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

Abstract

In a time dependent solution of electromagnetic field penetration into a conductor, it turns out that the impedance power series solution diverges sharply if the ratio conductor radius to skin depth a/δ exceeds 2.7, close to the mathematical irrational number e. This problem has previously been considered to be due to the role of inversion. However, in this paper we show that the power series solution may be derived analytically from Bessel function solutions where the latter does not show the divergence problem. Experimental results are presented and we find that the logarithmic slope of the ac resistance and reactance change close to the value a/δ=e in both models. Since both models are based on the solution of a diffusion-like equation then the divergence effect at e may be expected to occur in other diffusion processes.

Surface Wave, Skin Effect, and Per Unit Length Parameters of the Single-Wire Transmission Line at Low Frequency, for Nonmagnetic and Magnetic Wires

“…Those corrections are not only frequency-dependent but also approachdependent. They can be extracted from ( 28), ( 29), (33), (35), (40), (42), and are summarized below:…”

“…That is not the purpose of the present article, where the skin effect is a tool, not a target. Nonetheless, with respect to recent advances (2010's) regarding solid and tubular circular geometries, we may cite: [32], [33] for contributions to full time-domain skin-effect theory in homogeneous solid wires; [34], [35] for contributions to accurate numerical computations of skin-effect impedance in homogeneous cylindrical conductors; [36], [37], [38] for contributions to frequency-domain skin-effect impedance and field calculations in inhomogeneous cylindrical structures; and, also, [39], [40] for contributions to the problematics of the solitary conductor.…”

Influence of the Wire Spatial Structure on the Distribution of Product and the Peak Overpressure of Shockwave Generated by the Electric Explosion

“…According to the integral state of Maxwell's equations, the current density distribution, J(r), and the corresponding skin depth, δ, on the wire section are expressed by the Bessel function [23,31]. When studying the δ and the J(r) of wire in the electric explosion process, a new expression related to wire structure is needed.…”

Modelling the Impact of Ground Planes on Aircraft Transmission Cable Impedance

“…Skin depth, δ, is defined as the distance below the conductor surface where the current density has fallen to 1/e (~36 %) of the current density at the surface of the conductor, and underpins why the cable per-unit-length parameters are not constant for high frequencies [7]:…”