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

Harry wrote:
Hi Tim and Reg,

Thank you for your valuable information. Is there any website or
textbook that actually shows the step-by-step calculation of this magic
number which has been quoted so often in the cable industry?

You know most video cables and connectors have characteristic
impedance, 75 Ohms.

I am not afraid of math. I just like to understand the details of its
derivation.

-- Harry