Roy Lewallen wrote in
:

This might be useful:

King, in _Electromagnetic Engineering_, Vol. 1, says (pp. 467-8):

". . .the problem may be analyzed for wires sufficiently far apart and
so small in radius that rotational symmetry in the interior and on the
surface of each conductor is a good approximation, and then
generalized to closely spaced conductors of large radius simply by
writing

ae= a * sqrt(1 - (2a/b)^2)
be = b/2 * (1 + sqrt(1 - (2a/b)^2))

instead of a and b in formulas for the external impedance."

a is the wire radius, b the center-center spacing, ae is the radius to
be substituted and b is the spacing to be substituted. He's saying
that you can account for proximity effect just by substituting
equivalent wires of smaller diameter and closer spacing which have
idealized evenly distributed current around the outside.

He introduces the equation for ae earlier without proof, but
references a 1921 paper in an obscure publication, as a way to adjust
internal inductance for the proximity effect. The method is much
simpler than the approximate equations for added loss due to proximity
effect I've come across, so I suspect it's an approximation, but I
don't know the applicability limits. But you might try incorporating
it and see how it compares with measurements and with calculations
based on the more complicated equations.

Hi Roy,

Thanks again for the research.

I would have thought that current would distribute itself in the
conductors for the least voltage drop along the conductor due to
inductance and bulk resistance. That suggests that as the conductors are
brought very close together, one expects that Zo should decrease
smoothly(ie the Zo is a monotonically increasing funtion of distance
between conductors for all positive distance).

If I take acosh(be/(2*ae)), it has a minimum around be/(2*ae)=1.3.

In the expression ae= a * sqrt(1 - (2a/b)^2), ae goes to infinity as b
approaches 2a. This would drive acosh(be/(2*ae)) towards infinity as the
wires are bought together.

I see that when the wires are just touching (b = 2a), you end up
substituting wires of radius zero, spaced at half the actual distance.
I suppose that's reasonable.

I think there are probably limits for use of these formula.

The turning point in the region of (2*ae)=1.3 concerns me. I cannot
intuitively see an explanation for such a behaviour.

Owen