Preface ........................................................ xi
Authors ...................................................... xiii
Chapter 1 Statics and Quasistatics Electromagnetics:
Brief Presentation .............................................. 1
1.1 Introduction ............................................... 1
1.2 Maxwell's Equations ........................................ 2
1.3 Maxwell's Equations: Local Form ............................ 2
1.4 Maxwell's Equations: Integral Form ......................... 5
1.5 Maxwell's Equations in Low Frequency ....................... 9
1.6 Electrostatics ............................................ 11
1.6.1 Refraction of the Electric Field ................... 11
1.6.2 Laplace's and Poisson's Equations of the Electric
Field for Dielectric Media ......................... 15
1.6.3 Laplace's Equation of the Electric Field for
Conductive Media ................................... 16
1.7 Magnetostatic Fields ...................................... 16
1.7.1 Equation rot H = J ................................. 17
1.7.2 Equation div В = 0 ................................. 19
1.7.3 Equation rot E = 0 ................................. 20
1.7.4 Biot-Savart Law .................................... 20
1.7.5 Magnetic Field Refraction .......................... 22
1.7.6 Energy in the Magnetic Field ....................... 25
1.8 Magnetic Materials ........................................ 27
1.8.1 Diamagnetic Materials .............................. 28
1.8.2 Paramagnetic Materials ............................. 29
1.8.3 Ferromagnetic Materials ............................ 29
1.8.3.1 General Presentation ...................... 29
1.8.3.2 Influence of Iron on Magnetic Circuits .... 31
1.8.4 Permanent Magnets .................................. 32
1.8.4.1 General Presentation ...................... 32
1.8.4.2 Principal Types of Permanent Magnets ...... 39
1.8.4.3 Dynamic Operation of Permanent Magnets .... 39
1.9 Inductance and Mutual Inductance .......................... 41
1.9.1 Definition of Inductance ........................... 41
1.9.2 Energy in a Linear System .......................... 42
1.10 Magnetodynamic Fields ..................................... 43
1.10.1 Maxwell's Equations for the Magnetodynamic Field ... 44
1.10.2 Penetration of Time-Dependent Fields
in Conducting Materials ............................ 47
1.10.2.1 Equation for H ............................ 47
1.10.2.2 Equation for В ............................ 48
1.10.2.3 Equation for E ............................ 48
1.10.2.4 Equation for J ............................ 49
1.10.2.5 Solution of the Equations ................. 49
1.11 Fields Defined by Potentials .............................. 53
1.11.1 Electric Scalar Potential .......................... 53
1.11.2 Magnetic Scalar Potential .......................... 54
1.11.3 Magnetic Vector Potential .......................... 55
1.11.4 Electric Vector Potential .......................... 56
1.12 Final Considerations ...................................... 57
References ................................................ 57
Chapter 2 Ferromagnetic Materials and Iron Losses ............. 59
2.1 Introduction .............................................. 59
2.2 Basic Concepts ............................................ 59
2.3 Loss Components ........................................... 61
2.4 Iron Losses under Alternating, Rotating, and DC-Biased
Inductions ................................................ 66
2.4.1 Epstein's Frame and Workbench ...................... 67
2.4.1.1 Methodology for Iron Loss Separation ...... 70
2.4.1.2 Results for Two Different Iron Sheets ..... 72
2.4.1.3 Considering Eddy Current in Epstein's
Frame Corners ............................. 73
2.4.1.4 Improved Model for the Eddy Current
Losses .................................... 74
2.4.1.5 Results Verification by 3D FE Modeling .... 77
2.4.2 Single Sheet Tester ................................ 81
2.4.3 Rotational Single Sheet Tester ..................... 83
2.4.4 DC-Biased Inductions ............................... 93
2.5 Final Considerations ...................................... 99
References ............................................... 100
Chapter 3 Scalar Hysteresis Modeling ......................... 103
3.1 Introduction ............................................. 103
3.2 Preisach's Scalar Model .................................. 103
3.2.1 Magnetization in Terms of Everett's Function ...... 110
3.2.2 Identification of Everett's Function .............. 112
3.2.3 Results Obtained with Preisach's Scalar Model ..... 115
3.3 Jiles-Atherton Scalar Model .............................. 115
3.3.1 Original (Direct) Jiles-Atherton Model ............ 115
3.3.2 Inverse Jiles-Atherton Model ...................... 120
3.3.3 Jiles-Atherton Model Parameter Determination ...... 121
3.3.4 Results Obtained with the Jiles-Atherton Model .... 129
3.3.1 Modified Jiles-Atherton Hysteresis Model .......... 130
3.3.6 Determination of Parameter R in the Modified
Jiles-Atherton Model .............................. 134
3.3.7 Results of the Modified Jiles-Atherton Model ...... 134
3.4 Final Considerations ..................................... 136
References ............................................... 136
Chapter 4 Vector Hysteresis Modeling ......................... 139
4.1 Introduction ............................................. 139
4.2 Vector Model Obtained with the Superposition
of Scalar Models ......................................... 139
4.2.1 Model Principle ................................... 139
4.2.2 Identification of the Parameters of the Model ..... 141
4.2.3 Results of the Vector Model ....................... 141
4.3 Vector Generalization of the Jiles-Atherton Scalar
Models ................................................... 144
4.3.1 Vector Generalization of the Original Jiles-
Atherton Model .................................... 144
4.3.2 Vector Generalization of the Inverse Jiles-
Atherton Model .................................... 147
4.3.3 Some Aspects of the Jiles-Atherton Vector
Model and Results ................................. 149
4.4 Remarks Concerning the Vector Behavior of Hysteresis ..... 156
4.5 Final Considerations ..................................... 166
References ............................................... 168
Chapter 5 Finite Element Method: Brief Presentation .......... 169
5.1 Introduction ............................................. 169
5.2 Galerkin Method: Basic Concepts Using Real Coordinates ... 170
5.2.1 Equations and Numerical Implementation ............ 170
5.2.2 Boundary Conditions ............................... 178
5.2.2.1 Dirichlet Boundary Condition: Imposed
Potential ................................ 178
5.2.2.2 Neumann Condition: Unknown Nodal Values
on the Boundary .......................... 179
5.2.3 First-Order 2D Finite Element Program ............. 179
5.2.4 Example for the Finite Element Program ............ 184
5.3 Generalization of the FEM: Using Reference Coordinates ... 188
5.3.1 High-Order Finite Elements: General ............... 189
5.3.2 High-Order Finite Elements: Notation .............. 190
5.3.3 High-Order Finite Elements: Implementation ........ 193
5.3.4 Continuity of Finite Elements ..................... 196
5.3.5 Polynomial Basis .................................. 196
5.3.6 Transformation of Quantities: Jacobian ............ 198
5.3.7 Evaluation of the Integrals ....................... 199
5.3.1 Numerical Integration ............................. 203
5.5 Some Finite Elements ..................................... 206
5.5.1 First-Order Triangular Element .................... 207
5.5.2 Second-Order Triangular Element ................... 207
5.5.3 First-Order Tetrahedral Element ................... 208
5.5.4 Implementation Aspects ............................ 212
5.6 Using Edge Elements ...................................... 218
5.6.1 Magnetostatic Equation Using the Vector
Potential ......................................... 218
5.6.2 Brief Explanation of Edge Shape Functions ......... 219
5.6.3 Applying the Edge Element Shape Functions ......... 225
5.6.4 Implementing the First-Order Tetrahedron Edge
Element Shape Functions ........................... 232
5.6.5 Applying the Galerkin Method ...................... 237
5.6.6 Coding Tetrahedral Edge Elements .................. 238
5.7 Final Considerations ..................................... 240
References ............................................... 241
Chapter 6 Using Nodal Elements with Magnetic Vector
Potential ..................................................... 243
6.1 Introduction ............................................. 243
6.2 Main Equations ........................................... 244
6.2.1 Magnetostatic Governing Equation .................. 244
6.2.2 Defining Some Operations .......................... 244
6.3 Applying the Galerkin Method ............................. 247
6.4 Uniqueness of the Solution: Coulomb's Gauge .............. 249
6.5 Implementation ........................................... 251
6.6 Example and Comparisons .................................. 254
6.7 Final Considerations ..................................... 258
References ............................................... 259
Chapter 7 Source-Field Method for 3D Magnetostatic Fields .... 261
7.1 Introduction ............................................. 261
7.2 Magnetostatic Case: Scalar Potential ..................... 261
7.2.1 Main Equations .................................... 261
7.2.2 Hs Calculation: Edge Tree ......................... 262
7.2.3 Facet Tree ........................................ 264
7.2.4 Applying the Galerkin Method ...................... 267
7.2.5 Elemental Matrices: Evaluation, Notation, and
Array Dimensions .................................. 269
7.2.6 Considering Permanent Magnets ..................... 271
7.2.7 Boundary Conditions ............................... 272
7.3 Magnetostatic Case: Vector Potential ..................... 273
7.3.1 Main Equations .................................... 273
7.4 Implementation Aspects and Conventions ................... 275
7.4.1 Building the Facets ............................... 276
7.4.2 Building the Edges ................................ 276
7.4.3 Building the Edge Tree ............................ 277
7.4.4 Building the Conductor Facet Tree and
Calculating the Flux of J ......................... 278
7.4.5 Calculating Hs .................................... 280
7.4.6 Applying the Boundary Conditions .................. 281
7.5 Computational Implementation ............................. 282
7.5.1 Main Subroutines for the Scalar Potential
Formulation ....................................... 283
7.5.2 Main Subroutines for the Vector Potential
Formulation ....................................... 286
7.6 Example and Results ...................................... 288
7.7 Final Considerations ..................................... 292
References ............................................... 292
Chapter 8 Source-Field Method for 3D Magnetody namic Fields .. 295
8.1 Introduction ............................................. 295
8.2 Formulation Considering Eddy Currents: Time Stepping ..... 295
8.2.1 Governing Equations ............................... 295
8.3 Formulation Considering Eddy Currents: Complex
Formulation .............................................. 299
8.4 Field-Circuit Coupling ................................... 300
8.4.1 Basic Equations ................................... 300
8.4.2 Applying the Galerkin Method ...................... 304
8.4.3 Formulation Considering Eddy Currents and
Electric Circuit Coupling ......................... 307
8.5 Computational Implementation ............................. 309
8.6 Differential Permeability Method ......................... 315
8.6.1 Nonlinear Cases ................................... 315
8.6.2 Anisotropic Cases ................................. 320
8.7 Example and Results ...................................... 322
8.7.1 Eddy Currents, Circuit Coupling, Regular
Permeability ...................................... 322
8.7.2 Example of an Isotropic Nonlinear Case with
Differential Permeability ......................... 327
8.7.3 Anisotropic Magnetic Circuit ...................... 328
8.7.4 Scalar Hysteresis: A Didactical Case .............. 331
8.7.5 Vector Hysteresis Anisotropic Case: TEAM
Workshop Problem 32 ............................... 340
8.8 Final Considerations ..................................... 346
References ............................................... 346
Chapter 9 Matrix-Free Iterative Solution Procedure
for Finite Element Problems ................................... 349
9.1 Introduction ............................................. 349
9.2 Classical FEM: T-Scheme .................................. 350
9.3 Proposed Technique: N-Scheme ............................. 351
9.4 Implementation ........................................... 352
9.5 Convergence .............................................. 353
9.6 Implementation of N-Scheme with SOR ...................... 355
9.7 Applying the N-Scheme in Nonstationary Solvers ........... 358
9.8 CG Algorithm Implementation .............................. 361
9.9 Examples and Results ..................................... 365
9.9.1 Two-Dimensional Electrostatic Problem ............. 365
9.9.2 Three-Dimensional Nonlinear Case Using SOR
Technique ......................................... 367
9.9.3 Example with a Large Number of Unknowns ........... 369
9.10 Results and Discussion ................................... 372
9.11 Final Considerations ..................................... 372
References ............................................... 373
Index ......................................................... 375
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