RADC.TR40-30 1 "In-House ReportSeptember 1980

CIRCULAR ARRAY OF DIPOLESABOVE A PERFECTLY CONDUCTINGCYLINDER

Gregory Cruz, ILt, USAF

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SEP 1981D

4 AROME AIR DEVELOPMENT CENTERAir Farc* Systems CommandGriffiss Air Force Base, New York 13441

"81 ,

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RADC-TR-80-301 has been reviewed and is app.eoved for publicarion.

APPROVED;

WALTER ROTMAN, ChiefAnten.as and RF Components BranchElec:tromagnetic Sciences Division

A"ýOMW

- C. SCHELL, Chief-. ectromagnetic Sciences Division

-". -'-' 4. m

° "' i'°'°' ""FOR TH4E C0P14AND91t:I . • . -

. JOHN P. HUSSActing Chief, Plans Office

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20. Abstract (Continued)

>determining techniques for radiation pattern generation with very low side-lobes, optimum spacing, and a minimum number of array elements.

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Contents

1. INTRODUCTION 5

2. MATHEMATICAL ANALYSIS 8

2. 1 Self and Mutual Admittance 202.2 Numerical Results 22

3. CONCLUSION 33

REFERENCES 35

Illustrations

1. Dipc'e Near a Long Cylinder 6

2. Circular Arrays Around Cylinder. Optimization Approach 7

3. Perspective View of Array of Cylindrical Dipoles Surround-ing a Perfectly Conducting Cylinder 9

4. Perspective View of Dipole. Conductive Cylinder Geometry 9

5. Geometry for the Development of Integral Equations for aCircular Array Around a Perfectly Conducting Cylinder 10

6. Cylindrical Dipole, Showing the Method of Applying theBoundary Condition E z 0 at rd b 17

7. Array of Cylindrical Dipoles About a Perfectly ConductingCylinder, Simple Case 25

3

Illustrations

8. SR(n) = fQ(a) ReFS(n, a) for n 0 to 700 26

9. SAOn) = fQ(W) ImFS(n, a) for n = 0 to 70u 2610. BR(n) = fQ(a) ReFB(n, a) for n = 0 to 700 27

11. BI(n) = fQ(a)ImFB(n. a) for n = 0 to 700 27

12. BSUM(n) = V(SR(n) + BR(n))2 + (SI(n) + BI(n))2 = I BESS(n)Ifor n = 0 to 700 28

13. Self-Admittance for Active Element for Arrays of Sizem = k to 22 28

14. Mutual Admittance Between the Active Element(Dipole No. 1) and the Adjacent Dipole (No. 2) 29

15. Mutual Admittance Between the Active Element(Dipole No, 1) and the Adjacent Dipole (No. m) 29

16. Relative Currents for the Circular Array of Dipoles Abovea Perfectly Conducting Cylinder for Active ElementDipole No. 1, m 2, 4, 7, 10. 13, 16 30

i

ij4

Circular Array of DipolesAbove a Perfectly Conducting Cylinder

1. INTRODUCTION

Recent developments in low-inertia phase shifters and electronic switches1I

have aroused interest in large, circular arrays on conducting cylindrical surfaces

for electronic agile beam positioning or radar resource optimization, invariant

beam characteristics over the entire 360o azimuthal scan sector, and frequency-

independent beam pointing direction. A low-sidelobe, high-gain aperture designfor a circular array of dipoles surrounding a perfectly conducting cylinder requires

a knowledge and control of mutual coupling parameters in the array environment. 2

Carter 3 in 1943 outlined a rigorous solution of Maxwells' equations for a dipole

near a long cylinder (see Figure 1), and tabulated formulas for the radiation pat-

terns for three different circular array configurations. Carter used the reciprocity

theorem and the k 'own solution for scattering from a cylinder to obtain far-zone

(Received for publication 30 September 1980)

1. Provencher, J.H. (1910) A survey of circular symmetric arrays, PhasedArrax Antennas (Oliver and Knittel, Artech House, Inc.. Dedham, Mass.)page 292.

2. Hessel, A. (1970) Mutual coupling effects in circular arrays on cylindricalsurfaces-aperture design implications and analysis, Phased Array Antennas,(Oliver and Knittel, Artech House, Inc., Dedham, Ma-'.} age 273.

3. Carter, P.S. (1943) Antenna arrays around cylinders. Proceedings of IRE,Vol. 31, December.

5

z

-z',X/4

•Z'm-)Ji4

Figure 1. Dipole Near a Long Cylinder

patterns. Lucke 4 , used Green's function method to yield expressions for the

fields in terms of an integration irn the complex plane. The problem was also solved

by Harrington6. 7 by converting a three-dimensional radiation p,'oblein having cy-

lindrical boundaries into a two-dimensional problem by applying a Fourier trans-

form with respect to the cylinder axis, Z. Neif also used the Fourier transformmethod, and by means of v. limiting procvss, obtained an approximate form for a

dipole near a finite cylinder.

In this report, the problem of finding the electromagnetic field and admittance

or a circular array of dipoles near and parallel to an infinite conducting cylinder

4. Lucke, W. (1949) Electric Dipoles in the Presence of Elliptical and CircularCylinders, RepFNo.N 1, Project 188,tanford ReserchInstitute,Stanford, Calif.

5. Lucke, W. (1951) Electric dipoles in the presence of elliptical and circularcylinders, Journal of Applied Physics, Vol. 22, No. 1, January.

6. lHarrington, H. F. and LePage, W. R. (1949) A Study of Directional Antennas forDIF Purposes, Report 1, Department of Electrical Engineering, SyracuseUniversity, 9*yracust, NY, September.

7. llarrington, !. R . (1961) Time-Harmonic Electromagnetic Fields, McGraw-Hill,New York, N%, Chapter 3.

8. Neff, IH. P., Hickman, C. E., and Tillman, J.D. (1964) Circular Arra s AroundCylinders, Report No. 7, Department of Electrical Engineering, University (fTennesset', Contract AF19(628)-288, June.

Bb

has been mathematically formulated into integral equations via the Fourier trans-

form method, which relates the illumination function's voltages to the resultantcurrent distribution due to the mutual coupling. We use the computer to numerically

solve the m nonhomogeneous integral equations, using numerical approximations

for the integrrls and complex matrix techniques to solve the simultaneous equations.

Such an analysis should account for both the space wave with its grating lobe and

the creeping wave resonances on the curved surface, and have an impact on the

d••velopment of optimum methods for determining techniques for radiation pattern

generation with very low sidelobes, optimum spacing and a minimum number of

array elements (see Figure 2). We snail investigate the effects of mutual coupling

for a circular ar:-ay located a quarter of a wavelength above a perfectly conducting

cylinder having a radius of two wavelengths. This simple case was chosen to

easily investigate the m simultaneous integral enuationa and to lay the foundation

for future investigatio•ns into the more practical case of A perfectly conducting

cylinder with radiuei of approximately 10 wavelengths.

OS.IECTVVE TO OBTAIN 360-MEGREE SCAN CAPABILITY FORLOW SIDELOSE ARRAYS OF ELEMENTS

I. MUTUAL COUPLING EFFECTS I LOW SIDELOSE AZIMUTHAL I.ELEVATION PATTERN CONTROLZ.OPTIMUM ILLUMINATION PATTERN ?-STACKING OF CIRCULAR

FUNCTION 2.CREEPING WAVE PHENOMENA ARRAYSAPPLICATION" UNATTENDED RADAR, GROUND BASED TACTICAL RADAR

Figure 2. Circular Arrays Around Cylinder, Optimization Approach

I

2. MATHEMATICAL ANALYSIS

The circular antenna array to be considered (see Figure 3) consists of identi-

cal, parallel, cylindrical dipoles equally spaced around the circumference of acircle, which is concentric to the perfectly conducting cylinder which the array of

dipoles surroundo. Only center-ted dipoles with a halt-length ?,/4 and a radius b

will be used as elements. The axial center of each dipole is perpendicular to the

plane of the circle. Each of the elements of the array is thus in the same geomet-rical environment. The radius of the perfectly conducting cylinder is a, and theradius of the circular array of eipoles is r0 (see Figure 4). The expression ior

the vector potential A (r. #, z) has been developed for a circular array of m

dipoles located about a perfectly conducting cylinder. 9

For a current density j located above a perfectly conducting cylinder (see

Figure 5).

•2 X+k2•:u, l

For a unit dipole at (r 0. 00, Z0),

. 6(r - ro)6( 0 z - zo)Z (2)

Therefore, we need only be concerned with the A. component:

f(z) f IF(&, r, ) elOz do (3)

where i1

ii4

9. Fante, Dr. Ronald L. (private communication).

zw

Figure 3. Pernpective View of Ar'ray of Cylindrical Di-poles Surrounding a Perfectly Conducting Cylinder

CYLINDRICL SOUNDAR'WCONTAINING "tRENTELEMENTS

CONDUC;TIVE CYLt4OER

41DIPOLE RACIATrO

OW RADIUSab

Figure 4. Perspective View of Dipole, ConductiveCylinder Geometry

Figure 5. Geometry for the Development of Integral Equations for a CircularArray Around a Perfectly Conducting Cylinder

From Eqs. (3) and (4) we get

A z 9 j do e1 ie a (a, 0 e "~(5)

In cylindrical coordinates, the laplacian is

2LOA z 1(82 az) -A~Az r- + + (6)

zF ( r r'a z

Substitute Eqs. (6), (5), and (2) into Eq. (1) to yield

1 11110eiaz a2 2

10

Multiply both s:des of Eq. (7) by e in eQ'az thcra intcrat?

811 o •'8m 2 2 -m 2) amm-neda ( r -- + k-•- a d f

(a ) -a " 6(r " r0 d eI 6 (0 - ) f dz 6(z z.) eX dze r j u

-W -

(8)

Use the relations

[ Oif m nJf et iM " di o 2w ifm = n

f . , •"0 if m one• 2 2rif m = n

then Eq. (8) reduces to

ran +(k2_2 an 26(rr)e a Z (9)

For r > r 0

an -- e 0 e Hn (z 0 r) (10)

-in -i'Z

Next we derive continuity conditions at r r from Eq. (9). Integrate Eq, (9) from

r - to ro + and let c- 0.

11

"fOn di (+f r (k _T )ndr

r0-, r-0 -

-in# -.iaoz r 0+c"e e e

26 f (r r )dr2w if -0o)dr.

rO -C

therefore

/-3a (aa• -,o "~' -,-,'-8rPrne - in# e tna° o (12)\Or -ro+e \a r/o., 0

Substitute Eqs. (10) and (11) into Eq. (12) to yield

n~ 987 H n(1ýk r) - Cn [93- Jn(k *0 r]0 0

Dn H [ n(VkrT-o 0 (13)r o

0

define

Then rrum Eq. (13) we get

(n - n)HnI(x) -Cnjn'(x) 2r"x

Equation (14) is the first solution between Bno CnP and Dn. Since Az must be con-tinuous at r = r

Bn H (x) ('n JnX) + Dn Hn(x)

12

-~-~. " ----.

or

BnDn=fi * P.15)t

n n

Substitute Eq. (15) into Eq. (14) to yield

nx -J(x) =jj -

Define A J Wx H'(Y) - Jn(x)H (x) (Wronaktian), thennn n n

H Wr

"o n(X)C n 2 v X(16)

Also from Eq. (15).

-n o Jn(x)Bn"D = on (17)

Now we need to satisfy the boundary conditions at the surface r a of the cylinder,

which is that the electric field tangential on its cylindrical surface must be zero.

~~r ---- x vx •

II

jwc (ap

x (aA\ z -

r ( ar

a 2 Aa 2A !AF3

IX XXz+ ( a A_ - 1 z

I -araz r FF a -

therefore

=9zo 0 (18)[ z~jr=a

13

-I

and

2t 0 (19)

rea

But from the wave tequation

r Br ] T\2 + + k Ar 0

Therefore, Eq. (19) is equtvalent to

A2 + kAz (20)

z2

Upon substitution of Eq. (5) In Eqs. (18) and (20), we get

1 t a i:(1I J do(io)(im) e0 a e 021)

1 f da(k2 -a 2 )eiZ iz a eeimr 0 (22)

where

an = e a) + DnHn k a (23)

Equations (21) and (22) are both equivalent to requiring that Am = 0 at r a,

therefore

Cn Jn(y) + Dn1n(Y) 0

where

14

:': . ..... ; •:- >' ,, • , 4

r *

"-C J (Y) (24)

Upon substituting Eq. (16) into Eq. (24) and rearranging terms we get

.o H(X) / J~ly) J

Using Eq. (25) in Eq. (17) we get

Bn 2 ny) J WX)H (Y) - HNJ(Y) (26)n

Since ir X L 2i we get

B • n - Jn(X) Hn(Y) - n(X) Jn(y)) (27)

ioHnlX)

C = o r - (28)n 4

SoHn (x) Jn(y)Dn 4i Hn(y) (29)

where

X a 2 r

2- 2K '2-o aa

r =radius at which dipole is located0

a =radius of cylinder.

15

Now that B 1(a), Cn(a), and D no) are known, the Green's function is for r S r

1 40e ia(z'zo 0 in(#- o)n° f do e .; tin V- " kn r& 0+

and for r r

*0

2 FG l ea(z-z da tein% Bn Hn

-00 n= °0

Therefore for any arbitrary axial current density j (r, *, z)

GIif r 5 r•,,A .f !dvj (30)

z f G2 it r r

where

dv =volume of current density.

SFor a thin current filament we have the relationship I Z I R1 j where R is theradius of the current element. Then dv _ R H dz° and Eq. (30) becomes

A(r, 4, z) f dz / 0(zo) (31)= 1G2(r,,z) it r-r 0

the expression for the vector potential for a circular array of m dipoles located

about a perfectly conducting cylinder.

We now need to relate the currents on the circular array elements to the base

voltages to determine the self and mutual admittances of the circular array around

a perfectly conducting cylinder. It can be shown that the vector potential at any

point on the surface of antenna K (see Figure 6) is given by

10. Tillman, James D. (1966) The Theory and Design of Circular Antenna Arrays,Chapter 1, The University of Tennessee, Engineering Experiment Station.

16

}z

RADIUS b

Z%'/VI z Z

x

Figure 6. Cylindrical Dipole, Showing the Method ofApplying the Bcundary Condition Ez 0 at rd ý b

[Azzo- [Co Cos a + K sin 132)-rd b

where b is the radius of the dipole, -h - z nc h, h , VK is the base voltage of

the K element, and j 2r/A. Equating Eq. (32) to Eq. (31) wye get

S 1 /2 rG 21(r, , z) if t sr o _ fsi i jz-172 fInlG') "G ( Z ) if 'k Cos z3 i - .sn~

m- -L|/2 1

17

?ov-r L-r we 1,

In L/2 dino Irnlzo b o) do

rm;l -L/2 --

XBn(tw) H (v 2 r) [ con , + a-.in 1 (Izl (33)

where the Kth dipole is at (r. #. z). Thus, for an m-dipole circular array, we havein simultaneous equations to solve. Let z = w/20, then Eq. (33) becomes

Inl L/2 n-

Substitute Eq. (27) into Eq. (34) and let (m(z4) - rn cos -j then,

m L/2 rz go ia(W/23-zo) b ein(Ok+b/rOO~m)E A. f dz° cos Lo f doe e

m~l -L/2 -0 n-

r J~ ~(x y) - Hn(W)j (Y) v4V (5Hn(y)" H• ( k ar)] o 35

Since all dipoles are equidistant from the perfectly conducting cylinder. r rthen,

CV k' r H ~(x)

18

Thus Eq. (35) becomes

m L/2 d 2.o enlk+b!ro'm) doe'Am da° coo --L- do

: m~l -L/2 n - -

X Hl(x)H (y) H (4uVK

or

Fa AInIn Far e) do e~ g(o)

n m=l n--o "

X nl Hn() - n n ; K (36)

where

L/2 Wrz toE

g(f) f dz Cos -0 e0

-L/2

Equation (36) reduces to

m a in(ok+b/ro..m) o eL

SAm e do w2 2

F =, 2 -Im ~fn- l I• -•

j (Y) Hn(x) - 2V L

19

* m-

S1,-1.-.w.,MM . ..- - O.

which from symmetry becomes

m a in(#k0b/ro-m)f Cos LoA Am do M-Cos"-

2 2 4m=l n=O 0 C-

L

X[j nl nlX) -Hn(y) 1 -v C

2.1 Self and Mutial Admittmc.

Now, let

Tm(a) ein(#k+b/ro-#rn) Y

n=(I 0 L' "

L

H2 (X< [ 1~x (x) (Y

then I

M=I1 0

Now we shall consi`er the M•7 integral equations by taking a look at the c-quatlons

yielded for each base voltage Vk (K I. ... ni) simultaneouely. We get

A T +A2T + A T v1 11 2 12 rn Ilin ~.C I

A1 F.1 2 22 m 2nAmT~rn 2. • (38)

AITml +A 2Tm 2 + AmTm M

20

I

In a matrix notation, Eq. (38) to equal to

T1 1 T 1 2 ... Tim AI

T 1 2 T22 ... T 2 m A2 12

1 (39)

S• .9. • *

Tml Tn 2 Trm A0 m

where

Vk K

Therefore,

Given values for VV. V2 1 .... Vm and having numerically approximated the inte-

grals Tkm (m = 1 .... m; K I. ..... m), we compute the values for the A r's

by factoring the matrix T into the L - U decomposition or a row-wise permutation

of T and solving the systems of Eq. (37). Once the values for the Am's have been

found, we can then determine the mutual and self admittances of the circular array.Equation (33) gives the relationship between the currents and voltages of the array

elements. For an rn-element array,I

YliVI + YI2V2 + + YlkVk +. 4. Y 1 mVm 11

Y21VI +Y22V2+, +Y 2kVk+... +Y 2rVm 12

. .. (40)

tmhem2 2 ... mk1k MM m

where YlI' Y221 .... Ykkl ... Ym are the self admittance of the respective ele-ment, Y , is the mutual admittance between a and b. VK is the applied voltage on i

i the Kth~elf-ment, and IK is the curren nthKhelemen..I

For an array with a single active element. which for simplicity's sake we let

be element no. 1, every VK where K I is equal to zero. Thus, Eq. (16) becomes

21

Ii

12 XY 2V 1

* (41)

Im -- YmlVI

We let V1 • 1 and AK is the magnitude and phase of the current on element K.

Therefore,

A2 -Y 2 1

* (42)

Am -YmI

Thus. the coefficient A is the self admittance of the active element no. 1 and the

coefficient AK, where K is unequal to one, is the mutual admittance between thefirst and Kth element.

2.2 Nwumeu Reulta

The simple case of a = 2X and ro = a + X/4 (see Figure 7) was investigated

and the mutual and self admittances were determined through use of computerprograms to numerically solve the m nonhomogeneous simultaneous equations

(Eq. (37)). Equation (37) becomes

lvinOkb .VKL

A eBESS(n) (43)m~l n=O

22

.'-,-

where

kBESS(n) 2fdo Q(o) Rr4'S(n. a) + fdo Q(*) ReFB(n. a)

0

+ I ( do Q~o) lmFS(n. a) + 7 do Q(oi) ImFB(N. a) 4(44)

LT

2 2(y y2 ~2

ReFSKnO) J1 (x)J(x) n- n n (46) nn 2y

2J )K (x)Y (x)2 72)

ImnFS(n~a) J (xy (x) n -- -n,- J(n(y)yn(y) + Ynx)JIynY (4y )n (y) + 2(Y)(4 ).

IrnFB(no) = ! f±L____Y

S ý n~y)(49)

For Eq. (43) to be solvable, the expression

~ in(#k~b/ro-#m)BESnnOES~n (50)

must converge to zero for large n, otherwise, the appropriate coeffcients tor eachAm (m = 1. 2, ... m) cannot be determined. Let

SRt(n) JQ(*)ReFS(n. a) do

23

BR(n) - fQ()ReFB(n.a) da

Bl(n) = jQ(ar)ImFH(n.*) do

The coi tributions to BESS(n)I of SR and BI rapidly diminish for large order n.HR and St decrease very slowly in absolute ,.lue for l.hrge order n. Figures 8, 9,lb. and 11 show the values of Sfi, S1, BR and BI respectively for order n z 0 to700. BR and SI approach zero very slowly for large order n, while SR and SI con-verge rapidly to zero for large n. Figure 12 shows the value BSIJM(n) where

SUM(n) - 4(SR(n) + BR(n))" + (SJ(n) + BI(n))2 t IBESS(n)i

and Ok = Om 2 00 (worst case). BSUM is shown to slowly converge to zero for

large order n. Due to time limitation in computer usage, n = 0 to 700 was chosen;however, for more reliable results, n - 0 to 2000 may be required, especially for

the worst case #k = Om 00.

Once . BESS(n) were determined for n = 0 to 700, these values were used inEq. (50) to determine the coefficients for the A Is in the set of simultaneous non-

homogeneous equations (Eq. (37)). Each resultant matrix was found to be of the

form

"T1 Tm ... T2 Al vIT 2 T 1 ... T 2 AV 2

S.... A3 VA 3 (51)

S4 V4

Tm Tm-I ... T Vm i

Thus. due to symmetry, Eq. (37) need only be evaluated once for any m = i, 2 ...m to determine all coefficients for the Am'S in the equation.

IResults of the self and mutual admittances for a circular array of cylindricaldipoles a quarter or a wavelength above a perfectly conducting cylinder of radius

r - 2X were calculated. See Figure 12 for array sizes of in = 2 to 22. Figure 13

shows the self admittance for the active element, Figure 14 shows the mutual ad-mittance between dipole no. 1 and dipole no. m. Due to symmetry, these twoshould be the same. The inconsistency may be the result of not taking large enough

11. Amos, V. E., and Daniel, S.L. (1977) AMOSLIB, A Special Library Version,

Sandia Laboratories (SAND 77-1390).

24

order n in Eq. (50) to have BESS(n) approximately equal to zero. Figure 15 shows

the mutual admittance between dipole no. 1 and the adjacent dipole no. m. Fig-

ure 16 shows the relative currents for the dipoles for arrays of size m = 2, 4, 7,

10, 13, 16. Once again, for m a 12, asymmetries occur which may be due to not

taking high enough order n for the evaluation of BESS(n) In Eq. (50) for the deter-

rmination of the coefficients for the AmIS.

ZV 2O 1I Y

-h

Figure 7. Array of Cylindrical Dipoles About a Perfectly ConductingCylinder, Simple Case, a 2X, p -r a +X/4

25

0.003000 1 1

0.002000-

0.001000

0.0000001

0 100 200 300 400 500 600 700ORDER

Figure 8. SR(n) - IQ(*)ReFS(N, a) for n 0 to 700

0.000 1000.000000-.

-0.0001000

-0.000200

-0.000300

-0.000400

-0.000500

-o 00060oc,

-0.000700

-0.000800

-0.000900

-0.001000 •0 100 200 300 400 500 600 700

ORDER

Fik.,-e 9. Sl(n) IQ(a) ImFS(n, a) for n 0 to 700

26

.0.000100

-0.000200

-0.000100F/-0.000400

-0.000300

-0.000400

-0.000500

"-0.000600 0 100 200 300 400 500 600 700

ORDER

Figure 10. BR(n) - Q(a) ReFB(n, a) tor n 0 to 700

0.000010 1 -- ...r

0,00000 0-

-0.000010

-0.0000201"0 100 200 300 400 500 600 700

ORDER

Figure 11. BI(n) - Q(o) ImFB(n, a) forn 0 to 700

27

•A

0.001000

0.0009000.0008000.000600

0.000700

0.000600-D 0.000500

0.000400

0.0003 00

0.0002000

0.000000 400.000000ORDER

Figure 12. BSUM(n).= 2(SR(n) + + (SI(n) + BI(n))2

IBESS(n)l for n 0 to 700

36

36

,-J

~28

2 4 6 8 10 12 14 16 I 20 22 24

NUMBER OF ELEMENTS IN ARRAY

1 ,Figure 13. Self-Admittance for Active Element for Arrays of_ __Size m __2 to 22

• •t 28

.14

.12

2 ri

10

.10 ii- .t

(.04

.02

2 4 6 8 10 12 14 16 i8 20 22 24

NUMBER OF ELEMENTS IN ARRAY

Fig're 14. Mutual Admittance Between the Active Element(Dipole No. 1) and the Adjacent Dipole (No. 2)

.16

.14

x

I uta dmt .Betwe teAtiei(Dpl .06

S.04

.02o I- -L A I I. I I I I , I It

2 4 6 8 10 12 14 16 IS 20 22 24NUMBER OF ELEMENTS IN ARRAY

I1 ~ Figure 15. Mutual Admittance Between the Active Elerment,

(Dipole No. 1) and the Adjacent Dipole (No. m)

29

~. . .. , 7; , *1•

2 ELEMENT ARRAY

a ,2

F" 1.3 GOZ

I-

02

ELEMENT NUMBER

14I- 4 ELEMENT ARRAY

F'I.3 GHZ

I-

z

:EUJ

0

01

o

2 3 4

ELEMENT NUMBER

Figure 16. Relative Currents For the Circular Array ofDipoles Above a Perfectly Conducting Cylinder For ActiveElement Dipole No. 1, m = 2, 4, 7, 10, 13, 16

30

thtI 1 ELEMENT ARRAY

g e ,+ ./4F- 1.3 0HZ

I NI

0

S 2 3 4 6 0ELEMENT NUMBER

SI- 0 10 ELEMENT ARRAY

to" a. '/

F,1.3 0HZ

S~Figure 16. Relative Currents For the Circular Array of"Dipoles Above a Perfectly Conducting Cylinder For ActiveElement Dipole No. 1, m 2. 4, 7, 10. 13, 16 (Cont.)

3.

I 0 13 ELEMENT ARRAYSalk

fee a )+ 4Fa 1.3 2lZI r

0

:1 S

o0 4 a 10 ItELEMENT NUMBER

I - •16 ELEMENT ARRAY

0-2),toe a0a+/4

F,1.3 GWZ

IL-

0o 2 4 6 a 10 12 14 16

ELEMENT NUMBER

Figure 16. Relative Currents For the Circular Array ofDipoles Above a Perfectly Conducting Cylinder For ActiveElement Dipole No. 1, m =2, 4, 7. 10. 13, 16 (Cont.)

32

____ ____ ___ ____ ____ ___ ____ ___320

00000 00000

L CONCLUSION

It has been shown that numerical techniques for approximation of the solutionof the m nonhomogeneous simultaneous equations (Eq. (37)) provide useful and ana-lytical data in the determination of characteristics of a circular array above aperfectly conducting cylinder. From Eqs. (37) and (50), it is seen that the valueBESS(n) must converge to zero for large order n to determine the coefficients ofthe Am's in Eq. (39). Due to the symmetry of the coefficients, as seen in Eq. (51),the values Tlit T1 2 1 ... Tm need only be calculated once for a given array of sizem. Also, from the expression for BESS(n), Eq. (50), BESS(n) is independent of m,the number of dipole elements in the circular array being considered. BESS(n) is,however, dependent on the values of the radius of the cylinder and the frequency(Wavelength). Consequently, tables can be made of the BESS(n) values for n = 0to N, where N is of order n large enough to force BESS(n) to zero. These tableswould enable calculation of the mutual and self admittances for an array of arbitrarysize m about a perfectly conducting cylinder of radius r'. An understanding of theeffects of mutual coupling, taking into consideration both the space wave and thecreeping wave. is necessary in the synthesizing of an aperture distribution to give

the best fit to a specified radiation pattern, and for the determination of optimiza-tion techniques for circular array antennas,.i

33

- - -- - ~ -M ..~s~t4.rt~., ~P~-

Rf erences

1. Provencher, J. H. (1970) A survey of circular symmetric arrays, PhasedArray Antennas, (Oliver and Knittel. Artech House, Inc.. Dedham.--Mass.)page 292

2. Hessel. A. (1970) Mutual coupling effects in circular arrays on cylindricalsurfaces-aperture design implications and analysis Phased Array Antenna.s.(Oliver and Knittel, Artech House. Inc., Dedham, Mass. ) page 273

3. Carter, P.S. (1943) Antenna arrays around cylinders, Proceedings of IRE,Vol. 31. December.

4. Lucke, W. (1949) Electric Dipoles in the Presence of Elliptical and CircuzlarCylinders. Report No. 1. Project 188, Stanford Research Institute,Stanford. Calif.

5. Lucke, W. (1951) Electric dipoles in the presence of elliptical and circularcylinders. Journal of Applied Physics, Vol. 22, No. 1. January.

6. Harrington, R. F., and LePage, W. R. (1949) A Study of Directional Antennasfor DIF Purposes, Report 1, Department of Electrical Engineering. SyracuseUniversity, Syrause, NY, September.

7. Harrington, R. F. (1961) Time-Harmonic Electromagnetic Fields, McGraw-.F,11, New York, NY. Chapter 3.

8. Neff, H. P., Hickman, C. :!, and Tillman, J. D. (1964) Circular AraWaround Cylinders, Report No. 7, Department of Electrical-Engineering,University oTTennessee, Contract AF19(628)-288, June.

P. Fante, Dr. Ronald L. (private communication).

10. Tillman, James D. (1966) The Theory and Design of Circular Antenna ArraysChapter 1, The University o Tennessee, Engineering Experiment Station.

11. Amos, V. E., and Daniel, S. L. (1977) AMOSLIB, A Special Library VersionSandia Laboratories (SAND 77-1390).

35

PA

MISSIONOf

Rome Air Developmen Center

&WC plans and conducts research, *xploratoxV a&*6 -afanceddevolpsn Program Ir command, control, anf comunications(C3 ) actIvitlw., and in the C3 areas of i•ifomartion aciencesand .tZolig iie..--rheo principal teclftcol mussion wuare commnmications, electroumagnetic guidance. wxd- camtrol,surveillance of ground and aerospac- objets, Jl !-aý1g nc•mdata collection and handling, infoxm.tion system technology,ionoepheric propagation, solid state sciences, microwavephysics and electronic reliability, maintainability andcompantibility.

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