Numerical Analysis in Electromagnetics: The TLM Method

Chapter 4

The TLM Method in Matrix Form and the Z Transform

4.1. Introduction

In Chapter 2 we saw the various nodes that can be used in the traditional form of the TLM method. For example, the SCN node formed of 12 transmission lines, to which open circuit and short-circuit stubs, representing permittivities and permeabilities respectively, are added. Matched lines can be added in order to account for lossy media. This approach is effective for a simple medium whose parameters are independent of frequency. Nevertheless, complex media, such as dispersive or anisotropic media, whose parameters are dependent on frequency, require a different process in order to account for this variation. An interesting approach has been presented by Christopoulos [PAU 99], where the propagation of a wave in a vacuum is separated from the wave-matter interaction. The propagation is handled by the traditional TLM method [HOE 92], using 12-input nodes (vacuum), whereas the wave-matter interaction appears in the form of additional electromagnetic field sources. Since the interaction is causal, the Z Transform can be used to treat the problem. The advantage of this application is that convolution products can be avoided in the temporal processing of dispersive media. Moreover, in this case, the algorithm is unconditionally stable. In this chapter we are going to present the matrix formulation of Maxwell’s equations, then the application and implementation of the Z Transform, which will enable the simulation ...