Field Models in Electricity and Magnetism

The subject of computation in electricity and magnetism has so advanced in the past 40 years, since the advent of digital computers and thanks to the development of numerical methods, that today it urges and deserves adequate collocation also in curricula for electrical engineering. However, the time allotted to the subject in generalisnotverylargeinundergraduatestudies,wheremoreemphasisisstillusually attributed to circuits and systems than to ?elds. Moreover, ?eld models are generally not very popular among students, who are by far more familiar with circuit models. Even if one considers the quasi-static case, however, not only is electromagnetism fundamentalforpeopledealingwithelectricandmagneticdevices,butitprovidesthe basisfor,e.g.semiconductordevicedesign,bioengineeringapplicationsandsoforth. In the authors'opinion, therefore, time has come to present ?eld models in el- tricity and magnetism, in the frame of an introductory textbook to be used by senior undergraduateorgraduatestudentsintheareaofelectricalandcomputerengineering. Elementary electromagnetism, basic vector analysis and fundamentals of numerical analysis are assumed to be known subjects. Havingthisinmind,theauthorshavecollectedtheexperiencetheyhaveaccu- lated in teaching electromagnetic theory at various levels and in different countries; theyintendtoofferabookonappliedelectricityandmagnetism,describingthepr- lemsofcalculatingelectromagnetic?eldsandtheintegralparametersconnectedwith them in suf?ciently clear and short form. The aim is that of writing a textbook containing the necessary background, i.e. laws explaining electromagnetic phenomena, mathematical operators and equations as well as methods for electromagnetic ?eld calculation. The latter include both analytical and numericalmethods applied to the analysis as well as to the synthesis of electromagnetic devices.

A self-contained and up-to-date introduction to field modelling in electricity and magnetism Directly solves field problems from field equations in terms of vector quantities Comprehensive and including synthesis, in addition to analysis, of fields Based on a problem-solving-approach which offers a practical example solved by various methods

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Covering the development of field computation in the past forty years, Field Models in Electricity and Magnetism intends to be a concise, comprehensive and up-to-date introduction to field models in electricity and magnetism, ranging from basic theory to numerical applications.

The approach assumed throughout the whole book is to solve field problems directly from partial differential equations in terms of vector quantities.

Theoretical issues are illustrated by practical examples. In particular, a single example is solved by different methods so that, by comparison of results, limitations and advantages of the various methods are made clear.

The subjects of the synthesis of fields and of the optimal design of devices, which are growing in research and so far have not been adequately covered in textbooks, are developed in addition to more classical subjects of analysis.

Topics covered include:

• vector fields: electrostatics, magnetostatics, steady conduction;
• analytical methods for solving boundary-value problems;
• numerical methods for solving boundary-value problems;
• time-varying electromagnetic field;
• inverse problems;
• optimization.

Field Models in Electricity and Magnetism results from the cooperation of three authors in teaching electromagnetic theory at various levels and in different countries.

Contenu

1 Introduction

2 Vector fields

2.1 Basic operators and equations

2.1.1 Vector fields and operators

2.1.2 Definition of a vector field

2.1.3 Decomposition of a field

2.1.4 Scalar and vector potentials

2.1.5 Green's theorem

2.1.6 Green's formula

2.2 Electrostatic field

2.2.1 Maxwell's equations for electrostatics

2.2.2 Electrostatic potentials

2.2.3 Electrostatic energy

2.2.4 Field of a charged plane (rectangular domain)

2.2.5 Field of a point charge in R3 (spherical domain)

2.2.6 Field of a dipole of moment d q p = in R3

2.2.7 Field of a line charge in R3 (cylindrical domain)

2.2.8 Field of a surface charge on a sphere of radius a with density s

2.2.9 Energy and forces in the electrostatic field

2.2.10 Force between the plates of a capacitor

2.2.11 Force at the interface between two dielectric materials

2.3 Magnetostatic field

2.3.1 Maxwell's equations for magnetostatics

2.3.2 Magnetostatic potentials

2.3.3 Magnetostatic energy

2.3.4 Field of a line current in R3

2.3.5 Energy and forces in the magnetostatic field

2.3.6 Force on an electromagnet

2.3.7 Test problems

2.4 Steady conduction field

2.4.1 Maxwell's equations for conduction field

2.4.2 Potentials

2.4.3 Power loss

2.4.4 Analytic functions of complex variable

2.4.5 Field of a cylindrical conductor in R3

3 Analytical methods of solving boundary value problems

3.1 Method of Green's functions

3.1.1 Green's formula for electrostatics

3.1.2 Field of a point charge q surrounded by a sphere of radius a with U = 0

3.1.3 Field of a point charge q surrounded by a sphere of radius a with U = k

3.1.4 Field of a surface dipole distributed on a sphere of radius a with dipole densityt

3.1.5 Green's formula for two-dimensional magnetostatics

3.2 Method of images

3.2.1 Magnetic field of a line current in a slot

3.2.2 Magnetic field of an AC line current over a conducting half-space

3.3 Method of separation of variables

3.3.1 Magnetic field of a current uniformly distributed in a slot

4 Numerical methods of solving boundary value problems

4.1 Variational formulation in magnetostatics

4.2 Finite elements for two-dimensional magnetostatics

4.2.1 Discretization of energy functional

4.2.2 Local shape functions

4.2.3 Coefficient matrix and source vector

4.2.4 From potential to field

4.2.5 Magnetic field in a slot solved by the finite element method

4.3 Finite elements for three-dimensional magnetostatics

5 Time-varying electromagnetic field

5.1 Maxwell's equations in differential form

5.2 Poynting's vector

5.3 Maxwell's equations in frequency domain

5.4 Plane waves in an infinite domain

5.5 Wave and diffusion equations in terms of vectors E and H

5.6 Wave and diffusion equations in terms of scalar and vector potentials

5.7 Electromagnetic field radiated by an oscillating dipole

5.8 Diffusion equations in terms of dual potentials

5.9 Weak eddy current in a conducting plane under a.c. conditions

5.10 Strong eddy current in a conducting plane under a.c. conditions

5.11 Eddy current in a cylindrical conductor under step excitation

5.12 Electromagnetic field equations in different reference frames

6 Inverse problems

6.1 Direct and inverse problems

6.2 Well-posed and ill-posed problems

6.3 Fredholm's integral equation of the first kind

6.4 Case study: synthesis of magnetic field sources

6.5 Under- and over-determined systems of equations

6.6 Least-squaressolution

6.7 Classification of inverse problems

7 Optimization

7.1 Solution of inverse problems by the minimization of a functional

7.2 Constrained optimization

7.3 Gradientfree and gradient-based methods

7.4 Deterministic vs non-deterministic search

7.5 A deterministic algorithm of lowest order: simplex method

7.6 A non-deterministic algorithm of lowest order: evolution strategy

7.7 Numerical case studies

7.7.1 Identification of B-H curve of the iron region of a magnetic pole

7.7.2 Shape design of a magnetic pole (static optimisation)

7.7.3 Shape design of a magnetic pole (dynamic optimisation)

7.7.4 A multiobjective approach to the shape design of a magnetic pole