Finite Element Analysis of Electrical Machines

1 Introduction to Finite Elements.- 1.1 Introduction.- 1.2 The Galerkin Finite Element Method.- 1.3 Boundary Conditions.- 1.4 Magnetostatic Example.- 2 Nonlinear Problems.- 2.1 Introduction.- 2.2 Representation of the B-H Curve.- 2.3 The Basic Newton-Raphson Method for a Single Equation.- 2.4 Application of the Newton-Raphson Method to a First Order Element.- 2.5 Cubic Splines.- 3 Permanent Magnets.- 3.1 Introduction.- 3.2 Magnetization Models of Permanent Magnets.- 3.3 Current Sheet Equivalents.- 3.4 Example With Permanent Magnet.- 3.5 Intrinsic and Normal Hysteresis Characteristics.- 4 Eddy Current Analysis.- 4.1 Introduction.- 4.2 Eddy Currents and Skin Effect.- 4.3 Finite Element Example.- 4.4 Elliptical Description of Flux Density.- 4.5 Eddy Currents In Nonlinear Materials.- 4.6 Nonlinear Permeability Models.- 4.7 Coupling Finite Elements To External Circuits.- 4.8 Numerical Example.- 4.9 Modeling Considerations.- 5 Computation of Losses, Resistance and Inductance.- 5.1 Introduction.- 5.2 Computation of Eddy Current Loss.- 5.3 Losses In A Series Winding.- 5.4 Inductance and Reactance.- 5.5 Poynting Vector.- 5.6 Nonlinear Problems.- 6 Calculation Of Force And Torque.- 6.1 Introduction.- 6.2 Ampere's Force Law.- 6.3 The Maxwell Stress Method.- 6.4 The Virtual Work Method.- 6.5 Using Machine Models to find Torque.- 6.6 Errors in Force Computation.- 6.7 Convergence of Force.- 7 Synchronous Machines in the Steady State.- 7.1 Introduction.- 7.2 The Basic Configuration of a Synchronous Machine.- 7.3 Steady State Operation - Excitation Calculation.- 7.4 Modeling Considerations.- 7.5 Excitation Calculation.- 7.6 Computation of Steady State Reactances.- 7.7 Direct Axis Transient Reactance, X?d.- 7.8 Direct Axis Subtransient Reactance, X?d.- 7.9 Frequency Response Curves.- 7.10 Time Constants.- 7.11 Poynting Vector Method.- 8 The Induction Motor in Steady State.- 8.1 Introduction.- 8.2 Obtaining Steady State Parameters.- 8.3 Obtaining Reactances From The Frequency Response.- 8.4 Results.- 8.5 Using Frequency Domain Data With Time Domain Solutions.- 9 Time Domain Modeling of Induction Machines.- 9.1 Introduction.- 9.2 Electromagnetic and Mechanical Theory.- 9.3 Galerkin Formulation.- 9.4 Time Discretization.- 9.5 Linearization.- 9.6 Global System of Equations.- 9.7 Examples.- 10 Air-Gap Elements for Electrical Machines.- 10.1 Introduction.- 10.2 Description of the Method.- 10.3 Application.- 11 Axiperiodic Solutions.- 11.1 Introduction.- 11.2 Finite Element Formulation for Scalar Fields.- 11.3 Calculation of Steady State Forces.- 11.4 Calculation of Transient Forces.