NAT'L INST. OF STAND & TECH

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Reference

NBS

PUBLICATIONS

NBS TECHNICAL NOTE 1091

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

Interelement Interactions in

Phased Arrays:

Theory, Methods of Data

Analysis, and Theoretical

Simulations

Lorant A. Muth
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KESEARCH JNFOFJ,

Interelement Interactions in

Phased Arrays:

Theory, IVIethods of Data Analysis, and

Theoretical Simulations

I V

« 6 a

Lorant A. Muth

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

\—TTTT

7

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued December 1985
National Bureau of Standards Technical Note 1091

Natl. Bur. Stand. (U.S.), Tech Note 1091, 48 pages (Dec. 1985)

CODEN:NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1985

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

Page

I. Introduction 1 II. Theoretical Review 3

III. Methods of Data Analysis 10

IV, Experimental Procedure to Study Interaction Effects

in an Array Environment 12

V. Simulation Study 15

VI. Generalization to Two- and Three-Dimensional Arrays 21

VII. Summary, Conclusions, and Suggestions for Further Study 21

VI 1 1 . References 22

Tables 1-14 24-36

Appendix A Coordinate Independent Formulation of Scanning Theory 37

m

Interelement Interactions in Phased Arrays:

Theory, Methods of Data Analysis, and Theoretical Simulations

Lorant A. Muth

Electromagnetic Fields Division

National Bureau of Standards

Boulder CO 80303

We review theoretically the effects of multiple reflections and

mutual impedances in array environments and study possible methods of

far-field pattern data analysis to recover interaction effects. We use theoretical expressions derived earlier to calculate in a twoelement linear array the mutual -impedance matrix and effective

excitations of elements as functions of interelement separation

and n^,j^, the maximum mode number in the radiation pattern of the

elements. Generalizations to two- and three-dimensional arrays are discussed.

Key words: data analysis; effective excitations; interelement

scattering; multiple reflections; mutual impedance; phased arrays.

I. Introduction

To understand and to analyze interelement interactions in a phased array,

1 we need to know the far-field patterns of each element and of the full array.

In this study we examine methods of analysis that seek to provide adequate

information on interelement mutual -impedance and multiple-reflection processes

to be able to synthesize far-field patterns of phased arrays from subarrays.

We can then attempt to develop a procedure to obtain the far-field patterns of

very large arrays constructed of subarrays whose patterns are well known.

This problem is of current interest, and its solution, to our knowledge, has

not been attempted before.

In broad perspective the solution to the problem is seen as follows:

A. The effect of mutual impedances on element excitations in the array

environment has to be understood.

1 We will not be concerned here with the near-field measurement techniques and

the preliminary data analysis that leads to the knowledge of the far fields.

We assume that both the receiving and transmitting patterns of the elements

and of the array as a whole can be determined using existing standard nearfield measurement techniques.
B. The role played by multiple reflections and mutual impedances in producing elementary patterns in the open-circuited environment, which are different from free-space element patterns, has to be understood.

The two effects above are obviously related, but they are distinct. The first

effect describes the interactions between the elements when they are all

excited and radiating, and the second describes the interaction of a single

radiating element with all the other open-circuited elements. The difference

between these two effects can be understood immediately in the case of minimumscattering antennas [1,2], where the elementary patterns are exactly the freespace patterns, but mutual -impedance effects are still present when the array

is radiating. Both effects are simply a consequence of the fact that there is

electromagnetic radiation everywhere in space, and, therefore, voltages are induced at each element of the phased array. An open-circuited antenna will,

in fact, respond to incoming radiation according to its receiving pattern, but

it will also immediately reradiate this incoming excitation according to its

transmission and scattering patterns.

C. The open-circuited scattering matrix of each element that describes the

scattered radiation in terms of the incoming radiation has to be known.

The scattered radiation, when correctly combined with the free-space

pattern of the radiating element, will give rise to the elementary patterns in the array environment.

The details of the general discussion above have been treated extensively in

[1,2,3] and will not be repeated here.

D. The open-circuited scattering matrix of an element in an array environment has never been measured to our knowledge. It would be difficult to

perform such a measurement in a facility that is not set up to do so, but

in any case it would be costly to accomplish. Therefore, theoretical

modeling of the scattering matrix is needed. Only a simple model that

represents the phase shift of the signal as it enters and exits an opencircuited element can be written down immediately. Any further com- plexity in the scattering matrix will be due to the structural properties

of the element in question and cannot be modeled easily without
supporting measurements. One can hope that such effects will be negligible and modeling will not be required, but only real measurements would

shed some light on this matter.

E. As a consequence of the discussion in (D), it would be desirable to

formulate the problem of interelement interactions in such a manner that

the open-circuited scattering matrix does not have to be specified

explicitly. One needs to formulate the measurement and computational

procedures in terms of quantities that are easily measured or computed.

F. The problem of predicting the performance of large arrays from measure- ments performed on smaller subarrays made up of similar elements can be

solved if the dependence of the effects in (A) and (B) on distance and

the total number of elements in the entire array can be specified. Only

theoretical modeling can be accomplished here, since the entire array is

considered to be too large for deployment in existing measurement facilities. Essentially, the relevant effects fall off rapidly with distance;

hence, adequate modeling might be possible.

In the body of the report we examine some important fundamentals of analysis of array patterns and interaction effects. We seek to discover through

theoretical understanding and numerical simulations the information and mea- surements one needs to characterize interlement interactions in phased arrays.

First, we briefly review the theoretical fundamentals, emphasizing pattern addition with and without interaction effects. Second, we survey possible data analysis procedures using known information about the phase

relationships between elements of arrays. We then use numerical simulations

to test these procedures for numerical stability and for sensitivity to approximations. We conclude with a brief discussion of modeling of large arrays

from data obtained from subarrays.

II. Theoretical Review

In our theoretical framework to describe interelement interaction pro- cesses we rely extensively on the scattering-matrix formalism as developed in

[2] and [4], A relevant theoretical extension of scattering-matrix concepts
applicable to arrays has been developed in [3]. We will use freely the

material covered in these references, without going into detailed explanations

of concepts.

We will rely primarily on the far-field pattern-addition formulas that

give the resultant array pattern in terms of the individual patterns of the

elements of the array, the phase shifts in the individual patterns as a function of location of the radiating elements, and the excitations of the

elements [1,5]. For noninteracting elements, the pattern-addition formula is

simply

N -ik«D

FO(e,<t.) = fO(e,(j))

I aO e - -\ (1)

n=l

where the superscript zero indicates noninteracting elements, F_is the array

far-field pattern, X""^ ^^^ element pattern (assumed the same for all the

elements), a^ is the excitation applied to the nth element, D^ is the position

vector of the nth element, and _k_ i s the wave vector. The generalization of

this formula to n different but still noninteracting elements is straightforward,

N -ik'D

FO(e,<D) = I aO fO e "",

(2)

n=l

where f^ is the pattern of the nth element. In reality, of course, there

could be significant interactions among the elements of an array. This will

depend primarily on the individual element patterns and the relative locations

of the elements. If we denote the presence of interactions by the symbol ^,

then the array pattern is expressed as

N . . -ik«D

F(e,(D) =

I

a f^ e

' "",

(3)

n=l """

where f are now the elementary patterns in an open-circuited array environment, which, in general, are different from the element patterns in the non- interacting case: f now include the effects of interelement scattering. One

""ll A. can obtain the effective excitations a from the excitations a° applied to the

-n -n ^^ individual elements from
a = r(i)aO, (4)

where r(^) is the effective excitation matrix, and

Tr a = (3i» •••a. , ...a|^),

(5)

Q _ D Tr a - (ai, ...a., ...aj^),

where Tr denotes the transpose; here the effective excitation matrix r(z) is

given by [2] (X is the unit matrix)

Id) = y (i + I)^ (6)

where

W. . = (1 + z. .)&• •, 1 < i, j < N (6a)

is a diagonal matrix, and

z.. = z.^. a, §o) (6b)

are the elements of the mutual -impedance matrix |. As indicated, these quantities depend on the binary mode-mode mutual -impedance matrix Z [1,3], and on the open-circuit scattering matrix Sg [1,2].

We need to understand the structure of the elementary patterns f and the

effective excitation matrix Tin more detail to isolate effects accessible to

measurements. Consider that element n is radiating (with unit strength) and

all other elements are open circuited. Then the incoming radiation at element

m(*n) will induce currents that will radiate a pattern into all directions in

Mi)

space. We denote these induced patterns by f . The elementary pattern f

is then given by

N ,.. -ik-D

f,(e,4)) = f°(9,^) + X fl'Me,(J)) e ''^

(7)

m=l

This expression is similar to those in (1) and (2) and can be understood ac- cordingly. If we neglect all but the first of the multiple reflections

between the elements, we have in a first-order approximation that [1]

N f,. -ik.D

f,(e,(t)) = f°(e,^) + I

' f;'Me,4)) e " -''^

(8)

m=l
where the prime on the sum indicates that the nth induced pattern originating

from the radiating element is negligible. This is a higher order effect. One can write a first order approximation for the induced patterns emanating from

elements m^ in terms of the transmission characteristics s_and |, the modemode mutual -impedance matrix. For example, for two elements [1]

bP =-V2(l -^i'h I 12.

bj'^ = -V2(l = Sjo^^) |s.i,

where b are the coefficients of the induced patterns f , where element m

-m "^ -m

is radiating. Corresponding expressions in an N-element environment could

easily be derived [3], This, however, would not be really useful, since any

direct measurement of the induced pattern would yield the resultant pattern

and not an approximation of it. In general, these approximations do not allow

solving for the elements of §o without additional measurements. We can emphasize then that it is possible to obtain the coefficients of the scattered

fields, but not the elements of the scattering matrix, from a single pattern

measurement. Briefly, some important facts about the open-circuit scattering

matrix §o ^re that it satisfies, in general, the relationships [1,3]

So s, = s „, (10a)

and

bp = lo ap, (10b)

where s _ (s^ ) give the transmitting (receiving) characteristics of the

-ap -poc element, and b_ (a_) are the coefficients of the incoming (outgoing) waves,

-p -p

respectively. For reciprocal antennas a diagonal |o corresponds to introducing phase relationships into incoming waves, which then become time re- versed and outgoing. Under these conditions different spherical modes are not

mixed during scattering [1]. More detailed treatment of the scattering matrix

is beyond the scope of this study.

To predict far-field patterns of very large antennas from measurements

made on subarrays, the effective excitation matrix r emerges as an important

quantity. Equation (4) states that the effective excitations are linear
combinations of the excitations of the elements. For large interelement distances, or for large kD, r approaches the unit matrix, since the off-diagonal

elements z^- • approach zero in this limit. Thus,

A.

li ^^i

this, however, does not immediately imply that the effect of the array environment on the applied excitations can be approximated by ignoring the

contributions of all distant elements. In general, the condition of large

interelement distance is not satisfied for realistic arrays. Only measurements or a detailed understanding of environmental effects as contained in the

interaction matrix Twill reveal a valid approximation scheme. Environmental

effects have been treated quantitatively in [3]. Here the relevant facts are that each element in the array at successively larger distances from the

element under observation contributes less and less to altering the effective

excitation. In addition, the contribution of the array environment to a binary interaction is a higher-order effect that can be neglected in a firstorder approximation scheme. Therefore, we might be able to approximate the

elements of the effective excitation matrix with free-space mutual

impedances |. .. We can represent these ideas as

* (N) ~ (2) ^- (N-2)

;..'N-2) « -..(2)

(12)

where the " indicates that the quantity in question contains all interaction

effects, the superscript indicates the number of elements interacting, and

e. . and p.. . represent environmental contributions. In case this approximation

'J 'J scheme is invalid for a given array environment, a different approximation

scheme can be obtained by replacing all the 2s with some integer k > 3,

representing the number of elements undergoing significant interactions. The

specific value of k will depend on |o and f*^. A consistent criterion for
truncating the number of open-circuited elements whose contribution is

included to synthesize the elementary pattern of a radiating element can be

stated as

|ff ^e,4))| = |f^'^e,4))|, (12a)

(2) where f is the second-order induced-field pattern originating from the

radiating element n, and f^^' is the first-order induced-field pattern at the

open-circuited element m, located at_D^p. Condition (12a) will be satisfied

if D^P is large enough. More distant elements will contribute less than

(2)

f to the overall pattern and, hence, need not be included in a first-order

-napproximation scheme.

The mathematical formalism to isolate environmental effects has been presented in [3]. We need to solve (4) and (6) for a, assuming the I and a° are

known, using the cyclic-product decomposition of r. We rewrite (4) and (6) as

where

Q (W"-^ a) = aO (13)

Q = z + I. (13a)

In general, in an N-element array, a is given by

where Q.^-^ and Q_.^-^- are cyclic-product decompositions [3] of Q. It is instructive to work out a few examples. For N=2 we get

-1

- (2) ^1 " ^12 (Z22 + 1) ^2

ai = n Zl— 1 - (Zu + 1) Zi2(Z22 + 1) Z21

and

^ (2) 32 - Z2i(zn + 1)'^ ai

(^'^'^

a2 - zi Zl • 1 - (Z22+ 1) Z2l(Zll + 1) Zl2

We can write the first of these expressions as
(I)

1 + (zh + 1) Ui^ hi)

where n is the environmental effect. Careful examination of these equations

and the expansion for N=3 shows the recursive structure of fl. We can easily

verify that

^ Z22 + 1 Z33 + 1

where the superscripts indicate the number of elements in the environment and

the indices of the elements interacting, respectively. For four elements the

expansions get progressively more complicated. We merely write the expres2 sions at the next level of expansion of Q in (14). For i=l,

§:11 " ^1 " ^12 ^11:22 ^12:21 ^ ^13 ^11:33 ^13:31' ^14 ^11:44 ^14:41

^:11 " ^^11"^ ^^ " ^12 ^11:22 ^12: 21

"^ ^13 ^11:33 ^13 :3r ^14 ^11:44 ^14:41

(15)

Note that W^ = Zn + 1, which is the first term in Q-n*

The effective excitation a can be represented as

^°

?. (N-l)r ^ ^

^\ - —-—-—-I—fir)—' J^^"' (1^)

' 1 + (z.. + 1) 1 n.(^ ^Ui)

where n. (2,a.) and n. (z) represent environmental effects. Physical argument

suggests that 1 im a. exists; that is, the addition of a single element does

not significantly affect the array environment beyond some N. We intend to

use this fact when synthesizing patterns of very large arrays, where the

effects of distant elements are not measured and are unknown. However, we

will not examine this question further at this point.

2The general N-element expressions can be written down using the expansion

defined in Appendix B of [3],
III. Methods of Data Analysis

In this section we briefly examine methods of data analysis suggested by

the equations and discussion presented in the previous section. Our purpose

is to analyze simulated and real array data to isolate interaction effects

such as scattering, mutual impedance, and changes in the element patterns of

individual elements in the open-circuited array environment. We assume that

the standard methods of near-field scanning and data analysis will provide

far-field patterns that contain the effects we are trying to isolate and

study. If this is not the case, we will have to examine current scanning and

data-analysis techniques as applied to arrays anew.

We base our approach on one simple fact: all of the pattern equations in

the last section are linear functions of the element-excitation vectors a or a^ and also of the modal coefficients b_that generate the far-field patterns.

This suggests standard methods of analysis, such as least squares, wherein

linear coefficients are recovered from pattern data, and a given (assumed)

model of interaction between the elements of the array. Equation (3) is the

most general statement of array pattern synthesis; eqs (1) and (2) are special cases. For purposes of analysis we rewrite (3) as a linear system as follows:

-i k 'D fiCQi,-!)!) ... fj^(ei,(})i) e tNOi'^)!) e ^^.%

fi(e.,(t)j) ... f^(e.,(i)^.) e

y^ • A.

•^^-•^-x -ik'D.

f.,(e. ,(t).) e

-i k .D ln%^^n^ e -^^•5n

ai

'N

F(ei,(t)i)

F(e.,(t,j.)

[(Qn'^^^n^

(17)

We have required here that the resultant array pattern, represented by F(9,<t>) on the right, be known. In addition, the positions DLj of the elements must be

specified, as well as each elementary pattern f.. Equation (17) merely has to

be reinterpreted if the elements are considered noninteracting, but

distinct. Considerable simplification of (17) occurs if the elements are identical: according to (1) the array pattern can be divided by the element

pattern (provided it is nonzero), in which case the matrix in (17) reduces to

a matrix of the phase factors. In addition, one does not have to consider

10
each component separately. The solution of (17) gives the known excitations

applied to the elements.

Thus this simple and straightforward analysis can be used to decide

whether interaction effects are important: significant deviations from the

known excitations will indicate importance of interaction effects in the array

environment. One can then tarry the analysis deeper. Each elementary pattern

in (3) can be represented by a set of basis functions together with appropriate phase factors to indicate the position of the radiating element. A

matrix can now be constructed whose rows are

(1) (1) J2) -'^-'^-2

. . . p (2)

-^^--52

Pi G ... p. e

(18)

for each (9-, 4).), i = 1, M; j = 1, M' where the superscripts identify the

array elements, the p. are the far-field basis functions, L is the number of

modes radiated, and the other symbols have been previously defined. Each

point in the region of interest is represented by a row; the region need not

be a full sphere, although the conditioning of the matrix will be influenced

by the extent of the region. This analysis will yield coefficients that are

the products of the effective excitations (already determined in the previous

analysis) and of the transmission coefficients of each element. We can thus

obtain the transmission coefficients of each element in the array environment,

which is a function of the free-space transmission coefficients and of the

scattered fields.

A special combination of the previous avenues of analysis arises in the

case of one element radiating in the presence of another, open-circuited

element. Equation (8) gives a first-order approximation of the pattern

obtained in this situation, which in matrix representation is

(p) -ik«D2

fi (ei,<t.i) pr^e

. (p\ -ik'D

fi(e^.<t)-) pr^e

L^i -(^.H-f^

(9) -i!<*D2

. p] ^e

(2) "'i'-2 (2) "'!S*52 .... Pj 'e

(2) -1^52-]

P^ 'e

(2) ^!S'22

Pl e

f?'1 6^M»<Pw^ \ ) 6(2)e"'-*-' PI ^ .... s'2)e''-'-' p.; e .... B'2)e"'-'-' p, e

;(1)

k(2)

b^.J

(2)

(2)

(19)

11
Here we have omitted the right-hand side, which is exactly the same as in

(17). All symbols have been previously defined. We allow for an effective

excitation of the radiating element, which, however, should not be very

different from the applied excitation in a first-order approximation. This serves to check the validity of our model.

Assuming that the above analyses can successfully provide information on induced patterns and excitations in simple configurations, we can proceed to

obtain data to approximate the array environment of an N-element array.

IV. Experimental Procedure to Study Interaction Effects in an Array Environment

Briefly, the objective of the measurement procedure should be: 1) to

verify eqs (12) and (14) by measuring the mutual impedance between two elements in various array environments, and 2) to compare the measurements to

calculations. To accomplish this, both pattern measurement and port-side

excitation measurements are needed. Multiple-reflection effects between the

probe and the array environment, and on the port side, will have to be eliminated or corrected for during the analysis. The procedure is essentially the

same for arrays of microstrip panels, waveguides, or horns. The following

sequence of measurements is indicated:

A. Two-element study.

(1) Experimental measurements.

(a) Measure the free-space radiation pattern of both elements

under study. If the elements are known to be identical within

acceptable tolerances, then only one pattern measurement is

needed. The elements should be tuned so that no port -side re- flections occur when they radiate into free space.

(b) Measure the elementary pattern in the two-element array

environment. This involves open circuiting one of the elements

and exciting the other one. Ideally, elementary patterns for

both elements should be measured to have some estimate of the

variability within the array environment.

12
(c) Measure concurrently with (b) both the incoming and outgoing signals on the port sides of both antennas.

(d) Repeat steps (b) and (c) for various interelement distances

that represent the positions of elements in the full array.

Here, two-dimensional measurements are indicated if analysis of

planar arrays is desired.

(2) Important effects and computations.

(a) From (Ala) and (Alb) above, we can observe the effects of

interelement scattering and analyze the data for the coefficients of the main pattern as well as the induced patterns.

(b) From the data obtained in (Ale) and (Aid) above, the mutual

impedance matrix can be obtained as a function of interelement

distance in a two-element array. The measurements can be com- pared to computations of mutual impedance using various models

for interelement scattering, such as a minimum scattering model

and a model where the open-circuited scattering matrix is diagonal. These results can serve as an approximate binary interaction model in N-element arrays.

B. Three-element study.

(1) Experimental measurements.

(a) Measure the free-space radiation pattern of all elements

under study. If the elements are known to be identical within

acceptable tolerances, then only one pattern measurement is

needed.

(b) Measure the elementary patterns in the three-element en- vironment. This involves exciting one of the elements and open

circuiting all the others. Ideally, all elementary patterns

13
should be measured to have some estimate of the variability

within the array environment.

(c) Measure concurrently with (b) both the incoming and outgoing signals on the port sides of all antennas.

(d) Repeat steps (b) and (c) for various interelement distances. Two possibilities arise here: 1) only one of the

elements is repositioned; and 2) both of the open-circuited

elements are repositioned, but kept adjacent to each other. The

first of these procedures will show a larger effect than the

second.

(2) Important effects and computations.

(a) From (Bla) and (Bib) above, the effect of the opencircuited environment on the free-space element pattern can be

observed. More specifically, the results here can be compared

to the results in the two-element study to look for possible

convergence in both the elementary patterns and in the

coefficients of the scattered fields. This kind of analysis

will allow us to specify the size of the environment that must

be studied to accurately describe interelement interactions in

large arrays. In particular, we will be able to decide whether

nearest neighbor interactions or three-element subarrays are adequate to represent a full array. We might find that larger

subarrays are necessary to correctly approximate the

envi ronment.

(b) From (Blc) and (Bid) the mutual -impedance matrix in a three-element environment can be constructed. Results can be

compared to computed mutual -impedance matrices for various

models of interaction. By comparing the mutual -impedance data

obtained in the two cases indicated in (Bid), the relative

importance of open-circuited elements in the environment will

become apparent.

14
C. N-Element study.

In case the two- or three-element studies do not provide adequate information on convergence of elementary patterns and mutual

impedances, the measurement above should be repeated with larger

number of elements. We shall not outline here the steps involved as they are essentially the same as in (A) and (B) above.

V. Simulation Study

In our simulation study we had to satisfy three major concerns:

A. We verified that the computer codes were error free and, in fact,

performed the desired tasks. Naturally, we could not proceed without

first satisfying this requirement.

B. We ascertained that the matrix inversions required by the methods of

analysis were stable computationally, so that the results could be con- sidered reliable. To this purpose we have used the method of singularvalue decomposition [6], a powerful method of matrix inversion that automatically provides full information on the conditioning of a matrix.

C. We simulated some of the most important effects described in the previous sections. Primarily we were concerned with illustrating interaction phenomena using simple theoretical models for our array elements.

The following were studied:

(1) Two-element n-mode interactions. Here we calculated the

(a) mutual -impedance matrix as a function of interelement distance

in a two -element array,

(b) effective excitations as functions of interelement spacing

in a two-element array, and

(2) pattern synthesis and analysis for N>2 noninteracting elements.

15
In tables 1-4 (pp 24-28) we tabulated the elements of the mutualimpedance matrix for arrays of two identical elements, radiating with maximum

mode numbers nmax = 1, 2, 3, and 4, respectively. The zeroth (no multiple

reflections) and first order (only one reflection) approximations of each

quantity are also presented. (Note that the zeroth order approximation to

self impedances is 1.) In figures la, lb, and Ic the mutual impedance Z\2 and

the self-impedances z^ and Z22 together with their approximations are plotted

for n^g^ = 4, which is the most realistic case simulated. A few important

observations can be made.

1. The first-order approximations are significantly better than the

zeroth-order approximations for all but the largest separation of the

elements. This discrepancy increases with nmax. Since the zeroth-order

mutual impedance is the mutual impedance between minimum-scattering an- tennas [1], environmental effects are important even at relatively large

distances for a radiation mode as low as nmax = 4. The last entry in

table 4 will quickly substantiate this observation.

2. For small separation distances of the order of X all approximations

are largely in error, since the condition kD > nmax, defining the maximum

mode number in the antenna pattern, is almost violated. For such

separations the multiple-reflection effects due to the higher-order modes

are exaggerated.

In tables 5-8 we show a, the effective excitation vector as a function of

kD, when the two elements are excited in phase with unit amplitude. For n^^^^ = 4, the data are plotted in figure 2. The original excitation is modified by

the presence of mutual impedance and multiple reflections to result in a substantial change in amplitude and small induced imaginary components. As

expected, multiple reflections are important between closely spaced elements,

but the induced oscillations are still apparent at large separations, although

the effect diminishes.

In tables 9-12 we show the effective excitation vector as a function of

kD when the excitations of unit amplitude are 10 degrees out of phase. There

16
0.7

0.6

0.5

0.4

0.3

0.2

0.1

k

'-^^ I I I I I I I 3 5 7 9 11 13 15 17 19 21 23

kd

Figure la. The magnitude of the mutual -impedance 2^2 snd its

zeroth- and first-order approximations for n^^^ = 4

in a two-element array as a function of separation.

17
i-W

1.6

1.5

1.4

1.3

1.2

z 1.1

1.0

0.9

0.8

0.7

0.6

0.5

- z

o -o Z

11

(1)

11

I I I 3 5 7 9 11 13 15 17 19 21 23

kd

Figure lb. The magnitude of the self-impedance z^ and its firstorder approximation for n^^^ = 4 in a two-element array as a function of separation. The zeroth-order approximation is

identical ly unity.

18
1.7 r-t

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.6

22

(1)

22

0.5 ^^

N I I I I 3 5 7 11 13 15 17 19 21 23

kd

Figure Ic. The magnitude of the self-impedance Z22 and its firstorder approximation for n^^^ = 4 in a two-element array as a function of separation, the zeroth order approximation is

identical ^y unity.

19
eff

0.5 "-^

11 13 15 17 19

kd

21 23

Figure 2. The magnitude of the effective excitation vector in a twoelement array for n^^^ = 4 as a function of separation. The

real excitation is unity and is altered by mutual -impedance

effects.

20
seem to be no new qualitative features in the data when compared to the inphase simulation.

In the pattern analysis simulation study we posed the following question:

can the phased excitations of elements be recovered if both the array and element far-field patterns are known? We found that this inverse problem is

stable for exact data. The three parameters of the study were: 1) N, the

number of elements in the linear array situated on the z axis; 2) n^^^, the

maximum mode number radiated by the identical elements; and 3) NPOINTS, the

number of points where the pattern is assumed to be known and equally spaced

in the full range of 9 and (^ in spherical geometry. In table 13 we summarize

the values chosen in the parameter space. Pattern inversions were performed

for all possible combinations of values in the three dimensional parameter

space. In table 14 we present some of the condition numbers and residuals

obtained in the inversion of the pattern data as discussed in Section III.

The data clearly show that the inversion procedure is stable.

VI. Generalization to Two- and Three-Dimensional Arrays

Up to now we applied our analysis to one-dimensional arrays. The generalization to two- and three-dimensional arrays is of practical interest and

poses no new theoretical challenges or difficulties. Since the theory of

interactions has been formulated using binary interaction matrices [3], the

particular array geometry in which the two interacting elements are embedded

is of no consequence; if the free-space far-field patterns are known in frames

of reference fixed to the elements, the far-field patterns in other frames

rotated with respect to the fixed frames can be calculated according to well

known standard techniques [7]. In fact, using these ideas and a re- examination of antenna-scanning theories allows one to formulate a coordinateindependent scanning theory. This discussion is developed in somewhat more

detail in the appendix.

VII. Summary, Conclusions, and Suggestions for Further Study

In this study we have discussed methods of analysis of far-field patterns

of phased arrays both in the absence and presence of multiple reflections between elements. We have performed theoretical computations and obtained the

21
mutual -impedance matrix as a function of interelement spacing. We have

learned that the matrix inversions needed in the course of these computations

are stable as long-^ as the condition kd > n„,^ is not violated. We have also max

found that the first-order correction is an excellent approximation for the

mutual impedances. This eliminates the need for the costly matrix inversion,

unless extreme accuracy is demanded. For the self impedances, the first order

approximation was found to be inadequate. We have also calculated the effective excitation of elements as a function of position. The presence of a small induced oscillating component is observed even at large distances. Simulations to recover the element excitations from exact array far-field data

and elementary far-field data were successful.

The simulations described in this report can be considered to be phase I of a more complex study, wherein we would include

(a) more than two elements in the computation of mutual impedances and

effective excitations,

(b) the effect of multiple reflections on the elementary patterns,

and, finally,

(c) computations generalized to two- or three-dimensional arrays.

We conclude by noting that no new theoretical results are needed to ac- complish this more complex but more realistic phase II of our study.

VIII. References

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

by antennas. Ph.D. thesis. Polytechnic Institute of Brooklyn. Ann

Arbor, Michigan: University Microfilms, Inc; 1968.

[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; 1966. 83-114.

[3] Muth, L.A. A theory of mutual impedances and multiple reflections in an N-element array environment. Nat. Bur. Stand. (U.S.) Tech. Note 1078;

1985.

22
[4] Kerns, D. M. Plane wave scattering-matrix theory of antennas and an- tenna-antenna interactions. Nat. Bur. Stand. (U.S.) Monogr. 162; 1981.

[5] Hansen, R.C. Microwave scanning antennas, vol. II. New York: Academic

Press; 1966. 400 p.

[6] Golub, G. H.; Van Loan, C. F. Matrix computations. Maryland: The John

Hopkins University Press; 1983. 476 p.

[7] Edmonds, A. R. Angular momentum in quantum mechanics. New Jersey:

Princeton University Press; 1974. 146 p.

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

within the near-field region. IEEE Trans. Antennas and Propagat. AP30(1); 1982 January.

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to 1^

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00

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CM

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encn

r^ 00 en oi-H

t-H

i-H CM.-H CO

t-H t-H ID

t-H to

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t-H oCM CM

t-H CM 27
Table 5

The magnitude and phase (degrees) of the effective excitation

a for •n;<kD<7Ti for a two-element array excited with unit amplitudes

10 degrees out of phase for n^^^^ = 1.

kD ai(kD) a2(kD)

3.14

4.08

5.03

5.97

6.91

7.85

8.80

9.74

10.68

11.62

12.57

13.51

14.45

15.39

16.34

17.28

18.22

19.16

20.11

21.05

21.99 1.049 359.9

1.369 359.6

1.172 12.9

0.907 12.9

0.831 0.6

0.911 354.4

1.004 353.3

1.092 335.9

1.112 1.7

1.033 5.7

0.940 4.3

0.920 359.5

0.961 356.5

1.021 356.3

1.065 358.6

1.057 2.1

1.000 3.7

0.951 1.9

0.951 358.9

0.985 357.3

1.028 357.7

1.412 7.4

1.179 21.1

0.973 20.5

0.879 15.3

0.870 7.6

0.954 2.9

1.074 4.0

1.120 10.3

1.054 14.7

0.973 14.7

0.927 11.6

0.938 7.6

0.996 57.6

1.058 7.5

1.065 11.2

1.019 13.3

0.969 12.7

0.948 10.2

0.967 7.7

1.013 7.2

1.048 9.0

28
Table 6

The magnitude and phase (degrees) of the effective excitation

a for ix<kD<7-rt for a two-element array excited with unit amplitudes

10 degrees out of phase for n max

= 2.

kD ai(kD) a2(kD)

3.14

4.08

5.03

5.97

6.91

7.85

8.80

9.74

10.68

11.62

12.57

13.51

14.45

15.39

16.34

17.28

18.22

19.16

20.11

21.05

21.99

1.337 0.3

1.153 18.0

0.817 12.1

0.821 357.2

0.927 352.5

1.035 352.4

1.128 356.6

1.128 3.5

1.012 7.6

0.906 4.0

0.910 358.0

0.971 355.3

1.042 355.9

1.086 359.3

1.060 3.4

0.982 4.6

0.932 1.4

0.947 357.8

0.995 356.5

1.044 357.6

1.061 0.6

1.341 8.9

1.157 25.2

0.913 20.2

0.855 14.0

0.855 5.5

0.982 0.5

1.122 4.4

1.140 12.1

1.039 16.5

0.946 15.0

0.908 10.8

0.934 6.1

1.017 4.8

1.084 8.0

1.072 12.5

1.006 14.2

0.951 12.7

0.936 9.4

0.970 6.6

1.031 6.7

1.063 9.6

29
Table 7

The magnitude and phase (degrees) of the effective excitation

a for Tx<kD<7-n; for a two-element array excited with unit amplitudes

10 degrees out of phase for n max

= 3.

kD ai(kD) a2(kD)

3.14

4.08

5.03

5.97

6.91

7.85

8.80

9.74

10.68

11.62

12.57

13.51

14.45

15.39

16.34

17.28

18.22

19.16

20.11

21.05

21.99

1.453 356.0

1.094 16.8

0.955 7.6

0.898 5.9

0.873 357.1

0.975 353.7

1.057 355.6

1.098 359.7

1.050 4.2

0.972 4.1

0.935 1.1

0.952 357.6

1.004 356.7

1.045 358.2

1.052 1.0

1.012 2.8

0.970 2.0

0.957 359.8

0.978 358.0

1.015 357.9

1.037 359.4

1.441 0.7

1.153 27.04

0.985 16.61

0.953 16.44

0.865 11.52

0.933 4.28

1.028 4.5

1.096 8.0

1.070 13.3

0.995 14.0

0.951 12.3

0.938 9.0

0.983 6.7

1.033 7.5

1.056 10.0

1.027 12.5

0.984 12.4

0.960 10.8

0.966 8.6

1.001 7.7

1.032 8.8

30
Table 8

The magnitude and phase (degrees) of the effective excitation

a for 7t<kD<7-n; for a two-element array excited with unit amplitudes

10 degrees out of phase for nmax

= 4.

kD ai(kD) a2(kD)

3.14

4.08

5.03

5.97

6.91

7.85

8.80

9.74

10.68

11.62

12.57

13.51

14.45

15.39

16.34

17.28

18.22

19.16

20.11

21.05

21.99

1.298 12.6

0.938 5.6

0.747 358.3

0.925 1.7

1.019 354.7

0.983 352.1

1.147 358.1

1.085 3.3

1.026 5.7

0.903 5.0

0.928 357.2

0.978 356.4

1.027 356.1

1.088 359.4

1.042 3.2

0.990 3.7

0.936 1.7

0.954 357.8

0.997 357.1

1.035 357.7

1.058 0.5

1.338 19.4

0.998 14.6

0.829 15.9

0.951 14.0

0.986 4.7

0.902 2.0

1.160 5.8

1.093 12.4

1.044 14.6

0.953 15.6

0.911 9.6

0.954 7.0

1.001 5.5

1.088 7.9

1.054 12.5

1.009 13.3

0.958 12.8

0.940 9.1

0.978 7.3

1.021 7.1

1.061 9.6

31
Table 9

The magnitude and phase (degrees) of the effective excitation

a for icskD-i7ix for a two-element array excited with unit

amplitudes in phase for n^^^ = 1

kD ai(kD) a2(kD)

3.14

4.08

5.03

5.97

6.91

7.85

8.80

9.74

10.68

11.62

12.57

13.51

14.45

15.39

16.34

17.28

18.22

19.16

20.11

21.05

21.99

1.367 357.5

1.199 11.2

0.947 13.0

0.841 2.8

0.892 355.6

0.983 353.2

1.081 355.1

1.116 0.8

1.048 5.2

0.955 4.8

0.920 0.5

0.950 356.9

1.009 356.0

1.061 358.1

1.062 1.6

1.011 3.6

0.958 2.4

0.948 359.5

0.977 357.4

1.021 357.4

1.048 359.4

1.414 359.7

1.153 12.7

0.940 10.7

0.861 3.5

0.883 356.2

0.976 352.7

1.089 354.8

1.119 1.2

1.041 5.2

0.958 4.4

0.923 0.7

0.947 357.0

1.008 355.9

1.064 358.1

1.061 1.8

1.008 3.5

0.961 2.3

0.948 359.6

0.975 357.4

1.021 357.4

1.050 359.5

32
Table 10

The magnitude and phase (degrees) of the effective excitation

a for Ti<ikD<;7u for a two-element array excited with unit

amplitudes in phase for n^^g^ = 2

kD ai(kD)

' a2(kD)

3.14

4.08

5.03

5.97

6.91

7.85

8.80

9.74

10.68

11.62

12.57

13.51

14.45

15.39

16.34

17.28

18.22

19.16

20.11

21.05

21.99

1.335 358.3

1.189 16.1

0.863 13.1

0.817 359.9

0.900 353.3

1.012 351.9

1.119 355.6

1.135 2.5

1.033 7.2

0.922 4.9

0.905 359.1

0.960 355.5

1.030 355.4

1.083 358.6

1.068 2.8

0.996 4.6

0.938 2.2

0.940 358.5

0.984 356.5

1.037 357.2

1.062 0.0

1.341 0.9

1.123 16.9

0.874 9.7

0.845 1.7

0.876 354.0

1.009 350.8

1.135 355.5

1.134 3.2

1.021 6.9

0.930 4.4

0.908 359.7

0.948 355.6

1.031 355.1

1.088 358.7

1.065 3.1

0.993 4.3

0.943 2.1

0.940 358.7

0.981 356.4

1.040 357.1

1.064 0.2

33
Table 11

The magnitude and phase (deg^^ees) of the effective excitation

a for rt<|<0<7-: for a two-e'ement array excited with unit

anplitudes in phase for n^ = 3

kD ai(kD) a2(kD)

3.14

4.08

5.03

5.97

6.91

7.85

8.80

9.74

10.68

11.62

12.57

13.51

14.45

15.39

16.34

17.28

18.22

19.16

20.11

21.05

21.99

1.447 353.6

1.128 15.2

0.979 7.7

0.920 6.7

0.865 353.9

0.955 353.8

1.044 355.0

1.096 358.9

1.061 3.6

0.985 4.2

0.940 1.8

0.945 358.2

0.994 356.7

1.039 357.8

1.054 0.5

1.020 2.7

0.977 2.3

0.957 0.3

0.972 358.2

1.008 357.8

1.034 359.1

1.458 353.1

1.114 18.9

0.965 6.6

0.932 6.0

0.864 359.7

0.953 353.8

1.043 354.9

1.100 358.8

1.060 3.9

0.982 4.0

0.944 1.7

0.943 358.3

0.994 356.6

1.039 357.8

1.055 0.5

1.019 2.7

0.977 2.2

0.959 0.3

0.971 358.2

1.008 357.8

1.035 359.1

34
Table 12

The magnitude and phase (degrees) of the effective excitation

a for n<kD<7u for a two-element array excited with unit

amplitudes in phase for n^^g^^ = 4

kD ai(kD) , a2(kD)

3.14

4.08

5.03

5.97

6.91

7.85

8.80

9.74

10.68

11.62

12.57

13.51

14.45

15.39

16.34

17.28

18.22

19.16

20.11

21.05

21.99

1.320 10.5

0.956 6.0

0.755 2.6

0.933 2.4

1.003 354.5

0.957 352.1

1.142 357.0

1.092 2.5

1.041 5.3

0.922 5.8

0.919 358.1

0.967 356.6

1.016 355.8

1.086 358.6

1.050 2.7

1.001 3.6

0.944 2.4

0.947 358.3

0.988 357.2

1.028 357.4

1.059 0.1

1.319 11.6

0.984 4.6

0.808 2.8

0.938 3.4

1.002 354.8

0.929 351.5

1.168 357.1

1.086 3.1

1.031 5.1

0.935 5.1

0.914 358.5

0.965 356.6

1.014 355.6

1.093 358.7

1.046 2.9

0.999 3.4

0.949 2.3

0.945 358.5

0.987 357.1

1.029 357.3

1.061 0.1

35
Table 13

The parameter space in the array inversion study.

Simulations were performed for all possible combinations of the parameters

N

NMAX

NPOINTS*

2 4 10 20

1 2 3 4 10 15

Po Po+2 2po 3po

* Po = 2N+2, where N is the number of elements in the array

Table 14

Some condition numbers and residuals

obtained in the pattern-inversion simulations

N; NMAX; NPOINTS

2; 1; Po

2; 1; 3po

10; 1; Po+2

10; 10; 2po

20; 1; Po

20; 10; Po

20; 15; Po

Condition Number

1.6

1.6

2.5

2.3

2.7

2.7

2.1

Residual

^-6

2 X 10"

5 X 10"

3 X 10

6 X 10

9 X 10

1 X 10

1 X 10

-3

-4

-4

-3

-3

36
Appendix A

Coordinate Independent Formulation of Scanning Theory

The ideas presented in Section VI can be extended to formulate a coor- dinate-independent scanning theory. We achieve conceptual simplicity and

remove those features of the existing planar, cylindrical, and spherical

theories that are dependent on the choice of coordinate system in which the

measurements are taken and the data analysis is performed. What results is a formulation of antenna-antenna interactions specified in an interaction-coordinate system by the scalar distance d between the interacting antennas and by

two sets of Euler angles of rotations that describe the orientations of the

antennas with respect to initial orientations (the antenna-coordinate systems)

in which the transmitting (receiving) characteristics of the antennas are assumed known. The antenna-coordinate system is a reference frame fixed to the

antenna and moves with it as the antenna rotates. Physically, the interactioncoordinate system is the frame of reference in which the antennas interact.

Figures 3a and 3b illustrate these two coordinate systems. Figure 3a shows

the relationship of the test and probe antennas in the interaction coordinate

system. The antennas are situated at the origins of two coordinate systems

that can be brought to coincide with each other by a single translation; no rotations are involved. For simplicity we consider the antenna-coordinate

systems attached to the antennas to coincide with the set of axes shown in

figure 3a. In figure 3b the relationships between the interaction system

(broken lines) and the antenna systems (solid lines) are shown. If the

antenna characteristics are known in the antenna systems, they can be obtained

in the interaction system, as discussed in some detail below. The fact that a sequence of measurements takes place on a planar, cylindrical, or spherical

surface is an external constraint imposed by the needs of the experimentalist,

that is, by the technological limitations of the data gathering instrumentation and the efficiency requirements of the numerical inversion process. For

the moment we suspend these practical considerations to present an interaction

formulation of antenna scanning theory. As we shall see current scanning procedures can be easily incorporated into the interaction point of view.

37
Figure 3a. The interaction-coordinate system in which the interaction

between single-mode antennas is evaluated.

/

0^

Figure 3b. The interaction-coordinate system embedded in the usual

coordinate system of planar scanning. The antenna's

coordinate system moves with the antenna as it is rigidly

attached to it.

38
We can easily prove the following theorem:

A single measurement of a test antenna's performance at a given point in space

a distance d from the probe can be represented as

m = e^ ^(d) t, (Al)

where m (a complex scalar)- is the result of measurement registered by the

probe, £_ and t_ are column vectors of probe and test antenna coefficients,

respectively, and x(d) is the mode-mode coupling matrix, whose elements are coupling integrals evaluated for single-mode interactions (see eq (A6)

below). All quantities in eq (Al) are evaluated in the interaction coordinate

system. It is relatively straightforward to evaluate the matrix elements of

X, as discussed in [1,3]. In principle, x(d) is infinite dimensional; in

practice, however, only a finite number of mode coefficients will be specified

in _2_ and t_.

In the antenna coordinate systems, the probe and test antenna far-field

patterns f_ are given by

fjp) =ee,

and (A2)

respectively, where e_is an infinite dimensional column vector of spherical

far-field basis functions. Then the dot product

t; • it - i sijt (A3)

where |. . are matrix elements defined as

With these definitions the coupling integral,

2% I ^ -ik'd

J / f . f e " " d (cose) d4), (A5)

i" "^

describing antenna-antenna interactions [1,3] in spherical coordinates using

spherical basis functions can be written as in eq (Al), where

39
2u I -i k 'd

X^ • = / ./ lij e " " d(cose) d(t) (A6)

i"

which can be integrated exactly in the interaction coordinate system wherein

the antennas are always on the z axis. If the antenna-coordinate systems do

not coincide with the interaction coordinate system, the far-field coefficients in the interaction coordinates are given by

2^^^ = D (apy) B° (A7)

where g° are the coefficients specified in the antenna frame, and D (apy)

represents a finite rotation of Euler angles a,p,Y [7] between the antenna and

interaction frames. A similar expression holds for t .

A sequence of measurements m = (m^ ,m2, . . .m. , . . .m ) can then be represented as

m = P t°, (A8)

where the rows of the matrix P are given by

e°"^ g^Qp) x(d) D(Q^), (A9)

where Q (Q.) are Euler angles of rotation of the probe (test) antenna. The

far-field coefficients of the test antenna in the test antenna frame are then

obtained by taking the inverse,

tO = P"-^ m, (AlO)

if P exists.

Up to now no restrictions have been made as to how the sequence of mea- surements are to be made. If we wish to restrict ourselves to spherical

scanning, then d^- = d, and the Euler angles corresponding to each measurement

can be easily obtained in spherical geometry. For planar and cylindrical

scanning, d^- is not constant; again the Euler angles can be easily obtained.

We illustrate these ideas with a simple example. Let the test and probe

antennas be the set of dipoles

40
E° = (Pi Po Pi)

(All)

tj = (ti to ti)

giving in spherical coordinates the m = 1, 0, -1 components and, of course,

n = 1. We will make measurements along an arc, i.e., d^ = r, the radius of

the arc, such that only rotations about the x-axis will be needed to align the

antenna coordinates with the interaction coordinates. The rotation matrix

d\^(p) is then given by [7]

-2(1+ cosp) /J sinp 2 ^^ ' ^°^P)

^i'l^P^ = |-/^sinp cosp ^sinp j , ^^^2)

-2(1- cosp) -/J sinp 2" (^ "^ cosp)

and the interaction matrix is diagonal (no interaction between crossed

dipoles)

(A13)

with Ii(r) t Io(r).

The rows of P are then given by

If we make measurements at p = 0, p = 5- and at an arbitrary p, we get that

•k '^•k "k 'liPi loPo UPi

P(P;r) = I Iipt IiPo Iipt 1 , (A15)

.qi(P) q2(P) q3(P)

41
Po

where Po = -^ and

/2

qi(p;r) = IiPi + po (Ii - Iq) cosp sinp

^k

q2(P;r) = Po (Ii sin^p + Iq cos^p) (A16)

•k ^-k q3(P;r) = Pill - Po (Ii - Iq) cosp sinp.

For P" (p;r) to exist we must have det P # 0. One can show that

"] -k -k -k det P = - "2 Pi Po Po Ii (lo - Ii)^ cosp sinp * (A17)

except at p = 0, -^, tc, where measurements were already taken. Thus the third

measurement can be at any angle sufficiently distinct from previous measurement points, and the probe-interaction matrix P can be inverted and to

recovered.

For completeness we conclude by briefly commenting on the transformation

of the coupling integral expressed in Cartesian geometry as in [4] to the

spherical coordinate form as seen in eq (A5). Once the spectral quantities

are written in terms of far fields, the spherical representation can be

obtained by a simple coordinate transformation

k = k si ne cos(}) X

k = k sine sin<{) , k = constant

y

(A18)

whose Jacobian is k^ cosG sine, i.e.,

dk dk = k2 cose sinG ded(j) X y or (A19)

dk dk = - Ydydft) X y ' ' "^

where y = k cosG. A more detailed derivation of eq (A5) will show that the

factor Y in the Jacobian cancels due to the transformation to far-field

quantities using the reciprocity relationships (see [8]).

42
NB$-1T-4A JREV. 2-BC) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN-1091

2. Performing Organ. Report No. 3. Publication Date December 1985

4. TITLE AND SUBTITLE

Interelement Interactions in Phased Arrays: Theory, Methods of Data Analysis

and Theoretical Simulations

5. AUTHOR(S)

Lorant A. Muth

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

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

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10. SUPPLEMENTARY NOTES

[2^ 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)

We review theoretically the effects of multiple reflections and

mutual impedances in array environments and study possible methods of

far-field pattern data analysis to recover interaction effects. We use theoretical expressions derived earlier to calculate in a twoelement linear array the mutual -impedance matrix and effective

excitations of elements as functions of interelement separation

and n^^^j^, the maximum mode number in the radiation pattern of the

elements. Generalizations to two- and three-dimensional arrays are

di scussed.

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

data analysis; effective excitations; interelement scattering; multiple reflections;

mutual impedance; phased arrays.

13. AVAILABILITY

Q^ Unlimited

I I For Official Distribution. Do Not Release to NTIS

[^ Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

1^^ Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

48

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USCOMM-DC 6043-P80

U.S. GOVERNMENT PRINTING OFFICE: 1985—677-215/45073 REGION NO.

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