Resistor Thermal Noise

Noise from heat alone

Electrons in any resistor jiggle with thermal energy, producing a tiny random voltage across it. John Johnson measured it in 1926 and Harry Nyquist explained it. It depends on nothing but resistance, temperature and the bandwidth you listen through:

with $k_B = 1.380649\times10^{-23}\ \mathrm{J/K}$, $T$ in kelvin and $B$ in hertz. Dividing out the bandwidth gives the density that datasheets quote:

A useful anchor: 1 kΩ at room temperature makes about 4 nV/√Hz.

Why designers care

This is a floor no component choice can beat. A 10 kΩ volume control contributes roughly 1.8 µV over the audio band, 121 dB below a 2 V signal, which is harmless. The same resistance at the input of a moving-coil phono stage would sit only 49 dB below the cartridge's 0.5 mV, which is why those stages use resistances of a few ohms.

Good to know

Halving noise voltage (6 dB) takes a resistance four times smaller. Cooling helps very little at domestic temperatures since the dependence is on absolute temperature. The total input noise of an amplifier adds its own voltage and current noise to this figure.

References

1. J. B. Johnson, “Thermal Agitation of Electricity in Conductors,” Phys. Rev., vol. 32, pp. 97–109, 1928. APS
2. H. Nyquist, “Thermal Agitation of Electric Charge in Conductors,” Phys. Rev., vol. 32, pp. 110–113, 1928. APS