In Part 1 of this series (located here), I described how skin effect is all about current density. The units of current density are current/area², for example amps/cm². If we multiply current density by cross-sectional area, we get current.

So here are two propositions. They are well accepted in electronics, and are useful models, even if they are not exactly correct, as we shall see. They are also mutually exclusive!

At high enough frequencies, so that the skin effect is in play:

Proposition 1
There is a depth below the surface called the skin depth (sd). It is given by the formula:

 

where

ρ = resistivity (0.6787 uOhm-in)
ω = angular frequency = 2*π*f
μ = Absolute magnetic permeability of the conductor (3.192*10^-8 weber/amp-in)
f = frequency in Hz.

The skin depth defines a cylindrical cross-section between the surface of the (circular) conductor and the skin depth. This cylindrical cross-section has a higher resistance (per unit length) than does the full conductor, because the cross-sectional area of the cylinder is smaller. The current flows uniformly through this cylindrical cross-section. By calculating the area of this cylindrical cross-section, we can calculate the (increased) resistance of the conductor caused by the skin effect.

Proposition 2
The current density is highest at the surface and declines exponentially according to the (normalized) formula (I’ll explain what I mean by “normalized” in a moment):

where 

J = current density           
sd = skin depth
d = distance in from the surface
e = base of the natural logarithm (2.718)

These relationships are graphed in Figure 1. The graphs in Figure 1 are normalized. That means we have unit current density on the vertical axis and unit area on the horizontal axis. If we want to apply these (normalized) graphs to a real-world situation with, say, 20 ma of current through a conductor with an 8 mm² area, we would multiply the vertical axis by 2.5 ma/mm² and the horizontal axis by 8 mm². Equation 2 is normalized in the same manner. It applies to unit current and unit area.