Application of Spectral Techniques to Solving 2D Electrostatic Problems

Abstract

In pretty much all of the text treatment so far we have used spectral (Fourier/Laplace) techniques in the time domain. We expanded total solution in terms of summation of the exponential function e ^st where s is the complex frequency. In this chapter we carry the same principles to the space domain. Putting time a side, and assuming solution is a DC one, we apply spectral techniques to figure voltage (and electric field) across a 2D conductor slab with predefined boundary conditions. We effectively solve for Laplace’s equation (to be distinguished from Laplace’s transform) which governs the electrostatic potential in a zero-charge region, and subject to some boundary conditions. We use the method of separation of variables and decouple the problem into an x-dependent solution and a y-one. Again we use basis functions and evaluate Fourier coefficients by integrating the forcing boundary functions against the harmonics. We apply the method on a few examples, plot the 2D potential, and compare favorably with field-solver solutions. We wrap the chapter by showing how we can deal with multiple boundary conditions using superposition.