Scattering properties and mutual coupling of antennas with prescribed radiation pattern

IEEE TRAK~ACTIOSS ow .IXTESS.IR ASD PROPIGATIOK, TOL. AP-18, so . 6, SOVEMBEIL 19’70 741

tion, University of filarburg, Gelmany, Ma; 1909. an logarithmiscl~ periodischen Antennen”, inaugural disserta- [13] --, Elektt’omagnetische Strahlungsfelder. Berlin: Springer,

[IS] 11. Zuhrt, “Eine strenge Berechnung der Dipolantennen mit. [14] A. Erdelyi, dsyn tp fo t i c Erpansions. New I’ork: Dover, 1956. rohrformigem Quersrhnitt,” Frequenz, vol. 4, pp. 135-178, [15] J. Wolter, ”Theorie der Yagi-Antenne”, .Tachrichtentech. Z.,

1953.

1950. vol. 23, pp. 18C184, April 1970.

Scattering Properties and Mutual Coupling of Antennas with Prescribed Radiation Pattern

742 IEEE TRANE.ICTIOSS OK ANTEKSAS . ISD PROP.AG.iTIOX, KOTEXIBER 19‘70

played by t.he spherical modes in affecting t,he scattering and radiation characterist,ics of an antenna,. This net.\vork interpretation also facilitates the evaluation of mutual coupling among quite general a.ntennas, in t,hat the formulation for t.he (coupling) impedance matrix is reduced essentially to a network interconnection problem. The mutual impedance between the two ant.ennas is expressed explicibly in terms of the radiative and scatt.ering param- eters of ea.ch antenna when it is isolated in free spa.ce. It is shown that,, generally, the mut.ual impedance may be written as a, sun1 of two terms: a zeroth-order t.erm, dependent exclusively on the radiation patterns and a, second term involving t,he scattering properties as well as the antenna ra.diation patterns. Under certain conditions the mutual inlpeda.nce may be approximated by the zeroth- order term alone. This zeroth-order term is also identified as the mut.ua.1 impedance between two MS a,nt.ennas (with the same patterns as the given antennas) and which, therefore, may be computed by employing the various integral representations of [GI.

The necessary modifications in the formulation t.o in- clude N-element arrays are described briefly. It is also shown that., for an infinite planar phased array with identical elementary radiators, a similar zeroth-order approximation of the active impedance can be represented in the form of a grating lobe series [GI, [9]-[15]. More- over, the grating lobe series is shown to constit,ute an exact representation of t.he active impedance if and onl?- if the elemenkry radiators are AIS antennas.

11. SETWORK REPREGESTATION OF ANTENSAS

Employing t,he scat.tering representation described in [2]-[4] all the electromagnetic properties of a finit,e- dimensional lossless antenna are contained in the matrix relation

sua j Sa, b = &‘a = = _____ :_-_ [::I ; ,1[::-] ( l a )

tvh ere SST = 1 (1b)

and f denotes the complex conjuga,te tra.nspose. For sim- plicity, the scat,tering matrix S has been norma,lized to 1-ohm resist,ances. The a, and b, are column mat-rices, each conta.ining, respectively, incident and reflected xmve amplitudes at the N local accessible antenna ports; a$ and b, are infinite-dimensional column matrices denoting incident a.nd reflected wave amplit,udes of the spherical (cylindrica.1) modes in free space outside a ‘sphere (cylinder) complet,ely enclosing the antenna. An input a, at the antenna ports produces a radiat.ed xave b8 = &a,; the wave reAect.ed back into the antenna. ports is b, = Saaaa. A41ternatively, n-hen the N antenna ports are connected to matched receivers a, = 0. An incident \mve (from free space) a# gives rise t,o a wave b, = Su3ao at the antenna, ports and causes a wave bB = &,a, to be reflect,ed into surrounding space. Thus the N-dimensiona.1 sub-

the accessible antenna ports. d l i l e &,3tS3,,SB$ describe, respectively, t,he receiving, transmitting, and scattering properties of the ant.enna. The quantities Sa;7,S8, will be termed n1oda.l receiving and transmitting patterns, respectively, t,o dist,inguish them from conventional far- field pattern qua.ntit,ies wllich depend explicitJy on the angular coordinates. For n reciprocal antenna Sa8 = SB, ( A denotes the transpose). An N-element array of 1-port mtennas may be considered as a single N-port a.ntenna, in which case Sa, describes the mut,ual and self-coupling anlong the array elements. The mutual interaction among the e1ement.a of the arra,y ca.n also be expressed by the associated impedance matrix z

z = [l + SU,][1 - Sua]-l. ( 2 )

The correspondence betn-een the column nlat,rices a8,b8 and t,he electromagnetic fields in free space is established by the normalization of the (cylindrical) spherical modes and t.heir arrangement. in an infinite linear sequence in some agreed order. This is described in detail in [3] and [-I]; a concise summary of the ma.in results is presented in [GI. -4 genera.1 ant.enna., with scattering matrix as in ( la ) , is

not, uniquely defined by its receiving and transmitt.ing pa.ttern, since the unitary constraint. on S [i.e., (Ib)] is insufficient. to determine t.he scattering mat,rices Ssa and Suu wlml Sa$ and Sa, are prescribed. Since the radiation pattern is one of t,he most important. cllaracteristics of an a.ntetlna: i t is of interest to examine 1vha.t range of (scattering) propertries remain after a given radiat,ion pa,ttern is prescribed, and \-hat constraints might be im- posed on S while &ill maintaining the prescribed radiation pattern to yield canonical results of unique significance. One such constraint on S involves the scattering propert.ies of the antenna when its local ports are react.ively terminated [ 2 ] . The requirement. that for a particular set of reactive terminations the antenna does not scatter at. a,ll, i.e., is “invisible” from free space, leads t.0 an MS antenna [ 11-[GI. Should these react,ive terminat,ions formally be open circuits the antenna is t.ermed a canonical minimum- smttering (CAIS) antenna, and its scattering matrix is of t,he form‘ [2]-[G] [ 0 Saut t] s = ----I . (3)

I n (3) one observes t,hat, all the electromagnetic properties of a CMS antenna are completely and uniquely defined by its receiving or transmitting patt.ern. If the CATS antenna is reciprocal &, = SUB, t,hen the modal radiation patt,ern must be purely real. It may be shown that a pure real n1oda.l radiation pa.tt.ern constrains the pon-er pattern function to have point symmetry about the origin [?I.

Thus, by imposing the constraint of “invisibility” on open circuit., one obtains a class of ant,enna,s u-hich are defined by their pat.t,erns. Evidently such a constraint is

s,3a 1 - &a880

743

overly restrictive, since arbitrary radiation pat,terns cannot be realized by reciprocal CIIS a.ntenn9.s. In the following it, will be shorn-n that by imposing a diflerent t,ype of constraint on X one can obtain a class of reciprocal antennas completely defined by t,heir rdiation patterns which can be quite general. It, is convenient, t,o employ a. network represent,at,ion of a general a.nt,enna for this purpose.

Consider an N-port. CRIS antenna tx-it.h a scattering mat,ris

OM,lf ~ lmf : 0 - - * 0

(4) . , . I

. I . '

. I . :

This antenna is capable of radiat,ing ill spherical modes, a.nd to each local port t.here corresponds a dist,inct mode. Such an antenna may be considered as an array of de- coupled CMS antennas in which each array element radiates a single spherica.1 mode. A network represent,ation of the array is shown in Fig. 1 (a). Let S , V + ~ ~ , ~ - + . ~ ~ be the scattering matrix of a lossless ( N + N)-port network. Connecting the Jl ports of the netn-orlr t.o t.he ports of the array result*s in a new antenna as shown in Fig. 1 (b) . This antenna. has ,V local ports and a scatt,ering ma,trix

(3)

o i 0 where Sa, and S9$ are, respectively, X x A r and X x 31 nmtrices. Should the antenna, radiat.e a.n infinit,e number of modes, -+ r: . I n this case the network in Fig. 1 (b) Rill represent. a, general lossless antenna. In t.he following only reciprocal antennas n-ith a single local port will be considered for which S,, becomes a column nlatris to be denot,ed by s. lhrt.hern~ore, for simplicit,y, the antennas will be tuned t o free space and consequently &., = 0 (a scalar). Thus one may writ.e

0 ; E I 0 *..

(6)

. I .

. I . ,

M SPHERICAL MODE PORTS

r

1

M LOCAL PORTS

(a)

M SPHERICAL MODE PORTS

LOSSLESS N+M PORT NETWORK

'N-M, N+M

T

N LOCAL PORTS

(b) Fig. 1. Set.work representat.ion of general antenna.

SPHERICAL MODE PORTS

PORT LOSSLESS INTERCONNECTING NETWORK

Fig. 2. Network represent.ation of CMS antenna.

For a reciprocal CMS antenna with a single port the network representation of Fig. 1 (b) takes on a particularly simple form, as shown in Fig. 2. Here the lossless network consists of a bank of ideal tmnsformers ha.ving their primaries connect,ed in series with the input port and the secondaries connected to ports of those modes which the antenna radiates. The remaining mode ports are left open circuit,ed. For convenience, the modes for which .sln # 0 have been numbered consecutively from 2 to 1. (The index 1 is reserved for the local port). The turns ratios of t,he transformers are numerically equal to t.he nonvanishing elements of the modal radiation pat.tern s

= [ a ? $13 $14 ' * ' s l l ] ('7)

where .sln are all real and

To see that the network interconnection in Fig. 2 in- deed corresponds t.0 a. CIIS antenna, consider the antenna in transmission. Since the mode ports are in this case terminated in 1-ohm resistances, the impedance seen

7 4 JEEE TRANSACTIOXS ON .iSTENNAS .%ND PROPAGATION, SO\’EIlEER 19’70

looking into t.he local port is, in virtue of (S) , equal to 1 ohm, and the local port is matched. On the ot,her hand, open circuiting the local port produces an open circuit, a t each mode port, making the antenna ‘(invisible” to a.ny wave incident from free space. These properties define a CMS antenna.

Suppose one interposes a length of lossless tmnsmission line x-ith l-ohm characteristic impedance bet\\-een each tmnsformer in Fig. 2 and the corresponding mode port. Clea,rly such an antenna will have a modal radiation pa,ttern given by

= [SI? exp ( - h ) s13 exp ( 814

.exp ( - h ) - -SU exp ( - j 6 ) 1 (9)

where +&, - - - are phase shifts through each length of transmission line. But t,he SI, are arbitrary real numbers subject only to (a), and the 4, can be chosen arbitrarily. Hence s is an arbitrary moda.1 radiation pa.ttern. The net- \TOrk representation of such an antenna is shown in Fig. 3. The open-circuited transmission lines terminating t,he mode ports I, 1 + 1, I + 2, - - - have no effect on the radiation pattern. The turns ratios of the transformer are ta.ken to be positive and equal to the moduli of the sl,. (In Fig. 3 this would be equimlent to adding half wave transmission lines between a tra.nsformer and mode port whenever ~ 1 % < 0.) Wit.11 this change in notation, (9) is n-ritt.en

i= C ; sI.2 I exp ( -j+d 1 s13 I exp ( - j+3) . - / sll I exp ( - j+l )]

(10) where & in (10) and (9) ma.y differ by T.

By using straight.fomard circuit analysis the complete scattering matrix of the antenna in Fig. 3 mag be shown to be

s =

where D is a diagonal ma,trix

D =

:xp (-?a&)

0

SPHERICAL MODE PORTS A

c 2 3 4 5 f f + 1 i t 2 \

I I

Fig. 3. Net.work representation of canonical ant.enna.

is seen t.0 be a special case of a canonical antenna. Thus choosing s real one must have 4, = UT ( V = O , f l , s 2 , . - - ) for all -)I. 5 I, which makes the first I - 1 elements of D unity. If in addition C& = VK ( V = O,fl ,&2,. - .); n > I, the antenna. is a reciprocal CAIS antenna as in (3) . Clearly a canonical antenna with +n = VT ( V = 0 , s 1, k 2 , - * ) ; 11 > 1, is uniquely defined by its radiation pa.b tern, since in this case D is determined by the phase angles of s. Since s is arbitrary, one can const.ruct t,he scattering matrix of such an antenna, from any given modal radiation pattern.

In the preceding the scattering matrix of a canonical ant.enna bas been deduced from a network representation of a. reciprocal ChIS antenna. The same result can also be deduced in a more straight,forma,rd way. Consider a lossless matched reciprocal antenna u-ith one local port with

0

The &(n = 2,3, - - -,Z) are phase angles of the elements of s as in ( lo), while &(n = I + 1, I + 2,s-m) are phase shifts through the open-circuited transnlission lines.

An antenna with a scattering matrix given by ( l l a ) will be termed a ca.nonica1 antenna. A recbrocal CMS antenna SS+ + s,,s,,t = 188. (13c) .

Let t,he antenna be open circuited, and let, a wave as be incident from free space. Then, from ( la )

Carrying out the matrix n~ultiplication, one obt,ains

bg = sb + SSgag = (si + Sg3) ag. (15)

The scat,tering matrix SO

So = sii 4- Sflo (16)

defines the scatt.ering properties of the antenna when its local port is open circuit,ed. Combining (16) and (13b), and taking account of (13a) yields

Sos* = s. (17)

Equat,ion (17) shows an explicit relat,ionship between the modal radiat,ion patt.ern a,nd scatt.ering of an open-circuited ant.enna. So is a unitary matrix and the mdiation patt.ern must. be a solution of (17).

Given a prescribed lossless scatt.erer, one can construct a lossless antenna which, on open circuit, scatters in t,he prescribed manner. I n fact., an infir1it.e number of such antennas with different radiat.ion pat.terns can be con- structed. First consider t,he case in which So is diagonal. Physically this corresponds to an a.ntenna which, upon open circuit., scatters without int,roducing any coupling among the spherical modes. This will be the case, e.g., if the open-circuit,ed antenna appears like a lossless reciprocal spherically symmet,ric scatterer. Denoting the elements of s by I sln exp ( - j&) , (17) yields for $n (the phase angles of element,s of D)

+n = - 2@n. (18)

With these phase angles one can choose an arbitrary set, of nonnegstive numbers j sIn satisfying Cn I sl, I? = 1, and arrive at. the scatt,ering ma.trix of the canonical an- t,enna a.s in (11) with So = D. Hence a ca.nonica1 antenna is a.n antenna. which, on open circuit, ha.s a diagonal scat.tering mat.ris. To distinguish between an arbitrary matdled reciprocd one-port antenna, a.nd a canonica.1 ant,enna, the modal radiat,ion pattern of 6he latt.er will be denoted by u. Thus for a canonicsl antenna

s - -_I ---I-----

u D - ~ 6 e - i " ; $ ] (19)

where D is defined by the arguments of uln as in ( l l b ) . Consider now t.he case where So in (17) is not diagonal.

For m y given So which is unitary and symmet,ric, one can always construct a unit.ary scat,tering matrix S as in (12). T o show t,his, one invokes a well-known theorem which states that any symrnetric unita.ry matrix can be diagonal- ized by a rea.1 ort.hogona1 transformation. Thus So can be written

TSoT = D (20)

where T is a r e d orthogona.1 mat,rix whose columns are eigenvectors of So, and D is a diagonal unit,ary mat,rix with elements equal to the eigenvalues of So. Hence letting

s = Tu (21)

one can writ.e the scattering matrix in (12) as follows:

= [' I,___ 0][" _ _ I ---------- j[' --I 1 ---- ;] . (22)

0 : T U : D - U C 0 ,

Since

Q p [:-I-:] (23)

is also real and orthogonal, (22) sta.tes t,hat S is obtained from a canonical antenna by a real orthogonal tra.nsformation, i.e.,

s = QSc0 (24)

where S, is the scattering matrix of a canonical antenna. Clearly S is unita.ry whenever S, is unit.ary. Since T,D together with the arguments of uln a.re defined by So, and any set of nonnegative nunlbers 1 uln 1 satisfying En gln Iz = 1 may be chosen as the moduli of uln, one can alwa.ys construct a unit,ary S from any unitary So, as ma.s to be shown.

Inversely, for a.ny given modal radiation pat.tern s one can choose a.n arbitrary real orthogonal t,ra.nsformation T to find u = Ps. The arguments of uln can then be used to construct D, and (32) yields a representation of X for all possible matched reciproca.1 one-port antennas with the prescribed modal radiat.ion pat,tern. Choosing T = 1, one obtains a canonical ant.erlna w-hich will be uniquely defined by the given radiation pa.tt.ern only if i t radiates a.11 modes, i.e., none of the 1 q n I are zero. For if i cln I = 0 for some 71, the argument of the corresponding element of D is arbitrary. To circumvent, t.his h c k of uniqueness one can, e.g., set these arbitra.ry arguments to zero or, which is the sa.me thing, require the antenna on open circuit to be invisible to all modes which i t cannot radiate. In case the modal radiat.ion pattern is real, the elements of D must be unity for all 12. Consequently, D = 1, and one obtains a reciprocal CIIS ant.enna.

The canonical ant,enna can also be viewed in terms of the so-called chara.cteristic resonant modes of the scatterer (which in the present ca,se is t.he open-circuited antenna). If one expanded the electromagnetic fields in a coordinate system other t.han the spherical one could, formally, define a unitar: scattering mat.rix S' which would relate the incident to the reflected waves outside some surface enclosing the scat.t,erer. Whatever the nature of this new coordinate system one would expect. t h e old scattering va.ri- ables ag,ba t.0 t.ransform to the new scat,t,ering variables agf,bg' by a. unitary transforma.tion, since one must have

I aBf l 2 = I a3 :2, i bg' I l 2 = i b~ l 2 (25)

i.e., the incident and reflected powers must remain the same. Denoting the transformation matrix by G

b i = U b g (26a)

as’ = LTab (26b)

l iUt = 1. (26c)

In addition to the unitary constraint (26c) another constra,int on G arises from the requirement that the electromagnetic fields derived from aD,bg and ag’,bg’ both be consistent Tvith the Lorentz reciprocity theorem. It may be shown [l], [4] that this will be the case if for each solution of the field equations corresponding to the set aB(ab’) ,bg(bs’) there exists another solution derivable from the set2 bb* (bg*‘) ,ag* (aB*’). Thus, ma.king the replacement of variables aB* + bg, bg* --f as, ab*‘ + bo’, bg*’ + ab’ in (26a.) and (26b) one obtains I;r* = U , dlich, coupled with (26c) , states that G must be a real ort,hogonal mat,rix.

By making the identification T 4 U one ca.n view ( 2 2 ) as a transformat,ion of S from one co0rdinat.e system to another where So is diagonal. The antenna is a canonica,l antenna in the coordinate system in which So is diagonal, i.e., .the open-circuited antenna has a physical structure characterist,ic of this particular coordinate system. The eigenvectors of So may be considered a.s resonant modes of the open-circuited antenna. Analysis of scatt,erers in terms of such resonant modes was presented by Ga.rbacz [7]. Viewed in this way, the radiation pattern in (21) is a.n expansion in terms of these modes. If the ant,enna radiates all modes in the characteristic coordinate system it is uniquely defined by its radiation patkern. In the co0rdinat.e system in which So is not diagonal, one must knov,-, in addition to the radiation pa.ttern, the resonant, modes of the open-circuited antenna, Le., the tra.nsformation ma.tris T . If one is dealing with a CMS antenna t,hen it formally remains a CAIS a.nt.enna in all coordinate systems, as can be seen from ( 2 2 ) with D = 1.

111. ~ I U T U A L COUPLING BETWEEN Two AXTEXAS

A . Impeda.nce Description

The network represent.ation described in the preceding section can be used to adva.nta.ge in evaluating mutual coupling among antennas. Consider two antennas each ha.ving one local port, as shown in Fig. 4a. The antenrlas are taken to be of finite dimensions with sufficient sepa.ration to permit them to be enclosed by nonoverlapping spheres. Each ant,enna can be represented by a netu-ork similar to that in Fig. 1. For the two coupled antennas the network representations are shown in Fig. 4 (b) , and the scattering matrices of antenna 1 and 2, when isolated in free space are, respectively,

solution,” see [I]. The second set corresponds to t.he so-called “t.ime reversed

SPHERICAL MODE

PORT o1 bl

SPHERICAL MODE PORTS SPHERICAL MODE PORTS

1 2 3 N N - 1 2 I 2 3 N N - 1 =

LOSSLESS NETVtORK

t t

NEiWORK. LOSSLESS

LOCAL PORT --&

t ) O l b l O 2 b2

(b) Fig. 1. Yetwork representation of tno coupled antennas.

The S I and S? are referred, respectively, to origins 01 and 0, of Fig. 4(a), i.e., the spherical modes ent,ering into the scattering representation of each antenna are defined relative to it,s local coordinate system. The quantities rl and r2 are reflection coefficients, and the superscripts r,t refer, respectively, to the receiving and transmitting pattern of each antenna.

To avoid some notational difficulties, i t will also be assumed that. each ant,enna radiates and scatters only a finite nunlber of modes. Formally, the final results u-ill still hold if the llunlber of interacting modes is infinite. How- ever, except in special cases, numerical calculations can only be performed for a, finite number of interacting modes. Also the assumption, implied in Fig. 4(b), that each antenna has an equal number ( N ) of mode ports connected to it.s lossless net.n-ork involves no loss of generality. For if one ant.enna, say t,he first, radiates only X modes, JI < N , then one can a1wa.y-s consider the X - 31 open circuit.s as part of the lossless int,erconnect,ing network having all X p0rt.s connected. This is shown in Fig. 5 . The coupling impedance matrix 2 [see (‘211

of the txo a,ntennas can be obta.ined by considering the interconnection of t h e e netxorlrs, as shown in Fig. 6: a dissipative 2N port., consisting of all the local ports of two ident.ica1 coupled A-port CJIS antennas each wit11 a scattering nmtrix as in (4) , and two lossless networks with

W.%SYLKIWVSE(YJ A S D K A R S : SCBTTERIKG PROPERTIES AND UUTE.4L COEPLING O F ASTENNSS 747

I NEW NETWORK b(Sl ) W I T H N MODE

PORTS CONNECTED ORIGINAL

LOCAL PORT NETWORK( 5)

Fig. 5. Construction of new net,work w-it.h larger number of connected mode ports.

1 2 3 N 1 2 3 N

‘11 ‘I2 ‘13 “IN ‘21 ’22 ‘23 ‘2N L /

Y Y A

‘ 1 2 3 N ’ w w Fig. 6. Network interconnection for calculat.ion

of mutual impednnce.

sca,tt.ering matrices given by (27). The dissipative 2N port takes account, of the coupling among the spherical modes and is the same for all antennas radiating spherica.1 modes up to order N . T h e impedance matrix of this network will be denoted by Z2.+-

where Z is an N x N matrix. The N x N unit matrices arise because the N ports of each CMS a.nt.enna are de- coupled a.nd t,uned to free spa.ce (see (4) ) . Each spherica.1 mode is defined by a. single index, and t-he modes are ordered in a. linear sequence from 1 t.0 N as described in [SI. The same ordering scheme is used for each antenna.. The elements of 2, denoted by {il, a.re in fact mutua,l impedances between single-mode single-port reciprocal CMS antennas radiat.ing modes i and 1. They can be computed from the normalized vector mode funct,ions as described in [6] where i t is shown that3

0- ai?+jcc

da sin a

matrix Z?S in (29) is symmetric. S o t e t,hat even though { i l # {li, the coupling impedance

In (30) , D is a. fixed vector which defines the translat,ion of the coordinate syst.em (m,y?,z2) of antenna 2 relative to the coordinate system (zI,zJ~,z~) of antenna 1: as shon-n in Fig. 4 (a) ; k is the free-space propagation vector whose components are

X-, = (%/X) sin a cos y

X-, = (h/i) sin a sin 9

X-, = (27r/X) cos p.

The vector functions Fi(a,p) ,Fl CY,^) are proport,ional to the spherical mode funct,ions of [SI, [4], and [SI. Explicit forms for t.he Fi are [6]

Fi ( a g ) E ( j ) ne,m’(e,O) r

for E modes, and

a cos 1np - 9 0 - Prim ( C O S a ) aff sin nlp

for H modes. The P,’n(cos a ) are associated Legendre polynomials, and the normalization const.ant A T , , is given by

748 IEEE TR.bYS.iCTIONS Oh' .L?T'EhTAS AXD PROPAGATION, h'OVEMBER 1970

one readiiy obtains from (38) and (40) , Z12 = 221, as espect,ed. Thus, in this special case, one has

I n ( 3 5 ) , the scattering matrices So(') and Sac2) are related .Z[ 1 - t (1 - So@)) Z( 1 - So(1)) 21-1~~ (Qb) to X,# and S ~ B ( ~ ) by

- - 1 - 1 212 = 2 2 , = So(') = ~ sl(f)El(r) + SBB(l) (3Ga.) (1 - rl) (1 - r2) '?

I - rl .2[1 - t ( l - SO'")Z(l - So~?))Z]-lsl. (42c)

1 = - S p S p + S B p (3&) Wit.h the aid of the identity 1 - r2

[l - AI-' = 1 + A[1 - -41-1 (43) and describe the scattering properties of the antennas on

two e1ement.s of the coupling impedance matrix, i.e., 1

. open circuit (cf., (16)). From (35) one readily obt,ains ( 4 2 ~ ) may be mit ten

results are CMS antennas, this zeroth-order term can be expressed explicitly in terms of far-field pattern quantities of the

ment, in Fig. 4 (a,) , one has

zlp E

1 + r2 1 - r, - r2)2

2 2 2 = __ - S2(')Z(1 - So'") (isolat.ed) antennas. Thus, for the geometrical arrange- 1 -

-2c1 - $(l - SO@))Z(l - S0(1))Z]-1Sp (39) - 1 - - 2 S&Sl =

- 1 (1 - rl) (1 - r2) (1 - rl) (1 - r2) 2, =

(1 - rl) (1 - r,) El (7)

rrj?+jw

.Z[1 - $(l - So'")Z(l - So(l))Z]-l~z(t). (40) -Ir clp 1 sin eFl(e,,-). i?? (e,p) esp (--jk-D) dB. (45)

Equations (37)-(40) give the elements of the coupling The quantities Fl(6,p) and F2(6,p) are proport,iona,l to the impedance matrix Z, as in ( 2 ) , explicitJg in t.erms of the electric far field of xnt,enna 1 and 2, respectively, and are

normalized in accordance wit,ll

[= dp lz sin e i F1,2(B,q) i 2 (18 = 1 - I rl,? I?. (46)

The specification of the radiative properties of a.n a.ntenna by F(8,p) or the nodal quant,ity s is entirely equivalent, since the former can always be expanded in spherical vector mode funct.ions. The expansion coefficients in such a series representation are t,he elements of S. These may be readily computed from any given F(0,p) due to the ort,ho- normal character of the mode functions [SI. The function $ ? ( e , ~ ) appearing in the integrand of (45) is given by

= - [F, (T - + .eo] eo + [F?(T - ' , a r) 'PO] PO ("7)

and corresponds t.o t.he far field quant,ity obt.ained from F2(8,p) under the coordinat,e transformation 0 + T - 0, p + p + T (i.e., reflection t.llrough the origin). 4 s shown in [SI,

%(e,p) = F* (e*,q) (48)

whenever the modal radia.tion patt,ern is real. Thus, for antenna.s d l 1 identical radiat,ive properties, the precesing s:mmet.ry condit.ion permits the replacement of F1-F2 in (45) by a power pattern function 1 F(8,p) i 2 having point symmet.ry about the origin.

The formula for mutual impedance (44) can be written as a sum of two parts, i.e.,

z, = Z 1 p + Z1p. (49)

The Zl2(() depends both on the radiative and scattering properties of t,he antennas and mnishes identically only for CMS antennas. Even though a.n explicit expression for Z12(f) is available, t.he required matrix operations, particularly the t,aBing of the inverse, limit its usefulness to ant,ennas which can be characterized by a relatively small number of spherical modes. Physically this implies st,ructures whose dimensions are not too large in terms of wavelength Also, unlike it.s radiative properties, the knowl- edge of the scattering properties of an antenna is usually incomplete. Hence it appears worthwhile to examine conditions under which Z12(c) may be neglected.

First, one observes from (44) that whereas the zeroth- order term contains only a linear t.erm in Z,Zl2(e) will in- volve Z a t least to the third power. As shown in [4], a t,ypical element of Z,(il, decays with kD as 1/ ( kD)" . Hence the zeroth-order term will also decay a.s l / ( k D ) n while z G ( f ) will decay at least as l / (kD) 3n. Consequently, for suffi- ciently large separations

2, - Z12@), for large kD. ( 50)

Using similar reasoning, one also has from (42a) and (4%)

Since the approximations involved in (50) and (51 ) are roughly of the same order, Z,,@) will be a good approxima-

tion to mutual impedance whenever the mismatch intro- duced at the port of t,he radiat,ing ant.enna by the presence of the open-circuited antenna is small. From physical considerations, one ca,n see t.hat the sepa,ration between t.he antennas need not be unduly large for (50) to con- stitute a good approximation.

Anot.her criterion for neglect,ing zI2(O is afforded by the scattering propert.ies of each isolat.ed a.nt,enna. The matrices 1 - So(') a.nd 1 - So(?) entering into Z E ( e ) are a measure of scattered power of each open-circuited antenna. If the sum of t.he magnitudes of the eigen values of 1 - SO is much less than unit,y, the scat.tered power will also be small. In this case (44) can be expanded in a convergent matrix series by a repea.ted application of (43). Thus one h a.s

( 5 2 )

In (53), 2 1 2 can be considered t.0 arise from a.n infinite number of multiple reflections between the antennas. The zeroth-order term corresponds to a pair of CMS antennas and depends only on t.he radiation pa,tt.ern of each antenna,; all other t.erms depend on radiation patterns and scattering. Each higher order reflect.ion conbains Z to a progres- sively higher power. Hence, wit-11 a la.rger separation between a.ntennas, (52) would be expected to converge more rapidly. Also, since no mat.rix inversion is required in (5'3), for the purpose of numerical calculations such a representation would be more useful t,ban (44) , particularly if only the first t.erm in the mat.rix series con- stitut.es a significant contribution.

B. Admittance Description

It, has been shown that if scattering on open circuit can be neglect,ed for each isolat.ed ant,enna, then the elements of the coupling impedance matrix may be approximated by their zeroth-order t.erms. For some antenna structmes it may turn out. that although scat,tering on open circuit is not negligible, it. may be negligible for some other reactive t,ermination, eg. , a short circuit. This would be the case for apert.ure antennas in a perfectly conducting electric screen if the a.nt.enna ports were defined a t reference planes separat,ed by a multiple of half (guide) xmvelength from the aperture plane. To the ext,ent that a short circuit placed at the antenna port is reflect,ed as a short circuit a.t the aperture plane, the ant.enna may be approxinmted by an 91s antenna ("invisible" when its port is short circuited) [4], [SI. I n t.his ca.se mutual int,eraction between antennas is more convenient.ly expressed via a coupling admitkance matrix P = [g]-1. From an analysis very similar to the one leading to (42) one obtains (a.ssuming

750 IEEE TRAXSACTIONS ox ANTENNAS AND PROPAGATION, NOVEMBER 1970

reciprocal antennas) the admittance para.meters4

*Z[1 - $(XS(Z) - l)Z(S,(l) - 1)Z]-1s2

where the matrices SS(l) and Ss(2 ) are given by

a.nd describe the scattering properties of the antennas when their local ports are short circuited.

The corresponding zeroth-order a.pproximatior1s for follow from (53) by setting = Ss(z) = 1,

For ills antennas these become exact expressions. Such MS antenna.s differ from CMS antennas only by having an additional quart,er wavelength long lossless trmsmis- sion line connected at. the local port. Hence their model radiation patterns must be pure ima.ginary. Not,e t,hat, in general, the inverse of the zeroth-order approximation of the admittance matrix [F(o)]-l does not equal the zeroth- order approximation Z ( O ) of the impedance matrix, since the conditions S, = 1 and So = 1 cannot be simul- taneously satisfied.

IV. MUTUAL COUPLING AMONG ELEMENTS OF AN ARRAY

A. Finite Brrays

The preceding analysis of mutual coupling between two arbitrary antennas can be readily extended to anN-element a.rray. A formal expression for 2 may be obtained by employing a network interconnection technique similar to the

fectly conducting screen, the matrix elements in (53) must be 4 When applied to ant.ennas in a half space bounded by a per-

defined in terms of spherical mode functions appropriate to the half space 141, [6].

one s1~01r.n in Fig. 6 and solving an appropriat,e set of network equations. For an arbitrary t,hree-dimensional array in Fig. 7 (a) the network interconnect.ion is shown in Fig. 7 (b) . The voltages and currents at the mode ports may be arranged int.0 column ma.trices VJ,, where

vp = [ Vp1V,2Vy~ ’ - B”31 J (56a) - I, = [Ip1IpJv3* * * I p . ~ ~ ) , v = 1,2, * . . , N . (56b)

In (56), V p t ( I p t ) denotes the voltage (current) a t the vth array element and Zth mode port. To simplify notation it will again be assumed tha.t ea.ch array element radiates an equal number ( H ) spherical modes. The mode voltages and mode currents are related by an N M X ATM mode coupling matrix Z ( N M )

1 Z(l.2) 2(1,3) . . . z(1.X)

Z(1,Z) 1 z(2,3) . . . z(2,X) 1 1.e.,

Just as for two antennas, the elements of the (&I X 111) submatrices S V . p ) in (57) are mutual impedances between single-port and single-mode CMS antennas. These mutual impedances, denoted by {;z(~+), are now four-index quantities with upper and lower indices referring, respectively, to two t,ypical array elements and two typical modes. With D,, the vector distance between a.rray element Y and p [the positive direction of D,, is from v to p as in Fig. 7 (a)], one ha.s

{ i L ( Y . P ’ ) = 2 p P Fl”(QI,P) * @ l , ( c r , d a!2+jo?l

.exp (-jk-D,,) sin cr da (59)

where FI, and Fll are any of the mode funct.ions in (31). Sote that all mode functions must be referred to the local coordinate system of each array element. These local coordinat,e systems must be chosen such that they can all be brought, into coincidence by pure tra.nslations. By using straightforward circuit ana.lysis, one could now employ (58) together with the network representation in Fig. 7(b) to obt,ain a set of equations similar to (35) u-hich could then be solved for 2 . Unless the array has special symmetry, or (and) appropriate assumptions can be made concerning scat,tering by the array elements, the final expressions for 2, would be quite lengthy and shall not be derived here. If the mutual impedance between any two array elements can be approximat.ed by the zeroth-order

WrLYLKIWSKYJ ASD K A H S : SCATTERING PROPERTIES AXD MUTUAL COUPLING O F .WTENK.iS 75 1

+ - +-. +- +- + - -- +- +- +- +- +- + -

'I1 'I2 '13 V I M

+- +- +- +-

V I ' V 2 "V3 'vM 'PI vPp ',,g V pM 'NI 'N2 'N3 'NM -... -...-...- i

+ - "I t

v

f -

P

t -

"N

Fig. 7. Network interconnection for calculation of mutual coupling for an N-element array. (a) X-element array. (b) Network representation.

t,erm, then one has (assuming reciprocal elements)

3 5 2 r/2; jco

w - & F I ( ~ , ~ ) -P,(~,P) (1 - r m - r,) Sexp ( -J~-D, , ) sin a da, v # p (6Oa)

[Bob)

where F, a.nd F, a.re the normalized electric (far) fields, expressed in terms of the local coordina.tes of element p and V , respectively. If all elements in the array are reciprocal CMS antennas, then (60) is exact,.

B. Il?fi?lite Arrays i\Iutu.al coupling effects in infinit,e uniformly spaced

planar arrays excit.ed with equal amplihdes and a linear phase taper have been studied extensively in recent years. [9]-[14], [4]. A quant.it,y of central importance in con- nection with such arrays is the active impedance, i.e., the scan-dependent input impedance to a Dypical a.rray element when all elements are excit,ed. The functional dependence of the act,ive impedance on scan angles can be expressed in the form of a grating lobe series [9]-[14], n-hich involves t,he element patterns and array geometry. Even though it has long been recognized t.hat t.he grating lobe series representation is an approximation [lo], and more refined analytical techniques have yielded more accura,te results [14], no general criterion has been established for its validity. Recently i t has been shown that the act,ive impedance of a planar array of MS antenna.s is given precisely by a grat.ing lobe series, a result obtained without reference to the structure of the elements in the array [SI. The represent,ation for mutual impedance developed in Section 111 will now be employed to show that, conversely, a. grating lobe series representa.tion results only if t.he elementary radiators are AIS antennas.

Employing the not.at,ion of [e], the active impedance Q(a,/3) of a planar array with a recta,ngular grid and identical elements is given by a two-dimensional Fourier series

c o w

Q ( O ) = C C ZVpexp [ . i ( v a + ,PI1 (61)

where a,P are scan angles and 2, is the mutual impedance bebeen element n + v and 772 + p. This mutual impeda,nce ma.y be written as a sum of zerot,h-order term plus a term i?y,,(f) as in (49). If the array plane is taken as the x,y plane, the int,egral representation for the zeroth-order term in (60) can be writt.en in an a1ternat.e form, involving the components k,,k,, of the free-space propagation vect,or k. [6]. Thus

y--Oc ,=-m

9 z,, = h-(l - r)z

and

(62b)

where

P(k,,k,) = F(k,,k,) -$(kx,k,). ( 63)

The quantity z p p ( e ) depends on the radiation patterns and scattering and involves terms of the form given by the second t.erm on the right side of (44). Substituting (62) into (61), and employing the Dirac delta function repre-

752 IEEE TRAKS.\CTIOSS OK .\XTESSiAS AXD PROP.IGATION, KOYEXBER 1970

sent.ation

exp [- j (vz +- py)] = ( 2 ~ ) ~ 6(.c T . P I I , n r

yields

a m

where

Q C c

(65)

The last sum in ( 6 4 ) can be made tu vanish if and only if the array elements are CMS ante~m~s, jus t as Z12( t ) in (49) vanishes if and only if So = 1. For CMS antennas r = 0, and in virtue of (4s) P(k,,k,) reduces to a power pattern function (i.e., in the visible region k z 2 + k,2 < X.’). -41~0, the normalization condition in (4G) requires that

so that QCo) (a$) in (65) reduces t,o the grating lobe series given in [tj], apart from :L factor of 2. This difference arises, because unlike in (46), the int.egra1 of the power pattern function in [G] is normalized over a half space, i.e., 0 5 0 5 r / 2 and 0 5 p 5 2n. If the elements in the army are not C l I S antennas, t.hen the grating lobe series Q(0) ( a # ) may be considered a zerot,h-order approximation to the active impedance. Note that in this more general case F(kz,k,) is no longer a power pattern function, since the pattern symmetry constraint in (is) may not hold,

REFERENCES [ l ] C. G. hlontgomery, R. H. Ilicke, and E. 11. Pnrcell, Principles

of :llio.owa.ne Circuits, AUT. Had. Lab. Ser., vol. 8. New York: NcGraw-Hill, 1948, pp. 317-333.

[‘L] W . K. Kahn and H. Kurss, “hIinimum scattering antennas,” IEEE Trans. Antennas Propaoat., vol. A€’-13, DD. 671-675. September 1965.

[3] 11’. K. Kahn and 1V. JVasylkiwskpj, “Theory of coupling, radiation and scattering by antennas,” Proc. Symp. on Gen- e,nli.zd Xefworks, vol. 16. Brooklyn, X. Y.: Polytechnic Press, 1966, pp. 83-111.

:1] lV. Waqlkiwskyj, “A network theory of coupling: radiat.ion and scat.tering by antennas,” Ph.11. dissertation, Polytechnic 1nstit.nte of Brooklvn, Brooklvn, X. T., June 19GS.

[ j j A. G. Gately, Jr., ’u. J. It. Siock, and B. Itu-Schao Cheo, ‘‘A network dwcriDtiou for antenna Droblems.” Proc. IEEE. vol.

. ” . I - _

36, pp. 1181-li93, July 1968. [6] I$-. Wasylkiwskyj and IV. K. Kahn, ”Theory of mut1Ial cou-

pling among minimnm-scattering antennas,” IEEE Trans.

[i] It . J. Garbacz, ”Nodal expansions for resonance scat.tering phenomena,” Proc. IEEE, vol. 53, pp. 8.56-8G4, August 1965.

[SI R . Rasylkiu-skyj and W . K. Kahn, ‘.lIutual coupling and element efficiency for infinite linear arrays,“ Proc. IEEE, vol. 56, pp. 1901-19017, Xovember 1968.

[9] L. Stark, ”Radiation impedance of a dipole in an infinite planar phased array,” Radio Sci., vol. 1, pp. 361-378, March 1966.

[IO] H . A. Wheeler, “The grating lobe series for the impedance variat.ion in a planar phased array,“ IEEE Trans. Antennas Propagat., uol. AP-14, pp,. 707-711, November 1966.

[11] R. C. Hansen, Ed., :lfzcrowal]e Scanning dnieanas, vol. 2. S e a Tork: Academic Press, 1966.

[12] J. I,. Alk: and B. L. Diamond, ”Mut1d coupling in array antennas, M.I.T. Lincoln Lab., Lexington, Mass., Tech.

[13] L. Parad, “The real and reactive power of a planar array,” Rep. +24, October 4, 1066.

-41’-11, pp. 990-9q2, November 196.5. IEEE Trans. Antennas Pmpagaf . (Communications), vol.

ill] B. L. Diamond, ..-% generalized approach to the analysis of infinite planar arrays,” Proc. IEEE, vol. 36, pp. 1837-18.51:

[ l j] G. T. Borgiott.i, “A novel expression for the mutual im- November 1968.

pedance of planar radiating elements,” IEEE Z’ram. dntennas Propagat., vol. AP-16, pp. 32S-333, May 1968.

dntenms Propagat., VOL AP-18, pp. 2M-216, March 1970.

Documents

Scattering properties and mutual coupling of antennas with prescribed radiation pattern