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ELECTROMAGNETIC MODELING BY FINITE ELEMENT METHODS JOAO PEDRO A. BASTOS NELSON SADOWSKI Universidade Federal de Santa Catarina Florianopolis, Brazil
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Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-4269-9 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-260-6300; fax: 41-61-260-6333 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above.
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ELECTRICAL AND COMPUTER ENGINEERING A Series of Reference Books and Textbooks
FOUNDING EDITOR Marlin O. Thurston Department of Electrical Engineering The Ohio State University Columbus, Ohio
1. Rational Fault Analysis, edited by Richard Saeks and S. R. Liberty 2. Nonparametric Methods in Communications, edited by P. PapantoniKazakos and Dimitri Kazakos 3. Interactive Pattern Recognition, Yi-tzuu Chien 4. Solid-State Electronics, Lawrence E. Murr 5. Electronic, Magnetic, and Thermal Properties of Solid Materials, Klaus Schroder 6. Magnetic-Bubble Memory Technology, Hsu Chang 7. Transformer and Inductor Design Handbook, Colonel Wm. T. McLyman 8. Electromagnetics: Classical and Modern Theory and Applications, Samuel Seely and Alexander D. Poularikas 9. One-Dimensional Digital Signal Processing, Chi-Tsong Chen 10. Interconnected Dynamical Systems, Raymond A. DeCarto and Richard Saeks 11. Modern Digital Control Systems, Raymond G. Jacquot 12. Hybrid Circuit Design and Manufacture, Roydn D. Jones 13. Magnetic Core Selection for Transformers and Inductors: A User's Guide to Practice and Specification, Colonel Wm. T. McLyman 14. Static and Rotating Electromagnetic Devices, Richard H. Engelmann 15. Energy-Efficient Electric Motors: Selection and Application, John C. Andreas 16. Electromagnetic Compossibility, Heinz M. Schlicke 17. Electronics: Models, Analysis, and Systems, James G. Gottling 18. Digital Filter Design Handbook, FredJ. Taylor 19. Multivariable Control: An Introduction, P. K. Sinha 20. Flexible Circuits: Design and Applications, Steve Guhey, with contributions by Carl A. Edstrom, Jr., Ray D. Green way, and William P. Kelly 21. Circuit Interruption: Theory and Techniques, Thomas E. Browne, Jr. 22. Switch Mode Power Conversion: Basic Theory and Design, K. Kit Sum 23. Pattern Recognition: Applications to Large Data-Set Problems, SingTze Bow
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24. Custom-Specific Integrated Circuits: Design and Fabrication, Stanley L Hurst 25. Digital Circuits: Logic and Design, Ronald C. Emery 26. Large-Scale Control Systems: Theories and Techniques, Magdi S. Mahmoud, Mohamed F. Hassan, and Mohamed G. Darwish 27. Microprocessor Software Project Management, Eli T. Fathi and Cedric V. W. Armstrong (Sponsored by Ontario Centre for Microelectronics) 28. Low Frequency Electromagnetic Design, Michael P. Perry 29. Multidimensional Systems: Techniques and Applications, edited by Spyros G. Tzafestas 30. AC Motors for High-Performance Applications: Analysis and Control, Sakae Yamamura 31. Ceramic Motors for Electronics: Processing, Properties, and Applications, edited by Relva C. Buchanan 32. Microcomputer Bus Structures and Bus Interface Design, Arthur L. Dexter 33. End User's Guide to Innovative Flexible Circuit Packaging, Jay J. Miniet 34. Reliability Engineering for Electronic Design, Norman B. Fuqua 35. Design Fundamentals for Low-Voltage Distribution and Control, Frank W. Kussy and Jack L. Warren 36. Encapsulation of Electronic Devices and Components, Edward R. Salmon 37. Protective Relaying: Principles and Applications, J. Lewis Blackburn 38. Testing Active and Passive Electronic Components, Richard F. Powell 39. Adaptive Control Systems: Techniques and Applications, V. V. Chalam 40. Computer-Aided Analysis of Power Electronic Systems, Venkatachari Rajagopalan 41. Integrated Circuit Quality and Reliability, Eugene R. Hnatek 42. Systolic Signal Processing Systems, edited by Earl E. Swartzlander, Jr. 43. Adaptive Digital Filters and Signal Analysis, Maurice G. Bellanger 44. Electronic Ceramics: Properties, Configuration, and Applications, edited by Lionel M. Levinson 45. Computer Systems Engineering Management, Robert S. Alford 46. Systems Modeling and Computer Simulation, edited by Nairn A. Kheir 47. Rigid-Flex Printed Wiring Design for Production Readiness, Walter S. Rigling 48. Analog Methods for Computer-Aided Circuit Analysis and Diagnosis, edited by Takao Ozawa 49. Transformer and Inductor Design Handbook: Second Edition, Revised and Expanded, Colonel Wm. T. McLyman 50. Power System Grounding and Transients: An Introduction, A. P. Sakis Meliopoulos 51. Signal Processing Handbook, edited by C. H. Chen 52. Electronic Product Design for Automated Manufacturing, H. Richard Stillwell 53. Dynamic Models and Discrete Event Simulation, William Delaney and Erminia Vaccari 54. FET Technology and Application: An Introduction, Edwin S. Oxner
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Digital Speech Processing, Synthesis, and Recognition, Sadaoki Furui VLSI RISC Architecture and Organization, Stephen B. Furber Surface Mount and Related Technologies, Gerald Ginsberg Uninterruptible Power Supplies: Power Conditioners for Critical Equipment, David C. Griffith Polyphase Induction Motors: Analysis, Design, and Application, Paul L Cochran Battery Technology Handbook, edited by H. A. Kiehne Network Modeling, Simulation, and Analysis, edited by Ricardo F. Garzia and Mario R. Garzia Linear Circuits, Systems, and Signal Processing: Advanced Theory and Applications, edited by Nobuo Nagai High-Voltage Engineering: Theory and Practice, edited by M. Khalifa Large-Scale Systems Control and Decision Making, edited by Hiroyuki Tamura and Tsuneo Yoshikawa Industrial Power Distribution and Illuminating Systems, Kao Chen Distributed Computer Control for Industrial Automation, Dobrivoje Popovic and Vijay P. Bhatkar Computer-Aided Analysis of Active Circuits, Adrian loinovici Designing with Analog Switches, Steve Moore Contamination Effects on Electronic Products, Carl J. Tautscher Computer-Operated Systems Control, Magdi S. Mahmoud Integrated Microwave Circuits, edited by Yoshihiro Konishi Ceramic Materials for Electronics: Processing, Properties, and Applications, Second Edition, Revised and Expanded, edited by Relva C. Buchanan Electromagnetic Compatibility: Principles and Applications, David A. Weston Intelligent Robotic Systems, edited by Spyros G. Tzafestas Switching Phenomena in High-Voltage Circuit Breakers, edited by Kunio Nakanishi Advances in Speech Signal Processing, edited by Sadaoki Furui and M. Mohan Sondhi Pattern Recognition and Image Preprocessing, Sing-Tze Bow Energy-Efficient Electric Motors: Selection and Application, Second Edition, John C. Andreas Stochastic Large-Scale Engineering Systems, edited by Spyros G. Tzafestas and Keigo Watanabe Two-Dimensional Digital Filters, Wu-Sheng Lu and Andreas Antoniou Computer-Aided Analysis and Design of Switch-Mode Power Supplies, Yim-Shu Lee Placement and Routing of Electronic Modules, edited by Michael Pecht Applied Control: Current Trends and Modern Methodologies, edited by Spyros G. Tzafestas Algorithms for Computer-Aided Design of Multivariable Control Systems, Stanoje Bingulac and Hugh F. VanLandingham Symmetrical Components for Power Systems Engineering, J. Lewis Blackburn Advanced Digital Signal Processing: Theory and Applications, Glenn Zelniker and Fred J. Taylor
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87. Neural Networks and Simulation Methods, Jian-Kang Wu 88. Power Distribution Engineering: Fundamentals and Applications, James J. Burke 89. Modern Digital Control Systems: Second Edition, Raymond G. Jacquot 90. Adaptive MR Filtering in Signal Processing and Control, Phillip A. Regalia 91. Integrated Circuit Quality and Reliability: Second Edition, Revised and Expanded, Eugene R. Hnatek 92. Handbook of Electric Motors, edited by Richard H. Engelmann and William H. Middendorf 93. Power-Switching Converters, Simon S. Ang 94. Systems Modeling and Computer Simulation: Second Edition, Nairn A. Kheir 95. EMI Filter Design, Richard Lee Ozenbaugh 96. Power Hybrid Circuit Design and Manufacture, Halm Taraseiskey 97. Robust Control System Design: Advanced State Space Techniques, Chia-Chi Tsui 98. Spatial Electric Load Forecasting, H. Lee Willis 99. Permanent Magnet Motor Technology: Design and Applications, Jacek F. Gieras and Mitchell Wing 100. High Voltage Circuit Breakers: Design and Applications, Ruben D. Garzon 101. Integrating Electrical Heating Elements in Appliance Design, Thor Hegbom 102. Magnetic Core Selection for Transformers and Inductors: A User's Guide to Practice and Specification, Second Edition, Colonel Wm. T. McLyman 103. Statistical Methods in Control and Signal Processing, edited by Tohru Katayama and Sueo Sugimoto 104. Radio Receiver Design, Robert C. Dixon 105. Electrical Contacts: Principles and Applications, edited by Paul G. Slade 106. Handbook of Electrical Engineering Calculations, edited by Arun G. Phadke 107. Reliability Control for Electronic Systems, Donald J. LaCombe 108. Embedded Systems Design with 8051 Microcontrollers: Hardware and Software, Zdravko Karakehayov, Knud Smed Christensen, and Ole Winther 109. Pilot Protective Relaying, edited by Walter A. Elmore 110. High-Voltage Engineering: Theory and Practice, Second Edition, Revised and Expanded, Mazen Abdel-Salam, Hussein Anis, Ahdab ElMorshedy, and Roshdy Radwan 111. EMI Filter Design: Second Edition, Revised and Expanded, Richard Lee Ozenbaugh 112. Electromagnetic Compatibility: Principles and Applications, Second Edition, Revised and Expanded, David A. Weston 113. Permanent Magnet Motor Technology: Design and Applications, Second Edition, Revised and Expanded, Jacek F. Gieras and Mitchell Wing
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114. High Voltage Circuit Breakers: Design and Applications, Second Edition, Revised and Expanded, Ruben D. Garzon 115. High Reliability Magnetic Devices: Design and Fabrication, Colonel Wm. T. McLyman 116. Practical Reliability of Electronic Equipment and Products, Eugene R. Hnatek 117. Electromagnetic Modeling by Finite Element Methods, Joao Pedro A. Bastos and Nelson Sadowski
Additional Volumes in Preparation Battery Technology Handbook: Second Edition, H. A. Kiehne
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Preface This work is related to Electromagnetic (EM) Analysis based on Maxwell's equations and the application of the Finite Element Method (FEM) to EM low-frequency devices. New students in this area will find a didactical approach for a first contact with the FEM including some codes and many examples. For researchers and teachers having experience in the area, this book presents advanced topics related to their works as well as useful text for classes. Our text focuses on three complementary issues. The first is related to a didactical approach of EM equations and the application of the FEM to electromagnetic classical cases. The second one is the coupling of EM equations with other phenomena that exist in electromagnetic structures, such as external (electrical and electronic) circuits, movement and mechanical equations, vibration analysis, heating, eddy currents, and nonlinearity. The final issue is the analysis of electrical and magnetic losses, including hysteresis, eddy currents and anomalous losses. This book is intended primarily for graduate students but what must be pointed out is that more and more undergraduate students have been introduced to this area and this is the reason why efforts have been made to use a very didactical approach to the subjects presented in the book. Coupling and losses, advanced topics of the book, have been the objects of a great deal of scientific research in the last two decades and many related technical papers have been published in periodicals and at conferences. In spite of being active research topics, the content we have chosen is based on well-proven techniques. These may be applied without general restrictions. The book consists of the following chapters: • Chapter 1: A brief chapter on "mathematical preliminaries" is presented with the goal of recalling some useful algebra for the following chapters and establishing notations and language that will be used later.
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• Chapter 2: Maxwell's equations are described to provide didactical support for the following chapters. More classically, FEM is more commonly presented for mechanics and we consider that this brief review of EM is appropriate here. • Chapter 3: This chapter is devoted to an introduction to the FEM in a short presentation of the method. The goal is not to analyze this method very deeply (many books with this purpose are available) but to bring out the most important aspects of the FEM for EM analysis. It is a concise chapter in which virtually all the FE concepts are introduced and it is clearly shown how they should be linked in order to implement a computational code. • Chapter 4: After presenting the FEM, the method is applied to EM equations, pointing out their physical meaning and explaining in detail the particulars related to this area. Thermal equations are also included in this chapter. • Chapter 5: The coupling with electrical and electronic circuits is now presented. In this chapter much of our experience and advanced research work are extensively described. The formulation reaches advanced phenomena as in, for instance, linking EM devices to converters, whose topology is not known "a priori". It means that the dynamic behavior of the converter is taken into account (considering the switching of thyristors, diodes, etc., during operation) and calculated simultaneously with EM field equations. Eddy current phenomenon is also treated in "thick" conductors. • Chapter 6: Movement is an important aspect of EM devices; most of them (electrical machines, switchers and actuators) are subjected to mechanical forces and movement. In this chapter, methods for discretizing airgaps and for simulating the physical displacement are presented. In the final part of this chapter a method (based on 2D simulations) to take into account the skew effect in rotating machines is proposed. • Chapter 7: The interaction between electromagnetic and mechanical quantities is described. Many different and commonly employed methods are presented and compared. Here, again, a great deal of our
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experience, papers and results are brought together and can be viewed as a good synthesis of research performed by us and other groups. Also, results on vibrational behavior of EM structures (coupling mechanical equations with EM ones) are presented. • Chapter 8: This part of the book is dedicated to losses. Advanced studies on eddy current, anomalous and hysteresis losses are described. We may point out that the last subject, hysteresis, is (as far as numerical calculation and simulation of devices are concerned) now a topic of intensive study/research and has been the subject of many recent papers. In our text we present modeling for hysteresis and its implementation in a FEM code, using, as indicated above, proven methods. We hope that the book will provide reliable and useful information for students and researchers dealing with EM problems. Finally, we would like to express our sincere gratitude to many colleagues and friends who helped us to develop the works presented in this book. Without their support it would have been impossible to publish it. We would like specially to thank Dr. M. Lajoie-Mazenc (LEEI-Toulouse) and Prof. C. Rioux (Univ. Paris VI), our thesis advisors, who gave us the scientific background for our research and professional life; Prof. N. Ida (Univ. of Akron) for a long collaboration, multiple technical discussions and decisive help with editing this book; Dr. P. Kuo-Peng (GRUCAD-UFSC) for writing substantial parts of chapter 5; Dr. R. C. Mesquita (UFMG); Prof. J. R. Cardoso (USP) and their teams for continual collaboration and technical support; and Prof. A. Kost (T. U. Cottbus) for his cooperation and technical exchanges. Our deep thanks to the colleagues of GRUCAD-UFSC and the Department of Electrical Engineering of the Universidade Federal de Santa Catarina for their constant support and friendship. We are grateful to the CNPq and CAPES (Brazilian Government's scientific foundations) for their financial support of our research work. Joao Pedro A. Bastos Nelson Sadowski
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Contents Preface 1. Mathematical Preliminaries 1.1. Introduction 1.2. The Vector Notation 1.3. Vector Derivation 1.3.1. The Nabla (V ) Operato 1.3.2. Definition of the Gradient, Divergence, and Rotational 1.4. The Gradient 1.4.1. Example of Gradient 1.5. The Divergence 1.5.1. Definition of Flu 1.5.2. The Divergence Theorem 1.5.3. The Conservative Flux 1.5.4. Example of Divergence 1.6. The Rotational 1.6.1. Circulation of a Vector 1.6.2. Stokes' Theorem 1.6.3. Example of Rotational 1.7. Second-Order Operators 1.8. Application of Operators to More than One Function 1.9. Expressions in Cylindrical and Spherical Coordinates 2. Maxwell Equations, Electrostatics, Magnetostatics and Magnetodynamic Fields 2.1. Introduction 2.2. The EM Quantities 2.2.1. The Electric Field Intensity E 2.2.2. The Magnetic Field Intensity 2.2.3. The Magnetic Rux Density B and the Magnetic Permeability jU 2.2.4. The Electric Flux Density D and Electric Permittivity £ 2.2.5. The Surface Current Density J 2.2.6. Volume Charge Density p 2.2.7. The Electric Conductivity 2.3. Local Form of the Equations 2.4. The Anisotropy 2.5. The Approximation to Maxwell's Equations 2.6. The Integral Form of Maxwell's Equations 2.7. Electrostatic Fields 2.7.1. The Ele 2.7.1a. The Electric Field 2.7.1b. Force on an Electr 2.7.1c. The Electric Scalar Potential
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2.7.2. Nonconservative Fields: Electromotive Force 2.7.3. Refraction of the Electric Field 2.7.4. Dielectric Strength 2.7.5. Laplace's and Poisson's Equations of the Electric Field for Dielectric Media 2.7.6. Laplace's Equation of the Electric Field for Conductive Medi 2.8. Magnetostatic Fields 2.8.1. Maxwell's Equations in Magnetostatics 2.8.1a. The Equation ro/H= J 2.8.1b. The Equation divE = 0 2.8.1c. The Equation rotE = 2.8.2. The Biot-Savart Law 2.8.3. Magnetic Field Refraction 2.8.4. Energy in the Magnetic Field 2.8.5. Magnetic Materials 2.8.5a. Diamagnetic Materials 2.8.5b. Paramagnetic Materials 2.8.5c. Ferromagnetic Material a) General b)The Influence of Iron on Magnetic Circuits 2.8.5d. Permanent Magnets a) General Properties of Hard Magnetic Materials b)The Energy Associated with a Magnet c) Principal Types of Permanent Magnets d) Dynamic Operation of Permanent Magnets 2.8.6. Inductance and Mutual Inductance 2.8.6a. Definition of Inductance 2.8.6b. Energy in a Linear Syste 2.9. Magnetodynamic Fields 2.9.1. Maxwell's E namic Field 2.9.2. Penetration of Time-Dependent Fields in Conducting Material 2.9.2a. The Equation for H 2.9.2b. The Equation forB 2.9.2c. The Equation forE 2.9.2d. The Equation for J 2.9.2e. Solution of the Equations 3. Brief Presentation of the Finite Element Method 3.1. Introduction 3.2. The Galerki 3.2.1. The Establishment of the Physical Equation 3.2.2. The First-Order Triangle 3.2.3. Application of the Weighted Residual Metho 3.2.4. Application of the Finite Element Method and Solution 3.2.5. The Boundary Conditions 3.2.5a. Dirichlet Boundary Condition - Imposed Potential
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3.2.5b. Neumann Condition - Unknown Nodal Values on the Boundary 3.3. A First-Order Finite El 3.3.1. Example for Use of the Finite Element Program 3.4. Generalization of the Finite Element Method 3.4.1. High-Order Finite Elements: General 3.4.2. High-Order Finite Elements: Notation 3.4.3. High-Order Finite Elements: Implementation 3.4.4. Continuity of Finite Elements 3.4.5. Polynomial Basis 3.4.6. Transformation of Quantities - the Jacobian 3.4.7. Evaluation of the Integrals 3.5. Numerical Integration 3.6. Some 2D Finite Elements 3.6.1. First-Order Triangular Element 3.6.2. Second-Order Triangular Element 3.6.3. Quadrilateral Bi-linear Element 3.6.4. Quadrilateral Quadratic Element 3.7. Coupling Different Finite Elements 3.7.1. Coupling Different Types of Finite Elements 3.8. Calculation of Some Terms in the Field Equation 3.8.1. The Stiffness Matri 3.8.2. Evaluation of the Second Term in Eq. (3.72) 3.8.3. Evaluation of the Third Term in Eq. (3.72) 3.8.4. Evaluation of the Source Term 3.9. A Simplified 2D Second-Order Finite Element Program 3.9.1. The Problem to Be Solved 3.9.2. The Discretized Domain 3.9.3. The Finite Element Program
4. The Finite Element Method Applied to 2D Electromagnetic Cases 4.1. Introduction 4.2. Some Static Cases 4.2.1. Electrostatic Fields: Dielectric Materials 4.2.2. Stationary Currents: Conducting Mater 4.2.3. Magnetic Fields: Scalar Potential 4.2.4. The Magnetic Field: Vector Pote 4.2.5. The Electric Vector Potential 4.3. Application to 2D Eddy Current Problem 4.3.1. First-Order Element in Local Coordinates 4.3.2. The Vector Potential Equation Using Time Discretizatio 4.3.3. The Complex Vector Potential Equation 4.3.4. Structures with Moving Parts 4.4. Axi-Symmetric Application 4.4.1. The Axi-Symmetric Formulation for Vector Potential 4.5. Advantages and Limitations of 2D Formulations
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4.6. Non-linear Applications 4.6.1. Method of Successive Approximation 4.6.2. The Newton-Raphson Method 4.7. Geometric Repetition of Domains 4.7.1. Periodicit 4.7.2. Anti-Perio 4.8. Thermal Problems 4.8.1. Thermal Conduction 4.8.2. Convection Transmission 4.8.3. Radiation 4.8.4. FE Implementation 4.9. Voltage-Fed Electromagnetic Devices 4.10. Static Examples 4.10.1. Calculation of Electrostatic Fields 4.10.2. Calculation of Static Currents 4.10.3. Calculation of the Magnetic Field - Scalar Potential 4.10.4. Calculation of the Magnetic Field - Vector Potentia 4.11. Dynamic Examples 4.11.1. Eddy Currents: Time Discretizatio 4.11.2. Moving Conducting Piece in Front of an Electromagnet 4.11.3. Time Step Simulation of a Voltage-Fed Device 4.11.4. Thermal Case: Heating by Eddy Currents 5. Coupling of Field and Electrical Circuit Equations 5.1. Introduction 5.2 Electromagnetic Equations 5.2.1. Formulation Using the Magnetic Vector Potential 5.2.2. The Formulation in Two Dimensions 5.2.3. Equations for Conductors 5.2.3a. Thick Conductors 5.2.3b. Thin Conductors 5.2.4. Equations for the Whole Domain 5.2.5. The Finite Element Method 5.3. Equations for Different Conductor Configurations 5.3.1. Thick Conductors Connections 5.3. la. Series Connection 5.3.Ib. Parallel Connection 5.3.2 Thin Conductors Connectio 5.3.2a. Independent Voltage Sources 5.3.2b. Star Connection with Neutral 5.3.2c. Polygon Connection 5.3.2d. Star Connection without Neutral Wir 5.4. Connections Between Electromagnetic Devices and External Feeding Circuit 5.4.1. Reduced Equations of Electromagnetic Devices 5.4.2. Feeding Circuit Equations and Connection to Field Equations 5.4.3. Calculation of Matrices G,to G6
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5.4.3a. Circuit Topology Concepts 5.4.3b. Determination of Matrices G] to G6 5.4.3c. Example 5.4.3d. Taking Into Account Electronic Switches in the Feeding Circuit 5.4.4. Discre 5.5. Examples 5.5.1. Sim 5.5.la. A Didactical Example 5.5.1b. Three-Phase Induction Moto 5.5.1c. Massive Conductors in Series Connection 5.5.2. Modeling of a Static Converter-Fed Magnetic Devic 6. Movement Modeling for Electrical Machines 6.1. Introduction 6.1.1. Met 6.1.2. Methods with Discretized Airgaps 6.2. The Macro-Element 6.3. The Moving Band 6.4. The Skew Effect in Electrical Machines Using 2D Simulation 6.5. Examples 6.5.1. Thre 6.5.2. Permanent Magnet Motor 7. Interaction Between Electromagnetic and Mechanical Forces... 7.1. Introduction 7.2. Methods Based on Direct Formulations 7.2.1. Method of the Magnetic Co-Energy Variation 7.2.2. The Maxwell Stress Tensor Method 7.2.3. The Method Proposed by Arkkio 7.2.4. The Method of Local Jacobian Matrix Derivation 7.2.5. Examples of Torque Calculation 7.3. Methods Based on the Force Density 7.3.1. Preliminary Considerations 7.3.2. Equivalent Sources Formulation 7.3.2a. Equivalent Currents 7.3.2b. Equivalent Magnetic Char 7.3.2c. Other Equivalent Source Dist 7.3.3. Formulation Based on the Energy Derivation 7.3.4. Comparison Among the Different Methods 7.4. Electrical Machine Vibrations Originated by Magnetic Fo 7.4.1. Magnetic Force Calculation 7.4.2. Mechanical Calculation 7.4.2a. Calculation of the Natural Response 7.4.2b. Calculation of the Forced Response Direc Harmonic Regime
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7.4.2c. Calculation of the Forced Response Using the Modal Superposition Method 7.4.3. Example of Vibration Calculatio 7.5. Example of Coupling Between the Field and Circuit Equations, Including Mechanical Transients 8. Iron Losses 8.1. Introduction 8.2. Eddy Curre 8.3. Hysteresis 8.4. Anomalous or Excess Losses 8.5. Total Iron Losses 8.5.1. Example 8.6. The Jiles-Atherton Model 8.6.1. The JA Equations 8.6.2. Procedure for the Numerical Implementation of the JA Method 8.6.3. Examples of Hysteresis Loops Obtained with the JA Method 8.6.4. Determination of the Parameters from Experimental Hysteresi Loops 8.6.4a. Numerical Algorithm 8.7. The Inverse Jiles-Atherton Model 8.7.1. The Inverse JA Metho 8.7.2. Procedure for the Numerical Implementation of the Inverse JA Method 8.8. Including Iron Losses in 8.8.1. Hysteresis Modeling by Means of the Magnetization M Term. 8.8.2. Hysteresis Modeling by Means of a Differential Reluctivit 8.8.3. Inclusion of Eddy Current Losses in the FE Modeling 8.8.4. Inclusion of Anomalous Losses in the FE Modeling 8.8.5. Examples of Iron Losses Applied to FE Calculation graphy
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1 Mathematical Preliminaries 1.1. Introduction In this chapter we review a few ideas from vector algebra and calculus, which are used extensively in future chapters. We assume that operations like integration and differentiation as well as the bases for elementary vector calculus are known. This chapter is written in a concise fashion, and therefore, only those subjects directly applicable to this work are included. Readers wishing to expand on material introduced here can do so by consulting specialized books on the subject. It should be emphasized that we favor the geometrical interpretation rather than complete, rigorous mathematical exposition. We look with particular interest at the ideas of gradient, divergence, and rotational as well as at the divergence and Stokes' theorems. These notions are of fundamental importance for the understanding of electromagnetic fields in terms of Maxwell's equations. The latter are presented in local or point form in this work. 1.2. The Vector Notation Many physical quantities posses an intrinsic vector character. Examples are velocity, acceleration, and force, with which we associate a direction in space. Other quantities, like mass and time, lack this quality. These are scalar quantities. Another important concept is the vector field. A force, applied to a point of a body is a vector; however, the velocity of a gas inside a tube is a vector defined throughout a region (i.e., the cross-section
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of the tube, or a volume), not only at one point. In the latter case, we have a vector field. We use this concept extensively since many of the electromagnetic quantities (electric and magnetic fields, for example) are vector fields. 1 .3. Vector Derivation 1.3.1. The Nabla (V) Operator First, we recall that a scalar function may depend on more than one variable. For example, in the Cartesian system of coordinates the function can be denoted as
f(x,y,z) Its partial derivatives, if these exist, are
^?
&_9 §L
dx
dy
dz
The nabla ( V ) operator is a vector, which, in Cartesian coordinates, has the following components:
, . \dx
dy dz J
The operator is frequently written as d
d
a
V = i- — + 'j — +1 k — dx dx dx V7
where i , j , k , are the orthogonal unit vectors in the Cartesian system of coordinates. The nabla is a mathematical operator to which, by itself, we cannot associate any geometrical meaning. It is the interaction of the nabla operator with other quantities that gives it geometric significance.
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1.3.2. Rotational
Definition off the
Gradient, Divergence,
and
We define a scalar function U(x, y, z) with nonzero first-order partial derivatives with respect to the coordinates x, y, and z at a point M
dU dx
dU dy
dU dz
and a vector A with components Ax, Av and Az which depend on x, y and z; V is a vector which can interact with a vector or a scalar, as shown below:
A f- scalar product: V • A or divA (scalar) (Vector) 1 - vector product: V x A or curlA or rotA (vector) (Vector)
U (Scalar)
- product: VU or gradU (vector)
These three products can now be calculated:
or
dx
dy
rotA = curlA - V x A = det
or
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(1.1)
dz
dx
j d
k d
dy
dz
A rot A. = i'• I
z Z
dA
dA
dy
dz
dz
dx
dA,
+k
dy (1.2)
.
Jrr - i - + gradU dx
dy
+k
dz
(1.3)
After defining the gradient, curl, and divergence as algebraic entities, we will gain some insight into their geometric meaning in the following sections. 1 .4. The Gradient Given a scalar function U(x,y,z), with partial derivatives dU/dx , dU/dy , dU/dz , and dependent on a point M, with coordinates x, y, z, denoted as M\x, y, z] , we can calculate the differential of U as dU. This is done by considering the point M(x, y, z) and another point, infinitely close to M, M' (x + dx,y + dy,z + dz) , and using the total differential
... dU dU dU , dU = dx-\ dy + dz dx dy dz
(1.4)
gradU
U = const. Figure 1.1. The gradient is orthogonal to a constant potential surface. Defining the vector dM. — M' — M which possesses the components
= (dx,dy,dz) dU can be written as
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.dU .dU . = i O -f Ji / v + k obc
o \
GZ
. , . \ -f( i d x +Ji
or
= gradU-dM
(1.5)
As for the geometrical significance of the gradient, assume that there exists a surface with points M(x,y,z) and that on all these points, U = constant (see Figure 1.1). Hence, for all differential displacements M and M1 on this surface, we can write dU — 0 . From Eq. (1.5) we have
From the definition of the scalar product, it is clear that gradU and dM. are orthogonal. Assume now that the displacement of M to M" is in the direction of increasing U, as shown in Figure 1.2. In this case, dU > 0 , or gradU-dM>0 Note that the vectors gradU and fiflM form an acute angle. From the foregoing arguments we conclude that grad U is a vector, perpendicular to a surface on which U is constant and that it points to the direction of increasing U. We also note that gradU points to the direction of maximum change in U, since dU = gradU • dM. is maximum when dM. is in the same direction gradU. 1.4.1. Example off Gradient Given a function r, as the distance of a point M(x,y,z) from the origin 0(0,0,0) , determine the gradient of this function.
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U=U2 gradU
U=U1 Figure 1.2. Geometrical representation of the gradient.
The surface r — constant is a sphere of radius r with center at O(0,0,0), whose equation is 2
+y
2
2 +z
The components of gradr are: dr
x
a*" dr _ y dy r dr _ z dz r We obtain
If. grad r = — (i x + j r
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The magnitude of the gradient is
gradr
y Figure 1.3. Definition of the direction of gradr.
As for the direction of grad r, we define a vector OM = M — O, as shown in Figure 1.3. Noting that grad r = OM/r; where r is the distance (scalar) between M and O, we conclude that grad r and OM are collinear vectors. Therefore, grad r points to the direction of increasing r, or towards spheres with radii larger than r, as was indicated formally above. 1.5. The Divergence 1.5.1. Definition of Flux Consider a point M in the vector field A, as well as a differential surface ds at this point, as in Figure 1.4. We choose a point N such that the vector MN is perpendicular to ds. We call n the normal unit vector, given by the expression
n=
MN \MN\
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A vector ds, with magnitude equal to ds and direction identical to n is defined as
ds = n ds The flux of the vector A through the surface ds is now defined by the following scalar product
= A ds cosO
(1.6) N
A\e M
y
Figure 1.4. Definition of normal unit vector to a surface ds.
where 9 is the smallest angle between A and n. The flux is maximum when A and ds are parallel, or, when A is perpendicularly incident on the surface ds. Since ds is a vector, it possesses three components which represent the projections of the vector on the three planes of the system Oxyz (see Figure 1.5). Thus, ds has the following components dsx = dydz
dSy — dzdx dsz = dxdy With the components of A
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* 2
dsy= dxdz
0
x Figure 1.5. Components of the vector ds.
P'
dx
dz ,•'
S
Q'
dy , R — b.
A
^-
A/
\ \
^2
1
Figure 1.7. A tube of flux.
Because Sj and S2 are arbitrary surfaces, they can be approximated geometrically. It is clear that if S2 tends to Si and at the same time A2 tends to A i, the sum above tends to zero. Since the flux entering the tube is equal to the flux leaving it, we conclude that the flux through the closed surface, in this case, is zero. Utilizing the divergence theorem = cf A • ds = f divA. dv from which we note that divA. = 0. This leads to the conclusion that the flux is conservative (the flux in the tube is conserved). From the discussion here we conclude that when the flux is conservative, the divergence of the field is zero.
Figure 1.8. A radial vector field.
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Figure 1.9. Directions of A and ds for the field in Figure 1.8.
Figure 1. 10. A circumferential vector field.
1.5.4. Example of Divergence Consider a radial vector field as shown in Figure 1.8, and assume the magnitude of A is constant at all points on a sphere centered at M. To calculate the flux of the vector A through a spherical shell of radius R, we note that ds and A are collinear and in the same direction (as in Figure 1.9). We get
O = df A-ds = A Ads = A 0 ; thus, an augmentation of the charge in the volume occurs with time. Now, we will analyze, under local form, the Maxwell equations. • The equation
ro/H = J +
3D dt
expresses the manner by which a magnetic field can create a split into conduction current (associated with J) and a time variation of the electric flux density (associated with dD/dt). We assume first the situation in Figure 2.7, where there is no electric flux density, or, alternatively, the electric flux density is constant in time. Now the equation is ro/H = J . As we have seen in the previous section, H and J are connected by a rotation or curl relationship. The geometric relation between these quantities is demonstrated in Figure 2.7. The flux of the vector J is the conduction current. It is in general the dominant term in the relation while the term dD/dt, which will be discussed in more detail in subsequent paragraphs, is relatively small.
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Figure 2.7. Relation between conduction current density and magnetic field intensity.
• The equation
divB = 0 signifies, as shown in the previous chapter, that the magnetic flux is conservative. To understand this we can say that the magnetic flux entering a volume is equal to the magnetic flux leaving the volume. This relation corresponds to a condition which allows understanding of the field behavior and serves, in various cases, as an additional mean for determining the magnetic field intensity. However, Eq. (2.1) also established a relation between the magnetic field intensity H and J, and the same relation permits the determination of H as a function of J in a large number of practical cases. • The equation
rotE =
ae dt
is analogous to Eq. (2.1), showing that the time derivative of the magnetic flux density is capable of generating an electric field intensity E. The geometrical situation connecting these quantities is shown in Figure 2.8. Assuming that B increases as it comes out of the plane of Figure 2.8, the electric field intensity J is in the direction shown in Figure 2.8.
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Figure 2.8. Relation between the time derivative of the magnetic flux density and the electric field intensity.
•
The equation
divD = p demonstrates that the flux of the vector D is not conservative. We can easily imagine a volume in which there is a difference between the electric fluxes entering and leaving the volume. This situation is shown in Figure 2.9 where an electric charge is located at the center of a sphere. The flux traversing the volume is outward-oriented. D and p are related through the divergence, according to the relations shown in Chapter 1. The geometrical relation between the two quantities is shown in Figure 2.9. The flux of the vector D traversing the surface that encloses the volume V of the sphere is nonzero.
Figure 2.9. The nature of the electric flux.
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iron
Figure 2.10. Two types of anisotropic materials. The material on the left has grain-oriented structure while the one on the right is made of thin insulated sheets.
2.4. The Anisotropy It is possible to apply Maxwell's equations in various situations and in combinations of different materials. However, instead of discussing all possible applications, we prefer to present the equations through a general situation. For this purpose it is necessary to introduce the concept of magnetic anisotropy. Consider a material whose magnetic permeability is dominant in a certain direction. One such material is a sheet of iron with grain-oriented structure or thin plates made of sheet metal which form, for example, the core of a transformer, as in Figure 2.10. It is reasonable to assume that in both cases, the magnetic flux flows with more ease in the direction Ox. In the first case, this is due to the orientation of the grains and in the second due to the presence of small gaps between the layers of sheet metal. Assuming a field intensity H whose components Hx and Hy are equal to H and if, (J.^ and [iy are the permeabilities in the directions Ox and Oy respectively, we have B
x
=
and
We note that Bx is larger than By. In this case, there is an angle between H and B. If Hx=Hy, H forms a 45° angle with Ox. At the same
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time, B forms an angle different than 45° since Bx and By are different. We conclude that the relation
where )J, is a scalar, is not general since it does not satisfy the cases above. Because of this, we introduce the concept of the "permeability tensor" denoted by ki . In matrix algebra, a vector, for example B, is expressed as
B
*
The tensor HM| is a 3x3 matrix Vx
°
0
fly
0
0
where we have assumed, for the moment, that the off-diagonal terms are zero or that we have a diagonal tensor. The general expression B = ki H is, in matrix form,
X" By = B2
~flx
0
0
fly
0
0
0" 0
~HX~ Hy
flz_ Hz_
By appropriate matrix operations, we can write
We observe that when the material is isotropic, or if (\\,x = ji = jj,z = u,) the equation B = II (4, H assumes the scalar form B = jaH . We also
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observe that when the non-diagonal terms of ||(J,| are nonzero, there is an interdependency between variables. While in the example above, Bx depended only on Hx, now it may depend on all three components of H. In general, if the tensor jj, is not a diagonal tensor, we can write a more complex relation as
Besides the concept of anisotropy, which complicates the study of magnetic materials, we introduce another phenomenon, frequently encountered in electromagnetic devices. In these devices, the magnetic permeability is not constant but depends on the particular value of H in the magnetic material in question. This phenomenon is called "non-linearity." The general relation between B and H is now
2.5. The Approximation to Maxwell's Equations The complete set of Maxwell's equations is, for convenience, presented again: *\w\
rotH = J + — dt
(2.5)
divB = 0
(2.6)
=~ dt
/2 , it is evident that E2> El. Large field intensities (or gradients of potential) may exist in certain parts of the equipment. If these fields exceed allowable limits, they could cause harmful effects or damage to the equipment. We define now the dielectric strength K of an insulator. Consider an insulating material between two metallic plates separated by a distance / and subjected to a potential difference V, as shown in Figure 2.24. Due to the application of the potential V there is an accumulation of positive and negative charges on the two plates as shown. If we increase the potential V, a critical potential Vc is eventually reached at which the accumulated charge between the plates creates a current (or an electric arc)
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between the plates, penetrating or "breaking" the insulator. When this happens, the insulating properties of the material are lost. The dielectric strength is therefore defined as
K = ^-
(V/m)
(2.28)
K represents the maximum electric field intensity (and therefore the maximum potential difference per unit of length) an insulator can support without breaking down. Note that the units of K are the same as the units of the electric field intensity. Hence, returning to Figure 2.23, it is important that the highest field intensity in the equipment (in this case E2) does not exceed the dielectric strength of the material in which this field is encountered. In this sense, we observe that it is very important to know the electric fields in the equipment, in particular the high-intensity fields. A good, detailed knowledge of the field distribution allows the design of the device and optimization of its various dimensions so that the design is safe, compact, and done at a reasonable cost. Finally, we point out that an excessive electric field intensity not only damages equipment but can also be dangerous to personnel and to livestock that happen to be in the area of high field intensities. 2.7.5. Laplace's and Poisson's Equations of the Electric Field for Dielectric Media Assuming that in the domain under study there are no timedependent quantities, we can define an electric scalar potential V from which a conservative electric field intensity E = —gradV can be derived. This relation is valid for the electrostatic field because rctfE = 0, (rot(—gradV) = 0). Since rot(gradV) is always zero, this definition of the electric field intensity is correct. However, if dB/dt is not zero, we cannot use this definition. In the static case we have
di v D = p =p
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div e \gradV ) = — p which, in explicit form is
d
_
dV
o_
o
dx
dx
d
I _
r -
dy
dV
I
d
dV
dz
dz
(2.29)
I
dy
In two dimensions this equation becomes
d dV d dV — s -- 1 -- s — = -p dx dx dy dy
(2.30)
This is Poisson's equation and it defines the electric potential distribution in the dielectric domain where an electrostatic field exists. To solve this equation we must first impose the boundary conditions, or in other words, specify the potentials on the boundaries of the solution domain. In addition we must specify the geometry and the dielectric materials, as well as any static charge densities in the domain.
V
E
o
i
Figure 2.25. Electric field in a parallel plate capacitor due to a voltage difference on the plates.
If there are no static charges (p = 0) and a single dielectric material exists in the domain, the equation becomes
d2V2 dx
+
^ =0 dy
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(2.31)
This is a Laplace equation. In this case the source of the electric field in the domain of study is the boundary conditions through which potential differences are imposed. It must be pointed out that the analytic solution to this equation is extremely difficult for the majority of even the simplest realistic problems, and, in the case of complex geometries, practically impossible. For the time being we present the solution to this equation for a very simple problem. We use again the example of the parallel plate capacitor, where we wish to find the field intensity between the plates. Edge effects are neglected. The problem geometry is shown in Figure 2.25. The conditions on the boundaries of the geometry are V = Va at x = 0, and V — V^ at x — / . With the assumption that there are no edge effects, the problem is one-dimensional with variation in the Ox direction. Laplace's equation is therefore
d2v = 0 The solution to this problem can be written as V(x) = ax + b by direct integration. With the known boundary conditions we get
Va = a • 0 + b and
Vb =a.l + b which allows calculation of the constants a and fa. Substituting these in the solution we obtain
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With E= -gradV, we have in this case
^ .dV E = -i dx
Ex =
or
dV
dx
and, therefore,
E = F °~ F " I If Va>Vb, Ex is directed in the positive jc direction, or E is directed in the direction of decreasing potential, as required. 2.7.6. Laplace's Conductive Media
Equation
of the
Electric
Field
for
Here we use the "electric continuity" equation divJ — 0. Although this expression comes from an equation linked to magnetic cases, it deals with electrostatic fields and that is the reason why it is presented here. It is considered now that a potential difference is applied in conductive media. Using J =aE and E = -gradV we have divJ = divaE = diva (- gradV) = 0 or, as above
d l-r dV dx dx VJ
I
I
d ff dV dy dy VJ
I
T
d ft dV dz dz \J
n
— I)
— \J
/O
^9\
\£*.O£.)
which is a Laplace equation. Most of the considerations about this equation are similar to the situation with dielectric media, presented above. 2.8. Magnetostatic Fields The group of equations describing the magnetism in low frequency domain are
rom = J
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=0
dt In magnetostatics the quantities are independent of the time and we have
rotH = J
(2.33)
divB = 0
(2.34)
while the equation
ro/E = 0
(2.35)
does not play any role in this situation. The constitutive relations are
J=aE At first look, magnetostatic looks quite limited since the majority of devices have variable current sources, and/or have movement. However, when the structure is built in a way that we can neglect dB/dt in conductive materials, it is possible to treat it as a magnetostatic one. In other words, it is possible to study the structure at each position as a static one and, afterwards, compose the successive results in order to obtain the dynamic behavior of it. In addition we will present here the different types of magnetic materials, the expression of magnetic field energy and the concept of inductances. Although in some instances it will be necessary to use the notion of time, the results we obtain are static in nature.
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2.8.1. Maxwell's Equations in Magnetostatics 2.8. la. The Equation rotH = J This equation defines qualitatively and quantitatively the generation of H in terms of J. We recall that the same relation in integral form is
[' (rotH)-ds= *»j
f J-Js ^ij
(2.36)
where S is a surface on which H and J are defined. Using Stokes' theorem, the left-hand side of the expression can be written as
i(rotH)-ds = cf H • dl
(2.37)
where C is a contour, enclosing the surface S. The right-hand side of Eq. (2.36) represents the flux of the vector J crossing the surface S. This flux is the conduction current crossing S. That is <JH-dl =
(2.38)
which indicates that the circulation of H along a contour C encircling a surface S is equal to the current crossing this surface. Maxwell's equation rofH — J written in the form above is referred to as "Ampere's law." We look now at the application of this equation to the case of an infinite wire carrying a current / as shown in Figure 2.26. By choosing the surface Si as a circle of radius R, the application of Ampere's law is simplified:
(_H.d
=/
Since H and eft are collinear vectors in the same direction, the scalar product H • d[ is equal to the product of the magnitudes of H and eft. Because of the homogeneity of the material properties, H is identical at all points along Q and is not dependent on Q. Therefore, the integration becomes
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or
H2nR =
and
(2.39)
Figure 2.26. The use of Ampere's law to calculate the magnetic field intensity of an infinite wire.
Figure 2.27. The use of Ampere's law with an irregular contour.
We note here that the fact of choosing S such that di coincided with H facilitated the solution. Assume now that we choose the surface S shown in Figure 2.27. Since S2 is an irregular surface, we divide C2 into n segments and obtain
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This equation has n unknowns. It is important to point out that while Ampere's law is still valid, its application in this particular problem is practically impossible. An additional difficulty is the fact that, since H is always tangential to a circle whose center is the conductor, the scalar product Hn-din, changes to Hndlncos\pn) where the angle Bn, between Hw and d\ n, varies from point to point. We can choose still another type of surface, such as S3 shown in Figure 2.28, which does not contain the conductor. On this surface, the quantity
Jc-i Because the current crossing the surface S3 is zero, does this also imply that H = 0 ? In reality the field intensity H generated by the current does not depend on the surface we choose for the application of Ampere's law, hence, since / ^ 0 in the conductor, H is also not zero.
c, Figure 2.28. A contour that does not include the current.
In fact, we observe
in Figure 2.28 that
H j - ^ i >0
and that
H 2 • d\ 2 < 0 Only the sum of all sections yields zero, accounting for the zero net value. We point out that Ampere's law is always valid, but its application is not always a simple matter.
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An important aspect of the equation rotH. = J is apparent when we apply the divergence to both sides of the equation. By doing so, we obtain the equation divX = 0
(2.40)
which is the equation of electric continuity. This indicates that the conduction current, which is the flux of the vector J, is conservative; that is, a current entering a volume is the same as the current leaving the volume. 2.8.1 b. The Equation divB = 0 This equation is in a sense analogous to the equation divJ = 0 above. In this case, it is the magnetic flux that is conservative. We note that this equation does not indicate how B is generated; it only defines the conservative flux condition. We will see in subsequent paragraphs that the application of this condition provides a convenient relation for the solution of certain problems. 2.8.1 c. The Equation rotE = 0 This equation is a particular case of rotE> = —dB/dt and indicates how the electric field is generated due to the time variation of B. The fact that the curl of the electric field intensity is zero does not mean that in the magnetostatic case the electric field intensity is zero. There is no reason why an electric field external to the domain cannot be applied, which we can consider as constant. However, in the domain under study, with magnetostatics, we cannot have an electric field generated by devices contained within the domain. 2.8.2. The Biot-Savart Law The three equations above constitute the main relations of magnetostatics. We can attribute to Ampere's law, derived from rotH. = J , a certain prominence in relation to the other two equations since it relates the magnetic field intensity H to its generating source J. Although this law is valid in any situation, its application, in terms of solving practical problems,
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is limited to a few simple cases, unless we use approximations. One of these few applications with exact solutions is the example of an infinite wire, considered above, although considering an infinite wire is in itself an approximation.
Figure 2.29. Applying Biot Savart
The Biot-Savart law is an auxiliary expression for the calculation of H as a function of the current that generates it, but it is valid only in homogeneous material. It is necessary to note that Biot-Savart's law, conceptually, adds absolutely nothing to Maxwell's equations. We can view it as an algebraic variation to Ampere's law. This law was proposed by Biot and Savart as an experimental law. Biot and Savart's law was introduced relatively late in the development of field theory. This derivation is rather complex and involves electromagnetic quantities which we have not yet defined and we simply use it here as a given relation. To introduce Biot-Savart's law, we use Figure 2.29 where we wish to calculate the magnetic field intensity H at point P. This field intensity is generated by the current /, passing through a conductor of arbitrary shape. The Biot-Savart law is written in differential form as
47tr
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(2.41)
Figure 2.30. The use of Biot-Savart's law for the calculation of the magnetic field intensity of an infinitely long wire carrying a current /.
The wire is divided into small segments with which we can associate a vector dl, whose direction is the same as the current /. We now have to define a vector r as r = P — M. The summation of the vectors dH provides the field H generated by the current /, at point P. The direction of t/H is as indicated in Figure 2.29. One method of obtaining the direction of dH is to use the cross product dl x r . The magnitude of dH is given by Idl ' o 2 smv
4nr
(2.42)
where 0 is the angle between dl and r. Biot-Savart's law permits the calculation of H due to a conductor of irregular form. In this case we divide the conductor into a finite number of segments and sum the resulting values of dH vectorially. It is not difficult to write a computer program that performs these operations automatically. However, we can apply Biot-Savart's law in an analytic fashion only to a limited number of structures. As an example we look at the calculation of H generated by an infinite wire carrying current /. H is calculated at a point P at a distance R from the wire (see Figure 2.30). We note that dH is perpendicular to the plane of Figure 2.30. Its magnitude is Idl • ft 2 sitw
4nr
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or
dH =
4-Ti r
-costy
(2.43)
Noting that * tan 1000, the depth of penetration is much smaller. In this case, the considerations above are almost always valid.
E H B Figure 2.60. Relations between the electric and magnetic fields in a semiinfinite conducting block. The electric field intensity and the current density are perpendicular to the magnetic field intensity.
• As for the assumption that the fields are sinusoidal in nature, it is important to note that many electromagnetic devices operate under
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sinusoidal excitation. If the excitation is nonsinusoidal, we can decompose the excitation into sinusoidal components through the use of Fourier series. Each component will have different frequencies and, therefore, different depths of penetration. However, the fundamental frequency (the same frequency as the original excitation) possesses the largest depth of penetration since it has the lowest frequency. The rest of the harmonics, which normally have lower amplitudes, have lower depths of penetration because of their higher frequencies. Because of this, we normally use the same formula for nonsinusoidal fields as for sinusoidal fields. This is a valid assumption in a majority of practical cases; in other particular cases, where this is not valid, the situation must be considered on a case-by-case basis.
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3 Brief Presentation of the finite Element Method 3.1. Introduction In this chapter we will present, in a relatively brief and short manner, the Finite Element Method (FEM), which is an important tool in this work. It is not our intention here to present it in deep detail, because it can be found as main goal in some works listed in the bibliography section. Here only the practical aspects and concepts will be shown in a way that can be directly applied in the following chapters. The evolution of the finite element method is intimately linked to developments in engineering and computer sciences. Its application in a variety of areas, especially in the nuclear, aeronautics, and transportation industries, is testimony to the high degree of accuracy the method is capable of, as well as to its ability to model complex problems. Generally, in electromagnetics, the FEM is associated with variational methods or residual methods. In the first case, the numerical procedure is established using a functional that has be minimized. For each problem a particular functional has to be defined. It is worth mentioning that for the classical 2D problems, the functionals are well known, but for less usual phenomena a search for a functional is necessary, which can be a difficult task in some cases. Moreover, we do not work directly with the physical equation related to our problem, but with the corresponding functional. Contrarily, residual methods are established directly from the physical equation that has to be solved. It is a considerable advantage compared with the variational methods since it is comparatively simpler and easier to
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understand and apply. And certainly it is the main reason why nowadays most of FEM work is performed by using the residual method. The Galerkin method is a particular form of residual methods and it is widely used in Electromagnetism. This particular formulation is simple, practical to implement and, moreover, normally provides precise and accurate results. Because of these aspects we decided in this work to present solely the Galerkin method. Only the main points are here described, but the text is intended to be complete in order to furnish all the necessary elements and steps for its application. The text is presented for 2D cases, but for 3D problems the concepts are directly extended. In our experience, beginners have difficulties in understanding how the different concepts, formula, integration, etc. are connected. For this reason, in the following sections, we present two programs written in Fortran. They are listed and discussed and we believe that, in a practical manner, we show the way to implement two different types of finite elements. In the chapters ahead the application of the FEM for different electromagnetic problems is detailed, while in this chapter we emphasize the numerical method itself. 3.2. The Galerkin Method - Basic Concepts Because we are describing the finite element method in a relatively brief text, we have chosen to describe it using the Galerkin method applied to the electrostatic equation for dielectric media. All the following steps will be established for 2D domains and in this section we introduce the basic concepts of this method. 3.2.1. The Establishment of the Physical Equations The electric field intensity E is related to the scalar potential V as
E = -gradV And the Maxwell equation to be solved is divD = p
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(3.1)
Using D = sE and Eq. (3.1), we obtain
flf/'ve E = dive (-gradV) = p
or dive(-gradV)=p
or _ - s -
+ _- e
=-p (3.2)
3.2.2. The First Order Triangle In the FEM, the solution domain is subdivided or "discretized" in small regions called "finite element." For instance, in 2D applications, the domain can be discretized into finite area patches such as triangles. The points defining the triangles are the "nodes" or "degree of freedom" while the triangle itself is the "element." The assembly of elements is called "mesh".
Figure 3.1 . An element in a triangular element mesh.
In Figure 3.1 a generic triangle is shown. Because it is a first-order element the potential varies linearly within the triangle. For this type of element, the expansion of the potential is
V(x, y)= «j + a2x -I- a3y
(3.3)
This relation should be held at the nodes of the element. For the nodes in Figure 3.1, we get F, = «j + a2x{ + a^y\
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(3.4a)
V2 = a{ +a2x2
(3.4b)
V3=a,+a2x3+a3y3
(3.4c)
From these three equations we determine the required values of al, Ct2, and a3 by calculating the determinants in the following: AV
Yv
1 \ D
i y\
V2
x2
¥V
AV
3
~\)
1
1
^1
^1
y2
«2 - — 1
^2
^2
3 y?>\1
1
^3
J^3
1
X,
v{
1
*2
V2
^
1 x3
1 Xl D = 1 x2
y, y2
1
^3
^3
(3.5)
The value of D equals twice the area of the element as can be verified directly. Substituting the values of a{, a2, and #3 in Eq. (3.3) and simplifying the expressions gives 1
(3.6)
where Pi =
(3.7)
while the remaining terms: p2, ^2, r2, /? 3 , ^3, and r3 are obtained by cyclical permutation of the indices. Because E= -gradV, we have
dV dV = iEx+]Ey =-i-r--j-rox
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oy
(3.8)
and with the expression above we have
(3.9) The expression (3.6) can also be written as follows:
where —
y) = — These functions above are called "shape functions" and because the equations (3.4) must be verified, it is easy to observe that
since, for example, V{ = IF, +OF 2 +OF 3 . Moreover, (^j varies linearly between 1 at node 1 to zero at nodes 2 and 3. 3.2.3. Application of the Weighted Residual Method Now it is time to distinguish between the "exact solution" Ve and the solution obtained with the finite element method "V".
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
For the exact solution we have, from Eq (3.2): div(zgradVe ) + p = 0 However, the solution we obtain using the FEM is an approximation and different from the exact solution. When substituting this solution into Eq. (3.2) it generates a "residual" R: i(*) and <J>2(*) (defined the same way as Equations (3.11ac)), if we equate node 1 with node k and node 2 with node k +1. Therefore, instead of performing the integration node by node (as suggested by Eq. (3.16), we can integrate element by element. Furthermore, we can use the functions (j>( as the weighting functions. When we do so, the method is called the "Galerkin method". This represents a particular choice of weighting functions and therefore a particular weighted residual method. The Galerkin method is widely used in electromagnetics, while other types of weighted residual method are seldom used. For this reason, we will consider here only the Galerkin method.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
nodes
k-\
k
k+\ k+2
Figure 3.3. Sum of the weighting functions for element " n. " .
3.2.4. Application of the Finite Element Method and Solution The integrals on Q in Eq. (3.15) for the discretized domain become
£
f [KgradV'grad$n-p$n]da
=Q
(3.17)
n=\,N •""
where n represents a generic element and N is the number of elements in the solution domain. The evaluation of the integral in Eq. (3.17) for an element n follows. Let us recall that equation (3.6) is given by
-
y) = i=\ or J
—
~
D
From this, gradV is
i-
~
(3.18)
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Using the equations (3.11) we obtain
(3.19a)
(3.19b) .1 .1 =\ — 43 +J —
(3.19c)
Since Eqs. (3.18) and (3.19) are constants, the first term of Eq. (3.17) becomes for node 1,
and, noting that the integral on Sn equals the area of element n (that is, it equals D / 2 ), we get in matrix form
2D Extending the integral for the nodes n = 2 and H = 3, we obtain the elemental stiffness matrix v
\
2D
Symmetric Symmetric
2
Symmetric
q$ q-$
(3-20)
Vi
The assembly of the elemental matrices into a global matrix requires that the terms of this matrix be assembled in the lines and columns corresponding to the numbering of the nodes in the global mesh. The solution of the system is performed by any linear system solving technique (as Gauss Elimination) after inserting the boundary conditions into the
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
global system. For implementation purposes, a Fortran program is presented in section 3.3. Now, we evaluate the second term of Eq. (3.17) tnpds
(3.21)
Each of the functions (j); equals 1 at node i and decreases to zero at all other nodes of the element. For example, (j)j equals 1 at node 1 and zero at nodes 2 and 3, as shown in Figure 3.4.
Figure 3.4. Function (j), for a triangular element.
The evaluation of the integral in Eq. (3.21) corresponds to calculating the volume of the pyramid of height 1 shown in Figure 3.4. This gives
1
D
This, however, is only due to the function (j), . Performing identical calculations on <j)2 and (j)3 , we obtain the contribution due to charge density as
(3.22)
which is called the "source term" to be assembled on the right-hand side of the matrix system, since it does not depend on the unknown potentials and
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
on locations corresponding to the numbering of the three nodes. Equation (3.22) is also called the "source" term, since it generates electric fields in addition to those generated by the imposed potentials, called Dirichlet boundary conditions. If p = 0, the electric field is generated only by the Dirichlet conditions on the boundary. This is discussed in the following section. 3.2.5. The Boundary Conditions The boundary conditions are related to the first integral on the righthand side of Eq. (3.15), which is: cf, .WegradV • ds = 0
(3.23)
JL(s)
There are two types of boundary conditions we need to contend with: A
K
W.
imposed potential
unknow potential
Figure 3.5. Dirichlet boundary condition scheme (1D analogy).
Figure 3.6. Neumann boundary condition scheme.
3.2.5a. Dirichlet Boundary Condition - Imposed Potential Consider a physical configuration in which the potentials are known on part S, of the boundary. This is called a "Dirichlet boundary condition". When
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
we come to write the equations for the unknowns at the nodes of the mesh, the weighting functions Wk are only needed for the internal nodes of the mesh. At the Dirichlet boundary nodes the weighting functions are zero (see Figure 3.5). This condition assumes that Eq. (3.23) is satisfied. 3.2.5b. Neumann Condition - Unknown Nodal Values on the Boundary In certain cases, on part of the boundary S2 — L(S)— S^, the values of the potential are unknown. On this part of the boundary, Eq. (3.15) must be written and the weighting function in Eq. (3.23) is not zero. Moreover, because the integral in Eq. (3.23) is set to zero, we have
£gradV -ds = Q
(3.24)
Examining this expression, and taking into account the scalar product, we conclude that the electric field intensity E=-grad V must be tangential to the boundary S2 as shown in Figure 3.6. 3.3. A First-Order Finite Element Program In this section we present a first order FE program in the most elementary form possible to touch the essential aspects of the method. This program is intended to solve the equation
d dx
o
^
dV dx
d dy
11
-p
o
^
dV dy
— — /"\
—
jj
It was shown above that the Galerkin method conducts to the matricial contributions and the aim here is to present the implementation of these matricial terms. First, we define the variables involved: • NNO number of nodes • NEL number of elements • NCON number of boundary lines on which potentials are known
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
• NMAT number of dielectric materials • KTRI(NEL,3) array that indicates the node numbers of each element • MAT(NEL) indicates the material number of each element • RO(NEL) indicates the material static change of each element • PERM(NMAT) permittivities of the NMAT materials • X(NNO) x coordinates of the NNO node numbers • Y(NNO) y coordinates of the NNO node numbers • VI(NCON) imposed potentials on the NCON boundary lines • NOCC(NCON,20) node numbers at which the potential VI is imposed (maximum 20 nodes per equipotential line) • SS(NNOxNNO) global matrix of coefficients of the system of equations • W(NNO) vector of node potentials • VDR(NNO) vector of the right-hand side of the matrix A listing of the program, written in FORTRAN 77 is reproduced below: C
MAIN PROGRAM COMMOrWATA/KTRI(200,3),MAT(200),X(150),Y(150),PERM(10) *VI(10),NOCC(10,20),RO(200) COMMON/MATRIX/SS(150,150),W(150),VDR(150)
C
-— CALL ZERO TO NULL THE VARIOUS ARRAYS CALL ZERO
C
CALL INPUT TO READ DATA CALL INPUT(NNO,NEL,NCON)
C
CALL FORM TO FORM THE MATRIX SS CALL FORM(NEL)
C
CALL CONDI TO INSERT BOUNDARY CONDITIONS CALL CONDI(NCON,NNO)
C
—- CALL ELIM TO SOLVE THE MATRIX SYSTEM CALL ELIM(NNO)
C
-
CALL OUTPUT TO PRINT THE RESULTS
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
CALL OUTPUT(NNO,NEL) STOP END C C
SUBROUTINE ZERO COMMON/DATA/KrRI(200,3),MAT(200),X(150),Y(150),PERM(10) *VK10),NOCC(10,20),RO(200) COMMON/MATRIX/SS(150,150),W(150),VDR(150)
MAX1 = 150 MAX2 = 10 MAX3=20 DO 1 I=1,MAX1 W(I)=0. VDR{I)=0. DO 1 J=I,MAX1 1
SS(I,J)=0. DO2I=1,MAX2 DO2J=1,MAX3
2
NOCC(I,J)=0 RETURN END
C C
SUBROUTINE INPUT(NNO,NEL,NCON) COMMON/DATA/KrRI(200,3),MAT(200),X(150),Y(150),PERM(10) *VI(10),NOCC(10,20),RO(200) READ(5,*)NNO,NEL,NCON,NMAT C
READ THE MESH STRUCTURE DO 1 I=1,NEL
1 C
READ(5,*}KTRI(I,1),KTRI(1,2),KTRI(1,3),MAT(1),RO(I) READ NODE COORDINATES DO2I=1,NNO
2
READ(5,*)X(I),Y(I)
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
C
READ BOUNDARY CONDITIONS DO3I=1,NCON READ(5,*)VI(I) READ(5,*)(NOCC(I,J),J = 1,20)
3
CONTINUE
C
READ PERMITTIVITIES OF MATERIALS DO4I-1,NMAT
4
READ(5,*)PERM(I) RETURN END
C C
SUBROUTINE FORM(NEL) CONIMON/DATA/KTRI(200)3),MAT(200),X(150))Y(150),PERM(10) *VK10),NOCC(10,20),RO(200) COMMON/MATRIX/SS(150,150),W(l50),VDR(150) DIMENSION NAUX(3), 5(3,3) C
DO FOR NEL ELEMENTS DO 11=1,NEL N1=KTRI(I,1) N2=KTRI(I,2) N3=KTRI(I,3) NM=MAT(I)
C-
-
CALCULATE Ql, Q2, Q3, Rl, R2, R3 Q1=Y(N2)-Y(N3) Q2=Y(N3)-Y(N1) Q3=Y(N1)-Y(N2) R1=X(N3)-X(N2) R2=X(N1)-X(N3) R3=X(N2)-X(N1) XPERM=PERM(NM)
C
-CALCULATE DETERMINANT, TWICE THE AREA OF TRIANGLE DET=X(N2)*Y(N3)+X(N1)*Y(N2)+X(N3)*Y(N1) *-X(Nl)*Y(N3)-X(N3)*Y(N2)-X(N2)*Y(Nl)
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
COEFF=XPERM/DET/2. ROEL=-RO(I)*DET/6. C--
CALCULATE THE TERMS S(3,3) S(l,l)=COEFF*(Ql*Ql+Rl*Rl) S(1,2)=COEFF*(Q1*Q2+R1*R2) S(l,3)=COEFF*(Ql*Q3+Rl*R3) S(2,1)=S(1,2) S(2,2)=COEFF*(Q2*Q2+R2*R2) S(2,3)=COEFF*(Q2*Q3+R2*R3) S(3,1)=S(1,3) S(3,2)=S(2,3) S(3,3)=COEFF*(Q3*Q3+R3*R3)
C TERM
ASSEMBLE THE S(3,3) INTO THE MATRIX SS(NNO.NNO) AND SOURCE NAUX(l)=Nl NAUX(2)=N2 NAUX(3)=N3 DO2K=1,3 KK=NAUX(K) VDR(KK)=VDR(KK)+ROEL DO2J=1,3 JJ=NAUX(J)
2-
SS(KK,JJ)=SS(KK,JJ)+S(K,J)
1
CONTINUE RETURN END
C C
SUBROUTINE CONDI(NCON,NNO) COMMON/DATA/KrRI(200)3),MAT(200),X(150),Y(150),PERM(10) *VI(10),NOCC(10,20),RO(200) COMMON/MATRIX/SS(150,150),W(150),VDR(150) DO1I=1,NCON DO2J=1,20
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
NOX-NOCC(I,J) IF(NOX.EQ.O)GOTO 1 C
ZERO THE COEFFICIENTS IN LINE OF MATRIX SS DO3L=1,NNO
3 C
SS(NOX,L)=0. SET THE DIAGONAL TO 1. SS(NOX,NOX) = 1.
C
PLACE IMPOSED POTENTIALS IN THE RIGHT HAND SIDE VDR(NOX)=VI(I)
2
CONTINUE
1
CONTINUE RETURN END
C C
SUBROUTINE ELIM(NNO) COMMON/MATRIX/SS(36,36),W(36),VDR(36) C
GAUSSIAN ELIMINATION NN=NNO-1 DO 1 I=1,NN DO1M=I+1,NNO FACT=SS(M,I)/SS(I,I) VDR(M) - VDR(M)-VDR(I)*FACT DO5J=I+1,NNO
5
SS(M,J)=SS(M,J)-SS(I,J)*FACT
1
CONTINUE W(NNO)=VDR(NNO)/SS(NNO,NNO) DO7I=NN,1,-1 SUM=0. DO8J=I+1,NNO
8
SUM=SUM+SS(I,J)*W(J) W(I) = (VDR(I)-SUM)/SS(I,I)
7
CONTINUE RETURN
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END C C
SUBROUTINE OUTPUT(NNO,NEL) COMMON/DATA^TRI(200,3),MAT(200),X(150),Y(150),PERM(10) *yi(10),NOCC(10,20),RO(200) COMMON C
/MATRIX/SS(150,150),W(l50),VDR(150)
PRINT THE POTENTIALS AT THE NODES DO 1 I=1,NNO
1
WRITE(6,100)1, W(I)
100
FORMAT('NODE-',I3; POTENTIAL=',E10.4)
C
PRINT THE FIELDS IN THE ELEMENTS DO2I=1,NEL
C
O^II.GRADTO(^JCUIATETHEFELJDSORGRADENTS N1=KTRI(I,1) N2=KTRI(I,2) N3=KTRI(I,3) CALL GRAD(N1,N2,N3,EX,EY) EMOD=SQRT(EX*EX+EY*EY)
2
WRITE(6,101)I,EX,EY,EMOD
101
FORMATf ELEMENT-',I3,' EX=',E10.4,' £¥=',£10.4,' *EM=',E10.4) RETURN END
C C
SUBROUTINE GRAD(N1,N2,N3,EX,EY) COMMON/DATA/KrRI(200,3),MAT(200),X(150),Y(150),PERM(10) *VI(10),NOCC(10,20),RO(200) COMMON/MATRIX/SS(150,150),W(I50),VDR(150) Q1=Y(N2)-Y(N3) Q2=Y(N3)-Y(N1) Q3=Y(N1)-Y(N2) R1=X(N3)-X(N2)
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R2=X(NI)-X(N3) R3=X(N2)-X(N1) CALCULATE DETERMINANT, TWICE THE AREA OF TRIANGLE DET=X(N2)*Y(N3)+X(Nl)*Y(N2)+X(N3)*Y(Nl) *-X(Nl)*Y(N3)-X(N3)*Y(N2)-X(N2)*Y(Nl) EX=-(Ql*W(Nl)+Q2*W(N2)+Q3*W(N3))/DET EY=-(R1*W(N1) + R2*W(N2)+R3*W(N3))/DET RETURN END
3.3.1. Example for Use of the Finite Element Program Suppose that we wish to find the electric field intensity and the potential distributions within the geometry shown in Figure 3.7 where Cj = 5s o • A finite element mesh is shown in Figure 3.8 with element and node numbers shown. In this case, there is no static charge in the domain, and, therefore, RO(I) equals zero in all the mesh elements.
F=100
Figure 3.7. A simple geometry used to demonstrate the use of the finite element program.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
»
10
14
13
element number >node number 3
4
5
15
16
17
18
24
30
36
^x
10
0
Figure 3.8. Finite element discretization of the geometry in Figure 3.7. Circled number are element numbers, others are node numbers. Numbers on the axes are dimensions.
The various variables for this mesh are: NNO = 36 (number of nodes) NEL = 50
(number of elements)
NCON = 2 (number of equipotential boundary lines) NMAT = 2 (number of materials) The arrays KTRI, MAT and RO corresponding to the elements are: (1st triangle)
18
2
1
0.
17
8
1
0.
29
3
1
0.
28
9
1
0.
10
17
11
0.
(material 2, triangle 17)
29
35
36
0.
(triangle 50)
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
The array X and Y are given as: 0. , 10.
(Node 1)
2. , 10.
(Node 2)
10., 0.
(Node 36)
The boundary conditions are: 100.
(potential on upper boundary)
12345600....
(nodes with potential 100 V)
0
(potential on lower boundary)
31323334353600....
(nodes with potential 0 V)
Permittivities are given as: l . ( £ r of material 1) (£ r of material 2) The results obtained from this program are listed below: NODE-
1
POTENTIAL=. 1OOOE+03
NODE-
2
POTENTIAL- .1000E+03
NODE-
3
POTENTIAL- .1000E+03
NODE-
4
POTENTIAL-.1000E+03
NODE-
5
POTENTIAL- .1000E+03
NODE-
6
POTENTIAL-. 1000E+03
NODE-
7
POTENTIAL- .8010E+02
NODE-
8
POTENTIAL- .7960E+02
NODE-
9
POTENTIAL- .7740E+02
NODE-
10
POTENTIAL-.7145E+02
NODE-
11
POTENTIAL^ .7162E+02
NODE-
12
POTENTIAL^ .7612E+02
NODE-
13
POTENTIAL=.6120E+02
NODE-
14
POTENTIAL- .6089E+02
NODE-
15
POTENTIAL^ .5856E+02
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
NODE-
26
POTENTIAL .2274E+02
NODE-
27
POTENTIAL .2466E+02
NODE-
28
POTENTIAL .2565E+02
NODE-
29
POTENTIAL .2544E+02
NODE-
30
POTENTIAL .2453E+02
NODE-
31
POTENTIAL .OOOOE+00
NODE-
32
POTENTIAL .OOOOE+00
NODE-
33
POTENTIAL .OOOOE+00
NODE-
34
POTENTIAL .OOOOE+00
NODE-
35
POTENTIAL .OOOOE+00
NODE-
36
POTENTIAL .OOOOE+00
ELEMENT- 1EX= .OOOOE+00 EY= -.1020E+02 EM= .1020E+02 ELEMENT- 2EX= .2508E+00 EY= -.9950E+01 EM= .9953E+01 ELEMENT- 3EX= .OOOOE+00 EY= -.1130E+02EM=
.1130E+02
ELEMENT- 4EX= .1098E+01EY= -.1020E+02EM= .1026E+02 ELEMENT- 5EX= .OOOOE+00 EY= -.1427E+02 EM= .1427E+02
ELEMENT- 37EX= .4643E+OOEY=- 1307E+02 EM= .1308E+02 ELEMENT- 38EX= .1063E+00 EY=- 1343E+02 EM= .1343E+02 ELEMENT- 39EX= .2174E+01 EY=- 1135E+02 EM= .1156E+02 ELEMENT- 40EX= .4560E+00 EY=- 1307E+02 EM= .1308E+02 ELEMENT- 41EX= .3197E+00 EY=- .1137E+02EM= .1138E+02 ELEMENT- 42EX= .OOOOE+00 EY=- 1105E+02EM= .1105E+02 ELEMENT- 43EX= -.9604E+00 EY=- .1233E+02 EM= .1237E+02 ELEMENT- 44EX= .OOOOE+00 EY=- .1137E+02 EM= .1137E+02 ELEMENT- 45EX= -.4951E+00 EY=- .1283E+02 EM= .1284E+02 ELEMENT- 46EX= .OOOOE+00 EY=- 1233E+02 EM= .1233E+02 ELEMENT- 47EX= .1063E+00 EY=- 1272E+02EM= .1272E+02 ELEMENT- 48EX= .OOOOE+00 EY=- 1283E+02 EM= .1283E+02
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
ELEMENT- 49 EX= .4560E+00 EY=-.1226E+02 EM= .1227E+02 ELEMENT- 50 EX= .OOOOE+00 EY=-.1272E+02 EM= .1272E+02
Equipotential lines obtained for a similar case with the EFCAD finite element program (a much more sophisticated software package), using a larger number of nodes are shown in Figure 3.9. The electric field intensity lines are shown in Figure 3.10. As a final note we observe that the program presented above contains approximately 150 lines of code. This program can calculate most realistic problems and is much more flexible than analytic methods used before FEM methods were established and implemented.
Figure 3.9. Equipotential lines for the geometry in Figure 3.7.
Figure 3.10. Field intensity lines for the geometry in Figure 3.7
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
3.4. Generalization of the Finite Element Method The finite elements described in the previous sections are very simple, primarily because we allowed only linear variation between the nodes of the elements. There are more accurate finite elements, but their introduction requires some concepts which we will introduce in the following section. First, it is worth mentioning here that, as an example, for a ID element, with a linear variation of the potential between the nodes, two nodes are necessary. The potential varies as V(x) = a{ +a2x We used the two nodes to evaluate the two constants «, and a2 by satisfying this equation at the location of the two nodes:
V2 = a} +a2x2 Here, the approximation for the potential is a first-order polynomial approximation. If we wish to obtain better accuracy, we can use quadratic elements which have the following variation for potential:
V(x) = «j + a2x + a3x2
(3.25)
This approximation requires three nodes to determine the constants a\ , a2 , and a3 . Assuming an element with three nodes is given, the constants are evaluated from the following:
V2 =
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
where Fj , V2 , and F3 are the unknown potentials at the coordinates (x^ ) , (x2), and
The main point in the discussion above is that there is a relationship between the order of the approximation and the number of nodes defining the element. Although we used here a ID element, this is also true in 2D and 3D elements. For example, a quadratic variation of the potential in 2D element is
(x,y) \ = ai + a2X + a^y + a^xy + a^x
+2 a^y 2
(3.26)
This requires six nodes, such as a six-node triangular element. This will be discussed shortly. 3.4.1. High-Order Finite Elements: General Figure 3.11 shows some of the most commonly used finite elements in one, two, and three dimensions. The higher order (second- or thirdorder) elements are also called high-precision elements. There are many other finite elements but the elements shown in Figure 3.11 are the most commonly used in electromagnetic applications. Information about additional elements can be found in references in the bibliography section. To apply the finite elements shown here, it is first necessary to introduce some notation and relations, which we do in the following section. 3.4.2. High-Order Finite Elements: Notation To facilitate the definition of various finite elements, we introduce the idea of a "reference" or "local" element and the reference or local system of coordinates or space. Figure 3.12 shows an example and the relationship between the local and global systems of coordinates. The various relations needed to define an element are generated in the local system of coordinates because it is easier to do so. Then, a unique transformation is established which transforms the element from the local coordinate system into the global coordinate system. This transformation is accomplished by
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
the so-called "geometric transformation functions" or "mapping functions" or "shape functions" which express the real coordinates JC, y in terms of the local coordinates u, v .
a.
2 nodes, linear 3 nodes
^^ quadratic
'
6 nodes
J^l^ cubic
9 nodes,
quadratic
cubic
an 8 nodes
12 nodes
quadratic
cubic
IQnodesj
quadratic 20 nodes (
linear
quadratic
Figure 3.11. a. ID elements, b. Triangular 2D elements, c. Quadrilateral 2D elements, d. Tetrahedral 3D elements, e. Hexahedral 3D elements.
In Figure 3.12, the triangle in local coordinates is defined as
w>0
v>0
u + v 2
13 r
3_
>f det J dudv
^2 /3.
Since det J = Z), we have
E
~D
r\
-v
q2
The integrals give Vz and the final result is
2D
Symmetric Symmetric
r r
23
Symmetric
(3.67)
^3^3 +
This, of course, is identical to the elemental stiffness matrix we obtained in Eq. (3.20), Now that the elemental stiffness matrix has been calculated, we consider the source term which is 1-H-V
u
p det Jdudv
(3.68)
V
We have
l-w-v u
dudv =
pD
(3.69)
V
This is again identical to the expression in Eq. (3.22). This source term is independent of the potentials and therefore is part of the right-hand side of the system of equations.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
3.5. Numerical Integration When calculating, for example, the first term of Eq. (3.69), it is necessary to evaluate the following integral:
pZ>jjj£
V~v
2
V
(\-u-v)dudv=pD^ u
uv
which is the result shown in Eq. (3.69). It is clear from the example above that even for the very simple 2D element, the integration requires much work. This is in spite of the fact the det J is a constant. For second-order elements, the shape functions are much more complex and det J is not necessarily constant. Therefore, in general, it is practically impossible to evaluate the integrals analytically. Because of these reasons, the application of finite element codes is normally associated with numerical integration, and efficient integration algorithms feature prominently in fast and efficient finite element codes. Although any integral required for finite element calculations may be performed using the analytical expressions as above, it is not practical to calculate the integrals in this fashion. It is more common and more practical to use numerical integration methods of the type:
(3.70)
/=!
This means that the integrand K is not modified and therein lies the most attractive feature of these methods. In this form, r is the number of integration points, ut the coordinates of the integration point, and wt
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
weights associated with the integration points. The integral is reduced to a sum over a relatively small number of values as we shall see shortly. In general, integration over each type of finite element can be performed by different numbers of integration points r . Depending on the degree of the integrand terms, the number of points defines the accuracy of the integration. For each case, there is a number of integration points that provides exact value for the integral to be performed and it is useless to employ a larger number of integration points. Additional information on integration methods may be found in the bibliography section. Also, it is worth mentioning that the weights and integration points exist as tabulated values for all practical applications. As an example, for triangular elements with a single integration point (r = l), we get 1
1
1
U\ - —
Vi = —
Wi = —
1
3
l
3
l
2
Suppose that we wish to evaluate the first term in Eq. (3.68) f P (l - u - v)p det Jdudv = p det J P P (l - u - v)dudv In this case, the integrand is /(w,v) = l — u — v and with r = 1, we get
pD 6 This is an exact result because the polynomial approximation over the element is first order and r = 1 can handle it. Suppose now that we choose r = 3 ; that is, we choose to integrate using three integration points (see Figure 3.15). In this case the points and weights are
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Mi1 =
= 0
U-\ =— J 2
= —
V? = 0 3
2
_J_ 6 and we obtain
p det J r f
6l
2
/(«, v)dudv = pD 6
Figure 3.15. Three integration points for a triangle (one possible choice).
Figure 3.16. Three integration points for a triangle (a second choice).
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For, r = 3 it is also possible to use a different set of integration points (and associated weights) as shown in Figure 3.21:
1
2
1
Mi = — 1
M? = — 2
Mi = — 3
6
1 V =
' 6 _1
6
3
V 2 =-
6
1
_1 6
3
_2 3 _1 6
These points and weights give the same result. It is worth reiterating that the integration points and weights for any order elements are available in tables and need merely to be applied. There is rarely any need to find these points and weights. In table 3.1, the relation between polynomial order and number of integration points for triangular elements is shown. Order m 1 2 3 4 5 6
Integration points r 1 3 4 6 7 12
Table 3.1. Number of integration points for triangular elements.
3.6. Some 2D Finite Elements Putting together the concepts described in the previous sections, we can summarize the characteristics of a finite element. The finite element is described by: • The shape of the element (triangular, quadrilateral, etc.). • The coordinates of its geometric nodes.
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• The number of unknowns (or degrees of freedom). The element in Figure 3.17 has six nodes and therefore six unknowns. • The nodal variable (V in the example presented at the beginning of the chapter). • The polynomial basis of the element. For Figure 3.17, the polynomial basis is
v u2
uv v2
• The class or type of continuity: here only C considered.
continuity is
• The shape or mapping functions N(u,v) and its derivatives dN/du,dN/dv
(in 3D also dN/dp).
Also, the interpolation functions,
which in this text are the same as the shape functions (isoparametric elements).
(0,1)
(0,1/2)
(0,0)
(1/2,1/2)
(1/2,0)
(1,0)
«
Figure 3.17. Nodes of a quadratic triangular element.
The numerical integration table, indicating how the element is integrated. Conceptually, this step is independent of the finite element but, in practice, each finite element has its integration table and may be integrated differently.
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3.6.1. First-Order Triangular Element
(1,0)
(0,0)
(0,1)
0
«
X
Figure 3.18. Triangular element in local and global coordinates.
Node 1 2 3
[N]
[dN/du]
\-u-v u
_i
V
1 0
[dN/dv] -1 0 1
Table 3.2. Shape functions and their derivatives for the triangular element in Figure 3.18.
Polynomial basis: [1
II
V]
Note: analytical integration is both possible and recommended in this element. The coordinates of the nodes in the local coordinates are (0,0; Vfc,0; 1,0; VV/z; 0,1; 0, Vfe). The shape functions for this element are given in Table 3.3 and, because these are second-order, the element edges in the global coordinate system may be curved as shown in Figure 3.19.
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3.6.2. Second-Order Triangular Element 4V
47
u
0
x
Figure 3.19. Second-order triangular element in local and global coordinates.
Node
[TV]
1
- 1([ - 2t) 4ut - «(l - 2w) 4wv -v(l-2v) 4vt
2 3 4 5 6
[dN/du] l-4t 4(f-w) -l + 4w
[dN/dv] \-4t -4u
4v 0
4w -l + 4v
-4v
4(f-v)
0
where: ^ = 1 - u - v Table 3.3. Shape functions and their derivatives for 6-node, quadratic triangular elements.
Polynomial basis: [1
U
V
U2
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UV 1
The Jacobian J is given as
JT _
~
'dN_ du EC
y\ «-.)
y]=
-4u
dv Order m
Integration Points
1 2
1 3
5
7
«, 1/3 1/6 2/3 1/6 1/3
a l-2a a b \-2b b
v
i
w,
1/3 1/6 1/6 2/3 1/3
1/2 1/6 1/6 1/6
a a \-2a b b 1-26
9/80 0.066197076 0.066197076 0.066197076 0.062969590 0.062969590 0.062969590
(3.71) where: a— = V0.470142064; wiieie. w . T / viTJ£,VHJT ,
u—\j, = 0.101286507 iv.iz.ovjv /
Table 3.4. Integration points and corresponding weights for one, three, and seven-point Gauss-Legendre integration.
3.6.3. Quadrilateral Bi-linear Element
-1
(&
0
X
Figure 3.20. Quadrilateral bi-linear element in local and global coordinates.
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Node
[N]
[dN/du] (-l + v)/4
1
[dN/dv]
2 3
(l + v)/4 (l + v)/4
4
+ v)/4
Table 3.5. Shape functions and their derivatives for the quadrilateral bi-linear element.
Polynomial basis:
[l
u
v
uv]
For numerical integration, it is recommended that four integration points be used. This assumes exact integration up to a third-degree polynomial. The integration points are
=1 3.6.4. Quadrilateral Quadratic Element 4V 1
-1
u -1
0
X
Figure 3.21. A quadrilateral, quadratic finite element in local and global coordinates.
Because this element is quadratic, curvilinear edges in the global system of coordinates can be modelled.
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Node
[N]
3
- (l + w)(l - v)(l - u + v)/4 (l - v) (2w - v)/4 - (l + w)(« - 2v)/4 2 2 (l + w){l-v )/2 Tl-v V2 -(l + w)v
4
[ dN/du]
[dN/dv]
6 7 8
Table 3.6. Shape functions and their derivatives for the quadratic quadrilateral element in Figure 3.21.
Polynomial basis:
[l u v u2
uv v2
U2v HV 2 ]
To perform integration over this element the procedure used for the bi-linear element may be used here as well. The quadratic quadrilateral element is an "incomplete" element j f\ because the ninth term (u v ) in its polynomial basis is missing; its presence would make the polynomial expansion complete. The complete element requires an additional node at u = 0, v = 0 . However, the element as shown here is actually more often used than the "complete" element. 3.7. Coupling Different Finite Elements 3.7.1. Coupling Different Types of Finite Elements During the discretization process of a physical domain it is possible to use more than one type of element in the same domain, provided the continuity between elements is maintained. Figure 3.22 shows the use of triangular and quadrilateral elements.
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For each of the elements, the regular shape functions and integration are used separately without modifications. The coupling is done through the common node designation. This method is quite useful, especially when the geometry can be better described with a combination of elements. Another type of coupling between different elements is shown in Figure 3.23a. This, however, requires some attention because element (1) is linear while element (2) is a modified quadratic element. To match the two elements in Figure 3.23a, we start with element (2) shown in local coordinates in Figure 3.23b. Node no. 8 must be removed and, therefore, the quadratic element will be modified, since there are now only seven shape functions. To do so, we start with the eight shape functions of the regular quadratic element. These are
N
N
N
Nt]
Now, node no. 8 is assumed to be dependent and its potential is forced to equal the average between the potential at node no. 1 and node no. 7. This is accomplished by modifying the shape functions of the element as follows:
N
N4
N5
N6
N7+Ns/2\
Figure 3.22. Coupling of a triangular and a quadrilateral element.
Figure 3.23a. Coupling of linear and quadratic elements.
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1 t,
7
|/
6.
5 J k
,4
8s/ \ (
linear- —*
u^-
*-
2' t 3 quadratic
1
Figure 3.23b. Modifying the quadratic element.
Assuming now that in the process of defining the mesh, node no. 2 must also be eliminated (because another first-order triangular element is connected to the edge on which node no. 2 is placed), the shape functions become
N=[N}+N%/2 + N2/2
Ni+N2/2
N4
N5
N6
N7+Ns/2]
This method is commonly used when using the so-called adaptive methods of mesh generation. The main purpose of this seemingly more complicated method of discretization is to improve the solution while decreasing the amount of computation needed. 3.8. Calculation of Some Terms in the Field Equation In this section, we evaluate some terms that appear in electromagnetic field equations. These will be used in the following chapters when we explore physical phenomena. In support of the following sections, we use second-order triangular elements which were discussed in section 3.6. Consider the hypothetical general equation:
div(a grad U)+bU + c
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dx
+5= 0
(3.72)
where U is the unknown quantity, a, b , and c are constants and s is the source term. Applying Galerkin's method, the four terms of the equation above are transformed into matrices which will be assembled into a global matrix system before the solution can take place. 3.8.1. The Stiffness Matrix After applying the Galerkin method to Eq. (3.72) and separating the stiffness matrix from the boundary conditions, the first term in Eq. (3.72) results in
f a grad N
• grad N U dxdy
(3.73)
To perform the integration in the local coordinate system we write from Eq. (3.73)
f
a grad N
• grad N det J dudv U
(3.74)
^local
The integral in Eq. (3.74) is calculated numerically; its integrand is evaluated as follows. The Jacobian is
'dN_ J = du dN_ ~dv
[* y]
For the quadratic triangular element (using Table 3.3 and Eq. (3.71)), this is
l-4t
4(t-u)
1-4/
-4w
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4v
-
0
4w
0
- 4v
-1 + 4v 4(t -i
(3.75)
The product between the two matrices above gives the 2x2 Jacobian matrix. The determinant of the Jacobian is then easily obtained. The Jacobian is denoted as J=
J
\\
J
J
2\
J
\2
(3.76)
22 J
Next we calculate the term grad N (see section 3.4.7). This is
dN_ -1 du grad N = J 8N With the Jacobian above, we get grad N =
^22 ~J\: detJ ~ 2\ J
J\\
4(t-u) •- -4v - 4w • • • 4(/ - v) (3.77)
This gives a (2x6) matrix denoted as grad N =
dnx\
dnx2 •••• dnx^ .... dny6 (3.78)
and Eq. (3.74) becomes
dnx\ dny\ *hcal
•
*
dm\ dny\
U2 dnx2 ••• dnx§ a det J dudv C/3 dny2 ••• dny§
(3.79) This results in a (6x6) symmetric system of equations. To evaluate the expressions, we note that both grad N and det J depend on u and v,
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and, therefore, we use the integration points (u i , v,) for both expressions (see Table 3.4). After the integration, we obtain the stiffness matrix.
3.8.2. Evaluation of the Second Term in Eq. (3.72) Application of Galerkin's method to the second term in Eq. (3.72) gives
I
Nw • B0 ds
(4.23)
where Ln is the edge of the element in belonging to the domain boundary and Sn the surface of the element. The second term of the above equation, is a source term (not depending on A ) and it is
1 '3
O
which should be assembled on the right-side vector. The first term of (4.23) is related to the boundary conditions. There are different situations: • if the permanent magnet is in contact with a Dirichlet boundary condition n is equal to zero and it has no inference on the shape of the field. • if the boundary is over a Neumann condition line, and if BQ • dl is different from zero, we have a non-homogeneous Neumann condition; this term must be evaluated and assembled on the right-side vector. Such a calculation is explained below, for thermal analysis (see Eq. (4.85)). 4.2.5. The Electric Vector Potential In analogy to the magnetic vector potential, we can define the electric vector potential T related to the current density J by
Assuming E to be time independent, rot E = 0 and, with E = J/cr , we now have
rot — rot T = 0 a
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(4.24)
Comparing this with the formulation presented in the previous paragraph, the following equivalent relationships can be written:
A
[ty\
[
:;::; :;::; :;:"
which, using our notation becomes
~dN~ ~dx~ _ 1 dN ~ D
q\ QI ^3 .1
_dy
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^
^3_
(4.40)
4.3.2. The Discretization
Vector
Potential
Equation
Using
Time
We analyse here a two-dimensional (2D) case, for which the excitation current is time-dependent and where there are conducting materials, as shown in the example of Figure 4.13. A nonconducting magnetic circuit (i.e., a laminated core) and a piece P with nonzero conductivity, allow the generation of eddy currents in the direction perpendicular to the plane of the figure. The applied current density J $ , is also perpendicular to the plane of the figure, and is externally applied to the coil. Je is the induced current in block P . To formulate this problem we use the magnetic vector potential defined as B = rot A , where A = AVi , and k is the unit vector in the Oz direction, perpendicular to the plane of the figure. We also have J =Jk and J = J k . With the equation ro/H = J^, where
= J^- +Je
is the total
current density and v = I/// , we have
rot v rot A = Js +Je
(4.41)
Je Js
P
a=0 Figure 4.13. A magnetic circuit made of a nonconducting part and a conducting part. Eddy currents are generated in the conducting part of the circuit.
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Note that J e — oE, where E is the induced electric field intensity in the piece P and (J is its electric conductivity. With these we have
+A
')
symmetric (4.46)
For evaluation of this term using quadratic triangular elements see Eq. (3.79). b. The second term in Eq. (4.44). The term - oA(t + Af)/Af in Eq. (4.44), after applying Galerkin's method, becomes
s
-£/
i
N> A(t + At)dxdy
or, in the local element,
--jTv A/ -D-O
or -v
1-w-v u
Ij - M - V
U
'A\ Vpudv A2
A.
V
Performing the matrix product, we get
(l-M-v)2
g£)
-v W(l-M-v)
v(l - M - v)
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(l-W-V> W2 MV
(l-WMV V
2
(4.47)
The integration, performed term by term, gives
' 1 0.5 0.5" "4" o-D 0.5 1 0.5 A2 12Af 0.5 0.5 1 _A3.
(4.48)
For evaluation using second order elements, see Eq. (3.81). c. The third term in Eq. (4.44). This is similar to the second term, but with the known potential A(t) of the previous step included and the matrices should be multiplied. Unlike the second term, this term is a vector. It gives
'AI+ 0.5^2 +0.5^3
(0
oD 0.54+^2+0.543 12Af 0.54+0.5^2+^3
(4.49)
d. The fourth term in Eq. (4.44). This term contains the external current density J$ (t + A/), and, after applying Galerkin's method, is
J N*JS (t + &}dxdy = jj tVN* Js (t + A/)det J/dudv or
1- u - v dudv u V
which, upon integrations, gives
"1" 1
(4.50)
1 If second-order elements are used, the expression in Eq. (3.85) is obtained.
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After evaluation of the terms in Eqs. (4.46), (4.48), (4.49), and (4.50), the following matrix system is established: SS A(f + Af) = Q
(4.51)
where A(V + A?) is the vector of unknown vector potentials at time step (/ + A?) . Q is the right-hand side vector, containing the source terms resulting from applied currents Js , and induced currents of the previous step (Eq. (4.49)). The terms that depend on the unknown vector + A?) , must be assembled in matrix SS . In practice, the procedure for calculation evaluates the terms in Eqs. (4.46), (4.48), (4.49), and (4.50) for each element in turn. If for example Js = 0 and (7 = 0, only the terms in Eq. (4.46) will be nonzero. For a general case, the total contribution of an element in SS is
,
o-D 2,2
^2,3
~
12Af
3,2
"1
0.5 0.5" 1
0.5
0.5 0.5
1
0.5
(4.52)
where the S(i,j) terms in the left-hand side matrix are given by Eq. (4.46). The SS matrix is multiplied by the vector of the magnetic vector potentials at time step (t + A?). The source terms, placed in vector Q , are oD
12A/
AI + o.5A2 +0.5,43 0.5^ + A2 + 0.5^3
(0 +
(4.53)
+ 0.5^2 + ^3
Generally, in this type of problem, the external current source is timedependent. To establish a calculation procedure, we can assume an initial solution A(f -I- A/) = 0 for the first time step. The matrix system is assembled and a new solution A(? + A/) is obtained. With this result, the next step starts by calculating the new matrix system, where in the source expression, in Eq. (4.53), both vectors are calculated, modifying J to its
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value at the current time step. Continuing this process establishes the calculation procedure. Note that, for any step, it is possible to consider nonlinearity as well, by creating an iterative process for each time step. It is possible to apply the Newton-Raphson method as will be presented later. Two important aspects should be noted when using this method: a. The first aspect concerns the feeding of the device, that is: the method by which the current Js is applied to the device. Generally, electrical devices are fed by applying a voltage across their inputs. Current Js depends on the impedance of the structure. Therefore, a more useful method consists of considering the coupled calculation of the magnetic circuit and the external electrical circuit. This method will be presented soon. This calculation provides the current established in the electric feeding circuit and A as results. The formulation above is well adapted to situations in which the impedance of the external source is very high compared to the impedance of the device. Thus we can assume that the applied current, at steady state, has a well-defined shape. In effect, we assume feeding by a current source. b. The second aspect concerns the partial derivative of A with respect to time, which is approximated as (A(t + A^) — A(t}) I A?. The accuracy of A is acceptable for small values of A?. In the other hand, if A/ is relatively large, the 0 - algorithm is a better approach. This method will be also presented in Chapter 5. 4.3.3. The Complex Vector Potential Equation The purpose of this formulation, as was that of the previous section, is to solve Eq. (4.43). However, if the excitation is sinusoidal and the materials are linear, we can use the complex vector potential, which we denote as A . Denoting Js(tj the cosine current source with frequency G), gives
Js (t) = J
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or, using the complex notation j = V— 1
The system's response to this excitation is also at steady state, sinusoidal and out of phase, therefore,
where: A — Ae^a is the solution to Eq. (4.43) and a is the phase angle between a(t) and J s (/) . Equation (4.43) can be written, in this case, as
\
3
3 CM | 5 — v( — -— v ~ v(rA) a v ^ ^ v-~(rA) \ / =Jtr OZ dz (: 5z ) dr r dr
(4.62)'
v
and, finally,
AvM + A v f l + Af^L-y 5z 5z 5r 5r 5rv r J
(4.63)
The first two terms in Eq. (4.63) are similar to those of Eq. (4.61) for the Cartesian coordinates, if the substitution r = y and z = x is made. However, the term d/dr(yA/r)
creates an asymmetry in the elemental
matrix, when Galerkin's method is applied, because this term depends only on coordinate r. To eliminate this inconvenience we introduce a new variable A related to A :
A=rA
(4.64)
Equation (4.62) now becomes
d v dA
d v
dz r dz
dr r dr
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(4.65)
From here on we operate as with rectangular coordinates, taking r and z as y and X, respectively. Applying the Galerkin method to Eq. (4.65), we have
d v dA + d v dA drdz + dr r dr dz r dz
.n
(4.66)
The first term of this equation after integration by parts (similar to the integration performed in section 4.3), is
av dr r dr
drdz
-I-
* flr
Nl
, . v dN* dA . drdz- I drdz r dr r dr dr
Analogously, we obtain for the second term of the first integrand of Eq. (4.66):
LIS fa
r dz
drdz -
v dN* dA' drdz r dz dz
Adding these two terms, we get
rdr
dz
r dz
drdz-\ JS
v dN{ dA r dr dr
v dN* dA drdz r dz dz
(4.67) Recalling that
dA vdA vdA TT Hr = -v— = and H 2 = dz r dz r dr and defining the div operator in the z — r plane, as for the x — y plane,
,. . d . d div = \ — + j — dz dr the first intregral in Eq. (4.67) can be written as
drdz
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Applying the divergence theorem, we get
where n , the unit vector normal to line L is n = i nz + j nr . This expression is introduced in the equation above:
Following the procedure used to obtain Eq. (4.45a), we obtain the Neumann condition, since H x n = 0. For Dirichlet boundary condition, the shape function TV is equal to zero at the boundary part where the potentials are imposed. The second integral in Eq. (4.67) provides the elemental matrix. For an element / the relation becomes
-1,
v dNf dA i v dNl dA drdz r dr dr r dz dz
This is equivalent to Eq. (4.45b). In short form notation, we can write [ gradN* (vgradA)drdz (4.68) where we have replaced r by r§ , the centroid of element i. After some algebraic operations, we obtain, as for Eq. (4.45b):
2Drf]
r\r\ symmetric
symmetric
l
l
+
symmetric
r r
+ 23 #3^3 + (4.69)
The matricial contributions related to eddy currents (Eq. (4.48), (4.49) and (4.57)) should be also divided by TQ .
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The application of Galerkin's method to the source term Jt , yields NfJtdrdz
(4.70)
which is similar to Eq. (4.50). The permanent magnet source term does not have any change, either. 4.5. Advantages and Limitations of 2D Formulations All electromagnetic structures are three-dimensional in nature, and some precautions must be taken when 2D approximations to 3D problems are made. Many realistic problems in electrical engineering can be analyzed by 2D methods if appropriate cut planes are chosen. Note also that, generally, devices are built to avoid the generation of eddy currents, and therefore, we can often work with static methods, even if there are moving parts in the solution domain. Under certain conditions, a sequence of static solutions may provide an answer to the dynamic response of the system. When eddy currents are present, the 2D formulations presented above require that the currents be perpendicular to the cross-sectional plane over which the elements are generated. This implies that the currents flow from — oo to +00, that currents loops are closed at infinity and that all the conductor regions are connected and short-circuited. In many cases this is, at best, a poor approximation. For long structures, this approach may be correct under some conditions. An example for a good 2D approximation is the case of an induction motor, where the short-circuit bars connecting the extremities of the rotor, as shown in Figure 4.19a, are supposed to be a perfect short-circuit. In this case, we can assume, with little error, that the results calculated for the 2D domain in Figure 4.19b are satisfactory. However, the induction motor is a "bad" example in other aspects. Its operation involves many complex phenomena such as rotor speed, saturation, voltage fed windings and non-perfectly short-circuited rotor bars. A further difficulty is the very small air-gap, affecting accuracy of results. In other words, although the calculation of eddy currents is relatively simple and accurate, a complete, realistic and accurate analysis of an induction motor is one of the most complex of all electromagnetic devices and requires special techniques, as will be presented in subsequent chapters.
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Rotor plate Figure 4.19a. Approximation of a 3D problem geometry. Short-circuited bars in an induction machine are shown together with a section of the rotor plate.
Figure 4.19b. The two-dimensional plane used for analysis assumes the bars are infinitely long.
For axi-symmetric problems, the formulation in cylindrical coordinates is very efficient, since the eddy currents (flowing in the (p direction) are closed within the structure itself. This is taken into account by the formulation and requires no approximations. In effect, this is a simple solution to what would otherwise be a relatively complex 3D problem. One important precaution that must be followed concerns the mesh used for analysis of eddy current problems, especially in regions where the currents are significant. The depth of penetration is given by
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B
Figure 4.20. Discretization in eddy current domains. Smaller elements are used in eddy current regions.
This is normally very small, especially in ferromagnetic materials. For correct analysis, it is necessary to discretize eddy currents regions with small elements to obtain good precision, as indicated in Figure 4.20. For regions far from eddy currents, larger elements can be used. Normally, 1.5 to 2 elements per skin depth are required for the correct solution. 4.6. Non-linear Applications Generally, permittivity and conductivity can constants. However, permeability (or reluctivity) is magnetic field intensity. Ferromagnetic materials are B(H) curve, with permeability varying depending on
be considered as dependent on the characterized by a the location on the
B(H) curve. The assembly of the matrix SS(K,K) (K is the number of nodes) requires known values of // (or v) for each element /. However, how can we know the value of ju before the solution is obtained? The system is nonlinear and in order to find a solution, it is necessary to establish an iterative procedure. This process is discussed next. 4.6.1. Method of Successive Approximation This method is very simple in that we can use the normal computational procedure (for linear cases) in an iterative form. The process is as follows: (*F here is related to the scalar potential case, used as an example). a. Make an initial approximation, ^ = 0 for all nodes.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
b. Using the approximation for *F calculate the field intensity H in the element. c. Obtain ju from the B(H) curve based on the calculated H. d. Using ju calculate the contribution S(3,3) for the element as well as the source term (current or permanent magnet). As presented in section 4.2.3, the elemental contribution S(3,3) is for the scalar potential case: r r
\2
2D
+r
symmetric
q2q2 2r2
symmetric
symmetric
Using the same notation for the last term of Eq. (4.75), we have the generic Jacobian term J(n, k)
(4.77)
For use in this equation, JU and djU/dH
are obtained without any
particular difficulties from the material B(H) curve. From Eq. (4.77), we note that to obtain the complete Jacobian matrix, only the terms S(3,3) are needed, and these are calculated using the values of *¥[ in triangle / . The assembly of these terms (Eq. (4.77)) provides the global Jacobian, called here SJ . The matrix system to be solved is:
-» SJA^F-R
(4.78)
The right-hand side vector R is called a residual vector, because it -> originates from the matrix product SS ^F (from the previous iteration) that should be zero. It is not zero because the permeabilities used to construct
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
SS are approximations. The more the iterative procedure evolves, the —»
closer the permeabilities are to the final solution, and R and A ^F tend to zero. The Newton-Raphson algorithm can be summarized as a. Make an approximation to the vector *P as close to the solution as possible. b. Using the approximation for ¥ , calculate H and, from the B(H) curve obtain // and dju/dH2 . c. Calculate the elemental matrix terms 8(3,3) using ju . d. With S(3,3), d/J/dff2
and the potentials ¥ of the previous
iteration, calculate the Jacobian terms and the residual. e. Impose the Dirichlet boundary conditions and solve the system in Eq. (4.78). f. With the solution and the values of A T , obtain the new values of
g. Repeat steps b through f until the convergence criterion is satisfied. This method is very efficient, especially if the first approximation is close to the solution. A practical rule is to begin the Newton-Raphson loop after five or six iterations are performed with the Successive Approximation method. It is not necessary to dimension a new matrix, since the Jacobian matrix is topologically similar to the global matrix SS. The same space in computer memory can be used for evaluation of the Jacobian. For vector potential A applications, it can be written:
/?(*)= 5X*,/H /=!
where the general term S\kJ) is (as shown in section 4.2.4., Eq. (4.20))
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and Q\k] is the source term, which takes the following form according to the type of the magnetic field source (see section 4.2.4): a. For current sources:
b. For permanent magnets:
Both of these source terms are independent of A . Therefore, dQ/dAn = 0 and the general Jacobian term is
Ji(n,k)=S(k,n)-
/=i (4.79) The residual is R = SSA + Q
and it includes the source vector Q; A is here the vector potential of the previous iteration. When eddy currents are considered, we also have the contribution of equation (4.48):
" 1 0.5 0.5" "4" oD 0.5 1 0.5 A2 12 A/ 0.5 0.5 1 which depends on potentials A^, A2 and A3. The generic form for it can be written as
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where Ty is equal to 1 for k — I and equal to 0.5 for k =/= / . For the Jacobian we need to obtain dN(k)/dAn
dN(k) dAn
, which gives
crD T 12A/ *"
This term has to be assembled on the left hand side of the matricial system. When using second order elements the Jacobian is calculated as below. Here we present only the calculation of the Jacobian term corresponding to the elemental matrix where the reluctivity V varies as a function of the magnitude of B . The derivation with respect to An of line m of the elemental matrix (4.45b) is:
dAn
[ vgradNlmgradNAdxdy
-
dv dB2 - [ vgradN*m gradNn dxdy - [ gradN m gradNA —2 dxdy */ */ dA l
The first integral in this equation was evaluated earlier (see section 4.3.2). To calculate the second integral, note that
dAn
dAn
where
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dAn
B = gradNA
and
dA»
= gradNn
We have
dB' = 2 gradNn gradNA dAn and the second integral becomes
-2 [ gradNm gradNA — — gradN ngradNAdxdy */ dB 2 This expression is also evaluated by numerical integration. The Jacobian term is
r
J(m, n) = S(m, n) + £ Wtf(ut , vt )
(4.80)
/=!
where the term J(m,ri) is by:
r S(m,n) = i=\
and
/(«,-, v;) = -2Y WigradN'mgradNA-??-gradN'ngradNA
det[j/(- ]
We recall that in the expression above we use A from the previous iteration. The assembly of the matrix system is done analogously to that shown previously in this section. 4.7. Geometric Repetition of Domains Especially in electrical machines analysis, periodicity and antiperiodicity geometry and functioning aspects can considerably reduce the
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study domain of the device. For instance, when non-fractional windings are used, the study domain of a three-phase induction motor can be reduced to only one pole of the machine. Moreover, periodicity and anti-periodicity conditions are employed to take into account the rotor movement, as will be shown in chapter 6. 4.7.1. Periodicity Some problems have geometries that can be composed of a repetitive section of the domain to be analyzed, as for example, the geometry shown in Figure 4.21. In this case, the problem is "periodic", characterized by a geometric replication of the domain S. If there are coils and/or permanent magnets oriented in the same direction, the potentials on line C are identical to the potentials on line D.
! \. „
c•>
( ^
A C•> D
s
C•>
( I ]
I}
Figure 4.21. A periodic structure. The domain defined by tines A, B, C and D, is the repetitive domain. Only this part of the structure needs to be analyzed.
J
D Figure 4.22. Treatment of boundary nodes between neighboring domains in a periodic structure.
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1 t
c
IT)
$3>
A c
^
,_, 0
D
) (
| ]
S I?
Figure 4.23. An anti-periodic structure. The difference between this and Figure 4.21 is in the alternating directions of currents.
Instead of considering the whole structure, it is sufficient to analyze only the domain S. To do so, we treat the elements on the boundaries between neighboring domains as shown in Figure 4.22. When the elemental matrix S(3,3) for triangle T is generated, the contributions for the nodes i and j must be assembled in nodes i' and j'. This indicates, for line C of the domain, the presence of an identical domain to its left. It is not necessary to consider the nodes i and j in the matrix system; on the other hand, the value at node k does not change. When the system SS A = Q is solved, we set Aj = AJ' and A ; = A ,•'. The remaining nodes on line D are treated similarly. The elemental matrix and sources of triangle T are assembled as indicated below
linez'-» ~Su o line /-» *ji
*ti
line k —>
" t; /C/
IJ >CZ ti
Sjj o
s^
jk
s
kk_
and
"a" fiy L&_
This procedure is also valid for the Jacobian and the residual (NewtonRaphson method). 4.7.2. Anti-Periodicity Anti-periodicity is similar to the periodic case discussed above: we have geometric repetition of a domain, but the source (current or permanent magnet) has alternately opposing directions, as shown in Figure 4.23.
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We again consider only the domain S of Figure 4.23; for the assembly of elemental matrices and sources, the terms must be inserted in the locations for nodes /' and f, instead of nodes / and j, by the rule indicated below:
" stt -Sji line k -» ~ ki S
~S su
JJ
S
~ kj
"-a"
~Sik
-Sjk
and
S
-Qj -Qk .
kk
4.8. Thermal Problems Heating is a very frequent phenomenon on electromagnetic devices and, in many situations, the evaluation of temperature is necessary to avoid over-heating in structures. In our area, there are different sources for heating as, for example, Joule effects by eddy and conducting currents, magnetic hysteresis and also mechanical friction. In this section we will present briefly some topics of heating transmission, but for more detailed presentation, specialized references may be consulted. The FE implementation aspects are also shown. There are three different ways of heating transmission: conduction, radiation and convection. 4.8.1. Thermal Conduction Conduction is a process where the heating is transmitted inside a body or between different bodies having physical contact. The basic equation describing the thermal conduction is
c
dT
dt
h div(- k gradl) = q
(4.81)
where j
c is the thermal capability (J l(m
°C));
A is the thermal conductivity (W /(m°C)); T is the temperature (°C) ;
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-3
q is the thermal source volumetric density (W I m ) . For instance, q can be defined as Joule's effect source by q = J I <J , where J is the current density (for both, eddy or conducting currents, depending on the studied case). 4.8.2. Convection Transmission
Convection occurs when a fluid has contact with a heated solid body. There will be a constant movement where the heated particles (dilated by the contact) will be replaced by cooler ones. As main effect, heating is transmitted from the body to the fluid by the following equation -Ta)
(4.82)
OS *J
where h is the coefficient of heat transfer by convection (W l(m °C)) , T is the temperature at the heated wall and Ta is the temperature of the fluid at a point far from the wall. The quantity h depends on the fluid viscosity, thermal conductivity, density, velocity and specific heat as well as on the heated body superficial geometry. In practical applications, h is difficult to be evaluated and it is normally determined experimentally. 4.8.3. Radiation As seen before, normally for convection and conduction, at least two materials must be present in the system. This is not the case for radiation. A body emits electromagnetic waves. This radiation can reach another body. Part of these waves will be reflected and part will be absorbed by this second body. This last portion will be transformed into thermal energy. A body at temperature T radiates energy to another at temperature Ta , involving it, according to the following expression a
OS
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(4.83)
where y is the Stefan-Boltzmann constant and S is the emittivity of the body. Summarizing this section, we have the following equations to describe the heat transfer: Z^T1
a. c
dt
h div(- A, gradT) = q for thermal conduction; Z^T*
b. - A,
ds
n = h (T - Ta ) + £y(T4 - T* ) representing the transfer
by radiation and convection from a body whose external temperature is T to the ambient having the temperature Ta. This expression behaves as a source term interacting with the system by its boundary, meaning that heating can be dissipated or absorbed by the body depending on temperatures T and Ta. In some cases, for example in symmetry planes, without any heating transfer, we have the homogenous Neumann boundary condition
^n ds
=0
whose behavior is similar to any scalar potential seen before in this chapter (the gradient of temperature is perpendicular to the boundary). Also, in the part of the boundary where the temperature is imposed as T0, we have a Dirichlet boundary condition, as T 1
-T A
o
4.8.4. FE Implementation The implementation of the thermal formulation can be performed by applying the method of Galerkin to the above equations. Initially, when it is done only for thermal conduction Z^T1
c
h div(- A, gradT) = q
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the procedure is similar to the electromagnetic equation (4.43), related to eddy current problems in systems being fed by a current density. Therefore, the calculation of its three terms for a time-stepping procedure was already performed and, with the appropriate change of coefficients, for a first-order element they are, respectively, equations (4.46), (4.48 with 4.49) and (4.50). We point out that if we are interested in a steady-state regime (dT/dt = 0), the solution of the equation above give us the temperature situation when the thermal equilibrium is established. Also, the equation above handles homogeneous Neumann and Dirichlet conditions. However, having convection and radiation, the corresponding terms applied to the boundaries must be evaluated and added to the matricial system. To do so, and using Galerkin's method, the two terms below must be assembled in the 2D boundary Lcr having convection and radiation
\
h(T-Ta]Ndl+\
or
N hT + erT dl
L ( The term T
N hT
^ -L ( °
can be linearized by a Taylor's series around the point 7]_, . It
means that an iterative procedure must be employed since it behaves as a non-linear problem. 7]_j is the temperature known from the previous iteration and Ti is the temperature under evaluation and unknown. Notice that this iteration procedure must be applied for both steady-state and transient regimes. 7]_, is the temperature on the iterative procedure and not the temperature of the previous time step. Furthermore, the thermal conductivity A varies, even though not strongly, with the temperature. f_1 (7; - T^ }\dl -
or
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N(H + 4^1, )T,di -
NT^
+hTa+ efl*
Therefore, in order to simplify the expressions, we have two different integrals, as f NaTidl\ Nbdl l JLcr JLcr where
and b =
hT
a and b are constant, since Ta and 7}_j are known. Using a first order element on the boundary, having a length JL we have for the first term, which depends on Ti
u
[1 - w u\L du
or
aL ' 1 0.5
0.5 1
(4.84)
where Tfl and Ti2 are the temperatures on the two nodes on the boundary belonging to the element. For the second term, not depending on Ti, we have
\-u u
(4.85)
This last term is obviously assembled on the right-hand side of the matricial system.
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4.9. Voltage Fed Electromagnetic Devices Up to this point, we have assumed that the source of the magnetic field is a current density, i.e., the driving coil is fed by a current known before the simulation. However, in many cases, the system is voltage-driven and the current in the coil is unknown. To solve the problem under these conditions, the field equations and the coil voltage equation must be solved simultaneously. The voltage equation for the coil can be written as follows:
U = RI + — n6 dt *
(4.86)
where U is the voltage on the coil and R and n are, respectively, the coil resistance and number of turns; (f) (in Wb/m) is the magnetic flux generated in the solution domain and linked by the coil. As presented in previous sections, the current contribution of an element of the mesh is given by
JD
(4.87)
where J is the current density and D is equal to twice the surface of the element. A parameter K is now defined as the coil turn density (in turns I in ). If / is the current of one turn, we have J = Kl
With this, Eq. (4.87) can be written as
"1"
KD
and / is considered an unknown, as will be shown below. Notice that a term of P is equal to K D/6 .
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For the magnetic field equation, the flux, in terms of the magnetic vector potential, is given by (see section 4.2.4)
0 = A£- A£
(4.88)
where / is the depth of the device and Al and A2 are two magnetic vector potential values. As the zero magnetic vector potential value is normally present in the field evaluation as a result of the calculation or by means of Dirichlet boundary conditions, one can evaluate the magnetic flux with respect to the null value and then rewrite Eq. (4.88) as
where A is the potential at any point in the solution domain. Further, assuming that A is the average potential in an triangular first order element of the mesh having a specified current, we get
At + A.-+ALJ
A=-
-
-
3
where /', j and k are the nodes of the element. If n is the number of turns within the element, we get
K D n = K x element area = ^ Therefore, KDl
For the triangular element, we have, in matrix form
rub =
\cIV J^r4r n/ kIV r>/ L-fAs 6
and with
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6
KDt~
6
4
we have
Using Euler's scheme to the time derivative, we obtain
d nd— AA= Q — nd)* = Q — dt dt &
where A(t + A/1) and A(t) are, respectively, the magnetic vector potential values at times / + Af and t . Considering the circuit and voltage equations above, the global system of equations becomes SSO + AO + — N A? A/
-P
R
r AO + AO"
L i(t + AO
=
— N 0 ~A(f)~ +
1
-Lq o
.A/
.W.
(4.89) where N (related to eddy currents) results from assembling elemental matrices given by Eq. (4.48). Vector D(/ + A/) can describe the influence of a permanent magnet or another current-fed coil whose current is known at time t + At . Equation (4.89) is for 2D Finite Element modeling and in this way the end windings influence is not directly included in the modeling. Nevertheless, this can be considered by including an additional term L di/dt in the voltage equation (L representing a diagonal end winding inductance matrix). With this additional term, we can rewrite Eq. (4.89) as SSO + AO + — N -P ~A(f + A/)" Ar 1 i —Q R + — L _ !(/ + AO A/ A/
—N
0 "A(0" , A/ 1 l — Q — L JCO. .A/ A/
(4.90)
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Performing the products in equation (4.90) we get two equations; a. Field equation
At
N
+ A?) - Pl(t + AO = — NA(/) + D(/ + AO At
or
SS(/
At dA
which is referred to the equation rotvrotA + (T -- J^, =ro?vB0, dt recalling that BQ is the remanent induction of permanent magnets and J s is the current density of imposed current coils. b. Coil voltage equation J_
~At
At) + R + — L \l(t + At) = —QA(0 +—LI(0 + U(f + AO At \ At At
or
Q
AQ-A(Q' At
I(t + At) -!(/)' At
referred to Eq. (4.86) considering also the additional inductance L. Note that matrices P and Q are very similar. In the process of calculating the elemental matrices for elements having imposed current densities, we also calculate P and Q and assemble these in the appropriate locations in the global matrix. The lines of the global system corresponding to the currents (last lines) have terms out of the normal band (the band characteristics are preserved in the field part of the system). Therefore, in the solution process, some modifications were performed to take into account the last equations while still using the banded form of the solution.
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The simulation of voltage-fed electromagnetic devices as well as the modeling of their functioning coupled to external electric circuits, including static converters, will be presented with more detail in chapter 5. 4.10. Static Examples Some examples of finite element applications are presented in this section. To perform these calculations, the software package EFCAD (developed at the Universidade Federal de Santa Catarina, Brazil) is used. It is a general purpose software package, for 2D applications in static and dynamic electric and magnetic fields with specialized routines for analysis of electric machines and coupling of circuit equations with magnetic field solutions. Nonlinear, as well as static and transient thermal problems can be solved.
r
c
V=500*V
A
t E
E
Transformer
-±-
V=0
B
Figure 4.24. An electrostatic problem: the determination of the electric field in a domain with specified potentials on boundaries.
The software package has three important parts: • Pre-processor, in which the general data (geometry, field sources, boundary conditions) are furnished by the user. Additionally, the mesh is automatically generated from this input. • Processor, in which the finite element method is applied to the discretized domain.
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D
• Post-processor, in which graphic (equipotential lines) and numerical (flux, fields, forces, inductances) results are calculated and provided to the user. The following examples demonstrate the potential and versatility of the finite element method. 4.10.1. Calculation of Electrostatic Fields
Suppose that we wish to determine the electric fields in the region of a high-voltage substation with an electrically grounded piece of equipment such as the transformer shown in Figure 4.24. Assume that on lines A and B there are imposed boundary conditions (V = 500&F on line ,4and V = 0£Fon line B). Lines Cand D are chosen far from the transformer region, so that we can assume that the fields are approximately vertical and therefore almost tangent to lines C and D. This assumption allows the use of Neumann boundary conditions for the scalar potential on these two boundaries. Note that the application of Neumann conditions is performed by leaving these boundaries without any particular restriction or formulation. In other words, we do not need to do anything on them. The finite element mesh generated is shown in Figure 4.25a. After the finite element calculation, the post-processor displays the equipotential lines, indicated in Figure 4.25b. By visual inspection the regions with higher potential gradients (higher fields) are easily noticed.
Figure 4.25a. Finite element discretization of the solution domain in Figure 4.24.
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Figure 4.25b. Equipotential lines for the geometry in Figure 4.24.
As numerical results, the module provides the field values, which, if necessary, can be compared with the measured dielectric field strength of the material. These data can then be used for design and safety purposes. 4.10.2. Calculation of Static Currents We wish to obtain the current distribution in a conductor made of a layer of copper and a layer of aluminum, as shown in Figure 4.26. Two different approaches are shown. The first uses the scalar potential formulation, presented in section 4.2.2. Laplace's equation in this case is
d dV d dV a — +—a dx dx dy dy Va D
C
I 1.
^Eor J
B
cop per
aluminum
Vb
Figure 4.26. A static current problem: the determination of currents using the scalar potential with specified potentials on boundaries.
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In this formulation, the potentials V on lines A and B are specified as boundary conditions. Assuming that the field E (or J) is tangential to lines C and D, Neumann boundary conditions are used on these two boundaries. The mesh generated for this problem is shown in Figure 4.27a and the equipotential line distribution is presented in Figure 4.27b. This problem can also be treated with the electric vector potential using the formulation in section 4.2.5. The equation to be solved in this case is
Al^
dx a dx
JL-LfiZ^o dy (7 dy
Using this approach, we impose a current difference between lines C and D (for example 7 = 0 and 7 = Ia), meaning that the current crossing the conductors is Ia. Recall that these current values are given here in Amperes!meter, and they are obtained by dividing the actual current by the depth of the device (distance perpendicular to the study plane). In this case, on lines A and B, we have Neumann boundary conditions, meaning that J or E are perpendicular to these lines. The equipotential lines obtained after finite element calculations are shown in Figure 4.28, showing how the current flux is distributed in the domain. A higher current crosses the copper part of the conductor. In fact, using the numerical data provided by the program, the current in copper is approximately 60% higher than the current in aluminum. This is as it should be, since the conductivity of copper is approximately 60% higher than the conductivity of aluminum. The choice of formulation depends on the problem to be solved. If the device works with applied voltages, which then can be used as Dirichlet boundary conditions, the electric scalar potential should be used. On the other hand, if the current is known, the electric vector potential should be adopted.
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Figure 4.27a. Finite element discretization of the solution domain in Figure 4.26.
Figure 4.27b. Equipotential line distribution for the geometry in Figure 4.26.
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Figure 4.28. Equipotential lines for Figure 4.26 using the electric vector potential T. These are lines of current and are perpendicular to the potential lines in Figure 4.27b.
4.10.3. Calculation of the Magnetic Field - Scalar Potential Assume that in the air-gap shown in Figure 4.3a, there are ferromagnetic "poles" (created by slots in the ferromagnetic material in Figure 4.29) and we wish to determine the magnetic flux distribution in this region. Between lines A and B we impose a potential difference related to the magnetomotive force NI of the coil. On lines C and D, Neumann boundary conditions can be imposed by considering the field as tangential to these lines. In Figure 4.30a, the equipotential lines obtained after the finite element calculation are shown. In this case, the value of NI is very low and the structure does not reach saturation. The result in Figure 4.30b is for a high value of NI, and, since the magnetic flux is large, the material reaches saturation. Thus, we notice that some equipotential lines penetrate into the ferromagnetic material of the poles. To simplify field visualization, we repeat this analysis with the vector potential formulation. Instead of specifying the scalar potential difference between lines A and B, the magnetic flux (per unit of depth) difference between lines C and D is specified.
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This approach is similar to that of the previous example. Figures 4.30c and 4.30d show the flux obtained under linear and saturation conditions, respectively. Note that in Figure 4.30d a larger part of the flux crosses through air, since the teeth are saturated. The relative permeability in the highly saturated regions is approximately 10.
NI
A
D
C H
0 Figure 4.29. A magnetic field problem: calculation of the magnetic field intensity using the magnetic scalar potential. Boundary conditions are also in terms of the magnetic scalar potential.
For non-linear calculation with the scalar potential formulation, the Newton-Raphson method was used. However, the first five iterations were performed with the Successive Approximation method. The results from the Successive Approximation were used as an initial solution to the NewtonRaphson method. The solution required four Newton-Raphson iterations for convergence, which was obtained with a relative error of 0.001, that is: the value at all nodes of the mesh was within this error.
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Figure 4.30a. Equipotential lines for Figure 4.29 at low values of Nl (no saturation). The magnetic scalar potential is used.
Figure 4.30a. Equipotential lines for Figure 4.29 at high values of Nl (saturation). The magnetic scalar potential is used.
Figure 4.30c. Equipotential lines for Figure 4.29 at low values of flux (no saturation). The magnetic vector potential is used.
Figure 4.30d. Equipotential lines for Figure 4.29 at high values of flux (saturation). The magnetic vector potential is used.
4.10.4. Calculation of the Magnetic Field - Vector Potential
Figure 4.31 is related to an axi-symmetric structure, shownf in crosssection for visualization purposes. When a current is imposed, the magnetic forces created in the air gap attract the lower, mobile part, to the upper, stationary part of the structure. This attraction force between the two pieces is now calculated. For the calculation of this structure by EFCAD, the domain data shown in Figure 4.32 are provided to the software package. According to the conventions of this software the symmetry axis must be coincident with the Ox axis. Line A (Figure 4.32) is placed at some distance from the structure because it is known, a priori, that the field has a natural dispersion in the air gap region. A potential A = 0 is imposed on lines A,B,C since the magnetic flux does not cross these lines. On line D we also impose A = 0, since the
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axi-symmetry of the problem forces this condition. Figure 4.33a shows the mesh and Figure 4.33b shows the resulting flux distribution. The important numerical result in this problem is the attraction force on the mobile piece. This force is calculated on line E (Figure 4.32) using the Maxwell stress tensor method, (discussed in Chapter 7). Normally, the whole mobile piece must be enclosed by the line over which the force is calculated, but considering that the field is significant only on line E, it suffices to calculate the force on this line.
NI
1
X
X
1
Figure 4.31. An axi-symmetric structure. Two parts of a magnetic circuit are separated by a gap. The upper part contains a coil and is stationary. The lower one is free to move.
A
H
C B
E
t
D Figure 4.32. Solution domain for Figure 4.31 as supplied to the finite element program.
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Figure 4.33o. Finite element mesh for the solution domain in Figure 4.32.
Figure 4.33b. Magnetic flux distribution for the device in Figure 4.31.
(r=0 Js
a=0 Figure 4.34. A stationary conducting piece in front of an electromagnet. Eddy currents are induced in the piece due to a time-dependent magnetic field.
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4.11. Dynamic Examples 4.11.1. Eddy Currents: Time Discretization
Consider the geometry of Figure 4.34, which shows a stationary conducting piece in front of an electromagnet. Eddy currents, indicated by Je, are induced in the conducting piece due to the time variation of the external current Js, in the excitation coil. The conducting piece is ferromagnetic with jur = 1000 and cr = 10 (S I m). To show a problem that can be easily understood, we apply a current density with the waveform shown in Figure 4.35: it is a current pulse rising from zero to 2A/mm2 in 0.01 sec . The rising part is sinusoidal in shape; at times beyond 0.01 sec, the current density Js is constant at 2AI mm2. Figures 4.36a and b show the results of the calculation. The first one is for time equal to 0.01 sec, when the penetration of the flux is partial because of the eddy currents established in the conducting part. The second one, Figure 4.36b, shows the flux at time equal to 0.036 sec, when eddy currents are very small. The flux distribution is similar to the static case, with the external current density fixed at 2A /mm2 . The behavior of the field and eddy currents are as follows:
t(s) 0.010
0.036
Figure 4.35. The pulse used to drive the geometry in Figure 4.34.
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In Figure 4.37a the maximum negative potentials in the conducting piece are shown. They are directly related to the induced currents. These currents decrease after t = Q.Qlsec. By time / = 0.040sec, these currents have decreased significantly. Figure 4.37b shows the maximum values of the positive potentials of the whole structure; they correspond to the flux generated by the applied current. This curve tends to the flux value for the static case with a coil current density of J = 2A/mm2.
Figure 4.36a. Flux distribution in Figure 4.34 at t=0.01sec.
Figure 4.36b. Flux distribution in Figure 4.34 at t=0.036sec.
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0.030
0.010
Figure 4.37a. Maximum negative potentials in the conducting piece of Figure 4.34.
t(s) 0.010
0.030
Figure 4.37b. Maximum positive potentials in the conducting piece of Figure 4.34.
4.11.2. Electromagnet
Moving
Conducting
Piece
in
Front
of
an
Consider a nonferromagnetic conducting piece in front of an electromagnet fed by a constant current as shown in Figure 4.15. The behavior of the magnetic field for different velocities of the piece is required. Solutions are shown in Figure 4.38a through 4.38f. Figure 4.38a shows the solution at zero velocity. The field is identical to that for a static solution. In Figure 4.38b, c, d, e, and f, the field configurations for a moving conducting piece with velocities v = 1,5, 10, 20
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
and
30 m I s
respectively are shown. Asymmetries become more
pronounced as the speed increases.
Figure 4.38a. Solution for the geometry in Figure 4.15 at V — Qftl/ S .
Figure 4.38b. Solution for the geometry in Figure 4.15 at V = \Ttl IS .
Figure 4.38c. Solution for the geometry in Figure 4.15 at V = 5/W / S .
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Figure 4.38d. Solution for the geometry in Figure 4.15 at V = IQftl/ S .
Figure 4.38e. Solution for the geometry in Figure 4.15 at V = 20m/ s .
Figure 4.38f. Solution for the geometry in Figure 4.15 at V = 30w / S .
The following points should be noted in these solutions: a. Shape of the field: The eddy current is given by J = O\xS ; in front of the electromagnet (fed by a current perpendicular to the plane of the figure), the situation is that shown in Figure 4.39. The flux densities Bj and 62 are mainly established in the directions shown in this figure; from the equation for J , the eddy currents Jej and Je2
are
perpendicular to
the plane, as indicated. Figure 4.40 shows the three currents: J5 (the
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
applied current in the coil), Je\ and Je2 (induced currents in the moving piece).
B B
Jel Fl- -©—
*B
B
B
Figure 4.39. Current densities, magnetic field intensities, and forces in the moving piece in relation to the velocity.
The general shape of the magnetic fields generated by these currents is shown in Figure 4.40. Based on their summation, we conclude that the total field has the form indicated by the dotted line. This is consistent with the contours in Figure 4.38 a-f. b. Force due to the product J x B: The volumetric force density is given by f = J x B. Observing the vector directions in Figure 4.39, the forces FI and F2 are opposite to the velocity vector V of the conducting piece. c. Force due to Maxwells tensor: According to the contour plots obtained by the finite element method (Figures 4.38a-f) the fields have significant components tangential to the moving piece (Figure 4.41). Using the Maxwell stress tensor (presented in chapter 7), these components generate forces at an angle 20 with the normal direction. The forces are opposed to the piece movement as was also concluded above.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Figure 4.40. Relationship between applied and induced current densities.
normal
Figure 4.41. Force calculated using Maxwell's stress tensor.
4.11.3. Time Step Simulation of a Voltage-Fed Device As an example of the type of problems that can be solved using the formulation of the simultaneous resolution of field and voltage equations, consider the solenoid shown in Figure 4.42. This device is axi-symmetric and consists of two parts, made of iron. We assume the iron to have linear properties and neglect eddy currents in the iron. These restrictions are imposed to allow a comparison with analytical calculations. The numerical data used for this example are:
l\-\5mm\ / 2 = 3 0 w r a ; e = 2mm\
r\ = 20 mm; r^ - 25 mm', r^ = 32 mm', r^ - 38 mm n (number of turns) = 50; R (coil resistance) = 2 Q
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i
,1.
i ron
(i)
iron
t < 1 '3 i
coil ~*
(2)
h
/I
—^
^ 1
.3 J i J
« k-
/2
k
Figure 4.42. A voltage-fed solenoid.
A step pulse of 10 V (switched on at t=0 as shown in Figure 4.43) is applied on the coil.
10
Figure 4.43. The voltage source used to drive the circuit in Figure 4.42.
Since the air gaps are small, we can calculate the magnetic circuit. This gives
h\e + h2e = nl The flux conservation equations give #, = ^2;
B.S, = B2S2
or
^i0hlSl = ^i0h2S2
The radii TJ , r2 and r3 are dimensioned with the goal of making B{ — B2. Noting that 7tr\ = n(rl - r32) , we have: h{ = h2 = h, and
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
1e At steady state, / — V I R = 5 A . Using the numerical values above, we get h = 62.5kA/m The flux is
= 0.987 xlO~V&
=
The inductance is
Since the iron properties are linear and there are no eddy currents, the flux is proportional to the current /. Under these conditions, the inductance is constant during the transient state. The theoretical equation for /(/) for this RL circuit is _R
/ ( , ) =V! _ « , i or, numerically,
7(0 = 5(l -e-2026'3')
(4.91)
The results are shown in Figures 4.44 and 4.45. The first figure shows the flux distribution in the structure. The second shows the current in the coil as a function of time.
Figure 4.44. Flux distribution in the structure in Figure 4.42.
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0.000
0.001
0.002
0.003
0.004
0.005
0.006
Figure 4.45. Current in the coil of Figure 4.42 as a function of time.
Figure 4.45 is in agreement with Eq. (4.91) demonstrating the consistency of the formulation. In fact, the theoretical curve increases faster compared with the numerical results. The main reason for this is the fact that the real inductance is higher than the theoretical inductance because the flux crossing the solenoid (Figure 4.44) increases the value of the inductance compared to the theoretical calculation, in which this flux is not considered. 4.11.4. Thermal Case: Heating by Eddy Currents As thermal example, we will consider the case presented in 4.3.3, where eddy currents were calculated by a complex formulation. For the frequency / = 500 Hz (Figure 4.14e), from the post-processor we obtain the values of eddy currents in the conductive part. Because radiation and convection are boundary conditions for thermal problems we will consider only the conductive piece, as shown in Figure 4.46. Since there is a longitudinal symmetry, only half of the structure is presented in this figure. Because of the symmetry mentioned above and as there is no heat transfer through line D, the Neumann condition is here applied. The other three lines A (taking the whole upper line, including S{ and S2 parts), B and C are boundaries with radiation and convection. The outside temperature is 20°C and the convection coefficient h is equal to 10
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
(W /(m2°C)) . The other data for £, c and A are typical values for the iron. In this figure, for the regions Sl and S2, we obtained from the postprocessor the eddy current densities J\ and J2 , necessary to calculate the thermal sources as we will see soon. The field, as well as eddy currents, do not penetrate farther than S2 . Obviously, the shapes of the sources S{ and S2 are approximated by observing the eddy currents distribution on the post-processor sector. For more accurate results, it would be possible to establish, for each finite element, its own source by using the eddy current density obtained inside it.
B
D
C Figure 4.46. Conductive piece with boundary conditions and thermal sources.
The thermal sources q{ and q2 for Sl and S2 respectively, are calculated by the expression q — J /cr . As thermal transient effects are much longer than electromagnetic ones, we considered that a pulse of thermal excitation (q{ and q2} were imposed at regions 5, and S2 from the beginning. The simulation was performed for the time interval [0 ; 960] (seconds). We present in Figures 4.47a, 4.47b and 4.47c the distribution of temperature for the calculation times 2.4, 120 and 960 (seconds). It is possible to observe the evolution of the temperature distribution from the beginning (Figure 4.47a, heating is close to the thermal sources), to an intermediate stage (Figure 4.47b) until it reaches the steady state
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regime, when there will be no more changes of the temperature distribution with the time (Figure 4.47c). In Figure 4.48 the average temperature of the body is plotted as function of the time.
Figure 4.47a. Temperature distribution at time 2.4 (seconds).
Figure 4.47b. Temperature distribution at time 120 (seconds).
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Figure 4.47c. Temperature distribution at time 960 (seconds).
800-r
700-^
Temperature [° C]
600-^ 500-^ 400-^ 300-^ 200-^
100-
Time [s]
0 200
400
I 600
Figure 4.48. Average temperature evolution.
One can notice that by the time close to 900 seconds state regime for temperature is reached and there is good between the results obtained by time step procedure with the performed by the solver for the static case, where the term equation (4.81) is not considered.
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the steady agreement calculation dT/dt of
5 Coupling Field and Electrical Circuit Equations 5.1. Introduction Electrical machines are electromagnetic devices with very complex geometries and phenomena, having moving parts, magnetic saturation and induced currents. Therefore, their simulations by Finite Elements require some special considerations. Firstly, these devices are generally voltage-fed. Also, nowadays, it is quite common to find electrical machines fed by static converters and the field equations need to be written with the external electric/electronic circuits. A second aspect related to energy conversion is the rotor movement which must be simulated taking into account the torque. This chapter is devoted to special formulations and techniques used to simulate electrical machines and electromagnetic devices fed by external circuits or static converters. 5.2. Electromagnetic Equations Electrical machines are, generally, a set of conductors with an appropriate magnetic circuit where a magnetic flux, interacting with currents, is capable of generating mechanical forces. The conductors can be classified into two categories: • Thick or solid conductors where the current can be not uniform over their cross-section;
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
• Thin or stranded conductors, generally grouped in coils, where the current densities are normally considered as uniform on their cross-section. We will begin the development of equations related to conductors by introducing the magnetic vector potential, presented in section 4.2.4. 5.2.1. Formulation Using the Magnetic Vector Potential A formulation with the magnetic vector potential is here used, since it has a direct relationship with the magnetic flux (see section 4.2.4), which leads to an easy way to establish the coupled electric circuits-magnetic field equations. As shown in the previous chapters, the magnetic induction and the magnetic vector potential are related by:
B = rotA
(5.1)
If this expression is applied in Maxwell's equation (2.3), one obtains
rotE + —B = rot\ E + — A = 0 dt dt
(5.2)
With (5.2) an electric scalar potential V can be introduced as
d
EH— A = -gradV dt
(5.3)
With (5.3), the current density J can then be written as: A - gradV
a----
(5 4)
'
With equation (2.33) rotU = J
(5.5)
and writing the magnetic induction as:
where B0 is the remanent induction of permanent magnets and replacing (5.4) and (5.6) in (5.5) we have
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
=cr
- gradV
rot—B
(5.7)
5.2.2. The Formulation in Two Dimensions Electrical machines normally present complex geometries and, even using powerful computers, a two-dimensional approximation of the electromagnetic phenomena must be often made. As shown in the previous chapters, the magnetic induction is defined only in the Oxy plan and consequently the magnetic vector potential and the current density have only one component, as:
A = ^k
(5.8)
J = Jk
(5.9)
where k is the unit vector in the z direction. Equation (5.7) can be written as:
d_ \_dA_ dx
where
dA dt
i dx and
_d_ 1 dx p,
y
d 1 dy \i (5.10)
are, respectively, the x and y components Bg .
To solve (5.10), Dirichlet, Neumann and (anti) periodic boundary conditions must be imposed. 5.2.3. Equations for Conductors Two types of conductors are often present in electrical machines. They can be "thin" or "thick (massive)" conductors. We will start by presenting thick conductors, where eddy currents must be considered. 5.2.3a. Thick Conductors Figure 5.1 shows a thick conductor with section St and length t .
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Figure 5.1. Thick conductor.
Substituting J
of (5.4) into divj = 0, and noticing that A is
constant in z direction for this 2D formulation, we get
dV_
div
~dz
dz
dV_ dz
(5.11)
Therefore, we can define a scalar electric potential as (5.12) The voltage U( on the conductor is given by (5.13)
U. = f-
The total current in the thick conductor is obtained by integrating (5.4) over the section St. Noticing that from (5.12) gradV = V\, from (5.13) gradV = V\ = —Ut 11 and using Eq. (5.4) in z direction gives
L1 = \ Jds = - I a — ds +
l
CT —-ds
a
(5.14)
We introduce now in (5. 14) the definition of the d.c. resistance of the conductor, i.e., (5.15)
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Then, Eq. (5.14) can be written as
Rt
-V dt
ds
(5.16)
Finally, for the thick conductors, we have the two equations below:
]_dA dx i dx
]dA dy
dA dt
U, I
a — + a ^ - =0
— ds
(5.17)
(5.18)
This last equation expresses that the voltage over a thick conductor is related to a sum of the voltage drop over the d.c. resistance (/?///) and a
r
dA
voltage drop due to eddy currents Rt L G — ds . */ dt 5.2.3b. Thin Conductors Figure 5.2 shows a coil made of Nco
turns of thin conductors with
cross section s, serial connected. As already commented, in this type of conductor, the current density is considered uniform over the cross-section. We call If the current in a conductor.
Figure 5.2. A coil formed by thin conductors.
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With Eq. (5.18) in Eq. (5. 17), we get, for a thin conductor (where If is here replaced by If ) :
dx
l_dA |n dx
]_dA dA I f Ir -a— + —+ - a dy \i dy dt s s * dt
= 0(5.19)
As the induced current density a (3/4 / dt) is uniform over the crosssection S, we can write
1 dA - af — ds=J O ^v
. i_
£J _
The sub-matrices S/ have 0 or 1 as elements. Some considerations allow simplification of Eqs. (5.106) and (5.107): • In the tree construction procedure, the capacitors and the voltage sources have the priority for being considered as tree branches. If a capacitor would be a link, the voltage at its terminals could be calculated only as a function of the voltage sources and the voltage at the terminals of the tree branch capacitors; then 83, 84 and 85 are null. • Analogously, the inductors, the current sources and the windings have the priority for being considered as links. If an inductor belongs to a tree branch, its current could be determined only as a function of the
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independent current sources, the current in other inductors and the T T electromagnetic device winding currents. Therefore, 84 and 89 are nulls. In this method, the electromagnetic windings are forced to be a link. Consequently, there is no windings tree branch current ift in the circuit. So 85 , SIQ, 815, 820•> $25 are
nu
^ matrices-
• The analyzed circuit is used to feed an electromagnetic device. So the following assumption must be made: it is not possible to have loops containing only voltage sources and capacitors; then Sj must be null. Moreover, it is not possible to have cutsets with only current sources and T T inductors, and then 8^9 and 824 are zero. Taking into account these considerations, the matrix systems (5.106) and (5.107), become: v
0
wc
S
6
S
= - Sll
S
V wr v
w/
S
^ mi .VJ
S2
_
0
7
S8
12
S
17 S22
S
-i e
0
*bc
S
13
S
16 _S 21
0 " i14
18 S23
(5.108)
*br v
0 L 0_
wJ
l
i/ .l bc hr
\bl_
0 yr
S2
0 0
S6 rp
87 T
S8 0
Sn rr,
Sj2 T
mc
S16 S2i il mr rj,
rp
817 822 T
S13 S18 Sf4 0
T
S23 0
1l ml
(5.109)
\ . _J _
In addition to equations (5.108) and (5.109), it is necessary to express the voltage/current relationship of each passive element of the circuit. Then, for the resistors,
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0
mr
where
(5.110)
Rm
l
mr
is the sub-matrix of the tree branch resistors and Rm is the one
of the link resistors, defined as:
>!
0
••• 0"
•.-
o '-. o ; K.=
:
0
'•.
0
0
...
0
rn
r\
0
•••
0
0
'-.
0
i
; o '-. o 0
.-.
0 r*n_
For the capacitors
"it"
v 0 ~\d_ " ^7C Cm\dt nc _
_ ~Cb 0
(5.111)
where C^, and Cm are, respectively, the tree branch and link capacitors sub-matrices, defined as:
cfe =
"q 0 : 0
0 '-. 0 .-•
•-. 0 *•. 0
0" ! 0 cn
c.-
c\ 0 \ 0
0 .-. 0 \ 0 : 0 '-. 0 »• 0 cn
For the inductors
vl/
M
L / LW_
d ~*b / dt
(5.112)
where L^, is the tree branch inductors sub-matrix, Lw is the link inductors sub-matrix and Mw
the sub-matrix related to the mutual inductances
between the link and branch inductors. These matrices are defined as:
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m
m
m m m
m
m
/I
m m l
m m
m m m
m
m
/:
So, using equations (5.108) to (5.112), it is possible to construct automatically the voltage and state equations of any type of electrical circuit feeding an electromagnetic structure. a. Calculation of Glf G2 and G3 The matrices GI( G2 and G3 are calculated using Eq. (5.104). The derivatives of the state variables,
and
dt
dt
, must be written as
function of the state variables, V jjc and imj, the sources Ve and i ,• as well as function of the winding currents i m /. From Eq. (5.109) and (5.111) we have
_
l
d\i
bc =
+S
12 i w/ + S 17 i m/ (5.113)
where \mc and \mr must be expressed as function of the state variables. Using Eq. (5.108) and (5.111), \mc can be written as: l
mc
and imr calculated from (5.108), (5.109) and (5.111), gives:
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(5.114)
T ( Sg'wr
+S
T
13 i m/
+S
T
18 J w/
+S
T
23J7/
\
(5.115) Let us define: H^l
+ R^SgRjSl*
(5.116)
which, placed in (5.115) gives:
(5.117) Replacing (5.114) and (5.117) in the Eq. (5.113) and defining T
l =Cb + S 2 C m S 2
we obtain:
at
= -T, S 7 H, R m S 7 v i c +T 1 -T, 87!!, R m S 6 v e +T, (£17 -S7H! R w S 8 R fc S 18> /i m i
+Tr1^2-S7Hr1R;1S8R,SL)iy (5.118) For calculating d\m\ /dt , we use Eq. (5.108) and (5.112) m
at
u
at (5.119)
where iw , V br and vb[ should also be expressed as functions of the state variables. From (5.109)
dt
dt
and vw is obtained by combining (5.112) and (5.120).
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/(5.120) ciom
From (5.109) and (5.110), we have:
(5.121) and, from (5.108) and (5.109),
Lr = Rj(-S 6 v e -S7v,c -S8v J
(5.122)
Defining
H 2 = l + R 6 SjR«S 8
(5.123)
replacing (5.122) in (5.121) and taking into account (5.123) we have
(5.124)
+H2 R6S18iwl. Establishing
T2 =L m +S 1 4 LX 4 +MX 4 + S 1 4 M t t
(5-125)
and replacing (5.120) and (5.124) in (5.119) we obtain ff\
y
\ S
nil _ T^IIC
I
T
1
!
U~ID c^ D ~*c
c
\
Kr
1
rp—1^1
1
T*
T J — I D c*-* \
——--12 ip!3112 K Z>^8 K m^7 ~»12/ v ^c ~ *2 ^13M2 ^^H1 \/
1
T
"\
\
1
1
i TT1 -*•• d TT 1 ¥^ C! •* 13 ^ C C? I wr T1 * C TT 1 TlO^ \ISl-jllT ij & JVAOfi D O IV^ r nS/:o — O1111I/Vx, c — lO^ O1-2±10 i j z,
cT H T2—lo^13TJr~lij 2 K6
7
:
(5.126) Equations (5.118) and (5.126) written in matrix form have the following form:
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-'sfHf'R- 1 !
dt ^ml
\(
_1
V S 13 H 2
Tn-lc
T
_1
_1
T
2 S13H2 R^n
.*!»/.
dt J R
m ^ 6 -S\l)
I_I~IID cT T 2—IC^13^2 ^b^\8
(5.127) Finally, the matrices G, to G3 are obtained from (5.127) according to the form given by the Eq. (5.104), and resulting in:
Gi =
T-lfc TI-IU c^D-lc 1 2 \S13112 K 6 & 8 K ^ S 7 ~
-T2 S 13 H 2 R^,S13 (5.128)
with Dim G\ = [Nvar x A^var J, where N^
is the number of state
variables;
(5.129) with Dim G 2 = [Nvar x Nsources J, and where Nsources is the number of voltage and current sources from the electrical circuit. Finally, we have: r
3 =
~ S 7 H 1 R m S 8 R 6 S 18J -T2 S 13 H 2 ]
(5.130)
with Dim G 3 = \Nsources x Nwjncj \ and Nwind the number of windings in the electromagnetic device.
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b. Calculation of G4 , G5 and G6 The matrices G4 , G5 and G6 are calculated using Eq. (5.105). The voltage Uy in the windings of the electromagnetic structure, must be written as a function of the state variables \bc and \ml , of the sources ve and ij and of the windings current imi. From the Eq. (5.108) one can write: (5.131) Then, from (5.131), we obtain:
(5.132) Substituting the Eq. (5.124) in (5.132), we have the voltage on the electromagnetic structure windings: i
U
/
nr1 _ 1
_1
S
S
H
\
_i
y
£ 8 R w S 7/ v 6c ~ S 18 H 2 RZ>S13 'lml —1 T —1 i —1 T ~ S 16 + S 18 H 2 R ^ S 8 R w S 6/ v e~ S 18 H 2 R6S23 *j
=V~ 17 + 18 2
R
S
(
—1
T
-S18H2 R^S 18 i OTI (5.133) which can also be expressed as:
(5.134)
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Finally, the matrices G4 to G6 are obtained comparing Eq. (5.134) and (5.105): (5.135) where DimG 4 =[Nwind x Nvar]
(5.136) with DimG 5 = [Nwind x Nsources] G
6
=
-|_S18H2
R
6S
(5.137)
where Dim G6 - [Nwind x Nwind ] 5.4.3c. Example The circuit of Figure 5.7 is used as example but now the inductor L , which was a lumped parameter, is replaced by an electromagnetic device (W) modeled by 2D finite elements (Figure 5.13). 'cl
Ci Vc\
AMA
W C2 |!
Figure 5.13. Electromagnetic device W fed by an electrical circuit.
As seen previously, the equations linking the state variables of the external circuit and the currents in the electromagnetic device windings are given by
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
— dt
(5.138) (5.139)
where, in this particular example:
C
_
EC=[E]
lf=[iw]
2J
The main goal is now to determine matrices Gt to G6 as function of circuit parameters. Two methodologies will be used. The first approach is to determine the matrices Gj to Gg simply by Kirchhoffs voltage/current laws and voltage/current relationships of the passive components in the electrical circuit. The second one consists of using the methodology of automatic determination of these matrices already presented. O Determination of matrices Gj to G6 by Kirchhoff's laws Applying current Kirchhoffs law in the node 2 (Figure 5.13) we have: *R =icl+iw
(5.140)
*R= (5.142)
c\
(5.143)
at From (5.140), (5.141) and (5.142) we obtain: dV
C\ _
dt
=
y
V r\
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, -\
/CI/MI (5.144)
and from Figure 5.13 and Equations (5.141) and (5.143) we have: Z
c2 = *R
dt or —
vj.itj;
The voltage at the winding terminal is
(5.146)
t/y = Vcl
Writing (5.144), (5.145) and (5.146) under matrix form we have:
1" d dt
i
i T
i
i
RC• • • • > As^ ) and the currents in the slots \f . Notice that there is only one unknown current vector because the current in the stator keeps the same value through the slices.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
If the electrical machine is fed by a static converter, Eqs. (5.90) and (5.91) must be added to equations (6.17) to (6.20). The matricial system expressions are now (6.21) (6.22)
(6.23)
(6.24)
dt
X-G.X-G,!,J =G 2 E r
(6.25)
In similar way of section 5.4.4, if the time derivatives are discretized with Euler's scheme, equations (6.21) to (6.25) can be written in matricial form as 0
0 0
Psl
Pf2 Psk
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A.s\(t + At) A^2(f + At)
0
i
R -G 6 + -L
-G 4
-G 3
— 1-Gj At '.
i
A 5 £(? + At) lf(t + At) \(t + At)
0 0
0 0
0
0
o
0 0
_. 0 "r Ajl(0 0 A,2(0
0
0
0
0
:
1
~&iQsl 0
1Q
^*
2
0
1
'" A7Q^
0
0
1
+
A ci- (0
0 AfL 1 0 _l A/ .
i/(0 X(?)
G2E (6.26)
One observes that the number of unknowns strongly increases when the skew is considered. The number of slices k depends on the case under study; generally five slices yields good results. If the machine presents thick conductors in the rotor, as for instance in an induction motor, the conductor is generally skewed. Using the multislice representation of the rotor, each rotor bar will be represented as shown in Figure 6.11.
t Utsn
Figure 6.11. Skewed rotor bar n and its multi-slice model.
In this case, the current in the conductor n, Itn is J
tn = I tin = !t2n ='"
One needs also to impose the voltage Ufn on the bar n as (6.28)
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
To develop the equations for skewed thick conductors, let us consider the simplified case of a magnetic device with three conductors and four slices, as shown in Figure 6.12. conductor I .slice 4
r
$'=-
t
slice 3
conductor 2
conductor 3
Vt4l
t
Ut42
t
Ut43 }
Ut32
t
VtS3 }
t Uls,
t
slice 2
t
Ut2l
t Vt:2
slice 1
t utlj
t Ufn
A
1
t utl3 }
Figure 6.12. Three skewed conductors represented by four slices.
In this figure, the index / in voltage Uty represents the slice number and the index j the conductor number. For instance, voltage Ut\ becomes
U=U
U
+U
(6.29)
The vector U^ of the voltages on the three conductors Ufi,U(2,Uf$
can
be written as follows
'VA'
u,=
" 1 0 0 1 0 0 U,2 = 0 1 0 0 1 0 "a. 0 0 1 0 0 1
i o o ! i oo' o i o io i o o o i jo o i
U
/23
(6.30)
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
or, alternatively
u, =
U,2
" 1 0 0 1 0 0 1 0 0 1 0 0 " 1 0 0 1 0 0 1 0 0 1 0 = 0 0 0 1 0 0 1 0 0 1 0 0 1 (6.31)
where the vectors U tsl >U ts2 >U ts3 an(* ^^4
"tf/lf tf/12 _U«3_
"tf/2i" ; u^ =
^22
u
~tf/41~
~tf/31~
; *3 =
Pt23 _
are:
^32
; uto4 =
.^33 _
^/42 _t/ /43 _ (6.32)
For clarity, we recall the equations (5.76) to (5.79) derived in Chapter 5 for a single-slice structure
SSA + N — A-PI/—P'U, =D J dt '
(6.33)
2 =»
Using these equations, we obtain nl
1
«/ e (
!_ I
and
Now, with /Zj and h^, the Maxwell tensor can be easily applied. First, we examine Figure 7.5, where only the piece P is shown.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
h
Figure 7.5. Force evaluation on the piece P.
In the case of the airgap (1) we have 9 = 180° (angle between h\ and the normal) and a = 29 = 360° (angle between F\ and the normal); then, this force is downward oriented. For the airgap (2), 9 = 0° and FI is upward oriented. The total force acting on the body is Ft = FI — F\, which gives
With the values of h\ and h^ calculated above, Ft is easily evaluated. Moreover, let us calculate the ratio F^
ht F, because
§1 Si
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If\:
we obtain
By this expression, in such situations, we notice that the ratio of forces is inversely proportional to the ratio of surfaces. In fact, as Si is larger than ^2 ' ^2
=
-^1 (*^11 $2 ) is larger than F\ and the total force over the piece
P will be upward oriented. Now, in order to obtain a general expression for the Maxwell tensor, let us go back to the equation (7.8a) and consider a general case where the orientation of the normal unitary vector to the surface is generic. Then, we will obtain the general expressions of dFx and dFy which can be applied in a FE code:
dF = dFx\ + dFv} =
y
1+
(7.9) where
->• dY = dYxi + dYy\ Equation (7.9) is used to force evaluation when having the H components obtained from static and dynamic field calculations. However, complex formulation of Maxwell's stress tensor is much more complicated. We will present here the x component of the force when the complex vector potential formulation (see section 4.3.3) is used.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
The equivalent expression to dFx in (7.9) in complex variables is dTx + voR(Hxej(0t )R(HyeJa)t }dTy (7.10) where R(Hxe-'
) is the real part of Hxe
and Hx and Hy are the
complex components of the field. The term in square brackets in Eq. (7.10) is Ja*
-Hyeja*)
For two general complex numbers A and B and their conjugates A* and B*, the following properties apply:
and and the expression above becomes
(Hxej<s>t
-ffeja"
Performing the product and collecting terms, we get
Noting that the last term in brackets is imaginary, we get
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or
H
The second part of Eq. (7.10) is evaluated in similar steps:
'"" )= - *[(# y°' \Hyejt!"
or
(7
Denoting the real and imaginary parts of Hx real and imaginary parts of Hy
as hxr and hxj, and the
as hyr and hyj, respectively, we have
from Eq. (7.1 la)
H
2
2
and
Similarly, from Eq. (7. lib)
R\HxHy )= hxrhyr + hxihyi and
R(HXH
)=hxrhyr -hxihyi
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.lib)
2
2
Using the results in equations (7.1 la) and (7. lib) and applying them in Eq. (7.10), we obtain the following two expressions for the force dFx=dFxl + dFx2 • Continuous component
(7.12a) Frequency-dependent term (frequency is 2(0 )
= ^-(hl
-hit -h2yr +hfydTx + ^(hxrhyr
-hxihy^dTy (7.12b)
These expressions show that, using this formulation we obtain a continuous component of force which will be superimposed on another variable (frequency 2co) component. Equivalent expressions to (7.12a) and (7.12b) can be calculated for dFy of Eq. (7.9). The vector sum of the forces dFx and dFy , for both real and complex formulations, gives the total force over the body. As already commented upon in section 4.3.3, when the Complex Vector Potential Equations were introduced, this formulation can only be used when the excitation is sinusoidal and the materials are linear (saturation is not taken into account). Also, permanent magnets can not be present in the device. Because of this, the Complex Vector Potential formulation can not be employed in the simulation of many electromagnetic structures even though it can be very useful and practical for a number of cases. Now, we will proceed with the presentation of torque calculation by means of Maxwell Stress tensor from equations (7.9).
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
The electromagnetic torque can be obtained by: Te=l\(rxdF)dT
(7.13)
r where I is the depth of the domain (in Oz direction) and where r is the position vector linking the rotation axis to the element dT; as explained above, dF is generally calculated with (7.9). Numerically, the integration (7.13) is performed by the sum as:
Te=i^(dFyirxi-dFxiryi]
(7.14)
/'=!
where N is the number of elements with surface dT, dFy^ and dFxj are, respectively, the elemental forces acting on this surface along the y and X directions; rxj and ryj are the components of the position vector defined by the rotation axis and the middle point of dT. Although theoretically the surface F on which the force is evaluated can be any inside the airgap, in practical terms it is not appropriate for numerical implementation. For 2D cases, this surface is replaced by a line and to obtain good accuracy, this one should be a set of segments linking the middle points of the triangle's edges, as shown in Figure 7.6.
0 Figure 7.6. Suitable integration line for applying the Maxwell stress tensor.
As already commented, the use of triangles as shown in Figure 7.6, provides poor results and, consequently, inaccurate electromagnetic torque values. A good way to minimize and virtually eliminate such problems is to
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
use quadrilateral elements as for the moving band, as presented in chapter 6. In this case the surface F is chosen as indicated in Figure 7.7.
Figure 7.7. Surface for Maxwell tensor integration when using quadrilateral elements.
When using quadrilateral elements for torque calculation, the procedure is: a. proceed with quadrilateral element cuts as indicated in Figure 7.7.b; b. for all the elements, define the integration surface by segments crossing the middle of their edges; c. define the position line linking the rotation axis with the point placed in the middle of the segment dT ; d. calculate the torque with (7.14); c. make similar triangle cut using the second diagonal, as in Figure 7.7.c; f. repeat the steps b, c and d for this second cut; g. calculate the corresponding second torque with (7.14) and average it with the first one.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
If the integration line is placed in the interior of the moving band, the precision of the torque can be affected if the rotation step does not coincide with the discretization step, because the quadrilateral elements will be distorted. To avoid this, it is necessary to define a supplementary layer of quadrilateral elements in the airgap and proceed with the calculation along this layer without deformation. 7.2.3. The Method Proposed by Arkkio It consists in writing Eq. (7.13) as a function of FQ , the tangential force density component of the magnetic force F:
FQ can be expressed in terms of Br and BQ , respectively the radial and tangential components of the magnetic induction, as FQ =—BrBQ HO Substituting the equation above in the torque equation and remembering that dT - rdy gives
or
T e = — f r2BrBQd