contour matching using ant colony optimization and curve evolution
FE or Mode Matching Moment Method Analysis of Horn Ant
Transcript of FE or Mode Matching Moment Method Analysis of Horn Ant
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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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