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

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.
. . .

"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

Actually, the problem as I posed it is from an engineering point of
view, assuming that a particular constraint dominated the
problem...well, actually I had in mind stating it as wires that would
fit inside some diameter, but that complicated it more than I wanted as
an exercise for this group. Engineers should first ask what the goals
and constraints are, and the space the wires fit into may be one such
constraint. I suppose minimum copper cost would be the smallest wires
(that could handle the required power)!

A starting point for a more complete problem statement from an
engineering point of view might be, "Minimize total system cost,
expressed in net present value, of the system over its operating life,
under a particular set of performance and installation constraints."

Thanks, though, for your consideration of proximity effect. Without
that, I got the same answer as Roy, and appreciated Roy's inclusion of
the basis on which he made the calculation.

I'd point out that for coaxial cable construction, a particular D/d
minimizes the loss for a given D, and a different D/d minimizes the
electric field strength for a given power and therefore maximizes the
power handling capability of the line in the case where the line is
voltage-limited (generally low duty cycle pulse operation), and yet
another D/d minimizes the peak electric field strength for a given
voltage applied to the line. So long as the dielectric loss is
negligible and the dielectric is uniform, all those D/d ratios are
independent of dielectric fill. The D/d which minimizes loss is close
to the D/d which maximizes power handling capability of the line, if
the line is power-dissipation limited, but not exactly so because it
doesn't consider how the center conductor gets rid of its heat. The
minimum attenuation (if the inner and outer conductors are the same
smooth material and skin depth is small compared with the thickness of
each) is for D/d = 3.5911, which as others have pointed out yields
about 76.7 ohms with air dielectric, but if the dielectric is solid
polyethylene, it's about 51 ohms. If the center conductor is smooth
copper and the outer is corrugated aluminum, the minimum-attenuation
D/d increases. If the dielectric is foamed polyethylene, that pushes
the impedance back toward 75 ohms.

Cecil...if the line is limited in power handling by its temperature
rise (which is almost always the case, except for low duty cycle pulses
or line construction that's very different from what we commonly use),
the minimum attenuation configuration will also be the minimum net
dissipation configuration, and therefore close to the optimal power
handling capability. As an example of how cables we commonly use are
thermally limited, consider that solid-dielectric RG-213 at 40C ambient
and at 10MHz is rated at about 2kW. At 50 ohms, that's only 316 vrms
for a CW signal, and RG-213 is rated for use up to 5000 vrms, which
would 250 times as much power (so it better only come in little short
bursts that don't overheat the line). That "power handling is maximum
for Zo=30 ohms" thing is ONLY good if the line is limited by voltage
breakdown, not by temperature rise! It IS useful--in radar systems,
for example--but you must be careful to understand your application.

Cheers,
Tom

Reg Edwards wrote:
"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



Okay, Reg, then go read the material referenced by footnote 13. That's one
reason I included it. Maybe that way we won't need to rely on your memory.

John