This paper describes an analytical-numerical method for the skin and proximity effects in a system of two parallel conductors of circular cross section—a system very frequently encountered in various applications. The magnetic field generated by the current applied on each conductor is expressed by means of vector magnetic potential and expanded into Fourier series. Using the Laplace and Helmholtz equations, as well as the classical boundary conditions, the current density induced due to the proximity and skin effect is determined in each conductor. The resulting current density is expressed as a series of successive reactions. The results obtained are compared with those obtained via finite elements. Although the paper is theoretical, the considered problem has a practical significance, because transmission lines with round conductors are universally used. Besides, the results can be used to estimate errors when only the first reaction is taken into account, which gives relatively simple formulas.

The paper presents an analytical method for calculating impedances of rectangular bus ducts. The method is based on the partial inductance theory—in particular, the impedance of rectangular busbars in a three-phase system with a neutral conductor is described. The results of resistances and reactances of these systems of multiple rectangular conductors were obtained. Skin and proximity effects were taken into account. The measurements of the impedance of shielded and unshielded high-current busducts of rectangular conductors were also carried out. The magnetic field of the busbars was determined with several methods.

Hysteresis loops constitute the source of important information for the designers of magnetic circuits in power transformers. The paper focused on the possibility to interpret the phenomenological T(x) model in terms of effective field vs. magnetization dependence. The interdependence of anhysteretic curve and hysteresis loops was emphasized. The concept of the anhysteretic plane introduced at the end of the last century by Sablik and Langman was subject to a tangible interpretation within the hyperbolic model framework. A novel geometric interpretation of the “effective field” related to the concept of affine transformation was introduced. It was shown in the paper that minor hysteresis loops of grain-oriented electrical steel may be described with the proposed formalism.

Power and communication lines with round wires are often used in electrical engineering. The skin and proximity effects affect the current density distribution and increase resistances and energy losses. Many approaches were proposed to calculate the effects and related quantities. One of the simplest approximate closed solutions neglects the dimensions of neighboring wires. In this paper, a solution to this problem is proposed based on the method of successive reactions. In this context, the solution with substitutive filaments is considered as the first approximation of the true solution. Several typical arrangements of wires in single-phase communication lines or three-phase bus ducts are considered to detect the limits of applicability of the first approximation. The error of the first approximation grows with wire radius to skin depth ratio and wire radius to wire spacing ratio. When the wire radius to skin depth ratio is up to 1, and the gap between the wires is above the wire radius, the error is at a level of 1%. However, lowering the distance and/or skin depth leads to a much larger error in the first approximation.

The use of high voltage power cables in distribution and transmission networks is still increasing. As a result, the research on the electrical performance of cable lines is still up to date. In the paper, an analytical method of determining the power losses and the temperature distribution in the medium voltage cable was proposed. The main feature of the method is direct including the skin and proximity effects. Then the Joule law is used to express the power losses in the conductor and screen, and the Fourier-Kirchhoff equation is applied to find out the temperature distribution in the cable. The research was focused on a cable with isolated screen and return current in the screen taken into account. The proposed method was tested by using the commercial COMSOL software(5.6/COMSOL AB, Stockholm, Sweden) as well as by carrying out laboratory measurements. Furthermore, the results obtained via the proposed method were compared with those given in literature. The differences between the temperature values calculated by the analytical method, numerical computations and obtained experimentally do not exceed 10%. The proposed analytical method is suitable in prediction the temperature of the power cables with good accuracy.