"Roy Lewallen" wrote in message
...
Just about any antenna textbook will show you the calculation of a half
wavelength, infinitely thin dipole in free space. For that special case,
the answer ends up being simply 30 * Cin(2 * pi), where Cin is a
modification of the cosine integral Ci[*]. That's where the "magic number"
comes from. Kraus' _Antennas_ is just one of the many textbooks which give
the derivation for this. It takes about 3 pages and 23 equations for Kraus
to derive.

One assumption made in calculation of this value is that the current
distribution is sinusoidal, an assumption that's true only for an
infinitely thin antenna. For finite thickness wire, the calculation
becomes much more difficult. The radiation resistance changes only slowly
with wire diameter, however, so sinusoidal distribution is a reasonably
good approximation provided that the antenna is thin. Feedpoint reactance,
though, varies much more dramatically with both antenna length and
diameter. Calculating its value exactly requires solution of a triple
integral equation which can't be solved in closed form. That's why
computer programs are used to solve it numerically.

To include the effect of ground, you need to calculate the mutual
impedance between the antenna and its "image". If the antenna is about 0.2
wavelength above ground or higher (for a half wave antenna -- the height
must be greater if the antenna is longer), you can assume that the ground
is perfect and get a pretty good result. Below that height, the
calculation again becomes much more complicated because the quality of the
ground becomes a factor. If you're interested in the numerical methods
used, locate the NEC-2 manual (available on the web), which describes it.

If you're satisfied with approximate results, the work by S.I. Shelkunoff
provides formulas for free-space input impedance of antennas with finite
diameter wire which can be solved with a programmable calculator or
computer. They're detailed in "Theory of Antennas of Arbitrary Size and
Shape", in Sept. 1941 Proceedings of the I.R.E. The formulas for R and X
contain many terms involving sine and cosine integrals, which can be
approximated with numerical series. You'll find additional information in
his book _Advanced Antenna Theory_. For approximate calculations of mutual
impedance of thin linear antennas, see "Coupled Antennas" by C.T. Tai, in
April 1948 Proceedings of the I.R.E. Those also involve multiple terms of
sine and cosine integrals.

Before numerical calculations became possible, many very good
mathematicians and engineers devised a number of approximation methods of
varying complexity and accuracy. You'll find their works in various
journals primarily in the 1940s - 1960s.

The complexity and difficulty of the problem is why virtually all antenna
calculations are done today with computers, using numerical methods such
as the moment method.

In summary, here are your choices:

1. You can calculate the approximate radiation resistance but not
reactance of a thin, free-space antenna by assuming a sinusoidal current
distribution and using the method Reg described. To include the effect of
ground, you have to calculate or look up from a table the mutual impedance
between the antenna and its "image", and modify the feedpoint impedance
accordingly by applying the mesh equations for two coupled antennas. This
method of including the effect of ground becomes inaccurate below around
0.2 wavelength, if the antenna is over typical earth.

2. You can use various approximation methods to calculate reactance, and
resistance with better accuracy. But for the effect of ground, you're
still limited to being greater than about 0.2 wavelength high.

3. To accurately include the effect of real ground with low antennas,
and/or to get resistance and reactance values with arbitrarily good
accuracy requires numerical methods. A computer program is the only
practical way to do this. A very good basic description of the moment
method can be found in the second and later editions of Kraus' _Antennas_.

[*] Cin(x) = ln(gamma * x) - Ci(x), where gamma = Euler's constant, 0.577.
. . Ci(x) = the integral from -infinity to x of [cos(v)/v dv] = ln(gamma *
x) - (x^2)/(2!2) + (x^4)/(4!4) - (x^6)/(6!6). . .

Roy Lewallen, W7EL

Very interesting references. In particular Kraus' 2nd ed. pp 359 - 408.
Also Stutzman and Thiele, 1st ed. pp 306 - 374. Pretty much grad level,
needs effort, (for me) even though I took Stutzman and Thiele's antenna
course in 1998 (NCEE).

Regards,

Frank