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Magnetic Materials and 3D Finite Element Modeling

Book Description

Magnetic Materials and 3D Finite Element Modeling explores material characterization and finite element modeling (FEM) applications. This book relates to electromagnetic analysis based on Maxwell’s equations and application of the finite element (FE) method to low frequency devices. A great source for senior undergraduate and graduate students in electromagnetics, it also supports industry professionals working in magnetics, electromagnetics, ferromagnetic materials science and electrical engineering.

The authors present current concepts on ferromagnetic material characterizations and losses. They provide introductory material; highlight basic electromagnetics, present experimental and numerical modeling related to losses and focus on FEM applied to 3D applications. They also explain various formulations, and discuss numerical codes.

• Furnishes algorithms in computational language

• Summarizes concepts related to the FE method

• Uses classical algebra to present the method, making it easily accessible to engineers

Written in an easy-to-understand tutorial format, the text begins with a short presentation of Maxwell’s equations, discusses the generation mechanism of iron losses, and introduces their static and dynamic components. It then demonstrates simplified models for the hysteresis phenomena under alternating magnetic fields. The book also focuses on the Preisach and Jiles–Atherton models, discusses vector hysterisis modeling, introduces the FE technique, and presents nodal and edge elements applied to 3D FE formulation connected to the hysteretic phenomena.

The book discusses the concept of source-field for magnetostatic cases, magnetodynamic fields, eddy currents, and anisotropy. It also explores the need for more sophisticated coding, and presents techniques for solving linear systems generated by the FE cases while considering advantages and drawbacks.

Table of Contents

  1. Preface
  2. Authors
  3. Chapter 1 Statics and Quasistatics EIectromagnetics Brief Presentation
  4. 1.1 Introduction
  5. 1.2 Maxwell’s Equations Maxwell’s Equations
  6. 1.3 Maxwell’s Equations: Local Form Maxwell’s Equations: Local Form
  7. 1.4 Maxwell’s Equations: Iintegral Form Maxwell’s Equations: Iintegral Form
  8. 1.5 Maxwell’s Equations In Low Frequency Maxwell’s Equations In Low Frequency
  9. 1.6 Electrostatics
  10. 1.6.1 Refraction of the Electric Field
  11. 1.6.2 Laplace’s and Poisson’s equations of the electric field for dielectric media Laplace’s and Poisson’s equations of the electric field for dielectric media
  12. 1.6.3 Laplace’s Equation of The Electric Field Laplace’s Equation of The Electric Field FR Conductive Media
  13. 1.7 Magnetostatic Fields
  14. 1.7.1 Equation rot H = J
  15. 1.7.2 Equation div B = 0
  16. 1.7.3 Equation rot E = 0
  17. 1.7.4 Biot-Savart Law
  18. 1.7.5 Magnetic Field Refraction
  19. 1.7.6 Energy in The Magnetic Field
  20. 1.8 Magnetic Materials
  21. 1.8.1 Diamagnetic Materials
  22. 1.8.2 Paramagnetic Materials
  23. 1.8.3 Ferromagnetic Materials
  24. 1.8.3.1 General Presentation
  25. 1.8.3.2 Influence of Iron on Magnetic Circuits
  26. 1.8.4 Permanent Magnets
  27. 1.8.4.1 General Presentation
  28. 1.8.4.2 Principal Types of Permanent Magnets
  29. 1.8.4.3 Dynamic 0peration of Permanent Magnets
  30. 1.9 Inductance and Mutual Inductance
  31. 1.9.1 Definition of Inductance
  32. 1.9.2 Energy in a Linear System
  33. 1.10 Magnetodynamic Fields
  34. 1.10.1 Maxwell’s Equations for the Magnetodynamic Field Maxwell’s Equations for the Magnetodynamic Field
  35. 1.10.2 Penetration of Time-Dependent Fields in Conducting Materials
  36. 1.10.2.1 Equation for H
  37. 1.10.2.2 Equation for B
  38. 1.10.2.3 Equation for E
  39. 1.10.2.4 Equation for I
  40. 1.10.2.5 SoIution of the Equations
  41. 1.11 Fields defined by potentials
  42. 1.11.1 Electric Scalar Potential
  43. 1.11.2 Magnetic scalar potential
  44. 1.11.3 Magnetic vector potential
  45. 1.11.4 Electric vector potential
  46. 1.12 Final considerations
  47. References
  48. Chapter 2 Ferromagnetic Materials and lron Losses
  49. 2.1 Introduction
  50. 2.2 Basic Concepts
  51. 2.3 Loss Components
  52. 2.4 Iron Losses Under Alternating, Rotating, and DC‐Biased Inductions Iron Losses Under Alternating, Rotating, and DC‐Biased Inductions
  53. 2.4.1 Epstein’s Frame And Workbench Epstein’s Frame And Workbench
  54. 2.4.1.1 Methodology for Iron Loss Separation
  55. 2.4.1.2 Results for Two Different Iron Sheets
  56. 2.4.1.3 Considering Eddy Current in Epstein’s Frame Corners Considering Eddy Current in Epstein’s Frame Corners
  57. 2.4.1.4 Improved Model for the Eddy Current Losses
  58. 2.4.1.5 Results Verification by 3D FE Modeling
  59. 2.4.2 Single Sheet Tester
  60. 2.4.3 Rotational Single Sheet Tester
  61. 2.4.4 DC-Biased Induction
  62. 2.5 Final Considerations
  63. References
  64. Chapter 3 Scalar Hysteresis Modeling
  65. 3.1 Introduction
  66. 3.2 Preisach’s Scalar Model Preisach’s Scalar Model
  67. 3.2.1 Magnetization ln Terms of Everett’s Function Magnetization ln Terms of Everett’s Function
  68. 3.2.2 Identification of Everett’s Function Identification of Everett’s Function
  69. 3.2.3 Results Obtained With Preisach’s Scalar Model Results Obtained With Preisach’s Scalar Model
  70. 3.3 Jiles‐Atherton Scalar Model Jiles‐Atherton Scalar Model
  71. 3.3.1 Original (Direct) Jiles–Atherton Model Original (Direct) Jiles–Atherton Model
  72. 3.3.2 Inverse Jiles–Atherton Model Inverse Jiles–Atherton Model
  73. 3.3.3 Jiles–Atherton Model Parameter Determ ination Jiles–Atherton Model Parameter Determ ination
  74. 3.3.4 Results Obtained with the Jiles–Atherton Model Results Obtained with the Jiles–Atherton Model
  75. 3.3.5 Modified Jiles–Atherton Hysteresis Model Modified Jiles–Atherton Hysteresis Model
  76. 3.3.6 Determ ination of Parameter R in the Modified Jiles–Atherton Model in the Modified Jiles–Atherton Model
  77. 3.3.7 Results of the Modified Jiles–Atherton Model Results of the Modified Jiles–Atherton Model
  78. 3.4 Final Considerations
  79. References
  80. Chapter 4 Vector Hysteresis Modeling
  81. 4.1 Introduction
  82. 4.2 Vector Model Obtained With the Superposition of Scalar Models
  83. 4.2.1 Model Principle
  84. 4.2.2 Identification of the Parameters of the Model
  85. 4.2.3 Results of the Vector Model
  86. 4.3 Vector Generalization of the Jiles‐Atherton Scalar Models Vector Generalization of the Jiles‐Atherton Scalar Models
  87. 4.3.1 Vector Generalization of the Original Jiles–Atherton Model Vector Generalization of the Original Jiles–Atherton Model
  88. 4.3.2 Vector Generalization of the Inverse Jiles–Atherton Model Vector Generalization of the Inverse Jiles–Atherton Model
  89. 4.3.3 Some Aspects of the Jiles–Atherton Vector Model and Results Some Aspects of the Jiles–Atherton Vector Model and Results
  90. 4.4 Remarks Concerning the Vector Behavior of Hysteresis
  91. 4.5 Final Considerations
  92. References
  93. Chapter 5 Finite Element Method Brief Presentation
  94. 5.1 Introduction
  95. 5.2 Galerkin Method: Basic Concepts Using Real Coordinates
  96. 5.2.1 Equations and Numerical Implementation
  97. 5.2.2 Boundary Conditions
  98. 5.2.2.1 Dirichlet Boundary Condition: Imposed Potential
  99. 5.2.2.2 Neumann Condition: Unknown Nodal Values on the Boundary
  100. 5.2.3 First Order 2D Finite Element Program
  101. 5.2.4 Example for the Finite Element Program
  102. 5.3 Generalization of the Fem: Using Reference Coordinates
  103. 5.3.1 High-Order Finite elements: general
  104. 5.3.2 High-Order Finite elements: notation
  105. 5.3.3 High-Order Finite Elements: Implementation
  106. 5.3.4 Continuity of finite elements
  107. 5.3.5 Polynomial Basis
  108. 5.3.6 Transformation of Quantities: Jacobian
  109. 5.3.7 Evaluation of the integrals
  110. 5.4 Numerical Integration
  111. 5.5 Some Finite Elements
  112. 5.5.1 First-Order triangular element
  113. 5.5.2 Second-order triangular element
  114. 5.5.3 First-order tetrahedral element
  115. 5.5.4 Implementation Aspects
  116. 5.6 Using Edge Elements
  117. 5.6.1 Magnetostatic equation using the vector potential
  118. 5.6.2 Brief explanation of edge shape functions
  119. 5.6.3 Applying the Edge Element Shape Functions
  120. 5.6.4 Implementing the first-order tetrahedron edge element shape functions
  121. 5.6.5 Applying the galerkin method
  122. 5.6.6 Coding Tetrahedral Edge Elements
  123. 5.7 Final Considerations
  124. References
  125. Chapter 6 Using Nodal EIements with Magnetic Vector Potential
  126. 6.1 Introduction
  127. 6.2 Main Equations
  128. 6.2.1 Magnetostatic Governing Equation
  129. 6.2.2 Defining Some Operations
  130. 6.3 Applying the Galerkin Method
  131. 6.4 Uniqueness of the Solution: Coulomb’s Gauge Uniqueness of the Solution: Coulomb’s Gauge
  132. 6.5 Implementation
  133. 6.6 Example and Comparisons
  134. 6.7 Final Considerations
  135. References
  136. Chapter 7 Source-Field Method for 3D Magnetostatic Fields
  137. 7.1 Introduction
  138. 7.2 Magnetostatic Case: Scalar Potential
  139. 7.2.1 Main Equations
  140. 7.2.2 Hs Calculation: Edge Tree
  141. 7.2.3 Facet Tree
  142. 7.2.4 Applying the Galerkin Method
  143. 7.2.5 Elemental Matrices: Evaluation, Notation, and Array Dimensions
  144. 7.2.6 Considering Permanent Magnets
  145. 7.2.7 Boundary Conditions
  146. 7.3 Magnetostatic Case: Vector Potential
  147. 7.3.1 Main Equations
  148. 7.4 Implementation aspects and Conventions
  149. 7.4.1 Building the Facets
  150. 7.4.2 Building the Edges
  151. 7.4.3 Building the Edge Tree
  152. 7.4.4 Building the Conductor Facet Tree and Calculating the Flux of J
  153. 7.4.5 Calculating Hs
  154. 7.4.6 Applying the Boundary Conditions
  155. 7.5 Computational implementation
  156. 7.5.1 Main Subroutines for the Scalar Potential Formulation
  157. 7.5.2 Main Subroutines for the Vector Potential Formulation
  158. 7.6 Example and Results
  159. 7.7 Final Considerations
  160. References
  161. Chapter 8 Source-Field Method for 3D Magnetodynamic Fields
  162. 8.1 Introduction
  163. 8.2 Formulation Considering Eddy Currents: Time Stepping
  164. 8.2.1 Governing Equations
  165. 8.3 Formulation considering eddy currents: complex formulation
  166. 8.4 Field-Circuit Coupling
  167. 8.4.1 Basic Equations
  168. 8.4.2 Applying the Galerkin Method
  169. 8.4.3 Formulation Considering Eddy currents and Electric Circuit Coupling
  170. 8.5 Computational Implementation
  171. 8.6 Differential Permeability Method
  172. 8.6.1 Nonlinear Cases
  173. 8.6.2 Anisotropic Cases
  174. 8.7 Example and Results
  175. 8.7.1 Eddy Currents, Circuit Coupling, Regular Permeability
  176. 8.7.2 Example of an Isotropic Nonlinear Case with differential permeability
  177. 8.7.3 Anisotropic Magnetic Circuit
  178. 8.7.4 Scalar Hysteresis: A didactical case
  179. 8.7.5 Vector Hysteresis Anisotropic Case: Team Workshop Problem 32
  180. 8.8 Final Considerations
  181. References
  182. Chapter 9 Matrix-Free Iterative Solution Procedure for Finite Element Problems
  183. 9.1 Introduction
  184. 9.2 Classical Fem: T-Scheme
  185. 9.3 Proposed Technique: N-Scheme
  186. 9.4 Implementation
  187. 9.5 Convergence
  188. 9.6 Implementation of N-Scheme with SOR
  189. 9.7 Applying the N-Scheme in Nonstationary Solvers
  190. 9.8 CC Algorithm Implementation
  191. 9.9 Examples and Results
  192. 9.9.1 Two-Dimensional Electrostatic Problem
  193. 9.9.2 Three-Dimensional Nonlinear Case Using SOR Technique
  194. 9.9.3 Example with a large number of unknowns
  195. 9.10 Results and Discussion
  196. 9.11 Final Considerations
  197. References
  198. Index