Roy Lewallen wrote:

Define:

eta=120 Pi
k=2/lambda

reactance = (eta/(4*Pi)) (2 SinIntegral[k l] +
Cos[k l]*(2 SinIntegral[k l] - SinIntegral[2 k l]) -
Sin[k l]*(2 CosIntegral[k l] - CosIntegral[2 k l] -
CosIntegral[(2 k a^2)/l]));

where 'a' is the radius.


. . .

This is the formulation by S.A. Schelkunoff, which is one of many
approximations to the general problem of finding the reactance of a
simple cylindrical dipole of arbitrary length and diameter. The general
problem was attacked for decades by some very skilled mathematicians and
engineers including R.W.P. King, David Middleton, Charles Harrison, G.H.
Brown, D. D. King, F. G. Blake, M.C. Gray, and others. You'll find their
works scattered about the IRE (now IEEE), British IEE, and various
physics journals. The problem can't be solved in closed form, so all
these people proposed various approximations, some of which work better
in some situations and others in others. A good overview can be found in
"The Thin Cylindrical Antenna: A Comparison of Theories, by David
Middleton and Rolond King, in _J. of Applied Physics_, Vol. 17, April 1946.


Thank you for that. If by chance you have that as a PDF, perhaps you can
mail it to me. But if not, I'll try to get it for interest sake. I
needed this for a piece of work, but the work will have finished by the
time I get much more done. But at least I have a better understanding now.


. . .

Does anyone have any comments? Any idea if Balanis's work is more
accurate? It is more up to date, but perhaps its an approximation and
the amateur radio book is more accurate. (The ham book seems quite
well researched, and is not full of the voodoo that appears in a lot
of ham books).

As I mentioned above, some approximations are better in some
circumstances (e.g., dipoles of moderate diameter near a half wave in
length) and some in others (e.g. fat dipoles or ones near multiples of a
half wave in length). I don't know which is better for your particular
question. The easy way to find out is to get one of the readily
available antenna modeling programs, any of which is capable of
calculating the answer to very high accuracy, and compare this correct
answer with the various approximations you find published.

OK. I'm just a bit suspicious of computer programs some times, as
someone will have to choose an algorithm of some sort. But I assume you
are talking of something like NEC which breaks antennas into segments.

BTW, I'm also looking for an exact formula for input resistance of a
dipole of arbitrary length. I know is 73.13 Ohms when 0.5 wavelengths
long, but I'm not sure exactly how much it varies when the length
changes (I believe it is not a lot).

There is no exact formula for that, either. Calculating an exact answer
requires knowledge of the current distribution, which is a function of
wire diameter. Assuming a sinusoidal distribution gets you very close
for thin dipoles, but it's not exact. You'll find calculations based on
this assumption in just about any antenna text such as Balanis or Kraus.

Balanis has it, but leaves it as an integral, without simplifying like
he does for the real part. Yet the formuals for hte real and imaginary
parts look very similar. I might be able to attack it with a computer
algebra system - maths never was my strongest subject.

I thought I'd looked in Krauss and not found it, but perhaps it is
there. I think there is a relatively new version of Kraus, but my copy
is quite old.

But again, you can get extremely accurate results from readily available
antenna modeling programs.

OK, thank you for that.

Roy Lewallen, W7EL