On the Success of Electromagnetic Analytical Approaches to Full Time-Domain Formulation of Skin Effect Phenomena

Abstract

Abstract-Maxwell equations can be used to formulate an analytical full time-domain theory of skin effect phenomena in circular cylindrical conductors without any detour into the frequency domain. The paper shows how this can be done and concomitantly provides the means to determine the time-varying per unit length voltage drop along the conductor from a given time-varying conductor current. The developed relationship between voltage and current is not very complicated and led the authors to examine the reasons why it has never been utilized in transient analysis, nor given special emphasis in the literature. Those reasons are thoroughly examined and the conclusion is that the conditions required for the application of a purely time-domain skin effect theory are very restrictive.

Skin effect in the time and frequency domain—comparison of power series and Bessel function solutions

“…However, even though this transient approach gives the correct results at low frequencies and is consistent with Bessel function solutions it turns out that the series solution diverges if the ratio conductor radius to skin depth a/δ exceeds 2.7. This coincidentally is close to the mathematical irrational number e=2.718... [1,8]. This divergence has been considered previously to be a consequence of mathematical inversion but this has not yet been proved for the power series case.…”

“…Hence, where S, T o , T 1 ... etc are functions of time. After differentiating equation (6) and after some considerable algebra [9] which involves inversion of the power series for the current ('brute force approach', [12]) or using matrix inversion algebra [1,10] the emf becomes where μ=μ o μ r /(4π) and I ( m) is the mth derivative of the current. I is the current, R o and L o are the low frequency resistance and inductance respectively.…”

“…Conversely the internal inductance decreases with increasing frequency as 1/ f . This is the well known Skin Effect which has been studied for well over a century, [1,2]. Much of the interest in the early work concerned skin effect in the frequency domain mainly because of the dominance of analogue communications.…”

“…We first show that the time dependent solution can be derived from the Bessel function solution and also give an alternative method for multiplying the power series polynomials to obtain higher order terms. This, like in the matrix method [1], has the advantage of using computer techniques to obtain higher order terms in the series and thus examine its stability. We then investigate the behaviour of the impedance in the viscinity of a/δ=e both for the power series case and the Bessel function case and these theories compared with experimental measurements.…”

Magnetic charge, magnetic current, magnetic scalar potential, and electric vector potential

“…Note 2: In the case of imperfect wires, the result in (17) must be modified to include perturbation terms due to skin effect. Time‐domain skin effect terms require i ( t ) and all its time‐derivatives to be known .…”