Cấu trúc sóng chức năng trong điện lý thuyết P1

Spheroidal wave functions are special functions in mathematical physics which have found many important and practical applications in science and engineering where the prolate or the oblate spheroidal coordinate system is used. In the evaluation of electromagnetic (EM) fields in spheroidal structures, spheroidal wave functions are frequently encountered, especially when boundary value problems in spheroidal structures are solved using full-wave analysis.
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Cấu trúc sóng chức năng trong điện lý thuyết P1 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) Introduction1.1 OVERVIEWSpheroidal wave functions are special functions in mathematical physics whichhave found many important and practical applications in science and engineer-ing where the prolate or the oblate spheroidal coordinate system is used. Inthe evaluation of electromagnetic (EM) fields in spheroidal structures, spher-oidal wave functions are frequently encountered, especially when boundaryvalue problems in spheroidal structures are solved using full-wave analysis.By applying the separation of variables to the Maxwell’s equations satisfiedby either an electric or magnetic field, the spheroidal harmonics of electro-magnetic waves, corresponding to their spheroidal coordinate system, can beobtained. With prolate or oblate spheroidal coordinates, the separation of scalar vari-ables results in three independent functions: (1) the radial spheroidal function R$$&,2 INTRODUCTIONsion formulations. Theoretically, the formulation of these harmonics was welldocumented by J. A. Stratton et al. in 1956 [7] and C. Flammer in 1957 [l]. So far, there are only six coordinate systems in which the scalar Helmholtzequations are separable and solenoidal solutions of the vector Helmholtz equa-tions which are transverse to coordinate surfaces are obtainable: the rectan-gular , the circular-cylinder, the elliptic-cylinder , the parabolic-cylinder, thespherical, and the conical coordinate systems [S,91. The prolate and oblatespheroidal coordinate systems are two systems in which scalar wave equationsare separable but the vector wave functions are not separable because of thenonorthogonality of spheroidal radial functions. This causes a difficulty inobtaining rigorous solutions to those vector boundary value problems. In computing EM or physical quantities, the nonorthogonality is involvedin the method of eigenfunctional expansions, in which the linear equationsin the form of nonorthogonal summation are converted to a matrix equationsystem. The dimension of the matrix system is normally infinite and the con-vergence depends mostly on the magnitudes of the interfocal distance and theapplied frequency range. The convergence also depends on the representationof different kinds of spheroidal wave functions. Several kinds of spheroidalvector wave functions have been introduced by researchers [lo-121. However,most of them show fast convergence only for plane-wave scattering or far-fieldapproximations in free space. When the dielectric properties of propagationmedia are lossy, the convergences of some kinds of spheroidal vector wavefunctions become very slow. Furthermore, some kinds of spheroidal vectorwave functions converge much more slowly in the high-frequency region, al-though it is possible to use these vector wave functions in the low-frequencyregion. When used to solve practical problems, the most appropriate spheroidalvector wave functions are constructed by using the spheroidal scalar eigen-functions as the generating function and the orthogonal coordinate vectors(such as the unit vector in the rectangular coordinates) as the piloting vector[13]. Construction of the spheroidal vector wave functions is similar to thatdescribed by Tai [14] for orthogonal coordinates (such as the rectangular orspherical wave functions), except that the spheroidal vector wave functionscannot simply be separated to those of transverse electric (TE) or transversemagnetic (TM) modes. It is found in practical applications that this kind ofspheroidal vector wave function is very convenient in calculations involving in-tegral equation expressions, especially for caseswhere the electric or magneticsources have arbitrary shapes. One of the most convenient analytical methods of EM problems is theintegral representation of wave equations, in which the Green’s function is de-sirable. To obtain electromagnetic radiation due to an arbitrary current dis-tribution located in an inhomogeneous medium, the dyadic Green’s functiontechnique is usually adopted. If the source is of unknown current distribution,the method of moments, which expands the current distribution into a seriesof basis functions with unknown coefficients, can be employed. In this case, OVERVIEW 3the dyadic Green’s function is considered as a kernel of the integral. Therelated unknown coeffici ...