HANDBOOK OF ELECTRICAL TESTING KEMPE TK 401 .K32 1887 E&F.N.SPON ARTES 18370 1 VERITAS LIBRARY SCIENTIA OF THE UNIVERSITY OF MICHIGAN PLURIBUS-UNUM SQUAERIS PENINSULAM AMOENAM CIRCUMSPICE : 7 A HANDBOOK OF 297173 ELECTRICAL TESTING. 2082 BY arry obert H. R. KEMPE, = TECHNICAL OFFICER, POSTAL TELEGRAPHS ; MEMBER OF THE COUNCIL OF THE SOCIETY OF TELEGRAPH-ENGINEERS AND ELECTRICIANS; ASSOCIATE MEMBER OF THE INSTITUTION OF CIVIL ENGINEERS. FOURTH EDITION. (ADOPTED BY THE POSTAL TELEGRAPH DEPARTMENT.) E. & F. N. SPON, 125, STRAND, LONDON. NEW YORK: 35, MURRAY STREET. 1887. ... i " " NOTE. In the present Edition I have not only taken advantage, as far as possible, of the many friendly suggestions which have been made to me (especially by Dr. A. Muirhead) for the improve- ment of the original work, but have added a considerable amount of new matter, besides thoroughly revising the old. I have more particularly to thank Messrs. Elliott Brothers, Messrs. Latimer Clark, Muirhead and Co., Mr. B. Pell (of Messrs. Johnson and Phillips's), and Mr. P. Jolin, for the illustrations of apparatus which have been added. I am also greatly indebted to Messrs. W. T. Glover and Co. for per- mission to insert their Table (III.) of the Weights, Resist- ances, &c., of Pure Copper Wire, and to Mr. Herbert Taylor for the Table (IX.) of Data of recent Submarine Cables. H. R. K. ENGINEER-IN-CHIEF'S OFFICE, Od 3-21.38 ELECTRICIAN'S DEPARTMENT, GENERAL POST OFFICE. London, July 1887. ноздат SIMPLE TESTING RESISTANCE COILS GALVANOMETERS SHUNTS CONTENTS. : CHAPTER I. : CHAPTER II. CHAPTER III. CHAPTER IV. : : CHAPTER V. MEASUREMENT OF GALVANOMETER RESISTANCE : PAGE 1 : : : : : : : : 10 : 18 67 79 I CHAPTER VI. MEASUREMENT OF THE INTERNAL RESISTANCE OF BATTERIES 113 CHAPTER VII. MEASUREMENT OF THE ELECTROMOTIVE FORCE OF BATTERIES 137 ་ vi CONTENTS. CHAPTER VIII. THE WHEATSTONE BRIDGE • LOCALISATION OF FAULTS F • • CHAPTER IX. CHAPTER X. KEYS, SWITCHES, CONDENSERS, AND BATTERIES : CHAPTER XI. MEASUREMENT OF POTENTIALS • : : CHAPTER XII. MEASUREMENT OF CURRENT STRENGTH CHAPTER XIII. MEASUREMENT OF ELECTROSTATIC CAPACITY } : : CHAPTER XIV. THE THOMSON QUADRANT ELECTROMETER PAGE 188 : 242 270 : : : : CHAPTER XV. MEASUREMENT OF HIGH RESISTANCES CHAPTER XVI. MEASUREMENT OF RESISTANCES BY POTENTIALS 284 : 301 : 325 348 364 : 377 CONTENTS. vii CHAPTER XVII. LOCALISATION OF FAULTS BY FALL OF POTENTIALS CHAPTER XVIII. TESTS DURING THE LAYING OF A CABLE JOINT-TESTING : SPECIFIC MEASUREMENTS CHAPTER XIX. CHAPTER XX. · CHAPTER XXI. CORRECTIONS FOR TEMPERATURE CHAPTER XXII. LOCALISATION OF FAULTS OF HIGH RESISTANCE : CHAPTER XXIII. LOCALISATION OF A DISCONNECTION FAULT IN A CABLE CHAPTER XXIV. : : A : : : PAGE 386 : 396 402 408 414 428 439 A METHOD OF LOCALISING EARTH FAULTS IN CABLES 447 CHAPTER XXV. GALVANOMETER RESISTANCE 457 1 viii CONTENTS. ! [ CHAPTER XXVI. SPECIFICATION FOR MANUFACTURE OF CABLE.-SYSTEM OF TESTING CABLE DURING MANUFACTURE 461 MISCELLANEOUS CHAPTER XXVII. TABLES. : I. NATURAL TANGENTS .. 526 II. RESISTANCE OF A KNOT-POUND OF COPPER WIRE OF VARIOUS CONDUCTIVITIES, AT 75° FAHR. 528 528 III. THE RELATIVE DIMENSIONS, LENGTHS, RESISTANCES (AT 60° FAHR.), AND WEIGHTS, OF PURE COPPER WIRE IV. COEFFICIENTS FOR CORRECTING THE OBSERVED RESISTANCE OF PURE COPPER WIRE AT ANY TEMPERATURE TO 75° FAHR., OR AT 75° TO ANY TEMPERATURE V. COEFFICIENTS FOR CORRECTING THE OBSERVED RESISTANCE OF ORDINARY COPPER WIRE AT ANY TEMPERATURE TO 75° FAHR., or at 75° TO ANY TEMPERATURE • VI. COEFFICIENTS FOR CORRECTING THE OBSERVED RESISTANCE OF "SILVERTOWN" GUTTA PERCHA AT ANY TEMPERATURE TO 75° FAHR... 529 530 531 .. VII. COEFFICIENTS FOR CORRECTING THE OBSERVED RESISTANCE OF "WILLOUGHBY SMITH'S " GUTTA PERCHA AT ANY TEMPERATURE TO 75° FAHR. 532 VIII. MULTIPLYING POWER OF SHUNTS EMPLOYED WITH A GALVANOMETER OF 6000 О¤MS RESISTANCE 533 IX. MULTIPLYING POWER OF SHUNTS EMPLOYED WITH A GALVANOMETER OF 10,000 OHMS RESISTANCE 534 • X. STANDARD WIRE GAUGE 535 XI. ELECTRICAL AND MECHANICAL DATA OF RECENT SUBMARINE CABLES 536 INDEX .. : : 537 PAGE 490 A HANDBOOK OF ELECTRICAL TESTING. CHAPTER I. SIMPLE TESTING. 1. In order to be able to make measurements of any kind, it is necessary to have certain standard units with which to make comparisons. For example, in the case of length, or weight, we have as standards the foot and the pound. Some of the units are dependent upon two of the other units; the unit of "work,” for example, is the foot-pound, or the work done in raising a pound 1 foot high. Now in electrical measurements we require units of a like character. Those with which we have to deal chiefly are electromotive force, the unit of which is called the volt ; resistance, the unit of which is the ohm ; also we have the unit of current, which is dependent upon the two foregoing units, and which is called the ampère. 2. If the two poles of a battery be joined by a conductor a current will flow, and the strength of this current will vary directly as the electromotive force of the battery, and inversely as the total resistance in the circuit. This relation is known as "Ohm's law." the electromotive force is expressed in volts and the resistance in ohms, then the resulting current will be in ampères. If G FIG. 1. 000 3. Suppose now a battery of a resistance r and electromotive force E, a galvanometer of a resistance G, and a wire of a resistance R, be joined up in circuit, as shown by Fig. 1. By B 1 2 HANDBOOK OF ELECTRICAL TESTING. the foregoing law, the strength of current C, which will flow out of the battery and through the galvanometer, will be C E R + r + G * R+r+ The current, in flowing through the galvanometer, produces a deflection of its needle, which deflection will remain constant provided the electromotive force of the battery and also the resistances remain constant. If now R be a wire whose re- sistance we require to find, and which we can replace by another wire the value of whose resistance can be varied at pleasure, then by adjusting this latter so that the deflection of the galvanometer needle becomes the same as it was before the change of resistances was made, this resistance gives the value of our unknown resistance R. This method of testing, known as the substitution method, although exceedingly simple, is a very good and accurate one if a little ordinary care be taken in making it. Its correctness is only limited by the sensibility of the galvanometer to small changes of strength in the current affecting it, and by the accuracy with which the variable resistance can be adjusted. It should be mentioned, however, that for reasons which will become obvious when the subject of testing is gone further into, the resistance of the battery and galvanometer used in making a test of the kind should be small compared with the resistance being measured. 4. Next, suppose the galvanometer to have its scale so graduated that the number of divisions on it will, by the de- flection of the needle, accurately represent the comparative strength (C) of currents which may pass through it. Let the battery, galvanometer, and resistance be joined up as at first, then, as before, C = = E R+r+ G; or, E = C (R+r+ G). Now remove R, and insert any other known resistance p, in its place. Calling the new strength of current, C₁, then C₁ E p+r+G = ; or, E C₁ (p+r+G). But we have seen that E = C(R+r+G), therefore or C(R+r+G) - C₁(p+r+G), R+r+G= == √3(p+r+G) SIMPLE TESTING. 3 that is R = C (p + r + G) − (r + G). [1] Now, as we have supposed the deflections of the galvanometer needle to be directly proportional to the strengths of current which produce them, we may, instead of C and C₁, write in our formulæ the deflections of the galvanometer needle which those strengths produce. Calling, then, a the deflection obtained with the strength C, and a, that with the strength C₁, our formula [1] becomes R = (p+r+ G) − (r + G). [2] α In order to find R, it is necessary to know G, which is usually marked on the galvanometer by the manufacturer. r also must be known, but as it is difficult to determine its value accurately, it is best to use a battery whose resistance is very small in comparison with the other resistances in the circuit, and which may consequently be neglected; in this case we may write our formula Ꭱ ; (p + G) − G. a1 α [3] Having then obtained a with R and a, with p, we can find the value of R. For example. With a galvanometer whose resistance was 100 ohms, and a battery whose resistance could be neglected, we obtained with a resistance R a deflection of 20 divisions (a), and with a re- sistance of 200 ohms (p) a deflection of 30 divisions (a,). What was the unknown resistance R? R = 30 20 (200+100) 100 350 ohms. — = 5. Next, suppose it is required to find the resistance of a galvanometer. From equation [3], by multiplying up, we find that Ra = pa₁ + G α1 by arranging the quantities Gag Ga₁-Ga Rapa₁, = B 2 4 HANDBOOK OF ELECTRICAL TESTING. or therefore α G (a₁ − a) = Rа — pа₁, Ra - par G a1 α [4] If, then, with a known resistance R, we obtain a deflection of a divisions, and with a known resistance p we obtain a deflection divisions, we can determine G. of α1 For example. With a galvanometer (G) and a battery whose resistance could be neglected, we obtained with a resistance of 350 ohms (R) a deflection of 20 divisions (a), and with a resistance of 200 ohms (p) a deflection of 30 divisions (a,). What was the resistance of the galvanometer? G = 350 x 20 200 x 30 = 100 ohms. 30 - 20 6. Lastly, when the resistance of our battery is considerable, and it is required to find its value, from equation [2] by multi- plying up, we find - Ra = pa₁ + ra₁ + Ga₁ − r a — G a, by arranging the quantities or that is ναι a, — ra = Ra – pa₁ – Gα₁ + G a, - - r (a₁ − a) = R a − p a₁ — G (a₁ — a), par For example. r = Ra - pai а1 α G. [5] With a galvanometer whose resistance was 100 ohms (G), and a battery (r), we obtained with a resistance in circuit of 300 ohms (R) a deflection of 30 divisions (a), and with a resist- ance in circuit of 150 ohms (p) a deflection of 40 divisions (α₁). What was the resistance of the battery? r 300 X 30 150 x 40 40 - 30 100 200 ohms. = 1 SIMPLE TESTING. 5 7. The formulæ may be considerably simplified if we so adjust our resistances that one deflection becomes half the other, а1 or, in other words, if we make a = Formula [3] for deter- mining any resistance then becomes that is а1 2 R = º¹ (p+G) - G = 2 p + 2 G – G. α1 2. R = 2p+ G. 8. Similarly we should find that formula [4] for determining the resistance of a galvanometer becomes G = R-2 p; and formula [5] for determining the resistance of a battery, r = R(2p+G); R being in all cases the resistance which gives the small deflec- tion, and p being the smaller resistance which doubles it. 9. When the resistance we have to measure is very high com- pared with the resistance of the galvanometer and battery used for measuring, then in our equation α1 R = º¹ (p + r + G) − (r + G), α we may practically, especially when great accuracy of measure- ment is not required, put G as well as r equal to 0, in which case, R = α P. To measure a resistance according to this formula, we should first join up, as shown by Fig. 1, our battery, galvanometer, and standard resistance, as it is called, which in our formula is p; and having noted the deflection a₁, should multiply the latter by p; this gives us what is called the constant. K (the resistance to be determined) is then inserted in the place of p; a new deflection a is obtained, by which we divide the constant, and thus get the value of R. This method of measuring resistances is the one generally employed in taking tests for insulation resistance of telegraph lines, the standard resistance p being usually 1000 ohms. t 1 6 HANDBOOK OF ELECTRICAL TESTING. When the insulation resistances of several lines are to be measured, the constant would first be taken and worked out, and the several lines to be measured being inserted one after the other in the place of the resistance p, the deflections are noted; then the constant being divided by the several deflec- tions, the resistances are thus obtained. For example. With a battery, a galvanometer, and a resistance of 1000 ohms (p) in circuit, we obtained a deflection of 20 divisions (α1), then Constant 1000 x 20 - 20000. Taking away our resistance and inserting Wire No. 1, we obtained a deflection of 5 divisions "" 2, "" 3, "9 "" 4, 6 99 "" 12 "" 3 The resistance of our wires would then be No. 1, 20000 ÷ 5 = 4000 ohms. "" 2,20000 6 6 = 3333 "" "" "" 3, 20000 — 12 = 1666 4, 20000 ÷ 3 "" 6666 "" These results are the total insulation resistances of the wires, which may be of various lengths. To get comparative results, it is necessary to obtain the insulation resistance of some unit length of each wire, such as a mile. Now, it will be readily seen that the greater the length of the wire the greater will be the leakage, and consequently the less will be the insulation resistance, or, in other words, this resistance will vary inversely as the length of the wire. To obtain, then, the insulation resistance, or insulation as it is simply called, all we have to do is to multiply the total insulation by the length of wire. Thus, for example, if No. 1 wire was 100 miles long, its insulation per mile would be 4000 × 100 400,000 ohms. It is usual to fix a standard insulation per mile, and if the result is below that standard, the line is considered faulty. 200,000 ohms per mile is the standard adopted by the Postal Telegraph Department. 10. The rule of multiplying the total insulation by the mileage of the wire to get the insulation per mile is not strictly correct, more especially for long lines, as it assumes that the leakage is the same at every point along the line. This, how- SIMPLE TESTING. 7 • ever, is clearly not the case, as a little of the current leaking out at one point leaves a smaller quantity to leak out at the next. In fact, we really measure the last portion of the line with a weaker battery than we do the first. The true law is, however, somewhat complex, and will be considered hereafter.* 11. We have hitherto considered the galvanometer deflections to be directly proportional to the currents producing them, but in no galvanometer is this the case if the deflections are measured in degrees; in such a case they are proportional to some function of those degees, such as the tangent. Thus, if we were reading off the scale of degrees on a tangent galvanometer, that is to say a galvanometer in which the strengths of currents are directly proportional to the tangents of the angle of deflection which those currents produce, we should have to find the tangents of those degrees of deflection before multiplying and dividing. For example. If with a tangent galvanometer we obtained with our standard resistance of 1000 ohms a deflection of 20°, and with the unknown resistance (R) a deflection of 15°, we should have tan 20° x 1000 •364 × 1000 R = = tan 15° • 268 = 1358 ohms. When measuring the insulation resistance of a line of tele- graph, having taken the constant, we should join up our instru- ments and line, as shown by Fig. 2. In making a measurement FIG. 2. Insulated lated * Earth Earth of this kind, it is usual to have the positive pole of the battery to earth, so that a negative (zinc) current flows out to the line, as a zinc current will show best any defective insulation in the wire, a positive current having the effect, to a certain extent, of sealing a fault up, more especially if the defect is in any under- ground work which may be in the circuit. * See Appendix. 8 HANDBOOK OF ELECTRICAL TESTING. The foregoing method of measurement is, as a rule, sufficiently accurate for all practical purposes. Greater accuracy may, how- ever, be obtained with but little extra trouble by allowing for the resistance of our battery and galvanometer in the following manner:- Instead of multiplying the constant deflection by the 1000 ohms standard resistance, multiply it by 1000 plus the resist- ance of the galvanometer and battery, and having divided the result by the deflection obtained with the line wire in circuit, subtract from the result the resistance of the galvanometer and battery. For example. With a standard resistance of 1000 ohms, a tangent galvano- meter of a resistance of 50 ohms, and a battery of a resistance of 100 ohms, we obtained a deflection of 30°, and with the line wire in circuit a deflection of 10°. What was the exact insulation resistance of the line? tan 30° (1000+ 50+100) Insulation resistance f tan 10° (50 + 100) • 577 x 1150 150 = 3760 ohms. •176 When a large number of wires have to be measured for insulation daily, it is very convenient to have a table con- structed on the following plan:- EARTH READINGS. Constant Readings through 1000 ohms. 1º 2° 30 4° 20° 20852 10423 6945.0 5205.0 21° 21992 10993 7324.6 5489.5 22° 23146 11570 7709.3 5777.9 23° 24318 12155 8099.5 6070.2 21° 25507 12750 8495.5 6367.1 In this table the first vertical column represents the deflec- tions in degrees obtained with a tangent galvanometer through a standard resistance of 1000 ohms, and the top row of degrees SIMPLE TESTING. 9 The are the deflections obtained with the line wire in circuit. numbers at the points of intersection of a vertical with a hori- zontal column give the resistances corresponding to those deflections, these resistances being calculated from the formula tan constant reading × 1000 tan earth reading Insulation resistance } Thus the constant deflection, or reading, with the 1000 ohms. standard resistance being 22°, and the deflection with the line wire (the earth reading) being 2°, the resistance required is seen at a glance to be 11,570 ohms. Before proceeding to the more intricate systems of measure- ment, we will consider some of the instruments which would be used in making measurements such as we have described. 1 10 HANDBOOK OF ELECTRICAL TESTING. CHAPTER II. RESISTANCE COILS. 12. THE essential points of a good set of resistance coils are, that they should not vary their resistance appreciably through change of temperature, and that they should be accurately adjusted to the standard units, which adjustment ought to be such that not only should each individual coil test according to its marked value, but the total value of all the coils together should be equal to the numerical sum of their marked values. It will be frequently found in imperfectly adjusted coils that although each individual coil may test, as far as can be seen, correctly, yet when tested altogether their total value will be one or two units more or less than the sum of their individual values; because although an error of a fraction of a unit may not be perceptible in testing each coil individually, yet the accumulated error may be comparatively large. · The wire of the coils is either of platinum-silver alloy or of German silver; the former material has the advantage that its resistance changes but very slightly by variation of temperature; this variation not amounting to more than ·031 per cent. per degree centigrade. Platinum-silver is, however, rather ex- pensive, and consequently, where the highest possible accuracy is not of great importance, German silver, whose percentage of resistance variation per degree centigrade is 044, is used. Recently a new metallic compound called platinoid, which is a combination of tungsten, copper, nickel, and zinc, has been dis- covered by Mr. F. W. Martino. This alloy, besides being very inexpensive, has a lower co-efficient of resistance variation by change of temperature than even platinum-silver, this percentage being as low as 021 per degree centigrade; it is therefore likely to come into extensive use. The wire is usually insulated by two coverings of silk, and is wound double on ebonite bobbins, the object of the double winding being to eliminate the extra current which would be induced in the coils if the wire were wound on single. By double winding, the current flows in two opposite directions on the bobbin, the portion in one direction eliminating the in- ductive effect of the portion in the other direction. When RESISTANCE COILS. 11 ร่ wound, the bobbins are saturated in hot paraffin wax, which thoroughly preserves their insulation, and prevents the silk covering from becoming damp, which would have the effect of partially short circuiting the coils and thereby reducing their resistance. The small resistances are made of thick wire, the higher ones of thin wire, to economise space. When bulk and weight are of no consequence, it is better to have all the coils made of thick wire, more especially if high battery power is used in testing, as there is less liability of the coils to become heated by the passage of a current though them. The individual resistances of a set of coils are generally of such values that, by properly combining them, any resistance from 1 to 10,000 can be obtained. One arrangement in general use has coils of the following values: 1, 2, 2, 5, 10, 10, 20, 50, 100, 100, 200, 500, 1000, 1000, 2000, 5000 ohms. These numbers enable any resistance from 1 to 10,000 to be obtained, using a minimum number of coils without fractional values. With these numbers, however, it is a matter of some little difficulty to see at once what coils it is necessary to put into circuit in order to obtain a particular resistance; and as it is often necessary to be quick in changing the resistances, the following numbers are frequently used: 1, 2, 3, 4, 10, 20, 30, 40, 100, 200, 300, 400, 1000, 2000, 3000, 4000, which enables any particular resistance, that is required to be inserted, to be seen almost at a glance. The way in which the different coils are put in circuit is shown by Fig. 3. The ends of the several resistances, c, c, c, are connected between the brass blocks, b, b, b, Any of the p FIG. 3. b b b C coils can then be cut out of the circuit between the first and last blocks, by inserting plugs, p, as shown, which short-circuit the coils between them; thus, if all the plugs were inserted, there would be no resistance in circuit, and if all the plugs were out, all the coils would be in circuit. 12 HANDBOOK OF ELECTRICAL TESTING. 13. There are various ways of arranging the coils in sets; one of the most common is that shown in outline by Fig. 4, and in general view by Fig. 5. This form is much used in sub- marine cable testing. The brass blocks, here shown in plan, are screwed down to a plate of ebonite which forms the top of the box in which the coils are enclosed. The ebonite bobbins are fixed to the lower surface of the ebonite top, the ends of the wires being fixed to the screws which secure the brass blocks. The holes shown in the middle of the brass blocks are con- venient for holding the plugs that are not in use. FIG. 4. B C 10 100 1000 • A 1000 100 10 S 2 2 5 10 10 20 छ ון 5000 2000 1000 1000 500 200 100 100 @aaaOBBOI E FIG. 5. ELLIOTT BRD SIRAND It will be seen that six terminals, A, B, C, D, E, F, are provided; when we only require to put a resistance in circuit, the two terminals D and E would be used. The use of the other ter- minals and of the movable brass strap S, will be explained hereafter. 14. In using a set of resistance coils, one or two precautions are necessary. First of all, it should be seen that the brass shanks of the plugs are clean and bright, as the insertion of a dirty plug will not entirely short-circuit the coil it is intended to cut out. It is a good plan, before commencing to test, to give the plugs a scrape with a piece of glass or emery paper, taking care to rub RESISTANCE COILS. 13 off any grains of grit which may remain sticking to them after this has been done. When a plug is inserted, it should not be simply pushed into the hole, but a twisting motion should be given it in doing so, that good contact may be ensured; too much force should not be used, as the ebonite tops may thereby be twisted off in extracting the plugs. Care also should be taken that the neighbouring plugs are not loosened by the fingers catching them during the operation of shifting a plug. Before commencing work it is as well to give all the plugs a twist in the holes, so as to ensure that none of them are left loose. On no account must the plugs be greased to prevent their sticking, and their brass shanks should be touched as little as possible with the fingers.. 15. For taking the insulation resistance of a line in the manner described in the last chapter, such an elaborate set of coils is not of course wanted. A single coil of a resistance of 1000 ohms in a box with two terminals, to which the ends of the coils are attached, is all that is required. FIG. 6. B C 1000 100 10 10 100 1000 10 16. One of the most useful sets of coils for general purposes is represented in outline by Fig. 6, and in general view by Fig 7. The general arrange- ment of resistances, it will be seen, is the same as that shown by Fig. 4. Two keys, however, are provided (drawn D INF in Fig. 6 in elevation, for dis- tinctness). The contact point of the right-hand key is con- nected, as shown by the dotted line, with the middle brass block of the upper set of re- sistances, the terminal B' at the end of the key corresponding A INF 200 100 140 30 20 300 400 1000 200 3000 4000 E B in fact, when the key is pressed down, with the terminal B shown in Fig. 4. In like manner the terminal A' corresponds with the terminal A. In the place of the movable strap of brass between A and D (Fig. 4), a plug marked INF. (infinity) is provided, which answers the same purpose; an infinity plug is also placed at the first bend of the coils on the right hand of the figure. When we require simply to insert a resistance in a circuit, we should use the terminals A' and E, the left-hand key being pressed down when the deflection of the galvanometer needle is to be noted. The current can thus be conveniently cut off or 14 HANDBOOK OF ELECTRICAL TESTING. put on when required, by releasing or depressing the key. Care should be taken that the two infinity plugs are firmly in their FIG. 7. kumizin ELLIGSTAS STRAND LINDUN سبع places, to ensure their making good contact. For the same purpose the key contacts should be occasionally touched with emery paper or a fine file. 4 Another set of coils, known as the "Dial" pattern, is repre- sented in general view by Fig. 8; these will be again referred FIG. 8. ELLIOTT BROS STRAND, LONDON. to hereafter (Chapter VIII). In this pattern (as will be seen from the Fig.) ten brass blocks are arranged radially around a central circular block. One disadvantage of the arrangement is that it is difficult to clean the surface of the ebonite on which the brass blocks are mounted; in a somewhat similar pattern this disadvantage is got over by substituting a rectangular bar RESISTANCE COILS. 15 for the central circular block, and arranging five of the brass blocks in a row on one side and five on the other side of the same. By this arrangement a piece of rag can easily be passed between the blocks and the central bar, and the surface of the ebonite on which the blocks and bar are mounted be readily cleaned. SLIDE RESISTANCE COILS. 17. Fig. 9 shows the principle of this method of arranging Resistance Coils. The coils, which are generally all of equal value, are connected between brass blocks, as in Fig. 3, but instead of plugs being FIG. 9. B A inserted between the blocks to cut the various coils out of circuit, a sliding piece, B, is provided which can be moved along a rod with which it is in connection. The slider has a spring fixed to it which presses against the brass blocks; it is evident, then, that any required resistance can be inserted between A and B, that is between A and a terminal fixed to the end of the rod, by simply sliding the piece B along the rod. The object of arranging the coils in this manner is more par- ticularly to enable the ratio of AB to B C to be varied, whilst the sum of the two, that is to say the whole length, A C, remains constant; this is sometimes required to be done. These coils are sometimes set in a circle instead of a straight line, the contact-piece B being a spring forming a radius of the circle. This is a very compact and useful arrangement. 18. For some tests a long straight wire of German silver or other metallic compound is employed in the place of the resistance coils. It is important that this wire should be made of a perfectly uniform alloy, and should be of the same diameter throughout, so that its resistance may be directly proportional to its length; thus, if the slider were at the middle point of the wire, the resistance on each side should be exactly the same. If, as is sometimes the case, it is required to use a long wire of this kind, it would be inconvenient to have it straight; in such a case, therefore, the wire is wound spirally on a cylinder of ebonite or other insulating material, the two ends being con- nected to the metal axes, these latter being in connection with terminals. The sliding contact-piece is moved along parallel with the axes of the cylinder by a screw which gears with the • 16 HANDBOOK OF ELECTRICAL TESTING. cylinder, and which is therefore revolved by the handle which turns the latter; the contact of the slider with the wire is made when required by pressing the former with the finger. The arrangement, in fact, is a modified form of Jacobi's Rheostat. The Thomson-Jolin Rheostat. 19. This apparatus, which is shown by Fig. 10, has been recently devised by Sir William Thomson and Mr. P. Jolin, of Bristol, and is a valuable modification of the original rheostat of Wheatstone, the apparatus being entirely free from the defects which characterised the latter instrument. FIG. 10. " In the new rheostat the wire is guided between the cylinders, so as to be laid on them spirally, by means of a travelling nut on a long screw. The screw is turned by the handle, and carries a toothed wheel which gears into two other toothed wheels ; one of the latter turns one of the cylinders, and the other a loose shaft carrying the other cylinder; a spring fixed to this shaft acts on the last-named cylinder which surrounds it on the principle of the main spring of a watch. By this arrangement the wire is kept tightly stretched and the barrels can be turned backwards or forwards without the wire becoming slack. The guiding nut is also arranged to stop the motion of the screw shaft at each end of the range, and so prevent the possibility of over-winding; it also carries an index, which moves along a graduated scale and indicates the number of turns of wire on the insulating cylinder. The conducting cylinder and the wire are both of "platinoid," RESISTANCE COILS. 17 a metallic alloy which has been recently introduced, and which has properties which make it specially suitable for the purpose. It has very high electric resistance, very small temperature variation of resistance (as has previously been pointed out on p. 10), and it remains with its surface almost, if not altogether, untarnished in the air. On account of the last-named property, the contact between the wire and the conducting-cylinder, is as perfect as can be desired; and continuity of action, which was a great difficulty in the old Wheatstone instrument, is (according to Sir William Thomson) absolutely complete. 20. It is evident that a much finer adjustment of resistance can be obtained by the slide wire than by the slide resistance coils, but inasmuch as the length of the wire and the smallness of its diameter must be limited, it does not admit of very large variations of resistance being obtained. By combining, how- ever, a slide-wire resistance with plug resistance coils, this difficulty can be got over, though in tests which we shall describe it is preferable to use the slide coils. t 21. Slide resistance coils, though very convenient, are not absolutely necessary for varying the ratio of the resistances in the manner described; for it is evident that A B and B C could be two sets of resistance coils in which, to adopt the slide resistance principle, the resistances would have to be increased in one set and diminished in the other, or vice versa, care being taken that the same resistance is added in one set as is taken out in the other. 1 C 18 HANDBOOK OF ELECTRICAL TESTING. CHAPTER III. GALVANOMETERS. 22. For the class of tests in which it is required, by adjusting certain resistances, either to bring the needle to zero, or to the same deflection in making two measurements, as described on pages 1 and 2, a galvanometer having its scale graduated to degrees would be sufficient. It should be provided with an astatic pair of needles suspended by a cocoon fibre, the end of the latter being attached to a piece of metal connected to a 11111 FIG. 11. ! screw by the twisting of which the needles can be lowered down on to the coil, so that there would be no danger of the fibre being fractured when the instrument has to be moved about. Such an instrument is shown by Fig. 11. When the galvanometer is to be used it should be placed on a firm table, and the screw connected to the fibre turned until the needles swing clear of the coil. The instrument should then be placed in such a position that the top needle stands as nearly as possible over the zero points. It should next be carefully levelled by means of the levelling screws attached to its base, until the metal axis which connects the two needles together is exactly in the centre of the hole in the scale-card through which it passes. The adjustment of the needles to zero is much facilitated in the instrument by making the coil movable about the centre of the scale-card by means of a simple handle attached direct to the coil. The final touch can thus be given without shaking the needles, which would render exact adjustment difficult. In some galvanometers there is a scale graduated to degrees attached to the coil, so that the angle through which it is turned can be seen if required. This scale is employed when using the instrument as a Sine galvanometer. GALVANOMETERS. 19 THE SINE GALVANOMETER. 23. We before stated that the strengths of currents producing certain deflections are not directly proportional to those deflec- tions, but to some function of them, such as the tangent. In measuring strengths of currents by means of a sine galvanometer we proceed as follows:- The needle is first adjusted to zero. The current whose strength is to be measured is then allowed to flow, and a deflection of the needle produced. The coil is now turned round; this causes the needle to diverge still more with respect to the stand of the instrument, but the angle which it makes with the coil becomes less the farther the latter is turned, and finally a point is reached at which the needle is again parallel to the coil-that is, its ends are again over the zero points on the scale-card. The reason of this is, that the deflective action of the coil on the needle is always the same, provided the current strength does not vary, but the farther the needle moves from the magnetic meridian, the greater becomes its tendency to return to that meridian, and finally when the needle becomes parallel to the coil, the deflective force of the latter exactly balances the reactive force of the earth's magnetism. The strength of the current which produces the deflection of the needle will then be directly proportional to the sine of the angle through which the coil has been turned. The sine galvanometer, though but rarely used, is a very accurate instrument, in so far that its results are entirely inde- pendent of the shape of the coil, size of the needle, &c. The only precaution necessary is to see that when the needle is at zero at starting it is brought back exactly to zero. Indeed it is not absolutely necessary that the starting point be zero-the law of the sines holds good if the needle be at, say, 5° when com- mencing, but in this case, by the turning of the coil, the needle must be brought back to 5°, and not to zero. THE TANGENT GALVANOMETER. 24. The tangent galvanometer, which is perhaps the most useful and convenient instrument for general purposes, consists essentially of coils of wire wound in a deep groove in the cir- cumference of a circular ring, a magnetic needle being placed at the centre of the latter over a graduated circle. The length of this needle must be small compared with the diameter of the coils so as to ensure, as far as possible, the magnetic influence of the current on the needle being the same at whatever angle the P 0 2 20 HANDBOOK OF ELECTRICAL TESTING. J needle may be with respect to the coil. Theoretically to effect this result, the magnet should be a mere point, but this is of course impossible, and practically it is sufficient for the coil to be eight or ten times as large in diameter as the length of the needle. Upon the influence of the coil on the needle being the same, whatever angle the needle takes up with respect to it, depends the truth of the proposition, that the strengths of cur- rents circulating in the coil are directly proportional to the tangents of the angles of deflection of the needles. For a 6 or 7 inch ring, a needle about three-quarters of an inch in length is a con- venient size, and gives sufficiently accurate results for all practical purposes. The needle must be so placed that its central point is at the axis of the coils and also in the same plane with them. 25. The principle of the instrument is as follows: Let n s be the needle in its normal position, i. e. the position where it is parallel to the magnetic meridian and also parallel to the ring or coils. Let n, s, be the position. FIG. 12. 14 n. α Fa Ad a n the needle takes up when deflected by the action of the coils. Draw cd at right angles to n₁ s₁ making cn, equal to n₁d; draw a c and d a, each at right angles to cd; also draw n₁ a parallel to n s and n₁ a₁ at right angles to n s. Now the position which the needle takes up is due to the fact that the deflective action of the coils and the directive force of the earth's magnetism when resolved at right angles to the needle are equal and oppo- site in effect. The first of these forces f2, acts at right angles to n 8, and the second, fi acts parallel to n s; then if an, and a₁ n₁ represent the forces fi and ƒ, respec- tively, en, and d n, will represent the resolved forces at right angles to ni s₁, which forces are equal since equilibrium is produced; let their value be f. Now since a n₁ is parallel to n o, and a c parallel to n, o, the angle c n, a is equal to the angle aº°; * also since n, a, is perpendicular to n o, and nd is perpendicular to no the angle a, n, dis equal to the angle a. We consequently have S1 S therefore f = f₂ cos a°, and, ƒ = ƒ₁ sin aº, f₂cos a fi sin aº, = * 'Euclid,' book i., prop. 34. $ 1 1. Exis f GALVANOMETERS. 21 or sin a° £2 = fi = f₁ tan a°; CUs a but fi (the directive force of the earth's magnetism) is constant, therefore f₂ (the deflective force of the coils) is proportional to tan a°, that is to say, the current strength, C, in the ring or coils is proportional to tan a°, * or C = tan a° x a constant. 26. Fig. 13 shows a form of tangent galvanometer which is used by the Postal Telegraph Department. The mag- netic needle (which is of an inch long) has a long pointer of gilt copper, about five inches long, fixed at right angles to it; when the needle is parallel to the coil, each end of this pointer is over the zero of a graduated scale. One of these scales is divided to true degrees, and the other to numbers pro- portional to the tangent of those degrees, so that if we read off two deflections from the degrees scale, the other extremity of the pointer will indicate, approximately, numbers proportional to the tangents of those two degrees of deflection. Now as the strengths of currents producing certain deflections are proportional to the tangents of the degrees of those deflec- tions, if we read off from the degrees scale we must, as we have explained in Chapter I. (§ 11), reduce the degrees to tangents, from a table of tangents, before working out a formula which has reference to the strengths of currents. If, however, we read α *Professor J. P. Joule and Professor Jack point out in vol. vi. pp. 135, 147, and 151, of the 'Proceedings of the Manchester Literary and Philo- sophical Society,' that if the needle be of a considerable length, then if a be the angle of deflection, the magnetic length of the needle (generally about of the actual length), and d the magnetic diameter of the coil, the correction to be supplied to the tangent of the angle of deflection is (4 tan² a° — 1) 12 d2 sin 2 aº, which correction is additive at great deflections, and subtractive at small ones. At a certain deflection this correction vanishes, that is to say we have or 4 tan² aº 1 = 0, tan a = 14 - = tan 26° 30'. †The exact arrangement of this instrument is described in Chapter XXVII. Table I. # 22 HANDBOOK OF ELECTRICAL TESTING. FIG. 13. ILI HUWETTI off from the tangent scale, no reduction is necessary, and the numbers can be at once inserted in the formula. & 桌 ​* 21 1 i ! 2 • J a GALVANOMETERS. 23 To avoid parallax error, in consequence of the needle being elevated above the scale, a piece of looking-glass is fixed close to the tangent scale, so that when we look at the end of the needle and see that the reflected image of the pointer coincides with the pointer itself, we know that we are looking at the end of the pointer perpendicularly with the scale. As the instrument is generally only provided with a looking- glass near the tangent scale, it is necessary when reading off from the degrees scale to run the eye along the pointer to the looking-glass end and see whether the reflected image corre- sponds with the pointer at that end; if it does, we may be sùre that when we look at the degrees scale we do so correctly. 27. Before using the galvanometer it should be seen that the pointer has not become bent, but stands at right angles to the magnet, and that when suspended it turns freely. On no account should the magnet suspension be oiled, as quite the opposite effect to what is intended will be produced by so doing. Care should be taken that the scale is in its proper position, so that when the two ends of the pointer are over the zero points, the pointer stands at right angles to the coils. The correct setting of the position of the scale with reference to the coil is a mechanical adjustment which when once properly effected is not likely to alter; it is, however, as well to verify its correctness by means of a set square before the instrument is brought into general use. The pointer attached to the magnetic needle is very subject to accident, and is often badly adjusted. The correctness of the setting can be verified by placing the galvanometer so that the pointer stands at zero, and then send- ing a current through the coil first in one direction and then in the other. The deflections on either side of zero in this case should be equal; if they are not, the position of the pointer relative to the needle should be readjusted until the required equality of deflections on either side of zero is obtained. ~ Care should be taken when making this adjustment to place the instrument clear of any unequally distributed masses of iron, otherwise unequal deflections may be obtained although the pointer and magnet are correctly set. Angle of Maximum Sensitiveness. 28. In using the tangent galvanometer it is always as well to avoid obtaining either very high or very low deflections. The reason of this is, that any small change in the strength of a current traversing the galvanometer will produce the greatest effect on the needle when the latter stands at some deflection 24 HANDBOOK OF ELECTRICAL TESTING. which is neither very high or very low. The galvanometer is, in fact, most sensitive when the needle points, under the influence of a current, at that deflection. Thus, for example, suppose we had a current which produced a deflection of 5°, and this current was increased say byth, then the deflection would be increased to 5° 30', because tan 5°: tan 5° 30′ :: 1:1. Next suppose the needle stood at 80°, and the current was, as before, increased byth, then the deflection would be increased to 80° 54', for tan 80°: tan 80° 54' :: 1:10. Lastly, let us suppose the needle stood at 43°, then by the increase in the current the deflection would have changed to 45° 43', for In the first increase was tan 43°: tan 45° 43′ :: 1 : 1· case, then, when the deflection was low, the - 5° 30' 5° = 30'; . 氮 ​1 J 1 Į in the second case, when the deflection was high, 80° 54' — 80° = 54'; and in the third case, when the deflection was of a medium value, 45° 43' 43° = 2° 43'. What, then, is the deflection at which this increase is greatest? The point to be determined is, what deflection is increased most by any small alteration in the current producing that deflection? If C be a current giving a deflection of a₁°, and C, a current a little stronger, say, which increases this deflection to (a₁°+8°), we have to find what value given to a₁°, makes as large as possible when C and C, are very nearly and ultimately equal. We have therefore C: C₁ :: tan a₁° : tan (a₁+8°), C₁ tan (a₁° +5º) = C¹ tan a₂°. i F t • [A] Now we have to make 8° a maximum, supposing that the foregoing equation holds good. J [ E E GALVANOMETERS. 25 Since do is to be a maximum, tan 8° must also be a maximum. Now tan (a₁° + 8°) tan a₁+tan 8° 1 tan a₁° tan so C₁ tan a₁°, therefore α tan a₁° + tan 8° = tan a₁° (1 — tan a₁° tan 8°), therefore tan 8° (1+ C tan³α) = tan o̟ C₁ 510 (& − 1), therefore tan ai (-1) 1 C tan 8° = do [B] C₁ 1 C₁ 1+ tan2 a₁° + tan a₁ tan a C We have then to find what value of tan a, makes this fraction a maximum, and this we shall do by finding what value makes the denominator of the fraction a minimum. 1 tan ai O 1 Now 2 /C₁ + O tan a = (√tana,√tana,°) +2 √. C ai and this will be a minimum when 1 1 C₁ tan a₁ = 0, √tan a₁ that is, when C₁ 1 = tan² a₁0, or, tan a₁ = becomes equal to 1, but as C, and C are ultimately equal, therefore tan a₁° = √√1 = 1 = tan 45°. 29. We see then that in order to make the tangent galvano- meter as sensitive as possible we should obtain the deflection of its needle as near 45° as possible; 45° is in fact the angle of maximum sensitiveness. Every galvanometer has an angle of maximum sensitiveness, 26 HANDBOOK OF ELECTRICAL TESTING. although it is not the same in all. The angle can, however, be found experimentally (see 'Calibration of Galvanometers,' p. 46), and should be marked on the instrument for future reference. 30. If we require to adjust two currents in two different measurements so that they should be equal in both cases, it is evident that the needle of the galvanometer employed to measure them should in each case show the same deflection. In making the two measurements, we take the deflection obtained by one current as the standard, and then in making the second measurement we adjust the current until the same deflection is obtained. Now the accuracy with which this current can be adjusted depends upon the sensitiveness of the galvanometer to a change in the strength of the current, and we have seen that this sensitiveness is at a maximum when the deflection is 45°. If, therefore, we employ a tangent galvano- meter for such a test as that just mentioned, we should endea- vour in both measurements to bring the needle to 45°. 31. In what way can the property of the galvanometer be taken advantage of when comparing two deflections? We must in such a case endeavour to obtain both deflections as near 45° as possible. To do this we should have to get one deflection on one side, and the other deflection on the other side, of 45°. But then the question arises, should we get the deflections at an equal distance on either side, or one closer to the 45° than the other, and if so, should the higher or the lower deflection be the closer of the two? Now a little consideration will make it clear that if the two deflections in question are taken either near 0° or 90°, they will be much closer together than if they were taken near 45°, for the reason that the tangents of high or low deflections differ more widely from one another than do the tangents of medium deflections. But we have shown that when deflections are high or low, any increase or decrease in the strength of the current producing those deflections has less effect than when the de- flections are of a medium value. It is therefore evident that it is most advantageous to get the deflections as wide apart as possible. Let then tan e° represent the stronger, and tan øº the weaker current, and let one current be n times as strong as the other. We then have to find what values of 0° and 4° make a maximum, supposing that 0° — ø° tan 0° = n tan 4º. GALVANOMETERS. 27 If in the last investigation we substitute e°— 4° for 8°, 4° for a₁, and n for C₁ we can see that in order to get the required result we must make 1 tan ø° √ ñ and, since tan 0° = n tan 4º, n tan 0° = n. n If one current strength is to be twice as great as the other, then n = 2; consequently, and tan 0° = √ 2 = 1·41421 = tan 54° 44′ tan º = 1 √2 tan 5420, • 70711 = tan 35° 16′ = tan 351°. These then are the deflections that theoretically it is best to obtain in making a test with a tangent galvanometer in which one current is to be twice as strong as the other. But practi- cally we may make the deflections 55° and 3510, as these are more convenient to adjust to, and tan 55° is, within 1', exactly double tan 350. If we examine the theoretical deflections 54° 44' and 35° 16' it will be seen that and 54° 44' 45° = 9° 44', 45° 35° 16′ = 9° 44′, - or in other words, the angular deflections on either side of 45° are in this case the same. Let us then see whether they are so when n has any value other than 2. The angular deflection between 45° and 0° will be that between 45° and 4º, 0° - 45°, 45° — þ³, tan 0° 1 now tan (0° - 45°) 1 +tan 0° 1 and tan (45° - ¢°) = tan 6° ; 1 + tan 6° 28 HANDBOOK OF ELECTRICAL TESTING. but we know, since that tan 0° = √n and tan º tanp = 1 tan 0° 1 n that is 1 1 tan 0° tan 0° - 1 tan (45° — p³) 1 1 + tan 0° 1+ tan 0° that is or tan (45° — 4º) = tan (0° — 45°), 45° — ° = 0° - 45°; − 0° showing that these angular deflections are the same whatever be the value of n. This is a very useful fact, as it shows that when we are making a test in which two deflections are involved whose relative values are unknown, we should so adjust the resist- ances, &c., that the deflections are obtained, as near as possible, at equal distances on either side of 45°. To sum up, then, we have Best Conditions for using the Tangent Galvanometer. 32. When a test is made in which only one deflection is con- cerned, then that deflection should be as near 45° as possible. If there are two deflections to be dealt with, then these should be as nearly as possible at equal distances on either side of 45°. If one of these deflections is to be double the other, then 55° and 35° are the most convenient to employ. 33. Although it is usual to take the readings on the tangent galvanometer, starting with the pointer at the ordinary zero, i. e. with the needle parallel to the plane of the ring or coils, yet it is not absolutely necessary that this arrangement should be adopted; the instrument can be used when the needle in its normal position makes an angle with the plane of the ring. Under the latter conditions, however, the current strength will not be in direct proportion to the tangent of the angle. of deflection. 1 2 f GALVANOMETERS. 29 FIG. 14. Let the dotted line A B, Fig. 14, represent the plane of the coils, and let n s be the needle in its normal position, i. e. in the plane of the magnetic meridian; also let n₁s, be the position which the needle takes up under the influence of the current. Let ß° be the angle which the needle makes normally with the coils, and let a° + ß° be the angle through which the needle turns when deflected to the position n₁ 81. 1 Draw c d at right angles to n₁ 81, making c n₁ equal to n₁ d; draw ca and da, each at right angles to c d; also draw n₁ a parallel to n s, and n₁ a at right angles to AB. Now since a n is parallel to n o, and a c parallel to n₁ o, the angle ca n₁ is equal to the angle a +6°; also since n, a₁ is perpendicular '; to A o, and n d is perpendicular to no, the angle a₁ n₁ d is equal to the angle aº. We consequently have therefore or . $1 N A B f = fs cos a°, and f = fi sin (a° + ß°), fs cos a° = fi sin (a° + ß°), = f₁ £3 = f₁ sin (a° + ß°) * COS a sin a' cos ẞ° + sin ß° cos aº COS a = fi (tan a° cos p°+ sin p°) a [A] = ficos 6º (tan a° + (ta sin Bo cos Bo =ƒ₁ cos ß° (tan a° + tan 6º). so that cos B° being a constant quantity, the strength of a current is directly proportional to (tan a°+ tan 6º) which is the reading on the tangent scale (§ 26) if the figures on the latter * If the angle 6° had been on the right instead of the left hand side (as in Fig. 14) of the coils A B, the angle a° still being the angle A on₁, then we should have had sin (aº — ߺ) COS o * £3 30 HANDBOOK OF ELECTRICAL TESTING. T [ ! 2. are re-arranged so that the zero is at the division at which the needle points in its normal position. Fig. 15 shows a scale so re-arranged, the new figures being additional to the old ones; such a scale has been adopted in the tangent instruments used for testing purposes in the Postal Telegraph Department. FIG. 15. 90 0 100 110 120 130 140 150 160170180190 10 20 30 40 50 60 70 8020ION 0030 40 50 60 70 8 1000807060 5040 30 20 10 T 2 ļ 1 60 50 40 30 20 0 10 20 30 40 50 7080 I | I I Apart from the fact that the adoption of the foregoing "skew method of using the tangent galvanometer, gives an increased range to the instrument, a considerable increase of sensitiveness in the case of high deflections is also obtained by it, i. e. a current which would move the needle of the instrument through a given angle from the old zero, will move it through a much larger angle from the new or "skew" zero. zero. This, however, is only the case if the first angular deflection in question (the one from the old zero) exceeds a certain value, if it is less than this value, then the deflection for a given current will be less from the skew than from the old zero. Let go be the angular deflection obtained with a given current when the needle is deflected from the ordinary zero, then f = fi tan go; but if the needle had been at the skew zero, then with the same current we should have had f=f, sin (a²+fº) 1 cos a # h 2 J 1 1 } & 1 GALVANOMETERS. 31 • therefore tan co Suppose we have p° = sin (a° + ߺ) COS a 60°, and suppose the current to be of such a strength as to turn the needle through an angle of 120°, then in this case a° = 60°, and we consequently have sin 120° tan go = but sin 120° = sin (180° - tan co = 60°) cos 60° sin 60° cos 60° = tan 60°, ; = sin 60°, therefore, or °ޏ 60°; that is to say, the angle through which the needle would have been turned when the zero was 60° to one side of zero, would be twice what it would be if it had been deflected from the ordinary zero. The relative values of the deflections, with a given current, from the ordinary and from the skew zero, approach nearer to an equality in proportion as the deflections become smaller; at a certain point they become equal, and then the relative values become reversed, i. e. for the same current the deflection from the skew zero becomes less than the deflection from the ordinary zero. Let us determine at what point the deflections from the two zeros become the same. We have in this case sin (a° + B°) COS a sin a = tan a° o COS a therefore α sin (a° + ß°) = sin a°; if now the angle a° is negative, that is to say, if the angular deflection from the skew zero is less than the angle ẞ°, then we have or that is or sin (B° – a°) = sin a, ß° — a° = a°‚ ß° = 2 a°, a = B° To 12 32 HANDBOOK OF ELECTRICAL TESTING. that is to say, whatever be the angle 6° (the angular distance of the skew from the ordinary zero) then a current sufficient to move the needle a distance of from the ordinary zero would B° 2 move the needle the same distance from the skew zero. If the Bo 2 deflection from the old zero be less than then the deflection from the skew zero will be less still, so that there is no advantage in the use of the skew zero unless the deflections exceed B° 2 From what has been proved, it is obvious that the greater we make ẞ° the greater will be the deflection obtained with a given current, but there is a practical limit to increasing ẞ°, for the larger we make the latter the more does the deflective action of the coil tend to act in a direction parallel but opposite to the earth's magnetism, the consequence being that the resultant of the two forces is a comparatively small quantity, and the friction of the pivot, &c., prevents the needle from settling down to the true angle representing the force of the current. Under such conditions large errors in the readings may result. Were it not for this fact the instrument would increase in actual sensitive- ness up to the point at which ß° = 90°, at which point the needle would not move unless acted upon by a current exceeding in deflective force the intensity of the earth's magnetism; when the current exceeded this value the needle would swing com- pletely round through an angle of 180°. 34. What is the angle of maximum sensitiveness in the case of a tangent galvanometer with a skew zero? Referring to page 24, it is obvious, since the current strength is in proportion to tan atan 8°, that equation (A) on the page referred to becomes or tan (a₁° + 8°) + tan 6º C₁ C¹ (tan a° + tan 6º), 1 tan (q₂° + 8º) = ¦¹ (tan q¸° + tan ƒº) — tan ߺ. Now we have to make 8° a maximum, supposing that the fore- going equation holds good. Since 8° is to be a maximum, tan & must also be a maximum. Now i " [ t # 7 } } [ 拳 ​tan (a₁ + 8°) = tan a₁° + tan & 1 - tan a₁ tan d po, + tan B°) – tan 6, O² (tan o₂° ļ 1 GALVANOMETERS. 33 therefore tan a₁+tan 8° = [G² (tan a¸° + tan pº) — tan p°] ß° [1 − tan a,° tan 8°], therefore & tan 8 [1 + C¹ tan a₁ (tan a₁ + tan ẞ°) tan a tan 6° therefore tan 8° C₁ (tan a₁° + tan pº) (¦¹² − 1), (tan a₂° + tan 6º) C₁ (G¹² – 1 1 — tan a₁° tan ߺ + tan a₁ (tan a₁ + tan p°) 510 1 1 tan đi tan Bo + gta tan a₁. tan a₁ + tan ẞ° We have then to find what value of tan a₁° makes this fraction a maximum, and this we shall do by finding what value makes the denominator of the fraction a minimum. Let tan a₁° =¡a, then we have to determine what value tan ß° = b, and C₁ C = K, of a makes 1 — a b +ка a + b a minimum. Now 1 – ab a + b + Kα = K(a + b) 1 К '1 + b² +b²72 K a+b + 2 √ k (1 + b²) − b (x + 1), and this will be a minimum when 1 + b² 1 - a+b K = = 0, D 34 HANDBOOK OF ELECTRICAL TESTING. that is, when ! 1 + b² a+b= = K I but as C, and C are ultimately equal, C equal to 1, therefore, therefore therefore therefore that is or therefore or a+b= √1 + b², a²+b²+2ab1+ b², tan Bo 2 a b = 1 — a³, 1 — a² b = 2 a 1 - tan² a₁ 2 tan a₁ that is κ, becomes Y 1 = cot 2 a₁°, cot (90° — ẞº) = cot 2 a₁°, - p°) 90° – ß° = 2 a,° a1° = 45° B° N! Since B cannot be greater than 90°, or less than 0° (unless it has a negative value), we see that a must lie between the ordinary zero and 45° from it. In the case of a galvanometer where B° 60°, we have = 60° a1° = 45° = 15°, 2 that is (60° +15°), or 75°, from the skew zero. 35. In order that a tangent galvanometer when used in the ordinary way may give accurate results, it is obviously necessary that the magnetic needle, or rather the magnetic axis of the same, be strictly parallel to the magnetic plane of the coils, that is to say, the angle 6° must be equal to nothing. When the latter is the case, the angular deflection for a given current should be the same to whichever side of zero the needle is deflected. If it is found that these deflections are different, we can deter- mine from the two results what is the magnitude of the angle ་ V * ર GALVANOMETERS. 35 . p° (Fig. 14, page 29). Referring to this figure, let e° be the angular movement of the needle from its position of rest, then or 0₁° = a° + ß°, a° = 0₁° – ß° ; therefore from equation (A) (page 29), we can see that in this case fs = fi sin 0₁° cos. (01° — ẞ°)* - ß°)° If the same current is now sent in the reverse direction, and the angular movement of the needle from its position of rest is 02, we have ƒ3 = fi therefore sin 01° cos (01° - B°) sin 01° therefore cos e₁° cos ß° + sin ₁° sin ß° sin e cos (☺₂° + ß³)' sin 02 cos (0₂° + B°)' sin 0,0 cos e cos ߺ — sin 02° sin ß° = ẞo cot e₂ cos B° - sin ß°, therefore cot e₁° cos p° + sin p° therefore that is cot e₁°+tan ẞ° = cot 0° - tan ߺ, or 2 tan 6° = cot 0₂° - cot 0₁°, cot e₂° B° = tan - 1 cot e₁° 2 To make the instrument read correctly the graduated dial plate would have to be turned round through the angle ß, in the direction in which the needle moved when the largest of the two deflections was obtained; the zero point will then be correctly set, and the tangent of the angle of deflection taken from this zero will represent directly the current strength. When the needle is provided with a pointer, the simplest method of making the correction is to bend the pointer as ex- plained in § 27 (page 23), until equal readings are obtained, with the same current, on both sides of zero. A D 2 36 HANDBOOK OF ELECTRICAL TESTING. 36. A form of tangent galvanometer, which is in very general use for lecture and educational purposes, is shown by Fig. 16. This instrument is known as Gaugain's galvanometer, though actually, it is a modification by Helmholtz of the original FIG. 16. י P instrument of Gaugain. It was pointed out by the latter, that if the magnetic needle were suspended, not at the centre of the coil, but at a point on the axis at a distance from the centre equal to half the radius of the coil, then the chief error due to the magnetic needle not being infinitely short, disappears. Helmholtz improved upon this arrangement by placing a second coil, similar to the first, at an equal distance on the other side of the magnet; by this means, the error due to the centre of the magnetic needle not being truely at the point indicated by Gaugain, is got rid of. In order that the ratio between the diameter of the coil to its distance from the centre of the magnet may be preserved with reference to every turn of which the coil is composed, these turns should be wound on a conical surface as in the instrument shown by Fig. 16. It is pointed out by Clerk Maxwell, however, that such a method of winding * ' Electricity and Magnetism, by J. Clerk Maxwell,' vol. ii. p. 318. * t GALVANOMETERS. 37 is quite unnecessary, as the conditions may be satisfied by coils of a rectangular section, which can be constructed with far greater accuracy than coils wound on an obtuse cone. OBACH'S GALVANOMETER. 37. In this galvanometer, which is shown by Fig. 17, the ring instead of being fixed as in the ordinary tangent instru- ment is movable about an horizontal axis; by this means the FIG. 17. !!! deflective action of the ring on the needle can be reduced from the full effect (when the ring is in the usual vertical position) down to zero (when the ring is in an horizontal position), so that the instrument has a very wide range, a range which in practice is 100 times as great as that of an ordinary tangent 38 HANDBOOK OF ELECTRICAL TESTING. * galvanometer, thus enabling either weak or very powerful currents to be measured. The effect of setting the ring at an angle to the vertical position is as follows:- FIG. 18. In Fig. 18, let a b be the vertical position of the ring, and a₁ b₁, the latter when inclined at an angle 4°. Draw a₁c at right angles to a₁ b₁, and a₁d at right angles to ab, then the angle ca, d equals the angle °. a 4° a, а1 Now if c a₁, that is f, represents the mag- netic force of the ring when the latter is traversed by a current, this force being at right angles to the ring, then a₁d, that is fa, will be the resolved force at right angles to the vertical a b. We have then 3) or £2 £3 = sec 4º, ƒ2 = ƒ3 sec 4° ; that is to say, the magnetic force of the ring is equal to its deflective force on the needle multiplied by the secant of the angle at which the ring is set. But the magnetic force of the ring is in direct proportion to the current strength, and the resolved deflective force is in direct proportion to the tangent of the angle of deflection (a°) of the needle of the instrument. Hence the strengths of currents circulating in the ring are directly proportional to the tangents of the angles of deflection of the needle multiplied by the respective secants of the angles of inclination of the ring; or we may say C = tan a° x sec × a constant. It must be obvious that there are several ways in which the instrument can be used. In the first place it can be made use of as an ordinary tangent galvanometer, the ring being set at such an angle as would cause the deflections obtained to be brought as nearly as possible in the neighbourhood of 45° (the angle of maximum sensitiveness); the current strengths in this case would of course be directly proportional to the tangents of the angles of deflection. Again, the ring could be moved so that the same deflection of the needle is obtained with each current being measured; in this case, the current strengths will of course be directly pro- portional to the secants of the angles at which the ring had to be set in the different cases. Inasmuch as the adjustment of 1 } Į GALVANOMETERS. 39 the position of the ring is dependent upon the observation of the movement of needle, it is best to arrange that the latter shall point as nearly as possible at the angle of maximum sensitiveness, i. e. at 45°. The "equality" method of using the instrument consists in moving the ring until it is found that the angular deflection of the needle, and the angle through which the coil has been turned are the same; in this case we get C = tan 4° x secx a constant. As only a single angle has to be dealt with for a particular measurement, the products of tangents and secants can be calcu- lated beforehand and embodied in a table. In the ordinary tangent galvanometer, the deflective action of the ring acts in the same plane as that in which the needle turns; but in the Obach instrument, the deflective force, being at an angle with this plane, tends to make the needle dip when the ring is inclined. In order to avoid this tendency, the FIG. 19. Ba n Aq arrangement shown by Fig. 19 is adopted. The needle ns is fixed near to the upper end of a thin vertical axle a b, the lower end of the latter being provided with a cylindrical brass weight w. This weight offers but little additional momentum to the whole system round the vertical axis, whilst the movement 40 HANDBOOK OF ELECTRICAL TESTING. 7 round the horizontal axis is completely prevented. The alu- minium pointer p q, is situated in the same plane as the scale; the ends are flattened and provided with a fine slit, which serves as an index for reading the deflections; the bottom of the box in which the needle turns being blackened, the reading can be taken without parallax, and therefore very accurately. The magnetic needle ns, has a biconical shape, which entirely pre- vents the shifting of the magnetic axis from its original position, as was sometimes found to be the case with the old broad needles. Adjustments are provided by which the cocoon fibre f, serving to suspend the needle, can be raised or lowered, as well as accurately centred. In order to damp the oscillations of the needle, a shallow, cylindrical box, about 8 centimetres in diameter, and 11 centi- metres deep is provided; this box has two radial partitions which can be slid in or out; the axle of the needle passing through the centre of this box, carries a light and closely fitting vane. By sliding the partitions more or less into the box, various degrees of damping can be obtained; and if they are right in, the motion is practically dead beat. The scale over which the needle turns is provided with degree and also with tangent divisions. The scale fixed to the ring enables the inclination of the latter to be read to th of a degree; this scale is also engraved with secant divisions, so as to avoid the necessity of reducing the degrees to secants by means of a table. In order to enable the "constant" of the instrument, i. e. the deflection due to a given current, to be made the same at any place when the instrument is being used, an auxiliary magnet (seen in the figure) is placed at the side of the instrument; this magnet can be turned round an horizontal axis passing through its neutral point and the centre of the needle, and is at right angles to the diameter on which the ring is turned. This magnet does not affect the zero position, and moreover, if placed exactly vertical with its magnetic axis, it does not alter the original constant, which then only depends upon the horizontal terrestrial component, more or less modified by the surroundings; but if it is dipped, the horizontal force acting on the needle is either augmented or diminished, according to the direction in which the magnet is turned, and to the amount of dip given. The ring of the instrument, it should be mentioned, is of gun- metal, and serves for the purpose of measuring strong currents, whilst fine wire wound in a groove in the ring enables weaker currents also to be measured. The relative values of the deflective actions of the ring and of the fine wire upon it, are T 1 1 J E 1 1 GALVANOMETERS. 41 so adjusted that a current of 1 ampère through the ring gives exactly the same deflection as an electromotive force of 1 volt at the terminals of the fine wire. Since we have C = tan a° x sec 4° x a constant α we can easily see if any particular instrument is properly made and the scales correctly graduated; for if we pass a constant current through the ring, and set the latter at different in- clinations, then the products of the secants of the angles of inclination of the ring and the tangents of the corresponding angles of deflection obtained, should be the same in every case. METHOD OF READING GALVANOMETER DEFLECTIONS. 38. The reading of galvanometer deflections requires con- siderable method, in order that accurate results may be obtained in making measurements. Let A and B (Fig. 19A) be two contiguous division marks on the galvanometer scale. Now, by observation, we can always determine without difficulty whether the pointer lies exactly FIG. 19A. 2 1 BA BA BA 1111 Deflection=A. Deflection=A1. Deflection=A}. Deflection = A3. over A or over B, or whether it lies exactly midway between the two; and further, if it does not occupy either of these exact positions, we can judge without difficulty whether it lies nearest to A or to B. This is equivalent to saying that we can be certain of the magnitude of the deflection within a quarter of a degree. Thus, supposing the pointer stood between A and B, but nearer to A than to B, then we should call the deflection "Aд,” and supposing the deflection was actually very nearly 42 も ​HANDBOOK OF ELECTRICAL TESTING. equal to A, then A would be a quarter of a division, or degree, too much; if, on the other hand, the deflection was very nearly equal to Až, then A would be a quarter of a division, or degree, too little. In one case the error would be a plus one, and in the other a minus one, but in either case its maximum value would be only. We have, in fact, the rule that-if A be the smaller of two contiguous deflections A and B, then when the pointer is exactly over A, the deflection should be called "A"; if nearer to A than to B, then it should be called "A"; if exactly midway between A and B, it should be called "A"; and lastly, if the pointer is nearer to B than to A, then the deflection should be called "A3"-; thus, for example, if A and B (Fig. 19A) were the 57° and 58° division marks respectively on the scale ; then in case 1 the deflection would be taken as 57°, in case 2 the deflection would be taken as 571°; and again, in cases 3 and 4 the deflections would be taken as 5710 and 572° re- spectively. By keeping to these instructions, then, we can be sure of the magnitude of a deflection within 1 of a division or degree. 4 39. If we are making a measurement with a tangent galvano- meter and we read from the degrees scale, and if we have two deflections to deal with, one of which is to be a proportional part of the other (usually one-half), then after the first deflec- tion has been observed it has to be reduced to a tangent,* and then the latter being divided, say, by two, the corresponding deflection is ascertained from the tangent table; the resistances, &c., are then adjusted till the required second deflection is as nearly as possible obtained. If we find that the halved tangent does not exactly correspond to a deflection in the table, then we must take, say, the nearest deflection below the exact value, and then take care to adjust so that the deflection of the pointer is a little above that angle. Thus suppose the first deflection to be 58°, then the tangent of 58° is 1.6003, and = ·8001; 1.6003 2 now the nearest number below this in the table is 7954, which is the tangent of 3810; in adjusting the deflection, therefore, we should take care that we get it rather more than 3810. Degree of Accuracy attainable in reading Galvanometer Deflections. 40. If the galvanometer scale be so graduated that the number of divisions of deflection directly represent the pro- portionate strengths of the currents producing those deflections, * Table I. * " [ 每 ​GALVANOMETERS. 43 m then an error of, say th of a division in d divisions will represent a percentage error, y, in the strength of the current represented by d, which is given by the proportion 1 y::: 100: d, or y = m × 100 d per cent. [A] Y。: tan d¹º m If, however, the instrument be a tangent galvanometer and the deflection be read from the degrees scale, then an error of in d° will not represent an error of × 100 1º m 10 m in this case we must have the proportion d° tan d° :: 100: tan do, per cent., for or Y% = (tan d 10 tan d°) 100 ´tan d²° 10 m tan do tan do 1) 1 1100 per cent. [B] For example. If the deflection d were 46 divisions, then of a division error () would be an error, y, of Y × 100 46 = ·54 per cent. in the current strength represented by the deflection d; but if the deflection were 46°, then 1° error would be an error, yo, Y%= tan 461° tan 46° - - 1) 100 in the current strength. G of 1·0446 1 1.0355 1) 100 = •88 per cent. 41. In cases where we have two deflections to deal with, one of which, or the tangent of one of which, has to beth (usually) of the other, then after we have ascertained, as accurately as we can judge, the magnitude of the first deflection d, the latter (or the tangent of the latter) is divided by n, and then the resist- ances, &c., in the circuit of the galvanometer are adjusted until the deflection (or or the deflection corresponding to - N tan d n d) is obtained as accurately as possible. Now in adjusting to this 44 HANDBOOK OF ELECTRICAL TESTING. latter deflection we are liable to make a plus or minus error 1 d of th of a division or degree as in the first case, and as n (or tan-2) 1 m may itself contain an error due to d being th of a division or degree wrong in the first instance, the new deflection may be more than th of a division or degree out. What then is the "total possible percentage of error which may exist in the second deflection " m "? Now the absolute error which may be made in the two deflec- tions must be the same in both cases, viz., but the percentage value of the latter will be directly proportional to the value of the deflections; thus a division error in 50 divisions is a per cent. error, but a division error in 25 divisions is a 1 per cent. error; in fact, if y be the percentage error (corresponding to the absolute error) in d divisions, then ny will be the percentage error (corre- d 1 1 sponding to the absolute error in divisions. Now if d con- m → d N tains a percentage error then must also contain a percentage , n error y; consequently if we make a percentage error of n y when d already contains a percentage error contain a total percentage error, T, of T=y+ny = y (1 + n)* ; لا d in d n then must ท I or since we get T = × 100 γ d X 100 d (1 + n). [C] * Strictly speaking this is not absolutely correct, for it assumes that the second percentage should be calculated on on 1 d+ also. n m d - > whereas it ought to be calculated n 1 but as is small compared with d, the consequent error is small m 7 5 GALVANOMETERS. 45 (1) For example. m If d and were 58 divisions and division, respectively, and further, if the deflection d had to be halved, that is, if n = 2, then we should get T = 1 × 100 58 × 3 = 1.3 per cent. If we have to deal with degrees of deflection instead of divisions, then in the case of a tangent galvanometer we should have To = tan d - 1) 100+ tan do 1 (tan di – 1 ) 100 - - 1) = - ´tan d¹ tan d₂10 m 1m + 2 2) 100, [D] tan d° tan do where tan d° tan dio n (2) For example. If d,, and n, were 58°, 10, and 2, respectively, then we should have 1.6003 tan d₁o = •8001 (= tan 3810), 2 therefore 1.6160 To per cent. • 8026 + 2) 100 = 1.7 1.6003 ⚫7954 It may be pointed out that this last example shows the possible percentage of error which may occur when making a halved current test with the tangent galvanometer under the best possible conditions. Practically, therefore, we may say that under no possible conditions could the deflection error in a halved current test be regarded as being less than 12 per cent. As will be seen when we come to consider such tests, other sources of error are met with which still further reduce the degree of accuracy with which the tests can be made. 42. Although in formulæ [B] and [D] the function of the deflections has been taken as the tangent, yet the formulæ apply equally well in cases where the current strengths are propor- tional to any other function of the deflections. 46 HANDBOOK OF ELECTRICAL TESTING. CALIBRATION OF GALVANOMETERS.* 43. The deviations in degrees of the needle of a galvano- meter which is not of the tangent form are not generally pro- portional to any simple function of those degrees, yet it is easy to determine the relative values of the deflections in terms of the currents which would produce them, that is, to calibrate the scale. In order to do this, it is simply necessary to join up in circuit with the galvanometer, a battery, a set of resistance coils, and also a galvanometer, the values of whose deflections are known (a tangent galvanometer, for example). This being done, and the galvanometers being set so that their needles are at zero, we insert sufficient resistance in the circuit to reduce the deflection in one of the instruments to 1°, and then by means of a "shunt" (Chapter IV.) we also reduce the deflection of the needle of the second galvanometer to 1°. We now reduce the resistance in the circuit step by step so as to produce deflections of 1º, 2º, 3º, 4°, &c., from the needle of the galvano- meter whose scale is required to be calibrated. As each deflec- tion is obtained we observe and note the corresponding deflection on the tangent instrument. When the whole range of the scale (or as much of it as is considered necessary) of the instru- ment under calibration has been gone through, we can construct a table for use with it by writing down opposite the various degrees of deflection the tangents of the deflections which were obtained on the tangent instrument and which corresponded to the deflections in question. The table so constructed would be used precisely in the same way as would the table of tangents in the case of a tangent galvanometer, the use including, it may be remarked, the determination of the percentage value of an error in a deflection. It may also be remarked that the angle of maximum sensitiveness would be the deflection which was obtained when the needle of the tangent instrument pointed to 45°. THE THOMSON GALVANOMETER. 44. The accuracy with which measurements can be made depends chiefly upon the sensitiveness of the galvanometer employed in making those measurements. The Thomson re- flecting galvanometer supplies this requisite sensitiveness, and is the instrument which is almost invariably employed when great accuracy is required, and also when very high resistances have to be measured. * See also p. 76, § 73. i L GALVANOMETERS. 47 Description. 45. The principle of the instrument is that of employing a very light and small magnetic needle, delicately suspended within a large coil of wire, and of magnifying its movements by means of a long index hand of light. This index hand is obtained by throwing a beam of light on a small mirror fixed to the suspended magnetic needle, the ray being reflected back on to a graduated scale. This scale being placed about 3 feet distant from the mirror, it is obvious that a very small angular movement of the mirror will cause the spot of light reflected on the scale to move a considerable distance across it. The needle being very small, and being placed in the centre of a large coil, the tangents of its deflections are approxi- mately directly proportional to the strength of the currents producing them. D In Fig. 20, let L be a lamp which throws a beam upon the mirror m, which has turned through a small angle, and o reflected the beam on the scale at D. Let d be the distance through which the beam has moved on the scale from the zero point at L, and let 7 be the distance between the scale and a FIG. 20. P TP the mirror. Now the angle through which the beam of light turns will be twice the angle through which the mirror turns; this is clear if we suppose the mirror to have turned through 45°, when the reflected beam will be at 90°, or at right angles to the incident beam. If, then, we call a the angle through which the beam of light turns, will be the angle through α 2 а1 a2 which the mirror will have turned. Let and be the two 2 2 angles through which the mirror has been turned by two currents, of strengths C₁ and C₂ respectively, then C₁: C₂ :: tan а1 2 : tan 22 ; 2 therefore C₁: C₂ :: √1 + tan³ α, − 1. √1 + tan² α, 1 tan đi : tan az √1+ tan² being positive, as the angles are less than 90°. 48 HANDBOOK OF ELECTRICAL TESTING. 7 being the distance of the scale from the mirror, let d, and d₂ be the distances traversed on the scale by the beam of light, then d₁ d₂ tan α₁ = tan a₂ יך therefore therefore d₂2 1 + 1 1 + - 1 72 12 C₁: C₂ :: d₁ d2 2 2 C₁ : C₂ :: d₂ (√ F³ + d₁² − 1) : d₁ (√ T² + d²² − 1) ; when d₁ and d₂ do not differ largely, then we may take C₁: C₂ :: d₁: da; For but when this is not so a small error is observable. instance, suppose d₁ = 50, and d₂ = 300. According to the last formula this would show that one current is just six times as strong as the other, but by the correct formula, taking l divisions (which would be about its value), we find that C₁: C₂:: 300 (√1500² + 50º — 1500): 50 (√1500² + 300² – 1500), that is or C₁: C₂ :: 250 : 1485, 2 C₁: C₂ :: 50: 297; = 1500 so that when extreme accuracy is required we cannot take the strengths of currents as being exactly proportional to the number of divisions of deflection on the scale. The galvanometer, as usually constructed, consists essentially of a very small magnetic needle, about three-eighths of an inch long, fixed to the back of a small circular mirror, whose diameter is about equal to the length of the magnet. This mirror, which is sometimes a plano-convex lens, of about six feet focus, is suspended from its circumference by a cocoon fibre devoid of torsion, the magnetic needle being at right angles to the fibre. The mirror is placed in the axis of a large coil of wire, which completely surrounds it, so that the needle is always under the influence of the coil at whatever angle it is deflected to. A beam of light from a lamp placed behind a screen, about three feet distant from the coil, falls on the mirror, and is non s 1 FIG. 21. TERRAGEN ་་ Front Elevation. real size. E 50 HANDBOOK OF ELECTRICAL TESTING. reflected back on to a graduated scale placed just above the point where the beam emerges from the lamp. The scale is, as we have before said, straight, and is usually graduated to 360 divisions on either side of the zero point. It is not absolutely necessary that the working zero be the middle or zero point of the scale; it is a very common practice to adjust the instrument so that the reflected beam of light normally falls near the end of the scale; by this adjustment an extreme range of 360 × 2, or 720 divisions can be obtained. 46. The Thomson galvanometer is made in a variety of forms: Fig. 21 gives a front, and Fig. 21A a side elevation (with glass shade, &c., removed) of one very common pattern. FIG. 21A. | SALTAN FEAT. Side Elevation. (Shade removed.) real size. It consists of a base formed of a round plate of ebonite, provided with three levelling screws; two spirit-levels, at right angles to one another, are fixed on the top of this plate, so that the whole instrument can be accurately levelled: sometimes one circular level only is provided, but the double level is much the best arrangement. From the base rise two brass columns, between which a brass plate is fixed, rounded off at the top and bottom. Against the faces of this plate are fixed the coils (c, c, c, c) of the instrument. 1 $ J GALVANOMETERS. 51 The brass plate has shallow countersinks on its surface for the faces of the coils to fit into, so that they can be fitted in their correct places without trouble or danger of shifting. Round brass plates press against the outer surfaces of the coils by means of screws, and keep them firmly in their places. There are two round holes in the brass plates coinciding with the centre holes in the coils. The coils themselves, which are four in number, are wound on bobbins of thin insulating material, the wire being heaped up towards the cheek of the bobbin which bears against the brass plate. This heaping up is done in accordance with a law of Sir William Thomson, so as to obtain, as far as possible, a maximum effect out of a minimum quantity of wire. The edges of the coils are covered with shellac, so as to protect the wire from injury. Within the holes in the brass plate are placed two little magnets, n's and s n,* formed of watch-spring highly magnetised; they are connected together by a piece of aluminium wire, so as to form an astatic pair of needles. A small groove is cut in the brass plate, between the upper and the lower hole, for the aluminium wire to hang freely in. • An aluminium fan is fixed at right angles to the lower needle; this fan acts as a damper, and tends to check the oscillations of the needles and to bring them to rest quickly. In front of the top needle is fixed the mirror. It is suspended by a fibre attached at its upper end to a small stud which can be raised or lowered when required; when this stud is pressed down as far as it will go the needles rest on the coils, and the tension being taken off the fibre, there is no danger of breaking the latter by moving the instrument. One end of each coil is connected to one of the four terminals in front of the base of the instrument, the other ends being connected to one another through the medium of the small terminals placed midway on either side of the coils. The connections are so made, that when the two middle terminals on the base of the instrument are joined together the whole four coils are in the circuit of the two outer terminals, so that they all four act on the magnetic needles. 47. As it is often convenient to be able to couple up the four coils in different ways so as to vary their total resistance, in the instruments manufactured by the Indiarubber, Gutta- percha and Telegraph Works Company the ends of all the four * In the more recent instruments it is usual to have several small magnets placed one above the other at a short distance apart, in the place of a single magnet. 1 E 2 52 T HANDBOOK OF ELECTRICAL TESTING. - coils are connected to terminals in a manner designed by Messrs. March Webb and R. K. Gray, and shown by Fig. 22. This figure represents the base of one of these instruments. Lines are engraved on the ebonite base to show the routes followed FIG. 22. 7 6 5 4 Bottom Coil Τοπ Coil Bottom Coil Top Coil Level by the various coils. Arrows also are engraved alongside the lines to show the directions in which the currents must flow in order that all the coils may tend to turn the galvanometer needle in the same direction. There are five possible ways of coupling up all the coils together, so as in each case to produce a different resistance. The following will show the various methods: I. To obtain total resistance of all the coils in series, connect terminals 2 and 3, 4 and 5, 6 and 7. ¡ 1 T GALVANOMETERS. 53 II. To obtain ğ resistance, connect terminals 2 and 3, 2 and 5, 7 and 6, 7 and 4 III. To obtain resistance, connect terminals 2 and 3, 4 and 8, 1 and 5, 6 and 7. 16 IV. To obtain resistance, connect terminals 2 and 8, 1 and 3, 4 and 5, 6 and 7. V. To obtain resistance, connect terminals 1 and 3, 3 and 5, 16 5 and 7, 6 and 8, 4 and 6, 2 and 4. In each case the leading wires from the galvanometer must be connected to terminals 1 and 8. Referring again to Fig. 21; over the coils a glass shade is placed, from the middle of the top of which a brass rod rises. A short piece of brass tube slides over this rod, with a weak steel magnet, slightly curved, fixed at right angles to it. This FIG. 23. FIG. 24. Til milm magnet can be slid up or down the rod, or twisted round, as occasion may require. For fine adjustments a tangent screw is provided, which turns the brass rod round, and with it the magnet. Figs. 23 and 24 show modified forms of the instrument, · 54 HANDBOOK OF ELECTRICAL TESTING. I which, however, in general arrangement are similar to the pattern which has been described. In the more recent galvanometers manufactured by Messrs. Elliott Brothers, the brass plates, which, in the older instru- ments secured the coils in their places, are hinged to the frame, whilst the coils themselves are permanently fixed to the plates; by this arrangement the magnetic needles, with their mirror, fibre-suspension, &c., attachments, can be got at, if required, with the greatest facility. Altogether this improvement is one of the most convenient that has been made. About 5000 or 6000 ohms is usually the total resistance of the coils of these galvanometers. Fig. 25 shows a portable reflecting galvanometer which is very useful, especially for travelling purposes; the three legs are hinged at their junction with the lower part of the coil frame, so that they can be folded together, and thus made to occupy but little space. Owing to the instrument being pro- vided with but two coils (one in front of, and the other behind, the needle) its sensitiveness is not quite so great as that of the larger instruments with four coils, but for general purposes it is an excellent piece of apparatus. FIG. 25. FIG. 26. : { } # 48. We have said that the mirror is sometimes made of a plano-convex lens. This is done so as to obtain a sharp image of the spot of light on the scale. The width of the spot can be regulated by means of a brass slider fixed over the hole in the screen, through which the beam emerges from the lamp. A much better arrangement than the spot of light is now provided with most instruments. The hole through which the light emerges is made round, about the size of a sixpence, with I GALVANOMETERS. 55 a piece of fine platinum wire stretched vertically across its diameter. A lens is placed a little distance in front of this hole, between the scale and galvanometer, so that a round spot of light, with a thin black line across it, is reflected on the scale. This enables readings to be made with great ease, as the figures on the scale can be very distinctly seen. (The mirror in this arrangement may be a plane one.) When the spot of light only is used, it is necessary to partially illuminate the scale with a second lamp. The general appearance of a back view of the scale frame with the lamp placed in position, is shown by Fig. 26. Jacob's Transparent Scale. 49. The position of the ordinary form of scale for the Thomson's galvanometer is to a certain extent inconvenient, especially to near-sighted persons. Mr. F. Jacob has completely remedied this inconvenience by the arrangement shown in front view and cross section by Fig. 27. In this fig. B is a wooden scale-board with a longitudinal slot, as shown at C; P is the paper scale, cut FIG. 27. 34 35 36 39 38 29 40 Tumbull l l l l l l l l so that all the division lines reach the lower edge; A is a slip of plane glass with its lower half finely ground from one end of the slip to the other, on the side towards C: the scale is so placed that the lower end of the division lines just touches the ground part of the glass slip. The image of the slit with a fine wire stretched across it is focussed in the ordinary manner on the ground part of the glass, and will of course be clearly seen by the observer on the opposite side of the scale; as the line and printed divisions are in the same plane, there is no parallax; and a great increase in accuracy of reading the position of the hair line is obtained, owing to the greater ease of observing that two lines coincide when end on to one another, than when super- imposed; and further, from the circumstance that the room need not be darkened. The lamp and its slit is placed on one side and reflects the beam of light on to the galvanometer by a mirror or total reflection prism, and by means of two long plane 1 { 56 HANDBOOK OF ELECTRICAL TESTING. mirrors the actual distance between the galvanometer and scale is reduced, so as to have everything close to the observer's hand. The scale adopted is divided into half millimetres, and it is per- fectly easy to read to a quarter of a division, and with a hand magnifying-glass still further. This arrangement has been adopted in the testing-rooms of Messrs. Siemens Brothers and Co., at Woolwich, and gives great satisfaction. 50. In the testing-rooms at the Silvertown Telegraph Works, the scales employed are of large dimensions, being about 5 feet long, and are set at a distance of several feet from the galvano- meter. By this arrangement a greatly magnified image of the round spot of light with the black line across it is obtained, and the divisions on the scale being of correspondingly large dimensions, the readings can be made with great facility, and with very little fatigue to the eye. The only objection to the arrangement is the space which it necessarily occupies, but as it is not often that many instruments require to be set up in the same room, this need hardly be taken into account. To set up the Galvanometer. 51. It is essential, before proceeding to set up the instrument for use, to see that the ebonite base is thoroughly dry and clean, so that there may be no leakage from the wires to interfere with the tests taken. Indeed, it is as well to place the galvanometer and the other apparatus to be used on a large sheet of gutta- percha or ebonite, more especially if the room in which the tests are to be made is at all damp. Sometimes little ebonite cups are provided for the levelling serews of the instrument to stand in, which answers the purpose of insulating very thoroughly. The instrument should be set up on a very firm table in a basement storey. It is almost useless to test with it in an upper room, as the least vibration sends the spot of light dancing and vibrating to and fro. At all cable works the instrument is placed on a solid brick table built on the earth so that no vibration can possibly affect it. A suitable table being chosen, set the galvanometer in any convenient position, and adjust the levelling screws until the bubbles of the level or levels show the instrument to be perfectly level. Now remove the glass shade, and gently raise the stud at the top of the coils by squeezing the tips of the fingers between the head of the stud and the top of the brass plate in which it runs. If the stud is raised by a direct pull, there is almost a certainty 1 ! ། GALVANOMETERS. 57 On no of its coming up with a jerk and breaking the fibre. account must the stud be twisted round, except to get rid of any torsion which may exist in the fibre when it has been replaced after becoming broken. The stud being raised sufficiently high to allow the mirror to swing clear of the coils, replace the glass shade, screw the brass rod with the magnet, on to its top, and set the magnet about half-way up the rod, the poles being placed so as to assist in keeping the magnetic needles north and south. The scale lamp being lighted, place it in position on the scale stand, the edge of the wick being turned towards the brass slider which regulates the width of the beam of light. Having opened the slider to its full extent, the scale and lamp should be placed about 3 feet from the galvanometer, so that it stands parallel with the faces of the coils and so that a line drawn at right angles to the scale from the lamp-hole will pass through the centre of the galvanometer. The reflected beam of light should then fall fairly on the scale. If too high, this may be remedied by propping up the scale, and if too low, by screwing up the levelling screws of the galvanometer. Should the light be too high on the scale, it will be found an easier matter to prop up the scale than to lower the galvanometer by means of the levelling screws. The spot of light should now be set at the zero point on the scale by turning the regulating magnet by means of the screw; the spot should next be focussed, by advancing or retreating the lamp and scale, until a sharply defined image is obtained on the scale. The width of the slit may then be diminished, by means of the brass slide, until a thin line of light only is obtained on the scale. If the round spot of light with the line across it is used, the focussing must be made so that the black line is sharply defined. The position of the scale and galvanometer being once ob- tained, their positions on the table may be marked for future occasions, or, at least, the exact distance of the scale from the galvanometer noted, so that it can be placed right without trouble. The instrument being now ready for use, if it is not required to be sensitive, place the regulating magnet low down; if, on the contrary, it is required to be sensitive, place it high up. 52. To obtain the maximum sensitiveness :-Raise the magnet to the top of the bar, and then turn it half round, so that its poles change places. The magnet will now be opposing the earth's magnetism, and consequently will tend to turn the magnetic needles round. If the magnet is at the top of the rod, the 58 HANDBOOK OF ELECTRICAL TESTING. effect of the magnetism of the earth on the magnetic needles will be more powerful than the magnetism of the regulating magnet, and the needles will tend to keep north and south; but by placing the regulating magnet lower down, a point is reached where the earth's magnetism is just counteracted. Under these conditions the needles will stand indifferently in any position. By placing the regulating magnet about an inch higher than the position which gives this exact counteraction, the magnetism of the earth will be just sufficient to keep the magnets north and south, and consequently the spot of light at the zero on the scale, and at the same time leaves the magnets free to be moved by a very slight force. It will be noticed with the regulating magnet in this position, that in order to get the spot of light at the zero point, the magnet must be turned in the opposite direction to that in which it is required that the needles should move. It is not advisable to adjust the instrument too sensitively, because it is difficult then to keep the spot exactly at zero, as any slight external action may throw it a degree or two out. 53. The presence of iron near the instrument is not prejudicial to its correct working, so long as the metal remains stationary. The experimenter should, however, remove any keys or knives he may have about him, as they very much affect the galvanometer if he moves about much. These precautions may seem too minute, but as the very object of the Thomson galvanometer is to enable measurements to be made with accuracy, all likely causes of disturbance should be avoided. 54. If the fibre of the instrument by any chance gets broken, the top front plate must be unscrewed, when the coil which it secures can be removed, and the mirror and its appendages got at. Care should be taken, when replacing the fibre, that only a fine thread from the cocoon silk is used, or the sensitiveness of the instrument will be much diminished. The operation requires care, and must be done in a room free from draughts. When the ends of the fibre are passed through their respective holes, and tied, a small drop of shellac varnish may be dropped on them, which will prevent their becoming loose. It is as well to let the needles remain suspended for a time, so that the fibre may become stretched to its normal length before being used. The suspending stud should always be pressed down before removing the instruments. 55. A resistance box, containing three shunts, is provided with the galvanometer, of the values th, th, and th of the resistance of its coils, which values, as we shall show in the 999 GALVANOMETERS. 59 next chapter, enable us to reduce the sensitiveness of the galva- nometer to its 6th, 16th, and Tooth part respectively. 00 Fig. 28 shows a form of this shunt. By inserting a plug into one or other of the holes, the required shunt is inserted. The numbers are sometimes marked as th, th, and Tooth, instead of th, th, and th, thereby indicating that FIG. 28. FIG. 29. 00 999 7/ 3 999 real size elegant form than the foregoing. the particular shunt reduces the deflections of the needle to that particular fraction, but they have really just the same adjustment in both cases. The shunts are sometimes enclosed in a round brass box, as shown by Fig. 29, which is perhaps a more portable and The two broad strips of copper shown in Fig. 28 are used for the purpose of connecting the box with the galvanometer. The blank plug-hole is for the purpose of short-circuiting, which should always be done when the instrument is not actually in use. THOMSON'S DEAD-BEAT GALVANOMETER. 56. Great inconvenience and loss of time in testing often arise from the needle of the galvanometer not settling down at once to the angle of deflection it should take up when under the influence of a constant current, but oscillating to and fro several times before it finally comes to rest, and again acting in the same way when the current is taken off and the needle returns Į 60 HANDBOOK OF ELECTRICAL TESTING. towards the zero point. The object of the dead-beat galvanometer is to avoid these inconvenient oscillations. Fig. 30 shows the arrangement invented by Sir William Thomson for effecting this object. A is a brass tube, whose end a a, which is screwed, is closed by a piece of glass. B is a short piece of tube, which is screwed, FIG. 30. a b I I I 2 } A B ατ C α B screw. and whose end b b is similarly closed by a piece of glass. C is a third short piece of tube, into which the ends of A and B The length of this tube is such that when the whole arrangement is united together there is a very small space between the ends a a and b b; a small air-tight cell in fact is formed. Hanging midway inside C is a mirror m, with a magnetic needle fixed to it, as in the ordinary Thomson galvanometer. This mirror very nearly fits inside the tube, there being only just room for it to swing freely; it is suspended by a very short fibre. The space between a a and b b, although very small, is just sufficient to enable the mirror to turn through an angle large enough to give a good deflection of the spot of light on the scale. The complete arrangement is inserted in the centre of a single galvanometer coil, so that the mirror occupies the same position that it does in the ordinary galvanometer. Owing to the air inside the cell being so closely confined, the violent movement of the mirror is checked when it is acted upon by a current passing through the coils, and the consequence is that the mirror, instead of overshooting the mark and then recoiling, turns with a gradually decreasing velocity towards 3 GALVANOMETERS. 61 + its final deflection and ceases to move when the latter is reached. The same thing takes place when the current is cut off; in this case the spot of light moves back to zero and ceases to move at that point. The suspension fibre being very short, the mirror cannot turn so freely as the one in the ordinary galvanometer; its sensitive- ness is therefore not quite so great, but it is sufficiently so for most purposes for which the latter would be used. The fibre is very easily replaced when broken. One end being attached to the mirror, the other is passed through a small hole in the side of C, and is then drawn sufficiently tight to suspend the mirror inside the tube so that it does not touch the sides, a drop of shellac is then applied to the hole, which closes it and fixes the fibre. In some cases the cell is filled with paraffin oil, which still further tends to check the move- ment of the mirror. THE D'ARSONVAL DEPREZ DEAD-BEAT GALVANOMETER. 57. The main peculiarity of this instrument lies in the fact that, whereas in almost all galvanometers there is a fixed coil and a movable magnetic needle, in this galvanometer the coil is movable and the magnet-a massive compound horse-shoe of steel—is fixed. Fig. 31 represents the instrument itself, as manufactured by P. Jolin & Co., of Bristol. The steel magnet, made of three thin horse-shoes, each magnetised as strongly as possible, is firmly fixed to a metal base, with its poles upwards. Between the poles hangs the coil, which is rectangular in form and weighs only a few grains; it is held in its place by a thin silver wire above and another thin silver wire below. The coil is made by winding the wire on a continuous rectangular frame made of copper or silver as thin as possible; this frame, by the reactive effect of the induced currents which the move- ment of the coil sets up, causes the latter when deflected to come rapidly to rest. To reinforce the magnetic field, a strong compound magnet of cylindrical shape is arranged so that the laminations are in a horizontal direction, and so that its north pole comes opposite the south pole of the horse-shoe magnet; it is placed in the hollow of the suspended rectangular coil without touching it, and is firmly fixed; the coil is then free to turn in the very narrow space between the compound magnet core and the external magnet-poles; and it need hardly be added that this contrivance produces a very intense magnetic field. l'he current is led in by one of the silver suspension-wires, and 62 HANDBOOK OF ELECTRICAL TESTING. " of leaves the coil by the other. So far the arrangement precisely resembles that adopted in the well-known "siphon-recorder Sir W. Thomson, invented several years ago for the purpose of cable-signalling. A small mirror of 1 metre focus is fixed FIG. 31. ፡ ERINEGARMONDAYONGATES 135 P.JOLIN. & C CHAUVET to the suspended coil; a brass spring at the bottom keeps the suspending wires adequately stretched; and a screw-head at the top of the instrument serves both to regulate the tension in the wires and to let the coil down to a position of rest on the central iron cylinder, whenever the galvanometer is to be dismounted for removal to a distant place. The resistance of the coil is made from 150 to 750 ohms in the ordinary pattern of instrument. As there is no suspended needle, no external magnetic forces affect the zero of the instrument; and, since the position of the coil is determined solely by the elasticity of the suspending wires and the magnetic action of the fixed magnet on the current in the coil, it can be used in any position, and is inde- pendent of the earth's magnetic field. It can even be placed quite near to a dynamo-machine. The intensity of the magnetic 1 ! GALVANOMETERS. 63 field in which the coil is situated is such that whenever the galvanometer-circuit is closed-even through a considerable resistance the motion of the needle is dead-beat. It takes less than one second to come to rest at its final position of deflection, and when it returns to zero it does so with the most complete absence of oscillations. Altogether, the form of instrument is an extremely satisfactory one. THOMSON'S MARINE GALVANOMETER. 58. This instrument is specially constructed for use on board ship, where the rolling of the vessel and the constant movement of masses of iron about would render an ordinary reflecting galvanometer quite useless. Fig. 32 shows a side view of this instrument, the upper part being drawn in section. FIG. 32. A n B ի (CENTRIPETTEJO11 CCCC are the coils, which are similar in form to those employed in the ordinary Thomson's galvanometer; there is, however, but one set, of two coils, instead of two sets as in the latter instrument. The mirror, with the magnetic needle fixed to its back, is strung on a cocoon fibre in a brass frame. The fibre is fixed at one end, and at the other is attached to a spring, which draws t 64 HANDBOOK OF ELECTRICAL TESTING. the fibre tight. The frame slides in a groove between the coils, so that it can be drawn out for the purpose of repairing the fibre. A powerful directing horse-shoe magnet (not shown in the figure) embraces the upper parts of the coils, and serves to FIG. 33. overpower the directive effect of the earth's magnetism. This latter effect is still further rendered harmless by en- closing the whole system in a massive soft-iron case A A A A, a little window B being left through which the rays of light reflected by the mirror enter and return. For obtaining exact adjustment of the spot of light to zero, two little magnets, n and s, as broad as the mirror magnet is long, are provided; by turn- ing the pinion p these little magnets can be made to advance or retreat, and so act on the mirror magnet to make it turn in one direction or the other, as it is required. The resistance of this form of galvanometer (which is shown in general view by Fig. 33) is usually as high as 30,000 or 40,000 ohms. 59. The angle of maximum sensitiveness in a Thomson re- flecting galvanometer is, it is perhaps unnecessary to say, the largest deflection we can obtain, as the angle of deflection is but a very few degrees and, consequently, the true maximum angle can never be reached. 60. By turning the controlling magnet of the instrument so that the needle is turned through a large angle, the normal zero becomes at a considerable distance off the scale, and the sensi- tiveness of the galvanometer to changes in the current strength producing a deflection, can be made very great. Thus, supposing the needle to be normally at the ordinary zero, and suppose that a current caused it to deflect to 350 divisions, then an increase in the current of say 1 per cent. would increase the 350 × 101 deflection to , or, 353 ·5; that is, would increase it 100 3.5 divisions. If now the working zero had been 350 divisions to, say, the left of the ordinary zero, and if the current had been strong enough to produce a deflection of 350 divisions to the right of the ordinary zero, then the deflection would be equivalent to 350 +350, or 700 divisions, and an increase in the current of 1 per cent would increase the deflection to 1 M + 1 F י " C + 1 GALVANOMETERS. 65 700 × 101 100 or 707, that is to say, an increase of 7 divisions. If, lastly, the controlling magnet is turned so that the needle has a zero equivalent to, say, 2000 divisions to the left of the ordinary zero, that is an inferred zero, as it is called, of 2000, then, if the needle were deflected to the right by a current sufficiently strong to bring the deflection on the scale, and to give it a value of 350 to the right of the ordinary zero, the deflection representing the current would be 350 + 2000, or 2350 divisions, and an increase in the current of 1 per cent. 2350 × 101 would increase the deflection to 100 -, or, 2373.5, that is to say, an increase of 23 ·5 divisions. In actual practice it is often possible to use an inferred zero considerably greater than 2000, and with corresponding advantage. FIGURE OF MERIT OF GALVANOMETERS. 61. The "figure of merit" of any galvanometer may best be defined as the reciprocal of the strength of current which will produce one division or degree of deflection.* In order to find this current, we have simply to join up the galvanometer in circuit with a battery of a known electromotive force, and a resistance of a known value, and then note the deflection obtained; from this we can easily calculate the current required to produce 1 degree of deflection; thus, for example, if we had a tangent galvanometer which gave a deflection of 50° with a 10-cell Daniell battery, that is, with an electromotive force of 10 volts approximately, there being in circuit a total resistance of 1000 ohms, then the current producing this deflection would be 10 1000 = .01 ampère. The current which would be required to produce a deflection of 1° would obviously be tan 1° •01 × · 01 × tan 50° • 0175 1.198 =000146 ampère; 1 consequently the figure of merit is or, 6849. 000146' * It is preferable to define the figure of merit as being the reciprocal of the current rather than the current itself, inasmuch as by so doing we avoid the apparently contradictory statement, that a galvanometer with a high figure of merit is one which requires a low current to produce a perceptible deflection. F I 66 HANDBOOK OF ELECTRICAL TESTING. In the case of a Thomson galvanometer, we have simply to divide the current by the deflection obtained with the latter, since the deflections are approximately in direct proportion to the currents producing them. If we require to determine the figure of merit of a galvano- meter whose deflections throughout the scale are not pro- portionate to any ordinary function of the degrees of those deflections, then it is best to employ a sufficiently low electro- motive force and high resistance in circuit to obtain a few degrees of deflection only, and then to divide the current by this number of degrees; for on every galvanometer the first few degrees of deflection are almost exactly proportional to the currents producing them, although the higher deflections are not so. The "figure of merit" of a galvanometer has a considerable bearing upon the question of the degree of accuracy with which it is possible to make electrical measurements, as will be seen hereafter. SENSITIVENESS OF GALVANOMETERS. 62. A galvanometer with a high "figure of merit," that is, a galvanometer whose needle will deflect from zero with a very weak current, is not necessarily a highly sensitive instrument; by a sensitive galvanometer we mean one whose needle when deflected under the influence of a current will change its deflection perceptibly with a very slight change in the current strength. In many tests it is far more important that the galvanometer used be one of great sensitiveness rather than one with a high figure of merit. As a rule it is rarely that an instrument with a compass suspended or a pivoted needle is highly sensitive, unless indeed the pivoting is exceptionally good. Practically, it may be taken that for high sensitiveness the needle must be suspended by a fine fibre so that its movements may be perfectly free. 63. In order to check the oscillations of the galvanometer needle when the latter is either deflected under the influence of a current, or when it recoils after the current is taken off, Mr. J. Gott suggests that a small coil of wire should be placed under the galvanometer in circuit with a small battery and a key, the coil being in such a position that when a current passes through it a deflection of the needle is produced; by a proper manipula- tion (easily acquired) of the key, it will be found that the oscillations of the needle can, with such a contrivance, be checked in a few seconds, and much time (an important item in some tests) saved. [ ! J 3 1 [ ( 67 ) } こ ​CHAPTER IV. SHUNTS. 64. In making certain measurements we sometimes find that owing to the sensitiveness of the galvanometer, we are unable to obtain a readable deflection, from the needle being deflected up to the stops. We may reduce this sensitiveness by the insertion of a Shunt between the terminals of the instrument. This arrangement is shown by Fig. 34. If it is required to reduce the strength of current which ordi- narily passes through the galvano- meter to any proportional part of that current, we must calculate, from the resistance of the galvano- meter, what the resistance of the shunt should be to effect that purpose. FIG. 34. G R E Now if we call C the current passing through the gal- vanometer without a shunt, then on introducing the shunt, C will divide between the two resistances, the greater por- tion of the current going through the smaller resistance, and the smaller portion through the greater. Thus if we suppose the total current, which passes from one terminal of the galvanometer to the other, to consist of G+S parts, G of these parts will go through the shunt, and then S G+S G+S parts through the galvanometer; that is to say, the current going through the shunt will be G C G+ S' and the current going through the galvanometer, S C G+S F 2 68 HANDBOOK OF ELECTRICAL TESTING. If, in this last quantity, we put S = G, then current going through galvanometer will be G C C G+ G 2 Oia G Again, if we make S = வில் 2' current going through galvanometer will be 2 C C G G+ 2 105 10 G Once more, if S be made equal to 3' current going through galvanometer will be 3 C C G 4 Oi+ G+ Ꮐ Finally, if S be made equal to galvanometer will be n G - 1 current going through n 1 C G G+ n 1 N From this it is evident, that to reduce the current flowing through the galvanometer to its th part, we must insert a shunt whose resistance is galvanometer. 1 n 1 th part of the resistance of the n 1 * 65. In many galvanometers three shunts are provided, which enable us to reduce the strength of current flowing through the same to its th, th, or oth part. From what has been said, it will be evident that the resistances of the shunts neces- sary to produce these results will have to be respectively the th,th, and th part of the resistance of the galvanometer. 9 * Page 59. ず ​} | { J [ } 1 I SHUNTS. 69 We are thus enabled to reduce the sensitiveness of the galvano- meter to any one of these three proportions we wish. 66. Suppose now, in making a measurement, we placed a re- sistance box for a shunt between the terminals of the galvano- meter, and then adjusted it until we obtained a convenient deflection for the purpose we required; what deflection should we get on removing the shunt? Let us call C, as before, the current which passes through the galvanometer when no shunt is inserted, and let C₁ be the current which flows through it when the shunt is inserted, then the current which flows through the shunt will be C - C₁. Now the two currents will flow through the shunt and galvano- meter in the inverse proportion of their resistances, that is, therefore, C₁ : C - C₁ :: S: G, C = C₁₂ x G + S S Or expressed in words, we should say that the current which would flow through the galvanometer, when the shunt was Galvanometer + Shunt removed, would be times the strength of Shunt the current which flows when the shunt is inserted. This pro- portion is called the multiplying power of the shunt. 67. It will be noticed in a circuit like that shown by Fig. 34, that when a shunt having a resistance equal to that of the galvanometer is introduced between the terminals of the latter, it will not exactly halve the current passing through the instrument. If we used a tangent galvanometer, we should find, if the deflection without the shunt was 40 divisions on the tangent scale, the introduction of the shunt would not bring the deflection down to 20, but to some deflection greater than 20. The reason of this is, that the introduction of the shunt reduces the total resistance in the battery circuit, and consequently increases the strength of the current passing out of the battery. It is this increased current, then, which splits between the galvanometer and shunt, and not the original current. If it is required to make up for this decreased resistance caused by the introduction of the shunt, it is neces- sary to add in the battery circuit a compensating resistance equal in value to the amount by which the original resistance has been reduced. In order to obtain this, we must first consider the law of 70 HANDBOOK OF ELECTRICAL TESTING. The Joint Resistance of two or more Parallel Circuits. 68. If we have several wires whose resistances are R1, R2, R3.... respectively, then conductivity being the in- verse or reciprocal of resistance, their conductivities may be Now the joint conductivity represented by of 1 1 1 R,' R' R₂ any number of wires is simply the sum of their respective conductivities. Thus, two wires of equal conductivities, when joined parallel to one another, will evidently conduct twice as well as one of them; and in like manner, three wires will conduct three times as well as one. Similarly, two wires, one of which has a conductivity of 2, will, when combined with one which has a conductivity of 1, produce a conductivity of 2+1 or 3, for this is simply the same as joining up three wires, each having a conductivity of 1; and so with any number of wires. Therefore the joint conductivity of the several resistances, or of the multiple arc as the combination is called, will be 1 + 1 1 R₁₂ R₂+... and conductivity being, as we have said, the reciprocal of resist- ance, the resistance of the wires will be the reciprocal of this sum, or 1 1 1 1 + + + R₁ R2 R3 That is to say, the joint resistance of any number of wires joined parallel to one another is equal to the reciprocal of the sum of the reciprocals of their respective resistances. A particular case of these combinations is that of the joint resistance of two resistances, thus 1 1 1 1 + R₁ R₂ R₁ + R₂ R₁ R₂ or, the joint resistance of two resistances joined parallel to one another is equal to their product divided by their sum. 69. Applying the foregoing law, the resistance between the terminals of the galvanometer before the introduction of the shunt being G, that on the introduction of the shunt will be G.S. Or, as S is usually made some fractional value of G, G+S SHUNTS. 71 1 say the n 1 th part (whieh value would be used in reducing 1 n the sensitiveness of the galvanometer to th), this combined resistance will be G G G - n 1 n 1 G G 1 n G+ 1+ n 1 [1] n The resistance therefore to be added to the battery circuit will be . For example. G G G N — 1. n n [2] It was required to reduce the sensitiveness of a galvanometer, whose resistance was 100 ohms (G), to th. What should be the resistance of the shunt and of the compensating resistance? Resistance of shunt equals 1 100 X = 25 ohms, 5 - 1 and compensating resistance equals 3 100 X 5 - 1 5 = 80 ohms. It would be as well if the shunt boxes provided with galvanometers had compensating resistances connected with them, as calculation would be con- siderably simplified thereby in a large number of measurements. Fig. 35 shows how a set of shunts and compen- sating resistances may be adapted to any galvano- meter; we will consider how their values may be determined. Let S, S1, S2, be the shunts which can be con- nected to the galvanometer by inserting plugs at A, B, or C. Let r1 r2 r, be the compensating resistances, and let 2 Ꭱ. r₁ + r₂+r₁ = R₁₁ r₂ + r₁ = R2. FIG. 35. S: S₁ S, 2000 2000 2000 [3] C 帝 ​} 72 HANDBOOK OF ELECTRICAL TESTING. Now, what we have to do, is to find what values of S, S1, S2, and r₁, r₂, r₁, are necessary, so that when a plug is introduced either at A, B, or C, the resistance between D and E shall always be the same, whilst the necessary portion of the current is shunted off from the galvanometer. Let us first consider the shunt S and the compensating resistance which, in this case, will be R₁. When the shunts and compensating resistances are not in use, the resistance in circuit is of course G, and this value must always be preserved between D and E. 1 N Let the value of the shunt S required be th, then we know (page 68) that the resistance of S necessary to give this, is S G N 1 and from [2] (page 71) that the value of R₁ must be R₁ =G" = 1. - n 1 K ) t [5] We next have to consider what value to give to S, and R₂. * Let it be required, by means of these resistances, to reduce the deflection by th, then the value to be given to S₁ will be 1 i N1 G tri S₁ N1 1 to solve which we must know the value of r₁. Now the combined resistance of the shunt and G +r, we can see from [1] (page 71) is G+r1; n1 therefore the value required to be given to R2, in order to preserve the resistance between D and E, equal to G, when S₁ is connected, will be ог G+r1, F₂ = G - G + r₁ 2 n1 R₂ + 1 = G 1 = 1; n1 N1 ; [6] J T FL 1 1 SHUNTS. 73 but from [3], [4], and [5] (rages 71 and 72) R₁ = G2 — 1; R₂+ r₁ = R₁ N therefore, subtracting [7] from [6], we have ; [7] that is, or 1 "₁ = G 1 n n₂ = Gn1 - n N1 n1 N N NI 1 Պլ ni n r1 = G N1 n ni r₁ = G N nT n (n − 1) consequently the value of S, will be n N1 1 + n (n − 1) (n − 1) nı S₁ = G = G n1 1 n (ni 1)2 In like manner it could be shown that the resistance necessary 1th to give to S₂ and r₁ +r₂, to reduce the deflection to its th part would be 12 S₂ = G (n − 1) në n (n − 1)²' and r₁ + r₂ = or rz = G N G n(nz - n - n2 n (në − 1)' Finally we have from [3] and [5] N₂ r₂ = R₁ − (~1 + r₂) = G 1) n — 1 n (r₁ + r₂). 7 To summarise then, 1 S = G n- 1. 74 HANDBOOK OF ELECTRICAL TESTING. 1 : S₁ G (n − 1)nı n (n₁ — 1)² - · S₂ = G (n − 1) n₂ n (n₂ — ´1)² ² Sp S₂ = G G (2 (n − 1)n,, 'P n (n - 1)2° and for any other shunt S, The compensating resistances between the shunts will be n n1 n (n¸ — 1',' r1 = G n r2 = G and also we have or - n2 n (n₂— 1) r₁ + r₂ + ... . + r = G 2 41 ; N n Np - 1' (?1 + 2 + . . . + 1p-1). Tp N = G np n (n − 1) The last resistance r, beyond the last shunt will be n 1 r₁ = G 1 n (r₂+ r₂+...+rp). For example. It was required to provide a galvanometer with 'th, th, and Too shunts, and with corresponding compensating re- sistances arranged according to Fig. 35. What should be their values? We have therefore, n = 1000, n₁ = 100, n₂ = 10; n − 1 = 999, n¸ − 1 − 99, ` n½ − 1 = 9. - = Then S = G 1 999 G ×·001001, SHUNTS. 75 1 1 S₁ = G 999 X 100 G X ⚫010193, 1000 × 99 × 99 2 S₂ = G 999 X 10 = G ×·123333 1000 × 9 × 9 and "₁ = G 1000 - 100 = GX •0090909, 1000 × 99 1000 · 10 r₁ + r₂ = G 2 G x •11;' 1000 × 9 from which 12 = also 999 T₁ = G G (·11 — ·0090909) = G × •1009091 ; 1000 (1 + 2) = G (99911) G x 889. (r₁ +r½) = If the resistance of the galvanometer, for which these shunts and compensating resistances are to be provided, is 5000 ohms, then S = 5000 × ⚫001001 S₁ = 5000 × •010193 Sa = 5000 × 123333 r1 5000 X 0090909 = r₂ = 5000 × ⚫1009091 r2 5000 × 889 5.005 ohms. 50.965 "" 616.655 "" 45.455 = 504.545 4445.000 "" Fig. 36 shows how an ordinary Thomson galvanometer shunt box would be arranged with compensating resistances. 1 FIG. 36. ÷ 1000 100 The plug hole, when it has a plug inserted in it, connects the top left-hand brass block to the bottom left-hand block, and so leaves the galvanometer connected to the terminals of the shunt box without any additional resist- ance in its circuit. The connection between these brass blocks is shown by the dotted line in Fig. 35 (page 71). 1/4 real size 70. The accurate adjustment of ordinary shunts is often a somewhat troublesome operation, in consequence of the nume- rical values of the resistances of which the shunts are composed not being whole numbers; thus, supposing the resistance of the 76 HANDBOOK OF ELECTRICAL TESTING. galvanometer to be 5000 ohms, then the resistance of the th shunt would have to be 5000 ÷ 9, or 555·56; and, practically, this could not be adjusted to a greater degree of accuracy than one decimal place. Similarly, the th shunt should have a resistance of 500099, or 50 505, and the Tooth shunt a resistance of 5000 ÷ 999, or 5·005, both of which numbers are somewhat awkward to adjust exactly. n Now on page 71 (equation [1]) we saw that the combined G resistance of the galvanometer and its shunt was conse- quently to adjust theth shunt we may connect it to its gal- vanometer coil, and adjust it until the joint resistance of the two becomes equal to 500010, or 500 ohms. Similarly, theth shunt would be adjusted by connecting it to the galvanometer coil, and adjusting it until the joint resistance was found to be 5000100, or 50 ohms; lastly, in like manner we should adjust the rooth shunt until the combined resist- ance of the two became 5000 ÷ 1000, or 5 ohms. 71. We have shown in a previous chapter (page 48) that the deflections on the scale of a Thomson galvanometer, except when they are nearly equal, are not directly proportional to the current strengths which produce them, and that to compare them a formula must be used. If we wish to avoid the use of this formula we must adopt some method of avoiding widely different deflections. This we can do by using a variable shunt for the galvanometer, and with it obtaining either equal, or nearly equal, deflections for all measurements made in one set of tests. The graduated scale of any galvanometer, it should be recol- lected, is not only for the purpose of enabling the strengths of two or more currents to be compared by different deflections, but is also for the purpose of enabling any deflection which may be obtained to be reproduced when required. 72. It is best to obtain as high a deflection as possible, for then not only will a slight variation from the correct resistance of the shunt produce a greater number of degrees of variation from the deflection required, than would be the case if a low deflection was used (§ 60, page 64), but also a higher resistance being required for the shunt, a greater range of adjustment is given to it. 73. By the help of the points we have just considered we can graduate or calibrate (§ 43, page 46) the scale of a galvanometer. To do this, first calculate from the known resistance of the galvanometer, the resistance of shunts required to reduce the amount of current passing through the galvanometer when no Ů ,, ༢༤ནི ཏི Į 1 SHUNTS. 77 shunt is inserted, to 1, 1, 1, 1, &c., the amount passing when a shunt is inserted, then the resistance of the shunts necessary to reduce the current to 4 į, į, į †, · · n will, as we have shown, be 1, 12, 13, 1, . 'th 1 th of the resistance of the galvanometer. Now, as we have also shown, the insertion of shunts reduces the resistance of the circuit in which the galvanometer is placed; we must therefore also calculate the resistances necessary to be inserted in the circuit in order to compensate for the reduction of resistance which takes place when a shunt is inserted. These resistances will be respectively 1, †, 1. †, . . . . "-¹th 3, of the resistance of the galvanometer. The shunts and their compensating resistances being calcu- lated, to calibrate the galvanometer we proceed as follows:- The galvanometer, a resistance box, and a battery are joined up in circuit. The shunt, that is, the shunt equal in resist- ance to the galvanometer, is then inserted, together with the corresponding compensating resistance in the resistance box. Sufficient resistance is now added in the latter to bring the deflection down to, say, 1°; the shunt and compensating resist- ance are then removed, and as the resistance in circuit is the same as before, and also the whole of the current passing in the circuit now passes through the galvanometer, the strength of current affecting it is exactly double that which deflected the needle originally; the deflection of the needle, therefore, now represents a strength of current double that of the previous experiment. We next insert the shunt and its compensating resistance, and by again adjusting the resistance coils, obtain a deflection of 1°; on now removing the shunt and compensating resistance we get three times the strength of current passing through the galvanometer; the deflection obtained therefore will represent that strength, and so by inserting 49 ¹th n shunts one after another, and repeating the process described, we can get the deflections corresponding to strengths of current equal to 1, 2, 3, 4, . . n, and the scale can be marked corre- spondingly; or these deflections and the corresponding currents producing them can be embodied in a table, so that by referring 78 HANDBOOK OF ELECTRICAL TESTING. to the latter we can at once see the relative powers of various currents giving different deviations of the needle. 74. By the help of this method of calibrating a galvanometer we can determine its angle of maximum sensitiveness (§ 28, page 23). All we have to do is to obtain various deflections of the galvano- meter needle with various shunts and their corresponding com- pensating resistances, and in each case to increase the deflection of the needle slightly by reducing the compensating resistances by the same amount; then the required angle will be the one at which the diminution of resistance produced the greatest increase of deflection. 75. It is evident that if, in making a measurement, we want to reduce the_deflection of our galvanometer to a readable value, we can do so, either by placing a large resistance in the circuit of the instrument or, by introducing a shunt between its terminals. It is possible also, in certain cases, to produce the same effect by connecting a shunt between the poles of the battery, but this is not always advisable, as it interferes with the constancy of the latter. If the resistance of the battery and galvanometer in a simple circuit be very high it requires a very considerable increase of resistance in the circuit to produce an alteration in the deflec- tion of the galvanometer needle, whereas just the reverse is the case if a shunt be used to produce that effect. This fact is an important one, as it has a considerable bearing upon the accuracy with which measurements can be made. } + 1 ( 79 ) ! CHAPTER V. MEASUREMENT OF GALVANOMETER RESISTANCE. HALF DEFLECTION METHOD. 76. THE simplest method of determining the resistance of a galvanometer is perhaps the one we have already given on page 5 (§ 8). In this method it will be seen we joined up the galvanometer, whose resistance (G) was required, in circuit with a resistance p, and a battery of very low resistance, and having obtained a certain deflection we increased p to R, so that the current passing in the circuit became halved in strength, the resistance (G) of the galvanometer was then given by the formula GR 2p. If we were measuring the resistance of a tangent galvanometer, the deflections obtained should be such that the tangent of one deflection is half the tangent of the other, the precaution against having the deflections too high or too low being duly taken (§ 28, page 23). For example. With a tangent galvanometer whose resistance (G) was to be determined, and a battery whose resistance was very small, we obtained with a resistance in the resistance box (as the set of resistance coils is sometimes termed) of 10 ohms (p) a deflection of 58°, and by increasing the resistance to 120 ohms (R) the deflection was reduced to 381° (tan 38° = 1 tan 58°); what was the resistance of the galvanometer? G = 120 2 × 10 = 100 ohms. - 77. In measuring the resistance of an ordinary galvanometer by this method it would be necessary to know what ratio the deflections bore to the current strengths producing them, so that the resistances may be adjusted accordingly. A convenient arrangement is to employ a tangent gal- vanometer of a known low resistance in circuit with the 80 HANDBOOK OF ELECTRICAL TESTING. galvanometer whose resistance is required, and to take the readings from the tangent galvanometer, the resistance then obtained from the formula will evidently be the resistance of the two galvanometers together. If, then, we subtract from this result the known resistance of the tangent galvanometer, we get the resistance we are trying to obtain. If we have not a tangent galvanometer at hand, and if moreover we cannot tell what ratio the deflections bear to the current strengths producing them, we must of course employ a different method of testing. 78. In this, and indeed in all tests, it is important to consider what resistances and battery power should be employed to make the measurements, so that the greatest possible accuracy may be ensured. If we employ very high resistances to measure a low resist- ance, a considerable alteration in the former would produce but little alteration in the current flowing through the gal- vanometer, for the electromotive force being constant, this current, and consequently the galvanometer deflection, is dependent upon the total resistance in the circuit, and an alteration of several units in a large total practically leaves its value the same, but then a few units too much or too little inserted in a formula may make the result appear very much greater or less than its true value. Thus, in the test we have been considering, suppose the battery power had been such that we found it necessary to have the resistance p = 2000 ohms, and that to halve the deflection we found it necessary to increase p to 4100 ohms (R), this would make the resistance of the galvanometer to be, as we saw before, G = 4100 - 2 × 2000 = 100 ohms. Now, practically, if the resistance R had been made 4200 ohms the deflection would have been halved; whatever differ- ence there was would scarcely be appreciable. If now we work the result out from the formula we get G = 4200 2 x 2000 = 200 ohms, G or double what it ought to be. It is possible indeed that the error might be greater than this. The test, in fact, would be quite useless. In order to have the best chance of accuracy we should make our resistances as low as possible, for then a small change or error in the latter produces the greatest increase or decrease in the current, and consequently also in the deflection of the galvano- MEASUREMENT OF GALVANOMETER RESISTANCE. 81 meter needle, and, on the other hand, it produces the smallest error in the value of G, when the latter is worked out from the formula. In order to make R as low as possible it is evident that we must make p as low as possible. 79. What degree of accuracy is attainable in making the test? This is dependent upon the "total possible percentage of error which may exist in the second deflection" (§ 41, page 43). We have then to consider what error in the value of G the total error in the second deflection will cause. The error in G must be occasioned by the value of R being obtained incorrectly, this wrong value of R being due to an error made in reading the magnitude of the second deflection. If in the formula G = R − 2 p we make a mistake of, say, '₁ per cent. in R, then the resulting R percentage error, A', in G will be λ' = λ', 1G Now the accuracy with which we can adjust R is directly dependent upon the accuracy with which we can adjust (G+R) for the latter is the total resistance in the circuit of the galvanometer, and therefore any change or error made in the value of the galvanometer deflection (the second deflection) must be in direct proportion to the change or error made in (GR); consequently if we are liable to make an error of y' per cent. in the value of the second deflection, and an error of A' per cent. in R, then we must have 7::: R: G+ R " or d'₁ = = (G + R) y'; R but R G λ' = λ'₁ G' or, λ'₁ = λ' R' and therefore hence GR 2p, or, RG + 2p, - G 2 (G + p) y'; x = Ꭱ Ꭱ X = 2 (1 + 6) 1. [A] G 1 82 HANDBOOK OF ELECTRICAL TESTING. For example. In measuring the resistance of the galvanometer in the example given in § 76, page 79, it was known that the "total possible percentage of error (y') which could exist in the second deflection" could not exceed 1.7 per cent. (Example 2, page 45). What would be the percentage of accuracy (A') with which the value of G could be determined? λ = 2 (1+100) 1·7 = 3.7 per cent. A single cell of a battery is the lowest electromotive force that can be practically employed in making the test, but we may find that this one cell gives too low a deflection with the lowest value we can give to p, that is 0, and two cells too high a deflection; we should have, therefore, to employ two cells and then increase p until the proper deflection is obtained. Now on pages 80 and 81 it was pointed out that it is best to make p of a low value, so that the deviation of the needle from its correct position, when R is not correctly adjusted, may be as great as possible; but equation [A] (page 81), which represents the relative values of the errors X' and y', although it shows that the error X' is smallest when p is as small as possible, at the same time shows that we gain but little by making p very much smaller than G, for A' is only twice as great when p = 0, as it is when p = G. 80. Practically we may say therefore that the Best Conditions for making the Test are to make p a fractional value of G; and in the case of a tangent galvanometer the two deflections obtained should be as nearly as possible 55° and 35° (§ 32, page 28). Also as regards the Possible Degree of Accuracy attainable. If we can determine the value of the deflection of the galva- nometer needle to an accuracy of y' per cent., then we can determine the value of G to an accuracy of 2 (1+) per. cent. If p is very small, then x' = 2y'; y' ! MEASUREMENT OF GALVANOMETER RESISTANCE. 83 so that even under the best conditions for making the test the accuracy with which the value of G could be determined would be only one-half of the accuracy with which the deflections could be observed. 81. It must be understood that the resistance of the testing battery can only be neglected when it forms a small percentage of the total resistance in its circuit. If, then, the galvanometer to be measured has a low resistance, inasmuch as R will have to be proportionately small, the battery resistance can no longer be ignored without introducing an error; moreover, if R is made small, its range of adjustment becomes very limited. The test, therefore, is not suitable for measuring galvanometers whose resistance consists of a few units only. EQUAL DEFLECTION METHOD. FIG. 37. G 82. The theory of this method is as follows:-The galvano- meter whose resistance G is re- quired, a resistance p, and a battery E of very low resistance, are joined up in circuit, as shown by Fig. 37, a shunt S being between the ter- minals of the galvanometer; a de- flection of the galvanometer needle is produced. Let C be the cur- rent flowing out of the battery, then Ju eeeeee S Ꮅ E C Ꮐ Ꮪ p+ G+ S This current divides into two parts, one part going through S, and the other part through the galvanometer. It does this in the inverse proportion of the resistance of those circuits, the part, C₁, going through the galvanometer being E C₁ = GS p + G+ S X S ES G+ S S (G+p) + G p The shunt S is now removed; this causes the deflection of the galvanometer needle to be increased.. ρ is now increased to R, so that the deflection becomes the same as it was previous to the removal of the shunt or, in other words, so that the strength G 2 84 HANDBOOK OF ELECTRICAL TESTING. of the current passing through the galvanometer is C₁, then therefore E C₁₂ = R + G' R+G ES E S (G+p) + Gp ¯R+G By dividing both sides by E and multiplying up, we get therefore from which SR+GS = GS+Sp+Gp; Gp = SR – Sp, G = S = SR - P. Ρ For example. A galvanometer whose resistance (G) was required, was joined up in circuit with a resistance of 200 ohms (p), a shunt of 10 ohms (S) being between the terminals of the galvano- meter. On removing the shunt, it was necessary in order to reduce the increased deflection to what it was originally, to increase p to 2200 ohms (R). What was the resistance of the galvano- meter? G = 10 2200 200 200 = 100 ohms. 83. In making this test practically, we should proceed thus:- Join up the instruments, as shown by Fig. 38, taking care that the two infinity plugs are firmly in their places. Plug up the three holes between B and C, and remove the necessary plugs between D and B. Next remove plugs from between D and E, so as to introduce the resistance p. On the right-hand key being depressed the deflection of the galvanometer needle is obtained. The galvanometer should be gently tapped with the finger in order to see that the needle is properly deflected and is not sticking, as it is very liable to do, especially when a compass suspended needle is used. The oscillations of the needle may be arrested by a skilful manipulation of the key; slightly raising it when the needle swings under the influence of the current and again depressing it when it recoils. The needle being steadily deflected, and the precise resist- I { Į i H · MEASUREMENT OF GALVANOMETER RESISTANCE. 85 ance (p) in the box noted, the left-hand infinity plug must be removed, and the resistance between D and E increased until the deflection becomes the same as it was at first, and the resistance (R) being noted, the formula is worked out. FIG. 38. D B HE B₁ 84. What are the best values of S and p to employ in making a test like this? Should we make S and p of low, high, or medium values? The answer to these queries has an important bearing upon the accuracy with which the test can be made; and as we shall more than once have to consider questions of a similar kind, we shall in the present instance enter at some length into the problem. There are two quantities whose values we have to determine, viz. S and p; let us first consider what S should be, supposing R to be a given quantity and p to vary along with S. If we examine the formula we shall see that if we make S small, then an error of one or two units in the correct value of R will make a much greater difference in the formula than would be the case when there is the same number of units of difference with S large; thus to take a numerical example, suppose we had the following values in the formula: G = 5 500 20 = 120 ohms, 20 and suppose we made R 120 units too large, we should then have 620 — 20 G = 5 = 150 chms; 20 86 HANDBOOK OF ELECTRICAL TESTING. or an error of 150 120 had the following values:- 30 ohms. Next let us suppose we 500 · 400 G = 480 = 120 ohms, 400 and as before let there be an error of 120 units in R, we then have 620 400 G = 480. = 264; 400 or an error of 264 - = 120 144 ohms, and if S and p had been higher still we should have seen that the error would have been still greater. To put the case in another way; in the last example let us suppose the error in R had been, not 120 units, but 25 units; that is, make R = 500 + 25 = 525; we then find that G = 480- 525 400 400 = 150 ohms. The error in G, in fact, in the former case, where R was 120 units too large, was no greater than it was in the latter case, when the excess in the correct value of R was but 25 units. From this it must be evident that it is highly advantageous to make S as small as possible. Let us, however, put the matter in algebraical form; thus, let λ be the error in G, and let ø be the excess in the value of R which causes this error, then we have • R - P G+ λ = S R + $ − P = S P and G = S R - p ρ or, p = ρ SR + S G + S ; 1 : ! therefore by subtraction, λ = SL S&× G+S_ ø (G+S)¸ P S R Ꭱ From this we see that with a constant error ø, made in R, the corresponding constant error λ, made in G, will be as small as possible when S is very small, as indeed we before proved; but we also see that we gain but little by making S a very small fractional value of G, for the error would be only twice as great Į I I 1 [ MEASUREMENT OF GALVANOMETER RESISTANCE. 87 . with S = G as it would be if S were very nearly = 0. It would not do, however, to make S greater than G, for G + S increases very rapidly by increasing S. Practically, therefore, we may say-make S a fractional value of G. ::- We have next to determine what is the best value to give to p, supposing S to be a fixed quantity. Now if we put the equation in the form R - P G = S P R G=8(-1) S we can see that whatever value p has, R will have an exactly proportional corresponding value; thus to take the example we first had, viz. : G = 5 500 - 20 20 = 5 (500 - 1): 20 if in making the test we had made p = 2 × 20 40, instead of 20, then the value to which R would have required to have been adjusted would have been 2 × 500 1000, instead of 500. Further, if R had had this value, then an error of 20 units in R would have produced the same error in G as would the 10 units in the first case, when R was 500. At first sight then it might appear that it would not matter what value we gave to p. Let us, however, consider in what way the adjustment of R is effected. The means by which we adjust R is by observing the deflec- tion of the galvanometer needle, and seeing whether we have brought it to the deflection it had when p and S were the resist- ances in the circuit; when this deflection is correct we know that R`is correct. But the accuracy with which we can adjust R evidently depends upon the divergence of the needle from its correct position being as large as possible when R is not exactly adjusted, and if this divergence is greater when we alter R from 1000 to 1020 ohms than when we alter it from 500 to 510 ohms, then it is better so to arrange the value of p that R shall be 1000 ohms. Or in other words, if the error in R, corresponding to a con- stant error in G, produces a greater divergence of the needle from its correct position when R is large than when it is small, then it is better to have R large than small. 88 HANDBOOK OF ELECTRICAL TESTING. Now the current C producing the deflection of the galvano- meter needle is E C = = R+G' and if we suppose there to be a diminution an error ø, in R, then we have c, in C, caused by C − c = E R++G or but we know that therefore c = C – E R++ G' E C = R+G זי [ 1 1 9 E E Εφ C = R+G R++ G (R + 4 + G) (R + G) ' ļ or, since is very small, k Εφ c = C (R + G)² ; c, however, represents the absolute change from the correct current and as the latter is itself varied by the value of R, what we require to know is the relative change; this will be F 010 which equals Ꭼ Ꮮ (R + G)² E $ ÷ R+G R+G [A] But from page 86 we see that the constant error A, caused in G by an error in R, is or λ $ (G + S) Ꭱ $ λ R • = G+S' 1 Է 1 MEASUREMENT OF GALVANOMETER RESISTANCE. 89 substituting, then, this value of 4, we get C λ R λ [B] (G+S) (R+G) (G + S) (1 + R From this equation we see that in order to make c as large as possible, we must make R as large as possible; but it is evident that we increase c very little by making R much larger than G, for the reason we gave when we determined the ratio which S should have to G. We do not gain, then, anything as regards the sensitiveness of the arrangement by making R very large, but we gain as regards our power of adjusting R, for we can adjust a resistance with a much closer degree of accuracy when it consists of a large number, than when it consists of a small number, of units. It is therefore advantageous to make R as large as possible. Since when S, G, and p are given values, R must have a value dependent upon them; and since we have determined the value we must give to S, it follows that the value we should give to p must be such that R will be as large as possible. As we cannot make R larger than the resistance we can insert in the resistance box, we must not make p so large that R will have to exceed that value. From the equation we see that R - p G = S P S R. G+ S Theoretically, therefore, we must not make p larger than the value we can give to S G+ S R. The highest resistance we can practically give to R is 10,000 ohms; p, therefore, must not be larger than S G+S X 10,000 ohms. Thus, if we use a shunt whose resistance is th the resistance of the galvanometer, we must not make p larger thanth of 10,000, that is 1000, ohms. Equation [A] shows that the value of c is dependent upon the value of S, and that to make c large we should make S small. We previously proved, however, that there was another reason why S should be small, consequently we have a double reason why S should have a low value. ! .90 HANDBOOK OF ELECTRICAL TESTING. 85. What degree of accuracy is attainable in making the test? This, as in the last test, is dependent upon the value of the deflection error. We have, in fact, to consider what error in the value of G a definite error in reading the deflection of the galvanometer needle will cause. This we can determine from equation [B] (page 89). Let us, then, in this equation substitute percentages for absolute values, that is to say, let us have C C = of C, or, 100 C 100 and ג λ= of G; 100 then we get A'GR 100 100 (G + S) (R+G)' that is to say, For example. x = (1 + 8) (1 + 1 ) √. [C] In measuring the resistance of the galvanometer in the example given on page 84, it was known that the possible error y' in the current, due to the deflection being incorrect, would not exceed .88 per cent. (Example, page 43). What would be the percentage of accuracy (A) with which the value of G could be determined? x' = (1 + 100 ) ( 1 + 100 + 100 2200 .88 = .96 •88 ·96 per cent. 86. The practical results, then, that we have arrived at from these investigations are, that to obtain the Best Conditions for making the Test: First make a rough test to ascertain approximately the value of G. Having done this, insert a shunt (S) between the termi- nals of G, of a fractional value of the resistance of G. Next join up p in circuit with G and its shunt S, making p as large as possible, but not larger than highest resistance that can be obtained. S G+S R; R being the Insert in the circuit sufficient battery power of low resistance to bring the deflection of the galvanometer needle as nearly j 1 ↓ 1 1 MEASUREMENT OF GALVANOMETER RESISTANCE. 91 as possible to the angle of maximum sensitiveness (page 23), ad- justing p, if necessary, so that this angular deflection becomes exact, and note the exact value of p. Now remove the shunt and increase p to R, so that the increased deflection becomes the same as it was at first. Note R, and then calculate G from the formula. Possible Degree of Accuracy attainable. If we can determine the value of the galvanometer deflection to an accuracy of y' per cent., then we can determine the value of G to an accuracy (A') of λ' = S G (1 + )(1 + 1) G R y' per cent. If S is very small, and R very large, then x' = 7', so that under the best conditions for making the test, the accu- racy with which the value of G could be determined would be the same as the accuracy with which the value of the deflection could be observed. 87. In the practical execution of the test, inasmuch as there are only three resistances between D and B (Fig. 38) our choice of a shunt is limited from this source, but these three will usually be sufficient for most purposes. 88. The method we have described of making the test may be modified by making S or p the adjustable resistances instead of R, but in either of these cases it can be shown, by an inves- tigation precisely similar to the one we have made, that the proper values of the resistances should be those we have indicated. The test could also be simplified by making S = p, in this case we get G = S R S S = R – S ; such an arrangement, however, would not give the conditions for obtaining maximum accuracy. FAHIE'S METHOD. 89. If in the last test we make S the adjustable resistance, and make R = 2 p, we get G = S R-P = S 2 p - P P ୧ = S, 92 HANDBOOK OF ELECTRICAL TESTING. that is, the resistance of the shunt will be the resistance of the galvanometer. 90. The connections for making the test with the set of resist- ances shown by Fig. 38 would have to be so arranged that the resistances between D and E form the shunt, and those between D and C the resistances p and R. This arrangement, however, in consequence of there being so few plugs between D and C, is not a satisfactory one, as some difficulty would probably be found in adjusting the battery power and resistance R so as to obtain the deflection of maximum sensitiveness. With two sets of resistance coils, however, the test can easily be made. As in the previous method, it is best to make the resistance R as high as possible, for then any small change in the value of S produces the greatest movement of the galvanometer needle. The possible degree of accuracy attainable is the same as in the last test. 91. In order that satisfactory results may be obtained in the foregoing tests, it is necessary that the galvanometer be a sensi- tive one (page 66), otherwise even a moderate degree of accuracy cannot be assured. It is also very advantageous to arrange the resistances in connection with a key, as shown by Fig. 39. The key, K, it will be observed, in its normal position short FIG. 39. > I * G K : E еееев p.....s ----R------- circuits the right-hand resistance a, so that p is the only re- sistance in circuit; when however the key is depressed the short circuit becomes opened and p consequently becomes in- creased to R, whilst at the same time the shunt, S, becomes connected to the galvanometer; in practically making the test, therefore, what we have to do is to adjust p (or S, if ρ is a fixed quantity) until the deflection on the galvanometer remains the same, whether the key is up or down. As a break occurs when the key passes from the top to the bottom contact, during which break a slight movement of the galvanometer needle may take place, a preferable arrangement } 5 MEASUREMENT OF GALVANOMETER RESISTANCE. 93 is that suggested by Prof. Moses Farmer.* The key in this arrangement (Fig. 40) consists of two levers, L and 7; the latter is normally in contact with a stud joined to the junction FIG. 40. هو G I Δ E ееееее P........ ·--R · of the two resistances p and a. contact with 7, and at the same a When L is depressed it makes moment moves away the latter from its contact stud, thus practically no break takes place. THOMSON'S METHOD. 92. Join up the galvanometer g with resistances a, b, and d, and a battery of electromotive force E and resistance r, as shown by Fig. 41, and let a key be inserted between the points E FIG. 41. E d and B, so that by its depression these points can be connected together. First, let us suppose the key to be up and the points con- * Electrical Review,' Sept. 24th, 1886, page 316. * } ! J ! } 94 HANDBOOK OF ELECTRICAL TESTING. sequently disconnected. The current C, flowing through the galvanometer will then be E C₁ (a + b) (d + g) a+b+d+g E (a + b) +b) X a+b a+b+ d + g r (a + b + d + g) + (a + b) (d + g) a [1] Next, suppose the key to be depressed and the points E and B thereby to be connected together, then the current (C2) flowing through the galvanometer will be b d E α X C2 a g a + g g+ b + d + a + g E a (b + d) r (a + g) (b + d) + a g (b + d) + b d (a + g)° [2] Further, let us suppose the adjustment of the resistances to be such that • } we then get C₁ = C2, E (a + b) r (a + b + d + g) + (a + b) (d + g) E a (b + d) r (a + g) (b + d) + a g (b + d) + b d (a + g) by multiplying up and arranging the quantities we get [3] r [(a+b+g) (b + d) a + b g (b + d)] + b g (a+b) d + [d (b + g) +bg] (a + b) a = r [(a + b + g) (b + d) a + a d (b + d)] + a d (a + b) d + [d (b + g) + b g] (a + b) a; therefore bg [r (b+d)+(a + b) d] = a d [r (b + d) + (a + b) d]; that is, (b+d)+(a+b) ad = bg, or, g= a d A great advantage of this test is the fact of its being entirely independent of the battery resistance. It is also very easily made, as must be evident. } 1 T い ​fi MEASUREMENT OF GALVANOMETER RESISTANCE. 95 - In making the test practically, the connections would be made as shown by Fig. 42. The terminals E and B, would be joined by a short piece of thick wire. The other connections are obvious. A B FIG. 42. E لبا A B₁ The left-hand key (which is not shown in the theoretical figure) being first depressed and then kept permanently down, the right-hand key must be alternately depressed and raised, the resistance d, that is the resistance between A and E, being at the same time adjusted until the deflection of the galvano- meter needle remains the same whether the key is up or down. 93. We will now determine the best arrangement of resist- ances for making the test. What we have to do is to suppose that in the equation 9 = a d b there is a small but constant error in g, caused by a correspond- ing error in one of (the other quantities, let us say d, and then find what values of d and say, a, will cause the alteration of the deflection of the galvanometer needle produced on raising and depressing the key, to be as large as possible. Let A be the difference between the exact value of g and the value given it by the formula when we have d too large, and let the increased value of d be dɩ. We then have I + λ = a d₁; b 3 96 therefore HANDBOOK OF ELECTRICAL TESTING. F a d₁ = bg+bλ. We next have to determine what the alteration in the strength of the current passing through the galvanometer, produced by raising and depressing the key, is equal to. If in either equation [1] or equation [2] (page 94) we put b g a d 9. equal to a d, or b equal to then the resulting equation will give the current, C, which would flow through the galvanometer when the adjustment is exact; by doing this we get C E a r(g+a)+a(d+g) When the adjustment is not exact, the currents produced on raising and depressing the key will be obtained by equations [1] and [2] (page 94), and the difference between these two cur- rents relative to the current produced when exact equilibrium is obtained will give the relative current producing the altera- tion in the deflection of the galvanometer needle; hence we find C₂ - C₁ 1 + (b r (g+a) + a(d+g) { r(d₂+b) (g+a) + d₂b (g+a) +ga (d₂+6) b + a -a)} = · (d₁ + g + b + a) + (d₁ + g) (b + a 1 A (a d₁−bg) {r (d₁ + b) + d₁ (b + a) } {r (a + g) + a(d+g) } a{r(d₁+g+b+a)+(d₁+g)(b+a)} {r(d₁+b)(g+a)+d₁b(g+a)+ga(d₁+b)} but since a d₁ is very nearly equal to bg, we may without sensible error put a d₁ = ad = bg, or b = a d except where dif- g ferences are concerned; in which case we get C₂- C₁ C g (ad₁ - bg) a (a + g) (d+g) and since a d₁ = bg+b λ, and 02 1 · λ d g α = d we get λ G₂ = C = (a + 9) (1+0) = (a + 0) (1+8) g) (d g) [A] I 1 2 * ¿ 6 1 2 MEASUREMENT OF GALVANOMETER RESISTANCE. 97 C From this it is evident that, in order to make C2—C1 as large C2 C₂- C₁, it is It is evident also that, as regards increasing C useless making d very much larger, or a very much smaller, than g. If we make d about ten times as large, and a ten times as small, as g, we shall have good conditions for ensuring accu- racy, though as regards our power of adjustment, it would be advantageous to make d larger still if possible. as possible, we must make d as large, and a as small, as possible. From the equation bg= ad we see that g being a fixed quantity, and a as small as possible, we can make d as large as we like by making b as large as possible. 94. It may be pointed out, that when a is small and d and b large, we have the battery connecting the junction of the two greater with the junction of the two lesser resistances. 95. What degree of accuracy is attainable in making the test? This we can determine from equation [A]. Let us then, in the latter, substitute percentages for absolute values, that is, let C₂ — C₁ = of C, or, 2 C₂ - C1 100 100 and let λ λ= of g; 100 then we get X' g 100 100 (a+g) (1 + 2) g that is, For example. a λ x' = (1 + 2) (1 + 2) v. In measuring the resistance of a galvanometer by the fore- going method, the values of a, b, and d were 10, 100, and 300 ohms respectively. What was the resistance of the galvano- meter, and what was the possible degree of accuracy attainable? The smallest change in the value of the galvanometer deflection which it was possible to observe,* was 88 per cent. (§ 40 page 42). • *This is synonymous with "the degree of accuracy with which the value of the galvanometer deflection can be read" (page 42). H 98 HANDBOOK OF ELECTRICAL TESTING. 10 × 300 g= 100 30 ohms. x' = (1 + 10) (1 + 30) · 88 = 1•3 •88 1.3 per cent. To sum up, we have Best Conditions for making the Test. 96. Make a not greater than th of g, and make b not less than ten times as great as g, and preferably as much higher than g as possible, but not of such a high value that d, when exactly adjusted, has to exceed all the resistance we can insert between D and E (Fig. 42, page 95). Adjust d approximately, and then if necessary adjust the battery power, so that the final deflection is as nearly as possible that of maximum sensitiveness, and then, having exactly ad- justed d, calculate g from the formula. Possible Degree of Accuracy attainable. If we can read the galvanometer deflection to an accuracy of y' per cent., then we can determine the value of g to an accu- racy (A') of x' = (1 + 2) (1 + 2 ) √ If a is small and d large, then we get x' = y', per cent. so that under the best conditions for making the test, the accu- racy with which the value of G could be determined would be the same as the accuracy with which the value of a change in the deflection could be observed. 97. In the practical execution of the test with the set of resistance coils shown by Fig. 42 (page 95), the lowest value we could give to a would be 10 units, unless we improvised a resistance of less value, which it might be necessary to do. THOMSON'S METHOD WITH A SLIDE WIRE RESISTANCE. 98. The foregoing test is sometimes made by having a + b, a slide wire resistance (§ 18, page 15) d being a fixed resistance; in this case the slide would be moved along between A and C, } ! I r 琴 ​J I Į 1 ן } 1 MEASUREMENT OF GALVANOMETER RESISTANCE. 99 until the point is found at which the depression and raising of the key makes no alteration in the permanent deflection of the galvanometer needle. a As in the equation a d += b is merely the ratio of the resistances into which the total resistance a + b is divided, and as the resistances are directly proportional to the lengths of the wire on either side of the slide, it is sufficient for a and b to be expressed in terms of the divisions into which the length of wire is divided. Now as the total length, k, of the slide wire is constant, that is as a+b=k, or, b = k — αg therefore we must have y = d ( ) α k is usually divided into 1000 divisions, hence - g d - ¿ (1000 — a)- - For example. In the foregoing test, equilibrium was produced when d was 1 ohm, and a, 450 divisions; what was the resistance of the galvanometer? g=1 1 (10 450 1000 450 o) 450 = = ·85 ohm. 550 99. The best conditions for making the test in the case where a slide wire is used, are generally similar to those in the previous case, that is to say, we should require to have a small and d large. Now, the total resistance of a + b in the case of a slide wire would, under most conditions met with in actual practice, be small compared with g; consequently a would be small also. For this reason, therefore, it would not signify what were the relative values of a and b in making the test. But in order to make d large, it is obvious that a must be small compared with b; thus, if d is to be 10 times g, then a must be 10 times b. In the first case, when the test was made by adjusting d, it was pointed out that although there is an advantage in making d H 2. I 100 HANDBOOK OF ELECTRICAL TESTING. as large as possible, in so far that by so doing the range of ad- justment is made large, yet as regards the general sensitiveness of the whole arrangement, there is little, if any, advantage in making d greater than about 10 times g. In the case of the slide wire, where d is not the adjustable resistance, there is an actual disadvantage in making d excessively large, for the reason that, if we do so, we make a correspondingly smaller than b and, when this is the case, the error in g (when worked out by the formula g = ad) produced by the value of a being, b say, 1 scale division out, becomes comparatively large. Thus, if the slide wire scale were graduated into 1000 divisions (which is usually the case), it is clear that if the slider stood at, say, the “10” division mark on the scale, then an alteration or a mistake of 1 division would mean a change of 10 per cent. in the value of a, whilst if the slider stood at "100," then a change of 1 division would only mean a 1 per cent. change in the value of a. The change in a corresponding to a movement of 1 division, would obviously be less if the slider were near the centre of the scale, that is, near the "500" division mark, but in this case the increase in the range of adjustment would be more than com- pensated for by the reduced sensitiveness of the arrangement. The possible degree of accuracy attainable in making the test would be as follows:- Let there be an error A in g, caused by the slider being 8 divisions out of correct adjustment, then we have g + λ = d (1000 1000 a + s (a+8) or λ = d · 1000 a (100 a + 8 g = d - (a +8) a+d 1000-(a+d) 1000 α 1000-a d 1000 8 (1000 — a)² since 8 is very small. If we put percentages instead of absolute values, that is to = λ x d 100 1000 α (1000 -a), say, if we have 入​= of g 100 ľ λ = per cent. a (1000 - a) then we get 100000 8 } P 3 1 1 I 1 A ↓ MEASUREMENT OF GALVANOMETER RESISTANCE. 101 If the galvanometer is sufficiently sensitive to enable the position of the slider to be determined to an accuracy of 1 divi- sion, then 8 For example. = 1. In the last example, what would be the degree of accuracy, A', with which the value of g could be obtained, supposing that the position of the slider could be determined to an accuracy of 1 division (8)? λ 100000 × 1 450 (1000 450) G = •40 per cent. 100. The facility and accuracy with which all the foregoing tests (except the half deflection test) can be made may be greatly increased by the following device :-Instead of making the test with the galvanometer needle brought to the " angle of maximum sensitiveness" (page 23), make it with the needle brought approximately to zero by means of a powerful per- manent magnet set near the instrument. Under these condi- tions the galvanometer needle will be highly sensitive to any small change in the current strength. 101. In the case of Thomson's test with the slide wire, if the test is made by using a permanent magnet in the manner described, it is best to make d of a higher value than would otherwise be the case; for then, since the slider would have to be set near the centre of the wire, a greater range of adjustment is given to it, for 5 divisions near the centre portion of the wire (500 division mark) is equivalent to only 1 division near the 100 division mark. It is true that the arrangement is not quite so sensitive as when the slider has to be set towards the end of the scale; but still if sufficient sensitiveness be obtained, the small loss is more than compensated for by the advantage gained in having an increased range on the scale. 102. In order that satisfactory results may be obtained in the foregoing tests, it is necessary that the galvanometer be "sensi- tive” (page 66), otherwise even a moderate degree of accuracy cannot be assured. DIMINISHED DEFLECTION DIRECT METHOD. 103. This method, which has been generally described in Chapter I. (§ 5, page 3), is as follows:-The galvanometer G, a battery of low resistance, and a resistance p, are joined up in simple circuit; the deflection obtained is noted. Let this deflec- 102 HANDBOOK OF ELECTRICAL TESTING. tion be due to a current C₁, then calling E the electromotive force of the battery, we have E C₁ or, C₁ G+ C₁p = E. G+ P The resistance p is now increased to R, so that a new deflec- tion due to a current, C2, is produced; then we have hence C₂ E G+R' or, C₂ G+C₂ R = E; 2 or therefore C₁ G+ C₁P = C₂ G + C₂ R, 1 G (C₁ — C₂) = C₂ R - C₁ P₂ G = C₂R - C₁P C₁ – C₂ [A] In the case of a tangent galvanometer, if the deflections, D and d, are read from the tangent scale, then those deflections can be directly substituted for the quantities C1, C2, for in this case, then, we have D:d: C₁: C₂; G = dR-D p D-d DP. (1.) For example. [B] With a tangent galvanometer whose resistance G was required, and a battery of very small resistance, we obtained with a resistance of 10 ohms (p) in the circuit, a deflection of 60 divisions (D) on the tangent scale of the instrument; when the resistance was increased to 230 ohms (R) the deflec- tion was reduced to 20 divisions (d); what was the resistance of the galvanometer? G 20 × 230 – 60 × 10 60 · 20 = 100 ohms. If the readings are made from the degrees scale, then we must substitute the tangents of the deflections for the deflections themselves; the formula then becomes Ꮐ tan đ° R – tan D°p tan D° tan do [C] J ! 1 1 1 1 I 2 1 t F MEASUREMENT OF GALVANOMETER RESISTANCE. 103 - (2.) For example. In a measurement similar to the foregoing, the readings were made from the degrees scale of the instrument, and deflections of 50° (D°) and 213° (d°) respectively were obtained with resistances of 10 ohms (p) and 229 ohms (R) in the circuit. What was the resistance of the galvanometer? therefore G tan 50° 1.1918, = · tan 213° = •3990, • 3990 × 229 — 1·1918 x 10 1.1918. ⚫3990 = 100 ohms. 104. If in equations [B] and [C] we have p = 0, that is to say, if we make the test by having at first no resistance in the circuit except that of the galvanometer itself, then we get and GR d D - d tan do G = R - tan D° tan do [D] [E] 1 105. What are the "Best conditions for making the test?" and, what is the "Possible degree of accuracy attainable?" There are two points to be considered in the first question; one is-what value should p have? and the other-what should be the relative values of C₁ and C₂? Now we are liable to make an error in reading the value of C₁, or an error in reading the value of C₂, or again we may make errors both in C, and C₂, but inasmuch as the result of two errors would, of course, be greater than one only, it is advisable to make the test under conditions which ensure the result of the double error being as small as possible. Let us, therefore, in equation [A] suppose that there is a small error, c₂, in C2, and a small error, c₁, in C₁, the error c₂ being plus and c₁ minus, so that the resulting total error in G is as great as pos- sible; also let λ be this total error, that is, let us have G + λ = (C2 + c₂) R − (C₁ — c₂) p or λ= (C1 − ¢1) − (C₂ + €2) - - – (C₂ + c₂) R − (C₁ — c₂) P - G ; (C₁ c₁) (C₂ + C₂) - 104 + HANDBOOK OF ELECTRICAL TESTING. but G 2 C₂ R - C₁ P₂ C1-C2 or, R = G (C₁ − C₂) + C₁P C2 If we insert this value of R in the above equation, and multiply up, cancel, etc., then we get or, since C1 λ = (C1 C2 + C₂ C1) C2 [(C1 − c1) — (C₂ + C₂)] - (G+p); [F] and c₂ are very small, we may say λ C2 (C1 C1 C2 + C₂ C1 C₂) (G+p). From this equation we can see that if C₁ and C₂ have fixed values, then A varies directly as G+p, consequently in order to make as small as possible, we must make p as small as possible; but we can also see that there is no great advantage in making p very much smaller than G. We have next to consider what the relative values of C₁ and C₂ should be, p being taken as constant. In order to do this, we must assume C₁ to be constant, and then determine what value C₂ should have. We have then in equation [F] to find what value of C, makes λ as small as possible; to do this we require to make C₁ C₂ + C₂ C1 Į I I C₂ (C1 – C2) as small as possible by variation of C2. Now C1 C2 C1 C1 C2 - G0+ - 0) = 8 [C₁ = ²² + 0 (x+1) C2 (C1 – C2) C1 where k = ¹; and since C2 to make C₁ - C₂ ·2] +K+2 C2 is constant, what we have to do is [C₁ = 0₂+ O₂ (x + 1) + x + 2] as small as possible. Now C1 – C2 C2 K C₁ C₂ 1. [C₁ = C² + C₂ (x + 1) + x + 2 ] = C = C [1 - C₂ √ x + 1] (k+1) C₁- K C2 +2√x+1+x+2, 2 +1- K C₁ - C₂ 2 * 1 Tr 1 L 1 } P H 37 MEASUREMENT OF GALVANOMETER RESISTANCE. 105 and in order to make the latter as small as possible we must make 1 C₂ √ K + 1 2 К C1 – C2 as small as possible, that is to say, we must make it equal to 0, therefore 2 C₂ √ K + 1 1 = C₁ — С2 0, or, C₁ — C₂ = C₂ √√ k + 1, 2 from which we get C₂ (√√ k + 1 + 1) = C₁, or, C2 C₁ √x+1+1 [G] The greatest possible value which could have would be that which would result when both the errors c₁ and c₂ existed, these two errors being of equal value, or rather c₂ being as large as c₁. If the deflections are read in divisions, then c₁ and c₂ would be equal; but if the deflections are read in degrees, then will be larger than c₂, in proportion as C₂ is smaller than C₁. In the case where the greatest possible error can exist, that is, when c₂ C₁, or k = 1, then we have 2 C₁ C2 √2+1 2.4142 Practically we may make C₂ = 01 C₁ 3 for although this does not give the exact minimum value to λ, yet the difference between it and the actual minimum is very small, thus if C₁ C₂ " 2.4142 then from equation [F] we get λ = c1 C₁ C₁ + C₁ (C₂ 2.4142 C₁ 2.4142 but if 2.4142 (G+p) = &, 5·828 (G+p): C₁ C2 }} P 106 HANDBOOK OF ELECTRICAL TESTING. I then λ = C1 C₁ + C₁ 3 0+ (c₁ - 04) 3 (G+ p) = &, 6·000 (G+p); that is to say, the errors would be as 6.000 to 5.828, a difference which is of no practical importance. If the readings were made from the degrees scale of a tangent galvanometer, then the error c₁ would be larger than the error C2, in which case it would be actually an advantage to make C₂ equal to 1 in preference to making it equal to G 2.4142' C₁ 1 ; thus, if C₁ were, say, 3 times as large as c₂, then the best value to give 2 to C₂ would be C₂ C₁ √ 3 +1 +1 C₁ 3 The rule that C₂ should approximately equal may therefore be taken as the one which would enable satisfactory results to be obtained under all conditions. If the deflections, D, d, are read in divisions, then we must have D d 3 approximately. But if the deflections are in degrees, and we read from a tangent galvanometer, then we must have approximately. tan d° = tan Do 3 106. We have next to consider what is the "Possible degree of accuracy attainable" when p and G have any particular values; this we can ascertain from equation [F]. Let us, then, in this equation put percentages for absolute values, that is to say, let us have λ= ľ 100 of G, or, λ = 100 λ G I J ì + MEASUREMENT OF GALVANOMETER RESISTANCE. 107 then we get λ = C₂ (C1-C2) (C₂ c₂ + C₂ c₂) 100 (1+1). G [H] If the deflections are read in divisions, then the errors in both must be of the same absolute values; let each of these values be 1th of a division, then we must have m 1 λ x = (D+ d) 100 d (D − d) (1+ &). [I] For example. In example (1) (page 102) what would be the degree of accu- racy with which the test could be made? The deflections could be read to an accuracy of of a division. λ (60 +20) 100 20 (6020) (1 + 100) = 2.8 = 2.8 per cent. If the deflections are read in degrees from a tangent galvano- meter, then we must have λ' (tan D° 8₂+tan d° §) 100 tan d° (tan D° tan d°) - (1+1) per cent. where 8, and 8, are of the respective values &₁ = tan D¹º tan D, and, 82 = tan d¹º tan do, 978 m being the possible error in the deflections. For example. In example (2) (page 103) what would be the degree of accu- racy with which the test could be made? The deflections could be read to an accuracy of 1º. and = tan 5010 tan 50° = •0106, S2 = tan 22° tan 2120 = .0050; therefore λ' = (1.1918 x .0050+ ·3990 × ⚫0106) 100 ·3990 (1.1918-3990) 3.6 per cent. (1 + 100): 108 HANDBOOK OF ELECTRICAL TESTING. To sum up, then, we have Best Conditions for making the Test. 107. Make p as small as possible. Make R of such a value that when the deflections, D, d, are in divisions, then D d 3 approximately; and when the deflections are in degrees on a tangent galvanometer, then approximately. tan D° tan do 3 Possible Degree of Accuracy attainable. If the deflections are in divisions, and if we can read their value to an accuracy of th of a division, then we can determine the value of G to an accuracy, λ', of m 1 ท λ (D+ d) 100 d(D-d) 1 + 2) G per per cent. If the deflections are in degrees on a tangent galvanometer, then if we can read their value to an accuracy of th of a degree, we can determine the value of G to an accuracy, λ', of λ = where (tan D° ô, + tan đ° 8,) 100 tan do (tan D° m tan d°) 1 + 1 ) per cent. 81 = tan D1° - 82 — tan Dº, and, §₂ = tan¼° — tan dº. m DIMINISHED DEFLECTION SHUNT METHOD. 108. Referring to Fig. 43, this method is as follows:- The galvanometer G, whose resistance is to be determined, is joined up with a resistance R, a battery E, and a shunt S₁; the deflection obtained is noted; let this deflection be due to a current C₁, then (page 83) we have E S₁ C2 C₂ = = G (S₁ + R) + S₁ R’ MEASUREMENT OF GALVANOMETER RESISTANCE. 109 or C₁ G (S₁ + R) + C₁ S, R S₁ = E. The resistance of the shunt is now reduced to S₂, so that the galvanometer deflection is also reduced; let this new deflection be due to a current C₂, then we must have C₂ G (S₂ + R) + C₂ S₂ R 2 E; es $2 FIG. 43. G R E therefore C₂ G (S₂ + R) + C₂ S₂ R _ C₁ G (S₁ + R) + C₁ S₁ R that is $2 2 S₁ 1 G [C₂ S1 (S2 + R) — C₁ S½ (S₁ + R)] = S₁ S₂ R (C₁ — C2), from which we get G = S₁ S½ R (C1 – C2)' 2 C₂ S₁ (S₂+ R) - C₁ S₂ (S₁ + R)' or C₁ - C₂ G = C₂ (+豆 ​1 C₁ Ꭱ S₁ + 1 [A] R In the case of a tangent galvanometer, if the deflections, D and d, are read from the tangent scale, then we should have D-d G = d R α ( + 1 ) D ( + 1 ) 1 [B] R } 110 HANDBOOK OF ELECTRICAL TESTING. (1.) For example. With a tangent galvanometer whose resistance, G, was required, and a battery of very small resistance, we obtained with a shunt of 200 ohms (S₁), a deflection of 60 divisions (D) on the tangent scale of the instrument; when the shunt was reduced to 25 ohms (S2), the deflection was reduced to 20 divi- sions (d). The resistance, R, was 400 ohms. What was the resistance of the galvanometer? 1 60 - 20 G = 100 ohms. 20 ( + 25 400 60 (₂ (200 + 400 1 Į | T If the deflections are read in degrees, then in equation [B] we must substitute tan D° and tan d° for D and d respectively; we then get tan D° tan do - G = [C] tan d° 2 (515 + 1/1) tan D° R (+ 1 Ꭱ (2.) For example. In a measurement similar to the foregoing the readings were made from the degrees scale of the instrument, and deflections of 50° (D°) and 2130 (d°) respectively were obtained. The values of S1, S2, and R were 200, 25, and 380 ohms respec- tively. What was the resistance, G, of the galvanometer? 4 tan 50° = 1∙1918, tan 213° = •3990, therefore G • 3990 ( (12/5 +380) 1.1918 1 • 3990 = 100 ohms. 1 1.1918 (200 + 330) 109. If we make the test by having no shunt inserted when the first deflection is observed, that is to say, if we have S₁ 1 or, = 0, then equation [B] becomes S₁ = ∞, D-d G = d 1 1 D' [D] + R R ~ MEASUREMENT OF GALVANOMETER RESISTANCE. 111 and equation [C] tan D° tan do G [E] tan do ( 1 1 tan Do + Ꭱ R Further still, if we make R a very high resistance, that is, if in equations [D] and [E] we make simplifications and G = S₂ D 1 Ꭱ R (-1) d 0, then we get the [F] G = S2 S₂ (ttan do D° d° · 1). [G] 110. In order to determine the "Best conditions for making the test," and also the "Possible degree of accuracy attainable,' let us write equation [A] (page 109) in the form, 1 HO G ¢ ( + ) − C ( + 1 ) C₂ +1 C₁ - C₂ R Now this equation is similar in form to equation [B] (page 102) in the last test (Diminished deflection direct method), the only 1 difference being that we have instead of G, and ( 1 1 + Ꭱ G 1 + R and instead of R and p, respectively; and inasmuch as an 1 λ' per cent. error in is an A' per cent. error in G (though of G the opposite sign), we can see that the value of λ' must be expressed by an equation of the same form as equation [H] (page 107), that is to say, we must have λ' = 2 c₁) (C. c + C) 100 [1 + G (+1)] per C₂ (C1 – Ꭱ per cent. [H] We can see, therefore, from the investigations in the last test that we must have 112 HANDBOOK OF ELECTRICAL TESTING. Best Conditions for making the Test. 111. Make S, and R as large as possible* (§ 107, page 108). Make S₂ of such a value that when the deflections, D and d, are in divisions, then D d 3 approximately; and when the deflections are in degrees on a tangent galvanometer, then approximately. tan D° tan d° = 3 Possible Degree of Accuracy attainable. If the deflections are in divisions, and if we can read their value to an accuracy of th of a division, then we can deter- mine the value of G to an accuracy, λ', m 1 1 m λ = (D+ d) 100 G (3 G of 2 + 2) 100 [ 1 + 0 ( 3 + 1 ) ] pe [1 d (D − d) R per cent. If the deflections are in degrees on a tangent galvanometer, then if we can read their value to an accuracy of th of a degree, we can determine the value of G to an accuracy, X', of m ľ (tan D° 8, +tan d° 8₁) 100 tan d° (tan D° - tan d°) 1 + + G ( + 1 Ꭱ per cent. where 8₁ = 8₂ = tan d¹º - tan do. tan Do - 82 tan Dº, and, m m 112. It may be remarked, that in the foregoing methods unless the galvanometer under measurement has a high degree of "sensitiveness" (page 66), then even a moderate degree of accuracy in making the tests cannot be assured. * The investigations in the case of the last test prove that we should make + as small as possible; this, of course, is equivalent to making S₁ 1 S, and R as large as possible. 1 ( 113 ) CHAPTER VI. MEASUREMENT OF THE INTERNAL RESISTANCE OF BATTERIES. HALF DEFLECTION METHOD. 113. On page 5 a formula is given for determining the re- sistance r of a battery, viz. :— r = R − (2 p +G), where G is the resistance of the galvanometer employed to make the test, p a resistance which gave a certain current through the galvanometer, and R a larger resistance which caused the strength of this current to be halved. As this, though a simple, is a very good test, and is one which is very frequently made use of, a numerical example may prove of value. For example. With a galvanometer whose resistance was 100 ohms (G), and a battery whose resistance (r) was to be determined, we obtained with a resistance in the resistance box of 150 ohms (p), a deflection representing a current of a certain strength, and on increasing p to 600 ohms (R), we obtained a deflection which showed the current strength to be halved. What was the resistance of the battery? r = 600 − (2 × 150 + 100) : = 200 ohms. ; To avoid mistakes, it should be carefully observed that in working out the formula we "First double the smaller resistance to the result add the resistance of the galvanometer, and deduct this total from the greater resistance: 114. A very common method of making this test is to employ a galvanometer of practically no resistance, and to take the first deflection with no resistance in the circuit except that of the battery itself. In this case (2 p + G) = 0, so that r = R or the added resistance is the resistance of the battery. ↓ I 114 1 HANDBOOK OF ELECTRICAL TESTING. 115. If we compare the first method (§ 113) with the test for determining the resistance of a galvanometer described on page 79 (§ 76), we can see that the two are almost identical. In the one case we determine the resistance of the galvanometer, and in the other we determine the resistance of the battery plus the galvanometer, and then from the result deduct the value of the galvanometer. This being so, we can see that the Best Conditions for making the Test are obtained by making p + G a fractional value of r; to do which we should require a galvanometer of low resistance. As regards the possible degree of accuracy attainable, we can see from the galvanometer test referred to, that λε = 21+ 2 ( 1 + c ) √ ; r; 2: that is to say: Possible Degree of Accuracy attainable. If we can be certain of the value of the galvanometer deflec- tion to an accuracy of y' per cent., then we can be certain of the accuracy of the value of r within 2 (1 + G + =) y per cent. Or if we employ a galvanometer of low resistance, then we can be certain of the accuracy of the value of 2 within 2 y' per cent. If the galvanometer deflection be too high, i.e., above about 55° (page 28, § 32), with the lowest value we can give to p, then the galvanometer must be reduced in sensitiveness by being shunted, and the value of G in the formula will then be the combined resistance of the galvanometer and shunt, that is, the product of the two divided by their sum (page 70). THOMSON'S METHOD. 116. Fig. 44 shows the theoretical, and Fig. 45 the practical methods of making this test. The theory of the method is as follows: The galvanometer G, a resistance p, and the battery whose resistance r is required, are joined up in simple circuit with a shunt S between the poles of the battery; a deflection of the galvanometer needle is produced with a resistance p in the resistance box. The shunt is now removed; this causes the deflection to become larger; INTERNAL RESISTANCE OF BATTERIES. 115 p is then increased until the deflection becomes the same as it was at first. Let the new resistance be R, and let E be the electromotive force of the battery and C the current passing through the galvanometer. FIG. 44. محمد ееее P S G D Α' In the first case we have C = FIG. 45. 바 ​ C A B E S (p + G) r + S+p+ G ES X S S+p+ G r (S + p + G) + S (p + G)² and in the second case therefore E or E C r + R + G ³ r+R+G ES r(S+p+G) + S (p + G) By multiplying up and cancelling, r (p + G) = S (R − p), r = s R - ρ P + G E B' G I 2 116 HANDBOOK OF ELECTRICAL TESTING. For example. A battery whose resistance (r) was required, was joined up in circuit with a resistance of 200 ohms (p) and a galvanometer of 100 ohms (G), a shunt of 10 ohms (S) being between the poles of the battery. On removing the shunt it was necessary, in order to reduce the increased deflection to what it was originally, to increase to 3200 ohms (R). What was the resistance of the battery? r = 10 3200 200 200 + 100 100 ohms. P 117. The investigation for determining the best resistances to employ in making this test would be conducted in precisely the same manner as that given on page 85, et seq. For the equation is the same as r = S R ρ P+ G r = S (R + G) − (p + G) P + G which is the same kind of equation as the one in the test we have referred to, viz. :— Ꭱ G = SR-P. ρ ; Ρ and as in this case we proved that S was to be as small and R as large as possible, so from the preceding equation we should prove that S should be as small, and R + G as large, as possible. In order, therefore, to obtain the Best Conditions for making the Test, 118. First make a rough test to ascertain approximately what is the value of r. Having done this, insert a shunt (S) between the poles of the battery, of less resistance than r. Next join up p in circuit with G, with the battery, and with S its shunt §, making p + G not larger than (G+R); R being S, G the highest resistance that can be inserted in the circuit. The galvanometer needle being obtained at the angle of maximum sensitiveness, note the value of ρ. Now remove the shunt and increase p to R, so that the increased deflection becomes the same as it was at first. Note R and calculate r from the formula. INTERNAL RESISTANCE OF BATTERIES. 117 Possible Degree of Accuracy attainable. From the galvanometer test referred to, we can see that if we can determine the value of the galvanometer deflection to an accuracy of y' per cent., then we can determine the accuracy of r to an accuracy of (1+) (1++) per cent. G) Y G) 7' R+ 119. As we cannot in this test vary the resistance of the galvanometer so as to obtain the deflection at the angle of maximum sensitiveness, we must, if the deflection be too high with the highest resistances we can put in the circuit, reduce its sensitiveness by means of a shunt between its terminals; the value of G in the formula will then be the combined resistance of the galvanometer and its shunt. The constancy of a battery being much impaired by its being on a circuit of low resistance, it is not advisable to reduce the deflection of the galvanometer by making S very small. In fact S, although it should be lower than the resistance of the battery, should not, in this test, be made lower than we can help. Thus, if the resistance of the battery were about 200 ohms, it would be preferable to make S 100 rather than 10 ohms. Should the deflection of the galvanometer needle be too low, the only thing to be done is to use another which has a higher figure of merit. 120. A Thomson galvanometer answers very well for tests like this, as its figure of merit can always be made sufficiently low by placing a shunt inade of a short piece of wire between its terminals. 121. If we adjust p in the first place so that together with G it equals S, we get the simplified formula 1 = S R P S = R − p ; that is, the added resistance is the resistance of the battery.* Again, if we commence with no other resistance in the galvanometer circuit beyond that of the galvanometer itself, we get the simplification Ꭱ r = S G * Sabine's 'The Electric Telegraph,' p. 314. 118 * HANDBOOK OF ELECTRICAL TESTING. Lastly if we make S = G, then we get r = R. If we arrange the tests, however, so as to use these simplified formulæ, we are obliged to employ an arrangement of resist- ances which would not be at all advisable if we wish for accuracy, and it is very questionable whether any advantage is gained by adopting a simplification of a formula, in itself simple, at the expense of accurate testing. The arrangement of keys described in § 91, page 92, may obviously be applied to the foregoing tests with advantage; in fact, the key suggested by Professor Moses Farmer* was first applied by that gentleman to the last test mentioned, viz., that in which the resistance of a battery is given by the formula r = R. SIEMENS' METHOD. 122. Fig. 46 shows the arrangement of resistances, &c., for determining the resistance of a battery by Siemens' method. FIG. 46. A α B م B, C R G A C is a resistance on the slide principle (§ 17, page 15), R a resistance connected to the junction of the galvanometer G and the battery whose resistance r is required. The other end of R is connected to the slider B. Now it will be found that if B be moved towards A or towards C from a certain point midway between A and C, the current flowing through the galvanometer will be increased. It follows from this that if we put B near A and obtain a certain deflection, we can also obtain this same deflection by sliding B to a point near C. Let B and B₁ be these points, and let a be the resistance between A and B, b the resistance between B, and C, and p the resistance between B and B₁. Also let E be the electro- motive force of the battery, and r its resistance, and let C be the current deflecting the galvanometer needle. * Electrical Review,' I age 316, Sept. 24th, 1886. i A x² INTERNAL RESISTANCE OF BATTERIES. 119 Now when the slider is at B C r+a+ E R R (p+b+G) × R+p+b+G R+p+b+G ER (r + a) (R + p + b+G)+R (p+b+G)' and when the slider is at B₁ therefore therefore C Ꭼ Ꭱ (r + a + p) (R + b + G) + R (b + G); (r + a) (R+p+b+G)+R (p + b + G) (r + a + p) (R+ b + G) + R (b + G); from which or = (r + a) p + R p = p (R + b + G); r+a=b+G r = G+b-a. 1 In making this test, then, what we have to do is to note what are the values of A B (a) and B, C (b) when the same deflec- tions are obtained on the galvanometer, then from these values and the resistance of the galvanometer we can determine the resistance of the battery. 123. Another way of making the test is to find the point be- tween A and C which gives the least deflection; then a and b will be the resistances on either side of this point. 124. Let us now consider what are the "Best conditions for making the test." The points to be considered are, what are the best resistances to make R and A C, and also, at what point should we place the slider to commence with, that is, should we place it near one of the ends of A C or at some point nearer the middle of the latter? From the equation 1° = G+ b a it is clear that any error made in b or a will make an exactly corresponding error in r; in considering the problem, therefore, we have simply to determine what arrangement of resistances, &c., will cause any slight error in a or b, that is any slight movement of the slider, to make the greatest possible alteration in the current, that is in the deflection of the galvanometer needle. 120 HANDBOOK OF ELECTRICAL TESTING. Let us suppose the slider was at B for the first observation, and let us suppose that when the slider was at that point, a current C flowed through the galvanometer, and that when the slider was moved to B, the current was also C. Further, when the slider was moved a distance à beyond B towards, say, A, let us suppose the current was increased to C+ c. 1 We have then to determine what arrangement of resistances, C &c., will make as large as possible. Now C C ER (r + a) (R + p + b + G) + R (p + b + G)' and we know that consequently C r+ a = b+G; ER (r + a) (R + p + r + a) + R (p + r + a)' and by putting a λ for a, and p +λ for p, we get C + c or, ER (r + a − λ) (R + p + r + a) + R (p + r + a C₁, therefore c = C₁ − C ; с C 510 11 1; therefore C C A(R+p+r+a) (v + a − λ) (R + p +r+a) +R (p+r+a) or, since λ is a very small quantity, we may say or с λ (R+p+r+a) (r + a) (R + p +r+ a) +R (p. + r + a)' C C 11 R(p+r+a) λ r+a+ R + (p + r + α) [A] [B] We will first determine at what point the slider should be placed to commence with. Now if we show at what point it should be placed near A, we determine the point at which it should be placed near C, for INTERNAL RESISTANCE OF BATTERIES. 121 r+ a must equal G+b. What we have to do then is to determine the best value to give to a. To do this we must suppose the resistance AC to be constant, or since r and G are naturally constants, we must have that is, r+a+p+b+G; r+a+p+r+a, equal to a constant, say, K; therefore p+r+a = K − (r + a), therefore, by equation [A], we get C λ (R+ K − (r + a)) C (r + a) (R+ K − (r + a) ) + R (K − (r + a)) A(R+K (r+a)) r + a) (K − (r + a) ) + R K˚ From this we see that the smaller we make (r + a) the larger will be the numerator of the fraction. Also if r+a be less K 2 than (which it must be in the test), the smaller we make it the smaller will be the denominator of the fraction; * con- *This may be proved as follows:- (ra) (K− (r + a)) = (r + a) K − (r + a)²=- 2 K2 4 I² - ((r+a) - 5)². K\2 If in the latter expression we make K r+a = then (r + a) − 5 ) ² = = = 0, 2 which makes the expression as small as possible. K K\2 But if we make r + a either larger or smaller than then((ra) - 1) 2' 2 does not equal 0, but it has a plus value which increases in proportion as we K K make either (r+a) larger than or larger than (r+a); for although 2' 2 K ((r+ a) − 1 ) - value, still 2 in one case will have a positive, and in the other case a negative still ((r + a))*is positive in both cases. If, therefore, we make (r+a) smaller than K 2' the value of the expression referred to, and consequently the value (r+a) (K − (r + a) ), will increase in proportion. ↓ 4 122 HANDBOOK OF ELECTRICAL TESTING. sequently the smaller we make (ra), and therefore a, the larger will C -be. C It is best, therefore, to place the slider to commence with as near to one end of A C as possible. Next we have to determine what value we should give to A C. This we shall do if we determine what value should have. If we write equation [B] (page 120) in the form ρ C λ C 1 r+a+ 1 R 1 + p+r+a с is we can see that r, a, and R being constant, is made as large as possible by making p as small as possible; but we can also see that there is but little use in making p much smaller than a+a, or, as a ought to be small, in making it much smaller than r. Lastly we have to find what value it is best to give to R; this we can also determine from the last equation. We can see from the latter that, î, a, and p, being constant quantities, r, made as large as possible by making R as small as possible; but we can also see that we gain but very little by making R much smaller than r+a, or, as a ought to be small, by making it smaller than r. Actually of course we could not make R extremely small, for the reason that the battery and galvano- meter would then be practically short circuited and a readable deflection could not be obtained. Since 2 + a = G + b, a can only be made small by having G small; it is therefore best to have a galvanometer of as low a resistance as possible, or rather of a resistance not exceeding r. We proved that the slider should be as near one end of A C as possible. The end we can place it nearest to must evidently be the end to which the greatest resistance_is_connected; therefore, whichever value of r or G happens to be the greatest, at the end to which that larger value is connected should the slider be placed, to commence with. In order to determine the "percentage of accuracy attain- able" we must in equation [B] (page 120) put percentages A' and y' for the absolute values λ and c, that is to say, we must have ľ λ= of r, and, 100 ŕ C = of C, 100 | + 1 | T よ ​1 INTERNAL RESISTANCE OF BATTERIES. 123 in which case we get x' = [r+a+ per cent. R(p+r+r) R+p+r+a To summarise the results, then, we have Best Conditions for making the Test. 125. The slider at commencing should be as near as possible to that end of AC to which is connected the greatest of the values r and G. The value of A C should be not less than the value of the greater of the two quantities r and G. R should be lower than the greater of the two quantities r and G. The galvanometer resistance should not exceed r, and the deflection should be obtained at the angle of maximum sensi- tiveness. This can be done by varying K; but inasmuch as the latter should be lower than r, it is desirable to use a galvano- meter of such sensitiveness that R can be made sufficiently small without reducing the deflection too low. Possible Degree of Accuracy attainable. If we can be certain of the galvanometer deflection to an accuracy of y' per cent., then we can be certain of the value of r to an accuracy, λ', of λ [ Ꭱ r + a + R + p +r+a B (p +r+a) ] = 2° per cent. If R, a, and p are very small compared with r, then we get x' = y'. 126. As in previous tests, we should first determine the value of r roughly and then more exactly with the resistances properly arranged. 127. We have hitherto supposed AC to be a slide resistance, but it is not absolutely necessary that it should be so; the test can very well be made in the following manner: Referring to the figure, and supposing r to be greater than G, let the resistances p and b be ordinary ones and both capable of variation, and let the resistance a be done away with. Having connected R to B, that is, to the pole A of the battery, plug up all the resistance in b and adjust p and R till the deflection of maximum sensitiveness is obtained on the galvano- 124 HANDBOOK OF ELECTRICAL TESTING. meter. Care must be taken that the adjustment of p and R is so made that R is less and p greater than G. If the galvano- meter has a sufficiently high figure of merit, there will be no difficulty in doing this. 1 Next shift the connection of R from B to B₁ and proceed to adjust b and p until the original deflection is reproduced, the adjustment being made in such a manner that the same resist- ance is plugged up in p that is unplugged in b; then r = G + b. It must be noted that of the two quantities G and r the one which has the greatest resistance must be connected to p at B. In the case we have considered we have supposed that r was the larger quantity, but if G had been the larger of the two the position of G and r would have had to have been reversed, and the resistance of r would have been given by the formula r = G − b. The modus operandi of the test would, however, be precisely the same in the two cases. Two sets of resistance coils are evidently necessary to make this test, as it cannot be made with a single set of the ordinary kind (Fig. 6, page 13). MANCE'S METHOD. 128. This test is of a very similar nature to Thomson's method of determining the resistance of a galvanometer given on page 93. Fig. 47 shows the theoretical method of making the test. In the theoretical figure, a, b, and d are resistances, ga galvanometer, and E the battery whose resistance r is required. A key is inserted between the junctions of a with b`and d with r. By depressing this key the junctions are connected together. Let us first suppose the key to be up, then the current C₁ flowing through the galvanometer will be E a+b C₁ X r + d + (a + b) g a + b + g a+b+ g E (a + b) g (a + b + d + r) + (a + b) (d + r) [1] INTERNAL RESISTANCE OF BATTERIES. 125 Next suppose the key to be pressed down; then the current C₂ flowing through the galvanometer will be C₂ = E b d α X b + d +g) a bd +g+ a b + d r+ b d +g+a b + d E(b + d) a g (a + r) (b + d) + bd (a+r) + ar (b + d) FIG. 47. C BE a T [2] A K Now if the resistances be adjusted so that the deflection of the galvanometer needle remains the same whether the key is depressed or not, then equations [1] and [2] are equal; that is E (a + b) g (a + b + d + r) + (a + b) (d + r) E (b + d) a g (a + r) (b + d) + b d (a + r) + ar (b + d)' Now if we refer to "Thomson's galvanometer resistance test" on page 94, we can see that this equation is similar to equa- tion [3] on that page, with the exception that r and g`are interchanged. It must therefore be obvious, by the same development of the equation as that given on the page referred to, that r+add 126 忄 ​HANDBOOK OF ELECTRICAL TESTING. 129. The great advantage of this test is that the electro- motive force of the battery need only be constant during the very short interval of time occupied in depressing and raising the key. 130. In making the test practically the connections would be made as shown by Fig. 48. Terminals E and B' would be joined by a short piece of thick wire; the other connections are obvious. FIG. 48. D A B C E 0 J 1 Α B' The left-hand key puts the galvanometer on; this key must be depressed and held permanently down, and the right-hand key then alternately depressed and raised and the resistance d, that is the resistance between A and E, at the same time ad- justed until the deflection of the galvanometer needle remains the same whether the key is up or down. 131. Again referring to Thomson's galvanometer resistance test; it must be clear, by substituting r for g in the equations, that to obtain the Best Conditions for making the Test, Make a as low as possible and b as high as possible, but not so high that d when exactly adjusted would exceed all the resistance we could insert between D and E (see Fig. 48). Adjust d approximately and then, if necessary, adjust the resistance of the galvanometer shunt (which it will be necessary to employ) so that the final deflection is as nearly as possible that of maximum sensitiveness, and then, having exactly adjusted d, calculate r from the formula. 1 INTERNAL RESISTANCE OF BATTERIES. 127 * : Possible Degree of Accuracy attainable. If we can determine the value of the galvanometer deflection to an accuracy of y' per cent., then we can be certain of the value of r to an accuracy of (1+) (1+) y per cent. 132. In the practical execution of the test with the set of resistance coils shown by Fig. 48, the lowest value we could give to a would be 10 units, unless we improvised a resistance of less value, which it might be necessary to do. MANCE'S METHOD WITH THE SLIDE WIRE BRIDGE. 133. Mance's test is sometimes made by having a + b a slide wire resistance, d being a fixed resistance; in this case the slider would be moved along between A and C until the point is found at which the depression or raising of the key makes no alteration in the deflection of the galvanometer needle. "" For practically executing the test the apparatus known as the "Slide Wire or "Metre Bridge" may be used. This apparatus, which is shown by Fig. 49, is described in Chapter m, FIG. 49. d m₂ a S Ъ VIII. (The Wheatstone Bridge). The slide wire, a + b, which is 1 metre long, is stretched upon an oblong board (forming the base of the instrument) parallel to a metre scale divided throughout its whole length into millimetres, and so placed that its two ends are as nearly as possible opposite to divisions 0 and 1000 respectively of the scale. The ends of the wire are soldered to a broad, thick copper band, which passes round each end of the graduated scale, and runs parallel if 128 HANDBOOK OF ELECTRICAL TESTING. to it on the side opposite to the wire. This band is interrupted by four gaps, at m₁, r, d, and m₂. On each side of these gaps are terminals. In making the test under consideration, the gaps, m₁ and m2, are closed by thick copper straps. slider S makes contact with the slide wire by the depression of a knob on S. The The battery, r, a resistance, d, and a galvanometer, g, being joined up as shown, the slider S is moved along the scale, the knob being depressed at intervals, until the point is reached at which the depression makes no change in the permanent deflec- tion of the galvanometer needle. When this is the case, then, as in Thomson's galvanometer test (page 93), we have L α = d 1000 - α For example. In the foregoing test, equilibrium was produced when d was 1 ohm, and a, 450 divisions; what was the resistance, r, of the battery? r = 1 450 450 ⚫85 ohm. 1000 450 550 - 134. The best conditions for making the test are similar to those required for "Thomson's galvanometer test" (page 93), namely, we should make d larger than r, but not greater than about 10 times r. As a rule the complete slide wire bridge is furnished with but four resistance coils of 1 ohm each, so that the choice of a resistance to insert in d is limited, and it may not be possible to follow out the rule of "making d about 10 times as large as r." In this case the possibility of an accurate measurement becomes proportionately reduced below the highest possible standard, so that on the one hand a cell whose resistance is much less than one-tenth of an ohm, or, on the other hand, a cell whose resist- ance exceeds 4 ohms, cannot be measured with the highest possible accuracy. Strictly speaking (as has been pointed out) in order to ensure accuracy it is necessary that the resistance of the portion of the slide wire, a, be less than the resistance of the battery to be measured; but as the resistance of the whole length of the wire will not exceed one-tenth of an ohm, the resistance of the length, a, will practically be less than the resistance of the battery, unless, of course, this resistance is extremely low. # לן -- INTERNAL RESISTANCE OF BATTERIES. 129 The possible degree of accuracy attainable we can see from Thomson's galvanometer test (page 93) must be given by the equation λ = 100000 S a (1000 a) per cent. where 8 is the degree of accuracy in divisions to which the slider, S, can be adjusted. If we can adjust to an accuracy of 1 division, then 8 = 1. For example. In the last example, what would be the degree of accuracy, X', with which the value of r could be obtained, supposing that the position of the slider could be determined to an accuracy of 1 division (8)? λ 100000 × 1 450 (1000 — 450) - = .40 •40 per cent. 135. The facility and accuracy with which all the foregoing tests (except the half-deflection test) can be made may be greatly increased by the following arrangement: Use a galvano- meter with a high "figure of merit" (page 65), and instead of making the test with the needle brought to the "angle of maximum sensitiveness" (page 23), make it with the needle brought approximately to zero by means of a powerful per- manent magnet set near the instrument; under these conditions the galvanometer needle will be highly sensitive to any small change in the current strength. Another arrangement which may be very conveniently adopted is to employ a galvanometer with a high "figure of merit," and wound with two wires. One of these wires would be joined in circuit with the battery under test, &c., in the usual way; the other would be connected in circuit with a small battery and a set of resistance coils, the connections being so made that the currents through the two coils oppose one another. When the deflection due to the battery under test is obtained, the second battery and resistance coils are connected up, and then this battery is adjusted until the needle is brought to zero as nearly as possible. The test is then made, as in the case where a permanent magnet is used. 136. In the case of Mance's test with the slide-wire bridge, if the test is made either by using a permanent magnet in the way described, or by using a galvanometer wound with a double wire, it is best to make d as nearly equal to the resistance of the battery as possible (it should not be made less), as in this case, since the slider, S, will have to be set near the centre of K 130 HANDBOOK OF ELECTRICAL TESTING. the scale, a greater range of adjustment is given to it, for 5 divisions near the centre portion of the scale (500 division mark) are equivalent to only 1 division near the 100 division mark. It is true the arrangement is not quite so sensitive as it would be if the slider were set towards the end of the scale; but still, if we can employ a galvanometer with a high figure of merit, this small loss of sensitiveness is more than compensated for by the increased range which can be obtained on the scale. 137. In order that satisfactory results may be obtained in any of the foregoing tests, it is very necessary that the galvano- meter used be a sensitive" one (page 66), otherwise even a moderate degree of accuracy cannot be assured. DIMINISHED DEFLECTION DIRECT METHOD. 138. This method, which has been generally described in Chapter I. (§ 6, page 4), is as follows:- The battery whose resistance, r, is required, a galvanometer of resistance, G, and a resistance, p, are joined up in simple circuit; the deflection obtained is noted. Let this deflection be due to a current, C₁, then calling E the electromotive force of the battery, we have E C₁ r + G + p or, C₁(r+G) + C₁ p = E. The resistance, p, is now increased to R, so that a new deflection due to a current, C2, is produced, then we have C₂ = hence or therefore E r+G+R' or, C₂ (r+G) + C₂ R = E; C₁ (r+G) + C₁p = С₂ (r + G) + C₂ R, (~ + G) (C1 − C₂2) = C₂ R — C₁p; } → 1 7 that is r+G= C₂ R - C₁ p C₁ - C₂ 2 C₂ R - C₁ p - G. C₁- C₂ [A] If a tangent galvanometer is employed for making the test, then if the deflections, D and d, are read from the tangent scale INTERNAL RESISTANCE OF BATTERIES. 131 of the instrument, those deflections can be directly substituted for the quantities, C1, C2, for : D :d : : C₁ : C₂; in this case, then, we have dR-Dp - G. [B] D-d (1.) For example. With a tangent galvanometer whose resistance was 10 ohms (G), and a battery whose resistance, r, was required, a deflection of 60 divisions (D) on the tangent scale of the instrument was obtained, when a resistance of 10 ohms (p) was in circuit; when the latter resistance was increased to 230 ohms (R) the deflection was reduced to 20 divisions (d). What was the resistance of the battery? 20 × 230 60 × 10 1 = 10 = 90 ohms. 60 · 20 If the readings are made from the degrees scale, then we must substitute the tangents of the deflections for the deflections. themselves; the formula then becomes g tan d° R tan D° tan D° Ρ tan d° G. [C] (2.) For example. In a measurement similar to the foregoing the readings were made from the degrees scale of the galvanometer, and deflections. of 50° (D°) and 21° (d°) respectively were obtained with resistances of 10 ohms (p) and 229 ohms (R) in the circuit. The resistance of the galvanometer was 10 ohms (G). What was the resistance, r, of the battery? t tan 50° therefore 1.1918, tan 213 = ·3990, ⚫3990 × 229 ↑ = 1.1918 1.1918 × 10 • 3990 = 10 90 ohms. 139. If in equations [B] and [C] we have p = 0, that is to say, if we make the test by having at first no resistance in the K 2 132 HANDBOOK OF ELECTRICAL TESTING. circuit except that of the galvanometer and the battery itself, then we get and d r = R G D-d [D] tan d r = R - G. tan D tan d [E] 140. In order to determine the "Best conditions for making the test," and also the "Possible degree of accuracy attainable," let us write equation [A] in the form r = C₂ (R + G) − C₁ (p + G) C₁ - C₂ Now this equation is similar to equation [B] (page 102) in the "Diminished deflection direct method" of determining the resistance of a galvanometer, except that in the latter method we have the quantities R and p in the place of the quantities (R+G) and (p + G); consequently we can at once see from the investigation in the test referred to that we must have- Best Conditions for making the Test. 141. Make p as small as possible. Make R of such a value that when the deflections, D, d, are in divisions, then D d 3 approximately; and when the deflections are in degrees on a tangent galvanometer, then approximately. tan D° tan d° = 3 Possible Degree of Accuracy attainable. If the deflections are in divisions, and if we can read their value to an accuracy of th of a division, then we can determine the value of r to an accuracy, λ', of m ľ / (D+ d) 100 d (D − d) 1 + P+G) per cent. INTERNAL RESISTANCE OF BATTERIES. 133 If the deflections are in degrees on a tangent galvanometer, then if we can read their value to an accuracy of th of a degree, we can determine the value of G to an accuracy, λ', of λ'= where m ´tan D° 82 — tan d° 8₁) 100 tan d° (tan D° tan d°) (1+P+G); per cent. - &₁ = tan D¹º - tan D°, and, d = tan d¹º tan ď. m DIMINISHED DEFLECTION SHUNT METHOD. FIG. 50. 142. This method is shown by Fig. 50. The battery, ?, whose resistance is to be determined, is joined up in circuit with a resistance, R, a galvanometer, G, and a shunt, S₁; the deflection obtained is noted; let this deflection be due to a current C₁, then calling E the electro- motive force of the battery, we have (page 115) C₁ = or E S₁ r (S₁ +R+G) + S₁ (R + G)° 1 C₂ r (S₁ +R+G) + C₁ S₁ (R+G) – E. S₁ The resistance of the shunt is now reduced to S₂, so that the galvanometer Ꭱ go S G deflection is also reduced; let this new deflection be due to a current C2, then we must have therefore C₂ r (S½ +R+G) + C₂ S½ (R + G) – E; S2 = Ç₂r (S₂+R+G)+C₂S₂(R+G) _ С₁r (S₁+R+G)+C₂S₁(R+G) 2 that is, S₂ S₁ 1 → [C₂S₁(S₂+R+G) − C₁ S₂ (S₁+R+G)] = S₁ S₂ (R+G) (C₁+C2) from which we get r = 2 2 S₁ S₂ (R+G) (C₁ — C₂) C2 S1 (S2 + R + G) − C₁ S₂ (S1 +R+G)' - 134 HANDBOOK OF ELECTRICAL TESTING. or 1 C₁ - C₂ r = [A] 1 1 1 C2 + - R+G R+G In the case of a tangent galvanometer, if the deflections, D and d, are read from the tangent scale, then we should have 1 D-d 1 d + R+G 1 R+G [B] (1.) For example. With a tangent galvanometer whose resistance was 10 ohms (G), and a battery whose resistance, r, was required, we obtained with a shunt of 200 ohms (S₁), a deflection of 60 divisions (D) on the tangent scale of the instrument; when the shunt was reduced to 25 ohms (S₂) the deflection was reduced to 20 divisions (d). The resistance, R, was 710 ohms. What was the resistance of the battery? 60-20 1 = 90 ohms. 20 (12/15 + 710+10) 1 1 60 + 200 710 10. If the deflections are read in degrees, then in equation [B] we must substitute tan D° and tan d° for D and d respectively, we then get tan D° tan do 1 r = [C] tan do ( 1 + R + G) tan D° ( 1 R+G (2.) For example. In a measurement similar to the foregoing the readings were made from the degrees scale of the galvanometer, and deflections of 50° (D°) and 213° (d°), respectively, were obtained. The values of S1, S2, R, and G were 200, 25, 655, and 10 ohms, respectively. What was the resistance, r, of the battery? tan 50° = 1∙1918, tan 212° = •3990. 1 A ! ! 2 JJ ¿ * INTERNAL RESISTANCE OF BATTERIES. 135 therefore 1 1·1918 — ·3990 r= Ꭸ ; · 3990 (₂ + 25' 655+10. 10)-1-1918 1 1 +10) 90 ohms. + 200' 655- 655+10. 143. If we make the test by having no shunt inserted when the first deflection is observed, that is to say, if we have S₁ 1 = ∞, or, = 0, then equation [B] becomes S₁ ጥ 1 D-d 1 ( + x + Q) R+ G d and equation [C] tan Do ↑ = 1 tan do + 2 R+G - 1 D [D] R+G tan do tan D° [E] R+G Further still, if we make R a very high resistance, that is, if in equations [D] and [E] we make 1 O, then we get the R+G simplifications r = S. 8, -1) [F] and tan D° r = S2 tan do - 1). [G] 144. If we refer to the " Diminished deflection shunt method" of determining the resistance of a "galvanometer" we can see that equation [A] (page 109) in that test is almost precisely similar to equation [A] (page 134) of the present test, the only difference being that in the latter we have in the place 1 of consequently we must have— R' 1 R+G Best Conditions for making the Test. Make S, and R as large as possible. Make S₂ of such a value that when the deflections, D, d, are in divisions, then d D 3 136 HANDBOOK OF ELECTRICAL TESTING. approximately; and when the deflections are in degrees on a tangent galvanometer, then tan D° tan do = 3 approximately. Possible Degree of Accuracy attainable. If the deflections are in divisions, and if we can read their value to an accuracy of th of a division, then we can deter- mine the value of r to an accuracy, X', of 1 x' = (D+ d) 100 λ 90 m r per cent. d(D-d) [1+,+R+G)] per If the deflections are in degrees on a tangent galvanometer, then if we can read their value to an accuracy of th of a degree, we can determine the value of r to an accuracy, X', of tan D° 82 + tan d° 8₁) 100 λ' = tan d° (tan D° & = tan D¹º where m - - tan d°) 772 [1 1+r ( 1 1 + S1 R+ per cent. = tan d¹º tan do. tan D°, and, d₂2 145. In all the foregoing tests it is very necessary that the galvanometer used be a highly sensitive one (page 66), other- wise even a moderate degree of accuracy cannot be obtained. 146. Other methods of measuring the resistance of batteries will be referred to hereafter (see Index); these methods involve principles which can be more conveniently discussed later on. = | 7 ! ¿ 1 7 } r 1 វ 4 1 1 # 1 ( 137 ) CHAPTER VII. MEASUREMENT OF THE ELECTROMOTIVE FORCE OF BATTERIES. 147. THE methods of measuring or comparing the electro- motive forces of batteries are perhaps more numerous than any other class of measurements. Although no absolute standard of the unit of electromotive force (the volt) exists, yet there are several standards of known value with which comparisons may be made. STANDARD CELLS. WHEATSTONE STANDARD CELL. 148. This consists of an outer vessel containing a saturated solution of sulphate of copper; into this is placed a porous cell about 2 inches high, containing mercury with a few scraps of zinc dissolved in it; a cylinder of copper is placed in the copper solution, and connection is made with the zinc amalgam by a copper-wire dipping into it. These cells, although not suitable for continued use, can be relied upon to give a perfectly constant current for half an hour or so, in fact, for quite a sufficient time to enable any ordinary tests to be made; also the electromotive forces of any two of such cells may practically be relied upon as being equal. The porous tubes in these cells, after use, should be thrown into nitric acid for a short time, so as to dissolve any copper which may have become deposited in their pores; they must next be washed in water, and will then be ready for use again. The amalgam can be used over and over again. The electromotive force of a Wheatstone cell is approximately 1.079 volts. POST OFFICE STANDARD CELL. 149. A sectional view of this cell is shown by Fig. 51. The cell is formed with three chambers; in the right-hand one is placed a copper plate, C, immersed in a solution of sulphate of copper, and in the left-hand one is placed a flat porous pot, Z, containing a zinc plate and a semisaturated solu- tion of sulphate of zinc. The two chambers are called "idle ! 138 HANDBOOK OF ELECTRICAL TESTING. cells," as the copper plate and the porous pot and its contents are kept in them when the cell is not in use. The centre chamber contains a solution of sulphate of copper, and crystals of the latter are kept in a small compartment at the bottom to keep the solution concentrated. FIG. 51. When the cell is required for use, the copper plate and the porous pot are removed from their respective idle compartments and are placed in the centre chamber; the cell is then ready for work. When the cell is no longer required, the copper plate and porous pot are again replaced in their respective idle chambers, and whilst the cell is at rest any sulphate of copper which may have entered the pores of the porous pot becomes removed by the slow draining out of the sulphate of zinc solu- tion. The latter collects in the bottom of the idle chamber, from which it is occasionally emptied, the loss from the porous cell being made up by the addition of fresh solution. By this means the liquid in the porous cell remains clear and the zinc free from any deposit of copper. The porous cell is kept in position and raised above the bottom of the idle chamber by supports, one of which, a, is seen in the figure. The block b fixed to the cover prevents the latter being shut down unless the porous cell and its contents are removed to the idle chamber. When in thoroughly good condition this form of cell has an electromotive force of 1.079 volts approximately, but if it is in daily use the power is practically a little less than this; in the 1 T 1 K F ? I - 1 L 7 1 f K 25 Q I ELECTROMOTIVE FORCE OF BATTERIES. 139 Postal Telegraph Department the value is assumed to be 1.07, as being generally nearer the actual value. Although the foregoing cell will last for a considerable time without attention, yet it must not be imagined (as often seems to be the case) that it will keep up its power for an indefinitely long period. After a certain time, to be judged by experience, all its constituent parts should be thoroughly cleaned, the zinc plate scraped, &c. 150. The Wheatstone and Post Office standard cells, although they cannot be relied upon for extreme accuracy, are sufficiently correct for most purposes, and they have the advantage (which is common to all batteries on the Daniell principle) of not losing their power materially when worked through a low resistance. Practically, upon an emergency any form of Daniell cell may be used as a standard, the zinc plate being immersed in a semi-saturated solution of sulphate of zinc, and the electromotive force being taken as 1.079 volts. FLEMING'S STANDARD CELL. 151. This cell, devised by Dr. J. A. Fleming, is thus arranged:- 3 4 A large U-tube, about 2 inch in diameter and 8 inches long in the limb, has four side tubes (Fig. 52). The two top ones, A and B, lead to two reservoirs Z and C, and the bottom ones C and D are drainage- tubes. These side tubes are closed by glass taps. The whole is mounted on a vertical board, with a pair of test-tubes between the limbs. The left-hand reservoir S Z is filled with a solution of sulphate of zinc, and the right-hand reservoir S C with a solution of sulphate of copper. The electrodes are zinc and copper rods, Sulphate of Zinc SZ Zn- Pare Zinc Rod- Zme Sulphate... FIG. 52. and U R Sulphate of ComTYES Electrotype Copper tod Level of Cogtart Liquids C Zn and Cu, passed through vulcanized-rubber corks, P and Q, fitting air-tight, into the ends of the U-tube. The operation of filling is as follows:-Open the tap A and fill the whole U-tube with the denser zinc-sulphate solution; then insert the zinc rod and fit it tightly by the rubber cork P. Now, on opening the tap C the level of the liquid will begin to } 140 HANDBOOK OF ELECTRICAL TESTING. fall in the right-hand limb but be retained in the closed one. As the level commences to sink in the right-hand limb, by opening the tap B copper-sulphate solution can be allowed to flow in gently to replace it; and this operation can be so con- ducted that the level of demarcation of the two liquids remains quite sharp, and gradually sinks to the level of the tap C. When this is the case, all taps are closed and the copper rod inserted in the right-hand limb. Now it is impossible to stop diffusion from gradually mixing the liquids at the surface of contact; but whenever the surface of contact ceases to be sharply defined, the mixed liquid at the level of the tap C can be drawn off, and fresh solutions supplied from the reservoirs above. In this way it is possible to maintain the solution pure and unmixed round the two electrodes with very little trouble; and the electrodes, when not in use, can be kept in the idle cells or test-tubes L and M, each in its own solution. The electrodes are made of rods of the purest zinc and copper, about 4 inches long and inch diameter. The zinc found most suitable is made from zinc twice distilled and cast into rods; the copper is prepared by electro-depositing on a very fine copper wire, until a cylinder of the required thickness is obtained. The value of the electromotive force of the cell depends, to a considerable extent, upon the density of the solutions used. The latter should be as follows: For the zinc solution dissolve 555 grammes of chemically pure sulphate of zinc in 445 grammes of distilled water. This solution should have a specific gravity of 1.4 at 15° C. For the copper solution dissolve 83 grammes of chemically pure sulphate of copper in 417 grammes of distilled water. This solution should have a specific gravity of 1.1 at 15° C. Especial care must also be taken to lightly electrotype the copper rod with a fresh pure surface of new copper the instant before using. This is done in the small copper voltameter which the tube M forms, using a single Leclanché cell for the purpose. The pure zinc rod should be cleaned with new glass- paper. If these precautions are carried out the electromotive force of the cell will be 1.086 volts, which value will be correct within the ordinary ranges of temperature. CLARK'S STANDARD CELL. 152. A cell is formed by employing pure mercury as the negative element, the mercury being covered with a paste formed of mercurous sulphate in a thoroughly saturated solu- ELECTROMOTIVE FORCE OF BATTERIES. 141 tion of zinc sulphate; the positive element consists of pure distilled zinc resting on the paste. The best method of forming the paste is, according to Lord Rayleigh, as follows:-Rub up in a mortar 150 grammes of mercurous sulphate, 5 grammes of zinc carbonate, and use sufficient zinc sulphate solution (not supersaturated) to make a thick paste; leave the whole in the mortar for two or three days, occasionally pounding it up in order to allow the carbonic anhydride which forms to escape. Dr. A. Muirhead, who has had a very lengthened experience with the Clark cells, prefers to make the paste as follows:-A saturated solution of zinc and mercurous sulphates is prepared by heating in the saturated solution of zinc sulphate a portion of the mercurous sulphate, adding thereto a little free mercury to preserve the basicity of the mercurous salt; mercurous sulphate is then mixed into a paste with the solution so prepared. The mercurous sulphate can be obtained commercially; but it may be prepared by dis- solving pure mercury in excess in hot sulphuric acid at a temperature below boiling-point. The salt, which is a nearly insoluble white powder, should be well washed in distilled water, and care should be taken to obtain it free from the mercuric sulphate (persulphate), the presence of which may be known by the mixture turning yellowish on the addition of water. The careful washing of the salt (according to Clark) is a matter of essential importance, as the presence of any free acid, or of persulphate, produces an irregularity in the electromotive force of the cell for some time after charging. The paste is poured on to the surface of the mercury (which should have been distilled in vacuo); a piece of pure zinc is then suspended in the paste, and the vessel sealed up with marine glue (not paraffin wax). Contact with the mercury may be made by means of a platinum wire passing down a glass tube, cemented to the inside of the cell, and dipping below the surface of the mercury, or more conveniently by a small external glass tube blown on to the cell and opening into it close to the bottom. MUIRHEAD'S IMPROVED CLARK STANDARD CELL. 153. The usual forms of the Clark cell, especially when newly set up, are unsuitable for transport; the mercury, being free, is apt to leave the platinum wire contact when the cell is inverted or upset, and to fall through the paste into contact with the zinc rod, thereby either short-circuiting the cell altogether or destroying the value of its electromotive force. To remedy this defect Dr. A. Muirhead constructs the cell as shown by Fig. 53. 142 HANDBOOK OF ELECTRICAL TESTING. A is a flat closely-wound spiral of platinum wire (shown in plan in the lower part of the figure), which has been coated or amalgamated with pure mercury either by boiling it in the latter or by dipping the spiral, when heated red-hot, into mercury; the continuation of the wire is sealed into the glass cell, forming the outer connection. Zn is a rod of pure zinc supported by a cork, c, covered with cement. Inside the cell is placed a paste, p, composed of pure mercurous sulphate and a saturated solution of pure zinc sulphate. FIG. 51. FIG. 53. A STANDARD CELL MUIRHEAD BLONBON Zn Fig. 54 shows a very compact and useful form of this standard. The four terminals belong to two entirely distinct cells, the advantage being that the two cells may be used as a check one upon the other. A thermometer stands within the box, and the stem, being bent at right angles, lies in a groove across the top of the case. By this thermometer the temperature at the time of the reading can be ascertained. 154. The electromotive force of the two foregoing standard cells is, according to recent determinations by Lord Rayleigh, 143 1.454 volts at 15° C. The effect of change of temperature is to change the value of the force about 077 per cent. per degree C., that is to say, the electromotive force at a temperature of t° C. is 1·454 {1 — ·00077 (ť° — 15)} volts. 1.434 ELECTROMOTIVE FORCE OF BATTERIES. 143 The following table shows the electromotive force at various temperatures calculated from the foregoing formula :— Temp. ° C. B. A. Volts. Temp. • C. B. A. Volts. Temp. ° C. B. A. Volts. O123 HILO 0 1.471 1.470 22: 11 1.458 12 1.457 223 1.446 1.445 1.469 13 1.456 24 1.444 1.467 14 1.455 25 1.443 4 1.466 15 1.454 26 1.442 5 1.465 16 1.453 27 1.441 6 1.464 17 1.452 28 1.439 7 1.463 18 1.451 29 1.438 8 1.462 19 1.450 30 1.437 9 1.461 20 1.448 31 1.436 10 1.460 21 1.447 32 1.435 155. In order that the force in the Clark cells may preserve its value constant, care must be taken that the cells are not worked through a low resistance. It is necessary, therefore, in employing them, to take care that they are only used in circuits of a very high resistance, or for charging a condenser, or are balanced by a second battery, as in Clark's electromotive force test (page 181). DE LA RUE'S CHLORIDE OF SILVER CELL. 156. The chloride of silver cells of Mr. Warren de la Rue are said to be remarkably well adapted for standard elements. They will bear a considerable amount of agita- tion without their electromotive force being varied. Fig. 55 shows one of these cells. A is a glass vessel closed by a stopper of paraffin wax. The positive element consists of a cylindrical rod c of chemically pure zinc. The negative element is a cylinder В of chloride of silver, having a silver electrode b cast into it. This cylinder is usually enclosed in a bag of thin parchment paper. The solution for charging the cell is made by dissolving 23 grammes of pure sal- ammoniac in one litre of water. The electromotive force of the chloride of silver cell, according to some careful determina- tions made by Mr. F. H. Nalder, is 1·03 volts. FIG. 55. b A 1/2 full sizes 144 HANDBOOK OF ELECTRICAL TESTING. As in the case of Clark cells, the De la Rue battery when used as a standard must not be worked through a low resistance. ELECTROMOTIVE FORCE MEASUREMENTS. 157. To measure the electromotive force of a battery, we have to compare it with a standard of one or more cells, and having thus ascertained the relative values of the two, the electro- motive force of the battery, in volts, is obtained by an ordinary proportion sum. For example. The relative electromotive forces of a battery and 3 standard Daniell cells was found to be as 1·25 to 1; what was the electromotive force, in volts, of the battery? 1.25: 1 :: 3 x 1·079 : x; therefore 1 x 3 x 1.079 X = = 2.59 volts. 1.25 1 EQUAL RESISTANCE METHOD. 158. Let there be two batteries, whose electromotive forces E, and E2 are to be compared. Join up battery E, with a tangent galvanometer and resistance in simple circuit, as shown by Fig. 45 (page 115). All the plugs between A and C being inserted, the infinity plug between A and D being removed, and the connections being made, depress the right-hand key, and remove a sufficient number of plugs from between D and E to obtain a convenient deflection on, say, the tangent scale of the galvanometer. Note this deflection-let it be d, divisions; and also note the total resistance (R) in circuit—that is, the resistance between D and E, plus the resistance of the galvano- meter, plus the resistance of the battery (which must be deter- mined beforehand). Now remove battery E, and insert battery E, in its place, and if this battery has a different resistance to E₁, readjust between D and E so that total. resistance in circuit is the same as it was at first. Again note the deflection of the galvanometer needle-let it be de divisions. Then if C₁ be the current producing the deflection d₁, and C, the current produc- ing the deflection d₂, we must have by Ohm's law (page 1), E1 E2 C₁ = and, C₂ R' R ELECTROMOTIVE FORCE OF BATTERIES. 145 therefore E₁ : E, :: C₁: C2, or since d₁ and d₂ are directly proportional to C₁ and C₂, we must have For example. E₁: E₂ :: d₁: d2. - With a tangent galvanometer, whose resistance was 100 ohms, and battery Ě₁, whose resistance was 70 ohms, we obtained, with a resistance of 1830 ohms (total, 100 + 70 + 1830 = 2000), in the resistance box, a deflection of 50 divisions on the tangent scale of the galvanometer; and with battery E2, whose resist- ance was 50 ohms, we obtained, with a resistance of 1850 ohms (total, 100 + 50 + 1850 = 2000, as before), in the resistance box, a deflection of 40 divisions; then or as E₁: E2::50: 40, 1.25 to 1. If the deflections are read on the degrees scale of the tangent galvanometer, then d₁ and d₂ must be the tangents of the deflections. 1 In cases where the resistances of the batteries whose electro- motive forces are to be compared are very small, we may, by using a very high resistance, practically regard the total resist- ance in circuit as being the same, whatever battery we use. The deflections then obtained with any number of different batteries will represent their comparative electromotive forces. The galvanometer will, in this case, of course have to be one with a high figure of merit (page 65). 159. The "Best conditions for making the test," and the "Possible degree of accuracy attainable," are almost obvious; they are Best Conditions for making the Test. Make the resistances in the circuits as high as possible. Possible Degree of Accuracy attainable. If we can be certain of the value of the two deflections to accuracies of '1 and S2 per cent. respectively, then we can be certain of the relative values of the two electromotive forces to an accuracy of 8'₁ + 8'2 per cent. L 146 HANDBOOK OF ELECTRICAL TESTING. EQUAL DEFLECTION METHOD. 160. Join up as in last method, and having noted the deflec- tion and total resistance in circuit (R₁) with battery E₁, remove it and insert battery E₂ in its place. Now readjust resistance between D and E, until the deflection of the galvanometer needle becomes the same as it was at first. Note the resistance 2 in circuit (R2); then calling C the current, that is, E₁ E2 C and, C= R₁' R₂² : E2 E, E, R₂ R2, or the electromotive forces of the batteries are directly as the total resistances that are in circuit with the respective batteries. For example. With a galvanometer whose resistance was 100 ohms, and a battery E, whose resistance was 50 ohms, we obtained, with a resistance of 2350 ohms (total, 100 + 50 + 2350 2500), in the resistance box, a deflection of 40°; and with a battery E2, whose resistance was 70 ohms, it was necessary, in order to bring the galvanometer needle again to 40°, to have a resist- ance of 1830 ohms (total, 100 + 70 + 1830 resistance box; then 2000), in the or as 1 E₁: E2 :: 2500: 2000, 5 to 4. An advantage in this test is that it can be made with a gal- vanometer the relative values of whose deflections are unknown. The and the Best Conditions for making the Test Possible Degree of Accuracy attainable are the same as in the last test. WIEDEMANN'S METHOD. 161. In Fig. 45 (page 115) join the zinc pole of battery E₁ to D, as shown, and the other pole to the zinc pole of battery E2, whose other pole in turn is to be joined to C. Adjust the ELECTROMOTIVE FORCE OF BATTERIES. 147 resistance so as to obtain a high deflection on the tangent scale of the galvanometer. Let the current producing this deflection be C; then C E₁ + E₂ Ꭱ where R is the total resistance in the circuit. Now reverse battery E₂ (the weaker one) so that the two batteries oppose one another, we shall then get a smaller deflection due to a current C,; then C, E₁ – E2 Ꭱ From these two equations we get that is, E₁ C – E₂ C = E₁ C, + E₂ C,, E₁ E₂ C+C, CC,, or, substituting deflections d, d,, for current strengths C, C, E₁: E₂d+d, d — d. For example. E2 2 Two batteries E, and E₂ being joined up together in simple circuit, we obtained, by adjusting the resistance in the resistance box, a deflection of 72 divisions (d) on the tangent scale of the galvanometer; and with the same resistance in circuit we obtained, on reversing battery E2, a deflection of 8 divisions (d); then or as 1 - E₁ : E2 :: 72 +8:72 :: 80 : 64, 1.25 to 1. 8, If the deflections are read on the degrees scale of a tangent galvanometer, then d and d, must be the tangents of the deflections. 162. In order to make the test as accurately as possible under the last conditions, the resistance in the circuit should be so adjusted that the two deflections make approximately equal angles on opposite sides of 45° (§ 32, page 29). The more resistance it is possible to place in the circuit of the batteries the better, since the tendency of the latter to polarise is thereby reduced to a minimum. 163. Wiedemann's method is a very satisfactory one since it is absolutely independent of the resistance of the two batteries, L 2 148 HANDBOOK OF ELECTRICAL TESTING. thus one battery might have a resistance of a fraction of an ohm only and the other a resistance of several thousand ohms, yet this would in no way affect the correctness of the results, but to avoid errors due to polarisation it is necessary with some batteries to include several thousand ohms in the circuit; if the galvanometer used be one with a high figure of merit (page 65) this can always be done. 164. The "Possible degree of accuracy attainable" in making the test is greatly dependent upon the relative values of the two electromotive forces. Let us first suppose that the deflections are read in divisions, and let us suppose that there is a possible error & in both deflections. Now if we take both errors to be of similar signs, then we should have a total absolute error of 2 8 in the quantity (d+d,), but if one error were plus and the other minus, then we should have a total absolute error of 2 8 in the quantity (dd,). But the latter quantity must be smaller than (d+d,), therefore an absolute error 2 8 in its value must represent a greater percentage error in the relative values of E₁ and E, than would be the case if the same absolute error were in (d+d). As we must assume the resultant error to be the greatest possible, we must therefore take the error 2 8 to be in the quantity (d — d,). 2 Let, then, λ be the error in the relative values of E, and E2, that is in E, caused by, say, an error & in d, and an error б in d,, then we have 1 (d + 8) − (d, − 8) d+d, d-d, +28' d+d, d+d, d — d₁ + 28 E₁ (d + 8) + (d, −8) λ= E2 therefore E₁ λ = E₂ d+d, d-d, +2d d-d 2 8 (d + d) (d + d,) (d − d, +28)' or since 2 8 is very small we may say λ= 28 (d+ d) (d — d,)² If we put the percentage for the absolute value of A, that is if we have λ λ E of 100 E2 X' d + d, X 100 d – d, 1 ] L ELECTROMOTIVE FORCE OF BATTERIES. 149 then we get λ d+d, 2 8 (d + d) X 100 dd, (d — d‚)² · that is to say For example. 28100 dd, [A] In the example given on page 147 the deflections could each be read to an accuracy of of a division; what was the degree of accuracy with which the value of Ε1 could be determined? E, 2 N' = 2 × 1 × 100 72 - 8 = .78 per cent. If d, is small compared with d, then 28 λ d' We can see from equation [A] that unless d, is small compared with d, the accuracy with which the test can be made will be but small; for if d, approaches in value to d, then dd, becomes very small, that is λ' becomes large. In order that d, may be as much smaller than d as possible, E, and E, must be as nearly equal as possible; the test therefore will not be a satisfactory one unless such is the case. If d, is small compared with d, then 2 8 100 λ d or if we put the percentage instead of the absolute value of d, that is if we have &' of d, 100 then we get λ' = 28', so that under the best conditions for making the test the accuracy with which the value of E₁ 1 could be determined would E be but one-half the accuracy with which the higher deflection could be observed. € 150 HANDBOOK OF ELECTRICAL TESTING. 165. To determine the degree of accuracy attainable in the case where the readings are made from the degrees scale of a tangent galvanometer, we must in the preceding investigation substitute tangents for divisions of deflections. Thus we have tan (d° +8°) + tan (d,°+8°) tan (d°+8°) -tan (d,°+8°)' E λ E or tan d° + tan d‚° λ - tan do tan do If in this equation we put tan (d° + 8°) and tan (d° — 8°) tan (d° + 8°) + tan (d,° — 8°) tan (d° + dº) – tan (d,° — d°)* = tan do tan 8° 1 - do tan 8° tan do tan 8° 1+tan do tan so' we get λ 2 tan do [(tan d° + tan d,°) (1 + tan do tan d,°) + X] (tan do — tan d,°) (tan d° — tan d,° + Y) where X and Y are a number of factors of tan 8°. But since tan do is very small, we may put X and Y equal to 0, in which case we have λ= 2 tan 8° (tan d° + tan d,°) tan do tan d,° - 2 tan d° (tan d° + tan d,°) tan do tan do 1 + tan do tan d° X tan do tan do 1 X tan (d° — d,°)° If we put the percentage for the absolute value of λ, that is, if we have then we get ג 100 of E ג tan d° + tan d,° X E2 100 tan do tan do' 2 tan 8° 100° λ tan (d° — d,°)° [B] For example. In comparing the electromotive forces of two batteries by Wiedemann's method, the deflections obtained on the degrees scale of a tangent galvanometer were 71° and 18° respectively; ELECTROMOTIVE FORCE OF BATTERIES. 151 what were the relative electromotive forces of the batteries, and what would have been the degree of accuracy with which the value of could be determined? E₁ E2 could be read to an accuracy of 1°. The value of the deflections E₂:: tan 71° +tan 18° : tan 71° — tan 18°, E₁ : E2 or as that is, as also λ 2.9042 + ·3249 to 2.9042 — •3249, 1.25 to 1; 2 × tan 10 × 100 2 x •4363 tan (71° — 18°) 1.3270 = .65 • 65 per cent. Like equation [A] (page 149), equation [B] (page 150) shows that unless d, is small compared with do, the test cannot be made with a high degree of accuracy. To sum up, then, we have Best Conditions for making the Test. 1 2 166. To obtain satisfactory results, E, and E, should be as nearly as possible equal. As much resistance should be included in the circuit as possible. If the readings are made on the degrees scale of a tangent galvanometer, then the resistance in circuit should be so ad- justed that the deflections, as nearly as possible, make equal angles on opposite sides of 45° (§ 32, page 28). Possible Degree of Accuracy attainable. When the readings are in divisions, then 2 100 M Percentage of accuracy = d — d, where is the smallest fraction of a division to which the de- flections can be read. m When the readings are in degrees on a tangent galvanometer, then 10 2 tan 100 Percentage of accuracy = m tan (d° — d,°) 152 HANDBOOK OF ELECTRICAL TESTING. 10 where is the smallest fraction of a degree to which the deflec- tions can be read. 110 WHEATSTONE'S METHOD. 167. The most elegant method of comparing the electro- motive forces of batteries is that of the late Sir Charles Wheat- stone. Battery E, is joined up in simple circuit with a galvanometer and a resistance; a deflection of a° is obtained. The resistance is now increased by p₁, so that a new deflection, ß°, is produced. Battery E, is next joined up in the place of E₁, and the re- sistance in circuit is adjusted until the deflection obtained is aº, as at first. The resistance is now increased by p₂, so that the deflection is reduced to ß°, as in the first instance. Now from the "Equal resistance method" (page 144), we see that the total resistances, R₁ and R₂, in circuit, which were re- quired in the two cases to bring the deflections to a°, must be in direct proportion to the electromotive forces, E1, E2, of the two batteries. Also the total resistances, R₁ + P₁, and R₂ + P2, in circuit which were required in the two cases to bring the deflec- tions to ẞ°, must be in direct proportion to the electromotive forces, E1, E2. We therefore have or and or that is or 耳​は ​E₁: E₂ R₁: R2, E₁ R₂ = E₂ R1, 1 2 1 E₁ E₂ R₁ + P₁: R₂+ P2, 1 E₁ R₂+ E1 P₂ = E₂ R₁ + E2 P1 = E₁ R₂+ E2 P13 2 2 E1 P₂ = E2 P12 E₁: E₂ P1 P2. In fact, the electromotive forces of the batteries are directly proportional to the added resistances which, in both cases, were required to bring the deflections of the galvanometer needle from a down to ߺ. For example. 1 With a galvanometer and battery E, we obtained, with a resistance of 1950 ohms in the resistance box, a deflection of ELECTROMOTIVE FORCE OF BATTERIES. 153 54°, and by adding 2000 ohms (pi), a deflection of 34°. Battery E, being inserted in the place of E₁, a resistance of 1650 ohms was inserted in the resistance box, which brought the galvano- meter needle to 54° as at first, and by adding 1600 ohms (P2), the deflection was reduced to 34° as in the first instance; then or as E₁: E2: 2000 : 1600, 1 1.25 to 1. 168. In this and the preceding tests we have supposed that the electromotive forces of any two batteries were being com- pared, but it must be evident that by noting the deflections, resistances added, &c., as the case may be, with any number of batteries, their electromotive forces may all be compared. 169. We will now proceed to determine the "Best conditions for making the foregoing test." There are two points to be determined: first, what should be the resistances in circuit when observing the first deflections, and second, what proportion should the added resistances bear to the original resistances? When the test is executed, there are two or more sets of observations made, viz., one for each battery. But it will be found, on examination, that the proportion between the electro- motive forces, the original resistances, and the added resist- ances, is the same for every set; consequently, we have only to determine what relative values these quantities should have in any one set, then those in the others will be in the same proportion. For It will be convenient to consider first what proportion the added resistance should bear to the original resistance. this purpose we will suppose p₁ to be the former resistance. Now P₁ represents the electromotive force of the battery, and therefore in order that the test may be made as accurately as possible, it is necessary that we should be able to adjust or de- termine the value of P, as accurately as possible. In order to obtain the required value of P₁, we first adjust R₁ s› as to obtain the deflection a°, and then we increase R, by p₁ so as to obtain the deflection ẞ°; consequently, the accuracy with which we can obtain p₁ must be dependent must be dependent upon the accuracy with which we can read both the deflections, a° and ẞ°. 1 Let, then, the first deflection (a°) be due to a current, C₁, then we have C₁ = E₁ R₁' cr, C₁ R₁ = É₁. 154 HANDBOOK OF ELECTRICAL TESTING. When the current is reduced to C, by the addition of P1, get } then we therefore C2 = E₁ 1 or, C₂ R₁ + C₂ P1 = E₁ ; C2 R₁+ Pi or C₂ R₁ + C₂ P₁ = C₁ R₁, P₁ = R(-1) Now this equation is identical with equation [F] (page 111) in the "Diminished deflection shunt method" of determining the resistance of a galvanometer; consequently, we can see from the investigations there given, that p, would be most accurately obtained if C₂ = approximately; but when this is the case 515100 P1 = R₁C₁ 1 = 2 R₁; ; that is to say, the added resistance should be about double the original resistance. As regards the "Possible degree of accuracy attainable," we can see from equation [H] (page 111) in the test before referred to, that the percentage of accuracy, X', attainable must be λ (C1 C2 + C₂ c₁) 100* C₂ (C1 – C2) per cent. As it is the relative electromotive forces of two batteries E, which have to be determined, that is to say, the value of the Ε, percentage of accuracy with which the test can be made will be double the above. As regards the value for the original resistance there is little to be said. It does not affect the accuracy of the test, except * The expression [1+G( + 1}{})] R in the equation referred to [H] (page 111) becomes equal to 1 when S, and R are very high; this must be the case when equation [B] (page 109) becomes simplified into equation [F] (page 111). ELECTROMOTIVE FORCE OF BATTERIES. 155 in so far as the power of adjustment is concerned; this is evidently made as favourable as possible by making the resistance as high as convenient. We must have therefore Best Conditions for making the Test. 170. When making the observations with the first battery, make the original resistance as high as convenient, and make the added resistance as nearly as possible double this. Possible Degree of Accuracy attainable. When the readings are in divisions, then Percentage of accuracy = 1 918 (Dd) 200 d (D − d) 1 where is the smallest fraction of a division to which the deflections can be read. If the deflections are in degrees on a tangent galvanometer, then if we can read their values to an accuracy of th of a degree, we have Percentage of accuracy = (tan D° 8₂+tan d° 8) 200 tan d° (tan D° tan d°) 81 = tan D¹º tan D°, and, 82 tan d¹o = tan do. where 118 171. Wheatstone's test can be made with any form of galva- nometer, as it is not necessary that the values of the deflections in terms of the currents producing them be known, except for the determination of the "Percentage of accuracy attainable." If, however, the galvanometer be "calibrated" (page 46), this percentage can be determined. LUMSDEN'S OR LACOINE'S METHOD.* 172. This is an excellent method of determining the com- * This method was devised by Mr. D. Lumsden (Postal Telegraph Sub- marine Superintendent) in 1869, but the first description of the same appears to have been published by M. Emile Lacoine (Technical Director of the Ottoman Telegraphs) in the Journal Télégraphique of Berne' for January 25th, 1873, that gentleman having devised it independently of Mr. Lumsden. * 156 HANDBOOK OF ELECTRICAL TESTING. parative electromotive forces of batteries. The principle of the arrangement is shown by Fig. 56. E g FIG. 56. A Olle le eeeeee B First Method. The two batteries E1, E2 are joined up with their opposite poles connected together, and with resistances R, p in their circuit. A galvanometer g is connected between the points A, B. One of the resistances, say p, being fixed, the other, R, is adjusted until no deflection is observed on the galvanometer. this is the case we get the proportion When E, E2 R P 173. In order to understand why this is the case, let us examine the theory of the method; this may be explained by the help of Kirchoff's two laws,* viz. : 1. The algebraical sum of the current strengths in all the wires which meet in a point is equal to nothing. 2. The algebraical sum of all the products of the current strengths and resistances in all the wires forming an enclosed figure, equals the algebraical sum of all the electromotive forces in the circuit. 174. Supposing, at first, equilibrium not to be produced, then we have the following equations connecting the various current strengths, resistances, and electromotive forces:- C1 C C₂ = 0. Rege-E₁ = 0. c P C2 g c E2 E₂ = 0. From equation [1] we get, c₁ = c + C₂ ; * For the proof of these laws sce Chapter XXVII. [1] [2] [3] ELECTROMOTIVE FORCE OF BATTERIES. 157 therefore R (c + c₂) + gc – E₁ = 0. c₂ = B₂+go E₂ + gc; From equation [3] we get therefore 2 gc) P therefore therefore P ; R (c + E₂+90) + ge - E₁₂ = 0; до Rpc + RE₂+ Rgc+pgc-p E₁ = 0; PE₁ - RE₂ ρ Ει g (R+ p) + R p˚ If in this equation we put c = 0, then or that is, ρ Ε PE₁ - RE₂ = 0, 2 E, R E₂ Բ [4] E₁: E₂ :: R: p. 175. Let us consider what are the "Best conditions for making the test." What we have to determine is, what are the best values to give to R and p? Now, since E₁ and E₂ are definite quantities, the value given to R (supposing this to be the adjustable resistance) will be determined by the value given to p; we must therefore determine the value to give to the latter. The greater the accuracy with which we can adjust R, the greater will be the accuracy with which we can determine the E, value of that is, the relative values of E, and E. But the 1 E2 accuracy with which we can adjust R depends upon its range of adjustment being as great as possible, and this can only be the case when it has as high a value as possible. Thus, if R were 100 units, we could only adjust it to an accuracy of 1 unit in 100, or 1 per cent.; but if R were 10,000, then 1 unit in 10,000 represents an adjustment of 0 per cent. But it is + 158 HANDBOOK OF ELECTRICAL TESTING. • no use making R 10,000, unless a change of 1 unit in its value produces a perceptible deflection of the galvanometer needle. The best value therefore to give to R is the highest one in which a change of 1 unit from its correct value produces a perceptible deflection of the galvanometer needle. Since R is dependent upon the value given to p, what we require to know is the highest value to give to the latter quantity. Equation [4] shows the current, c, obtained through the galvanometer when equilibrium is not produced. If in this equation we put R - 1 in the numerator instead of R and then put RE₂ = p E₁, or, or, R PE₁ E₂ we shall get the current corresponding to the change of 1 unit in the correct value of R₁. Thus or – c = PE₁ (R − 1) E₂ g(R+p) + RP E₁ P E E₂ E₂ p E₁ Ρ E₁ 1 P E2 **(2+2)+357 g E2 [(+)+] E2 E₁ P [(1 + 1 ) + P ] - E 2 E₁ c [A] [B] Practically, the minimum readable deflection of a Thomson galvanometer (which is the best to employ in a test of this kind) is one division, and the reciprocal of the current pro- ducing this deflection is the figure of merit of the instrument (page 65). If, therefore, in the last equation we put for c the reciprocal of the figure of merit of the galvanometer, we can determine the highest value which can be given to p, E₁ and E₂ both being in volts. If we wish to get the exact value of p, we can do so by solving the quadratic equation; but, practically, we only require to get a rough idea of what the value of p may be, and this we may obtain by giving different values to p, and trying which of them nearly satisfies the equation. For example. Two batteries, whose electromotive forces E, and E₂ wcre 2 ELECTROMOTIVE FORCE OF BATTERIES. 159 2 known to be of the approximate values of 2:1 (E, being 1 volt), were to be tested by the foregoing method with a Thomson galvanometer whose resistance was 5000 ohms (g) and figure of merit 1,000,000,000: What was the highest value that could be given to p? p [5000 (1+1) + p] = 1,000,000,000 × 1, or p [7500 +p] = 500,000,000. = ρ 19,000 we shall be From this we can see that if we make very nearly right, for 19,000 [7500 + 19,000] = 503,500,000. With this value of p, the value which R would have when adjusted, would be 2 E, R 1 P 19,000 ײ = 38,000, E2 1 and with this value we could obtain a degree of accuracy equal to 1 38,000 × 100 = ·0026 per cent. Having then ascertained the value to give to p, suppose we actually made it 19,000, and further, we found that in order to get equilibrium as nearly as possible, we had to adjust R to 36,250 ohms, then the relative values of E, and E, would be or as 1 E₁: E2:: 36,250: 19,000, 1.9089 to 1, and we know this is correct within ⚫0026 per cent. From equation [A] (page 158) we can see that c is greatest when E, is larger than E₁. It is therefore best to so arrange the test that the resistance to be adjusted is the one in circuit with the strongest of the two batteries. Also we can see that the more the batteries differ in electromotive force the better, as the greater will be the value of p. Second Method. 176. In the example we have taken we have supposed the resistances of the batteries to have been so low that their values could practically be neglected in comparison with the high resistances R, p, which we were able to put in circuit. If, however, the batteries consist of a great number of cells of 160 HANDBOOK OF ELECTRICAL TESTING. high resistance, and also if the galvanometer be not a highly sensitive one, and consequently R and p have to be propor- tionately small, then we can no longer ignore the resistances of the batteries, and these must either be added on to R and eliminated in the following manner. Suppose the resistances of E, and E₂ to be r and tively, then when equilibrium is produced we have E₁: E₂ R+"₁P + 12 •1 1° ρο 2 respec- or E₁₂- E21 = E₂ RE₁ p. E, Ꭱ 71 2 1 [1] Now if we decrease p to p₁ and again obtain balance by de- creasing R to R₁, we get a second proportion, viz.— E₁ 1½ — E2 71 = E₂ R₁ — E1 P1· 1 12 By subtracting [2] from [1] we get or E, R – E₂ R₂ — E₁ p + E₁ P₁ = 0, E₂ (R − R₁) = E₁ (p − P₁) ; [2] that is or Ꭱ R-R, #12 E2 ρ P1 [A] - E₁: E₂:: R- R₁: p - P₁₂ E2 P1, a proportion in which differences of resistance alone appear. In fact (R- R₁) and (p — p₁) are merely the resistances which we subtracted from R and p, in order to get equilibrium a second time. For example. 2 Two batteries whose electromotive forces E, and E₂ were to be compared, were joined up in circuit with a galvanometer and two resistances as shown by Fig. 56, the resistance p being 500 ohms; in order to obtain equilibrium R was adjusted to 1050 ohms; p was then decreased to 300 ohms (p₁), and in order to again obtain equilibrium, R had to be reduced to 630 ohms (R₁). What were the comparative electromotive forces of the batteries? E₁ : E₂ :: 1050 – 630 : 500 – 300 420 : 200 + ELECTROMOTIVE FORCE OF BATTERIES. 161 or as 2.1 to 1. 177. The question now arises what are the best values to give to R₁ and p₁, or rather to p₁, for the value given to the latter will determine the value given to R₁. In order to work out the problem let us suppose, in the equation E₁ R – R₁ E Ρ P1 E there is a small error λ in E₂ caused by a definite error – ø in R₁, that is, let E₁ R — (R₁ $) +λ + λ = Ꭱ - Ꭱ, + φ [B] E₂ P - Pl P - Pi P - Pl By subtracting [A] (page 160) from [B] wẹ get φ λ P - P1 This shows that with the definite error ø, A is as small as possible when p₁ is as small as possible. A would be very great if p₁ approached in value to p, but it would be small when P1 is about equal to, and but little less if p, is made very much smaller still. Although, therefore, we should make P₁ small, there is but little advantage in making it very much smaller than ; in fact, there is an actual disadvantage, for when P₁ is very small, P₁ is proportionately small and its range of adjust- ment is correspondingly limited. P 2 '1 From equation [A] (page 158) we can see that in the present case the currents flowing through the galvanometer when equilibrium is not established, in consequence of R and R₁ being each 1 unit out of adjustment, are E₂ C1 E, (p + ~2) E₂ g (1+ E2 +p+r₂ E₁ and E C2 E (P1+82) (Pi+r) 1/4 [(1 + 1) +P₁+r₂] E2 9 2 : и 162 HANDBOOK OF ELECTRICAL TESTING. 1 respectively; and from these equations it is evident that if c, is a perceptible deflection when R is 1 unit out, c, will be a still more perceptible deflection when R, is 1 unit out, since R must be smaller than R; consequently the value we give to R₁ will not be limited by any considerations with regard to a perceptible deflection being obtained. As in the first test, c, and c₂ are both greatest when E, is larger than E2, the batteries should therefore be so arranged that this is the case. With regard to the Possible degree of accuracy attainable with this test, we can see first of all that R cannot be adjusted quite so accurately as in the case where the resistance of the batteries was negligible; we can, however, ascertain the exact degree attainable by putting pr₂ instead of p in equation [B] (page 158). Thus to take the example given on page 158, suppose the battery E, had a resistance of 5000 ohms (r2) approximately, then we should have or 2 (p + 5000) [5000 (1 + 1) + p + 5000] 1,000.000,000 × 2, = (p + 5000) [12,500 + p] = 500,000,000. If in this equation we make p = 14,000, we get (14,000+ 5000) [12,500 + 14,000]= 503,500,000, which is close to the correct value. In other words, if p does not exceed 14,000 ohms, we can be sure of the value of R within 1 unit. The degree of accuracy with which we can determine the value of E 1 E2 from the equation E₁ R - R₁ Ez P - Pl 2 depends upon the degree of accuracy with which we can adjust both R and R₁, and as the errors in either of them may be either or -, the greatest possible total error is that which will be produced by a + error in R, and a error in R₁, or vice versa. Let these errors be both 1 unit and let the corre- E₁ sponding error in be λ, then we have E₂ E, R + 1 − (R¹ − 1) R-R1 2 + λ + E2 0 01 · P1 P - Pl ELECTROMOTIVE FORCE OF BATTERIES. 163 and E, Fla R-R₁ E2 P - Pl therefore 2 λ P - Pl Since we require to know what percentage (\') of error this represents, we have λ= λ E, of 100 E, or E λ' = 100 λ E 200 E2 P - Pl E₁ [C] To take the example we have just considered, we see that the possible percentage of accuracy attainable, supposing p₁ to equal, is 'ג 200 = ⚫014 per cent. 14,000 7000 1 2 178. With a Thomson galvanometer of ordinary sensitiveness it is evident from the foregoing investigation, that if we have two batteries, one E₂ having an electromotive force of 1 volt or more, and E₁ an electromotive force of twice that value or more, we can without difficulty determine their relative electromotive forces to an accuracy of, at least, ⚫015 per cent.; and if the resistance of the batteries be very low we can be certain of the accuracy within, say, 003 per cent. 179. It is possible to get a still greater accuracy by employ- ing a set of resistance coils adjustable to th orth of a unit, for in this case we can make both R and R, low without losing the range of adjustment, whilst by making these quantities low we increase the value of the galvanometer deflection when exact adjustment is not obtained; this is only the case, however, when the resistances of the batteries and of the galvanometer are low. We can easily determine to what extent the degree of accuracy is increased by using submultiples of the units; first by ascertaining from equation [B] (page 158) what value p can E₂² have, being divided by 10 if R is adjustable to ths, and E₁c 1 M 2 164 HANDBOOK OF ELECTRICAL TESTING. 1 100 by 100 if R is adjustable to ths; and second by working out the value of a' from equation [C] (page 163) which gives the required percentage of accuracy. Of course when great accuracy is required, the test must be made by the method in which the resistances of the batteries are eliminated; it is no use making the test by the first method, since the accuracy attainable by having R adjustable to th orth of an ohm is more than counterbalanced by the error produced by not taking into account the resistance of the batteries. To summarise the results we have obtained, we have Best Conditions for making the Test. First Method. ΙΟ 180. First make a rough test to ascertain the approximate values of E₁ and E₂, then make p of such a value that E₂² P [8 (1 + 1 ) + P ] = B² ² E₁ E₁ c 1 approximately, c being the reciprocal of the figure of merit of the galvanometer, and E, the stronger of the two batteries, E₁ and E₂ being in volts. Second Method. Make p of such a value that P E (p + r₂) [9 (1 + 2) [12) + P+"] approximately. E₂2 E1 c Îf R is adjustable toth orth of an ohm, the right-hand E2 E₁₂ 2 E,2 side of the equation should be or respectively. E₁ 10 c E₁ 100 c på should be about equal to 2. In both methods E, should be the larger of the two batteries. Possible Degree of Accuracy attainable. First Method. Where resistance of battery is very small, ELECTROMOTIVE FORCE OF BATTERIES. 165 100 E2 Percentage of accuracy = X P Ει Second Method. 200 Percentage of accuracy = P - P1 Or, if p₁ is nearly equal to P1 2' X E2 E₁ Percentage of accuracy = 400 E X P E 181. A great point in these methods of determining the comparative electromotive forces of batteries, lies in the fact that both batteries are working under exactly the same con- ditions; moreover, if the resistances R and p are high there is but little tendency for them to polarise. If one of the batteries be a constant one, such as a Daniell, then by varying the values of R and p we can test how the other battery behaves when worked through different resistances. POGGENDORFF'S METHOD. 182. In this method one battery is balanced against the other. The method is shown by Fig. 57. In this figure E, and E, are the electromotive forces to be compared. R and p are adjustable FIG. 57. C 2 F E Cy G எண் T E2 C2 oooooo o RC 1 2 resistances, r₁ and r₂ being the resistances of the batteries. G is the resistance of the galvanometer. Before equilibrium is obtained we have C₁ + C₂ = C₂ = 0 [1] (12 + G) C₂+ R Cg3- E₂ = 0 [2] (r₁ + p) c₁ + Rc3 — E₁ = 0. [3] 166 HANDBOOK OF ELECTRICAL TESTING. By substituting the value of c₁ obtained from equation [1], in equation [2], and then again the value of c, obtained from equa- tion [2], in equation [3], we shall find that Cg C₂ = If we put c₂ (r₁+p) E₂-R (E,+ E₂) 1 R (r₂ + G + r₁ + p) + (1 + p) (½ + G)* = 0, we get (~1 + p) E₂ − R (E₁ — E,) = 0, E₂ (R+ r₁ + p) = E₁ R ; 1 [4] or that is E₁: E₂:: R+₁+p: R, [5] or E, 21+ p = 1 + [6] R 1 It will be observed that in order to get the ratio of E₁ to E₂ from this proportion, we must know the resistance r₁ of the battery E₁. If, however, we decrease p to p₁ and again get equilibrium by readjusting R to R1, we get a second proportion, viz., E₁: E₂: R₁ +r₁+ P₁: R₁, 2 1 19 P1 [7] and by combining the two proportions, r, is eliminated in the manner shown in the last test (page 160) and we get, or E, E2 FLE (R − R₁) + (p − P1) - Ꭱ - Ꭱ, - E₁ : E2 :: (R − R₁) + (p − p₁) : (R G R₁), a proportion in which differences of resistance alone enter. For example. 2 [A] Two batteries whose electromotive forces E₁ and E₂ were to be compared, were joined up in circuit with a galvanometer and two resistances as shown in Fig. 57. The resistance p being 200 ohms, it was necessary in order to obtain equilibrium to adjust R to 500 ohms. p was then reduced to 100 ohms (p1), and in order again to get equilibrium R had to be readjusted to 400 ohms (R₁), then or as E: E2: (500 - 400) + (200 — 100): (500—400); E₂:: 2: 1. ELECTROMOTIVE FORCE OF BATTERIES. 167 183. In making this test practically, the connections with the set of resistance coils shown by Fig. 6, page 13, would be as shown by Fig. 58. Having depressed the left-hand key, then, according to the example, we should take out the two 100 plugs between A and C, and proceed to adjust between D and E. This being done, we should insert one of the 100 plugs between A and C and readjust the resistance between D and E. 184. As only one of the batteries (the smaller) in this test has its electromotive force balanced, the other one should be a constant battery, whose electromotive force does not fall off on being worked continuously, such as a Daniell. 185. It is evident that the test can be made either by making p a fixed resistance and R an adjustable one, or by making R fixed and p adjustable. In order therefore to determine the Best conditions for making the test, one point for consideration will be-should R or p be the adjustable quantity? Now by a similar reasoning to that given in § 173, page 157, we can see that in either case the value of the adjustable resist- ance should be the highest one in which a change of 1 unit from its correct resistance produces a perceptible deflection of the galvanometer needle. FIG. 58. A H E D 1 A 1 1 If we refer to equation [6] (page 166) we can see that if E₁ 2 E₂ then r₁ +p must be equal R, and that according as E is greater or less than 2 E2, so will r₁ + p be greater or less than R. It is evident that the larger we make the adjustable resistance the greater will be the range of adjustment of which it is capable, therefore for this reason it follows that if E, is greater than 2 E₂ then r₁ + p should be the resistance in which the adjustment is effected, whereas if E, is less than 2 E₂ then R should be the adjustable resistance. 1 168 HANDBOOK OF ELECTRICAL TESTING. Now if R be the adjustable resistance, then inasmuch as the value which it will have will depend upon the value given to p, therefore we must determine the highest value we can give to p. Equation [4] (page 166) shows the current c₂ obtained through the galvanometer, when equilibrium is not produced. If in this equation we put R 1 in the numerator instead of R, and then put E1 E12 1 R+ r₁ + p R or, R = (~1+p) E2 E₁ E₂ we shall get the current, c2, corresponding to the change of 1 unit in the correct value of R. Thus (E₁ - E₂)² C2 = [1] (~1 + p) [(1 + G) E₂ + (~1 + p) E2], or (E₁ - E₂)² C2 [A] (r₁+p) | (½+G) E₁+(1+p) E₂ ] And if in this equation we make c₂ the reciprocal of the figure of merit (page 65) of the galvanometer, then the value of p which satisfies the equation will be the highest value which it should have; as explained in the last test, p can be obtained by trial. If p be the adjustable resistance, then what we have to determine is the value which R should have. To do this we must put p + 1 * instead of p in the numerator of equation [4] (page 166) and then put E₁ R + r₁ + p R R (E₁ — E₂) or, r₁+p E2 E2 we shall then get the current, c'₂, corresponding to the change of 1 unit in the current value of p. Thus c'₂ = or E₂2 R [(™½ + G) E₂ + R (E₁ — E₂)]' 2 R [(r₂ + G) E₁ + R (E₁ − E₂)] 2 1 E22 [B] from which, as in the previous case, R can be obtained by trial. *We put p + 1 in this case in preference to p— 1, simply in order to avoid giving c a minus value. The general result obtained, however, would be similar whether the 1 be plus or minus. ELECTROMOTIVE FORCE OF BATTERIES. 169 We have next to determine the value which should be given to R, or to p₁. Let us in the first instance take R₁ to be the adjustable resistance, then what we have to do is to find the proper value to give to P₁- If, then, we suppose in the equation E₁ (R − R₁) + (p − P1), − E₂ Ꭱ - Ꭱ or E₁ P - Pl = 1 + E2 - Ꭱ - ᎡᎥ' [2] [3] that there is a small error λ in error G in R₁; then we have E caused by a corresponding E2 E₁ +λ= 1 + E2 Ꭱ p = (P₁ − ¢ - ) R-R₁ [4] λ = but from [3] By subtracting [3] from [4] we get p − (P1 − p) Ꭱ . R R₁ R-R₁ = (1,1) (P); P - Pl R-R₁ φ .. R = R E2 -) (p therefore λ= φ E₁ E₁ — E₂/ :) (p = p₁) This shows that with the definite error p, λ is as small as pos- sible when p₁ is as small as possible. A would be very great if P₁ approaches in value to p, but it would be small when P1 is about equal to, and but little less if p, is made very small 2' indeed. As our range of adjustment of R, is limited by making P₁ very small, it is advisable not to make it smaller than 2 2 A similar investigation would have proved that if p₁ were the adjustable resistance, then R₁ should be made small, though not smaller than R 2* 186. From equation [1] (page 168) we can see that the test is impossible if E, and E, are equal, since c₂ = 0 with any value 170 HANDBOOK OF ELECTRICAL TESTING. ↑ we can give to the resistances.* We can further see that the more the batteries differ in electromotive force the better; and also that it does not matter materially which is the stronger of the two. 1 187. As regards the Possible degree of accuracy attainable, this depends upon the degree of accuracy with which we can adjust both R and R₁ (or p and p₁, if R and Ř₁ are the fixed resistances), and as the errors in either of them may be + or the greatest possible error is that which will be produced by a + error in R and a error in R, or vice versa. Let these errors be both 1 unit, and let the corresponding error in from equation [3] (page 169) '1 E₁ E₂ be λ, then we have E₁ +λ = 1 + P- P1 = 1 + E2 R-1-(R₂+1) P - P1 R - R₁ 2 S and E₁ P - Pl = 1 + or, R - R₁ = E₂ (p − P1); E2 Ꭱ - Ꭱ Ꭱ - R₁ E1 E₁ – E₂ therefore p - λ= P-P1 R-R₁-2R-R₁ (R- R₂ - 2) (R — R₁)' Ꭱ Ꭱ or, since RR₁ is very large, 1 2 (p − P1) P1 2 (p − P1) 2 Έ E2 X - (R − R₁)* - ρ P1 E2 (E₁₂ E₂). 2 Since we require to represents, we have know what percentage (A') of error this λ λ' E₁ of 100 E E2 or E2 200 (E₁ — E₂)² λ' = 100 λ X E₁ 1 P - Pl E₁ E2 In the case where p and p₁ are the adjustable resistances, we should get λ = ρ P - P₁ + 2 P1 2 R-R₁ R-R₁ R-R₁' and calling, as before, λ' the percentage of error, we get λ 200 E2 X R-RE · R₁^ E₁´ *This is not the case in Lumsden's test. ELECTROMOTIVE FORCE OF BATTERIES. 171 To sum up, then, we have Best Conditions for making the Test. 188. First make a rough test to ascertain the approximate values of E1, E2, r₁, and r₂; then if E, is less than 2 E₂, make p a fixed resistance, and of such a value that 71, 2 (r1 + p) [(r₂+G) E₁ + (rı + p) E₂] approximately. 1 (E₁ – E₂)² [A] C If R is adjustable to th of an ohm, then the right-hand side of the last equation should be 1 ´E₁ — E₂)² X * с c being the reciprocal of the figure of merit of the galvanometer, and E, and E2 both being in volts. 1 På should be about equal to 2. If E, is greater than 2 E, then make R a fixed resistance, and of such a value that R [(r½ + G) E¸ + R (E¸ − E₂)] = E₂² approximately. — C [B] If p is adjustable to th of an ohm, then the right-hand side of the last equation should be E₂² 2 2 с c being the figure of merit of the galvanometer, and E, and E₂ being both in volts. R₁ should be about equal to R 2* Possible Degree of Accuracy attainable. When R and R, are the adjustable resistances, then 200 (E₁ - E₂)² Percentage of accuracy = X P - Pl E1 E2 ; or if p₁ nearly equals 2 * § 177, page 163. 172 HANDBOOK OF ELECTRICAL TESTING. 400 (E, — E₂)². Percentage of accuracy = X ; P E₁ E2 When p and p₁ are the adjustable resistances, then Percentage of accuracy = 200 E₂ X R - R₁ 1 E₁ R or if R₁ nearly equals 2 Percentage of accuracy = 400 E2 X Ꭱ R E₁ 189. If the test is made by obtaining the result from formula [6] (page 166), the resistance r₁ of the battery being very small, then it is not difficult to see, from the investigation given in "Lumsden's test" (page 155) that when R is the adjustable resistance, 100 (E₁ - E₂)² Percentage of accuracy = X ρ Also we should make p of such a value that ρ 1 E₁ E2 approximately. p(GE₁+pE₂) (E, 1 E₂)² с When p is the adjustable resistance, then Percentage of accuracy 100 E2 X Ꭱ Ε, Also we should make R of such a value that approximately. R[GE, +RE, E₂)]-E," 2 FAHIE'S METHOD OF MEASURING BATTERY RESISTANCE. 190. It may be pointed out* that the foregoing test also affords a means of ascertaining the resistance r₁, of the battery E₁; thus from equations [5] and [7] (page 166) we can see that therefore R+r₁+p: R:: R₁++ P₁: R₁; R₁ R + R₁ г₁ + R₁ ρ = R₁ R + Rr₁ + Rp₁; 1 * See Sabine's 'The Electric Telegraph,' p. 323. ELECTROMOTIVE FORCE OF BATTERIES. 173 therefore or r₁ (R − R₁) = R₁ p − Rp1, R₁p - Rp1: r1 Ꭱ R-R₁ thus if we take the example given on page 166, in which we have = 400 R = 500 we get P = 200 P1 = 100 r₁ = r1 (400 × 200) – (500 × 100) 500 - 100 = 75 ohms. 191. A resistance test made in this way, however, would not be an accurate one if the resistance r₁ of the battery were small in comparison with the resistance p₁ (which is in the same circuit with r₁), for in this case the high value of the latter would swamp, as it were, the low value of r₁. If, however, as suggested by Mr. Fahie,* we commence the test by having no resistance at first in circuit with the battery E₁, that is to say, if we have p₁ equal to 0, then we can obtain more satisfactory results; in this case we get 1 r1 = R₁ p R - R₁ Ꭱ [A] 192. With regard to the Best conditions for making the test according to formula [A], the resistance R, is the resistance required to produce balance in the first instance and it can have but one value; R, however, is dependent upon p, so that what is required is the value which should be given to the latter quantity. Now from formula [A] we can see that the larger we make p the larger will be the value of R, and the larger we make the latter the greater will be its range of adjustment, consequently, as in the electromotive force test, we should give it the highest value in which a change of 1 unit from its correct resistance produces a perceptible deflection of the galvano- meter needle; this resistance we shall obtain by giving p such a value that (~1 + p) [(r½ + G) E₁ + (†1 + p) E2] 2 1 (E₁ — E₂)² C * See 'Electrical Review,' vol. xii., p. 203. 174 HANDBOOK OF ELECTRICAL TESTING. approximately,* c being the reciprocal of the figure of merit of the galvanometer. As regards the Possible degree of accuracy attainable, this we shall obtain, as in previous cases, by supposing that there is an error of + 1 in R and an error of - 1 in R₁, these errors causing a corresponding total error A in r₁: thus η + λ = and since 1 (R₁+ 1) p (R − 1) − (R₁ +1) R Ꭱ Ꮅ r1 = R – R₁ we get (R₁ + 1) p R₁ P λ = RR, 2 (R₁ + 1) p - R₁ — 2' P(R+R₁) R – R₁ (R − R₁ − 2) (R — R₁)' or since R - R₁ is very large, we may say but or 11 Fla Ez E2 p (R + R₁) (R+ (R — R₁)² ; R + p + r₁₂ [B] λ = R₁ + r₁ R₁ R 71 E₂ (r1 + p) and, R E₁ - E2 E₁ - E₂ E2 R-R₁ = Ε, ρ E₁ – E₂ E2 Bu1 = therefore R+R₁ = E₁ — E, (2×1+p), and, R - R₁ - and by substituting these values of R + R₁ and R - R₁ in equation [B] we get E₁ – E₂ λ E2 (2+1). Or if we call ' the percentage of error, then 1 or λ λ= of r1, 100 Ε, (2 E2 x100x - E₁ = E(+1) = λ 7'1 * Equation [A], p. 168. 100. 0 ELECTROMOTIVE FORCE OF BATTERIES. 175 193. The relative electromotive forces of the batteries, it may be pointed out, are given by the proportion E₁: E2 :: (R − R₁) + p : (R − R₁), which is the same as proportion [A], page 166, except that P1 is put equal to 0. To sum up, then, we have Best Conditions for making the Test. 194. Make p of such a value that (r₁ +p) [(r₂+G) E₁ + (~1 + p) E₂] (E₁ + E₂)² C approximately, c being the reciprocal of the figure of merit of the galvanometer. Possible Degree of Accuracy attainable. Percentage of accuracy E₁-E₂ (2 E2 E2 ( 1 + :): 100. FAHIE'S COMBINED METHOD OF COMPARING ELECTROMOTIVE FORCES AND MEASURING BATTERY RESISTANCE. 195. This is an extremely ingenious and elegant method, and although its application is rather limited it is well worth being noticed. The arrangement is a combination of Poggendorff's method of comparing electromotive forces (page 165) and Mance's method of measuring battery resistance (page 124). FIG. 59. B ееееееее مهما E еееее E A K Referring to Fig. 59, the following is the mode of making the test:-E is the stronger battery whose electromotive force is to be compared with the battery e, and whose internal resistance 176 HANDBOOK OF ELECTRICAL TESTING. is to be measured; d is a variable and a + b a slide, resistance, B being the slider by the movement of which the ratio of a to b can be varied. The key K being open, the resistance d is adjusted until the needle of the galvanometer shows that no current is passing through the latter; when this is the case, then, as in Poggendorff's method (page 165), we have E:e::r+d+a+b:a+ b. [1] Balance being thus obtained, the key K is alternately depressed and raised and the slider B moved until the latter is brought to such a position that the movement of the key K ceases to affect the galvanometer needle, as in Mance's test (page 124). Now, inasmuch as the battery e merely acts as a counteracting force to the current which in Mance's test would cause a permanent deflection of the galvanometer needle, it must be evident that when the movement of the key K ceases to affect g, then we must have or r = a d [2] or ad b d b r + d = a + d = & (a + b). Substituting this value of r + d in equation [1], we get d b E: e:: (a+b)+a+b: a + b, 5 Ee::d+b: b. [3] Equation [2], therefore, gives the resistance of the battery E, and equation [3] gives the relative electromotive forces of the two batteries. For example. The key K being raised, balance was obtained on the galvano- meter g by adjusting d to 200 ohms. When the key K was alternately raised and depressed, the balance on g was disturbed until the slider B was moved to the position at which b was equal to 100 ohms; the total resistance of the slide resistance a+b was 400 ohms, that is to say, a was equal to 300 ohms; then ↑ = 300 × 200 100 = 600 ohms, ELECTROMOTIVE FORCE OF BATTERIES. 177 and E: e:: 200 + 100: 100, or as 3:1. 196. The conditions for making this test so as to obtain accu- rate results must evidently be similar to those specified in the cases of Poggendorff's test and Manee's test made with a slide resistance. The nature of the method, however, is such that we cannot obtain the conditions which are best for the Poggen- dorff test without impairing the conditions necessary for making the Mance test accurately, so that practically we must arrange the resistances so as to suit the conditions necessary for making the latter satisfactorily; at the same time it may be pointed out that these conditions are such as to enable the Poggendorff test to be made with a considerable, though not with a very high, degree of accuracy. As in the case of Mance's test with a slide wire (page 127), the conditions required are that d shall be as large as possible, but not so large that the range of adjustment of the slider becomes excessively reduced. Now, practically, a slide resistance would not consist of more than about 100 coils; consequently if d were of such a value that the slider had to be set so that b was about 10 times as large as a (as would be the case when a slide wire is used), then the accuracy with which the latter could be adjusted would be extremely small, being only about 1 in 10, or 10 per cent. To make the test satisfactorily, therefore, it would be necessary to arrange so that the slider would have to come near the centre of its traverse, even though the sensitiveness of the whole arrangement became reduced in consequence. As long, however, as sufficient sensitiveness is obtained, that is to say, a sensitiveness such that a movement of the slider from its correct position to either of the contiguous coils produces a perceptible disturbance of the balance, then the nearer we can get the slider to the centre the better. It would not do, however, in any case to pass beyond the centre point; for in this case, although the error made in a by the slider being one coil out of adjustment is small, yet the error made in b becomes comparatively large. Now, in order that we may be able to get the slider near the centre of its traverse, it would be necessary that d should be approximately equal to r, but since, in order to obtain balance in the first instance, we must have E:e::a+b+d+r: a+b, N 178 HANDBOOK OF ELECTRICAL TESTING. or E d + r = 1 + e a + b d could not be made equal to r unless E = 1 + 1+ e 2r a + b² E(a+b) or, e = a+b+2 r Now if E and e were both fixed quantities and were not of such relative values that the above equation held good, then it would be impossible to obtain the conditions necessary for making the test favourably; the method of testing we are considering, however, would usually be employed for the purpose of measur- ing the electromotive force of a battery in terms of the electro- motive force of one or more standard cells whose number could be varied to suit any particular requirement; in such a case it would usually be possible to give to e the value which would enable the above equation to be satisfied. Thus, for example, suppose the resistance, r, of the battery E were estimated to be about 100 ohms, and suppose the slide resistance a + b consisted of 100 coils of 10 ohms each, that is, 1000 ohms in all, then we must have e .= E 1000 1000+ (2 × 100) E 10 ; 12 that is to say, the electromotive forces of the batteries E and e should be in the proportion of 10 to 12. Now, it is evident that if E were a battery of one or two cells only, then it would prac- tically be impossible to give to e the required value; but if E consisted of a considerable number of elements, 20 or 30 for example, then there would be no difficulty in adjusting e. From these considerations it must be evident that Fahie's method, although extremely ingenious and elegant, and in some special cases very convenient, is very limited in its application. 197. With respect to the Possible degree of accuracy attainable, this as regards the resistance test is directly dependent upon the accuracy with which we can adjust the ratio of a to b; thus if a+b consisted of 100 coils, then if the ratio of E to e were such that the slider when adjusted stood near the centre position of its traverse, the error caused by the slider being 1 coil out of position would be 1 in 50 in a, and 1 in 50 in b, consequently the total error would be 1 in 25, or 4 per cent. With n coils, in ELECTROMOTIVE FORCE OF BATTERIES. 179 fact, the Possible degree of accuracy attainable would be 1 in or, 100 × 4 n per cent. n To determine the degree of accuracy attainable in the electro- motive force test, we must suppose that d is 1 unit, and b 1 E coil, out of adjustment. If we call λ the error caused in then we must have e E +λ = 1 + e d+1 a + b' d+1 E or, λ = 1 + a + b e b b n n and since E d = 1+ e Ъ we get d+1 λ= 1+ 1 a+b b N d _ b (n + d) +ad · b ¯¯b [b (n − 1)—a]* If λ' be the percentage of error, then we have 'ג E b λ = of or, λ' = 100 λ 100 b + r therefore אג = 100 [b (n + d) + ad] (b + r) [b (n−1)—a] If the test is made under the best conditions, that is, if we have a = b, and d = r, approximately, then we get λ= 100 [b (n + r) +br] (b + r) [b (n − 1) — b] 100 (n + 2 v) (b+r) (n = 2) or since n is large, we may say X 100 (n+2r).. n (b + r} For example. In determining the relative electromotive forces, E and e, of two batteries by Fahie's method, the resistance, r, of E being ! N 2 # 180 HANDBOOK OF ELECTRICAL TESTING. approximately 100 ohms, a slide resistance having 100 coils (n) of 10 ohms each was employed. What was the greatest possible degree of accuracy attainable? λ = 100 [100 + (2 × 100)] 100 (500 + 100) = per cent. To sum up, then, we have Best conditions for making the Test. 198. Make E (a + b) a+b+ 2 r approximately, r being the approximate resistance of the bat- tery E. Possible Degree of Accuracy attainable. 100 [b (n + d) + a d} Percentage of accuracy = (b +r) [b (n − 1)— a] If a = b, and d 100 (n + 2 r) b, and d = r, and ʼn is large, then Percentage of accuracy = n (b + r) n in both cases being the number of coils of which the slide resistance is composed. 199. It may be as well to point out that Fahie's test cannot be made (except under very exceptional circumstances, rarely met with in practice) with a slide wire; for, as a rule, the latter has such an extremely low resistance that it would be impos- sible to obtain equilibrium in the first instance; the proportion E:e::r+d+a+b: a + b, which is necessary for equilibrium, could not, in fact, be satisfied unless the resistance of the battery E and the resistance d were both extremely small; in which case, moreover, the latter would have to be adjustable to a very small fraction of an ohm. CLARK'S METHOD. 200. This is a valuable modification of Poggendorff's method, and is shown in theory by Fig. 60. ab, which takes the place of R in Poggendorff's method (page 165), is a slide resistance; ELECTROMOTIVE FORCE OF BATTERIES. 181 3 E, is a third battery which is connected to a slider through a galvanometer G. Now if we suppose equilibrium to be obtained in both gal- vanometers, we must have from [5], page 166, E₁: E₂ 1 E2 and also r₁+p+a+b:a + b, α E₁: E₂::r+p+a+b:a; from which we get E2: E₂:: a+b: a. If then we take a + b to represent the electromotive force of the standard battery E2, a will represent the electromotive force of the battery E3. FIG. 60. FIG. 61. m Ем Ent E₂ Gr Cz T E ar Kr by K₂ Gz Cz Fg Ea щище G₂ In making this test practically, the battery E,, which would be the trial battery, being disconnected from the slide resist- ance, balance would be obtained with the standard battery E2 by adjusting p until no deflection is observed on the galvano- meter G₁. Eg would then be connected up and the slider moved until no deflection is observed on the second galvano- meter G3. 3 The great advantage of Clark's method is that both the standard and the trial battery are compared under the same conditions, that is, when no current is flowing in either of them; this is a great point, as errors due to polarisation are avoided. 201. It must be evident that if equilibrium is not produced with the trial cell, then the balance in the standard cell circuit will also be disturbed; it would therefore seem to be possible to 182 HANDBOOK OF ELECTRICAL TESTING. 1 dispense with the galvanometer G,, but inasmuch as the current which would flow through the galvanometer G₁ would only be a fraction of that flowing out of the battery E,, we should not be able to make a measurement with nearly such a degree of accuracy as we could if we employed the galvanometer G, which would be acted upon by the full force of the current. 202. To determine the best arrangement of resistances, &c., for making the test, let us suppose that there is a small error, A, in E, caused by a corresponding small error in a, and let us find what effect this error has upon the current which would flow through the galvanometer G. Supposing then that a, is the new value of a which causes this error, then, keeping in mind that a + b being a slide resistance is not altered by changing a, we have cr E3 + λ = +λ Ε2 01 a + b' [1] E3 + λ) (a + b) [2] αι Ez We next have to determine what the current flowing through the galvanometer, when equilibrium is disturbed, is equal to. Referring to Fig. 61, in which m, n, a, b, and g represent the resistances, and C1, C2, C3, K₁, and κ₂ the current strengths in the various circuits, we have by Kirchoff's laws (page 156) K2 C₂ + C₂+ C3 K₁ = 0 K1 C3 K₂ = 0 K2 C1 C2 0 c₁ m + K₁ α₁ + K₂ b₁ - 1 — E₁ = 0 E₂ = 0 C₂ N + K₂ A₂ + K₂ b₁ - E3 1 C3 G3+K₁ α₁ - Eg = 0. We know also that and E₁: E₂:: m+a₁ + b₁ : a₁ + b₁, E2 a₁ + b₁ = a + b. By finding then the value of c₁ from the first equation and substituting its value throughout the others, and then again ELECTROMOTIVE FORCE OF BATTERIES. 183 the value of c₂ from any other equation, and again substituting throughout and so on, and also substituting the value of E obtained from the proportion, and again the value of a₁ + b₁, we shall find that €3 11 a1 a + b E3 - E2 а1 E3E₂ a + b [3] G₂+ ar ( m n K + m + n m n a+b+ m + n If in this equation we substitute the value of a₁ given by equation [2], we shall get λ m n C3 а 1 (0₂+ m + n G3 + a+b+ m n m + n or as α, and b₁ are very nearly equal to a and b, we may say 1 λ C3 = m n a b+ m + n G₂+ n n a+b+ [4] m + n Ou examining this equation we see that to make c; as large` m n α b + as possible we must make m + n as small as possible, m n a+b+ m + n but we also see that it is no use making it much smaller than G3, as cg is but very little increased by so doing. m n alb+ Now the quantity m + n is the resistance a com- m n a+b+ m + n bined in multiple arc with the resistance b plus' m and n combined in multiple arc, consequently this quantity can never be greater 184 HANDBOOK OF ELECTRICAL TESTING. than a. As long therefore as a is smaller than G3, the highest values that can be given to the other resistances cannot make c3 less than > λ G3+ a make these resistances, we can never make c3 greater than whilst, on the other hand, however low we λ G3 The value therefore we give to a practically determines the sensitiveness of the system. But as a is only a portion of the slide resistance a + b, and as it may include the whole of the latter, as for instance when the slider is moved quite to the end of ab, the sensitiveness is practically dependent upon the value given to a + b. This must then be made as much lower than G, as may be desirable. It would not, however, do to have the resistance excessively low, for the following reason:— In order to get equilibrium on the galvanometer G2, it is necessary that the relation E₁: E2 :: r₁ +p+a+b:a+b, 1 r1 or a+b = 1 E, rit p 1 E, 2 rit p should hold good. This cannot be the case, however, if E₁ 1 E2 is greater than a+b; that is to say, if a + b is very small must be very small also; but to make the latter small r₁ + p E₁ E2 1 1 we must make E₁ large and r₁ + p small, but since r₁, the resist- ance of E,, will increase by increasing E, it may be impossible to do this. Practically we may say the resistance of a + b should be a fractional value of G3. 203. Let us now determine the Possible degree of accuracy attainable by the method. In equation [1] (page 182) we have supposed that an error λ has been caused in E, by a being out of adjustment; that is to say, from the slider being moved a little too far, so that a becomes a₁. If we call the distance the slider has been moved beyond its correct position, then we have E3 + λ = E₂ (α + $) a+b E₂ a + a+b Ε, φ a+b > ELECTROMOTIVE FORCE OF BATTERIES. 185 but E3 E₂ a a+b therefore E12 '2 λ= φ a+b' that is to say, the distance the slider is out of position re- presents directly the error A in E. The degree of accuracy therefore with which we adjust the position of the slider will be the degree of accuracy with which we can measure E¸. We have pointed out that if a + b is small, then α (b+ a+b+ m n m + n m n m + n 3 will be smaller still; if therefore G, is large compared with a+b, equation [4] (page 183) becomes λ C3 G3 If in this equation we put the value of λ, given above, we have or C3 Ó E2 G3 (a + b)' $ a + b C3 G3 Ez This equation enables us to determine what movement of the slider produces a perceptible deflection on the galvanometer. With a Thomson galvanometer of 5000 ohms' resistance and figure of merit = 1,000,000 (page 65) we should have, sup- posing E₂ to be 1 volt, φ a + b 5000 1,000,000,000 1 1 200,000' or a movement of the slider equal to 200000th of the length of a+b would produce a perceptible deflection; that is to say, we could determine the accuracy of an electromotive force Eg of about 1 volt to an accuracy of 200,000th. 186 HANDBOOK OF ELECTRICAL TESTING. To obtain this accuracy, however, it would be necessary to have the wire a + b graduated into 200,000 parts, each of which would be very small, unless indeed the wire were very long. If a lesser number of graduations were employed, we could practically subdivide each of them by noting what the galvano- meter deflections were when the slider stood, first at one division mark, and then at the contiguous mark. Suppose the slider stood at a distance a from the end of the slide wire, and a deflection due to a current c₁ was produced to one side of zero; and suppose that when the slider was moved 1 division forward, that is to a + 1, the deflection was on the other side of zero, or was produced by a current C₂. Then we have from equation [3] (page 183), since a and a + 1 are very nearly equal, a E3 – E₂ C1 = a + b K and a + 1 α E3 – E2 E2 E3 - a + b E2 a + b a+b C2 = K K ; therefore α E2 a C₁ E3 — C₁ E₂ C1 - C₂ E3 + C₂ E2 a+b a+b a + b diameter, is stretched upon an oblong board (forming the base of the instrument) parallel to a metre scale divided throughout its whole length into millimetres, and so placed that its two ends are as nearly as possible opposite to the divisions 0 and 1000 re- spectively of the scale. The ends of the wire are soldered to a broad, thick copper band, which passes round each end of the graduated scale, and runs parallel to it on the side opposite to the wire. This band is interrupted by four gaps, at m₁, a, x, and m2. On each side of these gaps, and also at B, C, and E, are terminals. In the ordinary use of the apparatus (Fig. 71), the wires from the battery are attached to the terminals B and E, and the galvanometer is connected between C and the slider A; by pressing down a knob this latter is put in contact with the wire. The conductor whose resistance has to be measured, and a standard resistance, are placed in the gaps at x and a respec- tively. The two gaps at m₁ and m₂ (Fig. 70) can either be bridged across by thick copper straps, or resistances of known values can be inserted in them; it is easy to see that these resistances are simply ungraduated prolongations of the slide wire. THE WHEATSTONE 215 • BRIDGE. 225. If we have no resistance in these gaps, then when we have equilibrium, 818 d d or, x = a ато d As is merely a ratio, we do not require to know the absolute values of d and b, but only their relative values, that is to say, we only require to know the lengths of the portions on either side of the slider A, and not the resistances of those portions. The length k of the slide wire is constant, that is, b+dk, or, dkb, therefore x = α k – b b b a († − 1); k but = 1000 millimetres, and b is usually called the scale reading, therefore we have For example. ?= (scale 1000 scale reading - 1). [A] The standard resistance a being 1 ohm, equilibrium was obtained when the scale reading was 510; what was the value of the unknown resistance x? x = 1 (1000-1) = •961 ohms. -1)= 226. It has been pointed out by Mr. Martin F. Roberts that equation [A] is the same as x = α a ({ reciprocal of scale reading × 1000 – 1), and that consequently, by the use of a table of reciprocals, cal- culations can be considerably simplified in working out the value of x. 227. Equation [A] is only true if the resistances between the ends of the slide wire and the terminals B and E are zero. But, although it may not appear so, it is by no means easy to make these resistances inappreciable; even the careful soldering of the ends of the wire to the copper straps introduces a resistance which is sufficient to affect very accurate tests. Referring to 216 HANDBOOK OF ELECTRICAL TESTING. Fig. 70, in which n₁ and n₂ are these resistances, we know that strictly speaking or that x = a قاه ? d + n z ; b + ni a 1000 + n + m² - 1). -1). scale reading + n₁ To make a strictly accurate test, then, we must know the values of n₁ and n₂ in terms of the equivalent length of the slide wire. These may be obtained in the following manner :- Having bridged across the gaps at m₁ and m, with thick copper straps, taking care that the surfaces in contact are scraped bright, insert known resistances at a and x, a being rather larger than x; then having obtained equilibrium, we have a (d + n₂) = x (b + n₁); now reverse a and x, and again obtain equilibrium. Let the new scale readings be b₁ and d₁; we then have x (d₁ + n₂) = α (b₁ + n₁). By multiplying up and arranging the quantities, we have and therefore that is a² ni therefore n1 an₂ = xb+ xN1 a d x n₂ = a ab₁ + an₁x d₁; ; a x b + x n₁ — a d a b₁ + a n₁ − x d₁' 18 x² n1 - di = x² baxda²b₁+ax d₂, a x (d¹ − d) + x² b — a²b₁ a² — x² In a similar manner we should find n₂ = a x (b − b₁) + x² d₁ — a² d - a2 x2 Or since that is, b+ d = b₁ + d₁ = 1000, dı d = 1000 – b, and d₁ = 1000 — b1, ! THE WHEATSTONE BRIDGE. 217 we have n1 and - a x (b − b₁) + x² b — a² b₁ аз a² - x² b x − b₁ a a X (1000 — b₁) x — · (1000 — b) a n2 α X For example. In order to determine n₁ and n₂, resistances were inserted at a and x equal to 3 and 2 ohms respectively. Balance was obtained when the scale reading b was 603. On reversing a and x, balance was obtained when the scale reading b₁ was 399. What were the values of n₁ and n₂? N2 n₂ = n1 (603 x 2)-(399 × 3) 3- 2 = 8 mm. (1000399) 2 (1000603) 3 — 3-2 1 = 11 mm. The value of x, then, would be given by the equation x = α ( 1000+9+11 scale reading + 9 1) = a( 1020 \scale reading +9 − 1). One 228. Although perfectly satisfactory results may be obtained with the metre bridge when the latter is properly made, and when the measurements are carefully carried out, yet consider- able trouble is often occasioned to inexperienced persons by results being obtained which are obviously erroneous. most frequent cause of error is that occasioned by imperfect con- tacts; great care should therefore be taken that the important connections, viz. those at the gaps, should be well made; this should be ensured by having the various surfaces in contact made clean and bright by scraping. Good contacts are best assured by having mercury cups at the gaps instead of screw terminals; care should be taken that the mercury in these cups is in good metallic contact with them, that is to say, it should wet the metallic surfaces. The mercury should also, of course, be in similar good contact with the ends of the wires or rods (the latter are usually attached to the standard resistances), which may be dipped into the cups. The amalgamation of the metallic surfaces is best effected by scouring the latter with emery paper, and then moistening them with a solution of nitrate of mercury. 218 HANDBOOK OF ELECTRICAL TESTING. 229. A form of bridge in which mercury cups are used in the place of terminals for the more important connections, is shown by Fig. 72. This apparatus is also provided with a commutator for reversing the resistances placed at a and x. This commuta- FIG. 72. zru Xx a ՂԱԼ 006 600 099 099 200 зро tor is formed of four mercury cups (seen in the centre of the figure) forming the corners of a square. These cups can be connected by means of the connector shown in the upper part of the figure. This con- nector is simply a short bar of ebonite with short copper rods at its extremities and at right angles to the latter; the ends of these rods are bent down so that they can dip into the cups when the arrangement is placed over the latter. If the connector is placed over the cups so that the ebonite bar is in the position shown by the dotted line, 1-2, then it will be seen that the left- hand cup at a is connected to the right-hand cup at m₁, and the right-hand cup at x to the left-hand cup at m2; if, however, the ebonite bar is in the position shown by the dotted line, 1,-2,, then the left-hand cup at a is connected to the left- hand cup at m₂, and the right-hand cup at x to the right-hand cup at m₁. Even if good contacts be assured, correct results cannot be obtained if the standard resistances are in- correct, or if the slide wire is not uniform in its resistance throughout its length. A metre bridge to be really useful, therefore, requires to be very carefully made. THE WHEATSTONE BRIDGE. 219 230. The form of standard resistance generally used with the metre bridge is similar to that shown by Fig. 73. The ends of the brass rods to the left of the figure dip into the mercury cups; the resistance itself is enclosed in a brass box and bedded in paraffine wax. The particular pattern shown is an arrangement devised by Professor Chrystal to show whether the temperature of the interior of the brass box is the same as that of the surrounding air. It consists of a thermo-electric couple with one junction outside and one junction inside the box; by connecting this couple (whose terminals FIG. 73. are seen on the upper part of the box) to a galvanometer of low resistance, no deflection would be produced if the two junctions, that is, the paraffine inside and the air outside the box, are at the same temperature. 231. The accuracy with which a test can be made, as in the ordinary form of bridge, depends upon the values of the various resistances, and amongst these upon the value given to k. In order to be able to vary the value of this quantity, the gaps at m₁ and m, are provided. As the resistances placed in these gaps are simply prolonga- tions of the slide wire, it is necessary that their values should be known in terms of equivalent lengths of the slide wire; that is, we must know how many millimetres of the wire they are equal to. This is best done in the following manner:— Close the gaps at m₁ and m₂ with the thick copper straps, and place resistances of known values at a and x. Adjust the slides so that equilibrium is produced, then or n₂ 1000 + m² + m² - 1), x = a (100 b + ni − x (b + n₁) + a(1000 + n₂ − b). Now insert one of the resistances, whose equivalent length m₁ in millimetres is required, at the left-hand gap, and again obtain equilibrium; calling the new scale reading b₁ we then have x (b₁ + n₁ + m₁) a (1000+ n₂ — b₁). = By subtracting the one equation from the other we get x (b − b₁) — xm₁ = a(b₁ — b), - xmı 220 HANDBOOK OF ELECTRICAL TESTING. that is, or For example. (b − b₁) (a + x) = m₁x, m₁ = (b − b₁) a + x X It being required to know how many millimetres of the slide wire a resistance m₁ was equal to, the scale reading b, with the two gaps closed, was 500 mm., and the scale reading b₁, with m₁ inserted, was 480 mm., the resistances at a and x being 6 and 4 ohms respectively. What was the value of m₁? M1 == (500480) 6 + 4 4 = 50 mm. If we have a and x equal, we get the simplification m₁ = (b − b₁) 2. There are other methods of determining the value of m₁, but the one given, besides being extremely simple, is very accurate, as it is independent of the quantities n, and n. The millimetre values of the resistances to be placed at m₁ and m₂ being thus determined, the value of x is given by the equation x = α (1000 ´1000 + n₁ + n₂ + m₁ + m² scale reading + n₁ + m₁ -1). 232. Let us now consider the Best arrangement of resistances, &c., for making a test with the metre bridge, under favourable conditions. Now a mistake of a millimetre in the position of the slider will make a much greater error in the result of a worked out from the formula, when the slider is near the ends of the wire than when it is near the middle. Thus, for example, suppose x was 1 ohm and a was also 1 ohm, then we should have the slider standing exactly at 500 if it were properly adjusted. Suppose, however, it was 1 millimetre out, then the apparent value of x would be x = 1 (1000- 1) = .996, that is, we make x, 1 — ·996, or ·004 ohms, too small. Next suppose a = 9 ohms, then for equilibrium the scale read- THE WHEATSTONE BRIDGE. 221 ing would be 900, and if we make a mistake of 1 millimetre we should have x = 9 (1000- − 1) = = •990; that is, we make x, 1 — ·990, or ⚫010 ohms, too small. Lastly let us suppose a = ohm, then the scale reading for exact equilibrium would be 100, and supposing there to be an error of 1 millimetre, we have = 1 (1000-1) = .9 + = ·989; that is, we make x, 1 — ·989, or ⚫011 ohms, too small. To summarise the results, then, we see that with a larger than x, error was ⚫010, or 1 per cent. equal to smaller than,, ( "" ⚫004, or ⚫011, or 1 "" The error, in fact, was smallest when the slider was at the middle of the wire. We must, however, determine whether the middle is really the point at which the error is least. Calling k' the resistance of the slide wire and its prolonga- tions m₁, m2, and b' the scale reading plus the prolongation m₁, let there be an error λ in x caused by an error - d in b', then - x + λ = a(−1) or λ = a(5-1) - z م ماه X. But therefore k' x = α a (k − 1), or, a α= k' b' ac 1 ; k' 1 b' 8 λ = x - 1 k' 1 b' 1- X k' S (b' −8) (k − b')' or since 8 is a very small quantity, we may say, k' S λ = x b' (k − b') [A] Now we have to make λ as small as possible; this we shall do, since x and k' are constant quantities, by making b' (k' —b′) as large as possible. 222 HANDBOOK OF ELECTRICAL TESTING. But k'2 2 b' (k' — b') = ¹²² - ('1' — b')', 4 and to make this expression as large as possible, we must make k' 2 b' as small as possible; that is, since b' must be positive, we must make it equal to 0, or I' k' b' ***** 0; that is, b' = 2 2 which proves the truth of the supposition. To obtain the slider as near to the middle of the wire as possible when equilibrium is produced, we must make a as nearly as possible equal to x. If in equation [A] we put ľ λ of x, and, b' = 100 we get 400 8 λ' = ; k' zia I' so that if when the slider is near to the centre of k' we can adjust the slider to an accuracy of 1 division (8), then if k' con- sisted of 1000 parts (as would be the case if there were no prolongations m₁, m₂), we could measure the value of x to an accuracy of 400 × 1 1000 4 per cent. 233. In order to make a measurement in this manner, as we have seen, it is necessary for a to be approximately equal to x. Now in many cases there would be no difficulty in arranging that such should be the case. Thus, for example, suppose it was required to measure the conductivity of a sample of wire, then in this case we should take a sufficient length of the wire to give a resistance approximately equal to a, and then having measured the exact length taken, we should ascertain its exact resistance by adjusting the slider until equilibrium was ob- tained. 234. If we wish the measurement to be made to a higher percentage of accuracy than can be made with the slide wire k alone, then we must add equal resistances, m, and m2, at each end of the wire so as to increase the value of k.. THE WHEATSTONE BRIDGE. · 222 Since therefore 400 S N' = k' 400 S k' ; λ so that if we wish to measure x to an accuracy, say, of 1 per cent., then we must make k' equal to 400 x 1 •1 = 4000; that is to say, we must add resistances m₁ and m, at each end of k, each equivalent to 1500 millimetres of the wire k. It must be recollected, however, that there will be no advantage in thus increasing the length of k, unless the figure of merit of the galvanometer employed is sufficiently high to enable a movement of the slider to a distance of 1 division from its correct position, to produce a perceptible movement of the needle. If the resistance to be measured is not one which admits of adjustment, then in order to obtain a satisfactory measurement we must add a resistance on to one or other of the ends of k, according as x is larger or smaller than a; or we may add resistances to both ends, their values being unequal. If in equation [A] (page 221) we put x = α b' a (1 − 1), or, k' = b' а + x [1] α then we get (a + x) 8 λ b' or if we put λ' λ= of x, 100 we have 100 (a + x) 8 100 (+1) 8 ג or, b' b' = b' x λ From this equation we can see that no matter what are the relative values of a and x still b' can always have a value which will enable x to be obtained to any percentage of accuracy λ'; that is, of course, provided the figure of merit of the galvano- meter be sufficiently high for the purpose. 224 HANDBOOK OF ELECTRICAL TESTING. For example. It is required to measure the exact value of a resistance x, whose approximate value is five times that of the resistance a; what must be the value of b' in order that the measurement may be made to an accuracy of 5 per cent.? The adjustment of the slider can be determined to an accuracy of 1 division. • 100+11 b' = = 240. •5 From equation [1] (page 223) we get 5 k = 240 (1 + 1) = 1440, consequently since k consists of 1000 divisions we must add a prolongation m₂ equal to not less than 440 divisions, on to k. We may of course make the prolongation larger than 440; in fact, in practice we should have to do so unless we had a resistance available of the exact required value; but it must not be too large, otherwise the position of balance for the slider would be at some point on m₂ instead of on the wire k. In fact, m₂ must not be greater than k x α If it should happen that in order to obtain a particular per- centage of accuracy it is necessary that b' should exceed k, then in this case it would be necessary to have a prolongation m₁ in addition to the prolongation m₂; the latter quantity in this case must not exceed (k + m² ) 2/2. X α In the last example we have supposed to be less than a. If, however, x is greater than a, then b' will probably have to be greater than k, in which case of course we should have to add the prolongation m₁ in the place of the prolongation m₂, the α X value of m₁ being such that it does not exceed k unless we also add a prolongation m₂ in addition to m₁, in which case m₁ must not exceed (k + m²) ==- α x We have seen that by means of m₁ and m₂-the values of which can be determined in the manner shown in § 231 (page 219)—we can theoretically arrange that the value of x can be assured to any required degree of accuracy, no matter what the relative values of x and a may be. This, however, can only be the case THE WHEATSTONE BRIDGE. 225 provided the figure of merit of the galvanometer is such as to enable the slider to be adjusted to an accuracy of 1 division. The figure of merit of the galvanometer, therefore, as in other tests, is the limit to the "Possible degree of accuracy attain- able." This limit can be determined from equation [2] (page 195) in the following manner :— Let A be the error in x, caused by b' being th of a unit out of adjustment, then we have α 1 x + λ = a ď + b' d' + a d'+ b'x+ n 19 or, λ = a n X = b' 1 1 b' - n n n and since a d'b'x, and is a very small quantity, we get λ = 1 a x b' + "; b x We have then from equation [2] (page 195) by putting d' = a a + x E b' C6 81 { (6' +9) (b' + g) + x + g} {r + a +x+ γα b' In order, therefore, that b' may be able to have the value necessary to ensure x being measured to the required degree of accuracy, the value of c must not be less than that given by the above equation. As the values of g, d, x, and r are mostly easily obtained in ohms, the value of b' corresponding to the number of divisions of which it would consist must be in ohms also;, likewise, will have to be the resistance, in the fraction of an ohm, correspond- ing to 1 division (or fraction of a division, if the slider can be adjusted to a closer accuracy than 1 division) of the wire k. For example. In the last example it was required to be known whether a galvanometer whose resistance was 1 ohm (g), and the reciprocal of whose figure of merit was ⚫0002 (c) would be suitable for the purpose of making the measurement in question. The resistance of the slide wire, which was divided into 1000 divisions (k), was 5 ohms; the resistance a was 1 ohm, and the resistance x, 5 ohms approximately. The actual value of the prolongation added to k was such as to make k' equal to 1560. The resistance 226 HANDBOOK OF ELECTRICAL TESTING. of the battery was 5 ohms (r), and its electromotive force 2 volts (E) approximately. •5 Since k = 1000, therefore = = .0005. 1000 Also (from equation [1], page 223) we have a k' 1 x 1560 b' = •5 × 260 = 260 divisions = a + x 1+5 = '13 ohms; 1000 therefore 1 +5 •13 C6 = 2 x .0005 X { ( (· 13 + 1) + 5 + 1 } { 5 + 1 + 5+ ⚫046 (11.65) (49.46) ⚫0008, 5 x 1 ·13 which is greater than ⚫0002, the reciprocal of the figure of merit of the galvanometer in question, consequently the latter instru- ment is well suited for the purpose for which it is required. 235. The resistance of the galvanometer employed in making a bridge test is an important point, especially as regards the measurement of small resistances. In the case of the ordinary bridge test, we can adjust within 1 unit, and in the case of the slide wire bridge, we can adjust within 1 millimetre of the wire; if then the galvanometers em- ployed in these cases are such that when we are 1 unit or 1 millimetre from exact equilibrium we obtain perceptible deflections of the needles, then we have what we require, what- ever the resistances of the galvanometers may be. In the ordinary form of bridge, where the adjustable resist- ances are not capable of being adjusted to a greater accuracy than 1 unit, a Thomson's galvanometer, such as that described in Chapter III. (page 46), and which has a resistance of about 5000 ohms, gives, under all circumstances, a very large deflection when the adjustment is only 1 unit from equilibrium. In the case of the slide wire bridge, however, where to be 1 millimetre from exact equilibrium means to be onlyth of an ohm, or even less, out, a galvanometer of such a high resistance as 5000 ohms would not be found to give a perceptible deflection. The reason of this is, that such a galvanometer is practically short circuited by the very low resistance it has between its terminals. • THE WHEATSTONE BRIDGE. 227 The question of galvanometer resistance is considered at length in Chapter XXV., and it is there shown that it is best that the instrument should have a resistance not more than about 10 times, or less than about th, Of course a (d + x) a + x in practice we cannot adjust the resistance to meet every par- ticular case, but the limits given are sufficiently wide to enable an instrument to be made which would prove satisfactory for most purposes for which the metre bridge is adapted; moreover, if a particular galvanometer does not prove to be suitable for a particular purpose, we can ascertain, by the help of the above rule, whether the cause is due to its resistance being too high or too low. It should be clearly understood that when we speak of the resistance of the galvanometer we mean the resistance of the instrument itself, and not the resistance in its circuit; thus, if according to calculation it were proved that the galvanometer resistance should be 1 ohm, then it would not be carrying out the rule if we took an instrument having a resistance of, say of an ohm, and added a resistance of of an ohm in its circuit, for this of an ohm would be an addition to the external circuit, and not an addition to the galvanometer itself. 4 Under no conditions should the battery be joined between A and C, and the galvanometer between B and E, for in such a case the battery current in passing from the slider to the wire would be liable to injure the surface of the latter. To sum up, then, we have Conditions necessary for making the Test to any required Degree of Accuracy. 236. The number of divisions of which b' must consist in order that x may be measured to an accuracy of λ' per cent. must be not less than 100 (+1) 8 λ' 8 being the number of divisions, or the fraction of a division, to which it is possible to adjust the slider. If prolongations are necessary, then my must not exceed α ՊՈՆշ and m₂ must not exceed (k + m²) 2. (k + m₂n) 12/23 a Q 2 228 HANDBOOK OF ELECTRICAL TESTING. The reciprocal of the figure of merit of the galvanometer must be not less than E S. a + x b' {~ (b' + 9 ) + x + 9} {r+a+x+ α γα b' where E is in volts and all the other quantities (including b' and d') are in ohms. Possible Degree of Accuracy attainable. Percentage of accuracy 100/a b'x 0 (+1) 8. MEASUREMENTS BY CAREY FOSTER'S METHOD. 237. This method, devised by Prof. Carey Foster,* consists in determining the value of the unknown resistance in terms of an equivalent length of the slide wire; this is effected in the following way:- FIG. 74. FIG. 75. C ຈ x x C E B E B Ni n n2 The resistance, x, whose value is to be determined, is placed in the left-hand gap (Fig. 74), and resistances r₁, 72, the ratio of whose values does not differ from unity more than does that of the resistance to be measured and the resistance of the whole slide wire, are placed in the two centre gaps; the right-hand gap is closed by a conductor without sensible resistance. The slider is now adjusted until equilibrium is obtained, and the reading b is noted. x is then transferred to the right-hand gap, and the left-hand gap is closed by a conductor without sensible resistance (Fig. 75); the slider is again adjusted and the reading by noted. Calling n the resistance of the portion of the copper strap between B and the left-hand end of the slide wire, and n₂ the Նշ * Journal of the Society of Telegraph Engineers,' vol. i. page 196. THE WHEATSTONE BRIDGE. 229 2 1 resistance of the portion of the strap between E and the right- hand end of the slide wire; also calling r₁ the total resistance between the points B and C, and r₂ the total resistance between the points C and E; finally, calling b and b₁ the respective resistances of the portions of the slide wire, in the two tests, and calling 7 the total resistance of the slide wire, we have and also 11 x + n₁ + b 512 r2 i = b + n₂ n₁ + b₁ b − b₁ + n₂ + x n₁ + b₁ 1 − b₁ + n₂ + ∞ n₁ + b₁ 1 − b₁ + n₂ + x +1; n₁ + b₁ + l − b₁ + n₂ + x 1 therefore x + n₁ + b ī − b + n z therefore x + n₁ + b + 1 = T= b + n₂ therefore x + n₁ + b + l − b + n₂ - 7 − b + n₂ therefore - l― b₁ + n₂ + x ; or lb + n₂ = l − b₁ + n₂ + x, b1 2 X b₁ — b. In order to make this formula useful we must know the resistance per millimetre of the slide wire, since b₁ and b on the scale represent not resistances but lengths. The simplest method of doing this is to take a test in the foregoing manner, giving the resistance x a known value, •1 ohm for example; in the latter case, since •1 = b₁ - b the difference between the two scale readings multiplied by 10 gives the number, v, of millimetres corresponding to 1 ohm resistance, and therefore when we make a test to determine an unknown resistance, x, we get X b - b₁ V The accuracy of the test depends upon the conductor with which the unknown resistance, x, is interchanged having prac- 230 HANDBOOK OF ELECTRICAL TESTING. tically no resistance; it should, therefore, be made of as massive and short a piece of copper as possible, and the connections should be made by means of mercury cups. * The great merit of Professor Foster's method lies in the fact, that the measurements are independent of the resistances of the various parts of the copper band. 238. Professor Foster points out that inasmuch as by his method the value of a resistance, x, can be determined in terms of a certain length of the slide wire, therefore if x be made a known resistance and the slide wire itself be formed of a portion of wire whose resistance per unit length is required, this latter resistance can easily be determined. Such a method would give very accurate results, and is as good as "Thomson's Bridge method, which was devised by Sir William Thomson for the same purpose, and is as follows:- THOMSON'S BRIDGE. "" 239. The arrangement of this bridge is shown by Fig. 76; its object is the accurate measurement of the resistance of a FIG. 76. еее едее 20 F portion of a conductor of low resistance, lying between two points, errors due to imperfect connections being avoided. In the Fig., B F is the conductor, the resistance b of the corre- sponding length, 1-2, of which requires to be determined. FE is a standard slide wire whose resistance per unit length is * As a rule the cups at each side of a gap are too small and are not put close enough together, the consequence being that a conductor used for bridging over a gap is comparatively long and has a sensible resistance. The cups ought to be of large dimensions and so close together as almost to touch, the bridge piece could then be made so massive and short as to be practically. of a negligible resistance. The ends of this piece should be quite flat, so as to lie closely in contact with the bottom of the cups. THE WHEATSTONE BRIDGE. 231 accurately known. Now when we have equilibrium we see from equation [A] (page 211) that we have a b (P1 + P₂ + K) + K P1 d (P₁ + P₂ + K) + K P2 ac by multiplying up and arranging we get (a d − b x) - Now if we have we get or 812 11 8 Pa K (x p₁ - a p₂) P1 P1 + P₂ + K that is, xp₁ = a P₂ ad-bx = 0, α b = d da; X from which we see that the value of b is independent of the resistance of any of the connections, provided the contacts at the points 1, 2, 3, and 4 are small compared with the resistances a, P1, P2, and x, which, by making these resistances high enough, will practically be the case. The points 1, 2, 3, and 4 should be knife edges, so that the exact distance between 1 and 2, and between 3 and 4 can be properly determined. MEASUREMENT OF THE CONDUCTIVITY RESISTANCE OF A TELEGRAPH LINE. Direct Method. 240. When, by means of the bridge, Fig. 63 (p. 191), we are measuring the conductivity resistance of a wire whose further end is not at hand, we should join one end to terminal C, put the further end to earth, put terminal E to earth, and then measure in the usual way. Loop Method. 241. It is always as well, however, when possible, to measure without using an earth, by looping two wires together at their further ends, the nearer ends being joined to terminals E and C respectively; this gives the joint conductivity resistance of the two. Errors consequent from earth currents, or a defective earth, &c., are thereby avoided. We cannot, however, by this means, 1 232 HANDBOOK OF ELECTRICAL TESTING. obtain the conductivity resistance of each wire separately. If, however, we have three wires at hand, we can by three measure- ments obtain the conductivity resistance of each wire, without using an earth. This is effected as follows: Let the three wires be numbered respectively 1, 2, and 3. First loop wires 1 and 2, at their further ends, and let their resistance be R₁. Next loop wires 1 and 3, and let their resistance be R2. Lastly, loop 2 and 3, and let their resistance be R. Supposing the respective resistances of 1, 2, and 3 to be 71, 72, and r3, we get r1+r₂ = R₁ 2 R1 r1+r3 = R₂ R2 r₂+r3 = R3. Now, since each of the wires is looped first with one and then with the other of the other two, it is evident that the sum of the three measurements will be the sum of the individual resistances of the three wires taken twice over, and conse- quently the three wires. If, then, we subtract R₁ from this result, the remainder must be the resistance of r3. Similarly, if we subtract R₂ from the same, the remainder will give us r2; and lastly, by subtracting R,, we get the value of r₁. R₁ + R₂+ R3 must be the sum of the resistances of For example. 2 The conductivity resistance of each of three wires, Nos. 1, 2, and 3 was required. Nos. 1 and 2 being looped, the resistance (R₁) was found to be 300 ohms. Nos. 1 and 3 looped gave a resistance (R₂) of 400 ohms. Lastly, Nos. 2 and 3 looped gave a resistance (Rg) of 500 ohms. Then:- added resistance of the three wires will be therefore, 300 + 400 + 500 2 600 ohms; Resistance (1) of No. 1 wire = 600 - 500 = 100 ohms. "" "" (r2) (3) 2 3 = 600 - 400 = 200 = 600 - 300 = 300 " "" By this device, then, we are enabled to eliminate all sources of error without making a greater number of measurements than would be required if we measured each wire separately, by using an earth. THE WHEATSTONE BRIDGE. 233 MEASUREMENT OF THE RESISTANCE OF AN EARTH. 242. By means of a method very similar to the foregoing we can, if we have two wires at our disposal, measure the resist- ance of the earths at the ends of the lines. The following is the way in which this can be done :- Let the two wires be numbered respectively 1 and 2. First loop the two wires at their further ends, and let the measured resistance of the loop be R₁. Next have No. 1 wire put to the earth at its further end, and measure the resistance, which will be that of the wire and earths combined; let this total resistance be R. Lastly, have wire No. 2 put to the earth at the distant station, and measure the total resistance, which we will call R5; then by adding R₁, R and R5 together, and dividing the result by 2, we get the sum of the resistances of the two wires and the earth; by subtracting from this result the resistances of the two looped wires the remainder will be the resistance of the earths. 243. By means of a test made in this manner we can deter- mine not only the resistance of an earth, but also the in- dividual resistance of two wires; for if we subtract R₁ from R₁+R₁+R5, the result will be the resistance of wire No. 2, 2 4 and if we subtract R, instead of R, then the result will be the resistance of wire No. 1. Such a test, however, although it eliminates errors due to defective earths, does not eliminate errors due to earth currents. But inasmuch as it is a test which is applicable when only two wires can be had, it is useful, since the earth current errors can be eliminated by a method which we shall investigate. MEASUREMENT OF THE INSULATION RESISTANCE OF A TELEGRAPH LINE. 244. In measuring the insulation resistance of a wire, the con- nections would be the same as for conductivity resistance, except that the further end of the wire, instead of being put to earth, would be insulated. 245. It sometimes happens that we require to find the insula- tion resistance of two sections of one wire, but we can only test from one end. Now, if we join several wires together, one in front of the other, it is evident that the total insulation resistance of the combination will diminish according to the number of the wires and according to the insulation resistance of each of them. 234 HANDBOOK OF ELECTRICAL TESTING. The law for the total resistance, in fact, will be the same as that for the joint conductor resistance of a number of wires joined up in multiple arc (page 70). That is to say, the total insulation resistance of any number of wires joined together will be equal to the reciprocal of the sum of the reciprocals of their respective insulation resistances. As a matter of fact, it is immaterial whether the wires be joined together one in front of the other or all be bunched together; the law of the joint insulation resistance is the same in both cases. * A B C Suppose, then, A C to be the wire which is required to be tested for insulation resistance from A in two sections, A B and BC. Let a be the insulation resistance of the section A B, and b the insulation resistance of the section BC; and suppose x to be the insulation resistance of the whole wire from A to C, then we have from which X ab a+b² ax b a x All we have to do, therefore, supposing we are testing from A, is first to get the end C insulated and to measure the insulation resistance; this gives us x. Next get the wire separated at B, and the end of the section A B insulated. Again measure the insulation resistance; this gives us a. Then from the two results b can be calculated. For example. The insulation resistance (x) of the whole wire, from A to C, was found to be 6000 ohms, and that from A to B (a), 24,000 ohms. What was the insulation resistance (b) of the section B? 24,000 × 6000 b = 24,000 - 6000 = 8000 ohms. 246. To obtain the conductivity resistance of one section of a wire when the resistance of the other section, and also of the whole wire, is known, we have only to subtract the resistance of the one section from the resistance of the whole section. The truth of this is obvious. * This is not the case if the insulation resistances are very low, as the resistance of the conductor then comes into question and modifies the result. THE WHEATSTONE BRIDGE. 235 MEASUREMENT OF THE CONDUCTIVITY RESISTANCE OF WIRES TRAVERSED BY EARTH CURRENTS. 247. When the conductivity resistance of a line of telegraph is measured by having the further end of the line put to earth, the presence of earth currents, that is to say, the currents set up by electrical disturbances over the surface of the earth, and also currents due to the polarisation of the earth plates, α renders the formula x = dz, when equilibrium is produced, incorrect. To obtain the true value of the resistance of the wire, therefore, a different formula is necessary. Equilibrium Method. 248. In Fig. 77 let E be the electromotive force of the testing battery, E₁ the electromotive force of the earth current, whose value will be + or - according to its direction, and let a, b, d, x, and r be the resistances of the various parts of the bridge; then FIG. 77. C B Ն α E CHE C₂ d Cs E C1, C2, C3, C4, and c, being the current strengths in the different branches, we have by Kirchoff's laws (page 156), when equi- librium is produced, the following equations connecting the resistances, current strengths, and electromotive forces: C1 - C4 C4 = 0 C₂ C3 0 C5 C3 C4 C₁ = 0 c₁ a c₂ b = 0 CA c₁ x c₂ d = ± E, c5 r + c d + c₂b E. + 236 HANDBOOK OF ELECTRICAL TESTING. J By elimination we obtain two values of c₁, one in terms of the battery E₁, and the other in terms of E, thus and E, CĄ C₁ = + X ad b E a (d+r)+b(a + r b 19 249. If we equate the two values of c4 we can get the relation between the two electromotive forces E and E₁, and thus obtain a method of determining the relative electromotive forces of the batteries, for we have E₁ E bx a d a (d+r) + b (a + r)° 250. From the latter equation we find a d E, X = ± a (d+r) + b (a + r) b E b a+r)]. To make this equation useful it is necessary that E, and E be known. If, however, we reverse the testing battery and again obtain equilibrium by readjusting d to d₁, we get a second equation, viz., + E₁ E we therefore have b x — a d b x a d₁ a (d₁ + r) + b (a + r) a (d + r) + b (a + r) By multiplying up we get + b x — a d₁ a (d1+r) + b (a + r) b x [a (d₁+r) + b (a + r)] + bx [a (d + r) + b (a + r)] − = 0; that is x = α = 0. ad [a (d₁+r) + b (a + r)] a d¸[a (d + r) + b (a +r)] d [a (d₁+r) + b (a + r)] + d₁ [a (d + r) b (a + r)] a (d₂+r) + b (a + r) + a (d + r) + b (a + r) a 2 (d + k) (d₂+k) (d + k) + (d₂ + k) k [A] | THE WHEATSTONE BRIDGE. 237 where For example. k b b b = [(1 + 1 ) + ]. 1 α In making a conductivity test of a wire in which an earth current existed, with the zinc pole of the battery to line equi- librium was obtained when d₁ was 8000 ohms. On reversing the testing current, equilibrium was obtained when d was 6000 ohms. The resistances a and b were 100 and 1000 ohms respectively, and the resistance, r, of the battery 200 ohms. What was the resistance, x, of the line? therefore X = k = [200 (1 + 1000) + 1000 (1+100) 叮 ​100 [2 (6000+3200) (8000+3200) 1000 (6000+3200)+(8000+3200) =3200, 3200 00]= =690·2 ohms. 2 1 in (d+k) (d₂+ k) It may be pointed out that the quantity (d+k) + (d₂+ k) equation [A] is the harmonic mean of the quantities (d+ k) and (d₁ + k). Various abbreviations of formula [A] have been suggested, but none of them are satisfactory except under certain condi- tions, and inasmuch as the formula is only required occasionally, the advantage of a simplification which at the best is only an approximation is a doubtful one. Mance's Method.* * 251. This method, devised by Sir Henry Mance, consists in making the observations as in the last test, but without reversing the current, the first observation being made with resistances a and b in the arms BC and BA of the bridge, and the second with these resistances changed to a₁ and b₁. In the first case, then, we have ± E₁ E in the second case ± E₁ E 1 b x ad a (d + r) + b (a +r)' b₁ x - α₁ d₁ 1 a₁ (d₁ + r) + b₁ (α₁ + r)' 1 * Journal of the Society of Telegraph Engineers,' May 8th, 1886. 238 HANDBOOK OF ELECTRICAL TESTING. L therefore bx - a d - b₁x — a₁ d₁ a (d+r)+b(a+r) a₁ (d₁ + r) + b₁ (α₁ + r) By multiplying up, and extracting x, we get 2 = α 1 1 ad [a₁ (d₁+r)+b₁ (a₁ +r)] − a₁ d₁ [a (d + r) + b (a +r)] b [a₁ (d₁ + r) + b₁ (α1+r)] b₁ [a (d+r) + b (a + r)] 1 ad[(a₁+b₁) r+ a₁ b₁] − a₁ b (b₁ + d₁ + r) — 1 a d₁ [(a + b) r + a b] - ab₁ (b + d + r) In practice Sir Henry Mance prefers to make b = a and b₁ in which case the formula becomes For example. X = 1 d (2 r + a₁) − d₁ (2 r + a) (d₁ + a₁) (d+a) = α₁, 19 In making a conductivity test of a wire in which an earth current existed the arms a and b of the bridge were each made equal to 100 ohms; equilibrium was then obtained when d was adjusted to 750 ohms. On altering a and b to 1000 each, balance was again obtained by making d₁ equal to 840 ohms. The resistance, r, of the battery was 200 ohms. What was the resistance of the line? 750 (2 × 200 + 1000) x = (840+1000) - - 840 (2 x 200 + 100) (750+100) = 636·4 ohms. With further reference to this test, see next chapter. Equal Deflection or "False Zero" Method. 252. Referring to page 194, if we suppose that there is the electromotive force E, in the branch x (Fig. 77, page 235), then equation [7] (page 172) becomes or say c5 [r (d₁ + x) + x (b + d₁)] c5 [(r (a + b) + a (b + d₁] — 1 [A] cε (bg - d₁ g + bx +b d₁) + E (d₁ + x) − E₁ (b + d₁). co ( a dit bài +bg+dng)+E(a+b) ; k' K = © k +E(d+ c) – En (b + c) ck" + E (a+b) I THE WHEATSTONE BRIDGE. 239 that is С6 C6 = E(d₁ + x) − E₁ (b + d₁) − E (a + b) K k" K + k' Now, supposing the electromotive force E is removed without altering r, and suppose at the same time that c and the other quantities remain unaltered, then we have K = C6 cε k' — E₁ (b + d₁) C6 k" that is E₁ (b + d₁) C6 = k" K + k' − therefore E(d₂+x) - E₁ (b + d₁) — E (a + b) K = − E₁ (b + d₁) ; - - therefore d₁ + x = (a + b) K. Or giving the value [A] of K, we have r (d₁ + x) + x (b + d₂) d₁ + x = (a + b) therefore therefore or 1 r (a + b) + a (b + d₁) r (a + b) (d1 + x) + a (b + d₁) (d₁ + x) = r (a + b)(d₁ + x) + x (b + d₁) (a + b); a (d₁+x) = x(a + b), X = a d₁ b If therefore we have a key so arranged that on depressing it a resistance equal to that of the battery is inserted in the place of the latter, then on adjusting the resistance d, until it is found that the deflection of the galvanometer needle is the same whether the key is up or down, we get the value of x at once from the above equation. In the practical execution of the test it would be necessary to short circuit the galvanometer at the moment when the battery key is depressed or raised, otherwise a violent movement of the needle would be produced by the static discharge from the cable. 1 240 HANDBOOK OF ELECTRICAL TESTING. 253. When the battery connections for measuring conduc- tivity are made, as shown by Fig. 63 (page 191), then in order to put the zinc current to line, we should put the cable or line to C and the earth to E. To put the copper to line we can either reverse the battery or put the cable to E and the earth to C, whichever is most convenient to the experimenter. MEASUREMENT OF THE CONDUCTIVITY RESISTANCE OF A SUBMARINE CABLE. 254. When we are measuring the conductivity of a submarine cable, which requires to be carefully done, the best method to adopt is the following: Put on the battery current for half a minute by pressing down the right-hand key (Fig. 63, page 191); at the expiration of that time, proceed to adjust the plugs, pressing down the left-hand key as required until equilibrium is produced; con- tinue to adjust, if the needle does not remain at zero, and at the expiration of half a minute note the resistance. Now reverse the battery connections, put on the current for half a minute; again measure, again reverse and measure, and so on until about a dozen measurements with either current have been taken. It will usually be found that about half the measurements made with the negative current are the same, and also half the mea- surements made with the positive current; these results may be taken as the correct measurements for d and d₁. 255. In order to reverse the current through the cable, we can either reverse the battery, or the line and earth, connec- tions (§ 253). There is an advantage in doing the latter, as by this means the galvanometer deflection due to, say, too much resistance being inserted between D and E (Fig. 63, page 191), is always on the same side of zero, although the direction of the current through the cable is reversed. Thus it is easy to see at a glance in every case, and without chance of a mistake, whether balance is out in consequence of too much or too little resistance being inserted. 256. The presence of earth currents can be detected when the line, galvanometer, and earth are joined to the resistance box, by pressing down the left-hand key alone. This will cause the galvanometer needle to be deflected if there are any currents present. A line is seldom, if ever, quite neutral in this respect. 257. It is almost immaterial what battery power is used in measuring conductivity; sufficient, however, should be used to obtain a good deflection on the galvanometer needle when equi- librium is not exactly produced. About 10 or 20 cells is a THE WHEATSTONE BRIDGE. 241 convenient number to employ. There is no danger of heating the resistance coils with such a power if the battery be a Daniell charged with plain water, or even a Leclanché, as their internal resistances are considerable. It would not be advisable, however, to use a Grove or a Bunsen battery, or a Daniell charged with acidulated water, as their heating power is great in consequence of their small internal resistances. ELIMINATION OF THE RESISTANCE OF LEADING WIRES. 258. In order to determine the exact resistance of the con- ductor of a cable, or coil of cable core, for example, it is of course necessary that the resistance of the wires leading from the testing- room to the tank in which the cable or core is placed, should be deducted from the total measured resistance. This involves a calculation which, although slight, still might be avoided with advantage, especially if a large number of measurements have to be made. At Messrs. Siemens' works, at Charlton, a very simple device is adopted which enables the resistance of the leading wire to be eliminated, thus rendering any deduction unnecessary. For this purpose a small supplementary slide wire resistance (§ 18, page 15) is connected in the arm A E of the bridge (Fig. 62, page 188); the leading wires (when con- nected to the bridge) being looped together at their further ends, and all the plugs being inserted in A E, the slide resist- ance is adjusted till balance is obtained on the galvanometer. The leads are now connected to the cable or core to be tested, and then balance is again obtained on the galvanometer by removing plugs from A E in the usual manner. This being done, the resistance unplugged in A E (allowing for the ratio of the arms AB, B C, of the bridge, if the two are unequal) obviously gives the exact value of the resistance required, since the resistance of the leads is balanced by the slide resistance. MEASUREMENT OF BATTERY RESISTANCE. 259. The resistance of a battery which consists of a large number of cells may in many cases be measured with a con- siderable degree of accuracy by means of the Wheatstone bridge, in the following manner :— Divide the battery into two equal parts, and connect the two halves together so that their electromotive forces oppose ore another; under these conditions the battery may be treated as an ordinary resistance, and measured as such. R 242 HANDBOOK OF ELECTRICAL TESTING. 1 CHAPTER IX. LOCALISATION OF FAULTS. 260. The theoretical methods of testing for the localities of faults are comparatively simple, but their practical application presents some difficulties. LOCALISATION OF A FULL EARTH FAULT. 261. The simplest kind of fault to localise is a complete fracture where the fault offers no resistance, and the conduc- tivity resistance at once gives its position. Thus, a line which was 100 miles long, and in its complete condition had a resist- ance of 1350 ohms, that is to say, a resistance of 1350 T00 = 13.5 ohms per mile, gave a resistance of 270 ohms when broken. Then distance of fault from testing station was 270 13.5 = 20 miles. LOCALISATION OF A PARTIAL EARTH FAULT. 262. When the fault has a resistance, the localisation becomes somewhat difficult. The following are the theoretical methods generally adopted (Fig. 78). FIG. 78. a C b_B Earth Ꮐ C b B f Earth Earth BLAVIER'S METHOD. 263. Let A B be the line which has a fault ƒ at C, A being the testing station. A first gets B to insulate his end of the Maol LOCALISATION OF FAULTS. 243 line. He then measures the resistance, which we will call 7, then therefore a + f = 1; f=l-a. [1] Next, B puts his end to earth, and A again measures. Let the new resistance be l₁, then bf a + ぴ ​b+f Calling L the resistance of the line, we have also t [2] therefore a+b = L; b = L - a. [3] Sub- From these three equations we have to determine a. stituting in [2] the values of ƒ and b obtained from [1] and [3], we get a + L+1-2a (1 − a) (L — a) — 1₁, = ,, therefore a² - 2 al₁ = LI - L₁ — 17₁; : from which, since a must be less than 1, and the root conse- quently negative, For example. a = 4₁ — √ (1 - 4) (L — 1). A faulty cable, whose total conductivity resistance when perfect was 450 ohms (L), gave a resistance of 350 ohms (1) when the further end was insulated, and 270 ohms (1) when the end was put to earth. What was the resistance of the conductor up to the fault? Resistance = 270 √ (350 — 270) (450 — 270) = 150 ohms. If the length of the cable were 50 miles, then conductivity per mile equals 450 = 9 ohms, and distance of fault from testing station consequently equals 150 50 9 = 16 miles. OVERLAP METHOD. 264. Two measurements are made, one by station A, and the other by station B, A and B insulating their end in turn. Thus resistance measured from A when B insulates, as before, is a+f=l. [1] R 2 244 HANDBOOK OF ELECTRICAL TESTING. Resistance measured from B when A insulates also Subtracting [2] from [1] and adding [3] therefore b+ƒ= l₂, a+b= L. [2]. [3] - a − b = l l₂, − 12, 2a=L+1 12 ; − L + 1 − 12 a = 2 For example. A faulty cable, whose total conductivity resistance when perfect was 450 ohms (L), when measured from A with the end at B insulated, gave a resistance of 350 ohms (); and when measured from B with the end A insulated, a resistance of 500 ohms (12). What was the resistance of the conductor from A to the fault? Resistance 450 + 350 – 500 2 = 150 ohms. 265. In making the foregoing test it is often found advan- tageous to introduce a set of resistance coils at the end of the cable nearest the fault, and to vary this until it is found that the measurements made at the two ends give the same results. The advantage of this arrangement is that if the same amount of battery power be used at the two stations the test current flowing out at the fault will be the same in both cases, conse- quently the fault is likely to remain constant and more uniform results be obtained. It is obvious that if r be the added resist- ance, then the resistance from either end (the resistance r being Itr taken as forming part of the cable) will be L being as in previous cases the total conductor resistance of the perfect cable. ? 2 PRACTICAL EXECUTION OF TESTS. 266. So far the testing is simple; the practical application, however, presents some difficulty. This is owing to the varia- tion of the resistance of the fault when the testing current is put to the cable, in consequence of this current acting on the copper conductor, and through the agency of the sea water LOCALISATION OF FAULTS. 245 covering it with a salt, which besides increasing the resistance of the fault, also sets up a current opposing the testing current. To make a proper test, then, it is necessary so to manipulate the testing apparatus and battery as to get rid of the polarisa- tion and resistance set up by the salt formed on the fault, and to measure the resistance at the moment this is done. following is known as :- LUMSDEN'S METHOD. The 267. The further end of the cable being insulated, the con- ductor is cleaned at the fault by applying a zinc current from 100 cells for ten or twelve hours, the current being occasionally reversed for a few minutes. A rough resistance test is then made with a copper current. A positive current is now applied to the cable for about one minute, using two or three cells for every 100 units of resist- ance which have to be measured. This coats the conductor with chloride of copper. The cable is now again connected to the resistance coils, and the battery and galvanometer connections made as shown by Fig. 63 (page 191), the zinc pole being to terminal B' and the copper to terminal E. The cable must be joined to C, and earth to E. Both keys being depressed, the galvanometer needle is care- fully watched and plugs inserted and shifted unit by unit, so as to keep the needle at zero; for the action of the negative current is to clean off the chloride of copper, and thereby to reduce the resistance of the fault. At a certain point this decomposition becomes complete, and the needle of the galva- nometer flies over with a jerk, showing that the disengagement of hydrogen has taken place at the fault, which enormously increases its resistance. The resistance in the resistance coils at that moment is the required resistance. The fault being once cleaned by the application of the 100 cells for ten or twelve hours, it is unnecessary on repeating the measurement, which should always be done, to apply the battery for so long a time; ten or twenty minutes, or even less, will generally suffice. When the measurement is made with the further end of the cable to earth, the same process of preparation can be employed. The rate at which the decomposition of the salts at the fault takes place, depends to a very great extent upon the strength of the current flowing out at the fault; now, if the latter be very near the end at which the test is being made, the resistance 246 HANDBOOK OF ELECTRICAL TESTING. between the testing battery and the fault will be so small that the changes at the latter will take place with great rapidity, and it would be a matter of great difficulty to adjust the resistance in the bridge quickly enough to follow up the change of resistance at the fault as it takes place. To avoid this difficulty the best plan is to insert a resistance between the bridge and the end of the cable; this will retard the changes by reducing the strength of the current flowing in the circuit. The value of this resistance will depend entirely upon circum- stances, and will be a matter of judgment with the person making the test, but in any case it should not be out of proportion to the actual conductor resistance of the cable. The amount of battery power used is also a matter dependent upon circumstances, but the higher the power it is found possible to use, the less will the effect of earth currents influence the accuracy of the test. The resistances employed in the arms AB, BC of the bridge (Fig. 63, page 191), will, to some extent, modify the rate at which the changes at the fault take place, and here again discretion must be used, as no definite rule can well be laid down. It might be imagined that a "slide resistance" (page 14) would be very advantageous for making a test of this kind, but practical experience shows that the plug resistances are preferable in many cases. The galvanometer with which this and the following test must be made, must be an ordinary astatic one (page 18) with fibre suspended or pivoted needles. A Thomson's reflecting galvanometer is quite useless for the purpose. Before making the test, A must of course arrange with B, or vice versa, at what time and for how long he is to insulate, put to earth, &c., his end of the cable. FAHIE'S METHOD. 267. Mr. J. J. Fahie, in a paper read before the Society of Telegraph Engineers,* has given the results of some very careful experiments and tests which he has made, bearing upon the subject of testing for faults. His method contains many valuable points, and is, in the author's words as nearly as possible, as follows:- The cable-current is eliminated by sending into the line the current of the opposite sign to that coming from it, and * 'Journal of the Proceedings of the Society of Telegraph-Engineers,' Vol. III., page 372. LOCALISATION OF FAULTS. 247 arranging the strength and duration of this current to suit the strength of the one from the cable. Thus, if the latter be strong and negative, put (say) sixty cells positive to line for a couple of minutes, and then note the condition of the cable- current; if it be still negative, but weaker, put the battery on again for a short time, and continue to do so until the galvano- meter needle indicates a weak positive current from the fault. If the latter be now left to itself and the cable put to earth through a galvanometer the needle will steadily, and as a rule leisurely, fall to zero and pass over to the other side, indicating a negative current again from the fault. While the needle is on zero the line is free and in a fit state for the subsequent test. If the cable-current be positive, put sixty cells negative on until the fault is depolarised; the effect in this case is more brief than in the other, the needle falling quickly to zero and crossing to its original position. Having once eliminated the current from the fault (and the operation very rarely exceeds ten minutes in the most obstinate cases) the cable can always be kept free by momentary applica- tions of the necessary battery pole. Thus, if the needle begin to move off zero in the direction indicating a negative current from the fault, a positive current applied for a moment will bring it back, and vice versa. In practice it is best to repolarise the fault slightly in the opposite direction, as a little time is thereby gained to arrange the bridge for a test. Having shown how to prepare the cable, the test will now be described. The bridge is arranged as shown by Fig. 79. P is the infinity plug; when this plug is removed the connection between the branch coils b and the resistance d is severed; K₂ is an ordinary key for putting the line to earth through the galvanometer G₂ or to the bridge as may be required. The rest needs no explanation. First ascertain by an ordinary test the approximate resistance of the faulty cable and leave it unplugged in d. Next allow the line to rest for a few minutes in order that it may recover itself from the effects of the current employed in this pre- liminary test, and then depress K₂, and observe the cable- current on the galvanometer G₂; let it be positive, open the key K₁, remove the plug P, and send a negative current from the testing battery of (say) sixty cells into the cable via the branch coils a, which should be plugged-in to avoid heating. When the cable-current has been repolarised a fact which may be ascertained by putting the cable to earth at intervals through G₂-arrange the bridge, close the key K₁, and, keeping the cable to G2, watch till the needle comes to zero; at that 248 HANDBOOK OF ELECTRICAL TESTING. 2 moment let K, fly back, and send a negative current through the bridge system, observing the instantaneous effect on the galvanometer G₁. If d be too great the needle will be deflected in a direction (say to the right) indicative of this, but imme- diately after it will rush across zero and up the other side of the galvanometer (to the left), showing that the cable current has again set in. If d be too small the needle will pass to the left, at first slowly, but immediately after with a bound. d is now adjusted, resistance is inserted or removed as required, and the eliminating process begun again. As d more nearly resembles the resistance of the cable, the first and instantaneous deflections after battery-contact become smaller; and, when d and the cable resistance are equal, the needle trembles over the FIG. 79. Ka 2 Cable B Ն K, CHCHE G ещ E. Earth zero-point for a moment, and then rushes over to the left under the influence of the cable current. • Should the current given off by the fault be negative, having arranged the bridge as before, repolarise the fault with a positive battery current, and, waiting till G₂ shows the cable free, proceed to test as before, but using a positive current instead of a negative. Should d be too great the needle of G₁ will be deflected in this case, first to the left and then to the right. Should it be too small the needle will move to the right, at first slowly, but immediately after with a rush. The galvanometer G, must always be ready, and not short circuited, else the first and instantaneous deflections after battery-contact will not be perceived. In practice it is found that when the cable current is positive it is easily eliminated by a negative current, but that when LOCALISATION OF FAULTS. 249 it is negative the operation with a positive current is more difficult. Indeed, it is better not to employ a positive testing current at all, except for a moment when it is required to eliminate a weak negative cable current. A positive current applied for a few seconds in this manner has only time to depolarise a fault, but when continued longer it seems to actu- ally coat the exposed wire with badly conducting substances, by which the total resistance is increased. It will be noticed that when the fault is depolarised by a positive current of any duration it does not recover itself for a long time. If a galvanometer be joined in circuit, its needle will remain at or near zero for a considerable time, occasionally oscillating feebly. The depolarisation by a negative current, on the other hand, lasts only a few moments. The whole of the foregoing observations do not appear to be applicable to every fault. Thus, when the fault has consider- able resistance in itself, or when more faults than one exist, it is not always possible to eliminate the cable current. Again, when the fault possesses resistance, the direction and strength of the cable current, when the distant end is alternately insulated and put to earth, do not always coincide. For example, a fault occurred on a six-mile piece of shore-end cable, which reduced the insulation resistance to about 2000 units absolute. Now, when the further end of this piece was to earth, a strong nega- tive current was often obtained, but when it was insulated the cable current was slight and positive. Again, when the fault is further off than about 150 miles, and the intervening cable perfect, the charge current interferes with the test. 269. The principal obstacle found in testing for faults is the presence of earth currents. If it were not for these there would really be but comparatively little difficulty in making satis- factory tests. But even earth currents would not create any serious difficulties, provided they kept constant in strength and‍ direction for any length of time; this, however, is unfortunately seldom the case, and it is often only by patient watching that a few seconds can be obtained when the cable is in a quiescent condition, and a test of correct value made. The earth current difficulty is especially met with in long cables, and it is not uncommon for days to pass without a satis- factory test being made. MANCE'S METHOD. 270. This method, devised by Sir Henry Mance, has for its object the elimination of the effects of an earth current in a 250 HANDBOOK OF ELECTRICAL TESTING. cable when making a resistance test. The general principle of this method has been described on page 237. As compared with the ordinary " Equilibrium Method" (page 235) it has the advantage that the polarisation current does not become changed, as it is liable to do when reversed currents are sent from the testing batteries; moreover, as the test can be made with a negative current only, the resistance of the fault does not alter materially, as it is liable to do when a positive current is applied. In making the test practically the inventor considers that the simplest plan and the one giving the best results is to have the resistances a and b (Fig. 77, page 235) of equal value; the 100 and 1000 pairs of proportion coils in the ordinary bridge would be used generally for the purpose. The test is com- menced by observing the resistance d with the smaller pair of coils, continuing the test until the resistance of the fault appears fairly steady, when, balance being obtained by adjusting d, the galvanometer is short circuited for an instant whilst changing the 100 coils to 1000, and then balance is again obtained by re-adjusting d to d₁. This operation should be several times repeated, and the pair of readings which seem most likely to be correct are then used for determining x from the formula. In working the method care should be taken that the battery is in good condition and that its resistance is not high. If the con- ductor is not broken and the fault is a small one, sufficient resis- tance should be added at the end nearest the fault to bring the latter near the centre (§ 265, page 244). The tests from In either side will then compare well with each other. arranging this, the resistance of the batteries must not be over- looked, and it is therefore desirable that all stations should use similar batteries with approximately the same internal resistance. When testing with the 1000 to 1000 proportion coils, the observations will generally, but not invariably, be higher than when using the 100 to 100 branches. This will depend on the earth currents existing at the time. The corrected result will, however, be approximately the same, although the readings may indicate an alteration of several hundreds of units in the resistance tested. The daily variations in the tests to a fault may of course be due to alterations in the fault itself, especially if it is a small one. The application of the correction will, however, at once show how much is due to the fault, and to what extent the tests are affected by other disturbing influences. Should the alterations be caused by the latter, there will be no material change in the corrected results. LOCALISATION OF FAULTS. 251 271. For the purpose of applying the test with ease and certainty Sir Henry Mance has devised a form of bridge specially adapted for the purpose. In this apparatus, which is shown by Fig. 80, a switch is provided for rapidly changing Fig. 80. TILL. LATIMER CLARK MUIRHEAD & CO LP 4. TULO the proportion coils from 100 to 1000, and vice versá; a set of single ohm slide resistances (page 15) is also added for the purpose of adjusting the main resistance (d and d₁) with rapidity. KENELLY'S LAW OF FAULT RESISTANCE. 272. When a cable which has become broken has its resistance measured in order to determine the locality of the break, the value of this resistance represents the resistance up to the fault plus the resistance of the fault itself. Now although by Lums- den's method (page 245) it is often possible to nearly eliminate the resistance of the fault, yet this cannot always be done. In a recent paper read before the Society of Telegraph-Engineers and Electricians,* Mr. A. E. Kenelly has pointed out as the result of numerous experiments, that when the current flowing does not exceed 25 milliampères (1256 ampère) the resistance of the fault in a broken cable varies inversely as the square root of the current passing, that is to say, for example, if we quad- ruple the current we halve the resistance. As a consequence of this law, it is shown that it is possible to determine what is the resistance of the cable up to the break, independent of the resistance of the break itself. 00 Let r be the resistance of the broken cable up to the fault, and f₁ and f₂ the resistances which the fault has when the 2 * Proceedings of the Society of Telegraph-Engineers and Electricians,' Vol. XVI., page 86. 252 HANDBOOK OF ELECTRICAL TESTING. currents passing are c₁ and c₂ respectively, then by the law stated we have therefore f1: f2 :: √ C₂: Naz fi √ cz 2 Let R, and R₂ be the total measured resistances when the currents c₁ and c₂ are passing respectively, then we have therefore R₁ = r+f₁ R₂ = ? +ƒ2; f₁ Ꭱ Ri f2 fr or therefore therefore or For example. : Bg-r √ C₂ R₁-r R₂ T R₂ √ C2 − r √ C₂ = R₁ √√ c₁r √ C₁₂ - r (√ 61 − √√ C₂) = R₁ √√ C₁ — R₂ √√ €2, R₁ √ C1 – R₂ √ C₂ ↑ = √ C1 √ C₂ The measured resistance of a broken cable when the current passing was 25 milliampères (c₁) was 435 ohms (R₁), but when the current was reduced to 9 milliampères (c2) the measured resistance was found to be 445 ohms (R₂2); what was the resistance (r) of the cable up to the break? therefore r = √√93, and, √√25 = 5, 435 × 5 – 445 × 3 5 - 3 420 ohms. It is obvious that the values of c₁ and c₂ might be determined by placing a low resistance galvanometer in circuit with the cable whilst the tests are being made, and noting the deflections 1 LOCALISATION OF FAULTS. 253 obtained in the two cases. The strengths of the current could be varied either by changing the battery power or by changing the resistances in the arms of the bridge, as in Mance's test. Mr. Kenelly prefers to adopt the latter method and to calculate the strengths of the current passing, instead of having a gal- vanometer in the cable circuit as suggested. In order to eliminate the effects of earth currents he balances to a false zero (§ 252, page 238). 273. Practice is required before any of the foregoing tests can be satisfactorily made. An artificial line, however, can easily be formed with resistance coils to represent the resistance of the line up to the fault, and a short piece of cable core which has been pierced with a needle for the fault itself. This piece of core should be immersed in a vessel of sea-water, using a piece of galvanised iron plate or wire for an earth. By this means a very fair idea of some of the difficulties encountered in testing for faults in cables may be obtained, and good practice made. JACOB'S DEFLECTION METHOD. 274. A disadvantage in using the Wheatstone bridge for measuring the resistances in the foregoing methods is the time it takes to arrive at balance, and the difficulty of seeing what is happening in the way of earth currents, polarisation, &c.; the determination of the resistance by deflection is, however, as rapid a method as can be desired, and allows of continuous observation of the behaviour of the fault. The only require- ments for the test are, the battery with a reversing switch,* a Thomson mirror galvanometer with a reversing key,* and a set of resistance coils. The battery, galvanometer and cable are first joined up in circuit, one pole of the battery and the further end of the cable being to earth; and the galvanometer being shunted by a shunt of very low resistance (a short piece of wire answers well for this purpose). The needle of the galvanometer is turned so that it has a large inferred zero (§ 60, page 64). The apparatus being thus joined up, the battery is switched on and one of the galvanometer reversing keys depressed so that the needle of the galvanometer turns in the direction necessary to bring the spot of light on the scale; by adjusting the shunt this deflection is brought to a convenient position. The gal- vanometer reversing key is now released, the battery is reversed by means of its switch, and then the second reversing key of the * Chapter X. } 254 HANDBOOK OF ELECTRICAL TESTING. | galvanometer is depressed so that the deflection of the galvano- meter needle is in the same direction as it was in the first instance. Since in one case the battery current is in the same direction as the earth current and in the other case it is opposing it, the two deflections will differ, but by a judicious adjustment of the shunt and of the magnitude of the inferred zero it may be arranged that both deflections come well within the range of the scale, the shunt being the same in both cases. These preliminaries being arranged, the shunt and the zero position must not be altered during the series of tests. A number of deflections are now taken with each current, and by a proper manipulation of the short circuiting key,* the oscillations of the needle can be checked so quickly that the value of the deflections can be determined within two or three seconds or less after the battery has been switched on: thus the behaviour of the fault can be carefully observed and the reliability of the readings with either current assured without any great difficulty. After the necessary deflections have been determined, the set of resistances is substituted in the place of the cable, and the deflections obtained are reproduced. Let d₁ and d₂ be the deflections obtained. Let E and e be the respective electromotive forces of the battery and of the earth or cable current. Let x be the resistance being measured. Let R, and R₂ be the resistances required to reproduce the deflections d₁ and d₂. 2 Lastly, let C, and C₂ be the currents producing the deflections d₁ and d₂; and let R be the resistance of the battery and shunted galvanometer. 2 Now when the deflections are taken on the cable we have { and E-te C₁ R + x E - e C₂ 2 R + x [1] [2] When the same deflections are taken with the resistance coils in the place of the cable, then we have = E R+R₁ * Chapter X. [3] i LOCALISATION OF FAULTS. 255 and consequently we have C₂ 2 = E+ e E R + R₂' E R+R₁ or or therefore We also have = R+ x E+e R+ x E R+R₁ 1 e R + x 1+ ; E R+R₁ e R+ x 1. ER+R₁ [4] or therefore therefore or E – e E R + x R+ R₂² 1 R+ x R+ R2 e E R+x R+ R₁ = 1 = 1 – R+x R + R₂ 1 1 + R+R2 (R+x) R+R₁ x = 2 (R+ R₁) (R + R₂) (R + R₁) + (R + R₂) = G Ꭱ 2, [A] that is to say, x equals the harmonic mean of (R+R₁) and (R+ R₂), minus R. In fact we have to add R to both R, and (R+R2), R2, take the harmonic mean of the results, and then subtract R from this mean. If R can be made so low as to be negligible, then of course the formula becomes considerably simplified, x being equal to the harmonic mean of R, and R. Although R could be determined by a separate measurement and then inserted in the formula, there is no absolute necessity for doing this, since we have actually all the data requisite to 256 HANDBOOK OF ELECTRICAL TESTING. determine x without knowing the value of R. From equation [3] we have C₁ R₁ = E - C₁ R, and from equation [2] 2 C₂ R₂ = E - C₂ R ; 2 therefore C₁ R₁ + C₂ R₂ = 2 E - R (C₁ + C₂)• Also from equation [1] and [2] we have therefore C₁ + C₂ = 2 E R+x C₂ R₁ + C₂ R₂ = R + x − R = x. C₁ + C₂ 2 Since the currents C, and C₂ are represented by the deflections d₁ and d₂ we have 1 X d₁ R₁ + d₂ R₂ d₁ + d₂ [B] an equation which is simpler than equation [A] and which does not require R to be known or to be made negligible, though in order to make the test with the greatest chance of accuracy it is advisable that R should not have a high value, for reasons which have been explained in § 78, page 80. If, however, equation [B] is made use of it would be necessary to make the zero of the galvanometer some point on, and not off, the scale, otherwise we should not know what are the true values of the deflections d, and d₂. By making the zero at the extreme end of the scale the range will be 700 divisions, which will generally enable sufficiently accurate tests to be effected. KEMPE'S LOSS OF CURRENT TEST.* 275. In this test, which is shown by Fig. 81, a battery E is permanently connected, through a galvanometer G₁, to one end A of the cable, the further end B being connected to earth through a second galvanometer G. * This test was first described by the Author in the second edition of the present work in the year 1881, but it was also independently devised by M. Emile Lacoine and described by him in the Bulletin de la Société Internationale des Electriciens' for April 1886. LOCALISATION OF FAULTS. 257 Let C, be the current sent through the galvanometer G₁, and let C, be the current received on the galvanometer G, then C. f f 0 = 0 + b + 5 + 6 + G = 0; or, 'ƒ G' ƒ Let the resistance beyond A be l, then b = a + f(b+G) ƒ+b+G C = a + &; (b+G); FIG. 81. A a CL B G G f E Earth also, as in the previous tests, let Earth a+b= L, or, b = L — a, then by substitution we get l3 = therefore that is or = a + C (L − a + G); C¸l¸ = C, a + C, (L + G) C, a, a (C, − C,) = C¸3 73 – C, (L + G), α = a C¸l3 − C, (L+ G) C. - C, Earth L in this equation is known, it being the conductor resistance of the cable when sound. l is easily determined, when the observations with the cable are completed, by joining up the galvanometer G, and battery E in circuit with a set of resist- ance coils, and then adjusting the latter until the deflection on the galvanometer G, is observed to be the same as it was when the cable was in circuit; the resistance in the resistance coils then gives the value of l¸.* * See § 3, page 1. Q 258 HANDBOOK OF ELECTRICAL TESTING. In order to determine C, and C, we must compare the deflec- tions they produce on the respective galvanometers with the deflections obtained on the same galvanometers from a standard current, such, for instance, as that given by a standard Daniell cell (1.079 volts) (page 137), working through 1079 ohms, that is a current of 1 milliampère. Supposing both stations are furnished with standard cells, then each station having noted the deflection obtained when in circuit with the cable, disconnects his galvanometer from the latter, and puts it in circuit with a standard cell and a certain definite resistance, say, 1079 ohms, including the resistance of the galvanometer. The deflection is again noted; then this deflection, divided into the deflection obtained when the cable was in circuit, gives the value of C, or C,, as the case may be. For example. In testing a cable by the foregoing test, the connections being made as in Fig. 81, station A obtained a deflection on his gal- vanometer equivalent to 2800 divisions; station B obtained a deflection equivalent to 1520 divisions. The deflection obtained by A on his galvanometer with a standard cell through 1079 ohms was 100 divisions, and the de- flection obtained by station B with a similar battery working through 1079 ohms was 95 divisions; then 2800 100 1520 = 28; C. = 16. 95 The value of 1, was found to be 280 ohms, and the values of L and G were known to be 345 ohms and 5 ohms respectively. What was the value of a? α = (28 × 280) — [16 × (345 + 5)] 28 - 16 = 186·7 ohms. If the cable had a conductivity resistance of 10 ohms per mile, then the distance of the fault from A would be 186.7 10 18.67 miles. A great advantage which this test possesses lies in the fact, that all the necessary observations with the cable can be made simultaneously, station A arranging with station B that at a definite time the observations are to be made on the galvano- LOCALISATION OF FAULTS. 259 meters; there is thus no chance of error from the fault changing its resistance between two independent observations, as might occur in the other tests. It has been assumed that this test has been made with Thomson galvanometers, and it is advisable if possible to employ them; the directing magnets in the instruments would, however, have to be placed very low down and very low shunts employed, otherwise the deflections obtained would be beyond the range of the scale. 276. It will sometimes be found that the cable is traversed by an earth current. The effects of this may be eliminated (as first suggested by Mr. Latimer Clark) by means of a compensating battery of one or two large-sized Daniell cells, inserted between the end of the cable and the galvanometer. The number of these cells used should be slightly in excess of that required to counteract the earth current, exact balance being obtained by means of a shunt inserted between the ter- minals of the battery. To effect this adjustment, previous to putting on the battery E, we should connect the galvanometer to earth, and then adjust the compensating battery shunt until no deflection is obtained. This being done, the battery E is connected up and the test made as if no earth current existed. It will seldom be found that a larger compensating battery than one or two cells is required to produce a balance, and if these be of a large size their internal resistance may practically be ignored. It is advisable to make the current from the testing battery flow in the same direction as the current which tends to flow from the compensating battery; thus, if the latter requires to be inserted, so that the zinc pole is connected to one terminal of G₁ and the copper pole to the end A of the cable, then the copper pole of the testing battery should be connected to the second terminal of G₁ and the zinc pole to earth. 1 Best Conditions for making the Test. The resistances of the battery E and galvanometers G and G₁ should be as low as possible. THE LOOP TEST. 277. When a faulty cable is lying in the tanks at a factory so that both ends of it are at hand, or when a submerged cable can be looped at the end farthest from the testing station with either a second wire, if it contains more than one wire, or with a second ↑ s 2 L F 260 HANDBOOK OF ELECTRICAL TESTING. cable which may be lying parallel with it, as is often the case, then the simplest and most accurate test for localising the position of the fault is the loop test. This test is independent (within certain limits) of the resist- ance of the fault, thus doing away with the necessity of cleaning and depolarising as would be necessary in the ordinary tests. There are two ways of making this test with the form of apparatus hitherto described. MURRAY'S METHOD. 278. Fig. 82 shows the theoretical and practical arrange- ments. p is the point where the two wires or cables are looped together at the further station, f being the fault. Let x be the resistance from C to the fault, y the resistance from E to the fault. Then BC being plugged up and A B (b) and E A (d) adjusted until equilibrium is produced, bxy = d xx. Let L be the total conductivity resistance of the whole loop, then therefore x + y = L, y = L − x. Substituting this value of y in the above equation, we get b (L−x) = d × x, from which x = L b b + d To obtain L, we should simply join up for the ordinary con- ductivity test, as shown by Fig. 63 (page 191). The fault in this case has no effect upon the test, provided it is not caused by the complete fracture of the cable; in the latter case the broken ends become covered with salts, which would make the resist- ance appear higher than it really is. When, however, the fault is due to a simple imperfection in the insulating sheathing, the ordinary conductivity test gives the correct result. 279. It is advisable to keep a record of the conductivity resist- ance, so that it can be ascertained without the necessity of making a measurement. 280. In the practical execution of this loop test, the connec- tions being made as shown by the figure, all the plugs between J LOCALISATION OF FAULTS. 261 B and C must be inserted; this is necessary, because the galvanometer connection is made on to the terminal B', which is the same as B, instead of on to C. The test could be made by placing the galvanometer on to C, but in that case we should lose the advantage of the key, which it is always best to use. The plugs being inserted between B and C, and the other plugs being in their places, we should remove, say, the 1000 plug from between A and B, and having pressed down the left-hand FIG. 82. INF D x Earth P Earthy A B C if INF Earth P. E B' Earth key, to put the battery current on, which should be a zinc (or negative) one as shown, we should adjust the plugs between D and E, pressing down the right-hand key as required until equi- librium is produced. The different resistances being inserted in the formula, is found, which being divided by the conductivity resistance per mile of the cable, gives the position of the fault. For example. A cable 50 miles long, whose total conductivity resistance was 450 ohms, that is, 9 ohms per mile, was looped with a second cable, which had the same length and conductivity 1 • 262 HANDBOOK OF ELECTRICAL TESTING. resistance as the first cable-the resistance of the loop being 450 × 2 = 900 ohms. The adjusted resistance in d to obtain equilibrium was 4000 ohms, b being 1000 ohms, then X = 900 (TO 1000 GOUD 1000 + 4000) 180 ohms. Dividing this by the conductivity per mile, which is 9 ohms, we get distance of fault from testing station 180 = 20 miles. In making a test of this kind it is advisable to use as high resistances as possible in b and d, because the greater these resistances are the greater will be the range of adjustment. 281. We know that the best galvanometer to employ would be one whose resistance does not exceed 10 times the joint resistance of the resistances on either side of it.* In practice, the resistances b and d would always be greater than the resistance of the looped cables, and the joint resistance of the two resistances would consequently never be more than one-half the resistance of the looped cables; if, therefore, we do not use a galvanometer with a resistance more than, say, five times the resistance of the looped cables, we may be sure that the con- ditions are very favourable for making an accurate test. The value which d should have depends upon the value given to b, and since the range of adjustment is large in pro- portion as d is large, therefore for this reason it is advan- tageous to make b as large as possible; but it is not advisable to make it higher than is requisite to obtain what may be con- sidered to be a sufficient range of adjustment, for by making b and d large the current which passes out of the battery becomes diminished, and consequently the effect on the galvano- meter will also be diminished. This can of course be compen- sated for by adding on extra batteries, but as the number of the latter may have to be inconveniently large, it is as well to avoid doing so, otherwise there is no limit to the values which may be given to b and d. It is possible to avoid making b and d high by making the latter resistance adjustable to a fraction of a unit. If the fault has a very high resistance the employment of high battery power is inevitable, as this high resistance is directly in circuit with the battery. In such a case, however, we may make b and d as high as we like, for, inasmuch as the current flowing out of the battery depends upon the total re- sistance in its circuit, the result of making b and d high is to * Chapter XXV. LOCALISATION OF FAULTS. 263 add but very little to the total resistance, unless indeed they are very excessive, which in practice can hardly be the case. To sum up, then, we have Best Conditions for making Murray's Loop Test. 282. Make b as high as is necessary to obtain the required range of adjustment in d; if b and d would in this case require to be excessive compared with the resistance of the loop, d must be adjustable to a fraction of a unit. Employ a galvanometer whose resistance is not more than about five times the resistance of the looped cables. Employ sufficient battery power to obtain a perceptible deflection of the galvanometer needle when d is 1 unit, or a fraction of a unit, out of exact adjustment. VARLEY'S METHOD. 283. This is shown theoretically and practically by Fig. 83. In this test BC (a) and AB (b) are fixed resistances, and EA (d) is adjusted until equilibrium is produced. Then, x and y being the resistances of the fault from E and C respectively, and therefore a (d + x) = by, y = L― x; a (d + x) = b (L — ∞), from which bL-ad X = b+ a If b = a, then L-d X = 2 For example. The two cables being of the same length and conductivity as in the last example, and b being equal to a, equilibrium was obtained by making d = 600; then x = 900 600 - = 150 ohms. 2 264 HANDBOOK OF ELECTRICAL TESTING. 284. It is necessary that the faulty one of the two looped cables be attached to E, or else it would be impossible to obtain equilibrium. If we were testing a looped cable, and after having joined it up we found that we could not obtain equili- FIG. 83. B идеи y P Earth E← x 1111 Earth A B C D VE C Earth TEHE INR P E Earth Β' brium, we may be sure that the fault lies between C and p. The cable must then be reversed, and a fresh test made. 285. The conditions for making this test with accuracy are not quite so simple as they were in Murray's test. In this case they are almost precisely similar to what they are in an ordinary bridge test, for the resistance dy takes the place of the resistance d in the latter test, and if we determine the best conditions for finding x we practically determine the best conditions for finding y, as the test is made in the same manner for determining either quantity. It is, however, always best to have the relative positions of the battery and galvanometer as indicated in the figure. For if LOCALISATION OF FAULTS. 265 1 1 the galvanometer took the place of the battery, and vice versa, it would be affected by any earth or polarisation currents which might enter at the fault, and this would render adjustment difficult. We have, then, Best Conditions for making Varley's Loop Test. 286. Make a as low as possible, but not lower than g x g+x Make b of such a high value that d when 1 unit out from exact adjustment produces a perceptible movement of the galvano- meter needle. A rough test would first have to be made to ascertain approximately the values of x and y, and then if necessary the resistances must be readjusted so that the above conditions are satisfied, and then exact adjustment of E A be made. Best General Conditions for making the Loop Test. 287. Although the loop test avoids errors due to earth currents it does not avoid errors due to cable currents, that is to say, currents set up by chemical action at the fault itself; this action causes a current to flow in opposite directions through the branches of the cable on either side of the fault, in other words, it causes a current to circulate in the loop. This current, although comparatively weak, yet is sufficient to cause errors which it is advisable to avoid if possible. Mr. A. Jamieson states that by balancing to a "false zero" (page 238) the above cause of error may be eliminated and a very considerable increase in the accuracy of the test be obtained. Correction for the Loop Test. 288. It sometimes happens that the resistance of the fault in a cable approaches the normal insulation resistance of the latter; then the position of the fault indicated by the loop test will not be its true position. The reason of this is, that the current flowing in a faulty cable has two paths open to it: one through the fault and the other through the whole of the insulated sheathing. The cable, in fact, possesses two faults: the actual fault, and the fault due to the conducting power of the insu- lating sheathing. This second or resultant fault, as it is called, in a homogeneous cable is equivalent to a fault in the centre of the cable of a resistance equal to the insulation resistance of the cable itself when in good condition. If the 266 HANDBOOK OF ELECTRICAL TESTING. 勇 ​cable is not homogeneous throughout, this resultant fault will lie away from the centre. Its position can be found, however, by the ordinary loop test when the cable is sound. We have then to determine the true position of the fault when the position and resistance of the resultant fault, the insulation resistance of the cable when imperfect, and the position of the fault indicated by the ordinary loop test, are known. The following shows how this may be done approxi- mately :- In Fig. 84 let A B be the cable joined up for the loop test, f being the actual fault, i the resultant fault, and f₁ the apparent position of the fault given by the loop test. Let P equal the resistance of i, that is, the insulation resist- ance of the cable when perfect; also let I equal the insulation FIG. 84. B a A B Z fi f resistance when the cable has a fault, which resistance is due to the joint resistances of the fault (which we will call c) and the insulation P; then I = Pc P+c PI ; whence c = Р P_I Τ Now it is evident that the position of f, with respect to i and f will depend upon the relative values of P and c: thus if P and c were equal, then f₁ would lie midway between i and ƒ; if P were greater than c, then f₁ would be nearer ƒ; or again, if P were less than c, then ƒ₁ would be nearer i. This being the case, we have the proportion p: (di distance between` ƒ₁ and i 1 Let distance A fi if₁ = ẞa; also let or → 1 : c : (di distance between f₁ and f B and A i = a, therefore distance distance fi f = x, then Px = c(6 − a), P x = PI P-I (B − a) ; f LOCALISATION OF FAULTS. 267 therefore I X = (B − a), P – I which gives us the position of the true fault beyond the apparent one. Or the distance of the fault from A will be B+ I P-I (B − a) = BP-a I P – I For example. In a looped cable, whose total length was 100 miles, and total conductivity resistance 900 ohms, the ordinary loop test showed the apparent position of a fault which existed in it to be 700 ohms from A, that is, B = 700. The position of the resultant fault given by the loop test when the cable was new was found to be 500 ohms from A, that is, a = 500. The insulation resistance of the cable when new was 3,000,000 ohms, and when faulty 600,000 ohms, that is, P = 3,000,000. f I = = 600,000. Where was the true position of the fault? Distance of fault from A = (700 × 3,000,000) – (500 × 600,000) 3,000,000 600,000 - = 750 ohms; that is to say, distance of fault beyond distance given by loop test was 750 — ß - = 50 ohms. Or, supposing the cable to have a resistance of 9 ohms per mile, the true distance of the fault beyond the apparent distance was 50 or 55 miles. 9 If the cable be homogeneous throughout, the resultant fault 268 HANDBOOK OF ELECTRICAL TESTING. } will appear in the middle of it. In this case a will equal where L is the total length of the loop. L 2' If we write the equation, BP-a I α Distance of fault from A P-I' in the form, I β-απ a Distance of fault from A I' 1 Р I we can see that if P is very large then get P 0, in which case we $ Distance of fault from A = ß, as in the ordinary loop test. 289. In order to make this test satisfactorily, it is necessary to know what are the insulation resistances of the cable when good and also when faulty, at the moment when equilibrium is obtained. Now, as will be shown in Chapter XV., the insulation resistance (P) of a sound cable alters in proportion to the time a current is kept on it; but the rate at which this alteration takes place is definite, and can be obtained by refer- ence to previous tests of the cable made when the latter was sound. The insulation resistance (I) of the cable when faulty cannot, however, be determined by any reference to previous tests; some plan of enabling it to be measured accurately is therefore necessary. A method suggested by Mr. S. E. Phillips enables this to be done in a very satisfactory manner. The whole of the testing apparatus is carefully insulated by being placed on a sheet of ebonite, or on insulated supports; the experimenter also stands on an insulated stand or a sheet of ebonite. The battery for making the loop test, instead of being connected directly on to the terminal of the resistance coils, is connected thereto through the medium of a second galvanometer. By noting the deflection on the latter at the moment equilibrium is obtained on the first galvanometer, and comparing it afterwards with the deflection obtained through a known resistance, we obtain the value of I plus the combined resistance of the resistances in the bridge, which quantity will, however, be insignificant compared with I, and need not be taken into account. LOCALISATION OF FAULTS. 269 A note should be made of the time at which the battery is connected to the instruments, and then, when the plugs are adjusted, equilibrium obtained, and the deflection on the second galvanometer observed, the time must again be noted, so that the period during which the battery current has acted may be known and the value of P correctly obtained. The method of determining the value of P will be considered hereafter. INDIVIDUAL RESISTANCE OF TWO WIRES BY THE LOOP TEST. 290. Mr. S. E. Phillips has pointed out that the loop test may be made very useful for determining the individual resistance of two wires, the leads in a cable factory, for instance, whose ends cannot be got at to connect to the testing apparatus. To do this, the further ends of the leads would be joined together, and the junction put to earth. It is evident, then, that the loop test applied to the wires would give the resistance of either of them to their junction. 270 HANDBOOK OF ELECTRICAL TESTING. 1 CHAPTER X. KEYS, SWITCHES, CONDENSERS, AND BATTERIES. SHORT-CIRCUIT KEYS. 291. Although the short-circuit plug-hole is convenient to avoid accidental currents being sent through the galvanometer when the various resistance coils, batteries, &c., are being joined up for making a measurement, yet a key which in its normal condition short circuits the galvanometer, is extremely con- venient and useful. Such a key is represented by Fig. 85. In its normal condi- tion the spring rests against a platinum contact, and, when pressed down, against an ebonite one. FIG. 85. FIG. 86. 1/4 real size. The two terminals of the shunt are connected to the terminal of the key, which in this and most keys are double, so as to enable the wires leading to the resistance coils, batteries, &c., to be conveniently connected to them. If it is required to keep the key pressed down for a lengthened period, a small piece of sheet ebonite or gutta-percha can be slipped in between the contacts, so as to prevent their making connection when the finger is taken off the key. Some keys of this kind are provided. with a catch (Fig. 86), which keeps the spring down when it is depressed. The advantage of the short-circuit key over the short-circuit plug may not seem obvious, but actual practice will soon show its value. KEYS, SWITCHES, CONDENSERS, AND BATTERIES. 271 } REVERSING KEYS. 292. Besides the short-circuit key, a Reversing Key is usually inserted in the galvanometer circuit, so that the deflections of the needle may always be obtained on the same side of the scale. A form of reversing key very commonly used is shown FIG. 87. FIG. 88. 1/4 real size. in elevation and plan by Figs. 87 and 88, and in general view by Fig. 89. The galvanometer terminals would be connected to the two end terminals of the reversing key, or, if the short-circuit key is inserted, to the terminals of the latter. By pressing down one or other of the springs, the current will pass through the galvanometer in one direction or the other. The two handles on either side of the two springs are for the purpose of clamping either of the latter down when required. Particular care should be taken, when procuring the key, FIG. 89. to see that the terminals, &c., are not fixed on the top of the ebonite pillars by means of bolts running right through them, as in such a case the advantage of the pillars is entirely lost, and the terminals might just as well be screwed direct into the base board. Care should also be taken that the contacts of the keys are clean, as when there are several contacts considerable resist- ance might be introduced into the circuit from their being dirty. 272 HANDBOOK OF ELECTRICAL TESTING. . 293. It is sometimes found in this form of reversing key that the springs fail to make the necessary contact when clamped down, owing to the loosening or wear of the cam employed to hold it down. Pell's Patent Self-locking Key, which is shown by Fig. 90, and which was designed by Mr. B. Pell, of the firm of Messrs. Johnson and Phillips, entirely overcomes this difficulty by dispensing with the cam altogether, and introducing a spring latch which, when the key is depressed, automatically catches FIG. 90. PELL'S PATENT RIMBAULTS and holds it with certainty in position until it is released, the movement, either in depressing the key or in releasing it being effected with one hand. Each latch is released by pressing the corresponding ebonite knob on the insulating pillar, as shown in the figure. The other advantages of this key over the old form, although not of so much importance, will be appreciated by all who take a pride in the appearance of their apparatus. The absence of the cams and their supporting pillars, besides improving the insulation, and allowing of the key being more easily cleaned, makes it look neater, and prevents the lacquered surface of the brass work being disfigured, as is invariably and unavoidably the case when the cam is used. A Short-circuiting Key is also made on the same principle, the spring in this case being somewhat stronger to prevent uninten- tional locking when the key is only gently tapped by the finger. REVERSING SWITCHES. 294. In addition to the reversing key for the galvanometer, a Reversing Switch for the testing battery is very useful: it need KEYS, SWITCHES, CONDENSERS, AND BATTERIES. 273 not, however, be such an elaborate one as that used for the galvanometer. Figs. 91 and 92 represent such a switch. It consists of four brass segments screwed firmly down to an ebonite base. Each segment is provided with a screw, to which to attach the testing wires. FIG. 91. FIG. 92. ! 1/4 real size In some cases each segment is supported on an ebonite pillar, which improves its insulation very much, and, indeed, would be absolutely necessary for some tests we shall describe. The poles of the battery would be attached to two opposite terminal screws, say A and A', and the leading wires to the two other screws, B and B'. To make the current flow in one direction, we should place the plugs between the segments A and B, and A' and B', and to make it flow in the other direction, between the segments A and B', and A' and B. If one or both the plugs are removed the battery current will be cut off altogether. It is always best, in order to do this, to remove both the plugs in preference to one only, for if the battery is not well insulated a portion of the current may still be able to flow out of the battery and disturb the accuracy of a test. Two other pieces of apparatus are necessary to form a very complete set, viz. a "Condenser" and a "Discharge key." CONDENSERS. 295. A Condenser is merely a Leyden jar exposing a large surface within a small space; those constructed for testing purposes are made of sheets of tin-foil placed in layers between thin sheets of mica coated with shellac. The alternate layers of tin-foil are connected together, so that sets are formed corre- sponding to the outside and inside coatings of the Leyden jar. A very convenient form of condenser, manufactured by Messrs. Warden, is shown by Fig. 93 (page 274). The layers of tin-foil and mica are placed in a round brass box with an ebonite top, on which are fixed the connecting T 1 274 HANDBOOK OF ELECTRICAL TESTING. terminals. These terminals are placed on brass blocks, the ends of which are in close proximity to one another, so that a plug can be inserted between them for the purpose of enabling the FIG. 93. apparatus to be short circuited. This should always be done when the condenser is not in use, so that any residual charge which may remain in it may be entirely dissipated. FIG. 91. • • The "electrostatic capacity" of these condensers is usually microfarad, the "farad" being the unit of electrostatic capacity. They are also made, however, so that several capacities can be obtained, by inserting plugs in different holes. Those having five different capacities (Fig. 94), viz. 05, 05, 2, 2, and 5 microfarads, enable any value ftom 05 to 1 to be obtained by inserting one or more plugs. It is very often extremely useful to be able to vary the capacity, so that it is better to have the latter form rather than the former, although it may be a little more expensive. • : I KEYS, SWITCHES, CONDENSERS, AND BATTERIES. 275 Fig. 95 shows another form of a divided condenser arranged in a brass box. A good condenser should not lose, through leakage, more than 1 per cent. of its charge in one minute. FIG. 95. 475 296. Condensers, like batteries, can be combined for "quan- tity" or in “series," and advantage may often be taken of this power of combination to obtain a large number of capacities from a small number of condensers. When condensers are connected together for "quantity" the capacity of the combination will be equal to the sum of the respective capacities of the several condensers. Thus, if we call F1, F2, F3, &c., these capacities, then the capacity of the combination will be F₁ + F₂+F3+.... 1 2 This may be expressed symbolically thus:- F₁ F'2 F3 When the combination is made in "series" (corresponding to the "cascade" arrangement of Leyden jars) the joint capacity of the series follows the law of the joint resistance of parallel circuits,* thus :— 1 1 1 1 + + F₁ F₂ 1 2 至る ​+. 3 * See Chapter XXVII. $ T 2 276 HANDBOOK OF ELECTRICAL TESTING. ! This may be symbolically expressed thus:-- F₁ - F₂ - F3 1 2 By following out these laws, if we had two condensers, F, and F2, we could obtain four different capacities, viz. F1, F2, F1 + F2, F₁ F₂ F₁ + F₂ and 1 With three condensers we could obtain fourteen different capacities, viz. F1, F2, F3, F₁ + F, F 1 2 F₁ + F₂ + F3, F₁ + F2 1 F2, F₁ + F3, F2 + F3, 1 F₁ F3 F₁ + F3 F2 F3 F2 + F3 F₁ + F2 F₂ F3 F₂+ F3' 1 F₁ F3 F, F, 1 2 2 F₂ + F₂+ 3 and 1 F₁ + F 3 1 F+F₂ 1 1 1 2 F₁ + + F₂ 2 F3 Any of these combinations may be expressed symbolically in the manner before shown; thus, for example, to take the F+ F2 F3 combination, this would be shown thus :- 2 F₂ + F3 F, F2. F's DISCHARGE KEYS. 297. To enable the discharge from the condenser to be read on a galvanometer a discharge key is necessary. This, like the other pieces of apparatus, is made in a variety of forms. Webb's Discharge Key. 298. Fig. 96 shows a pattern (designed by Mr. F. C. Webb), which is in very general use. It consists, primarily, of a hinged lever of solid make, pressed upwards by a spring and playing between two contacts. A vertical ebonite lever, hinged at its lower end, is fixed to the base of the instrument in the position shown. This lever has near its upper end a projecting brass tongue, which, when the lever is pressed forward (by means of a spring), hitches over the extremity of the brass lever. The end of the latter is cut away so as to form two steps; when the brass tongue on the vertical ebonite lever is hitched over the lower step then the KEYS, SWITCHES, CONDENSERS, AND BATTERIES. 277 brass lever stands intermediate between the top and bottom · contacts, and is insulated from both of them, but when the tongue is hitched over the top step then the brass lever is in connection with the lower contact. Again, when the ebonite lever is drawn back the brass lever is freed and springs up FIG. 96. • against the top contact step. If we suppose the brass lever to be hitched down on the lower contact step, then by pulling back the ebonite lever a little the brass tongue unhitches from the top step and hitches on the lower one, thus allowing the brass lever to spring up from the bottom contact but not to come in connection with the upper one; if, however (as before explained), the ebonite lever be pulled completely back then the brass lever rises in connection with the top contact. 299. When using this discharge key for the purpose of measuring the charge in a condenser, the connections to the galvanometer, &c., would be made as shown by Fig. 97 (page 278). On pressing down the key K, the two poles of the battery are put in connection with the two terminals A and B of the condenser C, and on releasing the key so that it comes in contact with the top contact, the two terminals of the condenser are put in connection with the two terminals of the galvanometer, which thus receives the discharge current through it. If we so arrange the connections that the top contact of the key, instead of being joined to the condenser through the galvanometer, is connected directly to it, and the galvanometer is placed between the back terminal of the key and the second terminal of the condenser; then on pressing down the discharge key we get the current charging the condenser through the gal- vanometer, whose needle will be deflected to one side of the zero point; and then, on releasing the key, we get the discharge deflec- tion, which will be of the same strength as the charge deflection, 278. HANDBOOK OF ELECTRICAL TESTING. but in the opposite direction to it. The first arrangement, given by the figure, is, however, the one generally employed. The discharge deflection on the galvanometer is only momentary, the needle or spot of light immediately returning towards zero. 1 300. In using the Thomson galvanometer (which is practically the only instrument of any use for the purpose) for measuring the discharge, the adjusting magnet must be put high up, if it is placed with its poles assisting the earth's magnetism, or low down if it opposes it, so as to make the needle swing slowly FIG. 97. B 웃으 ​G : S. K K A K enough to enable the deflection to be read on the scale. It is best to avoid making the needle swing very slowly, for then the spot of light will probably not return accurately to zero, but may be three or four divisions out. A little practice will enable a comparatively quick swing to be read to half a division, or even less. Kempe's Discharge Key. 301. A form of discharge key designed by the author is shown in plan and elevation by Figs. 98 and 99, and in general view by Fig. 100. It consists, like Fig. 96, of a solid lever, hinged at one end, and playing between two contacts attached to two terminals. ว KEYS, SWITCHES, CONDENSERS, AND BATTERIES. 279 66 CC Two finger triggers, near the other end of the lever, marked Discharge" and "Insulate," are connected to two ebonite hooks. The height of the hook attached to the finger trigger marked Discharge" is a little greater than that of the other hook, so that the lever stands intermediate between the two contacts when it is hitched against it. When the lever is pressed down against the bottom contact, the shorter of the hooks hitches it FIG. 98. FIG. 99. I 1/4 real size down. If in this position we depress the "Insulate" trigger, the lever is freed from its hook, and springs up against the second hook, thus insulating the lever from either of the contacts. The "Discharge" trigger now being pressed down, the lever springs up against the top contact. To the hook of the "Discharge" trigger there is a small FIG. 100. CATE piece of metal fixed which is broad enough to come in front of the second hook, so that if the "Discharge" trigger is de- pressed first it draws back both the hooks, and thereby, if the lever at starting be hitched to the bottom contact, allows the lever at once to spring up to the top contact. If, however, the "Insulate" trigger be depressed, only the hook attached to 280 HANDBOOK OF ELECTRICAL TESTING. that trigger is drawn back, allowing the lever to spring up against the second hook and be thereby insulated, as at first explained. Lambert's Discharge Key. 302. The arrangement of discharge key designed by Mr. Lambert and shown by Fig. 101, is a very good one, and possesses the advantage that the principal terminal is highly insulated when the key is in its normal condition, a point of importance in some tests. The two terminals seen at the front part of the key correspond to the top and bottom contacts of the keys previously described. The ends of two spring levers, provided with ebonite finger-knobs, are set over FIG. 101. these contacts; the other ends of the springs are fixed to a brass cross-piece provided with a terminal, the cross-piece being secured to an ebonite bracket fixed at the end of a stout ebonite rod. By this arrangement the terminal connected to the spring levers is insulated by the long ebonite rod as well as by the ebonite bracket by which the rod is supported on the stand. In manipulating the key, the left-hand lever, say, is first depressed, thus putting the back terminal in con- nection with the contact (corresponding to the bottom contact of the other forms of keys) beneath it. This lever is then released, and the right-hand lever depressed, thus putting the back terminal in connection with the contact (corresponding to the top contact of the other keys) beneath it. The only objec- tion to this form of key is the fact that it is possible to press both levers down at once, thus connecting together the back and the two front terminals; if this is done accidentally, then, as will be seen by reference to Fig. 97 (page 278), a direct circuit is formed by the battery through the galvanometer, which may result in the sensibility of the latter being altered through the violence of the deflection. Such an accident obviously cannot. possibly occur in the other forms of keys. KEYS, SWITCHES, CONDENSERS, AND BATTERIES. 281 The Lambert key is often provided with cams similar to those shown in Figs. 87, 88, and 89 (page 271), so that the spring levers can be clamped down if desired. Rymer Jones's Discharge Key. 303. An excellent form of discharge key has been devised by Mr. J. Rymer Jones, and is manufactured by the India Rubber, Gutta Percha, and Telegraph Works Company of Silvertown. The key is so constructed that (like Lambert's key) the principal terminal is left perfectly free during the period of "insulation," as shown in Fig. 102; the leakage Battery. FIG. 102. Cable. A B C' Galv: from this terminal is therefore confined to the ebonite support A B. The form of this support, a vertical section of which is shown by Fig. 103 (page 282), gives a very considerable length of surface over which any leakage must pass, it being in the present case 63 inches in a height of only 2 inches; while, since the portion A screws into the outer cap B, the former may be removed, when important tests are about to be made, * 282 HANDBOOK OF ELECTRICAL TESTING. and scoured with glass-paper, so as to secure the advantage of a fresh surface without disfiguring the outer polished surface. g B. FIG. 103. Cable. Ъ - Vertical section— 66 The movements for "Charge," Insulate," and "Discharge," will be readily understood from Fig. 102. ll' are ebonite rods; their brass prolongations c c', which move with them as one piece, have the under surfaces, where they rub against the platinum contacts b and g, tipped with platinum. When 7 is deflected to the left, the end of the rod r, attached to it, presses against l'—should the latter happen to be turned to that side -and carries it over in the same direction, first breaking contact at c' g, if previously made, and after- wards making contact at b c. Thus the "Battery Battery" and "Cable" ter- minals are connected together. To "Insulate" the cable terminal it is only necessary to move l back again towards the right, as in Fig. 102. To "Discharge," press towards the right. Should I not already be over to the right (as in the last position for "Insulate ") it will be carried over with l' and the contact at b c broken before c' and g come together. The rod r in fact prevents the galvanometer and battery terminals from both being put to the cable terminal at the same time. Length of insulating surface 63 inches. 304. Although not perhaps absolutely necessary, it is advisable to have a second set of resistance coils (which need not, how- ever, be of the bridge form) to act as an adjustable shunt for the galvanometer. 305. A simple form of galvanometer to enable the resistance of the Thomson to be quickly taken, is also useful. This, how- ever, can be dispensed with, as Mr. S. E. Phillips has pointed out that the resistance of the galvanometer can be determined by the very simple device of measuring the resistance of one of the shunts (the 4th preferably). To do this, the shunts will have to be removed from the galvanometer and connected up to the bridge as an ordinary resistance, the galvanometer itself being used in the usual manner. $ I KEYS, SWITCHES, CONDENSERS, AND BATTERIES. 283 Mr. Phillips suggests that the shunts should be enclosed under the glass shade, so as to ensure that they may have the same temperature as the galvanometer coils. As it is preferable to use a set of resistance coils as a shunt, a single resistance coil of the same wire and resistance as the gal- vanometer coils, might be permanently fixed to the galvano- meter stand under the glass shade; the resistance of this, measured by the help of the galvanometer, would at once give the resistance of the latter. If such a device were adopted, care would have to be taken that the coil is wound double on its bobbin, for otherwise it would affect the galvanometer needle when traversed by a current. 306. The form of bridge coil most generally employed with the Thomson galvanometer is that shown by Figs. 4 and 5 (page 12), the keys attached to the other form not being used. BATTERIES. 307. Besides the foregoing instruments, a battery of at least 200 cells is necessary. The form known as the Minotto is a convenient one, and is frequently used for testing. It consists of an earthenware (or more frequently of a gutta-percha) jar, about 8 inches high, at the bottom of which is placed a round plate of copper, resting flat. A strip of copper about three-quarters of an inch wide, coated with gutta-percha, is fixed to this plate, and brought up the side of the jar. Over this plate a layer of coarsely powdered sulphate of copper is placed; the jar is then filled nearly to the top with damp sawdust, and resting on this is placed a thick disc of zinc, provided with a terminal at the top. A series of these cells is coupled up in the ordinary manner. The Leclanché battery is used at some cable factories; it has the advantage of high electromotive force, but is not so constant as the Daniell, though if care is taken that it does not become accidentally short circuited through a low resistance it answers very satisfactorily, and requires but little attention. The Chlo- ride of silver battery of Mr. Warren de la Rue (page 143) is also now used to some extent for testing, especially on board ship, as it has the advantage of great compactness and portability. The batteries should be placed on well-insulated supports, in a dry situation, so as to avoid leakage, which interferes with the constancy of the current. 308. Besides the large battery, a single cell placed in a small box, with appropriate terminals outside, is required, whose use will be explained. 284 HANDBOOK OF ELECTRICAL TESTING. 1 ¡ [ CHAPTER XI. MEASUREMENT OF POTENTIALS. 309. Let E (Fig. 104) be a battery of which A and B are the free poles; then the free electricities at those poles will have equal but opposite potentials, and the difference of these P V A H FIG. 104. E potentials is the electromotive force of the battery. Thus, if V (that is the line PA) is the potential at A, then V (that is the line B Q) will be the potential at B, and the electro- motive force E of the battery will be E = V − (− V) = 2 V. Although the expression "poten- tials of the free electricities" is, V strictly speaking, more correct than potentials" simply, yet the latter is generally used as an abbreviation of the former, and we shall so use it unless the contrary is indicated. The potentials diminish regularly from one pole of the battery to the other, the potential at the middle of the battery being zero. 310. If the two poles are connected by a resistance A C B, as in Fig. 105, then the potentials will diminish regularly along A C B also, as shown in the figure, the potential at the middle (F) being zero as in the case of the battery. But the potentials at A and B will be less than they were previous to the joining of the poles by A CB, the amount of the diminution being dependent upon the value of the resistance ACB, and also upon the value of the resistance of the battery. These diminished potentials may be represented by, say, the lines pA (+ v) and B q (− v), respectively. 311. If the two terminals of a condenser are connected to any two points in the resistance, the electromotive force of the charge which the condenser will take will be directly propor- tional to the difference of the potentials, that is to the electro- } = MEASUREMENT OF POTENTIALS. 285 motive force, at those two points. Thus if the condenser were connected to A and B the charge it would take would have an electromotive force, E₁, of E₁ = v(v) = 2 v. If the points to which the condenser is connected were A and C, the electromotive force, E2, of the charge would be E2 = V V. Again, if the condenser were connected to C and B, the electromotive force, E,, of the charge would be E3 = ย - (− v) = v + v. It is easy to see that ▼ − v : ▼ + v :: a : b. If, therefore, we connect two condensers between the points P FIG. 105. • P a E F LV A CF 1 q A and C and the points B and C respectively, and adjust the resistances a and b, we could charge the condensers to any relative electromotive forces we please. 312. Although, strictly speaking, the diminution or fall of potential along the resistance ACB is represented by the line 1 { f 286 HANDBOOK OF ELECTRICAL TESTING. } 1. p F q (see small figure), the zero point being at F, yet we may generally with perfect correctness assume the zero to be at B and the fall of potential to be represented by the line p' B, and similarly with the fall from one pole of the battery to the other. In most cases it is convenient to consider the fall as taking place in this way, as we avoid having to consider the potentials as partly and partly quantities, which is liable to cause confusion in making calculations. · 313. We stated that if the poles of the battery are joined by a resistance, the potentials at those poles will be altered in value; they will, in fact, be reduced in proportion as the resistance is small or large. Now, when a current flows through the galvanometer, it does so in virtue of a difference of potential at its two terminals, and the strength of this current is directly proportional to the value of this difference; and conversely, if we note the difference in the strengths of currents passing through a galvanometer we shall know the relative values of the differences of potential at its terminals. It may at first sight, therefore, appear sufficient, in order to measure the relative values of the differences simply to connect the terminals of a galvanometer to the points at which the differences are to be noted, and then to observe the deflections obtained. But by connecting up a galvanometer in this way we should reduce the resistance of the portion of the circuit between those points, and the potentials at the poles of the battery would consequently decrease, therefore the potentials at the points where the galvanometer is connected would decrease also; the current then which would produce a deflection of the galvanometer needle, would be that due to the diminished potentials. If, however, the resistance of the galvanometer is very high compared with the resistances with which it is con- nected, then its introduction will produce no diminution in the potentials, and consequently its deflection, that is to say, the current passing through it, will be a true index of the value of the difference. If, therefore, we wish to theoretically consider what are the relative differences of potentials at any points in any particular arrangements of batteries and resistances, we have simply to suppose these points to be connected by a galva- nometer whose resistance is infinite compared with the other resistances, and then to determine the relative values of the currents which will flow through it in the several cases. 314. From what has been said we can see that, practically, it we connect a galvanometer to any two points at which a difference of potential exists, then the deflection obtained will accurately represent that difference of potential, provided the MEASUREMENT OF POTENTIALS. 287 galvanometer has a total resistance in its circuit very much greater than the resistance between the two points in question. 315. The quantity of electricity in a condenser depends directly upon the electromotive force of the charge, and the deflection obtained upon a galvanometer depends directly upon the quantity discharged through it; the discharge deflection obtained from a condenser, therefore (§ 300, page 278), other things being constant, will represent the electromotive force of the charge in it. It may be mentioned that this is only true if the discharge takes place through a comparatively low resist- ance, such as would be met with in an ordinary galvanometer, for then the whole discharge practically takes place instan- taneously; if, however, the discharge is effected through a very high resistance, such as a megohm (1,000,000 ohms) or more, then the discharge is gradual, and the deflection which would be obtained on the galvanometer would not be an accurate index of the electromotive force of the charge in the condenser. MEASUREMENT OF ELECTROMOTIVE FORCE BY LAW'S METHOD. 316. The electromotive force of a battery is the difference of the potentials at its poles, when those poles are free (§ 309, page 284); by successively charging a condenser, therefore, from two or more batteries, and noting the discharges on a galvanometer by the method described in § 300, page 278, we can very simply and quickly determine their comparative electromotive forces. Discharge deflections on a galvanometer whose deflections are truly proportional to constant currents, unless they are nearly equal, are not always proportional to the currents which produce them. It is therefore very desirable, in measurements such as these, in order to ensure accuracy, to adopt the method we men- tioned on page 76 (§ 71), viz. to obtain a uniform deflection by means of a variable shunt to the galvanometer. Thus, if we obtain two similar discharge deflections with two electromotive forces E, and E2, using shunts of the respective resistances S, and S₂; then, since the deflections are the same, the electro- motive forces are in the proportion E: E₂:: G+S₁. G+ S₂ E2 : S₁ S2 or as the multiplying power of the shunts, G being the resist- ance of the galvanometer; for if we multiplied the deflections we obtained, by these quantities, we should get the theoretical deflections we should have had if no shunts had been used. > 288 HANDBOOK OF ELECTRICAL TESTING. For example. With an electromotive force E, we obtained a discharge de- flection of 300 divisions on a galvanometer whose resistance G was 5000 ohms, using a shunt, S₁, of 1000 ohms, and with a second electromotive force, E2, also a deflection of 300 divisions, using a shunt, S2, of 2500 ohms; then 1000 5000+ 1000 E₁ : E2 :: that is, : 5000 + 2500 2500 E, E₂ 2: 1. It is not absolutely necessary that the same deflection be reproduced exactly, although calculation is saved by so doing; as long as the deflections are nearly equal they approximately represent the discharges. It is necessary, of course, that these to obtain the relative deflections be multiplied by strengths of the currents. For example. G+S S 1 With an electromotive force E, we obtain a discharge deflec- tion of 300 divisions on a galvanometer whose resistance G was 5000 ohms, using a shunt S₁, of 1000 ohms, and with a second electromotive force E₂ a deflection of 292 divisions, using a shunt S₂, of 2400 ohms; then 5000+ 1000 • E, E₂:: X 300: 1000 5000 + 2400 2400 × 292; that is, or as E, E₂ 1800 900 33, 21 very nearly. This method is very often the best one to employ, not only for discharge, but also for constant deflections, as it is sometimes inconvenient to have to continually adjust until the same de- flection exactly is reproduced. In certain cases, indeed, it would be impossible to do so, as will be seen hereafter. 317. It may be here mentioned that, in the case of discharge deflections, the fact that the resistance between the terminals of the galvanometer is varied by the introduction of shunts of different values, does not require to be taken into consideration. MEASUREMENT OF POTENTIALS. 289 CORRECTION FOR DISCHARGE DEFLECTIONS. 318. Mr. Latimer Clark, in a communication addressed to the Society of Telegraph Engineers,* points out an error caused by the use of shunts in measuring discharge deflections. It was found that if a certain discharge deflection were obtained with a shunt, then on removing the latter the dis- charge deflection obtained was larger than that given by G+S multiplying the original deflection by S After considerable research, the cause of the error was traced to the inductive action of the galvanometer needle on its coils. The movement of this needle set up a slight current, which opposed the discharge current, and consequently reduced its effect. This effect being more marked when the shunt was used, made the discharge deflection without the shunt to appear larger than it should. The formula for finding what would be the discharge de- flection obtained on the removal of the shunt, the discharge deflection without the shunt being given, may be thus arrived at:- First suppose the shunt to be inserted. Now in all problems in which a current from a condenser has to be considered, we may suppose the condenser to be a battery with a resistance infinitely great compared with the resistances external to it. Let E be the electromotive force of the charge, R the resist- ance of the condenser circuit, G the resistance of the galvano- meter, S the resistance of the shunt. Let the movement of the needle generate an opposing electro- motive force e; then calling C, a, and ẞ the respective current strengths in the galvanometer, condenser, and shunt circuits, we get the following equations: therefore therefore a-CB=0, ß = 0, or, ß = a − C aR+CG-E+e=0 a R+ẞS-E = 0; · aR+CG-E+e=0 a R+ (a - C) S — E = 0; a R=E-C G − e a (RS) = E+CS; * Journal of the Society of Telegraph Engineers,' Vol. II., page 16. U 290 HANDBOOK OF ELECTRICAL TESTING. then by division R R+S E-CG e ; E+CS # by multiplication and changing the signs we get (R+S) (CG + e) - RE-SE therefore - RE-CRS; (R + S) (C G + e) S (ECR) = 0. Next suppose the shunt to be removed, and let the strength of the current be C₁, and the new electromotive force generated by the movement of the needle be e₁, then C₁ = E e1 R+G therefore E = C₁ (R + G) + e̟ • Substituting this value of E in the last equation, we get (R + S) (C G + e) − S (C₂ (R + G) + e − C R) = 0. e1 Now e and e₁ will be proportional to the deflections of the needle, that is to say, to the strengths of current producing those deflections. They will also be proportional to the strength of the magnetism of the needle, which strength we will repre- sent by K. Then e = k C, Substituting these values we get от e₁ = K K C1. (R + S) C (G + k)−S (C₁ (R + G + k) − C R) = 0, 1 K (1 + 15) C (G + *) − 8 (0, (1 + G + ^ ) − 0) = Ꭱ | − = 0. Now, R is to be infinite as compared with S and G; therefore by putting R = ∞, we find that `C (G+ k) − S (C₁ — C) = 0, therefore C₁ = C = c(G + (G+ x + S) S To make this formula useful, we must determine the value of K. This can easily be done thus: Provide two condensers, one having exactly twice the capacity of the other. Charge the larger one with a sufficient battery MEASUREMENT OF POTENTIALS. 291 1 • power to obtain a discharge deflection (a,) of, say, 200 divisions on the scale, with a shunt inserted equal in resistance to the galvanometer. Now remove the shunt, and having charged the other con- denser from the same battery, note the discharge deflection (a2); let it be 204 divisions. It will be seen that the deflection we should have obtained with the larger condenser and no shunt would have been 2 ɑ2, and this is the theoretical deflection we should obtain when a₁ is multiplied by the multiplying power of the shunt corrected by the constant ; that is to say, therefore 2 a2 =α1 K (G+ k + G); Աշ G x=2G(2-1). To continue the example we have given, let us suppose G = 5000 ohms; then 204 K = 2 × 5000 − 1) = 200 = 200. For the particular galvanometer, then, we have been con- sidering, we say that when measuring discharge currents the multiplying power of any shunt (S) which may be used is G+ 200+ S S Suppose we have given the observed deflection without the shunt and also the observed deflection with the shunt, and we require to know what this latter ought to be in order to give us the true deflection compared with the first. Let the true deflection be A; then by the ordinary formula S C₂ = A (G + $). But when the error exists, S S S C₁₂ = c(G+x+8) From these two equations we get G+S S C = A (++) (+x+8); 8) G+k+ K+ S. • U 2 292 HANDBOOK OF ELECTRICAL TESTING. therefore or in words: c(G; C G+K+S G+S K A = c(+ + + + 5) = c(1 + 78), True deflection = G+ observed deflection (1 + K 4+ 5). G+ S It should be clearly understood that this formula is to be applied to the correction of the deflection obtained with the shunt, the deflection without the shunt being considered as the index of the current from the condenser. We may remark that this latter formula corresponds with that obtained by Mr. Charles Hockin, and given by Mr. Latimer Clark in the paper referred to. For practical use the formula C₁ = C = 1 (G+K+S) is the only one we should require, as by it we can at once determine from the deflection obtained without the shunt what the deflection with the shunt would be, or vice versâ. THE RELATION BETWEEN THE CURRENT, THE RESISTANCE, AND THE ELECTROMOTIVE FORCE, BETWEEN TWO POINTS IN A CIRCUIT. 319. In Fig. 106 let E be a battery of electromotive force, E, and resistance, x, joined up in circuit with a resistance r, and let G be a galvanometer having a resistance very much greater FIG. 106, E FIG. 107. e A B A B than the other resistances, so that it does not affect the flow of the current in the circuit, x+r. Now, the current, C₁, flowing through the galvanometer will be C₁ E r G x + r+G go Er X r + G xr+ x G + r G · MEASUREMENT OF POTENTIALS. 293 It is evident that this current must be due to the existence of an electromotive force, or a difference of potential, in some portion of the circuit in which the galvanometer is placed; and it is, moreover, evident that this electromotive force, or difference of potential, must exist between the points A and B, in the portion of the circuit external to G. Let e be this electromotive force (Fig. 107), then we have (since G is very much larger than the other resistances)- but therefore that is, e 11 C₁ = e G₁ + e α₁ = & G' Er xr + x G + rG' Er xr+ x G + r G' EG E +x+r G xr+xG+r G Xr but, since G is very great compared with the other resistances, therefore e 410 x r G E 0; x + r But by Ohm's law (§ 2, page 1), the current C, flowing out of the battery—that is, flowing through r-is therefore E C x + r that is to say, e C = /> or, e = Cr; (A) The difference of the potentials at two points in a resistance (in which no electromotive force exists) is equal to the product of the current and the resistance between the two points. 320. We will next consider the case where an electromotive force exists in the resistance through which the current is flowing, the strength of the latter, and the potentials at the two • ** 294 HANDBOOK OF ELECTRICAL TESTING. points, being partly due to an external electromotive force (as in the case just considered). In Fig. 108 let R be a resistance between the points A and B, and let E be an electromotive force in R, also let r be C R FIG. 108. FIG. 109. RE FIG. 110. Ꭱ C le BA! Ꮩ V T r B V V another resistance completing the circuit, this being either a single resistance or a combination of several resistances. Again, let us suppose there to be an electromotive force, e, in some part of r, and let C be the resultant current entering, say, at B‍and leaving at A. Let us first suppose, as in Fig. 109, that there is no electro- motive force in the resistance (or combined resistances) r, then by Ohm's law we have. E C1 R+r that is, or, c₁R+c₁ r = E, Ꭱ r = C₁ r = E c1 E - C1 R; but by the law (A) (page 293) we have therefore or - V₁' — V₁ = c₁r; V₁ V₁ = E c₁ R, – - V₁ - V₁' = c₁ RE. 1 C1 [1] Next let us suppose, as in Fig. 110, that we have in Ra current, c2, caused by an electromotive force, e, in some part of the resistance (or combination of resistances), r, then we have V2 — V2' = c₂ R. [2] Now if we take the case shown in Fig. 108, where the current C is produced by the two electromotive forces, then the re- spective potentials at the points A and B must be and V = V₁ + V₂ 1 2 V' = V₁' + V₂ MEASUREMENT OF POTENTIALS. 295 Therefore we have V − V' = (V₁ — V₁') + (V₂ − V₂'), - − 1 2 2 and by substituting the values of V₁ — V₁' and V₂ – V2 given in equations [1] and [2] we get V - V' = c₁ R - E + c₂ R = R (c₂ + €₂) — E; but we can see that C α = c₁ + C₂, therefore VV' = CRE, [3] which is similar to equation [1]. In the case we have taken we have supposed the electromotive forces (and consequently the currents c, and c₂) to act in the same direction, but we should have obtained an equation precisely similar to [3] had the electro- motive forces opposed one another, provided, however, the current due to the electromotive force e were less than the current due to the electromotive force E; if, however, the current due to the electromotive force e were greater than the current due to the electromotive force E, that is to say, if the current C acted against E, then we should have V - V' = CR+ E. [4] 321. The result, then, that we have arrived at by the fore- going investigation is, that- (B) The difference of the potentials at two points in a resistance in which an electromotive force exists is equal to the product of the current and the resistance between the two points, added to the electro- motive force in the resistance, this electromotive force being negative if it acts with the current, and positive if it opposes it. This law, we have seen, holds good whether the current in question is due only to the electromotive force in the resistance, or to an external électromotive force also. It should be remarked with reference to equation [3], that when e is greater than E then the potential V' becomes greater than the potential V, so that in such a case the equation should be written V' - V = CRE. In the case of equation [4], V is always greater than V. MEASUREMENT OF BATTERY RESISTANCE BY KEMPE'S METHOD. 322. Besides determining the electromotive force of a battery, we can also determine its internal resistance with great facility by means of a condenser. To do this, first charge the condenser 296 HANDBOOK OF ELECTRICAL TESTING. by means of the battery, and note the discharge deflection which we will call a; next insert a shunt, S, between the poles of the battery; again charge and discharge the condenser, and note the new deflection, which we will call B. Let e be the electromotive force between the poles of the battery when the shunt S is inserted, and let C be the current flowing, then by law (A) (page 293), we have e e = CS, or, C = = S Also, if E be the electromotive force of the battery, and r its resistance, we have 蠱 ​$ therefore C E S + r ; 0102 e E or therefore S+ 2 r eSer ES; = er = S (E− e), or E - e r = S e but we must also have therefore E: e::a: ẞ; r = s a В β * [A] To obtain accuracy it is advisable for the value of S to be such that the deflection ẞ is approximately equal to 3 For example. A battery whose resistance (r) was required was joined up with a galvanometer, condenser, discharge key, &c., as shown by Fig. 97, page 278. The condenser being charged and then discharged through the galvanometer (by depressing and then releasing the key K₂), a deflection of 290 divisions (a) was pro- duced. A resistance of 20 ohms (S) was then joined between the terminals of the battery, and the condenser again charged and discharged through the galvanometer, the value of the * The reason of this will be obvious from a consideration of the investiga- tions given in § 105, page 103, and § 110, page 111. 1 MEASUREMENT OF POTENTIALS. 297 deflection obtained being 105 divisions (6). What was the resistance of the battery? r = 20 290 105 G 105 = 35.2 ohms. It is evident that if S be adjusted till ẞ = then S. α 2' [B] An error in the foregoing kind of test may possibly arise from one measurement being made with the poles of the battery free, when no action goes on in it, and the second being made with it shunted, which may cause a falling off in its electro- motive force, as action would then be taking place; the accuracy of the test depends upon the electromotive force being constant in both cases. If the shunt S be connected to the battery by means of a key, then the second discharge deflection ẞ is best obtained by first pressing down the key K, (Fig. 97, page 278), then pressing down the key which connects the shunt to the battery, and then immediately afterwards releasing the key K2, and noting the deflection. Thus as little time as possible is allowed for polarisation to take place. 2 MEASUREMENT OF BATTERY RESISTANCE BY MUIRHEAD'S METHOD. 323. A very excellent modification of the foregoing method has been devised by Dr. A. Muirhead; it possesses the great advantage of being free from the source of error just mentioned. In this test (Fig. 111) the battery, galvanometer, and condenser are joined up in circuit with a key K,. The condenser C being short circuited for a moment, so as to dissipate any charge which may have been accidentally left in it, key K₂ is depressed; this causes a charge to rush into the condenser through the gal- vanometer, producing the same deflection as would be produced if the condenser, when charged from the battery direct, were dis- charged through the galvano- meter. The charge deflection (a) being noted, the key K₂ is kept per- manently down, so as to keep the condenser charged. By means of key K₁ a shunt (S) is now con- 1 S FIG. 111. G K, E ✓ R₂ HHHHHH nected between the poles of the battery; at the moment this 1 298 HANDBOOK OF ELECTRICAL TESTING. takes place the potential at the poles of the latter falls, and a reverse deflection of the needle of the galvanometer is produced. If we suppose this deflection to be due to an alteration of the potential from a to ẞ (the latter being the same quantity as that given in the previous test), its value, (, will be ૐ = a - ß, or, B = a. If, then, we substitute this value previous test, we shall get ß of ẞ in equation [A] of the t r = S α १ For example. 2 2 The shunt S having a resistance of 10 ohms, the deflection produced on depressing key K, was 310 divisions (a). K, being held down, K, was depressed, when a deflection, of 100 divisions (5)—in the reverse direction to a-was obtained. What was the resistance of the battery? 1 r = 10 100 310 - 100 = 4.76 ohms. As in the previous test, it is advisable to give S such a value α that is approximately equal to 3 1 As no polarisation of any extent takes place in the battery till some seconds after the shunt has been connected to the former by the key, and as the deflection takes place immediately the key is depressed, it follows that very accurate results will be obtained by this test. It may be remarked, however, that Pro- fessor Garnett has found that polarisation takes place in a battery in an extremely short space of time-in even the roboth part of a second; the amount is, however, of course very small. In Muirhead's test the time during which polarisation would tend to affect the accuracy of the test would be that occupied by the galvanometer needle in swinging from zero to the deflection, (, consequently the quicker the swing (consistent with accurate reading) the better. 000 MEASUREMENT OF BATTERY RESISTANCE BY MUNRO'S METHOD. 324. A modification of Muirhead's method has been suggested by Mr. J. Munro, which simplifies calculation, inasmuch as it gives the value of a by a single deflection. Key K, is first depressed, and then immediately afterwards 1 MEASUREMENT OF POTENTIALS. 299 2 key K₂ is also depressed; this gives a deflection 0, which is equivalent to the difference between the deflections a and ¿ in the last test. Key K, is now raised, leaving key K, down; and as soon as the galvanometer needle becomes steady, K₁ is depressed again, and the deflection ( read, then we have 1 ૬ R = S ន Ꮎ 1 1 As a slight interval of time may elapse between the depres- sion of key K, and key K₂, when obtaining the deflection (during which time the battery would be partially short cir- cuited), it would be preferable to make the test in the following manner :-Make the connections so that the front contact of key K₁ is joined on to the lever of key K, instead of on to the front contact of the latter, as in Fig. 111 (page 297); then in order to obtain 0, depress K, and keep it down, and immediately afterwards depress K2; the deflection observed in this case will be 0. Now raise key K₁, keeping key K, down, and when the galvanometer needle has become steady, depress K,, then the deflection obtained will be (. 1 2 Measurement of Polarisation in Batteries. 325. The amount of polarisation which takes place in a battery when the latter is short circuited may, if required, be easily ascertained in the following manner:-In Fig. 111 (page 297) let S be a short piece of wire of practically no resistance, then having short circuited the condenser C for a moment, depress key K2, and note the deflection d. Keeping K, down, depress K₁, and hold it down for a definite time, say one minute; at the end of the interval, release K₁, and note the deflection d₂ ; then the percentage of polarisation in the one-minute interval will obviously be — 100 (d₁ — d₂) d₁ Measurement of the Resistance of Batteries of Low Resistance. 326. In cases where it is required to measure the total resist- ance of a number of cells of extremely low resistance (secondary batteries, or accumulators, for example) by any of the foregoing methods, the heating effect produced by the current passing through the shunt S when the latter is connected to the battery by means of the key K2, would be liable to heat and damage the coils of which the shunt is composed. In such cases the 300 HANDBOOK OF ELECTRICAL TESTING. cells should be divided into two sets, one set having, say, one more cell than the other; the two sets should then be joined together so that their electromotive forces oppose one another. By this arrangement we practically obtain a battery whose electromotive force is equal to one cell only, but whose resist- ance is equal to that of all the cells; consequently the current generated can be but comparatively small, and would have but little heating effect. The contact in key K, should be made by means of a mercury cup. 2 327. When comparing large electromotive forces with small ones—as, for instance, 100 cells with 1 cell-by the condenser discharge method, the smaller force should be taken first; for a large charge usually leaves a residuum in the condenser, which may be greater than the small force, and which can only be thoroughly dissipated by leaving the condenser short circuited for some time. If the smaller force is measured first, then any residuum it may leave becomes entirely swamped by the larger force, and no increase of charge is added to the con- denser beyond what the force itself possesses. 328. Although the condenser practically becomes charged instantaneously, it is usual to keep the current on for a definite time; twenty seconds is the period very generally adopted; this ensures that the charging is complete. 329. When discharge currents are being measured, especial care must be taken to insert a shunt of small resistance in the galvanometer at first, as momentary currents are very liable to weaken the magnetism of the needles when these currents are strong. If this precaution is not taken, a set of measurements for one test may be rendered useless, as a comparison of measurements made before the magnets become weakened, with measurements taken after, would be obviously impossible, and much loss of time would result. ( 301 ) CHAPTER XII. MEASUREMENT OF CURRENT STRENGTH. 330. If we have a simple circuit, as shown by Fig. 112, then if we know the total resistance, R+G+r, of the same, and also the electromotive force, E, of the battery, we can at once determine the strength of the current flowing, for by Ohm's law we have C = E R+G+ r If the resistances are in ohms and the electromotive force of the battery in volts, then the resulting current will be in ampères. For example. The electromotive force, E, of a battery which produced a current, C, in a circuit whose total resistance, R₁, was 500 ohms, FIG. 112. 100 G ееееее was found by comparison with a Daniell cell to be 3.5 times as strong as the latter; what was the strength of the current, C, flowing in the circuit? E = 3.5 x 1.079 = 2.777 volts. C = 3.777 500 = .00755 ampères. 331. In the foregoing method of measurement, in order to determine the strength of the current, it was necessary to know both the resistance of the circuit and also the electromotive force producing the current. A direct determination of the latter can only be made by comparing it with a current of a known strength as follows:- ❤ 302 ·HANDBOOK OF ELECTRICAL TESTING. DIRECT DEFLECTION METHOD. 332. In this method a galvanometer is inserted in the circuit through which flows the current whose strength is to be measured. The resistance of the galvanometer should be suffi- ciently low not to appreciably increase the total resistance of the circuit in which it is inserted. The deflection produced by the current being noted, the galvanometer is removed and joined up in circuit with a standard battery (page 137) and a resist- ance; the latter is then adjusted until the deflection which was obtained in the first instance is reproduced; in this case, then, the current flowing must be equal to the current whose strength is required. If therefore we divide the electromotive force (in volts) of the standard battery, by the total resistance (in ohms) in its circuit, we get at once the required strength of the current, in ampères. The resistance of the standard battery requires of course to be included in the total resistance unless it is so small that it can be neglected. 333. The degree of accuracy attainable in a test of this kind is directly proportional to one-half the degree of accuracy with which the galvanometer deflection can be read, for, since two measurements have to be made, one with the current whose strength is required, and the other with the standard cell, there may be errors made in both of these. If the current to be measured is a strong one, so that it is necessary to shunt the galvanometer when obtaining a deflection, this shunt not being used when the deflection is reproduced with the standard cell, then in this case the result obtained by dividing the electro- motive force of the standard cell by the total resistance in its circuit, must be multiplied by the multiplying power of the shunt (§ 66, page 69, in order that the correct strength may be obtained. For example. In measuring the strength of a current, the deflection pro- duced on the galvanometer shunted with theth shunt, was 50°. The galvanometer (without the shunt) being connected up with a standard Daniell cell of 10 ohms resistance, and a set of resistance coils, it was found necessary to adjust the latter to 560 ohms in order to bring the needle to 50°; what was the strength, C, of the current to be measured? + C = 1.079 560 10 X 10.0189 ampères. MEASUREMENT OF CURRENT STRENGTH. 303 334. When a galvanometer is inserted in a circuit through which a current is flowing whose strength it is required to measure, it is very necessary that the resistance of the instru- ment be very low compared with the resistance of the circuit itself, otherwise the introduction of the galvanometer will reduce the current flowing, and the result obtained will not be the one required. Before making the test it would of course be necessary to ascertain whether the galvanometer available for use would meet the desired conditions. To make the test as accurately as possible it would be neces- sary that the galvanometer needle when deflected be as near to the angle of maximum sensitiveness (page 23) as possible. If the strength of current necessary to give this angle be found by joining up a standard cell and a set of resistances, and varying the latter until the required deflection is obtained, then we can always tell whether the instrument would be suitable for a particular purpose. Thus, for example, suppose the galvano- meter had a resistance of 1000 ohms, and the angle of maxi- mum sensitiveness were approximately equal to 60°, and sup- pose further that this deflection were obtained by 1 Daniell cell through a total resistance of 8000 ohms, that is, with a current 1.079 of = ·000135 ampères; then to measure such a current 8000 we must have the whole resistance of the galvanometer, viz., 1000 ohms, in circuit. If the instrument were shunted with the th shunt, then the resistance would be reduced to 100 ohms, and the current corresponding to 60° deflection would be ⚫00135 ampères; and again, if the 100th shunt were employed the resistance would be reduced to 1 ohm, and the current corre- sponding to 60° would be 135 ampères; thus we see that if it were required to measure a current of about 0135 ampères and it were necessary that a resistance no greater than 1 ohm should be inserted in the circuit, then it is evident that the galvanometer in question would not answer the purpose required, since a good deflection with ⚫0135 ampères of current would not be obtained if a lower shunt thanth were employed, which latter shunt would reduce the galvanometer resistance down to 10 ohms only. • It is preferable, when possible, to employ a galvanometer of high resistance shunted down, rather than one of low resistance not shunted down, since with such a galvanometer it is easier to measure the "constant" of the instrument accurately; or the high resistance of the latter, together with the high resist- ance which it would be necessary to place in circuit in order to get a readable deflection with one standard cell only, would 304 HANDBOOK OF ELECTRICAL TESTING. swamp, as it were, the resistance of the cell, which resistance need not then be taken into account, or at least need only be known approximately. With a galvanometer of low resistance, however, where a comparatively small resistance only would have to be introduced into the circuit in order to get the required deflection, the resistance of the cell would be required to be known accurately, as it would form an important item in the total resistance of the circuit. 335. The foregoing test has the advantage that it can be made with almost any form of galvanometer, for as only one deflection has to be obtained it is not necessary to know what proportions the various degrees of deflection which it is possible to have, bear to the currents which produce them. If, how- ever, a tangent galvanometer is employed to make the test, then it is unnecessary to reproduce the same deflection exactly, though it is advisable to make it approximately near to it. 336. Suppose that in the last example the test had been made with a tangent galvanometer, and the deflection obtained with the standard cell had been 54° instead of 60° (the deflection given by the current whose strength was required), then in this case the actual strength, C₁, of the current would be C₁ 1.079 560 + 10 X 10 X tan 60° tan 54° 1.079 560 + 10 X 10 X 1.7321 1.3764 = .0238 ampères. 337. If we use no shunt, or the same shunt when taking both the deflections, and, further, if we make the total resistance in circuit with the standard Daniell cell equal to 1079 ohms, then calling do the deflection given by the current, and d₁° the deflec- tion given by the standard cell, we have 1.079 tan do C₁ X tan do 1079 tan d₁° — 1000 tan d¸° ampères; or, further still, if by means of an adjusting magnet we can arrange that the deflection given by the standard cell through 1079 ohms equals 45°, then since tan 45° = 1, we must have C₁ = tan do 1000 ampères. T MEASUREMENT OF CURRENT STRENGTH. 305 CARDEW'S DIFFERENTIAL METHOD.* 338. This method, devised by Capt. Cardew, R.E., is a very satisfactory and useful one; its theory is shown by Fig. 113. The galvanometer G is wound with two wires, g and 91; the current C, whose strength it is required to measure is passed through the coil g₁ (which has a low resistance), and a standard E C FIG. 113. eeeeee G 9 C₁ eeeeee FIG. 114. battery E is connected in circuit with the second coil g and with an adjustable resistance R. The current being passed through the coil g₁, the resistance R is adjusted until the needle comes to zero. If we call n and n₁ the relative deflective effects, for the same current, of the two coils g and g₁, and if C₁ and c be the currents flowing through g, and g, respectively, then in order to produce equilibrium we must have c: C₁ n₁n, or сп C₁ n1 Now the current through c will evidently be E C = R+ g the resistance of the battery being included in R; therefore For example. E n C₁ X R+ g N1 The relative deflective effects of the coils g and g₁ were as 1000 to 1; the resistance of g was 100 ohms. The battery E was a 1-cell standard Daniell. In order to obtain equilibrium * • Journal of the Society of Telegraph Engineers,' Vol. XI., page 301. X 306 HANDBOOK OF ELECTRICAL TESTING. the resistance R had to be adjusted to 4800 ohms. What was the strength of the current C₁? C₁ = 1.079 4800 + 100 X 1000 1 =220 ampères. 339. The relative deflective effects of g and g₁ are easily ascer- tained by joining up a battery e and two resistances R, and R2, as shown by Fig. 114, and then adjusting until equilibrium is produced; in this case we have n: n₁ :: R₁ + g: R₂ + 91, +91, or n R₁ + g N1 2 R₂+91 340. As the accuracy with which a test can be made depends, amongst other things, upon the accuracy with which is known, n n1 the higher the battery power, e, it is possible to use, the better, since in this case the higher will be the values which can be given to R, and R2, and the higher consequently will be their range of adjustment; thus if we use sufficient battery power to enable a change of 1 per cent., that is to say, 1 ohm in 1000 ohms, or th of an ohm in 100 ohms in R₁ +g, to produce a perceptible movement of the needle, then we can obtain the value of to an accuracy of 1 orth per cent. n n1 10 • 1 The resistance of g₁ would have to be very small compared with the resistance of g, so that it would not add appreciably to the resistance of any circuit in which it is inserted. 1 341. As regards the Best conditions for making the test, this will be directly proportional to the relative values of the figure of merit (page 65) of the coil g, and the current to be measured, for it is evident that no matter whether equilibrium exists owing to there being no current flowing through the coils g and g₁, or to equal currents flowing, still the current which will deflect the needle 1 division will be the same in both cases; hence if the reciprocal of the figure of merit of the coil g₁ be, say, th of an ampère, then an increase of th of an ampère in the current C₁, no matter what the strength of the latter may be, will produce a deflection of 1 degree. It is evident, therefore, that the greater the strength of the current the greater is the degree of accuracy with which its value can be determined; thus if be the figure of merit of the coil g, and C, be the ¿ 1 1 10,000 10,000 MEASUREMENT OF CURRENT STRENGTH. 307 current to be measured, then the Percentage of accuracy attainable will be the percentage which c' is of C₁. To enable this percentage to be obtained, however, it would be necessary that the total resistance of the circuit of g be adjustable to a similar degree of accuracy; in order that this may be the case E must be of such a value that the number of units in R+g is not less than that which satisfies the equation ď Now, : C₁ 1 R + g¯C₁* = E R+g X n n₁ therefore or E c' C₁ X n E = 0333 X n For example. N1 n The reciprocal of the figure of merit of coil g₁ of the galvano- meter was ⚫0001 ampères (c'); the current to be measured was approximately 5 ampères (C₁); and the value of was ⚫001. What was the possible degree of accuracy attainable in making the test, and what would have been the lowest value which should have been given to E in order that this degree of accuracy might be attained? also Percentage of accuracy = 100 x .0001 = .02 per cent. ; ⚫5 •5 × 5 E ×·001 = 2.5 volts. • 0001 If, therefore, E consisted of 3 Daniell cells, the required value of R+g would have been obtained. To sum up, then, we have Best Conditions for making the Test. C,2 342. Make E not less than X C² 2, c' being the reciprocal c' n of the figure of merit of the coil through which the current to be measured is passed. x 2 308 HANDBOOK OF ELECTRICAL TESTING. 1 Possible Degree of Accuracy attainable. 100 c' Percentage of accuracy = C₁ KEMPE'S BRIDGE METHOD. 343. This method is a modification of the foregoing, and it has the advantage that it does not require a special form of gal- V FIG. 115. G V vanometer for its execution. It is shown in principle by Fig. 115. In making the test the resist- ance R is adjusted until no deflec- tion is observed on the gal- vanometer; when this is the case the current c from the battery must also be the current flowing through r, and again the current C₁ must also be the current flowing through r₁. Now since no current flows through the galvanometer, the potentials, V, V, on either side of it must be the same, hence if v be the potential at the junction of r and r₁, then by law (A) (page 293) we have and therefore or but ▼ − v = cr, V − v = C₁r₁; C₁r₁ = cr, C₁ c = 1 cr ; E R+ r 1 1 therefore For example. 1 Er C₁ r₁ (R + r) [A] The battery E consisted of a single standard Daniell cell. The resistances r and r₁ were 100 ohms and 1 ohm, respectively. Equilibrium was obtained on the galvanometer G when R was adjusted to 4000 ohms; what was the strength of the current C₁? 1.079 × 100 = .0263 ampères. 1 (4000 + 100) C₁ MEASUREMENT OF CURRENT STRENGTH. 309 344. Let us now consider the Best Conditions for making the Test. What we have to determine is,-1st, what should be the values of E and r? and 2nd, what should be the value of R? The values which E and r should have should be such that the deflection of the galvanometer needle is as large as possible when equilibrium is very nearly, though not quite, produced. Now if we regard R as a constant quantity, then the value which E must have will depend upon the value given to r, consequently we have to determine what the latter quantity should be. Practically the resistance r, would in all cases have to be of a very low value, and if we consider it to be so the problem to be solved becomes a comparatively simple one. We may regard the current c' producing the deflection of the galvanometer needle as due to a difference of two currents, C₁ being one, and the current produced by the electromotive force E, being the other. Let, then, c₁ and c₂ be the portions of these currents which flow in opposite directions through the galvanometer G, then if we suppose the deflection to be due to R being incorrectly adjusted to R +8, we have (supposing r, to be very small), C₁ r₁ (R + 8 + r) GR+GS+Gr+ Rr+rs C1 = G+ C₁r₁ (R+8) r R+8+ r and E 2 Er C2 R+S+ r G X + G GR + GS + Gr+ Rr +rd' r+G i a Er C₁ but since from equation [A] (page 308) we have = or, Er C₁r (R + r), r₁ (R+r) therefore C2 C₂ = Now, therefore c' = C₁ r₁ (R+r) 1 1 GR+G8+ Gr+ Rr+rd c' = C₁ - C₂ C₁ r₁ (R+d+r) 1 GR+GS+G r + R r + r d C₁ r₁ d C₁1 1 C₁r₁ (R+r) GR+GS+Gr+Rr+rd GR+GS+Gr+ Rr + rd 310 HANDBOOK OF ELECTRICAL TESTING. Or, since 8 is very small, we may say, C₁ r₁ d c' = C₁r, d GR+Gr+Br= R(G+r) + Gr From this equation we can see that r should be made small, but we can also see that there is but little advantage in making it much smaller than G. In fact, there is an actual disadvantage in making r extremely small, for this would necessitate E being made very large, which would be inconvenient. We have next to determine what is the best value to give to R. Now, the larger we make the latter, the greater will be its range of adjustment, consequently, as in previous tests, we should give it the highest value such that a change of 1 unit from its correct resistance produces a perceptible deflection of the galvano- meter needle. We have = 1 C₁ r₁ d R(G+r) + Gr and if in this equation we put 8 = 1 we shall get the current corresponding to a change of 1 unit from the correct value of R, that is c C₁ r₁ R (G+r)+Gr³ or, since r must be small, we may practically say from which c = C₁ RG C₁r1 R c' G [B] If then we make c' the reciprocal of the figure of merit (page 65) of the galvanometer, the value of R worked out from the equation will show the highest value that the latter quantity should have. But the value of R depends upon the value given to E; we must therefore determine what the latter should be. We have or Er C₁ r₁ (R+r)² E = C₁₁ (R+r), g MEASUREMENT OF CURRENT STRENGTH. 311 and substituting the value of R obtained from equation [B], we get ( C₁ 71 + c' G E = go or, as r is small compared with R, that is with 1, we may say For example. E = (C₁ r₁)² c' Gr It was required to measure the exact strength, C₁, of a current whose approximate strength was known to be 03 ampères. A Thomson galvanometer of 5000 ohms resistance (G) was employed for the purpose, its figure of merit being 1,000,000,000 (). The resistances of r and r₁ were 100 ohms and 1 ohm respectively. What should be the value of E in order that R may be as high as possible? E = (⚫03 × 1)² 1 1,000,000,000 = 1.8 volts; X 5000 X 100 that is to say, practically, E should consist of 2 Daniell cells. Assuming E to be equal to 2 volts approximately, then (from equation [B]) the value which R would have in order to obtain balance would be R = approximately. ·03 × 1 = 6000 ohms X 5000 1 1,000,000,000 345. In order to determine the Possible degree of accuracy attainable, let us suppose R to be 1 unit out of adjustment, and let λ be the corresponding error produced in C₁, then we have or λ= C + λ= Er Er r₁ (R − 1 + r) Er Er A = 5, (8 = 1 + r) − C₂ = 5, (8 = 1 + r) − r (8+r) (R − Er (R r₁ (R − 1 + r) (R + r)' 312 HANDBOOK OF ELECTRICAL TESTING. or, since R is large, we may say Er λ r1 r₁ (R+r)²³ but Er C₁ = r₁ (R+ r)' therefore Er λ= Er 2 ; or, (R+r)² = (Tr)² 2 C₁21. Fr (97)' -9²r Er Er If we call λ' the percentage of accuracy, then λ λ = of C₁, or, λ' 100 = 100λ C₁ 100 C₁ 1. Er If we take the values given in the foregoing example we have approximately ג 100 × ⚫03 × 1 = .015 per cent. 2 × 100 To sum up, then, we have Best Conditions for making the Test. 346. Make E the nearest possible value above (C₁r₁)² 2 where c' 1 c' G r is the reciprocal of the figure of merit of the galvanometer, and C₁ is the approximate strength of the current to be measured. The value which R will require to have will be C₁r1. R = c' G Possible Degree of Accuracy attainable. Percentage of accuracy = 100 C₁₂ 1 1 Er DIFFERENCE OF POTENTIAL DEFLECTION METHOD. 347. Fig. 116 shows the general principle of this method. A B is a low resistance through which the current, C₁, to be measured passes. A galvanometer, G, in circuit with a high $ MEASUREMENT OF CURRENT STRENGTH. 313 resistance, R, is connected between the ends of A B as shown, then, calling V and V, the potentials at A and B respectively, we have by law (A) (page 293) V – V₁ C₁ ጥ G FIG. 116. Ꭱ o O o O o O o00. V C T Vi B To determine V - V₁ all we have to do is to note the deflec- tion d on the galvanometer G, and then, having disconnected the latter, together with the resistance R, from A B, to join them in circuit with a standard cell of known electromotive force, E, and to obtain a new deflection d₁; we then have or V − V₁ : E :: d:d₁, 1 Ed √ − V₁ = d' so that E d C₁ r d₁ 348. In order that the test may be a satisfactory one the resistance G + R should be very high compared with the resistance r, so that the strength of C, is practically the same whether G +R is connected to AB or not; also r should be as low as possible, so that it may not appreciably add to the resistance of the circuit in which it is placed. In order, there- fore, that a good deflection may be obtained, the galvanometer G should be one with a high figure of merit (page 65); a Thomson galvanometer answers the purpose very satisfactorily. For example. In making a measurement according to the foregoing test the resistance r was th of an ohm, and the deflection obtained on G was 250 divisions (d). When G and R were connected to a standard Daniell cell in the place of being joined to AB, 霸 ​ 314 Į HANDBOOK OF ELECTRICAL TESTING. a deflection of 230 divisions (d₁) was obtained; what was the strength of the current C₁? C₁ 1.079 × 250 To × 230 = 11.7 ampères. As it is obviously advisable that the deflections obtained should both be as high as possible, the standard electromotive force E may have to be adjusted for the purpose, that is to say, it may have to consist of several cells. Instead of adjusting E only we may make the latter of any convenient high value, and then adjust R so that the required deflection is obtained; in this case if R₁ be the resistance when E is in circuit, we must have For example. C₁ Ed (R+G) r d₁ (R₁ +G)' In making a measurement according to the foregoing test the resistance of r was th of an ohm and the deflection obtained on G was 270 divisions (d); the resistances of G and R were 5000 ohms and 1000 ohms respectively. When G and R were connected to a standard Daniell cell, R had to be adjusted to 7000 ohms (R₁) in order to obtain a deflection of 300 divisions (d₁); what was the strength of the current C₁? C₁ 1·079 × 270 × (1000 + 5000) 1 × 300 × (7000 + 5000) = 4.86 ampères. Of course if the value of R₁ is made such that the deflections d and d₁ are equal, then E(R+G) r (R₁ + G) 349. From the extreme simplicity of the test it must be obvious that the "Best conditions for making the test" and the "Possible degree of accuracy attainable" must be as follows:— Best Conditions for making the Test. 1 Make R and R₁ of such values that the deflections obtained are as high as possible. Possible Degree of Accuracy attainable. ! Percentage of accuracy = 1001 (â + — ₁) 1 MEASUREMENT OF CURRENT STRENGTH. 315 1 m where is the fraction of a division to which each of the deflections can be read. DIFFERENCE OF POTENTIAL EQUILIBRIUM METHOD. 350. Fig. 117 shows the general principle of this method. A B is a slide wire resistance, s being the slider. A galva- nometer, G, and a standard battery, E, are joined up as shown, G FIG. 117. E A <---T G VAS. >D, B so that the latter tends to send a current through r₁ in a direc- tion opposing the current C₁. s is then slid along A B until the point is reached at which no deflection of the galvanometer needle is observed; when this is the case, then by law (A) (page 293) we have V − V₁ = С₁ ˜ı; 1 and by law (B) (page 295), since no current is flowing through the galvanometer, therefore V - V₁ = E; C₁r₁ = E, 1'1 or E C₁ 21 If the resistance of the whole length of wire A B be r ohms, and if it be divided into n divisions, then if the number of divisions between A and D be n₁, the resistance r₁ will be consequently we must have n1 r "1 ; n For example. En C₁ = rn 1 The electromotive force E consisted of 1 standard Daniell cell; the wire A B had a resistance of 1 ohm (r), and was 316 HANDBOOK OF ELECTRICAL TESTING. divided into 1000 parts (n). Equilibrium was obtained when the slider was set at the 750th division (n₁); what was the strength of the current C₁? 1.079,× 1000 1 x 750 = 1.44 ampères. 351. The conditions for making the test in the most satis- factory manner are comparatively simple. The nearer we have the slider to B, that is to say, the larger we make n₁, the smaller will be the percentage of error in the latter due to the slider being, say, I division out of position. As the position of the slider for equilibrium depends upon the value of E, the latter must be sufficiently great to enable n₁ to be as large as possible. The greatest theoretical value which E could have must be that which it would possess when n₁ = n, in which case we get C₁ E go or, E = C₁r. As it is only possible to adjust E by variations of 1 cell, we must take care that its actual value is less rather than greater than C₁r, otherwise it would be impossible to obtain equilibrium. It is also necessary that the figure of merit (page 65) of the galvanometer be sufficiently high to enable a perceptible move- ment of the needle to be obtained when the slider is moved a readable distance, d, from the position of exact balance. If we suppose the slider to be at D when equilibrium is produced, then the electromotive force which would tend to send a current through the galvanometer, supposing the slider to be displaced a distance 8, would be EX N₂ consequently the current c', passing through the galvanometer, will be ES 1 = G NI C₁rd G n ; if, therefore, we require to adjust the slider to an accuracy of 8, the figure of merit (-) of the galvanometer must not be less than G N 1 ES 1 MEASUREMENT OF CURRENT STRENGTH. 317 The percentage of accuracy, A', with which C, can be obtained must obviously be 100 S λ' = 21 or since E n En C₁ " or, n₁ = n1 r ni C₁r' therefore 100 C₁r d En For example. It being required to measure the strength, C₁, of a current whose approximate value is 1.5 ampères, a galvanometer of 500 ohms resistance (G), whose figure of merit is 1,000,000(1), is proposed to be employed for the purpose. The resistance of the whole length of the slide wire, which is divided into 1000 divisions (n), is 1 ohm (r); the position of the slider can be read to an accuracy of a division (8). What is the highest value that could be given to E? also to what percentage of accuracy could C₁ be determined, and what should be the figure of merit of the galvanometer in order that this percentage of accuracy may be attained? E = 1·5 × 1 = 1·5 volts; therefore we cannot make E greater than, say, 1 Daniell cell (1 volt approximately). 100 x 1.5 x 1x1 Percentage of accuracy = .075 per cent. 1 x 1000 To enable this percentage of accuracy to be obtained, the figure of merit () of the galvanometer must not be less than 1 500 × 1000 = 670,000; é 1.5 × 1 × 1/1/20 the figure of merit, therefore, of the galvanometer in question is sufficient for the required purpose. To sum up, then, we have Best Conditions for making the Test. 352. Make E the nearest possible value below C₁ r. The figure of merit of the galvanometer should not be less G n than C₁ r s 318 . HANDBOOK OF ELECTRICAL TESTING. TESTINGPossible Degree of Accuracy attainable. 100 C₁rd Percentage of accuracy = En SIEMENS' ELECTRO-DYNAMOMETER. 353. This apparatus, although it can be used for measuring ordinary powerful currents, yet has the special advantage that it enables rapidly alternating currents (such as are employed in the Jablochkoff system of electric lighting, for example) to be measured; such currents would give no indications on an ordinary galvanometer. The principle of the electro-dynamometer is based upon the mutual action of currents upon one another, i.e. upon the fact that currents in the same direction attract, and in opposite direc- FIG. 118. t A Ъ B CC tions repel, one another. Fig. 118 shows how the principle is applied. A B C D is a fixed wire rectangle, and a b c d a smaller one, suspended by a thread, t, within the larger, so that it can turn freely about its axis; the planes of the two are at right angles to each other. Now, if the two rectangles be connected together in the way shown, then a current entering at W₁, and passing out at W2, will traverse the two, and the current passing from B to W C will attract the current passing from a to d, and will repel the current passing from c to b. A We similar action takes place with reference to the current passing from D to A, consequently the smaller rectangle, under the influence of the forces, will tend to turn about its axis, in the direction in which the hands of a watch rotate. If the current enters at W,, and leaves at W 19 then, inasmuch as the directions of all the currents in the wires are reversed, the small rectangle must still tend to turn in the direction indicated. If one or both of the rectangles consist of several turns of wire, the turning effect for a given current will be proportionally increased. As the turning effect on the coil is produced by the action of the current through the fixed coil acting on the current MEASUREMENT OF CURRENT STRENGTH. 319 through the movable coil, and as the two coils are in the same circuit, it follows that if the current passing through the fixed coil is doubled, then the current passing through the movable coil is also doubled, consequently we have one doubled current acting upon another doubled current, and therefore we must have a quadruple deflective effect-in other words, the deflec- tive force tending to turn the movable coil will vary as the square of the current. The way in which this principle is utilised will be best understood by reference to Fig. 119 (page 320), which shows a general view of the Siemens Dynamometer. The apparatus consists of a rectangle of wire hung from a fibre whose upper end is fixed to a thumb-screw; the latter is provided with a pointer which can be moved round a graduated dial; one end of a spiral spring is also attached to the rect- angle, the other end being fixed to the thumb-screw. In this arrangement the number of degrees to which the pointer is directed evidently indicates the amount of torsion given to the spiral spring. To the rectangle also is fixed a pointer, the end of which just laps over the edge of the graduated dial. The rectangle encircles a coil consisting of several turns of thick, and a larger number of turns of thin, wire; the two ends of the thick wire are connected to terminals 2 and 3, and the two ends of the thin wire to terminals 1 and 3. Connection is made between the rectangle and the wire coils by mercury cups, into which dip the ends of the wire forming the rectangle. The base-board has three levelling screws; the level consists simply of a small pointed weight hung at the end of a rod (seen on the right of the figure), the pointed end hangs exactly over a fixed point when the instrument is level. 354. The method of using the instrument is as follows:- The wires leading the current whose strength is to be deter- mined are connected to terminals 1 and 3, or 2 and 3, according as a strong or weak current has to be measured. The current deflects the rectangle; the thumb-screw is now turned in the reverse direction to that in which the rectangle has turned, and torsion being thereby put on the spiral spring the rectangle is forcibly brought back towards its normal position-that is, at right angles to the coils, or to the position at which the pointer attached to the rectangle stands at zero on the scale. The number of degrees of torsion given to the spiral spring being then read off, the strength of the current is found by reference to a table supplied with each instrument. To construct this table a current of a known strength is sent through the instru- ment, and then the degree of torsion required to bring the rectangle back to zero is carefully noted. This being done, G 320 HANDBOOK OF ELECTRICAL TESTING. 1009 FIG. 119. MEASUREMENT OF CURRENT STRENGTH. 321 the currents corresponding to other degrees of torsion are easily calculated. The force of torsion varies directly as the number of degrees through which the spiral spring is twisted, whilst, as has been before explained, the deflective effect of the current varies directly as the square of the latter. In other words, if 4° be the number of degrees of torsion required to bring the rectangle back to zero when it is traversed by a current of C ampères, then if C₁ be the current which will correspond to any other degree of torsion 41°, we have or For example. °°: 02: C₁2; p° 41° C₂ = √ If 180° (°) of torsion were required to bring the rectangle back to zero when it was traversed by 47.57 (C) ampères of current, what current (C₁) would be represented by 80° (10) of torsion? C₁ 80 × 47.5 × 47·5 180 = 31.7 ampères. 355. Like galvanometers, the Siemens electro-dynamometer is not susceptible of great accuracy when the readings are very low; in fact the higher the readings are, the more accurate are the results obtainable. Thus, for example, 5º of torsion of the spring represents a current (in the instrument (No. 1009) shown by Fig. 119) of 7.93 ampères, whilst 5° more, that is 10° in all, represents a current of 11.23 ampères. In other words, a range of 5° of torsion only, represents a difference in the current of (11.237.93) 100 7.93 per cent. = 42 per cent. If, however, the current had been 66.38 ampères, which corre- sponds to a torsion of 350°, then 5° more of torsion, or 355° in all, represents a current of 66.86 ampères, consequently the range of 5° of torsion in this case represents a difference in the current of (66.8666.38) 100 66.38 per cent. = 72 ⚫72 per cent.; and a greater degree of torsion would have rendered the error still less. Every instrument is supplied with a table which shows the Y 322 HANDBOOK OF ELECTRICAL TESTING. current strengths corresponding to various angles of torsion; practically this table is different for every instrument, as it is almost impossible (nor is it necessary) to make two dynamo- meters alike. The table supplied with the instrument (shown by Fig. 119 (No. 1009)) is calculated so that the latter can theoretically be used for measuring currents varying from 1·05 to 66.86 ampères in strength. The thin wire coil is to be employed when currents of from 1.05 to 19.87 ampères are to be measured, and the thick wire coil for currents of from 3.54 to 66.86. The numbers of degrees of torsion representing various currents are all multiples of 5; thus the first calculation on the table (thick wire coil) is 1°, which represents 3.5 ampères of current; the next is 5°, representing 7.93 ampères; the next, 10°, representing 11.28 ampères; and so on. Practically the instrument cannot well be adjusted to a closer degree of accuracy than 5. The thin wire coil, having about three times the magnetic effect of the thick one, requires, for a definite current, that the number of degrees of torsion to bring the needle back to zero be about three times that which is required in the case of the thick coil; in other words, with the thin wire coil we can practically measure currents to about three times the degree of accuracy which is possible with the thick coil; but, on the other hand, the highest current which we can practically measure with the thin coil is about one-third only of the highest current which can be measured with the thick coil. The lowest current which can be measured consistent with a degree of accuracy equal to 10 per cent. is 5.76, for the next current below this on the table is 5.25, and therefore we have 5.25) 100 5.25 (5·76 = per cent. 10 per cent. nearly. If we require to be accurate within 1 per cent., then the lowest current we could measure would be 16·77, as the next current below this on the table is 16.60, and we therefore have (16.77 - 16.60) 100 16.70 per cent. = 1 per cent. nearly. - = C₁ 1) 100 Since the percentage of accuracy is equal to (C C₁) 100 C where C is a particular current, and C, the current next below it on the table, and since C²: C,²:: 4° 1° MEASUREMENT OF CURRENT STRENGTH. 323 where 4° and 1° are the degrees of torsion corresponding to the currents C, C₁, therefore (√-1) 100; Percentage of accuracy = (√ and as the smallest difference to which we can practically read is 5º, therefore Percentage of accuracy = $1° + 5° Φι - 1) 1) 100 = X', say. Therefore √1+ 5° + 1; 100 therefore 5° 12ג 1 + +1+ ; 10,000 50 therefore 5° 12ג 4 + 10,000 50; 50,000 $1° or, λ'² + 200 λ'' which shows us the smallest number of degrees of torsion which must be given to the spiral spring when measuring a current, in order that the latter may be measured to an accuracy of x per cent. For example. It was required to be able to measure currents of 10 ampères and upwards to an accuracy of 1 per cent., by means of an electro-dynamometer; how many degrees of torsion would the spiral spring be required to make? $1° 50,000 1 + 200 = 248°; showing that the electro-dynamometer must be so constructed that when currents of 10 ampères and upwards have to be measured, not less than 248° of torsion have to be given to the spiral spring in order to bring the needle back to zero. 356. From the construction and principle of the electro- dynamometer it must be evident that the accuracy of the Y 2 J 324 HANDBOOK OF ELECTRICAL TESTING. absolute results obtained by its means must depend entirely upon the torsion of the spiral spring remaining constant. It seems possible that change of temperature and frequent use might alter the value of the torsion, but this point does not appear to have been satisfactorily settled. The instrument might probably be made of more value if its coil were composed of a large number of turns of thin wire, shunted by a thick wire shunt. The latter would be used when measuring the strong currents, whilst the correctness of the instrument could be verified by sending a comparatively weak current through the unshunted coil. It is not often that powerful currents of an accurately known value can be had for the purpose of verifying the correctness of an instrument, though weaker currents are almost always obtainable. ( 325 ) CHAPTER XIII. MEASUREMENT OF ELECTROSTATIC CAPACITY. DIRECT DEFLECTION METHOD. 357. The simplest way of measuring electrostatic or inductive capacities is, with the same battery power, to compare the dis- charges from the unknown capacities with the discharge from a condenser of a known capacity; thus we note the discharge deflection a given by the standard condenser F, and then the discharges a1, a2, &c., given by the cables or condensers whose capacities F1, F2, &c., are required, in which case For example. a F: F₁: F₂ :: α : а₁: а2. 1 2 α1 A standard condenser had a capacity of microfarad, and gave a discharge deflection of 300, and two other cables or condensers, F1, F2, gave discharge deflections of 225 and 180 respectively, then : F₁: F₂ :: 300: 225: 180; 1 2 that is, F₁ = }} 225 300 = microfarad, and 180 F2 = 1/1/ • = microfarad. 300 If we use shunts and obtain the same deflection, then G+ S¸ G+ S₁ . G + S₂ : 1 F₁ : F₂ :: 2 S S₁ S2 358. In measuring the electrostatic capacity of a cable by this method, the connections for measuring the discharge from the cable would be made in the manner shown by Fig. 120 (page 326). The arrangements for measuring the discharge from the condenser would be those indicated by Fig. 97 (page 278). Then, as before, the capacity of the cable will be to the + 326 HANDBOOK OF ELECTRICAL TESTING. 1 capacity of the condenser as the discharge deflection of the one is to the discharge deflection of the other, or obtaining the same deflection by means of shunts, as the multiplying power of the shunts. FIG. 120. Cable K K K² HHHHH Earth 359. The capacity per mile will be the result divided by the mileage of the cable. 360. When a number of capacities of about the same value have to be measured, as, for instance, the capacities of two-knot lengths of cable core, a device may be adopted which consider- ably simplifies the operation. Let F be the capacity of the standard condenser whose discharge is D divisions, and let ƒ be the capacity of one of the lengths of cable, and d the discharge from the same. Then we have or F Now if we make D F:ƒ::D:d, f: = Fd Ꭰ D a submultiple of 10, then the value of d read off from the scale will give at once the value of f. Thus MEASUREMENT OF ELECTROSTATIC CAPACITY, 327 if F were a condenser of microfarad capacity, and we so ad- justed the galvanometer that this capacity gave a discharge deflection of a little over 333 divisions, then we should have. j d d f: 333/1 1000; so that if the discharge deflection reading from the cable con- sisted of three figures, a decimal point put before the latter would give at once the capacity of the cable; or if the reading con- sisted of two figures, then we must put a decimal point and a cypher. In the same way, if we had a condenser of 1 micro- farad capacity, we should adjust the galvanometer so as to obtain a deflection of 100 divisions, for then f = 1 d 100 d 100 SIEMENS' LOSS OF CHARGE DISCHARGE METHOD. 361. The principle of this method of measurement is that of observing the rate at which the charged condenser or cable, whose capacity is required, discharges itself through a known resistance, and calculating the capacity from a formula which we will now consider. The elements with which we have to deal are: capacity (farad), resistance (ohm), quantity (coulomb), time (second), and potential (volt). Let us suppose the cable or condenser has an electrostatic capacity of F farads, and is charged to a potential of V volts, so that it contains Q coulombs (equal to V F) of electricity, and is discharging itself through a resistance of R ohms during one second. The quantity of electricity in the condenser or cable at starting is Q coulombs. If now we take a very short interval of time t, we may con- sider the discharge, which really varies continually, to flow throughout that time t, at the same rate as it had at the com- mencement; and the smaller t is taken, the more accurate will be the result. Thus, since the quantity escaping is directly proportional to the potential driving it out, and to the time during which the escape occurs, and inversely proportional to the resistance 328 HANDBOOK OF ELECTRICAL TESTING.. through which the escape takes place, the quantity escaping will vary as V t Ꭱ V t R; that is it equals K, Ꭱ where K is a constant to be determined. Now the units are so made that a condenser of 1 farad electrostatic capacity charged to a potential of 1 volt, that is, containing 1 coulomb of electricity, will commence to discharge itself through a resistance of 1 ohm, at the rate of 1 coulomb per second. That is to say, =¹×¹K, therefore, K = 1. 1 = 1 x 1 1 The quantity escaping during the interval of time t in our problem is therefore V t Ꭱ The quantity remaining in the condenser will be Q V t R Q VFt FR = Q (1 − 1). - FR Again, since this is the quantity at the commencement of the second interval, that at the end will be [@(1 t t - t 2 [9 (1-FR)] [(1 − FR)] = Q(1 − ~'R); FR - and that at the end of the nth interval will be t Q (1 − + )" F FR = q. Let these n intervals of t seconds equal T, so that n t = T. Now we have seen that the smaller t is, the more accurate will our results be. Let us therefore make t infinitely small, and n infinitely great, so that n t still = T, we shall then get a perfectly accurate result, and the amount remaining at the end of time T will be q = Q(1 = = = n FR where n = ∞. MEASUREMENT OF ELECTROSTATIC CAPACITY. 329 To evaluate q put so that then T n FR || 1 f8 x = ∞ when n = ∞; Q = Q @[(1 1+ QC T * - FR when x = ∞; but when this is the case the expression within the square brackets is known to be equal to e,* thus therefore T FR = e Q T FR = Q log. गंन therefore T F = R loge q but Q VF V q v F V where v is the value of the potential corresponding to the value q of the quantity, thus T T F V R log. 2.303 R log, V v where, as stated at first, T is measured in seconds, F in farads, and R in ohms. Since V and v now appear in the form of a proportion, the unit in which they are measured is immaterial, although they were measured at the outset in volts. In practice R is usually measured in megohms (1,000,000 ohms), and consequently F will, in such a case, be measured in microfarads (1,000,000 farad). * Todhunter's Algebra, Fifth Edition, Chapter XXXIX. 330 HANDBOOK OF ELECTRICAL TESTING. For example. A fully charged condenser gave a discharge deflection of 300 divisions (V); after being recharged and allowed to discharge itself through a resistance of 500 megohms for 60 seconds (T), the discharge deflection obtained was 200 divisions (v). What was the capacity of the condenser? 60 F = 2.303 × 500 log 300 200 = ·295 microfarads. 362. In executing this test it is advantageous to make V and v bear a certain proportion to one another, for this will cause any small error in reading the value of v to produce as small an error as possible in the value of F when the latter is worked out from the formula. This may be proved thus:- Let us assume R to be constant, and let there be an error A in F caused by an error 8, in v and an error d₂ in V, the error & being plus and 8, minus, so that the total resulting error is as great as possible; we then have T T F+λ = R log. V - 8₂' ο + δι or, λ F V - d₂ 2 R log. v + d₂ but T F = therefore λ = F log. V – S₂ log. υ + δι R log. V V y' or, V BA T V = F log. Ꭱ V V - 82 U log. log. v v + & ₂ V 82 = F. FF- - log. – v + & ₂ log. (1 + 2) — log. (1 - 4). √ - 8₂ log. v + d₂ but if 8 and 82 are very small,* we get λ = F 81 (一​) v log. V v ; * Todhunter's Trigonometry, Third Edition, Chapter XII. MEASUREMENT OF ELECTROSTATIC CAPACITY. 331 1 If the deflections are taken on a Thomson galvanometer (as would practically be always the case), then d₁ = 8₂, so that we get 8. & ( + + + ) λ = F V log. v Now the value of v, which makes λ a minimum,* is v = V 3.59' * This may be determined in the following manner :- To make λ a minimum we must make a maximum. V log จ 1 1 Let the above expression equal u, and let we then get then V n u = V log. n n+ 1' 2 n d = (-1) {(n+1) - log. )=0 d n at a maximum; therefore or n+1 log. n = n log n = (n + 1) •4343. n The solution of this equation is best effected by the "trial" method, viz. by giving n various values until one is found which approximately satisfies the equation. If we make n = 3·59, we get (3-59 + 1) •55509 = = (³- .3.59 which is sufficiently close for the purpose. We have therefore ·4343 = ·55527, V V ▼ = vn, or, v n 3.59 € 332 HANDBOOK OF ELECTRICAL TESTING. V so that practically we may say-make v = V 3.5 We need not be particular, however, about making v exactly equal to as we could make it 50 per cent. greater or less 3.5' than this value without materially increasing . If the rate of fall were comparatively quick, there would be a positive V advantage in making v less than as the greater we make T 3.5' the less will any small error in its value affect the correctness of F, as must be self-evident. Now, if R is adjustable, it is clear that by making it large enough, we could make T large without reducing v too much. In the case of a cable, R, being the insulation resistance, is of course a fixed quantity; but when the measurement is being made with a condenser, any value may be given to R that is considered convenient. We therefore have Best Conditions for making the Test. V 363. Make v as nearly as possible equal to When it is 3.5° possible to adjust R, make the latter as high as convenient. Possible Degree of Accuracy attainable. 100 (+82) Percentage of accuracy F = V· 2.303 v log v If the deflections are read on a Thomson galvanometer (as would usually be the case) then 200 S Percentage of accuracy = F V' 2.303 v log บ where 8 is the fraction of a division to which each of the deflections V and v can be read. 364. When it is an ordinary condenser (whose insulation resistance would practically be infinite) that is to be measured, the connections would be the same as those given in Fig. 97, MEASUREMENT OF ELECTROSTATIC CAPACITY. 333 page 278, with the addition of the resistance, which would be inserted between the terminals of the condenser. The instantaneous discharge (V) can be taken without re- moving the resistance; for, since the latter would be extremely high, there would be no time for any of the charge to have leaked out through it during the small interval occupied by the lever of the key in passing from the bottom to the top contact. To take the discharge after the interval of time, having charged the condenser by pressing down the lever of the discharge key (Fig. 100, page 279), we should depress the "Insulate" trigger, which would take the battery off but not discharge the con- denser; then, after the noted interval of time, we should depress the "Discharge" trigger, which would allow the charge remaining to flow out, the deflection obtained from which gives us v. 365. To measure the capacity of a cable by this method, the connections would have to be those given in Fig. 120, page 326, and the way of making the test would be the same as has just been explained. R in this case would be the insulation resistance of the cable, which in this and the following method would have to be determined beforehand in the manner de- scribed in Chapter XV., page 368. Inasmuch as R in a cable is a variable quantity and is dependent upon the time a charge is kept in the cable, a mean value only can be given to it, and therefore this and the following test can only give the value of F approximately. SIEMENS' LOSS OF CHARGE DEFLECTION METHOD. 366. If the two terminals of a condenser are connected by a high resistance in the circuit of which a galvanometer is placed, and if the two terminals be also connected to a battery, then the condenser will become charged up, and the permanent deflection obtained on the galvanometer will represent the potential of the charge. If now the battery be taken off, a current will flow from the condenser through the resistance and the galvanometer, which current will continually decrease in strength as the condenser empties itself. But the current flowing at any particular moment will be represented by the deflection obtained at that moment, and this deflection will be the same as that which would be obtained if the condenser were kept continuously charged to the potential it had at that moment. The deflection obtained therefore on the galvanometer when the battery is connected to the condenser indicates the potential which the latter has when fully charged, and the deflection 334 HANDBOOK OF ELECTRICAL TESTING. after any interval of time after the battery has been taken off indicates the potential of the charge remaining; the capacity therefore is given by the formula T F m.f., D 2.303 R log d [A] in which D is the deflection obtained when the battery is on, and d the deflection obtained after T seconds, the battery being off during that time. R is the resistance through which the charge flows. It may be remarked that the deflection obtained when the battery is on is not affected by the presence of the condenser; it would be the same whether the condenser were connected up or not. 367. The connections for making a test of this kind would be as follows:-Referring to Fig. 97, page 278, the terminal of K₁, which is connected to the top contact of K2, should in the present case be connected through the resistance R to terminal A of the condenser; the other connections remain the same. 368. In the case of a cable where the flowing out of the charge takes place through the insulating sheathing, a galvanometer cannot be put in the circuit of the flow. To enable the fall of charge to be observed, therefore, a high resistance in circuit with the galvanometer is connected to the cable, and through this resistance a part of the charge passes. As it is only the rate at which the fall takes place that is required, it is quite sufficient, in order to observe this fall, that a part only of the charge be allowed to flow through the galvanometer. If we call R, the insulation resistance of the cable, and R₂ the resistance connected to it, then the total resistance through which the charge flows will be R₁ R2 R₁ + R₂ This quantity must be substituted in the place of R in equation [A], so that we have T F = R₁ R₂ m.f. D 2.303 log R₁ + R₂ d The resistance R2, it may be remarked, includes the resistance of the galvanometer. MEASUREMENT OF ELECTROSTATIC CAPACITY. 335 As in the first test, it is necessary that R₂, through which the discharge has to pass, be sufficiently great to prevent the flow from being too rapid. For example. A cable 30 knots in length being connected up, for making the test just described, with a galvanometer, and a resistance R2, of 4 megohms, the deflection obtained was 300 divisions (D). On taking off the battery the deflection after 30 seconds T) fell to 100 divisions (d); the mean insulation resistance R₁ of the cable was 10 megohms. What was the electrostatic capacity (F) of the cable? or F 2.303 X 30 10 x 4 300 log 10+ 4 100 = 9.55 m.f. 9.55 30 •318 m.f. per knot. 369. The connections for making this test would be as follows:-Referring to Fig. 120, page 326, the terminal of key K₁, instead of being connected to the top contact of the discharge key, would in the present case be connected to the cable through the resistance R2. 370. A great advantage which this test possesses over the first method (page 326) lies in the fact that it is correct either for long or short cables. Discharge deflections from long cables, or cables coiled in tanks, do not correctly represent their capacity, in consequence of a retardation which takes place in them and which causes the deflection of the galvanometer needle to be less than it would be if this retardation did not exist. By adopting the fall of deflection plan we avoid this cause of error; but, as we pointed out at the conclusion of the last test, since R₁ can only have a mean value, the value of F obtained from the formula will only be approximate. 1 THOMSON'S METHOD. 371. This is a very good method, and it can be applied to long cables, &c., with very accurate results. The following is its principle:- If we have two condensers containing equal charges of opposite potentials, and we connect the two together, the two charges will combine and annul one another, and if we then connect the two condensers, so joined, to a galvanometer, no deflection will 1 336 HANDBOOK OF ELECTRICAL TESTING. be produced, there being no charge left in either of the two. If, however, the charge in one condenser exceeds that in the other, then the union of the two condensers will not entirely annul their charges, but an amount will remain equal to the difference of the two quantities. This quantity will deflect the needle if the joined condensers be now connected to the galvano- meter, the deflection being to the right or left, according as the charge in the one or other of the condensers had the preponder- ance in the first instance. If then we know the capacity of one condenser, and we so adjust the potentials of the two that no charge remains when they are joined together, we can determine the capacity of the other condenser. Let Q₁ and Q₂ be the charges in each; then 1 2 Q1 Q₁: Q₂ :: V₁ F₁: V₂ F₂ 2 1 1 2 27 1 2 where F₁ and F₂ are the capacities of the two, and V₁ and V₂ the potentials of their charges. When Q₁ = Q₂ then or ·Cable لے Ки Ri 1 V₁ F₁ = V₂F2, k = √2 F 2 F₁ 1 V₁ 1 FIG. 121. んぅ ​Earth Earth A Fi A HHHHHH Earth 372. An important element in this test is the adjustment of the potentials V₁ and V2. Fig. 121 shows a method of making the test when it is a cable whose capacity has to be measured. 1 MEASUREMENT OF ELECTROSTATIC CAPACITY. 337 The poles of the battery are joined together by two resist- ances, R₁ and R2, connected to earth as shown. Then the poten- tials at the points of junction of the battery with the resistances will be in the proportion and since V₁: V₂:: R₁: R2;' 1 F₁ 1 2 = V. 1 2 F2, V₁ 1 * therefore R₂ F₁ = Pu2 F2⋅ Ꭱ, [A] 2, 373. In making the test practically, R₁ and R₂ are first ad- justed as nearly as can be guessed in the proportion of F, to F, 1 keys k₁ and k₂ are then depressed by means of the knob K; this charges the cable and the condenser. 1 K is now released so as to allow k₁ and k₂ to come in contact with their upper stops; as the two latter are joined together, the cable and condenser become connected to each other. Key k is now pressed, which allows any charge which may remain uncancelled to be discharged through the galvanometer G. If no deflection is produced, then R, and R₂ are correctly adjusted, but if not they must be readjusted until no discharge is obtained; F, is then calculated from the formula. For example. 1 A cable 500 knots long was joined up with a condenser of 20 microfarads capacity, and with resistance coils, according to Thomson's method of measuring electrostatic capacities. When R₁ and R₂ were adjusted to 500 and 4400 ohms respectively, no charge remained in the cable and condenser when the two were connected together. What was the capacity of the cable? or = 4400 1 500 × 20 = 176 m.f., 176 = ·352 m.f. per knot. 500 374. Fig. 122 shows a very convenient form of key, designed by Mr. Lambert, which enables the test to be made with the * Page 285. • Z از 338 HANDBOOK OF ELECTRICAL TESTING. 1 2 greatest facility. By pushing forward key button K the two keys k₁, k₂ (Fig. 121) are depressed, so that F, and F, become charged, and upon drawing K back, k, and k₂ are allowed to rise, thus causing the charges to mix; finally, by depressing k the galvanometer is brought into circuit. 2 1 In the most recent form of this piece of apparatus, on the depression of key k the cable F, becomes disconnected, so that only the condenser F₂ becomes connected to the galvanometer. By this arrangement any disturbing force which may cause the charge in the cable to vary slightly, and consequently to affect the galvanometer is prevented from acting. 375. If it were the capacity of a condenser that was to be measured, then the connections would be similar to those in FIG. 122. am R₂ k K R AKIT BROTHER Fig. 121, with the exception that the points there put to earth would in the present case be connected to the second terminal of the condenser. The resistances R₁ and R₂ may be formed of a slide resistance, the slider being to earth in the case of a cable test, or connected to the second terminal of the condenser in the case of a condenser test. F 2 376. As in the “Direct deflection method" (page 325), the test can be considerably simplified if we make (equation [A], R₁ page 377) a submultiple of 10, for then the value of R, read off from the resistance box will at once give the value of F Thus if F₂ were a condenser of, say, 5 of a microfarad, and if R₁ were 5000 ohms, then the capacity of F₂ can be read off directly from R₂ to four places of decimals. 2 2 1. 377. When a long cable has to be tested by this method Mr. A. MEASUREMENT OF ELECTROSTATIC CAPACITY. 339 Jamieson recommends that K be depressed for five minutes to charge, and then raised for ten seconds for mixing previous to depressing k. It is also advisable to take the mean of several tests made alternately with zinc to line and copper to line. 1 378. With regard to the "Best conditions for making the test" it is advisable that the capacity of the condenser F₂ be as nearly equal to F, as possible, so that the potentials to which the two have to be charged may not differ to any very great extent. For if a long cable has to be tested, then inasmuch as the latter would have to be charged to a potential of at least 5 Daniells so as to swamp, as it were, any local charge, the potential to which the condenser (if small) would have to be charged would be very great; this would be liable to cause an error, from the fact that with a very high potential a certain amount of the charge becomes absorbed, and this charge would cause a deflection of the galvanometer needle over and above that due to the simple inequality between the actual free quan- tities in the two capacities. This abnormal deflection might of course be mistaken as being due to an incorrect adjustment of R₁ and R₂. If F₂ is about a fifth of F₁ it will not be too small for the purpose of the test. 2 2 1 The values given to R, and R₂ should be as high as possible so that their range of adjustment may be sufficiently wide. The battery power should be sufficiently high to enable a perceptible discharge deflection to be obtained when R, (the larger of the two resistances) is 1 unit out of exact adjustment; this is best determined by experiment. We have therefore Best Conditions for making the Test. 2 1 379. Make F₂ as nearly equal to F₁ as possible. Make R₁ and R₂ as high as possible. Possible Degree of Accuracy attainable. 100 Percentage of accuracy = R₂ GOTT'S METHOD. 380. This method, devised by Mr. J. Gott, is shown by Fig. 123; it is executed as follows:— R₁ and R₂ are first adjusted as nearly as can be estimated in * Journal of the Society of Telegraph Engineers,' Vol. X., p. 278. This method, although independently devised by Mr. Gott, is practically identical with that of Sir William Thomson described in Vol. I., p. 397, of the same Journal. z 2 340 HANDBOOK OF ELECTRICAL TESTING. • 1 the proportion of F, to F2. The key K is then depressed and clamped down; this causes both the cable and condenser to become charged, since they are connected together in "cascade." After an interval of five seconds key k is depressed, and if a deflection is observed on the galvanometer G, this key is raised, key K is unclamped so that the latter is put to earth, and the condenser is short circuited by means of its plug for a few seconds. R₁ or R₂ is now readjusted, and the foregoing operations again gone through. When finally it is found that no deflection on the galvanometer is observed on depressing key k, then tr or 2 F₁: F₂:: R₁: R2, 1 2 F₁ 1 Ru2 tr It is obvious that we must have F2. : } Best Conditions for making the Test. 2 1 381. Make F₂ as nearly equal to F₁ as possible. Make R, and R, as high as possible. Possible Degree of Accuracy attainable. Percentage of accuracy = FIG. 123. 100 R₁ Cable Ri G Earth k K R2 F2 Gott's method is a very satisfactory one, and it possesses the advantage over that of Thomson of not requiring a well-insulated ѓ MEASUREMENT OF ELECTROSTATIC CAPACITY. 341 battery. The method is almost exclusively employed in the Cable Department of Messrs. Siemens' Telegraph Works, Charlton, a slide resistance of 10,000 ohms adjustable to 1 ohm R₁ being employed to give the ratio R2 DIVIDED CHARGE METHOD. 382. If a charged condenser has its two terminals connected to the two terminals of a second condenser which contains no charge, then the charge will become distributed over the two; and if the condensers be then separated, the quantities held by them will be directly proportional to their respective capacities. Thus, if Q₂ be the charge contained in a condenser whose capacity is F2, then if it is connected to a condenser or cable whose capacity is F₁, the quantity Q which will remain in F₂ will be From this we get F2 Q = Q2 F₁ + F₂ Q₂ F₁ = F2 Q 1 [A] 2 If therefore Q₂ be the discharge obtained from a condenser F₂ when full, and Q the discharge obtained from it when, after being charged from the same battery, it is connected for a few seconds to F1, then the capacity of F, is given by the above formula. For example. A condenser of microfarad capacity (F2), when fully charged, gave a discharge of 300 (Q₂). After being recharged and con- nected a few seconds to a piece of cable whose capacity F₁ was required, the quantity of charge remaining gave a discharge of 140 (Q). What was the capacity of the piece of cable? F₁ = 3 × Fi 300 140 140 = ·381 m.f. 2 383. The capacity which the condenser F₂ should have in order that the test may be made as accurately as possible, may be thus arrived at:— 1 — Sin Q Let there be an error λ in F₁ caused by an error and an error + 8 in Q₂, so that A is as great as possible; we then 342 HANDBOOK OF ELECTRICAL TESTING. K J have - Q₂+ d (Q-8) Q₂-Q+28 F₁ + λ = F₂ = F₂ 2 Q-8 but we know that F₁ = F2 Q₂ Q or, F₂ = F₁ Q Q2 Q ; therefore Q Q₂ − Q + 28. F₁ + λ = F₁ ; Q₂ Q Q-8 that is, λ == Q₂ - X Q or, since d is a very small quantity, we may say 2 Q₂ − Q + 2 8 Q-8 − 1 } = F, 1 (Q₂ + Q) S (Q₂ — Q) (Q — 8) (Q₂+Q) 8 (Q2 − Q) Q [B] We have then to find the value of Q which makes λ as small as possible. Now (Q₂+Q) & 8 J Q2 + Q 2 Q Q₂ - Q +3} = 2 (Q₂-Q) Q Q2 8 1 Q2 2 {²² = 2 [1 - 2√2 ]² + 2√3+3} Q2 Q Qz - Q and to make the latter expression as small as possible we must make [1– Q √2 Q₂ as small as possible; that is to say, we must make or Q √2 1 0, Q₂ Q = 1; therefore Q2 − Q Q √ 2 = Q₂ − Q, or Q (√ 2 + 1) = Q₂; MEASUREMENT OF ELECTROSTATIC CAPACITY. 343 that is Q2 Q₂ } Q 2.4142 √2+1 It was pointed out, however, in a similar investigation which we made in (§ 105, page 103), that practically we may say, make- Q = 3 2 or, in other words, the capacity of F₂ should be such that when it is connected to F, it should lose two-thirds of its charge. This is obtained, of course, by making F₂ equal to 2 1 F. 2 384. The connections for the practical execution of the test would be very similar to those shown in Fig. 97, page 278, but the condenser or cable under trial would be substituted in the place of the battery. When it is a cable whose capacity is being measured, then terminal B would be put to earth, and the wire shown as leading from B to the battery would be removed. The test would then be made in the following manner :- Key K, being pressed down so as to hitch on the "Insulate ” trigger (Fig. 100, page 279), the condenser C would be charged by touching the terminals A and B with the wires from the two poles of the battery. The "Discharge" trigger of the key then being depressed, the discharge Q₂ is noted. The key then being again placed at "Insulate," and the condenser again charged up by the battery, the key would be pressed down on to its bottom contact; this puts the condenser C in connection with the trial condenser or cable. The "Discharge" trigger then being pressed, the discharge Q is noted. The "Divided Charge" method, like that of Thomson or Gott, is very accurate when employed for measuring the capacity of long cables, or cables coiled in tanks. Best Conditions for making the Test. F₁ 2 385. Make F₂ as nearly equal to 1 as possible. 2 Possible Degree of Accuracy attainable. From equation [B] (page 342) it follows that Percentage of accuracy = 2 100 (Q₂+Q) 8 F₁ (Q₂ — Q) Q' 1 where & is the fraction of a division to which each of the deflec- tions Q and Q₂ can be read. } 1 344 HANDBOOK OF ELECTRICAL TESTING. 386. By a modification of the foregoing method, due to Dr. Siemens, a comparatively small condenser may be used for measuring the capacity of long cables, or of condensers of high capacity. It may be called SIEMENS' DIMINISHED CHARGE METHOD. If we connect a condenser to a charged cable, the latter loses the amount which the condenser takes up, and if the condenser be discharged and then again connected to the cable, and again discharged, and this process be repeated several times, the quantity in the cable can be definitely diminished as much as we like. The quantity removed each time, however, is not the same, but becomes less and less after each discharge. Let Q₂ be the quantity contained in the condenser, and Q, the quantity contained in the cable, when the two are charged full from the same battery. Then or • Q2 Q1 F₂: F₁, : 2 F Q₁ = Q₂ F₁₂ 2 Supposing now the cable to be completely charged, and the battery taken off, and the condenser to be empty, then, on con- necting the condenser to the cable, the charge the former will take will be 2 F. 1 X F 2 F1 F. 1 Q1 F₁ + F₂ 1 whilst the quantity remaining in the cable will be F. 2 Q₂ F₂ × F₁ + F₂ 2 1 Q2 F₁ + F 2 F₁ 1 1 Q1 F₁ + F₂ 2 On discharging the condenser and connecting it a second time to the cable, the charge it will take will be F 2 1 F₁ Q1 F1 F₂ F, 1 1 2 F₁ + F₂ ^ F₁ + F2 X = X 2 Q₂ F₂ × F₁ + F₂ X F 2 2 1 F₁ + F, 1 2 F Q2 F₁ + F 1+ ; consequently, after the nth application, the charge Q it will take will be F₁ N Q = Q; F₁+ F 1 2 MEASUREMENT OF ELECTROSTATIC CAPACITY. 345 1 therefore F₁ 1 F₁ + F2 N n/Q " Q2 from which 1 F₁ = F2 nj Q W Q₂ - w Q F N 1 For example. A condenser of 1.0 microfarad capacity (F2), when full, gave a discharge equal to 300 (Q₂). A cable whose capacity was required was charged from the same battery which was employed to charge the condenser. The latter was then alternately connected to the cable, removed and discharged 16 times (n); on the sixteenth occasion the discharge was noted, and it was found equal to 83 (Q). What was the capacity of the cable? 1.0 F₁ = 11·97 m.f. 16 300 83 1 387. In order to make this test as accurately as possible when it is applied to a cable, the repeated charges and discharges must be made with as little loss of time as possible, as during that time a leakage of the charge will be going on through the insulating sheathing of the cable; the accuracy of the test depends upon this leakage being nothing, or at least very small. 388. The connections for making the test would be similar to those employed in the foregoing one, and the practical execution would be the same with the exception that the trial condenser or cable, and not the standard condenser, would be charged from the battery, and in taking the repeated discharges the galvanometer would have to be short circuited. Best Conditions for making the test. 1 • 5552372 389. Make n equal to , approximately.* log. F, F 2 F₁ 1 * This may be proved as follows: In order to determine F, as accurately as possible from the equation = F₁ = n F₁ 1 Qz 1 Q1 that is. F₁ F₂ 1 Q (22) 1 鳝 ​346 HANDBOOK OF ELECTRICAL TESTING. Possible Degree of Accuracy attainable, Percentage of accuracy = 1 8 (Q₂+Q) Q₂ = −² 1 n Q (Q₂ñ — Q″) 1 100, where S is the fraction of a division to which each of the deflec- tions Q and Q₂ can be read. 2 we must determine 1 Let (22 ( 2 ) = 3 1 (2) as accurately as possible. 1 equal > and let there be a small plus error & in Q₂, and a small minus error & in Q, and let there be a corresponding error λ, in k, that is, let therefore Now therefore λι = 1 Q₂ Q1 kn k + ~₁ = (21 κ λ Q₂+ 1 x₁ = (²² = 3) = = = = λι 111, or, (Q₂ 10" - 5) d 1 or, Q₂ 21 = n — k = k k. k", or, Q₁ = Q₂ kn; = == [( but since d is very small, we get Q₂ λ1 = 7 n kn Q₂ 1 1 Q Q₂" Q₂ 18 To make λ, a minimum we must make 1012 Q₂ Q2 Q₂+8 HIR “7½ -"+1" - 1 8 7 (1 − " + 1). n Q₂ a minimum. Let then น *-*-*+1 k-n+ 7 n -n du = 1 [− n k − "log. k − (k −” + 1)] = 0 n at a minimum; therefore or - n log, k+1+k” = 0, log, k" + 1+k" = 0. 1 MEASUREMENT OF ELECTROSTATIC CAPACITY. 347 390. It may be remarked that when a cable is tested for electrostatic capacity at the factory, it is immaterial whether the test be made by charging the cable positively or negatively; but in the case where the cable is laid, it is advisable to make two tests (or sets of tests), one with a positive and the other with a negative charge, and to take the arithmetic mean of the two results. It is rarely, however, that the two latter differ to any material extent. or log k" + (1+k") ·4343 = 0. The solution of this equation may be obtained by the "trial" method, i.e. giving k" various values until one is found which approximately satisfies the equation. If we make k" equal to 27846 the equation will be very nearly satisfied, for log ⚫27846 = 1·4447628 ⚫5552372 (1 + ·27846) ·4343 = 5552352. and Now # 2 F₁ = (22) 75 1 F₁ F₂ 1 or, +(2)²=; F₁ therefore hence n For example. n (,,F)" log ⚫27846 F₁ log F₁ + F₂ F₁+F₂ = 7" = ·27846;` \ 5552372 F₁ ⚫5552372 1 10% F₁+ F₂ F₁ +F, log (F; +F) 1 It being required to measure the exact electrostatic capacity of a cable whose capacity was 12 microfarads (F₁) approximately, a condenser of 1 micro- farad (F2) was used for the purpose. How many times should the con- denser be applied to the cable in order that the test may be made with the greatest chance of obtaining an accurate result? ⚫5552372 n log (1271) •5552372 ⚫0347622 = 16. • 348 HANDBOOK OF ELECTRICAL TESTING. CHAPTER XIV. THE THOMSON QUADRANT ELECTROMETER. 391. This is a most valuable and useful instrument for accurately measuring potentials. FIG. 124. DESCRIPTION. Fig. 125 (page 349) gives a general view of the instrument. In the small figure to the right, n n is a thin needle of sheet aluminium, shaped like a double canoe-paddle. It is rigidly fixed at its centre to an axis of stiff platinum wire k (Fig. 124), in a plane perpendicular to it. At the top end of the wire a small cross-piece i is fixed, to the ex- tremities of which single cocoon fibres are attached. These fibres are fixed to small screws c and d, by the turning of which the length of the former can be altered. The small screws a and b enable the screws c and d to be shifted either to the right or left. Finally, by turning e, the screws a and b can be parted more or less, thereby separating the threads of suspension, and rendering the tendency of the needle to lie in its normal position more or less powerful. a c d b f9 m A little below the cross-piece i is fixed the mirror m, whose movements are reflected on a scale, as in a Thomson galvanometer (page 46). The platinum wire below the mirror passes through a guard tube t (Fig. 125), to prevent any great lateral deviation of the needle and its appendages, which might cause damage should the instrument receive any rough usage. The guard tube itself is fixed to the framework from which the needle is suspended. It will be seen in the figure that the needle is suspended, apparently, beneath four quadrants (q), A, B, C, and D. There are, however, four quadrants also below the needle, united to the top ones at their circumferences. The arrangement is in fact a round, flat, shallow box, cut into four segments. The alternate segments are connected together by wires as shown in the figure. THE THOMSON QUADRANT ELECTROMETER. 349 Now, if the needle is electrified and the quadrants are in their normal unelectrified condition, and are placed sym- metrically with reference to it, no effect will be produced on the needle. That is to say, the spot of light on the scale will be stationary exactly at the centre line. FIG. 125. A m LLLEI) g A B 71 C Wire O wire T]|[] — real size But if the quadrant D, and consequently A, be electrified, then an attraction or repulsion will be exerted on the needle, causing it to turn through an angle proportional to the potential of the electricity. As the angular movements are very small, the number of divisions of deflection on the scale will directly represent the degree of potential which the quadrants possess. " 350 HANDBOOK OF ELECTRICAL TESTING. We can also connect another electrified body to C and B; the needle will then move under the influence of both forces. To render the instrument of practical value, several conditions must be assured. Let us suppose the needle to be electrified. We stated that, at starting, the ray of light should point to the centre line on the scale. To ensure this, the quadrants must be symmetrically placed. This can be roughly done by hand, as means are provided for enabling the quadrants to slide backwards or forwards, and to be fixed by means of small screws, shown in the large figure. For obtaining the final position, one of the quadrants (B) is provided with a micro- meter screw (g), which enables a fine adjustment to be given to it. We must also have means of keeping the needle at one uniform potential for a considerable time. The needle itself could only contain a very small amount of electricity, and a slight escape of this would seriously lower the potential, and make comparative measurements useless; for it is evident that the whole principle of the instrument depends upon the potential of the needle remaining constant during the time a set of experiments are being made. To get over this difficulty a large glass jar, like an inverted shade, is provided, partially coated with strips of tin-foil (ƒ) outside. Înside the jar, to about a third of its height, strong sulphuric acid is placed. This answers a threefold purpose. It enables the air inside it to be kept quite dry, thereby very perfectly keeping those parts insulated which require to be so; secondly, it holds a charge of electricity (acting as the inner coating of the jar); and thirdly, it allows the charge to be communicated to the needle without impeding its movements. This latter is effected by means of a fine platinum wire, which is attached to the lower end of the thick wire which supports the needle and mirror. The fine wire dips into the acid, whose charge is thereby communicated to the needle. To keep this wire from curling up out of the acid, and also to steady the movements of the needle, a small plummet of plati- num is attached to the end of the wire, as will be seen in the figure. A thick platinum wire, fixed to the lower extremity of the guard tube t, and reaching nearly to the bottom of the jar, is for the purpose of enabling the latter to be charged, in a manner to be explained. So far, the jar answers the purpose of keeping the needle THE THOMSON QUADRANT ELECTROMETER. 351 supplied with electricity; but although this may prevent the potential from falling very rapidly, it will not prevent its doing so entirely. The Replenisher. 392. As the instrument is extremely sensitive to very slight changes of potential, some means are requisite by which any small loss can be easily supplied without there being any fear of putting in too much. This is effected by means of the "replenisher," whose principle we can explain by the help of the small cut to the left, in Fig. 104/25 A and B are two curved metal shields, one of which (say A) is connected to the acid in the jar and the other, B, to the framework of the instrument, and through it to the foil outside the jar. b and b are two metal wings insulated from one another by a small bar of ebonite, which is centred at s, so that it turns in a plane represented by the paper. The spindle is represented in the large figure by s, other parts being omitted for simplicity. It will be observed that the wings curve outwards. This is done in order that they may make a short contact in their revolution with springs c c and e e. c and c are connected together permanently, but are insulated from the rest of the apparatus. e and e are connected to the shields A and B respectively. Now let us suppose the wings to be rotated in the reverse direction to that in which the hands of a watch turn. As soon as the left-hand wing comes in contact with the spring c, at the lower part of the figure, the right-hand wing comes in contact with the other spring. The two wings being thus connected together, and under the influence of the shields, the electricity in A, which we will call positive, draws negative electricity to the wing close to it, and drives the positive to the other wing. On being rotated a little farther the wings clear the springs. and being thus disconnected, each retains its charge. Continuing the rotation, the right-hand wing, which had the positive charge communicated to it, comes in contact with the spring e of shield A, and the charge is communicated to the jar, the negative electricity in like manner on the other wing running to the outer coating of the jar. The shields are now in a neutral condition, as at first, and on continuing the rotation the process is repeated. Thus every turn increases the potential of the charge in the 352 HANDBOOK OF ELECTRICAL TESTING. jar, and by continuing the rotation we can augment this as much as we please. By reversing the motion we can diminish the charge, if we require to do so. The axis of the replenisher projects above the main cover, and is easily turned by the finger. The Gauge. 393. But we still require some arrangement by which we can see whether we have kept the potential constant. This is done by means of a small “ gauge." The gauge consists of two metallic discs having their planes parallel and close to each other. The lower of these planes, which will be seen dotted at the upper part of the figure, is in electrical connection with the acid of the jar from which it takes its potential. The upper disc is perforated with a square hole immediately over the centre of the lower disc. A light piece of aluminium, shaped like a spade, has the part corresponding to the blade fitting in this square hole. At the point where the handle would be joined to the blade this spade is hinged, by having a tense platinum wire fixed to it, which runs at right angles on each side of the handle and blade, and lies in the same plane as the latter. When the lower plate is electrified, it would attract the blade, thereby raising the end of the handle. So that if we notice the position of the end of the handle with respect to a mark, and see that it moves above or below it, we know that the elec- tricity of the lower plate is either overcoming the tendency of the light platinum wire to keep it up, or is unable to do so. If then we charge our jar to such a potential that the handle is situated close to the mark, and we keep it so, we know that the potential of the jar is constant. When we notice the handle sinking below the mark, we know that the potential of the electricity in the jar is falling; but a few turns of the replenisher will bring it up again. In the actual arrangement, the rung of the handle is formed of a fine black hair. Inside the handle there rises a small pillar, with two black dots on it. The sign of division represents this, the line being the hair which, by the movement of the spade blade, rises above or below the two dots, which of course would be almost quite close together. To enable the hair and spots to be seen distinctly, a plano- convex lens is placed a little distance off. Care must be taken, THE THOMSON QUADRANT ELECTROMETER. 353 in order to avoid parallax error, to keep the line of sight a normal to the centre of the lens. We spoke of the lower disc, which becomes electrified by the jar, and which acts on the spade blade. Now it is evident that if the distance between the plates be always the same, and the elasticity of the platinum axial wire be also the same, to get the hair between the two spots is to obtain the jar at a particular fixed potential. But we may require to get this potential, although the same whilst a certain set of experiments are being made, yet different for different series of experiments. This is provided for by enabling the lower disc to be lowered by screwing it round. The Induction Plate. 394. To enable high potentials to be measured, an "induction plate" is added. It consists of a thin brass plate, smaller in area than the top of the quadrant beneath it, and supported from the main cover by a glass stem. It is provided with an insulated terminal I. The use of the plate will be explained later on. 395. A flat brass plate covers the mouth of the jar, and is secured to it so as to be air-tight and prevent the entrance of moisture. A kind of lantern rises from the middle, which covers the mirror and its suspending arrangements, and above this a box with a glass lid protects the gauge. The front of the lantern is of glass, which allows the ray of light to fall on the mirror and be reflected back on the scale. Terminal rods or electrodes, in connection with each set of quadrants, pass through ebonite columns to the outside of the case, and have terminals attached to them. These electrodes can be pulled up and disconnected from the quadrants if necessary. A charging rod (seen in Fig. 125, page 349, to the left of the left-hand quadrant terminal) also is provided, which can be turned round on its axis. It has at its lower end a small spring, fixed at right angles to it. By turning this terminal rod round, the spring can be brought in contact with the framework from which the needle is suspended, and thereby, through the medium of the guard tube and the platinum wire attached to it, the acid in the jar can be charged. When this is done, the spring is moved away, so that no accidental leakage can take place through it. Various insulating supports are provided inside the jar and 2 A 354 HANDBOOK OF ELECTRICAL TESTING. $ 1 lantern. One supports the guard tubes and the adjusting screws of the needle; others support the quadrants. The whole arrangement is supported by a kind of tripod on a metal base, to keep it steady. There are also levelling screws, FIG. 126. and a level on the brass cover, to enable the instrument to be properly levelled, so that the axis of the needle may swing clear of the guard tube. H is a screw-capped opening through which acid can be introduced into the glass jar. Reversing Key. 396. Fig. 126 represents a reversing key which is specially adapted for use with the instrument. TO SET UP THE ELECTROMETER. 397. In setting up the instrument for use the following instructions* should be followed: The cover being unscrewed and lifted off and supported about 18 inches above the table, it will be observed that the stiff platinum wire to which the needle is attached just appears below the narrow guard tube enclosing it in the centre of the quadrants, and terminates in a small hook. The loop at the end of the fine platinum wire is to be slipped over this hook, so that the fine wire and plummet may hang from it. The wide guard tube, when in its proper position, forms a continuation of the upper guard tube, so as to enclose the fine platinum wire just suspended. It must therefore be passed upwards over the suspended wire, and neck foremost, until the neck embraces the lower part of the upper guard tube, where it must be fixed by the screw pin provided for the purpose; this pin is screwed in by means of one of the square-pointed keys, supplied with the instrument, fitting the square hole in its head. This being done, replace and fasten the cover, place the instrument on a sheet of ebonite or block of paraffin wax so as to insulate it, and level up by means of the circular spirit level on the cover. Next unscrew and lift off the lantern and, if necessary, adjust the four quadrants so that they hang properly in their places, with their upper surfaces in one horizontal plane. The needle and mirror which have been secured during transit by a pin passing through the ring in the platinum wire just above the * From instructions drawn up by the late Mr. W. Leitch. THE THOMSON QUADRANT ELECTROMETER. 355 guard tube, and screwed into the brass plate behind, must now be released by unscrewing this pin with the long steel square- pointed key, and placing it in the hole made for it in the cover just behind the main glass stem to prevent its being lost. The needle will now hang by the fibres. The two quadrants in front of the mirror should now be drawn outwards from the centre as far as the slots allow, by sliding outwards the screws from which they hang, and which project above the cover of the jar with their nuts resting upon flat oblong washers; a better view will thus be obtained of the needle. The surfaces of the latter ought to be parallel to the upper and under surfaces of the quadrants, and midway between them. This will be best observed by looking through the glass of the jar just below the rim. If the needle requires to be raised or lowered, it is done by winding up or letting down the suspending fibres, that is, by turning the proper way the small pins c, d (Fig. 124, page 348). The suspending wire which passes through the centre of the needle should also be in the centre of the quadrants. This is best observed when the quad- rants have been moved to their closest position. The fourth quadrant is moved out or in by the micrometer screw g (Fig. 125, page 349), with the graduated disc overhanging the edge of the cover. Á deviation of the suspending wire from its proper central position, as was explained at the beginning of the chapter, may be corrected by means of the small screws a, b, c, and d (Fig. 124, page 348). When proper adjustment is attained the black line on the top of the needle should be parallel to the transverse slit made by the edges of the quadrants when these are symmetrically arranged. The sulphuric acid may now be put into the jar. For this purpose, the strongest sulphuric acid of commerce is to be boiled with some crystals of sulphate of ammonia, in a florence-flask supported on a retort-stand over a jet of gas or other convenient source of heat. It is recommended to boil under a chimney, so that the noxious fumes rising from the acid may escape. To guard against the destructive effects of the acid in the event of the flask breaking by the heat, there should be placed beneath it a broad pan filled with ashes, or it should stand above a fire- place containing a sufficient quantity of cold ashes. A little sand put into the flask will lessen the risk of breaking, The object of boiling the acid is to expel the volatile acid impurities which will otherwise impregnate the air inside of the jar and tarnish the works. When cool, the acid may be best poured into the jar through a glass filler with a long stem inserted through the screw opening H (Fig. 125, page 349) provided for the purpose. 2 A 2 356 HANDBOOK OF ELECTRICAL TESTING. The stem of the filler should reach the bottom of the jar, to avoid splashing upon its sides or upon the works, and in removing it care should be taken that it is drawn out without its end touch- ing any of the brasswork. The acid may be poured in till the surface is about an inch below the lower end of the wide brass tube which hangs down the middle of the jar. It must at least reach the three platinum wires hanging from the works. 398. The instrument thus adjusted and charged with acid should be allowed to rest for some little time so that any films of moisture on the insulating portions of the apparatus may become absorbed. The scale should now be placed at the proper distance so that the reflected image is sharply defined and stands at the middle of the scale, that is, at 360; for the electrometer scale (unlike that of a galvanometer) is graduated from 0 to 720, 360 being the middle point. Care must be taken that the two ends of the scale are equidistant from the centre of the mirror. Next connect together the two electrodes of the quadrants and the induction plate electrode, by means of a piece of thin wire joined to the cover of the jar; also turn the charging rod so that it touches the framework of the platinum wire of the needle. Now charge the jar positively by means of a few sparks from a small electrophorus, the frame of the instrument being put to earth for the purpose, and afterwards disconnected. When the proper potential is reached, it is indicated by the lever of the aluminium balance rising; the charging rod should then be turned so as to disconnect the latter from the needle. The replenisher must now be used to adjust the charge exactly, so that the hair may stand between the black spots when observed through the lens. When the lever carrying the hair is at either extremity of its range, it is apt to adhere to the stop; in using the replenisher to bring it from either limit, therefore, it is necessary to free it from the stop by tapping the cover of the jar with the fingers. If the charge has caused the reflected image to be deflected from the middle of the scale, it may be brought back to that position by turning the micrometer screw which moves the fourth quadrant, and, if necessary, sliding out or in one or more of the other quadrants. The small percentage of the charge lost from day to day may be recovered by using the replenisher. Under ordinary conditions this loss will not amount to more than per cent. per day. The charge may suffer loss from several causes, the most prevalent being the presence of dust on portions of the appa- ratus inside the jar. Every portion should be carefully dusted THE THOMSON QUADRANT ELECTROMETER. 357 with a camel-hair brush, and especially the round induction plate beneath the aluminium balance. Loss may occur by shreds inside of the quadrants drawing the charge from the needle. It should be ascertained whether this takes place. Insulate alternately each pair of quadrants by raising the corresponding electrode, while the other pair are connected through their electrode with the cover. If the re- flected image in either case keeps moving slowly along the scale, for instance over 20 scale divisions in half an hour, the charge in the jar being at the same time kept constant by the use of the replenisher if necessary, the insulated pair of quad- rants is receiving a charge from the needle. In that case the inside of the quadrants may be brushed with a light feather, or camel-hair brush, after sliding them outwards as far as the slots allow, and securing the needle in the position in which it was fixed during transit; care being taken not to press upon the needle so as to bend it or the suspending wire. Without secur- ing the needle, each quadrant may be drawn outwards and brushed, while the needle is deflected away from it by the screws a. b (Fig. 124, page 348), or by any obvious means of keeping the needle deflected, care being taken not to strain the fibres. Another possible source of loss of charge is want of insulation over the portion of the glass jar above the acid. If the per- centage of the charge lost from day to day be so considerable as to require much use of the replenisher to recover it, the glass should be cleaned with a wet sponge, rubbed with soap at first, or with a piece of hard silk ribbon, wet and soaped at first, then simply wet with clean water, which may be drawn round the glass to clean every part of it. The ribbon being dried before a fire, may be used in the same manner to dry the glass. If everything fails to make the apparatus keep its charge, the cause is probably due to a defective glass jar, and this can only be remedied by the manufacturers. 399. The good insulation of the instrument being satisfac- torily accomplished, the symmetrical suspension of the needle by the fibres should be tested. The conditions sought to be realised are, that in the level position of the instrument the needle may hang with equal strain on the two fibres, and in a symmetrical position with regard to the four quadrants. It is plain that if these conditions be fulfilled the deflection produced by the same electric force in the level position of the instrument, will be less than it will be in any position of the instrument which throws the greater part of the weight on one fibre, or brings the needle nearer to any part of the inner surface of the quadrants than it is in its symmetrical position, which is its position of greatest 358 HANDBOOK OF ELECTRICAL TESTING. distance from all the quadrants. To make the test, the two quadrant terminals should be connected to the two poles of a single-cell battery, and the deflections produced upon the scale compared, while the instrument is set at different levels, by screwing one or more of the three feet on which it is supported. At each observation the extreme range, or difference of readings got by reversing the battery, should be noted. If the range diminishes as one side of the instrument is raised, the sus- pending fibre on that side must be drawn up, by turning very slightly the small pin c or d (Fig. 124, page 348), round which it is wound, and another series of observations taken in the same manner, beginning with the instrument levelled. Instead of drawing up one fibre, the other may be let down, to keep the needle midway between the upper and under surfaces of the quadrants, and after each alteration of the suspension it will be necessary to readjust the screws a, b (Fig. 124, page 348), to make the black line on the needle hang exactly midway between the quadrants when the needle is undisturbed by electricity. It will be observed also that the charge of the jar is lost by touch- ing these screws, unless the insulated key is used. They are reached without taking off the lantern by screwing out a vul- canite plug in the glass window in front of them. In deflecting the instrument much from its level position, the guard tube may be brought into contact with the wire hanging from the needle, and the movements of the latter be thus inter- fered with by friction. When the needle vibrates freely, it will be observed that the image comes to rest in any position to which it may be deflected, after vibrating with constant period and gradually diminishing range on each side of this position of rest. The occurrence of friction is shown by the needle coming to rest abruptly, or vibrating more quickly than proper. The reading obtained under these circumstances is, of course, of no value. The quicker vibrations obtained in using the induction plate must not be mistaken for vibrations indicating friction, from which they may be easily distinguished by their regularity. If, as may possibly happen, the process of observing the deflections at different levels, and drawing up the fibre on that side which is being raised while getting less sensibility, should only lead the operator to draw up one fibre till it bears the whole weight, while the other is seen to hang loosely, he should adjust them as nearly as he can by the eye to bear an equal share of the weight, and examine the position of the needle by looking through the glass of the jar just below the rim, the two quadrants in front of the mirror being drawn out, and the lantern taken off to let in plenty of light. He will probably THE THOMSON QUADRANT ELECTROMETER. 359 1 find that the needle leans slightly downwards relatively to the quadrants on that side which he was drawing up while getting smaller deflections. To correct this is a delicate operation, which should only be attempted by a very careful operator. Though perfect symmetry of suspension is aimed at, it is not essential to the utility of the instrument. If it be desired to make the correction, first secure the needle as during transit; take off the cover, and while it is held by a careful assistant, or properly supported in a position in which it may be levelled, remove the lower guard tube (the wide brass tube hanging down the centre) after screwing out the small pin in its neck. It will be observed that the upper and narrower guard tube consists of two semi-cylindrical parts united. The part in front may now be removed by taking out the two screws which fasten it at the top, and the platinum wire which carries the needle may be examined. If it has got bent it must be straightened; if not, it may be bent carefully just above the needle, so as to raise that end of the needle which was observed to hang lowest. If the cover be supported so that it may be levelled, the needle may be set free, and the operator may observe whether he has suc- ceeded in making it hang parallel to the surfaces above and below it. The needle must not, however, be allowed to hang by the fibres, while bending the platinum wire, or while re- moving or replacing the guard tubes. The works being replaced, the process of observing the deflections at different levels and adjusting the tension of the fibres should be repeated, with the view of getting minimum sensibility in the level position. The two unoccupied holes bored through the cover and flange of the jar are intended to receive the square-pointed keys, when not in use. GRADES OF SENSITIVENESS. 400. There are several ways of making the connections to the terminals of the quadrants, frame, and induction plate, so as to get various degrees of sensitiveness for measuring potentials of various strengths. 1st Grade. The following is the most sensitive arrangement, such as would be used for measuring the potential of a Daniell cell:- One pole of the battery would be connected, through the medium of a reversing key (Fig. 126, page 354), to one quadrant terminal, and the other to the frame of the instrument and to the second quadrant terminal. This, by reversing the key, L 360 HANDBOOK OF ELECTRICAL TESTING. would give about 50 divisions on either side of the 360, equal to 100 in all. 2nd Grade. Leaving one pole of the battery to the frame, the next degree of sensitiveness is obtained by disconnecting the pair of quad- rants that are connected to the frame, the electrode being raised for the purpose; the other connections must be the same as in the last case. By this arrangement the needle is acted upon by one pair of quadrants only. 401 By using the induction plate we may still further diminish the sensitiveness of the instrument. For instance, when we connect the pole of the battery to a pair of quadrants, those quadrants take the potential that it has; but if we connect it to the induction plate, then the charge in the quadrant below is only an induced one, and, since there is an interval between the plate and the quadrant, this induced charge will be small, and the effect on the needle proportionally small. Again, if we disconnect one pair of quadrants, and connect the wire from the battery to the induction plate and to the corresponding quadrants, then the charge will be partially bound. The effect on the needle will therefore be less still. The actual number of grades of sensitiveness with the induction plate are as follows:- 3rd Grade. One pair of quadrants connected to one pole of battery. Induction plate and second pole of battery connected to frame. Second pair of quadrants disconnected by raising electrode. 4th Grade. One pair of quadrants connected to one pole of battery, and also to induction plate. Second pole of battery connected to frame. Second pair of quadrants disconnected by raising elec- trode. 5th Grade. Induction plate connected to pole of battery. One pair of quadrants and second pole of battery connected to frame. Second pair of quadrants disconnected by raising electrode. 6th Grade. Induction plate connected to pole of battery. Second pole of battery connected to frame. Both pairs of quadrants discon- nected by raising the electrodes. " THE THOMSON QUADRANT ELECTROMETER. 361 402. We can in each of these cases interchange the terminals of the quadrants, that is to say, we can use the left terminal where we used the right, and vice versâ. 403. There is one more point to mention in connection with the instrument, and that is, that it may be found, on raising one of the electrodes to disconnect it from the quadrants, that the act of doing so causes the image on the scale to deviate a few degrees from zero in consequence of a charge being induced thereby. In the most recent form of instrument there is a small milled vulcanite head provided, by turning which the quadrants are connected to the frame, and the charge being thereby dissipated, the image returns to zero. When this is done the milled head must be turned back before commencing to test again. THE USE OF THE ELECTROMETER. 404. The electrometer can be used in every test where a con- denser is usually employed. In using the condenser we have to charge it, and then note its discharge on the galvanometer, which gives the potential. With the electrometer we have simply to connect to its terminals the wires which would be connected to the condenser, and the permanent deflection on the scale gives us the potential, which can be observed at leisure. Thus in measuring the resistance of a battery by the method given on page 295 (§ 322), we should first connect the battery wires to the electrometer (through the medium of the reversing key is best), note the deflection, then insert the shunt, again note the deflection, and calculate from the formula. The great value of the electrometer, however, lies in the fact of its enabling us to notice the continuous fall of charge in a cable, and not, like the condenser method, merely to determine what the potential has fallen to after a certain time. We can see with unfailing accuracy when the charge has fallen to one- half, or any other proportion we please. We see, in fact, exactly what is going on in the cable at any moment. The connections for such a test could not well be simpler. We charge the cable, connect it to the electrometer, the frame being to earth, and then notice the deflection as it gradually falls down the scale. We do not even require a battery, as we can charge the cable with a few sparks from an electrophorus. The degree of sensitiveness necessary for any particular cable we can, of course, only tell by experience. 362 HANDBOOK OF ELECTRICAL TESTING. Measurements from an Inferred Zero. 405. When very high resistances, such, for instance, as short lengths of highly insulated cable, are measured by the ordinary fall of charge method, the fall, even in a considerable time, would be so small that the test would be an unsatisfactory one, for the difference between the deflection at the beginning of the test, and that after the interval of time, could only be a small fraction of the whole length of the scale; and if the deflections are not accurately noted, still less can we be satisfied of the correctness of our result when worked out from a formula. By means of a plan suggested by Professor Fleeming Jenkin, however, such high resistances can be measured by the fall of charge method with considerable precision. Professor Jenkin's improvement consists in virtually prolong- ing the scale and counting the divisions from an inferred zero. An explanation of the method of making the test will best show what an inferred zero is. One pole of the battery being to earth, the other pole is connected to one pair of quadrants and to the framework of the instrument. The second pair of quadrants is connected to the cable. By joining for an instant the two pairs of quadrants together, the cable and quadrants take the same potential; therefore, at the moment of disconnecting them, the needle will be at zero. The potential, however, of the cable, and the quadrants connected to it, will fall, and the needle be deflected. Suppose, now, one cell connected to the electrometer gave 100 divisions deflection, and suppose the battery which charged the cable was 100 cells, then if the cable lost 1 per cent. of its charge, the charge remaining would be 99, and as the other quadrant, being permanently connected to the 100 cells, has the potential of 100, the difference between the two is 100 99 = 1 cell, which, as we have said, gives 100 divisions. The 2 per cent. loss would give 200 divisions, and so on, whereas by the method mentioned on the last page, if we get 300 say, at first, then 1 per cent. loss would only move the image down to 297, and 2 per cent. would move it down to 294. When all the charge is lost, the deflection would evidently be 100 x 100 = 10,000, which is the inferred zero. To obtain this zero for any particular battery, we should have to get the deflection from 1 cell and then determine, by the method given on pages 287 (§ 316) and 300 (§ 327), what the electromotive force of the testing battery is in terms of the 1 cell. Then by multiply- ing the 1 cell deflection by this value we get what we require. THE THOMSON QUADRANT ELECTROMETER. 363 The numbers representing the potentials we must evidently get by subtracting the deflections on the scale from the inferred zero. To obtain the full range of the scale we should, at starting, get the image on the actual marked zero, which is, as we have before said, at the end, and not at the middle of the scale. 406. It is possible to use the electrometer without having the acid of the jar charged. For this purpose one pair of quadrants should be connected to the needle; by this arrangement the needle becomes charged by the same electricity that charges the quadrants to which the needle is connected. It will be seen, however, that with this arrangement the deflections will not be directly proportional to the potentials producing them, as the action is similar to that which takes place in the case of an electro-dynamometer (page 318); the deflections, in fact, will be proportional to the squares of the potentials. The special advantage of the foregoing method of using the instrument is that it enables rapidly alternating potentials to be measured, as in the case with rapidly alternating currents through the electro-dynamometer. ť I + 1 364 HANDBOOK OF ELECTRICAL TESTING. 2 [ 1 CHAPTER XV. MEASUREMENT OF HIGH RESISTANCES. 407. The highest resistance which it is possible to measure by means of the Wheatstone bridge described at the commence- ment of Chapter VIII. (page 188), is 1,000,000 ohms. It is true that some bridges have another set of resistances in the top row, which will enable the ratio 10 to 10,000 to be used, and consequently a resistance of 10,000 × 10,000 10 = 10,000,000 ohms to be measured; but this is not often the case, and the values of resistances much greater than this frequently require to be determined. For this purpose a modification of the deflection method given in Chapter I., page 5 (§ 9), must be adopted. 408. Provide a single, and also about 100 constant cells. Find their respective electromotive forces by the discharge method given on pages 287 (§ 316) and 300 ($ 327). Thus, suppose the discharge taken from the 1 cell, which, as we have explained, should be taken first, gave a deflection of 300, the galvanometer shunt (S₂) being adjusted for this purpose to 560 ohms. Sup- pose also that the discharge from the 100 cells in the place of the 1 cell, gave a deflection of 302, with a shunt (S₁) of 6 ohms; then by multiplying the 302 by G+ S₁ S₁ we get the deflection we should have had if no shunt had been used; this will represent the electromotive force of the 100 cells. In like manner, by multiplying the 300 by G + S₂ S2 we get a number representing the electromotive force of the 1 cell. Taking the resistance of the galvanometer (G) to be 5000 ohms, and giving the other numerical values to the MEASUREMENT OF HIGH RESISTANCES. 365 quantities, the ratio of the electromotive force of the 1 cell to the electromotive force of the 100 cells would be or as 5000 + 560 560 5000 + 6 X 300: × 302, 6 2980: 252,000. If now we divide the greater number by the less, we get the value of the 100 cells in terms of the 1 cell. This value is 84·6, that is to say, the 100 cells are 84.6 times stronger than the 1 cell, and not 100 times. This might arise from some of the cells being defective, or imperfectly insulated. This does not matter, however, so long as we determine, as we have done, how much more powerful the 100 cells are than the 1 cell. Calculation may be saved in the foregoing measurement if we adjust the galvanometer, by means of the directing magnet, so that a convenient discharge deflection is obtained with the 1 cell when there is no shunt between the terminals of the instrument. The exact value of this deflection being noted, the discharge deflection from the 100 cells is next taken with the shunt (page 58, § 55); then the latter deflection multi- plied by 100 and divided by the first deflection, obviously at once gives the value of the 100 cells. 99 409. Having found the value of the 100 cells in terms of the single cell, we next proceed to join up the galvanometer, with a shunt, &c., between its terminals, in circuit with a resistance coil and the single cell, as shown by Fig. 127 (page 366). Put a resistance of 10,000 ohms in A B (a resistance of 10,000 ohms in a separate box is often used for this measurement), and having first inserted all the plugs in S, press down the short- circuit key, and proceed to remove some of the plugs, until a deflection of, say, 300 is obtained, then raise the key and see if the spot of light comes back to zero properly: if it does not, then by disconnecting one of the wires, see that the cause is not from the short-circuit key not making proper contact. If this has not the required effect, the adjusting magnet of the galvanometer must be slightly shifted, and, if necessary, put a little lower down, so as to make the needle a little less sensitive. After a few trials this will be satisfactorily done, and the spot of light will always come back to the zero point when no current is passing through the galvanometer. Let the deflection be 3014, the shunt being 7 ohms. 5000 + 7 Multiply 301.5 by which gives 215,700. 7 366 HANDBOOK OF ELECTRICAL TESTING. } ļ This is the deflection we should get through 10,000 ohms, with no shunt to the galvanometer. There is really in the circuit, besides the 10,000 ohms, the resistance of the 1 cell, and also the resistance of the galvanometer and shunt combined FIG. 127. 010,000 - S K A Ꮶ G (which will be practically 7 ohms), but this will be so small as to be of no consequence; it may, however, be added on to the 10,000 when working out the results, if preferred. Now, if we had used the 100 cells instead of the 1 cell, our deflection would have been 84.6 times as great as it was with the 1 cell. If, then, we multiply 215,700 by 84.6 we shall get the deflection obtainable with the 100 cells through a resistance of 10,000. This value will be found to be 18,248,000. Multi- plying this number by 10,000 we get the constant; this constant is obviously the theoretical resistance which would give a deflection of 1 division with the 100 cells. If it is required to use, say, 200 cells instead of 100 only, then in cases where galvanometer shunts of a fixed value (th, th,th), only, are available, it would be advisable to employ 2 cells instead of the 1 cell, for making the test, so as to cause the deflections to be of an approximately equal value (page 76, § 71); this would not of course alter the foregoing process of calculation in any way, it would only result in the numerical value of the "constant" being different. The actual number of cells used, it may be pointed out, has nothing to do MEASUREMENT OF HIGH RESISTANCES. 367 with the calculations; in fact, it is usual to speak of the 100, or 200, cells as the "battery" simply. A one-cell battery is used for producing the permanent deflection through 10,000 ohms, because 100 cells would deflect the spot of light off the scale with the lowest shunt that could be used; one cell happens to be a convenient electromotive force to employ, but, as pointed out, it might be preferable to use two, or even more, in certain cases. 1 It may be pointed out that the constant deflection with 1 cell through 10,000 ohms may usually be taken with the bo shunt in the place of a shunt of a particular numerical value (as in the foregoing example); this simplifies calculation, as we have then simply to multiply the constant deflection by 1000 G+ S instead of by S 410. The foregoing process is simplified by using a resistance of 1,000,000 (1 megohm) in the place of 10,000. The constant can then be found with the "battery" at once. 411. Having measured and worked out the constant (which is best done by the help of logarithmic tables *), we insert the resistance which is to be measured, in the place of AB, using the 100 cells in the place of the 1 cell. Having adjusted S till a deflection of 300, or near to 300, is obtained, note S and also the deflection. Let S be 2500, and deflection 298. Then the deflection without the shunt would be 5000 + 2500 298 X = 894. 2500 Dividing the "constant" by this number, we get 182,480,000,000 894 204,100,000 ohms, which is the value of the resistance. Practically, we may say the value of the resistance is 204,000,000 ohms, or 204 megohms, for inasmuch as we can only be certain of the values of the observed deflections to 3 places of figures, so we can only be certain of the worked out values to 3 places of figures. A great deal of time is often wasted in working out results to 5 or 6 places of figures when, in the observations necessary to obtain those results, it is possible to be certain of their value to 3 places only. * ' Chambers' Mathematical Tables' are those generally used. 368 HANDBOOK OF ELECTRICAL TESTING. MEASUREMENT OF THE INSULATION RESISTANCE OF A CABLE. 412. In measuring the insulation resistance of a cable, the constant having been taken in the foregoing manner, we should join up the galvanometer, shunt, short-circuit key, reversing key, battery switch, battery, and cable, as shown by Fig. 128.* FIG. 128. Cable G S K W K HHI Earth By having both a galvanometer reversing key and a battery switch the trouble of reversing the wires on the galvanometer, when the battery current is reversed, is avoided, as it can be done more readily by means of the key. The object of reversing the galvanometer connections when the battery is reversed is. to obtain the deflection always on the same side of the scale. 413. Both ends of the core of the cable must be trimmed by means of a sharp and clean knife, care being taken that the outer surface of the gutta-percha, which has been exposed and oxidised by the air, is completely cut away; the clean surface thus exposed should not be touched with the fingers. It is a good plan to paint the trimmed ends with hot paraffin wax (not oil). The ends being thus carefully insulated, and the further end left hanging free, so as not to touch anything, the nearer end of the cable must be connected, through the medium of the lead wire, to the terminal screw of the reversing key, as shown in *See also 'The Silvertown Compound Key for Cable Testing,' page 509. MEASUREMENT OF HIGH RESISTANCES. 369 Fig. 128, care being taken not to touch the trimmed end in doing so. The switch plugs being inserted, the reversing key which puts the zinc pole to the cable must be clamped down, and (the short-circuit key being depressed) sufficient resistance inserted in the shunt to obtain a deflection of about 300. At the end of a minute from the time the reversing key was clamped down, the exact deflection should be noted. 414. The deflection obtained, it will be found, is not a perma- nent one, but will gradually decrease as the current is kept on, falling rapidly at first, and then more slowly, until at length it becomes practically stationary; the continued action of the current, in fact, apparently increases the resistance of the dielectric. This phenomenon is known as Electrification, and its cause is not well understood; it seems to be due to some kind of polarisation.* The following shows the decrease in the deflection observed with a piece of cable core insulated with gutta-percha :- Minutes' Electrification. Deflection. O3TH LO SO ET CC 1 2 205 179 171 .. 4 161 .. 5 159 6 7 8 9 156 154 1:2 150 10 11 12 .. 13 14 15 118.5 147 145.5 144 143 142 415. Electrification is much more marked at a low than at a high temperature; thus in an actual experiment it was found that with a piece of core (insulated with gutta-percha) at a temperature of 0° C. the deflection fell from 240 to 75 in 90 minutes; whereas with the same piece of core at a temperature of 24° C. the deflection fell from 240 to 173 only, in the same time. * Although it is usually assumed that the decrease in the deflection is due to an increase in the resistance of the dielectric, it is very doubtful whether any such change in the resistance actually takes place; it is more probable that the diminution in the deflection is caused entirely by an opposing electro- motive force of polarisation, which force increases (but at a decreasing rate) in strength so long as the battery is kept on. 2 B 370 HANDBOOK OF ELECTRICAL TESTING. The rate at which the deflection decreases, also depends upon the nature of the insulating material; it is quicker in some kinds of gutta-percha than in others, being smallest in the best quality. In the case of gutta-percha, the rate of fall between the 1st and 2nd minute would average about 2 to 5 per cent. In indiarubber the decrease is very rapid, being as much as 50 per cent. between the 1st and 5th minute. 416. If the cable or insulated wire under test is quite sound, the electrification should take place perfectly regularly, that is to say, the deflection on the galvanometer scale should decrease steadily. An unsteady electrification, as a rule, is a sign that the insulation is defective. It sometimes happens, however, that the unsteadiness is due to the testing battery being in a bad condition, or not properly insulated; if, therefore, the electrification is such as to raise a suspicion that the insulation of the cable or insulated wire under test is not perfect, the battery should be looked to, to see whether it is in proper order. An unsteady electrification may also be caused by the ends of the cable or of the lead wire not being properly trimmed, or from their becoming damp. Before concluding, therefore, that the cable is faulty, these points should be attended to.: A third cause of unsteady electrification occasionally exists in factories; this is due to induced currents set up by the move- ment of the machinery in the proximity of the tanks in which the cable is coiled. When a cable is being tested on board ship, the rolling of the latter induces comparatively strong currents in the cable, and causes the galvanometer deflections to be very erratic. The effects of these currents, in both cases, may be completely got rid of by the simple device suggested by Mr. J. May, of the Telegraph Construction Company, of making the insulation test with both ends of the cable connected to the test- ing apparatus, instead of with one end only. 417. Although the deflections after the 1st and 2nd minute with a zinc current are usually all that is necessary when testing each of the lengths of core (about 2 knots) of which a cable is composed, or when testing a cable during manufacture, yet, when the cable is complete a more elaborate test requires to be made. 418. Now it is found with a good cable, that if the battery be taken off after electrification has proceeded for some time, and the cable be put to earth through the galvanometer, a continu- ally decreasing current will flow through the latter back from the cable.* Now if the deflection (called the "Earth Reading ") * Compare with note on page 369. MEASUREMENT OF HIGH RESISTANCES. 371 be noted exactly 1 minute after the battery current is taken off, then the value of this deflection added to the reading taken just at the moment when the battery was taken off (that is, the last electrification reading), will equal the deflection observed after 1 minute's electrification of the cable. And, again, the earth reading at the end of 2 minutes if added to the last electrifica- tion reading, will equal the deflection obtained after 2 minutes' electrification, and so on. Thus the “earth readings" obtained from the cable referred to on page 369 were as follows:- After 1 minute "" "" "" 2 minutes 3 44 4 5 19 93 ·· :: Earth Readings. 59 38 30 25 22 The last electrification reading (at the 15th minute), it will be seen, was 142; if we add to this the 1st minute earth reading viz. 59, we get 142+59=201, which is approximately the same as the 1st minute electrification reading, viz. 205. Again the last electrification reading added to the 2nd minute earth read- ing, viz. 38, gives 142+38=180, which is approximately the same as the 2nd minute electrification reading, viz. 179. If great care is taken to read the deflections at the exact termination of the minute intervals, the calculated and observed values will agree much more closely than in the actual examples just given. Considerable skill, however, is required in making the observa- tions, as the fall in the deflection being very rapid at first it may happen that the observed deflections are three or four divisions too much or too little, in consequence of the observa- tions being made a second too soon or too late. The relation between the electrification and earth readings, as has been be- fore stated, will only hold good if the cable is sound, and the accordance between the two may therefore be taken as an index of the good condition of the cable. It is not always the case however that the earth readings are noted. The process of manipulation for taking the earth readings is as follows:-A few seconds before the completion of the last minute for electrification (usually the 15th minute)* the last electrification reading is noted and the galvanometer short circuit key is raised, then exactly at the termination of the minute, one of the battery switch plugs is removed and placed * The fall in the deflection is so slow after about the 10th minute that the actual deflection at the exact termination of the 15th minute would be practically the same as it was a few seconds before that time. 2 B 2 372 HANDBOOK OF ELECTRICAL TESTING. I in the adjacent hole, so that the battery becomes disconnected, and the galvanometer terminal connected to earth. The gal- vanometer short circuit key being then depressed the current flows through the galvanometer, and the readings are taken at the exact termination of each successive minute, the 1st minute being counted from the time the battery was taken off. It is not usual to take more than 5 earth readings. Electrification readings are next taken with the copper pole of the battery connected to the cable. For this purpose the second reversing key of the galvanometer should be clamped down, and the first one released, so as to reverse the instru- ment; the plugs of the battery switch should then be inserted so that the battery sends its current to the cable in the reverse direction to that it did at first; this being done, the deflections on the galvanometer should be noted at intervals of a minute, as before, until the same number of readings are obtained. The readings in this case should be the same as those observed when the zinc pole was joined up, that is, provided the cable is sound, and also provided it is free from any absorbed charge when the current is put on. The current which causes the earth deflec- tions, however, continues for a considerable period, and therefore to render a cable neutral after it has been tested with any par- ticular current it requires to be put to earth for a certain time, which varies according to the length of the cable. If the latter is not more than 10 or 15 miles long, half an hour will usually be sufficient to render it neutral, but greater lengths require a proportionately longer time. It can easily be seen when the absorbed charge is got rid of, for if the cable is neutral no de- flection will be observed on depressing the short circuit key, but if a charge is still retained a slight constant deflection will be produced. 419. When the cable is put to earth great care must be taken that the short circuit key K (Fig. 128, page 368) is first raised, otherwise the whole static discharge (which is quite distinct from the current which causes the earth deflections) will pass through the galvanometer coils and the needles may be demag- netised or, at least, their magnetic power be altered. 420. Although it is advisable if possible to take a set of readings with a zinc and with a copper current, the cable being neutral in both cases, yet if time is an object the test with the copper current (which is usually made after the test with the zinc current) can be taken before the earth current due to the zinc test has ceased. In this case however the average read- ings will be higher than would be the case if the cable were neutral, in fact if we take the last of the earth readings, ob- MEASUREMENT OF HIGH RESISTANCES. 373 served in the case of the zinc test and we deduct it from the first minute electrification reading of the copper test, then the result should approximately equal the first minute electrification reading of the zinc test. Thus in the case of the cable the zinc readings on which were given on page 369, the electrification readings obtained with the copper pole of the battery connected to the cable, were as follows:- Minutes' Electrification. 12 4566 W 7 3 : : 9 10 11 12 Deflection. 227 : 200 189 182 178 174 171 .. 168 .. 165 163 161 159 157 156 155 : 13 14 15 :: Now the last earth reading taken in the case of the zinc test (page 371) was 22, and this deducted from 227 (the first copper electrification reading) gives 205, which is the first zinc elec- trification reading (page 369). In making the test in practice, as soon as the last earth reading of the zinc test is observed, the galvanometer short circuit key should be raised and the battery reversed, then one minute after this moment the first electrification reading should be noted. When a copper current test is made in the foregoing manner, that is to say with the cable not neutral, we cannot compare all the copper with all the zinc readings, as it would be necessary to make a deduction from each of the former; but inasmuch as these deductions would have to be less and less from each suc- cessive reading (for the earth current which causes the copper reading to average lower than the zinc reading is a continually decreasing quantity) and as we do not know at what rate the diminution takes place we cannot make the comparison; the uniformity of the electrification however and the approximate agreement between the first minute zinc reading, and the first minute copper reading minus the last zinc earth reading, is sufficient to indicate the condition of the cable under test. When all the electrification readings with the copper current 374 HANDBOOK OF ELECTRICAL TESTING. E are observed, a set of earth readings should be taken as in the case of the zinc current electrification test. The first earth reading added to the last electrification reading should, in this case, approximately equal the first zinc electrification reading. In the cable in question the actual earth readings observed were as follows: After 1 minute "" "" 2 minutes 3 4 5 "" Earth Readings. 50 28 21 17 14 It will be seen that in this case the first earth reading, viz. 50, added to the last electrification reading, viz. 155, is 205, which is the same as the first zinc electrification reading. 421. If there is not time to take readings both with the zinc and copper currents the zinc should be the one employed, as in the case of a fault it renders the latter very apparent, the copper current having the effect, to a certain extent, of sealing up a defect. The measurements being made, the resistance at the end of the first minute with the zinc current, and the percentage of electrification between the first and second minute and (in the case of a completed cable) also between the 1st and last (usually the 15th minute) should be worked out. It is not usual or necessary to carry the calculations beyond this. 422. When the cable is connected to the testing instruments by a long leading wire, then at the commencement of the test the end of the lead should be disconnected from the cable, and insulated; if any deflection is observable on the galvanometer when the battery current is put on, this deflection must be sub- tracted from the deflection obtained when the cable is attached to the lead. In making this correction care must be taken that the same shunt (if any) is connected to the galvanometer as will be employed when the cable is connected to the lead, or if no shunt is used with the lead the necessary allowance for this must not be forgotten to be made. The ends of the lead must be trimmed in the same manner as the ends of the cable. The practical way of noting down and working out these tests will be found in Chapter XXVI. 423. At the works of Messrs. Siemens & Co., Charlton, the method which has been described of testing the completed cable, is not generally adopted, the following test being preferred:- The testing battery is applied through a galvanometer to the MEASUREMENT OF HIGH RESISTANCES. 375 cable in the usual way, and readings for five consecutive minutes with the zinc pole of the battery to the cable are observed, the battery is then immediately reversed and five more minute readings taken; the battery is then again reversed, and so on until six sets of five minute readings have been noted, viz. three with a zinc and three with a copper current, taken alternately. If the cable is in good condition, then the last two sets of readings should be identical in value. 424. When a large number of cables have to be tested daily at a factory, any contrivances or methods for shortening calcu- lations are of great value. Now the use of shunts of different values for obtaining readable deflections on the galvanometer scale with different cables is continual, and the working out of the multiplying power of these shunts is a somewhat tedious operation when a large number have to be calculated. If the resistance of the galvanometer used for making the tests were constant, a small table could easily be calculated which would show the multiplying power of any particular shunt at a glance; but the resistance of a galvanometer varies considerably with change of temperature, and therefore under ordinary conditions a table of the kind cannot be employed. A very simple method of getting over this difficulty, due, it is believed, to Mr. Herbert Taylor, has been adopted in the testing rooms of the Telegraph Construction and Maintenance Company. The method is to have a small set of resistance coils directly in circuit with the galvanometer, so that the resistance of the latter can practically be always preserved the same. The resistance of the ordinary reflecting galvanometer usually averages between 5000 and 6000 ohms; by having the galvano- meter wound, therefore, so that in the hottest weather the latter value is never exceeded, and by having a set of resistance coils adjustable from 1 up to about 1000 ohms, the resistance in the circuit can always be kept up to 6000 under all conditions, and therefore a table giving the multiplying power of shunts for a galvanometer of 6000 ohms resistance can always be made use of. Tables of this description will be found at the end of the book. The tables also give the combined resistance of the galvanometer and shunt, which is sometimes required to be taken into account. 425. In the insulation testing of submerged cables the effects of earth currents are often to render the readings somewhat unsteady, so that considerable discrimination is required to determine whether the observed unsteadiness is due to this cause or to the existence of a fault. In the case of single cored 376 HANDBOOK OF ELECTRICAL TESTING. cables there is no method of eliminating these effects of earth currents, but if the cable is multiple-cored then Mr. F. Jacob points out that by a simple device the earth current difficulty can be entirely eliminated. This device consists in testing two of the cores at the same time, the second core being connected to the pole of the battery which in the ordinary insulation test is put to earth; the method is stated to give excellent results. 426. Mr. Jacob further points out that this method may be applied in other tests, those for capacity for example, it being only requisite to replace all the connections which are usually put to earth, by connections to the other core, the distant ends of the two cores of course being left separated and insulated. 427. As multiple core cables usually have not less than three cores, by making a series of tests in the manner indicated for conductor resistance tests in § 241, page 231, the individual insulation resistance of each wire can be obtained in a precisely similar way. If two separate cables which lie between the same termini are tested on Mr. Jacob's plan, the readings obtained will be much steadier than when each cable is tested separately in the ordinary manner, but they will seldom be absolutely steady, showing how local and variable the earth current changes are. In order to ascertain the individual insulation of each cable from a test of this kind, the approximate relative values of the insulation of each cable can be ascertained by balancing one cable against the other in a Wheatstone bridge, and then dividing the total observed insulation of the two in the proportion of these relative values. + ( 377 ) CHAPTER XVI. MEASUREMENT OF RESISTANCES BY POTENTIALS. 428. There are two distinct ways of measuring resistances by potentials:- 1st. By noting the fall of potential along a known resistance with which the unknown resistance is in connection. 2nd. By noting the rate at which a condenser, of a known capacity, loses its potential when it discharges itself through the unknown resistance. FALL OF POTENTIAL METHOD. 429. If we connect a battery to a resistance R + x, as shown by Fig. 129, the potential of the battery may be regarded as * falling regularly along the resistance, being full at a and zero FIG. 129. ν ннов 20 C E at c. The same would be the case if c and d were connected together instead of being put to earth. By similar triangles we have therefore or V:v::R+ x:x, V x = v R + vx, x (V − v) = Rv, * See Chapter XI., page 285, § 312. 378 HANDBOOK OF ELECTRICAL TESTING. from which v x = R √ - v [A] L V being the potential at a, and v the potential at b. So that, if R is a known resistance, we can-by observing the values of V and v-determine the value of x. For example. If R = 1000 ohms, V = 300, and v = x = 1000 200 300 - 200 200, then = 2000 ohms. 430. The relative values of the potentials can be measured by means of a condenser. To do this we should join up our condenser and galvanometer, as shown by Fig. 97, page 278, the only difference being that the terminals which are there represented as being in connection with a battery would, in the present case, be connected to the points a and d (or c) for determining V, and to b and d (or c) for determining v. The condenser discharges in the two cases give V and v. Another, and for most cases a preferable, method of measur- ing the potentials, is to insert a galvanometer between the point at which the potential is to be measured and the earth, there being in the circuit a resistance several thousand times greater than the resistance of the conductor of the cable. The permanent deflections in this case indicate the potentials (§ 314, page 286). 431. Instead of measuring the potential V, we can, if we please, at once determine the value of V v by connecting the wires from the condenser, &c. (or from the galvanometer and high resistance), to the points a and b; the deflection in this case at once gives us Vv. So that if we call v' this difference of potential, we get x = R · R. v' [B] 432. The conditions for making the test by formula [A] in the best possible manner are precisely similar to those in the case of the "Divided Charge Method" of measuring the electrostatic capacity of a cable or condenser (page 341); for equation [A] in this latter test is similar to (though not identical with) equation [A] (given above) of the test under consideration. We must, in fact, adjust R until we make v approximately equal to is to say, we must make R about half as large as x. V 3' that A ! 1 MEASUREMENT OF RESISTANCES BY POTENTIALS. 379 In the case of equation [B] the conditions are slightly dif- ferent, for here the quantity v' replaces (V — v), and although v' and (V — v) are equal, yet inasmuch as v' is the result of a single observation only, there can be but one error in it; con- sequently, to determine the best conditions for making the test, we must take equation [A], and assume an error & to exist in v only. Let A be the error in x caused by an error & in v, then x + λ = R 0+8 V − (v + 8) ³ but since 'x = R √ — v 2 - or, R = X V V v therefore λ = x - v v X 0+6 -1}= ▼ d) - (v + 8) - 1 } V S ; = x v{V − (v + d)} or, since d is a very small quantity, we may say λ = x V S v (V − v) Now we have to make λ as small as possible; this we shall do, since x, V, and 8 are constant quantities, by making v (V — v) as large as possible. But V2 4 v v) • (V − ) = 1 − ( − • ) ' ; - and to make this expression as large as possible we must make V 2 - v as small as possible; that is, since v must be positive, we must make it equal to 0, or V v = 0, 2 therefore But therefore V = 2v. v' = V - v, − In which case we get V' = 2 v — v = v. + x = R; 380 HANDBOOK OF ELECTRICAL TESTING. that is to say, in order to make the test as accurately as possible, we must make R approximately equal to x. 433. If, instead of introducing the unknown resistance x, and the known resistance R, between the points a and c, we join the pole a of the battery direct on to b, we can determine the value of x by simply noting V, and then inserting an adjustable resistance in the place of x, and altering it until we make the potential at b to be V, as at first, when of course x = R. Best Conditions for making the Test. 434. In the case of formula V x = R V - v Ꮳ make R approximately equal to In the case of formula 2 8 ia x = R R, make R approximately equal to x. Possible Degree of Accuracy attainable. [A] [B] In the case of formula [A], Percentage of accuracy S (V + v) 100 v (v - v) In the case of formula [B], L Percentage of accuracy = 8 (v + v') 100 ; v v' [C] ! where & is the fraction of a division to which each of the deflec- tions V, v, and v' can be read. LOSS OF POTENTIAL METHOD. 435. In Chapter XIII., page 329, an equation T F: V 2.303 R log 0 was obtained, where F was the electrostatic capacity, in micro- farads, of a condenser, or cable, the potential of whose charge 1 MEASUREMENT OF RESISTANCES BY POTENTIALS. 381 fell from V to v when it was discharged during T seconds through a resistance of R megohms. Now if F is the known and R the unknown quantity, then R = T ; V 2.303 F log V so that we can determine the value of a resistance by a capacity and loss of charge measurement. 436. The connections for making such a test would be pre- cisely similar to those given for determining electrostatic capacities by loss of charge (§ 364, page 332). If we were measuring the resistance of a short cable by this method, the discharge deflection V, compared with the discharge deflection obtained with the same battery from a standard condenser, would give us the value of F. For long cables, however, as we have before explained, this does not give correct results, so the capacity must be determined by other methods, Thomson's for example (page 335). 437. From (§ 362, page 330) it is obvious that we must have Best Conditions for making the Test. Make v as nearly as possible equal to V 3.5* Possible Degree of Accuracy attainable. 200 S Percentage of accuracy = R V 2.303 v log V where & is the fraction of a division to which each of the deflec- tions V and v can be read. GOTT'S PROOF CONDENSER METHOD. 438. An excellent method of determining the relative values of V and v in the foregoing test has been suggested by Mr. J. Gott. This method avoids the necessity of discharging the cable, and consists in applying what may be termed a "proof" condenser to the latter, and then measuring the discharge from the same. This condenser should be of small capacity, so as not to remove an appreciable portion of the charge from the cable; if this is the case, it is obvious that the discharge obtained from the condenser, after it has been connected for a few seconds to the cable at any particular time, will represent the potential which the cable has at that time. 382 HANDBOOK OF ELECTRICAL TESTING. 439. When the insulation resistance of a cable is measured by the foregoing methods, the result obtained is a mean of the resistances which the cable has at the commencement and at the end of the test, as electrification (§ 414, page 369) goes on the whole time the charge is falling. 440. Experimental results show that in the case of a cable whose core is insulated with gutta-percha, if the cable be charged 10 seconds before taking the discharge V, and again 10 seconds before insulating it preparatory to observing the discharge v, then the value of R after 1 minute, obtained from the formula, agrees with that obtained by the constant deflection method given in the last chapter (§ 414, page 369). 441. If we know the potential which the cable has when fully charged, and also its potential after a certain time, we can determine the potential it will have after any other time, in the following manner :- A charged cable loses equal percentages of its charge in equal times, that is to say-if, for example, 5 per cent. of its charge were lost during the first second, then five per cent. of what remained would be lost in the second second. Let V be the potential at first ; v "" after 1 sec.; V1 V2 "" "" "1 t₁ secs.; "" ta "" and let us suppose the charge loses th of its potential during n the first second; then the potential at the end of first second will be V V V = V = V n V and the potential at end of second second will be v V N but from equation [1] we get V n = √ - v therefore, substituting this value in [2] the latter becomes 2 V2 No. 2 Coil. =2.2291697 2.4684850 1.7606847 • log 2.0074 = 3026339 I⚫4580508 = log of ·2871 No. 3 Coil. log 169:5 =2.2291697 2.4684850 1.7606847 log 2.0069 = •3025257 (1 •4581590= log of ⚫2872 No. 4 Coil. log 171.5 =2·2342641 2.4684850 1.7657791 log 1.9990 •3008128 1·4649663 = log of •2917 } i ! $ I } ! [ + 1 472 HANDBOOK OF ELECTRICAL TESTING. CALCULATIONS FOR SHEET (C)-continued. log 168.5 No. 5 Coil. =2.2265999 2.4684850 1.7581149 log 1.9990 = ⚫3008128 No. 1 Coil. 167.5 1·4573021 = log of ⚫2866 7 Percentage of L088. No. 2 Coil. 169.5 76.25 log 91.25 1·9602329. log 100 =2. .3.9602329 log 167.5 = 2·2240148 1.7862181 = log of 54.48 77.25 log 92.25 = 1.9649664 log 100 3.9649664 log 169.5 = 2·2291697 1.7357967 = log of 54.42 No. 3 Coil. 169.5 75.75 log 93.75=1.9719713 log 100 = 2· 3.9719713 log 169.5 = 2.2291697 1.7428016 = = log of 55.31 : No. 4 Coil. 171.5 78.75 log 92.75 = 1·9673139 log 100 = 2. 3.9673139 log 171.5 = 2·2342641 1.7330498 log of 54.08 No. 5 Coil. 168.5 77.75 log 90.75 = 1·9578466 log 100 2. 3.9578466 log 168.5 = 2·2265999 1.7312467 = log of 53.86 SYSTEM OF TESTING CABLE DURING MANUFACTURE, 473 THE (D) TELEGRAPH COMPANY. INSULATION TESTS OF COILS AT 75° FAHR. CONSTANT. COILS. Date. No. of Coils. Length of Coils. Number of Cells. Resistance of Galvanometer. Battery Ratio. Discharge from Con- denser. Value Full of Bat- 1 Cell Bat- with tery tery. no with Shunt. Shunt. Deflection from 1 Cell through 10,000 ohms, with Shunt. Logarithm of Constant. Current from Coils before Test. --- No Shunt. Deflection after 1 minute Electrification. Deflection after 2 minutes Electrification. Shunt. Coil Lead. and Lead. Coil Coil. Lead. and Lead. Coil. Percentage of Electrification during second minute. Total resistance after 1 minute Electrification. Resistance per Knot. Remarks. 1 2 4 33 21 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1884. Apr. 3 1 knots. 1.9946 100 Dnls. ohms. divns. divns. 5460 173 170 98.27 152 11·1117541 divns. 151.5 | 148 divns. ohms. divns. divns. divns. divns. divns. divns. 780 3.5 megs. megs. 3.0 140 137 7.43 126 41 252.1 2 2.0074 39 " " 145.5 142 99 133 130 8.45 131.75 264.5 " 3 12.0069 " "9 99 " 148 144.5 137 134 7.27 129.47 259.8 " 4 1.9990 113 99 " " " 19 144 140.5 135 132 " 6.05 133 15 266.2 " 5 1.9990 19 = ཝ 36 141.5 138 "" "9 = 130.5 127.5 7.61 135.56 271.0 1 Signature. ** - HANDBOOK OF ELECTRICAL TESTING. 474 Y CALCULATIONS FOR SHEET (D). Insulation Resistance. · April 6th. log 17,000 = 4.2304489 log 173 = 2.2380461 1.9924028 = log of 98.27 = value of battery 7·0008677 log 10,020 × 1000 = log 152 = 2.1818436 11.1751141 = log 780 2.8920946 log constant 14.0672087 log 6240 = 3.7951846 Res. of galv. and shunt = 5.46 1 cell = 15 10.2720241 20.46 5460 + 780 780 6240 780 No. 1 Coil. 10.2720241 log 148 2.1702617 log 1·9946 = 8·1017624 = log of 126·41 megs. •2998558 8.4016186 log of 252-1 megs. No. 2 Coil, 10.2720241 log 142 = 2.1522883 log 2.0074 = 8·1197358 = log of 131.75 megs. ⚫3026339 8·4223697 = log of 264·5 megs. No. 3 Coil. 10.2720241 log 144.5 = 2.1598678 log 2.0069 = 8·1121563 = log of 129.47 megs. ⚫3025257 8.4146820 = log of 259-8 megs. SYSTEM OF TESTING CABLE DURING MANUFACTURE. -475 L 1 CALCULATIONS FOR SHEET (D)-continued. No. 4 Coil. } 10.2720241 log 140.5 = 2·1476763 log 1.9990 = 8.1243478 = log of 133.15 megs. • 3008128 8·4251596 = log of 266·2 megs. No. 5 Coil. 10.2720241 log 138 = 2.1398791 log 1.9990 = 8.1321450 = log of 135.56 megs. •3008128 8.4329578= log of 271.0 megs. Percentage of Electrification. No. 1 Coil. 148 137 log 11=1·0413927 log 100 = 2. 3.0413927 log 148 = 2.1702617 142 130 No. 2 Coil. log 12 = 1.0791812 log 100 = 2. 3.0791812 log 142 = 2.1522883 •9268929 .8711310 == log of 8.45 = log of 7·43 No. 3 Coil. 144.5 134 No. 4 Coil. 140.5 132 log 10.5=1·0211893 log 100 =2· log log 100 8.5= •9294189 3.0211893 2.9294189 log 144.5 = 2.1598678 ⚫8613215 = log of 7·27 log 140.5=2·1476763 .7817426 = log of 6·05 • 1 476 HANDBOOK OF ELECTRICAL TESTING. } 翼 ​CALCULATIONS FOR SHEET (D)—continued. No. 5 Coil. 138 127.5 log 10.5 = 1·0211893 log 100 = 2· 3.0211893 log 138 = 2·1398791 •8813102* = log of 7·61 די SYSTEM OF TESTING CABLE DURING MANUFACTURE. 477 (E) THE SPECIFICATION. CONDUCTOR, Weight per Knot, 107 lbs. INSULATOR. Weight per Knot, 150 lbs. Conductivity compared with Pure Copper, 96 per cent. Insulation Resistance at 75° Fahr., 250 megohms. TELEGRAPH COMPANY. MANUFACTURE OF. AT. RECORD SUMMARY OF TESTS OF COILS AT 75° FAHR. DETAILS OF COILS. SUBMARINE CABLE CABLE WORKS. RESISTANCE OF CONDUCTOR. INDUCTIVE CAPACITY. RESISTANCE OF DIELECTRIC. Difference from Date. No. of Coils. Length of Coils. Total Weight. Weight per Knot. Contract Weight in lbs. per Knot. Gutta- In In Yards. Knots. Copper. Gutta- percha. Total. Copper. Gutta- Total. Copper. percha. percha. + + Resistance per Knot. Percentage of Con- ductivity compared with Pure Copper. No. of Cells. Percent- age of loss after 10 secs. Electri- fication and 60 sects. In- sulation. Capacity per Knot. No. of Cells. Percentage of Elec- trification during second minute. Resistance per Knot. Remarks. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 18 19 20 31 32 3333 47 49 1884. lbs. lbs. lbs. Apr. 3 1 4047 1.9946 213 298 511 lbs. lbs. 106.79 149 40 256 19 lbs. lbs. lbs. lbs. lbs. ohms. •21 m.f. megs. .. •60 11.52 97.3 10 Dnls. 54.7 •2855 100 7.43 | 252.1 Dnls. 2 4073 2.0074 214 302 516 106.60 | 150·44 257·04] ⚫40 ⚫44 11.48 97.7 54.6 •2871 8.45 264.5 3 4072 2.0069 215 364 519 107.13 151 48 258 61 13 1.48 11.47 97.4 55.2 •2872 7.27 259.8 • 4 4050 1.990 214 299 513 107 05 149.57 256 62 ⚫05 •43 11.49 97.3 54·0 •2917 6*05 | 266•2 99 པ་ 5 4056 1.990 212 296 508 106.05 148·07 【254•12] ⚫95 1.93 11.62 9.72 53.7 •2866 7.61 271.0 " . Signature 478 HANDBOOK OF ELECTRICAL TESTING. TESTS OF THE CABLE. 533. As soon as one or more coils have been tested, the manu- facture of the cable is commenced; and as each coil is passed through the covering or "closing" machine, another is jointed on, the joint being made at such a time that at least twenty- four hours can elapse between the making and testing of the same. To ensure this necessary time intervening, as soon as one joint is passed through the closing machine the next should be made, so that there is a length of two knots of coil to be sheathed before the new joint is reached. The system of testing joints has been described in Chapter XIX. A form for entering the results of the tests is shown by Sheet F. In making a joint it is necessary to cut off a certain length from each coil. The amount of this length varies according to circumstances, but it is seldom more than a few yards. The order in which the coils are jointed together does not always correspond to the order in which they are tested at 75°, and therefore it is necessary to note down their consecutive order in a column provided on the test sheets for the purpose. In the case of a fault occurring in the cable, this information is of use in enabling an accurate measurement to be made. Sheets G, H, I, J, and K show the system of entering the tests as they are taken each day. The method of working out and entering the results will be understood from the examples given. With reference to columns 50 to 56 on Sheet J, as has been explained on page 233, § 245, the joint insulation resistance of a number of wires is equal to the reciprocal of the sum of the reci- procals of their respective insulation resistances. Column 53 contains, therefore, the reciprocals * of the values in column 52. These reciprocals are added together, and the results noted in column 54: the reciprocals of these numbers (multiplied by 10 million) give the values in column 55. Column 55 is obtained by comparing column 49 with column 54. 534. When a cable is of a considerable length it is usual, in order to save time, to manufacture the same in several lengths or "sections," so that several machines can be running at the * These are best obtained from tables (Barlow's are generally used). The numbers are multiplied by 10 million to avoid decimals. SYSTEM OF TESTING CABLE DURING MANUFACTURE. 479 same time. When the sections are completed they are spliced together so as to form one continuous length. The examples of tests given represent the tests of one section of the Main Cable. Final Tests. 535. On the completion of the cable special tests for insulation (page 370, § 417) are made. The general method of recording these special insulation and other tests, is shown on page 489, by Sheet L. 4 THE (F) 1 TELEGRAPH COMPANY, SECTION A. MAIN CABLE. 480 4 + HANDBOOK OF ELECTRICAL TESTING. JOINT TESTS OF COILS FOR. CABLE. Joint Made. Joint Tested. Leakage from Trough. Solid Core. Joint. Remarks. Time Con- elapsing secutive Accumu- between Order lation by Making Reduced of and Coils. Date. Date. Time. Time. Testing Joint. Full Potential. Potential Percentage Induced Leakage. Induced Accumu- lation by Leakage. Number of after 1 minute. of Loss. Discharge. Discharge. Cells for Testing Joint, 500 Dnls. 1st 2nd 1st 2nd min. | min. min. min. 1884. P.M. P.M. hours. divisions. divisions. divisions. divisions. | divisions. divisions. 1 Apr. 5 2.0 Apr. 6 4.0 26 200 114 3.0 150 8 15 145 10❘ 20 6 3.30 7 5.0 26 202 196 3.0 147 9 16 145 13❘ 22 "" Signature SYSTEM OF TESTING CABLE DURING MANUFACTURE. 481 1 i J CALCULATIONS FOR SHEET (G). log 1404 Length Manufactured. - 3·1473671 5 log 274.25 = 2.4381466 April 8th. 5.9926 : : : •7092205 = log of 5-1194 ⚫8732 CALCULATIONS FOR SHEET (H). Estimated Conductor Resistance. log 11.46 1.0591846 log 1·9952= • 2999864 1·3591710 = log of 22·86 Estimated Temperature. log 68.83 log 66.38 1 log 3 = 1·8377778 1·8220372 • 0157406 = log coeff. 573.* CALCULATIONS FOR SHEET (I). Inductive Capacity. ·4771213 log 1720 = 3·2355284 . log 100=2. 5.7126497 G+S +2 log 5560 = 3·7450748 1.9675749 log 165.5=2·2187980 1.9675749 •2512231 5460 + 100 100 5560 100 log 5·9926 = ⚫7776153 1.473€078 = log of ⚫2976 * Table V. • 21 482 HANDBOOK OF ELECTRICAL TESTING.. 1 CALCULATIONS FOR SHEET (I)-continued. Percentage of Loss. 165.5 90 log 75.5=1.8779470 log 100 = 2• 3.8779470 log 165.5 = 2.2187980 1.6591490= log of 45.62 CALCULATIONS FOR SHEET (J). Insulation Resistance. log 17,100 = 4.2329961 log 173 2.2380461 April 8th. 2·9949500 = log of 98.84 = value of battery log 10,020 × 1000= 7·0008677 log 152 = 2.1818436 11.1776613 = log constant log 1480 = 3.1702617 Res. of Galvanometer and Shunt = 5.46 1 Cell 14.3479230 log 6940 = 3.8413595 "" 10.5065635 = 15 20.46 log 164.5 = 2.2161659 8.2903976 G+S S 5460 + 1480 1480 6940 1480 log 1,000,000 = = 6· 2.2903976 2.2903976 log coeff.* 573° = •5778745 log 5·9926 = 1·7125231 = log of 51.585 •7776153 2·4901384 = log of 309·13 Estimated Insulation Resistance. log 270.3 = 2.4318460 log 1.9952= ⚫2999864 2.1318596 = log of 135·48 * Table VI. SYSTEM OF TESTING CABLE DURING MANUFACTURE. 483 } Percentage of Increase. 51.585 43.735 • log 7·850 = 8948697 t log 100 2. 1 : J !! 1 : 2.8948697 log 43.735 = 1·6408291 1·2540406 = log of 18·0 : } ! : ! 2 [ 2 1 水 ​1 484 HANDBOOK OF ELECTRICAL TESTING. > (G) THE TELEGRAPH COMPANY. SECTION A.-MAIN CABLE. DETAILS OF CONSECUTIVE ORDER OF COILS, LENGTH OF COMPLETED CABLE, ETC. Length of Circuit. Consecu- Original Date. Time. tive Order of Coils. Lengths of Coils. Lengths cut off in making Joints. Corrected Lengths of Coils. Total Length Revolutions of Drum. Remarks. Length of Length of of Core 274.25 in Circuit. 1 2 3 4 5 6 7 8 9 Revolutions = 1 Knot. 10 Cable Sheathed. Cable Un- sheathed. 11 12 1884. A.M. yards. yards. yards. knots. April 6 12.0 1 4047 yards. 4047 knots. 1.9946 286 knots. 1.0119 knots. ⚫9527 1 4047 7 4040 1.9911 7 2 4043 2 4041 2.0061 8111 3.9975 920 3.3546 ⚫6429 99 99 56 8 133 7 4051 3 4048 1.9952 12159 5.9926 1404 5.1194 ⚫8732 Signature SYSTEM OF TESTING CABLE DURING MANUFACTURE. 485 THE (H) TELEGRAPH COMPANY. SECTION A.-MAIN CABLE. CONDUCTOR RESISTANCE TESTS OF CABLE. Length of Circuit. Observed Tempera- ture in Tanks. Conse- cutive Date. Order of Cable Core (un- Coils. (sheathed). sheathed). Estimated Resistance of Conductor from Tests of Coils at 75° Fahr. Resist- ance of Total. Cable (sheathed). sheathed). Core (un- Leads. ductor and Leads. Measured Total Resistance of Con- Measured Total Resistance Estimated Calcu- lated Total Original Total of Con- Resistances Resistances Resist- Mean Tempe- ductor. per Knot of Coils ances of rature. Remarks. of Coils. 1 3 11 12 9 13 14 15 16 17 18 (Sheet B.) for Jointing. 19 when cut Cut Coils. 20 21 1884. No. Apr. 6 I knots. 1.0419 knots. knots. ⚫9527 1.9946 deg. Fahr. deg. Fahr. ohms. 58 ohms. ohms. ohms. ohms. ohms. ohms. 63 1.43 23.70 22.27 11.52 22.98 Not 22.98 60 cut. 1 11.52 22.94 22.94 7 2 3.3546 •6429 3.9975 58 62 1.42 45.84 44.42 11.48 23.03 45.97 58/ 8 7 • 5.1194 8732 5.9926 56 67 1.42 67.80 63.38 11.46 22.86 68.83 57 39 Signature 486 HANDBOOK OF ELECTRICAL TESTING. THE (I) TELEGRAPH COMPANY. SECTION A.-MAIN TABLE. INDUCTIVE CAPACITY TESTS OF CABLE. ༢. ཀ CONDENSER. CABLE. Conse- cutive Total Date. Order Length No. of of of Coils. Cells. Circuit. Resistance of Galvanometer. Capacity m.f. Immediate Discharge after 10 secs. Electrification. Discharge after 10 secs. Electrification and 60 secs. Insulation. Per- Shunt. centage Capacity Remarks. of Cable Shunt. Dis- charge. Lead. and Cable. Lead. Cable and Cable. Loss.* per Knot. Lead. 1 3 9 22 23 24 25 26 27 28 29 30 Lead. 31 32 33 34 Apr. 6 1881. No. knots. Dnls. 1 1.9946 10 5150 divns. ohms. divns. th 173 330 4.5 divi.8. divns. 173.5 169 divns. divns. divns. 4.0 85 m.farads. 81 52.10 •2859 7 2 3.9975 172.5 160 90 86 49.10 ·2864 "" "" "" "" "" "" "" 35 8 7 5.9926 5460 "9 3:34 172 100 170 165.5 94 90 45.62 2976 "" "" : * See p. 385 (§ 446). Signature SYSTEM OF TESTING CABLE DURING MANUFACTURE. 487 ! و THE Length of Circuit. Temperature. (J) TELEGRAPH COMPANY. SECTION A.-MAIN CABLE. INSULATION TESTS OF CABLE. CONSTANT. Date. Consecutive Order of Coils. Battery Ratio. Current Number of Cells. Resist- ance of Galva- nometer. Deflection Discharge from from from 1 Cell Condenser. Value through Logarithm Cable 1 Cell Full Bat- of Bat- with no tery with tery. 10,000 ohms, with 10 Shunt. of Constant. before Test + Shunt. To Shunt. 1 3 11 12 9 13 14 35 23 36 37 38 39 40 41 1884. April 6 No. knots. knots. knots. deg. F. | deg. F. Dhls. ohms. divisions. divisions. 1 1.0419 *9527 1.9946 58 63 100 5450 172 170 98.84 154 11.1807912 7 8 GO -I 127 3.3546 • 6429 3.9975 5.1194 .8732 5.9926 58 62 56 67 3253 100 5450 173 171 98.84 154 11.1833384 5460 152 11.1776613 "J Remarks. CABLE. Estimated Resistance of Cable from Tests of Coils at 75° F. Total Per- Deflection after 1 min. Electrifica- Deflection after Shunt. tion. Cable 2 mins. Electri- fication. Cable 42 43 Lead. and Cable. Lead.] and Lead. Lead. 44 45 46 47 divns. divns. divns. divns. divns. divns. 5200 3.0 176 173 2.5 163.5) 161 Cable. Electrification Percentage of during 2nd min. Total Resistance Measured Estimated Calculated centage of Resistance per Knot Resistance reduced to 75° F. reduced to 75° F. cut for Jointing. Total per Knot of Resistatice Coils at of Coils at 75° F. before 75° F. when cut for Jointing. Increase in Total Recipro- cals. Sums of Reci- Resistance Resistance from Tests procals. of Coils at during Manufac- ture. 75° F. 48 49 50 51 49 (Sheet D.) 52 53 54 55 56 megohms. megohms. megohms. megohms. X 10 miln. 2300 3.0 1.63 160 1480 3.5 168 2.5 164.5 3.0 6.9 151.5 149 7.5 73.184 166 154 6.4 51.585 136.80 272.86 252.1 megohms. 126.41 8.2 126.63 78972 292.55 264.5 131.81 75866 154838 64.583 13.3 309.13 270.3 135.48 73814 228652 43.735 18.0 Signature 488 HANDBOOK OF ELECTRICAL TESTING. Datc. Time. (K) THE MANUFACTURE OF. TELEGRAPH COMPANY. SUBMARINE CABLE AT. SECTION A.-MAIN CABLE. RECORD SUMMARY OF TESTS OF CABLE. CABLE WORKS. Length of Circuit. Consecutive Order of Coils. Length of Cable (sheathed). Length of Cable (unsheathed). Total Length of Core in Circuit. Temperature. RESISTANCE OF CONDUCTOR. INDUCTIVE CAPACITY. Resistance No. Cable. Core. per Knot. of Cells. Percentage of Loss after 10 secs. Electrification and 60 secs. Insulation. Capacity per Knot. No. of Cells. RESISTANCE OF DIELECTRIC. Resistance per Knot reduced to 75º Fahr. ance reduced to 75° Fahr. Total Measured Resist- ance from Tests of Coils at 75° Fahr. Calculated Total Resist- Total Percentage of In- crease in Resistance during Manufacture. Percentage of Electrifi- cation during 2nd min. Remarks. 1 2 3 11 12 13 14 18 22 33 34 35 51 50 55 56 49 1884. A.M. Apr. 6 12.0 1 1.0419.9527 1.9946 58 63 knots. knots. knots. deg. F. deg. F. ohms. Dnls. 11.51 10 7 CC "" 2 3.3546 6429 3.9975 58 62 11.50 49.10.2864 "" 8 33 7 5.1194 8732 5.9926 56 67 11.48 39 45.62.2976 m.f. Duls. mgohm.mgohm. mgohm. 52.10.2859 100 272.86 126.41 8.2 292 5573 184 64.583 13.3 "" "" 309.13 51 585 43.735 18.0 6.4 Signature. 6.9 7.5 Electrification steady. • SYSTEM OF TESTING CABLE DURING MANUFACTURE. 489 ! ། £ THE MANUFACTURE OF Length - 40·32 knots. (L) TELEGRAPH COMPANY. SUBMARINE CABLE AT CABLE WORKS. SECTION A.-MAIN CABLE. FINAL TEST. Conductor Resistance. June 3rd, 1884. Temperature. Total observed. Total of Coils at 75°. Observed.. Calculated. 450.17 ohms 463.68 ohms 611° Fahr. 61° Fahr. Inductive Capacity. Per Mile from Coils at 75°. 11.50 Cable. m.f. Con- denser Dis. immediate. After 1 min. Percentage of Loss. Total. Per Knot. 5520 + 15 172 × 10 162 X 144 X 15 5520 + 15 15 11.1 11.585 m.f. 287 m.f. Insulation Resistance. Constant. Battery 300 volts. 1 Cell thro. 10,000 + 20, S= 1000, G = 5520. rooo, 153 def. 5520 + 20 20 1 Cell Dis. 172. Battery Dis. 182 × S. on Cable, 560 ohms. Time. Zinc to Line. Earth Reading. Copper to Line. Earth Reading. After 1 min. 267 82 300 2 233 53 264 40 " ☺ ☺ ☺ ☺ "" 3 219 42 250 "" "" 4 213 35 241 "" "" "" 6 "" 9 "" "" 11 10 "" "" ☺ ☺ ☺ ☺ ☺ ☺ ☺ ☺ 207 30 234 60820 66 28 22 18 "" 204 229 "" 201 225 199 222 "" 197 218 "" 195 215 193 212 12 191 210 "" "" 13 189 208 "" "" 14 188 206 "" "" L 15 187 205 "" "" All readings steady. Resistance per knot at nor- mal temp. at end of let min. Zinc to Line. Do. reduced to 75º. Zinc to Copper Line. to Line. 1075 megs. Percentage of Electrification between 1st and 2nd min.) 13.1 12.5 370.8 Do. 1st and 15th min. 43.5 48.5 Signature I t ¡ 1 490 HANDBOOK OF ELECTRICAL TESTING. CHAPTER XXVII. MISCELLANEOUS. TO DETERMINE THE TRUE INSULATION AND CONDUCTOR RESISTANCES OF A UNIFORMLY INSULATED TELEGRAPH LINE. 536. On page 6 it was pointed out that the rule of multiply- ing the total insulation by the mileage of the wire to get the insulation per mile was not strictly correct. Now, although the leakage on a telegraph line insulated on poles is really a leakage at a series of detached points, and not a uniform leak- age, as in a cable, yet practically, and especially in the case of long lines, it may be considered as taking place uniformly, and consequently the solutions of problems dealing with cables also apply with considerable accuracy to land lines. We may there- fore consider the case in question by the help of the equations we have obtained in the investigations made in Chapter XXII. On page 434 we have an equation [12] 1 2 m 7 √ Ri + √ Re log - √ R₂ — √ R. and on the same page an equation [10] therefore m 1 : j √ R. R₂ 1 √R. Ri ; 2 m 2 r by substitution and transposition we get 1 r = Re 2 √ R. Ri √ Ri + √R. log √R-R₂ Re 2° Since 7 is the length of the line, and is the Conductor resistance per unit length, lr is the Total Conductor Resistance of the line, R. and R, being the respective total resistances of the line when the further end is to earth and when it is insulated. MISCELLANEOUS. 491 i Again we have (page 430) therefore m² 2.1 8 1 m 1 mi g⁰ √R.R therefore 1 ; 2 m 2 R. R by substitution and transposition we get J i ī 2 √ R. Ri loge √R; + √R. √ R; R Since 7 is the length of the line, and i is the Insulation resistance per unit length, is the Total Insulation Resistance of the line. 2 Τ To get the per mile results, we must, of course, in the first case divide the total by the mileage, and in the second multiply it by the mileage. By expanding the logarithm we may obtain approximate simplifications of the foregoing formulæ. We have √ R. 1 + √R; + √R. √ Ri loge = loge = R √ Re √ Re 1 √ R; 1 + x 2{x X3 + But therefore log. 1. XC log √R; + √R. √ R.R therefore = 2 √ R₂ {1+ √ Ri 1 Re lr = R₂ {1 + :{1+ + log. 1 + x √ Re if x = 1 x √ Ri x5 + +. 5 ...} 1 R 1 R.\2 + 3 R; 5 + . Ꭱ. ....}; 1/R.\2 1/R.\3 + 7\R +..} 3 R; 5 \R *Todhunter's Algebra, 5th Edition, page 337. , 3 492 HANDBOOK OF ELECTRICAL TESTING. and = 1 B. {1 – Bui R; 3 +1/+1(+1) + 1 R. R.\2 44 4.·()' - '" () (影​) 3 R. 45 945 If R, is not less than 5 times Re, then the formulæ R. ..}. } R. Ir = R, {1+} }, and { = R 1 - 1 } 423 {1- 3 R are correct within 1 per cent. 3 R If, however, R. is not more than 2 times R., then it would be necessary to take three of the terms given above in order to be correct within 1 per cent. In such cases the logarithmic formulæ would probably be but little more laborious to work out, and would, of course, give exact results. 537. A direct means of ascertaining the Insulation Resistance per mile of an insulated wire is the following:- As has been pointed out, on page 430, we have an equation g m² 2 Also on where r is the conductivity resistance per unit length, and i the insulation resistance per unit length, of the line. page 434 we have an equation 2.2 R. R₁ = m², where, as before, R, is the total resistance of the line when the further end is to earth, and R; the total resistance when the end is insulated. By combining these two equations we have or १ | i R. R₂ = p2 = ↑ ri, i = R, R.. r [A] If we take the unit length to be a mile, then r being the true conductor resistance per mile, i will be the Insulation Resistance per mile. 1 MISCELLANEOUS. 493 1 It will be seen that the mileage of the line does not come into the equation, this quantity being represented by Re go What we do, in fact, in order to obtain the true Insulation per mile of a line, is to multiply the total resistance of the line when its end is insulated, not by the absolute total conductivity divided by the absolute conductivity per mile, which is the same thing as the mileage, but by the observed total conductivity (i.e. the total resistance of the line when its end is to earth) divided by the true conductivity per mile. For example. The resistance of a line, 200 miles long, when the further end was insulated was 4000 ohms. When the end was to earth the resistance was 2400 ohms. The absolute conductor resistance of the wire, at the time the measurements were being made, was known to be 16 ohms per mile. What was the true insulation per mile of the line? i = 4000 × 2400 16 = 600,000 ohms. The value of i given by the ordinary rule would be i = 4000 x 200 800,000 ohms, = a result 200,000 ohms, or 33 per cent., too high. 538. It must be evident that what is ordinarily called the conductor resistance of a line is really the true conductivity resistance diminished by the conducting power of the insulators. Conductivity resistance, therefore, in the case of a land line can only be measured accurately in fine weather, when the insula- tion is very high. To obtain, then, the value of r from equation [A] it would be necessary to take a conductivity test in fine weather, and to note the temperature at that time; and then when an insulation test is made in wet weather, to observe the temperature, and from this correct the value of r previously obtained in the fine weather. 539. In the case of a submarine cable, the insulation resistance (when the cable is in good condition) is always so greatly in excess of the conductivity resistance that the true value of the latter is obtained at once by measuring the resistance of the cable when its end is to earth. Also the insulation per mile is practically equal to the total resistance when the end is insulated, multiplied by the mileage. 491 HANDBOOK OF ELECTRICAL TESTING. TESTING TELEGRAPH LINES BY RECEIVED CURRENTS. 540. The system of daily testing for insulation, described in Chapter I., page 6, and which was in general use on the lines of the Postal Telegraph Department, has been superseded by a system of testing by received currents, which possesses many advantages over the old method of testing. Every day at a definite time, currents from batteries, each of an approximately definite electromotive force, are transmitted over the different lines, or sections of lines, and the strengths of the currents received at the further ends are measured. It is evident that the strengths of these currents will vary with the amounts of leakage on the lines, that is with the state of their insulation; if then the battery power employed for transmitting the currents be constant, the strengths of the received currents observed from day to day will give an accurate knowledge of the condition of the lines. The way in which this general principle is practically carried out is as follows: Let A B (Fig. 145) represent the section of line to be tested, then to each end of the latter, resistances, R, R, of 10,000 ohms FIG. 145. Rr..... Ꭱ R2.. R.A BR 10,000 600 5.00 10,000 G 320 www 320 each are connected, together with a galvanometer G (whose resistance is 320 ohms) and a battery E (whose resistance is also approximately 320 ohms), as shown. Although the sections tested are not all of equal lengths or resistances, yet practically they are such that they may all be assumed to have a mean conductor resistance of 1000 ohms. Now it can be demonstrated mathematically that if thẹ resistances R, R, are very great, then a "resultant” fault* ƒ * See page 265, § 288. MISCELLANEOUS. 495 1 t મ (that is, the total insulation resistance of the line) will produce very nearly the same effect on the current received on the gal- vanometer G, whether this fault is at the middle, at the end, or at any intermediate point on the line. As a matter of fact the fault has the greatest influence when it is at the middle of the line, and the least influence when it is at either of the ends, but when the resistances R, R, are each about 10 times the con- ductor resistance of the line, then the difference in the two cases is practically very small. If then we assume, for convenience of calculation, that the resultant fault is at the middle of the line, we have E Eƒ C, X R₁ƒ R₁ + 1 f R₁ + ƒ 2 1 R₁² + 2 R₁ƒ' R₁ + ƒ where C, is the current received on the galvanometer G, E the electromotive force of the battery, and R₁ the total resistance on either side of the fault f. From this equation we get R₁2 1 f E E 2 2 Ꭱ, C R,2 C, 1 R1 Now the battery from which the current is sent consists of 50 Daniell cells, and if we take the electromotive force of a Daniell cell to be 1.07 volts approximately, we have We also have therefore E = 50 × 1·07 = 53·5 volts. = 320 + 10,000+ 500 10,820 ohms; R₁ f = = 1 53.5 2 10,820 × 10,820 × C, 10,820 1 ohms, • 00000045698 •00018484 C where C, is measured in ampères. If now we so adjust the galvanometer G by means of the directing magnet, that one milliampère (1th ampère) of 496 HANDBOOK OF ELECTRICAL TESTING. current gives a deflection of 80 divisions, then, if d be the de- flection given by any other current, we must have d 80 × 1000 = d x .0000125 ampères. From this last equation, then, we can obtain the strength (C) of the received current, in ampères, corresponding to any particular deflection; whilst from the previous equation, by inserting this value of C,, we can obtain the corresponding value off, that is the total insulation resistance of the line. For example. Suppose d = 136; then therefore C, = 136 × ・0000125 = .0017 ampères; f •00000045698 ·0017 1 = 11,909 ohms, ⚫00018484 or 11,900 ohms, approximately. 541. In order to save calculation, a table showing the values of C, and ƒ corresponding to the various deflections (d°), is pro- vided at each of the different test offices; this table is arranged as on p. 497. 542. In order that the station transmitting the currents may be able to ascertain whether his 50-cell battery is in proper condition, he can test its electromotive force in the following way:- The battery being joined up in circuit with the galvanometer and two of the 10,000 ohms resistances, the deflection is noted. Now if the 50 cells are in proper condition, their total electro- motive force would be 50 × 1·07 = 53.5 volts. Taking then the resistance of the battery to be 320 ohms approximately, and the resistance of the galvanometer being 1070 ohms,* the current deflecting the needle will be 53.5 x 1000 320+10,000+10,000+1070 = 2·5012 milliampères. * When this test is being made, the galvanometer resistance is 320 +750 = 1070 ohms; the 750 ohms is a resistance, connected to the instrument, whose use will be explained in describing the latter. MISCELLANEOUS. 497 Received Current Deflection on Outer Scale. Insulation Resistance. Received Current Deflection on Outer Scale. TABLE Insulation Resistance. * FOR CALCULATING INSULATION RESISTANCES. Received Current Deflection on Outer Scale. Insulation Resistance. Received Current Deflection on Outer Scale. Insulation Resistance. Insulation Resistance. Insulation Resistance. 198 Infinite. 169 31,800 140 13,100 111 6,920 82 3,830 53 1,980 197 1,350,000 168 30,500 139 12,800 110 6,780 81 3,750 52 1,930 196 595,000 167 29,300 138 12,500 109 6,640 80 3,670 51 1,880 195 379,000 166 28,300 137 12,200 108 6,510 79 3,600 50 1,830 194 277,000 165 27,200 136 11,900 107 6,380 78 3,520 49 1,780 193 218,000 164 26,300 135 11,600 106 6,250 77 3,450 48 1,730 192 180,000 163 25,400 134 11,400 105 6,120 76 3,380 47 1,690 191 152,000 162 24,500 133 11,200 104 6,000 75 3,320 46 1,640 190 132,000 161 23,700 132 10,900 103 5,880 71 3,240 45 1,590 Constant Deflection from Standard Cell to be 80 divisions on Outer Scale.† Needle should stand at Zero on the Outer ▷ Scale, both with and without the Controlling Magnet. Sending Battery should give 200 divisions through 20,000 ohms, with both plugs of the Galvanometer out. "Received Current Deflection" to be taken with left-hand plug in. M * A portion only of this Table is given. † Page 30. 498 HANDBOOK OF ELECTRICAL TESTING. But the adjustment of the tangent galvanometer should be such that 1 milliampère of current gives 80 divisions; conse- quently if the electromotive force of the 50 cells is equal to 53.5 volts, the deflection, d,, obtained should be such that that is, d₁: 80:: 2·5012 : 1; = d₁ = 80 × 2·5012 200.096 200 approximately. = 200 then is the deflection which should be obtained if the battery is in proper condition; if the latter is not the case, how- ever, then the power is brought up to its approximate proper value by adding on an extra cell or two until the deflection is increased to 200 as nearly as possible. It is seldom necessary, however, to do this in practice. 543. The measurement of the currents is effected by means of a tangent galvanometer of the pattern, and with the scale, shown on pages 22 and 30 respectively, and with the standard cell described on page 137, § 149. Fig. 146 shows in general plan the arrangement of the galvanometer. FIG. 146. Ос 320 OHMS น. A 12 TURNS 3TURNS E SHUNT 750 QHMS In this instrument there are three coils of wire, the one nearest the needle consisting of No. 35 copper wire wound to a resistance of 320 ohms. The other two coils are of No. 18 gauge-the one between C and D making three turns, and the one between D and E making twelve turns in the opposite MISCELLANEOUS. 499 F direction. The latter coils are for making rough tests of batteries. Let c be the current whose strength is to be tested, then if we connect the wires conveying this current on to terminals C and D, we get an effect c x 3 = 3 c. If we connect the wires to terminals C and E, the effect will be c × 12 − c × 3 = 9 c. Again, if we connect the wires to terminals D and E, the effect will be C x 12 = 12 c. Lastly, let terminals C and E be coupled together by a piece of wire, and let the connecting wires conveying the current to be measured be connected to terminals C and D, then the current c will split, and the amount going through the 12 turns of wire will be 3 1 C X = C 12 + 3 and the amount going through the 3 turns of wire will be 12 4 сх 12+ 3 5 The effect produced by the current going through the 12 turns of wire will be 1 C 12 × 12 = c ; and the effect produced by the current going through the 3 turns will be C 4 5 0 x 3 = c 3 = c12. 5 Therefore since the currents both affect the needle in the same direction, the joint effect will be 12 C C 12 24 5 = C = c 4.8. + 0 5 We can therefore obtain degrees of sensitiveness in the proportions 3: 4.89: 12 2 K 2 500 HANDBOOK OF ELECTRICAL TESTING. or 1:16:3: 4. These relative values are, however, only approximate. The resistances of the wires are practically nil. An adjusting magnet (as shown in Fig. 13, page 22) is set on the upper part of the instrument. 544. In testing the strength of a current in milliampères, the standard cell is connected to A and B, and both plugs are removed from the plug-holes; there is then in circuit a total resistance of 1070 ohms, viz. 750 + 320. As the electromotive force of the standard cell is 1.07 volts, the resulting deflection of the galvanometer-needle (which is adjusted by means of the adjusting magnet to 80 divisions on the outer scale) will be due to a current of 1.07 1070 = .001 ampère, or, 1 milliampère, and any other deflection obtained with any particular current, compared by direct proportion with the standard deflection, will give the strength of that current in milliampères. When the standard deflection is obtained, the standard cell is removed and the circuit from which the received current is to be measured is connected to terminal A, terminal B being put to earth. In order to enable the oscillations of the needle to be checked as quickly as possible, a key is provided, which short circuits the instrument on being depressed. TO DETERMINE THE INSULATION RESISTANCE OF A LINE WHEN THE STRENGTHS OF THE SENT AND RECEIVED CURRENTS ARE KNOWN. 545. The further end of the line being to earth, and 7 being the length of the line, we have from equation [2], page 430, by putting xl, 1, m Current sent C₁ ml ጥ [Ac-Be-N]; and from the same equation by putting x = 0, m Current received C 2 [ A - B; - B]: ; MISCELLANEOUS. .501 ! therefore C. 0°10° C ml A em² - Be- ml A - B but from equation [4], page 387, we have m σ +1 А B 1 AIR therefore m σ ተ 0°10 C C emi m σ or, A( − 1 ) = B (+1); m σ r +1 · = - ml m σ r σ • ~ ( ~ + 1) — 6 - ~ ( ~ — — 1), 2 by inserting the values of el, e-ml, and '[10] and [13], pages 434 and 435, we get √R; + √R. σ √RRR, R, √R. \√R.R; + 1)- 2 m given by equations Ri R √R; + √R₂ \/R¸R; + − R, - 1) − (√ R. — √/ R.) ( σ +1 1) √ R₂ R₂ 1) σ Cr σ (√ R₁ + √R.) (√ R R₂ Ri σ √R; + √ Ri √ Ri X (1 + 1). √R; — R. - √ Ri - Re 2/R. - R. The value of R, although it could be determined from this equation, would be represented by a somewhat complex fraction ; if, however, we have σ = 0, we then get Ri R; R. C2 C=1, or, R = B. - [B] ✔R; Re C.2 C,2° In which equation, C, and C, (being in the form of a pro- portion) may be measured in ampères or milliampères, or indeed in any multiple or submultiple of an ampère. + 502 HANDBOOK OF ELECTRICAL TESTING. For example. The resistance of a line when to earth at the further end was 1500 ohms (R.). The strengths of the sent and received currents were 2.8 and 2.6 milliampères respectively. What was the total insulation resistance of the line? 2.82 R₁ = 1500 10,908 ohms. 2.82 2.62 The measurement of the received current must be made by means of a low resistance galvanometer in order to avoid the introduction of the quantity o into the formula. 546. Having obtained R, the insulation per mile could be obtained in the manner shown on page 492, § 537; a simpler method of doing this is the following: If E be the electromotive force of the battery sending the current, then we have E² E C. = or, C2 R R2; by substituting this value in equation [B] we get R₁ E2 R. (C¸² — C¸²)° Again, for equation [A], page 492, we have R. i = R₁ =, or, R₁ = ir R.' where i is the true insulation resistance per mile of the line, and r its true conductivity resistance per mile; therefore ir E² or, i i = R¸¯ R. (C² – C¸²)' E2 r (C¸² — U¸²) (2) ; in which C, and C, are in ampères, E in volts, and i and r in ohms. If C and C, are measured in milliampères, then we have i = (EX 1000)2 r (C,² — C,²) E2 × 1,000,000 r (C,2-0,2) ohms. For example. The strengths of the sent and received currents on a line were 12 and 10 milliampères respectively, the sending battery being a 10-cell Daniell (10 volts approximately); the line had MISCELLANEOUS. 503 1 } an average estimated conductivity resistance of 14 ohms per What was the insulation per mile of the line? mile. 102 x 1,000,000 i 14 (122-102) = 162,000 ohms. 1 KIRCHOFF'S Laws. 547. These laws are two in number; the first is :- The algebraical sum of the current strengths in all those wires which meet in a point is equal to nothing. The truth of this law is almost obvious; thus, if we have, say, five wires meeting in a point, as shown by Fig. 147, then as the point A cannot be a reservoir, the sum of the currents C1, C2, approaching A must equal the sum of the currents C3, C4, C5, receding from A, that is or 4 C1 + C₂ = C3 + C₁ + C59 - C1 C2 C3 C4 C5 = 0. + C2—C3 CA CA FIG. 147. It may be as well, perhaps, to point out that although the quantities c₁, C2, C3, C4, C5, are partly positive and partly negative, yet they together constitute an algebraical "sum," for the equa- tion may be written ©1 + C2 + (− €3) + (− C4) + (− €5) = 0; the quantities C3, C4, and c5, in fact, are negative because the currents they represent flow in the opposite direction to the currents C1, C2-* * 548. The second law of Kirchoff is as follows: The algebraical sum of all the products of the current strengths *It is important that algebraical sum should not be confounded with arithmetical sum; the latter signifies a number of quantities connected by plus signs, whilst in the former the signs may be partly negative and partly positive, or, indeed, all negative. As a rule, when the word "sum" is used in stating a law, it is the algebraical sum which is meant. 504 HANDBOOK OF ELECTRICAL TESTING. 1 T and resistances in all the wires forming an enclosed figure, equals the algebraical sum of all the electromotive forces in the circuit. The truth of this law follows as a consequence from the laws we investigated on pages 292-295-viz.: (A) The difference of the potentials at two points in a resistance (in which no electromotive force exists) is equal to the product of the current and the resistance between the two points. (B) The difference of the potentials at two points in a resistance in which an electromotive force exists, is equal to the product of the current and the resistance between the two points, added to the electro- motive force in the resistance, this electromotive force being negative if it acts with the current, and positive if it opposes it. If we refer to Fig. 148, and we consider any closed circuit in it, then we can see that the sum of the differences of the FIG. 148. T Cs V V A CR B G E V Co potentials between the points in that circuit must be equal to 0; thus if we take the closed current formed by the sections A B, BC, CD, DA, for example, then it is evident that 4 (V₁ − V2) + (V2 − V3) + (V3 − V₁) + (V₁ − V₁) is the same as V4 V₁ − V₁ + V₂ − V2 + V3 − V3 + V₁ - V₁₂ which equals 0. 1 2 Now from laws (A) and (B) we have V₁ – V₂ = CR – e 1 √2 – V3 V3 V₁ = - = C3 r3 - (c44 — E) * V₁ − V₁ = − C₂ 12;* 4 - * These quantities are negative because the currents c, and c₂ flow in the reverse direction to the currents c, and c₂. MISCELLANEOUS. 505 # 1 ì * ፡ $ therefore, by addition, we get or CRe+ C3r3 C4 74+ E-C₂ r₂ = 0; - r2 : CR+C3 73 C₁ r4 C2 r2 = e — E, C373 which proves the law. 4 As in the case of Kirchoff's first law, we have, in the last equation, algebraical sums, for this equation may be written: CR + C3 r3+ (- €4 º4) + ( − C2 r₂) = e + (− E); C4 C4, C2, and E, in fact, are negative, because the currents in the sections (CD and D A) in which these quantities occur are in the reverse direction to the currents in the other sections (A B and B C). POLLARD'S THEOREM. 549. Let E be a battery of internal resistance r, which is shunted by a shunt S and is in circuit with a resistance R, then current through battery is FIG. 149. E E Ꮪ Ꭱ r + S+R E (S+ R) Sr+Rr+SR E(S+R) = Sr+R(S+r)' and current, C, through R, is есе ееееее Rece C E (SR) Sr+R(S+r) S+R S X S E ES Sr+R(S+r) S+ r ; [A] Sr + R S+ r that is to say, a battery E, having a resistance r and shunted by a shunt S, is equivalent to a battery of electromotive force Sr S E and internal resistance S+ r² S+ r 550. Now if we call e the electromotive force of the shunted ļ 506 HANDBOOK OF ELECTRICAL TESTING. battery, then we have or that is S e = E S+ "' Զ Sr er = E S + r Sr e: E:: : 2° S+ r It follows, therefore, from the theorem that the original electro- motive force, E, is to the reduced electromotive force, e, in the ratio of the original resistance of the battery to the shunted resistance of the same. A METHOD OF MEASURING THE RESISTANCE OF, AND THE CURRENT FLOWING THROUGH, ELECTRIC LAMPS WHEN BURNING. 551. This method is an adaptation of the methods given on page 312, § 347, and page 377, § 429, and is as follows:- FIG. 150. Ᏼ ཏི ཏོ ཏོནཏི B x For example. A resistance, R (Fig. 150), is in- serted in the circuit of the lamp whose resistance is to be measured, and then the potential, V, between the points, A and B, is measured. A similar measurement is then taken of the potential, V₁, between the ter- minals, C and D, of the lamp. We then have x = R . V V Suppose the resistance, R, were 1 ohm, and suppose that the discharge deflection obtained by the condenser from the points, A and B, were 250 divisions, there being no shunt to the galva- nometer; also suppose that the discharge deflection obtained from between the points, C and D, were 60 divisions, the galva- nometer, whose resistance was 6100 ohms, being shunted with a shunt of 200 ohms; then we have V = 250 6100+ 200 1 V₁ = 260 × = 200 8190; $ ! MISCELLANEOUS. 507 therefore 8190 x = 1 x 32.8 ohms. 250 If the discharge given by a standard Daniell cell (page 137) were 140, then we should have 250 140 Electromotive force between A and B = 1·079 × = 1·92. The current flowing, therefore, equals 1.91 = 1·92 ampères. 1 } } In cases where the current is powerful, and where it is not advisable to introduce so high a resistance as 1 ohm into the circuit, R could be made, say, th of an ohm. A METHOD OF MEASURING LOW RESISTANCES. 552. This method, like the foregoing, is merely an adaptation of the method given on page 371, § 429, and is shown in principle by Fig. 151. E is a single Daniell cell, R a resistance of 1 ohm, and BC the resistance, x, to be measured. Between B and C a Thomson galvanometer (page 46) in circuit with a resistance is connected. Now, taking the resistance of the cell E to be, say, 4 ohms, then if x beth of an ohm, the potential be- tween B and C will be approximately 5th of a volt, and the potential between A and B, th of a volt, con- sequently if we can measure these two potentials accurately we can de- termine the value of a resistance of 500 FIG. 151. G eeee R eeeeee 1th of an ohm to an equal degree of accuracy. Now a Thomson galvanometer, wound to about 5000 ohms resistance, will give a deflection of 100 divisions with one Daniell cell, there being in circuit a total resistance of 10,000,000 ohms. If there be no resistance in the circuit beyond that of the galvano- meter itself (5000 ohms) the deflection would be 100 X 10,000,000 5000 =200,000 divisions, representing an electromotive force, or potential, of 1 volt. approximately; hence 200 divisions would represent a potential 508 HANDBOOK OF ELECTRICAL TESTING. of Toth of a volt. We can easily, therefore, measure a potential of th of a volt. In order to make a measurement we should proceed as follows: The battery, resistances, etc., being connected up as shown in Fig. 151, and the shunt being removed from the terminals of the galvanometer, the resistance in circuit with the latter must be varied until a good deflection (about 300 divisions) is obtained. Let d₁ be this deflection, and let G and R₁ be the respective resistances of the galvanometer and the resistance in circuit with the latter; then if v₁ be the difference of potential between B and C, the current c, flowing through the galvanometer will be 1000 1 C1 = V1 R₁ + GⓇ 1 1 The galvanometer and the resistance in its circuit are now disconnected from B and C, and are connected to A and B, the Tooth shunt being joined up to the terminals of the instru- ment. The resistance in its circuit is then varied until a deflection d₂, approximately the same as d₁, is obtained; then if R₂ be this resistance, and if v₂ be the potential between A and B, and further if c₂ be the current producing the deflection d2, we have where g shunt. C2 = 2 V2 X 1 R₂ + g 1000' is the combined resistance of the galvanometer and We have therefore 1 1 I but .or C1 2 (R₂ + g) 1000 من (R1 + G) V1 : V2 :: x: R, V1 ac R' Ꭱ 212 V2 and as C1 C2 : d₁: d₂, or C1 آنان دية C2 d₂ == we get d₁ d2 (R₂ + g) 1000 (R₁ + G) X Ꭱ R' MISCELLANEOUS. 509 1 or For example. x = R d1 (R₁ + G) (R₂ + g) 1000 ˚ d₂ 000 The deflection obtained between the points B and C was equal to 320 divisions (d₁), there being a resistance of 8000 ohms (R₁) inserted in the circuit of the galvanometer. When the latter was connected between A and B, the Toth shunt was inserted, together with a resistance of 1200 ohms (R2); the deflection obtained was then equal to 310 divisions (d2). The resistance of the galvanometer was 5000 ohms (G), and the resistance, R, 1 ohm. What was the value of x? x = 1 320 (8000 5000) (1200+ 5) 1000 310 = •0111 ohms. 1 000 We are not, of course, necessarily bound to use the Tooth shunt, but in practice it would almost always have to be employed. 553. The degree of accuracy with which the test could be made would depend entirely upon the values of the deflections d₁ and d₂; and as we should endeavour to make them both as high as possible, that is to say, both as nearly equal as possible, the 8 200 Percentage of accuracy" would practically be where d is the fraction of a division to which each of the deflections could be read. 66 * d₁ THE SILVERTOWN COMPOUND KEY FOR CABLE TESTING. 554. This key, designed by Mr. J. Rymer Jones, and which is in general use in the testing rooms of the India Rubber, Gutta Percha, and Telegraph Works Company, Silvertown, is an excellent arrangement, and greatly facilitates the execution of the "Inductive capacity" and "Insulation" tests of insulated wires or of cables; it is particularly useful when a large number of wires have to be tested. The apparatus (Fig. 152) consists. of two keys, of the form shown by Figs. 102 and 103, pages 281 and 282, mounted on one base. Supposing the connections to be made as shown by the figure, then in order to measure the "discharge" from the cable, levers C and D are set in the positions shown. Lever B is now pressed to the left so that its projecting piece n comes in contact with lever A; the brass tongue of lever B is then in contact with b, so that the battery, whose zinc pole is joined to lever B, is connected to the cable. If now lever A is pressed over to the + 510 HANDBOOK OF ELECTRICAL TESTING. right, then lever B is also moved and the tongue of the latter consequently leaves b whilst the tongue of A comes in contact with a, and thus puts the cable in connection with the galvano- meter. As the second terminal of the galvanometer is connected to the piece cd, the circuit is completed to earth through d and the tongue of lever D. To measure the discharge from a condenser, one terminal of the former would be connected to the piece a b and the other terminal to earth; the manipulation of the levers would of course be the same as in the case of the cable. to Galy? to Galv FIG. 152. Cable Earth. To take the "Insulation" test (p. 368) of the cable, levers A and B would be set over to the right so that the tongue of lever A is in contact with a whilst the tongue of B is disconnected from b. The short-circuit key of the galvanometer being closed, lever C is now pressed over to the right, so that the tongue of lever C comes in contact with c, whilst the tongue of lever D becomes disconnected from d; the zinc pole of the battery thus becomes connected through c with one terminal of the galvano- meter, and as the other terminal is connected (through lever A and a) with the cable, the circuit is complete. The short-circuit key of the galvanometer is now depressed, and the deflection MISCELLANEOUS. 511 I noted in the usual manner (p. 369). As soon as the observa- tions are completed the short-circuit key of the galvanometer is raised, and lever D being pressed over to the left the battery becomes disconnected from the galvanometer terminal and the latter is connected to earth, so that the cable discharges itself. Particular care must be taken that the short-circuit key of the galvanometer is raised before lever D is pressed over to the left, otherwise the whole discharge from the cable will pass through the galvanometer coils, and the needles may either be demagnetised, or at least the "constant" of the instrument be altered. 555. The battery power with which the "Insulation" test is taken is much greater than that required for the "Inductive Capacity" test; consequently after the latter test has been made (with about 10 Daniell cells usually), the battery power has to be changed to the required larger amount. METHOD OF TESTING BATTERIES IN THE POSTAL TELEGRAPH DEPARTMENT. 556. One form of apparatus employed in the Postal Telegraph Department for battery testing is shown by Figs. 153 and 154. FIG. 153. FIG. 154. E R₁ B 100 200 400 R₂ evee B 1 It consists of two sets of resistance coils R1, R2, the former being in the direct circuit of a tangent galvanometer * G, and the latter being a shunt between the terminals of the battery x when the shunt plug S is inserted. The values of the resistance coils A, B, C, D, E and F, in R₁, are 1070, 3210, 4280, 8560, 17,120, and 34,240 ohms, respectively; that is, A, B, C, D, E, and F are in the proportion of 1: 3:4:8: 16:32. * This galvanometer is the same as that employed for making the daily morning tests by received currents (Fig. 13, page 22, and Fig. 146, page 498). 512 HANDBOOK OF ELECTRICAL TESTING. Electromotive Force Test. 557. The principle of the method of testing for electromotive force is as follows:- If the standard cell (page 137, § 149) were joined up in circuit with the tangent galvanometer, both plugs being out, then the deflection obtained would be that due to an electromotive force of 1.070 volts (the approximate electromotive force of the standard cell) acting through a resistance of 1070 ohms. If, say, five Daniell cells were in circuit, and also a total resistance of 5 × 1070 ohms, then the deflection obtained should be the same as that given by the standard cell, provided the total electromotive force of the five cells were five times that of the standard cell, or, in other words, if the average electromotive force per cell were 1.070 volts; and it is evident that if with a still larger number of cells there were placed in circuit a total resistance as many times greater than 1070 ohms as there are cells to be tested, then if the average electromotive force per cell of the battery were equal to the electromotive force of the standard cell, the deflec- tion obtained would be the same as that given by the latter. If the deflection were less, it would show that the average electro- motive force per cell of the battery must be proportionately less. For example. Suppose the standard cell gave a deflection of 25°, then if, say, a 30-cell battery with a total resistance in circuit of 30 × 1070, or 32,100 ohms, gave a deflection of 22°, the average electro- motive force per cell of the battery would be 928 volts, thus, 1.070 X tan 22° tan 25° 1.070 X • 404 • 466 = .928 volts. Now, if instead of the resistance in circuit being increased in exact proportion to the number of cells tested, it had been in- creased in a less proportion, then the deflection representing an electromotive force of 1.070 volts would be correspondingly higher. For example. If, when the 30 cells were tested, there were in the circuit, not 30 × 1070 ohms, but 12 × 1070 chms, then the deflection which would indicate that the average electromotive force per cell of the battery is 1.070 volts would be 493°, thus, tan 25° X 30 12 5 • 466 × = 1∙165 = tan 4910. 2 MISCELLANEOUS. 513 ! If, therefore, the total resistance in the circuit of the battery tested is made equal to 1070 × number of cells tested × %, [A] and if 25° is the deflection given by the standard cell through a total resistance of 1070 ohms, then 491° will be the deflection given by a battery whose average electromotive force per cell is 1.070 volts, and any deflection other than 491° will (by pro- portion of the tangents of the deflections) represent the actual electromotive force per cell of the battery. For example. If the deflection obtained were 40°, then the electromotive force per cell of the battery would be 767 volts, thus, 1.070 X tan 40° = 1.070 X tan 4910 • 839 1.171 = ·767. If the total resistance in the circuit of the battery tested is made equal to 1070 × number of cells tested × ‡, [B] then the deflections obtained will represent average electro- motive forces per cell which are double those which they represent when the resistance in circuit is that indicated by formula [A]. So that if formula [A] is applied when Daniell cells are tested, and formula [B] when Bichromates are tested, the range of deflections required in the two cases will be the same, since the electromotive force of a Bichromate battery is double that of a Daniell. 558. In order to facilitate calculation, tables constructed on the foregoing principles are employed; portions of these tables are as shown: TABLE I. Number of Cells to be tested. Coils to be placed in Circuit in R1. Daniells. Bichromates. Leclanchés. 5 3 A 10 15 35 2:42303 : 5: 6 B 8 C 10 A+ C 25 10 12 16 : 5: 15 18 and 20 B+C A+ D B+D A+ C + D 2 L 514 HANDBOOK OF ELECTRICAL TESTING. Observed Deflection (Dº). Equivalent Electromotive Force per Cell. Number and Description of Cells tested. TABLE II. Percentage of Fall from Normal Electromotive Force. Number and Description of Cells tested. Daniells. Bi. chromates. Leclanchés. Daniells. Bi- chromates. Leclanchés. 5 to 160. 5 to 80. 3, 6, 12, 18, 8, 16, 32, 40, 10, 20, 30, 24, 36. 48. 5 to 160. 5 to 80. 50, 60. 3, 6, 12, 18, 8, 16, 32, 40, 10, 20, 30, 24, 36. 48. 4910 1.070 2.140 1.612 : : 0.00 0.00 : 1.600 49° 1.051 2.102 1.584 1.80 1.80 50, 60. 4910 0.00 1.00 49° Observed Deflection (Dº). 4810 1.033 2.066 1.621 1.556 3.46 3.46 2.75 481° 1.600 .. 0.00 48° 1.015 2.030 1.593 1.529 5.14 5.14 ⚫44 4.44 48° 471° *996 1.992 1.565 1.502 6.92 6.92 2.19 6.12 4710 47° ⚫980 1.960 1.538 1.476 8.41 8.41 3.88 7.75 47° 461° ⚫963 1.926 1.612 1.511 1.451 10.00 10.00 5.56 9.31 461° 1.600 0.00 46° ⚫946 1.892 1.584 1.485 1.426 11.99 11.99 1·00 7.19 10.87 46° 451° ⚫930 1.860 1.557 1.459 1.401 13.09 13.09 2.69 8.81 12.44 451° 45° ⚫914 1.828 1.530 1·434 1.377 14.58 14.58 4.38 10.38 13.91 45° I J The underlined figures show the Normal Forces of the Cells. "Constant" with Standard Cell through Galvanometer with both plugs out to be 25°. MISCELLANEOUS. 515 The way in which these tables would be used would be as follows: The 25° constant deflection having been obtained correctly, the standard cell is removed from terminals B, and the battery to be tested joined in its place, resistances having been pre- viously inserted in resistance coils R₁, according to Table I. For example, if 35 Daniells are to be tested, the resistances to be inserted would be A, C, and D. The two plugs in the galvano- meter must still remain out so that the resistance of the latter (1070 ohms) is included in the circuit. • The deflection obtained being now noted, the electromotive force per cell of the battery is given by Table II.; thus if the deflection is 451°, the electromotive force per cell is 930, and the percentage of fall from the normal electromotive force is 13.09. 559. It will be observed that in the case of Leclanché batteries, the resistances to be placed in circuit and the deflections corre- sponding to the various electromotive forces, have to be taken in a somewhat different proportion from that adopted in the case of Daniell or Bichromate batteries, as the cells are made up in sets of 6, 8, and 10, and not in sets of 5, and moreover the normal electromotive force of a Leclanché is intermediate in value between a Daniell and a Bichromate battery; the general principle, however, upon which the resistances and deflections are arranged is similar to that adopted in the case of the latter batteries. 560. The accuracy of the method of testing electromotive force depends upon the resistance of the batteries being small in pro- portion to the external resistance, and this is attained by making the latter very large, so as to reduce the error beyond sensible limits. Resistance Test. 561. This test is made by the "Diminished deflection shunt method" described in Chapter VI., page 133. The resistance R₁ being very high, the resistance of the battery is given by formula [G], page 135, in the test referred to, that is to say we have tan D° x = R₂ tan d° - 1). For example. If by the insertion of a shunt R₂ of 25 ohms, the deflection D° of 45° were reduced to 23° (d°), then resistance (x) of battery 2 L 2 516 HANDBOOK OF ELECTRICAL TESTING. would be 35.0 ohms, thus, (tan 4510 x = 25 -1) = (tan tan 23° 25 ( 1.018 • 424 − 1) = 35.0 ohms. 562. To facilitate calculation, a table giving values of tan Do - 1) for various values of D° and do, is employed; hence tan do it is only necessary to multiply the corresponding quantity by R2, and the result is the total resistance of the battery. 563. In exceptional cases where an odd number of cells have to be tested for electromotive force, i.e. a number which is not included in Table I., the resistances inserted in R₁ are those cor- responding to the number in the table next above the odd number; thus if 13 Bichromates are to be tested, the resistances corresponding to 15 cells, viz. B and D, are inserted in R,. The deflection obtained having been noted, the result corresponding to that deflection in Table II. is multiplied by the even number of cells and divided by the odd number, the result being the electromotive force per cell of the battery. 564. It may be remarked that the range of the apparatus is considerable, it being possible to test from 5 to 160 Daniell cells, or 5 to 80 Bichromate cells, with an equal degree of accuracy, and with equal facility. DIRECT READING BATTERY TESTING INSTRUMENT. 565. In order to simplify the method of estimating the electro- motive force and resistance of batteries, and lessen the time necessary for the tests, a new instrument has recently been devised by Mr. A. Eden, which has partially superseded the foregoing apparatus, and which obviates the necessity for any calculation, or any reference to Tables. The theory of the instrument is as follows: Electromotive Force Test. 566. The constant of the galvanometer is so adjusted by means of the controlling magnet that a deflection of 80 divisions* on the skew tangent scale (page 30) is obtained with the standard cell (page 137, § 149) connected to the instrument; this deflection then represents the electromotive force of the standard cell. If now we place in circuit n cells, each having an electromotive force · * This deflection is taken because it is found to be the highest that can be obtained with certainty on the instrument. MISCELLANEOUS. 517 equal to that of the standard cell, and at the same time we add sufficient resistance in the circuit to make the total resistance n times as great as it was originally, then the deflection will still be 80 divisions, provided the cells are each equal in force to the standard cell. If, instead of increasing the total resistance in n x 80 circuit to n times its original value, we make it as great, 100 then it is obvious that the deflection given by the n cells will be 100 divisions provided all the cells are in good condition. If the force of the cells is less than their normal value, then the deflection observed will be lower than 100, and the value of this deflection will obviously directly represent the percentage value of the force; thus, if the deflection were 93 divisions, then this would mean that the cells have but 93 per cent. of their normal power. n x 80 100 Since the total resistance in circuit with n cells is times the resistance in circuit with the standard cell, and as this latter resistance is 1070 ohms (page 500, § 543), therefore this total n x 80 resistance is 100 × 1070 = n × 876 ohms. In order that a similar standard deflection (100 divisions) may be obtained with Bichromate and Leclanché batteries, shunts and compensating resistances (§ 568, page 519) are connected to the galvanometer when those batteries are being tested, so that the same resistance per cell in the case of the three kinds of batteries can be inserted in circuit. Resistance Test. FIG. 155. E 567. The general theory of this test is as follows:- Let r (Fig. 155) be the battery which can be shunted by a shunt, s, and R a high resistance which can be shunted by a shunt, S, and let G be a galvano- meter of negligeable resistance. First suppose both the shunts to be discon- nected, then, since R is very large, the current, C, through the galvanometer will be C = E R' 9 еде R ме S E being the electromotive force of the battery. Next suppose the shunts to be both connected up, then from 1) HANDBOOK OF ELECTRICAL TESTING. 518 + | t equation [A], page 505, we can see that if S and R are very great compared with 8 and r, the current, C', through the galvanometer will be or or 8 E 8 + r C' SR S+R If the currents in the two cases are the same, then we get ER E 8 + r Ꭱ SR S+R S S 11 S+ R s + r R = 1 + S 1 + S that is, R S ∞ 18 8 or ~ s R པ าง r = S Now r (the total resistance of the battery) is equal to the resistance per cell, r₁, multiplied by the number of cells of which the battery is composed; if, therefore, we make R directly pro- portional to the number of cells, as we do in the case of the electromotive force test, that is, if we make then we get or R. Ꭱ = n R₁, 8 s n R₁ nr 1 S R₁ r1 = 8 S From this it is obvious that if R, and S are constant quantities, MISCELLANEOUS. 519 1 then s multiplied by a constant will directly give the value of r₁ (that is, the resistance per cell of the battery), no matter what number of cells are being tested. In making the electromotive force test we insert in circuit a resistance n R₁, R₁ being of such a value that (as we have seen) the galvanometer deflection is 100 divisions if the battery is in good order. The value which it is preferable to give to S is that equal to the highest value which n R₁ will have; this in practice is the case when » is 60. As we have seen, the value of n R₁ is n × 856 ohms, that is, B₁ is 856 ohms, therefore the value of S should be 60 x 856, or 51,360 ohms. Since S is 60 times R₁, we get R, r₁ = 8 60 R1 60 that is to say, the resistance per cell of the battery being tested isth the resistance of the shunt. If therefore the resistances of which the shunt is composed are marked with values which areth of their actual values, then these marked values will give at once the resistance per cell of the battery under test. The theoretical values for S and R are only applicable when the battery resistance is inappreciable in comparison with the external resistance, and when the galvanometer is either of very low resistance, or double wound, so as to admit of one half of the coils being placed in circuit with R, and the other half in the shunt, S. 568. The use of a galvanometer which has a resistance of 320 ohms, and which is outside of the shunt, S, makes it necessary to compensate for the inaccuracies so introduced; this is done by making R₁ equal to 428 instead of 856 ohms, also by making S equal to 25,200 instead of 51,360 ohms, and by shunting the galvanometer by a permanent shunt of 320 ohms, thus reducing the resistance of the instrument to 160 ohms. As a further compensation, when the two shunts are connected up a resistance of 28 ohms is cut out of circuit from the n R₁ coils. These compensations are not based on any strictly theoretical basis, but are a compromise which, it is found, reduces the general error to practical limits. 569. The joining up of the shunts in the latest form of the apparatus is effected by means of a plunger key, so that the actual manipulation for the resistance measurement consists in adjusting the shunt, s, until it is found that the galvanometer de- flection remains unaltered when the key is depressed or raised. 570. It must be obvious that as the value of the shunt, s, is } 520 HANDBOOK OF ELECTRICAL TESTING. practically the same for every size of battery, the accuracy with which a test can be made varies according to the number of cells of which the battery is composed, but practicably sufficient accuracy is obtainable in all cases. 571. The actual form of the battery testing instrument embodying the foregoing principles, as most recently arranged, is shown by Fig. 156. In this fig., R are the resistances which FIG. 156. BATTY a! O ૩ & 5 6 1 9 8.1. B [DIRECT O DEF B GALVR 16 8 10 12 14 O 15 18 R 160 55 O O .6 3 5 4 40 35 30 28 1201 are inserted in circuit in proportion to the number of cells to be tested. B are the resistances for shunting the battery. b is a switch which can be turned to three different positions accord- ing as Bichromate (B), Daniell (D), or Leclanché (L) have to be tested. a is a plunger key which, on being depressed, con- nects up the shunts s and S (Fig. 155). c is a switch which, on being turned to the right, alters the connections in such a way that a half-deflection test* for resistance can be made as a check, if desired. COMBINED RESISTANCES. 572. PROBLEM-Required the joint resistance of the resistances a, b, c, d, and g, between the points A and B (Fig. 157). If we call R the resistance of the combined resistances between the points A and B, then what we have to do is to obtain an equation of the form E C5 r + R * This test follows from formula [G], page 135, if we put tan d for then r = tan D 2 S2, the S₂ in this case corresponding to the 8 in Fig. 155, page 517. MISCELLANEOUS. 521 " Now it is obvious that the value of R can be in no way depen- dent upon the value of r, hence in order to simplify the problem we may assume r to be equal to 0. By Kirchoff's laws (page 503) we have the following six equations, showing the connection between the resistances a, b, c, d, and g, the current strengths C1, C2, C3, C4, C5, and c, and the electromotive force E:- C5 C1 C6 CA - C1 C2 0 0 C3 + C6 C₂ = 0 c3 d + c₂ b − E = = 0 c₁ a c₂ b c3 d C4 X - cε 9 = 0 6 C6 g FIG. 157. Xx = 0. СА [1] [2] [3] [4] [5] [6] b C₂ T Co g C3 Сь E In order to determine the value of c, from these six equations we must first find the value of c₁ from, say, equation [1], and, substitute this value in the other equations, thereby getting rid of c₁; again in like manner, if we find the value of c₂ from, say, equation [3], and substitute throughout, we get rid of c₂, and so on. As it will be unnecessary to show all these substitutions, we shall confine ourselves to one or two only; thus from equation [1] we have C5—C1—C2 0, or, c₁ = C5 C2; therefore we get C4 C6 C5 + C₂ = 0 = [2] C3 + C6 C₂ = 0 C2 c3 d + c₂ b − E = cz a c₂ a c3 d C3 - C4 X - [3] 0 [4] c₂ b C6 cε g = 0 = [5] - C6 g = 0. [6] 盛 ​· 522 HANDBOOK OF ELECTRICAL TESTING. By continuing this process, we at length get E c5 a C6 a - có b C6 9 − (a + b) có b C6 = 0 b + d and E C5 C6 I c5 x + cε 9 - (d+ x) Co b = b + d = 0; therefore and c。 (a d + b d + b g + d g) = cz (a b + a d) − E (a + b) cε (bg+d g + bx+bd) = c5 (b x + d x) +E (d+ x). By dividing one equation by the other, c is eliminated, that is, we get or C5 a d + b d + b g + d g b g + d g + b x + b d - cz (abad) E (a + b) cz (bx+dx) +E (d + x)' E (ab+ad)(bg+dg+bx+bd)+(bx+dx)(a d+bd+bg+dg) (d+x) (ad+bd+bg+dg)+(a+b)(bg+dg+bx+bd) By dividing the numerator and denominator of the fraction below the thick line by a +x, we finally get C5 E. -6); g [(a + x) (b + d)] + a b (d + x) + d x ( a + b) g[(a + x) + (b + d)] + (a + b) (d + x) that is to say, d, m, } The combined resistance of the resistances, a, b, c, d, x,1 and g, between A and B g [(a + x) (b + d)] + a b (d + x) + d x ( a + b) g[(a + x) + (b + d)] + (a + b) (d + x) = ∞o, that is to say, if we remove It will be observed that if g = g, then we get Combined resistance } g[(a + x) (b + d)] = (a + x) (b + d). g [(a + x) + (b + d)] = (a + x) + (b + d) which is the joint resistance of (a + x) and (b + d). MISCELLANEOUS. 523. + If we have g = 0, that is to say if we join together the two points connected by g, then we get Combined) resistance T a b (d+x) + d x (a + b) ab d x a + b + ä + x 9 (a + b) (d + x) which is the joint resistance of a and b, added to the joint resistance of d and x. The truth of these simplifications is obvious. COMBINED CONDENSERS. 573. PROBLEM-Required the joint electrostatic capacity of two or more condensers joined up in "cascade." Let a, b, and c, f, Fig. 158, be the plates of the two condensers, then if we suppose these plates to be of equal size, and d₁ and d₂ to be the distances separating them, the respective capacities C₁ and C₂ will be in the proportion C₁ C₂ d₂ d₁, 2 or d₂ 1 C₁ d₁ C₂ Now the plates b and c, being joined together, may be con- sidered to be one plate as shown by the dotted line b c, Fig. 159; moreover as the latter plate is in no way connected with either + FIG. 158. k-dry fdr. + FIG. 159. ds dr. 00-0 f α Сх of the charging wires + and -, it practically does not affect the joint capacity of the arrangement; hence we can represent this joint capacity as being due to a condenser formed of the plates a and f, separated by a distance d₁ + d2. The capacity I 524 HANDBOOK OF ELECTRICAL TESTING. F Cx of the combination must therefore be given by the propor- tion : Cx: C₁:: d₁ d₁ + d₂, 1 L or C₁ d₁ 1 1 Cx d₁ + dz 1 d₂ 1 1 + + C₁d₁ C₁ 02 If we had a third condenser of a capacity C,, in the circuit of C₁ and C₂, then the joint capacity C's, of this condenser in combination with Cx must be 1 C'x = 1 1 1 1 1 + Сх + + C₂ + C3 and so on with any number of condensers. Hence we have the law: The joint electrostatic capacity of any number of condensers joined together in "cascade" is equal to the reciprocal of the sum of the reciprocals of their respective capacities. b. • 1 1 TABLES. J T } 526 HANDBOOK OF ELECTRICAL TESTING. 1 TABLE İ.-NATURAL TANGENTS. Degrees. Tangents. Degrees. Tangents. Degrees. Tangents. Degrees. Tangents. Degrees. Tangents. Degrees. Tangents. 1.00 ⚫0175 16.00 ⚫2867 31.00 • 6009 46.00 1.0355 61.00 1.8040 76.00 4.0108 1.25 ⚫0218 16.25 • 2915 31.25 • 6068 46.25 1.0446 61.25 1.8228 76.25 4.0867 1.50 • 0262 16.50 • 2962 31.50 • 6128 46.50 1.0538 61.50 1.8418 76.50 4.1653 1.75 ⚫0306 16.75 ⚫3010 31.75 .6188 46.75 1.0630 61.75 1.8611 76.75 4.2468 2.00 ⚫0349 17.00 ⚫3057 32.00 ⚫6249 47.00 1.0724 62.00 1.8807 77.00 4.3315 2.25 ⚫0393 17.25 ⚫3105 32.25 • 6310 47.25 1.0818 62.25 1.9007 77.25 4.4194 2.50 ⚫0437 17.50 ⚫3153 32.50 ⚫6371 47.50 1.0913 62.50 1.9210 77.50 4.5107 2.75 ⚫0480 17.75 ⚫3201 32.75 ⚫6432 47.75 1.1009 62.75 1.9416 77.75 4.6057 3.00 ⚫0524 18.00 ⚫3249 33.00 ⚫6494 48.00 1.1106 63.00 1.9626 78.00 4.7046 3.25 ⚫0568 18.25 ⚫3298 33.25 ⚫6556 48.25 1.1204 63.25 1.9840 78.25 1.8077 3.50 • 0612 18.50 ⚫3346 33.50 ⚫6619 48.50 1.1303 63.50 2.0057 78.50 4.9152 3.75 ⚫0655 18.75 ⚫3395 33.75 • 6682 48.75 1.1403 63.75 2.0278 78.75 5.0273 4.00 ⚫0699 19.00 ⚫3443 34.00 ⚫6745 49.00 1.1504 61.00 2.0503 79.00 5.1446 4.25 ⚫0743 19.25 .3492 34.25 ⚫6809 49.25 1.1606 64.25 2.0732 79.25 5.2672 4.50 ⚫0787 19.50 •3541 34.50 .6873 49.50 1.1708 64.50 2.0965 79.50 5.3955 4.75 • 0831 19.75 ⚫3590 34.75 • 6937 49.75 1.1812 64.75 2.1203 79.75 5.5301 5.00 ⚫0875 20.00 • 3640 35.00 ⚫7002 50.00 1.1918 65.00 2.1445 80.00 5.6713 5.25 ⚫0919 20.25 ⚫3689 35.25 ⚫7067 50.25 1.2024 65.25 2.1692 80.25 5.8197 5.50 ⚫0963 · 20.50 ⚫3739 35.50 ⚫7133 50.50 1.2131 65.50 2.1943 80.50 5.9758 5.75 .1007 20.75 ⚫3789 35.75 ⚫7199 50.75 1.2239 65.75 2.2199 80.75 6.1402 6.00 • 1051 21.00 ⚫3839 36.00 • 7265 51.00 1.2349 66.00 2.2460 81.00 6.3138 6.25 •1095 21.25 ⚫3889 36.25 ⚫7332 51.25 1.2460 66.25 2.2727 81.25 6.4971 6.50 •1139 21.50 ⚫3939 36.50 •7400 51.50 1.2571 66.50 2.2998 81.50 6.6912 6.75 •1184 21.75 ⚫3990 36.75 ⚫7467 51.75 1.2685 66.75 2.3276 81.75 6.8969 7.00 •1228 22.00 •4040 37.00 ⚫7536 52.00 1.2799 67.00 2.3559 82.00 7.1154 7.25 • 1272 22.25 •4091 37.25 ⚫7604 52.25 1.2915 67.25 2.3850 82.25 7.3479 7.50 • 1317 22.50 •4142 37.50 .7673 52.50 1.3032 67.50 2.4142 82.50 7.5958 7.75 •1361 22.75 •4193 37.75 ⚫7743 52.75 1.3151 67.75 2.4443 82.75 7.8606 TABLES. 527 8.00 • 1405 23.00 • 4245 38.00 •7813 53.00 1.3270 68.00 2.4751 83.00 8.1443 8.25 •1450 23.25 • 4296 38.25 ⚫7883 53.25 1.3392 68.25 2.5065 83.25 8.4490 8.50 •1495 23.50 ⚫4348 38.50 ⚫7954 53.50 1.3514 68.50 2.5386 83.50 8.7769 8.75 •1539 23.75 •4400 38.75 ⚫8026 53.75 1.3638 68.75 2.5715 83.75 9.1309 9.00 •1584 24.00 ⚫4452 39.00 ⚫8098 54.00 1.3764 69.00 2.6051 84.00 9.5144 9.25 • 1629 24.25 *4505 39.25 •8170 54.25 1.3891 69.25 2.6395 84.25 9.9310 9.50 •1673 24.50 *4557 39.50 ⚫8243 54.50 1.4019 69.50 2.6746 84.50 10.3854 9.75 •1718 24.75 •4610 39.75 ⚫8317 54.75 1.4150 69.75 2.7100 84.75 10.8829 10.00 • 1763 25.00 .4663 40.00 • 8391 55.00 1.4281 70.00 2.7475 85.00 11.4301 10.25 •1808 25.25 •4716 40.25 ⚫8466 55.25 1.4415 70.25 2.7852 85.25 12.0346 10.50 •1853 25.50 ⚫4770 40.50 •8541 55.50 1.4551 70.50 2.8239 85.50 12.7062 10.75 • 1899 25.75 ⚫4823 40.75 • 8617 55.75 1.4687 70.75 2.8636 85.75 13.4566 11.00 •1944 26.00 •4877 41.00 ⚫8693 56.00 1.4826 71.00 2.9042 86.00 14.3007 11.25 • 1989 26.25 •4931 41.25 .8770 56.25 1.4966 71.25 2.9460 86.25 15.2571 11.50 •2035' 26.50 ⚫4986 41.50 ⚫8847 56.50 1.5108 71.50 2.9887 86.50 16.3499 11.75 • 2080 26.75 ⚫5040 41.75 8925 56.75 1.5253 71.75 3.0326 86.75 17.6106 12.00 •2126 27.00 ⚫5095 42.00 9004 57.00 1.5399 72.00 3.0777 87.00 19.0811 12.25 •2171 27.25 •5150 42.25 .9083 57.25 1.5547 72.25 3.1240 87.25 20.8188 12.50 •2217 27.50 ⚫5206 42.50 .9163 57.50 1.5697 72.50 3.1716 87.50 22.9038 12.75 • 2263 27.75 ⚫5261 42.75 ⚫9243 57.75 1.5849 72.75 3.2205 87.75 25.4517 13.00 •2309 28.00 • 5317 43.00 .9325 58.00 1.6003 73.00 3.2709 88.00 28.6363 13.25 ⚫2355 28.25 ⚫5373 43.25 •9407 58.25 1.6160 73.25 3.3226 88.25 32.7303 13.50 •2401 28.50 ⚫5430 43.50 ⚫9490 58.50 1.6319 73.50 3.3759 88.50 38.1885 13.75 ⚫2447 28.75 • 5486 43.75 .9573 58.75 1.6479 73.75 3.4308 88.75 45.8294 14.00 ⚫2493 29.00 ⚫5543 44.00 .9657 59.00 1.6643 74.00 3.4874 89.00 57.2900 14.25 • 2540 29.25 ⚫5600 44.25 ⚫9742 59.25 1.6808 74.25 3.5457 89.25 76.3900 14.50 •2586 29.50 *5658 44.50 ⚫9827 59.50 1.6977 74.50 3.6059 89.50 114.5887 14.75 ⚫2633 29.75 •5715 44.75 ⚫9913 59.75 1.7147 74.75 3.6680 89.75 229.1817 15.00 •2679 30.00 •5774 45.00 1.0000 60.00 1.7321 75.00 3.7321 90.00 ∞ 15.25 ⚫2726 30.25 ⚫5832 45.25 1.0088 60.25 1.7450 75.25 3.7983 15.50 •2773 30.50 • 5890 45.50 1.0176 60.50 1.7675 75.50 3.8667 15.75 ⚫2820 30.75 ⚫5949 45.75 1.0265 60.75 1.7856 75.75 3.9375 528 HANDBOOK OF ELECTRICAL TESTING. TABLE II.*-RESISTANCE OF A KNOT-POUND of COPPER WIRE of various CONDUCTIVITIES, at 75° FAHR. Percentage ductivity. Percentage of Con- Resistance. of Con- Resistance. ductivity. Percentage Percentage of Con- Resistance. ductivity. of Con- Resistance. ductivity. 99.7 99.6 1200.3 1201.5 100.0 1196.7 97.5 1227.4 99.9 1197.9 97.4 1228.6 99.8 1199.1 97.3 1229.9 97.2 1231.2 97.1 1232.5 95.0 1259.7 92.5 1293.4 94.9 1261.0 92.4 1294.8 94.8 1262.4 92.3 1296.1 94.7 1263.7 94.6 1264.0 92.2 1297.4 92.1 1298.8 99.5 1202.7 97.0 1233.7 94.5 1266-4 92.0 1300-1 99.4 1203.9 96.9 99.3 1205.1 96.8 1236.2 1235.0 94.4 1267.7 91.9 1301.6 94.3 1269.1 91.8 1303.1 99.2 1206.4 96.7 1237.5 94.2 1270.4 91.7 1304.6 99.1 1207.6 96.6 1238.8 94.1 1271.8 91.6 1306.1 99.0 1208.8 96.5 1240.1 94.0 1273.1 91.5 1307.6 98.9 1210.0 96.4 1241.4 93.9 1274.5 91.4 1309.1 98.8 1211.2 96.3 1242.7 93.8 1275.8 91.3 1310.6 98.7 1212.5 96.2 1244.0 93.7 1277·2 91.2 1312.1 98.6 1213.7 96.1 1245.3 93.6 1278.6 91.1 1314.6 98.5 1214.9 96.0 1246.6 93.5 1280.0 91.0 1315.1 98.4 1216.2 95.9 1247.9 93.4 1281.3 90.9 1316.5 98.3 1217.3 95.8 1249.2 93.3 1282.7 90.8 1318.0 98.2 98.1 1219.9 95.6 1218.6 95.7 1250.5 93.2 1284.0 90.7 1319.4 1251.8 93.1 1285.4 90.6 1320.9 98.0 97.9 1222.4 95.4 1221.1 95.5 1253.1 93.0 1286.8 90.5 1322.4 1254.4 92.9 1288.1 90.4 1323.8 97.8 1223.6 95.3 1255.7 92.8 1289.4 90.3 1325.3 97.7 1224.9 95.2 1257.0 92.7 1290.8 90.2 1326.8 1328.2 97.6 1226.1 95.1 1258.4 92.6 1292.1 90.1 Resistance of "statute-mile-pound" equals resistance of "knot-pound” multiplied by ⚫752422. log ⚫752422 = 1.8764614. * See page 409, § 478. TABLE III.*—SHOWING THE RELATIVE DIMENSIONS, LENGTHS, RESISTANCES (AT 60° FAHR.), AND WEIGHTS OF PURE COPPER WIRE. LENGTH AND WEIGHT. LENGTH AND RESISTANCE. DIAMETER. AREA. RESISTANCE AND WEIGHT. B.W.G. B.W.G. No. Mils. Milli- metres. Square Inches. Square Millimetres. Pounds per Foot. Pounds per Yard. Pounds per Mile. Feet Yards Miles per Pound. per Pound. per Pound. Feet per Ohm. Yards per Ohm. Miles per Ohm. Ohms per Foot. Ohms per Yard. Ohms per Mile. Ohms per Pound. Pounds per Ohm. No. 0000 454 II 53 • 1619 IO˚44 •6239 1.872 3294 1.603 *5343 000 425 10.80 •1419 9.152 .5468 1.640 2887 1.829 • 6097 *0003036 19966 6656 ⚫0003464 17497 5832 3.782 3.314 00005008 •00005715 ·0001715 0001503 •2644 • 3018 OO 380 9.652 *I134 7°317 *4371 I'311 2308 2.288 • 7626 *0004333 13988 4663 2.649 00007149 *0002145 *3775 *00008027 12460 •0001046 9567 *0001636 6114 Oooo 000 340 8.636 ⚫09079 5.857 ⚫3499 1.050 1848 2.858 * 9526 ⚫0005412 11198 3733 2.121 ⚫00008930 ⚫0002679 •4715 ⚫0002552 3919 0 I 300 7.620 *07069 4.560 •2724 •8173 1438 3.671 • 1224 *0006952 8718 2905 1.651 * 0001147 *0003441 •6056 * 0004210 2375 I 2 284 7.213 ⚫06335 4.087 •2442 ·7325 1289 4.096 1.365 ⚫0007757 7814 2605 1.480 ⚫0001280 ⚫0003840 '6758 ⚫0005242 1908 2 3 250 6°578 *05269 3°399 * 2031 •6092 1072 4.925 I'642 *0009327 6498 2166 I*231 * 0001539 *0004617 •8125 *00075 79 1320 4 238 6.045 ⚫04449 2.870 •1715 ⚫5144 905.3 5.832 1.944 ⚫001105 5487 1829 1.039 ⚫0001822 ⚫0005467 ⚫9623 ⚫001063 940.8 34 5 220 5.588 *03801 2.452 • 1465 *4395 773.6 6.826 2°275 *001293 4689 1563 •8880 *0002133 *0006 399 I'126 *001456 686·9 6 203 5.156 ⚫03237 2.088 •1247 ⚫3742 658.6 8.017 2.672 ·001518 3992 1331 ⚫7560 ·0002506 ⚫0007515 1.323 ⚫002008 498-0 5 5 6 7 180 4'572 '02545 I'642 *09808 • 2942 517.8 IO' 20 3*399 *001931 3139 1046 *5944 *0003186 •0009558 i.682 *003249 307·8 8 165 4.191 • 02138 1.379 • 08241 •2472 435.1 12.13 4.045 ⚫002298 2637 879.1 ⚫4995 ⚫0003792 ⚫001138 2.002 ⚫004601 217.3 78 9 148 3.759 *01720 I'IIO •06631 • 1989 350'I 15 08 5.027 *002856 2122 707 3 *4019 ⚫0004713 *001414 2.488 *007108 140*7 9 10 134 3.404 ⚫01410 •9098 ⚫05435 • 1631 287.0 18.40 6.133 ⚫003485 1739 579.8 ⚫3294 ·0005749 ⚫001725 3.036 ⚫01058 94.54 10 II 120 3.048 '01131 7296 *04359 *1308 230°2 22.94 7.647 ⚫004345 1394 465.0 • 2642 *0007169 *002151 3.785 *01645 60.80 II 12 109 2.770 ⚫009331 • 6020 ⚫03596 • 1079 189.9 27.81 9.268 ⚫005266 1151 383.6 •2180 ⚫0008689 ⚫002607 4.588 ⚫02416 41.39 12 13 95.0 2.413 • 007088 • 4573 *02732 *08196 144 2 36.60 12. 20 •006933 874.3 291'4 * 1656 *001144 '003431 6°039 *04187 23.88 13 14 83.0 2.108 ⚫005411 ⚫3491 ⚫02085 ⚫06256 110.1 47.95 15.98 ⚫009082 667.3 222.4 •1264 ⚫001498 ⚫004495 7.912 ⚫07186 13.92 14 15 72.0 1.829 *004072 • 2486 *01569 *04708 82.86 63°73 21°24 *01207 502 2 167'4 *095 II *001991 *005974 10'51 • 1268 7.887 15 16 65.0 1.651 ⚫003318 •2141 ⚫01279 ⚫03837 67.53 78.19 26.06 ⚫01481 409.3 136.4 ⚫07751 ⚫002443 ⚫007330 12.90 •1910 5.234. 16 17 58.0 1473 ⚫002642 • 1705 •OI018 *03055 53 77 98.20 32 73 *01859 325°9 108.6 *06172 *003069 *009200 16.20 • 3014 3.318 17 18 49.0 1.245 ⚫001886 • 1217 ⚫007268 ⚫02180 38.37 137.6 45.86 ⚫02606 232.6 77.53 ⚫04405 ⚫004300 ⚫01290 22.70 ⚫5916 1.690 18 19 42.0 1.067 •001385 *08940 *005340 ⚫01602 28.19 187.3 62°43 *03547 170°9 56.96 *03236 *005852 *01756 30°90 I'096 *9124 19 20 35.0 21 32.0 ⚫8890 .8128 ⚫0009621 ⚫0008043 ·06207 003708 *05188 • • 01112 19.58 269.7 89.89 ⚫05108 149.4 49.80 ⚫02248 ⚫008427 ⚫02528 44.49 2.273 • 4400 20 *003100 *009299 16.37 322.6 107.6 ⚫06110 99.20 33°07 • 01879 • 01008 *03024 53°23 3.252 • 3075 21 22 28.0 • · 7112 0006158 23 24 25.0 6350 0004909 22.0 ⚫5588 ⚫0003801 03167 ⚫02452 ⚫03972 002373 001892 ·001465 • ⚫007119 *005676 ⚫004395 12.53 421.4 140.5 ⚫07981 75.95 25.32 ⚫01438 ⚫01317 ⚫03950 69.52 5.548 •1802 22 9'989 528.6 176.2 ΙΟΟΙ 60°54 20 18 *OI147 •01652 *04955 87.21 8.730 *1145 23 7.736' 682.6 227.5 •1293 46.89 15.63 ⚫008880 ⚫02133 ⚫06399 112.6 14.56 ⚫06869 24 25 20.0 •5080 0003142 *02027 • 001211 *003632 6.393' 825.9 275°3 * 1564 38.75 12.92 *007339 *02581 *07742 136°3 21.31 •04692 25 26 18.0 •4571 ⚫0002545 ⚫01642 ⚫0009808 ⚫002942 5.178 1020 339.9 • 1931 31.39 10.16 ·005944 ⚫03186 ⚫09558 168.3 32.49 ⚫03078 26 27 16.0 • 4064 '000201 I *01297 28 14.0 ⚫3556 ⚫0001539 29 13.0 *3302 *0001327 30 12.0 •3048 ·0001131 ⚫00993 c0856 ⚫007296 0004359 *0007749 *002325 ⚫0005933 0005 116 4.092 1290 430'I · 2444 24.80 8.266 • 004697 *04032 •1210 212.9 52.04 *01922 27 ⚫001780 3.133 1685 561.8 ⚫3192 18.99 6.329 ⚫003596 ⚫05267 •1580 278.1 88.77 ⚫01127 28 • 001535 2.701 1955 651.6 *3702 16.37 5°457 *003101 •06108 •1833 322.5 119.4 •008375 29 • ⚫001308 2.302 2294 764.7 •4345 13.95 4.650 ⚫002642 ⚫07169 •2151 378.5 164.5 • 006080 30 PURE COPPER weighs 555 lb. per cubic foot. The resistance of one mil-foot at 60° Fahr. is, according to Dr. Matthiessen, 10.323 ohms. Upon these data the above Table has been calculated. The resistance of Copper varies with the temperature at about 0.38 per cent. per degree Centigrade, or 0.21 per cent. per degree Fahrenheit. Stranded WIRES.-With a conductor of stranded wires of a definite length the total weight is greater, and the resistance is less, than with a similar number and equal length of wires not stranded. * This Table is abbreviated from one compiled by Messrs. W. T. Glover, wire makers, of Manchester, and is inserted by permission. Inches to millimetres, multiply by 25 3994. Feet to metres, ⚫3048. "" To convert Yards to metres, ⚫9144. Miles to kilometres, ⚫6214. "J Pounds to kilogrammes ⚫45359. " [To face p. 528. 4 TABLES. 529 TABLE IV.*—COEFFICIENTS for correcting the OBSERVED RESISTANCE of PURE COPPER WIRE at any TEMPERATURE to 75° FAHR., or at 75° to any TEMPERATURE. Tempera- ture in Degrees Coefficient. Fahr. Tempera- ture in Degrees Fahr. Coefficient. Tempera- ture in Degrees Coefficient. Tempera- ture in Degrees Coefficient. Fahr. Fahr. 100 •9484 82.5 ⚫9842 65 1.0214 47.5 1.0601 99.5 .9494 82 ⚫9853 64.5 1.0225 47 1.0612 99 •9504 81.5 •9863 64 1.0236 46.5 1.0623 98.5 ⚫9514 81 •9874 63.5 1.0247 46 1.0634 98 ⚫9524 80.5 • 9884 63 1.0258 45.5 1.0646 97.5 ⚫9534 80 ⚫9895 62.5 1.0269 45 1.0657 97 ⚫9544 79.5 ⚫9905 62 1.0280 44.5 1.0668 96.5 ⚫9554 79 ⚫9916 61.5 1.0290 44 1.0679 96 ⚫9564 78.5 ⚫9926 61 1.0301 43.5 1.0690 95.5 ⚫9575 78 • 9937 60.5 1.0312 43 1.0702 95 ⚫9585 77.5 ⚫9947 60 1.0323 42.5 1.0714 94.5 ⚫9595 77 •9958 59.5 1.0334 42 1.0725 94 •9605 76.5 ⚫9968 59 1.0345 41.5 1.0736 93.5 ⚫9615 76 ⚫9979 58.5 1.0356 41 1.0748 93 ⚫9626 75.5 ⚫9990 58 1.0367 40.5 1.0759 92.5 ⚫9636 75 1.0000 57.5 1.0378 40 1.0771 92 ⚫9616 74.5 1.0011 57 1.0389 39.5 1.0782 91.5 ⚫9656 74 1.0021 56.5 1.0400 39 1.0793 91 ⚫9666 73.5 1.0032 56 1.0411 38.5 1.0804 90.5 •9677 73 1.0042 55.5 1·0422 38 1.0816 90 .9687 72.5 1.0053 55 1.0433 37.5 1.0828 89.5 ⚫9697 72 1·0064 54.5 1.0444 37 1.0839 89 ⚫9708 71.5 1.0074 54 1.0455 36.5 1.0851 88.5 •9718 71 1.0085 88 ⚫9728 70.5 1.0096 87.5 ⚫9738 70 1.0106 87 ⚫9749 69.5 1.0117 86.5 ⚫9759 69 1.0128 53.5 1.0466 53 1.0478 52.5 1.0489 52 1.0500 51.5 1.0511 36 1.0862 35.5 1.0873 35 1.0885 34.5 1.0896 34 1.0908 86 •9769 85.5 •9780 85 ⚫9790 68.5 1.0139 68 1.0149 67.5 1.0160 51 1.0522 33.5 1.0920 50.5 1.0533 33 1.0932 50 1.0544 32.5 1.0943 84.5 •9801 67 1.0171 49.5 1.0556 32 1.0955 84 •9811 66.5 1.0182 49 1.0567 31.5 1.0966 83.5 ⚫9821 66 1.0193 48.5 1.0578 31 1.0978 83 ⚫9832 65.5 1.0204 48 1.0589 30.5 1.0990 * See page 416. 2 M 530 HANDBOOK OF ELECTRICAL TESTING. DBC $ TABLE V.*—COEFFICIENTS for correcting the OBSERVED RESISTANCE of ORDINARY COPPER WIRE at any TEMPERATURE to 75°, or at 75° to any TEMPERATURE. Temp. Co- Fahr. efficient. Logarithm. Temp. Co- Fahr. efficient. Logarithm. Temp. Co- Fahr. efficient. Logarithm. 100 99.5 •9501 99 9491 1.9772950 •9777491 ⚫9510 ⚫9782032 77 ·9958 1-9981836 54 1.045 0.0190722 76.5 • .9969 9986377 53.5 1.046 ⚫0195263 76 • ⚫9979 9990918 53 1.047 ⚫0199804 98.5 ⚫9520 ⚫9786573 75.5 98 ⚫9530 -9791114 97-5 •9540 •9795655 97 ⚫9550 •9800196 96.5 •9560 •9804737 74 73.5 1.003 •9990 9995159 75 1.000 0.0000000 74.5 1.001 ⚫0004541 1.002 ⚫0009082 ⚫0013623 52.5 1.048 ⚫0204345 52 1.049 ⚫0208886 51.5 1.050 ⚫0213427 51 1.051 ⚫0217968 50.5 1.053 ⚫0222509 96 •9570 •9809278 95.5 ⚫9580 73 •9813819 72.5 1.005 1.004 ·0018164 50 1.054 ⚫0227050 ⚫0022705 49.5 1.055 ⚫0231591 · 95 ⚫9590 •9818360 72 1.006 ⚫0027246 49 1.056 ·0236132 94.5 ⚫9600 ⚫9822901 71.5 1.007 ⚫0031787 48.5 1.057 ⚫0240673 94 •9610 ⚫9827442 93.5 •9621 •9831983 71 1.008 70.5 1.009 ⚫0040869 ⚫0036328 48 1.058 ⚫0245214 47.5 1.059 ⚫0249755 93 ⚫9631 ⚫9836524 70 1.010 92.5 ⚫9641 •9841065 69.5 1.012 ⚫0045410 ⚫0049951 47 1.060 ⚫0254296 46.5 1.061 ⚫0258837 92 •9651 •9845606 69 1.013 ⚫0054492 46 1.062 ⚫0263378 91.5 ⚫9661 •9850147 68.5 1.014 ⚫0059033 45.5 1.064 ⚫0267919 91 •9671 •9854688 68 1.015 ⚫0063574 45 1.065 ⚫0227460 90.5 ⚫9681 •9859229 90 •9691 •9863770 89.5 •9701 •9868311 ∞ ∞ ∞o 89 ⚫9711 ⚫9872852 66 67.5 1.016 67 1.017 66.5 1.018 1.019 ⚫0068115 44.5 1.066 ⚫0277001 ⚫0072656 44 1.067 ⚫0281542 ⚫0077197 43.5 1.068 ⚫0286083 ⚫0081738 43 1.069 • 0290624 88.5 -9722 ⚫9877393 65.5 1.020 ⚫0086279 42.5 1.070 88 ⚫9732 •9881934 65 1·021 0090820 42 1.071 87.5 ⚫9742 •9886475 64.5 1.022 ⚫0095361 41.5 1.072 • ·0295165 ⚫0299706 0304247 87 .9752 *9891016 64 1·023 • 0099902 41 1.074 ⚫0308788 86.5 ⚫9762 •9895557 63.5 1.024 • 0104443 40.5 1.075 • 0313329 86 ⚫9772 ⚫9900098 63 1.025 85.5 ⚫9783 ⚫9904639 62.5 1.026 ⚫0108984 ⚫0113525 40 1.076 ⚫0317870 39.5 1.077 ⚫0322411 85 ⚫9793 .9909180 62 1.027 •0118066 39 1.078 ⚫0326952 84.5 ⚫9803 •9913721 61.5 1·029 • 0122607 38.5 1.079 ⚫0331493 84 •9814 .9918262 61 1.030 • 0127148 38 1.080 ⚫0336034 83.5 •9824 •9922803 60.5 1.031 ⚫0131689 37.5 1.082 ⚫0340575 83 ⚫9834 ⚫9927344 60 1.032 ⚫0136230 37 1.083 ⚫0345116 82.5 ⚫9844 •9931885 59.5 1.033 ⚫0140771 36.5 1.084 ⚫0349657 82 .9855 •9936426 59 1.034 ⚫0145312 36 1.085 ⚫0354198 81.5 .9865 ⚫9940967 58.5 1.035 ⚫0149853 35.5 1.086 ⚫0358739 81 .9875 ⚫9945508 58 1.036 ⚫0154394 35 1.087 ⚫0363280, 80.5 •9886 ⚫9950049 57.5 1.037 ⚫0158935 34.5 1.088 ⚫0367821 80 ⚫9896 ⚫9954590 57 1.038 • 0163476 34 1.089 ⚫0372362 79.5 ⚫9906 •9959131 56.5 1.039 ⚫0168017 33.5 1.091 ⚫0376903 79 .9917 .9963672 56 1.041 •0172558 33 1.092 • 0381444 78.5 78 ⚫9927 .9968213 ⚫9937 •9972754 77.5 ⚫9948 ⚫9977295 55.5 1.042 •0177099 32.5 1.093 ⚫0385985 55 54.5 1.043 1.044 • 0181640 32 1.094 ⚫0390526 • 0186181 31.5 1.095 · 0395067 * See page 419. TABLES. 531 TABLE VI.*—COEFFICIENTS for correcting the OBSERVED RESISTANCE of “SILVER- TOWN " GUTTA-PERCHA at any TEMPERATURE to 75° Fahr. Co- Temp. efficient. Logarithm. Temp. Co- efficient. Logarithm. Temp. Co- efficient. Logarithm. 100 •1494 1.1744650 99.5 •1552 • 1909757 77 76.5 • 8589 1·9339572 -89229504679 54 4.937 0.6934494 53.5 5.128 7099601 99 • 1612 •2074864 76 • •9267 9669786 53 • 5.327 7264708 98.5 •1675 • 2239971 75.5 • •9627 9834893 52.5 O 98 •1740 •2405078 75 1.000 0.0000000 52 5.533 7429815 5.748 7594922 97.5 • 1807 •2570185 71.5 1.039 ⚫0165107 51.5 • 5.970 776002!) 97 •1877 • 2735292 74 1.079 ⚫0330214 51 • 6.202 7925136 96.5 •1950 •2900399 73.5 1.121 ⚫0495321 50.5 • 6.442 8090243 96 • 2026 ⚫3065506 73 1.164 • 0660428 50 6.692 .8255350 95.5 •2104 ⚫3230613 72.5 1.209 ·0825535 49.5 6.951 8420457 • 95 .2186 ⚫3395720 72 1.256 ⚫0990642 49 7.220 .8585564 94.5 • 2270 ⚫3560827 71.5 1.305 •1155749 48.5 • 7.500 8750671 94 •2358 • 3725934 71 1.355 • 1320856 48 • 7.791 8915778 93.5 •2450 •3891041 93 •2545 •4056148 70 70.5 1.408 1.463 •1485963 47.5 • • 1651070 47 8.093 9080885 8.406 9245992 92.5 •2643 •4221255 69.5 1.519 •1816177 46.5 • 8.732 9411099 92 •2746 •4386362 69 1.578 •1981284 46 • 9.070 9576206 91.5 •2852 •4551469 68.5 1.639 •2146391 45.5 • 9.422 9741313 91 •2962 •4716576 90.5 • 3077 •4881683 90 ⚫3197 • 5046790 67 68 1.703 67.5 1.769 1.837 • 2311498 45 • 9-787 9906420 •2476605 44.5 10.17 1.0071527 •2641712 44 10.56 ⚫0236634 89.5 • 3320 • 5211897 66.5 1.908 •2806819 43.5 10.97 •0401741 89 ⚫3449 •5277004 66 1.982 •2971926 43 11.39 0566848 88.5 ⚫3583 • 5542111 65.5 2.059 • 3137033 42.5 11.84 ⚫0731955 88 • 3722 • 5707218 65 2.139 • • 3302140 42 12.29 • 0897062 87.5 • 3866 • 5872325 64.5 2.222 •3467247 41.5 12.77 87 • 4016 ⚫6037432 64 2.308 ⚫3632354 41 13.27 •1062169 •1227276 86.5 •4171 ⚫6202539 63.5 2.397 • 3797461 40.5 13.78 • 1392383 86 4343 · 6367646 63 2.490 ⚫3962568 40 14.31 •1557490 85.5 •4501 ⚫6532753 62.5 2.587 •4127675 39.5 14.87 •1722597 85 .4675 • 6697860 62 2.687 • 4292782 39 15.44 •1887704 84.5 .4856 ⚫6862967 61.5 2.792 •4457889 38.5 16.04 •2052811 84 •5044 ⚫7028074 61 2.899 •4622996 38 16.66 • 2217918 83.5 • 5240 •7193181 60.5 3.012 •4788103 37.5 17.31 ⚫2383025 83 ⚫5443 ⚫7358288 60 3:128 •4953210 37 17.98 ⚫2548132 82.5 •5654 ⚫7523395 59.5 3.250 .5118317 36.5 18.68 ⚫2713239 82 ⚫5873 • 7688502 59 3.376 • 5283424 36 19.40 •2878346 81.5 ·6100 ⚫7853609 58.5 3.506 • 5448531 35.5 20.15 ⚫3043453 81 ⚫6337 • 8018716 58 3.642 .5613638 35 20.93 ⚫3208560 80.5 •6582 .8183823 57.5 3.783 •5778745 34.5 21.74 ⚫3373667 80 • 6837 .8348930 57 3.930 ⚫5943852 34 22.59 ⚫3538774 79.5 .7102 ⚫8514037 56.5 4.082 •6108959 33.5 23.46 ⚫3703881 79 .7378 • 8679144 56 4.240 •6274066 78.5 .7663 • 8844251 55.5 4.405 • 6439173 33 32.5 25.32 24.37 ⚫3868938 78 .7960 ⚫9009358 55 4.575 ⚫6604280 32 ⚫4034095 26.30 4199202 77.5 · 8296 •9174465 54.5 4.753 6769387 • 31.5 27.32 •4364309 * See page 419. 2 M 2 + 7 532 噔 ​HANDBOOK OF ELECTRICAL TESTING. TABLE VII.*-COEFFICIENTS for correcting the OBSERVED RESISTANCE OF "WILLOUGHBY SMITH'S" GUTTA-PERCHA at any TEMPERATURE to 75° FAHR. Temp. Co- efficient. Logarithm. Temp. Co- efficient. Logarithm. Temp. Co- efficient. Logarithm. 100 •1992 1.2992893 77 •8789 1.9439395 54 5.083 0.7061201 99.5 •2057 99 ⚫3132343 •2125 ⚫3273589 76.5 • 76 • •9077 9579423 .9375 9719713 53.5 5.284 ⚫7229628 53 5.492 ⚫7397305 98.5 + 98 97.5 97 •2194 •3412366 • 2266 •3552599 ⚫2340 •3692159 •2417 •3832767 74 1.080 75.5 • •9682 9859651 52.5 5.709 •7565600 75 1.000 0.0000000 52 5.934 ⚫7733475 74.5 1.039 ⚫0166155 51.5 6.168 ⚫7901444 ⚫0334238 51 6.412 ⚫8069935 96.5 • 2497 .3974185 73.5 1.123 ⚫0503798 50.5 6.665 ⚫8238002 96 •2579 •4114513 73 1.167 ⚫0670709 50 6.928 ⚫8406079 95.5 • 2667 ⚫4260230 95 • 2751 •4394906 94.5 •2841 •4534712 72.5 1.213 72 1.261 71.5 1.296 ⚫0838608 49.5 7.201 • 8573928 • 1007151 49 7.485 • · 8741918 •1126050 48.5 7.781 ⚫8910354 94 •2934 •4674601 71 1.363 •1344959 48 8.088 •9078411 93.5 ⚫3030 •4814426 70.5 1.417 ⚫1513699 47.5 8.407 •9246410 93 •3130 •4955443 70 1.473 ⚫1682027 47 8.739 .9414617 92.2 92 91.5 • 3232 •5094714 69.5 1.531 •1849752 46.5 9.084 •9582771 • 3338 •5234863 69 1.591 • 2016702 46 9.442 •9750640 ⚫3448 •5375673 68.5 1.654 •2185355 45.5 9.815 ⚫9918903 91 •3561 •5515720 68 1.719 •2352759 45 10.203 1.0087279 90.5 ⚫3678 • 5656117 67.5 1.787 •2521246 44.5 10.606 • 0255516 90 *3798 •5795550 67 1.858 •2690457 44 11.024 ⚫0423392 89.5 ⚫3923 •5936183 66.5 1.931 • 43.5 11 460 89 •4051 •6075622 66 2.007 43 88.5 •4184 • 6215917 65.5 2.086 88 87.5 87 • • 4321 6355843 65 2.169 ⚫4463 ⚫6496269 64.5 2.254 •4609 ⚫6636067 64 2.343 41 86.5 •4761 .6776982 63.5 2.436 ⚫3866773 • 2857823 • 3025474 •3193143 ⚫3362596 · 3529539 ⚫3697723 11.911 42.5 12.382 42 12.870 41.5 13.378 13.906 40.5 14.455 • 1432022 • ·0591846 ⚫0759842 ⚫0927908 •1095785 • 1263912 •1600181 86 • 4917 .6917002 63 2.532 •4034637 40 15.025 •1768145 85.5 85 84.5 •5417 •5078 ⚫7056927 •5245 ⚫7197455 ·7337588 62.5 2.632 ·4202859 39.5 15.618 •1936254 84 •5594 •7477225 83.5 .5778 .7617775 83 •59677757560 60 82.5 .6163 ⚫7897922 82 ⚫6365 ⚫8037984 81.5 • 6574 ⚫8178297 62 61.5 2.844 61 2.956 60.5 3.073 3.194 59.5 3.320 59 3.451 58.5 3.587 2.736 •4371161 39 16.235 •2104523 ⚫4539296 •4707044 38 ⚫4875626 •5043349 37 • 5211381 • 5379450 36 81 ⚫6789 •8318058 58 3.729 80.5 •7012 .8458419 57.5 3.876 •5547314 •5715924 .5883838 80 •72278589585 57 4.029 ⚫6051973 79.5 ⚫7480 .8739016 56.5 4.188 • 6220067 79 ⚫7725 .8878985 56 4.354 • 78.5 ⚫7978 •9018940 55.5 4.526 78 ⚫8240 •9159272 55 4.704 ⚫6724673 77.5 • 8510 •9299296 54.5 4.890 ⚫6893089 • 6388884 •6557145 38.5 16.876 17.542 37.5 18.235 18.954 36.5 19.702 20.480 35.5 21.288 35 22.128 34.5 23.002 34 23.910 33.5 24.853 33 25.834 32.5 26.854 32 27.913 31.5 29.014 •2945103 •3113300 • 3271349 • 3449422 ⚫3617656 · 3785796 ⚫3953788 •4121917 • 4290090 •4458065 •4626076 • 2272695 •2440791 •2609058 • 2777009 * See page 419. TABLES. 533 TABLE VIII.*-Of the MULTIPLYING POWER of SHUNTS EMPLOYED With a GALVANOMETER of 6000 OHMS RESISTANCE. Com- bined Resist- Logarithm Resist- ance of of ance of Shunt. Multiplying Galva- Resist- ance of Shunt. Logarithm of Multiplying Combined Resist- ance of Galva- nometer ance of Shunt. Power. nometer Power. and Resist- Logarithm Resistance of Multiply- ing Power. of Galva- nometer and Shunt. Combined and Shunt. Shunt. ohms. ohms. ohms. ohms. obms. ohms. 12345 74.1 9508642618 818.3 79.0 83.8 • 1000 8450980 857.2 • 88.7 3.7782236 1.0 75 1.9084850 3.4772660 2.0 80 1.8808136 3.3012471 3.0 85 1.8548402 3.1763807 4.0 90 1.8303769 3.0795430 5.0 95 1.8072508 93.5 3.0004341 100 1.7853298 6.0 98.4 2.9335581 7.0 110 1.7446450 108.0 2.8756399 8.0 120 1.7075702 117.7 2.8245619 9.0 130 1.6735185 127.2 10 2.7788745 10.0 140 1.6420488 136.8 11 2.7375504 11.0 150 1.6127839 146.3 6 7 8 9 12345 1100 • 8098626 929.6 1200.7781513 1000.0 1300·7493807 1068.5 1400 7231107 1135.7 • • • 1500 6989700 1200.0 16006766936 1263 2 1700 6560407 1324 7 1800 6368188 1384.6 19006188636 1443.1 20006020600 1500.0 2200.5713943 1609.7 24005440680 1714·3 2600-5195201| 1814·0 28004973306 1909-1 30004771213 2000·0 33004499718 2129.0 36004259742 2250·0 4000 3979400 2400.0 43003793780 2504 8 46003625579 2603-7 50003424227 2727.3 5500 3182929 2883.1 6000 3010300] 3000.0 65002840019 3120.0 70002688353 3230 8 75002552725 3333·3 80002430380 3428-6 85002319536 3517.2 90002218574 3600·0 95002126137 3677.4 • • 2.6998377 12.0 160 1.5854607 155.9 2.6651493 13.0 170 1.5598348 165.3 2.6320441 14.0 180 1.5357118 174.8 2.6031444 15.0 190 1.5129244 184.2 16 2.5751878 16.0 200 1.4913617 193.6 17 2.5489296 17.0 220 1.4513719 212.2 18 2.5241753 17.9 240 1.4149733 230.8 19 2.5007578 18.9 260 1.3816024 249.2 20 2.4785665 19.9 280 1.3508099 267.5 22 12.4373224 21.9 300 1.3222193 285.7 24 12.3996737 23.9 330 1.2828939 312-8 26 2.3650572 25.9 360 1.2461628 340.4 28 2.3330239 27.9 400 1.2041200 375.0 30 2.3031961 29.9 430 1.1747574 401.2 33 2.2620237 32.8 460 1.1461280 428.6 36 2-2244554 35.8 500 1.1139434 461.5 40 2.1789769 39.7 550 1.0722867 508.0 43 2-1477999 42.7 600 1.0413927 545·5 2.1187276 45.6 650 1.0131744 582.1 2.0827854 49.8 700 2.0378646 54.5 750 2·0043214 59.5 800 •9294189 705.9 100002041200 3750.0 1.9699189 64.3 850 •9062704 744-5105001962946 3818.2 ·8846085 782.6 11000 1890562 3882.3 46 50 55 60 65 70 1.9380892 69.2 900 85062 •9809755 617.6 ·9542425 666·7 • • • * See page 375 (§ 424). 534 HANDBOOK OF ELECTRICAL TESTING. I TABLE IX.*-Of the MULTIPLYING POWER of SHUNTS EMPLOYED with a GALVANOMETER of 10,000 OHMS RESISTANCE. Com- bined Resist- Logarithm Resist- Resist- Logarithm ance of of ance of ance Shunt Multiplying Galva- of of Multiplying Power. nometer Shunt. Power. Combined Resist- ance of Galva- Resist Logarithm Combined Resistance ance of of Galva- nometer of Multiplying nometer Shunt. Power. and and Shunt. and Shunt. Shunt. ohms. ohms. ohms. ohms. ohms. ohms. 1 3.0000434 1.0 2 3.6990569 2.0 3 3.5230090 3.0 75 2.1281838 74.4 950 1.0616905 863.6 80 2.1003705 79.4 1000 1.0413927 900.9 85 2.0742570 84.3 1100 1.0039303 982.1 3.3981137 4.0 90 2.0196487 89.2 1200 •9700368 1061.7 5 3.3012471 5.0 95 2.0264827 94.1 1300 •9391350 1140·4 6 3.2221092 6.0 100 2.0043214 99.0 1400 •9107769 1228·1 7 3.1552059 7.0 110 1.9633585 108.8 1500 ⚫8846065 1304·4 8 13.0972573 8.0 120 1.9259993 118.6 1600 ·8603380 1379.3 9 3.0461482 9.0 130 1.8916660 128.3 1700 •8377370 1453.0 10 11 17 18 30 50 3.0004341 10.0 140 1.8599100 138.1 2.9590848 11.0 150 1.8303747 147·8 12 2.9213396 12.0 160 1.8027737 157.5 13 2.8866208 13.0 170 1.7768721 167.2 14 2.8544796 14.0 180 11.7524753 176.8 15 2.8245597 15.0 190 1.7294206 186.5 16 2.7965743 16.0 200 1.7075702 196.1 2.7702888 17.0 220 1.6670282 215.3 2.7455085 18.0 240 1.6300888 234.4 19 2.7220708 19.0 260 1.5961741 253.4 20 2.6998377 20.0 280 1.5648351 272.4 22 2.6585137 22.0 300 1.5357159 291.3 24 2.6208299 23.9 330 1.4955864 319.5 26 2.5861544 25.9 360 1.4590573 347.5 28 2.5540563 27.9 400 1.4149733 384.6 2.5241796 29.9 430 1.3848158 412.2 33 2.4829169 32.9 460 1.3567739 439·8 36 2.4452582 35.9 500 1.3222193 476.2 40 2.3996737 39.8 550 1.2828898 521.3 43 2.3683950 42.8 600 1.2471546 556.0 46 2.3392354 45.8 650 1.2144362 610.3 2.3031961 49.8 700 1.1842858 654.2 55 2.2620194 54.7 750 1.1563472 697.7 60 2-2244467 59.6 800 1.1303338 740 7 65 2.1899004 64.6 850 1.1060108 784 1 70 2.1579315 69.5 900 1.0831840 825 7 11000 1800 ⚫8166095 1525·4 1900 •7967934 1596.8 2000 •7781512 1666.7 2200 ·7439371 1803.3 2400 • 2600 2800 7132105 1935 5 •6853972 2063·5 •6600520 2187.5 3000 •6368221 2307·7 3300 •6053377 2481·2 3600 ⚫5772364 2647·1 4000 •5440680 2857·1 4300 •5218675 3007·0 4600 •5015951 3150.7 5000 ⚫4771213 3333.3 5500 •4499690 3548.4 6000 •4259687 3750.0 6500 •4045705 3939 4 • 7000 ⚫3853509 4117.6 7500 •3679767 4285.7 8000 • •3521825 4444 4 8500 • 3377528 4594.6 9000 • 3245111 4736.8 9500 • 3123110 4871.8 10000 10500 · 3010300 5000.0 •2905646 5122.0 •2808266 5238.1 * See page 375 (§ 424). TABLES. 535. TABLE X.-STANDARD WIRE GAUGE.† Diameters. No. ¡No. Mils.* Differences. Millimetres. Diameters. Mils.* Differences. Millimetres. 0,000,000 500 12.70 23 24 000,000 464 36 11.78 24 22 00,000 432 32 10.97 25 20 0,000 400 32 10.16 26 18 4 2 2 N •610 -559 -508 2 - 457 000 372 28 9.45 27 16.4 1.6 • 417 00 348 24 8.84 28 14.8 1.6 +376 0 324 I 21 8.23 29 13.6 1-2 ⚫345 1 300 24 7.62 30 12.4 1.2 ⚫315 2 276 21 7.01 31 11.6 .8 •295 3 252 21 6.40 32 10.8 .8 •274 4 232 20 5.89 33 10.0 +8 •254 5 212 20 5.38 34 9.2 .8 •234 6 192 1 20 4.88 35 8.4 .8 •213 7 176 16 4.47 36 7.6 .8 · 193 8 160 16 4.06 37 6.8 .8 •173 9 144 16. 3.66 38 6.0 .8 •152 10 128 16 3.25 3.9 5.2 ·8 •132 11 116 12 2.95 40 4.8 •4 · 122 12 104 12 2.64 41 4.4 ·4 • 112 13 92 12 2.34 422 4.0 .4 • 102 14 80 12 2.03 43 $3.6 ·4 ⚫0914 15 72 8 1.83 44 3.2 ·4 ⚫0813 16 64 8 1.63 45 2.8 ⚫4 •0711 17 56 8 1.42 46 2.4 •4 ⚫0610 18 48 ∞ 8 1.22 47 2.0 •4 ⚫0508 19 40 8 1.016 48 1.6 •4 ⚫0406 20 36 4 •914 49 1.2 •4 ⚫0305 21 32 4 813 50 1.0 .2 ·0254 22 28 4 •711 * 1 Mil. = roth of an inch. + This gauge is the only legal standard wire gauge for the United Kingdom. : 533 HANDBOOK OF ELECTRICAL TESTING. TABLE XI.-ELECTRICAL AND MECHANICAL DATA OF RECENT SUBMARINE CABLES. Weight per knot (dry). CABLE. Date laid. Length. Res. per knot of Con- ductor at 75° Fahr. Specific Conductivity of Conductor. Res. per knot of Di- electric at 75° Fahr. Inductive Capacity per knot at nor- mal Temperature. Copper. Gutta Percha. Jute. Iron. Hemp. Asphalte, etc. Total. Туре. BRAZILIAN SUBMARINE COMPANY:- knots. ohms. per cent. megs. micro- farads. lbs. lbs. tons. tons. tons. tons. tons. Lisbon-Madeira Section 19 Oct. 1882 626.27 9.828 101.11 643 ⚫308 120 175 .844 13.523] •221 14.5 Shore end. 5.719 ⚫797 6.9 Intermediate. •181 • .710 108 .597 1.7 Main. Madeira-St. Vincent Section .. 31 March, 1884 1168.22 9.323 98.39 487 ⚫337 130 130 1.367 13.543 · 154 2.818 •152 15.0 Shore end. •511 3.6 Intermediate. .725 ⚫108 ⚫394 1.5 Main. 15.6 Shore end. St. Vincent-Pernambuco Section AFRICAN DIRECT COMPANY :- Bathurst-Sierra Leone Section Sierra Leone-Accra Accra-Lagos-Brass Brass-Bonny WEST AFRICAN COMPANY :---- Sierra Leone-Conakry Section Grand Bassam-Accra Accra-Cutanu Cutanu-São Thomé São Thomé-Principe 8 July, 1884 1862 36 4.939 96.57 497 ⚫338 250 250 4.7 Intermediate. 1.9 Main. 12 July, 1886 23 Aug. 1 Sept. 9 Sept.,, 462.14 9.373 97.86 492 ⚫339 1019.44 9.375 97.84 496 ⚫340 527.96 9.356 98.04 504 ⚫341 68.27 9.396 97.62 513 ⚫359 1.367 13.543] 15.0 Shore end. 130 130 •240 5.719 ⚫797 7.0 Intermediate. • 154 2.818 •511 3.6 Main. 29 June, 1886 27 July 6 Aug. 15 Aug. 21 Aug. St. Paul de Loanda Gaboon 15 Sept. 22 Sept.,, 70.71 11.364 98.07 241.30 11 364 98.07 214.95 11 347 98.22 486.04 11.362 98.09 126 26 11 311 759 64 11 344 176.49 11 352 98.19) 1100 629 •310 703 ⚫306 604 ⚫313 657 ⚫306 107 140 98.53 623 ⚫306 98.24 801 ⚫312 ⚫302 Wet in air. : : : 13.2 Shore end. 7.3 Intermediate. 1.5 Main. INDEX. ACCUMULATION joint test, Clark's, 402 Ampère, definition of, 1 A. Angle of maximum sensitiveness in galvanometers, 23, 78 Arc, multiple, 70 Astatic galvanometer, 18 B. BALANCE, Wheatstone's (see Wheatstone bridge) Batteries, 283 Clark's standard, 140 De la Rue's 143 "" Fleming's 139 "" Muirhead's 141 "" Post Office 137 Wheatstone's,, 137 Leclanché, 284 Minotto, 283 Electromotive force of, comparison of (see Electromotive force) ? 1 and 100 cells, 300 of low resistance, measurement of resistance of, 299` polarisation in, measurement of, 299 Resistance of, measurement of, 113 298, 299 method, 130 method, 133 by condenser method, 295, 297, deflection method, 4 diminished deflection direct 99 shunt electrometer method, 361 Fahie's method, 172, 175 half deflection method, 5, 113 Kempe's method, 295, 299 Mance's method, 124, 127 538 HANDBOOK OF ELECTRICAL TESTING. Batteries, Resistance of, measurement of, by Muirhead's method, 297 516 Munro's method, 298 Postal Telegraph method, 511, Siemens' method, 118 Thomson's method, 114 Wheatstone bridge method, 241 shunted, Pollard's theorem in, 505 Battery resistance, use of table for calculating, 516 testing apparatus, Eden's, 516 Bridge, Wheatstone's (see Wheatstone bridge) C CABLES, completed, tests of, 370, 374 C. compound, tests during laying of, 396 conductor resistance of, method of measuring, 240 -, corrections for effects of temperature on conductor and insulation resistance of, 414 earth readings on, 370 electrostatic capacity of, measurement of, 325 faults in, localisation of (see Faults) final tests of, 479 insulation of, measurement of, 368 laying of, tests during, 396, 397, 399 manufacture of, specification for, 461 "" tests during, 465, 478 single wire, tests during laying, 397, 399 Calibration or graduation of galvanometer scales, 46, 76 Capacity, electrostatic (see Electrostatic capacity) Cardew's method of measuring current strength, 305 Carey Foster's method of measuring low resistances, 228 Cells, standard, Clark's, 140 De la Rue's, 143 Fleming's, 139 Muirhead's, 141 Post Office, 137 Wheatstone's, 137 Charge, loss of (see Potential, loss of) Chloride of silver battery, 143 Chrystal's standard ohm, 219 Clark's accumulation joint test, 402 correction for condenser discharge, 289 electromotive force test, 181 fall of potential fault test, 377 method of eliminating earth currents, 259 · INDEX. 539 Clark's standard cell, 140 Coefficient for effect of temperature on conductor resistance, 414 Coils, for core of cable, tests of, 465 resistance, 10 Dial pattern, 14, 192 insulation resistance, 419 for cable testing, 12, 14, 192 Post Office pattern, 13 slide, 15 ? Varley's, 210 Combined capacity of condensers, 275, 523 conductivity resistance of parallel wires, 70 insulation resistance of parallel wires, 233 resistances, 520 .Compensating resistances for galvanometer shunts, 71 Compound cables, tests during laying of, 396 key for cable testing, 509 Condensers, 273, 523 -, battery resistance measured by means of, 295, 297, 298, 299 connections for discharge from, 278 corrections for discharge from, 289 electromotive force measured by means of, 287, 300 , joint capacities of, 275, 523 Conducting power of copper, effect of temperature on, corrections for, 414 Conductivity resistance, by Wheatstone bridge, 231 235, 237, 238, 259 " correction for effect of temperature on, 414, 422, 425 elimination of effects of earth currents in measuring, joint, of several wires, 70 of cables, method of measuring, 240 of three wires individually, 231 of two wires individually, by loop test, 269 per mile of telegraph lines, 490 , specific, 408 Constant for measuring high resistances, 366 insulation resistances, 366 morning tests, 5 Copper resistance, Mathiessen's standard of, 409 wire, effect of temperature on resistance of, 414 specific conductivity of, 408 W. T. Glover's table of, 410 Correction for condenser discharge deflections, 289 loop test, 265 tangent galvanometer, 21, 34 Corrections for temperature, 414 2 practical applications of, 426 540 HANDBOOK OF ELECTRICAL TESTING. 1 [ f ра Coulomb, definition of, 328 Cubic equation, example of practical use of, 452 Current, Resistance, and Electromotive force, between two points in a circuit, relation between, 292 Current strength, measurement of, 301 by Cardew's differential method, 305 Kempe's bridge 308 "9 difference of potential deflection method, 312 315 equilibrium 99 unit of, 1 direct deflection method, 302 Siemens' dynamometer, 318 Currents, earth, elimination of, effects of, in testing by Wheatstone bridge, 235, 237 • in testing, 259 received, table for calculating, 497 testing telegraph lines by, 494, 500 D. DAILY or morning table for calculating, 8, 497 tests of land lines, 8, 494 Dead-beat galvanometer, D'Arsonval-Deprez's, 61 Thomson's, 59 Deflections, galvanometer, degree of accuracy attainable in reading, 42 method of reading, 41 De la Rue's standard battery, 143 Deprez-D'Arsonval dead-beat reflecting galvanometer, 61 Dial pattern of resistance coils, 14, 192 Discharge deflections, connections for measuring, 278 ? correction for, 289 key, Kempe's, 278 Lambert's, 280 Rymer Jones's, 281 F. C. Webb's, 278 test of joint by, 404 Disconnection, partial localisation of, in cables, 439 total "" Dynamometer, Electro, Siemens', 318 99 439 E. EARTH current, to eliminate, in testing, 259 by Wheatstone bridge, 235, 237 • INDEX. 541 Earth faults, a method of localising, 447 readings, on cable, 370 table of, 8 resistance of an, to measure, 233 Eden's battery-testing apparatus, 516 Electric lamps, method of measuring the resistance of and currents flowing through, 506 Electrification, 369 • influence of temperature on, 369 Electrodynamometer, Siemens', 318 Electrometer, Thomson's quadrant, 348 362 > fall of charge in cable by, 361 gauge of, 352 grades of sensitiveness of, 359 induction plate of, 353 measurements from an inferred zero, by, replenisher of, 351 , reversing key for, 354 tests of joints by, 405 use of, 361 Electromotive force, Current, and Resistance, between two points in a circuit, relation between, 292 , measurement of, 137, 144 by Clark's method, 180 equal deflection method, 146 resistance method, 144 Fahie's method, 175 Law's method, 287 Lumsden's or Lacoine's method, 155, 159 Poggendorff's method, 165 Postal Telegraph 512, 516 Wheatstone's Wiedemann's table for calculating, 514 unit of, 1 "" 152 146 "" Electrostatic capacity, measurement of, 325 by direct deflection method, 325 divided charge method, 341 Gott's method, 339 Siemens' diminished charge method, 344 Siemens' loss of charge deflection method, 333 Siemens' loss of charge discharge method, 327 542 HANDBOOK OF ELECTRICAL TESTING. i Electrostatic capacity, measurement of, by Thomson's method, 335 -, specific, 413 F. FAHIE'S method of measuring battery resistance, 172, 175 False zero, 238, 265 testing for faults in cables, 246 Farmer's key for galvanometer and battery resistance tests, 93, 118 Fault resistance, Kenelly's law of, 251 Faults caused by disconnection, localisation of, 439 localisation of, 242 by Clark's fall of potential method, 386 combined resistance and discharge test, 403 Fahie's method, 246 Jacob's deflection method, 253 Kempe's loss of current method, 256 Loop test, 259 Lumsden's method, 245 Mance's 249 "" : Siemens' equilibrium method, 393 or Lacoine's equal potential method, 390 in coils of insulated wire, Jacob's method, 438 of high resistance, 428 Warren's Figure of merit of galvanometers, 65 Final tests of cables, 479 Fleming's standard cell, 139 Foster's, Carey, method of measuring low resistances, 228 G. 436 GALVANOMETER deflections, degree of accuracy attainable in reading, 42 ? astatic, 18 method of reading, 41 D'Arsonval-Deprez's dead-beat reflecting, 61 Gaugain's, 36 Helmholtz's, 36 Obach's, 37 sine, 19 tangent, 7, 19, 498 > best conditions for using, 28 correction for, 21, 34 principle of, 20 Thomson's reflecting, 46 dead-beat form of, 59 INDEX. 543' - Galvanometer, Thomson's reflecting, Gray and March Webb's arrangement of, 52 Jacob's transparent scale for, 55 lamp and scale for, 54 marine, 63 portable form of, 54 resistance of, 54 scale for, 56 Silvertown form of, 52 Galvanometers, angle of maximum sensitiveness in, 23, 78 calibration or graduation of scale of, 46, 76 figure of merit of, 65 for measuring currents, Post Office form, 498 method of adjusting, 75 resistance for best effect from, 457 Resistance of, measurement of, 79 method, 82 method, 89 sensitiveness of, 66 shunts for, 59 Gaugain's galvanometer, 36 by deflection method, 3 diminished deflection direct "" shunt equal deflection method, 83 "" "" half 5, 79 Phillips' method, 282 Thomson's "" 93,98 Gauge for electrometer, 352 Glover, W. T., table of resistances, etc., of copper wire, 410 Gott's electrostatic capacity test, 339 method of sealing up faults for testing, 456 proof condenser method of measuring resistances, 381 Gray, R. K., arrangement of reflecting galvanometer, 52 Gutta-percha, effect of temperature on resistance of, 369 electrification of, 369 specific inductive capacity of, 413 insulation of, 411 H. Halving deflection, resistance of battery by, 5, 113 HALF-CHARGE, fall to, 383 Helmholtz's galvanometer, 36 galvanometer, by, 5, 79 by Jacob's method, 438 High resistances, localisation of faults of, 428 ¿ + 544 HANDBOOK OF ELECTRICAL TESTING. High resistances, localisation of faults of, by Warren's method, 436 measurement of, 5, 364 by loss of potential, 380 Gott's proof condenser method, 381 I. INDIARUBBER, electrification of, 369 Individual resistance of three wires, 231 two Induction plate of electrometer, 353 269 Inductive capacity (see Electrostatic capacity) specific, 413 Inferred zero, 65, 362 Insulated wires, detection of faults in, by Jacob's method, 438 Warren's method, 436 Insulation, correction for effect of temperature on, 419, 424, 425 , joint, of several wires, 233 measurement of, 5, 7 by received currents, 494 tangent galvanometer, 8 transmitted and received currents, 500 Wheatstone bridge, 233 of cables, 368 of two sections of wire, 234 by Jacob's method, 375 -, per mile of telegraph lines, 490 specific, 411 standard of, for land lines, 6 table for calculating, 8, 497 J. JACOB's fault test, 253 method of measuring insulation of cables, 375 transparent scale for reflecting galvanometers, 55 Jenkin's method of measuring high resistances, 362 Joint capacities of condensers, 275, 523 conductivity resistance of parallel wires, 70 insulation "" Joints, testing of, at sea, 405 99 233 by Clark's accumulation method, 402 discharge method, 404 electrometer Warren's 405 436 Jolin's D'Arsonval-Deprez dead-beat reflecting galvanometer, 61 1 · ÷ T INDEX. 545 Jolin-Thomson rheostat, 16 Jones, Rymer, discharge key, 281 K. KEMPE, A. B., on the leakage of submarine cables, 428 Kempe's battery resistance test, 295, 299 current strength test, 295 -discharge key, 278 loss of current fault test, 256 Kenelly's law of fault resistance, 251 Key, compound, for cable testing, 509 discharge, Farmer's, for galvanometer and battery resistance tests, 53, 118 Kempe's, 278 Lambert's, 280 for Thomson's capacity test, 338 Rymer Jones', 281 F. C. Webb's, 276 reversing, 271 for electrometer, 354 Pell's, 272 short-circuit, 270 Kirchoff's laws, 156 proofs of, 503 L. LACOINE'S or Lumsden's electromotive force test, 155, 159 Siemens' fault test, 393 Lambert's discharge key, 280 key for Thomson's capacity test, 338 Lamps, electric, method of measuring the resistance of and current flowing through, 506 Land lines, measurement of insulation of, 6 standard of insulation for, 6 Laws' test for electromotive force, 287 Laying of cables, tests during, 396, 397, 399 Leading wires, elimination of resistance of, 241 Leclanché battery, 284 Loop method of measuring conductivity resistance, 231 test, 259 • correction for, 288 individual resistance of two wires by, 269 Murray's method, 260 Varley's 263 "" Phillips' method, 268 2 N 546 HANDBOOK OF ELECTRICAL TESTING. Loss of current fault test, Kempe's, 256 Low resistance batteries, a method of measuring, 299 resistances, a method of measuring, 507 measured by metre bridge, 213 Carey Foster's method, 228 Thomson's bridge, 230 Lumsden's, or Lacoine's, method of measuring electromotive force, 155, 159 system of testing for faults in cables, 245 M. MANCE's method of eliminating the effects of earth currents in conductivity tests, 237 testing for faults in cables, 249 resistance of battery test, 124 with slide wire bridge, 127 tests during, 465, 478 Manufacture of cables, specification for, 461 Marine galvanometer, Thomson's, 63 Matthiessen's standard of copper resistance, 409 Maximum sensitiveness, angle of, in galvanometers, 23, 78 Merit, figure of, of galvanometers, 65 Metre bridge, 213 Mile, insulation per, of lines, 490 Milliampère, 495 Minotto battery, 283 Morning, or daily tests of land lines, 8, 494 Muirhead's battery resistance test, 297 standard cell, 141 Multiple arc, 70 Multiplying power of shunts, 69 Munro's battery resistance test, 298 Murray's loop test, 260 OBACH'S galvanometer, 37 Ohm, definition of, 1 0. } standard, 219 Ohm's law, 1 One cell, 283 constant taken with, 364 P. PARALLAX error in galvanometers, method of avoiding, 23 Parallel wires, joint resistance of, 70 INDEX. 547 Partial disconnection in cable, localisation of, 439 Pell's reversing key, 272 Phillips, S. E., method of measuring the individual resistance of two wires by loop test, 269 galvanometer resistance, 282 -making loop test, 268 Platinoid, use of, for resistance coils, 10 rheostat, 16 Poggendorff's method of measuring electromotive forces, 165 Polarisation in batteries, measurement of, 299 Pollard's theorem of a shunted battery, 505 Portable reflecting galvanometer, 54 Postal Telegraph Department, galvanometer used by, 498 standard cell used by, 137 standard of insulation adopted by, 6 -, system of testing batteries, 515, 517 lines by received currents, 494, 500 Wheatstone bridge used by, 13 Potential, fall of, formulæ for, 382 measurement of, 284 resistances by, 377 loss of, 380 Gott's method, 381 Clark's test for fault by, 386 Siemens' "" >" equilibrium of, 393 equal, 30 Preece's fall of potential formula, 384 Proof condenser method of measuring resistances, Gott's, 381 Purity or conducting power of copper, effect of, on temperature corrections, 417 Q. QUADRANT electrometer, Thomson's, 348 Quantity, unit of, 328 R. RECEIVED currents, table for calculating, 497 testing telegraph lines by, 494, 500 Reflecting galvanometer (see Galvanometers) Replenisher of electrometer, 351 Resistance coils, 10 dial pattern, 14, 192 for cable testing, 12, 14, 192 Post Office pattern, 13 slide, 15 548 HANDBOOK OF ELECTRICAL TESTING. Resistance, Current, and Electromotive force, between two points in a circuit, relation between, 292 measurement of, by deflection, 3 half deflection, 5 fall of potential, 377 loss 380 ? Gott's method, 381 unit of, 1 Resistances, combined, 520 substitution, 2 Wheatstone bridge (see Wheatstone bridge) , compensating, for galvanometer shunts, 71 high, measurement of, 5, 364 insulation, "" 368 joint, of several wires, 70 low, a method of measuring, 507 measurement by metre bridge, 213 , Carey Foster's method, 228 Thomson's bridge, 231 Resultant fault, 265 Reversing keys, 271 for electrometer, 354 Pell's, 272 switches, 272 Rheostat, Thomson-Jolin, 16 Roberts, Martin, method of using metre bridge, 215 Rymer Jones' discharge key, 281 S. SCALE, and lamp, for Thomson's reflecting galvanometer, 54 galvanometer, graduation or calibration of, 46, 47 Jacob's transparent, for Thomson's reflecting galvanometer, 55 Silvertown form of, "" skew, for tangent galvanometer, 30 Sealing up faults for testing, Gott's method, 456 Sections, two, of wires, insulation of, 234 Sensitiveness, angle of maximum, in galvanometers, 3, 78 of galvanometers, 66 Short circuit keys, 270 Shunted battery, Pollard's theorem of, 505 Shunts, 67 galvanometer, 59 compensating resistance for, 71 method of adjusting, 75 multiplying power of, 69 "" 58 严 ​1 INDEX. 549 Shunts, galvanometer, table of, 75 Siemens' battery resistance measurement, 118 • electro-dynamometer, 318 electrostatic capacity by loss of charge measurement, 327, 333, 344 localisation of faults by potential, 390 or Lacoine's localisation of faults by potential, 393 –, telegraph works, method of testing completed cable at, 375 transparent galvanometer scale in use at, 55 Silvertown compound key for cable testing, 509 galvanometer scale, 56 reflecting galvanometer, 52 Sine galvanometer, 19 Single wire cable, test during laying, 397, 399 Skew scale for tangent galvanometer, 30 Slide resistance bridge, Varley's, 210 coils, 15 wire, or metre bridge, 213 battery resistance by, 127 galvanometer resistance by, 98 Small resistances, a method of measuring, 507 " measurement by metre bridge, 213 Carey Foster's method, 228 Thomson's bridge, 230 Smith, Willoughby, system of testing cables during laying, 399 Specific conductivity, 408 inductive or electrostatic capacity, 413 insulation, 411 measurements, 408 Specification for manufacture of cables, 461 Standard cell, Clark's, 140 2 De la Rue's, 143 Fleming's, 139 Muirhead's, 141 Post Office, 137 Wheatstone's, 137 of copper resistance, Matthiessen's, 409 of insulation for land lines, 6 ohm, 219 Substitution method of measuring resistances, 2 Switches, reversing, 272 T. TABLE for calculating battery resistances, 516 electromotive forces, 514 insulation resistances, 8 550 HANDBOOK OF ELECTRICAL TESTING. Table for calculating insulation resistances and strengths of received currents, 497 Tangent galvanometer, 7, 19, 498 angle of maximum resistance of, 23 best conditions for using, 28 corrections for, 21, 34 insulation resistance by, 8 principle of, 20 skew scale for, 30 Taylor, Herbert, galvanometer shunt tables, 375 Temperature corrections for conductor resistance, 414, 422, 425 insulation resistance, 419, 424, 425 ? effect on electrification, 369 of cable determined by conductor resistance, 421 Theorem, Pollard's, of a shunted battery, 505 Thomson's bridge, 230 electrostatic capacity test, 335 method of measuring battery resistance, 144 of, 52 galvanometer resistance, 93, 98 quadrant electrometer, 348 dead-beat form of, 59 reflecting galvanometer, 46 -, Gray and March Webb's arrangement lamp and scale for, 54 marine, 63 Thomson-Jolin rheostat, 16 portable form of, 54 "" by loop test, 269 Three wires, individual resistance of, 231 Two "" Transparent scale, Jacob's, for reflecting galvanometer, 55 U. UNITS, electrical, 1 V. VARLEY'S loop test, 263 slide resistance bridge, 210 Volt, definition of, 1 W. WARREN'S test for small faults in insulated wires, 436 Webb, F. C., discharge key, 276 March, arrangement of reflecting galvanometer, 52 Wheatstone bridge, 188 INDEX. 551 Wheatstone bridge, conditions for accurate measurements by, 192 conductivity resistance by, 231 obtained, 209 insulation " "" 233 measurement by, when exact equilibrium cannot be of wires traversed by earth currents, 235 method of connecting up, 191 slide wire or metre, 213 used by Postal Telegraph Department, 13 Varley's slide resistance, 210 Wheatstone's method of measuring electromotive force, 152 standard cell, 137 Wiedemann's method of measuring electromotive force, 146 Willoughby Smith's system of testing cables during laying, 399 Wires, copper, specific conductivity of, 408 temperature corrections for, 414, 422, 425 individual resistance of three, 231 two, by loop test, 269 joint resistance of, 70 ZERO, false, 238, 265 , inferred, 65, 362 Z. skew, of tangent galvanometer, 30 LONDON: PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, STAMFORD STREET AND CHARING CROSS. 20 Resevis 3-6-06 UNIVERSITY OF MICHIGAN 3 9015 06395 0805 A 518771 DUPL