Jim Kelley wrote:

Hi Ian,

Please know that my comment was never intended as a slight of anyone's
work. I simply don't presume to know anything about it other than to
observe that the citation appears to contradict the assertion that skin
depth is limitless and exponential in real conductors.

73, ac6xg

that's because the usual discussion of "skin depth" is cribbed from a
physics textbook, where the (not always explicitly said) assumption is
"in an infinite uniform plane of infinite depth with no other magnetic
fields"

In that restricted (but useful) case, you can model the current (for the
purposes of things like resistivity) as if it were uniform from the
surface to the skin depth.

In cases where the thickness of the conductor is "large" relative to the
skin depth, the error in using the rectangular layer of current
assumption is "small".

In cases where this assumption isn't valid (or, if you need higher
precision), then a more complete analytical formulation is needed. If
the conductor happens to be circular, then Bessel functions are surely
involved (differential equations in circular things almost always
involve Bessel functions and/or Hankel transforms). Since most of us
don't do Bessel functions in our heads, we use tables or lookups.

There's two sets of tables and graphs for round conductors: one is for
solid conductors; the other is for tubular conductors. Different
boundary conditions on solving the differential equations, so different
analytical solutions.

A 1998 paper by Gaba and Abou-Dakka gives all the equations and
background, and adds the details needed for stranded wires and cables
made of multiple substances (e.g. ACSR power lines).

There's also some analytical solutions for square and rectangular cross
sections, but they're pretty ugly, compared to the round conductors.

once you start talking multiple materials and dielectrics, it becomes
easier to do FEM (following the dictum of my father's differential
equations professor: useful differential equations should be solved
numerically, because the analytical solution is often harder and more
computation than the numerical one). (another good example of this is
calculating the field between two spherical electrodes)