Peter,

To satisfy yourself that a half-wave dipole automatically transforms
end-to-end wire resistance to an equivalent lumped resistance of half its
value located at the dipole centre, use program RJELINE3. It takes only a
few seconds.

Enter F = 10 MHz, Open-wire line length = 7.5 metres = 1/4-wave.

As everybody knows a 1/4-wavelength of line (a half dipole), behaves as an
impedance transformer.

Any value Zo of open wire line will do. But try Zo around 500 ohms with thin
wire such as 0.2mm diameter.

Terminate the line with 99999999 + j99999999 ohms, ie., open circuit just
like the dipole ends.

Loop-ohms per metre of the wire is one of the computed results.

Another computed result is exact line length in wavelengths.

Vary line length until it is exactly 1/4 wavelengths.

The input impedance of the 1/4-wave length of open-circuited line is also
calculated and displayed.

It will be found that at exact resonance (vary length or frequency very
finely) the input impedance of the line will be a pure resistance ( jXin =
0) equal to half of the of the line end-to-end wire resistance.

It is obvious exactly the same transformation occurs when the wire
resistance is replaced by a uniformly distributed radiation resistance.

If your own programs significantly disagree then consign them to the junk
box.

As you may have noticed I never support my stuff by citing the usual old
wives. Never come across, even heard of most of 'em. There are no references
except my tattered note books. I came across various useful relationship
around 1960 when researching into methods of locating faults on oceanic
phone cables. But I daresay Heaviside preceded me. I dug up much information
and designed fault locating and other test equipment but very little was
published beyond contract manufacturing information. There were two articles
in the house engineering journal. I worked alone with a small group of
assistants, a lab and a workshop. I did present a series of lectures
afterwards, twice in Europe. But it was all just in a day's work with
occasional trips aboard cable laying ships and at manufacturers. The nearest
I got to the States was Newfoundland and Nova Scotia. I then shifted in
succession to several entirely different fields of operations. But no
experience is ever lost.
--
Reg, G4FGQ