1 Electrical and Electromagnetic Applications.- 1.1 Introduction.- 1.2 Electromagnetic Theory.- 1.2.1 Maxwell's Equations.- 1.2.2 Electrokinetics.- 1.2.2.1 Hypothesis and Equations.- 1.2.3 Electrostatics.- 1.2.3.1 Hypothesis and Equations.- 1.2.4 Magnetostatics.- 1.2.4.1 Hypothesis and Equations.- 1.2.4.2 Interfaces.- 1.2.5 Magnetodynamics.- 1.2.5.1 Hypothesis and Equations.- 1.2.6 Discussion.- 1.3 BEM and Laplacian Potential (Electrokinetics, Electrostatics).- 1.3.1 Integral Equation and Discretization.- 1.3.2 Floating Potential Electrodes.- 1.3.3 Singular Points.- 1.3.4 Miscellaneous.- 1.3.4.1 Space Charge.- 1.3.4.2 Results.- 1.3.4.3 Capacitances.- 1.3.4.4 Forces.- 1.4 BEM and Magnetostatics.- 1.4.1 Homogeneous Unbounded Structures.- 1.4.2 Inhomogeneous Unbounded Structures.- 1.4.3 Miscellaneous.- 1.5 BEM and 2D-Magnetodynamics.- 1.6 Examples.- 1.6.1 3D-Analysis of a Potential Transformer.- 1.6.2 High Voltage Laboratory.- 1.6.3 X-Ray Tube.- 1.6.4 Conductors in a Ferromagnetic Slot.- 1.6.5 Rotating Arc Circuit Breaker.- 1.7 Conclusions.- References.- 2 Three-Dimensional Magnetostatic Field Analysis Vector Variables.- 2.1 Introduction.- 2.2 Basic Theory.- 2.2.1 Governing Equations.- 2.2.2 Uniqueness Conditions.- 2.3 Direct Integral Equation Formulations.- 2.4 Discretization of Boundary Surfaces and Variables.- 2.4.1 Interpolation of Normal Components.- 2.4.2 Interpolation of Tangential Components.- 2.5 Boundary Element Solution.- 2.5.1 Solution for the Flux Density.- 2.5.2 Solution for the Vector Potential.- 2.5.3 Example.- 2.6 Treatment of Kernel Singularities.- 2.7 Application to Interface Problems.- 2.7.1 Multi-region Formulation.- 2.7.2 Examples.- 2.8 Conclusion.- References.- 3 Electromagnetical Problems Taking Into Account External Power Sources.- Summary.- 3.1 Introduction.- 3.2 Boundary Element Formulations of Magnetic Field Problems.- 3.2.1 Fundamental Equation.- 3.2.2 Relationship Between Vector Potential, Currents, and Terminal Voltage.- 3.2.3 Boundary Element Formulation.- 3.2.4 Formulation of Sinusoidal Time-varing Field.- 3.3 Recent Developments in Magnetic Field Problems.- 3.3.1 Eddy Current Problems.- 3.3.1.1 Fundamental Equation.- 3.3.1.2 Boundary Element Formulation.- 3.3.1.3 Numerical Results.- 3.3.2 Non-linear Problems.- 3.3.2.2 Boundary Integral Equation Expression.- 3.3.2.3 Expressing Method of Hysteresis Curve.- 3.3.2.4 Treatment of Equivalent Magnetizing Current Density.- 3.3.2.5 Consideration of External Power Source.- 3.3.2.6 Examples of Application.- 3.3.3 Moving Sensor Problem.- 3.3.3.1 Formulation of Problem.- 3.3.3.2 Finite Element Region.- 3.3.3.3 Boundary Element Region.- 3.3.3.4 Circuit Equation.- 3.3.3.5 Application to a Magnetic Sensor.- References.- 4 Boundary Element Methods for Eddy Current Problems.- 4.1 Introduction.- 4.2 Eddy Current and Induction Problems.- 4.2.1 Basic Electromagnetic Field Equations.- 4.2.2 The Quasi-Static Approximation.- 4.2.3 Magnetic Vector and Scalar Potentials.- 4.2.4 The Impedance Boundary Condition.- 4.2.5 Fundamental Solutions.- 4.3 Basic Boundary Element Formulations for Eddy Current Problems.- 4.3.1 Direct Formulations.- 4.3.1.1 The Reduced Singularity or Mueller-type Formulation.- 4.3.1.2 Direct Use of the Boundary Integral Equations.- 4.3.2 Indirect Formulations.- 4.4 Boundary Integral Methods for 3-Dimensional Eddy Current Problems.- 4.5 A Boundary Integral Method of Minimum Order.- 4.5.1 The Basic Boundary Integral Equations.- 4.5.1.1 The H-? Formulation.- 4.5.1.2 Virtual Source Distribution.- 4.5.1.3 Boundary Integral Equations for the Virtual Sources.- 4.5.2 Multiply Connected Systems.- 4.5.3 Field at any Point in the Domain ?3.- 4.5.3.1 Magnetic Field Intensity.- 4.5.3.2 Electric Field Intensity.- 4.6 Discretization of the Boundary Integral Equations.- 4.6.1 Discretization of the Surface Integral Equations.- 4.6.2 Parametric Representation of Geometry and Functions.- 4.7 Singularity Evaluation.- 4.7.1 Integral Evaluation in the Sense of Cauchy's Principal Value.- 4.7.2 Computation of the Singular Integrals.- 4.7.3 Surface Magnetic Fields.- 4.8 Gauss Integration Methodology.- 4.8.1 Regular Sub-triangulation Scheme.- 4.8.2 Irregular Sub-triangulation Scheme.- 4.8.3 Test Results.- 4.8.3.1 Strongly Singular Integrals.- 4.8.3.2 Weakly Singular Integrals.- 4.8.3.3 Near Singular Integrals.- 4.9 Numerical Results for 3-D Eddy Current Problems.- 4.9.1 Conducting Sphere in a Uniform Time Harmonic Field.- 4.9.1.1 Problem Definition.- 4.9.1.2 Field Distribution.- 4.9.1.3 Power Loss.- 4.9.1.4 Effect of Gauss Integration Scheme.- 4.9.2 Finite Dimension Slab in a Uniform Time Harmonic Field.- 4.9.2.1 Problem Definition.- 4.9.2.2 Field Distribution.- 4.9.2.3 Treatment of the Singularity.- 4.9.3 Eddy Current Distribution in a Multiply Connected System.- 4.9.3.1 Problem Definition.- 4.9.3.2 Verification with Experimental Results.- 4.10 Conclusions.- 4.11 Appendix.- 4.11.1 Coefficients for the Boundary Integral Equations and the Magnetic Field Strength.- 4.11.2 Coefficients for the Electric Field Strength.- References.- 5 The Use of Boundary Element Finite Element Coupling Method in Electrical Engineering.- 5.1 List of Symbols.- 5.2 Field Problems in Electrical Engineering.- 5.3 Governing Equations.- 5.3.1 Scalar Potential Model.- 5.3.2 Vector Potential Model.- 5.3.3 3D-Problems.- 5.3.4 Saturation Effects.- 5.4 Numerical Analysis.- 5.4.1 Non Saturable Region.- 5.4.2 Linear Elements.- 5.4.3 Quadratic Elements.- 5.4.4 Saturable Regions.- 5.4.5 First Order Triangular Element.- 5.4.6 Isoparametric Quadrilateral Element.- 5.4.7 3D-Non Saturable Regions.- 5.4.8 3D-Saturable Regions.- 5.5 System Resolution.- 5.6 Application.- 5.6.1 Data Preparation.- 5.6.2 Resolution Process.- 5.6.3 Results.- 5.6.4 Magnetic Flux Calculation.- 5.6.5 Forces and Torques.- 5.6.6 Internal Values.- 5.7 Conclusion.- References.- 6 Hybrid Finite Element/Boundary Analysis of Electromagnetic Fields.- 6.1 Introduction.- 6.2 Electrostatics.- 6.2.1 Two-Dimensional and Three-Dimensional Analysis.- 6.2.1.1 Finite Element Formulation.- 6.2.1.2 Boundary Element Formulation.- 6.2.1.3 Hybrid Formulation.- 6.2.2 Axisymmetric Analysis.- 6.2.2.1 Finite Element Formulation.- 6.2.2.2 Boundary Element Formulation.- 6.2.3 Electrostatic Examples.- 6.3 Magnetostatics.- 6.3.1 Two-Dimensional Analysis.- 6.3.1.1 Finite Element Formulation.- 6.3.1.2 Boundary Element Formulation.- 6.3.1.3 Non-linear Formulation.- 6.3.2 Axisymmetric Analysis.- 6.3.2.1 Finite Element Formulation.- 6.3.2.2 Boundary Element Formulation.- 6.3.3 Magnetostatic Examples.- 6.4 RF Field Analysis.- 6.4.1 Finite Element Formulation.- 6.4.2 Boundary Element Formulation.- 6.4.3 Hybrid Formulation.- 6.4.4 RF Examples.- 6.5 Conclusion.- References.- 7 Applications in the Analysis and Design of Electrical Machines.- 7.1 Introduction.- 7.2 General Electromagnetic Problems.- 7.2.1 The Electromagnetic Field Equations.- 7.2.1.1 The Vector and the Scalar Potentials.- 7.2.1.2 The Case of the Two-Dimensional Plane Domains.- 7.2.2 Magnetic Energy.- 7.2.2.1 The Vector Potential Solution.- 7.2.2.2 The Scalar Potential Solution.- 7.2.3 Electromagnetic Forces and Torques.- 7.2.4 The Machine Parameters.- 7.3 The Boundary Integral Relations and the Boundary Element Method for Magnetic Field Problems.- 7.3.1 Some Mathematical Relations.- 7.3.2 The Scalar Potential Formulation.- 7.3.3 The Vector Potential Formulation.- 7.3.4 The Particular Solution - Scalar Potential Formulation.- 7.4 The Application of the Boundary Element Technique in Primitive Electrical Machine Problems.- 7.4.1 The Scalar Potential Problem.- 7.4.1.1 The Scalar Potential Problem on Axis d.- 7.4.1.2 The Scalar Potential Problem on Axis q.- 7.4.1.3 The Magnetic Energy and Electromagnetic Torque in the Scalar Potential Problem.- 7.4.2 The Vector Potential Solution.- 7.4.2.1 The Vector Potential Problem on Axis d.- 7.4.2.2 The Vector Potential Problem on Axis q.- 7.4.2.3 The Magnetic Energy and Electromagnetic Torque in the Vector Potential Problem.- 7.4.2.4 Conclusions.- 7.5 Calculation of the Magnetic Field and Parameters of Brushless D. C. Motors with Ceramic Permanent Magnets. The 2D-Problem.- 7.5.1 Calculation of Field Distribution Produced by the Stator Current, the Machine Parameters.- 7.5.1.1 The d-axis Problem.- 7.5.1.2 The q-axis Problem.- 7.5.2 The Electromagnetic Torque.- 7.6 Magnetic Field Computation in the D.C. Cylinder Type Brushless Servomotors Using the 3D-BEM.- References.