Skin Effect - Formula

Formula

The AC current density J in a conductor decreases exponentially from its value at the surface J_S according to the depth d from the surface, as follows:

where δ is called the skin depth. The skin depth is thus defined as the depth below the surface of the conductor at which the current density has fallen to 1/e (about 0.37) of J_S. In normal cases it is well approximated as:

.

where

ρ = resistivity of the conductor

ω = angular frequency of current = 2π × frequency

μ = absolute magnetic permeability of the conductor

A more general expression for skin depth which is more exact in the case of poor conductors (non-metals) at high frequencies is:

where is the electric permittivity of the material. Note that in the usual form for the skin effect, above, the effect of cancels out. This formula is valid away from strong atomic or molecular resonances (where would have a large imaginary part) and at frequencies which are much below both the material's plasma frequency (dependent on the density of free electrons in the material) and the reciprocal of the mean time between collisions involving the conduction electrons. In good conductors such as metals all of those conditions are insured at least up to microwave frequencies, justifying this formula's validity.

This formula can be rearranged as follows to reveal departures from the normal approximation:

At frequencies much below the quantity inside the radical is close to unity and the standard formula applies. For instance, in the case of copper this would be true for frequencies much below Hz.

However in very poor conductors at sufficiently high frequencies, the factor on the right increases. At frequencies much higher than it can be shown that the skin depth, rather than continuing to decrease, approaches an asymptotic value:

This departure from the usual formula only applies for materials of rather low conductivity and at frequencies where the vacuum wavelength is not much much larger than the skin depth itself. For instance, bulk silicon (undoped) is a poor conductor and has a skin depth of about 40 meters at 100 kHz (=3000m). However as the frequency is increased well into the megahertz range, its skin depth never falls below the asymptotic value of 11 meters. The conclusion is that in poor solid conductors such as undoped silicon, the skin effect doesn't need to be taken into account in most practical situations: any current is equally distributed throughout the material's cross-section regardless of its frequency.