268 Henin, Elsherbeni, and Al SharkawywhereV 1 = hP (i 1lj l J la i λ 2 − 1 )dλ 2 , (21)0V 2 = P i j l−1 (kdη d λ dJ ld J l − k 0η 0 λ 0J ′ l), (22)S 1 =0i = g,l≠ n[ hlHl=λ 2 0 a − hlH ]li λ 2 d a i = g,l= n (23)i[ hlJl=λ 2 0 a − hlJ ]lg λ 2 d a H ln i ≠ g,iandS 2 =0==[k0H l ′ jη 0 λ −0[k0J l ′ jλ −0k ]dH l J ldjη d λ di = g,l≠ ni = g,l= n (24)k ]dJ l J ld H ln i ≠ g,jη d λ dR 1 =0i = g,l≠ n[k0 η 0= H l ′ jλ − k ]dη dH l J ld i = g,l= n (25)0 jλ d[k0 η 0= J l ′ jλ − k ]dη dJ l J ld H ln i ≠ g,0 jλ dwhileR 2 =0i = g,l≠ n[ hlHl= −λ 2 0 a − hlH ]li λ 2 d a i = g,l= n (26)i[ hlJl= −λ 2 0 a − hlJ ]lg λ 2 d a H ln i ≠ g,iJ l = J l (λ 0 a i )J l (λ d a i ) ,J′ l = J l ′ (λ 0a i )J l (λ d a i ) ,J ld = J l ′ (λ da i )J l (λ d a i ) ,H l = H(2) l(λ 0 a i )J l (λ d a i ) ,H l ′ = H(2)′ l(λ 0 a i ),H ln = H (2)J l (λ d a i )l−n (λ 0d ig ) e −j(l−n)(φ ig−φ 0 ) ,P i = e jk 0ρ ′ i cos(φ′ i −φ 0) sin(θ 0 )
Progress In Electromagnetics Research, PIER 68, 2007269where the integers n, l =0, ±1, ±2,... ,±N i and i, g =0, 1, 2,... ,M.Theoretically, N i is an integer which is equal to infinity; however, it isrelated to the radius “a i ” of cylinder “i”, and type of the ith cylinderby the relation N i ≈ (1+2k i a i ). Equations (19) and (20) are then castinto a matrix form such as[ ] [ ][ ]V1 S1 R= 1 A. (27)V 2 S 2 R 2 CThe solution of the above truncated matrix equation yields theunknown scattering coefficients A in and C in .For the case of incident H-polarized TE z plane wave, the incidentmagnetic field is expressed in the (ρ i ,φ i ,z), cylindrical coordinatesystem as,Hz inci(ρ i ,φ i ,z) = H0 ∗ e jk 0z cos θ 0e jk 0ρ ′ i sin θ 0 cos(φ ′ i −φ 0)∞∑j n J n (k 0 ρ i sin θ 0 ) e jn(φ i−φ 0 ) , (28)−∞where H0 ∗ = H 0 sin θ 0 . One can easily express the field components foraTE z case, be applying the same procedure used to derive the fieldsfrom a TM z illumination, where the z component of the scatteredfield and the transmitted z component of the field inside the cylindermaterial have the same form as in equation (2) to equation (5). Theonly difference in the final expressions is in equations (21) and (22),where these two equations are to be replaced by,(V 1 = P i j l−1 kd η dJ ld J l − k )0η 0J l′ , (29)λ d λ 0V 2 = hP (i 1lj l J la i λ 2 − 1 )0 λ 2 . (30)dTherefore the numerical simulation of TE z polarization can be easilyobtained from the TM z simulation code.3. NUMERICAL RESULTSIn this section, sample numerical results are presented to proof thevalidity of the developed formulation for computing the radar crosssection(RCS) of an array of cylinders excited by an obliquely incidentTM z or TE z plane wave. For all configurations presented in this paper,the incident wave frequency was set to 300 MHz, and the echo width
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