Numerical Analysis in Electromagnetics: The TLM Method

176 Pages·2012·3.31 MB·English

by  Pierre Saguet

Numerical Analysis in Electromagnetics
Numerical Analysis
in Electromagnetics
The TLM Method

Pierre Saguet
First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
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The rights of Pierre Saguet to be identified as the author of this work have been asserted by /him in
accordance with the Copyright, Designs and Patents Act 1988.
____________________________________________________________________________________
Library of Congress Cataloging-in-Publication Data

Saguet, Pierre.
Numerical analysis in electromagnetics : the TLM method / Pierre Saguet.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-391-3
1. Electromagnetism--Mathematical models. 2. Time-domain analysis. 3. Numerical analysis. 4.
Electrical engineering--Mathematics. I. Title.
TK454.4.E5S34 2012
537.01'515--dc23
2012008582

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN: 978-1-84821-391-3
Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter1.BasisoftheTLMMethod:the2DTLM
Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.Historicalintroduction. . . . . . . . . . . . . . . . . . . . . . . 1
1.2.2Dsimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1.Parallelnode . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2.Seriesnode . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3.Simulationofinhomogeneousmediawithlosses . . . 9
1.2.4.Scatteringmatrices . . . . . . . . . . . . . . . . . . . . . . 11
1.2.5.Boundaryconditions . . . . . . . . . . . . . . . . . . . . . 14
1.2.6.Dielectricinterfacepassageconditions. . . . . . . . . . 15
1.2.7.Dispersionof2Dnodes. . . . . . . . . . . . . . . . . . . . 17
1.3.TheTLMprocess. . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.1.Basicalgorithm. . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.2.Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.3.Outputsignalprocessing. . . . . . . . . . . . . . . . . . . 24
Chapter2.3DNodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.Historicaldevelopment . . . . . . . . . . . . . . . . . . . . . . 29
2.1.1.Distributednodes . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.2.Asymmetricalcondensednode(ACN) . . . . . . . . . . 30
2.1.3.Thesymmetricalcondensednode(SCN) . . . . . . . . 31
2.1.4.Othertypesofnodes . . . . . . . . . . . . . . . . . . . . . 33
vi NumericalAnalysisinElectromagnetics
2.2.Thegeneralizedcondensednode. . . . . . . . . . . . . . . . 37
2.2.1.Generaldescription. . . . . . . . . . . . . . . . . . . . . . 37
2.2.2.Derivationof3DTLMnodes . . . . . . . . . . . . . . . 41
2.2.3.Scatteringmatrices . . . . . . . . . . . . . . . . . . . . . . 46
2.3.Timestep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4.Dispersionof3Dnodes. . . . . . . . . . . . . . . . . . . . . . 55
2.4.1.Theoreticalstudyinsimplecases . . . . . . . . . . . . . 56
2.4.2.Caseofinhomogeneousmedia. . . . . . . . . . . . . . . 60
2.5.Absorbingwalls . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.5.1.Matchedimpedance . . . . . . . . . . . . . . . . . . . . . 61
2.5.2.Segmentationtechniques . . . . . . . . . . . . . . . . . . 62
2.5.3.Perfectlymatchedlayers . . . . . . . . . . . . . . . . . . 62
2.5.4.OptimizationofthePMLlayerprofile. . . . . . . . . . 65
2.5.5.Anisotropicanddispersivelayers. . . . . . . . . . . . . 67
2.5.6.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.6.Orthogonalcurvilinearmesh . . . . . . . . . . . . . . . . . . 70
2.6.1.3DTLMcurvilinearcell. . . . . . . . . . . . . . . . . . . 70
2.6.2.TheTLMalgorithm . . . . . . . . . . . . . . . . . . . . . 73
2.6.3.Scatteringmatricesforcurvilinearnodes . . . . . . . . 75
2.6.4.Stabilityconditionsandthetimestep . . . . . . . . . . 78
2.6.5.Validationofthealgorithm. . . . . . . . . . . . . . . . . 79
2.7.Non-Cartesiannodes . . . . . . . . . . . . . . . . . . . . . . . 81
Chapter3.IntroductionofDiscreteElementsandThin
WiresintheTLMMethod. . . . . . . . . . . . . . . . . . . . . . . . 85
3.1.Introductionofdiscreteelements. . . . . . . . . . . . . . . . 85
3.1.1.Historyof2DTLM. . . . . . . . . . . . . . . . . . . . . . 85
3.1.2.3DTLM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.1.3.Applicationexample:modelingofap-ndiode. . . . . 100
3.2.Introductionofthinwires . . . . . . . . . . . . . . . . . . . . 105
3.2.1.Arbitrarilyorientedthinwiremodel . . . . . . . . . . . 106
3.2.2.Validationofthearbitrarilyorientedthin
wiremodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Chapter4.TheTLMMethodinMatrixFormandtheZ
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
TableofContents vii
4.2.MatrixformofMaxwell’sequations. . . . . . . . . . . . . . 124
4.3.CubicmeshnormalizedMaxwell’sequations . . . . . . . . 125
4.4.Thepropagationprocess . . . . . . . . . . . . . . . . . . . . . 127
4.5.Wave-matterinteraction. . . . . . . . . . . . . . . . . . . . . . 130
4.6.ThenormalizedparallelepipedicmeshMaxwell’s
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.7.Applicationexample:plasmamodeling. . . . . . . . . . . . 136
4.7.1.Theoreticalmodel . . . . . . . . . . . . . . . . . . . . . . . 136
4.7.2.ValidationoftheTLMsimulation. . . . . . . . . . . . . 139
4.8.Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
APPENDICES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
AppendixA.DevelopmentofMaxwell’sEquationsusing
theZTransformwithaVariableMesh . . . . . . . . . . . . . . . 147
AppendixB.TreatmentofPlasmausingtheZTransform
fortheTLMMethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Introduction
There are a number of modeling methods that are suitable for
solving problems in electromagnetism and analyzing the behavior of
certain media. In order to apply these methods the type of problem
must be specified and the boundary conditions must be clearly
determined and defined. Numerical or analytical solutions are then
carriedout.
Analytical solutions, which are already well established, were the
firsttobeapplied.Theyenabledanefficientresolutionofallproblems
relating to the majority of electromagnetic wave guiding systems.
However, these analytical methods remain limited, since, in these
cases, it is only possible to analyze structures with simple geometries
and which, in the majority of cases, have a certain degree of
symmetry.
For more realistic modeling of geometries and more complex
materials (indeed, complexity leaves little room for any analytical
resolution), we have numerical methods, which have become an
important element in the analysis of the behavior of various industrial
products.Theyhaveprogressedinparallelwithtechnologyandenable
electronic systems developers to have at their disposal all of the
necessary characteristics and data, which were difficult to obtain
through testing, in order to ensure the reliability of device operation
withoutanyaccompanyingperformancedegradation.
x NumericalAnalysisinElectromagnetics
Inthespecificcaseofelectromagnetism,therearevariousdiffering
numerical techniques, whose effectiveness depends on the problem
and on the desired results. These techniques can be classified
accordingtodifferentcriteria.
Classificationbasedonthetypeofequation
Firstly, we can classify numerical methods based on equation type.
Indeed, most models under consideration lead to differential or
integral mathematical equations. If the problem deals with
electromagnetic wave propagation, the equations which describe its
behavior (such as Maxwell’s and wave equations) can be expressed
usingtwomethods:differentialorintegral.
In order to solve these equations at any point in a finite space,
differential or integral methods are used to determine the values
required.
Classificationbasedontheapplicationdomain
A second classification which may be taken into account is the
domainwithinwhichtheequationstobesolvedaredefined.Intheory,
equations express the space and time variations in the scale of the
problem to be resolved (electromagnetic fields or potentials). Here we
are working in the time domain and the methods used are known as
“time-domainnumericalmethods”.
However, in the study of certain problems (notably in the area of
telecommunications), it is field cartography varying sinusoidally over
time or from a combination of multiple sinusoids, which is of interest.
In these cases, the electromagnetic characteristics of the majority of
materials can be expressed in a much simpler form, based on the
frequency of these sinusoidal signals. These equations are therefore
expressed frequentially and so the methods used to solve them are
knownas“frequency-domainnumericalmethods”.
The advantage of frequency methods is that they give rise to
equations which are more flexible and easier to simplify.
Introduction xi
Nevertheless, they are also limited, as they rely on signals always
beingsinusoidalorbasedonasinusoidalcombination.
In all cases a frequency representation can be obtained from a
signalbyusingaFouriertransformofthetimesignal.
In this book, we are going to look at the TLM (transmission line
matrix) method, which is one of the “time-domain numerical
methods”. These methods are reputed for their significant reliance on
computer resources. However, they have the advantage of being
highly general. We will focus our attention on the TLM method
which, since the pioneering article on TLM by P.B. Johns and
R.L.Beurle in 1971, has been intensively studied and developed by
many researchers. It has, therefore, acquired a reputation for being a
powerfulandeffectivetoolbynumerousteamsandstillbenefitstoday
from significant theoretical developments. In particular, in recent
years, its ability to simulate various situations, including complex
materials,withexcellentprecisionhasbeendemonstrated.
Thisbookconsistsofanintroductionandfourchapters.
Chapter 1 describes the basis of the TLM method in two
dimensions and enables different aspects of the method to be tackled,
aswellastheerrorsresultingfromspaceandtimesampling.
Chapter 2 is dedicated to a 3D analysis of the method. It maps out
themaintypesofnodescurrentlyusedbypointingouttheirrespective
advantages and disadvantages. This chapter also features the problem
of open structure simulation and the necessity of implementing
absorbing boundaries, including PMLs, which nowadays are used
universally.
Chapter 3 describes techniques which enable the simulation of
structures comprising passive and active discrete elements, as well as
thin metallic wires without the need to mesh these structures, which
would lead to memory problems. These techniques, as well as 3D
node and mesh flexibility, enable the simulation of a wide range of
problems where the properties of the surrounding medium are not
dependentonfrequencyandarethereforenotdispersive.
xii NumericalAnalysisinElectromagnetics
Chapter4demonstrateshowtosimulatedispersivemediausingthe
ZtransformwithintheTLMmethodinmatrixform.Thisrigorousand
unconditionally stable method makes the use of the TLM method
possibleinvirtuallyallcases.
Application examples are included in the last two chapters,
enabling us to draw conclusions regarding the performance of the
implementedtechniquesand,atthesametime,tovalidatethem.
Multi-scale problems which require the TLM method to be
combined with other methods will not be dealt with in this book in
spite of their undeniable interest. There are many papers dedicated to
thiswhichwouldrequirecollationintoasinglepublication.