3A. Duality Theorems

The next topic from homology theory which sheds light on the topological aspects of boundary value problems is that of duality theorems. Duality theorems serve three functions, namely to show

• (1) a duality between certain sets of lumped electromagnetic parameters which are conjugate in the sense of the Legendre transformation;

• (2) the relationship between the generators of the pth homology group of an n-dimensional manifold and the (n−p)-dimensional barriers which must be inserted into the manifold in order to make the pth homology group of the base manifold trivial;

• (3) a global duality between compatibility conditions on the sources in a boundary value problem and the gauge transformation or nonuniqueness of a potential.

In order to simplify ideas, the discussion is restricted to manifolds and homology calculated with coefficients in the field ℝ. The duality theorems of interest to us are formulated for orientable n-dimensional manifolds M and have the form

where Ω1 and Ω2 are manifolds having some geometric relation. In general, the geometric relationship of interest to us is that of a manifold and its boundary or the manifold and its complement. A complete development of duality theorems requires the calculus of differential forms, but we will merely state the relevant duality theorems without proof and give examples to illustrate their application. These duality theorems are a result of a nondegenerate bilinear pairing in cohomology classes and integration. The Mathematical Appendix covers details of the exterior product which leads to the necessary bilinear pairing, but for now we note that we note that the exterior product of a p-form and an n − p-form gives an n-form.

5A. Introduction

In this chapter we consider a formulation for computing eddy currents on thin conducting sheets. The problem is unique in that it can be formulated entirely by scalar functions—a magnetic scalar potential in the nonconducting region and a stream function which describes the eddy currents in the conducting sheets— once cuts for the magnetic scalar potential have been made in the nonconducting region. The goal of the present formulation is an approach via the finite elements to discretization of the equations which come about from the construction of the scalar potentials. Although a clear understanding of cuts for stream functions on orientable surfaces has been with us for over a century [Kle63] there are several open questions which are of interest to numerical analysts:

• (1) Can one make cuts for stream functions on nonorientable surfaces?

• (2) Can one systematically relate the discontinuities in the magnetic scalar potential to discontinuities in stream functions by a suitable choice of cuts?

• (3) Given a set of cuts for the stream function, can one find a set of cuts for the magnetic scalar potential whose boundaries are the given cuts?

In preceding chapters we have alluded to the existence of cuts, though we have not yet dealt with the details of an algorithm for computing cuts. The algorithm for cuts will wait for Chapter 6, but it is possible to answer the questions above. Section 5B gives affirmative answers to the first two questions by using the existence of cuts for the magnetic scalar potential to show that cuts for stream functions can be chosen to be the boundaries of the cuts for magnetic scalar potentials.

The title of this book makes clear that we are after connections between electromagnetics, computation and topology. However, connections between these three fields can mean different things to different people. For a modern engineer, computational electromagnetics is a well-defined term and topology seems to be a novel aspect. To this modern engineer, discretization methods for Maxwell's equations, finite element methods, numerical linear algebra and data structures are all part of the modern toolkit for effective design and topology seems to have taken a back seat. On the other hand, to an engineer from a half-century ago, the connection between electromagnetic theory and topology would be considered “obvious” by considering Kirchhoff's laws and circuit theory in the light of Maxwell's electromagnetic theory. To this older electrical engineer, topology would be considered part of the engineer's art with little connection to computation beyond what Maxwell and Kirchhoff would have regarded as computation. A mathematician could snicker at the two engineers and proclaim that all is trivial once one gets to the bottom of algebraic topology. Indeed the present book can be regarded as a logical consequence for computational electromagnetism of Eilenberg and Steenrod's Foundations of Algebraic Topology [ES52], Whitney's Geometric Integration Theory [Whi57] and some differential topology. Of course, this would not daunt the older engineer who accomplished his task before mathematicians and philosophers came in to lay the foundations.

The three points of view described above expose connections between pairs of each of the three fields, so it is natural to ask why it is important to put all three together in one book.

1A. Chains, Cochains and Integration

Homology theory reduces topological problems that arise in the use of the classical integral theorems of vector analysis to more easily resolved algebraic problems. Stokes’ theorem on manifolds, which may be considered the fundamental theorem of multivariable calculus, is the generalization of these classical integral theorems. To appreciate how these topological problems arise, the process of integration must be reinterpreted algebraically.

Given an n-dimensional region Ω, we will consider the set Cp( Ω) of all possible p-dimensional objects over which a p-fold integration can be performed. Here it is understood that 0 ≤ p ≤ n and that a 0-fold integration is the sum of values of a function evaluated on a finite set of points. The elements of Cp( Ω), called p-chains, start out conceptually as p-dimensional surfaces, but in order to serve their intended function they must be more than that, for in evaluating integrals it is essential to associate an orientation to a chain. Likewise the idea of an orientation is essential for defining the oriented boundary of a chain (Figure 1.1).

At the very least, then, we wish to ensure that our set of chains Cp(ω) is closed under orientation reversal: for each c ∈ Cp( Ω) there is also −c ∈ Cp( Ω).

This chapter serves two purposes. The first is to point to some applied mathematics, in particular the finite element method and corresponding numerical linear algebra, which belong in a book oriented towards computation. The second purpose is to point out the role of topology, namely simplicial homology of triangulated manifolds, in various aspects of the numerical techniques.

The chapter begins simply enough with an introduction to the finite element method for Laplace's equation in three dimensions, going from the continuum problem to the discrete problem, describing the method in its most basic terms with some indication of its practice. This leads naturally to numerical linear algebra for solving sparse positive-definite matrices which arise from the finite element method. The tie to previous chapters is that we would like to compute scalar potentials for electro- and magnetostatics. At a deeper level, there is a connection to a homology theory for the (finite element) discretized domain, so that useful tools such as exact homology sequences survive the discretization. We will see that in addition to everything discussed in the first chapters, the Euler characteristic and the long exact homology sequence are useful tools for analyzing algorithms, counting numbers of nonzero entries in the finite element matrix, and for constructing the most natural data structures.

The purpose here is to draw out some connections between the finite element method and the relevant homological tools. The reader interested in more on the finite element method is referred to [SF73].

The purpose of this chapter is to articulate the notion of a quasistatic electromagnetic system, and develop the topological aspects of the boundary value problems encountered in the analysis of quasistatic systems. The topological approach gives a new perspective on variational formulations which form the basis of finite element analysis.

2A. The Quasistatic Limit of Maxwell's Equations Maxwell's Equations.

Let S be a surface with boundary, V a volume in ℝ3, and note that ∂ denotes the boundary operator. The integral versions of Maxwell's equations are as follows.

If we let S′ = ∂V in Ampère's Law (2–3) and remember that ∂∂ = 0, then the field vector can be eliminated between Amp`ere's and Gauss’ law, (2–3) and (2–4) to reveal a statement of charge conservation:

This shows that conservation of charge is implicit in Maxwell's equations. When the surfaces and volumes S, S′, V, V′ are stationary with respect to the inertial reference frame of the field vectors, one can use the standard theorems of vector calculus to rewrite Maxwell's equations as follows.

Equations (2–5)–(2–8) are the differential versions of Maxwell's equations. Equation (2–9) is the differential version of the conservation of charge and can be obtained independently by taking the divergence of Equation (2–7) and substituting in (2–9), remembering that div curl = 0.

7A. The Paradigm Problem

The purpose of this chapter is to show how the formalism of differential forms reduces a broad class of problems in computational electromagnetics to a common form. For this class of problems, the differential complexes and orthogonal decompositions associated with differential forms make questions of existence and uniqueness of solution simple to answer in a complete way which exposes the role played by relative homology groups. When this class of problems is formulated variationally, the orthogonal decomposition theorem developed in Section MA-M generalizes certain well known interrelationships between gauge transformations and conservation laws (see [Ton68]) to include global conditions between dual cohomology groups. The orthogonal decomposition theorem can then be used to construct an alternate variational principle whose unique extremal always exists and can be used to obtain a posteriori measures of problem solvability, that is to verify if any conservation law was violated in the statement of the problem. A diagrammatic representation of the problem along the lines of [Ton72b] will be given and the role of homology groups will be reconsidered in this context. This of course will be of interest to people working in the area complementary variational principles.

In addition to the usual literature cited in the bibliography, the work of Tonti [Ton68, Ton69, Ton72b, Ton72a, Ton77], Sibner and Sibner [SS70, SS79, SS81] and [Kot82] have been particularly useful in developing the ideas presented in this chapter. From the view of computational electromagnetics, the beauty of formulating a paradigm variational problem in terms of differential forms is that the finite element method and Whitney form interpolation yield a discretization which faithfully reproduces all the essential features of the continuum problem.

The authors are long-time fans of MSRI programs and monographs, and are thrilled to be able to contribute to this series. Our relationship with MSRI started when Paul Gross was an MSRI/Hewlett-Packard postdoctoral fellow and had the good fortune of being encouraged by Silvio Levy to coauthor a monograph. Silvio was there when we needed him, and it is in no way an understatement to say that the project would never have been completed without his support.

The material of this monograph is easily traced back to our Ph.D. theses, papers we wrote, and courses taught at Boston University over the years. Our apologies to anyone who feels slighted by a minimally updated bibliography. Reflecting on how the material of this monograph evolved, we would like to thank colleagues who have played a supporting role over the decades. Among them are Alain Bossavit, Peter Caines, Roscoe Giles, Robert Hermann, Lauri Kettunen, Isaak Mayergoyz, Peter Silvester, and Gilbert Strang. The authors are also indebted to numerous people who read through all or part of the manuscript, produced numerous comments, and provided all sorts of support. In particular, Andre Nicolet, Jonathan Polimeni, and Saku Suuriniemi made an unusually thorough effort to review the draft.

Paul Gross would like to acknowledge Nick Tufillaro at Hewlett-Packard and Agilent Technologies for mentoring him throughout his post-doc at MSRI. Tim Dere graciously provided his time and expertise for illustrations. This book could not have happened without help and encouragement from Tanya.

6A. Introduction and Outline

In this chapter we consider a general finite element-based algorithm to make cuts for magnetic scalar potentials and investigate how the topological complexity of the three-dimensional region, which constitutes the domain of computation, affects the computational complexity of the algorithm. The algorithm is based on standard finite element theory with an added computation required to deal with topological constraints brought on by a scalar potential in a multiply connected region. The process of assembling the finite element matrices is also modified in the sense described at length in the previous chapter.

Regardless of the topology of the region, an algorithm can be implemented with 𝒪 (m30) time complexity and 𝒪 (m20) storage where m0 denotes the number of vertices in the finite element discretization. However, in practice this is not useful since for large meshes the cost of finding cuts would become the dominant factor in the magnetic field computation. In order to make cuts worthwhile for problems such as nonlinear or time-varying magnetostatics, or in cases of complicated topology such as braided, knotted, or linked conductor configurations, an implementation of 𝒪 (m20) time complexity and 𝒪 (m0) storage is regarded as ideal. The obstruction to ideal complexity is related to the structure of the fundamental group This chapter describes an algorithm that can be implemented with 𝒪 (m20) time complexity and 𝒪 (m4/3 0 ) storage complexity given no more topological data than that contained in the finite element connection matrix.

The advent of high speed computers has opened up a whole new range of possibilities for radio. If the RF signal can be adequately represented by a series of samples (at a rate that a computer can handle), standard operations such as mixing, filtering, signal synthesis and demodulation can all be handled as mathematical operations within the computer. Constructing systems that can handle the complex signal processing required by spread spectrum communications, radar and other more exotic RF systems is merely an exercise in computer programming. Since most of the processing will be done inside a computer, we have what is commonly termed software radio. Such radios can be extremely flexible and be instantaneously reconfigured to handle new forms of modulation and/or tasking. All we need is a suitable analogue-to-digital converter (ADC) to interface to the incoming analogue signal and a suitable digital-to-analogue converter (DAC) to produce the outgoing analogue signal. Obviously, any realistic implementation of software radio will involve many constraints, and issues such as sampling rate and quantisation error will need to be addressed. The following chapter introduces the basic ideas of digital RF techniques and their limitations.

The processing of digitised signals

We have already noted the utility of studying RF systems in terms of the complex signal exp(j2π ft). Since cos(θ) = ½ [exp(jθ) + exp(−jθ)], it is clear that the real signal s(t) = S cos(2π ft) will contain, in equal parts, contributions from frequencies f and − f.

The generation of a stable sinusoidal signal is a crucial function in most RF systems. A transmitter will amplify and suitably modulate such a signal in order to produce its required output. In the case of a receiver system, such a signal is fed into the mixer circuits for the purposes of frequency conversion and demodulation. A circuit that generates a repetitive waveform is known as an oscillator. Such circuits usually consist of an amplifier with positive feedback that causes any input, however small, to grow until limited by the non-linearities of the circuit. The feedback will need to be frequency selective in order to control the rate of waveform repetition. This frequency selection is often achieved using combinations of capacitors and inductors, but can also be achieved with resistor and capacitor combinations. In the present chapter, however, we will concentrate on feedback circuits based on capacitor/inductor combinations. We consider a variety of oscillator circuits that are suitable for RF purposes and investigate the conditions under which oscillation occurs. In addition, we consider the issue of oscillator noise since this can often pose a severe limitation upon system performance.

A particularly important class of oscillator is that for which the frequency can be controlled by a d.c. voltage. Such an oscillator is an important element in what is known as a phase locked loop. In such a system, there is a feedback loop that compares the oscillator output with a reference signal and generates a control voltage based upon their phase difference.

The following text evolved out of a series of courses on radio frequency (RF) engineering to undergraduates, postgraduates, government and industry. It was designed to meet the needs of such groups and, in particular, the needs of working engineers attempting to upgrade their skills. Thirty years ago, it appeared as if the fibre optics revolution would relegate wireless to a niche discipline, and universities accordingly downgraded their offerings in RF. In the past 10 years, however, there has been a renaissance in wireless and to a point where it is now a key technology. This has been made possible by the developments in very large-scale integration (VLSI) and CMOS technology in particular. In order to meet the manpower requirements of the wireless industry, there has been a need to upgrade the status of RF training in universities and to provide courses suitable for in-service training. The applications of wireless systems have changed greatly over the past 30 years, as has the available technology. In particular, there is a greater use of digital technologies, and antenna systems can often be of the array variety. The current text has been written with these changes in mind and there has been a culling of some traditional material that is of limited utility in the current age (graphical design methods for example). Material in the book has been carefully chosen to provide a basic training in RF and a springboard for more advanced study.

In propagating through free space, radio waves will suffer a reduction in amplitude as they spread outwards from the source. When a transmission must reach a large number of geographically dispersed receivers, such as in broadcast radio, there is little that can be done about this loss. If it is only required to reach one receiver, however, it is desirable to transmit all the energy to this one device. A structure for achieving this is known as a transmission line. Such structures will normally have a small uniform cross-section and can be constructed so that the loss along the line is extremely small. Transmission lines allow efficient and unobtrusive transmission over long distances. Two of the most common varieties of transmission line are the coaxial cable and the twin parallel wire. This chapter considers a simple lumped circuit model of such transmission lines and describes some important applications of these structures. The study of transmission lines leads very naturally to the concept of the reflection coefficient, a concept that provides an alternative description of impedances. Reflection coefficients generalise to the concept of scattering matrices which themselves provide an alternative means of describing multiport networks. The present chapter introduces the basic idea of the scattering matrix and shows howit can be applied to the design of small signal amplifiers at high frequencies.

The transmission line model

Figure 6.1 shows the construction of two important transmission lines, the coaxial cable and twin parallel wire.

Broadly speaking, radio frequency (RF) technology, or wireless as it is sometimes known, is the exploitation of electromagnetic wave phenomena in that part of the spectrum between 3 Hz and 300 GHz. It is arguably one of the most important technologies in modern society. The possibility of electromagnetic waves was first postulated by James Maxwell in 1864 and their existence was verified by Heinrich Hertz in 1887. By 1895, Guglielmo Marconi had demonstrated radio as an effective communications technology. With the development of the thermionic valve at the end of the nineteenth century, radio technology developed into a mass communication and entertainment medium. The first half of the twentieth century saw developments such as radar and television, which further extended the scope of this technology. In the second half of the twentieth century, major breakthroughs came with the development of semiconductor devices and integrated circuits. These advances made possible the extremely compact and portable communications devices that resulted in the mobile communications revolution. The size of the electronics continues to fall and, as a consequence, whole new areas have opened up. In particular, spread spectrum communications at gigahertz frequencies are increasingly used to replace cabling and other systems that provide local connectivity.

The purpose of this text is to introduce the important ideas and techniques of radio technology. It is assumed that the reader has a basic grounding in electromagnetic theory and electronics.