1. cylindrical-rectangular microstrip antenna

8/12/2019 1. Cylindrical-Rectangular Microstrip Antenna

1/6

194 IEEE TRANSACTIONS ONNTENNASND PROPAGATION, VOL. AP-31, NO. 1 , JANUARY 1983

. e S T E P P E D R E S O N AT O R---- E Q U I VA L E N T RECTANGULARRESONATOR

I I2'2 2 4 2'6 2' 3'0 3'2

F R E Q U E N C Y IN G H Z . -Fig. 2 . V S IR versus frequency plot.

VSWR versus frequen cy plots of the resonators shown in Fig. 1are given in Fig.2. The (1 :2) VSWR bandw idth of the m icrostripreson ator on a wedge-shaped dielectricis 28 percent and thatona stepped dielectric s 25 percent, whereas the bandwidth is13percent oranequivalent ectangular esonator.Themaximumheight for l the resonators was0.01 m . This indicates that thereis considerable improvement of bandwidth over that of a similarrectangular microstrip resonator. It may be mentioned hat hefeed point was in th e sa me position forll the resonators.

ACKNOWLEDGMENT

The authors gratefully acknowledge the cooperation extendedto them by Telec ommu nication Re search Cente r, New Delhi andthe Radar and Communicat ion Center,. I . T. Kharagpur for pro-viding HP network analyzer measurement facility.

REFERENCES

R . E . M u n s o n , "Conformal microstrip antennas and microstrip phasedarraks. IEEE Trans.AntennasPropagat . . vol. AP-22. no. 1 pp.7 4 7 7 . Jan. 1974.J . Q. Howell, "Microstrip antennas, IEEE Truns. Anrerrnas Prupu-g o t . . vol. AP-2.3. pp. 90-93, J a n . 1975.J . Watkins, "Circular resonant structures in microstrip." Eleerron.Le t r. . \ o l . 5 no. 21, pp. 524-525. Oct. 1969.A . G . Derneryd, A theorectical investigation of theectangular

microztrip antenna." lEEE Trans.AntennasPropagur.. vol . AP-26.pp. 532-535. July 1978.Y. . Lo er a / . . "Theory and experiment on microstrip antennas."I E E E Trans.AntennasPropagat. vol. AP-27. no. 2 , pp. 137-145.Mar. 1979.C . Wood . "lm pro>ed bandwidth of microstrip antennas using parasiticelements." Proc. Ins t . Elec . Eng. H . Microwaves ,O p t . Acolrsr.. \o l .

A G . Dernrr d e t a i . . "Broad-band microstrip antenna element and127. pp. 231-134.Aug. 1980.

array," I E E E Trans . Anrerlnas Pr opa gat . . bul. AP-29. no. I pp.11 111. J an . 1981P. S . Hall e r a i . , "Widebandwidthmicrostripantennas for circuitintegration." Elecrron.Lett . . vol . 15. no. 15. pp. 4 5 8 1 5 9 . J u l y 19.1979.

J . R. Jams5 er a i . . "Design of microstrip ntenna feeds. Part I:

E n g . H . Microwa l. e s . Op t . Acou s t . ,vol. 128,Feb. 1981. pp. 19-25.

Estimation ofradiation loss and design implications," Proc. n s r Elec.

Cylindrical-Rectangular Microstrip Antenna

CLIFFORDM. ROWNE

Absrracf-Resonant frequencies f, of a cylindrical-rectangular micro-strip antenna are theoretically calculated. omparison is made tof, for aplanar rectangular patch antenna, ncluding the simplest planar patchmodes having no field variation normal to the patch surface. The validitof using planar antenna patches to characterize microstrip antennas is

examined.

INTRODUCTION

In many applica tions pertainin g t o satellites, missiles, spcraft, and aircraft, conformal microstrip antenna patches asedMicrostrip antenna patches are placed above what may be char-acterized as a conducting plane with a dielectric substrate sept ing he patch from he conduct ing plane [ l ] However, oftenthis plane surface is either distortedor the antenna elements arintentionally placed on a curved surface. Thus to deter minecorrect modal field solution to the electroma gnetic cavity prolem, which can be used to find the radiation field solution, curvature should be taken into account. Here this is don e frectangular patch on a cylindrical surface. The assumption hatthe conduct ing patch and he conduct ing cyl inder (ground sur-face) act as electric walls, and that the open cavity ends act amagnetic walls is applied to t he analysis for obta inin g the fieand associated modal resonant frequencies [4]. his assumptionshouldbeparticu larly valid whe n using these fields fordeter-mining the radiation pattern for the limiting caseof thi n cavitieI? Q n) which are utilized for most microstrip antenna applica-tions. All of the analysis for simplicity also assumes that the mittivityE and permeabil i ty pare constant (homogeneousmfilling cav ity) and real (no dielec tric losses).

The eigenvalue equatio ns for resonan t frequen cies f , are nu-merically solved andexamined overa range ofdielectric ub-stratehicknesses k . Theseesonantrequencies f,c fo r hecurved cylindrical-rectangular antenna, representing a distorof a planar rectangular microstrip ntenna, re ompa red oresonant frequenciesfrR of the planar patch antenna in order assess the validity o f he com mon ly used assum ptio n hat coformally moun ted microstrip antenn as may be treated as plaThe results demonstrate that this assumption isgood for h thais small compared to the surface curvaturea , and that i t isex-cellent when considering excitation of the antenna with no field variation normal to the surface.

T H E O RY

The geome try of the cavity is shown in Fig.1 where Fig. l(ais a perspective drawing of a conductin g patchon a cylindricasurface, Fig. l(b ) is a cross section hrough he patch and normal to th e z-axis, and Fig. l(c) shows the cavity iso lated bytselfin cross section. The conducting patch and grounded cylindricalsurface are treated as electric walls and the magnetic walls ofcavity aredefinedbydroppingperpendiculars rom hepatch

Manuscript received August 5 , 981; revised June 29, 1982.The author is with the Electronics Technology D ivision, Naval Research

Laboratory, Washington, DC 20375.

U.S. Government work not protected byU.S. copyright.


2/6


3/6

9 6 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-31,NO. , JANUARY 1 9 8 3

H , = --A,~i(k,j)2R,(k,ip)i sing )os g) . ( l o e )wE1

Cavity resonant frequencies f for each mli mode are foundrom the equa t ion

nd (2c):

-

The TM to z field solution is constructed by choosingF = 0and A' = &$. Using (1) and (2) and applying magneticwall BC's,Jcan be wr i t t en as

Here Bmli is a constant for the mli th mode,m = 0, 1 , * e , and 1 =1 , 2, -, and R , satisfies (4) with p = k , ip. Equat ion(5) still ap-plies. Imposing the electricwall BC's on (5) requires

R u e ) p = a , a h = 0 , (14)

c1J,(kmi[a +h ] ) +c2Nu(kmi[a h ] ) = 0.n order thatl, c2 # 0 ,

Jdkm a)n'u(km i[a + h 11

-Ju(km i[ a + h ] Nu@, ia) = 0 > (1 6)which is obtained rom (15). Equa tion(16) produces n n-inite denumerable set of k , eigenvalues withi = 1 , 2, . r-

bitrarily setc2 = 1 and f in dc1 fro m (15a):

'1 =Crni=--N, km,u)/Ju k,ia). (17 )

The field solution is obtained from(4) and (13)

E = -i B,lik,i z),'(k,ip) cose$)OS 5 )U E

Cavity resonant frequenc ies are found from 12).

Inspection of the cavity field solutions in (10)for t h e TE t oz modes and (18) for the T M t o z mo4es sh2ws tkat the com-plexpowerdensity Poyntingvector) P = E X H* is purelyimaginary. Thus the t imeaverage power flowPav o u t of th e wallsis zero. This result is expec ted, howe ver, some comments are iorder.For pplicationswhere he cavity is used tomodel a.microstr ip antenna radiator, the fol lowing procedure can be uto obtain radiat ion. Firs t the cavi ty f ie ld solut ions are obtained

as t hasbeendonehere.Next he ieldsolutionsareused asHuygens ources t heopen wall (formedbyperpendicularsdropped from the antenna patch edges to the conduct ing surfbelow)boundaries.These wall fields allow foractual adiation(with the dropping at this stage of the magneticwall conditions).Finally he adiationproblemmaybe simplified by using th eequivalence principleto exp rejs the bou$dary fields alternativela,s radiatingelectriccurrent J = 6 X b ndmagneticcurrent

= E , X sources ( b subscript denote s bounda ry and is normal to wall which points outwa rd from avity).

Consider the limiting case of the c avity wh ereh 0. First ex-amine the TM, modal field solution. Equation (16) can be writ

ten asE@) = J,@)nr,(P + A P ) - u o , + A P Y t J ( P )= 0 (19 )

by let t ing p = ak, i and A p = hk, i. Using Taylor series expansions of theBessel func tions abo ut and retaining only irst ordterm s inA p , the left side of (19) can beimplified t o read

E P) = [ J u @ ) N , ' ( P ) - u (PWLJ(P ) lP ( 2 0 4

L

= - 4 0np

Equatio n (20b) was obtained from (20a) by identifying the terin brackets as the Wronskian ofJ , a n d N u .Combining (19) and(20) require thatp .+ 03. That is,a finitek, solution to th e adialBessel equation canno t be found for the limiting case ofA p / p0 or h/a 0. Thus for physical reasons theTM t o z field solutionwould not be used to obtain the general field solution.For com-pleteness, however, the TM toz field solution will be given fothis limiting ase. Referring t o (17) and (1 8), and setting3 = ln/2b

E,, = - ,li P COS (u ) OS (/3~); EO = E, = 0 (2 1 )WE

H# = Bmu' cos (@) sin (/3z); H p = H , = 0, (2 1 )

whereBml : is related to Bmlib y

Bmli' = -kmiBm~iRur(kmia). (22)

Next exam ine the TE, modal field case. Equ ation(8) can beexpressed as

= J,'(P)N,'(P + I -Ju (P + a P > N u ( P ) = 0. (23)Following the same procedure in going from (19)to (20a), theleft side of (23) becomes

D ( P ) = [J,'(P)A'u (P) -Ju (P)h : ' (P) l A P. (24)Utilizing (4) to el iminateJ , andN, in(24) transformsD ( p ) in to

u2 - 2

P 2

P

N P ) = [ J , ' (P)n 'u(P> -J , (P)Nu ' (P) l40> ( 2 5 )which by (20) becomes

2 p 2 - l ?D ( p ) = -

1 P34 0 .


4/6

TRANSAC TIONS ON ANTENN AS AND PROPAGATION, VOL. AP-3 1, NO. 1 , J A N U A RY 1 9 8 3 1 9 7

rom (23)D@) = 0. Imposing this constraint on (26) producestionsp = +u = kma /2 , if p ZO. Since m is an integer on t h eain 4 , ), p = u isa complete solut ion s tatement . For0, application of L'Hipital 's rule sh ows thatD(0) = 0 andis automatically satisfied. Sincep = 0 corresponds tom = 0,

u for all m on the integer domain.he TE t o z field solution for the limiting caseA p / p - 0 or

0, referring to(9) and (lo), is given by

(27a)

o = mI 'Pcos (@) sin (Pz); H p = 0i

ww-7

z = _2 Aml 'usin (u0)cos (Pz).wluI

we have usedk , = mn/2Oa, cmi = c, , and,I' = A,liUR, U). (28)nant frequencies using(12) are given by

hemodal ( f r ) m l are hesame orm as foun dfor a planarangularcavitywith k, = nn/h = 0 (i.e., n = 0 giving n ovariation n he x coordinatedirection) or he TE t o zes. This k, choice produces the same planar rectangular fieldtion functional form as we found in (27), the correspon denceg obvious if the limita - is applied and the identificationsy : 20a 2c, andp + x made (2c is the extent in the rectan-

r coordinatey direction).is interesting to note that the c orrec t ield solutions for thin

ties seen in (27) are obtaine d by solving the electrom agneticlem subject to the constraint thatl variation withp goes to; i.e., a / a p 0. The procedure of takinga/an 0 where isit vector normalto the conformal conduct ing surfaces of tenied t o microstrip antenna problems and leadsto good agree-t between theory and experiment[ 4 ] .

NUMERICALRESULTSor llustrativepurposes hedimensionsof hecylindrical-ngular cavity will be chosen to bee = 2 cm, b = 1 cm, 0 =The filling elative dielec tricconstant is set to E , = 5.0.

m h e k , eigenvalue solutions as determinedby 16), heli cylindrical-rectangularrequency eigenvalues fyc ared using( 1 2 ) and are shown in TableI for m = 0 , i = 1 -

nd = 1. The TE, li frequen cy eigenvaluesfyc re found in aar manner using(8) and (12) and are given in Table I1 for1 - , = 1 - , and = 0. Both tables providef,c for h/a =0.25, 0.50, 0.75, and 1.00.

he effect of curvature on a patch antenna'sf can be ascer-d by considering the inner radial surface of the cylindrical-

angularpatchantenna obeequal o hepatchareaof aar ectangularpatchantenna.The radial thicknessh is setl t o th e planar patch antenna's dielectric substrate thickness.ectangular dimensions are herefore x = h , y = 2Oa, and2b. Tables I11 and V give f r R for the rectangular patch for,ctively, h/a = 0.1 and 1.0,

TABLE IRESONANT FREQUENCIES F OR T H E T M , ~ ~ YL IN D RI CA L

RECTANGULAR CAVITY MODES

ha m O 1 2 3 4-

I

1

20.10 3

4

5

1

2

0.25 3

4

T

1

33.6812

67.11151oo.6080

134.1122

167.6219

13.8115

27,0187

40.3580

53.7310

61.1177

1.4825

33.8062

61.2257100.6a01

134.1665

167.6653

14.2642

27.2542

40.5162

53.8500

67 .?I30

8.1474

34.5333

67.5495100.8969

134.3290

167.7954

15.5424

2 1 . 9 4 9 3

4 0 . 9 a n

54.2054

67 LI;U

9 ,8668

35.5697

68.0858101.2567

134.5996

169.0121

17.4626

2?.0704

41.7606

54.7926

h l 97nh

12 .1789

36.9718

68.8296101.15Rb

134.9774

168.3149

19.8397

30.5721

42.6292

i5.6G46

h8 h?hq

14.7789

2 13.8156 14.1917 15.27336.92779.0098

0.50 3

5

12

0.75 3

4

5

1

2

20.5138

21.0194

33.6822

5.57249.5375

13.8139

18.1835

22.5916

4.7262

7.4856

20.6429

27.2159

33.8401

6.302910.0002

14.3395

18.4126

22.7928

5.4483

7.9932

21.4031

27 . ~ 1 4

34.3096

8.072611.2803

15.0377

19.1622

23.3870

7.1178

9.3591

22.61b1

28.7417

35.0790

10.289013.1466

16.5334

i0.3261

24.3486

9.1102

11.2583

24.2115

30.0166

36.1300

12.652515.3735

18.3943

21.8623

25.6408

11 . 1 5 ~ 1

13.3957

1.00 3 10.5948 10.9612 12.0074 13.5916 15. 506

13.8146 14.1002 14.9269 16.2303 17.9170

5 17.0867 17.3187 17.9996 1?.oa94 20.i3L4

Here m = 0-4, = 1, i = 1-5. h/a = 0 . 1 , 0 . 2 5 , 0 . 5 0 , 0 . 7 5 , and 1.00.0 =2 cm, b = 1 cm, e = 24" , and er = 5 .0 .

TABLE I1RESONANT FREQUENCIES FOR THE TEmIi CYLINDRICAL

RECTANGULAR C AVITY MODES

m . 1 2 3a

1

2

0.10 3

3

1

2

0.25 3

1

2

0.50 3

5

1

2

0.75 3

4

5

2

1.00 3

4

5

3.8117

33.7462

67.1455

100.6297

134.1285

3.5617

13.9070

27.065:

40.388153.7542

3.2029

7.5291

13.8246

20.3896

27.0236

2 . 8864

5.5236

9.4144

13.7656

18.1152

2.6013

4.5719

7.3312

10.4740

13.7212

7.6230

34.3685

67.4744

100.5464

134.2011

7.1111

15.2L70

27.7656

40.862:54.1107

6.29?1

9.5525

14.9569

21.1657

27.6119

5.5190

7.8925

1 0 . 8 9 0 ~

14.7470

18.8938

1. 545

7.0171

8.9682

11.6081

14.5818

11.4335

35.4334

68.0118

1 0 i . 2 0 6 6

134.5618

1 0 . 6366

1 7 . 2 6 1

28.7038

61.6L01

.54.098

9.2L.63

12.2701

16.6354

22.IrDb3

28.i6i4

7.9754

10.6914

13.c0:9

16.2170

20.084il

5,9525

9.4959

11 , 3300

1 j . j S j

15.9575

15.2426

36.8470

68.7571

101.7088

134.9391

lC.1262

I?. 649

30.4240

LZ.795555.5142

12.0673

15.3210

18 .8978

i4.0'13

20.8376

0.3760

13.4310

:5.6652

1t.2622

21.6723

9.0795

11.8GLR

13.8137

15.6580

17.7125

I ? . 0499

38.5892

69. 036

102.3508

135.4243

17.5892

22.6055

32.2770

'4.02.L

56.544'

1L.8769

18.442'

21.4561

26.012'

31.1452

12.7550

16.0361

:Koa76

20.6383

23.5902

: .1614

14.0fn4i

16.2729

18.2029

20.0036

Here m = 1-5, = 0, i = 1-5. Same cavity dimensions as in Table I.


5/6

198 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-31, NO. 1 JANUARY 1983

TABLE 111RESONANTFREQUENCIES OR THE TM li RECTANGULAR

CAVITY MODES FOR = 1 AND h/a = 0.1 WITHg = 1.675516IN (30)

m O 1 2 3 4

1 33.684993.921764.622351 37. 29334

2 67.119387.238517.5946713

3 100.6093000.6888200.9270001.3227301.87416

4167

5 167.6226067.6703467.e134868.0487766.38482

_ _ _ _

TABLE NRESONANT FREQUENCIES F O RT H E T M ~ ~ ~ R E C TA N G U L A R C AV I T Y

MODES FOR I = 1 AND h/a = 0.1 WITHg = 1.759292 IN ( 3 0 )

O O 1 2 3 41

1 33.68499 33.89981 34.52628 39.57175 36.97272

2 67.19375 67.22745 67.55062 68.08584 68.82815

~ _ _ I ~3609308143750613

434.11316 134.16728 134.32950 134.59944 334.97645

5 167.62260 167.66590 167.79574 168.01192 166.31L11

TABLE VRESONANT FREQUENCIES FORTHETM,~~RECTANGULARCAVITY

MODES FOR = 1 AND h/a = 1.0 WITH g = 1.675516 IN (30)

m O 1 2 3 4I

_ -I 4.740135. 20 29 06 9 . 3 n ~ o 4 1 2 .9 04 7 78 36.69@6 8

211

3 10.5992641.3232343.2805506.0127709.1952C4

4 13.8197:29 14.3 872 385.9691618.304368 21.1411779

5 17.09079':7.5526518.8712490.8864403.413884

.-

with g = 1.6755 16. Equation (30) gives f r R in GHz. For non-trivial TM, rectangula r patch field solutions= 1 , 2 , e , m= 0, 1*-, and 1 = 1, 2 , e-. These eigennumbers correspond exactly tothose in Table I. Comp arison of thef,c and f r R eigenvalues inTab le I and Tab le 111 fo rh/a = 0.1 shows the largest differencebeing 0.87 percent. Asi increases the difference is reduc ed, but asm increases, thedifference is increased.Thissame f r behaviorwith varyingi and nz occurs for h / a = 1 O in Tables I and V withth e largest difference being 4.4 pe rcent.For the TE, rectangularpat ch fields, having nontrivial solutions ati = 0, 1, .-,nz = 1 , 2 :

-..:2 = 0,

1,e - , the difference is about an order of magnitude

larger. Tables I1 and VI1 forh/a = 0.1 demon strate hat helargest f r difference is 5.2percent,andTab les I1 and IX forh/a = 1 O tha t thi s value is enlargedto 79.2 percent.

The TE, cylindrical-rectangular eigenvalues correspondothe two-dimensional resonating modes of the planar rectangularpatch antenna w hich have no field variation normal to the patchsurface.Thesemodes re ften til ized in planarmicrostripantenna analysis, and havef r R and f rc differing by less tha n5percent forh / a = 0.1.

Tables Iv, VI, VIII , and x provide f r R calculated using (30)with g = 1.759292 (h/a = 0.1) or 2.513274 (h/e = 1.0). Usingthese g values mean s hatanequivalent ectangularcavity tothe ylindrical-rectangular avity)hasbeen oundbecause naverage radius r = a + h/2 has been chosen for one side of thecylindrical-rectangular cavity. Onemightxpect f y R toeextremely close to f r c . Agreement is within0.01 1percentfor

TABLE VI

MODES FOR I = 1 AND h/a = 1.0 WITH g = 2 5 1327 4 IN (30)RESONANT FREQUENCIES FOR THE TM,liRECTANGULARCAVITY

m0 1 2 3 4

1. -_

i .iLGi35 5 . &3?GLZ 3 . , , L C 9 9 . OG4OL...,*, .. ,-,,-,I I . V I + V J *2.494811 7.955281 9 . 7 5 8 3 0 3 10.963614 13.038438

3 10.599261. 10.929716 11.865983 13.280550 15. 039051

43.819749 14.074792 14.813596 15.969161 17.458927

57.090799 17.297679 18.68 1803 18.871249 20.147558

TABLE VI1RESONANT FREQUENCIES FOR THE h l i RECTANGULAR CAVITY

MODES FOR I = 1 AND h / Q = 0.1 WITH g = 1.6755 16 IN (30)

m1 2 3 4 5

1

0 4.00089 8.00179 12.00269 16.00358 20.00447

1 33.75576 34.45973 35.60208 37.14245 39.03361

2 67.15492 67.51151 68.10 70 68.91 945 69.956.8 1

3 100.63101 1CO.87133 101.26727 101.81901 102.52402

4 13s .130q4 1 3 4 . 3 0 ~ ~ 4 1 34 .6 0 74 6 1 35 . 02 30 3 135.55546

TABLE VI11

MODES FOR I = 1 AND h/a = 0.1 WITH g = 1.759292 IN (30)RESONANT FREQUENCIES FOR THE TE,liRECTANGULAR CAVITY

1 2 3 4 51

0 3 . 8 1 0 3 8 7 . 6 2 0 7 5 11.&3 13 l> :Z415U i 9 . 05 28

1 33.73371 34.37324 35.41348 36 .E2048 38.55409

2 67.14384 67.46741 68.00329 68. 74649 69.6903Y

3 100.62562 100.84182 101.2G112 101.70201 102.3/12&2

434.12540 134.28768 134.55770 134.93483 135.41616

TABLE IXRESONANT FREQUENCIES FOR THE E,ziRECTANGULAR CAVITY

MODES FOR = 1 AND h/Q = 1.0 WITH g = 1.6755 16 IN (30)

m 1 2 3 4 5I

0 4.008949 8 .0017895 12.002685 16.003579 20.004474

1 5.219349 8.6754289 12.461896 16.350809 20.28 3328

2 7.806723 10.438697 13.747806 17.350858 21.097789

30.822066 12.850626 15.658045 1a.900383 22.389482

4 13.991361 15.613445 17.994873 20.877393 24.081735

TABLE XRESONANT FREQUENCIES FOR THE TE,liRECTANGULAR CAVITY

MODES FOR I = 1 AND h / Q 1.0 WITH g = 2.513274 IN (30)

I 2 3 41_ _ _ _

0 2.653263.334526.0017bY IU .bbYb5L 1 3 . 3 3 b 3 1 3

1 4.287542.300128.675L291.1631623.751063

2.214711.5670542.0267312.600?554.926321

3 10.4030881.3827552.85C 626 14.6CO.;SS 16. 702 313

43.Cb9E694.4294215.6134457.131.?678.91G535

_ _ _ ___-____

h / n = 0.1 and 4.4 percent or lz/a = 1 O for the TM, mode sFor the TE, modes, greement is within, espectively, 0.035percent and 19.5 percent forh/a = 0.1 and 1.0.

CONCLUSION

The ielddistributionwithinacylindrical-rectangularmicro,strip antenna has been determined using a cavity modal mode


6/6

ENSACTIONS ON ANTENNASNDROPAGATION, VOL. AP-31, NO. 1,ANUARY 1983 199

r he TE, and TM, mode s.Resona nt requenc y eigenvalueuations or hesemodeswere used t o calculate he eigen-quencies f r ) , li for some of the lowest order modes.Comparison f J . )mli forhe ylindrica l-rectangu lar casec, nd Cfv)mli for a planar rectangular microstrip antennaf i R ,ows n ssessment of heeffectcurvaturehas on resonantquency. Numerical results show that curvature changes f, b ys than about5 percent for a substrate dielectric thickness equalone tenth of the radius of curvature. The exact effect of curva-e on f will be depen dent on the particular choiceof antenna

rameters, but the results foundin this communication shoulda u seful guide when esigning conformal m icrostrip antennas.An equivalent ectangularmicrostripplanarantenna to helindrical-rectangularmicrostripntennaadeenefined.reementbetween f , ~nd f rc is better han0.035percent.

R E F E R E N C E S] K. R. Carver and J . W . link, Microstrip antenna technoloey, I E E E

Trans . Antennas Prop aga t . ,vol. AP-29. no. I , pp. 2-24, Jan. 1981.] Y . T. Lo, D . Solomon. and W . F. Richards, Theory and experiment

on microstrip antennas, IEEE Trans. Antennas Propagat. , vol. AP-27, no. 2 , pp. 137-145, Mar. 1979.

3 ] R . F . Harrington. Time-Harmoniclectronzagnetic Fields. NewYork: McCraw-Hill, 1961. p. 480.

] M. Abramowitz and I . A. Stegun. Handbook of Mathematical Func-tions with Formulas~Graphs , andklathematicalTables. Nat.Bur.Stand. Appl. Math. Ser., 1972. p. 1046.

] H. Saean. Boundary and Eigenvalue Problems i n Mahem atical Phys-i c s . . N e w York: Wile?. 1961, p. 381.

] R. V . Churchill. FourierSeries und Boundur: Value Problems. NewYork: McCraw-Hill, 1941, p . 206.

] C . M. Krowne. Cylindrical-rectangular microstrip antenna adiationefficiency based on cavity Q factor, IEEE Antennas Propagat. I n t .S J m p . Soc. D i g . , June 1981. pp. 11-13. (Note that in (7a ) of thispaper. k o j in the numerator needs a factor of 2. Same factor is neededfor R i n (7bl . )

Combined E - and H-Plane Phase Centers ofAntenna Feeds

PER-SIMONKILDAL, MEMBER, IEEE

Absrrucr-The feed efficiency,a irst approxim ation o the apertureiciency of a paraboloid or a conventional Cassegrain antenna, is useddefine uniquely a combined E- and H-plane phase center of the feedtern . A formula or numerical calculationof hecombined phaseter is presented, as well as heoretical results of he eedposition

erances and the efficiency oss due to differences in the principal plane

ase patterns.

I. INTRODUCTION

For designing aeflector ntenna ystem.t is impo rtantknow hefeedphasecenter since itdetermines he ocationthe feed relative t o th e focal point of the reflector.Too largeeviation betwee n he wo points causessevere defocusing ofe secondary radiation pattern. The phase center of an antennadeally the centerof a portion of a sphere on which the radia-n field of the a nten na has a constant phase[ l ] . 121. However,

Manuscript received March 29, 1982; revised June 29, 1982. This worksupported by the Norwegian Council for Scientific and Industrial Re-

ch.The author is with Electronics Research Laboratory, The Norwegianitute of Technology, 0. S. Bragstads Plass 6,N-7034,Trondheim-

H, Norway.

in man y cases the phase patterns are oscillatory,so tha t an idealphase center does not exist. A more practical definition of th ephase center has been given by Rusch and Potter [3, p. 1471as thecenterof he best nearlyconstant-phasespheres in aleast squares sense. A lthou gh the defin ition given by Rus ch andPotter is general enough to obtainacombined E- andH-planephase ce nter wit hin a given solid angle, practic al formu las weregiven only for nume rical calculatio n of a phase center in a singleprincipal plane. This communication presents a practical formulafor calculation of a combined E- and H-plane phase center, andis therefore an important supplement to R usch and Pot ter 's wo

The results n his comm unication are obtained by a slightlydifferent general definition.Theeedor araboloid rconven tional Cassegrain reflector ystem is located in suchway that the phase reference point of the feed pattern coincideswith he focal point of he reflector system. The phase centeris thendefined as thephase eferencepointwhichmaximizesthe eedefficiency, he atterbeing he irst-orderapproxima-tion o heapertureefficiency of theparaboloidor he Casse-grain system. This definition is equivalent to that of Rusc h andPotter [3] for proper hoiceofweighting unctions in theleast-squaresalculation.heesultingalculationormulafor th e overall phase center is therefore also very similar to th eformula for he principal plane phase center. The olerances onthe phase-c enter position (correspo nding o he axial toleranceson hepositionof he eed in the eflectorantenna)and heefficiency losses d ue o diffe renc es in the principal plane phasepat tern s are also discussed.

11. MATHEMATICAL FORM ULATION

We assume a far-field feed pat ter n, wh ich is linearly polariz edand determined by i ts E-plane pat ternA ( ) and H-plane pat ternc( ),

ccording to

where 2~ and i tcare unit vectors in th e direction of increasingpolar angle $ andazimuth angle t , respectively. Equation( 1 )assumes that A ( ) and C($) are given with respect to the samephase reference point 0 positioned at z = 0 and ha t p is thedistance rom 0 to he far-field poin t (Fig.1 ) . A ( ) and C($)can be transformed to a new phase reference point P positioneda t z = 6 : by the relations

~ ~ ( 1 :( ) e - i k 6 c o s 0 . c , ( l ) )=C( ) e - ik6cOS? (2)The n he ar field is given by( 1 ) if A & ( + ) nd C6($) replace

A ( ) and C($), respective ly, and ifp now is the distance fromP to the far-field point.

We consider the feed to be located in the primary focus of aparaboloidal antenna or in the secondary focus of a convention aCassegrain antennahyperboloid-paraboloidonfiguration) insuch a way that the pointP coincides with the proper focal point.Then he irstapproximation to heapertureefficiency of thereflector ntenna, xc luding he ffectsof blockage and ub-reflector diffraction in the Cassegrain, is

00 1S-926X/83/0100-01991 .OO 1983 IEEE

1. cylindrical-rectangular microstrip antenna

Documents