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

"Owen Duffy" wrote in message
...

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

I must be doing something wrong. 276log(2)=83 but looks like about 150 on
your graph. No?

John

"John KD5YI" wrote in
news:Z2srj.1036$qw4.677@trnddc02:

I must be doing something wrong. 276log(2)=83 but looks like about 150
on your graph. No?

Sorry, that expression should be 276log(2D/d) for a consistent meaning of D
and d. (Some people use wire radius and some, diameter... so D here is
centre to centre distance, and d is diameter.)

Owen

On Sun, 10 Feb 2008 00:08:33 GMT, Owen Duffy wrote:

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).

Hi Owen,

I have the following expressions found in "Reference Data for Radio
Engineers," 22-22 Transmission Lines:

Zo = 120 acosh(D/d)
Zo ~ 276 log(2D/d)
Zo ~ 120 Ln(2D/d)

where the symbol ~ means "approximately equal to."

73's
Richard Clark, KB7QHC

Richard Clark wrote in
:

On Sun, 10 Feb 2008 00:08:33 GMT, Owen Duffy wrote:

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).

Hi Owen,

I have the following expressions found in "Reference Data for Radio
Engineers," 22-22 Transmission Lines:

Zo = 120 acosh(D/d)
Zo ~ 276 log(2D/d)
Zo ~ 120 Ln(2D/d)

where the symbol ~ means "approximately equal to."

Thanks Richard.

Yes, I saw those in that publication.

ln(x) is a good approximation of acosh(x) for large x, so that explains
one approximation, and the log expression is just a scaling of the ln
expression.

I put some importance on the difference between the approximately equal
and unqualified equal signs.

If the acosh expression fully accounts for proximity effect, it is
interesting that it is independent of frequency, conductivity, and
permeability. It would be nice to find a derivation.

I think if you derive C and L from first principles, you don't get the
same curve as acosh... which suggests that acosh is account for something
that L and C from first principles (assuming uniform current
distribution) doesn't.

Owen

The acosh formula includes the effect of skin effect on inductance and
capacitance(*), while the simplified log formula is equivalent to
considering the currents concentrated along lines at the conductor
centers. Neither includes proximity effect.

Derivations of the acosh formula can be found in Johnson, _Transmission
Lines and Networks_, Sec. 3.8; Chipman, _Transmission Lines_ (Schaum's
Outline Series), Sec. 6.6; and other references.

I'm sure that a formula taking proximity effect into account would be
much more complicated, judging by approximate formulas I've seen for
calculating resistance change due to proximity effect.

From Chipman (p. 114), about the lack of published coverage of
proximity effect:

"Fortunately, the combination of circumstances that would require
accurate information about th eproximity 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 suraces 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

Owen Duffy wrote:
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

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