Sorry, I clicked Send before fixing up my typos in the Chipman
quotation. Here's how it's really written:

"Fortunately, the combination of circumstances that would require
accurate information about th proximity 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 surfaces 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

Roy Lewallen wrote in
:

Thanks Roy.

I was suspicious that proximity correction could ignore conductivity,
permeability, frequency and the actual diameter and distance.

Chipman paints a picture that it is pretty messy mathematically, and the
flow into Zo would be really messy.

What I do glean is that for D/d2, the error is small. That is a whole
lot better than the log formula which is poor for D/d10.

I have two immediate applications, one is a model for a 1:1 choke balun
(http://www.vk1od.net/balun/Guanella/G.1-1.htm) and the other is yet
another transmission line loss calculator, along the lines of my existing
calculator that is aware of about 100 standard transmission line types,
but this one allowing specification of an arbitrary two wire line
(http://www.vk1od.net/tl/twllc.htm).

For the balun cases, I am very interested in D/d2.

I think I will continue to use the acosh expression, and let it calculate
without limit on D/d, but qualify the results with a note that proximity
effect is not included in Zo, Zo is underestimated for D/d2 and the
error may be significant.

Roy, thanks for your time to research and type the notes up.

Owen

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.

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.

Roy Lewallen, W7EL

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