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ANTENNA LABORATORY

Technical Report No. 70

0 p405 543THEORETICAL BRILLOUIN (k-P) DIAGRAM FOR

MONOPOLE AND DIPOLE ARRAYS AND THEIRAPPLICATION TO LOG-PERIODIC ANTENNAS

byR. Mittra and K. E. Jones

Contract No. AF33(657)-10474Hitch Element Number 62405484

760D - Project 6278, Task 6278-01

April 1963

Sponsored byAERONAUTICAL SYSTEMS DIVISION

WRIGHT-PATrERSON AIR FORCE BASE, OHIOProject Engineer - James Rippin - ASRNCF-3

ELECTRICAL ENGINEERING RESEARCH LABORATORYENGINEERING EXPERIMENT STATION

UNIVERSITY OF ILLINOISURBANA, ILLINOIS j

V

A~~ (y,)A

Antenna Laboratory

[" Technical Report No. 70

I(4) THEORETICAL BRILLOUIN (k-P) DIAGRAM FOR

MONOPOLE AND DIPOLE ARRAYS AND THEIRAPPLICATION TO LOG-PERIODIC ANTENNAS,

by t~

I (/6) R. Mittra and K. E. Jones (/3)i•/

I (/0)Contract No. AF33f657ý-10474 (I•--/i

Hitch Element Number 62405484

1 ~~~760 D-Project 6278 Task 6278-01 (5 r.

1April 1963

Sponsored by

AERONAUTICAL SYSTEMS DIVISION

WRIGHT-PATTERSON AIR FORCE BASE, OHIO

Project Engineer - James Rippin - ASRNCF-3

I

I- Electrical Engineering Research LaboratoryEngineering Experiment Station

University of IllinoisUrbana, Illinois

Ii

Theoretical Brillouin (k-P) Diagram forMonopole and Dipole Arrays and theirApplication to Log-Periodic Antennas

R. Mittra+ and K. E. Jones++

SUMMARY[ by Carrel3 who has analyzed the LP dipole antenna.

His analysis of this particular problem is verythorough but his results are clearly not meant to

In this paper we derive e characteristic be applicable directly to other LP structures.equation for the complex propagation constant in a Some LP antennas may be represented in terms oftransmissio3iline periodically loaded with dipole radiating elements loading a transmission line inelements/ 6' sýuation is solved for real and a LP manner and a study of such non-uniformlyimaginary parts of the mode propagation constants, loaded lines has recently been reported by Mittra4

The effect of mutual impedance between antenna and Jones and Mittra5 . Again this line of attack,elements is included in the formulation and cal- although quite useful for some LP antennas, doesculation. Theoretical calculations are compared not cover all possible configurations. An approachwith experimental results reported by Mayes and which views the LP device as a tapered version of aIngerson. The Iness and limitation of the uniform periodic structure was first suggested byBrillouin Q diagram for analyzing LP Mayes, Deschamps and Patton6 . This work con-structures is discussed briefly. sidered periodic structures or gratings which have

the following propagation characteristics as afunction of frequency. The grating is assumed to

INTRODUCTION support bounded waves or surface waves up to acertain frequency, implying that the propagationconstant P along the structure is greater than k,During the recent years a new concept of the free space wave number, up to this frequency.

broadband antenna design has been introduced under As the frequency is increased further the wavethe name of Log-periodics or LP and the usefulness becomes fast, i.e., 1 becomes less than k. Nowof the concept ha• ýeen demonstrated by several the value of P relative to k gives the directionpractical designs of the same type. From the of the main lobe of the pattern through the simpledefinition of LP structures it is obvious that the relation.electromagnetic properties of an infinite LP

"structure repeat at f T where f is a reference Cos( = P/k for P < k0 0 CS / o

frequency and n is a positive or negative integer.However, this definition says nothing about thebehavior of the structure at the in-between fre- This picture although simple and quite useful•quencies. In order to be successful, a practical does not cover the case of all periodicantenna which necessarily has finite dimensions, structures but only those which support a wavemust not only have LP properties in the design characterized by a slowness factor which remainsrange but should also exh~ibit under continuous relatively unchanged as a function of frequency.sampling little variation in pattern and impedance Only the periodic structures belonging to the

Sthroughout this frequency range. 1:w exactly one helix family, which includes the zigzag, areis to assure this behavior when designing a LP known to possess such characteristics.antenna is a question which has not been answered 7satisfactorily to date. The LP design has largely A portion of a recent report by Mittra in-j been an art, an application of empiricism based on cludes a brief sketch on the use of the k-P orintuition, and not really a science. Brillouin diagram of periodic structures to pre-

dict the performance of its LP counterpart. AMost of the papers written on the subject are more elaborate discussion which generalizes their

experimental in nature and the theoretical devel- first work on the same topic has recently beenopment of the LP antenna design is very much in its published by Mayess et al. Oliner 9 has alsoinfancy, considered this aspect of the LP study in a recent

paper. It seems that viewing the LP as a taperedA significant theoretical work in this line is periodic structure is the only general approach

available so far for theoretically studying thelog-periodics. It must be emphasized, however,

+ University of Illinois, Urbana, Illinois that the arguments behind this approach so far++ Ground Systems Group, Hughes Aircraft Company, have largely been heuristic. A solid theoretical

Fullerton, California foundation and a more exact formulation for the LP

!S

structure has yet to be worked out. The authors no radiating properties. This is not very desir-I are currently looking into this problem in some able and the tapered version of the corrugated

detail. It is also quite important to study the surface is not very suitable for LP design with-limitations of this approach in order to estimate out some basic modifications. Further details onthe range of its usefulness, this structure may be found in an accompanying

paper, "The Letter Rack Antenna" by Mittra andAt present there are extremely few periodic 11

structures for which the complete k--P diagram,

"exhibiting both the real and complex roots has The third type of diagram for the stub-loadedbeen calculated. Moreover, none of the above has monopole array (see Figure 1c) has the exceptionalbeen related to the corresponding LP structures feature that its k-1 characteristic seems to beexcept on a qualitative basis. We propose to discontinous as a function of frequency.calculate the complete k-P diagram of the dipole Although the experimental diagram in Figure lc

loaded transmission and relate the calculations to does not show this, i diagram in F here

the results obtained by Carrel for the LP antenna. estso th , t is conjectured that thereIn aditon, e cmpar ou resltswithexpri- exists more than one mode on the structure althoughIn addition, we compare our results with experi - only one of these is perhaps predominant at any one

Smental measurements on the uniform periodic dipole feuny h wthn ftecrefo h

and monopole structures. These have been reported frequency. The switching of the curve from theby MyesandIngesono. efoe dicusingthe right hand side to the negative P region may well

by Mayes and Ingerson10. Before discussing the be due to switching of modes. At any rate this

dipole problem, however, we shall present a brief type of structure may also be tapered to give a

discussion on the k-P diagram and its significance w bpe rfor man altough the design atoteL nlss wideband performance although the design may be

to the LP analysis. quite critical.

E DIt should be realized that the solution ofTHE BRILLOUIN DIAGRAM AND ITS the source problem, which gives the relative

Smplitude coefficients of the various modes, is not

The Brillouin diagram of a periodic structure easy even for a periodic structure. Even if this

relates the propagation constant P along the problem is solved for the periodic case, thesolution cannot be easily related to the corres-

structure to the free space wave number k and hence sondion for aslP ructed th thereores

is often referred to as the k-P diagram. The ponding one for a LP structure. This therefore is

characteristics of the k-P diagram for a periodic one of the fundamental difficulties in applying

structure vary widely depending upon the nature of the k-P diagram of a periodic structure directly

the structure. As an illustration of some of the to the case of the corresponding LP

different types of diagrams, reference is made to one. Some investigations are now underway toward

Figure 1. One notes that the three examples shown resolving this problem.

have quite different characteristics, and althoughit my b posibl togrop dffernt eridicThe nultimode problem is quite prominent initthe uniform monopole and dipole arrays as we shallf structures into several different broad categories see later.it is not possible to draw hard and fast linesbetween them. The three examples shown may how-ever be considered as belonging to three typically now discuss the problem of the monopole and dipole

Sdifferent groups. The first one, the helix, maybe thought of as a constant slowness type of loaded transmission lines in detail.structure mentioned before. Figure la gives aIsketch of the solutions for two modes of propaga- DERIVATION OF THE CHARACTERISTIC EQUATIONtion which may exist simultaneously on the helix.

Out of these, the one on the left hand side turns The first step toward calculating the k-P3complex above a certain frequency called the turn- diagram is the formulation of the characteristic

I over frequency. The phase of the wave in the equation for P. This is to be outlined in theneighborhood of the turnover region is suitable following.for backfire radiation. When this structure istapered, it will support bound waves for smaller Consider a transmission line loadeddiameters of the helix and will eventually reach Cons with a tr on line bleda ego wee h wv o testutuewilperiodically wiha network describable by aaregion where the wave on the structure will admittance matrix [Y) also having periodic a

athatthe because of radiation. This is assuming properties, as shown in Figure 2. It is assumedthat th e am plitud e o f excita tion of th e se cond t a m h u u l a m t a c e w e h o e

mode represented by the right line is relatively that Ynmi the mutual admittance between the nodessmall in the region of interest. m and n is independent of m and n with in-ni heldconstant.

In the corrugated surface antenna, no complex It is obvious from the above that the entiresolutions of the type possessed by the helical s t is perious and he ab o d d.structure have been obtained. The only complex structure is periodic and has a period d.solutions found have a real part of equal to 7 and Writing the node equation say for node 1,theoretically, for an infinite structure, this o rating the node property f thesolution represents a filter type cutoff mode with one has using the symmetry property of thesoluionadmittances, viz. Ylq ff y1 -q

3

.oeV 2+(Y+J c kd)V Note that any row of the [Z] matrix m a y be13- k2 1 related to any other row by a simple shift. This

0 yof course follows from the periodicity property

+(-2 j cot kd + yll)V +(Yl+j cosec kd)V1 required here of the loading device. In order to1 0dV use Equation (4) one would like to calculate

Yeq in terms of the parameters of the [Z) matrix.+Y 1 3 V2 (1) To obtain the above, use the nth equation from the

13 2 matrix relation

where V's are the node voltages and yll is the (z]

m and obtainself admittance of the loading network for node 1.

From the neriodic nature of the structure we have,

V= + I +z I + z In 13 n-2 12 n-l 11 n

if-~-V = e-JI

0 + z 1 2 1In+l + z13 In+2 + -- (6)

-: +jaUse the periodicity property of the structure in

V p e-j(p-q) 3d connection with Equation (6) and obtainthen P e(2

V

q V-n V z d+ 2z12 cos Pd + 2z13 cos 2,d + -- (7)

where p and q are arbitrary. n

Using the above relation in Equation (I) we derive whereafter some simplification /

V -j(m-n) Pdm = e

V0 = (-2j cot kd +yll)+2(y1 2 +j cosec kd)cos Pd n

2 s 2If we start with the admittance Matrix [Y) we+ 2y13 cos 2 Pd -- (3) may similarly derive an alternate expression for

SVn/In, viz.,

or

jy eq ICos Pd = cos kd + -2-1 sin kdn=o-s2 =y + 2y cos Pd + 2y cos 2P"d + --- (8)

V 11 12 13(4) n

Yeq = Yll + 2y 1 2 cos Pd + 2y 1 3 cos 2dIt immediately follows from Equations (7) and

+ (8) that yeq defined in connection with Equation

Equation (4) is the desired characteristic equation (4) may be expressed as

for the configuration shown in Figure 2. Note that

if Ylm= 0, for m> 1, then Equation (4) reduces to 1the well-known periodically loaded transmission Yeq = z 1 + 2z12Cos Pd + 2z 13cos2d + -- (9)

line characteristic eauation.

Now consider a slightly different formulation This is an important relation which permits one to

in terms of the impedance matrix parameters of theloading network. Let the impedance matrix [Z) of evaluate the desired quantity eq in terms of the

the periodic loading structure be represented by impedance parameters of the loading structure. Itfollowq that the characteristic equation in terms

- 2 of the impedance parameters of the loading network

- 12 Z11 z12 z13 is

CosP• f= Cos kd -

S [Z]= - 1 3 Z1 2 Zll z12 z1 3 -- (5)

(z) sin 1 1d (10)-"-z 1 3 z 1 2 Zll z 1 2 z1 3 -- ii2253co~

I 4Equation (10) gives a suitable form of the It is not difficult to show that this pro-

characteristic equation for a dipole loaded cedure would yield exactly the same expression fortransmission line. This is because it is con- Yeq as given in Equation (9) if the inversionsiderably simpler to calculate the z-parameters eq

of an array of dipoles as compared to its must.admittance parameters. The Equation (10) has alsobeen independently derived by Laxpati.

For the dipole array the z-parameters may be

calculated by using the formula given below. TheAn alternative way of handling the problem is expression may be derived by using the induced-

I as follows. One may obtain the parameters of the efmto.Temta meac ewe w

[Y] matrix by directly inverting the EZ] matrix. ena istexpresse by

We shall illustrate the procedure by first con- a s is [Krus) by

sidering the simple case in which we assume that

Zi n = 0 for n > 2. Then [Z] becomes z2 60 e j2kh'd

12 1-cos 2kd Ku 0)-Ku)

0 z1 2 Z1 1 z 12 0-- + e-2jkh' [K(V )-2K(VI)] +2[K(uo')-K(u 1 )-K(V1 )]

[Z] 0 0 1z12 zll 1z12 0 0 (11) *

0 0 z z z 0 + 2K(uo') [1 + cos 2kh (14)

The star indicates the complex conjugate of theNow let expr~ssion in the braces. Here

K(x) = Ci (x) + j Si(x)

Y1 3 Y1 2 Y Y1 2 "-where K(x) and Ci(x) are the cosine and sine

SY = -- 1) integral functions of the argument x. also

[y)y 1 3 y1 2 y11 y12 y13 (2

-- y 3 y 2 y 1 ~1 y1 ~~u k[\~ 2I 4h - 2hýd 2o+ 2 ']

Using [Z] [Y] = [1] = unit matrix, one may derive Vo k I[d2 + 4h'2 + 2h']J the following approximate equations for the para-meter y11 and y12 in terms of the z and z12:

u =kd0

SZ 11 Yll + 2z 1 2 Y12 = 1 ( k

(13) k['+h? h

I z11 Y12 + z12 Yll =0

V 1 k 1 + h+

SNote that we have neglected the higher order y I'1. in

and also the higher order equations required to h'= half length of the dipole elements. d =I obtain the inverse of [Z]. With this approxi- spacing between the elements. Equation (14) is of

mation it is a simple matter to evaluate y 1 1 and the same type as used by Carrel except that becaumeof unequal lengths of the elements in the LP

Y1 2 in terms of 1and from Equation (13). antenna he had to use a more general formula

SThe above procedure may be extended to include the applicable to such cases.higher order mutual terms. That this method isonly an approximate one is fairly obvious. It may The self impedance z is calculated bybe verified, however, that the use of the admit-I tance parameters reduces the order of the charac- simply setting d = j•a in Equation (14), where a

teristic equation by one from that of the impedance is the radius of the antenna element.parameter equation of the type (10) when the samenumber of mutual terms are included and there may When the alternate dipoles in an array are

be some merit to using this procedure, transposed, z(t) the mutual impedances between theIn* Unpublished work

5

elements 1 and n in the resulting structure is Equation (14), say f(1), is a periodic function

simply given by of real V If 13is real, one would ex ct only afinite number of zero crossings of fQ in aperiod of , i.e., for o < p:S 2 I/d. It is not

( possible to make any such definite statements whenS(t) )n+ll n (15) P" is complex, however. In any case, whether or notln ln the modes with the complex ý's may be supported by

the structure may be answered only by solving a

e csource problem. The complex solutions of theThe characteristic equation for the alter- eigen-value equation of an infinite periodic

nately transposed structure is simply obtained by structure characterize approximate and alternative

modifying the z-parameters appearing in the representation of continous spectrum of spatial

.equation for P" according to Equation (15). harmonics and must be interpreted in this light.

One usually finds that the imaginary parts ofSOLUTION OF CHARACTERISTIC EQUATION many of these solutions which determines the

attenuation per cell is so large that they playIn this section we discuss the solution of little part in contributing to the current dis-

the characteristic equation for P Iffor a loadedliteprinctibigtoheurntdst hransmissiontline. LetusstartwithEquation 0)aoe tribution produced by a given source. In additionStransmission line. Let us start with Equation CL otiOheeaeawol ls fcope)ot

sto this there are a whole class of complex roots

and assume that all z which are physically inadmissible because they do

going through some algebra it is possible to de- not satisfy the consistency conditions for the

rive an equation in terms of cos npd which reads attenuation and phase constants. For details ITthis condition the reader is referred to OlinerOnce again it must be remembered that the complex

q+l_ solutions merely serve to represent the solution

Z C cos npd = o (16) to a source problem in approximate manner.

n=o

NUMERICAL CALCULATIONS AND

where C's may be expressed in terms of the z- COMPARISON WITH EXPERIMENTS

parameters. Using the transformation Single Mutual Term

t = e )uit is possible to derive an algebraicequation in t2 which reads Although computations have been carried out

for the k-P diagrams for several different para-meters of the dipole array, we shall only report

q /the calculations for a choice of dimensions whichZ 2n jtt (17) corresponds closely to one of the LP arraysSdnt = o , t =e (7

n=o investigated by Carrel.

Let us start with the discussion of the case

of one mutual impedance and assume that the effectThe coefficients of the above equation are com- of the higher order mutual terms is negligible.

plex numbers in general. The roots of this Assume further that the approximate Equation (13)

polynomial equation may be sought by using is adequate for calculating y 1 1 and y1 2 " For this

Snumerical techniques. The advantage of Equation approximate case, Equation (4) is quite convenient

(17) over Equation (16) is that whereas Equation to work with, and we neglect all the y I es for

(16) is transcendental, Equation (17) is 1g

algebraic and can be handled on the digital com- n > 2 in the expression for y The propagation

puter using standard library routines. eqconstant P may be calculated with little difficulty

SOne question immediately arises in connection and we get only a single solution for P under the

with Equation (15). In general, the above equation present approximation. Computations were made forf wthequaion(15.wIigeeraateraoveequtio

has (q+l') solutions where q depends on the number the following parameters

of mutual impedances taken to be non-zero. Does

this imply that as the number q is increased d 1 h = 10 h 177indefinitely the number of solutions become

infinitely many? The physical implication of this a = radius of dipole elementwill be that an infinite number of characteristic for both the unreversed and the reversed casesthe

modes may be supported by the structure. It is schematics for which are shown in Figure 3a and 3c

worthwhile examining this statement to see if it respectively. Note that the characteristics of

is true or not. This question may be answered by the monopole array over the ground plane (see

going back to Equation (14) which is in fact the Figure 3b) may be deduced from that of thesource of Equation (15). The left hand side of unreversed dipole array from symmetry considerations.

_6

The propagation constant 'j was calculated as a P2 might be considered as important above the"function of the free space wave number k. A plot

d, Fgre resonance point kh = 1/2. For the unreversed caseof kd vs P d s shown in Figur 4a and 4b for the

unreversed and reversed case, respectively. Recall comparison between Figures 6a and 4a shows that- that for the reversed case one changes the sign of single mutual solution essentially picks out the

the mutual impedance z in the expression for the P1 curve.characteristic equation.

The problem of deciding which solution is

Figures 4a and4 b also show the experimental more important than the other is not at all easycurves by Mayes and Ingerson1 0 . Note the close and in fact a satisfactory answer to this question

correspondence of the theoretical and experimental may be found only by solving a source problem.

curves for both the structures. The major However, the attenuation constant may be considereddeviation between the experiment and the theory as a fair guide for choosing a predominant solution

seems to occur in the neighborhood of the dipole out of several different ones. This criterion may

resonance where the experimental curve penetrates be misleading though because a particular mode,much farther towards a larger P. An explanation even though it has a relatively high attenuation

for this discrepancy will be offered later when may still be important close to the feed regionhigher order mutual terms will be considered, and may strongly influence the far field pattern.

All these considerations must be taken into

The results for the uniform case may be account before deciding in the case of multimodeapplied to the LP structure by using a propagation that a particular solution n most

perturbational analysis. One calculates the Pd predominant and it is best to put in a word of

for the local kd, where d now changes from cell to explanation saying what one really means by thatcell and uses the complex value of 3d to calculate statement.the attenuation and phase shift due to that cell.The results may then be compared with the corres- It is possible to conceive of many situationsponding values for the LP case. The above where it is difficult to pick out the pre-computations were carried out for the case of dominant mode in a multimode operation. As a

alternately reversed dipoles and they are plotted simple example let us reconsider Figure 6a. It isin Figure 5 which shows the feeder voltage certainly possible that both P and P2 are impor-

Samplitude and phase plotted as a function of dis- tant in the resonant region and that the experi-tance along the lines. The theoretical curves in mental measurements show a stronger influence ofFigure 4 were calculated using d = O.112h to P2 in this region. This is possibly why thecorrespond to Carrel's choice of parameters. TheI calculated points are plotted on Carrel's experi- measured curve show points in a region much slower

mental curves and close correspondence between the than those predicted on the basis of P solutiontwo is indeed striking, considering the approxi- alone. Experimental measurements reported bymations which were used to calculate the Hudock* on the monopole array exhibit the multi-theoretical points, mode type of propagation in that structure and his

T/ results also point to the fact that it is difficultThree Mutual Terms to separate the multiple modes in the neighborhood

of the resonance region.Computations were also carried out to study

the modification of the results when the higher We shall close this section with one furtherorder mutual terms are included. The dipole comment. Although we have not shown here thespacing d was now chosen to be 0.112h in order to other two solutions which were obtained for thecorrespond exactly to Carrel's case and other 3 mutual case, it seems that one of these extraparameters were kept unchanged, solutions may play a significant part in the

neightborhood of resonance. The third solution,We now use Equation (10) and include three may be excited with a comparable amplitude in this

mutual impedances, z 1 2 , z 1 3 and z14 which region and interfere with the P solution (for the

may be calculated by using Equation (14). The reversed case) to give a beat type of patterncharacteristic equation was solved on a computer which attenuates along the structure This hasand it was found that for most regions only two been observed by Mayes and Ingerson 0 duringsolutions branded P 1 and P 2 were admissible or their measurements on the dipole structure. At

I significant. These two solutions are shown in this stage, this explanation is merely a con-Figure 5a for the unreversed case and in Figure 5b jecture on our part, and further evidence is

for the alternately reversed case. Let us compare needed before the explanation can be considered asthese curves with the single mutual case. Notice convincing.first of all that the results for the single mutualcase (see Figure 4a and 4b) seem to constitute the However, one must not lose sight of one very

predominant part of the solutions for higher order important point in connection with this discussion.mutuals. As for instance P1 in Figure 6b is pre- Even if we know that mode 1 and 3, say, are

predominant in the lower part of the curve whereas * to be published

7

comparable in some region of frequency in the REFERNCES

uniform periodic case, this does not enable us tosay anything definite about the relative importance 1. DuHamel, R.H. and Isbell, D.E. "Broadband

of mode 3 as compared to mode 1 or even the Logarithmically Periodic Antenna Structures",existence of mode 3,in the LP case. No satis- IRE National Convention Record, Pt. 1, pp 119-

factory formulation is available as yet for 128, 1957.

answering these questions. This, in fact, is oneof the limitations of the approach which uses the 2. Isbell, D.E. "Log-Periodic Dipole Arrays",properties of the uniform periodic structure for Trans. IRE Vol. AP-8, pp. 260-7, May 1960.

predicting the performance of the correspondingLP one. 3. Carrel, R.L. "Analysis and Design of Log-

Periodic Dipole Antenna", Tech. Rep. No. 52

In concluding this section we observe that Antenna Laboratory, University of Illinois,there is a fairly satisfactory correlation of the Urbana, Illinois.

theoretical calculations with the experimentalresults. We note that multimode propagation 4. Mittra, R. "Theoretical Study of a Class of

exists in the periodically loaded transmission Logarithmically Periodic Circuits", Tech. Rep.

when the mutual terms are considered and that 59, Antenna Laboratory, University of Illinois.

this has not been reported by the experimentalobservers except in a very small frequency range 5. Jones, K.E. and Mittra, R. "A Study of

near the dipole resonance. Continuously Sealed and Log-PeriodicallyLoaded Transmission Lines", present at 1962

Fall URSI Meeting in Ottawa and to be pub-

CONCLUSION lished.

6. Mayes, P.E., Deschamps, G. A., and PattonW.T.The Brillouin (k-f) diagrams for the dipole "Backward Wave Radiation from Periodic

array have been calculated both for the unreversed Structures", PIRE, Vol. 49, No. 5, May 1961.and alternately reversed cases. The significanceof the k-3 diagram to the analysis of LP 7. Mittra, R. "Theoretical Study of a Class ofstructures has been indicated. Theoretical Logarithmically Periodic Circuits", Tech. Rep.calculations have been compared with experiments 59, Antenna Lpboratory, University of Ill1rois.with good agreement. Multimode propagation isobserved on the array structure; that this 8. Mayes, P.E. "Deschamps, G.A. and Patton, W.T.phenomenon may sometimes put a limitation on the "Backward-Wave Radiation from Periodicsimple perturbational approach of analyzing LP Structures", Tech. Rep. No. 60, Antennastructure is pointed out. Laboratory, University of Illinois.

9. Oliner, A.A. "Leaky Waves in ElectromagneticACKNOWLEDGEMENT Phenomena", Research Report No. PIBMRI-1064-

62 Polytechnic Institute of Brooklyn,SThe authors would like to acknowledge the August 1962.support of Hughes Aircraft Company throughcompany sponsorship, and Wright Patterson Air 10. Mayes, P.E. and Ingerson, P.G. "Near-FieldForce Base, Ohio through Contract AF33(657)10474 Measurements on Backfire Periodic Dipole

for sponsoring the investigation reported in this Arrays", Proceedings of 12th Annual Symposium,*paper. Numerical computation reported here were USAF Antenna Research, Allerton Park, Illiaoi%

carried out at the Numerical Analysis Section of 1962. Also Quarterly Progress Report No. 1* the Ground Systems Group, Hughes Aircraft Company. "Research Studies on Problems Related to

Thanks are particularly due to Dr. R. H. DuHamel, Antennas", Antenna Laboratory, University ofHead of the Electromagnetics Staff, Hughes Illinois, January 1963.Aircraft Company with whom one of the authors was

associated during the summer of 1962. We are also 11. Mittra, R. and Wahl, M. "The Letters Rackindebted to E. Hudock, formerly of Antenna Antenna", IRE Convention Record, (This issue).Laboratory, University of Illinois, for permissionto use some results from his forthcoming Technical

Report. Discussions with ProfessorsG. A. Deschamps, P. E. Mayes and Messrs. S. Laxpati,

P. E. Ingerson, E. Hudock, and other members of theAntenna Laboratory of the University of Illinoiswere always very helpful, and the authors are muchindebted for what they learned from thesediscussions.

V SF

i;

f. FIGURE CAPTIONS

Figure la Typical k-B diagrams for helix

Figure lb Typical k-B diagram for corrugated surface

Figure lc Typical k-B diagram for stub-loaded monopole array

Figure 2 Transmission line periodically loaded by a periodic loading circuit

Figure 3a Dipole array

Figure 3b Monopole array over ground

Figure 3c Dipole array with alternate elements transposed (schematic)

Figure 4a Theoretical k-B plots for dipole array (unreversed) or monopole

array over ground and comparison with experiment (Mayes and

Ingerson)

Figure 4b Theoretical k-B plots for dipole array (alternately reversed)

and comparison with experiment (Mayes and Ingerson)

Figure 5 Transmission line voltage amplitude and phase curves for LP

dipole array (theoretical) and comparison with experiment

(Carrel)

Figure 6 Theoretical k-B diagrams for dipole arrays with three mutual

impedances included - (a) unreversed case; (b) alternately

reversed case

II.II

9

144

- 0

10

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7 2 Ref. 5 line 2 Continuously Sealed Continuously Scaled

AF 33(657)-10474 DISTRIKUTRIO LIST

One copy each unless otherwise indicated

Armed Services Technical Information CommanderAgency Foreign Technology Division

Attn: TIP-DR Attn: TD-31Arlington Hall Station Wright-Patterson Air Force BaseArlington 12, Virginia (20 copies) Ohio

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Bedford, MassachusettsAeronautical Systems DivsionAttn: ASRNOF-3 CommanderWright-Patterson Air Force Base Air Force Missile Test CenterOhio (5 copies) Technical Library

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Ohio

2

National Aeronautics & Space Adm. DirectorAttn: Librarian Air University LibraryLangley Field, Virginia Attn: 3T-AUL-59-30

Maxwell Air Force BaseRome Air Development Center AlabamaAttn: RALTMGriffiss Air Force Base Commanding GeneralNew York White Sands Missile Range

Attn: ORDBS-(H-Tech Library RR-312Rome Air Development Center New MexicoAttn: RAWED, Mr. R. F. DavisGriffiss Air Force Base Commanding OfficerNew York Diamond Ordnance Fuse Laboratories

Attn: 240Commander Washington 25, D. C.Research & Technology DivisionAttný RTHR, Maj. J. Gregory DirectorBolling Air Force Base U. S. Navy Electronics LaboratoryWashington 25, D. C. Attn: Library

San Diego 52, CaliforniaOffice of Chief Signal OfficerEngineering & Technical Division Adams-Russell CompanyAttn: SIGNET-5 280 Bear Hill RoadWashington 25, D. C. Attn: Library (Antenna Section)

Walthazf 51,Massachusetts

Commanding OfficerU. S. Army Electronics R&D Activity Aero Geo AstroAttn: SELWS-ED Attn: Security OfficerWhite Sands, Missile Range, New Mexico 1200 Duke Street

Alexandria, VirginiaDirectorSurveillance Department NASA Goddard Space Flight CenterEvans Area Attn: Antenna Section, Code 523Attn: Technical Document Center Greenbelt, MarylandBelman, New Jersey

Airborne Instruments Labs., Inc.Commander Attn: Librarian (Antenna Section)U. S. Naval Air Tcst Center Walt Whitman RoadAttn: WST-54, Antenna Section Melville, L.I., New YorkPatuxent River, Maryland

American Electronic Labs

Material Laboratory, Code 932 Box 552 (Antenna Section)New York Naval Shipyard Lansdale, PennsylvaniaBrooklyn 1, New York

Andrew Alfred Consulting EngineersDirector Attn: Librarian (Antenna Section)Naval Research Laboratory 299 Atlantic AvenueAttn: Code 5200 Boston 10, MassachusettsWashington 25, D. C.

3

Amphenol-Borg Electronic Corporation ConvairAttn: Librarian (Antenna Section) Attn: Technical Library2801 S. 25th Avenue (Antenna Section)Broadview, Illinois P. 0. Box 1050

San Diego 12, CaliforniaBell Aircraft CorporationAttn: Technical Library Dalmo Victor Company

(Antenna Section) Attn: Technical LibraryBuffalo 5, New York (Antenna Section)

1515 Industrial WayBendix Radio Division of Belmont, California

Bendix Aviation CorporationAttn: Technical Library Dorne & Margolin, Inc.

(For Dept. 462-4) Attn: Technical LibraryBaltimore 4, Maryland (Antenna Section)

30 Sylvester StreetAntenna Systems, Inc. Westbury, L. I., New YorkManager DocumentationGrenier Field Dynatronics, Inc.Manchester, New Hampshire Attn: Technical Library

(Antenna Section)Boeing Airplane Company Orlando, FloridaAero Space DivisionAttn: Technical Library Electronic Communications, Inc.

M/F Antenna & Radomes Unit Research DivisionSeattle, Washington Attn: Technical Library

1830 York RoadBooing Airplane Company Timonium, MarylandAttn: Technical Library

M/F Antenna Systems Staff Unit Fairchild Engine & Airplane CorporationWichita, Kansas Fairchild Aircraft & Missiles Division

Attn! Technical LibraryChance Vought Aircraft, Inc. (Antenna Section)THRU: Bu AER Representative Hagerstown 10, MarylandAttn: Technical Library

M/F Antenna Section Georgia Institute of TechnologyP. 0. Box 5907 Engineering Experiment StationDallas 22, Texas Attn: Technical Library

M/F Electronics DivisionCollins Radio Company Atlanta 13, GeorgiaResearch DivisionAttn: Technical Library General Electric CompanyCedar Rapids, Iowa Electronics Laboratory

Attn: Technical LibraryConvair Electronics ParkFt. Worth Division Syracuse, New YorkAttn: Technical Library (Antenna

Section) General Electronic Labs., Inc.Grants Lane Attn: Technical LibraryFort Worth, Texas (Antenna Section)

C18 Ames StreetCambridge 42, Massachusetts

H

General Precision Lab., Division of Hughes Aircraft CorporationGeneral Precision, Inc. Attn: Technical Library

Attn: Technical Library (Antenna Section)(Antenna Section) Florence & Teal Street

63 Bedford Road Culver City, California JT

Pleasantville, New YorkITT Laboratories

Goodyear Aircraft Corporation Attn: Technical Library

Attn: Technical Library (Antenna Section)

M/F Dept. 474 500 Washington Avenue

1210 Massilon Road Nutley 10, New JerseyAkron 15, Ohio

U. S. Naval Ordnance Laboratory

Granger Associates Attn: Technical LibraryAttn: Technical Library Corona, California

(Antenna Section)

974 Commercial Street Lincoln LaboratoriesPalo Alto, California Massachusetts Institute of Technology

Attn: Document Room

I-T-E Circuit Breaker Company P. 0. Box 73

Special Products Division Lexington 73, Massachusetts

Attn: Research Library

601 E. Erie Avenue Litton Industries

Philadelphia 34, Pennsylvania Attn: Technical Library(Antenna Section)

General Mills Electronics Division 4900 Calvert Road

Attn: Dr. H. P. Raabe College Park, Maryland

2003 East Hennepin AvenueMinneapolis 13, Minnesota John Hopkins University

Radiation Laboratory

Grumman Aircraft Engineering Corporation Attn: Library

Attn: Technical Library 1315 St. Paul Street

M/F Avionics Engineering Baltimore 2, MarylandBethpage, New York

Lockheed Missile & Space Division

The Hallicrafters Company Attn: Technical Library (M/F Dept-Attn: Technical Library 58-40, Plant 1, Bldg. 130)

(Antenna Section) Sunnyvale, California

4401 W. Fifth AvenueChicago 24, Illinois The Martin Company

Attn: Technical Library

Hoffman Laboratories, Inc. (Antenna Section) Mail No. T-38

Attn: Technical Library P. 0. Box 179

(Antenna Section) Denver 1) Colorado

Los Angeles 7, CaliforniaThe Martin Company

John Hopkins University Attn: Technical Library (Antenna

Applied Physics Laboratory Section) .• , * U.-.

8621 Georgia Avenue Baltimore 3, Maryland

Silver Springs, Maryland

• I!

5

The Martin Company Northrop CorporationAttn: Technical Library (M/F NORAIR Division

Microwave Laboratory) 1001 East BroadwayBox 5837 Attn: Technical Information (3924-3)Orlando, Florida Hawthorne, California

W. L. Maxson Corporation Ohio State University Research

Attn: Technical Library Foundation(Antenna Section) Attn: Technical Library

460 West 34th Street (M/F Antenna Laboratory)New York 1, New York 1314 Kinnear Road

McDonnell Aircraft Corporation Columbus 12, Ohio

Attn: Technical Library Ohio State University Research(Antenna Section) Foundation

Box 516 Attn: Dr. C. H. Walter

St. Louis 66, Missouri 1314 Kinnear RoadColumbus 12, Ohio

Melpar, Inc.Attn: Technical Library Philco Corporation

(Antenna Section) Government & Industrial Division

3000 Arlington Blvd. Attn: Technical Library

Falls Church, Virginia (M/F Antenna Section)4700 Wissachickon Avenue

University of Michigan Philadelphia 44, PennsylvaniaRadiation LaboratoryWillow Run Westinghouse Electric Corporation

201 Catherine Street Air Arms DivisionAnn Arbor, Michigan Attn: Librarian (Antenna Ab)

P. 0. Box 746

Mitre Corporation Baltimore 3, MarylandAttn: Technical Library (M/F Electronic

Warefare Dept. D-21) Wheeler LaboratoriesMiddlesex Turnpike Attn: Librarian (Antenna Lab)Bedford, Massachusetts Box 561

Smithtown, New York

Northeastern UniversityAttn: Dodge Library Electrical Engineering Research

Boston 15, Massachusetts LaboratoryUniversity of Texas

North American Aviation, Inc. Box 8026, University Station

Attn: Technical Library (M/F Austin, Texas

Engineering Dept.)4300 E. Fifth Avenue University of Michigan Research

Columbus 16, Ohio Institute.-Electronic Defense Group

North American Aviation, Inc. Attn: Dr. J. A. M. LyonsAttn: Technical Library Ann Arbor, Michigan

(M/F Dept. 56)International Airport Radio Corporation of America

Los Angeles, California RCA Laboratories DivisionAttn: Technical Library

(W/F Antenna Section)Princeton, New Jersey

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I.'6

.I1

Radiation, Inc. Sperry Gyroscope CompanyAttn: Technical Library (U/F) Attn: Librarian (Antenna Lab)

Antenna Section Great Neck, L. I., New YorkDrawer 37Melbourne, Florida Stanford Electronic Laboratory

Attn: Librarian (Antenna Lab)Ramo-Wooldridge Corporation Stanford, CaliforniaAttn: Librarian (Antenna Lab)Canoga Park, California Stanford Research Institute Ii

Attn: Librarian (Antenna Lab)Rand Corporation Menlo Park, CaliforniaAttn: Librarian (Antenna Lab)1700 Main Street Sylvania Electronic SystemSanta Monica, California Attn: Librarian (U/F Antenna &

Microwave Lab)Rantec Corporation 100 First StreetAttn: Librarian (Antenna Lab) Waltham 54, Massachusetts23999 Ventura Blvd.Calabasas, California Sylvania Electronic System

Attn: Librarian (Antenna Lab)Raytheon Corporation P. 0. Box 188Equipment Division Mountain View, CaliforniaLibrary - J. PortschP. 0. Box 520 Technical Research GroupWaltham 54, Massachusetts Attn: Librarian (Antenna Section)

2 Aerial WayRepublic Aviation Corporation Syosset, New YorkApplied Research & Development

Division Ling Temco Aircraft CorporationAttn: Librarian (Antenna Lab) Temco Aircraft DivisionFarmingdale, New York Attn: Librarian (Antenna Lab)

Garland, TexasSanders AssociatesAttn: Librarian (Antenna Lab) Texas Instruments, Inc.95 Canal Street Attn: Librarian (Antenna Lab)Nashua, New Hampshire 6000 Lemmon Avenue

Dallas 9, TexasSouthwest Research InstituteAttn: Librarian (Antenna Lab) A. S. Thomas, Inc.8500 Culebra Road Attn: Librarian (Antenna Lab)San Antonio, Texas 355 Providence Highway

Westwood, MassachusettsH. R. B. Singer CorporationAttn: Librarian (Antenna Lab) New Mexico State UniversityState College, Pennsylvania Head Antenna Department

Physical Science LaboratorySperry Microwave Electronics Company University Park, New MexicoAttn: Librarian (Antenna Lab)P. 0. Box 1828Clearwater, Florida

!7I

Bell.,Tlephone Laboratories, Inc. Professor A. A. OliverWhippany, New Jersey Polytechnic Institute of BrooklynAttn: Technical Reports Librarian Microwave Research Institute

Room 2A-165 55 Johnson Streetr GBrooklyn 1, New YorkI Robert 0. Hansen

Aerospace Corporation U. S. Naval Ordnance LaboratoryBox 95085 Attn: Technical LibraryfLos Angeles 45, California Corona, California

Dr. Richard C. Becker Avco Corporation10829 Berkshire Research and Advanced DevelopmentWestchester, Illinois Division

Attn: Research Library/T.A. RupprechtDr. W. U. Hall 201 Lowell StreetRaytheon Company Wilmington, MassachusettsSurface Radar and NavigationOperations Raytheon Company

SBoston Post Road Missile and Space DivisionWayland, Massachusetts Attn: Research Library

Bedford, Massachusetts

Dr. Robert L. CarrelSCollins Radio Corporation Teledyne Systems, Incorporated

Antenna Section Attn: Technical Library/Antenna SectionDallas, Texas 1625 E. 126th Street

Hawthorne, CaliforniaDr. A. X. Chatterjee

Vice Principal and Head of the De- National Research Councilpartment of Research Attn: Microwave Section

Birla Institute of Technology Ottawa 2, CanadaP. 0. MesraDistrict-Ranchi (Bihar) India Sichak Associates

Attn: W. SichakUniversity of Dayton 518 Franklin AvenueResearch Institute Nutley, New JerseyAttn: Professor Douglas Hanneman300 College Park W. T. PattonDayton, Ohio 2208 New Albany Road

Cinn. Township

Technische Hochschule Riverton Post OfficeAttn: H. H. Meinke New JerseyS~Munich, Germany

Radio Corporation of America

NASA Goddard Space Flight Center Missile and Service Radar DivisionAttn: Antenna Branch Attn: Manager, Antenna Engineering

Mr. Lantz Skill CenterGreenbelt, Maryland Moorestown, New Jersey

Aeroneutic DivisionFord Motor CompanyFord Road - Attn: Mr. J. M. BlackNewport Beach, California

I

Advanced Development Laboratories, Inc. Avco CorporationHaines Street Electronic and Ordnance DivisionNashua, New Hampshire Attn: Technical Library

Cincinnati 41, OhioCornell Aeronautical Lab.Attn: Research Library Bendix Corporation 1Buffalo, New York Research Labs Division

Attn: G. M. PeaceFairchild Stratos Corporation Southfield (Detroit), MichiganAircraft-Missiles DivisionAttn: Technical Library (Antenna Section) Douglas Aircraft Co., Inc.Hagerstown 10, Maryland Attn: MSSD Technical Library

(Antenna Section)General Electric Company 3000 Ocean Park Blvd.Light Military Electronics Department Santa Monica, CaliforniaFrench RoadUtica, New York Emerson and Cuming, Inc.

Attn: E. J. LuomaGeneral Electric Company 869 Washington StreetGeneral Engineering Laboratory Canton, MassachusettsBldg 371, Room 478Schenectady, New York Radioplane Company

Attn: Librarian (M/F Aerospace Lab)Goodyear Aircraft Corporation 1515 Rancho Conelo Blvd.Antenna Department Newbury Park, CaliforniaLitchfield ParkPhoenix, Arizona Hughes Aircraft Company

Attn: Antenna SectionLaboratory for Electronics, Inc. Fullerton, CaliforniaAntenna Department1079 Commonwealth AvenueBoston 15, Massachusetts

Lockheed AircraftElectronic and Armaments System OfficeBurbank, California

Motorola, Inc.Western Military Electronics DivisionP. 0. Box 1417Scottsdale, Arizona

Philco CorporationAntenna Section3875 Fabian WayPalo Alto, California

Space Technology LaboratoryAttn: Research LibraryP. 0. Box 95100Los Angeles 45, California

F ANTENNA LABOUTTUY

TECHNICAL REPOTS AND MSRANDA INUE)

Contract AF33(616)-310

"Synthesis of Aperture Antennas," Technical Report No. 1, C. T. A. Johnk,October, 1954.*

"A Synthesis Method for Broadband Antenna Impedance Matching Networks,"Technical Report No. 2. Nicholas Yaru, 1iFebruary 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 S.Mayes, 1 June 1955 (CONFIDENTIAL).

"Coupling of Antenna Elements to a Circular Surface Waveguide," TechnicalReport No. 5. H. E. King and R. H. DuHamel, 30 June 1955.*

"Input Impedance of A Spherical Ferrite Antenna with A Latitudinal Current,"Technical Report No. 6, W. L. Weeks, 20 August 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,•A$*uly 1955.*

Contract AF33(616)-3220

"Effective Permeability of Spheriodal 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 ReportNo. 10. D. G. Berry and J. B. Kreer, 10 May 1956. AD 98615.

"A Technique for Controlling the Radiation from Dielectric Rod Waveguides,"Technical Report No. 11, J. W. Duncan and R. H. DuHamel, 15 July 1956.*

"Direction Characteristics of a U-Shaped Slot Antenna," Technical ReportNo. 12. Richard C. Becker, 30 September 1956.**

"Impedance of Ferrite Loop Antennas," Technical Report No. 13. V. H. Rumseyand W. L. Weeks, 15 October 1956. AD 119780.

"Closely Spaced Transverse Slots in Rectangular Waveguide," Technical ReportNo. 14, Richard F. Hyneman, 20 December 1956.I

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"Distributed Coupling to Surface Wave Antennas," Technical Report No. 15.

R. R. Hodges, Jr., 5 January 1957. 11"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. f-"A Note Concerning a Mechanical Scanning System for a Flush Mounted LineSource Antenna," Technical Report No. 18, Walter L. Weeks, 20 April 1957.

"Broadband Logarithmically Periodic Antenna Structures," Technical Report No.19. R. H. DuHamel and D. E. Isbell, 1 May 1957. AD 140734. II"Frequency Independent Antennas," Technical Report No. 20, V. H. Rumsey,25 October 1957.

"The Equiangular Spiral Antenna," Technical Report No. 21. J. D. Dyson, 115 September 1957. AD 145019.

"Experimental Investigation of the Conical Spiral Antenna," Technical ReportNo. 22. R. L. 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 ArbitraryCross Section," Technical Report No. 25, Robert L. Carrel, August 1957.

"Cavity-Backed Slot Antennas," Technical Report No. 26. R. J. Tector, 30

October 1957. i"Coupled Waveguide Excitation of Traveling Wave Slot Antennas," TechnicalReport No. 27, W. L. Weeks, 1 December 1957. i"Phase Velocities in Rectangular Waveguide Partially Filled with Dielectric,"Technical Report No. 28. W. L. Weeks, 20 December 1957. Ii"Measuring the Capacitance per Unit Length of Biconical Structures of Arbi-trary Cross Section," Technical Report No. 29, J. D. Dyson, 10 January 1958. jj

"Non-Planar Logarithmically Periodic Antenna Structure", Technical ReportNo. 30, D. E. Isbell, 20 February 1958. AD 156203.

"Electromagnetic Fields in Rectangular Slots," Technical Report No. 31, IIN. J. Kuhn and P. E. Mast, 10 March 1958. I

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I-s3

"The Efficiency of Excitation of a Surface Wave on a&Dielectric Cylinder,"Technical Report No. 32, J. W. Duncan, 25 May 1958.

"A Unidirectional Equiangular Spiral Antenna," Technical Report No. 33,SJ. D. Dyson, 10 July 1958. AD 201138.

"Dielectric Coated Spheroidal Radiators," Technical Report No. 34, W. L.Weeks, 12 September 1958. AD 204547.

"A Theoretical Study of the Equiangular Spiral Antenna," Technical ReportNo. 35, P. E. Mast, 12 September 1958. AD 204548.

Contract AF33(616)-6079

i Use of Coupled Waveguides in a Traveling Wave Scanning Antenna," TechnicalReport No. 36, R. H. MacPhie, 30 April 1959. AD 215558.

"On the Solution of a Class of Wiener-Hopf Integral Equations in Finite andInfinite Ranges," Technical Report No. 37, R. Mittra, 15 May 1959.

"Prolate Spheriodhl Wave Functions for Electromagnetic Theory," TechnicalReport No. 38, W. L. Weeks, 5 June 1959.

"Log Periodic Dipole Arrays," Technical Report No. 39, D. E. Isbell, 1 JuneI" 1959. AD 220651.

"A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," TechnicalI Report No. 40, S. Dasgupta and Y. T. Lo, 17 July 1959.

"The Radiation Pattern of a Dipole on a Finite Dielectric Sheet," Technical

Report No. 41, K. G. Balmain, 1 August 1959.

"The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problemin a Waveguide: Technical Report No. 42, R. Mittra, 1 October 1959.

"Impedance Properties of Complementary Multiterminal Planar Structures ,

Technical Report No. 43, G. A. Deschamps, 11 November 1959.

"On the Synthesis of Strip Sources," Technical Report No. 44, R. Mittra,

4 December 1959.

I "Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Wave-guides, Technical Report No. 45, C. H. Tang and Y. T. Lo, 11 March 1960.

"New Circularly Polarized Frequency Independent Antennas with Conical Beamor Omnidirectional Patterns," Technical Report No. 46. J. D.-Byson and P. 1.Mayes, 20 June 1960. AD 241321.

" "Logarithmically Periodic Resonant-V Arrays," Technical Report No. 47 * P. E.

Mayes and R. L. Carrel, 15 July 1960. AD 246302.1

4"A Study of Chromatic Aberration of a Coma-Corrected Zoned Mirror," Tech-nical Report No. 48, Y. T. Lo, June 1960.

"Evaluation of Cross-Correlation Methods in the Utilization of Antenna Systems,"Technical Report No. 49, R. H. MacPhie, 25 January 1961 I"Synthesis of Antenna Products Patterns Obtained from a Single Array," TechnicalReport No. 50, R. H. MacPhie, 25 January 1961.

"On the Solution of a Class of Dual Integral Equations," Technical Report No. 51,R. Mittra, 1 October 1961. AD 264557. [1"Analysis and Design of the Log-Periodic Dipole Antenna," Technical Report No. 52.Robert L. Carrel, 1 October 1961. AD 264558. i

"A Study of the Non-Uniform Convergence of the Inverse of a Doubly,InfiniteMatrix Associated with a Boundary Value Problem in a Wavegulie," TechnicalReport No. 53, R. Mittra, 1 October 1961. AD 264556, 11

Contract AF33(616)-8460

"The Coupling and Mutual Impedance Between Balanced lire-Arm Conical Log-Spiral IAntennas," Technical Report No. 54, J. D. Dyson, Jule 1962.

"An Investigation of the Near Fields on the Conical Equiangular Spiral Antenna," I

Technical Report No. 55, 0. L. McClelland, May 1962.

"Input Impedance of Some Curved Wire Antennas," Technical Report No. 56, 1C. H. Tang, June 1962.

"Polygonal Spiral Antennas," Technical Report No. 57, C. H. Tang. O.L.McClelland, June 1962.

"On Increasing the Effective Aperture of Antennas by Data Processing,"Technical Report No. 58, R. H. MacPhie, July 1962.

"Theoretical Study of a Class of Logarithmically Periodic Circuits,"Technical Report No. 59, R. Mittra, July 1962. 11"Backward Wave Radiation from Periodic Structures and Application to theDesign of Frequency-Independent Antennas," Technical Report No. 60.P. E. Mayes, G. A. Deschamps, and W. T. Patton, December 1962.

"The Backfire Bifilar Helical Antenna," Technical Report No. 61. W. T. Patton,September 1962.

"On the Mapping by a Cross-Correlation Antenna System of Partially CoherentRadio Sources," Technical Report No. 62. R. H. MacPhie, October 1962.

"On a Conical Quad-Spiral Array," Technical Report No. 63, J. D. Dyson,September 1962.

I

I 5"Antenna Impedance Matching by Means of Active Networks," Technical ReportNo. 64, S. Laxpati) R. Mittra, November 1962.

*Copies available for a three-week loan period.

** Copies no longer available

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