Uo in 
 
 MASTER COPY 
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 ANTENNA LABORATORY 
 Technical Report No. 59 
 
 ' 
 
 3V/ 
 
 THEORETICAL STUDY OF A CLASS 
 LOGARITHMICALLY PERIODIC CIRCUITS 
 
 by 
 
 Raj Mittra 
 
 Contract No. AF33(657)-8460 
 Project No. 6278, Task No. 40572 
 
 JULY 1962 
 
 Sponsored by 
 
 AERONAUTICAL SYSTEMS DIVISION 
 
 WRIGHT-PATTERSON AIR FORCE BASE, OHIO 
 
 ELECTRICAL ENGINEERING RESEARCH LABORATORY 
 
 ENGINEERING EXPERIMENT STATION 
 
 UNIVERSITY OF ILLINOIS 
 
 URBANA, ILLINOIS 
 
Antenna Laboratory 
 Technical Report No. 59 
 
 THEORETICAL STUDY OF A CLASS 
 OF LOGARITHMICALLY PERIODIC CIRCUITS 
 
 by 
 Raj Mittra 
 
 Contract AF33(657)-8460 
 Project No. 6278, Task No. 40572 
 
 July 1962 
 
 Sponsored by 
 
 AERONAUTICAL SYSTEMS DIVISION 
 WRIGHT-PATTERSON AIR FORCE BASE, OHIO 
 
 Electrical Engineering Research Laboratory 
 
 Engineering Experiment Station 
 
 University of Illinois 
 
 Urbana, Illinois 
 
ACKNOWLEDGMENT 
 
 The work reported in the first half of this report was carried out when 
 the author was employed by the Transport Division, Boeing Airplane Company, 
 Seattle, Washington, for the summer of 1961. Thanks are due to Mr. Dwight 
 Isbell for many fruitful discussions during the course of the author' s stay 
 at Boeing. The support provided by Aeronautical Systems Division, Wright- 
 Patterson Air Force Base, Ohio, through Contract AF33 (657)-8460 is gratefully 
 acknowledged. It is also a pleasure to thank the research personnel of the 
 Antenna Laboratory for many helpful discussions, and to acknowledge the help 
 of Mr. Marvin Wahl and Mr. Charles Liang for carrying out the numerical 
 calculations. 
 
CONTENTS 
 
 Page 
 
 1. Introduction 1 
 
 2 . Lumped Log Periodic Circuits of Foster Type 2 
 
 3 . Log Periodically Loaded Transmission Line 9 
 4. The Brillouin Diagram and Its Application to Tapered Structures 18 
 References 25 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/theoreticalstudy59mitt 
 
ILLUSTRATIONS 
 Figure Number Page 
 
 1 First order Foster type LP network 2 
 
 2 Equivalent circuit for a general first 
 
 order Foster type LP network 5 
 
 3 Typical plot for T. for a second order Foster LP network 9 
 
 in 
 
 4 LP loaded transmission line 10 
 
 5 Cell voltage amplitude and phase plots 
 
 for a loaded LP line 15 
 
 6 Amplitude and phase plots for a loaded 
 
 LP line 16 
 
 7 k-p diagram for a sinusoidally modulated 
 
 reactance surface (after Oliner and Hessel) 18 
 
 8 Simply periodic loaded transmission line 19 
 
 9 k-(3 diagram for a loaded transmission line 21 
 
 10 k-(3 characteristics for a lossy single 
 
 resonant shunt loading 22 
 
 11 k-(3 diagram for a LP dipole antenna 
 
 (calculations based on results by Carrell) 23 
 
 12 k-|3 diagram for the shunt cirrent in a 
 
 o 
 loaded line with additional 180 phase 
 
 shift per cell 24 
 
1. INTRODUCTION 
 
 Many log-periodic distributed circuits in the form of various antenna 
 
 12 3 
 structures have been investigated experimentally be DuHamel ; Isbell ; Carrel 
 
 4 
 
 Mayes ; and others. Theoretical investigations of LP circuits have been made 
 
 by DuHamel^ Carrel^ and Mayes ? et. al . DuHamel has analyzed a particular 
 lumped type of LP circuit composed of an infinite number of admittances in 
 parallel and has given closed form expressions for the representation of the 
 admittance of such circuits. In a later work he has also studied some general 
 properties of LP structures. Carrel has done a considerable amount of numeri- 
 cal work in connection with the LP antenna structure with a finite number of 
 elements. 
 
 The purpose of this paper is threefold: 
 
 a) to derive closed form expressions for lumped Foster type LP circuits. 
 
 b) to derive the characteristic equation for an infinite log-periodically 
 loaded transmission line and to discuss a method of solution of the 
 above equation. 
 
 c) to present a general approach towards studying a class of LP structures 
 in terms of the Brillouin (k-P) diagrams of the corresponding simply 
 periodic structures. 
 
2. LUMPED LOG PERIODIC CIRCUITS OF FOSTER TYPE 
 
 In the first part of this paper we shall derive a closed form expression 
 for the lumped Foster type LP network shown in Figure 1. 
 
 o 
 
 -o 
 
 .Y-r 
 
 .Yd 
 
 Y 2 ~S Y N " 
 
 Figure 1. First order Foster type LP network 
 
 We shall consider the first order circuit first and will subsequently extend 
 our analysis to the r order network. A first order LP Foster type of cir- 
 cuit will be defined as one which has the representation 
 
 Y in (S) = 23 
 
 (1) 
 
 n =-ooR + a SL + 
 
 where R ; L and C are constants and S = <T + jw ; or the complex frequency. 
 It is easily verified that Y. is log-periodic^ i.e.,, 
 
 Y. (as) = Y. (S) 
 
 in in 
 
 (2) 
 
 and a j. s identified as the log period of the network. An r order circuit 
 of the same type admits a general representation 
 
 Y. (n) (S) 
 
 m 
 
 r °o 
 23 23 
 
 q=l n=-°o R + a n S L + ■ — — 
 q q a n SC 
 
 (3) 
 
As a first step toward deriving a closed form expression for the first order 
 
 network we shall prove a Lemma ; followed by a theorem applicable to such 
 
 structures. 
 
 Lemma : 
 
 The driving point admittance function of a Foster type log-periodic 
 
 admittance network is pure real for two infinite sequence of frequencies 
 
 S = - iw a- and S = - iw a where w is a constant depending on the net- 
 o o o 
 
 work and n goes from -°o to +°° through integral values. 
 Proof: 
 
 From Equation (1) we obtain by substituting S = jw 
 
 Y. n (jw) = S 7—^— i r = ~T - 
 
 n=-°o R 
 
 • /„n T 1 \ w L / W v 
 + j {a wL _ \ o n==-~ / Q n w o\ 
 
 ° ^ (4) 
 
 = g. + jb. 
 
 in in 
 
 where 
 
 o ^/jEc > o w, l 
 
 Y may be re-written as 
 in 
 
 °o D oo w w a n 
 
 w L Y. = g. + jb. = 2 - j 2 — (5) 
 
 o in °in in 2 J w 1 .2 
 
 n=-°° 
 
 n 2 /nw W o \ n =-°°D 2 + fa n ^ °_ Y 
 
 o w a 
 
 Now we shall show that 
 
 b in = for w = a w , n goes from -°o to °o 
 
 n+1/2 
 
 and for w > = a ... . w 
 
 o 
 
First by substituting w = w in Equation (3) we get 
 
 oo 
 
 b. I = - 2 (6) 
 
 in 
 
 I w=w- n=-°° 
 o 
 
 2 / n -n'N 
 
 It is easily verified that b I =0 since the n = m term in the sum 
 J in|w=w 
 
 cancels with the n = -m term, and the n = term is identically zero. 
 
 From the log-periodic nature of Y. it is obvious that Y. =0 for 
 to * in in 
 
 w = a w where n goes from -°° to °° 
 
 o 
 
 1/2 
 Next put w = w a in Equation (5) obtaining 
 
 o 
 
 oo 
 
 -J S 
 
 , n=- 
 w=a ' w 
 
 o 
 
 (.^. a^j/ (.„.,(. 
 
 ■ i_ 
 
 2 
 
 (7) 
 
 Again it is verified that b. =0 since the term corresponding to n = m in the 
 sum cancels with the n = -(m+1) term. 
 
 It is a trivial step to extend the result to the negative frequencies by 
 using the relation 
 
 b. (w) = -b. (-w) 
 
 in in 
 
 Hence the Lemma is proved. 
 
 Theorem 1 : 
 
 A general first order log-periodic circuit of Foster type is equivalent 
 
 at its input terminals to a pure resistance network terminated by a purely 
 
 lossless two terminal log-periodic network with a driving point reactance 
 
 function of Foster type. 
 
 Proof: 
 
 From the form of Y. (S) given in Equation (1) we see that Y has denumei 
 in in 
 
 ably infinite numbers of poles and zeros in the complex S-plane. Also Y. (3) 
 
 in 
 
 is analytic on and in the neighborhood of the imaginary axis S = jw. Let 
 
Y. (jw) be a function which matches Y (jw) at denumerably infinite number of 
 in in 
 
 points, i.e. 
 
 Y in (jw r ) = Y. n (jw r ) for r = 1, 2, . . . oo 
 
 = 1, -2, ... -« 
 
 Where w 's may be located arbitrarily. Then from the well known sampling 
 theorem which may be extended for the case of non-uniform sampling, one may 
 prove 
 
 Y in ( J' W) - Y in ( J W) 
 
 Since Y. (j'w) is identical to Y (jw) for all w through analytic continuation 
 in in 
 
 we may show that 
 
 Y. (S) = Y. (S) 
 
 in in 
 
 Next we show that we can construct a Y. which matches Y. for a denumerably 
 
 in in 
 
 infinite number of points. 1 
 
 n+— 
 
 We have already shown that Y is pure real for S = - -ja w and S = - ja w 
 
 in o o 
 
 Y. = g n for S = - jci n w 
 in 1 o 
 
 j + g for S = - ja w 
 i Z o 
 
 n goes from -°o to °° (8) 
 
 Now let Y. be the driving point impedance of a network structure shown in 
 Figure 2. 
 
 Yfn 
 fin 
 
 -^ 9i 
 
 bi„U)-J 
 
 ^3 
 
 LOSSLESS 
 NETWORK 
 
 ZEROS AT S=tjw o a n 
 POLES AT S = tjw ft a r * l/2 
 
 Figure 2. Equivalent circuit for a general first order Foster 
 type LP network 
 
Then a little deliberation will show that 
 
 + n 4 
 
 Y. (jw) = Y (jw) for w = - o. w 
 
 in in o 
 
 and w = - ci w 
 
 n goes from -°° to °° (9) 
 
 and that the lossless LP network characterized by zeros and poles at S = 
 
 - jw a and S = - jw 1 respectively is a physically realizable Foster 
 o o 
 
 type log-periodic network. Hence the theorem is proved. It should be noted 
 
 that the turns ratio of the lossless transformer has to be determined through 
 
 one additional piece of information. 
 
 As a next step we derive a closed form for b. (1) which is the input 
 
 in 
 
 susceptance of the terminating lossless network described above. Its pole 
 and zero locations are given by: 
 
 zeros at w = - w a 
 o 
 
 and 1 n = -oo . . . -2, -1, 0, 1, 2, . . . °° 
 
 + n+- > , , , , , 
 
 poles at w = - a 
 
 Because of the nature of the location of the poles and zeros, it will be 
 found convenient to consider a variable W defined by 
 
 W = In w/w (11) 
 
 o 
 
 Then the location of poles and zeros, rewritten in terms of the new variable 
 become doubly periodic as seen from the following: 
 
 zeros at W = n r+ jm 77 n = -<» -2, -1, 0, 1, 2 ... oo (12a) 
 
 m = arbitrary integer or zero 
 
 poles at W = (n + -) r+ jm 77 n and m having the same range as in (12a) 
 
 (12b) 
 where J? = In a . 
 
 Functions exhibiting such double periodicity may be expressed in terms of 
 the Elliptic Functions. DuHamel was first to recognize this property of the 
 
LP Foster type networks. Now consider the infinite product representation for 
 the elliptic function sn(u, k) given by 
 
 sn(u,k) = ^ sin (JT u/2 K) n II- (13) 
 
 n R-q 211 " 1 '! fi - 2q 2 n cos (7r U/K) + q4n ~ l 
 
 n= l L 1-Q 2n J [i " 2q 2n_1 cos (77 u/kW^J 
 
 q = e" 7772 KVK , K = sn" 1 (l,k), K' = sn" 1 U,k'), k' ="]l-k 2 
 It is seen that the function has zeros at 
 
 •j — = jm7T - nlnq n = 0, 1, 2, ,,. . « m = 0, - 1, '- 2 (14a) 
 
 2K 
 
 and the poles are located at 
 
 -^ = jm77 - (n + i) lnq n = 0, 1, ...<», m = 0, - 1, - 2 . . . (14b) 
 2K 2 
 
 Hence we may construct a representation for b. (1) as follows: 
 
 , ln (i)-«.» (J-2p -, (15) 
 
 /j 2K In w/w , k) 
 
 C sn 
 
 where C is a constant as yet undetermined. It may be determined if b. (1) is 
 
 in 
 
 known for an additional value of w. However^ the constant C may be conveniently 
 
 absorbed in the transformer turns ratio r (see Figure 2). Equation (15) is of 
 
 the same form as obtained by DuHamel for a first order lossless Foster LP network. 
 
 It has now been shown that the general first order LP network may be 
 
 represented by a structure shown in Figure 2. We are now ready to state another 
 
 theorem concerning the general first order LP Foster network. 
 
 Theorem 2 : 
 
 The locus of the input reflection coefficient T. of a first order LP 
 
 in 
 
 network as a function of frequency is a circle. 
 
8 
 
 Proof : 
 
 The proof is based on the equivalent representation of the LP first order 
 
 Foster network, shown in Figure 2. Since the locus of I"! as seen at the pair 
 
 in 
 
 of terminals 3-4 is obviously a circle (unit circle) one immediately observes 
 
 that the plot of T. is measured at the terminals 1-2 must also be a circle 
 in 
 
 because of the property of bilinear transformation. This proves the theorem. 
 An expression for Y. for a general LP structure of first order may also 
 be derived from the equivalent representation. The expression is 
 
 2 /J2K . , ,\ 
 
 Jg 2 r sn \~r ln w/ v k ) 
 
 Y in = g l + , 2 7j2K- ; ~\ (16) 
 
 g 2 + j r sn i—fr- In w/w^ k) 
 
 The constant r is obtainable through the calculation of Y. for any single 
 
 in 
 
 convenient value of w. 
 
 We shall conclude this section with a brief comment on the r order LP 
 network of Foster type. By combining two or more first order networks in 
 parallel, a r order network is constructed. It is easily shown that the 
 input reflection coefficient is not necessarily a circle. A typical plot of 
 the input reflection coefficient for a second order network is shown in 
 Figure 3. 
 
Figure 3. Typical plot for T. for a second order Foster LP network 
 
 in 
 
10 
 
 3. LOG PERIODICALLY LOADED TRANSMISSION LINE 
 
 We shall now analyze the log-periodically loaded transmission line 
 structure. It is hoped that this will lead to an understanding of the 
 behavior of the LP dipole antenna by seeking extensions of the basic ideas 
 developed here. 
 
 Consider a LP loaded transmission line composed of lumped and distributed 
 circuits of the form shown in Figure 4. 
 
 Tx-LINE 
 
 2 NODE 
 
 Tx-LINE 
 La m 
 
 Figure 4. LP loaded transmission line 
 
 Note that the line length is scaled by a factor a in each cell and thus the 
 line length tends to zero for large n in the negative direction from the 
 zeroeth cell. Similarly the line length increases in the possible direction 
 as is to be expected. In addition, the loading admittances Y ' s and the 
 mutual admittances Y must satisfy the log-periodic property. This implies 
 that 
 
 Y (k a r ) 
 
 a, n+r 
 
 Y (k) = Y (k o- J 
 n, n+q p, p+q 
 
 (17a) 
 (17b) 
 
 k = 2^7^- = wave number. We shall now derive the characteristic equation for 
 the LP loaded line. To this end consider the node and derive the equation 
 
11 
 
 ° = '•• + Y -2.0 V -2 + Y -l,0 V ~ l + Vo VVl V l + Y 0,2 V 2 + ■" (18> 
 
 -1 
 Y„„ = Y „ + S Y n - j(cot kL + cot kL a ) 
 00 a0 0,m 
 
 -1-1 1 ' 
 
 Y = j (cosec kL a 6 • + cosec kL 5r )+ Y 
 
 5=1 for q = r and zero otherwise. If the structure is uniform periodic 
 r 
 
 a = 1 and if the mutual admittances Y are zero, Equation (15) reduces to 
 
 n, r ' 
 
 = j cosec kL V + (Y - 2j cot kL) V + j cosec kL V (19) 
 If P is defined as the propagation constant^ we have 
 
 ■jPL V n + 1 
 
 for any n and Equation (19) may be rewritten as 
 
 = j cosec kL e j L t (Y - 2j cot kL) + j cosec kL e~ j L 
 
 If the loading admittance Y is a pure susceptance i.e. Y = jb^ the 
 following equation may be derived for P: 
 
 cos PL = cos kL - b S ^ n kL (21) 
 
 Equation (21) is the well known characteristic equation for a uniformly periodic 
 loaded line with no mutual coupling admittances Y between the nodes. 
 
 p,q 
 
 Coming back to the LP structure^ we define 
 
 V l 
 
 T7- = Y(k) (22) 
 
 
 
12 
 
 Then using the LP properties of the structure it is readily shown that 
 
 V 
 
 g±± = Y 0<a P ) (23) 
 
 V 
 P 
 
 Equation (23) has the following implication. If -y(k) is known at all log- 
 periodic frequencies k a for all positive and negative q, then the ratio of 
 any two adjacent cell voltages are obtainable from this information for any 
 given k = k . If >/(k) is known for all k, then of course, we know the behavior 
 of the structure at all points at all frequencies. In the following we shall 
 detail a procedure for calculating ^(k) in terms of the circuit parameters of 
 the structure under consideration. 
 
 Use Equations (22), (23), and rewrite Equation (18) as 
 
 Y _ 2 y" 1 ^" 2 ) y" 1 ^" 1 ) + Y Q _ ± y' 1 ^- 1 ) + Y Q Q + Y Q ^(k) + 
 
 Y 2 Y °° Y(ka) 
 
 (24) 
 
 Equation (24) is the desired characteristic equation for y(K) . A solution of the 
 above equation will now be discussed. As a simplifying step assume that the 
 mutual admittances Y 's are negligibly small. Further, let the asymptotic 
 behavior of the loading elements be such that they satisfy 
 
 Y ^ for t large 
 
 a,t 
 
 for k finite. This has the implication that the elements away from the center 
 
 region and to the right hand side of it have negligible loading effect although 
 
 the same cannot be said in general about the loading aspect in the left hand 
 
 side. This is because the line length IP- also goes to zero for a large t 
 
 and the effect of Y may still be that of a uniform loading per unit length 
 
 of the line even if Y ■ > for large negative t as well. 
 
 at 
 
 Neglecting the mutual terms Yj^, Equation (21) may be simplified to 
 = JY~ 1 (k a_1 ) cosec(kLa _1 ) + 
 
 aO ' '• (25) 
 
 V 
 
13 
 which may be rewritten as 
 
 YCka" 1 ) = -j cosec kl^ 1 (2g) 
 
 Y (k) - j (cot kL + cot kIP- ) + jv (k) cosec kL 
 ao 
 
 for cell and in general 
 
 Y(ka- 1+r ) , -j cosec (kip' 1 ^) ^ 
 
 Y (k<i r ) - j (cot kLa + cosec klA +r ) + jv(ka r ) cosec (kLa r ) 
 a, r - • • 
 
 for cell ' r' . If r is large and positive Y > under the assumption stated 
 
 a, r 
 
 above. Under this condition Equation (27) asymptotically tends to 
 
 Y(ka" 1+r ) = -cosec ( kL a- 1+r ) (2g) 
 
 -cot (kLA r ) - cosec (kIP~ +r ) + -y (kd- r ) cosec (kLa r ) 
 
 which admits the solution 
 
 Y (ka r ) = e " JkLq ' (29) 
 
 for the outgoing wave. This may be easily verified by direct substitution. We 
 may now use this known solution in Equation (28) and work backwards using this 
 equation to calculate the various -y' s in the form of the sequence 
 
 Y(ka r-1 ), 7 (ka r ~ 2 ), ... v ( k ), ^(ka" 1 ), Y (ka" 2 ), ... Y^"" 1 ), ••• 
 
 It is easily verifiable that the process will converge to a limit as the index 
 
 m takes up increasing values. This is because the line length lfl-~ m ^0 for 
 
 a large m and |-y(k a )| >1. The phase also approaches the asymptotic behavior 
 
 very rapidly and it is possible to extrapolate the curve for the phase of -y 
 from the knowledge of its behavior for only moderate m' s. 
 
 The alogrithm outlined here yields all the Y(k ar )s and hence all the 
 desired ratios of V^/V . The current in the shunt element may also be obtained* 
 
14 
 
 The problem of solving for the voltage and currents in a LP loaded transmission 
 line of the type under consideration, may therefore be considered as solved. 
 Before we quote some numerical results based on the above procedure we 
 would like to mention one other point which has to do with the effect of end 
 reflection. To consider this effect, one starts with the solution 
 
 r ikL ar 
 ■y(k a ) =e for large r 
 
 instead of Equation (29). That Equation (30) is also a solution of Equation (28) 
 may be easily verified. Numerical calculations show that the solution corres- 
 ponding to this choice of asymptotic behavior in the right hand side does not 
 affect very much the nature of the solution in the left hand side of the so 
 called active region (region of high shunt current). The above is true for 
 a reasonably efficient circuit i.e„ one in which the current decays reasonably 
 well beyond the active region. In effect therefore we may conclude that the 
 end termination does not significantly affect the performance of an efficient 
 LP structure. 
 
 Numerical results will be presented below in terms of graphs for voltage 
 and current, etc for a log-periodically loaded transmission line. The cal- 
 culations are based on the Equation (28) developed above. The circuit parameters 
 chosen are: 
 
 Y = 1 
 ao / 
 
 a - 4 
 
 0.3 + j 
 
 lvi)| 
 
 and it is assumed that the mutual coupling between the nodes is zero. The 
 amplitude and phase distributions of the transmission line voltage and the 
 shunt current are shown in Figures 5 and 6. 
 
 The general behavior of the solution for such circuits may be described 
 as follows. Both the voltage and current decay rapidly beyond a certain 
 point along the line. The element current becomes small also in the left 
 
15 
 
 4 - 
 
 10 
 
 14 - 
 
 16 
 
 18 
 
 
 0° 
 
 
 50° 
 
 Ld 
 CO 
 
 < 
 
 X 
 Q_ 
 
 100° 
 150° 
 
 > 
 
 
 fe 
 
 200° 
 
 _l 
 UJ 
 
 
 
 250° 
 
 
 300° 
 
 
 350° 
 
 CELL NUMBER 
 H 1 1 h 
 
 -5 -4 
 
 UNLOADED LINE 
 
 LOADED LINE 
 
 Figure 5„ Cell voltage amplitude and phase plots for 
 a loaded LP line 
 
L6 
 
 10 
 
 -20 
 
 CELL NUMBER 
 
 300 
 
 200 
 
 100 
 
 - -100 
 
 200 
 
 300 
 
 400 
 
 500 
 
 -4 
 
 Figure 6„ Amplitude and phase plots for a loaded LP line 
 
17 
 
 hand side and has a peak in the middle, which is usually termed the active 
 region. The voltage tneds to a constant towards the left hand end. 
 
 The phase curve shows that there exists a slow wave in the left hand 
 side of the active region and that the effective phase constant^ defined by 
 the phase change per unit length, goes through a maximum in the neighborhood 
 of the active region. 
 
 Although the model chosen here is rather an oversimplified one ? it 
 should be noted that similar characteristics have been observed by Carrel 
 and others on the LP dipole antenna. 
 
 We shall close this section with one further comment concerning the 
 extension of the above work for a general case where the mutual admittances 
 are not zero. With one mutual term only, i.e., if the coupling is neglected 
 between all but the adjacent nodes^ the above procedure needs only a slight 
 modification. Beyond that one has to go through a slightly more complicated 
 process to solve the problem. However, there is no analytical difficulty in 
 doing this and the details will be left out here. 
 
4. THE BRILLOUIN DIAGRAM AND ITS APPLICATION TO TAPERED STRUCTURES . . 
 
 The Brillouin diagram describes the relationship between the wave number 
 k and the phase propagation constant per cell of a simply periodic structure. 
 In many cases^ a complete knowledge of the diagram in an appropriate range of 
 k, may be useful in predicting the behavior of tapered or- modulated periodic 
 structures derived from it. To illustrate the pointy let us consider the 
 case of a sinusoidally modulated reactance surface. The geometry of the 
 surface and its k-P diagram is shown in Figure 7. 
 
 LEGEND: 
 
 REAL P 
 
 REAL PART OF 
 
 COMPLEX £ 
 
 XU) 
 
 X(z) = X s [i + M cos(^p-)] 
 
 Figure 7. k-P diagram for a sinusoidally modulated reactanc 
 surface (after Oliner and Hessel) 
 
 Here X g is the unmodulated surface reactance, M is the modulation index and 
 a is the period of modulation. The ka-Padiagram for such a structure has 
 been calculated in detail by Oliner and Hessel . 
 
19 
 
 Now suppose the structure is tapered by gradually changing the period 
 along the z-direction. The behavior of the tapered structure may still be 
 predicted from the knowledge of the ka-Pa diagram of the constant periodic 
 structure, i.e., one with constant a. In the tapered structure, one expects 
 bound surface wave fields for the region for which the local ka le small. 
 As the local ka increases along the structure, there comes a region of 
 attenuation corresponding to the vertical, dashed line in the ka-Pa diagram. 
 Beyond this there is a propagation region again, which is followed by a leaky 
 wave region corresponding to the dashed line outside of the first triangle 
 of the k-P diagram. This general behavior repeats again for increasingly 
 higher values of ka. 
 
 From the above discussion it is seen that a general estimate of the 
 behavior of the tapered structure is possible in an approximate sense,, in 
 terms of the k-P diagram of its periodic counterparts . 
 
 To apply these ideas to the log periodically loaded transmission line, 
 consider its periodic counterpart, viz„ the simply periodic loaded line shown 
 below in Figure 8„ Y is the loading admittance and L is the length of the 
 transmission line for each cell. 
 
 
 L 
 
 
 L 
 
 
 L 
 
 
 L 
 
 
 ^0 
 
 I 
 
 Ya 
 1 
 
 1 
 
 a 
 
 Ya 
 
 i 
 
 L T" J 
 Ya 
 
 i 
 
 1 
 
 Ya 
 
 i 
 
 
 
 
 
 
 Figure 8. Simply periodic loaded transmission line 
 
 The characteristic equation for this structure was derived earlier in 
 Equation (20), which may be written as 
 
 cos PL = cos kL + ~ Y • sin kL 
 2 a 
 
 / 
 (30) 
 
 If Y is known a plot of kL versus Pl may be readily made using Equation (30), 
 Typical plots for various different types of Y are shown in Figures 9 and 10, 
 
20 
 
 With lossless resonant circuits as shunt elements, the k-P diagram 
 exhibits attenuation or stop bands, bounded by sharply defined values of 
 kL outside of which the attenuation a is zero. The case of a lossless shunt 
 loading circuit with two series resonances is shown by the dashed curve in 
 Figure 9. The structure essentially behaves like a band stop filter. How- 
 ever, when loss is added to the loading element in the form of a series 
 resistance, the corners of the lossless curve are rounded off as shown by the 
 solid curves in Figure 9. There are no absolutely zero attenuation regions 
 although high attenuation is still confined largely within bands centering 
 the series resonant frequencies of the shunting load. With a single resonance 
 type shunt load, the curve looks like the one shown in Figure 10, as of course 
 is to be expected. 
 
 Actual experimental measurements made on a periodic monopole and dipole 
 array made by Mayes show a behavior very similar to the multiple resonance 
 case described here. It is obvious, that, to obtain better correlation one 
 should perform the calculations with a more exact representation of the dipole 
 antenna rather than the simple one used here and should also include the effect 
 of mutual admittances. Calculations along this line have been planned for the 
 near future. 
 
 It is also of interest to compare the kL-PL diagram for dipole array 
 calculated from the results reported by Carrel on LP structure. The diagram 
 is shown in Figure 11. Local values of PL have been calculated from the 
 amplitude and phase plot of the transmission line voltage for the LP array. 
 It should be pointed out that because of considerably high attenuation in 
 the active region of the LP array, Carrel' s main region of interest was the 
 first resonance region and our calculations end there. This corresponds to 
 the single-mode type of operation of the array. The similarity between this 
 diagram and Figure 10 is indeed striking when one considers the general shape 
 of the curves. 
 
 It is possible to give a heuristic explanation of the performance of the 
 LP structure in terms of the k-P diagram given in Figure 10. The structure 
 supports slow waves (P>k) in the smaller kL regions and these are accompanied 
 by little attenuation. With increasing kL, the band of high attenuation is 
 reached. The phase Pl reaches a turning point and subsequently reverses to 
 
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24 
 smaller values. All this is borne out by the calculations (see for instance. 
 Figure 5) and measurements made on the LP structures of this type. 
 
 To explain the radiation in the backward direction by the LP dipole array 
 which have 180 phase reversals in the alternate elements one should consider 
 the phase plots of the currents rather than of the transmission line voltage. 
 The current phase plot is readily obtained by shifting the transmission line 
 voltage plot by 180 . This is shown in Figure 12. 
 
 -2T 
 
 Figure 12. k-P diagram for the shunt current in a loaded 
 line with additional 180° phase shift per cell 
 
 It is clear that near the attenuation regions A and C where the antenna is 
 close to 1/2 wave and 3/2 wave resonance and is an effective radiator, the 
 radiation pattern will favor the backward direction because of the nearness 
 of these regions to the P=k line. In the neighborhood of the point B however, 
 the antenna loading effect on the transmission line is negligibly small and 
 hence energy is not efficiently coupled to the radiating structure from the 
 feed structure. For the antenna this corresponds to 1-X. resonance region and 
 not much radiation will occur in the neighborhood of crossing B. 
 
 It should be emphasized again that this is merely an attempt to explain 
 the general picture of what happens in these structures and a more elaborate 
 analysis is planned for the future on the actual dipole loaded transmission 
 line. In an actual analysis the effect of the mutual coupling due to the 
 criss-cross feed has to be taken into account but it is believed that this 
 will not alter the general picture of the phenomenon which explains the 
 behavior of these structures. 
 
25 
 
 REFERENCES 
 
 1. R. H. DuHamel, and D. E. Isbell, "Broadband Log Periodic Antenna Structures, 
 IRE Convention Record, Pt. 1 } pp. 119-128, 1957. 
 
 2. D. E. Isbell, "Log Periodic Dipole Arrays," IRE Trans., Vol. AP-8, No. 3, 
 May 1960, pp. 260-267. 
 
 3. R. E. Carrel, "Analysis and Design of the Log Periodic Antenna," Tech. 
 Report No. 52, Antenna Laboratory, University of Illinois. 
 
 4. P. E. Mayes and R. L. Carrel, "Logarithmically Periodic Resonant V 
 Arrays," Tech. Report No. 47, Antenna Laboratory, University of Illinois. 
 
 5. A. A. Oliner and A. Hessel, "Guided Waves on a Sinusoidally Modulated 
 Reactance Surfaces," IRE Trans., Vol. AP-7, December 1959, pp. 5201-5208. 
 
ANTENNA LABORATORY 
 TECHNICAL REPORTS AND MEMORANDA ISSUED 
 
 Contract AF33(616)-310 
 
 "Synthesis of Aperture Antennas,," Technical Report No. 1 , C.T.A. Johnk, 
 October. 1954 .* 
 
 "A Synthesis Method for Broad-band Antenna Impedance Matching Networks," 
 Technical Report No. 2, Nicholas Yaru, 1 February 1955.* AD 61049. 
 
 "The Assymmetrically Excited Spherical Antenna," Technical Report No. 3 , 
 Robert C. Hansen., 30 April 1955.* 
 
 "Analysis of an Airborne Homing System/' Technical Report No, 4 , Paul E. 
 Mayes, 1 June 1955 (CONFIDENTIAL) . 
 
 "Coupling of Antenna Elements to a Circular Surface Waveguide," Technical 
 Report No. 5, H. E. King and R. H. DuHamel, 30 June 1955.* 
 
 "Axially Excited Surface Wave Antennas," Technical Report No. 7 , D. E. Royal, 
 10 October 1955.* 
 
 "Homing Antennas for the F-86F Aircraft (450-2500 mc)? ' Technical Report No. 8, 
 P. E. Mayes, R. F. Hyneman, and R. C. Becker, 20 February 1957, (CONFIDENTIAL) 
 
 "Ground Screen Pattern Range," Technical Memorandum No. 1 , Roger R. Trapp, 
 10 July 1955.* 
 
 Contract AF33 (616) -3220 
 
 "Effective Permeability of Spheroidal Shells," Technical Report No. 9, E. J. 
 Scott and R. H. DuHamel, 16 April 1956. 
 
 "An Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report 
 No. 10 , Do Go Berry and J. B. Kreer, 10 May 1956. AD 98615 
 
 "A Technique for Controlling the Radiation from Dielectric Rod Waveguides," 
 Technical Repo rt No. 11, J. W. Duncan and R. H. DuHamel, 15 July 1956.* 
 
 "Directional Characteristics of a U-Shaped Slot Antenna," Technical Report 
 No. 12, Richard C„ Becker, 30 September 1956.** 
 
 "impedance of Ferrite Loop Antennas," Technical Report No. 13 , V. H. Rumsey 
 and W. L. Weeks, 15 October 1956. AD 119780 
 
 "Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Report 
 No. 14, Richard F. Hyneman, 20 December 1956. 
 
Distributed Coupling to Surface Wave Antennas 8 Technical Report No. 15 , 
 Ralph Richard Hodges,, Jr., 5 January 1957,. 
 
 "The Characteristic Impedance of the Fin Antenna of Infinite Length," Technical 
 Report No, 16 , Robert L, Carrel , 15 January 1957 . * 
 
 "On the Estimation of Ferrite Loop Antenna Impedance/' Technical Report No. 17 , 
 Walter L, Weeks, 10 April 1957 .* AD 143989 
 
 "A Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source 
 Antenna," Technical Report No. 18 , Walter L. Weeks, 20 April 1957. 
 
 "Broadband Logarithmically Periodic Antenna Structures," Technical Report No. 19 ) 
 Ro Ho DuHamel and D. E. Isbell, 1 May 1957. AD 140734 
 
 "Frequency Independent Antennas/ Technical Report No. 20 , V. H. Rumsey, 25 
 October 1957 . 
 
 "The Equiangular Spiral Antenna/ Technical Report No. 21, J. D. Dyson, 15 
 September 1957. AD 145019 
 
 "Experimental Investigation of the Conical Spiral Antenna," Technical Report 
 
 No. 22, R„ Lo Carrel, 25 May 1957.** AD 144021 
 
 "Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technical 
 
 Report No. 23, E. J. Scott, 10 August 1957 . 
 
 "Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide," 
 Technical Report No. 24, J. W. Duncan and R. H. DuHamel, August 1957. 
 
 "The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary 
 Cross Section/' Technical Repor t No. 25, Robert L. Carrel, August 1957. 
 
 "Cavity-Backed Slot Antennas," Technical Report No. 26 , R. J. Tector, 30 
 October 1957. 
 
 "Coupled Waveguide Excitation of Traveling Wave Slot Antennas," Technical 
 Report No. 27, W. L. Weeks, 1 December 1957. 
 
 "Phase Velocities in Rectangular Waveguide Partially Filled with Dielectric," 
 Technica l Report No. 28, W. L. Weeks, 20 December 1957. 
 
 "Measuring the Capacitance per Unit Length of Biconical Structures of Arbitrary 
 Cross Section/ 8 Technical Repor t No. 29, J. D. Dyson, 10 January 1958. 
 
 "Non-Planar Logarithmically Periodic Antenna Structure," Technical Report No. 30 , 
 D. E. Isbell, 20 February 1958. AD 156203 
 
 "Electromagnetic Fields in Rectangular Slots," Technical Report No. 31 , N. J. 
 Kuhn and P. E. Mast, 10 March 1958. 
 
"The Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder, 
 Technical Repor t NOo_32_ L J, W Duncan, 25 May 1958 „ 
 
 "A Unidirectional Equiangular Spiral Antenna," Technical Report No, 33 , 
 Jo Do Dyson, 10 July 1958„ AD 201138 
 
 "Dielectric Coated Spheriodal Radiators," Technical Report No. 34 , W. L. 
 Weeks, 12 September 1958 . AD 204547 
 
 "A Theoretical Study of the Equiangular Spiral Antenna," Technical Report 
 No„ 35, Po Eo Mast, 12 September 1958, AD 204548 
 
 Contract AF33(616)-6079 
 
 "Use of Coupled Waveguides in a Traveling Wave Scanning Antenna," Technical 
 Report No„ 36, R„ H. MacPhie, 30 April 1959, AD 215558 
 
 "On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and 
 Infinite Ranges/ Technical Report No, 37 „ Raj Mittra, 15 May 1959. 
 
 "Prolate Spheroidal Wave Functions for Electromagnetic Theory , " Technical 
 Report No, 38 , Wo L. Weeks, 5 June 1959 
 
 "Log Periodic Dipole Arrays /' Technical Report No. 39 , D. £. Isbell, 1 June 1959. 
 AD 220651 
 
 "A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical 
 Report NOo 40,, S. Dasgupta and Y. To Lo„ 17 July 1959 „ 
 
 "The Radiation Pattern of a Dipole on a Finite Dielectric Sheet," Technical 
 Report No, 41, K» G. Balmain,, 1 August 1959c 
 
 "The Finite Range Wiener=-Hopf Integral Equation and a Boundary Value Problem 
 in a Waveguide," Technical Report No. 42, Raj Mittra, 1 October 1959. 
 
 "impedance Properties of Complementary Multiterminal Planar Structures, " 
 
 Technical Report No, 43 , Go A, Deschamps, 11 November 1959. 
 
 "On the Synthesis of Strip Sources," Technical Report No. 44 , Raj Mittra, 
 4 December 1959 . 
 
 "Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides," 
 Technical Report No, 45 ., Co Ho Tang and Y. To Lo, 11 March 1960 . 
 
 "New Circularly Polarized Frequency Independent Antennas with Conical Beam or 
 Omnidirectional Patterns," Technical Report No, 46, J„ D. Dyson and P. E. Mayes, 
 20 June 1960, AD 241321 
 
 "Logarithmically Periodic Resonant -V Arrays," Technical Report No. 47 , P„ E. Mayes 
 and Ro L. Carrel , 15 July I960, AD 246302 
 
"A Study of Chromatic Aberration of a Coma- Corrected Zoned Mirror," Technical 
 Report No, 48, Y„ To Lo ., June 1960 
 
 "Evaluation of Cross-Correlation Methods in the Utilization of Antenna Systems," 
 Technical Report No,, 49 . R Ho MacPhie,, 25 January 1961 
 
 "Synthesis of Antenna Product Patterns Obtained from a Single Array," Technical 
 Report No. 50 v R Ho MacPhie, 25 January 1961 „ 
 
 "On the Solution of a Class of Dual Integral Equations," Technical Report No. 51 , 
 Ro Mittra, 1 October 1961 . AD 264557 
 
 "Analysis and Design of the Log-Periodic Dipole Antenna," Technical Report No. 52 , 
 Robert L„ Carrel., 1 October 1961o* AD 264558 
 
 *°A Study of the Non-Uniform Convergence of the Inverse of a Doubly-Infinite 
 Matrix Associated with a Boundary Value Problem in a Waveguide," Technical Report 
 NOo 53, Ro Mittra, 1 October 1961 „ AD 264556 
 
 * Copies available for a three-week loan period „ 
 ** Copies no longer available „ 
 
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