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PUBLICATIONS

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NAT'L INST. OF STAND & TECH

A111DS Tbfi7Sfi

"<"»eA U O* NBS TECHNICAL NOTE 1078

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

A Theory of Mutual Impedances

and Multiple Reflections in an

N-Element Array Environment

P~QC

100

.U5753

No . 1078

1985

Lorant A. Muth
NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU

OF STANDARDS

LIBRARY A Theory of Mutual Impedances and (Xcioo

Multiple Reflections in an N-Element --uxici

Array Environment no .101 2

Lorant A. Muth

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

e

4?

J,. U.S. DEPARTMENT OF COlvlMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued February 1985
National Bureau of Standards Technical Note 1078

Natl. Bur. Stand. (U.S.), Tech Note 1078, 34 pages (Feb. 1985)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1985

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402
CONTENTS

Page

1. Introduction 1 2. N Coupled Antennas or Elements 3

3. Explicit Solution to a 4-Element Problem 8

4. Signal Paths and Multiple Reflections

in N-Element Arrays 11

5. General N-Element Interactions 15

6. Conclusion, Work in Progress and Future Plans..... 16

7. References 17

Appendix A Brief Review of the Scattering

Matrix Formalism 19

Appendix B Cyclic Product Decomposition of Matrix A 21

Appendix C Evaluation of the Coupling Integral 25

m

A Theory of Mutual Impedances and Multiple Reflections

in an N-Element Array Environment

L. Muth

Electromagnetic Fields Division

National Bureau of Standards

Boulder, Colorado

A general theoretical approach is formulated to describe the

complex electromagnetic environment of an N-element array. The

theory reveals the element-to-element interactions and multiple

reflections within the array. From the formulation, it is found

that the interaction between an excited element and an open-circuited element can be viewed as the sum of terms

describing all possible signal paths within the array environment

which start from the radiating element and terminate on the

element under observation. Within all paths except the most

direct one, multiple reflections between subgroups of elements

take place. The resulting solution is highly structured and

recursive and is discussed in detail in the text. Illustrative

examples are provided to facilitate understanding of these ideas.

Key words: array antennas; array environment; coupling; multiple

reflections; mutual impedance.

1. Introduction

In this paper we derive expressions for the mutual impedance between any

two elements in an N-element array environment, where a single element is

excited and N-l elements are open circuited. Knowledge of these mutual

impedances is necessary to accurately derive the radiation pattern of an Nelement phased array from the free-space patterns of single elements. Such

considerations are essential to accurately assess the performance of very

large phased arrays, since mutual coupling between elements might affect the

overall radiation pattern significantly.
The interaction between two antennas or two elements has been described

in [1,2] and [3], respectively. In these references the presence of multiple

reflections in the interaction is noted only briefly, and the observation is

made that under certain well-defined mathematical conditions the closed form

expression describing multiple reflections can be expanded in an infinite

series

The N-element interactions considered in this paper must be a complicated

generalization of two-element interactions, since any subgroup of M elements

can be thought of as a single element interacting with another element made up

of the remaining N-M elements. Specifically one can consider a single

radiating element interacting with the N-l open circuited elements in the

environment. Multiple reflections will then be present between the radiating

element and the environment, just as in the 2-element model. Since the

division of the N elements into a radiating component and an environment is

arbitrary, it immediately follows that there must be multiple reflections

within any subgroup of elements. However, there can be only a finite number

of arbitrary groupings of N elements, hence the total set of interactions

within N elements should be describable in a finite number of terms. We will

show that each of these interaction terms corresponds to a possible signal

path between the radiating element and the element under observation.

Multiple reflections then occur within signal paths between any subgroup of

elements.

To describe the above ideas mathematically we use the scattering matrix

formalism to relate the incoming and outgoing waves at each of the N elements. We then derive an associated (N-l) x (N-l) linear system whose

blocked elements are binary free space generalized current-current interaction

matrices A.., and the elements of the solution vector are the (N-l)

generalized currents describing the radiation pattern of each element. The

binary interaction matrices are constructed from the free-space mode-mode

mutual impedance matrices and the scattering matrices of the open circuited

elements. To exhibit the electromagnetic interactions present among the

elements of the phased array we solve this associated linear system in a manner that preserves the blocked matrix elements in the solution. The

solution is then easily interpreted as a sequence of direct and multiple

reflection interactions among the elements. To achieve this we developed a
novel cyclic decomposition of the interaction matrix in terms of its blocked

matrix elements A... and provide an inductive proof of the fact that the

solution of the linear system is given by such a decomposition. Using this

novel solution we study some elementary examples to demonstrate the effect of

the environment on free space radiation patterns. Finally, the mutual

impedances between elements z

q

- • are obtained in terms of the solutions of the

interaction matrix A. Some simple approximation schemes to the full solution

are suggested to reduce the computational requirements in obtaining the mutual

impedances between elements.

2. N Coupled Antennas or Elements

In Appendix A the basics of the scattering matrix formalism are reviewed

and the symbols used here are defined. In the discussion below we denote the

i-th antenna or array element by A., i = 1,N. The scattering matrix

formalism gives the following matrix equations when A^ is excited and A£

Am are open circuited. For element 1, (considering the reflectionless case

S = )

2 -1 -1' pa pa af3 x 2 -1 -1

» • • •

»

and the (N - 1) open circuited elements (i = 0) are described by

for I = 2,N. Here

(1)

(2)
¥h + w - #

]

< 3 >

and all symbols are defined in Appendix A.

In addition, the generalized voltages V are related to the generalized

currents I by an impedance matrix Z [3]

1 z z z *i2

••• Ln ••• nN

Z

xl

Z

Jt2 •*• * '•* ZxN

Z

N1

Z

N2 •** Z

Nx

(4)

with Z

Ak

= Zkr

.(M) Here Z. are free-space impedance matrices whose elements C|. are the

free-space mode-mode mutual impedance integrals [3] (closely related to the

coupling integral in [4]). Thus,

d

kl)

= 2 j

2 \ / e-J" kDcos e fj

k > f!<*> d(cos 9)

(k) where f. , i=l,2 is the far field pattern of the i-th mode (corresponding to some set of mode numbers nm) radiated by antenna k, D is the distance between

array elements k and x, and * denotes the complex conjugate. For a detailed

discussion of the method of evaluation of the above integral, see Appendix

C. From the above matrices we obtain the following system of equations:
V

l = Si6

)+

( V-l +

Il> +i

l

v *= s

ie

)+( ^

+ l-J- * =2 > N

(5)

V, - L = [s[

l)

- SJ^S^) (V. + I ) + s[

X)

(v. + 1.) -1 -1 ^ 8a afi J -1 -1' Ba 1 1

V - T = S^' (V + I ) I = 2 N

(6)

and

h

+

-

1 ! =

" (Z12^2

+ •" + Z1nIn>

^

+ ^ = " < ZnIl + - + Vn'' ' * =2 >'

(7)

In the last equation the diagonal term is missing on the right hand side, as indicated by the prime.

The equations for v

1 - i,, V, - L and V, + I, can be used to derive

an expression for I, in terms of i ^ and I , X = 2,N. Thus, from (5) and (6)

V _ T = S^MV + I ) + 2S^ /'i -1 -1 \ { -l -V ^Ba U

or0 S^i = -I +-fl - S^HV + I ) ^Ba \ -1 2 U \ M -l -V

where 1 is the identity matrix. We now define free space binary currentcurrent interaction matrices as

A,j 4(1 - S

o

1)

) 2

ij < 8 >

in terms of which the last expression, together with (7), give
(1)

ij = -{\ 2

1 2 +....+ ^ H

l H

) - s pa

i x (9)

Similarly, we get, in general, for the open circuited elements X

h

=

- ( Vi + - + ^nIn*'- * =2 ' n (10)

with the I term missing on the right side as indicated by the prime.

We can write the last expression as a (N-l) x (N-l) linear system

Hz

x

or, with I = A , I ,

A I = -I .

In sections 4 and 5 below we will show that the solution to this linear system

can be written in the form

I. = -Q., L,

-J Jl-l J = 2,N (12)

where

Q . n E wT . w .

Jl JJ Jl

and w. . represents the interaction of element j with itself in the presence of

the open circuited environment, and w., represents the interaction of the
radiating element 1 with element j in the presence of the open circuited

environment. It will be seen that these w matrices have a high degree of

structure that elucidates the interactions present in an N-element array. It will become apparent that Q., gives all possible interactions within the

signal paths in the array from element 1 to element j, including multiple

reflections between subgroups of elements along the paths.

In terms of Q., } the generalized currents I, in (9) and V, + I, and

V. + I. in (7) can be written as

!l

= " C1 " (A

12

Q

21

+ -+ ^V^te 1 !

• (13)

»i + h - ( zi 2 ^i + ..... + hu°n^h W( Vn + - WV *

= 2 - N - (14)

From equations (5), (13) and (14) the mutual impedances z,, = v,/i.. and

z , = v./i, are then given by

11

=

' -iT{I

l?

Q

2l

+ - + W [1 " (A

12

Q

21 + "• + 'inV 1 "^

(15)

(16)

Note that for N = 2 and Q, , = -1 the results in [3] are recovered. We have

now obtained expressions for z^, the mutual impedance between element i and

the radiating element "1" in terms of the free space and array binary currentcurrent interaction matrices A. and Q.,, respectively. As will be seen
below O., can also be interpreted as the total signal path interaction matrix

between the radiating element and element j. In the next section we show how

to obtain Q., .

For completeness we derive expressions for the incoming a). ' and outgoing

fo) "P

b

y

' waves in terms of the excitation current i^ of the radiating element.

Since from (10) and (12)

I* s 4

1J

- 6^ C} ^. Qjl

)'i

1 (17)

and (see Appendix A)

h U) _ cU) AD

one can easily show that

4" 4< s Vji>'ii (18)

J

tf

] 4 so*'<fW^i < 19 »

•J

which together with (13) give the incoming and outgoing waves at the Jt-th open

circuited antenna in terms of i-^, the excitation of the radiating element.

3. Explicit Solution to a 4-Element Problem

For 4 elements (element 1 radiating) equation (10) gives explicitly

1 (2)

I = - -fl - S

v ') [2 I, + Z I„ + Z Tj

-2 2

K o ' K 21-1 23-3 24-4 ;

I = - if 1 - S

(3)

) (Z ,1 + Z T + II) (20) -3 2^ o

J y 31-1 32-2 34-4 ;

I = . 1(1 . s

(4)

l (Z I + Z I + Z I 1 -4 2^ o

} K 41-1 42-2 43-3 J
which is a 3 x 3 system in the unknowns I , I , I

• I is treated as known. We have explicitly exhibited the form of A... Note that I. are vectors whose elements are given in terms of the modal coefficients of the

radiation pattern of the antennas (see equations (17), (18), and (19)). All

other quantities in (20) are matrices. Elements 2, 3, and 4 are open

circuited; if the j-th antenna is a minimum scattering antenna

(S^J ' = I) [1,7] then I. = in (20), i.e., the antenna does not scatter the

o -j

incoming radiation.

We can solve system (20) algebraically. To solve for I in terms of

I , for example, the subsystem I and I is expressed in terms of

I, and I ?

, which is then substituted into I , and finally, I is

obtained in terms of I . The result is:

_A

23 (1 " A

34

A

43 ) *32

+A

23 (1 " A34

A

43 ) *3aHzW -2

hi

-1

_A

23 (1 " A

34

A

43 ) *31

-1 + *23 (1 " ^Hz) ^4^41 r -

1 ! (21)

-1

_A

24 (1 " A43

A

34 ) Hz _A

24 (1 " HzHt) Hi

+^24 (1 ' A

43*34

)_

^43*32. +^4

(1 - A^A^)

- A

43

A

31

or

w

22-2

: "w

21-l

h - -<Wi (22)
where

-1 w

22 w21

Equation (21) or (22) gives the generalized current I ? at element 2 in terms of

I, , the generalized current at the excited element 1. To obtain I. , k # 1, in

general, one merely has to interchange all indices 2 and k. For example, let k =

3, then terms A^ and &oaKo become A~~ and 7L.iL-, respectively. Thus, all

I. can be written down after (21). Equation (22) generalizes to

Ik " -k -\l h W

where Q., = wj". w. ,, k = 2,4. One can easily generalize (23) to the case where

an element l other than element 1 is radiating. A mere interchange of indices I

and 1 will then yield a (the index 1 is now hidden in a and is not shown

explicitly). Thus,

where

-k Ta -I

(24)

Q

kl wkk

w

kA

for k,A = 1,4; X * k

Attempting an explicit solution of (10) for N > 4 leads one to unwieldy

algebraic manipulations. A general solution of (11) for any N has been developed

that gives Q . in terms of A , , and will be discussed in section 5 below. The

essential features of the general w . interaction matrices, however, are already

(4) (4) apparent in \r' and wA,' in (21). (Here we introduced the superscript notation

to indicate the total number of elements in the array.) Therefore, we now turn

our attention to both the mathematical structure and the physics contained in

(21). Naturally, these two distinct points of view are fundamentally connected.

10
4. Signal Paths and Multiple Reflections in N-Element Arrays

Equation (21) gives a highly structured detailed description of the

interaction between the open circuited element (I

?

) and the radiating element

(I,) ii. terms of sums of products of binary current-current interaction

matrices A... We denoted these sums in (22) by Wo? and v^. For the moment let

us treat v^ and v^ as independent entities describing how the radiating element

(denoted by the rightmost index) interacts with the element under observation

(denoted by the leftmost index). Then one can interpret the first term in each

as a direct interaction term or possibly a zeroth order interaction term if the

additional terms are small. Thus,

w

21

= hi

and w??

= 1 for all N, the number of elements in the array. It then follows

that

i< N > - . hl!

m (25)

could be a zeroth order approximation of the interaction between elements 1 and

2. This would strictly be true if all other elements in the array are minimum

scattering elements. The additional terms in (21) give the environmental

effects

Let us denote the individual environmental terms in w,,, and w

22

by

X?

, and Xo ?J and define ej ' as the environmental interaction operato

M-element environment, where M = N - 2. We can then write

*

(4)

" JL e

(2) A

*21 hk \5L hi

(26)

(4) . , (2) . *22 ' ^k \$. V

for £,k = 3,4. In addition there are the direct interaction or zeroth order terms

iU

?

= I and Ap-, in (21), which can now be written as

11
4 4

<*22 + \

*2k

e

ki' VI2

- "(Vl

+ I >k=3

*2k «# VIl • (27)

These relationships clearly point out how the radiating element and the element

under observation couple into the environment: the free space binary interaction

matrices of the element under observation operate on the environmental operators

and the product then operates on the free space binary interaction matrices of

the radiating element.

The terms in (21) can be given a graphical interpretation as shown in figure

1. There are direct interaction lines representing the A., and closed loops

representing the multiple reflections as expressed by inverses. Each diagram is

labeled by the operator it represents. There are five diagrams, which is the

total number of ways a line (representing an interaction) can be drawn from

element 1 to element 2 directly or via the two element environment. The

interaction of element 2 with itself can be represented similarly. For a larger

array these diagrams become progressively more complex.

Equation (27) puts the radiating element (I,) and the element (I

?

) on equ^l footing. Both elements interact with the element under observation and the

(?) environment in similar manner, i.e., e* ' is the same on both sides of the

equation. The minus sign, however, points to the fact that the open circuited

element 2 is radiating only because element 1 is excited; if we differentiate

(27) with respect to time we obtain a special case of Lenz's law [5],

(2) Further detailed examination of the environmental operators e/ ' in (26)

reveals a structure that is amenable to physical interpretation. We note that

(2) the leading terms in el' are inverses, i.e., they are structured exactly as

(4)

Qp/ in (22). Then we can write

> (3)

= y" 1 XI T M Ul Q\V = Yoo Yd * = 3,4 (28)

indicating, as in (22), that the remaining coefficients really represent

~(3) "(3)

currents I~ ' and l\ ' in a three element environment. This can be verified by

direct solution of (13) for a 3 element array. We see then that the interaction

12
1) h

.

-1

2) ^3(1 - A3

4 ^3) i^

-1

3) ^3(1 - ^34^3) h

3 ^\ l

-1

4) ^(1 - A

43

A3

4

) ^

-1

5) ^4(1 - A43A34) ^3^3!

Figure 1. Possible signal paths and multiple reflections in a four

element array.

13
of the environmental operator with the excited element simply gives the currents

in the environment. We can write

or

;(3)

-k

>(3)

1 = kl

-Q (3)

I (4)

= V e}

C) U K I* kl -1

.f3 \i A

JQ -1

-l e,<2>A

1=3 kx JQ'

.(2)

"kl

k = 3,4

(4)

(29)

Equation (21) can now be written as

^2 " h&2 *24^2 J-! 2 + lA21 " ^23^1 " ^4^1 Ul =

or w.

( 4

) = A~ - V /u Q^ for k = 1 2 2k *2k .L *2j jk » Tor K ^ ' J ^

(30)

Equations (22) and (30) can be generalized as J

N w > - A

Ak " \k (31)

where the prime indicates that j # A,k, and

W

(N) (N) (N) ,(N) _ w n kk -k

w

ki -jl

' u

Jk v 33 J Jk

j * X,k (32)

is the array binary current-current interaction matrix in an (N-l) element array

From (31) we can see that if the total number of terms in Q\,~ ' is t^_i

then w^, ' has

Jk

t

N - 1 + (N-2)t

N

_ lf N > 2 (33)

14
terms, each representing a unique path of interaction between elements

I and k. Each of these terms can be represented diagramatical ly, as illustrated

in figure 1 for N = 4. We note that the indices in the individual terms in (28),

as in (21), occur in pairs, i.e., the left index of 0.. equals the right index

JK

of A . as we go from right to left. Thus each term describes a continuous

sequence of interactions.

Since the terms grow both in number and complexity as the number of elements

N in the array increases, there is obviously a need for a general procedure to

(N) write down Q> ' in an N element environment. We turn to this problem next.

5. General N Element Interactions

In the previous section we discussed the structure of the binary array

current -current interaction matrices Q. . for a 4 element array, and the

expressions were generalized to N elements. However, the explicit form of the

(N) environmental interaction matrices or of the O

v

, ' matrices for the general case

still needs to be exhibited. In Appendix B we show that the solution to (11) is

given by

4

N> "= (Ajj)-1 A

:Jj

J-Z.H (34)

where A is the matrix defined in (11) and A..., the cyclic decomposition of the

matrix A about the element A.., is defined in (Bl). This expression was developed with the intent to keep the elements A,., which themselves are matrices, intact in the solution, rather than just treating (11) as a large

linear system to be inverted to obtain the solution. In this manner the physics

of interaction between the elements and the presence of multiple reflections is

explicitly exhibited. These were discussed in some detail in the previous

section.

Since the expansion as defined in (Bl) is rather complicated, we illustrate

it by writing the expansion of A in (11) for N = 3. Ap

2 = 1 is the (1,1)

element of the matrix and 1(11 ;44) = 1. Hence,

IB
1 *23 *24

hz 1 *34

A

42 4*3 1

1 - ^3^2:33^3:32 + A

24

M

22:44

M

24:42 ^ 35 ^

:22

where M22.33 is the submatrix of A^ = 1 obtained by deleting the first row and

column and is expanded in a similar manner about element A~~ = 1. Thus,

M

22:33

: I A 1 J -33

^

4

M

33:44

M

34:43

l " ^34^:

since M

_

3 3:44

= 1 and N

34:43

= A^ .

When the other terms in (35) are exhibited explicitly, the reader can easily

verify that we recover the terms in (21). Furthermore, the recursive properties

of the solution as discussed in the previous section become apparent from (34),

(35) and the more general proof in Appendix B. The assumption here, of course,

is that all inverses of the minors exist in (Bl). Physically this means that

there are multiple reflections among the elements whose indices appear in the

inverses. If the inverse did not exist then there could be no multiple

reflections. This could occur only if 2 or more elements occupied the same

space, which, of course, is not true by definition of an array. Special resonant

conditions that could lead to singular groupings of the interaction matrices are

not considered at this point.

6. Conclusion, Work in Progress and Future Plans

We believe that we have achieved an in depth understanding of the

electromagnetic interaction processes within an N-element array environment.

(X) Assuming that the transmitting characteristics of the elements S\ ' have been

measured and the open circuit scattering matrix S is known, we now have a basis

by which to calculate mutual impedances between elements and construct overall

radiation patterns of an N-element array taking mutual impedances into account.

Although the elements of S

Q

have never been measured, one can make certain

theoretical assumptions (see Appendix A and [3]) to perform sample

16
calculations. The problem of measuring S

Q

needs to be addressed seriously in the

future.

Computer codes to evaluate the A and Q matrices defined earlier need to be

developed. Computer codes to evaluate the mode-mode free space coupling

coefficients, which are the elements of A, are nearing the completion stage.*

Once these codes are in production mode we will compute

(1) Mutual impedances between elements for various inter-element spacings,

a) for various N (the number of elements)

b) and for various free space patterns for the elements.

(2) Elementary patterns in an array environment.

(3) The overall radiation pattern of an array.

After completion of this stage of theoretical exploration, we will start work on the inverse aspect of the problem of creating overall radiation patterns from

data obtained from subarrays. In addition, experimental work is being planned on

a large array to verify the theory presented in this study.

7. References

[1] Kahn, W. K.; Kurss, H. Minimum-scattering antennas. IEEE Trans, on Ant. and

Propag., vol. AP-13, No. 5: 671-675; 1965 September.

[2] Kahn, W. K.; Wasylkiwskyj , W. Coupling, radiation and scattering by

antennas. Proceedings of the Symposium on Generalized Networks at the

Polytechnic Institute of Brooklyn: 83-114; 1966.

[3] Wasylkiwskyj, W. A network theory of coupling, radiation and scattering by

antennas, Ph.D. thesis, Polytechnic Institute of Brooklyn, 1968, University

Microfilms, Inc., Ann Arbor, Michigan.

[4] Yaghjian, A. D. Efficient computation of antenna coupling and fields within

the near-field region. IEEE Trans, on Ant. and Propag., vol. AP-30, No. 1:

113-128; 1982 January.

*We will use a method different from that presented in [4]

17
[5] Bleaney, B. I. and Bleaney, B. Electricity and magnetism, Oxford University

Press, Oxford, U.K. 1959, 676 pp.

[6] Jackson, J. D. Classical electrodynamics, John Wiley and Sons, Inc., 848 pp.

[7] Muth, L. A. Progress Report to Rome Air Development Center, Hanscom Air

Force Base, Massachusetts, Report No. SR 723-56-83, 1983.

[8] Wasylkiwskyj , W.; Kahn, W. K. Theory, of mutual coupling among minimumscattering antennas. IEEE Trans, on Ant. and Propag., vol. AP-18, No. 2:

204-216; 1970 March.

[9] Kerns, D. M. Plane wave scattering-matrix theory of antennas and antennaantenna interactions, NBS Monograph 162, U.S. Dept. of Commerce, National

Bureau of Standards, Boulder, Colorado, 1981.

[10] Stratton, J. A. Electromagnetic theory, McGraw-Hill Book Company, Inc.,

N.Y., 1941, 615 pp.

[11] Rose, M. E. Elementary theory of angular momentum, John Wiley and Sons,

Inc., N. Y. 1957, 248 pp.

[12] Magnus, W. and Oberlettinger, F. Formulas and theorems for the special

functions of mathematical physics, Chelsea Publishing Company, N.Y., N.Y.,

1949, 172.

[13] Varga, Richard S. Matrix iterative analysis, Prenctice Hall, N.J., 1962,

322 pp.

18
Appendix A

Brief Review of the Scattering Matrix Formalism

In the scattering matrix formalism incoming waves a are related to outgoing

waves b by

b = S a, (Al)

where S the scattering matrix of the antenna (valid for a single antenna or an

a

r ray antenna) is given by

s s

f aa a(3 S = si (A2)

s s Pa pp

and

-(;:)-p' =(;:) \-p<

In the above a refers to the port side and p to the space side of the antenna,

S is the mutual coupling matrix of an array antenna or the input reflection

coefficient of a single antenna, the row vector S Q contains the receivihg

ap

coefficients, the column vector S contains the modal transmitting coefficients

p(X

and S 00 gives the scattering coefficients of the antenna: a (b 1 are incoming

PP

3 3 -a v -a ; 3 (outgoing) waveguide mode coefficients, and a (b ) are incoming (outgoing) wave

mode coefficients. In the main text we consider only single mode waveguides,

hence, a and b are scalars. The scattering matrix S is unitary [1,7], i.e.,

a a

3

S

+ S = I (A3)

where I is the unit matrix, which results in three independent conditions on the

elements of S [7,8]. In addition, one can show that S QQ can be written as

t t t pp

S„ = S - S n S' with S S =1, and S S„ = S Q . Sn is said to be the open

PP O pa ap pa ap

circuit (a = b or the current i=0) scattering matrix of the antenna which

*--a -a ; relates the incoming wave a n to the outgoing wave b. by b. = S a . For further

-p -p -p o-p

details the reader is referred to [1,2,7]. Theoretically all components of S

19
except Saa

are infinite dimensional, but in practice only finite dimensions can be considered. The level of truncation is in practice determined by the accuracy

or noise level of the measuring system.

Linear combinations of the a and b wave amplitudes (a, p understood) can be

formed to represent generalized voltages and currents [9]. Thus,

2 =kv + I)

b =-|(y - i) .

(A4)

We make extensive use of this representation in the text.

20
Appendix B

Cyclic Product Decomposition of Matrix A

Given a square matrix A of order N, whose elements labeled a, , k,A=l, are

of order 1 < n < N, then the cyclic product decomposition of matrix A about an element akk i, denoted by A ,,,, is defined inductively as*

A.

kk

,= I(kk':ap)a

kk . + £' I(kA:ap)a

kJl M^ , ;JU

M

kJ

^, , k, k' = 1,N (Bl)

Jkfck

1

where A. kk i is of order n, M, is a submatrix of A of order (N-l) obtained by

omitting the k-th row and A-th column of A, and K . , is the cyclic product

decomposition of M, about the element a^i (of order n) and

I(M.:ap) = (_ 1 ) k+^P+£ ( kA:aP)

} where a p is the set of al ] tne indices that were omitted from matrix A to form the submatrix M, , and e(kX:ap) = m, where

m is the number of times k > a and I > p for all a and p, and p is the

permutation index equal to the total number of times rows and columns have to

be permuted to have their indices in increasing order. Note that

initially ap is (N+1)(N+1). This complicated function of indices

I ( kJl:ap) merely has the effect of alternating the sign of successive

elements a, in each submatrix according to their row and column positions.

Note that if in the first step of the expansion k = k', then we obtain

products of terms with paired indices throughout the decomposition.

The definition (Bl) defines the cyclic product decomposition of matrix A

in terms of decompositions of submatrices of A. An important property of this

inductive definition is that a decomposition about an element a^i will result

in a sum of terms in each of which an element of column k

1 will appear as the

last factor on the right. We note, without detailed proof, the following

lemma:

Lemma 1: The cyclic product decomposition A: kk i is of the following form

A

:kk'

= a

kk' + J, B

j a

jk'

(B2)

j*

*The reader is referred to the example in Section 5.

21
where B- are matrices of order n obtained from the elements a^ of A. Thus,

the elements in column k' always appear on the right side in each term. The

importance of this fact will be apparent in the theorem proven below.

A brief examination of (Bl) should convince one of the correctness of

lemma 1. The first term of an expansion about a^i is an element of column k' by definition, and (except for a scalar factor in front of it) it stands by

itself. The column index k' is not changed in the definition, that is, the

first element of each successive decomposition of the

submat rices M, , Jtfk', will be an element from column k'

Theorem : Given a linear system Ax = 8 where x

=

(x-,» •••» Xw

)>

(3 = (B, , ..., p..) and A.. = a. ., i = 1,N; j = 1,N and x. and s. are

-1 -N ij ij -i -l n-dimensional vectors and a., are nxn matrices. Then

h 'V'

1 5:JJ

(B2)

where A is obtained by replacing the j-th column of A by 8. Thus, this

formula is seen to be a generalization of Cramer's rule to linear systems of

vector variables and matrix coefficients.

We shall prove by induction the above formula for j = 1 only, since the

result can be shown to follow for all j t 1 by simple relabeling of terms

after an even permutation of rows and columns of the linear system. The first

step in an inductive proof is to show that the result is valid for some

initial value of N, say N = 2. We will comment on this part of the proof at

the end of this section, leaving some easily obtainable details to the reader

to fill in.

Consider the linear system

a x + a X + ••• + <* X + ••• + a , vX =

P 11-1 12-2 U-l 1(N+1)-N+1 -1

a X

+ »«» + a , J =8

21-1 2(N+1)-N+1 -2

• • a

n-*l

+ '•• + a

M-*A

+ ••' + a

£(N+l)*N+l " ^l

a y + . . . + a Y+... + a y =

8 (N+l)l-l (N+1)A-Jl (N+l) (N+D-N+l -N+l

2?

(B3)
or Ax = p

where A = a , i = 1,N+1, j = 1,N+1 are the blocked square matrix elements

-ij ij

of A, and p., x- are vectors.

We form an NxN system from the above by omitting the first equation and

transferring the a-iXi» J = 2,N+1 terms to the right hand side. We then get

M

ll*

= £

- ?i2i • ( B4 )

Here a,, is a column vector whose elements are a.,, j = 2, N+l . The """

indicates that we are dealing with the NxN reduced system, and M-q is the

coefficient matrix of the reduced system and is also the submatrix of o,, of

the full (N+l) x (N+l) coefficient matrix. We now assume that the solution to

the reduced system is given by

X. = K] .. M„ ... j = 2, N+l (B5)

-j 11:jj -11:jj

where M is the matrix formed by replacing the j-th column of M by p

- a. x-i • We now permute column j of M so that it appears as the first column. We

denote the resultant matrix by P . Each exchange of columns introduces a

factor of (-1) or altogether (-1)^ J " 2

^ = (-1) J

. So,

x

. = VI'] ..P,, ..(-l) j

. (B6)

Since the right hand side has two terms

x. = n~] .. (L .. (-i)

j + (-i)

j+1

<! . . R„ ..X, (B7)

*j ll:jj -ll:jj 11:jj -ll:jj-l

where Q contains p, and R contains a,. On close examination we see that

R, , . . = M„ . ., of the full NxN system, where Mi ,• is the minor of ai ,•

,

-llijj -ljijl -^ iJ

and Q, , . . = M, . ., is the submatrix of a, . of A formed by replacing the first

-11 :jj -lj:jl lj

column of A by p.

23
Hence,

h

' Mn:jj^:jl ( - 1)j + Hn:jjVjl ( - 1)J+1

-V j = M+1 (B8)

and the other indices refer to the original full system. We now substitute

y. , j = 2,N+1 into the first equation of the full system

[a + E a m"

1 M (-l)

(j+1)

] x

= fl + E a M~! . . M, . . n (-l)

J+1

11 j ij ll:jj lj:jl -1 -1 lj ll:jj -lj :jl

•J or, by definition,

*:ll2fi *:ii < B9)

and

X

= (A )" A . (BIO)

-1 :11 -:11

We still need to show that the solution to a 2 x 2 system is given by our formula. Let

A =

-1

- 1 -1

Then ^

= [«n

- «

12 «

22 «^) (^ - «12 *22 |2

) according to the expansion

rule. This result can be easily verified by solving the 2x2 system directly

by substitution. Similarly the result can be verified for \ . The proof by

induction is now complete.

24
Appendix C

Evaluation of the Coupling Integral

The elements of the mutual impedance matrices Inx, between antennas I and

k in equation (4) are given by the mode-mode impedance integrals [3]

(with e time convention)

nn' ,m

2% 1 (k)

f a> J e (e,<b) J

, • -nm

v ,v/ e

n

i m

(9»<t>) e dx (CI)

where x = cose, D is the distance between antennas k and l located on the

z-axis, and e is the far field pattern of a single (even, odd) mode

U,o) . ~TM

r , n e « m given by [3] -nm r J

?nm ^»4>) ~ ~ N rVsVx) t S in m<fr' nm

T (C2)

where N

2 . 4u n(n + 1) (n + m)

!

nm

nm

2n + 1 (n - mj'l

gives the normal ization ( Anm

= lj

. if m = 0j

Anm

= 2 otherwise), P (x) are associated Legendre polynominals and the

surface component of the gradient V e v - r V (r = unit vector), or in

component form,

v

s " ^ ' r 59' r sin e 5<r

(C3)

Note that in (CI) only a single m index appears, explicitly stating that modes

with m*m' do not interact (the ^-integral vanishes). Coupling integrals

between far fields e = r x e and e or between e and e can also be -nm -nm -nm -nm -nm written analogously to (CI), but we will not do so, since the evaluation of

all possible coupling integrals proceeds along the same lines.

To evaluate (CI)

(1) we perform the <t> integral to obtain a factor of %

25
(2) we observe from the recursion relationships [10]

dP

" = 1 [(„ _ m+ i )(m+ n)p

m-l . Pm+1]

de 2 n n (C4)

^e^=i-se[(n- m + i)(n +m) P- 1 + P^] (C5)

m f[\~¥ 1 that f(x) = e • e. can be written as products of P P 7 , and from

-nm -n m

r

n n Rodriguez's formula [6]

/ , N m m/2 .Jt+m „

r ih\ dx

A+m

(C6)

one can easily see that f(x) is a polynomial in x, and hence, is identical to

a sum of Legendre polynomials P»(x)

n+n

mr f(x) , V Jf tM

I =

I n-n I

(C7)

where the prime on the summation indicates that only even or odd Z-s occur in

the sum.

The integral representation of spherical Hankel functions (for e" lwt

convention) [10]

hj^flcD) -rl

I P,(x) e+1kDxdx

i

<=° (C8)

allows one to write the coupling integral as

n+n

* JL= I n-n I

(C9)

26
This series can be easily evaluated if a are known.

The a. can be calculated theoretically from Clebsch-Gordon coefficients

using the coupling rule for spherical harmonics [11], but a simple numerical

procedure will also yield these coefficients. We generate f(x) on a discrete

set of points x. ,i=l,L, and write (C7) as a matrix equation

P (xj . . . P (xj . . . o^ r r V

p

( x , ) • • • p

n ( x , ) • • • o^ k

J r kJ

P (x

L

) • • • P x

(x

L

)

Since the Legendre polynomials are basic functions, the columns of the matrix

in (CIO) are essentially orthogonal, and the matrix will be well

conditioned. Then (CIO) can be easily inverted to obtain the a.. Numerical

experiments yielded values of a. to machine accuracy.

In large arrays the asymptotic behavior of £ , will give the

strength of interaction between distant elements. For x >> 1,1 the Hankel

functions behave as [6]

n x SL+1 ix

h U) (x) + (-1) ^- (Cll)

and, because of (C9), | C

n n

t m

l + 7 •

For x << 1,A, the real part of hj. (x)

J A

(x) -0 (C12)

and the imaginary part of hj. (x)

27
%(x) 1+1 (C13)

For X » x » 1 [12]

(C14)

This last relationship shows that for elements whose radiation is bandl imited

kD

to l ~ x = y- , where D is the distance between elements, the mutual

impedance between the highest modes present is of the order of unity. Hence,

the asymptotic behavior of n

5 (x) for x << 1 is not troublesome.

From the above considerations one can deduce that for identical elements

the off diagonal entries in (11) get progressively smaller. Hence, the linear

system could be diagonally dominant, in which case a solution always exists

[13], The condition for strict diagonal dominance for a complex matrix A of

order M is that [13]

a . . I j=l

ij

(C15)

where a^ ,• are the elements of A. It is easy to see from (C11)-(C14) that

there is an interelement spacing D

Q

at which the matrix A in (11) is

diagonally dominant. For x = kD

Q (C15) can be written as

1 > kD.

(C16)

where ^ is some constant such that < p. < ». At this point we will

consider only interelement spacings that satisfy (C16). The p^ could be

obtained theoretically from the CI ebsch-Gordon coefficients, i.e., from the

a. in (C9), in which case D

Q would be expressed as a complicated but exact

function of the mode numbers describing the element patterns.

Up to now we have assumed that the antennas in question are located on

28
Up to now we have assumed that the antennas in question are located on the z-axis. A two (or three) dimensionl generalization of the above procedure

is easily achieved. The line connecting the two antennas is considered to be

the z-axis, and the modes of the antennas expressed in the original reference

frame fixed to the antennas are are expressed in this new rotated frame. This

requires a change of basis, which is accomplished by the formula [11]

V9 '-*'' =

E DS,.<«f"T>Y Jh

(e.») (C17)

m

when a, p, y are Euler angles of rotation as defined in [11], and

D , (apy) is the rotation matrix in spherical basis. As an example, one can easily show that the field of an axial dipole, which is generated from

Y-iq(Q»<I>)» when rotated through p = - = , will be generated from the real

part of Y ni( e ><l>) + Y n _i(9><t>). These Y +

- ( e, <|>) are in fact the modes

present in the rotated frame.

29
NBS-114A IREV. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN-1078

2. Performing Organ. Report No. 3. Publication Date

February 1985

4. TITLE AND SUBTITLE

A Theory of Mutual Impedances and Multiple Reflections in an N-Element Array

Environment

5. AUTHOR(S)

L. Muth

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State. ZIP)

10. SUPPLEMENTARY NOTES

1\ Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

, bibliography or literature survey, mention it here)

A general theoretical approach is formulated to describe the complex

electromagnetic environment of an N-element array. The theory reveals the

element-to-element interactions and multiple reflections within the array. From the

formulation, it is found that the interaction between an excited element and an open-circuited element can be viewed as the sum of terms describing all possible

signal paths within the array environment which start from the radiating element and

terminate on the element under observation. Within all paths except the most direct

one, multiple reflections between subgroups of elements take place. The resulting

solution is highly structured and recursive and is discussed in detail in the

text. Illustrative examples are provided to facilitate understanding of these

ideas

.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

array antennas; array environment; coupling; multiple reflections; mutual impedance

13. AVAILABILITY

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fJ3) For Official Distribution. Do Not Release to NTIS

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