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Regarding David Bowman's 11/21/16 post for the series R,L, C
circuit, I can't get his result for the FWHM using his gain
function (G(f)). Using G(f) = 1/(1 + Q^2*(f/f_0 - f_0/f)^2) =
1/2, I get a quartic equation with the following four roots for
f:
f_1 = f_0*(1 + sqrt(1 + 4*Q^2))/(2*Q)
f_2 = f_0*(1 - sqrt(1 + 4*Q^2))/(2*Q)
f_3 = -f_0*(1 + sqrt(1 + 4*Q^2))/(2*Q)
f_4 = -f_0*(1 - sqrt(1 + 4*Q^2))/(2*Q)
For the upper edge of the FWHM, we require f_U > f_0. The only
root which does this is f_1, so:
f_U =f_1 = f_0*(1 + sqrt(1 + 4*Q^2))/(2*Q)
For the lower edge of the FWHM, we require 0 < f_L < f_0. The
only root which does this is f_4, so:
f_L = f_4 = -f_0*(1 - sqrt(1 + 4*Q^2))/(2*Q)
Therefore:
FWHM = f_U - f_L = f_0/Q (and not f_0*sqrt(1 + 4*Q^2)/Q which
was David Bowman's result) .
If my work is correct, and accepting that the Nyquist bandwidth
is B = pi*f_0/(2*Q), the ratio of B to the FWHM is given by:
B/FWHM = pi/2
This is the same result that John Denker gave in starting this
thread on 11/19/16.
What am I doing wrong?
Don Polvani