailing advantage 
 in the fact that the system is not so likely to become unbalanced 
 as when the V connection is used. 
 
 The relative transformer capacities required when the delta, 
 wye, vee, and tee connections are used for three-phase to three- 
 phase transformation may be determined by the following rela- 
 tions: For delta and Avye, the capacity of each of the three 
 
 transformers must be equal to one third of the total load in 
 kilovolt amperes. This, for any power factor, is equal to E r l 
 
 = EI± = — - El (where E r , E, I, and are the wye voltage, 
 
 V3 
 
55G 
 
 ALTERNATING CURRENTS 
 
 line voltage, line current, and delta current of the circuit), for 
 in the delta the primary coil current equals — ^ I and in the 
 
 wye the primary coil voltage equals E. The three required 
 
 Vb 
 
 transformers then have a combined kilovolt ampere capacity 
 of V3 El. If the windings are delta-connected, they must be 
 designed to produce full line voltage and carry a current equal 
 to the line current divided by V3, while if the windings are 
 wye-connected they must be designed to produce a voltage 
 equal to line voltage divided by V3 and to carry full line cur- 
 rent. In the tee connection the coils carry the same currents 
 as the line wires, but the voltage of one transformer having the 
 
 teaser winding is onlv 
 
 V3 
 
 times as large as the line voltage. 
 
 to El , while that of the main transformer must be equal to 
 
 The kilovolt ampere capacity of the teaser transformer is equal 
 V3 
 2 ~ 
 
 El. The combined capacity of the two transformers required 
 for an output of V3 El kilovolt amperes is therefore — El 
 
 In the vee connection the voltage in the primary windings of 
 each of the two transformers is E and the current I. The cur- 
 rents entering the windings at the outer ends of the vee are 
 equal to the two corresponding line currents, and these vectori- 
 ally add at the apex at a phase angle of 120°, which gives an 
 equal current for the third wire. Hence the combined capacity 
 of the two transformers should be 2 El. 
 
 It should be understood that in the case of the tee connection 
 or the vee connection the relations of the currents and voltages 
 in the transformers so connected are not necessarily the same 
 as the phase relations between line currents and line voltages. 
 In the case of the tee connection the current in the teaser 
 winding has the same angular relation to the voltage of that 
 winding as the line current lias to the line voltage, but as pre- 
 viously explained in this Article there is an inherent phase 
 displacement of the currents in the two halves of the main 
 transformer winding irrespective of the power factor of the 
 line. In the case of the vee connection the currents in both 
 transformers are out of phase with the respective transformer 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 557 
 
 voltages as explained earlier. Because of this condition the 
 total kilovolt ampere capacity of the transformers connected 
 in tee and in vee must be in excess of the total kilovolt amperes 
 supplied by the line or to the load. 
 
 When the three transformers are connected in wye in a tri- 
 phase system, a fourth wire may be connected to the neutral 
 point or common junction point, and part or all of the load 
 may be connected from the three independent wires to the 
 neutral wire.* This reduces the load voltage, as shown in Fig. 
 315, to the secondary coil voltage, while maintaining the voltage 
 between the three-phase wires at V3 times that value. When 
 the load is not perfectly balanced, currents will flow in the 
 neutral wire, and it is usual to make the neutral conductor of 
 
 
 ! S 
 
 
 
 K 
 
 E., 
 
 j 
 
 f» 
 
 :i 
 
 ■ n 
 
 / 
 
 
 i 
 
 E„ 
 
 r 
 
 
 
 i i 
 
 
 
 
 
 
 e 2 
 
 vf 
 
 Fig. 315 — Tri-phase Transformer Connections when a Neutral Wire is used. 
 
 the same size as the other three ; but as the phase voltages 
 are equal to V3 times the load voltages, and as the weight of 
 copper is inversely as the square of the voltage, the weight 
 of copper required for the transmission of a given power over 
 a given distance with a fixed loss of power is only ^ as great as 
 when the ordinary delta or wye arrangement is utilized with 
 the same voltage on the load, and these in turn require only 
 f as great a weight of copper as a single-phase or a four-wire 
 quarter-phase system with the same voltage between the con- 
 ductors of each phase. Consequently, the arrangement of Fig. 
 315 requires only ^ as great a weight of copper as a two-wire 
 single-phase or four-wire two-phase circuit with the same vol- 
 tage on the load. 
 
 The neutral point of a tri-phase system is likely to float from 
 the balanced center when the phases become unbalanced.! 
 That is, the voltage between each of the line wires and the 
 * Art. 100. t Art. 103. Examples. 
 
558 
 
 ALTERNATING CURRENTS 
 
 neutral wire is dependent upon the character of the load in the 
 several phases. 
 
 It is common practice to ground the neutral point of wye con- 
 nected secondary circuits, both for the purpose of reducing the 
 tendency of unbalancing and of reducing danger which might 
 threaten life or property through having exaggerated voltages 
 arising for any reason between the line conductors and ground. 
 Thus, for instance, if a single phase of a high voltage primary 
 circuit accidentally makes connection into the secondary wind- 
 ings where the neutral is grounded, it may be brought to earth 
 potential, and it cannot rise with reference to the ground poten- 
 tial above the potential of the secondary phase wire with which 
 it comes in contact. The neutral conductors of single-phase 
 three-wire secondary circuits are commonly grounded to ob- 
 tain similar protection. Variations of voltage between earth 
 and the line wires of an insulated system may also occur from 
 other causes than the illustration just given, but when a system 
 of conductors has a ground connection, whether from a wye 
 neutral or otherwise, the voltages between the ground and the 
 several line wires evidently depend upon the values of the vol- 
 tages between the wires themselves. Thus, if one line wire of a 
 wye or delta load is connected to ground, the voltages between 
 the ground and the other line wires are equal to the full line 
 voltage. When the wye neutral point is grounded, the voltage 
 between either phase wire and ground is equal to the voltage be- 
 tween the neutral point and the phase wire, which, in a balanced 
 three-phase system, is equal to the line voltage divided by V8. 
 
 When the neutral points of several generators or transform- 
 ers are brought into conjunction as by grounding, any third har- 
 monics of the voltages induced in the windings may cause third 
 harmonic cross currents to flow through the ground connections, 
 windings, and line wires, which currents could not exist except for 
 such connections. This condition may be controlled by inserting 
 a small amount of resistance in each conductor leading to ground. 
 
 Although grounding at special points may be made designedly 
 with good effect, accidental grounding of otherwise ungrounded 
 systems is generally objectionable, as it ma 3 r be the cause of 
 serious disturbances. This is particularly true of systems of 
 very high voltage. Even though this relation may not prove 
 serious at the point where accidental grounding occurs, the 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 559 
 
 effect is carried more or less directly, depending upon the com- 
 bination of wye and delta connections of the various step-up and 
 step-down transformers, over the whole distribution system and 
 may at one point or another be the cause of accident. It is 
 essential, then, that ground detectors be used on all insulated 
 portions of a distribution system, and that when grounding is 
 desired the ground connection be made with due reference to 
 the voltages and insulation strength over the whole system. 
 
 Disastrously high voltages may be caused under certain con- 
 ditions when the primary circuit of a transformer, having its 
 secondary winding connected into a long high voltage three- 
 phase transmission line of large electrostatic capacity, is open 
 circuited. Thus, suppose a group of three transformers are con- 
 nected in wye fashion to the primary and secondary conductors 
 of a transmission system, as illustrated in Fig 316. If the pri- 
 mary coil connected into lead A 1 is open-circuited for any rea- 
 son, and the sec- 
 ondary coil remains 
 connected into its 
 circuit, the voltage 
 between wires B 2 
 and C 2 may not be 
 seriously affected, 
 but the voltages be- 
 tween A 0 and B, 
 
 Aj A 
 
 Cl 
 
 Tc 2 
 
 Bf 
 
 
 
 
 
 L 
 
 2 ’ Fig. 316. — Diagram showing Three Transformers Con- 
 
 and C 2 and A 2 , may 
 become dangerously 
 high ; for now the 
 coil 0 2 A 2 acts as a high inductive reactance which is in series 
 
 nected in Wye Fashion in a Tri-phase System, with the 
 Primary Circuit of One Transformer Open. Illustration 
 of a Dangerous Condition. 
 
 with what may happen to be an equivalently high capacity re- 
 actance of the line conductor leading from A 2 to the load. This 
 would produce a condition of voltage resonance similar to that 
 assumed in some of the problems of Chapter VI,* and' an ex- 
 cessive voltage would appear between A 2 and 0 2 and also per- 
 haps elsewhere between the line wire connected with A 2 and the 
 other line wires. Evidently with the conditions, as illustrated, 
 the primary voltages will also be disturbed. 
 
 Various other conditions which produce objectionable results 
 may arise when making and breaking connections of trans- 
 * Art. 81 (Examples g to k) and Art. 82 ; also Art. 86. 
 
5G0 
 
 ALTERNATING CURRENTS 
 
 formers associated with polyphase circuits, so that circumspec- 
 tion must be exercised under such circumstances. 
 
 The balance of the load is a most important element in using 
 transformers with their secondary windings connected on single- 
 phase three-wire or on polyphase circuits. It may be readily 
 understood that with the two halves of the secondary winding 
 of one transformer connected to a single-phase three-wire system, 
 any unbalancing of the load will affect the balance of the second- 
 ary voltages respectively produced by the two half windings, 
 on account of the fact that greater magnetic leakage will affect 
 the more heavily loaded half winding, unless the two halves 
 are very closely associated on the magnetic circuit. It is there- 
 fore important to have each transformer, which is to be used 
 for this purpose, constructed with the halves of its secondary 
 winding so arranged that the magnetic leakage shall be ap- 
 proximately the same in each even if the load becomes unbal- 
 anced. When a core t} T pe transformer, with the primary and 
 secondary windings each equally divided between the two legs 
 of the core, is to be used to supply current to a three-wire sec- 
 ondary circuit, the half of the windings to be connected be- 
 tween either outside wire and the neutral wire should not be 
 placed on one leg of the core only, but should be composed in 
 equal parts of turns on each of the legs. When transformers 
 are connected in polyphase circuits a similar effect of unbalanced 
 magnetic leakage is produced by an unbalanced load, as may be 
 seen by reviewing the examples given earlier in the book.* 
 The effect of such unbalanced loads is to tend to overheat the 
 transformer coils, generating apparatus, and load appliances 
 on the phases carrying abnormal currents, and to cause unsat- 
 isfactory regulation and danger to the insulation over the phases 
 or circuits which have abnormal voltages. Not only do incan- 
 descent lamps suffer from the abnormal voltages, but the trans- 
 former coils, induction motor fields, and similar apparatus in the 
 circuits having excessive voltages are apt to have their core fluxes 
 go beyond the point of saturation, thus calling for excessive 
 exciting currents which may become, under such circumstances, 
 sufficiently largo to cause serious overheating. Many low 
 voltage secondary distributing systems supply three- wire loads 
 from the secondaries of transformers connected to polyphase 
 
 * Art. 103. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 561 
 
 primaries, and there have been frequent cases where excellent 
 plants of this kind have given most unsatisfactory service 
 because the systems have not been properly balanced. Thus, the 
 three-wire secondaries of the transformers may be unbalanced, 
 and in addition to this defect the sum of the loads of the trans- 
 formers attached to one phase of the primary supply system 
 may be very different from the loads called for on the other 
 phases. The line and generator IR drops will then vary in the 
 different phase circuits, causing the vector diagram of poly- 
 phase voltages to become skewed and the voltages of the 
 several phases at the points of power consumption to become 
 unequal. It is, therefore, essential for good service, when con- 
 necting transformers to compound primary or secondary circuits, 
 to first properly balance the load which is to be attached. 
 
 The connections of quarter and tri-phase transformers are 
 made circuit by circuit, in the same manner as earlier explained 
 for single-phase transformers. 
 
 141. Phase Transformation. — Connections for transforming 
 one polyphase system into another system with a different 
 number of phases may be readdy developed from the principles 
 set forth in earlier chapters. 
 
 Quite a number of commercial 
 devices for this purpose have 
 been proposed. The one most 
 commonly used is a T connec- 
 tion arranged as follows : In 
 Fig. 317, the primary wind- 
 ings of the transformers M and 
 M' are alike and are connected 
 to a quarter-phase circuit. 
 
 The secondary winding OA^ of M (sometimes called the Teaser 
 winding) is attached to the middle of the secondary winding of M 1 
 
 at the point 0. The secondary winding of M has —B times the 
 
 OR LOAD 
 
 Fig. 317. ■ — Diagram of Arrangement of Two 
 Transformers for converting Quarter- 
 phase into Tri-phase System and Vice 
 Versa. The Tee Connection. 
 
 number of turns that there are in the secondary winding of M' . 
 Then, the voltages impressed on the primary windings being 
 numerically equal and 90° apart in phase, the line OB in Fig. 
 314 represents the voltage between the points 0 and B 2 in Fig. 
 317, 00 that between 0 and C v and OA that between 0 and A 2 . 
 Voltage OA must be at right angles to OB and 00 , as the 
 
562 
 
 ALTERNATING CURRENTS 
 
 THREE-PHASE A 
 
 
 f Cl j 
 
 LESi 
 
 M/WWWWv 
 
 wwwwaaaT 
 
 ~lwvWW\AM 
 
 
 phases of the two primary circuits are 90° apart. Thus, the 
 voltages in i? 2 0ancl OC 2 being in phase with others and the 
 voltage in OA 2 being in quadrature with them, it is seen that 
 between the points A, B , and C three equal voltages are set 
 up 120° apart, since the number of inducing turns between 0 
 and A 2 bears the same relation to the number of inducing turns 
 between B 2 and C 2 , that the length of the bisector of an equi- 
 lateral triangle bears to one of its sides. The apparatus may 
 be used with equal facility for transforming two phases into 
 three phases or vice versa, by impressing two-phase voltages on 
 A X P X and B 1 0 1 and thereby producing three-phase voltages 
 between A 2 , B 2 and C 2 in the one case, or by impressing three- 
 phase voltages on A 2 B 2 , B 2 C 2 , and C 2 A 2 and thereby producing 
 quarter-phase voltages respectively in A X P X and B X C V 
 
 For some services it is desirable to use a six-phase system, and 
 to do this economically generally requires transformation from 
 quarter-phase or tri-phase transmission lines. Figure 318 shows 
 method whereby this can be done in a simple manner. Here 
 
 the secondary circuits of the three 
 transformers of a tri-phase primary 
 system are connected at opposite 
 junctions of the six junction points 
 R , M, S, 2F, T, P (Fig. 318) of 
 a mesh-connected six-phase load. 
 The load makes what is equivalent 
 to the combination of two tri-phase 
 systems. The dotted lines between 
 the load junctions R, 31, S, W, T, 
 and P represent the paths of the 
 current in a mesh load and the 
 broken diametral lines represent 
 the paths for a star load. In six- 
 phase systems mesh and star connections call for the same vol- 
 tages and currents in the individual branches of the load. 
 With the arrangement shown in Fig. 318 the neutral point of 
 the six-phase star is at the same potential as the middle point of 
 the secondary winding of each of the three transformers, and if 
 a neutral return conductor is added to the system, it may be 
 led to those three middle points. 
 
 In Fig. 319 each secondary winding is shown divided into 
 
 aaaaaa 
 
 MA/W\ 
 
 R 
 
 LOAD 
 
 P 
 
 M/WV\ 
 
 o 
 
 :u 
 
 N 
 
 Fig. 318. — Transformer Connections 
 for transforming Three Phases to 
 Six Phases. Opposition Arrange- 
 ment. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 563 
 
 two equal coils. Each set of 
 halves is connected in delta ar- 
 rangement, one delta being re- 
 versed compared to the other in 
 accordance with the connecting 
 diagrams shown in Fig. 319 a. 
 The corners of one delta are con- 
 nected in tri-phase mesh to three 
 points M, N, and P on the load, 
 while the other delta is connected 
 to the points P, S, and T. This 
 arrangement is commonly called 
 the Double delta connection. 
 
 Cl 
 
 MAMAWvJwA WW mT 
 
 B, 
 
 Ai 
 
 maammmJ 
 
 Fig. 319. — Transformation of 
 Three Phases to Six Phases by the 
 Double Delta Arrangement. 
 
 Figure 320 shows a similar arrangement, but with the two 
 
 Fig. 319 a. — Connection Diagram for Plan of Fig. 319. 
 
 Ci 
 
 B. 
 
 wmmvaaJ 
 
 Ai 
 
 sets of half coils connected in wye. The connection diagram 
 
 is shown in Fig. 320 a. 
 
 The primary windings of the 
 transformers may be in either delta 
 or wye in each of the arrangements 
 illustrated for the transformation 
 from three phases to six phases and 
 the reverse. Which arrangement 
 to select for either the primary 
 windings or secondary windings 
 depends upon the voltages desired 
 and the problems of insulation and 
 current-carrying capacities of the 
 copper in the transformers, second- 
 
 A/VWY 4vVW y 5 lA/VVWL _, 
 
 A/yw y' 3 A/wvy' s /wwy ' 
 
 '!S 
 
 '' N 
 
 T 
 
 Fig. 320. — Transformation of 
 Three Phases to Six Phases by the 
 Double Wye Arrangement. 
 
564 
 
 ALTERNATING CURRENTS 
 
 ary leads, and load. With a delta system the line current 
 equals V8 times the coil current for the balanced three-phase 
 system, with voltages in lines and coils equal; and with a wye 
 
 M 
 
 connection the line voltages equal V3 times the coil voltages, 
 with the currents in lines and coils equal.* For the balanced 
 six-phases, line currents are equal to coil currents, and line 
 
 voltages are equal to coil vol- 
 tages in both mesh and star, 
 provided line voltages are 
 measured between electrically 
 adjacent terminals. Figure 
 320 b illustrates the vector 
 relations of the line currents 
 and the coil currents for the 
 six-phase mesh. The line cur- 
 rent at corner M is equal to 
 the vector sum of current RM 
 and current SM (= — MS ) . 
 This is the current Mb. The 
 line current at corner S is 
 equal to the vector sum of 
 currents MS and JVS, and so 
 on at the other corners. 
 
 Figure 321 shows the tee method of connecting two trans- 
 formers with divided secondary windings for the purpose of 
 transforming from three to six phases. In this case the 
 primary winding may be connected to a quarter-phase circuit, 
 
 * Art. 100. 
 
 Fi«. 320 6. — Vector Diagram of Currents 
 in Six-phase Mesh. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 505 
 
 Ci 
 
 MWiwlvWVWVW 
 
 _/VW 
 
 
 VwWNAA 
 
 Fig. 321. — Transformation of Two 
 or Three Phases to Six Phases 
 using Two Transformers and the 
 Tee Connection. 
 
 in which case the primary windings of the two transformers 
 must be alike and the secondary winding of the teaser trans- 
 former must contain only .866 ^ = as many turns as the other 
 
 secondary winding, or to a three-phase primary circuit, in which 
 case both the primary and secondary 
 winding of the teaser transformer 
 must contain VB/2 times as many 
 turns as the corresponding winding 
 of the other transformer, as shown 
 in Fig. 321. Other combinations 
 may be used, but those given are 
 sufficient to indicate in general the 
 manner in which the transformation 
 is accomplished. 
 
 Transformation may also be ef- 
 fected from a polyphase circuit to a 
 single-phase circuit for the purpose 
 of distributing the load of a single- 
 phase circuit over the several phases of the polyphase circuit. 
 Each phase of a polyphase circuit is a single-phase circuit, but 
 when it is desired to distribute the load of a single single-phase 
 circuit over the three phases of a three-phase circuit this may 
 
 be done by connecting three trans- 
 formers, as illustrated in Fig. 321 a , 
 which also exhibits the vector diagram 
 of secondary voltages. It will be 
 observed that the connection of the 
 secondary winding of one of the 
 transformers is reversed in compari- 
 son with the normal connection for 
 a delta, and the connecting wire re- 
 quired to close the delta is omitted. 
 The vector diagram shows that the 
 secondary voltage is equal to twice 
 that of one transformer. The current 
 carrying capacit} r is equal to any one 
 
 of the transformers, and three 
 
 transformers are required to 
 Single-phase Load over Three Phases. transform th.6 full load, of two. 
 
 Cl 
 
 B, 
 
 Ai 
 
 Vwwww 
 
 wwwwv 
 
 VWAAAW 
 
56G 
 
 ALTERNATING CURRENTS 
 
 Moreover, the results are not to be recommended because the 
 arrangement transmits the pulsating power of the single-phase 
 circuit into the three-phase circuit and unbalances it. 
 
 To accomplish the corresponding result with a two-phase pri- 
 mary circuit, the secondary windings of the transformers should 
 be connected as for use with a common return wire, and the 
 single-phase load should be worked off the outside wires. 
 
 The transformation of single-phase into polyphase currents 
 by means of stationary transformers may be accomplished by 
 phase-splitting devices, such as by dividing the single-phase 
 circuit into two or more branches and obtaining currents of the 
 desired polyphase angles by inserting property proportioned 
 resistances and inductive and capacity reactances into the 
 branches. No satisfactory commercial method has been devel- 
 oped, where large units of power are required, which does not 
 include moving parts in the transformer, though phase-splitting 
 devices such as described are much used for single-phase inte- 
 grating meters and for starting small induction motors. 
 
 142. Transformation from Constant Voltage to Constant Cur- 
 rent. — - The effect of magnetic leakage has been shown in Art. 
 123. Remembering that the effect is like that produced by self- 
 inductance coils placed in the primary and secondary circuits, the 
 final vector diagram is the same as in the case of a transformer 
 working on an inductive secondary circuit, with an additional 
 correction applied to the angle of lag between the primary 
 voltage and current to account for the direct effect of the leak- 
 age on the primary circuit. Such a diagram shows that, as the 
 leakage is increased so that the angle of lag between the mutu- 
 ally induced voltages and the secondary current approaches 
 90°, the deficiency in the inherent tendency to regulate for con- 
 stant secondary voltage when constant primary voltage is im- 
 pressed, becomes so great that the secondary terminal voltage 
 tends to vary inversely with the current ; that is, the voltage 
 ordinates of the transformer regulation curve decrease with in- 
 creasing rapidity as the current increases. Such a transformer 
 should therefore tend to transform a variable current at con- 
 stant voltage into a constant current at a variable voltage. 
 This characteristic makes transformers with large magnetic 
 leakage desirable for use for providing constant current for 
 series arc lighting by transformation from a constant voltage 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 567 
 
 circuit. A transformer constructed for use in this way is 
 called a Constant current transformer. 
 
 When the lag angle between secondary voltage and current 
 becomes 90°, the transformer can of course do no work, conse- 
 quently it is impossible to get 
 very exact regulation in thus 
 transforming from constant 
 voltage to constant current, 
 but it is possible to arrange 
 the transformer so that the 
 percentage of leakage react- 
 ance can be varied when nec- 
 essary by partially closing a 
 shunt magnetic circuit by a 
 
 [G. 322. — Diagram showing a Simple Way 
 in which Magnetic Leakage can be Varied 
 in a Transformer Core. 
 
 proposed by Elihu Thomson 
 (Fig. 322). 
 
 Figures 323 and 324 show the results of a test of a small 
 transformer, in which the constant current regulation is wholly 
 due to fixed magnetic leakage. In the first figure, one curve 
 
 shows the efficiency 
 as a function of the 
 current in the sec- 
 ondary circuit, and 
 the other curve shows 
 the secondary termi- 
 nal volts as a func- 
 tion of the secondary 
 current. The sec- 
 ond figure has curves 
 A and B which show 
 for the same trans- 
 formers the watts in 
 the primary and sec- 
 ondary circuits as 
 functions of the sec- 
 ondary current. 
 The crosses on the curves in the two figures show the points 
 corresponding to normal load. Close regulation in the trans- 
 formation into constant secondary current from constant primary 
 
 
 
 ppfUClENCY 
 
 
 
 
 
 
 
 It* 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 0 2 4 6 s 10 12 
 
 AMPERES 
 
 Fig. 323. — Carves of Efficiency and Regulation as 
 functions of the Secondary Current in a Small Trans- 
 former having High Magnetic Leakage. 
 
568 
 
 ALTERNATING CURRENTS 
 
 voltage requires the 
 use of accessory 
 means, such as means 
 to vary the magnetic 
 leakage. 
 
 In well-built trans- 
 formers designed for 
 constant secondary 
 voltage, magnetic 
 leakage is not likely 
 to be of much mag- 
 nitude, and in fact it 
 can only be brought 
 to a large value by 
 making the space oc- 
 cupied by the pri- 
 mary and secondary 
 coils very large com- 
 pared with the cross section of the iron core, by using iron of a 
 low permeability, or by specially arranging leakage paths. In 
 Fig. 325 is shown a sketch of a modern constant current trans- 
 former in which it may be 
 observed that the magnetic 
 circuit is long and the space 
 within which the coils are 
 wound is large. The letters 
 (7, (Vindicate the laminated 
 iron core, the central limb 
 of which is embraced by the 
 primary and secondary coils 
 A and B. The coil A is 
 movable and its position is 
 determined by the balance 
 between the weight of the 
 counter-weighted coil and 
 the magnetic repulsion be- 
 tween the coils. Thus, if 
 the repulsion just balances the weight of the movable coil 
 and mechanism when certain currents flow in the two coils, 
 any change in the currents will cause the movable coil to 
 
 Fig. 325. — Constant Current Transformer. 
 
 0 2 4 6 8 10 12 
 
 AMPERES 
 
 Fig. 324. — Curves of Power Input and Output as func- 
 tions of the Secondary Current in a Transformer 
 having High Magnetic Leakage. A, Primary Kilo- 
 watts ; B, Secondary Kilowatts. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 569 
 
 move up or down. Instead of only one, both coils may be 
 made to move. 
 
 The transformer being connected to a secondary circuit com- 
 prising translating devices, such as arc lamps in series, if one 
 of the units of load is short-circuited for the purpose of remov- 
 ing it from operation, the current in the circuit will rise on 
 account of the reduced impedance. Thereupon, the repulsion 
 between the two coils is increased and the movable coil retreats 
 farther from the stationary coil, thereby causing an increase of 
 the magnetic leakage and a reduction of the secondary terminal 
 voltage. The counter-weight being suspended on an arc of 
 slightly decreasing radius, the coil promptly finds a position of 
 balance, with the secondary current at substantially its normal 
 value. If, on the other hand, additional load is added to the 
 circuit, the current will at once correspondingly decrease, the 
 repulsion between coils also decreases, the movable coil will 
 move toward the fixed coil, the magnetic leakage will be thereby 
 decreased and the secondary terminal voltage increased, and the 
 coil will find a new position of balance with the secondary cur- 
 rent again at substantially its normal value. 
 
 The full load secondary voltage evidently occurs when 
 the coils are nearly as close together as the construction per- 
 mits, and the load impedance is at the highest value through 
 which the transformer is designed to send the normal secondary 
 current; while at no load, which is the condition occurring 
 when the secondary circuit is short-circuited, the coils are at 
 the extreme ends of the central magnetic core. When a heavy 
 load is suddenly reduced, preliminary hand regulation may be 
 necessary with such a transformer to prevent unsafe currents or 
 voltages occurring while the movable coil is moving to its proper 
 position, but as a rule the devices are considered automatic. The 
 secondary terminal voltage when the secondary circuit is open, 
 that is, when the load impedance is infinite, should approach a 
 value equal to the primary voltage divided by the ratio of trans- 
 formation, as in such a case the coils will be at their nearest 
 approach to each other, since the secondary current is zero and 
 no repulsion exists between the coils. 
 
 The circle diagram for such a transformer when the load is 
 considered to have a power factor of unity and the ratio of 
 transformation is unity is shown in Fig. 326. For con- 
 
570 
 
 ALTERNATING CURRENTS 
 
 venience in this figure the negative resistance loci are used for 
 representing secondary quantities. In the case under consider- 
 ation the reactance and resistance of the equivalent circuit both 
 vary and in such proportion that the secondary current is main- 
 tained constant. The secondary current locus is therefore rep- 
 resented by the semicircle A, I 2 ", I 2 , I 2 , B.* The construc- 
 tion is otherwise of the same principles as shown in Figs. 277 
 to 281 inclusive. At full load the leakage reactance voltage 
 drop, represented by the line Q 1 B V is at the relatively low value 
 
 which occurs when the trans- 
 former coils are close together. 
 At no load, that is, the secondary 
 short-circuited, the leakage re- 
 actance voltage drop and the 
 resistance drop in the two coils 
 must combine to equal the im- 
 pressed voltage OE v The leak- 
 age voltage then has the value 
 of E X Q". The resistance drop 
 remains numerically equal to 
 K^Q V since the secondary cur- 
 rent is maintained constant and 
 the ampere-turns of the primary 
 winding therefore must stay con- 
 stant, except as the variation of 
 the magnetic leakage may affect 
 the losses and thus affect the 
 exciting component of the cur- 
 rent. That is, reducing the 
 series load by short-circuiting 
 the translating devices does not 
 decrease the primary current, but it causes the primary angle 
 of lag to increase and therefore reduces the power absorbed 
 from the primary supply circuit. The primary current is found 
 as in the constructions in the earlier figures referred to, and is 
 shown for one position of Q by the line S X I V where S 1 0 is the 
 exciting component. 
 
 If the load impedance is increased beyond the point required 
 to bring the two coils of the transformer to their position of 
 * Compare with Art. 70. 
 
 E, 
 
 Fig. 326. — Diagram showing Circular 
 Loci for Constant Current Trans- 
 former. 
 
MUTUAL INDUCTION, TRANSFORMED 
 
 571 
 
 nearest approach, which represents the condition of maximum 
 load, the primary and secondary currents will decrease along a 
 locus determined by the effects of a varying resistance^ and 
 constant reactance in a circuit.* 
 
 The equivalent impedance combination required to represent 
 the reactions of this type of transformer can be built up as in the 
 case of constant voltage transformers for any value of the load, 
 but the equivalent coil reactance is not fixed but varies with the 
 position of the movable coil, and the parallel impedance sup- 
 plying the exciting current should be connected at the position 
 indicated by JYS in Fig. 282, since the mutually induced vol- 
 tage varies widely. 
 
 Figure 329 shows a constant current transformer at A, which 
 has a double secondary circuit arranged to operate two series of 
 arc lamps. A plug switch is provided in each secondary ex- 
 ternal circuit, as shown at £, to be used for short-circuiting 
 either one at will. Other plug switches are provided at C and 
 D to be used for disconnecting the external secondary circuits 
 from the transformer and for disconnecting the transformer 
 from the primary supply circuit. An amperemeter is arranged 
 so that it may be introduced into either lighting circuit by 
 means of a switchboard plug, and a lightning arrester is con- 
 nected to each circuit. 
 
 143. Series Transformers. — When transformers are connected 
 in series with a line carrying a load as shown in Fig. 327 they are 
 usually called Series or Current transformers. Such transformers 
 in commercial service are usually of very small capacity, such as 
 for loads of from ten to fifty watts in the secondary circuit, and 
 are provided for service with electrical measuring instruments, 
 circuit breaker relays, and the like. The current in the second- 
 ary coil divided by the ratio of transformation takes a value 
 equal to the vector difference between the total primary current 
 and the exciting current component. The exciting current 
 varies with the load for the reasons next explained ; but in a well- 
 designed machine it is relatively small for all conditions of load 
 from a short circuit of the secondary terminals up to an imped- 
 ance in the secondary circuit which is considerably greater than 
 that of normal full load. The voltage across the primary 
 terminals equals the whole transformer impedance, including 
 
 * Art. 70. 
 
572 
 
 ALTERNATING CURRENTS 
 
 that of the secondary load, reduced to primary equivalents, multi- 
 plied by the primary current ; consequently, when the primary 
 current is constant, if the impedance of the load increases, the sec- 
 ondary current tends to fall, but the exciting component increases 
 so as to maintain the closed vector relation between secondary 
 ampere-turns, exciting ampere-turns, and total primary ampere- 
 turns. If the impedance of the load decreases, the secondary cur- 
 rent tends to increase and the exciting component decreases. 
 The tendency, therefore, is for the magnetic flux, and therefore 
 the secondary voltage, to vary so as to maintain the secondary 
 current in approximately fixed relation to the primary current. 
 
 The secondary mutually induced voltage equals the primary 
 
 < GENERATOR LOAD 
 
 WW 
 
 Fig. 327. — Connections for a Current Transformer. 
 
 mutually induced voltage divided by the ratio of transformation. 
 For the conditions of constant impedance in the secondary cir- 
 cuit, the secondary current is therefore substantially proportional 
 to the primary current and in nearly opposite phase, so long as 
 the maximum magnetic density in the transformer core is suffi- 
 ciently well below the point of saturation so that it may be 
 considered to be substantially proportional to the exciting 
 current. When the transformer secondary load is the current 
 coil of a wattmeter or an amperemeter, as is usually the case, 
 this condition is approximately realized. 
 
 If the load impedance is increased so that the saturation of 
 the magnetic core is high, a large portion of the primary current 
 may be used for exciting purposes, leaving a comparatively small 
 portion equal and opposite to the equivalent secondary current. 
 It is thus seen that the ratio of the secondary current to the 
 primary current may vary considerably in case the impedance of 
 the secondary circuit varies, especially if the latter is rather large. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 573 
 
 so that when a series transformer is used to supply current to the 
 coils of a measuring instrument, it should be used only with in- 
 struments with which it was designed to be associated, or the 
 instruments should be calibrated in association with the trans- 
 former. When more than one instrument coil (such as the 
 current coil of a wattmeter and the coil of an amperemeter) is 
 to be associated with a particular series transformer, the in- 
 strument coils should be connected in series with each other in 
 the transformer secondary circuit. The transformer should be 
 designed so that the magnetic density in the core is sufficiently 
 far below the point of saturation so that the exciting current is 
 small and approximately proportional to the impressed voltage 
 and hence to the secondary current, over the entire range of the 
 readings of the instruments from zero to their maximum 
 readings. 
 
 When the secondary circuit of such a transformer is opened, 
 the total current flowing through the mains into which the 
 primary coil is inserted, acts as an exciting current. The form 
 of the wave of this current is largely fixed by the constants of 
 the load attached to the mains, a small current transformer 
 having little effect upon it, so that the mutually induced vol- 
 tage and the secondary current may be very irregular in form if 
 the magnetization of the core is carried beyond the point of 
 saturation. This is the same condition as is shown in Fig. 264, 
 which gives the curves of current, magnetism, and induced 
 voltage in an induction coil through which a current of fixed 
 form flows. * The secondary voltage when the secondary circuit 
 is open depends upon the magnetic reluctance of the core, the 
 current in the main wires, and the ratio of the windings. As 
 the magnetic density in current transformers designed to be 
 used for measuring instruments is made very low through the 
 range of currents to be measured, it is evident that the second- 
 ary circuit voltage may rise to a high and dangerous value 
 when a large current flows in the mains, if the secondary circuit 
 is open, and in any event the magnetic flux is likely to rise 
 to an excessive density and cause injurious heating from the 
 hysteresis and eddy current losses. The secondary coils of 
 such transformers can be short-circuited at any time, however, 
 without danger, as they will not then take a greater current 
 
 * Art. 116. 
 
574 
 
 ALTERNATING CURRENTS 
 
 than that in the primary circuit divided by the ratio of trans- 
 formation. It is a safe rule never to open the secondary circuit 
 of an instrument transformer when current is 
 flowing, and it should be short-circuited when 
 connected to a supply circuit but not provided 
 with a load. 
 
 Figure 328 represents a commercial current 
 transformer complete in a neat iron case. 
 Figure 329 shows a current transformer at 
 F and a potential transformer at F connected 
 to a registering watt-hour meter. 
 
 When a wattmeter is used in association 
 with a current transformer and a potential 
 Fig 328. — Current transformer, a good deal of caution should be 
 Transformer in its observed in respect to its readings unless the 
 Case ' instrument has been calibrated in association 
 
 with the same transformers and with due regard to the power 
 factor of the load which it is to be used 
 to measure. The accuracy of the watt- 
 meter readings is dependent upon 
 the preservation of the same angular 
 relation between the current in its 
 current coil and the voltage impressed 
 on its voltage coil as exists between 
 the main circuit current and voltage ; 
 and also upon the transformation of 
 both current and voltage occurring in 
 fixed ratios. The potential trans- 
 former, being a small constant vol- 
 tage transformer, its secondary terminal 
 voltage differs in phase very nearly 
 180° from the phase of the primary 
 impressed voltage, but there is a very 
 small deviation from an exact 180° re- 
 lation which is caused by the magnetic 
 leakage and the primary IR drop. In 
 the current transformer, the secondary Fig _ 32 9 . _ constant Current 
 current differs substantially 180° from Transformer with Double See- 
 the phase of the primary current except 
 
 t 
 
 Tl 
 
 bu Ah 
 
 T TT-T. I LIGHTNING 
 
 JJG 13 ARRESTER 
 
 iJlU 
 
 C(y) C($) swishes (§)C (y)C 
 
 MMETEI 
 
 Q 
 
 f 
 
 * 
 
 
 =7 
 
 B 
 
 CONSTANT 
 
 CURRENT 
 
 TRANSFORMER 
 
 WATTHOUR 
 
 METER 4/ T 
 
 =-i- 
 
 R L 
 
 ? 
 
 )(|) 
 
 for the effect of the exciting current. 
 
 ondary Circuit connected to 
 Series Load. Also Instrument 
 Transformers. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 575 
 
 Each of these causes of inaccuracy in apparent lag angle is 
 quite small in good commercial transformers and has relatively 
 little effect on readings of the instrument when the load has a 
 high power factor, but even such small deviations may intro- 
 duce a large error into the readings when the power factor of 
 the load is small. 
 
 144. Impedance Coils, Compensators, etc. — The design of 
 Reactance coils, Impedance coils, or Choking coils, as they are 
 variously called, is carried out in very much the same manner 
 as the design of a transformer. An impedance coil consists of 
 a magnetic circuit with a winding of small resistance but large 
 inductance. It may be used in lieu of a rheostat to modify the 
 current in an alternating-current circuit and with less loss of 
 power than would be caused by the rheostat. The magnetic 
 circuit and winding are proportioned in exactly the same manner 
 as the magnetic circuit and the primary winding of a trans- 
 
 Coils of this type 
 
 former, using the formula E x = 
 
 are used for a variety of purposes where it is desired to throttle 
 the flow of current without the attendant loss of power which 
 always follows the use of resistances. 
 
 Thus where an arc lamp is used on a constant voltage alternat- 
 ing-current circuit, a reactance coil is ordinarily used to re- 
 duce the voltage from the voltage on the wires to that required 
 at the lamp and to aid in maintaining the stability of the flow 
 of current. Such an arrangement is illustrated in Fig. 330, 
 where the circuit 
 connections are in- 
 dicated at the left 
 hand, and the vector 
 diagram of voltages 
 measured across the 
 line, between the arc 
 terminals, and be- 
 tween the coil ter- 
 
 Y 
 
 REACTANCE COIL 
 
 
 (arc 
 
 Fig. 330. — Reactance Coil in Series with an Arc Lamp. 
 
 minals, are indicated at the right-hand. The voltages upon the 
 distributing circuits in a theater, or upon the feeders of any 
 plant furnishing alternating currents for incandescent lighting, 
 may be regulated by impedance coils. Figure 331 shows one 
 of the original Thomson impedance coils, in which the reactive 
 
576 
 
 ALTERNATING CURRENTS 
 
 Fig. 331. — An Early Type of Voltage Regulator in 
 which the Reactance of the Coil is varied- by Means 
 of a Movable Copper Shield. 
 
 effect is varied by moving a heavy copper shield A so as to 
 more or less inclose the winding B, instead of varying the 
 
 number of turns of the 
 winding included in the 
 circuit. This shield acts 
 like the short-circuited sec- 
 ondary of a transformer, 
 and therefore reduces the 
 apparent impedance of the 
 windings as it approaches 
 them. Figure 332 is a reg- 
 ulator for a constant-cur- 
 rent circuit such as 
 may be conveniently 
 used in series lighting 
 circuits, in which A 
 is the impedance coil winding and B is an iron core for the 
 magnetic circuit. By means of the regulating mechanism (7, 
 the core B is automatically thrust varying distances into A, 
 
 thus holding the load current con- 
 stant very much as is done in a 
 constant current regulating trans- 
 former.* 
 
 It is evident that the reactance 
 of an impedance coil is more eco- 
 nomical for use in reducing voltages 
 than a simple resistance, since pure 
 reactance in a circuit neutralizes 
 part of the impressed voltage with- 
 out absorbing power, though it has 
 the disadvantage of lowering the 
 power factor. 
 
 There is another type of induc- 
 tive apparatus which is much 
 used and which goes under 
 the name of Autotransformer or 
 Compensator. This consists of a single winding on a proper 
 magnetic circuit, to which single winding both the primary and 
 secondary circuits are connected. Figure 333 shows the connec- 
 
 Fig. 332. — Regulator for a 
 Current Circuit. 
 
 Constant 
 
 * Art. 142. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 577 
 
 tions of a 220-volt compensator which feeds two 110-volt sec- 
 ondary circuits. In this case the function of the compensator 
 is to equalize the voltage between the two secondary circuits re- 
 gardless of their relative loads. This purpose is fulfilled fairly 
 well, although when the two loads are unbalanced the voltages de- 
 livered by the two halves of the compensator are different, due 
 to the differences of drop caused by the local impedances of the 
 two halves with different currents flowing through them. 
 When the load is c 
 balanced, no current 
 flows through the 
 neutral wire which 
 is connected to the 
 compensator at 0 , 
 and the compen- 
 sator winding car- 
 ries Only sufficient Fig. 333. — Compensator used for Maintaining the Vol- 
 current to excite ta S es of a Three-wire Circuit. 
 
 the core so that a counter-voltage is produced which is equal 
 and opposite to the vector difference of the impressed voltage 
 and the IR drop caused by the exciting current. Upon unbal- 
 ancing the load more power is absorbed on one side of the 
 neutral wire than on the other, and a local current flows through 
 the neutral wire and part of the compensator winding. In this 
 case the halves of the compensator act as a transformer. Con- 
 sidering the case when one side, A, of the three-wire circuit is 
 fully loaded, and the other side, B, is without load, — then, 
 neglecting the compensator core losses and exciting current, 
 the current in wire 6r is zero and the current in wires E 
 and F is twice as great as the current in wires Q and D, since 
 the power delivered all comes from the primary circuit but the 
 delivery voltage is only one half as large as the primary voltage. 
 The compensator must operate as a transformer to transform 
 one half of the power required for the load from the B side of the 
 circuit to the A side of the circuit. The current in the neu- 
 tral wire therefore divides at 0 into two halves, which flow re- 
 spectively in opposite directions through the winding. A 
 similar condition exists for any degree of unbalance, so that 
 one half the current in the neutral wire flows in opposite direc- 
 tions through the two halves of the compensator winding when it 
 
 2 p 
 
578 
 
 ALTERNATING CURRENTS 
 
 is used to halve the primary voltage as in this instance, and the 
 wire of the winding need be made only large enough to carry 
 without overheating one half the current in the neutral wire at 
 the time of the maximum degree of unbalance that can occur in 
 the circuit concerned. A compensator therefore requires less 
 copper under these conditions than is required for a regular 
 transformer. 
 
 Figure 334 shows the connections of a 330-volt compensator 
 
 which supplies a 1100-volt secondary circuit or vice verm. In 
 
 this arrangement suppose 
 
 that a, b are the terminals 
 
 of the primary circuit, and b, 
 
 c of the secondary circuit. 
 
 Then the exciting current is 
 
 an element of the primary 
 
 current, and passes from a 
 
 to b. The induced voltages 
 
 between a and b, and b and 
 
 c are proportional to the num- 
 
 Fig. 334. — Compensator arranged to Trans- her of turns between the ter- 
 
 form Voltage from uoo to 330 Volts or m inals, as in a transformer. 
 
 Vice Versa. rr ,, , , • ,■ , 
 
 lhe secondary current is the 
 
 combination of two currents, flowing in ac and be, which 
 produce equal and opposite magneto-motive forces. Neglect- 
 ing the core losses and exciting current, the current in ac is 
 equal to the current in the secondary circuit divided by the 
 ratio of the turns in ab to the turns in cb, that is, divided by 
 the ratio of transformation, since the power delivered to the 
 secondary circuit all comes from the primary circuit. The 
 current in the windings between terminals cb must therefore 
 be less than the current in the secondary circuit in the ratio of 
 
 turns ab — turns cb ___ turns ab 
 turns cb turns cb 
 
 The current in the winding between c and b is therefore equal 
 to the difference between the secondary current and the primary 
 current. Thus, it is evident that the autotransformer acts very 
 much like a transformer ; and modified transformer diagrams and 
 arrangements of substituted impedance can be as readily worked 
 out for them as for transformers. The sizes of wire in ac and 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 579 
 
 Fig. 335. — Connection of Compensators to a Tri-phase 
 Circuit in Wye, — Step-down or Step-up. 
 
 be must be made with reference to the current in each part at 
 full load, and the two parts should be sandwiched together or 
 otherwise arranged 
 so as to keep down 
 magnetic leakage to 
 the limit required. 
 
 The magnetic circuit 
 is made of the same 
 materials and is sub- 
 ject to the same con- 
 ditions as are the 
 cores of transform- 
 ers, but for equal cur- 
 rent densities in the 
 windings the weight 
 
 of copper is less in the autotransformer by an amount proportional 
 
 to twice the weight of 
 the primary winding 
 divided by the ratio 
 of transformation. 
 
 Autotransformers 
 maybe connected in 
 wye to a tri-phase 
 circuit as shown in 
 Fig. 335, when the 
 secondary parts are 
 grouped around the 
 neutral point; or in 
 delta, as shown in 
 Fig. 336, though in this case, as can be seen from the figure, 
 the lowest secondary 
 voltage possible is 
 one half of the pri- 
 mary voltage. 
 
 Figure 337 shows 
 autotransformers con- 
 nected into a tri-phase 
 circuit for starting a 
 motor at lower vol- 
 tage than that of the 
 
 Fig. 336. — Connection of Compensators to a Tri-phase 
 Circuit in Mesh — Step-down or Step-up. 
 
 GENERATOR 
 
 Fig. 
 
 337. — Compensators connected for starting 
 Motor at Reduced Voltage. 
 
580 
 
 ALTERNATING CURRENTS 
 
 line. Here any one of three sets of secondary taps may be used. 
 The switching devices are so arranged that the autotransformer 
 is only in series during the starting period. Polyphase auto- 
 transformers can be built with the cores like those of polyphase 
 transformers. Regulation or starting taps can be taken from 
 the polyphase secondaries of banks of ordinary transformers used 
 in supplying load to motors or other apparatus. The connec- 
 tions can in such cases be made similar in principle to those 
 shown in Fig. 337 or sometimes as those in Figs. 335 and 336. 
 Where special transformers are used for supplying a motor this 
 arrangement is desirable, as the transformers thus perform the 
 dual functions of furnishing the proper operating voltage to 
 the machine and acting as a regulator or starter. 
 
 One or both of the secondary terminals of an autotransformer 
 may be arranged so that the position of connection to the 
 winding may be varied, and by that means the secondary 
 voltage may be caused to vary through any desired range, 
 while the primary current changes only so far as is required by 
 any change in the power absorbed b} r the secondary circuit. 
 The last step may bring the secondary circuit terminals into 
 connection with the primary circuit terminals, thus putting full 
 voltage on the secondary circuit, as, for instance, in the case of 
 autotransformers in three-phase circuits, the final step puts the 
 secondary terminals in connection with the primary terminals, 
 as shown by a, 5, c of Figs. 335 and 336. Autotransformers 
 are commonly used for voltage regulators in intermittent serv- 
 ice, as, for instance, in place of rheostats for starting alternat- 
 ing current motors. They are evidently lighter in weight per 
 kilowatt of energy transmitted, and are therefore desirable for 
 use when neither the primary nor secondary voltages are of 
 such a high value that it is unwise to have the primary and 
 secondary circuits electrically connected together. 
 
 Figure 338 shows a small autotransformer intended for vol- 
 
 tage regulation, which may 
 he used either to reduce 
 the voltage in the circuit 
 
 Fig. 338. — Auto transformer arranged telescop- 
 ically so that it can he used as a Regulator. 
 
 (in which case it is called 
 a Dimmer) or to raise it (in 
 which case it is called a Booster) depending upon how its internal 
 connections are made. The regulation is effected by pushing 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 581 
 
 Fig. 339. — Transformer Windings ar- 
 ranged in Shell Transformer of Given 
 Dimensions so that Leakage is large. 
 
 a part of the winding in or out of the solenoid formed by the re- 
 mainder. Numerous devices of this kind, which depend upon 
 moving the primary and secondary coils with reference to each 
 other, or the core with reference to both, are manufactured. 
 
 145. Calculation of Magnetic Leakage. — By properly plac- 
 ing the windings with respect to each other and to the magnetic 
 circuit, the magnetic leakage 
 of transformers may be re- 
 duced to reasonably small 
 values. Thus in Fig. 339 the 
 primary and secondary wind- 
 ings P and S are so placed 
 that a short-circuiting of mag- 
 netic flux along the path indi- 
 cated by the dotted lines is to 
 be expected ; but when the 
 windings are arranged as 
 shown in Fig. 340, the leakage is not as great, because of the 
 greater reluctance in the magnetic circuits of the leakage paths 
 caused by their greater length and lesser cross section; while if 
 each of the windings is divided and the primary and secondary 
 parts sandwiched together, the leakage maybe made very small. 
 
 The magnetic leakage may be 
 calculated with approximate 
 accuracy by the method indi- 
 cated below. 
 
 Since the currents in the 
 primary and secondary coils 
 of the transformer are in prac- 
 tical opposition of phase, their 
 magnetizing effects are oppo- 
 site. This tends to cause 
 lines of force to short-circuit 
 through the coils, as shown in Fig. 339, the tendency being 
 greatest at the plane where the coils touch each other, since 
 the magneto-motive force is there the greatest, and falling off 
 to zero at the outer edges of the coils, so that the magnetic 
 leakage will differ for each layer of wire in the coils. The 
 effect of leakage must therefore be calculated for each layer, 
 and the total effect may then be summed up. 
 
 Fig. 340. — Transformer Windings ar- 
 ranged in the Shell Transformer shown in 
 Fig. 339, but so that Leakage is reduced. 
 
582 
 
 ALTERNATING CURRENTS 
 
 C' 
 
 In Fig. 341 the ordinates of the line A 'BA" are proportional 
 to the total ampere-turns acting at any point to cause leakage 
 
 lines to pass through the coils P 
 and S. These ordinates are equal 
 to the number of turns in a coil 
 between the foot of the ordinate 
 and the outer edge of the coil, 
 multiplied by the current flowing 
 in the turns. The ordinate at the 
 outer edge of each of the windings 
 is evidently zero, and at the plane 
 between the windings reaches a 
 maximum equal to practically n 2 I 2 . 
 The number of the leakage lines 
 of force inclosed by any layer is 
 proportional to the corresponding 
 when these ordinates are respec- 
 
 A 
 
 1 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 1 
 
 ff> 1 
 *7 1 
 
 \ B / 
 
 
 1 
 
 1 
 
 
 ^ • 
 I 
 
 1 
 
 1 
 
 1 
 
 1 
 
 __ ^ 
 
 
 72' 2 , I 2 
 
 i 
 
 1 
 
 A' 
 
 
 A" 
 
 P s 
 
 Fig. 341. — Diagram for showing the 
 Relation of Transformer Leakage 
 Magneto-motive force to Magnetic 
 Leakage. 
 
 ordinate of the lines C DC" 
 tively equal to 
 
 1.25V2 ya 
 
 x = 
 
 l 
 
 where x is the desired ordinate, y is the mean ordinate of the 
 line A' BA" taken from the neutral plane at D to the point 
 under consideration, a = m n is the area of the coil between the 
 neutral plane and the point under consideration, n being the 
 length of the coil perpendicular to the plane of Fig. 342, and l 
 is the average length of the leakage 
 lines of force through the coils 
 (Fig. 342). The maximum value 
 of x falls at the outer edges of the 
 coils and is 
 
 _ 1.25V2V,a, 
 
 ** m 
 
 21 
 
 Fig. 342. — Diagram showing the 
 Dimensions of Leakage Paths in 
 a Transformer. 
 
 where A x is the total iron surface 
 presented to a coil from which leak- 
 age lines emerge ; and the average number of leakage lines 
 inclosed by the different layers at the instant of maximum 
 
 leakage is 
 
 1 . 2 -> d 2 I 2 A^ _ 
 
 2V2 l 
 
 ,45 # 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 583 
 
 The inductive effect of this leakage on the secondary winding 
 is equal to 
 
 /7T mf 
 
 V 2 7 rn 2 
 
 108 ; 
 
 and an equivalent effect is produced on the secondary voltage 
 on account of the leakage of lines of force through the primary 
 coil. If A is taken to represent the total area of iron presented 
 to both windings from which leakage lines emerge, the for- 
 mulas become 
 
 <J> = l- 25 n 2 I 2 A d E = V2 t m^J = ±n 2 2 I 2 Af 
 1 V2 1 10 8 10 8 Z ’ 
 
 where is the combined leakage voltage of the primary and 
 secondary coils, the former reduced to secondary equivalents. 
 The inductive effect due to magnetic leakage lias already been 
 shown to be in quadrature with the mutually induced voltages of 
 the transformer.* The effect of the leakage voltage is deter- 
 mined by the formulas or diagrams already given, f The formu- 
 las given above are for the parts of the coils under the iron only, 
 and the iron portion of the leakage paths was assumed to be of 
 negligible reluctance. Leakage around the ends of the coils 
 has some effect, and the leakage paths in the iron have a low 
 reluctance which cannot always be considered negligible. As 
 a result the formulas derived do not give the total magnetic 
 leakage exactly and though approximately correct they should 
 be used with discretion. 
 
 146. Current Rushes and Surges. — It was shown earlier that 
 the exponential term in the complete equation for an alternating 
 current in an inductive circuit is ordinarily negligible, but 
 under certain conditions its effect for a few periods after the cur- 
 rent is started in a circuit may be considerable. This question 
 was investigated by Fleming | and others § with especial refer- 
 ence to the action of transformers when first switched on to an al- 
 ternating current circuit. If a transformer is switched on to a 
 
 * Art. 124. 
 t Arts. 123, 124. 
 
 t Jour. Inst. Elect. Eng., Vol. 21, p. 677. 
 
 § Hay, On Impulsive Current Rushes in Inductive Circuits, London Elec- 
 trician , Vol. 33, pp. 229, 277, and 305. 
 
584 
 
 ALTERNATING CURRENTS 
 
 circuit, tlie current does not instantly assume the final form of 
 the wave, but comes gradually to its final form through a short 
 interval of time. The length of the interval and the magnitude 
 of the early current depend upon the instantaneous reactance 
 of the circuit, the frequency, and the point in the voltage wave 
 at which the connection is made. The instantaneous reactance 
 of the circuit depends upon the magnitude of the residual mag- 
 netism in the core at the instant of switching in the current, 
 and its direction compared with the instantaneous impressed 
 voltage at the instant of switching in. If the instant of switch- 
 ing on to the circuit is that at which the impressed voltage is 
 passing through zero, the current in the transformer is less dur- 
 ing the early interval than its final value ; while if, at the instant 
 of switching on, the impressed voltage is passing through its 
 maximum value, there may be quite an excess of current flow 
 through the circuit for a short time, on account of the relations 
 which exist between the instantaneous impressed and counter 
 electric voltages during the first half period. If the residual 
 magnetism has a direction in the core corresponding to the 
 magneto-motive force of the current at the instant of switching 
 in, the apparent reactance may be very small, and the instan- 
 taneous current rush be correspondingly large ; but if the resid- 
 ual magne'tism is in opposition to the magneto-motive force of 
 the current at the instant of switching in, the quick reversal of 
 this magnetism may give an apparent large reactance for the 
 instant and the current may rise gradually. The abnormal 
 state of the current can only exist for a very short time unless 
 the reactance of the circuit approaches a condition of resonance. 
 
 When the transformer circuits contain resistance, self-induct- 
 ance, and capacity, the disturbances upon opening or closing 
 the switching devices controlling the circuits may be excessive 
 and may cause the flow of very large momentary currents or 
 the generation of abnormal voltages.* Troublesome effects of 
 this nature are most apt to occur on high voltage transmission 
 lines and where large units of power are used. The excep- 
 tional care with which transformer terminals are insulated is in 
 part necessary on account of the unusual conditions that exist 
 upon opening or closing circuit switches or upon sudden shifts 
 of load. 
 
 * Art. 59. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 585 
 
 147. Methods of Testing Transformers. — The commercial 
 output rating of a transformer should be the number of kilo- 
 volt amperes it will supply to a load at unity power factor, 
 without heating beyond a specified rise of temperature, when 
 the rated full load voltages are used in the primary and second- 
 ary circuits, and when the voltage and current waves are ap- 
 proximately sinusoidal. Commercial efficiency and regulation 
 have already been defined.* The most important tests of trans- 
 formers are those for determining the temperature rise at full 
 load, and at any overload for which the transformer is de- 
 signed ; the regulation between no load and full load, or at other 
 proportions of load, as desired ; the core losses and the copper 
 losses, from which to compute the efficiency ; and the dielectric 
 strength of the insulation between the primary and secondary 
 coils and the coils and core. The efficiency and regulation 
 are often desired also for loads with power factors other than 
 unity. In addition to tests to determine those quantities it is 
 often desirable to obtain the values of the exciting current 
 under various conditions. 
 
 It is evident that having obtained the values of the coil re- 
 sistances and leakage reactances, the iron losses, and the value 
 and power factor of the exciting current, the more important 
 quantities related to the transformer operation except' the heat- 
 ing can be closely determined for various loads and power factors 
 by the use of the transformer circle diagrams and formulas. 
 
 All tests should be made with load currents and voltages of 
 rated value and frequency, and approximating to sinusoidal 
 form, that is, having ordinates not varying, at any instant, 
 more than 10 per cent from that of the equivalent sinusoid ; 
 with a room temperature at 25° centigrade or with the quantities 
 affected properly corrected ; with the barometer at 760 mm., 
 or with proper corrections for other conditions ; with the rise 
 of temperature of the transformer at the normal value for the 
 conditions of the test ; and witli the apparatus operating under 
 the normal conditions for which it has been designed and rated. 
 
 In this country the Standardization Rules of the American 
 Institute of Electrical Engineers are generally accepted as au- 
 thoritative.! In these rules are given detailed statements of the 
 
 * Arts. 123, 137. 
 
 t Trans. Amer. Inst. Elect. Eng , 1907, pp. 1797-1825, and Year Book, ditto , 1909. 
 
586 
 
 ALTERNATING CURRENTS 
 
 conditions under which tests of the electrical apparatus, in- 
 cluding transformers, should be made ; and they should be con- 
 sulted before making tests of the kind indicated above. The 
 following few paragraphs give in outline some of the simpler 
 methods for testing transformers. 
 
 Wattmeter Method. — A wattmeter may be placed in the pri- 
 mary circuit of a loaded transformer and another in the second- 
 ary circuit. Then the ratio of their readings is the efficiency of 
 the transformer. Or, a wattmeter may be used in the primary 
 circuit, and an amperemeter and voltmeter can be used to deter- 
 mine the output if the transformer is worked on a purely non- 
 reactive load, though it is always wisest to use a wattmeter 
 on alternating-current circuits for measuring power. The watt- 
 meter was early used by Fleming in an extended series of trans- 
 former tests, and found to be satisfactory. It has now become 
 the standard method for measuring electrical power. Before 
 taking the readings the transformers should be run at full load 
 until the temperature has become constant in all parts, which 
 requires from live to fifteen hours, depending upon the size. Also, 
 such other conditions as are specified earlier in the article and in 
 the rules referred to should be carefully observed. This method 
 requires a supply of power, and a secondary load for absorbing 
 it, equal to not less than the full load capacity of the apparatus, 
 or larger if the efficiencies at overloads are required. When very 
 large transformers are under test this may be impossible or un- 
 desirable, in which case one of the methods given later may be 
 adopted. In transformer manufacturing establishments it is 
 often possible to feed back the secondary test load into the load 
 circuits of the works. It is common in testing to obtain the 
 efficiency at |, 1, 1|, and sometimes 1-| the rated output. 
 For great accuracy these efficiencies should each be obtained after 
 the temperature has become constant. It is also not unusual 
 to make the efficiency tests for several power factors if the trans- 
 former is to be used on loads of various power factors. 
 
 Figure 813 shows a tri-phase transformer connected up with 
 tri-phase wattmeters in the primary and secondary circuits 
 as required for making an efficiency test, though ordinarily 
 two single-phase wattmeters would be used in place of each 
 tri-phase instrument. On the low voltage load side the watt- 
 meter current coils are attached to the secondaries of current 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 587 
 
 Fig. 313. — Tri-phase Wattmeters connected to a Tri- 
 phase Transformer for Testing. Connections are also 
 shown for Current and Potential Instrument Trans- 
 formers. 
 
 transformers (C. T 7 .), while on the high voltage primary cir- 
 cuit the instrument voltage coils are attached to the secondaries 
 of potential trans- 
 formers (P.IZ 7 .) in 
 addition to the cur- 
 rent coils being con- 
 nected to current 
 transformers. In 
 making the test, 
 two voltmeters and 
 amperemeters would 
 ordinarily be also 
 connected into two of the circuits of the primary and secondary 
 leads. 
 
 The method here described has the disadvantage of depend- 
 ing 1 for its result on the ratio of two instrument readings that 
 are nearly alike, and error is therefore introduced into the 
 efficiency measurement nearly in proportion to the errors 
 inherent in the instrument readings. Additional errors are 
 introduced when current and potential transformers are used. 
 This method, therefore, should only be used for purposes of 
 checking results of other methods or under circumstances 
 in which other and more accurate methods are not avail- 
 able. 
 
 Stray Power Methods for obtaining Efficiency. — A very 
 convenient method of measuring the efficiency of transformers 
 is to determine the various losses directly, and thence the 
 efficiency by calculation. The iron losses may be determined 
 by measuring with a wattmeter the power absorbed by the 
 transformer when the secondary circuit is open, and may be 
 considered to be constant for all loads with sufficient accuracy 
 
 for commercial purposes, since the core 
 
 magnetism 
 
 changes 
 
 but little with changes of load in constant voltage trans- 
 formers. The copper losses for any load are readily calculated 
 when the secondary and exciting currents and the primary and 
 secondary resistances are known. The exciting current may 
 be measured at the same time that the iron losses are de- 
 termined, by the insertion of an amperemeter into the primary 
 circuit with the wattmeter ; and the resistances may be 
 measured by a bridge or voltmeter and amperemeter method 
 
588 
 
 ALTERNATING CURRENTS 
 
 using direct currents. For a given load, the secondary current 
 is a fixed quantity. The efficiency is then, practically, 
 
 P <1 + P c + 1*1 P% + ( + In ) Pi 
 
 In 
 
 where P c represents the measured iron losses and s the ratio of 
 transformation. Due care must be used that the temperatures 
 are right, or else proper corrections must be made to the coil 
 resistances. The iron, loss will vary only slightly with the 
 temperature and need not ordinarily be corrected. 
 
 A still more convenient method, which may be readily used 
 in central stations for testing transformers, is to measure the 
 iron losses by a wattmeter, as explained above. The copper 
 losses may then be measured by short-circuiting the secondary 
 winding through an amperemeter, and adjusting the primary 
 voltage until the full load current, or any desired fraction 
 thereof, passes through the amperemeter. The reading of a 
 
 wattmeter in the 
 primary circuit is 
 now nearly equal 
 to the copper losses 
 for the tempera- 
 ture of the wind- 
 ings at the time, 
 since the voltage 
 and maximum 
 magnetic density 
 must be very small, and the iron losses are therefore almost or 
 entirely negligible. The exact copper losses may be determined 
 by measuring and correcting for the small iron loss. The tests 
 for the iron losses may often be conveniently made by using 
 the low resistance coil of the transformer as the primary coil 
 
 Fig. 314. — Transformer arranged for measuring Iron 
 Losses. Low Voltage Circuit used as Primary. 
 
 (Fig. 344). 
 
 This method of testing may be used with satisfaction where 
 numerous transformers must be tested, since the losses and 
 efficiency may be determined expeditiously and with the ex- 
 penditure of little power. When combined with a run of 
 several hours with full load current, the secondary circuit 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 589 
 
 being made up of impedance coils, the method proves econom- 
 ical for shop tests. 
 
 Ayrton and Sumpner early devised a method which is fre- 
 quently used in which two transformers of the same size and 
 make are opposed to each other, and which is often called the op- 
 position method. The method of connecting is shown in Fig. 345, 
 in which A and B are 
 the transformers to 
 be tested, with their 
 primary windings 
 connected in relatively 
 opposite directions to 
 the leads and their 
 secondary windings 
 in series with each 
 other. The voltages 
 of the secondary 
 windings are thus in 
 opposition. A trans- 
 former T is inserted 
 with its secondary 
 winding in circuit 
 with the secondaries 
 of A and B. By vary- 
 ing the resistance R, 
 the voltage of T may 
 be regulated so that any desired currents will pass through the 
 primary and secondary windings of A and B. Then the output 
 of T measured by the wattmeter W 2 will give approximate^ the 
 copper losses of A and B plus the loss in leads and instruments. 
 The power supplied to the primary circuits of A and B by the 
 alternator, measured by the wattmeter W v gives approximately 
 the iron losses of the two transformers. From the data thus 
 derived the efficiencies may be computed on the assumption that 
 the losses are equally divided between the transformers A and 
 B. Various modifications of this method may be made, such 
 as putting an autotransformer in place of T, or the two primary 
 circuits may be fed from two branches connected to the mains, 
 but with the voltage of one regulated above normal and the other 
 below normal, so that the difference will be sufficient to send 
 
 Fig. 345. — Transformers connected up for Testing by 
 the Feeding Back or Opposition Method. 
 
590 
 
 ALTERNATING CURRENTS 
 
 full load current through the short-circuited joint secondary 
 circuit. Wattmeters in the primary branches will then, together, 
 measure the entire losses of the two transformers. An ampere- 
 meter should he inserted in the primary or secondary circuit 
 by which to regulate the current. 
 
 Either the high or low voltage coil may be used as the pri- 
 mary coil in making transformer tests, as is convenient. The 
 last method described is largely used in manufacturing establish- 
 ments or when two transformers of similar make are to be tested. 
 
 It must be borne in mind that the maximum magnetic density 
 in the core which corresponds to any particular impressed 
 voltage as read by voltmeter is smaller when the voltage 
 wave is peaked and larger when the voltage wave is flat- 
 topped, than when the wave is sinusoidal,* and the core 
 loss is therefore less for a transformer run on a circuit with a 
 peaked voltage wave than when the voltage wave is flat-topped, 
 other things being equal, and correction must be made for that. 
 
 Regulation Tests. — The regulation of transformers which are 
 used in incandescent lighting is a matter of much moment, and 
 regulation tests are of almost equal importance to the tests of 
 losses and temperatures. An ordinary method of making regu- 
 lation tests is to place a voltmeter across the primary circuit and 
 another across the secondary circuit of the transformer tobe tested, 
 care previously having been taken that the transformer is at the 
 specified temperature and other conditions called for in the test- 
 ing rules heretofore referred to are observed. At no load, the re- 
 duced equivalent readings of the instruments should be equal, and 
 the numerical difference between the reduced readings at any 
 other load gives the drop of voltage corresponding to that load. 
 The reduced readings are gained by dividing the reading of the 
 voltmeter in the high voltage circuit by the ratio of transforma- 
 tion of the transformer. This method lies under the serious 
 disadvantage that the result to be measured is a small quantity 
 equal to the difference of the much larger observed quantities, 
 and the ordinary errors inherent in instrumental readings there- 
 fore are likely to introduce large errors in the value found for 
 the voltage drop. The drop of voltage, measured in this way, 
 includes the IR drop in the windings and the drop due to 
 magnetic leakage, both of which increase with the load. The 
 
 *Arts. 115, 134. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 591 
 
 magnetic leakage drop may be determined by subtracting from 
 the total drop, the value of the IR drop which is calculated 
 from measured resistances and currents. The scalar value of 
 the induced leakage voltage is obtained from the expression 
 
 e^-Vej-%? 
 
 when E d is the total drop measured and E R is the total drop in 
 the secondary voltage due to the resistance of the two coils. 
 Having obtained E L and E R , and the exciting current, for a 
 given voltage and load, the circular transformer diagram loci 
 may be drawn as earlier explained.* The transformer reactance 
 equals 
 
 A much more accurate regulation test may be made by using 
 two transformers of equal transformation ratios and one volt- 
 meter. The primary circuits are separately connected to the 
 supply mains, and the secondary circuits are connected together 
 on one side, terminals of like polarity being tied together. A 
 voltmeter of high resistance or an electrostatic voltmeter is con- 
 nected between the other legs of the secondary circuits. The 
 reading of this voltmeter with a load on one transformer, when 
 the other is unloaded, gives the total drop of voltage caused 
 by loading the former. 
 
 Regulation ratings ai'e usually made with non-reactive loads, 
 though the regulation for loads of lower power factor is often 
 very important. The regulation of a transformer is changed for 
 the worse by introducing inductance into the secondary circuit, 
 and for the better by introducing electrostatic capacity into the 
 secondary circuit, as has already been proved. f Regulation 
 tests on a reactive load are made in the same manner as ex- 
 plained above. The power factor of the load may be obtained 
 by dividing the wattmeter reading of the load by the product 
 of the secondary amperes and volts. 
 
 The efficiency and regulation of polyphase transformers can 
 be obtained by the same methods as have been explained for 
 single-phase transformer tests, using the methods given earlier 
 for measuring polyphase power and power factor. | 
 
 Heating . — The rated capacity of a transformer is determined 
 * Arts. 130, 132. t Art. 132. t Art. 104. 
 
59 2 
 
 ALTERNATING CURRENTS 
 
 mainly by the amount it rises in temperature clue to the heat caused 
 by internal losses of power. The rise of temperature is affected 
 by the temperature of the surrounding air, the barometric condi- 
 tions, and extraneous movements of air over the apparatus. There- 
 fore these conditions during a test should be made to accord with 
 the accepted standards.* The temperature of the conductors can 
 be accurately obtained by measuring their resistance while at the 
 temperature of the room in which the test is being conducted and 
 again after their temperature has become constant under load. 
 The change of temperature can then be determined by substitute 
 
 ing in the formula ? 2 - ^ when a is the tempera- 
 
 ture coefficient of copper with respect to zero degrees centigrade 
 and may be ordinarily taken as .0042, R 0 is the resistance of the 
 windings at zero degrees centigrade and is obtained from the 
 formula R 0 = i2 1 /(l + at 1 ), R x is the resistance measured at 
 the initial temperature t v t 1 is the temperature of the transformer 
 when the first resistance measurement was made, R 2 is the 
 resistance measured at the final tempei'ature, and f 2 — t 1 is the 
 rise of temperature to be determined. 
 
 The temperature of core, case, and insulation can be measured 
 by thermometers, as can also that of the windings as a check 
 upon the resistance temperature measurement. The bulbs of the 
 thermometers must be carefully protected from improper radia- 
 tion. In transformers with water circulation, air-blast, or forced 
 oil circulation the temperatures at inlet and outlet, and the 
 volumes of the water, air, or oil respectively introduced per 
 unit of time, should also be measured. From these data and the 
 specific heats of the cooling agent the amount of heat disposed 
 of by the special cooling device may readily be obtained. In 
 selecting a transformer, the amount of heat to be removed by 
 special cooling devices should be specified. No part of the trans- 
 former should exceed 50° centigrade rise in temperature for the 
 full load it is to carry, and from 5° to 10° lower is preferable, 
 referred to an initial temperature of 25° centigrade. 
 
 Dielectric Strength of Transformer Insulation . — The dielectric 
 strength of the insulation of Transformers is tested by applying 
 sufficiently high voltages between the primary coil and the core 
 or the secondary coil and the core connected together, to insure 
 
 * Trans. Amer. Inst, of Elect. Eng., 1907, pp. 1815-1818. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 593 
 
 an insulation factor of safety which will be a sufficient protection 
 against accident.* For transformers in which the high voltage 
 coil is for 550 to 5000 volts the accepted testing voltage for this 
 purpose is 10,000 volts when the secondary coil connects directly 
 with power consumption circuits. The general rule for other 
 transformers is that the test voltage shall be twice that of the 
 high voltage coil. The test voltage should be applied for the 
 period of one minute. When the test voltage exceeds 10,000 
 volts, great care must be exercised that the voltage be brought 
 gradually to its full value, and while being maintained no large 
 variations occur in it, as otherwise destructive oscillations 
 may be set up. 
 
 The testing voltage may be measured by a voltmeter in con- 
 nection with a potential transformer if necessary or by means 
 of a standard spark gap. Where high test voltages are used, 
 the connections must be made with great care to prevent a spark 
 occurring due to a break, which is especially apt, on account of 
 its own oscillatory character, to cause trouble. For this same 
 reason the spark gap, when used, should be in series with a 
 large resistance. The testing voltage should be sinusoidal, 
 under which conditions the stress imposed on the dielectric is 
 1.41 times the testing voltage observed by the voltmeter. The 
 ordinary commercial frequencies are all equally effective for 
 testing purposes. 
 
 Special testing transformers are often employed for testing 
 the dielectric strength of machines. These are designed with 
 relatively small electrostatic capacity and large impedance which 
 tends to reduce danger due to a breakdown of the dielectric 
 from the causes named above. 
 
 Determination of Wave Shape . — Various methods of checking 
 the shape of voltage and current waves used in testing have been 
 used. One of the oldest of these is by means of a Contact 
 maker revolved synchronously with the generating apparatus 
 and so arranged that instantaneous values of currents and volt- 
 ages may be obtained at any series of angles of advance during 
 a cycle ; but undoubtedly the most convenient and satisfactory 
 apparatus to use for the purpose is the Oscillograph. Briefly, 
 this consists of a looped ribbon wire, mounted so as to have a 
 relatively very high natural frequency of mechanical vibrations, 
 * Trans. Amer. Inst. Elect. Eng., 1907, pp. 1811-1815. 
 
594 
 
 ALTERNATING CURRENTS 
 
 which is placed within a strong constant magnetic field. 
 Through this loop is passed a current proportional at each in- 
 stant to the current or voltage wave that it is desired to trace. 
 The loop is then influenced by a torque proportional at each 
 instant to the value of the current within it, acting on one 
 half of it in one direction and on the other half in the opposite 
 direction. By a minute mirror attached at one edge to one 
 strand of the loop and the other edge to the other strand of 
 the loop, it is possible to cast a beam of light upon a fi lm 
 or screen which repeats very accurately a tracing of the wave 
 required. The subject of curve tracing is more fully discussed 
 later.* 
 
 Methods used in Some Historically Important Tests. — The con- 
 tact maker was used, in the early days before reliable watt- 
 meters had been invented, for tracing the voltage and current 
 curves of a transformer ; and from the data thus obtained 
 efficiency and other characteristics were calculated. Among 
 the historic investigations of this kind were those by Ryan 
 and Merritt,f John Hopkinson,| Mordey,§ and others. A large 
 calorimeter was sometimes used in which the transformer was 
 immersed for the purpose of obtaining its losses. || The power 
 in the transformer circuits was sometimes measured by the split 
 dynamometer or combinations of the three voltmeters and three 
 amperemeters method. 
 
 That the value of the iron losses is largely independent of 
 the load carried by a transformer was first conclusively proved 
 experimentally by Ewing. The same thing was also proved by 
 Fleming's experiments. Figure 346 is plotted from one of 
 Fleming’s tests made on a transformer of 4000 watts capacity. 
 The ordinates of line AB represent the differences of the power 
 in the primary and secondary circuits as measured by watt- 
 meters. The calculated copper losses are represented by the 
 ordinates of the line CD. The difference of the ordinates of 
 the lines AB and CD at any point is the iron loss for the par- 
 
 * Art. 157. 
 
 t Trans. Amer. Inst. E. E., Vol. 7, p. 1. 
 
 t Hopkinson’s Dynamo Machinery and Allied Subjects, p. 187 ; Elect. World, 
 Yol. 20, p. 40. 
 
 § Jour. Inst. E. E., Vol. 18, p. 008. 
 
 || Duncan, Electrical World , Vol. 9, p. 188 ; Roiti, La Lumiere Electrique, 
 Vol. 35, p. 528. 
 
MUTUAL INDUCTION, TRANSFORMERS 
 
 595 
 
 ticular load. The lines AB and CD are approximately paral- 
 lel, which shows that the iron losses were practically constant, 
 regardless of the load. Therefore the conclusion was drawn 
 that the stray power methods of testing transformers give 
 efficiencies which are entirely reliable. 
 
 OUTPUT IN SECONDARY WATTS 
 
 Fig. .'540. — Curves showing Results of a Classic Experiment for determining Iron 
 Losses with Varying Load. 
 
 Ewing’s very neat plan for proving this point was designed 
 to get at the matter directly. Two small transformers were 
 made up exactly alike, the cores of which consisted of insulated 
 iron wire wound into the form of a ring. Over this were 
 uniformly wound two layers of wire making a primary coil, 
 and another two layers making a secondary coil. In operating, 
 the primary and secondary coils were respectively connected 
 in series, but the two halves of each coil in one transformer 
 were so connected as to be in magnetic opposition (Fig. 347). 
 The core of one transformer was therefore magnetized and that 
 of the other was not. While the I 2 R losses at any load were 
 equal in the two, the transformer with magnetized core heated 
 more when put in operation than the other transformer, but 
 the temperature of the second was brought to equality with 
 that of the first by passing a direct current through the in- 
 sulated wire of the core. The power expended in heating the 
 second core by the direct current was thus equal to that ex- 
 pended in the first core due to iron losses. The equality of 
 temperature was determined by means of thermo-electric 
 couples embedded in the cores, which were connected in series 
 with each other through a galvanometer (Fig. 347). In this 
 experiment it was found that, after a balance of temperature 
 was once obtained, it was unaltered by any changes in the 
 loads of the transformers, thus showing that the core losses 
 
596 
 
 ALTERNATING CURRENTS 
 
 in the magnetized transformer were independent of the 
 load.* 
 
 These tests fully confirm the a priori deduction that the core 
 loss must be substantially independent of load in an ordinary 
 constant voltage transformer, which follows from the relations 
 of counter-voltage to impressed voltage. Neither the con- 
 clusions nor the experiments apply to transformers in tvhich 
 
 Fig. 347. — Ewing’s Apparatus for determining Iron Losses. 
 
 IR drop or leakage reactance drop absorb a considerable part 
 of the impressed voltage, however ; because in them the 
 mutually induced voltage and therefore the maximum magnetic 
 density in the core must vary with the load if the primary im- 
 pressed voltage is maintained constant, and the core losses 
 vary when the maximum magnetic density is varied provided 
 the cycles per second remain the same. 
 
 * Ewing and Klaassen, Magnetic Qualities of Iron, London Electrician , Vol. 
 32, p. 731. 
 
CHAPTER XI 
 
 SYNCHRONOUS MACHINES — ALTERNATORS. MOTORS, RO- 
 TARY CONVERTERS, FREQUENCY CHANGERS 
 
 148. Losses in an Alternator. The principal internal losses 
 of alternators are caused by: (1) T 2 R loss in the conductors 
 on armature and field magnet : (2) Eddy currents in armature 
 cores and field magnet : (3) Eddy currents in armature con- 
 ductors: (4) Hysteresis in armature cores: (5) Friction of 
 bearing and brushes, and air friction, often called Windage. 
 
 In well-designed direct-current dynamos, the pole pieces 
 usually cover not less than two thirds of the armature surface. 
 In alternators, the poles usually cover about one half of the 
 armature surface, or a little more.* This would make it appear, 
 upon a superficial examination, that the field ampere-turns, 
 and therefore the field losses, must be much greater in the 
 alternator. However, since alternator armatures are made 
 proportionally larger in diameter (usually exterior to a rotat- 
 ing field magnet) in order to give space for the windings 
 and to avoid excessive magnetic leakage, the proportional 
 excitation really required need not be much increased when 
 the magnetic circuit is well designed. In the same way, while 
 not much more than one half of the armature surface is 
 covered with wire in a single-phase machine, the surface for 
 winding is made much larger by increasing the diameter, 
 while the number of revolutions per minute of the rotating 
 part is not much reduced. Consequently, the voltage produced 
 in a given length of conductor is commensurable in the two 
 classes of machines running at equal speed. When windings 
 for two or more phases are wound upon the armature, the 
 entire surface is covered, and as a result, taking into account 
 the advantages in design named above, polyphase machines 
 can be made to give a larger output per pound of material, with- 
 out excessive heating, than is usual for direct current machines. 
 
 * Art. 21. 
 
 597 
 
598 
 
 ALTERNATING CURRENTS 
 
 The fact that the copper is usually divided among more 
 cores increases the length of wire on alternator field magnets 
 for a given magnetizing power, as compared with the field 
 magnets of direct-current machines. On the other hand, 
 the losses may be brought by careful designing to approach 
 the average values used in direct-current machines. For large 
 machines running from 200 to several thousand kilowatts 
 capacity, the combined armature and field copper losses in 
 good practice gradually reduce to a combined value of between 
 one and two per cent for the very large sizes. A certain line 
 of 5000 kilowatt machines built by a large manufacturer in this 
 country has a combined I 2 R loss of approximately one per cent, 
 while a similar line of 500 kilowatt machines has close to two 
 per cent I 2 R loss. The loss is about equally divided between 
 the field and armature windings. 
 
 Eddy currents in alternator cores are apt to cause greater 
 loss than in ordinary direct-current machines unless greater 
 precautions are introduced for preventing them. Thus, in 
 the armatures of direct current machines of considerable capac- 
 ity, the magnetic cycles in the core seldom reach twenty-five 
 complete periods per second. On the other hand, the com- 
 mercial frequencies for alternating currents are now 25 and 
 60 periods per second. The number of magnetic cycles per 
 second in alternator armatures is evidently equal to the fre- 
 quency of the induced voltage and current. Since the heating 
 in the core discs which is caused by eddy currents is pro- 
 portional to the square of tire induced voltage and therefore to 
 the square of the number of magnetic cycles per second, it 
 becomes particularly important that the discs composing al- 
 ternator armatures be thin and well insulated from each other. 
 Therefore, the iron oxide composing the mill scale on the sur- 
 face of the laminations is not always to be depended upon for 
 insulation, and the stampings may be covered with an insulat- 
 ing enamel. However, when some of the newer irons giving 
 low losses are used, the oxide proves sufficient. All burrs 
 caused by punching the discs or truing the surface of the core 
 should be carefully avoided or removed. 
 
 Eddy currents in the pole pieces are felt quite severely in 
 some types of alternators, but the loss caused by them can 
 always be brought within reasonable limits, in well-designed 
 
SYNCHRONOUS MACHINES 
 
 599 
 
 machines, by making the pole faces of laminated iron, which 
 is cast or dovetailed into the rotating spider frame. Tufting 
 of the magnetic flux in the polar space due to the effect of the 
 armature teeth is apt to be one of the causes of eddy currents 
 in the pole pieces. This trouble can be reduced by careful 
 design of the teeth. 
 
 The loss due to eddy currents in armature conductors of a fixed 
 size has a tendency to be greater for alternating than for direct- 
 current dynamos, on account of the more frequent and sudden 
 changes in the strength of the field through which the conductors 
 pass. However, since alternators are, in general, built for con- 
 siderably higher voltages than direct-current machines designed 
 for a similar duty, the conductors on the alternator armatures 
 are of proportionally smaller cross section. This reduces the 
 relative eddy currents to such an extent that they are not par- 
 ticularly noticeable except in very large or special machines. 
 The common practice of winding armature coils with copper 
 ribbons or bars set on edge (the broad side parallel with the 
 lines of force) also tends to decrease eddy currents. In very 
 large machines built to generate voltages not exceeding 2000 
 volts, armature conductors of a large cross section become es- 
 sential. In such cases they can be made up of insulated strips 
 connected in parallel. The uniform practice, however, of build- 
 ing alternator armatures with embedded conductors avoids all 
 great difficulty from eddy currents of this character, since the 
 relatively high frequency of the magnetic cycles in the core 
 makes it undesirable to saturate the core teeth and but little 
 stray magnetism from the field magnets gets into the slots. 
 
 The effect of hysteresis in iron core armatures is proportional 
 to the number of magnetic cycles per second, and is therefore 
 usually greater in alternator armatures, for a given magnetic 
 density, than in those of direct-current machines. There are 
 only two ways of decreasing the hysteresis loss per cycle and 
 per unit volume: (1) by reducing the magnetic density; (2) by 
 improving the quality of the iron used in the core. Reducing 
 the magnetic density serves to decrease the eddy current loss 
 also, and is therefore doubly advantageous. With the best 
 quality of ordinary soft steels, the magnetic density used in 
 the cores of alternator armatures and in the field cores varies 
 commonly from 10,000 to 12,000 lines of force per square centi- 
 
600 
 
 ALTERNATING CURRENTS 
 
 meter, and tends toward tlie former figure for machines having 
 frequencies of 60 cycles per second. In the teeth the density 
 runs up to from 15,000 to sometimes even 20,000 lines per 
 square centimeter. For machines of 25 cycles frequency the 
 magnetic densities tend to mark the higher values given. When 
 the new silicon and other alloy steels, which have very low hys- 
 teresis constants, are used, the magnetic densities for 60-cycle 
 machines also give values approaching 10,000 lines of force per 
 square centimeter or slightly more in all parts of the magnetic 
 circuit. In the case of these steels the low point of magnetic 
 saturation largely determines the density it is possible to use. 
 Halving the magnetic density tends to quarter the eddy current 
 loss for an equal number of cycles per second. In the same 
 way, the hysteresis loss varies nearly in the proportion of B 1S , 
 or in other words, halving the magnetic density decreases the 
 hysteresis loss per unit volume to one third. On the other 
 hand, the cross section of iron must be increased to decrease 
 the magnetic density per square centimeter. This makes the 
 careful selection and handling of the iron designed to enter 
 alternator cores of special importance. The total iron loss 
 varies, in commercial 60-cycle alternators of good design, from 
 1.5 to 2 per cent of the rated output capacity, and this is in- 
 creased about .25 per cent for alternators having a frequency 
 of 25 cycles per second. High voltage alternators are apt to 
 have slightly higher iron losses than those for low voltage on 
 account of the greater depth of teeth required on account of 
 the additional conductor insulation. 
 
 149. Armature Ventilation. Density of Current in Conductors 
 of Alternators, etc. — Since the hysteresis and edd}' current losses 
 are apt to be greater in alternator armatures, it is usually nec- 
 essary that more opportunity be given for cooling than is the 
 case in direct-current machines. The real effect of the velocity 
 of rotation upon cooling is dependent largely upon the design 
 of the rotating field magnet, or, if the field magnet is stationary, 
 upon the fanning arrangements made in the construction of the 
 armature; but for a given type of construction, the cooling 
 effect is roughly proportional to the velocity, as was early indi- 
 cated by the experiments of Rechniewski.* Consequently high 
 
 * Bulletin cle la Societe Internationale des Electriciens, 1892 ; Electrical 
 World , Vol. 19, p. 336. 
 
SYNCHRONOUS MACHINES 
 
 G01 
 
 surface velocity is desirable for the purpose of cooling. Internal 
 ventilation of rotating armatures is also rendered more effective 
 by I'eason of the high surface velocity. For the purpose of in- 
 ternal ventilation a revolving armature is virtually converted 
 into a centrifugal blower, which sucks in air at its center, along 
 the shaft, and ejects it from the surface. For this purpose the 
 openings in the spider which supports the core communicate with 
 the surface through radial ducts, as seen in Figs. 78 and 92. 
 The rotation of the armature then causes a continuous circula- 
 tion of air through the ducts, which is proportional in volume to 
 the surface velocity of the armature. Sometimes special wings 
 or projections are placed on a rotating armature which greatly 
 assist in ventilating the field magnet and also the armature. 
 These remarks apply with particular force to rotary converters, 
 which always have rotating armatures. When the field magnet 
 revolves, which is the case in all large size modern alternators, the 
 poles cause a vigorous circulation of air, which serves in lieu of 
 the blower action of a revolving armature, except in the case of 
 steam turbine driven field magnets, which are generally built 
 without projecting poles so as to give a cylindrical surface, and 
 such field magnets are provided with blower ducts similar in 
 character to those of revolving armatures. The stationary 
 armature has radial ventilation ducts between the laminations, 
 from ^ to inch in width and spaced from 1|- to 2|- inches apart. 
 When the field poles are properly arranged, large volumes of 
 air are forced through the ducts. Figures 80, 81, and 82 show 
 the ventilating ducts of stationary armatures quite clearly, 
 while Fig. 113 shows the adaptability of the rotating crown of 
 poles on a rotating field magnet to act as a fan. The blower 
 ducts of a steam turbine driven field magnet ordinarily deliver 
 air into corresponding ducts of the stationary armature of the 
 machine. The peripheral velocity of rotating field magnets 
 varies from 3000 to 20,000 feet per minute, the latter figure 
 being marked in certain high speed alternators of large capac- 
 ity, while the velocities near the lower limit are suited to low 
 speed small machines. 
 
 With all precautions to avoid excessive heating in alternator 
 cores, careful design is demanded to prevent them from heating to 
 a higher temperature than is desirable, especially when they are of 
 the modern high speed types. It then becomes a matter of some 
 
002 
 
 ALTERNATING CURRENTS 
 
 concern to determine the possible amount of energy which may 
 be expended in the armature conductors without placing an ex- 
 cessive additional burden upon the cooling surface. There is 
 no valid reason for admitting a higher temperature limit in al- 
 ternator armatures than is allowed in direct-current machines, 
 as was formerly done, and consequently, the radiating surface 
 per watt of the power to be dissipated should be the same, where 
 the ventilation and construction are similar. On account of 
 the large amount of heat caused by core losses which must be 
 radiated, it is not safe to allow an armature I 2 R loss of one 
 watt for each square inch of cooling surface. In common prac- 
 tice, for each watt of PR loss an average of from 1|- to 2 
 square inches of outside winding surface is allowed. This 
 constant is sometimes made much larger, but seldom smaller. 
 This makes the total watts to be radiated, about 2 to 6 watts 
 per square inch of the surface. 
 
 It will be appreciated that the velocity of an alternate!’ not 
 only determines the amount of blower action possible in a 
 machine of given type, but also determines the form of the 
 masses of material from which the heat must be extracted. 
 Thus, in a low speed alternator arranged for direct connection to 
 a reciprocating engine, the diameter of the armature must be 
 made large relatively to its length in the direction of the shaft. 
 While in a high speed machine, arranged for connection to a 
 steam turbine, the diameter of the armature is relatively less and 
 its length greater. This difference is caused by the different 
 numbers of poles required and the permissible circumferential 
 velocity. Further, the high speed machines, up to certain 
 limits of speed, as shown by Fig. 348*, have a better weight 
 efficiency and thus have a less volume of material per watt 
 loss. 
 
 The density of current in conductors wound upon iron core 
 armatures in good practice should usually not much exceed one 
 ampere for from 300 to 1000 or more circular mils cross sec- 
 tion. The density of current that is allowable in either arma- 
 ture or field windings varies quite widely with the dimensions 
 of the coils and the efficiency of their ventilation. 
 
 Since the field cores of alternators are usually quite thin, the 
 windings are often of a depth equal to as much as one half 
 *Behrend, Electrical Review , Yol. 45, p. 375 et seq. 
 
SYNCHRONOUS MACHINES 
 
 603 
 
 the thickness of the cores. At the same time this depth is 
 generally no greater than that on many direct current dynamo 
 field magnets. The same radiating constant can therefore be 
 safely used, that is, from .35 to .40 watt per square inch of the 
 outer surface of the winding when the armature rotates as in ro- 
 tary converters, and 1.5 to 3 watts per square inch when the field 
 magnet rotates — the latter value depending upon the speed and 
 
 SPEED IN REVOLUTIONS PER MINUTE 
 
 Fig. 318. — Relation of Speed to Weight in Alternators, for a line of 1000 Kilowatt, 
 3-Phase, 25-Cycle Alternators. 
 
 design of the field magnet. The ratio of winding depth to thick- 
 ness of core is widely variable, and the radiation constant can 
 wisely be based upon the external surface of the windings. The 
 field cores should therefore be made of sucb a length that the area 
 of the external surface of the windings on rotating field magnets, 
 in square inches, may be from ^ to f times the PR loss in watts. 
 The form of the field magnet, and therefore the effect of the 
 
604 
 
 ALTERNATING CURRENTS 
 
 iron in conducting away the heat in the field windings, has a 
 considerable effect on the allowable value of the constant, as is 
 also the case in direct-current machines. If the winding depth 
 exceeds two inches, it is likely to cause injurious heating in the 
 inner layers. The field windings are often made of edgewise- 
 wound coils of wide, thin copper strips. Strips of insulation 
 are wound with the copper strips, and the edges of the copper 
 strips are allowed to remain bare. This type of construction is 
 • not only mechanically strong but affords ready radiation of heat 
 from the coils. 
 
 The cross-sectional area of the field cores is given by the 
 
 formula, where <I> is the useful armature flux re- 
 
 quired from one pole, v is the leakage coefficient, and B the 
 magnetic density desired in the field core. The pole width is 
 determined by the mechanical and electrical conditions which 
 fix the pitch, and only the length of the pole faces may be al- 
 tered to vary B. The value of the coefficient of magnetic leak- 
 age is quite large in most types of alternators. It probably 
 averages from 1.15 to 1.45 in standard alternators with poles set 
 in a circle either without or within the armature. The calcula- 
 tion of this coefficient of alternators may be carried out upon the 
 same methods as those used for continuous current-machines,* 
 or it may be measured for a complete machine by the search coil 
 method; that is, a test coil may be arranged about a pole piece 
 and connected to a ballistic galvanometer. When the field 
 current is then suddenly reversed, the throw of the needle is 
 closely proportional to the magnetic flux in the pole. A coil 
 of equal turns may then be laid on the armature surface inclos- 
 ing an arc equal to the polar pitch and extending the full 
 length of the ‘armature coils. The throw is now proportional 
 to the useful flux from the pole. The ratio of the first to the 
 second reading is approximately proportional to the leakage 
 coefficient. The coefficient varies somewhat with different 
 armature currents and power factors on account of the mag- 
 netic reactions of the armature currents upon the poles. 
 By likewise using test coils placed around different parts of 
 the field core, the distribution of the magnetic flux may be 
 measured. 
 
 * Jackson’s Electro-magnetism and the Construction of Dynamos, p. 133. 
 
SYNCHRONOUS MACHINES 
 
 605 
 
 150. Determination of the Number of Armature Conductors. 
 
 — The formula for voltage induced in an alternator, 
 
 F 2 KSQVp 
 10 8 x 60 ’ 
 
 may be put into the form 
 
 yr _ 2 KS<l>f 
 10 8 ’ 
 
 since 
 
 Vp 
 
 60 
 
 is equal to the frequency f of the induced voltage.* 
 
 Taking a proper value for K, as already explained, gives the effec- 
 tive value of E. In a well-designed machine of the usual Ameri- 
 can types, the value of K is about 1.1, which is the value for a 
 machine which gives a sinusoidal voltage and in which the dif- 
 ferential action is negligible. The conditions of service usually 
 fix the values of E and/ in any particular case, and the equation 
 then contains two dependent variables, <E> and S. The ratio 
 of these is determined from the form and dimensions of the ar- 
 mature which it is desired to make. The number of poles is 
 limited by constructive considerations and the importance 
 of keeping the magnetic leakage within reasonable limits. On 
 the latter account the poles must not come too near together. 
 With this in view the frequency cannot be increased beyond a 
 certain limit by increasing the number of poles, without carry- 
 ing the periphery velocity of the armature beyond the safe limit. 
 Thus, suppose a machine is designed with a twenty-pole rotat- 
 ing field magnet, and its field magnet is designed to be driven 
 at the safe limit of velocity. If it should be desired to increase 
 the frequency of the voltage by 10 per cent, two poles must be 
 added to the field magnet and the revolutions kept constant, or 
 else the field magnet must be speeded up 10 per cent. Suppose 
 the latter is not permissible on account of mechanical safety; 
 then, if the poles are already as close together as economy 
 admits, in order to increase the number of poles the pitch circle 
 must be increased in diameter. This again calls for an in- 
 crease of the periphery velocity of the field magnet, the revolu- 
 tions per minute remaining constant. Hence in any type of 
 machine a limit of frequency may be reached which cannot be 
 safely exceeded. The limiting frequencies in the ordinary types 
 
 * Art. 20. 
 
GOG 
 
 ALTERNATING CURRENTS 
 
 of machines designed for commercial service are from 100 to 
 150 periods per second. In the old Mordey type the limit was 
 much higher, since the poles of this machine may be very close 
 together and cause no additional leakage, and structural con- 
 derations, only, set the limit. Commercial frequencies for elec- 
 tric lighting and power are all less than 150 periods per second, 
 and are usually 60 or less, so that all practical types of alter- 
 nators may be used in commercial service. The requirements 
 of wireless telegraphs, on the other hand, are now making a 
 demand for some relatively small alternators of special t} r pes 
 designed for generating currents of frequencies as great as two 
 or three hundred thousand cycles per second. 
 
 Fixing the frequency of a machine and the periphery velocity 
 and revolutions of the revolving part, fixes both the diameter 
 of the armature and the number of field poles. The diameter 
 of the armature fixes the space for armature slots. With the 
 slot space fixed and the dimensions of the slots determined 
 from the conditions of tooth saturation and core reluctance 
 desired, the value of iS is determined from the number of 
 insulated conductors of requisite area which can be properly 
 placed in the slots. The value of S being determined, the 
 length of the armature must be made such that the necessary 
 total magnetization may be set up in the field cores and 
 armature without forcing the magnetic density in the teeth 
 and cores to too high a value. Finally, the ratio of radiating 
 surface to I 2 M loss should be checked. 
 
 It is well to consider here the effect upon the output of an 
 alternator, of making a change in the number of armature con- 
 ductors. The voltage developed in the coils due to their cut- 
 ting lines of force is proportional to the number of turns in the 
 coils. The voltage of self-induction, on the other hand, is pro- 
 portional to the square of the number of turns in a coil. Hence, 
 increasing the number of armature turns beyond a certain limit 
 may actually decrease the output of the machine, and if carried 
 to a sufficient degree may make it even tend to regulate for 
 constant current instead of constant voltage. The following 
 argument, in which is approximately determined the number of 
 turns that will give maximum output, is of some value in giv- 
 ing a conception of the effect of the number of turns upon 
 output. In the discussion the effect of armature reactions on 
 
SYNCHRONOUS MACHINES 
 
 607 
 
 the field is assumed to be included in the inductive reactance 
 of the winding. Where the load is non-reactive the assumption 
 is justifiable.* If L l represents the effective or working self- 
 inductance for each armature conductor, including its propor- 
 tion of reactive effect on the field magnet, then the total 
 effective self-inductance of the armature is approximately 
 L— S 2 L V In the same way, if E 1 represents the voltage de- 
 veloped per conductor, the total voltage developed in the arma- 
 ture is E — SE V From the fundamental formula 
 
 VR 2 + 4 7 r 2 f 2 E 2 ' 
 we get EE 1 = E 2 - 4 7 j 2 f 2 EL\ 
 
 or IR = VE 2 - 4 EPEE 1 . 
 
 This may be put into the form 
 
 IR = V S 2 ]lf- 4 EP1 WE 2 , 
 
 supposing the external circuit to which the alternator is con- 
 nected to be non-reactive. The term IR is the active voltage 
 in the complete alternator circuit, and it is desirable that this 
 be a maximum for a given armature. Differentiating the equa- 
 tion with respect to S and solving for a maximum, we get the 
 following : 
 
 d(IR ) = 2 S(E 2 - 8 EfEEL 2 ) = Q . 
 
 dS 2 VS 2 E 2 - EEfU^L 2 
 
 hence E j 2 — 8 Ef 2 I 2 L 1 2 S 2 =0. Or IR is a maximum when 
 
 S = . 
 
 2 V2 7 rfIL l 
 
 In this it is assumed that the armature resistance is small com- 
 pared with that of the external circuit, which is always the 
 case in efficient working. 
 
 For E 1 may be substituted its value 
 
 E,= 
 
 2 K<i> f 
 ~ 10 8_ ’ 
 
 and the expression for S at maximum output becomes 
 
 a- K * - I-a, 
 
 10« tt/£, ILi 
 
 * Art. 155. 
 
608 
 
 ALTERNATING CURRENTS 
 
 where A is a constant depending upon the type and dimensions 
 of the machine. If K has a value of 1.1, the value of A is 
 practically 25 x 10 ~ 10 . 
 
 The final form of the variable portion of the expression giving 
 the maximum economical value of S is striking. Its numerator 
 is the total useful magnetization due to the field magnet which 
 passes through an armature coil, and its denominator is the 
 magnetization passing through the coil due to the current in 
 each of its own conductors. The fact is, however, that mechan- 
 ical limitations and limitations due to heating and regulation, 
 and the considerations of efficiency, prevent the number of con- 
 ductors in modern alternator armatures even approaching the 
 number given by this formula, when the machine is worked on 
 a non-reactive load. 
 
 When the external circuit is inductive, as is commonly the 
 case, the number of armature turns which gives a maximum 
 active voltage is smaller than when the external circuit is 
 non-reactive. If S' is the number of conductors giving the 
 maximum active voltage when the external circuit has self- 
 inductance _L C , and S is the number of conductors giving the 
 maximum voltage when the external circuit is non-reactive, then 
 
 In the latter expression S 2 L } is the self-inductance of the arma- 
 ture when wound with the proper number of turns to give a 
 maximum active voltage when the external circuit is non- 
 reactive. This becomes larger when there is an inductive 
 load, since the reactions of the armature upon the field are 
 greater.* When L e is greater than S 2 L V the right-hand mem- 
 
 S' 
 
 her of the expression for — becomes imaginary ; that is, the 
 ratio is an indeterminate. 
 
 It is impossible to put so many turns on commercial alter- 
 nator armatures as the criterion shows would give the greatest 
 output at unity power factor, since the question of regulation 
 
 or 
 
 * Alt. 152. 
 
SYNCHRONOUS MACHINES 
 
 609 
 
 in constant- voltage alternators demands that the fall of voltage 
 in the armature due to resistance and inductance shall be as 
 small as possible. Nevertheless where alternators are subject 
 to inductive loads, the self-inductance, reactive effect upon the 
 field, and resistance of the armature conductors are most im- 
 portant elements affecting regulation under the various operat- 
 ing conditions to be met in practice, as is shown clearlj" by the 
 relations just developed. 
 
 151. Armature Ampere-turns and Self-inductance of Alterna- 
 tors. — The self-inductance of an alternator armature may be 
 approximately estimated from the magnetic and electric data 
 of the machine. The reluctance of each magnetic circuit must 
 be calculated exactly as in the case of a direct-current multipolar 
 dynamo, — in order to determine the field windings, — using 
 
 the formula P = where l is the length, A the cross section, 
 /j.A 
 
 and /i is the permeability of the part of the magnetic circuit 
 
 Fig. 349. —Diagram for showing the Magneto-motive force required for Each Field 
 Core to set up the Required Magnetic Flux. 
 
 under consideration. The reluctance to be overcome by the 
 magneto-motive force of each field core belongs to that part of 
 the magnetic circuit which lies between the planes parallel to the 
 armature shaft in which lie the lines AA' and BB' in Fig. 349, 
 which is a section of an alternator with rotating armature. 
 Machines with rotating field magnets may be treated in the 
 
610 
 
 ALTERNATING CURRENTS 
 
 same manner. Calling that reluctance P, the ampere-turns 
 for each field core are in number, 
 
 nI=^- P 
 
 , <£ P 
 
 L _|_ 8 3 _|_ 
 
 1.25 1.25 
 
 $ P cb P 
 
 1.25 1.25 
 
 where ® f , <t> 8 , <!>„, and <!>,, and P f , P 8 , P a , and P t are respectively 
 the magnetic fluxes and reluctances of the field core and yoke, 
 air gap, armature body, and armature teeth lying between AA' 
 and BB'. 
 
 The reluctance in the different parts of the magnetic circuit 
 met by lines of force which are set up by the magneto-motive 
 force of the armature ampere-turns when the field magnet is ex- 
 cited, may be assumed to be equal to the reluctance in the same 
 parts of the circuit met by the lines set up by the field coils, except 
 for the part of the flux due to armature turns which leaks from 
 tooth to tooth in a multi-tooth armature. The number of Hues 
 of force set up in the portion of the magnetic circuit between 
 the planes A A' and BB' by a unit current in one armature coil, 
 
 the center of which is directly under a pole face, is 
 
 1.25 S, 
 2 P ’ 
 
 where S x is the number of conductors in the coil and is equal 
 to twice the number of turns in the coil, and P is the equiva- 
 lent combined reluctance of the magnetic path in the field mag- 
 net, armature, and air gap. Each one of these lines of force 
 in completing its circuit through another segment of the mag- 
 netic circuit must link another armature coil, so that we may 
 say that the number of lines of force set up by each pair of 
 
 coils is 
 
 2 P 
 
 The self-inductance of the pair of coils, there- 
 
 fore, so far as relates to the magnetic circuit through the field 
 core, is 
 
 
 1.25 S? 
 
 2 x P x 10 8 ’ 
 
 since S x is equal to the number of turns in two coils. 
 
 The self -inductance per phase, of the whole armature due to 
 this magnetic flux, is equal to L x multiplied by the number of 
 pairs of coils, when the armature windings of a phase are con- 
 nected in series, or 
 
 L = 
 
 1.25 Sfp 
 
SYNCHRONOUS MACHINES 
 
 611 
 
 when the coils are placed in one slot per pole per phase. When 
 the windings are distributed in several slots per pole per phase 
 the value of L is diminished. When the armature is connected 
 with the halves in parallel, this inductance is one fourth as great 
 as is given by this formula; but in the case of two similar arma- 
 tures built for the same voltage and output, the one connected 
 with the halves in parallel has twice as many conductors in 
 each coil as has the other armature, and their self-inductances 
 are equal. 
 
 The path of the lines of force set up by the armature coils 
 has been assumed to be the same as the path of those set up by 
 the field magnets, and the real effect of the armature magneto- 
 motive force upon the number of lines of force in the mutual 
 magnetic circuits is to increase or decrease the number that 
 would exist were the armature turns absent, rather than to set 
 up an independent magnetic flux. The extent of this effect 
 depends upon the lag of current in the armature, and the effects 
 of armature reactions and of self-inductance are therefore 
 closely related. In the case of machines with toothed armature 
 cores, as already said, the reluctance measured around the path 
 of the leakage lines of force set up by eacli group of conduc- 
 tors in a slot may be quite small. This is due to the effect of 
 the leakage from tooth to tooth. Consequently, the self-induct- 
 ance of the armature must be increased over the amount given 
 in the preceding formula by an amount equal to that caused by 
 the tooth to tooth leakage. This is made up of the self-induct- 
 ance caused by this leakage for each group of conductors in a 
 slot multiplied by the number of groups in the whole winding 
 of a phase. It is comparatively constant in value for different 
 armature currents, since the leakage path may be partially in 
 air, and the teeth, because of their small cross section relative to 
 other parts of the magnetic field circuit, are apt to be magnetized 
 to a comparatively high degree of saturation. 
 
 152. Armature Reactions in Alternators. — The armature re- 
 actions of alternators do not cause as serious consequences in 
 some respects as those of direct current machines, as the com- 
 mutator is absent, but their effect upon regulation is of prime 
 importance and demands first attention from the designer. 
 Consider first the conditions which exist in single-phase ma- 
 chines, or in polyphasers when the armature currents are 
 
612 
 
 ALTERNATING CURRENTS 
 
 unbalanced, which, as will be seen later in the discussion, pro- 
 duces much the same effect as is produced in single-phasers. 
 Then when the current of the armature is in exact phase with 
 the impressed voltage, the armature current has comparatively 
 little opportunity to affect the field magnetism. When the arma- 
 ture conductors are directly between the pole pieces, the instan- 
 taneous current is zero, and therefore at this point the armature 
 has no effect upon the field magnetism. When the coils have 
 
 the armature toward the right-hand with respect to the field poles. 
 
 moved through one half the pitch, a sheet of current at its max- 
 imum value flows directly under the pole faces. This current 
 has such a direction that its magnetic effect tends to crowd the 
 lines of force of the field magnet into the trailing tips of the 
 poles (Fig. 350). Hence the field is weakened on account of 
 the increased reluctance of the magnetic circuit. This effect is 
 probably not very marked in the usual forms of alternators, since 
 the armature ampere-turns are usually not large compai’ed with 
 the impressed magneto-motive force of the field magnets, but, to 
 
SYNCHRONOUS MACHINES 
 
 613 
 
 whatever extent the effect exists, it produces a periodic shifting 
 of the magnetic density across the pole face as the armature coils 
 move from pole to pole. The distortion and consequent weak- 
 ening of the field may be reduced by cutting a slot longitudi- 
 nally across the pole faces, or by carefully designing the pole 
 tips. In Fig. 350 the irregular line dd indicates the tendency 
 of the groups of conductors in a slot to distort the curve bb on 
 account of the fact that the magneto-motive forces of their 
 currents are not evenly distributed over the armature surface. 
 This effect is usually not large. 
 
 When the armature current is out of phase with the induced 
 voltage, the conditions are quite different. Suppose that the 
 phase of the current is retarded on account of self-induction. 
 When the centers of the coils are under the poles, the current 
 is not zero, but has an 
 instantaneous value 
 which depends upon 
 the amount of retard- 
 ation. This current 
 in a generator is in 
 such a direction that 
 its magnetic effect 
 opposes that of the 
 field, as illustrated in 
 Fig. 351, in which the 
 dot or arrow point 
 on the armature con- 
 ductors represents the current flowing toward the observer and 
 the cross or arrow feather represents the current flowing away 
 from the observer. As the coils move, the opposing effect of 
 armature reactions merges into the cross effect already indi- 
 cated, and the armature reactions therefore tend to cause a 
 periodic variation of the strength of magnetic flux. When the 
 machine under consideration is operating as a synchronous motor, 
 the current under the poles lagging behind the impressed voltage 
 but being reversed in phase with respect to the induced voltage, 
 evidently tends to strengthen the fields instead of to weaken 
 them. 
 
 If the current is in advance of the phase of the induced vol- 
 tage, the armature of a generator tends to strengthen the field 
 
 Fig. 351. — Diagram for showing the Effect of Armature 
 Reactions of a Single-phase Generator when the Cur- 
 rent in the Armature lags behind the Induced Voltage. 
 
614 
 
 ALTERNATING CURRENTS 
 
 magnetism when the coils are directly under the poles, as may 
 be understood by examining Fig. 352. On the other hand, a 
 motor armature with the current leading the phase of the im- 
 pressed voltage tends to weaken the fields. This tendency of 
 
 the armature current, 
 when in advance of the 
 induced voltage, to 
 strengthen the fields 
 could be taken ad- 
 vantage of to make an 
 alternating-current 
 generator completely 
 self-regulating, or 
 even self-exciting, 
 Fig. 352. — Diagram for showing the Effect of Armature through the action of 
 Reactions of a Single-phase Generator when the Cur- jp g armature CUITeilt 
 rent in the Armature leads the Induced Voltage. . 
 
 lhis, however, would 
 
 require the use of a condenser or other source of capacity react- 
 ance attached across the armature terminals to give the proper 
 lead to the current, which would not be practical in appli- 
 cation. 
 
 The quantitative effect of the armature current and the angle 
 of lag on the results produced by armature reactions is not read- 
 ily determined in single-phase machines. The effect is periodic 
 and depends for its relative instantaneous values upon the in- 
 stantaneous positions of the armature coils with respect to the 
 poles, and the corresponding instantaneous current values. The 
 latter depend upon the effective value of the current, the angle 
 of lag, and the form of current wave. Doubtless the relative 
 shapes of armature coils and pole pieces also enter the relation. 
 Moreover, the field frames are fairly large masses of iron, and 
 they do not respond rapidly to changes in their magnetic sur- 
 roundings ; this apparent magnetic inertia being caused by the 
 effect of eddy currents and the considerable inductance of the 
 field circuit, which tend to suppress rapid changes of the mag- 
 netic flux. It is therefore reasonable to assume in general that 
 the results of armature reactions are less marked than misfht be 
 inferred from a scrutiny of the variation of the instantaneous 
 values of armature magneto-motive forces acting in the polar 
 regions of the magnetic circuit. 
 
SYNCHRONOUS MACHINES 
 
 615 
 
 The instantaneous value of the back turns of each armature 
 coil at any moment depends upon the instantaneous value of 
 the current multiplied by a function of the instantaneous position 
 of the coil. If the current is sinusoidal, then the current at 
 any instant is proportional to sin a, and if the magnetizing 
 effect on the pole pieces caused by the current in the coil is 
 assumed to vary as a sine function of the position of the coil 
 with reference to its position when the induced voltage is zero, 
 then the magnetizing effect per ampere is proportional to cos a. 
 The back turns are then equal to 
 
 V2 nl sin(« — 0) cos a 
 
 at each instant, in which 0 is the angle of lag, I is the effective 
 value of the current in the coil, and n is the number of turns in 
 the coil. Expanding this expression reduces it to 
 
 7hl 
 
 (sin 2 «cos 0 — cos 2 a sin 0 — sin 0). 
 
 Vz 
 
 If 0 is zero, this is a periodic expression of twice the funda- 
 mental frequency and of zero average value. Averaging the 
 expression from a = 0 to a = 7 r, when 0 is not zero, gives the 
 
 value — -^isin 0. 
 
 V 2 
 
 The armature reactions in polyphase generators are materi- 
 ally different from those in single-phase generators, but consist 
 essentially of the sum of the effects of the several phases. 
 Thus, referring to Fig. 350, it was seen that when the current 
 of a single-phaser is in phase with the voltage, magnetism is 
 crowded into the trailing pole tips at each time of maximum 
 current, and resumes its initial position when the current falls 
 to zero. In the case of polyphase machines, in which the 
 wire, in effect, covers the entire surface of the armature, the 
 skewing effects which the currents in the armature coils of 
 the several phases of a balanced machine produce on the 
 magnetism of the field magnet come into action succes- 
 sively so as to maintain the total effect uniform when 
 the currents are equal and in phase with the induced vol- 
 tages of their respective circuits. When the currents lag 
 
bib 
 
 ALTERNATING CURRENTS 
 
 or lead, the skewing effect decreases and the back turns 
 become 
 
 V2 nl | sin (« — 6) cos « + sin [ a — 6 — ^-^jcos fu — ••• 
 
 m ) 
 
 . ( a 2(m — l)7r\ ( 2(m — l)7r' 
 
 + sm « — 6 L — cos a — 
 
 \ m J \ m 
 
 in which m is the number of phases. 
 
 This is zero and the back turns disappear when 6 = 0, since 
 
 sin «cos a + sin 
 
 2 7 r\ ( 2 ir\ 
 
 it I cos (a )+ ••• 
 
 m 
 
 J 
 
 + sin a — 
 
 2 ( m — 1 )7r 
 
 m 
 
 COS It — 
 
 2 (m — 1)7 r 
 
 m 
 
 = 0 . 
 
 Under these circumstances the skewing effect of the armature 
 magneto-motive force is a maximum. 
 
 When 6 is not zero, the expression becomes 
 
 nl r . o -of 2 77") • 0 ( 2 (m — 1)7 r 
 
 sin 2 a -f sm 2( a - + ••• + sin 2 ( a | — 
 
 m J 
 
 V2l_ 
 
 m 
 
 9 tt\ 
 
 cos 2 « + cos 2 ( a — - — )+ ••• + cos 2( te- 
 rn / 
 
 2 (m — l7r) 
 
 m 
 
 cos 0 
 sin 6 
 
 — m sin 6. 
 
 But 
 
 2 a + sin 2^a — ••• + sin 2 — ^) = 0, 
 
 sin 
 
 and 
 
 cos 
 
 m 
 
 2 ct + cos 2 (it — + ••• + cos2(^« — ^ = 0. 
 
 V m J \ m J 
 
 Therefore the expression for the armature magneto-motive 
 force has the constant value, 
 
 — rsin 6. 
 
 V2 
 
 This is obtained on the assumptions that the current is 
 sinusoidal and that each armature coil acts on the main magnetic 
 circuit with an influence proportional to cos a. Neither of 
 these assumptions is exactly met in most machines, and the 
 back turns therefore are more or less irregularly variable 
 through each period of the armature current. In case the cir- 
 
SYNCHRONOUS MACHINES 
 
 617 
 
 cuit is unbalanced, a pulsation is introduced by the differences 
 between the phases. 
 
 The effect of armature ampere-turns in a single-phase machine 
 is illustrated in Fig. 353, coils a, a, connected as in Fig. 350, 
 being indicated. (To simplify the figure, winding slots are not 
 shown.) Curve E is the induced voltage curve of the machine 
 plotted as a function of time beginning when the coils are 
 located neutrally in the magnetic fields as shown in the Figure, 
 that is, at the instant when the induced voltage is zero; and curve 
 c is a current curve assumed to be in phase with the induced 
 voltage. Curve d represents the magnitude of the magnetizing 
 effect which would be produced per ampere by a continuous cur- 
 rent flowing in the moving armature winding, which aids or 
 opposes the field magnetization, but differs in value with the 
 instantaneous positions of the coils, and is here plotted with 
 space abscissas corresponding to the time abscissas of curve E. 
 
 Fig. 353. — Curves showing the Magnetic Activity of the Armature of a Single-phase 
 Generator ivhen the Current is in Phase with the Induced Voltage. 
 
 This magnetizing effect or activity with a uniform current 
 flowing in the armature winding must evidently have a maxi- 
 mum value when the centers of the coils are directly under the 
 centers of the poles, i.e. when the coils are at the point shown 
 in the figure. When the centers of the coils are ninety elec- 
 trical degrees from the position shown in Fig. 353, only a skew- 
 ing effect results from the armature ampere-turns acting on the 
 field magnet, as is illustrated in Fig. 350. Multiplying the 
 ordinates of the current curve with the corresponding ordinates 
 of the curve of magnetic activity, the actual effect of the alter- 
 nating current in the armature coils may be obtained, as is 
 shown by the shaded areas oiftlined by curve k. It is seen that 
 the sum of the positive and negative effects is zero, but that they 
 produce the periodical skewing effect caused by the armature 
 
618 
 
 ALTERNATING CURRENTS 
 
 current upon the field magnetism as already explained. In the 
 same manner it is shown in Fig. 354 that if the current lags be- 
 hind or leads the voltage, alternate loops of the curve k are greater 
 than the others, and the field magnets are periodically either 
 weakened or strengthened. In this case, the effect of the back 
 turns is larger while the effect of the cross turns is smaller. 
 
 Fig. 354. — Curves showing the Magnetic Activity of the Armature as in Fig. 353, but 
 with the Current out of Phase with the Voltage. 
 
 Now, in the case of a balanced quarter-phase alternator, a 
 second curve of reaction exactly similar to k and one-fourth 
 the pitch, or 90 electrical degrees, from k , may be drawn as 
 shown in Fig. 355 to represent the action of the second set of 
 coils when the current is in phase with the induced voltage. 
 The vertical lines crossing Figs. 353, 354, and 355 are located 
 180 electrical degrees apart, as guides. The magnetizing effect 
 of one phase gradually increases while the magnetizing effect of 
 the other phase decreases, and the skewing effect becomes nearly 
 neutralized when the armature current and voltage in each 
 branch are in phase with each other. Similarly, when the cur- 
 rent lags or leads in the two branches of the two-phase circuit, 
 the armature magneto-motive forces add together so as to give 
 a more or less constant weakening or strengthening of the field 
 magnets, as shown in Fig. 355, where the ordinates of the hori- 
 
 Fig. 355. — Curves showing the Magnetic Activity of the Armature of a Quarter- 
 phase Generator. Currents in Phase with Induced Voltages. 
 
SYNCHRONOUS MACHINES 
 
 619 
 
 zontal line k" represent this effect. In Fig. 355 curves c and 
 c 1 represent the currents of the two phases ; d and d x are the 
 curves of activity of the coils of the two phases respectively ; 
 and k and k x are the reaction tendencies of the separate phases. 
 The poles are assumed to lie midway between the vertical broken 
 lines as in the previous two figures. In case the current leads, 
 the fields are strengthened, and in case it lags the fields are 
 weakened, as illustrated in Fig. 355 a. By the same process it 
 may be shown that the armature reaction in any polyphase 
 machine is practically constant when the system is balanced. 
 
 Fig. 355 a. — Curves showing the Magnetic Activity of the Armature as in Fig. 355, 
 but with the Currents out of Phase with the Induced Voltages. 
 
 The effect is possibly more simply explained by considering 
 the armature resultant magneto-motive force of a polyphaser as 
 producing a Rotating magnetic field, relatively to the armature 
 core, which core may either be stationary or rotate. This is 
 evidently permissible, as the armature windings and currents of 
 a balanced polyphase armature bear the same relations as those 
 in the primary circuit of a rotating field induction motor. The 
 creation of a rotating field is explained fully in the chapter 
 following.* It will suffice to say here that in a polyphase arma- 
 ture the magnetic fluxes of the phases grow successively to their 
 maximum values in a direction opposite to the direction of rota- 
 tion of the armature. Thus in a tri-phase armature a coil of 
 one phase, then the next phase to it against the direction of rota- 
 tion, and then the third in the same direction around the arma- 
 ture rise to maximum flux, thus giving an effect as though the 
 flux set up by the armature windings moved 360 electrical de- 
 grees with reference to a point on the armature during the 
 period of a cycle. Taking this view, and supposing first that 
 the armature is stationary and the field magnet revolves, it is 
 
 * Art. 185. 
 
620 
 
 ALTERNATING CURRENTS 
 
 evident that a rotating magneto-motive force may be considered 
 to be set up by the armature current having a speed equal to, 
 or in synchronism with, that of the mechanically rotating alter- 
 nator field magnet. This must be so since, when the alternator 
 field magnet moves through 360 electrical degrees of rota- 
 tion, it creates a complete cycle of currents in the several arma- 
 ture phases and hence the position of the resultant armature 
 magneto-motive force for the coils of each phase also moves 360 
 electrical degrees during the same period of time. When the 
 power factor is unity, the magneto-motive forces for the several 
 balanced phases substantially neutralize each other. When the 
 armature currents lag, their combined rotating magneto-motive 
 force lags in space with reference to the alternator field magnet, 
 causing a weakening of the field magnet by back magnetization. 
 When the currents lead, the armature magneto-motive force 
 steps forward in its space phase relation with the alternator 
 field magnet, and causes a strengthening of the latter. 
 
 Similar reasoning applies when the armature rotates and the 
 field magnet is stationary ; since, in that case, the magneto- 
 motive force set up by the armature currents may be considered 
 to stand in a fixed position with respect to the stationary field 
 magnet as the armature revolves, which fixed position depends 
 on the lag of the currents with respect to the induced voltages 
 in the branches of the polyphase windings. 
 
 The resultant magneto-motive force always rotates in rela- 
 tively the same direction about the armature as the relative 
 motion of the alternator field magnet, or against the direction of 
 rotation of the armature, as said before. This may be proved 
 by drawing the curves of phase currents and laying out the 
 space phase positions of the resultant armature magneto-motive 
 force and alternator magnet for a series of angular advances of 
 the armature or field. 
 
 The opposing and cross-magnetic effects of lagging armature 
 currents in alternating current generators, when operating 
 under usual conditions, cause the external characteristics * to 
 slope toward the horizontal axis. This effect must be added 
 to the slope of the characteristic caused by resistance and re- 
 actance in the armature. When the current leads, the voltage 
 tends to rise due to the strengthening of the field, a character- 
 
 * Art. 1 55. 
 
SYNCHRONOUS MACHINES 
 
 621 
 
 istic that is made valuable use of in maintaining uniform vol- 
 tage at the receiving ends of transmission lines through the use 
 of synchronous apparatus which is regulated to furnish a lead- 
 ing current.* The rise, however, may be made excessive by 
 the same effect when the conditions of the line and load are 
 abnormal. 
 
 It is evident that the effect of the armature reaction depends 
 for its relative instantaneous values upon the instantaneous 
 positions of the coils with reference to the poles, as already 
 pointed out ; and that it depends further for its actual instan- 
 taneous and average values on the current strength, the angle 
 of lag, the shape of the current curve, and the conditions of 
 balance in polyphase systems. The relative shapes of armature 
 coils and pole pieces also enter the relation, as do also the air 
 space and tooth reluctances, as well as the total reluctance of 
 the magnetic circuits. Since the effect of the reactions is 
 periodic in the case of single or unbalanced polyphase machines, 
 it is difficult to determine accurately its exact result in any 
 particular case, by any means except that of experiment. By 
 taking the instantaneous value of the single-phase current or 
 unbalanced polyphase current for a number of angular positions 
 of the armature, the average skewing and direct magnetizing or 
 demagnetizing effect of the armature reaction can, however, be 
 approximately determined. The reactions of polyphase ma- 
 chines, however, may be determined with a greater degree of 
 accuracy and ease for any load power factor when the sys- 
 tem is balanced, as has been shown by the previously given 
 equations. 
 
 153. Alternator Characteristics. — There are four curves which 
 exhibit particularly important relations between the functions 
 of alternators. These curves, which are called characteristics, 
 may be enumerated as follows : 
 
 1. The Saturation Curve or Curve of Magnetization. 
 
 2. The External Characteristic. 
 
 3. The Curve of Synchronous Impedance. 
 
 4. The Magnetic Distribution Curve and Voltage Curve. 
 
 154. Curve of Magnetization. — The curve of magnetization 
 shows the relation between the ampere-turns in the field wind- 
 ings and the total voltage induced in the armature. From the 
 
 * Art. 175. 
 
622 
 
 ALTERNATING CURRENTS 
 
 voltage induced in the armature, the value of <I> may be deduced 
 by means of the formula E = ’ P rov ided the value of K 
 
 is known.* The value of K cannot be determined exactly by 
 calculation, but may be ascertained by means of the fourth 
 curve named in the preceding article. The experimental 
 determination of the curve of magnetization is carried out 
 exactly as in the case of direct-current machines, substituting 
 for the direct-current voltmeter an instrument which is capable 
 of measuring alternating voltages. It is desirable that the 
 instrument used shall indicate the effective value of the voltage ; 
 hence, the measurements should be made by a voltmeter built 
 upon the principles of either the hot-wire instrument, the 
 electrostatic instrument modeled after the quadrant electrom- 
 eter, or the non-inductive form of high resistance electro- 
 dynamometer, arranged preferably for direct reading of voltage. f 
 All instruments used in alternating-current measurements 
 which depend for their indications upon electrodynamic action, 
 must be constructed with no masses of conducting metal about 
 them, or their constants will depend upon the frequency of the 
 current measured. This is due to the dynamic effects which 
 eddy currents, circulating in metallic masses, have on the cur- 
 rents in the moving parts of the instruments. If a voltmeter 
 has an appreciable inductance, its reading will also depend upon 
 the frequency, since the current flowing through it is inversely 
 proportional to VjR 2 -f- 4 tt 2 / 2 !?. From this it is readily seen 
 that if flu is not negligible in comparison with R< the current 
 flowing through the voltmeter when it is connected to a circuit, 
 and hence its indications, will be dependent upon the frequency. 
 The indications of an inductive voltmeter will always be 
 less when it is connected to an alternating-current circuit 
 than when it is connected to a continuous-current circuit of 
 equal effective voltage. The self-inductance of an electrody- 
 namometer arranged for use as an amperemeter is usually quite 
 small, but in some cases may reach a millihenry. An electro- 
 dynamometer which is arranged to be used as a voltmeter, 
 usually has a considerable non-inductive resistance in series 
 with the inductive working coils, so that its time constant is 
 quite small. 
 
 * Art. 20. 
 
 t Art. 24. 
 
SYNCHRONOUS MACHINES 
 
 623 
 
 If the alternator under examination is a special one and was 
 designed to be partially or entirely a self-exciting one, there is 
 some question of the comparative magnetizing effects of con- 
 tinuous and rectified currents. In general, however, the recti- 
 fied current in a self-excited field is doubtless always a wavy 
 one. The effective value of this, as indicated by an electrodyna- 
 mometer, is very nearly the same as the average value indicated 
 by an ordinary amperemeter. The average magnetizing effect 
 of the current is also practically equal to that of a continuous cur- 
 rent which gives the same indications on the instruments, though 
 the actual magnetic flux set up is always proportional to the 
 instantaneous value of the current divided by the magnetic circuit 
 reluctance, after due correction has been made for hysteresis 
 and the magnetizing effects of any eddy currents induced in the 
 field magnets by a pulsating excitation. A wavy current tends 
 to set up eddy currents in the iron of the magnetic circuit and 
 thus cause heating, but this result is not marked. 
 
 The exciting current of a separately excited alternator, the 
 type used almost exclusively in modern practice, may also be 
 caused to become wavy if the armature reactions are very large. 
 It has already been shown that the effect of the armature reactions 
 of a single-phase or unbalanced polyphase machine is a periodic 
 one ; and when the periodic effect becomes of sufficient magni- 
 tude, it causes fluctuations in the field magnetism, which react 
 upon the windings and throw the exciting current into pulsa- 
 tions. On account of this tendency of the exciting current to 
 become wavy and of the magnetic flux to vary locally in the pole 
 pieces, it is common to build the pole cores of laminations in 
 order to prevent the generation of excessive eddy currents in 
 them. 
 
 The general form of the curve of magnetization for an alter- 
 nator is similar to the form of the curve for a direct-current 
 dynamo. As it is not uncommon for alternators to have a some- 
 what larger reluctance in the air space than have direct-current 
 dynamos of the same size, the knee in the alternator curve is 
 sometimes not so abrupt as it is in the case of direct-current 
 machines (Figs. 356 and 357). For studying the details of 
 the design of the magnetic circuit, the curve may be re- 
 solved into component curves representing the air space, 
 frame, and armature. In this case the excitation compared to 
 
624 
 
 ALTERNATING CURRENTS 
 
 the magnetic flux 
 is a straight line 
 for the air space, 
 since the reluc- 
 tance of that part 
 of the magnetic 
 circuit is constant. 
 Figure 357 shows 
 the relation be- 
 tween exciting 
 current and ter- 
 minal voltage in 
 a particular alter- 
 nator when cur- 
 rent flows in the 
 armature. 
 
 155. External Characteristic. — An external characteristic is a 
 curve showing the relations between the armature current and 
 the terminal voltage which exist for a stated condition of field 
 
 Fig. 356. - 
 
 EXCITING CURRENT 
 
 - Type of Curve of Magnetization of an Alternator. 
 
 excitation. To ex- 
 perimentally deter- 
 mine an external 
 characteristic of an 
 alternator, it is ex- 
 cited by the method 
 for which it is de- 
 signed, so as to give 
 its normal voltage on 
 open circuit. The 
 volts at its terminals, 
 and the current and 
 kilowatts in the ex- 
 ternal circuit, are 
 measured with various 
 loads in the external 
 circuit of various 
 magnitudes but fixed 
 power factor. The 
 observations may be 
 plotted in a curve, 
 
 Fig. 357. — Relation of Terminal Voltage to Exciting 
 Current with Different Currents in the Armature. 
 
SYNCHRONOUS MACHINES 
 
 625 
 
 using volts as ordinates and kilovolt-amperes or kilowatts as ab- 
 scissas. In separately excited alternators, the curve cuts the ver- 
 tical axis at its highest point at no load, and then gradually falls, 
 the exciting current being held constant. The decrease of the 
 ordinates (drop in voltage) is caused by the effects of armature 
 resistance, armature self-inductance, and armature reactions. 
 
 The magnitude of the armature reactions is changed on 
 account of the lag or lead of the current, when there is re- 
 actance in the external circuit. If it is desired to quanti- 
 tively determine the effect of lag or lead on armature reactions, 
 curves may be taken with different values of reactance inserted 
 in the external circuit. The portion of the drop which is caused 
 by self-inductance and armature reactions when the machine is 
 supplying current to a load of unity power factor may be sepa- 
 rated from that caused by resistance by the formula, E? = E a 2 
 -P E In this case is the open-circuit voltage, E a is equal 
 to the terminal voltage for the current flowing plus IR a where 
 Ea is the armature resistance, and E, is the quadrature compo- 
 nent of the drop which results from self-inductance and arma- 
 ture reactions. By taking the characteristics of an alternator 
 when worked on circuits of different known resistances and re- 
 actances, the effect of armature reactions may be determined 
 for different values of the angle of lag, and such tests are quite 
 valuable for determining the purpose for which the machine is 
 best adapted. The self-inductance and reactions of some of 
 the early alternators were so great that the external character- 
 istic drooped to the X axis at a current not greatly exceeding 
 full load. Figure 358 shows the characteristics of two alter- 
 nators plotted to show the relations of terminal volts to arma- 
 ture current when the load is non-reactive. One (Curve A') is 
 of a standard type of alternator, having small armature self- 
 inductance and reactions, and the second (Curve B ) of a special 
 machine having large armature self-inductance and reactions. 
 Curves showing the relation of output and armature current 
 for the two machines are also shown. These are especially 
 enlightening as showing relatively how much lower the output 
 of the highly inductive machine is for the same proportions of 
 full load current. The two curves of machine B illustrate the 
 difficulty of obtaining good regulation or the economical utili- 
 zation of the copper and iron in the machine when the machine 
 
620 
 
 ALTERNATING CURRENTS 
 
 is highly reactive. As inductive reactance in the load has the 
 same effect upon these characteristics of operation as has react- 
 ance in the armature itself, it is evident from the figure that 
 inductive reactance should be avoided in both the machine itself 
 
 0 50 100 150 200 
 
 ARMATURE CURRENT IN PER CENT OF FULL LOAD CURRENT 
 
 Fig. 358. — External Characteristics. Machines worked on Non-reactive Loads. 
 Machine A with Small and Machine li with Large Armature Reactance. The Rela- 
 tions of Output to Current are also shown. 
 
 and its load so far as possible. The three following figures 
 (Figs. 359, 360, and 361) graphically indicate the effect of an 
 inductive load. Figure 359 shows the relation between voltage 
 drop and load power factor upon a generator of normal design 
 connected to inductive loads of different power factors. In 
 this case, the load was at the end of a transmission line. The 
 drops in the step-down transformers, line, and generator, and their 
 total are given for power factors ranging from 100 per cent to 
 20 per cent, inductive, when the full load rating of 250 kilovolt- 
 amperes are being delivered to the transmission line. Figure 
 360 shows the reduction in the kilowatts of capacity available 
 in this particular generator when used on the same range of 
 power factors and with a kilovolt-ampere output for the various 
 
SYNCHRONOUS MACHINES 
 
 627 
 
 Fig. 359. — Curves showing the Effects of the Power Factor of Inductive Loads or 
 the Regulation of an Alternator. 
 
 Fig. 360. — Curves showing the Relation of Inductive Power Factor to Kilowatts and 
 Kilovolt-amperes for the Alternator referred to in Fig. 359. 
 
628 
 
 ALTERNATING CURRENTS 
 
 power factors as shown by the upper curve. The lessening of 
 the capacity is clue to the fact that the output of the generator 
 is limited by the heating of the armature, and therefore when 
 the current contains a large quadrature component, the active 
 component must be relatively low. Figure 361 shows the field 
 exciting currents required by the alternator to supply normal 
 voltage to the transmission line at the same range of power 
 factors, when the alternator is operated so as to furnish 250 
 kilovolt-amperes, and it also shows the corresponding armature 
 core losses. 
 
 Eig. 361. — Curves showing the Variation of Field Exciting Current and Armature 
 Core Losses with the Power Factor of an Inductive Load for the Alternator referred 
 to in Figs. 359 and 360. 
 
 When the load has the effect of electrostatic capacity the 
 quadrature leading current tends to strengthen the alternator 
 field magnet and neutralize the effect of armature self-induct- 
 ance. It may, therefore, reduce or neutralize the drop in the 
 external characteristic, or cause it to rise, depending upon the 
 amount of the quadrature current flowing. 
 
 The external characteristic of a special, self-excited, shunt- 
 wound alternator is shown in Fig. 362. The droop in the curve is 
 a little greater than would occur for the same machine separately 
 
SYNCHRONOUS MACHINES 
 
 629 
 
 “2000 
 
 £1000 
 
 excited. This is due to 3000 
 the loss of exciting current 
 as the armature drop in- 
 creases, and the terminal 
 voltage decreases accord- 
 ingly- 
 
 The effect of armature 
 resistance and true self- 
 inductance can be shown 
 
 by drawing the loci of the p IG 352. — External Characteristic of a Special 
 alternator terminal voltage Self-excited Alternator. 
 
 — maintained constant by proper variation of the field excita- 
 tion — when the power factor of the load is varied, but the 
 load current is maintained constant. The construction is shown 
 in Fig. 363. 
 
 
 
 
 
 -■£2*2*4 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 13 
 
 ARMATURE AMPERE8 
 
 In this figure the circular arc EE x E^E^E i with the center at 
 0 is the locus of constant terminal voltage when the constant cur- 
 rent 01 flows but the power factor is varied. The triangle OXR 
 represents the voltage drops in the impedance of the armature 
 by reason of the flow of current OL Then the total voltage 
 generated, E', must be equal to the vector sum of OX and the 
 terminal voltage E. By geometrical construction it is seen 
 that the locus of E\ = OE', etc.) for various angles of lag 6 
 in the external circuit must be the circular arc E' E X E^E^E X 
 with its center at X. I 11 the figure, when 01 lags by the 
 angle 6 with respect to the terminal voltage, the latter is 
 represented by OE , the total voltage generated by OE ’, and 
 the impedance drop by EE', of which Ea is used in overcoming 
 self-inductance and aE' in overcoming resistance. The figure 
 shows the vector diagrams for lagging angles 6 and 0 V zero lag $ 2 , 
 and leading angles 0 3 and 0 X , where the thetas represent the 
 phase differences of the current and voltage in the external 
 circuit. It will be noted, in order that constant current may 
 flow with varying power factors, the load must be assumed to 
 change in its components of resistance and self-inductance or 
 capacity in such a way as to keep the impedance Z constant. 
 The line EX in the figure is assumed to be constant, which 
 is not absolutely true, as the reluctance of the self-induction 
 paths around the armature conductors will change with varying 
 values of theta. In this figure the values of total voltage in- 
 
630 
 
 ALTERNATING CURRENTS 
 
 duced, OE ' , OE^, etc., are due to cutting the total flux in the 
 air gap, which in turn is created by the vector sum of the field 
 ampere-turns and effective armature ampere-turns. The latter 
 
 Fig. 363. — Loci of Terminal and Total Voltages generated in an Alternator when 
 the Power Factor of the Load is varied but the Numerical Values of Terminal 
 Voltage and Current are maintained constant. 
 
 vary with the positions of the centers of the armature coils at 
 the instant they are carrying maximum current, as shown in an 
 earlier article, or approximately with sin 0.* This shows that 
 
 * Art. 152. 
 
SYNCHRONOUS MACHINES 
 
 631 
 
 when 9 is positive the field ampere-turns must not only create 
 sufficient flux to create the total voltage required, but must also 
 neutralize the effective armature-turns, but that when 9 is 
 negative the field ampere-turns directly add to the effective 
 armature ampere-turns to set up the required flux. A diagram 
 which expresses graphically the effect of armature-turns upon the 
 voltage generated 
 is shown in Fig. 
 
 364. T r ian g 1 e 
 OFC represents 
 the voltage triangle 
 of the external cir- 
 cuit, when the gen- 
 erator terminal 
 voltage is OC, the 
 angle of lag 9, and 
 the current 01. 
 
 Triangle CD A rep- 
 resents the internal 
 drops in the ma- 
 chine due to self- 
 inductance and re- 
 sistance, the total 
 drop being CA. 
 
 OA is the voltage that would be required to drive the cur- 
 rent 01 through an impedance equal to This voltage, 
 
 however, is less than would be generated by the same field 
 excitation on open circuit by reason of the demagnetization 
 caused by the current flowing in the armature coils. Repre- 
 senting the loss of voltage from this source by AB, then the 
 open circuit voltage is represented by OB. Or, a field excita- 
 tion which would create an armature voltage OB is required to 
 furnish a terminal voltage OC when current 01 flows at an 
 angle of lag 9 behind OC. The triangle OAB may then be 
 considered proportional to the ampere-turns of the machine, in 
 which OB is proportional to the useful field ampere-turns. AB 
 the armature ampere-turns, and OA the net ampere-turns effec- 
 tive in producing useful flux in the armature. It will be noted 
 that the total field ampere-turns must include the ampere-turns 
 
 Fig. 364. — Vector Diagram showing the Effect of Ar- 
 mature Magnetizing Ampere-turns upon the Voltage 
 generated in an Alternator. Armature Current lagging 
 behind Terminal Voltage. 
 
632 
 
 ALTERNATING CURRENTS 
 
 which set up leakage flux ; and these vary, since the leakage 
 increases with a positive lag of the current and decreases with 
 
 a negative lag. 
 Since OB does not 
 include ampere- 
 turns necessary to 
 set up leakage flux, 
 the figure is onty 
 approximate. Fig- 
 ure 365 is a diagram 
 showing- the effect 
 
 C D 
 
 Fig. 365. — Diagram showing the Effect of Armature 
 Magnetizing Ampere-turns upon the Voltage Gener- 
 ated iu an Alternator. Armature Current leading the 
 Terminal Voltage. 
 
 of armature magnetization when the current in the external 
 
 circuit leads the terminal voltage. 
 
 156. The Short-Circuit Current Curve and Synchronous Imped- 
 ance of an Alternator. — From the discussion preceding it is 
 evident that there is difficulty in separating the effects of true 
 self-inductance and armature reactions in an alternator, but 
 the combined effect of the two in connection with the armature 
 resistance may be studied to advantage by the use of the Short- 
 circuit current curve. This curve shows the relation between 
 the ampere-turns of the field magnet and the armature current 
 flowing when the armature terminals are short-circuited. The 
 curve may be obtained by placing an amperemeter in the field 
 circuit and short-circuiting the armature directly across its own 
 terminals except that a very low resistance amperemeter should 
 be placed in at least one of the leads. The field magnet may be 
 then excited by successively increasing values of exciting cur- 
 rent and the readings of the amperemeters observed and plotted, 
 this process being continued until the maximum safe value of the 
 armature current is reached. The machine should be driven 
 at normal speed. As in the case of the saturation curve, the 
 curves of rising and falling field excitation will differ on account 
 of the effect of hysteresis in the field magnet. In order to ob- 
 tain all the information desirable, a saturation curve for the 
 same excitations should also be obtained and should be plotted 
 on the chart with the short-circuit current curve. Figure 366 
 shows a short-circuit curve of an 1800-kilowatt, tri-phase ma- 
 chine, marked A. Curve B is the saturation curve of the 
 same machine, and the points R and S are the points for full 
 load rated voltage and current respectively. 
 
TERMINAL VOLTS 
 
 SYNCHRONOUS MACHINES 
 
 633 
 
 From these two curves an imaginary impedance may be 
 obtained corresponding with each armature current, which is 
 approximately equal to an actual impedance calling for the 
 same loss of voltage as is jointly caused by the armature resist- 
 ance, armature self-inductance, and armature reactions. This 
 is called the Synchronous impedance, and is sometimes defined 
 as the numerical ratio of the open-circuit voltage generated at 
 normal speed for a given field excitation to the short-circuit cur- 
 rent at the same speed and excitation. The term synchronous 
 impedance, however, can be better considered as designating the 
 
 7000 
 
 6000 
 
 5000 
 
 4000 
 
 3000 
 
 2000 
 
 1000 
 
 0 
 
 
 
 
 
 
 C/ 
 
 
 B 
 
 
 
 
 
 
 
 ^ / 
 
 
 r 
 
 
 
 
 
 J 
 
 r / 
 
 \ X 
 
 V 
 
 / 
 
 ✓ 
 
 
 
 
 
 
 
 
 ✓ 
 
 / 
 
 / 
 
 ' 
 
 
 
 
 
 
 
 / / 
 / / 
 
 x / 
 
 / | 
 
 / 
 
 / ' 
 ✓ 1 
 ✓ . 
 
 / 
 
 
 
 A 
 
 
 
 
 / / 
 / / 
 
 V' 
 
 / 
 
 / 
 
 / 
 
 
 
 
 
 
 
 A 
 
 v >.✓ 
 
 / 
 
 / 
 
 / ^00^ 
 
 
 K| 
 
 
 
 
 
 
 0 N M2000 4000 6000 8000 10000 12000 14000 16000 18000 
 
 AMPERE TURNS ON EACH FIELD COIL 
 
 Fig. 366. — Short-circuit Current Curve of an Alternator. A, Short-circuit Curve; 
 
 B, Magnetization Curve. 
 
 actual vector loss of voltage, due to armature resistance, self- 
 induction, and reactions, divided by the armature current flow- 
 ing. Its value, therefore, changes with the angle of lag of the 
 armature current on account of the resultant change in effective 
 armature magnetizing turns. 
 
 A curve which is useful in determining the excitation 
 required to give a desired terminal voltage or vice versa , when 
 a given current flows at a given angle of lag, is marked MGrL 
 in Fig. 366 and is sometimes called the wattless current char- 
 acteristic. The curve is constructed in this way: assume some 
 
 ARMATURE AMPERES PER PHASE 
 
634 
 
 ALTERNATING CURRENTS 
 
 armature current, say 50 amperes, and lay off the distance OM 
 equal to the number of ampere-turns per pole that are shown by 
 the short-circuit curve to be necessary to cause a short-circuit 
 current of the chosen value to flow through the armature cir- 
 cuit. Then lay off MN, which is the back-magnetizing turns 
 per pole caused by the armature current of 50 amperes when 
 flowing at an inductive lag angle of 90° and is computed for a 
 polyphase machine by the formula on page 616 and for a single- 
 phase machine by the formula on page 615. To do this the ar- 
 mature resistance must be considered negligible, which for 
 ordinary machines is admissible. Then ON represents the am- 
 pere-turns per pole in the field windings which are effective in 
 setting up useful flux, and I)N (which is the induced voltage 
 corresponding to field ampere-turns ON) represents the induc- 
 tive drop in the armature. The field reaction of a constant 
 quadrature-lagging current in the armature, and the self-induct- 
 ance set up thereby, are constant whatever the terminal voltage 
 may be, provided the effect of saturation may be neglected, and 
 therefore if the machine is loaded with pure inductive reactance 
 and the field current modified so as to keep the armature cur- 
 rent constant, the relation between terminal voltage and field 
 current will be expressed by curve MOL of Fig. 366, the offsets 
 HGr, PL , etc., being all parallel and equal to PM. Then, if 
 the reactive load is such that the terminal voltage is OK and 
 the ampere-turns per field spool are ON Gr will be one point on 
 the wattless current characteristic and GrH must equal and be 
 parallel to PM. The curve MGrL is evidently constructed by 
 drawing it through the extremities of lines drawn from curve 
 B equal and parallel to PM. For greater currents MGL is 
 further removed from B. 
 
 The line 00 represents approximately the ampere-turns re- 
 quired by the air space. 
 
 157. The Magnetic Flux Distribution Curve of an Alternator, 
 and Curves of Voltage and Current. — These curves may be ex- 
 perimentally determined by various methods. They consist of a 
 series of curves which are closely interrelated, but may be, and in 
 fact are likely to be, of quite dissimilar forms. The form of the 
 curve representing the wave of impressed voltage, or total vol- 
 tage induced in the armature of an alternator, is dependent upon 
 the distribution of the magnetic flux over the pole faces, and also 
 
SYNCHRONOUS MACHINES 
 
 635 
 
 on the arrangement of the armature windings. By design, either 
 of these may be given a controlling influence to the exclusion of 
 the other. A two-pole machine with closed-circuit, evenly dis- 
 tributed, Gramme ring armature winding, as shown in Fig. 30, 
 gives an excellent illustration of this, when it has two connections 
 180 degrees apart to two collector rings, thus connecting the 
 two halves of the winding in parallel between the rings. The 
 differential action, which occurs in the coils of such an arma- 
 ture, makes its curves of voltage almost independent of the 
 distribution of the magnetism over the pole faces, provided 
 the same total number of lines of force is cut by each con- 
 ductor per revolution; and the maximum value of the voltage 
 is entirely independent of the magnetic flux distribution. 
 This is shown by Fig. 367, where the full line in the left- 
 hand drawing shows the curve of voltage developed in such 
 
 Fig. 367. — Voltage Curves produced by a Two-pole, Gramme Ring Armature 
 Winding having Two Coils in Parallel and each Coil covering 180 Electrical De- 
 grees, when subjected to Magnetic Fluxes of Differing Distribution. Full Lines 
 represent Voltages and Dotted Lines Magnetic Distribution under the Pole Faces. 
 
 an armature winding with a uniform distribution of the 
 magnetic flux of the field, and the full line in the other draw- 
 ing shows the curve of voltage with the same total field greatly 
 distorted. The dotted lines in the drawing indicate the dis- 
 tributions of the magnetism over the pole faces. The unidirec- 
 tional voltage delivered by the armature when the windings 
 are connected to a commutator and the machine is operated 
 as a direct-current dynamo, is not affected by the distortion, 
 provided the brushes are always placed on the neutral plane 
 and the total magnetism passing through the armature re- 
 mains constant. When the machine has been converted into 
 an alternator by the addition of collector rings as stated, the 
 maximum instantaneous voltage is equal to the voltage devel- 
 oped in the direct-current machine, and is therefore independ- 
 
636 
 
 ALTERNATING CURRENTS 
 
 ent of the distribution of the magnetism,* and the form of the 
 curve representing the wave of voltage is but slightly altered, 
 as shown in Fig. 367. Now suppose the same machine to be 
 arranged with a single narrow coil on the armature. The 
 change in the magnetic distribution now not only changes 
 the form of the voltage curve proportionally, but also changes 
 in a marked manner the maximum value of the instantaneous 
 voltage. The difference between the curves of voltage devel- 
 oped by the broad coil and narrow coil armatures is due to 
 the effect of differential action in the broad coil armature. 
 
 It has already been shown that differential action occurs 
 to some degree in commercial alternators, but it does not 
 occur to a sufficient degree to make the form of the voltage 
 wave independent of the magnetic distribution. It is there- 
 fore true that the magnetic distribution largely influences 
 the form of the voltage wave, and the magnetic distribution 
 should therefore be carefully studied during the development 
 of a type of alternators. A proper study of the magnetic 
 flux distribution and of the arrangement of the armature 
 windings makes it possible to so design an alternator that 
 it will produce any desired form of voltage wave. 
 
 The angular relation between the curves representing the 
 magnetic flux distribution and the impressed voltage is inter- 
 esting. The ordinate of the curve of voltage at any instant is 
 proportional to the rate at which lines of flux are cut by the 
 armature conductors at that instant, the rate being taken alge- 
 braically. Consequently, the voltage is zero when the mag- 
 netization is all symmetrically threaded through the coils; that 
 is, when the algebraic net rate of cutting flux by the conductors 
 is zero. The voltage is a maximum when the rate of cutting 
 flux is the greatest ; that is, when the algebraic summation of the 
 number of flux lines threaded through the coils is a minimum. A 
 curve which represents the algebraic summation of the number 
 of flux lines threaded through the coils at each instant, therefore, 
 has an angular position which is 90° from that of the voltage 
 curve (Fig. 368). The form and dimensions of this curve of 
 magnetic linkages evidently depend upon the actual distribu- 
 tion of the magnetic flux and the arrangement of the armature 
 windings. When the voltage curve is irregular, and the angular 
 
 * Art. 20. 
 
SYNCHRONOUS MACHINES 
 
 637 
 
 relation is not made evident from the curve, as in Fig. 369, 
 the point at which the curve of magnetic linkages cuts the 
 X-axis may yet be easily found, 
 since it is directly under the center 
 of gravity of the voltage curve. 
 
 That is, since as manj r lines of flux 
 must be withdrawn from the coils 
 as are inserted, for each loop of the 
 curve, the summation of the ordi- 
 nates of the voltage curve on each 
 side of the crossing must be equal. 
 
 This curve, which shows the alge- 
 braic number of magnetic lines of flux 
 which are threaded at each instant 
 through the coils, may be deduced from a known curve of voltage. 
 Erect an ordinate to the voltage curve which bisects the area as 
 
 in Fig. 369 ; then 
 by means of ordi- 
 nates divide the half 
 areas into a number 
 of small areas. The 
 magnetism threaded 
 through the armature 
 coils is algebraically 
 equal to zero at the 
 instant represented 
 by the bisecting or- 
 dinate, and the al- 
 gebraic value of the 
 magnetic leakages 
 through the armature 
 
 Fig. 368. — Curves showing Relation 
 of Magnetic Flux Linkages and 
 Induced Voltage in an Alternator. 
 
 Fig. 369. — Alternator Voltage and Magnetic Linkage coils at anv other ill- 
 Curves. Area of Voltage Curve is bisected. J . 
 
 stant is proportional 
 
 to the area inclosed by the voltage curve between the corre- 
 sponding ordinate and the bisecting ordinate, since e = ~ and 
 
 do 
 
 $ = "Zedt, where e is instantaneous voltage, <fi magnetic flux 
 linkages, and t time. Therefore, the ordinates of the curve 
 representing the magnetic linkages through the coils are, at 
 the instants represented by the ordinates which divide the 
 
638 
 
 ALTERNATING CURRENTS 
 
 voltage curve into small areas, proportional to the area be- 
 tween the corresponding instantaneous voltage ordinate and the 
 bisecting ordinate. The full process is, therefore, as follows : 
 lay off on each ordinate a length from the X-axis proportional 
 to the area inclosed by the voltage curve between the respec- 
 tive ordinate and the bisecting ordinate. The points thus 
 found are points on the desired curve. It is evident that the 
 maximum ordinate of the curve comes at the instant when E 
 is equal to zero. The length of this ordinate is equal to <J>, the 
 maximum value of the magnetic flux linkages, and the scale of 
 the curve may thus be conveniently fixed. The loops may not 
 be symmetrical, but, with a fixed value of <E> and a fixed arma- 
 ture winding, the successive loops must always be exactly alike, 
 though they may be looked upon as alternately positive and neg- 
 ative, since the magnetic flux is alternately threaded through 
 the coils in opposite directions. 
 
 The corresponding curves of voltage and magnetic linkages 
 for various forms of voltage curves are shown in the accompany- 
 ing figures. In Fig. 369 the voltage curve is one experimen- 
 tally determined from a special alternator when working on a 
 series load of arc lights.* In Fig. 370 the voltage curve is an 
 
 Fig. 370. — Curve e is a Triangular 
 Voltage Curve ; Curve m, Magnetic 
 Flux Linkages. 
 
 Fig. 371. — Curve e, Sinusoidal Voltage 
 Curve; Curve in, Magnetic Flux 
 Linkages. 
 
 isosceles triangle, in Fig. 371 it is a sinusoid, and in Fig. 372 it 
 is a rectangle. The voltage curves of Figs. 373 and 374 are 
 respectively a flat-topped curve and a parabola. f 
 
 Since the induced voltage is proportional to the rate of 
 change of the number of lines of flux threaded through the 
 
 * Tobey and Walbridge, Stanley Alternate-Current Arc Dynamo, Trans. 
 Amer. Inst. E. E., Vol. 7, p. 367. 
 
 f Emery, Alternating Current Curves, Trans. Amer. Inst. E. E., Vol. 12, 
 p. 433. 
 
SYNCHRONOUS MACHINES 
 
 639 
 
 Fig. 372. — Curve e, Rectangular Voltage 
 Curve ; Curve m, Magnetic Flux Link- 
 ages. 
 
 Fig. 373. — Curve e, Flat-topped Vol- 
 tage Curve ; Curve in, Magnetic Flux 
 Linkages. 
 
 armature coils, the ordinates of the voltage curve are pro- 
 portional to the tangents of the curve representing the number 
 of magnetic linkages through the armature coils. Figure 375 
 shows a graphical construction for determining the voltage 
 curve from this magnetic flux 
 linkage curve. Ordinates Oa' , 
 
 Ob ' , and OA are, by construc- 
 tion, made proportional in 
 length to the tangents of the 
 angles made with the X-axis 
 by lines tangent to the mag- 
 netic curve at p v p 2 , and 0. 
 
 The point O' is a point taken 
 at any convenient position, and the lines O' a ' , O'b ', O' A are 
 drawn parallel respectively to aa , bb, and the tangent to the 
 
 Fig. 374. — Cur ve e, Parabolic Voltage Curve ; 
 Curve in, Magnetic Flux Linkages. 
 
 Fig. 375. — Construction showing Method of Determining Voltage Curve from th« 
 Magnetic Flux Linkage Curve. 
 
640 
 
 ALTERNATING CURRENTS 
 
 magnetic curve at 0. The points of intersection of horizontal 
 lines drawn from a b ' , etc., and vertical lines drawn from p v 
 p v etc., are points y> 2 ', etc., which are located on the re- 
 quired voltage curve. 
 
 Another directly useful magnetic flux distribution curve is 
 one showing the distribution of the lines of flux over the pole 
 faces. This curve is analogous to the magnetic flux distri- 
 bution curve of direct-current machines. It may be experimen- 
 tally determined by using a narrow test coil connected to a 
 ballistic galvanometer and fastened to the armature surface. By 
 rotating the field magnet or armature by equal small increments 
 through 360 electrical degrees, the galvanometer readings 
 plotted in the form of a curve will have ordinates proportional 
 to those of the curve of magnetic flux distribution. If the 
 number of turns of the test coil and the constants of the gal- 
 
 O 
 
 vanometer are known, the number of flux lines cut by the coil 
 are at once determinable. When this method is used, it is 
 convenient to wrap the test coil across the face of the armature 
 and around the core like a ring winding when the armature 
 construction permits this arrangement. The number of turns 
 needed in the test coil depends upon the density of the mag- 
 netic flux and the constants of the galvanometer. It can readily 
 be determined by trial. Another method is to make a narrow 
 test coil of pancake shape as long as the armature coils. This 
 is laid upon the armature surface. The field current is then 
 reversed and the galvanometer throw read. The field magnet 
 or armature is then rotated a distance equal to the width of the 
 coil and another reading taken, and so on through 360 electrical 
 degrees. The sides of the coil in this case should be very nar- 
 row so that all of the turns will inclose essentially all of the 
 magnetic flux passing within the boundary of the coil. 
 
 If the average distribution of magnetic flux over the pole 
 faces of a machine is known, it is evidently possible to approxi- 
 mately determine the form of the voltage curve which will be 
 produced b} r any particular arrangement of the windings. It is 
 also equally possible to determine the arrangement of the wind- 
 ing required to give any desired form of voltage curve. Again, 
 if a particular form of winding is desirable, the magnetic flux 
 distribution which is necessary to give a desired voltage curve 
 may be determined. This distribution may then be used as a 
 
SYNCHRONOUS MACHINES 
 
 641 
 
 guide in designing the width and shape of the pole faces. The 
 application of the magnetic flux distribution curve is illus- 
 trated in Fig. 376. The dimensions and form of the pole pieces 
 and of an armature coil belonging to an alternator are indi- 
 cated in the figure. The ordinates of the line ABCD repre- 
 sent the magnetic density in the air space. When the coil is in 
 the instantaneous position represented, the value of the voltage 
 is zero. As the coil moves, each conductor cuts lines of flux. 
 Suppose that in a frac- 
 tion such as a twelfth of a 
 period the coil has moved 
 from the position indi- 
 cated by the letters x, y , 
 to x ', y' . During this 
 motion each conductor 
 has cut a certain number 
 of lines of force, and the Fig. 370. — Diagram indicating Use of Magnetic 
 number cut by all the con- Flux Distribution Curve for determining Form 
 , . , of Voltage Curve, 
 
 ductors is approximately 
 
 proportional to the sum of the areas of the curve ABCD taken 
 from x to x' and from y to y' . The shorter the step taken, the 
 more accurate this becomes. The average voltage developed 
 during this interval is also proportional to the same area. 
 Consequently an ordinate which is numerically equal to the 
 area may be erected at the point corresponding to the inter- 
 mediate position of the coil during the increment motion, to 
 approximately represent the voltage at that point in the revo- 
 lution of the armature. This proceeding may be repeated step 
 by step through the half period, taking the algebraic summation 
 of the voltage developed in the two halves of the coil, and 
 the outline of the voltage curve which corresponds to the par- 
 ticular arrangement of armature conductors and flux distri- 
 bution is thus determined. 
 
 The curve representing the current wave also represents, 
 when taken to the proper scale, the curve of active voltage. 
 From preceding pages it is evident that the curves of active 
 and impressed voltages have the same forms and are superposed, 
 if the circuit in which they act is non-reactive. They have the 
 same form, also, when the circuit is reactive if the impressed 
 voltage is sinusoidal, provided the inductance or capacity 
 
642 
 
 ALTERNATING CURRENTS 
 
 causing the reactance is independent of the instantaneous values 
 of the voltage and the current and the armature reactions are 
 approximately uniform. The condition of a uniform inductance 
 can only hold when no iron is inclosed in any portion of the circuit. 
 In general this condition is not found in commercial service. 
 Even with a uniform inductance the curves of impressed and 
 active voltages will not coincide in phase, since phase coinci- 
 dence between them can occur only when the circuit is without 
 either inductance or capacity, or these exactly neutralize each 
 other. As the latter is also a condition not often found in 
 commercial service, it may be said that, in general, curves of 
 impressed and active voltages are neither similar in form nor 
 coincident in phase; but they are always, perforce, of the 
 same frequency, though usually composed of more than one 
 harmonic. 
 
 Fig. 377. — Diagram arranged to show the Angular Space Position of Alternator Pole 
 Cores with the Angular Time Value of Voltages or Currents. 
 
 The curves are usually plotted to rectangular coordinates; 
 magnetic densities, magnetic linkages, instantaneous voltages, 
 or instantaneous currents being plotted as ordinates, and elec- 
 trical degrees or time as abscissas. To more definitely locate 
 the phases it is not unusual to indicate the position of the pole 
 pieces by laying them off at the top or the bottom of the plot 
 (Fig. 377), a method used heretofore in this text. 
 
 To make a complete study of the magnetic flux distribution 
 in an alternator, the machine should be worked at various 
 loads under different conditions of current lag.* Since, at any 
 instant, the effect of the armature current on the magnetization 
 
 * Art. 155. 
 
SYNCHRONOUS MACHINES 
 
 t>43 
 
 of the pole pieces depends upon the positions of the armature 
 conductors as well as the strength of the current, the effect of the 
 magneto-motive force of the armature is evidently a variable, 
 and consequently the distribution of the magnetic flux will not 
 be constant in a single-phase machine. In other words, both 
 the cross and back magneto-motive force of the armature wind- 
 ing for any load vary continuously during each period, and 
 therefore the magnetic flux distribution varies with the position 
 of the armature. The variation due to this cause is probably not 
 very great in machines with multitooth armatures, and an aver- 
 age distribution may be assumed as satisfactorily representing 
 working conditions. In polyphase machines where the system 
 is balanced, the magneto-motive forces of the windings of 
 the several phases of the armature combine to form an approxi- 
 mately constant reaction; unbalancing, however, destroys the 
 uniformity of the resultant and the reactions vary in value, 
 depending upon the amount of unbalancing.* In machines 
 with only a few teeth, the effect of armature reactions and the 
 movement of the teeth across the pole faces is often sufficient 
 to cause regular 
 pulsations in the 
 field flux and ex- 
 tremely marked 
 distortions in the 
 voltage curve. 
 
 The pulsations are 
 sometimes suffi- 
 cient to materially 
 affect the exciting 
 current. Figure 
 378, Curve II, 
 shows an experi- 
 mentally deter- 
 
 n " j, , i Fig. 378. — Curve illustrating tne Variation of Field 
 mined curve of the Current caused by Poorly Designed Armature Teeth, 
 field current of a Curve I is the Irregular Voltage Wave, and Curve II 
 shows the Pulsations in the Field Current due to the 
 Reactions of the Teeth and the Armature Current. 
 
 nator having a very 
 
 deeply toothed armature core with one tooth per coil and a large 
 self-inductance in the armature. The machine was excited by a 
 
 * Art. 152. 
 
 single-phase alter- 
 
644 
 
 ALTERNATING CURRENTS 
 
 small shunt-wound exciter.* The very irregular wave of vol- 
 tage is shown by curve I. 
 
 Various methods may be used for experimentally determin- 
 ing the form of electric voltage or current curves. 
 
 1. The Oscillograph. — Much the most efficient method of 
 tracing alternating current and voltage curves is by means 
 of the Oscillograph, as originally worked out by Blondel and 
 Duddell, but now become one of the important measuring in- 
 struments for alternating current phenomena. This instrument 
 
 Fig. 379. — Diagrams showing the Essential Features of a Three-element Oscillo- 
 graph. I, View from the Side. II, View from Above. 
 
 in essential parts consists of a loop of wire ribbon placed in a 
 strong field of constant magnetic flux with the plane of the loop 
 parallel to the lines of force of the field. Figure 379 is a dia- 
 gram of the optical system of an oscillograph having three 
 independent loops placed at V V, and V. Sketch I is a side 
 view and II a view looking from above. The curve desired 
 may either be made visible upon the screen D or be photographed 
 on a sensitized film wrapped upon the drum T. First consider- 
 ing the visible record: an arc light at F throws a beam of light 
 
 D 
 
 * See Trans. Amer. Inst. E. E., Vol. 7, p. 374. 
 
SYNCHRONOUS MACHINES 
 
 645 
 
 through the condensing lens E to the prisms _P, _P, P , where it 
 is deflected in three beams and passes to mirrors fastened to the 
 loops at E, V, V, each mirror being fastened at one edge to one 
 strand of a loop and at the other edge to the other strand of the 
 same loop. The loops are so designed that the two wires 
 of the sides have a very rapid natural rate of vibration — some- 
 times as much as 10,000 vibrations per second, though more 
 frequently about one half that number. The beams of light 
 thrown upon the mirrors V, V, V pass through a long cylindrical 
 lens at C 1 and fall upon a long mirror shown at B which oscil- 
 lates on a horizontal axis. At the focus of C is placed a semi- 
 opaque screen D. When an alternating current, the curve of 
 which is to be determined is passed through a loop, the magnetic 
 reactions between the current in the two sides of the loop and the 
 constant field flux tend to turn the mirror at V to the left or 
 right, depending upon the instantaneous direction of the current. 
 Since the natural rate of vibration of the loop is much higher 
 than that of the current or its harmonics, the curve of which 
 is to be traced, the mirror changes its angular position substan- 
 tially proportionately to the instantaneous values of the current. 
 
 The beams of light from V, V, V move over the length of 
 mirror B. Mirror B is oscillated at one half synchronous speed 
 compared with the frequency of the current through the loop 
 wires. The beams of light are cast by B upon the semiopaque 
 screen at JD. Each is then subjected to the resultant of the 
 two mirror movements, i.e. a movement by B which is uniform 
 and is proportional to the angular advance of the voltage of 
 the circuit, and a movement at right angles thereto caused by 
 the mirror at V, V, V, which is at each instant proportional to 
 the strength of current in the loop wires. The beams of light 
 thus trace upon the screen curves in which the abscissas caused by 
 B are always proportional to the angular advance during a cycle 
 of the impressed voltage and in which the ordinates are propor- 
 tional at each instant to the current strength in the loop wires. 
 The curves are, therefore, of the same character as those obtained 
 by the ordinary method of plotting alternating current waves. 
 
 By using the several oscillograph loops at once and casting 
 the beams of light from their mirrors upon the same mirror B, 
 curves of three currents may be traced upon the screen at the 
 same time and their relative phase positions and forms compared. 
 
646 
 
 ALTERNATING CURRENTS 
 
 The wave cycles are repeated over and over at the frequency of 
 the impressed voltage, and for ordinary frequencies appear like 
 fixed curves on the semiopaque screen. To prevent the curve 
 from being shown double, the shutter S is driven in synchron- 
 ism with the vibrating mirror B in such a way as to cut off the 
 light from F during every other half oscillation of B. 
 
 When the photographic apparatus is to be used the beams of 
 light from V, V, V pass through the long cylindrical lens C 2 and 
 strike the photographic film placed upon T, (7 a and B being 
 moved aside. The drum T is rotated at a desirable speed and 
 the shutter S opens and closes at the beginning and end of one 
 of its revolutions. The movement of the drum obviously de- 
 scribes the abscissas of the curve. 
 
 Fig. 380. — Vibrating Mechanism and Electro-magnets of One Type of Oscillograph — 
 
 Cover removed. 
 
 Figure 380 shows the vibrators and magnets of a commercial 
 type of three element oscillograph. In this figure, Gr is one 
 of the openings for admitting the three beams of light to the 
 galvanometer or loop mirrors just inside. B is one of the elec- 
 tromagnets. The cells holding the loops, sometimes called 
 vibrators, one of which is shown at W. are immersed in a liquid 
 to deaden the natural vibrations. 
 
SYNCHRONOUS MACHINES 
 
 647 
 
 Figure 381 shows the vibrator element at A, which is made 
 of fine silver or bronze ribbon. The part between the bridges 
 B and B 1 is between the magnet poles and to that part is at- 
 tached the mirror. 
 
 To obtain curves of current, current transformers may be re- 
 quired, or sometimes the loop is shunted around a non-reactive 
 resistance device through which the current to be measured 
 flows. In measuring voltages it is in ordinary cases necessary 
 to use a constant voltage trans- 
 former to cut down the voltage 
 impressed upon the loop as well 
 as to place non-inductive resist- 
 ance in series in the loop circuit. 
 
 A forerunner of the oscillo- 
 graph is due to Gerard and may 
 sometimes be used for tracing 
 the voltage curves of an alter- 
 nator when no current flows in 
 the armature. The machine to 
 be tested is rotated at a very slow 
 speed, the field being excited in 
 the usual manner. The terminals 
 of the machine to be examined 
 are connected to a shunted d’Ar- 
 sonval galvanometer. The nat- 
 ural rate of oscillation of the 
 galvanometer bobbin is made 
 quite rapid, as compared with 
 the period of the voltage supplied 
 by the alternator at its slow 
 speed. Then the deflection of the needle at each instant will 
 be proportional to the instantaneous voltage. By moving a 
 sheet of sensitized paper before the galvanometer mirror which 
 throws upon it a beam of light, the paper being moved at right 
 angles to the mirror movement, the curve of voltage may be 
 permanently recorded.* 
 
 Figure 382 is a current curve traced by an oscillograph of a 
 current containing a marked peak caused by highly saturated 
 iron cores in coils in series with the circuit. Figure 383 is a 
 
 Fig. 381. — An Oscillograph Vibrator. 
 
 * See Gerard’s Le$ ons sur l' Electricite, Vol. 1, p. 565, 3d ed. 
 
648 
 
 ALTERNATING CURRENTS 
 
 Fig. 382. — Peaked Current Curve traced by an Os- 
 cillograph. 
 
 voltage curve of an 
 alternator which is 
 almost sinusoidal in 
 form. In both cases 
 the waves were taken 
 by means of the pho- 
 tographic attachment 
 to the instrument. 
 
 2. 3Iethods using 
 Contact Makers. — 
 A large number of 
 methods require the 
 use of a revolving 
 contact maker of 
 some kind, and they 
 therefore have much 
 in common. These 
 were much used in 
 the past and are use- 
 ful for special pur- 
 
 poses or where an os- 
 cillograph is not avail- 
 able. The principal 
 differences in the 
 methods relate to the 
 types of instruments 
 used to give the indi- 
 cations, and the con- 
 venience with which 
 the manipulations may 
 be made. Whether 
 current or voltage 
 curves are to be ob- 
 tained, only instanta- 
 neous voltage measure- 
 ments are made. For 
 the former, the instan- 
 taneous voltages ai’e 
 taken at the terminals 
 
 Fig. 383. — Sinusoidal Alternator Voltage Curve traced 
 by au Oscillograph. 
 
SYNCHRONOUS MACHINES 
 
 649 
 
 of non-reactive resistance, and the instantaneous currents are 
 readily deduced. 
 
 In one of the earlier methods, advanced and used by Joubert, 
 the terminals of the alternator armature, or of one bobbin of 
 the armature, are recurrently connected to a condenser in the 
 following manner : One armature terminal is connected perma- 
 nently to one terminal of the condenser; the other armature 
 terminal is connected to a rotating point which may be put into 
 brief connection with the free terminal of the condenser when 
 the armature is at any desired point in its rotation. At the 
 instant this contact is made, the condenser receives a charge 
 which is proportional to the instantaneous voltage between the 
 terminals. The charge may be measured by discharging the con- 
 denser through a ballistic galvanometer and the voltage computed 
 from the amount of charge and the capacity of the condenser. 
 By successively setting the contact maker so that the instant 
 of contact corresponds with various points in the revolution of 
 the armature, the correspond- 
 ing instantaneous voltages may 
 be thus measured and the curve 
 of voltage may be plotted (Fig. 
 
 384). The contact maker used 
 by Joubert was an insulated 
 pin set in the armature shaft, 
 against which a brush could be Fig. 384. — Approximate Curve of Voltage 
 made to bear at any point in the obtained by the Use of a Contact Maker. 
 
 revolution. A quadrant electrometer may be used in place of 
 the condenser and ballistic galvanometer ; in which case it is 
 desirable to introduce a condenser permanently in parallel with 
 the electrometer, to neutralize the effect of leakage in the test 
 circuit. 
 
 Joubert’s investigations made in 1880 resulted in the first 
 determination of the curve of voltage of an alternator. The 
 investigations of Duncan, Hutchinson, and Wilkes probably 
 produced the earliest series of experimental curves showing the 
 relations between the waves of voltage and current in circuits 
 of different kinds. The investigations of Searing and Hoffman 
 were probably the first made upon an alternator with iron in the 
 armature core. Their results showed the curve of voltage de- 
 veloped in a smooth-core drum armature with wide coils to 
 
650 
 
 ALTERNATING CURRENTS 
 
 approach a sinusoid. Ryan and Merritt, in 1889, did much 
 valuable work in studying the current and voltage relations in 
 a transformer. Various methods were used and desirable data 
 obtained by Blondel, Searing, Hoffman, Mershon, Duncan, Ayr- 
 ton, Pupin, and others. The last named used a method for 
 determining harmonics by means of resonance, while Professor 
 Ayrton proposed obtaining the harmonics by means of the vibra- 
 tions of a stretched wire. 
 
 The earliest and simplest contact maker was, as already 
 pointed out, simply an insulated pin set in the shaft of the 
 alternator furnishing the current for the test. With this was 
 a brush so arranged as to make contact with the pin at any 
 desired point in the revolution. This arrangement is often 
 inconvenient of application, and is likely to give rather irregular 
 results. The contact is likely to be variable in resistance and, 
 as the brush wears, the duration of contact varies. Each of 
 these variables introduces errors of greater or less magnitude, 
 depending upon the conditions of the test. Various refine- 
 ments of construction have been introduced by experimenters 
 in order that the defects of the contact makers may be eliminated. 
 If a contact of absolute uniformity were assured, special instru- 
 ments would not be necessary for taking the indications in 
 determining voltage and current curves, because the indications 
 of a sensitive electrodynamometer might then be directly used. 
 
 The contact maker was originally arranged for a single con- 
 tact, but it is frequently desirable to make simultaneous observa- 
 tions of several curves. This may be readily accomplished by 
 using a contact maker with the appropriate number of contact 
 disks on the same spindle. Then a satisfactory instrument, 
 such as an electrostatic voltmeter, may be used in each circuit. 
 
 Fig. 385. — • Contact Maker with a Flexible 
 Shaft. 
 
 Sometimes it is not conven- 
 ient to have the contact 
 maker attached to the dy- 
 namo shaft, in which case it 
 may be attached to a short 
 length of flexible shaft (Fig. 
 385), which may in turn be 
 attached to the djnamo 
 shaft, but the flexible shaft 
 must possess uniform tor- 
 
SYNCHRONOUS MACHINES 
 
 651 
 
 sional rigidity or the rate of rotation of the contact maker may 
 be unsteady, or the instrument may “ hunt,” and thereby in- 
 troduce error by causing variations of the contact duration. 
 When connection to the alternator cannot be conveniently 
 made, the contact maker may be driven by a synchronous motor 
 as has been done by Blondel, Siemens and Halske, and Fleming. 
 In this case, “ hunting ” by the synchronous motors is likely to 
 cause errors. 
 
 The contact maker shown in Fig- 385 indicates the attach- 
 ment of the contact brush to a scale, whereby it can be moved 
 to any desired angular position and there make contact once in 
 a revolution with the contact button on the revolving drum. 
 
 The plan of charging and discharging a condenser to measure 
 the instantaneous voltage is not convenient, and various other 
 expedients have been proposed and more or less used. A gal- 
 vanometer with a sufficiently great natural rate of vibration will 
 be steadily deflected by the succession of impulses which it re- 
 ceives when connected in circuit with a contact maker. This 
 deflection may be balanced by a steady voltage which is intro- 
 duced in the circuit in series with the galvanometer and contact 
 maker as indicated in Fig. 386. A telephone put in the place 
 
 - J o- 
 
 CONTACT MAKER 
 
 REVERSING KEY 
 
 © 
 
 GALVANOMETER 
 
 of the galvanom- 
 eter gives oral - 
 instead of ocular 
 notice of a bal- 
 ance. It is de- 
 sirable to place a 
 condenser in par- 
 
 all Q l with the gal ^ IG ' — Potentiometer Arrangement for Contact Maker. 
 
 vanometer or telephone when this arrangement is used. In 
 Fig. 386, cd represents a graduated rheostat with a moving 
 contact post e, V is a voltmeter, and B a battery or direct cur- 
 rent dynamo, to give a steady voltage which is greater than 
 the maximum instantaneous voltage to be observed. The con- 
 tact post c is moved along the rheostat until a balance is 
 produced, and the voltmeter reading is then equal to the instan- 
 taneous voltage corresponding to the particular setting of the 
 contact maker. The reversing key is introduced in the circuit 
 to make it equally convenient to explore the positive and nega- 
 tive loops of the alternating voltage. 
 
652 
 
 ALTERNATING CURRENTS 
 
 Any method enabling the use of a reflecting instrument in 
 circuit with the contact maker may be made continuously self- 
 recording by a proper disposition of the apparatus. In this 
 case, a beam of light is thrown upon the mirror, and its devia- 
 tion is recorded by means of a moving photographic film. In 
 order that the complete curve may be thus recorded, the contact 
 points must be caused to rotate continuously around the spindle 
 of the contact maker. Since the needle of the galvanometer 
 or electrometer which is used with the contact maker in this 
 case must rigidly follow the intensity of the current impulses, 
 the instrument must have little inertia and be truly deadbeat. 
 The vibrations of a telephone diaphragm with a mirror mounted 
 on it have been used to replace the deviations of a galvanometer 
 or electrometer needle. 
 
 158. Areas of Successive Curves of Alternating Currents and 
 Voltages. — In general, observations which cover one complete 
 period entirely define the curves of commercial alternating cur- 
 rents and voltages. Since there is no continuous transference of 
 electricity in one direction, the areas of successive loops of the 
 curves should be equal. In the voltage curves produced by 
 
 an 
 
 alternator, for instance, e — and = Cedt , where is 
 
 dt J 
 
 the total number of lines of flux passing into the armature core 
 and T is the time occupied by a cycle. If and Tare constant, 
 as would be the case for a symmetrical alternator with fixed 
 field magnetism and a rigid armature shaft, which is driven at 
 a uniform speed, the areas of the successive loops of the voltage 
 curve must be equal. On account of various irregularities in 
 the construction and working of alternators, experimentally 
 determined curves are not always uniform. In fairly large 
 commercial machines the differences are usually not greater 
 than might be caused by the errors of observation due to the 
 experimental determination ; and appreciable differences in the 
 areas of successive loops of the curves produced by mechani- 
 cally rigid machines, driven at a uniform angular velocity, are 
 not to be expected, except possibly when the machines have 
 armatures with their halves connected in parallel, and then 
 only when the magnetic circuits lack symmetry to a consider- 
 able degree. In the case of certain small eight-pole alternators, 
 Dr. Bedell found differences in the areas of the consecutive 
 
SYNCHRONOUS MACHINES 
 
 653 
 
 loops which are not explainable upon the ground of errors of 
 observation or of variable speed.* The curves given by two 
 of these machines in one complete revolution (four complete 
 periods) are shown in Fig. 387. The individual areas of the 
 
 loops are marked upon the figure. While these differ as much 
 as 25 per cent amongst themselves, the sums of the positive 
 and negative areas differ by no more than might be caused by 
 * Physical Review , Vol. 1, p. 218, 
 
654 
 
 ALTERNATING CURRENTS 
 
 experimental errors, while the angular speed of the prime 
 mover or of the exciter may cause similar deviations in the 
 curves. This apparently shows that irregularities in the mag- 
 netic circuits and in the armature windings may in some cases 
 cause differences in the successive loops of the curve developed 
 in 360 electrical degrees, but the algebraic summation of the 
 areas due to each revolution is zero. The latter must be true, 
 or there would be a continuous flow of electricity in one direc- 
 tion. The fact that the machines tested by Dr. Bedell had 
 notable structural weaknesses leads to the probability that the 
 springing of the shaft or other parts of the machine may have 
 caused the unusual result which he found. Indeed, it is not 
 uncommon where an alternator is badly aligned or the shaft is 
 slightly bent, in either the alternator or its exciter, to have such 
 irregularities set up in the voltage wave. Irregular angular 
 speed of prime movers or of the exciters may cause similar 
 variations of the areas. These may become so pronounced in 
 certain voltage loops occurring during the alternator revolu- 
 tions, as to be manifested to the eye by flickering of incandes- 
 cent lamps attached to the alternator circuit. Variation 
 between the areas of the successive loops of current curves may 
 be introduced by unsteady resistance or reactance, even when 
 the voltage curve is perfectly uniform. 
 
 159. Regulation of Alternators for Constant Voltage. — A sep- 
 arately excited alternator has, as already intimated, no inherent 
 tendency towards regulation. The regulation is usuall}' effected 
 by means of a rheostat in the field circuit of the shunt or com- 
 pound-wound exciter, a rheostat in series with the alternator 
 field winding, or both. The adjustment of these rheostats 
 may be performed by hand or through devices actuated by a 
 relay placed in shunt to the main circuit. In Great Britain 
 and Europe automatic devices have been used in large plants 
 for many years, and of recent years they have come into very 
 common use in this country. 
 
 A type commonly used in this country for controlling alter- 
 nator voltage is arranged to make brief successive periods of 
 short circuit around a part or all of the field rheostat of the ex- 
 citer, the periods of short circuit being of greater or less duration 
 depending upon whether the alternator voltage should be in- 
 creased or decreased. The short-circuiting periods repeat them- 
 
SYNCHRONOUS MACHINES 
 
 655 
 
 selves in rapid succession, varying the field current and giving it a 
 wavy form having a magneto-motive force of the value needed 
 to set up the required exciter voltage. The exciter field rheostat 
 is usually set so that when the short circuit around the rheostat 
 has been opened the alternator voltage will fall in from six to 
 eight seconds to about 80 per cent of normal full load value 
 when compound or interpole exciters are used, and to the no- 
 load alternator voltage when shunt exciters are used. The 
 delay with which the voltage falls to its lower value when the 
 rheostat comes into the exciter field circuit by the removal of 
 the short circuit is caused by the eddy currents and the self- 
 inductance causing an apparent electro-magnetic inertia in the 
 exciter and alternator field magnets. 
 
 At other than no load the short-circuiting contact of a regu- 
 lator of this type seems always rapidly moving. The smaller 
 the load and the greater the resultant tendency for the line 
 voltage to rise, the shorter must be the periods of the short 
 circuit of the exciter field rheostat. Figure 388 shows a dia- 
 
 D CURRENT 
 
 Fig. 388. — Voltage Regulator which short-circuits a Siugle Section of the Exciter 
 Field Rheostat and has One Set of Relay Contact Parts. 
 
 gram of one of the commercial types of such a regulator. The 
 regulator has a direct current control magnet A, an alternating 
 current control magnet B, and a relay C. The winding of the 
 magnet A is connected across the exciter circuit, and its movable 
 core is attached to one end of a pivoted lever which carries at 
 its opposite end a flexible contact part D which is pulled down- 
 ward by springs. The magnet B has two windings, one being 
 connected to a potential transformer and the other being a 
 
65b 
 
 ALTERNATING CURRENTS 
 
 compensating winding connected to a current transformer and 
 giving an opposing magnetizing effect proportional to the cur- 
 rent flowing in the alternator circuit. The movable core of 
 magnet B is attached to one end of a pivoted lever which is 
 counterweighted at its other end and carries a contact part 
 which cooperates with the contact part D already referred to, 
 securing what may be called a floating contact arrangement. 
 The relay C consists of a U-shaped magnet core carrying a 
 differential winding and provided with a pivoted armature 
 which controls a circuit contact arranged to close and open the 
 branch which short-circuits the exciter field rheostat. One 
 half of the differential winding is permanently connected across 
 the exciter circuit. The other half of the differential winding 
 is connected to the floating contact points, and is also con- 
 nected across the exciter circuit and neutralizes the magnet- 
 izing effect of the first half when the floating contact is 
 closed. A condenser K is connected between the contact parts 
 at the relay armature, to prevent serious sparking when the 
 contact opens. 
 
 Now, assuming the floating contact to be open, the relay 
 armature is drawn down and the field rheostat is introduced 
 in the exciter field circuit ; this lowers the voltage of the ex- 
 citer and thereby lowers the voltage of the alternating current 
 generator to which the exciter is connected. This weakens 
 both of the control magnets, and the floating contact closes, 
 whereupon the second half of the differential winding of the 
 relay is energized and neutralizes the relay excitation, and the 
 relay armature is released. This results in closing the relay 
 contact and short-circuiting the field rheostat. That increases 
 the excitation of the exciter and therefore of the alternator, so 
 that both exciter and main voltages increase. Both control 
 magnets are strengthened, the core of A is attracted downward 
 and the core of B is attracted upward so that the floating con- 
 tact opens, interrupting the circuit through one half of the 
 differential winding on (7, the relay armature is attracted, and 
 the exciter field rheostat is again short-circuited. As the vol- 
 tages rise in consequence of the last condition, the cycle of 
 operation begins again, and it is repeated over and over again 
 at a fairly high rate of vibration which maintains the alternator 
 voltage with reasonable precision at the value desired for all 
 
SYNCHRONOUS MACHINES 
 
 657 
 
 loads. The actual variation of voltage required to produce the 
 cycle of operations is very slight. 
 
 It is often desirable to have the regulator control the alter- 
 nator so as to keep the voltage constant at the point of con- 
 sumption rather than at the generating plant, and this may 
 be reasonably accomplished by the use of the combination of 
 current transformer and potential transformer indicated in Fig. 
 388. This obviates the use of “ pressure wires ” (that is, wires 
 which run from the center of consumption to voltmeters at the 
 generating station for the purpose of indicating the voltage at 
 the consumption center), which were much used in the early 
 days of electric lighting. 
 
 Where several exciters are used in parallel on the exciter bus 
 bars they may be connected as shown in Fig. 389. In this case, 
 
 Fig. 389. — Several Exciters connected to a Voltage Regulator of the Type in which 
 Part of the Field Rheostats are short-circuited. 
 
 the alternator field windings are connected in parallel, and care 
 must be taken that the regulation affects the voltages of all the 
 exciters alike, or the exciterloads will not divide equally. This 
 requires care in the adjustment of the rheostats, and of the exciter 
 field compounding where the exciters are compound wound. 
 When the exciters are of large capacity, the relay magnet controls 
 several sets of contact parts so that each exciter rheostat can be 
 operated by an independent set, or two or more rheostats con- 
 2 u 
 
658 
 
 ALTERNATING CURRENTS 
 
 trolled by separate contacts can be used for the field winding of 
 a single exciter. By such an arrangement the values of the cur- 
 rents and voltages can be reduced suitably to be 
 handled by the contacts without injury. 
 
 The device used for obtaining at the termi- 
 nals of the regulator a voltage which is propor- 
 tional to 1Z , where E is the generator 
 
 terminal voltage and IZ is the line drop, is 
 similar in operation to the arrangements used 
 for obtaining compensated voltmeter readings. 
 It is often desirable to show at the switch-board 
 the voltage at the point of consumption. The 
 station voltmeter is then connected in series 
 with the secondary windings of a voltage trans- 
 former and a current transformer which act 
 in opposition (Fig. 390). The transformers 
 being properly adjusted, the voltmeter shows 
 at all times the voltage at the center of con- 
 sumption. In such an arrangement several 
 terminals may be brought out from the sec- 
 ondary winding of the current transformer to a 
 switch like that shown in Fig. 391, and the ad- 
 justment of the apparatus to compensate for any 
 given line drop may then be made by changing 
 the secondary connections by moving the switch 
 lever and so changing the number of effective secondary turns 
 in the voltmeter circuit. The secondary winding of the cur- 
 rent transformer may be shunted by resistance 
 as at R 2 in Fig. 392, and the adjustment of this 
 resistance serves a corresponding purpose. To 
 get exact compensation in the voltmeter circuit, 
 
 E 2 should be replaced by resistance and react- 
 ance in the same proportions as the resistance 
 and reactance of the line for the drop over 
 which compensation is to be made. If the ratios 
 of transformation of the potential and current 
 transformer T and T' are alike, the ratio of the 
 line impedance to the impedance substituted in 
 the position of E 2 should be equal to the ratio of transformation, 
 and the voltage at the far end of the line will then bear the same 
 
 Fig. 390. — Diagram 
 of Connections for 
 a Compensated 
 Voltmeter. 
 
 Fig. 391. — Hand- 
 regulating Switch 
 for Controlling 
 Number of Turns 
 in Transformer 
 Secondary. 
 
SYNCHRONOUS MACHINES 
 
 659 
 
 ratio to the voltage measured by the voltmeter. If the ratio of 
 transformation of the current transformer differs from the ratio 
 of the potential transformer, then the shunting impedance 
 should be changed in a corresponding degree. The secondary 
 winding of the current transformer does not necessarily have 
 to be connected into the circuit of the regular voltmeter coil, 
 but can be connected to an auxiliary coil which is wound along- 
 side of or over the main coil. 
 
 When a regulator without a compensating coil is used, the 
 voltage acting in the circuit of the automatic regulator must be 
 caused to remain constant as the alternator current increases, 
 while at the same time the alternator voltage increases by a 
 sufficient amount to compensate for the fall of voltage in the 
 feeders, in case the regulator is intended to maintain constant 
 voltage at the center of consumption. In other words, E — IZ 
 must be kept constant, E being the alternator voltage, I the 
 current, and Z the impedance of the feeders. This may he 
 effected approximately as illustrated in Fig. 388, or as follows 
 (Fig. 392). The regulator is connected to the secondary cir- 
 cuit of potential transformer T, R 
 
 which is connected in parallel 
 across the feeders. The voltage 
 of the secondary of this trans- 
 former is proportional to the al- 
 ternator voltage E. A current 
 transformer T' is connected with 
 its primary winding in series with 
 the feeders. The voltage devel- 
 oped in the secondary winding of 
 this transformer can be adjusted 
 so as to be practically proportional 
 to IZ for all values of the current. 
 
 The secondary winding of this 
 transformer is connected in series 
 with the secondary circuit of the 
 first transformer, and in such a way that their voltages are in 
 opposition. Hence a voltmeter V connected across the ter- 
 minals of the two secondaries indicates a voltage which is pro- 
 portional to the voltage at the terminals of the feeder, or 
 E — IZ. If the automatic regulator is also connected across 
 
 1 
 
 p v ' 
 
 “CT 
 
 R 3 Jh 
 — fjjjj— •'Tnnnr— 
 
 
 SHUNT 
 
 Wwv 
 
 
 
 l/VW SER 
 
 
 T ,( 
 
 WV\A 
 
 [p V 
 
 q 
 
 
 
 
 Fig. 392. — Connections of Voltage and 
 Current Transformer to Alternator 
 Regulator to maintain Constant Vol- 
 tage at the Center of Distribution. 
 
660 
 
 ALTERNATING CURRENTS 
 
 the terminals of the two secondaries, it adjusts the excitation 
 of the alternator so that E — IZ is kept fairly constant regard- 
 less of the value of I. In the figure, S is the solenoid of the 
 regulator, R v is the resistance automatically controlled by the 
 solenoid to vary the excitation of the alternator, and R v R v R 3 
 are impedances in circuit with the regulator which are used 
 for adjusting it to give proper indications for various values 
 of I and Z. 
 
 Regulation of generator voltage by means of compound wind- 
 ing is difficult, as the number of series turns that give regula- 
 tion on a non-reactive load evidently may fail for a reactive load. 
 It is evident, therefore, that a compositely excited alternator 
 will only regulate on loads having the power factor for which 
 regulation of the machine was adjusted. The ratio of series 
 ampere-turns per field-magnet pole to armature ampere-turns 
 per coil which is required to give regulation for one alternator 
 of a fixed type will doubtless give equally satisfactory results 
 on machines of different capacities but of the same type' but 
 the marked differences in the magnitude of the effects of self-in- 
 duction and armature reactions in alternators of different types 
 make it impossible to fix any ratio that would even approxi- 
 mately cover all types of machines. The regulation of ordinary 
 alternators which are self-excited, or otherwise, can only be sat- 
 isfactorily effected by means of variable resistance regulators 
 placed in the exciting circuit, or by varying the exciter voltage. 
 This statement includes the self-excited alternators now on the 
 market in which the rectified current for the fields is obtained 
 from a special independent winding placed in the slots with the 
 ordinary windings of the armature. The regulation might be 
 effected by moving the brushes on the rectifying commutator, 
 if the machine was self-excited, but only at the expense of pro- 
 hibitive sparking and wear. 
 
 If it is desired to have an alternator give constant current, 
 the regulation maybe made inherent by designing the armature 
 reactions and self-induction to be so great that the current 
 cannot rise much above its normal value. The armature should 
 be wound to generate a voltage upon open circuit much greater 
 than that required for full load, and hence the current remains 
 near its full normal value up to, and even considerably beyond, 
 full load. Such machines can be worked on short-circuit with- 
 
SYNCHRONOUS MACHINES 
 
 G61 
 
 3,000 
 
 2,000 
 
 out injury; but if the circuit is opened, they are liable to injury 
 on account of the excessive open circuit voltage breaking down 
 the insulation. These machines are really 
 worked on a part of the characteristic which 
 is caused to be almost vertical on account 
 of the large self-inductance and reactions 
 of the armature ; that is, the number of 
 armature conductors is many times greater 
 than comports with obtaining the maximum 
 output per pound of copper and iron in the 
 machine. Constant-current alternators have 
 sometimes been used in the past for arc 
 lighting and are referred to here to show the 
 effect of their extreme design on regulation. 
 
 The external characteristic of such an alter- 
 nator is given in Fig. 393. This shows how 
 the terminal voltage increases as the current 
 decreases relatively little, while the external 
 circuit is changed from short circuit to an 
 equivalent resistance of between two and 
 three hundred ohms, the exciter voltage im- 
 pressed on the field windings remaining 
 constant. The curves are given for three 
 different excitations. 
 
 160. Feeder Regulators. — Feeder regula- 
 tors for use in plants where several circuits 
 are fed from one alternator or set of bus 
 bars are in quite common use, as already 
 intimated. The regulator may be a special 
 transformer (Fig. 394), with the secondary 
 windings CD in series with one feed wire, 
 and the primary winding AB connected 
 across the main bus bars. The voltage in- 
 duced in CD may be made to either aid or oppose the alternator 
 voltage by means of a reversing switch, X. The strength of the 
 voltage introduced into the feeder circuit by CD may be varied by 
 changing the number of turns of CD in the circuit, by means of 
 movable contact parts. For convenience, such contact parts are 
 usually arranged on the arc of a circle or upon a drum. The con- 
 tact slider at F may be divided into two parts and reactance 
 
 1000 
 
 500- 
 
 AMPERES 
 
 Fig. 393. — External 
 Characteristics of an 
 Alternator designed 
 to Produce Constant 
 Current. 
 
(362 
 
 ALTERNATING CURRENTS 
 
 inserted between the parts 
 to prevent excessive current 
 from flowing when a trans- 
 former coil is short-circuited. 
 Other arrangements are used 
 for this purpose and for re- 
 ducing sparking. 
 
 Devices in which the regu- 
 lation is effected by varying 
 the position of the primary 
 and secondary coils with re- 
 spect to each other, or of the 
 core with respect to both, are 
 also used. Thus, a regulator 
 with armature and field wind- 
 ings arranged like those of 
 an induction motor* (Fig. 
 395), with independent coil- 
 wound secondaries of the re- 
 Fig. 394.— Diagram of a Feeder Regulator, quired number of phases, may 
 consisting of Transformer with Secondary . , „ . . r i 
 
 Winding with Variable Turns connected Used foi polt phase feedei 
 in Series with the Feeder Circuit, and regulation. Each Set of Sec- 
 Primary Winding in Parallel with the Qnd coils of a pllase is 
 Feeder Circuit. J L 
 
 connected in series with a 
 line conductor of the feeder to be controlled. The primary wind- 
 ings are connected in mul- 
 
 tiple arc on the main poly- 
 phase circuit. The pro- 
 gressing or rotating mag- 
 netic field induced by the 
 primary windings, caused 
 by the magnetic fluxes of 
 the several phases rising 
 successively across the air- 
 space and at angles ad- 
 vancing around the polar 
 circle, sets up a voltage in 
 
 the secondary windings. Fig . 395. -Diagrammatic Representation of a 
 This voltage is constant Polyphase Induction Regulator. 
 
 * Chap. XII. 
 
 GENERATOR 
 
 BUS BARS 
 
 /, REGULATOR \ 
 
 
 PRIMARY 
 
 i 
 
 U MOVEABLE j 
 
 
SYNCHRONOUS MACHINES 
 
 663 
 
 in strength, but has a phase relation with the feeder voltage 
 dependent upon the angular position of the field magnet with 
 respect to the armature. Thus the voltage may directly add to 
 the feeder voltage, be subtracted from it, or combine in an 
 intermediate 
 angular position, 
 as shown in Fig. 
 
 396, where A 0 is 
 the generator vol- 
 tage of one phase 
 and AS, AB\ Fig. 396. — Voltage Diagram of a Polyphase Induction 
 AB",AB "' are the Regulator. 
 
 feeder voltages beyond the regulator for one phase for various 
 positions of the regulator field magnet with respect to its arma- 
 ture. One core carrying either the primary or the secondary 
 windings in such a regulator is mounted in a fixed position, 
 and the other is mounted in such a manner that it may be rotated 
 such part of a revolution as is needed for sufficiently influenc- 
 ing the feeder voltage. The core carrying the primary wind- 
 ings is usually made the movable core. 
 
 A single-phase regulator may be made in the same way, 
 using only one set of windings respectively on the primary and 
 secondary cores. In this case zero voltage will be produced 
 in the secondary winding when its coils are so moved as to be 
 parallel to the magnetic flux from the primary winding, and 
 will increase to maximum value when the coils are rotated to 
 the right or left from this point until they are at right angles 
 to the flux. Compensating short-circuited coils such as are used 
 in series motors * must evidently be used for a single-phase 
 regulator of this kind to prevent an excessive reactance in the 
 secondary circuit when its windings are not at right angles to 
 the flux from the primary winding. These compensating coils 
 are wound on the core with the primary winding but magneti- 
 cally at right angles thereto. The transformer characteristics 
 of these induction regulators makes their operation subject to 
 the condition that the vector difference between the primary 
 ampere-turns and the secondary ampere-turns plus the ampere- 
 turns of any compensating coil makes the exciting ampere-turns. 
 
 Autotransformers may also be used as voltage regulators, 
 
 * Art. 206. 
 
664 
 
 ALTERNATING CURRENTS 
 
 as has heretofore been pointed out ; * and a form of regulator 
 for single-phase circuits was commonly used in former years 
 which consists of two coils placed at right angles on the inner 
 barrel of a hollow cylindrical laminated core, one of the coils 
 being connected across the circuit and the other introduced in 
 series in the feeder concerned. A segmented rotatable core is 
 used to direct the magnetic flux set up by the primary winding 
 through the secondary winding or shunt it therefrom. 
 
 Such regulators are commonly made to vary the voltage of the 
 circuit to which they are attached over a range from 10 per cent 
 lowering to 10 per cent boosting. The primary winding must be 
 designed for the voltage of the main circuit, and the secondary 
 conductors must be capable of carring the full feeder current. 
 When the regulator is raising the voltage, the outgoing current 
 is smaller than the current in the main circuit leading to the 
 device, and when the regulator is lowering the voltage, the out- 
 going current is larger than the current in the circuit leading to 
 the device. 
 
 161. Connecting Alternators for Combined Output. — The con- 
 ditions required for successfully connecting alternators so that 
 their outputs may be combined are quite different from those 
 obtaining in the case of direct-current machines. In order 
 that the output of alternators may be added, it is evident that 
 the voltage waves impressed by them upon the circuit must be 
 in exact consonance. That is, the voltage waves must be of 
 equal period or In synchronism and also of corresponding phase 
 or In step with each other. If this is not the case, the machines 
 will be in opposition during all or a portion of the current wave. 
 The following discussions concerning combined output are 
 applicable to both single-phase and polyphase alternators. In 
 case of the latter, the voltage and current in the windings of a 
 single phase should be understood to be referred to, while the 
 power of the single phase should be multiplied by the number 
 of phases to get the total machine power. 
 
 161 a. Alternators in Series. — The alternators will be assumed 
 in this discussion to be constructed so as to give equal currents 
 at equal frequency. The form of the current waves will also 
 be assumed to approximate to sinusoids and the armatures to 
 have negligible reactive effects upon the field magnets. 
 
 * Art. 144. 
 
SYNCHRONOUS MACHINES 
 
 t>65 
 
 In Fig. 397 let the curves A and A' represent the voltage 
 waves measured respectively across the terminals of two alter- 
 nators with their armatures connected in series, the machines 
 being driven independently, but so as to give practically the 
 same frequencies and 
 voltages. The ordi- 
 nates of curve It are 
 the algebraic sums of 
 the corresponding or- 
 dinates of curves A and 
 A', and hence curve It 
 represents the resultant 
 voltage impressed on 
 the external circuit by 
 the two machines. 
 
 Curve C is assumed to 
 be the curve of lagging 
 current flowing in the 
 circuit. Assuming the 
 two machines to be 
 running synchronously, 
 but to be out of step 
 by an angle 2 </>, makes 
 <f> the phase difference between the resultant voltage wave and 
 either component wave. Finally, the current lags behind the 
 resultant voltage by an angle d, on account of self-inductance 
 
 in the circuit. The 
 work put into the 
 circuit by each 
 machine is propor- 
 tional to the alge- 
 braic summation of 
 the products of the 
 ordinates of the re- 
 spective voltage 
 
 Fig. 398. — Power Loops of Alternators producing the Vol- wave with the CU1'- 
 tages and Current shown in Fig. 397. , rT , , 
 
 ° s rent wave. The 
 
 total work done in the circuit is equal to the sum of the prod- 
 ucts of the ordinates of the current and resultant voltage curves. 
 Therefore, since the voltage wave of the lagging machine is 
 
 Fig. 397. — Curves of Voltages and Current show- 
 ing Unstable Conditions existing when Alternators 
 are Connected in Series. 
 
666 
 
 ALTERNATING CURRENTS 
 
 nearest the current wave, that machine furnishes more work to 
 the circuit than does the leading machine. The power loops 
 for the two machines are shown by the curves a and a' in Fig. 
 398. The power delivered to the circuit by one machine is 
 
 represented by the 
 height of the line xx 
 above the JT-axis, and 
 the power delivered to 
 the circuit by the other 
 machine is represented 
 by the height of the 
 line x'x' above the axis. 
 
 If the two machines 
 were rigidly connected 
 together, this condition 
 would continue indefi- 
 nitely. Assuming, how- 
 ever, that the machines 
 are driven by separate 
 t . „ , ,, u , „ . , belts, or attached to 
 
 Fig. 399. — Curves of voltages and Current snowing ’ 
 
 Unstable Conditions when Alternators are con- separate engines, the 
 nected in Series. lag'crinsf machine be- 
 
 OO O 
 
 ing the more heavily loaded, tends to fall farther behind its 
 more lightly loaded mate, and a still greater percentage of the 
 load is thrown upon it. At the same time, as is illustrated by 
 Fig. 399, this re- 
 duces the total 
 work done in the 
 external circuit, for 
 the total voltage 
 wave is now of less 
 height than it was 
 when the c o m - 
 ponent curves were 
 nearer 
 
 of phase. The 
 power loops for the condition of Fig. 399 are shown in Fig. 400. 
 The height of the line xx has decreased, and that of x' x' has 
 increased, but the sum of the heights is less than before. The 
 tendency of the lagging machine to fall farther behind may 
 
 a a 
 
 coincidence Fig. 400. — Power Loops of Alternators producing 
 Voltages and Current shown in Fig. 399. 
 
 the 
 
SYNCHRONOUS MACHINES 
 
 667 
 
 Fig. 401. — Voltages of Two Alternators connected 
 in Series, after they have the Stable Position of 
 Series Opposition. 
 
 continue until the voltage waves of the two machines approach 
 exact opposition to each other in the series circuit (Fig. 401). 
 When in opposition the machines are in stable equilibrium, but 
 are delivering no power to the external circuit. 
 
 If the machines were started with their voltage waves in exact 
 step, they would do equal work, but their equilibrium would be 
 unstable, and any disturbance of their relations would cause them 
 to tend towards opposition. It is therefore not possible to operate 
 alternators in cumula- 
 tive series on an induc- 
 tive circuit unless 
 they are rigidly united 
 by a mechanical coup- 
 ling. This result also 
 follows when the nor- 
 mal voltages of the 
 machines are different, 
 in which case the vol- 
 tage impressed on the 
 circuit when equilibrium is attained is the difference of the 
 machine voltages. Even if there were no reactance in the ex- 
 ternal circuit on which the machines were working, the resultant 
 voltage and current would have the same phase, and the ma- 
 chines would be in equilibrium; but the equilibrium would be 
 unstable, for after any disturbance of the operation of the 
 machines they would have no tendency to return to their 
 former operating state. 
 
 The condition of operation of two alternators connected in 
 series on an inductive circuit is also plainly indicated by means 
 of a vector diagram (Fig. 402). In this diagram the lines 
 represent quantities as follows: 
 
 Voltage of leading machine = OA. 
 
 Voltage of lagging machine = OA'. 
 
 Resultant voltage in circuit = OR. 
 
 Current in circuit = OC. 
 
 Power given to circuit by first machine = Oa x OC. 
 
 Power given to circuit by second machine = Oa’ x OC. 
 
 Total power given to circuit = Or x OC = (Oa + Oa ') x OC. 
 
 It is evident from the construction that if the angle ROC is 
 fixed by the conditions of the circuit, then the length of OR de- 
 
668 
 
 ALTERNATING CURRENTS 
 
 R 
 
 Fig. 402. — Vector Diagram of Voltages and 
 Currents for Two Alternators in the Unstable 
 Condition of Series Operation. 
 
 creases as the angle 2 <p increases, and that Oa decreases at the 
 same time; but Oa! increases for a time and then decreases at a 
 less rate than Oa , so that the machines tend to get farther 
 apart in phase. When = 90° — 0, the length of Oa vanishes, 
 
 the first machine gives no 
 power to the circuit, and 
 all the power is furnished 
 by the second machine. 
 When cf) approaches more 
 nearly 90°, or exceeds that 
 angle, as when the machines 
 are approaching opposi- 
 tion, one machine will run 
 as a motor if its prime mover is disconnected. Of course, when 
 OR decreases, if the resistance of the circuit is unaltered, the 
 current OC also decreases, but the relative outputs and phases 
 of the machines are not altered thereby. 
 
 In case capacity reactance could be placed in the series 
 circuit between the machines sufficient to cause the current to 
 lead the resultant voltage by an angle d, the condition is 
 represented by Fig. 403, in which the vectors are lettered as 
 in Fig. 402. In this case, 
 
 Oa is greater than Oa ' ; 
 that is, the leading machine 
 furnishes the greater 
 amount of power, and the 
 machines tend to come 
 together and run in cumu- 
 lative series. Similar de- 
 ductions can be drawn from 
 diagrams like Figs. 397 and 
 398, but with the current wave in the lead of the resultant voltage 
 wave. Since the alternator windings always contain self-induct- 
 ance, special insertions of capacity in the series circuit would 
 have to be made to get sufficient capacity reactance to produce 
 this result, and it may therefore be said that, in general, alter- 
 nators in series are not stable in operating. 
 
 161 b. Alternators in Parallel. — When the machines have 
 reached series opposition of phases, as explained above, their 
 voltages are in the proper relation to cause them to work in 
 
 Fig. 403. — Vector Diagram of Voltages and 
 Current of Two Alternators in Series when 
 working on a Load of High Capacity Reactance. 
 
SYNCHRONOUS MACHINES 
 
 669 
 
 parallel in delivering current to an external circuit attached be- 
 tween the lead wires connecting their terminals; for it will be 
 seen by reference to Figs. 401 and 404 that when machines A 
 and A' are in series opposition to each other, the points R and 
 S must at every instant be of opposite sign, and that, therefore, 
 the machines will deliver current through the load circuit m, 
 m , m. This is also the proper connection for direct current 
 generators arranged for parallel operation ; that is, the voltages 
 are opposed to each other in the series circuit through both 
 armatures. 
 
 - R m 
 
 Fig. 401. — Diagram of Two Alternators connected for Parallel Operation. 
 
 The operation of alternators in parallel was first achieved by 
 Wilde in 1868,* but this work was overlooked during the period 
 of development of the direct-current dynamo. In 1884 Dr. John 
 Hopkinson showed by mathematical analysis the practicability 
 of working them in parallel. This was done without a knowl- 
 edge of Wilde’s earlier experiments, and it led to some experi- 
 ments which were carried out by Hopkinson and Adams upon 
 De Meritens magneto machines, f These experiments fully 
 bore out Hopkinson’s deductions, but their practical bearing 
 was not fully appreciated until a few years later, when the 
 transformer system of alternating-current distribution was de- 
 veloped. 
 
 161 c. Synchronizing Current. — Successful parallel operation of 
 alternators depends upon their holding each other in synchronism 
 and step (by the motor action referred to later in the article), 
 even when the prime movers do not naturally synchronize. 
 The effort of the machines to do this causes the flow of a cross 
 current between them, which is commonly called the synchro- 
 nizing current. This component of the total current from a 
 machine may be considered as a series current circulating be- 
 
 * See Philosophical Magazine , Vol. 37, 4th series, 1869, p. 54. 
 t Adams, Jour. Inst. E. E., Vol. 13, 1884, p. 515. 
 
670 
 
 ALTERNATING CURRENTS 
 
 tween the machines considered as pairs connected to the station 
 bus bars. The remaining component of the current from each 
 machine joins in parallel with that of the other machines and 
 flows through the load. The total synchronizing current flow- 
 ing between a machine and the station bus bars may be re- 
 solved into two components, one in phase with the voltage which 
 causes the synchronizing current to flow, and the other lagging 
 90° behind the voltage. The former is usually quite small. 
 Thus, suppose in Fig. 405 that Oa and Ob represent equal 
 vector voltages of two generators A and B , taken in series cir- 
 cuit through the two armatures. The machines may be con- 
 sidered as connected to the same bus bars of a central station. 
 Curves a and b are sinusoids representing the voltage waves. 
 Assume for the present that the machines are carrying no ex- 
 ternal load and that the prime mover of B is not supplying 
 quite enough power to drive it unloaded while that of A is not 
 only able to drive A but also to aid B. Then machine B will 
 tend to fall back in phase or step and its voltage curve b will 
 lag with reference to the voltage curve a of the machine A. 
 As will be seen this will tend to cause a motoring synchronizing 
 current to flow through the series circuit comprising the arma- 
 tures of the machines. Thus, since a and b are slightly out of 
 opposition, there will be a resultant voltage Oq (curve g), tend- 
 ing to set up a series synchronizing current. The relations are 
 simply shown in the vector diagram in the figure, where the 
 outer ends of the lines representing the vector voltages and 
 currents corresponding to the curves are marked with the same 
 letters as the respective curves. The series current set up by 
 voltage q may be considered as made up of a component s, in 
 phase with q , and of another component u , in quadrature with 
 q , the former being the active and the latter the wattless com- 
 ponent. The component s has the same effect upon both a and 
 b during a complete period, i.e. each machine furnishes one half 
 the power required to drive s through the series circuit, hence it 
 can have no effect in tending to draw the trailing machine into 
 the proper phase relation with its mate; but the component u 
 (90° from j), which is dependent upon the self-inductance of 
 the series circuit, must, from its position, cause an accelerating 
 torque on the lagging machine (5), and a corresponding re- 
 tarding torque on the leading machine; and it thus tends to 
 
SYNCHRONOUS MACHINES 
 
 671 
 
 draw the machines into synchronism and step for proper parr 
 allel operation. It will be noticed by reference to the figure 
 that u is in phase with m and in opposition to and that m and 
 
 a 
 
 o 
 
 S-l 
 
 aa 
 
 o 
 
 a 
 
 rJl 
 
 c3 
 
 a 
 
 o — • 
 % © 
 
 ps « 
 
 a 
 
 o 
 
 bJD 
 
 £ 
 
 n are respectively the components of a and b , which are in op- 
 position to each other in the series circuit, and are therefore 
 impressed on the parallel circuit ; also that the motoring effect 
 
672 
 
 ALTERNATING CURRENTS 
 
 on B is proportional to Ou times On , while the retarding effect 
 on A is proportional to Ou times Om. If the machine B led the 
 machine A , then the synchronizing current would retard instead 
 of assisting the former, as the figure plainly shows. 
 
 The effect of the synchronizing current in dragging the ma- 
 chines into step depends upon its magnitude and relative phase ; 
 while its magnitude at each instant depends : (1) upon the 
 algebraic sum, or, what is the same thing, the arithmetical dif- 
 ference between the instantaneous voltages developed by the 
 machines ; (2) upon the reciprocal of the impedance of the 
 machine armatures. In other words, the instantaneous synchro- 
 nizing current flowing in series relation through the two arma- 
 tures, if the two machines are alike, is 
 
 i a = e ° + e *> 
 
 2 V B *+ 4 t t*PL? 
 
 where e a and e b are the instantaneous voltages of the machines 
 which are of ojiposite sign and equal when the machines are in 
 exact synchronism and step, and R a and L a are respectively 
 the resistance and the inductance of the armature circuit of 
 each machine including the leads from the bus bars. It is 
 here assumed that the effective voltages of the two machines 
 are numerically equal, which is an essential condition for the 
 synchronizing current to be practically wattless, and is the con- 
 dition in which it is aimed to run alternators in practice. 
 
 Now suppose the voltage curve of the machine B to lag be- 
 hind the phase of the voltage curve at the bus bars by an angle 
 fi. Let the effective values of these voltages be E. and the 
 corresponding maximum voltage be e m . At any moment the 
 instantaneous voltages are e a and e 6 , and 
 
 e a = e m sin « = v/2 E sin a, 
 
 e b = e m sin (« + 180° — /3) = V2 E sin (« -f 180° — $), 
 
 when considered to be acting in the series circuit. The in- 
 stantaneous voltage causing a synchronizing current to flow is 
 the algebraic sum of these, or 
 
 e a + e b — V2 E [sin a + sin (a + 180° — /3)]. 
 
 It is evident from Fig. 405 that this is a maximum at the in- 
 stant when e a and e b are equal and of similar signs, in which 
 
SYNCHRONOUS MACHINES 
 
 673 
 
 case a = .f /3. Tlien the maximum value of the voltage causing 
 a synchronizing current is found by substituting this value of a 
 in the expression for e a + e b and ( e a + e b ) m •= 2V2i?sin /3 . 
 
 The effective value of the voltage is then 2 E sin | /3, and 
 the synchronizing current is 
 
 I _ E sin j, /3 
 
 For smooth and successful working in parallel I s must become 
 sufficiently great, in case the machines tend to get out of step, 
 to pull the machines together before /3 becomes of appreciable 
 magnitude. Hence it is desirable that the denominator in the 
 expression for I s be reasonably small. In other words, arma- 
 ture impedance must be small to give smooth parallel work- 
 ing. Figure 405 shows plainly that the voltage ( Oq ) causing 
 I 3 is behind the phase of the voltage of the trailing machine 
 by an angle 90° — 1 /3. The synchronizing current (Z, = Oc ) 
 lags behind the voltage Oq by an angle 9 S (angle qOc ), the 
 
 tangent of which is 
 
 2tt fL a 
 
 Rn 
 
 Consequently the synchronizing 
 
 current is out of phase with the trailing machine voltage by an 
 angle of 90° + ( 9 a — 1/3) and out of phase with the leading 
 machine voltage by an angle of 90° — (0 8 -f | /3). A current 
 (wattless) which has a phase difference of 180° or 0° compared 
 with the voltage of a machine has the strongest effect in bring- 
 ing the machine into step by tending to accelerate it or to retard 
 it. This effect is to accelerate or retard the refractory machine 
 depending on whether the phase difference is 180° or 0°, which 
 in turn depends upon whether the machine is trailing or leading. 
 
 The component Ou( = 7^) of the synchronizing current just 
 found is 
 
 and therefore, 
 
 I = 
 
 1 * = la Sin 
 
 E sin i f3 2 i rfL a 
 
 V^ 2 + 4 ri 2 f 2 L a 2 vX 2 + 4 7 T 2 pL 2 
 
 or. 
 
 -^ = E sin -t /3 
 
 2 7rfL a 
 
 Ra + 4 7 r 2 PL 2 
 
 and, with a fixed value of R a , this will have a maximum value 
 when 
 
 R a = 2 7 rfL a or when tan 0 S = 1 and 0 S = 45°. 
 
 2 x 
 
674 
 
 ALTERNATING CURRENTS 
 
 The maximum possible value of I u where the machine cir- 
 cuit has a given resistance is therefore 
 
 1^ max = 
 
 E sin t /3 
 2 R a ’ 
 
 and the corresponding value of I„ the total synchronizing cur- 
 rent, is 
 
 T _ Esin 1/3 
 -*~8 — 
 
 V2 R a 
 
 The limits in the value of R a are fixed by considerations of 
 economy in construction and of efficiency, and the frequency 
 is fixed by conditions of operation ; L a is therefore the only 
 independent variable in the preceding equations. In order to 
 have the most sensitive mutual control, the self-inductance of 
 the armature circuit, which at its least value is always many 
 
 R 
 
 times larger than - ° ■ , must be as small as commercially pos- 
 
 2 7 TJ 
 
 sible. If it were possible to reduce the reactance to the value 
 of the resistance, the jerking of a refractory alternator into 
 phase would probably be too severe for good working, and the 
 stresses imposed on the machine might be injurious; but in 
 most commercial machines such trouble is not likely to exist on 
 account of the unavoidable magnitude of the self-inductance, 
 which cannot be reduced beyond certain limits. 
 
 The foregoing equations are developed on the hypothesis of 
 two equal alternators operated in parallel, but the conclusions are 
 equally applicable to two alternators of unequal internal resist- 
 ances and reactances and also to the conditions of many alter- 
 nators connected in parallel to the same bus bars. In either 
 case the impedance of the cross current circuit for any particular 
 machine is equal to the sum of the impedances of the arma- 
 ture and connecting leads of that machine in series with the 
 circuit composed of its mates in parallel and their connections. 
 The bus bar voltage may be considered to be unaffected by the 
 cross-current flow ; then the equations for i s and I s show that 
 their values are as large as the values given in the foregoing 
 paragraphs, for the reason that the resultant series voltage Oq 
 for one machine only acts in a circuit of impedance made up of 
 the armature and leads of only one machine. 
 
SYNCHRONOUS MACHINES 
 
 675 
 
 Fig. 406. — Vector Dia- 
 gram of Voltages and 
 Synchronizing Current 
 when One Alternator 
 in Parallel with Others 
 lags in Step. 
 
 The vector diagrams, Figs. 406 and 407, contrast the con 
 ditions of a leading and a trailing alternator with respect to 
 bus bars of fixed voltage OA. Figure 406 
 shows the position of the voltage OB of the 
 trailing alternator and the position of the 
 synchronizing current Oo which tends to 
 accelerate the alternator into step ; while 
 Fig. 407 shows the corresponding conditions 
 for an alternator with leading voltage OB 
 and synchronizing current Oc which tends 
 to retard the alternator into step. 
 
 In each’ of these figures, 
 
 OA represents the bus bar voltage in 
 phase and magnitude. 
 
 OB represents the machine voltage in 
 phase and magnitude. 
 
 OQ represents the resultant, or synchro- 
 nizing, voltage in phase and magnitude. 
 
 Oc represents the synchronizing current 
 in phase and magnitude. 
 
 A Ou represents its wattless, or phasing, 
 
 component. 
 
 /3 is the angle by which the machine 
 differs from its proper phase for parallel 
 working, OB 1 . 
 
 6, is the angle QOc by which the synchro- 
 nizing current lags behind the resultant 
 voltage. 
 
 The figures show plainly that if OA and 
 OB remain constant, then as /3 increases OQ 
 increases and at the same time Oc and Ou 
 increase. The product OB x Ou x cos \ /3 
 is proportional to the synchronizing torque 
 exerted on the machine. It is negative in 
 Fig. 406 and positive in Fig. 407, so that 
 hen One Alternator the torque is exerted on the machine and 
 m Parallel with others accelerates it in one case, and it is exerted 
 step. by £p e mac pi ne au q retards it in the other 
 
 case. For the conditions of OB = OA and a given value of /3, the 
 torque is a maximum when Ou x OB is a maximum, which 
 
 
 Fig. 407. — Vector Dia- 
 gram of Voltages and 
 Synchronizing Current 
 
676 
 
 Alternating currents 
 
 occurs when 2 n/L — E, or 6 S — 45°. In a 2200-volt 500-kilo- 
 watt machine giving a frequency of 60 periods per second and 
 having a resistance in the armature circuit of .5 ohm, it is 
 required that L = .0013 to give a maximum synchronizing 
 effect, which is smaller than the smallest value which is now 
 commercially attainable in any type of alternator of that size. 
 
 An inspection of the figures shows that if the synchronizing 
 current Oc led the resultant voltage OQ , there would be no 
 tendency for the machine to come into parallel step with the bus 
 bar voltage ; but this condition will not occur fn practice, since 
 it is impossible to build alternators without self-inductance in 
 the windings. 
 
 In the foregoing discussion, the usual condition in which a 
 prime mover tends to lag or race in speed is considered ; and 
 it is shown how a synchronizing cross current is set up through 
 alternators operating under such conditions, which current 
 tends to hold the generators at speeds giving a common fre- 
 quency for the voltage. This current causes a transfer of 
 power from a leading machine to its mates or from the mates 
 to a trailing machine. The current is therefore wattless with 
 respect to the system, except for PR losses. 
 
 162. Division of Load between Parallel Alternators. — Alter- 
 nators may be completely synchronized but yet fail to properly 
 divide the load of the external circuit between them, as the 
 driving torque of the prime mover of each machine may not 
 be adjusted to exactly overcome at the correct speed the resist- 
 ing moment of the load which that machine should carry. The 
 condition is illustrated in Fig. 408, where the conditions are the 
 same as assumed in the preceding article except that the ex- 
 ternal circuit is receiving power. In this figure, OT \ is the bus 
 bar voltage E which is assumed to be constant in length and is 
 used as the reference phase position with respect to time. The 
 vector of current I in the external circuit is represented by the 
 line OH , which is shown lagging behind the line voltage by an 
 angle 6 , and which is dependent for its value upon the char- 
 acter and value of the impedance of the external circuit. 
 
 Assume first that two exactly equal machines are connected 
 to the bus bars and are excited so as to give equal total vol- 
 tages. Further assume that the governors of the driving en- 
 gines are so adjusted as to give exactly equal torques. Then 
 
SYNCHRONOUS MACHINES 
 
 nrrrr 
 
 bt t 
 
 the voltage of the generators must be in exact opposition of 
 phase so far as the series circuit through the two armatures 
 are concerned, and no 
 series current will flow, but 
 they are in exact parallel 
 relation with reference to 
 the external circuit. 
 
 These relations, under 
 these circumstances, are 
 quite similar to that of two 
 equal direct-current gen- 
 erators delivering equal 
 loads to an external circuit. 
 
 The value and phase 
 position of the alternator 
 voltage required to furnish 
 the external power, P — 
 
 El cos d, may be obtained 
 as follows : lay out TM 
 parallel to OH and of such 
 a length that it equals 
 R a I, where R a is the re- 
 sistance of one of the 
 armature windings. At 
 right angles to TM lay off 
 MO of a length equal to 
 X a I, where X a is the 
 synchronous reactance of 
 one of the armature wind- 
 ings. Then CT=IZ a = E a 
 represents the voltage drop 
 in the parallel impedances 
 of the two armatures when current I flows. The generated 
 armature voltages must then be vectorially equal to OT plus 
 TC ', or E g = E + E a . When considered with reference to the 
 series armature circuit the total generated voltages may be laid 
 off as 00 and OC, as was done in Figs. 405, 406, and 407. 
 The load given out by each machine is, 
 ssrn TM 1 
 
 Fig. 108. — Vector Diagram for showing the 
 Division of Current and Load between Two 
 Equal, equally Excited Alternators connected 
 in Parallel. 
 
678 
 
 ALTERNATING CURRENTS 
 
 since 
 
 TM= H R Ji IR a and \ I = T ^-, 
 
 tin 
 
 and the total power generated by each machine is 
 
 i p — i p _i_ i nj? 
 
 Suppose, now, that the governors of the driving engines are 
 so adjusted that one has a greater torque than the other, but 
 so that the external load is not changed. The two machine 
 voltages will then fall apart in phase, say by the angle /3 (Fig. 
 408). Suppose also that the total voltages of the two genera- 
 tors are again equal in scalar value. Construct the isosceles 
 triangle OAB so that the base AB passes through the point C 
 and is at right angles to CC, and the sides OA and OB make 
 the required angle /3. OA and OB are the required generator 
 voltages and combine in parallel to send the load current I — 
 OH through the circuit composed of the line in series with the 
 two generator armatures in parallel. The resultant AB being 
 at right angles with the voltage OC it has no effect upon the 
 external current OH. Now draw the lines TA and TB and 
 erect the internal armature voltage triangles TNA and TSB. 
 The current generated by machine A is proportional to the 
 
 active side of TNA , or TN or I A — when E R is the 
 
 B a R a 
 
 active voltage lost in the generator copper. Likewise, the 
 total current flowing in generator B is proportional to TS or 
 
 I B = As the generators are similar and equal, the 
 
 b a B a 
 
 X 
 
 ratio — - is the same for both and for the two in parallel. 
 Ba 
 
 Scale TN and TS in terms of currents and then we have their 
 resultant TK equal to the total line current OH or double TOL 
 i.e. 1= I A + I B ; while their vector difference equals SN and 
 
 The total power gen- 
 
 . , . r SN — I B 
 
 the series current is J s = — = — - 
 
 erated by machine A is the product of the projection of TN on 
 OA with OA and of machine B the product of the projection of 
 TS on OB with OB. 
 
 For indicating the series relation, one of the generator vol- 
 tage vectors can be conveniently reversed. Thus, we obtain 
 
SYNCHRONOUS MACHINES 
 
 679 
 
 the parallelogram OAFB' which is similar to those of Figs. 
 
 405, 406, and 407. The resultant series voltage OF equals 
 
 BA , while the series current which would flow because of OF 
 
 . OG , . , . , , SN 
 
 is , which is equal to . 
 
 If OA and OB are not equal, the triangle OAB is no longer 
 isosceles, but the construction can be carried out in the same 
 manner as above. 
 
 However, instead Q 
 
 of going further 
 into this special 
 case of two alter- 
 nators, consider 
 the more general 
 case of one alter- 
 nator connected to 
 bus bars to which 
 are connected a 
 large capacity of 
 other alternators, 
 so that the bus bar 
 or external voltage 
 remains practically 
 constant in value. 
 
 In order to find the 
 current and power 
 that will be pro- 
 duced by a machine 
 having an induced 
 voltage E g of a 
 given scalar value 
 equal to the radius 
 of circle MM with 
 center at 0, but of 
 variable phase po- 
 sition, the follow- 
 ing construction 
 can be made; lay out OF, as in Fig. 409, equal to the bus 
 bar voltage E, and assume that the driving engine torque 
 is such that the generated voltage E g makes angle (3 with E. 
 
 Fig. 409. — Diagram showing Locus of Current for Con- 
 stant Bus Bar Voltage and Generator Voltage. 
 
680 
 
 ALTERNATING CURRENTS 
 
 Then OB=E g . The internal voltage is then B V. Lay out on 
 BV the internal voltage triangle BVN, in which BV=IZ a 
 is the total loss of voltage internally, and is composed of 
 the active and reactive voltages VN = IR a and NB = IX a , 
 respectively. In this case, as in those preceding, synchro- 
 nous impedance and reactance are used. The apparent vol- 
 
 E 
 
 tage or impedance triangle in which E g or — ? is the hypot- 
 enuse is OHB. The apparent impedances in this case are 
 composed of the self-inductances, field reactions, voltage re- 
 actions, and resistances encountered throughout the system 
 in absorbing E g . With any change of the driving engine 
 torque, which changes ft, the form of the triangle OHB 
 changes. In the figure with /3 = angle V OB the generator 
 current leads the bus bar voltage by the angle JVN, or 6 is 
 negative. 
 
 The side OH, however, must always be parallel to VN, the 
 active voltage line, and the angle OHB is always a right angle. 
 
 In finding the current locus as f3 varies or B moves along its 
 circular locus MM, the following symbols are used : 
 
 The generated voltage, OB = E g . 
 
 The bus bar voltage, 0 V = E. 
 
 The angle between E g and E, VOB — /3. 
 
 The internal voltage, BV— E z . 
 
 The internal active voltage, VN = E R . 
 
 The internal reactive voltage, NB — E x . 
 
 The armature current scaled to equal VN = I. 
 
 The internal angle of lag, 
 
 NVB = 6„ 
 
 The angle between I and E, NVI = 6. 
 
 The armature resistance = R a . 
 
 The synchronous armature reactance = X a . 
 
 The synchronous armature impedance = Z a . 
 
 The locus of the point B is determined by the formula for 
 the triangle OBV, 
 
 E g 2 = EJ - 2 E Z E cos [180° - (0 a - <?)] + E\ 
 
 where 6 is intrinsically positive or negative depending upon 
 whether it is to the right or left of OJ ; in the figure for point 
 B it is negative. When E and E e are of constant values and 
 
SYNCHRONOUS MACHINES 
 
 681 
 
 6 is variable, this is also the formula in polar coordinates of the 
 circular arc MM having the radius E g and the pole at V. To 
 find the locus of the current when 6 varies, the above formula 
 may be modified by multiplying both sides by cos 2 # a , which 
 gives 
 
 E g 2 cos 2 0 a = E R 2 — 2 E R (E cos d a )cos [180°— (9 a —9)~\ +E 2 cos 2 0 a . 
 
 This is evidently the formula in polar coordinates of a circle 
 having a radius E g cos 9 a and in which the initial line is on a 
 diameter ; also one of the polar coordinates E R , or taken to 
 E 
 
 proper scale it is which equals i, the current flowing in 
 
 R 
 
 the armature. Now, swinging E g over to the position OJ , the 
 triangle of internal voltages becomes VSJ, the angle JVS 
 being equal to 6 a . When E g is thus in phase with E , the in- 
 ternal drop E z — VJ and equals E g — E. Therefore, under 
 those conditions the active voltage is 
 
 E x '= VS = {E g — E') cos 0 a . 
 
 Likewise, when E g is swung around to OJ' , the active voltage 
 is 
 
 E r "= VT=(E g + E)cos0 a . 
 
 By this construction TVS is a straight line and its value is 
 TS = E R r + E R "= 2 E g cos 6 a . 
 
 One half of this or E g cos 9 a is the radius of the circular locus 
 for E r given above. Therefore, bisect the line TS , as shown 
 at O', and describe the side TNS by the radius O' T=E g cos 6 a . 
 When E g is at the position OJ the second coordinate is O'V, 
 and 
 
 O' V= O'S- VS= E g cos 0 a - [E g - E] cos 0 a , 
 or O' V — E cos 0 a , 
 
 which is the value given in the formula. V is evidently the 
 origin and TS the initial line for the circle TNS as given by 
 the formula. For the pair of coordinates VS and VO' the 
 angle included is 180°, which is correct as 180 ° —(0 a — 0) is 
 180° when E g — OJ. The circle TNS is then the locus of the 
 active internal voltage E R , which has its pole at V and its initial 
 
682 
 
 ALTERNATING CURRENTS 
 
 E 
 
 line TS ; or, if the scale is changed by a ratio equal to it is 
 the locus of the armature current. 
 
 The power given out is 
 
 P = El cos 0, 
 
 where 0 is the phase angle between E and I or is NVJ when E g 
 is at the position OB. The maximum power is reached when 
 E g has swung forward so that I takes the position VQ, where Q 
 is the point of tangency of the current locus to a perpendicular 
 to OV. At this point E g has swung to OB' , where /3—0 a . If 
 the driving torque is great enough to drive the armature farther 
 ahead, the machine will break from synchronism. The minimum 
 power is found when E g has fallen back so that I lies at the 
 position VL, which is perpendicular to OV = E. If the torque 
 of the driving engine is reduced so that E g falls still farther 
 back, current is driven through the armature by the voltage of 
 the bus bars and the machine runs as a synchronous motor. 
 
 Heretofore we have dealt with a machine in which the in- 
 duced voltage was kept constant. An equally important con- 
 sideration is the effect of the variation of this voltage. Assume 
 that the load, including the internal copper losses, and the bus 
 bar voltage are kept constant, but that the generator excitation 
 is varied. In order that the load may be kept constant, the 
 torque of the driving engine, as determined by the setting of 
 its governor, is kept constant. Evidently, so long as the driv- 
 ing engine gives constant torque and the machine is held to its 
 synchronous speed, the total power must be constant irrespec- 
 tive of variation of the excitation of the alternator field magnets 
 or other variables. In Fig. 410 the triangle OHB is similar to 
 the triangle OHB in Fig. 409. In this case the power given 
 to the bus bars is as before 
 
 P = El cos 6 . 
 
 The total power generated which is to be kept constant is 
 P t = E g I cos (/ 3 + 0 ), 
 
 or from the figure, 
 
 P t = EJ sin ( 8 + 90° - 0 a ). 
 
 Therefore, 
 
 P t = E g I fsin 8 cos (90° — # a )+ cos § sin (90° — 0 O )}, 
 
SYNCHRONOUS MACHINES 
 
 683 
 
 Fig. 410. — Voltage Locus and Diagram of a Generator running in Parallel when its 
 
 Excitation is varied. 
 
 or 
 
 P t = E g I ( sin 8 sin 6 a + cos 8 cos 0 a ~). 
 But from the triangle OVB it is seen that 
 E sin /3 = E z sin 8, 
 
 or 
 
 and 
 
 or 
 
 . c. E sin /3 E sin 8 
 sm o = — = t-, 
 
 E z IZ„ 
 
 E g = E z cos 8 + E cos /3, 
 
 s E a — E cos /3 
 
 cos o - — 2 — • 
 
 IZ n 
 
 The symbols used here are the same as in the discussion of 
 Fig. 409. ’ Substituting these values in the final formula given 
 for P t , there results 
 
 p t = EJ KEPA sin 0 a + E gI E g -Eco S l3 
 ±Zj n 1/Jn 
 
 COS 
 
684 
 
 ALTERNATING CURRENTS 
 
 or P, = E sin /3 sin 6 a + cos 9 a — ^ E cos /3 cos 0 a , 
 
 ^ a ^ a a 
 
 and P t = - ^ E cos (/3 + 9 a ) + cos 0 a . 
 
 Z a A,. 
 
 The variables in this expression are E g and /3. Multiplying 
 Z 
 
 through by - — to give a term containing only E g 2 , there 
 results 
 
 P t Zg _ 
 
 cos 6 n 
 
 E 
 
 cos 6, 
 
 E g cos (yS + 6 a ) -f E 2 . 
 
 If we add to each side of this 
 
 E 2 
 
 4 cos 2 6, 
 
 , we have 
 
 PtZa + E 2 
 
 cos 6 n 4 cos 2 6 n 4 cos 2 6 n cos 6, 
 
 E - - E n Eg COS + 0 a ) + E 2 , 
 
 which is the formula for a circle expressed in polar coordinates, 
 where the left-hand member is the square of the radius and where 
 
 E 
 
 the initial line passes through the center, — — is the coordi- 
 
 2 cos 6 a 
 
 nate between the initial point and the center of the circle, E g is 
 the variable coordinate, and (/3 + d a ) is the angle included be- 
 tween the two. Then, to construct the locus of E r when /3 is 
 varied by changing the excitation but the total generated elec- 
 trical power P, remains constant, take some point 0 (Fig. 410) 
 as the pole. Lay off E = OV. \ vertically for convenience. At 
 the angle 9 a to the right of 0 Play off the initial line XX'. 
 
 E 
 
 Measure off the distance — from 0 , giving 00' . With 
 
 2 cos 0 a 
 
 O' as a center, describe the arc TT having a radius equal to the 
 square root of the sum of the terms on either side of the polar 
 equation. This is the required locus of E g , which for the field 
 excitation that gives it the value OB , shown in the figure, leads 
 the terminal voltage OP by the angle /3. From the relation 
 
 given, it is seen that the perpendicular U O' — — tan 6 a , and 
 
 E 
 
 UO = —; therefore O' may be at once found by laying off the 
 
SYNCHRONOUS MACHINES 
 
 685 
 
 E 
 
 distance — tan 6 a on the perpendicular bisector of VO. The re- - 
 
 quired locus is then found as before by striking a circle having 
 the radius 
 
 PjZg , ^ 
 
 cos 0 a 4 cos 2 9 a 
 
 Since the total power generated in the windings is kept con- 
 stant, the output will be a maximum when E z (= VB), which 
 varies with the current and therefore with the copper loss, is at 
 a minimum. This occurs when E z lies on a radius of the locus 
 TT and therefore when BV extended passes through O', that 
 is, when the field excitation has a value which makes the in- 
 duced voltage equal to OB' and the drop of voltage in the 
 armature is therefore B'V. From the geometrical construction 
 of the figure it can also be seen that at this point E R takes a 
 position on OF" extended and hence that the external power 
 factor is unity. If the excitation be made less than when E g = 
 OB', the current is larger and leads the bus bar voltage E\ 
 and when the excitation is increased, as when E g = OB, the 
 current lags behind E. The lagging and leading components 
 are so related to the series circuits, comprised of the generator 
 under consideration and the other generators (which maintain 
 the constant terminal voltage E), that their effects tend to be 
 respectively those of leading and lagging motoring currents. 
 Thus, when the generator voltage is low, it tends to cause the 
 voltage of other machines connected to the line to fall off 
 and when high to increase. This is explained more fully 
 in the article dealing with the synchronous motor.* Substi- 
 
 XT2 
 
 tuting w for — H — — E g cos (/3 + 0 O ) + E 2 in the last 
 
 4 cos 2 6 a cos 0 a 
 
 equation given, an expression for P t may be found in which w 
 stands for the radius of the circle locus of E g with fixed power 
 generated (i.e. circle TT of Fig. 410, which may be called a 
 
 Power circle), thus : 
 
 P t = 
 
 E 2 \ 
 4 cos 2 6J 
 
 Draw a number of power circles concentric with TT with any 
 convenient radii, such as T' T' , T" T", and T"'T'". Then, for 
 any given or fixed voltage E g , the current and voltage relations 
 
 * Art. 167. 
 
686 
 
 ALTERNATING CURRENTS 
 
 are immediately determinable for any power. Thus, take E t 
 equal to OB , Fig. 410, and draw a voltage locus MM as in Fig. 
 409 ; then if the power increases from that corresponding with 
 radius O' B to that corresponding with radius O' B", B g swings 
 over from OB to OB" and the current swings forward from the 
 lagging position VN to the leading position VN" with respect 
 to the terminal voltage. 
 
 The curves of current for constant power and varying exci- 
 tation, when plotted with generator voltages as abscissas and 
 armature currents as ordinates, are sometimes called V-curves 
 from their shape.* They can readily be obtained from the 
 figure by giving E g a series of values. 
 
 Prob. 1. Given an alternator having an armature resistance of 
 1 an ohm and synchronous reactance of 4 ohms. Construct the 
 locus diagram (Fig. 410) of the generator induced voltage when 
 the bus bar voltage is 1000 volts and the generated power is 100 
 kilowatts. When the power is 50 kilowatts. When 25 kilowatts. 
 
 Prob. 2. Determine from the diagram of Prob. 1 the power 
 output of the generator when the total powers are 100, 50, and 
 25 kilowatts. 
 
 163. Maximum Possible Load and the Regulation of Prime 
 Movers in Parallel Operation. — An engine torque greater than 
 the resisting torque of the maximum possible electrical load, will, 
 of course, cause the machine to break from synchronism. With 
 very large machines, the excessive currents that result from 
 such a condition may cause injury to the armature conductors by 
 the excessive magnetic forces set up between the conductors by 
 the large currents which may occur before the circuit breakers 
 get open ; for at the ends of the armature core where the wind- 
 ings extend beyond the core the conductors are sustained in 
 position by insulating materials of comparatively small tensile 
 strength. This is similar to the case of transformers, f 
 
 The maximum possible load which a prime mover can deliver 
 to its alternator, which is operating in parallel with others under 
 the conditions explained in the last article where dealing with 
 Fig. 410, is readily determined from the construction of that 
 figure. Thus, with fixed field excitation, the generated voltage 
 OB swings to the left when increased power is supplied from the 
 
 * Art. 170. 
 
 t Art. 131. 
 
SYNCHRONOUS MACHINES 
 
 687 
 
 prime mover, because the increased driving torque tends to 
 make the machine race. The vector OB therefore cuts larger 
 and larger power circles ( TT \ T' T', etc.) as the engine power 
 increases until, if the power becomes sufficiently great, it lies 
 upon the initial line or radius XX ' . At this point it reaches 
 the largest power circle possible for the particular field excita- 
 tion of the alternator. Then the angle /3 = 180° — d a . Any 
 greater power from the prime mover causes the point B to move 
 toward smaller power circles ; the speed increases and the gen- 
 erator falls out of synchronism. Reducing the excitation may 
 likewise then throw the generator out of synchronism. Thus, 
 suppose the machine is running under constant power repre- 
 sented by the power circle T" T" . Then suppose that the 
 voltage is reduced to OB"" (Fig. 410), making the triangle 
 OVB "" of the bus bar voltage B = OV, B g = OB" ", and the in- 
 ternal voltage drop E z — B"" V. Any smaller excitation, giving 
 a smaller value of OB"", will cause the point B"" to touch a 
 smaller power circle and the engine will speed up, causing the 
 generator to break from synchronism. The excessive currents 
 caused to flow in either of these limiting cases, as indicated by 
 the large internal drop B""V in the latter and an even larger 
 drop in the former when the larger constant generated voltage 
 OB was used, are far beyond the capacity of the windings of 
 modern commercial alternators. Further, when the voltage 
 drop line approximates the position B""V, the reactive effect 
 upon the field magnets of the generators connected to the bus 
 bars would be very great and cause the break from synchronism 
 to occur before the voltage vector had been rotated to a point 
 where it reached the line XX'. It is evident, therefore, that 
 destructive results may follow an excessive lead imposed by the 
 prime mover, such as might occur through accident to the gov- 
 ernor, or great weakening of the field current. In studying 
 Figs. 408, 409, and 410, it is well to note that the armature 
 angle of lag 6 a has been made smaller than is likely to be found 
 in commercial machines. 
 
 When alternators not rigidly connected together by mechan- 
 ical means are operated in parallel, it has been shown above that 
 the load division is dependent on the relative torques which the 
 prime movers tend to exert upon the rotors of the generators to 
 which they are respectively attached, and consequently the 
 division of load between alternators operated in parallel is de- 
 
688 
 
 ALTERNATING CURRENTS 
 
 pendent on the speed regulation of the prime movers. If one 
 prime mover tends to lag in speed behind the other, it produces 
 less power than is the case if it tends to lead. As explained * 
 the lagging machine under these circumstances is relieved of load 
 by the leading machine. This may be considered to be due to 
 the accelerating effect on the lagging machine caused by the sj 7 n- 
 chronizing current, or to the resultant reduction of load on the 
 lagging machine and its consequent increase on the leading ma- 
 chine. The instantaneous torque of the leading machine may be 
 considered equal to that caused by its load sent to the external 
 circuit plus the torque exerted upon the lagging machine to keep 
 it in step, while the torque of the lagging machine (if of equal 
 internal impedance and equally excited) may be considered equal 
 to that of an equal load to the external circuit minus the torque 
 expended upon it by the leading machine. Evidently, when one 
 of two equal steam engines driving alternators in parallel has its 
 governor set for a higher speed than the other, it acts automati- 
 cally and takes more steam when held below its normal speed by 
 the more sluggish second engine, while the second engine, being 
 urged forward above its normal speed by the first engine, auto- 
 matically takes less steam. Thus, the torque of the first engine 
 is greater than that of the second. As the two engines are 
 running at the same or synchronous speed, the first must then 
 take a larger proportion of the load than the second. f It is 
 then seen that the division of load on alternators in parallel 
 must be accomplished by regulating the governors of the prime 
 mover. This is the ordinary practice. The governors of sev- 
 eral engines are not apt to regulate exactly alike, and the par- 
 allel operating alternators which are connected to them are apt 
 to divide their loads unequally, that is, one machine “ robs ” the 
 others, unless the governors are adjusted with great care. It 
 is sometimes desirable to fasten the governors of all the engines 
 except one at the point of full load and permit the governor of 
 the one engine to maintain the speed regulation for all and carry 
 the variation in load. Then when the speed tends to rise, the 
 governed machine tends to retard the others by absorbing a 
 cross current, that is, it shifts its load on to its mates and 
 thereby prevents their racing. When the speed tends to fall, 
 the governed machine tends to take on the excess load. 
 
 * Art. 161. t Art. 162. 
 
SYNCHRONOUS MACHINES 
 
 689 
 
 164. Effect of Form of Voltage Curve and Variation of Angu- 
 lar Velocity on Parallel Operation and Division of Loads. — The 
 
 able designer, C. E. L. Brown, early operated his own alterna- 
 tors with smooth iron armature cores in parallel with Ganz 
 alternators which have highly reactive pole-type armatures.* 
 The former machine gave a voltage curve which approximated 
 a sinusoid, while the curve given by the latter was quite irregu- 
 lar and peaked. Dr. Steinmetz also early operated machines 
 with smooth iron armature cores in parallel with others having 
 toothed cores, f These experiments showed that machines with 
 different voltage curves can be run together satisfactorily, as is 
 now very frequently done, but they require a considerably 
 
 Fig. 411. — Curves Voltage to Cause Flow of Harmonic Cross Currents between 
 Paralleled Alternators having Voltage Curves like A and B. 
 
 increased exchange of current as compared with the synchro- 
 nizing current of machines with voltage curves which are exactly 
 alike. For, since the unlike curves cannot coincide even when 
 the machines are in exact step, a current of more or less irregu- 
 lar form must be exchanged by the machines, and this is super- 
 posed upon the true synchronizing current. This superposed 
 current may have a very different frequency from that of the 
 machines and is not necessarily wattless. For instance, in Fig. 
 411 curves A and B represent the voltage curves of two alter- 
 nators operating in parallel, in exact step, and with the same 
 effective voltage. The instantaneous differences between the 
 two curves leave residual voltages forming the curve G which 
 acts to set up a cross current in the series circuit comprising 
 the two armatures. 
 
 * Jour. Inst. E. E ., Vol. 22, p. 600. t Electrical World, Vol. 23, p. 285. 
 
690 
 
 ALTERNATING CURRENTS 
 
 Also, higher harmonic cross currents which may flow when 
 polyphase armatures which are operating in parallel act much 
 as they do in polyphase induction motors.* The third harmonic 
 sets up a rotary armature magneto-motive force opposed to that 
 of the field magnet. It has already been pointed out that the 
 armature currents in a polyphase alternator tend to set up a 
 rotary field which reacts upon the alternator field magnet, f 
 When the cross currents have sufficient magnetizing force, they 
 may cause the machines to work unsatisfactorily in parallel. 
 
 The angular velocity of reciprocating engines used for driv- 
 ing dynamos is by no means uniform throughout the revolu- 
 tions, although the revolutions may be isochronous. Curves 
 showing the instantaneous crank velocities taken from engines 
 of well-known types are often quite irregular. If alternators 
 are driven from separate engines, this irregularity of angular 
 velocity may be the cause of more or less difficulty in parallel 
 working, and some machines may work quite satisfactorily in 
 parallel when driven from the same countershaft or from sepa- 
 rate hydraulic or steam turbines (which give a uniform angu- 
 lar velocity), when they will not do so if driven by separate 
 reciprocating engines. It is evident that when engines of 
 different types are used to drive alternators that it is desired 
 to parallel, the angular velocities may vary so differently for 
 the different engines that the synchronizing current may be 
 unable to hold the alternators together. It is therefore desir- 
 able to specify, in obtaining reciprocating engines for parallel 
 working, a minimum variation in angular velocity. This 
 variation should not cause a displacement of the rotor from its 
 normal position of more than two electrical degrees during the 
 time of one cycle. $ 
 
 Sometimes it is necessary to cause reciprocating engines to 
 move with their cranks in the same angular space relation, 
 though it is difficult to do this, and it results in the variation 
 in frequency due to variation in angular speed being transmitted 
 to other synchronous apparatus connected to the system. 
 These other machines are then apt to hunt.§ 
 
 165. Synchronizers and Synchronizing. — Mechanical imper- 
 fections in engine governors and machine pulleys cause slight 
 
 * Art. 198. f Art. 152. § Art. 173. 
 
 J Standardization Rules of Amer. Inst, of Elect. Eng., Rules 61-64. 
 
SYNCHRONOUS MACHINES 
 
 691 
 
 differences in the speeds of machines intended to run at equal 
 velocities. Consequently it is desirable to arrange some device 
 for determining the moment a machine is in synchronism with 
 one with which it is to be thrown in parallel. When the 
 alternators are to be connected in parallel, the terminals of 
 each may be connected directly to appropriate bus or main 
 conductors through convenient indicating instruments, switches, 
 and safety devices. Before switching a new machine upon the 
 bus conductors it should be brought to normal speed and to 
 the voltage of the other machines. Then at a moment when it 
 is in synchronism and in step with the voltage wave of the bus 
 bars, it may be switched into circuit without causing a disturb- 
 ance among the other alternators. 
 
 Fig. 412. — Simple Lamp Synchronizer, One Phase only shown. 
 
 Any device for indicating the synchronous relation is called 
 a Synchronizer. Its simplest form for low voltage machines 
 consists of one or more incandescent lamps in series, which are 
 connected as in Fig. 412. One terminal of a phase of the 
 alternator is connected directly to a bus conductor, while the 
 other is connected through the lamps to another bus. When 
 the voltage waves are equal but not in opposition, the lamps are 
 illuminated, and at the moment of opposition the illumination 
 dies out. If the frequencies of the alternator and the bus bars 
 differ materially, the flashes of illumination or “ beats ” are 
 quite rapid. As the frequencies approach synchronism, the 
 beats lengthen out, exactly as do the beats of two tones which 
 are approaching unison. The alternator should be connected to 
 the bus bars at an instant of no illumination during a period 
 
692 
 
 ALTERNATING CURRENTS 
 
 when the beats are fairly long. This indicates that the voltage 
 of the alternator is substantially in synchronism with that of 
 the circuit and that it is in proper step or angular phase posi- 
 tion. Continued darkness of the lamps under these circum- 
 stances can only occur when the machines produce the same 
 voltage and run at absolutely the same frequencies, which is 
 not a practical occurrence unless the machines are rigidly 
 connected together. 
 
 For polyphase machines the lamps may be connected to a 
 single phase, as shown in Fig. 412, since if one phase is in step 
 with the bus bars the others must be, if the machines are prop- 
 erly connected to the bus bar leads. When connecting a poly- 
 phase machine for the first time to the bus bars, much care must 
 be taken to give assurance that the various phases are in proper 
 relation, or disastrous results may follow from the flow of ex- 
 cessive currents caused by the reversed relation of one or more 
 phases of the machine with reference to their respective bus 
 bar phases. The correct connections can be obtained by plac- 
 ing synchronizing lamps, or equivalent devices, in each lead 
 from the alternator to the bus bars. When the connections 
 have been properly made, the synchronizers will show opposition 
 of voltage phase between each alternator lead and its respective 
 bus bar. Synchronizing lamps connected as in Fig. 412, placed 
 in each lead, should all be dark together. 
 
 Since alternators are commonly built for high voltages, it is 
 usual to use a transformer with the synchronizer lamps. The 
 primary winding of this transformer may be composed of either 
 one or two circuits. When it is composed of one winding, one 
 terminal of the alternator is connected to a bus bar through it, 
 the other terminal of the alternator being connected directly 
 to the other bus bar (Fig. 413). In this case the lamp on 
 the secondary circuit acts exactly as the synchronizer lamps 
 already described. When the primary winding is composed of 
 two circuits, one is connected between the bus conductors and 
 the other between the terminals of the alternator (Fig. 414). 
 In this case the proper phase relation for switching the alternator 
 into circuit may be indicated either by darkness or by illumina- 
 tion of the lamps, depending upon the connections of the pri- 
 mary windings of the synchronizer transformer. This arrange- 
 ment is advantageous, since it allows the use of a double-pole 
 
SYNCHRONOUS MACHINES 
 
 693 
 
 switch at the dynamo, while the previously described arrange- 
 ments require the use of single-pole switches. The latter ar- 
 rangement may be modified by using two separate transformers, 
 
 Fig. 413. — Synchronizer Lamp connected to the Secondary Winding of a Simple 
 
 Voltage Transformer. 
 
 the primary circuit of one being connected to the alternator 
 and that of the other to the bus bar circuit. The secondary 
 windings are connected together in series with one or two lamps. 
 
 L 
 
 Fig. 414. — Sychronizer Lamp connected to the Secondary Winding of a Transformer 
 having Two Primary Coils. 
 
694 
 
 ALTERNATING CURRENTS 
 
 If the secondary windings are connected directly in series, as in 
 Fig. 415, darkness of the lamps indicates the instant for con- 
 i necting the alternator 
 
 
 to the bus conductors. 
 _ If the secondaries are 
 cross-connected as in 
 Fig. 416, maximum 
 illumination of the 
 lamps indicates the 
 moment when the ma- 
 chines are in proper 
 step. In this country 
 the general practice 
 has been to connect 
 lamp synchronizers so 
 that darkness indicates 
 when the machines are 
 
 Fig. 415. — Synchronizer Lamps connected in the Sec- in step. 1 his has the 
 ondary Circuit of Two Transformers having their evident advantage 
 Secondary Windings in Series. 
 
 that darkness is a con- 
 dition which is more readily distinguished than the condition of 
 maximum brightness. This practice, however, is not always 
 
 Ai 
 
 b vVvWi/vV\KWvV/vV 
 
 -A/WV VvV-| 
 
 — o — o 
 
 followed. 
 
 In any of these 
 methods, the 
 lamps may be 
 replaced by a 
 sensitive high 
 resistance alter- 
 nating-current 
 
 amperemeter or 
 galvanometer 
 which is dead- 
 beat. 
 
 An ingenious 
 device for use as 
 a synchronizer, 
 which consists of 
 two electromag- 
 nets made with 
 
 L^WWVWWWA 
 
 — o — o 
 
 Fig. 416. — Synchronizer Lamps arranged with Two Trans- 
 formers, where One of the Secondary Windings is reversed 
 with Respect to the Other. 
 
SYNCHRONOUS MACHINES 
 
 695 
 
 iron wire cores, has been used sometimes. The windings oi 
 these are connected respectively to the alternator and the bus 
 bar circuit. Each magnet has placed in front of it an iron 
 diaphragm which emits a tone which has a pitch due to the 
 frequency of the current flowing in the winding. In front 
 of the magnets are placed resonators which magnify the sound 
 emitted by the diaphragms. When the two tones are not in 
 exact harmony, interference causes beats, and the synchronizer 
 emits an intermittent sound. If the speeds of the machines are 
 brought nearer and nearer to synchronism, the beats become 
 less rapid. At exact synchronism, the beats die out and a 
 clear tone results. The alternator may or may not be in step 
 when the clear tone is given, but a small machine may be 
 safely thrown into circuit, for the interaction of the current 
 waves will bring it into proper phase relations. This synchro- 
 nizer was designed for use with synchronous motors. It cannot 
 give satisfaction with large alternators or with alternators that 
 are furnishing current at constant voltage for incandescent 
 lighting, since placing an alternator in the circuit while it is 
 out of step is likely to momentarily disturb the voltage. A 
 visual synchronizer, having much the same principle, consisting 
 of a set of vibrating reeds, can also be constructed. 
 
 It is possible to do without a synchronizer when connecting 
 small alternators in parallel, provided they are driven at ap- 
 proximately equal speeds. In this case if the alternator is 
 brought to the proper voltage and is then connected to the bus 
 bars, the machine reactions will bring it into step. This plan 
 was practiced to some extent in one or two earlier American 
 plants, but it is vicious in its working unless inductance coils 
 of considerable magnitude are introduced in circuit with the 
 incoming alternator. Throwing machines on to the bus bars 
 without synchronizing usually causes a disturbance in the vol- 
 tage, and it also strains the armature even of a small alternator 
 on account of the sudden torque impressed upon it to bring it 
 into step, and in large machines is apt to cause destructive strains 
 in the ends of the armature coils. In order to prevent an undue 
 rush of current when alternators are thrown together while 
 slightly out of step, external inductance coils are sometimes 
 inserted temporarily in series between the machines and the 
 bus bars. 
 
696 
 
 ALTERNATING CURRENTS 
 
 A transformer having three legs and windings like the tri- 
 phase transformer of Fig. 294 may be used for operating the 
 synchronizing indicator. The windings of the two outer legs 
 are attached respectively to the bus bars and the machine, and 
 a voltmeter or lamp is connected across the middle winding. 
 At the point of synchronism the magnetic fluxes produced by 
 the two exciting coils are in parallel in the middle core and the 
 voltmeter shows a maximum reading. 
 
 An instrument which indicates whether the machine to be 
 synchronized is running fast or slow, as well as the condition 
 of step, is sometimes called a Synchroscope or Phase indicator. 
 One of the simplest arrangements of this kind consists of a disk 
 attached to the alternator shaft or that of a small synchronously 
 running motor. This disk is painted in alternately white and 
 black sectors, each sector having an angular width equal to the 
 polar pitch. In front of the disk is placed an arc lamp at- 
 tached to the bus bars through the necessary transforming 
 device. The ai'rangement is the same as that sometimes used 
 for measuring tire slip in an induction motor.* Then when 
 the illuminated sectors of the disk apparently stand still, the 
 frequency of the machine is the same as that of the bus bars. 
 When the illuminated sectors of the disk apparently move in 
 the direction opposite to the direction of the rotation of the 
 rotating part of the machine, the speed of the latter is too slow ; 
 and when the illuminated sectors apparently move in the direc- 
 tion of the rotating part, the speed of the machine is too fast. 
 The step phase when the bus bars and machine are in synchro- 
 nism is indicated by the position of the stationary sectors, but 
 the additional use of a lamp synchronizer is desirable. 
 
 In a three-phase machine the use of three lamps, as shown in 
 Fig. 417, may be adopted. In this case the three lamps, through 
 suitable transforming devices, are placed at the corners of a tri- 
 angle and a terminal of each connected to one of the three bus 
 bars, while the other three terminals are connected to one phase 
 wire of the machine. When the machine is in synchronism, 
 the lower right-hand lamp in the figure is in darkness while 
 the other two are equally bright. When the speed of the ma- 
 chine is slightly too high, the lamps glow one after the other, 
 giving the appearance of a rotation in one direction ; and when 
 
 * Art. 189. 
 
SYNCHRONOUS MACHINES 
 
 697 
 
 
 
 BUS BARS 
 
 
 
 
 
 
 17 
 
 SWITCH / 
 
 \ 
 
 V 
 
 l 
 
 
 LAMPS 
 
 Fig. 417. 
 
 ALTERNATOR 
 
 A Three-lamp Synchroscope for a 
 Three-phase System . 
 
 the speed is too low, the rotation of the lighting is in the other 
 direction. Various modified arrangements to accomplish this 
 result may be made. 
 
 Synchroscopes using the magnetic relations of a movable coil 
 to which a pointer is 
 attached are now gener- 
 ally used in commercial 
 plants and are eminently 
 convenient and satisfac- 
 tory. A common form 
 has an armature which 
 is somewhat similar to 
 the field magnet of a two- 
 coil, split-phase induction 
 motor without its iron 
 core. A two-pole field 
 magnet, 0, embracing 
 the armature may have 
 its coil connected to the 
 bus bars (Fig. 418), and 
 the split-phase armature winding, its two coils A and B wound 
 at right angles and its shaft pivoted between the pole pieces 
 
 of the field, may be connected 
 to the machine to be synchro- 
 nized, or vice versa. A pointer 
 is attached to the armature for 
 the purpose of indicating its 
 motion. When the currents in 
 the field winding and armature 
 windings are in synchronism, 
 the armature takes an angular 
 position with reference to the 
 field magnet which is deter- 
 mined by the phase difference 
 between the currents entering 
 the field and armature wind- 
 ings, the zero position being 
 assumed when the currents are 
 in step with each other. When the frequency of the machine 
 is greater than that of the bus bars, the armature of the syn- 
 
 LEADS TO 
 ALTERNATOR 
 
 wmrwrci 
 
 Fig. 418.- 
 
 U& flOOQPOQ J 
 ^LEADTa 
 
 Mi l {goo 
 BUS BARS 
 
 -Diagram of a Synchroscope of 
 Split-phase Type. 
 
698 
 
 ALTERNATING CURRENTS 
 
 chronizer rotates in one direction ; and when the frequency of 
 the bus bar voltage is the greater, the rotation is in the opposite 
 direction. Thus, when the machine is in exact step and syn- 
 chronism with the bus bar voltage, coil A takes a position with 
 its phase at right angles to the flux of the field. B is then 
 inert, its time phase being 90° from that of the field. When 
 the phases change, say 45°, A and B then equally tend to place 
 themselves at right angles to the field flux, and the armature 
 must take up a position with A 45° from its former position. 
 At 90° difference between the machine and bus bar phases, A is 
 inert and B takes a position at right angles to the field flux. 
 At 180° difference of phase A takes a position 180° from its first 
 position, and so on. Then if the machine voltage is in synchro- 
 nism but out of step with the bus bar voltage, the pointer takes 
 a definite angular position ; but if the machine frequency is 
 different from that of the bus bars, the pointer must make one 
 complete revolution in one direction for each cycle gained and 
 in the other direction for each cycle lost. 
 
 Considering the two coils A and B wound on the armature 
 at right angles to each other and traversed by currents in quad- 
 rature, then the effec f - of the windings is to set up a rotating 
 magnetic field which makes its complete rotation with a fre- 
 quency equal to the frequency /' of the currents in the coils.* 
 The field magnet directs a flux through the armature which 
 
 has the frequency / of the current through the field winding. 
 The armature tends to place itself in such a position that its 
 flux shall be at all times in the same direction as the flux of the 
 
 field magnet, but during one period of the field current the ro- 
 
 /' 
 
 tating field of the armature will rotate through ‘y 860°, where 
 
 /is the frequency of the current in the field coil and/' the fre- 
 quency of the currents in the armature coils. Therefore the 
 
 armature core will turn through an angle of fl — ^ \ 360° = 
 
 f-f 
 
 f 
 
 360° during each period of the field flux, which is equiv- 
 
 alent to saying that the armature will rotate /— /' revolutions 
 per second since the field flux makes/ complete cycles per sec- 
 ond. If /' is smaller than /, the armature core will rotate in 
 the same direction as the rotating magnetic field set up by its 
 
 * Art. 186. 
 
SYNCHRONOUS MACHINES 
 
 699 
 
 windings, but at a speed which is slower in the ratio of 
 
 /-/' 
 
 f 
 
 and if f is the larger, the armature will rotate in the opposite 
 
 f-f 
 
 direction, but also with the speed ratio 
 
 f 
 
 The bus bars 
 
 being permanently connected to one of the synchroscope cir- 
 cuits, the machine to be synchronized must be connected to the 
 other circuit, and the direction of rotation of the needle then 
 always indicates whether the incoming machine is running too 
 fast or too slow. When the pointer of the instrument stands 
 still, the two frequencies are equal. If the currents in A and 
 B stand respectively in phase with and 90° behind the voltage 
 impressed on their joint circuit and the field flux is 45° behind 
 the voltage impressed on the field magnet circuit, the pointer 
 will stand at zero when the voltages of bus bars and machine 
 are in both synchronism and step. The two differing phases 
 can be obtained for the armature coils by the use of inductance 
 and resistance as shown in the figure, or where the system is 
 polyphase, directly from the machine or bus bars. 
 
 Various other arrangements for accomplishing the result de- 
 scribed in the preceding paragraph are also possible. 
 
 Automatic synchronizers in which the main switch is closed 
 by the action of an electro-magnet with its coil in the synchro- 
 nizing circuit are sometimes used, though they have not fully 
 shown their desirability. The arrangement usually comprises 
 an electro-magnet in series with the synchronizer lamp circuit, 
 the armature of which either directly trips the main switch, 
 which closes by a spring, or closes a switch which in turn 
 causes a strong electro-magnet attached to the main switch to 
 become excited, and through the attraction of it the armature 
 closes the main switch. 
 
 166. Methods of Connecting Alternators in Parallel to Feeder 
 Circuits. — Switchboards for connecting alternators in parallel 
 are almost as various in details of construction as are the num- 
 ber of switchboards built, that is, each board is designed to 
 meet the conditions it must fill, although it is usual to follow 
 general standards. They are usually of much the same 
 character so far as relates to the underlying duty they are to 
 perform. Figure 419 is the diagram of a suitable board designed 
 by one of the large manufacturing companies for controlling 
 
700 ALTERNATING CURRENTS 
 
 Fig. 41!). — Switchboard Connections for operating Three Tri-phase Generators in Parallel. 
 
SYNCHRONOUS MACHINES 
 
 701 
 
 Fig. 420. — Cross Section of an Alternating-cur- 
 rent Switchboard. 
 
 tri-phase alternators. The 
 lower heavy lines repre- 
 sent the exciter bus bars 
 and connections, the upper 
 heavy lines represent the 
 alternating current bus 
 bars (which are often du- 
 plicated for safety but are 
 shown here in only one 
 set) and connections, and 
 the light lines show the 
 measuring instrument con- 
 
 nections. The feeders to 
 the load are shown at the 
 right, and the synchroniz- 
 ing apparatus and volt- 
 meters to the left. The details of the connections may be readily 
 obtained from a study of the figure. Additional generators and 
 feeders can be added to the system shown in the figure by ex- 
 tending the bus bars. Ordinarily all 
 high voltage is kept away from the face 
 of the switchboard, and the highest 
 voltage switches are usually situated 
 at a distance in fireproof cells and oper- 
 ated by suitable mechanical or electri- 
 cal transmission, the high voltage bus 
 bars being located in fireproof ducts. 
 Thus Fig. 420 shows a cross section of 
 a switchboard in which the high vol- 
 tage oil switches and the current and 
 voltage transformers for operating the 
 switchboard instruments are placed on 
 a framework at the rear of the board. 
 The instruments and controlling 
 levers are in front of the control 
 board. The switches may be 
 made to act as automatic circuit 
 breakers by adding suitable auto- 
 
 Fig. 421. — View of High Voltage Auto- 
 matic Switch. 
 
 matic tripping devices. A 
 switch or circuit breaker with 
 
702 
 
 ALTERNATING CURRENTS 
 
 its parts immersed in oil, called an oil switch or circuit breaker, 
 suitable for voltages of several tens of thousands of volts is 
 shown in Fig. 421, and in Fig. 422 is a cross section of the 
 same apparatus. The contact parts are immersed in oil in the 
 
 large tank and the con- 
 tact is opened and closed 
 through the intervention 
 of the wooden connect- 
 ing rod A. Movement 
 for operating the switch 
 is transmitted to the rod 
 A by means of the sole- 
 noid B and associated 
 links. The current for 
 the solenoid is controlled 
 from the control board. 
 A small electric motor 
 may be substituted for 
 the solenoid. The ter- 
 minals for leading current into 
 and out of the switch are shown 
 at TT , Fig. 421. 
 
 167. Alternators as Synchro- 
 nous Motors. — Any alternator 
 may be run as a motor, provided 
 it is brought up to synchronous 
 speed and into step before it is 
 thrown into circuit. The motor 
 will then run in complete syn- 
 chronism if left to itself. If it 
 is overloaded, or by other means 
 is thrown out of synchronism, it 
 will stop. In general, the action 
 Fig. 422. — Section of High Voltage of an alternator used as a S}-n- 
 Switch - chronous motor is quite similar 
 
 to that of an alternator operated in parallel with another. 
 A great disadvantage of single-phase synchronous motors 
 is the fact that they are not self-starting : but must be 
 brought up to speed before they will operate. The starting of 
 a single-phaser may be done by a small series-wound auxiliary 
 
SYNCHRONOUS MACHINES 
 
 703 
 
 motor made with laminated fields.* A small two-pliase motor 
 with a device for splitting the current into two phases may 
 be used, or the exciter of the alternator may be run as a motor 
 from a direct current source and used to bring the alternator 
 into synchronism. 
 
 A polyphase synchronous motor may be started by an ordinary 
 polyphase induction motor coupled to its shaft ; or it may be 
 self-started as an induction motor. In the latter case the ar- 
 mature of the synchronous machine becomes the primary portion, 
 and the field magnet the secondary portion, of the rotating field 
 induction motor. For starting, the field circuit is usually 
 opened. Current is permitted to flow from the main circuit 
 at reduced voltage through the armature windings, an auto- 
 transformer (Fig. 423) or a transformer having several voltage 
 taps from the secondary winding being ordinarily used for the 
 purpose of giving the reduced voltage. When the machine has 
 been brought up to speed by the torque exerted between the 
 eddy currents induced in the pole faces and the rotating magnetic 
 field set up by the currents in the armature windings, the field 
 switch may then be thrown on to the direct current exciter bus 
 bars, and the machine will continue to rotate as a synchronous 
 motor. When the field coils are large, the winding must be 
 opened in a number of places to prevent excessive voltages 
 being generated in them by the rotating armature flux. When 
 the rotor of the machine has been brought to as high a speed as 
 possible as an induction motor, it may jump into synchronism 
 before the field switch is closed to send direct current through 
 the field windings. The action is similar to that in large un- 
 loaded induction motors f with appreciable polar projections, 
 that is, the magnetic flux from the armature rotating but very 
 slightly faster than the projecting poles of the field magnet, 
 magnetizes the poles as it passes over them and they take up 
 synchronous speed, continuing to obtain magnetization from the 
 armature magneto-motive force. When the field switch is closed, 
 the magnetic flux already set up in the fields by the armature 
 reaction may be in the wrong direction, in which case the arma- 
 ture must fall back the distance of the polar pitch before the 
 regular field flux can produce forward operating torque. The 
 processes for starting rotary converters are given in greater 
 * Chap. XII. + Ibid. 
 
704 
 
 ALTERNATING CURRENTS 
 
 detail later,* and as they are essentially the same on the alter- 
 nating current side as for synchronous motors, many details con- 
 cerning starting the latter are deferred for the present. 
 
 Figure 423 shows the elementary connections for starting a 
 quarter-phase synchronous motor when an autotransformer con- 
 trolled by a cylinder switch is used. The rheostat in the motor 
 field circuit, the main switch, and measuring instruments are 
 not shown. The exciter may be driven mechanically by the 
 synchronous motor and should build up its voltage while the 
 
 Fig. 423. — Elementary Diagram of Connection for a Quarter-phase Synchronous 
 
 Motor. 
 
 synchronous motor is coming up to speed. In order to make 
 the machine field frame assume the function of the secondary 
 conductors of a short-circuited induction motor, it is sometimes 
 desirable to place copper conductors about the field poles in 
 such a manner as to make short-circuited paths (Fig. 442). 
 When the motors are very large, or are not designed to start as 
 induction motors without an excessive flow of current, they are 
 commonly brought into synchronism by starting motors, f as are 
 single-phase machines. 
 
 168. Relation of Voltages, Currents, and Power in Synchronous 
 Motors. Synchronous Impedance. — When a synchronous motor 
 is put in the circuit, a peculiar relation exists between the 
 strength of the field of the motor and the current in its arma- 
 ture. In direct-current motors, if the strength of the field is 
 
 * Art. 181. 
 
 f Chap. XII. 
 
SYNCHRONOUS MACHINES 
 
 705 
 
 slightly changed without altering any of the other conditions, 
 the speed of the motor changes inversely, and the current in 
 the armature remains practically unchanged ; but the speed of 
 a synchronous motor cannot change permanently, and, con- 
 sequently, upon first consideration, it would appear that the 
 strength of the field magnet of a synchronous motor must be 
 exactly adjusted, in order that the machine may operate satis- 
 factorily. This is not the case, however, as was seen in the 
 discussion of the parallel operation of alternators,* on account 
 of the effect which may be gained through variations of the 
 relative phases of the current and of the impressed and counter 
 voltages. The active voltage, which at any instant causes 
 current to flow through the armature of a motor, is equal to the 
 difference of the corresponding instantaneous values of the im- 
 pressed and counter voltages. If the field strength of a motor 
 is so adjusted that the value of the impressed and counter vol- 
 tages are equal, and the motor armature is brought by external 
 means into exact step with the impressed voltage curve, then, 
 when the motor is switched on the supply main, it will fall 
 back in phase with respect to the impressed voltage sufficiently 
 to permit the proper driving current to pass through the arma- 
 ture. Now suppose that at some instant the load is increased, 
 the difference of instantaneous voltage at that instant will be 
 insufficient to cause the larger current to flow through the 
 armature which is necessary to produce the torque for the new 
 load. The motor, therefore, falls back in its phase or armature 
 step without losing synchronism, if the load is not too great, 
 and then continues operating in synchronism, but with a greater 
 lag in step. When a motor lags in step behind the phase of 
 impressed voltage, its counter-voltage lags to an equal extent. 
 The armature current ordinarily takes an intermediate phase, 
 so that it is behind the resultant voltage but in advance of op- 
 position to the counter-voltage. 
 
 Were it not for the just described automatic adjustment of 
 the phases of the impressed and counter-voltages, it would be 
 necessary to adjust the field excitation of a synchronous motor 
 so that its counter-voltage would be less than the impressed 
 voltage and the range of load carried with a given excitation 
 would be limited to one value. The effects due to the auto- 
 
 * Arts. 162, 163. 
 
 2 z 
 
706 
 
 ALTERNATING CURRENTS 
 
 matic adjustment of phase, however, make it possible to fix the 
 field excitation once for all, and yet have the motor operate 
 on widely varying loads. It is even possible, on account of 
 the automatic adjustment of the voltage phases, to operate a 
 motor when its excitation is much greater or much less than its 
 normal value. 
 
 The adjustment is assisted by the effect of armature reactions 
 on the motor, in which a current lagging with respect to the 
 impressed voltage tends to strengthen the field magnet and a 
 current leading with respect to the impressed voltage tends to 
 weaken the magnet.* When a single motor is operated from 
 an alternator of about its own size, the automatic adjustment 
 of the machines is still more marked, since the current which 
 strengthens the field magnet of the motor tends to weaken that 
 of the alternator as the load is varied, and vice versa. A balance 
 for each load is reached when the two voltages have attained 
 such values that their vector difference or resultant is just suf- 
 ficient to drive a current through the circuit which gives a 
 torque sufficient to keep the motor running synchronously. 
 
 It is evident from the preceding paragraphs that the armature 
 current of a synchronous motor must have a quadrature com- 
 ponent which depends upon the phase difference of the im- 
 pressed and counter voltages and the internal angle of lag, and 
 it may readily be seen that a particular excitation of a syn- 
 chronous motor field magnet reduces the armature current to 
 a minimum (or makes the power factor a maximum) when the 
 motor is carrying a particular load. 
 
 The current which flows through the armature is not only de- 
 pendent upon the ordinary impedance, consisting of the resist- 
 ance of the windings and their self-inductances, but also upon 
 the field reactions caused by the quadrature component of cur- 
 rent. For convenience the effect of the field reactions is fre- 
 quently included and treated as part of the armature impedance 
 instead of as a factor directly affecting the value of the counter- 
 voltage of the motor. The impedance then obtained is called 
 synchronous impedance. As explained earlier f it may be de- 
 fined as the ratio of the voltage which would be developed at 
 the given excitation, if the machine was run as a generator at 
 synchronous speed without current in the armature, to the am- 
 
 fArt. 156. 
 
 * Art. 152. 
 
SYNCHRONOUS MACHINES 
 
 707 
 
 peres that would flow though the armature at the same excita- 
 tion when it is short-circuited. In the diagrams for the parallel 
 operation of alternators and those to follow for the synchronous 
 motor the machine voltages used are those that would be gen- 
 erated were field reactions absent, while to compensate for this 
 assumption, the synchronous impedance of the armature, which 
 includes the effect of reactions, has been used instead of the or- 
 dinary impedance caused only by the self -inductance and resist- 
 ance of the armature windings. The synchronous impedance 
 has also, for simplicity, been considered constant when the ex- 
 citation has been varied. As a matter of fact the field reactions 
 and the true reactance of the armature vary with the space posi- 
 tion of the armature coils with reference to the field magnet and 
 the time phase of the current, as well as upon the reluctance of 
 the leakage magnetic circuits, so that some error is involved in 
 assuming the synchronous impedance to be constant. Thus, it 
 is evident that the synchronous impedance is less for very high 
 excitations than for those of low values, since in the first case the 
 magnetic circuits are more highly saturated and a given arma- 
 ture magneto-motive force has less effect upon the field mag- 
 netic flux than in the second case. 
 
 In order to bring out more clearly the relation of voltages 
 and currents in the motor, recourse may be had to a diagram 
 in which relations of current and voltage are shown much as 
 in parallel working. It was shown* that in parallel working 
 the machines were held in step by a motor action, caused by 
 a wattless current flowing, and that if one machine was cut 
 off from its prime mover it would continue to run in synchro- 
 nism as a motor. In Fig. 424 let 01 be the current flowing 
 from a circuit through the armature of an alternator acting as 
 a motor, let SR be the drop of voltage in the synchronous 
 reactance, equal and opposite to the reactive voltage OL of the 
 armature, and let OS be the drop of voltage in the armature 
 resistance ; then OR is the resultant voltage IZ , required to 
 act on the synchronous impedance of the armature to pass the 
 current 01 through the circuit. Suppose the voltage OE x is 
 impressed at the armature terminals, and the motor is excited 
 to give a numerically equal voltage OE % ; then OE x and OE 2 
 must be in such phases with respect to each other as to give 
 
 * Art. 161. 
 
708 
 
 ALTERNATING CURRENTS 
 
 the resultant OR , while the components of voltage resolved 
 upon the current vector 01 must be such that the component 
 of OE x has the same direction as the current, and the compo- 
 nent of OE g is in opposition. To obtain this relation the arma- 
 ture voltage must fall back from direct opposition to the 
 impressed voltage by a space angle equal to 0 2 + 0 V In other 
 words, the armature conductors must take up a step relation 
 
 y 
 
 Fig. 424. — Diagram showing Relation of Voltages and Current in a Synchronous 
 Motor for a given Field Excitation and Armature Current. Impressed Voltage 
 Equal to Counter-voltage. 
 
 with respect to the field magnetism that is determined by the 
 vector values of E v E v and IZ. The work delivered to the 
 motor by the circuit is IE 1 cos 0 V and that utilized by the motor 
 armature in furnishing power and overcoming the magnetic and 
 frictional losses is IE 2 cos 0 2 ; while that lost, due to armature 
 resistance, is their difference, and is equal to 
 
 I 2 Z COS = OS X 01. 
 
 It will be seen that for small loads on the motor the current 
 with the assumed voltage may lead the impressed voltage, as 
 shown in Fig. 424, that is, 0 X is negative ; but that as the load 
 increases (and the length of OR therefore increases), the 
 motor counter-voltage is caused to swing farther back in step 
 compared with the impressed voltage, so that the current takes 
 a lagging position, as shown in Fig. 425. The construction 
 indicates that the current is always more than 180° in the lead 
 of the counter-voltage, when the impressed and counter vol- 
 tages are equal, and that the value of (# 2 + 8 x ) increases when 
 
SYNCHRONOUS MACHINES 
 
 709 
 
 the load on the motor is increased. The value of the current 
 I is one half greater in Fig. 425 than in Fig. 424, the input of 
 the motor IE 1 cos 0 1 is 50 per cent greater, and the output 
 IE 2 cos # 2 about 45 per cent greater. The latter is increased 
 in a smaller proportion, because the I 2 R loss increases directly 
 as I 2 . The construction of these figures makes it manifest that 
 the parallelogram of voltages OE v OE 2 and resultant OR de- 
 termines the values of 0 l and 0 2 so that each has a particular 
 value for each given value for the impressed and counter-vol- 
 tages and the magnitude of the motor load. 
 
 Fig. 125. — Vector Diagram of Voltages and Current when the Load is increased, hut 
 the Conditions are Otherwise as in Fig. 424. 
 
 The motor will continue to operate, as the load is increased, 
 until &2 has attained such a value that E 2 cos 0 2 x I becomes a 
 maximum ; then, if an additional load is put on the motor, the 
 corresponding increase of 0 2 will cause IE 2 cos d 2 to decrease, 
 and the motor will fall out of synchronism and stop, because 
 the maximum value of its torque is not sufficient to pull the 
 load. In the case under consideration (when the impressed 
 and counter voltages are equal) this will not occur with well- 
 designed alternators until a load much above that for which 
 the machine is rated is reached. 
 
 As was stated, the counter-voltage at which the motor is run, 
 and therefore its excitation, has an important bearing upon the 
 
710 
 
 ALTERNATING CURRENTS 
 
 stability of operation and the efficiency of transmission. If the 
 motor voltage is made larger than that of the generator, the 
 current and voltage relations may be shown by a construction 
 similar to that used in Fig. 424. If OR in Fig. 426 represents 
 the magnitude and direction of the resultant voltage, and the 
 impressed and counter voltages have magnitude OE x and OE v 
 then the parallelogram can be completed with the values of the 
 angles 6 l and 0 2 shown in the figure. In this illustration the 
 counter-voltage is greater than the impressed voltage. Now 
 
 Fig. 426. — Diagram showing the Effect of Variation of Excitation in a Synchronous 
 
 Motor. 
 
 suppose the motor excitation is changed so that the counter- 
 voltage is OE 2 having the same horizontal projection as OE v 
 while OR and the impressed voltage have the same values as 
 before ; then the phase relations are as shown by the lines 
 OE x \ OR , and OE 2 . The values of OE 2 cos 0 2 and OE 2 ' cos 0 2 
 are equal by the construction, since the points E 2 and E 2 are in 
 the same vertical line ; and since OR lias the same magnitude 
 and position in the two cases the current is the same. Also, 
 OE l cos dj and OE cos 0 X ' are equal ; but in the first instance 
 the counter-voltage is greater than the impressed voltage and 
 the current leads the impressed voltage, and in the second 
 instance the impressed voltage is the greater and the current 
 
SYNCHRONOUS MACHINES 
 
 711 
 
 lags. This construction shows (with a fixed value of impressed 
 voltage) that for every load on the motor, except that correspond- 
 ing to a zero angle of lag, there are two values of the excitation 
 which cause the same armature current to flow, the current lead- 
 ing the impressed voltage with the higher excitation and lagging 
 by an equal angle with the lower excitation ; hence an over-ex- 
 cited motor acts upon the line current very much like a con- 
 denser, and an under-excited motor acts like an inductance coil. 
 
 Figure 433 shows the armature current of a synchronous motor 
 operating under fixed load on a circuit of fixed voltage, plotted 
 as a function of the field-exciting current. This shows plainly 
 that the armature current assumes the same numerical value for 
 each of two excitations. The vector diagrams show that the les- 
 ser excitation corresponds to a current lagging and the greater ex- 
 citation to a current leading with respect to the impressed voltage. 
 
 The excitation at which the motor will do the most work 
 with a given current flowing, or will carry a given load with 
 the least current (and hence do it with the best power factor), 
 is, as is proved later (Art. 169), that which causes the current 
 to come into phase with the impressed voltage. In this case 
 the current for a given load is a minimum, and E 2 cos 0 2 is a 
 maximum for the conditions concerned. If the value of E 2 in 
 Fig. 426 is increased (by increasing the excitation of the 
 motor), either the value of I and the length of OR which is 
 proportional to it, or the impressed voltage, must be increased, 
 provided IE 2 cos 0 2 , which is equal to the motor load, is con- 
 stant. On the other hand, taking the vector rectangle OE 2 RE v 
 if E 2 decreases numerically while E x remains numerically con- 
 stant, the values of the angle 0 X and the current decrease until 
 the current and impressed voltage come into phase, after which 
 a further decrease of E 2 causes the current to increase again 
 and the angle of lag to increase in the positive direction. All 
 this is shown by the relations between E v /, and E 2 in the figure. 
 
 The value of IE 2 cos 0 2 (that is, the load carried when the 
 OR 
 
 current is , when Z is the synchronous impedance, and the 
 
 Z 
 
 counter-voltage is either OE 2 or OE 2 ~) is proportional to the 
 area of the rectangle OlqP , since 01 is proportional to I. Now 
 if the motor load is kept constant, but its field excitation is 
 changed, the corner of the corresponding rectangle must move 
 
715 
 
 ALTERNATING CURRENTS 
 
 from q, but the locus of the positions of the corner is a hyper- 
 bola with its origin at 0 and the rectangular axes x and y, since 
 the rectangles included between the axes and the ordinate and 
 abscissa for each point must be all of equal area. Consequently, 
 the point of the vector representing E 2 at the least current for 
 the given load must be in the rectangular hyperbola mm which 
 passes through q. This point, which is E 2 " for the given load 
 and the impressed voltage E v is found by laying off the hori- 
 zontalline equal in length to 0E 1 which will just reach between 
 the hyperbola and the line OR. This cuts OR at R ' , and the 
 
 sought for minimum current is 1= 
 
 9L 
 
 X 
 
 OR' 
 Z ’ 
 
 where X is 
 
 the synchronous reactance and Z the synchronous impedance. 
 The impressed voltage and current are, under these conditions, 
 in phase with each other. If a larger or smaller counter- voltage 
 than OE 2 " is used, such as 0E 2 or OE 2 ', the impressed voltage 
 and current are thrown out of phase, and the current in the cir- 
 cuit is, therefore, increased, if the impressed voltage remains of 
 the same numerical value, which causes increased PR losses and 
 armature reactions. The excitation of the motor which brings 
 the current and impressed voltage into step for a given load 
 depends upon the relation of the synchronous impedance of the 
 motor circuit to its resistance. When the angle i/r is 60°, that 
 is, when Z — 2 R, the counter-voltage is equal to the impressed 
 
 voltage at zero lag of the current, since ^ = :L — under con- 
 ditions of maximum output (Art. 169) ; and when Z is greater 
 or less than 2 R the counter-voltage is respectively greater or 
 less than the impressed voltage at zero lag. Putting X and Z 
 for synchronous reactance and impedance, then X = V3 R when 
 Z = 2 R, since Z — VR 2 + X 2 . In the machines which are now 
 commonly built, the synchronous impedance of the armature cir- 
 cuit is commonly larger than twice the resistance, and the max- 
 imum power factor and output are obtained in such synchronous 
 motors by an excitation which gives a counter-voltage that is 
 greater than the impressed voltage. 
 
 Figure 427 shows polar diagrams for various excitations at 
 which a small motor under constant load was provided with 
 power by a generator of exactly the same size and tj'pe as the 
 motor. Lines OE v OE v and OR are the generator, motor, and 
 
SYNCHRONOUS MACHINES 
 
 718 
 
 Fig. 427. — Vector Diagrams obtained from Experiments made with Two Small 
 Similar Machines, One Running as a Motor, the Other as a Generator. 
 
714 
 
 alternating currents 
 
 resultant voltages respectively acting in the circuit comprising 
 the two machines, and 01 is the current. These were obtained 
 from a diminutive pair of single-phase alternators connected 
 together and operated as motor and generator in a classic ex- 
 periment of Professors Ryan and Bedell. They illustrate the 
 vector relations of current and voltages for motors of all sizes. 
 The angle by which the current lags behind the resultant vol- 
 tage was obtained from the synchronous impedance of the arma- 
 ture circuit. It may be clearly seen from the diagrams that the 
 current swings from a position of large lag with reference to 
 the 'generator voltage at a small excitation of the motor as in 
 diagrams Nos. 1, 2, and 3 ; into phase with it, as between dia- 
 grams Nos. 4 and 5, the point of minimum current for the given 
 load on the motor ; and finally into a position of large lead 
 when the motor is greatly over-excited, as in diagrams Nos. 7 
 and 8. The numerical values of E v E v /, and E z and the an- 
 gular relation of E 1 and E v as observed, are shown in the figure. 
 The ratio of E z to 1 is the internal impedance of the motor. 
 
 Figure 428 shows the armature current for different motor 
 field excitations when the motor was under the same small con- 
 stant load. Curves I), F , and 
 Cr in the same figure show 
 plainly the tendency of the 
 armature reactions and leak- 
 age reactance to reduce differ- 
 ences between numerical val- 
 ues of motor and generator 
 voltages. Curve D repre- 
 sents the generator voltage, 
 and, although the generator 
 excitation was constant, the 
 generator voltage rises as 
 the motor voltage is in- 
 creased. This is due to the 
 reaction caused by the cur- 
 rent swinging from a lag to 
 a lead with reference to the 
 generator voltage. At the 
 same time the motor voltage, which is represented by curve F, 
 at first is larger and then grows smaller than would be the case 
 
 MOTOR FIELD CURRENT 
 
 Fig. 428. — Curves showing Relation of Vol- 
 tages and Armature Current for Constant 
 Load and Variable Excitation in a Syn- 
 chronous Motor. 
 
SYNCHRONOUS MACHINES 
 
 715 
 
 were no reactions present. The curve G- represents the motor 
 voltage considering reactions absent. The effect in the motor 
 is caused, as in the generator, by the current changing its lead 
 with reference to the motor voltage. 
 
 This series of experiments and the preceding discussions show 
 that there was some foundation for the statement of earlier ex- 
 perimenters, that alternators must have self-inductance in their 
 armature circuits if they are designed to be run in parallel. 
 The application of that statement to the case, however, is falla- 
 cious, since alternators operating in parallel should require 
 much less than the torque of normal load to hold them in step, 
 so that the synchronizing tendency of armatures with small in- 
 ductance is ample to make them run in parallel, and for either 
 parallel working or for operation as synchronous motors a small 
 armature impedance is of the greatest importance. 
 
 169. Locus Diagrams of Currents, Voltages, and Loads of the 
 Synchronous Motor. — In Fig. 426 a locus is given whereby the 
 motor excitation can be obtained when the load is constant and 
 the current varies. Suppose now that the current and im- 
 pressed voltage are maintained of constant values, then Fig. 429 
 results. The locus of the impressed voltage is a circle E X E 2 E", 
 with its center at 0. The current represented by 01 being con- 
 stant, OR , the resultant voltage, is constant, if it is assumed that 
 the synchronous impedance is constant. The latter assumption 
 is not exact for commercial machines on account of the effect of 
 saturation in the magnetic circuit with different excitations of 
 the field magnet as well as the effect of changes of the power 
 factor of the motor, but the assumption may prevail for the pur- 
 pose of the following exposition. Then in order that the im- 
 pressed and counter-voltages may always combine to equal 
 OR the locus of the counter-voltage E % must be the circle 
 E 2 E 2 'E 2 " with its center at R and its radius equal to E v The 
 geometrical construction of the figure shows that these relations 
 are necessary to fulfill the condition of constant current and im- 
 pressed voltage. When the excitation gives the counter-voltage 
 OE v the construction shows that the mechanical load is OT x 
 /0(= IE 2 cos d 2 , where d 2 is the angle between 0E 2 and _Z49), 
 which will be assumed to be the amount of power required to 
 drive the armature at synchronism without external load. The 
 current 01 leads OE x by a large angle so that there is a large 
 
716 
 
 ALTERNATING CURRENTS 
 
 leading reactive component of current equal to VI, while the 
 active component of 0E X QE X cos 6 V where 0 X is the angle between 
 E x and I x ) is equal to OW, and 0 W x 01 is in this instance only 
 sufficient power to furnish the I 2 R losses and the power re- 
 quired to overcome magnetic and friction losses. The output 
 of the motor is a maximum when E 2 equals 0E 2 ' (and is equal 
 
 Fig. 429. — Locus of Counter- voltage in a Synchronous Motor when the Current and 
 Impressed Voltage are Constant. The Current Vector is made the Initial Line. 
 
 to OT' x I O') for at that value of E 2 the active component of 
 the counter-voltage is a maximum, the vertical line E 2 T' be- 
 ing tangent to the arc E 2 E 2 E 2 . The construction shows 
 that with this excitation of the motor, the power is taken from 
 the circuit at unity power factor. This obviously must be true 
 under the conditions, since a maximum output with fixed losses 
 predicates a maximum input. As a similar construction may 
 be made with the current 1 of any fixed value, the motor ex- 
 
SYNCHRONOUS MACHINES 
 
 717 
 
 citation that gives unity power factor at any current gives 
 maximum output for that current. The minimum excitation 
 which will keep the motor running is when the counter-voltage 
 equals OE 2 " . At this point the power converted by the arma- 
 ture is again equal to OT x 10 and is just sufficient to drive the 
 machine without external load, while the impressed voltage OE " 
 again supplies an active component OW, and OIF x 01 equals 
 the power input required to supply all losses. The current now 
 has a heavy lagging reactive component TJI. The motor cannot 
 operate with the current 01, even without an external load, if the 
 excitation is either increased or decreased beyond the limits of 
 0E 2 and OE 2 and the impressed voltage is maintained constant; 
 hut it must be remembered that the armature reactions may con- 
 tribute to the excitation, and a motor, without external load, 
 may, therefore, run synchronously, even when not provided with 
 external direct current excitation. If, however, the motor is 
 coupled to a prime mover such as a steam engine, the counter- 
 voltage may take any position on the complete circle of which 
 E 2 E 2 E 2 is an arc. In this case when the motor voltage is near 
 the top or bottom of the circle locus so that its active component 
 is less than is needed to provide the load torque, it will draw 
 power partly from the electric mains and partly from the engine. 
 When the active component is on the right of 0 and equal to 
 or greater than — OT the machine becomes a generator work- 
 ing in parallel with the mains. The voltage locus of which 
 E X E^E" is an arc may also be drawn as a full circle; and when 
 a voltage vector is to the right of 0 power is delivered, but 
 when the vector is to the left of 0 power is absorbed. The 
 arc MM in the figure is the locus of E 2 when I equals ^ 01 ; 
 M'M' is the locus when I equals % 01; and 31" M" is the locus 
 in the limiting case which might occur if the armature reactions 
 could be negligible and the motor losses could be exactly sup- 
 plied mechanically through the motor pulley, in which case E l 
 and E 2 would have to always be equal and opposite. 
 
 A more generally useful locus is that of the current when 
 the impressed and counter voltages are maintained constant 
 and the load is varied. The construction is shown in Fig. 
 430, in which OE x is the fixed vector of impressed voltage E v 
 Consider first the case where the counter-voltage E 2 equals 
 E v Then when the counter-voltage is in the position OJ 
 
718 
 
 ALTERNATING CURRENTS 
 
 opposite to 0E V no current can flow as there is no resultant 
 voltage. This position is imaginary, as the machine will not 
 operate unless supplied with at least enough power to overcome 
 the losses. Assume now that the motor armature, and hence E v 
 falls back in step to the position 0E 2 , because of the applica- 
 tion of load. The resultant voltage is then OR, and the cur- 
 
 It 
 
 rent is 01 lagging behind OR by an angle = cos -1 —, where 
 
 Z 
 
 R is the armature resistance and Z the synchronous impedance. 
 The current is 1= When the point E 2 moves along the 
 
 Therefore, since the current maintains a constant angle i/r with 
 OR and varies in length in proportion to OR, the point I de- 
 scribes a circular locus 0E X GrB drawn on a diameter OAT ) . which 
 makes an angle equal to -v/r with the line 0E V The diameter 
 
 OB 
 
 of this circle is equal to — — , that is, the diameter of the locus 
 
 z 
 
 of R divided by the synchronous impedance. The diameter 
 OB is the theoretical value of OR that would exist under the 
 
SYNCHRONOUS MACHINES 
 
 719 
 
 hypothetical condition of OE, 2 in consonance with OF x and the 
 motor on the circuit with its induced voltage in series relation 
 with the impressed voltage, that is OB = E x + E 2 . The angle 
 DOB equals i/r, as Z is considered constant. The diameter of 
 the circle ODD equals 
 
 E x + E 2 _2E 2 _2E X 
 
 z ~ z ' 
 
 since E 2 — E x by the premises; and it makes an angle with OE x 
 which is 
 
 Y = cos 1 —- 
 
 Now suppose that E 2 is made OE 2 , which is less than E x by 
 such an amount that, when the two are in direct opposition, 
 the resultant voltage OR' in phase with OE x drives the cur- 
 rent 00 through the armature. By the geometrical construc- 
 tion it is seen that, when 0E 2 ' lags behind opposition to OE v 
 OR' describes the circular locus R'LJY, and therefore, the cur- 
 rent which, as before, must maintain a constant angular position 
 with respect to the resultant voltage and be proportional to its 
 length, must describe the circular arc OD'P. The diameter of 
 this circle is 
 
 R'N _ 2 E 2 
 Z Z 
 
 Again, suppose E 2 is made greater than E x and is given the 
 value shown by the line 0E 2 " . If E 2 " takes the position 0F 2 \ 
 shown in the figure, the resultant OR" drives the current OF 
 through the synchronous impedance. For this position of 0F 2 " 
 the machine must evidently be driven as a generator, as it is 
 driving' current against the impressed voltage F v It will not 
 run as a motor until 0E 2 " has fallen back sufficiently to cause 
 the current vector to take an angle of less then 90° from the im- 
 pressed voltage and have an active component sufficient to sup- 
 ply not less than the losses of the machine, other than field losses. 
 The locus of the resultant voltage OR" must by geometrical 
 construction, as before, be on the semicircle R"ST , and the cur- 
 rent locus in accord with the reasoning for the previous cases 
 is FD"H. By continuing this method the current loci for any 
 desirable number of counter-voltages may be laid off with the 
 respective loci of counter and resultant voltages. 
 
720 
 
 ALTERNATING CURRENTS 
 
 The input can be obtained from the product of current, 
 impressed voltage, and the cosine of the angle of lag between 
 the two. The maximum current input for the machine when 
 running as a motor with minimum excitation under which the 
 motor will run is slightly less than current OA which equals 
 
 The maximum possible current input for the machine 
 
 when running as a motor is when there is maximum excitation 
 under which the motor will run, and this can then be more than 
 
 current OD , and therefore more than 
 
 2 E, 
 
 The mechanical 
 
 load including core losses equals the input minus the armature 
 copper losses, or the total power transferred from the mains to 
 cause rotation equals the impressed voltage times the active 
 component of current minus the I 2 R losses, which can be read- 
 ily calculated. Therefore, we have the input, 
 
 Pi — E X I cos 0 X ; 
 
 and the total transfer of electrical energy by the armature into 
 mechanical energy and core losses, 
 
 P 0 — E 2 I cos d 2 = E X I cos 0 1 — T 2 R, 
 
 where d ^ and 0 2 are respectively the angles between impressed 
 and counter-voltages and the current. The mechanical power 
 given out is, as intimated, less than P 0 by an amount equal to 
 the mechanical and magnetic power losses caused by rotation. 
 
 In order to determine the current that is required at a given 
 power factor to supply a specific armature power P 0 , a locus of 
 currents for constant power may be constructed in accordance 
 with the foregoing principles. Thus, in Fig. 431 let OE x be 
 the direction of the impressed voltage, which is also made the 
 initial line. Assume now that a certain output P 0 , including 
 rotative magnetic and frictional losses, is being consumed by 
 the rotor or delivered from its shaft. Then the input is 
 E X I cos dj = P a + P 2 R, 
 
 where E 1 is the impressed voltage, I the current flowing, dj 
 the angle of lag between the current and impressed voltage, 
 and R the resistance of the armature. From this: 
 
 P- 
 
 EJcos d 1 _ 
 
 Po 
 
 R' 
 
 R 
 
SYNCHRONOUS MACHINES 
 
 721 
 
 and 
 
 t- 2 E x cos d, j 1 E cos 2 0 X _ 1 E x 2 cos 2 d 1 P 0 
 R + 4 i? 2 ~ 4 R 2 R' 
 
 lienee, 
 
 1 Ei cos $i ll E x 2 cos 2 #, R a 
 
 2 R M ^ if 
 
 ( 1 ) 
 
 From this expression for I when the armature power is P 0 it is 
 seen that there are two values of armature current for each 
 
 Fig. 431. — Diagram showing Current Loci in a Synchronous Motor for Constant 
 Power when the Impressed Voltage is Constant. 
 
 value of cos 6 X except one, and also that the currents are nu- 
 merically equal but of opposite algebraic signs for equal posi- 
 tive and negative values of cos 0 X . The values of the currents 
 
 ... • , i P 0 i E, 2 COS 2 0 X rp 
 
 contain imaginary terms when lo more 
 
 R R 2 
 
 clearly observe these conditions the locus of constant power, P 0 , 
 3a 
 
722 
 
 ALTERNATING CURRENTS 
 
 HMB is drawn. The locus is a circle having a radius equal to 
 
 when P 0 and 6 X equal zero. Then the current is in phase with 
 the voltage and flows through the windings of the armature in 
 accordance with Ohm’s law and all the energy is utilized in I 2 R 
 loss. Such a condition could only exist, of course, when the 
 rotor has its rotative losses supplied from external mechanical 
 sources, and has swung to such a position that the counter- 
 voltage forms a resultant with the impressed voltage of such 
 phase angle that the current is in consonance with the latter. 
 These conditions are only fulfilled when the counter-voltage is 
 at right angles with the current and of definite length. This 
 
 current ^ is laid off at 00 in the Fig. 431. When 0 1 only is 
 R 
 
 zero, the formula for current becomes 
 
 The larger value obtained by this formula is laid off as OB and 
 the smaller value as OH in the figure. The distances OH and 
 BO are evidently equal. 
 
 Now take any value of current I obtained for a given P a and 
 for any angle. Assume that 01 at lag angle 0 X is such a cur- 
 rent. Then draw the perpendicular I>K from a point I), mid- 
 way between 0 and C or H and B, and connect the points D 
 
 E 
 
 From the last formula it is seen that the current I equals — 1 
 
 R 
 
 and I. The length 01) = = ^ 00, and by construction 
 
 and 
 
 Also 
 
 2 R 
 
 By construction (2)F) 2 = ( PK) 2 + (FT/) 2 . 
 
SYNCHRONOUS MACHINES 
 
 723 
 
 Substituting in this the values of PK and KI and using the 
 value for current given in the formula for current in Equation 
 (1), then 
 
 E*n*-p s fvi -oo^' 
 
 + 
 
 1 E x cos 0, 
 R 
 
 ± 
 
 ll E 2 cos 2 6>, P 0 
 
 V 4 R 2 R 
 
 which reduces to 
 
 1 p 2 
 
 (Diy =-^y- 
 K J 4 R 2 
 
 Po 
 
 R 
 
 ( 2 ) 
 
 or 
 
 pi=hp = ± JI*1-^ 
 
 '4 R 2 R 
 
 ( 3 ) 
 
 Since I is assumed in the premises to he the current which can 
 supply the power P 0 at a given value for the angle 6 V the point 
 I so long as P 0 is constant must travel the locus HMBR which 
 
 must be a circle having a radius equal to 
 
 El 
 
 R 2 
 
 P 
 
 —2, with its 
 
 R 
 
 1 E, 
 
 center P at a distance equal to - ^ from the origin 0. 
 
 2 R 
 
 Any 
 
 power P 0 was assumed, hence by assuming a series of outputs, 
 the circular loci for the outputs are concentric circles such as 
 those marked M, M\ and M" . The largest locus possible has a 
 radius about equal to PO , P 0 then being only sufficient to supply 
 the no-load rotative losses. The smallest is the point P , where 
 the radius is zero, the power factor is unity and the greatest out- 
 put occurs. For this locus there is only one current OP , half 
 the input power is used up in I 2 R loss and the other half in 
 
 turning the armature since — = - and I=~^. The 
 s R 4 R 2 2 R 
 
 efficiency is thus less than 50 per cent. Commercial synchronous 
 motors could not approach this amount of load without causing 
 excessive and disastrous heating. The maximum efficiency oc- 
 curs when the copper losses equal the rotation losses, which 
 may usually be assumed without serious error to be of fixed 
 value. By projecting 01 upon the voltage line OE x and at right 
 angles, the active and quadrature components of the' current, 
 OF and 0(7, are obtained. By suitably changing the scale of 
 
724 
 
 ALTERNATING CURRENTS 
 
 the figure, 01, OF, and OGr , may be read off directly as total 
 kilovolt-amperes, kilowatts, and quadrature volt-amperes re- 
 spectively. The output may be determined for any locus by 
 marking the locus with the power assumed in its construction. 
 Since the radii of the current loci are scaled in amperes so that 
 
 
 FI 
 
 4 R 2 
 
 Po 
 
 R' 
 
 the radius of the locus circle depends upon the output, as F x 
 and R are constant. Therefore, loci for various outputs can 
 be readily laid out by making the radius 
 
 r — \l A — — ° 
 
 ' R 
 
 IE 2 
 
 where A is a constant equal to - — -L. 
 
 4 R 2 
 
 By combining Figs. 430 
 
 and 431 all of the information desired concerning the motor 
 voltages and phase angles is obtained for any output when the 
 voltage and power factor of the input are known. A figure or 
 construction similar to that of Fig. 410 may be made for a 
 synchronous motor having constant power output, variable 
 induced voltage, and constant impressed voltage. By a process 
 of reasoning similar to that used in discussing the division of 
 load in alternators,* it may be shown that the induced voltage 
 locus (constant power circle) is a circle having a radius 
 
 JA 
 
 and the polar coordinates E 0 and — 1 - ■ , 
 1 2 2 cos 6 a 
 
 where P 0 is the motor output, F 1 the impressed voltage, E 2 the 
 motor counter voltage, and 6 a the internal angle of lag. The 
 angle included between the coordinates is (# a — /3). The con- 
 struction is like that of Fig. 410, except that the angle O'OB is 
 (0 O — /3) instead of (# a + /3) and the locus radius is shorter 
 than 00', which throws the motor counter-voltage to the right 
 of the terminal voltage OB' and causes the loci TT, etc., to be 
 at the right of OV. 
 
 170. Curve showing the Relation of Armature Current to 
 Excitation. — The relation of armature current to field excita- 
 tion may be plotted for a motor operating under the conditions 
 considered above, namely, when driving a constant load, by 
 
 * Art. 162. 
 
 f F 2 P 0 Z a 
 
 4 cos 2 6. cos 6. 
 
SYNCHRONOUS MACHINES 
 
 725 
 
 taking the corresponding values of E l and I from a chart made 
 like Fig. 426, or like Fig. 431. This gives a curve like Fig. 
 432, which has two values of its abscissas for every value of the 
 ordinate except the lowest and highest, points A' and O' in the 
 figure ; or for each value of the armature current there may 
 be two values of the excitation, one giving a counter- voltage 
 
 FIELD EXCITATION 
 
 Fig. 432. — Calculated V-Curve for a Synchronous Motor, showing the Relation 
 between Field Excitation and Armature Current for a Constant Load. 
 
 greater and the other a counter-voltage less than the impressed 
 voltage, except at the points of minimum and maximum arma- 
 ture current, which correspond to but one excitation, as has 
 already been explained. Likewise there are two values of the 
 armature current for each value of the field excitation except 
 for the minimum and maximum excitations at which the given 
 
726 
 
 ALTERNATING CURRENTS 
 
 load can be carried, namely at tlie points I) and I)' in the figure. 
 The way in which Figs. 426 and 431 are constructed shows 
 that the smaller the angle -*/<-, or the smaller the armature self- 
 inductance, the less will be the difference in the two excita- 
 tions corresponding to any armature current ; and hence the 
 
 curve showing the relation 
 of excitation to current in a 
 machine having a large re- 
 actance compared with the 
 resistance is broad and 
 rounded, but the curve for 
 an armature having a small 
 reactance compared with the 
 resistance is sharp and nar- 
 row. In actual working, 
 only the lower part of the 
 closed curve of Fig. 432 is 
 available for use. This cor- 
 responds to curve C of Fig. 
 428 and the curve shown in 
 Fier. 433. These curves are 
 ordinarily called V-curves on 
 account of their shape. 
 Curves for loads smaller 
 than the one for which the 
 , „ - , r „ curve is shown in the figure 
 
 Fig. 433. — V-Curve of a Synchronous Motor ^ ° 
 
 Plotted from the Observed Value of Arina- 6Xp<Uld anti fall Outside ol 
 ture Current and Excitation when operating curve , while for larger 
 
 under Constant Load. , , , . , n V .. 
 
 loads they contract and fall 
 within it. At maximum load, represented by point D. Fig. 
 431, the curve reduces to the point P (Fig. 432). 
 
 In Fig. 430 it was seen that the counter-voltage, which is 
 proportional to the field flux, is also proportional to the radius 
 of the current locus. The latter is numerically equal to 
 
 10 12 u 
 
 EXCITATION 
 
 = OA when P, = E v 
 
 z 
 
 Hence if Z, the synchronous imped- 
 
 ance, has been determined, the counter-voltage for any current 
 locus is E 2 ccr Z , where r is the radius of the current locus 
 under consideration. By drawing the line OA in Fig. 431 so 
 that it makes the angle y]r with the line OE v it is possible to 
 
' SYNCHRONOUS MACHINES 
 
 727 
 
 determine a plot for the V-curve very readily for any output P 0 . 
 Considering the power locus H31B I to be for the load P 0 (Fig. 
 431), the current 01 laid out as an ordinate in Fig. 432 has an 
 abscissa proportional to AI, the radius of the current locus which 
 passes through I in Fig. 430. If the field reluctance is constant 
 and field reactions are included as part of the synchronous im- 
 pedance, the armature current locus may be obtained by drawing 
 an arc with the radius 01 in Fig. 431. Where it cuts the con- 
 stant power output locus HMBI a second time at I x the excita- 
 tion is proportional to the distance between A and I v In Fig. 
 
 _27 _27 
 
 431 it is seen that as 0 C = and, in Fig. 430, OA = , there- 
 
 ZJ 
 
 R 
 
 _z? 
 
 fore since BOA, Fig. 431, equals y\r, OA — — i in Fig. 431 is the 
 
 zj 
 
 horizontal side of the right-angled triangle OA C. The curve 
 of Fig. 432 is constructed from Fig. 431 as follows: the points 
 on the curve whose ordinates are equal to currents 01 and OI v 
 Fig. 431, are I x and B' in Fig. 432, with abscissas equal to AI 
 and AI' of Fig. 431, while the current OJ is represented by the 
 ordinate of point J with abscissa equal to AJ of Fig. 431. The 
 current 01 (Fig. 431) is so large compared with OB , which 
 represents the current when the armature IB drop is equal to 
 I B v that the efficiency would be very low. The useful part of 
 the curve of Fig. 432 is, therefore, only a short distance on either 
 side of A', the ordinate and abscissa of which are equal to OH and 
 AH of Fig-. 431. The maximum load for any excitation may 
 readily be determined from Fig. 431 ; thus, if the excitation is 
 proportional to AS, the largest load which can be carried is rep- 
 resented by the power locus M r , as this is the smallest power 
 locus that is cut by the current locus SS' having .4$ as a radius. 
 
 It must not be supposed that the armature voltages actually 
 rise in proportion to the amperes of field excitation, even for 
 constant reluctances, as indicated by the abscissas of Fig. 432, 
 because the quadrature current when the excitation is less than 
 AH (Fig. 432) strengthens the fields, and when greater than AH 
 weakens the fields; but this quadrature current calls for a pro- 
 portional decrease or increase of field current in the same amount 
 that would be called for if the reactions were zero and the coun- 
 ter-voltages were proportional to the abscissas in the figure.* 
 
 * McAllister’s Alternating Current Motors, p. 195. 
 
728 
 
 ALTERNATING CURRENTS 
 
 171. Application of the Parallel Operation and Synchronous 
 Motor Diagrams to Machines Wound with Any Number of 
 Phases. — The synchronous motor diagrams given in the pre- 
 vious articles are suitable for use for any single or polyphase 
 machines. In the case of single-phase machines the power in- 
 put, voltage, and current used in the diagrams are those meas- 
 ured in the supply mains by wattmeter, voltmeter, and am- 
 peremeter. For quarter-phase machines one full phase may be 
 conveniently considered using its voltage, current, and power 
 as in the single-phase machine. The powers must then be 
 doubled to obtain the values of motor input and output. Tri- 
 phase machines may be dealt with in the same manner, using 
 the voltage, current, and power in one branch for obtaining the 
 diagrams and then multiplying the inputs and outputs by three. 
 When the tri-pliase motor is wye connected, the branch voltage 
 
 is the line voltage, and when the motor is delta connected 
 
 V3 
 
 the branch current is — the line current. Six-phase mesli- 
 
 V3 
 
 connected motors may be treated in the same way as similarly 
 connected tri-phase machines, except that the power shown by 
 the single-phase diagrams must be multiplied by six to get the 
 entire power. For six-phase machines with a star connection, 
 the voltage, current, and power per phase are the same as in 
 the mesh connection. 
 
 Prob. 1. Given a three-phase wye-connected synchronous 
 motor operating under 2200 volts impressed voltage at its ter- 
 minals with a frequency of 60 periods per second. The syn- 
 chronous impedance per phase, when the frequency is normal, 
 is considered constant at a value of 5 ohms and is composed 
 of such proportions of resistance and reactance that the current 
 lags 75° behind the resultant voltage. Giving the vector of 
 impressed voltage a fixed position, draw the current loci for 
 100, 200, and 300 amperes ; the constant total armature me- 
 chanical power loci for 100, 200, and 400 horse power, assum- 
 ing that the mechanical and magnetic losses are constant at 12 
 horse power ; and the V-curves for the latter powers. Also, 
 giving the current vector a fixed position, draw the impressed 
 voltage and counter- voltage loci for currents of 100, 200. and 
 300 amperes. Complete the locus circles in the latter diagrams 
 
SYNCHRONOUS MACHINES 
 
 729 
 
 and mark the positions where the machine is a generator work 
 ing in parallel with the supply mains. 
 
 Prob. 2. Draw the V-curves for the generator and conditions 
 of Prob. 1, Art. 162, by the method given in that Article, when 
 the total powers are 100, 50, and 25 kilowatts. 
 
 172. Methods of testing Alternators. — The general defini- 
 tions of efficiencies, which have already been explained with 
 regard to transformers,* are applicable to alternators, although 
 in the case of alternators the power received by the field coils 
 is usually supplied from a direct current exciter circuit. The 
 principal causes of loss are the same as those in direct-current 
 machines ; but on account of increased frequencies of the mag- 
 netic cycles in the armature cores, the effects of eddy currents 
 and hysteresis are intensified. On this account particular care 
 is required in selecting and annealing the iron for the armature 
 cores, and in insulating the armature disks from each other. 
 Advantage is also taken of every opportunity for ventilation. 
 
 The conditions under which a test of alternator efficiency is 
 to be executed are usually specified with great care, including 
 the speed, the load, the voltage, the current, the rise of tem- 
 perature of windings, etc., corresponding to the circumstances 
 under which it is desired to have the machine work. Field 
 exciting current, rise of temperature of the bearings and the 
 collector rings, and other phenomena of the operation of the 
 machine are also ordinarily observed and recorded. The con- 
 ditions for making standard tests are given in the Standardiza- 
 tion Rules of the American Institute of Electrical Engineers. 
 
 1. Efficiency by Rated Motor and Stray Power Measure- 
 ments. — In case a transmission dynamometer is used to measure 
 the power absorbed by an alternator, the commercial efficiency is 
 equal to the electrical output shown by a wattmeter divided by 
 the input shown by the dynamometer readings plus the power 
 used for separately exciting the field magnet. When the ma- 
 chine is separately excited, as is the case in standard, modern, 
 commercial alternators, the energy supplied to the field mag- 
 net may be measured by amperemeter and voltmeter or by 
 wattmeter. The machine friction may be determined from 
 the dynamometer readings when the machine is run with its 
 field magnet unexcited and the external circuit open. The 
 
 * Art. 137. 
 
730 
 
 ALTERNATING CURRENTS 
 
 loss due to hysteresis and eddy currents in the armature core, 
 armature conductors, and pole faces may be approximately 
 determined from the dynamometer readings when the machine 
 is operated with its field magnet separately excited and the 
 external circuit open, and the total loss may be approximately 
 separated into its component parts by substituting in the for- 
 mulas * previously given, the power being indicated from the 
 dynamometer readings for two values of field excitation. The 
 I 2 R losses in field and armature conductors may be readily com- 
 puted from the respective hot resistances. In the special case 
 where the machine is self-excited by a rectified current, the 
 field current will be less than that calculated from the voltage 
 and resistance. In that case it is advisable to use an alternat- 
 ing-current amperemeter, such as an electrodynamometer, to 
 determine the effective current. From this the I 2 R loss may 
 be computed if the hot field resistance is known ; or, the field 
 loss may be directly determined with more assurance of 
 accuracy by a wattmeter in the field circuit. 
 
 Instead of a tnechanical transmission dynamometer, which is 
 generally of a low' degree of accuracy even if available, it is 
 convenient to mechanically connect the alternator under test 
 to an electric motor, of which the no-load losses and curve of 
 efficiency as a function of the load are known. The input of 
 the alternator under test can then be calculated from the elec- 
 trical input of the machine that is used as a driving motor 
 by deducting its losses. A machine used in this way as the 
 driving motor is commonly said to be “ rated ” when the rela- 
 tions between its outputs and inputs are known for all loads 
 and a number of speeds. It is desirable to connect the rated 
 motor by a flexible coupling to the alternator under test. 
 When the motor is connected to the test machine by belt, the 
 loss of energy caused by the belt is included in the power 
 delivered by the motor. This loss may be approximately deter- 
 mined by running the test alternator as a synchronous motor 
 with and without the belt and deducting from the difference 
 of the input readings the friction and windage losses of the 
 rated driving machine. During this test the rated machine 
 should be disconnected from the supply mains and be un- 
 excited. 
 
 * Art. 110. 
 
SYNCHRONOUS MACHINES 
 
 731 
 
 One of the methods of obtaining frictional or fixed losses is 
 sometimes called the retardation test. The machine under test 
 may be brought to full speed and then have the driving force 
 removed. The speed is then observed at certain intervals of 
 time. The friction loss is determined from the formula, 
 
 p 2 7r 2 k r r, 2 — r, 2 i 
 
 T 
 
 where P is the power in watts dissipated in friction, V x and T r 2 
 are the speeds in revolutions per second at the beginning and 
 end of the time interval of T seconds, and K is the moment of 
 inertia in kilogram-(meters) 2 of the rotating part. By exciting 
 the field magnet, core losses plus friction may be approxi- 
 mately obtained in this way. This method requires that the 
 weight and dimensions of the rotating part be known in order 
 that K may be determined, and even at the best is not very 
 accurate for ordinary dynamo tests. 
 
 It is usually best to measure the losses under conditions of 
 no load operation and compute the efficiency by taking the 
 ratio of the output under consideration to the output plus the 
 losses, rather than to measure the input and output and take 
 the ratio of their measured values. Windage, friction, and 
 core losses at no load, the field magnet being excited with the 
 normal excitation for the load concerned minus the armature 
 back turns for that load, may be measured by means of the 
 rated motor, and the PR losses may be computed after the hot 
 resistance of the windings has been measured. A correction 
 must then be made for the effect of armature reactions on the 
 distribution and density of magnetism in the armature core 
 which alter the core losses and also the eddy current and hys- 
 teresis losses in the pole faces of the field magnet. The magni- 
 tude of this correction depends upon the type of machine, but it 
 ought not to be large in well designed and constructed machines. 
 To obtain the armature resistance per phase of a wye-connected 
 tri-phaser, the measurement may be made from each line termi- 
 nal to the neutral point. Twice the resistance is obtained by 
 making the measurement from one line terminal to another. 
 For a delta-connected tri-phaser, two thirds of the resistance 
 per phase is obtained by measuring from one line terminal to 
 another. 
 
732 
 
 ALTERNATING CURRENTS 
 
 The load given out by the alternator under test can some- 
 times be utilized, at least in part, by being delivered to the 
 service mains of the plant in which the test is being made. 
 However, frequently all and usually part of the load must be 
 absorbed in metal or water rheostats. 
 
 A test of only poor accuracy can be made by using a rated 
 steam engine or other heat engine as the driving machine in 
 place of a rated electric motor, in which case the input to the 
 engine is obtained by means of indicator cards. 
 
 The efficiency of direct-connected steam engines and alterna- 
 tors, especially of steam turbo-generator sets, is often given in 
 terms of pounds of dry steam per kilowatt of output, in which 
 case the input is determined from the condensed steam at the 
 engine exhaust. Or when the efficiency is given in terms of the 
 ratio of power output to input, the input is measured by means 
 of a steam engine indicator, when this is possible. The output 
 is always to be measured in such cases by carefully calibrated 
 wattmeters. It is of course understood in the case of a steam 
 turbine set that the mechanical power delivered to the dynamo 
 is difficult to determine, but it may be approximately computed 
 by determining the no-load losses of the alternator and turbine 
 rotor by driving the alternator as a synchronous motor and 
 using the data thus measured as a basis for computing the 
 dynamo input from its output plus its losses. 
 
 In the case of water turbines the input to the turbine can be 
 determined by the head and rate of flow of the water used. 
 
 The following methods are especially adapted to shop testing 
 of the heating and efficiency of an alternator. 
 
 2. Feeding Bach Method for measuring Efficiency. — It is 
 sometimes desirable to make a test with actual full load, while 
 avoiding the use of a power dynamometer or the necessity for 
 supplying the full rated input from the shop circuits. With 
 this in view a modification of Hopkinson's method of testing * 
 may be made (Fig. 434). Two equal alternators are rigidly 
 coupled together in proper step for parallel working. Their 
 armatures are electrically connected together with a wattmeter 
 and an amperemeter in the circuit. The field magnets being 
 properly excited by a separate exciter, so that one machine will 
 act as a generator and the other as motor, the system may be 
 
 * Jackson’s Electromagnetism and the Construction of Dynamos, p. 256. 
 
SYNCHRONOUS MACHINES 
 
 733 
 
 driven by supplying only sufficient power to make up the losses 
 of the two machines. Assuming the armature and stray losses 
 of the two machines to be equal, and representing the wattmeter 
 
 
 /T JW 
 
 EXCITER 
 
 
 
 RHEOSTAT 
 
 Fig. 434. — Two Alternators connected up for an Efficiency or Heat Test by a Feed- 
 ing' Back Method. 
 
 reading by P e , the power supplied by P m , and the resistance of 
 the connections by i? 1 ; then the efficiency of the generator is 
 
 = P, , 
 
 if IfR s is the field loss of the generator. If the machines are 
 self-exciting, the power in the field circuits must be measured 
 and proper allowance made. The power supplied to make up 
 the losses may be measured by a rated direct-current or alter- 
 nating-current motor (as explained earlier in the article). 
 
 3. Efficiency by Rated Motor. — Where approximate determi- 
 nations of the various losses of conversion and of the commer- 
 cial efficiency are sufficient, the alternator tested may be driven 
 directly from a “rated” direct or alternating current motor 
 as explained earlier in the article. By means of amperemeter 
 
734 
 
 ALTERNATING CURRENTS 
 
 and voltmeter the power supplied to the motor may be deter- 
 mined with the alternator operated under such various condi- 
 tions as may be necessar}-' to determine the losses. In each 
 case the power supplied to the alternator is equal to the power 
 absorbed by the rated motor multiplied by the efficiency of the 
 motor given in per cent for its load in the particular test. The 
 motor may be rated by determining its efficiency by the stray 
 power method. If the motor efficiency for various loads is 
 determined by some exact method, the power transmitted to 
 the alternator may be determined with considerable exactness. 
 When it is desired to carry the accuracy of this method of 
 testing beyond a fairly good approximation, the efficiency at 
 various loads of the rated motor should be plotted, so that the 
 efficiency at any load may be readily read off. 
 
 4. Mordey s Method . — -A neat arrangement for testing a 
 single alternator with a stationary armature by a method akin 
 
 Fig. 436. — Unbalanced, Divided Armature Connection for making Efficiency or 
 Heat Test of Alternator with Revolving Field Magnet. Applicable to Single or 
 Polyphase Machine. 
 
 to Hopkinson's two-machine method was devised by Mordey.* 
 It is applicable to alternators which have stationary armatures, 
 and where the individual coils may be connected in the desired 
 combination. Such an armature may be divded into two parts 
 which are connected in such a way as to oppose each other 
 (Fig. 435). If one part gives a somewhat higher voltage than 
 the other, the first part will operate as a generator and the 
 second as a motor if the alternator is driven in the usual 
 
 * Testing and Working Alternators, Join-. Inst. E. E., Yol. 22, p. 116. 
 
SYNCHRONOUS MACHINES 
 
 735 
 
 manner. By properly adjusting the difference between the 
 voltages of the two parts, the current flowing in the machine 
 may be caused to have any desired value. The efficiency of the 
 machine is gained by measuring the power absorbed by the 
 machine operating as a self-contained motor-generator and 
 measuring its output by a wattmeter, the voltage coil of the 
 wattmeter being connected across the terminals of the motor 
 or generator coils and the current coil being connected 
 directly in the circuit. The power thus measured by the 
 wattmeter is evidently approximately one half of the total 
 energy of the machine; consequently the corrected efficiency is 
 
 2 P e 
 
 V 2 P e + P m + PR/ 
 
 where P m is the power supplied to the driving machine and P e 
 is the power read on the wattmeter. 
 
 The difference between the voltages developed in the parts 
 of the machine which is necessary to cause the desired current 
 to circulate may be caused by an unsymmetrical division of the 
 armature (Fig. 435), or by supplying a little additional vol- 
 tage to the generator by means of a transformer. The latter 
 may be supplied 
 from another alter- 
 nator operating in 
 synchronism with 
 the machine under 
 test or it may be 
 supplied directly 
 from the test ma- 
 chine (Fig. 436). 
 
 The transformer is 
 likely, however, to 
 introduce uncertain 
 elements of loss. 
 
 Figures 435 and 436 
 may be considered 
 to represent the windings of a single-phase alternator or one phase 
 of a polyphase alternator. In the case of polyphase machines 
 each phase is connected independently as in the case of the one 
 shown in the figures. In order that the alternator may be 
 
 Fig. 436. — Divided Armature Connection for Efficiency 
 and Heat Tests. Extra Voltage supplied by External 
 Transformer. Applicable to Single or Polyphase Al- 
 ternators with Revolving Field Magnet. 
 
736 
 
 ALTERNATING CURRENTS 
 
 tested for various loads, the secondary winding of the trans- 
 former shown in Fig. 436 should contain a number of voltage 
 taps, or a regulating impedance may be placed in series with 
 its primary winding. For unity power factor this impedance 
 should consist of resistance and for current lags of resistance 
 and reactance. Instead of a transformer an autotransformer 
 or compensator may be used. It will be noted that such 
 methods as this one in which the divided armature is used are 
 especially desirable for heat runs (that is, for operating the 
 machine for a period under full excitation and full load current 
 to determine the rise of temperature), as only a small fraction 
 of the full load power is required to obtain full heating effects, 
 but they are not so desirable for efficiency measurements, since 
 dividing the armature alters the armature reactions and changes 
 the core losses. 
 
 5. Divided Field Method of measuring Alternator Losses or 
 Efficiency. — As pointed out by Professor Ayrton,* the arrange- 
 ment preceding may be modified so as to apply to alternators 
 with either rotating armatures or rotating field magnets. In 
 this case opposite halves of the field magnet are magnetized in 
 opposite directions, and the armature is short-circuited through 
 an amperemeter (Fig. 437). By adjusting the relative excita- 
 tion of the halves of the field magnet by means of rheostats or 
 otherwise, a current of any desired value may be caused to cir- 
 culate in the armature. If the excitation is practically normal, 
 the measured losses of the machine when any current is flowing 
 will be approximately equal to those when the machine is oper- 
 ating normally on the same current, provided it may be assumed 
 that the losses in an alternator are appreciably equal when 
 driven as a generator and as a motor. This assumption is rea- 
 sonable for most instances in which the effects of armature re- 
 actions and leakage reactance are not large. The arrangement 
 here described is not applicable to machines with armatures 
 with the halves wound in parallel, since under the test condi- 
 tions an amperemeter connected between the terminals of such 
 an armature would not indicate the current circulating in the 
 armature coils. The plan does not put full voltage on the 
 armature circuit and therefore does not afford a complete test, 
 besides altering the effect of armature reactions. 
 
 * Jour. Inst. E. E„ Yol. 22, p. 136. 
 
SYNCHRONOUS MACHINES 
 
 737 
 
 Fig. 437. — Divided Field Connection for making Efficiency and Heat Tests. Appli- 
 cable to Single and Polyphase Alternators. 
 
 6. Motor -generator and Similar Methods for Measuring Effi- 
 ciency, Losses , and Heating of an Alternator. — Mordey also sug- 
 gested the following purely electrical method of testing alter- 
 nators which have stationary armatures.* The machine being 
 properly excited, one half of the armature winding of each 
 phase is connected as a generator to an external load R (Fig. 
 438). The other half of the armature is connected to another 
 
 Fig. 438. — Motor-generator Method of obtaining Efficiency, Losses, and Heating of 
 Single or Polyphase Alternator having a Rotating Field Magnet. 
 
 * Jour. Inst. E.E . , Yol. 22, p. 122. 
 
 3b 
 
738 
 
 ALTERNATING CURRENTS 
 
 alternator and driven as a synchronous motor. The total losses 
 under these conditions are evidently equal to the power ab- 
 sorbed by the motor half of the armature, minus the output of 
 the generator half of the armature ; that is, P m — P gl where P m 
 and P g are respectively the power absorbed by the motor and 
 the output of the generator. Since the output P a is the out- 
 put of only half the armature, but the losses thus determined 
 are those of the whole machine, the losses will be approximately 
 the same when the machine is run as a generator with an output 
 of 2 P g , provided generator losses and motor losses are the same. 
 The efficiency of the machine as a generator is therefore ap- 
 proximately 
 
 The input and output of the machine under test may be meas- 
 ured by wattmeters. 
 
 Another somewhat similar plan is to divide the stationary 
 armature into three divisions which are connected in series. 
 
 Fig. 439. — Synchronous Motor Method of testing Rotating Field Single-phase 
 Alternator. Armature divided into Three Parts. 
 
 Two of these are made equal and are connected in opposition. 
 The third consists of only a small portion of the armature. 
 The machine is electrically connected to another alternator and 
 operated as a synchronous motor under the influence of the 
 small armature division (Fig. 439). By a proper choice of 
 the number of coils composing the small division any desired 
 current may be sent through the machine to be tested. The 
 energy given to the machine represents the losses in the machine 
 
 V = 
 
 2 Pg + P m -Pg+l>R f 
 
 P a + Pm + I f l K{ 
 
SYNCHRONOUS MACHINES 
 
 739 
 
 when operated as a generator and producing the same current 
 with the same excitation. This method is also applicable to 
 determining the losses in machines with revolving armatures 
 the coils of which are all connected in series. In this case the 
 fields are excited with the poles on one half reversed (Fig. 440), 
 one half of the field being slightly stronger than the other. If 
 the armature is connected to that of another alternator of proper 
 voltage, it will run as a motor. The instantaneous counter- 
 voltage of the machine under test depends upon the relative 
 strength of the two halves of the field magnet, and by adjusting 
 
 Fig. 440. — Synchronous Motor Method of testing Single-phase Alternator with Ro- 
 tating Armature. Field divided into Two Parts. 
 
 this, with due reference to the impressed voltage which should 
 be of a relatively small value, the current flowing in the arma- 
 ture circuit may be given any desired value. Under these 
 conditions the losses in the test machine are equal to the power 
 absorbed, which may be measured by a wattmeter. The core 
 losses may be very much altered from their normal value by the 
 practice of this method. 
 
 When, in either of the cases mentioned heretofore, the opera- 
 tion of an alternator as a motor is predicated, it is assumed 
 either that the test machine is brought to synchronism with the 
 alternating source, or that the primary generator is started from 
 
710 
 
 ALTERNATING CURRENTS 
 
 a state of rest after the circuit connections are made with the 
 machine to be tested, in which case the duly excited test ma- 
 chine will start and run with the generator. 
 
 These methods of testing are not only sometimes convenient 
 in determining the losses and the efficiency of an alternator, 
 but the tests, according to several of the methods, are made 
 with the consumption of comparatively little power. This 
 makes the methods satisfactory for use in shop tests for deter- 
 mining the reliability in operation and the heating limits of 
 machines. Mordey suggested that the efficiency of an alternator 
 with stationary armature may be determined from a test of one 
 armature coil, but this cannot serve as a satisfactory shop test, 
 which requires a test of the complete machine. 
 
 7. Seating Tests. — The heating of an alternator should be 
 determined much as is done in the case of a transformer.* The 
 machine should be run at specified load under normal condi- 
 tions until the temperature has become constant. The temper- 
 atures of field and armature windings should be determined by 
 both resistance measurements and thermometers. The temper- 
 atures of the cores, collector rings, bearings, and other parts 
 must be measured by thermometers. After a machine has run 
 until thermometers on the stationary parts have reached con- 
 stant readings and is then stopped, the thermometers on venti- 
 lated stationary parts will again begin to rise. This is due to 
 the heat from interior masses of the constructive material tend- 
 ing by conduction to equalize the temperatures at the surface 
 and interior. The temperatures recorded for the test should be 
 the highest obtained. The heating test can sometimes be made 
 conveniently by alternately running the machine with the arma- 
 ture short-circuited, with the field flux of such a value that some- 
 thing over full load current flows, and then running it with open 
 circuit armature and a field flux which will give something over 
 normal no-load core losses (this can be determined by the power 
 input). The alternate periods of each kind of operation should 
 be short, i.e. only a few minutes. The short circuit current 
 and its period of flow should be so adjusted that the product of 
 the average I“R loss times the time of the runs is equal to the 
 kilowatt hours which would be expended by the rated full load 
 current flowing constantly during the total time of the test. 
 
 *Art. 147. 
 
SYNCHRONOUS MACHINES 
 
 741 
 
 Likewise the iron loss and its period of activity should be so 
 adjusted that its average value times the time of the runs is 
 equal to the kilowatt hours which would be expended by the 
 normal full load excitation during the period of the test. 
 
 8. Insulation Tests. — Dielectric strength tests should be 
 made as in the case of transformers.* The voltage applied 
 should follow the accompanying specification, which is in general 
 applicable to all electrical machinery. The voltage wave should 
 have an approximately sinusoidal form. In the case of field 
 windings, the voltage of the exciter circuit should be used in 
 obtaining the test voltage from the succeeding table, except 
 where the alternator is to be used as a synchronous motor and 
 is brought to speed as an induction motor, when the test voltage 
 should be 5000 volts, since high voltages are likely to be induced 
 in the field windings under such conditions. In testing the 
 dielectric strength of high voltage machines care should be ex- 
 ercised, as in the case of transformers, to bring the testing 
 voltage up very gradually. 
 
 It may be noted that high voltages in a large machine may 
 cause appreciable heating due to the dielectric loss. This 
 amounts to from 8 to 5 or more kilowatts in some of the largest 
 machines now in service while generating voltages of 10,000 
 volts and over. 
 
 TABLE OF DIELECTRIC STRENGTH. TEST VOLTAGES f 
 
 Rated Terminal Voltage ok Armature 
 
 Rated Output 
 
 Testing Voltages 
 
 Under 400 volts 
 
 Under 10 Kw. 
 
 1000 
 
 Under 400 volts 
 
 10 Kw. and over 
 
 1500 
 
 Under 800 and over 400 volts 
 
 Under 10 Kw. 
 
 1500 
 
 Under 800 and over 400 volts 
 
 10 Kw. and over 
 
 2000 
 
 Under 1200 and over 800 volts 
 
 All 
 
 3500 
 
 Under 2500 and over 1200 volts 
 
 All 
 
 5000 
 
 Over 2500 volts 
 
 All 
 
 Double the normal 
 
 
 
 rated voltage 
 
 9. Regulation. — The regulation of a constant voltage alter- 
 nator may be obtained by loading the machine to full load at 
 normal speed, voltage, and other normal full load conditions, 
 * Art. 147. 
 
 f Standardization Rules of American Institute of Electrical Engineers. 
 
742 
 
 ALTERNATING CURRENTS 
 
 and then removing the load without change of speed. The 
 regulation is the ratio which the difference between the voltage 
 at full load and the voltage at no load bears to the voltage at 
 full load. Knowledge of the regulation and the efficiency are 
 frequently desired for fractions of full load and at various 
 power factors. In the latter case full load is rated in kilovolt- 
 amperes, and the normal full load conditions of the test must 
 include the power factor concerned. Except where otherwise 
 denominated, however, testing is done with loads of unity power 
 factor. When the power factor is less than unity with current 
 lagging, the excitation must be larger than is required to 
 obtain equal voltage at unity power factor and equal arma- 
 ture amperes. The losses are then far greater at the lower 
 power factor, and both the efficiency and regulation are 
 depreciated. 
 
 In the combined regulation of direct-connected units composed 
 of an engine and constant voltage alternator, the speed is made 
 the normal value for full load and is supposed to be the same 
 for no load. However, when the load is suddenly removed, 
 there is a short fluctuation of speed. The speed regulation is 
 then commonly taken as the ratio of the greatest instantaneous 
 change of the speed at the instant the load is removed to the 
 normal full speed. The rapidity with which the load is re- 
 moved is apt to have an influence upon this ratio. The instan- 
 taneous values of the speed fluctuation can be obtained by a 
 sensitive speed-recording instrument, or the necessary data can 
 be obtained by a recording voltmeter or oscillograph connected 
 to a small magneto-generator driven from the engine shaft, in 
 which case the speed variation can be calculated from the vari- 
 ation in instantaneous voltage. 
 
 10. Wave Form. — The wave form of the voltage produced 
 by an alternator can he obtained by means of the oscillograph 
 or otherwise as already explained at some length.* 
 
 As said earlier, the conditions surrounding the measurements 
 of dielectric strength, regulation, and other quantities during a 
 test should, in America, be in accord with the Standardization 
 Rules of the American Institute of Electrical Engineers, to 
 which reference has already been given earlier in this article, 
 unless other specifications are specially provided. 
 
 *Art. 157. 
 
SYNCHRONOUS MACHINES 
 
 743 
 
 A Power-factor indicator is often of service in testing and also 
 for a permanent place on a switchboard. This may be built on 
 the same principle as the synchroscope explained earlier (Fig. 
 418), or it may be built like a two-phase induction motor. In 
 the latter case a current having the phase of the line current is 
 passed through the windings of one phase, and another having 
 the phase of the voltage is passed through the other, and the 
 torque of the armature is then a function of the phase angle. 
 Revolution of the armature being restrained by a spring, the 
 degrees of rotation of the armature are proportional to the 
 torque and therefore indicate the phase angle. A pointer at- 
 tached to the armature and arranged to play over a scale com- 
 pletes the essential parts of the instrument. 
 
 173. Hunting of Synchronous Motors and Other Synchronous 
 Machines. — All machines, such as alternating current generators, 
 synchronous motors, and rotary converters, which depend upon a 
 
 Fig. 441. — Synchronous Motor Diagram showing the Effect of Hunting. 
 
 series or synchronizing current to hold them instep, are subject to 
 vibratory irregularities in angular velocity which are given the 
 name of Hunting. In explanation of this effect consider a 
 synchronous motor of which the voltages and current relations 
 are illustrated in the diagram of Fig. 441. In this diagram the 
 impressed voltage is kept in a fixed position and is assumed 
 
744 
 
 ALTERNATING CURRENTS 
 
 to be of constant scalar value. The counter-voltage E 2 is also 
 assumed to be of constant scalar value. Now suppose that the 
 motor load has just been reduced from a higher value to one 
 where the normal relations of the voltages and current are repre- 
 sented by the parallelogram OE x RE 2 and the line 01, OR being the 
 resultant voltage and 01 the current. These relations will not, 
 however, immediately prevail, because in moving forward to 
 its new space-phase position the rotating part of the machine is 
 required to take a momentary speed somewhat greater than 
 that of synchronism. The added momentum gained by the 
 rotor then tends to drive it farther .ahead in space phase than 
 the normal position, so that the counter-voltage takes a position 
 such as OE 2 , which is too far advanced for steady balance, 
 and the resultant voltage and current vectors at the same time 
 move forward in phase and fall off in length as illustrated in 
 0R X and 01'. The component of the current in opposition 
 to the counter- voltage also falls off in length, and the resist- 
 ing moment of the load being then greater than the electro- 
 magnetic torque between rotor and stator, the extra momentum 
 is absorbed in the mechanical load and the speed is reduced 
 to normal. Now the resisting moment of the load still being 
 greater than the electro-magnetic torque, the rotor must fall 
 back. It then must for an instant slow its speed to below 
 that of synchronism and thus it gives up to the load some 
 of its momentum. When it has reached the space-phase posi- 
 tion where the counter- voltage for the load is normal, its speed 
 is, therefore, less than normal, and it drops back still farther, 
 as, for instance, so that the counter-voltage lies in OE 2 " . But 
 as it falls back the current increases and the electro-magnetic 
 torque becomes greater than the resisting moment of the load. 
 In the figure the limit to which the rotor has fallen behind 
 proper step is supposed to have been reached when the counter- 
 voltage is OE 2 " and the current 01 " . The electro-magnetic 
 torque while the rotor is at the retarded point is greater than 
 the load torque, 01" being above normal for the load, and the 
 armature tends again to gain speed in excess of synchronism and 
 step ahead to normal position. The excess momentum due to 
 excess speed may again carry the rotor beyond the normal 
 point for the load and the whole process be repeated over and 
 over again. If there were no losses caused by this vibration, it 
 
SYNCHRONOUS MACHINES 
 
 745 
 
 would continue indefinitely at full amplitude. There are, 
 however, frictional, magnetic, and electric losses. The frictional 
 losses of the rotor bearings and of windage absorb some of the 
 excess energy stored in the rotor which produces the vibratory 
 movement, just as frictional losses tend to absorb the energy 
 of a pendulum which has been set swinging. The hysteresis 
 loss and eddy, or short-circuit, currents in the iron core and 
 field windings caused by the changes of the current value act 
 in the same way. In connection with the latter, loss is also 
 occasioned by the inductive action of the armature magnetic 
 flux as it sweeps through the iron of the field poles because of 
 the superimposed vibratory motion of the rotor. If no new dis- 
 turbance occurs, this energy loss ordinarily causes the vibration 
 to rapidly reduce to zero, and the voltages and currents assume 
 and maintain their normal relation. In this case the rotating 
 part of the machine has the characteristics of a torsional pen- 
 dulum, of which the normal position gives counter-voltage in the 
 position OR 2 (Fig. 441), and which is drawn toward that posi- 
 tion by the electro-magnetic torque, but which possesses the 
 moment of inertia of the rotating part of the machine with re- 
 spect to the shaft. The normal position of rest of this torsional 
 pendulum travels in synchronism with the uniform theoretical 
 rotative speed of the rotor, and the pendulum vibration is super- 
 posed on that rotative speed. 
 
 A mechanical illustration of the conditions can easily be de- 
 vised. Thus, suppose a mass of given weight is attached to the 
 lower end of a spiral spring and is lifted at a uniform rate by 
 a rope attached to the upper end of the spring. Suppose now 
 that some of the mass breaks away and thus causes the remain- 
 ing mass to vibrate up and down. The spring is first shortened 
 due to the lessened weight, but the momentum given to the 
 mass causes it to pass beyond the normal position that it should 
 assume. The tension on the spring in pounds is then less than 
 the supported weight, and the mass falls back. When it 
 reaches normal position again, it is rising with less than the 
 speed of the rope and, because of its inertia or force of momen- 
 tum, it falls below the normal position. The spring is now 
 stretched so that its tension in pounds is greater than the 
 weight of the mass, which is therefore pulled back, and the 
 process may be repeated over and over. The molecular friction 
 
746 
 
 ALTERNATING CURRENTS 
 
 of the spring and the extra wind friction of the mass due to the 
 vibratory motions dampen down the amplitude of the vibrations 
 until they become zero. The product of the speed of the rope 
 by the weight of the mass is equivalent to the output of the 
 motor, or its rotor speed times the load torque. The weight is 
 equivalent to the rotor load torque ; the momentum of the mass 
 to the rotor momentum ; and the pull of the spring on the 
 mass to the electro-magnetic motor torque. 
 
 The time of vibration of the mass may be expressed 
 
 where M is its weight and K the compressibility of the spring. 
 Likewise the same formula is approximately applicable to the 
 motor, where M is the moment of inertia of the rotor about the 
 rotating axis, and K represents the reciprocal of the rate of 
 change of the electro-magnetic torque with reference to the dis- 
 tance of the rotor from its normal instantaneous position. It 
 is seen, therefore, that the time of vibration is increased with 
 increase in weight and diameter of the rotor and with decrease 
 of the rate of change of electro-magnetic torque per unit of dis- 
 tance that the rotor is away from its normal instantaneous posi- 
 tion. Now this latter factor depends upon the rate of change 
 of the component of the current vector which is in opposition 
 to the counter-voltage for the various positions of the rotor 
 with reference to its normal instantaneous position. A studj r 
 of the diagrams given in Figs. 424, 425, and 441 will show that 
 the angle -i ]r between the resultant voltage and current should 
 be small to give the greatest proportional change of electro- 
 magnetic torque with a swing of the rotor through a small 
 given angle. The armature impedance should evidently be 
 small, but also to make -</r small, 2 m fL should be small compared 
 with iL If R is very small compared to 2 7r/Z, the conditions 
 may permit the hunting to increase, eventually throwing the 
 armature out of step. This result may also occur through peri- 
 odic changes in the angular speed of the prime generators or of 
 the motor load, provided the period of such changes chances to 
 approach equality with the natural period of vibration of the 
 rotor about its axis as shown in the preceding formula. 
 
 If the field windings are of low resistance, they act in effect 
 like the short-circuited secondary of a transformer to prevent 
 
SYNCHRONOUS MACHINES 
 
 747 
 
 Fig. 442. — Amortisseurs for preventing Hunting. 
 
 The damping effect of these windings due to the currents 
 caused by the rotor vibrations may be understood when it is 
 remembered that the armature magnetic flux for a polyphase 
 synchronous motor is normally stationary with reference to the 
 poles of the field magnet in case the machine and line voltages 
 are in exact isochronism. Then when relative movements or 
 vibrations of the armature occur with respect to its normal 
 
 the sweeping back and forth of the armature flux; that is, they 
 reduce the apparent reactance of the armature and increase its 
 stability. Instead of lowering the resistance of the field wind- 
 ings, short-circuited squirrel cage conductors are sometimes laid 
 in the faces of the field poles, as shown in Fig. 442. These are 
 sometimes called Amortisseurs or Damping grids. They form 
 excellent short-circuited secondary paths for the production of 
 currents by the vibrating armature flux and the dissipation of 
 the energy of vibration by its conversion into heat. 
 
 Amo 
 
748 
 
 ALTERNATING CURRENTS 
 
 instantaneous positions with reference to the field magnet, the 
 magnetic flux of the armature currents acts upon the ainortisseur 
 windings and other metal parts of the field magnet exactly as 
 though their vibratory motion was the only one that existed. 
 In accord with the principles of electro-magnetic induction, 
 it is to be observed that the currents induced by this mo- 
 tion of the armature flux with respect to the damping grids 
 produce electro-magnetic torques tending to destroy the mo- 
 tion. This is similar to considering that the mass and spring 
 used in the earlier illustration are vibrated while the rope is 
 stationary, Avhich is entirely justifiable, as the vibrations have 
 no effect on the upward movement of the rope. The effect of 
 the amortisseur windings may also be compared to the damp- 
 ing effect which will be produced upon the weight by a 
 medium having greater frictional resistance than air, such as 
 would be the case if the weight were immersed in water or oil. 
 
 As noted earlier, amortisseur windings are of service in start- 
 ing polyphase synchronous motors by means of rotating field 
 induction motor torque. 
 
 174. The Synchronous Condenser. — It has been shown that a 
 synchronous motor causes a quadrature lagging current to flow 
 in the line when its field magnet is under-excited, and that it 
 causes a leading current to flow when it is over-excited. In 
 the above statement normal excitation is taken as that which 
 gives unity power factor to the input of the motor, for the load 
 under consideration. 
 
 This characteristic of the synchronous motor makes it valu- 
 able for use as a power factor regulator, and when so used it is 
 called a Synchronous condenser. 
 
 As an illustration of this service, consider a long transmission 
 line loaded at the receiving end with induction motors which 
 cause, during a certain period when their combined mechanical 
 load is equal to 500 kilowatts, a lagging power factor of 80 per 
 cent. Then the total kilovolt-amperes are 625 and the quad- 
 rature kilovolt-amperes are 375. The kilovolt-ampere triangle 
 for these conditions is marked ABC in Fig. 443. As the 
 sides of this triangle, AB, BC , and AC, respectively, equal the 
 impressed voltage multiplied by the active current, by the lag- 
 ging quadrature current, and by the total current, they may be 
 scaled to represent either currents or kilovolt-amperes. If it is 
 
SYNCHRONOUS MACHINES 
 
 749 
 
 C D 
 
 Fig. 443. — Diagram showing the Components of 
 Apparent, True, and Quadrature Power in a 
 Line supplying an Inductive Load ; and the 
 Rating of the Synchronous Condenser re- 
 quired to increase the Power Factor to Unity. 
 
 desired to increase the power factor of this circuit to unity, a 
 synchronous motor supposed for the present to be without 
 losses may be employed at the receiver end of the line, which 
 is excited so as to demand from the circuit 375 leading quadra- 
 ture kilovolt-amperes. 
 
 Since the motor’s input is 
 limited by the heating of 
 the armature conductors, 
 its rated capacity must 
 be at least 375 kilovolt- 
 amperes. Mechanically 
 it need not have this 
 rating, for if no load is 
 placed upon its shaft, the 
 only torque required is 
 that sufficient to over- 
 come losses of rotation. 
 
 If the voltage of supply 
 is assumed to be 10,000 
 volts and three phase, the quadrature current of the synchro- 
 nous condenser is 12.5V3 amperes. Since such a motor in fact 
 has some losses, a little true power must be supplied to it, which 
 is represented in the figure by BE — CD, so that the actual 
 
 kilovolt-amperes of the 
 motor armature are rep- 
 resented by CE. 
 
 If it is desired to in- 
 crease the power factor 
 from .80 to .95, the re- 
 lations are as shown in 
 Fig. 444. In this case 
 the angle of lag must be 
 reduced from 36° 45' to 
 18° 15'. To bring this 
 about, the synchronous 
 condenser must provide 
 approximately 211 kilovolt-amperes in leading quadrature, 
 which comes to 7 v/3 amperes in a three-phase circuit at 10,000 
 volts. The power losses of the condenser are assumed to equal 
 CD, the quadrature kilovolt-amperes necessary to be provided 
 
 C D 
 
 Fig. 444. — Diagram for determining the Rating 
 of an Unloaded Synchronous Condenser required 
 to increase the Power Factor of a Load. 
 
750 
 
 ALTERNATING CURRENTS 
 
 by the condenser are equal to DE , and the lagging component of 
 the kilovolt-amperes still in the line is equal to BF. The 
 rated loading of the condenser must be CE kilovolt-amperes. 
 
 Fig. 445. — Diagram showing Power or Current Relations in a Circuit containing a 
 Loaded Synchronous Condenser, which raises the Line Power Factor to Unity. 
 
 When it is desired to increase the power factor and at the 
 same time obtain mechanical power from the motor, the dia- 
 grammatic constructions are as in Figs. 445 and 446. In Fig. 445 
 the synchronous condenser is arranged to deliver 185 kilowatts 
 of mechanical power and is then supposed to have losses equal 
 
 Fig. 446. — Diagram showing Power or Current Relations in a Circuit containing a 
 Loaded Synchronous Condenser. The Line Power Factor is increased by the Con- 
 denser. 
 
 to 15 kilowatts. Therefore, to bring the circuit to unity power 
 factor, it must receive from the mains 200 = BE kilowatts of 
 power, 375 = DE — — BC of quadrature kilovolt-amperes, and 
 a total of 425 = CE total kilovolt-amperes. In Fig. 446 the 
 synchronous condenser is adjusted to only reduce the quadra- 
 ture power by 200 = DE kilovolt-amperes, although it receives 
 
SYNCHRONOUS MACHINES 
 
 751 
 
 the same true power as in Fig. 445. It will be noted that the 
 true power load of the condenser itself has an influence on the 
 angle of lag of the line current, for in Fig. 446 the reduction of 
 the lag angle would have been less by the angle FAE had the 
 motor run without mechanical load but had still provided 200 
 = DE quadrature kilovolt-amperes. It will also be noticed that 
 the total loading of the synchronous condenser of Fig. 446 is 
 only 282 kilovolt-amperes when receiving 200 = CD kilovolt-am- 
 peres true power and 200 = DE kilovolt-amperes quadrature 
 power. It is evident from the construction that the smallest 
 motor capacity is required for a given arithmetical sum of true 
 and quadrature volt-amperes when the two components are equal. 
 
 In general we have the loading required for a synchronous 
 condenser, 
 
 Ft = V/V + P q \ or T t = V7 a 2 + l q \ 
 
 where P t , P a , and I\ are the total, active, and quadrature volt- 
 amperes received by the machine, and I t , I a , I q , are the total, 
 active, and quadrature currents. 
 
 From the current locus of the synchronous motor, it may be 
 seen that the quadrature kilovolt-amperes for no mechanical load 
 on such machines make quite a large figure when the fields are 
 over-excited, and that these increase up to a certain point as 
 the load is increased (see current locus FGr"H \ Fig. 430). This 
 may also be seen by drawing a number of constant power cur- 
 rent loci for a synchronous motor as in Fig. 431, then for a given 
 over-excitation which gives a current locus such as SPS' in 
 that figure, the quadrature leading current increases with the 
 load, thus, the quadrature leading current increases from OR at 
 no load to OQ as the load increases from that represented by 
 the locus M" to the constant load locus that would pass through 
 the point P. For further increase of mechanical load the 
 quadrature current decreases. The line QP is parallel to OE v 
 and the point P is the tangent point between QP and SPS'. 
 It is therefore impossible to exactly neutralize the lagging cur- 
 rent of induction motors, when the loads vary, by the use of 
 loaded synchronous motors with fixed excitation. Thus, if the 
 synchronous motor is excited so as to supply the right quad' 
 rature current when driving its average mechanical load and 
 when the load of the induction motors on the line is average, it 
 
752 
 
 ALTERNATING CURRENTS 
 
 will supply more or less than enough if its mechanical load in- 
 creases or the load of the induction motors decreases, and vice 
 versa if the synchronous motor load falls off or that of the in- 
 duction motors increases. However, by setting the synchronous 
 motor excitation at the average value referred to, the actual 
 variation from the correct value, for commercial circuits, is not 
 likely to vary more than a few per cent. When necessary, the 
 excitation can be changed to more closely adjust the balance. 
 This is usually done by hand, but if the need is sufficiently great, 
 an automatic regulator can be constructed, using a phase-indi- 
 cating device in the main circuit to operate relays for control- 
 ling the field rheostat or exciter rheostat of the condenser. 
 
 Synchronous condensers are desirably placed on a part of a 
 transmission line near a group of apparatus, such as induction 
 motors, which cause the flow of lagging current. The con- 
 denser then performs a twofold service ; that is, it improves the 
 line regulation, and at the same time reduces the I 2 R loss in 
 the line for a given number of kilowatts transmitted. Thus, 
 suppose the line supplying the load diagrammed in Fig. 443 is 
 compensated to unity power factor. The line loss which was 
 proportional to (625) 2 becomes proportional to (500) 2 or it has 
 been reduced to close to of its original value. This saving in 
 power in itself would in many cases warrant the installation 
 of a synchronous motor to float without mechanical load on 
 the line. If it is not desired to reduce the line loss, the sav- 
 ing in copper would be nearly | of that which would be de- 
 manded without the condenser, for the instance mentioned. 
 If the total copper is large, the saving may be very material, 
 especially if the uncorrected power factor is very low. If the 
 synchronous condenser replaces other motors by carrying their 
 mechanical load and has a sufficient rating so that it can carry 
 an equal quadrature load (Fig. 446), the total kilovolt-amperes 
 rating required over that of the motors replaced is in the ratio 
 of 1 to V2. If the quadrature power to be neutralized is not 
 great, the increase may be but a small percentage of the power 
 of the total apparatus installed. 
 
 The line regulation may be improved by the use of the 
 synchronous condenser from two causes, first, by changing 
 the current flowing and bringing it into phase with the voltage, 
 and thus changing the IZ drop and bringing the IX component 
 
SYNCHRONOUS MACHINES 
 
 753 
 
 to a more favorable vector position ; and second, by the auto- 
 matic tendency in the synchronous motor to maintain constant 
 voltage. Thus, referring to the second cause, if the line 
 voltage drops, the motor will have a relatively higher over- 
 excitation and thus draw additional leading current from the 
 line, which will tend to strengthen the fields of the generating 
 apparatus and neutralize the effect of line inductive reactance. 
 If the line exceeds normal voltage, the counter-voltage of the 
 motor will be less in proportion thereto, and the motor will 
 draw less leading current from the circuit, thus leaving un- 
 neutralized inductive reactance in the line ; or if the rise of 
 line voltage is sufficiently great, the motor may draw lagging 
 current from the line, and thus directly add to the inductively 
 reactive element of the circuit impedance. 
 
 The first of these two effects is illustrated in Fig. 447, in 
 which OE represents the voltage vector at the receiver end 
 of the line, 01 the current vector delivered by the line at the 
 receiver end, and OE 0 the voltage vector at the generator 
 
 
 'O 
 
 0 
 
 I 
 
 0 
 
 jr 
 
 | LOAD OF UNITY POWER FACTOR 
 
 E 
 
 0 
 
754 
 
 ALTERNATING CURRENTS 
 
 end of the line. In the upper diagram of this figure, the 
 current lags behind the voltage, but gives the active component 
 OA which is proportional to the delivered power. The vector 
 EE' parallel to 01 represents the IR drop of the line, and 
 the vector E'E 0 in leading quadrature with 01 represents 
 the IX drop of the line. Hence EE 0 is the IZ drop and 
 OE 0 = OE 4- EE 0 is the generator voltage. When the same 
 power is delivered by the line at the same voltage, but the 
 power factor is brought to unity by means of a synchronous 
 condenser, the current takes the value OR shown in the middle 
 diagram of the figure, which is equal to OA of the first diagram. 
 In this case EE' and E'E 0 are shorter than in the first dia- 
 gram, in the ratio of the lengths and are respectively in 
 
 parallel and quadrature with OE. Under these circum- 
 stances the generator voltage OE 0 does not differ from the 
 receiving voltage OE by as many volts as in the first diagram. 
 Now, if the quadrature current of a synchronous condenser 
 causes the current to lead the voltage at the receiving end 
 sufficiently to bring the power factor to the same numerical 
 value as in the first diagram, the active component remaining 
 equal to OA , the current is illustrated by 01" in the bottom 
 diagram in the figure which has the same length as 01 in 
 the first diagram. The IZ drop is EE 0 , which has the same 
 length as EE 0 in the first diagram, hut its position now results 
 in still less difference between the numerical values of OE 0 
 and OE than in the second diagram. By still further increas- 
 ing the current lead, the generator voltage required to trans- 
 mit the power over the line may become smaller than the 
 voltage maintained at the receiver end. 
 
 Synchronous condensers are sometimes useful in generating 
 stations, where they care for the lagging current coming in on 
 the line. Synchronous motors of the ordinary type may be 
 used as synchronous condensers ; or rotary converters * may 
 be so used, in which case the true power output of the machine, 
 if any, is electrical instead of mechanical. 
 
 175. Converters. — Reference has already been made to the 
 possibility of converting a direct-current dynamo into a single- 
 phase alternator, and the fact that a machine so constructed, with 
 
 * Art. 175. 
 
SYNCHRONOUS MACHINES 
 
 755 
 
 commutator and alternating-current collector rings, may be used 
 as a rotary converter to convert alternating currents into direct 
 currents, or vice versa , or as a double-current machine to gen- 
 erate or absorb both alternating and direct currents at the same 
 time.* Similar combinations may also be applied to polyphase 
 machines. When such a machine receives alternating currents 
 through its collector rings, running as a synchronous motor, 
 and gives out direct currents through its commutator, it is 
 called a Converter or Rotary Converter. When it receives di- 
 rect current through its commutator, running as a shunt- wound 
 motor, and gives off alternating currents through its collector 
 rings, it is called an Inverted Converter or Inverted Rotary. 
 
 It is possible in this manner to make a quarter-phaser to 
 be used with separate circuits out of any direct-current ma- 
 chine with a closed circuit armature winding, such as the 
 Gramme or Siemens winding, by arranging four collector rings 
 on the shaft and connecting them to the armature windings at 
 points which are 90 electrical degrees apart. It is also pos- 
 sible to make a three-phaser out of a direct-current machine 
 by arranging three collector rings on the shaft, and connecting 
 them to the armature windings, at points 120 electrical degrees 
 apart. Such machines may be used as converters to transform 
 direct currents into polyphase currents or vice versa. Likewise, 
 a six-phaser can be constructed by using six collector rings con- 
 nected to the armature winding at points 60 electrical degrees 
 apart. 
 
 In Fig. 448 is shown a diagram of a multipolar machine with 
 the connections of the armature winding indicated from A to B. 
 The distance from A to B is 360 electrical degrees, so that the 
 taps to be used for a single-phase machine are those numbered 
 2 and 5. The left-hand tap marked 2 is the first connection 
 for the succeeding 360 electrical degrees of armature circum- 
 ference, and it must also be connected to ring 2 to complete the 
 circuit through the windings of the second 180°. For a quarter- 
 phase machine rings and taps 2, 1, 5, and 8 are used; for a tri- 
 phase machine the taps and rings to be used are marked 2, 4, 
 and 6 ; while for a six-phase machine they are 2, 3, 4, 5, 6, and 7. 
 
 Suppose now a commutator is attached to the windings of the 
 machine as shown in Fig. 448, with the brushes for connecting 
 
 * Art. 27. 
 
756 
 
 ALTERNATING CURRENTS 
 
 to direct-current circuits placed upon the commutation points 
 as indicated, then the machine so constructed may be used as a 
 converter. It will run in synchronism when fed with alternat- 
 ing currents through the collector rings, and its speed therefore 
 depends upon the number of poles in the field magnet and the 
 
 Fig. 448. — Diagrammatic Arrangement for showing Alternating and Direct-current 
 Connections for a Closed Circuit Armature. 
 
 frequency of the currents fed to it. Polyphase converters are 
 generally self-starting from the alternating-current end by the 
 effect of eddy currents set up in the pole pieces by the rotary 
 field which exists in the armature when polyphase currents are 
 passed through it. The starting torque may be increased, as in 
 
SYNCHRONOUS MACHINES 
 
 757 
 
 polyphase synchronous motors, by embedding short-circuited 
 copper bars, or amortisseur windings, across the pole faces. 
 After a converter fed by alternating currents is in synchron- 
 ism, its field magnet may be magnetized by the direct current 
 produced by itself and collected from its commutator. 
 
 In connecting the armature windings of converters to the 
 collector rings, the relative angles corresponding to the current 
 phases must be carefully distinguished as indicated above in 
 the description of Fig. 448. One complete revolution of an 
 armature in a two-pole field corresponds to one complete period 
 of the alternating current, and therefore 860 mechanical de- 
 grees corresponds to 360 electrical degrees; but in multipolar 
 machines an extent of rotation of the armature equal to twice 
 the angular pitch of the poles corresponds to one complete 
 period, so that, in general, the relation of electrical degrees to 
 mechanical degrees is p : 1, where p is the number of pairs of 
 poles in the field magnet. Two-pole converters evidently utilize 
 the whole of the armature winding with each collector ring con- 
 nected to a single point on the winding, and the same is true of 
 multipolar machines with two-path armature windings. If sin- 
 gle connections to the collector rings should be used in multi- 
 polar machines with multiple-path armature windings, a portion 
 only of the armature, corresponding to 360 electrical degrees, 
 would be occupied in utilizing the alternating currents ; the 
 armature capacity would therefore be not fully utilized and the 
 machine would be seriously unbalanced. To fully utilize the 
 armature in this case, each collector ring must be connected to 
 the winding at as many points as there are pairs of poles, the 
 points being 360 electrical degrees apart. In Fig. 448 the taps 
 and windings are shown only for 360 electrical degrees. Evi- 
 dently to complete the connections, the commutator and wind- 
 ings should extend over the entire 360 mechanical degrees and 
 there should be a total of 8 p taps like those shown in the 
 diagram, similar taps in each 360 electrical degrees of circum- 
 ference being connected to the same collector rings. In this 
 figure the collector rings for single, quarter, three, and six phases 
 are all connected to one armature winding for purposes of com- 
 parison. It should be understood that only the number of rings 
 necessary on the system for which the machine is designed to 
 be used are ordinarily placed on the armature shaft. 
 
758 
 
 ALTERNATING CURRENTS 
 
 In general, for any number of taps, the number of collector 
 rings equals the number of phases and the number of electrical 
 
 degrees between the taps of a phase is equal to 
 
 360 
 
 IT 
 
 where N is 
 
 the number of phases. In this case N for the single-phase sys- 
 tem must be taken as equal to 2, since there are really two 
 phases 180° apart in that system, and for the quarter-phase 
 system N must be taken as equal to 4, as the currents flowing 
 in the four wires differ in phase by 90°, when followed around 
 in order. 
 
 176. Ratio of Transformation in Converters. — Referring to 
 Fig. 448, it may be observed that when the machine is driven 
 as a synchronous motor by impressing alternating voltages upon 
 a set of rings, the field magnet, armature windings and commu- 
 tator constitute a direct-current generator. Consider a theo- 
 retical case first, where the armature windings are without re- 
 sistance and the voltage generated by the revolving conductors 
 is always equal and opposite to an alternating voltage supplied 
 from a single-phase system to the winding section between two 
 rings through which it is supplied, — the two rings considered 
 here being those numbered 2 and 5 in Fig. 448. Then, when 
 the armature is in the position shown in the figure with the 
 points 2 and 5 just under the direct-current brushes, the gen- 
 erated and hence the impressed alternating voltages are a max- 
 imum and equal and opposite. This is on the assumption that 
 there are no armature reactions and that the brushes are there- 
 fore upon magnetic neutral points. But if there are enough 
 coils in the winding and segments in the commutator to make 
 the direct-current voltage between the brushes of constant value, 
 the voltage generated, at the instant under consideration, is the 
 constant direct-current voltage between each pair of brushes, 
 which is equal to the maximum value of the alternating-cur- 
 rent voltage. This latter may be called e m . 
 
 By the assumptions the voltage generated in a section, say 
 AC, is to be opposite to the impressed voltage in the same sec- 
 tion. When the middle of this section is under brush m, the 
 voltage is zero because that generated in the two halves of the 
 section neutralize each other. It is again zero when the mid- 
 dle of the section is under brush n. Suppose that the gener- 
 ated and impressed voltages in this section are of sinusoidal 
 
SYNCHRONOUS MACHINES 
 
 759 
 
 wave shape, then, since the direct voltage is constant, the re- 
 lation between the direct and alternating voltages is 
 
 i? 2 =^ = .707 e m , 
 
 V2 
 
 where E 2 is the effective impressed alternating voltage. But 
 e m has been shown to be equal to the direct voltage, which 
 may be called E. Hence E^ = .707 E. 
 
 By the methods used in the construction of Fig. 33 * it was 
 shown that when a wide coil moves in a magnetic field so that 
 a sinusoidal voltage is induced in each conductor, the vector 
 voltages of the individual conductors form a vector diagram, 
 of which the resultant is the voltage between the coil termi- 
 nals, and is of such shape that the junction points of the com- 
 ponent voltages lie in the arc of a circle. The greater the 
 number of conductors, if uniformly distributed, or of slots if 
 they are placed in slots in a given width of winding, the nearer 
 the component vectors coincide with the circular arc, and the 
 wider the coil, the longer the arc, so that with a coil covering 
 180 electrical degrees it is a semicircle. Let the semicircle 
 OJGrCDKA of Fig. 449 represent the vector polygon of the 
 component voltages of the coil section AC in Fig. 448 and 
 semicircle ALFBHMO represent the similar polygon for sec- 
 tion CB which acts in parallel with AC and is therefore in 
 series opposition with respect to induced voltages. It is 
 assumed in this that the windings are so thoroughly distrib- 
 uted and the component voltage vectors are so short that the 
 vector polygon and the circumference of the arc coincide, 
 which for the purpose causes no appreciable error. The 
 diameter OA is then the alternating voltage between the 
 
 E 
 
 terminals of either section and equals — -• 
 
 V2 
 
 Suppose now the rings 2, 1, 5, and 8 only are connected in 
 Fig. 448, which results in a quarter-phase machine in which 
 the alternating-current coil sections are one half as long as a 
 single-phase section between 2 and 5. The voltage between 
 connectors 2 and 1 is equal to vector OC of Fig. 449 in length 
 and direction ; that between 1 and 5 is equal to CA ; that be- 
 tween 5 and 8 is equal to AB ; and that between 8 and the 
 
 * Art. 20. 
 
TGO 
 
 ALTERNATING CURRENTS 
 
 B 
 
 Fig. 449.- — Vector Diagram for showing the Relation of Direct and Alternating Vol- 
 tages in a Rotary Converter. 
 
 left hand connection, 2, is equal to BO. Hence the ratio of 
 voltage transformation for a quarter-phase converter is ex- 
 pressed by the relation 
 
 E = ^ = .5E, 
 
 4 9 
 
 where E± is the effective value of alternating quarter-phase 
 impressed voltage and E is the voltage between direct-current 
 brushes. 
 
 Likewise, if the three rings 2, 4, and 6 of Fig. 448 are used, 
 the tri-phase vector voltages are represented by the lines OD, 
 J)E , and FO in Fig. 449 and the relative direct and alternating 
 voltages are obtained from the expression, 
 
 E~ = ^r E=.%V2 E, 
 
 3 2V2 
 
 where E 3 is the effective value of tri-phase alternating voltage. 
 
SYNCHRONOUS MACHINES 
 
 7G1 
 
 In like manner the voltage relation in a six-phase converter 
 is found to be 
 
 E. = — — . 354 E, 
 
 2V2 
 
 where E 6 is the alternating six-phase voltage. For twelve-phase 
 converters the relation is 
 
 En = — ~^ 8 E= .183 E , 
 
 12 2V2 
 
 where E u is the alternating voltage, and E as before is the 
 direct-current voltage. 
 
 In general, it may be seen, from the construction of Fig. 449 
 that 
 
 E n = 
 
 E . 7 r 
 
 yl sm n ' 
 
 where E N is the alternating voltage per phase for N phases and 
 E is the direct-current voltage, since length 
 
 OA = E, = — . 
 
 1 V2 
 
 Continuing the assumptions heretofore made and neglecting 
 all losses, then, if the impressed alternating voltage and current 
 are in the same phase, we have 
 
 2 EIp=I’ — sin JVp, 
 
 V2 N 
 
 where E is the direct-current voltage, p is the number of pairs 
 of poles in the field magnet, N is the number of alternating- 
 current phases, I is the direct current in the windings of any 
 one of the armature phase sections, and I’ is the component of 
 alternating current in the same section. It will be noted that 
 the actual current flowing in a section is the resultant of I and 
 E . The two sides of the equation represent respectively the 
 total direct-current output and alternating-current input of the 
 machine. Hence, 
 
 2V2 I 
 
 N sin \ T 
 
 N 
 
7G2 
 
 ALTERNATING CURRENTS 
 
 Using the same subscripts for P as were used with E to 
 designate numbers of phases, we have 
 
 77 _ -x/2 T. T 7 _ 4~\/2 j-. j, _ j. j, _ 2V2 j 
 l 2 — V _ 7 , 7 3 — 7 , 7 4 — 7 , 7 6 — ^ , 
 
 3Va o 
 
 2 V2 
 
 and I' 9 = — — — 
 
 12 12 x .2u9 
 
 Z 
 
 Noting the manner in which the parallel currents join in the 
 leads of converters of different numbers of phases, we have the 
 direct current per lead, 
 
 Ii = 2I 
 
 for converters of all phases. The alternating current per lead 
 in a single-phase r is, 
 
 -Gt = 2 J 7 , = 2 V 2 1 = V 2 I L . 
 
 In a tri-phaser, the currents join in a lead at an angle of 120°, 
 and 
 
 J' 3i =V3Z 3 = 
 
 4V2 I= 2V2 
 3 3 
 
 For a quarter-phaser, since the currents from the two adjacent 
 coil sections join in a lead at the phase angle of 90°, 
 
 7'u = V2 J',= V2 J= A j 
 
 V2 
 
 In a six-phaser, the junction phase angle being 60°, 
 
 I' 6L =I' 6 =^I= ~I l . 
 
 3 
 
 And in a twelve-phaser, the junction angle being 30°, 
 
 I'm- 2x. 2597> b =^/-^| A 
 
 In the actual machine, where there are losses, an additional 
 alternating current must flow in each section of such a value 
 that, 
 
 P c = I c E'Np , 
 
 where P c is the watts loss due to the rotation of the armature, 
 I c is the additional current required, E' is the effective voltage 
 
SYNCHRONOUS MACHINES 
 
 763 
 
 in an alternating-current winding section, 2V is the number of 
 phases, and p is the number of pairs of poles in the field 
 magnet. Also as the current flowing in the armature causes 
 an IB drop, E in the formula given above for relations of 
 voltages and currents must be the total direct-current voltage 
 generated. If the impressed voltage and current are not in 
 phase, there will be an additional component of quadrature 
 current, I' q = T sin 0 — I a tan 6 , where I' is the total com- 
 ponent of alternating current, I a is the active component of 
 current heretofore dealt with alone, and 6 is the angle between 
 the impressed voltage and current in the section. The larger 
 current causes a greater IB drop in the direct-current voltage, 
 and as will be seen later causes increased reactions on the 
 converter field magnet. 
 
 177. Frequency and Voltage Limitations in the Converter. — 
 
 Direct-current generators usually have alternating-current fre- 
 quencies in the armature conductors which are between 8 and 
 25 cycles per second, while the frequencies of alternating-current 
 systems are usually from 25 to 60 cycles per second. Since the 
 product of number of magnet poles and speed is proportional to 
 the frequency of the conductor current in the usual direct-cur- 
 rent machine, and the speed is limited by the permissible circum- 
 ferential velocity of the commutator, which ordinarily should 
 not run at a velocity of over 3000 feet per minute, the number 
 of magnet poles must in general increase with the frequency. 
 The larger the number of poles, the less is the distance between 
 direct-current brushes on a commutator of given diameter and 
 hence the less the number of commutator bars of particular 
 width that can be placed in the commutator in the arc between 
 adjacent direct-current brushes. As the voltage which a direct- 
 current machine of given design can produce without excessive 
 sparking at the commutator is dependent, among other things, 
 upon the number of commutator bars between adjacent brushes 
 of opposite polarity, it is apparent that the higher the frequency 
 of the alternating current, the less is the possible safe direct-cur- 
 rent voltage on a converter. By careful designing it has been 
 found possible to build polyphase converters that operate satis- 
 factorily at a frequency of 60 periods per second with direct- 
 current voltages of from 500 to 600 volts, which is the max- 
 imum direct-current voltage ordinarily employed, and for lower 
 
764 
 
 ALTERNATING CURRENTS 
 
 voltages. Difficulty may be experienced in designing very 
 small machines which can be manufactured at a reasonable 
 cost for the maximum frequency and voltage named, on account 
 of the inherently small commutator. 
 
 The commercial converter, in its design, is a combination of 
 an alternator and a direct-current generator with a large com- 
 mutator. Figure 450 represents a typical 60-cycle, 600-volt 
 converter of American design. 
 
 Fig. 450. — A Typical Three-phase Converter. 
 
 At one end of the shaft of a converter there is ordinarily in- 
 stalled a small mechanical or magnetic device which will cause 
 the armature to move slowly back and forth lengthwise of the 
 shaft. If this was not done, the converter would rotate with- 
 out any material movement longitudinally, and grooves would 
 be worn in the commutator and bearings. The commutation 
 of the higher voltage converters is apt to be quite sensitive, 
 and if the machine is not in good order or there are disturb- 
 ances in the alternating-current line, it may “flash over,” that 
 is, short-circuit instantaneously from brush to brush. 
 
 When a converter is to be operated inverted (Art. 182) either 
 in regular service or at starting, it is usual to attach an auto- 
 
SYNCHRONOUS MACHINES 
 
 765 
 
 matic switch actuated by centrifugal balls on the end of the 
 shaft, for the purpose of automatically causing the main switches 
 to open in case the armature races and goes to too high a speed. 
 This is a precaution of great importance, since any defect arising 
 in the field excitation may result in the armature rapidly accel- 
 erating to a dangerous speed unless it is instantly disconnected 
 from the circuit.* 
 
 178. Heating of the Armature Conductors in Rotary Con- 
 verters. — Making the same assumptions as in the discussion of 
 the relations of voltages and currents,! the currents in the 
 direct and alternating circuits are as shown in the formulas 
 already developed, but they flow in opposite relative directions 
 in the coils since the alternating current produces, in connection 
 with the field magnet, the motor torque, while the direct current 
 produces the effect of a counter-torque. In Fig. 451 at the top 
 (a) is a diagrammatic representation of part of the field magnet 
 and armature of a quarter-phase converter developed. The ver- 
 tical lines at C and D represent two quarter-phase taps to two 
 of the four quarter-phase collector rings. For convenience 
 only three armature conductors or coils C, F , and D are shown, 
 though the armature is supposed to be completely wound with 
 distributed windings. Consider first the current which flows in 
 the middle conductor F of the section CD. The direct current 
 reverses in the conductor as it passes under the brush B (the 
 armature is supposed to move to the left), and again when it 
 passes under brush A. The current in the outside leads from 
 brushes A and B is 2 I, where the direct-current component in 
 section CD is I. The line NNX, drawn with respect to axis 
 XX, Fig. 451 (6), is the rectangular curve of this component of 
 the current in conductor F, where the abscissas are in electrical 
 degrees having the same scale as in Fig. 451, (a). The alter- 
 nating component of the current in conductor F, which is as- 
 sumed to be in exact opposition to the counter-voltage of the 
 windings and to be sinusoidal, is shown by the curve M, with 
 its maximum value occurring when F is under the center of 
 pole piece X, and the zero values of the cycle occurring when 
 F is under the brushes A and B. It has been shown f that in 
 a quarter-phase system the effective alternating current between 
 C and D of the section is equal to the current between the 
 * Art. 182. t Art. 176. 
 
766 
 
 ALTERNATING CURRENTS 
 
 direct-current brushes or I' = I. Hence, the maximum value 
 of current M is V2 /, where I is equal to one-half the current 
 entering the external circuit from brush B. Plotting the differ- 
 
 Fig. 451. — Diagrams for Representing the Current Distribution and Heating in the 
 Armature Coils of a Converter. 
 
 ence between curves M and NNF for the half period shown in 
 the figure gives curve PPP. This shows the resultant current 
 in conductor F, and it is equal in value and direction to current 
 
SYNCHRONOUS MACHINES 
 
 767 
 
 iV’iVTV’ just as the conductor approaches or leaves the direct-cur- 
 rent brushes but flows in the opposite direction therefrom when 
 the conductor is within the arc from 45° to 135°. Squaring the 
 ordinates of the current curve PPP, and the curve Q results, 
 the ordinates of which are proportional to the heat produced in 
 conductor F as it passes from B to A, and the area of which is 
 proportional to the average heating in F. The height of line 
 VW shows on the same scale the heating that would be pro- 
 duced in TP by the component NJSfN alone. The relative areas 
 of curves Q and VW for a half cycle, using the X-axis as the base 
 line, show the relative heating of conductor F when used in a 
 converter and in an equivalent direct-current generator. The 
 arrows in Fig. 451 (a) are of some service in obtaining a phys- 
 ical conception of the conditions — if the full line arrows rep- 
 resent the direction of the direct current component of current 
 between the brushes, the dotted arrows represent the direction 
 of the alternating component as the center of the section CD 
 passes from brush to brush. 
 
 Consider now conductor D at the end of section CD ; the rela- 
 tions are set forth in Fig. 451 (c). The alternating voltage 
 line M, marked M l , is as before, but conductor D does not re- 
 verse its direct current until 45° after the reversal in F , so that 
 the direct current curve for D is N X N X N V Then from 0° to 45° 
 of the trail of conductor F from brush B to brush A, the alter- 
 nating current component in D is in the same direction as the 
 direct current. This is shown by the curves and M v The 
 same condition is repeated when D starts its second loop at 180° 
 and at every other half period. The resultant of the direct and 
 alternating currents, in conductor D from —45° to 225°, is given 
 in the curve S'P l SP 1 and the heating curve is TQ X Q X T' Q v In 
 a quarter-phase converter the maximum resultant current in the 
 end conductors of a section is equal to 2 /as shown at S and *S", 
 and the heating is four times as great for the instant as that 
 which would be caused by current /, the heating effect of 
 which is shown by the height of the line VW as before. The 
 average heating in D is nearly as great as though I flowed in 
 it constantly. The condition of conductor C duplicates that 
 of D. 
 
 Conductors lying between F and D or F and C in the section 
 are subjected to heating effects, on account of the P Z R loss, 
 
768 
 
 ALTERNATING CURRENTS 
 
 which are intermediate between those of the edge conductors 
 C and D and the middle conductor F. 
 
 By making the width CD correspond with the angle covered 
 by a section between rings for any other number of phases, that 
 is, for tri-phases 120°, six-phases 60°, twelve-phases 30°, or 
 single-phase 180°, and using the currents determined in Art. 
 176, the relative heating effect in the conductors maybe deter- 
 mined by the foregoing process for any converter. By thus 
 obtaining and adding the heat loss for all the conductors of 
 a section, the average heat loss of the section is determined. 
 From this may also be obtained what may be called the 
 apparent resistance. The sum of the areas of the curves of 
 instantaneous resultant current squared for the conductors, 
 which have been referred to as heating curves, made to proper 
 scale and multiplied by the electrical resistance of the section, 
 gives the energy expended for heat losses in the armature 
 section per half cycle of the alternating current. Dividing 
 
 this by 7 r gives the average power, P', which goes into the heat 
 
 pi 
 
 losses. Then R' A = , where I is one half the current from a 
 
 P 
 
 direct-current brush, and R' A is the apparent resistance of the 
 alternating-current section. The apparent resistance of a direct- 
 ly 
 
 current armature circuit, R" A , is manifestly R' A multiplied by — , 
 
 which is the ratio of the winding arc between the direct-cur- 
 rent brushes to the arc between taps at the ends of an alterna- 
 ting-current phase section. Dividing R" A by 2 p, the number 
 of direct current paths through the armature winding, gives the 
 apparent resistance of the entire armature, or 
 
 NP' 
 
 7 ?" 
 
 7? — 7 A — 
 — 0 — 
 2 p 
 
 4 pi 2 
 
 In general, the heating under the conditions can be found in 
 
 this way : 
 
 I' = 
 
 2V27 * 
 
 where I' and I are the alternating and direct currents respec- 
 tively in an armature section, and N is the number of phases. 
 
 * Art. 176. 
 
SYNCHRONOUS MACHINES 
 
 769 
 
 This formula was derived under the assumption that the rota- 
 tive losses were zero and the angle of lag between the alternat- 
 ing voltage and current was zero. If, however, c per cent of 
 the total current component entering the conductor is used in 
 causing rotation, the left-hand side of the equation must be 
 
 increased by the ratio 
 
 and if the power factor is not unity, 
 
 it must be further increased by the factor — — , when 0 is the 
 
 cos 6 
 
 angle of lag. Hence the alternating current becomes for such 
 a general condition 
 
 I' — 
 
 2V2 I 
 
 N sin r ^~ 
 N 
 
 1 
 
 (1 — c) cos 6 
 
 From the curves in Fig. 451 it is seen that the component of 
 alternating current at any instant for any conductor between 
 commutator bars at a distance 8 from the middle conductor F, 
 when the angle of lag, 6 , is zero, is 
 
 i' c = V2 I' sin (« — 6), 
 
 where a is the angular advance of F from 0°. When the lag is 
 not zero this becomes 
 
 i' c = V2 I' sin (« — 8 — O'). 
 
 The component of direct current at any instant is /, where I 
 is one half the current flowing through a direct-current brush 
 to the external circuit. Therefore the instantaneous resultant 
 current in the conductor is 
 
 i R = I — V2 /' sin (« — 8 — O ') ; 
 
 and substituting the value of I' in terms of I, 
 
 41 1 
 
 ip — T — 
 
 N sin 
 
 7 T (1 — C) COS 0 
 
 ~N 
 
 sin (« — 8 — d). 
 
 Placing ^ • — j — = A , squaring, and integrating 
 
 tit ■ 7 r (1 — c) cos 0 
 
 Jy sin — v ’ 
 
 JSf 
 
 this function of i R with respect to a between the limits tt and 
 
770 
 
 ALTERNATING CURRENTS 
 
 0, and dividing by it to obtain the effective current,* we obtain 
 Ij ? = — 2 V = f f"[l - A sin (« - 3 - 0)] 2 d«, 
 
 7T ^0 7T ♦'0 
 
 or 
 
 i* 2 = / 2 
 
 ^ d 2 _ 4 4 cos (3 + 9 ) 
 
 nr 
 
 When 9 — 0, this is evidently minimum for 3 = 0 or the middle 
 point of the armature section, and maximum for the edges of the 
 section where 3 is a maximum. The heating of the conductor is 
 
 P' = RJ 2 
 
 A 2 j \ _ ^ 4 A cos (3 + 6 ) 
 
 where R c is the electrical resistance of a single conductor of the 
 section. To obtain the mean heating in the conductors of 
 the armature section assume that the single conductors are of 
 negligible width, and then, by integrating the right-hand term in 
 the brackets of the equation with respect to 3 between the limits 
 
 and — — and dividing by the width of the section, 2-^, the 
 mean heat is found to be 
 
 Pr. = RJ 2 
 
 Kf +i )-^4 C Jco f + > ] 
 
 and 
 
 \AN 
 
 • sin — cos 6 ■ 
 N 
 
 Substituting the value represented by A gives 
 
 8 1 
 
 P, = RJ 2 
 
 7V7-2 ' 2 7r (1 — e) 2 COS 2 0 
 
 iv 2 sim — v J 
 N 
 
 + i-SS 
 
 1 ■ 
 
 (1-c) . 
 
 When the rotation losses are considered zero and the power 
 factor is unity, this becomes 
 
 Pr = RJ‘ 
 
 W 2 sin 2 J 
 
 N 
 
 + 1 
 
 16“ 
 
 Trl 
 
 Since R c is the true resistance of the conductor, RJ 2 is the 
 heating per conductor that would occur if the machine were run 
 as a direct-current dynamo with an output per brush of 2 Z and 
 as P c is the mean heating per conductor when the machine gives 
 
 * Art. 5. 
 
SYNCHRONOUS MACHINES 
 
 771 
 
 out 2 1 amperes per direct-current brush when running as a con- 
 verter, the expression surrounded by brackets in the formulas 
 for P c is equal to the ratio of the power lost in heat in the two 
 cases. The rated capacity of a machine is inversely propor- 
 tional to the square root of its temperature rise, and therefore 
 the expression within the brackets may be considered as express- 
 ing the square of the ratio of the capacity of the machine when 
 running as a generator to its capacity when running as a con- 
 verter subjected to sinusoidal voltages. It will be noted that 
 through much the same method of reasoning it is possible to 
 find the ratio of the apparent to the true resistance of the 
 converter armature. 
 
 From the graphical constructions and the last formula it 
 may be observed that the capacity of a converter operated at 
 unity power factor, limited by the rise of temperature of the 
 armature, when compared with the capacity of the same 
 machine used as a direct-current generator, is as follows : single- 
 phase .85 to 1 ; quarter-phase 1.63 to 1 ; tri-phase 1.34 to 1 ; 
 six-phase 1.95 to 1 ; and 12-phase 2.21 to 1. Also, comparing 
 in the same way, the apparent resistances are respectively 1.38 
 to 1 ; .38 to 1 ; .56 to 1 ; .26 to 1 ; and .21 to 1. 
 
 If the small component of active current required to rotate 
 the armature is included in the calculations, the resistance 
 ratios will be slightly increased over those given. Also, if a 
 quadrature current flows in the armature, the increased current 
 will cause a greater I 2 R loss in itself and will cause a further 
 loss by reason of the shifting of the phase position of the 
 current. To solve the problem under those conditions the 
 values of c and 6 must be substituted in the general formula 
 for P c , or if in Fig. 451 a quadrature current flowed, it would 
 have to be combined with components JV and M and by reason 
 of its position would cause greater peaks of current and hence 
 greater heating ; and to obtain the heating effect of the total 
 current of a converter in which the current is out of phase 
 with the generated counter-voltage it is necessary to add fo M 
 an active component equal to that necessary to supply the 
 rotation losses, and a second component equal to the quadrature 
 current. The processes may then be carried through as 
 before. If the impressed and counter voltages are not both 
 sinusoidal, the heating is modified from that computed. 
 
772 
 
 ALTERNATING CURRENTS 
 
 It will be noted both from the formula for P c and Fig. 451 
 that the width of the alternating-current coil section directly 
 affects the capacity of the machine. It is evident then that 
 the type of the winding may materially influence the converter 
 capacity. The figure also shows that the heating is not uni- 
 formly distributed amongst the armature conductors, but that 
 part of each section is more heated than the remainder. 
 
 179. Armature Reactions of a Converter. — A consideration of 
 the basis of the construction of Fig. 451 will show that the mag- 
 netic reactions of the armature upon the field magnet in a con- 
 verter are small when no quadrature current flows. Thus, in 
 the figure, which is for a quarter-phase machine, the effective 
 alternating current component equals and is opposite to the 
 direct current component in any section, so that there can be no 
 resultant reactive effect when integrated through the duration 
 of a cycle. But it is noted that there is none the less a result- 
 ant current flowing of irregular double frequency wave shape 
 and irregularly disposed in the individual coils. This varying 
 current causes an alternate skewing of the field magnetic flux 
 forward and back as the directions of the current alternately 
 predominate in the coils. In a single-phase machine, having 
 alternating current coil sections 180 electrical degrees wide, the 
 heavy current that flows in a small part of an alternating current 
 section of the armature when the alternating currents in the 
 coils near the edge of the section are in the same direction as 
 the direct current, makes the reaction of double frequency 
 especially heavy. By reason of their changing the reluctance 
 of the field magnetic circuit these reactions may set up appreci- 
 able double harmonic vibrations in the current in the direct- 
 current leads. As the skewing is first one way and then the 
 other way, the brushes cannot be shifted to the point of field 
 strength which gives the least sparking, as in direct-current 
 machines. However, even single-phase converters are built 
 which give excellent satisfaction, while the smaller amount of 
 the resultant current variations in polyphase converters make 
 the effects mostly negligible. Further, induced currents in the 
 pole faces of the field magnet or in the amortisseur windings 
 cut down materially the variations in field flux. 
 
 A converter being equivalent to a synchronous motor, in 
 its effect on the alternating current supply circuit, when the 
 
SYNCHRONOUS MACHINES 
 
 “"5V 
 I i U 
 
 excitation is raised beyond the point giving unity power factor, 
 a leading quadrature current is drawn from the line as explained 
 in Art. 170. This is in such a position, as may be seen by 
 studying the relations shown in Fig. 451, that it tends to 
 weaken the field magnet. Indeed, the amount of the quadrature 
 current that flows in any synchronous machine is such that 
 the field magnet will have a strength which will enable the 
 resultant between the impressed and counter voltages to just 
 drive the armature current through the armature impedance. 
 Hence, changes in converter excitation make only small changes 
 in the direct-current voltage, when the impressed alternating 
 voltage is constant. 
 
 When a lagging current is drawn from the line due to under 
 excitation of the converter field magnet, the reactions strengthen 
 the field magnet. In polyphasers, the reaction due to the quad- 
 rature component of current is caused by flux from the armature 
 which is stationary with respect to the field magnet, i.e. the com- 
 bined ampere-turns of the several phases cause the armature flux 
 to revolve with respect to the windings against the direction 
 of the revolution of the armature and at equal speed, thus 
 causing this flux to maintain a fixed angular position with 
 reference to the poles of the field magnet.* 
 
 180. Voltage Control of Converters. Split-pole Converters. — 
 From the discussion of armature reactions in the previous article 
 it is evident that the direct-current voltage is approximately 
 proportional to the impressed alternating current voltage in any 
 particular converter. The direct-current voltage may then be 
 controlled by any method whereby the impressed voltage is 
 controlled, such as by cutting out or in active turns on the 
 transformers feeding a converter, or by using autotransformers 
 in which the secondary voltage may be varied, or using induc- 
 tion regulators f in the leads. 
 
 By placing reactive coils in series with the converter leads, 
 or when the converter is used on lines where there is other in- 
 ductance, the impressed voltage may be varied somewhat by 
 the same processes as where the synchronous condenser is em- 
 ployed.^; An illustration of this method is given in Fig. 447. 
 Or this action may be made automatic by winding a few series 
 compounding turns from the direct-current leads of the con- 
 
 f Arts. 159, 160. J Art. 174. 
 
 * Art. 152. 
 
774 
 
 ALTERNATING CURRENTS 
 
 verter on tlie field spools, and inserting reactances of proper 
 values in the alternator leads. A special alternator may be 
 connected to the converter shaft, with its own field magnet 
 having the same number of poles as the converter, and with 
 its armature coils of each phase connected in series with 
 those on the converter armature, thus composing a Synchro- 
 nous regulator. Such a machine is shown in Fig. 452. 
 The synchronous regulator is shown on the right-hand side of 
 
 Fig. 452. — Synchronous Regulator attached to a Converter. 
 
 the figure. By varying the field strength of the regulator, the 
 voltage delivered to the converter is varied. 
 
 As the direct-current voltage depends on the maximum value 
 and wave form of the alternating counter-voltage, anj r arrange- 
 ment whereby the wave shape of the counter-voltage can be 
 varied may be used to vary the direct-current voltage. Thus, 
 if each field pole is split parallel to the shaft into three narrow 
 poles, all having separate windings, the magnetic field distribu- 
 tion may be made peaked by strengthening the middle pole part 
 relatively to the others, or may be flattened by weakening the 
 
SYNCHRONOUS MACHINES 
 
 775 
 
 center part. If the outside pole parts are strengthened when 
 the middle part is weakened, or vice versa , the direct-current 
 converter voltage may be varied several per cent. When such 
 an arrangement is made, the machine is called a split-pole con- 
 verter. 
 
 As in any direct-current machine, the direct-current voltage 
 of a converter varies with the shifting of the brushes with refer- 
 ence to the field poles. In order to vary the voltage conven- 
 iently by this means, some machines are built with extra poles 
 whereby the magnetic flux instead of the brushes can be shifted. 
 Figure 453 is a diagram of such an arrangement. The auxil- 
 iary I may be strengthened when the poles NS are weakened 
 or vice versa, thus shifting the center of density of magnetic 
 
 Fig. 453. — Flux shifting Auxiliary Poles on a Converter. 
 
 flux. The auxiliary pole piece I is usually somewhat nearer 
 what is intended to be the trailing pole tip, thus leaving a fairly 
 wide space for commutation, and it is not an interpole for com- 
 mutation. 
 
 181. Some Features of Converter Operation. — A converter 
 may be started as a direct-current motor from the direct-cur- 
 rent side, or it may be brought to synchronism as a rotary field 
 induction motor, or again it may be brought to speed by some 
 external mechanical starting device. Except for the first-named 
 method, a converter is started like a synchronous motor, and is 
 paralleled in the same way. Figure 454 shows the connections 
 of two six-phase machines arranged to start as rotary field in- 
 duction motors. A switch for breaking up the length of the 
 shunt field windings is shown at the right hand of each con- 
 verter, for the purpose of preventing excessive voltages being 
 
776 
 
 ALTERNATING CURRENTS 
 
 induced in them during the starting process ; and a switch for 
 short-circuiting the series or compound field winding is at the 
 left hand. The direct-current switchboard panels shown in the 
 
 THREE PHASE 
 INCOMING LINE 
 
 Fig. 454. — Diagram of Connections for Six-phase Converters, including the High 
 Voltage Supply Line, the Transformers with Starting Taps, and the Direct-current 
 Switchboard Panels arranged for Electric Railway Circuits. 
 
SYNCHRONOUS MACHINES 
 
 777 
 
 figure are arranged for railroad work at from 500 to 600 volts. 
 The neutral point of the polyphase circuit is shown ungrounded. 
 Keeping in mind this figure, the following instructions, taken 
 from those issued by one of the large manufacturers of electrical 
 machinery, give an insight into the methods of operating this 
 kind of machines when arranged for self-starting by the alter- 
 nating current : 
 
 Close the oil circuit breaker on the high voltage side of the 
 transformer. Insert the voltmeter plug for the direct-current 
 voltmeter. Close the double-throw starting switch to the 
 starting position. The rotary converter should come up to 
 synchronous speed in about 30 seconds, and lock into step, in- 
 dicating this condition by a steady current on the alternating- 
 current side of the converter, and a continuous deflection of the 
 direct-current voltmeter. When the direct-current voltmeter 
 indicates the correct polarity, close the field break-up switch 
 so as to have the field rheostat in the field circuit. When the 
 direct-current voltmeter indicates reversal of polarity, reverse 
 the field break-up switch, thus reversing the shunt field and 
 connecting it directly across the armature. The voltmeter 
 indicator will swing back towards zero. When it reaches zero, 
 throw the field break-up switch to the lower position. If the 
 voltage now comes up with the right polarity, proceed as 
 directed above. If, however, the converter fails to slip a 
 pole, and the voltage again comes up with reversed polarity, 
 it is evident that the field induced by the alternating currents 
 in the armature is too strong. The starting switch should 
 then be opened for a moment, thus permitting the converter 
 to slow down somewhat. The starting switch should then 
 be closed again in the starting position. When the machine 
 is up to synchronous speed, and the direct-current voltmeter 
 shows correct polarity, throw the starting switch to the run- 
 ning position. Adjust the direct-current voltage to the proper 
 value. Close the series shunt switch (when there is a series 
 shunt circuit). Close the direct-current equalizer and negative 
 switches ; then close the direct-current circuit breaker and the 
 positive switch. 
 
 The following instructions for connecting up rotary converters 
 for the first time after they have been installed come from the 
 same source : 
 
778 
 
 ALTERNATING CURRENTS 
 
 “ Connect the alternating current leads from the same bus 
 bars to similar switches and collector rings as compared with 
 the converters already installed. Place the direct-current brush 
 holder in the same position with respect to the field poles as 
 that on the other converters, and run the positive, negative, and 
 equalizer leads through their respective switches to the positive, 
 negative, and equalizer bus bars of the other converters. See 
 that the field wires are brought out to the corresponding ter- 
 minals of the other converters and connected in the same way. 
 Be sure that the voltmeter lead from the positive terminal goes 
 to the positive voltmeter bus, and the negative to the negative 
 bus. After carefully checking all wiring and seeing that all is 
 clear, start the converter by closing the alternating-current 
 switch in the starting position. If the armature revolves in the 
 wrong direction, shut down and change the alternating-current 
 cables to the converter. If the converter is two-phase, reverse 
 the two leads of either one of the phases, if it is three-phase, re- 
 verse any two leads. After the converter has come to correct 
 speed, proceed as above for self-starting rotaries. In first start- 
 ing a converter it is necessary to synchronize all phases. This 
 can be done in a two-phase converter by placing a bank of lamps 
 equal to the voltage of the converter across the jaw and blade of 
 the switch on each side of each phase. If the converter is three- 
 phase, place a bank across each of the three switches. Now 
 synchronize as before, noticing the lamps. If they all indicate 
 the proper phase relation at the same time, the phases are wired 
 correctly and the switch may be closed at the proper time. If, 
 however, one set of lamps is bright, while the other is dark, it 
 will be necessary to change the main alternating-current leads, 
 either at the converter terminals or at the point of the switch 
 running directly to these terminals. If the converter is two- 
 phase, reverse the leads of either phase ; and if it is three-phase, 
 interchange any two leads. After this change is made, check it 
 by means of the lamps when starting again. After synchroniz- 
 ing the converter, put load on the direct-current circuit and test 
 the series field coil by short-circuiting it and noticing the direct- 
 current voltmeter. If this is in opposition to the shunt field, as 
 shown by the voltage increasing with the series coil short-cir- 
 cuited, reverse the lead connections to these coils. The con- 
 verter should now be ready for paralleling on both sides." 
 
SYNCHRONOUS MACHINES 
 
 779 
 
 The windings of a converter are of the same general type as 
 the windings on direct-current machines with reentrant arma- 
 ture coils. As there are likely to be many sets of direct-current 
 brushes on account of the multiplicity of poles on high fre- 
 quency converters, there are many direct-current paths through 
 the armature winding. An} r inequality of the resistances in 
 Taps to Collector Rings 
 
 Equalizer Connection 
 
 Fig. 455. — Equalizer Connections between Sections of the Armature Windings of 
 
 Converters. 
 
 these paths or in the strengths of the individual magnet poles 
 is apt to unbalance the currents in the paths and cause undue 
 heating. It is thus necessary to use equalizer connections 
 between points of equal potential in the windings, as shown in 
 Fig. 455. 
 
 The direct-current characteristics of a converter are very 
 much like those of a direct-current generator. Figure 456 shows 
 typical curves of regulation, commercial efficiency, and losses 
 for a machine of this type. Since there is little armature 
 
780 
 
 ALTERNATING CURRENTS 
 
 reaction, the field and all other losses except the armature cop- 
 per loss may be considered constant. 
 
 The efficiency of the converter is 
 
 IE 
 
 v Te+p + pr' 
 
 where I and E are the direct current and voltage, P is the fixed 
 power loss, including the field and rotation losses, and R is the 
 apparent resistance of the armature. The efficiency may be 
 measured by placing wattmeter’s in the alternating and direct 
 current leads respectively, and loading the machine to the 
 
 amount and at the power factor desired. Or the losses ma} r 
 be determined by running the motor unloaded at the excitation 
 giving minimum current, when the power input measured by 
 the wattmeters in the alternating-current lines plus the field 
 losses approximately equals P. The apparent resistance may 
 be obtained by measuring the actual resistance of the armature 
 and multiplying by the ratio of apparent to true resistance for 
 the number of phases of the machine tested.* From this the 
 PR losses can be calculated. Otherwise tests of this kind of 
 apparatus conform with that of other electrical machines. 
 
 The efficiencies of converters are higher than the efficiencies 
 
 * Art. 178. 
 
SYNCHRONOUS MACHINES 
 
 781 
 
 of direct-current or alternating-current generators of corre- 
 sponding size, on account of the greater output obtained for the 
 converters with given conditions of loss, as set forth in Art. 178. 
 
 When converters are feeding a three-wire direct-current sys- 
 tem, the neutral conductor of the direct-current side may be 
 connected to the neutral point of the alternating-current side. 
 For a quarter-phase converter, the secondary windings of the 
 supply transformers can be connected to the direct-current neu- 
 tral conductor at their neutral points. For a tri-phase con- 
 verter, the neutral point can be readily obtained from a wye or 
 tee connection of transformers and for a six-phase converter the 
 connections may be conveniently either wye or tee. 
 
 182. Inverted Converters : Double-Current Generators. When 
 a converter is driven by direct-current power and delivers 
 alternating current to the line, it is termed an inverted rotary 
 converter. When so run, the machine has features similar to 
 that of a direct-current shunt motor. Thus, the speed varies 
 inversely as the field strength and directly as the impressed 
 voltage. The alternating-current end acts like an alternating- 
 current generator, a lagging current delivered by the machine 
 causing a weakening of the field magnet, and a leading cur- 
 rent delivered by the machine causing strengthening of the 
 magnet. Therefore, if an inverted rotary finds variable alter- 
 nating inductive loads, the speed and hence the frequency are 
 apt to be dangerously variable. It is therefore common to use 
 an individual shunt-wound exciter for such a machine, which is 
 driven at a speed proportional to that of the inverted converter. 
 The field magnet of the exciter is designed with low magnetic 
 saturation, so that with slight increase of speed the exciter 
 voltage is largely increased, thus sending an increased current 
 through the inverted converter field windings and thereby main- 
 taining approximately normal speech If the armature of an in- 
 verted converter has little reaction on the field, and if the field 
 magnet is highly saturated, very little trouble need be experi- 
 enced. Mechanical and other devices may be used if necessary 
 to control the speed. The distribution of current among the 
 armature conductors and other characteristics for inverted con- 
 verters may be determined in the same manner as that developed 
 in the preceding discussion of the non-inverted converter. 
 
 In the use of inverted converters, and indeed in all alter- 
 
782 
 
 ALTERNATING CURRENTS 
 
 nating-current generating plants, it is desirable to have a Fre- 
 quency indicator attached to the lines. Such an instrument can 
 be made on the principle of a rotary field induction motor, where 
 a pointer is attached to the armature, which is restrained by 
 springs. The torque of the armature varies as the frequency and 
 hence a scale in frequency can be made for the pointer to travel 
 over. The frequency may also be determined by an instru- 
 ment depending upon the vibration of reeds and in other ways. 
 
 Double-current generators are sometimes of use in plants 
 connected to both direct and alternating current systems. The 
 double-current generator differs from the converter in that it 
 is driven by mechanical power and therefore both the alter- 
 nating and direct currents are in relatively the same direction 
 in the armature windings. If curve M in Fig. 451 is reversed, 
 its resultant with JSf gives the current distribution in the arma- 
 ture conductors of a double-current generator when the power 
 factor is unity and equal amounts of power are being delivered 
 to the direct current and the alternating-current circuits. It is 
 evident from the construction of Fig. 451 that reversing curve 
 M must result in a relatively great I 2 R loss, and a further in- 
 vestigation will show a larger I 2 R loss for a given output in a 
 double-current generator than in an equivalent converter for 
 the same output. 
 
 183. The Alternating-Current Motor-generator. — The motor- 
 generator consists of an alternating current motor driving a di- 
 rect current generator or vice versa , or another alternating cur- 
 rent generator, the two machines being mounted on a common 
 bed plate and shaft. In the latter case the generator in commer- 
 cial practice is usually of different frequency from that of the 
 motor, and the machine is called a Frequency changer. A problem 
 of importance is to be met in connecting up and paralleling the 
 latter machines when they are to run in parallel with each other. 
 Suppose, for instance, that the motor of a synchronous motor- 
 generator set is fed in parallel with the motors of other motor- 
 generators with currents having a frequency of 25 cycles per 
 second, and the generator delivers currents in parallel with the 
 generators of the same motor-generator sets, the currents having 
 a frequency of 60 cycles per second. The number of poles on 
 the two machines of each motor-generator set must have a ratio 
 equal to the ratio of their respective frequencies; thus if the 
 
SYNCHRONOUS MACHINES 
 
 783 
 
 25-cycle motor has 10 poles, the 60-cycle generator must have 
 24 poles. With the ratio of frequencies incommensurable, like 
 25 : 60 = 5 : 12, there is much embarrassment to conjointly meet 
 the mechanical and electrical requirements of good commercial 
 machines. Each field magnet must contain an even number of 
 poles, and the number of pairs of poles must be in the ratio of 
 5 : 12. This gives 10 poles and 24 poles for the respective field 
 magnets as the smallest numbers that will fit. The speed of the 
 motor-generator set with these numbers is 300 revolutions per 
 minute. The next polar relation is 20 : 48, which gives a speed 
 of 150 revolutions per minute and no exact ratio intervenes. 
 The choice of two incommensurable numbers like 25 and 60 for 
 the standard frequencies therefore causes inconvenience in this 
 situation. A modification to 24 and 60 would improve the re- 
 lations here and also elsewhere. 
 
 After the motor is running properly in synchronism and the 
 voltages of both motor and generator are right, the generator 
 phases may be out of step with those of the mains to which it 
 is to be connected, and it may become necessary to reverse the 
 polarity of the motor field magnet by the field switch and pos- 
 sibly continue opening or reversing the field circuit a number 
 of times in order that the motor may fall back successive angles 
 of 180 electrical degrees until the counter-voltages in the motor 
 and the terminal voltages in the generator hold the proper step 
 relations coincidently in their respective circuits ; or the re- 
 quired step relations may sometimes be obtained by reversing 
 the polarity of the generator field magnet. Even when the 
 best practicable point is reached, a large synchronizing current 
 may flow if the motor-generator does not have quite its designed 
 ratio of frequencies, i.e. 25 to 60, or if the angular relations of 
 the parts of the motor and the generator are not adjusted to ac- 
 curately give the proper time phase between their two voltages. 
 
 When a motor-generator set is being brought into parallel 
 operation with others, it must be brought up to speed (the 
 motor usually starting as an induction machine) and the motor 
 be connected to the supply circuit when the indications of the 
 synchroscope are favorable. The generator is then in syn- 
 chronism, but the synchroscope must still be used to show 
 whether the generator is in proper step with the generator bus 
 bars before it is connected to its circuit. 
 
CHAPTER XII 
 
 ASYNCHRONOUS MOTORS AND GENERATORS 
 
 184. Asynchronous Motors and Generators Defined. — Motors 
 called Synchronous are those in which the moving parts rotate at 
 such a velocity with reference to the fixed parts that an angular 
 distance equal to that of the polar pitch is passed over during 
 the time of a half cycle of the impressed alternating voltage, i.e. 
 the motor runs in Synchronism with the generating apparatus 
 which drives it, and it has the same speed for all loads pro- 
 vided the impressed voltage is of constant frequency. 
 
 Motors which do not have this characteristic, but vary the 
 relative speed between the fixed and movable parts more or 
 less independently of the frequency of the impressed voltage, 
 are called Asynchronous motors. Alternating current Induction 
 motors, Repulsion motors, Series motors, and others in which 
 there is no magnetic field set up by direct current are usually 
 of the latter class. Asynchronous generators are similar to 
 asynchronous motors in possessing no definite relation between 
 the armature speed and main circuit frequency. 
 
 185. Rotary Field Induction Motors. — The well-known prin- 
 ciples which cause the rotation of a disk of copper pivoted 
 above a rotating horseshoe magnet have been put into use 
 through the discoveries of Ferraris, Tesla, Hasel wander, Dob- 
 rowolsky, and many others. The arrangements proposed by 
 Tesla were doubtless the first direct applications of these prin- 
 ciples to commercial use, in which they now play a primary part 
 in the transmission and distribution of power.* An almost 
 simultaneous publication of a series of scientific experiments by 
 Ferraris shows the operation of similar apparatus, f and various 
 experiments of a similar nature or for a similar purpose are on 
 
 * A New System of Alternate-Current Motors and Transformers, Trans. 
 Amer. Inst. E. E., Vol. 5, p. 308. 
 
 t Electro-dynamic Rotation by Means of Alternating Currents, London Elec- 
 trician , Vol. 21, p. 86. 
 
 784 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 785 
 
 Fig. 457. — Apparatus for setting 
 up a Rotating Magnetic Field. 
 
 record. Each of these experiments caused an iron or copper 
 armature to rotate when placed within the region of a rotating 
 magnetic field, and these are the foundation of the now well- 
 known asynchronous induction motors. 
 
 186. A Rotating Magnetic Field. — If two pairs of coils are 
 placed at right angles on a laminated iron ring (Fig. 457), 
 with the connections so arranged that the coils of each pair are 
 in magnetic opposition in the ring, or again, if the two pairs of 
 coils are placed on two pairs of salient poles, or embedded in 
 the inside face of the ring, so that 
 the current in each pair of coils 
 tends to send magnetic flux directly 
 across a central core (7, the magnetic 
 flux set up in the core when a cur- 
 rent is passed through the coils may 
 be considered as the resultant of the 
 magnetization due to the two coils 
 or pairs of coils. As in a transformer, 
 if the resistance of the coils is rela- 
 tively small, the magnetic flux 
 threading those of each phase when alternating currents flow 
 in them must be such that it will create a counter- voltage 
 in the coils practically equal and opposite to the impressed 
 voltage. This must always be the case, whatever may be 
 the form of the impressed voltage wave ; consequently, if 
 the voltage wave is sinusoidal, the wave of magnetic flux 
 within the coils must be sinusoidal, because its rate of change 
 is sinusoidal. On the other hand, the wave shape of the 
 exciting current takes the form requisite to set up the re- 
 quired core flux, as in transformers,* and its wave form is, 
 therefore, dependent upon the variation in the reluctance of 
 the magnetic core caused by saturation and hysteresis, f The 
 resultant flux in any fixed direction across the central core 
 0 will vary in value as in the common core of a quarter-phase 
 transformer, but the special arrangement of the magnetic cir- 
 cuit causes it also to vary in direction in case the currents 
 in the two circuits are not in the same phase. Considering 
 a more or less conventional case where four coils, as in Fig. 
 457, are supposed to uniformly cover the whole ring, which is 
 
 t Art. 126. 
 
 * Art. 118. 
 
78G 
 
 ALTERNATING CURRENTS 
 
 equivalent to an elementary arrangement in a rotary field induc- 
 tion motor winding, the magnetic reluctance of the ring and 
 core C being assumed constant for all paths the flux may take 
 parallel to the plane of the paper, the following propositions 
 are approximately true. 
 
 If two equal sinusoidal voltages 90° apart in phase are im- 
 pressed upon the respective equal pairs of coils on the ring of 
 Fig. 457, the exciting currents must be sinusoidal to create the 
 required sinusoidal magnetic flux, the magnetic reluctance being 
 assumed to be constant, and the magneto-motive forces must, 
 therefore, be sinusoidal. Under these conditions at any 
 instant, the magnetizing force due to one pair of the coils is 
 H, = H m sin «, and the magnetizing force due to the other pair 
 of coils is, 
 
 = H m sin (a — 90°) = — H m cos a, 
 
 where H m is the maximum magnetizing force of either pair of 
 coils. Thus the two magnetizing forces differ in phase by 
 90°, and the resultant magnetizing force of the two pairs of 
 coils is then //, = Vi/j 2 + H 2 2 = H m , as illustrated in Fig. 458, 
 and is, therefore, constant in magnitude. The direction in 
 which this constant magnetizing force acts across the core 0 
 varies with «. When a — 0°, H, tends to lie in the plane of one 
 pair of coils, and when « = 90°, it tends to lie in the plane of 
 the other pair of coils. The magneto-motive force of each pair 
 of coils has a sinusoidal or harmonic variation, and the resultant 
 magnetizing force is the resultant of two harmonic variations 
 with 90° difference of phase. Such a resultant tends to have a 
 uniform magnitude and a uniformly rotating direction in one 
 plane. The instantaneous values of the resultant may, there- 
 fore, be diagrammatically represented, with approximate ac- 
 curacy under the conditions assumed, by the instantaneous 
 positions of a line of fixed length, rotating at a uniform rate 
 around one end, such as OH r in Fig. 458. In this figure the 
 vector of magneto-motive force H x of one pair of the coils of 
 Fig. 457 is considered to lie in the line AB , and to vary sinus- 
 oidally in magnitude, while the magneto-motive force R 2 of 
 the other pair of coils lies in the line CD , and also varies sinus- 
 oidally in magnitude. The combination of these two magneto- 
 motive forces in space causes the resultant R r , which as ex- 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 787 
 
 plained is, under the conditions, approximately constant in length 
 and of uniform coplaner rotation. In Fig. 457 the small 
 arrows show the direction of the magnetic fluxes at the instant 
 when the currents, and hence the magneto-motive forces, are 
 equal in the two branches of the two-phase circuit. As the 
 current in the coils of one phase dies out, and the other in- 
 creases to a maximum, the flux in C swings around 45° on the 
 center of C as an axis. When that point is reached, the current 
 is zero in one pair of coils, and the other pair furnishes the 
 entire flux in O. As the current now rises in the reverse 
 direction in the first pair of coils and falls in the other pair, 
 the flux in C is still caused to rotate by virtue of the rotating 
 direction of the resultant magneto-motive force. In this way 
 the direction of the flux in C uniformly rotates through 860° 
 during the time of each cycle of the current. This may be 
 readily illustrated by a series of diagrams similar to those 
 shown in Fig. 457, in which the magneto-motive forces are 
 represented by arrows for the range of angular advances during 
 a cycle of the currents. The arrows must be reversed in direc- 
 tion in a pair of coils when the current reverses, and should be 
 made approximately proportional in length to the instantaneous 
 currents. The instantaneous current for each value of a may 
 be taken from a pair of sine curves, drawn with one displaced 
 90° from the other. Y 
 
 If the maximum value 
 of the ampere-turns of 
 one pair of coils is greater 
 than that of the other 
 pair of coils instead of 
 the two being equal, as 
 above assumed, the mag- 
 nitude of the resultant 
 magnetizing force varies. 
 
 The rotating field, in this 
 case, may be diagram- 
 matically represented by 
 a uniformly rotating line, 
 which varies in length 
 so that its tip approxi- 
 mately traces an ellipse 
 
 Fig. 458.— Locus Diagram of a Uniform Constantly 
 Rotating Magneto-motive Force created by Two 
 Equal Harmonically Vibrating Magneto-motive 
 Forces spaced 90 Mechanical Degrees Apart. 
 
788 
 
 ALTERNATING CURRENTS 
 
 whose minor and major axes are respectively in the planes of 
 the stronger and weaker coils. If the windings of the coils 
 are similar, and the currents equal, but the phase difference is 
 not 90°, a variable field again results. 
 
 If the phases of the two currents are in unison instead of in 
 
 quadrature, l 
 
 H r = VT^ 2 + H* = V2 H m sin «. 
 
 This shows that when the two currents are in unison the mag- 
 nitude of II r varies with sin «, and therefore varies from 
 — V2 H m through zero to + V2 H m , hut the direction of H r re- 
 mains constant, since the instantaneous values of its two com- 
 ponents are always equal. Its direction evidently lies in a line 
 corresponding to the arrows in Fig. 457 (or 90° therefrom de- 
 pending upon the connections) if the two currents are equal, 
 and turned more or less from that position if the two currents 
 are unequal. The diagrammatic representation of the resultant 
 here is a line of fixed direction which harmonically varies in 
 length, the total range of variation being from — V2 H m to 
 
 + V2 X- 
 
 For any difference of the current phases between zero and 
 
 90°, both the magnitude and 
 direction of H r vary, and the 
 diagrammatic representation 
 is a rotating line with its tip 
 approximately tracing an el- 
 lipse. The ratio of the two 
 axes depends upon the phase 
 difference of the currents. 
 If the currents have 90° phase 
 difference, but the planes of 
 the coils are not 90° apart, 
 the effect on the resultant 
 magnetizing force is evidentlv 
 the same as if the conditions 
 were reversed. 
 
 If the impressed voltages are not sinusoidal, the value of the 
 resultant magnetizing force H r varies in a more or less irregular 
 manner, inasmuch as the different harmonics combine in differ- 
 ent ways. For instance, in a quarter-phase circuit the third 
 harmonics of the two magneto-motive forces produce a result- 
 
 A' 
 
 Fig. 459. — Locus of Rotating Field Vector 
 when the Exciting Current Wave is of 
 Distorted Form. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 789 
 
 / 
 
 1 
 
 \ \ 
 
 ! o 
 
 v 
 
 / / / 
 
 ant that rotates in the opposite direction from the rotation of 
 the resultant produced by the fundamentals, and the rotation is 
 three times as fast. The fifth harmonics produce a resultant 
 which rotates in the same direction as the resultant produced by 
 the fundamentals, but rotates five times as fast. The heavy line 
 in Fig. 459 illustrates the locus of the tip of the rotating field 
 vector produced by two mag- 
 neto-motive forces in time 
 and space quadrature, when 
 each consists of a funda- 
 mental and a third harmonic, 
 the latter located to cause 
 peakedness of the wave of B' 
 magneto-motive force. Fig- 
 ure 460 shows a correspond- 
 ing locus when the third 
 harmonic causes a flattening 
 of the top of the wave of 
 magneto-motive force. The 
 value of flux for a = 0°, 90°, 
 
 180°, and 270° is alike for the 
 two figures, and the dotted circles show the locus for sinusoidal 
 magneto-motive forces. 
 
 The same argument may be readily seen to apply to the re- 
 sultant magnetizing force due to any number of coils surround- 
 ing a core. Thus, an arrangement similar to that of Fig. 457, 
 except that it is applicable for obtaining a rotating field from 
 a tri-phase circuit, may be constructed by winding three coils 
 upon the ring and property connecting them to the three phases 
 of a triphase supply circuit. When equal coils are at equal 
 angular distances, and equal currents in the individual coils 
 differ in phase by an amount equal to the angular distances of 
 the coils from each other, the resultant magnetizing force is 
 always approximately uniform in magnitude under the condi- 
 tions assumed, and rotates at a uniform rate, provided the 
 currents are sinusoidal. The magnitude of the resultant is 
 
 ib In-iF;, 
 
 H r — 2 H m , where m is the number of phases, in accordance 
 
 with the theorems of simple harmonic motion. The correctness 
 of these deductions has been proved by experiment. 
 
 Fig. 460. — Locus of Rotating Field Vector 
 when the Exciting Current Wave is of Dis- 
 torted Form. 
 
790 
 
 ALTERNATING CURRENTS 
 
 The Germans call the rotating magnetic field Drehfelde, and 
 the polyphase currents which set up a rotating magnetic field 
 the Drehstrom, or rotating current. In the actual rotating field, 
 the magnetic flux does not necessarily vary with absolute con- 
 stancy in accord with the propositions given above, but it varies 
 even if the windings are uniformly distributed. This variation 
 is dependent upon the variation of the instantaneous values of 
 magneto-motive force and reluctance of each elementary width 
 of the magnetic circuit through the central core. Indeed, it is 
 possible for the resultant magnetic flux to vary in instantaneous 
 value as much as 40 per cent, as it rotates, even in the case of 
 properly distributed coils and balanced voltages, when there is 
 no closed secondary circuit acted on by the flux. The extent 
 of this variation depends upon the number of phases of the 
 windings and other variables. However, when secondary cir- 
 cuits on the central core are closed, the reactions are found to 
 smooth out the irregularities that occur, and the wave of result- 
 ant magnetic flux may be considered approximately sinusoidal 
 in distribution and of approximately constant value in magni- 
 tude and rate of rotation in those cases in which care has been 
 exercised to properly place the windings on the core, and when 
 approximately sinusoidal impressed voltages are used. 
 
 187. Action of a Short-circuited Armature Winding within a 
 Rotating Field. — If a drum core of laminated iron is suitably 
 pivoted within a ring on which coils carrying alternating cur- 
 rents are so situated that the resultant magnetic field rotates, 
 the pivoted core will be dragged into rotation by the magnetic 
 pull. If the pivoted core is a cylinder of copper, it will be 
 dragged into rotation by the reactions between the rotating 
 magnetic flux and the eddy currents which are developed in 
 the cylinder. The latter is directly analogous to the classic 
 experiment of Arago with a disk of copper pivoted before the 
 poles of a rotating permanent magnet. 
 
 In the case of either a solid core or Arago disk, the eddy 
 currents are not constrained in position and therefore take the 
 paths of least resistance. The result is that much of the effec- 
 tiveness of the currents in bringing about a rotation is lost, and 
 the efficiency of the device is small. If, in the disk experi- 
 ment, the disk is cut up into an indefinitely large number of 
 fine radiating wires which are connected together at their inner 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 791 
 
 and outer ends, the losses due to parasitic eddies may in a large 
 measure be done away with, and the efficiency of the device is 
 considerably raised. In the same way the drum core may be 
 made of laminated iron in order that the magnetic circuit shall 
 be of small reluctance, and embedded in this may be copper 
 wires which cross the face of the core and are all short- 
 circuited by copper 
 rings at the ends 
 (Fig. 461), making a 
 cage of conductors 
 like a squirrel cage. 
 
 These make con- 
 strained paths for 
 the induced currents, 
 and, if the core is 
 Sufficiently laminated F I( j. Til. — Induction Motor Armature having a Short- 
 , ,i circuited (Squirrel Cage) Winding. 
 
 and the copper con- 
 ductors are not too thick, the parasitic eddies are largely done 
 away with and the efficiency of such a motor may be made quite 
 large. This construction is the essential construction of what 
 are known as induction motors. 
 
 There has been more or less ambiguity in applying the desig- 
 nations of field and armature windings to the primary and sec- 
 ondary windings of induction motors, since either may rotate, 
 and both windings carry alternating currents. The following 
 definitions avoid all ambiguities in simple machines. The Field 
 magnet is the core upon which are placed windings connected to 
 the external circuit. The currents in the field windings are, 
 therefore, due to the impressed voltage of the external circuit. 
 The Armature is the part carrying conductors in which current 
 is induced by the revolving magnetism of the field magnet. 
 Since the current is inductively set up in the armature conductors, 
 these motors are called Induction motors. It is readily seen that 
 the induction motor is a transformer as well as a motor, the pri- 
 mary winding of which is on the field magnet, and the secondary 
 winding on the armature, and it is convenient to call the windings 
 receiving power conductively from the power supply circuit the 
 Primary windings, and those in which current is induced the 
 Secondary windings. The part that rotates in any alternating 
 current machine having a rotating part is often called the 
 
792 
 
 ALTERNATING CURRENTS 
 
 Rotor, while that which is stationary is called the Stator, irre- 
 spective of which is the primary or secondary part. 
 
 188. Counter-voltage induced in Primary Circuit. Exciting 
 Current. — Treating the primary winding of an induction motor 
 in the same manner as the primary winding of a transformer, — 
 which is permissible, — the following formula* gives the rela- 
 tion between counter-voltage, frequency, magnetism, and the 
 turns of the coils : 
 
 E' 
 
 1 1 A8 
 
 in which E\ is the induced voltage in the coil, n x the number 
 of turns in the coil, $ the maximum flux including the leakage 
 flux from one pole of the magnetic field, f the frequency of the 
 magnetic cycles, provided all the magnetism is included within 
 all the turns of the winding. This proviso, however, must be 
 modified in an ordinary induction motor, since the magnetic 
 density in the air gap may be assumed to vary as a sinusoid,! 
 and that condition requires that the number of lines of force 
 passing through the different turns of the coils shall also vary 
 as a sine function. This is illustrated in Fig. 462. In this 
 figure, overlapping distributed polyphase windings are laid in 
 slots in the internal face of the field magnet ; and polyphase 
 currents of a corresponding number of phases flowing in the coils 
 produce a resultant rotating magnetic flux distribution in the 
 air gap between field magnet and armature similar to that 
 shown in the figure for a particular instant by the dotted lines 
 in the air gap. The figure pertains to a four-pole field magnet. 
 
 Secondary windings are laid in slots in the external surface 
 of the cylindrical armature core, and both primary and second- 
 ary windings are laid down so as to uniformly cover the 
 winding surface of the core on which it is laid, as is usual in 
 commercial practice. The broken lines in the polar space and 
 the curves indicate the approximate magnetic distribution. 
 Consequently we have for the induction motor, 
 
 - 1 I cos uda 
 
 1 10 s 6 
 
 e . 
 
 where ^ is the value of « corresponding to the sine ordinate 
 
 * Art. 135. 
 
 t Art. 186. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 793 
 
 which is proportional to the number of lines of force passing 
 through the extreme turns of the winding concerned, when the 
 center of the magnetic field is exactly over the center turns of 
 the coil. 
 
 Fig. 462. — Diagram showing Primary and Secondary Windings of an Induction 
 Motor evenly distributed over Internal aud External Surfaces of Field Magnet 
 and Armature Core respectively, and also showing the Approximate Distribu- 
 tion of Magnetic Flux for a Given Instant. 
 
 For uniformly distributed windings in two phases, 9 = 90°; 
 and in three phases, 9 = 60° ; while for coils on salient poles 
 9— 0°. The values of E' for the field winding of the induction 
 motor then become 
 
 4 n^f 
 
 E' „ = and E' , 
 
 1,2 L3 
 
 _ 3V2 n&f 
 
 10 8 
 
 where E\ 2 is the induced voltage in the coils of one phase of 
 a quarter-phase motor field winding and E\ 3 is the induced 
 voltage, likewise, for a tri-phase motor field winding. Hence, 
 the voltage set up in a uniformly distributed field winding of a 
 
794 
 
 ALTERNATING CURRENTS 
 
 quarter-phase motor is, other things being equal, about 10 per 
 cent less than if the windings were in narrow coils ; and in a 
 tri-phase motor the deficit is nearly 5 per cent. The exact 
 ratios are 7 r : 2v/2 and 7r : 3. To give the same counter vol- 
 tage in the uniformly distributed field windings of an induction 
 motor arranged for two phases, requires about 6 per cent more 
 turns in the windings than when the same machine is arranged 
 for three phases. If the windings are placed on salient poles, 
 as was at one time done in quarter-phase motors, all the lines 
 of force pass through the windings, and the coils therefore act 
 as though they were very narrow; but the increased and irregu- 
 lar reluctance of the magnetic circuit caused bj r this construc- 
 tion more than destroys any advantage for ordinary motors per- 
 taining to this form of the winding. 
 
 The formulas given above may also be derived by consider- 
 ing the primary windings as stationary windings of an alternat- 
 ing-current generator with rotating field magnet in which the 
 magnetic flux from the poles has approximately a sinusoidal 
 distribution. For some purposes this method is useful.* It 
 is evident that the voltage set up in the windings by the rotat- 
 ing magnetism is exactly the same whether the magnetic flux 
 moves with the field core as in a synchronous machine or moves 
 through the core by reason of the varying resultant of two or 
 more magneto-motive forces as in the induction motor. The 
 voltage induced in a conductor depends, in other words, upon 
 the relative angular velocity of conductor and magnetic flux, 
 and it makes no difference how the relative velocity is ob- 
 tained. Therefore, the generator formulas apply to this case, of 
 
 2 7r/q<I> VP 
 10 8 x 60 
 
 sin a, f 
 
 in which e\ is the instantaneous induced voltage in the coil, V 
 the revolutions per minute, « the instantaneous angular position 
 of the coil in the magnetic field, and the flux per magnet 
 pole. In the case under consideration, 
 
 ^ _ 2 2 7 rrlcf) _ 2 rl<f> 
 
 o ’ 
 
 7T 2|) p 
 
 * Art. 20. 
 
 t Blondel, Notes sur la tlieorie dl^mentaire des appareils k champ tournant 
 La Lumiere Vlectrique, Vol. 5, p. 351 ; Jackson, Three-phase Rotary Field. 
 Electrical Journal, Vol. 1, p. 185. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 795 
 
 where r and l are the inner radius of the field core and its 
 length parallel to the motor shaft, <£ the maximum magnetic 
 density in the air space, and p the number of pairs of poles in 
 
 V f 
 
 the magnetic field. Since — = — , and the magnetism per 
 
 60 p 
 
 magnetic circuit in a multi-polar machine must be multiplied 
 by the number of pairs of poles to get the total number of lines 
 of force cut per conductor per revolution, the formula may be 
 written, generally, in the form 
 
 e\ = 
 
 _ 2 Trnp$>f 
 
 10 8 
 
 sin a ; 
 
 whence 
 
 e' m — 
 
 _ 2 7T» p \)f 
 
 10 8 
 
 •> 
 
 and 
 
 E\ = 
 
 1 1IJS 
 
 which is as in the transformer. This is the value of the vol- 
 tage developed in a narrow coil, but if the coil is spread over 
 a considerable area, the maximum voltage is less, as shown 
 earlier in the article. 
 
 For various reasons it is more convenient to stud}' induction 
 motors from the transformer standpoint, and we may consider 
 them as transformers with relative motion between the pri- 
 mary and secondary windings. 
 
 Since an air space must be made in the magnetic circuit to 
 allow such a motor to operate, it is evident that the quadrature 
 element of the exciting current of induction motors must be ma- 
 terially greater than that of ordinary transformers. In fact, 
 the no load current of some comparatively small motors of this 
 type, which show quite a high full load efficiency, is entirely 
 comparable to the full load current. To reduce the quadrature 
 current to a reasonable limit, every effort must be bent to 
 decrease the reluctance of the air space. As the armature 
 conductors are embedded in the primary and secondary cores, 
 it is possible to make the air space simply that required for 
 mechanical clearance ; and, by care in the workmanship, this 
 may be made quite small compared with the air space of 
 dynamos built according to the ordinary methods. 
 
796 
 
 ALTERNATING CURRENTS 
 
 The exciting current for an induction motor may be calcu- 
 lated for each circuit in exactly the same manner as that for a 
 transformer. It is composed of two components in quadrature : 
 
 (1) The active component, which is equal to the sum of the no- 
 load losses in the circuit measured in watts, divided by the volts 
 per circuit. The fixed losses, as in the transformer, vary but 
 little from no load to full load in well-designed motors. The 
 number of circuits is equal to the number of phases. The 
 total losses entering into the exciting current (or the no load 
 losses) are the core losses in the field magnet and armature ; 
 the I 2 R loss in the field winding which is due to the exciting 
 current ; a small I 2 R loss in the armature conductors which is 
 due to the secondary current required to run the armature 
 against friction and armature core losses ; and the friction loss. 
 The watts represented in the exciting current per electrical 
 circuit are equal to the total no load losses divided by the number 
 of phases. 
 
 (2) The magnetizing ampere-turns, which are in quadrature 
 with the active component of the exciting current, may be 
 calculated, as in the case of transformers, from the formula 
 
 nl— 
 
 
 1.25 
 
 , where P is the reluctance of the magnetic circuit 
 
 and nl represents the resultant ampere-turns. But the mag- 
 
 2 
 
 neto-motive force per phase is equal to — times the resultant 
 
 m 
 
 magneto-motive force.* Therefore, if n x is the number of 
 turns per phase which link each magnetic circuit, the actual 
 magnetizing component of current per circuit is 
 
 T Px$ 2 
 
 Ifj. — “= X , 
 
 rij x V2 x 1.25 m 
 
 where m is the number of phases. Which for a quarter-phase 
 machine becomes approximately 
 
 T _ P4> 
 
 " 2 ~ 1.75 V 
 
 and for a tri-phase machine, approximately 
 
 T = P($> 
 
 ^ 2.65 n x 
 
 * Art. 186. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 797 
 
 Since mechanical clearance between the armature and field mag- 
 net is an essential feature of a motor, the reluctance of the motor 
 magnetic circuit is much greater than that of a transformer of 
 corresponding capacity. This makes the magnetizing current 
 greater, as stated, increases the exciting current, and reduces 
 the power factor. 
 
 The total exciting current is equal to the square root of the 
 sum of the squares of its two components, or I e — V I c 2 + i^ 2 , 
 where I c is the active component of the current. 
 
 The ampere-turns on each magnetic circuit of an induction 
 motor are the resultant of the ampere-turns due to all the 
 phases. It is therefore evident, from previous deductions, that 
 the resultant ampere-turns in the magnetic circuit of an induc- 
 tion motor is --(nl^ 2), where n is as before the number of 
 
 turns belonging to each phase which link each magnetic circuit, 
 m is the number of phases, and I e is the no load or exciting 
 current in each phase.* Consequently if y ampere-turns are 
 required in the magnetic circuit, the winding in each phase 
 2 
 
 must furnish — times the whole, as was assumed earlier in the 
 to 
 
 paragraph in obtaining a value for 1^. For quarter-phase 
 
 2 2 2 
 
 motors, — = 1, and for tri-phase motors, — = -• 
 to mo 
 
 189. Motor Speeds, Slip, and Secondary Induced Voltage. — 
 
 The velocity of rotation of the resultant primary magnetic flux 
 depends upon the frequency of the current supplied to the 
 motor, and the number of pairs of poles in the field. In two- 
 pole machines, the number of rotations which the magnetic 
 flux makes per second, or the Field frequency, is equal to the 
 frequency of the primary voltage ; and, in multi-polar machines, 
 the field frequency is equal to the frequency of the primary 
 
 f 
 
 voltage divided by the number of pairs of poles, or — . The 
 
 P 
 
 number of pairs of poles which is referred to by the symbol p is 
 the number in the rotating magnetic field. This is equal to the 
 number of pairs of poles set up by the windings in the primary 
 
 cores with a smooth magnetic surface, but is equal to times 
 
 * Art. 186. 
 
798 
 
 ALTERNATING CURRENTS 
 
 the number of polar projections which would be necessary if 
 each phase coil was wound upon a projection, though in this 
 latter case the number of resultant magnet poles is the same as 
 where the smooth core is used. 
 
 The velocity of rotation of the secondary winding of an 
 induction motor can never equal the velocity of rotation of the 
 rotating magnetic flux (when the machine is running as an 
 induction motor), since the secondary conductors must be cut 
 by the lines of force of the rotating flux in order that voltage 
 may be developed in the armature conductors ; that is, the 
 rotating magnetic field must always have a relative velocity of 
 rotation with reference to the armature conductors. In any 
 machine, the relative velocity given in terms of revolutions per 
 minute is v—V— I 77 , where V and V are respectively the 
 number of revolutions per minute of the magnetic field and the 
 secondary conductors. This relative velocity measured as a 
 fraction of the rotating field velocity is called the secondary 
 Slip, and is small, seldom exceeding 5 or 10 per cent of the 
 speed of the motor for ordinary so-called constant speed 
 machines, and reaching these values only in small motors. 
 Slip is usually for convenience named as a fraction or per- 
 centage of the synchronous velocity V oi the machine, in which 
 
 case slip is s 2 = -p ; and as in a machine with a given number of 
 
 poles, the frequency of the current in the primary conductors 
 
 must be to the frequency in the secondary conductors as — * 
 
 f V 
 
 therefore s 2 = where f and / 2 are the two frequencies, re- 
 spectively. Since the current in the secondary winding must 
 be proportional to the work done by the motor, it must vary 
 with the load, if the secondary impedance is constant, and v 
 must increase as the load is increased. A little consideration 
 shows that, if the rotating field flux remains constant in value, 
 the variation of v with the load must be just sufficient to coun- 
 terbalance the drop of the secondary voltage caused by the 
 current flowing in the secondary conductors. 
 
 A variation of v demands a variation of V of equal magni- 
 tude, since J^is fixed by the frequency of the current delivered 
 to the motor ; consequently, the speed regulation of a rotary 
 field motor is directly dependent upon the loss of voltage in the 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 799 
 
 secondary conductors, if we neglect the effect of variable arma- 
 ture reactions and drop of voltage in the primary windings. 
 This is analogous to the case of direct-current shunt -wound 
 motors. 
 
 At starting, the relative velocity of the rotating field flux 
 with respect to the secondary conductors is evidently V, since 
 V is zero. The secondary current therefore may be very 
 great, and the starting torque may also be great provided the 
 armature reactions do not too greatly disturb the phase posi- 
 tion of the rotating magnetic flux. To avoid injury to the 
 secondary windings, in large motors, by the current at starting, 
 means must be taken to prevent its becoming excessive, exactly 
 as in the case of direct-current motors worked on constant 
 voltage. 
 
 From the relation V — F 7 = v, it is evident that the value of v 
 determines the frequency with which the rotating magnetic flux, 
 <F, cuts the secondary conductors, and it will vary from a value 
 v = V to v = 0 as the motor armature changes from a condition 
 of rest to a speed of synchronism. Slip is frequently named 
 in per cent of motor speed. In ordinary practice, the slip 
 varies, at full load, from a fraction of 1 per cent to 10 per 
 cent of V, having the smaller value in large machines. 
 
 As in a transformer, the voltage induced in the turns of the 
 secondary winding when at zero speed (7 r = 0) by the mutual, 
 rotating flux, is the same per turn as that induced per turn in 
 the primary winding, and its value is 
 
 r, V2 t r An^f 
 
 ^ 2 - 1(J 8 
 
 where <1? is the flux passing through botli primary and secondary 
 coils, i.e. the mutual flux per magnetic circuit, and A. is a 
 factor depending on the width of the coils per phase, being 
 unity for very narrow coils and less than unity when the coils 
 are distributed over material width, as explained in the previous 
 article. w 2 is the number of secondary turns and f the frequency 
 of the primary supply circuits. In a squirrel cage armature 
 the conductors within twice the width of the polar pitch may 
 be conveniently considered as composing one turn which is 
 wound of parallel conductors distributed over 180 electrical 
 degrees of the core surface and wound under two contiguous 
 
800 
 
 ALTERNATING CURRENTS 
 
 poles of opposite sign. For this case A becomes equal to — . 
 
 7T 
 
 Under these circumstances the formula becomes 
 
 -pi 2V2 n 2 <bf 
 
 E >= 10 » • 
 
 For other armature speeds than zero, the voltage evidently be- 
 comes s 2 E v as the rate at which the secondary conductors are 
 cut by the rotating field is proportional to s 2 , i.e. at standstill 
 the secondary voltage is proportional to the speed V, while 
 
 when running it is proportional to the speed v, but s 0 = — , 
 
 hence for any speed the secondary voltage is s 2 E 2 . E 2 , it will 
 
 be noted, is the standstill secondary voltage. 
 
 It is sometimes more convenient to know the voltage for a 
 conductor, rather than the resultant voltage for all the conduc- 
 tors in series or parallel; in such case the instantaneous maxi- 
 mum voltage is, for any slip s 2 , 
 
 _ 2 7rd>/s 2 _ 2 TrA>pv 
 
 e ‘lmc- 108 ~ 1Q 8 x 60 ’ 
 
 where <E> is the maximum value of the mutual rotating flux per 
 pole. The effective voltage per conductor is 
 
 p _ V2 7rfl>/s 9 _ V2 irA>pv 
 2c ~ 10 8 ~ 10 8 x GO ’ 
 
 since the voltage curves in the conductors must he sinusoidal 
 if the magnetism has a sinusoidal distribution in the air space 
 and the velocity is uniform, as has been assumed. 
 
 190. Currents, Torque, Impedance, and Magnetic Leakage in a 
 Rotary Field Induction Motor. — The reactance of an induction 
 motor is that due to the magnetic leakage between the primary 
 and secondary windings, as in a transformer. The reactance 
 due to one phase of the secondary windings when the armature 
 is at a standstill and the slip equals unity is 
 
 X 2 = 2 7 rfL v 
 
 where L 2 is the self-inductance of the secondary winding due to 
 leakage flux ; and where s 2 equals any fractional value other 
 than unity, X' 2 = 2 i rf 2 L v but f 2 = s 2 /, and 
 
 X\ - s. 2 X 2 = 2 t rfL 2 s 2 . 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 801 
 
 Therefore, the impedance of the secondary winding for any 
 slip is 
 
 ^2 = ( + S 2 2 X 2 2 )K 
 
 where B 2 is the resistance of the secondary winding. The cur- 
 rent which flows in the secondary winding is then 
 
 J __ S 2^2 _ ,S 2^2 TT-An 2 <j>fs 2 
 
 2 ^2 mZ 2 
 
 The symbols for current, resistance, reactance, impedance, 
 etc., have the same subscripts and meaning as used when in 
 dealing with the secondary windings of a transformer, but it 
 must be remembered that the impedance Z 2 varies as the 
 speed of the motor changes. 
 
 The primary current has a component, of which the magneto- 
 motive force is equal and opposite to that of the secondary 
 current, as in a transformer ; therefore, the primary current is 
 composed of the exciting current and the component required 
 to neutralize the mutual magneto-motive force of the secondary 
 current. 
 
 The angle of lag in the secondary circuit is 
 
 = tan -1 - 2 — 2 = cos 
 2 Bo 
 
 Bo 
 
 VB* + s* X* 
 
 The torque between the field magnet and armature is pro- 
 portional to the integrated product of the rotating mutual mag- 
 netic flux with the secondary current, or 
 
 T = KQIo cos 6o = jSTI? 
 
 
 Bo 
 
 V BJ + s’ z x* V M2 -is 2X 2 , 
 
 where K depends upon the construction of the machine and 
 the distribution of the flux and current at the surface of the 
 armature. From this the torque is found to be 
 
 T=K<t> 
 
 So Bo B .) 
 
 B*Vs*X*' 
 
 From the formulas given it is seen that the I 2 B 2 loss is 
 
 „ 27^2 
 
 P — b 2 ^2 7? - 
 
 ' B* + S*X , 2 2 
 
 Ts 2 E 2 
 
 K<$> 
 
 3 F 
 
802 
 
 ALTERNATING CURRENTS 
 
 Hence there results 
 
 when K can be considered to be constant, since E 2 is a direct 
 function of <t>. 
 
 From the last formula it is seen that, if the motor is attached 
 mechanically to a load having a constant resisting moment, the 
 copper loss and slip will bear the same ratio whatever may be 
 the changes in the secondary resistance or the primary voltage. 
 Thus, if the secondary resistance is lowered, the armature speed 
 increases slightly and the slip becomes less, the current becomes 
 somewhat less, but the cosine of the angle of lag decreases pro- 
 portionately so that the electro-magnetic torque continues to 
 just balance the mechanical torque of the load. 
 
 In a short-circuited secondary winding, s. 2 E 2 volts are all used 
 in driving the current through its impedance. The electrical 
 power converted into mechanical power by the motor can be 
 expressed by S 2 E 2 I 2 cos d 2 = E 2 I 2 cos d 2 — cos 0 2 , when 
 
 >%e 2 is a voltage that would be generated if the armature 
 ran at a slip of V — v = V' and S 2 = 1 — s 2 . An analogous 
 situation is created if the rotating flux is imagined to be station- 
 ary as in an alternator with rotating armature and the arma- 
 ture is mechanically driven at a speed V ; then there results a 
 voltage S 2 E 2 generated in the secondary windings if E 2 is the 
 
 voltage accompanying a speed V and *S 2 = — , as in the foregoing. 
 
 In the case of the motor the power delivered to the shaft 
 equals the torque times the velocity of rotation in angular travel 
 per unit of time, so that the mechanical power in watts is P 
 
 = 2 7T T. Transposing gives T= 2 7 The voltage S 2 E 2 
 
 does not actually appear in the motor as an electrical quantity, 
 but as a mechanical torque, and it is represented in the primary 
 circuit by the voltage drop whose active component, multiplied 
 by the component of primary current flowing because of the sec- 
 ondary current, equals the motor’s mechanical output in watts. 
 
 In a transformer the secondary external circuit provides a 
 path for a current which, by setting up a counter-magneto- 
 motive force, causes a current of equal and opposite-magneto- 
 motive force to flow in the primary windings. The vector 
 
ASYNCHRONOUS motors and generators 
 
 803 
 
 product of this current by the impressed voltage equals the 
 transformer output plus the copper losses of this component of 
 current. In an induction motor the resisting moment of the 
 load, by lowering the velocity of rotation of the armature, is 
 the cause of the flow of a current in the secondary windings 
 which, as in a transformer, demands current of equal and op- 
 posite magneto-motive force in the primary windings. 
 
 191. Vector Relations in the Rotating Field Induction Motor. — 
 From the preceding discussions it is evident that a diagram of 
 current and voltage loci may be drawn for an induction motor 
 similar to the corresponding diagram for a transformer. In 
 the transformer diagram showing the effect of internal reactance 
 and a non-reactive load (Fig. 277), the secondary current locus 
 is drawn upon the basis of a circuit containing fixed inductive 
 reactance and variable resistance.* In the rotary field induc- 
 tion motor the resistance of the secondary circuit is usually 
 constant when the motor is in normal operation, but the fre- 
 quency and voltage vary together, which results in an effect 
 equivalent to varying the resistance as in the case of a trans- 
 former, f Therefore, the induction motor diagram is similar 
 in principle to that of Fig. 277 for the transformer, but in the 
 motor diagram the diameter of the current locus is relatively 
 much shorter, on account of the greater magnetic leakage 
 caused by the air space ; and the motor current varies less in the 
 windings of small-sized motors from standstill, which is equiva- 
 lent to the condition of secondary terminals short-circuited in 
 the transformer, to nearly synchronism, which is equivalent to 
 secondary circuit open in the transformer, than is the case for 
 transformers. Figure 463 shows a diagram for an induction 
 motor which is similar to the transformer diagram of Fig. 277, 
 in which as before the ratio of transformation is for convenience 
 taken equal to unity, and reference should be made to the 
 method of construction of that figure. J The exciting current 
 SO is taken to be constant though it varies somewhat more in 
 both length and angle than in a transformer because of the 
 greater portion of the circumference of the current locus which 
 comes within the limits of no load and full load. The primary 
 power factor is a maximum when the primary current vector FS, 
 closing the triangle of which the other two sides are the exciting 
 * Art. 70 (a), Case 1. t Art. 70 (&), Case 5. \ Art. 129. 
 
804 
 
 ALTERNATING CURRENTS 
 
 current SO and tlie secondary current FO, is tangent at F to the 
 secondary current locus OFMX. For smaller primary currents 
 the exciting current component SO causes the lag angle between 
 the primary current and the vertical impressed voltage line OE 
 to be larger than at F , the point of tangency, and for larger 
 
 currents than FS the magnetic leakage of the primary and 
 secondary circuits causes the angle to be greater than at tan- 
 gency. At F" on OFMX is shown the current locus point of 
 maximum power input, as at this point the projection of the 
 primary current on the voltage vector OE has its maximum 
 value. The primary current at no load is SO in the diagram; 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 805 
 
 the exciting component, which may be considered constant 
 without serious error, and the component due to the secondary 
 current are at that time of very small value, the latter hav- 
 ing an active component just sufficient to drive the armature 
 against bearing friction and windage and to supply magnetiza- 
 tion losses. The component of the primary current required 
 to oppose the magneto-motive force of the no load secondary 
 current is added vectorially to the true primary exciting current 
 and is considered part thereof. The standstill or starting cur- 
 rent, when starting resistance or other current-reducing devices 
 are not used, is OF'". This is located on OFMX at the point 
 where the active voltage OQ'" shown in the voltage locus con- 
 struction is all absorbed in the resistance of the primary and 
 secondary windings. At this point the drop of impressed 
 voltage due to magnetic leakage is Q'"F. This is the same as 
 the short-circuit point in a transformer diagram.* The power 
 input, output, and all conditions of working may be obtained 
 from this locus, as has already been explained for the trans- 
 former,-)- or a different method described fully by McAllister, 
 which is sometimes more convenient, may be used for the pur- 
 pose. J Using Fig. 463, the triangle of primary voltages OEQ ' , 
 for any secondary current as OF', is similar by geometrical con- 
 struction to OF'c , where F'c is a line drawn from F' to c at right 
 angles to OX. The side OQ' of OFQ' is proportional to F'c. 
 The points a and b are laid off on F'c so that F'a is proportional 
 to 0 K, the impressed voltage drop due to the torque of the motor 
 load ; ab is proportional to KJ, the drop due to the secondary 
 resistance ; and be is proportional to J Q' , the drop due to the 
 primary resistance. The distance cd is equal to the active com- 
 ponent of the no load current. Then, since F'd is the active 
 component of the primary current, it is proportional to the 
 power input ; also, F'a is proportional to the power output ; 
 ab is proportional to the I 2 R loss in secondary resistance ; be is 
 proportional to the I 2 R loss in primary resistance ; and cd is 
 proportional to the no load power loss. The lengths of these 
 lines respectively multiplied by a value representing the im- 
 pressed voltage OF (that is, taken to a proper scale) are there- 
 
 * Art. 129. t Art. 130. 
 
 X McAllister, Sibley Journal of Engineering , Nov., 1904, and Alternating 
 Current Motors, 3d Ed., p. 105 et seq. 
 
806 
 
 ALTERNATING CURRENTS 
 
 fore equal to the motor input and output and the several losses. 
 The angle Q'KE is considered constant for all positions of Q\ 
 hence the angle Oac is constant for all values of F, and the two 
 straight lines drawn from 0 to F'" and 0 to M, through a and 
 l respectively, divide every vertical line dropped from any 
 point on the current locus to the line Sd extended into seg- 
 ments proportional to the power quantities. 
 
 F' a 
 
 From the construction it is seen that equals the efficiency 
 
 when the secondary current is OF' . The corresponding ratio 
 decreases to a value of zero when the motor is standing still or 
 running at no load, and increases to a maximum when F is at a 
 point on the locus where the tangent to the locus is parallel to 
 OF'". The length F'b is proportional to the torque measured 
 in terms of watts received by the secondary circuit divided by 
 the theoretical synchronous speed, and it grows from a value of 
 zero when the secondary current is zero to a maximum value 
 when F is at a point on the locus where the tangent is parallel 
 
 to OM. The speed is proportional to , since the speed equals 
 
 the output divided by the torque. The slip for varying loads is 
 
 proportional to since, as shown by the formulas,* the slip is 
 
 F b 
 
 proportional to the secondary copper loss divided by the torque, 
 provided K of the formulas remains constant. 
 
 When the impressed primary voltage, the no load primary 
 current, the no load primary angle of lag, the resistances and 
 standstill reactances of the primary and secondary windings, 
 and the synchronous speed of an induction motor are known, it 
 is possible to construct accurate curves of its performance from 
 no load to full load. In fact, test curves of motor performance 
 closely approximate those that are derived from the circle dia- 
 gram. In using the diagram it should be noted that the ex- 
 citing current is assumed to be the same in all the phases, and 
 that all currents represented are for a single phase, so that the 
 power and losses obtained from the products of current and vol- 
 tage vectors must be multiplied by m , where m is the number of 
 phases. The exciting current line S 0 should have an active com- 
 ponent equal to the no load losses divided by m and a quadra- 
 
 * Art, 190. 
 
ASYNCHRONOUS MOTORS ANI) GENERATORS 
 
 807 
 
 1 2 
 
 ture component equal to , . x — times the magneto- 
 
 n i X V2 X 1.25 m 
 
 motive force necessary to create a flux equal to that of the rotat- 
 ing field, as shown in Art. 188. 
 
 Prob. 1. Construct the circle diagram and from that draw 
 the curves — from no load speed to standstill — of efficiency, 
 power factor, torque, power input, copper loss, slip, speed, and 
 primary and secondary currents (considering the ratio of trans- 
 formation unity) as functions of the power output used as the 
 abscissas, for a rotary field two-phase induction motor having 
 the following characteristics : full load capacity rating 25 horse 
 power ; pairs of poles 6 ; connection quarter-phase ; impressed 
 volts per phase 440 between loads ; no load losses 2 kilowatts, 
 which are assumed fixed for all loads. The no load current is 
 15 amperes per phase and the no load power factor is 20 per cent. 
 The standstill, or short-circuit current, is 125 amperes, and the 
 standstill power factor is 50 per cent. The secondary winding is 
 also assumed to be quarter-phase and the resistances and stand- 
 still reactances of the primary and secondary windings to be equal. 
 Figure 463 indicates the process. The circle diagram may be 
 drawn and the currents laid out for a single phase, as suggested 
 above, using one-half of the total power as the full load output. 
 The standstill reactance for either the primary or secondary 
 winding may be obtained by dividing one half the product of 
 the voltage and standstill induction factor (sin 0) by the stand- 
 still current. The resistance of either winding is equal to one- 
 half of the product of the voltage times the standstill power 
 factor (cos 0 ) divided by the current. The diameter of the 
 E 
 
 locus is D = , where X , and X 0 are the standstill re- 
 
 Xi+X* 
 
 actances of the primary and secondary windings. The stand- 
 still current is very large, so that some error is introduced in 
 finding the diameter by using the reactances as here deter- 
 mined. More accurate methods will be given later. 
 
 192. Substituted Impedance for the Rotary Field Induction 
 Motor. — From the circle diagram and the formulas of the pre- 
 ceding article it is evident that equivalent impedance can be 
 substituted for the rotary field induction motor, as was done in 
 the case of the transformer. In this case the load must be 
 represented by non-reactive resistance having a value equal 
 
808 
 
 ALTERNATING CURRENTS 
 
 to the watts output per phase divided by the square of the sec- 
 ondary current reduced to the primary equivalent. This latter 
 is equal to the primary current per phase after the no load cur- 
 rent per phase has been vectorially deducted from it. Figure 
 464 shows the arrangement. In this figure R, and X f repre- 
 sent an impedance which will permit the exciting current to 
 flow; R x and X x are the resistance and reactance of the primary 
 coils ; R 2 is the resistance of the secondary coils ; X 2 is the re- 
 actance of the secondary coils at standstill ; s 2 is the slip ; and 
 R is the resistance that would be required when the slip is s 2 
 to dissipate power equal to the mechanical load of the motor 
 when connected into the circuit as indicated in the figure. The 
 
 a c e 
 
 Fig. 4G4. — Arrangements of Impedance which are Approximately Equivalent to the 
 Circuits and Load of a Rotary Field Induction Motor. 
 
 impedances of the primary and secondary windings should be 
 the impedances of one phase for each, and the secondary quan- 
 tities should be properly reduced to primary equivalents. The 
 coils arranged to absorb the exciting or no load current should 
 absorb a current comprising a quadrature component equal to 
 
 the — — — times the magneto-motive force of the coils of a 
 
 V2 x 1.25 n x 
 
 single phase, combined vectorially with an active component rep- 
 resenting — times the total no load losses, where m equals the 
 m 
 
 number of phases; this is of course equal to the no load current 
 measured in one phase. 
 
 The solution of these circuits by impedance and conductivity 
 formulas is the same as in a simple transformer,* except that 
 
 * Art. 133. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 809 
 
 the total motor output and losses are equal to the values given 
 by the formulas multiplied by the number of phases. The no 
 load current varies in value and phase position more than in 
 the case of a well-designed constant potential transformer, so 
 that it is necessary, where minute accuracy is for any reason 
 demanded, to transfer the no load circuit (Fig. 464) from ab 
 to cd. 
 
 The secondary induced voltage is all absorbed in the imped- 
 ance of the secondary windings. In terms of Fig. 464 in which 
 the secondary quantities are all reduced to terms of the primary 
 circuit, it is the voltage measured from c to e. It varies directly 
 with the slip and has a value at any slip s 2 which is s 2 E v in 
 which E 2 is the induced voltage corresponding to a frequency 
 of f periods per second for the cycles of inducing magnetism 
 and is the voltage measured from c to d in Fig. 464. The in- 
 duced voltage in the primary winding is also E 2 . From Fig. 
 464 it will be observed that 
 
 in which I x is the total current in the primary winding, 1^ is 
 the component in opposition to the induced secondary current, 
 I f is the exciting current, E x is the primary impressed voltage, 
 Z is the impedance of the motor circuit from a to b through c*, 
 e , /, and d, and Z s is the impedance from a to b through the 
 parallel path. 
 
 Also, obviously, 
 
 S 2 E 2 — E 2 (^2 4 * JS 2 ^ 2 ) 
 
 and E x — E 2 + I 2 (R t +jX 1 ) = / 2 (r i + - +j (A) 4- A 2 ) • 
 
 j 
 
 From which, 
 
 s 2 E 2 = h ( E 2 + s 2 2 - Y 2 2 )’’ 
 
 and SgAj = J 2 [(* 2 #! + R 2 Y + s 2 2 (A 1 + X 2 ) 2 ] J . 
 
 Therefore 
 
810 
 
 ALTERNATING CURRENTS 
 
 Also, the torque per phase is equal to 
 T = K'<t>n 2 I 2 cos 0 2 ; 
 
 10 8 _EL 
 
 but 
 
 
 V2ttAw 2 / 
 
 
 s 2^2 
 
 and 
 
 cos d 2 _ 
 
 W + S, 2 ^ 
 
 7L 
 
 Therefore, 
 
 T = K • S 2^2 2 ^2 _ X 
 
 R* + s 2 2 X 2 2 ( 
 
 s 2 EfR 2 
 
 h R, + ^ 2 ) 2 + «2 2 (^l + ^2) 2 ’ 
 
 in which K is a numerical constant. 
 
 The torque is therefore a function of the slip. If the resist- 
 ance of the armature circuit of a motor is varied, while E v R v 
 X x and X 2 are maintained unchanged, the torque will pass 
 through a maximum value when dT/ds 2 = 0, which value is 
 
 y 2 
 
 T = 1 K — 1 
 
 max 2 R 1 + [R* + (X 1 + X^f 
 
 This equation shows that the maximum torque of any induc- 
 tion motor is proportional to the square of the impressed vol- 
 tage, and that its value is independent of R 2 . The primary 
 current producing it is also independent of R 2 ; but the slip at 
 which it occurs, 
 
 R, 
 
 [^+(V 1 + X 2 )2]*’ 
 
 is directl} r proportional to R 2 . 
 
 If the armature core losses are relatively large and vary with 
 the slip, as would be the case if the armature core were made 
 of unlaminated hard cast iron, the apparent resistance of the 
 armature winding might be mostly the effect of core losses, 
 which vary with the slip, and the motor might then produce a 
 uniform torque (like a series-wound, direct-current motor sup- 
 plied with a constant current) over a considerable range of 
 speeds. 
 
 The resistance of the armature conductors is usually such 
 that the maximum torque comes at from 1 to 10 per cent slip, 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 811 
 
 or s 2 equals from .01 to .1 ; and in practice an external resist- 
 ance is usually introduced into the armature windings at start- 
 ing,* which serves both to increase the torque at starting and 
 to avoid the excessive rush of current which might otherwise 
 occur while the armature is stationary. Figure 465 shows the 
 relation of torque to slip for a motor when the armature circuits 
 have resistances of .02, .045, .18, and .75 ohm.f This shows 
 plainly that the torque can be caused to have a maximum value 
 at different slips from 100 per cent to 10 per cent of the field 
 
 Fig. 4(i5. — Torque Curves for a Rotary Field Motor with Different Resistances intro- 
 duced into the Secondary Winding. 
 
 frequency by gradually reducing the resistance of the armature 
 circuit from .18 to .02 ohm as the speed of the armature in- 
 creases. An increase of armature resistance above .18 ohm 
 brings the torque to a maximum value when the armature is 
 driven backwards by external mechanical power, thereby in- 
 creasing the slip beyond 100 per cent. 
 
 Induction motors are usually designed to run normally at a 
 speed which is between synchronism and the speed giving the 
 greatest torque. In designing them X 2 is made as small as 
 practicable, by reducing the magnetic leakage to a minimum, 
 and R 2 is then given such a value that the slip at normal full 
 
 t Steiumetz, Trans. Amer. Inst. E. E., Vol. XI, p. 700. 
 
 * Art. 196. 
 
812 
 
 ALTERNATING CURRENTS 
 
 load is sufficient to give a value of the torque which is from 
 one fourth to three fourths of its maximum value. Such 
 motors can therefore carry considerable overloads, as they 
 normally operate at speeds between synchronism and the speed 
 of maximum torque ; but if the resisting moment of the load is 
 increased beyond the maximum torque, the motor stops. In 
 this respect, induction motors differ from shunt- wound direct- 
 current motors operated at constant voltage, in which the 
 torque increases in direct proportion with the armature cur- 
 rent and therefore with the resisting moment of the load, 
 provided the total magnetic flux passing through the armature 
 remains constant and does not shift. Increasing the load on 
 a well-designed shunt-wound direct-current motor will not 
 ordinarily stop it until the armature windings are melted, or in 
 small motors until the drop of voltage due to current flowing 
 through the armature conductors is equal to the impressed 
 voltage. 
 
 Near synchronism the leakage reactance of induction motors 
 becomes negligible, and from the formula it is seen that the 
 torque varies almost directly as the slip. This is approximately 
 true through the ordinary range of slips from no load to full 
 load in commercial induction motors. 
 
 The torque of the armature when the speed is V revolutions 
 per minute and the output P is given in watts is equal to 
 
 T = — — in dyne-centimeters, 
 
 mn 
 
 P X 10 7 
 2 rrV 16.3 
 
 in gram-centimeters, 
 
 r " = 2^226 hl P° UDd ' feet - 
 
 Since the power equals 2 7 t times the product of torque and 
 speed, and it has already been shown that 
 
 s 0 E„Ro 
 
 p = 
 
 60 Z 2 2 
 
 therefore, 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 813 
 
 V V 
 in which S 2 = — r and s 2 = — , and consequently, since 
 
 V V 
 
 o V V- V 
 
 s *=v = ~r 
 
 1 V H 
 1 1 Sg, 
 
 P — J_ g 2 ) /i*. 2 ^2 
 
 60 K 2 + s 2 2 X 2 2 ’ * 
 
 which readies a maximum value at a lesser slip than the torque. 
 
 The sum of the output of the motor plus the armature 
 core losses and friction is equal to the continued product of 
 (1) the number of armature conductors, (2) the effective 
 motor load current in each conductor for the given output, 
 (3) the voltage which would be developed respectively in 
 each conductor if the armature were driven at its running speed 
 in an equal stationary field, and (4) the cosine of the angle of lag 
 of the armature current with respect to the induced voltage, or, 
 
 P — W 2 ^2r'^2 ^2c C0S ^2 c’ 
 but 
 
 J _ S 2-^2c . 
 
 2i 
 
 where n v L c , s 2 E 2c , ^20 X 2 c , and d 2c are the number of con- 
 ductors and the current, voltage, resistance, reactance, and 
 angle of lag for any conductor, the subscript c indicating that 
 the quantities are for a single conductor ; and 
 
 cos 9 2c — 
 
 Bo 
 
 Therefore, 
 
 also 
 
 vv+w 
 
 2c 
 
 P _ . 
 
 -^2 c — K — 
 
 1 
 
 
 where n 1 is the number of primary conductors, and therefore 
 
 1 F 2 >S. 2 s 2 B, 
 
 2c 
 
 E 2 S 2 s 2 R 2c 
 
 P = 
 
 V + S 2 X 2c ft 2 ( i? 2c 2 + S 2 X 2c) 
 
 This formula is not one which is ordinarily needed in the 
 design of a motor, but it plainly shows the effect on the output 
 
814 
 
 ALTERNATING CURRENTS 
 
 of a motor which is caused by varying any one of its construc- 
 tive details while the others remain unchanged. A very im- 
 portant deduction from the formula is that the torque and 
 output of an induction motor vary as the square of the primary 
 voltage, so that a machine which will carry an overload of 50 per 
 cent on its normal voltage will barely run at full load if the vol- 
 tage is reduced 20 per cent. The formula also shows that the 
 slip is inversely dependent on the square of the primary voltage. 
 
 The motor torque is shown from the equivalent impedance 
 circuits to be approximately, in gram-centimeters, 
 
 rp_ HI? 10 7 
 
 2ttV 16.3’ 
 
 where V is the revolutions of the motor secondary per minute 
 and Rl 2 2 is the motor rotative output. The slip is 
 
 , = _A_. 
 
 2 R 2 + R 
 
 The number of revolutions per minute the motor armature 
 speed is less than that of the rotating field is 
 
 v — V ^2 = PPX ^2 
 
 R + R 2 p R + R 2 ' 
 
 since the speed of the rotating field in revolutions per minutes. 
 
 V, is equal to PPX, where /is the frequency of the primary 
 P 
 
 circuit and p is the number of pairs of motor poles. Likewise, 
 the actual speed of the armature in revolutions per minute is 
 
 V = V( 1 — s ) = ^ ■ 
 
 2 P r + r 2 
 
 Methods of determining other characteristics are apparent from 
 the study of the induction motor formulas and Figs. 464 and 
 465. The processes used in dealing with substituted impedances 
 for the induction motor are very similar to those required for 
 the transformer. It will be remembered that to reduce sec- 
 ondary resistances and reactances to primary equivalents, they 
 must be multiplied by s 2 where s is the ratio of transformation. 
 
 l 3 rob. 1. Determine the substituted impedances for the motor 
 given in the Problem at the end of Art. 191. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 815 
 
 Prob. 2. From the substituted impedances obtain the pri- 
 mary current and power factor for full load, half load, and 
 quarter load for the motor of Prob. 1. 
 
 Prob. 3. From the substituted impedances of Prob. 1 find 
 the torque and slip at full load, half load, and quarter load. 
 
 193. Additional Formulas for Torque and Slip of a Polyphase 
 Induction Motor. Circle Diagram of Magnetic Fluxes. — In the 
 last article, the slip was expressed thus, 
 
 s = ^2 . 
 
 2 R 2 +R 
 
 Multiplying this by the secondary current I 2 , squared, there 
 results 
 
 I 2 R 0 
 
 82 i 2 2 r 2 + 1 2 R' 
 
 where the numerator represents the armature copper losses 
 and the denominator represents the same copper losses plus the 
 power output (72 should be of such value that this will in- 
 clude frictional losses) ; thei-efore, 
 
 where P c equals the armature copper losses and equals the 
 power transformed from the primary circuit into the secondary 
 circuit. The iron losses may be considered as all absorbed 
 from the primary circuit. 
 
 The value of torque stated in pound-feet is 
 
 T= 
 
 P n 10 * 
 
 2 77 - V ■ 226’ 
 
 where P a is the output. 
 
 P 0 = 7 2 72, where R is the equivalent secondary substituted 
 load impedance and P t = PR + Z 2 72 2 . Therefore, 
 
 R 
 
 P = P 
 
 n 1 
 
 R + 72/ 
 
 but V'=V- 
 
 R 
 
 60/ R 
 
 R 4- R 2 p R + R 2 
 P 0 and V in the formula for torque gives 
 
 - 117 £3 
 
 / 2 7T • 226 • 60 ' / 
 
 Substituting these values of 
 
 T= P- 
 
816 
 
 ALTERNATING CURRENTS 
 
 The secondary current of the motor when the armature has 
 no external load, expressed in primary equivalents, may be ob- 
 tained with approximate accuracy by vectorially subtracting the 
 primary current at no load from the primary current which flows 
 at any input under consideration. Likewise, the resistance of 
 a closed circuit secondary winding can be calculated by observ- 
 ing the primary power input and current when a low voltage is 
 applied to the primary windings and the armature is locked in 
 a stationary position. The difference between the calculated 
 primary I 2 R loss and the observed power is approximately the 
 secondary I 2 R loss. Dividing this difference by the square 
 of the observed current gives the approximate armature resist- 
 ance in primary equivalents. The current used should be less 
 than that for full load in order that the magnetization losses 
 may be negligible. However, correction can be made as in a 
 transformer.* 
 
 In an earlier article it was stated that the leakage reactance of 
 a polyphase motor decreases on account of the saturation of the 
 core teeth with the heavy currents of short circuit. In order 
 to obtain a value of X x + X 2 to be used in laying out the 
 diameter of the circular locus of currents, it is well to experi- 
 mentally determine the value when the current is at or near its 
 value for full load. This can be done by locking the armature 
 in a stationary position and applying a sufficiently reduced vol- 
 tage to the primary winding so as to cause the desired current to 
 flow. Then, if the observed voltage, current, and watts are E ' , 
 
 I r , and P\ we have 
 
 O' = cos -1 
 
 ET 
 
 where O' is the angle of lag. From which the reactive com- 
 ponent of the voltage is 
 
 E' sin O' = E' x = I'X' and X' = — , 
 
 where E' x is the component of the voltage produced by the 
 change of the leakage flux, and X' = X x + X 2 is the combined 
 leakage reactance of the primary and secondary circuits. Then 
 the diameter of the current locus for a machine having leakage 
 reactance X' when current F flows is 
 
 I) - 
 
 ET 
 
 E' sin O' ’ 
 
 * Art. 147. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 817 
 
 where E is the normal motor voltage; or, if the assumption is 
 made that the reactance of the primary and secondary circuits 
 are equal, — which will not ordinarily introduce a very large 
 error, — and we know the current, voltage, watts, and slip, and 
 hence 6 when the machine is running at the working load de- 
 sired, we can calculate the combined leakage reactance at that 
 load. Thus, 
 
 E sin d XT- v~\ 
 
 ■ — — — — (at + s 2 xy, 
 
 where X is the leakage reactance of either winding ; hence, 
 where s 2 is the slip 
 
 _ E sin 6 
 7(1 + s 2 ) 
 
 and the diameter of the current locus circle is 
 
 EI(1 4- s 2 ) 7(1 + s 2 ) 
 
 2 E sin 6 2 sin 9 
 
 In laying out the circle diagram, it is possible to consider 
 the diameter of the locus extended to the origin of the primary 
 current vector as being proportional to a magnetic flux, if the 
 power component of the exciting current is neglected. Thus, 
 consider OFCA, Fig. 466, the parallelogram of currents in a 
 motor when the primary current is OF = I v the secondary cur- 
 rent is OA = I 2 , and the exciting current is 00=1^ (neglect- 
 ing the power component of the exciting current for the present). 
 The impressed voltage is 0E V This is the phase diagram of 
 currents for a transformer with magnetic leakage * which is 
 similar in voltage and current relations to a polyphase induc- 
 tion motor. For convenience, consider that the primary current 
 sets up a magneto-motive force which sets up the mutual flux 
 through the primary and secondary windings, the leakage flux 
 about its own windings, and neutralizes any tendency of the 
 secondary current to set up leakage flux. The secondary cur- 
 rent and the component of voltage produced by the mutual flux 
 thus considered are in the same phase, while the leakage flux 
 about the primary conductors is due to the magnetic paths not 
 only across the teeth in the field core, but also across the teeth 
 of the armature core. This assumption is not necessary, but it 
 
 * Art. 124. 
 
 3g 
 
818 
 
 ALTERNATING CURRENTS 
 
 simplifies the conception without introducing error. Draw OB 
 upon 00 extended so that 
 
 OB = 4 = c p 
 
 10 R 1 
 
 where n l are the primary turns, <3?, the total flux, and R the 
 reluctance of the magnetic leakage and mutual magnetic paths 
 in parallel, or where 
 
 B M R XL R iL 
 
 [i M R\L + R M R,L + B XL R 2L 
 
 where R M is the reluctance of the path of flux which passes 
 through both windings, R XL is the reluctance of the path for 
 
 Fig. 466. — Circle Diagram for showing Relation of Fluxes, Slip Line, and Circle 
 Loci for determining other Characteristics of a Polyphase Induction Motor. 
 
 primary leakage flux which passes across the teeth of the field 
 core, and R 2L is the reluctance of the path for leakage flux 
 around the primary windings which passes across the teeth of 
 the armature core and around the air gap. OB then equals the 
 total flux set up in the magnetic circuits of the machine, and 
 it is produced by the magnetizing current 00. The leakage 
 component of this flux must be in phase with the total primary 
 current, if the iron loss angle is neglected ; and by properly 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 819 
 
 adjusting the flux scale in which OD and other fluxes are laid out 
 it may be represented in phase and magnitude by OF , where 
 
 OF= ^FILiLx = q> 
 
 10 R l 
 
 where <E> Z is the leakage flux and R L is the combined reluctance 
 of the leakage paths in parallel. The mutual flux must be the 
 vector difference of the two noted, or 
 
 ®M= ®t~®L=FI). 
 
 But the mutual flux <J> is cut by the secondary conductors and 
 sets up the secondary voltage E v which must be at right angles 
 to it. As the secondary voltage is, by the assumption, in phase 
 with the secondary current I v the latter also must be in quad- 
 rature with T > M . Therefore, as FC in Fig. 466 equals OA , the 
 angle DFO is always a right angle. Therefore, as F moves, 
 due to changes in the load and hence the primary and secondary 
 currents, it must describe an arc, OFMD. If there were no 
 losses this locus would be a complete semicircle. 
 
 The output when the secondary neutralizing component of 
 the primary current is OF is evidently F l x FP, or at proper 
 scale equals FP if there are no losses. A primary resistance 
 loss will reduce this because the induced primary and secondary 
 voltages will be smaller by the effect of the drop in the 
 
 primary windings. This means that FP, the mutual flux, must 
 be shorter in equal proportion. In the same way the secondary 
 output must be less by reason of the resistance drop in the sec- 
 ondary windings, and this may also be considered as equivalent 
 to the shortening of the mutual flux line. Suppose, now, the 
 motor is loaded until it comes to a standstill, and that the cur- 
 rents have increased until IP has moved to F' . Since the arma- 
 ture is doing no work, the secondary voltage set up by the 
 mutual flux FD is now all expended in driving the current F' C 
 through the secondary windings. The proportion of the in- 
 ductive effect of the mutual flux used in overcoming IR drop 
 
 per ampere is then measured by the ratio 
 
 F'D 
 
 OF 
 
 if the copper 
 
 loss of the exciting current 00 is neglected and the currents 
 are reduced for the ratio of transformation of unity as shown 
 in the figure. Since the same relations must hold for all cur- 
 
820 
 
 ALTERNATING CURRENTS 
 
 rents if the reluctances of the magnetic circuits are considered 
 to be constant, the proportion of the inductive effect of the 
 total flux which is absorbed by IR drop for any other current, 
 such as CF , is FD' where the triangle CFD' is made similar 
 to CF' D. But F is any point on the arc CFF' and angle FD' C 
 is constant, and therefore angle CD'D is constant for any posi- 
 tion of F on the locus CMD. Hence the point D' must trace the 
 arc of a circle CM" D, having the chord CD. If U is the center 
 of the chord, the perpendicular UN lies on the radius of the 
 circle CD'D , and, D being one point of the circumference, the 
 center of the circle is the intersection of a normal to F'D at D 
 with UN, since F'D must be tangent to the circle at Z>, as the 
 inductive effect of the mutual flux represented by F'D is just 
 used up in furnishing the IR voltages. The power lost in the 
 resistance of primary and secondary windings when any current 
 CF flows is then FR. If now the ordinates of the line ZZ' are 
 equal to the assumed constant waste of power in iron losses, 
 the losses in the primary winding by reason of the copper loss 
 caused by the flow of the magnetizing current OG , and the 
 mechanical frictional losses, then the net power given to the 
 shaft by the armature when the secondary current CF flows is 
 represented by the line RS. 
 
 The relative primary and secondary copper loss can be found 
 in this way : divide F'D, the standstill mutual flux which is all 
 used in generating the primary and secondary IR voltages, into 
 two parts, such that F'B is proportional to the flux used in 
 generating the primary resistance drop and BD to that used 
 in generating the secondary resistance drop. Then, the mutual 
 flux used in generating the IR drop for any other current such 
 as CF must be divided in the same proportions, and hence, FD' is 
 divided into FB' and B'D' . Now a triangle drawn between the 
 points C, B , and D is similar to a triangle drawn between the 
 points C, B', and D', since triangles CF' D and CFD' are similar. 
 Hence, as F varies in its positions, the angle made by lines 
 connecting C and B' , and B' and D must be equal whatever 
 may be the position of F. The locus of B' is then the arc of a 
 circle upon the chord CD. The center His the intersection of 
 the perpendicular UN with the normal to BD erected at center 
 point Q. This is evident, since B and D both lie on the arc and 
 the line BD is therefore a chord of the arc between B and D. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 821 
 
 For any primary current OF = /, supposing the scales in which 
 the lines are measured to be properly adjusted, we have the pri- 
 mary input equal to FP ; the secondary input nearly equal to TS ; 
 the secondary output equal to RS ; the primary copper loss equal 
 to FT ; the secondary copper loss equal to TR ; and the excita- 
 tion and frictional losses (assumed constant) equal to SP. 
 
 The total armature torque for any primary current OF may be 
 scaled as approximately equal to TS, since CFx FB=I 2 x 
 which is proportional to the torque. But, if the perpen- 
 dicular B' W = TS is dropped from B', the similar triangles 
 B' WB and CFB give the proportion OF : B' W: : OB : B'B, or 
 OF x B'B = B'W x OB. Therefore, as OB is fixed in length, 
 B' W = TS is proportional to the torque and can be scaled to 
 read torque directly. The load torque equals TS minus the 
 frictional and windage torque of the armature itself. 
 
 The slip for any primary current may be found as follows: the 
 
 secondary current I 2 ac $> M s v where s 2 is the slip, or s 2 ac 
 
 A 
 
 ac 
 
 OF 
 
 B'B' 
 
 But, as the triangle CFB' is similar for all positions of F, the 
 
 slip may be taken as approximately proportional to 
 
 OB' 
 
 B’B 
 
 There- 
 
 fore lay out the angle BKT equal to OB'B , then triangles OB'B 
 
 OB' BK 
 
 and LKB are similar and — — — = — — - . As KB is constant in 
 
 B'B KB 
 
 length, the slip is approximately proportional to the intercept 
 LK made from the fixed line JK by the line FB, wherever F 
 lies on the locus. 
 
 This diagram, in so far as the current locus is concerned, is 
 similar to that of Fig. 463, and the various quantities may be 
 determined in the same way in either, with the exception that 
 in Fig. 466 the fixed losses are subtracted from the measuring 
 line dropped from the end of the current vector, while in Fig. 
 463 the voltage is made a little larger and the fixed losses are 
 added to the same line. The circle diagram of Fig. 466, with 
 the approximate method of finding the losses, etc., was discov- 
 ered by Heyland, Behrend, and others early in the art of induc- 
 tion motor manufacture, but it is only approximate at the best 
 and in many instances is quite inaccurate. The diagram of 
 Fig. 463 is more accurate and is therefore preferable to use. 
 The point F' in both Fig. 463 and Fig. 466 shows the stand- 
 
ALTE RNATING CUHIl ENTS 
 
 • 822 
 
 still point ; the locus from F' to C represents the machine as a 
 motor for the entire range of speeds from standstill to theoretical 
 synchronous speed ; and the locus from F' to D represents the 
 machine as a frequency transformer, driven backwards (with 
 respect to the rotating magnetic field) from a speed of standstill 
 to infinite speed. 
 
 Either of these circle diagrams given for the induction motor 
 is lacking in absolute accuracy, since both the phase and the 
 scalar value of the induced voltage of the induction motor 
 change when the current changes, and this changes the phase 
 and scalar value of the exciting current as well as the radius and 
 position of the locus circle. Attention has already been called 
 to the fact that the reluctances of the leakage paths vary with 
 different currents. Suffice it to say, however, that the diagrams 
 given are sufficiently accurate for many practical purposes, and 
 if they are constructed upon the basis of the magnetic path 
 reluctances to be found at or near full load, they will ordinarily 
 approximate to accuracy from no load to well over full load. 
 
 194. The Rotary Field Induction Motor as an Asynchronous 
 Generator. — If a rotary field induction motor is driven above 
 synchronous speed by a source of mechanical power attached to 
 the shaft, while it is electrically connected to a conductor system 
 in which there are voltages which would run the machine as a 
 motor, it no longer acts in the capacity of a motor, but the arma- 
 ture generates electric power and delivers it to the supply lines. 
 In this case the primary circuit of the machine continues to 
 receive its quadrature current from the external supply system, 
 and the rotating magnetic flux is the same as if the machine 
 was operating as a motor. But the relative motion between 
 the primary and secondary circuits having been reversed, — the 
 secondary circuit now rotating faster than the rotating flux, — 
 the active component of current generated in the windings of 
 the armature is reversed, which results in the opposing compo- 
 nent of the primary current being reversed. This gives the 
 locus shown in Fig. 467. The upper half of the locus OTXT' 
 is the induction motor and frequency transformer locus and the 
 lower half the asynchronous generator locus. When sufficient 
 power is applied to the pulley to drive the machine at theoret- 
 ical synchronism, only the primary circuit exciting current SO 
 flows. When the armature speed is raised above synchronism. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 823 
 
 current flows in the armature windings, inducing an equal and 
 opposite component of current in the primary windings, which 
 passes out to the external circuit to which the machine is at- 
 tached. Since the magnetic flux has not been reversed, however, 
 the primary circuit of the machine must continue to draw from 
 the external circuit a component of lagging quadrature current 
 
 Fig. 467. — Locus of Currents in Induction Machine when Run as a Motor and as an 
 Asynchronous Generator. 
 
 sufficient to furnish the requisite magneto-motive force. This 
 is equivalent to a flow from the machine of a leading quadrature 
 current. Such a machine will evidently work in parallel witli 
 synchronous alternators or other apparatus which will furnish 
 this exciting current, but it will not work when connected to 
 circuits containing only resistance and self-inductance. The 
 exciting current may be obtained by placing condensers of suf- 
 ficient capacity across the circuit or by obtaining the required 
 leading current from a synchronous motor connected in the 
 
824 
 
 ALTERNATING CURRENTS 
 
 circuit.* The asynchronous generator function of the polyphase 
 induction machine is sometimes made use of on traction systems 
 in which the cars are driven by such motors. In this case, 
 when the motors are properly connected in tandem, f they can 
 be caused to pump power back into the line when running 
 down grade or being brought to a stop. 
 
 When an induction generator receives its quadrature exciting 
 current from synchronous generators of fixed speed, the current 
 delivered from the asynchronous generator is of the same fre- 
 quency as that of the synchronous generators, regardless of the 
 speed of the armature of the asynchronous machine, but the 
 output changes with the speed in the manner shown by the 
 locus diagram. However, when the quadrature exciting cur- 
 rent is provided by a condenser or a synchronous motor, no 
 synchronous machine of fixed speed being connected with the 
 circuit, the frequency of the voltage and current delivered by 
 the asynchronous machine varies with its armature speed. The 
 locus diagram of Fig. 467 is not applicable to the latter case. 
 
 If a circuit receiving power from an induction generator be- 
 comes short-circuited, the generator at once ceases delivering 
 power on account of the fact that the magnetic flux disappears 
 when the impressed voltage disappears. This is contrary to the ac- 
 tion of synchronous generators, which are self-reliant in magnetic 
 field and continue to deliver power to the short-circuited lines. 
 
 195. Some Features of Construction of Rotating Field Induc- 
 tion Motors. — The secondary windings of induction motors are 
 wound upon laminated drums or equivalent rings, which, with 
 the windings, usually constitute the rotating part or rotor. 
 The arrangement of the windings may be of several forms: 
 
 1. Squirrel-cage Form , in which single embedded bar con- 
 ductors are placed on the armature core and all connected to- 
 gether at each end by a copper ring, thus making a conductor 
 system similar in form to the revolving cylinder of a squirrel 
 cage (Fig. 461). The conductors are insulated from the core 
 
 2. Independent Short-circuited Coils. — In this form of wind- 
 ing the secondary conductors are of insulated wire wound in 
 independent short-circuited coils, or of insulated bars connected 
 by end connectors in such a way as to make independent short- 
 circuited coils. 
 
 * Art. 174. 
 
 t Art. 196 (6). 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 825 
 
 Fig. 468. — Elementary Diagram 
 of a Tri-phase Armature 
 Winding for a Secondary 
 Stator arranged to surround 
 a Primary Rotor having Eight 
 Resultant Poles. 
 
 3. Independent Coils short-circuited in Common. — Here the 
 coils are wound as in the preceding form, but instead of being 
 short-circuited independently, all the ends are brought to a 
 common point, or pair of points, one of which may be at the 
 front end and the other at the back end of the armature. 
 
 4. Coil-wound Armatures connected to External Devices. — 
 The windings of paragraphs 2 and 3 
 can be arranged for connection to ex- 
 ternal terminals or collecting rings, so 
 that external resistances may be in- 
 cluded in series for varying the torque 
 or slip. The connection may be so ar- 
 ranged that the windings form single 
 or any polyphase system of circuits. 
 
 It is evident that the pitch of the 
 coils of the second and third forms of 
 drum windings must be equal to an 
 odd number of times the pitch of the 
 rotating field poles, in order that the 
 voltage set up in the conductors may 
 be additive, and coils may therefore be diametral or chordal in 
 machines with an odd number of pairs of poles, but cannot be 
 diametral in machines with an even number of pairs of poles. 
 The actual number of coils is a matter of perfect freedom, pro- 
 vided it is a multiple of two or three, and 
 the connections of conductors is properly 
 made so that the surface of the armature 
 core may be uniformly covered. A tri- 
 phase winding for a stationary drum ar- 
 mature, which is intended to surround a 
 rotating primary structure having eight 
 poles, is shown diagrammatically in Fig. 
 Fig. 469. — Elementary 468, and a three-coil armature which is in- 
 
 Diagram of a Wound ^ en q e q t 0 revolve within a six-pole primary 
 Secondary Rotor to be . * 
 
 structure is shown in t lg. 4b9. 
 
 As said earlier, it is a matter of indiffer- 
 ence from the electrical standpoint whether 
 the primary or the secondary part rotates. From the mechan- 
 ical standpoint it is usually desirable to have the secondary part 
 rotate, as the primary windings, which are usually of higher 
 
 Surrounded by a Six-pole 
 Rotating Field of the 
 Primary Stator. 
 
826 
 
 ALTERNATING CURRENTS 
 
 voltage than the secondary, are then subject to less mechanical 
 vibration and can be connected directly to the supply mains 
 without the interposition of collector rings. Coil-wound sec- 
 ondary windings are in general of the same type as the arma- 
 ture windings of synchronous alternators, and, as explained, 
 may have their terminals short-circuited, or they may be con- 
 nected (through collector rings if the armature rotates) to out- 
 side devices suitable for starting or for regulation purposes.* 
 The primary windings of induction motors are almost always 
 arranged to produce more than two poles, in order to bring the 
 machine to a reasonable speed. The rotating field frequency 
 
 / 
 
 in revolutions per second is equal to — , where / is the fre- 
 quency of the alternating current and p the number of pairs of 
 poles in the magnetic field, and in revolutions per minute this 
 
 PA 
 
 becomes V= — — . We therefore have the following table of 
 
 p 6 
 
 motor speeds for the frequencies common in this country. 
 
 Relation of Number of Poles to Synchronous Speed in Induction 
 
 Motors 
 
 Number of Poles 
 of Motor 
 
 V , WHEN / = 25 
 
 Y , vritEN f = 60 
 
 2 
 
 1500 
 
 3600 
 
 4 
 
 750 
 
 1800 
 
 6 
 
 500 
 
 1200 
 
 8 
 
 375 
 
 900 
 
 10 
 
 300 
 
 720 
 
 12 
 
 250 
 
 600 
 
 16 
 
 187.5 
 
 450 
 
 20 
 
 150 
 
 360 
 
 24 
 
 125 
 
 300 
 
 This table shows the futility of attempting to build satisfactory 
 induction motors, intended for use on even the lowest fre- 
 quency commonly used in this country, with less than four 
 poles ; and on the higher frequencies not less than six to ten 
 poles are required to give reasonable speeds. Motors of 
 greater output than ten horse power should have a sufficiently 
 
 * Art. 19(5. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 827 
 
 large number of poles to give a field velocity which does not 
 exceed 750 or 800 revolutions per minute. A 
 
 The primary windings, in the 
 very early history of the art, 
 were sometimes placed directly 
 upon polar projections of the 
 field frame, as in Fig. 470, 
 which shows a four-pole two- 
 phase machine. In modern 
 machines the windings are ar- 
 ranged as embedded uniformly 
 distributed conductors in a 
 frame of uniform magnetic sur- 
 face, as in Fig. 471, which 
 shows a four-pole three-phase 
 machine. Conductors embedded in open slots give the most 
 approved arrangement, since embedding serves to reduce the 
 
 Fig. 471. — Diagram of Rotating Field Induction Motor, in Which the Stator has a 
 Tri-phase Four-pole Primary Winding, and the Rotor may be wound either as a 
 Squirrel Cage or Coil-wound Secondary. 
 
 Fig. 470. — Diagram of Salient Pole In- 
 duction Motor Primary Winding as 
 sometimes used in the Early History of 
 the Art. 
 
828 
 
 ALTERNATING CURRENTS 
 
 reluctance of the magnetic circuit and reduce magnetic leakage, 
 and therefore serves to increase the power factor of the motor. 
 Open slots are better than closed slots, as they offer greater 
 reluctance to the magnetic leakage paths and permit the use of 
 form-wound coils. Since the actual magnet poles are produced 
 by the resultant effects of the polyphase currents, it requires two 
 coil sections to produce each magnet pole in a two-phase rna- 
 
 Fig. 472. — Simple Ring Diagram for showing the connections of a Four-Pole Tri- 
 phase Wye-Connected Winding. 
 
 chine, and three coil sections in a three-phase machine. The 
 connections of the field coils may be traced out according to the 
 instructions given for connecting the armature coils of poly- 
 phase generators.* 
 
 The connections for a three-phase field winding are illustrated 
 diagrammatically by Fig. 472, which shows a wye-connected, 
 four-pole ring field. The arrows indicate how the magnetic 
 fluxes combine at a particular instant under the impulsion of the 
 magneto-motive forces of the tri-phase currents. It should be 
 
 * Art. 22. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 829 
 
 1 1 
 • I 
 
 s ! 
 
 NsX> 
 
 Fig. 473. — Diagram of a Simple Progressive Tri- 
 phase Eight-pole Winding such as is to be found on 
 the Primary and Secondary Windings of Fig. 474- 
 
 noted that the ring type of winding is used in this illustration 
 for the sake of simplicity in showing the connections. Wind- 
 ings embedded in the polar surfaces are now used almost alto- 
 gether, but the principle of the connections is the same as in the 
 simple diagram of the figure. The wye connection is often 
 preferred for three-phase field windings, as less voltage is im- 
 pressed on a coil, so that 
 
 fewer turns of wire are ' > 
 required, and the strain 
 on the insulation of each 
 coil is less. The circu- 
 lation of internally in- 
 duced third harmonic 
 currents is also pre- 
 vented. The total 
 weight of copper is 
 equal in wye and delta 
 connections. Figure 
 473 shows a development 
 
 of the eight-pole, wye-connected winding such as that of Fig. 474. 
 
 In the case of Fig. 474, the secondary winding terminals 
 are connected to rings so that regulating or starting resist- 
 ances may be inserted. 
 
 It is seen from the above discussion that the primary wind- 
 ings of rotating field induction motors are of the same forms as 
 the armature windings of synchronous alternators. The second- 
 ary windings may be of the short-circuited squirrel cage type or 
 be similar to the primary windings, but the numbers of slots in 
 the field and armature cores should differ, and should be of num- 
 bers which are prime to each other, to prevent recurrent varia- 
 tions of the resultant reluctance of the mutual magnetic circuit. 
 
 The frequency of the magnetic cycles in the iron of the 
 primary core is equal to the frequency of the current flowing 
 in the magnetizing coils, but in the armature it is equal to 
 the motor slip ; and the hysteresis and eddy current losses 
 per pound of iron are therefore many times greater in the 
 primary core than in the secondary core when the motor is 
 operating at ordinary speeds. In this respect the character- 
 istics of an induction motor are the reverse of those of a 
 synchronous alternator, in which the field loss consists in most 
 
830 
 
 ALTERNATING CURRENTS 
 
 part of the PR loss, while the armature loss is the sum of the I 2 R 
 and core losses, of which the latter may be the larger portion. 
 In the induction motor, the field core losses are large and the 
 
 Fig. 474. — Diagram of a Rotary Field Induction Motor having Eight Poles. The 
 Field and Armature Cores are both wound for Three Phases Wye connected and 
 the Secondary Winding has its Free Terminals connected to Collector Rings for 
 Use with Starting and Regulating Resistances. 
 
 armature core losses are scarcely appreciable in a well-designed 
 machine; it is therefore desirable to reduce the amount of iron 
 in the primary core to a small volume, keeping in mind the effect 
 of the several variables such as weight of copper, density of flux, 
 etc., which enter in the problem as they do in the design of a 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 831 
 
 transformer.* It will be noted that with a rotating armature 
 the winding space is smaller for the secondary than for the pri- 
 mary windings, but the voltage of the former is usually lower 
 than that of the latter, so that less space is used for insulation. 
 The smaller the number of slots per coil, the smaller is the re- 
 luctance of the leakage paths and hence the greater is the leak- 
 age reactance with its attendant disadvantages. It is therefore 
 common practice to make the rotor diameter large and the length 
 correspondingly short so that from 3 to 6 slots can be used on 
 the stator per coil side and from 3 to 7 on the rotor — the num- 
 ber depending upon the given conditions that must be fulfilled 
 in each particular motor design, such as voltage, speed, rated 
 load, etc. To reduce reactance the air space is of course made 
 as small as is mechanically practicable. 
 
 In all cases, especially where squirrel cage secondaries are 
 used, the number of primary core slots should be an uneven 
 multiple of the number of secondary core slots, as intimated, 
 in order that there may be no dead points in starting. If the 
 teeth of the primary core lie exactly opposite the teeth of the 
 secondary core, the magnetic reluctance of the paths through 
 the teeth will be very low compared with that of the wind- 
 ing spaces between the teeth. This causes exceedingly dense 
 tufts of magnetic flux to be set up between opposing teeth at 
 the instant when the)'' are facing each other. The tendency, 
 then, of each tooth of an opposing pair is to strongly 
 attract the other as the magnetic poles created by the flux 
 at their ends are of opposite sign. If at starting the tooth 
 to tooth attraction is stronger than the pull between the 
 secondary current and rotating magnetic flux, the motor will 
 not move. Using unlike numbers of slots on the two members 
 avoids this trouble. In a somewhat similar manner when 
 the armature is running unloaded and nearly at synchronism, 
 there is a tendency for the secondary core to jump into syn- 
 chronism with the rotating magnetic field and convert the 
 machine for the time into synchronous motor relations, the ar- 
 mature becoming the mechanically rotating field magnet with 
 permanent poles magnetized by the synchronously running re- 
 sultant rotating field. This is because the secondary current 
 when the speed nears synchronism at no load is exceedingly 
 
 * Arts. 135-137. 
 
832 
 
 ALTERNATING CURRENTS 
 
 small, only enough to drive the armature against the resisting 
 moment of the friction of bearings and air, and the attraction 
 between teeth under these circumstances sometimes shows a 
 larger torque than that between the rotating flux and the 
 bands of armature current. When this occurs, the former 
 prevails and the secondary core takes the same speed of ro- 
 tation as the rotating flux. While this condition continues, 
 the secondary current becomes zero and the windings act 
 somewhat to maintain the condition of synchronism as will be 
 seen later. The condition is not necessarily undesirable, though 
 it calls for a sudden change in primary current. Where a load 
 of any material size comes on to the motor, the secondary core 
 drops out of synchronism and the machine resumes its action 
 as an induction motor. This peculiarity is of material value 
 when synchronous motors are started as induction motors.* 
 
 196. Starting and Regulating Devices. — Several different ar- 
 rangements ma} r be used for starting polyphase induction motors. 
 
 1. Small machines are commonly connected directly to the 
 circuit without the intervention of any special starting devices. 
 This is not a safe proceeding for large machines, as when the 
 
 secondary circuit is at rest and the 
 primary windings are directly con- 
 nected to the supply circuit, the 
 machine is in the condition of a 
 transformer with the secondary 
 windings short-circuited, and is 
 liable to burn up before getting 
 under way. Some of the older 
 small motor armatures were ar- 
 ranged with two rows of con- 
 ductors, making two independent 
 squirrel cages (Fig. 475), one con- 
 siderably farther from the armature 
 surface than the other for the ostensible purpose of reducing the 
 starting current of the machines. 
 
 2. (a) Resistance in Field Circuit. — Resistance may be in- 
 serted in the circuits leading to the primary windings to be 
 used in much the same manner as starting rheostats are used 
 in starting direct-current, constant voltage motors. Rheostats 
 
 * Art. 167. 
 
 Fig. 475. — Squirrel Cage Secondary 
 with Double Row of Conductors. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 833 
 
 arranged in this way in each circuit must be manipulated simul- 
 taneously in all the circuits, and therefore must be mechanically 
 coupled. Besides starting rheostats similar to starting boxes for 
 direct-current motors, liquid rheostats, arranged to be varied by 
 dipping plates in a bath, have been used with early motors. 
 Three rheostats are required for a tri-phase motor, and two for 
 a quarter-phase motor operated on independent circuits. Quar- 
 ter-phase motors on three-wire circuits may be started with a 
 single resistance device inserted in the common wire, or by two 
 resistance devices inserted respectively in the independent wires. 
 
 The insertion of resistance in the field circuits of induction 
 motors serves to restrain the starting current by reducing the 
 voltage at the primary terminals. Tins, however, reduces in 
 the numerator of the expression * 
 
 T= K s 2 j?i 2 fi 2 
 
 (s 2 /fi + -Zf 2 ) 2 + s 2 2 (-^l + -^2 ) 2 
 
 and thereby greatly lowers the available torque per ampere and 
 the motor takes an excessive current from the supply mains to 
 accelerate a heavy load. In the past this plan was used quite 
 extensively by European manufacturers, especially for large 
 machines which could be started with the belt on a loose pulley. 
 This method has the disadvantage of wasting much energy 
 through the dissipation of heat in the rheostats, and has, there- 
 fore, except in special cases been abandoned in favor of the auto- 
 transformer and transformer methods described below. 
 
 (6) Variable Compensator or '•'■Autotransformer.'' — The vol- 
 tage at the terminals of the primary circuits may be reduced at 
 starting by introducing an impedance coil across the supply cir- 
 cuits and feeding the motor from variable points on its windings. 
 This arrangement may be caused to supply a large starting cur- 
 rent to the motor without interfering with the supply circuits, 
 but it has the same effect on the motor torque as resistance in 
 series with the primary windings. The losses by this method of 
 starting are much reduced, for the power required by the motor 
 for starting is delivered at the proper reduced voltage by trans- 
 former action at comparatively high efficiency. Figure 476 
 shows a compensator connected to a tri-phase motor with a no- 
 voltage release, which automatically opens the circuit when, for 
 
 * Art. 192. 
 
 3 H 
 
834 
 
 ALTERNATING CURRENTS 
 
 any reason, the supply voltage disappears. It contains also an 
 overload release. The lower row of terminals, designated the 
 starting side, are connected by proper switch blades or contactors 
 
 Fig. 476. — Diagram of Compensator Starting Device for Tri-phase Induction Motor, 
 with No-voltage and Overload Release. 
 
 with the middle row marked “ cylinder switch” when the motor 
 is to be started. This connects the motor with the low voltage 
 terminals of the autotransformer or compensator and the supply 
 
 Fig. 477. — Diagram of Compensator Starting Device for a Quarter-phase Motor with 
 No-voltage and Overload Release. 
 
 mains with the primary terminals thereof. When the motor 
 has reached its full speed for this connection, the central switch 
 cylinder blades or contactors are thrown over to the running side 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 835 
 
 and the motor is connected directly to the supply lines. When 
 very large motors are in use several steps can be made in this 
 operation by having several sets of taps on the compensator 
 coils and stepping from one to the next successively. Figure 
 477 shows a similar arrangement for a quarter-phase motor. 
 
 It is evident that these arrangements are purely starting 
 devices and not regulators, since from the motor formulas * it 
 is seen that the running torque is changed with the square of 
 the voltage; and, therefore, if it should be desired to drive a 
 load of given torque at reduced speed, reduction of the impressed 
 voltage by means of the compensator could only change the 
 speed through very narrow limits before the maximum motor 
 torque would be reduced to the opposing moment of the load. 
 Further reduction of voltage would result in the motor coming 
 to a standstill. 
 
 3. Resistance in Armature Circuits for Speed Regulation. — The 
 torque of the armature at starting may be made equal to the max- 
 imum running torque by inserting a definite amount of resistance 
 in the secondary circuits which increases the total armature re- 
 
 nal or starting resistance. The value of R e may be determined 
 
 where s 2m is the slip at maximum torque, and R t is the total 
 resistance through which the armature current flows. There- 
 fore to get a maximum torque when s 2m = l, requires that 
 
 reactances are determined from the circle diagram or by calcu- 
 lation. The total resistance used in a starting rheostat should 
 be much larger and arrangements should be made to reduce it 
 gradually in order that the motor may not start with a jerk. 
 
 As the maximum torque is usually designed to occur at a slip 
 between one and one third and two times that corresponding to 
 the normal full load torque, when running with the winding 
 resistance only in the secondary circuits, the armature and field 
 currents at maximum torque do not exceed twice the full load 
 currents, so that resistance inserted in the armature circuits 
 
 as follows: As already shown, f s 2m = 
 
 R t 
 
 [R* + (x 1 + x 2 y ]*, 
 
 R% + R e 
 
 = 1 or R e = [Rf + (X x + X 2 ) 2 ]* - R r The 
 
 [R^ + (X 1 + X 2 n h 
 
 t Art. 192. 
 
 * Art. 192. 
 
836 
 
 ALTERNATING CURRENTS 
 
 serves the double purpose of increasing the starting torque and 
 keeping the starting current within bounds. 
 
 This plan is largely used by many commercial companies. 
 The arrangement of the starting rheostat depends largely upon 
 the type of the secondary windings to which it is applied. 
 
 In secondary windings of the squirrel cage type the conductors 
 may be tipped at one end with tapered high resistance metal 
 strips, which come in contact with a sliding copper ring. At 
 starting, this ring may just touch the high resistance tips, and 
 as the machine speeds up the ring may be slid along until the 
 tips are cut out and the copper armature conductors are directly 
 connected together through the ring. If the armature revolves, 
 it is evident that the ring must be arranged to slide on a spline 
 on the shaft, and to be controlled by a grooved sliding collar 
 and loose lever. The same device may be used for secondary 
 windings with coils having a common short-circuiting point. 
 In this case one set of coil terminals are permanently connected 
 together, and the other set are connected into a rheostat of high 
 resistance metal strips, which may be short-circuited by a slid- 
 ing ring, as already explained. 
 
 If the secondary windings are arranged so as to have but one 
 coil for each phase, the introduction of resistance is very simple, 
 since only one resistance coil for each phase is required. In 
 this case, if the secondary core is stationary, the connections of 
 the rheostat are made directly into the secondary circuits, or 
 between the secondary circuits’ and one point of common con- 
 nection ; while if the secondary core revolves, collector rings 
 may be placed on the shaft, and stationary rheostats may be used 
 to control the resistance of the coils which are properly con- 
 nected to the rings, or the resistance may be placed inside the 
 armature spider and be controlled by a sliding collar and loose 
 lever. Figure 478 shows a tri-phase motor having three col- 
 lector rings for connecting up external starting and regulating 
 resistance to the secondary windings. The rheostat coils can 
 be connected in wye or delta. A quarter-phase winding would 
 preferably use four rings and two resistance devices. 
 
 By using this method the motor can be regulated for any 
 torque up to its maximum value for all speeds, from standstill 
 to the speed at which the machine drives its load when the sec- 
 ondary winding is short-circuited. The method is effective for 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 837 
 
 speed control, but has the disadvantage of absorbing much 
 power in the external resistance when the speed is made much 
 below that normal for the given load when the secondary wind- 
 ings are short-circuited without external resistance. Also, 
 with a given resistance inserted, the speed varies with the load. 
 These conditions are somewhat analogous to those of a direct- 
 current shunt- wound motor when the speed is controlled 
 through the introduction of external resistance into the arma- 
 ture circuit. 
 
 Fig. 478. — A Rotary Field Motor with Tri-phase Secondary Winding arranged with 
 Collector Rings for the Introduction of External Starting or Regulating Re- 
 sistances. 
 
 4. Commutated Armature . — The armature may be wound 
 with the coils so arranged that, instead of starting with all 
 conductors in cumulative series in each coil, a portion of the 
 conductors are connected in opposition to the others at the 
 start, and are then reversed and connected properly in series 
 with the others after the machine is in operation. The opposi- 
 tion arrangement affects the current both of the secondary and 
 primary circuits. 
 
 Again, the connectors, by means of which the armature con- 
 ductors are placed in series, may be made of high resistance 
 material, and these connectors may be excluded from the cir- 
 cuit when the conductors are short-circuited together after the 
 
838 
 
 ALTERNATING CURRENTS 
 
 motor is running at full speed. Devices of this character can 
 be used for giving two motor speeds through the range of torque 
 for which the motor is designed, but any additional speeds would 
 make cumbersome connections. 
 
 5. Commutated Primary Windings. — In order to enable a 
 motor to run at two speeds without loss in external secondary 
 resistance, the number of poles of the rotating magnetic field 
 may be changed by properly arranging the primary windings. 
 Thus, a motor having a primary winding for eight poles may 
 have its speed doubled by so connecting the primary coils that 
 the resultant flux has four poles. This, as may be seen by 
 a study of primary windings, requires quite a little shifting 
 of coil terminals, which becomes burdensome if the method 
 is used for obtaining more than two speeds. To obtain the 
 same motor torque at the double speed requires one half as 
 many primary coils per phase, since the number of poles of the 
 rotating field is cut in half, while the primary and secondary 
 magneto-motive force must remain constant to supply the neces- 
 sary torque, since the flux per pole is doubled but the polar 
 area is also doubled. Changing from wye to delta connection 
 for starting purposes is referred to on page 857. 
 
 6. Concatenation Control. — Where two like motors are con- 
 nected rigidly to the same mechanical load, as when two poly- 
 
 Fig. 479. — Diagram of Connections of Tri-phase Motors for Concatenation Control. 
 
 phase motors are used in driving an electric car, a speed control 
 may be obtained which is somewhat analogous to the series- 
 parallel control of two direct-current electric traction motors. 
 Figure 479 shows two motors, both having unity ratio of trans- 
 formation between their primarjr and secondary windings, con- 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 839 
 
 nected in this way, which is called the Tandem or Concatenated 
 connection, ready to start. Motor A is connected directly to 
 the supply mains ; its secondary windings are connected to the 
 primary windings of motor _B, which thereby is supplied with 
 current of frequency corresponding to the slip of motor A ; the 
 secondary windings of B are connected to an external set of 
 rheostats ; and the shafts of the two motors are rigidly joined 
 mechanically. The ratios of transformation of the motors are 
 equal, and the motors are otherwise alike. When motor A 
 begins to rotate, motor B is caused to rotate through the 
 mechanical connection. As the resistance in the secondary 
 circuits of motor B is cut out, the speeds rise until, when the 
 resistance is short-circuited, the motors are running at some- 
 thing under one half the synchronous speed pertaining to each 
 when supplied with voltage at the frequency of the mains. 
 They cannot run above half speed and carry any considerable 
 load, for in that case motor B would run above synchronism 
 with the current supplied to its primary windings from the sec- 
 ondary windings of A , and this would convert motor B into an 
 asynchronous generator * tending to force .current back into the 
 secondary circuits of A. The faster the motor armatures rotate, 
 the smaller is the slip of A and the lower the frequency of the 
 current it supplies to B. At a speed of revolution equal to 
 one half synchronous speed for motor A , the rotating flux of B 
 and its armature conductors would be in exact synchronism. 
 Then the only current that could flow from the secondary cir- 
 cuits of motor A would be that delivered to the primary circuits 
 of B , which, under the conditions named, would be the exciting 
 current necessary to set up the rotating magnetic field of B. 
 Therefore neither motor would produce useful torque at this 
 speed. Since the resisting moment of the load must be equaled 
 by the electrical torque, this state could not exist when any 
 considerable load is to be driven ; and, therefore, the speed of 
 rotation of the armatures when driving a load must be less than 
 one half normal speed by an amount which results in currents 
 flowing in the windings sufficient to produce the necessary 
 torque. At this point the slip in B (with respect to the field 
 frequency provided by current from the secondary circuits of 
 A) is sufficient to induce enough secondary voltage to drive 
 
 * Art. 194. 
 
840 
 
 ALTERNATING CURRENTS 
 
 current through its secondary winding. The slip in A (with 
 respect to the field frequency provided by current from the 
 mains) is sufficient to induce enough secondary voltage in its 
 secondary windings to drive the current through the imped- 
 ance of those windings plus as much more as is required to 
 provide the primary terminal voltage at B. 
 
 The primary current in each phase of A is the sum of a com- 
 ponent equal and opposite to the secondary current of B, the 
 exciting current of B , and the exciting current of A. The 
 currents in the secondary circuits of A are equal to those in 
 the primary circuits of B. The primary voltage is absorbed by 
 the drop due to the mutually induced voltage of A, the imped- 
 ance of the primary and secondary windings of A, the mutually 
 induced voltage in B , and the impedance of the primary and 
 secondary windings of B. 
 
 The induction motor locus diagram given in Fig. 463 can be 
 readily applied to this case by letting the current locus which 
 represents the current in the secondary windings of B have a 
 diameter equal to the voltage of the mains divided by the sum 
 of the leakage reactances of the two motors, and the exciting 
 current have a value equal to the combination of the exciting 
 currents of the two motors. The quadrature voltage in the 
 voltage locus diagram of Fig. 463 is then the sum of the 
 leakage drop voltages ; and the active voltage is the sum of 
 the drops through the counter-voltages and resistances of the 
 two motors. If the ratios of transformation of the motors are 
 not both exactly alike or the impedances of the two motors are 
 not equal, one motor will “ rob ” the other of load with danger 
 of its becoming overloaded; which is comparable to the results 
 in the case of two rigidly connected series direct-current motors 
 which have different armature resistances or field strengths. 
 
 As a result of this discussion the conclusion can be safety 
 drawn that the rotary field induction motor does not, in its 
 present form, lend itself readily to wide, economical variation 
 of speed. Where such speed variation is demanded, series or 
 repulsion motors, which are described in later pages, are more 
 naturally fitted. 
 
 197. Reversing Polyphase Motors. — Potyphase motors may 
 be reversed by reversing the direction of rotation of the rotat- 
 ing magnetic field. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 841 
 
 In quarter-phase motors with two independent circuits, re- 
 versing the terminal connections of either circuit will effect the 
 
 reversal of rotation, but reversing the terminals of both circuits 
 will not alter the direction of rotation. Quarter-phase motors 
 
 Fig. 480. — Curves for showing the Effect of Third and Fifth Harmonics on the Operation of a Quarter-phase Induction Motor. 
 
842 
 
 ALTERNATING CURRENTS 
 
 with three-wire connections may be reversed by interchanging 
 the connections of the outside conductors, leaving the common 
 return conductor unchanged. 
 
 The direction of rotation of tri-phase motors may be reversed 
 by interchanging the connections of any pair of leads. 
 
 198. Effect of the Form of Curves of Voltage. — The effect of 
 distorted curves of voltage upon the operation of induction 
 motors depends upon the number of phases. The harmonics of 
 three and five times the fundamental frequency are the only 
 ones which need be considered ; and indeed, that of three times * 
 the fundamental frequency is the only one which as a rule has 
 an appreciable influence. In single-phase motors the harmonics 
 should affect the magnetic field as they affect that of a trans- 
 former, so that peaked voltage curves should cause a decrease in 
 core losses, and the operation of the motor should not be other- 
 wise greatly influenced. In polyphase motors, however, the 
 harmonics may set up a rotating field of their own, which is 
 superposed upon the regular held, and may interfere with the 
 operation of the machine. 
 
 The harmonics with three times the fundamental frequency 
 belonging to the two circuits of a quarter-phase system have a 
 phase difference of 90° (Fig. 480), and these set up a superposed 
 rotating field in the induction motor which has a field velocity 
 of three times that of the main field. The figure shows that 
 the harmonics of ti'iple frequency, belonging to the two phases, 
 are reversed in relative position compared with the fundamental 
 waves. The field due to these harmonics rotates in the reverse 
 direction from that of the main field, and therefore tends 
 directly to decrease the torque of the motor and to increase the 
 slip. The field due to the harmonics of five times the frequency 
 rotates in the same direction as the main field, and its only dis- 
 advantageous effect is in causing eddy currents which may 
 slightly decrease the efficiency of the motor. The frequencies 
 of the harmonic curves are indicated in the figure by subscripts. 
 
 In tri-phase circuits the harmonics of triple frequency belong- 
 ing to the different currents are directly superposed in phase as 
 is shown in Fig. 481, and therefore the superposed field which 
 they cause in tri-phase induction motors is a stationary one 
 whose influence is only to decrease the efficiency by setting up 
 extra losses. The figure also shows that the harmonics of five 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 843 
 
 frequencies have 120° difference of phase and are in reversed 
 orders, so that they set up a reverse rotating field, and if they 
 are of much strength they may affect the torque. 
 
 199. Single-phase Induction Motors. — If the field magnet of 
 an induction motor is wound with one set of coils so that the 
 
 Fig. 481. — Curve showing the Effect of Third and Fifth Harmonics on the Operation of a Tri-phase Induction Motor. 
 
844 
 
 ALTERNATING CURRENTS 
 
 field poles are set up by a single alternating current flowing in 
 the coils, the poles will be stationary but alternating, and the 
 effects of electro-magnetic repulsion may be utilized for the pur- 
 pose of causing the armature to rotate. Thus, if a uniformly 
 wound short-circuited armature (such as is used for polyphase 
 induction motors) is started to revolving in a single-phase 
 alternating magnetic field, the balance of repulsions which ex- 
 ists when the armature is at rest is disturbed, and the armature 
 tends to continue its motion. To illustrate this, the condition, 
 of two coils in complementary positions with reference to one of 
 the poles may be considered. As the armature revolves, one 
 coil moves toward a position where it includes more lines of force 
 from the pole, and the other coil moves so as to exclude lines of 
 force. If the strength of pole is rising, the first coil will have 
 the larger current induced in it, since the rate of change of lines 
 of force through the first coil is proportional to the sum of the 
 rate of change in the strength of field and the rate of change of 
 flux linkages due to the motion, while the rate of change of lines 
 of force through the second coil is the difference of these two 
 rates. Thanks to the lag in the coil circuits, the currents in 
 both coils are in such a direction as to result in an attractive 
 force on the pole, but a much stronger force is experienced by the 
 first coil than the second. When the field is falling, the mag- 
 netic condition of the second coil is changing the more rapidly, 
 but the direction of the induced cui’rents in the coils is reversed 
 with respect to the direction of the inducing field, and the coils 
 experience a repulsive force with reference to the pole. The 
 effect during one complete period of the magnetic field therefore 
 tends to cause the armature to rotate in the same direction in 
 which it was started. The torque is a maximum when the posi- 
 tive product of current and magnetic flux is a maximum, which 
 is when the current lags behind the induced voltage by an angle 
 between 45° and 90°. The torque at an)' speed is equal to 
 the torque which would be given by a polyphase induction 
 motor of similar construction at the slip v = V — V minus the 
 torque which the polyphase induction motor would give in a 
 field of double the frequency, and with a slip proportioned to 
 V + T r '=2T r —v; where Y is synchronous speed, V' actual 
 speed, and v relative speed in revolutions per minute ; but with 
 the ordinary ratio of the resistance and inductance in the arma- 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 845 
 
 ture winding, the torque due to the latter slip is negligible at 
 such full load slips as are satisfactory in practice. In this case, 
 V need not be looked upon as a speed of rotation of a magnetic 
 field, but as the speed of the armature which keeps each con- 
 ductor at the same position with reference to a pole for any fixed 
 
 instant in each period of the magnetic flux. Hence V — 
 
 P 
 
 exactly as in rotary field machines. When the armature is 
 t stationary, v = V, and the two torques are equal and opposite. 
 A single-phase induction motor may therefore be designed in a 
 manner similar to the manner of designing a polyphaser in re- 
 spect to its operation after it has reached its normal speed, but 
 it requires special treatment in the design for the purpose of 
 making it self-starting, and it is both heavier and less efficient 
 for a given output. 
 
 The principal voltages induced are indicated in the dia- 
 gram of Fig. 482 for a bi-polar machine. The alternating 
 magneto-motive force 
 impressed by current 
 flowing from the ex- 
 ternal circuit through 
 the primary or field 
 coils is represented by 
 the arrow CD and the 
 designation N-S and 
 S-JV. The resulting 
 alternating magnetic 
 flux set up through 
 the armature core sets 
 up counter-voltages in 
 the secondary or arma- 
 ture windings, as in 
 any transformer. The 
 direction and position 
 of the voltage in the armature conductors are indicated by the 
 arrow points (dots) and feathers (crosses) marked in the figure 
 within the cross-sections of the conductors which are indicated 
 by small circles. This voltage is in time quadrature with the 
 primary flux. These conditions are as in a transformer with 
 large magnetic leakage and may be indicated by the ordinary 
 
 Fig. 482. — Diagram for indicating the Voltage gen- 
 erated in a Single-phase Induction Motor. 
 
846 
 
 ALTERNATING CURRENTS 
 
 transformer diagram.* The turns across the vertical diam- 
 eter have the highest voltage induced in them. (The arma- 
 ture, for convenience, is assumed to be of the squirrel cage 
 type and the primary windings to be distributed windings, 
 passing through the holes from the top to the bottom of the 
 figure.) This voltage decreases more or less sinusoidally until 
 the horizontal turns are reached, which are in a neutral posi- 
 tion with regard to N-S. When the armature revolves, vol- 
 tages as shown by the arrowheads and feathers indicated in 
 the inner circle are set up in the conductors by their mo- 
 tion through the primary field. These voltages are propor- 
 tional in strength to the field N-S and hence are in time phase 
 therewith. They reach a maximum value in the conductors 
 under the poles N-S and S-N and decrease more or less sinus- 
 oidally to zero in the turns 90° therefrom, which do not cut 
 the field flux. There are, therefore, two sets of voltages in the 
 conductors, the first created by ordinary transformer action 
 caused by variation of the field N-S, and the other by the 
 motion of the conductors through the lines of force of the same 
 field. These two voltages are at ninety degrees displacement 
 with reference to each other, both in time phase and space 
 position. Two components of current, conjointly composing 
 the armature current, will flow under the influence of these 
 two voltages. The fii’st reacts on the primary circuit like the 
 secondary current of a transformer, and has a time lag due to 
 leakage flux only ; while the second or “ speed ” current sets 
 up magnet poles at n-s and s-n , and as the reluctance of the 
 magnetic path is low, this component tends to take a time quad- 
 rature relation with its inducing’ voltage. Hence JY-S and n-s 
 tend to rise and fall nearly in time quadrature and exactly in 
 space quadrature. 
 
 When the armature is running at the speed corresponding to 
 theoretical synchronism for the frequency of the primary cur- 
 rent, each turn of wire on the armature in each revolution 
 cuts the same number of lines of force as are comprised in the 
 maximum value of the field JY-S in one cycle of the magnetic 
 flux, so that the transformer and speed voltages are equal. 
 Therefore, the fields JY-S and n-s are then of approximately 
 equal magnitude. Under these conditions N-S and n-s may be 
 
 * Arts. 124 and 129. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 847 
 
 considered to form an ordinary rotating field acting upon a short- 
 circuited armature, as in polyphase motors. When the speed 
 is lower than the theoretical synchronous speed, the speed vol- 
 tage is proportionately less; and at standstill the speed voltage 
 is zero. 
 
 The foregoing shows that it is possible to consider the single- 
 phase motor in the light of a motor-generator in which the 
 transformer component of current is a torque producing current 
 acted on by the poles n-s , and the speed current is a generated 
 exciting current which sets up the motor fields n-s. The 
 counter-voltage caused by the conductors cutting flux n-s tends 
 to oppose the effective transformer voltage in driving current 
 through the armature resistance. The product of counter- 
 voltage with transformer current equals the output of the 
 motor. If the load is changed, the motor speed must change 
 so that the difference between the transformer and counter 
 voltages will permit a transformer current to flow which will 
 give the requisite torque. 
 
 Again, the single-phase alternating field may be treated in a 
 different manner to get the same result more simply. An 
 alternating field stationary in position may be resolved into two 
 rotary fields, revolving in opposite directions, having the same 
 frequency as the stationary field, and of one half its magnitude 
 in strength.* This is similar to the principle of mechanics by 
 which a simple harmonic motion may be resolved into two uni- 
 form opposite circular motions of one half the amplitude. The 
 torque diagrams of each of these fields acting alone are shown 
 in Fig. 483, where 0 is the point of armature rest, and arma- 
 ture speed is counted from that point along the horizontal axis. 
 The curves A and A' are the torque curves that would be given 
 by either field acting alone, the torque due to one being in one 
 direction, and that of the other in the opposite direction. It is 
 evident that when the armature is at rest it has no tendency to 
 revolve, as the slip of the armature with respect to the two 
 fields is equal, and torques Ot and Ot' created by the two fields 
 are equal and opposite ; but if the armature is started in one 
 direction, for instance toward the right, the slip with respect to 
 field A decreases, the torque caused by it increases and tends 
 
 * Ferraris, A Method for the Treatment of Rotating or Alternating Vectors, 
 London Electrician, Vol. 33, p. 110. 
 
848 
 
 ALTERNATING CURRENTS 
 
 to continue the rotation, while the slip with respect to field A' 
 increases, and the torque caused by A! decreases. When the 
 armature speed becomes V, the torque caused by A is T, which 
 is due to a slip V— V=v; while the torque caused by A' is 
 T\ which is due to a slip in relative speed between armature 
 and field of V+ V = 2 V— v. From the relations of torque to 
 slip, which have already been discussed,* it is evident that the 
 torque caused by A' decreases as the relative speed increases. 
 If the differences between the corresponding ordinates of the 
 curves of torque A and A! are plotted in a curve, the actual 
 
 Fig. 483. — Motor Torque Diagram of a Single-phase Motor with the Field resolved 
 into Two Component Rotating Fields having Opposite Directions of Rotation. 
 
 torque curve M is given. The ordinates of this will give the 
 actual motor torque with respect to slip. From this curve it 
 is seen that the motor will work at no load at an almost syn- 
 chronous speed, and may then be loaded until the speed has 
 dropped to a point where the torque is at a maximum. If the 
 load exceeds this, the motor will stop. The curve M falls to zero 
 a little to the left of 4- T( because of the torque T' . However, 
 as the backward field is, at synchronism, cutting the armature 
 conductors at a rate of S' 2 = 2 /, the reactance (2 fX 2 ) against 
 that field is very great and the ordinates of T' (exaggerated in 
 the figure) are ordinarily negligible for working speeds. From 
 
 * Art. 192. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 849 
 
 the point + F to + V there should be a double frequency com- 
 ponent of current in the armature and a slight negative torque. 
 The ordinary values of resistance and inductance which are re- 
 quired in an efficient and economical design, make the effect of 
 A' so small at the speed of normal full load that the action 
 of A only need be considered. 
 
 If the armature should be started toward the left, instead of 
 the right, as here assumed, the conditions would be reversed 
 and the motor would operate under the torque line M'. Evi- 
 dently the formula from which the torque curve M (Fig. 483) 
 can be plotted is 
 
 T — K •b^’r^-2 1 _ 
 
 C^l + ^Y) 2 + s 2 2 (^1 + -Y 2) 2 
 
 _ X { - f — h) Ffp/Y 
 
 (2/Bj ~h^i + ^ 2 ) 2 + (2/-s 2 ) 2 (X 1 + X 2 ) 2 
 
 The second term becomes small for armature speeds approaching 
 synchronism. It will be appreciated, on account of the high 
 reactance (2 /— s 2 )X 2 , that the magnetic flux of this field which 
 actually passes through the armature windings is very small 
 when s 2 is small, so that if <£>' represents this flux, A>' = <3? — <Eq', 
 when <l> is the backward rotating component of the flux in the 
 field, and <Eq' is the leakage component of that flux. Evidently 
 <E> £ ' is always large and approaches closely to the value of <1> 
 when s 2 is small. 
 
 From these considerations it is seen that a single-phase motor 
 may be designed in the same manner as a polyphaser, and that 
 for equal output the ampere-turns upon the field magnet must 
 be equal to the resultant number on a polyphase motor. The 
 field winding may be arranged as in polyphase motors, cover- 
 ing the entire polar surface, as is shown in Fig. 484, for a four- 
 pole machine; but the differential action in this case reduces 
 
 2 
 
 the effectiveness of the winding in the proportion of 1 : as 
 
 7 r 
 
 lias already been shown in Art. 188. Consequently, the equa- 
 tion from which the field windings are determined becomes 
 
 xr/ _ x-V2 7 rn'A>f_ 2 V2 vn'^f _ 2 V2 
 
 — 108 10 8 
 
 The value of K may be increased and material saved by leaving 
 3 1 
 
850 
 
 ALTERNATING CURRENTS 
 
 space between the primary coils, as in Fig. 485, which shows 
 the windings for a two-pole field. 
 
 The efficiency of single-phasers is less than that of poly, 
 phasers, since the armature core losses are proportional to the 
 
 
 
 llllllllllllll 
 
 M 
 
 
 
 . 
 
 T 
 
 T i 
 
 
 Fig. 484. — Diagram of a Single-phase Induction Motor with Primary Winding dis- 
 tributed over the Entire Surface of the Field Core. 
 
 frequency of the main field instead of to the slip, and the I 2 R 
 losses are greater ; their slip for a given load and similar de- 
 sign is greater; and their maximum torque is slightly less 
 than that of polyphasers, as is shown by Fig. 483 ; but these 
 
 Fig. 485. — Diagram of a Single-phase Induction Motor with the Primary Winding 
 Grouped in Coils covering Part of the Field Core. 
 
 differences in well-designed machines should not be great. 
 The weight of single-phasers is larger than that of equal poly- 
 phasers, because the value of K is smaller. 
 
 200. Locus Diagram of the Single-phase Motor and Substi- 
 tuted Impedance. — As the single-phase motor is equivalent to 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 851 
 
 a transformer having constant reactance and varying load of 
 unity power factor, it may be represented by a current locus 
 diagram similar to that of a transformer or polyphase induction 
 motor. Thus Fig. 486 shows such a diagram, constructed and 
 lettered similarly to the diagram of Fig. 463, which represents 
 the condition of one branch or phase in a polyphase induction 
 motor. The only important element of difference is found in 
 
 Fig. 486. — Locus Diagram of a Single-phase Induction Motor. 
 
 the no load current SO. This shows an active component 
 which is greater in proportion, due to the larger armature 
 iron losses, and a wattless current which is greater in propor- 
 tion, due to the backward rotating components of the field. 
 
 The line OF' is as before the added component of secondary 
 current due to the load, but the total secondary current is 
 greater than this by some component SS' because of the arma- 
 ture current set up by the backward rotating component of the 
 magnetic flux. Referring to Fig. 483, the total armature cur- 
 rent found by adding the forward and backward components of 
 the current may be roughly considered, for a given load, to have 
 
852 
 
 ALTERNATING CURRENTS 
 
 the value F'S' (Fig. 486), and to have the relations to the im- 
 pressed voltage OF, the total primary current SF', and the phase 
 angles which are shown in the diagram. This is under the 
 assumption that the backward rotating component of the primary 
 current remains constantly of uniform value, since its function is 
 to set up the backward rotating component of the field magnetic 
 flux <!>'. The forward rotating component of the field current, 
 on the other hand, must not only have a no-load component S' 0 
 sufficient to set up its component of the rotating flux <E>, but 
 must also neutralize the mutual magnetic effect of the armature 
 current component OF', which is the t.orque-producing com- 
 ponent of the armature. Upon the same suppositions as used in 
 the diagram of Fig. 466, where the armature current and voltage 
 were considered to be in phase with each other, F' H is propor- 
 tional to the flux necessary to induce the voltage which drives 
 the current F'O through the armature. The diameter of the 
 locus semicircle is found in the same way as for the semicircle 
 representing a phase of a polyphase motor, as are the various 
 elements of performance. KJ represents the slip line con- 
 structed as in Fig. 466; the loci of copper loss may also be 
 shown as in that figure. 
 
 Impedances can be substituted for the single-phase motor 
 circuits, for the purpose of making computations, as was done 
 with the polyphase motor.* Referring to Fig. 464, the shunt 
 circuit representing the no-load losses should be connected at 
 cd instead of ab , in order that great accuracy may be obtained 
 in representing the conditions of a polyphase motor. In the 
 single-phase motor a similar arrangement is required for great 
 accuracy, and allowance must be made for the extra reactive 
 effects of the exciting currents ; but for most purposes it is suf- 
 ficiently accurate to have the no-load circuit connected from a 
 to b , which will take a current equal in phase and quantity to 
 the no-load current, and it is upon the basis of such an approx- 
 imation that the locus diagram Fig. 486 is drawn. 
 
 201. Starting Single-phase Induction Motors. — Since single- 
 phase induction motors are not per se self-starting, special start- 
 ing devices must be included in their design and construction. 
 This sometimes takes the form of what is called a Phase splitter. 
 The field is wound with two coils similar to the windings of a 
 
 * Art. 192. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 853 
 
 two-phaser, and at starting these are connected in parallel to 
 the circuit, one directly, and the other through dead resistance 
 or capacity. This throws the currents in the two coils into a 
 difference of phase, which may be accentuated by winding one 
 coil so that it has greater self-inductance than the other ; and 
 the machine then starts as a two-phaser. After the machine is 
 running, both coils are connected directly to the circuit, or one 
 coil is cut out, and the motor operates as a single-phaser. 
 This operation of “ phase spliting,” as applicable to such motors, 
 cannot give a large difference of phase between the currents in 
 the two motor circuits with a reasonably large power factor, and 
 consequently single-phase induction motors started in this way 
 must have either a very small starting torque or an unreasonably 
 small power factor at starting. Another method frequently 
 used in small motors, such as small fan motors, is to use Shading 
 coils. That is, the motor field magnet may be constructed with 
 salient pole pieces like an alternator field magnet, — the core of 
 course being laminated. Around a similar tip of each pole piece 
 is wound a short-circuited coil. The currents in the short- 
 circuited coils tend to oppose the growth of magnetic flux under 
 these pole tips, and hence it rises to a maximum first in the un- 
 wound tips, and then in the wound tips, causing the effect of a 
 two-phase winding. 
 
 A more satisfactory method is to start the motor as a repul- 
 sion motor,* but this requires a commutator and more expensive 
 mechanism. When this plan is adopted, the armature is wound 
 with an ordinary reentrant direct-current winding connected 
 in the usual manner to a commutator, and it is self-starting 
 with large torque when a short-circuiting connection joins the 
 brushes and the brushes are displaced from the neutral plane. 
 When the armature has come to full speed, an automatic device 
 may be used to short-circuit the commutator, and the machine 
 will then run as an induction motor. 
 
 202. Efficiency of Induction Motors and Methods of making 
 Tests. — Polyphase induction motors can be built to give about 
 the same efficiency as direct-current motors, and for somewhat 
 less cost on account of the absence of a commutator and the low 
 insulation required on the armature conductors, but with a 
 counterbalancing extra cost on account of the high grade and 
 
 * Art. 205. 
 
854 
 
 ALTERNATING CURRENTS 
 
 expensive sheet-iron stampings which are required for the field 
 magnet. The most satisfactory design, as already explained, 
 
 calls for a machine of 
 short axial length and 
 large diameter in order 
 that sufficient teeth 
 may be used to reduce 
 magnetic leakage to 
 small proportions. 
 This is well illustrated 
 in Fig. 487, which is 
 from the photograph of 
 the armature of a 300 
 horse-power, 25-cycle, 
 10-pole motor of stand- 
 ard design, having a 
 speed of 300 revolu- 
 tions per minute. The 
 length of the core is quite small compared to its diameter. 
 
 1. By far the quickest method of experimentally obtaining 
 the operating characteristics of an induction motor, including 
 efficiency, torque, regulation, and power factor, is by means of 
 the circle or locus diagram.* The no-load exciting current 
 can be obtained in phase position and scalar value by means of 
 amperemeters, voltmeters, and wattmeters when the armature 
 is running free. The current locus is fixed with approximate 
 accuracy by using the same instruments to determine the phase 
 position and quantity of the standstill current, using a reduced 
 voltage for the measurements and considering the current to be 
 proportional to voltage (or by obtaining it more accurately from 
 the reactances at full load). The vector difference between the 
 standstill and the no-load current is approximately the arma- 
 ture standstill current. The length and position of the vector 
 of the armature standstill current determines the current locus. 
 The resistance of the primary circuit may be measured, and 
 that of the secondary circuit may he obtained by dividing the 
 primary induced voltage (reduced to the secondary circuit by 
 the ratio of transformation) by the product of the computed 
 armature standstill current and the cosine of its angle of lag. 
 
 * Arts. 191, 192. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 855 
 
 Correction can be made for variations in the exciting current if 
 great accuracy is desired. Having thus laid out the locus, the 
 desired operating quantities may be read off for any load. 
 
 2. Direct Measurement. In testing the efficiency of these 
 motors, the output may be measured by a brake or transmis- 
 sion dynamometer, but the input must be measured by one of 
 the wattmeter methods explained earlier. The two-wattmeter 
 method is the best, but care must be taken to determine whether 
 the readings of the two instruments are additive or subtractive, 
 since the power factor of a partially loaded induction motor is 
 likely to be quite low and at no load may be only a few per 
 cent. The power factor is determined by taking simultaneous 
 readings of amperemeter, voltmeter, and wattmeter in one cir- 
 cuit, if the machine is balanced ; but if the circuits differ, read- 
 ings for each circuit must be taken. The power factor is then 
 the true watts divided by the apparent watts. This method 
 requires that the motor shall be operated with its full load if 
 the full load efficiency is to be obtained, and therefore may 
 prove inconvenient, and it does not give any way of separating 
 the losses. 
 
 3. Stray Power Method. — A method similar to that described 
 for testing transformers (Art. 147) is often more convenient and 
 satisfactory. By this plan the core losses, friction, and wind- 
 age losses are determined bj r measuring by wattmeter the power 
 which is absorbed by the- motor when running light under nor- 
 mal voltage and frequency. The losses may be of such value 
 that the field current flowing is considerable, especially as the 
 power factor is likely to be rather low, and the PR loss cannot 
 be neglected, but a correction can be made after the test for 
 copper losses is completed. To measure the copper losses, the 
 machine is locked so as to remain stationary, in which case the 
 armature serves the purpose of a short-circuited secondary cir- 
 cuit, and such a reduced impressed voltage is applied as to cause 
 any desired current to flow in the field winding. The wattmeter 
 readings give the PR losses for the current flowing, and the 
 losses for any other current may be at once calculated. A cer- 
 tain amount of core loss is included in this measurement, but 
 an approximate correction may be made on account of it by 
 considering its ratio to the total corrected core losses as the 
 1.6 power of the voltage applied in the copper loss test is to the 
 
856 
 
 ALTERNATING CURRENTS 
 
 1.6 power of the normal voltage. From these results the total 
 losses and the efficiency at any load may be calculated. 
 
 A motor running, without load, or with part load, on an un- 
 balanced circuit, is likely to absorb widely different amounts 
 of power in its coils ; one coil may even return power to the 
 circuit, while the others absorb the power required for opera- 
 tion plus that returned. In all such cases, the two-wattmeter 
 method of measuring the power gives the net power absorbed 
 by the machine. 
 
 4. Power Factor. — The power factor at any load may also 
 be calculated from the circle diagram or it may be obtained for 
 various loads by means of computations made from the readings 
 of amperemeters, voltmeters, and wattmeters. 
 
 5. Regulation and Torque. — The exact regulation of a ma- 
 chine may be determined by actual running tests under load, in 
 which the actual slip is measured, but the percentage slip may 
 be taken to be approximately equal to the percentage drop of 
 voltage in the armature windings. The starting torque can be 
 measured by clamping a lever upon the pulley, and measuring 
 the pull at the end of a fixed length of arm. For a machine 
 having an improperly divided starting rheostat associated with 
 the armature, this gives a value which is higher than the torque 
 against which the motor will start and run up to speed. In 
 such a machine, the standing torque and starting torque are dif- 
 ferent ; but in a machine having a properly arranged starting 
 rheostat associated with the armature, the maximum standing 
 torque and the maximum starting torque are equal, and are 
 practically equal to the maximum torque which the machine 
 can exert. The torque and slip are, of course, obtainable from 
 the circle diagram. All desired information in regard to the 
 operation of induction motors may be determined bj ? purely 
 electrical measurements, and to a high degree of accuracy for 
 commercial measurements. Using commercial amperemeters, 
 voltmeters, and wattmeters which have been properly calibrated, 
 the errors probably affecting the full load efficiency and power 
 factor need not exceed one per cent. 
 
 The operating characteristics of a 50 horse power, four-pole, 
 tri-phase, 60-cycle, 440-volt motor are given in Fig. 488. This 
 shows a drop in speed of about 5 per cent from no load to full 
 load, while the efficiency rises to about 88 per cent at slightly 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 857 
 
 over one half load. It is noted that the power factor drops 
 rapidly for loads smaller than about one half load. In large 
 machines the efficiency sometimes exceeds 95 per cent, while the 
 
 PER CENT OF FULL LOAD 
 
 Fig. 488. — Operating Characteristics of a Tri-phase 50 H. P. Induction Motor. 
 
 slip is from 2 to 4 per cent at full load. Figure 489 shows the 
 primary current per phase and the torque of a 5 horse power, 
 four-pole, 60-cycle motor, in per cent of their normal values at 
 full load, as functions of the armature speed. The figure there- 
 fore illustrates the conditions during starting, and especially the 
 change in operating characteristics caused by change of voltage 
 impressed upon the coils of a phase, obtained by changing the 
 connections of the field coils from wye to delta. 
 
 Figure 490 shows the curves of torque as a function of speed 
 and of current when a 25-cycle induction motor is operated from 
 various taps of an autotransformer starter. It is to be noted 
 that the starting current is reduced at serious expense of start- 
 ing torque, but that a starting torque equal to full load running 
 torque may be obtained with a current approximating to full 
 load current. The various taps of the autotransformer give 
 respectively 60, 70, 80, and 100 per cent of the full voltage, and 
 
858 
 
 ALTERNATING CURRENTS 
 
 0 10 20 30 40 60 60 70 80 90 100 
 
 SPEED IN PER CENT OF SYNCHRONISM 
 
 Fig. 489. — Curves of Primary Current and Torque of a Small Tri-phase Induction 
 Motor, with the Primary Coils connected in Wye and in Delta. 
 
 PER CENT OF FULL LOAD TORQUE 
 
 Fig. 490. — Torque Speed and Torque Current Curves of a 25-Cycle, Four-pole In- 
 duction Motor when started and run on various Taps of an Autotransformer. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 859 
 
 the starting torques corresponding to the respective taps are 
 therefore in ratio with each other as 36, 49, 64, and 100, that 
 is, as the squares of the respective voltages. The maximum 
 torques for the several voltages, which arise at 55 per cent of 
 synchronous speed in this machine, are in the same ratio, as 
 also are the torques corresponding to the several voltages at 
 any given speed. 
 
 203. Effect of Frequency. — -An examination of the formulas 
 relating to the design of induction motors shows that the fre- 
 quency of the current for which a machine is designed does not 
 greatly affect its efficiency, slip, power factor, or starting torque, 
 but that for a given speed the number of poles selected must be 
 directly related to the frequency. Increasing the number of 
 poles of a given machine reduces the cross section of each pole, 
 but the number of lines of force at each pole is equally reduced, 
 so that the magnetizing current is unaltered. Consequently, 
 induction motors of equal merit may be designed for all reason- 
 able frequencies, though magnetic leakage may interfere with 
 the operation when the poles become too numerous. 
 
 On the other hand, when a machine which has been designed 
 for a certain frequency is operated at another frequency, the 
 synchronous speed is changed in direct proportion to the fre- 
 quency, the percentage slip is practically unaltered, the starting 
 torque varies inversely from the frequency, and the efficiency 
 and power factor both vary in the same direction as the fre- 
 quency because the magnetic density is inversely proportional 
 to the frequency, as in transformers. 
 
 The frequencies which are commonly used with induction 
 motors cover a wide range. In Europe, 50 periods per second 
 is frequently adopted for tri-phase and single-phase motors ; 
 while in this country the frequencies of 60 periods per second 
 and 25 periods per second have been generally adopted. The 
 former is equally suitable for lighting and for stationary power 
 purposes. The latter was early adopted at Niagara Falls and is 
 used for plants where power service is of greater importance than 
 the lighting service. The ratio of these two frequencies being 
 12 : 5 makes frequency changing inconvenient by limiting the 
 numbers of pairs of poles practicable for use on the motor and 
 generator of the frequency changer, within a reasonable range 
 of speeds. The minimum number of poles available for use in 
 
860 
 
 ALTERNATING CURRENTS 
 
 the frequency changing between these frequencies is 5 pairs on 
 the 25-cycle machine and 12 pairs on the 60-cycle machine, giv- 
 ing a speed of 300 revolutions per minute. The next practicable 
 combination is 10 pairs of poles and 24 pairs of poles, giving a 
 speed of only 150 revolutions per minute, and therefore making 
 an unduly expensive machine. If the lower frequency is re- 
 duced to 24 periods per second while the other is maintained at 
 60 periods, rather more flexibility is obtained, as speeds of 720, 
 360, 240, 180, etc., are then available for frequency changers. 
 Nevertheless it would be of distinct advantage to electrical en- 
 gineering in this country if the two commonly used frequencies 
 were standardized at values more readily interconvertible. 
 This would be accomplished if the lower frequency were stand- 
 ardized at 30 periods per second.* 
 
 The frequency of 40 periods per second was adopted in 
 certain of the earlier power transmission plants in this country, 
 and still persists in a few. In most instances the frequencies 
 other than 60 periods per second and 25 periods per second 
 have disappeared in this country. 
 
 204. Polyphase Induction Motor with Exciting Current sup- 
 plied to the Armature. Motor with Unity Power Factor. — If 
 a polyphase induction motor has its armature coils wound as a 
 continuous reentrant winding, and has direct current supplied 
 to the coils through slip rings, so as to form as many magnetic 
 poles as there are poles in the rotating field of the field magnet, 
 the machine becomes a synchronous motor, and the lagging 
 current of the primary windings may be neutralized when the 
 machine is running in synchronism, or a leading current may 
 be drawn from the alternating-current supply mains, f 
 
 Now assume the simple case of a two-pole tri-phase induction 
 motor having a continuous reentrant winding tapped to a com- 
 mutator. Suppose that three brushes bear upon the commu- 
 tator, spaced 120° apart, as in Fig. 491, where B is the armature 
 winding, C the commutator, and d , d, d the brushes. Then, when 
 the armature is at rest, a rotating field may be created by current 
 in its windings by attaching the brushes to three-phase supply 
 mains. If the connections are properly made and the brushes 
 are set at the proper positions, this rotating field may be given 
 the same direction of rotation and space phase as the rotating 
 * Art. 183. t Art. 168. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 8G1 
 
 field set up by windings on the field magnet. If the two 
 magnetic fields are superposed and the direction of their fluxes 
 is the same, the exciting current in the field winding decreases, 
 since the total flux required to link the field winding is limited 
 by the amount necessary to set up a counter-voltage in the 
 primary winding, which is equal to the vector difference of 
 the impressed voltage and the IR drop in the winding. If 
 the current introduced through the brushes is increased by 
 raising the voltage impressed on the brush circuit, the exciting 
 
 Fig. 491. — Diagram of a Polyphase Induction Motor capable of High Power Factor. 
 
 current in the primary winding can be entirely replaced, and 
 further increases of current through the brushes tend to call 
 for a neutralizing leading current in the primary winding. 
 
 Since rotation of the armature does not affect the rotation 
 of the primary field, and the commutator maintains a uniform 
 relation of the magnetic axes of the armature winding with 
 respect to the brushes, the same time and space relations be- 
 tween the two magnetic fields exist, when the armature is run- 
 ning at anj r speed, as exist when the armature is standing still. 
 That is, whatever may be the speed of the armature, its wind- 
 
862 
 
 ALTERNATING CURRENTS 
 
 ing continues to be tapped through the medium of the brushes 
 and commutator at three fixed points in space. Therefore, 
 since the windings spanning between the brushes, in the three 
 segments X , Y , and Z , are at all times unchanged, except di- 
 rectly under the brushes as commutation goes on, the strength 
 and rotation of the field set up by the introduction of the tri- 
 phase currents through the brushes is practically unchanged 
 by changes in the speed of the armature. When the brushes 
 are placed directly under the field taps Gr , 6r, Cr, as shown in 
 Fig. 491, exciting current furnished through the armature will 
 tend to lag, and no advantage will be obtained in increased 
 power factor ; but by shifting the brushes slightly, and over- 
 exciting through the armature brushes, the counter-voltage of 
 the field windings may be made to take such a position with 
 reference to the impressed voltage that a leading current will 
 be drawn from the mains sufficient to neutralize the lagging 
 current to the armature, or even cause a resultant leading cur- 
 rent in the supply mains. 
 
 By joining the commutator bars by resistance A , the motor 
 may be run as an ordinary short-circuited armature induction 
 motor, while the exciting current is being supplied through the 
 brushes d, d , d. Evidently in this case part of the current 
 entering through the brushes is wasted in the parallel path 
 through A , but by making the length of armature coil be- 
 tween the commutators sufficiently small, the resistance of A 
 between brushes can be made great enough to reduce the cur- 
 rent through it to a reasonably small proportion. This gives 
 a motor with good starting torque and good regulation, and also 
 affords opportunity for control of the power factor. Various 
 modified arrangements for accomplishing the purpose may be 
 used ; thus, two separate windings may be placed in the arma- 
 ture slots, one to serve purely in the capacity of a commutated 
 field magnetizing winding, and the other a short-circuited wind- 
 ing, which makes unnecessary the use of resistance A shown in 
 the figure. The plan is equally applicable to single-phase and 
 polyphase machines. 
 
 After the motor has been started in this way upon the high- 
 starting torque and power factor, the brushes can be raised and 
 the commutator can be short-circuited, thus reducing the ma- 
 chine to an ordinary induction motor having all of the advan- 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 863 
 
 tages of low armature resistance, or it may be allowed to run 
 as a self-regulating motor of high power factor. In order to 
 start the motor through the armature, the requisite exciting 
 current can be obtained by connecting the brushes d, d, d to 
 the supply mains through the medium of an autotransformer, 
 or transformer having a suitable number of voltage regulation 
 taps, as shown at _F, Fig. 491. The effect of such lagging cur- 
 rent as flows to the transformer F can be neutralized by properly 
 placing the brushes. 
 
 205. Electromagnetic Repulsion and Repulsion Motors. — If a 
 
 coil of wire is held in an alternating magnetic field in such a 
 way that the lines of force pass through its turns, an alternating 
 voltage is set up in it which has 90° difference of phase from 
 the alternating magnetic flux. This in turn causes a current 
 in the coil, and the coil experiences a force at each instant tend- 
 ing to move it in the magnetic field, which is proportional in 
 magnitude and direction to the product of the corresponding 
 instantaneous values of current and magnetic flux, paying due 
 attention to their relative algebraic signs ; and the average force 
 for a cycle of flux is equal to the average of the instantaneous 
 torques during the period. If the coil could have no self- 
 inductance, and the phase of the current could therefore be in 
 quadrature with that of the magnetic flux, the average forces 
 during alternate quarter periods would be equal but in opposite 
 directions (compare Fig. 199), and the average force during 
 a whole period would be zero, so that the coil would have no 
 tendency to move ; but in all practical cases a coil, or even 
 a flat disk, must have some self-inductance, so that the current 
 lags behind the impressed voltage, and the current phase is 
 therefore more than 90° behind the phase of the magnetism. 
 In this case the instantaneous values of the force, when plotted 
 in a curve, give a figure similar to the dotted curve in Fig. 200. 
 The ordinates of the large loops represent a negative or repul- 
 sive force, and the ordinates of the small loops a positive or 
 attractive force, and the summation of the instantaneous forces 
 during a period has a finite negative value. This shows that 
 the coil experiences a repulsive force which tends to move it 
 out of the magnetic field. If the coil is pivoted, the force tends 
 to turn it into such a position that the lines of force of the field 
 do not thread through its turns. The conditions here set forth 
 
864 
 
 ALTERNATING CURRENTS 
 
 were first fully explained and illustrated in a remarkable lecture 
 by Professor Elihu Thomson.* 
 
 If an armature is wound with uniformly spaced short-cir- 
 cuited coils or conductors, the repulsive effects in the different 
 
 coils will balance each other when 
 the armature stands still ; but if 
 the coils have their independent 
 ends separately connected to the 
 opposite bars of a commutator 
 having as many bars as there are 
 sets of conductors in the arma- 
 ture, brushes may be so arranged 
 as to short-circuit each coil when 
 
 Fig. 492. — Diagram for Showing the ^ ^ a position to gi\e a force 
 Principle of a Repulsion Motor with in one direction. This arrauge- 
 Open Circuit Armature Coils. , . j , r> r 
 
 ment was suggested by Professor 
 Thomson,! and is illustrated in Fig. 492, and represents the 
 first type of repulsion motor. The motor is self-starting, and 
 runs by virtue of the repulsion between the magnetic field and 
 the coils, which, as they come into the active position, are short- 
 circuited by the brush connections. Such a motor is bulky, 
 inefficient, and expensive, since only a portion of the armature 
 can be made continuous^ effective. 
 
 A better method of arranging a repulsion motor is to use an 
 armature with a re-entrant winding and commutator like a 
 direct-current armature, the field magnet being laminated, as 
 originally developed by Anthony, Jackson, and Ryan. The ar- 
 rangement is as illustrated in Fig. 493. In this figure the lead 
 wire cc makes a short-circuiting connection between the brushes 
 b , b' which rest on the commutator to which the armature wind- 
 ing is attached. The brushes are usually set a little less than 45° 
 from the neutral plane. Then the alternating poles N-S induce, 
 by transformer action, currents in the armature winding which 
 is short-circuited through the brushes and conductor cc The 
 voltage induced in the armature winding is zero when meas- 
 ured between commutator' bars lying on a diameter perpendic- 
 ular to the magnetic axis of the inducing flux, and is a maximum 
 when measured between commutator bars on a diameter parallel 
 
 * Novel Phenomena of Alternating Currents, Trans. Amer. Inst. E. E.. Vol. 
 4, p. 160. t Trans. Amer. Inst E. E., Vol. 4, p. 160. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 865 
 
 to the flux. A large current flows through conductor cc when 
 the brushes are placed in the latter position, but the currents 
 flow in the armature conductors so that they neutralize each 
 other’s torque. When the brushes are placed in the position 
 at right angles, no current flows and therefore no torque is pro- 
 duced. Consequently, the brushes are placed in a compromise 
 position, giving a considerable current flow accompanied by the 
 jv oduction of a considerable torque. 
 
 A convenient method of determining the action in the repul- 
 sion motor is to resolve the magneto-motive force OP of the 
 field magnet into two quadrature components, one of which, 
 OM, is in line with the short-circuited brushes, and the other, 
 OF, at right angles thereto. The field windings are wound 
 through the holes or slots in the outer ring of Fig. 493, and 
 produce the magneto-motive force OP, proportioned to their 
 ampere-turns. Now, since the vector expression OP= OF + OM 
 holds true, we can replace these field windings by two sets in 
 space quadrature, but connected electrically in series, such that 
 one set, which we shall call the torque or F coils, creates the 
 magneto-motiye force OF, and the other, which we shall call 
 the mutual or M coils, creates the magneto-motive force OM. 
 
8G6 
 
 ALTERNATING CURRENTS 
 
 Assume first an ideal motor in which the F and M coils and 
 the armature winding are of negligible resistance. Remember 
 that the armature winding is of the closed coil variety and is 
 short-circuited between the brushes by wire cc. Assume also 
 that the reluctance of the magnetic circuit is uniform and the 
 magnetic leakage zero. If the armature is stationary and an 
 alternating voltage E is impressed across the extremities of the 
 F and M coil circuit, causing a current I to flow through the 
 circuit, an alternating magneto-motive force is set up along 
 the line MM\ and another along the line FF' by reason of the 
 M and F windings respectively. The flux from the M wind- 
 ings or poles induces in the armature windings bands of cur- 
 rents which complete their circuit from b through c to b\ The 
 armature windings having negligible resistance and leakage in- 
 ductance by assumption, this armature current I M flows with 
 
 negligible voltage drop. The value of this current is Im = ^Z 
 
 8 
 
 where s" is the ratio of transformation of the transformer thus 
 formed between the M coils and the armature windings or A 
 
 coils, or s" = — , where n A and n M are the equivalent numbers 
 n M 
 
 of turns in the A and M coils respectively. The resistance of 
 the M coils being also assumed to be negligible, I M in the A 
 coils acts to destroy the mutual reactance of the M coils, like 
 the current in the secondary circuit of any transformer; then, 
 there being no induced voltage drop in the A coils, the induced 
 voltage in the M coils is zero, which means that the magneto- 
 motive force of the M coils has been counterbalanced and 
 neutralized by that of the A coils. On the other hand, the 
 magneto-motive force caused by the current I in the F coils 
 does not set up current in the armature or A coils, because the 
 brushes b and b' rest on neutral points with reference to the 
 voltages induced by reason of the magnetic flux along FF' 
 caused by this F magneto-motive force. Hence, there must be 
 an induced counter-voltage in the F coils. This is E F = — IX F , 
 where I is the line current and X F the reactance of the F coils. 
 The impressed voltage E is thus all used in drop through the F 
 coils, in the ideal motor, there being no drop in the M coils, the 
 other part of the complete field circuit, since their reactance is 
 destroyed by the mutual effect of the A coils, and all resistance 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 867 
 
 is considered negligible. The voltage E M , across the M coils, 
 cannot be absolutely equal to zero, or E F be equal to — IX F , in 
 an actual machine, since neither resistance nor leakage reactance 
 can be entirely eliminated, but for our purposes of discussion 
 the assumption of an ideal motor is warranted. 
 
 The foregoing two effects are combined in the armature and 
 an alternating magnetic flux <3?^, emanates from the field mag- 
 net on the line FF and heavy sheets of alternating currents Im 
 flow through the armature coils and the conductor cc between b 
 and b' . Moreover, the time phases of <3>^ and I M are in opposi- 
 tion, since I M must be in phase opposition to the current / in the 
 inducing coils to accord with the laws of the transformer if 
 losses and magnetic leakage are negligible.* Therefore, the 
 armature current I M and field or F flux <3?^ are in phase relation 
 and space position to exert a torque upon one another. Now 
 suppose that the armature rotates under the influence of this 
 torque. Then the two sheets of cond uctors lying on the armature 
 between b and b' will generate a voltage E c between those points, 
 due to cutting the flux $ F , so that E c = kS<& F , where Jc is a con- 
 stant, and S is the speed of rotation in per cent of the frequency 
 /. But <3?^ is in phase with I. Idence E c and I are in the same 
 time phase and E c is in phase opposition with I M . The voltage E c 
 is alternating, because its instantaneous values are proportional 
 to <f) F at each instant, but it is also proportional to the speed of 
 the armature. The current component I c flowing through the 
 armature under the influence of E c creates a magnetic flux <3> c 
 which passes through the M coils and sets up in them a voltage 
 E u lagging 90° behind it and proportional to its rate of change. 
 E 
 
 But I c = —A., when X M is the reactance (mutual) offered on 
 
 Xju 
 
 account of the flux <3> c , and, therefore, since the circuit is as- 
 sumed to be purely reactive, I c lags 90° behind E c , and hence 
 is 270° behind or 90° ahead of I M . This makes E M ninety de- 
 grees behind E F = — IX F in time phase. But since E F is 90° 
 behind /, E M must be in phase opposition to I and the vector 
 product of the two must represent real power. 
 
 We now have two induced voltages E M and E F in quadra- 
 ture which cause the drop E in the field circuit or 
 
 E = -E'=- VE m * + E/. 
 
 * Chap. X. 
 
8G8 
 
 ALTERNATING CURRENTS 
 
 Likewise, we have two quadrature components of current h 
 the armature I M and / c , which are flowing in what is the equiv 
 alent of a transformer secondary to the M windings, and the 
 current through the brushes must be equal to their resultant, or. 
 
 I A = VIJ + I*. 
 
 The stator current is 
 
 j- E F E cos A 
 
 F 
 
 where A is of such value that tan A = — Evidently E F , E M , 
 
 E F 
 
 and I can be expressed graphically by means of locus diagrams. 
 Thus, let OE , Fig. 494, be the impressed voltage. Then since 
 
 Fig. 494. — Vector Diagram of Relations in a Repulsion Motor having Negligible 
 Resistance and Leakage Reactance. 
 
 the induced voltages E u and E F are in quadrature and their 
 vector sum equals — OE , they may be represented respectivel}' by 
 the vectors EQ and QO. The point Q travels from E at stand- 
 still, when E m is zero, to 0 , when E F is zero, along the semi- 
 circle OQE The current, I, in the stator is at right angles to 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 869 
 
 OQ - — E r , under the conditions assumed. It may therefore 
 be i presented by the line 01 for the particular value of E p 
 slum in the figure. The point I travels from X when the 
 
 _ Tl 
 
 armcure is at rest and the current I has the value — , to 0 
 
 X F 
 
 whe infinite speed has been reached and E F is zero. The sec- 
 ondly current is composed of the two components I M and / e . 
 As hs been explained, 1m sets up a magneto-motive force equal 
 and pposite to that of I in the M coils. It may therefore be 
 represented by the line OJ. The point J travels along the cir- 
 cula locus OJU with change in load, but alwaj^s takes such a 
 positon that JOI is a straight line. The scalar value of I M is 
 
 equi to I — , where s equals the ratio between the turns in the 
 8 
 
 M fild coils and the F field coils, or — , and s' is the ratio be- 
 
 n F 
 
 tweo the effective armature turns and the F turns, or — . This 
 
 n F 
 
 is erdent, since I and the mutually induced armature current 
 
 I u rust have the ordinary transformer relation = n M = A = 
 
 I n A s s" 
 
 Thi ratio is constant so long as the brushes are not moved, so that 
 if I races a semicircle, J must follow a similar locus. The di- 
 
 ame^r OU equals OX — . The second component of armature 
 s 
 
 curint / c , as can be gleaned from the preceding discussion, is al- 
 way in phase with E F and at right angles to E c . It is also pro- 
 
 _ JT 
 
 porDnal to I and the armature speed. Since I c = — £ , J= — — 
 
 X A X F 
 
 and he reactances are proportional to the square of the numbers 
 of trns in which the voltages E c and E F are induced, then 
 
 j n 2 
 
 -S =~ A ’ , and, as will be shown later, E c = E F s' S ; hence 
 I ™F n A 
 
 s 
 
 I c = — , S being the armature speed in per cent of the impressed 
 
 freqency. Being proportional to and set up by E c , I c may 
 be i presented by the line OK. The point K travels the semi- 
 
 E 
 
 circ; OKV \ which has a diameter OV equal to — f-, and reaches 
 
 K a 
 
 V a infinite speed of the armature. The resultant of OJ and 
 
870 
 
 ALTERNATING CURRENTS 
 
 OK is the total armature current, and it also, as seen by the 
 geometrical construction, traces a circular locus having a diam- 
 eter uv. 
 
 The power input is 01 x OE cos E 01 = IE cos 6 — E U I and 
 may be represented by IW in the figure. The torque, under 
 
 E I 
 
 the assumption made, is . Since the generated voltage E M 
 
 is proportional to the product S <fv, and d > F is the maximum 
 value of the flux from the E coils, and since <t> F is proportional 
 
 to E f , we have S x x tan (90° - 6) yo Zg. wer factor . 1 1 mav , 
 
 E F y J cos(90° — d) y ' 
 
 therefore, be represented by the line TE. The output under 
 
 the assumption of negligible losses is equal to the input. The 
 
 E E 
 
 angle of lag 6 equals cos^ 1 — tan -1 — £ • 
 
 E f E m 
 
 The relation of the voltages in the two sets of imaginary field 
 
 coils E m = EpSs may be found in this manner: — = ^ S = s'* S'. 
 
 E f n F 
 
 E 7t 
 
 and — 5 = — = — = s", from which E M = E F Ss. Substituting this 
 E M s 
 
 in the formula E— — V E^ 1 + E F 2 gives E F = — 
 E 
 
 Also 
 
 1 = — -- = 
 
 X F Xps/l + Sh 2 
 
 VI + w 
 
 Substituting the values of I c and I M , 
 
 obtained in terms of /, *S', s, and s', in the formula I A =V I 2 + I M 2 
 
 gives 
 
 Ia = 
 
 — Vs 2 + *S' 2 , and therefore 
 s' 
 
 Ia = 
 
 _Ej:_ 
 
 A>' 
 
 Vs 2 + S 2 — 
 
 EVs 2 + S 2 
 XpsW lTw' 
 
 From the above formulas and the diagram much information 
 concerning the operation of the repulsion motor may be ob- 
 tained. If the speed is such that S — 1, which is sometimes 
 called synchronous speed, since then 360 electrical degrees of 
 the polar pitch is turned through by the motor armature dur- 
 ing the time of one cycle of the impressed voltage, I A reduces 
 
 to 
 
 E 
 
 When s is also made unity by placing the brushes 45° 
 
 from the neutral plane, the primary current becomes I = — — 
 
 E 
 
 V2AV 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 871 
 
 By completing the circular arcs in the locus diagram the loci 
 are obtained for the machine driven backward, in which case it 
 runs as a generator. The conditions are illustrated by the 
 dotted halves of the circular loci of Fig. 494. 
 
 Fig. 495. — Vector Diagram of a Repulsion Machine as Generator and Motor when 
 Winding Resistance and Reactance are Neglected. 
 
 Figure 496 shows diagrammatically the windings of a repul- 
 sion motor when the stator coils are actually divided into parts. 
 In this case the brushes are directly under the windings marked 
 “ mutual coils.” At standstill the magneto-motive forces of the 
 armature and mutual field currents neutralize each other except 
 for the drops required to overcome resistance and leakage re- 
 actance. The sheets of current in the armature react upon the 
 unneutralized flux which is set up by the current in the “ torque 
 coils,” causing the torque which results in rotation. As in the 
 discussion of the simple repulsion motor, the voltage set up in 
 the armature conductors by the rotation of the armature is 
 (neglecting the iron loss angle) in phase with the torque flux 
 and hence with the field current. The component of armature 
 current caused by this flux lags nearly 90° behind this voltage, 
 
872 
 
 ALTERNATING CURRENTS 
 
 and hence its flux entering the mutual coils sets up a voltage in 
 those coils in quadrature with the voltage in the torque coils. 
 An extra current must flow through the stator circuit to oppose 
 the magneto-motive force of this new component of the armature 
 current. At synchronous speed the two stator fields become 
 equal, and beyond that speed the mutual field becomes the 
 stronger, for which reason the repulsion motor commutates 
 better below synchronous speed than above. The formulas and 
 diagram given for the simple repulsion motor apply also to this 
 modified type. 
 
 As explained, the usual forms of commercial repulsion motor 
 have field cores like those of induction motors. In order then 
 to prevent undue reactance in the armature circuit, caused by 
 the low reluctance of the iron path, special arrangements may 
 be used as shown in Fig. 497. In this case, in a two-pole 
 machine, the pair of mutually induced poles are at M and 
 while the torque poles excited by current in the armature are at 
 F and F ' . The two short-circuited brushes are directly under 
 the points M and M' and collect the mutually induced currents 
 in the armature, neutralizing the reactance in the field. The 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 873 
 
 two brushes under F and F' are connected in series with the 
 field coils and the mains. The armature currents in the short 
 circuit paths react on the torque poles F and F' and cause the 
 armature to rotate. The series current through the rotor sets 
 up the motor flux, which produces a torque by its action on the 
 current in the stator coils. The slxort-circuited induced rotor 
 
 Fig. 41)7. — Compensated Repulsion Motor, Compensating Current in Armature. 
 
 current opposes its magneto-motive force to that of the cur- 
 rent in the stator coils, reducing the reactance of the latter to 
 a reasonable value. At starting, the stator circuit has very 
 low impedance. When the machine is running, however, the 
 top and bottom bands of the rotor conductors cut the flux 
 set up vertically by the line current in the rotor. This creates 
 a voltage in phase with the line current, which in turn sends, 
 through the short-circuited rotor circuits, a current lagging 
 nearly 90° behind the line current. The flux from this current 
 passing into the stator at M or M' creates in the stator coils an 
 induced voltage lagging approximately 90° behind its own 
 phase, and hence in approximate opposition to the line current. 
 Therefore, the stator coils have set up in them a counter-vol- 
 
874 
 
 ALTERNATING CURRENTS 
 
 tage proportional to the speed of the rotor, which is similar in 
 effect to the counter-voltage in the armature of a direct or 
 alternating-current series motor. In this motor the excitation 
 of the field magnet is furnished by currents in the armature 
 conductors, which are at full line frequency when the rotor is 
 standing still ; at synchronism the frequency in each conductor 
 is fcero ; and above line synchronism the conductor frequency 
 again gains a value proportionate to the relative speed above 
 synchronism of the rotor with respect to the frequency of the 
 line current. As a result, at synchronism the power factor ap- 
 proaches unity, and above synchronism the motor may draw lead- 
 ing current from the 
 line. 
 
 Figure 498 shows 
 a cross section of a 
 single-phase motor 
 which starts as a re- 
 pulsion motor and 
 runs as an induction 
 motor. In this case 
 the brushes are mov- 
 able, and when the 
 motor comes to speed 
 the commutator is 
 short-circuited by a 
 ring of copper and 
 the brushes are raised from contact. The movements are 
 accomplished by means of a centrifugal governor on the shaft 
 within the armature spider. 
 
 Figures 499 and 500 show the normal performance of a 20 
 horse power, 60-cycle, six-pole motor of the type illustrated in 
 Fig. 498. As the repulsion motor commutates best at speeds 
 below the theoretical synchronous speed for its armature as an 
 induction machine, this combination of repulsion motor starting 
 and induction motor running gives an excellent form of single- 
 phase machine. 
 
 206. Series Alternating Current Motors. — Since a direct- 
 current series motor does not reverse its direction of rotation 
 when the current is simultaneously reversed in its field and 
 armature windings, it might be expected to run when supplied 
 
 Fig. 498. — Cross Section of a Single-phase Motor con- 
 structed to start as a Repulsion Motor and run as an 
 Induction Motor. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 875 
 
 with an alternating current. This is the case when the field 
 and armature cores are sufficiently well laminated to avoid ex- 
 
 Fig. 499. — Torque and Current Curves of Single-phase Induction Motor arranged to 
 start as a Repulsion Motor, such as is shown in Fig. 498. 
 
 cessive eddy currents and the electric and magnetic circuits are 
 so designed as to keep self-inductive reactance down to reason- 
 able limits. 
 
 In the case of the direct-current motor it is common to make 
 the fields strong to prevent skewing of the flux, with resultant 
 
 PER CENT OF FULL LOAD HJ 3 . 
 
 Fig. 500. — Performance Curves of Motor such as is shown in Fig. 498. 
 
87G 
 
 ALTERNATING CURRENTS 
 
 poor commutation, while the armature is made magnetically 
 weak for the same purpose, and in order to reduce the self- 
 inductance of the coils under commutation to a minimum. In 
 the alternating-current series motor the field magnet is made 
 magnetically weak to reduce its self-inductance to a minimum, 
 while the armature is made relatively strong as a magnet. The 
 self-inductance of the latter is reduced or overcome by means of 
 Compensating coils as described below. 
 
 The general construction of the motor, as frequently used, is 
 seen in Fig. 501. In this figure the field coils A are undistrib- 
 
 Fig. 501. — Diagrammatic Sketch of a Series Alternating Current Motor. 
 
 uted and surround the armature, the armature winding B is of 
 the continuous closed circuit type, and the compensating coils C 
 are distributed along the long pole faces and carry a current which 
 opposes the magneto-motive force of the armature current. 
 Sometimes the field windings are distributed, but this is more 
 or less disadvantageous, as it interferes with the best use of the 
 compensating windings. More often the ordinary polar projec- 
 tions are used with the compensating windings distributed over 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 877 
 
 Fig. 502. 
 
 -Diagram of an Inductively Compensated Series 
 Motor. 
 
 the pole shoes. For mechanical convenience every other pole 
 only need bear a field winding, the alternate ones merely 
 carrying the magnetic fiux. 
 
 The methods of connecting up the windings are shown dia- 
 grammatically in 
 Figs. 502 to 507. 
 
 In Fig. 502 is 
 shown an Induc- 
 tively compensated 
 series motor. In 
 this case the com- 
 pensation winding 
 distributed over 
 the pole faces is 
 short-circuited 
 upon itself, and 
 by the effect of 
 mutual induction 
 it has generated 
 within it a short-circuited current having approximately the 
 same magneto-motive force as that in the armature. In this 
 way the armature self-inductance is in large part destroyed. 
 
 The armature may 
 be considered 
 equivalent to the 
 primary winding 
 and the compensa- 
 tion winding to the 
 secondary winding 
 of a transformer, 
 and the transformer 
 diagrams previously 
 given apply to this 
 case.* 
 
 Figure 508 shows 
 
 Fig. 503. - Conductively Compensated Series Motor. the connections 0 f a 
 
 Conductively compensated series motor. Here the compensation 
 winding C is in series with the field winding A and the arma- 
 ture B . By properly proportioning the turns in winding O the 
 
 * Art. 124. 
 
878 
 
 ALTERNATING CURRENTS 
 
 motor may be Overcompensated or Undercompensated, i.e. C may 
 be made magnetically stronger or weaker than B. It will be 
 noted that less voltage will be available for the armature with 
 
 this connection 
 
 than when the in- 
 ductively compen- 
 sated arrangement 
 is used. 
 
 Figure 504 shows 
 the connections 
 when the field and 
 compensation wind- 
 ings are connected 
 in series with each 
 other and the req- 
 uisite current is 
 induced 
 in them by the ar- 
 mature acting as a transformer primary winding. In this case 
 full line voltage is available across the armature brushes. 
 
 Figure 505 shows the field windings A in series with the 
 compensation windings C arranged for electrical connection 
 
 Fig. 504. — Series Motor with Inductively Connected Field _ ^ nail v 
 
 and Compensation Windings. J 
 
 Fig. 505. — Series Repulsion Motor. Field and Compensation Windings Independent 
 of the Armature Winding. 
 
 with the supply circuit independently of the armature wind- 
 ings B. With this arrangement the stator and rotor cir- 
 cuit voltages can be controlled independently of each other. 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 879 
 
 A motor so con- 
 nected is sometimes 
 called a Series re- 
 pulsion motor. 
 
 An arrangement 
 somewhat similar to 
 the last is shown in 
 Fig. 506, where the 
 compensating coils 
 alone are connected 
 independently to the 
 line, and hence the 
 value of the com- 
 pensation Can be Fig. 506. — Series Repulsion Motor. The Compensation 
 varied at will with- Winding is connected to the Line independently. 
 
 out interfering with the voltage applied to the armature and 
 field windings, which are in series. 
 
 Figure 507 diagrams a true repulsion motor, similar to that 
 shown in Fig. 496, except that the current of the stator coils 
 
 setting up the torque 
 flux is obtained directly 
 from the armature. 
 
 Figure 508 shows a 
 conductively compen- 
 sated series motor such 
 as is illustrated in Fig. 
 503, except that a non- 
 reactive shunt D is 
 added around the field 
 windings for the pur- 
 pose of reducing the 
 impedance of the field 
 circuit. 
 
 It is evident that by 
 bringing the terminals 
 of the field winding A, the armature terminals J3, and the ter- 
 minals of the compensation winding C to a proper switching 
 device a motor can readily be given any of the connections 
 shown in Figs. 496, and 502 to 507. In this way a motor 
 may be started with the connection that gives best commuta- 
 
 
 Fig. 507. — Repulsion Motor with Torque Coils in 
 Series with Armature Circuit. 
 
880 
 
 ALTERNATING CURRENTS 
 
 tion, power factor, and torque, and may then be converted to 
 such other connections as will give the proper speed and best 
 power factor and commutation for the speed and load which 
 are to be maintained. 
 
 The compensation winding cannot absolutely neutralize the 
 effect of the armature magneto-motive force because of the mag- 
 netic leakage around the conductors of both the armature and 
 compensation coils, which is not mutual, but bridges across the 
 
 core teeth. Nor will it fully op- 
 pose the armature magneto-motive 
 C force in the mutual circuit unless 
 distributed entirely around the ar- 
 mature when the armature wind- 
 ing pitch is 180 electrical degrees. 
 Thus, take the inductively con- 
 nected compensation when the 
 Fig. 508.— Compensated Series Motor compensating current times the 
 
 with Non-reactive Shunt around ra tio of the compensating turns to 
 Field Windings. . . ° , 
 
 the armature turns is approximately 
 
 equal to the armature current, which gives equal and opposite 
 magneto-motive forces for the two. Then the distribution of 
 magneto-motive force in the compensating coils will be different 
 from that in the armature if the former is wound upon an arc equal 
 to the pole width while the latter covers 180 electrical degrees. 
 If the armature winding has a pitch equal to the width of the 
 pole face, the two magneto-motive forces can be made practi- 
 cally equal and opposite in their effects, and thej r therefore prac- 
 tically neutralize each other. Under ordinary construction, 
 then, the armature and compensation winding self-inductance is 
 not absolutely overcome, but it is reduced to reasonable values. 
 When the motor is to be used alternatively with direct currents 
 and alternating currents, conductive compensation should be 
 used, since otherwise with the magnetically strong armature 
 and weak field, excessive skewing of the field flux would occur 
 with the direct currents, with resultant bad commutation and 
 low maximum power. 
 
 The alternating-current commutator motor if sufficiently small 
 may be started by throwing it directly upon the line. In the 
 case of heavy motors, such as are used in railway work, it is 
 desirable to start by varying the supply voltage by means of 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 881 
 
 transformers or autotransformers having variable voltage taps 
 brought out from the secondary windings. When necessary to 
 improve the power factor, the field coil may be shunted by a 
 non-inductive resistance as shown at D in Fig. 508. 
 
 207. Vector Diagrams of Alternating Current Series Motors, 
 and Expressions for Voltage. — The vector diagram of the series 
 motor may be constructed as in Fig. 509. Consider the con- 
 ductively compensated type first. Here the current flowing is 
 
 represented by OA. The magnetic field flux, represented by 
 OM, lags behind this because of the iron loss angle and by reason 
 of the fact that there is a considerable amount of true power 
 consumed in the short-circuited armature coils lying under the 
 brushes during commutation. The voltage drop in the leakage 
 reactance of all the windings is represented by OC, which is in 
 quadrature with OA, and in their combined resistance by OD 
 which is in phase with OA. Thus OF may be considered as 
 the voltage drop in the local impedance. The line FG repre- 
 sents the voltage drop equal and opposite to OG', the voltage 
 generated in the field windings by the alternation of the field 
 flux, OM= <3>, through the field windings. The line OB is the 
 drop of impressed voltage equal and opposite to the voltage OB' 
 created by the armature conductors cutting the field flux <3? as 
 the armature rotates. Voltage OB' must be in phase opposition 
 
882 
 
 ALTERNATING CURRENTS 
 
 to the flux OM, as it tends to drive a current in opposition to 
 the current caused to flow by the impressed voltage. Combin- 
 ing these three voltage drops OF , FGr , and OB , gives OJ as 
 the impressed voltage for the current OA when the speed is 
 proportional to the voltage OB. The angle of lag is JO A, the 
 power input is OJ x OA x cos 6, and the torque is proportional 
 to OM x OA x cos B. 
 
 Suppose now that the current OA , Fig. 509, is kept constant 
 in value as the speed varies, a condition entirely possible in 
 traction work. Then T> = OM is constant ; and hence the vol- 
 tage OGr is constant. Now, suppose the impressed voltage is 
 reduced by means of a controller until the speed has lowered to 
 such a value that the induced voltage caused by the armature ro- 
 tation is — 0B 1 ; then the impressed voltage is OJ v the angle of 
 lag is increased, and the power input and output have decreased. 
 The torque, however, which is proportional to the current OA 
 times the flux fl>(= OM) times cos B, has remained constant. 
 At standstill the impressed voltage that maintains a current OA 
 is voltage OGr. By the construction it is seen that the straight 
 line G-J 1 JJ 2 extended to the right is the locus of the impressed 
 voltage vector for constant current and torque but variable 
 speed. If the speed increases beyond the original value, 
 considered, for instance, so that the armature counter-voltage 
 becomes — OB v the impressed voltage must become OJ 2 to 
 maintain the same current, and the angle of lag is decreased 
 and the input and output are increased. It will be observed 
 that the speed is proportional to the voltage on the line OB. 
 
 By considering the vertical component of OGr constant and 
 the angle S negligible in Fig. 509, a simple circular locus dia- 
 gram can be constructed for the compensated series motor when 
 operating at various loads (variable currents) under constant 
 voltage. The counter-voltage E c generated by the armature 
 conductors cutting the field flux may be considered as replace- 
 
 E 
 
 able by a variable resistance such that R c = —^-. Then the 
 
 current flowing 
 
 through the motor is / = 
 
 E 
 
 vl 2 +(«t rS 2 ' 
 
 where X is the uncompensated reactance, R is the true resist- 
 ance of the windings plus such additional amount as is required 
 to be substituted for iron and commutator losses to give an equiv- 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 883 
 
 alent power loss, and R c is the apparent resistance due to the 
 counter-voltage. But the reactance is constant, while the resist- 
 ance (i2 + Rc') varies when the current, and hence the speed, and 
 with it E c , varies. Therefore, we have a case of a circuit contain- 
 ing a constant reactance and variable resistance.* Then lay off 
 OR, the voltage E, Fig. 510, and at right angles with this lay 
 
 E 
 
 off OA such that OA = ~T. Then the semicircle OQA repre- 
 
 sents the required locus. At standstill R c is zero and the total 
 
 V 
 
 Fig. 510. — Current Locus of a Series Alternating Current Motor. 
 
 resistance is R. If, then, OQ x is the starting current, Q X T X rep- 
 resents the standstill losses. The power input for any current 
 OQ is _Pj= El cos 6 = OE x QT , which varies with QT. The 
 
 angle of lag is EOQ = 0 ; and tan 6— ^-~ = where E a is the 
 
 QT E a 
 
 active 'and E x the quadrature component of the impressed vol- 
 tage. Therefore tan (90° — 6) = — = . But X is con- 
 
 E r X 
 
 * Art. TO (a). 
 
884 
 
 ALTERNATING CURRENTS 
 
 stant and R is constant, but R c varies as the speed and the 
 counter-voltage E c . Therefore, a vertical line such as R V, 
 Fig. 510, can be erected and the speed and counter-voltage 
 will be approximately proportional to the intercept WV when 
 current OQ flows. WV at proper scale times the current then 
 equals the motor output. The torque is approximately propor- 
 tional to the current squared, under the conditions assumed. By 
 mean proportionals, OQ x 2 : OQ 2 : : 0T 1 : OT ; hence the torque 
 can be measured off directly on the line OA if the proper scale is 
 used. As the copper losses vary as T 2 , it is sometimes assumed 
 that all losses vary as P. In this case TU would be propor- 
 tional to the losses for current OQ , and UQ be proportional to the 
 
 Field Circuit. 
 
 The power factor of the motor may be increased if desired by 
 shunting the field circuit as shown in Fig. 508 at D. In this 
 case, the diagram of voltages becomes as shown in Fig. 511. 
 The current entering the motor terminals is OA. It divides 
 into two components assumed to be at right angles, through the 
 non-inductive shunt and the inductive field coils ; let the former 
 component be OQ and the latter OR. The magnetic field flux 
 OM= <I> is (neglecting the iron loss angle) in phase with OP, the 
 current in the field coil ; and the countei’-voltage OB' is in oppo- 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 885 
 
 sition to <f> ; while the field-induced voltage GF is at right angles 
 with <f>. Joining the local impedance drop OF with FG and 
 OB gives an impressed voltage OJ. The voltage locus is GJ 
 for constant current OA and variable speed. This arrange- 
 ment makes it possible by using a variable shunt resistance to 
 neutralize the lagging currrent and even draw leading current 
 from the line. Various other methods of controlling the phase 
 displacement between the impressed voltage and current are 
 omitted here for lack of space. 
 
 The diagrams just discussed (Figs. 509 to 511) apply 
 equally to motors having inductive compensation, since the 
 local drop in the impedance of the compensation windings is 
 furnished by the impressed voltage as in any transformer. 
 
 The vector diagram of an inductively excited motor such as 
 diagrammed in Fig. 501 is practically similar to that shown in 
 
 Fig. 512. — Vector Diagram of a Repulsion Motor or an Inverted Inductive Series 
 
 Motor. 
 
 Fig. 509. The only difference in the diagrams lies in the fact 
 that the secondary induced current is slightly more than 180° be- 
 hind the primary current on account of the losses in the winding 
 circuits ; hence its opposite component in the primary current 
 is slightly ahead of the total primary current ( OA , Fig. 509). 
 As the field induced voltage (OG' , Fig. 509) is at an angle to its 
 current equal to quadrature plus the iron loss angle, it is thrown 
 slightly to the left. The result is a slightly lower power factor. 
 With A and C of Fig. 504 connected to the line and the brushes 
 of B short-circuited there results a split coil repulsion motor. 
 
886 
 
 ALTERNATING CURRENTS 
 
 The vector diagram for such a motor may be drawn quite simi- 
 larly to that of Fig. 509, since the speed-induced counter- 
 voltage is transferred directly to the compensating or mutual 
 inducing coil C. Thus, assume that current OA of Fig. 512 is 
 flowing through the torque and mutual coils of Fig. 496. Cur- 
 rent flows in the armature, lagging slightly less than 180° from 
 OA , supposing the reluctance of the path of the flux which 
 links the mutual coils and armature coils is small. The voltage 
 set up by the armature rotation is in phase opposition to OM. 
 This induces in the mutual coil 0 a practically equal and oppo- 
 site voltage OC. The self-inductive voltage of the field GrF is 
 at right angles to OM. Then if OF is the total local drop of 
 voltage due to magnetic leakage and conductor resistance, the 
 impressed voltage is OJ , and the construction is, as said, sim- 
 ilar to that of Fig. 509. 
 
 The voltage induced in the field coils of a series motor by the 
 torque flux is 
 
 
 (i) 
 
 where n is the number of field turns, f the frequency of the 
 supply current, X s the reactance of the winding, and I the 
 motor current. 
 
 If the magnetic flux were constant instead of alternating, the 
 counter-voltage induced in the armature by its rotation, with 
 the brushes in the neutral plane, would be 
 
 F = a ~ 
 10 8 
 
 in which a is the number of conductors on the armature, is 
 the total flux per field pole, v is the number of revolutions of 
 the armature per second, p is the number of pairs of poles in the 
 magnetic field, and p' is the number of paths in parallel for cur- 
 rent to flow through the armature winding. But the flux is of 
 sinusoidal alternations instead of being constant, and, being 
 the maximum flux per pole, the counter-voltage set up in the 
 armature by its rotation is 
 
 E = 
 
 
 V2 x 10 8 P 1 
 
 ( 2 ) 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 887 
 
 The third voltage used in the diagrams represents that ab- 
 sorbed by resistance and leakage reactance and is 
 
 E,= IZ* ( 3 ) 
 
 in which Z t is the impedance from resistance and leakage react- 
 ance. 
 
 If R c is a resistance such that IR C = E c , then 
 
 7? _ E c a<S>v p_ 
 
 ' I V2 x 10 8 x I P 1 ' 
 
 (4) 
 
 Putting 2 tv for 
 
 00 
 
 a 
 
 n 
 
 and S for 
 
 v 
 
 T 
 
 gives from Equations (1) and 
 
 The motor input is 
 
 the output is 
 and the torque is 
 
 R c = -?- 7 wSX f . 
 
 irp' 
 
 Pi = El cos 0, 
 P 0 = PR C , 
 
 T RJ\ 
 
 27 TV 
 
 (5) 
 
 208. Commutation and Other Characteristics of Commutating 
 Alternating-current Motors. — The process of commutation in 
 alternating-current motors differs from that of direct-current 
 machines by reason of the voltages induced in the short-circuited 
 coils under the brushes, caused by the rapidly alternating char- 
 acter of the field flux passing through them. Thus, to prevent 
 sparking, it is necessary to neutralize these induced voltages to 
 reasonable values or reduce the resultant currents and yet per- 
 form the ordinary function of reversing the current in the coils 
 while short-circuited under the brushes. It will also be noted 
 that the value of the current in the short-circuited coils is de- 
 pendent upon two conditions at each instant of the primary 
 current cycle. Namely, the actual currents in successively 
 short-circuited coils depend upon the load and the instantaneous 
 value of the field flux. Thus, at the instant when the field 
 flux is a maximum, no current is induced in the short-circuited 
 coils due to its change, but the coils enter the condition of 
 short circuit bearing maximum line current ; and when the 
 field flux is zero, a maximum current is induced in the short- 
 circuited coil, but the line current is zero. 
 
888 
 
 ALTERNATING CURRENTS 
 
 The short-circuited coils being, with respect to the field wind- 
 ings, in effect transformer secondaries of very low resistance, 
 the short circuit currents become so great, if not reduced by 
 some special construction, as to seriously demagnetize the fields 
 and interfere with the operation of the motor by destroying its 
 torque. This, taken in connection with the liability to serious 
 sparking, makes the question of commutation of very serious im- 
 portance in designing alternating-current commutator motors. 
 
 One of the methods of reducing the short-circuit currents is 
 to introduce high resistance strips into the leads between the 
 commutator bars and the armature windings. This is useful ; 
 but leads of sufficient resistance, having large enough radiating 
 surfaces to dissipate the heat generated in them, are difficult to 
 insert in the confined space available. Therefore, though suit- 
 able when the armature is moving and the resistance strips are 
 in use only a small part of a revolution, they are liable to be 
 destroyed if the motor is stalled or fails to start promptly. 
 Nevertheless, this method has proved valuable. 
 
 Another method is to insert reactance coils in the commutator 
 leads and thus avoid the heating caused by the previous method, 
 but with the disadvantage of adding to the self-inductance of 
 the short circuit and thus making commutation difficult. Vari- 
 ous arrangements have been proposed of this nature. One of 
 them uses a differentially wound coil so that the line current 
 largely neutralizes the reactance in its path while the short- 
 circuited current sets up full reactive flux in the coil. 
 
 Another method uses a wide space between commutator bars 
 and a split brush between the halves of which is a reactance 
 coil. The two halves of the brush are so spaced that they 
 touch two adjacent bars, and hence the short-circuit current 
 must flow through the reactance. Special devices may be ar- 
 ranged so that the line current is not affected by the reactance. 
 
 The use of overcompensation with connections of the motor 
 windings in the relations of a series repulsion motor has been 
 found desirable for reducing the transformer voltage in the 
 short-circuited coils. The use of commutating interpoles placed 
 over the brushes has also been found advantageous. These 
 poles can have windings which are shunt-connected across the 
 brushes and through a variable autotransformer so that their 
 strength may be changed with the speed and load. A method 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 889 
 
 shown in Fig. 513 of connecting the commutating poles gives 
 one of the many possible arrangements. Thus, part A of the 
 commutating pole winding is shunted in series with resistance 
 B across the field coil, which gives a current strength in A pro- 
 portional to the field 
 strength. The part F 
 of the same winding 
 is connected in mag- 
 netic opposition to A 
 and receives current 
 from the secondary cir- 
 cuit of a small trans- 
 former 6f, the primary 
 winding of which is 
 connected across the 
 armature brushes and 
 which receives current in proportion 
 to the armature speed. Then at low 
 speeds, when the current to be reversed 
 in the armature conductors and the field 
 flux tending to induce heavy secondary 
 currents in the short-circuited coils 
 are large, the commutating poles are 
 strong. When the speed is high and 
 the armature current and field flux are 
 low, the commutating pole flux is low. 
 
 The reactance Vindicated in series with 
 the small transformer is required to ad- 
 just the value and phase of the current.* 
 
 However, since commutating poles 
 or overcompensation are effective to any 
 extent only when the armature is rotating and hence its con- 
 ductors are cutting the commutating flux, such schemes require 
 also the use of resistance, or an equivalent device, in the leads 
 between armature coils and commutator. 
 
 The commutation of repulsion motors is better when the 
 speeds are below synchronism than when the speeds exceed syn- 
 chronism, while the reverse is true of plain series motors whether 
 compensated or uncompensated. 
 
 * McAllister, Alternating Current Motors, p. 302. 
 
 Alternating-current Motor 
 having Commutating Poles 
 connected for Automatic 
 Strength Variation. 
 
890 
 
 ALTERNATING CURRENTS 
 
 The operating characteristics of a series alternating-current 
 motor are shown in Fig. 514. As in a direct-current series- 
 motor, the torque starts at a high value and decreases as the 
 speed increases and the current and field flux decrease. The 
 power rises rapidly to a maximum when the product of torque 
 and speed is a maximum and then decreases. The power factor 
 rises rapidly to nearly unity. 
 
 From the formula and diagram already developed (page 887) 
 
 -X" 7T ^ 
 
 it may be shown that tan 6 = = -IL , approximately, where 
 
 n c wSp 
 
 6 is the angle of lag in a series-motor. Then to reduce 0 to a 
 
 R.P. M. 
 
 Fia. 514. — Operating Characteristics of a Series Alternating-current Motor, Compen- 
 sated. 
 
 minimum, w, S and p should be as great as possible — a fact 
 which has been found true in practice. To keep the power 
 factor at the highest possible value, it is evidently desirable to 
 reduce the air gap to the minimum allowable by mechanical 
 considerations. 
 
 The frequency for which large series-motors are built in this 
 country is usually 25 cycles per second, though mam* smaller mo- 
 tors of the series and repulsion tj-pes are used on circuits having 
 a frequency of 60 cycles per second. Some European traction 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 891 
 
 circuits, on which commutating alternating-current motors are 
 used, have frequencies of 15 periods per second or even less. 
 
 Alternating-current motors having their field windings con- 
 nected in shunt with the armature can be built, though diffi- 
 culty is experienced in keeping the field flux and armature 
 current in phase ; and further the place for motors of this type 
 is occupied satisfactox-ily by the single-phase induction motor 
 or repulsion motor. 
 
 The efficiency of alternating-current commutator motors can 
 best be obtained by measuring the input b}^ a wattmeter and 
 the output by a rated generator or other mechanical absorbing 
 device. However, the losses can .be approximately determined 
 if desired. The copper losses can be calculated. The field 
 core losses cau be measured by a wattmeter when the field is 
 excited, the armature being removed. And the armature core 
 loss and other rotating losses can be approximately obtained by 
 driving the armature by a rated motor when the field is fully 
 excited for the load and speed desired. 
 
 209. Frequency Changers and Motor Converters. — Frequency 
 changers are much used in this country for transforming low 
 frequency currents to higher frequency where electric power 
 transmitted from a distance is needed for lighting. 
 
 One form of frequency changer, called a Motor-generator, 
 consists of a motor operated from the circuit of one frequency 
 and driving a generator connected with the circuit of the 
 other frequency.* The two machines are usually mounted 
 on one bed plate and have a common shaft, thereby making 
 a motor-generator set, and a starting motor is sometimes also 
 mounted on one end of the shaft. The numbers of magnet 
 poles in the field magnets of the two machines must be in 
 the ratio of the two frequencies. When two such machines 
 are expected to operate in parallel at both ends of the sets, it 
 is necessary that the alignment of the field magnets and arma- 
 ture coils shall be exactly corresponding in the two, or exces- 
 sive cross currents will flow. It is also necessary to synchronize 
 at both ends of the set, since the set may parallel correctly at 
 one end and yet be out of correct phase for paralleling at the 
 other end. In this case it is necessary to slip one or more poles 
 at the motor end by opening the field switch, or in some other 
 
 * Art. 183. 
 
892 
 
 ALTERNATING CURRENTS 
 
 manner bring the machine into paralleling phase at both ends. 
 The number of points in one revolution of the shaft of such a 
 machine at which paralleling can be accomplished at both ends 
 of the set is equal to the greatest common factor of the numbers 
 of pairs of poles respectively in the field magnets of the motor 
 and the generator. The generator terminal voltage of an un- 
 loaded set which is to be put in parallel with a loaded set will 
 also not be in exactly the same phase as the terminal voltage of 
 the loaded set, on account of the mechanical lag of the loaded 
 motor and the electrical lag of the loaded generator, even 
 though the motors have been brought into the correct relations 
 for paralleling the sets. 
 
 A Frequency converter as distinguished from a motor-gener- 
 ator consists, in its ordinary form, of a synchronous motor driv- 
 ing the armature of an induction motor backwards, i.e. against 
 its direction of rotation as a motor, the armature winding of 
 the synchronous machine and the field winding of the induc- 
 tion machine being connected to the supply mains of lower 
 frequency, while the armature winding of the induction ma- 
 chine is connected to the higher frequency circuit. The con- 
 nections are shown in Fig. 515. The power which must be sup- 
 plied by the synchronous motor in driving the induction 
 motor armature backward is equal to the torque exerted be- 
 tween the induction motor field flux and armature current 
 times the speed of the armature, while that transferred by 
 the induction motor itself is that which would be trans- 
 ferred if the armature were stationary and the same cur- 
 rent flowed. Therefore, if a is the frequency of the supply 
 mains and b is the frequency of the secondary mains, the in- 
 duction motor must supply j times the total power transformed 
 
 and the synchronous motor must supply 
 
 times the total 
 
 power. Then, if it is required to convert 200 kilowatts from 
 25 to 60 cycles, the induction motor field must transform 
 x 200, or 83^ kilowatts, while the synchronous motor must 
 supply 116-|- kilowatts to the armature shaft. The armature 
 losses must be supplied in equal ratio. The voltage given out 
 
 will be equal to - times the voltage that would be generated 
 a 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 893 
 
 in the armature windings if they were stationary. The capacity 
 of synchronous motor and induction motor field winding must 
 be then in the proportion given, but the capacity of the induc- 
 tion motor armature must be that of the entire output. The 
 losses of the armature tend to be high on account of the high 
 frequency of the magnetic cycles to which its core is subjected. 
 
 SUPPLY MAINS 
 
 Fig. 515. — Connections of a Frequency Converter. 
 
 A Motor-converter is a machine for converting compara- 
 tively high voltage alternating currents to comparatively low 
 voltage direct currents without the intervention of a transformer. 
 It consists of the combination of an induction motor and a rotary 
 converter as shown in Fig. 516. The armature of the motor 
 sends polyphase currents directly into the windings of the con- 
 verter armature, but slip rings are not needed, as the machines 
 are connected together mechanically. Suppose that the two 
 
894 
 
 ALTERNATING CURRENTS 
 
 machines have equal numbers of poles, then when the induction 
 motor armature reaches one half synchronous speed, the current 
 it sends to the synchronous converter armature has a frequency 
 equal to the frequency of the counter-voltage induced in the 
 converter armature. The frequency of the voltage induced in 
 
 RESISTANCE Fig. 516. — Diagram of a Motor-converter. 
 
 the armature conductors of the induction motor falls propor- 
 tionally with the armature speed, from the frequency of the 
 supply circuit to zero, as the armature increases in speed from 
 standstill to synchronism. The voltage likewise falls off dur- 
 ing this change of speed. At the same time the frequency of 
 the voltage induced in the armature conductors of the synchro- 
 
 O J 
 
ASYNCHRONOUS MOTORS AND GENERATORS 
 
 895 
 
 nous converter armature rises proportionally with the armature 
 speed, from zero to the frequency of the supply circuit, in the 
 same range of speed, and the voltage itself increases. At half 
 synchronous speed the induced voltages are of the same fre- 
 quency. If the speed tends to rise, the counter-voltage of the 
 synchronous converter armature tends to reverse the resultant 
 voltage in the circuit of the armatures. Therefore, if the in- 
 duction motor and synchronous converter are of the proper 
 relative proportions and have equal numbers of field poles, the 
 speed will be maintained equal to one half the speed corre- 
 sponding for the induction motor to synchronism with the 
 supply current. Part of the power converted by the synchro- 
 nous converter is received mechanically through its shaft from 
 the induction motor and part electrically from the induction 
 motor armature windings. If the machines have unequal num- 
 bers of field poles, the speed of the set is proportional to the 
 ratio of the number of poles on the induction motor to the sum 
 of the numbers of poles on the two machines. 
 
 In addition to permitting the use of high alternating voltages 
 without the use of transformers, the motor-converter has the 
 advantage of furnishing a low frequency to the rotary portion 
 of the apparatus. 
 
 210. The Mercury Vapor Rectifier. — The mercury vapor 
 rectifier or converter has come much into use for operating 
 series direct-current arc lamp circuits, and charging storage 
 battery circuits. It consists of an exhausted globe as shown 
 in Fig. 517. A small amount of mercury is placed in the tube 
 or globe, and when the latter is shaken, causing the mercury to 
 bridge across the terminals A and B, a current flows from the 
 alternating-current mains between the terminals. This frees 
 mercury vapor, and current flows alternately into the terminals 
 O and O' and out of the terminal A. By virtue of the peculiar 
 property of the vapor in connection with the terminal electrode, 
 current apparently cannot flow out of the terminals O and O', 
 so that it flows in on alternate half cycles. Consider an instant 
 when it is flowing into 0: It then passes from the iron or 
 
 carbon electrode, through the vapor, into the mercury at the 
 bottom, out of terminal A , through the storage battery E, 
 through the reactance coil F, and thence to the alternating- 
 current return main. Upon the next half cycle, the current 
 
896 
 
 ALTERNATING CURRENTS 
 
 flows into the rectifier at C ' , out at A, and thence through the 
 battery in the same direction as before, then through F' and out 
 to the other main. 
 
 A drop of voltage of about 15 volts occurs when current passes 
 
 through the tube, which 
 makes it quite efficient 
 for supplying series arc 
 lighting circuits, in which 
 the current is low and the 
 voltage high. For such 
 circuits the alternating 
 current is usually sup- 
 plied to the rectifier by a 
 constant-current trans- 
 former.* Polyphase 
 tubes are also used for 
 charging storage batteries 
 and similar purposes. In 
 a tri-phase tube three 
 positive electrodes similar 
 to C and C are used. 
 The secondary windings 
 of the supply trans- 
 formers are connected in wye. The free battery terminal is 
 connected to the neutral point and the three electrodes are 
 connected to the three wye terminals. 
 
 Electrolytic rectifiers are also used to some extent. They 
 are based on the fact that an aluminum electrode in a cell will 
 pass a current only in one direction (i.e. that which makes the 
 aluminum the anode) unless the voltage exceeds a certain criti- 
 cal figure. This peculiarity of aluminum is also made use of in 
 the electrolytic lightning arresters, now largely used on high 
 voltage transmission circuits. For this purpose what are equiv- 
 alent to long strings of aluminum cells are joined in series from 
 a line wire to ground or from one line wire to another. When a 
 current is passed through them, they quickly take a property of 
 very high resistance. This property gradually disappears with 
 time, when the arrester must be “ recharged,” by having a suit- 
 able voltage impressed upon it for a short time. 
 
 * Art. 142. 
 
 Fig. 517. — Single-phase Mercury Vapor Rectifier. 
 
CHAPTER XIII 
 
 SELF-INDUCTANCE, MUTUAL INDUCTANCE, AND ELECTRO- 
 STATIC CAPACITY OF PARALLEL WIRES. SKIN EFFECT 
 
 211. The Self-inductance of Parallel Wires. — The self-in- 
 ductance of two parallel wires hanging upon a pole line, con- 
 tained in a cable, or otherwise, has much influence on the 
 processes of long distance power transmission, long distance 
 telephony, and long distance telegraphy. In the ordinary 
 alternating-current systems for electric lighting and for trans- 
 mission of power over relatively short distances, the effects of 
 the self-inductance of the transmission lines are not particularly 
 serious, but even in those instances it is important to be able to 
 predetermine the nature and magnitude of the effects. 
 
 An expression for the self-inductance of two parallel wires 
 may be developed thus : 
 conductors A and A', 
 
 Fig. 518, form a circuit 
 of indefinitely great 
 length. Let I be the 
 amperes of current flow- 
 ing through the con- 
 ductors, r the radius o.f 
 each conductor meas- 
 ured in centimeters, and 
 d the distance between 
 the axes of the con- 
 ductors also measured 
 in centimeters, and as- 
 sume the current to be 
 uniformly distributed 
 over the cross section of 
 each of the conductors. 
 
 Also let fi and /// be respectively the magnetic permeability 
 of the medium surrounding the conductors and of the material" 
 composing the conductors. The intensity of the magnetic field 
 3 m 897 
 
 Suppose that two parallel cylindrical 
 
 / / 
 
 
 <— 1 crriT> 
 
 PLANE 
 
 
 
 i 
 
 d 
 
 PLANE 
 
 / 
 
 / 
 
 Fig. 518. — Diagram for studying Self-inductance 
 of Parallel Wires, consisting of Two Conductors 
 A, A' of Indefinite Length cut at Right Angles by 
 Two Imaginary Planes. 
 
898 
 
 ALTERNATING CURRENTS 
 
 (Ha) measured in dynes at a point outside of the conductor A, 
 at a distance a centimeters from its center and due to the 
 current in A , is 
 
 H = 
 
 21 
 10 a 
 
 ( 1 ) 
 
 This may be readily understood from the fact that the lines 
 of force due to the current in the straight conductor are circles 
 with their planes perpendicular to the conductor, and if the 
 magnetic force at any point in space at a distance a centimeters 
 from the conductor is F a dynes, the work done against this 
 force in moving a unit magnet pole around the conductor is 
 W = 2 7 raF a ergs. In this case the electromagnetic intensity, 
 and therefore the force exerted on a unit pole, is perpendicular 
 to the conductor, since the lines of force are circles with their 
 planes perpendicular to the conductor. Also, the work done 
 in carrying a unit pole around the conductor is equal to 
 
 W = 4:7m— in any case in which n is the number of times 
 
 10 J 
 
 the current is circled and the current is measured in amperes. 
 The symbol n stands for the number of turns in the coil if the 
 conductor concerned were a coil ; but in this case n = 1, and 
 therefore 
 
 2 iraF a 
 
 4 7 rl 
 10 ’ 
 
 or 
 
 F=H = 
 
 21 
 
 10 a' 
 
 The magnetic density (i? a ) in lines of force per square centi- 
 meter at the point a is therefore 
 
 & < 2 > 
 
 Now, considering a space cut out by two planes perpendicular 
 to the axes of the conductors and one centimeter apart (see 
 Fig. 518), within this space at a distance a centimeters from 
 conductor A a number of lines of force equal numerically to 
 
 B a da = d^ a =m^ 
 
 10 a 
 
 ‘will pass through a radial width da centimeters ; and the total 
 number of lines of force set up by the current in A which pass 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 899 
 
 through the area bounded on two sides by the surface of A and 
 the center of A' and on the other two sides by the planes, is 
 
 
 rd 2 filda 
 Ar 10 a 
 
 (3) 
 
 At any point p within conductor A and at a distance of b 
 centimeters from the center of A, the magnetic intensity will 
 be the same as though the current within a circle of radius b 
 centimeters were condensed at the center of the- conductor, as 
 far as the magnetic effect of the current in conductor A is con- 
 cerned. Since the magnetic effect at the point p of the uniform 
 layer of the current in the conductor outside of radius b centi- 
 meters will be zero, the strength of field at p may be written 
 
 2 I irb 2 _ 2 bl 
 
 105 X 7T? ~ ToV 2 ' 
 
 (4) 
 
 and the magnetic density at p is 
 
 (5) 
 
 Hence within the conductor and between the two planes one 
 centimeter apart, a number of lines of force numerically equal to 
 
 B h = 
 
 2 n’bl 
 10 r 2 ' 
 
 B b db = dA> b - - 
 
 2 p'bldb 
 10 r 2 
 
 will pass through a radial width db. But this flux does not 
 link with the whole current in the conductor A , but only with 
 7rb 2 
 
 the current equal to — - 1 amperes which is within the circle 
 
 having a radius of b centimeters. The product of the current 
 with the number of lines of force inclosed by it, giving the 
 magnetic linkages for current within the whole conductor, 
 therefore is 
 
 2 p'lbdb 1/J r6 . 
 
 «/o 7 rr 2 10 r 2 2 10 
 
 The self-inductance of conductor A given in henrys for a 
 centimeter of length is equal to 10 -8 times the number of lines 
 of force linking the current when it has a value of one ampere ; 
 and, summing the effects of the magnetic field without and 
 
900 
 
 ALTERNATING CURRENTS 
 
 within the conductor by adding (3) to (6) when I is one am- 
 pere, and multiplying by 10~ 8 gives 
 
 i=(2/*log^4-|/* , )l0-», (7) 
 
 which is the self-inductance in henrys per centimeter of con- 
 ductor length. 
 
 The effect of the return conductor A! in a single-phase cir- 
 cuit is to exactly double the flux which is linked or inclosed by 
 the current, and therefore the self-inductance per centimeter of 
 length of circuit is the same as the self-inductance for two cen- 
 timeters length of conductor. 
 
 Given in terms of common logarithms the self-inductance per 
 centimeter of length of conductor is 
 
 L= ^4.605 ti log- + O' 9 ; 
 
 and in terms of 1000 feet of conductor the self-inductance is 
 L x = ^.1404 n log - + .01524 fx'^j 10~ 3 henrys 
 
 or .1404 /a log - + .01524 /x ' millihenry. 
 
 r 
 
 When the conductors are of copper suspended in the air or 
 embedded in the usual insulating materials, fx — /x' = 1 and the 
 self-inductance per 1000 feet of conductor is 
 
 X = .1404 log- -f- .01524 millihenry. 
 r 
 
 As a rule the value of 2 log - derived from the distances 
 
 r. 
 
 apart of diameters of wires in ordinary overhead electric cir- 
 cuits is quite large compared with the term 4, but in the case 
 
 of conductors in underground cables the ratio - may be so 
 
 r 
 
 small that the constant term ^ comprises a large part of the 
 self-inductance. In underground cables, however, the con- 
 ductors are so close together that the uniformity of distribu- 
 tion of the currents over the cross section of each conductor 
 cannot be relied upon, as the conductors will influence each 
 other and the results may be modified. 
 
 The following table gives the resistances and self-inductances 
 of wires of different sizes at a number of distances apart. 
 
Table showing Self-inductances of Solid Cylindrical Copper Conductors at Various Distances Apart, 
 
 Measured from Center to Center 
 
 INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 901 
 
 
 © . 
 r^O 
 
 00 05 05 C 
 
 
 Cl 
 
 Cl CO 
 
 -H 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 co co o a 
 
 lo 
 
 c 
 
 05 CD 
 
 CO 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 05 O i-H i- 
 
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902 
 
 ALTERNATING CURRENTS 
 
 In the foregoing discussion of the self-inductance of two paral- 
 lel conductors the upper limit of integration in Equation (3) is 
 taken as 3, which carries the integration from the edge of con- 
 ductor A to the center of conductor A' for the magnetic link- 
 ages resulting from flux outside of the conductor A. This 
 is obviously correct for each thin cylindrical layer of current 
 in conductor A', and is therefore correct for the whole. 
 
 In case the solid cylindrical conductors are of different diam- 
 eters, the formula for the self-inductance per centimeter of 
 conductor length becomes 
 
 L = (/* log e — -f ^y^llg- 9 , (8) 
 
 V r i r 2 Z J 
 
 in which r x and r 2 are the radii of the two conductors. 
 
 The equation for parallel hollow cylindrical conductors is, 
 per centimeter of conductor length, 
 
 L = 
 
 i fl 2 1 f r, 2 — 3(V,) 2 4(/,) 4 i r, 1 
 
 /* lo ffe — + t Mi + rA~n~^T2 lo ^ 
 
 r x r 2 4 l r{ — {r\y [rp — (»i)yr r\ J 
 
 10- 9 , (9) 
 
 . 1 j rJ 
 
 3(V 2 ) 2 | 
 
 
 (^ 2 ) 2 [^ 2 -(^ 2) 2 ] 2 
 
 l0 ge^} 
 
 2 1 -> 
 
 in which /x, fi v /u, 2 represent the magnetic permeabilities respec- 
 tively of the medium surrounding the conductors and the two 
 conductors, r v r 2 represent the external radii of the two con- 
 ductors, r' v r\ represent the internal radii of the hollow 
 cylinders.* 
 
 The foregoing formulas relate only to cjdindrical wires, but 
 apply with an ample accuracy to conductors of other forms of 
 cross section, such as square or oval, provided the distance 
 between the conductors is large compared with their thickness 
 through. In the case of flattened conductors in underground 
 cables where the conductors are near together, these formulas 
 can he applied approximately in lien of satisfactory exact 
 formulas which have not been developed on account of the 
 complexity introduced in the integrations when the conductors 
 deviate far from cylindrical form. 
 
 If the currents are not uniformly distributed over the cross 
 section of the conductors, as may be the case if conductors A 
 
 * See Maxwell, Electricity and Magnetism, Art. 685. 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 903 
 
 and A! lie close together and are so thick that the magnetic 
 flux from each one sets up an appreciable redistribution of the 
 flux in the metal of the other, the inductive reactance com- 
 puted from the formulas may go quite wide of the truth if the 
 reactance is taken as 2 7 rfL and L computed from the formulas. 
 
 In case the alternating currents are of rather high fre- 
 quency and the diameters of the conductors are considerable, 
 the currents tend to forsake the central part of each conductor 
 and become of greater density toward its surface on account 
 of the so-called Skin effect,* and an approximation of the 
 self-inductance may then be made by means of formula (9). 
 When the frequency is sufficiently great to cause the currents 
 to concentrate substantially in a thin cylindrical film of the 
 metal at the surface, the inner and outer radii of the cylindrical 
 conductor become equal to each other. 
 
 Formula (7) may be written in the following form in case 
 the conductors are of the same radius and the same material 
 and are sufficiently far apart compared with their thickness 
 so that they do not affect the current distribution in each other, 
 
 L = C 2/xlog e ? + %')10- 9 , 
 
 r 2 
 
 in which v is a variable coefficient with limits from unity when 
 the currents are uniformly distributed over the cross section 
 to zero when the frequency is high enough to bring the 
 currents to the surface. 
 
 The minimum value for the self-inductance of a circuit 
 composed of two indefinitely long cylindrical conductors 
 obviously occurs when the conductors are separated by the least 
 practicable distance. If the current can be assumed to be 
 uniformly distributed over their cross sections, Equation (7) 
 reduces for the minimum value to 
 
 L = (2 ix, log e 2 + 1 n’) 10- 9 = (1.386 n + .5 /*') 10~ 9 , 
 
 and when and \jJ are equal to unity this becomes Z= 1.886 
 x 10" 9 henrys per centimeter of conductor length. This 
 reduces to L = 0 in the limiting case of the two conductors 
 very close together and carrying currents of such high fre- 
 quency that the currents concentrate wholly on the surface, 
 
 * Art. 212. 
 
904 
 
 ALTERNATING CURRENTS 
 
 since in that case (the conductors being so close together) the 
 currents in the conductors react on each other so that the cur- 
 rents in the tube conductors come to occupy positions in narrow 
 strips of the surfaces facing each other. 
 
 When the conductors of equal cross section are concentric 
 with respect to each other and the inner conductor is solid, a 
 similar integration shows that the self-inductance of the pair 
 of conductors per centimeter of length of the cable is 
 
 2 log. 
 
 R ( R 2 + r 2 \ 2 
 
 r V r 2 / 
 
 log. 
 
 where r is the radius of the inner conductor and R is the 
 radius of the inner cylindrical surface of the external con- 
 ductor. 
 
 In case the inner conductor is also tubular, the distribution 
 of the current is limited to the area of a hollow cylinder and 
 the self-inductance of the concentric cable per centimeter of 
 length of the cable then becomes 
 
 L = 
 
 2 log, 
 
 _i 
 
 r ( r 2 — rp 2 ) 2 
 
 R 
 
 log, - + R i log — - - (r x 2 + i^Xr 2 -^ 2 ) [ 
 
 r i /l i > . 
 
 10 ~ 9 , 
 
 in which r 1 and r are respectively the inner and outer radii of 
 the inner conductor and R 1 and R are the inner and outer radii 
 of the outer conductor. 
 
 Prob. 1. A pair of solid electric light wires are erected on a 
 pole line 15 inches apart center to center for a distance of 12 
 miles. The wires are of 0000 B. & S. gauge. What is the 
 self-inductance of the loop composed of these two conductors ? 
 What is the inductive reactance of the loop at a frequency of 
 25 cycles per second ? 
 
 Prob. 2. A concentric underground cable 3000 feet long is 
 composed of a solid No. 6 B. & S. gauge wire surrounded by an 
 external conductor composed of a spiral wrapping of fine wire 
 but giving the effect of a hollow cylinder of 300 mils internal 
 radius with the same conducting cross section as the inner wire. 
 The insulation between these two conductors is composed of 
 * Russell, Alternating Currents, Vol. 1, pp. 53-54. 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 905 
 
 oiled paper of which the permeability maj r be considered unity. 
 What is the self -inductance of the loop composed of these two 
 conductors? 
 
 It must be remembered that the foregoing formulas for the 
 self-inductance of parallel wires are applicable only to lengths 
 which are parallel for such large distances compared with the 
 distances apart of the conductors that the ends of the loops 
 need not be considered, and tliey are not applicable to coils or 
 to wires with frequent turns in them, inasmuch as the magnetic 
 
 2 I 
 
 force at the turns does not respond to the condition H a — 
 
 10 a 
 
 The equations are sufficiently accurate in their representation 
 of the facts to meet the requirements of most computations 
 with respect to overhead and underground electric light and 
 power circuits, telephone circuits and telegraph circuits ; but 
 may prove seriously in error if applied to wiring on switch- 
 boards or inside of buildings on account of tbe many turns 
 which may exist in such wiring, and also may prove seriously 
 in error if applied to circuits composed of conductors deviating 
 far from cylindrical form. Flattening the conductors tends to 
 decrease the self-inductance. Also dividing the total current 
 between two circuits, each having cylindrical conductors of 
 half the total cross section, reduces the reactive voltage in the 
 circuit because the current per circuit is reduced to a greater 
 degree than the self-inductance is increased. 
 
 Overhead wires of considerable thickness and the conductors 
 of underground cables of greater cross section than No. 6 B. 
 & S. gauge wire are usually made by stranding a sufficient num- 
 ber of smaller wires to afford flexibility while maintaining the 
 desired aggregate cross section. In applying the formulas to 
 such conductors it is usual to assume that they give the same 
 magnetic effects as smooth, cylindrical conductors of equal 
 cross section. The error introduced by this assumption is 
 difficult to determine for any particular instance, as it depends 
 upon the various proportions entering into the strand. It is 
 usually in the direction of a positive error, that is, it makes the 
 computed self-inductance somewhat too large, but the error is 
 usually quite small. 
 
 The foregoing formulas all deal with circuits composed of 
 parallel conductors completing a loop. When a conductor is 
 
906 
 
 ALTERNATING CURRENTS 
 
 vertical with respect to the earth and removed from parallel 
 conducting bodies, a flow of current may occur through the 
 effects of radiation or of electrostatic capacity. In this case 
 the self-inductance per centimeter of length of a cylindrical 
 conductor is of the form of 
 
 in which R is the geometric mean distance from each other of 
 the elements of current in the conductor and is equal to re-* or 
 r ( r being the radius of the wire), depending upon whether the 
 current is uniformly distributed over the cross section of the 
 conductor or is concentrated on its surface. 
 
 When the loop composed of two conductors carries currents 
 which are in different phases in the conductors, as is the case 
 with a loop composed of either pair of wires of a three-phase 
 circuit, the self-inductive voltage introduced into the loop is 
 not equal to the reactance of the loop multiplied by the current 
 in one of the wires. The currents in the two conductors being 
 not in the same phase, the maximum magnetic flux set up in 
 the loop is proportional to their vector sum instead of their 
 algebraic sum. The self-inductive voltage introduced in the 
 loop in such cases is therefore equal for a balanced circuit to 
 the loop reactance per unit length of conductor (as given above) 
 
 n is the number of phases. This voltage is in quadrature with 
 the vector representing the sum of the two currents. In the 
 case of a symmetrically placed three-phase circuit, a convenient 
 manner of computing the effect of self-induced voltage on the 
 regulation of the circuit is to consider the voltage per conductor, 
 which is equal to the conductor reactance multiplied by the 
 conductor current and acts in quadrature to the conductor cur- 
 rent. This voltage is opposite to the reactive drop along the 
 wire which is measured between the conductor and the neutral 
 point. 
 
 Prob. 1. The parallel conductors of a balanced three-phase 
 circuit are each composed of a No. 0 B. & S. gauge wire and are 
 located at the corners of an equilateral triangle so that they are 
 
 * Maxwell, Electricity and Magnetism , Arts. 685-693. Steinmetz, Transient 
 Electric Phenomena and Oscillations , Chap. VIII, Sec. III. 
 
 current per conductor and 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 907 
 
 36 inches apart, center to center. The length of each conductor 
 is 10 miles, and each carries a current of 50 amperes. What 
 is the value of the self-inductive voltage induced per con- 
 ductor ? What is the value of the self-inductive voltage in- 
 duced in a loop composed of two of the conductors ? 
 
 Prob. 2. In Prob. 1, suppose the voltage between conductors 
 at the receiving end of the line is 10,000 volts, what is the delta 
 voltage and also what is the wye voltage at the generator, 
 assuming that the onl} r causes contributing to line drop are 
 resistance and self-inductance ? 
 
 212. The Distribution of Current in a Wire. — The distribu- 
 tion of current over the normal cross section of a homogeneous 
 conductor along which it flows is uniform, provided the current 
 is steady, as this is the distribution that requires the smallest 
 voltage to cause a given current to flow. The proof of this 
 theorem is as follows: The total power lost in heating the con- 
 ductor is, according to Joule’s law, I 2 R, where R is equal to 
 
 Z, p, and A being respectively the length in centimeters, the 
 
 specific resistance in ohms per centimeter cube, and the area in 
 square centimeters of the conductor. Considering the con- 
 ductor to be divided into elementary filaments of equal area 
 and of resistance r, the current flowing in one of these is i 
 and the power lost in it is i 2 r. The total power lost in the 
 conductor is 2i 2 r, and the conditions afford the two simulta- 
 neous equations 
 
 2Z 2 r = im, 
 
 = I. 
 
 These conditions can be simultaneously fulfilled only when the 
 currents in the filaments are all equal to each other, and the 
 distribution of the current over the cross section of the con- 
 ductor is therefore uniform. If the filamentary currents were 
 not uniform, the values of ir for the different filaments would 
 differ from each other and from IR, which would cause a diver- 
 sion of current from one filament to another until the uniform 
 distribution was again reached. 
 
 The theorem of uniform distribution over the cross section 
 applies to steady currents as just shown, hut it does not apply 
 to alternating currents, as the effect of mutual induction be- 
 tween filaments comes in to disturb the distribution. Suppose 
 
908 
 
 ALTERNATING CURRENTS 
 
 a homogeneous cylindrical conductor divided into concentric 
 cylindrical elements of equal cross sections, then by the formulas 
 developed in the preceding article, 
 
 _2 ill- 
 
 
 <j> _ log 2 + 
 
 10 S r 10 
 
 I 
 
 which exhibits the flux within a radius d which is set up by a 
 steady current I flowing in the conductor, and shows that a 
 greater number of lines of force surround the central element of 
 the conductor than surround the elements nearer the surface, 
 when the current is uniformly distributed over the cross section 
 of the conductor. In fact, when the current is uniformly distrib- 
 uted over the cross section of the conductor, the element com- 
 posing the outside of the cylindrical conductor is surrounded 
 
 by ^ less lines of force than the central element. 
 
 When the 
 
 current flowing through the conductor is alternating, however, 
 the uniformity of distribution is disturbed because a counter- 
 voltage is set up in each element which is equal to — where 
 
 dt 
 
 <j> f is the flux which is set up by the current and surrounds the 
 filament under consideration. 
 
 Since (f) f increases from the outside of the wire towards the 
 central elements, unless the current is all concentrated on the 
 outer surface of the conductor, the counter-voltage is greatest 
 at "the center and least at the surface of the conductor. Con- 
 sequently there is a tendency for the current to forsake the 
 center of the conductor and to seek a place nearer the surface. 
 This tendency is directly proportional to the frequency of the 
 cycles of flux when the current is sinusoidal. This tendency is 
 opposed by the tendency of the current to a distribution which 
 will give the least loss of energy, and alternating current there- 
 fore distributes itself in a conductor so that these two tenden- 
 cies are balanced against each other, for which reason the current 
 becomes distributed over the cross section so that its density 
 increases from the center to the surface of the conductor. 
 This makes an increase in the actual resistance to the flow of al- 
 ternating current , and in the loss of energy caused by the current 
 flowing through the conductor. 
 
 The resistance i? a , which a straight conductor of length l ex- 
 poses to an alternating current, in ratio to the true resistance 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 909 
 
 (or resistance which the conductor exposes to a steady current) 
 R, may be calculated by the following formula in which R a 
 and R are given in ohms, f in periods per second, and l in 
 centimeters,* 
 
 R a _ X , 
 
 R 12 U0 SR) 
 
 2 tt/AV 
 
 . 10 v ) 
 
 JLr^fL Y+ etc 
 
 lSOUO 9 ^ 2440U0 9 i27 
 
 /X 4 f 2 tt/AV , p? (2 7rfA\ 6 , 
 
 180V 10 9 p 7 + 2440V 10 9 p 7 6 
 
 In the last expression A is the area in square centimeters of 
 the cross section of the conductor and p is the specific resist- 
 ance of the material measured in ohms per centimeter of length 
 and square centimeter of cross section (per centimeter cube). 
 
 When the material of the conductor is copper, fi = l, and the 
 formula may be written approximately for copper conductors, 
 in which p = 16 x 10~ 7 ohms for the centimeter cube, 
 
 ^ = 1 + 30(dyio- 9 ) 2 , 
 
 R 
 
 where cP is the number of circular mils in the cross section. 
 For other conductors the approximate formula is 
 
 •| a = l + 30^^10- 9 J, 
 
 in which p is the specific resistance of copper, p' the specific 
 resistance of the material composing the conductor concerned, 
 and p is the permeability of the conductor. When d 2 fp in 
 these approximate formulas exceeds 5 x 10 9 , the resistance 
 which a cylindrical conductor offers to alternating currents is 
 approximately equal to the true resistance of a tube of the same 
 material with its outside diameter equal to that of the conductor 
 and with the thickness of its wall equal to 2500/V/mils. 
 
 The following table gives the ratio of R a to R for various 
 values of the product d 2 f for cylindrical conductors, d being 
 the diameter in mils. 
 
 * Maxwell, Electricity and Magnetism , Vol. II, Chap. 13 ; Rayleigh , On Self- 
 Induction and Resistance of Straight Conductors, Philosophical Magazine , 
 May, 1886, p. 381. 
 
910 
 
 ALTERNATING CURRENTS 
 
 
 Product of Circular 
 
 Ha 
 
 Product of Circular 
 
 
 Mils and Frequency 
 
 It 
 
 Mils and Frequency 
 
 R 
 
 10,000,000 
 
 1.003 
 
 70,000,000 
 
 1.15 
 
 15,000,000 
 
 1.007 
 
 80,000,000 
 
 1.19 
 
 20,000,000 
 
 1.012 
 
 90,000,000 
 
 1.24 
 
 30,000,000 
 
 1.03 
 
 100,000,000 
 
 1.30 
 
 40,000,000 
 
 1.05 
 
 125,000,000 
 
 1.47 
 
 50,000,000 
 
 1.08 
 
 150,000,000 
 
 1.67 
 
 60,000,000 
 
 1.11 
 
 Infinite 
 
 GO 
 
 This table shows that with an increase of the frequency of 
 the circuit or of the diameter of the conductor, or of both, the 
 energy loss per ampere flowing in the conductor increases, and 
 that the energy loss may become very great (as though the 
 current wholly forsook the middle parts of the conductor and 
 confined itself to a thin exterior layer) when the product of the 
 frequency of the current and the square of the conductor 
 diameter becomes very great. On account of this apparent 
 concentration of the current in a superficial skin of the wire the 
 phenomenon is commonly called Skin effect. 
 
 The current density at any point within a conductor is shown 
 by Thomson to fall off in going from the surface inwards, in 
 the ratio of 
 
 where x is the distance of the point from the surface.* Gray f 
 points out that, however great the diameter of a wire may be, 
 its resistance to alternating currents of different frequencies 
 will never be less than the true resistance of a wire of the 
 diameter in centimeters given in the following table : 
 
 Frequency 
 
 Copper 
 
 Lead 
 
 Iron, h = 300 
 
 80 
 
 1.43 
 
 4.98 
 
 .195 
 
 120 
 
 1.17 
 
 4.08 
 
 .159 
 
 160 
 
 1.02 
 
 3.52 
 
 .138 
 
 200 
 
 .91 
 
 3.16 
 
 .123 
 
 * J. J. Thomson, Recent Researches in Electricity and Magnetism, pp. 260 
 and 281. 
 
 t Absolute Measurements in Electricity and Magnetism, Yol. II, p. 338. 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 911 
 
 The specific resistance of lead is not far from that of average 
 German silver. It is about twice that of iron and about twelve 
 times that of copper. The remarkably large skin effect indi- 
 cated for iron by this table is due to its relatively large magnetic 
 permeability. The tabular values should be proportional to 
 the square roots of the specific resistances of the different metals 
 and inversely proportional to the square roots of their mag- 
 netic permeabilities. For metals of a given specific resistance, 
 the tabular values should be inversely proportional to the 
 square roots of the frequencies. In view of the high frequency 
 harmonics that it is desirable to preserve in telephone currents, 
 the table indicates a reason why iron should be rejected as a 
 material for telephone conductors. It must be understood that 
 the permeability of iron wires depends upon the quality of the 
 iron and the magnetic density set up in its mass, and that 
 it may vary from a few tens of units to many hundreds of 
 units. The tabular value of 300 is taken as a median or rep- 
 resentative figure for iron subjected to rather low magnetiz- 
 ing forces. 
 
 The formula for the self-inductance of a cylindrical wire was 
 developed in the preceding article on the assumption of a uni- 
 form distribution of current over the cross section of the con- 
 ductor. The disturbance of this distribution by skin effect 
 reduces the self-inductance, the formula for self-inductance in 
 henrys per centimeter of length becoming 
 
 L a = 
 
 V lo g^ + /“' 
 
 1 1 ( 2 tt/^ Y | 13 
 
 2 tt/aMY 
 
 _2 48 \ 10 9 p J 8640 V 10 9 p J 
 
 f 2 7T 
 
 V 1( 
 
 — etc. 
 
 or 
 
 L a =L-/ 
 
 ■ 1 f 2 wffi'A V _ _ 13 _ ( 2 vfu'A Y 
 
 _ 48 V 10 9 p J 8640 V 10 9 p J 
 
 + etc. 
 
 lO- 9 , 
 
 10-9. 
 
 The limiting values of R a and L a are R a = R and L a = L for 
 steady currents or alternating currents of low frequency in con- 
 ductors of moderate thickness; and R a = cc and L a — 2 log - for 
 very high frequencies. 
 
 213. Mutual Induction of Parallel Distributing Circuits. — 
 
 Where two or more electric light or power circuits carrying 
 alternating currents run parallel to each other, they act in- 
 ductively upon each other, and in some cases the mutual indue- 
 
912 
 
 ALTERNATING CURRENTS 
 
 tion may cause considerable interference with the uniformity 
 of the voltage on the lines. The mutual inductance of any two 
 parallel circuits of indefinitely great length may be easily cal- 
 culated, provided the distances apart of - the different wires 
 composing the circuits are known. The mutual inductance of 
 the two circuits given in henrys is equal to 10 -8 times the 
 number of lines of force which pass through or link with one 
 of the circuits due to one ampere flowing in the other circuit, 
 provided there is no iron in the magnetic path ; and this num- 
 ber of lines of force is equal to the algebraic sum of the number 
 of lines of force embraced by the first circuit which would 
 be set up by the current in the individual conductors of the 
 second circuit taken separately. The method of Art. 211 is 
 therefore directly applicable to the calculation of the mutual 
 inductance of two long and parallel, narrow circuits. The 
 following examples represent the commonest arrangements of 
 circuits on pole lines. 
 
 Suppose that a, a' and 5, b' represent the conductors of two 
 circuits, and that the order of the wires is a — a' — b — b' , the dis- 
 tance apart center to center of the wires of circuit A is x, of 
 circuit B is y, and of the adjacent wires of the two circuits 
 ( a ' — b~) is z ; then if we consider the currents as concentrated at 
 the centers of the wires, which makes but an insignificant error 
 with the ordinary dimensions of conductors and circuits, and 
 consider the space between two planes perpendicular to the cir- 
 cuits and one centimeter apart, the number of lines of force 
 due to a current of one ampere in a', which pass through the 
 circuit B between the planes, is (Art. 211) 
 
 and the number of lines of force due to a current of one 
 ampere in a which pass through the circuit B between the 
 
 planes, is 
 
 C*+y+z 2 da 
 Uz 10 a 
 
 The total number of lines of force set up by the current of one 
 ampere in circuit A , which pass through the circuit B between 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 913 
 
 the planes, is <E> 0 4- 4> a <, the number which link through the B 
 circuit in a length of l centimeters is 
 
 Z($ a + 
 
 and the mutual inductance of the parallel circuits of length l 
 centimeters is 
 
 M= 1(4'. + K..) = 2 l A <r+£ _ j x + g + A 
 10 8 10 9 V 8 a * x + z J 
 
 _ ~lL Iqp- Q + *)0/ + z ) = 4 - 60 log (x + zYy + z) 
 10 9 c z(x-\-y + z) 10 9 5,10 z(x + y + z) 
 
 If x = ?/, this becomes 
 
 ■, (x + z') 2 4.60 Z. (x+z') 2 
 
 log - = Tor l0 *i» a (2UjTj ; 
 
 and if x = y = z, it becomes 
 
 jfff- 4 - 60 ^ i 4 
 - IQ 9 1Ogl0 3 
 
 575 l 
 10 9 ’ 
 
 where l is the length of the parallel circuits in centimeters. 
 
 The mutual inductance in millihenrys per 1000 feet of dis- 
 tance in which the circuits are parallel is, using common loga- 
 rithms, 
 
 M= .1404 log 
 
 M" -- .01752. 
 
 (x + z)(y + z) . 
 z(x + y + z) 
 
 M' = . 1404 log - (x + z ' )2 - ; 
 
 ° z(2 x + z) 
 
 Exchanging the order of the wires so that circuit A is be- 
 tween the conductors of circuit B, thus b — a — a' — b', changes 
 the formulas. Here the algebraic sum of the number of lines 
 of force set up by the circuit A which link with circuit B is 
 equal to the total number of lines of force set up by circuit A 
 minus the number passing backwards through b — a and a' — b'. 
 Suppose that distance a — a' is equal to x and distances b — a 
 and a ' — b' are each equal to y , then the total number of lines 
 of force set up by one ampere in a length of one centimeter of 
 circuit A , is 
 
 <D 0 =.4 log - 4- .2, 
 
 r 
 
 3 N 
 
914 
 
 ALTERNATING CURRENTS 
 
 (r being the radius of the conductor), and the number of lines 
 due to one ampere in circuit A, which pass between the planes 
 through the space b — a, is 
 
 = .2 log, V- + .1 - .2 log, _ .2 log. 
 
 r(x + y') 
 
 + -1 •> 
 
 
 xy 
 
 r(x + y) 
 
 ii 
 
 10 9 
 
 x -f y _ 9.20 l 
 y 10 9 -- y 
 
 TEtz-i JU ~ If V I X ~\- If 
 
 - ^ log, — ^ = - T ^ r log 10 
 
 T , Tvrn 9.20 1, 0 2.77 l 
 
 If x = y, M"= — — log 10 2 = - 
 
 10 9 
 
 The mutual inductance in millihenrys per 1000 feet of dis- 
 tance in which the circuits are parallel is, using common loga- 
 rithms, 
 
 M’= .2807 log 10 ^hJ/ ; M" = .08451. 
 
 y 
 
 If the circuits are not in the same plane, as, for instance, 
 they are arranged thus, 
 
 a — a ’ , 
 b-b', 
 
 and the distance a — a' is x, the distance b — b' is y , the dis- 
 tance a' — b' is z , the distance a' — b is w, a — b is v, and a — b' 
 is u ; then the formulas are 
 
 d>a = • 2 flog, - — log, -'W .2 log, - , 
 \ r r) v 
 
 4 >«• = - 2( l°g,^ - lo g,f) = - 2 lo £ 
 
 w 
 
 and 
 
 nr ^ i \ 2 Z, uiv 4.60 £ , uw 
 
 M = Tua (4> “ + *“■> = 105 ‘° g - ^ l0 *i. T7 ' 
 
 If one circuit is directly beneath the other and x = y, v = z. 
 then w = u = V.t 2 + z 2 , and the formula becomes 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 915 
 
 The mutual inductance in millihenrys per 1000 feet of dis- 
 tance in which the circuits are parallel is, using common loga- 
 rithms, 
 
 M= .1403 log — ; M' = .1403 log ; M" = .04225. 
 
 vz Z l 
 
 In case the arrangement is 
 
 a — b 
 V -a! 
 
 , one circuit being directly 
 
 above the other and the distance a — b equal to the distance 
 b' — a, that is, the wires are on the corners of a square with the 
 plane of each circuit tracing the diagonal, then the flux set up 
 by either circuit is tangential with the other and the mutual 
 inductance is zero, there being no linkages. 
 
 These results plainly show that the mutual inductance of two 
 circuits is entirely independent of the actual distances apart of 
 the conductors composing the circuits, but depends wholly 
 upon the relative values of the distances. The mutual induct- 
 ance of two circuits is a maximum when the circuits are exactly 
 superposed, in which case M=VL'L"=L, and decreases as 
 the distance between the circuits is increased in comparison 
 with the distance apart of the conductors of each circuit ; con- 
 sequently, mutual inductance between circuits on the same 
 pole line may be reduced by decreasing the distance apart of 
 the conductors of each circuit and increasing the distance apart 
 of the circuits. A better way to avoid annoyance from the 
 effects of mutual inductance in most cases is to transpose the 
 positions of the circuits with reference to each other at fixed in- 
 tervals of distance on the pole line, so that the inductive effects 
 of the circuits on each other are in opposition in different equal 
 parts of the line, and neutralize each other for the line as a whole. 
 If there are more than two circuits on a pole line and trans- 
 positions are needed, it is important to make the transpositions 
 so that the magnetic mutual reactions which occur between any 
 of the circuits all balance off and neutralize. For this purpose 
 it is necessary that the transpositions shall be made properly. 
 In any transposition interval, one line may be run straight 
 through, one should be transposed at the middle, one at a 
 quarter way from each end of the interval, one at the quarters 
 and the middle, etc. 
 
 The effect of mutual induction between two circuits is to set 
 up a voltage in one when the current in the other varies. If 
 
916 
 
 ALTERNATING CURRENTS 
 
 the current is a sinusoidal alternating one, this voltage is (Arts. 
 119 and 120) 2 7 rfMI, and the effect of an alternating current 
 in one circuit upon another circuit is easily determined if Afis 
 known. When the two circuits are fed from the same single- 
 phase alternator, the induction of one upon the other is in quad- 
 rature with the current in the first, and the phase relation of the 
 voltage induced in the second relative to the impressed voltage 
 therein depends on the current lag in the first. If this is zero, 
 the induced and impressed voltages are in quadrature, while 
 they are in opposition if the lag is 90°. The result is a dis- 
 placement of the voltage waves and a drop of voltage along the 
 lines. The effect of reversing the connection of one of the cir- 
 cuits to the alternator causes an interchange of the voltage 
 relations of the two circuits, but nothing else. 
 
 If the circuits are fed from different alternators the frequen- 
 cies of which are slightly different, the inductive voltage and 
 impressed voltage interfere in each circuit so as to form pulsa- 
 tions or beats, the frequency of which is equal to the difference 
 of the two alternator frequencies, and the amplitude of which 
 is the sum of the impressed voltage and the induced voltage. 
 This may cause a perceptible winking of incandescent lamps 
 connected to mutually inductive circuits carrying heavy cur- 
 rents of nearly the same frequency. 
 
 214. Effects of Self and Mutual Induction in Polyphase Cir- 
 cuits. — The effects of self and mutual induction in polyphase 
 circuits may be determined by using the principles set forth 
 in the preceding articles, provided the resultant effect of the 
 differing phases is always considered. In unbalanced circuits, 
 if mutual inductive effects are allowed to occur, they may in- 
 crease the defect in balance. Mutual induction between the 
 phases of one polyphase feeder may be avoided by suitably 
 placing the conductors. Thus, in the case of a four-wire two ' 
 phase feeder, the wires may he located on two cross arms so 
 that the planes of the conductors of the two phases form the 
 
 diagonals of a square, thus a where a , a are the conduc- 
 
 o a 
 
 tors of one phase and 5, b are the conductors of the other. In 
 this case there is no mutual induction between the circuits. 
 When this arrangement cannot be utilized with quarter phase 
 lines, transpositions must be resorted to. 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 91T 
 
 Three-phase circuits may he erected on insulators located so 
 that the three wires occupy the corners of an equilateral 
 
 triangle, thus a . The lines of force set up by the current 
 c b 
 
 in any one of the conductors are circles surrounding the conduc- 
 tor and are therefore tangent to the plane of the loop composed 
 of the other two conductors. Hence they do not link the loop 
 and no mutual induction occurs. When this arrangement can- 
 not be utilized, transpositions may be used to neutralize mutual 
 inductance. 
 
 When two or more feeders are near each other, as on the 
 same line of poles or towers, mutual induction between feeders 
 may be neutralized by transposing the wires of each feeder 
 spirally in such a way that the inductive actions in successive 
 lengths balance each other. With the equilateral triangle ar- 
 rangement of three-phase conductors, the spiraling of each 
 feeder composed of three conductors may be carried out with 
 regard onty to the effects of the feeders on each other and on 
 other neighboring circuits; but when the equilateral triangle 
 arrangement is not used, the spiraling of each feeder should 
 be accompanied by suitable transpositions to neutralize the 
 mutual induction occurring between the phases of the feeder. 
 
 The vector relations of the voltages in four-wire two-phase 
 circuits when the conductors lie in a plane (as when they are 
 mounted on the pins of the same cross arms on a pole line 
 or are supported one above another by suspension insulators) 
 are illustrated in Fig. 519. The assumed relative positions of 
 the wires are illustrated in the upper part of the figure, at AB, 
 and the vector relations in the left hand and lower parts. The 
 vector diagram is drawn for conditions of balanced load at the 
 receiver end of the line. 
 
 In the vector diagram of Fig. 519, 0E A and 0E B are the vol- 
 tages at the receiver in the respective phases 90° apart, and 
 OI A and OI B are the corresponding currents. The voltage .at 
 the generator 0E° in each circuit is equal to the vector sum of 
 the voltage at the receiver and components numerically equal 
 and opposite respectively to the IR drop in the circuit E' E, 
 the IX drop in the circuit E" E ' , and the voltage E°E" intro- 
 duced in the circuit by the inductive influence of the current in the 
 other branch of the two-phase circuit. The generator voltages 
 
918 
 
 ALTERNATING CURRENTS 
 
 in the two circuits are therefore 0E A and 0E B °. The effect of 
 the mutually induced voltage is therefore to increase the numeri- 
 cal difference and decrease the angular difference between the 
 genei’ator voltage and the voltage delivered to the receiver in 
 
 E°* 
 
 Fig. 519. — Diagram of Vector Voltage Relations in Four-wire, Four-phase Circuits 
 when the Four Wires lie in a Plane, as at AB. 
 
 the leading phase, and to decrease the numerical difference and 
 increase the angular difference between the generator voltage 
 and the voltage delivered to the receiver in the lagging phase, 
 compared with the relations when mutual induction is absent. 
 
 If E a ° is the line voltage at the generator for the leading 
 phase, E a the line voltage at the receiver, R A , L< and JSI the line 
 resistance, self-inductance, and mutual inductance, I A the cur- 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 919 
 
 rent, and Z , R. and X the impedance, resistance, and reactance 
 of the load in the same phase, then 
 
 Ia= ~z = ir+jx and 
 
 Also 
 
 E a = E A ° - I A R A - jcoLl A -jcoMTs 
 
 P o B a E a (R — jX') . coLE a (R — jX) — <bME a (X +jR) 
 
 A i2 2 + X 2 3 R 2 + X 2 
 
 J R +jX 
 
 = x A 0 -x A 
 
 Hence 
 
 e a °=e a 
 
 RR a + XX A + coMR . RX a - XR a - a)i¥X1 
 —+J — 
 
 R? + X 2 
 
 R 2 + X 2 
 
 ' RR a + XX, + CoMR . RX i - XR a - coMX ' 
 
 1 + TY>. . + J ~ 
 
 R 2 + X 2 
 
 R 2 + X 2 
 
 By the same process 
 
 e b ° = e b 
 
 ■ RR a + XX A — (o3IR ,RX a -XR a + coMX 
 1+ R 2 + X 2 +J R 2 + X 2 
 
 When the load is non-reactive, X = 0 and the equations for 
 E ° and E b ° become 
 
 e a ° =e a ( i + 
 
 e° = e b 1 + 
 
 R, — co3f . ,X 4 
 
 A 
 
 
 R 
 
 It may be observed from the equations as well as the diagram 
 that, when mutual induction exists between the phases, the 
 generator voltage is not balanced in case the voltage and cur- 
 rent at the load are balanced ; and conversely, if the generator 
 voltage is balanced, it is necessary to abate mutual induction 
 or else introduce special phase regulators in order to obtain 
 balanced voltage conditions at the load. Inasmuch as the IZ 
 drop of a distribution line is ordinarily much greater than the 
 mutually induced voltage between circuits on the same poles, 
 the unbalancing by mutual induction may not be very serious. 
 
 Figure 520 shows the effect of mutual induction in a three- 
 phase circuit with three wires in one plane, as when they are 
 erected on the same cross arm on each pole of a pole line, or 
 are suspended in a vertical plane by suspension insulators. 
 The lines OE ab , OE bc , and OE ca are balanced delta line voltages 
 
920 
 
 ALTERNATING CURRENTS 
 
 at the receiver, OE a , OE b , and OE c representing the correspond- 
 ing wye voltages from each line conductor to the neutral point, 
 and OI a , OI b , and OI c are the line currents. In this case the 
 mutual inductance of the middle conductor b on the loop com- 
 posed of conductors a and c is zero, provided the distance 
 between a and b is equal to the distance between b and c , since 
 
 Fig. 520. — Diagram for showing the Effect of Mutual Induction in a Three-phase 
 Circuit when the Wires are in One Plane, as at abc. 
 
 under those circumstances the flux set up through the loop a — c 
 by current in b is half upward and half downward. Conductor 
 c, however, possesses mutual inductance with respect to the loop 
 a — b, and conductor a has mutual inductance with respect to 
 loop b — c. In case conductor b is not exactly halfway between 
 a and c, mutual inductance also exists between conductor b and 
 loop a — c. 
 
 In a balanced three-phase circuit the mutual induced voltage 
 set up in a loop of two conductors, such as a and b , by current 
 in the third conductor is 
 
 -y V 2 2 7 rfMI L sin [a + 120° - (30° + 0)], 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 921 
 
 when the instantaneous delta voltage on the loop is propor- 
 tional to sin a. Therefore, if 9 = 0, that is, the receiver cur- 
 rent is in phase with the wye voltage at the receiver, the 
 mutually induced voltage is in the same phase as the delta 
 voltage at the receiver. If 9 is not equal to zero, the mutually 
 induced voltage differs in phase from the delta voltage at the 
 receiver by the angle 9. 
 
 If, in the arrangement of conductor's illustrated in the left- 
 hand part of Fig. 520, the conductor b is moved perpendicular 
 with respect to the plane of loop a — e, tire mutual inductance of 
 a with respect to b — c and of c with respect to a — b decreases 
 until b has reached a point at which distances a — b = b — c = c— a. 
 Under the latter circumstances, the conductors occupy positions 
 corresponding to the corners of an equilateral triangle, the lines 
 of force due to any one conductor are tangent to the plane of 
 the loop composed of the other two, and the mutual inductance 
 is zero. If conductor b is moved farther from the a — c plane, 
 the mutual inductance again increases, but the direction of the 
 fluxes of a and a through the loops b — c and a — b are reversed, 
 so that the mutually induced voltages are reversed. This is 
 obvious from the following considerations: If distance a — c is 
 #, distance a — b is y, and distance b — e is z, then the mutual 
 inductance of conductor a on loop b — c is 
 
 9 / T 
 
 M a = ^logA 
 
 10 M y 
 
 and the mutual inductance of conductor c on loop a — b is 
 
 M c = p-\og e -. 
 c 10 9 Se z 
 
 When b is nearer the plane of a — c than (distance a — <?) x sin 60°, 
 log - and log £ - are positive and therefore M a and M c are posi- 
 
 e y z 
 
 QC jC 
 
 tive; but when b is farther from the plane, log e - and log e - are 
 
 V z 
 
 negative and M a and M c therefore become negative. 
 
 In Fig. 520, EE' for each branch of the diagram represents 
 the line resistance and reactance drop and E'E Q represents the 
 mutually induced voltage, when the receiver voltages and cur- 
 rents are as shown. The vectors OE^ 0 , OE b ° , and OE c °. 
 
922 
 
 ALTERNATING CURRENTS 
 
 therefore, represent the delta voltages at the generator. In 
 this illustration the distance between conductors a and b is 
 taken equal to the distance between conductors b and c, the 
 three conductors being in a plane, and there is no mutual vol- 
 tage in the c— a loop. It will thus be observed that the voltages 
 at the generator end of the line must be unbalanced in case the 
 voltages at the receiver end are to remain balanced, unless 
 the conductors can be located on the apices of an equilateral 
 triangle or transpositions are resorted to. 
 
 215. Electrostatic Capacity of Parallel Conductors. — When 
 the voltage used for electrical transmission of power is high, the 
 amount of transmitted current is relatively small, especially 
 when the load is light, and the reactance drop in the line may 
 be largely affected by the electrostatic capacity between the con- 
 ductors. It is, therefore, necessary to investigate the magni- 
 tude of the electrostatic capacity between parallel conductors. 
 
 Electrostatic capacity of two neighboring conductors with 
 respect to each other may be defined as the ratio of the electro- 
 static charge on eacli to the difference of potential between the 
 conductors which is set up by the charges, one of which charges 
 is positive and the other of which is equal numerically but of 
 opposite kind. The capacity in farads is the ratio between the 
 charge measured in coulombs and the difference of potential 
 measured in volts. When the conductors are large parallel 
 plates whose distance apart is small compared with their super- 
 ficial dimensions, the capacity may be readily found on the 
 assumption that a charge distributes uniformly over them, 
 which is true except for portions near the edges. These con- 
 ductors being charged, and away from the influence of other 
 conductors, one carries a positive charge and the other carries 
 an equal negative charge. The electrostatic field between them 
 is uniform except near the edges, and the difference of poten- 
 tial measured from one to the other is equal to the intensity of 
 the field multiplied by the centimeters of distance between the 
 plates. The measure of the intensity of the field is the force 
 in dynes which would be experienced by a C. G. S. electrostatic 
 unit of charge moving from the plate of lower potential to the 
 plate of higher potential. 
 
 The force which measures the field intensity is numerically 
 equal to 1 /K times the electrostatic lines of force per square 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 923 
 
 centimeter in the dielectric (measured in a plane perpendicular 
 to the flux) ; and this flux is equal to 4 7 rcr, in which a is the 
 electrostatic charge per square centimeter on each plate, meas- 
 ured in electrostatic units, and K is the dielectric constant of 
 the insulator lying between the plates. Also Q = a A where A 
 is the area of each plate in square centimeters. Therefore, by 
 definition, the capacity of the condenser composed of the par- 
 allel plates is, in electrostatic units of measure, 
 
 C-Qa- Qa - KA _ 
 
 a E a F a d 4 7 rda a 4 ird 
 
 To transform from C. G. S. electrostatic units to C. G. S. 
 electro-magnetic units, with which the practical units conform, 
 it is necessary to divide by v 2 where v is the velocity of light, 
 and to transform from C. G. S. electro-magnetic units to farads 
 it is necessary to multiply by 10 9 since a C. G. S. electro-mag- 
 netic unit of capacity is TO 9 times as large as a farad. The 
 value of v has been experimentally proved to be substantially 
 3 x 10 10 . Consequently, the capacity of the parallel plates in 
 farads is 
 
 n __KA 10- 9 
 1.13 d ’ 
 
 A being the area of each plate in square centimeters and d the 
 distance between them measured in centimeters. For inch 
 
 KA IO - 9 
 
 measure the formula is C= 2.25— . 
 
 d 
 
 In the case of concentric cylinders of great length compared 
 with the radial clearance between them, one of which carries a 
 positive charge and the other an equal negative charge, the 
 surface density of the charge is uniform on each cylinder 
 except close to the ends. Considering the end effects negligi- 
 ble in long cylinders, the lines of force in the electrostatic field 
 are radial. Inasmuch as the radial force exerted on an electro- 
 static unit charge at any point P outside of a uniformly 
 charged cylinder of indefinitely great length is 
 
 ~ dynes, 
 
 Kr 
 
 where q is the charge in C. G. S. electrostatic units per centi- 
 meter of length of the cylinder and r is the radial distance in 
 
924 
 
 ALTERNATING CURRENTS 
 
 centimeters from the axis of the cylinder to the point P ; * and 
 also, inasmuch as the force inside of a uniformly charged 
 cylinder is zero, when the influence of the ends may be 
 neglected ; therefore the capacity in C. G. S. electrostatic 
 units of the concentric cylinders is 
 
 C -<?«- & - KI <1 = Kl 
 
 f r F a dr 2 qf ^ 2l0gX 
 
 */r r rj 
 
 in which and r 2 are respectively the outer radius of the inner 
 cylinder and the inner radius of the outer cylinder, and l is 
 the length of each cylinder in centimeters. Reducing the 
 equation to represent capacity in farads gives 
 
 C - 
 
 Kl 10- 9 
 1800 log E - 
 
 and, finally, reducing the equation to represent farads per 1000 
 feet of each cylinder and using common logarithms gives 
 
 r _ 7.35 K10~ 9 
 
 log w * 
 
 r i 
 
 When the cylinders are not concentric but the distance be- 
 tween centers is a centimeters, one cylinder being entirely sur- 
 rounded by the other, the last equation becomes 
 
 
 2 r 2 r 
 
 7.35 W10- 9 
 
 + r, 2 — a 2 \ 
 2 r r J 
 
 * } 2 r i ' 
 
 These formulas apply to an underground insulated conductor 
 surrounded by a lead sheath, in the usual construction of high 
 voltage cables for underground use. 
 
 * The force at P due to the charge on any element of length of the cylinder 
 
 is Qf ' 1 — — . Only the radial component of this is effective, as components 
 
 parallel to the cylinder neutralize each other, and the radial component is 
 
 2^ sin 0 = - — x — = The f orce d ue t0 t j ie 
 
 K(x> + r 2 ) K(x 2 + r 2 ) Vx 2 +r 2 jv(.r 2 +r 2 )i 
 
 entire cylinder of indefinitely great length is therefore 
 
 —7^ dynes. 
 
 K(x 2 + r 2 ) 2 Kr 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 925 
 
 In case the cylinders are side by side instead of one inside 
 the other, as in the case of two overhead line wires or two 
 separately insulated conductors lying side by side in the same 
 underground cable, the equation takes a different form. The 
 following development of the equation is simple. 
 
 Considering two indefinitely long parallel conducting fila- 
 ments A and A! (perpendicular to the plane of the page), Fig. 
 521, charged respectively with + q and — q electrostatic units of 
 charge per centimeter of length, the force experienced by a 
 unit charge at any point P in the electrostatic field of filament 
 
 9 
 
 A and distant r centimeters from the filament would be if 
 
 Kr 
 
 filament A' were removed, and the potential at the point due 
 to A is 
 
 =f 
 
 if 
 
 Also, the potential at P due to A' alone is 
 
 '•--JTfc*' 
 
 if r' is the radial distance of the point from the filament A ' . 
 The total potential at P due to the charges on A and A ' is 
 therefore 
 
 v p = V+ V' = 
 
 dr 
 
 2 q-. r' 
 
 =ic los -y 
 
 Therefore, if the point P moves, the potential V P at the 
 point P does not change provided the locus of P maintains the 
 
 ratio — constant ; and this locus is a circle with its center 0 
 r 
 
 (Fig. 521) located on the line joining A and A' extended and 
 with radius of length a which is a mean proportional between 
 the lengths OA and OA ' . If P 1 is the intersection of the 
 circle with the line OA', these conditions arise, 
 
 A'P 
 
 AP 
 
 - m, 
 
 OP , _ OA' 
 OA OP i 
 
92G 
 
 ALTERNATING CURRENTS 
 
 in which P is any point on the circle, P, is the point where the 
 circle intersects the line OA ! , and m is a constant. It therefore 
 follows that 
 
 OP OP, _ OA' A'P, A'P 
 
 OA OA OP, AP, AP m ' 
 
 and by division in the third and fourth ratios, 
 
 OP = A- P\ x AP, 
 
 1 A'P , - AP, 
 
 The equipotential surfaces around each one of the charged 
 filaments are cylinders of radius increasing from zero to infinity 
 
 Fig. 521. — Diagram showing a Locus of Equal Potential between Two Parallel 
 Charged Filaments A and A'. 
 
 as the point P is taken at increasing distances from the fila- 
 ment. These merge into a plane of zero potential when 
 A'P, = AP V P, being halfway between the two equally but 
 oppositely charged filaments. This plane touches infinity and 
 has the same potential (namely, zero) as space at infinite dis- 
 tance from the charges. 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 927 
 
 Introducing a cylinder of conducting material with its outer 
 surface coincident with the position of any one of the equipo- 
 tential surfaces will not alter the electrostatic field; and the 
 electrostatic field of two charged cylindrical parallel metal con- 
 ductors may be computed by considering the surfaces of each 
 as an equipotential surface surrounding a charged filament, the 
 
 Fig. 522. — Diagram for illustrating Method of finding the Mutual Potentials of Two 
 Charged, Cylindrical, Parallel Conductors. 
 
 center to center distance between tbe cylinders being the 
 center to center distance 00' between the corresponding equi- 
 potential circles (Fig. 522). Then, according to the preceding 
 paragraphs, the mutual potential at the surface of cylinder A is 
 
 D=^io g .r,, 
 
 and the mutual potential at the surface of the cylinder A' is 
 
 7 
 
 Therefore, the difference of potential in C. G. S. electrostatic 
 units is 
 
 K = v A - V A , = ii log r* = log, 
 K r,r, K 
 
 aa 
 
 1 4 
 
 since the relations proved by Fig. 521 are in Fig. 522, 
 
 r 2 _ A'P __ OP a d r„ _ AP' _ O' P' _ a ' 
 r x AP OA « :UU r 4 A’P' O' A! a r 
 
 The electrostatic capacity with respect to each other of the two 
 
928 
 
 ALTERNATING CURRENTS 
 
 parallel cylindrical conductors given in C. G. S. electrostatic 
 units is therefore 
 
 o. = i 
 
 Kl 
 
 2 l0g e 
 
 
 
 aa 
 
 aa' 
 
 where l is the length of each cylinder in centimeters. 
 
 It is desirable to convert this formula into one involving 
 only the radii a and a' and the center to center distance 3 of 
 the cylinders, by eliminating the distances a and This may 
 be easily done, since a + «' -f- d = 3. Then putting 
 
 gives 
 
 Whence 
 
 3 2 - a 2 - (V) 2 _ t 
 
 aa 
 
 — = b± V6 2 - 1. 
 
 aa' 
 
 c a = 
 
 Kl 
 
 21og e (b ± Vi 2 — 1) 
 
 When the cylinders are eccentric, as in the case under consider- 
 ation, the positive sign applies before the radical in the denom- 
 inator of the last expression. 
 
 To bring this to farads per 1000 feet of conductor in the 
 circuit and using common logarithms, this must be multiplied 
 by 7.35 x 10 -9 , so that 
 
 C 7.35 K 1Q~ 9 _ 3.68 K 10~ 9 
 
 1 21og 10 (6 +V6 2 - 1) log 10 (6 + VP- 1) 
 
 When the two cylinders are of the same diameter, a = a', 
 whence b = ^ — 1, 
 
 and 
 
 Therefore 
 
 b + VJ 2 - 
 
 
 3.68 Kl 0- 9 
 
 1.84 K 10~ 9 
 
 lo 
 
 olO 
 
 + 
 
 z a 
 
 m-')' 
 
 When the distance apart (center to center) of the conductors 
 is large compared with the diameter of each conductor, as for 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 929 
 
 instance if this ratio is as great as is usual with overhead line 
 ires, \j\ 
 
 wir 
 
 ^ — 1 approximates to and a closely ap- 
 proximate formula for the capacity in farads per 1000 feet of 
 conductor may be written 
 
 ^2 = 
 
 1.84 K 10 -9 
 
 log 
 
 10 ' 
 
 When the conductors are in air, the dielectric constant is 
 unity ; therefore for overhead line wires the capacity in farads, 
 measured from conductor to conductor, per thousand feet of 
 conductor, is usually obtained from the expression 
 
 ^2 = 
 
 1.84 x 10~ 9 
 
 i 3 
 
 log„- 
 
 Since log 6 (b + v/6 2 — 1) = cosh -1 b,* the exact formula for 
 the capacity in farads per thousand feet of conductor may be 
 written 
 
 c = 16.94 iT10- 9 
 1 cosh -1 b 
 
 which is easily computed with the aid of a table of the functions 
 of hyperbolic trigonometry, such as are contained in the Smith- 
 
 ,32 
 
 sonian Mathematical Tables. In this formula b = — - — 1 when 
 
 2 a 2 
 
 the conductors are of equal diameters, in which case, putting 
 c for the ratio of the center to center distance to the diameter 
 
 Q 
 
 of a conductor, c = — and b = 2 e 2 — 1. 
 
 2 a 
 
 Therefore, b + ^/b 2 — 1= 2 c 2 + 2 ci^c 2 — 1)^ — 1 = (<?+ Vc 2 — l) 2 , 
 and the capacity in farads per thousand feet of conductor is 
 
 c _ 8.47iT10- 9 
 
 2 cosh -1 c 
 
 Pender and Osborne have shown that for parallel cylindrical 
 conductors removed from the influence of other conductors, the 
 
 * McMahon, Hyperbolic Functions, Art. 19. 
 
930 
 
 ALTERNATING CURRENTS 
 
 distribution of the charge around the cross-section which is 
 required to keep the surfaces equipotential surfaces, gives a 
 surface density of the charge on each conductor which conforms 
 to an ellipse with its major axis in the line joining the centers 
 of the two conductors.* 
 
 The capacity with respect to ground of an overhead conduc- 
 tor may be deduced from the reasoning applied to two parallel 
 conductors. Suppose the conductor of diameter 2 a is erected 
 at a distance h above the surface of the earth and substantial!}' 
 parallel therewith, the distance h being many times larger than 
 a. The capacity of 1000 feet of this conductor, with respect 
 to a hypothetical equal conductor or image at a distance h 
 below the surface of the earth, would be 
 
 C = 
 
 3.68 xlO- 9 
 
 lo 
 
 2 h ’ 
 
 olO 
 
 in which K becomes unity because the dielectric is air. But 
 the surface of the earth corresponds with the plane of zero 
 potential which is halfway between two conductors, and the 
 difference of potential between the conductor concerned and 
 the earth with a given charge Q on the conductor is one half 
 as great as the difference of potential between the couductor 
 and its image. Therefore the capacity of the conductor with 
 respect to earth in farads per thousand feet of length is twice 
 the foregoing, or 
 
 7.35 x IQ- 9 
 
 C = 
 
 , 2 h 
 
 lo gio— " 
 a 
 
 The foregoing expressions for the capacity of cylindrical 
 conductors in concentric and excentric. relations are directly 
 applicable to the conditions of underground and overhead wires. 
 The formulas applying to cylinders one inside the other apply 
 not only to concentric cables, but are also useful for computing 
 the capacity between the conductor or conductors of a lead- 
 sheathed underground cable and the sheath. The formulas 
 are developed for the relations of two conductors with respect 
 
 * Pender and Osborne, The Electrostatic Capacity between Equal Parallel 
 Wires, Electrical World , 1010, Yol. 50, p. 067. 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 931 
 
 to each other, but the same reasoning may be extended to mul- 
 tiple conductor cables.* 
 
 The capacity current set up by each harmonic of voltage im- 
 pressed between the conductors is 
 
 I = 2 irfCEl, 
 
 where E is the effective value of the harmonic and l is the 
 length of the wire in thousands of feet. The total charging 
 current is the square root of the sum of the squares of all the 
 harmonic currents. The minimum charging current therefore 
 occurs when the impressed voltage is sinusoidal. 
 
 When dealing with polyphase circuits, the capacity between 
 conductors is computed by the foregoing formulas, just as for 
 single-phase circuits ; but the disturbing effect on the electro- 
 static field of the additional conductors and the relation of the 
 mesh current to the star current must be borne in mind when 
 computing the charging current flowing in a conductor. Con- 
 sidering a three-phase circuit with the conductors at the corners 
 of an equilateral triangle and balanced voltage, the electrostatic 
 capacity of one of the conductors with respect to the other two 
 is four thirds the capacity with respect to only one of the others ; 
 but the voltage impressed on this capacity is the voltage re- 
 action from the first conductor to the average or mid-voltage 
 
 of the other two, which is equal to -~-E if E is the line voltage. 
 
 o 
 
 Therefore, in a balanced three-phase circuit with equal capacity 
 for each pair of conductors, the charging current flowing in a 
 
 2 
 
 conductor is equal to -^-(2 TrfCEl^), when C is the capacity 
 ~\/ 3 
 
 per pair of conductors, E is the delta line voltage, and is the 
 length in thousands of feet of each conductor of the three-phase 
 circuit. This gives the same charging circuit per conductor of 
 the three-phase circuit as would be obtained by connecting, from 
 each conductor to the neutral point, a condenser of capacity 2 C 
 microfarads per 1000 feet of the line (not per 1000 feet of 
 wire), in which case the voltage impressed on each condenser 
 
 would be. -^ = , 
 
 V3 
 
 * Russell, Alternating Currents, Yol. I. Heaviside, Electrical Papers , Vol. I, 
 p. 42. 
 
932 
 
 ALTERNATING CURRENTS 
 
 216. Distributed Resistance, Self-inductance, Leakage Conduct- 
 ance, and Electrostatic Capacity. — The relations of resistance, 
 self-incluctance, and capacity which have heretofore been con- 
 sidered in this book have associated the quantities as though 
 they could be segregated in a circuit, and this is a sufficiently 
 accurate hypothesis for most conditions of low voltage com- 
 mercial circuits. In long distance transmission lines of very 
 high voltage, and in some high voltage machines, however, the 
 distributed quantities bring in additional effects which will 
 be briefly discussed here. The complete discussion of these 
 phenomena belongs more particularly to the special field of 
 power transmission, and not within the scope of this work, and 
 the following discussion therefore only gives the fundamental 
 relations. 
 
 In the case of two parallel uniform conductors, it is obvious 
 that each element of length of each conductor possesses a certain 
 self-inductance, and likewise that each element of length of 
 each conductor possesses a certain electrostatic capacity 
 measured between the conductors. These elements of self- 
 inductance and capacity being uniformly distributed in the 
 case of the uniform conductors, the distributed self-inductance 
 and capacity bear relations to each other in transmitting power 
 which are analogous to the mass and compressibility of a 
 vibrating uniform metal rod or string. 
 
 The drop of voltage along an element of length of a con- 
 ductor when a variable current flows is 
 
 de = Ridx + Ldx—, 
 dt 
 
 where R and L are the resistance in ohms and self-inductance 
 in henrys of the circuit per unit (such as a centimeter, a thou- 
 sand feet, or a mile) of length of conductor and dx is an element 
 of the length. Therefore with equal uniform conductors com- 
 posing the circuit, the rate of change of voltage between the 
 conductors as a function of distance along the circuit, that is, the 
 space-rate of change of voltage between the two conductors, is 
 
 de 
 
 dx 
 
 = Ri + L- 
 
 di 
 
 dt 
 
 ( 1 ) 
 
 Also, the loss of current caused by the uniformly distributed 
 capacity, and by leakage from conductor to conductor through 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 933 
 
 the insulation and by convection or corona effects, likewise 
 assumed to be uniform along the circuit, is for each element 
 
 di == Gredx -f C dx—, 
 dt 
 
 where G- and 0 are the conductance in mhos of the insulation* 
 and the capacity in farads per unit of length of conductor. 
 Therefore, the space-rate of change of current with distance 
 along a conductor is 
 
 — =#<; + < 7 —. ( 2 ) 
 
 dx dt 
 
 Differentiating equations (1) and (2) with respect to time 
 and then with respect to distance gives the following four 
 differential equations of the second order: 
 
 d 2 e _ di Pi 
 
 dx • dt dt dt 2 ' 
 
 d 2 i _g.de g d 2 e 
 
 dx • dt dt dt 2 ' 
 
 d 2 e _ di g Pi 
 
 dx 2 dx dx • dt' 
 
 (Pi _ de g d 2 e 
 
 dx 2 dx dx • dt 
 
 By eliminating 
 
 (Pi 
 
 dx ■ dt 
 
 from the second and third of these 
 di 
 
 and substituting the value of ~ given in equation (2), there 
 , , dx 
 
 results 
 
 LC~ + (RC+ LG~)— + RGe = ,:pe 
 
 dt 2 J dt 
 
 ( 3 ) 
 
 A similar process of elimination of 
 
 d 2 e 
 
 dx 2 
 
 from the first and 
 
 dx • dt 
 
 fourth of the group of four equations and substituting the value 
 
 of — from equation (1) gives 
 dx 
 
 LC^l + (RC+LG) c ^+ RGi = '^-- 
 
 dt 2 dt dx 1 
 
 ( 4 ) 
 
 If the impressed voltage is assumed to be sinusoidal at the 
 generator with a frequency corresponding to 2 irf — <a, equa- 
 
 * The insulation conductance in mhos is the reciprocal of the insulation re- 
 sistance measured in ohms. 
 
934 
 
 ALTERNATING CURRENTS 
 
 tion (3) may be solved in the following form, when neglecting 
 the transient state, 
 
 e x — M \e~ ax f A sin (a )t — bx + 7 ) \ 
 
 _l_ e -a(i-x ) ^ ft s [ n y) | . (5) 
 
 This shows that the voltage measured between conductors 
 at any point on the line is equal to the sum of a wave de- 
 creasing logarithmically as the length x of conductors between 
 the generator and point is increased and a wave decreasing 
 logarithmically as the length l — x of conductors from the 
 receiver at the far end of the line is increased. The former 
 wave may be looked upon as a disturbance or wave sent out 
 from the generator and the latter as the return wave produced 
 by reflection of the original wave at the far end of the line. 
 In this formula l represents the total length of conductor in 
 the chosen units and M, A , B, a, b , and 7 are quantities deter- 
 mined by the resistance, self-inductance, leakage conductance, 
 and electrostatic capacity of the line and the frequency of the 
 impressed alternating voltage. If the circuit is of infinite 
 length, the second term of the right-hand member of equation 
 (5) disappears for all finite values of x. 
 
 If all of the resistance, self-inductance, leakage conductance, 
 and capacity in the circuit have constant values which are in- 
 dependent of the current or voltage, the generator current after 
 the expiry of the transient state must as a consequence of the 
 symmetry of the equations be sinusoidal when the generator 
 voltage is sinusoidal, but the trigonometrical terms in the solu- 
 tion for current corresponding to equation (5) contain an addi- 
 tional term representing the angle of lag of the current. 
 
 Equation (5) shows that the voltage at any point is a sine 
 function with respect to time, since at a fixed point, x is con- 
 stant and t is the only variable. The period of this function 
 
 2 7T 1 
 
 is - — = - seconds, which is the period of the generator voltage. 
 
 ® / 
 
 On the other hand, the equation shows that the voltage at a 
 given instant of time varies along the line as a sine function of 
 x multiplied by a logarithmic decrement, and that the wave 
 length in the units of length in which x is measured is 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 935 
 
 That is, through the range of values of x extending from zero 
 units of conductor length (kilometers, miles, or otherwise as the 
 2 77 * 
 
 case may be) to units of length, the voltage goes through 
 o 
 
 a complete cycle of values which is of the form of a sinusoid 
 multiplied by a logarithmic factor. The cycle is repeated along 
 
 O 
 
 each additional units in length of conductor. 
 
 b 
 
 Since the voltage and current at any fixed point go through 
 sinusoidal cycles with a frequency cycles per second, and 
 
 1 7 T 
 
 2 77 * 
 
 points distant - — units of length from each other go through 
 
 b 
 
 their phases simultaneously, it is manifest that the voltage and 
 current cycles are in the nature of disturbances which travel 
 
 2 r 
 
 along the conductors at a rate which traverses — — units of dis- 
 
 b 
 
 2 77 * 
 
 tance in - — seconds. Hence, the velocity of propagation of 
 
 CO 
 
 the wave in distance traveled per second is 
 
 CO „ 
 
 v = 7 i =f\. 
 
 The solution of equations (3) and (4) which are most con- 
 venient for purposes of computation are obtained in terms of 
 the functions of hyperbolic trigonometry, as tvas early pointed 
 out by Kennedy and McMahon.* When the voltage and current 
 at any point on the line are sinusoidal functions of time, the 
 vector relations of e and i with respect to their derivatives for 
 a given value of t may be written 
 
 e = e m sin cot , 
 
 — • = e m a ) cos cot =jcoe , (6) 
 
 dt 
 
 § = - sin tot = — co 2 e ; (7) 
 
 dt 1 
 
 and correspondingly for the current,. 
 
 i = i m sin ( cot — 6 ), 
 
 * McMahon, Hyperbolic Functions, Art. 37. Pender has worked out comput- 
 ing formulas, using the solution with circular trigonometry. See Electrical 
 World, Yol. 56, p. 667. 
 
936 
 
 ALTERNATING CURRENTS 
 
 di . . 
 HI =J m -> 
 
 ( 6 ') 
 
 (7') 
 
 Substituting these values of the derivatives in equations 
 (3) and (4) gives a vector equation which is independent of t 
 and in which the values of e and i may therefore be taken as 
 either maximum or effective vector values, 
 
 and 
 
 = ( - co 2 L 0 + >(P C + L G) + P G) E 
 = ( P G A- j<oC) E ; 
 
 =(«+y»£>(ff+>c)i 
 
 Writing P for (E +ja>Ly(Gr +jcoC) 2 in the last two equa- 
 tions gives 
 
 d 2 E 
 dx 2 
 
 Pi 
 
 dx 2 
 
 and 
 
 = P 2 E 
 = P 2 I. 
 
 The solutions of these equations are 
 
 E = A cosh Px -f- B sinh Px, (8) 
 
 and 1— A! cosh Px + B' sinh Px.* (9) 
 
 To find the constants, put x = 0, when E and I become the 
 generator voltage E 0 and generator current I 0 . Therefore, 
 since sinh Pa* =0 and cosh Px = 1 when x = 0, A = E 0 and 
 A! = I 0 . Also, according to equations (1) and (6'), 
 
 = — (B +j(i>L)I= — ( B +jwL')QA' cosh Px + B' sinh Px), 
 dx 
 
 the negative sign being required here because of the vector 
 relations, (P + ja>L)I being a counter voltage measured in 
 the opposite direction from x; and by differentiation of equa- 
 tion (8), 
 
 — = AP sinh Px + BP cosh Px. 
 dx 
 
 Hence, AP sinh Px = — (P +ja>L)B' sinh Px , 
 
 and BP cosh Px = — (P 4 -jcoL)A' cosh Px. 
 
 * McMahon, Hyperbolic Functions, Art. 14. 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 937 
 
 Therefore 
 
 B _ _ R +jwL j , _ _ VR +,jcoL A , _ 
 
 P ^/G+jcoC 
 
 P A 7 ... : vg+j^c a = _ 
 
 R+jmL WR+jwL 
 
 ■Z t A'=-ZX, 
 
 and B' = — 
 
 A 
 
 Zi 
 
 % 
 
 z t 
 
 Z t being put for the impedance of the line per mile of con- 
 ductor, (R +jo)L')^( Cr +jco C)~'K 
 Consequently, equations (8) and (9) become 
 
 E = E 0 cosh Px — ZJq sinh Px , 
 
 and 
 
 1= P cosh Px — sinh Px. 
 Zi 
 
 If a; is measured from the receiver, when x = 0, A = E t and 
 A’ = in which E t and I x are the voltage and current at the 
 
 dE — 
 
 receiver. Also in this case — = + ( R +jcoL)I , and therefore 
 
 ax 
 
 - - E, 
 
 B= Z t I t and B r = —■ Hence the equations representing the 
 Zi 
 
 voltage and current at any distance x from the receiver, in 
 terms of the voltage and current at the receiver, are like 
 the foregoing except that E t and I x are substituted for E 0 and 
 I 0 and the negative sign before the second term of the right- 
 hand member changes to a positive sign. 
 
 Since 
 
 P = (R + j<*L) %(Gr +jcoC)%= \RGr — co 2 LC +j(i)(RC + LGr )\ *, 
 
 the foregoing equations contain hyperbolic functions of com- 
 plex arguments, which it is desirable to convert into functions 
 of real arguments.* Putting 
 
 \RGr — coPLC+jcoQRC + X£r)|* =a + j/3, 
 
 gives 
 
 E = E 0 cosh (u +j/3)x — ZJ 0 sinh (a + jf3~)x, 
 
 — — jE 
 
 1 = / 0 cosh (a -f- jfi)x — =5 sinh (a +jf, 3)cc. 
 
 Zi 
 
 * This is for the purpose of convenience in computation with the mathemati- 
 cal tables that are ordinarily available, but tables are being developed by 
 Kennedy and others which may make it more practicable to compute directly 
 from the hyperbolic functions with complex arguments. 
 
938 
 
 ALTERNATING CURRENTS 
 
 Since 
 
 sinh (ax + jfix) = sinh ax cos fix +j cosh ax sin fix* 
 
 and 
 
 cosh ( ax + j fix') = cosh ax cos fix -\-j sinh ax sin fix , 
 the foregoing convert to 
 E= E 0 cosh ax cos fix — Z l L 0 sinh ax cos fix 
 
 + j(E 0 sinh ax sin fix — Zjl^ cosh ax sin fix ), (10) 
 
 — — E 
 
 I — E cosh ax cos fix — ~ sinh ax cos fix 
 ° Z t 
 
 - _£> 
 
 +j(I 0 sinh ax sin fix — ■=? cosh ax sin fix'). (11) 
 Zi 
 
 These may be computed by the ordinary tables of functions 
 of circular and hyperbolic trigonometry, such as the Smithsonian 
 Mathematical Tables. A twenty-inch slide rule and a set of 
 tables make computation by this formula relatively easy. 
 
 The values of « and of fi may be computed from the original 
 expression which it has been convenient to represent by a +jfi, 
 that is, 
 
 f BG-< 0 2 LC+j<o(RC+LG-)\ 1 =a+ jfi , 
 from which comes 
 
 R Gr — <A?L C + ja>(^R C+ L Cr) = (« +^/3) 2 , 
 
 R a - gAL C + jco(RC + LG) = a 2 — fi 2 + 2 jafi, 
 and a 2 - fi 2 = RG - co 2 LC, 
 
 2 jafi =jco(RC+ LG). 
 
 Whence, taking positive and real values only, as being appli- 
 cable to the premises here under consideration, 
 
 a = — [ V( U 2 + co 2 L 2 )(G 2 + co 2 C 2 ) + (RG - a> 2 LC)f 
 V2 
 
 and 
 
 fi = - 4 ; r ^(R 2 + o> 2 L 2 )(G 2 + a> 2 C 2 ) - (RG - O > 2 Z<7 )] l - 
 V2 
 
 Of these terms, a is the coefficient a associated with x in the 
 exponential terms of equation (5). It is called the Attenuation 
 constant, since it determines the rate of decrement with distance 
 due to the exponential terms of the wave formula. The term 
 * McMahon, Hyperbolic Functions, Arts. 28-30. 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 939 
 
 /3 is the coefficient b associated with x in the sine terms of 
 equation (5), and is called by Fleming the Wave length constant 
 since the length of the space wave of voltage or current is 
 2 77- 
 
 equal to -g- • The unit in which the wave length is measured 
 
 depends on the unit of length corresponding to R , L , Cr, and 
 C. If these are measured in ohms, henrys, mhos, and farads 
 per mile of conductor, the last two being measured from con- 
 ductor to neutral plane, then the space wave length is 
 
 A. = — miles of conductor. 
 
 /3 
 
 The velocity of propagation of the space wave is then 
 
 v = ^ = /\ miles of conductor per second. 
 
 The quantity P = a +j (3 has been called the Propagation con- 
 stant of a circuit, since it is a complex quantity made up of the 
 attenuation constant and the wave length constant. 
 
 The imaginary roots of the equations on the preceding page 
 lead to solutions representing oscillatory phenomena. 
 
 To obtain the voltage and current at the receiving end of a 
 transmission line when the current and voltage at the sending 
 end are known, it is only necessary to substitute the total length 
 of a line conductor in the place of x in equations (10) and (11), 
 giving, 
 
 E 1 — R 0 cosh al cos (31 — Z t I^ sinli al cos /3Z 
 +j (P 0 sinh al sin /3l — Z t I 0 cosh al sin /3l), 
 
 I x = ^ = I (t cosh al cos /3l — ^ sinh al cos (31 
 
 
 
 +j[ I 0 sinh al sin /3l — n? sinli al cos (31 ), 
 
 ( 12 ) 
 
 (13) 
 
 E v I v and Z x being the voltage, current, and impedance at the 
 receiver. 
 
 Eliminating I 0 from (12) and (13) gives 
 
 
 
 Z ( sinh al cos (31 + Z)cosh al cos f3l + j{Z t cosh al sin j3l -1- Z x sinh al sin (31 ) 
 
940 
 
 ALTERNATING CURRENTS 
 
 and the ratio of / x to J 0 is 
 
 = ^ 
 
 I Z[ cosh cd cos (il + Z x sinh al cos [il + j(Z l sinh al sin (il + Z x cosh al sin /il) ’ 
 
 (15) 
 
 from which the generator voltage and current may be computed 
 if the characteristics of the line and the voltage, current, and 
 impedance of the receiver are given. 
 
 ... — !<} 
 
 It is to be observed that Z t is quite different from Z 0 = P & . 
 
 I, 
 
 The former depends only on the frequency of the current and 
 the electrical constants of the line conductors per unit length. 
 It is independent of the length of the line and of the character 
 of receiver into which the line delivers power. The impedance 
 Z 0 , on the other hand, depends on the length of the line and 
 the character of the receiver in addition to being dependent on 
 the constants of the line per unit length. For a line of infinite 
 length Z t = Z 0 , but for an ordinary line they bear a complex 
 relation to each other. A relation that is sometimes useful to 
 know is the ratio of the generator current when a short circuit 
 occurs at the receiver to the generator current in normal work- 
 ing. The ratio is 
 
 T {) _ tanh (PI + 6) 
 
 T tanh PI 
 
 in which P 0 and I 0 are the generator current respectively with 
 the receiver end of the line short-circuited and in normal work- 
 ing, and 6 = tanh -1 ^ 1 . 
 
 The effect of either a short circuit or an open circuit at anj r 
 point is readily observed from these equations. In the case of 
 a short circuit occurring with a length x of conductor between 
 it and the generator, P becomes zero at that point, and 
 
 E () cosh ax cos fix — Z,Z 0 sinh ax cos fix 
 
 +j(P 0 sinh ax sin fix— Zj^ cosh ax sin fix') = 0. 
 
 Therefore 
 
 h = 
 
 Z^fcosh ax cos fix +j sinh ax sin fix) 
 
 . t 9 
 
 Z/sinh ax cos fix +j cosh ax sin fix) 
 
 and 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 941 
 
 I — ZJ 0 
 
 - a ~x — - — . — 
 
 .Zi cosh az cos /ix + Zjsinhaxcos /to+A.Z'jsinhccrsin fix + ^cosh to; sin fix) 
 
 K 
 
 Zj(sinh ax cos / 3x +j cosh ax sin fix') 
 
 In case of an open circuit at distance x from the generator 
 /becomes zero at that point, and 
 
 _ 
 
 P cosh ax cos fix — sinh ax cos fix 
 Zi 
 
 +j(l 0 sinh ax sin fix— cosh ax sinh fix\ = 0. 
 
 ^ Z l J 
 
 Therefore 
 
 f _ A/ n (sinh ax cos fix +j cosh ax sin fix') 
 
 A)- , _ ; r , . _ ’ 
 
 and 
 
 E r = 
 
 Z;(cosh ax cos fix +j sinh ax sin fix) 
 
 K 
 
 (cosh ax cos fix +j sinh ax sin fix) 
 
 In an infinitely long cable I x = / (cosh Px — sinh Px) = I 0 e~ Px i 
 of which the real part is I(f~ ax and therefore I x = I 0 e~ ax . 
 
 As pointed out in an earlier paragraph, the space wave length 
 
 2 7 r 
 
 measured in conductor miles is when the resistance, in- 
 ductance, insulation conductance, and capacity are given in 
 terms of ohms, henrys, mhos, and farads per mile of conductor. 
 For conductors of the ordinary sizes and spacings used in the 
 long distance transmission of large amounts of power, and cur- 
 rents of the ordinary frequencies used for that purpose, a wave 
 length is many hundreds of miles when measured along the 
 line. For instance, an ungrounded 3-phase overhead line of 
 No. 0 copper wires spaced ten feet apart has resistance of ap- 
 proximately .52 ohm per mile of conductor, self-inductance 
 of approximately .0022 henry per mile of conductor, leakage 
 conductance negligible at ordinary altitudes if the line voltage 
 is below 125,000 volts, and capacity of .0078 xlO -6 farads per 
 mile of conductor. This gives for the value of the wave length 
 constant when/= 25 cycles per second or co = 157 radians per 
 second, fi = .00077 
 
 and for the wave length 
 
 2 7r 
 
 A = = 8150 miles of conductor. 
 
 fi 
 
942 
 
 ALTERNATING CURRENTS 
 
 The wave length in miles of line is one half as great as the 
 length in miles of conductor, since the constants are all given on 
 the assumption of an outgoingconductor and return. No over- 
 head transmission line has ) r et approached this length, or even 
 the length of 1925 miles, which are the miles of line required 
 to give a complete wave length if the frequency were 60 
 cycles per second on the foregoing line. 
 
 Consequently, it is possible to neglect the formulas which 
 recognize the effects of distributed resistance, self-inductance, 
 leakage, and capacity when computing long distance power 
 transmission lines of lengths within the limits of existing prac- 
 tice, and to make the computations on the assumption that the 
 fall of potential is uniform along the conductors. Leakage 
 being usually negligible, the effects of the electrostatic capacity 
 may be approximated by assuming the capacity of the conduc- 
 tors concentrated at certain points instead of distributed. For 
 short lines it is sufficient to consider the capacity all concen- 
 trated at the middle of the line ; but for lines having lengths 
 approaching the limits of the existing longest lines, it is some- 
 times essential to make a closer approximation to truth, and the 
 capacity may be assumed to be concentrated in three divisions, 
 one sixth of the whole capacity at each end of the line and the 
 remaining two thirds at the middle of the line. 
 
 The transmission of large amounts of power over under- 
 ground cables has not yet reached lengths of circuit approach- 
 ing the lengths in individual overhead lines, and the voltage 
 impressed on cables is usually less than 20,000 volts. The self- 
 inductance of the conductors in the usual underground circuit 
 is much less than the self-inductance of a corresponding over- 
 head circuit, on account of the closer spacing of the conductors ; 
 but the electrostatic capacity of the cabled conductors is much 
 greater than when the circuit is overhead, on account of the 
 closer spacing of the conductors and the relatively high 
 dielectric constant of the solid insulating material. An un- 
 grounded 3-phase cable of three symmetrically spaced No. 0 
 wires with rubber insulation having ^ inch thickness of wall 
 around each conductor has a resistance of approximately .52 
 ohm per mile of conductor, a self-inductance of approximately 
 .00065 henry per mile of conductor, leakage negligible, and 
 a capacity of approximately .0545 x 10 ~ 6 farads per mile of 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 943 
 
 conductor. The wave length constant when the frequency is 
 twenty-five cycles per second is therefore /3 = .001645 and the 
 wave length in miles of conductor is 
 
 This corresponds with 1910 miles in length of cable. This far 
 exceeds the lengths of circuit ordinarily found in underground 
 cables for heavy electric power transmission, hut in the case of 
 the longer underground cables used with voltages higher than 
 a few thousand volts, it is desirable to utilize the exact formu- 
 las for determining the effects of capacity. 
 
 In the case of long distance telephone service the situation 
 is different on account of the higher frequency of the current 
 harmonics which it is desirable to retain without great attenua- 
 tion, and also because it is desirable to adjust the relation of 
 resistance, self-inductance, and capacity so that the attenuation 
 and the velocity of propagation shall be substantially the same 
 for current harmonics of a considerable range of frequencies 
 (such as from 200 periods per second to 1800 periods per 
 second), in order that the intelligibility of the irregular 
 speech waves may be maintained. This requires what is 
 called a Distortionless circuit and the exact formulas are needed 
 in the computation of such circuits. 
 
 When leakage is negligible, the values of a and ft heretofore 
 given reduce to 
 
 If the inductive reactance a>L is at the same time negligibly 
 small compared with the resistance, these reduce to 
 
 approximately, since the square root of a -f b is very nearly 
 
 A = — = 3820. 
 
 /3 
 
944 
 
 ALTERNATING CURRENTS 
 
 equal to Va -| — when a is large compared with b , and when 
 
 2V a 
 
 leakage is negligible under these circumstances, 
 
 Also, leakage being negligible, if 2 wL is very large compared 
 with R 2 , 
 
 0 = coVCL, 
 
 1 
 
 and 
 
 « _ 
 
 0 V LC 
 
 Whence, by making wL for the various harmonics of a periodic 
 current very large compared with R , the attenuation and the 
 velocity of wave propagation of the harmonics are both made 
 independent of the frequency of the harmonics and dependent 
 solely on R , X, and C. 
 
 The latter condition may also be accomplished v r hen leakage 
 is not negligible, provided LG= RC , except that attenuation 
 and velocity of propagation are dependent upon R , L , 6r, and 
 C instead of R , X, and (7, only. In this case, the original ex- 
 pressions for a and 0 reduce to 
 
 «= -L [V(ze<7 + co 2 Lcy + a>\La - Rcy+<iRa-«) 2 Lcy]\ 
 V2 
 
 = y/RG, 
 
 /3 = - 1 [ V(i26^ + <o 2 LC) 2 + <d\LG- RC) 2 - (RG-a> 2 LCrf 
 
 V2 
 
 = coV LC. 
 
 The importance of this sort of treatment in the case of a 
 long distance telephone line may be recognized from the facts 
 that an overhead line of No. 8 B. & S. gauge wire spaced twelve 
 inches apart has resistance of 3.3 ohms per mile of conductor, 
 self-inductance of 2 x 10~ 3 henrys per mile of conductor, capac- 
 ity of .004 x 10 -6 farads per mile of conductor, and the insu- 
 lation resistance ought to be in the order of some megohms per 
 mile of conductor; and that it is desirable to maintain the 
 attenuation and velocity of propagation substantially the same 
 for harmonics of the voice currents within the range of 200 to 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 945 
 
 1800 cycles per second. The foregoing constants give wave 
 lengths which are but fractional compared with the length of 
 many long distance telephone circuits. 
 
 In the case of telephone cables, the relations are even more 
 impressive. With the usual lead-covered cables with loose dry 
 paper wrapping for the conductors, the constants for No. 19 
 B. & S. gauge conductors are per mile of conductor approxi- 
 mately 45 ohms, 5 x 10 -3 henrys, .035 x 10 -6 farads, and insu- 
 lation resistance of one half a megohm or more. 
 
 This gives a wave length of less than sixty miles of cable for 
 the harmonic corresponding to an angular velocity, &>, of 5000 
 radians per second (/= 800 periods per second, approximately), 
 which is about a mean value for the harmonics of voice currents 
 that are necessary to retain an approximately unaltered relative 
 magnitude during transmission in order that the transmitted 
 speech may be fully articulated by the receiver. For the har- 
 monic of frequency 1800 periods per second, the wave length 
 on this cable is less than thirty-five miles. Since telephony 
 through underground cables is now practiced over many tens of 
 miles, the transmission computations must obviously be executed 
 with the formulas which include all the features of the wave 
 transmission.* 
 
 The foregoing formulas relate to uniform circuits. They 
 may be applied to each uniform part of a non-uniform circuit, 
 provided the voltage impressed on the part concerned and the 
 vector impedance of the succeeding part are known; but changes 
 of character, especially if they are abrupt, cause reflection points 
 which greatly alter the distribution of voltage compared with 
 the distribution of voltage over a uniform section of the same 
 length. 
 
 217. Analogies of the Electric Circuit. — The deductions of 
 Chapter VI have shown very clearly that concentrated self-in- 
 ductance and concentrated capacity in a circuit may be made 
 to neutralize each other when a sinusoidal voltage is applied to 
 the circuit, and the self-inductance and capacity are constant. 
 In this case the self-inductance and capacity act in opposition, 
 so that at each instant energy is being restored or released in 
 
 * The simpler problems .of long distance telephone transmission are dealt 
 with admirably by Fleming in a book entitled The Propagation of Electric 
 Currents in Telephone and Telegraph Conductors. 
 
 3 p 
 
94 G 
 
 ALTERNATING CURRENTS 
 
 the magnetic field at exactly the same rate as energy is being 
 released or stored in the charge of the condenser. The self- 
 inductance and capacity may therefore be said to supply each 
 other’s demands, and the power delivered by a generator to the 
 circuit may be wholly utilized in doing work on a non-reactive 
 receiver such as incandescent lamps and in heating the wires of 
 the circuit. The vector power which is transferred back and 
 forth between the self-inductance and capacity may be many 
 times as great as that given to the circuit by the generator, 
 and volt-amperes at the terminals of the self-inductance and of 
 the condenser must then be proportionally greater than the 
 volt-amperes delivered by the generator. 
 
 This condition can fully exist only when the frequency of 
 
 the impressed voltage produces the relation 2 irfL = ^ or 
 
 2 7rfC 
 
 LO = and the natural period of the circuit is 2 tt^/LC . j 
 
 From the condition 2 rrfL— — - — it is seen that - = 2 7rVZ6 r . 
 
 2 irfC f 
 
 The natural period of discharge of the circuit is therefore equal 
 to the period of the cycles of impressed voltage, or, as may be 
 said, equal to the rate of the electrical vibrations impressed on 
 the circuit by the generator. This relation between the vibra- 
 tions of the line and of the generator is similar to that of a 
 vibrating tuning fork or string and a sounding board when the}* 
 are in resonance, and therefore the term Electrical resonance 
 lias, on account of the analogy, been applied to the electric 
 circuit. An electric circuit is said to be in resonance with an im- 
 pressed voltage when the natural period of the circuit is equal to 
 the period of the impressed voltage. When this condition exists, 
 the maximum current is caused to flow in the circuit by the 
 application of a given impressed voltage, the value of the cur- 
 rent in a resonant circuit from which no external work is sup- 
 plied being 
 
 If the self-inductance, capacity, and resistance are in series in 
 the circuit, it is evident that when the periods of the circuit 
 
 1 = 
 
 U 
 
 * Art. 69 (4) . 
 
 t Art. 58, Case (2). 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 947 
 
 and the impressed frequency are in resonance, the voltage be- 
 tween the terminals of the capacity ^ = ^ is a maximum, 
 
 since the circuit current is a maximum ; and the same is true of 
 the voltage between the terminals of the inductance. If either 
 the frequency, the self-inductance, or the capacity is changed in 
 value, the value of the current falls, and the condenser voltage 
 falls, unless the other elements are changed in value in such a 
 way as to continue the condition of resonance. A condenser in 
 a resonant circuit may be used as a transformer of voltage by 
 connecting non-reactive apparatus across its terminals, as has 
 been suggested by Blakesley, Loppe et Bouquet, Pupin, and 
 others. 
 
 If the self-inductance and capacity ai’e in parallel in the cir- 
 cuit, the voltage at their terminals cannot be greater than that 
 impressed upon the circuit minus the loss of voltage in the lead 
 wires; but when the circuit is resonant, the current furnished 
 to the circuit by the generator is at a minimum which is equal 
 to the power consumed by the circuit divided by T the impressed 
 voltage, while the current transferred between the inductance 
 and capacity is a maximum which may be a great many times 
 as great as the value of the generator current. 
 
 Resonant circuits in the hands of experimenters such as 
 Hertz, Lodge, and others have produced remarkable results, 
 which have led to great advances in our knowledge of elec- 
 tricity, while mathematical analysis of such circuits has led to 
 further discoveries. These results have caused some to expect 
 remarkable effects to be gained from the use of resonant cir- 
 cuits (or Tuned circuits, as they are sometimes called) for the 
 purposes of the electrical transmission of power. 
 
 Circuits which are installed for the transmission of power 
 over considerable distances (whether the wires are overhead or 
 underground) always contain capacity and self-inductance dis- 
 tributed along their lengths. It would be possible in such lines 
 to adjust the capacity and self-inductance so as to give reso- 
 nance, and the results to be gained from so doing may be ex- 
 amined through analogy. 
 
 A mechanical analogue of an electric circuit is shown in Fig. 
 523. This consists of a tube fitted with two plungers and 
 filled with a perfectly elastic fluid. The properties of this fluid 
 
948 
 
 ALTERNATING CURRENTS 
 
 may be used to represent electrical quantities according to the 
 analogies ; fluid velocity — electric current ; fluid pressure — 
 voltage ; inertia — self-inductance ; compressibility * — elec- 
 trostatic capacity ; frictional resistance — electrical resistance. 
 Now suppose the fluid to be without inertia and perfectly incom- 
 pressible ; then if plunger A is moved toward _Z>, a uniform cur- 
 rent is instantly set up in the whole length of the tube, the ve- 
 locity of which is equal (in proper units) to the pressure applied 
 to the plunger divided by the frictional resistance. If plunger 
 A is caused to move up and down harmonically, the plunger A' 
 
 D a 
 
 Fig. 523. — Illustration of a Mechanical Analogue of an Electric Circuit. 
 
 at the other end of the line will have an exactly equal syn. 
 clironous harmonic motion. This is analogous to the state of 
 an electric circuit without inductance or capacity. Figure 524 
 shows diagrannnatically the state of the circuit, the distance of 
 the broken line from the heavy line being equal to the current 
 at each point. The light full line shows the gradual fall of 
 pressure between A and A', caused by the resistance, and the 
 sudden fall of pressure at A f , caused by the external work done 
 by plunger A'. 
 
 If the fluid is compressible but has no inertia, it is evident that 
 the motion of the plunger at A! will be less than that at A, which 
 is analogous to the decadence of current as it flows along a circuit 
 having capacity, due to the quantity of electricity entering into 
 the electrostatic charge on the conducting wires. The move- 
 ments of the plungers are isochronous but not in synchronism. 
 
 * Compressibility of a fluid is the ratio of compression (change of volume) to 
 the pressure producing it, and electrical capacity is the ratio of the charge 
 (change of quantity) to the voltage producing it. 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 949 
 
 In this case the motion of the plunger A will exert its maximum 
 pressure when the fluid is most compressed, or at the end of its 
 stroke where its velocity is least. Hence the velocity of the 
 fluid at the genera- 
 tor (i.e. the genera- 
 tor current), which 
 is greatest at the 
 middle of the 
 stroke, leads the 
 
 pressure by 90° of Fig. 524. — Illustration showing Conditions of Currents 
 phase The mao'- an< * Pressures when the Pistons of Fig. 523 move in an 
 r " ” Incompressible Fluid which is without Inertia. 
 
 nitudes of the cur- 
 rent and voltage throughout the circuit are illustrated in Fig. 
 525, after the manner explained in connection with Fig. 524. 
 
 If the fluid has inertia but is incompressible, the velocities at 
 A and A' are equal ; that is, the current throughout the circuit 
 is uniform, but the pressure exerted upon piston A is greatest 
 when the acceleration is greatest, which is at the beginning of 
 the stroke where the velocity is least. Consequently the cur- 
 rent lags behind the pressure by 90°. This is analogous to the 
 electric circuit with self-inductance but no capacity. 
 
 If the fluid has both inertia and compressibility, the column 
 of fluid in the tube then takes upon itself the usual properties 
 
 of material elastic 
 bodies, and possesses 
 a natural rate of vi- 
 bration of its own 
 which depends upon 
 the dimensions of 
 the column and the 
 
 Fig. 525. — Illustration showing the Conditions of Cur- inertia atld COmpreS- 
 rents and Pressures when the Pistons of Fig. 523 move m-tj. r j.i a - i 
 in a Compressible Fluid having No Inertia. Slblllt y ° f the flulcb 
 
 The period of this 
 
 vibration, as is proved in elementary mechanics, is proportional 
 to the square root of the density divided by the elasticity, or to 
 the square root of the product of the inertia and compressibility. 
 Hence T = a V MK where a is a constant, M mass, K compressi- 
 bility, and T time of vibration. 
 
 In this case, if the plunger A (Fig. 523) is moved with a 
 sinusoidal velocity having a period of T seconds, which is the 
 
950 
 
 ALTERNATING CURRENTS 
 
 same as the natural period of vibration of the column of fluid, 
 the fluid will be thrown into vibrations which require one com- 
 plete traversal of the circuit to make a wave length. Hence, 
 if there is no power taken from the circuit, there are nodes or 
 points of no motion at a and a', and antinodes or points of 
 maximum motion at the plungers. Since the directions of mo- 
 tion in the two halves of a wave are opposite, the two plungers 
 move in opposite directions in the tube. As the velocity of the 
 fluid varies from node to antinode as a sinusoidal function, 
 the loss of power by friction is reduced to one half the value 
 which it has for an equal plunger velocity in the inertialess, 
 incompressible fluid. The velocity of propagation of the dis- 
 turbance through the fluid from plunger A to A' is equal to 
 
 - centimeters per second, where l is the length of column. 
 
 affMK 
 
 Since the velocity of movement of the fluid falls off from the 
 plungers toward the nodes, the pressure upon the fluid exerted 
 by the plungers must be proportionately multiplied at the 
 nodes, in order that the same power may be transmitted through 
 the fluid across the nodes as is applied at the prime plunger A. 
 The condition of pressure and velocity is diagrammatically rep- 
 resented in Fig. 526. If power is transferred to an outside 
 
 under the conditions here cited, require that the power shall he 
 transferred from one 'plunger to the other tvholly through the ab- 
 sorption and redelivery of the power by the fluid by means of the 
 effects of inertia and elasticity. The fluid must therefore have 
 a sufficient mass so that even at the slow velocity at the nodal 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 951 
 
 points its kinetic energy shall be sufficient to carry the energy 
 in the circuit across those points. 
 
 This analogue represents the conditions in the resonant elec- 
 tric circuit with distributed self-inductance and capacity. 
 Carrying in mind the analogue and the diagrammatic represen- 
 tation of current and pressure in Fig. 526, it is easy to draw 
 definite conclusions in regard to the effect of resonance on the 
 operation of electric circuits for the transmission of power. 
 
 The advantages of a resonant circuit for electrical transmis- 
 sion are then : (1) a gain of upwards of one half of the I 2 R loss 
 that would be caused by the transmission of an equal amount 
 of power at an equal receiving voltage over the same circuit 
 when out of resonance ; (2) more satisfactory regulation than 
 would be found in a non-resonant but reactive line, since the 
 difference in voltage between generator and receiver is equal 
 to current times resistance instead of current times an imped- 
 ance which is greater than the resistance. 
 
 The principal disadvantage of a resonant circuit for electrical 
 transmission is : a very large excess of voltage on the line at 
 certain points, or nodes of current, which excess decreases to- 
 ward the antinodes. If satisfactory resonance is to be gained by 
 adjusting the self-inductance and capacity of the circuit so that 
 the voltage at the nodes of current is no greater than ten times 
 that at the antinodes, the average voltage along the line must be 
 
 caused to be seven times that of the antinodes, using a 
 
 sinusoidal function. In other words, if the voltage which is 
 safe for use is limited by the insulation, we may say that the 
 average strength of insulation on the line must be seven times 
 as great as would be necessary at the generators. This enor- 
 mous increase of insulation must be made to save fifty per cent 
 of the I 2 R loss caused by the transmission of a certain amount 
 of power over a given line. A much more reasonable plan for 
 heavy power transmission would be to reduce the self-inductance 
 and capacity of the line to a minimum, avoiding resonance and 
 raising the generator voltage to 1.4 its previous value. Now 
 the same power could be transferred over the line with the 
 same resistance as before, the I 2 R loss being the same as when 
 the line was resonant, but the average strain on the insulation 
 would be only one fifth as great as in the resonant line. 
 
952 
 
 ALTERNATING CURRENTS 
 
 The highest voltage which can be economically used on cir- 
 cuits for the electrical transmission of large amounts of power 
 over long distances is generally conceded to be set at the limit 
 which may be properly insulated. If this is true, the preced- 
 ing paragraph shows that, with equal insulation, the generator 
 
 voltage may be safely made — = times greater on a non-resonant, 
 
 V2 
 
 long distance transmission line than that which is safe on a 
 resonant line, where X is the ratio of the maximum voltage to 
 the generator voltage on the resonant line. This shows that 
 the non-resonant line would be by far the most economical for 
 long distance heavy power transmission, even if it were com- 
 mercially possible to maintain resonance on service circuits. 
 For the distribution of power over short distances, the voltage 
 is usually quite low, and the insulation limit is not approached, 
 so that resonance might be introduced without adding to the 
 insulation ; but the reactions of transformers and motors on 
 the line make it practically impossible to keep the line in reso- 
 nance. Similar defects are seen in the propositions for using 
 resonant lines for various other classes of electrical transmission 
 except telephony. 
 
 These deductions in regard to resonance have been made 
 upon the assumption of sinusoidal currents. In practice these 
 are now seldom exactly realized, since iron-cored transformers 
 and motors, and tooth-cored alternators, introduce distortions, 
 and a circuit which is resonant for the fundamental wave is not 
 resonant for its harmonics. As the question of resonance now 
 rests, it does not enter into problems relating to ordinary elec- 
 tric power circuits in such a way as to modify practice except 
 as it in certain instances causes undue stresses on the insulation 
 through its accidental occurrence in connection with transient 
 effects caused by switching, arcing between wires or from wires 
 to ground, effects of lightning induction, and the like. 
 
 In telephony (which consists of the electric transmission of 
 power in very minute quantities under certain special condi- 
 tions), the voltages are relatively low and satisfactory insulation 
 for resonant transmission may be readily accomplished. It is, 
 therefore, good practice to load long distance telephone lines 
 with self-inductance by inserting coils at frequent intervals, for 
 the purpose of making the lines more nearly distortionless. 
 
INDUCTANCE AND ELECTROSTATIC CAPACITY 
 
 953 
 
 218. Corona. — One of the earliest observed manifestations 
 of electricity was the brush discharge between the electrodes of 
 an electrostatic machine or near a highly charged pointed body. 
 It was early recognized that this is accompanied by convection 
 effects whereby a current of electrically charged air is caused 
 to flow away from the body concerned, with a tendency to dis- 
 charge the body. This is illustrated by holding a lighted candle 
 near a point protruding from the electrode of an electrostatic 
 machine in operation, when it may be observed that the candle 
 flame is blown aside by the air current caused by the particles 
 of air receding from the electrode on account of repulsion after 
 they have become charged by contact. The phenomenon is ac- 
 companied by a silent transfer of electricity from one electrode 
 of the machine to the other by means of the air particles. 
 
 While the usual electric circuits of commerce were only of 
 relatively low voltage, it was thought that the phenomena of the 
 brush discharge and electrostatic convection might pertain only 
 to the extraordinary voltages of the so-called electrostatic 
 machines; but the development of high voltage power transmis- 
 sion has proved that those phenomena are associated with vol- 
 tages of only a few tens of thousands of volts and that they 
 occur in striking degree on aerial power lines when the voltage 
 exceeds a hundred thousand or more volts. Such lines, when 
 the voltage exceeds 125,000 volts, become luminous from the 
 discharge, and seem, when viewed in the dark, to be covered 
 with a brush discharge having the appearance of the correspond- 
 ing discharge between the electrodes of an electrostatic machine. 
 
 The discharge is established at a voltage which is inversely 
 related to the radius of curvature of the conductors on which 
 it is seated, and, in the case of aerial lines for the transmission 
 of power by alternating currents, it results in a serious loss of 
 power when the voltage much exceeds 125,000 volts and the 
 wires are of the usual transmission sizes such as from No. 6 to 
 No. 0000 B. and S. gauge. The loss is greater when the baro- 
 metric pressure of the atmosphere is low, and it is therefore 
 more strikingly observed in connection with high voltage lines 
 in mountainous regions. The phenomena thus denoted are 
 called Corona effects, on account of the luminous corona which 
 may be observed in a dark room around conductors on which 
 the phenomena are active. The phenomena are accompanied 
 
954 
 
 ALTERNATING CURRENTS 
 
 by the crackling or hissing sounds and the production of nitric 
 oxide and ozone (which may be observed by the odor) that are 
 recognized accompaniments of brush discharges at the terminals 
 of an electrostatic machine. 
 
 The amount of power lost by convection between power trans- 
 mission conductors of the usual diameters increases slightly 
 with the voltage up to a certain critical point, at which the 
 corona seems to be fuily established, and the ratio of power 
 lost to line voltage is much higher for voltages above the criti- 
 cal point. The critical voltage for ordinary transmission lines 
 at the usual central altitudes of this country seems to be about 
 125,000 volts, and the loss would be prohibitive on long lines 
 at higher voltages when conductors not exceeding one half inch 
 in effective diameter are used. The loss also increases with the 
 frequency of the cycles of the voltage. 
 
 These phenomena have been given much study, but no reli- 
 able laws have yet been formulated. The consensus of knowl- 
 edge may be obtained fairly well from various papers in the 
 Transactions of the American Institute of Electrical Engineers 
 for the years 1910 to 1912.* 
 
 It has not yet been conclusively determined whether or not 
 the effects of corona may occur in liquids or solids, but the 
 best opinion leans to the hypothesis that corona is a result of 
 ionization in a gas and requires a gaseous medium for its devel- 
 opment. There are, however, certain recognized effects of ex- 
 cessive electrostatic stresses in solid dielectrics which indicate 
 that some phenomena occur in solids which are at least similar 
 to corona effects. 
 
 * Steinmetz, Ryan, Merslion, Whitehead, Peek, and others. 
 
INDEX 
 
 A 
 
 Active current, 313. 
 
 Active voltage locus of transformer, 
 486. 
 
 Adams, operation of alternators in 
 parallel, 669. 
 
 Admittance, the reciprocal of im- 
 pedance, defined, 212 ; expressed as 
 a complex quantity, 214. 
 
 Admittances, combination of, 253-256. 
 
 Ageing, effect of, on iron in transformers, 
 509-510. 
 
 Air blast, cooling transformers by, 543- 
 544. 
 
 Air-cooled transformers, 539. 
 
 Air friction, losses in alternators due 
 to, 597. 
 
 All-day efficiency of transformers, 516. 
 
 .Alliance dynamo, 90. 
 
 Alloying materials used in transformers, 
 515. 
 
 Alternating circuit, current in, 316- 
 317 ; methods for measuring power 
 in, 344-358. 
 
 Alternating current, controlled by same 
 laws as direct current, 1-2 ; relation 
 between alternating voltage and, 
 2-3 ; period and frequency of, 3 ; 
 heating effect or power activity of 
 (Joule’s law), 3 ; determination of 
 effective value of, 4-6 ; frequencies 
 of, 20-22 ; instruments for measur- 
 ing, 43-45, 65-75. 
 
 Alternating-current curve, resolution 
 of, into its harmonic components, 
 35-42 ; determining the effective 
 ordinate of an, 42-43. 
 
 Alternating-current generator, 19. 
 
 Alternating-current motor-generators, 
 782-783. 
 
 Alternating - current motors, series, 
 874-881 ; commutation and other 
 characteristics of commutating, 887- 
 891 ; vector diagrams of series 
 motors, and expressions for voltage, 
 881-887. 
 
 Alternating flux, caused by current of 
 predetermined form, 438-440. 
 
 Alternating voltage, 2 ; instruments 
 for measuring, 43-45. 
 
 Alternators, 19 ; form of voltage 
 curve of, 22-27 ; measurement of 
 effective voltage developed by, 45 ; 
 comparison of voltages developed by. 
 
 and by direct-current dynamo, 46 ; 
 single-phase and polyphase, 55-60 ; 
 armature windings for, 77-119 ; field 
 excitation of, 106-108 ; composite 
 excitation of, 108-114; inductor, 
 117-119; calculation of effective 
 value of voltage of armature, 218- 
 220 ; losses in, 597-600 ; relation of 
 speed to weight in, 602-604 ; self- 
 inductance of, 609-611; character- 
 istics of. 621 ; regulation of, for 
 constant voltage, 654-661 ; con- 
 necting, for combined output, 664 ; 
 in series, 664-668 ; in parallel, 668- 
 669 ; synchronizing current of, 669- 
 676 ; division of load between 
 parallel, 676-686 ; maximum possible 
 load and regulation of prime movers 
 in parallel operation, 686-688 ; effect 
 of form of voltage curve and varia- 
 tion of angular velocity on parallel 
 operation and division of loads, 
 689-690 ; methods of connecting, 
 in parallel to feeder circuits, 699- 
 702 ; as synchronous motors, 702- 
 704 ; methods of testing, 729 ff. ; 
 efficiency by rated motor and stray 
 power measurements, 729-732 ; feed- 
 ing-back method for measuring effi- 
 ciency, 732-733 ; efficiency by rated 
 motor, 733-734 ; Mordey’s method 
 of testing, 734-736 ; divided field 
 method, 736-737 ; motor-generator 
 method, 737-740 ; heating tests, 
 740-741 ; insulation tests, 741 ; regu- 
 lation, 741-742 ; wave form of 
 voltage produced by, 742. 
 
 Amortisseurs for preventing hunting, 
 747. 
 
 Amperemeter (three) method of measur- 
 ing power, 356. 
 
 Amperemeters, alternating current, 43- 
 45, 65—75 ; arrangement of, for 
 
 measuring high voltages or large 
 currents, 76 ; alternating-current, for 
 testing alternators, 730. 
 
 Ampere turns, exciting, 446 ; of ar- 
 matures, 609. 
 
 Amplitude of scalar value of vectors, 11. 
 
 Analogies of the electric circuit, 945- 
 952. 
 
 Analytical method for solution of 
 problems, 261. 
 
 Angle of lag, 132-133 ; method of 
 measuring, 343. 
 
956 
 
 INDEX 
 
 Apparent power, 325. 
 
 Argument of vector quantity, 242. 
 
 Armatures, multipolar, 45 ; comparison 
 of voltages developed in alternator 
 and direct-current, 46 ; effect of 
 arrangements of windings on voltages 
 of, 47-54 ; classification of, 77 ; in- 
 sulating, 120-123 ; core materials, 
 123-124 ; ventilation of, 124-126 ; 
 measurement of self-inductance of, 
 145 ; curve showing relation of 
 current to excitation, 724-727 ; heat- 
 ing of conductors in rotary converters, 
 765-772; reactions of converter, 
 772-773 ; action of a short-circuited 
 winding within rotating field, 790- 
 792 ; of induction motors, 791 ; 
 coil-wound, connected to external 
 devices, in construction of rotating 
 field induction motors, 825 ; re- 
 sistance in circuits for regulation of 
 speed of induction motors, 835-837 ; 
 commutated, 837-838. 
 
 Armature conductors, determination 
 of number of, 605-609. 
 
 Armature cores and conductors, losses 
 in alternators due to eddy currents 
 and hysteresis in, 597. 
 
 Armature reactions in alternators, 611- 
 621. 
 
 Armature windings for alternators, 
 77-119. 
 
 Arnold, E., 80. 
 
 Asynchronous generator, 784 ; rotary 
 field induction motor as an, 822-824. 
 
 Asynchronous motors, 784 ff. 
 
 Attenuation constant, 938. 
 
 Automatic devices for regulation of 
 alternators, 654-661, 699-702. 
 
 Automatic switches, 764-765. 
 
 Automatic synchronizers, 699. 
 
 Autotransformers, 576-581 ; use of, 
 as voltage regulators, 663-664 ; for 
 regulating induction motors, 833-835. 
 
 Ayrton, measurement of self-induct- 
 ance, 145-146 ; measuring power 
 in alternating-current circuit, 356 ; 
 testing transformers, 650 ; testing 
 alternators, 736. 
 
 Ayrton and Sumpner, three-voltmeter 
 method of measuring power in al- 
 ternating-current circuit, 354 ; 
 method of obtaining transformer 
 efficiency, 589. 
 
 B 
 
 Balanced polyphase system, 359 ; 
 uniformity of power in, 367-368 ; 
 measurement of power in, 389 ff. 
 
 Ballistic galvanometer, effect on resist- 
 ances of shunting a, 163—165 ; use 
 of, in tracing curves, 649. 
 
 Barrel winding, 86, 102. 
 
 Bar windings, 78. 
 
 Bedell, testing alternators, 652-654. 
 
 Bedell and Crehore, Alternating Cur- 
 rents , cited, 301. 
 
 Behrend, relation of weight to speed in 
 alternators, 602. 
 
 Blakesley, T. H., application of graphi- 
 cal processes by, 257 ; measurement 
 of power, 316 ; three-instrument 
 method of measuring suggested by, 
 356. 
 
 Blondel, measurement of power, 395; 
 tracing curves, 650. 
 
 Blondel and Duddell, oscillograph of, 
 644. 
 
 Booster, 580. 
 
 Brown, C. E. L., parallel operation of 
 alternators, 689. 
 
 Brush alternator, 91. 
 
 Brushes of collectors, 103, 105. 
 
 Byerly, Fourier's Series and Spherical 
 Harmonics, cited, 27. 
 
 C 
 
 Calculation of magnetic leakage, 581- 
 583. 
 
 Capacity, electrostatic, 168 ; unit of, 
 168 ; specific inductive, 169 ; con- 
 ditions of establishment and termina- 
 tion of current in a circuit containing, 
 170-175 ; effect of, in a circuit, 185- 
 186 ; conditions of establishment 
 and termination of current in circuit 
 containing resistance, inductance, 
 and, in series, 187-199 ; vector 
 relations of current and voltage in 
 circuit containing resistance, self- 
 inductance, and, in series, 209-211. 
 
 Capacity circuit, definition, 279 n. 
 
 Capacity reactance, 212. 
 
 Capacity voltage, 177, 267. 
 
 Characteristics, alternator, 621. 
 
 Charge of a condenser, 169. 
 
 Choking coils, 575. 
 
 Chord-wound alternators, 78. 
 
 Chord-wound armature, methods of 
 applying the wires to, 93-102. 
 
 Circle, power, 685. 
 
 Circle diagram, for non-reactive sec- 
 ondary circuit, 4S6 ; where trans- 
 former load contains constant react- 
 ance and variable resistance, 490- 
 498 ; of magnetic fluxes in polyphase 
 induction motor, 816-822. 
 
 Circuit, effects of changing resistance 
 of a, 166-167 ; effect of introducing 
 resistance in a continuous current, 
 180-183 ; effects of self-inductance 
 and capacity in, 183-186 ; effect 
 on transient state in a, of self-induct- 
 
INDEX 
 
 957 
 
 ance and capacity combined, 186- 
 187 ; time constant of a, 206-208. 
 See Self-inductive circuit. 
 
 Circuits, solution of, by graphical 
 methods, 257 ff. ; classification of, 
 for solution, 261-262 ; series, 262-277 ; 
 parallel, 277-301 ; signification of 
 terms inductive, capacity, reactive, 
 and non-reactive circuits, 279 n. ; 
 series and parallel combined, 301- 
 309 ; mutual induction of parallel 
 distributing, 911-916; effects of self 
 and mutual induction in polyphase, 
 916-922 ; distortionless, 943 ; tuned, 
 947. 
 
 Circular loci for constant current 
 transformer, 570. 
 
 Classification of circuits, for solution by 
 graphical methods, 261-262. 
 
 Coefficient, of self-induction, 135 ; of 
 mutual induction, 450. 
 
 Coils, current, 345 ; voltage, 345 ; 
 reactance of, causing errors in watt- 
 meter readings, 346-349 ; impedance, 
 467, 575; reactance, 467, 575; 
 
 choking, 575 ; shading, 853 ; com- 
 pensating, 876. 
 
 Coil windings, 78, 85. 
 
 Coil-wound armatures connected to 
 external devices, in construction of 
 rotating field induction motors, 825. 
 
 Coincidence of voltage and current 
 phases, 325. 
 
 Collector rings, 19, 47, 102-106. 
 
 Combination of admittances, 253-256. 
 
 Commutated armature, 837-838- 
 
 Commutated primary windings, 838. 
 
 Commutation of commutating alternat- 
 ing-current motors, 887. 
 
 Commutator, rectifying, 114-117. 
 
 Compensated alternator, 113. 
 
 Compensating coils, 876. 
 
 Compensators, 576-581 ; variable, for 
 regulating induction motors, 833- 
 835. 
 
 Complementary vectors, 246. 
 
 Complex expressions, addition and 
 subtraction of, 244 ; multiplication 
 and division of, 244-246. 
 
 Complex quantities, 17 ; involution 
 and evolution of, 249-251 ; differen- 
 tiation and integration of, 251-252 ; 
 logarithms of, 252-253. 
 
 Complex quantity, vector analysis and 
 the, 241-244. 
 
 Composite excitation of alternators, 
 107-114. 
 
 Compound-wound machines, 108, 110. 
 
 Concatenation control, induction mo- 
 tors, 838-840. 
 
 Condenser, the term, 169 ; dielectric 
 of a, 169 ; the charge of a, 169 ; I 
 
 curves of charge and discharge of a, 
 171-173; energy of a charged, 175- 
 176 ; synchronous, 748-754. 
 
 Condenser circuit, definition, 279 n. 
 
 Condensers, total impressed voltage of 
 circuit when connected in series, 276 ; 
 current when connected in parallel, 
 297-298. 
 
 Condenser voltage, 177, 267. 
 
 Conductance, of a circuit, 214 ; of an 
 admittance, 253. 
 
 Conductively compensated series motor, 
 877. 
 
 Conductor losses in transformers, 64. 
 
 Conductors, density of current in, of 
 alternators, 602-604 ; electrostatic 
 capacity of parallel, 922-931. 
 
 Connecting constant voltage trans- 
 formers, 546-561. 
 
 Connecting up armature windings, 
 58-60. 
 
 Constant current transformers, 468, 567. 
 
 Constant voltage transformers, 468 ; 
 constructive features of, 531-546. 
 
 Constants, determination of value of, 
 in Fourier’s Series, 33-35; hys- 
 teresis, 408-409. 
 
 Construction of rotating field induction 
 motors, 824-832. 
 
 Contact makers, 593 ; methods of 
 using, in determining form of voltage 
 or current curves, 648-652. 
 
 Converters, rotary, 83, 754-758 ; ratio 
 of transformation in, 758-763 ; fre- 
 quency and voltage limitations in, 
 763-765 ; heating of armature con- 
 ductors in rotary, 765-772 ; arma- 
 ture reactions of, 772-773 ; voltage 
 control of, 773 ; split-pole, 773 ; 
 synchronous regulators, 774 ; features 
 of operation, 775-781 ; inverted, 
 781-782 ; mercury vapor, 895-896. 
 
 Cooling, of transformers, 538-544 ; 
 of alternator armatures, 600. 
 
 Copper-band contactors, 105. 
 
 Copper conductors, self-inductances of 
 solid cylindrical, 901. 
 
 Copper losses in transformers, 519-520. 
 
 Core magnetization, 483-486. 
 
 Core materials in armatures, 123-124. 
 
 Core type transformers, 532-534. 
 
 Corona effects, 953-954. 
 
 Counter voltage, 61 ; induced in pri- 
 mary circuit of induction motor, 792- 
 795. 
 
 Current, conditions of establishment 
 and termination of, in circuit con- 
 taining resistance and self-inductance 
 in series, 146-149 ; effect on rise 
 and fall of, of eddy currents and of 
 hysteresis, 165-166 ; establishment 
 and termination of, in circuit con- 
 
958 
 
 INDEX 
 
 taming capacity and resistance in 
 series, 170-175; establishment and 
 termination of, in a circuit containing 
 resistance, inductance, and capacity 
 in series, 187-199 ; measure of the 
 growth of, by the time constant, 206- 
 207 ; 'general equation for, in a cir- 
 cuit, 222-224 ; in a circuit when any 
 periodic voltage is impressed, 224- 
 226 ; irregular, and voltage waves 
 expressed as complex quantities, 226- 
 229 ; envelope of vector when condi- 
 tions in circuit vary, 237-240 ; in 
 series circuits under varying condi- 
 tions, 276 ; in parallel circuits, 297- 
 298; active or energy, 313; wattless 
 or quadrature, 314 ; magnetizing, 
 476 ; curves of voltage and, 634 ff. ; 
 distribution of, in a wire, 907-911. 
 
 Current coil, 345. 
 
 Current curve, methods for deter- 
 mining form of, 644-652. 
 
 Current rushes, 583-584. 
 
 Current surges, 583-584. 
 
 Current transformers, 571-575. 
 
 Currents, relations between voltages 
 and, in polyphase systems, 370 ff. ; in 
 rotary field induction motor, 800-803. 
 
 Curve, alternating voltage, 2 ; of 
 squared instantaneous ordinates, 4 ; 
 harmonic, produced by rotating 
 vector, 9-10 ; form of voltage, of an 
 alternator, 22-27 ; method of finding 
 the harmonics of a periodic, by 
 Fourier’s Series, 35-38 ; of voltage in 
 separate slots of progressive distrib- 
 uted windings, 48, 50, 51 ; satura- 
 tion, or magnetization, 621-624 ; 
 external characteristic, 621, 624-632 ; 
 of synchronous impedance, 621, 
 632-634 ; magnetic distribution, 621, 
 634 if. ; voltage, 621, 634 ff. ; short- 
 circuit current, 632 ; showing rela- 
 tion of armature current to excita- 
 tion, 724-727. 
 
 Curves, equations of, by Fourier’s 
 Series, 27-33 ; table of characteristic 
 features of different forms of alternat- 
 ing-current and voltage, 43 ; of rise of 
 charge and discharge in a condenser, 
 171-173 ; of hysteresis, and of current 
 and voltage distorted by hysteresis, 
 404-407 ; of hysteresis and eddy 
 current losses in transformer irons 
 and steels, 509-515 ; areas of succes- 
 sive, of alternating currents and vol- 
 tages, 652-654. 
 
 Curves of torque of induction motor, 
 857, 858. 
 
 Curves of voltage, effect of form of, on 
 operation of induction motors, 842- 
 843. 
 
 Cycles of frequencies of alternating 
 currents, 20-21. 
 
 Cyclic curves, of eddy currents, 425 ; 
 of iron loss, 476-477. 
 
 Cyclic hysteresis curves, 404. 
 
 Cylindrical conductors, eddy current 
 loss in, 418-421. 
 
 D 
 
 Damping grids, 747. 
 
 De la Tour, Moteurs Asynchronous 
 Polyphases, cited, 52. 
 
 Delta connection, 368-370, 552-554. 
 
 Delta winding, 57, 364. 
 
 De Meritens armature, 92. 
 
 Deptford Station transmission plant, 
 91, 105. 
 
 Dielectric constant, the, 169. 
 
 Dielectric hysteresis, 407-408. 
 
 Dielectric of a condenser, 169. 
 
 Dielectric strength, of insulating ma- 
 terials, 121-123 ; of transformer 
 insulation, 592-593. 
 
 Dielectric strength tests of alternators, 
 741. 
 
 Differentiation and integration of com- 
 plex quantities, 251. 
 
 Dimmer, 580. 
 
 Direct-current dynamo, conversion into 
 a double-current machine, 82-83. 
 
 Direction coefficient of an expression, 
 242. 
 
 Disk armatures, 90-92. 
 
 Distortionless circuit, 943. 
 
 Distributed coils, 27 ; effect of, on 
 armature voltage, 47-54. 
 
 Distributed resistance, 932. 
 
 Distributed windings, 54, 78, 85-90. 
 
 Distribution of current in a wire, 907- 
 911. 
 
 Divided circuits, transient transfer of 
 electricity in, 162-165. 
 
 Divided field method of measuring al- 
 ternator losses or efficiency, 736- 
 737. 
 
 Dobrowolsky, 784. 
 
 Double-current generators, 782. 
 i Double-current machine, 47. 
 
 I Double delta connection, 563. 
 
 Drehfelde, 790. 
 
 Drehstrom, 790. 
 
 Drum winding, 78. 
 
 Dry-core transformers, 539. 
 
 Du Bois, H., electro-magnet designed 
 by, 145. 
 
 Duncan, measurements by, 145 ; testing 
 transformers, 594 ; tracing curves, 
 649. 
 
 Dynamo, voltage developed by a direct- 
 current, compared with that de- 
 veloped by an alternator, 46 ; convert- 
 
INDEX 
 
 959 
 
 ing direct-current into double-current 
 machine, 82-83. 
 
 Dynamometer, measuring power by a 
 split, 357-358. 
 
 E 
 
 Eddy currents, effect of, on rise and 
 fall of current in self-inductive 
 circuit, 165-166 ; effect of, in watt- 
 meter frame, 350-351 ; cyclic curves 
 of, 425-427 ; magnetic screening due 
 to, 427-433 ; effect of, on form of 
 primary current wave, 480-481. 
 
 Eddy current losses, in core of a trans- 
 former, 64 ; from magnetic hysteresis, 
 400-404 ; measurement of, 410-417 ; 
 computations of, 417-424; in cylin- 
 drical conductors, 418-421 ; in sheets 
 of rectangular cross section, 421-424 ; 
 curves of, in transformer irons, 512- 
 515; in transformers, 518-519; in 
 alternators, 597. 
 
 Effective value, of alternating current 
 and voltage, 4-6 ; of an irregular 
 curve of voltage or current, 42-43. 
 
 Effective volts and amperes, measure- 
 ment of, 43-45. 
 
 Efficiencies of transformers, 515-523. 
 
 Efficiency, of alternators, 729 ff. ; of 
 converters, 780-781. 
 
 Electrical degrees, 21. 
 
 Electrical resonance, 946. 
 
 Electric circuit, analogies of the, 945- 
 952. 
 
 Electricity, transference of, 149-152 ; 
 transient transfer of, in divided cir- 
 cuits, 162-165. 
 
 Electro-dynamometers, 65-71, 344-346, 
 357. 
 
 Electrolytic rectifiers, 896. 
 
 Electromagnetic instruments for meas- 
 uring alternating voltages and cur- 
 rents, 43-44. 
 
 Electromagnetic repulsion, 863-874. 
 
 Electrometer, quadrant, 73-74. 
 
 Electrometer method for measuring 
 power in alternating circuit, 351-353. 
 
 Electro-motive force of self-induction, 
 128. 
 
 Electrostatic capacity, of parallel con- 
 ductors, 922-931 ; relations of resist- 
 ance, self-inductance, and, 932 ff. 
 
 Electrostatic instruments for measur- 
 ing alternating voltages and currents, 
 44-45, 73-74. 
 
 Electrostatic wattmeter, for measuring 
 power in alternating circuit, 353-354. 
 
 Emery, alternating-current curves, 638. 
 
 Energy, stored in a magnetic field 
 associated with an electric circuit, 
 153-162; of a charged condenser, 
 
 175-176; of mutual induction, 453- 
 456. 
 
 Energy current, 313. 
 
 Energy losses caused by magnetic 
 hysteresis, 400-404. 
 
 Equalizer connections between sections 
 of armature windings, 779. 
 
 Equations of curves by Fourier’s Series, 
 27-33. 
 
 Equivalent impedances, use of, in 
 solving transformer problems, 501- 
 502. 
 
 Equivalent resistance and reactance, 
 
 221 . 
 
 Equivalent resistance of a circuit, 441- 
 442. 
 
 Equivalent sinusoids, 319. 
 
 Ewing, .1. A., experiments by, 403, 404, 
 407 ; hysteresis tester, 414; table of 
 ratio of magnetic force in a plate to 
 magnetic force at surface, 430-431; 
 experiments by, to determine iron 
 losses in transformers, 595-596. 
 
 Ewing and Klaasen, cited, 596. 
 
 Ewing’s apparatus, 596. 
 
 Excitation, of alternators, 106-114; 
 relation of armature current to, 724- 
 727. 
 
 Excitation angle, 481. 
 
 Exciters, 20, 106. 
 
 Exciting ampere turns, 446. 
 
 Exciting current, form of, required to 
 set up a given core magnetization, 
 433-437 ; of transformer, 475-477 ; 
 for an induction motor, 796-797. 
 
 Exponential terms, 200-204. 
 
 External characteristic, alternator, 621, 
 624-632. 
 
 F 
 
 Farad, the unit of capacity, 168. 
 
 Feeder circuits, connecting alternators 
 in parallel to, 699-702. 
 
 Feeder regulators, 661-664. 
 
 Feeding-back method, of testing trans- 
 formers, 589 ; for measuring efficiency 
 of alternators, 732-733. 
 
 Ferranti alternator, 90, 91. 
 
 Ferraris, 784. 
 
 Field current, wavy alternator, 116, 117. 
 
 Field excitation, relation of armature 
 current to, 724-727. 
 
 Field frequency of induction motor, 797. 
 
 Field magnet, defined, 791. 
 
 Field windings for armatures, 77 ff. 
 
 Fleming, Alternate Current Transformer 
 in Theory and Practice, cited, 33 ; 
 transformer tests, 583 ; determining 
 iron losses, 594-595 ; Propagation of 
 Electric Currents in Telephone and 
 Telegraph Conductors by, 945 n. 
 
960 
 
 INDEX 
 
 Form factor, of current curve, 7 ; de- 
 pendence of iron losses of transformer 
 on, 506. 
 
 Foucault currents. See Eddy currents. 
 
 Fourier’s Series, equations of curves by, 
 27-33 ; determination of the value 
 of the constants in, 33-35 ; finding 
 the harmonics of a periodic curve by, 
 35-38. 
 
 Frequencies, commercial, 20-22. 
 
 Frequency, of alternating current, 3 ; 
 effects of changes of, on transformers, 
 507-509 ; in converters, 763-765 ; 
 effect of, on induction motors, 859- 
 860. 
 
 Frequency changer, 782, 891. 
 
 Frequency converter, 892-893. 
 
 Frequency indicator, 782. 
 
 Friction, similarity between magnetic 
 and dielectric hysteresis and, 407-408 ; 
 losses in alternators due to, 597. 
 
 Functions, alternating and pulsating, 
 319-320. 
 
 G 
 
 Galvanometer, effect of shunting a 
 ballistic, 163-165. 
 
 Ganz and Co., alternator, 115-116. 
 
 Generators, synchronous, 19 ; multi- 
 polar, 20-22 ; double-current, 782 ; 
 asynchronous, 784. 
 
 Gerard, cited, 354 ; device for tracing 
 voltage curves of an alternator, 647. 
 
 Gilbert, defined, 400. 
 
 Graphical indication of effect of induc- 
 tive load, 626-628. 
 
 Graphical solutions of problems, 257 ff . ; 
 in series circuits, 262-275; in series 
 circuits, conclusions, 277 ; in parallel 
 circuits, 277 If. ; in parallel circuits, 
 conclusions, 297-299 ; in series and 
 parallel circuits combined, 302-309. 
 
 Gray, distribution of current in a wire, 
 910. 
 
 H 
 
 Harmonics, spherical, 28-33 ; method 
 of finding, of periodic curve, 35-38 ; 
 variation in impedance offered to 
 current and voltage, of different 
 frequencies, 229-236 ; effect of, in 
 waves of voltage and current upon 
 operation of a transformer, 505-507. 
 
 Harmonic voltages and currents, 11. 
 
 Haselwander, 784. 
 
 Hay, investigation of current rushes 
 by, 583. 
 
 Hayward, Vector Algebra and Trigo- 
 nometry, cited, 243. 
 
 Heating, testing transformers as to, 
 591-592 ; tests of alternators by. 
 
 740-741 ; of armature conductors 
 in rotary converters, 765-772. 
 
 Heating effect of alternating current, 3. 
 
 Heaviside, Electrical Papers, cited, 931. 
 
 Hedgehog transformer, 517. 
 
 Henry, defined, 136, 452. 
 
 Hobson, Plane Geometry, cited, 49. 
 
 Hoffman, tracing curves, 649. 
 
 Holden, hysteresis tester, 415—416. 
 
 Hopkinson, John, 594 ; operation of 
 alternators in parallel, 669. 
 
 Hospitalier, cited, 354. 
 
 Hot wire instruments for measuring 
 alternating voltages and currents, 
 44, 71-73. 
 
 Houston and Kennedy, article by, 33 n. 
 
 Hunting of synchronous machines, 
 743-748. 
 
 Hutchinson, tracing curves, 649. 
 
 Hysteresis, effect of, on rise and fall of 
 current in a self-inductive circuit, 
 165-166; energy losses caused by 
 magnetic, 400-404 ; curves of, 404- 
 407 ; similarity between magnetic 
 and dielectric, and friction, 407-408 ; 
 constants of, 408-409 ; measurement 
 of energy absorbed by, 410—417 ; 
 form of exciting current in circuit 
 with and without, 433-437 ; effect 
 of, on form of primary current wave, 
 479-480. 
 
 Hysteresis loop, 403. 
 
 Hysteresis losses, in core of a trans- 
 former, 64 ; measurement of, 410- 
 417 ; curves of, in transformer irons, 
 510-512; in transformers, 518-519; 
 in alternators, in armature cores, 597. 
 
 Hysteresis testers, 414-416. 
 
 Hysteretic angle of advance, 435, 442, 
 481. 
 
 I 
 
 Impedance, of alternating-current cir- 
 cuit, 149, 211-214; in circuit con- 
 taining capacity and resistance in 
 series, 175 ; in circuit containing 
 resistance, inductance, and capacity 
 in series, 197; polygons of, 212; 
 expressed as a complex quantity, 
 213-214 ; variation in, offered to 
 current and voltage harmonics of 
 different frequencies, 229-236 ; syn- 
 chronous, 633, 706-715; in rotary 
 field induction motor, 800-801 ; sub- 
 stituted, for rotary field induction 
 motor, 807-815; locus diagram of 
 single-phase motor and substituted, 
 850-852. 
 
 Impedance coils, 467, 575. 
 
 Impedance methods, solution of parallel 
 circuits by, 299—301 : solution of 
 
 series-parallel problems by, 309-310. 
 
INDEX 
 
 961 
 
 Impressed voltage, 129 ; in self-in- 
 ductive and capacity circuits, 179- 
 180 ; solution of series-parallel prob- 
 lems by method of, 309-310. 
 
 Indicator, power-factor, 743; frequency, 
 782. 
 
 Inductance, conditions of establish- 
 ment and termination of current in 
 circuit containing resistance, capacity, 
 and, in series, 187-199 ; mutual, 449- 
 453 ; leakage, 468. 
 
 Induction, mutual, of parallel dis- 
 tributing circuits, 911-916; effects 
 of self and mutual, in polyphase cir- 
 cuits, 916-922. 
 
 Induction coils, 61-62 ; primary and 
 secondary, 62. 
 
 Induction factor, 343. 
 
 Induction instruments for measuring 
 alternating-current circuits, 75. 
 
 Induction motor, 65, 784, 791 ; rotary 
 field, 784-785 ; counter voltage in- 
 duced in primary circuit of, 792- 
 795 ; exciting current for, 796-797 ; 
 speeds, 797-798 ; slip, 798 ; secondary 
 induced voltage, 798-800 ; currents, 
 torque, impedance, and magnetic 
 leakage in a rotary field, 800-803 ; 
 vector relations in the rotating field, 
 803-807 ; substituted impedance for 
 the rotary field, 807-815; formulas 
 for torque and slip of a polyphase, 
 815 ff. ; circle diagram of magnetic 
 fluxes, 816-822 ; the rotary field, as 
 an asynchronous generator, 822-824; 
 squirrel-cage form of windings, 824 ; 
 independent short-circuited coils, 824; 
 features of construction of rotating 
 field, 824 ff. ; starting and regulating 
 devices, 832 ff. ; reversing polyphase, 
 840-842 ; effect of form of curves 
 of voltage on operation of, 842-843 ; 
 single-phase, 843-850 ; locus diagram 
 of single-phase, and substituted im- 
 pedance, 850-852 ; starting single- 
 phase, 852-853 ; efficiency of, and 
 methods of making tests, 853-859 ; 
 effect of frequency, 859-860 ; poly- 
 phase, with exciting current supplied 
 to the armature, 860-863. 
 
 Inductive circuit, definition, 279 n. 
 
 Inductive circuits of equal time con- 
 stants, connected in series, 276 ; 
 connected in parallel, 297. 
 
 Inductively compensated series motor, 
 877. 
 
 Inductive reactance, 212 ; effect of, 
 in secondary external circuit, 490- 
 492. 
 
 Inductor alternators, 77, 117-119. 
 
 Instruments for measuring alternating 
 voltages and currents, 43-45. 
 
 3 Q 
 
 Insulation, of armatures, 120-123 ; 
 dielectric strength of insulating 
 materials, 121-123 ; in transformers, 
 544- 545 ; dielectric strength of trans- 
 former insulation, 592-593. 
 
 Insulation Tests of alternators, 741. 
 
 Inverted converters, 755, 781-782. 
 
 Involution and evolution of complex 
 quantities, 249-251. 
 
 Iron, constants of magnetic hysteresis 
 related to, 409 ; quality of, for trans- 
 formers, 509-515. 
 
 Iron losses, 221, 412 ; due to hysteresis, 
 402-404; measurement of, 410-414; 
 cyclic curve of, 425-427 ; effect of, 
 on apparent resistance and reactance 
 of a circuit, 440-442 ; cyclic curve of, 
 476-477 ; dependence of, upon form 
 factor, 506 ; in transformers, 519- 
 520 ; experiments to determine, in 
 transformers, 594-596 ; in alter- 
 nators, 599-600. 
 
 Irregular current and voltage waves 
 expressed as complex quantities, 
 226-229. 
 
 Irregular rotary field, 787-788. 
 
 Irregular waves, examples of, 28-32. 
 
 J 
 
 Jackson, Electromagnetism and Con- 
 struction of Dynamos , cited, 23, 33, 
 47, 107, 120, 128, 135, 604, 732; 
 Elementary Electricity and Magnetism, 
 cited, 43, 65, 68; paper on “Three- 
 phase Rotary Field,” cited, 794. 
 
 Joubert, tracing curves, 91 ; use of 
 contact maker by, to determine form 
 of voltage and current curves, 649. 
 
 Joule’s law, 3. 
 
 K 
 
 Kapp, Dynamos, Alternators, and Trans- 
 formers, cited, 48, 123. 
 
 Kapp alternators, armature cores in, 
 123. 
 
 Kelvin, alternating kilowatt balance, 
 71 ; electrostatic voltmeter, 73-74. 
 
 Kennelly and McMahon, 935. 
 
 Kilo-volt-amperes, 325. 
 
 Kirchoff’s law of current flow, 366. 
 
 Kirchoff’s Laws, 262. 
 
 L 
 
 Lag angle, 132-133 ; measuring, 343 ; 
 relation of wattmeter readings to, in 
 balance three-phase circuit, 397-399. 
 
 Laminated core of transformer, 64. 
 
 Lamp synchronizers, 691-694. 
 
 Lap windings, 80, 102. 
 
902 
 
 INDEX 
 
 Lead and lag, 11, 129. 
 
 Leakage, magnetic, 446, 451 ; magnetic, 
 in transformers, 465-468 ; calculation 
 of magnetic, 581-583 ; in rotary field 
 induction motor, 800. 
 
 Leakage conductance, 932. 
 
 Leakage flux, 451. 
 
 Leakage inductance, 468. 
 
 Leakage reactance, 468. 
 
 Lenz's Law, 128. 
 
 Liquid rheostats, 833. 
 
 Load, maximum possible, in parallel 
 operation of alternators, 686-687. 
 
 Load reactances of transformers, 498- 
 501. 
 
 Locus diagrams, of currents, voltages, 
 and loads of the synchronous motor, 
 715-724 ; of single-phase motor and 
 substituted impedance, 850-852 ; 
 testing induction motors, 854. 
 
 Locus of current in a transformer when 
 load reactance is varied and resist- 
 ance is kept constant, 498-501. 
 
 Logarithms of complex quantities, 
 252-253. 
 
 Loppe et Bouquet, cited, 301, 423. 
 
 M 
 
 McAllister, Alternating Current Motors, 
 cited, 727, 889 ; vector relations in 
 rotary field induction motor, 805. 
 
 McMahon, Hyperbolic Functions, cited, 
 929, 935, 936, 938. 
 
 Magnetic distribution curve, 621, 634 ff. 
 
 Magnetic field, energy of self-induced, 
 153-162; rotating, 619, 785-790. 
 
 Magnetic flux distribution curve of an 
 alternator, 634 ff. 
 
 Magnetic fluxes, circle diagram of, in 
 polyphase induction motor, 816-822. 
 
 Magnetic hysteresis, energy losses 
 caused by, 400-404. 
 
 Magnetic leakage, 446, 451 ; in trans- 
 formers, 465-468 ; diagram of trans- 
 former with, 468-475; calculation 
 of, 581-583 ; in rotary field induction 
 motor, 800. 
 
 Magnetic linkages, 152 ; energy stored 
 in, 158-160. 
 
 Magnetic screening, due to eddy cur- 
 rents, 427-133. 
 
 Magnetic vane instruments, 74-75. 
 
 Magnetism wave, relation between 
 form of, and form of induced voltage, 
 437—138. 
 
 Magnetization, core, 483-486 ; curves 
 of, 514, 621-624. 
 
 Magnetizing current, 476. 
 
 Magnetomotive force, 427. 
 
 Mathematics, employment of, by 
 engineers, 241-242. 
 
 Maxwell, Electricity and Magnetism by, 
 cited, 902, 906, 909. 
 
 Measurement, of alternating voltages 
 and currents, 43-45 ; of self-induct- 
 ance, 136-141 ; of power factor, 
 343 ; of angle of lag, 343 ; of power 
 in alternating circuit, 344-358 ; of 
 power in polyphase systems, 389- 
 399 ; of energy absorbed by hystere- 
 sis, 410-417. 
 
 Mechanical analogue of an electric 
 circuit, 947-948. 
 
 Mellor, J. W., Higher Mathematics, 
 cited, 33. 
 
 Mercury vapor rectifier, 895-896. 
 
 Merriman and Woodward, Higher 
 Mathematics, cited, 33. 
 
 Merritt, testing transformers, 594. 
 
 Mershon, testing transformers, 650. 
 
 Mesh winding, 57, 364. 
 
 Mhos, unit for measuring admittance, 
 
 212 . 
 
 Microfarad, the, 168. 
 
 Mordey, testing alternators, 594, 734- 
 736, 737. 
 
 Mordey alternator, 90, 91. 
 
 Motor-converter, 893-895. 
 
 Motor-generator, 782-783, 891-892. 
 
 Motor-generator method for measuring 
 efficiency, losses, and heating of an 
 alternator, 737. 
 
 Motors, alternators, as synchronous, 
 702-704 ; relation of voltages, cur- 
 rents, and power in synchronous, 
 704-715; asynchronous, 784; series, 
 784 ; induction, 784, 791 ; repulsion, 
 784, 863-874 ; rotary field induction, 
 784-785; series alternating current. 
 874-881 ; inductively and conduc- 
 tively compensated series, 877 ; com- 
 mutation of commutating alter- 
 nating-current, 887. 
 
 Multiphase machines, 55-58. 
 
 Multipolar armature, 45 ; with wind- 
 ing distributed in four slots, 48, 49, 
 51. 
 
 Multipolar generators, 20-22. 
 
 Murray, Differential Equations, cited, 
 148, "l74, 181, 188, 190. 191, 196, 222, 
 460. 
 
 Mutual inductance, 449-453. 
 
 Mutual induction, 62, 443-449 ; coeffi- 
 cient of, 450 ; energy of, 453—156 ; 
 transfer of electricity by the effect of, 
 456-463 ; of parallel distributing 
 circuits, 911-916. 
 
 N 
 
 National alternator, 93. 
 
 Neutral point, 56; of wye connection, 
 365. 
 
INDEX 
 
 963 
 
 Non-inductive circuits, connected in 
 series, 276 ; power loops in, 320. 
 
 Non-reactive circuits, 279 ; current in, 
 when connected in parallel, 297 ; 
 expenditure of power in, 312. 
 
 O 
 
 Oersted, discovery by, 1. 
 
 Oil, transformer, 539. 
 
 Oil-cooled transformers, 539. 
 
 One-circuit windings, 82. 
 
 Open-circuit current, 488. 
 
 Open delta connection, 552-554. 
 
 Opposition method of testing trans- 
 formers, 589. 
 
 Oscillograph, 593-594, 644-648. 
 
 Over-compensated series motor, 878. 
 
 Over-compounded machines, 108, 110. 
 
 P 
 
 Parallel, mutual inductance in coils in, 
 459-463 ; alternators in, 668-669 ; 
 division of load between alternators 
 in, 676-686 ; maximum possible 
 load and regulation of prime movers 
 in operation of alternators in, 686- 
 688. 
 
 Parallel circuits, 261 ; solutions of 
 problems in, 277-299 ; solution of, 
 by the impedance methods, 299-301 ; 
 solution of combined series and, 302- 
 309. 
 
 Parallel conductors, electrostatic capac- 
 ity of, 922-931. 
 
 Parallel distributing circuits, mutual 
 induction of, 911-916. 
 
 Parallel operation, of alternators, 668- 
 669 ; effect of form of voltage curve 
 and variation of angular velocity on, 
 689. 
 
 Parallel wires, self-inductance of, 897- 
 907. 
 
 Parshall and Hobart, diagrams of al- 
 ternator windings, 80 n. 
 
 Pender, computing formulas, 935 n. 
 
 Pender and Osborne, electrostatic capac- 
 ity of parallel conductors, 929-930. 
 
 Period of alternating current, 3. 
 
 Permeability, effect of, on self-induct- 
 ance, 137-138 ; curve of, of trans- 
 former steels, 514-515. 
 
 Perry, John, article by, 33 n. 
 
 Phase, 8. 
 
 Phase diagram, 12, 260, 443-449. 
 
 Phase difference, 9. 
 
 Phase indicators, 696-699. 
 
 Phase splitter, 852. 
 
 Phase transformation, 561-566. 
 
 Picou, quoted, 116. 
 
 Plant efficiency, 339. 
 
 Polar coordinates, 13-15. 
 
 Polar curve, method of obtaining effec- 
 tive voltages from the, 13-15. 
 
 Polarity of windings, determination of, 
 547-548. 
 
 Polar surfaces, 46. 
 
 Pole armature, 92-93. 
 
 Pole face, width of, 54-55. 
 
 Polygons, of voltages, 131, 149, 179, 
 209-210 ; of impedance, 212 ; of 
 vectors, 260. 
 
 Polyphase alternators, 55-60 ; method 
 of connecting up, 58-60 ; method of 
 compounding, 113. 
 
 Polyphase circuits, effects of self and 
 mutual induction in, 916-922. 
 
 Polyphase induction motor with ex- 
 citing current supplied to the arma- 
 ture, 860-863. 
 
 Polyphase machines, 58. 
 
 Polyphase systems, defined, 359 ; bal- 
 anced, 359 ; uniform power in, 367- 
 368 ; relations between currents 
 and voltages, 370 ff. ; measurement of 
 power in, 389-399. 
 
 Polyphase transformers, 524-531. 
 
 Potentiometer arrangement for contact 
 maker, 651. 
 
 Power, expended in a circuit on which 
 a sinusoidal voltage is impressed, 
 312-315; in alternating circuit, 316; 
 expended in circuit when voltage 
 and current are single-valued periodic 
 functions, 317-319; apparent, 325; 
 true, 325 ; vector, 333 ; quadrature, 
 334 ; methods for measuring in al- 
 ternating circuit, 344-358 ; measure- 
 ment of, in polyphase systems, 389- 
 399. 
 
 Power circle, 685. 
 
 Power curves, produced by sinusoidal 
 voltages and currents are double 
 frequency sinusoids wdth axes dis- 
 placed, 323-324. 
 
 Power factor, 325 ; method of measur- 
 ing, 343 ; relation of wattmeter 
 readings to, in balanced three-phase 
 circuit, 398-399 ; relation of, to 
 capacity in transformers, 498-501 ; 
 relation of, to capacity, 500-501 ; of 
 induction motors, 856. 
 
 Power-factor indicator, 743. 
 
 Power factors, table of reactive factors 
 and, 342. 
 
 Power loops, 320-323 ; quadrature 
 components of, 340-341. 
 
 Power relations, expression of, by means 
 of vectors, 333-339. 
 
 Pressure wires, 657. 
 
 Primary current wave, effects of vari- 
 able reluctance, of hysteresis, of 
 eddy currents, and of current in the 
 
9G4 
 
 INDEX 
 
 secondary winding upon form of, 
 477-483. 
 
 Primary windings, 791 ; commutated, 
 for regulating induction motors, 
 838. 
 
 Prime movers, regulation of, in parallel 
 operation of alternators, 686-688. 
 
 Progressive windings, 81-82, 102. 
 
 Propagation constant, 939. 
 
 Pupin, testing transformers, 650. 
 
 Q 
 
 Quadrant electrometer, 73, 351. 
 
 Quadrature components of power loops, 
 340-341. 
 
 Quadrature current, 314. 
 
 Quadrature power, 334. 
 
 Quarter-phase currents, 359-361. 
 
 Quarter-phase transformers, 524. 
 
 R 
 
 Rated motor, testing alternators, 729- 
 734. 
 
 Ratio of transformation in a trans- 
 former, 463-465. 
 
 Rayleigh, article by, cited, 909. 
 
 Reactance, of alternating-current cir- 
 cuit, 149, 197, 212 ; of circuit con- 
 taining capacity and resistance in 
 series, 175; capacity, 212 ; inductive, 
 212 ; expressed as a complex quan- 
 tity, 213-214 ; equivalent or working, 
 221 ; of coils, errors in wattmeter 
 readings due to, 346-349 ; of a cir- 
 cuit, effect of iron losses on the ap- 
 parent, 440-442 ; leakage, 468. 
 
 Reactance coils, 467, 575. 
 
 Reactions, alternator armature, 611- 
 621 ; armature, of a converter, 772- 
 773. 
 
 Reactive circuit, definition, 279 n. ; 
 expenditure of power in, 312-313 ; 
 power loops in, 320-321. 
 
 Reactive factor of a circuit, 342-343. 
 
 Reactive factors, table of power factors 
 and, 342. 
 
 Reactive volt-amperes, 334. 
 
 Rechniewski, experiments of, 600. 
 
 Reciprocal of a vector expression, 248. 
 
 Rectifier, mercury vapor, 895-896 ; 
 electrolytic, 896. 
 
 Rectifying commutator, 114-117. 
 
 Regulation, of transformers, 522-523 ; 
 of alternators, 654-661, 741-742; 
 
 of induction motors, 832 ff., 856. 
 
 Regulation tests for transformers, 590- 
 591. 
 
 Regulators, voltage, 575-581 ; feeder, 
 661-664 ; synchronous, 774. 
 
 Relays, table of polarized, 143. 
 
 Reluctance, effect of variable, on form 
 of primary current wave, 477—479. 
 
 Repulsion, electromagnetic, 863-874. 
 
 Repulsion motors, 784 ; series, 878- 
 879. 
 
 Residual magnetism, effect of, on rate 
 of change of magnetic field in self- 
 inductive circuit, 166. 
 
 Resistance, effect of introducing, in a 
 continuous current circuit, 180-183 ; 
 conditions of establishment and 
 termination of current in circuit 
 containing inductance, capacity, and, 
 in series, 187-199 ; vector relations 
 of current and voltage in circuit 
 containing self-inductance, capacity, 
 and, in series, 209-211 ; equivalent 
 or working, 221 ; effect of iron losses 
 on apparent, of a circuit, 440 ; in 
 field circuit of induction motors, 832- 
 833 ; in armature circuits for speed 
 regulation of induction motors, 835- 
 837 ; distributed, 932. 
 
 Resonance, condition of, 234 ; electrical 
 946. 
 
 Reversing polyphase motors, 840-842. 
 
 Revolving armatures, 77, 92. 
 
 Rheostats, starting and regulating, for 
 induction motors, 832-833. 
 
 Ring armature, 92. 
 
 Ring winding, 78. 
 
 Roessler, on dependence of iron losses 
 of transformer upon form factor, 
 506. 
 
 Roiti, cited, 594. 
 
 Rotary converters, 83, 755 ; heating of 
 armature conductors in 765-772 ; 
 connecting up, 777-778. 
 
 Rotary field induction motors, 784- 
 785 ; currents, torque, impedance, 
 and magnetic leakage in, 800-803 ; 
 vector relations in the, 803-807 ; 
 substituted impedance for the, 807- 
 815 ; as an asynchronous generator, 
 822-824 ; features of construction, 
 824 ff. 
 
 Rotating-field alternators, 77, 125-126. 
 
 Rotating magnetic field, 619, 785-790; 
 action of a short-circuited armature 
 winding within a, 790-792. 
 
 Rotating vector, 257-258. 
 
 Rotor, defined, 791-792. 
 
 Rushes, current, 583-5S4. 
 
 Russell, Alternating Currents, cited, 430, 
 904, 931. 
 
 Ryan, experiment showing effect of 
 magnetic leakage, 466 ; tracing trans- 
 former curves, 594. 
 
 Ryan and Bedell, synchronous motor 
 experiment, 714. 
 
 Ryan and Merritt, testing transformers, 
 650. 
 
INDEX 
 
 965 
 
 s 
 
 Saturation curve, 621-624. 
 
 Scalar value of two forces, 8. 
 
 Screening action due to eddy currents, 
 427-433. 
 
 Searing, tracing curves, 649. 
 
 Secondary induced voltage of induction 
 motor, 798-800. 
 
 Secondary winding, effect of current in, 
 on forms of primary current waves, 
 481-483. 
 
 Secondary windings, 791. 
 
 Self-excited alternator, 107, 
 
 Self-inductance, 134-136; the henry the 
 unit for measuring, 136 ; of a short 
 coil, 136-137 ; of a circuit containing 
 variable permeability, 137-138 ; ex- 
 amples of values of, 141-146 ; rate 
 of expenditure of work in circuit 
 containing, 183-185 ; vector relations 
 of current and voltage in circuit con- 
 taining resistance, capacity, and, in 
 series, 209-211 ; of alternators, 609— 
 611 ; of parallel wires, 897-907 ; 
 relations of distributed resistance, 
 leakage conductance, electrostatic 
 capacity, and, 932-945. 
 
 Self-induction, 62, 128-130 ; coefficient 
 of (self -inductance), 135. 
 
 Self-inductive circuit, vector diagrams 
 of voltage relations in, 130-132 ; 
 current in, 146-149 ; effect of eddy 
 currents and of hysteresis on rise and 
 fall of current in, 165—166 ; high 
 voltage generated on breaking a, 
 166-167. 
 
 Separately excited alternator, 107. 
 
 Series, mutual inductance in coils in, 
 456-459 ; alternators in, 664-668. 
 
 Series alternating current motors, 874- 
 881. 
 
 Series circuits, 261 ; graphical and 
 analytical treatment of problems 
 relating to, 262-275; conclusions in 
 regard to, 275-277 ; solution of, com- 
 bined with parallel circuits, 302-309. 
 
 Series motors, 784 ; vector diagrams of, 
 and expressions for voltage, 881-887. 
 
 Series repulsion motor, 878-879. 
 
 Series transformers, 75-76, 571-575. 
 
 Series-wound alternator, 107. 
 
 Shading coils, 853. 
 
 Shape of voltage and current waves 
 used in testing transformers, 593-594. 
 
 Sheet-iron plates, eddy current loss in, 
 421^24. 
 
 Shell type transformers, 532, 534-535. 
 
 Short-circuit current curve, 632. 
 
 Short-circuited coils, independent, in 
 windings of rotating field induction 
 motors, 824. 
 
 Short-circuiting of terminals of second- 
 ary windings, 49.5 — 198. 
 
 Shunt, use of. in amperemeters, 72-73, 
 76. 
 
 Shunt-wound alternator, 107. 
 
 Siemens electro-dynamometer, 66—67. 
 
 Single-phase induction motors, 843- 
 850 ; locus diagram of, and sub- 
 stituted impedance, 850—852 ; start- 
 ing, 852-853. 
 
 Single-phase machines, 55. 
 
 Sinusoidal voltage in a self-inductive 
 circuit, 133-134. 
 
 Skin effect, 903, 910. 
 
 Slip of induction motor, 798, 815, S56. 
 
 Slip rings, 19. 
 
 Smithsonian Mathematical Tables, 93S. 
 
 Solenoid, self-inductance of a, 136—137. 
 
 Sparking, avoidance of, at rectifying 
 commutator, 115—117. 
 
 Specific inductive capacity. 169. 
 
 Speeds of induction motor, 797-798. 
 
 Split dynamometer methods of measur- 
 ing power, 357-358. 
 
 Split-pole converters, 773. 
 
 Squirrel cage winding, 791 ; of induc- 
 tion motors, 824. 
 
 Standardization Rules of American 
 Institute of Electrical Engineers, 
 585—586, 742. 
 
 Standing torque, 856. 
 
 Stanley alternator, 118-119. 
 
 Star-connected systems, 361-363. 
 
 Starting devices, polyphase induction 
 motors, 832 ff. 
 
 Starting single-phase induction motors, 
 852-853. 
 
 Starting torque, 856. 
 
 Star winding, 58. 
 
 Stator, defined, 792. 
 
 Steel, constants of magnetic hysteresis 
 related to, 409 ; for use in trans- 
 formers, 509-515. 
 
 Steinmetz, cited, 13, 423, 506, 906; 
 experiments by, 122, 407 ; experi- 
 ments showing energy loss due to 
 hysteresis, 403 ; hysteretie angle of 
 advance of, 435. 442 : parallel opera- 
 tion of alternators, 689. 
 
 Step, voltage waves in. 664. 
 
 Step by step method of testing for 
 hysteresis loss, 416. 
 
 Stray power methods, for obtaining 
 transformer efficiency, 587-590 ; of 
 testing alternators, 729—732 : of 
 
 testing induction motors, 855-856. 
 
 Substituted impedance, for the rotary 
 field induction motor, 807-815 : locus 
 diagram of single-phase motor and, 
 850-852. 
 
 Sumpner, 354, 356. 
 
 Surges, current, 583-584. 
 
966 
 
 INDEX 
 
 Susceptance, of a circuit, 214 ; of an 
 admittance, 253. 
 
 Swenson and Frankenfield, table . of 
 hysteresis constants compiled by, 409. 
 
 Swinburne, Hedgehog transformer of, 
 517. 
 
 Switchboards for connecting alternators 
 in parallel, 699-702. 
 
 Synchronism, voltage waves in, 664. 
 
 Synchronizers and synchronizing, 690- 
 699. 
 
 Synchronizing current of alternators, 
 669-676. 
 
 Synchronizing lamps, 691-694. 
 
 Synchronous condenser, 748-754. 
 
 Synchronous generators, 19. 
 
 Synchronous impedance, 621, 632-634, 
 *706-715. 
 
 Synchronous machines, 597 ff. 
 
 Synchronous motors, alternators as, 
 702-704 ; relation of voltages, cur- 
 rents, and power in, 704-715 ; locus 
 diagrams of currents, voltages, and 
 loads of, 715-724 ; application of 
 diagrams to machines wound with 
 any number of phases, 728-729 ; for 
 testing alternators, 738-739 ; hunt- 
 ing of, 743-748 ; asynchronous motors 
 contrasted with, 784. 
 
 Synchronous regulator, 774. 
 
 Synchroscopes, 696-699. 
 
 T 
 
 Table, characteristic features of dif- 
 ferent forms of alternating-current 
 and voltage curves, 43 ; effect of 
 distributed coils on armature voltage, 
 52 ; specific resistance of insulators, 
 122 ; formulas for dielectric materials, 
 122 ; polarized relays, 143 ; self-in- 
 ductance of armature in place, 145 ; 
 power factors and reactive factors, 
 342 ; hysteresis constants, 409 ; 
 Ewing’s, of ratio of magnetic force 
 at various depths in a plate to 
 magnetic force at surface, 431 ; 
 of iron and copper losses in trans- 
 formers, 520 ; of dielectric strength 
 of alternators, 741 ; relation of 
 number of poles to synchronous 
 speed in induction motors, 826 ; 
 self-inductances of solid cylindrical 
 copper conductors, 901 ; distribution 
 of current, 910. 
 
 Tandem connection, induction motors, 
 838-840. 
 
 Teaser winding, 554, 561. 
 
 Tee connection of transformers, 554- 
 555. 
 
 Teeth, armature, 94, 124 ; of rotating 
 field induction motors, 831. 
 
 Telephone service, long distance, 943- 
 945. 
 
 Tensor of vector quantity, 242. 
 
 Terminals of transformer, insulation 
 of, 545. 
 
 Tesla, rotary field motor applications, 
 784. 
 
 Testing, transformers, 585-586; alter- 
 nators, 729 ff. ; induction motors, 
 853-859. 
 
 Thompson, S. P., 80, 81. 
 
 Thomson, Elihu, 538, 567 ; on electro- 
 magnetic repulsion, 864. 
 
 Thomson, J. J., formulas showing 
 extent of magnetic screening in 
 stampings, 430 ; current density at 
 any point within a conductor, 
 910. 
 
 Thomson alternating-current ampere- 
 meter, 74. 
 
 Thomson-Houston alternator, 93. 
 
 Thomson impedance coils, 575-576. 
 
 Three-instrument methods of measuring 
 power, 354-358. 
 
 Three-phase machine, 56-57. 
 
 Three-phase systems, methods of con- 
 nection, 361—367 ; measurement of 
 power in, 389, 391. 
 
 Time constants of a circuit, 206-208 ; 
 examples of, 208-209. 
 
 Tobey and Walbridge, article by, cited, 
 638. 
 
 Todhunter, cited, 368. 
 
 Torque, in rotary field induction motor, 
 801 ; of polyphase induction motor, 
 815 ff. ; standing and starting, of 
 induction motors, 856. 
 
 Torsion head, 66-67. 
 
 Transference of electricity, in self-in- 
 ductive circuits, 149-152 ; transient, 
 in divided circuits, 162-164; by the 
 effect of mutual induction, 456- 
 463. 
 
 Transformation, ratio of, in converters, 
 758-763. 
 
 Transformer oil, 539. 
 
 Transformers, fundamental principle 
 of, 61-64 ; definition, 64 ; losses 
 in operation of, 64-65 ; mutual 
 induction by means of, 443 ; diagrams 
 of, 443-449 ; ratio of transformation 
 in, 463—465 ; magnetic leakage in, 
 465—168 ; constant current, 46S ; 
 constant voltage, 468 ; diagrams of, 
 with magnetic leakage, 468—475; 
 exciting current, 475-477 ; effects 
 of variable reluctance, of hysteresis, 
 and of eddy currents on form of 
 primary current wave, 477-481 ; 
 forms of primary current waves as 
 affected by current in secondary 
 winding, 481-483 ; core magnetiza- 
 
INDEX 
 
 967 
 
 tion, 483-486 ; circle diagram for 
 non-reactive secondary circuit, and 
 active voltage locus, 486 ; circle dia- j 
 gram where transformer load con- 
 tains constant reactance and variable 
 resistance, 490 ; locus of current 
 when load reactance is varied and 
 resistance is kept constant, 498-501 ; 
 use of equivalent impedances in solv- 
 ing problems, 501-505 ; effect of 
 harmonics in waves of voltage and 
 current upon operation, 505-507 ; 
 effects of changes of frequency and 
 voltage, 507-509 ; iron and steel for, 
 509-515; alloying materials, 515; 
 efficiencies of, 515-523 ; Hedgehog, 
 517 ; U. S. table of iron and copper 
 losses, 520 ; regulation of, 522- 
 523; polyphase, 524-531; construc- 
 tive features of constant voltage 
 transformers, 531 ff. ; core type and 
 shell type, 532-537 ; cooling, 538- 
 544; insolation in, 544-546; connect- 
 ing constant voltage, and features 
 of their operation, 546 ff. ; polarity 
 of windings, 547 ; open delta on V 
 connection, 552 ; tee connection, 554 ; 
 teaser windings, 554, 561 ; phase 
 transformation, 561-566; double delta 
 connection, 563; transformation from 
 constant voltage to constant current, 
 566 ; series or current transformers, 
 571-575; reactance coils, impedance 
 coils, or choking coils, 575-576; 
 autotransformers or compensators, 
 576-581 ; calculation of magnetic 
 leakage, 581-583 ; current rushes 
 and surges, 583-584 ; methods of 
 testing, 585 ff. ; wattmeter method, 
 586-587 ; stray power methods, 
 587 ; regulation tests, 590-591 ; heat- 
 ing tests, 591-592 ; dielectric strength 
 of insulation, 592-593 ; determina- 
 tion of wave shape in testing, 593- 
 594 ; methods used in historically 
 important tests of, 594-596. 
 
 Transient state in a circuit, effect of 
 self-inductance and capacity on, 
 186-187. 
 
 Tri-phase transformers, 524-527. 
 
 True power, 325. 
 
 Tuned circuits, 947. 
 
 Turbine generators, rotating field for, 
 125-127. 
 
 Two-circuit windings, 80. 
 
 Two-phase machine, 55-56 ; method 
 of converting a direct-current ma- 
 chine into, 83. 
 
 Two-phase system, voltage curves of, 
 360 ; methods of connection, 360- 
 361 ; measurement of power in, 389- 
 391. 
 
 U 
 
 Undercompensated series motor, 878. 
 
 1 Underground cables, transmission of 
 power over, 942-945. 
 
 Uniformly rotating magnetic field, 786- 
 788. 
 
 V 
 
 Variation of angular velocity, effect of, 
 on parallel operation of alternators 
 and division of load, 684-690. 
 
 V connection of transformers, 552-554. 
 
 V-curves, 686. 
 
 Vector, envelope of the current, when 
 conditions in circuit vary, 237-240. 
 
 Vector analysis and the complex 
 quantity, 241-244. 
 
 Vector diagram, 260. 
 
 Vector diagrams, representing voltage 
 relations in an inductive alternating- 
 current circuit, 130-132 ; showing 
 voltage and current relations in a 
 charged condenser, 176-180; of alter- 
 nating-current series motors, 881- 
 887. 
 
 Vector formulas in solving transformer 
 problems, 502-504. 
 
 Vector polygon, 12, 260. . 
 
 Vector power, 333. 
 
 Vector quantities, 8. 
 
 Vector relations, of current and voltage 
 in circuit containing resistance, self- 
 inductance, and capacity in series, 
 209-211; in rotating field induction 
 motor, 803-807. 
 
 Vectors, 8 ; rotating, 9 ; relations 
 of their components and, 15-17 ; 
 addition and subtraction of, 244 ; 
 multiplication and division of, 244- 
 246 ; complementary, 246 ; recipro- 
 cals of, 248 ; involution and evolu- 
 tion of, 249-251 ; differentiation and 
 integration of, 251-252 ; graphical 
 combination of, 257 ; expression of 
 power relations by means of, 333- 
 339. 
 
 Vector triangle, 12. 
 
 Ventilation, armature, 124-126 ; of 
 armature alternators, 600. 
 
 Versor of an expression, 242. 
 
 Vibrator, oscillograph, 647. 
 
 Vinculum, use of, 255. 
 
 Voltage, alternating, 2 ; resolution of 
 irregular waves of, into their har- 
 monics, 33-35 ; instruments for 
 measuring alternating, 43-45 ; com- 
 parison of that developed by an 
 alternator and that developed by a 
 direct-current dynamo, 46 ; effect 
 of arrangements of windings on that 
 of armatures, 47-54 ; of self-indue- 
 
968 
 
 INDEX 
 
 tion, 128 ; impressed, 129 ; genera- 
 tion of high, by breaking a self- 
 inductive circuit, 166-167 ; capacity 
 or condenser, 177, 267 ; effects of 
 changes of, on transformers, 507- 
 509. 
 
 Voltages, vector diagrams representing, 
 130-132 ; vector relations of, in 
 circuit containing resistance and 
 inductance, 149 ; relations between 
 currents and, in polyphase systems, 
 370. 
 
 Voltage coil, 345. 
 
 Voltage control of converters, 773. 
 
 Voltage curve of alternator, 22-27, 621, 
 634 ff. ; determining the effective 
 value of an irregular, 42-43 ; methods 
 for determining form of, 644-652 ; 
 effect of form of, on parallel operation 
 and division of loads, 689-690. 
 
 Voltage limitations in converters, 763- 
 765. 
 
 Voltage regulators, 575-581, 654-661. 
 
 Voltage transformer, 75-76. 
 
 Voltage waves, irregular current and, 
 expressed as complex quantities, 
 226-229. 
 
 Volt-amperes, 325 ; reactive, 334. 
 
 Voltmeters, alternating-current, 43-45, 
 65-75; ar\ ,'angement of, for measur- 
 ing high voltages or large currents, 76. 
 
 W 
 
 Warburg, on energy losses due to 
 hysteresis, 403-404. 
 
 Warren alternator, 119. 
 
 Water-cooled transformers, 542-543. 
 
 Wattless current, 314. 
 
 Wattmeter, alternating-current, 65-75, 
 313 ; arrangement for measuring 
 high voltages or large currents, 75-76 ; 
 for measuring power in alternating 
 circuit, 344-351 ; errors in readings, 
 due to reactance of coils, 346-349 ; 
 correction of readings, on account of 
 power absorbed by instrument, 349- 
 350 ; effect of eddy currents in frame, 
 350-351 ; electrostatic, 353-354 ; for 
 measuring power in polyphase sys- 
 tems, 389 ff. 
 
 Wattmeter method of testing trans- 
 formers, 586-587. 
 
 Watts expended in a circuit, 325. 
 
 Wave form of voltage produced by an 
 alternator, 742. 
 
 Wave length constant, 939. 
 
 Waves, examples of irregular, 28-32. 
 
 Wave shape, determination of, in test- 
 ing transformers, 593-594. 
 
 Wave windings, 81-82. 
 
 Wavy field current in alternator, 116, 
 117. 
 
 Westinghouse alternator, 93, 126. 
 
 Westinghouse rotating field magnet, 
 126. 
 
 Weston alternating-current voltmeter, 
 67-69. 
 
 Weston wattmeter, 350. 
 
 Wilde, operation of alternators in 
 parallel, 669. 
 
 Wilkes, tracing curves, 649. 
 
 Williamson, Differential Calculus, cited, 
 242. 
 
 Windage, losses in alternators due to, 
 597. 
 
 Windings, armature, 47 ff. ; single and 
 polyphase, 55 ; mesh or delta, 57 ; 
 star or Y, 58 ; methods of connecting 
 up, 58-60 ; drum and ring, 78 ; 
 bar and coil, 78 ; distributed, 78, 
 85-90 ; lap, 80 ; two-circuit, 80 ; 
 wave or progressive, 81-82 ; one- 
 circuit, 82 ; barrel, 86 ; polarity of, 
 547-548 ; teaser, 554, 561 ; amor- 
 tisseur, 747 ; armature, in rotating 
 field, 790-792 ; primary and second- 
 ary, 791 ; of induction motors, 
 824-832. 
 
 Wood, H. P., cited, 373. 
 
 Working resistance and reactance, 221. 
 
 Y 
 
 Y-box for measuring power, 395. 
 
 Y-connection, 361, 363; for more than 
 three phases, 36S-370. 
 
 Y-winding, 58. 
 
 Z 
 
 Zipernowsky alternator, arrangement of 
 commutator in, 115-116. 
 
 Printed in the United States of America, 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Date Due 
 
 i ..... _ 
 
 
 
 
 MAR 2 41< 
 
 48 
 
 
 
 g|p1 9'S 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L. B. Cat. No. 1137 
 
u 
 
 54537 
 
 f