Cấu trúc sóng chức năng trong điện lý thuyết P9

EM Eigenfrequencies in a Spheroidal Cavity Computation of eigenfrequencies in EM cavities is useful in various applicat ions. However, analytical calculation of these eigenfrequencies is severely limited by the boundary shape of these cavities. In this chapter, the interior boundary value problem in a prolate spheroidal cavity with a perfectly conducting wall and axial symmetry is solved analytically.
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Cấu trúc sóng chức năng trong điện lý thuyết P9 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) 9EM Eigenfrequencies in a Spheroidal Cavity9.1 INTRODUCTIONComputation of eigenfrequencies in EM cavities is useful in various applica-t ions. However, analytical calculation of these eigenfrequencies is severelylimited by the boundary shape of these cavities. In this chapter, the interiorboundary value problem in a prolate spheroidal cavity with a perfectly con-ducting wall and axial symmetry is solved analytically. By applying Maxwell’sequations to the boundary, it is possible to obtain an analytical expression ofthe eigenfrequency fnSo using spheroidal wave functions regardless of whetherthe parameter c is small or large. An inspection of the plot of a series of fnSa values (confirmed in [64])indicates that variation of fnSo with the coordinate parameter < is of the formfn&) = fnS(0)[ 1 + g(l)@ + gt2)/c4 + gc3)/t6 + 9 l] when c is small. By lfitting the fnSo, 5 evaluated onto an equation of its derived form, the first fourexpansion coefficients - g(O), g(l), gc2) and g13) are determined numericallyusing the least squares method. The method used to obtain these coefficientsis direct and simple, although the assumption of axial symmetry may restrictits applications to those eigenfrequencies f72Sml, where m’ = 0. 245246 EM EIGENFREQUENCIES IN A SPHEROIDAL CAVITY9.2 THEORY AND FORMULATION9.2.1 Background TheoryThe prolate spheroidal body under consideration is shown in Fig. 9.1. In viewof the fact that Mathematics handles only vector differential operations in theprolate spheroidal coordinates in accordance with the notations used in thebook by Moon and Spencer [9, pp. 28-291, a temporary change of coordinatesis necessarv. The new notation used is shown in Fig. 2.1. a Fig. 9.1 Geometry of the spheroidal cavity. As noted by Moon and Spencer (91,the vector Helmholtz equation is morecomplicated than the scalar counterpart, and its solution using the variable-separation principle may sometimes cause new problems. This is especiallytrue in rotational systems like that of the spherical coordinates or spheroidalcoordinates. In spheroidal coordinates, the solving of vector boundary valueproblems is further complicated by the fact that the vector wave equationis not exactly separable in spheroidal coordinates. Although another moregeneral analysis has been performed using the vector wave functions, formedby operating on the scalar spheroidal wave functions with vector operators,the validity of the results obtained is doubtful. In view of these limitations, THEORY AND FORMULATION 247several assumptions are made in the formulation of the current boundaryproblem in order to provide a truer, more accurate picture.9.2.2 DerivationWith axial symmetry assumed, it is possible to separate the field componentsinto Et, Eq, and I$ for the TM mode and Ht, &, and E4 for the TE mode. First, the TM mode is considered. With axial symmetry, I74 can be as-sumed simply as Hc#l F(c, W(c, 7). = (9 .1) By applying the Maxwell equations dB VxE = ---, (9.2a) dD VxH = dt, (9.2b)and using the formulation of V x X in the spheroidal coordinates where vxx = (9.3) d2K2 - v2) (9.4a) Q77= 4(1 - 72) ’ d2(C2 v2) - (9.4b) iI< = 4K2-l) ’ 94 = %(1 - v2>(C2 - l), ...