Jim Lux wrote:
. . .
Lest you think I am nit picking here.. take a piece of venerable RG-8
style coax, with the AWG13 inner conductor (0.072" diameter, 1.83 mm).
The skin depth at 1.8 MHz (per the above post) is 0.18mm, so the wire is
10 skin depths across, so it's probably a reasonable assumption.

However, let's take something a bit smaller, like RG-8X or RG-58 type
coaxes, which have a inner conductor on the order of 0.9mm. Now, you're
talking only 4-5 skin depths, and the assumption of an infinite plane
probably doesn't hold.

That would be nit picking unless very high accuracy is required.

As Jim said, the current density actually decays from the surface in an
exponential manner. The skin depth is the depth at which it's dropped to
1/e its density at the surface. If a conductor is infinitely thick, the
total loss is exactly the same as if the current density was uniform to
the skin depth and zero below. So this approximation is widely used when
it can be assumed that the conductor is at least several skin depths
thick. A rigorous calculation for a round wire really requires a
computer, since it involves evaluating complex Bessel functions, and I
believe that closed form equations for many other wire shapes don't
exist at all.

But there are two levels of approximation you can make with the
assumption that the current is all flowing in a uniform layer. If you
calculate the cross sectional area of the ring of current, you come up
with (from simple geometry)

Area = pi * delta * (OD - delta)

where OD is the outer diameter of the wire, and delta is the skin depth.
The material's bulk resistivity is divided by this area to find the
wire's resistance per unit length.

If the diameter is much greater than the skin depth (OD delta), an
even simpler approximation can be and is often made:

Area ~ pi * delta * OD

I assume this is the infinite diameter assumption Jim mentions.

If you use this infinite diameter assumption, the error in the
calculated resistivity of a copper wire 0.9 mm diameter at 1.8 MHz is
5.4% (compared to a rigorous calculation). This error isn't a big deal
for most purposes. But by simply using the first rather than the second
equation for area, the error drops to less than 0.1%. You're still using
the approximation that the current is flowing in a uniform layer one
skin depth thick, so the entire calculation can easily be done on a
pocket calculator in a minute or so.

Roy Lewallen, W7EL