Electromagnetic Field Near Conducting Half-Space: Theory and Application Potentials (Lecture Notes in Electrical Engineering, 1070) 3031384229, 9783031384226

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Lecture Notes in Electrical Engineering 1070

Yuriy Vasetsky Artur Zaporozhets

Electromagnetic Field Near Conducting Half-Space Theory and Application Potentials

Lecture Notes in Electrical Engineering Volume 1070

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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Yuriy Vasetsky · Artur Zaporozhets

Electromagnetic Field Near Conducting Half-Space Theory and Application Potentials

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

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

ISSN 1876-1100 ISSN 1876-1119 (electronic) Lecture Notes in Electrical Engineering ISBN 978-3-031-38422-6 ISBN 978-3-031-38423-3 (eBook) https://doi.org/10.1007/978-3-031-38423-3 © 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

Contents

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located Near Conducting Body with Flat Surface . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 General Solution of Wave Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Problem Statement and Basic Equations . . . . . . . . . . . . . . . . . 1.2.2 General Solution of the Wave Problem for Contour with Alternating Current in the Form of a System of Emitting Current Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 General Solution of Quasi-Stationary Problem for Arbitrary Contour with Alternating Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Small Parameter to Take into Account the Displacement Current Density . . . . . . . . . . . . . . . . . . . . . . 1.3.2 3D Solution Problem for Dielectric Half-Space in the Quasi-Stationary Approximation . . . . . . . . . . . . . . . . . . 1.3.3 3D Electromagnetic Field in Conducting Medium in the Quasi-Stationary Approximation . . . . . . . . . . . . . . . . . . 1.4 Features of the Formation of the 3D Quasi-Stationary Electromagnetic Field in the System “Arbitrary Spatial Current Contour–conducting Half-Space” . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Main Feature of the Distribution of the 3D Electromagnetic Field in the Conducting Half-Space . . . . . . 1.4.2 Field Souses and Conditions on the Interface Between Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Complex-Value Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Examples of Electric Field Intensity and Surface Charge Density Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 4

6 9 10 11 16

19 19 23 26 27 30 30

v

vi

Contents

2 Approximate Mathematical Models for Analysis of Alternating Electromagnetic Field of Sources Near Conducting Body . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Electromagnetic Field of Spatial Contour with Current Near Flat Surface of Conducting Body with Perfect Skin Effect . . . . . . . . 2.3 Asymptotic Expansion Method for Calculation of Three-Dimensional Quasi-Stationary Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Expansion of the Potentials and Field Vectors into Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Estimation of the Asymptotic Series Expansion Errors . . . . . 2.3.3 Choice of the Number of the Asymptotic Series Terms . . . . 2.3.4 Comparison of Exact and Approximate Calculations of 3D Electromagnetic Field for Specific Contour Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Mathematical Model of a Locally Two-Dimensional Electromagnetic Field Near the Current Contour . . . . . . . . . . . . . . . . 2.4.1 Conditions for the Possibility of Using the Model of Locally Two-Dimensional Electromagnetic Field . . . . . . . 2.4.2 Expressions for Calculating the Electromagnetic Field and the Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Penetration of Non-uniform Sinusoidal Electromagnetic Field into Conducting Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Electromagnetic Field in a Conducting Half-Space—A General Feature of the Non-uniform Electromagnetic Field Penetration into Conducting Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Penetration of Non-uniform Electromagnetic Field in the Case of Strong Skin Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Comparison of Decay of Non-uniform and Uniform Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Non-uniform Electromagnetic Field at the Interface Between Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Influence of the Small Parameter Value to the Field Penetration Law with the Strong Skin Effect . . . . . . . . . . . . . 3.4 Impedance Boundary Condition of Non-uniform Electromagnetic Field Penetration into Conducting Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 34 34

41 41 46 51

55 57 58 62 66 66 69 70

71 76 77 79 83

85 90 91

Contents

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing Near Conducting Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 General Analytical Solution for a Three-Dimensional Pulsed Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Asymptotic Expansion Method for Pulsed Three-Dimensional Quasi-Stationary Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Solution for Pulsed Electromagnetic Field in Dielectric Half-Space as Asymptotic Series . . . . . . . . . . . . 4.3.2 Cutoff Values of Frequency and Time Interval in the Asymptotic Series Expansion Method of Pulsed Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Integral Indicators to Take into Account Restrictions on the Frequency and Time Interval of Current Pulses . . . . . 4.3.4 Taking into Account Limited Time Intervals for Current Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Electromagnetic Field of the Standard Current Pulses Flowing Near Conducting Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Exponentially Decaying Current Pulse . . . . . . . . . . . . . . . . . . 4.4.2 Current Pulse Represented by the Difference Between Two Decaying Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Decaying Oscillating Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Possibility of Application the Asymptotic Expansion Method to Determine the Three-Dimensional Pulsed Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Comparison of the Results for the Integrand in Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Comparison of Approximate and Exact Calculation Methods for System with Current Contours . . . . . . . . . . . . . . 4.6 An Example of Using the Analytical Method for Calculating Three-Dimensional Electromagnetic Field for Systems with Pulsed Current Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

93 94 94 97 97

101 105 106 107 107 110 112

114 114 116

118 122 122

Chapter 1

Electromagnetic Field of Arbitrary Spatial Current Contour Located Near Conducting Body with Flat Surface

Abstract In the chapter, the studies are aimed at the exact analytical solution of a three-dimensional quasi-stationary problem, which is formulated in a fairly general setting. It is necessary to find the electromagnetic field of an arbitrary spatial contour with an alternating current, located above a conductive magnetizable half-space, in which eddy currents are induced. Restrictions are not imposed on the geometry of the contour with current and its orientation relative to the interface of dielectric and conducting media, the electrophysical properties of media and the field frequency. A linear task is considered, which, based on the principle of superposition, can easily be extended to the general case of an arbitrary system of the contours, that is, an arbitrary location of external field sources, as well as an arbitrary dependence of the current on time using the Fourier transform in time. The analytical solutions for the vector and scalar potentials, electric and magnetic field intensities are defined both in dielectric and conducting media. The main feature of quasi-stationary electromagnetic field formation for system with plane interface between the dielectric and conducting media is determined—the components of electric intensity and current density which are perpendicular to boundary surface are not available (equal to zero) in the conducting medium. This property holds true for any spatial configuration of the initial system of current and for any time dependence of external field sources. The physical reason for the absence of vertical components of the current density and electric field intensity in the conducting half-space is the appearance of a distributed electric charge on the interface surface, the field of which in the conducting half-space completely compensates for the vertical component of the external induced electric field. At the surface in the dielectric area the vertical component of the electric field intensity is twice the vertical component of the known induced field of the sources. Keywords 3D quasi-stationary electromagnetic field · Exact analytical solution · Arbitrary current contour configuration · Conducting half-space · Eddy currents

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Y. Vasetsky and A. Zaporozhets, Electromagnetic Field Near Conducting Half-Space, Lecture Notes in Electrical Engineering 1070, https://doi.org/10.1007/978-3-031-38423-3_1

1

2

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

1.1 Introduction In this work for solving the three-dimension problems the analytical methods are developed. These methods, despite the widespread successful use of numerical methods of calculation, allow to obtain results for a wide range of objects where specialized analytical or numerical-analytical methods remain effective. Such objects include, in particular, systems, the geometric features of which determine the different nature of the change the electromagnetic field in space—a rapid change in the field near source of field or near the interface between media and much slower in another part of space with larger volume [1]. An important task from a theoretical point of view is the possibility to obtain, on the basis of exact analytical solution, the general features of the physical processes, which are difficult to establish, and even more to justify their general nature on the basis of the numerical solution of a large number of specific tasks. The analytical solution of the problem was considered in many works, in which, nevertheless, restrictions were imposed on the geometry of the system or on the electrophysical properties of the conducting medium. For example, in [2, 3] the solution is obtained for current conductor which is located parallel to the boundary of media; the problem for the conductor located in vertical plane is solved in [4]; the contour with arbitrary configuration and perfect (ideal) skin-effect is examined in [5]. The complete analytical solution of the linear quasi-stationary problem given in [6, 7] is based on known analytical expression for the field of the emitting harmonic current dipole placed above the plane boundary of media [8–10]. In addition, without loss of generality contours are presented as a serial system of emitting dipoles [11]. A feature of study in this chapter is that it is based on the exact analytical solution obtained for the general problem of a three-dimensional quasi-stationary field. In particular, the main properties of the formation of electromagnetic field with a flat interface have been established in [6]. So, the present chapter is consistent presentation the analytical solving of the problem of 3D quasi-stationary electromagnetic field generated by arbitrary spatial current contour taking into account eddy current in the external conducting body with plane surface. The following objectives are to achieve the aim: to solve the problem in all space; to simplify the expressions under the condition of closed current contour; to determine some important peculiarities of the distributions of 3D electromagnetic field and its sources. Let us consider an arbitrary contour in a non-conducting medium with relative dielectric permittivity εe . The alternating current I0 flows along this contour located near conducting body with plane boundary having conductivity γ and relative magnetic permeability μ. The electrophysical parameters within the dielectric and conducting media are not variable in space and in time. The initial current contour is shown in Fig. 1.1 by solid line. As considered, the dimensions of the contour are less than the dimensions of the plane section of body surface. It gives a possibility to use the model of current contour above the conducting half-space. The points M and Q mark respectively the source point of the field on the contour with current I0 and

1.2 General Solution of Wave Problem

3

Fig. 1.1 The calculation model

the observation point at which the value of the electromagnetic field is determined. A contour element with length dl is represented as directed elementary segment t M dl, where t M is unit tangent vector to the contour at the point M. The axis z is oriented perpendicular to the media interface in the vertical direction. Since the end result is to obtain analytical solution of quasi-stationary problem for which the density current satisfies the continuity condition ∇ · j = 0, it is necessary to consider a closed contour with unchanged current I0 (t) along this contour. In addition, the two conditions are to be satisfied. Firstly, the wavelength of electromagnetic field λ should be much greater than any representative dimension L of the electromagnetic system, i.e. λ >> L. Secondly, the displacement current density j D is ignored in the conducting medium in comparison with the conducting current density j [12]. In the dielectric medium only the displacement current takes place. In this case, the displacement current density is taken into account and the wave phenomena are disregarded. Additional restrictions that take into account the absence of displacement currents in conducting medium and their presence in dielectric one will be formulated below, based on boundary conditions.

1.2 General Solution of Wave Problem In [12] it is noted that the considered class of electromagnetic fields can be described both with the help of the quasi-stationary field equations and on the basis of general wave equations. In the latter case, the transition to quasi-stationary formulation is performed on the basis of the obtained solution of the wave problem. It is this approach that is used in this chapter, which is due to the following two circumstances. Firstly, to obtain a solution of the quasi-stationary task, the known solution of the wave problem [11] is used, in which the field of the contour with current is considered as a superposition of the fields of a system of current dipoles. Secondly, the presence of electric field intensity in the dielectric medium causes the corresponding displacement currents to flow in it. In the quasi-stationary setting, displacement currents in the entire space are usually neglected. However, since there are no conduction currents in the dielectric medium, in this formulation, it is required justification of condition for the current density at surface in different media. This issue is considered on the basis of the exact solution of a three-dimensional quasi-stationary problem.

4

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

1.2.1 Problem Statement and Basic Equations The problem in terms of complex amplitudes of electromagnetic field is formulated as follows. In the general case the problem is described by Maxwell equations for the ˙ magnetic flux density B˙ and vectors of electric intensity E˙ and magnetic intensity H, ˙ total current density ˙j0 + ˙jt = ˙j0 + ˙j + ˙j D that includes electrical displacement D, the density of the current from the external sources in the elements of contour ˙j0 , ˙ conduction current density ˙j and displacement current density ˙j D = iωD: 

˙ = ˙j0 + ˙j t ; ∇× H ˙ = −i ω B; ˙ ∇×E

˙ = 0; ∇·B ˙ = 0, ∇ ·D

(1.1)

where i is the imaginary unit. As it is often used the complex-value amplitudes are marked with a dot over the corresponding symbols. For the linear problem, the free charge in a piecewise homogeneous medium can be concentrated only on the interface with corresponding surface charge density σ . Then the last equation of system (1.1) that describes the field out of the boundary surface has no charge. As noted in [12], the class of phenomena under study can be presented both by the equations for quasi-stationary field and by general wave equations. Because the work uses the known analytical solution of 3D wave problem for the field of emitting harmonic current dipole, at first the wave problem is formulated to determine the vector potential for current dipole. Then the quasi-stationary problem is solved to find the analytical solution. Using the complex / electrical conductivity γ˜ = γ + i ωεε0 and relative complex permittivity ε˜ = γ˜ (i ωε0 ), the constitutive relations in addition to system (1.1) are written as z>0: z 0 and z < 0, respectively, γ˜e = i ωεe ε0 , γ˜i = γ . ˙ and scalar φ˙ potentials as The vector A ˙ ˙ = ∇ × A; B

˙ ˙ = −∇ φ˙ − i ωA E

(1.3)

are introduced. The Lorenz gauge condition ˙ − k 2 φ˙ = 0 i ω∇ · A is used. Here k 2 = ω2 μμ0 ε˜ ε0 = −i ωμμ0 γ˜ . The last condition in the dielectric and conducting media has the form:

(1.4)

1.2 General Solution of Wave Problem

z>0: z 0 : ΔA Δφ˙ e + ke2 φ˙ e = 0; ˙ i + ki2 A ˙ i = 0, Δφ˙ i + ki2 φ˙ i = 0. z < 0 : ΔA

(1.6)

Making an assumption that the conductor is infinitely thin, the current density in (1.6) is written using the Dirac delta function j0 = I0 δ(r M − r)t M .

(1.7)

At the interface of media, the boundary conditions for the tangential and normal components of the electromagnetic field vectors are satisfied. Furthermore, no-field condition should be set at infinity:     + ˙ −H ˙ − = ˙j S ; ez × E˙ + − E˙ − = 0, ez × H

(1.8)

    ˙− = 0; ez · B˙ + − B˙ − = 0, ez · ˙j+ t − jt

(1.9)

˙ A(∞) = 0,

(1.10)

where ˙j S is surface current density, the field vectors at the interface on the sides of positive and negative values of the axis z are marked with the subscripts «+» and «−», respectively (Fig. 1.1). Regarding to the last condition in (1.9), which follows from the first Maxwell‘s for the normal equation in (1.1), it should be pointed out that the boundary   + condition ˙ −D ˙ − = 0 is usually used components of the electric displacement vector ez · D instead, which is direct consequence of the electrostatic Gauss theorem. Taking into account constitutive equations (1.2) both boundary conditions are equivalent. The use of the boundary condition for the current density, rather than for the electric displacement vector, is associated with the analysis of electromagnetic processes in the system with conducting medium, although this is not important.

6

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

1.2.2 General Solution of the Wave Problem for Contour with Alternating Current in the Form of a System of Emitting Current Dipoles The electromagnetic field of the entire current contour is defined as the superposition of the fields of elementary current dipoles t M dl (Fig. 1.1) distributed along the contour. For obtaining the field of the entire contour, it is sufficient to represent the solution as a contour integral of the found expressions for the dipole field. For observation point Q we will apply the local coordinates x, y, z associated with source point M(0, 0, h) in the contour. The center of the coordinate system (point M0 ) is located on the interface of media, the vertical axis passes through the source point M(0, 0, h). The coordinate x is reckoned in the direction parallel to vector t|| , which is the projection of unit tangential vector t to interface / | | surface. The direction of the axis x is determined by the unit vector e|| = t|| |t|| |. In addition to the local Cartesian coordinate system, we will also use the local cylindrical system with the coordinates ρ, θ, z shown in Fig. 1.2. Equations for potentials (1.6) and boundary conditions (1.8)–(1.10) for the field of the current dipole remain the same with the difference that the current element I˙0 δ(r M − r)t M dl acts as source j0 . An arbitrary orientation of the current element relative to the flat interface between media is considered. In this case, the unit vector can be represented by the sum of two projections t = t⊥ + t|| , where t⊥ is the projection onto the vertical axis. The analytical solution of 3D problem for harmonic dipole I˙0 tdl is found in [8, 9, 11]. The efficiency using for this purpose of two-dimensional Fourier transform in coordinates is shown in [11]. The expression for the direct Fourier transform in coordinates of two variable function f (x, y) has the form Fig. 1.2 The local coordinates of observation point Q associated with current element

1.2 General Solution of Wave Problem

1 f (ξ, η) = 4π 2

∞ ∞

∗

7

f (x, y) · e−i(ξ x+η y) d xd y,

(1.11)

−∞ −∞

where i is the imaginary unit for the given transform. The multiplication of the function transform by operators i ξ and i η corresponds to differentiation with respect to coordinates x and y, respectively. Using the integral transformation (1.11) for Eqs. (1.6), the next one-dimensional equations for vector potential transform are derived instead of three-dimensional equations: z>0: z 0) and conducting media (z < 0) are the following [11]: ˙ ∗e A =

μ0 I˙0 4π 2

 l

exp(−qe |z − z M |) exp(−qe (z − z M1 )) ∗ t + Y ∗ t · e e + Y ∗ t dl; t− t1 + Ye|| z || || ⊥ e1 e2 2qe 2qe

(1.15) ˙ ˙ i∗ = μ0 I0 A 4π 2

 l



   ∗ Yi|| t|| + Yi1∗ t · e|| ez + Yi∗2 t⊥ dl.

(1.16)

8

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

Fig. 1.3 The element of contour and its mirror reflection from a flat interface

Here t and t1 are the unit vectors tangential to the initial contour at point M and to the contour mirrored from interface at point M1 , respectively. The element of mirrorreflected contour is shown in Fig. 1.3 in the area where z < 0. The projections of vectors t and t1 on vertical axis are the same in magnitude but opposite in direction t1⊥ = −t⊥ . Their projections on the interface surface are identical in magnitude and direction t1 || = t|| , i.e. t = t⊥ + t|| , t1 = −t⊥ + t|| . The functions in the integrand in (1.15) in dielectric medium at z > 0 have the following form: μe−qe (z−z M1 ) ; μqe + qi (μγ˜i − γ˜e )e−qe (z−z M1 ) ; = i ϑ cos ψ (μqe + qi )(qe γ˜i + qi γ˜e ) qi γ˜e e−qe (z−z M1 ) . =− qe (qe γ˜i + qi γ˜e )

∗ Ye|| = ∗ Ye1 ∗ Ye2

(1.17)

In conducting medium at z < 0 the integrands in the solution (1.16) for the Fourier transforms are as follows: μeqi z−qe z M ; μqe + qi μ(μγ˜i − γ˜e )eqi z−qe z M ; Yi1∗ = i ϑ cos ψ (μqe + qi )(qe γ˜i + qi γ˜e ) μγ˜i eqi z−qe z M Yi2∗ = . qe γ˜i + qi γ˜e Yi∗ =

(1.18)

To derive the solution in terms of physical variables, the inverse Fourier transform in coordinates should be implemented according to the expression:

1.3 General Solution of Quasi-Stationary Problem for Arbitrary Contour …

∞ ∞ f (x, y) =

f ∗ (ξ, η) · ei (ξ x+η y) dξ dη

−∞ −∞ ∞

⎡

f 1∗ (ϑ)⎣

=

9

π

⎤ f 2∗ (ψ)ei ϑρ cos(ψ−θ ) dψ ⎦ϑdϑ,

(1.19)

−π

0

where any component in (1.15), (1.16) in the general case is expressed as f ∗ (ξ, η) = f 1∗ (ϑ) f 2∗ (ψ). For these functions, integration over (1.19) gives [13] f ∗ (ξ, η) = f 1∗ (ϑ) :

∞ f (x, y) =

⎡ f 1∗ (ϑ)⎣

= 2π

⎤ ei ϑρ cos(ψ−θ ) dψ ⎦ϑdϑ

−π

0

∞

π

f 1∗ (ϑ)J0 (ϑρ)ϑdϑ;

0

f ∗ (ξ, η) = f 1∗ (ϑ) cos ψ :

∞ f (x, y) =

⎡ f 1∗ (ϑ)⎣

0

∞ = −2π cos θ

π

⎤ cos ψeiϑρ cos(ψ−θ ) dψ ⎦ϑdϑ

−π

f 1∗ (ϑ)J1 (ϑρ)ϑdϑ.

(1.20)

0

where J0 (·), J1 (·) are the zero and first order Bessel functions of the first kind, respectively. Specific expressions for the vector potential of the wave electromagnetic field after performing the inverse Fourier transform according to (1.20) are presented in [11]. However, since here the quasi-stationary field is analyzed, the expressions in physical coordinates for such fields can be presented after the necessary additional transformations that take into account the features of the quasi-stationary formulation of the problem.

1.3 General Solution of Quasi-Stationary Problem for Arbitrary Contour with Alternating Current As noted, in the quasi-stationary approximation, the displacement current is neglected in comparison with the conduction current. Quite correctly for local values, such comparison can be performed in the conducting medium. In the dielectric medium, there is only displacement current, and the density of this current is uniquely related to the presence of an alternating electric field ˙j D = i ωεe ε0 E˙ and cannot be equal

10

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

to zero if the electric field intensity is not equal to zero. Therefore, the neglect of displacement currents in the dielectric area requires special justification. Comparison of local values of conduction current density and displacement current density can be performed not only in conducting medium, and also on the interface of media. As will be shown, it is possible to introduce a small parameter and, using the well-known solution of the wave problem, select components in the equation for the vector potential that determine the magnitude of the potential component of the electric field intensity. In this case, we will take into account the displacement currents in the dielectric area, but we will neglect the wave phenomena. Such approach in the quasi-stationary approximation will make it possible to find in the dielectric medium not only a solution for the distribution of the magnetic field ˙ but also a solution for the scalar potential and, accordingly, the electric B˙ = ∇ × A, field intensity. Taking into account the displacement currents makes it possible to find ˙ e , but also for analytical dependences not only for the non-potential E˙ eV = −i ωA the potential component E˙ e P = −∇ φ˙ of the electric field intensity E˙ e = E˙ eV + E˙ e P [7] in the transition from the wave to the quasi-stationary problem.

1.3.1 Small Parameter to Take into Account the Displacement Current Density As follows from (1.5), if the displacement current is taken into account in the dielectric medium, then in order to solve the problem in both conducting and dielectric media it is enough to determine the vector potential distribution.  − ˙ = 0 (1.9) the normal components of electric As follows from ez · ˙j+ − j t t intensity at the boundary satisfy the next condition: | |/ | | / |ez · E + | |ez · E − | = |γ˜e | |γ˜i | 0:

∇ · Ae = 0.

11

(1.22)

The components of electric intensity which are normal to the boundary are: ez · E− = 0,

/ ez · E+ = σ εe ε0 .

(1.23)

Under this formulation, the scalar potential φe in a dielectric medium is undetermined. The scalar potential is not available in the gauge condition (1.22). The normal electrical field component at plane boundary in (1.23) is determined by unknown surface charge density σ . Then using only vector potential, the electric intensity in (1.3) in region z > 0 can be evaluated to an accuracy of a potential summand. The absence of the potential component is insignificant if it is necessary to know the distribution of the magnetic field induction B = r otA or the electromotive force induced in closed contour. However, finding the electric field intensity taking into account the potential component becomes necessary, for example, when determining the local power characteristics of the action of the electric field, the energy flux density of the electromagnetic field ∏ = E×H, and in a number of other applications. The authors of [14] note this characteristic property of the field conjugation problem in quasistationary statement and reveal that the solution is single-valued at complementary conditions.

1.3.2 3D Solution Problem for Dielectric Half-Space in the Quasi-Stationary Approximation The wave phenomena in dielectric medium are neglected when qe = ϑ in (1.17).In the quasi-stationary approximation the condition (1.21) is taken into consideration and the functions in (1.17) are as follows: exp(−ϑ(z − z M1 )) ; w(ϑ)   γ˜e ϑ + μqi γ˜e ∗ exp(−ϑ(z − z M1 )) 1− ; = Ve1∗ − Pe1 = i cos ψ γ˜i w(ϑ) γ˜i μϑ γ˜e ∗ γ˜e qi exp(−ϑ(z − z M1 )) = − Pe2 =− , (1.24) γ˜i γ˜i ϑ2

∗ ∗ Ye|| = Ve|| = ∗ Ye1 ∗ Ye2

where w(ϑ) =

1√ 2 μϑ + qi =ϑ+ ϑ + i ωμμ0 γ . μ μ

(1.25)

The vector potential in the region z > 0, subject to (1.15), (1.24) is represented by:

12

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

  ∗ ˙ ∗e = A ˙ ∗e P , ˙0 +A ˙ ∗eV + A ˙ ∗e P = A ˙ ∗1 + A ˙ ∗2 + A A

(1.26)

˙ ∗0 and A ˙ ∗1 correspond to the first two summands in (1.15); A ˙ ∗2 is determined where A ∗ ∗ ∗ ˙ by functions Ve|| and Ve1 in (1.24); Ae P is proportional to small parameter (1.21) and ∗ ∗ and Pe2 in (1.24). determined by functions Pe1 ˙ ∗ in the expression for vector In quasi-stationary approximation the summand A eP ˙ ∗ must be considered potential can be ignored. At the same time, the component A eP to determine the scalar potential ϕ˙e∗ . Using (1.20), integrals for first two summands in (1.15) are tabular [13] and expressions in physical coordinates have the following appearance ∞

e−ϑ|z−z M | 1 J0 (ϑρ)ϑdϑ = π ; 2ϑ r

0

∞

e−ϑ(z−z M1 ) 1 J0 (ϑρ)ϑdϑ = π . 2ϑ r1

(1.27)

0

In turn, application of inverse Fourier transformation for composed in (1.15) and (1.16) in which the terms connected with displacement currents are excluded allows to find result of integration in the form of analytical dependences only on the angle ψ. ∞ Ve || = 2π 0

e−ϑ(z−z M1 ) J0 (ϑρ) ϑdϑ; w(ϑ) ∞

Ve1 = −2π cos θ 0

e−ϑ(z−z M1 ) J1 (ϑρ) ϑdϑ. w(ϑ)

(1.28)

As a result, substituting in (1.17) instead of Fourier images their expressions in physical coordinates (1.27) and (1.28), we receive expressions in the form of quadratures of the vector potential for the three-dimensional electromagnetic field in dielectric area. Expression for vector potential at z > 0 takes a form: ˙ ˙ eV = μ0 I0 A 4π

 l

   t t1 1 − + Ve || t|| + Ve1 t · e|| ez dl. r r1 π

(1.29)

It follows from (1.28) that for the expression in parentheses, instead of two functions Ve || and Ve1 , it suffices to use single function G e [6]

1.3 General Solution of Quasi-Stationary Problem for Arbitrary Contour …

∞ Ge = 2 0

e−ϑ(z−z M1 ) J0 (ϑρ) dϑ, w(ϑ)

13

(1.30)

e for which the equalities are true in local coordinates Ye || = −π ∂G ,Ye1 = π ∂∂Gxe . ∂z In the third term of the integrand (1.29) there is no vertical component t⊥ of the tangent vector. This allows to represent the vector potential as follows:

˙ ˙ eV = μ0 I0 A 4π

 

  t t1  − − ez × t × ∇G e dl. r r1

(1.31)

l

Let’s simplify the result of integration of the third term in Eq. (1.31) using the condition of closed contour with current. To do this, we will take into account the following relations: / 2  2 x Q − x M1 + y Q − y M1 ; ρ = ρ Q M = ρ Q M1 =     ez × t M = ez × t M1 ;     ∇ Q G e r Q M1 = −∇ M1 G e r Q M1 , dl M = dl M1 .

(1.32)

It is shown here that the  coordinates of the vector r1 = r Q M1 are the argument of the function G e r Q M1 (Fig. 1.3). The bottom indexes mark the points on which the corresponding operations are performed. Replacing the differentiation of the function G e with respect to observation points Q to differentiation with respect to source points M1 is accompanied by a sign change in front of the function [15]. As a result, taking into account that the circulation of the gradient of the scalar function is equal to zero, for the third term in (1.31) we have 

  ez × t M × ∇ Q G e dl M =

l





   t M1 ez · ∇ Q G e + ez (t M1 · ∇ M1 G e ) dl M1

l1

 =

t M1 l

∂G e dl M . ∂z Q

(1.33)

Finally, omitting the notation of points, the equations for the vector potential and ˙ in the dielectric medium (z > 0) are simplified magnetic field induction B˙ = ∇ × A and written as following ˙ ˙ eV = A ˙0 +A ˙1 +A ˙ 2 = μ0 I0 A 4π

  l

μ0 I˙0 B = B+B+B = − e 4π 0 1 2 .

.

.

.

  l

 t t1 ∂ Ge − − t1 dl; r r1 ∂z

  t × r t1 × r1 ∂G e dl. − − t1 × ∇ r3 ∂z r13

(1.34)

(1.35)

14

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

It is easy to check by direct verification that function G e in (1.30) satisfies  the Laplace equation ΔG e = 0. In addition, for any function in the form of t M1 f r Q M1 the following integral taken along the closed contour is equal to zero: 

   ∇ Q · t M1 f r Q M1 dl M1 = −

l1



   t M1 ∇ M1 · f r Q M1 dl M1 = 0.

(1.36)

l1

Consequently, in quasi-stationary approximation, the expression (1.34) for vector potential AeV in the dielectric medium satisfies the continuity condition ˙ eV = 0. ∇ ·A

(1.37)

This result is in agreement with (1.22) for alternative formulation of conjugation problem. The solution of general three-dimensional problem for magnetic field in quasi-stationary formulation can be represented as three summands, i.e. by field of current contour, field of the current contour mirrored from interface and the third term for taking into account the electrophysical properties of the medium and the current frequency. Below we determine the scalar potential φe and electric intensity E˙ e in region z > 0. On the basis of (1.5) allowing for (1.37), the scalar potential is determined by the ˙ e P containing small parameter (1.21). Furthermore, as summand of vector potential A ˙ e P has a unique component perpendicular seen from (1.15) and (1.24), the potential A to the interface of media. Then the Fourier transform of scalar potential φ˙ e∗ can be written as:    ∗ ∗  ∂ Pe2 ∂ Pe1  I˙0 (1.38) φ˙ e∗ = t · e + · e (t z ) dl. || 4π 2 γ ∂z ∂z l

The inverse Fourier transform (1.19) gives the following expressions for the functions presented in (1.38): 2π ∂ Pe1 = cos θ ∂z μ ∂ Pe2 = −2π ∂z

∞ 0

ϑ + μqi exp(−ϑ(z − z M1 ))J1 (ϑρ)ϑdϑ ; w(ϑ)

∞ qi exp(−ϑ(z − z M1 ))J0 (ϑρ)dϑ .

(1.39)

0

To use the condition of contour closure, the subsidiary function G e1 is introduced:

1.3 General Solution of Quasi-Stationary Problem for Arbitrary Contour …

G e1

For this function: φ˙ e =

2π = μ

∂ Pe1 ∂z

=

∞ 0

ϑ + μqi exp(−ϑ(z − z M1 ))J0 (ϑρ)dϑ. w(ϑ)

∂ G e1 . ∂ x M1

15

(1.40)

Then the scalar potential is represented by:

 

    ∂G e1 ∂G e1 ∂G e1  I˙0 ∂ Pe2 t1 · e|| + + (t1 · ez ) − (t1 · ez ) dl. 4π 2 γ ∂ x M1 ∂ z M1 ∂z M1 ∂z l

(1.41)     The last equation takes into account that t · e|| = t1 · e|| and (t · ez ) = −(t1 · ez ). The integral along the closed contour of the expression in the square brackets in (1.41) is equal to zero. From this, the following expression for scalar potential is obtained after some transformations: φ˙ e = i ω

μ0 I˙0 4π

 (t1 · ez )G e dl.

(1.42)

l

The potential component (curl-free component) of electric intensity is:  μ0 I˙0 ˙ ˙ Eep = −∇ φe = −i ω (t1 · ez )∇G e dl. 4π

(1.43)

l

As seen, the potential component of electric intensity E˙ ep as well as the scalar potential φ˙ e is equal to zero if the current contour has no sections perpendicular to the boundary surface. ˙ e along with potential The vortex (divergence-free) component E˙ eV = −i ωA component E˙ ep = −∇ φ˙ e gives the total electric intensity: 

  t1 t ∂G e μ0 I˙0 − − t1 − (t1 · ez )∇G e dl E˙ e = E˙ eV + E˙ ep = −i ω 4π r r1 ∂z l 

  t t1 μ0 I˙0 ∂ Ge ∂ Ge − − t || − (t1 · ez ) eρ dl = −iω 4π r r1 ∂z ∂ρ l    t t1 μ0 I˙0 − − ez × [t1 × ∇G e ] dl =E˙ 0 + E˙ 1 + E˙ 2 . = −i ω (1.44) 4π r r1 l

Here eρ = e|| cos θ + e y sin θ is unit vector along the radial direction of the local cylindrical coordinate system (Fig. 1.2).

16

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

Note that the third term in the integration function depending on the properties of the conducting medium and field frequency has no component perpendicular to the interface at any configurations and orientations of the initial current contour. It is seen from (1.44), both the vortex and potential parts of the electric intensity have component in a direction perpendicular to the plane media interface caused the vertical sections of the initial current contour. However, taking them into account together gives the absence of the vertical component of the electric field intensities in conducting medium for the term in (1.44).

1.3.3 3D Electromagnetic Field in Conducting Medium in the Quasi-Stationary Approximation For the quasi-stationary field in the conducting half-space, in contrast to the dielectric medium, neglect of displacement currents in comparison with conduction currents does not lead to uncertainty in determining the electric field intensity. The scalar potential φ˙ i from the Lorentz calibration condition (1.5) is found from the known ˙ i , determining the potential component E˙ i P of distribution of the vector potential A ˙ i − grad φ˙ i . the electric field E˙ i = E˙ i V + E˙ i P = −iωA The small parameter (1.21) is disregarded in a conducting half-space z < 0. Then the Fourier transform functions in (1.18) are determined as: exp(qi z − ϑ z M ) ; w(ϑ) exp(qi z − ϑ z M ) μ; Yi1∗ = i cos ψ w(ϑ) μ Yi2∗ = exp(qi z − ϑ z M ) . ϑ

∗ Yi|| =

(1.45)

The inverse Fourier transform with respect to (1.20) gives: ∞ Yi || = 2π 0

eqi z−ϑ z M J0 (ϑρ) ϑdϑ; w(ϑ) ∞

Yi1 = −2π μ cos θ 0

eqi z−ϑ z M J1 (ϑρ) ϑdϑ; w(ϑ)

∞ Yi2 = 2π μ

eqi z−ϑ z M J0 (ϑρ)dϑ.

(1.46)

0

Using designations (1.46), the expression for vector potential in terms of physical coordinates in region z < 0 takes the form:

1.3 General Solution of Quasi-Stationary Problem for Arbitrary Contour …

˙ ˙ i = μ0 I A 4π 2





   Yi|| t|| + Yi1 t · e|| ez + Yi 2 t⊥ dl.

17

(1.47)

l

We simplify (1.47) by elimination of the second term in the integrand. To do this, the condition of contour closure is used and the next function is introduced: ∞ Gi = 2 0

exp(qi z − ϑ z M )J0 (ϑρ) dϑ . w(ϑ)

(1.48)

For this function Yi1 = π μ ∂∂Gx Mi1 and then (1.47) can be rewritten as: ˙i A μ0 I˙0 = 4π 2

   

  ∂ Gi ∂ Gi  ∂G i t · e|| + + Yi2 t⊥ dl. −π μ (t · ez ) ez + Yi|| t|| + π μ ∂ xM ∂zM ∂z M l

(1.49) The circulation of a gradient for scalar function is equal to zero. Then the expression for the vector potential of electromagnetic field in conducting medium takes the form, where each term in the integrand is due to the current element of the corresponding direction: ˙ ˙ i = μ0 I0 2 A 4π

⎫ ⎧  ⎨ ∞ qi z−ϑ z M ∞ qi z−ϑ z M ⎬ e J0 (ϑρ) e J0 (ϑρ) ϑdϑ + t⊥ qi dϑ dl. t|| ⎭ ⎩ w(ϑ) w(ϑ) l

0

0

(1.50) The scalar potential in the conducting half-space is determined by the Lorentz / ˙ i (μμ0 γ ). gauge condition (1.5) φ˙ i = −∇ · A Using local cylindrical coordinates, from (1.50) we find μ0 I˙0 4π ⎧ ⎫  ⎨ ∞ qi z−ϑ z M ∞ qi z−ϑ z M ⎬   e J1 (ϑρ) 2 e J0 (ϑρ) 2 ×2 ϑ dϑ + (t · ez ) qi dϑ dl. − t · e|| cos θ ⎩ ⎭ w(ϑ) w(ϑ)

˙ = ∇ ·A i

l

0

0

(1.51) Let us exclude from the written expression the term associated with the sections of the contour that have a non-zero tangent component t|| to the media interface. To do this, we use subsidiary function

18

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

∞ G i2 = 2 0

eqi z−ϑ z M J0 (ϑρ) ϑdϑ. w(ϑ)

(1.52)

˙ can be written as Then the expression for ∇ · A i ⎧ ⎞ ⎤⎫ ⎡⎛ ⎪  ∞ q z−ϑ z  ⎪ ⎨  ∂G ! M J0 (ϑρ) 2 ⎥⎬ e i ⎟ i2 t · e + ∂ G i2 t · e  + ⎢⎜ ∂G i2 + − q dϑ t · e ⎠ ⎣⎝ z z ⎦⎪dl || i ⎪ ∂ x ∂z ∂z w(ϑ) ⎩ ⎭ M M M 0 l ⎫ ⎧ ∞ ⎪  q z−ϑ z  ⎪ ⎬ ⎨ M J0 (ϑρ)  e i μ I˙ t · ez dϑ dl. = 0 0 2i ωμμ0 γ ⎪ ⎪ 4π w(ϑ) ⎭ ⎩ 0 l

˙ = ∇ ·A i

μ0 I˙0 ×2 4π

(1.53)

Now the expression for scalar potential takes the final form: φ˙ i = −i ω

μ0 I˙0 4π

 (t · ez )G i dl.

(1.54)

l

From (1.54) it follows, in particular, that for flat contours lying in a plane parallel to the interface between the media, i.e. in the absence of contour sections with non˙ i = 0 and φ˙ i = 0 respectively. For such zero normal component t⊥ , we have ∇ · A contours, the electric field intensity in the conducting medium E˙ i = E˙ i V + E˙ i P = ˙ i − grad φ˙ i does not contain the potential component and the electric field −i ωA ˙ i is determined directly by the distribution of the vector potential intensity E˙ i = −i ωA in space. Finally, we will find an equation for the electric field intensity in the area z < 0 for the general case of the spatial configuration of current contour and its arbitrary orientation relative to the flat interface between the media. ˙i− After some conversions of the total electric intensity E˙ i = E˙ i V + E˙ i P = −i ωA ˙ grad φi the following expression is derived: ! ! E˙ i = E˙ i V + E˙ i P = E˙ i ||1 + E˙ i⊥ + E˙ i ||2 − E˙ i⊥ ⎫ ⎧ ⎪  ⎪ ∞ q z−ϑ z ⎬ ⎨ ∞ eqi z−ϑ z M J (ϑρ) M J1 (ϑρ)   e i μ0 I˙0 0 t|| 2iω ϑdϑ + t · ez eρ ϑdϑ dl. =− ⎪ ⎪ 4π w(ϑ) w(ϑ) ⎭ ⎩ 0 0 l

(1.55)

Note that the equation E˙ i⊥

⎫ ⎧  ⎨ ∞ qi z−ϑ z M ⎬ ˙ e J0 (ϑρ) μ0 I0 2 qi dϑ dl = t⊥ ⎭ ⎩ 4π w(ϑ) l

(1.56)

0

includes with different signs as components of the E˙ i V and E˙ i P of the total electric intensity and, therefore, is absent in (1.55). For this reason, each of the components E˙ i||1 and E˙ i||2 in (1.55), in contrast to the components of the electric field intensity E˙ i V and E˙ i P , cannot be considered as field intensities (at least because the integration

1.4 Features of the Formation of the 3D Quasi-Stationary Electromagnetic …

19

is not necessarily carried out over closed contours) and only in sum give electric field intensity in the conducting half-space. The division into two components in (1.55) reflects the effect on the field intensity of differently oriented sections of the initial contour with current.

1.4 Features of the Formation of the 3D Quasi-Stationary Electromagnetic Field in the System “Arbitrary Spatial Current Contour–conducting Half-Space” 1.4.1 The Main Feature of the Distribution of the 3D Electromagnetic Field in the Conducting Half-Space It follows from Eq. (1.55) that in the conducting medium the electric field intensity vector, and the current density vector, does not contain a component perpendicular to the media interface. Zero values are realized not only on the surface, as required by the fulfillment of the boundary condition (1.23), for which the small parameter (1.21) can be ignored in the conducting medium. Zero values apply to the entire conducting half-space. This result is valid for any configurations of current contours and their orientations relative to the media interface; it is also valid for any values of the electrophysical properties of the conducting and dielectric media and does not depend on the field frequency. The limitation is the use of a mathematical model of the conducting half-space, which, as is known, is possible for conducting bodies provided that the penetration depth of the electromagnetic field is small compared to the characteristic dimensions of the body. We emphasize that the obtained result about the absence of a vertical component of the electric field intensity in the conducting medium was a consequence of solving the wave problem of field theory. Let’s show that such conclusion for the quasi-stationary problem has more general grounds [6]. From Maxwell’s Eqs. (1.1), written in the form of dependences of vector fields on time, it follows that the electric field intensity in conducting medium satisfies the hyperbolic type homogeneous equation: ΔEi − μμ0 γ

∂Ei = 0. ∂t

(1.57)

The vertical component of electric field intensity is equal to zero not only at conducting side of the interface. The condition of finiteness of the field at infinitely distant points provides zero-value of this component of electric field at infinity. As a result, for the vertical component of electric field in the region z < 0 we obtain the problem for the hyperbolic type homogeneous equation with zero boundary conditions:

20

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located … ⎧ ⎨ ΔE

∂ Ei z = 0; ∂t ⎩ E i z (z = 0) = 0, E i z (∞) = 0. i z − μμ0 γ

(1.58)

The solution of this problem for a steady-state sinusoidal process (process “without initial conditions”) in half-infinite region has only zero value [16]. The field component will be also absent at any instant of time under zero initial conditions. Since we are considering a linear problem, based on the principle of superposition, the conclusion on the absence of vertical components of the electric field intensity and current density in the conducting medium extends to a more general case. First, the conclusion is valid not only for current flow along a specific contour, but also for any given distribution of the initial current in the dielectric medium. Secondly, since an arbitrary time dependence of the current can be represented by its frequency spectrum, the obtained result is also valid for any time dependence of the initial current. The general nature of the found feature of the field distribution makes it possible to define it as the main feature of the formation of the three-dimensional quasi-stationary electromagnetic field for arbitrary system of currents of spatial configuration flowing near conducting magnetizable half-space. In connection with the considered feature of the field distribution, it is also necessary to note a well-known fact from the theory of reflection and refraction of electromagnetic waves from a flat interface between media, when the optical densities of the media in which they propagate are significantly different (for example, air and metal). In this case, the angle of refraction of the plane wave is practically equal to zero, and the direction of propagation of the refracted wave almost completely coincides with the direction of the normal to the surface [17]. The consequence is that the directions of the intensity vectors of both electric and magnetic fields are oriented only in directions parallel to the boundary surface, and the vector fields themselves on the surface and inside the optically dense medium are related by the relation obtained by M.A. Leontovych in studying the propagation of radio waves [18]. It is called the approximate Leontovych’s boundary condition or also the Shchukin–Leontovych–Rytov condition. The problem being solved here differs in two respects from the problem of propagation of the electromagnetic wave: firstly, the conclusion was obtained for the problem in quasi-stationary setting, secondly, and this is most important, in contrast to the wave problem, the field of external sources in generally is non-uniformity, its non-uniformity is not limited at all. For this reason, only the electric field intensity has zero component in the direction perpendicular to the interface between the media. It will be shown later that as the sources of the external field move away from the interface, the vertical component of the magnetic field vector will also tend to zero. And then the directions of the vector fields in the quasi-stationary and wave problems will completely correspond to each other, since in both cases the uniformity field of external sources is considered. It is possible to confirm the validity of the conclusion made about the main feature of the distribution of the quasi-stationary electromagnetic field by comparing the

1.4 Features of the Formation of the 3D Quasi-Stationary Electromagnetic …

21

Fig. 1.4 The induced current flow pattern calculated by the numerical method: a in a vertical plane passing through the center of the circular contour; b and c in the plane z = −h 0 and z = −2R parallel to the interface

results of calculating the three-dimensional electromagnetic field by the considered and independent methods. Let’s present the results of a numerical calculation using the Comsol software package for a specific electromagnetic system, which also has an applied value [19]. In Fig. 1.4, the arrows show the direction and magnitude of the current density ji = γ Ei obtained as a result of the numerical solution of the problem in the Comsol software package for an external three-dimensional electromagnetic field created by one or two circular contours with current located in a plane perpendicular to the interface. Figure 1.4a shows the picture of current flow in a vertical plane that passes through the center of the contour. Figure 1.4b, c illustrate the pictures of current flow in a plane parallel to the boundary surface. ˙ 0 causes the The alternating external magnetic field of the currents B˙ 0 = r ot A appearance of an induced electric field and eddy currents in the conducting halfspace. With the chosen orientation of the contours, the intensity of the induced ˙ 0 has a non-zero component ez · E˙ 0 electric field of the initial current E˙ 0 = −i ωA that is perpendicular to the interface between the media. However, in the conducting medium, this component of the electric field intensity turns out to be equal to zero. The quantitative results of numerical and analytical calculations are shown for an external field of one contour (Fig. 1.5). The radius of the contour is R = 0.05 m, the minimum distance from the contour to the interface is h0 = 0.02 m, the electrophysical properties of the medium |are| as follows: μ = 1, γ = 105 1/(Ω·m), field frequency is f = 103 Hz, current is | I˙0 | = 103 A. With such parameters, the field penetration / / depth is δ = 2 (ωμμ0 γ ) = 0.05 m. In contrast to the analytical method, in the numerical calculation, the current contour was selected in the form of a conductor with a square cross-Sect. 1.2r × 2r at r = 0.004 m. In the numerical calculation, the problem was solved in a limited area, the dimensions of which significantly exceed the contour radius. The component of the electric field intensity parallel to the flat interface is due not only to the presence of contour sections that have a current direction parallel to the surface. The influence of these sections of the contour is displayed by the first component in (1.55). For a three-dimensional field of a spatial contour, the

22

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

Fig. 1.5 Electromagnetic system with a circular current contour located in the plane perpendicular to the interface

horizontal component of the electric field intensity is also due to the presence of contour sections normal to the surface. The second component in (1.55) is related to them. For the considered calculation model (Fig. 1.5), both components of the electric field intensity are present. Figures 1.6 and 1.7 illustrate the results of the calculation of the field distribution and explain its structure. Figure 1.6 corresponds to the distribution of the electric field intensity along the x-axis directly under the circular contour with current at y = 0 in a plane that is parallel to the surface at a depth z = −0.01 m. On this line, the electric field intensity and, accordingly, the current density ˙ji = γ E˙ i = j˙x ex have a non-zero component only along the x-axis. Figure 1.6a shows the distribution of the relative value of the modulus of the complex-value amplitude of the field intensity, normalized to the maximum value of the field, which is realized at x = 0. Figure 1.6b illustrates the distribution along the same direction of the phase difference between the current I˙0 and the field intensity E˙ x or current density j˙x , which change with time according to a sinusoidal law. It can be seen that the phase shift does not remain unchanged. Therefore, the maximum values of the field intensity at different points are reached at different times.

Fig. 1.6 Distribution of the sinusoidal electric field intensity along the axis x at y = 0, z = −0.01 m: a normalized amplitude value; b phase displacement between electric field intensity E˙ x and contour current I˙0

1.4 Features of the Formation of the 3D Quasi-Stationary Electromagnetic …

23

Fig. 1.7 Distribution of the electric field intensity along the axis x at y = 0.03 m, z = −0.01 m: a normalized amplitude value; b phase displacement between components of electric field intensity E˙ i and contour current I˙0

On the same surface z = −0.01 m, but at points not lying in the plane of the circular contour, the field intensity and current density are no longer directed only along x-axis. In general, they have two non-zero components ˙ji = γ E˙ i = j˙x ex + j˙y e y . In Fig. 1.7 similarly to the previous figure, but with y = 0.03 m, the distribution along the x-axis of the relative values of the amplitudes of these components of the field intensity (Fig. 1.7a) and the corresponding phase shifts (Fig. 1.7b) are shown. The dots denote the values that were found using the calculation in the Comsol package. A feature of the field distribution at points that are not located in the plane of the contour is that the phases of two mutually perpendicular field components differ one from other. This indicates the presence of “elliptical polarization” of the field in the region where y /= 0. As can be seen from Fig. 1.7, the results of the analytical and numerical methods are consistent with each other. However, in this case, the numerical method failed to achieve sufficient calculation accuracy, which is associated with the need to use a relatively coarse computational grid to solve a three-dimensional problem, especially for points near the interface between media. At the same time, the obtained results testify to the reliability of the analytical solution of the problem and the correctness of the conclusion about the zero values of the vertical components of the electric field intensity and current density in the conducting half-space.

1.4.2 Field Souses and Conditions on the Interface Between Media The distributions of electromagnetic field sources, currents and charges makes it possible to establish the physical causes of the features of the distributions these characteristics the conducting half-space.

24

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

Let’s represent the vector potential as a potential that is determined only by ˙ =A ˙0 +A ˙ 0 is the vector potential ˙ c , where A currents flowing in the entire space A ˙ c is the vector potential determined due to the current of the initial contour in (1.34), A by the induced currents in the z < 0 area: ˙ c = μμ0 A 4π



μ0 γ E˙ i dV + r 4π

Vi



j Sm d S. r

(1.59)

S

Here, both the conduction current density γ E˙ i and the magnetization current density (μ − 1)γ E˙ i in the volume, and the surface magnetization current density j Sm = ˙ − at the media interface are taken into account. −(μ − 1)ez × H Equation (1.59) is valid if the Coulomb calibration is valid for the vector potential in the entire space, both in the dielectric and in the conducting areas, that is, the ˙ 0 +A ˙ = 0 continuity condition is satisfied [15]. Obviously, both components A ˙c ∇ ·A ˙ also separately satisfy this condition. (Note that Ac generally differs from the sum ˙ 2 in (1.34) at z > 0, and A ˙1 + A ˙0 + A ˙ c differs from Ai in (1.50) for z < 0 by A the magnitude of the gradient of the scalar function, which is due to different gauge condition). With the chosen calibration, the electric field intensity is represented by the sum of the field intensities E˙ = E˙ σ + E˙ j , where the scalar potential φ˙ σ and the associated potential term of the electric field intensity E˙ σ = −grad φ˙ σ are due to electric charges [20], which in the considered model are concentrated at the interface between media with surface density σ˙ : φ˙ σ =

1 4π εe ε0



σ˙ d S. r

(1.60)

S

˙ The  second component gives the intensity of the induced electric field E j = ˙ c . It is determined by the all currents flowing in the system. ˙0 +A −i ω A ˙ =A ˙0 + A ˙ c , as follows from An essential circumstance in the representation A ˙c (1.48), is the equality to zero of the vertical component of the vector potential A and, accordingly, the equality to zero of the component of the electric field intensity ˙ c . The normal component of the total field can appear only in connection with −i ωA the flow of the initial current I˙0 of the contour in the direction perpendicular to the interface between the media. Such a situation is possible for spatial configuration contours or for planar contours lying in a plane that is non-coplanar to the media interface. The normal components of the electric field intensity at opposite sides of the interface are caused by the surface charges and have equal absolute values and opposite signs: E˙ σ+z = − E˙ σ−z .

(1.61)

1.4 Features of the Formation of the 3D Quasi-Stationary Electromagnetic …

25

Besides, in conducting medium the vertical component of the field of the electric charges at the interface is compensated by the intensities of the induced electric field, i.e. E˙ σ−z − iω A˙ − 0z = 0.

(1.62)

The surface density σ˙ of electric charge is determined from the boundary condition σ˙ = εe ε0 E˙ z+ . Taking into account (1.61), (1.62) and continuity of the normal ˙− component of the vector potential A˙ + 0z = A0z , we finally have: σ˙ = E˙ z+ = −2i ω A˙ 0z . εe ε0

(1.63)

Equation (1.63) is valid not only for harmonic currents, but also for an arbitrary time dependence of the initial contour current. In addition, based on the principle of superposition, the result remains true for any initial system of closed currents: σ ∂ A0z , = E z+ = −2 εe ε0 ∂t μ0 where − ∂ A0z (z=0,t) = − 4π ∂t

∂ I0 (t) ∂t

* l

(1.64)

(t·ez ) dl. r

Formerly the relation (1.64) was derived in [5] for perfect (ideal) skin-effect δ → 0, where the current in conducting medium flows only along flat surface and naturally does not have a component normal to it. Now this relation has a more general character. To determine the normal component of electric fields intensity in dielectric area at the interface between the media, it is sufficient to know only vertical A0z . component of the induced electrical field of the external sources ∂ ∂t Thus, we can conclude that the physical reason for the absence of the vertical component of the current density and electric field intensity in the conducting halfspace is that the electric charge distribution is formed on the conducting surface, the field of which completely compensates for the external induced electric field. This charge creates in the conducting area an additional electric field which has also a component of the intensity vector directed parallel to the interface between the media. The electric charge distributed on the interface with the surface density σ˙ , together with the given initial currents and the found distribution of induced currents and magnetization currents, create a complete system of sources of a quasi-stationary electromagnetic field and thereby completely determine the magnitude of the field in the entire space.

26

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

1.4.3 Complex-Value Poynting Vector The found analytical solution of the problem can be used not only to find the electromagnetic field, but also to determine other characteristics, including the distribution ˜ = 1 E˙ × H, where the complex-value of the electromagnetic energy flux density ∏ 2 ˜ is determined by the complex amplitude of the electric field Poynting vector ∏ intensity E˙ and the complex conjugate amplitude of the magnetic field intensity H. In many technological devices, the electrophysical processes is associated with the thermal energy released in the metal, the source of which is the electromagnetic field. With strong skin effect, we can speak about the surface power density of the released heat, the value of which is determined by the density of the electromagnetic energy flux from the dielectric medium to the conducting body through the interface between the media. As it is known, the real part of the vertical component of the Poynting vector at the interface, taken with the “−” sign, gives the flux density of the period-averaged power pz inside the conducting body, which is released in the form of Joule heat in the surface layer and determines the surface density of the released thermal power in body: pz =

1 ˜ = 0) · ez ). Re(−∏(z 2

(1.65)

In the last equation, the complex-value Poynting vector at the interface of the media is determined by the values of the electric and magnetic field intensities at z = 0, and to find the pz , it is sufficient to the vector product    calculate  of the projections of the field intensities E˙ e|| = E˙ e − E˙ e · ez ez and He|| = He − He · ez ez at z = 0: 1  pz = − Re ez 2 1  = − Re ez 2

  · E˙ e (z = 0) × He (z = 0)   · E˙ e|| (z = 0) × He|| (z = 0) .

(1.66)

Let us take into account that the sum of the first two terms in (1.44) of the electric field at the boundary surface gives only the vertical component     t t1 μ0 I˙0 μ0 I˙0 (t · ez )ez ˙ ˙ 2 − dl. dl = −i ω E0 + E1 = −i ω 4π r (z = 0) r1 (z = 0) 4π r (z = 0) l

l

(1.67) Otherwise, the third term of the electric field intensity does not have the component perpendicular to the interface between the media. ˙ where By introducing the designation E˙ 2 = ez × P,

1.4 Features of the Formation of the 3D Quasi-Stationary Electromagnetic …

˙    ˙P = −i ω μ0 I0 t1 × ∇ G˙ e dl, 4π

27

(1.68)

l

and substituting (1.68) into (1.66), the real part of the vertical component of the complex-value Poynting vector pz becomes ⎫ | |2 ⎧  ⎨ ⎬ | | ˙     μ0 I0 1 t1 × ∇ G˙ e dl · He|| , pz = Re P˙ · He|| = −ω Im 2 ⎩ ⎭ 2 16π

(1.69)

l

where vectors in (1.69) correspond to the observation point at z = 0.

1.4.4 Examples of Electric Field Intensity and Surface Charge Density Distributions As can be seen from (1.63), (1.64), for a given geometry of the contour with alternating current, the determination of the normal component of the electric field intensity at the interface between media in the dielectric area and the surface density of the electric charge is not difficult. Let us illustrate the distribution of these quantities with two examples. In the first case, we consider the previous model as a circular contour with current located in a plane perpendicular to the interface between the media (Fig. 1.5). For the selected previous parameters of the system Fig. 1.8a shows the results of calculating the electric field intensity in the form of the lines E z+ = const. Figure 1.8b shows the distribution of the normal component E z+ field along straight lines on the boundary surface parallel to the plane of the contour. (In this subsection, quantities without a dot above the symbol denote the modulus of the complex-value amplitude of the corresponding quantity). The second example gives current contour configuration typical for technological systems. Central section of current contour l1 is located parallel to the plane surface of the conducting body. The sinusoidal current is supplied by two parallel conductors l2 , oriented perpendicularly to the plane of the central section of the contour (Fig. 1.9). In one design or another, such configurations are used, for example, in high frequency induction heat treatment of metals [2, 21, 22], installations for processing of metals under the action of high intensity electromagnetic field and high density currents [23–25], devices for electromagnetic forming or high-speed forming technology using pulse magnetic field [26–28]. A strong skin effect occurs in conducting elements of this equipment, in which the current and electromagnetic field are concentrated in a thin skin layer. Let us analyze the distribution of the electric intensity in dielectric area at the interface of media (z = 0) and find the surface charge density.

28

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

Fig. 1.8 Distribution of the vertical component of the electric field strength of a circular contour on the interface between media in the dielectric region: a lines E z = const; b changing the E z+ component along the x coordinate Fig. 1.9 Current contour with separated horizontal and vertical sections

For the considered initial spatial contour the electric intensity at interface in the dielectric half-space has both the vertical component E z+ and the tangential component E|| to the surface (which coincides with the tangential component in conducting medium). Since at z = 0 the distances from the observation point to the source points on the initial and mirror reflected contours are the same r = r1 as well as the projections of the tangent vectors t1 || = t|| , the sum of the first two terms in (1.44) does not create component of the field parallel to the plane surface. Tangential component E˙ || for current contour shown in Fig. 1.8 can be represented as the sum of two contour integrals along sections l1 and l2 : μ0 I˙0 E = E + E = iω 4π || ||1 ||2 .

.

.

 l1

μ0 I˙0 ∂G e dl − i ω t || ∂z 4π

 (t1 · ez ) l2

∂ Ge eρ dl. ∂ρ

(1.70)

Note, each separate term in (1.70) does not satisfy the continuity condition and cannot be regarded as a separate electric intensity. As a result of integration along the vertical elements of the contour, we obtain the following expression for the term E˙ ||2 in (1.70):

1.4 Features of the Formation of the 3D Quasi-Stationary Electromagnetic …

∞ exp(−ϑ h) μ0 I˙0 ˙ E||2 = i ω [J1 (ϑρ1 )e1 − J1 (ϑρ2 )e2 ]dϑ, 2π w(ϑ)

29

(1.71)

0

where / | h is| the distance from the central part of the contour to the surface, e1,2 = ρ1,2 |ρ1,2 |. The value of the field given in (1.71) must be added to the electric intensity E˙ ||1 due to the current flow through the central horizontal section. For the example considered, we also find the surface charge density. As the distribution of electric charge is determined only by sections of the contour which are perpendicular to the interface, in this example the distribution of charge will be the same for any configuration of the central part of the contour. So, for the chosen geometry of the contour, the vertical component of the vector potential and, correspondingly, the surface charge density can be presented by means of simple algebraic expressions: iωεe ε0 μ0 ˙ σ˙ = −2 I0 4π

∞ + 

z 2M + ρ22

−1/ 2

−1 2 ,  − z 2M + ρ12 / dz M

h

1 2  h + h 2 + ρ22 / i ωεe ε0 μ0 ˙ = I0 ln 1 2 .  2π h + h 2 + ρ12 /

(1.72)

 2  2 2 Here ρ1,2 = x Q − x1,2 + y Q − y1,2 , where (x1 , y1 ) and (x2 , y2 ) are coordinates on the plane surface of the two vertical conductors location along which the current is directed to the central section of the contour and from it, respectively. The results of / the calculation according to (1.72) are shown in Fig. 1.10 in the form of lines σ σm = const, where σm is the maximal value of surface density of the distributed electric charge. Fig. 1.10 Distribution of the surface charge / √density for / a h = 10 2

30

1 Electromagnetic Field of Arbitrary Spatial Current Contour Located …

1.5 Conclusions 1. Based on the well-known solution of the wave problem for emitting current dipoles and application of the closed current contour condition, the analytical solution is presented for three-dimensional quasi-stationary alternating electromagnetic field of the current contour located near the plane surface of conducting body. The solution is obtained without restrictions on the contour configuration, electrophysical properties of the media and the field frequency. In addition to the results presented earlier, the solution for the scalar potential and the electric intensity in the entire dielectric half-space is found based on the use in the quasi-stationary approximation the displacement currents in the dielectric region. Expressions in the form of quadratures are obtained for determining the vector and scalar potentials, magnetic induction and electric field intensity in the dielectric area (1.34), (1.42), (1.35), (1.44) and conducting area (1.50), (1.54), (1.55). 2. As a revealed peculiarity of the distribution of quasi-stationary electromagnetic field for the systems with plane interface between the dielectric and conducting media, the components of electric intensity and current density which are perpendicular to boundary surface are not available (equal to zero) in the conducting medium. The result holds true for any spatial configuration of the initial system of current and for any time dependence of external field sources. 3. The physical reason for the absence of vertical components of the current density and electric field intensity in the conducting half-space is the appearance of a distributed electric charge on the interface surface, the field of which in the conducting half-space completely compensates for the vertical component of the external induced electric field. Accordingly, on the surface in the dielectric region, the vertical component of the electric field intensity is twice the vertical component of the induced field of the sources. The electric charge creates both the vertical component of the electric intensity and the component parallel to the media interface.

References 1. Vasetsky Yu (2010) Asymptotic methods for solving electrodynamics problems in systems with bulky curvilinear conductors. Naukova dumka, Kyiv, 271 p. ISBN 978-966-00-0937-2 2. Acero J, Alonso R, Burdio JM, Barragan LA, Puyal D (2006) Analytical equivalent impedance for a planar induction heating system. IEEE Trans Magn 42(1):84–86. https://doi.org/10.1109/ TMAG.2005.854443 3. Tsitsikyan GN (1997) Electromagnetic field of a linear conductor with a current, that parallel to a boundary plane «the air medium—conducting half-space». Elektrichestvo 12:55–61 4. Makarov VM (1987) Vector potential of curvilinear wire laying in vertical plane above ground. Electron Model 9(2):41–45

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Chapter 2

Approximate Mathematical Models for Analysis of Alternating Electromagnetic Field of Sources Near Conducting Body

Abstract Approximate mathematical models for calculating three-dimensional electromagnetic fields are developed and analyzed on the basis of the exact analytical solution found for the problem of determining the three-dimensional quasi-stationary field of arbitrary external sources near conducting half-space. From the standpoint of the general electromagnetic field, the simplest model of the perfect skin effect is considered. It is shown that the normal component of the electric field intensity and the surface density charge on the surface are completely determined by the normal component of the induced electric field of external sources. For the case of small but finite field penetration depth the method of expand exact analytical expressions into asymptotic series has been developed to based on the introduction of a small parameter that generalize the concept of strong skin effect. The strong skin effect is understood as its extended definition, when not only the penetration depth of the alternating field is small compared to the characteristic dimensions of the conducting body, but the ratio of the penetration depth to the characteristic dimensions of the entire electromagnetic system is also small, including the distances between external field sources and conducting body. Unlike the original expressions, which contain improper integrals from special functions, the specific expressions found are bounded asymptotic series, where each term is found by calculating only one-dimensional contour integral. It is established that the required accuracy is achieved by using the first few terms of the series, the number of which depends on the value of the introduced small parameter. It is substantiated that the possibility of further simplification of calculations using the model of a locally two-dimensional field, which allows the replacement of contour integrals by simple algebraic expressions. The mathematical model is valid when determining the electromagnetic field in the area near initial conductor with current near interface of media. Calculations are limited to systems with a small angle of inclination of the contour sections to the interface, which is due to the neglect of the electromagnetic field component associated with the current component normal to the flat surface. Keywords 3D quasi-stationary electromagnetic field · Exact and asymptotic solutions · Strong skin effect · Locally two-dimensional approximation

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Y. Vasetsky and A. Zaporozhets, Electromagnetic Field Near Conducting Half-Space, Lecture Notes in Electrical Engineering 1070, https://doi.org/10.1007/978-3-031-38423-3_2

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2 Approximate Mathematical Models for Analysis of Alternating …

2.1 Introduction The previous chapter presents, obtained in [1, 2], a general analytical solution to the problem of determining a three-dimensional quasi-stationary electromagnetic field created by arbitrary current contour flowing near conducting half-space. In this solution, no restrictions were imposed to the configuration of external field sources, electrophysical properties of the medium and the field frequency. At the same time, the use of simpler approximate mathematical models solves a number of important problems. From a methodological point of view, it is essential to determine the place of the used approximate model within of the exact solution. The determination of actual errors of approximate models becomes possible. In some cases, approximate models make it possible to obtain a clear physical representation of the electromagnetic processes. Approximate models developed on the basis of the obtained exact solution make it possible to use justified assumptions. Finally, approximate models require smaller amount of calculations of three-dimension field. All these circumstances indicate the importance of the analysis and development of approximate mathematical models for the calculation of three-dimensional electromagnetic fields. These questions are present in this chapter in the study of approximate mathematical models of three-dimensional sinusoidal electromagnetic fields. The place of the simplest model of the perfect (ideal) skin effect in the general exact solution of the problem is determined. A method of asymptotic series expansion for exact expressions is developed based on the introduction of a small parameter, which extends the concept of a strong skin effect to diffusion of a non-uniformity electromagnetic field into conducting half-space. The general property of a stronger decay of non-uniformity electromagnetic field in conducting medium compare to uniform one is established on the base of exact solution. The possibility of further simplification of calculations using the model of a locally two-dimensional field is substantiated.

2.2 Electromagnetic Field of Spatial Contour with Current Near Flat Surface of Conducting Body with Perfect Skin Effect Let us first consider the formation of an electromagnetic in the case of perfect /field / skin effect, in which the field penetration depth δ = 2 (ωμμ0 γ ) is negligible compared to any characteristic dimensions L of the electromagnetic system. This situation is realized for fast pulsed or high-frequency processes in electrically conductive bodies with high electrical conductivity. Mathematical models with perfect skin effect are often used, for example, in the analysis of electromagnetic forming process of thin-walled shells [3–5] and in the equipment of superstrong currents [6].

2.2 Electromagnetic Field of Spatial Contour with Current Near Flat Surface …

35

/ In the case of perfect skin effect δ L → 0, for calculation of field outside the conducting body it is necessary to solve the corresponding stationary problem for ideally conductive body of the same shape [7, 8]. Under a given distribution of current density j0 of the sources or the magnetic field of these sources H0 , the statement of the boundary value problem is reduced to setting the Maxwell’s equation for the magnetic field intensities H and to the condition of absence of the normal component of the field at the surface: rotH = j0 ,

divH = 0,

H · n = 0,

(2.1)

where n is the unit vector outer normal to the surface of the conducting body. The boundary value problem (2.1) completely determines the existence of a unique solution. For analysis of electromagnetic field in surface layer of conducting body, the known model of diffusion of uniform field into conductive half-space, usually, holds true [6]. In such a model, the initial physical quantity is the tangential component Hτ of the field (Fig. 2.1a), whose local value for an arbitrarily shaped body is determined from the solution of outer problem (2.1). The penetration of uniform electromagnetic field into conducting half-space ˙i (Fig. 2.1b) is described by the known distribution of the electric E˙ i and magnetic H intensities. ˙i = H ˙ τ e− pz . E˙ i = E˙ τ e− pz , H

(2.2)

The field vectors are related by the Leontovich‘s approximate impedance boundary condition ] [ ˙τ . E˙ τ = ς ez × H

(2.3)

Fig. 2.1 Perfect skin effect model—(a); diffusion of uniform field Hτ into conducting half-space— (b)

36

2 Approximate Mathematical Models for Analysis of Alternating …

/ / / √ Here p = i ωμμ0 γ is propagation constant, δ = 1 Re( p) = 2 (ωμμ0 γ ) is the penetration√depth of /a uniform field into conducting half-space [9]. Surface impedance ς = i ωμμγ γ in this case connects the values of the field vectors ˙ τ , but also not only at the interface between dielectric and conducting media E˙ τ , H ˙ i. in the entire conducting half-space E˙ i , H The current flow in a thin surface layer allows us to introduce the surface current density ˙j S : ˙j S =



0 −∞

˙jdz =



0 −∞

γ E˙ τ e− pz dz =

E˙ τ . ς

(2.4)

The surface current density j S is connected with tangential component of the magnetic field Hτ and must satisfy the condition of continuity: ˙j S = ez × H ˙ τ , div˙j S = 0.

(2.5)

In this work, the subject of study is, first of all, a system with flat interface between media and closed current contour as the initial source of the field. The magnetic field above the surface of the conducting body for z > 0 is caused by the initial current contour I˙0 and by the mirror reflection current contour I˙1 = I˙0 (Fig. 2.2) [10]. ˙ in an arbitrary point (z > 0), which is created by The magnetic field intensity H the two current contours, is the following ) ˙  (t × r t1 × r1 ˙ = − I0 ˙ = 1 r ot A dl. − H μ0 4π r3 r13

(2.6)

l

Taking into consideration the fact that in the case when the observation point Q is at the flat interface between media (z = 0) the vectors which connect point Q with Fig. 2.2 Element of arbitrary contour with current I˙0 located near conducting half-space, and the mirror reflection of this contour element with current I˙1 from the interface between the dielectric and the conducting media

2.2 Electromagnetic Field of Spatial Contour with Current Near Flat Surface …

37

the elements tdl and t1 dl are r = ρ + hez , r1 = ρ − hez , the magnetic field intensity can be presented in terms of coordinates of only the initial contour in the form ez × (t × r) h t|| − ρ (t · ez ) I˙0 I˙0 ˙ ez × ez × H(z = 0) = dl = dl. (2.7) 3 2π r 2π r3 l

l

From (2.7), it immediately follows that the normal component of the magnetic field intensity is equal to zero, H · ez = 0. This indicates that (2.6) is the solution of the stated problem (2.1) for flat interface between media. Expression (2.6) is a special case of the solution of quasi-stationary problem presented in the previous chapter [for the perfect skin effect, only the first two terms must be left in (1.34), (1.35)]. From (2.7) we find that the surface current density is directly determined by the field of external sources: ⎡ ⎤ ˙0 t × r I 2 ˙ = 0) = ez × ⎣− ˙ 0, j˙ S = ez × H(z dl ⎦ = ez × r ot A (2.8) 3 2π r μ0 l

˙ 0 is vector potential of the field of the initial current. where A For perfect skin effect, both the magnetic and electric fields are zero inside conducting body. However, not only the magnetic field turns out to be non-zero in the outer region, the electric field can also differ from zero. The correct approach to the study of the perfect skin effect must to consider not only as the formation of a magnetic field, but as electromagnetic field. The total electromagnetic field can be considered as a field in vacuum created by both primary and secondary sources [11]. At the interface between media, not only does a surface electric current flow, but there are a free electric charge with a surface / density σ˙ and a polarization charge with a surface density σ˙ p = −σ˙ (εe − 1) εe on the interface. Conduction current ˙j flows in the volume of the surface layer, and the magnetization of the medium is taken into account by the magnetization current density (μ − 1)˙j. The surface current density for these components are ˙j S and ˙τ (μ − 1)˙j S . In addition, a jump change in the magnetization vector from (μ − 1)H in the lower half-space to zero in the upper one causes the introduction of the magne˙ τ . It follows tization current with surface current density j Sm = −(μ − 1)ez × H ˙ ˙ ˙ from (2.5) that (μ − 1)j S = (μ − 1)ez × Hτ = −j Sm , i.e. the surface density of the distributed magnetization current is opposite in sign to the other component of the surface magnetization current density ˙j Sm . Therefore, in the model with secondary sources, only one component ˙j S remains. All sources: the initial current contour I˙0 , surface current with density ˙j S and distributed surface charge σ˙ +σ˙ p —together in vacuum in throughout space, including the area z < 0 where the field is equal to zero, create the electromagnetic field equivalent to the field in the system with real media.

38

2 Approximate Mathematical Models for Analysis of Alternating …

A simple analytical solution of the problem under consideration makes it possible to obtain a clear view of the field formation of all sources. Still, only the magnetic field has been considered in limited area of space above the surface of conducting body at z > 0, to determine which it was enough to introduce a “fictitious” current of the mirror reflected circuit. In this case, the vector potential is written as ) ˙ (t t1 ˙ 1 = μ0 I0 ˙ =A ˙0 +A dl. (2.9) − A 4π r r1 l

In reality, the current, in addition to the initial contour, flows only over the surface of the conducting body. The surface current density does not contain a component perpendicular to the interface between the media. As a result, the vector magnetic potential of all currents in all space, including the area z < 0, is the following ˙0 +A ˙ S = μ0 I0 A˙ ' = A 4π

 l

μ0 t dl + r 4π



jS d S. r

(2.10)

S

˙ 0 and A ˙ S satisfies It can be seen from (2.9) and (2.10) that each of the terms A ˙ ˙ the continuity condition ∇ · A0 = 0 and ∇ · A S = 0, while the vector potentials ˙ and A P' differ from each other both in magnitude and in the area of definition. A ˙ is valid in the upper half-space, and the potential A P' is valid in the The potential A entire space. Since both vector potentials at z > 0 determine the same magnetic ˙ the potentials may differ by the gradient of the scalar ˙ = μ0 H, P' = r ot A field r ot A ' ˙ P ˙ function A − A = ∇ ψ. ˙ S · ez = 0, from which it follows From (2.10), the obvious equality follows A that the vertical component of the total vector potential is determined only by the P' · ez = A˙ 0z and will be different from zero if current of the contour source A˙ ' z = A the contour source has sections directed perpendicular to the interface between the media. In this case, due to the continuity condition, the normal component A˙ 0z does ˙− not change when passing through the interface A˙ + 0z = A0z . In the area of the conducting medium z < 0, the magnetic field intensity is equal ˙ = 1∇×A P' is P' = 0. This implies that in this area the vector field A to zero H μ0 P' = −∇ φ˙' can be introduced for it. From the potential, and a scalar potential A ' P condition ∇ · A = 0, the scalar potential φ ' satisfies the Laplace equation, and its normal derivative at the interface between the media z = 0 − 0 is determined by the normal component of the vector potential. This implies that for the scalar potential φ˙' at z < 0, the Neumann problem [12] is valid, the solution of which exists and is determined up to an arbitrary constant ⎧ ⎨z < 0 :

Δφ˙' = 0; ˙' ⎩ z = 0 − 0 : ∂ φ = − A˙ 0z . ∂z

(2.11)

2.2 Electromagnetic Field of Spatial Contour with Current Near Flat Surface …

39

˙ is also The electromagnetic field, in addition to the magnetic field intensity H, ˙ − ∇ φ, ˙ where ϕ˙ is the scalar characterized by the electric field intensity E˙ = −i ωA potential, which in this case is determined by the electric charges on the surface. The electric field intensity, like the magnetic field intensity, is equal to zero P' − ∇ φ˙ = 0 in the region z < 0. Performing the operation di v for E˙ = −i ωA P' = 0, we find that the scalar the last expression and again taking into account ∇ · A potential ϕ˙ also satisfies the Laplace equation. The last statement also follows from the conception that the scalar potential φ˙ is the potential of electric charges concentrated on the boundary surface. The absence of an electric field allows, similarly to the previous case, to formulate the boundary value problem for the scalar potential φ˙ ⎧ ⎨z < 0 :

Δφ˙ = 0; ˙ ⎩ z = 0 − 0 : ∂ φ = −i ω A˙ 0z . ∂z

(2.12)

Comparing (2.11) and (2.12) it follows that the scalar potentials are related by a simple relationship ˙ φ˙' = iωφ.

(2.13)

The same spatial distribution of potentials, expressed by equality (2.13), is, in fact, a consequence of the absence of both electric and magnetic fields in conducting medium at z < 0. In a particular case, for a flat contour with current that lies in a plane coplanar to the boundary surface, the vertical component of the vector potential is equal to zero A0z = 0. Hence, solutions of boundary value problems (2.11) and (2.12), which are regular at infinity, give φ ' = 0, φ = 0. The first equality means that at z < 0, the vector potential of the surface current density is equal in magnitude and opposite in ˙ 0 = −A ˙ S ). The direction to the vector potential of the initial current (z < 0 : A equality to zero of the scalar potential φ˙ indicates the absence, in this case, of charges on the interface between the media. Let us now consider, from the standpoint of secondary sources, the vector and scalar potentials in the area above the interface between the media z > 0, where the initial current I˙0 flows along the contour in the general case of a spatial configuration. In the area z > 0 for the term of the vector potential in (2.10) associated with ˙ S (x, y, z) = A ˙ S (x, y, −z). the surface current ˙j S , the symmetry condition is valid A Then, taking into account that A' (x, y, −z) = −∇φ ' |x,y,−z , the vector potential over the conducting surface can be represented as follows ˙0 +A ˙S =A ˙ 0 (x, y, z) − A ˙ 0 (x, y, −z) − ∇ φ˙' |x,y,−z , P' = A A

(2.14)

where each of the terms is considered known. From (2.14) it can be seen that ∇ φ˙' is ˙ −A P' = ∇ φ˙' . the potential function by which the vector potentials differ A

40

2 Approximate Mathematical Models for Analysis of Alternating …

Since the scalar potential φ˙ is due to electric charges distributed on the boundary surface, it satisfies the Laplace equation not only in the area of the conducting body, but also at z > 0. In this area, the component of the electric field intensity, normal to ˙ the interface between the media, created by the surface charge E˙ σ+z = − ∂∂zφ is equal and opposite in sign to the vertical component of the field of electric charges at the boundary under the surface, where it is compensated by the intensity of the external induced electric field E˙ σ+z = − E˙ σ−z = −i ω A˙ 0z .

(2.15)

This means that in the area z > 0 for the scalar potential of electric charges we have the following Neumann boundary value problem ⎧ ⎨z > 0 :

Δφ˙ = 0; ˙ ⎩ z = 0 + 0 : ∂ φ = iω A˙ 0z . ∂z

(2.16)

Finally, the last characteristic of the electromagnetic field, which characterizes the system under consideration from the standpoint of secondary sources, is the is easily from the surface density of the electric charge σ˙ . Its value / ( determined ) boundary condition σ˙ + σ˙ p = σ˙ − σ˙ (εe − 1) εe = ε0 E˙ σ+z − E˙ σ−z . In turn, the vertical component of the electric field intensity at the media interface is defined as the sum of the field of electric charges and the external induced electric field E˙ z+ = E˙ σ+z − i ω A˙ 0z . Taking into account (2.15), we have σ˙ = E˙ z+ = −2i ω A˙ 0z . εe ε0

(2.17)

The resulting expression (2.17) completely repeats the expression (1.51) found in the previous chapter. However, there the result was obtained using the solution of the wave problem and its consequence in the quasi-stationary approximation on the zero value of the vertical component of the current density in the conducting half-space. Here we present the complete solution of the problem for the perfect skin effect, where the induced current flows only over the surface of the body and the vertical component of the current density is basically absent. The coincidence of the expressions also confirms the conclusion that expression (2.17) is valid regardless of the specific electrical conductivity and relative magnetic permeability of the medium. The approximate mathematical model of the perfect skin effect and diffusion of the field into conducting half-space does not allow us to indicate the limits of applicability of this model and estimate the calculation error. To answer these questions and take into account the features of the distribution of a non-uniform electromagnetic field, it is necessary to use a more accurate mathematical model.

2.3 Asymptotic Expansion Method for Calculation of Three-Dimensional …

41

2.3 Asymptotic Expansion Method for Calculation of Three-Dimensional Quasi-Stationary Electromagnetic Field Let us consider the case of a rather strong skin effect in conducting body with flat interface between the media. However, in contrast to the previous subsection, where the perfect skin effect was analyzed, it is believed that the field penetration depth is a limited value compared to the characteristic distances in the electromagnetic system. In this case, the strong skin effect is understood as its extended definition, when not only the penetration depth of the alternating field is small compared to the characteristic dimensions of the conducting body, but the ratio of the penetration depth to the characteristic dimensions of the entire electromagnetic system is also small, including the dimensions of the current contour and distances from the contour to the interface between the media. The ratio of penetration depth to the characteristic dimensions of electromagnetic system is believed to be a small parameter, but not necessarily to tend to zero. As a basis, we put the exact solution of the electromagnetic field problem for the sinusoidal current of an arbitrary contour, taking into account eddy currents in conducting half-space. The basis for the development of a specialized method is a rather significant amount of necessary calculations on the obtained exact analytical expressions, which is associated with the definition of contour integrals of functions containing improper integrals from special functions. The calculation becomes especially laborious for inverse problems of field theory and for solving problems of optimizing electromagnetic systems. Such problem occurs, for example, when creating devices for heat treatment of metal products by the induction method. In this case, one of the conditions is the creation of an inductor configuration that provides the necessary distribution of the field and currents in the heated metal product [13–15]. In addition, the presence of an approximate asymptotic solution makes it possible to analyze the features of the non-uniform external field penetration into conducting medium.

2.3.1 Expansion of the Potentials and Field Vectors into Asymptotic Series In the dielectric area, where the alternating current flows along external contour ˙e = A ˙0 +A ˙1 +A ˙ 2 (1.34), the (Fig. 1.1), the vector potential in the Lorentz gauge A ˙ e = B˙ 0 + B˙ 1 + B˙ 2 (1.35) and the electric induction of the magnetic field B˙ e = ∇ × A field intensity E˙ e = E˙ 0 + E˙ 1 + E˙ 2 (1.44) are presented by the sum of three terms. The first two summands do not depend on the properties of the conducting medium and they are determined by the alternating current of the initial contour and the current of the contour mirrored from the interface. The third summands in all vector fields ˙ 2, B ˙ 2 , E˙ 2 , as well as the scalar potential φ˙ e (1.42), are completely determined by a A

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2 Approximate Mathematical Models for Analysis of Alternating …

single function G e ˙ ˙ 2 = − μ0 I0 t1 ∂ G e dl; A 4π ∂z l ) ( μ0 I˙0 ∂G e ˙ dl; t1 × ∇ B2 = 4π ∂z l

μ0 I˙0 E˙ 2 = i ω 4π



μ0 I˙0 φ˙ e = i ω 4π



ez × [t1 × ∇G e ]dl; l

(t1 · ez )G e dl.

(2.18)

l

The function G e depends on the field penetration depth δ = the function can be represented as follows: ∞ Ge = 2 0

e−ϑ(z−z M1 ) J0 (ϑρ Q M ) √ dϑ =2 ϑ + μ1 ϑ 2 + i ωμμ0 γ

∞ 0

/

2 ωμ0 μγ

=

√ 2 | p|

and

e−ϑ(z−z M1 ) J0 (ϑρ Q M ) /( ) ( √ )2 dϑ. (2.19) 2 ϑ ϑ+ + i μδ2 μ

From (2.19) it is seen that the numerator of the integrand depends on the components of the vector r1 = r M1 − r Q connecting the observation point Q and the source point M1 on the mirrored contour. The denominator of the integrand includes separately the product of the field penetration depth by the relative magnetic permeability δμ, and the relative magnetic permeability μ. Since a strong skin-effect is being considered, a small parameter that follows directly from (2.19) is a value that is defined as the ratio of two dimensional values: the value δμ and the distance r1 μδ μ ε1 = √ = . | p|r1 2r1

(2.20)

For a nonmagnetic medium (μ = 1), this parameter coincides with the ratio of the field penetration depth to the distance r1 . For ferromagnetic materials, the value of the parameter ε1 may be much larger. But in this case we will assume that the introduced parameter remains small, for example, for higher frequencies than for nonmagnetic media. When integrating along the contour in (2.18) the parameter ε1 is a variable depending on the position of the source point. Therefore, we will also use a single parameter ε, for which the field penetration depth is normalized to a common characteristic size λ, for example, to the smallest distance between the observation point in the dielectric area and the source points on the mirrored contour:

2.3 Asymptotic Expansion Method for Calculation of Three-Dimensional …

μδ μ , ε=√ = √ λ ωμ0 μγ 2λ

ε1 = ε

λ . r1

43

(2.21)

To use an approximate method for calculating electromagnetic field in dielectric half-space, it suffices to find an approximate representation of the function G e that determines the potentials and vectors of the field. For the analysis to be performed, we introduce a dimensionless variable μδ μ ϑ = √ ϑ, χ=√ ωμμ0 γ 2

(2.22)

and represent the expression for G e as follows: ( ) ( ) ∞ exp − cos β1 χ J0 sin β1 χ ε1 ε1 2 dχ . Ge = √ w (χ ) i 1

(2.23)

0

Here the dimensionless function w1 (χ ) in denominator of the integrand is χ w1 (χ ) = √ + i

/

( 1+

χ √ μ i

)2 .

(2.24)

The geometric representation of the quantity β1 included in the factors of the numerator in (2.23) is explained in Fig. 2.3. This is the angle between the vertical axis and the direction of the vector from the source point M1 to the observation point Q. Peculiarities of improper integral (2.23) and limitation ε < 1, basing on Laplace’s approach to estimation of functions of this kind, allow us to substantiate the use of asymptotic expansion for the function G e [16] and thereby simplify the analysis of the main electrophysical processes. Fig. 2.3 Mutual location of source and observation points

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2 Approximate Mathematical Models for Analysis of Alternating …

The variable χ with respect to which / the integration is performed varies in the range from 0 to ∞ and the multiplier 1 w1 (χ ) can be expanded into a power series with respect to χ within the convergence radius χ ≤ χc . As will be shown, the value χc is not less than one χc ≥ 1. At χ /ε1 ≥ 1, the numerator of the integrand in (2.23) decreases rapidly, and at large values of χ /ε1 it changes faster than any power function. The integral for each term of the power series exists even outside the convergence radius. The magnitude of the improper integral (2.23) for small ε1 is determined mainly by the behavior of the integrand near χ = 0. Despite the fact that the integral for each term of the series exists, the series composed of integrals of the terms of the expansion is divergent for any ε1 [16]. In order to be able to use the resulting series, you need to limit it to a fixed number of the terms N . In this case, the function G e is replaced by a function with an error, which can be made arbitrarily small by choosing a sufficiently small value of ε1 . Such the series with expansion of the integrand in our case in powers of χ is called the asymptotic series of Poincare type [17]. The usefulness of such asymptotic series is determined by the fact that the error of the cutoff series does not exceed the first discarded term of the series and, therefore, quickly tends to zero at ε1 → 0. It should be noted that each term of the asymptotic series is determined with an error, the value of which depends on the value of the small parameter and the number of the series term. Therefore, the estimate of the accuracy of the expansion in terms of the value of the first discarded term is also approximate. This implies the need to also evaluate the error of each member of the series. To obtain the asymptotic series, we represent the factor w1−1 in (2.24) as follows: /

)2 ( 1 + μχ√i − √χi 1 = )( )2 . ( w1 1 − 1 − μ12 √χi

(2.25)

Expanding the numerator and denominator (2.25) into a Taylor series, the function w1−1 can be represented as [ n ]( ) ( )n ( )2n+1  ∞ ∞ ∞    χ χ 2n χ 1 = an (μ) √ =− c2n √ + b2k c2(n−k) √ . w1 i i i n=0 n=0 n=0 k=0 (2.26) Here, b2n , c2n are the coefficients of the expansion into Fourier series with respect to the parameter √χi for the following functions, which are included in the numerator and denominator of expression (2.25)

2.3 Asymptotic Expansion Method for Calculation of Three-Dimensional …

/

(

1+

χ √ μ i

)2

(

∞ 

χ = b2n (μ) √ i n=0

)2n =1+

∞ 

(−1)

45

( ) ( ) − 3)!! 1 2n χ 2n ; √ 2n!! μ i

n−1 (2n

n=1

(2.27) ( 1− 1−

1 1 μ2

)(

χ √ i

)2 =

( )2n  ) ( ) ∞ ( χ 1 m χ 2m 1− 2 c2n (μ) √ = , (2.28) √ μ i i n=0 m=0

∞ 

where it is considered that (−1)!! = 1. The convergence radius of series (2.26) is defined as the smaller of the convergence radius χc1 of function (2.27) in the numerator and χc2 of function (2.28) in the denominator. They can √ be determined based on the fact that the convergence radius of the functions 1 + x and (1 + x)−1 is equal to one [18]. Hence, the convergence / / radius/ of the numerator has χc1 = μ ≥ 1, and the denominator has χc2 = 1 1 − 1 μ2 ≥ 1. The comparison shows that the convergence radius of √ the entire series lies within 1 ≤ χc ≤ 2. For non-magnetic media μ = 1, the denominator in (2.25) takes on the constant value equal to one. The series contains only the expansion terms of the numerator and χc = 1. It is important that even outside of the convergence domain of the function w1−1 , the integral (2.23) of the product of power and exponential functions exists even outside the convergence domain. For each term, we obtain the limited value [19] G (n) e1

2 =√ i

∞ 0

(

χ an (μ) √ i

)n

( ) ( ) sin β1 cos β1 exp − χ J0 χ dχ = ε1 ε1

(

ε1 = 2(−1)n an (μ) √ i

)n+1

(n) ( ) 1 n+1 ∂ . r1 n ∂z r1

(2.29)

As the series composed of integrals from the terms of integrand expansion G (n) e1 is divergent for any ε1 it is necessary to restrict it to a fixed number of terms N . In this case, G e is replaced by the function G eN with an error. G e ≈ G eN =

N  n=0

G (n) e1 =

( )n+1 ( ) ε1 ∂ (n) 1 , 2(−1)n an (μ) √ r1n+1 n ∂z r1 i n=0

N 

(2.30)

where the multiplier, √ which contains small parameter is associated with constant propagation p = i ωμμ0 γ by the ratio ε√1 ri1 = μp . Now let us use the given asymptotic expansion of the function G e and write down the final expressions for the vector potential, magnetic field induction, as well as the scalar potential and electric field intensity in the dielectric half-space, replacing the corresponding terms (1.34), (1.35), (1.42) and (1.44) with the function G e (2.19) by

46

2 Approximate Mathematical Models for Analysis of Alternating …

its approximate value in the form of an asymptotic series G eN =

N  n=0

G (n) e1 (2.30)

⎤ ( )n+1 (n+1) ) ( N  ˙ μ t t1 ∂ t1 ⎦ ˙ e = μ0 I0 ⎣ dl − − A (−1)n 2an (μ) dl ; n+1 4π r r1 p ∂z r 1 n=0 ⎡

l

l

(2.31) ⎤ ⎡ ( )n+1 (n+1) N  ˙0 ( t × r t1 × r 1 ) μ ∂ t × r I μ 0 1 1 ⎣ dl − B˙ e = − − (−1)n 2an (μ) dl ⎦; 3 n+1 4π r3 p ∂z r13 r 1 n=0 l

l

(2.32) φ˙ e = −

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

(2.33)

l

( )n (n) ) ( N μ ∂ I˙0  t t1 t1 × r 1 μ0 I˙0 dl + − ς E˙ e = −i ω (−1)n 2an (μ) e × dl. z 4π r r1 4π n=0 p ∂z n r13 l

l

(2.34) Here ς = γp is the surface impedance, the coefficients an (μ) are defined in (2.26)– (2.28). In the approximate expressions (2.31)–(2.34) the restriction is imposed. At the μ = r1 √ωμ must observation point in the dielectric area, the parameter ε1 = √μδ 0 μγ 2r1 be small for any point of the current contour. The possibility of obtaining results, the accuracy of calculations, and the relationship of the small parameter with the properties of non-uniformity external field are considered in the course of further presentation. Thus, unlike the original expressions, which contain improper integrals from special functions in integrands, the found expressions in (2.31)–(2.34) are limited series, where each term is found by calculating only one-dimensional contour integral. One-dimensional integrals are similar to the corresponding integrals for calculating the field of linear currents, the calculation and analysis of which can be easily performed using known methods.

2.3.2 Estimation of the Asymptotic Series Expansion Errors Let us represent the function G e by the sum of the limited number of its first N terms G (n) e1 and the remainder R N

2.3 Asymptotic Expansion Method for Calculation of Three-Dimensional …

Ge =

N 

G (n) e1 + R N (μ, ε1 ) = G eN + R N (μ, ε1 ).

47

(2.35)

n=0

The remainder R N depends on both the number N and the quantities μ, ε1 . Therefore, when studying the approximate method of calculation, it is necessary to determine not only the influence of the small parameter value, but also the number of terms of the asymptotic series. The error R N (μ, ε1 ) is connected not only with the limitedness of the series, but also with the fact that each term of the asymptotic series (2.30) is determined with an error, the presence of which is due to integration outside of the convergence radius of the integrand. Asymptotic series (2.30) is divergent with the feature generally characteristic of asymptotic series, that when the number of series terms increases, the error first decreases, reaching a minimum, and then the addition of new terms only increases it. Let us analyze how the series terms number N influences to the approximate value of the modulus of function |G eN | in comparison with the exact value |G e | with regard to the value of the small parameter ε1 . In the points located over mirrored current contour element (Fig. 2.2) at ρ Q M1 = ρ = 0 (r1 = z + h), the value J0 (ϑρ) = J0 (0) = 1 takes its maximum value in (2.19) and (2.23). In these points the improper integral in (2.19) may be received in analytical form for nonmagnetic medium (μ = 1) [18]. In this case, the function G e is written as ( √ [] [ (√ [ i i π ε1 ε2 G e = √ H1 − N1 − 1, ε1 ε1 i 2 i

(2.36)

where H1 and N1 are the Struve function and Neumann function, correspondingly. (In more general case μ /= 1 and ρ /= 0 to calculate the values of the function G e , it is necessary to use the methods of numerical integration). ( ) ∂ (n) 1 For ρ = 0, μ = 1 the factor ∂ z n r1 in (2.29) takes on the value equal to ( ) ( ) 1 ∂ (n) ∂ (n) 1 (−1)n n! = = . ∂z n r1 ∂z n z + h (z + h)n+1

(2.37)

Then the terms of the asymptotic series are G (n) e1

( = 2n!an

ε1 √ i

)n+1 .

(2.38)

/[ ] √ For the chosen observation points r1 = z + h we have ε1 = 1 (z + h) ωμ0 γ . Substituting (2.38) into the limited sum of terms (2.30), we find the approximate N  value in the form of the asymptotic expansion G eN = G (n) e1 at the considered n=0

point (ρ = 0, z). This value must be compared with the exact result (2.36).

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2 Approximate Mathematical Models for Analysis of Alternating …

Figure 2.4 shows the dependencies of the moduli of the exact function |G e | and the approximate function |G eN | on the parameter ε1 for different number N of series terms taken into consideration. As one can see, by small ε1 the most exact results are obtained when greater number of the terms of the asymptotic series is taken into consideration. With increasing the value of ε1 the deviation from the exact values increases and the deviation may reach its maximum when higher series terms are taken into consideration. This feature is more evident while relative errors of the / modulus of the approximate functions Δ N = |R N | |G e | are compared (Fig. 2.5). For the dependencies shown in the case of small ε1 the most accurate approximation includes four series terms (N = 4). When the value of parameter ε1 increases, at some value the calculation error starts to be greater for N = 4 that for N = 2. Further, for even greater values of parameter ε1 the error increases, and the most accurate approximation includes less asymptotic series terms, namely N = 1. At the same time the biggest error in this range of ε1 value appears by the approximation with the biggest number of series. Fig. 2.4 The approximate and exact values of the modulus of function |G e | for different number N of series terms at μ = 1 in the observation point ρ = 0, z = 0

Fig. 2.5 Influence of the number N of the series terms on the relative / error Δ N = |R N | |G e | of the approximate calculation of the function |G e | by μ = 1 in the observation point

2.3 Asymptotic Expansion Method for Calculation of Three-Dimensional …

49

Fig. 2.6 Relative error / Δ N = |R N | |G e | depending on the number N of terms in the series (μ = 5; ρ = 0)

It follows that for each ε1 there is a value for the number of terms in the series for which the error will be minimal. In this case, the smaller the value of the small parameter ε1 , the greater the number of terms of the asymptotic series can be used. This feature shown in Fig. 2.6 (for this case the exact values were calculated by numerical integration). With ε1 decreasing (for example, with increasing a field frequency or for materials with higher electrical conductivity, or at points most remote from the interface), the minimum error decreases and minimum is achieved when the number of the asymptotic series members increases. This feature determines the usefulness of divergent asymptotic series. For limited power asymptotic series (2.30) of the Poincare type [17] the error does not exceed the first rejected member and, therefore, quickly approaches zero at ε1 → 0. However, each term of an asymptotic series is determined with an error, whose value depends on the value of the small parameter and the number of the series term. So, the estimate of the expansion accuracy in the magnitude of the first rejected term is approximate and this also requires an estimate of the error of each term of the series. As before, we will estimate the error for the case ρ = 0 and, respectively r1 = z + h, when the small parameter ε1 at the chosen height of the observation point takes the largest value. To obtain estimate of the error of the series terms, we divide the integration interval in (2.29) into two intervals, namely (0 ÷ 1) and (1 ÷ ∞). Within the first interval (0 ÷ 1), the corresponding series (2.26) in the integrand ( )n ∞  1 = an (μ) √χi converges, and the integrals (2.29) of each term of this series w1 n=0

are known [18]. Within the second interval (1 ÷ ∞), the general expression (2.23) is valid. As a result, we write as Ge =

∞ 

2δ1 G˙ (n) e + √ i n=0

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2 Approximate Mathematical Models for Analysis of Alternating …

) ( ) ( ∞ exp − χ ε1 2 χ dχ + √ dχ exp − ε1 w1 (χ ) i 0 1 ( [ ∞ n   2an (μ)n! n+1 1 2δ1 = 1 − e−1/ε1 + √ . (2.39) (√ )n+1 ε1 k k!ε i 1 n=0 k=0 i

2 =√ i

1

(

χ an (μ) √ i

)n

The result of integration of each member of the series within the entire range (0 ÷ ∞), including the outside domain of the convergence radius, is represented by ˙ (n) ˙ (n) the values G˙ (n) e1 (2.38). The difference between the values G e1 in (2.38) and G e in (2.39) turns out to be  1 2an n! n+1 −1/ε1 ˙ (n) . G˙ (n) e1 − G e = (√ )n+1 ε1 e k!ε1k k=0 i n

(2.40)

The relative value of the error Δn for term of the series with number n is determined by the expression n | |  1 | ˙ (n) | e−1/ε1 |G e1 − G˙ (n) | k!ε1k e k=0 | | = Δn = . n | ˙ (n) |  1 −1/ε |G e | 1 1−e k!εk k=0

(2.41)

1

The relative error of calculation of each series term at n → ∞ tends to infinity. When the number n of series term is increased, the error of determination of this term is increased too, although its relative contribution to the total sum is decreased. The relative values of the error for the first ten terms of the series at different values of the parameter ε1 are shown in Fig. 2.7. Fig. 2.7 The relative value of the error for asymptotic series terms

2.3 Asymptotic Expansion Method for Calculation of Three-Dimensional …

51

Let us also estimate the remainder |δ1 | in (2.39): ∞ |δ1 | = 1

| | | √χ | i |

) ( ) ( ∞ exp − εχ1 χ | dχ < dχ = εe−1/ ε ≈ εΔ0 . (2.42) exp − / )2 | ( ε 1 | 1 + 1 + μχ√i | |

As one can see, the magnitude |δ1 | introduces an error smaller than the replacement ˙ (0) of the main term G˙ (0) e by G e1 , the relative error of which is the smallest. For this reason, the value |δ1 | can be neglected, and this confirms that the relative error can be estimated using (2.41).

2.3.3 Choice of the Number of the Asymptotic Series Terms When using the asymptotic method, an important practical task is the optimal choice of the number of terms of asymptotic series depending on the value of the small parameter ε1 . The choice can be based on the analysis of the value of the remainder R N (μ, ε1 ). Two approaches can be used here: the first is based on the estimation of the last considered term of the series, the second is based directly on the estimation of the remainder R N (μ, ε1 ) at given allowable error. Choosing the number of terms to estimate the error of the last member of series. The possibility of applying the first method is due to the fact that R N (μ, ε1 ) does not exceed the last rejected term of the series and, however, itself is determined with a certain error. In this case, if a certain series term is calculated with an error exceeding a certain allowable value, for example, if the error becomes comparable to the value of the series term, then taking into account such term does not increase the accuracy of calculation. Moreover, if the relative error in determining the value of any series term increases with increasing its number, then the assessment of the achievement, by any term, of the relative error limit will determine the limit value of the number of series terms. The number of series terms should be limited to the value of N at which their further increase leads only to an increase in the total error. It follows that the number of terms of a series should not exceed the value at which the relative error of the last series 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 (μ, ε1 ) ≤ C N .

(2.43)

Having compared the data presented in Fig. 2.7, with the dependences in Fig. 2.6, we can see that the minimum error of the approximate calculation is achieved when

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2 Approximate Mathematical Models for Analysis of Alternating …

the number of the series terms meets the condition C N ≈ 1. This confirms the assumption that the estimate of the optimal number of the series terms can be performed on the basis of the estimate of the error of the last considered term. The limit number of the series terms found from (2.41) for different allowable values of the relative error Δn for the last considered term of the series at n = N is shown in Fig. 2.8. As we can see from Fig. 2.8 when the value of the small parameter ε1 decreases, there is a rapid increase in the possible number of the asymptotic series terms. However, the value of the considered terms of the series at large values n is very small. This fact is illustrated by the data in Fig. 2.9, which shows the value of the boundary terms, which are obtained by fulfilling condition (2.43). As Fig. 2.9 shows, 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 and restrictions on the accuracy of the calculation are necessary. Choosing number of the terms for a given error of asymptotic series. The approach is to estimate the error of the whole asymptotic series Δ N (N , ε1 ).

Fig. 2.8 The limit number of the asymptotic series terms (μ = 1)

Fig. 2.9 The values of the boundary terms of the asymptotic series selected under condition (2.43)

2.3 Asymptotic Expansion Method for Calculation of Three-Dimensional …

53

Fig. 2.10 Choosing the number of series terms for Δmin = 10−3 , Δmax = 10−1

Assume that it is sufficient that the calculation be performed with an accuracy at which the error does not necessarily have to be less than the specified Δmin . Furthermore, the approximate method of calculation, whose accuracy depends on the value of the small parameter ε1 being used, we then define the error limit Δmax > Δmin , the excess of which indicates the inadmissibility of using this method. Consider, as an example, a nonmagnetic medium / | | μ = 1 for which we develop a series of dependences Δ N (N , ε1 ) = |R N | |G˙ e | as functions of the number of considered series members for different values of a small parameter. At the same time we choose specific error limits, for example, Δmin = 10−3 and Δmax = 10−1 . The dependencies Δ N (N , ε1 ) are shown in Fig. 2.10. The Figure highlights two curves, among others. One curve, which is shown by a solid bold line at εmin = 0.18, corresponds to the dependence for which the minimum value of the error is equal to the limit value Δ N (N , εmin ) = Δmin and which is realized for the number of series terms N = 6. At lower values of the small parameter ε1 < εmin , the error limit value Δmin will be implemented when the number of terms of the series does not exceed the set value, i.e. N ≤ 6. The Figure shows the dependencies when we can be limited to smaller number of series terms: ε1 = 0.15 N = 4; ε1 = 0.1 N = 2. The other curve in Fig. 2.7, which is shown by a bold dotted line at εmax = 0.5, corresponds to the dependence for which the minimum value of the error is equal to the limit value Δ N (N , εmax ) = Δmax and which is realized for the number of series terms N = 2. With the values of the small parameter ε1 > εmax , for any number of series terms, the calculation error exceeds Δmax . This proves that it must not be to use the approximate method for such values of the small parameter at the chosen limit value of the error Δmax . In the intermediate range of values of the small parameter εmin < ε1 < εmax , the minimum achievable calculation errors are already within Δmin < Δ N < Δmax . In

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2 Approximate Mathematical Models for Analysis of Alternating …

this case, these minimum errors are realized for the number of members of the series, which also do not exceed the maximum number N = 6 corresponding to the value of the number of members of the series at ε1 = εmin . The main result of this analysis is that when using the method of asymptotic expansion, the number of terms of series can be limited to a relatively small number, which is determined by the allowable accuracy of the calculation (relative error). The number of expansion terms is determined by the value of small parameter ε1 = h √2πμf μμ γ , which depends on the electrophysical parameters of the conducting 0 medium μ, γ , the field frequency f and the minimum distance from the current contour to the interface h. For a specific conducting material, the obtained results allow us to indicate the required number of expansion terms depending on the field frequency and the minimum distance h. Consider an example where the conducting medium is aluminum with electrophysical properties μ = 1, γ = 3.7 × 107 Ω−1 · m−1 . Setting specific values of the small parameter ε1 , including the set εmin = 0.18 and εmax = 0.5, we find a dependence of the field frequency on the minimum height of the current-carrying conductor at ε1 and respectively for a specific number of series terms, at which the minimum error for the given ε1 is realized. Such dependences are shown in Fig. 2.11. When performing practical calculations, these dependences allow us to choose the required number of the asymptotic series terms and specify the estimate of the calculation accuracy. Usually the material and field frequency are known. Further, taking into account a geometrical configuration, it is necessary to define only the minimum distance from contour to conducting surface and given the presented dependences to find calculation parameters. In that case, if a current dependence on time differs from sinusoidal, it is possible to use a frequency spectrum of the current and for each frequency (or for characteristic frequencies) to apply the described approach. Fig. 2.11 The dependence of the field frequency on the minimum height of the current contour at the given number of series terms for aluminum (Δmin = 10−3 ,Δmax = 10−1 )

2.3 Asymptotic Expansion Method for Calculation of Three-Dimensional …

55

2.3.4 Comparison of Exact and Approximate Calculations of 3D Electromagnetic Field for Specific Contour Configuration The value of the small parameter ε1 depends on the distance r1 and, accordingly, changes depending on the relative position of the point M1 on the mirrored contour and the observation point. For specific contour the parameter ε1 takes its greatest value at the smallest distance r1 , when the observation point is located at the interface between the media on vertical axis passing through the point on the contour. Therefore, the largest error will occur when the field is calculated at the interface between media. In this regard, we compare the results of calculations using exact and approximate expressions for the field on the surface of the conducting half-space. The analysis of the calculation errors of the sinusoidal field depending on the value of the parameter ε1 for specific points M1 not related to the configuration of the contour was studied in sufficient detail in this chapter above. Therefore, here we will compare the calculation of three-dimensional electromagnetic field for the model of the electromagnetic system with circular current contour shown in Fig. 1.5 and previous sizes. The electrophysical properties of the medium correspond to those / of aluminum γ = 3.7 1 · 107 (Ω · m), μ = 1, frequency is variable. The calculations was carried out for all components of the electric and magnetic field intensities according to exact (1.34), (1.35), (1.42), (1.44) and approximate (2.31)–(2.34) expressions [20]. The results are presented for the normalized | | compo˙ k∗ = ±| E k∗ | exp(i φ Ek ) nent values of the complex-value amplitudes of the electric E | | and magnetic H˙ k∗ = ±| Hk∗ | exp(i φ H k ) fields, where k = x, y, z. The argument of the complex-value amplitude shows the phase displacement angle / relative to the phase / of the contour current I˙0 within the limits −π 2 ≤ φ ≤ π 2 (the “–” sign in front of the complex-value amplitude modulus is equivalent to phase change by π ). μ | I˙ |ω The normalized values of the field intensities are defined as follows E˙ = 0 4π0 E˙ ∗ , ˙ ˙ = | I0 | H ˙ ∗. H 4π h 0 In Fig. 2.12 for the point on the surface x = 0, y = 0 closest to the contour, nonzero normalized values of the components of the electric E˙ x∗ and magnetic H˙ y∗ field intensities are presented depending on the value of the param/( ) √ eter ε = μ r1 min 2π f μμ0 γ , where in this case the minimum distance of all r1 is r1 min = h 0 . Modules of complex-value amplitudes for calculation by exact (solid curves) and approximate (dashed curves) expressions are shown in Fig. 2.12a, c. The arguments for the complex-value amplitudes are given in Fig. 2.12b, d. From the presented comparison results, it can be seen that for the values of the small parameter ε ≤ εm = 0.3, the results for the moduli of the field vectors practically coincide. For the arguments of complex-value field vectors the deviation of the results occurs at a slightly smaller value of the small parameter, and also at values close to the indicated value. In this case, for example, for aluminum, the calculation with sufficient accuracy can be carried out for the field frequencies f ≥ f m = 380 Hz.

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2 Approximate Mathematical Models for Analysis of Alternating …

Fig. 2.12 Comparison of exact and approximate calculations of field intensities in system “circular current contour –conducting half-space” at the point on the surface closest to the contour x = 0, y = 0

For other observation points on the interface between media with larger minimum distance to the contour r1 min > h 0 , a similar limiting value of the small parameter also takes place. This circumstance is illustrated in Fig. 2.13 for point on the surface x = 0.025 m, y = 0.01 m for which the minimum distance to the contour increases to the value r1 min = 0.018 m. At this point all components of the field intensities are not equal to zero. Therefore, Fig. 2.4 shows the results of comparing calculations for all field components except for the component E˙ z , which is completely determined only by the induced electrical field of external sources (1.64).

Fig. 2.13 Comparison of exact and approximate calculations of field intensities in system “circular current contour –conducting half-space” at the point on the surface x = 0.025 m, y = 0.01 m

2.4 Mathematical Model of a Locally Two-Dimensional Electromagnetic …

57

From the point of view of the possibility of using computationally simpler asymptotic expansion, the main conclusion is that for all components of the electromagnetic field, the results practically coincide with the calculation using exact expressions up to the value of the small parameter εm ≈ 0.3. In addition, it is essential that the introduced small parameter, which combines several quantities, is the single parameter that indicates the limiting value for the application of approximate asymptotic calculation method. So, for the considered point on the surface, which is at the greater distance from the contour r1 = 0.018 m > h 0 = 0.01 m, the limiting value of the field frequency decreases to the value f m = 117 Hz.

2.4 Mathematical Model of a Locally Two-Dimensional Electromagnetic Field Near the Current Contour Despite the significant simplification of the general three-dimensional problem when using the asymptotic expansion, the calculation from the found expressions still requires the calculation of contour integrals. This does not create fundamental computational difficulties, however, when solving some problems, especially problems of optimizing and synthesizing systems of a complex spatial configuration, the amount of calculations increases significantly and multiparameter search becomes quite laborious. It takes place in the case when it is necessary to ensure intensive interaction of current contours with conducting medium, for which it is necessary to determine the electromagnetic field in limited area of space near small distance between contour and the conducting body. Here, the possibility of further simplification of calculations appears, which consists in the local replacement of a three-dimensional electromagnetic field by a two-dimensional one [19]. In such mathematical model, when determining the electromagnetic field, the contour integrals in the dielectric area are replaced by integrals along an infinite straight line parallel to the flat interface between the media. Such integrals are calculated separately, and as a result, the terms of asymptotic series instead contour integrals contain only simple algebraic expressions. The problem is to study the conditions of local replacement of the external source field in the form of a curvilinear contour with alternating current located near the interface of the media with a straight line of current. The possibility of such replacement is connected with the analysis of two circumstances. The first is related to the influence of parameters that characterize the curvilinear nature of the real contour. The second is due to the possible presence of elements of spatial contours that are not parallel to the interface of media.

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2 Approximate Mathematical Models for Analysis of Alternating …

2.4.1 Conditions for the Possibility of Using the Model of Locally Two-Dimensional Electromagnetic Field It is possible to replace the field of curvilinear current contour with the field of rectilinear current filament if certain conditions are occurring. For contours, observation points should be located at such distances r⊥ from the single contour for which the following parameters turn out to be small [21]: εd1 =

r⊥ r⊥ , εd2 = , R T

εd3 =

r⊥ , rl

(2.44)

where R, T are the radii of curvature and torsion of the contour line, respectively, rl is the characteristic size of the contour itself, which is understood as the minimum distance between a given point of the contour and any other point of it, provided that the internal distance along the curve has an order equal or greater than R or T . It was shown in [21] that the influence of parameters (2.44) on the field distribution near the contour line take place already in terms of the first order in these parameters. This circumstance must be taken into account when using the mathematical model of the field of a rectilinear conductor with current. In this case, the field can be calculated with sufficient accuracy only in the area near of the conductor. In contrast to single contours, the analyzed problem contains a conducting body with flat interface between the media. In this case, the solution is represented by the sum, where the initial contour and its mirror reflection from the flat boundary surface, and, accordingly, two distances r⊥ and r1 ⊥ , as shown in Fig. 2.14a for a conductor section parallel to the interface. The condition of small parameters (2.44) must be satisfied for both distances. The possible presence of contour sections directed at an angle to the surface imposes restrictions to the magnitude of the components of the tangent vector to the contour t = t|| + t⊥ (Fig. 2.14b). An infinite rectilinear current filament located in dielectric half-space cannot be located at an angle to the surface. Therefore, the

Fig. 2.14 Features of the mathematical model: a—distances from the observation point both to the conductor and to its mirror reflection from the interface of media; b—the presence of vertical sections of the contour

2.4 Mathematical Model of a Locally Two-Dimensional Electromagnetic …

59

full length of the tangent vector t or t1 , which are included in the corresponding expressions, is replaced by its horizontal projection t|| to/ the | |interface between the media that is straightened along the unit vector e|| = t|| |t|| |. Accordingly, the tilt angle α can only slightly differ from the zero value. The error associated with the replacement of the vector t by its projection t|| can be estimated from the change in the length of the vectors. For small tilt angles α, the relative error will simply be / Δ|| = 1 − cos α ≈ α 2 2.

(2.45)

Note that the value Δ|| does not depend on the parameter ε1 . 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. Let’s perform the estimation of the error as the ratio of the additional terms of the electric field to the value caused by the horizontal projection of current. We take into account the tangential component of the electric field intensity on the interface between the media. This choice caused by the fact that the tangential component on the surface is the same for the dielectric and conducting areas. Its presence is associated with the flow of current in conducting body, the manifestation of electromagnetic forces and thermal processes. In the general expression for the electric field intensity (1.44), the sum of the first two terms on the surface gives only the vertical component, and the third term E˙ e2 has only the component of the field tangent to the surface. In order to separate the component fields associated with the current flow both parallel to the boundary surface along the component t || of the tangent vector and in the direction normal to the surface along t⊥ , we present the electric field intensity at z = 0 as follows | ] [ ∂ G˙ e || ∂ G˙ e μ0 I˙0 ˙ ˙ ˙ t || dl Ee (z = 0) = E || 1 + E || 2 = i ω − (t1 · ez )eρ 4π ∂ z |z=0 ∂ρ l √ [ ( ρ) ( ρ )] μ0 I˙0 i ωμμ0 γ 2 = iω + (t · ez )eρ f e2 ε, dl. t|| f e1 ε, 4π μ h h l

(2.46) Since the field intensity on the surface is considered, the expressions are / (√in (2.46) ) / √ 2h , where written using instead of ε1 the parameter ε = μ h ωμμ0 γ = μδ h is the height of the contour element location; ρ is the distance from the vertical axis of the local cylindrical coordinate system associated with the source point M on the contour. For the specific source point on the contour M, the value of the parameter ε remains unchanged for all points Q on the surface. The functions included in the contour integral are

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2 Approximate Mathematical Models for Analysis of Alternating …

( / ) f e1 ε, ρ h =

∞ 0

( / ) f e2 ε, ρ h =

∞ 0

) ( ) ( exp − χε J0 ρh χε χ dχ ; w1 (χ )

(2.47)

) ( ) ( exp − χε J1 ρh χε χ dχ . w1 (χ )

(2.48)

These functions determine the contribution to the value of the contour integral of the sections of the contour with, respectively, parallel and perpendicular current directions. | ( / )| | f e1 ε, ρ h | and shows the dependence of the modulus of functions | In( Fig./2.15 )| / | f e2 ε, ρ h | on the relative distance ρ h = tgβ1 at μ = 1 and for different values of the parameter ε. As can be seen from the presented dependencies, areas with different current directions participate differently in creating the tangential component of the electric field intensity on the surface of the body. The horizontal component of the current gives the largest contribution directly below the current element in the direction coinciding with the direction of the current. The contribution to the total magnitude of the field rapidly decreases with increasing distance from the point M0 of the smallest distance from the current element to the surface. The second term in (2.46) has a different effect to the creation of the electric field intensity on the surface. The largest contribution from the vertical component of the current is realized at a certain distance ρ = ρm from the point M0 . This is explained by the fact that this summand of the electric field intensity is related to the field of the surface electric charge, which compensates the normal component field for the normal component of the induced external electric field. The vertical section of the current contour creates a contribution to the value of the tangential component of the electric field intensity, which is directed in the radial direction to the axis z and has the same value at the points of the circle ρ = const. We estimate the calculation error by the value of the ratio of individual components of the integral function in (2.46). Since these components have significantly different Fig. 2.15 |The( values / of )| the | f ε, ρ h | and functions | ( / e1 )| | f e2 ε, ρ h | at the interface of media

2.4 Mathematical Model of a Locally Two-Dimensional Electromagnetic …

61

Fig. 2.16 To determine the error associated with the neglect of the electric field intensity of normally oriented sections of the current: a—the ratio of the functions | f e2 (ε, ρ = ρm )| and | f e1 (ε, ρ = 0)|; b—errors estimate Δ⊥ according to (2.49) and Δ|| according to (2.45)

dependences on the points on the surface, for the relative error indicator we will choose the values of the components at the points where the corresponding functions take their maximum values: ρ = 0 for the function f e1 and ρ = ρm for the function f e2 . In this case, we will get Δ⊥ = K (ε)tgα,

(2.49)

t·ez m )| where K (ε) = | |fe2fe1(ε,ρ=ρ , tgα = t·e . (ε,ρ=0)| || Note that the multiplier K (ε) varies slightly in a wide range of parameter ε. Figure 2.16a illustrates this circumstance. As it can be seen, in the selected range the multiplier takes values within K (ε) = 0.386 ÷ 0.337. It follows from this that the relative error of neglecting the flow of current in the direction perpendicular to the boundary surface is determined primarily by the angle α of inclination of the contour. From Fig. 2.16b it can be seen that neglecting the field component due to the flow of current in the vertical direction gives a large error Δ⊥ . In this case, the value of the relative error is practically independent of the value of the small parameter ε. For strong skin effect, when ε ≤ 0.3, a single dependence can be used for estimation at small tilt angles α.

Δ⊥ ≈ 0.366 α.

(2.50)

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2 Approximate Mathematical Models for Analysis of Alternating …

Fig. 2.17 Local coordinates in a plane normal to rectilinear current filament

2.4.2 Expressions for Calculating the Electromagnetic Field and the Poynting Vector In the considered model of locally two-dimensional field, specific expressions for calculating the vector potential in the dielectric area the induction of the magnetic field and intensity of electric field can turn out to be useful. We write such expressions in the local Cartesian coordinate system associated with a specific point on the current contour. The real curvilinear contour is locally replaced by a rectilinear current filament, in which the direction of the current coincides with the direction of the projection of the tangent vector to the contour onto / | |flat interface between the media t|| and is characterized by a unit vector e|| = t|| |t|| | (Fig. 2.17a). One of the coordinates is the length of the straight filament, measured from the point selected on the contour. In the plane normal to the rectilinear axis, local coordinates z, ξ are introduced in the to the flat interface directions determined by the unit vectors ez and eξ : perpendicular / ez and perpendicular to the curvilinear contour eξ = t × ez |t × ez | = e|| × ez . Since in the considered mathematical model the source of the external field is the current filament, which is parallel to the interface, the electric field intensity is ˙ e . For rectilinear infinitely completely determined by the vector potential E˙ e = −iωA long conductor, the integrals included in the expressions for the vector potential (2.31) are easily calculated. By introducing the designation ∂ (n+1) ∂u n−1 = n+1 u n (ξ, z) = ∂z ∂z

∞ / −∞

dl|| ξ 2 + (z + h)2 + l||2

.

(2.51)

The expression for the vector potential near conductor with current can be written as

2.4 Mathematical Model of a Locally Two-Dimensional Electromagnetic …

[ ( )n+1 ] N 2 2  ˙0 e|| μ + + h) I μ ξ (z 0 n ˙e = A − (−1) 2an (μ) un . ln 2 4π p ξ + (z − h)2 n=0

63

(2.52)

˙ = r ot A ˙ in local coordinates has two The induction of magnetic field B components, namely: ˙e ˙e ∂A ∂A eξ + ez = B˙ eξ + B˙ ez B˙ e = − ∂z ∂ξ

(2.53)

$ μ0 I˙0 e|| (z + h)eξ − ξ ez (z − h)eξ − ξ ez B˙ e = − − 2 − 2π ξ 2 + (z + h)2 ξ + (z − h)2 ( )n+1 ( )% N  μ ∂u n ∂u n n eξ − ez . − (−1) an (μ) p ∂z ∂ξ n=0

(2.54)

or

The functions u n (ξ, z) are defined by successive differentiation of the function u 0 (ξ, z) = − ξ 22(z+h) with respect to the coordinate z, therefore +(z+h)2 2(z + h) ; ξ 2 + (z + h)2 2 4(z + h)2 + n = 1; u 1 = − 2 [ ]2; ξ + (z + h)2 ξ 2 + (z + h)2

n = 0; u 0 = −

12(z + h) 16(z + h)3 n = 2; u 2 = [ ]2 − [ ]3; ξ 2 + (z + h)2 ξ 2 + (z + h)2 n = 3; u 3 = [

96(z + h)2 96(z + h)4 ]2 − [ ]3 + [ ]4; ξ 2 + (z + h)2 ξ 2 + (z + h)2 ξ 2 + (z + h)2 12

.......

(2.55)

Substituting (2.55) and their derivatives in (2.52) and (2.54) it is easy to obtain specific expressions for the vector potential and induction of magnetic field. These expressions are greatly simplified for points on interface of the media directly under the conductor with current, when z = ξ = 0, and r1 = h takes the minimum value. In this case, for the components of vectors parallel to the surface, we have % & ( )2 ( )( )3 ˙0 1 ( μ ) μ 1 μ 1 μ 1 I 0 − 2 − 3 2− 2 + ... ; A˙ e = π h p h p h μ p % & ( )2 ( ) ˙0 μ μ 1 I μ 1 0 + 2 + ... . B˙ eξ = − 1− πh h p h p

(2.56)

(2.57)

64

2 Approximate Mathematical Models for Analysis of Alternating …

Calculation of the energy flux density averaged over a period of electromagnetic field through surface from dielectric to conducting medium pz (1.65) is also greatly simplified when using the model of the locally two-dimensional field. ) ( iω 1 1 pz = Re(−∏ · ez ) = − Re At B ξ = 2 2 μ0 ⎧ ⎫ ] | |2 [ N ( )n+1 | I˙0 |  ⎪ ⎪ μ i ω μ 0 ⎪ ⎪ n ⎪ ⎪ (−1) · 2a u (ξ, 0) × ⎪ ⎪ n n ⎪ ⎪ ⎨ 2π 2 ⎬ p n=0 = = Re [ ] | ( )n+1 N ⎪ ⎪  | ⎪ ⎪ 4h μ ∂u (ξ, z) n ⎪ ⎪ n | ⎪ ⎪ − (−1) · 2an ⎪× 2 ⎪ | ⎩ ⎭ ξ + h 2 n=0 p ∂z z=0 [ | |2 ( √ μ0 ω| I˙0 | h2 2μ = · + .... , √ 2π 2 2 ωμμ0 γ (ξ 2 + h 2 )2

(2.58)

We note that in expression (2.58) for the energy flux density of the field pz directly under the contour with current at ξ = 0, there will be only terms containing h to even degree. For well-conducting non-magnetic media, when ε 0. The axis z is oriented perpendicular to the interface surface in the direction of the single vector ez . For an 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 expression for the electric field intensity E˙ i is given in (1.55). It can be written as following ∮ [ ] μ0 I˙0 E˙ i = − t|| F1 (ρ, θ, z) + (t · ez )eρ F2 (ρ, θ, z) dl. 2i ω 4π

(3.1)

l

where ω is cyclic frequency, i is imaginary unit, μ0 is permeability of vacuum. Here the local cylindrical coordinates (ρ, θ, z) with its unit basis vectors (eρ , eθ , ez ) are used (Fig. 3.1). The center of the coordinate system is located 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 values of local coordinates depend on the position of the source point M during integration along the contour. ˙ i in conducting half-space follows The expression for the magnetic field intensity H ˙ from the Maxwell equation ∇ × E˙ = −iωμμ0 H ˙ ˙ i = μ0 I0 H 4π μ

∮ [ l

[ ] ] | | [ ] |t|| | ez sin θ ∂ F1 + e|| × ez ∂ F1 − (t · ez )eθ ∂ F2 dl. ∂ρ ∂z ∂z

(3.2)

The functions F 1 (ρ, θ, z) and F 2 (ρ, θ, z) in (3.1) and (3.2) are as follows ∫∞ F1 (ρ, θ, z) =

exp(qz)

exp(−ϑ z M )J0 (ϑρ) ϑdϑ ; w(ϑ)

exp(qz)

exp(−ϑ z M )J1 (ϑρ) ϑdϑ , w(ϑ)

0

∫∞ F2 (ρ, θ, z) = 0

(3.3)

3.2 Electromagnetic Field in a Conducting Half-Space—A General Feature …

73

√ √ where, as before q = ϑ 2 + p 2 , p = i ωμμ0 γ is propagation constant, w(ϑ) = ϑ + q/μ, J 0 (·) and J 1 (·) are Bessel functions of the first kind of zero and first orders. Since the decrease of the field with respect to depth is considered, the functions that depend on the coordinate z are distinguished by a separate factor in (3.3). The distribution of the field in the surface layer, depending on the coordinate z that is perpendicular to the boundary surface, is reflected by the presence in the √ z ϑ 2 +i ωμμ0 γ qi z integrands of the factor e = e . To study features of the electromagnetic field formation associated with the penetration of non-uniform field into conducting half-space, it is advisable to introduce dimensionless parameters, whose values are due to the form of expressions (3.3). In this case, analogous Chap. 2, we use dimensionless integration/variable χ = √ / /√ / ϑμ ωμμ0 γ and take into account that pz = 2i z δ, where δ = 2 (ωμμ0 γ ) is the penetration depth of a uniform ( field into ) conducting half-space( [26]. As)a result, the functions F1 (ρ, θ, z) = μp f 1 δz , ε0 , β and F2 (ρ, θ, z) = μp f 2 δz , ε0 , β will be expressed in terms of dimensionless parameters, where ( ) ( ) χ cos β χ sin β ∫∞ ( ) exp − J 0 ε0 ε0 z χ dχ ; f 1 , ε0 , β = K , χ · δ δ w1 (χ ) 0 ( ) ( ) ∞ β χ sin β ∫ (z ) ( z ) exp − χ cos J 1 ε0 ε0 χ dχ , f 2 , ε0 , β = K , χ · δ δ w1 (χ ) (z

)

0

⎛

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

)

(3.4)

(3.5)

/√ Here the parameter ε0 = μδ 2r0 is proportional to the ratio of the penetration depth δ to the distance r 0 from the field source at a point M on the contour to the body surface at a point Q0 (Fig. 3.2). The denominator w1 (χ ) in the integrands (3.4) is determined in (2.24). The expressions (3.1)–(3.3) describe the penetration of the electromagnetic field of arbitrary contour with current into conducting half-space and in the general case they differ from approximate description of the penetration of uniform field. As it follows from (3.3) and (3.4), the distribution of any component of the electric and magnetic field intensities in the skin-layer, depending on the coordinate z, is associated with exponential function K(z/δ,χ) (3.5) in the integrands. The factor / [ / ( √ )]2 1+ χ μ i in the exponent affects to the field decrease law. If the influence [ / ( √ )]2 μ i is absent, it corresponds to the decrease law of of the second term χ

74

3 Penetration of Non-uniform Sinusoidal Electromagnetic Field …

Fig. 3.2 Depth distribution of the normalized amplitudes for the components of the electric E x∗ and magnetic Hx∗ , Hy∗ , Hz∗ intensities in conducting half-space for non-uniform three-dimensional field created by the specific system in the form of circular current contour near the flat interface between media

(/ the uniform field. Since Re

[ / ( √ )]2 1+ χ μ i

) > 1, the decrease of the non-

uniform electromagnetic field created by the current contour is always faster than that of the uniform field. Taking into account the superposition principle, this conclusion will be valid for any system of initial closed contours and therefore is valid in the general case of arbitrary external field. Thus, faster decrease of non-uniform electromagnetic field as compared to uniform field is general feature of the electromagnetic field formation at its diffusion into conducting half-space. Let us consider the influence of the parameter ε0 on the field penetration law, that is, the effect of the distance between the external field sources and the body surface in comparison with the penetration depth (at μ = 1). The parameter ε0 also characterizes the field non-uniformity, since the closer the current contour is to the surface, the more non-uniform field is at its surface. This is reflected in the influence

3.2 Electromagnetic Field in a Conducting Half-Space—A General Feature …

75

( ( ) ) of the parameter ε0 on the dependences of the functions f 1 δz , ε0 , β and f 2 δz , ε0 , β with respect to coordinate z in (3.4). Let, for example, the sources of the external field are remote at a considerable distance from the surface of conducting body and, accordingly, for all points of the 1. In) this case, in (3.4), due to the presence of the exponential function contour ( ε0  / exp −χ cos β ε0 , the value of the integrands turns out to be insignificant when χ cos β > ε0 . That is, the value of improper integrals (3.4) at small values ε0 is mainly determined by the behavior of the integrand near χ = 0 the lower limit of integration. This means that when integrating in (3.4), the influence of the factor / [ / ( √ )]2 1+ χ μ i will slightly differ from the case when this factor is equal to one. Therefore, if ε0  1, then the decrease in the field from its local value on the surface at the point Q0 will be close to the decrease in the uniform / field. [ / ( √ )]2 If the parameter ε0 is not small the influence of the factor 1 + χ μ i is much more. In this case, the elements of the contour as a source of the external field are located closer to the interface between the media and the decrease of the electromagnetic field will occur according to a different law with larger decrease rate in depth. We also note that when integrating along the contour, it is necessary to take into account that ε0 and β depend on the position of the point M. The parameter ε0 takes the greatest value when the point M is at the smallest distance / / from the interface. A specific example when the penetration depth δ = 2 (ωμμ0 γ ) is comparable to the dimensions of the contour illustrates the general conclusion of threedimensional field decrease. An additional argument for the validity of the conclusion can also be a comparison of the results of calculating the decrease of non-uniform electromagnetic field, performed using the obtained analytical expressions and using the numerical method in the Comsol package [21]. The calculation was performed for the model of electromagnetic system shown in Fig. 1.5 with 3D electromagnetic field. The radius of the circular contour located in a plane perpendicular to flat interface is R = 0.05 m, the minimum distance from the contour to the interface is h0 = 0.02 m, the electrophysical properties of the medium are as follows: μ = 1, γ = 105 1/(Ω m). In contrast to the analytical method, in the numerical calculation, the current contour was selected in the form of a conductor with square cross-section 2r × 2r at r = 0.004 m. In the numerical calculation, the problem was solved in a limited area, the dimensions of which significantly exceed the contour radius. Different values of the field penetration depth and, accordingly, /√ 2h 0 are obtained by choosing the field the values of the ratio δ/R or εm = μδ frequency. The results of calculating the electric and magnetic field intensities are shown in Fig. 3.2a,c. In the upper group of figures, arrows show the distribution of the vectors of induced current density ji = γEi whose vertical component is equal to zero. The curves in the figures below show the change with depth for component of the amplitude of the electric field intensity, normalized to the value of the field at the surface

76

3 Penetration of Non-uniform Sinusoidal Electromagnetic Field …

| |/ | | E x∗ = | E˙ i x | | E˙ i x (z = 0)|. The coordinate value in conducting medium is normalized to the value of the field penetration depth. The dotted lines show the decrease of the uniform field. Solid lines correspond to analytical calculations. Individual points marked with squares correspond to the results of numerical calculations. The bottom row of figures shows dependences for different components of the magnetic field intensity, also normalized to the amplitude values of the| corresponding field | compo|/ | nents at the surface of the conducting medium Hk∗ = | H˙ ik | | H˙ ik (z = 0)|, where k = x, y, z. Note, in contrast to the electric field, the vertical component of the magnetic field intensity in conducting half-space in this case of a three-dimensional field is not equal to zero. It is seen that with a decrease in the penetration depth δ in comparison with the radius of the contour R or with the distance h0 , the penetration law both electric and magnetic fields approaches the slowest decrease of uniform field. Immediately below the contour at x = 0, y = 0, where the contour section most closely approaches to the interface, decrease is more pronounced than at x = R. This is explained by the fact that at x = R, the contour sections are at greater distance from the surface, and therefore the non-uniformity of the external field distribution near the surface is less than in the case when x = 0. Dashed curve in Fig. 3.2b for vertical component of magnetic field has a conditional meaning, since in the approximate model of the diffusion of uniform field the component of the magnetic field intensity normal to the surface is equal to zero. However, for diffusion of a three-dimensional non-uniform field, this component is nonzero. For the considered system on the plane x = 0, the field component H˙ x is equal to zero, and therefore in Fig. 3.2c the corresponding curve is missing.

3.3 Penetration of Non-uniform Electromagnetic Field in the Case of Strong Skin Effect The general feature of faster decay of non-uniform field in comparison with uniform one is the basis for analyzing the decrease of the field, when introduced parameter is small value ε < 1, not necessarily going to zero. This parameter depends on the position of the source point M on the contour. This section deals with arbitrary electromagnetic systems for which the maximum value εm = max(ε) of all ε is a small parameter.

3.3 Penetration of Non-uniform Electromagnetic Field in the Case of Strong …

77

3.3.1 Comparison of Decay of Non-uniform and Uniform Fields To confirm that the penetration law of non-uniform field is approached the exponential decrease of uniform field, let us compare f 1 (z/δ,ε0 ,β) and f 2 (z/δ,ε0 ,β) / the[functions / ( √ )]2 in (3.4), taking into account the factor 1 + χ μ i in the exponent with the same functions f 10 (z/δ,ε0 ,β) and f 20 (z/δ,ε0 ,β), but provided that the factor is taken to be equal to one, which corresponds to decrease of uniform field. For the electric field intensity, these functions are related to the directions of the initial current parallel and perpendicular to the interface. For the magnetic field intensity, the corresponding functions that follow from (3.2) can be similarly considered. As shown in Chap. 2, sections with different directions of the current involve in different ways in the creation of the tangential component of the electric field intensity at the body surface. The horizontal component of the current gives the largest value of integrand in (3.1) just below the current element. On the contrary, the largest value from the vertical component of the current is realized at a certain distance from the point M 0 in the radial direction at a distance ρ approximately equal to the height at which the contour element is located. Both field components decrease depending on the depth of penetration into the conducting body. The curves in Fig. 3.3a show values of the modules of functions dependents on the depth for direction of the current parallel to the interface at μ = 1: solid lines correspond to the function |f 1 (z/δ,ε0 ,β|, dashed lines correspond to the function |f 10 (z/δ,ε0 ,β|. The results are given for the case β = 0 where the function |f 1 (0,ε0 ,β| at the surface takes the largest values. The curves for different values of the small parameter ε0 are obtained by choosing the corresponding values of the height h above the surface on which the contour element is located. The comparison confirms the statement about the insignificant influence of the functional dependence of the integration variable in the exponential function ε0 . The quantitative / for[small / ( √ )]2 values of the deviation that arise when the factor 1 + χ μ i is replaced by one are shown in Fig. 3.3b in the form of a relative deviation value Δ1 = ||f 1 | − |f 10 ||/|f 1 |. Similar results are also valid for the term of the integrand in the contour integral (3.1) related to the vertical direction of the current. Relative values of functions f 2 (z/ δ,ε0 ,β), f 20 (z/δ,ε0 ,β) and the values of their deviation Δ2 = ||f 2 | − |f 20 ||/|f 2 | are shown in Fig. 3.4. In this case, the observation point is selected near the maximum value of the function |f 2 (0,ε0 ,β)| at the interface at ρ = h, (β = π /4). From the presented calculations, it can be seen that with decrease in the value of the small parameter ε0 , the error from replacing the factor in the exponential by one rapidly decreases, approximately inversely proportional to the ε02 . Similar results turn out to be valid for the magnetic field intensity. The following conclusion can be made from this. With a strong skin effect, when the maximum value of the introduced parameter for contour points ε0 = εm is small,

78

3 Penetration of Non-uniform Sinusoidal Electromagnetic Field …

Fig. 3.3 Comparison of the decrease of non-uniform and uniform fields for functions f 1 and f 10 corresponding to the direction of the current parallel to the interface between media for the small parameter ε0 < 1 and μ = 1

Fig. 3.4 Comparison of the decrease of non-uniform and uniform fields for functions f 2 and f 20 corresponding to the direction of the current perpendicular to the interface for the small value of the parameter ε0 < 1 and μ = 1

(a)

(b)

the electromagnetic field decrease from the local value on the surface, approximately according to the penetration law of uniform field. Since the penetration law depends on the value of the parameter εm , the concept of strong skin effect can be extended from the point of view of the possibility to use the penetration law of uniform field. The skin effect can be considered as strong when the product of the relative magnetic permeability and the penetration depth μδ is small not only with respect to the characteristic dimensions of conducting body, but also of the entire electromagnetic system, including the distance from the surface of body to the external sources. ˙ ˙ ˙ Taking into / account the boundary conditions (z = 0 : Ee|| = Ei|| , He|| = ˙ i⊥ ) and the expressions for the field intensities in the dielectric ˙ e⊥ μ = H ˙ i|| , H H half-space (1.35), (1.44), the approximate expressions in the conducting half-space take the following form ∮ ] [ μ0 I˙0 ez × t1 × ∇G e |z=0 dl; E˙ i ≈ e pz E˙ i (z = 0) = e pz E˙ e|| (z = 0) = e pz i ω 4π l

(3.6) [ / ] ˙ e|| (z = 0) + H ˙ i ≈ e pz H ˙ i (z = 0) = e pz H ˙ e⊥ (z = 0) μ , H

(3.7)

3.3 Penetration of Non-uniform Electromagnetic Field in the Case of Strong …

79

where, using the designation in (1.35), the components of the magnetic field intensities at the dielectric surface are as follows [ ] ˙ 0 (z = 0) + B˙ 1 (z = 0) ˙ e|| (z = 0) = 1 B H μ0 )⎫| ( ∮ ⎧ [ ] ∂2Ge ∂G e || I˙0 t|| × ez − t⊥ × ∇ + | dl; 4π ∂ z2 ∂z z=0 l ) ⎫| ⎧ ( ˙ ∮ | ∂2Ge ∂ Ge ˙ e⊥ (z = 0) = I0 − t|| × ∇ ez || dl. H 2 4π ∂z ∂z z=0

(3.8)

l

The expressions (3.6)–(3.8) presented as two factors are approximate only in relation to the dependence on the coordinate z. On the surface at z = 0, they take into account the non-uniformity of the electromagnetic field and give the values of the field intensities without restrictions on the value of the parameter ε0 . The next two questions are related to the introduced extended concept of the strong skin effect. First, what is the difference at the interface the intensities of the non˙ i (z = 0) in (3.6)–(3.8) from the values of the tangent uniform field E˙ i (z = 0) and H ˙ ˙ components Eτ and Hτ for the model of the perfect skin effect. Second, what is the error replacing the penetration law of non-uniform field with the penetration law of uniform one, depending on the value of the small parameter.

3.3.2 Non-uniform Electromagnetic Field at the Interface Between Media At points on the surface (z = 0), general expressions for electromagnetic field are simplified. An even greater simplification, in particular using asymptotic expansion, can be obtained in the case of strong skin effect, when the external field sources are far from the surface of the conducting body. Moreover, the final results allow us to draw conclusions of some general nature. Expressions (2.34), (2.32) for the intensities of the electric and magnetic fields of external source in the form of a single contour with current in arbitrary point of dialectic/ area under the condition of strong skin effect, when the parameter ) (√are valid ε1 = μδ 2r1 is small ε1 < 1. For clarity we repeat these expressions ) ( )n (n) ∮ ( ∮ N I˙0 ∑ ∂ t1 × r 1 t t1 μ μ0 I˙0 − ς dl + (−1)n 2an (μ) e × dl; E˙ e = −i ω z 4π r r1 4π p ∂z n r13 n=0 l l ˙e H ⎤ ⎡ ( ) ( )n+1 (n+1) ∮ ∮ N ∑ t×r I˙0 ⎣ t1 × r 1 ∂ t1 × r 1 ⎦ μ n dl − =− − (−1) 2an (μ) dl . 4π p r3 ∂z n+1 r13 r13 n=0 l

l

(3.9)

(3.10)

80

3 Penetration of Non-uniform Sinusoidal Electromagnetic Field …

/√ / √ Here ε1r1 i = μ p, where p = i ωμμ0 γ is propagation constant, r1 is vector from the arbitrary observation point in the dielectric/half-space to the source point on the mirror reflected contour (Fig. 2.2), ς = p γ is surface impedance. The distribution of the electromagnetic field (3.9), (3.10) on the surface (ε1 = ε0 ) is not limited to members of the asymptotic series, for which the law of penetration of uniform field is valid, which is often used when constructing mathematical models. First, consider the electric field intensity. On the interface in the dielectric halfspace, the first term in (3.9) has only the normal component, which is completely determined by the induced electric field of external sources. On the contrary, the second term has only the tangential component E˙ || (z = 0). In turn, the terms of asymptotic series E˙ || (z = 0) in (3.9) contain derivatives of the expression for the magnetic field intensity created by the current contour mirrored from the interface. We transform the expression in such a way as to have the magnetic field intensity of the initial current flowing in dielectric medium. Taking into account the relations between the values for the initial and mirror reflected contours t = t|| + t⊥ , t1 = t|| − t⊥ , r(z = 0) = ρ + z M ez , r1 (z = 0) = ρ − z M ez (Fig. 2.2) we have (−1)n

⎧ ⎨ ∂ (n) ⎩ ∂ zn

∮ ez × l

⎫| | t1 × r 1 ⎬|| dl | ⎭| r13

=−

⎧ ⎨ ∂ (n) ⎩ ∂z n

∮ ez ×

z=0

l

⎫| | t × r ⎬|| dl | ⎭| r3

.

(3.11)

z=0

Note that the expression, which is contained in the individual terms of the asymptotic expansion, ∮ t×r I˙0 ˙ H0 = − dl 4π r3

(3.12)

l

represents the magnetic field intensity of the initial current contour. The tangential component of the field to the surface is the following ] [ ˙ 0|| = −ez × ez × H ˙0 . H

(3.13)

˙0 = Substituting expressions (3.11) and (3.12) into (3.9), taking into account ez ×H ˙ ez × H0|| the tangential component of the electric field intensity at the interface between media can be written by expressing it through the tangent component of the known field of external sources and derivatives of the field with respect to the coordinate z. ( )n ⎧ (n) ⎫| | μ ∂ ˙ 2an (μ) ez × H0|| || . E(z = 0) = E =ς n p ∂z || || n z=0 n=0 n=0 .

N ∑

.

N ∑

(3.14)

3.3 Penetration of Non-uniform Electromagnetic Field in the Case of Strong …

81

Based on the principle of superposition, expression (3.14) will also be valid in ˙ 0 created by a system of sources, with the the case of an external magnetic field H restriction associated with their distance from the interface. ˙e = H ˙ e1 + H ˙ e2 (3.10) at the interface Similarly, the magnetic field intensity H can be represented as a limited sum, including the external magnetic field and its derivatives with respect to the coordinate z. ˙ e1 gives the magnetic field in the dielectric half-space at perfect The first term H ˙ e1 has a zero value of the normal skin effect, when δ → 0 and accordingly H component of the magnetic field intensity at the interface ⎤ ⎡ ˙0 ∮ ( t × r t1 × r 1 ) I ˙ e1 (z = 0) = − ⎣ dl ⎦ H − 4π r3 r13 l

∮ I˙0 ez × (t × r) ˙ 0|| (z = 0). = ez × dl = 2H 2π r3

(3.15)

l

˙ e2 depends on the electrophysical properties of the medium The second term H and the field frequency. Here, each term of the series contains as a multiplier the derivative of the corresponding order with respect to the vertical coordinate from the magnetic field intensity of the mirror-reflected contour with current. In contrast to ˙ ˙ ˙ has non-zero both tangential the electric field ) the term He2 = H(2|| + H2⊥ ) ( intensity, ˙ e2 and normal H ˙ e2 ez components to the interface. ˙ 2|| = −ez × ez × H ˙ 2⊥ = ez · H H ( )n+1 N ˙ ∑ μ P 2|| (z = 0) = − I0 (−1)n 2an (μ) ez H 4π n=0 p ⎧ ⎫| ⎨ ∂ (n+1) ∮ e × (t × r ) ⎬|| z 1 1 × dl || ⎩ ∂ z n+1 ⎭| r13

;

l

z=0 ⎫| ⎧ ( ) N n+1 ⎨ ∂ (n+1) ∮ e · (t × r ) ⎬|| ∑ ˙0 μ I z 1 1 P e⊥ (z = 0) = − (−1)n 2an (μ) ez dl || H ⎭| ⎩ ∂z n+1 4π n=0 p r13 l

z=0

(3.16) ˙ e2 in such a way that instead of the magnetic field created by the We rewrite H current of the mirror reflected contour, we have the magnetic field intensity of the initial current contour. As before, we take into account the relations between the quantities characterizing the points of the initial and mirror-reflected contours. As a result, the expression for the tangential component of the magnetic field intensity takes the form

82

3 Penetration of Non-uniform Sinusoidal Electromagnetic Field …

˙ || (z = 0) = H

N +1 ∑ n=0

˙ || n H

( )n ⎧ (n) ˙ ⎫|| μ ∂ H0|| | =− 2an−1 (μ) | | p ∂ zn n=0 N +1 ∑

.

(3.17)

z=0

˙ e1 (z = 0) (15) and H ˙ 2|| (z = 0) are combined. The numbers n Here, the terms H of the asymptotic series terms are replaced by n + 1; accepted a−1 = −1. In the designation, the index of belonging to the dielectric half-space is omitted, since the tangential components of the magnetic field intensity are continuous at the interface between the media. Let us now consider the component of the magnetic field intensity perpendicular to the interface. For a magnetizable conducting medium, this component has a discontinuity on the surface—the normal component of the magnetic field induction ˙ e⊥ = μμ0 H ˙ i⊥ . is continuous μ0 H When transforming, it is necessary to take into account that the expressions in curly ˙ e⊥ differ in sign from similar transformations brackets (3.16) for the component H ˙ (3.11) for the component H2|| ⎫| ⎧ ⎨ ∂ (n+1) ∮ e · (t × r ) ⎬|| z 1 1 dl || (−1)n ⎭| ⎩ ∂z n+1 r13 l

z=0

⎧ ⎫| ⎨ ∂ (n+1) ∮ e · (t × r) ⎬|| z =− dl || ⎩ ∂ z n+1 ⎭| r3 l

. z=0

(3.18) As a result the expression for the normal component of the magnetic field intensity on the surface in the dielectric half-space takes the form ˙ e⊥ (z = 0) = H

N ∑ n=0

˙⊥n = H

( )n+1 ⎧ (n+1) ˙ ⎫|| μ ∂ H0⊥ | 2an (μ) | n+1 | p ∂z n=0

N ∑

.

(3.19)

z=0

So, the expressions (3.14), (3.17) and (3.19) allow, without solving additional equations, to directly find the electromagnetic field at the interface, having only the known distribution of the field of external sources at the boundary. The presence of derivatives indicates that the electromagnetic field on the surface is determined not only by the value of the field of external sources, it also depends on the nonuniformity of the external field. This feature is related to the dependence of the distribution law of the induced field and currents in the conducting medium on the degree of remoteness of the external field sources compared to the field penetration depth. The analysis of the calculation errors of the sinusoidal field using asymptotic method at interface was studied in sufficient detail in Chap. 2. This area was chosen due to the fact that the parameter ε1 = ε0 takes the largest value at the interface between the media and, accordingly, the calculation error is the largest here.

3.3 Penetration of Non-uniform Electromagnetic Field in the Case of Strong …

83

3.3.3 Influence of the Small Parameter Value to the Field Penetration Law with the Strong Skin Effect 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 expressions (3.4) in the asymptotic series, where for small ε0 the Taylor series expansion of the factor in the integrand is used near the zero value of the integration variable χ. Unlike the function G e (2.23), for the asymptotic expansion of improper integrals (3.4) it is necessary to use not only the expansion in a power ( series ) of the function w1−1 (χ ), but also the expansion of the exponential function K δz , χ (3.5). Taking into / [ / ( √ )]2 account, except one, next term in the expansion of the factor 1 + χ μ i , approximate expression for exponential function (3.5) will be as follows ⎛ ⎞ [ / )2 ( )2 ] ( (√ z ) √ z √ z χ 1 χ ⎠ ≈ 1 + 2i · exp⎝ 2i 1+ 2i , (3.20) ex p √ √ δ δ 2μ2 δ μ i i where it is considered that the ratio z/δ does not exceed several units. Taking into account (3.20), the functions f 1 (z/δ,ε0 ,β) and f 2 (z/δ,ε0 ,β) in (3.4) can be approximately represented as following (below we use the combined designation f 1,2 (z/δ,ε0 ,β) for the two functions) f 1,2

(z δ

)

, ε0 , β ≈ ex p

(√

] [ √ z 1 z) f 1,2 (0, ε0 , β) + 2i · 2i k1,2 (0,ε0 , β) , δ δ 2μ2 (3.21)

where k 1,2 (0,ε0 ,β) differ from f 1,2 (0,ε0 ,β) by the presence of additional factor ( / √ )2 χ i in the integrands (3.4). ( / √ )n ∞ / ∑ After substitution 1 w1 = an (μ) χ i , the functions k 1,2 (0,ε0 ,β) and n=0

f 1,2 (0,ε0 ,β) can be represented as expansion in asymptotic series, similarly Ge for z = 0 in (2.29), (2.30). ⎧ N ⎪ √ ∑ ⎪ ⎪ f 1 (0, ε0 , β) = i an f 1, n , ⎪ ⎪ ⎨ n=0

N ⎪ √ ∑ ⎪ ⎪ ⎪ an f 1, n+2 , ⎪ ⎩ k1 (0,ε0 , β) = i n=0

where

f 2 (0, ε0 , β) =

N √ ∑ i an f 2, n ; n=0

N √ ∑ k2 (0, ε0 , β) = i an f 2, n+2 , n=0

(3.22)

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3 Penetration of Non-uniform Sinusoidal Electromagnetic Field …

f 1, n

) ( ) ( ∫∞ ( / √ )n+1 χ sin β χ cos β J0 dχ ; χ = i exp − ε0 ε0

f 2, n

) ( ) ( ∫∞ ( / √ )n+1 χ sin β χ cos β J1 dχ . χ = i exp − ε0 ε0

0

(3.23)

0

To obtain the final expressions, it is sufficient to use expressions (3.14), (3.17) and (3.19) of the expansion into asymptotic series of the electric and magnetic intensities at the interface. In this case, for the additional term containing k 1,2 (0,ε0 ,β), the same expressions will be valid, in which the values of the degree of functions and derivatives change from n to n + 2. Besides, since in (3.20) only one additional term of the series is taken into account, the functions k 1,2 (0,ε0 ,β) must also contain only one term of the expansion. With the same exactness the functions f 1,2 (0,ε0 ,β) can contain no more than three terms of the series. As a result, using the value of the field intensity at the interface (3.14), the expression in which the factor in the exponential function differences from one will be as following P i ≈ 2e pz ς ez E | ⎧ ) (2) P || ( )2 ( P 0|| || ∂ H0|| | 1 ∂ H μ μ P 0|| − 1− × H | + | 2 | p ∂z p 2μ ∂ z2 | z=0 z=0 | ⎫ ( )2 (2) P | pz ∂ H0|| | μ + | p 2μ2 ∂z 2 |

(3.24)

z=0

Similarly, using the values of the components of the magnetic field intensities at the boundary (3.17), (3.19) and the expansion of the exponential function (3.20), we can also write approximate expressions for the decrease of the non-uniform magnetic field in the conducting half-space. ˙ i|| H

| ⎧ ˙ 0|| || ∂H ˙ 0|| − μ ≈ 2e pz H | p ∂z |

z=0

) (2) ˙ || ( )2 ( ∂ H0|| | 1 μ 1− | p 2μ2 ∂z 2 |

+ z=0

| ( )2 ˙ 0|| || pz ∂ (2) H μ | p 2μ2 ∂z 2 |

⎫ ; z=0

(3.25) | ) (3) P || ( )2 ( P 0⊥ || ∂ H0⊥ | 1 2 pz μ ∂ (2) H μ 1− ≈− e − | + | 2 2 p p ∂z | p 2μ ∂z 3 | z=0 z=0 z=0 | ⎫ ( )2 | (3) P pz ∂ H0⊥ | μ + . (3.26) | p 2μ2 ∂z 3 | ⎧

P i⊥ H

+

| P 0⊥ || ∂H | ∂z |

z=0

˙ i || , directed parallel For the components of the electromagnetic field E˙ i || = E˙ i and H to the interface between the media, the deviation from the penetration law of uniform

3.4 Impedance Boundary Condition of Non-uniform Electromagnetic Field …

85

field takes place for the terms of series proportional to the second-order derivative of the field intensities at the surface. The deviation for the component of the magnetic ˙ i ⊥ occurs for the term of series proporfield intensity perpendicular to the surface H tional to the first-order derivative. The next term is proportional to the third-order derivative of normal component of the external magnetic field. This is due to the absence of local value of the field at the surface in (3.19) which already contains a common factor proportional to the value ε0 . As follows from (3.24), (3.25), the deviation of the penetration law of non-uniform electromagnetic field in conducting medium from the penetration law of uniform one appears when the small parameter is taken into account in the second power ε02 . For normal component of magnetic field the term of series (3.26) proportional to the third-order derivative is also proportional ε02 . This conclusion is in agreement with the calculation results shown in Figs. 3.3b and 3.4b. In addition, it follows from takes (3.24)–(3.26) that the maximum value of the modulus of the |(additional / ) term ( / )|| | √ place at the maximum value of the function | pz exp( pz)| = | 2z δ exp z δ |, which is realized at –z = δ. This value also agrees well with the ratio z/δ in Figs. 3.3b and 3.4b when the deviation reaches its maximum value. As can be seen from (3.24)–(3.26), for all components of the electromagnetic field with strong skin effect, the first additional term of asymptotic series is determined by the value of the same parameter ( )2 ( )2 μ pz ε z . ∼ 2 p 2μ μ δ

(3.27)

Here, ε is estimated by the minimum distance from the contour points to the interface between the media and, accordingly, it has the maximum value of all ε0 . The estimate (3.27) takes into account only the difference between the field penetration laws. The total relative error associated with the use of the model of perfect skin effect will be much more, since this model also does not take into account the field non-uniformity at the interface.

3.4 Impedance Boundary Condition of Non-uniform Electromagnetic Field Penetration into Conducting Half-Space In the case of a strong skin effect the expressions found for the electric and magnetic field intensities at the interface between media, which take into account the finite value of the parameter ε0 , make it possible to extend the concept of the impedance boundary condition to the penetration of non-uniform field into conducting halfspace. The impedance boundary condition establishes the relation between the tangential components of the electric and magnetic field intensities at the interface. We will

86

3 Penetration of Non-uniform Sinusoidal Electromagnetic Field …

not impose a limiting condition for the uniformity of the field at the boundary, which is usually used in mathematical models. Moreover, provided that the value of the small parameter ε0 does not necessarily tend to zero, the aim is to establish a relationship not only between the values of the field intensities at the boundary, but also between the derivatives with respect to the vertical coordinate of different orders of the external field sources. In addition, a comparison is made with calculations using exact expressions and the permissible limits of small parameter are established up to which the generalized impedance boundary condition is valid [21–23]. Let us establish series of electric ∑ N the relationship for each term of the∑asymptotic ˙ || (z = 0) = N H ˙ || n fields from (3.14) E˙ || (z = 0) = n=0 E˙ || n and magnetic H n=0 the( series term n simultaneously indicates the and (3.17). Note that the number[ of / )] n √ ]n [ / 2λ = μ (| p|λ) and also the order degree of small parameter εn = μδ of derivatives from external sources field with respect to the vertical coordinate, λ is common characteristic size, for example, the smallest distance between the observation point on interface and the source points. Comparison of the series terms (3.14) and (3.17) with the same number n shows that they are interconnected by the following relation ˙ || n . an−1 (μ)E˙ || n = −an (μ)ς ez × H

(3.28)

/ The first several expansion coefficients an in the power series of the function 1 w1/ defined ) in (3.6) have the values a−1 = −1, a0 = 1, a1 = −1, a2 = ( 1 − 1 2μ2 , .... Given these values for the first three terms of the series, we obtain ( ) n = 0 ε0 : ( ) n = 1 ε1 : ( ) n = 2 ε2 :

˙ || 0 ; E˙ || 0 = ς ez × H ˙ || 1 ; E˙ || 1 = ς ez × H ) ( 1 ˙ || 2 , · · · · · · ς ez × H E˙ || 2 = 1 − 2μ2

(3.29)

It can be seen that the Leontovych’s approximate impedance boundary condition (2.3) ) is valid up to the first two terms of the asymptotic series. Starting from n = ( [25] 2 ε2 , the ratio changes. Nevertheless, relationship between the terms of the series for the tangent components of the field intensities at the interface still exists. In this regard, it can be argued that expression (3.28) generalizes the impedance boundary condition to wider domain of non-uniformity electromagnetic field diffusing into the conducting body. The generalized impedance boundary condition, which connects the field intensities on the interface z = 0 and is not limited to the first two terms of the series, can be also represented as follows ˙ || = ς T˙ = E˙ || − ς H

( )n ⎧ ⎫| (n) | ε n∂ ˙ λ 2(an + an−1 ) √ e × H0|| || . n z ∂z i z=0 n=2

N ∑

(3.30)

3.4 Impedance Boundary Condition of Non-uniform Electromagnetic Field …

87

Start the summation from n = 2 indicates, that terms for n = 0 and n = 1 are equal to zero. As follows from (3.30), the first nonzero term of the series of the functions T˙ is proportional to ε2 , and the argument of complex-value amplitude T˙ is equal to / −π 4. ) 2 ⎧ ⎫| ( (2) | ε 1 n∂ ˙ 0|| | . λ T˙ ≈ 2|ς | 1 − e × H √ z | 2 2 2μ ∂z i z=0

(3.31)

The function T˙ introduces only for tangential components of field intensities. However, the generalized impedance boundary condition also implies the presence of nonzero normal component of the magnetic field intensity. The presence of exact and approximate solutions of the problem allows not only to determine the field intensities for specific current contour configuration, and also to investigate the features of the generalized impedance boundary condition: calculate ˙ compare the results of exact and approximate approaches and analyze the function T, the influence of the parameter ε to the value of the function T˙ for different components of the electromagnetic field. Earlier in Chap. 2 the results of exact and approximate calculations of threedimensional electromagnetic field in the dielectric area were presented in the case when the external electromagnetic field is created by a circular contour with current I˙0 whose plane is perpendicular to the boundary surface of the conducting body. Now we present similar results for the same model of the electromagnetic system (Fig. 1.5) with the same values of the electrical properties of the conducting medium. The results of calculations for the function T˙ and field/intensities ) are presented (√ 2rmin , which corredepending on the maximum value of the parameter ε = δ sponds to the minimum distance rmin between the contour and the observation point on the surface of the| conductive body. Components of complex-value amplitudes | of a function T˙k∗ = |T˙k | exp(i φk ), where k = x, y, were investigated for modules and arguments of the function. for the modules are presented in the form | results | |/The | of relative values Tk∗E = |T˙k | | E˙ k |. This normalization allows, depending on the parameter ε to determine the relative effect of the non-uniformity of the electromag˙ which for the model of the uniform field netic field on the value of the function T, penetration is equal to zero. Figures 3.5, 3.6 and 3.7 show the values of different components of the function T˙ and the intensities of the electromagnetic field at different points on the surface, depending on parameter ε. In all figures, the solid curves correspond to the calculation by exact analytical expressions, the dotted curves correspond to the calculation by the approximate asymptotic method. At the point x = 0, y = 0 where the distance r0. = rmin from the contour to the surface is the smallest, the electric field intensity E = E˙ x ex and, accordingly, the function T˙ = T˙x ex are directed along the axis x. At this point, the intensities of electric E˙ x and magnetic fields H˙ y on the surface are largest. The values of Tx∗E in dependence on the small parameter ε at this point is shown in Fig. 3.5a. A characteristic feature is

88

3 Penetration of Non-uniform Sinusoidal Electromagnetic Field …

˙ of generalized impedance boundary condition: a—modules of relative T ∗ = T Fig. kE | Functions | | |/3.5 |T˙ | | E˙ | complex-value amplitudes; b—argument of the component T˙x k k

Fig. 3.6 Modules of the normalized electromagnetic field intensities as function of small parameter ε: a—components of electric field intensity; b—components of magnetic field intensity

| | Fig. 3.7| Modules of the normalized components of the function |T˙k |—(a) and its relative values | | | / Tk∗E = |T˙k | | E˙ k |—(b)

the approximately quadratic dependence on ε, which confirms the conclusion for the generalized impedance boundary condition with strong skin effect. The influence of the higher terms of the series to the value of the argument of the function T˙x is more significant. This can be seen from Fig. 3.5b for the same point on the surface.

3.4 Impedance Boundary Condition of Non-uniform Electromagnetic Field …

89

If the observation points move away from the point closest to the contour on the surface, the electromagnetic field intensity decreases. In addition, at points on the surface that do not coincide with the axis x, all components of the electromagnetic field intensity are nonzero. The values of the field components can differ significantly from each other. This is illustrated in Fig. 3.6, which shows the values of the tangent components of the electric and magnetic field intensities. In the figure, the magnitude ˙ μ | I˙ |ω ˙ = | I0 | H ˙ ∗. of the intensities are normalized as follows E˙ = 0 4π0 E˙ ∗ , H 4π h 0 Despite the significant difference in the values of the components of the field, the components of the function T˙ that characterizes the generalized impedance condition for non-uniform field differ much less. As can be seen from Fig. 3.7a, the absolute values of the various tangent components of the function T˙ have values of the same order. The figure shows the normalized values, which are determined similarly to the μ | I˙ |ω electric field intensity T˙ = 0 4π0 T˙ ∗ . In this case of the specific model of the electromagnetic system, at the point of the surface we have relations | E˙ x | > | E˙ y |, | H˙ y | > | H˙ x | for the components of the field intensities. However, for the components of the function T˙ we have the opposite inequality |T˙x | < |T˙y |. This feature is appropriately reflected in the relative value of the components of the function T˙ (Fig. 3.7b)—larger values of |the |components Tk∗E occur for smaller value components of the electric field intensity | E˙ k∗ |. In all cases, the possibility of applying for non-uniform electromagnetic field the simplified model of penetration into conducting half-space of uniform electromagnetic field is performed only for fields that are characterized by very small value parameter ε. This implies the need to take into account the non-uniformity of the field in the development of methods for calculating electromagnetic fields using the impedance boundary condition. Comparison of the results obtained by analytical exact and approximate methods allows us to make conclusion about the allowable values of the parameters at which the use of simpler asymptotic approximation is acceptable. The error of the results of the electromagnetic field / calculation method depends on the / approximate )by the (√ / 2 2rmin = μ ωμ0 γ rmin that combines the eleccomplex parameter ε = μδ trophysical properties of the medium, field frequency and the minimum distance from field sources to the surface of the conductive body. As can be seen from Fig. 3.6, calculations of field intensities can be performed when the small parameter does not exceed the value ε = 0.3. In Figs. 3.5 and 3.7 the results of calculations of the ˙ of the generalized impedance condition are also presented depending on function T the same complex parameter ε. It can be seen that sufficient accuracy is provided here for a slightly smaller range of values of the small parameter ε ≤ 0.2. This is due to the fact that the approximate expressions of the function do not take into account the first two terms of the asymptotic series, whose contribution to the value of field intensities is the largest.

90

3 Penetration of Non-uniform Sinusoidal Electromagnetic Field …

3.5 Conclusions The exact analytical solution of the three-dimensional problem of quasi-stationary electromagnetic field in the system “current contour of arbitrary configuration— conducting half-space” allows to obtain some general substantiated consequences of the field formation. These consequences are as follows. 1. It has been established that non-uniform electromagnetic field, upon penetration into conducting half-space, decreases in depth always faster than uniform field. Quantitative characteristic of the field decrease rate can be considered the parameter proportional to the ratio of the penetration depth of uniform field to the distance from external sources to the interface between dialectic and conducting media. With decrease in this parameter, the field is decreased slower, tending to the slowest decrease of uniform electromagnetic field, when the quantitative parameter tends to zero. 2. From the point of view of the possibility of using the penetration law of uniform field the concept of strong skin effect can be extended. The skin effect can be considered strong when the penetration depth is small not only with respect to the characteristic dimensions of conducting body, but also of the entire electromagnetic system, including the distance from the surface of body to the external sources. 3. In the case of strong skin effect in its extended interpretation the non-uniformity of the electromagnetic field affects both the values of the field intensities at the interface between the media and the field penetration law into conducting body. The effect of field non-uniformity at the boundary surface is pronounced in the fact that the electric and magnetic field intensities, in addition to local values, contain derivatives of the external sources field with respect to the coordinate perpendicular to the interface. 4. The found analytical expressions for the field intensities in the form of asymptotic series make it possible to generalize the Leontovych’s impedance boundary condition to the diffusion of non-uniform field into conducting half-space. The mathematical model of the uniform field penetration into conducting medium to study the penetration of the non-uniform electromagnetic field is valid up to the introduced small parameter in the first degree. The same limitation is valid when using the Leontovych’s approximate impedance boundary condition. 5. The expressions found in the asymptotic approximation generalize the Leontovych’s impedance boundary condition to the case when non-uniform electromagnetic field penetrates into the conducting medium, which is formulated for each term of asymptotic series. The analysis of the function introduced as generalization of the impedance boundary condition to the case of non-uniform electromagnetic fields confirmed the conclusion that the approximate Leontovych’s impedance condition is valid only for the first two terms of the asymptotic series expansion. Comparison of the results obtained by exact and approximate methods allowed to find the value of small parameter in which the use of simpler asymptotic approximation is acceptable and to conclude that its allowable limit value

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for introduced function is slightly smaller than the same value for fields on the surface of the conducting half-space.

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Chapter 4

Three-Dimensional Pulsed Electromagnetic Field of Current Flowing Near Conducting Half-Space

Abstract In this chapter the study of pulsed three-dimensional electromagnetic fields is based on the obtained accurate analytical solution to the problem of the field of arbitrary alternating initial sources near conducting half-space with account to eddy current in conducting body. For the pulse current, the obtained solution is a frequency spectrum of the electromagnetic field created by a current with a set frequency spectrum. Time dependencies can be obtained by performing the inverse Fourier transform. In this case, the solution is represented by triple improper integrals, that, despite of the analytical form of the expressions, has some calculation difficulties. To simplify the computational procedures under the condition of strong skin effect asymptotic calculation method is developed, which takes into account the features of the pulsed process. For the calculation of three-dimensional pulsed electromagnetic fields the integrands are represented by bounded asymptotic series, in each term of which the dependences on coordinates and time are calculated separately using well-known simple expressions. Since the applied expansion into the asymptotic series is valid in the high-frequency interval of the spectrum, then for the terms of the series, the lower boundaries of the frequency spectrum and, accordingly, the maximum values of the time intervals from the pulse start are determined. Based on comparison of the results of exact and approximate calculations for nonuniform field at the interface between the media the proposed choice of the limited time interval for calculating electromagnetic field using the asymptotic method is justified. As an example, some results of the application of the developed calculation methods in the field of high-density pulsed current technology to change the mechanical properties and control the stress–strain state of metal products are given. Keywords Three-dimensional pulsed electromagnetic field · Exact analytical solution · Asymptotic expansion method · High-density pulsed current technology

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Y. Vasetsky and A. Zaporozhets, Electromagnetic Field Near Conducting Half-Space, Lecture Notes in Electrical Engineering 1070, https://doi.org/10.1007/978-3-031-38423-3_4

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4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

4.1 Introduction Attention to the study of pulsed electromagnetic fields is due to the need to take into account the influence of geometric and electrophysical factors on the course of processes, in the device elements of which a strong skin effect is appeared. It is enough to point out, for example, the technology of high-density pulsed currents to change the mechanical properties and control the stress–strain state of metal products [1–3], devices for high-speed forming technology using pulse magnetic field [4–6]. Here, the search for the geometry of electromagnetic systems and the optimization of their parameters is of particular interest, which is associated with the significant laboriousness of computational procedures. Therefore, an urgent task is the development of methods for solving problems of the electromagnetic field, which provide high accuracy with a moderate amount of necessary calculations. Among them, analytical and numerical-analytical approaches standout. Considering that with a strong skin effect, the current and the field are concentrated in a thin surface layer of the conductor, approximate calculation methods are often used to determine the electromagnetic field [7], primarily using the impedance boundary condition [8, 9]. The most advanced methods are those based on the perturbation method [9, 10]. The developed approaches are used in modeling electrodynamics problems taking into account the geometric and physical properties of real boundary surfaces [11, 12]. A distinctive feature of this chapter is the use of an exact analytical solution for a three-dimensional sinusoidal field of an arbitrary system of external sources located near the conducting half-space [13, 14]. The solution is based on the known analytical solution of the problem for an emitting current dipole near the interface [15, 16]. In the quasi-stationary approximation, a closed contour of arbitrary configuration located in a nonconducting nonmagnetic medium, without loss of generality, was represented by a serial system of dipoles with a constant initial current I˙0 along the contour. The purpose of this chapter is to develop a theory for solving three-dimensional problems of a pulsed electromagnetic field, taking into account eddy currents in conducting half-space. The exact and approximate solution of the problem is presented. The exact solution has no restrictions on the external field configuration, electrophysical properties of the medium, time dependence of external field. The simplified approximate solution is based on an expansion in asymptotic series and is limited by the initial time interval of the pulse current [17].

4.2 General Analytical Solution for a Three-Dimensional Pulsed Electromagnetic Field The found general solution for the complex-value amplitudes of the threedimensional quasi-stationary problem of the electromagnetic field for the system “arbitrary spatial contour with sinusoidal current-conducting half-space” also solves

4.2 General Analytical Solution for a Three-Dimensional Pulsed …

95

the problem for the pulse current in general formulation. In this case considering the obtained expressions for the potentials and field vectors as their frequency spectrum, it is sufficient to apply the inverse Fourier transform to determine the corresponding quantities. If the initial current I˙0 (i ω) is considered as the frequency spectrum of a nonsinusoidal current I0 (t), the expressions (1.34), (1.42), (1.35), (1.44) in dielectric area and (1.50), (1.54), (1.55) in conducting one for potentials and vectors of electromagnetic field give the frequency spectrum of the corresponding characteristics. In dielectric half-space, the potentials in the Lorentz gauge and the field vectors contain terms that depend on frequency in different ways. ˙e = The frequency spectrum of the first two terms of the vector potential (1.34) A ˙1+A ˙ 2 and the magnetic field induction (1.35) B˙ e = r ot A ˙ e = B˙ 0 + B˙ 1 + B˙ 2 ˙0+A A is determined only by the frequency spectrum of the contour current I˙0 (i ω), and therefore for these terms the time dependence repeats the time dependence of the current pulse I0 (t). In the case of an perfect skin effect, when penetration depth equal to zero δ → 0 for all frequencies of the spectrum, the first two terms completely solve the problem of vector potential and magnetic field induction distributions. If the condition of the perfect skin effect is not satisfied, then it is necessary to ˙ 2 and B˙ 2 , which, in addition to the current I˙0 (i ω), are take into account the terms A determined by single function G e (i ω) (2.19) depending on the frequency. In this case, the frequency spectrum of the vector potential and magnetic field induction differ from the frequency spectrum of the current ˙ e (i ω) = A ˙1 +A ˙ 2 = μ0 ˙0 +A A 4π

∮ [( l

) ] t t1 ˙ ∂ V˙ A (i ω) − I0 (i ω) − t1 dl; r r1 ∂z

(4.1)

B˙ e (i ω) = B˙ 0 + B˙ 1 + B˙ 2 ) )] ( ˙ ∮ [( t × r t1 × r1 ˙ μ0 ∂ V A (iω) dl, (4.2) =− − × ∇ − t I (iω) 0 1 4π r3 ∂z r13 l

where V˙ A (i ω) = I˙0 (i ω)G˙ e (i ω).

(4.3)

Using expressions (4.1) and (4.2) for the frequency spectrum, the time dependencies for current pulses can be found by performing the inverse Fourier transform [18] Ae (t) =

μ0 4π

∮ [( l

) ] t t1 ∂ V A (t) I0 (t) − t1 − dl; r r1 ∂z

(4.4)

96

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

Be (t) = −

μ0 4π

∮ [( l

) )] ( t × r t1 × r1 ∂ V A (t) I dl, − × ∇ − t (t) 0 1 r3 ∂z r13

(4.5)

where 2 V A (t) = π

∫∞

[ ] cos(ωt)Re V˙ A (i ω) dω.

(4.6)

0

The expressions for the frequency spectrum of the scalar potential φ˙ e and the electric field intensity E˙ e = E˙ 0 + E˙ 1 + E˙ 2 , , in contrast to (4.1) and (4.2), contain a common factor i ω. For the first two terms of the electric field intensity E˙ 0 + E˙ 1 , the presence of a multiplier i ω is associated with the dependences of these terms on the derivative of the contour current with respect to time. The frequency spectrum of the scalar potential φ˙ e and the electric field intensity term E˙ 2 differs from the frequency spectrum of the time derivative of the current. The expressions for these characteristics of the electromagnetic field have the form φ˙ e = μ0 E˙ e (i ω) = − 4π

∮ [(

μ0 4π

∮

(t1 · ez )V˙φ (i ω)dl;

(4.7)

l

) ] [ ] t t1 ˙ ˙ i ω I0 (i ω) − ez × t1 × ∇ Vφ dl, − r r1

(4.8)

l

where V˙φ (i ω) = iω I˙0 (i ω)G˙ e (i ω).

(4.9)

The time dependences of the scalar potential φe (t) and the term E2 (t) are found as a result of the inverse Fourier transform of the function V˙φ (iω), which, as can be seen, differs from V˙ A (i ω)V˙ A (i ω) ∮ μ0 φe (t) = (4.10) (t1 · ez )Vφ (t)dl; 4π Ee (t) = −

μ0 4π

∮ [( l

where

)

l

] [ ] t t1 ∂ I0 (t) − − ez × t1 × ∇Vφ (t) dl, r r1 ∂t

(4.11)

4.3 Asymptotic Expansion Method for Pulsed Three-Dimensional …

2 Vφ (t) = π

∫∞

[ ] cos(ωt)Re V˙φ (i ω) dω.

97

(4.12)

0

Similarly, using the found solution in the form of complex-value amplitudes, one can also write down general expressions for the potentials and vectors of the electromagnetic field in the conducting half-space. Expressions (4.7), (4.8), (4.10) and (4.11) together with (4.9) and (4.12) in general case provide the analytical expressions in the form of quadrature for the calculation of pulse electromagnetic fields. At the same time, the solution represented by the triple improper integral and, moreover, the presence of the complex-value magnitude of the current frequency spectrum implies an additional integral procedure of the direct Fourier transform. The noted circumstances show that the simplification of calculations is an important task for the efficient use of the given analytical approach. Such simplification can be realized with the use of asymptotic expansion, which is performed for the problem being considered in the case of the strong skin effect in extended formulation.

4.3 Asymptotic Expansion Method for Pulsed Three-Dimensional Quasi-Stationary Field As shown in Chap. 2 of this book, the application for the electromagnetic field the asymptotic series expansion method with respect to the introduced small parameter ε1 = √μδ allows simplifying calculations in the case of a strong skin effect and 2r1 also formulating some general features of electromagnetic field distributions. At the same time, the asymptotic expansion method is approximate research method, therefore, for a pulsed field, it is necessary to analyze the possibility of using it, since the condition for the parameter to be small is not satisfied for the entire frequency spectrum. In this case, exact analytical expressions for the electromagnetic field can serve as a standard for comparison and finding the errors of the approximate method.

4.3.1 Solution for Pulsed Electromagnetic Field in Dielectric Half-Space as Asymptotic Series Let us consider the application of the asymptotic expansion method for the electromagnetic field in dielectric half-space z > 0. Here, the complex-value amplitudes of the potentials and field vectors, or, in other words, their frequency spectra for nonsinusoidal quantities, are entirely determined through the function G e (i ω), which is included in the integrands of the linear contour integrals.

98

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

The potentials and field vectors were represented as the corresponding asymptotic series (2.31)–(2.34) using the expansion of the function G e (i ω) G e (i ω) ≈ G eN =

N ∑

G (n) e1 (i ω)

n=0

N ∑ gn (μ, γ , r1 ) . = (i ω)(n+1)/ 2

(4.13)

n=0

It is essential that each term of the asymptotic series is represented as a product of two functions, one of which depends on the frequency, and the other depends on the properties of the medium and the coordinates of the vector r1 . Here the functions gn are as follows (

μ gn = gn (μ, γ , r1 ) = (−1) 2an (μ) √ μ0 μγ

)n+1

n

( ) ∂ (n) 1 . ∂z n r1

(4.14)

The frequency spectrum of each term of the asymptotic series is determined not only by the frequency spectrum of the function G (n) e1 (i ω), but also by the frequency spectrum of the current I˙0 (i ω). For this reason, in analytical expressions, similarly to (4.6) and (4.12), we introduce the following functions for the terms of the series: • for vector potential and magnetic field induction V˙ An (iω) = I˙0 (i ω)G (n) e1 (i ω),

(4.15)

• for scalar potential and electric field intensity V˙φn (iω) = iω I˙0 (i ω)G (n) e1 (iω).

(4.16)

The task in both cases it is necessary to determine the functions V An (t) and Vφn (t) from given time dependence of the current I0 (t). Taking into account the simple frequency dependence of each term in series (4.14), we will solve the problem in two stages. First, we find the time dependence under the action of unit current pulse T (t)—Heaviside function ⎧ T(t) =

0,

t < 0;

1,

t ≥ 0.

(4.17)

At the next stage, using the Duhamel integral, we will find functions V An (t) and Vφn (t) for an arbitrary dependence of the current on time I0 (t). As the integrand (4.14) in the improper integral of the inverse Fourier transform does not satisfy the boundedness condition, we will use the Laplace operator s instead of the operator i ω. / Since the Laplace transform of unit impulse is T (t) ÷ T (s) = 1 s, then the (n) Laplace transforms of the products T (s)G (n) e1 (s) and sT (s)G e1 (s) for each term of

4.3 Asymptotic Expansion Method for Pulsed Three-Dimensional …

99

the expansion into asymptotic series are the following ⌃ An (s) = T (s) · G (n) e1 (s) =

gn ; (n+3)/2 s

(4.18)

⌃φn (s) = sT (s) · G (n) e1 (s) =

gn . s (n+1)/2

(4.19)

The inverse Laplace transform for functions (4.18) and (4.19) are known [18], it gives power functions of time. As a result, the time dependence for the approximate value of the functions λ A (t) and λφ (t) (the functions V A (t) and Vφ (t) under the action of the single current pulse T(t)) will take the form N ∑

λ A (t) =

λ An (t) =

n=0 N ∑

λφ (t) =

N ∑ n=0

λφn (t) =

n=0

[

N ∑ n=0

[

gn ( n+3 ) t (n+1)/2 ;

(4.20)

2

gn ( n+1 ) t (n−1)/2 ,

(4.21)

2

where [(·) is the gamma function ( [

k+1 2

)

⎧( / ) ⎪ (k − 1) 2 ! at k = 1, 3, 5, . . . ; ⎨ √ = 2k /π2 (k − 1)!! at k = 2, 4, 6, . . . ; ⎪ √ ⎩ π at k = 0.

(4.22)

The functions λ An (t) and λφn (t) for the zero expansion term (n = 0) differ significantly from each other. The function ⌃ A0 (t) ∼ t 1/ 2 is bounded at t → 0. On the contrary, the function ⌃φ0 (t) ∼ t −1/ 2 increases indefinitely. Therefore, we consider the final expressions separately. Since for all λ An (0) = 0, then for function V A (t) which determines the vector potential and magnetic field intensity, we apply the Duhamel integral in the following form [19, 20] ∫t [ V A (t) = λ A (0)I0 (t) + 0

=

N ∑ n=0

] | dλ A (ξ ) || I0 (τ ) dτ dξ |ξ =t−τ

n+1 ( / ) gn (μ, γ , r1 )P(t), 2[ (n + 3) 2

(4.23)

where ∫t Pn (t) = 0

(t − τ )(n−1)/ 2 I0 (τ )dτ .

(4.24)

100

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

The expressions (4.23), (4.24) represent an approximate value in the form of asymptotic series of the integrand (4.6) in the contour integrals (4.4) and (4.5). Let us consider the possibility of using the Duhamel integral to calculate the scalar potential and electric field intensity. In this case, it is necessary to limit the form of the dependence of the current on time in such a way as to exclude unlimited field growth. To carry out the analysis, we will use another form of the Duhamel integral. For it, the derivative is applied not to the function associated with the action of the unite impulse, which leads to singularity of the integrand, but in the form in which the derivative of the current on time is used. First, we will analyze only the zero term of the asymptotic series n = 0, since a certain difficulties arise for this term. For this term we use the Duhamel integral with the following form [21] ∫t [ Vφ0 (t) = λφ0 (t)I0 (0) + 0

] | d I0 (ς ) || λφ0 (τ ) dτ . dς |ς=t−τ

(4.25)

The first term will be equal to zero if there is no current jump, i.e.I0 (0) = 0. In this case, without loss of generality, we will assume that near t = 0 the current changes with time according to an arbitrary power law I0 (t) ≈ at k , where k > 0. Then from (4.25) near t = 0 it follows gn / ) Vφ0 (t) = ak ( [ (n + 1) 2

∫t

(t − τ )k−1 τ −1/ 2 dτ .

(4.26)

0

By the mean value theorem expression (4.26) can be written as gn / ) ξ k−1 Vφ0 (t) = ak ( [ (n + 1) 2

∫t 0

gn / ) ξ k−1 t 1/ 2 , (4.27) τ −1/ 2 dτ = 2k ( [ (n + 1) 2

where the point ξ is in the interval 0 ÷ t. For small values t, we have lim Vφ0 (t) ∼ t→0

t k−1+1/ 2 = t k−1/ 2 . The function Vφ0 (t) and, accordingly, the scalar potential and the electric field intensity at t → 0 will have a limited value if the/current changes at the initial moment according to the law I0 (t) ≈ at k when k > 1 2. So, the function Vφ (t) can be defined by the following expression with a limitation on growth current I0 (t) at the initial time: Vφ (t) =

N ∑ n=0

where

Vφn (t) =

N ∑ n=0

1 ( / ) gn (μ, γ , r1 )Q n (t), 2[ (n + 1) 2

(4.28)

4.3 Asymptotic Expansion Method for Pulsed Three-Dimensional …

∫t [ Q n (t) = 0

101

] | d I0 (ς ) || (n−1)/ 2 τ dτ . dς |ς =t−τ

(4.29)

For the electric field intensity Ee (t), one must also take into account the limitation associated with the presence of the first two terms in (4.8). With these terms taken into account, the exclusion of the unbounded growth of the function Ee (t) gives more severe constraint than the restriction for the third term. It is necessary that the change in the initial current occurs no faster than according to a linear law, i.e.lim I0 (t) ∼ t k , t→0

with k ≥ 1. The specified limits on the rate of change of the initial current concern not only the initial moment of time, but must also be carried out at any intermediate time interval. Knoepfel [22] also points out incorrect physical consequences when using models with a external field jump at the initial moment of time. Therefore, in the future, we will assume that its change in time satisfies the indicated restrictions. Thus, instead of triple improper integrals in the expressions for potentials and field vectors in the dielectric half-space, there are only contour integrals. The integrands are represented by bounded asymptotic series, in each term of which the dependences on coordinates and time are calculated separately using well-known simple expressions. This allows to significantly reduce the required for the calculation of three-dimensional pulsed electromagnetic fields, taking into account eddy currents in an conducting half-space.

4.3.2 Cutoff Values of Frequency and Time Interval in the Asymptotic Series Expansion Method of Pulsed Electromagnetic Field A feature of impulse processes is that the frequency spectrum of the signal is represented by all frequencies. At the same time, the application of approximate calculation method using asymptotic series is possible then if the parameter is small μ ε1 = r1 √ωμμγ = √μδ < 1. This parameter relates the field penetration depth and 2r1 the distance between the observation and source points on the mirrored reflected contour. Therefore, the method is not applicable for the entire frequency range, and this circumstance should be taken into account when developing approximate method for calculating pulsed electromagnetic field. The size limit of the small parameter is connected with the field frequency limit which means that the frequency must be bigger than some limit value f > fm =

μ , 2π μ0 γ r12 εm2

where εm is the chosen permissible value of the small parameter.

(4.30)

102

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

The second limitation is related to the fact that each term of the asymptotic series is determined with error that increases with decreasing frequency and increasing number of the series term. This circumstance determines the limitation of the number of terms in the series. To estimate the relative error Δn in the frequency domain of each member G (n) e1 of the asymptotic series for the function G e , we will use the found expression (2.46), namely n | | ∑ 1 | (n) | e−1/ε1 |G e1 − G (n) | k!ε1k e k=0 | | = Δn = . n | (n) | ∑ 1 −1/ε |G e | 1 1−e k!εk k=0

(4.31)

1

Expression (4.31) can be considered as equation for small parameter with given allowable error Δn . We will denote by εn , in contrast to εm , the value of the small parameter, which is associated with specific member of the series, and the limiting frequency corresponding to the found εn by f n . The series term with number n can be applied within frequencies, which are defined similarly to (4.30) f > fn =

μ . 2π μ0 γ r12 εn2

(4.32)

Let us analyze the cutoff frequencies f m and f n , at which it is possible to expand into an asymptotic series. The cutoff frequencies can be / conveniently analyzed with the use of dimensionless values of frequency f ∗ = f f b . The chosen value of the basic frequency f b = )−1 ( 2 should be such, that the penetration depth in the conducting medium π h μμ0 γ is equal to the characteristic dimension of the electromagnetic system which is the distance of the contour element with current to the interface of media δ = h. In this case, the value of the normalized frequency is related to the value of the small parameter as f∗ =

f μ2 = ( / )2 . fb 2ε2 r1 h

(4.33)

For example, for the chosen values of the small parameter εm = 0.3 and εm = 0.2 at points on the interface directly below the contour element (r1 = h), the corresponding normalized frequencies will be f m∗ = 5.56 and f m∗ = 12.5. Figure 4.1 shows the normalized values of the cutoff frequencies depending on the number of the series term n for three values of the permissible relative errors Δn . The cutoff frequencies are obtained at μ = 1 for the point ρ = 0, z = 0 for which, due to the minimum value of the distance, the cutoff frequencies take the maximum values. For this point, the part of the frequency spectrum of the pulse that is not taken into account will be the largest.

4.3 Asymptotic Expansion Method for Pulsed Three-Dimensional …

103

Fig. 4.1 Normalized cutoff frequencies for terms of the asymptotic series at the point ρ = 0, z = 0 and μ = 1

The degree to which the frequency spectrum in approximate calculation is determined by number of terms that can be taken into account, i.e. on the cutoff frequency f n∗ for the end term of the asymptotic series, the value of which depends on the required accuracy for this series term Δn . Since the frequency spectrum is also limited by the frequency f m∗ , then as the frequency decreases, the calculation of the function G e will be carried out less and less accurately, up to the frequency f m∗ , which is the maximum for the asymptotic expansion for the chosen maximum value of the small parameter εm . For the maximum frequency f m∗ , the number of terms in the ∗ ∗ < f m∗ < f n=N series is determined from the condition f n=N +1 . From Fig. 4.1 it can be seen that N is the closest value to the right of the intersection of the dependencies for f m∗ and f n∗ . For a current pulse, limiting the frequency spectrum in the low-frequency range limits the time interval over which the field can be calculated. For example, when a unit current pulse flows along the contour, the presence of low frequencies in the spectrum leads to an unlimited growth of functions λ An (t) at t → ∞, while the vector potential and magnetic field intensity should tend to a constant value. / The t ≈ 1 expressions will be valid until a certain characteristic point in time c / ( )f c , for √ √ which, in fact, the previous condition is satisfied ε1 = μ tc 2πr1 μ0 μγ < 1. The assessment of the allowable / time intervals can be performed as follows:/for the entire calculation t ≤ tm = 1 f m and for each term of the series t ≤ tn = 1 f n . The normalized time is related to the value of the small parameter as t ∗ = t f b = ( / )2 ( / )2 2 ε1 μ r1 h . For example, for εm = 0.3 and εm = 0.2 the values of the considered time interval at r1 = h are respectively tm∗ = 0.18 and tm∗ = 0.08. Choosing smaller value εm for more accurate calculation reduces the time interval 0 ≤ t ≤ tm . At points in space with a larger distance r1 > h, the calculation accuracy turns out to be higher than at r1 = h. The cutoff values of the normalized time tn∗ depending on the permissible relative errors Δn in determining the terms of the asymptotic series are shown in Fig. 4.2 for different series terms. The dotted lines correspond to the limiting values of the normalized moments of time tm∗ , which are determined by the chosen admissible value of the small parameter εm . During studying the number of terms of the series that provide the necessary accuracy of the calculation (Figs. 2.10 and 2.11), it was shown that to ensure accuracy

104

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

Fig. 4.2 Normalized maximum moments of time tm∗ for two values εm and time tn∗ depending on the permissible error Δn at the point ρ = 0, z = 0

Δmin = 10−3 , the number of terms of the series does not exceed N = 6. In the Table 1 for this number of the terms of the series it is given the normalized values of cutoff frequencies for different number n of series term and for three permissible relative calculation errors of asymptotic series terms determination. The cutoff frequencies have been obtained at μ = 1 for the point with the minimum distance r1 = h and, therefore with maximum values of cutoff frequencies. In Table 4.1 calculated values of limiting frequencies and time intervals for some numbers n of the series terms at three permissible errors Δn and minimum distance r1 = h for non-magnetized media μ = 1 are presented. It can be seen from the table that with increase in the number of the series term, the limiting values of the frequency increase rapidly and, accordingly, the time interval during which this specific term of the asymptotic series can be taken into account rapidly decreases. This property limits the scope of asymptotic series to high-speed current pulses. On the other hand, since the current pulse usually changes rapidly and reaches maximum values for a short time, the electromagnetic field is determined at this most important stage. Table 4.1 Cutoff values of the frequency and time interval for the terms of the series at r1 = h n

Δn = 1

Δn = 0.5

Δn = 0.25

f n∗

tn∗

f n∗

tn∗

f n∗

tn∗

0

0.24

4.16

0.60

1.66

1.30

0.77

1

1.41

0.71

2.62

0.38

4.48

0.22

2

3.58

0.28

5.89

0.17

9.16

0.11

3

6.74

0.15

10.4

0.096

15.2

0.066

4

10.9

0.092

16.0

0.062

22.6

0.044

5

16.1

0.062

22.8

0.044

31.3

0.032

6

22.2

0.045

30.7

0.033

41.2

0.024

4.3 Asymptotic Expansion Method for Pulsed Three-Dimensional …

105

However, the accuracy of the calculation during limited time interval is not the same. A sufficiently strong dependence of the cutoff moments of time for the series terms indicates that the closer to the beginning of the current pulse, the more accurately the calculations of the three-dimensional electromagnetic field are performed.

4.3.3 Integral Indicators to Take into Account Restrictions on the Frequency and Time Interval of Current Pulses Since for pulsed electromagnetic field, when using the asymptotic expansion method, not the entire spectrum of the signal is taken into account or, on the other hand, calculations are performed for a limited period of time, it is necessary to have indicators that reflect the noted circumstances. Such indicators can be based on two integral properties that take into account a limited part of the entire total current impulse. The first integral indicator takes into account that in the approximate calculation method the low-frequency part of the current pulse spectrum is neglected. If the frequency spectrum of the current pulse is specified I˙0 (i ω), then the ratio of the root mean square values of the amplitude of the frequency spectrum can be used to quantify the part of spectrum S f , i.e., the energy indicator of spectrum is considered [ [ | ∫∞ / |∫∞ | | | | | |2 2 | |˙ | √ S f = √ | I˙0 (i2π f )| d f I0 (i2π f )| d f ,

(4.34)

0

f min

where f min is either f m referring to the chosen permissible value of the small parameter εm , or takes one of the values of minimal frequency f n for the asymptotic series terms. The second approach is the selection of an indicator that takes into account the fact that the calculation can be carried out over the limited period of time. The corresponding indicator can be based on the part of the charge that is carried by the current pulse during the considered limited period of time, in relation to the full value of the transferred charge of the pulse [ [ | ∫tmax / |∫∞ | | | | √ |I0 (t)|dt, St = √ |I0 (t)|dt 0

(4.35)

0

where, similarly to (4.34), the notation tmax means that the maximum time interval can take the value tm or tn for the last term of the asymptotic series that taken into account. In expression (4.35), in order to take into account the processes in which

106

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

the direction of the current changes, the current modulus is used, i.e., the charge that crosses the cross section of the conductor, regardless of its direction. Integral indicators are calculated on examples for specific current pulses.

4.3.4 Taking into Account Limited Time Intervals for Current Pulse At the initial time of the impulse, there are no restrictions on the number of terms in the asymptotic series. The expediency of their limitation can be determined only, for example, by the chosen minimum calculation error Δmin . So, from Fig. 2.10, as noted, at Δmin = 10−3 it is sufficient seven terms (N = 6). With the passage of time from start of the impulse, the accuracy of the series terms decreases the faster, the larger the number of the series term. When the time reaches the cutoff value for a specific series term (for example, according to Table 4.1), this series term should be excluded from the field calculation. Finally, when the time from the start of the pulse reaches the value t = tm , the calculation stops. For example, if is chosen εm = 0.3 and the normalized value of the moment of time is tm∗ = 0.18, then according to Fig. 2.10 the limiting number of terms of the series will be N = 2 or N = 3 , and the estimate of the calculation accuracy gives approximately Δ N ≈ (2 ÷ 3) · 10−2 . However, the accuracy estimate is given for the corresponding cutoff frequency, / and the cutoff time is related to frequency only in order of magnitude as tm∗ = 1 f m∗ . In the following examples, this circumstance will be considered in more detail. The calculation accuracy can be significantly increased if we take into account that the Duhamel integrals in expressions (4.25), (4.29) reflect the principle of superposition. For example, in the expression ∫t [ Vφ (t) = 0

[ the small increment

|

] | d I0 (ς ) || λφ (τ ) dτ , dς |ς=t−τ

d I0 (ς ) | ⌃(τ ) dς |ς=t−τ

(4.36)

] dτ begins to act at the moment of time τ ,

and not from the start of the impulse. Therefore, it is sufficient to take into account the restriction on the time interval τ ÷ t for each term of the series. For definiteness, we will call this method : “t − τ approximation” [23]. Such calculation algorithm, in addition to increasing the accuracy of the calculation, makes it possible to take into account, for example, the situation when the action of an impulse begins not from the moment of time t = 0, but at a later moment. In addition, it is necessary to exclude jumps of functions at the moments of time tn when the cutoff values for the terms of the series are reached.

4.4 The Electromagnetic Field of the Standard Current Pulses Flowing Near …

107

To avoid a jump, one must take into account that the error for each term of the series increases gradually from zero to a certain maximum value. There will be no jumps if each term is taken into account with a certain weight function K (ε1 , n), the value of which depends on the small parameter for a given moment of / value of the (∗ ) / μ ∗ ∗ ∗ ∗ time ε1 t − τ , μ, r1 = r ∗ (t − τ ) 2. When t ∗ − τ ∗ 1

Δn ≤ 1

.

(4.37)

Choosing a linear relationship is not necessarily optimal. When optimizing calculation the real dependence of the error with respect to value of small parameter can be taken into account, which in turn depends on interval (t − τ ) in accordance with expression (4.31).

4.4 The Electromagnetic Field of the Standard Current Pulses Flowing Near Conducting Half-Space The obtained solutions to either general problem of calculating the electromagnetic field, or approximate solution on the basis of the method of asymptotic expansion allow the use of them for arbitrary time dependencies of the currents. However, during the investigations the mathematical models for “standard” current pulses are used. These “standard” pulses are considered to be the following: an exponentially decaying current pulse; pulse represented by the difference between two decaying exponents, exponentially decaying oscillating pulse. For such current pulses, let us analyze the major features of the use of approximate analytical approach to the calculation of the electromagnetic field. Time dependencies of the current can be/ conveniently analyzed / with the use of dimensionless values of frequency f ∗ = f f b and time t ∗ = t tb .

4.4.1 Exponentially Decaying Current Pulse The edge interval of the exponentially decaying current pulse of the initial contour is equal to zero. In this case, the time dependence is determined only by the value of

108

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

a decay coefficient α: I (t) = Im · I ∗ (t),

I ∗ (t) = exp(−αt) = exp(−α ∗ t ∗ ),

(4.38)

/ where α ∗ = α f b . The frequency spectrum of the current pulse (4.38) is known [18] I˙∗ (iω) =

1 1 1 , = α + iω f b α ∗ + i ω∗

(4.39)

where ω∗ = 2π f ∗ . From (4.34) it follows that the value of the extent of taking into account the signal spectrum, while the approximate calculation method is being used for the exponentially decaying function, is determined by an analytical expression / Sf =

1−

∗ 2 2π f min . ar ctg π α∗

(4.40)

As it can be seen, the integral indicator of using the spectrum for different f min depend on the decay coefficient. The dependencies of value S f on normalized quantity α ∗ are shown in Fig. 4.3. The figure shows that for more speedy exponential decay of the current pulse the indicator of taking into account its spectrum increases. A dashed curve illustrates the value S f realized when the calculations are limited by time tm and corresponds to the boundary value εm = 0.3. At this time moment the series members at n ≥ 3 should be eliminated from the calculations, and overall accuracy is determined by the left members. At substituting (4.38) into expression (4.35) for another indicator, we obtain its value ) ( ∗ . St = 1 − exp −α ∗ tmax

Fig. 4.3 Pulse spectrum constraint indicator S f for a pulse I ∗ (t ∗ ) = exp(−α ∗ t ∗ ) at Δn = 1, r1 = h, μ = 1

(4.41)

4.4 The Electromagnetic Field of the Standard Current Pulses Flowing Near …

109

Fig. 4.4 Charge indicator St for a pulse I ∗ (t ∗ ) = exp(−α ∗ t ∗ ) at Δn = 1, r1 = h, μ = 1

Figure 4.4 shows the dependence of the indicator St on the decay coefficient α ∗ for the selected number of taken into account terms of the series n = N , as well as the dependence St for the previous value of the small parameter εm = 0.3. It can be seen that the dependences S f and St are qualitatively similar. They allow to compare the influence of restrictions when using the asymptotic calculation method for impulse process. However, the quantitative values of the indicators differ significantly, so in the future we will use only one indicator associated with restrictions on the frequency spectrum of the current pulse. As the impulse (4.16) grows up to the maximum value by jump, this impulse impossible to use for determine the electric field intensity. Although this restrictions do not apply to the vector potential and magnetic field intensity. In this case, time integrals Pn (t) (4.24) are represented as [14] ∫t Pn (t) =

[ )] ( n+1 −αt −(n+1)/2 (n−1)/ 2 , −αt , I e γ − τ = I (t ) (−α) 0 (τ )dτ m 2

0

(4.42) where γ (β1 , β2 ) is an incomplete gamma function. The product in brackets can be shown as a series and the following expression can be used instead the special functions Pn (t) = Im e−αt t (n+1)/2

∞ ∑ k=0

(αt)k ). ( k! k + n+1 2

(4.43)

110

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

4.4.2 Current Pulse Represented by the Difference Between Two Decaying Exponents The pulses being considered are characterized by the finite value of the edge duration of the initial current pulse of the contour (Fig. 4.5a). The law of time changes of such a pulse can be described by the following dependency [ ] I0 (t) = I1 (t) − I2 (t) = Im exp(−α1 t ) − exp(−α2 t ) [ ] = Im exp(−α1∗ t ∗ ) − exp(−α2∗ t ∗ ) .

(4.44)

The expressions for the normalized values will be written as follows ( ) I ∗ (t ∗ ) I0 t ∗ = Im · , Im∗

( ) [ ] I ∗ t ∗ = Im∗ exp(−α1∗ t ∗ ) − exp(−α2∗ t ∗ ) .

(4.45)

Here and below, it is assumed/that α1∗ < α2∗ . Maximum value of I ∗ (t ∗ ) is equal to one. In this case, the expression 1 Im∗ represents the maximum value of the difference of 2 / α1 ) ∗ the exponent, which is reached at the moment of time tmax = ln(α . α2 −α1 The longer the front length compared to the duration of the entire pulse, the less high frequencies [are represented in the frequency ] spectrum. For example, the current pulse I ∗ (t ∗ ) = 4 exp(−50t ∗ ) − exp(−100t ∗ ) , shown in Fig. 5a, has the frequency spectrum with the high frequency range being considerably narrower than for each component of the whole pulse (Fig. 5b). Figure 4.6 illustrates the frequency spectrum of the pulse (4.45) depending on the decay coefficient α1∗ , which in the first place determines the whole pulse duration at different values of α2∗ affecting the pulse edge duration. The calculations have not been performed for each series term, as in Fig. 4.3, they have been performed for

[ ] Fig. 4.5 Current pulse I ∗ (t ∗ ) = I1∗ (t ∗ ) − I2∗ (t ∗ ) = 4 exp(−50t ∗ ) − exp(−100t ∗ ) : a—time dependencies; b—amplitude-frequency responses

4.4 The Electromagnetic Field of the Standard Current Pulses Flowing Near …

111

Fig. 4.6 Indicator of considering the pulse spectrum I ∗ (t ∗ ) = [ ] Im∗ exp(−α1∗ t ∗ ) − exp(−α2∗ t ∗ ) at εm = 0.3

limited whole pulse interval tm∗ = 0.18, which is determined by the chosen value εm = 0.3. It should be mentioned that in the case of relatively small values of the decay coefficient α2∗ , the indicator S f considerably differentiates from one. Therefore, the application of the method of asymptotic expansion is possible, but only up to time, when current value cannot still be considered small. Here it is possible to take into account not the whole of the current pulse, but only some of its properties, for example, reaching the maximum value by its electromagnetic field. The pulse (4.44) is a sum of two exponents, so the expressions for the intensity of the magnetic field are also determined by the sum of two components, whose time depending functions are similar to those noted in (4.42) and (4.43). Since the current pulse (4.44), as opposed to (4.38), does not change by jump at the initial moment, the calculation of the electrical field intensity for such pulse does not cause the peculiarity at t → 0. For this case we give only expression for the electric field intensity. In the expression (4.28) functions Q n (t) (4.29) can be shown as follows [ ( ) n+1 , −α2 t Q n (t) = −Im e−α2 t (−α2 )−(n+1)/2 γ 2 ( )] n + 1 , −α1 t . −e−α1 t (−α1 )−(n+1)/2 γ 2

(4.46)

Instead of special functions their representation as a power series can be useful. For this (4.46) can be used in a following form

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4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

[ Q n (t) = −Im t

(n+1)/2

α2 e

−α2 t

∞ ∑ k=0

] ∞ ∑ (α2 t)k (α1 t)k −α1 t ( ( ) − α1 e ) . k! k + n+1 k! k + n+1 2 2 k=0 (4.47)

4.4.3 Decaying Oscillating Pulse Let us consider the standard current pulse (Fig. 4.7) which depending on time changes according to the law: I0 (t) = Im I ∗ (t), I ∗ (t) = exp(−αt) sin βt = exp(−α ∗ t ∗ ) sin β ∗ t ∗ ,

(4.48)

/ / where α ∗ = α f b , β ∗ = β f b . The expression for the frequency spectrum of normalized values is: I˙∗ (iω) =

β∗ 1 β = . f b (α ∗ + i ω∗ )2 + (β ∗ )2 (α + i ω)2 + β 2

(4.49)

At the initial moment, the current jump is absent. Therefore, for such pulse, the results of calculating the intensities both magnetic and electrical field will be correct. For determining the values of integral indicator S f taking into account the limitation of the frequency spectrum of the signal, we can chose, as previously, the variant with the biggest error at the expansion into the asymptotic series, when r1 = h. The values of the small parameter are chosen εm = 0.3 and μ = 1. The dependencies of the indicator S f on the decay coefficient α/∗ at different values of the ratio of oscillation frequency to the decay coefficient β α are shown in Fig. 4.8. As the / dependencies in / Fig. 4.8 show, the indicator S f is the greater, the greater the values β α < 1, the pulse decays completely during the oscillation ratio β α. For / period. At β α > 1, the value of the indicator S f proves to be bigger, than of that Fig. 4.7 Decaying oscillating current pulse

4.4 The Electromagnetic Field of the Standard Current Pulses Flowing Near …

113

Fig. 4.8 Indicator S f of spectrum in the case of decaying oscillating pulse (εm = 0.3, r1 = h, μ = 1)

/ without current oscillations. It can be explained by the fact that in this case β α > 1 the frequency spectrum contains a wider high frequency range comparing with the exponentially decaying pulse. For the pulse shape (4.48) the analytical representation of time integrals is also possible. Here, it is convenient to use a complex-value symbolic form for describing the pulse (4.48). In this case, time dependencies Pn (t) (4.24) of the function V A (t) (4.23) determining the vector potential and the magnetic field intensity can(be )found as an imaginary part of appropriate complex-value expressions Pn = Im P˜n [ )]⎫ ⎧ ( ( ) n+1 , −ηt , Pn (t) = Im P˜n = Im Im e−ηt (−η)−(n+1)/2 γ 2

(4.50)

where η = α − iβ. As in (4.43), the last expression can be represented as a series Pn (t) = Im t

(n+1)/2 −αt

e

( / ) ∞ k ∑ t |η| sin βt − kar ctgβ α ) ( . k! k + n+1 2 k=0

(4.51)

At last, let us find time dependencies Q n (t) (4.29) for the series terms of the function Vφ (t) (4.28) determining the intensity of the electrical field. Repeating the transforms performed for obtaining expression (4.50) for real variables and using the complex value η = α − iβ, we can find expression with the use of special functions [ )]⎫ ( ⎧ n+1 , −ηt , Q n (t) = Im −Im ηe−ηt (−η)−(n+1)/2 γ 2 or as a series

(4.52)

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4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

Q n (t) = −Im t

(n+1)/2 −αt

e

( / ) ∞ k ∑ t |η|k+1 sin βt − (k + 1)ar ctgβ α ) ( . k! k + n+1 2 k=0

(4.53)

Finally, let us note that the calculations of time integrals in (4.24) and (4.29) are not difficult. Although, the advantage of analytical approaches allows the thorough analysis of the influence of different factors on the distribution of the electromagnetic fields, as well as promotes the well-grounded development of mathematical models for investigating the complex pulse electromagnetic systems.

4.5 Possibility of Application the Asymptotic Expansion Method to Determine the Three-Dimensional Pulsed Electromagnetic Field Since in the asymptotic expansion method the lower frequency limits increase with increase the number n of the series term, then in the initial period the largest number of the terms in the series is taken into account and the field can be calculated most accurately. The validity of the proposed estimate of the time intervals for the integrands in the contour integrals was made without sufficient theoretical justification. This refers to the assumption that limiting the frequency spectrum of the current pulse to frequencies that exceed the lower limit f > f m , simultaneously limits / the time interval from the start of the pulse t < tm , that can be estimated as tm = 1 f m . A similar assumption is also made regarding the limited time for taking into account each term of the asymptotic series. The validity of such assumptions is analyzed based on a comparison of the calculations by approximate and exact methods for functions V A (t) and Vφ (t).In addition, the examples confirm the expediency of using the “t − τ approximation”. Since the value of the small parameter ε1 is a function of the integration point along the contour, the calculation results for the contours of a specific configuration are compared. It is shown that the limitation is determined mainly by the maximum value of the small parameter, when the distance between the source and observation points is minimal.

4.5.1 Comparison of the Results for the Integrand in Contour Integrals Figure 4.9 illustrates a comparison / of the exact and approximate normalized values of the function V A∗ (t ∗ ) = V A (t) I0 max for an exponentially decaying current pulse, which at the initial moment t = 0 takes on a maximum value by jump. At the point on the surface under the contour element at ρ = 0, r1 = h and μ = 1 for the chosen permissible value of the small parameter εm = 0.3 the limit

4.5 Possibility of Application the Asymptotic Expansion Method …

115

Fig. 4.9 Comparison of approximate and exact calculations of the function V A (t ∗ ) for current pulse ∗ ∗ I ∗ (t ∗ ) = e−α t in observation point ρ = 0, z = 0 and μ = 1

value of the considered time interval turns out to be tm∗ = 0.18. The figure shows the dependence function V A∗ (t ∗ ) with regard to normalized time t ∗ for exact calculation, approximation by asymptotic series with different number of series terms N at Δn = 1 and with application “t − τ approximation”. Calculations with the choice of different number of the series terms were not limited by the time for individual terms. In this case, there is a tendency for the calculation error, which was noted for the field, which changes in time according to sinusoidal law (Fig. 2.5). For example, in the beginning more exact values will be for N = 4, and with growth t ∗ the error increases much faster, than for N = 2. In all cases, the permissible maximum duration of the calculation time interval is much / less than tm∗ = 1 f m . Application in asymptotic expansion smaller time interval in “t − τ approximation” gives results with considerably bigger accuracy. In this case, the calculation is performed with /sufficient accuracy up to the point in time / that corresponds to the estimate tm∗ = 1 f m , tn∗ = 1 f n∗ . Therefore, for specific calculations, however, only using “t − τ/ approximation”, / there is no need to limit the calculation to times less than tm∗ = 1 f m , tn∗ = 1 f n∗ . Figure 4.10 gives comparisons√exact and approximate calculations in observation points, for which distances r1 = (h + z)2 + ρ 2 are larger than minimal value r1 = h. In Fig. 4.10a the observation points are located at different heights relative to the interface between the media. In Fig. 4.10b, on the contrary, the observation points are located on the interface, not under the contour element. In both cases, the calculation ( / )2 by the approximate method can be carried out up time moment t ∗ = tm∗ r1 h greater than for the point located at the shortest distance. During this time the errors is smaller than for minimal distance between observation point and source point on mirror reflected contour. In Fig. 4.10b the difference almost absent. The results of comparing exact and approximate calculations of the normalized / values of the function Vφ∗ (t ∗ ) = Vφ (t) (I0 max f b ) at a point on the surface z = 0, ρ = 0 for some current pulses with finite value of the edge duration are shown in Fig. 4.11. It is seen that the approximate method of asymptotic expansion gives very insignificant deviations in comparison with the exact one in the time interval from the beginning of the current pulse action to the limiting value tm∗ = 0.18. For the pulses in Fig. 4.11b, c the values practically coincide over the entire time interval.

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4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

Fig. 4.10 Comparison of approximate and exact calculations of the function V A (t ∗ ) for current ∗ ∗ pulse I ∗ (t ∗ ) = e−α t : a—observation point located on vertical axis; b—observation point located on interface

∗ Fig. 4.11 Comparison of approximate ( ) and exact calculations of the function V A (t ) for current ∗ ∗ pulse I ∗ (t ∗ ) = Im∗ e−α1 t − e−α2 t at ρ = 0, z = 0,μ = 1

Qualitatively, the dependence of the function Vφ∗ (t ∗ ) on time differs from the same dependence of the function V A∗ (t ∗ ). First, the maximum Vφ∗ (t ∗ ) occurs earlier than the maximum of the current pulse. Secondly, with an increase in the rate of current rise at the pulse front, the maximum value of the function Vφ∗ (t ∗ ) also increases. Finally, the value of the function Vφ∗ (t ∗ ) changes sign over time, which is due to the presence of the time derivative of the current.

4.5.2 Comparison of Approximate and Exact Calculation Methods for System with Current Contours Since the small parameter ε1 is function of points on the contour, the permissible calculation time interval also changes when integrating along the contour. For the entire contour, it is advisable to choose the minimum value of the permissible time interval, which corresponds to the minimum distance between the mirrored contour

4.5 Possibility of Application the Asymptotic Expansion Method …

117

Fig. 4.12 Dependence of the vector potential on time at point on the media interface closest to the circuit with pulsed current I0∗ (t ∗ ) = exp(−50t ∗ )

and the observation point. In this case, the accuracy of the field calculating for the entire contour will be higher than for the contour point with the greatest ε1 . In Fig. 4.12 the results for the vector potential when pulsed current I0∗ (t ∗ ) = exp(−50t ∗ ) flows along an elliptical contour are presented. The contour geometry and orientation relative to the boundary surface are given by the following parametric equations x = a cos θ cos α, y = b sin θ, z = H − a cos θ sin α,

(4.54)

where the parameter θ changes within 0 ÷ 2π ; α is the tilt angle of the plane in which the contour lies relative to the interface between the media; H is the height of the location of the ellipse center relative to the boundary surface. /The relative / sizes of the semi-axes of the ellipse and tilt angle are chosen a H = 1, b H = 2 and α = 60◦ . The contour is shown)in Fig. 12a. The vector ( / potential is determined at the point Q x H = 0.5; 0; 0 marked in the Figure with a cross, where / the distance from the contour to the interface is minimal h ∗min = r1∗ min = h min H = 0.134. At this point, for the chosen geometry of the contour, the vector potential has only a tangent component, parallel to the axis y. The current pulse at the moment of time t = 0 jumps to its maximum value and then decreases according to the exponential law (4.38). The calculation results obtained by approximate expressions (4.4) and (4.23) applying “t − τ approximation” are shown in Fig. 4.12b with solid curve. Individual points show the results of calculations using exact analytical expressions (4.4) and (4.6). The normalized values of the vector potential are determined in accordance 0 Im A∗Q . It can be seen from the given dependences that with the expression A Q = μ4π the values of the vector potential calculated by the approximate expression coincide with the calculation by the exact expressions from the beginning of the pulse action to its normalized value t ∗ ≈ 0.2. At this time interval, the calculation accuracy is higher than for the function V A∗ (t ∗ ) at the point located closest to the interface between the media (Fig. 4.9).

118

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

Fig. 4.13 Dependence of the electric intensity on time at point on the] media interface closest to [ the circuit with pulsed current I0∗ (t ∗ ) = 4 exp(−50t ∗ ) − exp(−100t ∗ )

The electric ] contour (Fig. 4.13) with pulsed current [ field intensity of the circular I0∗ (t ∗ ) = 4 exp(−50t ∗ ) − exp(−100t ∗ ) was determined at the interface between the media at the point closest to the contour. The contour, as before, is located in a plane perpendicular to the boundary surface./ The ratio of the circle radius R to the distance H from its center to the surface is R H = 0.833. In this case, the minimum distance of the contour from the surface is equal to r1∗ min = 0.167. It can be seen that for the selected current pulse, the induced electric field decays before the maximum time value tm∗ = 0.18 is reached, and throughout the entire transient process, the exact and approximate values practically coincide. Comparison of the results of calculating the pulsed electromagnetic field by approximate and exact expressions confirms the validity of the choice of time intervals during which the use of approximate asymptotic method is permissible. In this case, the required amount of calculations by the approximate method is approximately an order of magnitude less than by exact expressions. Thus, we can conclude that the developed approach using the method of asymptotic expansion turns out to be useful not only for substantiating the general provisions on the formation of a quasi-stationary electromagnetic field, and also is an effective method for calculating the distribution of three-dimensional field.

4.6 An Example of Using the Analytical Method for Calculating Three-Dimensional Electromagnetic Field for Systems with Pulsed Current Contours Finally, we will give an example of possible use of contour’s model with the pulse current flowing near conducting half-space in technological process.

4.6 An Example of Using the Analytical Method for Calculating …

119

Recent scientific and experimental studies have shown that the treatment of metallic materials, including welds by pulsed electric current and pulsed electromagnetic field has a positive effect on reducing the stress–strain states of metal structures [1, 24, 25]. Pulsed electromagnetic field creates a magnetic pressure in the metal, which is close to the yield strength of the material, which reduces stress levels and increases the plasticity of the metal. Another means of influence is the use of the effect of electro-plasticity, which occurs provided that the current density in the metal is not less than a certain value, which, for example, for aluminum alloys reaches 109 A/m2 . Achieving the required level of current density using the noncontact induction method only in a limited volume of the weld is characterized by relative simplicity of use and low energy consumption [3]. It is possible to ensure the flow of a current of significant density in the metal using a non-contact method of introducing energy using the induction method. In this case, the intensity of the induced electric field and, accordingly, the current density are proportional to the rate of change of the field in time. Therefore, it is expedient to use a fast-flowing pulsed electromagnetic field. In this case, in a metal with high electrical conductivity, the field and current exist in a thin surface layer, and if the thickness of this layer is much less than the thickness of the metal sheet, a strong skin effect takes place. It is in this case that the model of a pulsed current flowing near conducting half-space can be used. For specific calculations, a system was chosen (Fig. 4.12a), for which the maximum density of the induced current is localized near the point closest to the surface of the metal sheet. The field calculations were carried out using exact and approximate methods for calculating a three-dimensional electromagnetic field for a current pulse of a circular contour of radius R located in plane perpendicular to the interface of media. The geometric dimensions are still normalized to the distance from the center of the circular contour to the plane H = R + h 0 , where h 0 is the smallest distance from the contour to the surface. We choose the dependence of the current on time in the form of the difference between two decay exponents [ ] I0 (t) = Im exp(−α1 t ) − exp(−α2 t ) .

(4.55)

The choice of dependence (4.55) is due to the shape of the current pulse of real sources used in devices for the electrodynamic processing of welds (for example, the current pulse of the source in [26]). Their pulse front, which determines the maximum values of the induced current density, is quite well described/by expression (4.55) at α2 = 2α1 . In this case, the maximum current is Imax = Im 4. Specific values of α1 and α2 will be estimated from the condition of attenuation of the induced current in conducting sheet of thickness D. Let’s define the duration (of the / pulse )/ front as = ln α2 α1 (α2 − α1 ). the achievement of the maximum value of τ p = tmax / Then, to estimate the characteristic frequency f p = 1 τ p , the field attenuation condition / / in conducting sheet of thickness D can be represented as D = kd δ = kd τ p (π μμ0 γ ), where it is enough to choose kd = 3. At D = 3 × 10−3 m and

120

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

Fig. 4.14 Dependences of the current density on time at different depths from the surface of an aluminum sheet under the action of current pulse I 0 (t)

/ −1 −1 α2 α1 = 2 for aluminum sheet (γ = 3.7 × 107 Ω an estimate of / 2 m ), we obtain 2 −4 τ = D π μ γ k ≈ 1.5 × 10 s and the decay the required the pulse front time p 0 d / / / coefficients α1 = α2 2 = ln 2 τ p ≈ 1 τ p = 6.67 × 103 s−1 . To obtain specific results of the field decay, the dependences of the current density at different depths directly under the circular contour with current of radius R = 5×10−2 m and the distance h 0 = 10−2 m were calculated (Fig. 4.14). The amplitude of the initial current pulse is chosen such that the current density on the sheet surface is 109 A/m2 , that is, the value at which the electro-plasticity effect takes place. The presented data indicate that for the chosen decay constant values for sheets with a thickness D ≥ 3−5 mm, it is sufficient to apply the model of conducting half-space. The external electromagnetic field of circular current contour creates annonuniform three-dimensional field that decreases in all directions along the plane of the metal sheet relative to the point closest to the loop. Such a feature in Fig. 4.15 is illustrated by the calculated dependences of the current density on time at the points marked with crosses on the surface (Fig. 4.13).The calculations are performed by the method of asymptotic expansion. The limit of the considered time interval from the beginning of the impulse at εm = 0.3 is tm = 2600 μs. From the point of view of using the effect of electro-plasticity to influence the mechanical properties of a material, the results on the distribution of the electromagnetic field in metal sheet can be useful for determining the boundaries of the area in which these changes occur, as well as for studying the effect of currents with a lower density. On the line of intersection of the plane in which the contour is located with the interface between the media, the vector of the induced current density is directed only along this line—it has only the x-component. Therefore, Fig. 15a shows the time dependence of this current density component, which changes direction of flow at a certain point in time. On the parallel line, which in this case is 0.6 cm away from the first one (Fig. 15b), the current density vector also has the component perpendicular to the contour plane. Therefore, the figure shows the dependence of the absolute value of the current density vector |j|. With the selected system geometry and pulse parameters, the contour current amplitudes should have a fairly high value Imax = 83.8 × 103 A.It is possible to reduce the current amplitude at reaching the same value of the maximum current density in the aluminum sheet by reducing the distance h 0 between the contour and

4.6 An Example of Using the Analytical Method for Calculating …

121

Fig. 4.15 Dependences of the current density on time for points on the surface of aluminum sheet under the action of current pulse I 0 (t)

the surface conductive medium. Figure 4.16 shows the calculated dependences for the same contour with current impulse that has the previous parameters, but with the distance to the surface reduced twice h 0 = 0.5 · 10−2 m. The dependences of the current density are determined on the interface between the media on the line y = 0 at the same points as in the previous example. The calculations were performed using the approximate asymptotic method. The limit value of the time from the beginning of the impulse at εm = 0.3 in this case is four times less than in the previous version and is tm = 660 μs. It can be seen that in this case, the required amplitude of the contour current pulse is more than halved, which now equals to Imax = 38.7 × 103 A.However, the size of the area with increased current density also decreases. In addition, we note that the decrease in the distance h 0 is limited by the fact that the cross section of the conductor for current pulse with sufficiently large amplitude must be of the appropriate value, which is determined, for example, by the allowable heating temperature of the conductor material. A complete correct statement of the problem should include the real section of the conductor of the contour, in which the skin effect is taken into account. The problem must be solved at a given current pulse repetition rate, the presence of a cooling system, and other factors. The solution of all Fig. 4.16 Dependences of the current density on time for points on the surface of aluminum sheet at the minimum distance from the contour to the surface h 0 = 0.5 cm

122

4 Three-Dimensional Pulsed Electromagnetic Field of Current Flowing …

these issues is beyond the scope of the mathematical model of the used electromagnetic system. At the same time, the obtained results make it possible to determine the main characteristics of the current pulse and ways to solve the problems that arise.

4.7 Conclusions The presented analytical expressions for calculating the potentials and intensities of pulsed electromagnetic field are based on the exact analytical solution of the threedimensional quasi-stationary problem for calculating the field in the system “current contour—conductive half-space”. There are no restrictions on the geometry of the contour, the electrophysical properties of the medium and the frequency, and hence the dependence of the pulse current and the field on time. Since for pulsed field the exact analytical solution in the frequency domain involves the inverse Fourier transform and its obtaining is associated with calculating triple improper integrals, then under the strong skin effect it is reasonable to use the asymptotic expansion method for pulsed processes. It is shown that the calculation of the vector potential and the magnetic field strength can be performed to change the current in jumping manner. On the contrary, in calculating the scalar potential and electric field intensity, the finite rise time of the external field must be taken into account. Due to in the method of asymptotic expansion the values of the lower frequencies cutoff increase with increasing number of the series term, the field in the initial period can be calculated most accurately. Since the current pulse usually changes most rapidly and reaches its maximum values over a relatively short period of time, so during this, most important stage, the electromagnetic field is determined. Comparison of the results of calculating the pulsed field by exact and approximate methods shows a high accuracy of calculations using the asymptotic expansion within the proposed time interval. Taking into account a significant reduction in the amount of calculations, the approximate method is effective method for calculating the distribution of the three-dimensional pulsed field.

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