Electromagnetics and Calculation of Fields

Intended for undergraduate 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 equations - which for most practical calses cannot be solved analytically.

I. The Electromagnetic Field and 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 the Gradient, 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 of Operators to More than One Function.- 1.9. Expressions in Cylindrical and Spherical Coordinates.- 2. The Electromagnetic Field and 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 the Equations.- 2.2.3. The Equations in Vacuum.- 2.2.4. The Equations in Media with ?=?0and ?=?0.- 2.2.5. The Equations in General Media.- 2.2.6. The Integral Form of Maxwell’s Equations.- 2.3. Approximations to Maxwell’s Equations.- 2.4. Units.- 3. Electrostatic Fields.- 3.1. Introduction.- 3.2. The Electrostatic Charge.- 3.2.1. The Electric Field.- 3.2.2. Force on an Electric Charge.- 3.2.3. The Electric Scalar Potential V.- 3.3. Nonconservative Fields: Electromotive Force.- 3.4. Refraction of the Electric Field.- 3.5. Dielectric Strength.- 3.6. The Capacitor.- 3.6.1. Definition of Capacitance.- 3.6.2. Energy Stored in a Capacitor.- 3.6.3. Energy in a Static, Conservative Field.- 3.7. Laplace’s and Poisson’s Equations 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 Introduction to 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 Equations in 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-Savart Law.- 4.4. Boundary Conditions for 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 between Magnetic and Electric Circuits.- 4.7. Inductance and Mutual Inductance.- 4.7.1. Definition of Inductance.- 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 Equation in Terms of the Magnetic Scalar Potential.- 4.10. Properties of Soft Magnetic Materials.- 5. Magnetodynamic Fields.- 5.1. Introduction.- 5.2. Maxwell’s Equations for the Magnetodynamic Field.- 5.3. Penetration of Time 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 Losses in 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. Visualization of Penetration of Fields as a Function of Frequency.- 5.6.6. The Voltage Transformer.- 6. Interaction between Electromagnetic and Mechanical Forces.- 6.1. Introduction.- 6.2. Force on a Conductor.- 6.3. Force on Moving Charges: The Lorentz Force.- 6.4. Energy in the Magnetic Field.- 6.5. Force as Variation of Energy (Virtual Work).- 6.6. The Poynting Vector.- 6.7. Maxwell’s Force Tensor.- 6.8. Examples.- 6.8.1. Force between Two 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 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 Propagation and High-Frequency Electromagnetic Fields.- 7.1. Introduction.- 7.2. The Wave Equation and 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 of Waves in Materials.- 7.3 1. 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 of Plane 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 a Cavity.- 7.7.3. Energy in a Cavity.- 7.7.4. Quality Factor of a Cavity Resonator.- 7.7.5. Coupling to Cavities.- II. Introduction to the Finite Element Method in Electromagnetics.- 8. Introduction to the Finite Element 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 Variational Method — Basic Concepts.- 8.5. The Variational Method — Extension to 2D.- 8.5.1. The Variational Formulation.- 8.5.2. Calculation of the 2D Elemental Matrix.- 8.6. Generalization of 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 of Finite Elements.- 8.6.5. Polynomial Basis.- 8.6.6. Transformation of Quantities — the Jacobian.- 8.6.7. Evaluation of the Integrals.- 8.7. Numerical Integration.- 8.7.1. Evaluation of the Integrals.- 8.7.2. Basic Principles of Numerical Integration.- 8.7.3. Accuracy and Errors in Numerical Integration.- 8.8. Some Specific Finite Elements.- 8.8.1. 1D Elements.- 8.8.2. 2D Elements.- 8.8.3. 3D Elements.- 8.9. Coupling Different Finite Elements; Infinite Elements.- 8.9.1. Coupling Different Types of Finite Elements.- 8.9.2. Infinite Elements.- 8.10. Calculation of Some 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 Simplified 2D 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 Element Method: Some Static Applications.- 9.1. Introduction.- 9.2. Some Static Applications.- 9.2.1. Electrostatic Fields: 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 Variational Method.- 9.3.1. The Variational Formulation.- 9.3.2. Functionals Involving Scalar Potentials.- 9.3.3. The Vector Potential Functionals.- 9.4. The Finite Element Method.- 9.5. Application of Finite 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 the Matrix System.- 9.7. Axi-Symmetric Applications.- 9.8. Nonlinear Applications.- 9.8.1. Method of Successive Approximation.- 9.8.2. The Newton-Raphson Method.- 9.9. The Three-Dimensional Scalar 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. Galerkints Residual Method: Applications to Dynamic Fields.- 10.1. Introduction.- 10.2. Application to Magnetic Fields in Anisotropic Media.- 10.3. Application to 2D 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 of the 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 Hexahedral Edge Element Shape Functions.- 11.3. Construction of the Shape Functions.- 11.4. Application of Edge 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 Waveguides and 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 Current Calculations (TEAM Problem 21).- 11.6.4. Calculation of Resonant Frequencies (TEAM Problem 19).- 12. Computational Aspects in Finite Element Software Implementation.- 12.1. Introduction.- 12.2. Geometric Repetition of Domains.- 12.2.1. Periodicity.- 12.2.2. Anti-Periodicity.- 12.3. Storage of the Coefficient 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 of Dirichlet Boundary Conditions.- 12.5. Quadrilateral and Hexahedral Elements.- 12.6. Methods of Solution of the Linear System.- 12.6.1. Direct Methods.- 12.6.2. Iterative Methods.- 12.7. Methods of Solution for 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 a Finite Element Program.- 13. General Organization of Field Computation Software.- 13.1. Introduction.- 13.2. The Pre-Processor Module.- 13.2.1. The User/System Dialogue.- 13.2.2. Domain Discretization.- 13.3. The Processor Module.- 13.4. The Post-Processor Module.- 13.4.1. Visualization of Results.- 13.4.2. Calculation of Numerical Results.- 13.5. The Computational Organization of a Software Package.- 13.5.1. The EFCAD Software.- 13.6. Evolving Software.- 13.6.1. The Adaptive Mesh Method.- 13.6.2. A Coupled Thermal/Electrical System.- 13.6.3. A Software Package for Electrical Machines.- 13.6.4 A 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.