Cecil:

[snip]
What got me going on this subject is months ago, someone
asked what would be the feedpoint impedance of an infinitely
long dipole in free space. Reg said it would be about 1200
ohms. Since that figure is obviously related directly to Z0,
it got me to thinking about the similarity of dipoles to
transmission lines. In fact, Balanis, in his 2nd edition
"Antenna Theory" illustrates how a dipole is created by
gradually opening up 1/4WL of a transmission line. That's
on page 18. The current distribution on the dipole after
unfolding is the same as the current distribution on the
transmission line stub before unfolding.
[snip]

Well, as we all "know" the current wave on a dipole antenna is exceedingly
close to sinusoidal, but not [exactly] sinusoidal, because if it were
exactly sinusoidal it wouldn't be radiating. That [small] difference
between the actual current distribution on an antenna and the actual current
distribution on a transmission line is the [tell-tale] residual that
separates us from an exact analytic expression for the driving point
impedance of a dipole.

Interesting stuff...

Our man Reg [Edwards, RIP] always said that... "an antenna is just a lossy
transmission line." And of course that is what it looks like approximately.

Heh, heh... everyone always wanted to know the "mathematical" formulae and
theory behind Reg's compact programs. He tantalized us all with a peek or
two at some selections of his Turbo Pascal source code, but essentially left
us all wondering... "How does he do that?"

Apparently, at least one could infer so from his comments, Reg often modeled
antennas as "lossy" transmission lines in some of his
"nutcracker/lightweight" programs. Did he use the Heaviside/Kelvin
formulae? I wonder...

It would be interesting to compare how closely the input impedance of a 1/2
wave lossless feed line of appropriate Zo (Say 600 Ohms?) terminated in a 73
Ohm resistor would approximate that of a "real" dipole. At resonance it
would be 73 Ohms at least.

Such a comparison should be simple to check using EZNEC numerical readouts
for the dipole and comparing to the numerical results obtained from the
formula (1) for the input impedance of the terminated line.

[snip]
For transmission line analysis, we begin with simple lossless
line formulas and then add complexity such as losses per unit
length. For what we call near lossless feedlines, we often
ignore the losses or at least consider them to be secondary
effects. Going where angels fear to tread, I thought, why can't
these same principles be applied to dipole antennas with
admittedly reduced accuracy? Or as one of the r.r.a.a gurus
said: "A wrong answer is better than no answer at all." :-)
[snip]

In many transmission line modeling programs [such as the ones used by the
designers of xDSL modems who, unlike radio amateurs, need models that range
from DC and on up over many decades of frequency range.] the fundamental
transmission line parameters R, L, C, G are often replaced by [empirically
derived] functions of frequency that represent "perturbations" from the
constants to mimic skin and proximity effects. Both R and L are
simultaneously affected by skin and proximity effects. Of course both
Heaviside and Kelvin knew of these effects but could not include them in
their simple derivations.

Speaking of the "L" parameter and proximity... I thought the article by
Gerrit Barrere KJ7KV in the most recent QEX was interesting because he
points out that a large fraction of the "L" parameter in transmission lines
results from the mutual inductance between and because of the proximity of
the two conductors not the individual self inductance of the conductors.

This is not obvious when looking at the "standard textbook"
presentation/derivation of the Heaviside/Kelvin formulation for a
differential section of transmission line. Such standard textbook
derivations almost universally begin with a lumped differential model
consisting of series R, series L, shunt C, shunt G per unit length with no
mention of mutual inductance. In fact of course the "standard" model is
"equivalent" to a model that explicitly exhibits the mutual inductance,
[Because Leq = L1 + L2 + 2M] but it is much more physically satisfying to
see the transmission line inductance presented the way Barrere did.

Thoughts, comments?

--
Pete K1PO
Indialantic By-the-Sea, FL

"Cecil Moore" wrote in message
. ..
Peter O. Brackett wrote:
Zo(Z) = ZL[(cos(theta) - jZ*sin(theta))/(Zcos(theta) - jsin(theta))] (2)

[Aside: Apart from the fact that line parameters L,C are also implicit
in the wavelength, Cecil is this right?]

What got me going on this subject is months ago, someone
asked what would be the feedpoint impedance of an infinitely
long dipole in free space. Reg said it would be about 1200
ohms. Since that figure is obviously related directly to Z0,
it got me to thinking about the similarity of dipoles to
transmission lines. In fact, Balanis, in his 2nd edition
"Antenna Theory" illustrates how a dipole is created by
gradually opening up 1/4WL of a transmission line. That's
on page 18. The current distribution on the dipole after
unfolding is the same as the current distribution on the
transmission line stub before unfolding.

For transmission line analysis, we begin with simple lossless
line formulas and then add complexity such as losses per unit
length. For what we call near lossless feedlines, we often
ignore the losses or at least consider them to be secondary
effects. Going where angels fear to tread, I thought, why can't
these same principles be applied to dipole antennas with
admittedly reduced accuracy? Or as one of the r.r.a.a gurus
said: "A wrong answer is better than no answer at all." :-)

My thoughts didn't go to solving for Z0 as you did above.
Using the well known Z0 formula for a single wire transmission
line above ground, we get Z0=600 ohms for #14 wire 30 feet above
ground and it certainly bears a resemblance to an infinite dipole
made of #14 wire 30 feet above ground. Putting a differential
balanced source in the middle of the single-wire transmission
line would result in a balanced feedpoint Z0 impedance of 1200
ohms. 1/2 of this dipole resembles a 1/4WL stub. An infinite
dipole is, of course, a traveling wave antenna.

This is getting long but I think you can see where it is going.
Make each 1/2 of the dipole equal to 1/4WL and we have the
standard standing wave antenna. Analyze the 1/2 dipole as a
lossy 1/4WL stub with a Z0 of 600 ohms not differentiating between
radiation loss and other losses. (For this purpose, we are not
interested in analyzing the radiation.) Hence, the earlier
lossy stub where the impedance looking into the stub was 50
ohms and the Z0 was 600 ohms.

Now quoting Balanis again, page 488 and 489:
"The current and voltage distributions on open-ended wire
antennas are *similar* to the standing wave patterns on open-ended
transmission lines." "Standing wave antennas, such as the dipole,
can be analyzed as traveling wave antennas with waves propagating
in opposite directions (forward and backward) and represented by
traveling wave currents If and Ib in Figure 10.1(a)."

Figure 10.1(a) is very similar to the graphic depicting
a single-wire transmission line over ground whe

Z0 = 138*log(4D/d) D=height, d=wire diameter
--
73, Cecil http://www.w5dxp.com