7/28/2019 24388173

1/9

Nathan Ida Joao P.A. Bastos

Electromagnetics andCalculation of FieldsWith 300 Illustrations

Springer-VerlagNew York Berlin H eidelberg London ParisTokyo Hong Kong Barcelona Budapest

7/28/2019 24388173

2/9

Contents

PrefacePart I. The Electromagnetic Field and M axwell'sEquations1. Mathematical Preliminaries

1.1. Introduc tion 11.2. The Vector Notation 11.3. Vec tor Derivation 2

1.3.1. The Nabla (V) Operator 21.3.2. Definition of the Grad ient, Divergence and Curl 2

1.4 The Gradient 31.4.1. Example of Gradient 5

1.5. The Divergence 61.5.1. Definition of Flux 61.5.2. The Divergence Theorem 81.5.3. Conservative Flux 91.5.4. Exam ple of Divergence 11

1.6. Th e Curl 121.6.1. Circulation of a Vector 121.6.2. Stokes' Theorem 141.6.3. Exam ple of Curl 17

1.7. Second Order Op erators 181.8. Application of Operators to M ore than One Function 191.9. Expressions in Cylindrical and Spherical Coordinates 20

2. The Electromagnetic Field and Maxwell'sEquations2.1. Introduction 222.2. M axwell's Equations 22

2.2.1. Fundam ental Physical Principles of the ElectromagneticField 23

ix

7/28/2019 24388173

3/9

x Contents2.2.2. Point Form of the Equations 292.2.3. The Equations in Vacuum 322.2.4. Th e Equ ations in M edia with e=eo and |i.=(io 342.2.5. Th e Equations in General M edia 352.2.6. The Integral Form of M axwell's Equations 37

2.3. Approximations to Maxw ell's Equations 432.4. Units 45

Electrostatic Fields3.1. Introduction 473.2. The Electrostatic Charge 473.2.1. The Electric Field 48

3.2.2. Force on an Electric Charge 483.2.3. The Electric Scalar Potential V 49

3.3. Nonconservative Fields: Electromotive Force 533.4. Refraction of the Elec tric Field 553.5. Dielectric Strength 593.6. The Capacitor 61

3.6.1. Definition of Capacitance 613.6.2. Energy Stored in a Capacitor 643.6.3. Energy in a Static, Con servative Field 64

3.7. Laplace's and Poisson 's Equa tions in Term s of the Electric Field 653.8. Examples 67

3.8.1. The Infinite Charged Line 673.8.2. The Charged Spherical Half-Shell 703.8.3. The Spherical Capacitor 713.8.4. The Spherical Capacitor with Tw o Dielectric Layers . . . 72

3.9. A Brief Introduction to the Finite Elem ent M ethod: So lution of theTwo-Dimensional Laplace's Equation 743.9.1. The Finite Elemen t Technique for Division of a Dom ain 753.9.2. The Variational Method 773.9.3. A Finite Element Program 803.9.4. Exam ple for Use of the Finite Element Program 84

3.10. Tables of Perm ittivities, Dielectric Strength and Con ductivities 88

4. Magnetostatic Fields4.1. Introduction 904.2. Maxw ell's Equations in M agnetostatics 91

4.2.1. The Equation Vx H= J 914.2.2. The Equation V B =0 934.2.3. The Equation Vx E= 0 93

7/28/2019 24388173

4/9

Contents xi4.3. The Biot-Savart Law 944.4. Boun dary Conditions for the Magn etic Field 964.5. M agnetic Materials 98

4.5.1. Diam agnetic M aterials 994.5.2. Param agnetic M aterials 1004.5.3. Ferromagn etic M aterials 1004.5.4. Permanen t M agnets 104

4.6. The Analogy Between M agnetic and Electric Circuits 1154.7. Inductance and Mu tual Inductance 119

4.7.1. Definition of Induc tance 1194.7.2. Energy in a Linear System 1204.7.3. The Energy Stored in the M agnetic Field 1224.8. Examples 1234.8.1. Calculation of Field Intensity and Inductance of a Long

Solenoid 1234.8.2. Calculation of H for a Circular Loo p 1254.8.3. Field of a Rectangular Loop 1274.8.4. Calculation of Inductance of a Coaxial Cable 1284.8.5. Calculation of the Field Inside a Cylindrical Conductor . . 1294.8.6. Calculation of the Magnetic Field Intensity in a Magnetic

Circuit 1304.8.7. Calculation of the Magnetic Field Intensity of a SaturatedM agnetic Circuit 133

4.8.8. Magnetic Circuit Incorporating Permanent Magnets . . . . 1354.9. Laplace's Equation in Term s of the M agnetic Scalar Poten tial . . 1384.10. Properties of Soft M agnetic M aterials 140

5. Magnetodynamic Fields5.1. Introduction 1425.2. Max well's Equations for the M agnetodynam ic Field 1435.3. Penetration of Time Depen dent Fields in Cond ucting Materials 146

5.3.1. The Equation for H 1465.3.2. The Equation for B 1475.3.3. The Equation for E 1475.3.4. The Equa tion for J 1485.3.5. Solution of the Equations 148

5.4. Eddy Curren t Losses in Plates 1535.5. Hysteresis Losses 1565.6. Exam ples 160

5.6.1. Induced Currents Due to Chang e in Induction 1605.6.2. Induced Currents Due to Changes in Geom etry 1635.6.3. Som e Further Exam ples of Skin Depth 165

7/28/2019 24388173

5/9

xii Contents5.6.4. Effect of Movem ent of a Magnet Relative to a Flat

Conductor 1695.6.5. Visualization of Penetration of Fields as a Function of

Frequency 1715.6.6. The Voltage Transformer 172

6. Interaction Between Electrom agnetic andMechanical Forces6.1. Introduction 1756.2. Force on a Conductor 1756.3. Force on Moving Charges: The Lorentz Force 1786.4. Energy in the M agnetic Field 1806.5. Force as Variation of Energy (Virtual W ork) 1826.6. The Poyn ting Vector 1846.7. Maxwell's Force Tenso r 1886.8. Exam ples 195

6.8.1. Force Between Tw o Conducting Segments 1956.8.2. Torque on a Loop 1986.8.3. The Hall Effect 2006.8.4. The Linear Motor and Generator 2026.8.5. Attraction of a Ferromagnetic Body 2056.8.6. Repulsion of a Diam agnetic Body 2066.8.7. Magnetic Levitation 2076.8.8. The Magn etic Brake 209

7. Wave Propagation and High FrequencyElectromagnetic Fields7.1. Introduction 2127.2. The W ave Equation and Its Solution 215

7.2.1. The Time Dependent Equations 2157.2.2. The Time Harm onic Wave Equations 2207.2.3. Solution of the Wave Equation 2227.2.4. Solution for Plane W aves 2227.2.5. The One-Dimensional W ave Equation in Free Space and

Lossless Dielectrics 2237.3. Propagation of Waves in Ma terials 2277.3.1. Propagation of W aves in Lossy Dielectrics 2277.3.2. Propagation of Plane Waves in Low Loss Dielectrics . . . 2297.3.3. Propagation of Plane Waves in Cond uctors 2307.3.4. Propagation in a Conductor: Definition of the Skin Dep th 232

7.4. Polarization of Plane W aves 233

7/28/2019 24388173

6/9

Contents xiii7.5. Reflection, Refraction and Transmission of Plane W aves . . . . 235

7.5.1. Reflection and Transmission at a Lossy Dielectric Interface:Norm al Incidence 236

7.5.2. Reflection and Transmission at a Conductor Interface:Norm al Incidence 2397.5.3. Reflection and Transmission at a Finite Conductivity

Con ductor Interface 2407.5.4. Reflection and Transmission at an Interface:

Oblique Incidence 2417.5.5. Ob lique Incidence on a Condu cting Interface:

Perpendicular Polarization 2427.5.6. Oblique Incidence on a Conducting Interface:

Parallel Polarization 2447.5.7. Oblique Incidence on a Dielectric Interface:

Perpendicular Polarization 2457.5.8. Oblique Incidence on a Dielectric Interface:

Parallel Polarization 2487.6. W aveguides 249

7.6.1. TEM , TE and TM Waves 2497.6.2. TEM Waves 2517.6.3. TE Waves 2517.6.4. TM W aves 2527.6.5. Rectangular W aveguides 2537.6.6. TM Modes in W aveguides 2537.6.7. TE Modes in W aveguides 256

7.7. Cavity Resonators 2587.7.1. TM and TE Modes in Cavity Resonators 2597.7.2. TE Modes in a Cavity 2617.7.3. Energy in a Cavity 2617.7.4. Quality Factor of a Cavity Resonator 2637.7.5. Co upling to Cavities 263

Part II. Introduction to the Finite Element Method inElectromagnetics8. The Variational Finite Element Method: SomeStatic Applications

8.1. Introduction 2658.2. Some Static Ap plications 266

8.2.1. Electrostatic Field s: Diam agnetic Materials 2668.2.2. Stationary Cu rrents: Conduc ting M aterials 2678.2.3. M agnetic M aterials: Scalar Potential 269

7/28/2019 24388173

7/9

xiv Contents8.2.4. The Magnet ic Fie ld : Vector Po ten t ia l 2708.2.5 . The Electr ic Vector Potential 275

8.3. The Var ia t iona l Method 2768.3.1. The Var ia t iona l Formula t ion 2778 .3 .2 . F un ct io na l Invo lv ing Sca lar Po ten t ia ls 2798.3.3. The Vector Po ten t ia l Funct iona ls 283

8.4. The Fin i t e E lemen t Me th o d 2878 .5 . Appl ica t ion of Fin i te Elements wi th the Var ia t io n a l Me th o d . . . 291

8.5 .1 . Appl ica t ion to the Elec t ros ta t ic Fie ld 292U.5 .2 . Appl ica t ion to the C a s e of Sta t ionary Curren ts 2958.5.3 . Applicat ion to the Magnet ic Fie ld : Sca lar Po ten t ia l . . . . 2968.5.4 . Applicat ion to the Magnet ic Fie ld : Vector Po ten t ia l . . . . 2968.5.5 . Applicat ion to the Elec t r ic Vector Po ten t ia l 298

8 .6 . Assembly of the Matr ix System 2998 .7 . Ax i -Sy m met r i c Ap p l i ca t io n s 3018 .8 . No nl inear App l ica t ions 308

8.8.1. M e t h o d of Su ccess iv e Ap p ro x ima t io n 3098.8.2. The N e w t o n - R a p h s o n M e t h o d 309

8.9. The Three-Dimensional Sca lar Po ten t ia l 3138.9 .1 . The Firs t Order Tet rahedra l E lement 3148.9.2 . Applicat ion of the Var ia t iona l Method 3158 .9 .3 . Model ing of 3 D Pe rman en t Mag n e t s 315

8 .10 . Examples 3168.10 .1 . Calcu la t ion of Elec t ros ta t ic F ie lds 3178.10.2. Calculat ion of Sta t ic Cu rren ts 3188.10.3. Calculat ion of the Magnet ic Fie ld : Sca lar Po ten t ia l . . . 3208.10.4. Calculat ion of the Magnet ic Fie ld : Vector Po ten t ia l . . . 3228 .10 .5 . Three -Dim ensiona l Calcu la t ion of Fie lds of

Perman en t Mag n e t s 324

9. Galerkin's Residual M ethod: Applications toDynamic Fields9.1. Introduction 3269.2. Application to Magnetic Fields in Anisotropic Media 3279.3. Application to 2D Eddy Current Problems 332

9.3.1. First Order Element in Local Coordinates 3329.3.2. The Vector Potential Equation Using Time Discretization 3369.3.3. The Complex Vector Potential Equation 3439.3.4. Structures with Moving Parts 3479.3.5. The Axi-Symmetric Formulation 3499.3.6. A Modified Complex Vector Potential Formulation for

Wave Propagation 3529.3.7. Formulation of Helmholtz's Equation 354

7/28/2019 24388173

8/9

Contents xv9.3.8. Advan tages and Limitations of 2D Formulations 357

9.4. Higher Order Isoparametric Finite Elements 3599.4.1. The Second Order Triangular Isoparametric Element . . . 3609.4.2. Application to the New ton-Raphson M ethod 3649.5. Tw o Three-Dim ensional Isoparametric Elements 3669.5.1. The Second Orde r Tetrahedron 3669.5.2. The Linear Hexahedron 368

9.6. Exam ples 3709.6.1. Eddy Currents: Tim e Discretization 3709.6.2. M oving Conducting Piece in Front of an Electromagnet 3739.6.3. M odes and Fields in a W aveguide 3769.6.4. Resonant Frequencies of a M icrowave Cavity 378

10. Computational Aspects in Finite ElementSoftware Implementation10.1. Introduction 38110.2. Geom etric Repetition of Dom ains 381

10.2.1. Periodicity 38110.2.2. Anti-Periodicity 383

10.3. Storage of the Coefficient Matrix 38410.3.1. Symmetry of the Coefficient Matrix 38410.3.2. The Banded M atrix and Its Storage 38410.3.3. Com pact Storage of the Matrix 386

10.4. Insertion of Dirichlet Boundary Conditions 38710.5. Quadrilateral and Hexahedral Elements 38810.6. Methods of Solution of the Linear System 390

10.6.1. Direct Methods 39010.6.2. Iterative M ethods 39410.7. M ethods of Solution for Eigenvalues and Eigenv ectors 39710.7.1. The Jacobi Transformation 39710.7.2. The Givens Transformation 40010.7.3. The QR and QZ Methods 401

10.8. Diagram of a Finite Element Program 403

11. General Organization of Field ComputationSoftware11.1. Introduction 40611.2. The Pre-Processor M odule 407

11.2.1. The User/System Dialogue 40711.2.2. Domain Discretization 408

11.3. The Processor Module 413

7/28/2019 24388173

9/9

xvi Contents11.4. The Post-Processor Mo dule 414

11.4.1. Visualization of Results 41411.4.2. Calculation of Num erical Results 416

11.5. The Computational Organization of a Software Package . . . . 42111.5.1. The EFC AD Software 42211.6. Evolving Software 425

11.6.1. The Adaptive Mesh Method 42511.6.2. A Coupled Thermal/Electrical System 43011.6.3. A Software Package for Electrical M achines 43411.6.4 A System for Simu ltaneous Solution of Field Equations

and External Circuits 43811.6.5. Computational Difficulties and Extensions to Field

Com putation Packages 44411.7. Recent Trends 444

Bibliography 446Su bject Index 454