Hi Dave,
The answer is yes.
The Antenna Engineering Handbook, 1st edition by Jasik; later editions by
others, McGraw Hill, various dates, show these relations quite well graphically
but, omit the precise formulae.
This topic is covered extensively in Chapter Five of "Antennas" by Kraus,
McGraw Hill, 1950. A book well worth having in this hobby.
The formula (10-62) you seek is in Chapter 10 on page 262 but, it is way too
complicated to type. If you can't find the info elsewhere, I can scan it for
you sometime.
73 & Good evening,
Marv WC6W
http://wc6w.50webs.com
--- On Sun, 7/27/08, Dr. David Kirkby <david.kirkby@onetel.net> wrote:
> From: Dr. David Kirkby <david.kirkby@onetel.net>
> Subject: [Amps] Thoughts on half-wave dipole
> To: "'AMPS'" <amps@contesting.com>
> Date: Sunday, July 27, 2008, 12:10 PM
> 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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