Abstract
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.
Highlights
Power and communication lines with wires of circular cross-section are very often used in electrical engineering
Λ(ρ, φ; ξ, γ) = −γ ξ n−1 (γ) cos nφ n=1 and is interpreted as eddy current density induced at point of polar coordinates (ρ, φ) in a unit radius round wire aligned with z axis and having propagation constant γ due to a parallel current filament with the current π crossing point (ξ, 0)—see Figure 3
It was calculated via Equation (18) and compared with that given in work [2], Table 1, as well as with that obtained via finite element method (FEMM software, version 4.2, 64-bit 21 April 2019 by David Meeker, MA, USA, http://www.femm.info/wiki/HomePage) with very fine mesh
Summary
Power and communication lines with wires of circular cross-section (often called round wires) are very often used in electrical engineering. There were many approaches with semi-analytical solutions, where fields are expressed as infinite series, yet the coefficients in the series have to be determined numerically by solving a system of linear algebraic equations. Dlabač and Filipović [18] solved the integral equation for two round parallel wires by expressing the solution in the form of power series; this required series truncation and solving a linear system of algebraic equations. The most general approach leads to the Fredholm integral equation of the second kind It can be solved numerically, e.g., via the collocation method, like in [19]. The method is generalized for lines with multiple wires In this approach, the solution with substitutive current filaments is regarded as the first approximation of the true solution. Several typical lines with round wires are considered as examples to check the performance of the method as well as the limits of applicability of the first approximation
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