Reg,

Here is the analysis of an 8 radial system. All other parameters
the same:

0.4 m -- Radial Z = 65.4 - j 58.1 -- Efficiency 11.2%
0.6 m -- Radial Z = 48.7 - j 55.4
0.8 m -- Radial Z = 39.8 - j 29.6
1.0 m -- Radial Z = 34.2 - j 23.5
1.2 m -- Radial Z = 30.2 - j 19.2
1.4 m -- Radial Z = 27.2 - j 16.1
1.6 m -- Radial Z = 24.9 - j 13.7
1.8 m -- Radial Z = 23.0 - j 11.7 -- Efficiency 19.4%
2.0 m -- Radial Z = 21.3 - j 10.0
2.2 m -- Radial Z = 19.9 - j 8.5
2.4 m -- Radial Z = 18.7 - j 7.1
2.6 m -- Radial Z = 17.7 - j 5.9
2.8 m -- Radial Z = 16.7 - j 4.6
3.0 m -- Radial Z = 15.9 - j 3.5
3.2 m -- Radial Z = 15.2 - j 2.3 -- Efficiency 22.4%
3.4 m -- Radial Z = 14.7 - j 1.1
3.6 m -- Radial Z = 14.3 + j 0
3.8 m -- Radial Z = 14.0 + j 1.1
4.0 m -- Radial Z = 13.9 + j 2.2 -- Efficiency 23.2%
4.2 m -- Radial Z = 13.9 + j 3.2
4.4 m -- Radial Z = 14.1 + j 4.2
4.6 m -- Radial Z = 14.4 + j 5.0
4.8 m -- Radial Z = 14.7 + j 5.7
5.0 m -- Radial Z = 15.2 + j 6.3 -- Efficiency 23.0%
5.2 m -- Radial Z = 15.6 + j 6.7
5.4 m -- Radial Z = 16.1 + j 7.1
5.6 m -- Radial Z = 16.5 + j 7.3
5.8 m -- Radial Z = 16.9 + j 7.4
6.0 m -- Radial Z = 17.2 + j 7.5
6.2 m -- Radial Z = 17.5 + j 7.5 -- Efficiency 22.5%
6.4 m -- Radial Z = 17.8 + j 7.5
6.6 m -- Radial Z = 18.0 + j 7.4
6.8 m -- Radial Z = 18.1 + j 7.4
7.0 m -- Radial Z = 18.3 + j 7.4
7.2 m -- Radial Z = 18.4 + j 7.3
7.4 m -- Radial Z = 18.4 + j 7.3
7.6 m -- Radial Z = 18.5 + j 7.2
7.8 m -- Radial Z = 18.6 + j 7.2
8.0 m -- Radial Z = 18.6 + j 7.2 -- Efficiency 22.4%

Note that after 4 m the antenna efficiency starts to drop,
which is not the case for large numbers of radials.

Now I have to figure out the best way to model a 100
radial system.

Frank
================================================== ===

Frank,

Thank you very much for the results on 8 radials.

As expected, it seems that for small numbers of radials the resonance
effects are begining to appear again.

I have ideas as to why this should happen. It's to do with the
geometry of the system and the fact that the ends of the radials are
not terminated with true open-circuits when calculating input
impedance. ( As is assumed by program Radial3.)

I look forward to receiving results for 100 or more radials which may
allow me to improve, in the mathematical model, the function of N
which describes the system's input impedance in terms of the number N
of radials.

----------------------------------------------------------------------
-----------------------

But, let's face it, a good understanding of what's going on under the
soil surface, without investigating input impedance at say 25 MHs, and
without investigating input imedance at soil resistivities of say 2000
ohm-metres, will be known only crudely.

Is it all worth the trouble?

After all, we already know enough quite enough about radial systems at
HF to design one which will work good enough, performance-wise, to
keep anybody happy. ( BL&E's work, as good as it may be, does not
apply at HF.)

Just lay one or two dozen radials, in ordinary soils, with lengths
equal to about half antenna height. Which is a good enough
rule-of-thumb for anybody who doesn't expect to win contests because
he has the advantage of 0.05 S-units.

And extremely few people know what their local soil resistivity is
within +/- 40 percent. It's largely guesswork!

But please keep up the good work with NEC4 in which I have great
confidence.
----
Reg, G4FGQ