Chapter 9 Nyquist theorem A standard LCR circuit satisfies dI 1 L + RI + dt C
Z Idt = V (t),
(9.1)
when all the elements are in series. For small elements, thermal fluctuations may generate random potentials whose effects can be understood from the fluctuation dissipation theorem. (ω) 1 where Z(ω) = R + i(ωL − ωC ) is the impedance of the circuit. The Use FDT : I(ω) = VZ(ω) fluctuating emf generated by thermal fluctuations will generate a current whose magnitude can be estimated by the equipartition theorem. We get
1 hq 2 i 1 = kB T 2 C 2
1 1 LhI 2 i = kB T, 2 2
.
If we compare with the oscillator case, I here is analogous to v for an oscillator. The response function therefore connects charge q and voltage V with χ(ω) =
−Lω 2
1 , + C1 − iωR
so that χ00 =
Now I(ω)I(−ω) = ω 2 q(ω)q(−ω) = ∴ q(ω)q(−ω) =
(−Lω 2
ωR . + C1 )2 + ω 2 R2
V (ω)V (−ω) 1 2 + (ωL − ωC )
R2
V (ω)V (−ω) . (ωL2 − C1 )2 + R2 ω 2
F.D.T says χ00 =
ω ωR ω hV (ω)V (−ω)i S(ω) ⇒ = . 1 2 2 2 2 2kB T 2kB T (ωl2 − C1 ) + R2 ω 2 (−Lω + C ) + ω R
Therefore R=
1 hV (ω)V (−ω)i 2KB T
This is Nyquist’s theorem. 1
(9.2)
SMB/NESM
Lecture 8
The voltage fluctuation is controlled by T and R, independent of ω. One gets white noise in a circuit. If the rms voltage is measured over a band width Ω, then Z +Ω 2 hV (t)i ≡ hV (ω)V (−ω)idω = 4kB T R Ω. −Ω
Names: For any signal V (t), the Fourier transform of the auto correlation function is V (ω)V (−ω). This is called the “spectral density” or the “power spectrum” of V(t). For a sensitive measurement of current or voltage, one has to take care of this noise either by lowering temperature or by taking low R or both. Similar conditions apply for balances for measuring mass. (E.g. Sensitivity of a pendulum for Gravitational are detection.) Problem: 1. Derive the Nyquist theorem for (i) LR circuit, and (ii) RC circuit.
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