electromagnetics and calculation of fields 2nd ed.
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electromagnetics and calculation of fields 2nd ed. - back
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Electromagnetics and Calculation of Fields

    Intended for advanced undergraduate or first-year graduate students of electrical engineering,  this introduction  to electromagnetic fields emphasizes the computation of fields as well as the development of theoretical  relations.  The first part thus presents the electromagnetic field and Maxwell's equations with a view toward  connecting the disparate applications to the underlying   relations, while the second  part presents computational  methods of solving the equation - which for most practical cases cannot be solvedanalytically.
After a chapter on the basis of  vector calculus, the discussion  begins with the electromagnetic field and Maxwell's equations;  subsequent chapters treat the special cases of electrostatic and magnetostatic phenomena and introduce electrodynamics and induction.
    The subsequent discussion of wave propagation and high-frequency
fields emphasizes such practical matters as propagation in lossy dielectrics, waveguides, and resonators.
    The second part begins with a discussion of the finite-element method as a general design tool. Variational  techniques are introduced and used to formulate finite-element solutions for static applications.
In subsequent chapters the finite-element method is adapted for solving dynamic problems using Galerkin's residual method, and hexahedral edge elements are introduced.
    The text concludes with discussions of  software issues associated with finite-element techniques for field applications, solution methods and program implementation.

Second Edition
Spring Verlag, New York, August 1996, 570 pages
ISBN 0-387-94877-5

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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 ofthe Gradient, Divergence, and Curl
   1.4. The Gradient 
      1.4.1.Example of Gradient
   1.5. The Divergence 
      1.5.1.Definition ofFlux 
      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 than One 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
      2.2.2.Point Form ofthe Equations 
      2.2.3.The Equationsin Vacuum 
      2.2.4.The Equationsin Media with ?=?0 and ?=?0 
      2.2.5.The Equationsin General Media 
      2.2.6.The Integral Form of 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: Electromotive Force
   3.4. Refraction ofthe Electric Field
   3.5. Dielectric Strength 
   3.6. The Capacitor
      3.6.1.Definition ofCapacitance
      3.6.2.Energy Storedin a Capacitor 
      3.6.3.Energy in a Static, Conservative Field 
   3.7. Laplace's andPoisson's Equationsin 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 SphericalCapacitor
      3.8.4.The SphericalCapacitor with Two Dielectric Layers
   3.9. A Brief Introductionto the Finite Element Method: Solution of
          the Two-Dimensional Laplace 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,Dielectric Strength, 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
      4.5.4.Permanent Magnets 
   4.6. The Analogy betweenMagnetic and Electric Circuits 
   4.7. Inductance andMutual Inductance 
      4.7.1.Definition ofInductance
      4.7.2.Energy in a Linear System 
      4.7.3.The Energy Stored in 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 Heating of a Conducting Block

      5.6.4.Effect of Movement of a Magnet Relative to a Flat
      5.6.5.Visualizationof 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 (Virtual Work) 
   6.6. The Poynting Vector 
   6.7. Maxwell's ForceTensor 
   6.8. Examples
      6.8.1.Force betweenTwo Conducting 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 ofa Ferromagnetic Body
      6.8.6.Repulsion of a Diamagnetic Body 
      6.8.7.Magnetic Levitation
      6.8.8.The Magnetic Brake

7. Wave Propagation and High-FrequencyElectromagnetic Fields
   7.1. Introduction 
   7.2. The Wave Equationand Its Solution 
      7.2.1.The Time Dependent Equations 
      7.2.2.The Time Harmonic Wave 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 Incidence on a Conducting Interface: 
                Perpendicular Polarization 
      7.5.6.Oblique Incidence on a Conducting Interface:
                Parallel Polarization
      7.5.7.Oblique Incidence on a Dielectric Interface:
                Perpendicular  Polarization 
      7.5.8.Oblique Incidence on 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 aCavity 
      7.7.3.Energy in a Cavity
      7.7.4.Quality Factor of a Cavity Resonator
      7.7.5.Coupling to Cavities

Part II. Introductionto the FiniteElement Method in Electromagnetics

8. Introductionto the FiniteElement Method
   8.1. Introduction 
   8.2. The Galerkin Method- Basic Concepts 
   8.3. The Galerkin Method- Extension to 2D 
      8.3.1.The Boundary Conditions
      8.3.2.Calculation of the 2D Elemental Matrix 
   8.4. The VariationalMethod - Basic Concepts 
   8.5. The VariationalMethod - Extension to 2D 
      8.5.1.The Variational Formulation 
      8.5.2.Calculation of the 2D Elemental Matrix 
   8.6. Generalizationof the Finite Element Method 
      8.6.1.High-Order Finite Elements: General 
      8.6.2.High-Order Finite Elements: Notation 
      8.6.3.High-Order Finite Elements: Implementation 
      8.6.4.Continuity ofFinite Elements 
      8.6.5.Polynomial Basis 
      8.6.6.Transformation of Quantities - the Jacobian 
      8.6.7.Evaluation ofthe Integrals
   8.7. Numerical Integration 
      8.7.1.Evaluation ofthe Integrals
      8.7.2.Basic Principles of Numerical Integration 
      8.7.3.Accuracy and Errors in Numerical Integration 
   8.8. Some SpecificFinite Elements Elements Elements Elements 
   8.9. Coupling DifferentFinite Elements; Infinite Elements
      8.9.1.Coupling Different Types of Finite Elements 
      8.9.2.Infinite Elements
   8.10. Calculation ofSome Terms in Poisson's Equation
      8.10.1.The Stiffness Matrix
      8.10.2.Evaluation of the Second Term in Eq. (8.130)
      8.10.3.Evaluation of the Third Term in Eq. (8.130) 
      8.10.4.Evaluation of the Source Term
   8.11. A Simplified2D Second-Order Finite Element Program 
      8.11.1.The Problem to Be Solved
      8.11.2.The Discretized Domain
      8.11.3.The Finite Element Program 

9. The Variational Finite ElementMethod: Some Static Applications
   9.1. Introduction
   9.2. Some Static Applications 
      9.2.1.ElectrostaticFields: Dielectric Materials
      9.2.2.Stationary Currents: Conducting Materials 
      9.2.3.Magnetic Fields: Scalar Potential 
      9.2.4.The Magnetic Field: Vector Potential
      9.2.5.The Electric Vector Potential
   9.3. The VariationalMethod 
      9.3.1.The Variational Formulation 
      9.3.2.Functionals Involving Scalar Potentials
      9.3.3.The Vector Potential Functionals 
   9.4. The Finite ElementMethod 
   9.5. Application ofFinite Elements with the Variational Method 
      9.5.1.Application to the  Electrostatic Field 
      9.5.2.Application to the Case of Stationary Currents 
      9.5.3.Application to the Magnetic Field: Scalar Potential 
      9.5.4.Application to the Magnetic Field: Vector Potential 
      9.5.5.Application to the Electric Vector Potential 
   9.6. Assembly of theMatrix System 
   9.7. Axi-SymmetricApplications
   9.8. Nonlinear Applications 
      9.8.1.Method of Successive Approximation 
      9.8.2.The Newton-Raphson Method 
   9.9. The Three-DimensionalScalar Potential 
      9.9.1.The First-Order Tetrahedral Element
      9.9.2.Application of the Variational Method
      9.9.3.Modeling of 3D Permanent Magnets 
   9.10. Examples
      9.10.1.Calculation of Electrostatic Fields
      9.10.2.Calculation of Static Currents 
      9.10.3.Calculation of the Magnetic Field: Scalar Potential 
      9.10.4.Calculation of the Magnetic Field: Vector Potential 
      9.10.5.Three-Dimensional Calculation of Fields of Permanent

10. Galerkin's Residual Method:Applications to  Dynamic Fields
   10.1. Introduction
   10.2. Application toMagnetic Fields in Anisotropic Media
   10.3. Application to2D Eddy Current Problems 
      10.3.1.First-Order Element in Local Coordinates
      10.3.2.The Vector Potential Equation Using Time Discretization
      10.3.3.The Complex Vector Potential Equation
      10.3.4.Structures with Moving Parts
      10.3.5.The Axi-Symmetric Formulation 
      10.3.6.A Modified Complex Vector Potential Formulation for
                 Wave Propagation 
      10.3.7.Formulation of Helmholtz's Equation 
      10.3.8.Advantages and Limitations of 2D Formulations
   10.4. Application ofthe Newton-Raphson Method 
   10.5. Examples
      10.5.1.Eddy Currents: Time Discretization
      10.5.2.Moving Conducting Piece in Front of an Electromagnet
      10.5.3.Modes and Fields in a Waveguide 
      10.5.4.Resonant Frequencies of a Microwave Cavity

11. Hexahedral Edge Elements -Some 3D Applications
   11.1. Introduction
   11.2. The HexahedralEdge Element Shape Functions
   11.3. Constructionof the Shape Functions
   11.4. Application ofEdge Elements to Low-Frequency
            Maxwell's Equations
      11.4.1.Static Cases 
      11.4.2.Listing of the Matrix Construction Code 
      11.4.3.Modeling of Permanent Magnets 
      11.4.4.Eddy Currents ? the Time-Stepping Procedure
      11.4.5.Eddy Currents ? The Complex Formulation
      11.4.6.The Newton-Raphson Method 
      11.4.7.The Divergence of J and Other Particulars
   11.5. Modeling of Waveguidesand Cavity Resonators
   11.6. Examples
      11.6.1.Static Calculations (TEAM Problem 13)
      11.6.2.A Linear Motor with Permanent Magnets 
      11.6.3.Eddy CurrentCalculations (TEAM Problem 21)
      11.6.4.Calculation of Resonant Frequencies  (TEAM Problem 19)

12. Computational Aspects in FiniteElement Software Implementation
   12.1. Introduction
   12.2. Geometric Repetitionof Domains
   12.3. Storage of theCoefficient Matrix
      12.3.1.Symmetry of the Coefficient Matrix
      12.3.2.The Banded Matrix and Its Storage
      12.3.3.Compact Storage of the Matrix
   12.4. Insertion ofDirichlet Boundary Conditions
   12.5. Quadrilateraland Hexahedral Elements
   12.6. Methods of Solutionof the Linear System 
      12.6.1.Direct Methods 
      12.6.2.Iterative Methods
   12.7. Methods of Solutionfor Eigenvalues and Eigenvectors 
      12.7.1.The Jacobi Transformation
      12.7.2.The Givens Transformation
      12.7.3.The QR and QZ Methods
   12.8. Diagram of aFinite Element Program 

13. General Organization of FieldComputation  Software
   13.1. Introduction
   13.2. The Pre-ProcessorModule
      13.2.1.The User/System Dialogue 
      13.2.2.Domain Discretization 
   13.3. The ProcessorModule 
   13.4. The Post-ProcessorModule 
      13.4.1.Visualization of Results 
      13.4.2.Calculation of Numerical Results 
   13.5. The ComputationalOrganization of a Software Package 
      13.5.1.The EFCAD Software
   13.6. Evolving Software 
      13.6.1.The AdaptiveMesh Method
      13.6.2.A Coupled Thermal/Electrical System
      13.6.3.A Software Package for Electrical Machines 
      13.6.4A System for Simultaneous Solution of Field Equations
                 and External Circuits
      13.6.5.Computational Difficulties and Extensions to Field
                  Computation Packages
   13.7. Recent Trends


Subject Index


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