On the convergence of matrix elements in planar and waveguide problems

In many integral equation formulations, impedance matrix elements derived for a moment method solution may not be computable. This happens when the summations involved tend to diverge because of a poor choice of basis and testing functions. In cases amenable to the spectral representation, these phenomena can be predicted and avoided, as shown in the paper. The development for many planar structures is shown in a general manner and guidelines for the selection of basis and testing functions for the generation of convergent matrix elements are deduced. The guidelines are applied to the Galerkin formulation, where the decay rate of the spectral basis functions has a double effect on the integrand. Finally, the analysis is applied to waveguide problems, where the full three-dimensional spectral Green's dyad is used. These principles are worked out for transversal slots in waveguides, where divergence has been observed in the past. A stable formulation is then derived and results are presented.