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Re: [Phys-L] weighting in the wings ... damped harmonic oscillator ... bandwidth ... algebra ... bug hunting



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
Adjunct Faculty, Physics, Retired
Anne Arundel Community College
Arnold, MD 21012