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 of dispersive media.

4.2. Matrix form of Maxwell’s equations

Maxwell’s equations and the constitutive relations of the matter can be expressed in a matrix form, which is convenient for formulating the problem:

[4.1]

[4.2]

(a) expresses the relation in the vacuum, and (b) the wave-matter interaction.

The currents are given by:

[4.3]

and are the electrical and magnetic susceptibility tensors, respectively (without dimension); and are the electromagnetic and magnetoelectrical coupling tensors; J_ef and J_mf are the free electrical and magnetic currents; and are the electrical and magnetic conductivity tensors and and reflect an eventual magnetoelectrical interaction. All of the tensors are (3 × 3).

The notation “×” in equations [4.2] and [4.3] in the remainder of this chapter expresses convolution in the time domain.

4.3. Cubic mesh normalized Maxwell’s equations

In this section, the mesh of the elementary 3D TLM node cell is such that Δx = Δy = Δz = Δl (Figure 4.1).

The normalizations are as follows:

[4.4]

where is the free space impedance

The normalized current sources J_ef and J_mf are also obtained:

[4.5]

which leads to normalization on the electrical and magnetic conductivity:

[4.6]

The spatial and temporal derivatives are also normalized:

[4.7]

Δt is the time step and is the normalized time.

By applying these normalizations in 3D TLM, with the following expression is obtained:

[4.8]

The first part of equation [4.8] refers to propagation in a vacuum, and can be solved using the traditional TLM method. The second part deals with wave-matter interaction, which behaves in the same way as the sources.

4.4. The propagation process

The propagation process is directly based on the equivalence between the model deduced from the circuit laws and that computed by Maxwell’s equations. The formulation for 3D TLM is based on the symmetrical condensed node (SCN), which is a node with 12 different polarizations represented by the 12 corresponding voltage pulses Vⁱ = [V₁ V₂ …. V₁₂] ^T on its six arms (see Figure 2.3). The six electromagnetic field components (E_x, E_y, E_z, H_x, H_y and H_z) are defined at the center of the cell formed by this node.

The “total field” vector F = [V_x V_y V_z i_x i_y i_z]^T is defined by:

[4.9]

[4.10]

This total field vector enables the components of the electromagnetic field to be expressed based on the voltage pulses Vⁱ, by taking into account the normalizations defined above.

The propagation process occurs in three essential stages:

– Definition of the external excitation vector at node F^R from incident voltage pulses and free excitation sources F_f:

[4.11]

This vector is expressed in simple matrix form, , where:

[4.12]

– Evaluation of the total fields: the left hand side of relation [4.8] with these definitions is written 2 F^R − 4 F. The relation between the “total field” vector F and the “excitation” vector F^R is obtained by solving the general equation [4.8]. This is performed by using the Z transform. It is expressed by the matrix [t(z)], which represents the matrix of transmission coefficients.

We can then write:

[4.13]

– Computation of reflected pulses: the reflected pulses are obtained by:

[4.14]

where:

and the matrix [R_t]:

The process is summarized in the following organigram (Figure 4.2):

The role of the [P] matrix in this organigram is to arrange incident voltage pulses V in order to obtain the vector . It should also be noted that the components of the reflected pulses vector V^R are renumbered.

4.5. Wave-matter interaction

The term on the right-hand side of equation [4.8] involves convolution products. In order to deal with this problem, the Z transform is used which enables the products of convolution to be passed to simple products. A bilinear transformation is used, of the form:

[4.15]

If s = j, ω represents the derivative, we obtain:

[4.16]

where z⁻¹ is the time shift Δt.

Under these conditions, the second part of relation [4.8] can be written based on the Z transform, as follows:

[4.17]

where:

Relation [4.8], including these transforms, is written:

[4.18]

or:

[4.19]

In matrices and there are some constant elements and others which are dependent on frequency. In order to cause the frequency dependence, we make use of the causality of elements in order to use the shift in time.

Thus, the expression z⁻¹ F transfers the value for F at the previous instant. This is written:

[4.20]

[4.21]

where and are frequency independent, which then gives us:

[4.22]

[4.23]

Following further development of the computations, we arrive at the following expression:

[4.24]

where:

From these last relations, the organigram of the process is established, including all of the permittivity, permeability and conductivity tensors (Figure 4.3).

In order to model the media under investigation, by knowing the general tensors and it only remains for expressions [4.20] and [4.21] to be re-used in order to determine the terms , and (for example, see section 4.7).

4.6. The normalized parallelepipedic mesh Maxwell’s equations [LOU 04]

A variable step mesh is often used for structures with inhomogeneities, i.e. the application of a thin mesh in high field gradient areas and then a coarse mesh in the rest of the structure.

For the development which follows, the following normalizations are used:

where Δl is the spatial step used in the process.

Thus, the following normalized values are obtained:

where is the free space impedance.

Time is also normalized:

The tensor values , and that need to be taken into account in the process of the method are based on the elementary dimensions (u, v, w) of the TLM cell.

Maxwell’s equations are solved by considering relation [4.2]:

– First equation: − rot .

The curl is written:

[4.25]

After further development (Appendix A), the final expression for the equation becomes simpler and is written in the following form:

[4.26]

– Second equation: rot .

Using equation [4.2], the curl is expressed in the following form:

[4.27]

and, in this case, equation [4.8] is written as:

[4.28]

The new tensor parameters are given by (Appendix A):

Once the tensor parameters have been computed, we then have an identical situation to that of the uniform mesh.

4.7. Application example: plasma modeling [MOU 06]

4.7.1. Theoretical model

By way of example, we will discuss the case of non-magnetized plasma. Its permittivity tensor is given by:

where refers to the relative permittivity of the plasma.

The electrical susceptibility tensor of the plasma is expressed 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 representation of the computation process, which enables the plasma to be integrated, occurs through solving the equation (equation [4.21]), having applied the Z transform to the expression for its susceptibility tensor. Indeed, this solution enables the parameters of the plasma, which must be considered in the algorithm of this Z transform technique for the TLM method, to be determined. This computation is developed in Appendix B. To summarize, the following parameters are obtained:

where:

The solution obtained using the Z transform technique for the TLM method is written in the following form:

where:

For the plasma, the following is obtained:

In order to evaluate the term , we make use of the phase-variable states technique. We obtain:

where:

This computation process is summarized by the organigram in Figure 4.4.

4.7.2. Validation of the TLM simulation

Validation of the propagation simulation concept is performed on a structure using a parallelepipedic resonant metallic cavity (Figure 4.5). Firstly, the resonant modes are determined by simulating an empty cavity, which is thus considered to be a reference cavity. Then, cavities filled with plasma are simulated and their responses compared with that of the reference cavity.

The TLM method has been designed in order to deal with media with permittivity and permeability such that εr≥1 and μr≥1. When this is not the case, the convergence of its algorithm is not guaranteed.

In the plasma example, the condition regarding permittivity is not fulfilled (see [4.29]). In order to compensate for this problem, a dielectric, or static constant can be added to the plasma, which thus presents a constant permittivity ε_r>1 and which compensates for the value of the permittivity brought into play by the plasma in order to avoid this type of divergence. This value for static permittivity is applied during testing of the reference cavity in order to isolate the response obtained by the plasma effect.

In order to achieve simulations which enable the plasma to be modeled, cavities with the dimensions 23 × 17 × 60 µm³ have been used. For these cavities, a static permittivity with a value of 2 has been used. At the same time, several plasma types have been tested. Indeed, the plasma is characterized by its plasma angular frequency ω_p and the frequency of collisions between particles v_p. For the micrometric dimensions chosen, the plasma samples tested are as follows:

This choice for the value v_p is justified by the approximation, which is often used in low-pressure plasma studies, enabling the expression for the relative permittivity of the plasma to be simplified. It is expressed in the form:

[4.29]

The simulation of these cavities is achieved using a network of transmission lines with a base mesh Δl of 1 µm and a time discretization step Δt of 1.667 × 10⁻¹⁵ s. Each simulation uses 200,000 iterations. Excitation is a Dirac signal.

For each cavity, the response obtained is compared with the reference cavity, as well as the expected theoretical values.

For this series of cavities, the resonant modes researched were 101, 102 and 103.

The responses from the filled cavities for plasma #1, #2 and #3 samples are given in Figures 4.6 to 4.8. Each response is compared with the reference cavity corresponding to this series.

The resonant frequencies of each resonant mode of all of the cavities are given by the formula:

[4.30]

The comparison of resonant frequency values for each cavity, obtained by simulation, compared to those computed using traditional formulae, enable the relative errors resulting from these simulations to be estimated. The value of these errors does not exceed 1%. This low simulation error rate confirms the good modeling of wave propagation for the TLM method and the Z transform.

4.8. Conclusion

The application of the Z transform in the TLM method is extremely useful for two main reasons. On one hand, it enables the rigorous simulation of physical dispersive media (causality property) without the need to deal with problems from convolution, which are always memory hungry. On the other hand, the process is unconditionally stable. It can, of course, be used for non-dispersive media, although, historically, the traditional method is always used.