Resistor Noise

Due to the random thermal motion of charge carriers, we observe thermal noise, voltage and current noise in resistive circuit elements that is proportional to absolute temperature.

Modeling Thermal Noise in Resistors

Consider the noisy resistor in Fig. 1. We can model this as a noiseless resistor in series with a voltage noise generator, or a noiseless resistor in parallel with a current noise generator. Since we cannot predict the voltage or current noise at any point in time, we instead specify the noise voltage or current power:

${\overline {v_{n}^{2}}}=4kTRB$

(1)

${\overline {i_{n}^{2}}}=4kTGB={\frac {4kTB}{R}}$

(2)

Where $k$ is Boltzmann's constant, equal to $8.617\times 10^{-5}\mathrm {\tfrac {eV}{K}}$, and $B$ is the observation bandwidth. Thermal noise is white noise, hence the noise generators have white power spectral densities, as shown in Fig. 2. Just like any small signal, the shape of the power spectral density can be shaped by the frequency response of the rest of the circuit.

For example, a resistor with $R=10\mathrm {k\Omega }$ at $T=20^{\circ }C=293K$, the voltage noise power is:

${\overline {v_{n}^{2}}}=4kTRB=1.62\times 10^{-16}\cdot B$

(3)

Which gives us:

$v_{\mathrm {n,rms} }={\sqrt {\overline {v_{n}^{2}}}}={\sqrt {4kTRB}}=1.27\times 10^{-8}\cdot {\sqrt {B}}$

(4)

For a measurement bandwidth of $B=1\mathrm {MHz}$, we get $v_{\mathrm {n,rms} }=13\,\mathrm {\mu V}$. This is the smallest voltage, or the minimum detectable signal we can resolve across this resistor with this bandwidth, i.e. the smallest voltage we can distinguish from the thermal noise of the resistor.