Finite Element Method to Model Electromagnetic Systems in Low Frequency E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Numerical modeling now plays a central role in the design and study of electromagnetic systems. In the field of devices operating in low frequency, it is the finite element method that has come to the fore in recent decades. Today, it is widely used by engineers and researchers in industry, as well as in research centers. This book describes in detail all the steps required to discretize Maxwell's equations using the finite element method. This involves progressing from the basic equations in the continuous domain to equations in the discrete domain that are solved by a computer. This approach is carried out with a constant focus on maintaining a link between physics, i.e. the properties of electromagnetic fields, and numerical analysis. Numerous academic examples, which are used throughout the various stages of model construction, help to clarify the developments.
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Table of Contents
Cover
Table of Contents
Title Page
Copyright Page
Introduction
1 Equations of Electromagnetism
1.1. Maxwell’s equations
1.2. Behavior laws of materials
1.3. Interface between two media and boundary conditions
1.4. Integral forms: fundamental theorems
1.5. Various forms of Maxwell’s equations
2 Function Spaces
2.1. Introduction
2.2. Spaces of differential operators
2.3. Studied topologies
2.4. Relations between vector subspaces
2.5. Vector fields defined by a vector operator
2.6. Structure of function spaces
3 Maxwell’s Equations: Potential Formulations
3.1. Introduction
3.2. Consideration of source terms
3.3. Electrostatics
3.4. Electrokinetics
3.5. Magnetostatics
3.6. Magnetodynamics
4 Formulations in the Discrete Domain
4.1. Introduction
4.2. Weighted residual method: weak form of Maxwell’s equations
4.3. Finite element discretization
4.4. Discretization of weak formulations
References
Index
Other titles from ISTE in Numerical Methods in Engineering
End User License Agreement
List of Tables
Chapter 2
Table 2.1. Synthesis of function spaces of vector operators and adjoint operat...
Chapter 3
Table 3.1. Summary of the equations to be solved in electrostatics for the sca...
Table 3.2. Summary of equations to be solved in electrostatics for the vector ...
Table 3.3. Summary of equations to be solved in electrokinetics for the scalar...
Table 3.4. Summary of equations to be solved in electrokinetics for the vector...
Table 3.5. Summary of equations to be solved in magnetostatics with the scalar...
Table 3.6. Summary of the equations to be solved in magnetostatics with the ve...
Table 3.7. Summary of the equations to be solved in magnetodynamics with the e...
Table 3.8. Summary of equations to be solved in magnetodynamics with the magne...
Table 3.9. Summary of equations to be solved in magnetodynamics with the elect...
Table 3.10. Summary of the equations to be solved in magnetodynamics with the ...
Chapter 4
Table 4.1. Electrostatics, scalar potential V formulation; vector operator and...
Table 4.2. Electrostatics; vector potential P formulation; vector operator and...
Table 4.3. Electrokinetics, scalar potential V formulation; vector operator an...
Table 4.4. Electrokinetics; vector potential T formulation; vector operator an...
Table 4.5. Magnetostatics, scalar potential φ formulation; vector operator and...
Table 4.6. Magnetostatics; vector potential A formulation; vector operator and...
Table 4.7. Magnetodynamics, imposed electrical quantities, A-V electrical form...
Table 4.8. Magnetodynamics, imposed electric quantities, electric formulation ...
Table 4.9. Magnetodynamics, imposed magnetic quantities, electric formulation ...
Table 4.10. Magnetodynamics, imposed magnetic quantities, electric formulation...
Table 4.11. Potential formulations; properties of strongly and weakly verified...
Table 4.12. Facet elements; succession of nodes of the tetrahedron in Figure 4...
Table 4.13. Summary of the properties of the node, edge, facet and volume elem...
Table 4.14. edge–node incidence matrix of a tetrahedron
Table 4.15. Facet–edge incidence matrix of the tetrahedron in Figure 4.4(a)...
Table 4.16. Volume–facet incidence matrix of a tetrahedron
Table 4.17. Discrete domain, function spaces and vector operators
Table 4.18. Correspondence between the function spaces in the continuous domai...
Table 4.19. Function spaces of fields or pote ntials in the continuous domain ...
Table 4.20. Matrix , example of Figure 4.8
Table 4.21. Matrix ; example of Figure 4.8
Table 4.22. Matrix ; example of Figure 4.10(a)
List of Illustrations
Chapter 1
Figure 1.1. M(H) characteristic of a ferromagnetic material
Figure 1.2. a) Characteristics of the most common hard ferromagnetic materials...
Figure 1.3. Definition of a vector field based on its normal and tangential co...
Figure 1.4. Normal and tangential components of fields B
1
, H
1
, B
2
and H
2
Figure 1.5. Faraday’s law implementation example
Figure 1.6. Illustration of Ampère’s law: conductor carrying a current...
Figure 1.7. Flux tube: law of conservation of the flux
Figure 1.8. Illustration of Gauss’ law
Figure 1.9. Representation of an electrostatic problem
Figure 1.10. Representation of an electrokinetic problem
Figure 1.11. Representation of a magnetostatic problem
Figure 1.12. Example of studied domain in magnetodynamics with a source term J...
Chapter 2
Figure 2.1. a) Example of a connected domain; b) disconnected domain
Figure 2.2. a) Simply connected domain; b) not simply connected domain
Figure 2.3. a) Contractible domain; b) non-contractible domain
Figure 2.4. Decomposition of space L2(Ω) without a priori on the topology of d...
Figure 2.5. Decomposition of space L2(Ω) for a contractible domain Ω (simply c...
Figure 2.6. Decomposition of space L2(Ω) for a not simply connected domain Ω w...
Figure 2.7. Decomposition of space L2(Ω) for a not simply connected domain Ω w...
Figure 2.8. Decomposition of space L2(Ω) for a not simply connected domain wit...
Figure 2.9. Not simply connected domain with disconnected boundary: graphical ...
Figure 2.10. Series of function spaces of grad, curl and div operators
Figure 2.11. Contractible domain: graphical representation of function spaces ...
Figure 2.12. Example of path γ for the gauge v.η = 0
Figure 2.13. Tonti diagram in the general case
Chapter 3
Figure 3.1. Example of contractible domain Ω with the various notations employ...
Figure 3.2. a) Domain with a not simply connected boundary Γm; b) introduction...
Figure 3.3. Geometry with a local source quantity inside the studied domain
Figure 3.4. Geometry with a local source quantity, represented by a density q ...
Figure 3.5. Geometry studied for the calculation of fields βs and αs...
Figure 3.6. Introduction of a cut in the geometry studied in Figure 3.5
Figure 3.7. Geometry studied for the calculation of support fields λsl and χsl...
Figure 3.8. Cylindrical coordinates of support fields λ
sl
Figure 3.9. Geometry studied for the calculation of support fields ξsl and ηsl...
Figure 3.10. Geometry studied in electrostatics when the source terms are impo...
Figure 3.11. Geometry studied in electrostatics in the case of an internal ele...
Figure 3.12. Tonti diagram in electrostatics
Figure 3.13. Simplified geometry studied for electrokinetics
Figure 3.14. Electrokinetics: example of multisource geometry with 3 gates (N ...
Figure 3.15. Tonti diagram in the case of electrokinetics
Figure 3.16. Geometry studied for magnetostatics having as source terms the fl...
Figure 3.17. Studied geometry for magnetostatics when the source term is a cur...
Figure 3.18. Studied geometry for magnetostatics when the source term is a per...
Figure 3.19. Study of a permanent magnet: discontinuity of the normal componen...
Figure 3.20. Magnetostatics problem: studied geometry in the general case
Figure 3.21. Tonti diagram for magnetostatics
Figure 3.22. Basic geometry for the magnetodynamics study
Figure 3.23. Geometry studied in magnetodynamics: electric quantities imposed ...
Figure 3.24. Geometry studied in magnetodynamics: magnetic quantities imposed ...
Figure 3.25. Tonti diagram for magnetodynamics
Chapter 4
Figure 4.1. Example of subdomain Ω′ ⊂ Ω with a transition zone...
Figure 4.2. First order node elements: a) case of a tetrahedron; b) 2D case, t...
Figure 4.3. a) Edge elements of a tetrahedron; b) orientation of an edge with ...
Figure 4.4. a) Facet elements of a tetrahedron; b) facet orientation
Figure 4.5. Set Se{i} of nodes of Figure 4.2, connected to the node “i” via an...
Figure 4.6. Denomination of nodes j, k, l, m of a tetrahedron and of the edge ...
Figure 4.7. Sequence of discrete function spaces
Figure 4.8. Mesh composed of two adjacent tetrahedrons with numbering of nodes...
Figure 4.9. Edge tree in the thick line, composed of edges 1, 3, 6 and 9
Figure 4.10. a) Two adjacent tetrahedrons with facet orientation; b) correspon...
Guide
Cover
Table of Contents
Title Page
Copyright Page
Introduction
Begin Reading
References
Index
Other titles from ISTE in Numerical Methods in Engineering
End User License Agreement
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Series Editor
Piotr Breitkopf
Finite Element Method to Model Electromagnetic Systems in Low Frequency
Francis Piriou
Stéphane Clénet
First published 2023 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2023The rights of Francis Piriou and Stéphane Clénet to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2023943094
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-811-5
Introduction
As calculation tools are reaching increasingly high performances, numerical modeling has developed significantly in all sectors of society. It can be used to predict the evolution of a given structure or device starting from an initial state, study physical phenomena by accessing quantities that are not measurable or develop virtual prototypes in order to improve a design process. Applied physics, and in particular low-frequency electromagnetism, which is the subject of this book, are no exceptions. Nowadays, high-performance simulation software is available for students, engineers and researchers. A prerequisite for making the best use of a tool, even in the field of computation, is obviously a good knowledge of its foundations and principles. In this context, it seemed interesting to propose a book that may grasp, under the best conditions, the path leading to building these numerical models.
The modeling of electromagnetic phenomena relies on two partial differential equations, known as Maxwell’s equations:
These two equations should be completed by behavior laws that describe the reaction of media to electromagnetic fields, which are associated with physical phenomena such as dielectric polarization, electric conduction and ferromagnetism. Finally, for a proper formulation of the problem, boundary conditions should be added, for either a finite or infinite studied domain. Although it may appear simple, this problem, composed of several equations, has no analytical solution, except for the case of elementary geometry, with linear behavior laws.
As the exact solution to the problem is not available, there are two possibilities for reaching an approximation of this solution:
Formulate hypotheses on the geometry, the behavior laws of the materials and the spatial distribution of electromagnetic fields. The objective is to make the analytical solution possible. This approach requires the “model builder” to have very deep, expert knowledge on the studied system, to be able to formulate the “right hypotheses”. If the latter are not valid, there is a very high risk of reaching a low-quality solution, which is very far from the exact solution of the initial problem. Moreover, this approach is not always possible if complex phenomena, such as nonlinearities, are predominant.
Or, reformulate the initial problem in discrete form, leading to a system of differential algebraic equations. An approximation of the exact solution is then obtained at the cost of a significant amount of computation, which can be readily processed by the computers that are available nowadays. This reformulation, requiring few or almost no hypotheses, is obtained by implementing numerical methods. In the field of electromagnetism, the most widespread such method is the finite element method.
This book focuses on the second approach, often referred to as “computational electromagnetics”, providing a detailed description of the implementation of the finite element method in low-frequency electromagnetism. Our purpose is to explain the process starting from equations verified by electromagnetic fields in the continuous domain, in order to arrive at a system of equations that will be solved using a computer. This process, often called “discretization”, will be conducted with a permanent concern for maintaining a link between physics, i.e. the properties of electromagnetic fields, and numerical analysis, through the finite element method.
Furthermore, this book is mainly addressed to students, engineers and researchers in the field of electrical engineering. They will be able to better understand the intricate details of (open-source or commercial) software that models the behavior of electromagnetic fields. They will thus have the possibility of better using these tools and therefore have a good knowledge of their limits. This book is also addressed to students, engineers and researchers in the field of numerical analysis who are interested in better understanding the links between numerical methods and physics in the field of electromagnetism.
Even though this book offers few pieces of information on numerical implementation, it provides all the elements required for understanding the theoretical foundations. It also allows us to conceive the link between physics and numerical methods and therefore between the applications and the software used.
The above-stated Maxwell’s equations allow for the study of all electromagnetic phenomena. For certain low-frequency applications it is, however, possible to derive them in a “static” or “quasi-static” state. Under certain hypotheses, these simpler problems lead to solutions that are equal or very close to those that would have been obtained using the full Maxwell equation system. After discretization using a numerical method, they can be used to obtain smaller size systems of equations that are easier to solve due to their mathematical properties.
Approximations by problems under a static or quasi-static state are widely used in many domains such as power grids, electrical machines, power electronics and non-destructive testing. This book focuses in particular on three static problems, namely electrostatics, electrokinetics (when electric charges travel at constant speed, the fields do not depend on time) and magnetostatics. In the quasi-static state, Maxwell’s equations can be written in the magnetoquasistatic form (more often referred to as “magnetodynamics”) or in the electroquasistatic form. In this quasi-static case, our focus will be on magnetodynamics. On the contrary, electroquasistatic problems will not be considered, but the developments remain similar to those used in the case of magnetodynamics.
This book has four chapters, each corresponding to a stage of the process leading to the discretization of Maxwell’s equations.
The objective of Chapter 1 is to formulate various problems in the static state and the magnetodynamic state, and then to solve them. For each problem, the equilibrium equations are written, as well as the behavior laws and the boundary conditions on the electromagnetic fields. A review of the properties of these fields also highlights their behavior at the interface between two media, and the nature of their integral forms. A key point of this chapter is the definition of electric and magnetic quantities, referred to as “source terms”, which are at the origin of the creation of electromagnetic fields. These terms can be located inside the studied domain (electric charges, inductors, permanent magnets) or imposed on the boundary of the domain (electromotive or magnetomotive forces, current density or magnetic flux).
Chapter 2 is dedicated to the introduction of functional spaces associated with vector operators: gradient, curl and divergence. As these operators are used when writing the equations of static and quasi-static problems, they can be used to define the functional spaces to which various electromagnetic fields belong. An analysis is conducted on the properties of functional spaces and in particular on the images and kernels of the vector operators in relation to the topology of the studied domain. These properties lead quite naturally to the notion of scalar and vector potentials, widely used as intermediary for solving static and quasi-static problems, which will be introduced in Chapter 3. The notion of gauge is also presented, which imposes the uniqueness of a field when defining by a single vector operator. Gauge conditions will therefore be very useful to impose the uniqueness of potentials so that the problem is properly posed. They are also used in the construction of source terms.
Chapter 3 focuses on the potential-based formulations for static and magnetodynamic problems. In the case of static problems, the introduction of these potentials allows for the reduction of the number of unknowns, passing from two unknown fields to only one unknown potential. This potential can be a scalar or a vector quantity. For each problem, two formulations in terms of potential referred to as “scalar” or “vector” are obtained. In magnetodynamics, two potentials are used in close relation to those introduced for static problems. Two formulations known as “electric” and “magnetic” are then deduced.
These potentials are not necessarily introduced in a direct manner, requiring instead a reformulation of the source terms of the initial problem, located either inside or on the boundary of the domain. The first part of this chapter is dedicated to this reformulation. The number of sources is often limited, facilitating a focus on the essential, which is a systematic method for imposing source terms. However, as shown by the examples presented, the methodology is readily applicable to problems with a greater number of sources, using the superposition theorem (even though the behavior laws are not linear).
Chapter 4 is dedicated to the discretization of formulations of static and magnetodynamic problems. Successful completion of this discretization requires first of all finding the proper spaces of approximation within which the approximate solutions will be sought. These spaces must have a finite dimension for implementation on a computer. In the case of the finite element method, the spaces of approximation are defined from a mesh, which is obtained by splitting the studied domain into elements of simple shapes (tetrahedron, hexahedron, prism, etc.). A field is then perfectly defined by a vector, whose entries are the coefficients of the basis of the approximation space. The entries of this vector are then the degrees of freedom to be determined. These spaces of finite dimension must be included in the functional spaces to which electromagnetic fields belong. This means they must meet the properties introduced in Chapter 2. This condition leads to physically acceptable field approximations in the sense that they verify the continuity conditions. Whitney finite elements are currently the most commonly used, and they generate spaces that allow for a real physical interpretation of the degrees of freedom, which are then fluxes, circulations and densities. Moreover, imposing gauge conditions is natural, as well as the calculation of source terms. It is important to note that, in this book, our developments are limited to first-order finite elements for a tetrahedron-based mesh. Very similar approaches can be applied with higher-order functions and elements of other shapes (hexahedron, prism, pyramid, etc.). The introduction of discrete forms of fields in the potential-based formulations of static and magnetodynamic problems does not allow us to directly build a system of equations. Our objective is to use the weighted residual method, allowing for the transformation of the initial problem, based on local equations, into a problem based on integral equations. For each of the potential-based formulations developed in Chapter 3, the weighted residual method is used in association with Whitney finite elements in order to build a system of equations to be solved, which is then the numerical model we seek.
1Equations of Electromagnetism
1.1. Maxwell’s equations
Maxwell’s equations can be written in a general form as follows (Stratton 1941; Ida 2020):
The vector fields denoted by E, B, H and D represent, respectively, the electric field, the magnetic flux density, the magnetic field and the electric displacement field. Electric current density J and electric charge density ρ can be considered source terms.
Finally, it is common to adopt the quasi-static approximation for electromagnetic devices operating at industrial frequencies. In this case, the term ∂D/∂t in equation [1.2] can be considered negligible. Equation [1.2] can then be written as:
If the divergence operator is now applied to equation [1.5], considering the properties of vector operators (divcurlH = 0), the following property can be deduced for the electric current density:
1.2. Behavior laws of materials
Maxwell’s equations, as presented above, are independent of the media. But electric fields (E, J, D) and magnetic fields (H, B) are related by behavior laws.
1.2.1. General case
It can be noted that in vacuum these behavior laws are linear and have the following form:
where ε0 represents the electric permittivity constant (ε0 = 10−9/36π F/m) and μ0 is the magnetic permeability constant (μ0 = 4π10−7 H/m). They are linked by the classical relation: ε0μ0c2 = 1, where c represents the speed of light in vacuum.
On the contrary, electromagnetic fields in media interact with their environment. These interactions also depend on fields of different physical natures, such as temperature T or mechanical stress σm. In this case, the behavior laws become significantly more complex, and the following relations can be written:
These various functions may depend on the history of the material. As an example, in the case of ferromagnetic materials, the value of the field B at the time t depends not only on the value of the field H at time t, but also on its previous values in the time interval [0, t]. This phenomenon is known as magnetic hysteresis. Some materials, used for their electric properties linking fields D and E, also exhibit this hysteresis phenomenon. They are referred to as ferroelectric materials.
Behavior laws are often at the origin of the links between various physics domains. This is the case with coupling the equations of electromagnetism with the equations of thermodynamics and mechanics. For example, electric current density J, for a given value of the electric field E, decreases as a function of temperature due to thermal energy, which tends to reduce the conductivity of the material. As a first approximation, conductivity σ(T) can be written as a function of temperature:
In this expression, ΔT = (T – Tref) with T > Tref, σref is the conductivity at temperature Tref and α is a temperature coefficient of the considered material also dependent on Tref. However, in many applications, multiphysics couplings can be considered negligible in a first approach. In this case, the behavior laws can be written in a simplified form:
1.2.2. Simplified forms
Even in the form presented in equations [1.13], [1.14] and [1.15], the behavior laws in material media may be relatively complex (anisotropy, hysteresis). It is nevertheless often possible to simplify them without affecting the precision of the results, which will be shown in the following section.
1.2.2.1. Dielectric materials
Consider the behavior law of dielectric materials written using relation [1.13]. Introducing the electric polarization vector , which depends on the electric field E, the following relation can be written (Ida 2020):
In the case of ferroelectricity, polarization follows a hysteresis loop when the electric field varies as a function of time. However, the hypothesis of isotropy and linearity is acceptable for many dielectric materials. Polarization can therefore be considered to be directly proportional to the electric field strength. It can then be written as follows:
where χe represents the electric susceptibility of the material. Grouping equations [1.16] and [1.17] leads to:
This expression involves εr, a dimensionless number, which represents the relative permittivity of the considered material. Introducing permittivity ε = ε0εr, the commonly used behavior law of dielectric materials is obtained:
1.2.2.2. Conductive materials
For conductive materials, assuming thermal effects are negligible, electric current density is proportional to the electric field. The electrical behavior law is then expressed as:
where σ is the electrical conductivity.
1.2.2.3. Magnetic properties of materials
The magnetic properties of materials can be expressed using relation [1.14]. Similar to the dielectric materials, a magnetic polarization vector, denoted by , can be introduced (Bozorth 1993; Benabou 2002). Using this vector, it is possible to express the magnetic flux density in the following form:
In this expression, the magnetization vector M can also be introduced, posing . Equation [1.21] can then be written as:
Equations [1.21] and [1.22] contain a first term corresponding to the magnetic flux density created in vacuum (μ0H) and a second term, respectively, or μ0M(H), corresponding to the response of the material medium to the external magnetic field.
There are various behaviors leading to the following classification:
diamagnetic materials that, when subjected to a magnetic field, create a magnetization that opposes the external field. In this case,
equation [1.22]
can be written as:
where χm represents the magnetic susceptibility with χm < 0 and is of the order of −10−5. It can be noted that this reaction magnetization is very weak for most materials used in electrical engineering. As an example, the magnetic susceptibility of copper is equal to −1.18 10−5;
Paramagnetic materials that, when subjected to a magnetic field, create a very weak magnetization in the same direction as the external field. In this case, the expression
[1.23]
is unchanged, but χ
m
>0 may range between 10
−3
and 10
−5
. As an example, magnetic susceptibility of molybdenum is equal to 1.05 10
−4
.
Ferromagnetic materials that, when subjected to a magnetic field of several hundred amperes per meter, may create a magnetization
M
(
H
) of the order of 10
6
A/m. Moreover, in the absence of an external field, they may present a remanent magnetization. Considering their exceptional magnetic properties, these materials are understandably used in the field of conversion of electromagnetic energy.
For industrial frequencies, with the exception of ferromagnetic materials, it is commonly accepted that the vacuum behavior law (see relation [1.8]) is perfectly suited for the modeling of the magnetic behavior law of material media.
1.2.2.4. Ferromagnetic materials
As shown in Figure 1.1, for ferromagnetic materials, magnetization describes a hysteresis loop when the magnetic field varies alternatively. In this figure, Mr represents the remanent magnetization, Μs represents the saturation magnetization and Hc represents the coercive field. It can be readily shown that the energy dissipated as losses, during a loop, is equal to the loop area. It can also be noted that magnetization M(H) varies with temperature, and beyond the Curie temperature the material exhibits paramagnetic behavior. Finally, ferromagnetic materials often have anisotropic magnetic properties, which means that their behavior varies according to the direction of the applied magnetic field.
Figure 1.1.M(H) characteristic of a ferromagnetic material
Depending on the value of the coercive field intensity Hc, two families of magnetic materials can be identified:
Soft magnetic materials, for which H
c
is below several hundred amperes per meter. They are mainly used for concentrating and driving the circulation of the magnetic field. The material then behaves as a “good magnetic conductor”.
Hard magnetic materials, for which the coercive field intensity H
c
is very high. These materials are used as permanent magnets.
1.2.2.4.1. Soft magnetic materials
In order to represent soft ferromagnetic materials, it is possible, for certain applications, to ignore the anisotropy and also the hysteresis phenomenon. This new behavior law is then deduced from the M(H) characteristic, shown in Figure 1.1, by considering the anhysteretic curve (one-to-one curve experimentally measured according to a standard). In this case, magnetic flux density can be written as:
or by introducing a nonlinear magnetic permeability μ(H) = μ0(1 + χm (H)):
Finally, it is also possible to take into account only the linear part of the magnetic characteristic. This yields:
1.2.2.4.2. Permanent magnets
As for the hard magnetic materials, used as permanent magnets, they are of various types. For example, we can mention iron–nickel–aluminum–cobalt (Alnico) alloys, rare-earth-based alloys (samarium-cobalt (SmCo) and neodymium–iron–boron (Nd–Fe–B)) and ferrites.
Figure 1.2(a) shows the useful characteristics (part of the hysteresis loop that is exploited when the material “operates” as a permanent magnet) of the materials commonly used as permanent magnets.
This figure shows that except for the case of Alnico-type alloys, the useful characteristic of permanent magnets can be represented by a linear characteristic, as shown in Figure 1.2(b). This is:
where μA represents the magnetic permeability of the permanent magnet (close to μ0) and Br the remanent flux density.
Figure 1.2.a) Characteristics of the most common hard ferromagnetic materials used as permanent magnets; b) simplified representation
1.3. Interface between two media and boundary conditions
Before studying the behavior of materials at the interface between two media with different properties, it is important to recall the definition of a vector field based on its normal and tangential components at a point on a surface. Considering a vector, denoted by u (see Figure 1.3), at a point M of a surface Γ, and n the normal vector to the surface at this point, it can be decomposed into its normal component un and tangential component ut as follows:
where un = (n.u)n is the normal component, and ut = n ∧ (u ∧ n) is the tangential component (the operator “.” denotes the scalar product and “∧” denotes the vector product).
NOTE.– For the sake of simplicity, when conditions are imposed on the tangential component, it is preferable to use the term u ∧ n. Indeed, it can be verified that if u ∧ n = 0, then ut = 0. Similarly, if two vectors u1 and u2 have equal tangential components, this is equivalent to having u1 ∧ n = u2 ∧ n.
Figure 1.3.Definition of a vector field based on its normal and tangential components
Having set these definitions, the next section describes how they are used to define the continuity conditions between two media and the boundary conditions.
1.3.1. Continuity conditions between two media
The five already defined vector fields, E, H, B, D and J, have certain properties when passing from one medium to another (Ida 2020). These properties are derived from equations [1.1], [1.3], [1.4], [1.5] and [1.6].
1.3.1.1. Electric and magnetic fields
It can be shown that if the curl of a field is defined, then the tangential component of this field is continuous at the interface between two media that may have different characteristics. Since the curl of the electric field is defined by equation [1.1], it can be deduced that:
where Ekt (k ∈ {1,2}) represents the component of the electric field tangential to the interface. This result shows that at the interface between two media with different properties, the tangential component of the electric field is conserved.
For the magnetic field, as defined by equation [1.5], similar to the case of the electric field, the following can be written:
This relation shows that at the interface between two media the tangential component of the magnetic field is conserved.
1.3.1.2. Electric displacement field, magnetic flux density and current density
Likewise, it can be shown that if the divergence of a field is defined, then its normal component is continuous at the interface between two media with different physical properties.
The divergence of the fields D, B and J is defined, and this property is then applied. For the electric displacement field, based on equation [1.4] and in the absence of surface charge density, the following expression can be written:
where Dkn (k ∈ {1,2}) represents the normal component of the electric displacement field on the interface.
Concerning the magnetic flux density, based on equation [1.3], the following expression can be written:
where Bkn (k ∈ {1,2}) represents the normal component of the magnetic flux density. Therefore, at the interface between two media, the normal component of the magnetic flux density is conserved.
As already noted for the electric displacement field and the magnetic flux density, equation [1.6] leads to the relation:
therefore at the interface between two media the normal component of the current density is conserved.
1.3.1.3. Refraction of field lines
In order to alleviate the developments, this section considers a two-dimensional (2D) case, limited to the pair of fields composed of the magnetic field and the magnetic flux density {H,B}. The conclusions that will be drawn are, however, valid for the three-dimensional (3D) case and can be extended to the case of the {E,D} and {E,J} pairs that verify the same conservation conditions as the {H,B} pair.
Figure 1.4.Normal and tangential components of fields B1, H1, B2 and H2
As shown in Figure 1.4, let us consider an interface Γ between two magnetic materials denoted by 1 and 2. The behavior law of the materials, assumed to be isotropic and linear, is given by relation [1.26] with permeabilities μ1 and μ2. Under these conditions, the magnetic fields H1 and H2 are, respectively, collinear with magnetic flux densities B1 and B2. Finally (see Figure 1.4), the projection of the fields onto the two axes “n” and “t”, corresponding to the normal and tangential components, meets the properties expressed by [1.30] and [1.32].
α1 and α2 are the angles made with the normal n directed from medium 2 to medium 1 by the two pairs of fields {H1,B1} and {H2,B2}, respectively.
An elementary calculation, based on the continuity properties of the normal and tangential components [1.30] and [1.32] and on the behavior law [1.26], leads to the relation:
where represents a refractive index such that .
Let us now consider the following case: if μ1→ ∝, then , tgα2 ≈ 0 and α2 = 0. This implies that the pair {H1,B1} is normal to the surface and therefore Ht = 0. On the contrary, if μ1 → 0, then , tgα2 → ∝ and α2 = π/2. In this case, the pair {H1,B1} is tangential to the surface and the component Bn = 0.
When considering the pairs {E1,D1} and {E2,D2} with the behavior law [1.19] or the pairs {E1,J1} and {E2,J2} with the behavior law [1.20], the same conclusions are reached. Under these conditions, the refractive index is equal to ε2 / ε1 and σ2 / σ1, respectively.
As an example, let us consider the case of a conductive material whose conductivity has a finite value σ2. If it is brought into contact with another conductive material, whose conductivity σ1 tends to infinity, then and the tangential component of the electric field strength E and of the current density J at the interface is equal to zero. In this case, the interface can be considered a gate for the current density.
On the contrary, if the conductive material, of conductivity σ2, is in contact with an insulating material, whose conductivity is σ1 = 0, then and the normal component (of E and J) to the interface between the two media is equal to zero. This interface will be considered a wall.
These considerations will be very useful in the following section, particularly when boundary conditions imposed at the boundary of a domain are imposed.
1.3.2. Boundary conditions
For the study of electromagnetic systems, a well-posed formulation of the problem requires imposing spatial boundary conditions to the fields. For an infinite domain, these conditions are applied to infinity. In the case of numerical simulation, the domain is often limited to a part of the space. In this case, boundary conditions should be imposed at the boundaries of the domain. These boundary conditions may be derived either from symmetry conditions of the problem or from properties of the materials that are in contact with the boundary (see section 1.3.1). For example, if the boundary is in contact with a highly insulating material, then the normal component of the current density is imposed to zero. To have a physical meaning, these conditions always relate to the conservative (normal or tangential) component of the concerned field. Therefore, if a condition applies to the magnetic field, it concerns the tangential component. On the contrary, in the case of magnetic flux density, it relates to the normal component.
However, in the context of problems evolving in time, a generally imposed condition is that the value of fields at the initial instant t = 0 is equal to zero.
Taking into account the notations introduced after equation [1.28] for the normal and tangential components, the boundary conditions on the fields E, H, D, B and J can be written, for a large number of applications, as follows:
These conditions, known as “homogeneous boundary conditions”, can be interpreted as follows:
Equation [1.35]
indicates that the tangential component of the electric field is equal to zero on the boundary Γ
e
and therefore the electric field
E
is normal to this surface. Using the expression introduced at the end of
section 1.3.1.3
, this boundary can be considered a gate for the field
E
. These are gate-type boundary conditions.
Equation [1.36]
leads to the same interpretations for the magnetic field intensity
H
on the boundary Γ
h
. It can also be considered a gate for the magnetic field.
Equation [1.37]
indicates that the normal component of the magnetic flux density is equal to zero on the boundary Γ
b
. As already seen in
section 1.3.1.3
, this condition requires the boundary to behave as a wall for the magnetic flux density
B
. These are referred to as wall-type boundary conditions.
Equation [1.38]
leads to the same interpretations for the current density, i.e. Γ
j
behaves as a wall for the current density
J
.
Equation [1.39]
, similar to
equations [1.37]
and
[1.38]
, shows that the boundary Γ
d
behaves as a wall for the electric displacement field
D
.
It can be shown that relations [1.1] and [1.35] imply equation [1.37]. Similarly, relations [1.5] and [1.36] imply equation [1.38]. On the contrary, the reverse is not true and depends on the topology of the domain (see Chapter 2).
1.4. Integral forms: fundamental theorems
The above-stated Maxwell’s equations provide local information on electromagnetic fields. The integral form of these equations leads to general theorems that are commonly used in electromagnetism. These theorems can be used to connect local quantities (vector fields) and global quantities such as the electromotive force “e”, the current density flux “I”, the magnetic flux “ϕ”, the magnetomotive force “fm” and the total charges “Q”.
1.4.1. Faraday’s law
Given a rigid loop Cs, the boundary of a surface denoted by S (see Figure 1.5), consider equation [1.1] that connects the electric field and the magnetic flux density. Integrating the equation over the surface S yields:
Using the Stokes theorem and inverting the operator differentiated with respect to time with the surface integral (which is possible, as the loop is assumed rigid), the following can be written:
Figure 1.5.Faraday’s law implementation example
The left-hand side term of this equation corresponds to the electromotive force “e” induced in the loop, and the right-hand side term corresponds to the time derivative of the magnetic flux through the surface S (denoted by ϕ). This leads to Faraday’s law:
1.4.2. Ampère’s law
As shown in Figure 1.6, consider a conductor carrying a current density J and a surface S bounded by a contour Cs. Based on equation [1.5], using the same approach as for Faraday’s law, the following can be written:
Using the Stokes theorem, the term on the left-hand side is replaced by the circulation of the magnetic field along the contour Cs. The term on the right-hand side, which represents the flux of J, is therefore equal to the value of the electric current (denoted by “I”) flowing through the surface S. This relation leads to Ampère’s law, namely:
Figure 1.6.Illustration of Ampère’s law: conductor carrying a current
1.4.3. Law of conservation of the magnetic flux
Equation [1.3] provides information related to the behavior of the magnetic flux density, i.e. it is divergence free. In order to analyze this property, consider the case of the domain Ω, of boundary Γ = Γh1 ∪ Γb ∪ Γh2, defined in Figure 1.7. A magnetic flux density B flows through this domain. The boundary condition on the boundaries Γh1 and Γh2 is [1.36], and on the lateral boundary Γb it is [1.37]. This is known as the flux tube.
Figure 1.7.Flux tube: law of conservation of the flux
Calculating now the volume integral over the domain Ω of equation [1.3], we obtain:
Applying to this equation Ostrogradski’s theorem, also known as the “divergence theorem”, we have:
Magnetic flux density is therefore a conservative flux vector field. This means that the magnetic flux flowing through a closed surface (in this case, the surface Γ of the domain Ω) is equal to zero. In the studied example, decomposing the boundary Γ (Γb, Γh1, Γh2), the following can be written:
Considering the boundary conditions on the lateral surface Γb (B.n = 0), it can be deduced that the incoming flux through Γh1 (see the orientation of the normal vectors in Figure 1.7) is naturally equal to the outgoing flux through Γh2.
It is important to note that the divergence of the current density is also zero (see equation [1.6]). Under these conditions, it has the same properties as the magnetic flux density, i.e. it is a conservative flux vector field. This reflects the fact that electric current is conserved all along a conductor.
1.4.4. Gauss’ law
This section focuses on equation [1.4] that links the electric displacement field to the electric charge density ρ. To study the properties of this equation, consider the domain Ω of boundary Γe enclosing a charge density ρ (see Figure 1.8).
Figure 1.8.Illustration of Gauss’ law
Let us calculate, for equation [1.4], the volume integral over the domain Ω. This yields:
Applying Ostrogradski’s theorem for the divergence operator leads to:
The term on the right-hand side corresponds to the total charges Q inside the domain, therefore:
As for the term on the left-hand side, it corresponds to the electric flux ϕe through the surface of the domain Ω. This reflects Gauss’ law, i.e. the electric flux through a closed surface is equal to the total charges Q enclosed by the volume defined by this surface.
1.5. Various forms of Maxwell’s equations
Depending on the given problem, in the context of low-frequency electromagnetism (see section 1.1), it is possible to simplify the initial model defined by equations [1.1], [1.3], [1.4] and [1.5]. Static and quasi-static problems are then identified. Concerning static problems, our focus is on studying the problems of electrostatics, electrokinetics and magnetostatics. As far as quasi-static problems are concerned, this book focuses only on magnetoquasistatics, commonly referred to as “magnetodynamics”. The following section studies these various forms and introduces, for each of them, the boundary conditions and the notion of source term.
1.5.1. Electrostatics
Electrostatics aims to study, within a given domain Ω, the distribution of the electric field and of the electric displacement field in the presence of static source terms. The study is conducted at electrostatic equilibrium; therefore, the problem to be addressed is stationary in time. As an example, Figure 1.9 shows a domain of permittivity ε0 inside of which there is a subdomain Ω1 of permittivity ε1. On the boundary, there are two types of boundary conditions, Γdk (see equation [1.39]) with k ∈ {1,2} and Γek (see equation [1.35]) with k ∈ {1,2}. It is important to recall that Γdk represents a wall for the electric displacement field and Γek is a gate for the electric field. The two gates, Γe1 and Γe2, are in contact with electrodes and .
For this example, the source term can be the circulation fs of the electric field strength E along an arbitrary path γ12 (see Figure 1.9) linking the two electrodes:
At the surface of the electrodes, located on the boundary of the domain Ω, the electric displacement field has the following property:
where σs is the surface density of charges on the boundary with the electrode and n is the outgoing unit normal vector. The expression of the amount of charges Qσ on each electrode is:
In this case, Maxwell’s equations (see equations [1.1]–[1.4]) in electrostatics and in the absence of electric charge density within the domain lead to solving two equations:
Figure 1.9.Representation of an electrostatic problem
The solution to these equations aims to find a curl-free electric field [1.54] that verifies equation [1.55] via the dielectric behavior law [1.19] and also the boundary conditions on the boundary of the domain, defined in Figure 1.9.
1.5.2. Electrokinetics
Electrokinetics studies the distribution of the electric field and of the current density in a conductor in the presence of charges in motion, when the speed of these charges is constant.
As an example, consider the set-up represented in Figure 1.10. The conductive domain Ω is composed of a main region of conductivity σ1 surrounding two subregions of conductivity σ2 and σ3. On the boundary of the domain, there are two wall-type boundaries for the current density (Γj1 and Γj2) and two other gate-type boundaries for the electric field Γe1 and Γe2. The boundary conditions on these boundaries are defined, respectively, by equations [1.38] and [1.35].
Two types of source terms can be applied on the boundaries Γe1 and Γe2:
the first is an electromotive force, denoted by “e”, which corresponds to the circulation of the electric field on a path γ
12
(see
Figure 1.10
) inside the domain, linking the two surfaces Γ
e1
and Γ
e2
, such that:
the second consists of imposing the current density flux, denoted by I, to the surfaces Γ
e1
and Γ
e2
. Its expression is:
where k ∈ {1,2}.
In this context, Maxwell’s equations can be written as:
which can be completed by the electric behavior law [1.20] and the boundary condition [1.35] on the boundaries Γek and [1.38] on the boundaries Γjk.
Figure 1.10.Representation of an electrokinetic problem
1.5.3. Magnetostatics
The magnetostatics problem aims to study the distribution of the magnetic field H and of the magnetic flux density B for source terms that are time invariant. In this context, the distribution of the current density, denoted by J0, is assumed to be known, unlike in the case of electrokinetics.
For the study of magnetostatics, the general case is considered, as illustrated in Figure 1.11. Given a domain Ω of boundary Γ such that: Γ = Γb1 ∪ Γb2 ∪ Γh1 ∪ Γh2. The boundaries Γbk (k ∈ {1,2}) are of wall type for the magnetic flux density (see equation [1.37]). On the contrary, Γh1 and Γh2 represent a gate for the magnetic flux density (see equation [1.36]).
Figure 1.11.Representation of a magnetostatic problem
The studied domain, of permeability μ0, contains a ferromagnetic material of permeability μ [1.26] and source terms, hence a permanent magnet (denoted by “PM” in Figure 1.11) and a conductor carrying a current density J0 that is also referred to as inductor. Between the two gates, Γh1 and Γh2, it is possible to impose a magnetomotive force (or a magnetic flux through both of them).
The following section details the various source terms with the associated equations:
the magnetomotive force
f
m
that is imposed between the two boundaries Γ
h1
and Γ
h2
is defined by:
where γ12 represents a path through the domain Ω linking the boundaries Γh1 to Γh2, as shown in Figure 1.11;
the magnetic flux, denoted by ϕ, can be imposed on the two surfaces Γ
h1
and Γ
h2
such that:
where k ∈ {1,2}. When the two boundaries, Γh1 and Γh2, are separated by surfaces of type Γb, the incoming flux through Γh1 is equal to the outgoing flux through Γh2 (see equation [1.47] related to the law of conservation of the magnetic flux). The two source terms fm and ϕ are exclusive, in the sense that they cannot be imposed simultaneously;
an inductor, carrying a current density
J
0
. In the case of multi-wire winding, by knowing the intensity I of the current through a conductor, the current density
J
0
is defined by:
where Sc represents the cross-section of the wire conductors and n is the unit normal vector of current density whose direction corresponds to the geometrical orientation of the conductors;
a permanent magnet, characterized by its behavior law. A simplified characteristic is generally used, as shown in
Figure 1.2(b)
, which can be written in the form of
equation [1.27]
. It can be easily verified that the coercive field
H
c
has the following expression:
Based on equations [1.27] and [1.63], the magnetic field strength H
