How many Radials Do I Need?

The following question was posted on the QRPL list a few days ago:

I plan on some portable operations(camping, hiking, etc…) and I was
wondering what are the optimum length of radials for verticals, or
counterpoise for End-fed half waves wires?  Frequencies are 40 through 10
meters (including 30,17, and 12 meters).

I seem to remember something about at least 2 or 3  ¼ wavelengths laid on
the ground.  I just want to get a consensus of the group here about what
seems to work best. 

My response may be of interest to some members of CARC:

I am not sure whether the attachment will make it through the reflector (it’s only 21KB), but it’s worth a look if it does.

It is Fig 30 from Ground System as a Factor in Antenna Efficiency, which is the seminal paper by Brown, Lewis and Epstein published in Proc IRE vol 25, #6 June 1937.

This is the paper where the 200 radial comes from, although they ran out of wire at 113 radials as you can see from the graph.

This shows that with two radials at 1/8 lambda, received field intensity is slightly less 50% down on theoretical maximum. Received power goes as the square of intensity, so two radials are slightly less than 6db down on theoretical maximum, i.e. less than 1 S-point.

If you go up to 15 radial at 1/4 lambda to get to something like 148/196 intensity, i.e. 2.5dB down.

Given that 15 @ 1/4 lambda represents a practical maximum for temporary operation, you have to work pretty hard to win the 3.5 dB relative to the two 1/8 lambda wires.

Two other points of note are that you get within 1dB of maximum with
60 radial at 1/8 lambda and if you go up to 113 radials 1/10 lambda seems to be about 1dB down. However I do worry how accurate the top left corner of the graph is.

I would of course be interested in knowing whether there is later (definitive) evidence that contradicts George Brown’s paper, but the conclusion that very few radial are all this is needed seem startling compare to the advice of the seasoned DX community, which I am sure disheartens operators of ordinary means.

Stewart/G3YSX