D

#### Dave

- Jan 1, 1970

- 0

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

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.

(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 wavelength 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

david <dot> kirkby <at> onetel <dot> net