Technique for Extension of Small Antenna Array Mutual ...

National Aeronautics and Space AdministrationLangley Research Center • Hampton, Virginia 23681-0001

NASA Technical Paper 3603

Technique for Extension of Small AntennaArray Mutual-Coupling Data to LargerAntenna ArraysM. C. BaileyLangley Research Center • Hampton, Virginia

August 1996

Printed copies available from the following:

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Summary

A technique is presented whereby the mutual inter-action between a small number of elements in a planararray can be interpolated and extrapolated to accuratelypredict the combined interactions in a much larger arrayof many elements. An approximate series expression isdeveloped, based upon knowledge of the analytical char-acteristic behavior of the mutual admittance betweensmall aperture antenna elements in a conducting groundplane. This expression is utilized to analytically extendknown values for a few spacings and orientations to otherelement configurations, thus eliminating the need tonumerically integrate a large number of highly oscillat-ing and slowly converging functions. This paper showsthat the technique can predict very accurately the mutualcoupling between elements in a very large planar arraywith a knowledge of the self-admittance of an isolatedelement and the coupling between only two elementsarranged in eight different pair combinations. Theseeight pair combinations do not necessarily have to corre-spond to pairs in the large array, although all of the indi-vidual elements must be identical.

Introduction

The electromagnetic performance (e.g., impedance,radiation pattern, and scanning characteristics) of anarray antenna sometimes is influenced significantly bythe combined mutual interaction of the local fields of thearray elements. A good antenna design would include anaccurate account of these interaction effects.

The radiation characteristics of an array antenna canbe obtained from a knowledge of the properties of asingle element, the array geometry, and the combinedinteraction of all elements of the array. The combinedmutual interaction of all elements can be obtained fromthe scattering matrix in which the size of the matrix isdetermined by the number of elements in the antennaarray (e.g., an antenna containingN elements would berepresented by anN × N scattering matrix). The complexcoefficients of the scattering matrix can be obtained fromthe complex admittances of all element pair combina-tions of the antenna array, either by calculations for ele-ments that can be adequately modeled or bymeasurements for more complicated elements.

Since these interactions are typically a near-fieldphenomenon, a theoretical formulation will usuallyinvolve the numerical evaluation of quite complicatedintegrations, resulting in significant computational timefor an array of many elements. The measurement ofmutual interactions between all pair combinations withina large array can also become very laborious and timeconsuming. A technique is presented that utilizes the

characteristic analytical behavior of the couplingbetween two elements, in conjunction with known valuesof coupling for a few configurations, to accurately pre-dict the mutual coupling for any configuration of theseelements in an array.

Method

The mutual admittance between two copolarizedapertures in a common, flat, perfectly conducting groundplane is postulated as

(1)

The geometry of two rectangular apertures is shown infigure 1, whereR is the center-to-center spacing of theapertures,φ is the relative angular orientation of the aper-tures, andk = 2π/λ, with λ being the operating wave-length in free space. The polarization direction of theapertures is assumed such thatφ = 0 represents couplingin the magnetic field direction (i.e.,H-plane) andφ = π/2represents coupling in the electric field direction (i.e.,E-plane).A1, A2, …, A8 are complex coefficients to bedetermined from known values ofY12 for pairs of aper-tures with known relative positions.

The reasoning that influenced the postulated form ofequation (1) follows:

(a) Previous observations have shown that the mutualcoupling between widely spaced apertures in acommon ground plane varies inversely as the sepa-ration distance in theE-plane and inversely as thesquare of the separation distance in theH-plane,with a phase variation that is linear with separation.

(b) The fields near a radiating source vary inversely asthe cube of the distance away from the source.

(c) For a fixed separation, the mutual coupling variesin a sinusoidal manner with relative angularposition.

These basic analytical properties are also illustratedby the closed-form expression for mutual admittancebetween dominant-mode circular apertures (ref. 1). Itis anticipated that the mutual admittance between anyotherpair of apertures would exhibit similar character-istics versus separation and angular position; thus the

Y12 A11

kR------

2A2

1kR------

3+ cos2φ

=

A31

kR------

A41

kR------

2A5

1kR------

3+ + sin2φ+

A61

kR------

A71

kR------

2A8

1kR------

3+ + sin22φ

e jkR–+

2

postulated expression in equation (1) is expected to applyto a wide variety of elements in planar arrays. However,the coefficients in equation (1) must be determined foreach unique type of element pair.

In postulating the form of equation (1), the sinusoi-dal functions were chosen such that theH-plane andE-plane mutual-coupling values could be determinedindependently. If one knew the complex mutual admit-tance between two elements for two spacings in theH-plane (φ = 0), the coefficientsA1 and A2 could bedetermined. Likewise a knowledge of the mutual admit-tance for three spacings in the E-plane(φ = π/2) wouldbe adequate for determining the coefficientsA3, A4,andA5. The first five terms are usually sufficient forapproximating the mutual admittance for other elementpair combinations because of the cylindrical interpola-tion for a specified value ofR by the sin2φ and cos2φ fac-tors. The inclusion of three additional terms (withmultiplying factor sin22φ) yields even more accurateresults for values of 0< φ < π/2. Eight distinct element-pair combinations are used to establish a set of eightindependent equations to be solved for the eightunknown coefficients by matrix inversion. These paircombinations do not necessarily have to include theH-plane andE-plane, although these would be logicalchoices.

In obtaining sample data for the evaluation of coeffi-cients in equation (1), some rather loose (althoughlogical) restraints on the selection of element-pair combi-nations should be observed in order to obtain realisticand accurate interpolation/extrapolation values. No morethan three element spacings for the same value ofφshould be used for sample data (and no more than twosample spacings forφ = 0). Sample data for very closespacing (where the dominant terms are 1/R3), sampledata for large spacing (where the dominant termsare 1/R), and sample data for intermediate spacingsshould be used.

The coefficients in equation (1) could be determinedby either calculations or measurements. Calculations areappropriate for elements that can be adequately modeledanalytically, such as waveguide-fed apertures or cavity-backed slots. However, more complicated elements donot lend themselves easily to straightforward analyses;therefore, a measurement approach may be more appro-priate in some cases.

If measurements are used to determine the coeffi-cients, the simplest approach is to measure the reflectioncoefficient and coupling coefficient (i.e., scatteringmatrix) for a pair of elements (i.e., a two-element array)for various spacings and positions. The mutual admit-

tances can then be determined from the relationshipbetween the admittance and scattering matrices.

(2a)

(2b)

where [S] is the complex scattering matrix, [I ] is theidentity matrix, [y] is the complex normalized admit-tance matrix, and []−1 indicates matrix inversion. Thecoefficients of the admittance matrix in equations (2a)and (2b) are normalized to theith individual modal-characteristic admittance for a waveguide of the samecross-sectional dimensions as theith aperture,yij = Yij /Yi,whereYi is the modal-characteristic admittance for theith aperture and where subscriptsi and j refer to theithrow andjth column of the matrix. The measured mutualadmittance between two identical apertures can then bedetermined from

(3)

whereS11 andS12 are the measured complex reflectionand coupling coefficients, respectively, of the two-element array.

Results

For purposes of this paper, calculations based uponthe numerical evaluation of integral expressions for cir-cular, rectangular, and square apertures are used to deter-mine the coefficients in equation (1) and to evaluate theaccuracy of the utility of equation (1). The integralexpressions for mutual admittance of circular and rectan-gular apertures can be obtained from the general formu-lation in reference 2. The computer program used forcircular aperture results is documented in reference 3,and the computer program used for rectangular andsquare apertures is documented in reference 4.

Figures 2 and 3 show the arrangements of elementsused for determination of the complex coefficients inequation (1) for rectangular and circular apertures. Thegeometries for figures 2 and 3 were selected to satisfytheloose constraints mentioned earlier. These arrange-ments were found to give very accurate interpolated/extrapolated results. It was also found that these specificarrangements were not critical to obtaining accurateresults. Some reasonable deviations from these geome-tries yielded insignificant deviations in the final results,provided that the loose constraints were met. The mutualadmittance between the element at the origin and each ofthe other eight elements was calculated from the respec-tive computer programs, RWG and CWG, by numericalintegration. These calculated values were used to solve

S[ ] I[ ] y[ ]–[ ] I[ ] y[ ]+[ ] 1–=

y[ ] I[ ] S[ ]–[ ] I[ ] S[ ]+[ ] 1–=

Y12

Y1-------- y12

2S12–

1 S11+( )2 S122–

---------------------------------------= =

3

the corresponding set of eight simultaneous equations forthe coefficients. These coefficients were then used inequation (1) to predict the mutual admittance betweentwo elements for a wide range of spacings and orienta-tions, and comparisons were made with correspondingresults obtained by direct numerical integration.

Figures 4 through 9 show a comparison betweenrectangular aperture mutual admittance obtained byequation (1) and the values obtained by numericalintegration. The sample data points are those used forinterpolation/extrapolation. These data show both thereal and imaginary parts of the complex mutual admit-tance between two identical apertures as a function of theaperture spacing at specific orientation anglesφ. Fig-ures10 through 14 show the results for circular aper-tures. It is readily evident that very accurate results canbe obtained for any orientation and spacing of apertureswith eight sample data points.

Interpolated/extrapolated results obtained from fivesample data points (H-plane andE-plane only and coeffi-cients A1 through A5 in eq. (1)) for circular apertureswere indistinguishable from those in figures 10 through14. A very slight, although still insignificant, deviationwas observed in theφ = π/4 plane results when usingonly the five sample data points for rectangular aper-tures; therefore, use of all eight coefficients is recom-mended if sufficient sample data are available.

Figures 15 through 18 show a comparison betweeninterpolated/extrapolated mutual-coupling results withthose obtained by direct numerical integration. Veryaccurate results are indicated for extrapolation to spac-ings of 15 wavelengths and greater and for very low cou-pling levels, although the sample data spacing waslimited to about 5 wavelengths or less, implying that thetechnique could be used for very large arrays of manyelements. The results using direct numerical integrationfor the square apertures exhibits an oscillatory behaviorfor wider spacings, which indicates possible errors in thecalculations. This accurate extrapolation to large spac-ings was indeed verified in the expanded plots in fig-ures19 through 22, where the computer codes were mod-ified for improved accuracy in the numerical integration,with a corresponding increase in computation time. Withimproved numerical accuracy, the amplitude of the oscil-lations became smaller and closer to the extrapolateddata. These results illustrate an important aspect of thepresent technique. As the spacing between array ele-ments increases, the numerical-integration approach

becomes less accurate and more computationally inten-sive; however, the algebraic extrapolation techniquemaintains extreme accuracy without increasing computa-tion time.

Concluding Remarks

An algebraic interpolation/extrapolation techniquefor predicting the mutual interaction between elements ofa large planar phased-array antenna has been presented.The technique has been shown to be extremely accuratealthough it only uses eight sample data points. Thetechnique yielded very accurate results for large ele-mentspacings and low coupling levels, as opposed tonumerical-integration techniques that exhibit oscillatoryerrors and large increases in computation time. Althoughthe technique has been verified only for dominant-modecircular and rectangular aperture elements andtheinterpolation/extrapolation equation may take on aslightly different form for other elements or apertureswith different modes present, this general approachshould be applicable to a wide variety of array elementtypes. The present extrapolation technique may also pro-vide significant advantages over methods of direct mea-surement of coupling between widely spaced elementsbecause extreme care must be exercised in order toobtain accurate measured results for low coupling levels.An attractive feature of the present technique is that itutilizes and extrapolates from data for strongly coupledelements that are closely spaced, the very situation wheremeasurement methods or numerical integration methodscan provide it with accurate sample data.

NASA Langley Research CenterHampton, VA 23681-0001May 21, 1996

References1. Bailey, M. C.:Closed-Form Evaluation of Mutual Coupling in

a Planar Array of Circular Apertures. NASA TP-3552, 1996.

2. Bailey, M. C.:Analysis of Finite-Size Phased Arrays of Circu-lar Waveguide Elements. NASA TR R-408, 1974.

3. Bailey, M. C.: CWG: A FORTRAN Program for MutualCoupling in a Planar Array of Circular Waveguide-FedApertures. NASA TM-101614, 1989.

4. Bailey, M. C.: RWG: A FORTRAN Program for MutualCoupling in a Planar Array of Rectangular Waveguide-FedApertures. NASA TM-4723, 1996.

4

Figure 1. Geometry of two identical apertures in ground plane.

E

R

y

x

φ

Figure 2. Rectangular element arrangement for determination of admittance coefficients.

x

ya

b

1.5a

2.5a

5a

1.5b

2.5b

5b

5

Figure 3. Circular element arrangement for determination of admittance coefficients.

x

y

d

1.5d

2.5d

5d

1.5d

2.5d

5d

Figure 4. Mutual admittance between two rectangular apertures (0.8λ × 0.4λ). φ = 0.

.00025

0

–.00025

–.000500 1 2 3 4 5

Center-to-center spacing, λ

Mut

ual a

dmitt

ance

, mho

s

Numerical integration

Sample points

Interpolation (real)

Interpolation (imaginary)

6

Figure 5. Mutual admittance between two rectangular apertures (0.8λ × 0.4λ). φ = π/8.

.00025

0

–.00025

–.000500 1 2 3 4 5


Mut

ual a

dmitt

ance

, mho

s




Figure 6. Mutual admittance between two rectangular apertures (0.8λ × 0.4λ). arctanφ = 1/2.

.00050

.00025

0

–.00025

–.000500 1 2 3 4 5


Mut

ual a

dmitt

ance

, mho

s


Sample points



7


.00050

.00025

0

–.00025

–.000500 1 2 3 4 5


Mut

ual a

dmitt

ance

, mho

s




Figure 8. Mutual admittance between two rectangular apertures (0.8λ × 0.4λ). φ = 3π/8.

.00050

.00025

0

–.00025

–.00075

–.00050

0 1 2 3 4 5Center-to-center spacing, λ

Mut

ual a

dmitt

ance

, mho

s




8


.00050

.00025

0

–.00025

–.00075

–.00050


Mut

ual a

dmitt

ance

, mho

s


Sample points



Figure 10. Mutual admittance between two circular apertures (0.6λ diameter).φ = 0; H-plane.

.0003

.0002

.0001

0

–.0001

–.0002

–.0003

–.00040 1 2 3 4 5


Mut

ual a

dmitt

ance

, mho

s


Sample points



9

Figure 11. Mutual admittance between two circular apertures (0.6λ diameter).φ = π/8.

.0003

.0002

.0001

0

–.0001

–.0002

–.0003

–.00040 1 2 3 4 5


Mut

ual a

dmitt

ance

, mho

s




Figure 12. Mutual admittance between two circular apertures (0.6λ diameter).φ = π/4.

.0003

.0002

.0001

0

–.0001

–.0002

–.0003

–.00040 1 2 3 4 5


Mut

ual a

dmitt

ance

, mho

s


Sample points



10

Figure 13. Mutual admittance between two circular apertures (0.6λ diameter).φ = 3π/8.

.0003

.0002

.0001

0

–.0001

–.0002

–.0003

–.00040 1 2 3 4 5


Mut

ual a

dmitt

ance

, mho

s




Figure 14.Mutual admittance between two circular apertures (0.6λ diameter). φ = π/2; E-plane.

.0003

.0002

.0001

0

–.0001

–.0002

–.0003

–.00040 1 2 3 4 5


Mut

ual a

dmitt

ance

, mho

s


Sample points



11

Figure 15. Mutual coupling between two circular apertures (0.6λ diameter).

–20

–25

–30

–35

–40

–45

–50

–55

–60

–65

–70

–750 321 654 10987 14131211 15


Mut

ual c

oupl

ing,

dB


Sample points

Interpolation and extrapolation

Circular apertures (0.6λ dia.)

φ = π/2

φ = π/4

φ = π/8

φ = 0

Figure 16. Mutual coupling between two circular apertures (0.8λ diameter).

–25

–30

–35

–40

–45

–50

–55

–60

–65

–70

–80

–75


Mut

ual c

oupl

ing,

dB


Sample points


Circular apertures (0.8λ dia.)

φ = π/2

φ = π/4

φ = π/8

φ = 0

12

Figure 17. Mutual coupling between two rectangular apertures (0.8λ × 0.4λ).

–10

–15

–20

–25

–30

–35

–40

–45

–50

–55

–60

–65

–75

–70


Mut

ual c

oupl

ing,

dB


Sample points


Rectangular apertures (0.8λ × 0.4λ)

φ = π/2

φ = π/4

φ = π/8

φ = arctan(1/2)

φ = 0

Figure 18. Mutual coupling between two square apertures (0.6λ × 0.6λ).

–15

–20

–25

–30

–35

–40

–45

–50

–55

–60

–65

–700 321 654 10987 14131211 15


Mut

ual c

oupl

ing,

dB


Sample points


Square apertures (0.6λ × 0.6λ)

φ = π/2

φ = π/4

φ = π/8

φ = 0

13

Figure 19. Amplitude ofH-plane coupling between two square apertures.

–60

–61

–62

–63

–64

–65

–66

–67

–68

–69

–707.5 8.0 8.5 9.0


Mut

ual c

oupl

ing,

dB


Numerical integration (improved accuracy)

Extrapolation

Square apertures (0.6λ × 0.6λ) H-plane coupling (φ = 0)

Figure 20. Phase ofH-plane coupling between two square apertures.

15

10

5

0

–5

–10

–157.5 8.0 8.5 9.0


Phas

e re

lativ

e to

ext

rapo

late

d da

ta, d

eg



Square apertures (0.6λ × 0.6λ) H-plane coupling (φ = 0)

14

Figure 21. Amplitude ofE-plane coupling between two square apertures.

–45

–46

–47

–48

–49

–5012.5 13.0 13.5 14.0


Mut

ual c

oupl

ing,

dB



Extrapolation

Square apertures (0.6λ × 0.6λ) E-plane coupling (φ = π/2)

Figure 22. Phase ofE-plane coupling between two square apertures.

15

10

5

0

–5

–10

–1512.5 13.0 13.5 14.0


Phas

e re

lativ

e to

ext

rapo

late

d da

ta, d

eg



Square apertures (0.6λ × 0.6λ) E-plane coupling (φ = π/2)

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August 1996 Technical Paper

Technique for Extension of Small Antenna Array Mutual-Coupling Data toLarger Antenna Arrays WU 505-64-52-04

M. C. Bailey

L-17553

NASA TP-3603

A technique is presented whereby the mutual interaction between a small number of elements in a planar array canbe interpolated and extrapolated to accurately predict the combined interactions in a much larger array of many ele-ments. An approximate series expression is developed, based upon knowledge of the analytical characteristicbehavior of the mutual admittance between small aperture antenna elements in a conducting ground plane. Thisexpression is utilized to analytically extend known values for a few spacings and orientations to other element con-figurations, thus eliminating the need to numerically integrate a large number of highly oscillating and slowly con-verging functions. This paper shows that the technique can predict very accurately the mutual coupling betweenelements in a very large planar array with a knowledge of the self-admittance of an isolated element and the cou-pling between only two elements arranged in eight different pair combinations. These eight pair combinations donot necessarily have to correspond to pairs in the large array, although all of the individual elements must beidentical.

Antennas; Phased arrays; Mutual coupling 15

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