You're correct, I neglected proximity effect. And the wires are close
enough that it's a factor. What is the approximate expression you used?

Roy Lewallen, W7EL

Reg Edwards wrote:
As he says, Roy's resulta are very approximate. That's mainly because
he neglected proximity effect between the wires.

A better approximation is obtained by incorporating an approximate
expression for proximity effect. However, this makes differentiation
of the loss formula with respect to wire diameter ridiculously
tedious. So I found minimum loss by plotting a graph with a pocket
calculator and searching for it.
. . .

Provided skin effect is fully operative, ie., skin depth is about
1/6th wire diameter or less, proximity effect increases wire
resistance by dividing normal skin-effect resistance of a single
straight wire by K :

K = SquareRoot( 1 - Square( D / S ) )

where D is wire diameter and S is centre-to-centre wire spacing. Note
that resistance increases towards infinity as the pair of wires
approach contact with each other. This is confirmed by precision
measurements.

To minimise line attenuation for any given wire spacing, maximise U
with respect to D :

U = D * InvCosh( S / D ) * SquareRoot( 1 - Square( D / S ) ) .

-----
Reg, G4FGQ

Provided skin effect is fully operative, ie., skin depth is about
1/6th wire diameter or less, proximity effect increases wire
resistance by dividing normal skin-effect resistance of a single
straight wire by K :

K = SquareRoot( 1 - Square( D / S ) )

where D is wire diameter and S is centre-to-centre wire spacing.
Note
that resistance increases towards infinity as the pair of wires
approach contact with each other. This is confirmed by precision
measurements.

To minimise line attenuation for any given wire spacing, maximise U
with respect to D :

U = D * InvCosh( S / D ) * SquareRoot( 1 - Square( D / S ) ) .

====================================

Roy, having given a little more thought to it, I think that by
differentiating U with respect to D and equating dU/dD to zero, things
will then simplify and the value of the ratio S /D, and hence Zo, will
be obtained directly.

If you still have enough enthusiasm I leave it to you to perform the
differentiation.
----
Reg.