ESSA TR ERL 161 -ITS 103 A UNITED STATES DEPARTMENT OF COMMERCE PUBLICATION *>Vt $ o« S ESSA Technical Report ERL 161 ITS 103 U.S. DEPARTMENT OF COMMERCE Environmental Science Services Administration Research Laboratories An Analysis of the Shunt-Fed Log-Periodic Monopole Array Antenna PITT WILLIS ARNOLD BOULDER, COLO. MARCH 1970 ESSA RESEARCH LABORATORIES The mission of the Research Laboratories is to study the oceans, inland waters, the lower and upper atmosphere, the space environment, and the earth, in search of the under- standing needed to provide more useful services in improving man's prospects for survival as influenced by the physical environment. Laboratories contributing to these studies are: Earth Sciences Laboratories: Geomagnetism, seismology, geodesy, and related earth sciences; earthquake processes, internal structure and accurate figure of the Earth, and distribution of the Earth's mass. 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C. 20402 Price 65 cents AN ANALYSIS OF THE SHUNT-FED LOG-PERIODIC MONOPOLE ARRAY ANTENNA by Pitt Willis Arnold B.S., University of Illinois, 1959 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Electrical Engineering 1969 111 ABSTRACT An analytic method is here introduced enabling one to insert the physical dimensions of a shunt-fed log- periodic monopole array antenna into a properly arranged computer program and obtain reasonably accurate infor- mation about the radiation and impedance characteristics of the antenna from the resultant computer output. The method is related to a similar analysis of dipole type antennas but the complexity of the feed system of the shunt-fed structures causes the matrix relations to be more complex than those of the dipole antenna. Computed and experimental results are compared and the "optimized feed" case is discussed at some length. IV ACKNOWLEDGMENTS The efforts of Li Hie C. Walters, who programmed the matrix relations herein in the NBS-ESSA CDC 3800 computer, helpful discussions with Dr. Mark T. Ma, and the use of the experimental facilities at ESSA Research Laboratories are gratefully acknowledged. TABLE OF CONTENTS PAGE ABSTRACT iv ACKNOWLEDGMENTS v LIST OF TABLES vii LIST OF FIGURES vii SECTIONS I INTRODUCTION 1 II THE MATRIX RELATIONS 5 III THE COMPONENT MATRICES 9 A. The Z Matrix and Relation a Between Base-Fed and Shunt- Fed Monopoles 9 B. The Y and Z Matrices 19 s c C. The Y f Matrix 21 IV EXPERIMENTAL AND COMPUTED RESULTS ... 30 V CONCLUSIONS ' 38 BIBLIOGRAPHY 39 FIGURES 42 VI LIST OF TABLES PAGE Y' and V 1 Calculations for Case A 3 . . . 27 n n LIST OF FIGURES 1 Photograph of a Wire-Outline Shunt-Fed Log-Periodic Monopole Array 42 2 Photograph of a Shunt-Fed Log-Periodic Monopole Array with Cylindrical Monopoles 43 3 Geometry of Log-Periodic Monopole Arrays Showing Two Adjacent Monopoles 44 4 Diagram of a Two-Element Shunt-Fed Array Illustrating Equivalent Circuit Elements and Node Current Relations 45 5 Geometry and Notation Used in Mutual Impedance Calculations 46 6 Resonant Impedance of a Shunt-Fed Monopole as a Function of Feedpoint Height with Height to Radius Ratio as Parameter ... 47 7 Current Distribution on a Shunt-Fed Monopole vs. Probe Height with Feed-Point Height as a Parameter 48 8 Current Distribution on a Grounded Monopole vs. Probe Height. Current Induced by Adjacent Shunt-Fed Monopole of Equal Height. Feed-Point Height is Parameter . 49 vii 9 Input Impedance vs . Frequency of a Shunt-Fed Monopole and a Base-Fed Monopole with a Matching Network .... 50 10 Locus for Z and V Determination for o p Case A3 51 11 Locus for Z and V Determination for o p Case C 3 52 12 Measured and Calculated Radiation Patterns for Case C3 in Azimuth Plane 53 13 Measured and Calculated Radiation Patterns for Case D in Azimuth Plane 54 14 Input Impedance of Wire-Outline Monopole Array vs. Frequency for Case A3 55 15 Input Impedance of Wire-Outline Monopole Array vs. Frequency for Case C3 56 15A A Comparison of Calculated Impedances for Case C 3 57 16 Average Characteristic Impedance and VSWR of Cases A and B. A Comparison of Measured and Calculated Data 58 17 Input Impedance of Cylindrical Monopole Array vs. Frequency for Case D 59 18 Computed Self Impedances for Monopoles as a Function of Electrical Length, in Radians, for Constant h/a 60 Vlll INTRODUCTION The shunt-fed log-periodic monopile array antenna, two types of which are shown in Figures 1 and 2 , is an array of grounded monopoles whose heights, spacings , and effective diameters increase from the front or feed end of the structure by a fixed proportionality constant. Properly functioning antennas of this type exhibit essentially frequency independent radiation patterns and impedance loci within their operating frequency range. Their usable frequency range has a lower limit defined approximately by the frequency where the rear, or tallest, monopole is one-quarter wavelength in height and an upper limit defined by the frequency where the shortest element is three-sixteenths of a wavelength in height. Their radiation patterns are usually single lobed, with the maximum radiation along the line of the monopoles in the direction of the smaller elements . The phasing required for this direction of radiation is produced by a combi- nation of factors , the net result being an effective reduction in phase velocity, in the feed system, between the monopoles near their resonant frequencies. The beamwidth of the radiation pattern depends on the number of radiating monopoles and on the choise of log-periodic parameters . 1 2 Wickersham's ' "Ladder" antenna was one of the first shunt-fed log-periodic monopole arrays built. It used metal blades as monopoles which were trapezoidal in shape like the monopoles of the Wire-Outline antenna 3 developed by Ore, and shown in Figure 1. These two types are quite similar. The antenna pictured in Figure 2 was developed by the author for a high-power HF radar application where 10 of these antennas are arrayed together. It is a one-hundredth scale model and oper- ates between 700 and 2500 MHz to simulate the 7-25 MHz full-scale antenna. The shunt-fed type of monopole array with cylindrical monopoles appears well suited for this application. All three types of shunt-fed arrays mentioned above use metal cylinders, which are fastened to the monopoles at a fixed percentage of a monopole' s total height and are coaxial to the feed rod, thus forming a coupling capacitor. The feed rod is connected to the center conductor of a coaxial feedline at a point just in front of the first monopole and its capacitor. The outer conductor of this coaxial line is connected to the ground plane near the same point. Most of the models built have had this coaxial line come in beneath and perpendicular to the metal ground plane, although it might be arranged to come in perpendicular to the feed rod and parallel to and contacting the ground plane. The geometry of the array is given in Figure 3 showing two monopoles of an array. The envel- ope of the element tips makes an angle a with the ground plane and intersects it at the virtual vertex. The constant T relates the heights, spacings and widths, or effective diameters, of adjacent monopoles ; t and a are related to a, the spacing constant, by a=i (1-t) cota. These constants are those commonly used to describe log 4 periodic dipole arrays as in the work of Isbell and in the analysis of Carrel. With the exception of an investigation by Hudock and Mayes into the near- field properties of uniformly- periodic monopole arrays, little, if any, analytical work has been published on these antennas. No attempts to relate their behavior to the log-periodic dipole arrays have been made nor has any analysis been made of them in terms of network theory. This lack of definitive analyses together with some unusual experi- mental results, obtained during the earlier development of the antenna in Figure 2 prompted this analysis. The unusual experimental results showed that a feed net- work parameter, specifically the capacitor lengths y , should be varied in other than log-periodic fashion to obtain the best or "optimum" input impedance vs. fre- quency. These results were obtained without prior know- 7 ledge of Ore's similar findings. The method herein, of necessity, follows the lead g of Carrel, who first analyzed the transposed feeder log-periodic dipole array using circuit theory and matrix algebra. The antenna impedance matrix is derived as Carrel derived it, but an additional two-element passive network is connected between the monopoles and the transmission line. This two-element network causes the base-fed monopoles used in the computer program to appear as an approximation to shunt-fed monopoles , both with respect to input impedance and current phase shift as viewed from the coupling capacitor connection points , or nodes . The matrix algebra is presented first, showing the relations between the matrices to be individually described later. The manner of calculation of the antenna impedance matrix and the relation between base- fed and shunt- fed monopoles are then discussed. The method for calculating the elements of the network and related matrices is then described, followed by a description of feeder admittance matrix calculation. Finally, the computed and measured radiation patterns and impedance loci , for the antennas in Figures 1 and 2 are presented and discussed. The impedance "optimization" method is also discussed. II THE MATRIX RELATIONS The circuit shown in Figure 4 represents two shunt- fed monopoles and their associated transmission line. Application of elementary circuit theory permits exten- sion of the following to N monopole elements. The passive elements connected between the unprimed nodes and the primed nodes, as well as the passive ele- ments connected between the unprimed nodes and the ground are assumed to have no interconnection other than that shown and thus can be represented in the matrix rela- tions by a simple 2, or N , element diagonal matrix. The elements shown as resistors represent 2, or N, inter- coupled base-fed, gapless monopoles and thus are repre- sented by a 2X2, or NXN, matrix. The transmission line is represented by an admittance matrix having only diago- nal and sub-diagonal entries, since there are only direct connections between a node and its immediate neighbors. Matrix quantities are indicated by a bar above the vol- tage, current, impedance, or admittance symbol. At the unprimed nodes, a summation of currents gives i' = I + I (1) a a s ' or i in matrix notation, i 1=1+1 (2) a a s v ' or I = Y V + Y V . (3) a a a s s v ' Since V and V are the same, (3) may be written as a s I = (Y +Y )V . (4) a a s a Since we wish to introduce the element impedance matrix and solve for the element currents, we introduce the relation V = Z I (5) a a a into (4) to obtain I = (Y +Y ) Z I = (U+Y Z )I , (6) a a s a a s a a ' where U is the unit matrix. Moving now to the primed nodes and again summing currents we obtain I. = I- +.1 , (7) in f a ' or, using matrix notation, I I. = I, + I . (8) in f a Since I £ = Y £ V , (9) and V. = V + V = Z I +ZI , (10) f c a c a a a ' (7) becomes i _ i _ i I. = Y,[Z I +Z I ] + I = [U+Y^Z ]I + Z I . (11) in f c a a a a f c a a a _ i Substituting the relation for I , or (6) , into (11) , we a obtain i ln = j[u + y f z c ][u + Y s z a ] + z a }i a . (12) The Column matrix I. has only one non-zero element, in 2 the input current to the antenna array, which we con- veniently assign a unit value. All the quantities in the large brackets of (12) may be calculated or are known. Therefore, we may solve for I , the column matrix of element currents, by matrix inversion. Let the bracketed quantities of (12) be called X. Then (12) becomes I. = XI . (13) in a _-l Pre-multiplying each side by X gives 5 = x"'i. n . (14) Since the values of I are now known, we may solve for the input voltage, which will give us the input imped- ance/ since the input current is assumed to be one ampere. The relation used is V f - !v 5+ v a ] + h\h ■ (15) from which we take V f to obtain the impedance. Equa- tion (14) is programmed in a computer to obtain element currents , from which radiation patterns may be obtained with an additional computation. The input impedance of the antenna is then obtained from (15) . The Z , Y„ , Z / a s c and Y f matrices must first be calculated, preferably with the computer, and the methods of calculating them are treated next. Ill THE COMPONENT MATRICES A. The Z Matrix and Relation Between Base-Fed and a Shunt-Fed Monopoles. The description to follow closely parallels that 9 of Carrel and uses the method of mduced-emf as applied to unequal length dipoles by H. E. King. The follow- ing assumptions and/or approximations are made: (1) A symmetric sinusoidal current distribution is assumed over the dipole formed by a monopole and its image in the ground plane. This is reasonably valid where a monopole is less than one-half wavelength long, the accuracy being greatest for quarter wave and shorter mono- poles. Accuracy is improved by avoiding fre- quencies where any monopole is within a few percent of one-half wavelength. Greater accuracy can be obtained, by following 11 12 13 Cheong and Cheong and King ' with three- term formulation of the currents . (2) In the calculation of mutual impedances, the elements are assumed to be inf initesimally 9 thin; i.e., the current at a cross-section of the actual dipole has been replaced by an average current concentrated at the center of the cross section. (3) The mutual term involves only the two elements considered; i.e., the intervening elements are neglected. This assumption is actually im- plicit in (2) above. In the limiting case of zero element thickness, the current in the first dipole induces a voltage across the terminals of the second but no current along it, since the inductance per unit length of an infinitesimally thin dipole is infinite. Since there is no induced current, there is no voltage across any other dipole from the induced current in the second dipole, and therefore no secondary action. (4) The self -impedances are calculated from the same formula as the mutuals by approximating the self-impedance of a dipole of radius a as the mutual impedance of two infinitesimally thin dipoles spaced a distance /2a apart. Figure 5 shows two parallel symmetrical dipoles, of half-lengths hi and h 2 , separated by distance s, 10 with element dz at coordinate distance z from the x-y plane. The distances ro , ri, and r 2 are distances from fixed points on one dipole to a typical element on the other. The mutual impedance between the two antennas in Figure 5 may be defined by 221 = ITToT ' (16) where V 2 1 is the open circuit voltage at the terminals of antenna 2 caused by a center current li(0) at the terminals of antenna 1. The voltage V21 is given by ' i_r h 2 V21 = T ,7ni / E„ I 2 (z)dz , (17) 2 where E is the z component of electric field intensity at the location of antenna 2 caused by the current of antenna 1, when antenna 2 is removed. The current dis- tribution on antenna 2 is assumed to be sinusoidal and is given by I 2 (z) = I 2m=v sin B(h 2 -|z|) , (18) where 3 is the free-space propagation constant. The expression for the z component of electric field caused by a sinusoidally distributed current in antenna 1 is given by 1 1 E = -j30l! Zi J a max C21H + e" jSr2 _ 2 cos gh ie - j3r ' ri r 2 r . (19) Inserting (18) and (19) into (16) and (17) gives the mutual impedance referred to the base of the antenna , Z12 = Z 2 i = j30 Ii I 2 max max Ii(0)I 2 (0) t 1 ■ I sin $(h 2 - z ) -JBri ri r^ r2 2 cos 3h ie -^ r ° r 2 r dz (20) and with the assumption of sinusoidal currents the maximum currents are related to the base currents by Ii (0) = Ii sin 3hi , 1 ^max I 2 (0) = I 2 sin $h 2 , z max and from Figure 5 (21) r = Vs* + z 2 , ri = ^s 2 +(h!-z) 2 , (22) r 2 = ^s 2 +(hi+z) 2 . Therefore, (20) may be rewritten as 12 Zi2 = J30 esc 3hi esc 3^2 i sin $(h 2 -|z|) -h 2 rJ3r x ri r^ r2 2 cos eh ie -^ r ° r 2 r dz (23) Integration of (23) gives an expression for the mutual impedance in terms of sine and cosine integral functions 1 2 60 cos w 2 -cos w -|e jWl [K(u )-K( Ul )-K(u 2 )] + e" jWl [K(v )-K(vi)-K(v 2 )] + e jWz [K (uj ) -K (m ) -K (v 2 ) ] + e~ 3 ™ 2 [K(vJ)-K(vi)-K(u 2 ) ] + 2K(w )[cos w x +cos w 2 ]J , (24) where * denotes the complex conjugate of the expression in the braces and K(x) = Ci(x) + jSi(x) ■jr a f jt **if s 7 x * • (25) The other quantities in (24) are defined below. 13 (26) uo = gVs 2 +(hi+h 2 ) 2 - (h!+h 2 ) , v e = 6k 2 +(hiih 2 ) 2 + (h!+h 2 ) , uj = ^s 2 +(h r h 2 ) 2 - (hi-h 2 ) , vj = 3 Up+ThT^haT 1 + (h!-h 2 ) / Ui = 3 V s2+h i 2 ~ hj , vi = 3 V s2+ hi 2 + hi , u 2 = 3 Vs 2 +h 2 2 - h 2 , v 2 = 3 Vs 2 +h 2 2 + h 2 , wi = 3 (hi+h 2 ) / w 2 = 3(h 1 -h 2 ) , w = 3s The mutual and self-impedances of a log-periodic monopole array may then be calculated by suitably, pro- gramming the lengths , spacings and length to radius ratio in a digital computer. The formulation presented above yields the self and mutual impedances of an array 14 of dipoles. Dividing all values obtained by 2 gives the Z matrix for a monopole array. The Z matrix obtained is for base-fed monopoles a having no gap at the feedpoint. How can we relate the base-fed monopole to the actual monopole used, one which is grounded and fed at some distance above the base? At first, the method of impedance calculation suggested by 17 Morrison and Smith was tried, wherein the portion of the antenna between the feedpoint and the ground is assumed to be a shunt inductor and the portion above the feed- point is assumed to be a base-fed structure of height equivalent to the height above the feedpoint. This proved to be less than satisfactory because of the diffi- culty of calculating the inductive reactance of the por- tion of antenna below the feedpoint. It was decided instead to measure the actual impedance vs. frequency of shunt-fed monopoles for several different feedpoint heights and monopole diameters, and use a network that would best approximate the impedance of a shunt-fed monopole when used with the base-fed monopole. Figure 6 gives the resonant impedance of shunt-fed monopoles of three different height to radius ratios as a function of feedpoint height H/h . This information allows the design of a network which will match the approximately 15 34 ohm resonant impedance of a base- fed monopole to the resonant impedance of a shunt-fed monopole of specified feedpoint height. This is not enough, since we must also simulate the phase shift between the input current to the shunt-fed monopole and the radiation current. Also we must approximate the impedance vs. frequency characteristics of the shunt-fed monopole. The desired phase shift information was obtained from Figures 7 and 8 by the following reasoning. For continuity of current at the feedpoint of an isolated shunt- fed monopole we may assume that the input current I . is equal to I +1 , where I is the in s a s current flowing toward the earth in the monopole shown in Figure 6 and I is flowing upward from the feedpoint. With these assumed current directions, a sensing probe near the antenna will sample I -I or I -I . Call this c s a as difference I . Figure 7 shows this measured difference current as a function of probe height z for two dif- ferent feedpoint heights . Figure 8 , which shows the measured current of a parasitically excited monopole of the same height and radius as the monopole in Figure 7 is of the assumed sinusoidal form of the base- fed monopoles of our Z formulation. The radiation current of the shunt- fed monopole, or I , is assumed to a be of this form also. Referring again to Figure 7, for 16 the case of H/h=0.126 and taking the readings for z=llcm, we note that there is little phase change above z=llcm and that the magnitude of I seems to be varying as in Figure 8. We then assume that we are measuring I only a at z=llcm and according to Figure 8 I has reached one- half its maximum at z=llcm. We therefore conclude that near the base, or even at the feedpoint, the I of the -i56° antenna in Figure 7 is . 7e J . The measured feedpoint - i 18 ° current I =1 -I is 0.69e J . We wish to know I. in mas in terms of I in order to establish at least the direction a of phase shift which our equivalent circuit should introduce. Adding the two relations I =1 -I and ' mas I. =1+1 , we obtain I. =21 -I . Substituting the mas' mam 3 -i 82° values deduced above, we obtain I. =0.96e J ' which lags in — 2 — I by 26°. A similar treatment of the H/h=0.088 case shows I. lagging I by 30°. The required circuit is in 3. seen to be one which will introduce a phase lag between I and I. which is dependent on feedpoint height H, and therefore dependent on the required tranformation ratio between a base-fed monopole' s resonant impedance of about 34 ohms and that of a shunt-fed monopole of feedpoint height H. A two-element network which has about the correct phase shift characteristics and whose impedance vs. frequency characteristic, when matching a base-fed monopole to a lower impedance, is a fair approximation to the impedance of a shunt-fed monopole 17 is a shunt inductance series capacitance L network. For practical feedpoint heights and monopole height to radius ratios, a reduction to a value between 8 and 12 ohms is usually called for, as experimental data will later show. Therefore, this form of matching network will suffice in most cases. The degree of approximation to the impedance vs . frequency characteristic of a shunt-fed monopole, with h/a=14 and H/h=0.126, obtained with an L network is shown in Figure 9 . Better agreement is obtained below f , the resonant frequency, than above f . This is perhaps fortunate since the monopoles or dipoles in front of the resonant one are usually more of a factor in log-periodic monopole and dipole structures than are those behind it. It should be stressed at this point that the shunt inductor and series capacitor of the L network are not intended to simulate the reactance below the feedpoint as in the much earlier work of Morrison 18 and Smith, but, instead, are intended to make the base- fed monopole, whose impedance is easily computer simu- lated, approximately equivalent to the shunt- fed mono- pole. Resonance in a shunt-fed monopole occurs at about the same electrical length as it does in the base-fed monopole, but the locus (see Figure 9) has the form of an anti-resonant circuit. The measured locus in 18 Figure 9 might be more closely duplicated analytically by considering it as a lossy, tapped transmission line having varying characteristic impedance and attenuation along its length, but such an analysis would little suit our purpose here due to its complexity. B. The Y and Z Matrices, s c The Y matrix is seen to be a diagonal matrix con- sisting of the shunt inductors of L networks, which match the resonant impedance of a base-fed monopole of known h/a ratio to the resonant impedance of a shunt-fed monopole of the same h/a and known H/h. Any single monopole may be calculated first at its resonant fre- quency and the rest are then calculated by using the scaling constant t. For example, the monopole in Figure 9 has a resonant impedance, in the shunt-fed configuration, of 9 ohms. This resonance occurs at a frequency where h~0.226A. The base-fed monopole having the same h/a also resonates where h~0.226X and has an impedance of 33.5 ohms. With these two values known we may compute Y from s (27) y = -L- t/ R2 " R s jR 2 ™ Ri With R 2 =33.5 and Ri=9, a value of -j.049 mhos is obtained. 19 The remainder of the Y values are then easily calcu- s J lated from Y s = TY . (28) n n-1 The Z matrix is also a diagonal matrix of imaginary terms, but it consists of two parts, one of which is the required L network series capacitor. This may be calcu- lated for a resonant monopole from -jX„ = -j/(R 2 -Ri)Ri , (29) e where R2 and Ri are the resonant impedances of the base- and shunt-fed monopoles respectively. For Figure 9, -jX =-jl4.9 ohms. The rest of the values may be ob- e tained from Z c = tZ c . (30) e n e (n+l) The other part of the Z matrix is the physical coupling capacitor, C , which is in series with C and is cal- c n n culated from Z = -jz cot 3y n , (31) c o n 20 where y is the physical length of the air dielectric coupling capacitor, and Z is the characteristic imped- o ance of the coaxial capacitor. C. The Y f Matrix. A brief discussion of the three regions of a log- periodic structure may be helpful before describing the method of calculation of the Y f matrix values. 19 Carrel calls these three regions the transmission region, the active region, and the unexcited region. The front or transmission region contains monopoles which are shorter than their resonant lengths, thus they absorb little energy from the line and in conjunc- tion with their feed capacitors serve only to load the feedline and reduce its impedance and velocity of propa- gation. The energy entering the active region encounters monopoles which load the line with resistive impedances, thus absorbing energy and radiating it into space. The remaining energy enters the unexcited region and since this region contains mostly reactive impedances, the energy is reflected back into the active region. The amount of energy entering the unexcited region is usu- ally quite small in a successful design. The upper frequency limit of a log-periodic structure is reached 21 when there is no longer a transmission region on the antenna, and the lower frequency limit is reached when there is no longer an unexcited region on the antenna. For calculating the Y f matrix components, I assume that the entire antenna is being operated in the transmission region and that the components of the Z matrix, which we assume in our matrix formulation are lumped at the monopole f eedpoints , can be assumed as distributed along the feed rod. In fact, we assume that the entire antenna is a tapered transmission line with uniform impedance between monopoles , whose induct- ance per unit length and capacity per unit length may be obtained, for each section, from Z L n " T* < 32 > and c n = rV • (33) o p n c where the nth section is between monopoles n and n+1, Z is the characteristic impedance of a section, and °n V is the velocity of propagation of all sections. Before proceeding with the calculation of the Y f matrix, we must obtain the V and Z quantities. This was p o ^ ^ n 22 first done experimentally for all the cases later treated mathematically and experimentally. These measurements of V and Z are shown, for two cases, in Figures 10 and 11. (V and Z may also be calculated using formulas developed from the V and Z measurements, which will be ^ p o given later.) Figure 10 will be used for the following explanation of V and Z calculation. It shows a F p o circular locus of impedances on a Smith chart whose nomi- nal impedance is 50 ohms. (All Smith charts in this thesis will have a nominal impedance of 50 ohms unless otherwise specified.) This locus is for a monopole array having 6 monopoles and 5 sections of transmission line, each with a different impedance. The higher meas- ured resistive impedance, or R2 , has a value of about 125 ohms, and the lower value, or Ri , is about 49 ohms. We may then assume that the average Z of the equivalent line is Z = /R 2 Ri = 7 8.2 ohms . (34) o avg At the higher resistive impedance point, we assume that the line is one-half wavelength long and accordingly transforms the 150 ohm load impedance by the ratio Z /Z . Then we have 05 01 23 Z /Z = R /R 2 = 1.2 (35) O5 Oi li and \/"z Z = Z =78.2 ohms . (36) * o o o 5 1 avg Solving (35) and (36) we obtain Z and Z values of Oi o 5 71.5 ohms and 85.8 ohms, respectively. We then calcu- late the other sections ' impedances from n-1 Z^ = Z o (1.2) 4 . (37) °n 0l The V may also be calculated from the locus of Figure 10 provided certain precautions are taken. It has been found that the most reliable V values are obtained by P using the lower frequency resistive crossover, about 120 MHz in Figure 10 , and by assuming that this is the fre- quency where the line is an electrical quarter wave. The frequency where this line is a physical quarter wavelength is 188 MHz. Taking the ratio of the electri- cal to physical quarter wavelength frequencies and multi- o 8 - plying by 3X10 gives a V of 1.92X10 m/sec. If the c higher frequency resistive crossover is used, and is assumed to be the electrical half-wavelength frequency, a lower V is obtained, which is not consistent with the P 24 low frequency approximation being made, since the coupling capacitors no longer have capacity values directly pro- portional to their physical lengths. We now have the Z values and V , which we assume o p n r is constant for all sections. We must then remove the shunting effect of the series combination of capacitors making up the Z matrix. This is done, how- ever, for a very low frequency where the coupling capac- itor will have a value of lO" C c = 3Z — y n P icofarads i ( 38 ) n c o where y is the length of a capacitor in meters. (We also assume that the Y and Z values are infinite at s a these frequencies.) The series capacitor of the L network may be calcu- lated from c e ■ zrt- farads ' (39) n on where X is the calculated reactance value at f and n o co =2Trf . One-half of the series combination of the o o above two capacitors plus one-half the series combina- tion of the (n+l)th monopole's capacitors is assumed to 25 be distributed along the nth section of line. This combination, converted to a per unit length capacity, must then be subtracted from C n to yield a new quantity ti * C .which will be used with the calculated value of L n n i i _ to obtain new values of Z and V for the Y x matrix. op t n *n The diagonal or self-admittance terms will be Y. = -jY cot B's - jY cot 3's ,/ (40) f J o n J o, n-1 nn n n-1 and the off diagonal terms will be Y- = jY esc 3's / 41 f , , J o n n,n+l n where Here s is the space between the centers of the nth and n ^ (n+l)th monopoles and 3 is 2tt/A. A tabulated calcula- tion for the case treated in Figure 10 is given in Table 1. The last line of this table is the half-sec- tion between the first node, where the input impedance is calculated, and the actual input terminal of the antenna. The true input impedance is obtained from 26 w to < « o fa. to o H En < D U <: - 04 rQ H ^■N CSl » vO LO LO e o O O O O - o -fl O O O O >> • • • • • s_^ O O O 00 vo CO LO O o c CN vO CN LO O rH H - D. ON 00 00 r^. r^ rH X! > • • • • • • en O O O O rH oo 00 00 00 00 c •"-s O O 00 O - o c o rH rH rH rH rH Nl hJ w co CO CO CO j 1 u ^N ^^^^ CO in O LO ^j^p* £3 • • • • • • n^ .C .H LO o\ CN O c O CN ro sf r^ o\ O - o ^ / rH rH rH rH rH rH Nl ti - H U /-% 1 B H «* CO 00 rH r^ - c "»**, • • • • • • u fe LO H 00 co ^■N + CM s vO rH CN r^ LO O c • • • • • • H PX4 l-» 00 00 00 00 vO O a ~tf « 00 LO r~- . rH rH rH c C o o u o + ,—N 00 -0- a> CN CO LO c e P4 LO rH rH «* CN QJ 0) a U u ^— ■ ' H rH CN 00 /— N H e B r^ LO LO LO VD r~- o ^^ • • • • ■ • CN pC< CN ^. 1 O 1 1 O 1 1 1 o 6 H rH rH rH rH rH Nl X X X X X X II X CN O 00 r^ vO CM - c ^-^ • • • • • • J r^ o\ CN ) E e (j>,£) - K^ I n e n x(l-cos 3h n ) ,(47) n=l where I is the element current, a complex number; X n ' e n is the element distance from the virtual vertex, meas- ured along the array; is the azimuthal angle with respect to the array line; and h is the height of an element. The agreement seems to be fairly good with respect to -3db beamwidth and front to back ratio. The calculated beamwidth is broader than the measured one. This characteristic is even more evident in Fig- ure 13, which compares measured and calculated beamwidths for Case D. (This again is a typical pattern.) A broader calculated H-plane beamwidth is in agreement 22 . . with the results of Carrel, ' which is not surprising, since the Z matrix formulation is the same as Carrel's, a 23 Both Carrel and Cheong " cite the sinusoidal approxi- mation as the reason for errors in the radiation patterns . Figure 14 shows the measured and calculated input impedance for Case A3. The average impedance is 32 slightly lower for the computed data and the VSWR is slightly higher. Figure 15 compares measured and calcu- lated impedances for Case C 3 . The agreement here is seen to be quite good both with respect to average impedance and VSWR. The calculated data here is ob- tained using a Y f matrix formulation based on the meas- ured locus in Figure 11. Figure 15A compares the com- puted data in Figure 15 with computed data obtained from a Y_ matrix formulation based on (44) and (45) . Cases C3 and D were both tested in this way, and the agreement between both methods of Y_ matrix formulation was quite good. Figure 16 summarizes all the Case A and Case B data. In general, the computed average impedance is lower and the computed VSWR is higher than the corresponding measured values. Figure 17 compares the measured and computed impedances of Case D. Again, we see that the computed average impedance is lower and the computed VSWR is higher than the corresponding measured quantities. An isolated shunt-fed monopole having the same feed capaci- tor and support arrangement as the monopoles in Case D had to be measured to obtain the resonant impedance for this case, since the effective diameters of these mono- poles are not constant over their entire length, and 33 Figure 6 could not be used. It is interesting to note that the resonant impedance obtained for an H/h of . 1 was 11.5 ohms which is the same as for a uniform cross- section monopole with H/h =.1 and h/a=30 (h/a for Case D is 60) . All experiments attempting to match this antenna, to a reasonable VSWR with respect to 50 ohms, without the outer vertical capacitor supports were unsuccessful. The reason remains unexplained. It was the antenna of Case D which provided the experimental evidence leading to the technique, using a t'>t for determining capacitor lengths, for optimizing the impedance match. A description of the experimental process of impedance optimization may be of interest. After selecting the log-periodic parameters of t=0.82 and a=0.15 from the dipole data of Carrel and adding the outer capacitor supports, the following additional initial parameters were selected; y =s , , H =0.1h , and v 2 n n-1' n n Z =51 ohms. This yielded a maximum VSWR of 3.2 with c J o respect to 63 ohms. Raising H to 0.12 h improved the average VSWR slightly but did not change the average impedance. With Z =34 ohms, y =s , and H =0.12 h , c c J n n-1 n n o the nominal impedance dropped to 43 ohms, but the VSWR rose to 8; H was decreased to 0.1 h , and the average ' n n 3 VSWR dropped, but a VSWR of 8 persisted at the lower 34 frequencies. y was now decreased so that y =(s -,)/2. ^ J n J n n-1 The average VSWR remained about the same, but the higher VSWR values were now at the higher frequencies. It appeared as if the capacitor lengths should be changed by some other ratio than the log-periodic constant t. With the front capacitor assumed to be yi=s =xsi and the rear capacitor assumed to be yi i=sio/2, a constnat T , = (yi/yii) 1/10 =(2TSi/s 1 o) 1//10 =2 1/10 T was calculated. 2 ' x=0.8785«0 . 88. The use of this ratio together with a slight increase in all capacitor lengths resulted in a VSWR of less than 2 with respect to an average imped- ance of about 44 ohms. See Figure 17. At this time Fred R. Ore sent me a copy of his report on the Wire-Outline monopole array with which he obtained "optimum" impedance match by step-tapering his feed rod to obtain less proportionate capacity at the rear of the antenna. A model was built, the antenna in Figure 1, to see if the method used on the antenna in Figure 2 would work with Ore's Wire-Outline mono- pole array. The successful result of this experiment is Case C3 which suggests that the length taper and the feed-rod diameter step-taper used by Ore are approximately equivalent, although the length taper is more easily implemented. 35 o n Why does this length taper improve the impedance match? The answer is connected with the rate of taper of Z values calculated or measured when operating the entire structure in the transmission region. A comparison of the values calculated for Cases A 3 and C3 is enlightening. For A3, an unsuccessful design, Z =71.5 and Z =85.8, * 01 05 while for C3 , a successful design, Z =73.6 and Z = * 01 05 103.0. The ratio of Z to Z is 1.2 in the first case 05 01 and 1.4 in the second case. Of even greater interest are the cutoff frequencies of the equivalent II or T networks for both cases. For Case A3 section 4 has a cutoff frequency of 780 MHz, and Case C3 has a cutoff frequency of 843 MHz. It so happens that section 4 is in the active region of the antenna at 720 MHz, which suggests that Case A3 may be very close to a stop-band situation, especially when one considers that the loading of the line within the active region may cause an even lower cutoff frequency due to the lower V . A recent 24 publication by Ingerson and Mayes tends to support this supposition. A possible explanation for the consistently lower characteristic impedances obtained in Cases A and B is suggested by Figure 18. The matching networks for these cases and for Case C3 were calculated for a trans- formation from 3 3.5 ohms to whatever resistance value 36 a given feedpoint height H required. The calculated self -impedance is 27.5 ohms, which is about 20 percent 25 less than the 33.5 ohm value obtained from King's second order approximation curve, the dashed locus in Figure 18. 37 V CONCLUSIONS A method of analysis , related to the Log-Periodic 5 Dipole analysis of Carrel, for the analysis of shunt- fed log-periodic monopole arrays has been presented. Calculated data has been compared with experimental data and a reasonable degree of correlation obtained. The method of analysis suggests an explanation for the success of experimental impedance optimization. The key to the success of the analysis is in the calcula- tion of the feeder admittance matrix which it is shown may be calculated entirely from physical dimensions of the antenna. 38 BIBLIOGRAPHY i. A. F. Wickersham, Jr., "Recent Developments in Very Broadband End-Fire Arrays," Proceedings of the I. S.E.j April 1960, pp. 794-795. 2. A. F. Wickersham, Jr. et al. a "Further Developments in Tapered Ladder Antenas , " Proceedings of the I. S.E.j January 1961, p. 378. 3. F. R. Ore, "A Coaxial Fed Unidirectional Log- Periodic Monopole Array," Collins Radio Company ., Cedar Rapids, Iowa, August 1961. 4. D. E. Isbell, "Log Periodic Dipole Arrays," Antenna Laboratory Technical Report No. 39j University of Illinois, Urbana, Illinois, June 1959. 5. R. L. Carrel, "Analysis of the Log-Periodic Dipole Array," Antenna Laboratory Technical Report No. 52, University of Illinois, Urbana, Illinois, October 1961. 6. E. Hudock and P. E. Mayes, "Near Field Investiga- tion of Uniformly Periodic Monopole Arrays," Trans. IEEE, Vol. AP-13, No. 6, November 19 65, pp. 840-855. 7. Ore, op. cit., p. 10. 39 8. Carrel, op. cit. 3 pp. 21-38. 9. Ibid. 3 pp. 26-33. 10. H. E. King, "Mutual Impedance of Unequal Length Antennas in Echelon," IRE Transactions 3 Vol. AP-5, No. 3, July 1957, pp. 306-313. 11. Weng-Meng Cheong, "Arrays of Unequal and Unequally Spaced Dipoles , with Application to the Log- Periodic Antenna," Scientific Report No. 14 (series 3) Contract AF19 ( 628) -2406 3 Craft Labora- tory, Harvard University, Cambridge, Mass., December 19 66. 12. Weng-Meng Cheong and Ronold W. P. King, "Arrays of Unequal and Unequally Spaced Dipoles," Radio Science 3 Vol. 2, No. 11, November 1967, pp. 1303- 1314. 13. Weng-Meng Cheong and Ronold W. P. King, "Log- Periodic Dipole Antenna," Radio Science 3 Vol. 2, No. 11, November 1967, pp. 1315-1325. 14. Carrel, op. ait. 3 pp. 31-33. 15. P. S. Carter, "Circuit Relations in Radiating Systems and Application to Antenna Problems," Proceedings of the I.R.E. 3 Vol. 20, June 1932, pp. 1004-1041. 16. Carrel, op. cit . 3 p. 27. 40 17. J. F. Morrison and P. H. Smith, "The Shunt Excited Antenna," Proceedings of the I.R.E., Vol. 25, June 1937, pp. 673-696. 18. Ibid. 19. Carrel, op. ait. 3 pp. 39-77. 20. Ore, op. ait. 3 Figure 7. 21. S. Uda and Y. Mushiake, "Yagi-Uda Antennas," Maruzen Co., Ltd., Tokyo, 1954, p. 19. 22. Carrel, op. cit . , p. 102. 23. Weng-Meng Cheong, op. cit. > pp. 10-15a - 10-15b. 24. P. G. Ingerson and P. E. Mayes, "Log-Periodic Antennas with Modulated Impedance Feeders," Trans, IEEE, Vol. AP-16, No. 6, November 1968, pp. 633-642. 25. R. W. P. King, "The Theory of Linear Antennas," University of Harvard Press, Cambridge, Mass., 1956, p. 169. 41 42 < 2 2 W Eh "Z 2 « CO rij & J W C ^1 ft o o Cn 2 ° Q 2 S o a i- <: u u H H Q PS O Q H 2 « H w J ft >H i u u O B tl H H Q S w |i4 1 Eh 2 5 a CO en •H 43 co UJ _i o Q. O UJ o < Q < I co CO ->■ > w < cr or m c "5* < \ < c "aa UJ _i o - CL o <=* o Q " o c £ QCL ^? UJ V a. II 1 < c o ^v 3 o a >- a: o o h- «~ Ul i Al o -1* UJ II s> o T JZ 1- v? " ro c co c a> ^>» i_ II \ 3 b >« o> 44 Ll) _l | => O UJ CM < C\J Q- CVJ X CO UJ o o H < CO UJ p: _l uj UJ ^ 1 UJ o 1 <: UJ 1- < t- z> u_ o o or O :> < cr e> < Q U_ 45 to < 3 o _) < o LU o < Q Ld a. < 3 I- 3 Q UJ CO 3 is O < > or \- o UJ CD CD Ll 46 60 55 / / h /o = 75 / 50 45 1 40 35 30 /h/ a = 30 Ri. "in 25 20 15 h /o = 14 2a — Cylindrical ■^ Monopole 10 / / « R n 5 / / i H \ — // / ' / 7 7 / / / / / / / / / / 0.2 0.3 H/h Figure 6. RESONANT IMPEDANCE OF A SHUNT-FED MONOPOLE AS A FUNCTION OF FEEDPOINT HEIGHT WITH HEIGHT TO RADIUS RATIO AS PARAMETER. 47 \ 2 S Q- \ „ S a: T5~ >i >1 T 1 ^ >l CO A C\J y £ ,00 X„ ] o 6 -C Jt X ■*-" <1 ■£\-" ' *» ^" ir> co r— ao co 1^f /"id ™ Z 3 ^A d s M r\ ■** ■KJ / J /_ r > ^1_--* ^xr; *«j-^» N CD CO X CD Ld X 0> LJ C\J CD 11 O O CL CO > f\l X III 2 1 O a. < X Q. CO < < CO z T en ce \- h- 7> co O Q Q. 1 h- Q 7* Ld Ld Ld a: Li_ cc 3 T CJ 1- $ K a> w. zy o> 9pnj!u6Dy\| juejjno 9Ai;D|9y 48 I) \ go S u. \ \ * X-*i \ S \ \ 1 1 s I — L. \ t E 0> o s to 0) Q_ \ CvJ £ °* \ 1 - \ \ 2 \ e o_ |-« — rvi — *■ £ s \ K 1_ ^T] (0 S5 " 3^ O o ]J] X I . ]T[ T^ II <^ - 1 ~J~_ 1 " ~^T^ u r (D 6 .c X s^^ & 1 N \ // 00 00 // o d -rCt^L- £ \ T/^nH N CD IjU X QQ O CC CL X CD UJ X u en CO CO > U- 3 O _c UJ _) UJ NJ O 1 J_ O o n 5 o <£ o o o 7> *■ -> O a UJ Q 2 Q UJ cr — 1 U_ UJ O cr (0 I i— Ll) Z ^ z> < < X cr CO o < X CD h- ~3 m < UJ — T ir h- >- eo UJ H Q 2 Q O H- UJ CL 2 ll ! 1 Q cr rv- Q 7* UJ UJ 00 UJ or cr IU3Jj.no aAnoiay 49 IMPEDANCE OR ADMITTANCE COORDINATES h/a = 14 for both monopoles H/h = 0.126 for shunt -fed monopole Shunt -fed monopole. Base fed monopole with matching network. {Network shown above) Figure 9. INPUT IMPEDANCE VS. FREQUENCY OF A SHUNT-FED MONOPOLE AND A BASE FED MONOPOLE WITH A MATCHING NETWORK. VALUES ARE NORMALIZED TO 9 OHMS. 50 IMPEDANCE OR ADMITTANCE COORDINATES Figure 10. LOCUS FOR Z AND V p DETERMINATION FOR CASE A 3 ANTENNA FEEDER IS TERMINATED IN A 150 a RESISTOR. 51 IMPEDANCE OR ADMITTANCE COORDINATES Figure II. LOCUS FOR Z AND V p DETERMINATION FOR CASE C 3 . ANTENNA FEEDER IS TERMINATED IN A 150 ft RESISTOR. 52 130 -140 150 -160 -170 180 170 160 Measured Calculated X- X 150 140 130 Figure 12. MEASURED AND CALCULATED RADIATION PATTERNS FOR CASE C 3 IN AZIMUTH PLANE. (Eg vs. ^ for = 90°) ARRAY VERTEX AT<£ = 0° FREQUENCY IS 720 MHz 53 - 30 -20 -10 < ) 10 20 30 ^^S^~/\ ^"^^^ *^s ,>» w ^y 0.9 - v^ >o< j NTx • -40 S/ N / — 0.8 \ / -C | s\ /Ik t0.7 a> | 35 -50 / 1 1 \x / A y L— | 0.6 \ ^ 1 \ ** X-\ 0.5 -i \ • 1 / I £C / \ \ \ \\\ \ \ 1 *\ -60 / 1 V / \ ,JU- 0.4— L sJ\- 0.3 — /___ \ 1 1 \ \ 1 \ -70 VJi 0.2 -f-7 y / V- A \ -80 \\V'o.i yV/ f\- — \ \ \ • 1 -90 p— ~^ f • \* '^"\c <*' ^^j ^^^^W-^» ■> ^"^ -100 /)^°- 2 "Y\ -110 i~~ 0l ~\ -120 / — 0.4 — V / / •130 -140 -150 -160 -170 180 170 160 150 O O Measured X- — — — — X Calculated 140 130 40 50 60 70 80 90 100 110 120 Figure 13. MEASURED AND CALCULATED RADIATION PATTERNS FOR CASE D IN AZIMUTH PLANE. (Eg vs for = 90°) ARRAY VERTEX AT=0° FREQUENCY IS 943 MHz 54 IMPEDANCE OR ADMITTANCE COORDINATES T = T Z co = 52 Ohms H n /h n = O.I26 © Experimental Points X Calculated Points Figure 14. INPUT IMPEDANCE OF WIRE -OUTLINE MONOPOLE ARRAY VS. FREQUENCY FOR CASE A 3 . VALUES NORMALIZED TO 50 OHMS. 55 IMPEDANCE OR ADMITTANCE COORDINATES r =0.8 Z co = 42.8 Ohms H n /h n = 0.126 O Experimental Points X Calculated Points Figure 15. INPUT IMPEDANCE OF WIRE-OUTLINE MONOPOLE ARRAY VS. FREQUENCY FOR "OPTIMIZED" FEED (Case C 3 ) VALUES NORMALIZED TO 50 OHMS. 56 IMPEDANCE OR ADMITTANCE COORDINATES X Calculated data with Vp and Zon obtained by experiment. O Calculated data with Vp and Zon calculated. Figure I5A A COMPARISON OF CALCULATED IMPEDANCES FOR CASE C 3 ( Vp and Zo data for Yf matrix calculations obtained in different ways) 57 UJ «MSA CO UJ UJ a. o UJ o UJ _l e> z < o UJ OQ UJ a MSA in UJ ui (T o ui Q ui _i o z < Q UI > CT> oo r— < UJ U CO c o UI a 3 UJ O - u. u. o tr O $ H- CO co > o Q o 2 Q < w UJ < o UI UJ < UJ tr 2 U. UJ > u. < O CD a> w 3 o> u. 58 IMPEDANCE OR ADMITTANCE COORDINATES T =0.88 Z = 33.5 Ohms Hn/hr, = 0.1 O Experimental Points X Calculated Points Figure 17. INPUT IMPEDANCE OF CYLINDRICAL MONOPOLE ARRAY VS. FREQUENCY FOR CASE D. VALUES NORMALIZED TO 50 OHMS. 59 IMPEDANCE OR ADMITTANCE COORDINATES O Q Computed Herein for h/a = l4 A A King's 2nd Order Approx. for h/a = l6.5 Figure 18. COMPUTED SELF IMPEDANCES FOR MONOPOLES AS A FUNCTION OF ELECTRICAL LENGTH, IN RADIANS, FOR CONSTANT h/a. (IMPEDANCES NORMALIZED TO 50 SI) 60 GPO 858 -052 PENN STATE UNIVERSITY LIBRARIES llllllillllllllllllllll A0DDD7201iab7