Forgotten? How can we forget a "fact" we learned wasn't true in the
first place?

According to the many references I have, the equation you quote is a
simplified equation that's valid for a single wire over a perfect
conducting ground plane, where the height is a very small fraction of a
wavelength (i.e., radiation is negligible). Even when you ignore the
relatively poor conductivity and the permittivity of real ground, the
equation is certainly not valid if the wire is high enough for
significant radiation to take place. There are several reasons for this:

1. The field shapes become different from the shapes assumed in deriving
the equation.
2. Radiation would make the impedance complex rather than purely real.
3. The voltage between the conductor and ground depends on the path
taken to measure it, so "characteristic impedance" takes on a whole
different meaning, if it has any at all in this context.

There is, of course, also the problem of ignoring the finite
conductivity of real ground, which will likewise impact the angle of the
impedance.

It's surely tempting to take a nice, simplistic equation like this and
build from it a whole theory of how things work. The seductive thing
about it is that it seems to work, sort of, for some special
applications. But it's a house of cards, and is at its root based on
invalid assumptions. So all the wonderful theories that follow from it
are fatally flawed and not to be trusted.

As apparently the only person on this newsgroup to have "learned" this
"fact", it would serve you well to un-learn it. That is, if you're
really interested in discovering how things really work rather than
clinging to possibly mistaken notions about how they do.

Roy Lewallen, W7EL

Cecil Moore wrote:

Apparently, a lot of the otherwise knowledgeable people
on this newsgroup have forgotten that the formula for
the characteristic impedance of a single-wire transmission
line is 138*log(4h/d) where h is the height of the wire
above ground and d is the diameter of the wire. There's
no difference between that single-wire transmission line
and a lot of ham antennas. That single-wire transmission
line radiates just like an antenna.

1/2WL of #16 wire 24 feet in the air has a Z0 of 600 ohms.

If that center-fed dipole were terminated at each end with
a 600 ohm load, it would be a traveling-wave antenna with
a feedpoint impedance of 600 ohms. Take away the loads and
there's a match to 50 ohm coax at the feedpoint.

The only difference in those two antennas is that removing
the loads turned the antenna into a standing-wave antenna
and reflections are arriving back at the feedpoint, lowering
the feedpoint impedance.

Any coil installed in a standing wave antenna is going to
be subjected to both forward and reflected currents. There
is no hope of understanding the current in a loading coil
without understanding the component currents flowing both
directions through the loading coil.
--
73, Cecil, W5DXP