Report #389942-2-T e J;7 (/%,Integral Equations with Reduced Unknowns for the Simulation of Two-Dimensional Composite Structures M.A. Ricoy and J.L. Volakis November, 1987 Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, Michigan 48109-2122 389492-2-T = RL-2566

Abstract A new set of integral equations with reduced unknowns is derived for the modeling of two-dimensional composite scatterers. In accomplishing this reduction the scatterer is first simulated with thin curvalinear layers of material. The traditional integral equations corresponding to each inhomogeneous layer are then manipulated in a manner allowing the identification of a new equivalent current component to replace two of the traditional ones accross the layer. A major effort in this study was devoted to a moment method implementation of the new compact set of integral equations with special attention given to the analytical evaluation of the diagonal and near diagonal elements of the impedance matrix. Several scattering patterns are presented as computed with the new compact set of integral equations. These are further compared with measured data and computations using alternative analytical techniques.

Contents 1 Introduction 9 2 Review of Standard Integral Expressions 12 3 Development of Compact Set of Integral Equations 23 4 Solution of the Compact System of Integral Equations 27 4.1 Formal Solution............................. 27 4.2 Selection of Basis and Weighting Functions.............. 33 4.3 Definition of Coordinates............................. 34 4.4 Explicit Forms of Impedance Elements for Numerical Integration..38 4.4.1 Element Fi, Rectangular Cell:.................... 39 4.4.2 Element Fj, Circular Cell:........................... 39 4.4.3 Element F?, Rectangular Cell:.................... 40 4.4.4 Element F2, Circular Cell:....................... 40 4.4.5 Element F:........................... 40 4.4.6 Element F4, Rectangular Cell:................. 41 4.4.7 Element F4, Circular Cell:....................... 41 1

4.4.8 Element F5, Rectangular Cell:.................. 42 4.4.9 Element F5, Circular Cell:.................... 43 4.4.10 Element F6:........................... 43 4.4.11 Element FT, Rectangular Cell:................. 44 4.4.12 Element Fi, Circular Cell:.................... 44 4.4.13 Element Fj, Rectangular Cell:................. 45 4.4.14 Element Fs, Circular Cell:........................... 45 4.4.15 Element F9........................... 45 4.5 Evaluation of the Singular Impedance Elements............. 46 4.5.1 Element F.......................... 47 4.5.2 Element:.. 48 4.5.3 Element F^:.. 49 4.5.4 Element F... 50 4.5.5 Element F....................... 53 4.5.5 Element F^:........................... 53 4.5.6 Element F:........................... 56 4.5.7 Element F:........................... 60 4.5.8 Element F:........................... 61 4.5.9 Element:........................... 62 5 Far Field Computation 65 6 Description of the Computer Code QRCOMB 69 7 Code Validation 72 2

A Appendix 92 3

List of Figures 2.1 Arbitrary Cylindrical Structure in Free Space.................14 4.1 Discretization of the layers comprising the scatterer..............29 4.2 Coordinate frame used in the evaluation of impedance elements which require integration over a rectangular cell............... 35 4.3 Coordinate frame used in the evaluation of impedance elements which require integration over a circular cell................ 36 7.1 Ez backscatter echowidth of a perfectly conducting half-plane; comparison of results computed via the compact integral equations and a high frequency method for various values of resistivity, R..... 74 7.2 Hz backscatter echowidth of a perfectly conducting half-plane; comparison of results computed via the compact integral equations and a high frequency method for various values of resistivity, R..... 75 7.3 Hz backscatter echowidth by a 1.7 wavelengths wide and 0.01 wavelengths thick dielectric strip having 6r = 7.4 + il.11 and yar = 1.4+i.672; comparison of high frequency, moment method and measured results.................................. 76 4

7.4 Hz backscatter echowidth of a 2.A x.05A material strip with sr and PJr as indicated; comparison of results computed via the compact integral equations and a high frequency method............ 77 7.5 Backscatter echowidth of a 3"x.0544" material (Er = 5. + i.5, r = 1.5 + i.5) layer..................................... 78 7.6 Ez backscatter echowidth of a 1.A x 0.25A perfectly conducting rectangular cylinder; comparison of high frequency method and compact integral equation solutions........................ 79 7.7 Ez backscatter echowidth of a.05A thick material (r = 5. +i.5, /r = 1.5 + i.5) layer upon a 1.A x 0.25A perfectly conducting rectangular cylinder; comparison of compact integral equation solution with finite element method................................. 80 7.8 Ez backscatter echowidth of a 1.A x 0.25A perfectly conducting cylinder coated by a.05A material (Er = 5. + i.5,,r = 1.5 + iO.5) layer; comparison of solutions obtained via compact integral equations and the finite element method........................... 81 7.9 Hz backscatter echowidth of a 1.A x 0.25A perfectly conducting rectangular cylinder; comparison of high frequency method and compact integral equation solutions........................ 82 7.10 Hz backscatter echowidth of a.05A thick material (Er = 5.+i.5,,r = 1.5 + i.5) layer upon a 1.A x 0.25A perfectly conducting rectangular cylinder; comparison of compact integral equation solution with finite element method................................ 83 5

7.11 Hz backscatter echowidth of a 1.A x 0.25A perfectly conducting cylinder coated with a.05A thick material (er = 5. + i.5, r = 1.5 + i.5) layer; comparison of solutions obtained via the compact integral equations and the finite element method......................... 84 7.12 Backscatter echowidth of indicated microstrip geometry.......... 85 7.13 Hz backscatter echowidth of a l.A-per-side perfectly conducting equilateral triangular cylinder; comparison of solutions via the traditional and compact integral equations..................... 86 7.14 Hz backscatter echowidth of a l.A-per-side perfectly conducting equilateral triangular cylinder coated with a.05A thick material (er = 4) layer; comparison of solutions via the traditional and compact integral equations................................. 87 7.15 Ez backscatter echowidth of a partially coated circular perfectly conducting circular cylinder of radius R = 1.OA. The coating is over half of the cylinder's surface, as shown, and its dielectric constants are er= 5. +i.5& pr = 1.5 +i.5........................ 88 7.16 Hz backscatter echowidth of a partially coated circular perfectly conducting cylinder of radius R = 1.OA. The coating is over half of the cylinder's surface, as shown, and its dielectric constants are er- 5. + i.5& r- =1.5 + i.5............................................ 89 7.17 Ez backscatter echowidth of the shown coated and uncoated wedgecircular geometry............................. 90 6

7.18 Hz backscatter echowidth of the shown coated and uncoated wedgecircular geometry............................. 91 7

List of Tables 4.1 Criteria Used for Integration of Impedance Cells........... 35 8

Chapter 1 Introduction Conventional approaches for the numerical computation of the electromagnetic scattered fields by inhomogeneous dieletric scatterers entail the formulation of a system of integral equations for the determination of the induced polarization current densities. These can then be used in the radiation integral for the computation of the scattered field. Assuming an e-i"" time convention, the electric (J) and magnetic (J*) polarization current densities are traditionally defined [1] as J -ikoYo(er -1)E, J* = -ikoZo(r - 1)H (1.1) where er and Pir are the relative permittivity and permeability, ko is the free space wave number, ZO = 1I/Yo is the free space intrinsic impedance and E and H denote the electric and magnetic fields within the dieletric medium. For the general three dimensional case, (1.1) involves six independent current components which must be determined usually via a numerical solution of the pertinent integral equations. However, when (1.1) are applied to a two dimensional composite scatterer, with TE or TM plane wave incidence, only three unknown current components are required for the complete characterization of the scatterer. Thus, in the numerical imple 9

mentation of the pertinent integral equations 3N unknowns must be determined if N denotes the number of cells comprising the discretized scatterer. As is well known, the required computation time for the solution of a system via a matrix inversion method is proportional to the cubic power of the number of unknowns. Thus, any reduction in the unknowns translates into a substantial improvement in the computational efficiency/economy of the solution algorithm. In this report we derive a new set of integral equations as applied to two dimensional composite scatterers involving a reduced number of unknowns. Particularly, it is shown that the complete modeling of the scatterer can be accomplished with two equivalent current components over its cross section versus the three current components usually required with the traditional approach [3,4,5,6,7,8]. In accomplishing this reduction we first subdivide the scatterer into thin inhomogeneous curved layers of material. The integral equations corresponding to each layer are then manipulated through various integrations-by-parts, differentiations and rearrangements. This allows the identification of a new equivalent current component to replace two of the traditional polarization currents accross the length of the layer. Namely, the axial and one of the transverse components (to the cylinder) of the polarization current are effectively replaced by a single mathematically equivalent current component. The normal component of the polarization current at the two terminations of the layer, though, cannot be eliminated. However, assuming that the extent of each layer is long, the presence of these last components do not add appreciably to the total system of unknowns. Thus, the presented modeling formulation allows a reduction of the unknown current components from 3N to es 10

sentially 2N. It will be seen, though, that this reduction in unknowns is achieved at the expense of complexity in the resulting integral equations. To demonstrate the validity and applicability of the derived integral equations to cylindrical scatterers of arbitrary cross section and composition, we have considered the numerical implementation of these via the method of moments. In this implementation we employed pulse basis expansion functions and point matching primarily for the purpose of simplifying the derivation of the matrix elements. Special attention is given in this report for the evaluation of the matrix elements and particularly the diagonal (self cell) and near-diagonal ones. As usual, the integrals defining these are associated with integrands involving the green's function which is singular at the self cell and must be evaluated via analytical means. This is accomplished by employing the small argument expansion of the green's function to obtain an integral that can be evaluated analytically. By keeping sufficient terms in this expansion we have attained an extremely accurate evaluation for the diagonal and near diagonal matrix elements. As will be seen, such evaluations are of particular importance in the case of vanishing thin adjacent layers. An example of such a situation is a perfectly conducting surface on a dielectric layer. Several scattering patterns are presented using the developed code, described above, for a variety of composite structures. These are also compared with measured data and computations via alternate numerical and analytical methods. 11

Chapter 2 Review of Standard Integral Expressions In this chapter, we derive the standard integral expressions for the fields generated by the presence of two dimensional electric and magnetic currents. From Stratton [9], the most general expressions for the electric and magnetic fields are (in terms of Hertz potentials) -. -. d2- d E = vv. n- - -v x fn* (2.1) dt2 (dt _ d2- d H = VV * II- II* + eV x n. (2.2) In the above equations, e = er6o, p = prOo and the Hertz potentials II and 11* are given by iZ / n=- JG (r ) dv' (2.3) H* =iY f*G3 (r, ) dv', (2.4) where J and J* denote the electric and magnetic currents, respectively. In addition, G3d(ri,?) is the three dimensional free space green's function and V' is the volume occupied by the currents. Employing the e-"t time convention, (2.1) and (2.2) 12

become: E = VV. H + k2H + ikZV x l* (2.5) H = VV 1* *+ k2I* - ikYV x (2.6) with k2 = w2-ye, kZ = wy, and kY = we. If we are interested in the field scattered by a generalized volumetric material in free space (see figure2.1), Maxwell's equations within the material take the form: VXE = iwuH = iwo - J* (2.7) VXH = -iweE = iwo + J (2.8) where J* = -iwo (r - 1) E and = -iweo (E, - 1) H are the equivalent electric and magnetic currents. Thus, the volumetric material can be replaced by equivalent currents acting in free space. This allows the field expressions to be written solely in terms of free space parameters, E =VV. * + koH + ikoZoV x I* H8 = VV. H* + koH* - ikoYoV x = ~ I JG3d dv' -. I* = iYo JG3 ( r)dv'. (2.9) Expanding the above field terms we find that d2 d2 d2 d d d2 d2 d2 d d ( ~I, + n2 y + dy dz I + ikoZo n* - ikoZo- ) y d dx y2 d ydz dz - dx 13

y Single Curvalinear Layer C C' Figure 2.1: Arbitrary cylindrical structure in free space. Figure 2.1: Arbitrary cylindrical structure in free space. x 14

+ (~Hx~+ d2 d2 d ikZ0- A H~y + koH + -Hz + i-koZoyll* - iooll dz dy dz2 x dy x/ (2.10) and Hjs = k2ll*~+ II*~ 0xdX2X dx dy " d2d d A\ II*_ - ikoZ0-H-I + ikoZo-ll-i 'y dxdz zdy dz, (d2 d2 d 2 d d A + 1 1* + k,2ll* + -XI ~H* - ikoZo-H7x ikZO-H ~ \+ ddI X 0 dy2~ Y+ dy dz dz +iodx,/y d2_d_ d2 d d A + (d[Xl* flr* + k2]ll* + _H* - dxko~ ~l ikoZo-llH Z. +zd X dzdy dz0 ik0Zo-17I1d dy x (2.11) Substituting the full expressions for the Hertz potentials, we further obtain ES i ko / J[0 k 0 Jv iko I' (dy Es=- Z iJ d k0Jv ko dzdx d2] ___ -I y +Jiz dX2J + dxdy dxdz) G d(F, i) dvI * d JZ - - I dy d2] d2 )d~(,)d/~(~ + j dydz ~Jdd H + ~2] Gd (F i —) dv'- JY d2d2 d 2d dX2j dxdy Zdxdz I rv+Vz d G3d (F dj - Gd (F, fq) dv' - Z - d (F, iF') dv' - Gydz d (, fq) dv' Hxs =ZYO ko I k( 2k + Hy x +Jv*k~ +,i4)d' f (J — - Jz~~ H ko J( dydx ~ d +*.... )dv +Iv kdz dxj.dzd2 2 [ 2~] ddJx* + dy JzLkodZ2) Gd (i? d I) Gd (F, i-') dv' G3 (F, f') dv'. In order to specialize these equations to the two-dimensional case, all derivatives with respect to z must be equated to zero since in the two dimensional case there 15

is no field variation in the Z' direction. After doing so, we obtain Es izo x ko I E's izo y ko I [2 + +d dd) G3d (i? ~ v - J * J~;3d (j; -~ v (d2 d2rd + 2+ d \G3d (F,9)v+ 3d (F -d dydx +Jy [ko dy, rJ))dv IvJz dx~ Es=iZoko Jz Gd (F F') dv'I - J ( - -J4 - Gd (F r-') d v' Hs= kIV(Jx* [k 0 ]+ jvdd* )G d(F, ') dv'~ f Jz 7G rd( i') dv' Hys= il0f ko Iv (dr ixdydx 2. 2 d + jy* k 0 +dy2 9. A Gd (F, i;') dv' - J i, Jd+GGdr(, F'*) dv' iYk0Jz, JGd (F iF) dv' + I JY7 -i - G I (i;* ) dv'. (2.13) Since J and P are independent of z', the integration over z' can be carried out as G r(F, -f') dz' L ~ ~xx)2+yyI2+(z~) — dz' - 47r x - x') + (y - y/)2 + (z -z2 00 eik +(7~Z-)2 -]oo 4 r p2(-'2 - i o'(kp 4r - GH(lF)(k (2.14) where Ho(') (kop) denotes the Hankel function of the first kind and zeroth order and Gd (F, F)is the two dimensional Green's function. Substituting (2.13) into (2.14), we obtain izo d2. d2 Es Jx k 2 + x ko I 0 dX2 + j y 7; -d-y izo d2 d 2' Es Jx + Jy k2 + y ko I dydx L 0 dy2i G2d (-,'d~ A'Jz -Gd (, F) dA' G rd(F F) dA' + IA'p +Gd (r?* F') dA' 16

E: = iZoko J zG (r, if)dA'- J - -J y- G (r, i) dA' ' J: ~2 + 2 d2 d k J ( [ 2] + Y d) G2 (, ) dA' +, J G2d ( dA' Yz = iYoko JzG (,) dA + ' (J+ -J Gy ( ') dA'. (2.15) We can now derive integral equations for the solution of J and J*. It will suffice to consider development of the Ex and Ez integral equations, since the evolution of the remaining integral equations parallels these two cases. Rewriting Ex from (2.15) in a slightly different manner, we have E:x= ko JA Q+2 + yxj q Jdxdyj i ~ ( [ d2 dy2 2 IA| J d G2 ' ( dA (2.16) (k2 + d- 2 + d (, ') = -(- F) (2.17) and assuming that the field i s measured within the volu metric material, (2.16) becomes iZ0 do d d 2 d2 E J(- k')dA'o + -GodA ko - J ( )dA + kd y2 y2 dxdy )dA r dG2 (', )dA'. JA ddA. (2.18) It is now observed that - k ~ | - - dA (2.17) Z ko Jk dOdy Eo (-Jx6(r F- f*)) dA' Es + zo i x ko k 17

= E + ~ (-k0Y0(r - l)(E C + E)) = E + (e- 1)(EC~ + E) = (Exnc + E;) - E'nc rEtot- Ex. (2.19) Furthermore, recalling that Jx = -ikoYo(e,.r- l)Ett, we find Etot = iZo (-l1)ko x = RJ, (2.20) where R is usually referred to as the resistivity. Substituting (2.19) and (2.20) into (2.16) we obtain ko JA' dy2 Ydxdy JA' dy (2.21) The Ey, Hx, and Hy equations are obtained in a similar manner. We note, however, that for the Ez and Hz components, we must also make the substitutions: Ez = Etot - EFnc = RJz - Enc (2.22) H Ht~t -HC = R*J - HznC. (2.23) Summarizing, we have E7c = d.RJ+ d ]Gd( fA+fJ,y2d &y ko dy2 dxd) A' + JA Z dy E rR + f [ d2 7dd ] G2d (f)dA ' J J dG2d (ff)dA EY I JRJy J + JY G (r, r*) dA - J* -- -dA E~'C = RJz - ikoZo JzG (r) r) dA + dx dy G ( ) dA 18

Y d d. d H IR*J+ ']o JJdd )a- dGdA, H rR C JA' dy2 J dxdy (f)dA JZ dy iYo d da] Hy,., D.* ^ 12df^ 1f\ nj A2d/ T- ^ (/ ' I [ + + J~2 G (r, ) dA' +, Jz c ldA R* -koo JA y ddx J dx = R*J -ikoYo JzG (i, i) dA - - G2 (r ) dA'. (2.24) The above form a set of integral equations applicable to dielectric/magnetic cylinders of arbitrary cross section and material composition. Their direct numerical implementation, however, can become cumbersome for scatterers associated with curvalinear surface material boundaries. Therefore, it is instructive to obtain a set of integral equations suitable for modeling curvalinear layers of material. Such a set must then utilize the local directions of the layer over the integration. We denote these directions as s' and n', where s' is tangent to the curvalinear layer at (x', y'), the integration point and n' is the corresponding normal to the layer so that s' x n' = z'. Furthermore, we may express the location of each integration point in the new coordinate frame (s', n', ) by (s', n'). Similar transformations may be introduced for the observation point so that the directions (x,,z) are transformed to ((s,n,i) and the coordinates (X,y) to (s,n). We now proceed to implement the above variable substitutions in the integral equations (2.24). Again, attention will be confined to the Ex integral equation with the understanding that similar steps apply for the integral equations involving the other components. Suppose we have some arbitrary cylindrical material configuration which has been partitioned into thin layer where the observation point is within the material and is defined by the coordinates (s, n). Since it is immaterial in which direction we orient the x and y axes, it is to our advantage to orient them so that x =, 19

and y = n. From (2.24) EZo d+ dJ-7 G G2d ) ds'dn'+~ JdG ds'dn' Eo J=r [dn2 dsdn J dn (2.25) The integrals in the above equation are over s' and n', it is necessary that the corresponding integrands be expressed solely in terms of these variables rather than s and n. The second integrand in (2.25) is already in this form and it is required to only consider the first integrand of (2.25). We have d( 2 d2 dG2H') (kp) d [-ik (Js( 5 ) + M ) n) (p. n)()(kop)] dn [io (Jt(n s' ) + Jonl(n n')) (/p s.)(, ) (2.26) 4dsdn However, (d. s')H l)(p s)ko )( s') x p) =-z ( As' x p) d z (p x s') p ( {s' x z) =-p n' (2.27) 20

and n. n')(p ) - ( -. n)(s. n) A (n x s) * (n' x p) -z (n' x p) A = z (x n') (n' x z) p.s (2.28) implying that J dn2 J dsd2 ) dn [ (Jn'(P ' s)- Js'(p n')) H()(kop) d ( (rG ) -,dG2 (r) dn Jn' ds' i dn' ). (2.29) Thus, Einc may be written as Einc = iZ d ( Es = erRJs+ - - Jnlko JA' dn + / J dG () ds'dn'. JA' dn -G,) - dG2 r, ds'~ dn') (2.30) As discussed earlier the integral equations for the other components follow in a similar manner. Introducing the definitions J = ZoJ, R = R/Zo, and R* = R*/Yo, the complete set of integral equations in terms of localized variables is Einc J i r d (j dG2 (r, ) - JdG ds(dn ErR koErR dn dn' ds' 1 2dG + 1_ J*dG ds'dn' ER A' Z dn ERn i f d (y, dGd (dFG) dG'd(n, )) IJn + is - dn n ds /dn' kr A dn' ds' 21

1 f dG~2d (ii)dd ec.R A'Z ds ZoInc ik fd( Go ~ ~)-, U R 8 _ _ __ _ _ _ _ddd (, r '! ds' G r) ds'dn'n 1f dG2 (f G~ /IR J' dn' dsdn ZoHnc idIG2d~f _ 2dG(llf J*ds') ds'dn' /I rR* ko/iL rR*JAdn \S1 dn' J s + - 'jz dGsd(l)i ds'dn' YLrR* JAn Z0Hy" ik f ZGd (idG~ ds'dn'd PrR* Jn + kogft* JA'*/ d' Jn s 1 f ('i, d 2d (,i) d~ j ~ +t A k< dn d-s ds'dd'n'n2.1 22

Chapter 3 Development of Compact Set of Integral Equations The integral equations developed in the previous chapter involve a total of six unknown current components which reduce to three provided we assume an Ez(Hz = 0) or Hz(E\ = 0) incidence as is usually the case. In particular, for the Hz incidence case the three relevant integral equations are _ -__ / d ( dG2d(G ') ( dG (,' ErR kS OErR IA'dn k dn' ~ Jn ds')) ds'dn 1 I J* dG2d ds'dn' eR z dn En l+ -J +J d is, d Gds'dn\ R +k A' ds 8dn' J ds'n 1rR [ + E IA d (iyrd,d ErR A' ds Z iHnc i*d Jz -/ Jz G2 (; ) ds'dn' R* ' dn' ds' As will be shown in this chapter, it is possible to reduce that number of unknowns in the above equation set by introducing a known equivalence between electric and 23

magnetic currents [2]. Before proceeding with the application of this equivalence it is necessary to first introduce certain modifications in the integral equation (3.1). Noting the identies (d 2d (j? pt) (ds' d2d (F iF')' ds'J ds' (dn) dbs' ( ) d ds (3.2) and employing integration by parts in the integrals of (3.1) involving i,, we find that k0rIA dI- ds)ds'dn' koerR IA dn ds' 'd'n lbIA dJ (,, ds'dn' z koerR? [ / dG" (r, F)dn' I endpoints JIc dn fdJni dGd (j, i~ ds'dn' k0&rR IAdcs' dn ____ d2d (F- iF') k0ir I - dG cbr dn' endpoints + [ dJn' dGd (rF' ds'dn' k06rR IA' ds' dn = R 'C JiG~ (r* Fv dn' endpoints ~ dflG"d(r iF.) ds'dn'. (3.3) Substituting these expressions back into (3. 1) and introducing the equivalence [2] jz* =Jz* iw e obtain ErR is - kO L~d' (dG rFF))d'n is [ d (is F') ds'dn krR IA Idn' d _____ iIdGd (F dn/) en dp oint s +ko0&rR I '~ dn d 24

~?'inc 2 d ' JG" (, E~n _ i f -d (dG d(FG ')) \ - = Jn + R Js d ' — d ds'dn' 6r k JAd fl ( ds n / J1 dG r ds'dn' i| JnG, f) dn' I endpoints kJErRS ds Zo-H c -iko fa jGd - Z R* ]A1 1 r,dG2 ) G Js, ds'dn' R* IA' dn' + I Jn G (r, k) dn ) di endpoints. (3.4) To remove the remaining component of J, in the En integral equation we first diffentiate that equation with respect to s, multiply by -i/ko and then add it to the Hz integral equation. The resulting equations are Einc Js! J' d )dGds'dn' rR krRJ A dn' dn 1 dG2 ( ) ds'dn' rRJA' dn R* oCeeR cds o s kd tko ej d[1 ld (1 d 1 d21 - ikoJ[1 i d (I) d k d2]G2d ( ) d + J n'[ * k(2ds ) k~ 2] G2d (F, F?) dn'j in (3.6) Clearly these involve only two unknowns throughout the cross section of the inhomogeneous cylinder, namely J, and the equivalent magnetic current J*. In 25

addition, there remains a presence of the Jn, components only over the outer surface of the cylinder coincident with the n direction. When considering a layer simulation of the structure, these Jn, components are in fact the currents at the two ends of the inhomogeneous layer in the direction normal to the layer. It should be obvious that since the bulk of the unknowns is certainly within the cross section of the cylinder, the presence of the above Jn do not add appreciably to the total system unknowns. Thus, with the derivation of (3.6) we have in essence reduced the system unknowns from 3N to 2N. Considering that the computer time required for the numerical solution of a system is proportional to the cubic power of the number of unknowns, equations (3.6) call for a tremendous increase in the efficiency of the intended numerical solution in comparison with that required for a solution of (3.1). However, it should be noted that we have achieved such an efficiency by increasing the complexity of the resulting integral equations. As will be seen in the next chapter, the numerical implementation of the higher order derivatives appearing in (3.6) must be evaluated with extreme care especially at the self cell. In the next chapter we consider in some detail the numerical implementation of the compact set of equations (3.6). 26

Chapter 4 Solution of the Compact System of Integral Equations 4.1 Formal Solution Although the compact set (3.6) may be solved as is, it proves more convenient to utilize the En integral equation in (3.4) as an additional auxilliary equation to be enforced at the strip edges. This, of course, does not bring about an increase in the number of unknowns. Before proceeding with the details of the numerical implementation of (3.6) the following definitions are made in an effort to simplify the discussion on the subsequent operations. Namely, we define g9i(r) = g( = ZHnc() i dE'nC(r) iEtnc() d ( 1 R * () koe r(F)R() ds ko ds Er()R()) g3() E= (0 y6 / r(rjR(r) = 1 dG2D(,i ' ) '~(W(~ dn 27

1 1 d d 1 1 d R*(r- ko ds E.(rjR(rj d 9 ko=~rM(r( d-. ~: J f(rr)- kd R ) -ds k r(R( s dG2D(2 ' f 1 dG2D() (4.1) /3e R (rj ds Furthermore, introducing the subscript q = 1/2 to denote leading/trailing edges, the new system augmented with the En integral equation of (3.4) may be written as ( ) [ s/ df ( ")] JdA~ IA' Jz ()f f (rF)dA '+ Jn, (i) [i(- 1) fi(r )] dn' (4.2) 2( = JZ( A+ ) [ df n ] dA'J+ JZ () (-i ko) f2(r, Mi)dA'+J Jn'(Ia) [(-i)f2, r )] dn (4.3) df3(i? ~)] dA'-F[ J() f( 1d f [( ) 93( = jn(+j ) [ d dA'+JA ) [-f J)dAcn' () f3( if) dn JA1 [ko dn' JA' kJc [ ko J (4.4) where (4.4) will be utilized only for generating the additional equations needed for the determination of Jn(r) at the terminations of the layers. Assuming that the material has been partioned into thin layers, let each layer be further subdivided into a discrete number of expansion cells (totalling N for the entire body), whose area is denoted by A, i = 1, N (see fig4.1). The currents Js and J may now be expressed by a series of expansion functions with constant coefficients, viz., N WJ(j) = E [KsiPipd(r + AJsi(')] i=l N J (i) = E [Kpp 2d(i) + AJzi(r)] i=l Pi2d(r { hi (r), F E Ai ~ A(4.5) 10, r#Ai 28

Y termination surface elements of length C N+1 C N+3 th cell of area A Ii cell of area A j N+2 termination surface elements O- X Figure 4.1: Discretization of the layers comprising the scatterer. Figure 4.1: Discretization of the layers comprising the scatterer. 29

where h d(r) is the basis function corresponding to the th cell and AJsi(r) with AJZ7(r-) have been introduced only to make the equation exact. Let now Nedge be the number of expansion cell edges which lie along layer edges, denoted by Cii = 1 Nedge. Note also that since in the solution of (4.2) to (4.4), we require knowlege of Jn(f) only along layer edges, we may replace Jn(r) by Jnde(r). The current component Jed9e(r) may subsequently be expressed as a series of expansion functions with constant coefficients (over Ci = -,lede), augmented by a difference term, viz., Nedge Jedge( E [niPl( + J (r)] i=l Pld(_ ' = (4.6) o, ~ Ci It is clear that expressions (4.5) and (4.6) contain 2N+Nedge unknowns. These can now be substituted into (4.2) - (4.4) and the resulting equations may be rearranged so that the terms involving AJ are collected on the left hand side of each equation while the remainder of terms are gathered on the right hand side. In the application of the method of moments for their solution, it is then customary to multiply both sides of the equation by some weighting function. The integral equations can be subsequently integrated over the width of the weighting function which usually spans the extent of a single cell. Appropriate weighting functions are W2d(r) { w2d(') r e A, Wi? (4.7) o re Ai for the case of (4.2) - (4.3) and W l ) = V (4.8) W( _ Ci 30 r 30

for the case of (4.4), where wOd, (wI!d) is an arbitrary function corresponding to the ith cell(edge). When these are now employed in the procedure duscussed above and the weighted integrals involving AJ are set to zero, we obtain GI =U~d Ks~ + E~ F'K8 ~ + y~v1qF~j*. + Z~Vedge Fj3,edge C> u~K*- + Z~v J7AK + z;~: Fj~IKZ+ F Ne IL Ul ~dI~edge + N 7 EN FS* edge WeKdge ~ Z-~~Ksj+ZjFKi 31 in; i= 1,2,.. Neg (4.9) where, upon definition of the operators L~?d, Ltd, Mld, and Mjd as we have L~d(f) = W~d(rl')f (idA Lld(f) = W~Id(rf f(fldn M~d(f) = A 2d (i;*)f)A M~d(f) = ~ hd(F9f(,dn, G2 = L~d(gi) G= L d(92) G. = L (93 Uid = L?d (h?d) Ud= L d( d = LdM~d (f1(li) 31 (4.10)

F3 ij4 4 1' — tj 5 Z33 ~j7 F.6 1' — F7.8 F? 9 r —~ = (t A )) ko dn' ) = Lt, (-iko/f(r )) LdMld ((-1 )f2(r, )) = LdMjd ((-l)f3(r- r)) ko f( (4.11) Equation (4.9) may be also written in matrix form as G1 [p1] [F2] [F3]) ( K, G2 = [F4] [F5] [F6] Kz G3 [F7] [F8] [F9] Kdge where Fall are the impedance matrix elements given in (4.11) i (4.12) and Fij are given by 1ij F1ij + 7yjui:5 5 2d F, + YjUd Fij = F + ju t3 S (4.13) with yij denoting the kronecker delta function. A solution for the current expansion coefficients K1s, KI and I~Kdge may now be found via standard matrix inversion techniques. However, before this can be accomplished it is necessary that all integrals in (4.11) be first evaluated numerically or analytically. Specifically an analytic evaluation will be necessary for the self cells and a numerical one for most of the other cells. 32

4.2 Selection of Basis and Weighting Functions To simplify the evaluation of the matrix elements we will employ a rather simple form of basis and weighting functions. Specifically we will employ pulse basis for the expansion of the current, implying hid(r-) = 1, h d(r = 1 (4.14) and the weighting functions will be set to wld(r) = 6(r- ri), i2d(w ) = 6(- i), (4.15) where, as shown in Fig. 4.2, ri denotes the centroid of the ith cell or the center of the ith edge element. Using the above weighting and expansion functions, the elements of the excitation vector become G =gl(i) = E lC(r) G2 ZoHtc(i) i dEC(ri) iEnc(ri) d1 2R(r) kor())R(i) ds ko ds e()R )) G =g93(I) = E () (4.16) er(rj)R(rj) Also the impedance elements may be summarized as ~1 A o dn' _=ri Fi=. li(r-, ) Fi4 = IA i ()-1)q ko fl(~/, ~) F f. =j (-df2(,Fi?)) 33

5; = J+ (-ikof2(r)) F^ = -w) F. |,(1)qf ri )a 7 f (i df ) (r ) = JAi ko dn' f ) F, - IA (-f3(ri r)), = V+J+ | k )!f3(7ri, ) (4.17) Two different integration schemes will be considered for the evaluation of (4.17), depending upon whether a rectangular or uniformly curved (circular) cell is assumed. In the former case, the cell evaluations may be done with high degree of accuracy for all values of p, where, as usual, p is the distance from the point of observation to the point of integration. However for the latter case (uniform curvature) the cell evaluations will be in certain cases more approximate depending upon the magnitude of p and the size of the arc subtended by the cell of integration. Cells of non-uniform curvature (e.g. elliptical or parabolic) are not considered in this report. The type of integration employed will vary according to p as summarized in Table 4.1. 4.3 Definition of Coordinates In the evaluation of the impedance elements requiring integration over rectangular cells, it is convenient to introduce the geometry shown in Fig. 4.2. The coordinates/components of the labeled points/vectors are the ones which are required in the subsequent analysis. It is further necessary that some of these be defined with respect to two coordinate frames in the case when a singular evalua 34

Regime Straight Edge(Rect. Cell) Circular Cell kop > a1 (2-dim) 3pt. Simpson's rule 2-dim. 3pt. Simpson's rule al > kop > a2 (2-dim) 5pt. Simpson's rule 2-dim. 5pt. Simpson's rule a2 > kop analytical evaluation 2-dim. 5pt. Simpson's rule/analytical self cell analytical evaluation approx. analytical evaluation Table 4.1: Criteria Used for Integration of Impedance Cells y A n / I Is A 0 -- - - - - - - - - - - -01 x Figure 4.2: Coordinate frame used in the evaluation of impedance elements which require integration over a rectangular cell. 35

Y j2 'D r P \j3 Pj4 r 0 Figure 4.3: Coordinate frame used in the evaluation of impedance elements which require integration over a circular cell. q 36

tion of the element is required. One of these coordinate frames is associated with directions (s', n', z) and has its origin at 00. With this coordinate frame as our reference, the following definitions will apply (the z directed component is suppressed because of the assumed two dimensional geometry): (Sobs, nobs) (s', n') (si, ni) (sj, nj) (sj - j/2, nj -rj/2) (sj - bj/2, nj + rj/2) (sj + Sj/2, nj + rj/2) (sj + &j/2, nj - rj/2): and n' components of the vector r ' and n' components of the vector r' ' and n' coordinates of Pt:' and n' coordinates of P s' and n' coordinates of Pj s' and coordinates of P s' and n' coordinates of Pj ' and n' coordinates of Pj4..~ and hn coordinates of Pj3 The other coordinate frame required in the analysis is that associated with same directions (.s', n', z) but with origin shifted to Pi. With this coordinate frame as our reference, we introduce the following additional definitions: (sj - -j/2- si, nj - Tj/2 - ni) = (xl, yl): s' and n'coordinates of Pj3 (sj - j/2 - si, nj + rj/2 - ni) = (x2, Y2): s' and n'coordinates of Pj2 (sj + 8j/2 - si, nj + rj/2 - ni) = (x3, y3): S' and 'coordinates of Pj3 (sj + 5j/2 - si, nj - rj/2 - ni) = (4, y4): S' and n'coordinates of Pj4 P1 ()2 + (yl)2 P2 = /(2)2 + (y2)2 37

P3 /(x3)2 + (y3)2 P4- (X4)2 (4)2. In the evaluation of the timpedance elements requiring integrationion over circular cells, we introduce the geometry shown in Fig. 4.3. In this situation, it proves convenient to define a cylindrical coordinate system (pj, 9' Z) with origin 0', the local center of curvature of the jth cell, and axis 9' = 0, which contains the point Pj. The following definitions apply with this coordinate frame as our reference: (Pobs, Oobs) (p', ') (pi, Oi) (pj, O) (pj -,pj/2, -AOj/2): (pj + Apj/2, -Aj/2): (pj + Apj/2, AOj/2): (pj - Apj/2, Aoj/2): Al p pA p p, p p, p p and and and and and and and and o0 components of the vector r 0' components of the vector r' 0' coordinates of Pi 9' coordinates of Pj A' coordinates of Pj1 9' coordinates of Pj2 9' coordinates of Pj3 0' coordinates of Pj4. 4.4 Explicit Forms of Impedance Elements for Numerical Integration To perform the numerical integrations, a 5x5 grid of sample points is generated for each cell. The parameter p given in Table 4.1 above is taken to be the distance from the point of observation to the closest point on the sample grid. Below we derive the final expression for the elements Fij as employed in their numerical 38

implementation: 4.4.1 Element Fj', Rectangular Cell: =~ ko dn' I8-6j/2 1n —r,/2 ko dn' -s + js+Si/2 kf1(r n)If n +flj T/2d -1 j+ j83+/2 ri [dG f, j~ ] l'fd+T/2ds 6/2k0o f)R t dnj 2. s +5j/2 -dG[ r-k 1 fl nflj+Tj /2 - /'j+ I.d.2 7(s7f4\I A f _ s 833 dn rPr2 rJ n=nj. 1- /2 1 f\\1i'=jJ +r,/ -s- 13j /2 46(-RF)~ ik)o'(ko))A.J, /2d' 4.8 I 4 p n21-idfk(p),ds) =kf -i r~prip3Apn/2 j+Ap,/22 sf AO,/2 -i iGnfi' 1 '=,+A3/ J-O/ ke()Rf) n HT=TiJo p'p-Ap3/82 p3+ 8p/2 r~6/ _ _ _ __ _ _rd, ' dI dp ]prij2 -O/ kR6r'i) R( )\ dn 46r */2-r - 4-4AGElmen kirulr~i)~~ C(el(h. ~H1(0P) 1P=j AJ/ - +Apj /2 A~O /2 _ f =i i - j p7 AA9j/2 1 rd)I~~ 4P (0'd.)H((k ))__ddp A~y/2 4- i)- i)[(',s?pH'(k' ] pjp/2 +Ap /2 [Pi+ fl 1'p + Ap'__ Jr Apj/2 Jo ri- ~ p-pj/2 (ol + -)A1(kP)) __2 AO 2k lrrdO'dp' 4.9 39

4.4.3 Element F~, Rectangular Cell: fS +T32~r/ =],6, /2 I-r/2 fii, iV')ds'dn' [si +6 /2 fn3 +rj/2 1 dG(rf i;)) = s6, /2 1n-r,/2 &rif)R(rit) dn d'n js.+S,.7/2 f..+rLj+/2 1 -iko( HA A k~) 'n =ps6/ nr/ 6 f)~~) n)H1(kop p ds'dn' 4.0 J 6 p+p/2 fArj /2 1i (d4, ~ Ji p+Ap,/2 jA +ry/2 1ik [Akoi Al1j~~~) ~ jppp/ O/ k (s nt)H(1)(kop)] _ pdO'dp' (4.21) 4.4.5 Element F3:icla el IjAp/ fj+j/2 j~) 40

4.4.6 Element Fi~, Rectangular Cell: J dfSj2JrkT/ if()'dd s,-S,/2 n.,-r,/2 -dn' j3.+S3/2 1 -1d s,7+6,j/2 [1I d + 18-6,/2 [kg2ds I j 1 d, n =nj +,r /2 r-'f - # - # -u r r )- - ds' Er(r-*i) R (r-*i ds ( I r=ri - nf=nj —rj/2 s +6 /2 1 r-Hl'(kop I. dsI=njI+Trj/2 s-Sj/2 R *( r'i ) 4 n=n - 2 I. s,+c5/2 1 d +s-8j/2 ko7is I -iko ( A _ A I n'= nj +,rj / 2 p S)H("(kop) ds 1 1 -4 1 -# -0 n =nj -,rj /2 Er(ri)R(r'i) 4 r=ri I s+63/2 11 + ki ()I 9 i (2k(, A )2H A)(kA)) ___ (kp n=n+j s+ 6,2 26r (1 -#) + ko p 5 p nkpdln~j-j/ s,-6,/2 Si s+5,/2 -i d + 865/2 ~, - I ( A _ A I n'= nj +,rj / 2 S)H")(kop) ds' I p 1 n =nj -,rj /2 er(ri)R(r'i) r=ri fsj+6j/2 j ( A. A2( A A2' H______ I 4-(6)Ri2 ki - _JJ p.j) 2 kop (4.23) 4.4.7 Element Fi~, Circular Cell: = J j f AO/2 -df2( -, j~v) ('Od fAO/2r Iq p fP+Ap3 /2 ~AO /2 ]Lf ri, ] =P-Lp/ dO' + ]-p/2] f2(ri, i")dOdp' 41

4.4 25 I+r/2 -i (p 1 ( *(pj^/2 JAj/2 4 kR *(r:) +Er(ri, ) P) ~ p' (k~p)'-' ~'': -" / +A O,/2 - d ( (op)), /2 +i p(.S- ( H d(kop) dPp +-Azoj/2 4,(,.i),R(ri) 4 P ('i * ) ' ' kop),'" -'"' + +AOj/2 pj+Apj/2 1 (((*S)2-(. )2)H (koP)) ddp J-^j/2 p-Apj/2 4e\(rf) e(r-i )R(), rOrr 3'j/ 464 )R )k(4.24) 1.8 Element /i, Rectangular Cell: = j + | (-ijo)A(., )dA' +/ A +1 J /2 nj+ 2 d f( / + / (-io) I(- ) - Go.(rF)(q)) ( ) n' + s-6/2 n-rAj/2 4 [o k d (-)) ) dr) ( 0r ( ') ' ' ' =s;+6;/2 fjn;+T/2 -iko iH()(kop)=I ds'dn' 4 + sj6+/2 ni+Tr3/2 ko d ( i. ) 4k ((.)H)(op))= dr ' +sJ 6 /2 nj+j"(iko) k/r(r)R(r) 4 (ko ((is )2 - (is )2) H) 1)(koP)) Jsj-6;/2 -nA- /2 4 krj (i) )R() 4op i i = 7 + sj+6/ 2 n/,rj2 (-ikof(1)i7 ) kodskdnn (-4k) + i-o)- ds'dn 1.-S;/2 J,-r,/2 v ( — k)o1(op) 2 R' ^-'k 4+ / ) (:ko) G - o()1- d'dn' "-6/2 e-T(r2 -jzt(r) 4 ds'dn' 42

s.-6,/2 n 3-r /2 -__d'n + I,8/ n+r/ 6p~i(~). S)II1)rop) dsdn + I sp+c,/p -k 8j6j/2 flrj / 2 4,6r ( r-i)R (-i~) kop ). ds'dn' (4.25) 4.4.9 Element E, Circular Cell: A53j = ij ~iI+ 2J '(-ik~f2(*I~ i') (p'dO'dp') I +.O,/2 AP3 +Lo/2 k0(1ri\T()1 -AG,/2 Jp-Apj /2 4 R (rfj) 6r(rj)R(r-)) noOP r~'dOdp I+L\9,/2 Pi +Apr3/2 1 d ( I " + -zj8/2 ]p-Ap, /2 4- d 'E:;<\ \r1\Or/ffti +AIO'~j/2 JPi+LA~pj /2 - ko0 (( A.)2 ~M H '(ko p) ' 4.4.10 Element p'dO'dp' (4.26) 6 (_I)qf2(-l Fi'j = ri i;*)dn' nj +,rj /2( _1)qf2( -1 -*)dn/ -,rj / 2 ri inj +,rj 12 1 - -rj /2 (_I)q -*(rj) - Ild kgds I d -4 ri)R(r-*) ds Cr( 1 - d G (r- i —*) -- - dn 1 2 I r=ri koEr(ri)R(r'i) j I -Rj 2*(ri) 4 n.,.+-ri/2 11 d ( - -rj k0 ds 6(riF)Ri -iko (P- - S-) H ") (ko p).. I ri 4 1 r=ri dn' I (1I)q+l 1 o /~ k2opd o~( ro(,s8 -(3.f)Ja + fI'(~1 +l k2-~) (-k(j. p )2H )(kop)) dn' = - +,j 2i(-l) (1I ( A. A)2 p s -. -. r=ri - H(l)(kop)-. — dn/ — + 0 r=r, Er(ri)R(r'i) t 43

+ jnp~2i~1) (s.S)H,')(kop)) f__ f dn' 441Elmnt, F+,RetanulrCll f.+6/2 f/2 _ji fsA A F) dA A (dkop Js- /2n i,/ 1 n + p~~fl = n + 7J2,,n(4 27 ]Srj- /2 4ervf3(rj, i) Ino=n-r;/2 i,-7 / kof3r(ri)r)Ls rin=n-r/ s3c5/2ko e d )Rn) '4 d )~'(oP)lcs Ji s+6 /2 1iT/ r1..3 s,-S,/2 46ri~)R~f) rkS iS)Hds'(kP)dn' / (.8 FT P+\P/2 [Oj / 2 ko df dn'i Jp,-L.7 /2 IA /2 k nfdpn(p'd 'dp' = 192 A8, / 2 niUrk')"'t)1Tcbsd [p+p/2f0/ - G (dG f i ')' ds'dp I I-A2ko3/2r kei)R(r')i cs 92 '(P )H(kop)) _dO' +r~33~~i2 R(ri/ pp3 p -8j/2 /2r k Lri)~j r=,i)HjkP) n__ndO'dp' 44

AJL\G/2 46(FRFj (p'$ - )H"')(kop)) ];pj:;:::2 dO' rp+A i/ ri,/ -1i Ip-A. + Ip,-Ap3/2 ]A03/2 4e(~RF)(A* S)H("(kop))_ dO'dp' (4.29) 4.4.13 Element Fi~, Rectangular Cell: = js(I~rfl/2 I, ni;/2()fdA, ')d'n = [8+832lfff+T/ ri (-'dsdG(n')\ ddn f 8jc5/2 n-,+j./ 2 1 [ = Is,8,/2 In,ri,/2 e~)~~ -dG$r..)H(k)J ddn = j3+3/ ~4k~~i~)Ri~ [~*.~I4'(p] ds'dn' (.0 = f6+A /2 -frGi/2 1i i (dGs~i) pd'p Jp-~32IA32Er(r )Rk~ \r s ri - I 8. p+L6p/2 nf Lr6/2 1 Fiko 1 JP+IJ2jA6/ k ( p S)H()(kop)__ pds'dp' (.1 + (1)(k(p1)sf(n' f4)dn' =P'i +Aj /2h+OT32 ~ (GFF) n 45

.j+rj/2 _i(_i)q -iko () ( Jjr-Tj/2 koer(ri)R(ri) 4) dn' r ---ri =nj 3j /2 () [(P. ) H- (kop) ] dn' (4.32) n j-rj/2 4&r(rF)-R() 4.5 Evaluation of the Singular Impedance Elements In the subsequent derivations, each of the impedance elements will be expressed in terms of four generic integrals. These are defined as Jo((a, ) lim / Ho()(koVt2 u2)dtdu e —+O J0 Ji(a,,/) = lim H (1)(k0ot2 2)dt ~ ---+ 0 J2(a, ) - lim HO) (kot2 /)dt,-+o dp J J3(a, ) lim Ho (k ef —O dp2 (4.33) and their analytical evaluation is given in the appendix. In the computations that follow, use will be made of the substitutions = s'-Sobs (4.34) n = n'-nobs (4.35) These imply the identities ds ds' (for integration over primed coordinates) dn = dn' (for integration over primed coordinates) dH(1) -dH(1) dsobs ds dH(1) -dH(1) =. 0 (4.36) dnobs dii 46

4.5.1 Element F. =2i aJ +83+6/2 4n/=)Rn[oj i)H ')ko)}j2 sF,+5,/21 niH (1)(kop) 1 l+s/ 8/2 4k0r(r-*) R (ri i) [rcri~ n/=j-,j/ Utilizing the expansion dH(1) d___ dH(1 dn_ n~' is d~b+ (i n flobs = _(A. A) o- + A s')_ the above equation may be rewritten as n nl bs' + (n (i.1) ji+i2 c ob) lb)2)) 4koii )Rii [[ +(k dS HO)2+(n') -k ni)] Sos'::.+6.n/ n obns+d,/ 234kOer(rj)R(rFj 8/2 obs [Jdj cm n~ni-nj/2-nbj/2 (n.SI 4ko6r(ri) ~(ri) [.U')(-p3 - S'Ikop) -H8'(k24) H f~1)(koi)] Intrducig th subtituions(4.3) an (4.(4.37)av 47

4.5.2 Element F 2: 23 = Jsi+6 /2Jn3 +T3j/2 -iko HA)(d. A'n jn+)Hj(3r/2{1 ')(kop)} ds'dn' dj6 2 nj =,j/ d~o, it nb ds' kp) dsdn' = -6 i(.n')l Js46 r(-/2 ri Tn - + - ]02 s sI8/ n+r/ d H H)(o(1'-)2k + (n'- -bs dn'. dn 46r/2i -)Rj / ) Jn,-1r/ L )8=8 -6r/ Wgith eplyn the substiution nn = nds n n - we may wrieepesda - inh t) SiH1(~ i2):=l+2n iFf H()1n)(k/on [ l)kVT7 I)> dii2 n1- 0/-i8,6/s 4 [J1(r3jy T) r-J (x2,y /) - Jn(4,y) Jirxi/yi)] 48

()()[J1 (Y3, X3) -1 J(Y2, X2) -1 J(Y4, X4) + J1i(Y1, X1)] (4.38) 4.5.3 Element EF = F7 4& sR~~){p n)H H~)kp} dn' (~1q+ fljT32 H0 ~(kop) d 4ko$r(rj)R(ri) Jn-l)/2 dn r?=r (_lq~fnj +rT3/2 { ( A)d d 4koEr( r#i) R rf* -) / (_I~q * (A ) A n+Ti/2d f1 d Ho')(V 4koer(r-i~)R (ri) Jn-rj/2 dsobs, r + (1)(i, ~ HO') (koP (sri + (n.1)4ko6r(r7i)R(ri) dn' HO(')(kop) dn' (q + (-1)qS3/2 - -Sobs)2 + (n' - flobs )2)}- - dn' - si)2 + (n'- - i2 For this case, the appropriate substitutions are = Sj + (_l)qS/2 - 8ob8 ii = I' - n2obsThese allow F~ to be written as F3 = ( (n -H's( vj~ ~~ 4kej ~r~i r) ~j-T,r/2-nob d9 0 + n1)( i' [H(1(koyS ()PS/2 - S,)2 + (n' - n' nj-2/ 49

Thus for leading edges we may write F3'- [J2(Y2, X2) -J2(Yli X1)] n [I(11)(kkpp2) I +4 k oe, r(j)R(rj) 0 while for trailing edges the appropriate expression is 3 4k-Frr-j)R (-j)[J2(Y3, X3) - J2(Y4, X4)] + ) n H4')( kOP3) - H~l)(kOP4)] (4.39) +4ko r(ri)Rri) 4.5.4 Element F> /n/ n=n, +T3/2 r*(f kr-f)R(ri) /j s+6J /2 1 d1 dG(r-, FY) n=,+/ + J2S/ e~)( ds' I ns,+6,/2j-2 J 33+83/od2H l)kop) dI' ['kA2O4R* r- T'i) R(ni =cbs - J /2 n T,2cs IS,+6j/2 d iHdO(') dH((kko1npn+)/ s + ds'TL.\-\ bs~I__ mt63+8/2 L -4 kOOVj)~r - rSrb ) nflnj bS)2)fI] i~/ H j j(kzdI ds o + (n'P -d ds L- ( n nj -r l -s86/2 4R*[r-*(i) ) n0~ r /2fb. Jfri 50

= jSj/2-s i H -- )ko 2 d,j-6j,/2-, [4R*(r) -+n)/2-nJ -i = 4R*() [Jl(x3, ys3) - Jl(X2, y2) - Jl(x4, y4) + Jl(xl, Y1)]. In order to simplify Int2, we first rewrite d2H(')/ds2 as d2 H /' d d L' dsb5 dnb\ = A(,.) dd _ ') + (. A,) d ds2 od dn ds dn o = - ) ds~__ 2(sA. A)(s. A d (A. A' ) (i SOsds' dn obs dn ob, Employing this expression in Int2, we obtain 4ko2er( R() s'i-6j/2 d2 t,-/2 - i(. s)2 fsi+6S/2 F dH ()(kop) 'l+ sT/2 ] n'=nj-+Tj/2 4ko-er-)R(2) sj-6j/2 ds' dsobs r=J/ '=.-./2 i(s n)(s. s') s+6/2 d dHol)(kop) ds' 2_()() -/2 ds dns '=2d ] - n'=nj-,j/2 i(s An').2 s,+6,/2 4ko6r(rj)kr j-Sj/2 d2Hol)(kop) =n,/2 dnobs =rs - [Is - S 2 (f 1 I n/=nh+r,/21 = 4k (fi)rH o) [ /* -H)(k - a )2+( n'- )2)}. J 4 k o~r (r.i^fll(r+T) I2 s os,+ _/ is 2n') (s s') 2koer(rFi)R(ri) [ (p n')H(l)(ko/(s' - Sobs)2 + (n' - nobs)2)}r= t-3/2 Lri nl" nj-,r/.i =,-8, is. in')2 sj +~8f/2 dn2 n j __ +'/2 i(.H'~[+~/ ~U(l)(ko/(, o,)+ (n, - no ) 2) d +4k( 2er)ii) R(-i*i) i,/2 ob 2 (ko I(0,' ++T/.2 0 ''-^J 6-To/2 s- r~r_ nl~n - drj. 51

In the last term of the above expression, we further make the substitutions (4.34) and (4.35) to yield Int2 i( 8.)2 [x3H(l)(k)p - 2H(1) (koP2) - 4~)(O + x~H'I-)(kopi)] 4k06r (ri)R(rfji )P3 P2 P4 Pi +s2k0sr( i Y)Ri 3 [H~)(kop) -Y~H(~')(koP2) Y4H(l)(kop) + Y ~)(oi i(.+i2)[ 1 '(o3) - 1H' kp2 ~H')kp4 4ko6r(i) R (ri) _P3 P2 P4 Pii P( /22 P4 Pir/2no] i(+ Y,~'(k~3 ~~H (')(kopV2) - j2~)(kd4) j( A. A/)2 s n [J3(X3, Y3) - J3(X2, Y2) - J3(X4, Y4) + J3(Xl, YOI 4k 26 r (r-*i) R Similarly, for Int3 we have (with the usual definitions for ~3 and n-) nl=nj+-rj/2 s.+6 /2 Int3= z d d H (1) (kop ds' )r=ri -6j12 Q ds ri)R(r-) ds Ir( i J nl=nj -,rj /2 -i(.s') d ( I___ - k0 ~ erfi)R(ri~). ninj+rj/ ' 3=S,+5j/2 [ [ ~'~(0~(I - Sb)2 ~ (n' - flob)2)ff:::::]_=,S/ 7% n=nj -4kg2ds '\6r(i R(Tt) s.t') d ( 1 + k0 &rij)R(r'i) - 4k s (1 [H~l)(koP3) - HO~l)(koP2) - 14'l)(koP4) +Ho)kp) { Si L61 128:obs d] flflj +-,/2-nobs - [H~l)(koP3) - HO(')(koP2) - HO(')(koP4) + HO(')(kopi)] 52

~ sk n' (1 FJ2(X3\ - J2(XY2) - J2(X4, Y4) + J2(XlYl)] Hence, Fi4 can be expressed as = (rj [J1 (X3, Yi) - J1 (X2, Y2) - J1 (X4, Y4j) +.J1(XI, Yl)] + i(s n' [J3(X3, Y3) - J3(X2, Y2) - J3(X4, Y4) ~ J3(Xl, Y1)] 4kO6r (rj)R(rFj) ~ 3 1 4kko)R )p XH(l)(koP2) -X4H~l(kOP4) + ~lH(1)(kopi)] -') [Y. H1()(koP3) - 1-H ~(o2) - ~- H1' (op4) ~2koeFr(i;r&:;;)L j3P2 P4 PiJ ~ 2 ' [HO~)(kOP3) - HO~l)(koP2) - HO~l)(koP4) + HM')(kop,)] + 2i i') 6~i(~)[J2(X3, Y3) -J2(X2, Y2) - J2(X4, Y4) + J2(xi, ylA4.4O) 4.5.5 ElementE: + s-6/2 In,rj /2 kdsRi)j d = m ~18+i/2Jo)Tj2 d'(ko'ddsd ~ js,6.2 j,+rd 2 -1 id2H ri)(kip) ds'dn s,-8 /2n)-06/24koer()R)id)2s2rf=i Ji s+56j/2 nm-+T,r/ 2 -1 d ('\dH~r-(kop) d'n + fd-'dsn'f =S +ni.+, mt2 n +,r mt23. Th sbsituton (43)anI435lo t1) tkop be writtn a +t L-j2m~r1H~O( os (n'd-fnb))ffd'f 53

- f[ s,~~~+6 /2-ob8, Jn +-r /2-nbSH'(kVTi2)d4 4R (f t Si -&/-Bb ob-rs/-03 ~ - si632s./n+~32~sH~ (k ovf,-s~2iT 2)dgdni 4R*(r-i~) s-6,/2-os~ J,j2nb - 4R*i~) Jo(X3, Y3) - JO(X2, Y2) - JO(X4, Y4) ~ JO(X1, Y1)]. TsipiyItwe first expand d 2H(l)/d S2 as d2H(1) [ d ~ dfb] L dos = A dHA + 2( AA1 A.A) + ( A. A I9)dH ds ob nobsjio) kdn'I = - [S 2 +6,2 f n +s/ d2s)(k dSs'ndn'2 A(. d) H 83+1/2[+T/ dH (A(ko)dds'dn +(.s+6,2 sn+r/ d dH H(k1p) s'dn 4ke.VR~~ Ld, sobs/2dn dnIb8 dnfdnfdo ~I)2 +6 /2 njlb)2}/ _____________ddn 4k /2~)~~ ~-' -ros2 n ri '=,6/ 8 [[~)ki~ - d d)2 + (n' -) ds)d)] +2 O_(A. A=)(-.A/) n'-6rtn -T3/ + ~ jsi~Si/2 dsld/ H H)(oi(s' - s~tb-)2 +d (n' -nb)d 2d' 4 ko~r(r i)R(rj),-S,/2 [tj nb8/)2f ~fl-/ 54

For the first and second terms, we use (4.34) and (4.35) to express mnt2 as A A)2 f fl+TIj/2-nobs [dH'(L /T2l8=3~_S+63/2-sobsdi Int2 = ke(-Ri)~ n T/-lb d? ) (ko, ~ -S,,/2-sobsA 2koer(i)r(i) 8 =Sj/2/2nfl'-lJ-Tj/ O jEr )2~ L,5/2Sb F d 6 d23os ~ -1A0 A A,) flJ- - 'n j _2 4k0 A1i)2~~ { S3-S32-Sobs dii n=n, -mr /2 -nobs fi + -nC ) H1(o3 - Ho'(k"p) -k Hd9kp4 c)kp) 4k06r(rj)R(rFj) 2-sb 0 + ~[J2 (Y3,yX3) - J2 (Y2,yX2) - J2 (Y4,yX4) + J2 (Y17yX1)] mt3 jih (0('Ak1) 3/2 ny+)(o/2) dH (1)(koP) ds'dn' kopi 2k Li r/2 Jn,-m/ -1 d 1 si +6j /2 n.,. +,rj /2 d d 1. ') _ _ ( A. A ') I s n HO(') (ko p) ds'dn' er(ri)R(r'i) -6j /2 j -,rj /2 ds ft- -0 A/Af + r /2 )1 1 s=s3+Sj/2 H') d1 nr,/29 Sb)2 + (n' - nb rridn' 4k0 ds Er6rfi? Rri Tj/ I=\6/ ____ A ( i ts +6 /2 r/=j+j/ (F R()Ig / LHO(')(k oI(S' - Sob)s ~obsr- rl'=# + ds' (55' (d 1 rriRrif [ r/2 _,ss+6n/2-sTjb A A___d k, ~ ) n,+r,/2-ni8n-1- / - l (d [j+S/-obs s+j2sb +H(.1 71')o-;)i~(j;; j' /2 fl 4k0 ds Ker(2 [3= j -(' 2 -Sob ) A, +,/2-sobs nt0V=TI +j -r/2 - n~b. *-. ') d - 4k0 dsi (&r+i(i) [J (Y3, X3) - JI (Y2, X2) - Jl (Y4, X4) + Jl (Yl, Xi)] + -I -i [J (X3, Y3) -.Jl(X2, Y2) - Jl (X4, Y4) + Jl (X 1, Yl)] 4k0 ds rr)Rri 55

Hecthe expression for bcoe 'Yj+ 4R(o [JO(X3, Y/3) - JO(X2, Y2) - JO(X4, Y4) + JO(Xl, Yl)] + ( -I) [J2(Y3, X3) - J2(Y2, X2) -J2(Y4, X4) + J2(Yl, Xl)] 4kosr(ri)Rf(rtj) +4(~)(.- [J2(X3, Y3) -.J2(X2, Y2) -J2(X4, Y4) +.J2(Xl, Yl)] + n [Ho"1)(koP3) - Ho")(koP2) - HO(')(kOP4) -I Ho')(kopi)] *. ') d + 4ko ds s. n) d + 4ko ds (6~it)~i))[J1(Y3, X3) - Jl (Y2, X2) - Jl (Y4, X4) + J1(Yl, Xl)] (&(14(-) [Jl(X3, Y3) - Jl(X2, Y2) - Jl(X4, Y4) + Jl(Xl, Yl)]; 4.5.6 Element F: n. +,r. /2 6 i 3 _I)q, F. t3 rj /2 ( i - { kO6r(i,)R 1(I +jn,+rj/2 (_1)q+ 1 d ( 1 -,r/2{'2 H~l)(kop)- -dn' Jn ---/ 2 4R*(r-i) 0 rr + n-ri/2 4kg~r(ri)R(ri~) ds dG(r- fV) dn dbs r~r dn' dH'0ko) dn' dbs f-ri dn' + [nj +rj/2 i(-1)q+l d 1n3-T3j/2 4kg2 cs = Intl + nt2 + Jut3. Making the substitutions g = Sj+ (_1)qSj 2- S0b8 ii= nl' - nOb ( I 56

for Intl yields mt1 IH~(kI +(1)S/ -Sb)2 + (n' - flob)2> ~dn' 4R(),-r,/2 j(_1)q fn j+Tj/2-nobs - H(l)(koVs=FTVO2)dnz 4R*(F ) Jn-r,/2-n, { ~)[Jl (Y2, X2) - J (Yl, Xl)]; leading edge 4R* (r-i 4~~)[J1 (Y3, X3) - Jl (Y4, X4)]; trailing edge. To simplify Int2, we first expand d H~l)/ds as d'HHl)) ddH d d2H' = (A. A ds~b + (A. Al')( (A d. Ad/)_ (A A')2 nb dS sb dnods 5 b8 A) dnobs8 2t2= k (Af)R.t L:AdH o)( A dnl' ( A ds n+r2 dsHn')(ks) dn' snd 4keobsl n-~/ dsbobsof A A d2H') d +d/ ddH' (k1) dn' (1 2ke dsR~~ 2 s n /2 dn' ds~b s n=InA)2 niz /2d dl)(kop) dn' +4k26rfi)R(rj) jn-i-r/2 dn' dn0b f# 4k r 2I2fr/ ds2 dn'4ko-er (r)R(rFj) ]n —,/2 ds2 H ob(0s~ +r~)6/2-Sb)rin?o2J 57

+ *- n s ' { d H(1) + ( 1)b632 - Sobs )2 + (n' - nb 2 dkr)~ri S obBs~o~inos2JJ q( A /2 j _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _1n'=nj -rj /2 (i n) dn~8H'(o/ (1q6j/2 -Sb)2 + (n' - n b)2)J -I + 922fn3+r/4 d HS'(b(s,+flbsb) 4ko'r (ri)R(r')L ]nrr/2 d~ ~rqi2 -~~2 n dln '-r/ i(_I~q~l (1 nA=n2 + 7,r/2 8 I)2 { (d tt'H')kI(s (11) S/2-Sobs2- +ob (n' - n' nob) )I 2J ni n. j = - ~ +( d1S2 - 8obsob 4k r(rj)R(rij) -H0/2 os #* 2rP S') H (1)(ko v ij) n+r/ ~ob, b +AJfllj./flbsf 2f A AI 6 ))+)2 + ( ') P' 0 (kop2 -j+/ n 'H1 b b 4k* r irz'(r)L ryin 1jr + = j + H_~(kop2) -S ob~koiI; edngeg 4keseallo I)R2to ) [xp2ese as k *~ (ri)R2' oJbX) 3y, X4)]~ 58

)) s 3H(l)(kop X4H(1)(kp) L- - (kop3) - (4) + k(:-)- [f H(l) (kop3)- 4 H(l)(kop4)]; trailing edge. 4k0or(r,)R(r,) P3 P4 Similarly, for Int3 we have (with the usual definitions for s and n) inti —1)+l d ( 1 dn,+rj/2 A t) d )() 4Iknds (S ( R(i)IS- ) /2 ( HO( (kop ) - /2- dn i ')d(^ +/2 d Hoi)o + (-1)q/2- Sobsj+ ( -nob r= if-l)(l( As') d( 1 + { ni+TJ2 + (n - ' ) (4d ( 1R [H)(kol(s.+(- + (j/- 2 - s)2 + (n'- - ni)) = 4ko~ d5 ~(,)R(r,) ) -[J2(Y2o-, xoJ(yo, +2 )d + 4ko ds,. r(r,) H(r (k) n=n,-T,/2 (A dA ( iT) J )2-n" kop; leading edge - S* ') d (1 S ob d 4ko ds \er(ri)R(ri) L T/ - 4ko s Ke,(f~)R(i,) [J(y3, x3) - J2(y4, x4)] s nV ) d 1 - 1 F (1)j / 2 +~' h'i)R ( 1 )) [Z(1p3- H(1)(kop4)]; trailing edge. Hence, the expression for F becomes J76= S 4R*(\) [J(y2,x2) - Jl(yi,xi)] +42. ) (r [J3(y, x2) - Ja3(y, Xl)] 59

* x.H~l)(ko) - lH oi 2ko06r'-fi)R(rFt) P2i + iGHl'(kP2 4k0&r (rj)R~(r) _PioP2 -Pi + k(6r2 d(is) [J2(Y2, X2) -J2(Yl, Xl)] -i(s.iV) d + 2k b (6(i-(~) [H"')(koP2) - 1H4')(kopi)]; leading edge i -4Ri*( [Jl (Y3, X3) - Jli(Y4, X4)] + [J3(Y3 X3) - J3(Y4, X4)] 4kgEr ()r~i ) + n fs)( s X32H' (kHop3)oP2k0&r(i-)R-Nz p p + I-.t2 3Hl(')(kop3) - Y4H,4') (koP4) 4k r~i)r~i)L P3 P4 + 4kt) db [J2(Y3, X3) - J(~ 4 )l i(s.in') d +4k 0 ds (rii)(1)) HP)(kOP3) - Ho')(kOP4)], trailing edge. (4.42) 4.5.7 Element F 7.: = S+6 /2 1 Fr/ATj/ Js-6,/2 46~fr( f r)i~)kP}..~,~2cs fit/2 -[dHO~l)(kop) ] 'n+~2ds'. 8Js,6/2 4kocr (ri3r~i~ I b __ Utilizing the expansion ds =d(H~l) ( Hflld = 8 8)ds sb8d 60

=H -() dH) ds' we obtain Fj7 =sQ.' n~ +S/ $34koe r-*i~)R (ri~) 18-83/2 d HO')( b o(S - -Sobs) 2 + (n' - nobsY)2)} ds' ndnn-bs/ (. _________ + (*R(*)[Hc '(koV"s - 2+(n' -nni,+2/ 4koF (rj-~(rj L JS/=S 7. 63 /2J n'-=n -,r/2 If we now introduce the substitutions (4.34) and (4.35) we obtain 4kocr(FiR (ri~) lJ-6j12-Sobs L~ J n =nj -,rj/2 -nobs Ir-r+ S _8'! [H()(koP3) - H(1)(koP2) - H(1)(koP4) + H~l)(kopi)] 4koEr(~j)R?(rj) Finally, introducing the pre-defined generic integrals, F~ becomes 7 [J2(X 3, Y3) - J2 (X2, Y2) - J2 (X4, Y4) +.J2 (X1,Yl)] + k4)()[H"')(koP3) - H"'~(koP2) - HO~l)(koP4) + HP()(kopi)] (4.43) 4.5.8Eement 8: 4.5.8 Elemet $3 = sjs+8, /21n, +-r,/2 iko 4() ( )lP -s)H1z')(kop)} ds'dn' si +8j /2 nj +,rj /2 - i -83 /2 -,rj / 2 4e,(rj0(rj) si +6 /2 nj +,rj / 2 - i ri ri -61/2 -,rj / 2 4s,( *)-k-*) {+Ho(")}_k p ds'dn' sHO") k 0Si- S5obs)2 + (ni - flobs)2)} ds'dn'. 61

Recalling that ds ds snd = _- A'. A/_ ( A)____ S ds' sndn' Fijmay expressed as = j [H~l(ko (s' -S,)2 +(ni' - fli)2)] ds' s' fljTj/ r0 1s=s+6/ +4E(ri)R(r'j) Inyj/2 nH kVs s) n =n,-6,r/2 d' Further, employing the substitutions I S = $ -i n = n - we obtain -nfsr) [3+j/S [')kvZi2] nn+,rj/2-ni ) n,+',r, /2-n H() TF ]sj+6j /2-s, - Jix3y3.i~2,2 - J~x4y4 -I- Jx12,yi) 4er(ri+)R(rFs) -/2n [Jl(y3,xY3) - Jl(X2,xY2) - Jl(X4,xY4) + J1(X1, x1)]. (4.44) 4.5.9 ElementF: -9 nm +,rj/2 (_1~q+l F~j yij + In-,2 )(S* (.)H~l)(kop)}. dn' 62

_ ________ fni +Tr/2 dHO(1)(kop) dn' =T +4koer(r;i)R~(ii) In —rj /2 dbS r=ri 4k(_i)R(q n,-r,/2 = ffl,+Tj/2 dHP)(kop) dn +4koer (r;) R (rj;) In-,rj/2 dso b8ob7r?7 =(_1)q(.' jni-+ Ti /2 {d P)k (A3 + (d)&2-Sb) n lb)) r +4ko~r(rjt)R(ir) n,-rj /2 obs5obs r*j R ri) [H ')(k (/)22 ~2 +4ko6r(F )Robs (n'nri) = _ q+ l (-1)AJ A 50b s./ f [nO+T /2-nfobs d / +4kocr(i;;)R(r;; j 2osd * iV r f (1) o~(a+(ki)qS/2 -bs )2 + (n' - nos)2)] dTn Tj/ +4ori)R?( F) 1H0(k fl2flj -T,/ 4 ko~r~r +4kr'j r(- nii)R-i;j ) 63

+ 4k( )(n ) HoI')(kOP2) - Ho')(kopi)]I while for trailing edges we have t3 -ij +[J2(Y3, X3) - J2(Y4, X4)] +4koe~r(rj)R?(it) + 4kH4R(( [4) (koP3) - HO(1)(koP4)] (4.45) 64

Chapter 5 Far Field Computation In this chapter we evaluate the scattered electric field produced by the equivalent currents computed in the previous chapter. Rewriting (2.18) in terms of (s, n, z) coordinates we have -EsiJ I (ri- ( i)dA' + IA -( + G2d r ), ) dA' ko ' Io ' dn2 dsdn A dG2d (, )dA'. Z dGn dA'. (5.1),A dn If the point of observation lies in the far zone, the first term of (5.1) is identically zero. Furthermore, substituting (2.29) inton (5.1), we may write i z d ( dG2d (, i.) G2d (rj)) k JA' dn kj dn' ' ds' ) IA Z d A' (5.2) Using integration by parts to modify the Jn, term, (5.2) may be written as i A dJi dG2d (if) dG2d (, )dA Es is - T - ~ Jn' --— / -- — dn' endpoints ko A' dndn' Io dn / aGd (2, ~)A + dJn dG2d r dA'J J d dA' K ds' dn J, dn i dG r Z G dG (Jz dA' - jnlJl1q dn' - ko A' ddn' ko cdn ' dn (5.3) 65

where q = 1/2 for leading/trailing edges and again J = J - ko d,. Expanding (5.3) we have k0 IA' J iko) Hzl)4(kop) dA' Es s) (p * )(p n) + ko(p.- n)(p )Ho(kop) dA' ko 4, y j().ko -H)(. ) -iko ( -4 (p' )H (ko p)dn'- J 4 (p' n)H (kop)dA'. (5.4) Since the point of observation is in the far zone, the term containing H(l)(kop)lp becomes negligible in comparison with the remaining ones and (5.4) may be simplified to E = HPo(1)kdA' s 4 j| J8'(p' p*)(p. * ')H~)'(kop)dA' 1 - iko - 4 Jnt(-1l)(p n ()I(kop)dn' + j J*(p f n)H(l)(kop)dA'. (5.5) Furthermore, (s, n) may be set to s = -q$, n = Q, where (e, q$) are the cylindrical coordinates of the point of observation. Because this point is in the far zone, p L g and we may thus write -ko - 1 k -E = — 4 J ( n)) (kop)dA' - Jn (-1 H) Hl(ko p)dn+ JH)(kop)dA' (5.6) We now recall that the asymptotic expressions for the zeroth and first order Hankel functions are given by / H ()kop) e~ (ko e-/4-)e-iko (x' cos+y' sin) (5.8) rV rop H ()(koP) \/r ei(koq-1r/4-7r/2)e-iko(xpcosk+y'sin4) (5.8) 66

Substituting (5.8) (with p replaced by e) and the current expansions (4.5) and (4.6) into (5.6),we find E = e(ko/ 4) (Q. n)e-iko (' cos +y sin)dA _ eikoeI4) S oz. e- (, cosq+y, sin 87re pi J Ad i=l - 8= oei( e+|/a) 5 _,[ (-~dn. (5.9) -koei( ei(ko/4) Ki eiko (' cos q+y + sin si i —1 e - ik(ooL+7r/4) k ed gef (co('cos+'sin+)n (5.10) - ge ] d (5-1)edn. Es= e(o/4) I [cos (. i) + sin (y ni)] e-iko('c ')dA ei EK- ie 0 )dA, 8ir7ri=1 JA, e-8 (koe /4) yS Kj (-1)qe-k~o( +Y'i)dn'. (5.11) j=i with the rectangularach cell of integration is now approximation, the remainitangu integrals in (5.11)we may write evaluated as s +Employ/2ni +/2 eiko (x' cos,'+y, sin )ds'd n' i-6/2 ni-r/2 67

s, +6/2 jn,+r/2 e-iko {['~.jn(& ~]cos k+[s ('.)+n'(#L'.j)] sin dsn s,-6/2 n3-'r/2 e iko s'(i.~)cosO +(S'.~) sin 0] SI=S +6 /2 e -~ ik[o i )cos k+(f'~ sino~] n'=ni+,r,/2 -io[(A,'. A) cs$+ (A,'. A) si q] i'=j/j2ik (t )COS q + (Ai' A) s n q]n=nj-Tri/2 = eko {[,(,.)n (f.)]cs k[s (,i~)ni,.) sin o xnysi e-iko 6,/2[(Si, ~) cos k+(.Si.) sin o] - ikosi /2[(S, i~ cos qS+(g,.g) sin k] -zk [.i COS coq A) sinq$ e -ikort /2[(fA,.) cos q+(,i~i,.)sin q] - eikorT/2 [(Aii -.) cos 0+ (Ai if) sin O] -iko n(i.) co qCSO+ (ni~ _ ) sin /0] = ri e 1e-iko (xi cos k+yi sin 'k) (5.12) '~i vi where = kS/[(.~ i)COSq+ (, A)sq] =i kori/2 [(A x) co n~i y) sin q$] (5.13) Substituting (5.12) into (5.11) we obtain the far zone scattered field E =ei(koo Nr4 1 OAAsin sin vi eiko (xi cos qS+yi sin Es e ZKs2(cos x n) + sin Kz- 1 6 e-, / 1 sedge - i(ko e+rI4) S edge(_l)qTJ si / iko (x, cosq5+y3 sin$) (.4 8ikgj=1 1/3 The echo width is now given by = 27r~o lrn IEscatI2 e-400 jEincJ2 (COk ONA(Asin i)) -iko(xi cos +yi sinq0) 5[K3i (co _n~ it) + sin q(y. ni) - K*] e~~~i i 4 i=1 Z ir i V N edge fin.\ 2 nedg iko(xjcos ~yj inO 68

Chapter 6 Description of the Computer Code QRCOMB A computer code was written to implement the numerical solution of equation (4.12) for the most general case of an inhomogeneous composite cylinder of arbitrary cross section. Either principal polarization of incidence can be specified with this code. The code, to be referred to as QRCOMB, provides the user with the equivalent current distribution and the echowidth of the modeled structure. The code is written in FORTRAN and is self-contained, requiring only the basic system supplied functions. The sequence of steps executed by the code are: (1). Reading of input file (2). Geometry generation/discretization (3). Compution of R parameters for each cell (4). Computation of matrix elements (5). Computation of input vector for the given observation angle (6). Equivalent current solution via matrix inversion (7). Application of current tapering 69

(8). Computation of echo width for given observation angle (9). Repetition of (5) through (8) if generation of the backscatter pattern was required Repetition of (8) if bistatic pattern was requested (10). Output echowidth data. Currently, the code accepts inputs for rectangular and circular layers. The input process is accomplished in the subroutine GEOQRC where each layer is first subdivided into discrete cells of equal width. Subsequently a 5x5 sampling grid of points is generated for each cell with the exterior sampling points outlining the cell boundary. GEOQRC also provides the geometrical and material specification of all layer edges in addition to sorting the edges common to more than one layer. The constitutive parameters of each cell are assigned in accordance with a set of input data referred to as "tapering specifications". These input specifications allow an (eri jPr) profile to be imposed over a particular portion of the scattering body. This profile describes the manner in which Er and jar vary in the region between two points on the structure which have specified values of Er and Pr. A directional derivative of the R parameter is also computed for each cell via a finite difference approach. The matrix element evaluations are carried out in the subroutine MTXQRC via analytical or numerical means (see Table 4.1 for the pertinent regimes). To perform the analytical evaluations, the coordinates of the of the integration cell are first transformed to local coordinates as described in Chapter 4. The matrix 70

elements are then expressed in terms of the generic integrals given in the Appendix. The numerical integrations are carried out using the expressions of chapter 3. Each integral over an area is evaluated via a two dimensional 5pt. or 3pt. Simpson's rule, depending on the distance between the point of integration and the point of observation. In the same way, one dimensional integrals are evaluated utilizing a one dimensional 5pt. or 3pt. Simpson's rule, with the comments made above pertaining here as well. The remaining tasks are performed directly within the main program. The current tapering is accomplished by multiplying each current element by a coefficient between zero and one. This feature is included to simulate two dimensional structures that are infinite in one direction. This code also incorporates the capability to model a structure placed upon an infinite ground plane by adding the contribution due to the image wave incident upon the structure at an angle of 360 - Oinc, Finally, the code provides an input option whereby an offset in decibels to be added to each computed value of echo width to adust the output quantity to a three dimensional cross section for comparison with other data. This adjustment in the output data is particularly useful for comparisons with experimental data corresponding to an elongated three dimensional target. 71

Chapter 7 Code Validation In this section we present a sequence of two-dimensional geometries modeled by the compact integral equations and, where possible, contrasted with results obtained via alternate formulations. The last include a traditional integral equation formulation, a finite element method [10] and several high frequency techniques [12], [11]. The modeled geometries include: 1. perfectly conducting half-planes (figs 7.1,7.2); 2. single strips of various constitutive parameters (figs 7.3,7.4,7.5); 3. Partially and fully coated perfectly conducting rectangular cylinders, (figs 7.6,7.7,7.8,7.9,7.10,7.11,7.12); 4. perfectly conducting triangular cylinders with and without material coating (figs 7.13,7.14 5. Partially coated perfectly conducting circular cylinders (figs 7.15,7.16); 6. perfectly conducting wedge-circular cylinders with and without material coating (figs 7.17,7.18). 72

The specific geometrical details associated with each scattering plot are included in the corresponding figure. As seen, the agreement between the compact integral equation results and those obtained via alternative methods is always excellent, thus demonstrating that the compact set of integral equations may be implemented in a robust manner to handle a wide variety of scatterer configurations. 73

cm GTD 8 v R=.I g ~ —A ---- R=.01 '0.00 30.00 60.00 90.00 120.00 150.00 180.00 Rngle in Degrees Figure 7.1: Ez backscatter echowidth of a perfectly conducting half-plane; comparison of results computed via the compact integral equations and a high frequency method for various values of resistivity, R. C3 -. —I 74

u R --- R=.1 -s — R=.01! --- R=.001 -D C, — '- / I, II.U.CM C) L3S '0.00 30.00 60.00 90.00 120.00 150.00 180.00 Rngle in Degrees Figure 7.2: Hz backscatter echowidth of a perfectly conducting half-plane; comparison of results computed via the compact integral equations and a high frequency method for various values of resistivity, R. 75

High frequency Compact integral equations x Measured Data A1 0 H - N CM I I H 0.00 15.00 30.00 45.00 60.00 75.00 90.00 RNGLE IN DEGREES Figure 7.3: Hz backscatter echowidth by a 1.7 wavelengths wide and 0.01 wavelengths thick dielectric strip having ~r = 7.4 + il.11 and Mr = 1.4 + i.672; comparison of high frequency, moment method and measured results. 76

er =4, gr = 1 (h.f.) 0 er=4+i.4, r =1.5+i.1 (h.f.) 8 0 -4 e =1.5+i.1,! =4+i.4 (h.f.) r Compact integral equation cn QO H /.-, I o U1 -Ln '0.00 15.00 30.00 45.00 60.00 75.00 90.00 RNGLE IN DEGREES Figure 7.4: Hz backscatter echowidth of a 2.A x.05A material strip with Er and,r as indicated; comparison of results computed via the compact integral equations and a high frequency method. 77

LI' 1 0" long slab at 9.6 GHz 0.0544"1 Lf 0)' 11CO (f) C%j. LL I UC0 a V E-polarization H-polarization LW; I-n nn nfl ftl ft 111 - -- OUUU-U.U-J.U.S U.UU b0.00 90.00 RNGLE IN DEGREES GD STRIP Figure 7.5: Backscatter ecliowidth of a 3"x.0544" material (elr = 5. + i-~1 = 1.5 + i.5) layer. 78

I, -Compact integral equations -- High frequency 8 _ J. — CO rll1- -- q l ) I. I. I -90.00 -60.00 -30.00 0.00 30.00 60.00 90.00 Rngle in Degrees Figure 7.6: Ez backscatter echowidth of a l.A x 0.25A perfectly conducting rectangular cylinder; comparison of high frequency method and compact integral equation solutions. 79

, Compact integral equations - e- Finite element method cn r — Ci UM (.N I 2I -90. - 30.00 0.00 30.00 60.00 90.00 oT, T t-,.oi ---- H I 1-90.00 -60.00 -30.00 0.00 30.00 60.00 90.00 Angle in Degrees Figure 7.7: Ez backscatter echowidth of a.05A thick material (er = 5. + i.5,r = 1.5 + i.5) layer upon a 1.A x 0.25A perfectly conducting rectangular cylinder; comparison of colr pact integral equation solution with finite element method. 80

CD C: ^ r- ' I I I Compact integral equations --- Finite element method C- - C L -.0 -,.I.J WI o C3 1 _ '-90.00 -60.00 -30.00 0.00 30.00 60.00 90.00 Rngle in Degrees Figui 7.8: Ez backscatter echowidth of a l.A x 0.25A perfectly conducting cylinder coatel by a.05A material (,~ = 5. + i.5,p,r = 1.5 + i0.5) layer; comparison of solutions obtained via compact integral equations and the finite element method. 81

8 O I I I i i ' i Compact integral equations | O High frequency ~ - ' ---- — J: S ^ 2 t |. 25. —= 0 rLLJ '-90.00 -60.00 -30.00 0.00 30.00 60.00 90.00 Rngle in Degrees Figure 7.9: Hz backscatter echowidth of a l.A x 0.25A perfectly conducting rectangular cylinder; comparison of high frequency method and compact integral equation solutions. 82

C) Compact integral equations - - Finite element method c) C) 0L r-I, I',:I. -.25X t=-" 0C) 0 Cm c) -90.00 -60.00 -30.00 0.00 30.00 60.00 90.00 Rngle in Degrees Figure 7.10: Hz backscatter echowidth of a.05A thick material (er 5. + i.5,p,r = 1.5 + i.5) layer upon a 1.A x 0.25A perfectly conducting rectangular cylinder; comparison of compact integral equation solution with finite element method. 83

Compact integral equations -0 - Finite element method 0r C3.0 LLl.. X | f --- ' -.|.. Iu).25X. T '-90.00 -60.00 -30.00 0.00 30.00 60.00 90.00 Rngle in Degrees Figure 7.11: Hz backscatter echowidth of a 1.A x 0.25A perfectly conducting cylinder coated with a.05A thick material (er = 5. + i.5,/r = 1.5 + i.5) layer; comparison of solutions obtained via the compact integral equations and the finite element method. 84

&5r = 5, CZ C)l C* C) CM C) 9).0'~ 10.0 Li cm LLJ c* I) C) cC) Ca ' -90.00 -60.00 -30.00 0.00 30.00 60.00 Ang1e in Degrees 90.00 Figure 7.12: Backscatter echowidth of indicated microstrip geometry. 85

- -e- - Traditional integral equations l l '0.00 60.00 120.00 180.00 240.00 300.00 360.00 Angle in Degrees Figure 7.13: Hz backscatter echowidth of a l.A-per-side perfectly conducting equilateral triangular cylinder; comparison of solutions via the traditional and compact integral equations. equations. 86

o o n -- - I - l --- I-l- ---- I - I - I - Compact integral equations -e- Traditional integral equations cm D -x - - " - ~M 0 - 1. '. 1. 1. 1. Wc) 0.00 60.00 120.00 180.00 240.00 300.00 360.00 Rngle in Degrees Figure 7.14: Hz backscatter echowidth of a lA-per-side perfectly conducting equilateral triangular cylinder coated with a.05A thick material (e, = 4) layer; comparison of solutions via the traditional and compact integral equations. 87

_ I I ' I ' I ' I I I II ('I ~.5 r-o ac LJc W, — I I= Io I Compact integral equations --- Finite element method - -.05X C: '0.00 I I I I I I I I. I I 30.00 60.00 Rngle 90.00 in Degrees 120.00 150.00 180.00 Figure 7.15: Ez backscatter echowidth of a partially coated circular perfectly conducting circular cylinder of radius R = 1.OA. The coating is over half of the cylinder's surface, as shown, and its dielectric constants are Er = 5. + i.5& /r, = 1.5 + i.5. 88

03 in *-0l CM 0 CY Lc C3 I C=; c: Wt - c I~~ I ~ - a I I I I I I I I I I' I I I I I Compact integral equations -e- Finite element method?======^===^^ ) cL I0.0 I I I I I I I I I I I 10 30.00 60.00 Rngle 90.00 in Degrees 120.00 150.00 180.00 Figure 7.16: Hz backscatter echowidth of a partially coated circular perfectly conduct 'g cylinder of radius R = 1.OA. The coating is over half of the cylinder's surface, as shon v, and its dielectric constants are er = 5. + i.5& jr = 1.5 + i.5. 89

cv= CT CT)L I -- - I -- -- - --- -180.00 -120.00 -60.00 0.00 60.00 120.00 180.00 Rngle in Degrees Figure 7.17: Ez backscatter echowidth of the shown coated and uncoated wedge-circular geometry. 90

C3p C2 CY on -D -Hq 4-a C 0 Wc: C3 C= v C2 CY) I. ^ rs Pk 0% &JPt 0% -rbU.UU -120.00 -60.00 0.00 60.00 120.00 180.00 Rngle in Degrees Figure 7.18: Hz backscatter echowidth of the shown coated and uncoated wedge-circular geometry. 91

Appendix A The four singular expressions Jo(a,) = lim j 1 H )(koJt2 + u2)dtdu e —. 0 Ji(ca,/) lim J H(')(ko/t2~ +32)dt d fa H(1) J2(, ) lim 0 (ko0 )dt J3(a,/) lim d2 j H1)(kot2 + /dt J- Hdt (A.1) may be evaluated analytically when the argument of the Hankel function is small. Utilizing the small term expansion of Hol to 0(4) in the above expressions for J1,J2 and J3, we have Jl(a,/ ) = lim J H 1)(koJt2 + 2)dt = o + I1 + 12 + I3 + I4 e — i ia Io = -[-ir + 27- ln4], I1 = -[ 21n ko 2+-2+ arctan (!I arctan T [ 3 a) 92

ia I4 - 327r - za + 3 2 r (koa)4 5 2(koa)2 (ko/3)2 17 221 3 + (ko #)4][~-~- y Y-1ln2 +1n ko~[f c+/3j2 16 # #)41 ( I (ko/3) 2(ko a)2 5 0k)4"~ L 15 a koK arta3 15 \ 3 + 5(o ))J (A.2) J2(a, /3) di, d12 d13 d14 = lim~ 1o H0l(ko t+ 3)dt =d~ d3dI2 -ikoa 2 a 7r koa act /3]-' +d13 d/3 ikoca ikocx 87r [7+ 1 —y~1n2]ko/, -ko3 In k0 & - 3 ko/3(# arctan[(koa) + (ko #)2] -r 2 +- [I.n 2~ n ko a2/2ko/3, 8iko 2#a (koa)2] (A.3) J3(a, /3) d211 d21J2 d#21 d021 limd/3J{H~l)(ko t2 + M3)dt = + ikoa -2 ir /32+a2 ' d 212 id21J3 d/32 d/32 0k'a 'i 7r -+I-,y+ln2, 7r.2 ik 2a'I -0 - - In ko 7r -2 a/ arctan-~ I i koc r (koa)2 ~ 3(0)2l rim 87r 3 j 12 651 36 j Y-I nk +___a 8 acanik.c - - ~ (ko/3)2 ac an 8ir [3 a /3 2(koa)2 1 27 (A.4) 93

Combining these results, we obtain the following analytic expressions for Ji(a, /3), J2(oa, P) and J3(ca,/3): Ji (ax3 =lim jHO(') (ko t2 +32)dt Fa (koa)] [a (koa)4 +3 [4 6] 480ir 10+ + Ii- 2 3 + 30 arct an 3) =a d3 jHO(') (ko %t2+/32)dt )a)2 _ (ko$#2 62 +(koa)4+ + 160+ (koa)2 (ko 0)2 48 +(kofl3 +32 2(koa )2 (ko/3) 2+ 11(ko/3)4]3 ~ 2] (P ik 2a/3 ik~a, 96ir Zii 3 F' 7-in2- - + In k0 2+# 22J (koa)_2 k p +3 k/) - 8 fi [(koa) )2 21 i I2 (kop3)4]. + 5(ko #) 2 + - i2 -(ko p)2 ~ I- 3 J ir L 12] arct an - (p J3(o, /3) =lim~ jHO(')(k o t2 +32)dt ____ ik7r/33[(ke 33)o 9iko [(koaxy ~ 11(ko3)2] -k 12- Ik arctana 9 i r 3 /3 ik'ax _ (koa)2 +(koo)2] (p Finally, the expression for Jo may be obtained by substituting the small term expansion of HId to 0(2) and utilizing cylindrical coordinates to perform the integration. This gives Jo(c, /3) =lim HO' ko dd rff -In2A- lIkoV r 2 2n F2/2 [2 (koa )2-i-(ko/3)2] 94

i\ ( _ a i&2[ (kce)2], + [1- 1 arctan + [- 2 arctan 12 / ir[ 12 ] ica [(koa)2 + (ko ])2 + [ 18 — (A. 8) ir 18 Hence, although higher order terms for Ho' were retained in the evaluation of J3, Jo displays the same degree of approximation as J3, with terms to 0(2) in a and /. 95

Bibliography [1] R.F. Harrington, Time-Harmonic Electromagnetic Fields, New York: McGraw-Hill, section 3.11, 1961. [2] P.E. Mayes, "The equivalence of electric and magnetic sources," IRE Trans. on Antennas and Propagation, vol. AP-6, pp. 295-296, July 1958. [3] J.H. Richmond, "Scattering by a dielectric cylinder of arbitrary cross section shape," IEEE Trans. on Antennas and Propagation, vol. AP-13, pp. 334-341, May 1965. [4] J.H. Richmond, "TE-wave Scattering by a dielectric cylinder of arbitrary cross section shape," IEEE Trans. on Antennas and Propagation, vol. AP-14, pp. 460-464, July, 1966 [5] D.H. Schaubert, D.R. Wilton, and A.W. Glisson, "A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies," IEEE Trans. on Antennas and Propagation, vol. AP-32, pp. 77-85, January 1985. [6] D.K. Langan and D.R. Wilton, "Numerical solution of TE scattering by inhomogeneous two-dimensional composite dielectric/metallic bodies of argitrary cross section", presented at the 1986 National Radio Science Meeting, Philadelphia, PA, 1986. [7] E.H. Newman, "TM scattering by a dielectric cylinder in the presence of a half plane," IEEE Trans. on Antennas and Propagation, vol. AP-33, pp. 773-782, July 1985. [8] E.H. Newman, "TM and TE scattering by a dielectric/ferrite cylinder in the presence of a half plane," IEEE Trans. on Antennas and Propagation, vol. AP-34, pp. 804-813, June 1986. [9] J.A. Stratton, Electromagnetic Theory, New York: McGraw-Hill, section 1.11, 1941. 96

[10] J.Jin and V.L. Liepa, "Application of hybrid finite element method to electromagnetic scattering from coated cylinders", accepted for publication in IEEE Trans. on Antennas and Propagation. [11] J.L. Volakis, "High frequency scattering by a thin material half-plane and strip", submitted for publication to Radio Science. [12] M.I. Herman and J.L. Volakis, "High frequency scattering from polygonal impedance cylinders and strips", accepted for publication, IEEE Trans. on Antennas and Propagation. 97