Numerical Simulation of Eddy Currents and the Associated Losses …khollaus/Thesis/PhDTheses.pdf · 2002-09-24 · simulation of eddy currents in three dimensions and the prediction

Numerical Simulation of Eddy Currents and the Associated Losses in Laminated Ferromagnetic

Materials by the Method of Finite Elements

submitted as

Theses

to obtain the academic degree of

Doctor of technical sciences

at the

University of Technology of Graz

by

Dipl.-Ing. Karl Hollaus

Supervised by

Ao. Univ.-Prof. Dr. Oszkár Bíró

Graz, September 2001

To my mother

For her trust and support all over the years.

Contents

Contents Introduction 1 1 Aspects in the Simulation of Eddy Currents in thin Conductive Sheets 4 1.1 Separation of the Fields in Laminations 4 1.2 Analyses of an Overall field distribution 5 1.3 Application of Potential Formulations to the Eddy Current Problem 8 2 Estimation of 3D Eddy Currents in Laminations by an

Anisotropic Conductivity and a 1D Analytical Model 10 2.1 One Dimensional Analytical Model 10 2.2 Numerical Examples 12 A. Conductive Cube 13 B. C-magnet with a Coil 15 3 A FEM Formulation to Treat 3D Eddy Currents in Laminations 18 3.1 Local Field in the Laminates 18 3.2 Numerical Examples 20 A. Conductive Cube 20 B. Large Magnetic Circuit 23 4 Preisach Model 25 4.1 Introduction 25 4.2 Classical Preisach Model of Hysteresis 27 4.2.1 Relationship between Mathematical Model and Magnetics 28 4.2.2 Geometric Interpretation of the Preisach Model 29

Contents

4.2.3 Description of the Functionality of the Classical Preisach Model 29 4.2.4 The wiping-out property of the classical Preisach model 33 4.2.5 The congruency property of the classical Preisach model 33 4.2.6 First Order Reversal Curves 34 α'β'f 4.2.7 Determination of the Preisach Distribution Function ( )α,βµ 35 4.3 Numerical Implementation of the Classical Preisach Model 36 4.4 Hysteresis Energy Losses of the Classical Preisach Model 37 5 Hysteresis Modeling by the Finite Element Method Considering Eddy Currents 41 5.1 Methods to Incorporate the Hysteresis – Nonlinearity in the Finite Element Method 41 5.1.1 Methods based on Material Coefficients 41 5.1.1.1 Differential Magnetic Permeability diffµ 42 5.1.1.2 Modified Constitutive Law 43 5.1.1.3 Fixed Point Technique 44 5.1.2 Magnetic Field Quantities based Methods 45 5.2 Treatment of the Hysteresis Non-linearity by BH – Curves 45 5.2.1 Techniques to handle the BH – Curves 48 5.2.1.1 Direct Technique 48 5.2.1.2 Store Technique 49 5.3 Formulations of an Eddy Current Problem with Hysteresis 50 5.3.1 H - Formulation of an Eddy Current Problem with Hysteresis 51 5.3.2 A - Formulation of an Eddy Current Problem with Hysteresis 54 5.4 Time Stepping Method 58 5.4.1 Time Stepping Method for the H – Formulation 60 5.4.2 Time Stepping Method for the A – Formulation 61 5.4.3 Reverse Mode of the Preisach Model for the A – Formulation 62

Contents

5.4.3.1 Reverse Mode with Sectioning an Interval 62 5.4.3.2 Reverse Mode with regula falsi 63 5.6 Numerical Simulations 63 6 Alternative Method in the Identification of the Preisach Model 78 6.1 Preisach Distribution Functions 79 6.2 Simulation Model 80 6.3 Definition of the Objective Function 81 6.4 Choice of Optimization Strategy 82 6.4.1 Simulating Annealing Algorithm 82 6.4.1.1 Metropolis Algorithm 83 6.4.1.2 Cooling Schedule 83 6.5 Simulation Results 84 7 Preisach Model versus Complex Permeability 87 7.1 Material Properties 87 7.2 Constitutive Law 88 7.3 Nonlinear Complex Permeability 89 7.4 Approximated Hysterese Loops 90 7.5 Numerical examples 91 Conclusions 96 References 98

Introduction

Introduction

The numerical simulation of low frequency electromagnetic fields in three-dimensional arrangements has become the focus of research in computational electromagnetism in the last decade. This is well documented by the proceedings of the Compumag and IEEE CEFC conferences published in IEEE Transactions on Magnetism over this period.

If the material characteristics are nonlinear, differential approaches (the method of finite elements (FEM) and of finite differences) are preferable over integral methods. Among differential techniques, the versatility of FEM in modeling complex geometries with curved boundaries makes it the mainstream method of numerically simulating low frequency electromagnetic phenomena.

One of the most important tasks of low frequency computational electromagnetism is the simulation of eddy currents in three dimensions and the prediction of the losses due to them. The FEM has proved to be an excellent tool for this. The methods of computing 3D eddy currents in linear media can be considered to be established by now. The possible formulations in terms of vector and scalar potentials are well documented. Even if eddy currents occur in isotropic ferromagnetic materials with pronounced saturation effects, no theoretical difficulty arises in applying FEM. All these methods have found their way into commercial simulation packages and it can be stated that three-dimensional eddy current problems in isotropic materials can nowadays be solved routinely.

In view of the above, it is all the more striking that the problem of eddy currents in ferromagnetic laminations, a case of considerable practical importance, has not been treated as frequently in the literature and it is still far from being a routine task of computational electromagnetism. This can be explained by the fact that modeling each of the laminations leads to extremely high computational costs, although the rapid development of computer architectures may even allow such a brute force approach, at least in order to verify approximate techniques. These latter are, however, clearly necessary to facilitate practical application in routine engineering tasks.

The purpose of laminating iron cores being the minimization of eddy current losses, the laminations are devised to be parallel to the expected magnetic field. Aside from edge effects, one-dimensional models are sufficient to account for losses due to fields parallel to the laminations, the so called iron losses. Therefore, considering non-linearity and even hysteresis is relatively simple provided the magnetic field is known [1-7].

The above methods do not provide a possibility of computing the two- or three-dimensional magnetic field distribution in the laminations. Attempts to achieve this are rather scarce in the literature and they use either two-dimensional models or magnetic network models [8-11].

Although the main flux in electrical devices is usually parallel to the laminations, unavoidable leakage fields have considerable normal components to the sheets causing large eddy current loops (leakage losses). Three-dimensional models accounting for the laminar nature of these eddy currents have been developed either using a single component current vector potential [12-14] or an anisotropic conductivity [15-17]. In these works, the non-linearity is neglected (with the exception of [12] using a time harmonic approximation of the non-linearity) and hysteresis losses are not taken into account.

1

Introduction

In the first chapter the magnetic fields occurring in laminated iron are separated into main and leakage field to simplify the considerations used to develop approximate techniques. The feasibility of an overall field distribution is studied extensively on a small but representative numerical example. The application of the different potential formulations to facilitate the computation of an eddy current problem is discussed briefly.

To circumvent the high computational costs of modeling each laminate to compute the eddy current distribution in three dimensions in conducting laminations, a simple method is presented in chapter 2. The laminar nature of the eddy currents due to the magnetic leakage field is considered by applying an anisotropic conductivity with zero or very low value in the direction normal to the laminations. This yields an overall field distribution serving as basis. To this end, the finite element method is utilized with different potential formulations. In a second step, the much smaller eddy current loops caused by the main magnetic flux parallel to the laminations are taken into consideration by a one-dimensional analytical model assuming an exponential variation of the tangential field quantities in each laminate. A comparison shows that the average value of the tangential components of the magnetic field intensity as well as of the current density across the laminations are approximately equal in a laminated model and in the corresponding anisotropic model within each lamination. Only linear material properties are considered.

An obvious shortcoming of the approach in chapter 2 is that the eddy current loops in the sheets are not modeled correctly, leading to errors at the edges of the laminates. Therefore, a method is described in chapter 3 to overcome this limitation. The first step is the same as in chapter 2: a three-dimensional model of the core treated as a bulk of anisotropic material with low conductivity in the direction normal to the lamination is carried out. Each laminate is taken into account in the second step by writing the field components in it as fundamental solutions of the diffusion equation so that the averages of the tangential electric and magnetic field components over the thickness of the sheet equal the averages obtained in the first step.

For a comprehensive analysis of the eddy current distribution in thin ferromagnetic sheets hysteresis has to be considered. A very common and wide spread way to model the hysteresis by finite elements is the Preisach model. Therefore, the basics of the classical Preisach model are summarized in chapter 4 as far as its incorporation in the finite element method is concerned. First, hysteresis non-linearity is discussed in general. The classical Preisach model is introduced and its properties are shown. Next, the numerical implementation is demonstrated in general. Finally, the determination of hysteresis losses related to the classical Preisach model is derived. Both the model itself and the associated hysteresis losses are represented in a way that their implementation in a computer code is straightforward.

Chapter 5 starts with a brief review of various already existing methods using the Preisach model by the finite element method. Then, a novel technique to treat the hysteresis non-linearity is presented and the way to incorporate the Preisach model in the finite element method is shown. A small one dimensional finite element model of second order is used to study the behavior of the method efficiently. The eddy current problem to be solved uses the magnetic field intensity or the magnetic vector potential as solution variable. The technique makes use of the treatment of a univalent non-linearity to achieve a stable solution in the time stepping scheme. It computes the future BH - curves starting from a known and stable solution in the integration points of the finite elements. The solution of the subsequent time step has to be situated on these curves. Experience has shown that such a method incorporating the hysteresis non-linearity by the permeability works only with the differential permeability. To accelerate the computations, the BH - curves are kept in the store as long as possible. For both formulations, the time stepping scheme is described in detail. In case of the magnetic vector potential the reverse mode of the Preisach model is discussed. Numerous numerical simulations complete the chapter.

2

Introduction

The identification of the Preisach distribution function is an essential problem in dealing with the Preisach Model. A method for the identification of the classical Preisach model is proposed in chapter 6. It exploits the one dimensional finite element model described in chapter 5. A classical Preisach model is sought which fits best the specific losses and the magnetization curves of a material with the aid of a stochastic optimization method (simulated annealing). It is assumed that the data are representative for the material. The results obtained confirm the capability of the method to arrive at a suitable classical Preisach model. Three different Preisach distribution functions which are typical for soft magnetic materials are tried to fit to a certain material.

It is well known that the Preisach model suffers from its extremely high demand on memory as well as from its expensive handling. Simulations of the eddy currents of electrical devices with laminated ferromagnetic cores by the finite element method coupled with the Preisach model to consider the hysteresis are almost impossible. An attractive way to deal with hysteresis non-linearity in case of time harmonic excitation of an eddy current problem is to introduce an effective material representing the ferromagnetic material. This is the subject of chapter 7. The fictitious material describes the relationship of the field quantities approximately. Two methods to create an effective magnetization curve on the basis of the Preisach model of different soft magnetic materials are presented. The complex magnetization curves obtained are applied to a simple one-dimensional finite element model. The losses in a sheet are calculated once by a transient analysis with the Preisach model and once by the approximate technique applying effective magnetization curves. A comparison of the results shows a very satisfactory accuracy of the approximating technique and extremely high time and memory savings.

3

1 Aspects in the Simulation of Eddy Currents in thin Conducting

1 Aspects in the Simulation of Eddy Currents in thin Conductive

Sheets

The purpose of laminating iron cores is the minimization of eddy current losses. The laminations are devised to be parallel to the expected magnetic field. Therefore, the simulation of the eddy current losses in thin laminations are of considerable practical importance.

Unfortunately, this is still far from being a routine task of computational electromagnetism. This can be explained by the fact that modeling each of the laminations by the finite element method leads to extremely high computational costs and thus, does not represent a viable solution. These models may serve to verify approximate techniques at best. Approximate techniques are clearly necessary to facilitate practical application in routine engineering tasks.

The computational effort of the approximate techniques to simulate the eddy currents in a laminated iron core should not be substantially higher than that necessary to analyse eddy currents in the corresponding isotropic iron core treated as a bulk, i.e., neglecting the individual laminates. The aim of the models to be developed is to simulate the eddy current losses in laminated iron cores as accurately as possible at a minimum of computational cost.

Experience has shown that, to split up the whole simulation in two subsequent steps represents a suitable way. In the first step, the impact of the magnetic leakage field is considered by treating the iron core as a bulk leading to an overall field distribution explained comprehensively in the following sections 1.1 and 1.2. Then, the lamination is considered by perturbing the overall field distribution by means of analytical models in the second step. Two possible analytical models are introduced in chapters 2 and 3.

1.1 Separation of the Fields in Laminations To simplify the subsequent considerations, the magnetic flux density and the eddy current

density occurring in a single lamination are separated into two parts (Fig. 1.1). Although the main magnetic flux density in electrical devices is usually parallel to the

laminations, unavoidable magnetic leakage fields (magnetic stray fields) B have considerable normal components with respect to the laminations causing large eddy current loops confined to flow essentially parallel to the laminations.

s

sJThe main magnetic flux density which is parallel to the lamination induces extremely

narrow eddy current loops . mB

mJ

4


mJ

sJ

sB

mB

Fig. 1.1. Single lamination with different types of eddy currents. The iron losses are caused by the main magnetic flux density whereas the leakage losses

occur due to the magnetic leakage fields B . mB

s

1.2 Analyses of an Overall field distribution

The large eddy current loops due to the magnetic leakage fields can be simulated by a single component current vector potential [12, 13] or an anisotropic conductivity [15, 16]. This results in an overall field distribution.

T

To substantiate this assertion, some investigations are carried out on a small conductive cube arranged symmetrically in the center of a cylindrical coil which carries a sinusoidal current. The model is described in detail in section 2.2 A. Conductive cube.

Both a three-dimensional eddy current analysis assuming an anisotropic conductivity with zero or very low value in the cross direction and a computation on a model taking account of each laminate by prescribing zero normal current density at the interfaces between the sheets are carried out.

5


0.465 T≡

a)

0.497 T≡

b) Fig. 1.2. Arrow plot of the imaginary part of the magnetic flux density on the surface of the Conductive cube, relative permeability rµ = 1 000, T, Φ - Φ formulation.

a) Anisotropic cube (conductivity perpendicular to the conducting plane is 1 000 times less than that of the conducting plane

b) Laminated cube

A comparison of the results demonstrates that, the overall distributions of the magnetic field as well as the eddy currents coincide quite well with each other within the volume and also on the surface. Figs. 1.2 to 1.3 illustrate the above assertions using the example of the Conductive cube defined in section 2.2 A.

6


a)

b) Fig. 1.3. Scalar plot of the imaginary part of the eddy current density on the surface of the Conductive cube, relative permeability rµ = 1 000, T, Φ - Φ formulation.

a) Anisotropic cube (conductivity perpendicular to the conducting plane is 100 times less than that of the conducting plane)

b) Laminated cube

The eddy current losses obtained from the anisotropic analysis are essentially lower than that of the exact model (see Table I, section 3.2). This is explained by the fact that the anisotropic model does not take account of the local field in the laminates.

7


1.3 Application of Potential Formulations to the Eddy Current Problem

Isoparametric hexahedral nodal finite elements [18, 19] with twenty nodes and quadratic

shape functions are applied to model the geometry of a problem. To simulate the eddy currents of the overall field distribution vector potentials and scalar potentials are used.

Nonconducting region: σ = 0 µ = µo

Je

Eddy currents: J ≠ 0

Conducting region: σ > 0 µ > µo

Φ;

Ar T, Φ;

A; Ar, V

Fig. 1.4. Scheme of an eddy current problem.

The computations are carried out by applying the T, Φ - Φ, A - Φ or the ,V -

formulation. In case of the T, Φ - Φ formulation the reduced current vector potential T is used in the eddy current carrying region, whereas the reduced magnetic scalar potential Φ is employed throughout the problem region. By applying the A - Φ formulation the total magnetic vector potential A is used in the conducting region, the total magnetic scalar potential Φ is applied in the non-conducting region. In case of the ,V - formulation the reduced magnetic vector potential and the electric scalar potential V are employed in the conducting region, the reduced magnetic vector potential appears also in the non-conducting region [20].

rA rA

rA rA

rA

The scalar potentials Φ and V are represented by means of nodal elements anyway. The vector potentials T, A as well as are approximated by means of nodal or edge elements. In case of nodal elements all components of the potential functions are continuous. The Coulomb gauge is enforced implicitly ensuring the uniqueness of the vector potentials and resulting in an excellent numerical stability [20]. The use of the nodal finite elements to approximate the vector potentials leads to difficulties in some specific problems [21]. These problems are avoided if the vector potentials are approximated by edge finite elements. Experience has shown that it is more favorable not to gauge these vector potentials and to

rA

8


solve the arising singular systems by iterative methods like the Incomplete Cholesky Conjugate Gradient technique leading to robust solutions. The numerical stability of these formulations are comparable to that of the corresponding Coulomb gauged approaches provided the consistency of the right hand side of the equations is ensured.

9

2 Estimation of 3D Eddy Currents in Laminations by an Anisotropic Conductivity and a 1D Analytical Model

2 Estimation of 3D Eddy Currents in Laminations by an

Anisotropic Conductivity and a 1D Analytical Model

The computation of eddy current losses in laminated iron by the finite element method is a particularly challenging task. To simulate the field distribution exactly to predict the eddy current losses accurately each laminate has to be modeled individually. However, this leads to extremely high computational costs.

To overcome this unpleasant limitation, much effort has been undertaken using either a single component current vector potential [12, 13] or an anisotropic conductivity [15, 16]. These models consider the large eddy current loops due to the magnetic stray field normal to the lamination and, thus, reproduce an overall field distributions only.

On the other hand, taking account of the small eddy current loops caused by the main magnetic flux parallel to the laminations, only 1-D models [2, 5] or 2-D models [8, 11] have been developed up to now. These models fail to describe the 3-D eddy current distribution.

To circumvent the drawbacks of the models mentioned, a simple method is developed which combines their advantageous features. The method computes the eddy current losses in laminated non-saturated iron cores based on the finite element method. The laminar nature of the eddy currents due to the magnetic leakage field is considered by applying an anisotropic conductivity with zero or very low value in the direction normal to the laminations. This yields an overall field distribution serving as basis. In a second step, the much smaller eddy currents loops caused by the main magnetic flux parallel to the laminations are taken into consideration by perturbing the overall field distribution with a one-dimensional analytical model. Thus, the method avoids to model each lamination individually. Non-linearity is neglected. 2.1 One Dimensional Analytical Model

Apart from edge effects, a one dimensional model should be sufficient to consider the small eddy current loops due to the main magnetic flux. Therefore, in a second step, a rather simple, one dimensional analytical model perturbing the overall field distribution obtained by assuming an anisotropic conductivity will be tried. The investigations described in section 2.2 allow to assume that the mean value of the tangential components of the magnetic field intensity as well as of the current density across the laminations are approximately equal in the laminated cube and in the anisotropic cube within each lamination (see also Figs. 2.1 and 2.2). To keep the model as simple as possible, all other components of the field quantities are neglected. The variation of tangential field quantities in the i laminate is simply assumed to be

zH yJ

th

(s)fH(s)fH(s)H i

2i2z

i1

i1zz

)()()()( += (2.1) and

(s)fJ(s)fJ(s)J i2

i2y

i1

i1yy

)()()()( += , (2.2)

10


respectively, where s is the coordinate normal to the lamination ( )( )ss is1i δδ ≤≤− , sδ the

thickness of a lamination and δ the penetration depth ωµσ/2 (ω is the angular frequency, σ is the conductivity of the laminations and is the imaginary unit). and

, as well as, and denote the tangential components of the field quantities on the surfaces in the lamination. and are the complex functions

j )( i1zH

)( i2zH )( i

1zJthi

)( i2zJ

(s)i )f1( (s)f i

2)(

( )

( )

+

−

+=

δδj1sinh

δsiδj1sinh

(s)fs

s

i1

)( (2.3)

and

( ) ( )

( )

+

−−−

+=

δδj1sinh

δsδ1ij1sinh

(s)fs

s

i2

)( , (2.4)

respectively.

The assumption (2.1) must satisfy the interface condition

)()( i2z

1i1z HH =+ (2.5)

between two arbitrary adjacent laminations. These tangential components are unknown and have to be determined.

The equations stating the relationship between the solution obtained by applying an anisotropic conductivity and the approximation is

∫ ∫==− −

si

1i

i

1iz

szai

s

izm

s

s

s

(s)dsH1(s)dsH1Hδ

δ

δ

δδδ )( )(

)( . (2.6)

Here, denotes the anisotropic solution and ) the mean value in the lamination of the laminated model. Using (2.1), the second integral in (2.6) yields

(s)H zai( izmH thi

[ ]F(s)HF(s)HH i

2zi1z

izm

)()()( += , (2.7) where is the complex function F(s)

( )

( ) ( )

++

−

+

=

δδ

δδ

δs

s

j1sinhj1

1j1coshF(s) . (2.8)

Applying (2.6) - (2.8) to all laminations, results in a linear equations system with one more

unknown as equations obtained. The missing condition can be derived by assuming that the

11


mean values of the tangential component of the current density of the laminated model as well as of the anisotropic solution in the first lamination, for instance, are also equal. Using the first Maxwell equation

yJ

JH =curl (2.9)

and neglecting all other components except and in Cartesian coordinates, can be obtained from as

(s)H z (s)J y (s)J y

(s)H z

(s)Js(s)H

yz −=∂

∂ . (2.10)

Applying (2.1), (2.2) and (2.10) yields a simple relationship for the first lamination

(s)fJ(s)fJ(s)fH(s)fH i2

12y

i1

11y2

12z1

11z

)()()()()()( −−=′+′ . (2.11) In this and are the derivatives with respect to s of and , respectively. (s)f1′ (s)f2′ (s)f1 (s)f2

Since the anisotropic solution is available, the mean value of the right hand term of (2.11) is also known. The second constitutive equation is (2.2). 2.2 Numerical Examples

To illustrate the accuracy of the proposed approach, it is benchmarked against the solution obtained by modeling each lamination individually. A cube immersed in the field of a cylindrical coil carrying a sinusoidal current serves as a simple example. To analyze a more practical problem for employing the introduced model, a C-magnet with a coil is treated.

All available symmetries of the problem regions are utilized to reduce the computation and modeling effort. To be able to compare the results meaningfully, the examples below are modeled as follows. For the entire problem region of the laminated model as well as of the anisotropic model, the same finite element mesh is applied except perpendicular to the lamination within the conducting region.

For the laminated problems, the T, Φ - Φ formulation is applied only, because only this formulation offers the possibility to model the laminations by setting the tangential component of T to zero on boundaries between the sheets. Both scalar and vector potentials are approxiamted by nodal finite elements.

A cylindrical coil external to the conducting region carrying time harmonic current with a frequency of 50 Hz serves as excitation. The source field is computed with the aid of the Biot-Savart´s Law.

The material properties are linear, the conductivity of the laminations and in the conducting plane in the anisotropic case is 5.8751*106 S/m, the conductivity perpendicular to the conducting plane in the anisotropic case is varied between 100 and 10 000 times less than this value, the relative permeability is varied between 100 and 10 000 for both the laminated and for the anisotropic case. The thickness of the laminations is 1mm in both examples below.

12


A. Conductive Cube

The cube is arranged symmetrically in the centre of the cylindrical coil (inner radius =7.1 mm, outer radius =11.0 mm and height h=20.0 mm) carrying a current of 6*10

ir

ar7 AT

assumed to be real, the rotation axis of the coil coincides with the z axis of the Cartesian coordinates and the x axis is perpendicular to the laminations. The length of the edge of the whole cube is 10.0 mm (Fig. 2.1).

All in all, an essentially better agreement of the overall field distribution with the field distribution obtained by modeling each lamination individually is obtained by applying the T, Φ - Φ formulation compared with A, V - Φ and A - Φ formulations. The best agreement is achieved, if the conductivity perpendicular to the conducting plane is chosen to be zero which means the use of a single component current vector potential. Fig. 2.2 illustrates that the proposed method is capable to deal with the problem and that it is sufficiently accurate. These computations are carried out applying the T, Φ - Φ formulation.

# plane of symmetry

x

y

z

5 mm

5 mm

1 mm

#

#

#

2 mm

2 mm

5 mm

0

Fig. 2.1. Model of the conductive cube.

x (mm)

Laminations modelled

Approximation

Anisotropic conductivity

2.52.01.51.00.50.0 4.5 5.04.03.53.0

0.6

0.7

0.8

0.9

1.0

1.1

1.2

a)

13


-

-

x (mm)


Approximation


0.2

0.1

0.0

0.2

0.1

0.5

0.4

0.3

2.52.01.51.00.50.0 4.5 5.04.03.53.0

b)

-

-

-

-

x (mm)


Approximation


2.52.01.51.00.50.0 4.5 5.04.03.53.00.2

0.15

0.1

0.0

0.05

0.15

0.1

0.05

c)

-

-

-

-

x (mm)


Approximation


2.52.01.51.00.50.0 4.5 5.04.03.53.02.0

1.0

1.5

1.0

0.5

0.0

0.5

d) Fig. 2.2. Tangential components of the field quantities along the line y = 2.0 mm and z = 2.0 mm parallel to the x-axis in the Conductive cube; relative permeability = 1 000, the conductivity perpendicular to the conducting plane is 10 000 times less than that of the conducting plane. a) z-component of the real part of the magnetic flux density Re(Bz(x)) b) z-component of the imaginary part of the magnetic flux density Im(Bz(x)) c) y-component of the real part of the eddy current density Re(Jy(x)) d) y-component of the imaginary part of the eddy current density Im(Jy(x))

14


The eddy current losses summarized in Table I in a laminated cube are computed once with each laminate modeled, once from an analysis with a bulk anisotropic conductivity and by the method with 1D analytical correction. The thickness of the laminates is chosen to be δ =1.0 mm or δ =0.5 mm. The relative permeability is to be rµ =103 or rµ =104. The investigations are carried out by a conductivity ratio tσ / nσ equals 103. Only the T, Φ - Φ formulation is used here. The improvement achieved by the present method is clearly visible. The 1D analytical correction yields a power loss in good agreement with that of the model with each lamination taken into account.

TABLE I

COMPARISON OF THE EDDY CURRENT LOSSES WITH DIFFERENT METHODS (mW)

Thickness of the laminates 1.0mm 0.5mm

Relative permeability

rµ


Laminates modeled

1D analytical correction

Laminates modeled


1 000 0.8203 2.5228 2.3278 1.0648 0.8125

10 000 0.3514 1.8759 2.0245 0.6812 0.6078

B. C-magnet with a Coil

A cylindrical coil carrying a current of I=9*104 AT assumed to be real is wound around a c-shaped iron core with a large air gap (Fig. 2.3). The laminated core consists of 40 laminations. Only the T, Φ - Φ formulation is used for the anisotropic as well as for the laminated model.

40 mm

40 mm

20 mm

40 laminations1mm

# plane of symmetry

40 mm40 mm 50 mm

I = 90 MA

z

x

y ri = 29 mm

ra = 59 mm

140 mm

#

#

250 mm

170 mm

40 mm

10 mm

80 mm

80 mm

0

20 mm

5 mm

Fig. 2.3. Model of the C-magnet with a Coil.

15


Fig. 2.4. Finite element discretisation of one fourth of the C-magnet with a Coil with anisotropic conductivity

-

-

-

-

-

-

-

-

-

-

z (mm)


Approximation


0.0 8.0 10.06.04.02.0 20.018.012.0 14.0 16.0

0.2

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.0

a)

-

-

-

-

z (mm)


Approximation


0.0 8.0 10.06.04.02.0 20.018.012.0 14.0 16.00.04

0.01

0.02

0.03

0.04

0.030.02

0.01

0.0

0.06

0.05

b)

16


-

-

-

z (mm)


Approximation


0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.0 8.0 10.06.04.02.0 20.018.012.0 14.0 16.0

c)

-

-

-

-

z (mm)


Approximation


0.0 8.0 10.06.04.02.0 20.018.012.0 14.0 16.00.2

0.05

0.0

0.05

0.1

0.15

0.15

0.1

d) Fig. 2.5. Tangential components of field values along the line x = 110.0 mm; y = 10.0 mm parallel to the z-axis in the C-magnet with a coil; relative permeability = 10 000, the conductivity perpendicular to the conducting plane is 1 000 times less than that of the conducting plane. a) y-component of the real part of the magnetic flux density Re(By(z)) b) y-component of the imaginary part of the magnetic flux density Im(By(z)) c) x-component of the real part of the eddy current density Re(Jx(z)) d) x-component of the imaginary part of the eddy current density Im(Jx(z))

17

3 A FEM Formulation to Treat 3D Eddy Currents in Laminations


An overall field distribution of laminated iron can be obtained by using an anisotropic conductivity described in chapter 1. The method in chapter 2 corrects this solution to consider the lamination by assuming an exponential variation in the cross direction within each laminate. An obvious shortcoming of this approach is that the eddy current loops in the sheets are not modeled correctly, leading to errors at the edges of the laminates [22]. The method to be introduced attempts to overcome this limitation.

First, a three-dimensional finite element model of the core treated as a bulk of anisotropic material with low conductivity in the direction normal to the lamination is analyzed. This yields an overall field distribution serving as a basis. Similarly, to the method of chapter 2 each laminate is taken into account in the second step by writing the field components in it as fundamental solutions of the diffusion equation so that the averages of the tangential electric and magnetic field components over the thickness of the sheet equal the averages obtained in the first step. 3.1 Local Field in the Laminates

Each laminate is assumed to consist of rectangular portions. One such portion is shown in Fig. 3.1 with a local coordinate system attached to it. The components of the electric and magnetic field, E and H are written in the form of a fundamental solution of the diffusion equation

0EE =+∆− ωµσj , 0HH =+∆− ωµσj (3.1) where ω is the angular frequency, µ is the permeability and σ is the conductivity.

x

y

z

δ

a

b

Fig. 3.1. A rectangular portion of a single laminate.

Hence, the six field components Ex, Ey, Ez, Hx, Hy and Hz are linear combinations of the functions

18


zz

by2nby2n

ax2max2m

mn

mn

γγ

ππ

ππ

sinhcosh

sincos

sincos

//

//

(3.2)

where the separation equation

0jb2na2m 2mn

22 =+−+ ωµσγππ )/()/( (3.3) is satisfied for each value of the integers m=(0),1,2,... and n=(0),1,2,... . The index 0 is used in the cosine terms only. The number of coefficients in each expansion is 8 for each m and n. For example, Ex is written as

∑ ∑== =0m 0n

x zyxE ),,(

( ) ++ by2nax2mzEzE mnccsxmnmn

cccxmn // ππγγ coscossinhcosh

( ) ++ by2nasx2mzEzE mncssxmnmn

cscxmn // ππγγ incossinhcosh

( ) ++ by2nax2mzEzE mnscsxmnmn

sccxmn // ππγγ cossinsinhcosh

( ) by2nax2mzEzE mnsssxmnmn

sscxmn // ππγγ sinsinsinhcosh + (3.4)

and similar expressions hold for the other five field components. It is sufficient to introduce the 16 coefficients for Ex and Ey, since the rest can be easily written from the conditions

0=⋅∇ E and HE ωµj−=×∇ . (3.5)

These 16 coefficients are determined from the conditions that the tangential field components Ex, Ey, Hx and Hy averaged over the thickness of the laminate are the same as in the overall field distribution. This assumption is justified by the results of chapter 1 and 2.

The averages of the tangential components are expanded in two-dimensional Fourier series. Denoting the overall solutions obtained from the 3D eddy current analysis with anisotropic conductivity by the index 0, the expansion of the average of E0x, for example, has the form

∑ ∑=∫= =− 0m 0n

2

2x0 dzzyxE1 /

/),,(

δ

δδ

by2nax2mEby2nax2mE csxmn

ccxmn //// ππππ sincoscoscos +

by2nax2mEby2nax2mE ssxmn

scxmn //// ππππ sinsincossin ++ (3.6)

and similar expansions are determined for the averages of E0y, H0x and H0y, i.e. altogether 16 Fourier coefficients are computed for each value of m and n.

It is now easy to write the Fourier expansions of the same averages of the fundamental solutions written in the form (3.4) for Ex, Ey, Hx and Hy. By comparing the coefficients, the 16 unknowns , , , , , , , , , , , , ,

, and can be determined from

cccxmnE ccs

xmnEsssymnE

cscxmnE css

xmnE sccxmnE scs

xmnE sscxmnE sss

xmnE cccymnE ccs

ymnE cscymnE css

ymnE sccymnE

scsymnE ssc

ymnE ccxmnE , cs

xmnE , scxmnE , ss

xmnE , ccymnE , cs

ymnE , scymnE ,

ssymnE , cc

xmnH , csxmnH , sc

xmnH , ssxmnH , cc

ymnH , csymnH , sc

ymnH and ssymnH which are known from the

overall solution. Once the expansions (3.4) are known for all components of E, the power loss can be

computed by simple integration. The effort needed to compute the Fourier coefficients of the averaged fields in (3.6) is negligible as compared to that of the finite element analysis.

19


3.2 Numerical Examples

The same problem as in section 2.2 A, a cube immersed in the field of a cylindrical coil carrying a sinusoidal current serves as a simple example in order to investigate a few parameters and to compare the different methods. To demonstrate the capability of the present method, a large magnetic circuit excited by a race track coil is treated, too. In order to reduce the computation and modeling effort, all available symmetries of the problem regions have been exploited. The vector potentials are approximated by nodal or edge finite elements. A. Conductive Cube

The geometry is repeated in Fig. 3.2. Only the T, Φ - Φ formulation is used here. The source field was computed with the aid of a single component current vector potential T. Further details can be found in section 2.2 A.

mmin Dimensions

A34.8x10

50Hz

3x100.1

S/m65.875x10

=

=

=

=

I

fr

µ

σ14.2

10

10

I 20

1

22

y

z

x

10

Fig. 3.2. One eighth of the finite element discretisation of the Conductive Cube in the field of a cylindrical coil.

The eddy current losses in a laminated cube are computed once with each laminate modeled, once from an analysis with a bulk anisotropic conductivity, once by the method with 1D analytical correction described in chapter 2 and by the present method. The thickness of the laminates is chosen to be δ =1.0 mm or δ =0.5 mm. The relative permeability is chosen to be

rµ =103 or rµ =104. The investigations are carried out by a conductivity ratio tσ / nσ equals 103. Only nodal finite elements are used. The results are summarized in Table I. The improvement achieved by the expansion technique is clearly visible, the present method yields a power loss in good agreement with that of the model with each lamination taken into account. All in all, a better agreement with the exact solution can be achieved with the expansion technique than with method using a 1D analytical correction.

20


TABLE I

COMPARISON OF THE EDDY CURRENT LOSSES WITH DIFFERENT METHODS (mW) Thickness of

the laminates 1.0mm 0.5mm

Relative permeability

rµ


Laminates modeled


Expansion technique

Laminates modeled


Expansion technique

1 000 0.8203 2.5228 2.3278 2.7432 1.0648 0.8125 1.0421

10 000 0.3514 1.8759 2.0245 2.0178 0.6812 0.6078 0.6327

A comparison of the number of iterations required by the Incomplete Cholesky Conjugate Gradient (ICCG) technique to solve the anisotropic model by nodal and by edge finite elements is shown in Table II. It can be observed that the number of iterations required by nodal finite elements increases considerably with the conductivity ratio tσ / nσ , whereas the iterations keep almost constant in case of edge finite elements. The iterative method is stopped when a normalized residual falls below a certain desired value. For instance, the results in Table II relate to a normalized residual of . To be able to carry out a meaningful comparison the problem is modeled in the same way for both types of finite elements. Therefore, the number of nodes is 8281 for both cases. The number of equations of the entire equations system to be solved is 7830 for edge finite elements and 7905 in case of nodal finite elements. The relative permeability

1410−≤ε

rµ equals 103.

TABLE II

COMPARISON OF THE NUMBER OF ICCG-ITERATIONS Conductivity Ratio

/ tσ nσ 100 200 500 1 000 10 000 100 000

Nodal finite elements 140 154 202 259 538 814

Edge finite elements 239 247 247 252 250 252

The effect of the ratio tσ / nσ in the anisotropic model on the results is also investigated. The results are shown in Fig. 3.3. The relative permeability rµ equals 103. The losses obtained by the anisotropic model only vary with the conductivity ratio tσ / nσ enormously. This means that the true losses cannot be obtained with sufficient accuracy utilizing an anisotropic model only. On the other hand it can be seen easily that the method proposed is able to reproduce the true losses in the laminated cube by means of the field distribution obtained by an anisotropic model and subsequent correction of the method quite well. The losses evaluated by the expansion technique converge with respect to the conductivity ratio rapidly and they are more or less independent of this ratio.

21


0,0

1,0

2,0

3,0

4,0

5,0

6,0

1,0E+01 1,0E+02 1,0E+03 1,0E+04 1,0E+05

Conductivity ratio

Expansion technique withlaminates 1.0mm thick


Anisotropic model

Laminates modeled1.0mm thick

Laminates modeled0.5mm thick

Fig. 3.3. Losses in the Conductive Cube obtained with different methods.

Fig. 3.4 shows the losses in the individual laminates computed by modeling each laminate and by the present method. The agreement is very satisfactory for the two thicknesses. The relative permeability used is rµ =103 and the conductivity ratio used is tσ / nσ =103.

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0 1 2 3 4 5 6 7 8 9 10-th laminate


Laminates modeled 1.0mmthick


Laminates modeled 0.5mmthick

Fig. 3.4. Comparison of the losses in each laminate in the Conductive Cube.

22


B. Large Magnetic Circuit

The second example represents a large magnetic circuit with an air gap (Fig. 3.5). A race track shaped coil is wound around the large core carrying a sinusoidal current of I=68 500 A with 50 Hz. The yoke is separated from the core by a small air gap and is completed by magnetic channels to suppress the leakage fields. The material properties are linear, the conductivity in the conducting plane in the anisotropic model is tσ =2,5x105 S/m, the conductivity normal to that plane has been chosen to be 106 times less than tσ . The relative permeability has been assumed to be rµ =103 as an average value in the iron. The arrows in the iron (see Fig. 3.5) indicate the normal direction with respect to the laminations. The vector potentials are approximated by edge finite elements.

Yoke

Core

Coil

1056660

120

314

160

160

606

190

J

314

712

720

1450

mmin Dimensions

A46.85x10

50Hz

30x10 1.

S/m52.5x10

=

=

=

=

I

fr

µ

σ

Fig. 3.5. One eighth of the finite element discretisation of the conducting region with the race track shaped coil of the Large Magnetic Circuit with anisotropic conductivity and with its overall dimensions shown.

The aim of the following calculations is to estimate the total eddy current losses in the iron

core. Depending on the thickness of the laminates, 1.0mm or 0.5mm, the number of sheets is as high as 190 or 380 in the yoke. Therefore, modeling each laminate is obviously not a solution. Table III shows the eddy current losses and the CPU times required on the same personal computer for different formulations applied and different thicknesses of laminates used by the expansion technique compared with those resulting from the anisotropic model.

23


TABLE III

COMPARISON OF THE LOSSES (W) AND THE CPU TIMES (S) Anisotropic model Thickness of the laminates

0.5mm Thickness of the laminates

1.0mm

Formulation Losses in W

CPU-time in s

Losses in W

CPU-time in s

Losses in W

CPU-time in S

T, Φ - Φ 175.0 72840 178.9 404.5 469.9 204.0

rA ,V - rA 201.0 41058 324.0 8850.6 575.2 4432.4

Some details of the finite element models with anisotropic conductivity and different potential formulations used to simulate the losses in the Large Magnetic Circuit are listed in Table IV.

TABLE IV COMPARISON OF THE FINITE ELEMENT MODELS WITH DIFFERENT FORMULATIONS

Formulation Number of finite elements

Number of equations Number of iterations with ICCG

T, Φ - Φ 33810 211940 17266

rA ,V - rA 28196 359100 5697

24

4 Preisach Model

4 Preisach Model 4.1 Introduction

Hysteresis is encountered in many different areas like magnetic hysteresis, ferroelectric hysteresis, gas absorption and so on. A clear and rigorous representation which does not depend on a certain physical nature is obviously required. Therefore, the concept of a transducer (Fig. 4.1) in the control theory with an input ( )tu and an output shall be adopted [23].

( )tf

( )tf( )tu

HT

Fig. 4.1. Hysteresis transducer.

A transducer whose input-output relationship is a multibranch non-linearity (Fig. 4.2) shall

be called hysteresis transducer HT. Branching occurs each time when the input u passes a extremum.

( )t

u

f

Fig. 4.2. Trajectory of a multibranch non-linearity.

Mostly, looping is expected to be the essence of hysteresis. In fact it represents a special case of branching and happens whenever the input ( )tu varies between the same two extreme values.

The memory of hysteresis nonlinearities can be subdivided in a local or non-local memory. In case of an HT with local memory (Fig. 4.3) the future output ( )tf is determined uniquely by the current state of the output ( 0)=tf and the time behavior of the future input . This means that the number of possible branches of a HT with local memory starting from an arbitrary point in the input output plane is one for increasing input as well as decreasing input.

( )tu

25

4 Preisach Model

u

f

Fig. 4.3. Input output relationship of a hysteresis transducer with local memory.

Experimentally observed crossing and partially coincident minor loops (Fig. 4.4) contradict the property of HT with local memory.

u

f

Fig. 4.4. Crossing and partially coincident minor loops.

The output of a HT with non-local memory is additionally influenced by the memory contents. This mathematical tool detects and accumulates significant extrema of the input

( )tf( )tu

in the memory. Due to the nature of the memory the possible number of branches of a HT with a non-local memory is infinite (Fig. 4.5).

u

f

Fig. 4.5. Input output relationship of a hysteresis transducer with non-local memory.

26

4 Preisach Model

Some restrictions shall be made for the sake of simplifying the mathematical representation of hysteresis. The output should not depend on the rate of changes of the input u . For a rate independent hysteresis non-linearity the inputs

( )tf ( )t( )tu1 and ( )t2u in Fig. 4.6 and Fig. 4.7

result in the same input-output trajectory fu − shown in Fig. 4.2.

1M

1m

2M1u

t

*

*

*

Fig. 4.6. Input ( )t1u .

2u

t

1M

1m

2M*

*

*

Fig. 4.7. Input ( )t2u .

Second, only scalar hysteresis shall be considered. Actually scalar hysteresis does not exist in reality. The term scalar relates to the special case when the input varies only along one arbitrary line in space.

Since the intention is to analyze hysteresis in electrical devices, the HT is part of a system. Consequently, its input is not predictable a priori and mathematical models are needed. The notion of HT in magnetics means an infinitesimally small particle. The size of a particle is large enough to treat the material properties as a continuum and small enough to consider the essentials which are of importance for the design of electrical devices. 4.2 Classical Preisach Model of Hysteresis

Based on some hypotheses the model was developed and published by Ferenc Preisach in 1935 [24]. The model has been regarded as a purely physical model for a long time. At the beginning of the seventies, the Russian mathematician Krasnoselskii recognized the

27

4 Preisach Model

mathematical meaning first and thus the associated general validity for hysteresis non-linearity.

The model from the mathematical point of view consists of an infinite set of simple rectangular hysteresis loops shown in Fig. 4.8. The operator has “up” and “down” switching values of the input at α and β , respectively. Operators for which βα ≥ is valid are considered exclusively. Only these operators are physically meaningful. The output of the operators shall assume ( )t = 1+uαβγ or ( ) 1tu −=αβγ only.

The fundamental operator represents a hysteresis non-linearity with local memory. When the input ( )tu increases monotonically the branch is followed, whereas, the branch is traced by a monotonically decreasing input

abcde edfba( )tu .

u

uαβγ

β α

a cb

d ef1+

1−

Fig. 4.8. Rectangular hysteresis loop (elementary operator).

Along with the set of elementary operators an arbitrary distribution function ( )βαµ , , frequently called the Preisach distribution function (PDF) in literature, will be assumed. The PDF acts as a weighting term of the specific elementary operators, see (4.1). The symbol Γ stands for the Preisach hysteresis operator. Simplified, the output of the Preisach hysteresis operator represents the sum over all weighted specific elementary operators.

( )tf

( ) ( ) ( ) ( )∫∫=Γ=

≥βααβ βαγβαµ ddtu,tutf (4.1)

It is surprising that the hysteresis operator Γ in (4.1) has a non-local memory as a collective

property of a system consisting of infinite elementary operators with local memory. 4.2.1 Relationship between Mathematical Model and Magnetics

The physical interpretation in magnetics is possible but it holds some difficulties. An elementary operator can be seen as a magnetic particle [25]. The Preisach distribution function corresponds to some statistical distribution function. The output value of the hysteresis transducer represents the magnetization or the magnetic flux density. The magnetization or the magnetic flux density are a collective property of the magnetic particles.

One elementary operator does not state anything about either its location or its shape. The interpretation of asymmetric loops ( )βα −≠ is downright unsatisfactory. Due to the

28

4 Preisach Model

interaction of the particles among each other the asymmetry is caused. The size of the asymmetry serves as a measure of the interaction. Almost all operators of the entire set are asymmetric. This is no problem for the mathematical model because it is consistent with experimentally observed asymmetric minor loops. A hysteresis operator based on a set of symmetric elementary operators is not capable to simulate asymmetric minor loops. 4.2.2 Geometric Interpretation of the Preisach Model

The description of the Preisach model is essentially facilitated by its geometric interpretation. An elementary operator ( )βα , (see Fig. 4.9) an be seen as a point in the plane with the coordinates α and β , respectively. The quantities α and β denote the “up” and “down” switching values of the elementary operator.

Since only operators which comply with βα ≥ are regarded because of physical reasons, the αβ -plane is subdivided by the line βα = in a useful half and an unusable one. All operators in the region βα < are omitted.

A further restriction of the feasible area can be made without limiting the feasibility of the mathematical model for practical applications. Measurements of hysteresis loops substantiates the assumption of an outermost loop comprising all possible minor loops. This means for the model to be developed that only operators in a “limiting triangle” T determined by the point ( 00 )βα , are used ultimately.

β

α

( )βα ,

( )00 βα ,βα =

T

Fig. 4.9. “Limiting triangle” T in the Preisach plane.

Due to the fact that in the majority of practical cases of hysteresis the value of positive saturation equals minus the negative value of saturation the simplification that 00 βα = shall be made and, thus, a right triangle will be obtained. 4.2.3 Description of the Functionality of the Classical Preisach Model

Next, the functionality of the classical Preisach model CPM (hysteresis non-linearity with non-local memory) will be described. To this end, the input of the hysteresis operator is set to

29

4 Preisach Model

a negative value so that the output of all elementary operators represented by the “limiting triangle” are in the “down” position, i.e., the output equals T 1− . Now the input of the hysteresis transducer is monotonically increased. Thereby the output of all elementary operators with an “up” switching value α less than the current input u switch the output to

. The horizontal line 1+ u=α in Fig. 4.10 splits T into a set ( )tS + of elementary operators

β

α

u=α

( )tS +

( )tS −

( )tL

Fig. 4.10. Subdivision of the limiting triangle in a set ( )tS + and a set ( )tS − by the interface . ( )tL

with the output ( ) 1+=tuαβγ and a set ( )tS − consisting of elementary operators with the output ( ) =tuαβ 1−γ . The line u=α moves upward and forms the interface between the

two sets and

( )tL( )tS + ( )tS − .

After the maximum of the input u (see Fig. 4.11) has been achieved the input is decreased. All operators in the “up” position whose “down” switching value

1

β is greater than the current input u are turned back to the “down” position. This results in the second link of the interface in Fig. 4.12 which is vertical and is connected to the hypotenuse (tL ) βα = . The vertex of the interface belongs to the input maximum u and the current input u . ( u,1 )u (tL ) 1

1u

β

α

( )tS +

( )tS −

( )tL

Fig. 4.11. Input maximum . 1u

30

4 Preisach Model

β

α

u=β

( )tS +

( )tS −( )tL( )uu ,1

Fig. 4.12. A second link of the interface ( )tL with vertex ( )uu ,1 .

In case the input ( )tu is increased again it passes through a local minimum and the vertex in Fig. 4.12 becomes a fix point

2u( u,u1 ) ( )21 u,u in Fig. 4.13 in the Preisach plane αβ

representing the local input maximum u and the local input minimum . The final link of the interface is attached to the hypotenuse

1 2u( )tL βα = . It is horizontal and moves upward.

( )tL

( )tS +

( )tS −

β

α

u=α

( )21 u,u

Fig. 4.13. Interface with the vertex ( )tL ( )21 u,u representing the local input extrema u and . 1 2u

Following the previous line of reasoning the input function ( )tu in Fig. 4.14 , for instance, would lead to a corresponding subdivision of the Preisach plane represented in Fig. 4.15. The interface is a staircase line shaped by the vertices ( )tL ( )1+νν m,M . The vertices in turn are caused by the sequence of dominant input extrema of the input function u .

( )1+νν m,M( )t

31

4 Preisach Model

t

u 1M3M

2m4m

**

**

Fig. 4.14. Input function ( )tu .

( )tS +

( )tS −

β

α

1M

3M

2m 4mu=α

( )tL

Fig. 4.15. Subdivided Preisach plane by a staircase interface ( )tL with the vertices ( ) . 1+νν m,M

The output of the given HT is of course determined by the integral (4.1) which can be

rewritten in the following adequate form for the present situation emphasizing the nature of the non-local memory:

( )tf

( ) ( ) ( ) ( )

( )( )

( )∫∫∫∫ −∫∫ ==−+≥ tStS

dd,dd,ddtu,tf βαβαµβαβαµβαγβαµβα

αβ . (4.2)

In this way the past leaves its mark on the future output ( )tf . This HT has obviously a non-

local memory. Which means that a different shape of the interface ( )tL results in general in a different future output even if the input function ( )tf ( )tu is the same from certain time instant.

It can be easily reconstructed by means of equation (4.2) and due to the symmetry condition assumed at the beginning, namely that for the negative and the positive saturation, and

of the output , the following relation holds:

−f+f ( )tf

−+ −= ff . (4.3)

32

4 Preisach Model

4.2.4 The wiping-out property of the classical Preisach model

Here, another property of the memory should be introduced which elucidates additionally its essence.

Let us have a closer look at the input function ( )tu in Fig. 4.14. The particular feature is the monotonically decreasing sequence of input maxima ,...M,M 31 and the monotonically increasing sequence of input minima ,...m,m 42 which yields to the typical βα − diagram in Fig. 4.15.

Next, we assume that the input u is increasing monotonically so that it exceeds even the local maximum in Fig. 4.15 which results in a modified 3M βα − diagram represented in Fig. 4.16. Some of the vertices have been wiped out. Thus, a part of information about the past history has been erased. A quite analogous consideration can be made with a monotonically decreasing input u . Now it becomes clear why the memory registers only an alternating sequence of dominant input extrema. All other input extrema are wiped out. In particular the memory does not store the entire input history.

( )tS + ( )tS −

β

α

( )tL1M

2m

u=α

Fig. 4.16. Staircase interface ( )tL after wiping out some of the vertices. The wiping-out (deletion) property is confirmed mostly by experiments. The shapes of

major hysteresis loops do not depend on how they have been achieved. 4.2.5 The congruency property of the classical Preisach model

To complete the description of the properties of the memory, the congruency property will be explained. If the input is varying between the same two extreme values, and u , congruent loops demonstrated in Fig. 4.17 are traced independent of the recorded history indicated in Fig. 4.18 by the different interfaces

u 1u 2

( )tL and ( )tL* . They are only displaced along the - axis. f

33

4 Preisach Model

u

f

1u 2u

Fig. 4.17. Congruent loops.

β

α

( )tL

( )tL*

1u

2u

Fig. 4.18. Possible interfaces ( )tL and ( )tL* when the input u varies

between the two extreme values and u . 1u 2

This congruency property contradicts experimentally observed hysteresis properties of

almost all real existing materials. 4.2.6 First Order Reversal Curves β'α'f

In the following the meaning of first order reversal (transition) curves will be explained. To this end, the hysteresis transducer is brought in the state of negative saturation. In the next step the input of the hysteresis transducer is increased monotonically up to an arbitrary value

u'α (see Fig. 4.19). The input-output trajectory of the HT follows the limiting ascending

branch. The corresponding output value of the hysteresis transducer to be achieved is . Next, the input u is reversed and decreases monotonically to a certain value . The trajectory to be followed is the first order reversal curve. At the end the output of the hysteresis transducer achieves . The first order reversal curves are attached to the limiting ascending branches. Fig. 4.20 shows the

'αf'β

f

''βαfβα − diagram due to the first order reversal curve.

34

4 Preisach Model

f

u'α'β

''f βα 'fα

Fig. 4.19. First order reversal (transition) curve.

β

α

'β

'α

( )',' βαT

( )tS +

( )tS −

Fig. 4.20. βα − diagram due to a first order reversal (transition) curve.

4.2.7 Determination of the Preisach Distribution Function ( )βα,µ

For the determination of Preisach distribution function ( )βαµ ,

'βαf, the first order reversal

curves are needed. To derive the relation between ' and ''βαf ( )βαµ , the following difference will be used.

( )( )∫∫=−

','''' ,2

βαβαα βαβαµ

Tddff (4.4)

The triangle T( ',' )βα in (4.4) is represented in Fig. 4.20. To make the subsequent explanation clear, the boundaries of the integral in (4.4) are written in detail.

( )( )

( )∫

∫=∫∫

= =

'

'

'

','T,,

α

ββ

α

βαβαβαβαµβαβαµ dddd (4.5)

35

4 Preisach Model

Differentiating of (4.4) twice (first with respect to 'β and then with respect to 'α ) and taking account of (4.5) yields

( )''2

1',' ''

2

βαβαµ βα

∂∂

∂=

f. (4.6)

Equation (4.6) shows how the Preisach distribution function ( )βαµ , is related to experimentally measured first order reversal curves . ''βαf

4.3 Numerical Implementation of the Classical Preisach Model

Based on the facts discussed till now one would assume that the implementation of the Preisach model needs, first, the differentiation of experimentally measured data according to (4.6) which emphasizes substantially the noise which always occurs in the context of measured data and, secondly, to carry out the double integral in (4.2) which is very time consuming.

Luckily, both really cumbersome drawbacks can be circumvented. To this end, the integral in (4.2) will be rewritten in the similar form ( ) ( ) ( )

( )∫∫+∫∫−=+ tS

ddddtf βαβαµβαβαµ ,2,T

, (4.7)

whereby the first integral in (4.7) equals the value of positive saturation:

( )∫∫=+

T, βαβαµ ddf . (4.8)

Then, the second integral of (4.7) will be split up in a sum of integrals carried out over the individual trapezoids Q represented in Fig. 4.21. In general, the number n and the shape

of trapezoids vary with time t . k ( )t

( )tQk

( )( )

( )( )

( )∑ ∫∫=∫∫=+

tn

k tQtS k

dddd1

,, βαβαµβαβαµ (4.9)

The integrals over the trapezoids in turn will be decomposed in differences of two integrals carried out over corresponding triangles. Along with (4.4) the equation (4.10) will be obtained.

( )( )

( ) ( )kkkkkk

k

mMMmMMtQ

ffffdd ,, 21

21

,1

−−−=∫∫−

βαβαµ (4.10)

Considering equations (4.8) to (4.10) the equation (4.7) becomes

( ) ( ) ( )( )∑ −+−+−=−

=

+−−

1

1,,,, 11

tn

kmMtuMmMmM nnnkkkk

ffffftf . (4.11)

36

4 Preisach Model

Equation (4.11) is valid for a vertical final link of the interface between the positive set ( )tS + and negative set . A similar relation to (4.11) can be derived which holds for the case with a horizontal final link.

( )tS −

( )tS +

( )tS −

β

α

( )1−kk m,M

( )tu

( )kk m,M

kQ

Fig. 4.21. Subdivision of ( )tS + in trapezoids Q . k

The values of the first order reversal curves are determined experimentally or by

evaluating the integral (4.2) when the anlytical form of the Preisach distribution function ( )1, −kk mMf

( )βαµ , is known in discrete points of an imaginary grid in the βα − diagram corresponding to vertices of dominant input extrema ( )( )1−k,k mM . Experience has shown that it suffices to approximate the values in points within the grid linearly.

( )1, −kk mMf

4.4 Hysteresis Energy Losses of the Classical Preisach Model

The prediction of the hysteresis losses in electrical devices is very important for the design. Therefore, an accurate determination of the hysteresis losses is needed.

For a long time only the computation of the hysteresis energy losses caused by cyclic time variation was known. In this special case the energy conversation principle is valid and the f u

Fig. 4.22. Hysteresis energy losses caused by cyclic time variation

are proportional to the enclosed dark area.

37

4 Preisach Model

hysteresis energy losses are proportional to the enclosed area of the input-output trajectory (see Fig. 4.22). However, the Preisach model is also capable to calculate the hysteresis losses for arbitrary time variations.

In magnetics, the infinitesimal energy supplied to the transducer in Fig. 4.1, wherein and stand for

uf H and B , is

BHW δδ = . (4.12) In (4.12) H means the magnetic field intensity and M stands for the magnetization.

Next, the hysteresis energy losses shall be derived by means of the Preisach model. The hysteresis energy losses W of a rectangular loop shown in Fig. 4.8 for one cycle are cycle

( )βα −= 2cycleW . (4.13)

A comparison of Fig. 4.8 with equation (4.13) demonstrates that no energy losses occur as the horizontal links are traced, because the horizontal links are fully reversible. Whereas the vertical links are irreversible and, therefore, the switchings between the output values and

cause the energy losses. Since the switchings are symmetric what energy losses concerns, the losses in (4.14) of one switching can be equated to one half of the losses W caused by a full cycle.

1−1+

q cycle

βα −=q . (4.14)

The product ( ) ( )tuαβγβαµ , can be seen as the losses of a transducer with the output ( )βαµ ,± . Considering additionally that a hysteresis non-linearity is represented by a

continuous set of weighted elementary rectangular loops the losses caused by all loops come into question represented by

QΩ (dark area) in the Preisach plane in Fig. 4.23 are

calculated by the following two dimensional integral.

( )( )∫∫ −=Ω

βαβαβαµ dd,Q (4.15)

Ωβ

α

( )tS +

( )tS −

Fig. 4.23. Hysteresis energy losses caused by all rectangular loops in the

dark area Ω due to an arbitrary input variation.

38

4 Preisach Model

Equation (4.15) rests on two facts. First, the hysteresis energy losses can be calculated of a

rectangular loop for arbitrary input variations and, secondly, a hysteresis non-linearity can be represented by the Preisach model as a superposition of rectangular loops.

In Fig. 4.23 a typical arbitrary shape of Ω is shown. It can easily be seen that Ω can be subdivided in trapezoids. The surface of a trapezoid in turn can be observed as a difference of two triangles.

Now, the energy losses of an arbitrary triangular area is derived (see Fig. 4.24). In the following some simplifications of (4.15) are carried out that facilitate the determination of the hysteresis energy losses Q on the bases of first order reversal curves. To this end, the function ( ) ( )βαβα ,F− is differentiated twice.

( ) ( )[ ] ( ) ( ) ( ) ( )βαβαβα

αβα

ββαβαβα

βα ∂∂∂

−+∂

∂−

∂∂

=−∂∂∂ ,F,F,F,F

22

(4.16)

The function ( )βα ,F in (4.16) is one half of the difference along the first order reversal curve.

( ) ( ''' ff,F βααβα −=21 ) (4.17)

A simple manipulation of (4.16) and taking account of (4.19) along with (4.6) the integral in (4.15) carried out over the triangle ( )−+ uu ,T in Fig. 4.24 yields

( ) ( ) ( ) ( ) ( )∫−∫−=+

−

+

−

−+−+−+−+

u

u

u

udu,Fd,uFu,uFu,uu,uQ ααββ . (4.18)

β

α

+u

−u

( )+− uuT ,

kΩ

Fig. 4.24. Hysteresis energy losses caused by all rectangular loops in the triangle T . ( )−+ u,u

Thus, the determination of the hysteresis energy losses ( )−+ u,uQ dissipating as the triangle is swept has been simplified enormously, because, in (4.18) only one dimensional

integrals have to be evaluated and, secondly, the energy losses ( −+ u,uT )

( )−+ u,uQ are expressed in terms of experimentally measured or calculated first order reversal curves (see also (4.17)).

39

4 Preisach Model

The total hysteresis energy losses associated with an arbitrary input variation represented by the area Ω in Fig. 4.23 of a Preisach operator is simply the sum of the hysteresis energy losses over the corresponding trapezoids

ΩQ

kQ kΩ shown in Fig. 4.24.

∑== ΩΩ

n

1k k

QQ (4.19)

40

5 Hysteresis Modeling by the Finite Element Method Considering Eddy Currents

5 Hysteresis Modeling by the Finite Element Method

Considering Eddy Currents

The aim of this chapter is to develop methods capable of simulating the eddy current distribution in ferromagnetic materials. Hysteresis has to be considered, the associated eddy current losses and hysteresis losses should be computed. The model of the hystersis non-linearity has to be accurate and suitable for incorporation in the finite element method. A small memory demand and guaranteed fast convergence of the hysteresis model incorporated in the finite element method are required features. To facilitate the analysis of practical problems, the computations have to be carried out at a minimum expense of time. Therefore, a small effort in handling of the hysteresis model is inevitable. The Preisach model fulfills these requirements at best as demonstrated by the fact that it is by far the most widely used hysteresis model. Therefore, the Preisach model is employed here, too. 5.1 Methods to Incorporate the Hysteresis – Nonlinearity in the Finite

Element Method

To consider the hysteresis of ferromagnetic materials accurately and efficiently, the incorporation of the hysteresis model in the finite element method is an essential point. The methods for it can be subdivided in those taking account of hysteresis non-linearity by means of material coefficients i.e., depending on the formulation to be used, the magnetic permeability µ or the reluctivity ν , and in those describing hysteresis directly with the aid of the magnetic field quantities magnetic field intensity H and magnetic flux density B or magnetization M . 5.1.1 Methods based on Material Coefficients Investigations have shown that models which consider the hysteresis non-linearity based on the total magnetic permeability

HB

tot =µ (5.1)

do not lead to numerically stable solutions. This is confirmed by many authors in the related literature, see also Fig. 5.1. In (5.1), scalar quantities are regarded only for the sake of simplicity. Herein totµ means the total magnetic permeability, B the magnetic flux density and H stands for the magnetic field intensity. The reason of the numerical instability is due the fact that the total magnetic permeability totµ may be negative or even infinite. However, the differential permeability diffµ is positive and bounded.

41


H

BB∆

( )HB

totµ

H∆

diffµ

H

B

Fig. 5.1. Total and differential permeability . totµ diffµ

5.1.1.1 Differential Magnetic Permeability diffµ The differential magnetic permeability

dtdHdtdB

dHdB

diff //

==µ (5.2)

is always a positive and finite number whose application leads to a stable numerical solution. This is discussed in numerous papers, for instance, in [26]. It is worth to note that the differential magnetic permeability diffµ stems from the differentiation with respect to time.

In the numerical realization, the differential quotient in (5.2) is approximated by the corresponding finite difference quotient

HB

diff ∆∆

≈µ . (5.3)

For instance, Dubre et al [26] introduce the differential permeability or the differential

reluctivity diffµ

diffν by differentiating the constitutive laws

µHB = (5.4) or

BH ν= (5.5) with respect to time. Hence,

Ht

µBt diff ∂

∂=

∂∂ (5.6)

and

42


Bt

Ht diff ∂

∂=

∂∂ ν (5.7)

is obtained. The feasibility is demonstrated for instance in [26, 27] for 2D field problems leading to material tensors for the differential magnetic permeability [ ]diffµ and the differential reluctivity [ ]diffν , respectively. The problems are formulatd by means of the magnetic vector potential or by the magnetic scalar potential Φ . In fact, the implemented algorithm dealing with the hysteresis non-linearity by the time stepping method uses the average value

A

( )( )

( )( )

( )( )1idiff

l

idiffl

idiff µµ21µ

−+= . (5.8)

The index i in (5.8) denotes the time step and the superscript l relates to the nonlinear

iterations. Another proposal has been made by Jack and Mecrow in [28]. A set of curves consisting of

BH -loops along with the associated peak values of the magnetic field intensity H and “recoil lines” are determined from the Preisach model. The three quantities define a surface in 3D. The “recoil lines” are first order reversal curves [23]. Since the model considers only the value of H of the last reversal and the current value of H it fails an exact simulation of general transient problems. The Dufort-Franklin-algorithm with reluctivity damping has been applied [29] for the time stepping scheme. The reluctivity damping

( ) ( ) ( ) (i1i1i r1r )ννν −+= ++ (5.10) represents a relaxation of the material coefficients. In (5.10) ν denotes the reluctivity, the time step in a time stepping scheme and

ir is the relaxation factor. It has been found that

values of r from 0.02 to 0.35 produce stable results for a large range of excitations with acceptable accuracy. For each finite element or integration point the current state of the magnetic field along with the reversal point have to be stored. 5.1.1.2 Modified Constitutive Law

Henrotte [30] introduced the new constitutive law

c'HB'H +=ν . (5.11) In (5.11) means the coercive field of the outermost c'H BH -loop and 'ν is the reluctivity calculated as the secant of B and H related to . In principle, this constitutes a coordinate transformation to avoid that

c'H'ν becoming infinite except 0B = , (see Fig. 5.2).

43


B

Hc'H

'ν

H

B

Fig. 5.2. BH -loop and coercive field . cH'

5.1.1.3 Fixed Point Technique

The fixed point method (FPM) is an alternative method developed by an Italian research team and has been applied successfully. This is well documented in various publications, for instance [31, 32, 33].

The idea is the following. A particular value for the magnetic permeability µ or for the reluctivity ν is used, namely, FPµ or FPν , which remains constant for all integration points throughout the time steps. The choice of FPµ or FPν obeys a certain criterion [33]. The nonlinear relationship

(HBB = ) (5.12) is decomposed into a linear part HFPµ and a nonlinear residual R :

RHB FP += µ . (5.13) Depending on the formulation applied, (5.13) or

SBH FP +=ν (5.14) is exploited. In case of the - formulation, the magnetic flux density A B is obtained from the magnetic vector potential. The magnetic field intensity H is linked to B via the Preisach model by (5.12). The new residual can be updated by

( ) ( )( ) ( )lFP

ll BBHS ν−= . (5.15) The subscript l in (5.15) relates to the nonlinear iterations. The procedure is quite similar when H is the primary variable as in T-formulations.

There are some outstanding advantages. The system matrix of the finite element method remains the same over all time steps. Therefore, its decomposition has to be performed only once. The convergence is guaranteed even in case of inflection points without a priori information. It does not need the derivative of the BH - curve, thus, piecewise interpolation suffices.

44


The disadvantage is the considerably high number of nonlinear iterations required to achieve a convergent solution for one time step. 5.1.2 Methods based on Magnetic Field Quantities

These models incorporate the magnetic field values directly in the governing equations of the field problem and thus, the model gets by without determining the magnetic permeability µ or reluctivity ν . Two magnetic field quantities based models are mentioned here, representative for many existing models. Both models approximate the hysteresis non-linearity.

In [34] the hysteresis non-linearity is recorded by measuring of dc BH - loops. Loops within them are approximated. The method is applied for time harmonic and transient simulations. The hysteresis non-linearity is represented by means of a local memory.

In the second model [35] the hysteresis characteristic is described by the outer most BH - loop. Minor loops and the initial curve are deduced from the outer most loop by a simple analytical expression. The progress of the magnetic field intensity is recorded. Only a hysteresis non-linearity with local memory is considered. 5.2 Treatment of the Hysteresis Non-linearity by BH - Curves

A novel alternative technique to achieve fast convergence of the time stepping scheme is presented here. Let us assume that the solution at an arbitrary point in the problem region is sought for the next time step. The solution must lie either on a branch of the BH - curve valid for increasing magnetic field intensity, H ( BH - curve, positive branch), or on a branch of the BH - curve valid for decreasing magnetic field intensity, H ( BH - curve, negative branch), see Figs. 5.3 and 5.4. Both branches are known a priori and can be easily computed from the Preisach model taking account of the corresponding history. Therefore, the hysteresis non-linearity is reduced to a univalent function from the current solution point of view.

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

-1000 -750 -500 -250 0 250 500 750 1000Magnetic field intensity H (A/m)

Stationary BH - loopBH - curve,negative branchBH - curve,positive branch

Fig. 5.3. Stationary BH - loop and BH - curves from the Preisach model inside the area restricted by the limiting triangle.

45


-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

-1000 -750 -500 -250 0 250 500 750 1000

Magnetic field intensity H (A/m)

BH - curve,negative branchBH - curve,positive branch

Fig. 5.4. Positive and negative BH - curve from the Preisach model inside the area restricted by the limiting triangle.

The BH - curves are determined in discrete points and within them linear interpolation is applied. To take care of the case that a solution lies temporarily outside the area restricted by the limiting triangle of the Preisach model [23], the branches of the BH - curves are extended up to an almost arbitrarily large scale. The slope of the extensions has been chosen slightly smaller than that within the permissible area and adjacent to the limit to ensure fast convergence as depicted in Fig. 5.5.

-4

-3

-2

-1

0

1

2

3

4

-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000

Magnetic field intensity H (A/m)

BH - curve,negative branchBH - curve,positive branch

Fig. 5.5. BH - curves from the Preisach model inside and outside the area restricted by the limiting triangle.

Experience has shown, that branches of the BH - curves can be convex or in the first part concave and later on convex. In [36] it has been elaborated extensively how to proceed in order to achieve a stable scheme of nonlinear iteration steps.

46


Only the case when the magnetic field intensity H is known from the solution, i.e., the case of - formulation, and the magnetic flux density H B - is sought is treated here. The procedure for the - formulation is similar and can be found in [36]. A

In the i-th iteration step, the magnitude of yields from the iH 1iB + BH - curve, sketched in Fig. 5.6. From the secant the new permeability i1i HB /+ 1i+µ is computed. In fact, this is convergent without underrelaxation provided the BH - curve is monotonous and convex (i.e.

and ∂ ). 0HB >∂∂ / 0H <B 22 ∂/

Unstable choice forStable choice forB

B

B

H H H

i

i+1

i i+1

ii+1 i+1µ µ

µ

Fig. 5.6. Scheme of nonlinear iteration step for convex BH - curve. To ensure convergence without underrelaxation even in the case where the BH - curve is

partially concave (i.e. ), as shown in Fig. 5.7, a slight modification has to be undertaken in this section. The magnitude is multiplied by the previous value of the permeability

0HB 22 >∂∂ /iH

iµ which yields . From the iB BH - curve is calculated, the new permeability is the secant .

1iH +

1ii HB +/

HHHH

B

B

B

B

T

i+1

i

i+1 i T

Unstable choice for

Stable choice for µ

µ

µ

i+1

i+1

i

Fig. 5.7. Scheme of nonlinear iteration step for concave - convex BH - curve.

47


The iteration process is terminated once the variation in the permeability between two iteration steps becomes small enough.

To apply the procedure for a stable scheme to the BH - curves calculated from the Preisach model each branch is observed separately and their initial point is moved in the point of origin. The procedure for negative branches which are in the third quadrant in the BH - plane is analogous to that for positive branches in the first quadrant.

The permeability calculated as the secant with respect to the initial point of the branches represents a differential permeability

HB /diffµ .

A few points for the branches of the BH - curves have to be calculated only once for all nonlinear iteration steps. The cost involved is small compared with that needed when the points of the BH - curve are calculated for each nonlinear iteration step directly from the Preisach model. Moreover, the treatment of the hysteresis non-linearity does not need a differentiation of the BH - curve and it is convergent even in case of inflection points. 5.2.1 Techniques to handle the BH-Curves

To update the material characteristics, i.e., to compute the BH - curves from the Preisach model in all integration points after each time step in the time stepping scheme requires a lot of time. An idea to minimize the computational effort is to store the BH - curves as long as the related magnetic field intensity H does not reverse. An alternative technique to handle the hysteresis non-linearity efficiently is presented and compared with the time consuming one. All techniques have in common that they get by without relaxation of the material properties, i.e. the relaxation factor r in (5.10) equals 1.0. The permeability diffµ is derived in such a way that it is always a positive and finite number. In case of the - formulation the approximation

H

( ) ( )( i1idiff hht

1tB

tB

−∆

=∆∆

≈∂∂

+µ ) (5.16)

for the time stepping scheme is used. The time instant is represented by i in (5.16). Analogous consideration can be made for the formulation with the magnetic vector potential

. A 5.2.1.1 Direct Technique

The BH - curves are determined from the Preisach model after each time step. Therefore, the direct technique is very time consuming. However, it is accurate and few sampling points of the BH - curves suffice assuming small time steps in the time stepping scheme because the solution is located at the beginning of the branches of the BH - curves. Consequently, it is advisable to arrange many sampling points in the first part of the branches and few for the rest to achieve high accuracy on one hand and on the other hand to minimize the computational effort. In case of the direct technique, the relation (5.16) becomes

( )refdiff hht

1tB

−∆

=∆∆ µ , (5.17)

see also Fig. 5.8.

48


H

B( )HB

hrefh

B∆diffµ

refb

b

Fig. 5.8. Direct technique and differential permeability . diffµ

5.2.1.2 Store Technique

The BH - curves are determined from the Preisach model and kept in the memory as long as the magnetic field intensity H does not reverse. Writing the equation (5.16) specially for the store technique, follows

( ) ( )( )i1idiff hht

1tB

−∆

=∆∆

+µ . (5.18)

H

B

( )2ib +

( )1ih +

( )ib

( )ih ( )2ih +

( )1ib +B∆diffµ

( )HB

refh

refb

Fig. 5.9. Store technique and differential permeability . diffµ

Using this method, the computation of the eddy current losses is very sensitive with respect to the number of sampling points used for the

eddyPBH - curves. The reason is that,

contrary to the direct technique, the solution may also lie at the end of the branches when

49


using the store technique. Therefore, too few sampling points at the end of the branches yields a poor approximation of the BH - curves, i.e., the differential permeability diffµ deviates from the true one. The value of the approximate differential permeability diffµ is smaller than the true one.

J

5.3 Formulations of an Eddy Current Problem with Hysteresis

The eddy current problem, i.e., the low frequency subset of Maxwell’s equations, neglecting

the displacement current density, t∂

∂D , is the starting point of the formulations to be derived.

Hence, the system of partial differential equations governing in a region observed are

JH =curl , (5.19)

tcurl

∂∂

−=BE (5.20)

and

0div =B . (5.21) Additionally, the constitutive laws

EJ σ= , (5.22) or

JE ρ= , (5.23)

HB µ= , (5.24) or

BH ν= . (5.25) are valid. As before, in (5.19) to (5.25) means the magnetic field intensity, is the electric current density, stands for the magnetic flux density and E for the electric field intensity. The electric conductivity

HB

σ , the electric resistivity ρ , the magnetic permeability µ and the magnetic reluctivity ν are material properties.

The hysteresis non-linearity is considered by the classical scalar Preisach model [23]. Special attention is paid to the incorporation of the hysteresis non-linearity to achieve equations systems which, solved by a time stepping scheme, deliver stable solutions.

The eddy current problem with hysteresis is formulated once with the magnetic field intensity and once with the magnetic vector potential A as solution variable to cover two different types of problems regarding the excitation which prescribes either a certain current or a certain voltage. The resulting partial differential equations with appropriate boundary

H

50


conditions leading to a unique system are solved by the method of Galerkin for finite elements using nodal elements of second order.

Since the purpose is to study the behavior of the hysteresis non-linearity incorporated in the finite element method considering the eddy currents, a very simple one dimensional model suffices. Suitable boundary conditions allow to exclude the non-conducting region. First, the eddy current problem is posed in three dimensions and then reduced to the one-dimensional case. 5.3.1 - Formulation of an Eddy Current Problem with Hysteresis H

The one dimensional model with appropriate boundary conditions is shown in Fig. 5.10. For instance, it may serve to simulate the field distribution across one lamination approximately.

yeH th=

y

x0

d

⊗J

yeH th=

σ

Fig. 5.10. One dimensional model with the magnetic field intensity Has solution variable and appropriate boundary conditions.

From Ampere’s law (5.19) and Faraday’s law (5.20), the partial differential equation

HH µρt

curlcurl∂∂

−= (5.26)

is obtained in the conducting region after considering (5.23) and (5.24). On the boundary, the tangential component

( ) tHnHn =×× (5.27) is prescribed.

To solve the partial differential equation (5.26), the method of Galerkin for finite elements is employed. To this end, the unknown field value H is approximated by nodal vector shape functions as

( ) ( )( )

∑=

+++=≈noden

1kzkzkykykxkxkD

n hhh NNNHHH . (5.28)

51


In (5.28), the term H considers the inhomogeneous boundary conditions (5.27), whereas the sum represents the field within the conducting region with homogeneous boundary conditions. The vector shape functions can be written as

D

xkxk N eN = (5.29a)

ykyk N eN = (5.29b)

zkzk N eN = . (5.29c)

Here represents a scalar nodal shape function. To minimize the error due to the approximation, the residual is weighted by the same vector shape functions over the conducting region

kN

cΩ

( ) ( ) 0dt

curlcurlc

nni∫

Ω

=Ω

∂∂

+ HHN µρ , (5.30)

whereby means iN

xii N eN = (5.31a) or

yii N eN = (5.31b) or

zii N eN = . (5.31c) After exploiting the vector identity

( )∫∫∫ΓΩΩ

Γ⋅×−Ω=Ω dcurldcurlcurldcurlcurlcc

nVUVUVU ρρρ (5.32)

the double derivation of the unknown field value ( )nH in (5.30) can be avoided:

( ) ( ) ( )( )∫∫ΓΩ

=Γ⋅×−Ω

∂∂

+ 0dcurldt

curlcurl ni

ni

ni

c

nHNHNHN ρµρ . (5.33)

Since the shape functions have to fulfill the required homogenous Dirichlet boundary conditions

iN

0nN =×i , (5.34)

52


the surface integral in (5.33) vanishes. Hence,

( ) ( )∫Ω

=Ω

∂∂

+c

0dt

curlcurl ni

ni HNHN µρ (5.35)

is obtained.

The problem is a scalar one, hence

yyh eH = (5.36) is valid and also one dimensional, thus

0y=

∂∂ (5.37a)

and

0z=

∂∂ (5.37b)

holds. The unknown field value h in (5.36) is approximated by scalar shape functions y

( )

( )

∑=

+=≈noden

1kkykyD

nyy Nhhhh . (5.38)

In (5.38) considers inhomogeneous Dirichlet boundary conditions. Thus, yDh

0dxht

Nhx

Nx

d

0x

nyi

nyi =

∂∂

+∂∂

∂∂

∫=

)()( µρ (5.39)

is valid in the conducting region and cΩ

( ) ( ) tn

y h0xh == (5.40a)

( )( ) tn

y hdxh == (5.40b) on the boundary . Γ

After integrating (5.39) and taking account of the known inhomogeneous Dirichlet boundary conditions (5.40a) and (5.40b), the description of the model leads to a system of ordinary differential equations

[ ] ( ) ( )( )[ ] ( ) ( ) ( )tttt hrhhBhA =∂∂

+ ( ) . (5.41)

53


The vector in (5.41) represents the unknown nodal values of ( )th ( )nyh in (5.39), the mass

matrix comprises the non-linear material property indicated by its dependence on .

( )( )[ th ]B( )thFinally, the system of inhomogeneous ordinary differential equations (5.41) will be solved

by a time stepping scheme. The fact that, the hysteresis non-linearity in terms of the magnetic permeability µ appears

in the mass matrix supports its incorporation in the finite element method in the context of a time stepping scheme, because the time derivative in (5.39) will be approximated by

( )yrefydiffn

y hht

1ht

−∆

≈∂∂ µµ )( , (5.42)

see also Fig. 5.3. Thus, the hysteresis non-linearity will be incorporated in terms of the differential magnetic permeability diffµ in the equations systems to be solved.

5.3.2 A - Formulation of an Eddy Current Problem with Hysteresis

In many practical problems one is confronted with excitations in which the time behavior of the voltage is given, i.e., the magnetic flux Φ is prescribed. The formulation with the magnetic vector potential can cope with this tasks at best. The simple model is one-dimensional and is sketched in Fig. 5.11 with the boundary conditions selected.

A

zeA ta= zeA ta−=

z

x0

d

⊗Φ

σ

Fig. 5.11. One dimensional model with the magnetic vector potential A as solution variable and appropriate boundary conditions.

A magnetic vector potential can be introduced by considering (5.21) A

AB curl= . (5.43) Taking account of (5.43) and Faradays law (5.20) yields

54


0AE =

∂∂

+t

curl . (5.44)

The vector field t∂

∂+

AE in (5.44) is curl free and, thus, it can be represented as a gradient of

an electrical scalar potential V .

gradVt

−=∂∂

+AE (5.45)

Since only problems with constant electric conductivity σ are considered, the right hand side of (5.45) can be set to zero and the electric field intensity is determined simply by E

t∂∂

−=AE . (5.46)

Inserting (5.46) in Amperes law (5.19) leads to the partial differential equation

0AA =∂∂

+t

curlcurl σν (5.47)

valid in the conducting region. To achieve a unique magnetic vector potential , the tangential component

A

( ) tAnAn =×× (5.48)

is prescribed.

To solve the boundary value problem (5.47), (5.48) the Galerkin finite element technique is applied. To this end, the magnetic vector potential is approximated by nodal vector functions:

A

( ) (

( )

∑ +++=≈=

noden

1kzkzkykykxkxkD

n aaa NNNAAA ) (5.49)

In (5.49), the term considers the inhomogeneous boundary conditions (5.48), whereas the sum in (5.49) represents the magnetic vector potential within the conducting region with homogeneous boundary conditions. This leads to

DAA

( ) ( ) 0d

tcurlcurl

c

nni =Ω

∂∂

+∫Ω

AAN σν . (5.50)

Applying the vector identity (5.32) to equation (5.50) yields

( ) ( ) ( )( ) 0dcurldt

curlcurl ni

ni

ni

c

=Γ⋅×−Ω

∂∂

+ ∫∫ΓΩ

nANANAN νσν . (5.51)

The shape functions fulfill the required homogenous Dirichlet boundary conditions (5.35) iN

55


and the surface integral in (5.51) vanishes. Hence,

( ) ( ) 0dt

curlcurlc

ni

ni =Ω

∂∂

+∫Ω

ANAN σν (5.52)

is obtained.

The problem is a scalar one, hence

zza eA = (5.53) is valid, and also one-dimensional, thus (5.37) holds. and one dimensional, i.e., (5.37) holds. The unknown quantity in (5.53) is approximated by scalar shape functions

za

( )

( )

∑=

+=≈noden

1kkzkzD

nzz Naaaa . (5.54)

The term in (5.54) takes account of inhomogeneous Dirichlet boundary conditions. Thus, zDa

0dxat

Nax

Nx

d

0x

nzi

nzi =

∂∂

+∂∂

∂∂

∫=

)()( σν (5.55)

is valid in the conducting region . On the boundary, inhomogenous Dirichlet boundary conditions

cΩ

( )( ) tn

z a0xa == (5.56a)

( ) ( ) tn

z adxa −== (5.56b) are prescribed to enforce a certain magnetic flux Φ shown in Fig. 5.11, which yields

( ) ( ) ( ) ( )2Φdxa0xa n

zn

z ==−== . (5.57)

The boundary value problem (5.55), (5.56) has one major drawback. The hysteresis non-

linearity is considered in (5.55) in terms of the total magnetic reluctivity totν and, contrary to the H - formulation, in the stiffness matrix. In this case it is not straightforward to introduce the differential reluctivity, as it was easy to use the differential permeability in the H - formulation (see (5.42)). To overcome this problem, the difference of the solutions in two consecutive time steps is built leading to a differential magnetic reluctivity diffν , see Fig. 5.12. This idea is based on the time stepping scheme to solve the boundary value problem (5.55), (5.56). To this end, the equation (5.55) is written for the time instant 1t

56


0dxat

Nax

Nx

d

0x

n1zi

n1z1i =

∂∂

+∂∂

∂∂

∫=

)()( σν (5.58a)

as well as for the subsequent one ttt ∆+= 12

0dxat

Nax

Nx

d

0x

n2zi

n2z2i =

∂∂

+∂∂

∂∂

∫=

)()( σν . (5.58b)

Hence, the difference of (5.58a) and (5.58b) yields

( ) 0dxaat

Nax

ax

Nx

d

0x

n1z

n2zi

n1z1

n2z2i =

−

∂∂

+

∂∂

−∂∂

∂∂

∫=

)()()()( σνν . (5.59)

A small modification of the first expression in brackets in (5.59) leads to

( ) ( ) Hhhbbax

ax 1y2y1y12y2

n1z1

n2z2 ∆−=−−=−−=

∂∂

−∂∂ νννν )()( . (5.60)

Comparing Fig. 5.8 and Fig. 5.12, H∆ in turn can be rewritten as follows.

( ) ≈−−=∆−=∆−=∆− 1y2ydiffdiffdiff

bbBB1H ννµ

)()()( ndiff2zdiff

n1z

n2zdiff a

xa

xa

x ∂∂

=

∂∂

−∂∂

= νν (5.61)

Modifying also the time derivative of the second expression in brackets in (5.59) yields

( ) ( ))()()()( ndiff1z

ndiff2z

n1z

n2z aa

taa

t−

∂∂

=−∂∂ . (5.62)

t2t1t0t

( )nza

( )ndiff2za

( )ndiff1za

( )n2za

( )n1za

( )n0za

Fig. 5.12. The time function of the magnetic vector potential of one component A

regarding the time stepping scheme.

57


Inserting (5.59) and (5.62) into (5.55) results in

∫∫==

∂∂

=

∂∂

+∂∂

∂∂ d

0x

ndiff1zi

d

0x

ndiff2zi

ndiff2zdiffi dxa

tNdxa

tNa

xN

x)()()( σσν . (5.63)

The corresponding boundary conditions are

( )1t2t

ndiff2z aa0xa −==

(5.64a)

( )

2t1tn

diff2z aadxa −==

. (5.64b)

The magnetic vector potential ( )n

diff1za is known on the right hand side in (5.63). After integrating (5.63) and considering of the known inhomogeneous Dirichlet boundary

conditions (5.64a) and (5.64b), the description of the model leads to a system of inhomogeneous ordinary differential equations

( )( )[ ] ( ) [ ] ( ) ( )tttt raBaaA =∂∂

+ . (5.65)

The vector in (5.65) represents the unknown nodal values of in (5.63), the stiffness matrix

( )ta ( )ndiff2za

( )( taA)

)][ comprises the non-linear material coefficient as indicated by its dependence on a . (t

Finally, the system of inhomogeneous ordinary differential equations (5.65) will be solved by a time stepping scheme. 5.4 Time Stepping Method The system of inhomogeneous ordinary differential equations with nonlinear material properties obtained by the finite element Galerkin technique shall to be solved for transient time behavior as indicated in Fig. 5.13 by the time stepping method.

The duration T of the analysis is subdivided in n time steps. This leads to n+1 discrete time instants

ttt i1i ∆+=+ , (5.66) i=0, 1, 2, ..., n (5.67) with the time step assumed to be constant. Hence, t∆

nTt =∆ (5.68)

is valid.

58


( )tu

t

u

it 1it +

t∆

iu

1iu +

Fig. 5.13. Discretization of the time function ( )tu and linear approximation within two time steps.

The solution of the differential equations is determined only in the time instants t , within them linear interpolation is assumed. The time derivative of an unknown function

i

( )tu is approximated by the finite difference quotient

( )i1i uut

1tu

−∆

≈∂∂

+ . (5.69)

A quantity not differentiated with respect to the time t is replaced by ( ) ( ) i1i u1utu Θ−+Θ≈ + . (5.70)

The value is chosen from the interval Θ [ ]010 ., . Certain values of Θ are well known time stepping methods. 0=Θ is called forward Euler method, 21 /=Θ Crank-Nicholson,

Galerkin and Θ backward Euler [19]. 32 /=Θ 01.=In the subsequent sections the time stepping schemes for the H and A - formulation are

derived. Quantities in squared brackets mean matrices and quantities in braces are vectors. The index stands for a certain time instant, the superscript l belongs to a nonlinear iteration at the time instant i and the index D means Dirichlet boundary conditions. Quantities at the time instant i are unknown and at the time instant i

i

1− they are known. For the sake of clarity, the superscript representing the nonlinear iterations of the known time instant is omitted. Distinction is drawn between quantities which vary with the iterations and those which remain constant because this can be exploited to save computation time. The resulting equations systems to be solved are very small for the model used, so that they can be solved directly for each nonlinear iteration.

The Gaussian quadrature is used to set up the element matrices for the finite element method and this is why the material coefficients (magnetic permeability µ or reluctivity ν ) are stored in the integration points. At the time instant , the material properties are updated in an iterative procedure till two stopping criterions are fulfilled simultaneously. The criterions are the average change of the material coefficients

i

( ) ( )

( ) meanmean

1l

1ll

εµ

µµ≤

−−

−

(5.71a)

59


or

( ) ( )

( ) meanmean

1l

1ll

εν

νν≤

−−

−

(5.71b)

and the maximal change

( ) ( )

( ) maxmax

εµ

µµ≤

−−

−

1l

1ll

(5.72a)

or

( ) ( )

( ) maxmax

εννν

≤−−

−

1l

1ll

(5.72b)

in the integration points. The values %1mean =ε and %5=maxε have been used. Since the

solution vector belongs to the material coefficients ( )( )liu ( )( )1l

i−µ or ( )( )1l

i−ν , the material vector

or ( )( )liµ ( )( )liν is calculated once the stopping criterion is met and serves as initial condition

for the next time step 1i + . 5.4.1 Time Stepping Method for the H – Formulation

Carrying out the integration in (5.39) and considering (5.40) and (5.42) yields a homogeneous system of ordinary partial differential equations.

[ ] [ ] ( ) 0~~~~=

∂∂

+ ht

h BA (5.73)

The tilde above quantities in (5.73) means that Dirichlet boundary conditions are included. [ ] [ DAAA , ]~

= (5.74a) [ ] [ DBBB , ]~ = (5.74b) Dhhh , ~

= (5.74c) Applying (5.69) and (5.70) to (5.73) and considering (5.74) yields to a nonlinear system of algebraic equations for each unknown time step.

[ ] ( )[ ]( )( ) ( )( ) ( )[ ] [ ]( ) ( ) −

∆+Θ−−=

∆+Θ −−

−1i1i

li

1li h

t11hh

t1 BABA

[ ] ( )[ ]( )( ) ( ) ( )[ ] [ ]( ) ( 1iD1iDDiD1l

iDD ht

11hht

1−−

−

∆−Θ−−

∆+Θ BABA ) . (5.75)

60


The stiffness matrix [ remains constant for the whole simulation. The mass matrix ]A [ ]B on the left hand side depends on the current solution vector h . A closer look at the right hand side of (5.75) shows that it can be split up into a part which varies with the nonlinear iterations l and into one which remains constant:

[ ] [ ]( )( ) ( )( ) ( ) 21l

1li

1li rrh

t1

+=

∆+Θ −−BA . (5.76)

5.4.2 Time Stepping Method for the A – Formulation

The integration of (5.63) and consideration of (5.64) leads to a non homogeneous system of ordinary partial differential equations.

( )[ ] [ ] ( ) diff1diff2diff2diff2 abt

at

aa ~~~~~~~∂∂

=∂∂

+ BA . (5.77)

The tilde above the quantities in (5.77) means that Dirichlet boundary conditions are considered: [ ] [ DAAA , ]~

= , (5.78a) [ ] [ DBBB , ]~ = , (5.78b) diffDdiffdiff aaa 111 ,~ = , (5.78c) diffDdiffdiff aaa 222 ,~ = . (5.78d) Applying (5.69) and (5.70) to (5.77) and considering (5.78) yields a nonlinear system of algebraic equations for each unknown time step:

( )[ ]( )( ) [ ] ( )( ) ( )[ ]( ) [ ] ( ) −

∆+Θ−−=

∆+Θ

−−−

1idiff21il

idiff21l

idiff2 at

11at

1a BABA

( )[ ]( )( ) [ ] ( ) ( )[ ]( ) [ ] ( ) +

∆−Θ−−

∆+Θ

−−−

1idiffD2D1iDidiffD2D1l

idiff2D at

11at

1a BABA

( )( ) ( )( ) ( )( ) ( )( )(1idiffD1idiffD11idiff1idiff1 abababab

t1

−−−+−

∆). (5.79)

It is easy to see that, the mass matrix [ ]B remains constant throughout the whole simulation. The stiffness matrix on the left side depends on the current solution vector [ ]A diff2a . The right hand side of (5.79) can be split up into a part which varies with the nonlinear iterations l and into one which stays constant:

( )[ ]( )( ) [ ] ( )( ) ( ) 21l

1l

idiff21l

idiff2 rrat

1a +=

∆+Θ −− BA . (5.80)

61


5.4.3 Reverse Mode of the Preisach Model for the A – Formulation

At the - formulation the Preisach model in Fig. 5.14 can be treated in the forward mode to update the material properties. Entering the model for one nonlinear iteration step ( and for one integration point with the available magnetic field intensity

H)l

( )lh , the corresponding value of the magnetic flux density b is directly obtained in one step taking account of the history. ( )l

( )lh ( )lbPM

Fig. 5.14. Use of the Preisach model, forward and reverse mode.

Contrary to that, at the - formulation the output A ( ) of Preisach model is given and the corresponding input

lb( ) has to be sought iteratively. The search algorithm is stopped once the

deviation of the approximation b from the true value

lh( )lr

( )lb is small enough. It has been found out that, is a suitable value for the criterion 6

RM 10−=ε

( ) ( )RM

llr bb ε≤− . (5.81)

The subscript r in b means the iteration index. This algorithm is called reverse mode. It can be demonstrated that any

( )lr

BH - curve derived from the classical Preisach model is strictly monotonic. Therefore, sectioning of an interval and regula falsi lead to a stable and unique solution. 5.4.3.1 Reverse Mode with Sectioning an Interval

The algorithm starts with the input value 0=0h which bisects the interval ( )maxmax h,h +− sketched in Fig. 5.15 and leads to the output b . Let us assume that b is less than the true value

0 0( )lb . In this case h is chosen which bisects the interval ( for the

second trial. Now, b is obtained which is greater than maxh.501 = )maxh+0,

1( )lb . Therefore, the interval

is bisected which leads to b , and so on till the stopping criterion is fulfilled. Obviously, this algorithm is very slow because a new step width is always the half of the previous one only.

( )maxh., 500

2

62


( )lh1h

2h

0h

maxh−

maxh+

0

maxh.501b

2b0b

( )lb

maxb+

0

maxb−

Fig. 5.15. Reverse mode with sectioning an interval.

5.4.3.2 Reverse Mode with regula falsi

The principle of regula falsi is straightforward therefore, Fig. 5.16 serves as an explanation only. This method is well suited for this task because it is much faster than sectioning an interval. However, the regula falsi method cannot compete with just taking a point from a curve which is the case for the method proposed with BH - curves.

maxB

maxB−

maxH

maxH−H

B

( )lb

( )lh2h 1h

2b

1b

Fig. 5.16. Reverse mode with regula falsi. 5.6 Numerical Simulations

The one-dimensional models used for a ferromagnetic sheet are shown in Figs. 5.10 and 5.11. Appropriate boundary conditions are prescribed on the surface for both formulations. A typical soft magnetic material with a coercive field of mA80H c /= and a remanent

63


magnetization of is utilized [37]. The maximal magnetic field intensity in the Preisach plane is h , the limiting triangle is subdivided uniformly with a mesh width of

2r mVs650M /.=

A01000.max =mA010h /.

m/=∆

Hz050.=

. Linear interpolation is assumed within the mesh. Investigations have shown that the formulation with the magnetic vector potential requires a finer mesh than that with the magnetic field intensity . The frequency of the sinusoidal excitation is . The conductivity of the material is . All simulations are based on homogenous initial conditions. The simulations are two periods long, each period is subdivided into 200 time steps. The thickness of the sheet is modeled by 4 finite elements of second order. The

A

A /H

f Vm10250 7. ⋅=σ

BH - curves are approximated by 29 sampling points and linear interpolation.

H

1=

Bp =

yB y

H

The abbreviation “A” means that the magnetic vector potential is exploited, “H” that the magnetic field intensity is applied, “WOE” stands for “without eddy currents”, “WE” means “with eddy currents”, “Hyst.” that the material with hysteresis and “I. C.” that the material with the related initial curve is analyzed. The index “

A

p ” stands for “peak value” and “ ” means the “root mean square value”. Both values are the arithmetic mean across the thickness of the sheet at a time instant.

rmsd

The thickness of the sheet for the first simulations is mm0d . . To carry out a meaningful comparison of both formulations, boundary conditions are prescribed so that the peak value of the magnetic flux density is about T81. . Only one period is represented in the figures because the steady state is approximately achieved after one half of the period.

First, the influence of hysteresis and eddy currents on the shape of the time functions magnetic flux density and magnetic field intensity in the middle of the lamination at

is investigated. The results are shown in Fig. 5.16 and 5.17. Due to the saturation, the magnetic flux density with the - formulation is flattened and due to the hysteresis a characteristic delay is observed. In case of eddy currents, it can be easily seen that all curves of the magnetic flux density demonstrate an appreciably larger delay.

Hmm50x .=

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,0Time (ms)

Mag

netic

flux

den

sity

By

(T) A, WOE

A, WEH, WOEH, WE

Fig. 5.17. Magnetic flux density with respect to the time with and without yB

eddy currents in the middle of the sheet mm50x .= . The magnetic field intensity with the - formulation has typical maxima. All curves yH A

64


with eddy currents show a delay compared to the field quantity on the surface.

-1000,0-800,0-600,0-400,0-200,0

0,0200,0400,0600,0800,0

1000,0

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,0Time (ms)

Mag

netic

fiel

d in

tens

ity H

y (A

/m)

A, WOEA, WEH, WOEH, WE

Fig. 5.18. Magnetic field intensity with respect to the time with and without yHeddy currents in the middle of the sheet mm50x .= .

Further characteristic results are summarized in Table I.

TABLE I COMPARISON OF THE FORMULATIONS WITH AND WITHOUT

EDDY CURRENTS AT A PEAK VALUE OF THE MAGNETIC FLUX DENSITY = 1.8T. pB

Formulation Eddy Currents

hystP

(W/kg) eddyP

(W/kg) rmsB

(T) pH

(A/m) rmsH

(A/m) Number of Iterations

Computation Time (s)

With 4.152 7.426 1.475 783.8 526.1 656 2.0 H

Without 4.157 0.0 1.533 785.0 555.1 600 1.8

With 4.260 4.367 1.273 811.1 433.3 604 2.2 A

Without 4.261 0.0 1.274 811.5 432.5 625 2.4

Next, the influence of the formulation with the corresponding boundary conditions on the behavior of the time functions is studied. The time functions of the transient analysis on the surface, , and at a distance of mm00x .= mm250x .= from the surface are shown in Figs. 5.19 to 5.21. All simulations consider eddy currents. Although the time function of the prescribed magnetic flux Φ of the - formulation is sinusoidal, the magnetic flux density

deviates from the sinusoidal progress, see Fig. 5.19. It is remarkable that leads close to the surface with respect to Φ . The behavior of of the H - formulation is typically flattened due to the saturation effect of the hysteresis non-linearity. Similar observations can be made for the magnetic field intensity in Fig. 5.20. The current density of the - formulation is almost sinusoidal whereas the current density of the H - formulation shows pronounced maxima, see Fig. 5.21.

AyB yB

zJ

yB

yH A

65


-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,0Time (ms)

Mag

netic

flux

den

sity

By

(T) A, Hyst., x=0.0mm

A, Hyst., x=0.25mmH, Hyst., x=0.0mmH, Hyst., x=0.25mm

Fig. 5.19. Magnetic flux density with respect to the time yBwith hysteresis.

-900-750-600-450-300-150

0150300450600750900

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,Time (ms)

Mag

netic

fiel

d in

tens

ity H

y (A

/m)

0

A, Hyst., x=0.0mmA, Hyst., x=0.25mmH, Hyst., x=0.0mmH, Hyst., x=0.25mm

Fig. 5.20. Magnetic field intensity with respect to the time yH with hysteresis.

66


-1,5E+6

-1,0E+6

-5,0E+5

0,0E+0

5,0E+5

1,0E+6

1,5E+6

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,0Time (ms)

Cur

rent

den

sity

Jz

(A/m

2)

A, Hyst., x=0.0mmA, Hyst., x=0.25mmH, Hyst., x=0.0mmH, Hyst., x=0.25mm

Fig. 5.21. Current density with respect to the time zJwith hysteresis.

The initial curve of the hysteresis non-linearity and its approximation by 29 sampling points and linear interpolation used subsequently as ordinary non-linearity is shown in Fig. 5.22. The related penetration depth

ωµσδ 2= (5.82)

is shown in Fig. 5.23. In (5.82), ω stands for the angular frequency and µ for the total permeability.

0,00,20,40,60,81,01,21,41,61,82,0

0 200 400 600 800 1000Magnetic field intensity H in A/m

Mag

netic

flux

den

sity

B in

T

Initial curveApproximation

Fig. 5.22. Initial curve of the hysteresis non-linearity and the piecewise linear approximation.

67


0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0Magnetic flux density B in T

Pene

trat

ion

dept

h in

mm

Fig. 5.23. Penetration depth of the approximated initial curve. The coincidence of the magnetization curves shown in Figs. 5.24 and 5.25 obtained once with the hysteresis non-linearity and once with the initial curve for different formulations is excellent. Therefore, it can be concluded that a simulation of the time functions of a problem with hysteresis can be approximated by the related problem with an ordinary non-linearity, i.e., initial curve.

0,00,20,40,60,81,01,21,41,61,82,0

0,0 200,0 400,0 600,0 800,0 1000,0Magnetic field intensity Hp (A/m)

Mag

netic

flux

den

sity

Bp

(T)

A, Hyst.H, Hyst.A, I. C.H, I. C.

Fig. 5.24. Magnetization curves of the 1.0mmd = thick sheet.

68


0,00,20,40,60,81,01,21,41,61,82,0

0,0 100,0 200,0 300,0 400,0 500,0 600,0 700,0Magnetic field intensity Hrms (A/m)

Mag

netic

flux

den

sity

Bp

(T)

A, Hyst.H, Hyst.A, I. C.H, I. C.

Fig. 5.25. Magnetization curves of the d 1.0mm= thick sheet with respect to the root mean square value of the magnetic field intensity H .

In general, the approximation of the time functions with hysteresis by the time functions with the ordinary non-linearity of the associated initial curve is very satisfactory. This is substantiated by the simulations shown in Figs. 5.26 to 5.31 and in Table II. The eddy current losses in case of the - formulation show an appreciable difference. The associated hysteresis losses may be reconstructed in the simulation with the ordinary non-linearity with the aid of Figs. 5.32 or 5.33. The approximation by the initial curve is in case of the - formulation better than those in case of the - formulation.

H

AH

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,0Time (ms)

Mag

netic

flux

den

sity

By

(T) H, I. C., x=0.0mm

H, I. C., x=0.25mmH, Hyst., x=0.0mmH, Hyst., x=0.25mm

Fig. 5.26. Approximation of the magnetic flux density yB in case of the - formulation. H

69


-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,0Time (ms)

Mag

netic

flux

den

sity

By

(T) A, I. C., x=0.0mm

A, I. C., x=0.25mmA, Hyst., x=0.0mmA, Hyst., x=0.25mm

Fig. 5.27. Approximation of the magnetic flux density yB in case of the - formulation. A

-900-750-600-450-300-150

0150300450600750900

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,Time (ms)

Mag

netic

fiel

d in

tens

ity H

y (A

/m)

0

H, I. C., x=0.0mmH, I. C., x=0.25mmH, Hyst., x=0.0mmH, Hyst., x=0.25mm

Fig. 5.28. Approximation of the magnetic field intensity yH in case of the - formulation. H

70


-900-750-600-450-300-150

0150300450600750900

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,Time (ms)

Mag

netic

fiel

d in

tens

ity H

y (A

/m)

0

A, I. C., x=0.0mmA, I. C., x=0.25mmA, Hyst., x=0.0mmA, Hyst., x=0.25mm

Fig. 5.29. Approximation of the magnetic field intensity yH in case of the - formulation. A

-1,5E+6

-1,0E+6

-5,0E+5

0,0E+0

5,0E+5

1,0E+6

1,5E+6

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,0Time (ms)

Cur

rent

den

sity

Jz

(A/m

2)

H, I. C., x=0.0mmH, I. C., x=0.25mmH, Hyst., x=0.0mmH, Hyst., x=0.25mm

Fig. 5.30. Approximation of the current density zJ in case of the - formulation. H

71


-8,0E+5

-6,0E+5

-4,0E+5

-2,0E+5

0,0E+0

2,0E+5

4,0E+5

6,0E+5

8,0E+5

0,0 2,5 5,0 7,5 10,0 12,5 15,0 17,5 20,0Time (ms)

Cur

rent

den

sity

Jz

(A/m

2)

A, I. C., x=0.0mmA, I. C., x=0.25mmA, Hyst., x=0.0mmA, Hyst., x=0.25mm

Fig. 5.31. Approximation of the current density zJ in case of the - formulation. A

Some selected results at a peak value of the magnetic flux density of the second period are summarized in Table II. The computation in case of the initial curve is essentially faster than those with hysteresis.

T81Bp .=

TABLE II COMPARISON OF THE FORMULATIONS ONCE WITH THE HYSTERESIS NON-LINNEARITY

AND ONCE WITH THE INITIAL CURVE AT A PEAK VALUE OF THE MAGNETIC FLUX DENSITY = 1.8T. pB

Formulation Non-linearity

hystP

(W/kg) eddyP

(W/kg) rmsB

(T) pH

(A/m) rmsH



Hysteresis 4.152 7.426 1.475 783.8 526.1 656 1.8 H

Initial Curve 0.0 6.684 1.450 785.0 540.0 526 0.3

Hysteresis 4.260 4.367 1.273 811.1 433.3 604 2.4 A

Initial Curve 0.0 4.344 1.273 811.6 450.6 568 0.3

The total specific, the specific hysteresis and the specific eddy current losses with their approximation for both formulations are shown in Figs. 5.32 to 5.34.

72


0,0

2,0

4,0

6,0

8,0

10,0

12,0

14,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0Magnetic flux density Bp (T)

Tota

l los

ses

(W/k

g) A, Hyst.H, Hyst.

Fig. 5.32. The total specific losses with respect to the peak value of the magnetic flux density . pB

0,00,51,01,52,02,53,03,54,04,55,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6Magnetic flux density Brms (T)

Hys

tere

sis

loss

es (W

/kg)

A, Hyst.

H, Hyst.

Fig. 5.33. The specific hysteresis losses with respect to the root mean square value of the magnetic flux density . B

The specific eddy current losses in Fig. 5.34 are reproduced by the approximating technique accurately.

73


0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0


Eddy

cur

rent

loss

es (W

/kg) A, Hyst.

H, Hyst.A, I. C.H, I. C.

Fig. 5.34. The specific eddy current losses losses with respect to the peak value of the magnetic flux density . pB

Next, the time behavior of the specific losses is analyzed. The specific hysteresis losses decrease from the first to the second period shown in Fig. 5.35 for both boundary condition problems, whereas the specific eddy current losses, see Fig. 5.36, increase in the same time.

0,0

1,0

2,0

3,0

4,0

5,0

6,0


Hys

tere

sis

loss

es (W

/kg)

A, Hyst., 1. PeriodA, Hyst., 2. PeriodH, Hyst., 1. PeriodH, Hyst., 2. Period

Fig. 5.35. The specific hysteresis losses of the first two periods.

74


0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0


Eddy

cur

rent

loss

es (W

/kg) A, Hyst., 1. Period

A, Hyst., 2. PeriodH, Hyst., 1. PeriodH, Hyst., 2. Period

Fig. 5.36. The specific eddy current losses of the first two periods.

The eddy current losses in turn can be approximated by the initial curve for both formulations with sufficient accuracy, see Figs 5.37 to 5.38.

0,00,51,01,52,02,53,03,54,04,55,0


Eddy

cur

rent

loss

es (W

/kg) A, Hyst., 1. Period

A, Hyst., 2. PeriodA, I. C., 1. PeriodA, I. C., 2. Period

Fig. 5.37. Approximation of the specific eddy current losses of the first two periods in case of the - formulation. A

75


0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0


Eddy

cur

rent

loss

es (W

/kg) H, Hyst., 1. Period

H, Hyst., 2. PeriodH, I. C., 1. PeriodH, I. C., 2. Period

Fig. 5.38. Approximation of the specific eddy current losses of the first two periods in case of the H - formulation.

Subsequently, the specific losses of a 0.5mmd = thick sheet are compared with those of a thick one. The specific hysteresis losses are almost the same for both thicknesses

and the same formulation, see Fig. 5.39. They are appreciable higher for the - formulation compared to those of the - formulation with respect to the root mean square value of the magnetic flux density .

mm0.1d =A

HrmsB

0,00,51,01,52,02,53,03,54,04,55,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6Magnetic flux density Brms (T)

Hys

tere

sis

loss

es (W

/kg) A, 0.5mm

A, 1.0mmH, 0.5mmH, 1.0mm

Fig. 5.39. The specific hysteresis losses of a 0.5mmd = and a 1.0mmd = thick sheet of both formulations.

The specific eddy current losses increase with the thickness of the sheet drastically shown in Fig. 5.40. They are essentially higher for the H - formulation than for the - formulation. A

76


0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0


Eddy

cur

rent

loss

es (W

/kg) A, 0.5mm

A, 1.0mmH, 0.5mmH, 1.0mm

Fig. 5.40. The specific eddy current losses of a 0.5mmd = and a 1.0mmd = thick sheet of both formulations.

In Table III results obtained by the different techniques to treat the BH - curves are summarized. The most accurate solution is obtained by the direct technique with 29 sampling points. A comparison shows that the results of the direct technique with 10 sampling points agree with the accurate solution excellently, but the computation is essentially faster. This is true for both formulations. The simulations with the store technique are faster than those with the direct technique but more sampling points are required to achieve a similarly good accuracy.

TABLE III

COMPARISON OF THE TECHNIQUES TO TREAT THE BH – CURVES ONCE WITH 10 AND ONCE WITH 29 SAMPLING POINTS AT A PEAK VALUE OF THE MAGNETIC FLUX DENSITY = 1.6T. pB

Formulation BH – Curves Technique

Number of Sampling

Points hystP

(W/kg) eddyP

(W/kg) rmsB

(T) pH

(A/m) rmsH



10 2.909 4.944 1.264 532.49 350.99 666 1.1 Direct

29 2.909 4.944 1.264 532.49 350.99 666 2.2

10 2.890 2.923 1.303 528.79 363.06 518 0.4 H

Store 29 2.906 4.775 1.263 531.90 350.57 564 0.5

10 2.907 3.439 1.132 532.15 305.27 563 1.0 Direct

29 2.907 3.439 1.132 532.14 305.27 561 1.8

10 4.359 3.487 1.284 831.89 430.42 562 0.5 A

Store 29 2.918 3.421 1.134 532.65 305.71 597 0.5

77

6 Alternative Method in the Identification of the Preisach Model


An essential part in dealing with the Preisach Model (PM) is the identification of the Preisach Distribution Function (PDF). The Classical Preisach Model (CPM) exhibits the congruency and the wiping out (deletion) property. The correct way to identify the CPM is the measurement of the first order reversal curves [23]. Experience has shown, that almost no material exhibits the congruency property. Thus, the CPM may serve as an approximation only. In this case the question can be posed, which data should be exploited to identify the CPM to achieve the best possible approximation? The major BH-loop [38, 39, 40] (especially the saturation magnetization, the coercive field and remanent magnetization of this loop [41]), symmetric minor BH-loops with the virgin curve [42, 43], see Fig. 6.1, or arbitrary trajectories in the BH-plane (Fig. 6.2) have been used to identify a reasonable PM [44, 45].

H

B

Fig. 6.1. Major and symmetric minor BH-loops with virgin curve.

B

H

Fig. 6.2. Arbitrary trajectory in the BH-plane.

78


Technical characteristics in terms of the magnetization curve and the total specific losses of sheets frequently used in electrical devices are provided by the manufacturers [46]. Just this information is exploited in the method proposed to identify the PDF wanted. It is assumed that the data are representative for the material. To demonstrate the capability of the method one arbitrary material for electrical sheets has been selected and its PDF of the CPM has been determined.

One possible way to identify a PM are neural networks which has been used widely [40, 44, 45]. On the other hand, the PM can be represented by the PDF. In [40], the identification of the parameters of the PDF is also investigated. To facilitate the determination of the parameters of the PDF in this case, an optimization procedure can be exploited [38, 39]. Therefore, an objective function is required.

The goal is to identify optimal values of the parameters of the PDF. In general, in the context of optimization, design parameters, , are to be chosen to minimize the objectives . In our case, the problem is to approximate the curves describing the

dependence of the total specific losses on the peak magnetic flux density

x( )xif

totP B , that of the

peak magnetic flux density B on the peak magnetic field intensity H as well as that of the peak magnetic flux density B on the root mean square value of magnetic field intensity (see solid curves in Figs. 6.4, 6.5 and 6.6). 6.1 Preisach Distribution Functions

Three different types of PDFs typical for soft magnetic materials are investigated [47, 25]. The PDF1 in (6.1) defines a Gauss-Gauss distribution function [37], PDF2 in (6.2) is called Factorized-Lorentzian distribution function [48] and PDF3 in (6.3) represents a Factorized-Lorentzian distribution function plus a term multiplied by the delta function ( )βαδ − which is valid only along the hypotenuse of the limiting triangle [49].

( ) +

+−

−−

−

=2βαCexp

2βαCexp

!122βαC

CC,p 11

12

1

3211 βα

+

+−

−−

2βαCexp

2βαCexpCC 54654C

+

+−

−−

2βαCexp

2βαCexpCC 87987C

+−

−−

2βαCexp

2βαCexpCC 1110121110C (6.1)

( )( )[ ] ( )[ ]22

22

22 11

N,pββαα

βα−+−+

= (6.2)

79


( ) ( )

+

+

−+

++

−+

= f

e1

N

11

N,p2

322

3

3

2

3

3

313

αβαδ

βαβ

βαα

βα (6.3)

In (6.2) and (6.3) is a normalization constant. The quantities C , , µN µ µN µα , µβ , and in the relations (6.1) to (6.3) are treated as degrees of freedom, i.e., they serve as design

parameters in the optimization procedure.

ef

x 6.2 Simulation Model

The Epstein frame is frequently used to extract the data needed in this context. In one lamination of the iron core of the measurement system the field quantities vary approximately only across the sheet. Therefore, a one-dimensional finite element model (see Fig. 6.3) suffices to simulate the field distribution in a lamination.

zeA νA= zeA νA−=

z

x0

d

⊗Φ

σ

Fig. 6.3. One dimensional model of one lamination with

the appropriate boundary conditions. Since the voltage in the exciting coil is prescribed which entails a sinusoidal magnetic flux, , the magnetic vector potential, , is used as solution variable because the appropriate

boundary conditions can be prescribed easily. The thickness of the sheet is , its conductivity is and the frequency of the excitation is .

Φ A0.5mm=dHz.050m/S*. 710250=σ f =

The magnetic flux density B in the cross section of the sheet can be calculated approximately as

dA2

SB νν =

Φ≈ . (6.4)

80


In (6.4) Φ is the magnetic flux, the sampling point of the boundary values of the magnetic vector potential, S and are the cross section and the thickness of the sheet. The boundary condition are successively increased starting from very small field values up to the saturation of the given material whose PDF is wanted.

νAd

thν

Configurations of the design parameter leading to smaller maximal value of the magnetic flux density of the associated PM than 1.4T are rejected. Otherwise, the sampling values of the boundary condition are chosen uniformly distributed between 0.2T and .

xmaxB

maxB 6.3 Definition of the Objective Function

All the objectives ( )xif

best

to be required in the identification problem are best fit objectives because certain values v should reach reference quantities )( x ( )xrefv as close as possible.

Since three objectives have to be treated simultaneously, a multi objective optimization has to be solved. Among various methods to set up a scalar objective function the weighting of objective techniques is very common and indisputably the simplest one. A suitable scalar function is defined by transforming the multi objective optimization problem to a scalar objective optimization problem by weighting the different objectives

in such a way that the global minimum of

( )xif

( )xf

( )xif ( )xf occurs at the optimal design parameter configuration.

( ) ( ) ( ) ...** ++= xxx 2211 fwfwf (6.5)

The task of the weighting terms in (6.5) is to normalize the terms , i.e., to bring them in about the same size of order as well as to put different emphasis on the objectives

.

iw ( )xii fw *

( )xifTo restrict the search to a feasible region of the parameter space, lower and upper bound of

the individual parameters, i.e., box constraints, are adjusted simply by testing. In the present example to consider all sampling points ν of the boundary conditions (see

also 6.2 Simulation Model) the objective function ( )xf can be set up in the least square sense:

( ) ( )( ) ( )

+

−+

−

++= ∑∑

==

n

12

spec

2

speccalc2

n

12

spectot

2spectotcalctot

1321 B

BBw

PPP

wwwwn

1fν ν

νν

ν ν

νν

,

,,

,

,,

ˆˆˆ

**x

( )

−∑=

n

12

specrms

2specrmscalcrms

3 B

BBw

ν ν

νν

,

,,* (6.6)

In (6.6) “calc ” stands for “calculated”, “ ” for specified and the index “ rms ” means

that the magnetic flux density belongs to the magnetization curve where the magnetic field intensity is given in terms of root mean square values.

spec

The three specified curves are given by sampling points and within them a linear approximation is used.

81


The number of sampling values of the boundary conditions is 17n = . To determine ,

is used in the specified curve. spectotP ,ν

calcB ,ˆν

To make the value of the objective function ( )xf independent of the weighting terms and of the number of sampling values of the boundary conditions, the objective function

in (6.6) is additionally normalized with respect to the sum of the weighting terms and to the number of sampling points n .

iw

iwn

( )xf

6.4 Choice of Optimization Strategy

Since no prediction about the properties of the objective function can be made, it seems to be advisable to use a stochastic strategy instead of a deterministic one [39]. Contrary to the deterministic strategies, stochastic strategies end up for the same optimization problem in more or less different final parameter configuration for different computational runs.

Stochastic strategies are zero order methods and hence get along with the function value of the objective function only, one does not have to worry about the continuity of the objective function. Box constraints, for instance, can be treated easily by replacing not feasible solutions by newly computed feasible ones.

Examples for stochastic strategies are the evolution strategies and the genetic algorithms imitating the evolutionary behavior of nature as well as the simulating annealing algorithm which takes advantage of the analogy between the cooling process of liquids and the localization of the global minimum.

Stochastic strategies start with one or more initial parameter configurations, search their path through the parameter space randomly and end up in one final parameter configuration, which is expected to be located close to the global minimum. Another important feature which discriminates them from deterministic strategies is that they accept deterioration in the objective function during the minimum seeking process. This fact enables stochastic strategies to escape local minima. In contrast to that, deterministic strategies often get captured in local minima which is a major drawback.

Strategy parameters have an important influence on the convergence behavior. Their optimal values depend on the specific problems. Fortunately, the strategy parameters can be set for a greater class of problems.

Stochastic strategies are rather simple to implement, stable in convergence, relatively robust in their application to completely different problems and provide the user with reliable results. However, stochastic strategies suffer from their extremely high number of objective function evaluations needed.

Simulated annealing (SA) exhibits all advantages of stochastic strategies and has proven its reliability in numerous different optimization problems in electrical engineering. Therefore, SA has been chosen for the identification of a suitable PDF. 6.4.1 Simulating Annealing Algorithm

The simulated Annealing Algorithm (SAA) imitates the cooling process of a molten solid. The cooling process has to be carried out slowly to avoid that imperfections (displacements, inclusions, etc.) corresponding to a higher energy state than the minimal energy state (ground state) are confined in the structure of the crystalline lattice. The Boltzmann probability

82


TkEE

BB

oldnew

ep−

−

= (6.7) determines the probability that a new configuration with a higher energy state than that of the previous one at the temperature

newE

oldE T exists. In law (6.7) k is the Boltzmann constant.

B

The following analogy between the physical system and SAA can be drawn. The energy E represents the objective function to be minimized and the configurations of the physical system correspond to the design parameters .

( )xfx

6.4.1.1 Metropolis Algorithm

The Metropolis Algorithm which is the heart of SAA decides whether a new configuration in the parameter space x is accepted or rejected. If the new value of the objective function

is less than the previous one , is accepted anyway. The Metropolis Algorithm facilitates the acceptance of a deterioration in the objective function during the optimization process. This enables SAA to escape local minima of the objective function

in the parameter space x . In case leads to a higher value of the objective function than , a Boltzmann like probability

new

newf

(xfnewf

oldf newx

newx

( )xf

)oldf

cTff oldnew

ep−

−= (6.8)

is calculated and compared with the value determined randomly from the intervall [0,1] of uniformly distributed numbers. In (6.8) c is a normalization constant, normally set equal to

, and

equalp

01. T is a quantity which is equivalent to temperature of the physical analogue. If holds, the deterioration is accepted otherwise rejected. From (6.8) it can be easily

seen that the probability to accept a deterioration decreases with decreasing p<pequal

T . The algorithm applied possesses an adaptation of the stepsize in such a way that the

probability to accept new configurations in the parameter space is close to fifty percent. x 6.4.1.2 Cooling Schedule

The starting value of the temperature like quantity T is determined by a small number of initial configurations of the design parameters . The quantity x T is decreased in the most outer loop of the algorithm as

Toldnew f*TT = . (6.9)

The reduction factor in (6.9) is chosen from the interval Tf [ ]0.8,0.9 .

83


6.5 Simulation Results

To achieve a stable transient simulation, one period subdivided in 100 time steps was found to be sufficient. The solution is extracted from the second half of the period. Two finite elements are used to model the 0.5mm thick lamination.

The limiting triangle in the Preisach plane is subdivided in 300 sections for the maximal value of the magnetic field intensity h m/Amax 3000= for PDF1 and PDF2. For PDF3 1000 subdivisions are required for h to achieve a reasonable solution. This might be due to the structure of the PDF3, which possesses an additional term valid only along the hypotenuse of the Preisach triangle, see (6.3).

m/A2000max =

The strategy parameters used are shown in Table I.

TABLE I STRATEGY PARAMETER

Strategy Parameter

Number of initial Configurations

iniN

Number of temperature steps

TN

Number of loops at constant temperature

LTN

Number of loops at constant stepsize

LSN

Temperature reduction factor

Tf

Value 100 35 10 10 0.8

The optimal parameter configuration and the related optimal value of the objective function of the three PDFs are shown in Table II. The weighting terms in (6.6) are equal to 1.0.

optx

optf iw

TABLE II

COMPARISON OF DIFFERENT OPTIMAL PREISACH DISTRIBUTION FUNCTIONS PDFS Preisach

Distribution Function

Design Parameter

optx Objective Function

optfNumber of

Function Calls

PDF1

0.45065=1C , , 0.325596=3C

3-5.910898E=4C , , 2-1.244421E=5C

0.690397=6C , , 2-2.171817E=7C

3-.42532E5C8 = , , 0.268686=9C

0.5159=10C , C , 4-1.293646E=11

0.216506=12C

2-2.561316E

38600

PDF2

0.162974=2N , , 0.529548=2α

96.34014=2β

2-1.471151E

10600

PDF3

41.921847E −=31N

44.539454E −=32N

,

,

79.37867=3α , , 38.5282=3β

120.1671=e , 5-1.167147E=f

3-1.10334E

21100

84


The optimal identified solutions are compared with the objectives in Figs. 4 to 6. In general, the results obtained by the Preisach model with PDF3 are considerably better than those by PDF1 and PDF2. Although the magnetization curves obtained by PDF1 and PDF2 deviate from the objectives, nevertheless, the total losses are reproduced very satisfactorily.

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0Peak magnetic flux density (T)

ObjectivePDF1PDF2PDF3

Fig. 6.4. Specific total losses of the soft magnetic material.

0,0

0,5

1,0

1,5

2,0

10,0 100,0 1000,0 10000,0Peak magnetic field intensity (A/m)


Fig. 6.5. Magnetization curve of the soft magnetic material in peak values.

85


0,0

0,5

1,0

1,5

2,0

10,0 100,0 1000,0 10000,0RMS magnetic field intensity (A/m)


Fig. 6.6. Magnetization curve of the soft magnetic material (peak value of the magnetic flux density versus root mean square value of the magnetic field intensity ). B rmsH

The specific total losses corresponding to the identified optimal Preisach models are separated in hysteresis and eddy current losses in Fig. 6.7. The hysteresis losses are about three times higher than the eddy current losses .

totP

hystP eddyP

hystP eddyP

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0Peak magnetic flux density (T)

Objective (tot)PDF1 (hyst)PDF2 (hyst)PDF3 (hyst)PDF1 (eddy)PDF2 (eddy)PDF3 (eddy)

Fig. 6.7. Specific total losses separated in hysteresis and eddy current losses totP hystP eddyP

versus peak value of the magnetic flux density . B

86

7 Preisach Model versus Complex Permeability


Large electrical devices with laminated cores made of ferromagnetic materials are of great practical importance. Therefore, an efficient approximate technique to simulate the eddy current distribution and the associated losses, i.e. eddy current and hysteresis losses, with the aid of the finite element method is indisputably needed for practical applications.

A very common and wide spread way to model the hysteresis is the Preisach model described in chapter 4. It is well known in this context that the PM suffers from its extremely high demand on memory as well as from its expensive handling, especially in the reversed mode [50], see also section 5.4.3.

A transient analysis in terms of a time stepping procedure of a problem with time harmonic excitation does not represent a suitable solution either, because of its time consuming behavior.

One possible way to solve the problem above represents the harmonic balance finite element method (HBFEM) introduced by Yamada and Bessho [51] in the late 80’s in which all quantities are expanded in truncated Fourier series. A major drawback is that it is not clear a priori how many coefficients have to be considered. The system to be solved increases with the number of coefficients of the truncated harmonic series [52]. This may be the reason, why this method is not frequently used.

An attractive way to deal with ferromagnetic materials with significant hysteresis and pronounced saturation in case of time harmonic excitation of an eddy current problem is to introduce an effective material representing the ferromagnetic material. This fictitious material describes the relationship of the field quantities approximately. Numerous models treating nonlinear materials by effective materials which are constant throughout the time have been developed and applied succesfully, see [53] to [56] for instance. However, the generalization of the models to materials with hysteresis are rather scarce, see [28], [57] to [59], [60] and [61].

Therefore, two different ways to derive a nonlinear complex permeability from classical Preisach models are shown. 7.1 Material Properties

The CPM [23] describes the hysteresis nonlinearity by a set of elementary rectangular loops with different up and down switching levels. The state of the loops can be only +1 or –1 and they are weighted by the Preisach distribution function which determines a certain material uniquely. These distribution functions for three different soft magnetic materials will be assumed to be known, see [37], [48] and [49]. The Preisach distribution function represents the starting point of the derivation of the complex magnetization curves. The magnetic properties of the materials are represented by their BH-loops and their initial curves in Fig. 7.1. The three curves will be referred to as Mat.1, Mat.2 and Mat.3. Dynamic effects are neglected.

87


-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

-1250 -1000 -750 -500 -250 0 250 500 750 1000 1250Magnetic field intensity (A/m)

Mag

netic

flux

den

sity

(T)

Mat.1Mat.2Mat.3

Fig. 7.1. BH-loops of the three soft magnetic materials with initial curves. 7.2 Constitutive Law

The complex formalism can be exploited favorably for problems with linear material properties and time harmonic excitation in the steady state, because they have to be solved only once to obtain the exact solution. To preserve the advantage of the complex formalism for nonlinear problems too, equivalent material characteristics have to be introduced. In this equivalent material the field quantities are assumed to be sinusoidal and obey criterions, described below. A constitutive law

HB µ= (7.1) is introduced. In the relation (7.1) B represents the magnetic flux density and H the magnetic field intensity, respectively. They are complex numbers. Without hysteresis, a real number suffices for the magnetic permeability µ to represent the original material [53]. Due to the phase shift in materials with hysteresis between the magnetic flux density and the magnetic field intensity in the time domain and due to the saturation effect, the magnetic permeability

( )tB( )tH

µ has to be complex and a nonlinear function of the field quantities to be able to approximate the relationship between ( )tB and ( )tH . The solution by this technique is of course not exact. It is not feasible for obtaining the time functions of the field quantities in strongly saturated domains. However, it is capable to provide the designer of electrical devices with valuable quantities like specific losses or even field quantities in certain cases [61] at a very small expense.

Since the nonlinear complex magnetization curve has to be calculated only once, a very fine discretisation of the limiting triangle in the Preisach plane can be carried out.

88


7.3 Nonlinear Complex Permeability

The task is to devise a criterion to derive an effective material in order to obtain solutions as accurate as possible by the approximate technique.

Subsequently, two possibilities to create an effective material are discussed. To create a complex permeability, only the magnetic field density B is assumed to be sinusoidal and given, because this can be expected in most practical problems. Prescribing the time behavior of B implies that the PM has to be employed in the reversed mode. Fundamental harmonics (FH) The first method is based on a very simple idea to derive a complex permeability µ by considering only the fundamental harmonics of the magnetic field quantities. Equivalent hysteresis losses (EHL) In the second approach, a complex permeability µ is derived to yield the same hysteresis losses as the real material. The hysteresis losses are proportional to the enclosed area of the BH-loop. The area of the approximating ellipse can be calculated as

hystP hystP

µϕµπ sinHP 2hyst −~ . (7.2)

Here, H and µ

µ

mean the moduli of the magnetic field intensity and of the magnetic permeability, ϕ is the phase angle between B and H . The second condition used is

max

max

HB=µ (7.3)

which ensures that the maximal value of H is retained.

From (7.2) and (7.3) it can be easily seen that the losses determine the phase angle hystP µϕ

whereas the saturation is responsible for the modulus µ . Figs. 7.2 and 7.3 show the complex permeability derived for the three materials.

89


0,0E+00

1,0E+03

2,0E+03

3,0E+03

4,0E+03

5,0E+03

6,0E+03

7,0E+03

8,0E+03

9,0E+03

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0Magnetic flux density Bmax (T)

Mat.1, FHMat.2, FHMat.3, FHMat.1, EHLMat.2, EHLMat.3, EHL

Fig. 7.2. Modulus of the relative permeability of the three soft magnetic materials.

-40,0

-35,0

-30,0

-25,0

-20,0

-15,0

-10,0

-5,0

0,00,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

Magnetic flux density Bmax (T)

Mat.1, FHMat.2, FHMat.3, FHMat.1, EHLMat.2, EHLMat.3, EHL

Fig. 7.3. Phase angle of the relative permeability of the three soft magnetic materials. 7.4 Approximated Hysteresis Loops

The constitutive law (7.1) describes in the time domain an ellipse approximating the original hysteresis loop. Figs. 7.4 and 7.5 represent the ellipses obtained by the approximating techniques FH and EHL together with the corresponding original hysteresis loop of the two materials Mat.2 and Mat.3.

90


-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

-1000 -750 -500 -250 0 250 500 750 1000Magnetic field intensity (A/m)

Mat.2FHEHL

Fig. 7.4. Approximating ellipses and the original hysteresis loop of the material Mat.2, . T81.Bmax =

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

-1500 -1000 -500 0 500 1000 1500Magnetic field intensity (A/m)

Mat.3FHEHL

Fig. 7.5. Approximating ellipses and the original hysteresis loop of the material Mat.3, . T41.Bmax =

7.5 Numerical examples

To validate the accuracy and efficiency of the approximating technique, the eddy current and the hysteresis losses in thin ferromagnetic sheets (0.5mm and 1.0mm thick) have been computed by a one-dimensional finite element model for the three materials. The frequency of the excitation was Hzf 50= . In the transient case, the magnetic vector potential is applied. The transient simulations are two periods long, each period is subdivided in 400 time steps. For the complex analysis simply the magnetic field intensity

A

H is used as the solution variable. Appropriate boundary conditions for the two formulations are prescribed on the surface.

For the 0.5mm thick sheet only two finite elements (FE) have been required in both cases. However, the number of required FEs had to be increased drastically for the 1.0mm thick sheet in the transient case, especially for material Mat.3. This might be due to the structure of the PDF of Mat.3, which possesses an additional term valid only along the hypotenuse of the

91


Preisach triangle [14]. To achieve a stable transient simulation, 20 FEs were found to be sufficient. In the complex analysis 4 FEs are used.

The time functions of the transient and the approximate field quantities at a distance of from the surface are shown in Figs. 7.6 to 7.8 for the material Mat.1. The

thickness of the sheet is 1.0mm. mm.x 250=

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

0 5 10 15 20 25 30 35 40

Time (ms)

TransientFHEHL

Fig. 7.6. Magnetic flux density of the material Mat.1 at mm.x 250= .

-800,0

-600,0

-400,0

-200,0

0,0

200,0

400,0

600,0

800,0

0 5 10 15 20 25 30 35 40

Time (ms)

TransientFHEHL

Fig. 7.7. Magnetic field intensity of the material Mat.1 at mm.x 250= .

92


-6,0E+05

-4,0E+05

-2,0E+05

0,0E+00

2,0E+05

4,0E+05

6,0E+05

0 5 10 15 20 25 30 35 40

Time (ms)

TransientFHEHL

Fig. 7.8. Current density of the material Mat.1 at mm.x 250= .

In Figs. 7.9 to 7.14 the specific eddy current and hysteresis losses are shown. represent the average maximum value of the magnetic flux density across the sheet. In general, the losses are reproduced by the approximating technique very satisfactorily. The eddy current losses increase extremely in the 1.0mm thick sheet compared to that in the 0.5mm thick sheet.

maxB

The complex magnetization curves are approximated by 29 sampling points and linear interpolation. The maximal field values of the complex magnetization curves are restricted by the limiting triangle in the Preisach plane. The corresponding upper bound in case of the materials Mat.1 and Mat.2 is T81.Bmax = and of Mat.3 T41.Bmax = . Therefore, a linear expansion of the curves is used. This is the reason for the deviation in Fig. 7.13.

The approximation by EHL is slightly better than that by FH.

0,00,51,01,52,02,53,03,54,04,55,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0


Transient, Mat.1, 0.5mmFHEHLTransient, Mat.1, 1.0mmFHEHL

Fig. 7.9. Hysteresis losses of the material Mat.1.

93


0,00,51,01,52,02,53,03,54,04,55,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0



Fig. 7.10. Eddy current losses of the material Mat.1.

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0




0,0

1,0

2,0

3,0

4,0

5,0

6,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0




94


0,0

0,5

1,0

1,5

2,0

2,5

3,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6




0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6




95

Conclusions

Conclusions

The overall field distribution to estimate the eddy current losses in laminated core has been studied in chapter 1. The eddy current losses obtained from the anisotropic analysis are essentially lower than that of the exact model because the anisotropic model does not take account of the local field in the laminates. However, the overall field distribution represents a good basis for approximate techniques to consider the lamination too.

The method introduced in chapter 2 can obviously be used for a rude approximation with sufficient accuracy to deal with three dimensional, linear, time harmonic eddy currents in laminations. The estimated eddy current losses obtained by this method are in good agreement with the true losses and represent also a substantial improvement compared with those computed by the corresponding anisotropic model. Investigations show that less accuracy in the field distribution is achieved with large relative permeability (µr ≥ 5 000) and near the interfaces between conducting and non-conducting region. Since it considers only one component of the field quantities, the correction of the overall field distribution taking account of the laminations neglects edge effects. The additional condition required in one sheet that the average of the tangential component of the current density in the cross section in the anisotropic model equals that in the approximate technique impedes the general applicability of the method.

To overcome the drawbacks of the model introduced in chapter 2 an improved method has been presented in chapter 3 to compute the eddy current losses in non-saturated laminated iron subjected to a sinusoidal magnetic field. The truly three-dimensional distribution of the field is taken into account by a 3D finite element eddy current analysis with an anisotropic conductivity of zero or very low value assumed in the cross direction perpendicular to the laminates. This field and eddy current distribution is corrected within each laminate using the fundamental solution of the diffusion equation there and determining the coefficients from the condition that the averages of the tangential field components across the sheets are the same as those obtained from the 3D analysis. The eddy current losses obtained agree well with those provided by the analysis of a 3D model wherein each laminate is modeled individually. The computational effort of this method is, however, substantially lower, since the discretization level in the finite element model using an anisotropic conductivity is comparable to that in a non-laminated bulk conductor and the correction yielded by the Fourier analysis needs a negligible amount of arithmetics. The effect of some parameters on the accuracy has been investigated on the simple model of a conducting cube. Results presented for a large magnetic circuit demonstrate the capability of the method in improving the accuracy of the loss computation.

One way to incorporate the classical Preisach model in the finite element method by means of the material properties has also been presented. A novel, efficient and resource saving method to treat the hysteresis non-linearity to achieve stable simulations by means of BH–curves has been introduced. Different formulations have been analyzed and the approximation of the hysteresis non-linearity by the associated initial curve has been investigated. Various aspects in the context of hysteresis and a thin sheet has been studied.

An alternative method to identify the classical Preisach model, i.e. the Preisach distribution function, of a certain material has also been developed. Unlike already existing various identification procedures, the method presented exploits the specific losses and magnetization curves with the aid of a stochastic optimization strategy (simulated annealing). The accuracy of the approximating technique is very satisfactory. A major drawback of the method introduced is the extremely high computational costs due to the stochastic optimization strategy and due to the simulation model used. For each configuration of the design

96

Conclusions

parameters, the Preisach model has to be calculated and a relatively large number of sampling values of the boundary condition have to be considered to achieve a good fit of the model to the material investigated. An irregularly subdivided Preisach plane may yield substantially shorter computation times without loosing accuracy. Since the suspected global optimum in the parameter space is already localized after about one third of the steps by the stochastic strategy, it is advisable to continue the optimization process from here with a deterministic strategy because of their faster convergence in the vicinity of the solution.

A comparison of the Preisach model with the related effective magnetization curve in the computation of the eddy current and hysteresis losses in thin ferromagnetic sheets of three different materials has shown that the accuracy of the approximating technique is very satisfactory and the time and memory savings are extremely high. A single time step in the transient analysis needs about the same number of nonlinear iterations as the corresponding complex computation. The complex simulation is about 9500 times faster then the transient one for the simple example treated. Only a complex number instead of the PM has to be stored for an integration point of the finite element method.

97

References

References [1] Nikola Rajakovic, Adam Semlyen, "Harmonic domain analysis of field variables related to eddy current

and hysteresis losses in saturated laminations", IEEE Transactions on Power Delivery, vol. 4, pp. 1111-1116, April 1989

[2] Francisco de Leon, Adam Semlyen, "Time domain modelling of eddy current effects for transformer transients", IEEE Transactions on Power Delivery, vol. 8, pp. 271-280, Jannuary 1993

[3] E. J. Tarasiewicz, A. S. Morched, A. Narang, E. P. Dick, "Frequency dependent eddy current models for nonlinear iron cores", IEEE Transactions on Power Systems, vol. 8, pp. 588-597, May 1993

[4] J. G. Zhu, S. Y. R. Hui, V. S. Ramsden, "Discrete modelling of magnetic cores including hysteresis eddy current and anomalous losses", IEE Proceedings-A, vol. 140, pp. 317-322, July 1993

[5] Jian. G. Zhu, S. Y. R. Hui, V. S. Ramsden, "A Generalized Dynamic Circuit Model of Magnetic Cores for Low- and High-Frequency Applications – Part I: Theoretical Calculation of the Equivalent Core Losses Resistance", IEEE Transactions on Power Electronics, vol. 11, pp. 246-250,March 1993

[6] L-L. Rouve, F. Ossart, T. Waeckerlé, A. Kedous-Lebouc, "Magnetic Flux and Losses Computation in Electrical Laminations", IEEE Transactions on Magnetics, vol. 32, pp. 4219-4221, September 1996

[7] P. Holmberg, A. Bergqvist, G. Engdahl, "Modelling eddy currents and hysteresis in a transformer laminate", IEEE Transactions on Magnetics, vol. 33, pp. 1306-1309, March 1997

[8] L. Luk, G. E. Dawson, A. R. Eastham, "Finite element and difference analysis of losses in laminated material", IEEE Transactions on Magnetics, vol. 24, pp. 2679-2681, November 1988

[9] G. Bertotti, A. Boglietti, M. Chiampi, D. Chiarabaglio, F. Fiorillo, M. Lazzari, "An improved estimation of iron losses in rotating electrical machines", IEEE Transactions on Magnetics, vol. 27, pp. 5007-5009, November 1991

[10] L. R. Dupré, R. van Keer, J. A. A. Melkebeek, "On a Mathematical Model for Electromagnetic Losses in Electric Machinary", Mathematical Modelling of Systems, vol. 1, pp. 63-74, 1995

[11] L. R. Dupré, J. A. A. Melkebeek, R. van Keer, "On a numerical method for the evaluation of electromagnetic losses in electric machinary", International Journal for Numerical Methods in Engineering, vol. 39, pp. 1535-1553, 1996

[12] A. G. Jack, B. C. Mecrow, "Calculation of three-dimensional electromagnetic fields involving laminar eddy currents", IEE Proceedings, vol. 134, Pt. A, pp. 663-671, September 1987

[13] T. W. Preston, M. A. Timothy, "Eddy current distribution in segmented stator core laminations", presented at 25th Universities Power Engineering Conference, Aberdeen, UK, 12-14 September, Proceedings, vol. 2, pp. 631-634, 1990

[14] B. C. Mecrow, A. G. Jack, "The Modelling of Segmented Laminations in Three Dimensional Eddy Current Calculations", IEEE Transactions on Magnetics, vol. 28, pp. 1122-1125, March 1992

[15] Viviane Cristine Silva, Gérard Meunier, Albert Foggia, "A 3-D Finite- Element Computation of Eddy Currents and Losses in Laminated Iron Cores Allowing for Electric and Magnetic Anisotropy", IEEE Transactions on Magnetics, vol. 31, pp. 2139-2141, May 1995

[16] Peter Hahne, Rainer Dietz, Bernd Rieth, Thomas Weiland, "Determination of Anisotropic Equivalent Conductivity of Laminated Cores for Numerical Computation", IEEE Transactions on Magnetics, vol. 32, pp. 1184-1187, May 1996

[17] Viviane Cristine Silva, Gérard Meunier, Albert Foggia, "A 3D Finite-Element Computation of Eddy Currents and Losses in Stator End Laminations of Large Synchronous Machines", IEEE Transactions on Magnetics, vol. 32, pp. 1569-1572, May 1996

[18] H.R. Schwarz, „Methode der finiten Elemente,“ B.G. Teubner, Stuttgart, 1984. [19] O.C. Zienkiewicz and R.L. Taylor, „The finite Element Method,“ vol. 1 and 2, Fourth Edition, McGraw-

Hill, London, 1991 [20] O. Biro and K.R. Richter, "CAD in Electromagnetism", in P. W. Hawkes (ed.), Advances in Electronics

and Electron Physics, vol. 82, pp. 1-96, Academic Press, 1991 [21] O. Bíró, “Edge element formulations of eddy current problems,”Computer methods in applied mechanics

and engineering, vol. 160, pp. 391-405, 1999. [22] K. Hollaus, O. Bíró, “Estimation of 3D eddy currents in conducting laminations by an anisotropic

conductivity and a 1D analytical solution”, COMPEL, vol. 18, no. 3, pp. 494-503, 1999. [23] I. D. Mayergoyz, "Mathematical Models of Hysteresis," Springer-Verlag, New York, 1991. [24] F. Preisach, "Über die magnetische Nachwirkung," Zeitschrift für Physik, vol. 94 1935. [25] E. Della Torre and F. Vajda, "Physical Basis for the Parameter Identification in Magnetic Materials,"

IEEE Transactions on Magnetics, vol. 32, pp. 4186-4191, September 1996.

98

References

[26] Luc R. Dupré, Johan Gyselinck and Jan A. Melkebeek, "Complementary Finite Element Methods in 2D Magnetics Taking into Account a Vector Preisach Model," IEEE Transactions on Magnetics, vol. 34, pp. 3048-3051, September 1998.

[27] L. R. Dupré, R. van Keer and J. A. A. Melkebeek, "Complementary 2-D Finite Element Procedures for the Magnetic Field Analysis Using a Vector Hysteresis Model," Int. J. numer. Meth. Engng., vol. 42, pp. 1005-10231, 1998.

[28] A. G. Jack, and B. C. Mecrow, "Methods for magnetically nonlinear problems involving significant hysteresis and eddy currents," IEEE Transactions on Magnetics, vol. 26, pp. 424-429, March 1990.

[29] A. G. Jack, and R. L. Stoll, "A class of methods for solving two-dimensional linear and non-linear steady state and transient problems," COMPUMAG Conference Grenoble, September 1978.

[30] F. Henrotte, A. Nicolet, F. Delince, A. Genon and Pr W. Legros, "Modeling of ferromagnetic materials in 2D finite element problems using Preisach’s model," IEEE Transactions on Magnetics, vol. 28, pp. 2614-2616, Sept. 1992.

[31] O. Bottauscio, M. Chiampi, D. Chiarabaglio, and M. Repetto, "Preisach-type hysteresis models in magnetic field computation," Physica B, vol. 275, pp. 34-39, 2000.

[32] O. Bottauscio, M. Chiampi and D. Chiarabaglio, "Advanced Model of Laminated Magnetic Cores for Two-Dimensional Field Analysis," IEEE Transactions on Magnetics, vol. 36, pp. 561-573, May 2000.

[33] M. Chiampi, D. Chiarabaglio, and M. Repetto, "An accurate investigation on numerical methods for nonlinear magnetic field problems," J. Magn. Magn. Mat., vol. 133, pp. 591-595, 1994.

[34] K. Muramatsu, N. Takahashi, T. Nakata, M. Nakano and Y. Ejiri, "3-D Time-Periodic Finite Element Analysis of Magnetic Field in Non-Oriented Materials Taking into Account Hysteresis Characteristics," IEEE Transactions on Magnetics, vol. 33, pp. 1584-1587, May 1997.

[35] S. R. Naidu, "Time domain model for the ferromagnetic core, including the effects of hysteresis and eddy currents," IEE Proc. A, vol. 138, pp. 44-50, 1990.

[36] Biro, O. and Preis, K.: „Finite element calculation of time-periodic 3d eddy currents in nonlinear media“, in T. Honma (ed.), Advanced Computational Electromagnetics, Selected papers of the 3rd Japan-Hungary Joint Seminar on Applied Electromagnetics in Materials and Computational Technology, Budapest, Hungary, 10-13 July, 1994, pp. 62-74, Elsevier, 1995.

[37] A. S. Akpinar and S. A. Nasar, "Parametric analysis of ferroresonance, including hysteresis effects," Electric Machines and Power Systems, vol. 18, pp. 473-493, 1990.

[38] B. Brandstätter, C. Magele, C. Ragusa and M. Repetto, "Use of Stochastic Optimization Procedures in the Identification of the Generalized Moving Preisach Model," Proceedings of the ISEM Conference, May 12 - 14, 1997, Braunschweig

[39] P. G. Alotto, C. Eranda, B. Brandstätter, G. Fürntratt, C. Magele, G. Molinari, M. Nervi, K. Preis, M. Repetto and K. R. Richter, "Stochastic Algorithms in Electromagnetic Optimization," IEEE Transactions on Magnetics, vol. 34, pp. 3674-3684, September 1998.

[40] H. H. Saliah and A. Lowther, " The use of neural networks in magnetic hysteresis identification," Physica B, vol. 233, pp. 318-323, 1997.

[41] P. Andrei and A. Stancu, "Identification Method Analyses for the Scalar Generalized Moving Preisach Model Using Major Hysteresis Loops," IEEE Transactions on Magnetics, vol. 34, pp. 629-635, May 1998.

[42] E. Della Torre and F. Vajda, "Parameter Identification of the Complete-Moving-Hysteresis Model Using Major Loop Data," IEEE Transactions on Magnetics, vol. 30, pp. 4987-5000, November 1994.

[43] M. L. Hodgdon, "Mathematical theory and calculations of magnetic hysteresis curves," IEEE Transactions on Magnetics, vol. 24, pp. 3120-3122, November 1988.

[44] H. H. Saliah and A. Lowther, "Magnetic Material Property Identification Using Neural Networks," ACES, vol. 2, pp. 44-49, July 1997.

[45] A. A. Adly and S. K. Abd-Ed_Hafiz, "Using Neural Networks in the Identification of Preisach-Type Hysteresis Model," IEEE Transactions on Magnetics, vol. 34, pp. 629-635, May 1998.

[46] Surahammars Bruk, "Non-oriented Fully Processed Electrical Steels," Sweden [47] G. Bertotti, "Hysteresis in Magnetism," Academic Press 1998. [48] G. Bertotti, "Dynamic Generalization of the Scalar Preisach Model of Hysteresis," IEEE Transactions on

Magnetics, vol. 28, pp. 2599-2601, September 1992. [49] L. R. Dupré, G. Bertotti and J. A. A. Melkebeek, "Dynamic Preisach Model and Energy Dissipation in

Soft Magnetic Materials," IEEE Transactions on Magnetics, vol. 34, pp. 1168-1170, July 1998. [50] N. Takahashi, S. Miyabara and K. Fujiwara, "Problems in Practical Finite Element Analysis Using

Preisach Hysteresis Model," IEEE Transactions on Magnetics, vol. 35, pp. 1243-1246, May 1999. [51] S. Yamada and K. Bessho, "Harmonic field calculation by the combination of finite element analysis and

harmonic balance method," IEEE Transactions on Magnetics, vol. 24, pp. 2588-2590, September 1988. [52] H. de Gersem, H. Vande Sande and K. Hameyer, "Strong coupled multi.harmonic finite element

simulation package," COMPEL, vol. 20, pp. 535-546, 2001.

99

References

[53] G. Paoli, O. Bíró, G. Buchgraber, "Complex Representation in Nonlinear Time Harmonic Eddy Current Problems," IEEE Transactions on Magnetics, vol. 34, pp. 2625-2628, September 1998.

[54] J. D. Lavers, ’’Finite element solution of nonlinear two dimensional TE-mode eddy current problems,’’ IEEE Trans Magn. Vol.19, pp.2201-2203, Sept. 1983.

[55] Y. Du Terrail, Jean-Claude Sabonnadiere, P. Masse and J.L.Coulomb, ’’Nonlinear Complex Finite Elements Analysis of Electromagnetic Field in Steady-State AC Devices,’’ IEEE Trans. Magn. Vol.20, pp.549-552, July 1984.

[56] D. Labridis and P. Dokopoulos, ’’Finite Element Computation of Eddy Current Losses in Nonlinear Ferromagnetic Sheaths of Three-Phase Power Cables,’’ IEEE Trans. on Power Delivery Vol.7, pp.1060-1067, July 1992.

[57] D. O’Kelly, "Flux penetration in a ferromagnetic material including hysteresis and eddy-current effects," Journal of Physics D (Applied Physics), vol. 5, pp. 203-213, 1972.

[58] M. S. El-Markaby, T. H. Fawzi, M. T. Ahmed, "Approximate treatment of nonlinear eddy currents problems," IEEE Trans. Magn. Vol.18, pp.1737-1739, Jov. 1982.

[59] B. Leutke-Daldrup, "Comparison of exact and approximate finite-element solution of two-dimensional nonlinear eddy-current problem with measurements," IEEE Trans. Magn. Vol.20, pp.1737-1739, Sept. 1984.

[60] H. Hedia, P. D. Henrotte, A. Genon, W. Legros, "Determination of equivalent material characteristics for parasitic effects in electrical apparatus," IEEE Transactions on Magnetics, vol. 32, pp. 1168-1170, September 1996.

[61] D. Lederer and A. Kost, "Modelling of Nonlinear Magnetic Material using a Complex Effective Reluctivity," IEEE Transactions on Magnetics, vol. 34, pp. 3060-3063, September 1998.

100

Acknowledgments

First, I’d like to thank Prof. Oszkár Bíró from the University of Technology Graz at the “Institut für Grundlagen und Theorie der Electrotechnik” (IGTE) for supervising and reviewing the Ph.D. thesis. I’m also very grateful to him for having initiated the research of this topic and for securing a grant of the „Fonds zur Förderung der wissenschaftlichen Forschung“ (FWF), Vienna, Austria. Otherwise, it wouldn’t have been possible for me to carry out this work.

Also I’d to thank Prof. Kurt Preis, head of the IGTE, for providing all necessary facilities and for fruitful discussions.

I’m also very grateful to Prof. Maurizio Repetto from the Politecnico di Torino, Italy, Dipartimento Ingegneria Elettrica Industriale, for his scientific support during the work and also for his readiness to review the theses.

I’m also grateful to the FWF and to IGTE for helping me to have the opportunity to attend the COMPUMAG conferences in Sapporo, Japan, in 1999, and in Evian, France, in 2001.

Last but not least, I’d like to express many thanks to all colleagues at the IGTE for many discussions and especially for the great atmosphere.

Documents

Numerical Simulation of Eddy Currents and the Associated Losses …khollaus/Thesis/PhDTheses.pdf · 2002-09-24 · simulation of eddy currents in three dimensions and the prediction