2024 | Book

# Interactions Between Electromagnetic Field and Moving Conducting Strip

Authors: Ihor Kondratenko, Yuriy Vasetsky, Artur Zaporozhets

Publisher: Springer Nature Switzerland

Book Series : Lecture Notes in Electrical Engineering

2024 | Book

Authors: Ihor Kondratenko, Yuriy Vasetsky, Artur Zaporozhets

Publisher: Springer Nature Switzerland

Book Series : Lecture Notes in Electrical Engineering

The book combines two interrelated lines 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 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 products using the induction method of heating in a transverse magnetic field. The inverse problems are solved to determine the inductor configuration as flat and spatial current contours for providing the necessary temperature distribution of moving metal strips. To achieve uniform heating of strips across the width using inductors in the form of flat current contours parallel to the strip surface, 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. A more uniform temperature distribution during high-frequency induction heating is achieved by using inductors in the form of current contours of the required spatial configuration.

The book is intended for researchers, postgraduate students, and students specialized in theory and calculations of electromagnetic fields and induction heating installations.

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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.

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.

Abstract

The research in this chapter is dedicated to issues related to the use of single-phase transverse magnetic field inductors in induction heating devices for thin moving metal strips. The used mathematical models in the research of the three-dimensional electromagnetic field of the induction system allow to apply analytical calculation methods. There are no restrictions on the thickness and width of the metal strip or the frequency of the field. The current load is modeled using conductors in the form of a current layer. The analytical method is based on a two-dimensional integral transformation of the magnetic field and current load. The analytical solution, taking into account the velocity of the conducting medium, is found for the magnetic field in the area of the metal strip and the area without eddy currents. Calculations are made for the current density in the area of the moving 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. Numerous calculations were performed with using the developed methodology, which allowed to conclude that the minimum heating non-uniformity for single-phase inductors is approximately 15%.

Abstract

The chapter discusses electromagnetic systems for thermal treatment of metal strips using the induction heating by eddy currents. A mathematical model of the heating device is used for analysis, which includes an inductor to generate an alternating magnetic field and the metal strip moving in this field. Flat-shaped inductors without magnetic cores are considered. To solve the three-dimensional electromagnetic field problem considering the movement of the conducting medium, an analytical method based on representing the current load as a two-dimensional harmonic series expansion is applied. To solve the problem, an analytical method and mathematical models are used to investigate the electromagnetic interaction between field sources and the conducting body. The temperature distribution in the strip is obtained by solving heat transfer equations with internal heat sources. Specific temperature calculations are performed using numerical methods. The analysis begins with the general problem of heating systems using inductors in the form of flat contours of arbitrary shape. Then, the results for specific contour shapes of canonical forms such as rectangles, rhombuses, and ellipses are presented. For these systems, the distribution of the electromagnetic field, current density, and temperature in the strip has been determined. The equivalent resistance, efficiency coefficient, and power factor of the devices have been calculated. The influence of the winding height of the inductor on these parameters has been analyzed. It has been found that the considered flat geometrical inductors do not provide uniform heating of finite-width strips. To achieve uniform heating across the width of the strips, combinations of current contours are proposed, where the geometric dimensions are determined by the size and electro-physical parameters of the metallic strips.