A CLOSED-FORM SOLUTION TO ANALYZE RCS OF CAVITY WITH RECTANGULAR CROSS SECTION

Transcription

1 Progrss In Elctromagntcs Rsarch, PIER 79, , 2008 A CLOSED-FORM SOLUTION TO ANALYZE RCS OF CAVITY WITH RECTANGULAR CROSS SECTION L. Xu, J. Tan, and X. W. Sh Natonal Ky Laboratory of Antnna and Mcrowav Tchnology Xdan Unvrsty NO. 2 SOUTHTaba ROAD X an, Shaanx, P. R. Chna Abstract In ths papr, a st of formulas to analyz th scattrng from opn-ndd rctangular cavty s prsntd on th bass of Shootng and Bouncng Ray (SBR) mthod. By analyzng th ray paths nsd th cavty, th Physcal-Optcs (PO) ntgraton on th aprtur s carrd out n a clos form. Usng closd-form soluton, th Radar Cross Scton (RCS) of cavty n hgh frquncy can b studd sntntously and accuratly. All th paks and nulls n th RCS plot of cavty ar prdctd succssfully wth th formulas dducd n th papr, and a 3-D scattrng pattrn of rctangular cavty s smulatd by th proposd mthod. 1. INTRODUCTION Th problm of lctromagntc scattrng from opn-ndd cavts has bn studd ntnsvly by varous computatonal lctromagntc tchnqus for many yars [1 5]. Ths problm srvs as a smpl modl of duct structurs such as Jt ngn ntaks or antnna wndows mbddd n mor complx bods [1, 6 8], and much rsarch has bn carrd out on analyss of both radar cross scton and lctromagntc puls couplng [9 12]. At th low frquncy nd,.., cavts wth opnng lss than a wav-lngth, a rgorous ntgral quaton can b usd [1, 5]. For an aprtur opnng on th ordr of svral wavlngths, on has to rsort to hgh-frquncy approxmatons [4]. Th shootng and bouncng ray (SBR) mthod s proposd to study th scattrng from cavty for dcads [3, 4, 9], and t has bn wdly usd n th smulaton of lctromagntc wav scattrng from cavty. Th valdty and accuracy of SBR mthod s provd n many paprs avalabl. In ordr to analyz th scattrng from cavty upon complx objct, som hybrd mthods of SBR wth

2 196 Xu, Tan, and Sh othr computatonal lctromagntc tchnqus ar dvlopd. Th SBR mthod s know as a powrful RCS analyss mthod bcaus t can analyz th RCS charactrstcs for arbtrary shap opn-ndd cavts, and t s also wdly usd f th spac nsd th cavty s not homognous or th lctrcal dmnson of th cavty s larg [3]. Whl much rsarch has bn dvotd to th problm that shap of th cavty s arbtrary, lttl rsarch has bn don on th acclraton of SBR mthod to analyz th cavty n larg sz. Ownng to th rrgular postons of xt rays, th physcal-optcs ntgraton cannot b asly carrd out n orgnal SBR [3]. Although th SBR approach can prdct RCS for cavty n any shap, t s tmconsumng and unpractcal to ralz modlng of th physcal problms for cavty n larg sz. Thr ar also othr mthods that can b usd to calculat th scattrng from cavty, such as wavgud modal approach, but t s cumbrsom f th lctrcal dmnson of th cavty s vry larg. Th prsnt papr dscusss th charactrstcs of ray tracng nsd th cavty wth typcal shap such as rctangular cross scton, whch s appld wdly n th flds of lctromagntc. A closd-form soluton for analyss th RCS of rctangular cavty s dvlopd basd on th SBR mthod. Th formulas ar of concs xprsson and can b ralzd asly by computr. Th charactrstc of rctangular cavty s studd, such as th paks and nulls n th RCS plot of cavty, and th scattrng pattrn for rctangular cavty s prsntd, whch s dffcult to b ralzd by th orgnal SBR or othr mthods. 2. PREDICTION METHOD In th SBR mthod, th ncdnt wav s dvdd nto a st of ray tubs. By th approach of ray tracng and physcal optcs ntgral, th scattrng from cavty s calculatd. A dtald procdur of th SBR can b found n [3]. On th bass of a study on ray tracng, th postons of ray bouncd on th wall of cavty and th locaton of ray xt ar analyzd, and th ruls of ray path nsd th cavty ar nvstgatd. Th proprty of ray bouncng n rctangular cavty s dscussd n ths papr, and th rlaton btwn ray ncdnt and ray xt s dducd. A closd-form soluton for 3-D problms of scattrng from rctangular cavty s prsntd. Th procdur s bord and complx, and som mportant rsults ar lstd hr.

3 Progrss In Elctromagntcs Rsarch, PIER 79, Th backscattrd fld can b carrd out by th follows [3]: E = jk 0r [ˆθA θ +ˆϕA ϕ ] r Aθ = jk [ 0 Exm cos ϕ + E ym sn ϕ A ϕ 2π (E xm sn ϕ + E ym cos ϕ ) cos θ m Hr, I m s dfnd as follows, I m = mth xt ray tub dxdy jk 0(î m+ŝ m) r ] I m jk 0ŝ m r m (1) (2) In Eq. (1), m s th ndx of ray tub. E xm and E ym ar th x and y componnts of th outgong fld on th aprtur. Th drcton vctors of th ncdnt rays and xt rays for mth ray tub ar dnotd by î m and ŝ m rspctvly, and th cntral ray n mth xt ray tub hts aprtur of cavty at pont r m =(x m,y m ). Eq. (2) s th Fourr transform of th ray tub shap functon (normalzd wth rspct to th ray tub ara), and t can b carrd out as follows: I m =( x m y m ) jk 0(î m+ŝ m) r m [ sha k 0 ( mx + s mx ) x ] [ m sha k 0 ( my + s my ) y ] m 2 2 (3) Whr, sha(x) = sn x x ( x m y m )=ara of th xt ray tub. In ordr to gt a prcs rsult, th dmnson of ray tub s supposd to tnd to zro, and Eq. (3) can b carrd out wth a nw xprsson: I m =( x m y m ) jk 0(î m+ŝ m) r m (4) Th gomtry of rctangular cavty s shown n Fg. 1, and Fg. 2 s th local coordnat systm at th rflctd pont on th wall of cavty.

4 198 Xu, Tan, and Sh Fgur 1. Gomtry for rctangular cavty. Fgur 2. Local coordnat systm at rflct pont. Accordng to th rflcton laws of plan wav ncdnc on a dlctrc ntrfac, som spcal ruls for ray bouncd n rctangular cavty can b found: { î = (sn θ cos ϕ, sn θ sn ϕ, cos θ ) (5) ŝ = î 2(î ˆn)ˆn ê s =2(ê ˆn)ˆn Γ // ê // +Γ ê (6) Whr, î, ŝ, ê and ê s ar th drcton vctors of th ncdnt wav, rflctd wav, ncdnt lctrc fld and rflctd lctrc fld rspctvly. Th subscrpt and rprsnt th componnts of

5 Progrss In Elctromagntcs Rsarch, PIER 79, paralll or prpndcular polarzaton accordngly. If th wall of cavty s prfct conductor, thn Eq. (6) can b rwrttn nto: ê s =2(ê ˆn)ˆn ê (7) Consdr on of th ncdnt ray tubs, gvn th locaton of ncdnt ray ht on th aprtur, a st of rflctd ponts n cavty and th drcton of th xtng ray can b obtand wth th Gomtrc Optcs (GO) mthod. Th dtald procss of ray tracng s bord and lngthy so only som rsults ar dscrbd hr. In ordr to carry out Eq. (1), th rlatonshp of ncdnt ray tub and xt ray tub should b dtrmnd at frstly. Gvn th ncdnt fld, th ray paths n th cavty can b found by ray tracng and th ruls proposd abov, and th fld ampltud of th xt rays on th aprtur of cavty s got basd on GO mthod. Tabl 1 prsnts a summarzaton of th rlaton btwn ncdnt ray and ray xt. As Tabl 1. Rlatonshp of ray ncdnt and ray xt on th aprtur. N 0x s odd N 0x s vn N 0y s odd N 0y s vn x [0,B x ] x [B x,a] x [0,B x ] x [B x,a] y [0,B y ] y [B y,b] y [0,B y ] y [B y,b] x out = x + a B x s x = x s y = y s z = z x out = x+a+b x s x = s y = y x s z = z x out = x + B x s x = s y = y x s z = z s x x x out = x B x s x = s y = y x s z = z y out = y + b B y s y = s x = x y s z = z z y out = y+b+b y s y = s x = x y s z = z y out = y + B y s y = s x = x y s z = z y y out = y B y s y = y s x = x s z = z s y s z th axs of cavty s paralll to th Z axs n Cartsan coordnats, th locaton of ray ncdnt and ray xt can b rprsntd by (x, y) and (x out, y out ) rspctvly. Th paramtrs N 0x, N 0y, B x and B y n Tabl 1 ar dfnd as follows: ( 2h N 0x = Round a ) x + 1 (8) z

6 200 Xu, Tan, and Sh ( 2h N 0y = Round b ) y + 1 (9) z ( B x = N 0x a ) x 2h (10) z ( B y = N 0y b ) y 2h (11) z Onc th ncdnt rays hav bn dfnd, th mpact pont of ach ray on th nnr wall or th bottom of th cavty wll b dtrmnd, and th paramtrs N 0x, N 0y, B x and B y can b carrd out, whch wll confrm th xt ray by th rlatonshp shown n Tabl 1. For xampl, consdr th cas n whch N 0x s odd and N 0y s vn, whl ncdnt ray hts th aprtur of cavty at pont (x, y), f x [0,B x ] and y [0,B y ], thn thn xt ray can b confrmd as followng quatons: x out = x + a B x y out = y + B y ŝ =( x, y, z ) ê s =( x, y, z) As mntond abov, thr should b nough ray tubs launchd nto cavty to gt an accurat rsult n SBR mthod, and th szs of ray tub must tnd to zro ( x 0, y 0), whch mans that th sum n Eq. (1) can b rplacd by an ntgral opraton, so Eq. (1) s rwrttn as: [ Aθ A ϕ ] 2h 2π jk 0 z = jk 0 S [ Eoutx cos ϕ + E outy sn ϕ (E outx sn ϕ + E outy cos ϕ ) cos θ jk 0î ( rout r) dx out dy out (12) Whr, subscrpt out mans th componnts rlatd to ray xt. Bcaus th postons of xt rays ar nonunformly dsprsd ovr th aprtur, th ntgraton n Eq. (12) cannot b carrd out drctly. Howvr, th ncdnt rays ar launchd unformly. If w can carry out Eq. (12) wth th ncdnt fld on th aprtur nstad of outgong fld, th ntgraton may b solvd out. As sn from Tabl 1, th rlaton btwn r =(x, y) and r out =(x out,y out ) can b concludd as follows, ] x = ±x out + C1 y = ±y out + C2 (13)

7 Progrss In Elctromagntcs Rsarch, PIER 79, Whr, C1 and C2 ar constant corrspondng. That s to say, dx out dy out can b rplacd by ±dxdy, whch mpls that th ntgral rgon may b convrtd to th aprtur of cavty. Th sgn ± wll b dtrmnd by th rlatonshp of ncdnt ray and xt ray, whch s shown n Tabl 1. In ordr to gt a closd-form soluton of Eq. (12), th aprtur of cavty s dvdd nto four subrgons whch ar shown n Fg. 3. Fgur 3. Subrgons of ntgral rgon. As an xampl, th papr carrs out th ntgral n Eq. (12) whn N 0x and N 0y ar both odd. Th soluton for othr cass can b dducd n a smlar procdur, and th papr dosn t lst thm hr. Lt th ntgral part of Eq. (12) on rgon S 1 b dnotd by I s1, thn: x cos ϕ y sn ϕ I s1 = ( x sn ϕ y cos ϕ ) cos θ jk 0( xx+ yy) S1 jk 0[ x(x+a B x)+ y(y+b B y)] dxdy x cos ϕ y sn ϕ = ( x sn ϕ y cos ϕ ) cos θ jk 0( xa+ yb) B x B y sha(k 0 x B x )sha(k 0 y B y ) (14) x cos ϕ + y sn ϕ I s2 = ( x sn ϕ + y cos ϕ ) cos θ jk 0[ x(a+b x)+ yb] (a B x )B y sha(k 0 y B y ) (15) x cos ϕ + y sn ϕ I s3 = ( x sn ϕ + y cos ϕ ) cos θ jk 0[ x(a+b x)+ y(b+b y)] (a B x )(b B y ) (16)

8 202 Xu, Tan, and Sh [ I s4 = x cos ϕ y sn ϕ ] ( x sn ϕ y cos ϕ ) cos θ jk 0[ xa+ y(b+b y)] B x (b B y )sha(k 0 x B x ) (17) Obvously, th ntgral n Eq. (12) s th sum of four ntgrals n thos subrgons. Onc th ntgral s carrd out, Eq. (12) s solvd n a closd-form. Th formulas for othr cas ar lst as follows. 1). If N 0x s odd and N 0y s vn, x cos ϕ y sn ϕ I s1 = ( x sn ϕ y cos ϕ ) cos θ jk 0( xa+ yb y) B x B y sha(k 0 x B x ) (18) x cos ϕ + y sn ϕ I s2 = ( x sn ϕ + y cos ϕ ) cos θ jk 0[ x(a+b x)+ yb y] (a B x )B y (19) x cos ϕ + y sn ϕ I s3 = ( x sn ϕ + y cos ϕ ) cos θ jk 0[ x(a+b x)+ yb] (a B x )(b B y )sha[k 0 y (b B y )] (20) x cos ϕ y sn ϕ I s4 = ( x sn ϕ y cos ϕ ) cos θ jk 0( xa+ yb) B x (b B y )sha(k 0 x B x )sha[k 0 y (b B y )] (21) 2). If N 0x s vn and N 0y s odd, [ x cos ϕ + y sn ϕ I s1 = ( x sn ϕ + y cos ϕ ) cos θ ] jk 0( xb x+ yb) B x B y sha(k 0 y B y ) (22) x cos ϕ y sn ϕ I s2 = ( x sn ϕ y cos ϕ ) cos θ jk 0( xa+ yb) (a B x )B y sha[k 0 x (a B x )]sha(k 0 y B y ) (23)

9 Progrss In Elctromagntcs Rsarch, PIER 79, x cos ϕ y sn ϕ I s3 = ( x sn ϕ y cos ϕ ) cos θ jk 0[ xa+ y(b+b y)] (a B x )(b B y )sha[k 0 x (a B x )] (24) x cos ϕ + y sn ϕ I s4 = ( x sn ϕ + y cos ϕ ) cos θ jk 0[ xb x+ y(b+b y)] B x (b B y ) (25) 3). If N 0x s vn and N 0y s vn, x cos ϕ + y sn ϕ I s1 = ( x sn ϕ + y cos ϕ ) cos θ jk 0( xb x+ yb y) B x B y (26) x cos ϕ y sn ϕ I s2 = ( x sn ϕ y cos ϕ ) cos θ jk 0( xa+ yb y) (a B x )B y sha[k 0 x (a B x )] (27) x cos ϕ y sn ϕ I s3 = ( x sn ϕ y cos ϕ ) cos θ jk 0( xa+ yb) I s4 = (a B x )(b B y )sha[k 0 x (a B x )]sha[k 0 y (b B y )] (28) x cos ϕ + y sn ϕ ( x sn ϕ + y cos ϕ ) cos θ jk 0( xb x+ yb) B x (b B y )sha[k 0 y (b B y )] (29) Thn, Eq. (12) can b carrd out n a clos form as shown blow. Aθ = jk 0 z I (30) A ϕ whr, 2h 2π jk 0 I = I s1 + I s2 + I s3 + I s4 Th proposd mthod s summarzd blow, 1) Input th gomtry sz of cavty (a b H) and th drcton of ncdnt wav î; 2) Calculat th valus of N 0x, N 0y, B x and B y by th quatons Eqs. (8) (11);

10 204 Xu, Tan, and Sh 3) Slct a st of ntgral formulas accordng to whthr N 0x and N 0y ar odd or vn; 4) Carry out Eq. (30) wth th ntgrals solvd by stp 3. Wth th closd-form soluton basd on SBR mthod, th scattrng from rctangular cavty can b smulatd succnctly. Wthout worryng about tm-consumng, th mthod s abl to dal wth th problms n hgh frquncy. 3. RESULTS AND ANALYSIS As th valdty of SBR mthod s tstd n many paprs avalabl [3, 4], and th purpos of th papr was to provd a mthod to analyz th lctromagntc scattrng from rctangular cavty n hgh prcson and fast spd, so th objctv of th prsnt smulatons was to vrfy th valdty of nw mthod and nvstgat th charactrstcs of rctangular cavty. Th scattrng pattrn of rctangular cavty s vsualzd to show th advantag of proposd mthod SBR mthod Formualton RCS(dB squar wav lngth) Thta(Dgrs) Fgur 4. Comparson btwn th rsults tradtonal SBR mthod and formulaton, ϕ =45 o Frstly, th rsults n tradtonal SBR and closd-form formulas ar compard n Fg. 4 to vrfy th corrctnss of th dvlopd mthod. Th xampl s slctd from th ltratur [4], whch studd a rctangular cavty wth a 10λ by 10λ squar cross scton and 30λ n lngth. Excllnt agrmnts can b found btwn two curvs. Th rsults show an ntrstng phnomnon that almost all nulls and paks n curv tak plac whr th valu of 2h a x z s an ntgr. Rfrrng to th closd-form xprsson, t can b xpland vsually. In Eq. (8), B x

11 Progrss In Elctromagntcs Rsarch, PIER 79, tnds to zro as 2h a z s clos to an ntgr numbr. That s to say, accordng to Tabl 1, narly all ray tubs xt cavty n th drcton of ncdnt wav or n th contrary drcton. It can b prdctd xactly that paks wll tak plac at θ =13.26,35.26 and so on, whn th ray xt cavty n th drcton of ncdnt ray. Thr wll b a null at θ =25.24,43.31 whn ray xt cavty n a contrary drcton. x Fgur 5. Pattrn of RCS θθ for rctangular cavty. Thn, a pattrn of scattrng from cavty s shown n Fg. 5. Bnftd from th soluton n closd-form, smulaton of RCS pattrn n 3-D bcoms practcabl. In ordr to avod som spcal phnomna for cavty wth squar cross scton, a rctangular cavty wth dmnsons a = 10λ, b = 15λ and H = 40λ s slctd, and th papr calculats th RCS n paralll polarzaton. As complcatd procdur of ray tracng n SBR mthod, pattrn analyss of cavty s a burdnsom work for tradtonal SBR. Wth th closd-form formulas proposd hr, t taks fw sconds to gt th rsults. Fg. 6 shows th projcton of pattrn to th x-o-y plan. Du to th symmtry of gomtry, th mag shows symmtrcal charactrstc wth th varaton of ϕ. As mntond abov, t can b prdctd that thr ar four nulls at ϕ =0 and thr nulls at ϕ =90. Fg. 6 shows that th sz of cavty n wdth plays an mportant rol n th rgon from ϕ = 16 to ϕ = 16, and th sz of cavty n hght s mportant n th rgon from ϕ = 78 to ϕ = 100. Th mpact of gomtry sz on scattrng charactrs of cavty can b studd n

12 206 Xu, Tan, and Sh Fa(Dgr) Thta(Dgr) Fgur 6. Projcton of RCS θθ pattrn to X-O-Y plan. Fgur 7. Pattrn of RCS θθ for rctangular cavty n sphr coordnats.

13 Progrss In Elctromagntcs Rsarch, PIER 79, dtal wth quatons from Eq. (8) to Eq. (11). In ordr to gv a vsual undrstandng of RCS pattrn for cavty, Fg. 7 plots th normalzd RCS pattrn n sphr coordnats. 4. CONCLUSIONS A st of formulas was proposd to calculat th lctromagntc scattrng from a rctangular cavty. Ths approach s basd on th SBR mthod by takng nto consdraton 1) launchng nough ray tubs nto th cavty, 2) trackng th ray path nsd th cavty, and 3) dtrmnng th GO fld assocatd wth ach ncdnt ray. Th rlaton btwn ncdnt rays and xt rays s found by nvstgatng th charactrstcs of ray launchd nto th cavty. Th sum of th scattrd fld du to ach ndvdual xt ray tub s rplacd by an ntgral of ncdnt rays on aprtur of cavty, and a 3-D closd-form soluton basd on SBR to analyz th hgh frquncy backscattrng from opn cavts ar carrd out. Although thr ar othr analytcal solutons that may b possbl for rgular shaps, thy bcom cumbrsom f th lctrcal dmnson of th cavty s larg. Usng th proposd mthod, th problm of cavty wth rctangular cross scton can b solvd n hgh spd and prcson, and t maks scattrng pattrn smulaton for cavty possbl. Th locatons of paks and nulls n th scattrng ar prdctd accuratly wth th formulas proposd n th papr. Th scattrng pattrn of rctangular cavty s vsualzd by th closd-form formulas, and som ntrstng charactrstc s prsntd. Du to th assumpton that th sz of ray tub n SBR tnds to nfntsmal, a hghr prcson of rsults s xpctd. REFERENCES 1. Huddlston, P., Scattrng from conductng fnt cylndrs wth thn coatngs, IEEE Trans. Antnnas Propagat., Vol. 43, , Altntas, A., P. H. Pathak, and M. C. Lang, A slctv modal schm for th analyss of EM couplng nto orradaton from larg opn-ndd wavguds, IEEE Trans. Antnnas Propagat., Vol. 36, 84 96, Lng, H., R. C. Chou, and S. W. L, Shootng and bouncng rays: calculatng th RCS of an arbtrarly shapd cavty, IEEE Trans. Antnnas Propagat., Vol. 47, , 1999.

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