Appendix B

Treatment of Plasma using the Z Transform for the TLM Method

For a non-magnetized plasma, the permittivity tensor is given by:

where refers to the relative permittivity of the plasma.

The electrical susceptibility tensor of the plasma is written in the form:

Within the technical context of the Z transform for the TLM method, the plasma can be handled as a purely dielectric medium. Its conductivity tensor is null and its susceptibility tensor is reduced to:

The Z transform is applied to this tensor of the form:

This amounts to treating the element:

It is expressed in the form:

With the transform above, this becomes:

The simplification reduces this element to:

It can be expressed in the following form, which is practical for the following computations:

where:

and:

The equation (see equation [4.21]) corresponding to the susceptibility tensor must be solved:

This equation is reduced for the plasma to its electrical susceptibility tensor:

This solution amounts to identifying in the second part of this equation.

We set the following:

By developing this equation, we obtain:

Hence, by identification, we have:

This system of equations is solved in order to determine the tensors enabling the plasma to be represented in the TLM method according to the Z transform technique. We can choose to keep just the “dispersive” terms in the expression for the tensor by taking: .

The researched solution is thus the following:

Since b₀ = 1, b₁ = 2 and b₂ = 1, we obtain:

We recall that the solution (see [4.24]) obtained using the Z transform technique for the TLM method is expressed in the following form:

For the plasma, we obtain: