Self-induction in two long parallel conductors connected to sinusoidal voltage source

Abstract

The paper deals with the self-induction of two parallel infinitely long conductors of arbitrary cross section connected to an ideal sinusoidal voltage source. The conductors do not move. A quasi-stationary behavior is assumed, the displacement current is neglected and the permeability of the conductors and their surroundings equals the permeability of vacuum. The subject matter of the paper is the calculation of impedance, which in the theory of circuits with lumped elements can replace 1 m of conductors. The definition of inductance and the formulae for its calculation that form part of textbooks on physics and electromagnetism have been the same for such a long time that nobody doubts their correctness. Only the recently published method for the calculation of current density in long parallel conductors allows calculating accurately the equivalent impedance and self-inductance, and also evaluating critically the present knowledge.

Transient Current Density in a Pair of Long Parallel Conductors

“…The published methods, with the exception of those in [1,3,24], are approximate, with the current density error unknown. The self-inductance of two conductors cannot be calculated without accurate current density [2] and thus approximate formulae continue to be used for the calculation of self-inductance of two coaxial conductors and two circular cross section conductors [2]. According to these formulae, inductance does not depend on conductor resistivity and voltage source frequency, in contrast to the inductance calculated using accurate current density.…”

“…At the instant t = 0, the conductors are connected to the source of sinusoidal voltage U(t) =Û sin ωt,Û = 1 V•m −1 , f = 10 5 Hz. The inductance L of the conductor pair can be established using the formula [2,15]…”

“…At time t = 0, the conductors are connected to the sinusoidal voltage source U(t) =Û sin(ωt + α),Û = 1 V•m −1 , the frequency f = ω/(2π) and the period T = 1/ f . This example is for f = 1 kHz, the same as in [2], Variant 4; however, in Figure 6 in [2], the results of the calculation of current density in conductors are only given for the steady state, i.e., for t → ∞. Figure 2 illustrates the dependences of current density on x and t for a constant y in the conductors A 1 and A 2 .…”

“…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].…”

“…Either there is a constant current density in the conductor cross section [3] or the loop is replaced by a circuit with lumped parameters. The circuit is a series connection of a resistor with the resistance R and an inductor with the inductance L. The quantities R and L are specified prior to the calculation of transient current, which is problematic [2]. Using the proposed method, it is therefore possible to solve problems that have been solved in a steady state (e.g., [4][5][6][7]).…”