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The mixed-potential electric-field integral equation is used in conjunction with the Galerkin's method and complex image theory for analyzing a transmission line with multiple strips embedded in different layers of a multilayered uniaxially anisotropic dielectric substrate. The two-dimensional Green's functions for the scalar and vector potentials are analytically obtained in the space domain due to the approximation of its spectral-domain version with complex images, thus avoiding lengthy numerical evaluations. Double integrals involved in the computation of Galerkin's matrix entries are quasi-analytically carried out for the chosen basis functions, which are well suited to the problem.

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In this paper, Galerkin's method in the Hankel transform domain is applied to the determination of the resonant frequencies, quality factors, and radiation patterns of circular microstrip patch resonators. The metallic patches are assumed to be embedded in a multilayered substrate, which may contain uniaxial anisotropic dielectrics, magnetized ferrites, and/or chiral materials. The numerical results obtained show that important errors can be made in the computation of the resonant frequencies of the resonators when substrate dielectric anisotropy, substrate magnetic anisotropy and/or substrate chirality are ignored. Also, it is shown that the resonant frequencies of circular microstrip resonators on magnetized ferrites can be tuned over a wide frequency range by varying the applied bias magnetic field. Finally, the computed results show that the resonance and radiation properties of a circular microstrip patch on a chiral material is very similar to those of a circular patch of the same size printed on a nonchiral material of lower permittivity.