I had to read between some invisible lines to figure this out. I was
finally able to get numbers that matched the results Beers obtained in
the article (August 1987 Ham Radio). I'm sure someone with more
spreadsheet savvy could automate this completely.
Choose upper and lower center frequencies (band centers).
Calculate the geometric mean frequency.
Pick a capacitor value (220 pF on 160M to 50 pF on the higher bands).
Assume trap Q of 200.
Make the "arbitrary" frequency equal to the upper frequency. That one
was a head scratcher for a few days.
Calculate:
D (Eqn. 2)
Resistance (Eqn. 3)
Inductance L (Eqn. 7 - note should be 10^6 in the numerator to get uH).
t (Eqn. 8)
Z (Eqn. 9)
Rs (Eqn. 10)
Xs (Eqn. 11)
Choose an antenna height and wire diameter, both in the same units such
as inches.
Calculate the impedance Z0 of the "transmission line", aka the antenna
wire. H is height above ground, d is wire diameter. (Eqn. 14).
Now, what you're trying for, is to find two lengths (A and B in degrees,
or S1 and S2 in feet), such that U (Eqn. 20) is as small as possible,
AND the sum of reactances of the two segments equal Z0. Equations 1, 15
and 17. Remember, X2 is negative, so X = X1 + X2.
I don't think there's any need to go longer than 90 degrees for each
segment.
I made a spreadsheet to help with this. Pick a B. Run A from 1 to 90
degrees. Calculate U and X for each A.
Look at the U values where it changes from negative to positive. If X is
close to Z0, insert rows to step A in .1 degrees. Then change B by tenths.
If your X value isn't close to Z0 when U is very small, change B and run
it again.
The first thing to do after building the spreadsheet is to plug in
Beers' numbers and see if your results match his.
Z0 = 575 Ohms (Eqn. 14, 12 AWG at 20 feet).
Use C of 50 or 51.6 pF, depending on which set of numbers you're trying
to match.
Good luck!
Dave AA3EE
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