Interactions Between Electromagnetic Field and Moving Conducting Strip (Lecture Notes in Electrical Engineering, 1111) 3031482735, 9783031482731

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Lecture Notes in Electrical Engineering 1111

Ihor Kondratenko Yuriy Vasetsky Artur Zaporozhets

Interactions Between Electromagnetic Field and Moving Conducting Strip

Lecture Notes in Electrical Engineering Volume 1111

Series Editors Leopoldo Angrisani, Department of Electrical and Information Technologies Engineering, University of Napoli Federico II, Napoli, Italy Marco Arteaga, Departament de Control y Robótica, Universidad Nacional Autónoma de México, Coyoacán, Mexico Samarjit Chakraborty, Fakultät für Elektrotechnik und Informationstechnik, TU München, München, Germany Jiming Chen, Zhejiang University, Hangzhou, Zhejiang, China Shanben Chen, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, China Tan Kay Chen, Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore Rüdiger Dillmann, University of Karlsruhe (TH) IAIM, Karlsruhe, Baden-Württemberg, Germany Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Gianluigi Ferrari, Dipartimento di Ingegneria dell’Informazione, Sede Scientifica Università degli Studi di Parma, Parma, Italy Manuel Ferre, Centre for Automation and Robotics CAR (UPM-CSIC), Universidad Politécnica de Madrid, Madrid, Spain Faryar Jabbari, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA Limin Jia, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Janusz Kacprzyk, Intelligent Systems Laboratory, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Alaa Khamis, Department of Mechatronics Engineering, German University in Egypt El Tagamoa El Khames, New Cairo City, Egypt Torsten Kroeger, Intrinsic Innovation, Mountain View, CA, USA Yong Li, College of Electrical and Information Engineering, Hunan University, Changsha, Hunan, China Qilian Liang, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX, USA Ferran Martín, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain Tan Cher Ming, College of Engineering, Nanyang Technological University, Singapore, Singapore Wolfgang Minker, Institute of Information Technology, University of Ulm, Ulm, Germany Pradeep Misra, Department of Electrical Engineering, Wright State University, Dayton, OH, USA Subhas Mukhopadhyay, School of Engineering, Macquarie University, NSW, Australia Cun-Zheng Ning, Department of Electrical Engineering, Arizona State University, Tempe, AZ, USA Toyoaki Nishida, Department of Intelligence Science and Technology, Kyoto University, Kyoto, Japan Luca Oneto, Department of Informatics, Bioengineering, Robotics and Systems Engineering, University of Genova, Genova, Genova, Italy Bijaya Ketan Panigrahi, Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, Delhi, India Federica Pascucci, Department di Ingegneria, Università degli Studi Roma Tre, Roma, Italy Yong Qin, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Gan Woon Seng, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Singapore Joachim Speidel, Institute of Telecommunications, University of Stuttgart, Stuttgart, Germany Germano Veiga, FEUP Campus, INESC Porto, Porto, Portugal Haitao Wu, Academy of Opto-electronics, Chinese Academy of Sciences, Haidian District Beijing, China Walter Zamboni, Department of Computer Engineering, Electrical Engineering and Applied Mathematics, DIEM—Università degli studi di Salerno, Fisciano, Salerno, Italy Junjie James Zhang, Charlotte, NC, USA Kay Chen Tan, Department of Computing, Hong Kong Polytechnic University, Kowloon Tong, Hong Kong

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Ihor Kondratenko · Yuriy Vasetsky · Artur Zaporozhets

Interactions Between Electromagnetic Field and Moving Conducting Strip

Ihor Kondratenko Institute of Electrodynamics National Academy of Sciences of Ukraine Kyiv, Ukraine

Yuriy Vasetsky Institute of Electrodynamics National Academy of Sciences of Ukraine Kyiv, Ukraine

Artur Zaporozhets General Energy Institute National Academy of Sciences of Ukraine Kyiv, Ukraine

ISSN 1876-1100 ISSN 1876-1119 (electronic) Lecture Notes in Electrical Engineering ISBN 978-3-031-48273-1 ISBN 978-3-031-48274-8 (eBook) https://doi.org/10.1007/978-3-031-48274-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The main purpose, which is set in the book, combines two interrelated directions of research. One of them is devoted to the development of the theory for solving a certain class of three-dimensional electromagnetic field problems of the three-dimensional electromagnetic field, taking into account eddy currents in a moving conducting magnetizing body. Preference is given to the development of the analytical solution methods of the three-dimensional quasi-stationary problem of field conjugation in the system: “a contour of an arbitrary spatial configuration with an alternating current is conducting body with a flat boundary surface.” The choice of analytical calculation methods is due to several reasons: the recently obtained analytical solution of threedimensional problems in a fairly general formulation, which may be useful in the field of theoretical research; the ability to obtain the general patterns of the formation of a three-dimensional field and the analysis of physical processes; the use of analytical approaches, where the numerical methods are not effective enough; and the use of a set of exactly solvable problems that can serve as a benchmark for comparison in the development of other methods. The second direction refers to the development of mathematical models for solving applied problems, which involve the use of developed methods for calculating the electromagnetic field and their characteristics. The main application of calculation methods is aimed at solving problems of heat treatment non-ferrous and ferrous metal strips using the induction method of heating in a transverse magnetic field. In particular, research is aimed at determining the geometry and the parameters of heat treatment of metal products, which provide a given character of heating and temperature distribution in the conducting magnetizable strip. Chapter “Mathematical Models of Electromagnetic Interaction of Field Sources with Conducting Body” deals with mathematical models for studying the electromagnetic interaction of field sources with a conducting half-space. The mathematical models are based on the presented exact analytical solution of the field conjugation problem, which has no restrictions on the geometric configuration of the external field sources, the properties of the media, and the frequency of the field used. In general, the problem is complicated in the computation sense, especially when solving inverse v

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problems of field theory. Therefore, the approximate methods of calculation which allow considering the most essential geometrical, electro-physical, and heat transfer properties of electromagnetic systems are presented. The approximate model is based on the exact solution’s expansion into asymptotic series. An even simpler mathematical model of a locally two-dimensional electromagnetic field is used in the case of a close location of the field sources and the conducting medium and a small inclination angle of the inductor conductors relative to the surface of the conducting body. It is shown that the value of this energy differs significantly for sections of the strip passing under the edge of the inductor contour and the rest of it. The mathematical model of heat transfer is substantiated, in which the temperature is uniform throughout the thickness, and the process is considered adiabatic in the longitudinal directions. In Chapter “Configuration of Spatial Iron-Free Inductors for High-Frequency Induction Heating of Metal Strips”, studies on solving inverse problems of finding the inductor’s geometry are formulated as parametric optimization problems in a certain class of current contours of spatial configurations. The investigations focus on determining the geometric configuration of iron-free electromagnetic field inductors in the form of current contours intended for heat treatment of metal strips, the induction heating carried out when they move in the transverse high-frequency field. It is substantiated by the expediency of using inductors in the form of current spatial contours with edges raised above the surface. The optimal configurations of spatial inductors are found for the following important practical heating conditions: the linear density of the electromagnetic energy flux does not exceed the specified maximum value; does not fall below the specified minimum value; and has minimum deviation from the average value at a certain width. Methods for achieving uniform heating of non-ferrous and ferrous metal strips over the entire width and in the local area are analyzed. The research in Chapter “Electromagnetic Systems of Transverse Magnetic Flux with Ferromagnetic Core for Induction Heating Devices” is dedicated to issues related to the use of single-phase transverse magnetic field inductors with magnetic cores in induction heating devices for thin moving metal strips. The used mathematical models, there are no restrictions on the thickness and width of the metal strip, the frequency of the field, and strip speed. The analytical method is based on a twodimensional integral transformation of the magnetic field and current load. The solution is found for the electromagnetic field in the whole space and the current density in the conducting strip. In addition to the distribution of the electromagnetic field and current density, specific expressions are obtained for energy and force characteristics. An analysis of the attractive forces of ferromagnetic strips to the magnetic core of the single-phase inductor is conducted. It is established that the electrodynamic stabilization of the strip position in the center of the air gap is achieved in the range of intermediate frequencies (400–800 Hz), depending on the magnitude of the voltage (current) supplied to the inductor. The minimum heating non-uniformity for single-phase inductors is approximately 15%. Chapter “Electromagnetic Systems with Iron-Free Inductors for Induction Heating of Moving Strip in Transverse Magnetic Field” discusses electromagnetic systems for the thermal treatment of metal strips using induction heating by eddy

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currents. A distinctive feature of this research is the simultaneous solution of electromagnetic and thermal problems in a three-dimensional setting, taking into account the constant velocity of the moving conductive object. Both one-sided and two-sided inductors are considered. The induction heating system includes both the heating element (inductor and heated object) and its electrical power source. This allows to determine the equivalent resistance of the heating element and perform the analysis of the electrical characteristics of the entire system. First, a general problem is considered for heating systems using inductors in the form of flat contours of arbitrary shape. Then, the results for specific canonical contour shapes such as rectangles, rhombi, and ellipses are presented. For these configurations, the possibilities of achieving a uniform temperature across the width of the strip are analyzed, including the combined use of multiple contours with different geometries. To achieve uniform heating across the width of the strips, it is advisable to use combinations of current contours, where the geometric dimensions are determined by the size and electrophysical parameters of the metal strips. The chapter concludes with an investigation of the influence of the finite height of the current-carrying conductors of the inductor. Kyiv, Ukraine August 2023

Ihor Kondratenko Yuriy Vasetsky Artur Zaporozhets

Contents

Mathematical Models of Electromagnetic Interaction of Field Sources with Conducting Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematical Model for Calculation of Electromagnetic Field Created by Sources Near Conducting Half-Space . . . . . . . . . . . . . . . . . . . . 2.1 Exact Analytical Solution of the Three-Dimensional Problem . . . . . 2.2 Asymptotic Expansion Method for Approximate Calculation of the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Approximate Mathematical Model of the Diffusion Electromagnetic Field into Conducting Half-Space for Calculation of the Field Near Current Contour . . . . . . . . . . . . . . . 3 The Electromagnetic Energy Flux During High-Frequency Induction Heating of Moving Conducting Strips . . . . . . . . . . . . . . . . . . . . . 3.1 Small Parameters for the Formulation of an Approximate Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Electromagnetic Energy Flux Density into Heating Metal Strip . . . . 4 Assessment of the Main Parameters of the Heating Process . . . . . . . . . . . . 4.1 Temperature Stabilisation Along Thickness . . . . . . . . . . . . . . . . . . . . 4.2 Heat Transfer Along Surface of Metal Strip . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Configuration of Spatial Iron-Free Inductors for High-Frequency Induction Heating of Metal Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Mathematical Model of the Electromagnetic System for High-Frequency Heating of Metal Strips . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Main Assumptions at High-Frequency Heating of Metal Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Geometric Parameters of the Spatial Configuration Contours . . . . . .

1 2 4 5 8

13 19 20 22 27 27 28 31 33 37 37 39 39 41

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2.3 Some Examples of Calculating the Distribution of Active Power Flow Through the Surface of a Metal Strip . . . . . . . . . . . . . . . 3 Geometric Parameters of Spatial Contours with Current (h = var, R = const) for Leveling Heating of Metal Strips . . . . . . . . . . . . . . . . . . . . . 3.1 Estimation of the Ratio of the Height of the Contour Edge to the Height of the Central Part According to the Condition of Heat Release Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Geometric Parameters, Provided that the Temperature Across the Width Does not Exceed the Specified Value . . . . . . . . . . . . . . . . . 3.3 Geometric Parameters in Case, When the Temperature Does not Exceed the Set Maximum Value . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Contour Configurations in the Case of Minimal Deviation at a Given Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Some Examples of Inductor Geometry for Leveling Heating of Metal Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Flat Elliptical Contours (h = const, R = var) . . . . . . . . . . . . . . . . . . . . . . . . 5 Features of the Application of the Asymptotic Method for the Study of Induction Heating of Limited Width Conducting Strips . . . . . . . . . . . . . 5.1 Mathematical Model for Calculating the Electromagnetic Field of the Current Contour Over Conductive Strip of Limited Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Geometric Parameters of the Electromagnetic System for Induction Heating of a Metal Strip of Limited Width with Minimum Temperature Non-uniformity Along Its Width . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic Systems of Transverse Magnetic Flux with Ferromagnetic Core for Induction Heating Devices . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Main Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Magnetic Field of Transverse Magnetic Flux Inductors with Ferromagnetic Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Energy Characteristics and Electromagnetic Forces in Multi-pole Single-Phase Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Induction Heating of Strips in Single-Phase Inductor . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic Systems with Iron-Free Inductors for Induction Heating of Moving Strip in Transverse Magnetic Field . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 General Solution of the Electromagnetic and Temperature Problem During Heat Treatment of a Moving Strip by Iron-Free Inductors as Arbitrary Planar Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Heat Treatment of Moving Strip by Iron-Free Inductors as Current Contours of Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Rectangular Current Contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Current Contour in the Form of a Rhombus . . . . . . . . . . . . . . . . . . . . 3.3 Current Contour in the Form of an Ellipse . . . . . . . . . . . . . . . . . . . . . . 4 Influence of the Finite Height of the Winding Cross-Section on the Energy Characteristics of the Inductor . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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About the Authors

Prof. Ihor Kondratenko is working as Head of Department of Electromagnetic Systems, Institute of Electrodynamics, National Academy of Sciences of Ukraine. He graduated from the Faculty of Electroacoustics, National Technical University of Ukraine, Kyiv Polytechnic Institute (1978) and post-graduated in Institute of Electrodynamics (1986). He obtained his Ph.D. in 1987 and Dr. Sc. in 2005. He has been Corresponding Member of the National Academy of Sciences of Ukraine since 2015. He received the State Prize of Ukraine in the field of science and technology in 2018. He is laureate of V. Khrushchev (2010) and G. F. Proskura (2014). The main profession activity during from 1979 to the present time is associated with the Institute of Electrodynamics, National Academy of Sciences of Ukraine. He has published six monographs in electrical engineering, over 260 articles in international and national journals and at conferences. Prof. Yuriy Vasetsky is working as Chief Researcher in Electromagnetic Systems Department, Institute of Electrodynamics, National Academy of Sciences of Ukraine. He graduated from the Electric Power Faculty in Moscow Power Engineering Institute (Technical University) (1973) and post-graduated in Institute of Electrodynamics (1979). He obtained his Ph.D. in 1981 and Dr. Sc. in 1995. He is laureate of G. F. Proskura Award of NAS of Ukraine. The main profession activity during from 1979 to the present time is associated with the Institute of Electrodynamics, National Academy of Sciences of Ukraine. In additiona, he worked as a professor in Universities in Ukraine: Kyiv Commercial-Economical University (1999–2003, course of lectures in physics), Physic-technical Institute of the Kyiv National Technical University (2001–2003, course of lectures in electromagnetic theory), Chair of Applied Physics and Chair of Electrical Engineering and Lighting Engineering in National Aviation University (2003–2019, course of lectures in Electrodynamics and High Voltage Technology). Dr. Artur Zaporozhets is working as Leading Researcher in the General Energy Institute of the National Academy of Sciences of Ukraine. He graduated from the Applied Physics Department of the National Aviation University in 2013. He obtained xiii

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his Ph.D. in 2017 and Dr. Sc. (Eng.) in 2022. He is a laureate of the Medal “For work and achievement” (2022), Award of the NAS of Ukraine for young scientists (2022), Award of the Verkhovna Rada of Ukraine to young scientists (2021), Award of the President of Ukraine for young scientists (2019), and others. He has published more 200 scientific works, among them 14 books in Springer, over 50 articles in international peer-reviewed journals and 60 conference proceedings. His current research interests include energy informatics, power equipment diagnostics, environmental monitoring, algorithms and data structures, big data, data processing.

Mathematical Models of Electromagnetic Interaction of Field Sources with Conducting Body

Abstract The chapter deals with mathematical models for studying the electromagnetic interaction of field sources with a conducting body. In the general case, the presented exact analytical solution of the field conjugation problem on a flat interface between media can be used as a mathematical model for finding the electromagnetic field. The solution has no restrictions on the geometric configuration of the external field sources, the properties of the media and the frequency of the field used. The approximate model is based on the expansion of the exact solution into an asymptotic series. The model is valid for processes in which the product of the field penetration depth and the relative magnetic permeability of the conducting medium does not exceed the distance between the field sources and the media interface. An even simpler mathematical model of a locally two-dimensional electromagnetic field is valid in the case of a close location of the field sources and the conducting medium. The model can be used to study processes with strong interaction between field sources conducting body, for example, in induction heating devices of conducting bodies. Mathematical models are considered for induction devices of heat treatment by a high-frequency field of moving conducting strips, the thickness of which significantly exceeds the field penetration depth. It is assumed that the field is created by an inductor without a ferromagnetic core in the form of current contour in the general case of a spatial configuration. Using the model of the locally two-dimensional field, the value of the surface density of energy flux into the metal strip is analyzed. It is shown that the value of this energy differs significantly for sections of the strip passing under the edge of the inductor contour and the rest of it. The mathematical model of heat transfer is substantiated, in which the temperature is uniform throughout the thickness, and the process is considered adiabatic in the longitudinal directions. Keywords 3D electromagnetic field · Approximate mathematical models · Induction heating · Heat treatment of moving strips

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. Kondratenko et al., Interactions Between Electromagnetic Field and Moving Conducting Strip, Lecture Notes in Electrical Engineering 1111, https://doi.org/10.1007/978-3-031-48274-8_1

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Mathematical Models of Electromagnetic Interaction of Field Sources …

1 Introduction The chapter combines two interrelated lines of research. One of them is devoted to the development of methods for calculating the three-dimensional electromagnetic field in the general case, taking into account eddy currents in a conducting body. Preference is given to the theory developed in [1, 2] of the analytical solution of the threedimensional quasi-stationary problem of field conjugation in the system: “a contour of an arbitrary spatial configuration with an alternating current is conducting body with a flat boundary surface”. The developed theory is based on an exact analytical solution of the electromagnetic field problem for radiating current dipoles, taking into account the influence of conducting half-space [3–5], as well as a system of such dipoles distributed along the contour [6]. The found solution has no restrictions on the configuration of the external field sources, the electro-physical properties of the media and the frequency of the field. The second direction refers to the development of mathematical models for solving applied problems, which involve the use of developed methods for calculating the electromagnetic field and their characteristics. The main application of calculation methods is aimed at solving problems of high-frequency induction heating of metal products [7–9]. At the same time, the calculation method proposed in the chapter can also be used in developments in the field of technology for changing the properties of materials under the influence of high-density currents [10, 11], in devices on high-speed electromagnetic forming technology [12–15], and in other applications. Despite the presence of analytic expressions, the amount of necessary calculations remains significant. The problem becomes especially time consuming for inverse problems of field theory and for solving optimization problems of electromagnetic systems that arise in the development of specific devices. Therefore, it is expedient to develop approximate calculation methods based on the exact solution. When a high-frequency electromagnetic field interacts with conducting body, a strong skin effect takes place, in which the current and electromagnetic field are concentrated in a thin surface layer. / / A perfect skin effect takes place, if the field penetration depth δ = 2 (ωμμ0 γ ) at cyclic frequency ω in the conducting medium with conductivity γ and relative magnetic permeability μ is negligible compared to any characteristic dimensions L of the electromagnetic system. At the same time, the calculation method proposed in the chapter can also be used in developments of technology for changing the properties of materials under the influence of high-density currents [10, 11], in devices on highspeed / electromagnetic forming technology [12–15], and in other applications. When δ L → 0 for calculation of field outside the conducting body it is necessary to solve the corresponding stationary problem for perfect conducting body of the same shape [16, 17]. Locally electromagnetic field distribution in surface layer of conducting body is the same when uniform field penetrates into conducting half-space [18, 19]. To take into account the small but finite penetration depth of the electromagnetic field the most advanced methods are those based on the perturbation method [20, 21]. An effective technique for determine the field in the dielectric region outside the

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conductor is to use the impedance boundary condition [16, 22], which establishes a connection between the components of the electric and magnetic fields tangential to the surface. Modeling of electrodynamics problems, taking into account the geometric and physical properties of boundary surfaces, can be based on the use of the impedance boundary condition [23, 24]. A detailed analysis of the application of various numerical methods using the impedance boundary condition in electrodynamics problems is presented in the book [21]. The exact solution made it possible to proposed justified approximate methods for solving problems based on the asymptotic expansion of the exact expressions [25, 26]. One of main problems of induction heating of metal strip is to ensure of certain temperature of the strip moving across alternating electromagnetic field of the inductor [7]. The temperature distribution is determined, first of all, by the distribution of Joule heat releases associated with the flow of induced currents in conducting medium. The search for the geometry of electromagnetic systems according to a given condition for induction heating of conducting bodies is a inverse problem formulated as an optimization parametric problem for a certain class of configurations of contours with current located above conducting magnetizable half-space. Such formulation for high-frequency heating of metal strips is valid if the strip thickness is at least several times greater than the field penetration depth [27, 28]. In practice, it is usually necessary to carry out the most uniform heating of the product over a certain surface width [30]. In the general case the problem is complicated in the computation sense. Therefore the approximate asymptotic methods of calculation which allow considering the most essential geometrical, electrophysical and heat-transfer properties of electromagnetic systems are rational. Main objective is the analysis of electromagnetic and heat-transfer parameters of electromagnetic system of metal strips high-frequency induction heating for definition of a possibility to use of the approximate mathematical models and to determine of conditions under which electromagnetic and thermal problems can be considered separately. To do this, we analyze the conditions for using the asymptotic expansion method for the field created by the inductor as a current circuit located above the conducting half-space. This makes it possible to determine analytical estimates of the geometric parameters of the system for the uniform distribution of thermal energy generated in the strip moving in the field of the inductor. The distribution of thermal energy determines the temperature distribution in the strip. At the same time, the temperature distribution in the metal strip is affected by significantly different sizes in the transverse direction and along the surface, as well as the uneven nature of the transfer of thermal energy during the process of moving the strip under different sections of the inductor. The task will be simplified if the heat transfer along the surface of the strip is negligible. All these factors are taken into account in the developed mathematical models [31, 32].

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Mathematical Models of Electromagnetic Interaction of Field Sources …

2 Mathematical Model for Calculation of Electromagnetic Field Created by Sources Near Conducting Half-Space The exact and approximate mathematical models of the high-frequency quasistationary electromagnetic field, which is formed near conducting body with flat surface, are based on the following system: an arbitrary configuration contour with current I0 (t) located above conducting magnetized half-space in which eddy currents are induced. In Fig. 1, the field source contour is shown as a solid curve in the upper half-space z > 0. The conducting medium has electrical conductivity γ and relative magnetic permeability μ. The linear problem in a piecewise homogeneous medium is considered. Since in the quasi-stationary formulation, which does not take into account wave phenomena, the current density satisfies the continuity condition ∇ · j = 0, the contour is closed and the current I0 (t) does not change along the contour. Since steady sinusoidal processes of certain cyclic frequency ω are considered, then all quantities that change according to the sinusoidal law will be written in the form of complex-value amplitudes. These quantities are vector and scalar potentials, ˙ e , φ˙ e , electric and magnetic field intensities in both areas: in the dielectric half-space A ˙ e , where the closed contour l with current I˙0 is located, and in the conducting E˙ e , H ˙ i , φ˙ i , E˙ i , H ˙ i , where the eddy current with density ji = γ E˙ i flows. half-space A As it is often used the complex-value amplitudes are marked with a dot over the corresponding symbols. The problem easily extends to the general case of an arbitrary system of contours, that is, an arbitrary external field and to an arbitrary dependence of current on time I0 (t) using a Fourier transform over time. Fig. 1 Initial contour with current I 0 located above conducting half-space and its mirror reflection from a flat interface

2 Mathematical Model for Calculation of Electromagnetic Field Created …

5

2.1 Exact Analytical Solution of the Three-Dimensional Problem (a) In dielectric half-space the exact expressions for the potentials and the field intensities are determined by single function Ge [2] ˙ ˙ e = μ0 I0 A 4π

∮ ( l

φ˙ e = i ω

) t t1 ∂G e − − t1 dl; r r1 ∂z

μ0 I˙0 4π

(1)

∮ (t1 · ez )G e dl;

(2)

l

) ∮ ( t t1 μ0 I˙0 − − ez × [t1 × ∇G e ] dl; E˙ e = −i ω 4π r r1

(3)

l

)] ( ˙ ∮ [ t × r t1 × r1 ∂G e ˙ e = − I0 dl, H − − t × ∇ 1 4π r3 ∂z r13

(4)

l

where function Ge using dimensionless values are determined by following improper integral 2 Ge = √ i

) ( ) ( ∮∞ exp − χ cos β1 J0 χ sin β1 ε1 ε1 0

w1 (χ )

dχ .

(5)

The denominator w1 (χ) in the integrands (5) is written as χ w1 (χ ) = √ + i

/

( 1+

χ √ μ i

)2 .

(6)

Here J 0 (·) is Bessel function of the first kind of zero order, i is the imaginary unit. The axis z is oriented perpendicular to the interface surface in the direction of the single vector ez . For arbitrary spatial contour, the unit tangent vector to the contour t = t∥ + t⊥ has nonzero projections onto the vertical direction t⊥ = (t · ez )ez and onto the interface between the media t∥ = t − (t · ez )ez . The geometric quantities included in expressions (1)–(5) are shown in Figs. 1 and 2. The elements tdl of the initial contour and t1 dl of the mirror reflection contour relative to the interface are located at the points M and M 1 respectively. Projections of tangent vectors onto the vertical axis are equal in absolute value and opposite in their directions (t1z = −tz ), and the projections t∥ and t1∥ onto the plane of interface between media are equal in their lengths and directions t1∥ = t∥ , i.e. t = t∥ + tz , t1 = t1∥ + t1z = t∥ − tz . The vectors r = (zM − z)ez + ρ and r1 = (zM1 − z)ez + ρ = −(zM + z)ez + ρ (the vector ρ

6

Mathematical Models of Electromagnetic Interaction of Field Sources …

Fig. 2 Location the contour element tdl and its mirror reflection t1 dl relative to the observation point Q

is the projection of vector r or vector r1 onto interface) determine positions of points M and M 1 relative to the observation point Q. The angle β 1 shows the /orientation (√ ) 2r1 is of the vector r1 relative to the vertical axis. The parameter ε1 = μδ connected with penetration depth δ and distance r 1 between points M 1 and Q. The first and second terms in (1), (3) and (4) describe the solution of the problem for perfect skin-effect when δ → 0 [1]. At the value of the penetration depth other than zero the impact of electro-physical properties of the medium is taken into account by the third term. (b) In conducting half-space the expression for electric field intensity is the following [1, 26] ∮ [ ] μ0 I˙0 2i ω E˙ i = − t∥ T1 + (t · ez )eρ T2 dl. 4π

(7)

l

˙ i in conducting half-space follows The expression for the magnetic field intensity H ˙ ˙ from the Maxwell equation ∇ × E = −i ωμμ0 H ˙ ˙ i = μ0 I0 H 4π μ

] ] ∮ [ [ | | [ ] |t∥ | ez sin θ ∂ T1 + e∥ × ez ∂ T1 − (t · ez )eθ ∂ T2 dl. ∂ρ ∂z ∂z

(8)

l

In Eqs. (7) and (8) the local cylindrical coordinates (ρ, θ, z) with its unit basis vectors (eρ , eθ , ez ) are used (Fig. 3). The center of the coordinate system is located

2 Mathematical Model for Calculation of Electromagnetic Field Created …

7

at point M 0 intersection of the vertical axis with the interface. The angle θ is defined relative to the axis directed along the unit vector e∥ = t ∥ /t. The expressions T1 (ρ, θ, z) and T2 (ρ, θ, z) can be written as functions of dimensionless variables ( ) ( ) χ sin β χ cos β ∮∞ ( ) exp − J 0 εS εS p z χ dχ ; K ,χ · T1 (ρ, θ, z) = T1 , ε S , β = δ μ δ w1 (χ ) 0 ( ) ( ) χ cos β β ∮∞ ( (z ) J1 χ sin εS p z ) exp − εS T2 (ρ, θ, z) = T2 , ε S , β = χ dχ; K ,χ · δ μ δ w1 (χ ) (z

)

0

(9) ⎛ ⎞ / )2 ( (z ) √ z χ ⎠. K , χ = exp⎝ 2i 1+ √ δ δ μ i

(10)

/√ Here the parameter ε S = μδ 2r S is proportional to the ratio of the penetration depth δ to the distance r S from the field source at a point M on the contour to the body surface at a point Q0 (Fig. 3). Presented the analytical solution also has no restrictions on the contour configuration, electrophysical properties of the media and the field frequency. To determine a value of electromagnetic field it is need to calculate the double integral of special function. This solution allows also to establish some essential features of field formation and to develop reasonable approximate calculation methods.

Fig. 3 Relative position of the current element tdl and the observation point Q(ρ, θ, z) in conducting half-space z < 0 in a cylindrical coordinate system

8

Mathematical Models of Electromagnetic Interaction of Field Sources …

2.2 Asymptotic Expansion Method for Approximate Calculation of the Electromagnetic Field The presented exact analytical expressions show that the solution directly depends on / the( two )similar parameters both in / the observation points of dielectric ε1 = √ √ 2r1 and conducting ε S = μδ 2r S areas. For the observation points μδ located at the interface the distances between appropriate points are equal one another r1 = r S and therefore the introduced parameters are also equal ε1 = ε S = ε. If these parameters are small ε1 < 1 and ε S < 1 then this circumstance can be used to significantly simplify the calculation expressions. The concept of a strong skin effect can be considered in an extended formulation when the product value of relative magnetic permeability and field penetration depth μδ is small compared not only with the characteristic dimensions of the conducting body, but also with the characteristic distances between external field sources and the body [25]. (a) Asymptotic approximation in dielectric area z ≥ 0 The magnitude ε1 is the complex parameter that combines the electromagnetic properties of the medium, the frequency of the field, and the distance between the source and observation points. The condition of smallness of the parameter determines the / limitation to totality of the quantities. For example, the frequency/f( = ω 2π for ) given other quantities must be large than the limit value f > f m = μ 2π μ0 γ r12 εm2 , where εm is the chosen permissible value of the small parameter. The largest limitation of the frequency f√ m occurs for observation points located at the boundary surface where r1 = r S = h 2 + ρ 2 and the f m achieves the maximum value at the point directly below the contour element with current r1 = r S = h. The function G e (5), with the help of which the potentials and the electromagnetic field vectors in the dielectric area are determined, can be represented as an expansion into an asymptotic series in the small parameter ε1 [25]. In this case, the function G e is expanded into asymptotic power series of the Poincaré type [33, 34] with an error that can be made arbitrarily small by choosing ε1 → 0. For the single contour with current of initial external field the asymptotic series for the function G e takes the following form G e ≈ G eN =

N ∑ n=0

G (n) e1

N ∑

(

ε1 = 2(−1) an (μ) √ i n=0 n

)n+1

(n) ( ) 1 n+1 ∂ , r1 n ∂z r1

/ where an (μ) are the Taylor series coefficients of the function 1 w1 ) ( / n √ ∑∞ i . n=0 an (μ) χ Taking into account (11), expressions (1)–(4) can be written as follows

(11) =

⎤ ⎡ ( )n+1 (n+1) ∮ ) ∮ ( N ∑ ˙ μ t t ∂ t I μ 1 1 0 0⎣ dl − − (−1)n 2an (μ) dl ⎦; (12) A˙ = n+1 4π r r1 p ∂ z r 1 n=0 l

l

2 Mathematical Model for Calculation of Electromagnetic Field Created …

φ˙ = −

( )n (n) ∮ N μ ∂ I˙0 ∑ (t1 · ez ) n (−1) 2an (μ) dl; ς 4π n=0 p ∂z n r1

9

(13)

l

) ∮ ( t1 μ0 I˙0 t ˙ E = −iω − dl 4π r r1 l

+

I˙0 ς 4π

N ∑ n=0

( )n (n) ∮ t1 × r1 μ ∂ e × dl; z n p ∂z r13

(−1)n 2an (μ)

(14)

l

⎡

˙ ∮ ( t × r t1 × r1 ) ˙ = − I0 ⎣ − H dl 4π r3 r13 l

⎤ ( )n+1 (n+1) ∮ ∂ t1 × r1 ⎦ μ (−1)n 2an (μ) dl . − n+1 p ∂z r13 n=0 N ∑

(15)

l

/√ / √ i = μ p, where p = i ωμμ0 γ is Here, it is taken into account that ε1 r1 / propagation constant, ς = p γ is surface impedance. A feature of asymptotic series is that they are divergent series. With an increase in the number of terms in the series, the error in the approximation of the function first decreases, reaching a minimum, which depends on the value of the small parameter. After that the approximation error increases. This feature is illustrated in Fig. 4a for / relative deviation Δ N = |G eN − G e | |G e | [32].

/ Fig. 4 The calculation errors in asymptotic method: a relative deviation Δ N = |G eN − G e | |G e | between exact and approximate values of function G e at ρ = 0, μ = 5; b the relative error of asymptotic series terms

10

Mathematical Models of Electromagnetic Interaction of Field Sources …

In addition, each term of the asymptotic series is determined with an error, the magnitude of which depends on the value of the small parameter and the number n of the term in the series. Therefore, for each term of the series, there is also a limitation on the lower value of the frequency f ≥ f n , which increases with increasing number of the series term n. In this regard, with an increase in the value of the small parameter, the total number of considered terms of the series N decreases and, accordingly, the total approximation error increases. The relative error estimate for each term of the series can be determined at ρ = 0. In this case, there is only the maximum value of the error and at the same time greatly simplifies the expressions for its estimation. The relative value of error Δn for the term of series with number n is determined by the expression Δn =

e−1/ε1 1−

∑n

1 k=0 k!ε1k ∑ e−1/ε1 nk=0 k!ε1 k 1

.

(16)

The relative error of calculation of each term at n → ∞ tends to infinity. The larger the number of the series term, the greater the error in its determination, although its relative contribution to the total sum decreases. The value of the relative error for the first ten terms of the series at different values of the parameter ε1 is shown in Fig. 4b. The choice of the number of the series terms N can be made by the magnitude of the error of its last term. The number of the series terms should not exceed the value at which the relative error of the last term does not exceed a given value C N , for example, equal to one. In this case, the condition for determining the number of series terms n = N can be written in the form Δn (ε1 ) = C N .

(17)

Figure 5a–c show the results of calculating the number of the asymptotic series terms using (16) at C N = 1 for conducting media with parameters corresponding to brass (μ = 1, γ = 1.25 × 107 Ω−1 m−1 , Fig. 5a), aluminum (μ = 1, γ = 3.7 × 107 Ω−1 m−1 , Fig. 5b) and magnetizable material, for example, steel in the condition of magnetic saturation, typical for induction heating (μ = 30, γ = 106 Ω−1 m−1 , Fig. 5c) [29]. As it seen from Fig. 5a, b at higher frequencies and for materials with higher electrical conductivity the possible number of the asymptotic series terms are enough big. For such number series terns the calculation accuracy can significantly exceed the required or even reasonable level. Calculations without further limitation of the number of series terms are unnecessarily complicated. For magnetizable material (Fig. 5c), the number of the asymptotic series terms is much smaller and, accordingly, the achievable calculation accuracy is lower. Unlike the original expressions containing improper integrals of special functions, one-dimensional integrals for each term in (12)–(15) are similar to the known integrals for determining the field of the contour with current. Such representation of solutions is convenient not only for determining the characteristics of the field, but

2 Mathematical Model for Calculation of Electromagnetic Field Created …

11

Fig. 5 The number of the asymptotic series terms at C N = 1 for some metals: a brass μ = 1, γ = 1.25 × 107 Ω−1 m−1 ; b aluminum μ = 1, γ = 3.7 × 107 Ω−1 m−1 ; c magnetic materials μ = 30, γ = 106 Ω−1 m−1

can also be used to find the distribution of the energy flux density in the area above the conducting medium ˜ = ∏

1˙ E × H. 2

(18)

˜ is determined by the complex ampliHere the complex-value Poynting vector ∏ ˙ tude of the electric field intensity E and the complex conjugate amplitude of the magnetic field intensity H. The vertical component of the Poynting vector at the surface between the media pz determines the flux density of the period-averaged power into the conducting body. This energy is released in the form of Joule heat in the surface layer of conducting body, which can be found using expressions (14) and (15) [ ]) 1 ( ˜ = 0) · ez ) = − Re ez · E˙ e∥ (z = 0) × He∥ (z = 0) , pz = Re(−∏(z 2

(19)

where E˙ e∥ (z = 0) and He∥ (z = 0) are the tangent components of vectors to the surface. (b) Diffusion of non-uniform electromagnetic field into conducting medium in the case of strong skin effect ( / ) The presence of multiplier K z δ, χ in the functions (9) for determining electromagnetic field in conducting medium indicates the general feature of faster decay of non-uniform field in comparison with uniform one [26]. For strong skin effect in its extended formulation the introduced parameter is small value ε S < 1 for all points of external source field. In the case the difference in the laws of decreasing the field intensities for non-uniform and uniform fields is insignificant. /Taking into ˙ i⊥ ) ˙ e∥ = H ˙ i∥ , H ˙ e⊥ μ = H account the boundary conditions (z = 0 : E˙ e∥ = E˙ i∥ , H

12

Mathematical Models of Electromagnetic Interaction of Field Sources …

and the expressions for the field intensities in the dielectric half-space (3), (4) the approximate expressions in the conducting half-space take the following form E˙ i ≈ e pz E˙ i (z = 0) = e pz E˙ e∥ (z = 0),

(20)

[ / ] ˙ i ≈ e pz H ˙ i (z = 0) = e pz H ˙ e∥ (z = 0) + H ˙ e⊥ (z = 0) μ . H

(21)

At z = 0 the expressions (12)–(15) are greatly simplified and for field intensities the expressions can be written as [25] E˙ ∥ (z = 0) =

N ∑

E˙ ∥n = ς

n=0

˙ ∥ (z = 0) = H

N +1 ∑

˙ ∥n = − H

n=0

˙ i⊥ (z = 0) = H

N ∑ n=0

( )n { (n) }| | μ ∂ ˙ ez × H0∥ || ; 2an (μ) n p ∂z z=0 n=0

N ∑

˙ i⊥ n H

( )n { (n) ˙ }|| μ ∂ H0∥ | 2an−1 (μ) | | p ∂z n n=0

N +1 ∑

}| ( ) { N ∑ ˙ 0⊥ || an (μ) μ n+1 ∂ (n+1) H = 2 | | μ p ∂z n+1 n=0

(22)

;

(23)

,

(24)

z=0

z=0

˙ 0 is the magnetic field intensity of external sources in dielectric medium where H ˙ 0∥ and H ˙ 0⊥ are the tangent and normal components of the field, at the interface; H respectively; it is accepted a−1 = −1. As can be seen at strong skin effect the electromagnetic field is determined only by the known distribution of the field of external sources and its derivatives with respect to the vertical coordinate z at the boundary without the need to solve additional equations. The distribution of the electromagnetic field at the interface between the media is great importance. The electromagnetic field on this surface defines such characteristics as the energy flow of the electromagnetic field into the conductive body, the surface density of Joule heat release in the surface layer, the magnetic pressure on the surface of the body. Let us perform a quantitative assessment of the influence of the small parameter value ε to the change in the penetration law of the electromagnetic field into conducting half-space. The analysis is carried out based on the expansion of expresseries, where for small ε the Taylor series expansion sions (7), (8) in(the / asymptotic ) of the factor K z δ, χ (10) in the integrand is used near the zero value of the integration variable χ. Taking into account only two terms in the series expansion, the expressions for the field intensities turn out to be | ˙ 0∥ || ∂ H μ ˙ 0∥ − E˙ i ≈ 2e ς ez × H | p ∂z | {

pz

z=0

2 Mathematical Model for Calculation of Electromagnetic Field Created …

13

| } ) (2) ˙ || ( )2 ( ˙ 0∥ || 1 ∂ H0∥ | μ ∂ (2) H 1− + ; | +d | p 2μ2 ∂z 2 | ∂z 2 | z=0 z=0 | { ) (2) ˙ || ( )2 ( ˙ 0|| || 1 ∂ H0|| | ∂ H μ μ ˙ i|| ≈ 2e pz H ˙ 0|| − 1− H | | + 2 | p ∂z p 2μ ∂z 2 | z=0 z=0 | } | (2) ˙ ∂ H0|| | +d ; | ∂z 2 | z=0 | | { | (2) ˙ ˙ 0⊥ || ∂ H ∂ H μ 2 0⊥ | pz ˙ i⊥ ≈ − e H | − | p ∂z | p ∂ z2 | z=0 z=0 | } ) (3) ˙ || ( )2 ( ˙ 0⊥ || 1 ∂ H0⊥ | μ ∂ (3) H 1− + . | +d | p 2μ2 ∂z 3 | ∂z 3 | z=0

(25)

(26)

(27)

z=0

Equations (25)–(27) show that for all components of the electromagnetic field with strong skin effect, the deviation of the penetration law of non-uniform electromagnetic field from the penetration law of uniform one is determined by the value of the same parameter proportional to the value ε2 ( )2 ( )2 pz μ ε z . d= ∼ 2 p 2μ μ δ

(28)

Maximum value of the modulus of the takes place at the maximum |(√additional / ) term ( / )|| | value of the function | pz exp( pz)| = | 2z δ exp z δ |, which is realized at –z = δ, where the field itself decreases by a factor of e. Therefore, with a strong skin effect, calculations are usually performed without taking into account the change in the law penetration for non-uniform field.

2.3 Approximate Mathematical Model of the Diffusion Electromagnetic Field into Conducting Half-Space for Calculation of the Field Near Current Contour The previous section presents the mathematical model for calculating the threedimensional electromagnetic field of the system with contours of arbitrary configuration, which uses the asymptotic expansion method. Despite a significant simplification compared to the original general three-dimensional formulation, the expressions still contain integrals along the contours with current. When it is necessary to ensure intensive interaction of current contours with conducting medium, the contours are located as close to the surface as possible and it is enough to determine the electromagnetic field in limited area of space at small

14

Mathematical Models of Electromagnetic Interaction of Field Sources …

distance from the contour. Such problem arises, for example, in devices for highfrequency induction heating of metal bodies. In this case locally the field approximately is the same as two-dimensional field created by infinite straight conductor with current for which only algebraic expressions describe the electromagnetic field [31]. The application of the model with local replacement of a three-dimensional field by a two-dimensional one can be the most effective in solving the problems of finding the geometry of inductors for high-frequency heating of bodies and optimizing these devices. We will consider the electromagnetic field at observation points at the interface between media or near of it in the dielectric area. We also assume that the distance r from the contour to the observation point is small compared to the characteristic size of the contour D (Fig. 1) and to the radius of curvature of the contour R near the point Q. An additional limitation is that the distance from the contour with current to the interface h also turns out to be small compared to the characteristic dimensions of the contour D and to/the radius / / of/curvature / R, that is, in the case under/consideration, the parameters r R, r D, h R, h D and, as a consequence, r1 R, r1 D are small h, r, r1 « R, D.

(29)

In Fig. 6 the local coordinate ξ is perpendicular to the unit vector of tangent t. The in the directions determined by the unit vectors local coordinates z, ξ are introduced / /| | ez and eξ , where eξ = t × ez |t × ez | = e∥ × ez , the unit vector e∥ = t∥ |t∥ | coincides with the direction of the projection t∥ of the tangent vector t onto flat interface between the media. Local replacement of a real contour by the rectilinear conductor is possible, if the vector t is parallel to the boundary of metal body. If the segment of contour is not parallel to the boundary, the inaccuracy takes place and the error will increase with raising the slope of the tangent. For the rectilinear indefinitely conductor the integrals in Eqs. (12) and (15) are easily calculated. Using the expression Fig. 6 Local coordinates in the plane normal to the circuit with current

2 Mathematical Model for Calculation of Electromagnetic Field Created …

∂u n−1 ∂ (n+1) u n (ξ, z) = = n+1 ∂z ∂z

∮∞ −∞

15

[ ] z+h ∂ (n) √ , = −2 n 2 ∂z ξ + (z + h)2 ξ 2 + (z + h)2 + l 2 dl∥

(30) where l|| is the length along the straight conductor, the equations for vector potential ˙ = 1 ∇ ×A ˙ are given by and magnetic field intensity H μ0 ( )n+1 ] [ N μ ξ 2 + (z + h)2 ∑ μ0 I˙0 t n ˙ ˙ ln 2 Ae = Aet t = − (−1) 2an (μ) un ; 2 4π p ξ + (z − h) n=0 ˙ { ˙ e = − I0 t (z + h)eξ − ξ ez − (z − h)eξ − ξ ez H 2π ξ 2 + (z + h)2 ξ 2 + (z − h)2 ( )n+1 ( ) N ∑ μ ∂u n ∂u n eξ − ez . − (−1)n an (μ) p ∂z ∂ξ n=0

(31)

(32)

For current filament, which is parallel to the interface, the electric field intensity is ˙ e . Hence the expression (19) for the determined by the vector potential E˙ e = −iωA vertical component of the Poynting vector at the surface pz = 21 Re(−π · ez ) appears ) iω ( ˙ Re At H ξ 2 ⎧ ⎫ ] [ N ( )n+1 ⎪ ⎪ μ ⎪ μ0 i ωI02 ∑ ⎪ n ⎪ ⎪ (−1) · 2an u n (ξ, 0) ⎪ ⎪ ⎪ ⎪ 2 ⎨ 2π ⎬ p n=0 . = Re [ ] | ( )n+1 N ⎪ ⎪ ∑ ⎪ ⎪ 4h μ ∂u n (ξ, z) || ⎪ ⎪ n ⎪ ⎪ (−1) · 2an ⎪ ⎪ | ⎩ × ξ 2 + h2 − ⎭ p ∂z z=0 n=0

pz = −

(33)

Equations (31)–(33) can be basis of approximate computational models of electromagnetic systems, for which the mentioned above conditions take place. There are two sources of the errors applying considered approximate mathematical model. The first error is coursed by the replacement of a real bending conductor with a rectilinear infinitely long conductor with current. The second error is associated with the presence of the contour sections with the direction of the current perpendicular to the interface between the media. Let us estimate the magnitude of these errors. (a) The error for bending conductor Since the error depends on how far the observation point is located from the contour compared to the bending radius of the conductor, the comparison is made for different values R/h. Due to the fact that the result is presented as a series, the analysis is performed by estimating the error both for each term of the series and by determining the total error.

16

Mathematical Models of Electromagnetic Interaction of Field Sources …

To estimate the error of each term of the series let us compare the approximate calculation of the vector potential obtained by the approximate expression (31) and by the expression of the asymptotic approximation (12), in which there is no restriction is made for circular contours on the radius of curvature of the contour. The comparison / with different ratios R/h. In this case, the ratio R h → ∞ corresponds to the model of the locally two-dimensional field. The comparison is illustrated in Fig. 7 for the first four terms n = 0, 1, 2, 3. For each term, depending on the coordinate ξ , it is given the ratio of the corresponding term of the series (12) and the values of the same term of the series (31) at the point ξ =z=0 Dn (R, ξ, z = 0) = u n (0, 0)

∂ (n+1) ∂z (n+1)

∫ l

t 1 ·t Q dl r1

u n (0, 0)

.

(34)

In Fig. 7 ratio (34) is shown for four values R/ h = ∞, 10, 5, 3. Dependencies for R/ h → ∞ at point ξ = 0 pass through the value Dn (∞, 0, 0)/u n (0, 0) = 1. They are shown by solid curves. Dependences Dn /u n for the other extreme case

Fig. 7 Comparison of the first four series terms for the vector potential at different values R/ h at δ = 1.4 × 10−3 m, h = 0.025 m, μ = 1 (solid curve—R/h → ∞; circle—R/h = 10; multiplication symbol—R/h = 5; dotted curve—R/ h = 3)

2 Mathematical Model for Calculation of Electromagnetic Field Created …

17

R/ h = 3 are shown by dotted curves. The results for intermediate values R/ h = 10 and R/ h = 5 are shown by corresponding marks. From the comparison of dependencies in Fig. 7 corresponding to different terms of the series, it can be seen that the largest error occurs for the first term of the series n = 0, whose contribution to the total amount is the largest. The relative error of each term of the series decreases with increasing n. Since the value of each term decreases with growth n, it can be concluded that in order to ensure the necessary accuracy of calculating the field near the conductor, the real contour geometry must be taken into account only for the first terms of the series. For subsequent terms, the approximate calculation method can be used. For significant radii of curvature compared to the height of the contours above the conducting medium, the error may turn out to be insignificant for all terms of the series. The total error (in contrast to the error in calculating each member of the series) is illustrated by the example of calculating the vertical component of the electromag˜ = 0) · ez ) (19). netic energy flux density in conducting medium pz = 21 Re(−∏(z The calculations were carried out for the electro-physical parameters of the medium corresponding to brass (Fig. 8a) and steel (Fig. 8b) with/a value μ = 30 typical for the induction heating [29]. In figures the values pz (R, ξ ) pz0 (∞, 0) depending on the coordinate ξ are presented, where pz (R, ξ ) is calculated by expressions containing the operation of integration along the contour; pz0 (∞, 0) is the value for rectilinear infinitely long conductor at ξ = 0. / From Fig. 8, in particular, it follows that for the contour with R h = 10 the error does/not exceed 3.5% for steel and even less for brass material. At the same time, at R h = 3, it is already equal to approximately 20%, and calculations must be carried out without resorting to simplified model of the locally two-dimensional field. It should be noted that contours with such small ratio are rarely used, in particular, in the devices for induction heating of metal sheets.

Fig. 8 Comparison of the vertical component values of the electromagnetic energy flux density in conducting medium at different values R/ h: a brass δ = 1.4 × 10−3 m, h = 0.025 m, μ = 1; b steel δ = 4.1 × 10−4 m, h = 0.04 m, μ = 30 (solid curve—R/h → ∞; circle—R/h = 10; multiplication symbol—R/h = 5; dotted curve—R/ h = 3)

18

Mathematical Models of Electromagnetic Interaction of Field Sources …

(b) The error coursed by presence of contour sections directed at an angle to the interface The infinite rectilinear current cannot be located at an angle to the surface. For this reason the contour locally is replaced by horizontal filament directed along projection t∥ of the tangential vector t to the interface. The error of this replacement can be estimated from the change in the length of the vectors. For small tilt angles α (Fig. 2), the relative error is the following | |/ / Δ∥ = 1 − |t∥ | |t| = 1 − cos α ≈ α 2 2.

(35)

Another source of error is due to the neglect of the magnitude of the electromagnetic field associated with the flow of current in the vertical direction. The sections of initial contour directed at an non-zero angle α to interface is course of additional eddy currents in conducting half-space and as a result additional electric field intensity that is parallel to interface between the media. As follows from Eqs. (7), (9) and (10) the component of the field tangent to the surface associated with the initial current that flows in both directions. The electric field intensity at z = 0 is the follows E˙ e (z = 0) = E˙ i (z = 0) = E˙ ∥ 1 + E˙ ∥ 2 ∮ ] μ0 I˙0 [ t ∥ T1 (0, ε S , β) + t⊥ eρ T2 (0, ε S , β) dl. = −2i ω 4π

(36)

l

The influence for any contour section of vertical direction of initial current compare to current parallel to interface we estimate by ratio of individual components in the integrand (36). Δ⊥ =

|T2 (ε S , β = βm )| tgα. |T1 (ε S , β = 0)|

(37)

/| | Here tgα = |t⊥ | |t∥ |, the values β = βm and β = 0 are chosen accordance the maximum values/ of the functions |T2 (ε S , β)| and |T1 (ε S , β)|,where the angle βm close to βm = π 4. As it was shown in [35] the first factor in (37) weakly depends on the parameter m )| ≈ 0.38. ε S . In the case of a strong skin effect ε S < 1, it takes the value |T|T21(ε(εSS,β=β ,β=0)| Therefore, the relative error of neglecting the field created by vertical component current of initial contour is determined primarily by the angle α of inclination of the contour section Δ⊥ ≈ 0.386α. Figure 9 shows the two components of the error.

(38)

3 The Electromagnetic Energy Flux During High-Frequency Induction …

19

Fig. 9 Relative errors Δ∥ and Δ⊥ depending on the angle of the tangent inclination to the contour α

Thus, it is possible to use the analytical methods to calculate the electromagnetic field of the system in which the sources of the external field are contours with current located near the conducting half-space. Depending on the geometry of the electromagnetic system, the electro-physical parameters of the medium and the frequency of the field, different approaches can be used: the general solution that has no restrictions on the characteristics of the system; asymptotic expansion method in case of strong skin effect; locally two-dimensional approximation for the systems with contours of small curvature and small value of the inclination angle of the tangent.

3 The Electromagnetic Energy Flux During High-Frequency Induction Heating of Moving Conducting Strips In this subsection, one of the problems of high-frequency induction heating is considered, namely, high-frequency heating of conducting strips. The main attention is paid to the problems of calculating the electromagnetic field and the electromagnetic energy flux into the conducting strip. One of main problems of induction heating of metal strip is to provide a certain temperature of the strip moving across alternating electromagnetic field of created by inductor [7]. In the general case the problem is complicated in the computation sense since it is necessary to take into account geometrical, electro-physical and heattransfer properties of electromagnetic systems. Therefore, here the most significant features of systems are taken into account and approximate calculation methods are used. As in this part the high-frequency induction heating of strips is analyzed it is considered that field is generated by a coreless inductor, fulfilled in the form of a contour generally of a spatial configuration with alternating current I˙0 (Fig. 10) [30,

20

Mathematical Models of Electromagnetic Interaction of Field Sources …

Fig. 10 The model of electromagnetic system

31]. The conducting strip of thickness d moves with a speed v = v y e y in the direction of the axis y. Due to the flow of eddy currents in the strip, thermal energy is released.

3.1 Small Parameters for the Formulation of an Approximate Mathematical Model When high frequencies are used, the field penetration depth δ is small compared to the strip thickness d, and the induced current is distributed only in a thin surface layer. When the following condition is true 1 δ ε1 = = d d

/ 2 ≤ 0.3, ωμμ0 γ

(39)

the mathematical model of the inductor as contour with current located above the conducting half-space is applicable. The conductors of the inductor are located at a height h relative to the surface of the body, much greater than the field penetration depth. In this case, the strong skin effect condition in the extended formulation is valid. As shown above, under the condition of smallness of the introduced parameter ε2 = ε S =

1 μ δμ = √ ≤ 0.3, h h ωμμ0 γ

(40)

the solution is simplified, and it can be represented as asymptotic series for the potentials and field vectors (12)–(15), as well as for the density of the active power flux pz inside the conducting body (19). Each term of the series is a function of

3 The Electromagnetic Energy Flux During High-Frequency Induction …

21

the field of the original contour with current. Their expressions in the form of onedimensional integrals can be easily calculated. As the field frequency increases, the influence of the higher terms of the series decreases, and in practice it is sufficient to confine of only a few first terms. Further simplification of the calculation model is related to the distance between the contour and the points of the area in which it is necessary to determine the field. In induction heating installations, the inductor is placed near the surface of the metal strip, and for the area of space near the conductor, the following condition is true / ε3 = r⊥ D « 1,

(41)

where D is the characteristic size of the contour, for points on the surface, r⊥ = ( 2 )1 2 h + ξ 2 / . In addition, we will assume that the slope angle α (Fig. 2) of the tangent t to the contour is small; this allows us to neglect errors (35) and (37) and consider the current directed parallel to the flat boundary surface. These limitations are typical for high-frequency induction heating of metal strips. The restrictions make it possible to describe the electromagnetic field using the model of the locally two-dimensional field. The introduced small parameters make it possible to analyze the characteristic values of the electromagnetic and thermal parameters of high-frequency induction heating of metal strips and to substantiate the computational mathematical model. In this case, the accuracy sufficient for performing estimated calculations is achieved by taking into account only the first term of the expansion into asymptotic series with respect to the parameter ε2 . Then in the model of the locally two-dimensional field at z = 0 from (31)–(33) we have ˙ h ˙ = − iωμ0 I0 t √ μ ( ) E˙ = E˙ t t = −i ωA 2 π i ωμμ0 γ ξ + h 2 √ h I˙0 2η =− eiπ / 4 t; (42) 2 π ξ + h2 ] [ ) ( 2 ξ − h 2 eξ + 2ξ hez heξ μ I˙0 ˙ ˙ ˙ +√ H = Hξ eξ + Hz ez = − ; (43) ( )2 π ξ 2 + h2 iωμμ0 γ ξ 2 + h2 pz =

1 I02 η ·( )2 , 2 2 2π h 1 + ξ 2/ h2

(44)

/ 0 is real part of the surface impedance value. where η = Re(ζ ) = ωμμ 2γ Expressions (42)–(44) are obtained under the assumption that there is no movement of conducting medium in magnetic field. Let us evaluate the legitimacy of such assumption for the processes of induction heating of moving metal strip. To do this, let’s compare the electric field intensity at the interface z = 0 due to the alternating ˙ (42) and the electric field intensity associated with motion magnetic field E˙ = −iωA

22

Mathematical Models of Electromagnetic Interaction of Field Sources …

˙ Let us make a comparison for the maximum in the magnetic field E˙ v = μ0 v × H. values of the intensities of the field components tangential to the surface. It can be seen from (42) that E˙ = E˙ t t takes the maximum value directly under the contour at ξ = 0. The component of the electric field intensity E˙ v , tangential to the surface, taking into account (43) will be √ ξh I˙0 2 2η −iπ 4 ˙ ˙ Ev (z = 0) = μ0 v × ez Hz (z = 0) = −v × ez ( )2 e / . πω ξ 2 + h2

(45)

It can be seen from the last expression that directly under the conductor of the contour the intensity E˙ v equal to zero E˙ v (ξ = 0, z = 0) = 0. The electric field intensity, due to the movement of conducting/medium in magnetic field, reaches its √ 3 where it turns out to be equal to maximum value E˙ v max at the point ξmax = h E˙ v

max

√ I˙ζ 9 2 −iπ / 4 . = −v × ez √ e π ωh 2 8 3

(46)

To compare the field intensities, we introduce a parameter equal to their ratio | | |E˙ v | 9 v . εv = | | = √ |E˙ | 8 3 ωh

(47)

If εv « 1, the distribution of the electromagnetic field and induced currents in the system under consideration will be the same as at the speed v = 0. So, for example, at v ∼ 1 m/s, h ∼ 3 cm, typical for induction installations for the heat treatment of metal strips, already for frequencies ω ≥ 2π · 50 rad/s the ratio is εv ≤ 0.07, and the calculation of electromagnetic fields can be carried out without taking into account the movement of the strip. Note that for high-frequency induction heating, the expression uses field frequencies that are significantly higher than the specified value. Therefore, for this process, the determination of the electromagnetic field can be performed without taking into account the speed of the conducting strip.

3.2 Electromagnetic Energy Flux Density into Heating Metal Strip Consider the process of transferring electromagnetic energy to a volume element of a metal strip ΔV = ΔxΔyd, which moves at the speed v. Let at the moment of time t the volume element be at the point with coordinates x, y = y0 + vt, where y0 is the coordinate in which the considered volume is located at the moment of time t = 0. It is believed that the heating process begins in the area where the field is absent at

3 The Electromagnetic Energy Flux During High-Frequency Induction …

23

y → −∞. By the moment of time t the volume element will receive energy ∮t ΔW (x, y) = ΔxΔy −∞

ΔxΔy pz (x, y0 + vt) dt = v

∮y pz (x, y) dy.

(48)

−∞

The volumetric energy density w(x, y), received by the element of the strip, which has reached the position with coordinates x, y, turns out to be 1 ΔW = w(x, y) = ΔxΔyd vd

∮y pz (x, y) dy.

(49)

−∞

The distribution of the amount of released heat along the width of the strip is characterized by a value that can be called the linear density of the field energy flux transferred to the heated strip ∮y P(x, y) =

pz dy.

(50)

−∞

The total amount of electromagnetic energy transferred during the entire heating process is characterized by the linear density flux of all released energy P(x) = P(x, ∞). The resulting temperature of a certain section of the tape depends on the value P(x), and in the case of insignificant heat transfer, this value directly determines the temperature of the corresponding section of the strip. Therefore, in the future, the main attention will be paid to the distribution of the value P(x) over the width of the strip. As follows from (44), the characteristic size of the area within which the electromagnetic energy is transferred to the metal is the height h of the contour element above the media interface. During the movement of the strip, the transfer of electromagnetic energy to the medium occurs unevenly. Heat generation is negligible away from the contour with current and increases sharply when a portion of the conducting medium passes under the contour. Figure 11 shows the change in the relative value of the electromagnetic energy flux density (44) along coordinate ξ perpendicularly to the direction of the current flow of the contour. The value of the distance marked in the figure ξ/ h = 0.64, at which the energy flux density is halved, characterizes the area within which electromagnetic energy is released near the conductor. The characteristic size, which will be used in what follows, is also the value at which the modulus / √ of the derivative with respect to the 5h ≈ 0.45h. coordinate has the greatest value ξmax = 1 The linear density of the transferred energy P(x) depends on the characteristic time during which the corresponding strip element was under the contour. Therefore,

24

Mathematical Models of Electromagnetic Interaction of Field Sources …

Fig. 11 Change of the energy flux density in the direction perpendicular to the direction of the contour current flow

the value P(x) depends on the orientation of the contour section relative to the direction of speed v. In this case, the influence of the contour geometry turns out to be different for sections far from the edges (point C in Fig. 10) and near the edge of the contour, where the direction of the tangent to the contour is parallel to the speed vector (points A and B in Fig. 10) [31]. Let us estimate the value separately for the two specified sections of the contour. (a) The linear density of the field energy flux transferred to the metal strip far from the edge of the contour The influence of the contour sections located far from the element of the heated strip / is insignificant, which is determined by the small value of the parameter ε3 = d D « 1. In this case, when integrating in (50), we can neglect the curvature of the contour near the chosen point and assume that the field is created only by a straight conductor (Fig. 12). The actual contour geometry is shown in the figure as a dotted curve. Fig. 12 On the determination of the linear density electromagnetic energy flux transferred to metal strip far from the edge of the contour

3 The Electromagnetic Energy Flux During High-Frequency Induction …

25

By integrating, we find ∮∞ P(C) =

pz (xC , y(ξ, β))dy = −∞

I02 η , 2π h C cos β

(51)

where y(ξ, β) = yC − cosξ β . It can be seen that the amount of released thermal energy will be the greater, the greater the angle β between the vectors of the tangent t at a given point of the contour and the speed v of the strip. Note that it follows from (51) that in order to ensure the total amount of heat released uniformly over the width of the moving strip, it is necessary to choose the gap profile between the contour and the surface of metal strip from the condition h C cos β = const.

(52)

Condition (52) means that heat release uniform in width in the strip can be achieved either under the straight sections of the flat contour, or under the curved sections of the spatial contour with a correspondingly changing distance to the surface of the metal strip. (b) Linear density of field energy flux transferred to the metal strip near the edge of the contour The expression (51) for the linear heat release density P(x) and, accordingly, condition (52) will not be valid for sections of the moving strip under the contour near its edge (the coordinate x is in a small neighborhood of the coordinate x A in Fig. 10). Here, when integrating in expression (50), it is necessary to take into account the finite radius of curvature R and the dependence of the angle β on the coordinate y. To perform estimates, we will replace the real geometry of the contour with an arc of circle of constant radius R equal to the radius of curvature of the contour at the point A (Fig. 13). The dotted curve in the figure corresponds to the actual configuration of the contour, and the solid one shows semicircle of radius / R. The value P(x) is determined near the edge of the contour, when the ratio Δ R is small, and therefore the error associated with the difference between the geometry of the contour and the arc of the circle will be insignificant. The integration of the electromagnetic energy flux / / density in (50) will be carried out within the azimuthal angle −π 2 ≤ φ ≤ π 2, taking into account the insignificant contribution of the far sections of the contour to the value of the linear density of the transferred energy. Figure 13 shows the Cartesian coordinates x ' , y ' , which are measured from the center of curvature, the value Δ is ' measured from the edge of the contour in the direction ) coordinate x . ( of 'the ' The distance ξ and coordinate y Q of the point Q ξ, y Q , expressed in terms of the introduced parameters, are the following ξ=

R+Δ − R, cos φ

y Q' = (R + Δ)tgφ.

(53)

26

Mathematical Models of Electromagnetic Interaction of Field Sources …

Fig. 13 On the determination of the linear density electromagnetic energy flux transferred to metal strip near the edge of the contour

Substituting these values into (50), we find expression for the linear density of the transferred energy flux near the contour edge P(x A + Δ) in the form P(x A + Δ) =

) I02 η(R + Δ) ( ∗ J h , Δ∗ . 2 2 π hA

(54)

/ / Here h ∗ = h R, Δ∗ = Δ h, the function J (h ∗ , Δ∗ ) is (

∗

J h , Δ

∗

)

∮π / 2 =

[ 0

cos2 φdφ )2 ]2 . ( 1−cos φ 2 ∗ cos φ + −Δ h∗

(55)

In a particular, directly under the edge of the contour at Δ = 0, one can obtain a ∗ 2 simple estimate for the integral / (55). Indeed, given that (h ) « 1 and performing the change of variables z = h ∗ on the angle φ, we find

2 h∗

sin φ2 , without taking into account the dependence

√ ∗ ( ) ∮1/ h √ √ ( ∗2 ) ( ∗ ) 3π dz ∗ ∗ +0 h . J h , 0 ≈ 2h ( )2 = h 8 1 + z4 0

(56)

4 Assessment of the Main Parameters of the Heating Process

27

Substituting the found value into (54), we write expression for the linear density of the released heat in the conducting medium after it passes under the edge of the contour with current at the point A A P(x A ) =

3I02 η R 1/ 2 . 8π h 3/ 2 A

(57)

Comparison of (57) and (51) shows that with increasing height, the heat release near the edge decreases faster than away from it. In addition, near the edge, it is necessary to take into account the radius of curvature of the contour, with the growth of which the amount of transmitted electromagnetic energy increases.

4 Assessment of the Main Parameters of the Heating Process 4.1 Temperature Stabilisation Along Thickness The representative size in a transversal direction to strip is penetration depth of electromagnetic field to metal δ. For high-frequency heating the penetration depth δ is usually considerably less than the strip thickness. Due to the thermal conductivity the released heat is extended in the depth of the metal. If this process is enough quickly, after a certain time the uniform temperature will be established on a thickness strip. Let’s compare the characteristic time of temperature stabilization on the thickness τd to the characteristic time of strip heating τ p . As in actual practice the thickness of strip d is much less than distance h the temperature rise ΔT in any point of medium can be estimated from expression ΔT (z, t) =

Q z2 e− 4at , z ≤ 0, √ cρ πat

(58)

where Q is surface density heat, thermal diffusivity a = λ/cρ determined through specific heat capacity c, thermal conductivity λ and density ρ, t is time. If after a certain time the temperature of the given volume of a metal strip becomes homogeneous with thickness, in the absence of a heat emission it will be equal to ΔT∞ = cρQd . The estimation of stabilisation time of the homogeneous temperature τd we will determine from the following condition: ΔT (0, τd ) = ΔT∞ . From here the estimate cρ d 2 of the temperature stabilisation time is τd ∼ πλ . The characteristic time of strip heating τ p can be estimated as time of the metal strip section transit under conductor. The following parameter is used for a conclusion, whether the homogeneous temperature in the process of the heating is established

28

Mathematical Models of Electromagnetic Interaction of Field Sources …

Table 1 Parameter εd

Thickness d, m

Aluminium

Brass

Steel

10–3

1.2×10−2 1.8×10−3 0.11 1.6×10−2

4×10−2 6×10−3 0.36 5.5×10−2

8.4×10−2 2×10−3 0.75 1.1×10−1

3 × 10–3

⎧ cρ d 2 v cos β ⎪ ⎪ , far from edges (point C) ⎨ τd π λh . = εd = 2 ⎪ τp ⎪ ⎩ cρ d v , near to points A and B π λD

(59)

Table 1 contains values εd for strips by thickness of d = 10−3 m and d = 3 × 10 m (the line separates values corresponding to the upper and lower expressions in (59) correspondingly). Following values of conductor dimensions and velocity are typical for the induction heating: D = 0.2 m, h = 3 × 10−2 m, cos β = 1, v = 10−1 m/s. The parameters εd are given for the following materials: −3

aluminium c = 8.8 × 102 J/(kg K), ρ = 2.7 × 103 kg/m3 , λ = 2.1 × 102 W/ (m K); brass c = 3.8 × 102 J/(kg K), ρ = 8.5 × 103 kg/m3 , λ = 85.5 (W/m K); steel c = 4.6 × 102 J/(kg K), ρ = 7.8 × 103 kg/m3 , λ = 45.4 W/(m K). Obviously that practically always the temperature becomes homogeneous with thickness already during the strip transit under the corresponding sections of contour. Therefore in mathematical models of induction heating the temperature of metal strips can be accepted everywhere homogeneous for thickness except for the sections transiting at present time under the contour with current.

4.2 Heat Transfer Along Surface of Metal Strip The released heat energy is transferred with the motion of strip in the direction of velocity vector. On the other hand, the strip temperature directly under the contour strongly increases, and because of the originated temperature gradient there are the heat flow caused by thermal conductivity. In this case the problem consists in the comparison of two processes of the heat transfer: thermal conductivity and transition of heat by motion of metal. Parameters of the processes of heat transfer are different for the sections near intermediate points C and at edges of the contour near to points A and B (Fig. 10). Therefore we will consider these sections separately. (a) Heat transfer near intermediate points of the contour During transition under contour, the element of the strip volume ΔV = ΔxΔyd which has reached the coordinate y receives energy, equal to (48). This thermal energy leads to temperature growth

4 Assessment of the Main Parameters of the Heating Process

29

Fig. 14 The directions of heat flows under intermediate points of the contour

ΔW (x, y) = cρΔxΔyd(T (x, y) − T0 ),

(60)

where T0 is temperature before heating. Comparing (48) and (60), we obtain 1 T (x, y) − T0 = cρ dv

∮y pz (x, y(ξ )) dy.

(61)

−∞

The temperature gradient and medium motion cause the corresponding thermal flows. Figure 14 qualitatively illustrates directions of flows of thermal energy related to thermal conductivity U1 and heat transfer by medium motion Uv1 . The heat flow due to the thermal conductivity is directed in opposite direction to strip motion, and is equal to U1 (x, y(ξ )) = −λ

λΔx dT Δxd = − pz (x, y(ξ )). dy cρv

(62)

The maximum value U1 max is reached directly under the contour at ξ = 0. Let’s compare U1 max to a maximum of heat flow transferred by motion of heating metal Uv1 = cρ vΔxd(T (∞) − T0 ). In order to describe the heat transfer by the thermal conductivity in the direction along the velocity vector in comparison with the heat transfer by the motion let’s introduce the following parameter ε L1

| | | U1 max | z (x, y(ξ = 0)) | = λ · ∫p∞ | =| . | Uv1 cρ v −∞ pz (x, y) dy

(63)

Considering (44), the resultant expression for ε L1 is given by ε L1 =

2λ . π cρ h v

(64)

Table 2 in the first raw contains values of parameter ε L1 for aluminium, brass and steel strips at h = 3 × 10−2 m and motion velocity v = 10−1 m/s.

30

Mathematical Models of Electromagnetic Interaction of Field Sources …

Table 2 Parameters ε L1 and ε L2

Aluminium

Brass

Steel

ε L1

1.9 × 10−2

5.6 × 10−3

2.7 × 10−3

ε L2

2.6 × 10–1

7.6 × 10–2

3.6 × 10–2

In all cases near intermediate points of the contour even at the velocity in some cm/s heat transfer by thermal conductivity caused by gradient of temperature is significantly lesser in comparison with the heat transfer due to the metal motion. (b) Heat transfer near contour edges Near the edges of the current contour the temperature gradient is directed perpendicularly to the velocity vector v. Heat input happens on the long section. To get the estimates, we will consider that heat is generated uniformly along section, equal to the characteristic size of contour D. The amount of heat brought to the strip section of a small length Δy and a width of 2ξ will be ∮ξ ΔWv2 =

pz (ξ )ΔyΔt dξ ,

(65)

−ξ

where Δt = D/v—section heating time. Quantity of heat, given away by thermal conductivity is λdΔy ΔW2 = − v

∮D/2 −D/2

∂ T (ξ, y) dy. ∂ξ

(66)

The direction of thermal energy flow is shown in Fig. 15. The temperature linearly grows in process of motion along coordinate y T (ξ, y) − T0 =

Fig. 15 The directions of heat flows under edge point of the contour

pz (ξ ) y, ξ ≥ 0. cρ dv

(67)

5 Conclusions

31

Therefore we get ΔW2 =

λ Δy D 2 ∂ pz (ξ ) · · . cρ 2v 2 ∂ξ

(68)

The ratio of the transferred energy due to heat conduction ΔW2 to the energy ΔWv2 , that reached the elements of the strip with a width of 2ξ , is the following ∂ pz (ξ )

ΔW2 D λ ∂ξ · · ∫ξ . = ΔWv2 2v cρ −ξ pz (ξ )dξ

(69)

Taking into account (44) the factor in (69) becomes F(χ ) = ∫ ξ

−ξ

∂ pz (ξ ) ∂ξ

pz (ξ, h)dξ

=

h 2 (1

+

χ 2 )2

16χ ) ], [ ( · χ + 1 + χ 2 ar ctg(χ )

(70)

where χ = ξ/ h. The greatest magnitude ΔW2 = ΔW2 max will be at χ = χ/max , where the √ z (ξ ) has maximum value. In this point χmax = 1 5 and then derivative ∂ p∂ξ F(χmax ) = 2.6. The dimensionless parameter is equal to ε L2 =

D λ W2 max · = 1.3 · . Wv2 cρ vh 2

(71)

In Table 2 in second raw the values of the parameters ε L2 are presented. As it seen in most cases the heat transfer due to the thermal conductivity near the edges remains negligible in comparison with heat income, caused by Joule dissipation of an electromagnetic energy. Only at rather small velocities thermal flows can appear comparable, especially for materials with high value of thermal diffusivity (aluminium, cuprum, etc.).

5 Conclusions The choice of mathematical model for studying the electromagnetic interaction of field sources with conducting body essentially depends on the electromagnetic and thermo-physical characteristics of the process. When there are no restrictions on the geometric configuration of the external field sources, the properties of the medium, and the frequency of the field, the exact analytical solution of the problem of field conjugation on flat interface of the media is used. The approximate model based on the expansion of the exact solution into asymptotic series is valid for processes in which

32

Mathematical Models of Electromagnetic Interaction of Field Sources …

the product of the field penetration depth and the relative magnetic permeability of the conducting medium does not exceed the distance between the field sources and the media interface. Even simpler mathematical model of the locally two-dimensional electromagnetic field is valid in the case of close location of the field sources and the conducting medium, which usually takes place for processes with strong interaction between the field sources and the conducting body, for example, in devices for induction heating. Based on the fulfilled estimates of main parameters of heating of moving metal strips in high-frequency field of the ironless conductors, the next conclusions follow, which can be used for the elaboration of mathematical models of considered processes: (1) The temperature can be considered homogeneous in thickness in any point of metal strips out of the area directly under the current contour. (2) In directions along surface of heated metal strip the heat transfer due to the thermal conductivity is negligibly small in comparison with the heat transfer, caused by the medium motion and the heat income due to the Joule dissipation of electromagnetic energy. In this sense in directions along surface the heating process it is possible to consider as adiabatic. Temperature in any point of moving strip (with the restrictions introduced in the chapter) will be ΔT (x, y) = T (x, y) − T0 =

P(x, y) , cρ vd

(72)

∫y where P(x, y) = −∞ pz (x, y)dy is linear density of electromagnetic energy flux caused by Joule dissipation. These conclusions are illustrated below by comparison of the calculations results for the brass strip temperature, performed with assumption of adiabatic heating and taking into account the thermal conductivity. The strip with thickness of d = 3 × 10−3 m and width of 0.6 m is heated up, moving under a flat circular contour with radius of a = 0.25 m. Distance between contour and the brass strip is h = 0.04 m. The frequency of current is f = 104 Hz. Calculations have carried out for two velocity values of the strip: v = 0.25 m/s and v = 0.01 m/s. With the chosen input data, the parameters ε L1 and ε L2 have the values shown in Table 3. From Table 3 it follows that along the surface of the strip for velocity of v = 0.25 m/s the process is adiabatic. At the same time for a strip velocity of v = 0.01 m/s it is impossible to consider the process as adiabatic and for the determination of distribution temperature it is necessary to consider thermal conductivity of strip. Table 3 Parameters ε L1 and ε L2 for brass

v, m/s

εL 1

εL 2

0.25

1.7 × 10−3

4.3 × 10−2

0.01

4 × 10−2

1.1

References

33

Fig. 16 The distribution temperature on strip width on distance of y = 0.5 m from the centre of the circular inductor: a v = 0.25 m/s; b v = 0.01 m/s

The above conclusions are proven to be true by the calculations of the distribution temperature on the strip width on distance of 0.5 m from the centre of the circular contour. In Fig. 16 the solid curves represent the results obtained with application of asymptotic computational method [32] with assumption of adiabatic heating. The dotted curves correspond to the data of joint solution of the electromagnetic and thermal problems obtained by method in which no restrictions are put on the parameters of heating [30, 32] The comparison shows that at the strip velocity of v = 0.25 m/s there is a good match of the results of two approaches and in this case the condition of adiabatic heating is really satisfied. For a strip velocity of v = 0.01 m/s as it follows both from estimates of parameters and from the results of the calculations (Fig. 15b), it is impossible to consider the heating as adiabatic process. The calculations in this case should be performed taking into account the joint influence of the heating by eddy currents and the heat transfer related to thermal conductivity and the motion of the medium.

References 1. Vasetskyi, Yu.M., Dziuba, K.K.: An analytical calculation method of quasi-stationary threedimensional electromagnetic field created by the arbitrary current contour that located near conducting body. Tech. Electrodyn. 5, 7–17 (2017). https://doi.org/10.15407/techned2017. 05.007 (Rus) 2. Vasetsky, Yu.M., Dziuba, K.K.: Three-dimensional quasi-stationary electromagnetic field generated by arbitrary current contour near conducting body. Tech. Electrodyn. 1, 3–12 (2018). https://doi.org/10.15407/techned2018.01.003 3. Sommerfeld, A.: Lectures on Theoretical Physics, vol. III, Electrodynamics. Academic Press, New York (1952). ISBN 148321429X, 9781483214290

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4. Sommerfeld, A.: Lectures on Theoretical Physics, vol. IV, Partial Differential Equations in Physics. Academic Press, New York (1949). https://doi.org/10.1016/B978-0-12-654658-3.X50 01-0 5. Grinberg, G.A.: Selected Topics in the Mathematical Theory of Electric and Magnetic Phenomena. Published by Academy of Sciences of SSSR, Moscow–Leningrad (1948) 6. Tozoni, O.V.: The Method of Secondary Sources in Electrical Engineering. Energiya, Moscow (1975) 7. Rudnev, V., Loveless, D., Cook, R., Black, M.: Handbook of Induction Heating, 772 p. Taylor & Francis Ltd., London. https://doi.org/10.1201/9781315117485 8. Lucía, O., Maussion, P., Dede, E.J., Burdío, J.M.: Induction heating technology and its applications: past developments, current technology, and future challenges. IEEE Trans. Industr. Electron. 61(5), 2509–2520 (2014). https://doi.org/10.1109/TIE.2013.2281162 9. Acero, J., Alonso, R., Burdio, J.M., Barragan, L.A., Puyal, D.: Analytical equivalent impedance for a planar induction heating system. IEEE Trans. Magn. 42(1), 84–86 (2006). https://doi.org/ 10.1109/TMAG.2005.854443 10. Babutsky, A., Chrysanthou, A., Ioannou, J.: Influence of pulsed electric current treatment on corrosion of structural metals. Strength Mater. 41(4), 387–391 (2009). https://doi.org/10.1007/ s11223-009-9142-3 11. Gallo, F., Satapathy, S., Ravi-Chandar, K.: Melting and crack growth in electrical conductors subjected to short-duration current pulses. Int. J. Fract. 16, 183–193 (2011). https://doi.org/10. 1007/s10704-010-9543-0 12. Psyk, V., Risch, D., Kinsey, B.L., Tekkaya, A.E., Kleiner, M.: Electromagnetic forming—a review. J. Mater. Process. Technol. 211(5), 787–829 (2011). https://doi.org/10.1016/j.jmatpr otec.2010.12.012 13. Gayakwada, D., Dargara, M.K., Sharmaa, P.K., Purohitb, R., Ranab, R.S.: A review on electromagnetic forming process. Procedia Mater. Sci. 6, 520–527 (2014). https://doi.org/10.1016/ j.mspro.2014.07.066 14. Batygin, Y., Barbashova, M., Sabokar, O.: Electromagnetic Metal Forming for Advanced Processing Technologies. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-74570-1 15. Schneerson, G.A.: Fields and Transient Processes in the Equipment of Superstrong Currents, 416 p. Energoatomizdat, Moscow (1992). ISBN 5-283-03959-5 16. Landau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media, 475 p. Elsevier (1984). https://doi.org/10.1016/B978-0-08-030275-1.50024-2. 17. Knoepfel, H.: Pulsed High Magnetic Fields, 372 p. Wiley, Canada (1997). ISBN 0471885320, 9780471885320 18. Simonyi, K.: Foundation of Electrical Engineering, 865 p. Elsevier (1963). https://doi.org/10. 1016/C2013-0-02694-1 19. Polivanov, K.: Theory of Electromagnetic Field. Imported Pubn (1984). ISBN 978-0828527477 20. Rytov, S.M.: Calculation of skin effect by perturbation method. J. Exp. Theor. Phys. 10(2), 180–190 (1940) (Rus) 21. Yuferev, S., Ida, N.: Surface Impedance Boundary Conditions: A Comprehensive Approach, 412 p. CRC Press (2018). https://doi.org/10.1201/9781315219929. 22. Leontovich, M.A.: On the approximate boundary conditions for electromagnetic field on the surface of highly conducting bodies. In: Propagation of Electromagnetic Waves, pp. 5–20. USSR Academy of Sciences Publ., Moscow (1948) (Rus) 23. Berdnik, S.L., Penkin, D.Y., Katrich, V.A., Penkin, Yu.M., Nesterenko, M.V.: Using the concept of surface impedance in problems of electrodynamics (75 years later). Radio Phys. Radio Astron. 19(1), 57–80. https://doi.org/10.15407/rpra19.01.057 24. Liu, X., Yang, F., Li, M., Xu, S.: Generalized boundary conditions in surface electromagnetics: fundamental theorems and surface characterizations. Appl. Sci. 9(9), 1891–1918 (2019). https:// doi.org/10.3390/app9091891 25. Vasetsky, Yu.: Nonuniform electromagnetic field at the interface between dielectric and conducting media. Progr. Electromagn. Res. Lett. 92, 101–107 (2020). https://doi.org/10.2528/ PIERL20050802

References

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26. Vasetsky, Yu.M.: Penetration of non-uniform electromagnetic field into conducting body. Electr. Eng. Electromech. 2, 43–53 (2021). https://doi.org/10.20998/2074-272X.2021.2.07 27. Demirchyan, K.S., Kuznetsov, V.F., Boronin, V.I.: Surface Effect in Electric Power Devices, 280 p. Nauka, Leningrad (1983) (Rus) 28. Semiatin, S.L., Zinn, S.: Elements of Induction Heating: Design Control and Applications. ASM International (1988). ISBN 13: 9780871703088 29. Slukhotsky, A.E., Nemkov, V.S., Pavlov, N.A., Bamuner, A.V.: Induction Heating Equipments, 325 p. Energoizdat, Leningrad (1981) (Rus) 30. Vishtak, T.V., Kondratenko, I.P., Rashchepkin, A.P.: Induction heating of a strip by current circuits of canonical forms. Tech. Electrodyn. 1, 63–68 (2003) 31. Vasetsky, Yu., Mazurenko, I.: Geometric parameters of electromagnetic systems for highfrequency induction heating of metal strips. Tech. Electrodyn. 5, 9–15 (2009). https://previous. techned.org.ua/article/9-5/st8.pdf (Rus) 32. Vasetskyi, Y., Mazurenko, I.: Approximation mathematical models of electromagnetic and thermal processes at induction heating of metal strips. Comput. Probl. Electr. Eng. 1, 45– 50 (2011). https://ena.lpnu.ua:8443/server/api/core/bitstreams/8f110656-3314-458e-a3b2-749 d52f42ca3/content 33. Nayfeh, A.H.: Introduction to Perturbation Techniques, 536 p. Wiley-VCH (1993). ISBN 9780471310136 34. Smirnov, V.I.: A Higher Mathematics Course, Complex Variables, Special Functions, vol. 3, part 2, 700 p. Pergamon Press, Oxford (1964). ISBN 978-0080136226 35. Vasetsky, Yu.M.: Simplified mathematical model of three-dimensional electromagnetic field of arbitrary current system near conducting body. Tech. Electrodyn. 3, 3–8. https://doi.org/10. 15407/techned2020.03.003 (Ukr)

Configuration of Spatial Iron-Free Inductors for High-Frequency Induction Heating of Metal Strips

Abstract The chapter focuses on finding the geometric configuration of iron-free electromagnetic field inductors in the form of current contours intended for heat treatment of metal strips, the induction heating of which is carried out when they move in transverse high-frequency field. Inverse field theory problems are solved using approximate methods in a given class of contour configurations, as parametric optimization problems. It is substantiated the expediency of using inductors in the form of current spatial contours with edges raised above the surface, for which the significantly lower heating temperature non-uniformity across the width of the strip is achieved compared to the traditional approach, when using the flat contours with current. The optimal configurations of spatial inductors are found for the following important practical heating conditions: the linear density of the electromagnetic energy flux does not exceed the specified maximum value; does not fall below the specified minimum value; has minimum deviation from the average value at a certain width. Methods for achieving uniform heating of non-ferrous and ferrous metal strips over the entire width and in the local area are analyzed. It is shown that with uniform heating over the entire width of the tapes, the edges of the contours of the optimal configuration should be raised from the surface to greater distance with lower slope, compared with the optimal geometry of the contours for heating the local area of the strip. Keywords High-frequency induction heating · Configurations of iron-free inductors · Inverse problems · Uniform heating of moving metal strips

1 Introduction One of the main problems of the induction method of heat treatment of metal strips is to provide a certain temperature distribution of the heated strip [1–3]. The metal strip is heated during its movement in the alternating transverse electromagnetic field of the inductor. The given temperature regime is determined, first of all, by

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 I. Kondratenko et al., Interactions Between Electromagnetic Field and Moving Conducting Strip, Lecture Notes in Electrical Engineering 1111, https://doi.org/10.1007/978-3-031-48274-8_2

37

38

Configuration of Spatial Iron-Free Inductors for High-Frequency …

the distribution of Joule heat releases associated with the flow of eddy currents in conducting medium. Among the most promising methods that have recently been considered is heating using a coreless inductor made in the form of a conductor, which forms a contour that is powered by a high-frequency source. The advantage of the high-frequency method of induction heating is the possibility of achieving a high density of transferred energy, which is practically impossible to obtain at low frequencies [4]. This chapter deals with high-frequency induction heating of conducting strips. The small field penetration depth compared to the thickness of the metal strips makes it possible to use the conducting half-space model [5, 6]. Such model is also successfully used to solve other problems using both alternating and pulsed fields, the diffusion of which into conducting medium manifests itself only in a thin surface layer of metal. Examples are the processing of materials by high-density currents [7, 8], electromagnetic forming technology [9–11] and others. The temperature distribution in the strip is determined, first of all, by the amount of transferred active energy of the electromagnetic field, which determines the amount of heat released equal to it. The temperature distribution is also affected by factors such as the thermal conductivity of the strip and heat transfer from the surface by radiation [1, 12]. Only in the case of an insignificant influence of these factors, the temperature distribution will be determined by the value of the transferred thermal energy. Based on the results of Chap. 1, the study assumes that the temperature is uniform over the thickness of the strip, and the process is considered adiabatic in the longitudinal direction. In this case, the local values of the strip temperature are determined by the local amount of the transferred electromagnetic energy. The strip temperature is considered to be relatively low, which makes it possible to ignore heat transfer by radiation. The main purpose of the study is to search for the geometry of inductors in the form of spatial contours with current, provided that a given (usually uniform over the strip width) total heat release in moving conducting medium. The problems under consideration are essentially synthesis problems in the electromagnetic field theory. In this chapter, the inverse problem is formulated as an optimization parametric problem for a certain class of contour configurations. In addition to practical use in the field of induction methods of heat treatment of metals, such studies are of independent importance for the analysis of the distribution of an electromagnetic field in an electromagnetic system with sources of an external field of a certain geometry. In addition to practical use in the field of induction methods of heat treatment of metals, such studies are of independent importance for the analysis of the distribution of electromagnetic field in electromagnetic systems with of an external field sources of a certain geometry. The solution of the inverse problem, taking into account the joint manifestation of thermal and electromagnetic processes in moving conducting media with threedimensional structure of fields, is an important and, at the same time, computationally difficult problem. Here it is justified to use approximate calculation methods [13, 14], for example, asymptotic ones, which are presented in Chap. 1 and allow taking into

2 Mathematical Model of the Electromagnetic System …

39

account the most significant geometric, electrical and thermal features of electromagnetic systems. In this case, it is possible to find the main geometric parameters of the electromagnetic systems under given conditions for the distribution of thermal energy released during the movement of the strip in the field of the inductor. The found geometry of the systems and the temperature distribution can be used as the initial ones when using more accurate, but also more time-consuming calculation methods.

2 Mathematical Model of the Electromagnetic System for High-Frequency Heating of Metal Strips 2.1 The Main Assumptions at High-Frequency Heating of Metal Strips The metal strip with thickness d of electrical conductivity γ and relative magnetic permeability μ moves at a speed v in the alternating magnetic field of the inductor (Fig. 1). The inductor is modeled by a contour in the general case of a spatial configuration with variable distance h between the contour points and the surface of the ˙ conducting body. The alternating / / current with complex-value amplitude I0 flows along the contour. Here δ = 2 (ωμ0 μγ ) is field penetration depth, where ω is / / cyclic frequency, and it is true the ratio ε1 = δ d < 1 (2 ÷ 3). This allows to apply the model of conducting half-space. In addition at high-frequency induction heating of metal strips, usually the / ε = δμ h, ε following three dimensionless parameters appear to be small: 2 3 = / / h D, ε4 = v (ωh), where D is the representative size of the inductor contour. As shown in Chap. 1 under the condition of smallness of the parameter ε2 , , the solution for the field above the conducting medium, including points at the surface, can be written as an asymptotic series for the electromagnetic field (1.12–1.15) and for the active power flux density pz into the conducting body (1.19). The terms of the series are functions of the current field of the contour and their expressions in the form of one-dimensional integrals can be easily calculated. As the field frequency increases, the influence of the higher terms of the series decreases, and in practice it is sufficient to keep only a few first terms. In induction heating installations, the inductor is located near the surface of the metal strip, and for points at the surface the parameter ε3 turns out to be small. In addition, the angle of inclination α (Fig. 2) of the tangent to the contour relative to the surface, even for spatial contours, usually does not exceed a few degrees. These limitations, which are characteristic of high-frequency induction heating of metal strips, make it possible to use a model of locally two-dimensional field to describe the electromagnetic field. Expressions for the complex-values of potentials and vectors of the electromagnetic field, as well as for the complex-value of the

40

Configuration of Spatial Iron-Free Inductors for High-Frequency …

Fig. 1 The geometry of the electromagnetic system for determining the main parameters of the heating process

∏ Poynting vector ~ , are simplified. The terms of asymptotic series do not contain contour integration operations. In particular, the active power flux density into the conducting body takes the form (1.33). ⎡ ⎫ ⎧ ( )n+1 ⎤ μ N ⎪ ⎪ n 2 ∑ ⎪ ⎪ (−1) · 2an μ0 iωI0 ⎢ ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ p ⎦ ⎣ ⎪ ⎪ 2 ⎪ ⎪ 2π ⎪ ⎪ n=0 ⎪ ⎪ u (ξ, 0) ⎪ ⎪ n ⎬ ⎨ ) ( ⎤ ⎡ ( ) n+1 ~ , pz (ξ ) = Re −π(z = 0) · ez = Re μ n ⎪ ⎪ ⎪ ⎪ N (−1) · 2an ⎥ ⎢ ⎪ ⎪ ∑ ⎪ ⎢ 4h p ⎥⎪ ⎪ ⎪ ⎪ − | ⎥⎪ ⎪ ⎪ 2 + h2 ⎪ ⎪ ×⎢ | ⎦ ⎣ ξ ∂u (ξ, z) ⎪ ⎪ n ⎪ ⎪ n=0 | ⎭ ⎩ | ∂z z=0 (1) where ξ is the local coordinate which is directed perpendicular to the tangent to the[ contour ]and parallel to the interface between the media, u n (ξ, z) = (n) z+h −2 ∂∂z n ξ 2 +(z+h) 2 .

2 Mathematical Model of the Electromagnetic System …

41

Fig. 2 Comparison of calculation results for 3D and locally 2D models

At the induction heat treatment of metal strips the condition ε4 x0 , the linear density of the released thermal energy, and hence, according to (2), the resulting temperature decreases rather quickly. The temperature non-uniformity in the interval (−xm ≤ x < xm ) is determined by this particular area of the heated metal strip. In this case xm < a and for the example under consideration, the degree of nonuniformity is less than for the problem (6, 7) at xm = a, but still remains significant.

50

Configuration of Spatial Iron-Free Inductors for High-Frequency …

Fig. 7 Projections of contours to the plane x Oz (a) and dependencies P ∗ (x) (b) when the temperature does not exceed the set maximum / value. 1 − g h 0 = 1.4, n = 1.55; / 2 − g h 0 = 1.5, n = 2; / 3 − g h 0 = 1.75, n = 2.7

However, as can be seen from the figure, the degree of non-uniformity essentially depends on the value x0 < a and falls very quickly with decreasing x0 . The most acceptable distribution of heat releases is represented by dependence 3 in Fig. 7b.

3.4 Contour Configurations in the Case of Minimal Deviation at a Given Value Most often, in practice, it is necessary to carry out the most uniform heating of the product over a certain surface width [12]. In this case, it is immediately necessary to minimize the deviations of the linear density of heat release at a given width, without requiring the fulfillment of condition (7) or (9). Figure 8 shows the results of the research when the optimization condition is fulfilled at different widths xm for c = a. As can be seen from the dependences found, the value of the maximum deviation |ΔP|max of heat release decreases sharply with a decrease in width xm compared to the radius of the current contour (curve 3).

3 Geometric Parameters of Spatial Contours with Current …

51

Fig. 8 Projections of contours to the plane x Oz (a) and dependencies P ∗ (x) (b) in the case of minimal deviation at a given value. 1 − g/ h 0 = 1.5, n = 4.3, xm = 0.25 m; 2 − g/ h 0 = 1.8, n = 3.4, xm = 0.23 m; 3 − g/ h 0 = 2, n = 3.2, xm = 0.21 m; 3 − g/ h 0 = 2, n = 3.2, xm = 0.21 m

Figure 9 shows the minimum possible deviations of the linear density of heat release ΔP ∗ for a given class of contour geometry. The obtained dependences make it possible to find the necessary transverse dimensions of the contour according to a given level of non-uniformity of heat release. At the same time, the parameters n and g/h0 provide the minimum possible deviation values |ΔP|max for the selected a and xm or the minimum radius a for a given level |ΔP|max and width xm , at which these deviations are allowed.

3.5 Some Examples of Inductor Geometry for Leveling Heating of Metal Strips The configurations of inductors in the form of contours with current, optimized for certain conditions, make it possible to confirm not only the correctness of the assessment of conditions (5) imposed on the geometric parameters to ensure temperature equality at the extreme points of the strip P(x A ) = 2P(xC ), but also the condition h C cos β = const for the constant linear density of the active energy flux for arbitrary points away from the edge of the contour. So, as an example in Fig. 10a–c for

52

Configuration of Spatial Iron-Free Inductors for High-Frequency …

Fig. 9 The minimum possible deviation of the linear density of heat is released ΔP ∗

different optimization conditions, as well as for significantly different material properties and the acting field frequency, the found gap profiles/and the corresponding relative values of the heat release linear density P ∗ = P(x) P(0) are shown. Curves 1 correspond to the profile, which ensures the minimum deviation P(x) of heat release from the value P(0) on the width within −xm ≤ x ≤ xm . Curves 2 correspond to the condition when the linear density of heat release does not exceed the value at the central point P(x) ≤ P(0) at the same width of the strip. As can be seen from the presented dependences (Fig. 10c), relation (5) can indeed serve as an estimate of the constancy of the linear density of the active energy flux in sections away from the contour edges. The considered methods for optimizing the geometry of the current contours make it possible to provide the necessary character of the distribution of the resulting temperature of the heated strips. However, during heating, the temperature distribution is not uniform due to the fact that different areas of the surface pass under the corresponding sections of the contour at different times. This fact is illustrated in Figs. 11 / and 12, which show the relative temperature increase of the strip T ∗ = ΔT (x, y) ΔT (0, ∞). Figure 11 shows the temperature distribution when the strip is heated by a spatial contour, shown by curve 3 in Fig. 4. Figure 12 illustrates the/temperature distribution of the strip heating by a current contour with parameters g h 0 = 1.75 and n = 2.7, which makes it possible to obtain a more uniform resulting heating temperature across the surface width than that shown in Fig. 11. It can be seen that heating by inductors in the form of current-carrying circuits is accompanied by the appearance of temperature gradients along the direction of the strip motion during heating. This circumstance should be taken into account in connection with the possibility of the appearance of thermomechanical stresses in the metal [17].

3 Geometric Parameters of Spatial Contours with Current … Fig. 10 Optimization of the linear energy flux density across the width of the strip x m = 0.2 m for inductor with parameters a = b = 0.25 m, h 0 = 0.02 m. 1–γ = 1.25 · 107 Ω−1 · m−1 , μ = 1, f =/ 2 · 103 Hz, a/c = 0.8, d h 0 = 8.4, n = 3.82; 2–γ = 1.0 · 106 Ω−1 · m−1 , μ = 10, f = 105 Hz; a/c / = 1, d h 0 = 1.75, n = 2.70

53

54

Configuration of Spatial Iron-Free Inductors for High-Frequency …

Fig. 11 Temperature distribution during strip heating by a current circuit with parameters g/ h = 0.75 and n = 1.5

Fig. 12 Temperature distribution during strip heating by a current circuit with parameters g/ h = 1.75 and n = 2.7

4 Flat Elliptical Contours (h = const, R = var) It follows from (5) that not only the use of spatial contours with raised edges can be effective, but also the use of flat contours with a relatively small radius of curvature at a point A [18]. In this regard, we consider elliptical contours with semiaxes b < a

4 Flat Elliptical Contours (h = const, R = var)

55

(Fig. 1), for which the release of Joule heat near the edges decreases. Let us find the geometric parameters of the contours for which condition (5) is satisfied for xC = 0. The minimum radius of/curvature of an elliptical contour will be at the point A where its value is R = b2 a. Substituting this value into (5), we find an estimate of the ratio of the lengths of the semiaxes of the ellipse, at which the linear density of the released thermal energy is the same at the central point x = 0 and under the edge of the contour x = a 4 b = a 3

/

h . a

(12)

Figure 13 shows the results of calculation by the method of asymptotic/expansion using three-dimensional mathematical model for two values of the ratio h a. Curves 2 show the relative values of the linear density of the transferred energy with the ratio of the semiaxes of the ellipse in accordance with (12). It can be seen that the heat releases at the points x = 0 and x = a practically coincide. The distribution of the released thermal energy over the width of the strip becomes much more uniform compared to a flat round contour (h = const, a = b), the calculated data for which are presented by curves 1. Despite the simpler configuration compared to the spatial contours, it should be noted that there is an increased heat release when moving away from the edge of the contour. This behavior is explained by the difference from zero of the angle between the direction of the current at a particular point of the contour and the velocity vector of the strip (β /= 0 in Fig. 1). This feature is seen from the nature of the dependences of the value of P(x) at x < x A (curves 2). The non-uniformity of heat release across the width of the strip using flat elliptical contours can be significantly reduced within the limits somewhat smaller than the

Fig. 13 Linear density of released thermal energy for the inductor of elliptical form

56

Configuration of Spatial Iron-Free Inductors for High-Frequency …

transverse dimension of the contour. To do this, the ratio of the semiaxes of the ellipse must be chosen smaller than that which ensures the fulfillment of condition (12). Curves 3 in Fig. 13 correspond to the ratio of the semiaxes of the ellipses, which are 80% of the ratio of the semiaxes chosen in accordance with (12). One can see a significant decrease of non-uniformity of the linear density of the released energy, including when the contour is located closer to the surface of the strip and, accordingly, the absolute values of the heat release intensity increase.

5 Features of the Application of the Asymptotic Method for the Study of Induction Heating of Limited Width Conducting Strips Real bodies always have limited dimensions, so the mathematical model of contours with current located above the conducting half-space considered so far has a welldefined area of application. It is valid if the sections of the contour are located from the edges of a flat conducting body at a sufficiently large distance. In a number of problems of practical importance, the distances from the system of currents that create the electromagnetic field to the edge of the conducting body become commensurate with the distance from the contour to the flat interface between the media. A similar problem arises, for example, when developing devices for highfrequency heating of metal sheets of limited width, heating sections of conducting products near their edges [19, 20], determining the electromagnetic stray field of technological devices, studying the transverse edge effect [4, 21, 22], and others. In such cases, a mathematical model should be used, which takes into account the features of the flow of induced currents at the edge of the electrically conductive medium. The fundamental issues of electromagnetic phenomena of the transverse edge effect in the plane-parallel field approximation were studied in [23]. The equality to zero of the secondary field at the side borders of the surface is substantiated and an analysis of the main electromagnetic processes is carried out. The authors of [24] and [25] studied the spatial distribution of electromagnetic fields in planar induction devices and conducting media of limited width, and determined the limits of applicability of the plane-parallel field approximation. The solution obtained in [25] corresponds to the physical model formulated in [24] of “periodic alternation of inductors” at the edge of the conducting surface. The inductors are located on both sides of the conducting surface, providing the required conditions on the side surface. However, the expressions obtained are presented in the form of infinite series, which makes it difficult to analyze and determine the conditions for applying approximate models for thin strips under the condition of the strong skin effect.

5 Features of the Application of the Asymptotic Method for the Study …

57

5.1 Mathematical Model for Calculating the Electromagnetic Field of the Current Contour Over Conductive Strip of Limited Width In this work, to take into account the limited width of the conducting strip, the method of mirror reflections relative to normal planes passing through the edges of the conducting medium [25] (Fig. 14) is also used. At the edge of the strip, the total normal component of the current density inside the metal should be equal to zero. To ensure this condition, the currents in the reflected contours must change their direction relative to the current of the original contour. Theoretically, in the presence of a strip bounded on two sides, there will be an infinite number of reflections. However, the contour is located close enough to the interface between the media, and the strongest electromagnetic field exists only in a relatively narrow spatial region near the contour. Therefore, the field of mirrored contours must be taken into account if the contour is simultaneously and sufficiently close to the edge of the strip. This is the peculiarity of the study in this work. With this assumption, it is sufficient to take into account only the field of one reflected contour, as shown in Fig. 14 [26]. The use of such model for calculating the field near the edge of the strip is possible under the following conditions for the smallness of the parameters h, r, r1 , rk a, the temperature distribution in the strip is close to rectangular contour. When b < a, the temperature distribution in the strip changes significantly. The center of the strip is heated more intensely, reducing the overall non-uniformity of heating across the width. Therefore,

106

Electromagnetic Systems with Iron-Free Inductors for Induction …

Fig. 6 Strip temperature distribution at a = 0.25 m, b = 0.5 m for current contour in the form of a rhombus

(T-T0), K 150

100

50 0.43

0 -0.24

0.11 -0.12

0.00

y, m

Fig. 7 Strip temperature distribution at a = 0.5 m, b = 0.25 m for current contour in the form of a rhombus

-0.21 0.12

0.24

x,m

-0.54

(T-T0), K 100 75 50 25 0.43

0 -0.24

0.11 -0.12

0.00

y, m

-0.21 0.12

0.24

x, m

-0.54

a combination of rectangular and rhombus-shaped current contours would contribute to improving the uniformity of heating across the width of the moving strip.

3.3 Current Contour in the Form of an Ellipse The distribution of current in an elliptical contour (Fig. 8) i = Im w F (x, y) is described by the function: ( F(x, y) = δ

) y2 x2 + − 1 [Θ (x + b) − Θ(x − b)] [Θ(y + a) − Θ(y − a)]. b2 a2 (52)

Let’s use the relationship [22]:

3 Heat Treatment of Moving Strip by Iron-Free Inductors as Current …

107

Fig. 8 Current contour in the form of an ellipse

δ ( f (x)) =

∑ i

) ( 1 δ x − xi , | f '(xi )|

(53)

where x i —first-order roots of the function f (x). Then the components of the current load in Cartesian coordinate system can be represented as follows:

jSmy

jSmx

) ⎤ ⎡ ( b√ 2 2 δ x− a −y ⎥ ⎢ x a ⎢ ( )⎥ = − Im w / ( ) ⎦; ⎣ √ b b2 b4 2 2 2 a −y x 1 − a 2 + a 2 +δ x + a ( ) ⎤ ⎡ b√ 2 2 δ x− a −y ⎥ ⎢ y a ⎢ ( )⎥ = Im w / ( ) ⎦. ⎣ √ b a2 a2 2 2 2 2 +δ x + a −y x b2 − 1 b2 + a a

(54)

The function Fkn (17) expressing the integral transformation of the spatial distribution of the current contour in the form of the ellipse is given by the function: ⎫ √ ( ) ⎬ 2 − y2 √ a b 2 − y 2 ch (q y) dy / ( = −4 μ0 ⎝ pk a sh p a k n ⎭ ⎩ y 2 b2 − a 2 ) + a 4 a 0 ⎫ ⎞ ⎧ √ ∫b ⎨ ⎬ ( ) √ 2 2 b −x a / ( + qn b sh qn b2 − x 2 ch ( pk x) d x ⎠. (55) ) ⎭ ⎩ x 2 a 2 − b2 + b4 b ⎛

Fkn

⎧ ∫a ⎨

0

By integrating the component of the magnetic field induction Bz1 at z = 0 over the area bounded by the current contour, we can determine the magnetic flux:

108

Electromagnetic Systems with Iron-Free Inductors for Induction …

∫b

a b

√ b2 −x 2

∫

ϕm =

Bz1 d x d y, −b

− ab

√ b2 −x 2

and using Eqs. (35) and (36), we can find the complex impedance ∫ ∞ ∞ ) ( a√ 4 i ω w ∑ ∑ Fk n = b2 − x 2 ch px d x. (Mk n − 1) sh qn τ d k=−∞ n=1 η1 k n qn b b

zm

0

(56) The given expressions for the elliptical contour are simplified for a circular current contour. In this case, when a = b = r , Eq. (55) takes the form: ⎡ Fkn = −

4 μ0 ⎢ ⎢ ⎢ pk r ⎣

∫r 0

) ( √ √ r 2 − y 2 sh pk r 2 − y 2 ch (qn y)dy

⎤

⎥ ⎥ ∫r √ ) ( √ ⎥. 2 2 2 2 +qn r − x sh qn r − x ch( pk x) d x ⎦

(57)

0

The performed analysis using the presented calculation method has shown that elliptical current contours allow for a more uniform heating of the strip across the width compared to rectangular, rhomboidal, and circular current contours. Here are the results of calculations for the temperature distribution across the width of the strip moving with constant speed in the magnetic field created by elliptical current contour, where the most uniform heating is achieved. Figure 9 shows the distribution of temperature increment (T −/T0 ) across half the width of the strip is shown at the exit of the heating zone (x = τ 2). The following geometric dimensions of the ellipsoidal contour are considered:a = 0.265 m, b = 0.13 m. The calculation was performed for strip with width of 0.6 m and thickness s = 1 mm, moving at a speed of 0.1 m/s at the distance h = 0.02 m from the current contour. The magnitude of the current in the contour equal to I = 1000 A, the frequency is f = 2 kHz, and the number of turns is w = 4. At the given parameters, the electromagnetic system has active power of 38.5 kW, with efficiency η = 88.5% and power factor cos φ = 0.31. Figure 9 shows the temperature values at individual characteristic points. The temperature distribution shows the significant variation, indicating considerable nonuniformity across the width of the strip and the possibility of local overheating in certain areas of the strip. To improve the uniformity of heating across the width of the strip, it is possible to use a combination of multiple current contours. Let’s consider two coaxial current contours of elliptical shape above the strip. The resulting magnetic flux is the superposition of the magnetic fluxes of both contours. In each contour, there is an induced EMF:

3 Heat Treatment of Moving Strip by Iron-Free Inductors as Current …

109

(T-T0), K 160

139

120

115

112

80 40 0 0.0

0.1

0.3 y, m

0.2

Fig. 9 Temperature distribution across the width of the strip as a result of heating using the elliptical inductor

∫ E 1 = −i ω w1

(

) Bz1 + Bz2 d s;

S1

∫

E 2 = −i ω w2

(

)

(58)

Bz1 + Bz2 d s,

S2

where S1 and S2 , w1 and w2 are the areas and number of turns of the current contours, Bz1 and Bz2 are the magnetic induction at z = 0 of the first and second contours respectively. The contours can be connected aiding or out-of-phase according to the magnetic flux. When they are connected aiding, the magnetic inductions in (58) have the same sign, while in the case of out-of-phase connection, they have opposite signs. The complex impedance of the series equivalent circuit of the combined two contours is determined as follows zm =

E1 + E2 . Im

The induced current densities are also determined by the these superposition jx m = jxm1 + jxm2 ; j y m = j ym1 + j ym2 .

(59)

The volumetric heat dissipation PW (44) is calculated based on the obtained current density distribution. The temperature distribution in the strip, moving in the magnetic field of two coaxial elliptical current contours connected in series, at x = τ , is shown in Fig. 10. The geometric dimensions of the contours and the number of turns are chosen as follows: a1 = 0.275 m, b1 = 0.15 m, w1 = 4, a2 = 0.18 m, b2 = 0.15 m,w2 = 1. The remaining parameters correspond to the example shown in Fig. 9.

110

Electromagnetic Systems with Iron-Free Inductors for Induction …

(T-T0), K 250 195

200

183

150 100 50 0 0.0

0.1

0.2

0.3 y, m

Fig. 10 Temperature distribution across the width of the strip as a result of heating using the two coaxial elliptical contours

The electromagnetic system composed of two contours has active power 63.3 kW, efficiency η = 91% and power factor cos φ = 0.34. The non-uniformity of temperature distribution across the width in the case of using combination of two contours is 7%, which is significantly lower than heating the strip with single current contour (21%). The last example demonstrates the potential of using current contours for induction heating devices for flat metal products. By carefully selecting the geometric dimensions of the current contours, it is possible to achieve uniform heating of the strip across its width, meeting the requirements of heat treatment technology. At the same time, the energy efficiency of such induction systems remains high, which is characteristic of induction heating devices in the transverse magnetic field.

4 Influence of the Finite Height of the Winding Cross-Section on the Energy Characteristics of the Inductor Above, the influence of the configuration of the current contour on the uniformity of heating across the width of the strips was investigated. Real windings have finite dimensions of cross-sectional, which has a certain impact on the energy characteristics of the induction heating device. The effects of the finite height of the winding cross-sectional are examined in this subsection. In contrast to the previously discussed one-sided arrangement of the inductor relative to the conducting strip, here the investigation is performed for the case of symmetrically placed, identical current contours of arbitrary polygonal configuration relative to the strip. According to the geometry of the considered electromagnetic system, the center of the Cartesian coordinate system is placed at a point on the strip equidistant from the contours. In this case, the Ox axis is aligned with the edge of

4 Influence of the Finite Height of the Winding Cross-Section …

111

Fig. 11 Calculation model of the system with winding of real cross-section

the strip and is measured from the reference point, where the influence of the coil’s field is negligible and through which the Oy axis passes (Fig. 11). With such a choice of coordinate axes, it can be considered that the heating process of the strip begins when it moves from the position of the Oy axis. When the opposite current contours are connected aiding, on the plane z = 0 the symmetry relationships for the magnetic field induction are satisfied: / ∂ Bz ∂z = 0,

Bx = 0,

B y = 0.

(60)

By representing the cross-section of the current contour as a rectangle with a height h 1 and width c1 , we can determine the current density in the contour with the following functional dependence: ) ⎡ ( ⎤⎡ ( h c1 ) ⎤ Θ z− + Θ x − x 1 ⎢ ⎥ Im w ⎢ 2 ⎢ 2 )⎥ ei ωt . )⎥ ( j= ( ⎦⎣ c1 ⎦ h h 1 c1 ⎣ −Θ x − x1 − −Θ z − − h 1 2 2

(61)

112

Electromagnetic Systems with Iron-Free Inductors for Induction …

In the further discussion, we will consider the winding under the condition c1 = 0, which significantly simplifies the obtained analytical solutions and makes the analysis more intuitive [23]. For such winding, instead of (61), we have the expression: [ ( )] ) ( h h Im w Θ z− δ(x − x1 ) eiωt . − Θ z − − h1 j= h1 2 2

(62)

For a closed current contour composed of straight segments, the components of current density jx and j y can be represented as follows N ) [ ( )] [ ( ) ( )] Im w ∑ ( k j sinϕ j δ x − k j y − b j Θ x − x j − Θ x − x j+1 h 1 j=1 )] ) ( [ ( [ ( ) ( )] h h × Θ y − y j − Θ y − y j+1 × Θ z − − Θ z − − h 1 ; (63) 2 2

jxm =

N ) [ ( )] [ ( ) ( )] Im w ∑ ( sinφ j δ x − k j y − b j Θ x − x j − Θ x − x j+1 h 1 j=1 )] ) ( [ ( [ ( ) ( )] h h × Θ y − y j − Θ y − y j+1 − Θ z − − h1 . Θ z− 2 2

j ym =

(64)

( ) Here N is the number of sides of the polygon; x j , y j are coordinates of the polygon’s x j+1 −x j ; sinφ j = vertex projections on the plane z = 0; b j = y j k j − x j ;k j = y j+1 −y j y

/

−y

j+1 j . (x j+1 −x j )2 +( y j+1 −y j )2 In the area occupied by the current contour, the following equation holds for the component Bz of the magnetic field induction:

( ) ∂ jy ∂ jx ∂ 2 Bz ∂ 2 Bz ∂ 2 Bz − . + + = −μ 0 ∂x2 ∂ y2 ∂ z2 ∂x ∂y

(65)

By averaging Eq. (65) over the height, we obtain (

∂ Bz− ∂ Bz+ − ∂z ∂z _

where Bz =

1 h1

∫h 1

(

) = −μ0 h 1

∂ jy ∂ jx − ∂x ∂y

(

) − h1

_ _ ) ∂ 2 Bz ∂ 2 Bz + . ∂x2 ∂ y2

(66)

Bz dz is average value of magnetic induction within the height of

0

the contour; the “+” sign corresponds to the value of the function for the winding in the upper area (z > 0), while the “–” sign denotes the winding / below the strip (z < 0). In real induction heating systems, the ratio ε = h 1 d < 1 is a small quantity. Neglecting second-order terms of smallness ε2 > L u + 2h, where L u is the maximum linear size of the contour along the Ox axis. Let’s extract the component of the magnetic field induction Bz from Eqs. (2) and (3) and perform a two-sided Laplace transform [24] with respect to the coordinates x and y, taking into account the periodic variation of the magnetic field induction. As a result, instead of the three-dimensional equations for the originals, we obtain second-order one-dimensional equations for/the representation / of the magnetic field 2 ≤ z ≤ Δ 2 and/ in the area not induction. Inside the conducting strip −Δ / / occupied by the current winding |z| ≥ h 2 + h 1 , Δ 2 ≤ |z| ≤ h 2, the equations and expressions for the constants η12 and η22 do not differ from those obtained earlier in (7–9) when considering the strip heating with one-side inductor in the form of a flat contour. The solutions of Eqs. (7) and (8) are well known and can be expressed as sums of exponential functions multiplied by constant coefficients. The constant coefficients are determined by the boundary conditions at the interfaces, the conditions for the limitation of the magnetic field induction on infinity and the symmetry conditions

114

Electromagnetic Systems with Iron-Free Inductors for Induction …

(60). In the area occupied by the current contour, we will use the conditions (66) and (67), which in transformed values take the form ∂ Bˆ zm2 ∂ Bˆ zm1 − = g + h 1 η12 Bˆ zm1 ; ∂z ∂z

(71)

Bˆ zm1 = Bˆ zm2 ,

(72)

where the index “1” corresponds to the magnetic field above the current contour, and the index “2” corresponds to the air gap below the contour, 4μ0 Im w ˆ )( ) G; g=( −2 1 − e pτ 1 − e−2qd [ ⎛

( ) ( )]⎞ ch px j+1 sh qy j+1 ( ) ( ) ) ⎟ ⎜ pk j N ( ∑ −ch px j sh qy j ⎟ p − k j q sinφ j ⎜ ⎜ ˆ [ ( ×⎜ G= ) ( )] ⎟ ( )2 ⎟. sh px j+1 ch qy j+1 ⎠ pk j − q 2 ⎝ j=1 ( ) ( ) −q −sh px j ch qy j

(73)

By supplementing Eqs. (71) and (72) with the interface / conditions at the boundary between the conducting strip and the air gap (z = Δ 2), which require the equality of the normal components of the magnetic field induction and their derivatives with respect to z, we can find the solutions for the transformed values of the normal component of the magnetic induction: in the area above the inductors ( Bˆ zm1 ), in the gap between the strip and the inductor ( Bˆ zm2 ), and in the strip ( Bˆ zm3 ). Bˆ zm1 = −

( 2η1 1 +

η1 h 1 2

g )

) ( 1 − Me−η1 (h−Δ) −η1 z e ; h e−η1 ( 2 +h 1 ) (1 − h 1 L M)

(74)

( η1 z ) g e−η1 2 ) e − M e−η1 (z−Δ) ; 2η1 1 + η12h 1 (1 − h 1 L M)

(75)

1− M g e−η1 ( 2 − 2 ) ) ( ( Δ ) ch(η2 z), η1 h 1 2η1 1 + 2 (1 − h 1 L M) ch η2 2

(76)

h

Bˆ zm2 = −

(

h

Bˆ zm3 = −

Δ

sh(η2 Δ/ 2) − μη1 ch(η2 Δ/ 2) e−η1 (h−Δ) , L = η12+η . where M = μμ00 ηη22 sh(η 2 Δ/ 2) + μη1 ch(η2 Δ/ 2) 1 h1 To determine the components of the magnetic field induction Bx and B y , we can use the expression obtained for the component Bz and the equations ∇ · B = 0 and jz = ∇ × H|z = 0. After transforming these equations using (9), we obtain the expressions that describe the desired functions. Finding the expressions is similar to the presented derivation in Sect. 1 of the Chap. 2 of this book.

4 Influence of the Finite Height of the Winding Cross-Section …

115

By taking the limit of the current contour height approaching zero (h 1 → 0), the expressions (74–76) reduce to the expressions obtained under the assumption of infinitely thin current layer. This allows for an easy analysis of the influence of the current contour height on the energy characteristics of the inductor. The originals of the obtained solutions can be determined using the theorem of decomposition for a meromorphic( function )with infinitely many simple poles ( ) determined by the zeros of functions 1 − e−2 pτ and 1 − e−2qd in Eq. (73). Thus, the normal component of the magnetic field induction in the air gap is calculated using the formula Bzm2

) h( ∞ ∞ Im μ0 w ∑ ∑ G kn e−η1kn 2 eη1kn z − eη1kn Δ Mkn e−η1kn z pk x+qn y ( ) e =− , ηkn1 h 1 τ d k=−∞ n=−∞ 2η − h L M (1 ) 1kn 1 + 1 kn kn 2 (77)

where π , k = −∞, . . . , ∞; τ π qn = in , n = −∞, . . . , ∞, d pk = ik

(78)

ˆ pk , qn ),η1kn , η2kn , Mkn , L kn are determined according to the expressions G kn = G( provided above, taking into account Eq. (78). To analyze the influence of the finite height of the current coil on the energy characteristics of the inductor, we will use a series equivalent circuit, which consists of a series connection of the active resistance of the current contour r1 , the reactive resistance xσ of the leakage flux within the height of the coil, and the complex input impedance z m that takes into account the influence of the air gap and the conducting strip. To determine z m , we/calculate the magnetic flux through the area bounded by the current coil for z = h 2 and the corresponding electromotive force (EMF). The ratio of the induced EMF to the current in the contour determines the complex input impedance: ( ) G kn 1 − Mkn e−η1kn (h−Δ) ) ( ηkn1 h 1 ∞ ∞ 1 + 2η (1 − h 1 L kn Mkn ) 2 ∑ ∑ 1kn 2 i ω w μ0 zm = ) . ( N τd p x +q y p x +q y k=−∞ n=−∞ ∑ e k j+1 n j+1 − e k j n j pk + qknj j=1

(79)

The active resistance of a multi-turn coil r1 can be assumed to be equal to the resistance of the coil for direct current.

116

Electromagnetic Systems with Iron-Free Inductors for Induction …

r1 =

∏e ρ1 w , t

(80)

where ∏e —equivalent perimeter of the current contour, ρ1 —specific resistance of the conductors, t—effective cross-sectional area of the coil. Such representation of the active resistance of the current contour is justified when the windings are made of multi-strand wire (Litz wire), but it requires further clarification for massive conductors. To calculate the resistive losses within the height of the coil xσ , taking into account the finite dimensions of the current path, we can use the well-known method for calculating losses in electrical machines, described in [25]. Following a similar approach, we have the expression: xσ = ω w 2 μ0 λ ∏e ,

(81)

where λ=

2c12 ln(c1 + h 1 ) + h 21 − 2c1 h 1 − 2c12 ln(c1 ) . 8h 21

(82)

The current in the windings of the coil, when the upper and lower windings are identical and connected in series, can be calculated using the formula: Im =

√

/ 2U [2(z m + r1 + i xσ )].

(83)

The total power consumed from the network S, power factor cosφ, and heating efficiency η of the induction heater are determined according to the following expressions: / / ∗ S = U I , cosφ = Re(S) |S|, η = Re(z m ) (Re(z m ) + r1 ).

(84)

The quantitative influence of the height of the contour to the energy parameters of the inductor is determined by the multiplier in the denominator of Eq. (79). Increasing the height of the contour leads to a decrease in the complex impedance of the inductor and a significant reduction in the electromagnetic power flux into the heated strip. Figure 12 shows the dependence of the change in active power P of two-sided inductor on the height of the current contour. The inductor consists of two three-turn coils connected in series and connected to a current source of 1000 A at frequency of 2000 Hz. The width of the strip is 0.6 m, and the thickness is 2 mm. The current coils have a square shape with a side length 0.36 m. The coordinates of the vertices of the squares projected onto the plane z = 0 are as follows: (0.3, 0.3), (0.6, 0.5), (0.8, 0.25), (0.5, 0.05). The specific electrical conductivity of the strip corresponds to the conductivity of regular structural carbon steel at the temperature of 900 °C, which is above the Curie point. Therefore, the magnetic permeability is equal to the magnetic constant μ0 . In addition to the general trend of decreasing electromagnetic power

5 Conclusions Fig. 12 Active electromagnetic power of induction heating system with double-sided inductor depending on the height of the winding

117

Р, kW 60.0 40.0 20.0 0.0 0

0.02

0.04

0.06

0.08

0.1 h1,m

Fig. 13 Power factor of induction heating system with double-sided inductor depending on the height of the winding

flux with increasing coil height, there is also the decrease in the power factor of the inductor (Fig. 13). This, to some extent, is due to the increased leakage inductance within the height of the current contour, as described by Eq. (82). From the presented results, it can be concluded that maximum energy characteristics are achieved with the minimum possible height of the current winding.

5 Conclusions The developed method for calculating electromagnetic fields and energy characteristics of systems consisting of alternating current contours and a moving metal strip in the magnetic field offers the advantages of accurate analytical solution to the problem. The proposed approach for the quasi-stationary electromagnetic fields has no limitations on the thickness and width of the heated strip, the frequency of the field, the speed of strip movement, and the geometry of the planar inductors. The method allows for multivariate calculations of systems to find the most effective design solutions and operating modes of installations in terms of providing the required heating characteristics using eddy currents.

118

Electromagnetic Systems with Iron-Free Inductors for Induction …

The inductors in devices for induction thermal processing of thin metal strips, designed as planar current contours of canonical shapes (rectangle, rhombus, circle, ellipse), and positioned in a plane parallel to the surface of the strip, do not provide uniform heating across the finite width of the strips. To achieve uniform heating across the width of the strips, it is advisable to use combinations of current contours, where the geometric dimensions are determined by the size and electro-physical parameters of the metal strips. Further improvement of heating devices can progress towards the application of inductors with spatial configurations. The corresponding calculation methods can be based on the approach presented in Chap. 1 of this book, expanding it to systems with finite-thickness metal strips and use of double-sided inductors.

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