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İki parçalı rezistif ve kondüktif şeritlerden düzlemsel dalgaların kırınımı

Plane-wave diffractıon by two-part resistive and conductıve strıps

  1. Tez No: 39105
  2. Yazar: ORHAN BIÇAKÇI
  3. Danışmanlar: DOÇ.DR. EREN ERDOĞAN
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Dalgalar, Kırınım, Şeritler, Waves, Diffraction, Bond plates
  7. Yıl: 1993
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: Belirtilmemiş.

Özet

ÖZET İki parçalı bir düzlemsel şeritten düzlemsel elektromagnetik dalgaların saçılması Spektral İterasyon Tekniği (SIT) kullanılarak; şeridin rezistif, kondüktif ve empedans özelliği göstermesi durumlarında incelenmiş ve üçüncü mertebeye kadar ardışık kırınımlar için uniform alan ifadeleri yüzey dalgalarının muhtemel katkılarımda içerecek biçimde elde edilmiştir. Rezistif ve kondüktif şeritlerin uygun bir kombinasyonla aynı geometrideki empedans şeridine denk olduğu gösterilmiş ve empedans şeridi için elde edilen bazı bistatik kırınım grafikleri verilmiştir.

Özet (Çeviri)

SUMMARY PLANE- WAVE DIFFRACTION BY TWO-PART RESISTIVE AND CONDUCTIVE STRIPS 1. Introduction A two part impedance strip, shown in Fig.l, is a suitable model for studying the radiation and scattering of electromagnetic waves in the pres ence of a discontinuity in the material properties of a surface of finite width, or of composite structures on aircrafts, satellites, etc. As is well known, at high frequencies the total diffracted field by a strip or slit can be expressed as the sum of singly and multiply diffracted fields [1]. A part of the fields diffracted at one of the edges illuminated by the incident plane wave propagates along the upper and lower faces of the strip and gives rise to secondary diffractions at the other edge. These waves, named as the doubly diffracted fields, propagate, in turn, on both faces of the strip back to the first edge to excite triply diffracted fields [l]. The mechanism continues likewise, creating higher order diffracted fields. M Z, v / ; s / > j />/> u'=eI, t....2..... Nl -ti ?1 Fig.l. Geometry of the diffraction problem. According to the geometrical theory of diffraction (GTD), the primary diffracted fields from a two-part impedance strip can be evaluated easily by \ising the known canonical solutions related to an impedance half-planeand to a two-part impedance plane. However, when one is concerned with the multiply diffracted fields, the classical GTD approach which consists of employing the known edge diffraction coefficients repeatedly with the incident field being the field previously diffracted from the other edge fails because of the non-ray-optical behaviour of the interacting fields. So does, equivalently, any attempt to obtain non-uniform results for such fields, the expressions becoming invalid for angles of incidence and observation close to grazing, which is the most troublesome forward scattering region. Sim ilar reasoning obviously holds for resistive and/or conductive structures of the same kind. Essentially a spectral domain method, The Spectral Iteration Tech nique (SIT) recently developed by Büyükaksoy et al. [1],[6], is a conve nient way to overcome this difficulty. One of the two main objectives in this work is to provide, using SIT just mentioned, uniform diffraction coefficients for the doubly and triply diffracted fields associated with a two-part impedance strip including the possible surface wave contribu tions, also in a uniform manner. The other is to demonstrate, as proposed and discussed in detail in [2], [4], that a combination strip, that is, a cer tain superposition of two resistive and conductive strips (two-part strips, in our case) so formed as to be mathematically identical to an impedance strip of the same geometry, does in fact simulate an impedance strip in case of multiple diffractions. It is well known that an electrically resistive surface is characterized by the relations [2] nAE\± = 0, (la) nA(nAE) = -RnAH\±, (lb) and a magnetically conductive surface by the relations n A H\± = 0, (2a) nA(nAH) = R*nAE\± (2b) where n is the normal unit vector directed into the region (+ ), R(R*) is the resistivity(conductivity) of the resistive(conductive) surface. For an impedance surface with equal face impedances one has [2], [3] nA(nAE±) = ±ZnAH± (3) where Z denotes the surface impedance. (The conditions for the validity of (la-3) should never be forgotten or overlooked [2], [3]. For example, the surface impedance above may be (and is, in general) dependent on the angle of incidence [3]. Not always, therefore, has one the right to take Z in (3) as characteristic to the surface, except perhaps at one critical frequency [3]. Even at that frequency, the other side of the surface does vinot have to behave as an impedance surface, and in case it should, the impedances of the sides do not have to be equal [3].) Again, it is known that a resistive surface supports only electric cur rents, and a conductive surface only magnetic currents given by J = nAH\t,J* = -nA E\±, (4) respectively; whereas an impedance surface is capable of supporting both currents. This gives the idea that, combined properly, the former two surfaces can simulate an impedance surface(strip). This“proper”combi nation can be easily shown to be [4] JW-i;a-f;*- = -L (5) assuming R,R* (hence Z) to be isotropic. 2. Statement of the Problem In this work, the above-mentioned equivalency is demonstrated for the two-part strip shown in Fig.l., illuminated by a plane wave whose electric field is assumed to be parallel to the z axis. Since the boundary conditions (la-3) are very similar, so will be the problems connected with them. For that reason, only the calculations for the two-part resistive strip, using SIT, are given explicitly in the work. For the other two structures, except some necessary explanations made, only the final results are provided. In what follows in this section, we give the results for the impedance strip; since these allow, as will be detailed later, for the corresponding expressions for the resistive and conductive strips to be extracted with rather not much effort. In the expressions below, those terms with superscripts (l,j) show the transformation SW- *SW for j = 1, SW- »SR for j = 2 ; and those with (2, j) show the transformation SR^SW for j = 1, SR-+SR for j = 2. i) Secondary diffraction by M Vom-V^+VT, (6a) VIIOMW'Wj l } y/2^ H^(k cosh) H+(k cos <f>0)H^(k cos <f>0) [H+jkPj)]2 eikP^ #+ ( JbPi ) Pi (Pi - cos <£0 )(Pi + cos fa ) #+ (fcPi ) Pi (Pi - cos 4>0 ){P1 + c r / / 1 e'fcpl sin ^i ± yl - -fi v 1 - cos ^i /t - > L J V&pi (66) ısı tix \kzos<p\) ti\\k cos q>Q)ii 2 [kcoscpo) \H+(k)}2 eike> 1 f [l-^C-H^l + cos^))] x + X H+{k)}2 eike> 1 f [l-^j-khjl + cosfa))] H+(k) y/İc% 1 + Pi 1 (Pi + cos fa )(cos <^o + cos fa ) (COS Çİ>o - Pi )(cOS ^>o + COS <j)\ ) (Pi - cos <^o)(Pi + cos fa) } r.. 11 /: -1 e**“1 1 1 1 eikpi > sin^T7=-Vl-cos^J7=,y<0. (6c) ii) Secondary diffraction by O Vmo = Vjgi + 1#L (7a) r(l) X^(r^) ~ T^(M) e**'4 Ci - C2 sin<^ 2tt Ci H?{k cos ^)fl£”(ft cos <£) x x WC*^)]5 oik Pıtı H+{kPx ) H~(k cos <f>0 )(P, - cos ^)P, (P, + cos çio ) Cı sin <t>o T (76) y/1 - cos <^o 1 etfcr v/l + PT \/ftr' vinm Çj H^{kcos<p)H2 («cos 9) /7 \19 1 il-f. Jkex H}(k) H~(k cos fa)(l + P1)y/kl[ ' \\-T{-Ux{\-cos<$>))\ | [l-^-H^l + cos^p))] + COS^o) f [l-.F(-fcli(l-cos^))] [l-.T(-Hi(l + c( \ (cos <j) - P\ )(cos <$ + cos ^>o) (-Pi + cos <^o)(cos <f) H [l-^(-Hı(l-Pı))] U v/1-cos^l (Pa+cos^oXPx-cos^)/^8111^^ V2 eifcr iii) Tertiary diffractions by N and M, y<o. (7c) VMON - vMON“*”^MOJV“+”vMON“*”vMONi (8a) ^ok^^2)~-^(M) e''*'4 Cı - C2 sin <f>o sin ^2 x X sphi C1C2 E^ {k cos fa)E^{k cos fa) [H+(kP1)]2[H+(kP2)]2 eiktiPieike2p2 #+(kP2)If2+(fcPi) PjP2(Pi - P2XP1 + cos<£o) IT %/Î^Pl Cı Vl + cos <£o\A - cos fa \/l + Pi =«'*r2 (P2 -cos<£2) -v/ferâ' (86) t/(i,2) /, x TTifh/r\ 1 Ci - C2 sin fa sin fa 7TZ ÇjÇ2 İ2a (KCOS<po)-n2 (KCOS^2J [fff(lb)]2 e'“^ f [1 ~ ^(-H2(l - cos fa))} H+(fc) Pi (Pi + cos ^0)(1 + P2) I (Pi - cos fa)(P2 - cos <f>2) [1_^(-H2(1-P2))] [i-^-^a-P!))]! (P2 - Pi)(P2 - cos fa) + (Pi - P2)(Pi - cos fa) J (P2-PlX^2-COS^2)İ”(P] 2 1.*- riiAlliZL^ ± 2Cı C2 \/l - cos fa VI + cos <^o J H+(kPx) ^/H2~ y/kr^' ( [H+(kPj)}2 eiM* eikr> IXvMON.(2'1) u a \ TTi(M\ l C1-C2 sin ^0 sin ^2 in C1Ç2 -“1 (k cos (po)H2 (k cos fa) [H+(k)]2 e'H*p* f[i-jr(-Hi(l-.,”t(^2-A)(P2 + coS^o) [H+(k)}2 eike>p* f [1 - ^(-Hi(l - J #+(*) P2(l + Pi )(P2 - cos ^) I (P2 - Pi)(P2 + cos [l-^-H^l-PQ)] [l-^-^q + cos^o)] V (Pi - P2)(Pi + cos <j>0) (Pi + cos <£o)(P2 + cos <j>Q) J 1T, ^ CiVl - cos^2\/2(l + cos <^0) ikli £ikr2 [HÎ(kP2)f e ^ H+(kP2) v^y^z“ (8d) v2>2)^ ^^ 7-ru »^ v^e 7 Ci ~ C2 sm ^0 sin <j>2 7TJ// OC2 -Hi {k COS <Po)H} [k COS $2) Hl{k)H+(k) e'*£l e''*'» X (1 + Pj)(l + P2) VH7 V^ ; f[l-.T(-fcli(l + cos &Q)] [l-^-Hxq-Pi))]) X 1 1 + cos <^o Pj - 1 J ^ f JF(-H2(1 - P2)) - ^(-H2(l - cos<^2)) 1 \ (P2 - COS <j)2){P\ + COS <£o) J f 1 1 eJkr* 1± 1, 2CiC2\/l - cos <f>2y/l + cos <f>\ \/^r2 ' (8c) and v'MOM - fj^OA^ ”+“ KMOM + vMOM + KMOM (9a),(1,1) V£'oM<ru<h)~-vtW) ”[Ht(kP1)}4 eT'4 Ci - C2 sin (^0 sin <j)i 2v^7r C4 #! (k cos <f>o)H1 (kcos<j>i)“i2*Pı/ı [İJ+^Pj)]2 Pf(Pl + cos ^0)(Pi + cos ^) 1=F (96) v/T^PT Ci Vl + cos 4>\ \A + Pi \/l + cos <£0 Jkri fkr\ 'T/(ı,2),, >. rrift^ 1 C1-C2 sin^osin<^i 7re Çj i/j {kcos(po)H1 (kcosç>i) [g1+(ikPı)]2[gı+(fe)12 eiWl e'Wl [ffı+(fcPı)]2[ff,+ (fc)12 e1”' e'*^* Ht{kPı)H+{k) V^“Pı(l + Pı)2 f f(-k£x(l + cos fc)) - ^(-Mi(l - Pi)) I \ (Pi + cos ^1 )(Pi + cos fa ) J I”1 dV1 + cos <f>i V^i1 + pi)\/l + c°s <£o (9c) 1± v(2,i),,.. TTir*^ l C1-C2 sin ^0 sin ^i. yMOM(rı^ı) ~ -17 (Mj - - - - >?..,, -z m Çı H1 (kcos(/>o)H1 (k cos fa) x + [H+(k)}2( [1-^-^(1 + Pi))] [l-.T(-M1(l + cos6t))] #+ ( jfe) 1 2Pi (Pi - cos fa ) (cos <£o + Pi )(cos fa - Pi ) fi _ T(-kPJl - P, Yll 1“«*J\*i ) ^-^(-^(l-P!))]) e'Wl 2Pi (Pi + cos fa ) J Pi ( 1 + Pi )(Pi + cos fa )(Pi + cos fa ) y/T^PT 1 [H+(kP1)}2 eike> eikr> Ci y/l + cos & i/2(l + cos fa) H+(kPı) y/Je&l y/Srl' (9d) v(2,2),, s rTj /, ^ V^e t7r/4 Ci - C2 sin ^0 sin fa yMOM(rl,0l) ~ Ü (M), >2 g-,,, sn-n 7T 7TJ/Z Çf f/j (fcCOS^oJİ/j (fccOS^>i) [H}(k)]* ( e«'w» V / [1 - ^(-2Hi )] [if+(fc)]2 \V^7 l(l+Pi)(l-cos^o) [l-^-H^l + cos^o))], {l-^(-k£1{l-P1))} + (cos <^o + Pi)(l - cos fa) + (Pi + cos fa)(l + Pi X { JF(-H!(1 + cos^i)) - T{-Ux{\ - P] 1± Pl+COS^(l + Pj)2 1 - 1 e.-tn > 2(2 y/1 + cos </>a y/1 + cos ^0 J \/fcri ' < (9c) XIiv) Tertiary diffraction by O V”r,“ - V(1,1) 4-V(1'2) 4-V(2,1) 4-V(2'2) VOMO - VqMO + VOMO ”+“ VOMO + KOMO' (10a) r(l,l) V£}&(r,*)~-Cr«(0) e^4 (C1-C2)2 sin ^0 sin ^ 2^2^ C5 H+(kcos<l>o)Hî(kcos4>o) ei2fcPı^ı iT+(fe cos (fjH^ik cos <j>)Pf(Pi - cos <£o)(Pi - cos <£) (106) 1± 4 _ifcr r(l,2) «(^)^'(O)- 1 (Ci - C2)2 sin 4>q sin ^ x 7H (j H*(k cos 4>0)H2 (kcOS(f)o) fflJ^^l-F,))-;,-^!-»,,))),ikPı£ı H+(k cos <£)#- (Jfe cos <£)Pa(l + Pı)2(Pı - cos<£0)(Pi - cos, X 1± %/t^~p7 (10c) v^'1)^^ rrim 1 (Ci ~ C2)2 sin^psin^ K°Mo(r,0J~ ^ (U;7Tİ C:4 H+(k cos <j>o)H;(k cos fa) [i^(fc)]2 e'«T ffr(fc)]2 elfcP^ ff^(Jb) jff+(Jb cos <£)#2-(fe cos <^)Pi(l + Pi )(Pi -cos<£) [l-jTÇ-fcl^l + P,))] [l-^-H^l-cos^p))] 2Pj (Pa + cos ^0 ) (cos (j>Q - Pj )(cos <f)Q + Pi ) [1-^-^(1-^))]! 2Pı(P1-cos<?i»û) J ”[J?-(tPi)l2 eitr (lOd) IT i(A GV2(l + Pi) Xllr(2,2)“ [H?(k)\* V2e-i7r/4(Cl_C2)2 sin 4>o sin 4> H^(k cos <f>a)H 2 (kcos^>o) [H; 4^|^(-W,(l-Pi))-^-Mi(l-cos^))} 1 Ht{k cos <f>)H2-(k cos <f>)(l + Pı)2(Pı - cos#) -{ [l-f{-2Ux)) [1-:F(-Mi(l- cos <*”))] + (l+Pi)(l + cos^o) (cos^o -P,)(cos^0 +1) (l + POCA-cos^o) J V\/H7,»fcf, \ 2 r 1 1 *ikr 1=F 1 2Ci y/kr In the expressions (6a-10e), we have defined ^1,2 0,2 = Zo where Zq is the intrinsic impedance of free space; and J°i,2 = y/l ~ 1/CiV Finally, ^(z), the Fresnel integral, is given by ulmv dt. sT* K(v)=Jk2-v2; K(o)=-k. kcos0o 0 -k Fig. 2. Complex «/-plane. y<o. (10e) (11) (12) (13) Rev XI nif]2(ı/) are“split”functions resulting from the Wiener-Hopf factor ization of ^,2(^ = ^0,2 + 1 (14) and are regular and free of zeros in regions 3m{//} 5m{^Ia} in the complex //-plane cut as shown in Fig.2. The branch for the square-root function K (u) is chosen to give +k for v = 0, k being the wave number for free space. (?*, <j>) shows the position of the observer in the cylindirica! polar coordinates, with origin located at O and <t> measured from the positive x axis. It is evident from (6a-10e) that the expressions for the impedance strip are each composed of two terms, one representing the contribution from the resistive strip, the other from the conductive. To show this it suffices to use (5) in (6a-10e) and then pick the individual terms to re- obtain results for the resistive and conductive strips. Conversely, (5) can be used in these latter two sets of results, which then add up to give (6a-10e), as expected. Apparently, to each term above there corresponds a dual term con tributing to the total field observed; as, for instance, Vono to Vomo or, Vnom to Vmon-, etc. These can be obtained from (6a-10e), simply letting ^1,2 -» li,\, fa, 2 -> 7T - 4>2,\,<!>-> ir -<j> and Ci,2 -» C2,i. In the final section, bistatic patterns for the doubly and triply dif fracted fields and the total field are provided. Fig.5-a(Fig.6-a) and Fig.5- b(Fig.6-b) show the variation of the doubly (triply) diffracted fields versus observation angle in cases where ?;2 > ?/i and ?/.; < ?/i, respectively, with ?7i held fixed. It is observed that for ?/2 > ??i (?/2 < i]i) the amplitudes of the doubly and triply diffracted fields increase (decrease) with increasing values of 772. The effect of increasing the relative strip widths is visible in Figs.5-3a,b and Figs.5-4a,b in the form of a noticeable -corresponding- increase in the oscillatory behaviour of the diffracted fields. Finally, the total field patterns for different ifthli ratios are presented (Fig.5-5). These, in fact, do not visibly differ from those that could be plotted with only the first order diffraction terms retained; since the second or higher order diffracted fields have considerably lower levels. xiv

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