ADAlI2 616 CALIFORNIA UNIV LO6 ANGELES INTERA1TED ELECTOAT-ETC F/f 20/14HZCROSTRIP DIPMAS ON CYLINDRICAL STRUCTURCS.W1UDKE $I N 0 ALEXOPOULOS. P L USLEWU4I DAA29-19-C-0O5O

UNCLASIFIED UCLAEG-U*.IA APO-I5965. 12-f L ML

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MICROCOPY RESOLUTION TEST CHART

NATIONAL BUREAU OF STAN bAARI', A A

NEL*%M i ut .,ufS-14

"MICROSTRIP DIPOLES ON CYLINDRICAL STRUCTURES"

by

N. G. Alexopoulos, P.L.E. Uslenghi andN.K. Usunoglu

Sponsored by:

U.S. Army Contract DAAG 29-79-C-0050

82 2 2vo-3 2 2

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S T. I . , ,, (andSubtitle. 6FI W I ITO R T & P ERIOD CO V ERED

Microstrip Dipoles on Cylindrical Structures LaboratrReoLaboratory Report

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Microstrip Dipoles

Cylindrical Structures

20 ABSTRACT (Cmtinue on roerse olde Ifnecooeoey d iddenllfy 6y block "1b016

An electric dipole tangent to the outer surface of a dielectric layer which

coats a metallic cylinder is considered. Exact expressions are obtained forthe electromagnetic field produced by the dipole, both inside the coatinglayer and in the surrounding free space. Asymptotic results are derived for acylinder whose diameter is large compared to the wavelength. Arrays of elem-entary dipoles are discussed.

DID I r 1473 EDITION OF I NOV65 IS OBSOLETES/N 00LF 0144601 SECURITY CLASSIFICATION OF THIS PAGE (lihea boff 3elm

Microstrip Dipoles on Cylindrical Structures ( "

N. G. AlexopoulosElectrical Engineering Department

University of California at Los AngelesLos Angeles, CA 90024

P. L. E. UslenghiDepartment of Information Engineering

University of Illinois at Chicago CircleChicago, IL 60680

N. K. Uzunoglu

Department of Electrical EngineeringNational Technical University of Athens

Athens TT-147, Greece

Accession For

, DTIC T -, ,

Avpil I- Codes

Dist E pecial

A A 9(*)iResearch supported by U. S. Army Research contract DAAG29-79-C-0050

1. Abstract

An electric dipole tangent to the outer surface of a dielectric

layer which coats a metallic cylinder is considered. Exact expressions

are obtained for the electromagnetic field produced by the dipole, both

inside the coating layer and in the surrounding free space. Asymptotic

results are derived for a cylinder whose diameter is large compared to

the wavelength. Arrays of elementary dipoles are discussed.

2. Introduction

icrostrip antennas and arrays have received increasing attention

in the scientific literature during the past few years, largely as a

consequence of advances in printed circuit technology. The state of the

art, in both theoretical and experimental studies, is sumarized in the

book by Bahl and Bhartia [I) and in the special issue [2), which contains

two exhaustive review papers on these subjects [3,41. The geometries of

microstrip antennas are not conducive to easy analytical treatment; for

example, rectangular and triangular microstrip patch antennas may be

studied by combining function-theoretic methods with ray-tracing tech-

niques [5,6,7]. Therefore numerical treatments, based e.g. on the moment

method [8], have been extensively adapted, especially for computation of

input Impedance and mutual Impedance [9,10].

Although most studies carried out so far have dealt with planar

substrates, from a practical viewpoint it is very Important to consider

the case of printed antennas and arrays on curved surfaces, especially

on portions of cylinders, cones or spheres. Together with a companion

work on spherical structures (111, this paper presents a detailed study

of dipoles on a dielectric-coated cylindrical structure. The exact.9Jfield produced by an electric dipole tangent to the outer surface of

_ 0 - -- -, . 7 ,

2

the coating layer is given in Section 3; the results are specialized to

the radiated far field and to the surface field. An asymptotic analy-

sis for the radiated equatorial field due to a longitudinal dipole is

given in Section 4, for a thin substrate and a cylinder whose diameter

is large compared to the wavelength. Preliminary results for arrays of

such longitudinal dipoles are presented in Section 5. The time-depen-

dence factor exp(-imt) is omitted throughout.

3. Exact Solution

Consider an infinitely long, perfectly conducting cylinder of

radius p - a coated by a uniform layer of constant thickness D - b - a,

permittivity cc 0 and permeability uo , and imersed In free space

(see fig. 1).

Let us introduce a cylindrical coordinate system p,$,z with the

z axis on the axis of the cylinder. The primary source t an electric

dipole located at r = (p 0 ,o'oZ) where Po . b , and whose electric

dipole moment is

4we-y-- c ,(1).

where Z is a unit vector and k = wr/- is the free-space weve-0 0

number. The source strength of Eq. (1) corresponds to an incident (or

primary) electric Hertz vector

ikRe- cR -rI, (2)

where r = (p,#,z) is the position of the observation point. It should

be noted that with the primary source normalized as In Eqs. (1,2), the

electric dyadic Green's function has dimensions of Un1 , whereas the

3

field is measured n a-2

The total (incident plus scattered) electric field Is given by:

E(r) - 4wkGI (r;) •^ a < P.1 b:to - "-0 (3)- 4thkG(II orar tnc p p b.

The electric dyadic Green's functions G In the coating layer and

G (II)in the surrounding medium may be obtained by the method described

by Tai [12], as amended in [13,14]. It should be noted that disagree-

ments on the singular term which appears in Eq. (6) below (see e.g. [15])

are of no relevance here, because the O term does not contribute to

the field generated by a dipole tangent to the cylinder (i.e., • = 0).

After imposing the boundary conditions at the perfectly conducting

surface p - a and across the dielectric interface p - b , as well as

the radiation condition at p " , it is found that-

=:a- -0 1 (J~n rM(1!(ur-) +cM

L n-0 n e " J o

C + u () +, ] 1-!!(3 ) (-,r)

r.(l)( + OM H, . -.

-ie(x (r~- G =o(r;r ) +C'r ;r-o) , (5)

C- (riro)a" -k 2p6(r r) +G(Co( -, (p< o 6

((4

G r u) r ) "- ,I (U)

+ MW (u,r)I+aM (U (p ) ,)

-go.T) -- n -o

4

GWr-. du In~ 1:[~!(u'.r)MK (-aur ) +

~0 - OWnu'0 n 0,0 0 0

+ NM1 (u,r)M (3 (-Ua,r )I pcp)--en'n -*ar 0

a 0

I T rG ,(rxo 2 fte 4 la!:n(u~r

+C MM (u"r)1 (3) (-u..r ) +. b N1(3) (U,rE) +(9)ztonT J 40 1 0

c M(3) (u,r)]N (UO Jo

where

MW U'D- vxrZ(J)(lp)COU(fl,).±UZ4lenhiln stnJ

e .iuz[~ Z()n~snn - a Z(J)(flp)cos(n);] (

P cn p 3 sin

N~)(ur) - 1 nj (u,rE)a

ex n -(e1)

a ~ i lu . Z(J)(np)cos(n#); Z J(ip)sin(n#); +

u 1 sin Cosn

+ f2zOj)(q~p)co(fl.);]

nux7yj u k* - ck - Ak (12)

j-1 or 3, Z W x)J ft W and Z)W - %. W(x are the Bessel

function and the Hankel function of the first kind, 6(r - _t) Is the

three-dimensional Dirac delta-function, To M 1 and T a 0 - 2, and

the Integral path along the real u-axis passes below the points

u - k, u - k and above the points u - -k, u - -I (see fig. 2).

The various coefficients which appear in Eqs. (4) and (9) are

given by (the prime means derivative with respect to the argument of

the primed function):

1), n Ot, -a() (13)n H 1 ) '(ta) nM(a

r 21rAn K - 2n-2 (14)

ne b6 H 1 l)(b)nn

Y 2fluyna 2Cb H (rDb) (1 - 2) 9 (15)

(rib) 2n-- . + ne An, (16)

b n~ + 2 n o (17)bn HMI (rib) Nn 2 RI) (rb)

n

C - 2YnB (18)n Nn 2 H(1(0) Cn

where

Ync Jn (C b) + H ()(Cb) .y Jn(b) + b (1 , (19)

ayna 2 n H(1 ) 0(b) (20)a -+ Ynab n

rB - 8b +:j YBo - nn (Tb) , (21)enl

....... MEO

6

-(n) 2v rn. (22)

an ra170 b (1 2 nyo

The above formulas can be considerably simplified In particular

case. Consider, for example, an axially-oriented dipole ( -

located on the substrate (P° W b); the total electric field on the

substrate and n the equatorial plane z to is:

E-pI-b F- 2 T coon - 0 ) duy 6 (23)

9 z[K wkn- t°

8:20

In the more general case when the dipole at r a (b, opZo) is

tangent to the substrate but not necessarily axially oriented, i.e.

A A^L A" + c ""zz, c " ;o M 0, (24)

then the electric field at any point r in free space is:

E~) E + E A (25)

*1 0 Cj ~

where

1 d k k211-2 e-tuz n os

wkb mO H )(0~) e,o sini.n

n n (nb) n2u2 2 ynB 4; 1)1 (3) (ur) +

V2 ~2 2 2 on-

+ no - Yn a ( _ - ]) N()3)b - (un" "-" (26)

1 du 2n-2 e- nnE"web d. n-o 81 (rib) *,o sin-Elln

7

'-CrA .(u.3 ± - - (2 ,fl (27)

In the far f eld (p + -), the Integral* in Eqs. (26-27) may be

asymptotically evaluated by the method of stationary phsee, the station-

ary point being at u - k cose where 0 - arcos(s/r) Is the usual

polar angle in spherical coordinates. If we wrte the radiated field as

Ikr

2 k re ( ) (P .. (28)

then the dimensionless far-field coefficients So1 ad S are-.

511 = S1 ' + Sll1(29)

21",. ( -c 2 - -- ° .WE -.- (A911S A*)*

2 -ik 0 caoW (D +e 3 (30)

where

- kbsine

A 6 - i I(..)n oon(,.s#)g (32)

nn

A#i (e: 1) Isla* (-iL)% n V a(*o **) (33)3 ,. ( 0

S

n(1 o2) 2cotO 2n no

nH ( 1) ' -Q )$ - ) (34 )

Sne H (~1)(C) J i ( 0 0n_- y(. ) .r "- (' *o

n

(35)

ctr' 0 C 1 C co2) n oCn(

an f(f -kcosO* Inpriuaibeveht j 8 adS 1are even and odd functions of 0 -o ) , respectively, an it mst be

by reason of symetry. Also, S11 -0When C - 1. If the cylind-

rical structure were absent, the dipole at the origin (p0 0)

and axially oriented (R = *) mould yield S11 *m sinS, Sj 0,

as expected.

4

I.D

!

p

9

4. Asyptotic Expansions for Thin Substrate

We limit our considerations to the far field produced by a dipole

on the substrate and parallel to the z axis, in the equatorial plane

0 - , so that $ -i* - 0. We assume

kb>> 1, Ik1DI << ; (36)

the second inequality means that the coating layer is electrically

thin. Then

Z(kia) : Z (k1 b) - klDZn(k 1 b) , (37)

z'(ka) 8 z'(kb) + k1 )D Z( -b)

where Zn - jn or HM; substitution into (32) yields, for 6 - w/2:

2ikD T n(_,)n coSn(# - ()(S ilo ) W S - k . ° . .. . ( 3 )

Set:

#0

(39)no M -ikD,

and observe that n0 would be the relative surface impedance of a

thin substrate with magnetic permeability equal to that of free space.

Then:

10

Sr 1n (-Lle rcan

-r n R 1 (kb) - inA u 1 (kb)

n on

-21n e- coosV dv (40)wk[n(1) (kb) -n (T1) (kb)] sin TV

where the contour C is along the real axis and just above It In the

complex v-plane, from - to + *The Integral in (40) is stmi-

lar to the one studied by Goriainov [161 In relation to plane-imve

scattering by a cylinder. Following [16), we set

1!V 3w ,e 2= I i- - 21e 2 siniv (41)

In the integrand of Eq. (40), so that

S - S I + 2

whereI

S1 -~ J N(k) dv, (42)

IV3w

-S 21 e ITCosv dv, (43)2 ikb Jc m(kb)uinwv

with

P1 (kb) - ul)(kb) - in U(1 1(kb). (44)

The integral Sl has a stationary point, as is seen by using Debye's

expansion for HM1 In (44); the integral 82 does not have a stationary

point. Assume

IV kbl > I 1/ 31 '(5

179

then, in a region about the origin n the -p"ane (for details see,

e.g. (17,181):

rr I kbS1 2- n 'u ' dv

(4~6)

e I i[v(.2+$+ arcco~ A-kb) 2 -v2 +i~+ exp1 +0 + arccos ) - -(kb) v2 +

the first integral in (46) has a stationary point at vs. - -kbsiau,

whereas the second integral has no stationary point. A stationary phase

evaluation of S is therefore obtained by considering the stationary

phase contribution due to the first term In the Integrand of Eq. (46)s

21kDcos(4 - $0) e-ikbcos(* - #0)

1 -_ coa0- 00•(47)

On the basis of Eq. (36 the denominator in (47).may be replaced by

unity, so that

-ikbcos(*, - 4,o )S -- 2ikDcos( -)e 0) (48)

At the point of stationary phase, condition (45) Is satisfied if

which defines the region of free space into which direct radiation by

the dipole occurs. Thus, Eq. (48) is valid in the "Illuminated region"

defined by (49), as shown in Fig. 3.

The far field in the penumbra and shadow regions, where nequality

(49) does not hold, Is obtained by letting

12

v kb+ut a k )1 3 1 (50)

into zq. (44), so that:

kDM,(kb) . , [w(t) + --v(t) . (51)arm

where wI(t) is Airy's function in Fock's notation [18]. The poles

of (40) n the complex v-plane are the zeros of M,(kb), i.e.:

v a - . (52)

Since this last ratio is large compared to unity,

kDt a Z tos 06 (53)

where the zeros tos (s 1,2,...) of wi(to- 0 are wall tabulated*

Since Inv > 0 at t., we may rewrite Eq. (40) as:

If W (34)

+ f ( . - ) ai k b ( t

where

~M(9t) , (55)

and the generalized Fock function

I(,p a dt (56)Jr v1 (t) + Ow(t)(

Is well known, and can be evaluated e,.g by residues at the poles (53);

the contour r starts at Infinity In the angular sector w • arg t v 1/39

13

passes between v - kb and the pole of the Integrand nearest the origin

(i.e. t t). and ends at Infinity in the angular sector

ir/3 > arg t > - w/3 (see fig. 4).

The approximation (54) Includes only the first two creeping waves,

which complete less than one complete turn around the cylinder; the

geometric interpretation of the two terms in Eq, (54) Is shon in fig. 5.

t

14

5. t-ras of Lontitudinal Dipoles

An axially oriented dipole at angular position 0o on the sub-

strate produces the far-field pattern of Eq. (48) in the Illuminated

portion of its equatorial plane. Consider an array of n such dipoles

with angular separation a between dipoles, i.e. the total array angle

is (n - l)a (see fig. 6). The far-field point of observation is In

the illuminated region of all dipoles if

w+ (n - 1)a + r2(kb)-4/ 3 < # < - /r(kb) -4 /3 (57)

Under limitation (57), dipoles fed with equal amplitude and progressive

phase shift 0 yield the pattern:

an_ e-ikbcos(# - is) + ibtStotal -2kD O-k - (58)

6. Concluding Remarks

The basic analysis for studying the behavior of printed circuit

antennas on cylindrical structures has been presented herein. Numeri-

cal results pertaining to current distribution and other antenna charac-

fsr

15

References

[1] 1. J. Bahl and P. Bhartia (1980), Microstrip Antennas, Artech

House, Dedham, MA.

[2] IEEE Trans. Antennas Propag. (1981), special issue on microstrip

antennas and arrays (D. C. Char%~, editor), Vol. hP-29, No. 1.

[3] K. R. Carver and J. W. Mink (1981), "Microstrip antenna technio-

logy", IEEE Trans. Antennas Propag., AP-29, 2 - 24.

[41 Rt. J. Mailloux, J. F. Mcllvenna and N. P. Kernveis (1981),

"Microstrip array technology", IEEE Trans. Antennas Prapag.

AP-29, 25-37.

[5] D. C. Chang and E. F. Kuester (1981), "Total and partial reflec-

tion from the end of a parallel-plate waveguide vith an extended

dielectric slab", Radio Science 16, 1 - 13.

[6] D. C. Chang (1981), "Analytical theory of an unloaded rectangular

microstrip, patch", IEEE Trans. Antennas Propag., AP29 54-62.

[7] E. F. Kuester and D. C. Chang (1981), "Resonance and Q proper-

t ties of Isosceles triangular patch antennas of 600 and 900

vertex", National Radio Science Meeting, Los Angeles, CA.

[8] E. H. Newman and P. Tulyathan (1981), "Analysis of Microstrip

antennas using moment methods", IEEE Trans. Antennas Propag.,

AP-29, 47-53.

[9] 1. E. Rana and N. G. Alexopoulos (1981), "Currant diatribution

and input Impedance of printed dipole*", IEEE Trans. Antennas

Propag., AP-29, 99 -105.

[101 N. G. Alexopoulos and I. E. Rana (1981), '"Mtual impedance compu-

tation between printed dipoles", IEEE Trans. Antennas Propag.,

AP-29, 106-111.

[11] N. G. Alexopoulos and P. L. E. Uslenghi (1981), "Microstrip

dipoles on spherical structures", National Radio Science Meeting,

Lo Angeles, CA.

[12) C. -T. Tai (1971) Dyadic Green's Functions in Electronantic

Theory, Intext, Scranton, PA.

[13] C. -T. Tai (1973), "Eigen-function expansion of dyadic Green's

functions", Math. Note 28, AFWL, Kirtland AFB, NN.

[141 C. -T. Tai (1976), "Singular terms in the egen-function expan-

sIon of dyadic Green's function of the electric type", Math. Note

65, AFWL, Kirtland AFB, NM.

[15] R. E. Collin (1981), "Dyadic Green's functions-Current views on

singular behavior", session on Scattering and Diffraction, 20th

General Assembly of the International Union of Radio Science,

Washington, D.C.

[16] A. S. Goriainov (1958), "An asymptotic solution of the problem

of diffraction of a plane electromagnetic wave by a conducting

cylinder", Radotekhinka i Elektronica 3, 23 - 39.

[17] G. N. Watson (1966), A Treatise on the Theory of Bessel Functions,

2nd ed., Cambridge University Press, Cambridge, England.

(18] J. J. Bowman, T.B.A. Senior and P.L. R. Uslenghi (1969),

Electromalnetic and Acoustic Scattertim by Simple Shapes, North-

holland, Ansterdam.

dipole source

region I (ee.,p.) (potion r.)

region 11 (E4.4) ;v;~;"

Figure 1 Geometry of the Problem

p Imu

__k____kRe

-k1 It%- kW *

;igure 2 Path of Integration Along the Rew"-axis *

Figure 3 Geometric Interpretation of Coadition (4S)

Jmv

H m/

ro

eigure 4 Contour r and Poles In the Comspfx P-plan*

x

(b)Figure 5 Geometric Interpretation of th* Creeping-

wave Terms In Equation (54); (a) lmt;(b)~-

Figure 6 Geometry for Circumferential Array