Thermal noise

Thermal noise

Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the electronic noise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium.

Thermal noise in an idealistic resistor is approximately white, meaning that the power spectral density is nearly constant throughout the frequency spectrum (however see the section below on extremely high frequencies). When limited to a finite bandwidth, thermal noise has a nearly Gaussian amplitude distribution.

History

This type of noise was first measured by John B. Johnson at Bell Labs in 1926. He described his findings to Harry Nyquist, also at Bell Labs, who was able to explain the results.

Noise voltage and power

Thermal noise is distinct from shot noise, which consists of additional current fluctuations that occur when a voltage is applied and a macroscopic current starts to flow. For the general case, the above definition applies to charge carriers in any type of conducting medium (e.g. ions in an electrolyte), not just resistors. It can be modeled by a voltage source representing the noise of the non-ideal resistor in series with an ideal noise free resistor.

The one-sided power spectral density, or voltage variance (mean square) per hertz of bandwidth, is given by



\bar {v_{n}^2} = 4 k_B T R

where kB is Boltzmann's constant in joules per kelvin, T is the resistor's absolute temperature in kelvins, and R is the resistor value in ohms (Ω). Use this equation for quick calculation, at room temperature:



\sqrt{\bar {v_{n}^2}} = 0.13 \sqrt{R} ~\mathrm{nV}/\sqrt{\mathrm{Hz}}.

For example, a 1 kΩ resistor at a temperature of 300 K has



\sqrt{\bar {v_{n}^2}} = \sqrt{4 \cdot 1.38 \cdot 10^{-23}~\mathrm{J}/\mathrm{K} \cdot 300~\mathrm{K} \cdot 1~\mathrm{k}\Omega} = 4.07 ~\mathrm{nV}/\sqrt{\mathrm{Hz}}.

For a given bandwidth, the root mean square (RMS) of the voltage, $v_\left\{n\right\}$, is given by



v_{n} = \sqrt{\bar {v_{n}^2}}\sqrt{\Delta f } = \sqrt{ 4 k_B T R \Delta f }

where Δf is the bandwidth in hertz over which the noise is measured. For a 1 kΩ resistor at room temperature and a 10 kHz bandwidth, the RMS noise voltage is 400 nV. A useful rule of thumb to remember is that 50 Ω at 1 Hz bandwidth correspond to 1 nV noise at room temperature.

A resistor in a short circuit dissipates a noise power of



P = {v_{n}^2}/R = 4 k_B \,T \Delta f.

The noise generated at the resistor can transfer to the remaining circuit; the maximum noise power transfer happens with impedance matching when the Thévenin equivalent resistance of the remaining circuit is equal to the noise generating resistance. In this case each one of the two participating resistors dissipates noise in both itself and in the other resistor. Since only half of the source voltage drops across any one of these resistors, the resulting noise power is given by



P = k_B \,T \Delta f

where P is the thermal noise power in watts. Notice that this is independent of the noise generating resistance.

Noise current

The noise source can also be modeled by a current source in parallel with the resistor by taking the Norton equivalent that corresponds simply to divide by R. This gives the root mean square value of the current source as:



i_n = \sqrt - 1}

where f is the frequency, h Planck's constant, kB Boltzmann constant and T the temperature in kelvins. If the frequency is low enough, that means:



f \ll \frac{k_B T}{h}

(this assumption is valid until few terahertz at room temperature) then the exponential can be expressed in terms of its Taylor series. The relationship then becomes:



\Phi (f) \approx 2 R k_B T.

In general, both R and T depend on frequency. In order to know the total noise it is enough to integrate over all the bandwidth. Since the signal is real, it is possible to integrate over only the positive frequencies, then multiply by 2. Assuming that R and T are constants over all the bandwidth $\Delta f$, then the root mean square (RMS) value of the voltage across a resistor due to thermal noise is given by



v_n = \sqrt { 4 k_B T R \Delta f },

that is, the same formula as above.