Electromagnetic Applications

ISBN-13: 9783642836824 / Angielski / Miękka / 2011 / 234 str.

Carlos A. Brebbia

Opis

General Applications of BEM to electromagnetic problems are comparatively new although the method is ideally suited to solve these problems, which usually involve unbounded domains. The present volume comprises contributions by eminent researchers working on applications of boundary elements in electromagnetic problems. The volume deals with the solutions of Maxwell's equation for three-dimensional as well as two-dimensional cases. It also discusses combination of BEM with FEM particularly in the case of saturated media. Some chapters specifically deal with the design of electromagnetic devices. The book is essential reading to those engineers and scientists, who are interested in the state of the art for electrical and electromagnetic application of boundary elements. It is also an important reference for those engineers who are working on the design of electromagnetic components many of which can be advantageously carried out using BEM.

Spis treści

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.