Numerical Computation of Potential in Unbounded Two-Dimensional Regions using Schwarz-Christoffel Transformation

摘要

Finite difference is a method of choice for educators to demonstrate and compute potential field of two-dimensional geometries, such as integrated circuit planar resistors. It is simple and can be readily programmed by undergraduate students. It is also very accurate and its accuracy can be easily controlled by changing the grid size. Finite difference method (FDM), however, has two serious limitations. One, it can not be easily applied to unbounded regions such as integrated circuit (IC) microstrip lines. And two, the FDM computes potentials at predetermined grid points only. Unlike the finite element method (FEM), it does not generate potential functions that can be used to interpolate for potentials at the points that are not located at the grid, or to use these functions in determining other quantities based upon the computed potential such us the field intensity. This paper describes a hybrid method based upon conformal transformations, including the Schwarz-Christoffel (S-C) transformation (without having to compute the transformation functions) to map the original boundaries, including those at infinity, to a bounded region and only then applies a numerical method based on finite differences. This paper also describes a method that is a combination of the FDM and the FEM to generate the potential functions after the FDM has been applied. The combined method retains the simplicity and accuracy of the FDM. Yet it, like the FEM, provides potential functions that can be used for interpolation as well as post-processing of potential. Testing theses approaches by means of an example for which exact solution is obtainable, the hybrid method and the combination of the FDM-FEM are applied to determine the electrical potential at a specific point in the field of an IC microstrip line. In both cases the results are in agreement with analytically derived results. The approach we have developed is simple, readily applied by undergraduate students, yet accurate and thus of use in professional engineering work.