Doing some simple maths on a the simple model of a (lossless) double tuned
vertical I got this result :

1. Starting from a (lossless) single tuned vertical you get

            +------+
            |      |
           +-+   -----        where L = loading coil
         L | |   ----- C            C = antenna capacitance
           | |     |                R = radiation resistance
           +-+    +-+  R            Z = antenna impedance
            |     | |
       -----+     +-+
 Z =               |
       ------------+

If the reactance L (XL) cancels the reactance of C (XC) the antenna
impeadnce Z equals the radiation resistance R
so wL = 1/(wC) (where w = 2 x Pi x frequency)

2. 'Upgrade' the antenna to a double tuned vertical :

            +------+--------+
            |      |        |
           +-+   -----     +-+
        L' | |   ----- C   | | L'
           | |     |       | |
           +-+    +-+  R   +-+
            |     | |       |
       -----+     +-+       |
 Z =               |        |
       ------------+--------+

Instead of 1 inductor L we now have 2 inductors L' of the double value (2L)
Since the values of C and R are unchanged is wL' = 2/(wC)

Further XC = 1/(jwC) = -j/(wC)  (where j = square root of -1)
and     XL' = jwL' = j2/(wC) = -2XC

Calculating Z gives

Z = XL + (XL x (XC + R)) / (XL + XC + R)      -> extract XL

Z = -2XC x (1 + (XC + R) / (R - XC))          -> equalize to (R-XC)
Z = -2XC x (R - XC + XC + R) / (R - XC)       -> R - XC + XC + R = 2R
Z = -2XC x (2R) / (R - XC)

for practical antennas will XC be in the range of 500 to 5000 Ohm while R
will be well below 1 Ohm so we can say that R - XC = -XC

Z = -2XC x (2R) / (-XC)                       -> -2XC / -XC = 2
Z = 4R

So the antenna impedance of a resonant double tuned vertical will be 4
times the impedance of a single tuned vertical.

But what to do with the 'loss resistance' (RL) in the real world ?
Assume a simple model where both tuned sections share the same
ground-system. In that case RL would be in series with R and this would
mean that a double tuned vertical would not only quadruple the radiation
resistance but also the loss resistance, resulting in an unchanged ratio
between radiation resistance and loss resistance.

From the above I expect that a multiple-tuned vertical will only outperform

a simple vertical if each tuned section has at least its own independent
groundsystem. This would also mean that you need a certain distance between
the tuned sections in order to get these independent grounds.

Comments welcome ...

73, Rik

[email protected]
Villadreef 14  B-3128 Baal  BELGIUM   (JO20IX)