Hello Owen,

In electromagnetic waves & antennas (Orfanidis, paragraph 9.5), they
calculate the electrostatic capacitance of a two wire conductor
assuming the wires as equipotential surfaces (so no E-field components
parallel to the conductor surface). This implies non-uniform charge
distribution for close distance (so proximity effect). Also when you
look in the graph, the charge centers are closer to each other then
the center to center distance of the parallel wires.

From the capacitance you can directly calculate the characteristic
impedance based on c0 propagation speed.

The exact solution (based on the capacitor approach) comes with the
Acosh function. So I think that (some or all) proximity effect is
taken into account. Off course they assume all charge on the surface,
so skin depth diameter of wire.

The formula with the "ln" is just an approximation for the "acoh"
function when D/d is large. The acosh formula gives zero for touching
wires. This is correct for the assumption that all charge is on the
surface (so no magnetic field will penetrate the wires).

Hope this will help you a bit.



Best regards,

Wim
PA3DJS
www.tetech.nl (Dutch)


Owen Duffy ha escrito:
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