[Amps] Thoughts on half-wave dipole

[Dr. David Kirkby]

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.