Reg Edwards wrote:
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The characteristic impedance of an infinitely long wire is Zo.

If we cut the line and measure between the two ends we obtain an input
impedance of twice Zo. Which is the answer to our problem.

Zo is a function of wavelength, conductor diameter and conductor
resistance R where R includes the uniformly distributed radiation
resistance. On a high Zo line the radiation resistance is small
compared with Zo and the only effect of the radiation resistance is to
give Zo a small negative angle. Which when estimating Zo can be
ignored. (It is conductor resistance which at HF gives Zo of ALL
lines a very small negative angle).

This assumption is correct only when the transmission line conductors
are closely spaced. That isn't at all true for the halves of a dipole.

In the problem posed, the current is also uniformly distributed along
the low-loss line and radiation resistance is not the value we are
familiar with and what we might do with it.

And so we get approximately -

Rin = 120 * ( Ln( Wavelength / 2 / d ) - 1 )

At a wavelength of 2 metres and a conductor diameter of 10mm the input
resistance = 433 ohms.

I cannot guarantee the above formula to be correct. But is it low
enough for you? ;o)

I can't see how it can possibly be correct. Unless I'm mistaken, you've
completely ignored the effect of radiation in calculating the radiation
resistance. It sure makes the calculation a lot simpler, though!

Mr Wu calculates radiation resistance which is not the same as input
impedance unless correctly referenced. It is usual in technical papers
to calculate Radres at one end of the antenna. Or it may be the
distributed value. I havn't the time to find and study the full text.
From past experience, with me, it usually ends up as a wild goose
chase.

It depends on the author. Kraus uses feedpoint resistance and radiation
resistance interchangeably when loss is assumed to be zero. It's
traditional in AM broadcasting to give radiation resistance referred to
a current maximum. The conclusion is that radiation resistance can be
referred to any point along an antenna you wish, which means that it's
essential to state what point you're using as a reference.

Roy Lewallen, W7EL