Richard Fry wrote:

Reg,

The 1937 Brown, Lewis and Epstein IRE paper "Ground Systems as a Factor
in Antenna Efficiency" include an analysis of the currents in radial
ground systems, along with equations and graphs for it in various
configurations. All you need to do to apply them to a system of raised
radials is to modify these basic equations.

Of course, you will have to read the paper first to do that (wink,
nudge). But then you might also see why knowledge of ground conductivity
was unimportant to the conclusions of this paper, and refrain from
saying so in the future.

RF

Unfortunately, the mathematical analysis in that paper was found to be
in error. A search of the literature shows that quite a number of people
worked on this problem well after publication of the BL&E paper. Some
notable work was done by J.R. Wait and W.A. Pope of the Radiation
Physics Laboratory, Defence Research Branch, in Canada. Two papers in
particular give equations for the impedance of radial systems which
appear to be valid -- "The Characteristics of a Vertical Antenna With a
Radial Conductor Ground System", Appl. Sci. Res. B, Vol. 4, 1954; and
"Input Resistance of L.F. Unipole Aerials With Radial Wire Earth
Systems", Wireless Engineer, May, 1955. The equations involve multiple
integral equations which can't be solved in closed form. In papers I've
read which do involve equations which can be solved in closed form, even
approximately, the results have deviated greatly from BL&E's measured
results, making the accuracy of the method doubtful. This holds true for
Reg's program, also, which apparently depends on some simplifying
assumptions which aren't valid.

NEC-2, which is readily available in numerous forms, does about as good
a calculation as any of elevated radial systems. Its major limitation,
in my opinion, is the inability to deal with stratified ground. Of
course, even if it could handle stratified ground, the user would
somehow have to determine the properties and locations of the various
strata. NEC-4 can, in addition to NEC-2's capabilities, include buried
radials in its models. A few tests show reasonable agreement between it
and BL&E's results. Incidentally, the equations in the first Wait and
Pope paper I mentioned resemble those used in NEC-4, but I haven't
studied them in enough detail to determine if they are indeed the same.

Elevated radial systems have been somewhat controversial, with some
indications that modeled results don't imitate actual results very well,
particularly at low frequencies. But there's very little really good
measurement data available to make a valid judgement. Besides the
possibility of stratified ground, some people have reported difficulty
in maintaining equal currents in near-resonant elevated radial wires in
real installations. This would have a substantial effect on a system,
and would definitely cause deviation between modeled and measured results.

There's considerable work to be done in this field, but what really
needs to be done is the making of good, well documented and carefully
done measurements of elevated radial systems -- not more calculations
based on invalid assumptions.

Roy Lewallen, W7EL