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 Thermal noise

# Johnson–Nyquist noise

Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the noise generated by the thermal agitation of the charge carriers (the electrons) inside an electrical conductor in equilibrium, which happens regardless of any applied voltage.

## History

This type of noise was first measured by John B. Johnson at Bell Labs in 1928[1]. He described his findings to Harry Nyquist, also at Bell Labs who was able to explain the results.[2]

## Noise voltage and power

Thermal noise is to be distinguished 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 conductor medium (e.g. ions in an electrolyte), not just resistors. It can be modeled by a voltage source in series with the noise generating resistor. The root mean square (rms) of the voltage, vn, is given by

$v_{n} = \sqrt{ 4 k_B T R \Delta f }$

where kB is Boltzmann's constant in joules per kelvin, T is the resistor's absolute temperature in kelvins, R is the resistor value in ohms, and Δf is the bandwidth in hertz in which the noise is measured.

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 the noise power transfer to the circuit 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. Also the noise is white noise, equal throughout the frequency spectrum.

## Noise in decibels

In communications, decibels(dBm) are often used. Thermal noise at room temperature can be estimated as:

$P_\mathrm{dBm} = -174 + 10\ \log(\Delta f)$

where P is measured in dBm. For example:

Bandwidth Power
1 Hz -174 dBm
10 Hz -164 dBm
1000 Hz -144 dBm
5 kHz -137 dBm
1 MHz -114 dBm
6 MHz -106 dBm

## 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:

$i_n = \sqrt {{ 4 k_B T \Delta f } \over R}$

Thermal noise is intrinsic to all resistors and is not a sign of poor design or manufacture, although resistors may also have excess noise.

## Thermal noise on capacitors

Johnson noise in an RC circuit can be expressed more simply by using the capacitance value, rather than the resistance and bandwidth values. The rms voltage noise on a capacitance C is

$v_{n} = \sqrt{ k_B T / C }$

independent of the resistor value, since bandwidth varies reciprocally with resistance in an RC circuit.[3] In the case of the reset noise left on a capacitor by opening an ideal switch, the resistance is infinite, and the formula still applies; however, now the rms must be interpreted not as a time average, but as an average over many such reset events, since the voltage is constant when the bandwidth is zero. In this sense, the Johnson noise of an RC circuit can be seen to be inherent, an effect of the thermodynamic distribution of the number of electrons on the capacitor, even without the involvement of a resistor.

The noise is not caused by the capacitor itself, but by the thermodynamic equilibrium of the amount of charge on the capacitor. Once the capacitor is disconnected from a conducting circuit, the thermodynamic fluctuation is frozen at a random value with standard deviation as given above.

The reset noise of capacitive sensors is often a limiting noise source, for example in image sensors. As an alternative to the voltage noise, the reset noise on the capacitor can also be quantified as the charge standard deviation, as

$Q_{n} = \sqrt{ k_B T C }$

Since the charge variance is kBTC, this noise is often called kTC noise.

## Noise at very high frequencies

The above equations are good approximations at low frequencies. In general, the power spectral density of the voltage across the resistor R is given by (be careful this is power spectral density of an stochastic voltage signal in V2 / Hz don't confuse with the power in watts):

$\Phi (f) = \frac{2 R h f}{e^{\frac{h f}{k_B T}} - 1}$

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:

$f << \frac{k_B T}{h}$

(this assumption is valid until few gigahertz) 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 Δf, 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.

## References

1. ^ J. Johnson, "Thermal Agitation of Electricity in Conductors", Phys. Rev. 32, 97 (1928) – the experiment
2. ^ H. Nyquist, "Thermal Agitation of Electric Charge in Conductors", Phys. Rev. 32, 110 (1928) – the theory
3. ^ R. Sarpeshkar, T. Delbruck, and C. A. Mead, "White noise in MOS transistors and resistors", IEEE Circuits Devices Mag., pp. 23–29, Nov. 1993.