ELECTROMAGNETIC MODELING BY FINITE ELEMENT METHODS JOAO PEDRO A. BASTOS NELSON SADOWSKI Universidade Federal de Santa Catarina Florianopolis, Brazil
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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.
y
Figure 1.6. A closed surface in Cartesian coordinates: definition of divergence.
we obtain
= Axdydz + Avdxdz + Azdxdy
(1.7)
Note that the flux can be defined only when the direction of n is defined. In the case of a closed surface S, n always points to the outward direction of the volume enclosed by the surface S. 1.5.2 The Divergence Theorem Consider the surface of a rectangular box whose sides are dx, dy, and dz, and parallel to the planes of Oxyz . The area of the lower face PQRS is dxdy (see Figure 1.6), and its vector ds is
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From Eq. (1.7), the flux of A crossing this surface is - Azdxdy
(1.8)
On the upper face, P'Q'R'S', an analogous expression is found. On the upper surface the normal to the surface is positive, and the component Az of the vector A, is augmented by dAz. Therefore, A2 on the upper face is
dA, dz dz
Az + dA = A and the flux is
V
(1.9)
Az +—-dz \dxdy dz )
The sum of the two fluxes gives dA. dz
dv
(1.10)
where dv is the volume of the rectangular box. Using the same rationale on the other two pairs of parallel surfaces, yields the expression
dx
dy
y
dA.
dy
dz
+
dz
dv
Observing that dAv dx
«
= divA
The expression for the flux becomes -divAdv
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(1.11)
With dCD = A • ds , the integration over the whole surface S gives the total flux 0) = <j" A-flfs = [divAdv
(1.12)
This equality between the two integrals means that the flux of the vector A through the closed surface S is equal to the volume integral of the divergence of A over the volume enclosed by the surface S. 1 .5.3. Conservative Flux Consider a tube of flux, such that the field of vectors A defines a volume in which the vectors are tangent to the lateral walls, as shown in Figure 1.7. S^ and S2 are arbitrary surfaces which section the flux tube. S3 is the lateral surface of the tube section. We denote Ab A2, and A3, the vectors A on surfaces Si, S2, and S3, as indicated in Figure 1.7. To simplify the discussion we assume the vectors \i and nj are in opposite direction and A2 and 112 are in the same direction. Now the different fluxes can be calculated: = cf A- ds = [ divAdv
JS
= A 3 -ds3
W
=0
The total flux is therefore
L
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*-— "-""""
s,
A,
I
/
/\
i_^ ->
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
div\ * 0 We now calculate the flux of a different vector field A, through the lateral surface of the cylinder shown in Figure 1.10.
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Figure 1.11. Directions of A and ds for the field in Figure 1.10.
Examining the transverse cross-section (Figure 1.11) we note that the vectors A and ds are perpendicular. Consequently, from the divergence theorem we conclude that in this case, divA = 0. Observing these two examples, we note that in the first case, vectors of the field are literally divergent at point M and, therefore, divergence of the field is different than zero. In the second example, vectors of the field do not depart but rather suggest rotation or a vortex, divergence. In this case the divergence of the vector is zero.
the the the not
1.6. The Rotational 1.6.1. Circulation of a Vector The circulation of a vector field A along a contour L (Figure 1.12), between points P and Q is given by the line integral (1.13) where dl is a differential length vector along the contour. If A and dl are parallel, as in Figure 1.13a, the circulation is maximum. However, if A and d\ are perpendicular to each other, the circulation is zero. In the first case, the vector A circulates along the contour L, while in the second case, where A is perpendicular to the contour, it does not circulate along the contour L. If the vector A is the gradient of a scalar function, A = gradU we have
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CPQ =
Figure 1.12. Definition of circulation of a vector A along contour L.
maximum C Figure 1.13a. Maximum circulation.
c =o Figure 1.13b. Zero circulation. Assuming that dl is equal to dM in Eq. (1.5) we get
dU =gradU • d\
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and the circulation is
Thus, if A.=gradU, the circulation along the contour depends only on the initial and final points of the contour (P and Q), not on the contour itself. If the contour L is closed, or, if Q and P coincide, the circulation of A is zero. It remains to be verified if all vectors can be identified by a gradient. To do so, suppose that a general vector has components dU/dx , dU/dy ,
dU/dz . Note that
dy \ dz )
d2U dydz
dzl^dy
d2U dzdy
or that 32U dydz
dzdy
This is valid for any gradient. Writing A.=gradU, gives
x —
dx
'
A -dU v— ' dy
-dU z— dz
if
d2U
d2U _ d (dU}
d (dU]
dydz
dzdy
dz ^ dy )
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—
dy\dz)
—u
we get the condition
dy
dz
Analogously for the indices x, y and x, z we obtain
dAy.•*
dA-£_ _
dz
dx
dAvy
QAxY
dx
dy
Q
=0
These three terms are the components of the rotA [see equation (1.2)]. Therefore, if A = gradU , the rotA is identically zero. 1.6.2. Stokes' Theorem Assuming that the flux of a vector B is conservative, it is possible to define a vector A so that B = rotA , since divB = 0. By substitution we get div rotA = 0. This quantity, div rotA = 0 is identically zero for any vector A, and the definition B = rotA is correct for any vector B if = 0 . This theorem will show an important relation for this vector field A. Assume that there is a closed contour defining a surface. This surface is divided into small areas s. We examine one infinitely small rectangle, with sides dx, dy, parallel to the axes of the coordinate system (Ox and Oy) as in Figure 1.14. The circulation of A=(A ,A ,A ) along the contour L which encloses the area s can now be calculated. This is done separately for each section of the contour. •
Between points P and Q
or j = Axdx
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Figure 1.14. A small surface defined by a closed contour. Between points R and S, the vector A is
A = \AX + dAx, Ay + dAy, Az + dAz) where
Ax + dAx = Ax and, therefore, the circulation is
The sum of C, and C2 is dA
dy In a completely analogous manner, the sum of the circulations along lines SP and QR is
dA i ^ — y dxdy
dx
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The sum of CJi2 and C3,4 gives the total circulation along the rectangle PQRS. Denoting this circulation as C2, we have
dy )
dx
dxdy
However, since B = rotA , the expression in brackets is equal to Bz: Cz = Bzdxdy Analogously, Cx and Cy can be calculated on rectangles parallel to Oyz and Ozx, to obtain
Cx=Bxdydz Cy = Bydxdz The total circulation on a curve composed of these three differential rectangles is [see Eq. (1.7)] C = Cx + Cy + Cz =
B • ds — I rotA • ds
This expression was obtained through the calculation of the circulation of A, and, therefore, we can write
c[ A-d[= I B-ds = f rotA-ds J/>
»O
(1.14)
**J
This is
cf A-d[= \ rotA-ds TjLrf
(1.15)
*tJ
Carrying out the calculation above over the whole surface of the global domain validates this result in a global sense. Therefore, we conclude that the flux of a vector A through the open surface S is equal to the circulation of the vector A along the contour L, encircling the surface.
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Figure 1.15. A vector field A with constant magnitude at distance R from point M.
1.6.3. Example of Rotational Consider the field of vectors A, as indicated in Figure 1.15. Assume that at a constant distance /?, the magnitude of A is constant. Defining a surface 5" such that it is enclosed by the circle of radius R, we can calculate the circulation of A along the contour L. A and dl are collinear vectors, and we have
cf A • d\ = cf Adi = A(f dl = 2nRA JL JL JL Since this circulation is nonzero, we conclude from Stokes1 theorem that rot A. is also nonzero. Because the vector rot A is perpendicular to A (by definition of the cross product), we observe that the geometrical positions of A and rot A are as shown in Figure 1.16. Another example is shown in Figure 1.17. Choosing the same surface S, and noting that A and d\ are orthogonal vectors, gives
{£ and we conclude that rot A = 0.
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rotA
Figure 1.16. Geometric representation of the rotA
Figure 1.17. A radial vector field.
In this particular vector field, there is no rotation or vortex. On the other hand, the field is divergent. 1.7. Second-Order Operators It is possible to combine two first-order operators on scalar functions t/and vectors functions A. The possible combinations are:
divgradU
(1.16)
rot gradU
(1-17)
grad divA
(1.18)
div rotA.
(1.19)
rotrotA
(1.20)
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Note that, for example, rot divA , cannot exist by definition since the rotational operates only on vectors, while divA is, by definition, a scalar. We now calculate, as an example, the operator div gradU. Given:
.dU .dU , dU Jrr gradU = i- + j- + k dx dy dz we can write JJTT div gradU =\\ i' -- h j• -- hkI — • gradU (^ dx dy dz J
JTT
or, after performing the dot product:
div gradU =
d2U
d2U
d2U
dx2
dy2
dz2
Defining the Laplace operator (or, in short, the Laplacian) as
dy
we have
div gradU = V2U
(1.22)
The calculation of rot gradU and div rot A represents operations which are identically equal to zero, without any particular condition being imposed on U and A. Finally, Eq. (1.18) and (1.19) are generally used together such that
V 2 A = gra d div A -rot rot A where V A is called the "vector Laplacian" of A. This is written as
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(1.23)
V2A = i V2AX + j V2Ay + k V2AZ
(1.24)
where, for example, the component in the Ox direction is |
+
Sy
dx'
dz'
These operators define second-order partial differential equations, and constitute a very important domain in mathematics and physics. Written in an appropriate form, they describe the phenomena of diffusion of fields, either electromagnetic (electric or magnetic fields), or mechanical (diffusion of heat, flow of fluids, etc.) 1.8. Application of Operators to More than One Function If U and Q are scalar functions and A and B are vector functions, all dependent on x, y, and z, we can show that:
gradUQ = U gradQ + Q gradU
(1.25)
divUA = UdivA + gradU • A
(1.26)
div(A x B) = - A • rotB + rotA • B
(1.27)
rotUA = UrotA + (gradU)x A
(1.28)
Figure 1.18. Cylindrical coordinate system, its relation to the Cartesian system, and components of vector A.
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1.9. Expressions in Cylindrical and Spherical Coordinates
The expressions below give the principal operators in cylindrical and spherical coordinates, applied to a vector function A and a scalar function U. a.
Cylindrical coordinates r,
dU +13U dU gradU = r -> +
id
dA
i
-r dr
r d(f)
rotA = r\
\
dz
/
_
(1.30)
dz
b. Spherical coordinates R,0,(f)
1 dAz
(1-29)
(Figure 1.19)
.
_ .
dz
dr
X
/
r
dr (1.31)
d2U i 1 dU i I d2U i d2U 2
r dr
2
2
(1.32)
Figure 1.19. Spherical coordinate system, its relation to the Cartesian system, and components of a vector A.
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ac/ ,TT =nR - +16U t_l_ gradU +0 -- + d> -dR R d RsinQ d
3 \R (n2A, ]+ \ /vA = —1- — K ^ ^/
1--
5 (A0s /. . sm ^°
v d-33) '
1 RsinO d(/> (1.34)
R^sinQ
R(
8R
8AR 8Q (1.35)
d2u
i 2
dR
2
2
R sin ed>
2
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«
2
i
a
R sine ^0
0 (1.36)
2 2.1. Introduction In this chapter a brief presentation of Maxwell's equations is done, assuming that the bases of Dedromagnetics (EM) are already known. The main idea here is to recall the principal concepts necessary to the work developed afterwards. Besides, it is interesting to define and establish the notation of the main physical quantities that will be used in the following chapters. Electromagnetics (EM) can be described by the equations of Maxwell and the constitutive relations. The theory of EM took a long time to be established, and it can be understood by the fact that the EM quantities are "abstract" or, in other words, can not be "seen" or "touched" (contrarily to most others, such as mechanical and thermal quantities). Actually, the majority of the EM phenomena were established by other scientists before Maxwell, such as Ampere (1775 -1836), Gauss (1777-1855), Faraday (1791-1867), Lenz (1804-1865) among others. However, there was some incompatibility on the formulation and Maxwell (1831-1879), by introducing an additional term (in 1862) to the Ampere's law, could synthesize the EM in four equations. The genius of this man brought the EM to a very simple formalism, kept mainly by only four equations. The physical possibility of this group of equations (along with constitutive ones) is so high that very distinct phenomena (like p. e. microwaves and permanent magnet fields) can be precisely described by it. While the formalism and the basic concepts
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of the EM are relatively simple, realistic problems can be very complicated and difficult to solve. In fact, when, p. e. complicated geometries, nonlinearity, many non-statics field sources, etc, appear together (or sometimes even alone) it is virtually impossible to find analytical solutions for such problems and that is the main reason why numerical methods have become widely used tools in Electrical Engineering nowadays. In this work, we will be interested in low-frequency phenomena and the contents of the chapters are mainly focused on this part of the EM. As a matter of fact, Maxwell's equations do not make distinction between low and high frequency, as already mentioned above, but bringing them to practical applications, it is possible to adapt them to these two situations. Moreover, when describing low frequency problems, generally, the Maxwell's equations can be divided into two groups: electrostatics and magnetics and, an important aspect: they can be treated independently. The diagram below (Fig. 2.1) is representative of the EM, and its possible (and most common) division.
Electromagnetics (Maxwell's Equations)
Electromagnetics Low Frequency
Electromagnetics High Frequency (Waves)
Fig. 2.1. EM division for physical applications thus, in each block of this diagram, the four Maxwell's equations are adapted to the corresponding physical situation.
2.2. The EM Quantities Maxwell's equations are a set of partial differential equations in space and time applied to electromagnetic quantities. When they interact with
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material, the equations can assume nonlinear forms. We attribute properties of principles or postulates to Maxwell's equations.
E
Q
Figure 2.2. Electric field due to a charge Q, or an equivalent charge distribution.
The electromagnetic quantities involved in Maxwell's equations are: • The electric field intensity E • The electric flux density or electric induction D • The magnetic field intensity H • The magnetic flux density or magnetic induction B • The (surface) current density J • The (volume) charge density p In addition to these we define: • The magnetic permeability \\, • The electric permittivity 8 • The electric conductivity a The significance of each of these quantities is considered next. To do so we assume here that the notions of electric charge and electric current are known.
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2.2.1. The Electric Field Intensity E An electric charge or an assembly of electric charges Q, stationary in space, has the property of creating an electric quantity in space, called an electric field intensity E as shown in Figure 2.2. The electric field intensity is a vector quantity and obeys the rules of vector fields. The manner in which the electric field intensity E is calculated will be shown in subsequent sections. 2.2.2. The Magnetic Field Intensity H Suppose that the charge or assembly of charges in Figure 2.2 are not stationary in space but move at a given velocity. In this case, a magnetic field intensity H is generated as shown in Figure 2.3.
Charge -*•
Figure 2.3. A moving charge and the magnetic field intensity it generates.
A moving charge or charges lead to the idea of the electric current. This is the ultimate result of the vector field H, whose calculation we will present shortly. If this movement of charges occurs in a conducting wire (as it is the case in a majority of real situations), the electric field intensity is practically nonexistent since the electrons move between vacant positions in the atoms of the conducting material and the net sum of charges is essentially zero. Later on we will see that variations in electric field intensities also generate magnetic field intensities.
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2.2.3. The Magnetic Flux Density B and the Magnetic Permeability ju. Since B is a vector field, we can define a flux O crossing a surface S
as
O= f B-ds JS The flux
Consider two media with identical geometries which have different permeabilities jl, and \12 such that \\.l > \JL2 as in Figure 2.4. Suppose that, by external means, we create a magnetic field intensity H in both materials and that H is constant throughout the cross-section S. Then, and B2 =
Figure 2.4. Two materials of different permeabilities maintain different magnetic flux densities for the same field intensity.
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The fluxes
, and
and
® = BS =
where B{ and B2 are the magnitudes of B, and B2 perpendicular to S, and constant on it. We obtain
Note that the larger is the permeability of the medium, the larger are the magnetic flux density and the flux passing through its cross-section S. In other words, B is called "magnetic flux density" or "induction" since this quantity expresses the capacity to induce flux within a medium. As in the example above, a high flux density is associated with a high permeability |J, . Using the literal meaning of the terms "induction" and "permeability," we can say that a large flux is "induced" in a medium and that the medium is highly "permeable" to flux. The permeability of air is |I0 =471 -10" Henry/meter. 2.2.4. The Electric Flux Density D and Electric Permittivity 8
There is a direct parallel between the pairs D, 8 and B, JJ, , shown above. D is also called "electric induction" and it plays an important role in Gauss' theorem, as will be presented soon. There are, in spite of similarities with magnetic quantities, a few salient differences between them. The first difference is the fact that 8 varies little between materials, in contrast to the permeability fJ, . In useful dielectric materials 8 varies no more than by a factor of 100 while the variation in (J, can often attain factors of 10
or
more. A second observation is that, in general, when solving problems with electric fields and electric flux densities, we are especially interested in the electric field intensity, while in magnetics the magnetic flux density assumes
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a predominant role in the analysis of the phenomenon. The permittivity of air is S0 =8.854-10~ 12 Farad/meter.
_ — /) 0 <— 0
s Figure 2.5a. Straight conductor Figure 2.5b. Conductor dimensions
2.2.5. The Surface Current Density J Consider a straight conductor with a uniform cross-sectional area S and a current / crossing the cross-section in the direction indicated in Figure 2.5a. We define a unit vector u perpendicular to the surface S. The mean (average) surface current density crossing the area S is given by
If we assume that the surface S is small, the current density J can be considered to be constant over the surface. We define a vector which has a magnitude equal to J and its direction is given by u
The calculation of the flux of J crossing the surface S defines the current /, since
/={„ J-rfs where ds is the differential area. In many cases, J varies throughout the cross-section. 2.2.6. Volume Charge Density p Assuming that a number of charges Q occupy a volume Vol, a uniform volume charge density is defined as
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= _Q_
Vol Nonuniform charge distributions can similarly be taken into account observing that
where dv is the volume differential. 2.2.7. The Electric Conductivity a
In general, when analyzing electric field problems we distinguish between two types of materials: dielectric or insulating materials and conducting materials. Insulating materials are characterized by their permittivity 8 and their dielectric strength (which will be discussed later). Conducting materials are characterized by their conductivity a. The latter expresses the material's capacity to conduct electric current. We define the relation J=tfE
which is Ohm's law in point or local form. In the case of a linear conductor of length / and cross-sectional area S, as in Figure 2.5b, this assumes the familiar form shown below. We have seen earlier that
s and we state that the electric field in this case is
where V is the electric potential difference on this section of the conductor. Substituting these two relations in the equation J = crE we have
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L-
L
s~°7
or
where / /(jS is the resistance R of the conductor. We now have V = RI which is Ohm's law in a more common form. The difference between these two forms of expressing Ohm's law, is that the first, J = <J E , is a "local form" expression. It defines a quantity at any point in space. On the other hand, in the form V = RI , it is necessary to introduce the dimensions of the conductor (S and /) making this an "integral" form of Ohm's law. The relation, J = a E , as well as B = (J,H and D = c E are called constitutive equations or constitutive relations and are used in addition to Maxwell's equations. They describe the relations between field quantities based on the electric and magnetic properties of materials (8 , [i , and G ). 2.3. Local Form of the Equations Maxwell's four equations are as follows:
(2.1)
— dt
divB = 0 rotE = divD = p
(2.2) --
(2.3) (2.4)
from these equations, we can define a fifth relation. Applying the divergence on both sides of Eq. (2.1) gives
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div(rotH) = divj + div v
'
dt
or, using the fact that div(rottl) = 0, we have Q = divJ + —(divV)
ar
'
Utilizing Eq. (2.4) gives ,. divJT =
dp — dt
This equation is called the electrical continuity equation. We observe that in general, dp/dt is zero and therefore we normally obtain divJ = 0. This is significant in that the flux of the vector or, similarly, the conduction current is conservative. In other words, the current entering a given volume is equal to the current leaving the volume. In fact, in practically all electromagnetic devices, the current injected into the device is equal to the current leaving it. When this does not happen, there is an accumulation of charges in the device, or a certain amount of charge is extracted from the device. This is shown schematically in Figure 2.6. Assuming that /2 < /,, we have an accumulation of charges in the volume V. As a consequence, dp/dt^Q
in this volume. The negative
sign of the expression
Figure 2.6. Accumulation of charges in a volume due to nonzero divergence of the current density.
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signifies that the sum of the flux of J is negative (/2 < /,)and it augments the volume charge density with time. In fact, applying integration over the volume and utilizing the divergence theorem gives:
JS(V) or
i r -
dQ
— 1\ + + 17 = 1
-dt
2
The first term is therefore negative (see section 1.5.3), resulting in dQj dt > 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
<2-7)
= p
(2.8)
There is an interdependency between all variables in the equations and, therefore, a unique solution. For practical purposes, it is often useful to simplify these equations based on the conditions of operation. One of the most important
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simplifications we can make is to neglect the displacement current density term dD/dt. Obviously for static cases this term is zero. If it is not the case we notice that this term, appearing in (2.5), can be related to the equation (2.8). At first look, one can imagine that as (2.8) is a static equation that its derivative is zero. However, if p depends on the time, the derivative dD/dt is not zero, in the same way that a time varying J can create a H(/). That is the link between (2.5) and (2.8). We can consider the following physical situation. In Fig. 2.11 we have two conductive plates and the potential difference V between them varies. V can depend on time, and the resulting E too. Of course, the electrical field E created is calculated using the equation (2.8) (classical case of conductive parallel plates capacitor where p is related to V).
air study E(f) domain.
Fig. 2.11. Electric field E(/)
•V(t)
calculated by Equation (2.8).
If in our study domain we are proceeding with magnetic calculations, in the equation (2.5) we should consider, formally, the term dD/dt. However, let us proceed with a quantitative analysis: at frequency of the order of 105 Hz, and with an electric field intensity of the order of 105 V/m, the displacement current density is of the order of 10" AI mm2 . This is much smaller than conduction current densities which are of the order of 1 A/mm2. For this reason we can neglect parasitic capacitances in wires. We cannot neglect this term if, in the domain under consideration, we have large capacitors and large electric fields, or if the involved frequencies are very high. In these cases, the displacement current can be large. However, this constitutes a sufficiently unique case to be treated separately.
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Now, we will examine the relationship between the equation (2.5) and (2.7). Suppose initially that we are considering conductive media. The time variation of B (dB/dt) creates E and, by the constitutive equation J =CTE, defines J , which, in this case will be called Je . Therefore in the equation (2.5) J is composed of two parts: an external imposed Ji and Je, and the equation (2.5) becomes (neglecting here dD/dt):
rot H = Ji + Je There is another common linking case between (2.5) and (2.7). Suppose that our study domain is placed in a free space instead of a conductive medium as above. In this situation J is zero and dB/dt creates E by (2.7) and D, by D = S0E . The equation (2.5) becomes
„ dD rotH = — dt which can be written as
rotH =SQ
dE dt
This expression, together with (2.7) or
rotE = -I
dt
forms the set of equations related to electromagnetic waves, which normally operates at high frequency. Concluding this analysis, it was possible to show that for the most common and practical low frequency situations, the displacement current density dD/dt is much smaller than J and generally can be neglected. Of course, for particular and less usual cases, an appropriate analysis should be performed. This approximation allows decoupling the system of Maxwell's equations into two independent set of equations:
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The first one consists of the equation below: rotH = J
divB = 0 dB dt
with
B = llullH
which have the properties needed to treat magnetic problems. The second system, with the remaining equations representing the electrostatic system is divD = p and
In a classical and more formal approach we can define the three inequalities of the "electrotechnic" or "quasi-static" domain: a. let us consider a generic physical quantity G depending on time (t) and space r as G = G\r,t). We call dG/dt the time derivative and dG/dx, dG/dy,
dG/dz the space derivatives, which will be noted as
VG for simplicity. The "quasi-static" approximation is expressed by:
c dt
VG
for practically all the points of the study domain; c is the speed of light. We recall that in the low-frequency cases the wavelength generated in the
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system is much larger than the dimensions of the device under consideration and therefore it has virtually no influences on the device itself. However the space derivatives can be intense. One can imagine, for example, that in an airgap of electrical rotating machine the fields are very high and with small geometric dimensions, it will create very high values for VG. fa. the velocities (v) of the materials are much smaller than the speed of light
v «c c. in terms of energy density the magnetic fields are preponderant compared to electrical fields. Using the expressions of energy density we have for the air:
l^
«
6
2 '
1B 2 u,
or
and B
1
E
noticing that c = 1/-J80|J,0 .If we are considering different materials with S and \Ji , c should be replaced by the local velocity c'= l/^SfJ, ; this will not change the final results of this analysis. This set of inequalities is compatible with practically all the cases with which we normally deal in low-frequency studies. Again, for particular situations, an adapted and appropriate analysis has to be performed. Now, we examine the term (dD/dt) using the above considerations
dD dt
= °o
dE_ «s0c dt
or
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VH
dD dt
VH
noticing the use of inequalities presented in sub-sections "a" and "c" above. Therefore in quasi-statics we have equation (2.5) approximated by
ro/H = J This analysis can be a subject of further discussions. The quasi-static phenomena can exist even for relatively high frequencies as, for example, 10 GHz. The corresponding wavelength is close to 30 mm and for microelectronics design this frequency is sufficient to define a quasi-static problem. 2.6. The Integral Form of Maxwell's Equation The Maxwell's equations, under local form, consist in a powerful set of equations, since under this form, they are valid at any point. Also, having short notations, it is easy, under algebraic aspects, to manipulate and adapt them to different physical situations. However, when it is necessary to apply them to a realistic case, the notions of volume, surface and line (directly linked to the physical device itself) should be taken into account. In such cases their generality (local form) is adapted to a particular case, and, of course, limited to it. As example, the Maxwell's equations adapted to an electrical machine will not be appropriate to analyze a different device. On the other hand, by using them in integral form, normally we are able to calculate and obtain the quantities we need. In practice, for almost every case, we use the integral form of Maxwell's equations in order to solve the corresponding problem. For doing so, we use the theorems of the divergence and Stokes (presented in chapter 1) to obtain the equations integral form. For the first equation rotH = J we calculate the flux through an open surface S in both sides as
f rotH -ds = \ J • ds JS *S
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and using Stokes' theorem
The left-hand side is the circulation of H along the closed line L(S) delimiting S; the right-hand side is the conductive current involved by the line L(S). The equation above, under this form, is known as Ampere's law. The second equation divB = 0 is integrated in a volume V as
Using divergence's theorem, it gives cfe-
[rotE-ds -- \
•k
and
We will show that the left-hand side is the e.m.f. (electromotive force whose unit is Volts); the right-hand side, which must be manipulated with caution depending on the physical situation, under the majority of cases becomes
dt *
dt
And we obtain "Faraday's law"
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f e.m,f.J
«# — dt
Finally the fourth equation divD = p is written as
f divDdv= and
f
ds -q
known as Gauss' theorem meaning that the flux of D through the closed surface S(V) is equivalent to the electrical charge q contained in V. 2.7. Electrostatic Fields For simplicity, we start presenting the second and simpler set of equations, related to the electrostatic fields, whose main equations are, for isotropic media
divD = p
(2.9)
D = eE
(2.10)
We make here the assumption that in dielectric materials we can use the relation D = s E , where S is a scalar. The concept of relative permittivity S , is defined as:
_ 8 £
0
where 8 is the permittivity of a material and 8 Q is the permittivity of free space (80=8.854x10
—12
F / m) . His normally sufficient to specify the
permittivity of a material through £r; when it becomes necessary to use the permittivity 8 of the materials, we use the relation s = S Q£r •
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2.7.1. The Electric Charge
Electric charge is of considerable interest in Electrical Engineering. However, we will not dwell here on its applications but rather on its use in defining some concepts of major importance and utilization relevant beyond the electrostatic field problem. 2.7.1 a. The Electric Field
Consider a charge or an assembly of static charges q. It was shown in the previous chapter that q can generate an electric field in the space around it. We have also seen that the equation div D = p , becomes, in integral form, the expression for Gauss' theorem. (2.11) where S is the surface enclosing the volume V, which contains the charge q. In this case, we choose for the volume V a sphere with radius r. Assuming free space (s = E Q ) , we have
Since E and ds are collinear vectors at any point of 5", we obtain
noticing that E is the same at any point of S and therefore independent of it. And
(2.12) To understand the meaning of the vector E, we define the vector r as r =P-0 as shown in Figure 2.12. The unit vector u in the direction of r can be obtained from u = r/r. With this notation, E, which is associated with the charge q through the divergence, can be written in the following form:
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qr
Figure 2.12. Vector r
(2.13)
Figure 2.13 Force on q'
2.7.1 b. Force on an Electric Charge
It has been observed experimentally that the force F exerted on a charge q under the action of an electric field intensity E is given by the expression F = #'E
(2.14)
Assuming that the electric field intensity E is generated by a charge q, as in Figure 2.13, we have
F=
47isr 3
(2.15)
The magnitude of F is
F= This expression is known as "Coulomb's law," according to which the force is directly proportional to the product of the charges, and inversely proportional to the square of the distance between them. 2.7.1 c. The Electric Scalar Potential V From the force defined in the previous paragraph it is possible to calculate the work dW performed by this force by the expression
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where dl is the displacement of the charge q under influence of the force F. We are particularly interested in the work per unit of the charge q '. Viewing it in this manner allows the definition of an expression related to work, but with the idea of representing the capacity of the charge q to perform work. We have
Defining the work per unit charge as Fand using F = q'E gives
The reason for introducing the negative sign will become evident shortly. Assuming that the electric field intensity varies along a trajectory, the energy per unit charge in moving the charge from /j to /2 can be expressed as
V2-V} =-
Figure 2.14. Change displacement.
Consider now an example consisting of a point charge, as shown in Figure 2.14. We wish to move the test charge q' from a position vl to a position r2. Moving q' away from q and denoting Vl the potential at r^ and Vz the potential at r2 gives
Choosing the path dl — dr gives: r • d\. — rdl — rdr . Upon evaluation of this integral we have
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A ' 47tS 0
(2 16)
'
'
We note that the negative sign introduced in the expression dV = —E • dl comes from the fact that the right-hand side of this expression is negative. This implies that Vl is larger than F2. Effectively, the adopted convention is that the potential close to the source is larger than the potential at a larger distance. The electric field points from the higher potential to the lower. The electric field intensity is, therefore, divergent in relation to the charge q. On the other hand, the value of the integral depends only on the initial and final points on the trajectory. In the case of a closed contour we have (2.17) This can be easily verified using Figure 2.15a. Note that where E is perpendicular to dl we have E • d\ = 0 . On the other two parts of the path, the direction is radial. On one part of the path E • d\. > 0 while on the other the magnitude is the same but negative in sign. This occurs on any arbitrary contour, as shown in Figure 2.15b, where any segment dl can be decomposed into a radial and a tangential component. This decomposition allows simple evaluation of the scalar product E • dl, and produces a zero sum whenever the contour is closed. E
dl
E
Figure 2.15a. A closed contour in an electric field. The closed contour integral of static E is zero.
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Figure 2.15b. Decomposition of an arbitrary contour into radial and tangential components.
The integral
represents the work per unit charge. In the example above, when a charge is moved in such a way that it returns to the starting point, the total work performed is zero. In this case, we can say that the field is conservative. This indicates that, if the charge returns to the starting point, there is no energy lost or gained in the process. In other words, the electric field intensity E can be derived from a scalar potential V as
E = -gradV By calculating the circulation of both sides of this relation we obtain an expression for the absolute potential as
Taking into account the definitions in Chapter 1 [Eq. (1.5)] gives
where Kis a constant. In the case of a point charge q, and setting d\. = dr , we obtain
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V=1000 V=500
V=500
E
V=0
E 0
a.
b!
Figure 2.16. Two examples of different potential references that produce the same electric field.
r-dr+K or
By setting, for instance, F = 0 for r = oo, the constant K is zero. This gives (2.19) It is always necessary to define a value for the constant K, since different values of V can generate the same field intensity E. This is reflected from the examples in Figure 2.16a and 2.16b. Assuming E is constant in the direction Ox and existing between the plates of the device, we get
dx
therefore Ex = AF// . In both cases, Ex = 500// ; however, the values of F are different. We conclude that in order to define F uniquely in the whole domain it is necessary to fix F at some point in this domain. This value of the potential is a reference potential.
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2.7.2. Nonconservative Fields: Electromotive Force In addition to the conservative electrostatic field there is another type of field, called "nonconservative." Suppose there is a closed current loop due to a conductor with contour C and enclosing a surface S. From Ohm's law:
E = — —J cr L where R — L/aS
(2.20)
represents the resistance and L is the length of the
conductor. The circulation of E along the circuit is
R ~L where the relation / = JS is used. As explained above, the circulation of E along the closed contour is zero because it is a conservative electrostatic field. Therefore,
This indicates that the current due to a conservative field is zero. To see this, we use the example in Figure 2.17. A block of material is located between the two charged plates. Because of the electric field between the plates, there is an instantaneous movement of charges in the block as shown; however, this electrostatic field is not capable of maintaining a current. The charges simply move to a new location where they are again stationary. Sustaining a current requires a source of energy since a current is a continuous movement of electrons in a circuit. This movement is impeded by the resistance of the circuit and is characterized by dissipation of energy as heat (Joule's effect). The energy must originate in a nonconservative electric field. As an example, the chemical reaction in a battery can produce
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a nonconservative force by which the electrons can circulate in a loop, until the energy in the battery is exhausted. Thus, the electric field intensity is the sum of a conservative electrostatic field intensity E, and another field intensity E f.
E, =E/ +Ef Again using Ohm's law in Eq. (2.20) we get
Figure 2.17. Placement of charges due to an electrostatic field. A current cannot be sustained under these conditions.
Figure 2.18. Conservative and nonconservative electric fields in a circuit. Only the nonconservative field causes flow of current.
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These electric fields are shown in the circuit of Figure 2.18, where R represents the total resistance of the circuit. Integrating the expression above along the circuit gives
f ' d l =I — Since E/ = —gradV , the first integral is zero and we are left with cf Ey -d\ =
We define dl
(2.21)
as the electromotive force of the battery (emf). U is present between the terminals of the battery and the vector Ef is indicated in Figure 2.18. The quantities U and E/ are related through the internal chemical reaction of the battery
f -cK
(2.22)
Therefore, we can view the equation U = RI as evidence to the existence of a nonconservative electric field Ef .
Figure 2.19a. Refraction of the electric field at an interface between two different materials.
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E,
C(S) 2n
Figure 2.19b. A contour enclosing the boundary used to evaluate the tangential components of the electric field intensity.
2.7.3. Refraction of the Electric Field The electric field as it passes from one material to another undergoes a change in direction at the interface between the two materials. This effect is called "refraction," and is similar to refraction of light rays passing between two materials with different indices of refraction. Figure 2.19a shows two materials with different permittivities: 8 l in material 1 and e 2 in material 2. We assume that a uniformly distributed static charge density exists on the boundary between the two materials. Since this charge is distributed on the interface it is called a surface charge density p s. Because the fields are static in time, rotTL = 0. To evaluate this we define a small surface S shown in Figure 2.19b, infinitely close to the interface. For this reason El and E2 are considered constant on S. Using Stokes' theorem, we have
f The limiting value of the circulation contribution of the smaller sides (perpendicular to the interface) is zero and we obtain
= f Ej - d l + f
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where C\ and C2 are the parts of the contour in materials 1 and 2 respectively. We note that
and analogously
If C, and C2 are equal, we obtain Eltldl-E2tl dl = •
*-
or
Elt=E2t
(2.23)
indicating that the tangential components of the electric field intensity are conserved. We turn now to the equation divD = p , with which we associate an infinitesimal volume at the interface as shown in Figure 2.20. Integrating this equation over the volume and using the divergence theorem gives
where q is the total charge enclosed in the volume. This charge is located on the surface Sf within the cylinder: q=psSf
(2.24)
In other words, the flux of the vector D is divided in two parts; because the flux on the lateral surface of the cylinder tends to zero, we have, for an infinitesimally flat cylinder
o.*=f D , . * + f D 2 .
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where Si and S2 are the surfaces of the bases of the cylinder. Since
and analogously
•s, Figure 2.20. A contour enclosing the boundary used to evaluate the normal components of the electric flux density.
We have, using Eq. (2.24)
Since in this case S{ — S2 = S- , we obtain (2.25) or 0/7
o/7 82-^9 —"i-^i
r\
Ps
/O O£\ l^.^Jo)
Thus, the change in the normal component of the electric flux density passing through the surface is equal to the surface charge at the interface between the two materials. In the particularly common case, when there are no static charges on the interface (p5 = 0), we get E
\t =E2t
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©
Figure 2.21. Relations between the components of the electric field intensity at the boundary between two different dielectrics.
Using Figure 2.21 and the relations above we can write the following expressions
and
E, tone, _ E2H _ D2n/e2 ln
From this, we obtain to/70
E
(2.27)
The larger the change in material properties, the larger the angular change between E, and E2. However, we must point out that the variation in 8 between dielectric materials is rather small.
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As an example, the maximum ratio between permittivities of air and insulating mineral oil (these two materials are frequently used in transformers) is no higher than 4. Figure 2.22 shows the angular change in the electric field intensity in a structure with two dielectric materials. In material 1, S r = l and in material 2, £r =4.
This plot was obtained with the EFCAD computer
program. (EFCAD is a finite element computer software designed for numerical solution of electromagnetic field problems).
Figure 2.22. The electric field in a geometry with two dielectric materials. The material at center has a permittivity four times higher that the surrounding material.
2.7.4. Dielectric Strength In many devices subjected to large variations in potential, in particular in high-voltage equipment, the expression for the electrostatic field E = —gradV assumes a very important role. We look at this role in Figure 2.23. Suppose that one part of the equipment is at ground potential (V= 0). Another part of the device is at a high voltage V = Va . We can calculate the electric field intensity as E = -gradV and therefore, as a good approximation
and Ei =
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E, E,
Figure 2.23. A schematic representation of a high-voltage device with lowand high-intensity electric fields.
Figure 2.24. Definition of dielectric strength. The electric field at which the insulator breaks down is defined as the dielectric strength of the insulator.
With /! > /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
l
— R
andA
R
cos§A = — r
we have
dl = R sec <> d
and
r=
R cosfy
Substituting these expressions in Eq. (2.43), yields upon simplification
TT
The limits of integration, — 7C/2 and +71/2, are the angles (|) corresponding to the limits — oo and + oo of the wire. As a result, we obtain
H = I/2nR This value is identical to that calculated by Ampere's law for the same situation. The example given above has a didactic benefit. The utilization of Ampere's law is much simpler, and we should employ it whenever possible. However, there are many cases in which Biot-Savart's law is the appropriate tool to be employed. It is important to point out that beyond these two laws there are no other analytical methods for computation of the field H as a function of J for general applications. Numerical methods alone can determine H in most realistic geometries.
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2.8.3. Magnetic Field Refraction In a manner similar to the electric field intensity E, the magnetic field intensity H also undergoes an angular change in passing from one material to another, if the two materials have different permeabilities. Consider two materials which possess permeabilities jn, l and jn2 respectively, as in Figure 2.31. The relation between Ql and 02, is obtained by using the equations
rotH = 0
and
divB = 0
assuming that there are no currents on the boundary between the two materials. Utilizing these equations in a manner analogous to that done for the equations rotE - 0 and divD = 0 in the section 2.7.3, we obtain equivalent results
Figure 2.31 . Boundary conditions for the magnetic field intensity at the boundary between two materials with different permeabilities.
• continuity of the tangential component of H H
\t = H2t
• continuity of the normal component of B B
\n ~ B2n
Similarly we observe that TT
H In
TT
and tan92 =
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2t
H 2n
and
tan9! =H2n tan6 2 Hln Recalling that H = B/|I , we obtain tanOi
Ui
(2.44)
M-2
Figure 2.32. The magnetic field intensity relations at the boundary of a highpermeability material and free space.
As we will see in following sections, there are materials with very large permeabilities and materials with low permeabilities. We can obtain important relations based on these permeabilities. Suppose for example, that (J,2 = |U0 , (4,, = 1000m and that Ql is 85°. Then
,
Q
tanO 7 =
tan 85° 1000
or
9 2 =0.65° This effect is demonstrated in Figure 2.32. Note that, in contrast to electric fields, the angular change is much larger. As an example, in passing the boundary between iron (fa,} ^lOOOfO-QJ and air (|J,2 = Ho)' me
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magnetic field H undergoes an angular change such that, in air, it is practically perpendicular to the iron. Figures 4.8a and 4.8b show the angular change of electric fields (s 2 /Sj =5) and magnetic fields (jo,2/|Ll, =1000). These examples were obtained through the use of the EFCAD program.
Figure 2.33a. Change in the electric field intensity due to a dielectric material with relative permittivity of 5.
Figure 2.33b. Change in the magnetic flux density due to a magnetic material with relative permeability of 1000.
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2.8.4. Energy in the Magnetic Field
Figure 2.34a. A material with permeability JJ,, a magnetic field intensity H and a magnetic flux density B.
Figure 2.34b. A solenoid that produces an equivalent magnetic field replaces the magnetic field intensity.
If a magnetic field intensity H exists in a given material (and, in consequence a magnetic flux density B), there is a magnetic energy associated with this field. A certain amount of energy will be required to generate the flux density B in a given time, say between zero and T. To establish the expression for magnetic energy, consider the situation in Figure 2.34a, where a material with permeability jj,, a field intensity H, and a flux density B is shown. For the purpose of this discussion, we replace the magnetic field by a small solenoid, of length 6/ and cross-section 85 (Figure 2.34b) such that the magnetic field intensity generated by the solenoid is equivalent to the field intensity H. The current is defined here as a linear current density J, expressing the current per unit length (A/m). In this case, / is equal to
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J$l . Writing Ampere's law, and assuming the field outside the solenoid to be zero (since we use "long solenoid" approximation), gives
H • eft = f Hdl = I = [ Jdl
(2.45)
From this, H=J. In the other hand, the electric energy that must be supplied to the solenoid is
where V is the voltage applied to the solenoid. Note that the various quantities vary simultaneously as t => [0,T ]
(time)
j => [0, J ]
(linear current density)
h => [0, H \
(magnetic field intensity)
b => [0, B\
(magnetic induction)
/ => [0, /]
(current in the solenoid)
<j) => [0,<1) ]
(magnetic flux)
Since / = 78 / and
dt
dt
dt
we get
=^ r\j*i—ssdt= J Krjdb&tos dt Substituting j' = h , gives
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Since 8/85 is the volume of the solenoid, we obtain an expression for the volumetric energy density as
vv =
hdb
(2.46)
In general, magnetic materials are nonlinear and the magnetic energy needed to generate a flux density B in the material, corresponds to the shadowed area in Figure 2.35.
b
Figure 2.35. Energy needed to generate a magnetic flux density in a material with given B(H).
For materials in which (4, is constant (linear materials), we have
"v=nf
hdh =
H'
Using the relation B = [iff, this expression can be written in the following equivalent forms
w. =
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HB = B2 2 ~2}i
(2.47)
2.8.5. Magnetic Materials We first define the relative permeability of a material as
where
fJ,
is
the
|J,0 =471x10" H/m
actual
permeability
of
the
material
and
is the permeability of air (actually free space).
Therefore \Jir of air is equal to 1. There are basically two types of magnetic materials: • Soft magnetic materials: these include diamagnetic, paramagnetic, and ferromagnetic materials. • Hard magnetic materials: permanent magnets. It is not our objective here to analyze the microscopic structure of these materials. We will only look at their behavior from a macroscopic point of view. Magnetic materials possess small domains (macroscopically they have dimensions of the order of 10~3 to lO^m) called "magnetic domains" or "Weiss domains," composed of various molecules of the material in question. Initially, these domains possess magnetic fields directed to arbitrary directions (see Figure 2.36), however, when a large external field H is applied, the magnetic fields of the domains tend to align with the external field. When the external field is removed, in the second case, the domains maintain their alignment. The cumulative effect of the magnetic domains form the remanent flux density of the permanent magnet. In soft magnetic materials, the situation is approximately the same, however, the application of a weak external magnetic field in the direction opposite to the internal field is sufficient to eliminate the remanent flux density, as opposed to the situation in permanent magnets.
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H
Figure 2.36. Magnetic domains before and after alignment with the external magnetic field.
2.8.5a. Diamagnetic Materials Diamagnetic materials have a relative permeability slightly lower than 1. Notable materials in this group are: mercury, gold, silver, and copper. Copper, for example, has a relative permeability \\,r = 0.999991; the other materials possess \ar of the same order of magnitude. In practice we can consider these materials as having \\,r equal to 1. One effect of diamagnetism is illustrated in Figure 2.37. A diamagnetic material is placed under the influence of a uniform field. Since \\.r < 1, more of the flux passes through air than through the material, since air is a more permeable material. This causes a force that tends to repel the diamagnetic body from the source generating the field. However, because the permeability is very close to 1, this effect is very small and difficult to measure.
Figure 2.37. A diamagnetic material in a magnetic field. Flux lines tend to pass through free space because it has lower reluctance than the diamagnetic material. The diamagnetic material is repelled by the field.
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2.8.5b. Paramagnetic Materials Paramagnetic materials possess a relative permeability \ir slightly larger than 1. One example of this type of materials is aluminum, for which \Jir =1.00000036. Therefore, as with the diamagnetic materials, we can consider the materials as having \lr — 1 for most practical purposes. In general, the effect due to paramagnetism is negligible. 2.8.5c. Ferromagnetic Materials a) General Ferromagnetic materials possess relative permeabilities much larger than 1. As we shall see shortly, these are materials of extreme importance in electromagnetic devices due to their large relative permeabilities )Ll r . As an example, iron with 0.2% impurities has a relative permeability of about 6000. A few iron alloys reach a relative permeability of 106 . It is interesting to note that when a ferromagnetic material is placed in a hot environment, and when the temperature passes over a critical value, called "Curie temperature", the ferromagnetic material changes its magnetic behavior to that of a paramagnetic material. Each material has its specific Curie temperature. For iron this value is approximately 770°C. Another characteristic of this type of materials is that the relative permeability \\,r, depends on the magnitude of the magnetic field intensity |H| in the material. This phenomenon, called "nonlinearity" can be explained with the aid of Figure 2.38. The figure shows a magnetic circuit made of a material with high permeability. A ferromagnetic sample is inserted in this circuit. We wish to obtain the magnetic characteristics of the material. We will see later that H = NI/l and noting that B = <&/S , we can measure the current / applied to a coil and the flux O passing through the material.
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Figure 2.38. Magnetic circuit used to obtain the magnetic characteristics of a ferromagnetic sample.
For low values of /,
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begins to diminish (a2 «*i);
hence the
•_J£ / Figure 2.39. The B(H) curve obtained for the sample in Figure 2.38. Only the curve in the first quadrant is shown.
Figure 2.40. Permeability curve corresponding to Figure 2.39.
Figure 2.40 shows the permeability curve. In practice, at the beginning of the B(H) curve, a varies (not shown). However, many properly designed electromagnetic devices work at high flux densities, close to saturation (therefore far from the beginning of the curve), and we can often neglect this perturbation in (I . If we subject a ferromagnetic material to the influence of a magnetic field, we obtain the situation shown in Figure 2.41. With [ir »1, the magnetic flux is strongly attracted by the ferromagnetic material since it is a highly permeable material. In this case, the ferromagnetic material is physically attracted.
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Figure 2.41. Ferromagnetic material in a magnetic field. Flux lines pass through the material and the material is attracted to the source of the field.
Figure 2.42. A current surrounded by a high-permeability magnetic circuit.
b) The Influence of Iron on Magnetic Circuits An infinitely long wire, carrying a current 7, creates a field in the surrounding space, which, as we have seen, has a magnitude H = Ij2iir (r is the distance of a point from the wire). /, the current in the wire, can be called a magneto-motive force (mmf) since it is capable of generating a magnetic field. Consider a second situation as shown in Figure 2.42, where the same wire is surrounded by a ferromagnetic material with high relative permeability [ir. Assume now that this material has a physical gap. To calculate the magnetic field we use Ampere's law
Choosing a contour L that coincides with the field H, and dividing the contour into // in iron and /g in the gap, the expression is
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f
g
Assuming that the fields are constant in their respective regions, we obtain Hflf+Hglg
=1
(2.48)
There are two unknowns in this equation, requiring the establishment of a second relation. This relation is obtained from the secondary consideration that the flux Oy in iron is identical to the flux <&g in the gap
where Sf and Sg are the cross-sectional areas of the iron and the gap (perpendicular to the plane of the figure). Assuming that these two have approximately the same magnitude, we have / / • = — #.
(2.49)
using (2.49) in (2.48) gives Hg = - -^-l f +1g
(2.50)
With \if » |J,Q , we get Hg = I jig . If the circuit is in deep saturation, this approximation is not valid. Obviously for small airgaps, very intense fields are created. In conclusion, we observe the following phenomena due to the presence of ferromagnetic materials in magnetic circuits: • Modification of the shape of the magnetic field and conduction of flux to regions where it is required.
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• Generation of high magnetic field intensities in gaps. Notice that magnetic forces depend on the field intensity squared. This is one of the reasons for the interest in high-permeability materials. • The circulation of the magnetic field in the ferromagnetic material is negligible if fj,y » JLIQ , since Hf «Hg (Eq. 2.49). We conclude that, in practice, all the mmf is applied in the gap, where
Hg-dlanl
(2.51)
assuming here that instead of a single wire there are n wires. 2.8.5.d. Permanent Magnets In this section, permanent magnets will be presented with the goal of solving practical magnetic circuits where they are inserted. Some more advanced concepts (reversible and irreversible demagnetization) are presented in Chapter 8. a) General Properties of Hard Magnetic Materials We have already classified magnetic materials as follows: • Soft magnetic materials: materials which, when the applied magnetic field is canceled, do not retain a significant remanent flux density (or remanent induction). These materials are "passive" to the presence of the magnetic field. If the external field varies in magnitude or direction, the same change takes place in the material with essentially no retardation effect. All the materials studied in the previous sections were soft magnetic materials. • Hard magnetic materials: materials that retain a significant remanent flux density when the external field passing through the material is removed. These hard magnetic materials are called permanent magnets. The magnetic properties of a permanent magnet are described with the help of Figures 2.43a and 2.43b.
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The magnetic circuit in Figure 2.43a is made of a ferromagnetic material with high permeability. A hard magnetic material is placed in the
gap-
Figure 2.43a. A hard magnetic material sample in the gap of a highpermeability magnetic circuit.
H.
Figure 2.43b. The B(H) curve (hysteresis loop) for the sample in Figure 2.43a.
We assume that this material has not yet been subjected to the effect of any magnetic field. The field generated inside the magnet, is proportional to /, since H = nl/l. The flux density B in the magnet is proportional to the measured flux
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a. Initially, increasing the current / causes the magnetic field intensity to change from zero to H1. The curve OM1 is called the "initial magnetization curve". b. Decreasing H from H1 to zero (M1 to M2), or decreasing / to 0. At this stage the intrinsic characteristic of hard magnetic materials appears, since the magnet retains a considerable remanent flux density, indicated as B0 in Figure 2.43b. We note that, at this point, where / = 0, we have a flux in the magnetic circuit due to the remanent flux density B0 of the magnet. c. Increasing the current in the opposite direction, B passes through zero at Hc (interval M2 - M3). At Hc, the flux due to the permanent magnet is identical in magnitude but opposite in direction to the flux of the coil, canceling the flux in the magnetic circuit, and therefore B = 0. The field intensity Hc is called the "coercive field intensity" or "coercive force". d. Continuing the cycle (M3, M4, M5), we again create a magnetic flux, but the remanent flux density BQ at point M3 is opposite in direction to BQ at point M2. This cycle is called the hysteresis loop, whose internal area is significant for hard magnetic materials. Soft ferromagnetic materials also posses hysteresis curves, but the area described by the loop is relatively small. Nevertheless, even small, the hysteresis loop is associated with losses in the material, as we will see in Chapter 8. A magnet is defined by its B(H) curve in the second quadrant (interval M2 to M3 in Figure 2.43b), which is reproduced in Figure 2.44. This curve, in general, is known or is supplied by the manufacturer, and indicates the remanent flux density BQ and the coercive field intensity Hc, as well as the manner in which B and H vary between these two points. Its purpose is to define how the magnet's flux density varies as a function of the internal field intensity of the magnet. We can now explain why the magnet is represented in the second quadrant: in Figure 2.45 a field line of the magnet in air is emphasized.
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Figure 2.44. The B(H) curve of a permanent magnet.
ft Figure 2.45. The field of the magnet. A field line in air is shown.
Applying Ampere's law, we calculate the field intensity H along the contour /, where lt and le, are the internal and external parts of the contour respectively d\. - 0 (no currents)
f HJ dl + f *»i
**z»
This analysis assumes that the fields are uniform on the two parts of the contour (/, and le) respectively. We get
Note that HI must have an opposite direction to He , along any contour (total field circulation must be zero). On the other hand, considering the conservation of flux, the magnetic flux densities Bi and Be , must be in
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the same directions in relation to the contour of circulation. In other words, since in air, B and H have the same direction (B = H-ow, me directions of B and H in the magnet must be opposite to each other. The magnetic field intensity H and the magnetic flux density B in the magnet and the region that it encloses are indicated in Figure 2.46, showing that in the interior of the magnet H and B are opposite in direction. B must be the same throughout the tube of flux because of the conservation of flux. In addition, it is evident that the apparent permeability of the magnet is negative, since H<0 and B>0 (Figure 2.44, in the second quadrant).
Figure 2.46. The magnetic field intensity and magnetic flux density inside and outside a magnet. The magnetic field intensity inside the magnet is opposite, in direction, to the magnetic flux density.
b) The Energy Associated with a Magnet It is obvious that if a magnetostatic field exists, there must also be an energy associated with this field. Provided that the field is defined in a volume V, this energy is given by the expression W = (-HBdv *1
(2.52)
This expression is also valid for a permanent magnet whose relative permeability is generally close to 1. We point out that this question is somewhat controversial and some authors consider it an open subject of research. However, the presentation below is generally well accepted. We consider the volume V to be a tube of flux with an internal volume V, and an external volume Ve, as shown in Figure 2.47. The flux in this tube is
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. If ds is the cross-section of the tube and dl an elemental distance along the tube's length, we can write dv — dsdl . The energy is now
2 *i
Hdl Bds
2
Substituting £/O for its value in the integral, we get
2
*,
2
J
/e
2
where lj and le are the internal and external contours of circulation.
Figure 2.47. A flux tube inside and outside the magnet.
Since there are no currents in the system, we find from Ampere's law that W = 0, which indicates that the sum of the internal energy (in volume VJ and external energy (in volume VJ is zero. Extending this result over the whole magnet, using an assembly of flux tubes, we have
We = -Wt From this, the internal energy in the magnet is given by Wif1 = - [ BHdv
2*1
and the free energy (external) is, therefore,
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Wee=--
BHdv
(2.53)
This energy depends on the product BH at the operating point of the magnet and on the volume of the magnet. In general it is of interest to concentrate this energy in a specific area. Two examples of energy due to magnets are shown in Figures 2.48a and 2.48b. In Figure 2.48a the energy due to the magnet is distributed in space and, therefore, in the volume surrounding the magnet the volumetric energy density is low. In Figure 2.48b, we will assume that the magnet is identical to that in Figure 2.48a even though the magnetic field of the magnet depends on the surrounding materials (the operating point of the magnet is influenced by the surrounding materials). Now the energy is concentrated in volume Vg, which is much smaller.
Figure 2.48a. A magnet in an open magnetic circuit. The energy is distributed throughout the space.
Figure 2.48b. A magnet in a magnetic circuit. Energy is concentrated in the volume of the gap.
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The energy density in this volume is high; the energy of the magnet does not disperse as in the first example. For a given volume of the magnet, the energy depends only on the product BH. There is a point on the characteristic curve B(H) of the magnet at which this product is maximum, representing the operating point of the magnet at which it can produce maximum energy. At points B = BQ (and, therefore, H = 0) or B = 0 (and, therefore, H = Hc), the BH product is zero and the magnet does not produce any external energy. The first case B = BQ is represented in Figure 2.49 where a magnet is embedded in a magnetic circuit without a gap. The permeability |0,y of the circuit is considered to be infinite. Considering the fields Hf and H, to be constant in the iron and magnet, we get from Ampere's law that H A f + ///// = 0. Therefore, provided \ls » \IQ , we get Hf = 0 , and Hf = 0, which correspond to B = BQ on the magnetization curve. However, this is a purely hypothetical situation, since Jly has a finite value and there is some field dispersion through the sides of the magnet. The field, therefore, is not constant throughout the magnet. This, however, does not change the fact that the flux density B of the magnet is very close to B0. Also, the permeability of the magnet is low and it creates a reluctivity to its own flux.
Figure 2.49. A permanent magnet in a closed magnetic circuit.
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Figure 2.50. A permanent magnet in a magnetic circuit including a gap. The second case, where H = Hc (and B = 0) represents a situation
where the effect of the magnet is canceled, for example, by the magnetic field of a coil in the magnetic circuit in which the magnet is inserted. This case was shown in Figure 2.43a. The operating point of the magnet is point M3 in Figure 2.43b. We now examine an example of magnetic field calculation due to a permanent magnet, as in Figure 2.50. A magnet is inserted in a magnetic circuit with (lr »1. The magnetic circuit has a gap as shown. Writing Ampere's law, and assuming that the fields are constant in their respective domains, we have for the circulation of H along the contour L Hm Lm +
HfLf+HgLg=0
With / / , « 0 , we have
HmLm=-HgLg The conservation of flux gives
or
Dividing Eq. (2.55) by Eq. (2.54) gives
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(2.54)
Bm Hm
(2.56)
Eq. (2.56) describes Bm/Hm as a function of the dimensional factors of the structure.
Figure 2.51. Load line of a permanent magnet. The intersection of the load line with the B(H) curve is the operating point of the magnet.
The value is negative as it should be since the magnet operates in the second quadrant. In fact, the value Bm /Hm represents, in the B-H plane, a straight line. This line is called a "load line" (or "operating line"). The intersection of this line with the characteristic curve of the magnet provides the points Bm and Hm at which the magnet operates, as a function of the dimensions of the magnetic circuit (see Figure 2.51). The angle a in Figure 2.51 is evaluated from the relation
tana =
Bm Hm
J
m
From the properties of the tangent function, we can use the angle |3 directly
P = arc tan From Bm and Hm, we can determine Hg by multiplying Eq. (2.54) by Eq. (2.55)
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BmHm SmLm = -\iQffg
SgLg
Denoting Vm the volume of the magnet, such that Vm = SmLm and Vo — Sg Lg the volume of the gap, we obtain
Hn =
n mJ-f m vVm
D /?
^0
V
(2.57)
g
We note here that Hg is larger as the product BjHj
is larger (for this
reason we are interested in operating at (BH)max which is normally indicated by the manufacturer) and as the ratio VmjVg is larger (therefore we are interested in increasing the volume of the magnet and using small gaps). c) Principal Types of Permanent Magnets A good permanent magnet is required to have a high coercive field intensity Hc, as well as high remanent flux density BQ . A high coercive field intensity is important because it does not allow the magnet to be demagnetized and a high BQ is normally associated with the capacity of the magnet to produce high magnetic fields in magnetic circuits in which it is inserted. Until 1930, usable magnetic materials were the Chrome-Tungsten and Chrome-Cobalt alloys. Their major problem was a low coercive field intensity Hc (Hc < 20,000 A/m). In 1940, the Alnico alloys (Fe-f Al+Ni+Co) appeared, whose BQ is approximately 1 T and which have a coercive field intensity of Hc > 50,000 A/m . This type of magnet is still used extensively, especially in applications where an operation at high temperatures is required. In 1947, with the appearance of ceramic ferrite magnets (SrFe12O19 or BaFe12O19), the use of magnets became widespread since these are inexpensive and possess high Hc (Hc =100,000 A/m). Despite the fact that BQ is low (BQ « 0.4 T), their properties prevail and today these magnets are utilized extensively. Another important property of
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ceramic magnets is the fact that they are nonconducting, and therefore are the preferred magnetic element for application at high frequencies (since there are no induced currents circulating in the magnet). In 1974, permanent magnets made of rare earth elements were introduced. The Samarium-Cobalt magnets (SmCo5 with BQ w 0.8 7" and Hc «600,000 A/m
and
Sm2Co17
with
BQ «1T
and
Hc « 600,000 A/m) represented a revolution in this domain since, in addition to possessing high coercive fields, they have a high value of BQ . However, due to very complex fabrication processes and difficulties in obtaining the raw materials for the magnets, their prices continue to be very high. This does not diminish the considerable interest in these materials. Another type of rare earth permanent magnets were introduced later: Neodymium-Iron-Boron magnets (Nd2Fe14B with Br=\.2T and Hc = 800,000 A/m). In spite of their high performance, these magnets were very sensitive to high temperatures and their performance was greatly affected by them. Therefore, using these magnets required much attention and they could be employed only in relatively low-temperature application. However, much progress has been accomplished in recent years and now they are more resistant to heating. They have become the most commonly used magnet in high performance devices, noting that Samarium-Cobalt magnets are still very expressive. Developments in high-energy permanent magnets continue with the objective of improving their characteristics as well as increasing their energy density. Figure 2.52 shows the approximate B(H) curves of the principal permanent magnets available. The magnetic characteristics of the magnets described above are given in Table 2.1.
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1 - Alnico 2 - Ferrite 3 - Samarium-Cobalt 4 ~ Neodymium-Iron-Boron
800
600
400
200
H(kPJm)
Figure 2.52. B(H) curves for some permanent magnets.
d) Dynamic Operation of Permanent Magnets The utilization of a permanent magnet requires some precautions to ensure that the magnet is not demagnetized during normal operation. As an example we assume the curve in Figure 2.53 to be the curve for the permanent magnet. The most commonly used magnets (ferrite, Sm-Co, Alnico) have a differential permeability (tana =Ba/Ha), very close to that of air (|ir =l). If we operate at point Plt the magnet preserves its remanent flux density J?Q . However, if we operate at point P2, it loses its remanent flux density BQ and, instead, the magnet has a new and lower remanent flux density B$2 • This may cause the performance to fall below an acceptable limit. If the operating point is changed to point P3, the magnet loses its remanent flux density completely. Therefore, it is important to avoid operating points below P1 in Figure 2.53.
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Table 2.1. Typical properties of permanent magnets.
Material at (20°Q
*oM
Hc[KA/m]
(BH}ma\KJ/m*
Alnico Ferrite Sm-Co Ne-Fe-B
1.25 0.38 0.9 1.15
60 240 700 800
50 25 150 230
H
Figure 2.53. Dynamic operation of a permanent magnet.
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Figure 2.54. Two electric circuits in a magnetic path. The flux linking the two circuits is O;2 and the flux linking only C, is O;/.
2.8.6. Inductance and Mutual Inductance 2.8.6a. Definition of Inductance Consider the situation shown in Figure 2.54. Two electric circuits C2 and C2 are located in the presence of ferromagnetic materials. We do not consider any losses in the ferromagnetic material. A time-dependent current 7j flows in circuit Q. Because the frequency is sufficiently low, losses due to time dependency of the current /j can be neglected (these losses will be examined later). The current /2 in C1 generates a magnetic flux. Part of this flux links with circuit C2. This flux, denoted O,2 , is the flux generated by Cj in C2. We assume that the ferromagnetic circuit is linear so that, for example, if the current 72 is doubled O,2 is also doubled. Since circuit C2 has N2 turns, the "flux linkage" is defined as the number of times circuit C2 links the flux O 1 2 , or, N2Q>\2. For a linear circuit, a proportionality factor K between the flux linkage and the current /2 can be defined as K = N2Q>n/1\ . This factor is called the mutual inductance between Cj and C2 and is denoted as M12:
Mn = N2 ^H
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(2.58)
In addition, circuit C1 contains Nj turns and produces a flux linked with itself. This flux is denoted On . We can therefore define a "self inductance" or simply "inductance" of Q as (2.59) If the behavior of the magnetic materials is nonlinear, the proportionality factor between ®12 and \i or CDU and 7j is not constant. In this case inductance is still defined the same way. With this loss of proportionality, M12 and Lj vary with the value of /^ and, therefore, are functions of 7j . 2.8.6b. Energy in a Linear System In the system discussed in the previous section, neglecting ohmic losses, the potential U^ corresponding to the emf at the terminals of circuit Cj is equal to
1
dt
n
dr
l
The electric power associated with Q is
at and the magnitude of the energy (meaning the energy needed in circuit in order to generate the magnetic flux On in the inductor) is
Assuming that the system is linear, or that there is no saturation in the magnetic circuit, Lj is constant and we get W = -LL2
2
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(2.60)
This expression represents the magnetic energy necessary for the coil in C3 to generate the magnetic field responsible for the flux CD,, . Assuming now that circuits C} and C2 carry currents I\ and /2 , respectively, we can write
=
{
1
(J ~> 2 —
dt
dt
d(N2®22) -dt
i
h
d(N2$>l2) dt
In a manner analogous to the calculation above, we can write the total energy in the two circuits as
W = W = ( 1 ' ' 2 [ ((#,*, , ) +rf(Ar,4) 2 1))/, + f 2 2 '*' 1122 (rf(Af 2 0 22 ) + d(N212 ))
Noting that O
N2^>\2 — ^12^1
= L! i and N2^>22
=
an<
^
^1^21
=
^21^2 »
as
we
^
as
LI2 , we obtain
- f •" or 1
-Li/I
2
1
2
+-L 2 / 2 + L 1 ' * ( M n I } d I i + M ] 2 I 2 d I ] )
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(2.61)
It is easily verified that
1 f' /2 (M 21 -M12X/i/2 -/jrf,) (2.62) The fact that we called one circuit Cj and the other C2 is purely arbitrary. The magnetic energy is not altered if the notation is reversed. This implies that, in Eq. (2.62), M]2 must be equal to M21, since otherwise the energy will be altered. Thus:
M21 - M12 = 0 and M2l = M12 This result is universally applicable. Now, the second integral on the righthand side of Eq. (2.62) is zero, and we obtain
w = 1 7 2 + L2h2+ M 2/1/2
i'' \
'
(2 63)
'
This expression can be generalized for any number of circuits as
W/
(2-64)
2.9. Magneto dynamic Fields After discussing electrostatic and magnetostatic fields, we now turn to time-dependent phenomena. From the decoupling of Maxwell's equations, already presented, the equations for magnetodynamic fields are rotH = J
(2.65)
divB = 0
(2.66) r5R
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(2.67)
together with the constitutive relations:
J=aE The difference between these equations and the equations for magnetostatics is in that dQ/dt is not zero. The electromagnetic phenomena associated with the magnetodynamic field can be very complex. On the one hand, a new variable (time) has been introduced. On the other hand, the problems are, in general, three-dimensional in nature. This is because, as we will see shortly, the magnetic flux density B and the electric field intensity E generated by the flux density variation are in different planes. Due to these facts, magnetodynamic problems are, for the most part, difficult to solve. In most cases we must use approximations in order to solve the problem and, in general, we need to adapt the solution methods to the characteristics of the problem. Here we will treat some of the more important magnetodynamic effects that occur in devices under time-dependent excitation or devices that contain moving conducting parts in a static magnetic field. The issue of solution of the complex magnetodynamic problems will be discussed later, when numerical methods will be presented. 2.9.1. Maxwell's Equations for the Magnetodynamic Field The equations rottl = J and divB = 0 were introduced in the previous sections and we made considerable use of both. We recall that the first equation, in integral form, is usually referred to as "Ampere's law" and that the second indicates the conservative nature of the flux. Both were part of the set of equations needed for the static magnetic field and they are fully applicable here. The equation that is relevant to the magnetodynamic domain is
rotE =
dB -dt
which indicates that the flux density B time variation generates an electric field intensity E. First, we represent this equation in integral form. To do so,
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we consider a surface S where E and B are defined and apply the integration:
= - f -- ds ft fit
f
JS
Using Stokes' theorem gives (2.68)
cf
•fc
An example of this relation is shown in Figure 2.55a and 2.55b. Consider a ferromagnetic cylinder such that the flux density B is constant throughout its cross-section. The magnetic flux density B is time-dependent and we use the cross-section of the cylinder for the surface S in Eq. (2.68). In addition, a conducting loop encircles the cylinder but the two are electrically isolated. This loop forms the contour C in Eq. (2.68). From the expression
dt we note that the variation in B generates an electric field intensity E in the loop. This electric field intensity is proportional to — dB/dt , as shown in Figure 2.55b.
•u Figure 2.55a. A ferromagnetic cylinder with a constant, time-dependent, magnetic flux density throughout its cross-section. A conducting loop surrounds the cylinder.
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Figure 2.55b. Relations between the magnetic flux density variation, the electric field intensity and the emf in the loop.
The circulation of E along contour C leads to an electromotive force, which can be measured by a voltmeter as a voltage U. We have, therefore,
U= We now turn our attention to the other term in Eq. (2.68), namely, A ds
dt
First, we note that in this case, B depends only on time, since it is assumed that it is constant throughout the cross-section S. Therefore, we can write
dt
dt
The integration over the surface S and the differentiation with respect to time are independent operations. Interchanging between the two operations gives
r c» , d - I 1 ds =
•o d/
dt
dt
Equating the two terms of Eq. (2.68) gives
U =-
dt
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(2.69)
B
Figure 2.56. The use of laminated cores to reduce losses. The laminations are placed in planes perpendicular to the flow of induced currents.
This is known as "Faraday's law" in recognition of Faraday who was the first to observe this phenomenon. The negative sign in this expression is a simple consequence of the original Maxwell's equation. To demonstrate this, we use Figure 2.55b and note that if the loop is viewed as a short circuit, a current circulates in the loop (since J=crE). This current, in turn, generates a magnetic field intensity inside the material it encircles. This magnetic field intensity is opposite in direction to the variation dB/dt. Hence, the generated field in Figure 2.55b penetrates into the plane of the figure, reducing the increase in B, whose direction is out of the plane. The direction of the electric field intensity has been established by Lenz and the rule that defines it is known as "Lenz's law". This is defined as: "the induced current is such that the flux generated by itself tends to reduce the flux variation that generates this induced current". We observe here that Faraday's and Lenz's laws are contained in Maxwell's equation rotE = -dB/dt. The-example above can be extended further; because the loop is short-circuited, the current in the loop is called an "induced current" (as opposed to a source current) since it is induced by the time variation of B. In reality the electric loop of Fig. 2.55b is not required to establish this current. If the material of the cylinder is conducting, there are currents generated in the material itself. It is evident that we can associate losses with these currents through Joule's effect. This causes most of the energy used to
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generate B to be converted into heat. In other words, if the initial field in the material is zero, the induced currents in the material are such that they impede the increase in B. The net effect is that the total flux density in the material (the sum of the external flux density B and the flux density generated by the induced currents) is very low. Is this effect desirable? The answer depends on the application. If we wish to heat up the material, the answer is positive. In an induction motor, the induced currents are necessary to generate torque. In these two examples, the generation of induced currents is necessary for the functioning of the device. An example in which induced currents are not desired is the transformer. The purpose of the transformer is to transfer energy from its primary coil to its secondary coil (with or without change in potentials). Any energy lost as heat in the ferromagnetic material reduces the efficiency of the transformer. If this loss is excessive, the transformer cannot function adequately. To overcome this difficulty and reduce the induced currents, it is common to build the transformer with laminated cores. This reduces the effective conductivity of the material by breaking the paths through which currents can flow. The laminations are insulated from each other and are placed such that they are perpendicular to the planes in which induced currents tend to flow, as shown in Figure 2.56. There is a subtle distinction between induced currents that are useful and those that represent losses. Whenever the induced currents degrade the performance of a device, they tend to be termed as "eddy currents" and are associated with "eddy current losses" which heat the structure through Joule's effect. Induced currents that are beneficial or necessary for the function of the device are simply called induced currents. It is important, however, to note that the two currents are physically the same phenomenon and are described by the same Maxwell's equation: rctfE = — 2.9.2. Penetration Conducting Materials
of
Time-Dependent
Fields
in
The penetration of fields in arbitrarily shaped materials (either ferromagnetic or non-ferromagnetic) is a very complex problem. To solve this problem we assume that the materials are linear and isotropic for
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electromagnetic analysis. Based on this we establish a second-order, timedependent partial differential equation using Maxwell's equations. The penetration of fields in conductors is then found as a solution to this equation. 2.9.2a. The Equation for H Using the equation ro/H = J , and applying the rotational to both its sides, gives
rot ro/H = rot J
(2.70)
Using Eq. (1.23), the left side of Eq. (2.70), is
rot rot H = grad div H - V2H
(2.71)
Assuming that the permeability is constant in the medium, we have
which leads to
-V2U
(2.72)
As for the right-hand side of Eq. (2.70), rotJ , we have
roti = rotaE = a rotE = -a — =P-
(2.73)
Substituting Eqs. (2.72) and (2.73) into Eq. (2.70) gives
V 2 H=aji— dt
(2.74)
2.9. 2b. The Equation for B Substituting H = B/ja, on both sides of Eq. (2.74), we obtain —
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(2.75)
2.9.2c. The Equation for E Starting with the expression rotE = —dB/dt
and applying the
rotational to both sides we get
8B rot rotE = -rot — dt
(2.76)
Assuming that there are no static charges in the domain and that s is constant, the equation divD = p becomes
divE = 0
(2.77)
Using Eq. (1.23) for the left-hand side gives
rot rotE = grad divE - V2E which, with Eq. (2.77) is
rot rotE = -V 2 E
(2.78)
For the right-hand side of Eq. (2.76), we have
dE d _ d d _ -rot— = rotB = urotH = -u—J =-ua dt dt dt dt
dE — dt (2.79)
Substituting Eqs. (2.78) and (2.79) into Eq. (2.76) gives
r5F V 2 E=CTU— dt
(2.80)
2.9.2d. The Equation for J The equation for J is obtained from rotE = —dB/dt by substituting
J/<7 for E
i
_
an
— rotJ =-IL a ^ dt
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Using the rotational on both sides,
d TT rot rotJ = -ua —rotn. dt
(2.81)
The left-hand side, using the relation div J = 0 is
rot rot3 = grad divX - V2J = -V2 J
(2.82)
and, therefore, <3T
V 2 J=aj4,—
(2.83)
dt
This equation can also be obtained by simply substituting J/CT for E in Eq. (2.80). 2.9.2e. Solution of the Equations We note that Eqs. (2.74), (2.75), (2.80), and (2.83) have the general form
5P V 2 P=aji—
(2.84)
dt
where the vector P represents H, B, E, or J in the respective equations. We now seek a general solution, applicable to all four vectors. Eq. (2.84), in explicit form for the x component of the vector P is
d2P~ dx2
+
d2Px dy
;r- + 2
d2Px L
dPx
dz
T = ^cr-rL 2
(2.85)
&
The solution of this equation is difficult to obtain. To understand the complexity of the phenomenon involved we remember that, on one hand, this equation was obtained by assuming linearity and isotropy of materials and, on the other hand, that there are two more equations, identical to (2.85), for the y and z components of P. For this reason, and to be able to analyze this equation, we limit ourselves to a simpler case. First, we assume
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that all fields are sinusoidal in nature. Using the current density J as a variable, in complex notation, we have
(2.86) where j = V— 1, co = 2nf is the angular frequency, / the frequency, and J0 the amplitude of the sinusoidal current density J. We can also write C/J
_
if(\ f
— = ya>J 0 e y dt
_
=/coJ
(2.87)
Substituting this in Eq. (2.83) yields
V2 J - y'a^ico J = 0
(2.88)
Using a variable 8 such that
(2.89) Eq. (2.88) becomes
V 2 J-|(j = 0
(2.90)
This equation is still difficult to solve and we need to simplify the problem further; we assume that the conductor is a semi-infinite block with the surface on the xy plane as shown in Figure 2.57. Consider now a sinusoidal electric field intensity E0, coincident with the Ox direction throughout the surface of the conducting block.
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Figure 2.57. An electric field intensity, parallel to the surface of a semi-infinite conducting block.
Since the tangential component of the electric field intensity is continuous, the same electric field intensity exists immediately below the surface, in the conductor. This implies the existence of a current density
J0=dE0. With these assumptions, J has only a component in the x direction and varies only in the z direction. Eq. (2.90) assumes the following simplified form:
(2.91) The solution to this equation is Jx(z,t) = JQe~z^ cos(cof-z/5)
(2.92)
In this equation the following are immediately evident: • The amplitude of the current density is JQ€ ' • The phase of the current density is - z/5 . As z increases, or as fields penetrate deeper into the conductor, the amplitude of J decreases and the phase changes. The amplitude decreases
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exponentially. The term 8 is called "depth of penetration" or "skin depth". When z = 8 , or when the field has penetrated to a depth 8 , we get
J Q c :—O /O
r / HTTT — J Q I & — U.J/t/0
At this point (z = 8 ), the value of J is 37% of J0. For practical purposes, J can be neglected for z = 38 or larger. ®-
y
JC
Figure 2.58. Relations between the amplitudes of the current density at two locations along the z axis.
From the expression
8 =
it is obvious that the higher the frequency the smaller the depth of penetration of the field. Similarly, the larger the permeability (in ferromagnetic materials) or the conductivity, the smaller the depth of penetration 8 . As for the phase — z/8 , it varies with z. If we define a reference phase as zero at the surface of the conductor (z = 0) we can write: cos (»/ — 0) . At another point below the surface (z^O), we have cos\G)t — a ) . By way of example, suppose that:
• at z = z\ , CO/ - z\ /S =0 and J\ =
~ZI'
• at z = z2, co/-z 2 /8 =180° and J2 =J0e~Zl/d
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(-l)
Figure 2.58 shows the values of J at z = Zj and z = Z2 . The vectors J are in opposite directions ( 1 3 2 *s obviously smaller than J j due to the exponential attenuation in amplitude). For these geometrical and temporal considerations the following expression for J:
J(z, t) =
cos(cor - z/5 )
describes damped harmonic motion, as for example, in the case of a string, fixed at one end and subjected to a varying motion at the other end (see Figure 2.59). (The effects of reflection of waves from the fixed point cannot be considered in this comparison). The vectors Jt and J2 mentioned above and their relationship are also shown in Figure 2.59. The considerations above are valid for H, B, and E as well since Eqs. (2.74), (2.75), (2.80), and (2.83) have the same coefficients.
Figure 2.59. Wave motion of a string with one end fixed. The other end is moved up and down.
The depth of penetration 8 is the same for the fields above: as one decays due to penetration in the conductor, the others decay at the same rate. The only point that must be remembered is that H and B are perpendicular to J and E as defined by the rot relation rottl = J (see Figure 2.60).
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These results were obtained by applying a series of assumptions and simplifications. The question is, how good are these results in real applications? The answer to this question is given from two points of view: • In the semi-infinite block assumed in Figure 2.57, calculation of the depth of penetration 8 for copper, at 60 Hz (a = 5.7 x 107 S/m; fj, = |J,0], gives 5 =8.6mm. This dimension is small compared to most low-frequency electromagnetic devices (motors, transformers, power transmission cables, etc.). This justifies the assumption that the dimensions of the device are large compared to the depth of penetration and the error in using the definitions above is small. If we were to establish the correct (and much more complex) differential equations for a given structure, taking into account its geometry, we would almost certainly get different values for 8 . However, intuitively, we can sense that these results would not be much different than those obtained from the simplified solution. If the dimensions of the structure are small compared to 8 , the calculation above must be reevaluated. We note, however, that the calculation above is for copper, for which JJ, = \JLO and at a low frequency of 60 Hz. For this reason the depth of penetration 8 is relatively large. In a linear ferromagnetic material, where \\,r > 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
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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\
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
(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
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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:
(3.12)
To establish a numerical procedure we force R to be zero using the following operation:
f WRdQ = 0
(3.13)
where W is a "weighting function" and Q represents the domain in which the condition is enforced. In our case, the expression in Eq. (3.13) is
f W[div(s gradV)+ p]dQ = 0
(3.14)
Using Eq. (1.26) we get
UdivA = divUA - A • gradU and
UdivAdQ =
divUAdQ -
A • gradUdQ
Applying the divergence theorem (section 1.5.2) to the first term on the right-hand side we obtain
f UdivAdQ = A, MA • ds - f A • gradUdQ where S'(Q) is the surface enclosing the domain Q . Returning now to Eq. (3.14), we substitute U = W and A=£grad¥. With these, Eq. (3.14) becomes
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
j W [div (e grad K)+ p ]dQ = (tfFe grad V-dsJQ JS(Q) |s gradV• gradWd£l+ \Wp dQ. JQ JQ (3.15) This is set to zero. The first term on the right hand side is related to the boundary conditions of the problem and will be discussed soon. We merely comment here that, because the numerical procedure imposes that the sum of the integrals on Q be zero, the integral on •S'(Q) must also be zero. Eq. (3.15) is commonly called the "weak form" of the formulation. The origin of this terminology is in the fact that in Eq. (3.14) there are second-order derivatives, while in Eq. (3.15) there are only first-order derivatives, resulting in a "weaker" order of derivation which is easier to handle in terms of numerical techniques. Next, the concept of discretization will be linked with the weighted residual method. For didactical purposes we will use here an analogy with ID elements, where an element is a segment of line and the nodes are the points delimiting the segment; obviously, there are only two shape functions corresponding to the two nodes. Equation (3.13) for the discretized domain Q is written as:
(3.16) k=1, A
There Wk is the weighting function for node k , K is the total number of unknown nodes, and Qfc is the partial domain to which node k belongs. This corresponds to K equations, for K unknown potential values at the K nodes in the solution domain.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
* wt
k-l
k
k+l
k+2
Figure 3.2. Weighting functions Wk and ^#+1 •
The weighting functions are established as shown in Figure 3.2, where the weighting functions W^ and W^+\ corresponding to the nodes k and k +1 in element n are shown. From Figure 3.2 we note that the weighting function Wk acts strongly at node k where it equals 1 and decreases linearly away from the node, becoming zero at nodes k — 1 and k + 1. Similarly, W^^ equals 1 at node k + 1 and decreases to zero at nodes k and k + 2. Since Eq. (3.16), represents a sum only on element n in Figure 3.2, the situation in Figure 3.3 arises. That is, the sum of the weighting functions in the element is evaluated. From Figure 3.3 we observe that the functions Wk and W^+\, in element "n" are identical to the functions <j>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)
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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)
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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
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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
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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 =
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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
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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
(3.27)
The approximation within the triangle can be written in terms of the shape functions N(u,v) as
x(u, v) =./% (u, v)jCj + N2 (u, v)x2 + N3 (u, or, in matrix form as
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(3.28)
x(u,v)= [N{ (u,v)
N2 (u, v) N3 (u,v) ]
(3.29)
(0,1)
(0,0)
u
(1,0)
0
Figure 3.12. A finite element defined in a local system of coordinates and mapping to the global system of coordinates.
For first-order triangles, the shape functions in local coordinates are N{(u,v) = l-u-v
N2(u,v) = u
And Eq. (3.29) becomes
x(u, v)= [l - u - v
N3(u,v) = v
xl
u
v]
X
2
(3.30)
For the node at the origin of the local system of coordinates (u = 0, V = 0) we get
x(o,o)=[i-o-o
o o]
x2
For the node at (w = 1, v = 0) we have
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xl x(l,0)=[l-l-0
0] x2
1
L*3
and, similarly, for the node at (u = 0, v = l)
Jt(0,l)=[l-0-l
0
l]
X
2
X
3
Identical transformations apply to the y coordinates:
.'y* y(u, v)= [l - u - v
u
v]
(3.31)
This means that the functions N{ , N2 and N3 are valid for x and y . The net effect is that node (w = 0, v = 0) is mapped onto (jCj , y{ ) , node (w = l,v = 0) is mapped onto (^25^2)' an<^ tne n°dc at (w = 0,v = l) is mapped onto v*^,.)^). As an example, suppose we map the centroid of the triangle which, in the local element, is located at u = 1/3, v = 1/3 . In the global element these become
,(1/3,1/3)=
" 11-i-I 1 3
3
or
and similarly for the y coordinate,
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1
r
3
3_
l
Thus, for any point (u, v), there corresponds a unique point (x, y). An important property related to the coordinates transformation is the "Jacobian" matrix J:
dx
dy
du dx dv
du dy dv
(3.32)
The transformation is only possible if this matrix is not singular. To evaluate the Jacobian, we calculate the terms of the matrix using Eqs. (3.30) and (3.31):
du
= [-1 1 0]
dv
= [-1 o i] L
J
or
dx _ "~~~ .A"^
"""" *V *
3w
dx ~dv
^3
(3.33a)
-*i
and, analogously,
—=yi-y t du
dy
— dv
(3.33b)
With these, the Jacobian is
J=
and its determinant is
-y\)
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(3.34)
The determinant equals twice the area of the triangle. The Jacobian is zero if the area of the triangle is zero (for example, the three nodes of the triangle are on a single line). This is obviously not an acceptable finite element and should be avoided. 3.4.3. High-Order Finite Elements: Implementation The 2D approximation for potential in a first-order triangle is (from Eq. (3.10))
or, in matrix form, (3.35)
where, as shown before,
with the following values:
For any internal point in the triangle, the values of §} , (|)2 , and (j)3 vary depending on the location of the point in relation to the three nodes. The functions (j)j , (j) 2 , and (j)3 are called "interpolation functions". These interpolation functions are written in global coordinates. Interpolation functions in global coordinates are often used for simple elements such as
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the first-order triangular element described above. For most elements, but in particular for higher order elements, it is much easier to define first the interpolation functions in local coordinates and then map them using mapping functions in the global domain. The interpolation functions in local coordinates will be denoted here A r *(w,v). Theoretically, the mapping functions, denoted 7V(w,v), are different than the interpolation functions N*(u,v). However, in practice they are most often chosen to be the same, defining an isoparametic mapping process. From now on, we will refer to both functions as N(u, v). In the example above, we have
(3.36)
The value we obtain for V(u,v) or V\x, y) for any specific values of u and V or the corresponding values of X and y are the same as will be shown next. Suppose u = 1/4, V = 1/2 . By direct evaluation, we get
v(*,y)=
4 2
V* (3.37)
The coordinates are transformed as
4/4,1/2) =
i_!_! 4
2
I
4
I
2
= -(Xl+x2+2x3) (3.38a)
and
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(3.38b) The interpolation function <j)j is
Replacing the value of pl , q{ , and fj as defined in Eq. (3.7) and using the coordinates x, y from Eqs. (3.38a) and (3.3b), gives
—
which, after some algebra, gives
^(x,y)=Performing similar calculations for (j)2 and (()3 , we get
<
1 l>2(^> ; ) = -
a°d
9 3 (*>>>) =4
And, finally, we get
(3.39) a result which we obtained in Eq. (3.37) in a much simpler way. We note that
V(u = 1/4, v = 1/2) = Even though this is only an example and cannot be viewed as general proof, this property is always valid.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
3.4.4. Continuity of Finite Elements An element is said to have a C° continuity if the variable approximated over the element (in this case, V(x, y)) is continuous across the interface between adjacent elements. These elements are also called Lagrange elements. An element is said to have C continuity if both the variable and its first derivative are continuous across inter-element interfaces. Similarly, a C continuous element means that the variable and its first n derivatives are continuous across element interfaces. For such elements, the derivatives at the nodes are also unknowns, representing additional degrees of freedom. In electromagnetics, derivatives are often discontinuous at the interface between two different materials. For example, in the discussion above we used the potential as the variable. Its derivatives are the electric field intensity (E = gradV }. It is therefore appropriate to use C elements to ensure continuity of the potential V and allow discontinuity of the electric field intensity at interfaces between two dielectric materials with different permittivity. Because the need for discontinuous derivatives is common in electromagnetics, the Lagrange ( C ° ) elements are most commonly used. It is possible to use C elements in domains without material discontinuities, but the complication in defining these types of elements limits their use. For this reason, we will use only C elements in this work. As mentioned before, a finite element is called "isoparametric" when the shape functions (the geometric interpolation functions) are identical to the interpolation functions. The isoparametric elements are the most commonly used elements in FE codes. However, if we decide to use, for example, triangular elements with straight edges, the element can be mapped with linear functions while, for reasons of accuracy, we may wish to use quadratic functions for the interpolation functions. Because the order of the mapping function is lower than that of the interpolating function, the element is "subparametric". The opposite can also be implemented: mapping functions with order higher than the interpolation function in
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
which case the element becomes "overparametric or hyperparametric". While subparametric mapping is common, overparametric mapping is not. 3.4.5. Polynomial Basis Another fundamental characteristic of finite elements is their polynomial basis. To see what the polynomial base of a finite element is, recall that for a linear triangle, the approximation for V\x, y) is
al + a2x + a3y
(3.40)
The same relation holds in the local coordinates V(u, v) = ai +a2u
(3.41)
with the following property:
which means that V has the same value when calculated through Eq. (3.40) or (3.41) for any corresponding point, as was shown in section 3.4.3. The expression in Eq. (3.41) can be written as
w,v)=[l w v ]
(3.42)
and the vector
[l
«
v]
is called the "polynomial basis" of the element. We will consider now the six-node quadratic triangle shown in Figure 3.16b, for which the potential V(x,y) is given in Eq. (3.66). The corresponding equation in local coordinates is
W, v) =
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(3.43)
and the polynomial basis for this element is u
[l
v u2
uv v2
(3.44)
This basis has six terms and is called a "complete basis" since all combinations of u and v are present in the expansion. Consider now the first-order rectangular element in Figure 3.lie. The element has four nodes and the approximation in local coordinates is K(w, v) = a{ + a2u + «3 v + a4uv
(3.45)
The polynomial basis is
[l u
v uv]
(3.46)
Now the basis has only four terms and is obviously an incomplete basis 2 2 since the terms u and v are absent. In conclusion, we point out that each finite element has its own polynomial basis. 3.4.6. Transformation of Quantities - the Jacobian In actual solution of a problem, the derivatives dV/dx,dV/dy, dV/dz
and
are also required, in addition to the potentials V. The normal
method of obtaining these derivatives is to calculate the Jacobian in the local system of coordinates and then to transform the derivatives into the global system of coordinates. This transformation is facilitated by the Jacobian through the following relation:
' d~
"a"
dx
dy
dz
du
du dx dv dx dp
du dy dv dy dp
du dx dz d dv dy dz d dp ,dz_
d
dv
d
.8P.
This is denoted in short form as
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(3.47)
du =
(3.48)
On the other hand, we can also write
du dx du dy du dz
dx d_ dy d_ dz.
dv dx dv dy dv dz
dp dx dp dy dp dz
du (3.49)
dv
which is denoted as ~
du
(3.50)
In 2D calculations we write
J=
and
detJ (3.51)
The Jacobian will prove to be a useful tool and we will make much use of it. As an example, we calculate the Jacobian J for a first-order triangle:
dx J = du dx dv
dy du dy dv
du [x(u,v)
d_
y(u,v)]
dv
Taking into account the shape functions as in Eq. (3.29), we get
J = du
d_ dv
We obtain
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N2(u,v)
(3.52)
^ "\ r oN] I
x
^ Ar ^ "* 7" dN*j cW-j Z J du du
du dNi
dN2
. dv
dv
\ y\~ (3.53)
dv _
or
d(u) d(v) du c d(u) d
-u-v) du -u-v) dv
dv
y\ (3.54)
dv
This gives •^1
"- 1 1 0" X -1 0 1_ X2
^1
y2 = 3 y3
y\
(3.55)
and
J-l = detJ
x
\-
(3.56) X
2 ~ x\
where det J equals twice the area of the triangle as mentioned before.
(0,10)
(10,0)
x
Figure 3.13. Triangle in global coordinates.
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(0,1)
(1,0)
u
Figure 3.14. Triangle in local coordinates.
3.4.7. Evaluation of the Integrals The change of variables makes the integration relatively easy. In fact, in some cases it would be next to impossible to perform the integration without the use of local coordinates with the corresponding transformation to global coordinates. The integration of a function f\x) over an element in global coordinates, obtained through the use of the local element, is
f f(x)cbcdydz = f f(x(u, v, /?))det J dudvdp *x
(3.57)
*u
where for simplicity, the function f\x, y, zj is denoted as
f(x).
A very simple example is now shown. Suppose that the function f(x,y) = 1 has to be integrated on the triangle in Figure 3.13. In global coordinates, we have
f Idxdy = lx
10x10 2
= 50
Using the local coordinates in Figure 3.14, the integral is f Idet J dudv
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As discussed above, det J equals twice the area of the triangle, and we write
100
ir
-V
= -100 = 50
Even though this is a particular, and very simple example, Eq. (3.57) is a valid method of evaluating the integral through the use of local coordinates. To clarify some of the concepts discussed above we will evaluate now the stiffness and source matrices for the electrostatic problem introduced at the beginning of the chapter. The electrostatic governing equation is
divD = p or
div(s gradV) + p = 0
(3.58)
The problem is considered in two dimensions using first-order triangular elements. First, to conform to the common notation in finite element calculations, we denote:
That
is,
Nl(x,y) = l(x,y),
N2(x,y
N3 (x, y) = ^3 (x, y) . With this notation we can write
"cW"
dx gradN(x,y) = dN Noting that
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and
dx dN_ dy
r-l
'dN_ du dN_ dv
(3.59)
and observing Eq. (3.54), we have
~dN~
du dN
i
i
0"
-1 0
1
X
A
.dv_ With J
given in Eq. (3.56), obtain
~dN~
dx _ i y2-y* dN ~ D x3 — x2
y*-y\ y\-y^ Xj — ,x3
x2 — xl
(3.60)
.&. Using the notation in Eq. (3.7), we have
'dN'
dN
l_ D
03
(3.61)
Applying the Galerkin method we calculate the elemental stiffness matrix (see Eq. (3.17)) using
f gradN* -8 gradV ds
(3.62)
where St is the area of element /. Using the interpolation functions (which are identical to the shape functions), we have (3.63)
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which is a compact notation for
N3(x,y)]
With these, gradV(x,y) becomes
'dN_ gradV(x, y) = gradN(x, y) V = dx V dN_ dy
(3.64)
Using Eq. (3.61), we get
gradV(x,y) = —
(3.65)
V, This is in accordance with Eq. (3.9). Now substituting Eq. (3.61) and (3.65) in Eq. (3.62), we get
f ]~ gradN* • s gradN V det Jdudv
(3.66)
or
8
~<1\
r
<12
r
.11
r
D2
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\~
2
3.
12 r
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.
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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
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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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
• 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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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,
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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
N<-NUbdetJdudv
(3.80)
•Woe/
Noting that U(u,v)= N(p,v)U
i/,v) ....
we
get
N6(u9v)]
"
u, With the aid of Table 3.3, this gives
4ut
4ut ....
4vt]detJductv
U2
local
4vt (3.81) This is also a 6x6 symmetric system. The procedure for integration is the same as indicated in the previous section. For the equations above, exact integrations are obtained for three integration points. However, for a code which can be extended for different terms, using seven integration points is recommended.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
3.8.3. Evaluation of the Third Term in Eq. (3.72) The third term in Eq. (3.72) results in the following:
f
N* - gradx N U c det J dudv
where gradx NU
(3.82)
denotes the derivative of U in the JC direction. Using
Eq. (3.78) in Eq. (3.82) gives
4ut
\dnx\
~\ c det Jdudv
4vt (3.83) Unlike the previous matrices, this matrix is nonsymmetric, but the integration is done as in the previous two sections. 3.8.4. Evaluation of the Source Term The source term is represented by the fourth term in Eq. (3.72). This term gives
I"
N*s det J dudv
(3.84)
or
-t(l-2tj
4ut
s det Jdudv
•Woca/
4vt (3.85) This vector does not depend on the unknowns and is therefore part of the right-hand side of the system of equations for the element.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
3.9. Program
A Simplified
2D
Second-Order
Finite
Element
Now that the basic concepts necessary for finite element calculations were introduced, we present an example using a simple finite element program to bring together the different concepts discussed before. 3.9.1. The Problem to Be Solved Starting with Eq. (3.72) and retaining only the first and fourth term,
we get
div(a grad U)+s = Q
(3.86)
where the first term is the term leading to the stiffness matrix and the second is the source term. This equation may represent many physical situations but in our case it may represent electrostatic applications:
div(s grad V) + p = 0
(3.87)
This equation was used at the beginning of the chapter. However, rather than solving this equation again, we introduce now, briefly, another electromagnetic equation which is similar to Eq. (3.87). This equation describes magnetostatic problems in 2D applications and is written as
div(y grad A)+J = 0
(3.88)
where V is the reluctivity of the material ( V — I/ ju ), A is the magnitude of the z component of the magnetic vector potential A (in this approach, the magnetic vector potential has only a z component). J is the magnitude of the z component of current density (which also has a single component in the z direction). A and J are perpendicular to the x — y plane. A is related to B as
B = rot A
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
(3.89)
B
i d —
ax 0
j d— dy 0
k" d — dz A_
.dA dy
_ •
I.dA — cbc
(3.90)
The governing equation is
With H = vB , we get
Substituting Eq. (3.90) into the equation above and retaining only the z direction componentes, we obtain Eq. (3.88). Much more information and discussion on this equation and its solution will be given in the following chapter. For now, we simply view this as an additional use of the code presented below. The example to be solved here is shown in Figure 3.24.
Figure 3.24. Conducting bar carrying a current density J .
In this case, the magnetic flux (related to B) flows in a block of material with permeability equal to 10//0. The conductor carrying the current (related to J) has permeability //0 (such as copper or aluminium).
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Solving Eq. (3.88) for the magnetic vector potential A , we obtain the components of the magnetic flux density by finding the derivatives dA/dx and dA/dy (see Eq. (3.90)). 3.9.2. The Discretized Domain The discretized domain is shown in Figure 3.25. The bar is 60x60 mm and the nodes are numbered horizontally starting on the bottom as shown. There is a total of 72 elements out of which 8 are source elements (elements no. 29, 30, 31, 32, 41, 42, 43, and 44). The current density is J = 10 A/m2 and is perpendicular to the plane. Second-order triangular elements are used throughout with the node numbers as shown. For example, element no. 1 is defined by nodes no. 1, 2, 3, 16, 29, and 15. Nodes on the edges of the elements are placed midway between corner nodes. For example, X2 = (x3 + xj/2 and y2 = (y3 + yJ/2 . The boundary conditions in this case are A = 0 on the outer boundary, which means that the magnetic flux density has no normal components on the outer boundaries. The following nodes represent boundary lines and will be assigned zero magnetic vector potential values: [1,2,3,4,5,6,7,8,9,10,11,12,13] [14,27,40,53,66,79,92,105,118,131,144] [157,158,159,160,161,162,163,164,165,166,167,168,169] [26,39,52,65,78,91,104,117,130,143,156]
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
' ^ y (mm)
1 2 3 4
5 6 7 8 9 10 11 12 13
'0
'20
'10
30
40
50
'60
Figure 3.25. Discretization of the domain in Figure 3.24.
3.9.3. The Finite Element Program A listing of the second-order finite element program, written in standard FORTRAN, is shown below. Reading of the program is advised so that the reader may understand the implementation of the concepts described above. The main arrays and variables are: NNO - Number of nodes. NEL - Number of elements. NF - Number of lines with imposed boundary conditions. KT(NEL,6) - Node numbers in the element. NMAT(NEL) - Material index for the element. SOURCEL(NEL) - Value of the source in the element.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
X(NNO), Y(NNO) - Coordinates of the nodes. NUMBMAT - Number of different materials. PROP(6) - Property index of the materials. AA(NNO,NNO), BB(NNO) - Left and right-side global system of the equations. NCD(NF,20) - Node numbers with Dirichlet boundary conditions. Program Listing program Secdemo c Call Data to read the problem data call Data(nno,nel,nf) c Call Zero to set matrices to zero call Zero(nno) c Call Form to create the matrix system call Form(nel) c Call Condi to insert Dirichlet boundary conditions call Condi(nf.nno) c Call Elim to solve the matr. system by Gauss Elim. method call Elim(nno) c Call Sort to furnish results call Sort(nno,nel) stop end c subroutine Data(nno,nel,nf) common /mesh/ nmat(200),kt(200,6),ncd(10,20),sourcel(200) common /real*/ vi(10),prop(6),x(300),y(300) character arq*20 c Data reading write(6,'(a$)')' File Name=' read(5,'(a20)')arq open (9,file=arq,form='formatted') c nno-number of nodes; nel-number of elements read(9,'(i3)')nno read(9,'(i3)')nel do i=l,nel c kt-element nodes; nmat-material; sourcel-source read(9,'(6i4,i3,el2.4)')(kt(i,j),j=l,6),nmat(i),sourcel(i) enddo do i = l,nno c x,y-coords. of the nodes read(9,'(2f8.4)')x(i),y(i) enddo c — nf-number of boundary (Dirichlet) conditions read(9,'(i3)')nf do i=l,nf
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
c---------------vi-imposed potential; ncd-nodes with this potential read(9,'(f7.2)')vi(i) read(9;(20i4)')(ncd(ij)j=l,20) enddo c---------------numbmat-number of materials; prop-constit. property value read(9,'(i3)')numbmat do i=l,numbmat read(9,'(e9.4)')prop(i) enddo return end subroutine Zero(nno) common /matrixl/ aa(200,200),w(200),bb(200) do i=l,nno bb(i)=0. w(i)=0. doj=l,nno aa(ij)=0. enddo enddo return end C*"~"""~ •—----—«-——--—--« — ---------- — -~---w--— ----- ---__—-
subroutine Form(nel) common /mesh/ nmat(200),kt(200,6),ncd(10,20),sourcel(200) common /realx/ vi(10),prop(6),x(300),y(300) common /matrixl/ aa(200,200),w(200),bb(200) common /shap/ dnu(6),dnv(6),xp(6),yp(6),xj(2,2),rn(6), * dnx(6),dny(6) dimension n(6),r(6,6),s(6),u3(3),v3(3),w3(3) c---------------Data for 3 integration points, sufficient for this equation (table 3.4) data u3 /0. 166667,0.666667,0. 166667/ data v3 /0.166667,0.166667,0.666667/ data w3 /0.166667,0.166667,0.166667/ c ............... Loop on elements do i=l,nel c---------------Arrays n, xp and yp created as internal numbering c ............... of the element i doj=l,6 enddo nm=nmat(i) xprop=prop(nm) ss=sourcel(i) doj=l,6 xpQ)=x(n(j))
ypO)=y(n(j)) enddo c---------------Stiffness r(6,6) and source s(6) elemental c---------------matrices set to zero doj=l,6 s(j)=0.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
c
c
c
c c
c
dok=l,6 r(j,k)=0. enddo enddo Integration points loop doit=l,3 w=w3(it) u=u3(it) v=v3(it) Call Dnxy to calculate grad N, and det(J) call Dnxy(u,v,detj) doj=l,6 sG)=s(j)+m(j)*ss*detj*w do k-1,6 Calculation of the Stiffness matrix r(j,k)=r(j,k) + (dnxQ)*dnx(k)+dny(j)*dny(k))*xprop*detj*w enddo enddo enddo Assembling the global system doj = l,6 jj=n(j) Source vector on the right side bb(jj)=bbGj)+sG) dok=l,6 kk=n(k) Stiffness matrix on the left side aa(jj,kk)=aa(jj,kk)+r(j,k) enddo enddo enddo return end
subroutine Dnxy(u,v,detj) common /shap/ dnu(6),dnv(6),xp(6),yp(6),xj(2,2),rn(6), * dnx(6),dny(6) c Calculation of dN/du and dN/dv (table 3.3) t=l.-u-v m(l)=-t*(l.-2.*t) rn(2)=4.*u*t rn(3)=-u*(l.-2.*u) rn(4)=4.*u*v m(5)=-v*(l.-2.*v) m(6)=4.*v*t dnu(l) = l.-4.*t dnu(2)=4.*(t-u) dnu(3)=-l.+4.*u dnu(4)=4*v dnu(5)=0. dnu(6)=-4.*v dnv(l) = l.-4.*t
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
c
c c
dnv(2)=-4.*u dnv(3)=0. dnv(4)=4.*u dnv(5)=-l.+4.*v dnv(6)=4.*(t-v) Calculation of Jacobian terms xj(l,l)=0. xj(l,2)=0. xj(2,l)=0. xj(2,2)=0. doi=l,6 xj(l,l)=xj(l,l)+dnu(i)*xp(i) xj(l,2)=xj(l,2)+dnu(i)*yp(i) xj(2,l)=xj(2,l)+dnv(i)*xp(i) xj(2,2)=xj(2,2)+dnv(i)*yp(i) enddo Calculation of det(J) derj=xj(l,l)*xj(2,2)-xj(2,l)*xj(l,2) Calculation of derivatives DN/dx and DN/dy doi=l,6 dnx(i)=(+xj(2,2)*dnu(i)-xj(l,2)*dnv(i))/detj dny(i) = (-xj(2,l)*dnu(i)+xj(l,l)*dnv(i))/detj enddo return end
subroutine Condi(nf,nno) common /mesh/ nmat(200),kt(200,6),ncd(10,20),sourcel(200) common /realx/ vi(10),prop(6),x(300),y(300) common /matrixl/ aa(200,200),w(200),bb(200) do i=l,nf do j=1,20 noc=ncd(i,j) if(noc.ne.0)then c Zero is placed in all terms of the line do k=l,nno aa(noc,k)=0. enddo c 1. in diagonal and imposed value on the right side vector aa(noc,noc)=l. bb(noc)=vi(i) endif enddo enddo return end (*-._-____---_-____-_,-__—-__—______-—___________._________.__..________._
c
subroutine Elim(nno) common /matrixl/ aa(200,200),w(200),bb(200) Applic. of Gauss Elimination method to solve AA*w=bb nn=nno-l do i=l,nn do m=i+l,nno
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fat=aa(m,i)/aa(i,i) bb(m)=bb(m)-bb(i)*fat do j=i+l,nno aa(m,j)=aa(m,j)-aa(ij)*fat enddo enddo enddo w(nno)=bb(nno)/aa(nno,nno) do i=nn,l,-l som=0. do j = i+l,nno som=som+aa(i,j)*w(j) enddo w(i) = (bb(i)-som)/aa(i,i) enddo return end subroutine Sort(nno.nel) character q*l common /mesh/ nmat(200),kt(200,6),ncd(10,20),sourcel(200) common /realx/ vi(10),prop(6),x(300),y(300) common /matrixl/ aa(200,200),w(200),bb(200) common /shap/ dnu(6),dnv(6),xp(6),yp(6),xj(2,2),rn(6), * dnx(6),dny(6) dimension n(6) open (11 ,file='sort'.form - 'formatted') write(6,'(a)')' Set Printer' read(5,'(a)')q c Potentials printing do i=l,nno write(6,'(a,i3,a,el0.4)')' Node-',i,' Potential=',w(i) write(ll,'(a,i3,a,el0.4)')' Node-',i,' Potential=',w(i) enddo c Printing of potentials and derivatives c in the element centroids ub = l./3. vb=l./3. do i=l,nel doj=l,6 n(j)=kt(ij) xp(j)=x(n(j)) ypli)=y(n(j)) dnx(j)=0. dnyQ)=0. enddo call Dnxy(ub,vb,detj) potcen=0. dnxv=0. dnyv=0. doj=l,6 potcen=potcen+m0)*w(n(j)) dnxv=dnxv+dnxO)*w(n(j))
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
dnyv=dnyv+dny (j)* w(n(j)) enddo write(6)1(a,i3,a,el0.4,a,el0.4,a)el0.4)') *' Elem-',i,' Pot=',potcen,' dV/dx=',dnxv,' dV/dy=',dnyv Write(ll)1(a,i3,a,el0.4,a,el0.4,a,el0.4)1) *' Hem-'.i,' Pot=',potcen,' dV/dx=',dnxv,' dV/dy=',dnyv enddo return end
Some remarks are in order here: •
The program listed above is a very simple, elementary program intended to demonstrate some of the concepts discussed in this chapter such as shape functions, numerical integration, matrix assembly, and the like. For this purpose, the two most important subroutines are FORM and DNXY.
•
In subroutine FORM, the loops on the elements and numerical integration are performed. Assembly of the stiffness and source matrices are also done in FORM.
•
Subroutine DNXY calculates the Jacobian, its inverse, its determinant, and the derivatives dN/dx and dN/dy.
•
It would be possible to simplify this program further by using subparametric mapping instead of the isoparametric mapping used in the current program. This would make the edges of the elements in global coordinates, straight lines (rather than curvilinear) and, because the Jacobian is simplified, the evaluation of matrices would also be simpler.
•
Insertion of boundary conditions is performed after the global matrix has been fully assembled.
•
The solution of the system of equations is done using a standard Gaussian elimination method.
•
The matrix AA(NNO.NNO) is assumed to be fully populated and nonsymmetric. The solution here takes no advantage of either the sparse or the symmetric nature of the matrix. In more
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
elaborate finite element programs both symmetry and sparseness are exploited to increase speed and efficiency. •
The results listed by SORT are the potentials at the nodes of the mesh, and the derivatives dA/dx and dA/dy at the centroids of the elements. The potentials at the centroids are also calculated.
The data entry for program SECDEMO is listed below (see subroutine DATA to follow the sequence of input data). Input Data Listing 169 72
1 2 3 16 29 15 1 29 28 27 14 1 15 1 3 4 5 18 31 17 1 31 30 29 16 3 17 1 5 6 7 20 33 19 1
.OOOOE+00 .OOOOE+00 .OOOOE+00 .OOOOE+00 .OOOOE+00
81 80 79 66 53 67 1 55 56 57 70 83 69 1 83 82 81 68 55 69 1 5758597285712 85 84 83 70 57 71 2 59 60 61 74 87 73 2 8786857259732 61 62 63 76 89 75 1 89 88 87 74 61 75 1 63 64 65 78 91 77 1 91 90 89 76 63 77 1 79 80 81 94 107 93 1 107 106 105 92 79 93 1 81 82 83 96 109 95 1 109 108 107 94 81 95 1 83848598111972 1111101099683972 858687100113992 1131121119885992 87 88 89 102 115 101 1 115 114 113 100 87 1011 89 90 91 104 117 103 1
.OOOOE+00 .OOOOE+00 .OOOOE+00 .1000E+07 .1000E+07 .1000E+07 .1000E+07 .OOOOE+00 .OOOOE+00 .OOOOE+00 .OOOOE+00 .OOOOE+00 .OOOOE+00 .OOOOE+00 .OOOOE+00 .1000E+07 .1000E+07 .1000E+07 .1000E+07 .OOOOE+00 .OOOOE+00 .OOOOE+00
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
(Number of nodes) (Number of elements) (Nodes, material, source)
(Elements with source)
(Elements with source)
141 142 143 156 169 155 1 .OOOOE+00 169 168 167 154 141 155 1 .OOOOE+00 .0000 .0000 .0050 .0000 .0100 .0000 .0150 .0000 .0200 .0000 .0250 .0000 .0300 .0000 .0350 .0000 .0400 .0000 .0450 .0000 .0500 .0000 .0550 .0000 .0600 .0000 .0000 .0050 .0050 .0050 .0100 .0050 .0150 .0050
.0500 .0550 .0600 4
.0600 .0600 .0600 .00
(Last element) (Node coordinates)
(Last node) (Number of boundary conditions) (Value of first boundary condition)
1 2 3 4 5 6 7 8 9 10 11 12 1 3 0 0 0 0 0 0 0 (Nodeswith Ist boundarycond.) .00 157 158 159 160 161162 163 164 165 166 167 168 169 0 0 0 0 0 0 0 .00 14 27 40 53 66 79 92 105 118 131 144 0 0 0 0 0 0 0 0 0 .00 26 39 52 65 78 91 104 117 130 143 156 0 0 0 0 0 0 0 0 0 2 (Number of materials) 7958E+05 7958E+06
(Permeability of \S
material)
After the solution we obtain the following output. Note that the derivatives printed dV/dx and dV/dy correspond directly to — By and Bx, respectively (Eq. (3.90)). If we were solving Eq. (3.87) instead, the derivatives would correspond to — Ex and — Ey.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Output Data Listing Node-1
Potential .OOOOE+00
Node- 2
Potential .OOOOE+00
Node- 3
Potential .OOOOE+00
Node- 4
Potential^ .OOOOE+00
Node- 5
Potential^ .OOOOE+00
Node- 6
Potential .OOOOE+00
Node- 7
Potential .OOOOE+00
Node- 8
Potential^ .OOOOE+00
Node- 9
Potential .OOOOE+00
Node-10
Potential^ .OOOOE+00
Node- 11
Potential = .OOOOE+00
Node-165 Potential^ .OOOOE+00 Node-166 Potential^ .OOOOE+00 Node-167 Potential = .OOOOE+00 Node- 168 Potential .OOOOE+00 Node-169 Potential .OOOOE+00 Elem-1
Pot= .3451E-04
dV/dx= .5217E-02 dV/dy= .1052E-01
Elem-2
Pot = .3451E-04
dV/dx= .1052F-01 dV/dy= .5217E-02
Elem-3
Pot= .8235E-04
dV/dx= .4114E-02 dV/dy-.2537E-01
Elem-4
Pot= .1394E-03
dV/dx= .9637E-02 dV/dy= .2139E-01
Elem-5
Pot= .1111E-03
dV/dx= .9600E-03 dV/dy= .3429E-01
Elem- 6
Pot= .2193E-03
dV/dx= .4800E-02 dV/dy= .3409E-01
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Elem-7
Pot= .1053E-03
dV/dx=-.2782E-02 dV/dy= .3227E-01
Elem- 8
Pot= .2318E-03
dV/dx=-.2978E-02 dV/dy= .3611E-01
Elem- 9
Pot= .6819E-04
dV/dx=-.4887E-02 dV/dy= .2062E-01
Bern-10 Pot= .1694E-03
dV/dx=-.8865E-02 dV/dy= .2614E-01
Elem-11 Pot= .1739E-04
dV/dx=-.5217E-02 dV/dy= .5217E-02
Elem-68Pot=.llllE-03
dV/dx=-.9600E-03 dV/dy=-.3429E-01
Oem- 69 Pot= .1394E-03
dV/dx=-.9637E-02 dV/dy=-.2139E-01
Elem- 70 Pot= .8235E-04
dV/dx=-.4114E-02 dV/dy=-.2537E-01
Hem- 71 Pot= .3451E-04
dV/dx=-.1052E-01 dV/dy=-.5217E-02
Elem- 72 Pot= .3451E-04
dV/dx=-.5217E-02 dV/dy=-.1052E-01
From these results, we note the following:
•
The central node of the mesh (in the middle of the conductor, number 85) has the highest potential value in the solution domain as expected.
•
Symmetrically located nodes have identical potential values.
•
Symmetrically located elements (such as no. 34 and no. 39) have the same value of derivatives (fields).
These observations are simple but useful indications that the software is working properly. Whenever a code is used, it is advisable that at least a simple verification as indicated here be performed.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Figure 3.26. Plot of equal potential lines provided by EFCAD.
The results from SECDEMO were verified with results from EFCAD which has been used satisfactorily for many years. The plot of constant potentials provided by EFCAD is shown in Figure 3.26.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
4 The Finite Element Method Applied to 2D Electromagnetic Cases 4.1. Introduction In this chapter we will apply the FEM concepts presented in the previous chapter to EM cases in 2D. We will use the Galerkin method as support to do so. In fact, nowadays there is a strong tendency to apply this method rather than the variational one, even for static cases which were classically formulated with the variational method. The main reason is related to the fact that with the Galerkin method we can establish the numerical formulation directly from the physical equation defining the phenomenon, instead of an associated functional which may or may not be easy to formulate. We will present two sets of problems, beginning with the classical static cases. The second part is related to eddy currents. Axi-symmetry and non-linearity will also be explained. 4.2. Some Static Cases
The cases presented here are considered as static in time. Even though, at first glance, this might appear to be an important limitation (because most devices have moving parts, or operate under time-dependent conditions), there are many situations where a dynamic problem can be studied as a composition of static cases. Here we establish, using Maxwell's equations, the second-order partial differential equations (Laplace's and
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
A
V=Va I
c
D
V=Vb
B
Figure 4.1. A problem solved using the electric scalar potential. The solution domain includes dielectric materials and charges with specified voltages on the boundaries A and B .
Poisson's equations) associated with specific physical phenomena. Instead of specifying the equations in terms of field variables, we use the scalar and vector potentials as primary variables. 4.2.1. Electrostatic Fields: Dielectric Materials Consider the situation shown in Figure 4.1, where the domain under study contains several dielectric media with different permittivities "s ". On lines A and B potentials Va and V^ are imposed. Assume that a static charge q defined by a volume charge density p exists in the interior of the domain. Since rotlL — 0, a scalar electrical potential V (Volts), linked to the electrical field intensity E by (4.1)
is defined. Also, because
div D = p
(4.2)
and using the constitutive relation
D = eE we have =p
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(4.3)
or, in terms of the potential, div 8 (- gra d V) = p which, in explicit form in two dimensions (2D), is
d
dx
8
dV
dx
+
d
dy
8
dV
dy
(4.4)
= -p
This is Poisson's equation which describes the potential distribution in a domain. If the domain contains only one material, the permittivity can be considered constant, and we get
d2V i d2V dx2 dy2
(4.5)
In practical electrical engineering problems, the charge density is frequently zero. In this case, Eq. (4.4) is transformed into Laplace's equation:
d dx
8
dV dx
I
d dy
8
dV _ —U dy
Here, the sources of the field are the boundary conditions imposed as potentials (on lines A and B). In contrast, in Eq. (4.5), the field can also be generated by the charge density p . The calculation of the matricial terms of the general equation (4.4) was already obtained in the previous chapter. They are, for a first order element:
d dV d dV 1 8 : dx dx dy dy
• For the term —8
r r
\ \ 0102 + r\r2 symmetric <7 2 # 2 + r2 r2 2Z) symmetric symmetric 8
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r r
23
(4.6)
Using the notations defined previously, this matrix must be assembled in the left-hand side of the matricial system, since it is multiplied byK • For p Y 1 1
which should be assembled in the right-hand side vector of the matricial system. 4.2.2. Stationary Currents: Conducting Materials The physical situation considered here is shown in Figure 4.2. The potential difference (Va — Vb) establishes the currents in the composite medium with conductivities (J,, a 2 and a 3.
V=Vu
V=Vh
B
Figure 4.2. Currents in a domain made of materials with different conductivities. One appropriate method of solution is the electric scalar potential.
To define an equation for this situation we use the relation E = —grad V, and the electrical continuity equation, presented in Chapter 2
Q
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(4.7)
With the point form of Ohm's law, J = aE, we have diva E = diva (- grad V) = 0 which, in explicit form, is
d dV d dV _ —a— + —a — = 0 dx dx dy dy
(4.8)
If there is only one material in the domain, with constant conductivity, Eq. (4.8) becomes
d2V T dx2
'
d2V _ T" — ^ dy2
(4.9)
Solving for V, allows the calculation of E and, using Ohm's law, allows the calculation of J. The matricial contribution term of equation (4.8) is similar to (4.6)
ID
symmetric symmetric
q2q2 +r2r2 symmetric
<12<13 +r2r3 q% #3+73 r%
to be assembled on the left hand side of the matricial system. 4.2.3. Magnetic Fields: Scalar Potential The geometry relating to this situation is shown in Figure 4.3a and 4.3b. We assume that the magnetomotive force of the coil (Figure 4.3a) is established between lines A and B. This is a good approximation if the permeability of the magnetic circuit is high.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
NI
Figure 4.3d. A magnetic circuit with a gap between lines A and B. Because of the high permeability, only the gap is considered.
A
D
Figure 4.3b. The gap with magnetic scalar potentials enforced on lines A and B. Lines C and D are Neumann boundary conditions.
Thus, we only need to consider the air-gap domain shown in Figure 4.3b where the magnetic field is generated by the potential difference on the boundaries. There are no currents in the domain (J = 0), therefore, it is possible to define a scalar magnetic potential *¥ (the units for this scalar potential are Ampere-turns) related to the magnetic field H by (4.10) The validity of this definition is established by the following:
and, therefore,
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Note that if J ^ 0 , this relation is inconsistent, since rot(— grad ^P) is always equal to 0. Using the following: B = \iR
(4.11)
<#vB = 0
(4.12)
we get
div(\i H) = div (i(- grad V) = 0 which can be written as
dx
dx
+ — |J, -
dy
dy
_ =0
//110X
(4.13)
The permeability )a, commonly, is not constant because almost all electrical devices have magnetic paths that contain nonlinear materials. The matricial contribution is also similar to (4.6), where 6 is replaced by (J-. 4.2.4. The Magnetic Field: Vector Potential If we wish to calculate the field in a domain with current sources (as in the example of Figure 4.3a, including the coil and the magnetic circuit), the scalar potential formulation shown above is not applicable because, using this formulation, J is required to be equal to 0. If J =£ 0 , it is possible to use the vector potential A, related to B by B = ro/A
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
(4.14)
Figure 4.4. Definition of a two-dimensional domain. Current densities and the magnetic vector potential are perpendicular to the plane. The magnetic flux density is parallel to the plane.
In 2D problems, the vectors J and A have only one component, perpendicular to the plane Oxy in Figure 4.4. Denoting i , j , and k the orthogonal unit vectors in directions Ox, Oy and Oz, we have
To establish the formulation for this case, we use the equations (4.15) (4.16)
H = vB
where v is the magnetic reluctivity (v = l/|-l). We have now rot vB = rot v rot A = J where B = rot A is, in the 2D case,
i d_ = det dx 0
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
j d_ dy 0
k d_ dz A
(4.17)
With this, the magnetic flux density is written as =1- -- Ji- — dy cbc
(4.18)
or
dy
dx
Substituting Eq. (4.18) in (4.17), and since in 2D cases there are no variations in the Oz direction, we have
i
J
k d
dy vdA dx
dz
d
d
det — dx vdA [dy
0
or, equating terms for the Oz component
d
dA
d
dA
—v 1 v— = - J dx dx dy dy
(4.19)
This is Poisson's equation for the two-dimensional magnetic vector potential. Now consider the line in Figure 4.5, where we assume that the potential A does not vary along the line. Defining a local system of coordinates Oxy, and assuming that the direction Ox coincides with the line, we have dA/dx = 0, since A is constant. Therefore, By = dA/dx = 0.
However,
in the direction
Oy,
A
can vary
and Bx = dA/dy can be nonzero. From this, we conclude that the magnetic field is parallel to the equipotential lines of A .
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Figure 4.5. The magnetic flux density is parallel to equipotentials of A .
B
Figure 4.6. In two-dimensional geometries, magnetic flux is given per unit depth.
In 2D fields, it is possible to attribute a very important physical meaning to the vector potential. First, note that, because the structure is two-dimensional, the magnetic flux is given per unit depth of the structure, as shown in Figure 4.6. The flux per unit depth is <j) =Bl(J¥b/m). If the actual flux is needed, it is necessary to multiply this value by the depth of the structure. An example is shown in Figure 4.7, where the xy plane and the orthogonal surface S of the actual device are indicated. At point (1), A = A{ and at point (2), A = A2, both in the Oz direction. Therefore, <|> = f B-ds= LrotA-ds using Stokes' theorem, we obtain
We calculate the circulation of A on contour C, in two parts. On the parts of the contour where A is perpendicular to fifl the circulation is zero; on the others, assuming that A is constant and parallel to dl (2D approach), we have
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
where P is the depth of the device. The difference between the potentials A{ and A2 gives the magnetic flux (in Wb/m) because
4=
Figure 4.7. A contour in a 2D geometry used to show the interpretation of the magnetic vector potential as the difference in flux per unit depth.
A=Q
A=A\ Figure 4.8. A geometry with two equipotential lines. The region between the two lines defines a flux tube.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Figure 4.9. Magnetization curve for a permanent magnet.
Note that the flux is defined by the potential difference and not by the absolute values of A , meaning that, as was the case in the example of Figure 2.16, it is necessary to fix one potential as a reference value and only then is the remaining value defined. Consider now the example of Figure 4.8, where a potential A = 0 is imposed on the domain boundary. After computing the potential distribution in the domain, the potential A is known everywhere in it and we can obtain the equipotential line A = A\, as indicated in Figure 4.8. Because this line is a field line, the region included between the boundary (^4 = 0) and this equipotential line ( A — Al) is a flux tube, whose flux per unit of length is (A\ — 0) / P . Thus, the drawing of a number of equipotential lines provides an efficient method for visualization of the magnetic field distribution in the domain. Now that the physical meaning of the vector potential is explained, we present the matricial contribution for the two terms of the governing equation (4.19). As for the first case shown, they are V
2D
0101 +n r i symmetric symmetric
and
JD
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0202 + r 2 r 2 symmetric
0203 + r 2 r 3 0303 + r3r3
(4.20)
Suppose now that the domain contains permanent magnets (Figure 4.9) and that their magnetic characteristic curves are given by B = //H + B0
(4.21)
where ju is the magnet permeability and BQ is the remanent flux density. Notice that some authors prefer to express the above equation as B = //H , where the right-hand side product takes into account the remanent induction. We prefer using Eq. (4.21) since it brings us to well established numerical procedures. BQ is related to the coercive field intensity Hc by
This is a good approximation for the commonly used magnets (such as ferrite, Samarium-Cobalt and Neodymium-Iron-Boron magnets) whose relative permeability is close to 1. Writing H in Eq. (4.21) explicitly, we have
H=-(B-B O ) ^ which, together with the relation rot H = 0, gives rot — (B-B 0 ) = 0
(4.22)
V since, for a permanent magnet, J = 0 . This equation becomes 1 rot A - rot — 1 BQ „ = 0„ rot — ju ju
Its first term was already calculated and the corresponding matricial contribution is given by (4.20). Let us evaluate the second one, writing firstly BQ as a function of its x and y components: B
0 ~
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Now
k
d_
rot — Bn =
~d~z
0 The term in z direction is
_d
BO,
dx v ^
d j
B(
dy
Which can be written, for simplicity, as
where
When applying Galerkin's method we have
following the steps presented in the Chapter 3, it is equal to: e I divWBftds - If grad W • B0e ds f
or, applying the theorem of divergence
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For the discretized domain it becomes >w • 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<»T
HoE B = //H o J = oE
where T is given in A/m and P is the depth of the device. Using this notation, the difference between 7J and T2 is, in this case, the current between the lines on which T = T\ and T = T2 , as indicated in Figure 4.10. In the preceding case B was parallel to the equipotential lines. Here, J is parallel to the equipotential lines. This formulation is very useful when, instead of the potential difference between lines C and D (section 4.2.2) the current difference between lines A and B is known. This current is imposed as boundary conditions (i.e., T = T{ on line A and T = T2 on line B ), making
T-T2
=
T2
Figure 4.10. A problem that can be solved using the electric vector potential. The electric vector potential is specified on lines A and B as Dirichlet boundary conditions.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Note that in the magnetic case the magnetic field can be generated by currents (as in Figure 4.8) because the basic equation is ro/H = J. In electric cases, the field can only be generated by a potential difference imposed on the boundaries, since the formulation is based on the equation rot E = 0 . Laplace's equation for this problem is similar to Eq. (4.19),
a i dr d i ar _
--+ --- = 0 dx a dx dy (j dy
(4.25)
having also matricial contributions analogous to (4.6). 4.3. Application to 2D Eddy Current Problems The computation of the matrix terms up to this point has been relatively simple, primarily because the integration was independent of coordinates. In the following paragraphs we present the formulation of equations involving eddy currents, whose numerical formulation is more complex. The additional complexity is due to the integration, which depends on the system of coordinates used. More details on this topic may be found in section 3.7. 4.3.1 . First-Order Element in Local Coordinates For didactical purposes, we will briefly recall some topics of Chapter 3. Instead of using the global system of coordinates (i.e., Oxy), it is possible to define the element in a "local" or "reference system" of coordinates. This simplifies the algebraic operations involved in the integration. The triangle or "local element" is defined in the M,v plane instead of in the "real" x, y plane as shown in Figures 4.11 and 4.12. The relationship between these two triangles is defined by a set of "geometric transformation functions" or simply "mapping functions" JV(w,v), which, for first order triangular elements are:
\-u-v N2(u,v) = u
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(4.26a) (4.26b)
(4.26c)
Ri
l
u
Figure 4.11. A finite element defined in a local system of coordinates.
0 Figure 4.12. The same element after mapping to the global system of coordinates.
With these functions, the abscissae X are written as
[
N
-/V2
P 1
= W, (w, v)c, +N2(u, v)x
(4.27)
With the functions in Eq. (4.26), this becomes
x = [l - « -v u v]
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(4.28)
Analogously, we can write for y
y\
(4.29)
= \L-u-v u v]
The same geometric transformation functions are used as "interpolation functions" for the vector potentials A in the triangular element:
= \L-u-v u v]
The functions NI , N2
(4.30)
and A^ given in Eq. (4.26) describe linear
distributions, for both coordinates and potentials. The potentials in the global element vary linearly as
corresponding, in local coordinates, to
A(u, v) = flj + a2u + a3 Taking into account the direct transformation between the points in the u, v and x, y planes, the expressions above can be written as
A(U,V)= [l u v]
The matrix P is called the "polynomial basis" of the element P = [l u v]
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Using the notation a for the coefficient vector,
a = a2
we get
A(u,v) = Pa We use the simplified notation,
x(u, v) = NX
(4.31)
y(u,v) = Ny
(4.32)
A(U,V)=NA
(4.33)
for the matrix products in Eqs. (4.28), (4.29), and (4.30), where by convention, N is an array corresponding to the geometric transformation functions (Eqs. (4.31) and (4.32)) and the interpolation potential function (Eq. (4.33)). These are the same for isoparametric elements. The integration of a function f(x, y) over the global element can be performed using the local element as
^f(x9y)dxdy
= £ f(x(u,v\y(u,v))det Jrfudv
(4.34)
where S, is the surface of the local element and det J/ is the determinant of the Jacobian of the coordinates transformation
dx
dy
du dx
du dy
dv
dv
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(4.35)
The partial derivatives are calculated using Eqs. (4.28) and (4.29) X,
du
= [-1 10] L
J
and, analogously,
dy
dy
The Jacobian is
(4.36) The determinant of J/ is equal to twice the surface of the triangle, or equal to D, with the notation defined in Chapter 3. At this point it is useful to calculate the gradient of N, as it is frequently used in subsequent calculations
grad N - dx dN dy
Noting that
'dN~ dx du - du dN dx ,dv_ _dv
dy ~dN~ du dx dy dN dv. _dy_
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and using Eq. (4.12) for the Jacobian matrix, we can write
~dN~ ~dx~ dN
-t,r
_dy _
~dN~ (4.37)
dN ,dv_
The inversion of the matrix in Eq. (4.36) gives J,
JJ-l
/
1
as
y$ -y\ y\- yi
detJ/
\ ~X3
x
X
2 ~ x\
(4.38)
and
dN ~du dN dv
-1 1 0 -1 0 1
(4.39)
Performing the product of Eq. (4.38) by Eq. (4.39), we obtain for Eq. (4.37)
dx dN
_ 1 ~Z>
[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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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 <5
rotE =
(rot A)
ar
'
or
8t
This is,
dA E + — = grad V Considering that E is generated only by the time variation of B , we get gra d V = 0 , and dA
E =
dt
andj Aiep = —a
dA
dt
where all variables are in the Oz direction. With these, Eq. (4.27) can be written as
rotvrotA + cr
dA dt
Jv=0
(4.42)
According to section 4.2.4, in a 2D case, the first term of Eq. (4.42) can be written as
d dx
v
dA dx
d
dA v— dy dy
where A is the component of A in the Oz direction and is, therefore, a scalar. This expression is equal to div v grad A
d dA v dx dx
d dA v— = -div v grad A dy dy
Thus, Eq. (4.42), with all vectors in the Oz direction, can be written as
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
dA div v grad A-a — + Js - 0 dt
(4.43)
The time derivative can be expressed (for more detail see section 5.4.4) as dA _A(t + At)-A(t)
dt ~
At
where At is a time step and A(t + At) — A(t) is the change in A between time steps (t + At) and t. Equation (4.43) can now be written as r-
_, *,
A(t + At)
div v grad A(t + At)-& —
At
A(t)
r
,
. .
A
+ a -^- + J,(t + At) = Q At (4.44)
Now we apply the Galerkin procedure to Eq. (4.44), where the weighting functions are the functions TV already defined in the previous section. For simplicity, the formulation is performed term by term: a. The first term in Eq. (4.44). This is
f N'div v grad A(t + At) ds
(4.45)
This term, after application of the divergence theorem, results in two integrals: • The first integral, accounting for boundary conditions, leads to v grad A(t + At) • n = 0 or, explicitly,
( 8A(t + A/) SA(t .dA(t + At)} (. . \ J j-(inx+Jny)=0n dx With B = rot A , we get
or
Bxn = 0
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(4.45a)
meaning that, for the Neumann condition, B and n are co-linear. For Dirichlet boundary conditions, in the corresponding boundary lines, N is equal to zero (see section 3.3.1), and the magnetic induction is parallel to the boundary. • The second integral provides the terms of the elemental matrix: - f grad N* (v grad A(t +
*w
where St indicates that the integral has been applied to element i of the mesh. Noting that A can be expressed as A(u, v) = NA (from Eq. (4.33)), we get - f grad N* v grad NA(t + &t)dxdy
(4.45b)
Using the local element, we obtain - LL
grad N * v grad NA(t + A/) det J / dudv
which, using Eq. (4.40) is
~4\ n ~
-ft
r
42
2
v 4l 2 r\
42 YI
'A{ 43 *2 r^
detJ,
A.
As there is no dependence on u and v and det J / = D , we get
41 n 42
r
_43
r
2
3_
r
-^ ~AI~
v Q\
Q'9
Q*\
~b
r2
^3
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r\
A2 _^3_
The integration is performed on the surface of the local element and is equal to 1/2. For Eq. (4.45b), the result is
2Z)
\2
\\
v
4 l('
r r
r r
+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
/vv grad A e jco ' - — ejo) l
Js ej(0 ' = 0
(4.54)
or
div(vgrad A)- qjcoA + Js = 0
(4.55)
We have assumed here that the phase angle of the current source is zero. This is often a practical consideration and is commonly used, but having more than one source, the phase should be introduced in the other sources, by decomposing the current density in real and imaginary parts. When applying Galerkin's method, the first and third terms in Eq. (4.55) get the same results as in Eqs. (4.46) and (4.50), respectively. However, the second term must be re-evaluated. The elemental matrix for the second term is given by
-crja>( N* Adxdy
(4.56)
The integrand of this expression is similar to that of Eq. (4.47), and using the same calculations we get
" 1
ajcoD 0.5 12
0.5 0.5" 1
0.5 0.5
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0.5 1
2 A
(4.57)
This elemental matrix has imaginary terms and must be added to the matrix in Eq. (4.46) to form the global complex matrix SS. The solution of the system
SS A = Q results in the complex vector A. This type of formulation has an important advantage: solving a single matrix system we obtain the real and imaginary parts of A and, therefore, its magnitude and phase angle in relation to Js. However, we can not include nonlinearity. If ferromagnetic materials are present, it is necessary to know, a priori, if the excitation current is low enough to avoid nonlinear effects such as saturation in the structure under study. If these conditions are not satisfied, the time discretization formulation of the previous section must be used. For the latter, the time required for computation is longer (because an iterative calculation is performed for each step), the time discretization method is the only way to obtain correct results for nonlinear problems of this kind. The main results obtainable with the complex variables formulation are: • Penetration effects can be seen graphically, as shown in Figure
4.14. • Impedance calculations, as seen from the exterior source, are possible. Noting that (R + JGoLj / = JCG<J>, we obtain L and R from the expression
CO
where <j) is equal to A multiplied by the depth of the structure and / is equal to Js multiplied by the cross-sectional area of the coil.
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• The real part L is the inductance of the magnetic circuit. The inductance decreases as CO increases, since the magnetic flux decreases due to lower field penetration (the reluctance increases). • The imaginary part R/CO represents the resistive term. This is a typical characteristic of eddy currents, induced in conducting media. R increases with CO. The global losses through eddy currents in the domain are given by RI2 2
The eddy current density Je is given by
Je=Re(-crjcoAeJO}t) This allows the calculation of average losses by local integration 1
f\
T 2.
where S is the cross-section of the conductive piece. Averaging over a cycle gives
7re = " i Jf a / 2 * 2
This must be equivalent to the quantity RI /2 shown above. If the integration is performed only over the surface St of an finite element (part of S), we can obtain the local heating in this element.
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Figure 4.14a. Field distribution at 0 Hz. Figure 4.14b. Field distribution at 50 Hz. The field penetrates freely through the conductor.
Figure 4.14c. Field distribution at 100 Hz.
Figure 4.14d. Field distribution at 250 Hz.
B
B P
Figure 4.14e. Field distribution at 500 Hz.
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Figure 4.15. A magnetic structure with a moving piece. Eddy currents are induced in the moving piece due to velocity.
4.3.4. Structures with Moving Parts Consider the case of a magnetic brake, where a piece P, placed in a magnetic field, moves at a velocity V, as shown in Figure 4.15. This piece is uniform in shape and only the part under the magnetic field influence is shown. Eddy currents depend on the velocity of the piece P through the expression E = V x B . As V and B are in the plane of the figure, E is perpendicular to this plane:
[
>
^A
+ vxB
dt
\
)
where the derivative of A can be described either by complex formulation (if it is linear and the excitation is sinusoidal) or through time discretization. If the excitation current is constant, we have a static situation (even if velocity V exists), because, for constant V, the potential A is also constant in the structure. Using the components of V:
and B:
.dA
. X
J
V
**
'
qy
.dA «l yx
obc
the product v x B, for this two-dimensional case (the product is perpendicular to the plane of the figure, in the k direction), is
xB=- —
dx**
-—
dyVy
Eq. (4.43) becomes
div(ygradA)-a\—vx +—v \ + Js=Q ^ dx dy ^ J
(4.58)
Applying Galerkin's method to the first and the last terms, we obtain the expressions in Eqs. (4.46) and (4.50), respectively. The second term containing the speed vx is calculated below. We have
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- f Nl<j — vxdxdy Ox
i
which can be written in the following form
- ovx [ Nf — Adxdy */ dx Using the local element and the expression in Eq. (4.40), which defines
dN/dx, I-M - v u v
-av.
-ov,
u
det J
4 dudv A2 vql
vq2
^
which, after integration, becomes
(4.59)
Analogously, the term with velocity component v v is
(4.60)
The sum of Eqs. (4.59) and (4.60) provides the velocity terms, and these terms must be added to the global matrix SS . For equivalent expressions for second-order elements, see Eq. (3.83). One important remark is that, contrary to other formulations we presented previously, the matrices in Eqs. (4.59) and (4.60) are not
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symmetric. Generally, computation programs take into account the symmetry of the global matrix SS, storing only half of this matrix in memory. However, in this case the whole matrix must be stored. In practice, analysis with velocity terms normally needs a separate solver. If the excitation current is time-dependent, the potential A is timedependent as well. In the term for Je of the basic equation we now have to add " — a&A/dt", which, depending on the formulation chosen, is either Eq. (4.48) or Eq. (4.57). 4.4. Axi-Symmetric Applications In electrical engineering, there are many important geometries with axial (or rotational) symmetry. This includes many applications of solenoids (relays, valves). The formulations presented above were all based on rectangular coordinates. The use of two-dimensional description was based on the assumption that the structure has no geometric variation in the direction perpendicular to the plane of study and that the fields are constant in this direction. While this treatment is a good approximation to many problems, in the case of solenoids and other circular geometries, it is not a convenient approach. In fact, these problems are three-dimensional in nature. However, if the geometry possesses an axial (or rotational) symmetry, the problem can be treated as two-dimensional, using the methods described in the previous sections, provided some simple modifications in the numerical procedures are made. These problems are called "axi-symmetric", and are studied in the cross-sectional plane P, shown in Figure 4.16a, using cylindrical coordinates. Figure 4.16a shows a half solenoid for visualization purposes. Figure 4.16b indicates the cross-sectional plane and the axis of symmetry used to define the problem. The same cross-sectional plane, in Cartesian coordinates, would describe an infinitely long, conducting bar with current /, as shown in Figure 4.17, which is very different from the solenoid problem in Figure 4.16.
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Figure 4.16a. Analysis of an axi-symmetric problem. A short solenoid is shown in cross-section.
Figure 4.16b. The solution domain is an axial cross-section through the axi-symmetric geometry.
Figure 4.17. Equivalent cross-section in Cartesian coordinates. Unlike the axisymmetric geometry, this implies an infinite, straight current layer.
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For all the static cases, except to the vector potential one (which will be presented shortly), in the axi-symmetry we should consider the domain Q with its three-dimensional geometry. For example, the matricial contributions of equation (3.17) are calculated from
f n=\,N
*W
- p(/>n ^Q = 0 ftfl
Supposing that the elements have small surface compared to the whole domain, as shown in figure 4.18
Figure 4.18. Small triangle for axi-symmetric scheme.
and that dQ. = rdcpdrdz , it is possible to consider that dQ = 2nr0drdz , where r0 is the baricenter of the triangle, or r0 — (r\ +r2 + 7*3 )/3 where r\ , TI , and r^ are the radii of the nodes. Therefore, the equation above becomes
lTor0 f [sgradV.grad
Thus, for the axi-symmetric case, all the contributions are calculated exactly as cartesian cases but multiplied by 2;zr0 . However, for the vector potential case, the approach is different, as we will see below.
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4.4.1 . The Axi-Symmetric Formulation for Vector Potential First, we establish the equation rotfl = 3 1 in cylindrical coordinates. Jt
(4.61)
In this system B = rot A has two components
B,r = -2L ^
and
oz
B, = z «•» ^
r or
where A has only a component in the (p direction. This is also the direction of J? . With these conditions, Eq. (4.61) becomes
5z x
' /
^3
/3 >\
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.
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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
(4.71)
q^3 +
e. Assemble S(3,3) in the global matrix SS(^, K) , and the source terms in the matrix Q . f. Impose the Dirichlet boundary conditions. g. Solve the system SS T = Q . h. Compare *P with *¥ of the previous iteration. If the error criterion indicates that the convergence has not yet been obtained, go to step b. Repeat steps b through h until the solution converges. Steps b through g constitute the normal, or linear solution. It is obvious that if the material property (permeability in this case) is constant, the solution converges in one iteration. With nonlinear material properties, the solution for ^P is convergent after a number of iterations. Generally, this method is convergent, but, sometimes, the iterative procedure is slow. However, it is very simple and easy to implement, since it is based on linearization of the nonlinear dependency. 4.6.2. The Newton-Raphson Method This method is an extension of the Newton method, adapted to matrix systems of equations. Compared to the successive approximation method, it converges faster because the solution is found by tangent functions rather than by a simple linearization of the B(H) curve. The basis of this method is a Taylor series, truncated after its first term. For a column matrix P, which is a function of a vector X, the truncated Taylor series is, for X close to \m
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AP
Ax
(x~xm)
where [AP/Ax] is the Jacobian of P at xm. If, for example, P and X have two components
"AP]_ _AxJ
^Pl
a*n
dP2
dx2 dP2 dx2_
_dXl
If xm is not too far from the solution of P, we find that X m + j in the relation
Ax m is a better approximation to the solution than Xw. The Newton-Raphson method consists of solving the matrix system in the following form AP m
Ax m
(4.72)
The unknown vector is now Ax m + i, and X w + j is easily obtained from (4.73) after the matrix system in Eq. (4.72) is solved. In our case, using the scalar potential, the column matrix P is the matrix product SS ¥ where X has been replaced by *F . This matrix product, called here P j ( k ) , is obtained by assembling the terms below, obtained from Eq. (4.71) above. 3
..
The general term of the Jacobian [AP/Ax] corresponds to the derivation of Eq. (4.74) related to the mesh node potentials. Performing this derivation in Eq. (4.74) we get
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a
..
W c/T rt
•^-L/
1
2) -
z Ay
a.. (7/7
atr2
2 u:r
3
=
(4.75)
2
2
Note that // depends on H and // depends on *¥n. We need to 2 / calculate d// /o^n , which can be written in the form (see section 3.2.4)
dH2
2 D<
(4.76)
1=1
With the notation S(n,k)-(]Ll/2D\qnq^
+rnr^), Eq. (4.76) can be
written as
dH2 _ 2 2D ^
~"^
l
'7
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
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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
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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 .
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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
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(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;
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• 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
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=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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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=<5 s * dt dt which cancels with the third term in the left-hand side; Eq. (5.19) then is reduced to:
l_dA i dx
dx
l_dA
(5.20)
dy M- &
Supposing that the total surface of the coil is given by Sf
]_dA dx i dx
= Ncos , we get:
\_dA_ dy \i dy
(5.21)
The voltage U, at the terminal of the winding can be written as: Uf = NCOU, = N,R,I, + NCOR, {a ^ds
=
(a ^-ds = N^ —lf fit rr Sc ° a s ** dt (7
c
c
aso
f
+-NCO s^- = c
ni
„ dA s
f
dt
or
Uf=Nco—If+Nc—\ a s Sf
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—<
(5.22)
The first term in Eq. (5.22) is the voltage drop over the coil resistance
Rf.
The second term is the voltage induced in the coil. We will simplify Eq. (5.22) by introducing Rf
— ds,
D T + U s =RI
(5.23)
dt
f
Finally, in the thin conductors region, the equations become:
ldA
\dA
(5.24)
r-J/ = o
dy
dx
Uf =
(5.25) f
5.2.4. Equations for the Whole Domain According to the above presented, the set of equations for an electromagnetic device presenting magnetic materials, permanent magnets, thick and thin conductors, is
]_dA_ dx i dx
\dA dy
-a
dA
N, • f H
dt
* t// ^
=
a i #0y
cbc u,
a i""C
dy u,
(5.26)
+ Rf 1 a — as
'#, a/
J
J dt
J
A/"c ^ dA ^ Sf ^sf dt
(5.27) (5.28)
In equation (5.28) we have introduced an additional term Ldl f jdt which allows the end-winding inductances to be taken into account.
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5.2.5 The Finite Element Method After applying the Galerkin method to equations (5.26), (5.27) and (5.28) (where the weighting functions N are the geometric transformation functions, as presented in Chapter 2), a set of matrix equations is obtained, as follows:
SSA + N — A - P / /J - - P - U , =D
(5.29)
Q1 — A + R'I, = U,
(5.30)
Q — A + R I] . + L — l f} =UJf dt dt
(5.31)
dt
Vectors A , Iy , U^ , I, , U, are related to: • The magnetic vector potential at the nodes; • The currents and voltages of thin conductors windings; • The currents and voltages of the thick conductors. Matrix R is the d.c. resistance of the thin conductors windings and L is the matrix of the end-windings inductances. R' is a diagonal matrix containing the d.c. resistances of thick conductors. Matrices SS , N , P,
P',
D, Q and Q' are obtained by
assembling elemental matrices, whose terms are
a. SS(k,j)= I gradN[v gradN j ds JSj
(5.32)
where Sf indicates the mesh element /surface and V = 1/fl . This term was already deduced and given in equation (4.45b).
b N(kJ)=sGNkNjds This term was also presented in section 4.3.2.
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(5.33)
(5.34) S
' JS if node k belongs to the region of winding j , or
P(£,y) = 0 if node k
(5.35)
is elsewhere. In (5.34), the coefficient Nco; ISfj
when
multiplied by / ,• gives J ; , the current density in winding ;. With this, one can verify that (5.34) was already given in section 4.3.2 d.
-Nds
(5.36)
if node k is in the region of winding j, or equal to zero elsewhere.
e. D(k) = I — \gradNk x B0 1 • Vds ;) = [ij i oCOK N;ds y
(5.37)
(5.38)
if node k belongs to the region of winding j, or otherwise equal to zero. S.Q'(k,j)=^RtkckNjds
(5.39)
if node k belongs to the thick conductor k, or equal to zero elsewhere. 5.3. Equations for Different Conductor Configurations Different combinations of thin and thick conductors windings can be found in electrical devices. We will first consider thick conductors, supposing that the thin conductor windings are independently fed. Afterwards we will investigate the way that thin conductors windings are connected. Finally, the equations for devices fed by static converters will be presented.
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5.3.1. Thick Conductors Connections
It is virtually impossible to describe all types of possible series/parallel connections. Nevertheless, the most common ones found in practical applications will be described. There are two types: series and parallel connections. 5.3.1 a. Series Connection
Figure 5.3 shows four thick conductors which could, for instance, represent a coil. Using two-dimensional formulation, Rext and Lext could be, respectively, a resistance and an inductance related to the third dimension, as for example, the end winding parameters. Utj is an external voltage feeding source. If the coil is short-circuited, obviously U^ = 0.
Figure 5.3. Series connection of thick conductors.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Ut\ » •••' k/4 an " Ai> •••> A 4 represent, respectively, the voltages represent, respectively, the voltages and currents of the four conductors. According to Figure 5.3 and adopted conventions, we have:
uff =tf,i-tf,2+tf, 3 -tf,4+*-Ai+A-:ki dt
(5-40)
Itl=-I,2
(5.41)
/,2=-'/3
(5-42)
/, 3 =-/,4
(5-43)
In the regions of thick conductors, equations (5.29) to (5.31) can be written as:
SSA + N—A-P'U, =0 dt '
(5.44)
Q'
(5.45)
dt
C 2 U,+C 3 I,=E,
(5.46)
The unknowns are then the vector potential A on the mesh nodes, the voltages U, and currents I, on the thick conductors. Also, we have for equations (5.45) and (5.46) the following new matrices generalized for a number n of conductors:
0 0 - 0
" - 1
0 - 1 0
.»
0 0 - 1
••• 0
0
0
0
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.»
0
-
(5.47)
~Rtl
0
0
•••
0 "
0
R t2
0
•••
0
0
0
Rt3
•••
0
0
0
0
R'=
(5.48)
Rtn (5.49)
0 = null vector
" 1 - 1 1 C2 =
• • •- 1 "
0
0
0
-
0
0
0
0
•••
0
0 0 0
(5.50)
• • •0 _
R ext + L ext — dt
0
0
•••
0
1
1
0
•••
0
0
1
1
•••
0
0
0
0
•-•
1
C3 =
(5.51)
0
E,=
(5.52)
0
0
5.3.1 b. Parallel Connection Figure 5.4 shows four parallel connected thick conductors. /„ ... //4 and Un
... Ut4
are, respectively, the currents and voltages on the
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conductors. Using 2D formulation, r and t' are, respectively, the resistance and inductance associated to the third direction, as end-winding parameters.
Ut4 r
R*x
f
r
f
r
f
Figure 5.4. Parallel connection of thick conductors.
From the scheme of Figure 5.4 one can write:
d
n + 7/-J A ) + L P Yext f — /2 t3 + I fHI dt (5.53)
d ,
(5.54)
n
= 2r(/r2
d_ dt r \ *»d I./4A)+2t'— dt
(5.55)
+ 7/3 + 7I4)+ C//
(5.56)
For the region of parallel connected thick conductors, equations (5.29) to (5.31) become:
SSA + N—A-P'U,t =0 dt
(5.57)
Q'—A + CU.+R'L =0 dt
(5.58)
C 4 U , + C 5 I , =E,
(5.59)
The terms of equations above were defined in the previous section, except:
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c ^4
" 1 0 0 -1 1 0 0 -1 1
•••0" ••• 0 ••• 0
000 ext ^
II \J
r = ^
(5.60)
••• 1 ext ^
ext
sV £*t
. -I-1--J * ^ iT ^v
0
0
0
0
d — ^
ext ^
ex,
d / I/* t / 0* ^A "T~ jvv — J/ J 2r + 2t'— dt
0
* dt ,d_ dt ,d_ dt ,d_ dt . (5.61)
The unknowns are the same as above. Very often, in electrical machines, we have the conductors parallelconnected and short-circuited as in the case of squirrel-cage induction motors, shown in figure 5.5.
Figure 5.5. Parallel connected and short-circuited conductors.
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In solving such problems, normally (anti) periodicity is employed (see section 4.7). In this way, only a partial set of conductors must be simulated. As an example, such a situation is given in Figure 5.6.
Figure 5.6. Parallel-connected and short-circuited conductors (reduced study domain).
From scheme of Figure 5.6, we have:
•-/
(5.62)
dt '
2rI2+2t'-I2-Ut2+Utl=Q dt
"
(5.63)
r--/ 3 -(/, 3 +t/, 2 =0 dt
(5.64)
''—-I 4 -U 4 +U 3 = 0 dt ' '
(5.65)
where 7j to 74 are the currents in the components r and C belonging to the connection circuit of thick conductors, as shown in Figure 5.6. The factor / , appearing in (5.62) is equal to 1 or -1 for, respectively, periodic and anti-periodic domains.
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It is possible to write (5.62) to (5.65) in matrix form, considering n conductors, as:
C 6 I + C 7 U, =0
(5.66)
where:
f
d
0
0
0
0
d 2 r + 2^'— dt
0
0
0
0
0
0
~di
* d 2r + 2^'— ••• dt
0
.» 2
(5.67)
0 .
,d ~di_
"V h (5.68)
i
_ n_
"-1 0 0 1 -1 0 0 1 -1 = 0
0
noticing that U^ = [Uti
0
... /" ... 0 ••• 0
(5.69)
••• -1_ Ut2
f//3
••• Utn\
, the vector of the
voltages over thick conductors was previously defined (see section 5.2.5). Observing Figure 5.6, currents /, to /4 are related to currents Itl to /M by the next matrix:
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I, =
- 1 1 0 - 1 0 0 / 0
112 1,3
0 0 1 0 1 1 0 -1
h h
(5.70)
L'4J
where superscript T denotes a transposed matrix. Multiplying (5.66) by C7 and considering that C6 is a diagonal matrix having identical terms, we obtain
C 6 C£l + C*C 7 U,=0
(5.71)
Using (5.70), we have
C6I,+C?C7U,=0
(5.72)
Finally, equations (5.29) to (5.31) in the region of this connection type and for thick conductors not externally fed, can be written as:
SSA + N—A-P'U,t =0
(5.73)
Q'— A + C,U, dt
(5.74)
dt
=0
(5.75) recalling that the unknowns are the vector potential on the nodes, the currents and the voltages on the conductors. One can notice, in the equations above, that there are no electrical external excitations. Those will be present in the thin conductor coils section described next. 5.3.2. Thin Conductors Connections Depending on the electromagnetic device, the thin conductor windings can be fed by independent voltage sources or by a set of polyphase sources.
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5.3.2a. Independent Voltage Sources Here, the vector \J f of equation (5.31) will be formed by voltages of different sources. Therefore, the matrix system describing this case comes from the equations (5.29)-(5.31), (5.44)-(5.46), (5.57)-(5.59) and (5.73)(5.75):
SSA + N — A - P IJf - P ' U , =D dt
(5.76)
Q'—A + QU, +R'I, =O dt
(5.77)
[C2 or C4 or Cy^U, +[C3 or C5 or C6]lt = [E, or O] (5.78) Q — A + Wf+L — If =VfJ J dt dt J
(5.79)
where C2 and C3 will be employed if the thick conductors are series T connected. If they are connected in parallel C4 and C5 or C y C y and C6 (this latter for cage configuration) will replace them. The right-hand side vector of (5.78) is O in the case of parallel connected and short-circuited thick conductors. In equations (5.76) to (5.79), the unknowns are the magnetic vector potential on the mesh nodes, the currents and voltages on the thick conductors and the currents on the thin conductor windings Iy. In electrical machines, the thin conductor windings are normally connected and it is possible to distinguish three main types of connections: star (with and without neutral) and polygon connections. 5.3.2b. Star Connection with Neutral In this case, the voltages to be inserted in the vector U /• in (5.79) are the phase to the neutral ones.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
5.3. 2c. Polygon Connection Here, the phase-to-phase voltages U a are applied. The use of them in vector U s allows the determination of phase currents / s directly. The line currents 1
can be calculated from:
-I
. J
(5.80)
5.3.2d. Star Connection without Neutral Wire Supposing that there are n windings star connected without neutral, there will be n — 1 linearly independent currents. Moreover, the feeding voltages are, again, the voltages between phases U a . We will call If
the
vector of the (n — l) linearly independent currents. The relationship between Iy and Iy is:
"1
0
0 0
1
0 0 1
0
••• ••• •••
i
0 0 0
*
*
(5.81)
; I f — rl 4
o ... i -1 -1 -1 ••• -1 0
0
On the other hand, the relationship between the phase to phase voltages vector U /£ and the phase voltage vector U /• is: '\ 0 0 1 0 0 ¥'Uf= 00 0
0
0 • • 0 • • 11 •••• • 0
•
0 0 0
• o r
-l" "0 0 0 • 1 0 0 •• 0 -1 u = 1 -1 • 0 -1 U , = - 1 1 00 ••••
• 1 -1.
/ -
1 1 1 •• o
1 11
\JjL=-KUjL
i. (5.82)
Replacing (5.81) and (5.82) in (5.76) - (5.79) we have a particular system of equations for this connection as:
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SSA + N—A-PFI*Jf -P'U, =D
(5.83)
Q'—A + QU, +R'I, =0 dt
(5.84)
dt
[c2 or C4 or C^CJU, + JC3 or €5 or C6]lt
(5.85)
= [E, or O] F r Q—A + F r RFl}+F r LF—l}=-KU y L
(5.86)
We point out that the equation system from (5.83) to (5.86) is reduced in one equation compared with the system described by equation (5.76) to (5.79), since the «th current is not calculated. This last one can be determined by:
k=l 5.4. Connections Between Electromagnetic External Feeding Circuits
Devices and
Nowadays, with the evolution of semi-conductors and highperformance electronic components, electrical machines are increasingly driven by static converters. One way to simulate the coupling between electronic circuits and devices consists in determining the fluxes and winding inductances for performing calculations based on lumped parameters representing the magnetic part of the set. However, this procedure lacks generality. One restriction is the non-linearity and, moreover, the impossibility of considering eddy currents established on conductive parts. We will present a methodology allowing one to perform the coupling and the simultaneous resolution of field equations and feeding circuits.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
It can be noticed that such a formulation does not require the a priori knowledge of the sequences of the static converter. It means that the method considers the feeding circuit topologic changes automatically when the electronic components (thyristors, diodes, etc...) change their conducting state. In order to simplify our notation, some simplifications will be considered in the equations already presented. 5.4.1. Reduced Equations of Electromagnetic Devices Admitting that the thick conductors are not externally fed (corresponding to the majority of practical situations) and that they are connected and short-circuited, we can write (5.76) to (5.79) as:
SSA-fN— A - P I , = D dt J
(5.88)
Q — A + R IJJ. + L — I , = U , dt dt
(5.89)
where the eddy current of the (non-laminated) conducting parts are represented in the second term of (5.88), as presented in section 4.3.2. The source terms of (5.88) and (5.89) are related to the permanent magnets and the voltages applied on the thin conductor windings; here these sources are supposed independent. 5.4.2. Feeding Circuit Equations and Connection to Field Equations The general equation of an electrical circuit to be connected to an electromagnetic device is written in a state form using the state variable approach:
— X = G1X + G 2 E C +G 3 I /
(5.90)
where: • X is the vector of capacitance voltages and inductance currents, which are the state variables;
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
• Ec
is the vector of voltages and currents feeding the electrical
circuit; • Iy, as already defined, is the vector of thin conductor winding currents. The matrices Gj,
G2 and 63 depend on the topology of the
electric circuit and a methodology to obtain them will be presented soon. The vector Uy of the voltage applied on the device windings terminals can be expressed by:
Uy = G4X + G 5 E C + G6If where 64,
(5.91)
G5 and G£ also depend on the circuit topology. The
coupling between the electromagnetic structure and the feeding circuit is made by equalizing (5.91) with (5.89). We can also write, using (5.90):
SSA + N — A - P I , = D
(5.92)
Q —A + l R - G e J l y + L — I / - G 4 X = G 5 E C
(5.93)
— X - G 1 X - G 3 I / - =G 2 E C dt
(5.94)
dt
J
The unknowns are now the vector potential on the mesh nodes A, the currents in the windings 1^ and the state variables of the feeding circuit X . The entry data are the permanent magnet sources given by D and current and/or voltage sources of the feeding circuits Ec. Now, we will describe a procedure to evaluate matrices G| to G4. 5.4.3. Calculation of Matrices G, to G6 The coupling between electromagnetic structure and state equations is performed through their common quantities. As shown in previous sections, they are the current in the device and the voltages applied to it.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
The static converter may have switches (thyristors, transistors, diodes, etc.) which change their conducting status according to the conception of the circuit. To implement a calculation procedure, it is suitable that the equations system be determined in an automatic manner, without knowing a priori the operation sequences of the converter. For didactical purposes we will start the presentation with a brief review of the graph theory. 5.4.3a. Circuit Topology Concepts
A linear graph is composed by a set of branches and a set of nodes. The branches are connected by the nodes. An electrical circuit can be associated to a linear graph, where circuit elements are replaced by branches. If the branches have an orientation, commonly taken from the electric current flux, the graph is called an oriented graph (Figure 5.7). Ci
R
C2
Figure 5.7. Electric circuit and its corresponding oriented graph.
The orientation of the branches is chosen accordingly with a receptor convention, as presented in Figure 5.8. 1
Figure 5.8. Example of a circuit element orientation.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
From the oriented graph, it is possible to define several trees, which are considered as oriented sub-graphs, containing all the initial graph nodes and a number of branches sufficient to connect all the nodes, without forming a closed loop. Figure 5.9 presents an example of a tree associated to the electric circuit presented in Figure 5.7. The branches that belong to a tree are called tree branches and those which do not belong to a tree are called links. For example, for the graph given in Figure 5.7 branches 1, 2 and 3 are tree branches and branches 4 and 5 are links.
Figure 5.9. A tree associated to the electric circuit.
a. The Fundamental Cutset Matrix A cutset is a set of branches of a connected graph in the way that the removal of the set of branches divides the initial graph into two nonconnected sub-graphs. The restoration of these two non-connected subgraphs results in a connected graph again (Figure 5.10).
Figure 5.10. Cutsets of the graph given in Figure 5.7.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
A. fundamental cutset cuts only a single tree branch. The fundamental cutset orientation is randomly chosen to match the orientation of the tree branches defining it. The number of fundamental cutsets, ncf, is equal to the number of nodes (n) minus one \ncf = n — Ij. In Figure 5.11 we have the fundamental cutsets for Figure 5.9.
1
m
Figure 5.11. Fundamental cuts I, II and III.
The fundamental cutset matrix Kc describes the presence of tree branches in a fundamental cut and their orientation related to this cut. The lines of Kc correspond to the cutsets and the columns to the tree branches. The terms of the fundamental cutset matrix are defined as:
{
1 -1 0
if the branch j belongs to the fundamental cutset /' with the same direction, if the branch j belongs to the fundamental cutset / with opposite direction, if the branch j does not belong to the fundamental cutset ;
where: the dimension of matrix Kc is Dim Kc = (n - l)x a n — l: the number of fundamental cutsets; a: the number of branches. The matrix Kc can be partitioned in two sub-matrices considering first the tree branches and then the links, as
Kc2]
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(5.95)
The dimensions of Kc, and Kc2 are: DimKq
=(n-l)x(n-l)
Dim Kc2 = (n - l)x (a - n +1) For the case of Figure 5.9, the fundamental cutset matrix is given by: 1
2
3
4
I
"1
0
II
0
1
0 0
_J
1
III 0
0
1 -1
0
1
5
0"
According to Kirchhoff's current law, the currents sum on a node is zero. A generalization of this law establishes that the sum of all the currents in a cutset is zero. Therefore
Kc2]
= [0] l
m (5.96)
By definition, KCj is the unity matrix, so the tree branch currents can be expressed as linear combinations of link currents: [iJ = -Kc2[iJ
(5.97)
where: i^ J : matrix of currents in the tree branches; im J : matrix of currents in the links. b. Fundamental Loop Matrix A loop is defined as follow: starting at an arbitrarily selected node, a closed path is traced through the graph returning to the original node without passing through any intermediate node more than once. For
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
example, in Figure 5.12, branches (1, 2, 3, 4) , (2, 5) and (1, 3, 4, 5) form a loop. A fundamental loop is a loop that contains one and only one link. Example: (2, 5) and (1, 2, 3, 4). The orientation for a fundamental loop is arbitrarily chosen to coincide with the orientation of the corresponding link. The number of fundamental loops, riy, is equal to the number of the branches minus the number of nodes increased by one {ny =a — n + lj. The number of fundamental loops is equal to the number of the graph links. Figure 5.12 shows the fundamental loops corresponding to the graph of Figure 5.7.
Figure 5.12. Fundamental loops IV e V.
The fundamental loop matrix B/ is defined as follows: its lines correspond to the fundamental loops and the columns are related to the branches of the graph. The terms of this matrix are equal to:
{
1
if the branch j belongs to the fundamental loop / with the same direction,
-1
if the branch j belongs to the fundamental loop / with the opposite direction,
0
if the branch j does not belong to the fundamental loop.
For the dimension, we have:
Dim B/ = (a - n + l)x a a — n + l: Number of fundamental loops (number of links). The matrix B/ can be divided in two sub-matrices considering, firstly, the tree branches and then the links: /! B/ 2 ]
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(5.98)
where the dimensions of B/( e B/2 are: Dim B/j = (a - n + l)x (n -1) Dim B/2 = (a - n + l)x (a - n +1) For the case of the circuit of Figure 5.7, the fundamental loops matrix is given by: 1
IV V
2 1
0
3 1
-
1
4
1 0
5
1
0
0
1
The Kirchhoff's voltage law establishes that the sum of voltages around any closed loop is zero. Then:
[B/i B/ 2 ]
= [0] L v mj
(5.99) By definition, the matrix B/2 is unitary. Consequently, the link voltages can be expressed as linear combinations of tree branches voltages: [vJ = -B/,[vJ
(5.100)
where: \m : matrix of the voltage across the links; Vb : Matrix of the voltage across the tree branches. c. Relationship between matrices Kc2 and B/, Let us recall these two matrices obtained from the network presented in Figure 5.7:
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
1
2 3 IV "-1 1 1 " B/i = V 0 -1 0 4 5 1 0" i 1 Kco = III[|_-1 0 One can easily observe that B/,=-Kc 2 r
(5.101)
The generation of circuit equations in an automatic manner depends on the determination of the matrices Kc2 or B/,. The matrix Kc2 can be determined from the incidence matrix using the Welsch's algorithm as will be presented shortly. d. Incidence Matrix For a graph with n nodes and a branches, the terms of the incidence matrix Az are defined as:
1 -1
if the branch j leaves the node /; if the branch j enters the node /;
0
if the node / does not belong to branch j.
where:
Dim A/ = n x a For example, for the graph of Figure 5.7, the incidence matrix is given by
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
1 (nodes) 1 "-1 2 1 3 0 4 0
2 0 0 1 -1
3 -1 0 0 1
4
5
0 1 -1 0
0" 0 1 _1
(branches)
Notice that every column of A/ has exactly two non-zero elements ( + 1 and -1). This is a general property for any linear graph since each branch is incident in two nodes. The number of non-zero terms in a line indicates, for each node, the number of branches leaving or entering the node. Obviously, each line should have, at least, two terms different from zero, or, otherwise, the branch would have at least one disconnected terminal. e. Welsch's algorithm The Welsch's algorithm is used to obtain matrix Kc2 automatically (see Eqs. (5.95)-(5.97)) relating tree branches currents and loop currents. Now, let us describe this algorithm. For a given ixj incidence matrix A/, starting with the first column, scan in all columns the first non-zero element aiy in row i, which was not considered yet; • if such a term exists, the scanned column j is associated to a tree branch. Then, reduce all the other non-zero elements of this column to zero by adding or subtracting the lines corresponding to these non-zero elements with line i one. In that way, aiy becomes the unique non-zero element in column j; • if that term is not encountered, the scanned column;' is associated to a link. In this case no operation is performed. This procedure is repeated until the last line of matrix A/ is set to zero. A new matrix A/' is obtained by neglecting this last line and by reorganizing the matrix A/', considering first the columns corresponding to tree branches and then the columns corresponding to the links. Matrix A*'1 is then partitioned as follows:
Ai'^Ai1! A/' 2 ]
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(5.102)
where: Dim Ai'i = (n - l)x (n -1) (tree branches); Dim A/' 2 = (a - n + l)x (n -1) (links). Finally, the matrix Kc2 is obtained by: Kc2 = A/',rxA/'2
(5.103)
To illustrate this algorithm, let us consider the circuit of Figure 5.7. Its incidence matrix is: 1 (nodes)
A/ =
2
3
4
5
(branches)
1 "-1 0" 0 -1 0 1 0 2 1 0 0 1 1 0 -1 3 0 1 0 -1 4 0 -1
• step 1: the first non zero element at column ;=1 is located at line i=l; branch 1 is then a tree branch. The other non null terms at column ;=1 are set as zero by the sum: line 2 = line 2 + line 1
1 (nodes)
2
3
4
5 (branches)
"_1
A/'=
0 -1 0 0" 1 0 0 0 _j 1 0 -1 1 0 0 _l 1 0 -1
• step 2: in Az'1, the first non-zero term of column ;=2 is located at line 1=3 and this line is considered for the first time; then branch 2 is a tree branch. The non-zero terms in column j=2 are set as zero by the following operation: line 4 = line 4 + line 3
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
1 2 3 4 5 (branches) - 1 0 - 1 0 0 0 0 - 1 1 0 0 1 0 - 1 1 0 0 1 - 1 0
(nodes) 1
Ai'=
• step 3: the first non-zero term of column j=3 is at line i=l, but this line was already considered (step 1). Then the next non-zero term in column j—3 will be considered (term a/23 located at line i=2); so branch 3 is a tree branch. The non-zero terms at column;=3 are set as zero with: line 1 = line 1 - line 2 line 4 = line 4 + line 2
1 "-1
2 0 0 0 0 1 0 0
(nodes)
A/'=
4 5 ((branches) 0 -1 0 " -1 1 0 0 -1 1 0 0 0 3
The last line is then null; the first three columns represent the tree branches and the last two columns the links. Matrix A? is now written, neglecting the last null line
- 1 0 0 -1 0" 1 Ai"= 0 0 - 1 0 1 0 -1 Ai12
A/'i
Finally, the matrix Kc2 is obtained by using the Eq. (5.103), resulting in
Kc=A/xAi
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
I
1 = -1
and the matrix B/, is calculated using Kc2 according to the Eq. (5.101)
<1 =
-1 0
Notice that matrices Kc2
1 -1 and B/,, obtained previously from the
topological analysis of the graph are identical to those calculated from the incidence matrix and using the Welsch's algorithm. 5.4.3b. Determination of Matrices G, to G6 The construction of matrices Gj to G6 depends only on the electric circuit topology. In order to determine these matrices automatically, the state variable approach is used because it is well adapted to many numerical methods of analysis. Moreover, the state variable method can be extended easily to the study of non-linear networks. So the state equations of the electric circuit feeding an electromagnetic device can be expressed as follows:
d_ dt
+ G2
(5.104)
LVJ (5.105)
where the state variables are the capacitor tree branch voltages (v^ c ) and the inductor link currents (i m /); \e and i ,- correspond to the voltage and current sources; im/- is the current in the electromagnetic structure windings. The first letter of the subscript of the state variable (b or m } is related to a tree branch (b ) quantity or a link (m ) quantity, and the second one (c, I, j , etc.) corresponds to a circuit element. In the state variable approach, it is convenient to find a tree in which the tree branches are, in priority, independent voltage sources and capacitors, and the links are, in priority, independent current sources and inductors. Thus, the procedure begins with the selection of a tree with the
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
following order of preference for tree branches: independent voltage sources (e), capacitors (c), resistors (r), inductors (I), electromagnetic device windings (i) and independent current sources (j). In other words, it is preferable to have the maximal number of voltage sources and capacitors in the tree. In the same way, it is preferable to have the minimal number of inductors and current sources in the tree. This procedure guarantees the unicity of the chosen tree. Such a tree is called a normal tree. Using the normal tree, Eq. (5.100) can be detailed as: ^ me
8j
S2
83
S4
S5
v
v
S6
S7
S8
S9
S10
v
^
v
z
mr
v v
w/
- - Sll S
m/
,VJ .
i6
C? _ 3 ^9 i1
and Eq. (5.97) as: ~ijcr 1 ^e
1
^c • br
82
j
S
12 817
S
C! ^99 ^«.£f
C ^9^ Xrf*J
CT
LJ O A
Sy
r c [j T
c1%rO
bl
oj j
cj -/
.
c^ ij c
c^ Sin
l
. l
.bi _
J
J
O
1U
13 S18
S
14 S15 S19 S20
C? ^94. A-T^
v
_^
-{l
-
v^11 11
&T c
Sj2
Sjy
^1
mc
$22
imr
cJjr1 1 c rJj 1O c r& O^ ilml 1J 1O ^J j o •*• C "^ C
c*7" c^ *^ 1 ^ i5 OA 1J
£\)
(5.106)
^
C! ^ 9^ ^«/ __
CT
1O
v
(5.107)
mi • c^ >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
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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).
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
/(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.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
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 —
v r\
#C2
#C2
r ~o
/te2
z>
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•<-'2
RC^-*2_J
"
1 1
Vcl
y L.
^/ =[io]
d L c2_
+ cl
+[o]M+[o]pw]
[1
RC
C
\E\j \ L-^J
1RC2]
l L'wJ \i 1
I
J
(5.147) (5.148)
Comparing, respectively, (5.147) and (5.148) to (5.138) and (5.139), one can notice that
1 G
i =
RCi 1 RC2
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1 RCi 1 RC2
G, =
Q o
G 4 = [ l 0] G 5 =[0] G6 = [0]
© Automatic determination of matrices Gj to Ggwith the proposed method According to Eq. (5.100), the relationship between the link voltages and the tree branches voltages is: (5.149) where the matrix B/j is obtained from the incidence matrix A,. using Welsch's algorithm. In our example, the link voltages are, respectively, the resistor R voltage (V^ ) and the winding IV voltage The tree branches voltages are, respectively, the feeding voltage E and the capacitors C\ ty c\) and C^ (^2) voltages. Therefore (5.149) can be also expressed as: _
VR
yw_
-i
i i
. 0 -1 Oj
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£ ~
(5.150) V
c2_
Comparing (5.150) to (5.108), we can deduce:
s 6 =-i S 7 = [ l 1] S16=0 S n = [ - l 0] All other submatrices of Eq. (5.108) are null. Analyzing the normal tree obtained from the circuit we can deduce that: R£ = JO], since there are no tree branch resistors in this example.
Q 0
0 C2
Cm = [O], since there are no link capacitors in the circuit. Matrices L£, Lw
and Mw are zero because there are no lumped
inductances in the circuit. Finally, from Eqs. (5.128), (5.129), (5.130), (5.135), (5.136) and (5.137), we obtain:
G, =
f~<
_
1
1
I RC2
1 RC2
RC,
1 RC,
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G, =
c
0
o]
As expected, the matrices above are equal to those obtained by means of Kirchoff's laws application. 5.4.3d. Taking Into Account Electronic Switches in the Feeding Circuit For a circuit without switches, the matrices Gl to G6 stay constant during the whole simulation. However, when the circuit has switches (as for static converters), these matrices should be calculated for all operation sequences of the electronic converter. In fact, each sequence of operation corresponds to an equivalent circuit and, as consequence, variations of matrices G, to G6 occur. The previously presented method to determine matrices Gt to G6 remains valid, since the switches can be modeled by low valued resistances (0.1 Ohm) in its conducting operation and by high valued resistances llx 10 Ohm] in its open state. These switches can be diodes, thyristors and transistors. The commutation mechanism of these different electronic switches can be summarized as follows: • The diode conducts when the voltage at its terminals becomes positive and stops conducting when the current in it is null. • The thyristor conducts when it receives a trigger pulse and when the voltage at its terminals is positive. It stops conducting when its current becomes zero.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
• The transistor can be controlled for either turn on or turn off mode. Thus, the switching mechanism can be divided into two classes: the spontaneous commutation and the controlled commutation. The switching period, as well as the switch turn on and turn off times are defined by the user. So the controlled commutation does not represent any problem for implementation. The spontaneous commutation of the switches occurs at the zero crossing of the voltage across their terminals. This instant has to be determined accurately because it may introduce numerical problems and consequently errors in the solution of the equations. So, to determine this instant, at any simulation time, the switches voltage signs are observed. If a switch voltage sign inversion is detected, the program returns to the previous simulation time. Then, from that point, the step length is reduced until the threshold value representing the zero crossing point is reached. 5.4.4. Discretization of the Time Derivative The solution of Eqs. (5.92), (5.93) and (5.94) must be performed by using a time step-by-step procedure. The discretization of the time derivatives can be done by the 9 — algorithm. This method allows us to describe a variable, for instance X, at the time (/ + A^) as: l _e)^-XO+edt dt
where X\t)
is the value of X
at time t,
A/ (5.151)
A/ is the time step and
9 < 9 < 1. With 9 = 9 we obtain, as a particular case, the Euler's scheme and for 9 = 9,5 we have the Crank-Nicholson algorithm. Notice that the names are related to the same general procedure, changing with the value of 9 and some authors use different values of 9 between 9 and 1 . Now, we apply this algorithm to Eq. (5.92). Let us first write it for times / and
SS(f)A(f)+N~A(/)-PIy(f) = D(f)
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(5.152)
SS(t + A/)A(/ + A/)+ N— A(/ + Af)- PI y/• (/ + A/) = D(r + A/) <#
(5.153) In the equations above, we suppose that the matrix SS evolves from a time step to the next one. It can happen by two different possibilities: the ferromagnetic non-linearity and/or the relative movement between the moving and the fixed parts of the device, as it will be examined in the next chapter. Also, due to the movement, the permanent magnet remanent induction x and y (or r and 9 ) components can change. On the other hand, the matrices N and P are constant, as can be seen in Eqs. (5.33) and (5.34). The following step in using the 0 — algorithm consists in multiplying (5.152) by (l -0 ) and (5.153) by 0
(5.154)
— (5.155) Then the equations above are added — dt
— dt
(5.156) Now, using (5.151), the second term of the left hand side of Eq. (5.156) can be written as:
N
dt
dt
=N
(5.157)
Replacing (5.157) in (5.156), leaving the / + A/ terms on the left-hand side and passing to the right-hand side those depending on t (as they are
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known from the previous step) as well as the source terms related to permanent magnets, we obtain: J_
At j
At (5.158) Applying the same procedure in (5.93) and (5.94), we have the following matrix system:
At
O
N
9[R-G 6 ]+—L O
1 (0-l)SS(0+-N
-6G-
-0G 4 —1-GGi At
(i-e)p
O
(1-9)64 o
— At
x(0
(5.159) where 1 is the unitary matrix and all the right-hand side terms are known.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
According to our experience, 6 values different from 1 do not produce accuracy improvements if the time step A/ is small (typical value is \-4 1x10 seconds or smaller). But the computational work can be performed in a much easier way if Euler's scheme (9 = 1) is applied, which can be seen below when comparing (5.159) with (5.160):
-P
—N At
-Gi
O
MN At
O
At
At O
O
O
A«)
O
I/O +
1 _J At
O
—1-«
x(?) (5.160)
If the simulation takes into account static converters, the time step is calculated in the following way: the voltages at the switches terminals are calculated at each time step; these values are multiplied by the corresponding previous step voltages; if the product is negative, a voltage inversion is detected. In this situation, a new time step is established by reducing the original A/. If necessary, this procedure is repeated until the voltage zero crossing is reached. The Gauss elimination method is used for solving the linear equation system, because it is not symmetric and also because strong terms variation occur due to the external circuit (conducting and non-conducting states of the switches). In fact, this method is very robust and reliable for such applications.
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5.5. Examples Here we will present some examples of the above formulations. Two types of simulations are shown: simulations where the voltage sources applied to the windings are known and simulations taking into account the external feeding circuits. The emphasis of the discussion below is on the physical meaning and typical applications of the presented formulations, rather than strictly quantitative results. 5.5.1. Simulations with Known Voltage Waveforms 5.5.la. A Didactical Example Figure 5.14 shows the study domain of a structure where the stator has a thin conductor winding disposed in four slots. Anti-periodic boundary conditions are imposed in the two radial lines delimiting the domain; Dirichlet boundary conditions (A=0) are imposed in the outer stator limit. In the two right stator slots the conductors are entering into the plan and in the two left slots the conductors are leaving the study domain. Two thick or massive aluminum made conductors are placed in the rotor and, firstly, we will consider them as short-circuited.
Figure 5.14. Electromagnetic device with massive (thick) aluminum bars.
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The stator winding is fed by a sinusoidal voltage. As the aluminum bars are connected and placed along the quadrature axis, the currents in the bars should have the same amplitude but opposite directions. For didactical purposes, notice that the direct axis is placed between the two exciting stator coils, where the major magnetic flux is established, while the quadrature one is displaced 90 electrical degrees of the direct axis. Obviously, the quadrature axis is coincident with the exciting coils. Figure 5.15 shows the induced current densities for the time step when Figure 5.14 was obtained. One can observe that, effectively, the magnitude of induced currents are the same, but they present opposite directions as expected. This result is similar to what we would obtain with classical 2D simulations, from which all the massive conductors are intrinsically supposed to be shortcircuited, as described in section (4.3.2).
Figure 5.15. Distribution of induced currents in the interconnected aluminum bars.
Now, the bars are supposed not connected, although the same voltage is applied. The resulting magnetic flux is shown in Figure 5.16, graphically very similar to Figure 5.14 but having different numerical results. For modeling it, we impose a large value for the interconnecting resistance r of Figure 5.4 ( I x l 0 6 0 / w i ) .
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Figure 5.16. Device with isolated aluminum bars.
In this case, the total bar currents (obtained by the integration of the current density over the conductor surface) are null. However, the current densities are not zero and there should exist current densities in opposite directions inside the bars. That can be seen in Figure 5.16 and in Figure 5.17 as well. One can see that an extreme (maximum or minimum) value of current density exists close to the radial airgap and also on the lateral sides of the bars. As above mentioned, such results can not be obtained with classical 2D formulations when the bars are supposed to be short-circuited.
Figure 5.17. Current densities induced in not-connected bars.
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5.5.1 b. Three-Phase Induction Motor A second example related to thick conductors is shown in Figure 5.18. We consider a four-pole induction motor with the corresponding flux plot. In this case, the simulation is performed under locked rotor condition when applying three-phase sinusoidal voltages at the stator terminals.
Figure 5.18. Magnetic flux distribution in a four-pole induction motor.
The rotor massive bars are here modeled considering the ring connection, as in Figures 5.5 and 5.6. Notice in Figure 5.18 that the largest part of the flux is placed at the periphery of the rotor because of strong induced currents established in the bars. 5.5.1 c. Massive Conductors in Series Connection
Figure 5.19 presents a transformer where the primary winding has thin wire conductors and it is voltage-fed. The secondary winding is shortcircuited and it is made of four thick conductors serial connected, as shown in the figure.
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Figure 5.19. Transformer with thin conductors in the primary winding and massive series conductors in the secondary winding.
In the simulation, the resistance Rexl of Figure 5.3 takes into account the "head" conductors resistances according to the third dimension and the voltage UtT is zero. Figure 5.20 shows the calculated current waveforms established in the two windings when voltage is applied on the primary. As can be seen, the secondary currents have the same magnitude but with different directions depending on the location, because both sides are seriesconnected. Again classical formulation (all the conductors would be implicitly short-circuited) will not provide correct results.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
0.000
0.002
0.004
0.006
0.008
0.010
Figure 5.20. Established currents in the transformer of Figure 5.19. 5.5.2. Modeling of a Static Converter-Fed Magnetic Device
An example of the combined field and circuit equations simulation (Eq. (5.160)) is the fly-back converter presented in Figure 5.21. The input supply d.c. voltage Ein is 100 Volts , the output voltage is 24 Volts and the switching frequency is 20 kHz. il
;„!
Q1?l
t
iCs
JUUL -VDS
Figure 5.21. Fly-back converter with ferrite transformer. / : primary winding current. 12: secondary winding current.
VDS : transistor voltage. Ds: snubber diode. Cs: snubber capacitor (lO x 10" Farads).
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Rs: snubber resistor (270 Ohms). D0: output rectifier diode. C0: output filter capacitor (lOQQmicro Farads). R0: load resistor (9.5 Ohm). The transformer shown in Figure 5.21 was constructed using an axisymmetric ferrite pot core. Equations (5.159) and (5.160) are still valid for axisymmetric representations of electromagnetic devices by changing the matrix terms SS, N, P and Q . The primary and the secondary windings of the transformer used in the converter have respectively 16 and 8 turns. Two types of winding arrangements were tested as shown in Figure 5.22.
Figure 5.22. Different arrangements of the transformer windings: vertical and horizontal splits.
Arrangement A consists of primary and secondary windings superposed on the pot core. In arrangement B, the primary and secondary windings are split up into two separate groups. The simulation and experimental results are shown in Figures 5.23 and 5.24. It can be noticed that the leakage flux (and consequently the leakage inductance) in arrangement B is larger than the corresponding one of arrangement A . This specific point would be very difficult to be quantified by analytical approaches. This leakage inductance increasing
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leads to a higher value of transistor peak voltage in arrangement B (Figure 5.24b) compared to structure A (Figure 5.23b).
Voltage VDS (100/div)
^Current \i (2A/div)
Current I, (2A/div)
J
(
I Time (20ns/div)
Figure 5.23a. Simulation result for winding arrangement A.
Voltage VDS (lOOV/div)
\
(Current I2
2A/div
Current I, 2A/div Time (10ns/div)
Figure 5.23b. Experimental result for winding arrangement A.
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Time (20ns/div)
Figure 5.24a. Simulation result for winding arrangement 8.
Voltage VDS (lOOV/div)
V ^Current I2 2A/div
Current I, 2A/div Time(10ns/div)
Figure 5.24b. Experimental result for winding arrangement B.
For the above examples the movement was not taken into account, and this will be considered in the next chapter.
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6 Movement Modelingfor Electrical Machines 6.1. Introduction Assuming that the device under analysis has mobile parts, it is necessary to take into account the movement. This can be performed by using the term vxB where v is the velocity and B the magnetic induction, as seen before in section 4.3.4. However, this method is suitable only when the mobile part is invariable along the movement direction. If it is the case, with the spatial discretization of the field equations, the matrices created by the finite element method are non symmetrical (see Eq. 4.59 and 4.60) and the use of special techniques as "up-winding" can be necessary to avoid numerical problems. As mentioned above, this method is useful only for particular structures whose the mobile part has simple geometry, as for example, electromagnetic breaks. More general methods must consider the movement in the spatialtemporal discretization of field equations. For achieving it, two referential systems are used: the first one related to the fixed part and the second one to the mobile part. The field equation (5.10) is applied to both and the relative movement is taken into account in the airgap, an ideal place to do it since it is non-ferromagnetic, non-conductive and without sources. This technique is well adapted to complex structures as electrical machines, since, as mentioned above, there are no restrictions to the mobile parts. We can classify these methods in two categories: with discretized airgaps or non-discretized ones. 6.1.1. Methods with Non-Discretized Airgaps A first approach consists in modeling the airgap by boundary integral methods or similar ones. However, it seems that their efficiency is not good
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for very thin airgaps. Some researchers couple the fixed and mobile meshes by using Lagrange multipliers. It is also possible to use analytical solutions in the airgap and, by means of this technique, perform the coupling between fixed and mobile parts. The airgap becomes a region called macro-element since it takes into account all the nodes surrounding it. 6.1.2. Methods with Discretized Airgaps Here, the coupling between the fixed and the mobile part is made by finite elements placed in the airgap. Two techniques can be found: one is by the use of the slipping line and a second one is the moving band. With the first one, the coupling between the fixed and the mobile meshes is made along a line and the displacement must correspond to an integer number of elements. The difficulty with this method is that the rotation step should be necessarily linked to the mesh. However, with the moving band, the coupling is performed with a layer of elements and this restriction can be avoided. 6.2. The Macro-Element This method can be applied if the structure has (anti)-periodic boundary conditions (see section 4.7) in the boundaries F e F1 as shown in Figure 6.1, where the dark region corresponds to a macro-element. The airgap, where the macro-element is placed, is a linear medium without sources. The vector potential A equation in this region is:
rotrotA = Q
(6.1)
The Eq. (6.1) above can be solved analytically. With polar coordinates (r and /?) , well adapted to rotating electrical machines, its solution at any point inside the macro-element has the following form: A(r, p )= C0 In r + d0 + f) [an cos^p )+ bnsin(pn?> $£nrPn + dnr~Pn ] n=\
where p is the number of pole pairs of the machine.
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(6-2)
r,
r,
Figure 6.1. The macro-element.
For the coupling, the macro-element must be connected to the finite elements of the rotor and the stator parts. This is made by writing the vector potential continuity equations. With this aim, expression (6.2) is decomposed as the sum of the two functions that follow:
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\P»
*.v
Pn
~\pn]
„=! /"1
I'D ~Pn r
(6.3)
where Rr and R^ are the internal and external radii of the macro-element as shown in Figure 6.1. The constants a]0 - - M l n .-.b2n are determined by verifying the vector potential continuity between the macro-element and the finite elements on the boundary
(6 4)
Nr
-
1=1
where Ai , A} and Nr , Ns are respectively the vector potential at the nodes and the number of nodes in the internal and external boundaries of the macro-element. Fi (P ) and F (|3 ) are the shape functions which are approximated by a Fourier series as:
a Fi (P ) = ~
a
°° +
a
in COS(P"P ) + T ,-n««G>«P )
(6.6)
For the case of first order triangular elements, these shape functions are defined as: Ffp
= l if P = P/ and *;P
= 0 if P < PM or P > p /+1
(6.7)
It is also possible to demonstrate that the coefficients OC /0 ,OC /n ,. . .,y in are
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(6.8)
2
[cos(/?/i>gf)-cos(/7/i)gM)
[
cos(/w/?f)-cos(/7yi>g/+1)]
(6.9) fV-in
=:
A-A+i
&-A-1
(6.10) where P0=-— From Eqs. (6.3), (6.4), (6.5), (6.6), (6.8), (6.9) and (6.10) the expression of the vector potential at an internal point of the macro-element is: R.Y" Nr
11 2 In MV —
P» f R \Pn
n=\
pn
Ns
ajo Inj
• Z l Aj' j=
R
a/n
cos(pn/l)+yinsin(pn/3)]
fj^ \P"
r
[ajn
cos(pnp)+yjnsin(pnp)]
2 In
(6.11) The term SS(i, j) of the matrix SS of Eq. (5.29) for the macroelement is given by
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in
-In
J£ -hJl
4//o
\c'J
/?
\e J
°°
+—z "o£ ^c
/
pn
C
2
e
rjr-(
e 'y"]
~(UJT f u.''
awQ
^J
^"1 Tr/?
"-f-fT KJ 1I «• J U,) J-p /
D \P"
—
/.«
(
+
-t '
pn
P"
~
^ e, ^P«
^ e 1
^r
(6.12) where
c = Rr and
c = Rs
and
e = Rr
and
e'= 7?c and
je[Nr+l,Nr+Ns]
The number of harmonic terms used for the Fourier coefficients has influence on the precision of the analytical shape functions. According to our experience, 100 to 200 terms assure good accuracy. The macro-element is treated as a single particular finite element. The terms SS(i,j} of Eq. (6.12) are assembled in the matrix SS as normally done for classical finite elements. However the computation time is bigger since the macro-element needs successive calculations to obtain the Fourier coefficients. Moreover, as all the nodes of the macro-element are connected, the band width of matrix SS is normally large, increasing the calculation time even more.
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6.3. The Moving Band The principle of the moving band is illustrated in Figure 6.2, recalling that any angular displacement can be handled by this technique. For an effective implementation, we adopted quadrilateral elements obtained by triangular elements as will be presented below. It is possible to adopt this type of finite element, since the numbers of divisions in both, internal and external sides of the moving band, are necessarily the same. Actually, a mesh with "diagonal" cuts (see Figure 6.3) would be primarily obtained, but it has been clearly demonstrated that, with such procedure, precision problems arise. In fact there is a lack of contribution equilibrium on the nodes and, consequently, a lack of vector potential accuracy. These errors are largely amplified when calculating the potential derivatives, i.e., the magnetic induction.
Figure 6.2. Principle of the moving band.
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Figure 6.4 shows, an airgap meshed with rectangular elements, pointing out that these elements are obtained from triangular ones, as shown in Figure 6.5.
4
3
Figure 6.3. Triangular mesh, typical for thin airgaps of electrical machines.
Figure 6.4. Rectangular mesh.
1
2
1
2
1
2
Figure 6.5. Quadrilateral element decomposition.
As indicated in this last figure, during the Finite Elements numerical procedure, the integration over a quadrilateral elemental is performed twice, according to the two diagonal decompositions. To take it into account, the contributions are divided by 2. In fact, the matricial contributions are obtained by integration over the elements and it is performed twice here. With an angular rotation of the rotor, which does not match the side of a mesh element, a moving band deformation occurs, as in Figure 6.6.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
4
1
3
2
Figure 6.6. Moving band element deformation.
However, the application of the quadrilateral elements on the situation of Figure 6.6 can generate precision problems, since a pair of elements is much flatter than the other, as shown in Figure 6.7. In such cases, the quadrilateral element is replaced by only a pair of triangular elements, which means that only a diagonal cut is used. The choice of the adequate pair is based on a quality factor q. Considering a,b,c the triangle edges, q is given by:
-3
„
,
\
abc (a + b + c)
where S = ^
'-
—/
(6.14)
•a 4
-
0/ ^
1
3
/ + 1
2
1
2
Figure 6.7. Deformed quadrilateral element cutting. For an equilateral triangle, q is equal to 1 and it is the highest
possible value. Normally, we consider that the higher it is, the better is the triangle. In Figure 6.7, the second pair of triangles has the higher quality factor. In this case, only the second pair is considered and obviously, the division by 2 (as shown for Figure 6.5), is not applied. The implementation and principle of the moving band is also based on the dynamic allocation of (anti) periodicity boundary conditions.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Let us suppose an initial position as shown in Figure 6.2a, on which r\ e IV are the boundaries having (anti) periodicity. As the mobile part turns, the moving band elements are deformed, as indicated in Figure 6.2b. During the displacement, one observes that a modification of the moving band mesh minimizes the deformation. This operation implies the redefinition of nodes/elements belonging to the moving band as well as the bandwidth change of matrix SS in Eq. (5.29), which should be reevaluated for all positions. Moreover, as shown in Figure 6.2c, new nodes are created at the boundary. The expression indicating the number of supplementary nodes Nsupi is:
supl = int N MB
N
where N^g
(6.15)
is the number of nodes on one side of the moving band, CXm
is the rotation angle and OLSD is the adopted machine domain angle. Regarding the matricial system to solve, these additional nodes do not create any difficulty, since they are not taken into account in the solving procedure, as presented before in section 4.7. In fact, they are associated by (anti) periodicity conditions to the corresponding nodes located on the new boundary F2'. Therefore, with this technique, there is no creation of new unknowns. This dynamic allocation of boundary conditions makes the moving band method very efficient and operational. The procedure continues as illustrated in Figure 6.2d and any rotor displacement can be performed. When the rotation angle CLm is larger than the domain angle a translation is necessary. The mobile part is shifted to the initial position if the boundary conditions are periodical and the procedure follows as normal. If the problem is anti-periodical, an inversion is applied on the sources located in the rotor.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
6.4. The Skew Effect in Electrical Machines Using 2D Simulation If the movement is inserted in the field modeling, it is also possible to consider the rotor and stator skew for electrical machines simulation as shown in Figure 6.8. Skewing reduces the harmonic content of currents, electromotive forces and torques. Regarding the magnetic field modeling, skewing implies that the currents are not purely perpendicular to plan Oxy, as shown in Figure 6.8 (b), and consequently the two-dimensional approximation cannot be used directly.
(b)
Figure 6.8. Schematic electrical machine stator; a) unskewed stator slots; b) skewed stator slots.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
An approximation to this 3D problem is to represent the continuous structure by a set of slices (Figure 6.9) in which the current has only z components. This representation is called as multi-slice model. Figure 6.10 shows schematically how a slot and the corresponding current density are approximated. slice k 2ndslice Ist slice
Figure 6.9. The multi-slices model used to represent skewed slots.
Y
Figure 6.10. A slot current approximation.
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In the functioning of the electrical machine, the same effect of stator skewing is also obtained by rotor skewing. If the rotor (instead of the stator) is skewed it allows the use the movement techniques previously presented in this chapter. At each time calculation, the rotor is turned k times by an angle 8(r|) to construct the finite element matrices related to the k slices. This angle is calculated on the supposition that the slices are uniformly displaced following relationship below
8fa) =
fa-l)y/(£-!)
(6.16)
where y is the slots skew angle and T| = l,2,..., k the slice number. For simplicity, let us firstly suppose that there are not thick conductors in the electromagnetic device. When the skew effect is considered for k slices, the field/voltage equations (see Eq. (5.88) and (5.89)) become SS, 1 A j l -P j l I / =D j l
(6.17)
SS,2A52-P,2l=D*2
(6.18)
(6.19) 51
dt
sl +
2
dt
s2+
'"
+
^sk
sk+ at
f+
at
f
f
(6.20) As one can see from the equations above, matrices SS, P, Q and D must be constructed for the k slices. For instance, SS^j and P5j are, respectively, matrices SS and P for slice 1 (named si). The unknowns of the problem are now the magnetic vector potentials at the nodes of the different meshes of the slices (A 5 j, A52 > • • • > 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)
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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)
(6.34)
[c 2 orC 4 orC^C 7 ] U, + [C3 orC 5 or C6] lt =
orO] (6.35)
dt
+L — l=
(6.36)
^
The multi-slice equations corresponding to the first and second equations above (Eqs. (6.33) and (6.34)) are A
,. Asl
dt
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. _ rp ,l I , _rp1 , TT . _u T| ,
sl f J
sl^tsl~ sl
if. q7\
(0.0/)
As2 -Ps2lf -P',2 Uts2 = Vs2 (6.38)
-\sk -Psklf -P'sk Utsk =Vsk
(6.39)
and
, =0
at
(6.40)
~ at
'=0
at
(6.42)
where N^ , Q' sk and P'5^ are, respectively, matrices N, Q1 and P' for slice k , observing that the current in the thin and thick conductors, I f and \t , keep the same value through the slices. Equation (6.35) is related to the way that the thick conductors are connected (serial, parallel or parallel and short-circuited for a squirrel-cage) as presented in section 5.3.1. Suppose that, for simplicity, they are all connected in a squirrel-cage form. Equation (6.35) then becomes , =0
Using (6.31), the equation above now becomes
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" 1 0 0 1 0 0 1 0 0 1 0 0" C 7C7 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1J T
u <*4
or
+C 6 I, =0
(6.43)
+ C 6 I, =0
(6.44)
Generalizing for k slices one has
For the other connections, similar results are obtained substituting T Cy Cj , G£ and 0 by C2 , C3 and Ef (in the case of serial connections) or by 04,05 and Ej if the thick conductors are parallel connected. As Eq. (6.36) is not related to thick conductors, its multi-slice version is equal to Eq. (6.20) and the final set of equations is
- A5l -Psllf -P'5l
=D
(6.45)
(6-46)
+N 5jk -
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(6.47)
^ + R ' I , =0
Q',2
A
(6.48)
*2 + ClU t o 2 + R'I, = <>
(6-49)
=0
(6.50)
(6.51)
—A
O
—A
RT
L —I (6.52)
6.5. Examples 6.5.1. Three-Phase Induction Motor The goal of this first example is to present the moving band technique described in section 6.3. We consider again the three-phase induction motor shown in Chapter 5 simulated at locked rotor condition. Now, we simulate this machine at no-load and fed by three phase nominal voltage. The modeling of the movement is performed using the moving band technique, implemented in the software EFCAD. Figure 6.13 shows a mesh detail of the airgap, indicating the moving band.
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Figure 6.13. Airgap mesh detail of a three-phase induction motor.
A plot of the field distribution at no-load is shown in Figure 6.14. We can notice that, contrarily to the locked rotor situation presented in the previous chapter, the magnetic flux penetrates more intensively in the rotor.
Figure 6.14. Field distribution in the three-phase induction motor.
The calculated and measured currents in the stator windings are given in Table 6.1 and it shows the good accuracy of the presented method. Table 6.1. Induction motor phase currents at no-load. Calculated Current [A] 10,782
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Measured Current [A] 10,72
6.5.2. Permanent Magnet Motor In Figure 6.15 we present the structure, mesh and field distribution in a four-pole permanent magnet machine constructed by the research group of M. Lajoie-Mazenc of the LEEI/ENSEEIHT in Toulouse. Because of geometric and magnetic symmetries, the calculation domain is one-fourth of the machine, using anti-periodic boundary conditions. The stator three-phase windings are star (Y) connected. The no-load generator operation of the machine at constant speed is shown in the next two figures. Figure 6.16 presents the phase to neutral voltages and Figure 6.17 the phase-to-phase voltages calculated, respectively, for straight and skewed slots. One observes the influence of skewing in reducing the harmonic content in the voltage waveforms. In the next simulation the generator is short-circuited at 0.0 seconds. Figure 6.18(a) shows the transient regime current for the non-skewed machine and Figure 6.18(b) for the skewed one. Figures 6.19 were obtained when the generator is operating at shortcircuit but now with the neutral wire connected. Comparing the results of Figure 6.18 with those presented in Figures 6.19, even for the skewed machine, one observes the presence of other harmonics generated by the use of a neutral wire in the star connection. For these simulations, the motor velocity is imposed. However, in many applications, as for instance, in a starting transient regime of an electrical motor, the movement is not a priori known and must be also calculated. To do so, it is necessary to take into account the mechanical torque and evaluate the relative position between stator and rotor. This will be described in the next chapter.
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Figure 6.15. Permanent magnet machine: materials, mesh and noload magnetic field.
m
0 00
0 05
0 10
0.15
0.20
0.25
0.30
(a)
Figure 6.16. Phase to neutral voltages for the permanent magnet generator: (a) straight stator slots; (b) skewed slots.
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0.00
0.05
0.10
0.15
0.20
0.25
0.30
(b)
Figure 6.17. Phase-to-phase voltages for the permanent magnet generator: (a) straight stator slots; (b) skewed slots
[A] 5025-
oo
o.i
0.5
0.6
0.0
O.I
02
0.3
Q.»
0.5
0.6
(b)
Figure 6.18. Current in the permanent magnet short-circuited generator Y-connected without neutral wire: (a) straight stator slots; (b) skewed slots.
00
0.1
0.2
0.3
04
0.5
Figure 6.19. Current in the permanent magnet short-circuited generator Y-connected without neutral wire: (a) straight stator slots; (b) skewed slots.
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0.6
7 Interaction Between Electromagnetic and Mechanical Forces 7.1. Introduction A precise analysis of an electrical rotating machine requires the study of the interaction between mechanical and electrical quantities and the electromagnetic torque plays a fundamental role in the corresponding energy conversion. When the rotor speed com is unknown, it is calculated by a mechanical equation as:
(7.1) where Jm
is the rotor inertia moment, Dm
is the friction damping
coefficient, Te is the electromagnetic torque and TL is the load torque acting on the machine axis. Also, the rotor angular position a m is obtained from the angular speed COW by:
— a w =co w
at
(7.2)
With the rotor angular displacement calculated by (7.1) and (7.2), and with the moving band technique presented in the previous chapter, it is possible to determine the behavior of electrical machines taking into account the electromechanical transients. Then, it is necessary to calculate with
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accuracy the electromagnetic torque obtained from the solution of field equations. There are different methods, based in several formulations, to evaluate the torque. They can be classified in two categories: a. methods based on the direct results furnished by the vector potential equation; b. methods based on the force density over the magnetic material surfaces. In the first category, the following methods are included: • The co-energy variation method; • The Maxwell stress tensor method; • The method proposed by Arkkio; • The derivation of the local Jacobian matrix method. In the second category, here called "methods based on the force density evaluation", there are several techniques using the formulations based on equivalent sources: currents, magnetic charges and combination of magnetic charges and currents. In this same category we include a particular formulation deduced from the energy derivative. 7.2. Methods Based on Direct Formulations 7.2.1. Method of the Magnetic Co-Energy Variation With constant current and for a bidimensional domain the torque can be obtained by the derivation of the magnetic co-energy W by length unity:
T =
dW' dam 7=cte
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(7.3)
where i is the length (placed in the direction perpendicular to the study plan) of the device and am the angle defining the relative position between rotor and stator. Numerically the derivation in (7.3) is approximated by the following equation
Ta=£-
Aa
(7.4) I=cte
where Aa is the angular rotation step. On the other hand, the magnetic co-energy expressions are: • For permanent magnets: 'H
ds
(7.5)
Sa _ 0
with B = \iaH
+ BQ . In Eq. (7.5) Sa is the region where the permanent
magnets are located,
\\,a is the magnetic permeability and B0 the
remanent induction of the magnets. • For other materials and air: n
\BdH ds
W' =
(7.6)
Sr
where Sr
represents regions without permanent magnets and where
The co-energy of the whole domain S = Sa + Sr is given by the sum
of (7.5) and (7.6) W'=Wa+W"r
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(7.7)
The electromagnetic torque is then calculated by (7.7) and (7.4) applied to the whole structure for two successive positions OLm and (ptm — Act). As mentioned above, with this technique, it is necessary to have constant current and therefore, it is not appropriate when the electrical machine is simulated, for example, by converters as seen in Chapter 5. 7.2.2. The Maxwell Stress Tensor Method Maxwell's stress tensor is one of the most efficient general methods for the calculation of forces on bodies under the influence of magnetic fields. It leads to expressions that allow the computation of forces on such diverse structures as the rotor of an electric machine or the moving piece of a relay. The calculation of forces through Maxwell's tensor is used extensively in computer programs for numerical computation of fields. To apply it, the fields must be known and a computation method must be capable of providing these data. This is perhaps the reason why, on one hand it is commonly used in conjunction with numerical methods and, on the other hand, it was very rarely used during pre-computer periods when the computation of fields was difficult or impossible. Also of importance is the fact that Maxwell's tensor facilitates understanding of the relation between the magnitude and direction of forces as function of the magnetic fields that generate the forces.
Figure 7.1. Body under magnetic field delimited by the surface F .
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For the practical application of Maxwell's tensor, suppose that a body occupies a volume V and that the magnetic field intensity H is known on the surface F enclosing the body. It is also required that this body be located in air or within a material with permeability (J, = JIQ . Figure 7.1 shows such a body, where dT is a differential area of F ; n is a unit vector normal to the surface, and we define dT = dTn . Maxwell's tensor is given by (7.8a) or, in the form of force density
aT
(7.8b)
This gives the force dF on a differential area, on which we know the field intensity H. Integrating over the surface F provides the total force applied to the body. This force can be viewed as acting at the center of gravity of the body. We analyze now a problem in the Oxy coordinate system such that the axis Ox is parallel to the differential surface dT of the body, as shown in Figure 7.2. Assuming that H is constant over this differential surface and using the orthogonal unit vectors i and j associated with the axes Ox and Oy , we can write
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Figure 7.2. Magnetic field on the surface dT parallel to Ox.
Substituting these expressions in Eq. (7.8b) gives
dr
y
2
With some algebraic manipulation, we get
u : i r^O / rj2
or, equating components
dFx =
From these expressions, the following observations are in order: • Hx = 0, Hy > 0 ; results in dFx - 0 with dFy > 0 and, therefore, only a normal force to the surface exists (Figure 7.3a);
• Hx = H
^ 0 ; we have the case where dFy = 0 and dFx ^ 0 ;
only a tangential force exists (Figure 7.3b);
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• Hx ^ 0; Hy = 0 ; results in dFx — 0 and dFy < 0; a normal compression force exists (Figure 7.3c). A ^F k
H
i
0=45°
«/*
r
777, b.
a.
e=9o»
0
H , , fc,
c.
Figure 7.3. Various possible force components, depending on the direction of H. a) Normal tension force; b) tangential force; c) normal compression force.
We evaluate now the magnitude of dF:
dF = or using the expressions of dFx and dFy above, upon simple algebraic transformations, we get
or
The direction of dF is obtained from the tangent of the angle 0 , which H forms with the normal. This is
tane^ =
Hy
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The angle a , which dF forms with the normal, is
tana = Again, using the expressions of dFx and dFy we obtain
H dT tana = ——'-—-— . JL -it \sf>±.
2H H =— ^±* J. JL Y" <*• JL •%!
2 Dividing numerator and denominator by Hy and using the expression of
tan 6 we obtain tana =
2tan9 __ — = tan 29 2 l-tan 9
Therefore, we conclude that, knowing H (meaning that Hx and Hy were determined) it is easy to evaluate the magnitude of dF and its direction. In fact as demonstrated above, the angle that dF forms with the normal is twice the angle of H with this direction. This explains the cases shown in Figure 7.3. Notice also that if H penetrates in the body (9 = 180°), the force takes the angle of 360°, i.e., a tension force occurs in the same way as H exits the body (9 = 0° ). As a simple application of the Maxwell's tensor, we present the magnetic circuit of Figure 7.4. In this circuit we consider that the magnetic field is constant in the airgaps (1) and (2) and there is no flux crossing the two lateral oblique surfaces. Also, we consider that the iron permeability is much larger than JJ,Q .
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iron
(2) h2 nl
\e irom
(1)
*l|
1'
Figure 7.4. Magnetic circuit with piece P.
To obtain the values of h\ and h^ in the airgaps (1) and (2) respectively, we apply the equations of field circulation and flux conservation, which gives
= nl =4>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.
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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.
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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).
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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
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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.
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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
(7.15)
HO 0 Considering that r is defined inside the airgap and this latter is limited by the internal radius rr and the external one rs, theoretically the torque should not vary with the radius, and we can write
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rs
Te(rs-rr} = \Tedr
(7.16)
r
r
Replacing (7.15) in (7.16) we obtain
:.Te=—rl
r \rBrBQds (7.17)
where Se is the airgap surface limited between radii rr and rs . Numerically the integration in (7.17) is performed by the following sum:
x r / * r / A ) A/
T e =—-
(7.18)
where Ne is the number of elements placed between the radii rr and rs , Brj and BQJ are the radial and tangential induction components in the element / and Sej is the element area. It is convenient to use, as radius TI , the barycenter of the element i . One can notice that, with this method, the definition of a specific integration line is no longer necessary, since the sum in Eq. (7.18) is made over the surface between rr and rs . 7.2.4. The Method of Local Jacobian Matrix Derivation Based on the virtual work principle, the local Jacobian matrix derivation, proposed by J. L. Coulomb, allows the torque evaluation by means of the next relationship:
Te=lZ
T
W
n=\
1
d
H
i d
J'1 5-^-JH+ f B-dH-detJ" 1 5— e o e
J (7.19)
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in which J is the Jacobian matrix and det J its determinant (as presented in Chapters 3 and 4). The sum is performed in the airgap and for the Nvd elements placed between the fixed and mobile parts that can be virtually deformed when the mobile part turns. Equation (7.19) can be also written as:
i
d
1 / \?
1 d'
• J'1 . — jB + -(B) 2 detJ~ 1 — *s
3
/^ \
2
/
^
d
(7.20) As presented in Chapters 3 and 4, for bidimensional cases, with first-order elements, J is given by:
'd_ du /=!
(7.21)
dv
where / is a node with coordinates */,>'/ and ^Vf- is the geometric transformation function identical to the interpolation function for isoparametric finite elements. As already shown previously, for first-order elements the terms N are: (7.22)
N2=u
(7.23) (7.24)
The derivatives related to 0 in (7.20) are easily calculated by writing the coordinates X and y of the nodes in a polar coordinates system. The final expression of the torque by this method can be expressed as:
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11=1
B2
o
4n0
4^
D
°
with: = ($3*3 ~
where x\ , . . . , y^ are the coordinates of the nodes 1 , 2 , 3 of triangle n and 5/ =1 if the node / belongs to the mobile part and 8, = 0 if the node is placed on the fixed part. It is possible to show that the Eq. (7.25) corresponds to the integration of the Maxwell stress tensor along a line linking the middle points of the triangle edges, as already indicated in the section 7.2.2 above. With this method, the definition of an integration line is no longer necessary since it is done intrinsically in (7.25). 7.2.5. Examples of Torque Calculation All the above methods present the same results if the rotation does not imply on a deformation of the moving band elements. However, in our experience, if there are element deformations, the technique furnishing the best results is the Maxwell stress tensor method, with the torque calculated on a second layer of quadrilateral elements (which are not distorted) as presented in section 7.2.2. In Figure 7.8 we have the calculation domain and the corresponding flux distribution of a permanent magnets machine designed and constructed by the research group of M. Lajoie-Mazenc of the LEEI/ENSEEIHT of Toulouse. Because of the geometric and electromagnetic symmetries, the study domain is half of the machine, using anti-periodic boundary conditions. For
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
this application, the static torque was calculated with two series-connected phases fed by continuous current. The cogging torque was also evaluated. The results obtained by the Maxwell tensor method and by experimentation are shown in Figure 7.9, with very good agreement.
Figure 7.8. Permanent magnet motor.
2—1 Torque [N.m]
2-i Torque [N.m] Static
1-
0-
-226
60
76
100
Angle [degrees] ••"O"*' iwo""""*J
50 I..
75 100 Angle [degrees]
Figure 7.9. Calculated and measured static and cogging torque.
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Figure 7.10. Permanent magnet motor with polar piece.
">-] Torque [N.m]
10
~\ Torque [N.m
lie' ' ' Angle [degrees]
Figure 7.11. Calculated and measured static torque for two phases in series and fed by a continuous current.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
A second example is shown in Figure 7.10. For this machine, the simulation domain is even smaller than the previous one, and it takes a fourth part of the machine using anti-periodic conditions. This structure was also designed and constructed at the LEEI/ENSEEIHT of Toulouse. The static torque was calculated for two phases in series fed with a continuous current, and the corresponding waveform is presented in Figure 7.11. Again, for calculating the torque, the Maxwell stress tensor method was employed, using a quadrilateral elements layer as seen above. These results point out that the movement band associated with the Maxwell tensor method provides accurate results, in spite of the complexity of the two simulated structures. Other examples, not shown in this text, confirm the above statement. Nevertheless, we remark again that, for obtaining good torque results, the mesh should be well refined. 7.3. Methods Based on the Force Density These methods are based on the use of equivalent sources for the magnetic materials or on the magnetic energy derivation. After some preliminary considerations, these two methods will be successively presented. 7.3.1. Preliminary Considerations In the air, the vectors B and H are related by: rt
H =—
(7.28)
Ho where |10 is the air permeability. In an isotropic magnetic material, we have H =—
or
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(7.29)
n
H=-
(7.30)
where |J, and (J,r are, respectively, the real permeability and relative one. Also, to take into account the magnetic material, we can define the magnetization vector by: M = — -H
(7.31)
Using Eqs. (7.30) and (7.31) we can express the magnetization as a function of the field by: (7.32) or as a function of the induction as: ( \ "\ R M= 1- — — I HjHo
(7.33)
The relationships between magnetic induction and fields in a material 1 having permeability \JL and magnetization M, and a material 2 having permeability |j,0 (Figure 7.12), are obtained by the conservation of normal induction components as well as tangential field conservation (assuming there are no superficial currents between the two materials), as: B
2n = B}n
(7.34)
H2I = Hlt
(7.35)
which, associated with (7.31) and (7.28) give:
M
= ^--H
or
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=Bln H
Mln=H2a-Hln
(7.36)
and Mlt= — Hi
Ho
Ho
Ho
or 1
(7.37)
Figure 7.12. Interface conditions between material 1 and 2.
The two above equations can be written as a function of the relative permeability |J,r of material 1 using the expressions (7.32) and (7.33):
H2n=»rHln
(7.38)
B2l=—Blt H,
(7.39)
7.3.2. Equivalent Sources Formulations These methods are based on replacing the magnetic material (where we wish to calculate the forces) by a non-magnetic one having, in its interior, a volumetric distribution of field sources and presenting, over its surface, a superficial distribution of field sources. These distributions can be current or charge distributions, or even a combination of both.
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7.3.2a. Equivalent Currents According to this method, the magnetic material with permeability [I is replaced by a non-magnetic one having, on its surface, current densities J 5 equal to:
xn
(7.40)
where n is the unitary vector normal to surface S involving the material. When calculating the magnetic field with such current distributions, the induction B in the whole system remains the same, but the magnetic field H, in its interior is modified. The superficial force density f - can be calculated using Laplace's
law: (7.41) where B5 is the induction at the surface of the material. Figure 7.13 illustrates the equivalence in this particular 2D case.
.k JS)
Js = M, A n
Figure 7.13. Equivalent current distribution for a homogeneous material with permeability (4, .
Using the interface conditions given by Eq. (7.36) and (7.37) and also the relationships (7.40) and (7.41), we obtain the following expression for the superficial force density: (7.42)
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where k is the unitary vector tangent to surface S at the considered point. The global force is calculated by integrating this expression along the surface S involving the material. 7.3.2b. Equivalent Magnetic Charges Here, the magnetic material with permeability \i is replaced by a non-magnetic material having a superficial distribution of magnetic charges with density ps at surface S involving the material given by: (7.43)
From this equivalent charge distribution, the magnetic field H in the whole system remains the same, contrarily to the magnetic induction Bt, which inside the system is different from the real one. With this formulation, we can calculate a superficial distribution of force density fs as given below: fs = ps Hs
(7.44)
where Hs represents the magnetic field on the material surface S. Figure 7.14 indicates the equivalence for this particular 2D case. Mo
Figure 7.14. Equivalent charge distribution for a homogeneous magnetic material with permeability (J, .
Hs
For evaluating the superficial force density, we can adopt the field on the surface as the average between field H2 on the external part
of S and field Hj, internal to it.
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Using Eqs. (7.36), (7.43) and (7.44), we obtain the following expression for the superficial density of force fs :
(7.45)
Also here, the global force is obtained by the integration of the force density fs along surface S of the material. 7.3.2c. Other Equivalent Source Distributions Other equivalent source distributions can be deduced by combining the magnetic charge and current distributions. A very interesting one consists in combining the superficial charge density p5 ps=nB2
(7.46)
with the superficial current distribution with density J5 equal to:
Js = — nxB 2 Ho
(7.47)
With these distributions, the field and induction stay the same outside the medium, noticing that in the interior of the magnetic material both are null. This method furnishes a superficial force distribution of density fs equal to:
f
/
\B c
I
rf T 1
= n . B - ^ + nx-2-
xB
(7.48)
In the equation above B5 is the induction on the surface of the material, which is the average value between the internal induction (which is zero) and the external one B2 (which does not change in the real system). Developing (7.48), we have the following superficial force density applied to the material surface:
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(n
(7.49)
As before, the global force results from the integration over surface S of the material. It is interesting to point out that the above equation is identical to the Maxwell tensor expression. 7.3.3. Formulation Based on the Energy Derivation The global force acting on a magnetic non-deformable medium can be calculated by: v
Fx=
-- W 6 jc
(7.50)
which gives the force X component as a relationship between the energy variation for a virtual displacement 8 x at constant flux. If the medium is isotropic and without currents, the energy variation $W , during the displacement at constant flux, can be expressed as a function of the permeability variation 8|4, by:
(7.51) where F is the volume of the material on which the force is calculated. The permeability variation 8(J, can be originated from non-homogeneities and/or mechanical deformations. If this last one is neglected, permeability variation 5 JO, for a displacement 8.x is given by: Sp, = -^5x
dx
the
(7.52)
This relation is only valid if the permeability varies in a continuous manner in the study domain. Under these conditions (with non-deformable isotropic medium, without currents), the force Fx acting on this volume, according to (7.50), (7.51) and (7.52) is given by:
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du
9
U2dv
(7.53)
Analogously, we obtain the forces on the remaining directions. The global force is then expressed by: Tj2
F =- J v
2
grad\i dv
(7.54)
If the medium does not have permeability discontinuities, this relationship can be applied on the whole volume V. From it, we obtain the volumetric force as:
H2 tv=-—grad\JL
(7-55)
If in the volume V there are volumetric current densities J v, it is necessary to add a term to the right-hand side of Eq. (7.55). The volumetric force is now given by:
H2 f v =- —gra4i+J v x(nH)
(7.56)
£*
When calculating forces on a volume V involved by an external surface S and where the permeability is not continuous, this volumetric distribution is not sufficient, since it is necessary to add the forces on the discontinuities. To perform this, some authors (Stratton, Woodson & Melcher, Carpenter) propose an approach in which the surface has a fictitious volume where the permeability varies continuously. To evaluate the force on a part Se of this surface (Figure 7.15a) it is necessary to integrate (7.56) over a fictitious volume Ve (Figure 7.15b). These authors demonstrate that Eq. (7.56) can be presented under a tensor divergence form. With the theorem of the divergence it is possible to calculate the force acting on Ve by integrating, over the surface delimiting Ve , the following expression:
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f 5 =(nB)H-i(BH)n
(7.57)
,r n Figure 7. 15. Fictitious Volume Ve for force density calculation by the energy derivation.
To obtain the force on Se, the thickness of the volume Ve is enforced to tend to zero. The integral is then calculated on the surfaces Se+
and Se_
(Figure 7.15c) whose respective normal vectors are n
and n . As these vectors n obtain:
and n
have opposite directions, we
(u • B)H - i (7.58) where n , coincident with n , is normal to surface S . Enforcing surface Se to zero, we have the superficial force density as: fc = (n-B 2 )H 2 -I(B 2 H 2 )n-(n.B 1 )H 1 -i(B 1 H 1 )n (7.59)
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
7.3.4. Comparison Among the Different Methods Four formulations for evaluating force distribution on a magnetic medium have been presented: • equivalent currents method • equivalent magnetic charges method • superficial charges and currents method • energy derivation method With the adopted hypotheses (linear and homogeneous medium), all these methods are only associated to the concept of superficial force density. Considering Eqs. (7.34), (7.35), (7.38) and (7.39), we can reproduce the corresponding expressions in Table 7.1. Table 7.1. Superficial Force Components Methods Equivalent Currents
Tangential Component
Normal Component
Equivalent Charges
Equivalent Charges and Currents
Energy Derivation
0
For comparison, we present in Figure 7.16a an upper piece of a magnetic material being attracted by a permanent magnet. In this same figure, the distributions of the different superficial force densities acting on the magnetic material are also shown. These densities are indicated by vectors whose length are proportional to their density. The results are presented for static field calculations, using quadrilateral finite elements, as described above for the Maxwell stress tensor.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
As can be seen, each formulation has a particular force density distribution.
Figure 7.16. Magnetic structure, a) Magnetic field distribution. Magnetic force distribution by the methods: b) equivalent currents; c) equivalent charges; d) equivalent charges and currents; e) energy derivation.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
The global force from the different densities presented in Figure 7.16 is obtained by integration of the force densities on the edge of the finite elements located on the interface between iron and air. The global forces obtained with the four formulations and the force obtained by the Maxwell stress tensor can be observed in the Figure 7.17 for different meshes. l-orce [N]
a-
1000
2000
3000 Numher of Elements
4000
Figure 7.17. Global force as function of the number of mesh elements for the methods: A) equivalent currents; B) equivalent charges; C) equivalent superficial charges and currents; D) energy derivation; E) Maxwell stress tensor.
From Figure 7.17, we notice that the formulations based on energy derivation (Figure 7.17d), equivalent charges (Figure 7.17b) and combined equivalent charges and currents (Figure 7.17c) present, regardless of the mesh, very close results. On the other hand, the formulation based on equivalent currents (Figure 7.17a) shows very different results compared to the other three methods, when the mesh is not dense. However, for a refined mesh, the global force of this last formulation tends to a value close to the one obtained from the others. The force calculated by the Maxwell stress tensor, is shown in the curve E in Figure 7.17. In software EFCAD, the force calculation by means of the Maxwell stress tensor was extensively validated. This last result was determined by using an integration line passing in the middle of the airgap, i.e., between the magnetic piece and the
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
permanent magnet. As noted above, this result is identical to those obtained by the method based on combined superficial charges and currents. The relatively small difference between them in the beginning of the curve is explained by the mesh, which is not very dense. The results show that, for a good mesh, all of the methods based on the force density distribution conduct to the same global force value. The use and implementation of methods based on force densities to calculate electromagnetic torque and global forces requires a bigger computational effort compared to methods based on "direct formulations". On the other hand, the local force density is necessary when we wish to analyze the vibrational behavior of electrical machines generated by electromagnetic causes. 7.4. Electrical Machine Vibrations Originated by Magnetic Forces If the mode and frequency of a magnetic force wave coincides with the mode and natural frequency of a machine stator (resonance condition), even a relatively small excitation force can produce very intense vibrations. In this type of study, we generally assume that the rotor vibration can be neglected, since its natural frequencies are very high. We also consider that the stator mechanical deformations are elastic. In such cases, it is common to consider a unidirectional coupling between the magnetic and mechanical equations. In other words, with a finite elements model we calculate the magnetic field configuration and the forces acting on the stator. These forces are applied to a mechanical finite elements software which will give the mechanical accelerations and deformations. Then, as basic hypotheses, it is considered that the motor deformations are small and that the ferromagnetic laminations are weakly magnetostrictive in the way that the magnetic field is not influenced by mechanical deformations.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
7.4.1. Magnetic Force Calculation As mentioned above, for electrical machines vibrational studies in general, the origin of the vibrations is supposed to be located in the stator. Therefore, we need to determine the magnetic forces distribution along the stator internal periphery. To simplify this complex problem, it is common practice to integrate the force density on the machine teeth and suppose that it is concentrated in the central interior part of it. Assuming this, we can choose any of the four formulations presented in Table 7.1, since they conduct to similar values of the global force. After obtaining the radial and tangential components of the global force acting on the teeth as function of the time, we employ a Fourier analysis for the harmonic decomposition, since the mechanical posterior analysis is performed in the frequency domain. 7.4.2. Mechanical Calculation We will consider as known and adopt here the dynamic equation of a mechanical discrete system (already discretized in structural finite elements). We remark that it is not our goal here to develop elasticity and continuous media equations deeply. The readers could find more information in a wellestablished and specialized bibliography. The equation mentioned above is:
SSm—+ Kmq = Vm q + Cm-q dt2 & where: SSm
is the mass global matrix;
Cm
is the damping global matrix;
KOT
is the rigidity global matrix;
F^
is the equivalent forces vector in the nodes;
q
is the mesh node displacement vector.
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(7.60)
Eq. (7.60) can be solved in order to obtain the natural response and the forced response as well; it will be described below. 7.4.2a. Calculation of the Natural Response The determination of the natural response is necessary to obtain the natural frequencies and vibration modes of a structure. This analysis is important for a posterior dynamic analysis because the knowledge of fundamental modes and natural frequencies helps in characterizing the dynamic response and behavior of the device. Assuming that there are no forces acting on the structure and neglecting the mechanical damping, Eq. (7.60) becomes:
2 a SS ^q + K m
mq
=0
(7.61)
Supposing also that the system response is harmonic, or (7.62) where j — V—1 , CO = 2nf
and / is the deformation frequency, we can
write: *2
dt2^
(7.63)
Replacing (7.62) and (7.63) in Eq. (7.61) we obtain: (7.64) which is an eigenvalue problem. The solution of (7.64) furnishes CO and q , respectively the natural vibration frequencies and modes of the mechanical structure.
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7.4. 2b. Calculation of the Forced Response Directly in Harmonic Regime With this calculation, it is possible to analyze the vibrations. The choice of the method to solve the Eq. (7.60) depends on the type of excitation Fm . For electrical machines in steady state regime, the magnetic forces are periodical and we can define the force vector as:
Fm = j:¥ke k=\
(7.65)
where FA is the complex amplitude vector corresponding to the k harmonic of the equivalent force applied to the mesh nodes. Analogously, the vector q of displacements is:
q=Iq*e y ' t o ' k=\ where q^
(7.66)
is the complex amplitude vector corresponding to the k
harmonic of nodal displacement. The complex vector q^ is obtained by solving, for each harmonic, the following complex equation:
(7-67) Finally, having the nodal displacements we can evaluate the accelerations by:
d
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7.4.2c. Calculation of the Forced Response Using the Modal Superposition Method As seen before, to obtain the response of a mechanical structure, it is necessary to solve Eqs. (7.67) and (7.68), for all the frequencies, one by one. However we can consider another method, using the technique of modal superposition. Let us observe that in Eq. (7.60), the matrix SSW is diagonal, but the matrices Cm and Kw
are non-diagonal. This means
that there is a coupling between equations. By the modal superposition it is possible to decouple the equations by diagonalizing the mass, rigidity and damping matrices, although this last one requires some additional considerations. This method consists in transforming the actual displacements q presented in the Eq. (7.60) in modal displacements by the following operation:
q = Pmr
(7.69)
where r is called modal displacement vector and Pm is called the modal matrix, which is constructed with the eigenvectors obtained from the solution of (7.64) and here called u, i.e., (7.70)
With (7.69) we write (7.60) as: 2 SS
r m P m —T +C w P m —r + K m P m r = Fw 2
dt
ft
(7-71)
Multiplying both sides of (7.71) by the transposed modal matrix P have: —S c Pw aQ^l » m rP/ w — m
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T
, we
l"7 ( ' . 79\ /£)
P r K" fP r
^
~— ~
(7.74)
Pm*m F =flm
(7.75)
1
and Eq. (7.71) becomes:
ss^-yr + c^— r + k m r = f, dt dt2 where the matrices SSm and km
(7.76)
are diagonal. In order to make Cm
diagonal and the equation system (7.76) decoupled, we consider the matrix Cw as proportional to the matrix K.m or SS OT , or still a combination of both. This hypotheses is not far from reality, because there are well defined vibration modes for all the resonance frequencies, which points out a proportional damping. Now, considering that the structure is excited by a forces set of the same frequency CO ^, but with many magnitudes and phases, and assuming a response of the same form, Eq. (7.76) becomes, in the frequency domain: (7.77) where G m
(co /j) is a term of the so-called mechanical structure transfer
matrix, which can be expressed as:
1
/ \ N m(®h)= I
G
ssmtwf '
../..*
(7.78)
2
co/, "
+ Jcml®h
I' »,'J where N
is the number of modes, u\ is the modal coordinate in the
k response position i', u, is the modal coordinate in the excitation position
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k associated to mode r; and ssmf, CO/ and cmj are respectively the mass, the natural frequency and the damping coefficient of mode /. From Eq. (7.77) the frequency response to each harmonic can be calculated taking into account the viscous damping of each mode. 7.4.3.Example of Vibration Calculation As example of vibration originated by magnetic forces, we have a variable reluctance motor, whose calculation domain and field plot is shown in Figure 7.18.
Figure 7.18. Calculation domain and flux distribution for a variable reluctance motor.
The method of superficial charges and currents was adopted for evaluating the stator teeth forces. Its distribution is presented in Figure 7.19, where the arrow lengths indicate the force density.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Figure 7.19. Force density distribution on a stator tooth.
As stated before, the global force is supposed to be concentrated in the central part of the tooth. The temporal evolution of the global radial force is shown in Figure 7.20. ForcefNJ
0.
-200.^ -300.: -400.^ -500.: Timefms]
-600. 0.
0.5
1.0
1.5
2.0
Figure 7.20. Temporal evolution of a stator tooth radial global force.
Having the radial and tangential global force components, the Fourier analysis procedure is applied. For this example, the harmonic spectrum for the radial component (Figure 7.20) is presented in Figure 7.21.
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mj. -
1 2
Force[N]_
3
40..
4 5
6 7/
20.-
I n n
11
0 -
250
750 500
1250 1000
1750
1500
'n 10
1
2250
2000
2500
Frequency[Hz] Figure 7.21. Harmonic spectrum of the radial magnetic force.
For the natural response, the natural frequencies and vibration modes for the motor stator are determined. Some associated modes and frequencies are shown in Figure 7.22. By comparison between magnetic force frequencies spectrum and the natural frequencies, it is possible to determine the frequencies where the vibration peaks will occur. Comparing the force modes with the natural ones, it is possible to predict the type of forced deformation occurring on the machine when excited by a specific force harmonic. For example, Figure 7.21 indicates that the magnetic forces can excite several vibration modes, but in particular, the mode presented in Figure 7.22b (natural frequency of 1242 Hz) which is very close to the fifth force harmonic. Moreover, the fifth force harmonic shape (Figure 7.23e) has a trapezoidal shape, tending to move the structure "diagonally". In Figure 7.22b, although it is somewhat difficult to visualize, there is also a similar movement created by the vibration at 1242 Hz. In this way, it is possible to predict (even before performing the calculation of the forced response) that there will be a vibration peak with the frequency of 1250 Hz and that the resulting deformation will correspond to the ones of the 1242 Hz mode.
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(b)1242[Hz]
(a)1125[Hz]
(e)1938[Hz]
(d) 7 831 [Hz]
(f)3065[Hz] Figure 7.22. Natural vibration frequencies and modes for the reluctance motor.
The magnetic forces spatial modes distribution is presented in Figure 7.23. These force modes are obtained by linking the arrows related to the radial and tangential components of force.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
(a)250[Hz]
(b)500[Hz]
(e)1250[Hz]
'*"****' ''.** (f)1500[Hz] Figure 7.23. First modes and frequencies of stator teeth magnetic forces.
The accelerations calculated punctually on the structure, as well as the measured ones, are presented in the Figure 7.24. In this example, only the first 12 force harmonics were used. One can notice a good agreement between the results.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Acceleration[m/s ] 9.8
125dHz
8.58 7.35 (5.13
4.0
3.68 2.45
750 IE
1.23
i L 0
250
500
750
1000 1250 1500 1750 2000 2250
2500
2750
3000
Frequency[Hz]
(a) Acceleration[m/3*] IU
9
1250 Hz
- Q
8 o 7
-
6
-
5
-
4
-
o
1750 Hz
^ 2 1 n
2250 Hz
750 Hz
2750 Hz
', ,
,.L
1000
JUL,
„ ,,. i i J ,rJu...l. .1 j i JL ..1 . L 2000 3000 4000 5000 Freqnency[Hz]
Figure 7.24. Acceleration as function of the frequency at a stator point, a) Calculation, b) Experimentation.
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7.5. Example of Coupling Between the Field and Circuit Equations, Including Mechanical Transients The next example takes into account many phenomena including: the coupling between field and feeding circuit equations, the interconnection between massive rotor conductors, torque calculation and movement. It is a single-phase squirrel cage induction motor with two poles, having 24 stator and 28 rotor slots. Because of the symmetry, the smallest possible domain, using anti-periodic boundary conditions, is half the machine. The stator presents two thin conductor windings, the principal one (/?) and the auxiliary one (a) whose magnetic axes are perpendicular to each other. Figure 7.25 shows the calculation domain for this machine.
Figure 7.25. Calculation domain for the single-phase induction motor simulation.
The motor starting is performed by an external electrical circuit presenting time-dependent functioning characteristics. This electrical circuit is shown in Figure 7.26. The principal and auxiliary windings are represented by blocks Wp and Wa respectively.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
C
tt A/W v
(0 -r
WD
Figure 7.26. Feeding circuit of induction motor stator windings.
The resistance R\t), connected in parallel to the capacitor C, is a PTC (Positive Temperature Coefficient device), whose resistance value varies with the temperature. This component assumes a small resistance value at starting (4Q) and a large one (10&QJ with temperature increasing due to the current circulation. The voltage v(t) is applied on the external circuit; /p (t) and ia (t) are the currents in the principal and auxiliary windings, respectively. The principal purpose of the PTC is to command the auxiliary winding operation during the motor starting. At this stage, a large current circulates on the principal winding and, because the PTC has low resistance, the auxiliary winding current flowing into the capacitor is practically zero. Some fractions of second after starting, the PTC resistance increases considerably. The capacitor C starts working in order to improve the motor efficiency. With this, the current on the auxiliary winding decreases significantly. The time when R(t) changes its value from 4Q to 10AQ (0.3 seconds) was obtained experimentally and used in the simulation procedures. The matrices G j to G 5 related to the feeding circuit of Figure 7.26, as well as the vector E
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
, can be obtained from the circuit tree shown in
Figure 7.27. For didactical purposes, we will present the analytical determination of these matrices, which can also be automatically determined by using the methodology presented in the Chapter 5.
c
AAAA W
wp\ \vp -Va •-C-I.
Figure 7.27. Induction motor electrical circuit tree (full lines). From Figure 7.27, we have the following elements:
a. Branch elements: v and C b. Link elements: R((), Wp and Wa Equations (5.90) and (5.91), for this particular case become
— v c = G 1 v c + G 2 v + G3
= G 4 v c +G 5 v + G where vc is the only state variable of the external feeding circuit. Analyzing this electrical circuit, we have:
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(7.79)
(7.80)
d (7.81) (7.82)
C
R(t)C
dt
(7.83)
vp=v
(7.84)
v 4- V = V c ^ va
Then, (7.79) and (7.80) can be rewritten as:
1
d_
L me.
V
dt
"0"
-1
c
v c +[G]v +
o J'
(7.85)
\'p'
(7.86)
™t
"1" "0 0" v+ 0 0 I'a. 1
We will now present some results related to the induction motor starting. In this simulation, no load torque is considered. The sinusoidal applied voltage frequency is 60 Hz and its r.m.s. value is 115 Volts. The calculated principal and auxiliary winding current waveforms are shown in Figure 7.28. The relative error between the measured and experimental values of the total line current ie =ip +ia in steady state condition was 4%. 20-,'/'. IAJ
Time [s] -20-
oo
o.a
0.6
Figure 7.28. Currents in the principal (/ j and auxiliary \ia) windings of the motor.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
The instantaneous electromagnetic torque curve is shown in Figure 7.29. This result was obtained with the Maxwell stress tensor integrated on a quadrilateral elements layer outside the moving band, as presented in section 7.2.2. Torque [N.m]
0.0
0.1
0.2
0.3
0:4
OiS
Figure 7.29. Instantaneous electromagnetic torque. 400 -i
200-
Time [s] 0.0
0.2
I
0.4
0.6
Figure 7.30. Speed of the induction motor as function of the time.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
The resistance R\t) variation effect, occurring at 0.3 seconds, can be observed in these figures. One can see a particularly strong current variation in the stator windings. The curve of velocity as a function of the time is presented in Figure 7.30. As the motor operates at no-load situation, the speed at steady regime is close to the synchronous one. One may notice, on the speed and torque curves, the presence of oscillations with frequency which doubles the feeding one at steady regime. This doubled frequency is characteristic of single-phase induction motors functioning. According to the classical machines theory, the magnetomotive force of each stator winding can be decomposed into two components: the first one turning in the same direction of the rotor and the other turning in the opposite direction. Each one of them induces its own rotor current component. Also, as seen from the stator, two magnetomotive force waves appear: one turning in the same direction of the field created by the stator and the other in the opposite direction. The interactions between flux waves and the magnetomotive forces turning at opposite directions produce, at high rotor speeds, torque pulsations having double frequency, compared to the feeding one, but not creating average torque. However, higher frequency operation increases the iron losses in the motor. The next chapter is dedicated to iron losses.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
8 Iron Losses 8.1. Introduction The evaluation of iron losses in electromagnetic devices is presented in this chapter. This is a multi-disciplinary subject, involving Physics, Material Engineering and Electrical Engineering. We are here more interested in the generation of different losses by means of macroscopic models to be used in Electrical Engineering. After presenting the different iron losses components, two ways of using these models in Finite Elements field calculations will be discussed. The first one is based on a priori knowledge of elemental inductions obtained from a field distribution, evaluated when field sources (permanent magnets and currents) are known. In some cases, it is an approximation because the iron losses can influence the current waveforms, especially when high induction levels exist in "magnetically short-circuited" devices as transformers, where airgaps are absent. When the influence of the iron losses over the current waveform is important, the losses must be directly integrated into the field calculations. This is a very difficult research problem and, even nowadays, a challenge to researchers. Such a methodology, allowing the integration of iron losses models in finite elements calculations, will be also presented in this chapter. The iron losses can be calculated by the composition of three losses: • eddy currents • hysteresis
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
• anomalous losses These three losses components will be introduced in the following sections. 8.2. Eddy Current Losses
We saw in section 2.9.1 that in a conductive material submitted to time-varying fields, loops of induced currents, or eddy currents are created. Suppose now that the material is laminated, for the purpose of impeding the main flow of the eddy currents. We will see shortly, that in spite of this, there are losses due to Joule's effect. Figure 8.1 shows a thin plate or lamination made of a ferromagnetic material. The plate is subjected to a flux density B, parallel to the plate in the Ox direction. The dimensions lx and /,, are much larger than d, which is the plate thickness. This means that the plate is thin and low-amplitude induced currents will not significantly affect the externally generated flux density B. Under these conditions, we can assume that the induced current density J does not depend on x or y. This is shown in Figure 8.2a and 8.2b in which the principal component of J is in direction Oy. From J = 0E, the same is true for E . The equation rotlL = — dB/dt applied to this case assumes the form
h
B Figure 8.1. A thin lamination in a magnetic field.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Figure 8.2a. The current density in the thin plate of Figure 8.1.
r©—•
dt Figure 8.2b. Relation between the magnetic induction variation and the induced current density.
det
dx
dy
dz
0
Ey
0
dt 0 0
and, therefore,
oEy _ dBx dz
dt
The solution to this equation is _ y Z
dB
x
dt
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Since J and E can not have discontinuities, Ey = 0 at z — 0 (at the middle of the lamination, see Figure 8.2b) and the constant k is zero; the solution is
Ey(z) = —*-z
(8.1)
The power dissipated in the plate by Joule effect is PE = ^aE*dv
(8.2)
where V is the volume of the plate (V = lxlyd ). Applying Eq. (8.1) in (8.2) gives D
"f =cr
"
(dBA2 Jt
dt
\2\ dt ) * y
(Watts)
12
(8.3a)
or, in terms of power density i2 / ^n
\2
(Watts/m3)
12 ^ dt )
(8.3b)
The time average value of the losses is given by:
12 T
dt
(Watts/m3)
(8.3c)
where T is the period of induction B . Another useful expression is the energy density per cycle, which can be obtained by multiplying (8.3c) by T:
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dt In the case when
5^
(Joules /m3)
(8.3d)
varies sinusoidally with time as
Bx=Bmcosart, the derivative dBx/dt = -6)Bmsino)t
and (8.3b)
takes the form:
(Watts/m)
12 T
(8.4a)
For sinusoidal B waveforms, the average value per cycle of the power /%
density is obtained using the average value of sin G)t, 1/2:
,2
Pe=
a)2B2
(Watts/m3)
(8.4b)
From this expression we note the following: • Pe depends on d ^ (depends quadratically on the thickness of the plate). 2
• Pe depends on CO (quadratic in frequency). • Materials with low conductivities (J have small losses. 8.3. Hysteresis
The turning of an electron around the nucleus is similar to an electrical current on a ring loop. From Ampere's law, an electrical current creates a magnetic field. Then, internal magnetic fields, more or less oriented, are present in all materials. In some substances this effect is weak, but in others, as ferromagnetic materials, this effect is well pronounced. The effect of moving electrons is the magnetization M ('Amperes/'m). The magnetic induction B is the sum of the effects of a magnetic field H and the magnetization of the material, as given by Eq. (8.5), already presented in section 7.3.
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(8.5)
Several electrons "spins" oriented in the same direction create magnetic domains, which are delimitated by domain iua//s. The term magnetic domains was introduced by Weiss in 1906 as a hypothesis, but in 1949 Williams, Borzort and Shockley demonstrated experimentally their existence. The material magnetization process is accomplished by the magnetic domains movement and rotation as illustrated in Figure 8.3. In the magnetic domain movement region, two phenomena occur: the reversible and the irreversible movement of the domains, corresponding, respectively, to a reversible and an irreversible magnetization.
Magnetic Domain rotation region.
Irreversible ^ Movement Region
Magnetic Domain movement region
Reversible Movement Region
200
-0,25 J
300
400
500
600
H[A/m]
Figure 8.3. Experimental Initial magnetization curve.
The saturation of the material is reached when all the "spins" are aligned in the same direction, i.e., when the magnetization vector has only one direction and value M5 , corresponding to a saturation induction B5 . The hysteresis losses are associated to the magnetic domains movement and rotation, and they also depend on the material grains
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composition and size. Figure 8.4 shows a B(H)
curve obtained
experimentally when a ferromagnetic material is submitted to a low frequency ( / ) magnetic field intensity H. In these results a low frequency is used to ensure that eddy current losses are negligible and only the magnetic hysteresis phenomena are present. As H varies, the material passes through the loops of the hysteresis curve. This means that part of the energy in the device is consumed in the cycling through the hysteresis curve. Denoting Pfj (Watts) the power associated to the hysteresis curve, one can write the energy Wfj consumed in a cycle as:
(Joules)
(8.6)
As seen in section 2.8.4., the volumetric energy density W^ , is given by the following relation: (8.7)
The total energy in a block of material having volume V is therefore WH=WhV. We now look at the graphical representation of Eq. (8.7). Because it is an integral, its representation on the B(H) curve is the surface shown in Figure 8.5. If the curve is traced in the positive direction, this area is positive. If it is described in the opposite direction, this surface becomes negative, as shown in Figure 8.6. The difference between the positive and negative areas corresponds to the internal area of the hysteresis loop. The same consideration applies to the whole hysteresis loop. We conclude that the cycle area represents the total volumetric energy consumed in one loop.
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1,4 -
Magnetic domain rotation region
H[A/m1 -600
400
-400
600
Hysteresis loop ..Magnetic domain rotation region -1,6 J
Figure 8.4. Experimental hysteresis curve, obtained at low frequency (1 Hz).
tB
dB
T o
H
Figure 8.5. Differential magnetic energy density.
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Figure 8.6. Difference between positive and negative energy density. The power due to hysteresis is, therefore,
PH=AVf
(Watts)
(8.8)
where A is the internal area of the hysteresis loop. The dissipated energy per cycle is
WH=AV
(8.9)
As a didactical example of losses calculation, suppose we have a transformer core, made of an iron alloy with the B(H) curve as shown in Figure 8.7. A sinusoidal induction is presented in the core whose volume is 100 cm . Other parameters are:
2-
200
-200
-200
200
(b)
(a)
Figure 8.7. (a) B(H) curve for a transformer core, (b) Approximation of the B(H) curve by a parallelogram.
; Bm=1.5T
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The losses due to eddy currents are given by Eq. (8.4b) p e
=
03 2 #2
H
w
24
Qyatts i m3 x
Using the numerical values above (in MKS) gives
Pe= 133,240 (Watts I m3) To obtain the total power in the core, the power density is multiplied by the volume and
P £ =13.3
(Watts)
To calculate the losses due to hysteresis, we must evaluate the area of the loop in Figure 8.7(a). We will make an approximation here and suppose that this area is given by the surface of the parallelogram with dotted lines of Figure 8.7(b), i.e.: A = 3 x 200 = 600 (Joules
In?)
The power given by Eq. (8.8) is
PH = A Vf = 600x 10~4 x 60 = 3.6
(Watts)
The total iron losses are PT=PE+PH =16.9 (Watts) This means that 16.9 (W) of the power injected into the transformer is dissipated by losses in the iron core. Another way to calculate the hysteresis losses is by using the Steinmetz's equation, an empirical relationship established in 1892 by Steinmetz. This equation was obtained by means of experimental results for values of Bm between 0.2T and 1.5T. The equation is:
Wh = f]B^
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(Joules/m3)
(8.10)
In Eq. (8.10) the parameters 77 and a must be obtained from experimental results, as the ones given in Figure 8.8. 308
wh
= 129.365,
1.6740
231 154 -
Experimental points
77 -
0.00 0.00
0.25
0.50
0.75
1.00
1.25
1.50
Figure 8.8. Experimental plot of hysteresis losses as a function of the magnetic induction.
Equation (8.10) can be used only under sinusoidal induction variations. To calculate hysteresis losses for non-sinusoidal induction waveforms, i.e., presenting reversals A/? as shown in Figure 8.9, Lavers et al. (in 1978) proposed the following equation, based on Steinmetz's equation: (Joules I m )
(8.11)
/=
Figure 8.9. Non-sinusoidal induction as a function of time and associated hysteresis loop.
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The inconvenience in using (8.11) is that the position of minor loops originated by reversals on the hysteresis loop is not considered. Also, the efficiency in using (8.10) and consequently (8.11) resides in the proper determination of the model parameters. The best way to calculate hysteresis losses is to evaluate precisely the hysteresis loop surface. For this, mathematical models to describe the hysteretic relationship between B and H (as for instance Preisach's or the Jiles-Atherton models) will be presented later. In evaluating iron losses, samples of the material under analysis are normally tested on the Epstein's frame. To obtain only the hysteresis losses as a function of the induction (as presented in Figure 8.8) the measurements must be made at very low frequencies to ensure negligible influences of 2 eddy currents which, as given in Eq. (8.4b), are proportional to CO . After low frequency measurements, a test is made at the nominal operation frequency of the device which will be constructed with the iron sheets under analysis (normally 50 or 60 Hz). If the hysteresis losses determined at low frequency are added to the theoretical eddy current ones the result is lower than the measured losses. The reason for this is the existence of other losses components which are also a function of the frequency. These additional losses are called anomalous or excess losses. The mechanism of origin of these losses has been studied over the last fifty years but the theory presented by Bertotti in 1985 is the most accepted and adopted. 8.4. Anomalous or Excess Losses
Bertotti in 1983 and 1984 defined a new physical entity: the magnetic object (MO). Under this concept the magnetic domain wall movement can not occur in an isolated manner. These movements dislocate other domain walls and they are all correlated in the same correlation region. To each correlation region corresponds a MO. Suppose a magnetic sheet with conductivity <j, cross section S, submitted to an alternate induction B(t) with frequency / and amplitude
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Bm . We will consider that «om (/) is the number of MO taking place in the magnetization process. Let us also define Hex (t) the excess field, which is the part of the external field needed to compensate the field originated by the currents created by the MO movement. The mean value of the excess losses originated by the nom (t) MO is given (in Watts/m3) by:
r«W^*
(8.12)
Bertotti also supposes that the excess field Hex (f) is proportional to the velocity of variation of the local flux variation d(f>jdt
induced by the MO
moving. This proportionality is given in Eq. (8.13) where G is the MO friction coefficient. (8.13)
at The velocity of variation of the global flux SdB(t}jdt
results from the
contribution of the nom (t) active MO. This is given by Eq. (8.14)
,.
at
o
.
at
With (8.13) and (8.14) we have (8.15) om
Let us now verify the relationship between the number of active MO ( «om (t) ) and Hex (t) . Experimental results obtained by Bertotti for several crystalline magnetic materials, as Fe-Si sheets, show that there is a linear relationship between Hex and the number of active MO as, for instance, in Figure 8.10.
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Fe-Si 3% NO
Fe-Si 3* GO
150 100
100
/' 1/Vo o/
50
Vo = 0.07 A/m 10
5
20
H«(A/m)
10
15
Hcx(A/m)
Figure 8.10. Variation of the number of active MO as a function of the excess field in Fe-Si sheets with oriented and non-oriented grains (extracted from Bertotti 1985).
Observing Figure 8.10, we can write the next equation between «om(0
and
#ex(0 = (8.16)
where V0 is defined in Fig. 8.10. Using Eq. (8.12), (8.15) and (8.16), Fiorillo and Novikov (1990) show that the mean value of the anomalous or excess losses is:
dB(t} dt
1.5 dt
(Watts In}
(8.17a)
(Joules I m )
(8.17b)
or
dB(t) dt
1.5
dt
If the induction has a sinusoidal waveform with frequency / and amplitude Bm , the anomalous losses can be expressed, alternatively, as:
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(Watts /m5)
Pex = %.
(8.18)
8.5. Total Iron Losses Experimental results obtained for iron sheets at a fixed frequency and for several induction values are shown in Figure 8.11. As already noted, the hysteresis losses (curve W^) are determined at very low frequencies. The eddy current losses (curve We] are calculated, now at the rated feeding frequency (50 Hz in this example), using the theoretical equations presented in section 8.2. Finally, the anomalous or excess losses curve is obtained by subtracting from the total iron losses experimental curve ( W t } the sum of curves ( W ^ ) and (We). With this method also called losses segregation or losses separation we can analyze the behavior of each component and their influence on the total losses. An additional advantage in using the segregation losses method, mainly for electrical engineers, is that we do not need to determine explicitly the variables G and VQ (related to metallurgical material properties) presented in the anomalous losses coefficient, because these losses are obtained by simple algebraic difference. 385 -I
346.5 -
W, [J/m3] = 196.585J 723»
308-
Wh [J/m3] = 129.365J 6MO
269.5
We[J/m3]=48.59Bm2
231 192.5 154
115.5-
77 38.5
0.000 0.25
0.50
0.75
1.00
1.50
Figure 8.11. Total iron losses at 50 Hz ( W t ) and losses separation into eddy current ( W e } , hysteresis ( W f r } and anomalous losses (Wex e
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From the previous equations for hysteresis, eddy currents, and anomalous losses, respectively Eqs. (8.11), (8.3c) and (8.17), we obtain the total losses expression:
I "i i
a65
i i
B
-ZAfi >_
m /=! r/
dB(t) dt
craf2 ' 12
,T(dB(t)} , dt *[ dt )
1.* ^
(8.19)
(J( wles 1 1
In 2D, the magnetic induction B is:
and the two-dimensional version of (8.19) can be written as:
^EAB;
W,=r}\B,,
+
12
/=! 1.5
dt
dt
dt )
(Joules I m )
^
(8.20)
In Eq. (8.20) the time derivatives are calculated as a function of the components of the induction vector, i.e.,
dt
dB(t)
1.5
3/4
dt where A/ is the time step; &BX and ABy are, respectively, the variation of the x and y components of the induction during the time interval A/.
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The amplitude of the induction | Bw | is calculated with the corresponding x and y components. An alternative way to express the iron losses is in (Joules/kg). In this case, Eq. (8.20) must be divided by mv, the iron volumetric density -5
(kg I m ). The total losses in (Joules/kg) are:
t/kg
mv 12
dt
)
1.5
dt
dt
(Joules I kg) (8.21a)
The total iron losses can also be written in (Watts/kg) by multiplying Eq. (8.21a) by (1/T)
t/kg
1 m
Bm
1+
dt
0.65 ^ "i=l i dt
mv 12 T
dt
)
dt +
(Watts I kg} (8.21b)
When Eq. (8.20) or (8.21) are used to predict iron losses, we need to know the spatial and temporal flux density distribution over the period T in each finite element of the mesh. This is the reason why, in the introduction of this chapter, it was called an a priori method. 8.5.1. Example Figure 8.12 shows the structure of a three-phased permanent magnet machine constructed at the LEEI/ENSEEIHT, of Toulouse, France. In this machine, the mobile part is the external one. It is made with a 22 poles (permanent magnets) external rotor and the stator has 18 slots. Considering this number of poles and slots, the field calculation must be done on onehalf of the machine. The iron losses were calculated with Eq. (8.21b) over a
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range of speeds with the stator open-circuited. In this case, the iron losses are due to the permanent magnet excitation and the rotation of the rotor. This is an example where this a priori methodology can be employed since the iron losses do not influence the magnetic field sources. The parameters used in these calculations are given in Table 8.1.
Figure 8.12. Structure and no-load flux distribution due to permanent magnets. Table 8.1. Parameters used in calculation of iron losses
1 —TI m
a
0.0371
L58
v
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1 j2 _L^L_ mv 12 0.187xlO"3
l m
O.llxlO'
Iron Losses (W) 14012010080604020-
200
400
600
800
1000
1200
Speed (rpm) Figure 8.13. Permanent magnet machine iron losses as a function of the rotor speed.
The calculated and measured results are shown in Figure 8.13. An acceptable agreement between calculated and measured results can be observed in spite of: • the incertitude about the permanent magnet characteristics • the incertitude in knowing precisely the parameters used in the calculation and given in Table 8.1. • the imprecision in the hysteresis losses modeling by Eq. 8.11. The preferable process should be to evaluate the hysteresis losses directly by the B(H) loop. In such a case, a theoretical model is needed. Several hysteresis models have been developed in the last hundred years. Among them, the most commonly used and known are the Preisach and the Jiles-Atherton methods. We use the second one here because it is based on the physical behavior of the magnetic materials and, moreover, because it is the simpler one to implement on numerical codes.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
For further discussion on these and other models recommended references are Ivanyi and Mayergoyz papers and books. 8.6. The Jiles-Atherton Model In the original Jiles-Atherton (JA) model, the magnetization M is decomposed into its reversible component Mrev and its irreversible component Mfrr , corresponding respectively, to reversible and irreversible magnetic domain deformation (see Fig. 8.3). The relationship between these two components and the anhysteretic magnetization Man (ideal case without losses) is based on physical considerations of the magnetization process. 8.6.1. The JA Equations The equations of the JA method are:
M - Mirr + Mrev
(8.22a)
Mrev=c(Man-Mirr)
an
coth
dHe
kd
(8.22b)
H0e
a
a
He
where a, c, k and the saturation magnetization Ms
(8.22c)
are parameters
which have to be determined from an experimental hysteresis loop; 5 is a directional parameter and takes the value + 1 for dH/dt > 0 and -1 for dH /dt < 0 . The term H e is called the effective field or the Weiss mean field, and is the sum of the external (applied) field H and the molecular field , i.e.:
He = H+aM
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(8.23)
where a must be experimentally determined. We also introduce the effective flux density Be , which is related to the effective field by
Be = \i0He = \1Q(H + aM)
(8.24)
and let us recall Eq. (8.5) of the magnetic induction: (8.25)
As one can observe, the magnetic field H is the independent variable in the JA model (as well as in Preisach's and in the majority of hysteresis models). We will present a procedure to obtain B from the magnetic field but firstly we need to manipulate the JA equations to obtain a differential equation in terms of dM/dH. Let us substitute Eq. (8.22b) in (8.22a): M = Mirr + c(Man - Mirr ) and differentiate this last equation by H:
dM _ dMirr dH dH
+
(dMan _ dMirr^ \ dH dH
We will now describe the terms dMan /dH and dM^ a. Term dMan /dH dMan dH
dMan dHe dHe dH
but, from Eq. (8.23)
dHe , dM - = l+cc dH dH
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/dH
and then
dM an dH
dHe \
(8.27)
dH
b.Term dMirr/dH dMirr dH
dMirr dHe dHe dH
dMirr dH.
dM ~dH
(8.28)
Now substituting Eq. (8.27) and (8.28) in (8.26) gives
dM dH
dM I r r l dM\ an 1+a +c l+a dH dH. dH, dH}
and, isolating
dM dH
dMir dM an ~dH, dH, dM***' an* __rv i I . /-»\ /-y /^ \A C Li IA ^/ dH. //„ (l-c)
dM
dH
dMirr dM irr f ' 1+a dH dH.
1I
1
(8.29)
where: dMirr jdHe is given by Eq. (8.22d) dMan /dHe is obtained by differentiating Eq. (8.22c), that is
dM an
dH.
(3
a
a ^
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(8.30)
Another important relationship is obtained from the substitution of Eq. (8.22a) in (8.22b):
Mirr =
(8.31)
(l-c)
Before presenting the numerical procedure, we point out that for low values (typically less than 0.1) of the argument (He/a) in the coth function, equations (8.22c) and (8.30) must be rewritten to avoid problems in a practical implementation of the JA method. To find a version of Eq. (8.22c) to small (He Id) values, we will expand the function coth using a Taylor series:
,.,„ / x coth(He/a) =
3 (Held) (He/a) 2(Hela)5 J + ^-^—i-1-£— — + -±—Z—'— + 945 45
1
Substituting the equation above in Eq. (8.22c) and retaining only the two first terms, we have:
Man(t) =
1
(He I a)
(Held)
a
He(t) la
and Eq. (8.30) becomes:
8.6.2. Procedure for the Numerical Implementation of the JA Method The numerical procedure shown in Figure 8.14 lead us to calculate an induction waveform knowing the field as a function of time. In this figure the pairs H(t) , B(t) and H(t + &t),B(t + A/) are, respectively, the field and inductions at time t and / + A/. The integration of the differential equation dMI dH is performed by using Euler's scheme, i.e.:
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
dM ~dH • Knowing H(t) and B(t) from previous time step • For the given actual time step magnetic field H(t + AO , calculate:
A// = H(t + AO -H(t)
M(/) =
If
a
< 0.1
If ^> 0.1 a
Mirr(t) = If
a
If
coth-
M(t)-cMan(t} l-c
< 0.1
dMan _ Ms 1-coth' dHe a
0.1
Man(t)-Mirr(t) dH
dM dH
(1 =
dMirr
c}
dHe
1 o.cdMan
\cdMan dHe a(l t A//
Figure 8.14. Numerical implementation of the JA method.
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a He(t}}
8.6.3. Examples of Hysteresis Loops Obtained with the JA Method Figure 8.15 presents the evolution of the anhysteretic (Man)
and
the first magnetization curves as function of the field. One can notice the difference between the anhysteretic magnetization (where the magnetic domain mechanism is not considered) and the first magnetization curve (where the deformation of the domains due to the reversible and irreversible magnetization are taken into account). xio r
1.6-
MfA/mJ 1 .4-
1.21.0-
Anhysteretic magnetization
o.e0.60.4-
First magnetization 0.2-
HfA/mJ 0.0-
0
2000
4000
6000
8000
10000
Figure 8.15. Anhysteretic and first magnetization curves.
In Figure 8.16 the irreversible and reversible components of the magnetization are shown. We recall that the reversible magnetization is related to low H values (see scale). With the magnetic field increasing, the irreversible magnetization becomes preponderant.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
xio c
-2
-1 xio
XiO
(a)
(b)
Figure 8.16. Irreversible (a) and reversible (b) magnetizations. The M(H)
as well as the B(H) curve are given in Figure 8.17. For these
hysteresis loops, the JA method parameters are:
• Ms =1.6xl0 6 (Aim) • 0 = 1000 (Aim) • k = 1000 (Aim) a = 0.001 xio MfA/mJ
i-
—
1
mi-j-
HfA/mJ -2-
-1
I 0
xio 1 '
-1
Figure 8.17. M(H) and B(H) loops.
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XIO'
The influence of the parameters on the hysteresis loops are shown in Figures 8.18, 8.19, 8.20 and 8.21. 2.0-
B[T] 1 .5-
k=3000 k=500
0.0-
-0.5-^ -1.0-1.5-
H[A/m] -2.0
1
-1.5
' ' ' I ' ' ^r -1.0
I ' • ' ' i ' • ' ' I '
-0.5
0.0
0.5
1.0
1 .5
X104
Figure 8.18. Hysteresis loops for different k values.
2.0
B[T] 1 .5-
a=0.0005 a=0.002 0.0-
-0.5-
-1.5-
HfA/mJ -2.0-1.5
-1.0
-0.5
0.0 X104
0.5
1.0
Figure 8.19. Hysteresis loops for different OL values.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
1.5
B[T]
a=1000
a=500 0.5-
0.0-
-0.5-
H[A/m] 1
-2.0-
I ' ' ' ' I ' -l.O -0.5
-1.5
0.0
0.5
1.0
1.5
X104
Figure 8.20. Hysteresis loops for different a values.
2.0
B[T]
1.51.00.5
O.OH
c=0.005 c=0.3
-0.5 -1 .0-
-1.5-
H[A/m] 1
-2.0-
-1.5
-1.0
-0.5
I ' ' ' ' I ^^' ' ' I '
0.0 X10 4
0.5
1.0
Figure 8.21. Hysteresis loops for different c values.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
1.5
From these results one observes that: • Variations of variable k dilate the loop and change the coercive field. • Changes in parameter a modify the loop rectangularity and change the remanent induction or magnetization. • Variations on a modify the loop shape. • Changing c modifies the initial magnetization which is related to the reversible magnetization. To evaluate the hysteresis losses we calculate the surface A of the cycle. With the numerical procedure presented in Figure 8.14, this surface can be determined by the parallelogram rule: (JoulesIn?) 1 2 Using this last equation, surface A of the B(H) curve of Figure 8.7 was calculated, giving now A=558.42 (Joules/m3). This is, of course, more precise than the value previously calculated in section 8.3 and, for practical material hysteresis loops, the integration procedure is recommended. 8.6.4. Determination of the Parameters from Experimental Hysteresis Loops From an experimental B(H)
hysteresis loop, the five JA model
parameters (Ms , &, #, a and c} can be obtained. As we saw at the end of the previous section, each parameter has influence on certain points of the hysteresis loop. We will use these particularities to determine them. The first step is to transform the B(H) loop in a M(H)
one, using
Eq. (8.5). The following step is to identify on the hysteresis loop the particular points and the different associated variables indicated in Figure 8.22.
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2.0
-2.0
-1.5
-1.0
-0.5
1.5
X10*
Figure 8.22. Particular points on the hysteresis loop for determination of JA parameters.
In
this figure,
variables
X
are
differential
susceptibilities,
i.e.,
X = dM/dH. • Determination of parameter Ms Parameter Ms
is the magnetization value when the material is
strongly saturated. • Determination of parameter k From Eq. (8.22d):
dMirr
Man-Mirr
dH
k5
And using lrr
dH
dMirr dHe dHe dH
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We get, accordingly to Ivanyi:
dM,.irr
M
an-Mirr
dH
(8.32)
kS
Recalling Eq. (8.26)
dMirr dH
dH
(dMan dH
TC
dMirr dH )
And substituting Eq. (8.32) in equation above yields
dM dH
=
(l-c)(Man-Mirr) . dM + c- an kd - a (Man - Mirr) dH
(8.33)
The parameter k is determined from the susceptibility
at the
coercive field Hc . Notice that in this case 6 = +1:
dM dH H=HC
(l~c)[Man(Hc)-Mirr] k-a[Man(Hc)-Mirr]
=
|c
dH
As the magnetization M is equal to zero at the coercive field, Eq. (8.31) becomes Mirr =
-cMan(Hc)
(l-c)
and then, Eq. (8.34) gives us
(l-c)
a+
l-c dMan(Hc) -cdH
Notice that parameter k can be calculated if
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(8.35)
This parameter is obtained using the remanent magnetization
Mr
of Figure 8.22. At this point 6 = — 1 and the field is zero. For M = Mr, Eq. (8.31) becomes M
_ Mr -cMan(Mr) (1-c)
(8.36)
irr =
In an analogous manner for determining parameter k, we will write the susceptibility y^r at the remanent magnetization as
dM dH M=Mr
(\-c}[Man(Mr)-Mirr]+cdMan(Mr} dH -k-a. \Man (Mr) - Mirr (8.37)
With Eq. (8.36) in Eq. (8.37) we have
Mr=Man(Mr)
1 dMan(Mr}
a
(8.38)
With a, k and c known, a can be calculated from Eq. (8.38). • Determination of parameter a When M = Mm , Eq. 8.31 becomes:
Mirr =
Mm-cMan(Mm) (1-c)
(8.39)
Parameter a is obtained from the data at the hysteresis loop tip, with the assumption that at high saturation levels the hysteretic and anhysteretic susceptibilities are equal, or:
dM dH H = H m
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dM an
dH
m
And, with Eq. (8.39), the relationship equivalent to Eq. (8.34) is
m
.
(I-C)
dM dH
m
Mm-cMan(Mm) -cMan(Mm)
k-a
(1-c)
After some algebraic operations Mm can be written as
Mm=Man(Hm)-
(8.40)
Determination of parameter c As indicated in section 8.6.3, c is associated to the reversible magnetization. At the beginning of the initial magnetization curve we have Mirr = 0 and dMirr I dH = 0. With this, Eq. (8.26) becomes
Xin ~
dM dH
dH
(8.41)
Similarly to the results presented in section 8.6.1, for low H values the anhysteretic magnetization can be expressed by
,,
,,
,, H + aM
As M = 0 in the starting region of the initial magnetization curve, we have
H Man(t) = Ms— 3#
(8.42)
Substituting the derivative of equation above into (8.41), the parameter c of the JA model is obtained by
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
m
(8.43)
Ms 8.6.4a. Numerical Algorithm
Parameters k, a, a and c are determined by solving iteractively Eqs. (8.35), (8.38), (8.40) and (8.43). The secant method is employed in the algorithm shown in Figure 8.23 to calculate a and a. The procedure presented in Fig. 8.23 is repeated until k, a, a and c are obtained with good precision. 8.7. The Inverse Jiles-Atherton Model In the original JA method, the magnetic induction B is obtained if the magnetic field H is known. However, when using a magnetic vector potential formulation in a time step by step procedure, B is obtained directly. Then, using B as the independent variable, an inverse method, allowing the determination of H from the field history is better adapted here. A method of this type can be obtained from the original JA method, and will be now presented. 8.7.1. The Inverse JA Method To obtain the inverse JA method, we will firstly write
dM _ dM dB ~dH~ dB dH Recalling Eq. (8.25) one has
dM
dM {844)
dH
dB
dB
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(
dM
• Calculate k
dMan(Hc)_\Ms dH
k=
Man(Hc) Man(Hc)
2a H
1-c
dMan(Hc) dH
(1-c)
• Calculate a
a a.Mr dMan(Mr) _ Ms ~~dH ~ a
Man(Mr)\Man(Mr) a M
2a 1 aM (
X
f(a)=Man(Mr)-Mr+-
Use the secant method to evaluate (X (n is the iteration number)
a n _2L =a n _il
"
"
««=««-!-• ®if |(a w -a n _i)/a w |> admissible error return to (D; else a =an • Calculate a
Ho = H'm o.H m + ' "•"/«
a
He
(d) = Man(He)-MmUse the secant method to evaluate a (n is the iteration number) a
n-2 = an-\
a
n-\ =an a
a
n =an-\
•Calculate c
:
n-\ ~ an-2 r^—(
if \(an~ an-\)/ an\
>
f -§(an-l)
admissible error return to ®; else a = an
Figure 8.23. One iteration of the numerical algorithm to calculate JA method parameters.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
and, with Eq. (8.29), Eq. (8.44) becomes
(\-c)dMirr JUQ dHe
dM dB
|
c dMan HQ dHe
llc
dHe or, using the effective flux density Be (Eq. (8.24)) |
C (845)
With equations (8.24) and (8.22d), the new term dMirrjdBe
in the last
equation is M
irr
Notice that the term dMan/dHe , necessary in Eq. (8.45), was already given by Eq. (8.30). 8.7.2. Procedure for the Numerical Implementation of the Inverse JA Method Numerically, we can calculate a magnetic field waveform knowing the flux density as a function of time, as presented in Figure 8.24. Analogously as presented in section 8.6.2, A/ is the time step, and the pairs H(t\B(t) and H(t + At),B(t + Af) are, respectively, the field and inductions at time t and (/ + A/) . Here, the directional parameter 5 takes the value +1 when dB/dt>0
and -1 when dB/dt<0. For the
integration of the differential equation dM/dB shown, Euler's scheme is employed.
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(Eq. (8.45)), as previously
• Knowing B(f) and H(t) from previous time step • For the given actual time step magnetic induction B(t + A/) , calculate:
coth-
1-c dH If
la
dMan __ Ms
o.l
dB dM
(1-,
dM
Figure 8.24. Numerical implementation of the inverse JA method.
8.8. Including Iron Losses in Finite Element Calculations We will now include iron losses in the modeling of the magnetic field by Finite Elements Method in a "strong coupling" way. Let us firstly recall the reduced equations of electromagnetic devices (Eqs. (5.88) and (5.89)) given in Chapter 5. The system is voltage-fed and, for simplicity, we do not consider conductive massive parts here (i.e., term NdfA/<# = 0):
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
(8.47)
SSA-PIy = D
+L — lfJ = U dt
dt
(8.48)
As already presented, matrix SS in the equations above is a function of the magnetic reluctivity V = 1 / ju . If the hysteresis effects are neglected, only positive values for the reluctivity exist. However, when considering a hysteretic B(H) relationship, negative reluctivity values are presented, as can be seen in Figure 8.25. This negative values give origin to numerical problems when solving the above equations. 20000-
-20000
-1.0
-0.5
0.0 X10 4
0.5
1 .0
Figure 8.25. Magnetic reluctivity as function of H for the case of a hysteretic B(H) relationship.
To avoid working with negative reluctivity values a first alternative is to use Eq. (8.5) and express B = //Q(^ + M) instead of the expression B = juH used for Eq. (8.47) and (8.48). A second way to handle this problem is to work with a differential reluctivity. These two alternatives will now be presented.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
8.8.1. Hysteresis Modeling by Means of the Magnetization M Term With this expression, the magnetic permeability in the whole study domain is supposed to be //Q . For the different magnetic materials, different magnetization vectors M are associated. Analogously to the development presented in section 4.2.4, the equation
rotR = J with B = //Q (H + M) becomes
rot
=J
Using the relationship between the magnetic vector potential A and B (B = rot A ) we have
= J + rotM
(8.49)
where VQ = I///Q . The differences between Eq. (8.49) and (4.17) are the reluctivities ( VQ or v) and the new term rctfM . The elemental Finite Element contribution for the left-hand side term of Eq. (8.49) is similar to the one already given in Chapters 4 and 5, obtained by changing v for VQ and resulting in Eq. (8.50) below. In the same way, the elemental contribution related to the term rotM is similar to those of permanent magnets in section 4.24. For first order triangular Finite Elements we can write these elemental contributions as r r
\\ symmetric 2D symmetric
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W2 +r\r2 #2#2 + r2r2 symmetric
0103 + rlr3 #2^3 + r2r3 q-^ q^
(8.50)
qlMy-rlMx q2My -r2Mx
(8.51)
All the terms presented in the equations above have been defined in Chapter 4, except Mx and My which are, respectively, the x and y components of the magnetization vector M. After assembling these elemental contributions, we get a system similar to the one given by Eq. (8.47) and (8.48) as:
SS0A-PI/=DM
(8.52)
Q — A + R I Jr + L —j+I fJ =U Jf ^ j* dt
(8.53)
and
where matrices
are related to Eq. (8.50) and (8.51)
respectively. Using Euler's scheme to discretize the time derivatives, Eq. (8.52) and (8.53) become (8.54)
_1_ At
R + _LL (8.55)
Equations above are solved iteratively at each calculation time, furnishing I f (t + At) and A(/ 4- At), respectively, the currents in the device windings and the magnetic vector potential at the nodes. Knowing A(/ + At), the inductions B(/ + At) are calculated, the inverse JA method (see Fig. 8.24) is used to evaluate the magnetization M(/ + Af) and vector D^(/
+ A^) is constructed. This procedure is repeated until
convergence is achieved (see section 4.6).
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
8.8.2. Hysteresis Modeling by Means of a Differential Reluctivity The relationship between vector dR and dB can be written as
dH = |vrf|dB where v^/
(8.56)
is called as reluctivity tensor. For isotropic materials in two
dimensional approaches, v^
'vd 0
takes the following form
0 v
(8.57)
and Eq. (8.56) becomes (8.58)
where
vd =
1
dH dB
(8.59)
dB
is called differential reluctivity, which is always positive as illustrated, for instance, in Fig. 8.26. This figure shows the differential reluctivity plot for the same B(H) hysteretic relationship from which Fig. 8.25 was obtained. An alternative way to express the differential reluctivity Vj is obtained by
A H ) , i.e.
multiplying Eq. (8.59) by
AH-^AH _ AH AB
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400000
HfA/ml -0.5
0.0 X10 4
0.5
Figure 8.26. Differential reluctivity as function of H.
On the other hand, using Euler's scheme to represent the derivatives, Eq. (8.58) becomes AH = VjAB
(8.61)
where AH = H(t + A/) - H(/), AB = B(f + A/) - B(0. Eq. (8.61) can also be written in terms of H(/ + A/) and H(f) as
As in the previous section, we can write Ampere's law for the time instant t +& rotH(t + A/) = J(t + A/) With Eq. (8.62), the above expression becomes rot[vd AB + H(0] = 3(t + A/)
(8.63)
Using B(/ + A/) = rot\(t + A/) and B(t) = rotA.(t), Eq. (8.63) is written as
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
rot{vd [rotA(t + At) - rot\(t)] + H(t)} = J(V + At) or
rotvdrotk(t + At) =
rotvdrotA(t) - rotH(t) (8.64)
Comparing Eq. (8.64) with (8.49) allows us to deduce the elemental Finite Element contribution for the left-hand side of the equation above as well as for the rotH.(t) term. They are similar to the ones presented in section 8.8.1 with the replacements: • Vj replaces VQ in Eq.(8.50) and the corresponding elemental contribution becomes r r
2D
\\ symmetric symmetric
r r
l2 r r 22 symmetric
?1?3 + rlr3 4233 + r2r3 q^3+r3r3
(8.65)
• Hx (t) replaces Mx and Hy(t) replaces My in Eq. (8.51) and we have the following source contribution:
q2Hy(t)-r2Hx(t) q3Hy(t)-r3Hx(t)
(8.66)
Notice that for the term rotvdrotA.(t) the elemental contribution is the same as Eq. (8.65) but, as A(f) is known from the previous time step, the product of Eq. (8.65) by the vector A(/) is performed and, as a source vector, assembled on the right-hand side. The assembling of the elemental contributions results in the following matrix equation system for a voltagefed electromagnetic device:
Af )A(f
At) =
At)A(t) + DH (t) (8.67)
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
R + —L A/
1
1 A/
(8.68) where matrices SS^
and vector T)fj are obtained by assembling Eq.
(8.65) and (8.66). Equations (8.67) and (8.68) are solved iteratively. For each time step, the following procedure is repeated until convergence is obtained: • With A(t + A/) the induction B(/ + A/) is calculated for each mesh element; • For each component of the induction, Bx(t + At)
and
Bv (t + A/), the inverse JA method is used to calculate Hx (t + A/) and
• As H x (/) and H y (t) are known from the previous time step, and &H y are evaluated and v^ is calculated with Eq. (8.60); Matrix SSd(t + Af) is recalculated and Eqs. (8.67) and (8.68) are solved. According to our experience, the method based on the use of the differential reluctivity furnishes better convergence rates than the method using JUQ and M presented in section 8.8.1. For instance, let us present in Figure 8.27 the hysteresis loops obtained by the two methods mentioned. The results are virtually the same, but the computational time for the second method (differential reluctivity) is 2.2 times smaller than for the first one. Moreover, in an easy way, the last method enables including the other iron losses components in the Finite Element modeling as shown next.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
1.8 -| Flux density, B [T]
6000
-6000
-1.2 -
• • Simulated with /4, ™+—~•*- Simulated with vj
-1.8 -I
Figure 8.27. Hysteresis loops for the two methods.
8.8.3. Inclusion of Eddy Current Losses In the FE Modeling According to Eq. (8.3d), we can express the energy density (Joutes/m3) dissipated by eddy currents in iron sheets of conductivity cr and thickness d in a time interval Af as: R \2
12
^u
(8.69)
With Euler's scheme for time derivatives, Eq. (8.69) becomes
_j2
12 •&> A/
(8.70)
One can also express the energy dissipated by eddy currents by the magnetic energy:
We =
(8.71)
where He is a magnetic field associated to the eddy currents in the iron sheets. Eq. (8.70) and (8.71) allow us to obtain He as
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
ad1 AB 12 At
(8.72)
Supposing that the total magnetic field in the device H j- (t + At) is given by the sum of the hysteretic field H(t + At) and the field originated by eddy currents, we have:
Hr (/ + At) = H(> + A/) + Ue(t + A/)
(8.73)
With Eq. (8.73), Ampere's law is now rotHT (t + At) = 3(t + At)
or rot[H(t + At) + Ue (t + At)] = J(f + A/)
(8.74)
with Eqs. (8.62) and (8.72), the equation above is written as
AB =
ro/j[v^AB + H(0]
At)
1 **L\l
which, using the magnetic vector potential, becomes
rot<
od' 12 At
rotA(t + A/) (8.75)
When the Finite Elements Method is applied to Eq. (8.75), the voltage-fed device equations are similar to Eqs. (8.67) and (8.68); the difference is in the matrix SS^, whose equivalent reluctivity expression is now
l2At The interactive solution is made in the same way as presented previously.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
8.8.4. Inclusion of Anomalous Losses in the FE Modeling Starting from Eq. (8.17), we can write the dissipated energy (Joules/m3) due to anomalous or excess losses during a time interval Af as
dB 1.5 dt
dt
or
dB
-0.5
dt
dt
dt
For a numerical procedure, the equation above becomes
AB
-0.5
AB
dB
(8.76)
As for the eddy currents case, one can express the energy dissipated due to the anomalous losses during time interval Af as
Wex = where H^
(8.77)
is a magnetic field associated to the anomalous losses. As made
previously, Eq. (8.77) and (8.76) lead us to obtain
AB
-0.5
H^
AB A/
or
AB
(8.78)
Suppose now that the total field Hy (t + A^) is the sum of the field due to the non-linear field (including hysteresis) H(f + A/), the field due to eddy
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
currents in the sheets H e (/ + A/) and the field related to the anomalous losses He:c(^ + A?) , i.e.,
HT(t + A/) = H(t + A/) + He(t + AO + H^t
+ A/)
Writing Ampere's law for the total field H^ (/ + A/) results rot[R(t + AO + He(t + AO + Hex(t + A/)] = J(f + At) (8.79) With Eqs. (8.62), (8.72) and (8.78), Eq. (8.79) can be written as
rot (8.80) and using the magnetic vector potential Eq. (8.80) becomes
rot<
A,°-5lAB|0-5
(8.81)
rotA(t + A/) - rotA(t)
Using the Finite Elements Method, the stiffness matrix SS^ in Eq. (8.67) will be now a function of the terms in the first bracket of Eq. (8.81). As previously commented, Eqs. (8.67) and (8.68) are solved iteratively. The term |AB in Eq. (8.81) is obtained by the difference between the induction value at the actual iteration and the one at the previous time step B(Y). 8.8.5. Examples of Iron Losses Applied to FE Calculations A sinusoidal voltage fed Epstein frame was chosen to illustrate the inclusion of iron losses into 2D Finite Element magnetic field calculations.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Figure 8.28 shows the induction distribution in the retained study domain of one-fourth of the transformer domain.
Figure 8.28. Induction distribution and calculation domain of an Epstein's frame.
In Fig. 8.29 the calculated current waveform is plotted when only the magnetic hysteresis is considered. In the same figure is presented the simulated current when the eddy currents in the sheets are also taken into account. The third current waveform is obtained when hysteresis, eddy currents and anomalous losses are simulated simultaneously. One observes that, as expected, the inclusion of the different losses components increases the harmonic components of the current circulating in the device windings.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
0.4
0.3-
0.20.10.0
-o.i-IT -0.2-0.3-n -0.4
0.270
0.275
0.280
0.285
0.290
0.295
0.300
Figure 8.29. Current waveforms when considering the different iron losses components: fa) only hysteresis losses; (bj hysteresis and eddy current losses; (cj hysteresis, eddy current and anomalous losses.
Figure 8.30 shows the corresponding simulated B(H) characteristics when all the different losses components are taken into account. Observing Fig. 8.30 one notices, from curves a) to c) that the inclusion of the different losses components leads to larger B(H) loops. The methodology that takes into account the losses in Finite Elements calculations was validated by comparing simulated and experimental results. For a specific set of iron sheets, the calculated (thin line) and measured (thick line) current waveforms at low frequency operation are shown in Fig. 8.31. In this result the Epstein's frame was also fed by a sinusoidal voltage and only magnetic hysteresis losses were considered. Other results are also given in Fig. 8.32 and Fig. 8.33, now for an operation frequency of 50 [Hz]. In these cases, in addition to the magnetic hysteresis, also eddy current and anomalous losses were successively considered in the modeling. One can observe in Fig. 8.33 that the B(H) loop becomes larger with the inclusion of the different losses components, as expected.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
1.5-
BfTJ
i.o0.5-
o.o-0.5-
-1.0-
HfA/mJ -1.5-300
-200
-100
100
200
300
Figure 8.30. B(H) characteristics taking into account: (a) only hysteresis losses; (b) hysteresis and eddy current losses; (c) hysteresis, eddy currents and anomalous losses. Measured current
l s T
-
Time, t [s] Figure 8.31. Measured and calculated current waveforms at 1 [Hz].
From these results one can notice that the simulation results, when considering all the losses, are in very good agreement with the experimental ones.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Hysteresis, eddy currents and anomalous losses
-*-*-Hysteresis only -W-M-Hysteresis and eddy currents losses Time / fsj
Figure 8.32. Measured and calculated current waveforms at 50 [Hz].
1.2 -, Flux density, B [T]
Hysteresis, eddy currents and anomalous losses
-600
0.9.
-400
400 600 Field strength, H[A/m]
Measured BH loop
-1.2
Hysteresis only Hysteresis and eddy currents losses
Figure 8.33. Measured and calculated current waveforms at 50 [Hz].
We would like to point out that the inclusion of iron losses in field calculations in the case of complex devices, as electrical machines, is still a challenge because not only alternating fields are present. To include rotational fields, which are present in electrical machines, the losses must be evaluated by vectorial models. This is a current and complex research domain still under investigation.
Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.
Bibliography
Books on Electromagnetics, Numerical Field Calculation and Other Techniques 1. M. A . Plonus, Applied Electomagnetics, McGraw-Hill, New York, 1978. 2. J. D. Kraus, and K. R. Carver, Electromagnetics, 2nd ed., McGraw-Hill, New York, 1973. 3. J. A . Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1952. 4. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, Addison-Wesley, Inc., Reading, Mass., 1956. 5. R. Plosney and R. E. Collin, Principles and Applications of Electromagnetic Fields, McGraw-Hill, New York, 1961. 6. P. Lorain and D. R. Corson, Electromagnetism, W. H. Freeman, San Francisco, 1978. 7. C. W. Steele, Numerical Computation of Electric and Magnetic Fields, Van Nostrand Reinhold Company, New York, 1987. 8. P. P. Silvester and R. P. Ferrari, Finite Elements for Electrical Engineers, Cambridge University Press, Cambridge, 1990. 9. J. C. Sabonnadiere and J. L. Coulomb, Finite Element Methods in CAD, Springer Verlag, New York, 1989. 10. S. R. H. Hoole, Computer-Aided Analysis and Design of Electromagnetic Devices, Elsevier, New York, 1989. 11. D. A. Lowther and P. P. Sylvester, Computer Aided Design in Magnetics, Springer-Verlag, New York, 1986. 12. J. R. Brauer, What Every Engineer Should Know About Finite Element Analysis, 2nd ed., Marcel Dekker, New York, 1993. 13. N. Ida, Engineering Electromagnetics, Springer-Verlag, New York, 2000. 14. J.D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, New York, 1991.
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15. A. Ivanyi, Hysteresis Models in Electromagnetic Computation, Akademiai Kiado, Budapest, 1997. 16. A. Bossavit, Computational Electromagnetism, Academic Press, San Diego, 1998. 17. N. Ida, J. P. A. Bastos, Electromagnetics and Calculation of Fields, Springer-Verlag, New York, 1997. 18. M. L. James, S. M. Smith and J. C. Wolford, Applied Numerical Methods for Digital Computation, Harper & Row, New York, 1985. 19. J. Pachner, Handbook of Numerical Analysis Applications, McGrawHill, New York, 1984. 20. R. Courant and D. Hilbert, Methods of Computational Physics, Vol. 2, Interscience, 1961. 21. K. J. Binns and P. J. Lawrenson, Analysis and Computation of Electric and Magnetic Field Problems, Pergamon Press, 1973. 22. E. Durand, Electrostatique - Les Distributions, Masson et Cie., Paris, 1964. 23. E. Durand, Electrostatique - Problemes Generaux Conducteurs, Masson et Cie., Paris, 1966. 24. E. Durand, Magnetostatique, Masson et Cie., Paris, 1968. 25. G. Dhatt and G. Touzot, The Finite Element Method Displayed, Wiley Interscience, Chichester, 1984. 26. K. H. Huebner, The Finite Element Method for Engineers, Wiley Interscience, John Wiley & Sons, Inc., New York, 1975. 27. C. S. Desai and J. F. Abel, Introduction to the Finite Element Method, Van Nostrand Reinhold, 1972. 28. O. C. Zienkiewicz, The Finite Element Method in Engineering, Third Edition, McGraw-Hill, London, 1977. 29. S. G. Mikhlin, Variational Methods in Mathematical Physics, Macmillan, New York, 1964. 30. A. Arkkio Analysis of induction motors based on the numerical solution of the magnetic field and circuit equations, Acta Polytechnica Scandinavica, Helsinki, 1987 (Thesis). 31. Balabanian, N., Bickart, T., Electrical Network Theory, John Wiley & Sons Inc., 1969.
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32. Chua, L., Lin, P., Computer aided analysis of electronic circuits, Prentice Hall Inc., Englewood Cliffs, New Jersey, 1975. 33. H. H. Woodson, J. R. Melcher, Electromechanical dynamics, Part II Fields, Forces and Motion, John Wiley and Sons, New York, 1968. Papers on Nonlinear Methods 1. P. P. Silvester and M. V. K. Chari, "Finite element solution of saturable magnetic field problems," IEEE Transaction on Power Apparatus and Systems, Vol. 89,1970, pp. 1642-1651. 2. J. P. A. Bastos and G. Quichaud, "3D modeling of a nonlinear anisotropic lamination," IEEE Transactions on Magnetics, Vol. 21, No. 6, Nov. 1985, pp. 2366-2369. 3. J. Penman and A. M. A. Kamar, "Linearization of saturable magnetic field problems including eddy currents," IEEE Transactions on Magnetics, Vol. MAG-18, No. 2, March, 1982, pp. 563-566. 4. C. S. Holzinger, "Computation of magnetic fields within threedimensional highly nonlinear media," IEEE Transactions on Magnetics, Vol. MAG-6, No. 1, March 1970, pp. 60-65. 5. J. R. Brauer, R.Y. Bodine and L. A. Larkin, "Nonlinear anisotropic three-dimensional magnetic energy functional," Proceedings of the Compumag Conference, Santa Marguerita, Italy, May 1983. 6. J. R. Brauer, "Saturated magnetic energy function for finite element analysis of electrical machines," IEEE PES Winter Meeting, New York, 1976, paper C75, pp. 151-156. 7. J. P. A. Bastos, N. Ida, R. C. Mesquita, "A variable local relaxation technique in non-linear problems," IEEE Transactions on Magnetics, Vol. 31, No. 3, May 1995, pp. 1733-1736. 8. A. Kost, J. P. A. Bastos, K. Miethner, L. Janicke, "Improvement of nonlinear impedance boundary conditions," IEEE Transactions on Magnetics, Vol. 38, No. 2, March 2002, pp. 573-576. Papers on Solution Methods 1. J. A. Meijerink, and H. A. van der Vorst, "An interative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix," Mathematics of Computation, Vol. 31, No. 137, January 1977, pp. 148162.
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2. R. L. Stoll, "Solution of linear steady state eddy current problems by complex successive over-relaxation," IEE Proceedings, Part. A, Vol. 117, 1970, pp. 1317-1323. 3. J. K. Reid, "On method of conjugate gradients for the solution of large sparse systems of linear equations," in Large Sparse Sets of Linear Equations, Reid, J.K., Ed., Academic Press, 1971, pp. 231-252. 4. J. K. Reid, "Solution of linear systems of equations: direct methods (general)," in Lecture Notes in mathematics, Barker, V.A ., Ed., SpringerVerlag, Berlin, 1976, No. 572, pp. 102-130. 5. A. Jennings, "A compact storage scheme for the solution of symmetric simultaneous equations," Computer Journal, No. 9, September 1969, pp. 281-285. 6. J. A. George, "Solution of linear systems of equations: direct methods for finite element problems," in Lecture Notes in Mathematics, Barker, V. A ., Ed., Springer-Verlag, Berlin, 1976, No. 572, pp. 52-101. 7. J. A. George, "Sparse matrix aspects of the finite element method," Proceedings of the Second International Symposium on Computing Methods in Applied Science and Engineering, Springer-Verlag, 1976. 8. F. G. Gustavson, "Some basic techniques for solving sparse systems of equations," in Sparse Matrices and Their Applications, Rose, D.J. and Willoughby, R. A ., Eds., Plenum Press, New York, 1972, pp. 41-53. Papers on Magnetostatic and Electrostatic Formulations 1. P. Hammond, and T. D. Tsiboukis, "Dual finite-element calculations for static electric and magnetic fields," IEEE Proceedings, Vol. 130, Part. A, No. 3, May 1983, pp. 105-111. 2. M. V. K. Chari, M. A. Palmo and Z. J. Cendes, "Axisymmetric and three dimensional electrostatic field solution by the finite element method," Electric Machines and Electromechanics, Vol. 3, 1979, pp. 235-244. 3. M. V. K. Chari, P. P. Silvester and A. Konrad, "Three-dimensional magnetostatic field analysis of electrical machines by the finite element method," IEEE Transactions on Power Apparatus and Systems, Vol. PAS100, No. 8, August 1981, pp. 4007-4019. 4. Z. J. Cendes, J. Weiss and S. R. H. Hoole, "Alternative vector potential formulations of 3-D magnetostatic problems," IEEE Transactions on Magnetics, Vol. MAG-18, No. 2, March 1982, pp. 367-372.
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5. O. W. Andersen, "Two stage solution of three-dimensional electrostatic fields by finite differences and finite elements," IEEE Transactions on Power apparatus and systems, Vol. PAS-100, No. 8, Aug. 1981, pp. 3714-3721. 6. J. R. Cardoso, "A Maxwell's second equation approach to the finite element method applied to magnetic field determination," International Journal of Electrical Eng. Education - IJEE, Manchester, July 1977, Vol. 24, pp. 259-272. Papers on Eddy Current Formulations 1. C. J. Carpenter and M. Djurovic, "Three-dimensional numerical solution of eddy currents in thin plates," IEE Proceedings, Vol. 122, No. 6, 1975, pp. 681-688. 2. M. L. Brown, "Calculation of 3-dimensional eddy currents at power frequencies," IEE Proceedings, Vol. 129, Part A, No. 1, January 1982, pp. 46-53. 3. C. S. Biddlecombe, E. A . Heighway, J. Simkin and C. W. Trowbridge, "Methods for eddy current computation in three dimensions," IEEE Transactions on Magnetics, Vol. MAG-18, No. 2, March 1982, pp. 492-497. 4. P. Hammond, "Use of potentials in the calculation of electromagnetic fields," IEE Proceedings, Vol. 129, Part. A, No. 2, March 1982, pp. 106112. 5. T. W. Preston and A. B. J. Reece, "Solution of 3-dimensional eddy current problems: the T-Q method," IEEE Transactions on Magnetics, Vol. MAG-18, No. 2, March 1982, pp. 486-491. 6. C. R. I. Emson, J. Simkin and C. W. Trowbridge, "Further developments in three-dimensional eddy current analysis," IEEE Transactions on Magnetics, Vol. MAG-21, No. 6, Nov. 1985, pp. 2231-2234. 7. P. I. Koltermann, J. P. A . Bastos and S. R. Arruda, "A model for dynamic analysis of AC contactor," IEEE Transactions on Magnetics, Vol. 28, No. 2, March 1992, pp. 1348-1350. 8. R. C. Mesquita, J. P. A. Bastos, "3D finite element solution of induction heating problems with efficient time-stepping," IEEE Transactions on Magnetics, Vol. 27, No. 5, Sept. 1991, pp. 4065-4068. 9. N. Ida, "Three Dimensional Eddy Current Modeling," Review of Progress in Quantitative Nondestructive Evaluation, D. O. Thompson and D. E. Chimenti, Eds., Plenum Press, New York, Nov. 1987, Vol. 6A, pp. 201-209.
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10. H. Song and N. Ida, "An eddy current constraint formulation for 3D electromagnetic field calculation," IEEE Transactions on Magnetics, Vol. 27, No. 5, Sept. 1991, pp 4012-4015.
11. R. D. Pillsbury, "A three dimensional eddy-current formulation using two potentials: the magnetic vector potential and total magnetic scalar potential," IEEE Transactions on Magnetics, Vol. MAG-19, No. 6, 1983, pp.2284-2287. 12. R. L. Ferrari, "Complementary variational formulation for eddy-current problems using the field variables E and H directly," IEE Proceedings, Vol. 132, Part A, No. 4, July 1985, pp.157-164. 13. P. Hammond, "Calculation of eddy currents by dual energy methods," IEE Proceedings, Vol. 125, No. 7, 1978, pp.701-708. 14. C. R. I. Emson and J.Simkin, "An optimal method for 3-D eddycurrents," IEEE Transactions on Magnetics, Vol. MAG-19, No. 6, 1983, pp. 2450-2452. Papers on Formulations with Velocity Terms 1. N. Ida, "Modeling of effects in eddy current applications," Journal of Applied Physics, Vol. 63, No. 8, April 15, 1988, pp. 3007-3009. 2. N. Ida, "Velocity effects and low level fields in axisymmentric geometries," in COPEL - International Journal for Computation and Mathematics in Electrical Engineering, Vol. 9, No. 3, September 1990, pp. 169-180. Papers on Computation Systems and CAD 1. N. Ida, "PCNDT - An electromagnetic finite element package for personal computers," IEEE Transactions on Magnetics, Vol. 24, No. 1, January 1988, pp. 366-369. 2. P. Masse, J. L. Coulomb and B. Ancelle, "System design methodology in CAD programs based on finite element method," IEEE Transactions on Magnetics, Vol. MAG-18, No. 2, March 1982, pp. 609-616. 3. M. L. Barton, V. K. Garg. I. Ince, E. Stemheim and J. Weiss, "WEMAP: a general purpose system for electromagnetic analysis and design," IEEE Transactions on Magnetics, Vol. MAG-19, No. 6, November 1983, pp. 2674-2677. 4. B. Ancelle, E. Gallagher and P. Masse, "ENTREE: a fully parametric preprocessor for computer aided design of magnetic devices," IEEE Transactions on Magnetics, Vol. MAG-18, No. 2, March 1982, pp. 630-632.
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Papers on General Finite Element Method 1. J. T. Oden, "A general theory of finite elements I: Topological Considerations," International Journal for Numerical Methods in Engineering, Vol. 1, No. 2, 1969, pp. 205-221. 2. J. T. Oden, "A general theory of finite elements II: Applications," International Journal for Numerical Methods in Engineering, Vol. 1, No. 3, 1969, pp. 247-259. 3. M. J. Tuner, R. W. Clough, H. C. Matin and L. J. Topp, "Stiffness and deflection analysis of complex structures," Journal of Aeronautical Science, Vol. 23, 1956, pp. 805-823. 4. M. E. Gurtin, "Variational principles for linear initial-value problems," Quarterly of Applied mathematics, Vol. 22, 1964, pp. 252-256. Papers on Coupled Problems 1. Sadowski, N., Lefevre, Y., Lajoie-Mazenc, M., Bastos, J.P.A., "Calculation of transient electromagnetic forces in an axisymmetrical electromagnet with conductive solid parts," COMPEL, Vol. 11, N.I, March 1992, pp. 173176. 2. J.P.A. Bastos, N. Sadowski, R. Carlson, "A modeling approach of a coupled problem between electrical current and its thermal effects," IEEE Transactions on Magnetics, Vol. 26, No. 2, march 1990, pp. 536-539. 3. A. Arkkio, "Finite element analysis of cage induction motors fed by static frequency converters," IEEE Transactions on Magnetics, Vol. 26, No. 2, March 1990, pp. 551-554. 4. Strangas, E.G., Theis, K. R., "Shaded pole motor design and evaluation using coupled field and circuit equations, IEEE Transactions on Magnetics, Vol. MAG-21, N. 5, September 1985, pp. 1880-1882. 5. Piriou F., Razek, A., "Coupling of saturated electromagnetic systems to non-linear power electronic devices," IEEE Transactions on Magnetics, Vol. 24, N. 1, January 1988, pp. 274-277. 6. Bouillault, F., Kladas, A., Piriou, F., Razek, A., "Coupled electric-magnetic model for a synchronous machine associated with a static converter," ICEM Proceedings, pp. 453-456, Pisa, 1988. 7. Lombard, P., Meunier, G., "Coupling between magnetic field and circuit equations in 2D," Proceedings of the International Workshop on Electric
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and Magnetic Fields from Numerical Models to Industrial Applications, pp. 7.1-7.6, Liege, September 1992. 8. N. Sadowski, R. Carlson, S.R. Arruda, C.A. da Silva, M. Lajoie-Mazenc, "Simulation of single-phase induction motor by a general method coupling field and circuit equations," IEEE Transactions on Magnetics, Vol. 31, N. 3, May 1995, pp. 1908-1911. 9. Kuo-Peng, P., Sadowski, N., Bastos, J.P.A., Carlson, R., Batistela, N.J., Lajoie-Mazenc, M., "A general method for coupling static converters with electromagnetic structures," IEEE Transactions on Magnetics, Vol. 33, N. 2, March 1997, pp. 2204-2009. 10. P. Zhou, S. Stanton, Z. J. Cendes, "Dynamic modeling of three phase and single phase induction motors," Proceedings of IEEE Int. Electric Machines & Drives Conference, 1999. 11. J. R. Brauer, B. E. Mac Neal, F. Hirtenfelder, "New Constraint Technique for 3D Finite Element Analysis of Multitum Windings with Attached Circuits," IEEE Transactions on Magnetics, November 1993. 12. A. M. Oliveira, P. Kuo-Peng, M. V. Ferreira da Luz, N. Sadowski, J.P.A.Bastos, "Generalization of coupled circuit-field calculation for polyphase structures," IEEE Transactions on Magnetics, Vol. 37, No. 5, September 2001, pp. 3444-3447. 13. A. M. Oliveira, P. Kuo-Peng, N. Sadowski, M. S. de Andrade, J. P. A. Bastos, "A non-apriori approach to analyze electrical machines modeled by FEM connected to static converters," IEEE Transactions on Magnetics, Vol. 38, No. 2., March 2002, pp. 933-936. 14. R. Carlson, C. A. da Silva, N. Sadowski, Y. Lefevre, M. Lajoie-Mazenc, "The effect of the stator-slot opening on the interbar currents of skewed cage induction motor," IEEE Transactions on Magnetics, Vol. 38, No. 2, March 2002, pp. 1285-1288. Papers on Movement Modeling in Field Calculation 1. B. Davat, Z. Ren and M. Lajoie-Mazenc, "The movement in field modeling," IEEE Transactions on Magnetics, Vol. MAG-21, No. 6, November 1985, pp. 2296-2298. 2. A. A. Abdel-Razek, J. L. Coulomb, M. Feliachi and C. Sabonnadiere, "Conception of an air-gap element for the analysis of the electromagnetic field in electric machines," IEEE Transactions on Magnetics, Vol. MAG-18, No. 2, March 1982, pp. 655-659.
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3. J. Bigeon, J. C. Sabonnadiere, J. L. Coulomb, "Finite element analysis of na electromagnetic brake," IEEE Transactions on Magnetics, Vol. MAG19, No. 6, November 1983, pp. 2632-2634. 4. T. Furukawa, K. Komiya, I. Muta, "An upwind Galerkin finite element analysis of linear induction motors," IEEE Transactions on Magnetics, Vol. MAG-26, No. 2, March 1990, pp. 662-665. 5. S. J. Salon, J. M. Schneider, "A hybrid finite element - boundary integral formulation of Poisson's equation," IEEE Transactions on Magnetics, Vol. 17, No. 6, November 1981, pp. 2574-2576. 6. S. J. Salon, J. M. Schneider, "A hybrid finite element - boundary integral formulation of Poisson's equation," IEEE Transactions on Magnetics, Vol 18, No. 2, March 1982, pp. 461-466. 7. D. Rodger, H. C. Lai, P. J. Leonard, "Coupled elements for problems involving movement," IEEE Transactions on Magnetics, Vol. 26, No. 2, March 1990, pp. 548-550. 8. Ki-sik Lee, M. J. De Bortoli, M. J. Lee, S. J. Salon, "Coupling finite elements and analytical solution in the airgap of electrical machines," IEEE Transactions on Magnetics, Vol. 27, No. 5, September 1991, pp. 39553957. 9. M. V. F. da Luz, P. Dular, N. Sadowski, C. Geuzaine, J. P. A. Bastos, "Analysis of a permanent magnet generator with dual formulations using periodicity conditions and moving band," IEEE Transactions on Magnetics, Vol. 38, No. 2, March 2002, pp. 961-964. 10. D. Deas, P. Kuo-Peng, N. Sadowski, A. M. Oliveira, J. L. Roel, J. P. A. Bastos, "2-D FEM modeling of the tubular linear induction motor taking into account the movement," IEEE Transactions on Magnetics, Vol. 38, No. 2, March 2002, pp. 1165-1168. Papers on Interaction between Electromagnetic and Mechanical Forces 1. Sadowski, N., Lefevre, Y., Lajoie-Mazenc, M., Bastos, J.P.A., "Sur le calcul des forces magnetiques," Journal de Physique III, Vol. 2, N. 5, pp. 859-870, May 1992. 2. C. J. Carpenter, "Surface - integral methods of calculating forces on magnetized iron parts," IEE - Monograph, N. 342, August 1959, pp. 19-28. 3. T. Tarnhuvud, K. Reichert, "Accuracy problems of force and torque calculation in FE-Systems," IEEE Transactions on Magnetics, Vol. 24, No. 1, January 1988, pp. 443-446.
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4. G. Reyne, G. Meunier, J. F. Imhoff, E. Euxibie, "Magnetic forces and mechanical behaviour of ferromagnetic materials. Presentation and results on the theoretical, experimental and numerical approaches," IEEE Transactions on Magnetics, Vol 24, 1988, pp. 234-237. 5. W. Miiller, "Comparison of different methods of force calculation," IEEE Transactions on Magnetics, Vol 26, No. 2, March 1990, pp. 10581061. 6. Sadowski, N., Lefevre, Y., Lajoie-Mazenc, M., Cros, J., "Finite element torque calculation in electrical machines while considering the movement," IEEE Transactions on Magnetics, Vol. 28, N. 2, March 1992, pp. 14101413. 7. C.G.C. Neves, R.Carlson, N.Sadowski, J.P.A.Bastos, N.S. Soeiro, S.N.Y. Gerges, "Vibrational behavior of switched reluctance motors by simulation and experimental procedures," IEEE Transactions on Magnetics, Vol. 34, N. 5, September 1998, pp. 3158-3161. 8. C.G.C. Neves, R.Carlson, N.Sadowski, J.P.A.Bastos, "Experimental and numerical analysis of induction motor vibrations," IEEE Transactions on Magnetics, Vol. 35, N. 3, May 1999, pp. 1314-1317. 9. C.G.C. Neves, R.Carlson, N.Sadowski, J.P.A.Bastos, N.S. Soeiro, S.N.Y. Gerges, "Calculation of electromagnetic-mechanic-acoustic behavior of a switched reluctance motor," IEEE Transactions on Magnetics, Vol. 36, N.4, July 2000, pp.1364-1367. Papers on Hysteresis and Iron Losses 1. J.D. Lavers, P.P. Biringer, H. Hollitscher, "A simple method of estimating the minor loop hysteresis loss in thin laminations," IEEE Transactions on Magnetics, Vol. 14, No. 5, September 1978, pp. 386-388. 2. F. Preisach, "Uber die magnetische Nachwirking," Zeit. Phys. 94, 1935, pp. 277-302. 3. K. Atallah, Z. Q. Zhu, D. Howe, "An improved method for predicting iron losses in brushless permanent magnetc DC drives," IEEE Transactions on Magnetics, Vol. 28, No. 5, September 1992, pp. 2997-2998. 4. A. Boglietti, O. Bottauscio, M. Chiampi, M. Pastorelli, M. Repetto, "Computation and measurement of iron losses under PWM supply conditions," IEEE Transactions on Magnetics, Vol. 32, No. 5, September 1996, pp. 4302-4304. 5. M. Chiampi, D. Chiarabaglio, M. Repetto, "A Jiles-Atherton and FixedPoint Combined Technique for Time Periodic Magnetic Field Problems with
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Hysteresis," IEEE Transactions on Magnetics, Vol. 31, No. 6, November 1995, pp. 4306-4311. 6. F. Ossart, V. lonita, "Convergence de la methode du point fixe modifiee pour le calcul de champ magnetique avec hysteresis," The European Physical Journal Applied Physics, AP 5, 1999, pp. 63-69. 7. V. lonita, B. Cranganu-Cretu, D. loan, "Quasi-Stationary Magnetic Field Computation in Hysteretic Media," IEEE Transactions on Magnetics, Vol. 32, No. 3, May 1996, pp. 1128-1131. 8. P. Alotto, P. Girdinio, P. Molfino, "A 2D Finite Element Procedure for Magnetic Analysis Involving Non-Linear and Hysteretic Materials," IEEE Transactions on Magnetics, Vol. 30, 1994, pp. 3379-3382. 9. K. Muramatsu, N. Takahashi, T. Nakata, M. Nakano, Y. Ejiri, J. Takehara, "3-D Time-Periodic Finite Element Analysis of Magnetic Field in NonOriented Materials Taking into Account Hysteresis Characteristics," IEEE Transactions on Magnetics, Vol. 33, No. 2, March 1997, pp. 1584-1587. 10. D.C. Jiles, J.B. Thoelke, M.K. Devine, "Numerical Determination o of Hysteresis Parameters for the Modeling of Magnetic Property Using the Theory of Ferromagnetic Hysteresis," IEEE Transactions on Magnetics, Vol. 28, No. 1, January 1992, pp. 27-35. 11. G.Bertotti. "General Properties of Power Losses in Soft Ferromagnetic Materials," IEEE Transactions on Magnetics, Vol. 24, No.l, January 1988. pp.621-630. 12. D. C. Jiles and D. L. Atherton, "Theory of the magnetisation process in ferromagnets and its application to the magnetomechanical effect," Journal Phys. D: Appl. Phys. 17, 1984, pp. 1265-1281. 13. D. C. Jiles and D. L. Atherton, "Theory of ferromagnetic hysteresis," Journal of Magnetism and Magnetic Materials 61, 1986, pp. 48-60. 14. G. Ban, G. Bertotti, "Dependence on peak induction and grain size of power losses in nonoriented SiFe steels," J. Appl. Phys. 64 (10), November 1988, pp. 5361-5363. 15. G. Bertotti, "Physical interpretation of eddy current losses in ferromagnetic materials. I. Theoretical considerations," J. Appl. Phys. 57 (6), March 1985, pp. 2110-2117. 16. G. Bertotti, "Physical interpretation of eddy current losses in ferromagnetic materials. II. Analysis of experimental results," J. Appl. Phys. 57 (6), March 1985, pp. 2118-2126.
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17. G. Bertotti, "General properties of power losses in soft ferromagnetic materials," IEEE Transactions on Magnetics, Vol. 24, No. 1, January 1988, pp. 621-630. 18. N. Sadowski, M. Lajoie-Mazenc, J. P. A. Bastos, M. V. F. Luz, P. KuoPeng, "Evaluation and analysis of iron losses in electrical machines using the Rain-flow method," IEEE Transactions on Magnetics, Vol. 36, No. 4, July 2000, pp. 1923-1926. 19. N. Sadowski, N. J. Batistela, J. P. A. Bastos, M. Lajoie-Mazenc, "An inverse Jiles-Atherton model to take into account hysteresis in time-stepping finite-element calculations," IEEE Transactions on Magnetics, Vol. 38, No. 2, March 2002, pp. 797-800. 20. P. I. Koltermann, J. P. A. Bastos, N. Sadowski, N. J. Batistela, A. Kost, L. Janicke, K. Miethner, "Nonlinear magnetic field model by FEM taking into account hysteresis characteristics with M-B variables," IEEE Transactions on Magnetics, Vol. 38, No. 2, March 2002, pp. 897-900.
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