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Electromagnetics and Calculation of FieldsChinese edition This reprint: 565
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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
Field
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
4.5.3.FerromagneticMaterials
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
Solenoid
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
Circuit
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
Conductor
5.6.5.Visualizationof
Penetration of Fields as a Function of
Frequency
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
8.8.1.1D Elements
8.8.2.2D Elements
8.8.3.3D 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
Magnets
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
12.2.1.Periodicity
12.2.2.Anti-Periodicity
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
Bibliography
Subject Index