Null Steering and Maximum Gain in Electronically Scanned ...and places a number of independent nulls in the radiation pattern has already been presented. The elements used were idealized

AFCRL.72-00831 FEBRUARY 1972PHYSICAL SCIENCES RESEARCH PAPERS, NO. 478

(MICROWAVE PHYSICS LABORATORY PROJECT 5635

AIR FORCE CAMBRIDGE RESEARCH LABORATORIESL. G. HANSCOM FIELD, BEDFORD, MASSACHUSETTS

Null Steering and Maximum Gain inElectronically Scanned Dipole Arrays

C.J. DRANE, JR.J.F. Mc ILVENNA

Approved for public release; Jistribution unlimited._

AIR FORCE SYSTEMS COMMANDUnited States Air Force

UnclassifiedSecurity Classificaio

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Air Force Cambridge Research Laboratories (LZ) UnclassifieaL. G. Hanscom FiEl 2k GROUP

Bedford, Mass. 0!730s REPORT TITLE

NULL S'IEERING At D MAXIMUM GAIN IN ELECTRONICALLY SCANNEDDIPOLE ARRAYS

4" OESCRIPTIVE NOTES (Type of OW rmid aadma .•lnue =sJ

Station ReportS AUTNORIS) (FrWI 109. frid&a ama. ;4814001)

C.J. Drane, Jr.J. F. McIlvenna

W. REPORT DATE 7. TOTAL NO OF PAGES 7& NO. OF REFS

1 Febraary 1972 38 24SA, CONTRACT OR GRANT NO. go. ORIGINATOR*S REPORT MUMIEI S)

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A DOo SUMELEMENT 681305

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Approved for public release; distribution unlimited

II. SUPPLEMENTARf NOTES ILSPONISORING MILITARY ACTIVITYAir Force Cambridge ResearchI Laboratories (LZR)

TECHOTHERL.G. Hansoom FieldBedford, Mass. 01730

11. ABSTRACT

A technique that simultaneously maximizes the gain of an antenna arrayand places a number of independent nulls in the radiation pattern has alreadybeen presented. The elements used were idealized point sources. In thepresent paper, the previous analysis is extended to cover arrays of thin wireelements in which interelement mutual coupling and scanning effects areinclud( d. Sample calculations and computed radiation patterns demonstratethat an appreciable amount of pattern and sidembe control can be obtainedwith on;y a small sacrifice in gain.

DD, FORM 1473INOV 65 Unclassified

Scurity Classification

UnclassifiedSerity Casification

tWO LINK A LINK 9 LINK CKEY IV)IOR I

11_1.9 " WT ROlE WT ROLE WT

Antenna gainMaximum directivity or gainMutual couplingAntenna arraysRadiation pattern nullsSidelobe control

Unc assifiedSecurity Clussi ficstion

Abstract

A technique that simultaneously maximizes the gain of an artenna array and

places a number of independent nulls in the radiation pattern has alruady becn

presented. The elements used were idealized point sources., In the present paper,

the previous analysis is extended to cover arrays of thin wire elements in which

interelement mutual coupling and scanning effects are included. Sample calcula-

tions and computed radiation patterns demonstrate that an appreciable amount of

pattern and sidelobe control can be obtained with only a small sacrifice in gain.

r iii

Contents

1. INTRODUCTION 1

2. DESIGN TECHNIQUES FOR ARRAYS OF WIRE ELEMENTS 2

3. THE CONSTRAINT METHOD 7

4. SOME EXAMPLES 9

5. CONCLUSIONS

ACKNOWLEDGMENT 21

REFERENCES 23

APPENDIX A: Derivation of a Relation for tne Directive Gain of anArray of Electrically Thin, Straight Wires With MutualCoupling Effects Included 25

Illustrations

1. Broadside Beamp 102. Resultant, Patterns for Two-Constraint Case Plotted in 100

Increments in Scan Angle 11

3. Beam Scanned to 1000 14

4., Beam Scanned to 1100 14

V

Illustrations


6. Beam Scanned to 1i 0 0 15






A1. 12-Element Array, 5-Segments per Element 29

Tables

1. Comparison of Unconstrained (U), One Constraint (C 1) and -woConstraint (C2) Array Properties 12

VI

Null Steering G.id Maximum Gain in

Electronically Scanned Dipole Arrays

1. INTRODUC.TION

In a recent paper (Drane and Mcllvrnna, 1970), the authors presented a

technique for maximizing the gain of ar. antenna array, wP ile at the same time

placing one or a number of independent nulle andfo" a'-'-lobes in the far-field

radiation pattern. * Some p:oblem F.reas requiring such a t.ehnique include the

elimination of jammer interference in radio-racdr comm 'imtim ns links, the re-

duction of ground reflections in sited antennas and th. minimizaticn of inter-

antenna interference effects in multi-antenna enviroixnents. The vimple computa-

tions in that earlier paper were restricted to arrays Lf idealized nn,'enna elements,

that is, isotropic radiators. It was pointed aut there, hcýwever, that the "maximum

gain - null placin" .--chnique would be •pp .icable even with arr•-c of e,,ýments

(dipoles, for example) when intdrelernent mutual coupling effects must, of neces-sity, be considered. Since that time, a particular method of design for arrays ofwire elements (Str. it and Hirasawa, 1968 a.id 1960) which elegantly and %.ractically

accounts for mutual coupLing effects has been .ombined with the afore.en'.ioned

maximum gain - constraint technique. These design methods are di,,ouafed belbw

together with some examples of practical interest.

(Received for publication 31 Ja-iuary 1972)*Several papers on this and closely related subjects have apr.eared (Nemit, 1969;Sandrin and Glatt, 1970; Riegler and Compton, 1970; Pierce. 1970; Hesse] andSureau, 1971; Adams and Strait, 1970; Sanzgiri, etal, 197 1).

2

2. DESIGN TECHNIQUES FOR ARRAYS OF WIRE ELEMENTS

The method summarized below is due ta Strait and Harrington and has been

reported in detail by Strait and Adams (1970), Chao and Strait (197 1), Strait and

Hirasawa (1970a), Hirasawa and Strait (197 la, 197 lb), and Strait and Chao (1971).

With it, the designer can determine such properties as the actual current distribu-

tions along the antenna elements, input and driving-point impedances and the radi-

ation and scattered field patterns oi arrEa,,s of parallel, staggered or even arbitrarily

bent wireE., The elements may be arbitrarily spaced, loaded or unloaded, lossy or

lossless, center-fed or arbitrarily excited at points along their length, and they

may be placed in any geometric arrangement. Even radiating Alements with wire

junctions such as crossed dipoles (Chao and Strait, 1971) can be handled. Most

importantly, in every case mutual coupling effects are included. Detailed handbook-

typ( -- ,ports with computer programs for a wide variety of array design problems

are available by Strait and Hirasawa (1969b, 1970b), Chao and Strait (1970), and

Mautz and Harrington (1971). Only the highlights of the technique are presented

here; they lead to aa expression for array gain with mutual coupling effects included.

Starting froni the integral equation for a thin wire, one considers each antenna

in the array to be subdivided into a number of segments, K (ten per wavelength has

proven to be sufficient for far-field considerations), v- ose end-points are treated

as the terminals of a multi-terminal pair network. The goal is to relate every

terminal pair to every other terminal pair through a mutual impedance matrix.

•The two boundary conditions, that the tangential component of the electric fieldI vector E must be zPro on the surface of each conducting element and that the axial

current must go to zero at the ends of the E itenna, are included in the analysis.

Introducing column vectors I and V, each with K components that ai-e respec-

tively the segment currents and voltages,

• • and

VT

where superscript T denotes the vector transpose operation, allows one to repre-

sent the array problem in terms of circuit theory format, that is,

V ZI,

3

where Z, a (K x K) - element, symmetric complex matrix, is the mutual impedance

matrix relating the voltage and current for any segment to the voltage and current

for every other segment.

Methods for calculating the elements in Z for any array geometry are available

(Strait and Hirasawa, 1969a). It has been uhown that

Zkn = jW Mo A0 n '1 k 4, (nk)

+ I•-- [,p(n', k+) - 0b (n- k'- (n+, k) + •b(n-, k')]

where

k) f exp [ -j 3R kn(Z')•b(n,k) ff 4i~ljA R(Z) dz',

'&nJAf R k(z')n

and 1n is the length of the nth segment, Rkn Z4) is the distance between the kth

and nthnsegments and the superscripts + and - (in n+ and n-), denote the starting

point and termination, respectively, of the nth segment. The quantity u, is the

operating angular frequency, 0 is the corresponding wave number, and Eo and 1o

are the universal electric and magnetic constants. The only other approxima-

tions made in this analysis are that the current and charge are assumed to be con-

stant over the length of each segment. Calculations and comparisons with other

mutual coupling methods have shown that such assumptions are not overly restric-

tive (Strait and Hirasawa, 1969a). Resistive or reactive loading and element

losses can be included simply by adding a load matrix Z to the Z matrix defined

above (Hirasawa and Strait, 197 1a).

Given an array geometry and a set of driving voltages, the segment currents,

w .th all mutual effects included, are found quite simply from

I V= 'Iv ='V

where Y is the (KXK) admittance matrix. With these segment currents, all the

parameters of interest to the array designer can be obtained. For example, the

far-zone vector potential for z-directed current elements has only a z-component

given by (Strait and Hirasawa, 1967a)

KS(0,) p[-jr] exp[j(x sinecos+Sin00)],

4 7r n=l (I)

4

and the far-field amplitude pattern is

In thase expressions xn, Yn and zn are the Cartesian coordinates of the center ofthe nth segment while r. 0 and 0 denote the polar coordinates of the far-field

observation point.

A non-restrictive assumption that simplifies computations is to take all seg-ment lengths to be equal. Defining an angle-dependent row vector F(0, 0) with K

elements, that is,

FIn(0.) =exp[j (x nsin0co,,k +Ynsin0sin +zncos 0) 1--5n-K,

and replacing I by W leads to a matrix form for Az0 viz.,

Az (, iK° [F(8,0)YV], (2)

where

K0 = fxp[-jrl ,4Wr

where 6 is the length common to all segments. One need only use the relations

above ana some standard definiticns to develop an expression f.:r antenna gain

which includes the effects of mutual couplitg (Strait and Hirasawa, 1960a).

Recall that power gain for an. antenna is defined to be

G = 4 I !wer radiated in a particular direction (3)to-fTjweFr input TSi array

The power radiated in a particular direction (00, j0) is .:-1ated to the far-field

amplitude pattern byr2/E (0 0)/2

P (o0 = r / Z %.no i

where 17 is the impedance of free space, In terms of the inatrix expressions

above,

5

P(OO / [a+ (- )+ (FY)V1 sin2 0,

where superscript + denotes the combined operations of complex conjugation and

transposition. The denominator term in Eq. (3) can be handled by multi-port

circuit theory techniques and is expressible in terms of the D driving-point voltRges

vd and currents id a, fStrait and Hirasawa, 1969a)

D

P = Real (vdid*).d=l

Let ID and VD represent D-element vectors of the driving-point currents and

voltages. In vector notation

FT - Real (VD ID)

Let YD represent the (D x D) complex driving-point admittance matrix formed

from Y by selecting only those elements that correspond to driving points (Strait

and Hirasawa, 1969a). Because of the symmetry in YD,

PT = VD (Real (YD)) VD

and Eq. (3) becomes

V+ TFi)•(•) V(4

G =K VD (FR W (4

D(Real(YD))VD

where

KI = sin 20 (A 0°611 4 7rwo

the numerator matrix (F)+ (FY) is a (K x K)-element matrix and the 'e.-ominator

matrix, Real (YD), is a (D X D)-element matrix.

In the authors' earlier paper (Drane ;7 J McIlvenna, 1970). the parameter

optimized was directive g6in in contrast to -.he po~rer gain discussed above. The

difference between the two gains lies only in the denominator term of Eq. (3).

Directive gain utilizes the total power radiated, that is,

1' 62''

PRad. = / (0,0) sin OdOd4,f f0 0

instead of the total power input to the ar-ay. When the radiating elements are loss-

less (the usual assumption), the two gain formulations are identical. In any case,

however, directive gain is an upper bound for the power gain and is therefore often

used as a comparative indicator of array performance. It can be calculated from

the expressions for P(Q, 4) given earlier. (See Appendix 4, for details. )

Note that, as yet, no assumptions about the number of Led points have been

made. If for example, every segment is fed, then D = K, VD = V, and both the

numerator and denominator matrices in Eq. (4) are (K x K) -ilement matrices.

Usually, however, D < K, and, in fact, a very common array configuration has

each antenna element center-fed, in which case, D = N. This center-fed assump-

tion also piovides some simplification in the numerator of Eq. (4). For perfectly

conducting center-fed elements, the only non-zero entries in V are at the N drivirg

points themselves. One may then delete all the elements in the numerator matrix

(FY) (FY) which do not correspond to these feed peints, denoting the resulting

(N x N) deleted matrix as:,

_+

A = (FY)D (FY)D•

For example, in a two-element array with five segments per element (K = 10), the

center-feed points correspond to segments 3 and &, and one need retain only four

of the original 100, that is (K x K) elements of tne numerator matrix (FY) (FY),

namely, those with corresponding pairs of indices (3, 3), (3, 8), (8, 3) and (8, 8).

Thus, the center-fed assumption reduces the '!mensions of the numerator and

denominator matrices of Eq. (4) from (10 x 10), that is (K X K) to (2 x 2), that is,

(N x N). The final form foi, the power gain of an N-element array of center-fed

thin wires, with mutual effects accounted for, can then be wi itten as

V D AV D VD AV DG = K1 K-- K1 ., (5)

VD• Real (YD) V D VD B VD

The numerator and denominator matrices in Eq. (5) are both Hermitian, A is

a one-term dyad, B is positive definite, hence the maximum gain can be written

down immediately as (Struit and Hirasawa, 1969a)

Gmax K1 (FY)D (Real 'Y)) (FY)D, (6a)

7

IIand the voltages that produce it are given (to within a constant) oy

maxVD (Real D (FY)D (6b)

Equations (5) and (6) serve as the starting points for ,he maximum gain-constraint

technique.

3. THE CONSTRAINT METHOD

The technique for placing nulls in the far-field radiation pattern, while simul-

taneously maximizing gain, follows the steps outlined in the authorst earlier paper

(Drane arn. McIlrcnna, 1970). Placing nulls in the directions { qi, Oi ,

i - 1, 2, .... , M, with M < (N - 1), requires that E (8, ) (or eoiivalently Az(.9, ))be set to zero in the M debired directions. For the center-fed wires discussedearlier, we therefore generate a. set of M linear, homogeneous constraint equations

of the form

(F(Oifi)Y)DVD .0, i = 1, 2, .. M.I

Each of these equations consists of the inner product of a D-element angle-

dependent row vector

(F (0i# i •)i D

henceforth referred to as the constraint vector, and the column vector of driving

point voltages, VD. These constraint equations are not directly solved for the

di ing voltages. Instead, their effect is introduced into the g".n -xpression

through a transformation as follows., Each one of the M constraint vectors is

considered to be a particular row in a constraint matrix which thus has M rows

and D columns. In the case of the center-fed wires being considered here, D = N.

The rectangular (M x N) constraint matrix is riade into a square (N x N) matrix,

C, by adding ,N - M) rows. These additional rows can be any collection of N-

element inderendent vectors. For example, the s'.mplest such collection could be

(N - M) rows or columns of the (N x N) identity matrix.. The (01 xN) matrix C is

then orthogonalized, row by row, using any appropriate method (Guillemin, 1949);

and the resulting (N X N) matrix, denoted as P, contains all the constraint informa-

tion and is now introduced into the gain expression via the transformatic,

f -- V cr VZPD.DU

Thus, from Eq. (5),

V A VG=K 1 V+ Vc

where

[A =PAP+ =P(FY)L (FY)D P' aca

and

Bc = P D

Following the arguments outlined earlier, we discard the first M rows and columnsin Ac and Bc, leaving the abridged (N - M) x (N - M) element matrices Aaand Ba,

discard the first M entries in V (denoting the resultant ab' ;aged vector V d

and finally discard the first M entries in a. (denoting thd resultant vector a.). Thus,

the final quadratic form representation for array gain, with all the effects of

mutual coupling and pattern constraints included is

+X

Va a aVr = K I V+- IaVVa aV

the mnaximum constrained gain is

jnax zK 1 (aaBa 1aa) - Gmax

and the corresponding driving voltages are found from

max p+BIa

where Pd is an (N - M) Y N-element matrix formed by deletion of the first M rows

in the (N x N)-element matrix P.

These results verify the claim made in our earlier work that, even when

mutual coupling is included, the form of the gair relation and the properties of the

numerator and denominator matrices are unchanged. *

*An alternate constraint technique has been proposed by Adams and Strait (1970).

9

1!94. SOME EXAMPLES

One of the most common linear array configur .tions utilizes X/2 elements

uriformly spaced at X/2 intervals. This geometry was used for the sample com-

putations in this paper. There were 12 thin, straight, wire elements, oriented in

the z-direction and spaced along the x-axis, with 5 segments per element. Strait

and Hirasawa (1969b, 1970b), Chao ard Strait (1970), and Mautz and Harrington

(1971) provide the computer programs to consider other geometries and element

types.

One of the problems ideally suited to the maximum gain - constraint technique

is the elimination of interference, jamming or unwanted signals through control of

the radiation pattern. Unconstrained maximum gain radiation patterns are charac-

terized by the presence of a relatively high secondary peak, for example 13 dB

below the main peak. These high secondary peaks make such arrays susceptible

to jamming. One can use constraints in several different ways to reduce these high

peaks; two possibilities are discussed in the examples below.

One corrective technique requires the use of a single constrar,t and replaces a

secondary peak of the unconstrained maximum gain pattern with a null. A secondapproach is to use two closely spaced nulls to reduce the radiation level over an

angular sector that includes the secordary peak. The first approach is useful when

the angular location of the jamming source is exactly known., The second is more

suitable when the location of the interfering source is not accurately known or when

one desires to control a broader region in the pattern perhaps because the jamming

source is of some s-rificant angular extent. In both cases, the null or nulls

should be maintained fixed in space even while the main beam is st anned all the

way from broadside to endfire. Because of this wide range in scan angles, it is

especially important to include mutual coupling effects.. Figure I demonstrates

the two techniques with the unconstrained broadside pattern of a 12-element, X/2 -

spaced array of thin-wire dipole elements, X/2 long. All patterns are normalizedto facilitate ,omparison. The secondary peak of the unconstrained maximum gain

pattern at 0 = .760 is eliminated by the first technique or it can be reduced by the

second technique., Nct'. that the pattern structure is relatively unchanged, except

in the vicinity of the enforced nulls. This demonstrates the localized pattern

control available with the constraint method. The 3 dB beamwidths of all three

patterns are almost identical, while the beamwidth between nulls of the constrainied

patterns increases only slightly. For the two-constraint case, the design criteria

were as follows. The main beam should be scannable in the principal H-plane, but

for every scan angle in that plane, the gain in the direction of the scan angle is

maximum, while the radiation level is held at or ielow -30 dB over a 40 angular

sector centered at • = 760. This minimal radiation level sector was created by

10

Ode

UNCONSTRAINED-I CONSTRAINT

- 2 CON3TPAINTS

IO I'a-2010

030 60 1T6 90 120 150 ISO# (degrees)

Figure 1. Broadside Beams

using two nul's, fixed in place (at 6 = 740 and 780) for all scan angles, The

resiltant patterns, for this two-constraint case plotted at 100' increments in the

scan angle, are shown in Figure 2., Note in Figure 2(a) the well defined trough

in the patterns, which is at least 40 wide at every scan angle and in which the highestradiation level is -30 dB for the broadside pattern and at least -35 dB for the

other patterns, Figure 2(b'y shows the gradual and expected broadening of the mainbeam and the buildup of the endfire lobes as scan angle increases, until at 0 = 1800o

the pattern is bifurcated. This splitting of the main beam into two equal lobes ischaracterstic of the X/2 spacing chosen for this example.

Complete information for the unconstrained, one-constraint and tivo-constraintcases is presented in Table 1. Corresponding radiation patt(.,rn plots are shown in

FFigures 1, and 3 through u . d

iExamining Table n(a), wf e ( that as mentioned earlier, constrained gains are

less than or equal to the unconstr ain cas. An interesting observation is the

small amounits of ge:.' that must be surrendered to obtain a desired degree of

pattern eontro]W In the one-constraint case, the largest loss occurs at broadsideand represents only a 5 percent reduction in gain. In the two-constraint case, the

largest loss is about 4 percent again at broadside, and in both the one- and two-

constraint cases, the iain loss is less than abopt 2 percent over the range of other

scan anglesb,

11

(a)

I!I

o v 40 90 120

Lte Rps~road uced 1 Pý

Figi Resutp nt F abtte cpy

Figure 2. Resultant Patterns for Two-Constraint Case Plotted in 100Increments in - Angle

12

0 0 r-0t *NU. C ~ 0 0 0 0 00c 0i , - g- 4 Nb -W'

0 00000 0

o 9 V: 4! 00 M. T , O OO

0 U 0 oaao d °0•°° ° •

g °~

000000 00000

S S

a~0 % t. 4 It t- 0 0 U

0m IV cc 0 0 P3 I- -- c- d 0 0 0 0 0 0

oc

- o 40 -0t)Nc 0000

~ ! 0 C! 1 G¶

u•

co 9 ?I4w w v T m ' - c

o ~~- cbN' -o 0 00'; . .- - -o r. 0) .4 a* w co w t- 0-. -

o; -4C NC C ;C ;

U0

o 0 C4 0 00 0 0 0 0 000 m

u V ~ 0 0 IOI0 C; 0 0 0

o 0o

06 C; 0 C ;00 N~ 2 ~ 00 oO 001o0d0

m 0 t- D n C4l toCoC 00 0 0 0.OW 0

0 - 0

#~ N Mf

__ _ __ __ _ __ __ __ _

CM X -0aC C

M0 0 c c o CD 0 0 ! 0 '0 a! 0, o -0W0 I s I . .C;6C;o0o

r- ~~ 0~f en) m 1 . c o fr 0 )000

13

C' c- 0. 0.O 1-.01 1 O

w -.4" 0 0 r.C

'. 0 0 00 0 a00nQ0 0o 0 0 o

p4 0 CD M C's* e a~~ 0OC.6q........ 0 00 00 00 00000 0C! C. C'0s

- - ~0U P! 9 ýC . 1 C0 o oc. 0 e. o

0t 0 0el t t t . 0 00 0 0 0000-~~~~ - - -t f -f - -t . -. G -. - 0 co 0 ~ f f t ~ t 6

..C' 0 0 00 00 0 o

-4 m N m

00 t-. t-c 0 M0 M. 0 - CE00mw m w000 0 %o o 0o 0 000U.4~~c 0 CD.0t~ N4 Wtt

!...2 0 00a0 0 .0 030,m0 0000000000

o C; C; C 0 C

O 0 et0 0 om mcc

t: 9 9 !C !C 7 %2.ý & '

m -l I 0 00'., o w .0 0 P4 00 00000. . . 0 0 0 000 0 0 0n0 ;C -N

0 0 0ooo 0 000'! 9 !0 !C 0 !C 6 az

0 W ~0 CoN 0 C S 0 0 Dvct- 0D 000 0" co

*0 . NN

0 C.N N c 0 0 00 000 0 0 0 0 0 -0 00

u .4. A a

0 C

oz w

14 o iOdaU uNCONSTRAINED

- I CONSTRAINT2 CORSTRAINTS

-20id

Si I i|.L

| Io

0 30 60 6 90 120 150 180

*(degrees)

Figure 3. Beam Scanned to 1000

OdB

S. . . .- U N C O N S T R A IN E D

-I CONSTRAINT2 CONSTRAINTS

- IOdB

-20dB

i I 'I-30dr, I

0 30 60 76 90 120 150 I80

* (degrees)

Figure 4. Beam Scanned to ii10

15

041

I.. - IUNCONSI RAINEDI - ICONSTRAINT

.. 2 CONSTRAINTS

030 60 l 90 120 150 180

# (degrms)


OdB -... UNCONSIRAtNED !

SI CONSTRAINTS ...2 CONSTRAINTS

-IOdB

*1 Vi' ' ':

-30dB 8i

30 60 71 90 20 150 180

16 (d0 r90s


16

OM

.. UNCONSTRAINED

I CONSTRAINT-" - 2 CONSTRAINTS

-IOd8

-2-i

|i Ill I J |1i x

0 30 60 76 90 I0 150 80* (degrees)


U.ACONSTRAINED- I CONSTRAINT

2 CONUTRAINTS

-10dm

SI

0 30 60 90 120 150 too


17

0 .. UNCONSTRAINED

0I CONSTRAINT2 CONSTRAINTS

-20dB -- /. I,I

0 - 30 60 76 90 2CO*(degrees)


-- -.-. UNCONSTRAINEDI CONSTRAINT

- 2 CONSTRAINTS

-lode

if'

-2008 r

-. 3Od8 BS30 6,0 90 Q20 150 IS0

*(degrees)


18

----. UNCONSTRAINEC-I COASTRAINT*.... 2 CONSTRAINTS

-l0de

0 0 60 1`6 90 120 150 180

Figure 11, Beam Scanned to 1800

We also see that for some scan angles, that is 900 to 1100, the gain in the

two-constraint case is3 higher than in the one-constraint case. This is simplyexplained. Controlling the value of the secondary peak to -30 dB as done in the

two constraint vase represents less of a constrainit on the pattern behavior at these

angles than d.)es forcing that same peak value to be zero, as is done in the one-

constraint case. It thus does not necessarily follow that the gain decrease is

directly proportional to the number of null constraints used. Only when the con-

I" ~straints are used to perform the sam.re function on the pattern will the gain decrease

i ~be proportional to the numb-,' of constraint6. For example, replacing two peaks

with nulis causes more gain loss than replacing one peak with a null, and replacing

three peaks with nulls causes even more of a gain loss, and so on. This type of

dependence, linking gain loss to number of enforced nulls, is the only one that canbe identiFiedc,

Gocnera)ly speakinci, the decrease in gain value caused by pattern constraints

is proporti ontrollii the value of the u co ned pattern a the immed-ate vicinty

aef the enforced nulfsr For example, as seen above and er Figure d, at broadside

Che constraint caeg.on includes a sngifhcant secondary peak of the unconstrasned

pattera and tre, gain loss is largesm ; when the unconstrained pattern value in the

uistraint regnu o becomes smaller, the gain loss is lesspa Fwit example, at 140r

19

and 1600, the unconstrained pattern already has a null at - 760, implying that the

one constraint case is in actuality no constraint at all. There is therefore no loss

in gain, the amplitude and phase distributions are almost identical and the uncon-

strained and one constraint patterns are quite similar (see Figures 7 and 9).

An examination of the voltage amplitude distributions in Table l(b), (each

distribution is normalized to the voltage for its first element) reveals that these

maximum gain voltages, constrained and unconstrained, are all non-uniform and

most are non-symmetric about tht center of the array. The only two symmetric

distributions (the broadside and endfire unconstrained cases) correspond to the only

two radiation patterns that are symmetric about the broadside angle 0 = 900. Note

that in some cases the voltage required for element 12 differs significantly from

the voltages for all other elements. It also appears that for certain scan angles,

for example 1500, the difference between the unconstrained and constrained ampli-

tude distributions is small. As was true for the earlier nomparison of the corre-

evonding gainvalues, the explanation here is that at some scan angles the uncon-

strained pattern values are either very small in the constraint region, or the

direction for one of the unconstrained nulls is nearly the same as that of one of the

constraints, The constraint condition therefore requires only a slight perturbation

of the unconstrained voltage distribution in these cases. Finally, note that in no

case is the dynamic range of the driving-point voltages excessive. Compared to

the ten-to-one variation usually taken as the practicaý lir, il of realizauility, therange for both constrained and unconstrained optimum distributions is less than

about two-to-one, and the distributions should offer no construction problems., The

phase distributions foi these maximum gain arrays can be considered to consist of

the superposition of several components. The first is the uniform progressive

phase taper that steers the beam to the desired scan angle and is common to all

electronically scanned arrays, const-ained or unconstrained, The second tapcr is

that required to maximize the gain. And, in the constrained cases, an additional

taper is required to place the pattern nulls as prescribed. Table l(c) shows the

unconstrained and constrained normalized phase distributions, after tle beam

steering phase distribution (shown for reference in Table l(d)) has been removed.

As in the case of the voltage distri'.utions, note here also that the phase distribu-

tions are non-uniform and that most are non-symmetric. Finally, keep in mind

that the examples discussed in this paper, and indeed in most expected applications,

even though the designer directly controls the sidelobe level in only some local

region of the radiation pattern, additional control, as a direct result of the applica-

tion of these constraints will not become necessary over the remainder o^ the

sidelobe region (Sanzgiri and Butler, 197 1). This is due to the ge_.,.ral phenomenon

that reasonably low sidelobes are an intrinsic concomitant to maximizaton of gain.

20

5. CONCLUSIONS

We conclude that even with mutual coupling accounted for, the maxinirm gain -

constraint technique provides r'ealistic and practicai solutions to array design

problems requiring localized pattern control. The technique is applicable to prac-

tically all arrays cf wire elements, regardless of their geometric arrangement a, d

their electrical loading characteristics. The required relations and oquations can

be cast in matrix form and are ideally suited to computer manipulation, making

them useful in adaptive or real-time situations requiring rapid pattern reconfigura-

tion.

21

Acknowledgments

The authors wish to express their appreciation to Mrs. Evelyn Jones of the

Experimental Fabrication Section (SURSF) at AFCRL. for all her help in providing

the radiation pt ttern models used in Figure 2.

I

23

References

Adams, A. T. and Strait, B.J. (1970) Modern analysis methods for EMC, IEEEEMC Symposium Record, pp 383-393.

CK o, H. 11. and Strait, B. J. (1970) Computer Programs for Radiation and Scatter-ing by Arbitrary Configurations of Bent Wires, Scientific Report No. 7 onContract F19628-68-C-01890, Report No. AFCRL-70-0374.

Chao, H. H. and Strait, B. J. (1971) Radiation and scattering by configurations ofbent wires with junctions, IEEE Trans. AP AP-19 (No. 5):71.

Drane, C. J., and Mcllvenna, J. F. (1970) G-.ir maximization and controlled nullplacement simultaneously achieved in aerial array patterns, AFCRL ResearchReport 69-0257, June 1969. Subsequently published in The Radio and ElectronicEngineer, 39 (No. 1):49-57.

Gnillemin, E.A. (1949) The Mathematics of Circuit Analysis, The TechnologyPress, Cambridge, Mass.

Hessel, A., and Sureau, J. C., (1971) On the realized gain of arrays, IEEE Trans.AP AP-19 (No., 1):122-124.

HI trasawa, K. and Strait, B.J. 0(97 la) Analysis and Design of Arrays of LoadedThin Wires by Matrix Methods, Scientific Report No. 12 on Contract F19628-68-C-0180, Report No. AFCRL-71-0296.

Hirasawa, K ,nd St'ait, B.J, (197hb) On a method fcr array design by matrixitiversion, IEEX Trans. G-AP AP-19 (No, 3):446-447.,

Lo, Y. T., Lee, S. W., and Lee, Q. H. (1966) Optimization of iirectivity andsignal-to-noise ratio of an arbitrary antenna array, Proc. IEEE 54:1033-1045.

Magnus and Oberhettinger (1949) Functions of Mathematical Physics, New York,pp 30.

Mautz, J. R. and Harrington, R. F, (1971) Computer Programs for CharacteristicModes of Wire Objects, Scientific Report No. 11 on Contract F19628-68-C-0180,R,'port No. AFCRL-71.-0174.

Preceding page blank

24

Nemit, J. T. (1969) Reduction of Phased Array Sidelobes Toward the Ground byPhase Control, Report No. 1609P, Wheeler Labs. Inc., Smithtown, N. Y.

Pierce, J. N. (1970) Receiving Array Coefficients That Minimize Interference andNoise, AFCRL-71-0079.

Riegler, R. L. and Compton, R. T. (1970) An Adaptive Array for InterferenceRejection, Report No. 2552-4, Ohio State University.

Sandrin, W.A. and Glatt, C. R. (1970) Computer aided design of optimal linearphased arrays, The Microwave Journal, pp 57-64.

Sanzgiri, S.M. and Butler, J. K. (1971) Constrained optimization of the perfor-mance indices of arbitrary array antennas, IEEE Trans. AP AP- 19:493-498.

Strait, B.J. and Adams, A. T. (1970) Analysis and design of wire antennas withapplications to EMC, IEEE Trans. EMC EMC-12:45-54.

Strait, B.J. and Chao, H. H. (1971) Radiation and Ecattering by wire structures,IEEE Intl. Convention Digest, A.I Y.

Strait, B.J. and Hirasawa, K. (19G8) Array desigi, by matrix methods, IEEE Trans.AP. AP-17:237-239, Mar. 1969, or Report AFCRL-68-0542, Oct. i968, AD678-089.'-

Strait, B. J., and Hirasawa, K. (1969a) Applications of Matrix Metb ods to ArrayAntenna Problems, Scientific Report No. 2 on Contract F19628-6 -C-0180,Report AFCRL-69-0158, AD 687 -48 1.

Strait, B. J. ana Hirasawa, K. (1969b) Computer Programs for Radiation, Recep-tion and Scattering by Loaded Strafht Wires, Scientific Report No. 3 on ContractF19628-68-C-0180, Report No. AFCRL-69-0440.

Strait, B.J. and Hirasawa, K. (1970a) On long wire antennas with multiple excita-tions awl loadings, IEEE Trans. ATI AP-18:699-700.

Strait, B. J. and Hirasawa, K. (1970b) Computer Programs for Analysis andDesign of Linear Arrays of Loaded-Wire Antennas, Scientific Report No. 5 onContract F19628-68-C-0180, Report No. AFCRL-70-0108.

Tai, C. T. (1964) The optimum directivity of uniformly spaced broadside arraysof dipoles, IEEE Transactions G-AP AP-12:447-454.

I

25

Appendix A

Derivation of a Relation for the Directive Gain of an Array of ElectricallyThin, Straight Wi res With Mutual Coupling Effects Included

It was pointed out in the text that the expressions for directive gain and power

gain differ only in the form of the denominator term. Power gain used the total

power input to the array while directive gain uses the total power radiated. These

are the same when the system is lossless. In all other cases, directive gain serves

as an upper bound on the power gain and is often used as a performance criterion in

array design. This latter expression is given in general by

21 rPa r 4"P(0,0) sin Od4dO,BRad. 47

0 0

where P(6, 0), the power pattern of the array, can be expressed as

P(0,4) 0) E( 2 =g KAz(A , )(0 sin2 0

and K = (rwm0 )2 /Io.. This integral has been evaluated by Tai (1964) and Lo et al

(1966) for isotropic sources and short dipoles. For the case of electrically thin,

z-directed wires, spaced along the x-axis, Az (0, 4) is given by

26

K

Z . Itni eXp [I 3 (xn sin Gcos + zn cos 0].

nl1

Taking all segments to be of equal length leads to

K KlfP *j.a (Al1)BRad. 370 41 )' EIbnim

n=1 m=1

where21 v

b b exp j (Dn sin 0cos +Zn cos 0)]sin3d~dO0,

0 0

and

Dnm =3(xn-x ), E fi(zn-Z)•nm n mn nm n In

The bnm terms depend only on the array geometry and in no way are modified by

the inclusion or exclusion of mutual coupling effects. But, the In quantities do

indeed depend for their values on mutual coupling considerations. In matrix nota-

tion, Eq. (Al) becomes

= I(o~~ V jWoI -- (A2)Rad 0\4 I+I -- / (Y j Y) V. (A2)

The integration on b in the expression for bnm is readily performed (Jahnke

and Emde, Pg. 149), that is,

2wr

f exp L jDnm sin0cosJldO = 27rJ (0'G).n sin 0).

0

Thus

bnrn 2 Jo (Dn sin 0) exp [j Enm cos 0] , in 3 0 dO.

0

This integral may be evaluated as follows. Note that about 0 =7r/2, sin3 0 and

Jo (Dnm sin 0) are even functions of 0, while sin (E cos 0) is odd. Hence, we0 nm nmcan discard the imaginary part of bn and, using the evenness of the real part, we

have

I

27

bm f Jo (Dnm sin 0) cos (Enm cos 0) sin30 dO. (A3)

0

Since

sin3 0 = sin 0 (1 - cos 2 0)

and

cos (E cos0) 0 (-1) (Enm Cos 0) 2i

Ln E (2i)!i=0

w e c a n w r ite 02o00 (1i(E )2i 1 /2

b = Z (E)'ni, f J (Dnm sin 6) (cos 0)2i sin 0 dO

i=O 0

I J (Dnmsin 0)(cos 0)2i+2 sin 0 d0

I0Integrals of this same form were treated by Lo et al (1966) using the relation

(Magnus and Oberhettinger, 1949)

10 2 ?vr (v+li) (Z2v+1 Jv+;4+l(),R1, e I

J, (zsin0) sin) (0cos d Rev> -z1

l Thus

S•(-1)1(]Enrm)2 2i I/2(i+1/ 2) J 1+ 1/2 (Dnrm)

iibnm= E (20)' i+ 1/2

ii=° (Dnrm)

2 i+1/2r(i+3/2) J i+32(D)nm

"(D )i+3/2nm

S~Recall that

Sr(i+3/2) (i+ 1/2) F (i+ 1/2),

(2i)0 = i(1+2i) = F[2(i+1/2)],

28

and hat

2z- 1

r(2z) 2 2-r(z) r'_z+ 1/2)1s /2

"Lhus

o 2i m i Ei+ 2 )+nm)bnz 2~ K. /+2

10 2 1! (D )'

2 (i+ 1/2)J 1 +3 2CDn,) ] (DC)i+3/2 m(Dmi+312 ,m n (Ut)

and -- 2/3.

Note that

Dm = D

E Emn TL-

J i+ 1/2 (Dmn) =('1)i+ 1/2 Ji+ 1./2 (Dnrm)

i+3/2 nJi+ 3/2 (Dmn) -li / i+ 3/2 (n)

hence bmn bn and ý is a real, symmetric matrix. Both m and n range from

1 to K, where K is the total number of segments in an N-element array. K can be

rather large; for example, in the arrays considered in this paper, 12-elements

with 5 segments per element, K = 60. Fortunately, however, one needs to calcu-

late all 1800 of the bmn elements only for the most general array geometries, that

is, wires of arbitrary length and shape, arbitrarily spaced and located. In most

arrays of practical interese, significantly less calculation is required. For equally

spaced arrays of N straight wires, with S segments per wire, only NS or K of the

1800 elements in ý are distinct. To see this, refer to Figure Al taking N = 12

elements and S = 5 segments per element. Each of the twelve antenna elements

gives rise to 25 bmrn terms. These (5 X 5) - element square submatrices are

situated symmetrically about the main diagonal in 0; hence the submatrix diagonal

terms are known to be 2/3. Because of the symmetry in T, only 10 of the remaining

20 terms in any main diagonal submatrix can be distinct. One such submatrix, that

associated with the first element, is shown below.

29

2/3 b 12 b 1 3 b 14 b 15

b 12 2/3 b2 3 b2 4 b2 5

b1 3 b2 3 2/3 b3 4 b3 5

b 14 b 24 b34 2/3 b4 5

b 15 b2 5 b3 5 b4 5 2/3

Each one of the 10 required b. terms depends on both a Dnm and an E nu term.If the wire is straight and parallel to the z-axis, all 5 segments on that wire have

the same x-coordinate and Dmn a 0 for all 10 of the desired elements in •1 Iff inaddition, all segments are chosen to be equal, then

E 12 =E 2 3 =E34 =E 4 5

E 13 = 24 = 35

E 14 E E25•

z

5 10 15. 60

4 9 14 59

3 8. 13 058

2 7 12 57

-I" 6 II. 56 -

S- d

Figure Al. 12-Element Array, 5-Segments per Element

30

Hence,

b12 0 b2 3 w b 34 b 4 5

b 1 3 ' b 24 ' b 3 5

b14 z b2 5 .

and only 5 elements are needed to completely specify . Its final form is

2/3 b12 b13 b14 b15

b12 2/3 b12 b 1 3 b1 4

1= b13 b12 2/3 b12 b13

b14 b1 3 b12 2/3 b12

b1b b 14 b1 3 b 12 2/3

If all the elements in the array are identical, then alR of the 12 main diagonal sub-matr~ces ar- identical to P, and 300 elements in • are determined with only fourb calculations.

n nThe off-diagonal submatrices in • depend on the array spacing. Each pair ofelements in the array gives rise to a 25 term bubmatrix and for a K-element array,

there are 66 of these. In any one of these submatrices, if the Lvo antenna elementsin question are straight and parallel, all Dn terms are the same and have thevalue P]d where d is the interelement separation. As in T, there are only 5 distinctEnm values, hence once again only 5 calculations are needed to determine all 25

terms in any of the submatrices. The submatrix, 02' representative of the firsttwo array elements, is shown below. It has the same type of symmetry and row

regularity as F,, sometimes called a Toeplitz property.

b16 b17 b18 b 19 bl' 10

b17 b16 b17 b18 b19

=2 b 18 b 17 b1 6 b 1 7 b 18

b 19 b 18 b 17 b 16 b 17

bl, 10 b19 b18 b17 b16

31

Similar arguments show that the submatrices repre3entatie of the remaining

array elements and element one, that is 031 94' 912# have the same form,

symmetry ar-d regu. rity as •2 and that there are only 5 distinct terms in each.

When the array is equally spaced, each of the remaining 48 submatrices in

Sis identical to one of the 62' %3 submatrices discussed above. Thus,

for an equally spaced array of straight parallel wires with equal segments, • con-

sists of only 12 distinct submatrices, and in these, only 60 terms are distinct; it

has the same Toeplitz property as its component submatrices.

• .' {I902 03 $14 5 h6 97 A8 "' -12

-~ A 0061•2l3 05 As16 07 ... -Oll

063 92 Al P2 03 94 A5 06 ... 1110

The cyclic p•7zperty o. ( and its submatrices is also present in the generalized

impedance matrix L for this type of array (Strait and Hirasawa, 1969a), since

elements in both t and Z depend orly on intersegment distances.

The 60 distinct elements in fare the first row blm, 1 :_ m :-- 60, or the first

column terms. These 60 terms are associated with four combinations of Dnm and

E nm:

(a) D =nm 0

(b) nr 0, Enri 0

(c) nm =0, nl•,

(d) D 0, Enm• 0.

These can be used to simplify the bnm expression in Eq. (A4). Case (a) has been

discussed earlier and constitutes the main diagonal terms in •where b 2/3.

Case (b) is the situation in 11 of the 60 required brim elements. 1hey are bla

where a = 6, 11, 16,. ., 56.

32

Referring to the general brm expression in Eq. (A4), note that when Enm 0, an

but the first (i - 0) term disappears and

b. 1/ (1 JI/2(D 1~a) J ~3/2 (D la2 (Dla) 1/ 2 (Dal)3 /2

for these 11 terms. But,

and = 2D, sin Dia

i (D~

1I/2 la) • -•- DZ

and

F2D2.•-I sin D1 cos D1J3/2 la 2 D"

Therefore,

b2 sinDlI& (D l2 _ 1) + cos Da (A5)la (Dla)2 la

Note that Case (b), with D. * 0 and Enm N 0 corresponds to elements of zero

length, that is isotropic or short dipoles, and has been treated by Tai (1964) and

Lo et al (1966). Equation (A5) agrees with their results. Case (c), Dnm 2 0,

E nm 0 arises in calculating 4 of the required 60 bnnm elements. They are

bi$ -2, 3,4, 5. Looking at a general series representation for Jv (x), that is,

( 1)r xv+2 r

Jv (x) = r! 2v+2 rr(v+r+ 1)r=o

we see that

Jv (x) 1asx -0.

x v 2 v r(v+ 1)

Hence, if in Eq. (A4) D rn = 0, we have

S(-.1)' (E Ib)

b (2i+-1) (1+3/•) (A 6)

i=o

33

where we have used the fact that

71/2 1

2 2 1+11(i+1/2) P(i-1) - 2r'(2i+1)

The remaining 44 of the 60 required brnm terms must be calculated from Eq. .A4).

It is probably easier to simply use Eq. (A4) for all 60 bnrn terms when programming

a computer for the calculations. In this case, fifteen terms of the series, Eq (A4),

will provide sufficient eccuracy, that is on the order of 10-8, as long as the inter-

element spacing is no smaller than X/3. Calculations based on these results agreed

with those based on the power gain formulation for the 12-element, lossless arrays

considered in this paper.

Equations (A2) and (M4), togethei with the symmetry arguments and simplified

b forms, provide the basis for calculating the directive gain of an array of

equally spaced straight wires with mutual coupling effects included. The final

formiula for this gain is given by-,

G V+ (FY)+ (FY)V sin e 0D V + (Y+ VY)V

Documents

Null Steering and Maximum Gain in Electronically Scanned ...and places a number of independent nulls in the radiation pattern has already been presented. The elements used were idealized