Electrothermics E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 288

Veröffentlichungsjahr: 2012

Beschreibung

This book concerns the analysis and design of induction heating of poor electrical conduction materials. Some innovating applications such as inductive plasma installation or transformers, thermo inductive non-destructive testing and carbon-reinforced composite materials heating are studied. Analytical, semi-analytical and numerical models are combined to obtain the best modeling technique for each case. Each model has been tested with experimental results and validated. The principal aspects of a computational package to solve these kinds of coupled problems are described. In the first chapter, the mathematical tools for coupled electromagnetic and thermal phenomena are introduced. In Chapter 2, these tools are used to analyze a radio frequency inductive plasma installation. The third chapter describes the methodology of designing a low frequency plasma transformer. Chapter 4 studies the feasibility of the thermo inductive technique for non-destructive testing and the final chapter is dedicated to the use of induction heating in the lifecycle of carbon-reinforced composite materials. Contents 1. Thermal and Electromagnetic Coupling, Javad Fouladgar, Didier Trichet and Brahim Ramdane. 2. Simplified Model of a Radiofrequency Inductive Thermal Plasma Installation, Javad Fouladgar and Jean-Pierre Ploteau. 3. Design Methodology of A Very Low-Frequency Plasma Transformer, Javad Fouladgar and Souri Mohamed Mimoune. 4. Non Destructive Testing by Thermo-Inductive Method, Javad Fouladgar, Brahim Ramdane, Didier Trichet and Tayeb Saidi. 5. Induction Heating of Composite Materials, Javad Fouladgar, Didier Trichet, Samir Bensaid and Guillaume Wasselynck

Leseprobe

Electrothermics

Edited by Javad Fouladgar

First published 2012 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:

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Library of Congress Cataloging-in-Publication Data

Electrothermics/edited by Javad Fouladgar.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-242-8

1. Thermoelectric apparatus and appliances. 2. Thermoelectricity. I. Fouladgar, Javad.

TK2950.E43 2012

537.6′5-dc23

2011052451

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-242-8

Table of Contents

Introduction Induction Heating: Principles and Applications

I.1. Principle

I.2. Characteristics

I.3. Power supply

I.4. Industrial potential of induction

I.5. Bibliography

Chapter 1 Thermal and Electromagnetic Coupling

1.1. Introduction

1.2. Electromagnetic problem

1.3. Thermal problem

1.4. Magnetothermal coupling

1.5. Solving the electromagnetic and thermal equations

1.6. Conclusion

1.7. Bibliography

Chapter 2 Simplified Model of a Radiofrequency Inductive Thermal Plasma Installation

2.1. Introduction

2.2. Plasma and its characteristics

2.3. Modeling a plasma installation

2.4. Calculating charge impedance

2.5. Generator model

2.6. Conclusion

2.7. Bibliography

Chapter 3 Design Methodology of a Very Low-Frequency Plasma Transformer

3.1. Introduction

3.2. Different types of very low-frequency applicators

3.3. Simplified analytical model for analysis and preliminary design

3.4. Nonlinear model

3.5. Plasma stability in the transitory and sinusoidal states

3.6. Advanced inductive plasma transformer model

3.7. Plasma initialization

3.8. Conclusion

3.9. Bibliography

Chapter 4 Non Destructive Testing by Thermo-Inductive Method

4.1. Introduction

4.2. Principles of the thermo-inductive method

4.3. Basic thermo-inductive technique theory

4.4. Application of the thermo-inductive method to inspect massive magnetic steel components

4.5. Comparison with infrared thermography

4.6. Applications on composite materials

4.7. Conclusion and general instructions

4.8. Bibliography

Chapter 5 Induction Heating of Composite Materials

5.1. Introduction

5.2. Composite materials

5.3. Lifecycle of composite materials

5.4. Induction and the lifecycle of composite materials

5.5. Identifying the physical properties of composite materials by experimental methods

5.6. Homogenization techniques

5.7. Heating composite materials by induction

5.8. Setup model

5.9. Influence of the folds’ orientation

5.10. Difficulty of the electrothermal coupling

5.11. Validating the electrothermal model

5.12. Conclusion

5.13. Bibliography

Index

Introduction

Induction Heating: Principles and Applications1

I.1. Principle

Induction heating is a technique that can heat a material without direct contact with the electrical energy source. It consists of placing the object to be heated in a time-varying electromagnetic field and dissipating the energy induced inside the object as heat [DEV 00] (Figure I.1). It is appropriate for materials that are electrical conductors or semiconductors. It is used for many thermal processes, especially for the fusion or thermal treatment of metals [DEV 00].

Induction heating is based on four physical phenomena:

– creating an electromagnetic field using a solenoid with an electric field as described by Ampère’s law;

– electromagnetic induction, as expressed by the Faraday–Lenz law;

– heat production by Joule effect; and

– heat diffusion by conduction, as expressed by Fourier’s law.

In their differential form, these laws are expressed by the Maxwell equations and the heat transfer equation.

Figure I.1.Induction installation schematic

I.2. Characteristics

A general characteristic of induction heating is the non-uniform power distribution in the thickness of the material to be heated. This phenomenon, known as the skin effect, arises from the fact that the inducted currents create fields opposed to the source fields, and leave the object’s core to “group” in a zone known as the skin depth. In this layer near the material’s surface, 63% of the inducted currents and 87% of the generated power are concentrated (Figure I.2).

Figure I.2.Skin depth

The skin depth can be deduced from Maxwell’s equations. For a semi-infinite plate, it is given by:

[[I.1]]

where is the magnetic permeability of the object to be heated (H.m−1), is its electric conductivity (.m)−1, and is the frequency of the induction generator. Of these three parameters, only the frequency can be modified by the user. Therefore, the frequency is a tool to change the skin depth.

For a given material and a given type of heating, the skin depth is the most important parameter for defining the heating frequency. Thus, melting steel (i.e. heating a volume of a conductive, ferromagnetic material) requires weak frequencies, of the order of a few kHz. But, to maintain an inductive plasma at 10,000 K, greater frequencies of the order of a few MHz are required. Table I.1 groups skin depth as a function of frequency for some materials traditionally heated by induction.

Table I.1.Skin depth of various materials

I.3. Power supply

The power supply technology depends strongly on the heating frequency. The main power supply technologies are the following:

– Direct link to the network. In this case, the inductor is directly linked to the network transformer that regulates current and power.

– Thyristor converters with frequencies between 100 Hz and 10 kHz.

– Transistor converters with frequencies of up to 2 MHz.

– Triode converters with frequencies of up to 5 MHz.

In an application of induction heating, the impedance of charge is often strongly inductive. We then use an impedance matching unit to improve the power factor and to obtain a nearly resistive charge with respect to the generator.

The efficiency of transistor generators is of the order of 90%. However, triode generators have an efficiency of less than 70% due to the losses in the triode’s anode.

I.4. Industrial potential of induction

Induction heating was used for decades (and is still used today) for the fusion, soldering, elaboration, and the surface treatment of metals (especially steel). It is, however, almost absent in other industrial sectors, although it presents numerous advantages and a great potential for innovation in many fields.

In our opinion, the absence of induction in these sectors is not only due to the ignorance of users and manufacturers of non-metallic materials regarding induction, but also due to the lack of mastery in modeling and experimentation of the induction experts in these fields. This mastery begins by making simulations of these materials in an induction heating process, which must be applied to the real cases to be validated and modified. Since 1987, the LRTI (Laboratoire de Recherche en Techniques Inductives, which later became a member of the IREENA) has worked on innovative applications of induction heating. It has developed an analysis and conception methodology for the control of these applications, at the levels of both academic research and industrial collaboration. This book takes up some of these research activities.

Induction heating creates coupled electromagnetic and thermal phenomena (and sometimes flow). Chapter 1 of this book presents the methodology and mathematical tools required to overcome the difficulties linked to these problems, not only in a general context but also in applications with low-conductivity or anisotropic materials [TRI 00, RAM 09].

In Chapter 2, a simplified model of an installation of the inductive radiofrequency plasma is discussed. This model combines a simplified flow model and axisymmetric models of the electromagnetic and thermal equations. These equations are finally coupled to the circuit equations of the generator to obtain a complete tool of analysis of the installation. To complete this chapter, in situ measurement methods are introduced to validate the global model [PLO 97].

The design methodology of a low-frequency plasma transformer is discussed in Chapter 3. A nonlinear one-dimensional model is introduced to size the plasma and calculate its maintenance voltage and temperature. Then, plasma setup methods and the influence of the magnetic circuit’s number of arms are analyzed using a coupled three-dimensional (3D) model [MIM 96].

In Chapter 4, the feasibility of a new technique of non-destructive testing (NDT) based on the principles of induction heating is studied. This new technique combines NDT by Eddy currents and infrared thermography. A tool to guide the design is introduced and the default detection conditions of this technique are studied. This chapter ends with general instructions and perspectives for the industrial development of the technique [RAM 09].

The last chapter of this book tackles the problem of induction heating of composite materials. These materials, especially used in the aeronautical industry, have multiscale, multilayer, heterogeneous, and anisotropic characteristics. Homogenization techniques are described in this chapter to calculate the global characteristics of these materials. A thorough study is made with regards to understanding the circulation of inducted currents within these materials. Coupled 3D anisotropic and multilayered systems are then introduced to calculate the electric field and the temperature distribution [TRI 00, BEN 06, WAS 10].

I.5. Bibliography

[BEN 06] BENSAID S., Contribution à la caractérisation et à la modélisation électromagnétique et thermique des matériaux composites anisotropes, PhD thesis, University of Nantes, France, December 2006.

[DEV 00] DEVELEY G., “Chauffage par induction électromagnétique, principe”, Techniques del’ingénieur, Référence D5935, February 2000.

[MIM 96] MIMONE S., Contribution à la conception, à la modélisation et à la réalisation d’un transformateur à plasma inductif annulaire basse fréquence, PhD thesis, University of Nantes, France, November 1996.

[PLO 97] PLOTEAU J.P., Contribution à la simulation électrique, électromagnétique et thermique d’un ensemble générateur et applicateur de plasma inductif: Validation des modèles par des mesures globales et locales, Thesis, University of Nantes, France, July 1997.

[RAM 09] RAMDANE B., Contribution à la modélisation tridimensionnelle de la technique thermo-inductive de contrôle non destructif: Développement d’un outil de conception, d’analyse et d’aide à la décision, PhD thesis, University of Nantes, France, 2009.

[TRI 00] TRICHET D., Contribution à la modélisation, à la conception et au développement du chauffage par induction des matériaux composites, PhD thesis, école Doctorale Sciences pour l’Ingénieur de Nantes, France, January 2000.

[WAS 10] WASSELYNCK G., TRICHET D., RAMDANE B., FOULDAGAR J., “Interaction between electromagnetic field and CFRP materials: a new multiscale homogenization approach”, IEEE Transactions on Magnetics, vol. 46, no. 8, pp. 32777–3280, 2010.

1 Introduction written by Javad FOULADGAR.

Chapter 1

Thermal and Electromagnetic Coupling1

1.1. Introduction

In induction heating of conductive and semi-conductive materials, electromagnetic, thermal, and sometimes flow phenomena are present simultaneously. These phenomena are governed by Maxwell’s equations, the heat transfer equation, and the Navier–Stokes’ equation, respectively.

In most cases, these equations are mutually coupled by one or more parameters. The electromagnetic and thermal equations are linked by electromagnetic conductivity, magnetic permeability, and induced power. The coupling between the thermal equation and that of flow is done by the density and fluid velocity, and the link between the electromagnetic and flow equations is due to electromagnetic forces.

These phenomena are often governed by partial differential equations where the unknown is either a scalar or a vector quantity, dependent on spatial coordinates and time. In most cases, numerical methods are used to solve these equations. Nevertheless, in certain simple cases and for the understanding of the involved physical phenomena, analytic methods are also used.

In this chapter, we present different local formulations of the phenomena acting in the context of induction heating. For ease of reading, we limit ourselves to the electromagnetic and thermal equations and deal with the problem of flow in Chapter 2, in which we study inductive plasma.

We also study the problem of solving these equations using analytic, semi-analytic, and numerical methods. Particular attention is given to two numerical methods: the finite volume methods and the finite element method, which will be applied to solve the electromagnetic and thermal equations using iterative coupling algorithms.

Finally, we present a computational package’s architecture using all these methods to define the geometry, to solve partial differential equations, and to analyze the results.

1.2. Electromagnetic problem

1.2.1. Local formulation of the electromagnetic problem

Generally speaking, an electromagnetic device can be represented with the help of Figure 1.1.

Figure 1.1.Device configuration

This device consists of non-conductive regions (air or isolating materials), ferromagnetic and/or conductive materials, and sources of magnetic fields (inductor). The whole device forms the domain of study with boundary .

1.2.1.1. Maxwell’s equations

The set of electromagnetic phenomena can be described by six values that depend on the space variable x and time t. They can be written as:

– H: magnetic field (A/m);

– E: electric field (V/m);

– B: magnetic induction (Tesla);

– D: electrical induction (C/m2);

– J: conduction current density (A/m2);

– : free electric charge density per unit volume (C/m3).

These different quantities are linked by Maxwell’s equations, which describe the laws of electricity and magnetism in their most general form [DUR 68, FOU 85]:

Ampère’s law:

[1.1]

where represents the displacement current .

Faraday’s law:

[1.2]

Magnetic flux conservation law:

[1.3]

Gauss’ law:

[1.4]

Equations [1.1] and [1.2] express the coupling and evolution of electric and magnetic quantities whereas equations [1.3] and [1.4] show flux conservation.

From equations [1.1] and [1.4] we obtain the conservation law of electric charge:

[1.5]

1.2.1.2. Interaction between electromagnetic waves and materials

Maxwell’s equations are general and apply to all environments. To predict the behavior of electromagnetic phenomena, we must add relations specific to particular environments to these equations: the constitutive relations. These relations are formulated as follows:

[1.6]

1.2.1.3. Vector and scalar potentials

Magnetic and electric fields in Maxwell’s equations can be represented using vector or scalar potentials. The advantage of this representation lies in the simplification of these equations and their adaptation to different devices. The potentials most often used in the literature are the following:

– Magnetic vector potential:

From the flux conservation law:

[1.7]

where A is the magnetic vector potential.

– Electric vector potential:

In regions without volume charges or in the regions where the volume charges are time independent, equation [1.5] becomes:

[1.8]

where T is the electrical potential vector.

– Electric scalar potential

By combining equations [1.2] and [1.7], the field E can be written as a function of A and the gradient of a scalar potential as:

[1.9]

where V is the electric scalar potential.

– Magnetic scalar potential

In devices where the displacement current D/t is negligible, the magnetic field H can be written as a function of T within one gradient by combining equations [1.1] and [1.8]:

[1.10]

where, is the magnetic scalar potential and T0 the field created by the source current.

Figure 1.2.Boundary between two media

1.2.2. Boundary conditions

1.2.2.1. Boundary conditions between two different media

The electromagnetic fields are discontinuous at the boundary of two different media (see Figure 1.2), and are therefore no longer differentiable. The relations between the electromagnetic values at the interface, called boundary conditions, can then be written as [FOU 85]:

[1.11]

[1.12]

[1.13]

[1.14]

where n is the unit vector normal to the surface, directed from media 2 toward media 1, and s and K are the charge density per unit surface area and the current density per unit length, respectively.

1.2.2.2. Boundary conditions at the domain’s limits

The system composed of Maxwell’s equations and the constitutive laws has an infinite number of solutions. Consequently, to ensure the uniqueness of the solution, boundary conditions are imposed on the domain limits.

[1.15]

From equation [1.1], we can deduce that:

[1.16]

And on boundary B, we impose boundary conditions of the form:

[1.17]

From equation [1.2], we can deduce that:

[1.18]

1.2.3. Functional spaces

Solving Maxwell’s equations (which are partial differential equations) requires the introduction of a mathematical structure capable of accepting that sort of equation: functional spaces of scalar and vector fields [BOS 98, DUL 96]. Let be the studied domain of boundary . L2 and L2 are the spaces of scalar functions and vector functions, respectively, which are square-integrable in . On these two spaces, we define the scalar product of two functions:

[1.19]

From these two functional spaces, we can build subspaces in which we can seek solutions to Maxwell’s equations.

In electromagnetism, we principally distinguish between three differential operators that are subspaces of L2 and L2 [BOS 98]:

[1.20]

The subspaces associated with the boundary conditions are then as follows:

[1.21]

To complete the set of functional spaces, two subspaces (denoted by W3B and W3H) are introduced. They are the images of W3B and W3H by the div operator. Thus, we can define the Tonti diagram that represents a series of subspaces. Passing from one space to another is carried out by using a differential operator [BOS 88].

1.2.4. Tonti diagrams

Maxwell’s equations can be described in two dual formulations: Faraday’s and flux conservation laws on the one hand, and the Ampère–Maxwell and Gauss theorems on the other. The Tonti diagram shows this duality [BOS 88]. It consists of a four-level, two-column diagram (Figure 1.3).

Figure 1.3.Tonti diagram

The equations are read vertically on both sides of the diagram, but the constitutive laws are read horizontally. They are linked by the Hodge transformation in differential geometry [BOS 91, REN 97].

To take the temporal derivative into account, we have introduced a third dimension to Figure 1.4. The derived field potentials are also introduced (scalar potentials and vector potentials). They have a predetermined position in the Tonti diagram, and are naturally placed downstream of fields that can be derived from them.

Figure 1.4.Tonti diagram with temporal variations

1.2.5. Different formulations of the electromagnetic field

All electromagnetic problems can be represented by the diagram in Figure 1.4. For example, the combination of the back plane with the plane linking the forward left and the back right columns represents the magnetodynamic problem. Hereafter, the principal formulations used to study electromagnetic problems will be detailed using Tonti diagrams.

1.2.5.1. Magnetostatic formulation

In a magnetostatic problem, we consider the study of phenomena in a stationary state. This means we cancel all the temporal derivatives in Maxwell’s equations. Figure 1.5 shows a typical magnetostatic problem.

Figure 1.5.Magnetostatic problem

Generally speaking, such a problem consists of an inductor with current density J, a permanent magnet with remnant induction Br, a ferromagnetic region (saturated or not), all enveloped by a box of air with boundary . From the hypotheses described previously, we deduce the new expression for Maxwell’s equations:

[1.22]

[1.23]

1.2.5.1.1. Magnetostatic formulation in magnetic scalar potential

For a stationary state, equation [1.10] becomes:

[1.24]

where, T0 is the source field.

From equations [1.3] and [1.24], we obtain the formulation in magnetic scalar potentials:

[1.25]

1.2.5.2. Magnetostatic formulation in magnetic vector potentials

By substituting the induction field B obtained from equation [1.7] in equation [1.1], we obtain the formulation in magnetic vector potentials.

[1.26]

with boundary conditions:

[1.27]

[1.28]

For a geometrically simple inductor, we can analytically calculate the magnetic field from T0 created by JS. In other cases, the source field is determined by numerical methods [HEN 07, REN 96]. To find the null divergence of the source current density, the term JS is expressed by the rotational of a source field T0:

[1.29]

1.2.5.3. Magnetodynamic formulation

In this case, the phenomena are no longer assumed to be stationary and the inductor’s current is time varying. Figure 1.6 shows such a configuration.

Figure 1.6.Magnetodynamic problem

In this case, if we ignore displacement currents, Maxwell’s equations are written as:

[1.30]

[1.31]

With the following constitutive relations:

[1.32]

1.2.5.4. Magnetodynamic formulation in A-V

By combining equations [1.9], [1.30], [1.31], and [1.32], we obtain the formulation in magnetic vector potential A and electric scalar potential V [BIR 89b, MEU 91].

[1.33]

[1.34]

1.2.5.5. Magnetodynamic formulation in T–T0–

In this formulation, we assume that:

[1.35]

By combining equations [1.8], [1.31], [1.32], and [1.35], we obtain the formulation in electric vector potential and magnetic scalar potential [BIR 89b, MEU 91]:

[1.36]

[1.37]

1.2.5.6. Formulation in H-[DUL 96]

From the Ampère equation and neglecting the displacement currents, we obtain:

[1.38]

the rotational of which yields the following expression:

[1.39]

This equation, associated with adequate boundary conditions, has a unique solution. To take non-conductive environments into account, we must couple the preceding equations with a formulation in scalar potential [LEM 99]. In this case, the magnetic field can be written as:

[1.40]

In the regions where a current source J exists, the created field is given by:

[1.41]

where, T0 is the source field.

For non-conductive regions, the formulation is thus:

[1.42]

and the boundary conditions are:

[1.43]

1.2.5.7. Uniqueness conditions

In the formulations shown above, the magnetic vector potential A or the electric vector potential T is not unique because their divergence is not defined. Different gauge conditions are proposed to ensure their uniqueness. When we solve equations [1.33] to [1.37] using the finite element method, the following gauges may be used:

– Solving the system of equations by iteration: it was demonstrated that when the system of equations is solved using an iterative method, such as the conjugate gradient method, the gauge condition is implicitly set by recurrence of the iteration [REN 97, REN 96, KAM 93].

1.2.6. Time harmonic form

1.2.6.1. Maxwell’s equations in the time harmonic form

In the time harmonic form, the operator is replaced with j, where is the angular frequency of the source current.

The Maxwell’s equations are then rewritten as:

[1.44]

[1.45]

The time harmonic form is also called the phasor form. In this case, the quantities would be the amplitudes and the phases of the electric and magnetic fields.

1.2.6.2. Electromagnetic power

Maxwell’s equations can be written as:

[1.46]

Multiplying the first equation by H* (complex conjugate of H), let us take the complex conjugate of the second equation and multiply it by E. After subtracting one from the other, we find that:

[1.47]

Then, after integrating over an arbitrary volume [DEL 91]:

[1.48]

Equation [1.48] is the equation for the Poynting vector. The coefficient of is introduced in all fields because the fields are expressed as amplitude crests. Each of the terms has the following physical significance:

represents the source power emitted in volume V.

represents the electromagnetic power induced in V. The real part of this term is the power transformed into heat by Eddy losses.

represents the electromagnetic power transformed into heat by dielectric losses.

represents the difference between the mean time values of stored magnetic and electric energies, in V.

represents the power that volume V exchanges with its surroundings. It is the Poynting vector’s flux.

1.3. Thermal problem

The electromagnetic formulations discussed above allow us to calculate the power density created by the induced currents. This power is, for induction heating, the starting point for solving the thermal problem.

The principal modes of heat transmission are conduction, convection, and radiation.

Conduction corresponds to a heat transfer between two points within a solid under the influence of a temperature gradient. Its behavior is described by the following Fourier relation:

[1.49]

where, T represents the temperature in K and the material’s thermal conductivity in W.m−1.K−1.

The material’s thermal behavior is ruled by the following heat balance:

[1.50]

where, p is the generated power density (heat source or induced power), is the density and Cp is the specific heat. The first term in the equation describes the power density exchanged by the volume, the second the power density induced in the volume, and the last the variation of the internal energy density.

If the object to be heated is moving with velocity V with respect to the reference frame, the heat equation takes the following form:

[1.51]

The boundary conditions, at the borders of the domain in which the heat equation is solved, are generally obtained from the following three conditions:

Imposed temperature (Dirichlet condition):

[1.52]

Imposed thermal flow (Neumann condition):

[1.53]

Thermal exchange with the ambient environment through two different ways. Exchanges by convection:

[1.54]

where, h is the convection coefficient expressed in (W.m−2.K−1).

Exchanges by radiation:

[1.55]

with the emissivity and b the Stefan–Boltzmann constant (5,6710−8 W.m−2.K−4).

1.4. Magnetothermal coupling

The thermal source term introduced in the heat diffusion equation is none other than the power density per unit volume p dissipated by Eddy losses. This term defined in section 1.2.6.2 is given by:

[1.56]

Because electromagnetic phenomena evolve very quickly compared to thermal phenomena, the electromagnetic model is considered to be in the steady state during the magnetothermal coupling.

For the thermal problem, the study is conducted based on the steady or harmonic state. In the transient state, the temperature’s evolution is obtained step-by-step in time. However, for the harmonic state, complex representation is used. The temporal derivation operator is replaced by j.

For large variations in temperature, the electrical conductivity also depends on temperature. The (T) term is thus another coupling element between electromagnetic and thermal phenomena. We then speak of strong coupling between the two phenomena. In this book, the iterative algorithm described in Figure 1.7 has been used to solve strong coupling problems. In this algorithm, T represents temperature and A the magnetic vector potential. For other electromagnetic formulations, we must replace A with the sought variables in the formulations. If the nonlinearity of the equations is strong enough, we must use a relaxation coefficient on both T and A. If we call this coefficient r, U the variable to determine, and k the iteration number, we relax the value of U at the kth iteration as per the following formula:

[1.57]

Figure 1.7.Electromagnetic and thermal coupling algorithm

1.5. Solving the electromagnetic and thermal equations

The electromagnetic and thermal equations are partial differential equations, where the unknown is either a scalar or a vector quantity dependent on spatial coordinates and time. Numerical methods are often used to solve these equations. Nevertheless, in certain simple cases and for the comprehension of the involved physical phenomena, analytic methods are interesting.

1.5.1. Analytic methods

1.5.1.1. Transient state

In the one-dimensional case, the general form of the electromagnetic and thermal equations is as follows:

[1.58]

where, z is the propagation direction of the wave U and, p, represents internal sources.

For example, the one-dimensional formulation of magnetic potential A is:

[1.59]

and for the formulation in H, we obtain:

[1.60]

As for the thermal equation, it can be written as:

[1.61]

1.5.1.2. Harmonic state

For a sinusoidal variation, the general form is given by:

[1.62]

where with as the penetration depth of wave U.

For the electromagnetic problem, the penetration depth is given by:

[1.63]

For the thermal problem:

[1.64]

For these analyses, the electromagnetic wave was assumed to always be sinusoidal. Furthermore, since the period of the electromagnetic waves is much less than the thermal time constant, the instantaneous induced powers are replaced with their effective values.

The general solution of [1.62] without the source term can be written as:

[1.65]

where, and or A and B are two integration constants to set with the boundary conditions.

The .e−z term represents the incident wave and the .ez term represents the reflected wave.

For a semi-infinite device, there is no reflected wave. The solution is then given by:

[1.66]

For electromagnetic waves, this equation can be written as:

[1.67]

The induced power density is then:

[1.68]

where, P0 is the power per unit volume at the material’s surface.

The solution to the transient equation [1.61] is obtained through separation of variables. In some cases, it is also obtained using Laplace transforms [JAN 09]. These solutions are often used for the study of the thermal problem’s transient state. We shall discuss the solution to this equation in Chapter 4.

1.5.2. Semi-analytic methods

Semi-analytic methods combine analytic and numerical methods. There are a large number of semi-analytic methods in the literature. We can quote, for example, the boundary integral method, the method of coupled circuits or the shell element formulation. In this book, we show the shell element formulation and the method of coupled circuits.

The first method is well suited to solving the electromagnetic equations in 3D when one of the dimensions of the studied system is much less than the other dimensions. This is the case, for example, for metal or composite plates a few millimeters thick and a few meters wide. This method is also used in the systems where the electromagnetic skin depth is very weak compared to the dimensions of the studied material.

The method of coupled circuits, also called “moment method” (MM), is very close to physical considerations and consists of expressing the mutual and self-inductances between multiple inductors, the object to be heated, and eventually a magnetic circuit. For geometrically simple, yet common coils, these inductive terms may be given by analytic expressions.

1.5.2.1. Shell elements and surface impedance methods

When heating highly conductive materials by induction, the skin depth is very weak, especially for high electromagnetic frequencies. We can then replace these conductive parts with boundary conditions on the fields [SAK 92].

The electromagnetic phenomena within the conductor are replaced with a surface current layer K. From the expression of J, defined by equation [1.30], this current layer is expressed by:

[1.69]

with

[1.70]

The current layer implies a discontinuity in the magnetic field’s tangential component at the non-conductor/conductor interface, which can be expressed as:

[1.71]

By substituting the expressions described in equations [1.69] and [1.70] into [1.71], an expression called the surface impedance condition, linking the electric and magnetic fields, is obtained:

[1.72]

This condition can be combined with the formulations in the other parts of the system to obtain electromagnetic fields.

1.5.2.2. Generalized shell element formulation of a conductive plate

The proposed generalized shell elements method consists of a numerical solution obtained throughout the studied domain, including on the thin region borders, and an analytic solution that takes into account the variation of unknowns through the thickness of the thin region. The latter is expressed as a function of the solution on the boundaries of the thin region.

Consider an infinite plate of thickness e with two tangential fields H1s and H2s acting on its upper and lower surfaces (Figure 1.8). The analytic solution to Maxwell’s one-dimensional equations is given by the following relation:

[1.73]

Figure 1.8.Shell elements problem notation

The electric field E at the plate’s surface is given by:

[1.74]

The electric field on the plate’s surfaces 1 and 2 is then:

[1.75]

These relations give a generalized surface impedance of the form:

[1.76]

The solution on the boundary surfaces is given by:

[1.77]

In this case, the equation to solve is then:

[1.78]

with the boundary conditions at the plate surface expressed by the surface impedances of equation [1.76]. Equation [1.78] will be solved numerically.

To use the generalized shell elements in reduced scalar potential the conditions are [GUE 94]:

– region thickness very small compared to its other dimensions;

– region permeability near to that of the neighboring regions;

– no presence of multiply connected regions;

– if the skin effect is large, the source field must rather be tangential to the thin region’s surface.

The generalized shell element formulation developed in [GUE 94] uses the hypothesis that the thin region’s physical properties are isotropic. For anisotropic materials, we must extend this formulation to the cases of thin anisotropic regions with a tensorial electric conductivity [BEN 06].

1.5.2.3. Moment method

The principle of the MM consists of fractioning a set of N conductors (and/or a charge) into multiple n distinct inductor elements, within each of which current density is assumed to be constant. The discretization can be regular or not, depending on the conductor type, its dimensions, or the predictable current distribution inside it [CHE 95]. Generally speaking, for coils made of conductors with arbitrary cross-sections, this method can be presented by the Figure 1.9.

The whole elementary coil obtained in this way forms a mutually coupled device, which can be modeled using an equivalent electrical scheme.

1.5.2.3.1. Interactions between circuit elements

So far and in the absence of magnetic circuits, the equivalent electrical scheme of Figure 1.9 is equivalent to an associative parallel/series network of resistances and self- and mutual inductances (non-exhaustive representation of the mutual).

[1.79]

Figure 1.9.MM principle: fractioning of inductor elements

Figure 1.10.MM electrical schematic

It is also conceivable to apply the MM’s principle of mutuality to a similar network, taking the different parasite capacities between the elements into account [GUO 01].

Finally, knowing the modeling frequencies, solving the problem using this method simply consists of expressing the various inductances numerically [TAB 95] or analytically. By omitting the preceding indices from the formulation, we can define the law of conservation of magnetic flux going through a surface S, bounded by a closed loop C, by Stokes’ theorem. We can then calculate as a function of the magnetic vector potential A.

[1.80]

Generally speaking, the magnetic vector potential in an arbitrary point m, generated by a filiform inductor of length l with current I, is expressed by the following relation:

[1.81]

The vector potential is thus collinear to the inductor, and ||r|| represents the distance between a point on the inductor and point m. For a current assumed to be unitary, formulas [1.79] and [1.80] yield a direct analytic solution to the self- and mutual inductances in a vacuum or in air.

1.5.2.3.2. Axisymmetrical case

Let us consider a filiform coil, in air or vacuum, 0 (Figure 1.11