2. Lateral skin effect

As mentioned in the introduction, the skin effect in thin conductors can be separated into a lateral problem and a depth-penetration problem. The lateral problem is, however, of a different nature for conductors with no sharp edges (such as a hollow elliptic cylinder) on the one hand, and for conductors with sharp edges, on the other. Since condition (2) for a thin strip does lead to sharp edges (points of infinite curvature) at x =  a, both for the thin ellipse and the thin rectangle, the current concentration towards the ends is more important, which results in a different law of increase of resistance. In this section we treat successively (a) thin conductors in general, (b) thin conductors with no sharp edges, (c) thin conductors with sharp edges.

Instead of the three dimensions a, b and appearing in (2) and (3), it is convenient to introduce the two dimensionless ratios

and

(4)

the lateral problem is obtained when b/a is made strictly zero, as the limit of (2). On the other hand, at frequencies where becomes large compared to unity, the lateral skin effect reaches its asymptotic behaviour. This means that the linear density i(x) and the external electromagnetic fields have high-frequency limiting values which are independent of frequency. The corresponding behaviour will be called asymptotic lateral, and thus assumes a frequency range such that

(5)

which is of course compatible with (2).

When the frequency increases further, so that (3) no longer holds, the true current density (A/cm2) begins to vary along the thickness coordinate, but the linear density (A/cm) and the external fields keep their lateral asymptotic values, because of the separation noted earlier of the lateral problem from the depth-penetration problem.

For thin conductors, the pattern of the variation of resistance with frequency is markedly different from that for massive conductors. There are in fact two distinct phases of increase (the lateral effect and the depth effect), obeying different laws and separated by a large frequency interval corresponding to (5), where the lateral effect has already reached its asymptotic state while the depth penetration has not yet come into play. Moreover, the linear density and the external fields reach their asymptotic behaviour in the first phase and remain unaltered during the second phase.

For any thin conductor with no sharp edges, such as a thin hollow elliptic cylinder of moderate eccentricity, the asymptotic linear density is finite at every point, and so is the lateral asymptotic resistance. The law of resistance increase therefore has the form shown in fig. 2: the first lateral phase AB is followed by a long stationary interval BC corresponding to (5), where the resistance keeps its constant lateral asymptotic value, the second phase (depth penetration) is CD and the resistance ultimately increases as the square root of the frequency.

Although the problem of the thin hollow elliptic cylinder is rather academic, its lateral asymptotic behaviour (in the phase BC) is so elementary and illuminating that it deserves a short discussion. Since the external magnetic field satisfies Laplace'146;s equation and has no component normal to the ellipse, the lines of force are homofocal ellipses. The linear current density (along the periphery of the ellipse) is given directly by the discontinuity of the tangential component of the magnetic field and is thus inversely proportional to the distance along the normal between two adjacent ellipses of the family. In particular, the linear density is independent of the thickness 2h of the cylinder, even if the latter is variable. By contrast, the asymptotic resistance (the constant ordinate of BC of fig. 2) depends on the thickness, because the true density i/2h (A/cm2) in an element of area 2hds (ds = element of length along the boundary) produces a dissipated power

(6)

involving h. In particular, if the thickness is chosen proportional to the lateral asymptotic density, i/2h is a constant in (6), just as at d.c., and the asymptotic value of R/R0 is 1. Since the resistance is a non-decreasing function of frequency, it must remain constant in the whole lateral phase, for the law of thickness variation adopted. For the hollow elliptic cylinder, the corresponding law defines the conductor as the interior between two homothetic ellipses, which means that the ratios of the major and minor axes are equal. In such a conductor, the increase of linear current density towards the ends of the major axis is exactly compensated by the increased thickness, so that the true current density remains uniform. As a trivial particular case, there is no lateral skin effect in a hollow circular cylinder of constant thickness.

The above discussion, leading to the resistance behaviour of fig. 2, was specifically restricted to thin conductors with no sharp edges, and does not apply to the flat strip, whether rectangular or elliptic. This is because the lateral asymptotic linear current density becomes infinite at the edges (x =  a) of the strip, so that the dissipation (6), and hence the resistance, is infinite for any thickness law h(x), unless h also becomes infinite at the edges, which is inconsistent with the assumption of a thin conductor. In fact the asymptotic linear density for a strip of any thickness is

(7)

where

(8)

is the average linear density, and I the total current.

Expression (7) is well known but will be derived again in this article. For a rectangular section, the linear density thus varies from the uniform density (8) at d.c. to the asymptotic density (7) at high frequencies. For the elliptic section, however, it is the true density i/2h which is uniform at d.c. Since the conductor of fig. 1 has the variable thickness 2h(x), with

(9)

the linear density at d.c. is not uniform, but is given by

(10)

so that the variation from d.c. to high frequencies in the elliptic case is much stronger.


Fig. 2. The resistance R as a function of the frequency for a thin hollow elliptic cylinder: AB is the lateral phase, BC the lateral asymptotic value, CD the phase of depth penetration. R0 is the value of R for zero frequency. The curves are only qualitative.


Fig. 3. As fig. 2 but now for a thin flat conductor. Here the curve BCE represents the asymptotic behaviour.

Since the lateral asymptotic resistance of a flat strip is infinite, the law of resistance increase must be of the form qualitatively shown in fig. 3. At the end of the lateral phase AB, the resistance reaches its asymptotic behaviour BCE, from which it deviates in accordance with CD when depth penetration comes into play. Curve BCE tends to infinity in accordance with a law still to be discovered, but certainly more slowly than the square root of the frequency since it is dominated by the latter behaviour at the end of phase CD. Finally, the lateral law AB, and its asymptotic behaviour BCE, are different for a thin rectangle and a thin ellipse, whereas the ultimate square-root law (at the end of phase CD) is the same in both cases.

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