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Spheroidal Wave Functions in Electromagnetic Theory Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong Copyright  2002 John Wiley & Sons, Inc ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic) Spheroidal 6.1 Antennas INTRODUCTION In this chapter, a solution of EM radiation from a prolate spheroidal antenna, excited by a voltage across an infinitesimally narrow gap somewhere around the antenna center, is obtained Three specific cases are considered: an uncoated antenna, a dielectric-coated antenna, and an antenna enclosed in a confocal radome The method used is that of separating the scalar wave equation in prolate spheroidal coordinates and then representing the solution in terms of prolate spheroidal wave functions A simplified version of the solution, after taking account of the fact that the antenna is symmetrical in the &direction, can then be used to obtain the electric and magnetic fields The Mathematics code written allows a user to model each of the three types of antenna radiation problem discussed in this chapter The type of antenna used in this chapter is a prolate spheroid excited by a slot cut through the spheroid Axial symmetry prevails regardless of the location of the slot In aircraft applications, the antenna used is generally mounted on the nose of the aircraft, and the type of antenna used is often a slot antenna Therefore, it is possible to model this antenna configuration as a slot antenna mounted on a spheroid The effects of a protecting coating layer or radome on such a configuration can also be investigated and considered in the optimized operation 145 146 6.2 6.2.1 SPHEROIDAL PROLATE Antenna ANTENNAS SPHEROIDAL ANTENNA Geometry A perfectly conducting prolate spheroidal antenna excited by a specified field over an aperture on its surface and immersed in a homogeneous, isotropic medium is a convenient introductory problem It is also assumed that the surrounding medium is nonconducting and nonmagnetic To simplify the situation, it is assumedthat symmetry about the axial direction prevails The geometry of the prolate antenna is shown in Fig 6.1 The semimajor and semiminor axes of the spheroid are designated a and b, respectively, and the interfocal distance d, as indicated in previous chapters In general, the surface of the spheroid is < = 51, and the excitation gap can be located anywhere, say at = ~0 6.2.2 Maxwell’s Equations for the Spheroidal Antenna For the symmetrical situation (i.e., a/&$ = 0), Maxwell’s equations in free space, relating E and H, take the form (6.la) hrE77 = -GT’ a(P%J (6.lb) (6.1~) (6.ld) hrH,, a(Pw = -jClowpdSy achvH,) w-QH< -icowhth,E4, > at drl = (6.le) (6.lf) where p = dJ(1 - q2)(S2 - 1)/2, ht and h, have been defined in Chapter Also, ~0 and ~0 are the permittivity and permeability of free space, respect ively From Eqs (6.la) to (6.lf) it is observed that the problem may be split into two parts If the applied field on the aperture has only an Eq component, the excited magnetic field has only the H$ component and E+ = On the other hand, if the applied field has only an E4 component, the excited electric field has only an E+ component and H& = PROLATE SPHEROlDAL iv=+1 I b : I : I i T Fig 6.1 Prolate spheroid model of an antenna ANTENNA 147 148 SPHEROIDAL ANTENNAS Here we consider only the former case; that is, the applied field has only an I$, component and is considered in the subsequent analysis Then, following Schelkunoff [35], we set H4=p’ A (6 2) where A is an auxiliary scalar wave function from Eqs (6.la) and (6.lb), we obtain Et = E rl = By substituting that A satisfies Eqs - to be defined later &icowd2 Jcrz - l)( -(Q) )( #)=( ‘“i, (6.49) where is the zero matrix; ’ (g; g g: I;;), nodd (g! g j, neven w> =’ \ with X being either A, C, D, E, G, H, J, K, L, 0, P, or Q; ( (Y) CyOn CyZn CY4n l ** )‘T nodd - { CC with Y being either B or F; and Yin w C ( Zr i ( 22 Y3n 23 24 C 25 &j Ysn l me)‘, l l ** )‘, l l )’ , n even nodd n even with being either W, M2, N2, or M3 The matrix equation (6.49) can then be solved twice, for the even and odd cases The truncation number here is taken to be 30 This means that there are a total of 120 coefficients (30 each for A$, MI, Ni, and MI) to be computed By considering the odd and even casesseparately, there are 60 coefficients to be computed for each case Since there are four equations for every value of r, for the odd case, r will have to range from to 28 in steps of in order to have sufficient equations For the even case, r will have to range from to 29 in steps of 6.4.5 Mathematics Code The final main module in the package SpheroidAntenna.nb is for the spheroidal antenna enclosed in a confocal radome This module is called Radome[ul-, 178 SPHEROIDAL ANTENNAS u2,, u3-, erl,, er2-, er3,, l,, vO-, Vapplied-1 and they are specified as follows: There are nine arguments l ul: the radial coordinate representing the boundary between the spheroidal antenna and region I l u2: the radial coordinate representing the boundary between regions I and II l u3: the radial coordinate representing the boundary between regions II and III l erl: the relative dielectric constant of region I l er2: the relative dielectric constant of region II l er3: the relative dielectric constant of region III l I: the semi-interfocal distance l ~0: the angular coordinate on which the excitation gap lies l Vapplied: the applied gap voltage This module can be summarized as follows: Based on the input arguments, the parameters cl, ~2, and c3 are calculated Using the values of cl, ~2, and ~3, the expansion coefficients c”(c) and the eigenvalues A,, (c) are computed for each ci (i = 1, 2, and 3) using the support modules Getdmn and EigenCommon, respectively The spheroidal radial functions of the first, second, third, and fourth kinds and their derivatives are evaluated subsequently This is achieved through the supporting modules PSpheroidRl, PSpheroidR2, PSpheroidR2forsmallc, and PSpheroidR2forlargec These are computed for all combinations of c and t except for the following combinations: c2 and cl, c3 and 52, cl and &, and cg and 51 Next, the spheroidal angular function of the first kind is computed via the PSpheroidS module With the radial and angular functions computed and tabulated, it is then possible to compute the expansion coefficients using a submodule called p[n-1 In this submodule, the supporting modules V[m-,n-,q-y-1 and NormFactor[m-,n-,q-] are called to compute the angular function and the normalization factor, respectively To compute the expansion coefficients required in Eq (6.45), the matrix equation in Eqs (6.42a) to (6.42d) is formed for the odd case using PROLATE SPHEROIDAL ANTENNA ENCLOSED IN A CONFOCAL RADOME 179 LHSl and RHSl and for the even case using LHS2 and RHS2 The solutions for both casescontain AL., M., Nz, and A4.i Therefore, A4, is extracted from the solutions and then combined to form the table Mcoeff l In finding the solutions to the matrix equation, the supporting module called IterativeRefine[A-,b-,N-] was called Prior to computing the & pattern, the expansion coefficients are first checked to remove those coefficients with magnitudes that are lessthan 10? The reason for this is again that these coefficients are too insignificant to affect the computation of the H+ pattern and are removed to save computation time l 6.4.6 A table for the H# pattern is then computed by calling the supporting module Hphi[q-, coeff-, theta-] Results and Discussion As in previous chapters, the Mathematics code written for the radome case needsto be verified However, there is no available result in existing literature u2-, u3,, erl,, er2,, er3,, to compare with Therefore, the Radome[ul-, I-, vO-, Vapplied-] code is used to compute the radiation patterns for the uncoated and coated prolate spheroidal antennas The results are presented in Fig 6.15 As expected, the Radome[ ] code produces almost exactly the same results as the Uncoated[ ] or Coated[ ] codes Since both codes have been verified earlier, this implies that the Radome[ ] code is accurate, too Figure 6.16 shows the effect of different radome materials on the radiation pattern of prolate spheroidal antennas Observed carefully, the top figure shows that increasing the dielectric constant sharpens the pattern The bottom pattern showsalmost no variation in the pattern at all becausethe radome is placed lo& away from the antenna At this distance, both the radome and the radiated fields are almost spherical, and the effect on the pattern is minimal In addition, the radome has a thickness of 0.5X2, and transmission line theory implies that the radome acts almost as if transparent to the waves, due to the matched impedance This is desirable because the purpose of radomes is to protect the antenna from environmental effects while avoiding affecting its operation (1371 Figure 6.17 shows something similar, but the antenna is made slightly fatter, with 51 = As in Fig 6.16, the radome material affects only the radiation pattern at a nearer distance Figures 6.18 and 6.19 show the effects on the radiation patterns of the distance at which the radome is placed In both figures it can be observed that the effect of radome is significant only when placed very close to the antenna (e.g., s/x0 = 0.25) For a thin antenna (