PROCEEDINGS OF THE I.R.E.

BIBLIOGRAPHY

(1) C. R. Burrows, "Radio propagation over plane earth," Bell. Sys.Tech. Jour., vol. 16, pp. 45-75, January, 1937. (Available powerfrom a receiving antenna, p. 48.)

(2) H. A. Wheeler, "Radio Wave Propagation Formulas," HazeltineReport 1301W, May, 1942. (The isotropic antenna with effectivearea of one radian circle.)

(3) H. A. Wheeler, "Radiation Resistance," Hazeltine Report1491W, May, 1942. (Radiation resistance in terms of directivegain relative to isotropic antenna.)

(4) J. C. Slater, "Microwave Transmission," McGraw-Hill BookCo., New York, N. Y., 1942. (Wave impedance of space and ofparallel-strip transmission line, pp. 93-100, 182.) (Wave im-pedance of rectangular waveguide, pp. 184-185; formulas lack-ing a factor 2.) (Directive gain defined relative to mean sphericalradiation, pp. 215-216.) (Effective area of mean spherical radia-tion, equal to one radian circle, pp. 243-245.) (Directive gain interms of the area relative to one radian circle, pp. 260-264.)(Rectangular waveguide equivalent to an infinite lattice or flatarray of images, pp. 284-285.) (Radiation resistance of small di-

pole in rectangular waveguide, pp. 288-289; with end reflector,pp. 296-298.) (Coupling of a dipole in a waveguide, pp. 291-296.)

(5) S. A. Schelkunoff "Electromagnetic Waves," D. Van NostrandCo., New York, N. Y., 1943. (Radiation resistance of a smalldipole, pp. 133-134.) (Available power from a receiving antenna,pp. 136, 512.) (Radiation resistance of half-wave dipole, pp. 144-340.) (Wave impedance of parallel-strip line, p. 243.) (Wave im-pedance of rectangular waveguide, pp. 316-319.) (Directivegain defined relative to isotropic antenna, pp. 335-336; of a smalldipole, p. 337.) (Directive gain in terms of the area relative toone radian circle, pp. 360, 365.) (Radiation resistance of an an-tenna in a waveguide, pp. 494-496.)

(6) F. E. Terman, "Radio Engineers' Handbook," McGraw-HillBook Co., New York, N. Y., 1943. (Mutual impedance of anten-nas, pp. 776-782.) (Flat array of diagonal antennas, p. 852.)

(7) H. T. Friis, "A note on a simple transmission formuila," PROC.I.R.E., vol. 34, pp. 254-256, May, 1946. (Available power froman antenna in terms of its effective area. Isotropic antenna havingan area of one radian circle.)

(8) Simon Ramo, "Introduction to Microwaves," John Wiley andSons, New York, N. Y., 1946 (Parallel-strip transmission line, pp.101-103, 113-1 14.)

Coupled Antennas *C. T. TAIt, STUDENT, I.R.E.

Summary-The integral equation governing the current distribu-tion on two coupled antennas has been solved. The method used isan improvement on the work originally formulated by King andHarrison. As a result of this improvement, the general solutionpertaining to the antenna problem reduces to the conventional oneobtained from transmission-line theory, when the two antisym-metrically driven antennas are closely coupled to each other.Numerical values of the self- and mutual impedances based uponthe present work have been computed. The result is compared withthose obtained by Carter based upon the so-called e.m.f. method,assuming a sinusoidal distribution of the currents.

INTRODUCTIONr HE PROBLEM of finding the current distribu-

tion and impedance characteristic of a center-driven antenna is, in general, a problem of how to

find a solution of the three-dimensional vector waveequation that satisfies the specified boundary condi-tions. Unless the body of the antenna as a whole can bewell defined by one appropriate co-ordinate in some co-ordinate system-as, for example, a prolate spheroid'-no general method2 is so far available in the sense thatthe solution would satisfy the boundary condition atevery part of the body, including, for instance, the endsurfaces of a cylindrical antenna or those of a biconicalantenna.

Hallen's vector potential method3 in dealing with thecylindrical antenna is a very satisfactory one because

* Decimal classification: R125.1. Original manuscript received bythe Institute, May 21, 1947. The research reported in this documentwas made possible through support extended Cruft Laboratory,Harvard University, jointly by the United States Navy, Office ofNaval Research, and the United States Army Signal Corps, underONR Contract N5ori-76, T.O.I.

t Cruft Laboratory, Harvard University, Cambridge, Mass.J. A. Stratton and L. J. Chu, "Steady-state solutions of electro-

magnetic field problems," Jour. Appl. Phys., vol. 12, p. 230; March,1941.

2 J. Aharoni, "Antennae," Oxford University Press, Oxford Eng-land, 1946.

3E. Hall6n, 'Theoretical investigations into the transmitting andreceiving qualities of antennae," Nova Acta Uppsala, vol. 77, pp.1-44; November, 1938.

the end effect in such a formulation is negligible,4 whilemathematically it permits reduction of the analysis intoa one-dimensional form. Moreover, this method is es-pecially appropriate for handling the problem of cou-pled antennas.The present work is an improvement on the niethod

originally formulated by King and Harrison.5 The im-provement is twofold. In the first place, a proper distri-bution function has been chosen in expanding the inte-gral equation as was done in the case of a single antenna,6and secondly, the term corresponding to the contribu-tion of the vector potential by the second antenna istreated as part of the main integral instead of as a cor-rection term. Results derived from the present methodshow that, in the case of two coupled antennas drivenantisymmetrically, the solution reduces exactly to theconventional one obtained from transmission-linetheory, when the two antennas are sufficiently close tosatisfy the conditions of line theory.

GENERAL EQUATIONSThe general formulation of the problem has been dis-

cussed in detail.' Two coupled antennas of identical sizeare considered in this paper. To simplify the discussion,the internal or surface impedance of the antennas is alsoassumed to be negligible. With the arrangement shownin Fig. 1, the z component of the vector potential at thesurface of each of the two antennas (viz., A1, and A2,)satisfy the following differential equations:

+2Al_ = 021222 (1)

' L. Brillouin, "The antenna problem," Quart. Appl. Math., vol.1, pp. 201-204; October, 1943.

b R. King and Charles W. Harrison, Jr., "Mutual and self-im-pedance for coupled antennas," Jour. Appl. Phys., vol. 15, pp. 481-495; June, 1944.

6 R. King and D. Middleton, "The cylindrical antenna; currentand impedance," Quart. Appl. Math., vol. 3, pp. 302-335; January,1946.

4871948

PROCEEDINGS OF THE I.R.E.

a2A2s,Z+ 32A22= 0

az2

with

IA0 h e- jj9rll Y h

Alz= -°I Ii dz' + - 1'2'47r _h ril 47r h

hor h e- jor22A2z = '2z,' dz'

47r-h r224Ar _h

e- j rl2dz'

r12

e- j#r2l-dz'.r2l

Fig. 1-Two coupled antennas.

The following notation is used:

/0 = 4,r X 107 henry/meterco'

ril = \/(z' - ZI)2 + a2

r12 = N/(~Z2,-Zl1)2 + b2

r22 = (Z2'-z2)2 + a2

r2l= (Z'-

(2) By means of the superposition theorem, the solutionfor two antennas driven by two given voltages V10 andV20 always can be obtained from the solutions for sym-metrically and antisymmetrically driven pairs by writ-ing V10=(V,+Vs), V20=(V8-Va). Therefore, (1), (2),(3), and (4) may be specialized to these two cases with-

(3) out loss in generality. The superposition theorem and(4) the notation are illustrated graphically in Fig. 2.

SYMMETRICALLY DRIVEN IDENTICAL ANTENNASFor two identical antennas driven symmetrically, one

has

'lz = T2z; A1z = A2z. (6)With (6), (1) and (3) or (2) and (4) reduce to the follow-ing form:

02A2+:Z 2Az= O

8z2

Az, dz' +- rw ___dz'.4r h ril 4TrJh r2

The solution of (7) is

-jA =- (Cl cos ,Bz + 'V. sin ,B z )c

(7)

(8)

(9)

where one of the two arbitrary constants of the generalsolution has been determined to satisfy the discontinuityof the scalar potential V. at the driving point of each an-tenna. The second constant C1 is to be determined laterwhen the boundary condition that the current mustvanish at the ends of the antenna is imposed. Equating(8) and (9), one obtains the integral equation of I,:

*h /e-joril e-ifrl2\(5) J TZI- + ) dz'

-h \ ril r2 /

-j4ir=- (Ci cos z+ 'V. sin B z )

Rwhr

where

(10)

where a is the radius, and h is the half length of each an-tenna.

VoI It V2 = Vstttl vs + Va

RC = o-= 1207x ohms.

The two integrals in (8) have been combined to form oneintegral in (10) in order to emphasize the fact that thekernel of the integral equation is

+ e-iPrl2ril r12Va

Vs +V20 VIO - V2o22

Fig. 2-Schematic diagram to illustrate the theorem of superpositionas applied to two coupled antennas.

(11)

It is at this point where the new method diverges fromthe old one, in which the integral due to the distant ac-tion of the second antenna was treated as a correctionterm, while the function e-izlll/rll alone was regarded asthe kernel of the integral equation. Equation (10) hasthe same form as that encountered in the problem of asingle antenna except that the kernel of the integralequation involves two terms. A method of solving an

488 April

Tai: Coupled Antennas

equation of this type has been discussed in detail,5 andwill not be repeated here. The final expression of thefirst-order solution for I,8 is

j27rVJrz8 =

Rc*ab

rh e- ifrl2Sb(Z)-= sin I 2t dzl-h rT2

(17)

(12)

where the constant Tab and various functions are definedas follows:

Fo(z) = cos fz; Go(z) =sin ,| z

Fo, =Fo(z)-Fo(h); Goz=Go(z) -Go(h)rh e-ir1l

Fiz=*ab,Fz- Foz0 dz'-h ril

='Ifab(cos 1Z-cos lh)-Ca(Z)+Ea(z) cos ,Bhfwh e- ifrii

Giz=TabGoz- Go dz'

-Tab(sin I | z |-sin 3h)1-Sa(z) +Ea(Z) sin h(13)

F1l =Fl(z) -Fl(h); G1z=G,(z)-Gj(h)rh e- jfr12

P1(z) = - Foz, - dz'= -Cb(z)+Eb(Z) cos,Bh-h r1 2

rh e-irj2Qi(z) = - Goz I dz'= -Sb(z)+Eb(z) sin 5h

-h rl2

Plz=Pl(z)-Pi(h); Qiz= Qi(z) -Ql(h)

[Ca(Z) +Cb(Z)] sin h- [S.(z) +Sb(z)] cos ,hifab(Z)= sin,(h- jz) (14)

T| 'iab(O) for Bh-<7r

TIabl(5Tab(h ) | for ,Bh>-2

The functions C(z), Cb(z), Sa(Z), Sb(z), Ea(z), andEb(z) are defined by the following definite integrals:

rh e- iffrilCa(Z) = Ccos8z' dz';-h ril

j2irVaIza

=

Rc''ab'

with

ril = (zI - Z)2 + a2;

(18)

r12 = \(z Z)2 + b2.

It is to be noted that the functions Fo(z), Go(z), P1(z)and Q1(z) are the same as previously defined,5 6 while thefunctions F1(z) and Gj(z) can also be expressed in termsof similar functions elsewhere defined.5 Consequently,(12) can be rearranged to contain these old functions inorder to facilitate the numerical evaluation of (12).This modified form of (12) is given in the appendix,where the relations between the new functions and theold ones are outlined.

ANTISYMMETRICALLY DRIVEN IDENTICALANTENNAS

For two identical antennas driven antisymmetrically,one has

Tlz = - JT2z Al =- A22. (19)

The integral equation in I becomes

rh / e-3r11 e- iP12z dz'

Jh \ rll r12/

-j47r(ClcosPoz+ 'Vasini3IzI).

Re (20)

As a result of the change of sign in the kernel, every func-tion involving b reverses its sign, whereas those involv-ing a do not. The final expression of the first-order solu-tion for Iza follows:

sin R(h-| zI )+ {Go(h) [F1z'-Pz] +Foz [Gl'(h)-Ql(h) ]-Fo (h) [Gl ,'-Q i,z']-Gotz[F'(h)-PI (h) ]

, .-1 . (21)Fo(h)+ -, [F'(h) - Pl(h) ]

~~~~~~~~~a'I'

h r h ei jfri2Cb(z) = cos Z dz'-h -~~~hrl12

S (z) = Jhsin ,B | e-dz';-h rl

The functions Fo(z), Go(z), Pj(z), and Q1(z) remain the(16) same as before. The functions F1'(z) and Gl'(z), however,

appear in place of F1(z) and Gj(z), since Tab is replacedby ''ab, which is given by

[C4(z) Cb(z) ] sin fh3- [S(Z) -Sb(z) ] COSO(S''ab/(Z) = )(22)

1- ~~~~~1sin #(h- z I) J+-bGo(h)[Fl+Pl]+Foz [Gl(h) +Ql(h)]-Fo(h) [Giz+Qiz]-Goz[Fi(h)+Pi(h) lI

Fo(h)+- [Fi(h)+P1(h)]LTa

rh e- ifrilEa(z) = dz';

Eh ril

rh e-ifrn2Eb(Z) = - dzi

-h 'rI2

4891948

PROCEEDINGS OF THE I.R.E.

'ab'(0) for (3h<-tJ!ab = 2 (23)

1Iab(kil) for Ah>--

The symmetrical and antisymmetrical impedances areobtained by setting z=0 in (12) and (21), and findingthe ratio V8/108 and Va/1oa, respectively. This gives

2'r

-jRcJab'27r

substituting in (30) and (31). The resultant current onantenna 2 is the difference (IZJ-Ita) obtained from (12)and (21).

CLOSELY COUPLED ANTENNASTwo antennas are said to be closely coupled if the

separation b between the two satisfies the following con-dition:

b2 << h2.

1Cos (3k + ~~[Fi(k) + Pl(kt)]

TIab

(32)

1 J (24)sin (h + { [Fl(O) + P1(O)] sin Oh- [G1(o) + Qi(O)] cos (h + [G,(k) + Qj(h)] I-

cos (h + [F'((k) - PO(k)]1

a b'. (22)

sin kh + / { [Fl'(O) - P1(O)I sin #h- [G1'(o) - Qi(O)] cos,Bh + [G1'(k) -Q(h)]ITab'ab

For two identical antennas driven by two arbitrary volt-ages V1o and V20, the relation between the input cur-rents and the exciting voltages are

V10 = T10Z11 +120Z12V20 = 120Z11 + 110Z12

(26)(27)

where the self-impedance, Zul and the mutual impedanceZ12 are defined by (26) and (27) and are related to Z8 andZ. according to the following equations:

Z8 + ZaZ11 = - ;Z - Za

Z12 =2

(28)

If the second antenna is a parasitic antenna loaded atcenter with an impedance ZL, one replaces V2o by-ZLI2o. The input impedance for Vio is then

Z122 2ZZa + (Z8 + Za)ZL

ZL +Zll 2ZL +Zs +ZaThe values of V. and Va in terms of Z8, ZA, ZL, and V1oare given below,

Without loss of generality, the discussion will again becarried on in two separate cases; namely, the sym-metrical and the antisymmetrical.

Case 1. Symmetrically Driven Antennas

If one defines Ja(Z) and Ib(Z) according to the follow-ing equations,

Ca(z) sin 3h- Sa(Z) cos Phsin3(k- I zI )

''b(Z) - Cb(z) sin (3h - Sb(Z) cos Ph(sin ,(h-|Iz )

(33)

(34)

then (14) can be written into

Ta bb(Z) = *a(Z) + * b(Z). (35)

For a2 and b2<<h2, the formulas of Ca(z), Cb(z), Sa(z), andSb(z) tabulated in Appendix II can be used in (33) and(34). It can easily be verified that

for ,Bhk- 2

7rfor ph3k~-

[Z8(Za+ZL)1Vs = r ] FoL2Z,Za + (ZJ + Za)Z -

Va Za(Ze+ ZL) v.

2Z,Za + (Z8 + Za)ZLI

(30)with

(3t)

The resultant current in antenna 1 is then the sum

(I,.+I,.) obtained from (12) and (21) with V. and V.

2hDa= 2 In -;

a

2hQb= 2 In -.

b (37)

The parameter S2. was first introduced by HalI6n in hisstudy of the integral equation for a single antenna. Thefunction 'J was introduced by King and Middleton in

(36)2*.(O) - Q,,, + 0, = 2T,(O) + 12a - Ob

*abx x

2*a h - - Oa + Ob = 2*b h - - + Oa - 12b4 4

490 April

Tai: Coupled Antennas

their recent work on the same problem. Equation (36) isvery useful to determine the numerical values of *ab, asit is possible to make use of data already computed bythe authors mentioned above.

Case 2. Antisymmetrically Driven Antennas

Two closely coupled antennas driven antisym-metrically are equivalent to two open-end sections oftwo-wire line in series with each other and two genera-tors Va. Using the same formulas for CQ(z), Cb(z), etc.,(23) in this case reduces simply to

btab = Qa - Qb = 2 In -.

a

The solution of I,a then reduces ultimately to

Ij27rVa sin,B(h - IIZ|)Iz=Rc=ab/ cos O3hjV,, sin0(h - Iz I)Rline cos ,Bh

with

(38)

(39)

bRiine = 120 ln -e

a

Terms corresponding to higher orders than the first areidentically vanishing. It is recalled that the term Rlime isprecisely the characteristic impedance of the parallel-wire line subjected to the condition that b2>>a2. Equa-tion (39) therefore coincides with the solution derivedfrom the line theory.7 TIhe proximity effect is, of course,neglected at the very beginning, where the rotationalsymmetry of the current distribution was assumed inderiving the formula of A for the vector potential on thesurface of the conductors. This effect can be taken intoconsideration by substituting the effective spacing

2 (1 + +/1I (2ab

for b in the original equations for the antisymmetricalcase.8

EXTENSION TO n-COUPLED ANTENNAS

The method of analysis of two coupled antennas canbe extended to n-coupled antennas provided that thesimultaneous integral equations can be reduced to thesame type described above. This sets up a limit to thegeometrical configuration of the antennas as well as theway of excitation. For three identical coupled antennasarranged at the corners of an equilateral triangle, theproblem can be solved completely no matter how theantennas are excited. By the method of symmetrical

7 R. King, 'Transmission-line theory and its application," Jour.Appi. Phys., vol. 14, p. 577; November, 1943.

8 R. King, "Electromagnetic Engineering," McGraw-Hill BookCo., New York, N. Y., vol. 1, p. 468; 1945.

components, three voltages of arbitrary magnitudes andphases can be decomposed into three sequences of volt-ages as shown schematically in Fig. 3. Each sequence is

IV, V3 V8, VAF Va Z PVa |Va PVO2~+ 1pVO + v

Fig. 3-Three different types of excitation occurring in theproblem of three coupled antennas.

then analogous to the symmetrical or antisymmetricalcomponent treated previously. For the zero sequence,the integral equation corresponding to (10) is

:h /( +2 e- )r2Iz, + 2 )dz'

J_z \ ril r12/-j4ir

- (Ci cos z+ .V8 sin , | z I ).Rc2

(40)

The integral equation corresponding to the positive ornegative sequence isfIh / e e- irI2 e- ifr13

i/ +p + p2 ) dz'Jh \ '1rl2l r13/

.-j47r- i (C1 Cos z+ iVa sin( Iz I )Re

(41)

where p is the phase factor ei(2T17) which satisfies theequation

1 + p + p2 = 0. (42)

By means of (42) and the relation r12 =r13, (41) can bereduced tofh /(e-jri-e-ere2\JIZ ( - Zdzi-h \ril r12

-j47r= i (C1 Cos /Z+ 'Va sin 3 1z ) ...Rc2

(43)

which is identical with (20). The method of evaluating(40) and (43) is the same as described before, and willnot be repeated.When the three antennas are closely coupled and ex-

cited by a sequence of voltages of equal amplitude but ofa phase difference of ei(2rl3) (that is, by a positive se-quence or a negative sequence), the system forms a three-phase transmission line. Accordingly, we may expectthat a certain type of transmission-line equation can bederived from the equations of the potentials. The deriva-tion of these line equations and a detailed analysis ofthem will be treated in a separate paper to be publishedlater, where the general problem of n-phase transmissionline and of two-phase multiple-wire transmission linewill be discussed.

1948 491

PROCEEDINGS OF THE I.R.E.

Numerical Computations

In order to give a quantitative discussion concerning

the impedances, or, more essentially, the current distri-bution of two coupled antennas, it is necessary to knowthe nature of several functions that occur in the expres-

sions for current and impedances. To illustrate the char-acteristic of these functions, two sets of curves have been

computed corresponding to two distinct values of an-tenna length, namely, h =X/4 and h =X/2.For h =X/4, it can be shown, by substituting (13),

(14), and (15) into (12), that the expression of I, can bereduced to the following form:

j27rV. [K1 cos i3z+K2 sin ZX-Ca(Z) -Cb(z)]Rc*8 [F1(h) - Pl()](

11:z20

<5-\ Fig. 4-The C0(z) func-tion, h = X/4.

Fig. 5-The Cb(z) func-tion, h=X/4.

-0.25 -0.20 -0.15 -0.10 -0.5 0) 0.5 0.10 0.15 0.20 0.25

71| 0III ImCbCj

,I

492 A pritl

iTai: Coupled Antennas

where K1 and K2 are two complex constants defined as tions, Ca(z) and Cb(z). The latter can be computed infollows: terms of some sine integrals and cosine integrals. The

K= 2I' + Ea(h) + Eb(h) -Sa(h)- Sb(h) (45) formulas are given in Appendixes I and II. Figs. 4 and 5

K2 = Caz(h) + Cb(h). (46) show two typical sets of curves for different values of aand b. The value of the imaginary part of Ca(z) or Cb(z)

It is obvious that the first-order solution for the cur- is practically independent of a or b. There is an over-allrents on antennas may be considered as a superposition change of about 1 per cent when a/h changes from 10-4of two sinusoidal functions and two nonsinusoidal func- to 10-1.

20

1tz20

'5

to

Fig. 6-The Sa(z) func- Re-a( )tion, h=X/2. A|

Fig. 7-The Sb(z) func-tion, h=X/2.

4931948

PROCEEDINGS OF THE I.R.E.

For hl=X/2, (12) reduces to K1'= 2Ts + Ea(h) + Eb(h) + Ca(h) + Cb(I) (48)j27rV. -K1' sin 3 z -K2' cos IZ-Sa(Z)-Sb(z)l K2' = Sa(h) + Sb(h). (49)

IzR=-L -'+F1(k)+P1(k)- (47) The fractions Sa(z) and Sb(z) for h =X/2 are shown in

Figs. 6 and 7. The parameters 'f. and gJa for different val-where K1' and K2' are two constants defined as follows: ues of a and b are plotted in Figs. 8 and 9.

Fig. 8-The parameters'I, and *., h=X/4.

Fig. 9-The parametersT' and Ta, h =X/2.

25

20

15

to

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

494 A pril

1948 Tai: Coup,

To describe the current distribution on two coupledantennas, it is convenient to treat the symmetrical and

/'Fig. 10-First-order cur- /

rent distribution for

,,' Ir~~~

hx/4.~ ~ ~

/l X_

-10 -7.5 -5

I: SINGLE ANTENNA, 1l. 15J: TWO COUPLED ANTENNAS S)

nf=15, b0.OOI3m= SINUSOIDAL DISTRIBUTION (

Fig. 11.-First-order cur-rent distribution for

h =x/2.

lIecd Antennas 495

antisymmetrical cases separately. In fact, these twocases may be regarded as two extreme conditions with

-2.5 0 2.5 5 7.5 10(10-3 MHOS)

-2.5 0 2.5 5-15 -12.5 -10 7.5 -5(104 MHOS)

49'PROCEEDINGS OF THE I.R.E.

the characteristics of an isolated antenna lying betweenthem; thus, suppose one starts with two closely coupledantennas, driven antisymmetrically by two equal andopposite voltages. The system is then equivalent to atwo-wire line. The current is known to be sinusoidal, orvery close to sinusoidal, if the attenuation along the lineis small. By separating these two lines, the currentsgradually depart from the sinusoidal distribution. As thewires are further separated so that their separation isinfinite, the distribution approaches that of an isolatedantenna. Suppose that one of the exciting voltages nowhas its polarity reversed, and that the two antennas arethen brought close together. The current distributionwould then change from that of an isolated antenna towhat would appear on two symmetrically driven anten-nas. The whole cycle therefore represents a completepicture involved in the problem of two coupled anten-nas.The above reasoning suggests that, to study the cur-

rent distribution on two coupled antennas, the simplestway is to compare three types of distribution cor-responding to (a) two closely coupled antisymmetricallydriven antennas (line current), (b) isolated antenna, and(c) two closely coupled symmetrically driven antennas.

OHMS150

The curves representing these three cases are shown inFigs. 10 and 11, where G(z) and B(z) are two real func-tions defined according to the following equation:

Iz = V[G(z) + jB(z)]. (50)

In drawing the line current, it has been assumed thatthe attenuation is small but not identically zero. Fortwo copper wires with Q 15 and b/X = 0.01, the magni-tude of B(0), i.e., the amplitude of the cosine function inFig. 10, will be equal to about 0.78, or 78 times as greatas the one drawn there.

Because of the importance of the knowledge aboutimpedance in the design of an antenna system, the sym-metrical and antisymmetrical impedances of two cou-pled antennas have been computed for the case h=X/4and h =X/2. The self and mutual impedances as de-fined by (28) have also been evaluated. These curves areshown in Figs. 12, 13, 14, 15, 16, 17, and 18. It is inter-esting to compare the numerical result obtained here forthe mutual impedance between the half-wave dipoles

Fig. 13-Symmetrical and antisymmetrical reactances. j5oh =ir/2.v~ %P.C- W. -.b/A0 -

Fig. 12-Symmetrical and antisymmetrical resistances. ,6oh = ir/2.

496 A pril

Tai: Coupled Antennas

with that of Carter9 and Brown,10 who computed thiscoefficient by assuming a sinusoidal distribution of cur-rent on the two dipoles and obtained the result from aquite different approach. The curves are shown in Fig.19. It is significant that the curve computed based uponthe present theory oscillates up and down around that ofCarter's.

CONCLUSION

The difference between the newly proposed method ofsolving the problem of coupled antennas and the oldermethod lies in an improved mathematical approach tothe solution of the integral equation. The verification bythe present method of results obtained from line theoryfor closely spaced antisymmetrically driven wires is asignificant confirmation of the validity of the newmethod. An extension of this method leads us to a rigor-

OHMS

Fig. 14-Symmetrical and antisymmetrical resistances. Boh=ir.

I P. S. Carter, "Circuit relations in radiating systems and applica-tions to antenna problems," PROC. I.R.E., Vol. 20, pp. 1004-1042;June, 1932.

10 G. H. Brown, "Directional antennas," PROC. I.R.E., vol. 25,pp. 78-145; January, 1937.

ous formulation of the problem of the n-phase transmis-sion line and that of the single-phase multiwire trans-mission line. The analysis is useful to compute the inputimpedance of many antenna systems, including thefolded dipole, triple-folded dipole, H antenna, and thecorner-reflector antenna.

ACKNOWLEDGMENTThe writer wishes to acknowledge his indebtedness to

Ronold King for suggesting this problem and supervis-ing this work.

Fig. 15-Symmetrical and antisymmetrical reactances. 30h =r.

APPENDIX IGeneral Formulas for Ca(Z), Sa (z) and Ea(z)The following notations are used in these formulas:

112=h+z; ,u1=k-Z

R2= ,u22+a2; Ri=,il2+a2; Ro= z2+a2crx=f Icox= x sin,ucix= J d,u; SiX= si-l dA

JA Io

(51)

1948 497

00

S<<

~~~~~~~~~

~~~~~~20002

"IX

X,~~~~~~~~~~~~~~~~10"s,X2

xI/

~%..

AP20

1000

900

~~800

700

.%..

-R8

1~~~

~~~~60

0

RSI

5~~~00

_R__

20

15~~~~~~~

0.40.

60.8

~~a=1.0

40/X

o100

12~~~~~00020.

6-

..0bA

Fig.

17-S

elf-

impe

danc

e.f3

0h=00

Fig.

18-M

utua

lim

peda

nce.

,Boh=7

r.

499Tai: Coupled Antennas

cos2 1z[Ci(R2+12)+CiI(Ri+ii) Cuv X-= ( 2+V2)-12cos (u2+v2)12du; v=fla (55)

- Ci#(R2- P2)- Cig(Ri- i)-jSio(R2+Y2)o

-jSi0(Rj+y,)+jSif(R2- Y2)+jSi!(Ri-yi)] (52) Suv X= X(u2±v2>2 sin (u2+v2)112du; v=3a (56)

+- sin Oz [Si# (R2+Y2)-Si0(Rl+yJ) o

+SiO(R2- 12)-Si0(Rl-y,)+jCi3(R2+P2) The formulas for Cb(z), Sb(z), Eb(z) will be the same as

-jCi0(Ri+y1)+jCio(R2-Y2)-jCi3(Ri-y1)] (52), (53), and (54), except that b is substituted for a.

OHMS

Fig. 16-Self and mutual impedance. ,Boh = 7r/2.

Sa(Z) =2 cos (z [SiP(R2+1A2)+Si3(Ri+y,)+Si3(R2- 1A2)+SiO(Ri- 11) - 2Si3(Ro+z)- 2Sif(Ro- z) +jCi1(R2+112)+jCio(Rl+ 111)+jCiO(R2-112)+jCil(Ri-111)-j2Ci#(Ro+z) -j2Ci#3(Ro-z) ]

sin 13z[CiI3(R2+IA2)-Cio3(Rl+l± ) '(5)- Ci3(R2- /12) +Cif3(Ri 1) - 2Cii(Ro+z)

+ 2Ci3(Ro- z) -jSii(R2+ A2)+jSiO(Rj+yj)+jSif3(R2-112) -jSil(Rl-1IA)+j2Sif(Ro+z) -j2Sifl(Ro-z) ]

Ea(z) =Cuv 0,42+CUV 13 SV-jSUV 13A2-j Suv O13A (54)

where the integrals Cuv X and Suv X are defined as fol-lows:

APPENDIX II

Approximate formulas of Ca(Z)), Sa(Z), and Ea(z) subjectedto the condition a2<<h2:

Ca(Z) -- COS Oz[E[t2O(h+z)+Ci2O(h-z)+jSi2O(h+z)+jSi2I3(h-z) ]+2 sin 3z [Si2f(h+z)-Si2i3(h-z)-jCi213(h+z)+jCi2f(h-z)] (57)

+ cos 3z sinh- l z+ sinh- ]_ a a

Sa(Z)- 2 COSzz[Si2o (h+z) +Si (h1-z)-2Si2oz-jCi2 (h+ z) -jCi2 (h-z)+ 2jCi2z ]

+2 sin Oz[Ci212(h+z)-Ci213(h-z)-2Ci213z (58)+jSi2o(h+z) -jSi2(h-z) - 2jSi213z]

1948

PROCEEDINGS OF THE I.R.E.

h+z h-z-sin 13z sinh- I - sinh'I

_ a a

-2 sinh'-Ia-

Ea(Z) * -Ci(h+z) -Ci,B(h-z) -jSii(h+z) -jSif3(h-z)h+z h-z

+sinh-I +sinh I (59a a

RI, ab=

z-=

reference 6, are related to the functions Fj(z) and Gj(z)defined in this paper by the following equations:

Fl(z) = ('a b - Qa)Foz + F1H(Z)

G1(z) = (fa b -Qa)Goz + GiH (Z).

(60)

(61)

It is to be noted that F1(h) = FQH(h), G1(h) = GlH(h) asFo0 and Go0 are equal to zero when z = h. By substituting(60) and (61) into (12), one obtains the following equa-tion for I,8, where only the first-order terms are retained:

(2-Oa

sin O(k-| z)h- z) I)- Go(h) [FlH,+Pl]+Fo,[Gl(h)+QiQ(h)]-Fo(h) [GlH+G1,]-Go [FIH(h)+PUz(h)l]2-'kasnb~ h-Iz h 1i LR1J LHlWb!Wb (62)

Fo(h) ±-1 [F1(h)+Pl(h)]FtabFor Z,, one obtains the following expression:

(63)

where Ci X is defined as

fx( -cos U)I( ~~~du,

APPENDIX III

I,, and Z., in (12) and (24), are expressed in terms offunctions previously defined.

The functions Fl1H(z) and GlH(Z), defined in footnote

90 r IT 'I

Cl)

0

80

70

60

50

40

30

20

o0

where the following notations were used in the previousnapers: jt- ~~~~~FlH(z) =all'+ja2I ]

Pl(h)=Cli+jCI . 64

F1H(O) sin ,3h-G1H(O) P1h+Gl(h) =C,I+jC.liiP1(0) sin Th-Q1(0) cos iTh+Ql(h) =D11+jD1IJ

In case of antisymmetrically driven antennas, onechanges .ab into T'ab and reverses all the signs of P andQ functions in (62) and (63).

I80

ANGLE 120

_____ 0 60

w-J

0 0z

-60

IGLE -120

b

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Fig. 19-Mutual impedance of two half-wave dipoles (comparison with Carter's computation).

r ~~~~~~~~~~~~~~1rcos 31h+ [FlH(h)+Pl(h)]

iRcTab Tab

27r L [2- sin ,h+- {[FlH(O)+Pl(O)] sinikh-[GlH(O)+Ql(O)I cos 3h+GlH(h)+Ql(h) I j

500 April