I know this is a bit off-topic. but possibly someone here has some
thoughts. I've posted it to rec.radio.amateur.antenna and
sci.electronics.design, but someone here might know. I've corrected a
couple of typos that appeared on the newsgroup post.
--------
I wish to know if the reactance of a dipole that is physically 0.5000
wavelengths in length depends on the diameter of the wire or not.
I know a dipole 0.5 wavelength long is not resonate, but inductive so
you need to shorten it a few percent to bring it to resonance. I know
the length at resonance depends on wire diameter.
But I'm interested if the reactance does very with wire diameter when
the antenna is physically 0.5 wavelengths long, which means it will be
somewhat inductive.
A book published by the ARRL by the late Dr. Laswon (W2PV)
Lawson J. L., “Yagi Antenna Design”, (1986), The American Radio Relay
League. ISBN 0-87259-041-0
has a table of reactance vs the ratio K (K=lambda/a, where a is the
radius) for antennas of 0.45 and 0.50 wavelengths in length. I've
reproduced that table below.
The first column (K) is lambda/a
The second column (X05) is the reactance of a dipole 0.5 wavelengths in
length.
The third column X045 is the reactance for a dipole 0.45 wavelengths in
length.
K X05 X045
-------------------------
10 34.2 23.1
30 36.7 6.4
100 38.2 -14.1
300 39 -33.6
1000 39.6 -55.5
3000 40 -75.7
10000 40.4 -98.1
30000 40.6 -118.6
100000 40.8 -141.1
300000 41.0 -161.8
1000000 41.1 -184.4
What one notices is:
1) Reactance for 0.45 lambda is very sensitive to radius, varying by
more than 200 Ohms as K changes from 10 (fat elements) to 1000000 (thin
elements).
2) The value for a dipole 0.5 lambda in length changes much less (only 6
Ohms), but it *does* change.
3) For infinitely thin elements (K very large), the reactance of a
dipole 0.5 lambda in length looks as though it is never going to go much
above 41.2 Ohms. Certainly not as high as 42 Ohms.
Now I compare that to a professional book I have:
Balanis C. A., “Antenna Theory – Analysis and Design”, (1982), Harper
and Row. ISBN 0-06-0404458-2
There is a formula in Balanis' book for reactance of a dipole of
arbitrary radius and length, in terms of sine and cosine integrals. It's
hard to write out, but the best I can do gives:
Define:
eta=120 Pi
k=2 Pi/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 a2)/l]));
where 'a' is the radius.
(It's in Mathematica notation)
What is interesting about that is that for a length of 0.5 lambda, the
reactance does not depend on diameter at all - it is fixed at 42.5445
Ohms. So two different books give two quite different results.
Numerically evaluating the above formula gives this data.
K X05 X045
-------------------------
10 42.5 35.7183
30 42.5 15.5269
100 42.5 -6.79382
300 42.5 -27.1632
1000 42.5 -49.4861
3000 42.5 -69.8555
10000 42.5 -92.1784
30000 42.5 -112.548
100000 42.5 -134.871
300000 42.5 -155.24
1000000 42.5 -177.563
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).
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).
Dave G8WRB.
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