Dave wrote:
Oops, I made a couple of mistakes the
Dave wrote:
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/lambda
k = 2 Pi / lambda,
not 2 / lambda.
You can possibly see that when the length is 0.5 lambda, the sine term
in there is always zero, so the radius 'a' has no effect on the reactance.
What is interesting about that is that for a length of 0.5 lambda, the
reactance does not depend on wavelength at all - it is fixed at
42.5445 Ohms. So two different books give two quite different results.
Sorry, I mean the reactance does not depend on radius when the dipole is
0.5 wavelengths in length.
First, as you point out one book is using an approximation where the
other may be using calculated data. I believe the approximations start
by assuming a perfectly sinusoidal current distribution, which may not
be entirely correct, but does make the math easier. The ham radio book
would be more likely to use output from an antenna modeler, since
antennas are an area where theory gets damn complicated damn quick, but
the modelers can do a pretty good job if you treat them right.
Second, check to see if they're both using the same equivalent circuit
-- if one is looking at parallel equivalent reactance and the other
serial, that would account for the difference.
Third, check to see if the ham book is giving you figures for a dipole
way out in free space, or one that's mounted some known wavelength above
some known ground, or that has some real resistivity in the wire, or
some other 'real world' assumption that a pure theory book may not
bother with.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
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