LC Voltage Noise

The circuit in Figure 1.1 consists of a coil of inductance L and a capacitor of capacitance C. What is the rms noise voltage across AB at temperature τ in the limit where

a) τ is very large?

b) τ is very small?

Figure 1.1

SOLUTION

Write the Hamiltonian H for the circuit:

(1)

Figure 1.2

where Q is the charge on the capacitor. This is the Hamiltonian of a harmonic oscillator of frequency  whose energy levels ε_n are

(2)

The average energy in the circuit is given by (with β ≡ 1/τ).

(3)

The average energy is equally distributed between the capacitance and the inductance:

(4)

where V is the voltage across the capacitor (between points A and B; see Figure 1.2). We then have

(5)

a) In the classical limit hω << τ:

(6)

We could equally well have derived the classical result by using the equipartition theorem. For the single degree of freedom, there is an average energy τ, which, as noted, is divided between the capacitor and inductor, so

(7)

as found in (6). The mean square noise voltage is

(8)

b) If hω >> τ then (6) becomes

(9)