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.

Power lines are often placed in ground or sea water, which are weakly conductive media. In the paper, a new analytical formula, taking into account the proximity effect in a twin line placed in a weakly conductive medium, is derived, and the effect of the conductive medium is considered. In the first step, one of the wires is replaced by a current filament, and the solution is sought for magnetic vector potential around the filament. In the next step, an analytical formula for eddy currents induced in a long straight conductor of circular cross-section placed near to the current filament in the extensive conductive medium is found by using the method of separation of variables. The correctness of the formula is checked by comparison with the results obtained via other methods like finite and boundary element methods. Then, the effect of various parameters on the eddy current distribution is tested. Next, the proximity effect in a twin symmetrical line is considered, and the effect of the conductivity of the surrounding medium is investigated. The results indicate that the conductive medium weakens the proximity effect, but in typical cases (ground and sea water), the effect is very small.

In this paper, a non-linear programming method allowing for the optimization of the structure of high-current circuits that supply resistance-arc furnaces was presented. In the case of resistance-arc furnaces, two types of asymmetries most often occur: structural and operational ones. The structural asymmetry is related to the construction of a bifilar high-current busduct, which leads to the so-called short network. Knowing the parameters of the high-current busduct allows one to determine the operating characteristics of the arc furnace. It is also necessary to know the energy consumed in individual steps of the arc furnace operation. The method proposed in this paper makes it possible to establish guidelines for the modernization of a short network in order to eliminate asymmetry. The presented method was verified on a real object by conducting experimental tests on a furnace with a power of 12 MVA. Experimental tests were first carried out for a furnace with asymmetry, and then, by conducting simulation tests, guidelines for changing the design of the short network were determined. The measurements carried out after the modernization of the short network confirmed that the furnace was in a symmetrical operating condition and confirmed the correctness of the calculation method proposed in this paper.