On graduating twenty years back, in 1984, my first job was as a research engineer working on computational electromagnetics (CEM) at the National Institute for Aeronautical Systems Technology (as it was then called) of the Council for Scientific and Industrial Research (CSIR) in Pretoria, South Africa. It was an exciting time to be working in this field. Although a number of methods had already been successfully introduced, including the three which will be discussed in detail in this book, major advances were being made in all of these methods, and the power of desktop computers was growing in leaps and bounds. No commercial programs (or codes, as they are generally called) were then available for RF problems, but some US government-sponsored codes, in particular the NEC-2 code, were becoming available for general use.

The 1980s saw the final decade of the Cold War, which in some areas (such as Southern Africa) was far from cold. New military technologies, in particular stealth, were driving CEM to address progressively more electromagnetically complex problems. However, when the ColdWar ended, far from CEM work coming to a halt, new commercial markets, such as the rapidly developing market in mobile telephony and personal communication systems, and the proliferation of electronic systems in motor vehicles, continued to drive the technology forward at breakneck speed throughout the 1990s. This was also due to the widespread availability of cheap and progressively more powerful personal computers as a crucial enabling technology.

CEM has now reached a modicum of maturity, with a number of powerful methods available, able to solve problems of real engineering interest at radio frequencies, and with a number of commercial codes available.

Printed antenna and microstrip technology: a brief review

Microstrip patch antennas are an example of a large class of modern antennas known as “printed antennas.” Microstrip was originally developed in the early 1950s as a transmission line, and the first publication on using this structure as a radiator appears to have been by Deschamp in 1953 [1, Section 1.1]. Almost twenty years then passed until the first patent of the modern microstrip antenna was registered in 1973 by Munson, although the structure was independently discovered in at least one other location.

Microstrip antennas are generally constructed using the same photo lithographic process using to create printed circuit boards. In their simplest form, radiation is due primarily to energy leaking out of the cavity formed by the patch located close to a ground plane; physically, the patch is simply a very wide microstrip line. For the basic rectangular patch, the radiation from two opposite sides reinforces, whereas that from the other two sides cancels. The patch is usually supported on a dielectric substrate of some form, primarily for structural reasons. Typical materials are Teflon and glass-reinforced plastics, as used in printed circuit board technology. Typical material properties for these are ∈r in the range from 2–2.5, and tan δ from 0.0004–0.002. High-∈r substrates such as alumina ceramics produce physically small patches, but with very limited bandwidth. Typical material properties in this case are: ∈r 9.7–10.3, tan δ ≈ 0.0004. For some applications, plastic foam substrates have been used. These materials (sometimes using cheap materials such as expanded polystyrene tiles) have properties close to free space: ∈r ≈ 1.05, and tan δ ≈ 0.0008.

In this final chapter, we discuss a selection of more advanced topics, primarily relating to the finite element method. However, as will be seen, a linkage to the method of moments will be established, and perhaps rather less expectedly, the finite difference time domain method will also emerge as a special case of a finite element time domain treatment, so amongst other purposes, the chapter serves to draw together these three apparently quite different methods.

We will start by considering a very important extension of the vector elements, namely higher-order elements. Following this, the stationary functional formulation for deterministic (driven) problems will be outlined. In the preceding chapter, an eigenvalue problem was used to illustrate the FEM in two dimensions; in this chapter, a deterministic three-dimensional problem will be discussed, namely the analysis of waveguide obstacles. Finite element analysis is ideal for this problem, and good results have been obtained by a number of workers. Results for two waveguide problems computed using FEM codes incorporating higher-order elements will be shown. Then, a hybrid FEM/MoM formulation, which has proven very powerful for specialized applications, will be introduced, and an application to radiation exposure assessment near a base-station antenna will be presented. Following this, time domain finite element analysis is briefly discussed.

We conclude the chapter with a discussion on two issues which impact on efficiency. Firstly, sparse matrix storage schemes are briefly outlined, and secondly, error estimation and the use of mesh adaptation based on this is discussed.

Introductory comments

With the theoretical background now established, one is in a position to start using commercial and public domain MoM programs intelligently. In this chapter, we will discuss primarily the application of the commercial code FEKO for antenna modelling, but will also discuss the use of the public domain code NEC-2 in this regard. Other than FEKO, few commercial programs (other than some proprietary NEC-2 extensions) provide good support for modelling thin-wire antennas, the topic of this chapter; such antennas are still very widely used indeed. For commercial programs, material is usually available to assist novice users to get started with the codes. Hence we will not describe the basic concepts of entering the geometry of the problem, including the source, and specifying parameters such as operating frequency and radiation patterns, since these vary from program to program, indeed quite often from release to release, and are usually quite well documented by the suppliers. However, in the case of NEC-2, some comments are in order.

NEC-2 is a “card driven” program, dating back to the days of “decks” of punched cards. A NEC model is described by a geometry file, usually with a. nec extension. An example is given in Fig. 5.1. If using NEC in this form, one must obtain a copy of the usual manual [1].

Throughout this book, checking convergence numerically has been continually emphasized. However, we have not discussed the more theoretical issues of whether the underlying numerical formulations are indeed convergent, in the sense that the

approximate numerical solution fN of the continuous operator equation Lf = g has the property fN → f as N → ∞. The aim of this appendix is to give a brief summary of the current status of this – which readers may be surprised to learn is far from a closed subject.

With the FDTD, the Lax equivalence theorem (discussed in Chapter 2) provides us with confidence that refining the FDTD mesh will indeed result in a convergent solution. With the FEM, work in applied mechanics has provided a rich set of convergence results – although we should note that convergence for high-frequency electromagnetics problems is often in terms of the energy norm, as discussed in Chapter 10. This is a slightly weaker statement of convergence, since the energy norm does not satisfy all the properties of the norm. Also, these proofs are usually in terms of interpolation error; as has been noted, dispersion (or pollution) error is a different problem specific to the differential equation based solvers, but can usually be controlled by adequate meshing. (Integral equation formulations using exact Green functions do not suffer from this problem of cumulative error resulting from dispersion error [1, p. 200].)

However, with the MoM, the problem has been studied somewhat less, presumably since the Green function is specific to electromagnetics. Rather surprisingly, only one form of operator has been rigorously shown to be convergent.

Introduction

The finite difference time domain method, usually referred to as the FDTD, is a particular implementation of a general class of methods known as finite difference techniques. The FDTD is so widely used in the CEM community that although finite difference methods cover a wide spectrum of complexity and accuracy, it is the FDTD which is almost always implied in CEM when finite differences are mentioned.

Finite difference methods are numerical methods in which derivatives are directly approximated by finite difference quotients. The general class of such methods is the most intuitive numerical approach, and was the first to be extensively developed by the scientific computing community. To this day, it probably remains the most universally applicable numerical technique and the one most widely used for scientific computation. As just discussed, for dynamic problems in CEM, the most popular is the FDTD method. The opening discussion in this chapter will discuss finite differences in general, before moving on to the specifics of the FDTD.

At this point, a general comment about the philosophy underlying the mathematical treatment of the computational algorithms in this book would be in order. Although we endeavor not to be “sloppy” mathematically, the emphasis in this book is in presenting well-known methods for well-known problems in CEM, rather than on the basic mathematical requirements of the methods, as one would expect to find in an applied mathematics text, for instance. An example of the type of issue which we will gloss over, at least initially, is the differentiation of discontinuous functions, which requires the generalized (weak) derivative, properly the field of functional analysis.

Introduction

Modelling stratified media is an important application of the MoM. A stratified medium is one consisting of homogeneous layers of material, each layer having different electromagnetic properties. This includes the general category of printed antennas, of which microstrip is the best known. (Microstrip technology is discussed in more detail in the next chapter.) It also brings with it the problem of dealing with dielectric materials. Central to this is the issue of the Green function for the problem. The MoM relies on an appropriate Green function as the “field propagator.” Due to its perceived complexity, the topic of stratified media is generally regarded as an advanced one, and the coverage tends to be highly theoretical, and frequently impenetrable without lengthy study. One reason for this is that historically, analysis focussed on the problem of a dipole above a dielectric half-space. There are a number of complex issues which this raises, requiring quite sophisticated analytical techniques to understand, in particular for the asymptotic cases where interesting radiation physics can be extracted. However, the analysis of a very important special case, namely the grounded single-layer microstrip line (or patch antenna), can be undertaken without undue complexity, at least for most practical cases where the substrate is relatively thin.

In this chapter, a static analysis of a microstrip transmission line is first undertaken, to demonstrate the basic principles of the spectral domain and the derivation of the Green function. Following this, the dynamic analysis is introduced, and the Sommerfeld potentials derived from first principles. Although the work in this chapter is certainly not original, being based on a synthesis of the literature – in particular [1] – the presentation in the present format does not appear to have been thus undertaken in other works to date.

Introduction

The method of moments – MoM – was one of the first numerical methods to achieve widespread acceptance in electronic engineering for the analysis of antennas and scatterers. It is generally defined as a method for reducing an integrodifferential equation to a set of linear equations. The origins of the method are old; as was already indicated in Chapter 1, some of the early work was done over a century ago. One of the widely used integral equation formulations still used for the analysis of thin wires (that due to Pocklington) was first presented in 1897 (although he used a series expansion method, rather than the modern segmentation approach). The first publications in the antenna and propagation professional literature were in the early 1960s, and some of the canonical papers (those of Harrington, Richmond, Mei and Andreasen) appeared at much the same time as Yee's paper. The specific name “method of moments” was introduced by Harrington in his early work, and the name caught on quickly; this was perhaps unfortunate, since the name has a slightly different meaning in contemporary applied mathematics. In that field, and also fields such as computational mechanics, the term “method of weighted residuals” is generally used for what has become known as the MoM in radio-frequency engineering. Another term widely used in other fields of engineering is “boundary element method”; for highly conducting structures, this term and the MoM as used in electromagnetics are synonymous.

For graduate level courses, the following are suitable exercises. Most have been tested over the years by the author in a classroom environment. The approximate time required by a student to complete the assignment is also indicated, to assist in planning. This must be treated as only a guideline; it can change significantly, perhaps by as much as a factor of two either way, depending in particular on the programming ability of students or their familiarity with a particular code. The times given are for code development from scratch, and are based on the time the author and/or typical students have spent developing the routines or models; if some existing material is made available to the students, these can be greatly reduced. A number of MATLAB files,. pre files etc. are available to assist readers.

Chapter 2

1D FDTD analysis

1. Write a program to implement the 1D FDTD analysis of a transmission line, as discussed in this chapter. In particular, repeat the results given for the single-frequency source (Fig. 2.6), and also for the wideband source (Figs. 2.20, 2.21 and 2.22). Also investigate the effects of other termination conditions, such as a matched load. [20 hours]

Chapter 3

2D FDTD analysis

1. Repeat the TEz scattering analysis discussed in this chapter using longer (in time) pulses and shorter pulses. Explain the time domain results obtained with each of these. Keep the grid size at 800 × 400 and M = 1024 so that run-times remain minutes rather than hours. [20-30 hours]

2. Modify the code to compute TMz scattering from a cylinder. Does the TMz polarization also show creeping waves? [10 hours]

3. […]

INTRODUCTION

The design of microwave circuits and systems has its origins in an era where devices and interconnect were usually too large to allow a lumped description. Furthermore, the lack of suitably detailed models and compatible computational tools forced engineers to treat systems as two-port “black boxes” with graphical methods. The most powerful of these graphical aids, the Smith chart, dates from the 1930s, an age where slide rules dominated. Although Smith charts today are perhaps less relevant as a computational aid than they were then, RF instrumentation, for example, continues to present data in Smith-chart form. It also remains true that visualizing certain operations in terms of the Smith chart can inform design intuition in rich ways that modern computational aids may unfortunately bypass. This chapter thus provides a brief history and derivation of the Smith chart, along with an explanation of why a particular set of variables (S-parameters) won out over other parameter sets (e.g., impedance or admittance) to describe microwave two-ports.

THE SMITH CHART

Introductory presentations of the Smith chart are frequently devoid of any historical context, leaving the student with the impression that it sprang forth spontaneously and fully formed. This impression, in turn, makes many students feel mentally deficient if they are unable to appreciate instantly the subtle beauty, logic, and power that the chart must “obviously” possess. The real story, though, is that the Smith chart is the result of cumulative incremental refinements spanning about a decade.