Advances in Electronics and Electron Physics, Volume 82

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 82

EDITOR-IN-CHIEF

PETER W. HAWKES Centre National de la Recherche ScientiJique Toulouse, France

ASSOCIATE EDITOR

BENJAMIN KAZAN Xerox Corporation Palo Alto Research Center Palo Alto, California

Advances in

Electronics and Electron Physics EDITEDBY PETER W. HAWKES C E M ESILaboratoire d 'Optique Electronique du Centre National de la Recherche Scienrijque Toulouse, France

VOLUME 82

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers Boston San Diego New York London Sydney Tokyo Toronto

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CONTENTS

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CONTRIBUTORS . . , PREFACE. . . . . . .

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vii ix

1

CAD in Electromagnetism OSZKAR BIROAND K. R. RICHTER

............ ................. ............................. 111. Eddy Current Fields . . . . . . . . . . . . . . . , , . . . . . . . . . IV. Waveguides and Cavities. . . . . . . . . . . . . . . . . . . . . . . . V. Galerkin’s Method . . . . . . . . . . . . . . . , . . . . . . . . . . . VI. Application of the Finite Element Method . . . . . . . . . . . . . Acknowledgments.. . . . . . . . . . . . . . . . . . . . . . . . . . . References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Introduction. .

1

..

4

11. Static Fields

Introduction . . . . . . . . . . . . . . Signal Processing in Speech Coding Speech Coding Systems. . . . . . . . Future Research Directions . . . . . Acknowledgments. . . . . . . . . . . References . . . . . . . . . . . . . . .

........ ......... , . . . . . . . . ......... ......... ......... .

95

97

Speech Coding VLADIMIR CUPERMAN

I. 11. 111. IV.

20 38 43 63 95

... . , . .,. ... ... ...

..... ...,. ..... ..... ..... .....

97 110 147 188 189 189

Bandgap Narrowing and Its Effects on the Properties of 197 Moderately and Heavily Doped Germanium and Silicon SURESH C. JAIN,R. P. MERTENS, A N D K. J. VANOVERSTRAETEN I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11. Calculations of BGN in n-Type Silicon and n-Type Germanium . . . . . . . . . . . . . . . . . . . . . . . . . 111. Impurity Concentration Fluctuations and Band Tails . IV. EfTect of Bandgap Narrowing on Optical Properties. .

.

197

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203 232 244

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vi

CONTENTS

V . Summary of Important Results . . . . . . . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Rectangular Patch Microstrip Radiator-Solution by Singularity Adapted Moment Method E . LEVINE.H . MATZNER. AND S . SHTRIKMAN I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. The Spectral Domain Presentation . . . . . . . . . . . . . . . . . . I11. The Moment-Method Formulation . . . . . . . . . . . . . . . . . . IV. Two-Dimensional Solution . . . . . . . . . . . . . . . . . . . . . . V . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A . Far fields in Two Polarizations . . . . . . . . . . . . Appendix B. Evaluation of Typical Matrix Elements . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some Recent Advances in Multigrid Methods JANMANDEL I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. The Fundamental Multigrid Algorithim . . . . . . . . . . . . . . I11. Preconditioning by Multigrid . . . . . . . . . . . . . . . . . . . . IV . Methods Based on Space Decomposition . . . . . . . . . . . . . V . Multigrid in Elasticity . . . . . . . . . . . . . . . . . . . . . . . . VI . Multigrid for Mixed Problems . . . . . . . . . . . . . . . . . . . VII . Multigrid for High Order and Spectral Methods . . . . . . . . . VIII . Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . IX . Multigrid and Parallel Computing . . . . . . . . . . . . . . . . . X . Some Other Multigrid Developments . . . . . . . . . . . . . . . XI . Multigrid Software . . . . . . . . . . . . . . . . . . . . . . . . . . AppendixA . PLTMG . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. MADPACK . . . . . . . . . . . . . . . . . . . . . . Appendix C . MUDPACK . . . . . . . . . . . . . . . . . . . . . . AppendixD. MGDl . . . . . . . . . . . . . . . . . . . . . . . . . Appendix E. MGOO. . . . . . . . . . . . . . . . . . . . . . . . . . Appendix F. AMG . . . . . . . . . . . . . . . . . . . . . . . . . . AppendixG. BOXMG . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

268 270 270 272

277

277 279 286 302 317 317 319 324

327 328 329 347 349 357 359 360 363 364 365 365 365 366 366 367 368 368 368 368 379

CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors' contributions begin.

OSZKAR BIRO (I), Institute for Fundamentals and Theory in Electrical Engineering, Graz University of Technology, Kopernikusgasse 24, A-8010 Graz, Austria

VLADIMIRCUPERMAN (97),Communication Sciences Laboratory, School of Engineering, Simon Fraser University, Burnaby, BC, Canada V5A IS6 SURESH C. JAIN(197),Delft Institute of Microelectronics, TU Delft, Postbus 5053,2600 GB Delft, The Netherlands

E. LEVINE(277), ELTA Electronics Industries Ltd., PO Box 330, Ashdod, 77102 Israel JAN MANDEL(327), Computational Mathematics Group, University of Colorado at Denver, Denver, C0802O4

H. MATZNER(277), Department of Electronics, Weizmann Institute of Science, PO Box 26, Rehovot, 76100 Israel R. P. MERTENS (197),IMEC, Kapeldreef 75,B-3030 Leuven, Belgium K. R. RICHTER(l), Institute for Fundamentals and Theory in Electrical Engineering, Graz University of Technology, Kopernikusgasse 24, A-801 0 Graz, Austria S. SHTRIKMAN (277), Department of Electronics, Weizmann Institute of Science, PO Box 26, Rehovot, 76100 lsrael

R. J.

VAN

OVERSTRAETEN (197), IMEC, Kapeldreef 75, B-3030 Leuven,

Belgium

vii

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PREFACE Computational methods, coding, semiconductors, and antenna arrays are the subjects of this latest volume of Advances in Electronics und Electron Physics, in which we try to maintain the traditional broad coverage while highlighting particular themes in occasional topical volumes. The first of the contributions on computation is concerned with computeraided design in electromagnetism, where the widespread availability of ever faster and better computing facilities has revolutionized the field. 0. Biro leads us through the various methods that are being used or developed for studying static fields, eddy current fields, and cavities. The amount of research on these problems is enormous, as the annual COMPUMAG Conference proceedings show, and this survey of developments is very useful. The other chapter on computation, by J. Mandel, is concerned with multigrid methods, the literature of which is growing explosively. The author concentrates on recent developments in a wide range of fields. After examining the mathematical fundamentals, the special needs of elasticity, mixed problems, and spectral studies are explored in detail. There are sections on eigenvalue problems and on parallelization; a useful survey of available software concludes this up-to-date review of an important class of methods. Speech coding is intrinsically of great interest and also of considerable social and economic importance. The field is too vast to be covered usefully in a single review; the chapter by V. Cuperman is mainly concerned with a new class of speech coding systems that has emerged in the past few years, known as “analysis-by-synthesis.” The author does, however, devote considerable space to the many other types of coding that are employed, including vector quantization. The subject has important ramifications for society, not only for telephony but also for vocal recognition and vocal synthesis for the handicapped. Rectangular microstrip patches are specialized radiators used in printed antenna arrays. Although they are difficult to analyze, E. Levine, H. Matzner, and S. Shtrikman offer a means of solving the design problems associated with these devices, particularly in accelerating the convergence of the calculation. Finally, we have a long chapter on doping in semiconductors, by S. C. Jain, R. P. Mertens, and R. J. van Overstraeten. The band gap of silicon and germanium is considerably narrowed by heavy doping, and the authors examine this in great detail. After critically examining the various theoretical approaches, they consider fluctuations in impurity concentration and their ix

PREFACE

X

effect on the optical properties. This survey brings together a great deal of scattered information on these important questions. It remains only for me to thank all the authors for taking such trouble over their contributions and to list articles promised for forthcoming chapters in this series.

FORTHCOMING ARTICLES Neural Networks and Image Processing Image Processing with Signal-Dependent Noise Residual Vector Quantizers with Jointly Optimized Codebooks Parallel Detection Ion Microscopy Magnetic Reconnection Vacuum Microelectronic Devices Sampling Theory ODE Methods Nanometre-Scale Electron Beam Lithography The Artificial Visual System Concept Dynamic RAM Technology in GaAs Corrected Lenses for Charged Particles Foundations and Applications of Lattice Transforms in Image Processing The Development of Electron Microscopy in Italy The Study of Dynamic Phenomena in Solids Using Field Emission Invariant Pattern Representations and Lie Group Theory Amorphous Semiconductors Median Filters Bayesian Image Analysis Magnetic Force Microscopy Theory of Morphological Operators Kalman Filtering and Navigation

J. B. Abbiss and M. A. Fiddy H. H. Arsenault C. F. Barnes and R. L. Frost P. E. Batson M. T. Bernius A. Bratenahl and P. J. Baum I. Brodie and C. A. Spindt J. L. Brown J. C. Butcher Z. W. Chen J. M. Coggins J. A. Cooper R. L. Dalglish J. L. Davidson G. Donelli

M. Drechsler M. Ferraro

W.Fuhs N. C. Gallagher and E. Coyle S. and D. Geman U. Hartmann H. J. A. M. Heijmans H. J. Hotop

xi

PREFACE

3-D Display

Applications of Speech Recognition Technology Spin-Polarized SEM Finite Topology and Image Analysis Expert Systems for Image Processing The Intertwining of Abstract Algebra and Structured Estimation Theory Electronic Tools in Parapsychology Image Formation in STEM Phase-Space Treatment of Photon Beams Low Voltage SEM 2-Contrast in Materials Science Languages for Vector Computers Electron Scattering and Nuclear Structure Edge Detection Electrostatic Lenses Scientific Work of Reinhold Rudenberg Metaplectic Methods and Image Processing X-ray Microscopy Accelerator Mass Spectroscopy Applications of Mathematical Morphology Focus-Deflection Systems and Their Applications Echographic Image Processing The Suprenum Project Knowledge-Based Vision Electron Gun Optics Spin-Polarized SEM Cathode-ray Tube Projection TV Systems

n-Beam Dynamical Calculations Thin-film Cathodoluminescent Phosphors Parallel Imaging Processing Methodologies Diode-Controlled Liquid-Crystal Display Panels Parasitic Aberrations and Machining Tolerances Group Theory in Electron Optics

D. P. Huijsmans and G. J. Jense H. R. Kirby K. Koike V. Kovalevsky T. Matsuyama S. D. Morgera

R. L. Morris C. Mory and C. Colliex G. Nemes J. Pawley S. J. Pennycook R. H. Perrot G. A. Peterson M.Petrou F. H. Read and I. W. Drummond H.G. Rudenberg W. Schempp G. Schmahl J. P. F. Sellschop J. Serra T. Soma J. M. Thijssen 0. Trottenberg J. K. Tsotsos Y. Uchikawa T. R. van Zandt and R. Browning L. Vriens, T. G . Spanjer and R. Raue K. Watanabe A. M. Wittenberg S. Yalamanchili Z. Yaniv M. I. Yavor Yu Li

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ADVANCES IN ELECTRONICS A N D HLECIRON PHYSICS. VOL 82

CAD in Electromagnetism OSZKAR BIRO AND K . R . RICHTER Institute for Fundamentals and Theory in Electrical Engineering Graz University of Technology. Graz. Austria

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Static Fields . . . . . . . . . . . . . . . . . . . . . . . . . . A. Differential Equations and Boundary Conditions of Static Fields . . . . . . B. Scalar Potential Descriptions of Static Fields . . . . . . . . . . . . . C. Vector Potential Descriptions of Static Fields . . . . . . . . . . . . . I11. Eddy Current Fields . . . . . . . . . . . . . . . . . . . . . . . A. Differential Equations, Boundary and Interface Conditions of Eddy Current Fields B. Potential Descriptions of Eddy Current Fields . . . . . . . . . . . . . C. Coupling Eddy Current and Static Magnetic Fields . . . . . . . . . . . IV . Waveguides and Cavities . . . . . . . . . . . . . . . . . . . . . A. Differential Equations and Boundary Conditions of Waveguides and Cavities . . B. Potential Descriptions of Waveguides and Cavities . . . . . . . . . . . V . Galerkin’s Method . . . . . . . . . . . . . . . . . . . . . . . A . Weak Formulations . . . . . . . . . . . . . . . . . . . . . . B. General Description of Galerkin’s Method . . . . . . . . . . . . . . C. Application of Galerkin’s Method to Potential Formulations . . . . . . . V1. Application of the Finite Element Method . . . . . . . . . . . . . . . A . A summary of the Finite Element Method . . . . . . . . . . . . . . B. Analysis of an Iron Cored Choke Coil . . . . . . . . . . . . . . . C. Analysis of a Plate Beneath a Coil . . . . . . . . . . . . . . . . . D . Analysis of Transient Eddy Currents in a Conducting Brick . . . . . . . . E . Analysis of Anisotropic Waveguides . . . . . . . . . . . . . . . . F. Analysis of a Dielectric Loaded Cavity . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

i 4 4 9 15 20 21 25 29 38 39 40 43 43 46 49 63 64 68 75 80 85 92 95 95

I . INTRODUCTION The rapid development of computer hardware in recent years provides vast resources for the design of electromagnetic devices. Comparable progress in CAD software has to follow in order to exploit these possibilities . The advent of CAD methods revolutionizes the design procedure by bringing the analysis of the electromagnetic field in the devices to the foreground. which provides an insight into their operation far superior to that obtainable by traditional network considerations .

.

Copyright i( 19Yl by Academic Press Inc All rights 01 reproduction rn any form reserved ISBN~ - I z - o I ~ ~ R ~ - ~

2

OSZKAR BlRO AND K. R. RICHTER

The field analysis methods of two-dimensional models can be regarded as established, successful CAD software packages are commercially available. The situation, however, is different with three-dimensional models: their use is by far less widespread. The reason is no longer the considerably higher memory and CPU-time requirement, since this is not essential in an age of cheap memory and fast computers. The main problem is the scarcity of robust and reliable numerical field analysis methods. The present work attempts to make up for this shortage. A largely unified field analysis approach is proposed for various types of problems: static fields, eddy current fields and general electromagnetic fields. The uniformity is attained by using similar potential functions to describe the field in each particular case. This allows for a great degree of generality with respect to material properties, since the continuity of the potentials is sufficient to ensure the satisfaction of the interface conditions on surfaces where the material characteristics change abruptly. The Coulomb gauge is invariably applied to ensure the uniqueness of the vector potentials, which are necessary in three-dimensional analysis. This results in great numerical stability and in the lack of any spurious solutions when the finite element method is employed. The robustness of the methods is shown by some illustrative examples. It is the feeling of the author that the analysis methods presented in this work can serve as a basis for some general purpose three-dimensional CAD software packages in the near future. The analysis of electromagnetic field problems is based on Maxwell's equations (Maxwell, 1864). These are partial differential equations stating the relationships between the field vectors E (electric field intensity), D (electric flux density), H (magnetic field intensity), B (magnetic flux density) and J (electric current density) as well as the scalar p (electric charge density):

V

x H =J

+-,aD at aB

VXE=--

at

V*B=O,

(3)

V*D=p.

(4)

Further relationships between the field quantities are defined by the constitutive equations

B = pH, D = EE, J = oE.

(5)

(6) (7)

CAD IN ELECTROMAGNETISM

3

The permeability p, the permittivity E and the conductivity CT describe the properties of the medium, and in the simplest homogeneous, isotropic and linear case they are constant scalar values independent of the fields. In the general case, however, they may vary in space when the medium is inhomogeneous, they may be tensor quantities describing an anisotropic medium, and they may even be field dependent resulting in nonlinearity. The electromagnetic field can be completely represented in terms of two of the field vectors (e.g., E and H), the remaining quantities are obtainable from the constitutive relationships in Eqs. ( 5 ) to (7). This way of representation involves six scalar functions (the three components of each field vector), a description that proves to be redundant. Indeed, the introduction of so-called potential functions allows the electromagnetic field to be represented in terms of a lower number of scalar functions. This potential representation is of further advantage because the potential functions can always be chosen to be continuous, while the field vectors are discontinuous whenever the material properties change abruptly. In particular, the continuity of the representative functions is profitable in a numerical context and is, therefore, of great importance in CAD methods. The complete set of the Maxwell equations (1) to (4) represent the fairly general case of electromagnetic waves in nonlinear, anisotropic media. Most practical problems do not warrant a treatment of such an extensive scope, and some simplifying assumptions may be applied. Three particular sets of simplifications will be treated in the present work, which, however, cover a wide range of problems of practical importance. The equations of static fields are arrived at by neglecting any variation in time. This case will be treated in Section 11, with which inhomogeneous, nonlinear media will be allowed. Static approximations are useful in most high voltage applications, investigations of the static behaviour of electrical machines, and many transmission line problems. When induced conductive currents are considered with the neglected displacement currents, the equations of eddy current fields are obtained. This case of the quasi-stationary limit will be discussed in Section 111, also with inhomogeneous, nonlinear material properties considered. The quasistationary approximation is satisfactory in all power frequency applications involving metallic structures. Especially, the analysis of losses in electrical machines, some nondestructive testing problems and also the prediction of the transient behaviour of metallic components in fusion reactors require eddy current fields to be computed. The third special case considered here is constituted by electromagnetic waves confined to more or less closed regions filled with a linear but possibly anisotropic, inhomogeneous medium. In contrast to the previous cases requiring the solution of differential equations, this problem involves the

4

OSZKAR BIRO AND K . R. RICHTER

finding of particular frequencies at which nonzero fields may exist. The corresponding equations will be treated in Section IV. They are useful in analyzing waveguides, cavities and resonators at high frequencies. The relevant potential functions will be introduced in each of the above cases, and the differential equations and boundary conditions will be written for them. Special attention will be given to the uniqueness of the potentials, since this is of extreme importance from a numerical point of view. The numerical solution of the differential equations in electromagnetism constitutes the central problem of CAD. Since the use of computers is indispensable in solving this analysis problem, it is usually referred to as Computer Aided Analysis. The first step in the numerical solution of the partial differential equations discussed in the first four sections is to reduce them to a system of algebraic equations, of ordinary differential equations or to a matrix eigenvalue problem. Several techniques are known for the execution of this task, the most widespread ones being the method of finite differences (Mitchell and Griffiths, 1980),variational principles (Mikhlin, 1964)and Galerkin’s method (Galerkin, 1915). Since, in the opinion of the author, the last provides the most flexible approach and its scope covers all topics treated in the present work, Galerkin’s method is singled out and its brief general description as well as its application to the partial differential equations in electromagnetism is presented in Section V. The most powerful method for the numerical realization of Galerkin techniques is the Method of Finite Elements. Omitting its extensive treatment, which can be found in many excellent books (e.g., Zienkiewicz, 1977), the application of a special form of nodal finite elements is presented by means of several examples of Computer Aided Analysis in Section VI. 11. STATIC FIELDS The neglect of the time variation of the field quantities eliminates the interdependence between the electric and the magnetic field. The models so derived describe the electrostatic field, the magnetostatic field and the static current field. In Section II.A, a summary of the differential equations and typical boundary conditions is given in these cases. Section I1.B is devoted to the description of static fields by means of scalar potentials, while vector potential formulations are introduced in Section 1I.C.

A. Diferential Equations and Boundary Conditions of Static Fields The differential equations derived from the Maxwell equations under the assumption of static conditions are given in this subsection for the electro-

5

CAD IN ELECTROMAGNETISM

static, the magnetostatic and static current fields. The typical boundary conditions are also presented and discussed. The region of interest where the fields are to be computed is invariably denoted by R bounded by the closed surface r. This bounding surface is subdivided into disjunct sections with different types of boundary conditions prescribed on them. 1. Static Electric Field

Under static conditions, the Maxwell equations (2) and (4) along with the constitutive relationship of Eq. (6) yield the description of the static electric field: VxE=O, (8) V - D = y,

inn,

(9)

D = EE.

(10)

The solution of Eqs. (8)-( 10) is sought in R. In writing the boundary conditions, n denotes the outer normal of the relevant surface with respect to the region Q. Two types of boundary conditions are of practical importance. On the part rEof the surface r bounding the domain Q, the tangential component of the electric field intensity is known and expressed as a given magnetic surface current density: E x n = K,, on rE. (1 1) E is constituted by metallic electrodes and/or planes In most cases, the surface r of symmetry with the electric field intensity normal to them (in both cases there are no magnetic surface currents; K, = 0). The boundary condition (1 1) may also model electric double layers (magnetic surface currents; K, # 0).If r, is made up of nE disjunct sections (as in the case when several electrodes are present), then further (nE - 1) integrals have to be specified: either integrals of D over (nE- 1) surface sections (charges) or integrals of E along lines connecting one section of r, to the remaining (n, - 1) sections (voltages):

61,

D e n d r = Q , , i = 1 , 2,..., a,-1 E.dl

=

U , , i = 1,2 ,..., nE - 1,

or (12)

JCE1

denotes the ith section of r E and CE,is the curve connecting the where rEi surface rE, to the section r,,, (Fig. 1). O n the part r, of the bounding surface r (with r, + r, = r),the normal component of the electric flux density is given as a surface charge density D . n = - Ps,

on

r,.

(1 3)

OSZKAR BIRO A N D K. R. RICHTER

6

/D. ndT=Q3

'

€2

FIG.1. The scheme of an electrostatic problem.

The surface rD is often made up of surface sections with the normal component of the flux density being zero (usually symmetry planes where the surface charge density is zero, ps = 0). It may also model known surface charge densities. A typical arrangement of the electrostatic problem is shown in Fig. 1. The solution of Eqs. (8)-( 10)with the boundary conditions (1I)-( 13) yields unique E or D. Indeed, let us assume that two solutions exist. Consequently, their difference E''), D'O) satisfies

V

x E'O' = 0,

(14)

V D(O)= 0,

(15)

in !2, D(0)= EE(0), E(O) x n = 0,

on r,,

D ( O ) . n d T = O , i = 1 , 2 y . . . y n E -1 S , i

LE,-

(16)

E ' o ' - d l = O , i = 1,2 ,...,nE-1,

(17)

or (18)

D(O) n = 0, on rD. (19) The difference field E(O), D'O) will be shown to be zero, a fact proving the uniqueness of the solution. The vanishing of the difference field will be

7

CAD IN ELECTROMAGNETISM

demonstrated by showing that the quantity

is zero. Since the permittivity E is positive in Eq. (16),this does in fact imply that the difference field is zero. In view of Eq. (14), E'O) can be written as

E(0) = -Vv("J.

(21)

Now, using some vector algebra, W(O)can be rewritten as

-I

.

.

y(0)V D(O)dQ-

Vp"0) D(O)dQ=

In

i

.

V(O)D(O) n dr.

(22)

The volume integral on the right-hand side is zero in view of Eq. (15), and Eq. (19) implies that the surface integral vanishes on r,. According to the boundary condition (17),V'O' is constant on ,-l . Its value can be chosen to be zero on the surface r,,,, so we end up with

where VI0' is the constant value of V'O' on the surface section yl0) =

I,. -

E'O' dl,

rEi. Obviously (24)

so the conditions (18) do in fact imply that W'O' is zero, i.e., the solution of Eqs. (8)-( 10) with the boundary conditions (1 1)-( 13) is unique. Note that no assumption of linearity has been made for the constitutive relationship (lo), i.e., the proof holds for the nonlinear case, too. 2. Static Magnetic Field Neglecting the time derivative term in Eq. (1) and further writing Eqs. (3) and ( 5 ) , the equations of the magnetostatic field are obtained as

VxH=J,

-

V B = 0, B = pH,

(25)

in R,

(26) (27)

In ferromagnetic materials the permeability p is strongly dependent upon the field vectors making the magnetostatic problem a nonlinear one. The boundary conditions of practical significance are again of two types. On the part r, of the bounding surface r,the tangential component of the magnetic field intensity is assumed to be known and given as a surface current

OSZKAR BIRO AND K.R. RICHTER

8 density

H x n=K, onr,. The surface rHis in most cases constituted by boundaries of highly (infinitely) permeable iron parts or symmetry planes with the tangential component of H vanishing (K = 0). Known surface currents are also modeled by this boundary condition. Similarly to the electrostatic case, further integrals need to be specified if H r consists of several disjunct parts. The number of necessary specifications is (nH - 1) if there are n H sections; either integrals of B over (nH - 1) surface sections (fluxes) or integrals of H along lines connecting (nH - 1) of the sections of r H to a particular section (magnetic voltages) must be gven: B * d S = Y ; , i = 1 , 2,..., n H - 1

or

I H t

H dl = Umi,i = 1,2,. . . , n H- 1,

(29)

where the meaning of r H i and CHiis similar to that of rEi and CEiin the static electric case. On the rest rB of the bounding surface r, the normal component of the magnetic flux density is assumed to be given as a magnetic surface charge density

Ban = -Pmsr onrB(30) This boundary condition can be used for modeling surfaces parallel to flux lines (pmS= 0), or known flux distributions generated by fictitious magnetic surface charges. Figure 2 shows a schematic layout of a typical magnetostatic problem. It can be shown in a way similar to the electrostatic case that the magnetic field vectors H or B can be obtained in a unique way as the solutions of Eqs. (25)-(27) with the boundary conditions (28)-(30). 3. Static Current Field By taking the divergence of Eq. (25), a continuity equation stating the source-free property of the current density in the static case is obtained. Supplementing this with Eqs. (8) and (7), the description of a static current field is obtained that is totally analogous to that of the charge-free static electric field with J written instead of D and Q instead of E :

VxE=O, V.J=O, J = aE.

(31)

in R,

(32) (33)

9

CAD IN ELECTROMAGNETISM

rH:Hxn=K

re: B .n=-P ms

rH 1

FIG.2. The scheme of a magnetostatic problem.

In view of the analogy mentioned, the typical boundary conditions are briefly mentioned only. On r,, the boundary condition is identical with Eq. (1 1): Ex n

= K,,

r,,

(34)

I,,

(35)

on

with the necessary supplementary conditions

6,1.

J .dS = I i , i

=

1,2,..., nE - 1 or

E e d l = Ui, i = 1,2,. ..,nE - I .

The surface integrals here specify the total current leaving the relevant sections of r,. On r, (as before, r, + r, = r),the normal component of the current density is given: J . n = -J,,

on

r,.

(36)

A static current problem is schematically drawn in Fig. 3. The uniqueness of E or J as the solution of Eqs. (31)-(33) with the boundary conditions (34)-(36) can be shown as before. B. Scalar Potential Descriptions of Static Fields

The most economical way of describing static fields is by means of scalar potentials. This is directly possible in the case of electrostatic and static current fields since the nonrotational property of E allows it to be derived as

10

OSZKAR BIRO A N D K . R. RICHTER

rE: Exn=K, r,: J. n=-J,

‘El

FIG.3. The scheme of a static current problem.

the gradient of a scalar. This will be expounded in Section II.B.1 for the electrostatic case, a discussion that is directly valid for the static current field in view of the existing analogy. In the magnetostatic case it is necessary to split the field into a known rotational and an unknown nonrotational part in order to be able to introduce a scalar potential. Section II.B.2 is devoted to the treatment of this problem. 1. The Electric Scalar Potential

The differential equations and boundary conditions on the electric scalar potential will be presented for the case of the electrostatic field only. As it is well known, the electric field intensity can be written as the gradient of a scalar V in view of Eq. (8):

E = -VV.

(37) This formulation ensures the satisfaction of Eq. (8) and, using the constitutive relationship (lo), Eq. (9) yields a second order differential equation for V : - V (e VV) = p,

in a,

(38)

a generalized Laplace-Poisson equation. The boundary condition (1 1) allows the specification of V on the surface r, if the additional conditions (12) refer to the voltages of the surface sections.

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11

Indeed, on each disjunct section rEi of rE, a scalar function VOican be defined as follows. Let us choose an arbitrary point P, on each surface section rEir and specify

VOi= 0, In each other point P on

at 4 .

(39)

rEi,VOiis defined as

where C, is an arbitrary curve lying in rEi and connecting P to P,. The value of the integral is independent of the choice of the curve; otherwise the boundary condition ( 1 1 ) would contradict the nonrotational property of E. On electrodes and planes of symmetry where K, = 0 (i.e., in most practical cases) VOi= 0. The negative gradients of the functions VOievidently satisfy the to boundary condition (1 I), so VOidefines the electric scalar potential V on rEi the extent of a constant value b: V

=

Voi+ 6,

on

rEi.

(41)

The constant V,, can be chosen to be zero, thus specifying the electric scalar The rest of the (nE - 1) values potential to vanish on the surface section I-,,. are given by the voltages specified in Eq. (12). Indeed, the line integrals along the curves CEiare the differences of the potential values at the endpoints of the curves. In summary, the boundary condition (1 1) is formulated for the electric scalar potential V as V = U,,

onr,,

(42)

where the known function U, equals the the sum of the function VOiand of the voltage q on the ith section of I-,. The condition (42) is a Dirichlet boundary condition. The boundary condition (1 3 ) constitutes a Neumann boundary condition for V: n * EVV= ps,

on

rD.

(43)

In summary, the electric scalar potential satisfies the differential equation (38) with the boundary conditions (42) and (43). The uniqueness of V as the solution of this boundary value problem is assured if V is specified at an arbitrary point in Q, since the uniquely defined electric field intensity E determines V up to a constant. This specification is evident if rEis present (i.e., Dirichlet boundary conditions are given). In the case when rD = r, i.e., Neumann boundary conditions only are specified, the value of V should be set to zero at an arbitrary point in 0.

12

OSZKAR BIRO AND K. R. RICHTER

2. Magnetic Scalar Potentials The static magnetic field intensity cannot be derived as the gradient of a scalar since its curl is not zero. It is, however, possible to split it into two parts as

H

= H,

+ HM,

(44)

where H, is constructed so that it satisfies V x H, = J.

(45)

The remaining part H M is then nonrotational and can therefore be written as the negative gradient of a magnetic scalar potential 0:

H, = -V@. (46) The scalar function 0 is called the reduced magnetic scalar potential, since it describes a part of the magnetic field only. There are many possibilities for the construction of H, from the known current density J. The most attractive one seems to be to choose H, as the magnetic field due to J in free space. This field is calculated in a well known way by means of Biot-Savart’s Law:

where r is the vector pointing from the source point to the field point. The numerical execution of the integrals in Eq. (47) is well established for all practically important distributions of the current density, so H, can be regarded as known. The above formulation evidently satisfies Eq. (25). The differential equation for the reduced scalar potential is provided by Eq. (26) with the constitutive relationship (27) taken into account:

V * ( p V@)

=V

- pH,,

in Q.

(48) Similarly to the electrostatic case, this is a generalized Laplace-Poisson equation. The boundary conditions (28)-(30) can be formulated for the magnetic scalar potential exactly as in the electrostatic case. The potential CD satisfiesthe Dirichlet boundary condition 0=0,,, onr,, (49) where the function CDo can be determined by integrating the function n x (K - H, x n) and taking the given magnetic voltages into account. It is again assumed that the conditions specifying the magnetic voltages in Eq. (29) are in effect.

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13

The boundary condition (30) is a Neumann boundary condition for the reduced magnetic scalar potential 0 :

The uniqueness of Q, an also be ensured by setting it to zero at some point. This is only necessary if no Dirichlet boundary conditions (49) are specified. The above formulation of the static magnetic field is satisfactory if the permeability of the medium is not very high, and so the magnitude of the resultant magnetic field H does not much differ from that of H,. Otherwise, IHI

(,Aat + v at

+ rnc*

nw d r

for any function w,

= 0,

(328)

V.(pv+)~dR-

( n * p V + -pms)wdr JrB*

(-n*. pV$

+ n A .V x A)wdT = 0,

for any function w.

+ j r n c q f rAJl

(329) Again, Galerkin's equations can be written by replacing the potentials with the expansions (316)-(318) and the functions w and w with the functions fi andf,, respectively. In order to achieve symmetry, use is made of the Green's identities (293), (303),(304) and (315) as well as of the fact that the expansion functions satisfy the Dirichlet boundary conditions (3 19)-(321). Further, the integral coupling the magnetic vector potential and the magnetic scalar potential in Eq. (327), i.e., the second term in the fifth surface integral, is transformed by the identity

where CAJlis the curve bounding the surface rnCJl + F A # , which is the interface between the regions with A and If this surface is closed, the curve integral does not arise; otherwise any part of the curve CA*bounds either a surface with n x fi zero (r,or r B A , see Eq. (319))or one where the approximating function +(") satisfies the Dirichlet boundary condition (181) (on r H J l ) . Denoting this and using n* = nA,the identity (330) becomes latter curve by CHAJ,

+.

s

rncJl + tiIy.

(- VI,P

x n9). fi d r =

s

+(")V x f i . n,dr rncJl + TAIL.

+ JcHA* +ofi

*

dl.

Finally, Galerkin's equations have the following form:

(331)

57

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s

(v x A(”).n,)i

r n c , + rAQ

dr-

I,, v q - ~v i +

dR = -

LL

i = n,

jrB+

pmsfid r ,

1, n2 + 2,...,n. (334)

This set of ordinary differential equations can be summarized in the matrix form

[‘oA AAA 0

;j[;]+FiA;];[;]=p]? A,,

BAA” BA” :

(335)

where the elements of the matrices can be taken from Eqs. (332)-(334). Both the matrix A and the matrix Bare readily seen to be symmetric. However, since the diagonal elements of A,, are negative, the matrix A is not positive definite even if the zero rows and columns are disregarded, whereas the matrix B does have this property. b. Galerkin ’s Equations for the T,$-A-$ Formulation The differential equations, boundary and interface conditions for the T,$-A-$ formulation have been written in Eqs. (203) to (227) in Section III.C.4. The symmetry of the operator L, turns out to be ensured only if Galerkin’s method is applied to the time derivatives of the differential equations (204), (205) and (206), of the boundary conditions (208), (212), (214), (216), and (227) and of the interface conditions (219), (220),(222),(224) and (225). The Green’s identities for the vector potentials T and A in the differential equations (203) and (205) are given for the magnetic vector potential A in Eqs. (303) and (304). For the current vector potential T they are completely analogous. The relevant identity for the magnetic scalar potential $ in the differential equation (206) has been written in Eq. (293). For the time derivative of the differential equation (204), the following identity will

58

OSZKAR BIRO AND K. R. RICHTER

be used:

The expansions used for the approximation of the potentials satisfy the Dirichlet boundary conditions

2 &fk,

T x T(")= TD +

(337)

k=l

2

A z A(")= AD +

Akfk.

(339)

k=n2 t 1

This is achieved again by using expansion functions that obey the homogeneous Dirichlet boundary conditions n x fk = 0,

on

-

fk n = 0,

rH,,

on I-,,

r&,and r,,,,, r H A , rneA

and

rAs,

(3401 (341)

on rHc and r H $ , (342) and by constructing the functions T O , $, and AD to safisfy the Dirichlet boundary conditions of the problem: fk

= 0,

nx TD

TD

= 0,

- n = 0,

on I ', and on

rHc

n x AD = K,

on

rkA,

AD n = 0,

on

rHA

$D

=

r,,,,

on r H c and

rncA,

and and

rH#

(343) (344) (345) (346)

rAs.

(347)

Hence, the satisfaction of the Dirichlet boundary conditions (209), (2 lo), (21l), (213), (215), (218), (221) and (226) by the approximating functions T'"), $(") and A(") is ensured. The continuity of the magnetic scalar potential on is provided by the use of the same expansion for $ in the the surface rnc$ regions R, and R,, , The weak formulation consists of three equations, one for each potential. They are written for the differential equation (203), for the time derivatives of the differential equations (204)-(206). They take care of the Neumann boundary conditions (207), (211a) and (219a), the time derivatives of the Neumann boundary conditions (208) (212), (214), (216) and (227), the inter-

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59 face condition (223) and the time derivative of the interface conditions (219), (220), (222), (224) and (225):

v x -1v

1 a

x T - V-v

+IrE(:.

. T + -a- ( p ~ )at

x T xn).wd,+jr~li(:V.T)(w.n)dT

function w,

(348)

a

-VXjQtZA[

" (L

-VxA

+V-

-V.A)

wdR

a

1 -VxA

xn+-K

a

1 -VxA

FT ~n,--xn~+V-xn,

+ jrHA[-'(/'

+6;teA[-'(p

) " (L ] ) ]. w d r )

a*at

1

.wdT

at

for any function w. (350)

OSZKAR BlRO AND K.R. RICHTER

60

Similarly to the previous formulations, Galerkin's equations can be derived by writing the expansions (337)-(339) instead of the potentials and replacing the weighting functions w and w with the expansion functions fi and J , respectively, Symmetric forms are obtained when the identities (303),(304), (293) and (336) are applied as appropriate, and the homogeneous Dirichlet boundary conditions (340)-(342) on the expansion functions are taken into account. The integrals over the surfaces r " c A and r A * containing the scalar potential I) in Eq. (350) can be transformed in a way similar to Eq. (331):

where, on the surface r n C A , n# stands for n, and nA for n,. The curve CHA+ is that part of the boundary of the surface r n c A + F A + , which is also a boundary of the surfaces r,, or r H $ . Galerkin's equations can then be brought to the following form using n, = -nA on the surface r n f A and nJI = -nA on r A + :

1

['-(V x T ' " ) ) - ( V x f i ) + - (1V - T ( " ' ) ( V - f i ) ] d R

Rc

CT

a -'lnnA[;

0

-(V x A'"') (V x fJ

1 + -(V c1

1

A("')(V fi) dR

61

CAD IN ELECTROMAGNETISM

The matrix form of these equations can be written as

where the elements of the matrices can be obtained from Eqs. (352)-(354). The matrices are obviously symmetric with the diagonal elements of the submatrix B A A negative. In contrast to the A,V-A-$ formulation, the time derivatives of the elements of the matrix B are also present if, in the nonlinear case, they are time dependent. 3. Galerkin s Method for Waveguides und Cavities

The differential equations of the two potential descriptions of waveguides and cavities described in Section IV.B.l and 2 are completely analogous: therefore, Galerkin's equations will only be presented for the method using the magnetic vector potential A and the electric scalar potential V. Similarly to the eddy current case, the symmetry of Galerkin's equations turns out to be achievable only if the electric scalar potential V is treated as the time derivative of a modified scalar potential t', V

(356)

= jwv,

resulting in the symmetry of the operator LA. Let us introduce the relative permeability and permittivity tensors [ p , ] and [ E ? ] and the relative permeability p , corresponding to the constant p in the second term of the differential equation (243) as (357)

(358) (359) where p,, and E,, are the permeability and permittivity of free space. Further, introducing the free space wave number k, as

k, =

mzz,

(360)

the differential equations (243) and (244) can be rewritten as V x [pr]-' V x A

- V -1V Pr

- A - k;[&,]A - k;[t;,] VV = 0, k i V * [ & , ] A+ k i V * [el] VV = 0.

in Q,

(361) (362)

Similarly, the Neumann boundary condition (245) has the form

-

n ( - k ; [ & , ] A- k t [ c , ] Vv) = 0,

on

r,.

(363)

OSZKAR BIRO AND K.R. RICHTER

62

The eigenvalues to be determined are the values of the squared free space wave number, i.e, in effect, of the frequency. The proper vector identity for the differential equation (362) has the form

(k;V.[&,]A+

~ ; V . [ & , ] V U ) W ~ R = (-k;[&,]A-

~;[E,]VU)*VW~R

Jn

- $r

(-k;[&,]A

- k;[&,] VU). n w d r .

(364) Again, the approximations of the potentials have to satisfy the Dirichlet boundary conditions, which are a!! homogeneous for this kind of problems: ni

A z A'"' =

C Akfk,

(365)

k=l

where the Dirichlet boundary conditions (239), (240) and (242) are satisfied by the expansion functions

r,, on r,, on r,.

n x fk = 0,

on

fk n = 0, jk

= 0,

(367) (368) (369)

The weak formulation of the problem involves two equations, one for A and one for u. They take care of the differentia! equations (361),(362) and the Neumann boundary conditions (24l), (245) and (246): 1 JQ(V x [/A,]-' V x A - V-V Pr

- A - k;[cr]A

- k;[&,] Vu) w dR

for any function w, (370)

Jn(k;V

- [cr]A + k ; V .

S,

( - k;[&,]A

+

[&,]Vu)wdn

-

- k;[&,] Vu) nw d T = 0,

for any function w. (371)

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CAD IN ELECTROMAGNETISM

Galerkin's equations can be obtained from here by replacing the potentials with their approximating expansions in Eqs. (365) and (366) and the weighting functions w and w with the expansion functions fi and fi, respectively. Applying the identities (303),(304) and (364) as well as the fact that the expansion functions fi and fi satisfy the homogeneous Dirichlet boundary conditions (367)-(369), the following symmetric forms are derived: Jn {(V x A'"'). [p,]-'(V

x fi)

1 + -(V 11,

- A'"))(V - fi)}dQ

( [ E , ] A ' " ' + [ E , ] V U ( " ' ) . V ~ ; ~ Q =iO=,n , + l , n , + 2 ,...,n.

-kij n

(373) In a matrix form, this generalized eigenvalue problem can be written as

:I[:]

: :-[ : A

:$]

I:[

=

(374)

The matrices A,, and the entire matrix B are easily seen to be symmetric and positive definite. This ensures that the eigenvalues are nonnegative. Since ( n - n l ) rows and columns of the matrix A are zero, (n - n , ) zero eigenvalues are obtained. However, all other eigenvalues are positive, corresponding physical modes. VI. APPLICATION OF

THE

FINITE ELEMENTMETHOD

An essential problem of Galerkin's method is the selection of the expansion functions that satisfy the Dirichlet boundary conditions and constitute an entire set for the approximation of the potentials. The most flexible way is the use of the Finite Element Method. Excellent treatment of the method is given by Zienkiewlcz (19771, so only a few outstanding features will be summarized in Section VI. A. The finite element solution of a nonlinear, static magnetic problem, of the three-dimensional model of a choke coil is presented in Section VI. B. A time-harmonic eddy current problem involving a coil over an aluminum plate with a hole is solved in Section V1.C. The transient eddy current problem of a conducting brick with a hole in a homogeneous, exponentially decaying magnetic field is tackled in Section V1.D. The dominant modes and the corresponding dispersion characteristics of two ferrite loaded anisotropic waveguides are determined in Section VI. E. Finally, Section V1.F is devoted to the problem of a dielectric loaded cavity.

64

OSZKAR BIRO A N D K. R. RICHTER

A . A Summary of the Finite Element Method

The Finite Element Method is based on a division of the studied region Q into small subregions, so called, finite elements. The potential functions are approximated by low order polynomials within each finite element. Elements of various shapes are in current use, triangles and quadrilaterals being the most popular ones in two dimensions and tetrahedra, prisms and hexahedra in three dimensions. In the particular variation used exclusively in the present work, the elements are defined by means of nodes. Some typical two- and three-dimensional elements are shown in Fig. 7. Special interpolation polynomials, called element shape functions, are used for the approximation of the potentials within each element. One element shape function N f ’ is associated with each node in an element; it assumes the value one at this node and is zero at all other nodes: N f )=

at the node k, at other nodes.

1 0

(375)

b 19 10

13

1

9

5

1

C d FIG.7. Typical finite elements. (a) three-noded triangular element. (b) eight-noded quadrilateral element. (c) ten-noded terahedral element. (d) twenty-noded hexahedral element.

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65

The element shape functions are conveniently defined in a local coordinate system associated with each element. For example, a two-dimensional, eightnoded, rectangular element is shown in the local as well as the global coordinate system in Fig. 8. The element shape functions can be written as

Nf’(t,?) = +

ttk)(l

+ qqk)(ttk + qqk -

for corner nodes

((k

=

f 1, q k

=

f l),

Nf)(t,q =)!d1 - t2)(l+ q q k ) , for midside nodes N f ) ( t ? q= )

-

q2)(I

(tk

= 0, q k =

& I),

= 0,

f l),

(376)

+ ttk)?

for midside nodes

(qk

t k

=

where ( t k , q k ) are the local coordinates of the node k. The shape functions represented by Eq. (376) are shown in Fig. 9. They are quadratic polynomials and are easily seen to satisfy the conditions (375). In three dimensions, three local variables t, q and i are involved. A transformation between the local and global coordinates of an element with n:’ nodes can be defined with the aid of the locally defined shape functions X ( t 7 v],

c) =

nf)

nf xkNf’(