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FE/Mode-Mat ching/Moment MethodAnalysis of Horn Antennas

Rairicr Bungcr, arid Fritz Ariidt 'hlirrowave Depa rtment, University of ben i enr.0. ~~ 330 44 0Kiifstriner Str. N W 1, D-28331Breinen, Germany

A b s t r u c t ~ A h y b r id m e t h o d is p r e s e n t e d f o r t h e r i g o r o u s a n a ly s i s of h o r n a n t en n a s . T h ei n t er i o r h o r n t a p e r s t r u c t u r e of a r b i t r a r y c ro s s - se c t i o n is d e s c r i b e d b y t h e f a s t F E / m o d e -m a t c h i n g m e t h o d c o m b i n e d w i t h t h e g e ne r a li z ed s c a t t e r i n g m a t r i x t e c h n i q u e . For t h e e x t e r i o rh o r n p r o b l e m i n cl u d in g t l re r a d i a t in g a p e r t u r e , a n E F I E f o r m u l a t i o n i s u s e d w h e r e t h e e i ge n -v e c t o r s of t h e w a v e g u i d e t a p e r a n d t li e b a s i s f u n c t i o n s of Rao are c h o s e n a s b a s i s f u n c t i o n s .R e s u l t s for a c o n ic a l h o r n s h o w g o o d a g r e e m e n t w i t h m e a s u r e m e n t s . T h e f l ex ib il it y of t h ep r e s e n t e d i n e tl i o d is d e m o n s t r a t e d a t t h e e x a m p l e of a c o n i c a l r i d g e d w a v e g u i d e h o r n .

I . INTRODUCTIONOR T t w rigo roiis niialyxis of horn antcnnaa incl t iding the outer horn surface, many tcchniqiws ar c avai lable[I]171.The intcsgral equation 121, finitodiffcrcrnce 131, m d comIrinw1 i n t egrd-equat ion hnindary finite

h i i e i i t [SI solutions are flexible hut the high ininierical effort irsually rcrqiiired for acciirate rcsiilts oftenl imits their prar t i i :al appl icabi l i ty. The modernatkt i ing techniques of [GI, 71 are efficient, the outcr surface,however, is restrictrd to a spher ical segment [G ] or to s t ruct i r ren with symmetry of revolut ion 171. Th eI iybr id mod i~rnatcl r ing ntegral winat ion appronch 111, [41, where the inter ior flaring is modrlrcl hy th e niodomatching techniqi ic and t l i e exter ior prohlem is dv sc r i l d by the in t cgr; l l eqi ia t ion method is judged to bean nilnqna te coniprtimisc! bc tw im efficitincy and I l rxi l ii l ity. Th e me thod has h c ~ n i t l icr to app i i rd tn ci rcular[ I ] and rcct,angiilar inrns 111 wliere analyt ical snlut ions for t.lic: cigonvalue lirohlem arc availablr!.This paper extends the hybr id modcmatching int rgral eqiratiori approa:h to the inclns ion of horns ofarhi t ,rary inter ior cross-section and exterior s u r face gm m et r y (cf. Fig. 1) . Th e t t d i i i i qne fn r tlie interiorhorn taprr is bawd on the efficient finitoelemcnt(FE)/rrio~le-n~at~tiiiignctthorl 181 which al lows fnr arbitrarywaveguide cross-sections . For t he genera l adm i t t ance ma t r ix of an arbi t r ary hoimlnl r egion, KIRCI IHOFF'Sprinciple [I]eads to a n electric f i e l d integral equat ion ( E F I E ) formulat ion which is mlvecl numeriridly bythc incthod of moment s . For rtrfercrrce purposes, f i rs t a conical horn is calculatcxl . The flnxibilify of the:prcsuitcd inot,tiud is demonsl.ratedat t he example of a conical r idged waveguide horn (Fig. 1). The tlitxiryis verificcl with availatilc measirroinents o r refi:reoce ~:alculat ions .

Fig. 1. Ridgrd wav rg l i i d f 3 hnrn a i r t rnna

0-7803-4478-2/98/$10.000 1998 IEEE1710

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11 . 'rl1ECJRY'Tlie caiitirc rcgion is siihrlividcd into t W O rcsgions / an d I! 191 wliich are scpa ra t rd by a pi:rf~.c.tly nndtir:t.ings~trf;u:c!Sa. nforcing the cxmtinui ty of the tangent.ial niagnet ic fisld on tlici aperture yields th r followingcx~iiatir~iiur the niagnet ic surface cur , ' rmt drnsi ty:

I I ~If!.,,(M) + I I & , ( M ) , on s., (1 )w h e r e HI;: , H [ . ~ ( M )ncl fI;!,,(M) arc! thci tarigential ini:idi!tit magnc*tii: field i n rrtgion I, the l incaroptmitor for thr: tangcntinl acatterrcl niagni:tic liold i n region I arid the lincar opctratur for the langcntialscat t i .r rd iungiwti i : fidd in region [ I .

T h e niagnrtic: surface cu rre nt c1ensii:y is approxinia tn l by a n u n h e r of Q linear indepeiidcnt haqis hinctionsM, 111 S

vv= 1

M ( r )= rk, VqMv(r), r (i n s,, (2 )whe re thc! I; nre th c ex pa tisi o~ i oc!fflciimls for th e magnot ic surface ci r rrcnt dim ity . i lenutw t h o frm spaceinrprdai ice. Introducing a set of P = 80 l inear independent tes t funct ions W , un S resnlts in a xystern ofl inrar cqi iat ions

[ I $ ] = [Tiq+ TZ] Vqj]wit.li

an d t hr innrr products dcf inrcl

Th e cnlci i lat ion of t,ha iulniit ,txnc:c inatrix fur the oittrr giximetry is basc:d un K IRCIIIIO F F s principle [l ]iising the el is t r ic ai i rfacc current denai t icn on S en d S (where Sa is th e extcrior horn sirrfacc) an d t h e rnag-i ll it ic snr la cc curre nt densi t ies on S T h e m agnet ic and clrct r ic mrface cnrrcnt drnsi t ics are approximatedIiy equat ion (2 ) a n d

SJ(r)= I . J,(r), r on S. o r ~ c . (4 )

.=1After int roducing a se t of R l inear independent tes t funct ions U, on S. and Ss, a system uf linear eqnat.ionsrcsults 111:

[D,"I IVVl = [ClK l+ [Cl I . 1 3 (sa)with

(51:)(5d)

1T,!, =- ci', (r); ot ro t1 J . r')Ge(r, ') d s ' ) ,3s S A . t S .

D Z = (Ur(rLn Mq(r)) .Vrr-tors [I;) ~ n dI, ] dc:note R column 01 their corrcaponding matr ices fo r a eacital.iun with a single modc jonly. Using the linearity of t he opera to i ' I " and i n t roducing th e mat r ix

[ F L ] =[(uv, (r);-nxJ. lr)) l , P = 1...4, (6)

171 I

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Fig. 2 : Conical horn antenna . Compa r i son wi th rneasurenient s 171

the i :xpri rss ion for thc ad mit ta nce ma tr ix for region II rcsol ts , s imilar to [ I ] :[z = [F i n ] TA]-' [DE sz] (7)

Eqiiat,ion(7 ) is iiasically t,he elcctric field equat ion, EFIE.Th e dcrivatc i l equat ions a llow t o ~o lv e h e r a d i a t h n prohlom of a wavrgui ile ap er tur i ! wi th the c r o wsection of the last scc.tiun of t he in t er ior horn t ap er i ncluding the outer gconietry of t he horn . In the next

step of tha analysis , we combine t he s ca t t e ri ng m a t r i x of t he t ape r [E] wi th the s cat t er ing mat r ix of t h eape r t u r e . As a r r sul t , we ohta in th e overa l l modal s r a t t er ir ig m at r ix of t he complet e s t ructure (asxxn fromt he f e d waveguide) , arid t he forward an d ba r kw ar d p r opaga ti ng modes in th e last stxtiiin of t he horn t aper .For th e EFIE, we t ake the m t of th e -uz ei of region I which are calculated hy th e FE mcthot l [ 8]and RAO'S mis funct ions 1111 HS: basis funct ions for t hc m agne t ic an d cl r r t r i c siirfacc: currunt donsitica,rcwpc!ctively. Tlir: us c of t t A O ' s haxis funct ions for both the elaclr ic ond the rnagi iet ic surface wrrcnt dens i tywould rosiilt i n a ni imcrical ly s ingula r niat ,r ix [T '+T "] . Because of the or thon orma li ty of t h s -U , x e i hasisf i inct ions , t he ad mi t t ance mat r ix [T']and t he inr ideot cu r r ent vector s [Ii"']re given arialytirally. The Retof th e P = Q est fiinctions W , s t he set of basis funct ions for the magnet i c surface cur rwt dens i ty , M,( G A I . E I ~ K I N - M e t h o d ) .T h e wt of R = S tes t funct ions U, s the se t of basis funct io ns for the el rct r ic surfaceciirrcnt dcnsit.y, J , .Th c 1-m at r ix clcrnrnts are t ransformcd into the well known mixed pot ,ent ial form i n o r i h t o h ave a1 / R s ingulari ty only . Th e r emaining nimwr i r a l i n t egra t ions are evaluated in nornia1izr:d area cix)rdinnt.es[ l l ] siug a t h m - aiid a seven point rule [IZ]. he cor r ect evaluat ion of t he S-m at r ix e l cmeot s r r iIu i rm the6 . r p r a t i un of th e M I I I I C ~ o i n t i n t l ~ g r s ln to a C A l I C l l V principal value term an d a revidrlc ti!rni in tlle u~IIa1way.

111. RESULTSFirst a circrilar h o r n s t r i icture of 171 (Fig. 2) is invcwtigated. G ~ i dgrerment w i th m easurei iwi t s r ryor tcdi n [7 ] m ay he s t a t ed .

T o d(!rnonstratc thct flexihility of the prmc.nt.ivl me tho d, a r idged horn s t r i ict i r rc (Fig. 1) is invistigatcrl.The r idgod horn taper wct ion has h t x m opt imiuvl acco rding to a high handwicl th of th i : i nput w t i i r i i l o s s[SI. I n order to i:nahlc rcf i : rmre calculat ionf i wi th the tmundary contour f iphi! r icd wavi : expansion (SWE)in(i~li.-iiiat.cliiiigi!ctiniqui! [7], thi: oute r con tou r of thr: h w n is s haped according to th e inncr skctch of Fig. 3.l l i e result is pliittcd ii i Fig. 3. Also g w d ag ri x*n ien t m ay he s t a t ed .

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0-5

-1 0-1 5-2 0-2 5-30-3 5

20 40 CO 120 14 0 I G0 180

Fig.3: Ridged wavrgi i ide horn. Reference calculations by 1711V. CONCI.IISION

A iisefiil hybrid FE/fvlM/monient method technicliie is presented for tile r igorous analysis of horn antennnr,of arb itrary intitrior crcrs-.w:tioii arid exter ior surface. Gcmd agrccnient with ii:P~rencevaliim i s o l ~ t a i n r d .REFERENCES

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