Examination, Clarification, and Optimization of the Green's Function/$Z$ -Matrix Models and Calculations for Rectangular Planar Microwave Circuits

In this paper, we examine the Green's function and
$Z$
-matrix models commonly used in calculations for microwave planar circuits. We prove, for the first time, the pointwise convergence of the Green's function and the interchangeability of the integration and infinite summation used in deriving the
$Z$
-matrix model. We also show the validity of interchanging the order of the double summation of the
$Z$
-matrix. In a related vein, we point out some potential problem of performing series rearrangements in evaluating infinite sums. Through computational analysis, we will demonstrate that a stable and efficient algorithm for the Green's function and
$Z$
-matrix calculations can be optimally obtained for all practical boundary port configurations of rectangular planar circuits.