electromagnetics and calculation of fields -1st ed.
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Electromagnetics and calculation of fields - back
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Electromagnetics and Calculation of Fields

   This introduction to electromagnetics emphasizes the computation of electromagnetic fields and the development of  theoretical relations.
   Beginning  with the idea that  Maxwell's  equations are  primary,
the author avoids the lengthy of electrostatics and magnetostatics that are customary in texts on electromagnetism. Thus,after a chapter on the basics of vector calculus,  the discussion  begins with  the electromagnetic field and Maxwell's equations; the  two following chapters then present the special cases of electrostatic and magnetostatic  phenomena.  Dynamics is introduced in chapter 5, and electromagnetic induction in chapter 6. The  discussion of wave propagation and high-frequency fields in chapter 7 emphasizes such practicalmatters as propagation  inlossy dielectrics, waveguides, and  resonators. The remaining four chapters discuss computational  techniques.  mulate finite element solutions for static applications. In chapter 9, the  finite element  method is adapted for solution of dynamic  problems using Galerkin's residual method. Chapters 10  and11  discuss software  issues  associated  with  finite element techniques for field applications, solution methods, and program implementation.

Springer Verlag, New York, August 1992, 458 pages

ISBN 0-387-97852-6, 0-540-97852-6

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Part I. The Electromagnetic Fieldand Maxwell's Equations

1.  Mathematical Preliminaries
   1.1. Introduction 
   1.2. The Vector Notation
   1.3. Vector Derivation
      1.3.1.The Nabla Operator
      1.3.2.Definition of theGradient, Divergence and Curl
   1.4 The Gradient
      1.4.1.Example of Gradient 
   1.5. The Divergence 
      1.5.1.Definition of Flux 
      1.5.2.The Divergence Theorem
      1.5.3.Conservative Flux 
      1.5.4.Example of Divergence 
   1.6. The Curl 
      1.6.1.Circulation of a Vector
      1.6.2.Stokes' Theorem 
      1.6.3.Example of Curl
   1.7. Second Order Operators 
   1.8. Application ofOperators to More thanOne Function 
   1.9. Expressions inCylindrical and Spherical Coordinates 

2.  The Electromagnetic Fieldand Maxwell's Equations
   2.1. Introduction
   2.2. Maxwell's Equations
      2.2.1.Fundamental Physical Principles of the Electromagnetic Field
      2.2.2.Point Form of theEquations
      2.2.3.The Equations in Vacuum 
      2.2.4.The Equations in Media with ?=?0 and ?=?0 
      2.2.5.The Equations in General Media
      2.2.6.The Integral Formof Maxwell's Equations 
   2.3. Approximationsto Maxwell's Equations
   2.4. Units 

3.  Electrostatic Fields
   3.1. Introduction 
   3.2. The ElectrostaticCharge
      3.2.1.The Electric Field 
      3.2.2.Force on an Electric Charge 
      3.2.3.The Electric Scalar Potential V 
   3.3. NonconservativeFields: ElectromotiveForce 
   3.4. Refraction ofthe Electric Field 
   3.5. Dielectric Strength 
   3.6. The Capacitor 
      3.6.1.Definitionof Capacitance 
      3.6.2.Energy Stored in a Capacitor
      3.6.3.Energy in a Static, Conservative Field
   3.7 Laplace's and Poisson'sEquations in Terms of the Electric Field 
   3.8 Examples
      3.8.1.The Infinite Charged Line
      3.8.2.The Charged Spherical Half-Shell 
      3.8.3.The Spherical Capacitor 
      3.8.4.The Spherical Capacitor with Two Dielectric Layers
   3.9 A Brief Introductionto the Finite Element Method: Solution of the
        Two-Dimensional Laplace's Equation 
      3.9.1.The Finite Element Technique for Division of a Domain 
      3.9.2.The Variational Method
      3.9.3.A Finite Element Program 
      3.9.4.Example for Use of the Finite Element Program 
   3.10. Tables of Permittivities,DielectricStrength and Conductivities

4. Magnetostatic Fields
   4.1. Introduction 
   4.2. Maxwell's Equationsin Magnetostatics 
      4.2.1.The Equation ??H=J
      4.2.2.The Equation ?.B=0
      4.2.3.The Equation ??E=0
   4.3. The Biot-SavartLaw 
   4.4. Boundary Conditionsfor the Magnetic Field 
   4.5. Magnetic Materials 
      4.5.1.Diamagnetic Materials 
      4.5.2.Paramagnetic Materials 
      4.5.3.Ferromagnetic Materials 
      4.5.4.Permanent Magnets
   4.6. The Analogy BetweenMagnetic and Electric Circuits
   4.7. Inductance andMutual Inductance
      4.7.1.Definition of Inductance 
      4.7.2.Energy in a Linear System
      4.7.3.The Energy Storedin the Magnetic Field 
   4.8. Examples 
      4.8.1.Calculation of Field Intensity and Inductance of a Long
      4.8.2.Calculation of H for a Circular Loop 
      4.8.3.Field of a Rectangular Loop
      4.8.4.Calculation of Inductance of a Coaxial Cable 
      4.8.5.Calculation of the Field Inside a Cylindrical Conductor
      4.8.6.Calculation of the Magnetic Field Intensity in a Magnetic
      4.8.7.Calculation of the Magnetic Field Intensity of a Saturated
                Magnetic Circuit 
      4.8.8.Magnetic Circuit Incorporating Permanent Magnets 
   4.9. Laplace's Equationin Terms of the Magnetic Scalar Potential
   4.10. Properties ofSoft Magnetic Materials 

5. Magnetodynamic Fields
   5.1. Introduction
   5.2. Maxwell's Equationsfor the Magnetodynamic Field
   5.3. Penetration ofTime Dependent Fields in Conducting Materials 
      5.3.1.The Equation for H 
      5.3.2.The Equation for B 
      5.3.3.The Equation for E 
      5.3.4.The Equation for J 
      5.3.5.Solution of the Equations 
   5.4. Eddy Current Lossesin Plates
   5.5. Hysteresis Losses
   5.6. Examples
      5.6.1.Induced Currents Due to Change in Induction 
      5.6.2.Induced Currents Due to Changes in Geometry 
      5.6.3.Inductive Heatingof a Conducting Block
      5.6.4.Effect of Movement of a Magnet Relative to a Flat
      5.6.5.Visualization of Penetration of Fields as a Function of
      5.6.6.The Voltage Transformer

6. Interaction Between Electromagneticand Mechanical Forces
   6.1. Introduction 
   6.2. Force on a Conductor 
   6.3. Force on MovingCharges: The Lorentz Force 
   6.4. Energy in theMagnetic Field 
   6.5. Force as Variationof Energy (VirtualWork) 
   6.6. The Poynting Vector
   6.7. Maxwell's ForceTensor
   6.8. Examples
      6.8.1.Force Between TwoConducting Segments 
      6.8.2.Torque on a Loop
      6.8.3.The Hall Effect
      6.8.4.The Linear Motor and Generator
      6.8.5.Attraction of a Ferromagnetic Body 
      6.8.6.Repulsion of a Diamagnetic Body
      6.8.7.Magnetic Levitation 
      6.8.8.The Magnetic Brake 

7. Wave Propagationand High Frequency Electromagnetic Fields
   7.1. Introduction
   7.2. The Wave Equationand Its Solution
      7.2.1.The Time Dependent Equations
      7.2.2.The Time HarmonicWave Equations
      7.2.3.Solution of the Wave Equation 
      7.2.4.Solution for Plane Waves
      7.2.5.The One-Dimensional Wave Equation in Free Space and 
                Lossless Dielectrics
   7.3. Propagation ofWaves in Materials 
      7.31. Propagation of Waves in Lossy Dielectrics
      7.3.2.Propagation of Plane Waves in Low Loss Dielectrics
      7.3.3.Propagation of Plane Waves in Conductors
      7.3.4.Propagation in a Conductor: Definition of the Skin Depth
   7.4. Polarization ofPlane Waves
   7.5. Reflection, Refraction,and Transmission of Plane Waves 
      7.5.1.Reflection and Transmission at a Lossy Dielectric Interface:
                Normal Incidence
      7.5.2.Reflection and Transmission at a Conductor Interface: 
                Normal Incidence
      7.5.3.Reflection and Transmission at a Finite Conductivity 
                Conductor Interface 
      7.5.4.Reflection and Transmission at an Interface: 
                Oblique Incidence 
      7.5.5.Oblique Incidenceon a Conducting Interface: 
                 Perpendicular Polarization
      7.5.6.Oblique Incidenceon a Conducting Interface: 
                 Parallel Polarization
      7.5.7.Oblique Incidenceon a Dielectric Interface: 
                 Perpendicular  Polarization
      7.5.8.Oblique Incidenceon a Dielectric Interface: 
                 Parallel Polarization
   7.6. Waveguides
      7.6.1.TEM, TE and TM Waves
      7.6.2.TEM Waves
      7.6.3.TE Waves 
      7.6.4.TM Waves 
      7.6.5.Rectangular Waveguides 
      7.6.6.TM Modes in Waveguides 
      7.6.7.TE Modes in Waveguides 
   7.7. Cavity Resonators
      7.7.1.TM and TE Modes in Cavity Resonators 
      7.7.2.TE Modes in a Cavity
      7.7.3.Energy in a Cavity 
      7.7.4.Quality Factor ofa Cavity Resonator 
      7.7.5.Coupling to Cavities 

Part II. Introduction to the FiniteElement Method in Electromagnetics

8. The Variational Finite ElementMethod: Some Static Applications
   8.1. Introduction 
   8.2. Some Static Applications
      8.2.1.Electrostatic Fields: Dielectric Materials 
      8.2.2.Stationary Currents: Conducting Materials 
      8.2.3.Magnetic Fields: Scalar Potential 
      8.2.4.The Magnetic Field: Vector Potential 
      8.2.5.The Electric Vector Potential 
   8.3. The VariationalMethod 
      8.3.1.The Variational Formulation
      8.3.2.Functionals Involving Scalar Potentials
      8.3.3.The Vector Potential Functionals
   8.4. The Finite ElementMethod 
   8.5. Application ofFinite Elements with the Variational Method
      8.5.1.Application to the  Electrostatic Field 
      8.5.2.Application to the Case of Stationary Currents
      8.5.3.Application to the Magnetic Field: Scalar Potential 
      8.5.4.Application to the Magnetic Field: Vector Potential 
      8.5.5.Application to the Electric Vector Potential
   8.6. Assembly of theMatrix System 
   8.7. Axi-SymmetricApplications 
   8.8. Nonlinear Applications
      8.8.1.Method of Successive Approximation
      8.8.2.The Newton-Raphson Method 
   8.9. The Three-DimensionalScalar Potential
      8.9.1.The First Order Tetrahedral Element 
      8.9.2.Application of the Variational Method
      8.9.3.Modeling of 3D Permanent Magnets 
   8.10. Examples
      8.10.1.Calculation of Electrostatic Fields 
      8.10.2.Calculation of Static Currents
      8.10.3.Calculation of the Magnetic Field: Scalar Potential 
      8.10.4.Calculation of the Magnetic Field: Vector Potential 
      8.10.5.Three-Dimensional Calculation of Fields of Permanent

9. Galerkin's Residual Method:Applications to Dynamic Fields
   9.1. Introduction
   9.2. Application toMagnetic Fields in Anisotropic Media 
   9.3. Application to2D Eddy Current Problems
      9.3.1.First Order Element in Local Coordinates 
      9.3.2.The Vector Potential Equation Using Time Discretization 
      9.3.3.The Complex Vector Potential Equation 
      9.3.4.Structures with Moving Parts 
      9.3.5.The Axi-SymmetricFormulation 
      9.3.6.A Modified Complex Vector Potential Formulation for 
               Wave Propagation 
      9.3.7.Formulation of Helmholtz's Equation 
      9.3.8.Advantages and Limitations of 2D Formulations
   9.4. Higher Order IsoparametricFinite Elements 
      9.4.1.The Second Order Triangular Isoparametric Element 
      9.4.2.Application to the Newton-Raphson Method 
   9.5. Two Three-DimensionalIsoparametric Elements 
      9.5.1.The Second Order Tetrahedron 
      9.5.2.The Linear Hexahedron
   9.6. Examples
      9.6.1.Eddy Currents: Time Discretization
      9.6.2.Moving ConductingPiece in Front of an Electromagnet 
      9.6.3.Modes and Fields in a Waveguide
      9.6.4.Resonant Frequencies of a Microwave Cavity

10. Computational Aspects in FiniteElement  Software Implementation
   10.1. Introduction 
   10.2. Geometric Repetitionof Domains
      10.2.1. Periodicity
      10.2.2. Anti-Periodicity
   10.3. Storage of theCoefficient Matrix
      10.3.1. Symmetry of the Coefficient Matrix
      10.3.2. The BandedMatrix and Its Storage
      10.3.3. Compact Storage of the Matrix 
   10.4. Insertion ofDirichlet Boundary Conditions
   10.5. Quadrilateraland Hexahedral Elements
   10.6. Methods of Solutionof the Linear System
      10.6.1. Direct Methods 
      10.6.2. Iterative Methods 
   10.7. Methods of Solutionfor Eigenvalues and Eigenvectors
      10.7.1. The JacobiTransformation
      10.7.2. The GivensTransformation 
      10.7.3. The QR andQZ Methods
   10.8. Diagram of aFinite Element Program

11. General Organization of FieldComputation  Software
   11.1. Introduction 
   11.2. The Pre-ProcessorModule
      11.2.1. The User/System Dialogue 
      11.2.2. Domain Discretization 
   11.3. The ProcessorModule 
   11.4. The Post-ProcessorModule 
      11.4.1. Visualization of Results 
      11.4.2. Calculation of Numerical Results 
   11.5. The ComputationalOrganization of a Software Package 
      11.5.1. The EFCAD Software
   11.6. Evolving Software 
      11.6.1. The Adaptive Mesh Method 
      11.6.2. A Coupled Thermal/Electrical System 
      11.6.3. A SoftwarePackage for Electrical Machines

      11.6.4 A System for Simultaneous Solution of Field Equations 
                  and External Circuits
      11.6.5. Computational Difficulties and Extensions to Field
                   Computation Packages
   11.7. Recent Trends 


<>Subject Index 

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