Maximizing Secondary Q


I have been making a lot of calculations on the Q of the secondary coil with
consideration of self capacitance recently, and have made a few findings.
Could any of you tell me if I've reinvented the wheel here?

Based on the following equivalent to the sencondary coil, I have derived a
more complete formula for the Q of the secondary coil. 

      toroid                 coil
---------| |----------/\/\/\/-------uuuu--------------
|              |                                     |   |   
|              |----------| |------------------------|   |
|                     self C                          |    
|                                                       |  

The formula is

Q = (Xl/R) - ((Xl^2 + R^2)/(R*Xcs))

Xl = inductive reactance
Xcs = reactance of the coil's self capacitance
R = RF resistance of the wire

Also, based upon the previous formula, I derived a formula to find the optimum
frequency for a coil to operate at (resulting in maximum Q). I substituted the
actual electrical value and frequency variables for the reactance variables.
Instead of Xl, I used 2*pi*F*L, and so on. I took the derivative of the
formula (dQ/dF), and solved for frequency (F), where the derivative equaled
zero (Q reached its peak). I got the following formula.

F = 1,000,000/(2*pi*root(3*L*C))

F = optimum frequency in hertz
L = secondary inductance in henrys
C = self capacitance in picofarads

I was rather surprised when I found this, because it looks so similar to the
formula for resonance. Please let me know what you think of this! Let me know
if I didn't consider any other SIGNIFICANT factors. Also, would any of you
like to try this on your coil? Please post the results.

I have created a page on my web site that gives a more detailed explanation of
how I came to my findings.

Matt Behrend