Electromagnetic Field Near Conducting Half-Space

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.

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.

The study is based on the exact analytical solution for the general conjugation problem of three-dimensional quasi-stationary field at a flat interface between dielectric and conducting media. It is determined that non-uniform electromagnetic field always decreases in depth faster than uniform field. The theoretical conclusion is confirmed by comparing the results of analytical and numerical calculations. The concept of strong skin effect is extended to the case when penetration depth is small not only compare to the characteristic body size, but also when the ratio of the penetration depth to the distance from the surface of body to the sources of the external field is small parameter. For strong skin effect in its extended interpretation, the influence of external field non-uniformity to electromagnetic field formation both at the interface between dielectric and conducting media and to the law of decrease field in conducting half-space is analyzed. It is shown, at the interface the expressions for the electric and magnetic intensities in the form of asymptotic series in addition to local field values of external sources contain their derivatives with respect to the coordinate perpendicular to the interface. The found expressions made it possible to generalize the approximate Leontovich’s impedance boundary condition for diffusion of non-uniform field into conducting half-space. The function that generalizes the impedance boundary condition in the case of diffusion of non-uniform field into conducting body is proposed. Based on comparison of the calculations results by the exact and approximate methods for a specific model of an electromagnetic system, the permissible maximum value of the introduced small parameter is established. The difference between the penetration law for the non-uniform field and the uniform one takes place in the terms of the asymptotic series proportional to the small parameter to the second power and to the second derivative from the external magnetic field intensity at the interface with respect to the vertical coordinate.

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 non-uniform 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.