More for the fun of pondering it than for any immediate practical
reason...

If I make a balanced two-wire transmission line from round wires, and
I'm constrained to have the wire center-to-center spacing some
particular value (say one inch, or five centimeters, or whatever), what
wire diameter should I use to get the lowest matched-line loss? What
impedance line does that give (assuming air dielectric)?

Clearly for a given wire diameter, the wider the spacing the lower the
loss up to the point where radiation plus dielectric loss becomes
significant, but I can imagine situations where you want to limit the
wire spacing and get low loss.

Cheers,
Tom

Unless I goofed up the math (a distinct possibility), the answer is that
you want a Z0 of about 144 ohms, which is a spacing/diameter ratio of
about 1.81.

This was pretty easy to solve via differentiation using the
approximation Z0 ~ 120 * ln(2S/d) where S = the center-to-center wire
spacing and d is the wire diameter. The answer I got was 120 ohms, or d
= 2S/e, or S/d ~ 1.36. Unfortunately, this spacing is too close for the
approximation to be accurate, so the answer wasn't good. When I used the
full inverse hyperbolic cosine formula for Z0 and took the necessary
derivative, I found the resulting equation too messy to solve in closed
form. So I went brute force and used a simple program to iterate and
spot the maximum. It looks to me like you need to maximize d * Z0 in
order to minimize the loss (see below).

Don't trust my answer without some checking -- I did it pretty quickly
so there's lots of room for error.

It's good to have an opportunity to dust off the neurons once in a
while. Thanks.

------

Loss per unit length = attenuation constant alpha = 1/2 * (R/R0 + G/G0)
nepers/m. With air dielectric, G ~ 0, so alpha = R/(2*R0), where R = AC
resistance per meter (both wires) and R0 = (assumed purely real) Z0.
Assuming skin effect is fully developed, R ~ k / d where d = wire
diameter and k depends on frequency, resistivity, and permeability of
the wire but not on d, S, or Z0. So alpha = k / (2 * d * R0). This shows
that minimizing alpha (loss) requires maximizing d * R0, which appears
to occur when R0 ~ 144 ohms (S/d ~ 1.81).

-----

Roy Lewallen, W7EL

K7ITM wrote:
More for the fun of pondering it than for any immediate practical
reason...

If I make a balanced two-wire transmission line from round wires, and
I'm constrained to have the wire center-to-center spacing some
particular value (say one inch, or five centimeters, or whatever), what
wire diameter should I use to get the lowest matched-line loss? What
impedance line does that give (assuming air dielectric)?

Clearly for a given wire diameter, the wider the spacing the lower the
loss up to the point where radiation plus dielectric loss becomes
significant, but I can imagine situations where you want to limit the
wire spacing and get low loss.

Cheers,
Tom

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.

At HF when skin effect is fully effective, and neglecting dielectric
loss in comparison with conductor loss -

For a fixed wire spacing, as wire diameter increases, the wires get
closer together and proximity loss eventually increases faster than
ordinary loss decreases due to the increase in diameter.

Thus minimum loss occurs at a smaller diameter and a greater
Spacing/Diameter ratio. The Ro at which minimum loss occurs is
independent of both frequency and wire conductivity. Results are -

Ro = 177 ohms. Spacing between wire centres is 2.29 times wire
diameter.

Which demonstrates that mathematics is vastly superior and takes
priority over practical experiments and making measurements.

From an engineering point of view, K7ITM asked the wrong question. He
should have asked, for a given wire spacing, what wire diameter
minimises the cost of the copper. Or something like that.

Many years back a similar sort of calculation was done for coax. Coax
does not suffer from proximity effect. It's easier to work out. The
answer was 75 ohms. That's how 75 ohms became the standard
comunications Ro. There are many millions of miles of the stuff. The
Chinese are now making even more of it.
----
Reg, G4FGQ

Reg Edwards wrote:
Many years back a similar sort of calculation was done for coax. Coax
does not suffer from proximity effect. It's easier to work out. The
answer was 75 ohms. That's how 75 ohms became the standard
comunications Ro.

I vaguely remember something about efficiency Vs power
handling capability being the difference in the 75 ohm
standard and the 50 ohm standard. Is that right?
--
73, Cecil http://www.qsl.net/w5dxp

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Cecil Moore wrote:
Reg Edwards wrote:

Many years back a similar sort of calculation was done for coax. Coax
does not suffer from proximity effect. It's easier to work out. The
answer was 75 ohms. That's how 75 ohms became the standard
comunications Ro.


I vaguely remember something about efficiency Vs power
handling capability being the difference in the 75 ohm
standard and the 50 ohm standard. Is that right?
--
73, Cecil http://www.qsl.net/w5dxp

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Page 5-15 of The ARRL UHF/Microwave Experimenter's Manual says:

"Consider that both power handling capability and cable losses vary with Zo.
It has been shown that cable losses are minimum at a characteristic
impedance on the order of 75 [Ohms], while power handling capability is
maximum at a Zo of about 30 [Ohms]."

(The book used the Greek symbol rather than [Ohms])

The quoted passage is in a chapter by Dr. Paul Shuch, N6TX, Professor of
Electronics, Pennsylvania College of Technology. At the end of the quote, is
an indication to see footnote 13 which is:

"Moreno, Theodore, Microwave Transmission Design Data, Dover Publications,
1948."

73,
John

"John - KD5YI" wrote
Cecil Moore wrote:
Reg Edwards wrote:

Many years back a similar sort of calculation was done for coax.
Coax
does not suffer from proximity effect. It's easier to work out.
The
answer was 75 ohms. That's how 75 ohms became the standard
comunications Ro.


I vaguely remember something about efficiency Vs power
handling capability being the difference in the 75 ohm
standard and the 50 ohm standard. Is that right?
--

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Page 5-15 of The ARRL UHF/Microwave Experimenter's Manual says:

"Consider that both power handling capability and cable losses vary
with Zo.
It has been shown that cable losses are minimum at a characteristic
impedance on the order of 75 [Ohms], while power handling capability
is
maximum at a Zo of about 30 [Ohms]."

(The book used the Greek symbol rather than [Ohms])

The quoted passage is in a chapter by Dr. Paul Shuch, N6TX,
Professor of
Electronics, Pennsylvania College of Technology. At the end of the
quote, is
an indication to see footnote 13 which is:

"Moreno, Theodore, Microwave Transmission Design Data, Dover
Publications,
1948."

73,
John
============================================

It is unreliable to use ARRL and similar publications as Bibles. They
are written by amateurs for amateurs and tell only a sufficient
fraction of the whole story. Phrases such as "It has been shown that
..... " arise. They also refer to UHF/Microwave when LF and HF are of
interest. At microwave frequencies the dielectric loss cannot be
considered negligible.


For minimum attenuation, air-spaced coax Zo = 75 ohms and D/d = 3.6
For solid polyethylene Zo is smaller.


Confusion about the value of Zo which maximises power handling
capabilty arises because coax cables have different shapes and
materials to support the inner conductor. Even though the dielectric
may be considered lossless its presence affects matched line loss.


If my memory serves me correct, for maximum power handling I think 50
ohms refers to air-spaced coax and 30 ohms or thereabouts refers to
solid polyethylene dielectric. Or it may be the other way about. It
will be different again for a different dielectric permittivity.


For a coax line used as a tuned circuit, eg., when short-circuited,
maximum impedance at resonance occurs when Zo = 132 ohms and D/d ratio
= 9.1


And just to add a little more to the confusion, whether the outer
conductor is solid or braided also makes a small difference.
----
Reg, G4FGQ

Cecil Moore, W5DXP wrote:
"I vaguely remember something about efficiency versus power handling
capability being the difference in the 75 ohm standard and the 50 ohm
standard."

That seens exactly right. It`s the reason you would use 75-ohm Zo cable
for TV distribution at low-power where you want to minimize loss but
there is no danger of too much voltage flashing over the cable.

Terman in his 1955 edition on page 106 says:
"---in an air-insulated coaxial line of given outer radius b, Q will be
maximum when the inner conductor has a size such thet b/a = 3.6.
(b=inner radius of outer conductor in concentric line, and a=outer
radius of inner conductor in concentric line) corresponding to Zo = 77
ohms. These are also the proportions for minimum power loss----."
However the maximum power that can be transmitted without exceeding a
given voltage gradient occurs when b/a 1.65, giving Zo = 30 ohms."

So for minimum loss you would want Zo of about 75 ohms and for maximum
power capability you would want 30 ohms. I suspect Zo=50 ohms is a
compromise between power handling capability and reasonable loss.

Best regards, Richard Harrison, KB5WZI

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.