Inverse Problems in Electric Circuits and Electromagnetics (Mathematical and Analytical Techniques with Applications to Engineering)

339 Pages·2006·15.91 MB·English

by  N.V. Korovkin

Inverse Problems in Electric Circuits
and Electromagnetics
Inverse Problems in Electric Circuits
and Electromagnetics
V.L, Chechurin
Saint Petersburg State Polytechnical University
St. Petersburg, Russia
N.V. Korovkin
Saint Petersburg State Polyteclinicai University
St. Petersburg, Russia
M. Hayakawa
Tlie University of Electro-Communications
Cliofu City, Tokyo
Springer
N.V. Korovkin
Saint Petersburg State Polytechnical University
St. Petersburg, Russia
V.L. Chechurin
Saint Petersburg State Polytechnical University
St. Petersburg, Russia
M. Hayakawa
The University of Electro-Communications
Chofu City, Tokyo
Inverse Problems in Electric Circuits and Electromagnetics
Library of Congress Control Number: 2006932802
ISBN 0-387-33524-2 e-ISBN 0-387-46047-0
ISBN 978-0387-33524-7
Printed on acid-free paper.
© 2007 Springer Science+Business Media LLC
All rights reserved. This work may not be translated or copied in whole or in part without
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Contents
Preface.
Chapter 1 Inverse Problems in Electrical Circuits and
Electromagnetic Field Theory 1
1.1 Features of inverse problems in electrical engineering 1
1.1.1 Properties of inverse problems 6
1.1.2 Solution methods 12
1.2 Inverse problems in electric circuits theory 18
1.2.1 Formulation of synthesis problems 18
1.2.2 The problem of constructing macromodels
(macromodeling) of devices 26
1.2.3 Identifying electrical circuit parameters 29
1.3 Inverse problems in electromagnetic field theory 33
1.3.1 Synthesis problems 35
1.3.2 Identification problems 43
References 45
Chapter 2 The Methods of Optimization of Problems
and Their Solution 47
2.1 Multicriterion inverse problems 47
2.2 Search of local minima 59
2.3 Search of objective functional minimum in the presence
of constraints 68
2.4 Application of neural networks 85
2.5Application of Volterra polynomials for macromodeling 98
2.6 Search of global minima 105
VI Contents
2.6.1 The multistart method and cluster algorithm 107
2.6.2 "Soft" methods 109
References 119
Chapter 3 The Methods of Solution of Stiff
Inverse Problems 121
3.1 Stiff inverse problems 121
3.2 The principle of quasistationarity of derivatives and
integrals 136
3.3 Using linear relationships for solving stiff inverse
probems 151
3.4 The problems of diagnostics and the identification
of inverse problems in circuit theory 156
3.4.1 Methods of identification of linear circuits 159
3.4.2 Error of identification problem solution 161
3.5 The method of stiff diagnostics and identification
problems solutions 168
3.5.1 AppHcation of the principle of repeated
measurements for solution of electric
circuits' identification problem 168
3.5.2 Definition of linear connections between
parameters of circuit mathematical models 169
3.5.3 Algorithm and results of electric circuits'
identification problem solution using repeated
measurements 172
3.6 Inverse problems of localization of disturbance sources
in electrical circuits by measurement of voltages in
the circuit's nodes 181
References 191
Chapter 4 Solving Inverse Electromagnetic Problems
by the Lagrange Method 193
4.1 Reduction of an optimization problem in a stationary field
to boundary-value problems 193
4.2 Calculation of adjoint variable sources 202
4.3 Optimization of the shape and structure of bodies in
various classes of media 213
4.4 Properties and numerical examples of the Lagrange method 220
4.4.1 Focusing of magnetic flux 221
4.4.2 Redistribution of magnetic flux 223
Contents Vll
4.4.3 The extremum of electromagnetic force 229
4.4.4 Identification of substance distribution 231
4.4.5 Creation of a homogeneous magnetic field 234
4.5 Features of numerical optimization by the Lagrange method 239
4.6 Optimizing the medium and source distribution in
non-stationary electromagnetic fields 242
References 249
Chapter 5 Solving Practical Inverse Problems 251
5.1 Search for lumped parameters of equivalent circuits
in transmission lines 251
5.2 Optimization of forming lines 262
5.3 The problems of synthesis of equivalent electric
parameters in the frequency domain 275
5.4 Optimization of current distribution over the conductors
of 3-phase cables 285
5.5 Search of the shape of a deflecting magnet polar tip
for producing homogeneous magnetic field 296
5.6 Search of the shape of magnetic quadrupole lens polar
tip for accelerating a particle 301
5.7 Optimum distribution of specific electric resistance
of a conductor in a magnetic field pulse 306
References 315
Appendices
Appendix A A Method of Reduction of an Eddy Magnetic
Field to a Potential One 317
Appendix B The Variation of a Functional 323
Index 325
Preface
The design and development of electrical devices involves choosing from
many possible variants that which is the best or optimum according to one or
several criteria. These optimization criteria are usually already clear to the
designer at the statement of the design problem. The methods of optimization
considered in this book, allow us to sort out variants of the realization of a
design on the basis of these criteria and to create the best device in the sense
of the set criteria.
Optimization of devices is one of the major problems in electrical engi
neering that is related to an extensive class of inverse problems including
synthesis, diagnostics, fault detection, identification, and some others with
common mathematical properties. When designing a device, the engineer ac
tually solves inverse problems by defining the device structure and its pa
rameters, and then proceeds to deal with the technical specifications followed
by the incorporation of his own notions of the best device. Frequently the so
lutions obtained are based on intuition and previous experience. New meth
ods and approaches discussed in this book will add mathematical rigor to
these intuitive notions.
By virtue of their urgency inverse problems have been investigated for
more than a century. However, general methods for their solution have been
developed only recently. An analysis of the scientific literature indicates a
steadily growing interest among scientists and engineers in these problems.
As a result, there has been an increase in the number of publications of new
methods of solution of inverse problems as well as their active application in
practice. It is essential that methods of solution of inverse problems find ap
plication not only for the development of new devices, but also for the mod
ernization of existing equipment with the purpose of improving its character
istics or the extension of its operational life.
Inverse problems that are significant for practical purposes are, as a rule,
solved numerically. The increase in efficiency of computers has allowed us
to put into practice many new and effective methods of solution of inverse
problems. Therefore we have focused the book on an account of methods
oriented towards numerical solutions. We have not included analytical
methods because they are not so effective for optimization of designs of real
devices or physical properties of materials. Furthermore, an exposition of
analytical methods would substantially expand the volume of this book and
would result in excessive complication.
X Preface
Inverse problems in the theory of electric circuits and in electromagnetic
field theory have some particular features. We, however, notice numerous
common features of these problems that allow their exposition to be com
bined within the limits of one book. In particular, the solution of inverse
problems in the theory of circuits and in field theory is based on the same
mathematical apparatus, namely, methods of solution of incorrect problems
and methods of optimization. This material is presented throughout the book
from a general point of view.
Together with well-known methods of solution of inverse problems that
have been widely used in practice, we have described some methods based
on ideas borrowed from various areas of science. Generic methods of mini
mization of functionals, or methods based on the application of neural net
works can be treated among them. The method of Lagrange multipliers ex
plicitly considered in the book was found to be very effective for the solution
of optimization problems in electromagnetic field theory.
In the first chapter a classification of inverse problems is given with an
analysis of their properties, and we describe the basic methods for the nu
merical solution of inverse problems in electrical engineering. In the second
chapter methods of searching for local and global minima of functionals are
discussed, as well as methods of searching for the minima of functionals in
the presence of constraints on the parameters to be optimized. In the third
chapter we discuss methods of solution of inverse problems in the theory of
electric circuits. Special attention is given to the solution of stiff inverse
problems, in particular problems of identification that are characterized by
the presence of measurement errors. The fourth chapter is concerned with a
systematic account of the Lagrange Method of continuous multipliers, which
is applied to optimization in an electromagnetic field characterized by quite a
large number of parameters. The fifth chapter presents examples that demon
strate the solution of practical inverse problems in the theory of electric cir
cuits and in electromagnetic field theory, thus illustrating the effectiveness of
methods considered in this book.
Alongside the classical methods of solution of inverse problems in electri
cal engineering, up-to-date methods have also been investigated by the au
thors themselves within the limits of their scientific activities, as well as by
their colleagues: A.Adalev, A.Potienko, T.Minevich, A.Plaks, M.Eidmiller
and others. The authors have also used the results of scientific studies of re
search workers of the faculty of Fundaments of Theoretical Electrical Engi
neering in the State Polytechnic University, St.-Petersburg (Russia) con
cerned with the solution of problems involving the analysis of electric
circuits and electromagnetic fields. These include using the method of the
scalar magnetic potential for the analysis of direct current magnetic fields
(K.S.Demirchian). The authors are very grateful to professor K.S.Demirchian
and professor P.A.Butyrin for useful discussions and valuable advice that
Preface XI
helped improve the book and the understanding of distinctive features of the
solution of inverse problems in electrical engineering.
The authors express their appreciation to Mr. R.Hogg, Mr. D.Bailey, and
Mr. M.Repetto for helpful discussion of optimization problems in electro
magnetics, as well as to Mr. Kh.Partamyan and Mr. B.DeCarlo for their help
during the writing of the book.
The authors thank professors I.G.Chernorutski and E.B.Soloviova, whose
scientific ideas have helped us with the preparation of this work.
The authors believe that this book will be useful for engineers, scientific
researchers, postgraduate students and students major in electrotechnology,
electrical power systems, and other specialties.
It is hoped that this book will promote further interest in inverse problems
in electrical engineering among university students, lecturers and research
workers.