Introduction

There are a number of modeling methods that are suitable for solving problems in electromagnetism and analyzing the behavior of certain media. In order to apply these methods the type of problem must be specified and the boundary conditions must be clearly determined and defined. Numerical or analytical solutions are then carried out.

Analytical solutions, which are already well established, were the first to be applied. They enabled an efficient resolution of all problems relating to the majority of electromagnetic wave guiding systems. However, these analytical methods remain limited, since, in these cases, it is only possible to analyze structures with simple geometries and which, in the majority of cases, have a certain degree of symmetry.

For more realistic modeling of geometries and more complex materials (indeed, complexity leaves little room for any analytical resolution), we have numerical methods, which have become an important ement in the analysis of the behavior of various industrial products. They have progressed in parallel with technology and enable electronic systems developers to have at their disposal all of the necessary characteristics and data, which were difficult to obtain through testing, in order to ensure the reliability of device operation without any accompanying performance degradation.

In the specific case of electromagnetism, there are various differing numerical techniques, whose effectiveness depends on the problem and on the desired results. These techniques can be classified according to different criteria.

Classification based on the type of equation

Firstly, we can classify numerical methods based on equation type. Indeed, most models under consideration lead to differential or integral mathematical equations. If the problem deals with electromagnetic wave propagation, the equations which describe its behavior (such as Maxwell’s and wave equations) can be expressed using two methods: differential or integral.

In order to solve these equations at any point in a finite space, differential or integral methods are used to determine the values required.

Classification based on the application domain

A second classification which may be taken into account is the domain within which the equations to be solved are defined. In theory, equations express the space and time variations in the scale of the problem to be resolved (electromagnetic fields or potentials). Here we are working in the time domain and the methods used are known as “time-domain numerical methods”.

However, in the study of certain problems (notably in the area of telecommunications), it is field cartography varying sinusoidally over time or from a combination of multiple sinusoids, which is of interest. In these cases, the electromagnetic characteristics of the majority of materials can be expressed in a much simpler form, based on the frequency of these sinusoidal signals. These equations are therefore expressed frequentially and so the methods used to solve them are known as “frequency-domain numerical methods”.

The advantage of frequency methods is that they give rise to equations which are more flexible and easier to simplify. Nevertheless, they are also limited, as they rely on signals always being sinusoidal or based on a sinusoidal combination.

In all cases a frequency representation can be obtained from a signal by using a Fourier transform of the time signal.

In this book, we are going to look at the TLM (transmission line matrix) method, which is one of the “time-domain numerical methods”. These methods are reputed for their significant reliance on computer resources. However, they have the advantage of being highly general. We will focus our attention on the TLM method which, since the pioneering article on TLM by P.B. Johns and R.L. Beurle in 1971, has been intensively studied and developed by many researchers. It has, therefore, acquired a reputation for being a powerful and effective tool by numerous teams and still benefits today from significant theoretical developments. In particular, in recent years, its ability to simulate various situations, including complex materials, with excellent precision has been demonstrated.

This book consists of an introduction and four chapters.

Chapter 1 describes the basis of the TLM method in two dimensions and enables different aspects of the method to be tackled, as well as the errors resulting from space and time sampling.

Chapter 2 is dedicated to a 3D analysis of the method. It maps out the main types of nodes currently used by pointing out their respective advantages and disadvantages. This chapter also features the problem of open structure simulation and the necessity of implementing absorbing boundaries, including PMLs, which nowadays are used universally.

Chapter 3 describes techniques which enable the simulation of structures comprising passive and active discrete elements, as well as thin metallic wires without the need to mesh these structures, which would lead to memory problems. These techniques, as well as 3D node and mesh flexibility, enable the simulation of a wide range of problems where the properties of the surrounding medium are not dependent on frequency and are therefore not dispersive.

Chapter 4 demonstrates how to simulate dispersive media using the Z transform within the TLM method in matrix form. This rigorous and unconditionally stable method makes the use of the TLM method possible in virtually all cases.

Application examples are included in the last two chapters, enabling us to draw conclusions regarding the performance of the implemented techniques and, at the same time, to validate them.

Multi-scale problems which require the TLM method to be combined with other methods will not be dealt with in this book in spite of their undeniable interest. There are many papers dedicated to this which would require collation into a single publication.