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



I'm confused by some of the details in what David Bowman
wrote on 11/19/2016 08:44 PM, but overall the note was
helpful. It helped me clarify my thinking and my writing
on the following points:

You may wonder, why are we even talking about area under the
curve? If you take the classic Nyquist formula for the Johnson
noise, it is just

⟨|V|^2⟩ = 4 kT R B [1]

If you slavishly apply the formula, all you need is the
bandwidth, not the gain or the area under the curve.

However, as usual, it pays to understand where the formula
comes from. I claim the real physics says:

⟨|V|^2⟩ = ∫ 4 kT R |G|^2 df [2]

integrated over circular (not angular) frequency. Now in some
ideal world where the voltage gain G is unity within a passband
of width B and zero everywhere else, then equation [2] reduces
to [1]. However, in the other 99.999999% of the cases, you
actually have to do the integral. Figure out what's happening
at each frequency, and then add up all the contributions.

This is relevant to the RLC circuit, because the circuit does not
merely "select" the noise voltage components within the bandwidth
of the resonance; it also amplifies those components by a factor
of Q or so, just due to stuff sloshing around at resonance. Even
though it's a passive circuit, it still has loads of gain.

Equation [1] will give the wrong answer for |V|^2, wrong by a
huge factor, on the order of Q^2.

The point of my previous note was that even after you figure out
that you need to look at the area under the curve, you can't
approximate it by height times FWHM, because there is a lot of
weight in the wings.

Here is some evidence that my calculation of the area under the
curve is correct, and my general way of looking at things is correct.
Let's check the average /energy/ in the capacitor. It had darn well
better be ½ kT, independent of all circuit parameters.

This is calculated and discussed in a newly added subsection:
https://www.av8n.com/physics/rlc.htm#sec-thermo

There's also a new discussion of bandwidth, corner frequency, and
other frequency-like quantities at:
https://www.av8n.com/physics/rlc.htm#sec-freq-like

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