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.

Yes, it does.


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.

yes, it does vary


K X05 X045
-------------------------

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
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).


Balanis is giving the usual closed form expression for self Z.. I think
the original is from Schelkunoff or King.. I don't have my copy of Kraus
in front of me so I can't check.

Perhaps Lawson is using a different approximation?

Some formulas make the assumption of a sinusoidal current distribution,
others are more refined.




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).

73.1 +j42.5 to be more accurate..

"exact" as in analytical expression with no error? Or good to less than
a percent? Accounting for resistance of the element?


As a practical matter, I use NEC for this kind of thing (which does take
into account resistance, etc.) You can set it up to zap out a table
that you can then interpolate into, for instance.

However, there are a variety of formulas that one can use. I suggest
taking a look at Orfanidis's book
http://www.ece.rutgers.edu/~orfanidi/ewa/

Chapter 16 is probably the one you want. Figure 16.3.1 for instance.
As you noted, X varies a lot more slowly for fat elements (which is to
be expected.. ). Chapter 22 is also quite handy. Equation 22.2.10 is
the expression for Z11, which is an integration of F(z), given in
22.2.11. The author makes the point "In evaluating the self impedance
of an antenna with a small radius, the integrand F(z) varies rapidly
around z = 0. To maintain accuracy in the integration, we split the
integration interval into three subintervals, as we mentioned in Sec.
21.10"



He has matlab procedures and functions for most of them.. imped.m is
probably the one you want.