Finite-difference solution of the transverse Laplace´s equation in unbounded planar transmission lines

A new method for the finite-difference solution of the transverse Laplace´s equation is presented. The method uses special arctangent transformations to map the unbounded domain into a rectangle suitable for the application of the finite-difference method. The properties of these transformations are highlighted. Equations for the electrostatic potential and the per-unit length capacitances of stripline and microstrip planar transmission lines are derived in the transformed domain, and then discretized with the finite-difference method. The Gauss-Seidel over-relaxation algorithm is used to solve the discrete equations. The characteristic impedance of the aforementioned lines is computed in terms of their per-unit length capacitances. Results for the characteristic impedance as a function of the center conductor width are shown. These results are compared with results computed with Wheeler´s (1978) formulas, and excellent agreement between the results of the two methods is found