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