Electromagnetic Modeling by Finite Element Methods

0824742699_01__SCLZZZZZZZ_.pdf......Page 1
ELECTROMAGNETIC MODELING BY FINITE ELEMENT METHODS......Page 2
PREFACE......Page 9
CONTENTS......Page 12
BIBLIOGRAPHY......Page 17
1.2. THE VECTOR NOTATION......Page 18
CONTENTS......Page 0
1.3.1. THE NABLA (V) OPERATOR......Page 19
1.3.2. DEFINITION OFF THE GRADIENT, DIVERGENCE, AND ROTATIONAL......Page 20
1 .4. THE GRADIENT......Page 21
1.4.1. EXAMPLE OFF GRADIENT......Page 22
1.5.1. DEFINITION OF FLUX......Page 24
1.5.2 THE DIVERGENCE THEOREM......Page 26
1.5.3. CONSERVATIVE FLUX......Page 28
1.5.4. EXAMPLE OF DIVERGENCE......Page 30
1.6.1. CIRCULATION OF A VECTOR......Page 31
1.6.2. STOKES' THEOREM......Page 34
1.6.3. EXAMPLE OF ROTATIONAL......Page 37
1.7. SECOND-ORDER OPERATORS......Page 38
1.8. APPLICATION OF OPERATORS TO MORE THAN ONE FUNCTION......Page 40
1.9. EXPRESSIONS IN CYLINDRICAL AND SPHERICAL COORDINATES......Page 41
2.1. INTRODUCTION......Page 43
2.2. THE EM QUANTITIES......Page 44
2.2.2. THE MAGNETIC FIELD INTENSITY H......Page 46
2.2.3. THE MAGNETIC FLUX DENSITY B AND THE MAGNETIC PERMEABILITY JU.......Page 47
2.2.4. THE ELECTRIC FLUX DENSITY D AND ELECTRIC PERMITTIVITY 8......Page 48
2.2.6. VOLUME CHARGE DENSITY P......Page 49
2.2.7. THE ELECTRIC CONDUCTIVITY A......Page 50
2.3. LOCAL FORM OF THE EQUATIONS......Page 51
2.4. THE ANISOTROPY......Page 56
2.5. THE APPROXIMATION TO MAXWELL'S EQUATIONS......Page 58
2.6. THE INTEGRAL FORM OF MAXWELL'S EQUATION......Page 63
2.7. ELECTROSTATIC FIELDS......Page 65
2.7.1 A. THE ELECTRIC FIELD......Page 66
2.7.1 C. THE ELECTRIC SCALAR POTENTIAL V......Page 67
2.7.2. NONCONSERVATIVE FIELDS: ELECTROMOTIVE FORCE......Page 72
2.7.3. REFRACTION OF THE ELECTRIC FIELD......Page 75
2.7.4. DIELECTRIC STRENGTH......Page 79
2.7.5. LAPLACE'S AND POISSON'S EQUATIONS OF THE ELECTRIC FIELD FOR DIELECTRIC MEDIA......Page 81
2.8. MAGNETOSTATIC FIELDS......Page 84
2.8. 1 A. THE EQUATION ROTH = J......Page 86
2.8.2. THE BIOT-SAVART LAW......Page 89
2.8.3. MAGNETIC FIELD REFRACTION......Page 93
2.8.4. ENERGY IN THE MAGNETIC FIELD......Page 96
2.8.5. MAGNETIC MATERIALS......Page 99
2.8.5A. DIAMAGNETIC MATERIALS......Page 100
A) GENERAL......Page 101
B) THE INFLUENCE OF IRON ON MAGNETIC CIRCUITS......Page 104
A) GENERAL PROPERTIES OF HARD MAGNETIC MATERIALS......Page 106
B) THE ENERGY ASSOCIATED WITH A MAGNET......Page 110
C) PRINCIPAL TYPES OF PERMANENT MAGNETS......Page 116
D) DYNAMIC OPERATION OF PERMANENT MAGNETS......Page 118
2.8.6A. DEFINITION OF INDUCTANCE......Page 120
2.8.6B. ENERGY IN A LINEAR SYSTEM......Page 121
2.9. MAGNETO DYNAMIC FIELDS......Page 123
2.9.1. MAXWELL'S EQUATIONS FOR THE MAGNETODYNAMIC FIELD......Page 124
2.9.2. PENETRATION OF TIME-DEPENDENT FIELDS IN CONDUCTING MATERIALS......Page 128
2.9. 2B. THE EQUATION FOR B......Page 129
2.9.2D. THE EQUATION FOR J......Page 130
2.9.2E. SOLUTION OF THE EQUATIONS......Page 131
3.1. INTRODUCTION......Page 138
3.2.1. THE ESTABLISHMENT OF THE PHYSICAL EQUATIONS......Page 139
3.2.2. THE FIRST ORDER TRIANGLE......Page 140
3.2.3. APPLICATION OF THE WEIGHTED RESIDUAL METHOD......Page 142
3.2.4. APPLICATION OF THE FINITE ELEMENT METHOD AND SOLUTION......Page 146
3.2.5A. DIRICHLET BOUNDARY CONDITION - IMPOSED POTENTIAL......Page 149
3.3. A FIRST-ORDER FINITE ELEMENT PROGRAM......Page 150
3.3.1. EXAMPLE FOR USE OF THE FINITE ELEMENT PROGRAM......Page 157
3.4. GENERALIZATION OF THE FINITE ELEMENT METHOD......Page 162
3.4.2. HIGH-ORDER FINITE ELEMENTS: NOTATION......Page 163
3.4.3. HIGH-ORDER FINITE ELEMENTS: IMPLEMENTATION......Page 168
3.4.4. CONTINUITY OF FINITE ELEMENTS......Page 171
3.4.5. POLYNOMIAL BASIS......Page 172
3.4.6. TRANSFORMATION OF QUANTITIES - THE JACOBIAN......Page 173
3.4.7. EVALUATION OF THE INTEGRALS......Page 176
3.5. NUMERICAL INTEGRATION......Page 181
3.6. SOME 2D FINITE ELEMENTS......Page 184
3.6.1. FIRST-ORDER TRIANGULAR ELEMENT......Page 186
3.6.2. SECOND-ORDER TRIANGULAR ELEMENT......Page 187
3.6.3. QUADRILATERAL BI-LINEAR ELEMENT......Page 188
3.6.4. QUADRILATERAL QUADRATIC ELEMENT......Page 189
3.7.1. COUPLING DIFFERENT TYPES OF FINITE ELEMENTS......Page 190
3.8. CALCULATION OF SOME TERMS IN THE FIELD EQUATION......Page 192
3.8.1. THE STIFFNESS MATRIX......Page 193
3.8.2. EVALUATION OF THE SECOND TERM IN EQ. (3.72)......Page 195
3.8.4. EVALUATION OF THE SOURCE TERM......Page 196
3.9.1. THE PROBLEM TO BE SOLVED......Page 197
3.9.2. THE DISCRETIZED DOMAIN......Page 199
3.9.3. THE FINITE ELEMENT PROGRAM......Page 200
PROGRAM LISTING......Page 201
4.2. SOME STATIC CASES......Page 212
4.2.1. ELECTROSTATIC FIELDS: DIELECTRIC MATERIALS......Page 213
4.2.2. STATIONARY CURRENTS: CONDUCTING MATERIALS......Page 215
4.2.3. MAGNETIC FIELDS: SCALAR POTENTIAL......Page 216
4.2.4. THE MAGNETIC FIELD: VECTOR POTENTIAL......Page 218
4.2.5. THE ELECTRIC VECTOR POTENTIAL......Page 226
4.3.1 . FIRST-ORDER ELEMENT IN LOCAL COORDINATES......Page 228
4.3.2. THE VECTOR POTENTIAL EQUATION USING TIME DISCRETIZATION......Page 234
A. THE FIRST TERM IN EQ. (4.44).......Page 236
B. THE SECOND TERM IN EQ. (4.44).......Page 238
D. THE FOURTH TERM IN EQ. (4.44).......Page 239
4.3.3. THE COMPLEX VECTOR POTENTIAL EQUATION......Page 241
4.3.4. STRUCTURES WITH MOVING PARTS......Page 246
4.4. AXI-SYMMETRIC APPLICATIONS......Page 248
4.4.1 . THE AXI-SYMMETRIC FORMULATION FOR VECTOR POTENTIAL......Page 251
4.5. ADVANTAGES AND LIMITATIONS OF 2D FORMULATIONS......Page 254
4.6.1. METHOD OF SUCCESSIVE APPROXIMATION......Page 256
4.6.2. THE NEWTON-RAPHSON METHOD......Page 257
4.7. GEOMETRIC REPETITION OF DOMAINS......Page 263
4.7.1. PERIODICITY......Page 264
4.7.2. ANTI-PERIODICITY......Page 265
4.8.1. THERMAL CONDUCTION......Page 266
4.8.3. RADIATION......Page 267
4.8.4. FE IMPLEMENTATION......Page 268
4.9. VOLTAGE FED ELECTROMAGNETIC DEVICES......Page 271
B. COIL VOLTAGE EQUATION......Page 274
4.10. STATIC EXAMPLES......Page 275
4.10.1. CALCULATION OF ELECTROSTATIC FIELDS......Page 276
4.10.2. CALCULATION OF STATIC CURRENTS......Page 277
4.10.3. CALCULATION OF THE MAGNETIC FIELD - SCALAR POTENTIAL......Page 280
4.10.4. CALCULATION OF THE MAGNETIC FIELD - VECTOR POTENTIAL......Page 282
4.11.1. EDDY CURRENTS: TIME DISCRETIZATION......Page 285
4.11.2. MOVING CONDUCTING PIECE IN FRONT OF AN ELECTROMAGNET......Page 287
4.11.3. TIME STEP SIMULATION OF A VOLTAGE-FED DEVICE......Page 291
4.11.4. THERMAL CASE: HEATING BY EDDY CURRENTS......Page 294
5.2. ELECTROMAGNETIC EQUATIONS......Page 298
5.2.1. FORMULATION USING THE MAGNETIC VECTOR POTENTIAL......Page 299
5.2.3A. THICK CONDUCTORS......Page 300
5.2.3B. THIN CONDUCTORS......Page 302
5.2.4. EQUATIONS FOR THE WHOLE DOMAIN......Page 304
5.2.5 THE FINITE ELEMENT METHOD......Page 305
5.3. EQUATIONS FOR DIFFERENT CONDUCTOR CONFIGURATIONS......Page 306
5.3.1 A. SERIES CONNECTION......Page 307
5.3.1 B. PARALLEL CONNECTION......Page 309
5.3.2. THIN CONDUCTORS CONNECTIONS......Page 314
5.3.2B. STAR CONNECTION WITH NEUTRAL......Page 315
5.3.2D. STAR CONNECTION WITHOUT NEUTRAL WIRE......Page 316
5.4. CONNECTIONS BETWEEN ELECTROMAGNETIC DEVICES AND EXTERNAL FEEDING CIRCUITS......Page 317
5.4.2. FEEDING CIRCUIT EQUATIONS AND CONNECTION TO FIELD EQUATIONS......Page 318
5.4.3. CALCULATION OF MATRICES G, TO G6......Page 319
5.4.3A. CIRCUIT TOPOLOGY CONCEPTS......Page 320
A. THE FUNDAMENTAL CUTSET MATRIX......Page 321
B. FUNDAMENTAL LOOP MATRIX......Page 323
C. RELATIONSHIP BETWEEN MATRICES KC2 AND B/,......Page 325
D. INCIDENCE MATRIX......Page 326
E. WELSCH'S ALGORITHM......Page 327
5.4.3B. DETERMINATION OF MATRICES G, TO G6......Page 330
A. CALCULATION OF GLF G2 AND G3......Page 334
B. CALCULATION OF G4 , G5 AND G6......Page 338
5.4.3C. EXAMPLE......Page 339
5.4.3D. TAKING INTO ACCOUNT ELECTRONIC SWITCHES IN THE FEEDING CIRCUIT......Page 344
5.4.4. DISCRETIZATION OF THE TIME DERIVATIVE......Page 345
5.5.1 A. A DIDACTICAL EXAMPLE......Page 349
5.5.1 C. MASSIVE CONDUCTORS IN SERIES CONNECTION......Page 352
5.5.2. MODELING OF A STATIC CONVERTER-FED MAGNETIC DEVICE......Page 354
6.1.1. METHODS WITH NON-DISCRETIZED AIRGAPS......Page 358
6.2. THE MACRO-ELEMENT......Page 359
6.3. THE MOVING BAND......Page 364
6.4. THE SKEW EFFECT IN ELECTRICAL MACHINES USING 2D SIMULATION......Page 368
6.5.1. THREE-PHASE INDUCTION MOTOR......Page 377
6.5.2. PERMANENT MAGNET MOTOR......Page 379
7.1. INTRODUCTION......Page 382
7.2.1. METHOD OF THE MAGNETIC CO-ENERGY VARIATION......Page 383
7.2.2. THE MAXWELL STRESS TENSOR METHOD......Page 385
7.2.3. THE METHOD PROPOSED BY ARKKIO......Page 398
7.2.4. THE METHOD OF LOCAL JACOBIAN MATRIX DERIVATION......Page 399
7.2.5. EXAMPLES OF TORQUE CALCULATION......Page 401
7.3.1. PRELIMINARY CONSIDERATIONS......Page 404
7.3.2. EQUIVALENT SOURCES FORMULATIONS......Page 406
7.3.2A. EQUIVALENT CURRENTS......Page 407
7.3.2B. EQUIVALENT MAGNETIC CHARGES......Page 408
7.3.2C. OTHER EQUIVALENT SOURCE DISTRIBUTIONS......Page 409
7.3.3. FORMULATION BASED ON THE ENERGY DERIVATION......Page 410
7.3.4. COMPARISON AMONG THE DIFFERENT METHODS......Page 413
7.4. ELECTRICAL MACHINE VIBRATIONS ORIGINATED BY MAGNETIC FORCES......Page 416
7.4.2. MECHANICAL CALCULATION......Page 417
7.4.2A. CALCULATION OF THE NATURAL RESPONSE......Page 418
7.4. 2B. CALCULATION OF THE FORCED RESPONSE DIRECTLY IN HARMONIC REGIME......Page 419
7.4.2C. CALCULATION OF THE FORCED RESPONSE USING THE MODAL SUPERPOSITION METHOD......Page 420
7.4.3.EXAMPLE OF VIBRATION CALCULATION......Page 422
7.5. EXAMPLE OF COUPLING BETWEEN THE FIELD AND CIRCUIT EQUATIONS, INCLUDING MECHANICAL TRANSIENTS......Page 428
8.1. INTRODUCTION......Page 434
8.2. EDDY CURRENT LOSSES......Page 435
8.3. HYSTERESIS......Page 438
8.4. ANOMALOUS OR EXCESS LOSSES......Page 445
8.5. TOTAL IRON LOSSES......Page 448
8.5.1. EXAMPLE......Page 450
8.6.1. THE JA EQUATIONS......Page 453
8.6.2. PROCEDURE FOR THE NUMERICAL IMPLEMENTATION OF THE JA METHOD......Page 456
8.6.3. EXAMPLES OF HYSTERESIS LOOPS OBTAINED WITH THE JA METHOD......Page 458
8.6.4. DETERMINATION OF THE PARAMETERS FROM EXPERIMENTAL HYSTERESIS LOOPS......Page 462
8.7.1. THE INVERSE JA METHOD......Page 467
8.7.2. PROCEDURE FOR THE NUMERICAL IMPLEMENTATION OF THE INVERSE JA METHOD......Page 469
8.8. INCLUDING IRON LOSSES IN FINITE ELEMENT CALCULATIONS......Page 470
8.8.1. HYSTERESIS MODELING BY MEANS OF THE MAGNETIZATION M TERM......Page 472
8.8.2. HYSTERESIS MODELING BY MEANS OF A DIFFERENTIAL RELUCTIVITY......Page 474
8.8.3. INCLUSION OF EDDY CURRENT LOSSES IN THE FE MODELING......Page 478
8.8.4. INCLUSION OF ANOMALOUS LOSSES IN THE FE MODELING......Page 480
8.8.5. EXAMPLES OF IRON LOSSES APPLIED TO FE CALCULATIONS......Page 481
BIBLIOGRAPHY......Page 486