The acosh formula includes the effect of skin effect on inductance and
capacitance(*), while the simplified log formula is equivalent to
considering the currents concentrated along lines at the conductor
centers. Neither includes proximity effect.

Derivations of the acosh formula can be found in Johnson, _Transmission
Lines and Networks_, Sec. 3.8; Chipman, _Transmission Lines_ (Schaum's
Outline Series), Sec. 6.6; and other references.

I'm sure that a formula taking proximity effect into account would be
much more complicated, judging by approximate formulas I've seen for
calculating resistance change due to proximity effect.

From Chipman (p. 114), about the lack of published coverage of
proximity effect:

"Fortunately, the combination of circumstances that would require
accurate information about th eproximity effect factor for distributed
internal inductance occurs rather rarely in transmission line practice.
The most unfavorable situation would be a parallel wire line with solid
circular conductors, the facing suraces of the conductors being
separated by only a few percent of a conductor radius, operating at a
frequency to have a/[delta] have a value near 2. These conditions make
the distributed internal inductance comparable in magnitude to the
distributed external inductance, with a proximity factor that might be
as small as 0.8 or 0.85. There is no recognized basis for making an
accurate analysis of the total distributed inductance of a line for such
a case.

.. . . When the facing conductor surfaces are at least a conductor
diameter apart (s/2a = 2), the distributed internal inductance will be
less than 20% of the total distributed inductance, and the proximity
effect factor will me not less that 0.87. . . Proximity effect can then
not modify the total distributed inductance value by more than about 2%,
and the factor need be known only very roughly. . ."

(*)And even this is a high-frequency approximation which assumes that
the conductors are at least several skin depths thick. Expressions for
line inductance without this assumption involve Bessel functions, which
I assume would also appear in expressions for Z0.

Roy Lewallen, W7EL

Owen Duffy wrote:
I have found two common expressions for the Zo of a two wire line in
space.

One expression is Zo=276log(D/d).

The second is Zo=120acosh(D/d).

I have been searching for information on whether the acosh expression
takes into account proximity effect. Because it does not consider
conductivity or permeability of the conductors, one wonders if it does.

Laport has a graph that shows the log expression and a proximity
corrected line which turns out to be a very close fit to the acosh line
over the range that he plots. The curves are compared in
http://www.vk1od.net/balun/Ruthroff/R07.png .

I have another reference, Marchand (1947), which gives the log expression
then, the acosh expresssion and says "The hyperbolic cosine is obtained
because the currents are attracted to one another and become more and
more confined to the inside surface as the wires are brought closer
together."

This suggests (states) that the acosh expression (fully) accounts for
proximity effect on Zo.

Comments?

Owen