Current density in a group of long parallel conductors

Abstract

An original method is proposed for the calculation of current density in a finite number of long parallel solid conductors of arbitrary cross section. Some pairs of the conductors under examination can be connected to a voltage source; they are active. The other conductors are passive. The currents, voltages and magnetic fields are assumed to be quasi-stationary and the displacement current is neglected. The permeability of the conductor material is constant and equals the vacuum permeability. No method has so far been published that would allow an exact calculation of current density in all the cases that satisfy the above assumptions. The application of the method is demonstrated by solving an example with two active conductors, one of which is of circular and the other of rectangular cross section, while a third passive conductor is of rectangular cross section.

Transient Current Density in a Pair of Long Parallel Conductors

“…The essence of the method for calculating current density as published in [1] consists in the conductors being replaced by the partial conductors A i , ∀i. The symbol ∀ means that the quantity after it takes the values 1, 2, .…”

“…is the centre of the rectangle A i . The partial conductors A i , which form a(A 1 ) and a(A 2 ), must be numbered such that the graph with N vertices (X i , Y i ) and N − 1 edges is a path graph [1,8]. It is assumed that, in the partial conductor A i , ∀i, there is a constant resistivity i = (X i , Y i ) and that the current density J i (t) does not depend on x and y.…”

“…According to Coufal [1], the current densities J i (t), ∀i, in conductors A 1 and A 2 are a solution to the system of N − 1 ordinary differential equations…”

“…The source voltage is assumed to be dependent on time t. Due to induction, the current density in the conductors is a function of t too. It is only recently that a method has been published for accurately calculating current density in a group of long parallel conductors [1]. The method has thus far been applied to the calculation of steady-state current density in a loop connected to a sinusoidal voltage source [2].…”

“…• adjustment of the method of current density calculation proposed in [1] to a form suitable for numerical calculation of transient current density; • presentation of results of the calculation of transient current density and transient current; • analysis of transient current when connecting on and short-circuiting a voltage source; • results of the calculation of the dependences of maximum current density, maximum current and the time of achieving steady state on the distance of circular cross section conductors, the frequency of sinusoidal voltage source and the resistivity of conductors.…”