Richard Fry wrote:

For the sake of discussion, below are two pastes from the same NEC
model using the demo version of EZNEC v. 5.0 -- which rather well
support my earlier post that the radiation resistance (NOT the
impedance) of an electrically short monopole is a function of its
electrical length, and not the loss resistance of the r-f ground and/
or the loading coil.

. . .
EZNEC calculated the radiation resistances of these two cases to be
0.14 ohms and 0.17 ohms, respectively -- fairly close, but not exact.
Perhaps Roy could comment on the reason why their agreement using NEC/
EZNEC is not better.

Sorry, I can't tell without seeing the EZNEC description file. If you'll
attach the .EZ file to an email message to me, I'll be glad to answer
your question. I wasn't able to get a radiation resistance that high at
that frequency for a 3 meter vertical of any diameter, so there's
something in the model which isn't immediately apparent.

Those wanting a good resource for the measured results for monopoles
of less than 1/8 electrical wavelength might try to locate the paper
by Carl E. Smith and Earl M. Johnson titled PERFORMANCE OF SHORT
ANTENNAS, published in the October, 1947 edition of the Proceedings of
the I.R.E.

The equation for the radiation resistance of short antennas given in
that paper is independent of the resistive losses in any loading coil
or r-f ground system.

And the same fundamental equations are used by modeling programs. The
problem is that interaction between the antenna, an abbreviated ground
system, and the Earth can modify the radiation resistance as well as
adding loss resistance. You might try modeling a few short verticals
with a few radials just above ground, and looking at the gain with
various radial systems. You'll find that the gain change doesn't exactly
correlate with the feedpoint resistance change when you assume a
constant radiation resistance. This isn't a shortcoming of the modeling
program, but a real effect. I doubt you'll find much about it in
pre-computer age texts, though, because it's probably a very tough, or
maybe impossible, manual calculation.

Roy Lewallen, W7EL