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 integro-differential 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.

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.

These sites, which include a number of commercial companies, were correct as of the date of publication — websites do change from time to time. This list is far from exhaustive, but gives a flavor of the variety of CEM products on offer, as well as the international technology base in this regard.

Ansoft Corporation

A Pittsburgh, USA-based company specializing in commercial FEM code suites, recently bought out by ANSYS (who market FEM codes for many fields of engineering application).

URL: http: // www.ansoft.com/

Antenna Magus

An antenna expert design system, interfacing seamlessly with FEKO and MWS.

URL: http://www.antennamagus.com/

Applied Computational Electromagnetics Society

An organization supporting the development, validation, and distribution of numerical EM modelling codes. Presently hosted by the University of Mississippi. Contains a number of very useful CEM links, including links to the public domain code NEC2.

URL: http: // aces.ee.olemiss.edu/

Computer Simulation Technology

Based in Darmstadt, Germany, this company specializes in commercial Finite Integration Technique (largely FDTD) code suites, in particular MWS.

URL: http: // www.cst.de/ or http: // www.cst-world.com/

COMSOL

A Swedish company, their main product is Multiphysics. The name is self-explanatory: a multi-physics FEM solver.

URL: http:// www.comsol.se/

EMSS (Electromagnetic Software and Systems)

Originally based in Stellenbosch, South Africa, this company now also has a German branch and US offices. Their main product is FEKO.

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. 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.

Introduction

The finite element method (FEM) is one of the best-known methods for the solution of partial differential equations in applied mathematics and computational mechanics. It is a method for solving a differential equation subject to certain boundary values, and in its modern form originated in the field of structural mechanics during the late 1950s; the first specific usage of the term “element” is due to no lesser a person than Courant. In common with the MoM, its historical antecedents are far older than this, in this case dating back to the nineteenth century and the variational methods first described by Lord Rayleigh. It is very widely and routinely used in structural mechanics today, as well as in computational fluid dynamics, computational thermodynamics, the numerical solution of Schrödinger's equation, field problems in general, and of course, in electromagnetics.

An historical aside — Courant and the finite element method

The finite element method as presently accepted can be credited to Courant — whom we have already encountered in the context of the Courant limit for the FDTD method. The published version of his 1942 address to the American Mathematical Society contained an appendix added after the talk, to show by example how variational methods could be put to wider use in potential theory. He used piecewise linear approximations, on a set of triangles which he called “elements” — and thus the method was born [1, p. 5].

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 20 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 used 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 εr in the range from 2—2.5, and tan δ from 0.0004—0.002. High-εr substrates such as alumina ceramics produce physically small patches, but with very limited bandwidth. Typical material properties in this case are: εr 9.7—10.3, tan δ ≈ 0.0004. For some applications, plastic foam substrates have been used.

The helix antenna discussed in the previous chapter used a new type of element to model surfaces. The theory underlying this is described in this chapter. The basic theory is quite complex, and general implementations are especially challenging. However, by choosing a suitable problem, it proves possible to undertake a limited implementation of a three-dimensional scattering problem, using a basis function defined on a triangular patch known as the RWG element. This is named after Rao, Wilton and Glisson, who introduced the element in their classic 1982 paper [1]. It represented a new type of element, the vector or edge-based element, and a closely related class of element was also under development for finite element applications at that time, although it would be some years before the connection was fully appreciated. (This will be pursued in more detail in the later coverage of the FEM.) The RWG element underlies the surface treatment of modern codes such FEKO (although not NEC), and some examples of using existing codes (in particular FEKO) to compute scattering from more general surfaces will further illustrate this.

We will also see that not only can perfectly (or highly) conducting structures be efficiently modelled using surface currents, but also homogeneous dielectric and/or magnetic regions, using fictitious equivalent currents. (We will even briefly describe how inhomogeneous bodies can be modelled using volumetric currents, but note at the outset that this is not one of the strong points of the MoM.)

Six years after the first edition was prepared, it was clear that a revised edition was in order. Continued advances in computational electromagnetics, new capabilities in commercial codes, the continual increase in computational resources, challenging new problems and a new generation of research students and engineers required new material.

Since the first edition appeared, several trends can be noted in the field. Firstly, in terms of commercial companies, there has been a significant shake-out in the market. Whilst not pretending to offer encyclopedic coverage of the large number of commercial codes available, the three codes whose application is discussed in this book, viz. CST, FEKO and HFSS, have further established themselves as amongst the market leaders in their regions of application during this period. These codes have evolved continuously, and this evolution is reflected in places in this revised edition. Secondly, whilst no fundamentally new techniques have been introduced (either in the field in general or in commercial codes in particular), a large number of additional features, improvements and enhancements have continued to extend the utility of these packages. Thirdly, after more than two decades of continual increase in CPU clock speeds in personal computers (which now dominate computational engineering), the last few years have seen clock speeds not only stagnate, but in some cases actually decrease. However, Moore's law has continued to hold sway, but in terms of multi-core and multi-processor systems. Exploiting parallelism has become essential to benefit from new hardware.

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 12. 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.

This book is designed to serve as an introduction to computational electromagnetics for radio-frequency applications. It assumes the reader has completed typical undergraduate courses in electromagnetic field theory, and has some basic knowledge of antenna design and microwave systems.

For readers in a hurry, who already know which of the techniques discussed they would like to learn more about, it is possible to go directly to the relevant chapters, but it would nonetheless be useful first to read the introductory chapter. For those in a hurry, but who need first to find out which method (or methods) to use, this chapter is essential reading.

For readers who intend working through most of the book, it would be best to work through it in the sequence presented, although the chapters on the Sommerfeld formulation and practical applications thereof could be omitted without interrupting the sequence of presentation. A more detailed outline of the book may be found in Section 1.12; this will also assist readers to locate rapidly the parts of the book of interest to them.

At the end of each chapter, a list of references linked to the chapter topic is presented, for further reading and study.

Introduction

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 user manual [1].

Introduction

RF MEMS are key enabling components within a number of applications where the level of RF power can reach significant values (e.g. above 1 W). Examples include reconfigurable matching networks, for optimised power-added efficiency (PAE) amplifiers; tuneable or switchable filters; and reconfigurable routing networks that can include switching matrices. Contrary to non-RF MEMS technologies that have already reached industrial maturity (e.g. pressure sensors accelerometers, etc.), the operational principle of RF MEMS devices is, in general, based on physical contact (be it ohmic or capacitive), between constitutive mechanical parts, and are characterised by the coexistence of biasing and RF signals. Exceptions to this include variable inductors and RF-coupled cantilever inverted-microstrip filters. This is unique to RF MEMS and becomes of increasing importance as the RF power level increases. The RF power transfer through the device strongly depends on the quality of the physical contacts, with the associated RF voltages and RF currents affecting the electromechanical action – by yielding unwanted phenomena (e.g. self-actuation, stiction and contact wearing). Moreover, other and more conventional related effects (e.g. self-heating and electromigration), as shown in Fig. 8.1, must be considered. With device sizes of approximately a few square millimetres and made of suspended movable metal and/or dielectric parts, it becomes evident that RF power-handling optimisation must be based on a comprehensive modelling approach, covering the underlying coupled thermo-electromechanical phenomena, and be driven by innovative design and technological approaches. Because of this complex scenario, the topic of RF power handling has become the subject of great interest in the research community and one of the top priorities for industrial applications. In this chapter, the state-of-the-art will be surveyed from the perspectives of the fundamental related phenomena, such as electromigration, self-heating and self-actuation. Throughout, the focus will be on capacitive switches, varactors and metal contact switches.

Introduction

Radio frequency microelectromechanical systems (RF MEMS) can offer unsurpassed RF performance over more conventional solid-state electronic devices and can help to implement advancements within a broad range of applications; from ubiquitous smart sensor networks to mobile handsets. Moreover, they can substantially reduce the size, weight and cost of reconfigurable subsystems; making this an important enabling technology for the twenty-first century.

MEMS technologies are already firmly established within high-volume commercial markets. Examples include inertial sensors/accelerometers (e.g. used in car airbag sensors, gaming accessories and mobile handsets), disk drive read/write heads, ink-jet printer nozzles, microphones and digital light projectors. In contrast, MEMS for RF applications has been relatively slow to move out of the laboratory and into commercial products. Indeed, the first RF MEMS papers started to appear over three decades ago. For example, a truly landmark paper was published on electrostatically actuated cantilever-type ohmic contact switches back in 1979 [1]. Over the past decade, however, a raft of interesting components and circuits has been demonstrated. Some of these developments have been reviewed from the perspective of enabling technologies [2], while the real founding principles have been described in some detail within the established textbooks by Santos [3], Rebeiz [4] and Varadan et al. [5]. More recent articles of noteworthy merit have also appeared on technologies, testing, reliability and applications associated with general RF MEMS [6–8].

Introduction

The reliability of RF MEMS switches constitutes a key issue for their commercial success and, thus, warrants an intense research effort. In the case of capacitive switches, the reliability is practically limited by dielectric charging. More precisely, it comes from the fact that when a capacitive RF MEMS switch is in the down position, i.e. touching the dielectric, there is a high electrical field across this dielectric. This can cause charging of the dielectric, which further results in a change in the pull-in voltage Vpi and may severely limit the functionality of the device (e.g. cause stiction).

Currently, many capacitive switches have been tested in excess of 108 cycles, operating with various voltage waveforms and with acceptable degradation. To increase the device lifetime, several excitation waveforms have been investigated; such as bipolar, two-step unipolar, etc. All these efforts, although prolonging lifetime, have not contributed significantly to our understanding of dielectric charging. The aim of this chapter is to present existing knowledge for lifetime improvement and an understanding of dielectric charging, as well as to analyze the charging processes and injection mechanisms within dielectrics. Finally, having these in mind, this chapter will attempt to examine the issue of dielectric charging from a different perspective.

Introduction

Reliability can generally be defined as ‘the probability that an item will perform a required function under stated conditions for a stated period of time’. The term ‘probability’ indicates that one deals with probabilistic models and statistical methods. The term ‘required function’ assumes that one has a specification of satisfactory operation (i.e. it includes the definition of failure). The term ‘stated conditions’ includes the total physical environment (i.e. mechanical thermal, environmental and electrical conditions). Finally, the term ‘stated period of time’ gives one the concept of required lifetime; this mostly depends on the application [1].

In general, MEMS are very robust and reliable, and several systems have successfully reached maturity. However, placing RF MEMS into commercial products is proving more challenging. At the beginning of 2000, publications on RF MEMS indicated that switch technologies were very promising, but that “the trade-off is in contact lifetime” [2], “adequate lifetime has to be demonstrated” [3], they “suffer from reliability problems” [4] and “device reliability is a key factor in the ultimate insertion of RF MEMS devices into operational systems” [5]. A survey undertaken on publications on RF MEMS in 2005 showed that only 5% of the papers reported reliability data, and most of these data were very limited. Although research is on-going and the number of institutes working on RF MEMS is constantly increasing (e.g. EU's NoE AMICOM), there is, at this moment, still no RF MEMS capacitive switch to be found in any wireless communication system. Why is it taking so long to solve the reliability problems of RF MEMS, and especially for capacitive RF MEMS switches?