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



On 11/24/2016 10:07 AM, Donald Polvani wrote:
Specifically, I solved:

G(f) = 1/(1 + Q^2*(f/f_0 - f_0/f)^2) = 1/2 [1]

Different questions lead to different answers:
-- Equation [1] describes the current in the resistor, whereas
-- I was looking at the voltage across the capacitor.

One dead giveaway: Look at the asymptotes. For my filter,
the square of the voltage gain goes to unity at low frequencies,
and falls off in proportion to 1/f^4 at high frequencies.
Equation [1] does neither of those things.

As remarked previously, data visualization helps a lot.
It's not just for experimental data; it also helps
make sense of theoretical calculations.

In this case, see e.g.
https://www.av8n.com/physics/rlc.htm#fig-rlc6

==========

Tangential remark: Equation [1] could also be thought of
as the voltage across the ideal (noiseless) part of the
resistor, but since that is unobservable, this interpretation
is not the best.

A small suggestion: For today it doesn't matter, but in
the future, in equation [1], it might be better to write
the LHS as G /squared/. It is more conventional to
reserve G for the /voltage/ gain. Today we happen to
be most directly interested in the square of the voltage
gain, but in other situations the voltage gain itself is
of tremendous interest, and G is the conventional symbol
for that. There is always temptation to redefine symbols
so as to streamline the notation, but such streamlining
risks sowing confusion.