Appendix A

Development of Maxwell’s Equations using the Z Transform with a Variable Mesh

Maxwell’s equations are written (without considering sources) as follows:

– First equation: − rot

For normalized field values, we obtain:

Knowing that Δl = 2 c Δt, and following simplification, we obtain:

In order to be able to use the traditional TLM algorithm, the term is added to both sides of each equation:

We can see that each part of this set of equations is simplified and can be integrated into the classic TLM process. It only remains for us to develop the second part. For example, for the first line, we obtain:

Knowing that , this second term is then expressed in the following form:

We define the new parameters based on the elementary dimensions of the TLM cell:

For the other terms, we follow the same process of development and obtain:

These new terms are taken into consideration in the 3D TLM algorithm.

In this case, the final version of [4.26] is more simply expressed as:

– Second equation:

In the same way, the first line is expressed using normalizations:

As with the first Maxwell equation, a term is added to each part, which gives:

Hence we deduce the general parameter values:

and in this instance, equation [4.28] is written: