3. Aspects of the mathematical treatment

In order to avoid certain duplications in the analysis of the elliptic and rectangular strips, we denote the thickness of the strip at the abscissa value x by 2h(x), so that h is the constant b in the rectangular case, and the variable (9) in the elliptic case. To permit a coherent normalized frequency to be used, both for the elliptic and the rectangular sections, we introduce the variable

(11)

which is proportional to the square of the first parameter (4). We thus have the following notations:

 

Ellipse

Rectangle

 

D.c. resistance per unit length

R0

(12)

Normalized frequency

k

 

The classical skin-effect equation for a massive conductor is

(13)

with

(14)

where u stands for any axial component (electric field, current density, or vector potential) and where denotes the 2-dimensional Laplace operator. In the case of the circular cylinder, the magnetic lines of force are concentric circles at all frequencies, and there is no exchange of flux between the conductor and the surrounding dielectric. As a consequence, the internal current distribution can be studied alone, as a solution of (13) with circular symmetry, and the external field is usually disregarded. By contrast, for elliptic and rectangular sections, the magnetic field has a non-zero normal component penetrating into the conductor (except at high frequencies), so that the internal problem is not separable from the external one. Since  = 0 in the dielectric, the external problem satisfies (13) with  = 0, which is a Laplace equation, and the external and internal solutions are connected by boundary conditions. For thin conductors, only the Laplace equation and the boundary conditions remain, which produces a considerable simplification.

At a large distance D from its centre, a conductor carrying a total current I produces a magnetic field of tangential component , and hence a magnetic flux per unit length proportional to , which tends to infinity with D. Since the external problem depends on the position of the return conductor, which cannot be relegated to infinity, because of the preceding difficulty a coaxial return conductor is generally assumed of large, but finite, radius D, concentric with the go conductor. This makes negligible the proximity effect of the return conductor, but the arbitrary constant D appears in all inductance expressions.

Any skin-effect impedance has the nature of the impedance of a (distributed) RL circuit. In particular, it is known from circuit theory that the inductance L is a monotonically decreasing function of the frequency and takes its minimum value, at infinity. Since there is no internal magnetic field at high frequencies, is the only natural definition of the external inductance. Moreover, is then a minimum-reactance impedance, and the constant D disappears in this difference. In the following, we always evaluate the reduced impedance

(15)

where R0 is the d.c. resistance (12). The external inductance is the one related to the capacitance C per unit length of the pair of conductors by where c is the velocity of light. For a thin strip (ellipse or rectangle) we have:

(16)

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