An exact expression for the noise voltage across a resistor shunted by a capacitor

The mean square voltage developed across a noisy resistor (R ) with a white spectral density shunted by a capacitor (C) is known to be (kT/C), where k is Boltzmann´s constant and T the absolute temperature of R. A puzzling aspect of this simple result is that the value of the resistor (which is the real source of noise) appears to play no part and gives no clue on the outcome in the limiting cases of R→0 or C→0. The difficulty is resolved by first noting that due to quantum effects, the simple white noise model breaks down at frequencies of 10 ⁹ Hz or so. Using a quantum-corrected expression for the spectral density, the problem of the RC network has been reworked exactly. It is verified that the new expression yields correct results for all values of the parameters. A Taylor series expansion of the output in terms of the noise equivalent bandwidth of the RC circuit shows that the old result (kT/C) is very accurate for bandwidths of up to tens of GHz. Numerical results are illustrated graphically