Electromagnetics and Calculation of Fields[Djvu格式]: Electromagnetics and Calculation of Fields.rar

Preface v
Part I. The Electromagnetic Field and Maxwell's
Equations
1. Mathematical Preliminaries
1.1. Introduction 1
1.2. The Vector Notation 1
1.3. Vector Derivation 2
1.3.1. The Nabla (V) Operator 2
1.3.2. Definition of the Gradient, Divergence, and Curl 2
1.4. The Gradient 3
1.4.1. Example of Gradient 5
1.5. The Divergence 6
1.5.1. Definition of Flux 6
1.5.2. The Divergence Theorem 8
1.5.3. Conservative Flux 9
1.5.4. Example of Divergence 11
1.6. The Curl 12
1.6.1. Circulation of a Vector 12
1.6.2. Stokes' Theorem 14
1.6.3. Example of Curl 17
1.7. Second Order Operators 18
1.8. Application of Operators to More than One Function 19
1.9. Expressions in Cylindrical and Spherical Coordinates 20
2. The Electromagnetic Field and Maxwell's
Equations
2.1. Introduction 22
2.2. Maxwell's Equations 22
2.2.1. Fundamental Physical Principles of the Electromagnetic
Field 23
2.2.2. Point Form of the Equations 29
2.2.3. The Equations in Vacuum 32
2.2.4. The Equations in Media with e=eo and jlu=|Jo 34
2.2.5. The Equations in General Media 35
2.2.6. The Integral Form of Maxwell's Equations 37
2.3. Approximations to Maxwell's Equations 43
2.4. Units 45
3. Electrostatic Fields
3.1. Introduction 47
3.2. The Electrostatic Charge 47
3.2.1. The Electric Field 48
3.2.2. Force on an Electric Charge 48
3.2.3. The Electric Scalar Potential V 49
3.3. Nonconservative Fields: Electromotive Force 53
3.4. Refraction of the Electric Field 55
3.5. Dielectric Strength 59
3.6. The Capacitor 61
3.6.1. Definition of Capacitance 61
3.6.2. Energy Stored in a Capacitor 64
3.6.3. Energy in a Static, Conservative Field 64
3.7. Laplace's and Poisson's Equations in Terms of the Electric Field 65
3.8. Examples 67
3.8.1. The Infinite Charged Line 67
3.8.2. The Charged Spherical Half-Shell 70
3.8.3. The Spherical Capacitor 71
3.8.4. The Spherical Capacitor with Two Dielectric Layers ... 72
3.9. A Brief Introduction to the Finite Element Method: Solution of the
Two-Dimensional Laplace Equation 74
3.9.1. The Finite Element Technique for Division of a Domain . 75
3.9.2. The Variational Method 77
3.9.3. A Finite Element Program 80
3.9.4. Example for Use of the Finite Element Program 84
3.10. Tables of Permittivities, Dielectric Strength, and Conductivities 88
4. Magnetostatic Fields
4.1. Introduction 90
4.2. Maxwell's Equations in Magnetostatics 91
4.2.1. The Equation VxH=J 91
4.2.2. The Equation VB=0 93
4.2.3. The Equation VxE=0 93
4.3. The Biot-Savart Law 94
4.4. Boundary Conditions for the Magnetic Field 96
4.5. Magnetic Materials 98
4.5.1. Diamagnetic Materials 99
4.5.2. Paramagnetic Materials 100
4.5.3. Ferromagnetic Materials 100
4.5.4. Permanent Magnets 104
4.6. The Analogy between Magnetic and Electric Circuits 115
4.7. Inductance and Mutual Inductance 119
4.7.1. Definition of Inductance 119
4.7.2. Energy in a Linear System 120
4.7.3. The Energy Stored in the Magnetic Field 122
4.8. Examples 123
4.8.1. Calculation of Field Intensity and Inductance of a Long
Solenoid 123
4.8.2. Calculation of H for a Circular Loop 125
4.8.3. Field of a Rectangular Loop 127
4.8.4. Calculation of Inductance of a Coaxial Cable 128
4.8.5. Calculation of the Field Inside a Cylindrical Conductor . . 129
4.8.6. Calculation of the Magnetic Field Intensity in a Magnetic
Circuit 130
4.8.7. Calculation of the Magnetic Field Intensity of a Saturated
Magnetic Circuit 133
4.8.8. Magnetic Circuit Incorporating Permanent Magnets .... 135
4.9. Laplace's Equation in Terms of the Magnetic Scalar Potential . . 138
4.10. Properties of Soft Magnetic Materials 140
5. Magnetodynamic Fields
5.1. Introduction 142
5.2. Maxwell's Equations for the Magnetodynamic Field 143
5.3. Penetration of Time Dependent Fields in Conducting Materials . 146
5.3.1. The Equation forH 146
5.3.2. The Equation for B 147
5.3.3. The Equation for E 147
5.3.4. The Equation for J 148
5.3.5. Solution of the Equations 148
5.4. Eddy Current Losses in Plates 153
5.5. Hysteresis Losses 156
5.6. Examples 160
5.6.1. Induced Currents Due to Change in Induction 160
5.6.2. Induced Currents Due to Changes in Geometry 163
5.6.3. Inductive Heating of a Conducting Block 165
5.6.4. Effect of Movement of a Magnet Relative to a Flat
Conductor 169
5.6.5. Visualization of Penetration of Fields as a Function of
Frequency 171
5.6.6. The Voltage Transformer 172
6. Interaction between Electromagnetic and
Mechanical Forces
6.1. Introduction 175
6.2. Force on a Conductor 175
6.3. Force on Moving Charges: The Lorentz Force 178
6.4. Energy in the Magnetic Field 180
6.5. Force as Variation of Energy (Virtual Work) 182
6.6. The Poynting Vector 184
6.7. Maxwell's Force Tensor 188
6.8. Examples 195
6.8.1. Force between Two Conducting Segments 195
6.8.2. Torque on a Loop 198
6.8.3. The Hall Effect 200
6.8.4. The Linear Motor and Generator 202
6.8.5. Attraction of a Ferromagnetic Body 205
6.8.6. Repulsion of a Diamagnetic Body 206
6.8.7. Magnetic Levitation 207
6.8.8. The Magnetic Brake 209
7. Wave Propagation and High-Frequency
Electromagnetic Fields
7.1. Introduction 212
7.2. The Wave Equation and Its Solution 215
7.2.1. The Time Dependent Equations 215
7.2.2. The Time Harmonic Wave Equations 220
7.2.3. Solution of the Wave Equation 222
7.2.4. Solution for Plane Waves 222
7.2.5. The One-Dimensional Wave Equation in Free Space and
Lossless Dielectrics 223
7.3. Propagation of Waves in Materials 227
7.3 1. Propagation of Waves in Lossy Dielectrics 227
7.3.2. Propagation of Plane Waves in Low-Loss Dielectrics : . . 229
7.3.3. Propagation of Plane Waves in Conductors 230
7.3.4. Propagation in a Conductor: Definition of the Skin Depth 232
7.4. Polarization of Plane Waves 233
7.5. Reflection, Refraction, and Transmission of Plane Waves .... 235
7.5.1. Reflection and Transmission at a Lossy Dielectric Interface:
Normal Incidence 236
7.5.2. Reflection and Transmission at a Conductor Interface:
Normal Incidence 239
7.5.3. Reflection and Transmission at a Finite Conductivity
Conductor Interface 240
7.5.4. Reflection and Transmission at an Interface:
Oblique Incidence 241
7.5.5. Oblique Incidence on a Conducting Interface:
Perpendicular Polarization 242
7.5.6. Oblique Incidence on a Conducting Interface:
Parallel Polarization 244
7.5.7. Oblique Incidence on a Dielectric Interface:
Perpendicular Polarization 245
7.5.8. Oblique Incidence on a Dielectric Interface:
Parallel Polarization 248
7.6. Waveguides 249
7.6.1. TEM,TE, and TM Waves 249
7.6.2. TEMWaves 251
7.6.3. TEWaves 251
7.6.4. TMWaves 252
7.6.5. Rectangular Waveguides 253
7.6.6. TM Modes in Waveguides 253
7.6.7. TE Modes in Waveguides 256
7.7. Cavity Resonators 258
7.7.1. TM and TE Modes in Cavity Resonators 259
7.7.2. TE Modes in a Cavity 261
7.7.3. Energy in a Cavity 261
7.7.4. Quality Factor of a Cavity Resonator 263
7.7.5. Coupling to Cavities 263
Part II. Introduction to the Finite Element Method in
Electromagnetics
8. Introduction to the Finite Element Method
8.1. Introduction 265
8.2. The Galerkin Method - Basic Concepts 266
8.3. The Galerkin Method - Extension to 2D 277
8.3.1. The Boundary Conditions 278
8.3.2. Calculation of the 2D Elemental Matrix 279
8.4. The Variational Method - Basic Concepts 281
8.5. The Variational Method - Extension to 2D 284
8.5.1. The Variational Formulation 284
8.5.2. Calculation of the 2D Elemental Matrix 289
8.6. Generalization of the Finite Element Method 291
8.6.1. High-Order Finite Elements: General 292
8.6.2. High-Order Finite Elements: Notation 293
8.6.3. High-Order Finite Elements: Implementation 296
8.6.4. Continuity of Finite Elements 298
8.6.5. Polynomial Basis 298
8.6.6. Transformation of Quantities - the Jacobian 300
8.6.7. Evaluation of the Integrals 302
8.7. Numerical Integration 306
8.7.1. Evaluation of the Integrals 306
8.7.2. Basic Principles of Numerical Integration 307
8.7.3. Accuracy and Errors in Numerical Integration 311
8.8. Some Specific Finite Elements 313
8.8.1. ID Elements 314
8.8.2. 2D Elements 315
8.8.3. 3D Elements 318
8.9. Coupling Different Finite Elements; Infinite Elements 323
8.9.1. Coupling Different Types of Finite Elements 323
8.9.2. Infinite Elements 325
8.10. Calculation of Some Terms in Poisson's Equation 327
8.10.1. The Stiffness Matrix 327
8.10.2. Evaluation of the Second Term in Eq. (8.130) 329
8.10.3. Evaluation of the Third Term in Eq. (8.130) 329
8.10.4. Evaluation of the Source Term 330
8.11. A Simplified 2D Second-Order Finite Element Program .... 330
8.11.1. The Problem to Be Solved 330
8.11.2. The Discretized Domain 332
8.11.3. The Finite Element Program 333
9. The Variational Finite Element Method:
Some Static Applications
9.1. Introduction 343
9.2. Some Static Applications 343
9.2.1. Electrostatic Fields: Dielectric Materials 343
9.2.2. Stationary Currents: Conducting Materials 345
9.2.3. Magnetic Fields: Scalar Potential 346
9.2.4. The Magnetic Field: Vector Potential 347
9.2.5. The Electric Vector Potential 352
9.3. The Variational Method 353
Contents xv
9.3.1. The Variational Formulation 354
9.3.2. Functional Involving Scalar Potentials 355
9.3.3. The Vector Potential Functional 359
9.4. The Finite Element Method 362
9.5. Application of Finite Elements with the Variational Method . . . 366
9.5.1. Application to the Electrostatic Field 367
9.5.2. Application to the Case of Stationary Currents 370
9.5.3. Application to the Magnetic Field: Scalar Potential .... 370
9.5.4. Application to the Magnetic Field: Vector Potential .... 371
9.5.5. Application to the Electric Vector Potential 373
9.6. Assembly of the Matrix System 373
9.7. Axi-Symmetric Applications 375
9.8. Nonlinear Applications 383
9.8.1. Method of Successive Approximation 383
9.8.2. The Newton-Raphson Method 384
9.9. The Three-Dimensional Scalar Potential 387
9.9.1. The First-Order Tetrahedral Element 388
9.9.2. Application of the Variational Method 389
9.9.3. Modeling of 3D Permanent Magnets 389
9.10. Examples 390
9.10.1. Calculation of Electrostatic Fields 391
9.10.2. Calculation of Static Currents 392
9.10.3. Calculation of the Magnetic Field: Scalar Potential ... 394
9.10.4. Calculation of the Magnetic Field: Vector Potential ... 396
9.10.5. Three-Dimensional Calculation of Fields of Permanent
Magnets 398
10. Galerkin's Residual Method: Applications to
Dynamic Fields
10.1. Introduction 400
10.2. Application to Magnetic Fields in Anisotropic Media 401
10.3. Application to 2D Eddy Current Problems 405
10.3.1. First-Order Element in Local Coordinates 405
10.3.2. The Vector Potential Equation Using Time
Discretization 409
10.3.3. The Complex Vector Potential Equation 417
10.3.4. Structures with Moving Parts 420
10.3.5. The Axi-Symmetric Formulation 422
10.3.6. A Modified Complex Vector Potential Formulation for
Wave Propagation 425
10.3.7. Formulation of Helmholtz's Equation 427
10.3.8. Advantages and Limitations of 2D Formulations 430
10.4. Application of the Newton-Raphson Method 432
10.5. Examples 434
10.5.1. Eddy Currents: Time Discretization 434
10.5.2. Moving Conducting Piece in Front of an
Electromagnet 437
10.5.3. Modes and Fields in a Waveguide 440
10.5.4. Resonant Frequencies of a Microwave Cavity 442
11. Hexahedral Edge Elements - Some 3D
Applications
11.1. Introduction 445
11.2. The Hexahedral Edge Element Shape Functions 448
11.3. Construction of the Shape Functions 456
11.4. Application of Edge Elements to Low-Frequency
Maxwell's Equations 460
11.4.1. Static Cases 461
11.4.2. Listing of the Matrix Construction Code 464
11.4.3. Modeling of Permanent Magnets 466
11.4.4. Eddy Currents - the Time-Stepping Procedure 466
11.4.5. Eddy Currents - The Complex Formulation 468
11.4.6. The Newton-Raphson Method 469
11.4.7. The Divergence of J and Other Particulars 471
11.5. Modeling of Waveguides and Cavity Resonators 472
11.6. Examples 473
11.6.1. Static Calculations (TEAM Problem 13) 474
11.6.2. A Linear Motor with Permanent Magnets 475
11.6.3. Eddy Current Calculations (TEAM Problem 21) 477
11.6.4. Calculation of Resonant Frequencies
(TEAMProblem 19) 480
12. Computational Aspects in Finite Element
Software Implementation
12.1. Introduction 484
12.2. Geometric Repetition of Domains 484
12.2.1. Periodicity 484
12.2.2. Anti-Periodicity 486
12.3. Storage of the Coefficient Matrix 487
12.3.1. Symmetry of the Coefficient Matrix 487
12.3.2. The Banded Matrix and Its Storage 487
12.3.3. Compact Storage of the Matrix 489
12.4. Insertion of Dirichlet Boundary Conditions 490
12.5. Quadrilateral and Hexahedral Elements 491
12.6. Methods of Solution of the Linear System 493
12.6.1. Direct Methods 493
12.6.2. Iterative Methods 497
12.7. Methods of Solution for Eigenvalues and Eigenvectors 500
12.7.1. The Jacobi Transformation 500
12.7.2. The Givens Transformation 503
12.7.3. The QR and QZ Methods 504
12.8. Diagram of a Finite Element Program 506
13. General Organization of Field Computation
Software
13.1. Introduction 509
13.2. The Pre-Processor Module 510
13.2.1. The User/System Dialogue 510
13.2.2. Domain Discretization 511
13.3. The Processor Module 516
13.4. The Post-Processor Module 517
13.4.1. Visualization of Results 517
13.4.2. Calculation of Numerical Results 519
13.5. The Computational Organization of a Software Package .... 524
13.5.1. The EFCAD Software 525
13.6. Evolving Software 528
13.6.1. The Adaptive Mesh Method 528
13.6.2. A Coupled Thermal/Electrical System 533
13.6.3. A Software Package for Electrical Machines 537
13.6.4 A System for Simultaneous Solution of Field Equations
and External Circuits 541
13.6.5. Computational Difficulties and Extensions to Field
Computation Packages 547
13.7. Recent Trends 547
Bibliography 549
Subject Index 559

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