Comments on the Design of Log-Periodic DiDole Antennas

IEEE TRAKSdCTIONS ON ANTENKAS AKD PR0P.4GA4TIOX, VOL. AP-21, NO. 3, MAY 1973 303 Comments on the Design of Log-Periodic DiDole Antennas GIUSEPPE DE VITO AND GIOVANNI B. STRACCA Abstract-The design procedure of the log-periodic dipole (LPD) antennas is discussed. The results of the calculations carriedout by the use of the more recent and accurate methods of the analysis for the arrays of unequal dipoles, are compared with those already available in the literature. The calculations refer to the gain of LPD antenna as a function of Za, the characteristic impedance of the exciting transmission line, of h/a, the ratio between the halflength and the radius of dipoles, of T and u, two parameters which describe the geometry of the LPD antenna. The values of gain are lower than those given by earlier calculations. The degradation of the gain is more consistent, when the values either of 20 or h/a areincreasedwithrespect to Zo = 100 s2 and h/a = 125, which areparametersoften used in the literature.themanydiagrams included in this paper for different values of T, u, Zo, and h/a can be useful in a more accurate design of an LPD antenna. Confumation of the computer data has been obtained by the tests performed on various scale models and on a full scale antenna. T I. INTRODUCTION HE DESIGN procedure discussed in this paper is limited to the analysis of the gain of a single pla.ne log-periodic dipole (LPD) antenna in free space. The latter is a linearly polarized dipole array having a periodic structure which renders t.he ant,enna somewhat frequency independent. St.udies onlog-periodic st.ructures and LPD antennas began about, 13 years ago [l], [a]. The mathematical analysis of Carrel [3],[4], which took into account the mutual coupling between dipole e1ement.s and presented a step-by-step procedure for t,he design on an LPD ant.enna has been available since 1960. This analysis enables one t.o design the LPD antenna for a. wide range of input impedance, ga.in, bandwidth and size values, and t,o verify voltage and currents at t,he input terminals of t,he elements and a.t any feed line section. However, experimenta.1 results obtained by the authors and other experimenters [5], have shown considerable discrepancies from the design ta.rgets defined according to the criteria available in t,he literature [4], [SI. For example gain degmdations ranging from 2 to 3 db, as well as strong deformations of the radiation patterns and of the VSWR have been found in practice. The need of a more accurate design procedure, particularly for large highpower HF antennas with a high impeda.nce feed line, is quite evident to these authors. The reasons why previous result,s may not be satisfa.ctory, a,re discussed in Section III-A. A new design procedure has been developed and is presented in the sa.me f0rma.t as in [43 to permit ease of comparison; i.e., diagrams, similar to those shown [4, figs. 8 and 111, are present,ed and compared. The procedure here presented is based on a more sophisticated mat,hematical ana.lysis, which determines a. more accurate dipole current distribution. This analysis follows the method, recently proposed by King, Cheong, and other authors [7]-[lo], to study linear dipole arrays with unequal length and unequally spaced dipoles. The results of t.hese calculations are shown and discussed in Sect.ion IV in diagrams which pernlit the design of LPD antennas mit.hfeed line impedance Zo, ranging from 50 to 500 D and for three values of the rat.io h/a between the dipole half-1engt.h h and its radius a. Experiments have beenperformed in order to verify the results of the calculations in order to ascert.ain the accuracy obt,ainable wit,h the design procedure presented here. As shown in Sect,ion V, t.he agreement between the ca.lculat,ions and t.he experiments is very good. 11. MAIN PROPERTIES OF LPD ANTENNAS MAIS DEFINITIONS AND The LPD antenna consists of a copla.nar array of unequal length and unequa.ily spaced parallel linear dipoles, fed at their center by a parallel-wire transmission line of characteristic impedance Zo. This line is terminated in an impedance ZT, often equal to Zo. The 1engt.hs 1, = Zh, of the dipoles are a.rranged in such a way t.hat their ends lie on two straight lines, which intersect the antenna - axis at the point 0 (t,he antenna apex) and with the same angle a. The spacing d, between the dipoles, their lengths I,, their radius a, and their dist.ance R, from t.he apex are arra.nged in a geomet,ric progression: In order to describe the geometry of the structure a second paramet,er is needed in addit,ion to the logarithmic decrement r, i.e., either the angle a or t,he spacing ratio u, which is a function also of r and CY: Manuscript received June 10, 1972; revised December 4, 1972. G. De Vito is with the Complernenti Elettronici, Milan, Italy. G. B. Stracca is with the Istituto di Elettrotecnica ed Elettronica, University of Trieste, Trieste, Italy. d, 1-7 (T=-=- 26 4 cot CY.

..- 304 IEEE TRANSACTIORS ANTENNAS ON AND PROPAGA!ITON, MAY 1973 For a description of t-he properties and of t.he operation of Dhe LPD antenna the reader may consult the literature on the subject [1>[4], [7], [SI. It should be noted that at each frequency only a small portion of the antenna radiates. Such a region is called the active region [4]. In the ideal case where the active region could be reduced to only one dipole (i.e., the resonating dipole of length 1 = X/2), the operating bandwidth B = fmax - fmin would be given by the difference of the resonance frequency of the first and of the last dipole; i.e., when N is the number of dipoles, the ratio fmjfmin wouldbe, in this case, equa.1 to the ra.t.io: In fact this is not true, because in the active region there are N, = N1 + Nz dipoles (Nl dipoles shorter than X/2 dipole and N z dipole larger than the resonant dipole) ; fmax/fmin is t,herefore smaller than (3) and can be written: In (4), the factor I)B (< 1) has the meaning of a bandwidth efficiency (it coincides wit.h the inverse of Carrel s factor Ba,.). When VB is known, it. is possible to calculate the number of dipoles as a function of fmax/fmb and qb: = 1 + log ax/^^> 1% (fmax/vbfmin) =I+ 1% (1/r) 1% (1/r) Efficiency I)B (or the fact,ors K1 and K2) should be evaluated for each pa.ir of va.lues r, u. A criterion of evaluation of VB was given in [4], considering an active region limited to t.he dipoles whose current level, at the \ dipole input, is 10 db lower than the value at. the input of the dipole,wheresuch current. ismaximum,a.nd by assuming K1 = 0.5 (i.e., hrl = 0). Results for 20 = 100 0, h/a = 125, a,ccording to such criterion, are shown in [4, fig. 81. The corresponding values of K,, K2, and I)B evaluated as a function of r for the u values, which result in maximum gain following [4], me shown in curve a of Fig. 1. Curve b of Fig. 1 gives the K1, K2, and I)B values given in [6, fig. 4.41, for (3 (7..:. 1,.. _~. 1,.. t...,.l_-... I!.,~_!,..,...-,..!.....,. -. ~.. :..~~.,_. 1 I, 1. 1.9.8 -G Fig. 1. K,, Kf, and VB = K2/K1 values versus corresponding optimum 7 and u pairs (2, = 100 (1, h/a = 125): curve a is criterion of [4]; curve b is criterion of [6]; curve c is criterion given in section IV. the same values of r, u, Zo, and h/a. Curve c gives the K1, Kz, a.nd I)B va.lues evaluat.ed by following the criterion discussed in Section IV and for the u values, which result in maximum gain (for the same values of Zo and h/a). It has to be pointed out that, t.he choice of N, = 0 does not cha.nge very much the overall number of active dipoles N, = N1 + Nz, but it changes substantially t.he length L of the antenna and the number of longer dipoles, which have a st,rong influence on the cost of the antenna. In fact, L is given by: and depends in a releva.nt way on N,. The optimiza.tion of the antenna should consist in finding the L and AT pair of values, which result in the wanted design t,argets (gain, bandwidth, and VSWR) and choosing the pair which results in t.he best cost compromise. 111. COMMENTS ON &IATFIEbfATICAL ANALYSIS OF LPD ANTENNAS Both in Carrel s [4] and King s [lo] ana,lysis, a system of equations is writt.en to re1at.e t.he currents I,(O) at the input. of the dipole elements to the current lo(0) at the input terminals of the wholeandenna. Solution of this equation system requires the knowledge of the elements of two admittance matrixes: t.he { YA] matrix ofat-port network,composed by t,he N dipoles mutually coupled to each other, and t,he { YL) mat,rix of another N-port net,work, composed by the feed line, whose terminals pairs are the feed points of t,he dipoles. The e1ement.s of t,he { YL) matrix are t.he same in both the analyses. The two analyses differ, as shown in Section 111-A and B for the derivation of t.heelements of t,he {FA] matrix (i.e., t,he input. admittance of t,he N dipoles and their mutual admittances). In fact the { YA) depends on the current,s distribution I, (2) on the dipoles, which are governed by a set of integra.1 equations. Different appr0ximat.e mays are used in the two methods of analysis

DE VITO AND STFLACCA: DESIGN OF LOG-PERIODIC DIPOLE 305 to solve such integral equations, which lead to different { YA} elements. When the { YA) and { YL) elements are cdculated, it is possible to solve the matrix equation of t.he system and to calculate the input current of the dipoles and the input impedance of the antenna at any frequency. When the input currents I,(O) of the dipoles are known, it is possible to derive the approximate current distribution and, in successive steps, the radiation field, the directivity and the gain. A. Comments on Mathematical Analysis and Design Procedure of Carrel To derive the elements of the matrix { YA 1, Ca.rre1 starts from the hypothesis of a sinusoidal current distribution in the dipoles and assumes therefore, for the elements of the mat.rix {ZA) = { YA}-~, the input impedances and mutual impedances derived from the well known first approximation theory of the antennas. In [4], details of the procedure are not given; only the results of the calculations are shown in some particular case. The main results are given in [4, figs. 8 and 111. Curves of [4, fig. 111 are reproduced in this paper as Fig. 2 and give all t,he pairs of values of r and u which result in a given Go. In fact, N and L (and not 7 and u) govern the cost of the antenna, but they are not given in diagrams of [4, fig. 111. It is possible, however, by determining the act,ive region length with the help of [4, fig. SI, to calculate N and L for any pa.irs of 'I and u and, by means of a cutand-try process, to find the best compromise for N and L, which allows to obtain the design targets (Go a,nd fmsx/fmin>. Alt,hough in a single dipole antenna the sinusoidal distribut.ion hypothesis affects only in a negligibleway the gain comput,ation, the situation may be subst.antially different in the LPD array, where there are many dipoles cont.ributing to the radiation field a.nd where amplitnde and phase of currents at the dipoles inputs depend on the true impedancespresented t.0 the feed line and on the mutual impedances. The sinusoidal distribut.ion seems particularly poor, when large power LPD antennas are considered, having low Q dipoles and high Zo. B. Comments to King's Analysis Method G(dB/iso) 12 11 10 9 a? 6 5 4 3 10 11.12 63.14.15.16.17.18.19.20.2l.22 6 Fig. 2. Comparison of GO versus u for given values of 7(Z0 = 100 Q and h/a = 125): curve a is values calculated in [4], and curve 6 is values calculated in this paper A T = 8, ZT = Zo. chosen, may be writ,ten for the qth dipole: + IC, cos -- cos -. (9) (2 Oh) 2 In this way, because t.he vector pot,ential A,, on the qth-dipole axis may be alsoexpressed as a function of two sinusoida.1 terms of amplit,ude AI, and AZ,, the integral equation, whichgives t,he relationship between the unknown current distribut,ion I,(z) on the various dipoles and A,,(z), is transformed in a set of algebric equat,ions. For example, by equating the A,, values to the values of the int.egra1 expression at 4 different values of z. The unknown variables are the amplitude IAJB,,Ic,, and the amplitude AI, of the first t,erm of Az,; t.he amplitude APq of the second term may be found as a funct,ion of the vohage Vo at the input. The indicated procedure allows one to transform the system of N integral equations, which has to be solved in this case, in a system of 4N algebraical equations with 4B unknown variables (i.e., the three IA,,IB,,Ic, amplit,ude of t.he currents, and AI, of AB at each dipole). In this pa.per a short account is given of the computations made in order to obtain a more accurate rela.tionship between the gain and the geometry of t.he structure, With the aim to obtain a set of computed diagrams, ITT. TENTtlTIVE DESIGN PROCEDURE which could be more useful to designers than the existing ones, the analysis method due to King, Cheong, and The analysis method described in section III-B has other workers [7]-[10]was used for the calculation of been used to verify which modifications are introduced, the dipole current distribution. A current distribution, by new calculat.ions, in the diagrams of [4], reproduced obt.ained summing only three sinusoidal t.erms, gives on curves a of Fig. 2, for one LPD a.nt,enna in free space, accurate results, at least when 2 h/a >> 1, i.e., when the nith 20 = 100 Q and h/a = 125. The results of the approximation involved in neglecting the effect of the new calculations, for the same values of Zo, h/a and N, terminal cha,rges (end effect) and of t,he capa.cit.ance a.re given in the diagrams b of Fig. 2. The large end terminabetween the input terminals have a small influenceon tion is equal to the transmission line characteristic the results. Such current dist.ribution, as a function of the impedance. dipolehalf-length h of the phase propagation constant By comparing curves a and b of Fig. 2, the reader may p and of the z coordinate, when as z axis the dipole axis is easily observe that the first approximation analysis leads

,- 306 IEEE TRANSACTIOXS ON ~TENNAS AND PROPAGATION, MAY 1973 %. applications, to a more optimistic result. This means that a more expensive ant,enna is needed t.0 obt.ain the wanted design target for the gain. Both curves a and b of Fig. 2 show t.ha.t for each Go there exists an optimum pair of values u and r. For the same value of T the curves a exhibit larger values of Go than those of the curves b. Differences in t-he gain values seem more important for the smaller values of 7 (for example, approximately 2 db for r = 0.82). However, it has to be pointed out that a number of dipoles N = 8 is too small, when T becomes larger. The following diagrams of Figs. 3-6 have therefore been calculated with 12 dipoles, if 7 < 0.94, and with 16 dipoles, if r 2 0.94, in order to simulate better the limiting case of an ideal very long antenna. These diagrams allow one to check t.he influence of 2 0 and h/a (i.e., of the Q of dipoles). The behavior of Go versus u, with r as parameter, is shown for Zo = 50 Q (Fig. 3), Zo = 100 Q (Fig. 4), Zo = 150 Q (Fig. 5), Zo = 500 Q (Fig. 6), respectively. In each figure,curves are presented for h/a = 125, h./a = 340, and h/a = 2700. Different values of h/a have been considered in order to take int,o account the most common where t.he h/a is high as for low power LPD and low as for high power antennas. If we compare the curves a of Fig. 2 with those of Fig. 4 (i.e., 20 = 100 Q, h/a = 125), we can observe t.hat the gain degradation, with respect to the calculation of [4], becomes less important for very large values of 7 than what, it results by comparing curves a and b of Fig. 2. This result is due to the larger number of dipoles taken into account for diagrams of Fig. 4 and not to a better accuracy of curves a of Fig. 2. Furthermore, it may be observed t.hat the curves of Fig. 4 present a posit,ion of the maximum, shifted towards higher values of u -4th respect to curves a of Fig. 2. These resu1t.s are summarized in diagram of Fig. 7, where the va.lues of GO, corresponding to the ma,ximum of the diagrams of Fig. 3-6, are presented as a function of Zo with 7 as parameter. All diagrams presented here have been calculat.ed assuming i?1 = 2 (i.e., two dipoles behind the X/2 dipole),,except for the diagram of Zo = 500 Q. When Zo = 500 9, for the smaller value of T(T < 0.84), LVl = 4 hasbeen used in calculation, in order to avoid consistent variations of gain and of other characteristics with frequency. This influence is clearly shown in Fig. S, where the behavior of Go, as a function of t.he frequency of operation, is shown for u = 0.16, T = 0.78 (i.e., for a very lorn value of 7). Some remarkable gain variations, with respect to the values shown in the diagrams, may be obtained by changing the frequency of operation, for high Zo (see Fig. S). This effect is responsible for the irregular sha.pe of the curves of the diagrams of Fig. 6. Such results have to be taken carefully into account, particularly in the design of high power antennas, where high 20 values are generally used to reduce the volt,a,ge gradient betmeen the line wires. Fig. 8 and t,he comparison of Fig. 2 (curves b) and Fig. 3 @/a = 125) show also that the eva.luation of K1 G(dB/iso) Fig. 3. Values of GO versus u, for given values of h/a, 20 = 50 Q, ZT = 20 (12 dipoles for T < 0.94, 16 dipoles for 7 2 0.94) and the following values of T: curve A- = 0.96; curve l3- = 0.94; curve C-T = 0.92; curve D- = 0.9; curve E--r = 0.86; and curve F-T = 0.82. 2-..,..IO.ll 12..13.I4.15.16-1L-I. d-.-j _-I..17.18.I9.20.21.22 6 Fig. 4. Values of GO versus u, for the givenvalues of h/a, 0 = 100 o, ZT = 20 (12 dipoles for T < 0.94, 16 dipoles for T 2 0.94) and the following values of T: curve A 7 = 0.96; curve B--r = 0.94; curve C-T = 0.92; curve D--r = 0.9; curve E--r = 0.86; and curve F-T = 0.82..~-.. I.. :.-..........._......_ 4 - F.ya:=...-.......,. -~ 125,..... _.... I, 3. ~. -+- :Vi.=-340,..,.,...... I..... ; I - %I 2704... --!.,.. 2 L -.,.... ~., i-;ldl. I. 10.I 12.a.14 65 16.17.18.19.20 21.22 6 Fig. 5. Values of GO versus U, for given values of h/a, ZO = 150 a, ZT = 20 (12 dipoles for T < 0.94, 16 dipoles for T 2 0.94) and the following values of T: curve A 7 = 0.96; curve = 0.94; curve C 7 = 0.92; curve D--r = 0.9; curve E-T = 0.86; and curve F-T = 0.82.

DE VITO dyd STR4CCA: DESIGN OF LOG-PERIODIC DIPOLE 307 G(dB/iso) 5-4. Fig. 6. Values of GO versus c, for given values of kla, 20 = 500 fl, ZT = ZO (12 dipoles for T < 0.94,16 dipoles for T 2 0.94) and following values of T: curve A 7 = 0.96; curve B-T = 0.94; curve C 7 = 0.92; curve L)1 = 0.9; curve E 7 = 0.86, and curve F-T = 0.82. 2-,, ~. I. >. I 50 100 200 300 400 - _ -. 500 1 I I ZO(fl) Fig. 7. Go versus 20 for given values of hla and following values of 7 (U optimum): curve A 7 = 0.96; curve B-r = 0.94; curve C 7 = 0.92; curve D 7 = 0.9; curve E 7 = 0.86; and curve F----7 = 0.82. f " fn-1 f N -2 Fig. 8. Behavior of GO versus frequency of operation where fx is resonating frequency of longest dipole, for 2 0 = 500 n, hla = 125, T = 0.78, u = 0.16, ZT = 20. Fig. 9. Behavior of Go versus overall length L, as function of number N of dipoles for fmax/fmin = 4, Zo = 100 $2, hla = 125. Length L is normalized with respect to X,. and Kz, i.e., VB or the number ATa = N1+ IVZ of the dipoles of t.he active region, is somewhat critical. If one t.akes, as in [4], N1 = 0, a remarkable degradation of gain and of t,he input impedance may be often found pa.rticularly for higher Zo values and lower T values. In this paper, curves of Fig. 1, which give the active region for h/a = 125 and Zo = 100 il and u opt.imum, have been evaluated wit.h the following criteria. 1) As far as N1 isconcerned, the degradation of the ga.in has been observed, when t,he frequency of operation is moved (depending on t.he value of T, 20, and h/a) between the frequencies of resonance of dipoles number N and number N - 3. hginimum frequency has been taken as the frequency of operation for which the gain is 0.5 db lower t.han maximum gain; 2) As far as IV, is concerned, it has been observed that when T < 0.92 the gain decreases slowly by decreasing Nz, only as fa,r as Nz is smaller t,han a critical value, and de- creases abruptly when AV2 is larger. Such a critical value has been taken for the Nz value. When T 2 0.92 t,he extension of the active region has been t,aken up as t,he Np value, which results in a 0.25 db loss of gain with respect to maximum gain. Curves of Fig. 1 give the behavior only for 20 = 100 Cl and h/a = 125. By increasing 20, t,he number N 1 of dipoles is more critical. By increasing h/a (i.e., t.he Q of the dipoles), the number Nz of dipoles becomes lower. By taking into account the curves of Fig. 3-6 and the N1 and N z values, found in t.hisway, it is possible to calculate the values of the ga.in as a function of t,he overall length L of the antenna and the overall number N of the dipoles which are t.he parameters governing the cost of the antenna. Fig. 9 is an example in the case of fmax/fmin = 4, 20 = 100 n, h/a = 125. v. EXPERIMEhTAL RESULTS In order to check the h t approximation analysis of [4] and t,he one described in t.his paper, antenna models have been made mit,h the characteristics indicated in

308 IEEE TkANSACTIONS ON ANTENNAS AND PROPAG.4TION, MAY 1973 TABLE I DIMENSIONS OF EXPE-NTAL MODELS Frequency Number z, of L of test 21 k2 u dipoles (mm) (MEtz) (mm) (mm) h/a (Q) 7 100 0.8 0.14 12 395 700 26.5125 308 100 0.88 0.16 12 605 620 76 308 125 100 0.94 0.18125 308 1211225101122 420 0.146 0.93 12 84 480 336151 125 neither on the accuracy of the mathematical approach nor on the computer program, but on the practical difeculties experienced in manufacture and testing a high impedance model. VI. CONCLUSIONS Diagrams have been presented, which show, for different values of 20 and h/a, the relationship between gain and antenna geometry for a free-space LPD antenna. Such diagrams allow, as confirmed by the experimental tests performed on some antenna models of different, geometry, a more precise and more complete evaluation of LPD antenna structures than results previously available in the technical literat,ue. They can, t.herefore, be used for a more accurate design of such kind of antennas. Fig. 10. Far-field radiation pattern (H plane) for models of Table I. Comparison between computed and experimental values (circles). Curve a: T = 0.8; u = 0.14; Zo = 100 Q; h/a = 125; ZT = 20. Curve b: 7 = 0.88; u = 0.16; 20 = 100 Q; h/a = 125; ZT = 20. Curve c: T = 0.94; u = 0.18; 20 = 100 3; h/a = 125; ZT = 20. Curve d: 7 = 0.93; u = 0.146; Zo = 420 Q; h/a = 125; ZT = 20. Table I. The frequency of operations has been selected in order to attain both an easier construction and an easier testing of the models. The parameters of the first. three models have beenselected in order to checkt.he results of both the analyses in three more interesting different point,s of diagrams of Fig. 2, curves a, and Fig. 3 @/a = 125) (i.e.,for 2 0 = 100 a). The thee values of r have been selected to check low, intermediate, and high 7 values; the corresponding u values are those which result in the maximum of Go, respectively. The fourth model has been made with higher 20 value (i.e., 20 = 420 Q). In order to check the theoretica.1 gain, t.he far-field radiation pattern was calculated for each model, following the procedure indicated in Section 111-B and compared yith the measured far-field radiation pattern. Results for the four models are shown in Fig. 10. Agreement is very good, mainly for 20 = 100 8, and gives confidence about the precision of the method of calculation. The experiment,al results for the higher impedance model (curve d) are slightly worse. The reason probably depends ACKNOSVLEDGMENT The authors wish to thank Dr. Ing. V. Moretti for his assistance and contribution in the computer calculation of the diagrams presented here. All the calculations have been performed on t.he high speed digital computer, IBM 7044, of the Centro di Calcolo of the University of Triest,e, by using with some modifkations, the computing program GINRAY, which was studied by Mr. Parker in strict collaboration with Mr. Rodoni, both of the Australian Post Office, in the cooperative m-orking effort with Complementi Elettronici, which lead to the design and installation at Darwin, Australia, of a 500 kw carrier power LPD a.ntenna, for application t.0 RF broadcasting in t.he frequency band from 5.95 to 26.2 MHz. REFERENCES 111 R. H. Du Hamel and D. E. Isbell, Broadband logarithmically periodic antenna structures, in 1957 IRE Nat. Cmv. Rec., pt. 1, pp. 119-128. D. E. Isbell, Long periodic dipole arrays, IRE Trans. Antennnas Propagat., vol. AP-8, pp. 260-267, May 1960. R. L. Carrel, An analysis of the log-periodic dipole antenna, Dresented at 10th Ann. S-D. USAJ? Antenna Res. Dev. hog. Oct. 4, 1960. 141 -, The design of log-periodic antennas, in 1961 IRE Int. COW. RK., v01. I, pp. 61-75. 151 C. C. Bantin and K. G. Balmain, Study of compressed logperiodic dipole antennas, IEEE Trans. Antennnas Propagat., VO~. AP-18, PP. 195203, Mar. 1970. [61 C. E. Smith, Log-periodic antenna design handbook, Smith Electronics, Inc., Cleveland, Ohio, Res. Rep., 1966. 171 W. M. Cheong, Arrays of unequal and unequally spaced elements, Ph.D. dissertation, Harvard University, Cambridge, Mass., 1967. W. M. Cheong and R. W. P. King, Log-periodic dipole antenna, Radio Sn., vol. 2, pp. 1315-1326, Nov. 1967. - Arrays of unequal and unequally spaced elements, Rad& Sci., vol. 2, pp. 1303-1314, Nov. 1967. R. W. P. King, R. B. Mack. and S.S. Sandler, Arraus of Culindn cal Di&ks. New York: Cambridge Uhiv. Press. 1968. 1111 A. Boswell, Log-periodic dipole arrays, Marmi Rev., no. 3 pp. 225-231, 1970. [la] A. G. Roederer, Calculation of the electromagnetic field radiated by a log-periodic dipole antenna, Philips Res. Rep., VO~. 23, pp. 175-188, 1970. [13] R. H. Kxle, Mutual coupling between log-periodic dipole aneennas, IEEE Trans. Antennas Propagat., vol. AP-18, pp. 15-22, Jan 1970. -

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