Re: Capacitor-feedback for low noise

Zigoteau wrote:


If you want to calculate the noise you get from an arbitrary circuit,
then you need a model for the noise behavior. The thermal noise of an
impedance Z(f) can be modeled by a Thevenin equivalent circuit, where
the voltage source in series with Z(f) is random with a spectral
density of 4kTRe(Z(f)) V2/Hz. Equivalently, its thermal noise can be
modeled by a Norton equivalent circuit, where the current source in
parallel is random with a spectral density of 4kTRe(1/Z(f)) A2/Hz.

Yes, the physics behind it is summarized in the fluctuation-dissipation theorem of statistical mechanics, which says that any mechanism that can dissipate energy has associated fluctuations at finite temperature. If this weren't so, you could make heat flow spontaneously from cold to hot.

The usual way to derive the Johnson noise formula for a resistor is to use classical equipartition of energy, which predicts that any single degree of freedom, e.g. the charge on a capacitor, has an RMS energy of kT/2. Classical equipartition is a very general consequence of statistical mechanics, and even in a quantum treatment, it can be shown to hold for frequencies << kT/h, about 6 THz at room temperature. (The high-frequency correction is due to the Planck function rolloff.) Since E=CV**2/2, kT/2 of energy corresponds to voltage Vrms = sqrt(kT/C), and charge Qrms = CV = sqrt(kTC).

If you have a parallel RC, isolated from the rest of the universe, this fluctuation must be maintained in equilibrium by the resistor noise--otherwise, the initial sqrt(kTC) would just discharge through the resistor. This must be true regardless of the values of R and C. Therefore, the open-circuit thermal fluctuations of the resistor, in the bandwidth of the RC, must equal sqrt(kT/C) volts; since the noise BW is 1/(4RC) (noise BW = pi/2* 3 dB BW), the open-circuit resistor noise voltage density is sqrt[(4RC)*(kT/C)] = sqrt(4kTR), which we all know and love.

You have to work a little harder to make this demonstration completely rigorous, e.g. by showing that the fluctuations have to be flat with frequency, but this is the idea. It can also be shown directly from statistical mechanics applied to a semiclassical electron gas model of metallic conduction, but I don't know how that derivation goes.

Cheers,

Phil Hobbs
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