1. Introduction

Prof. Dr. V. Belevitch is the Director of MBLE Laboratoire de Recherches, Brussels.

The classical skin effect is a well-known phenomenon, a convenient reference being the recent survey by H. B. G. Casimir and J. Ubbink. In a wire of circular cross-section, the radial distribution of the current density is a Bessel function of argument proportional to the square root of the frequency At d.c., the density is uniform, whereas at high frequency the current is approximately concentrated in a peripheral sheet of thickness

(1)

( = permeability,  = conductivity,  = angular frequency; practical electromagnetic units), called the skin depth. Correspondingly, the resistance R per unit length increases from the d.c. value , at first in proportion to the square of the frequency, but ultimately it tends to infinity as the inverse of the skin depth, and hence as the square root of the frequency.

Qualitatively, the preceding results hold for massive conductors of any cross-section with a smooth boundary, the only difference being that the tendency of the current density to concentrate towards the surface is more marked at the points where the curvature is greatest. For instance, in a conductor of elliptic cross-section, the density at the ends of the major axis will ultimately be larger than at the ends of the minor axis.

The effect is particularly marked when the eccentricity of the ellipse is very large, and an interesting problem is the limiting case of a very thin elliptic cylinder (fig. 1a), which almost reduces to a flat strip of width 2a. The problem of major practical importance is, of course, the thin rectangular strip (fig. 1b) widely used in printed circuitry and in microwave applications. It so happens, however, that the elliptic case is amenable to analytical treatment whereas the rectangular case can only be treated numerically. Consequently both cases deserve to be considered, with the hope of deducing front the theory of the thin ellipse some results which hold, at least qualitatively, for the thin rectangle.

Because the skin-effect problem in thin conductors involves two dimensions of different orders of magnitude, a thickness 2b, and a lateral dimension 2a, with

(2)

it automatically splits into two almost unrelated problems. At frequencies such that

(3)

the current density is still uniform along the thickness coordinate, so that the only problem is the lateral distribution of the linear current density (A/cm), to be written as i(x), along the width coordinate: i(x)dx thus designates the total current between x and x + dx. The problem corresponding to the restriction (3) characterizes the lateral skin effect, and is the only one treated in this paper, because the depth penetration, occurring at much higher frequencies, is well known.

 

(a)

(b)

Fig. 1. a) Cylindrical conductor with elliptic cross-section. b) Cylindrical conductor with rectangular cross-section.

The plan of the article is as follows. A first section is devoted to a qualitative physical discussion of the lateral skin effect in thin conductors and the resulting increase in resistance with frequency. Although all the mathematical derivations are concentrated in the last section, some general remarks of a mathematical nature are necessary at the beginning and are included in the second section. A summary of the results is then presented: complete analytical results are given for the thin ellipse and are all original, to the best of our knowledge; the relatively meagre existing information on the thin rectangle is reviewed and a minor addition is made. In the following section, the impedance (both for the ellipse and the rectangle) is characterized by its poles (2) and zeros, which brings deeper additional information on its behaviour. Finally, an approximate treatment of the high-frequency behaviour of the impedance for the rectangle is given in the Appendix.

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