Integral Equation Solution for Microstrip Structures at Low Frequencies

Abstract

Microstrip StmCNrCS have bcen shldied extensively using various types of full wave analysis techniques. However, these techniques are having difficulties because they usually involve the solution ofa very large system of linear equations. In this approach, a symmetrical form of electric-field spatial-domain Green's function [I] different from [Z] and 131 is applied. Funhcr, the numerical solution af Maxwell's equalions at low frequencies is plagued with numerous problems. Because of the discrepant frequency dependence of the solenoidal and irrotational components of the current when the frequcncy tends to zero, a working numerical method has to include this Helmholtz decomposition and ascribe the requisite frequency dependencies lo the solenoidal and irrotational camponents of the current. This decomposition is achieved by selecting the loop-tree basis [4], IS]. The use of the loop-tree basis, followed by frequency normalization, solves the problem of singular malices at low frequencies. However, if an iterative solver is used, the iteration count is usually very large and may even diverge for some problems. To overcome this problem, a method of transformation of the matrix equations [6] is also applied