Johnson–Nyquist noise

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Noise voltage and power[edit]



\overline {v_{n}^2} = 4 k_\text{B} T R






v_{n}  = \sqrt{\overline {v_{n}^2}}\sqrt{\Delta f } = \sqrt{ 4 k_\text{B} T R \Delta f }


A resistor in a short circuit dissipates a noise power of

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


P = k_\text{B} \,T \Delta f


Noise current[edit]


i_n = \sqrt {{4 k_\text{B} T \Delta f } \over R}.


Noise power in decibels[edit]


P_\mathrm{dBm} = 10\ \log_{10}(k_\text{B} T \Delta f \times 1000)


P_\mathrm{dBm} = 10\ \log_{10}(k_\text{B} T \times 1000) + 10\ \log_{10}(\Delta f)





Bandwidth  (\Delta f ) Thermal noise power Notes
1 Hz −174 dBm
10 Hz −164 dBm
100 Hz −154 dBm
1 kHz −144 dBm
10 kHz −134 dBm FM channel of 2-way radio
15 kHz −132.24 dBm One LTE subcarrier
100 kHz −124 dBm
180 kHz −121.45 dBm One LTE resource block
200 kHz −121 dBm GSM channel
1 MHz −114 dBm Bluetooth channel
2 MHz −111 dBm Commercial GPS channel
3.84 MHz −108 dBm UMTS channel
6 MHz −106 dBm Analog television channel
20 MHz −101 dBm WLAN 802.11 channel
40 MHz −98 dBm WLAN 802.11n 40 MHz channel
80 MHz −95 dBm WLAN 802.11ac 80 MHz channel
160 MHz −92 dBm WLAN 802.11ac 160 MHz channel
1 GHz −84 dBm UWB channel

Thermal noise on capacitors[edit]


\overline {v_{n}^2} = k_\text{B} T / C

v_{n} = \sqrt{ k_\text{B} T / C }.

Thermal noise in the resistor accounts for 100% of kTC noise.




Q_{n} = \sqrt{ k_\text{B} T C }.

Since the charge variance is k_\text{B} T C, this noise is often called kTC noise.


The kTC noise is the dominant noise source at small capacitors.

Noise of capacitors at 300 K
Capacitance  \sqrt{ k_\text{B} T / C } Electrons
1 fF 2 mV 12.5 e
10 fF 640 µV 40 e
100 fF 200 µV 125 e
1 pF 64 µV 400 e
10 pF 20 µV 1250 e
100 pF 6.4 µV 4000 e
1 nF 2 µV 12500 e

Generalized forms[edit]

The 4 k_\text{B} T R voltage noise described above is a special case for a purely resistive component for low frequencies. In general, the thermal electrical noise continues to be related to resistive response in many more generalized electrical cases, as a consequence of the fluctuation-dissipation theorem. Below a variety of generalizations are noted. All of these generalizations share a common limitation, that they only apply in cases where the electrical component under consideration is purely passive and linear.

Reactive impedances[edit]

Nyquist's original paper also provided the generalized noise for components having partly reactive response, e.g., sources that contain capacitors or inductors.[4] Such a component can be described by a frequency-dependent complex electrical impedance Z(f). The formula for the power spectral density of the series noise voltage is

S_{v_n v_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Z(f)].

The function \eta(f) is simply equal to 1 except at very high frequencies (see below).

The real part of impedance, \operatorname{Re}[Z(f)], is in general frequency dependent and so the Johnson–Nyquist noise is not white noise. The rms noise voltage over a span of frequencies f_1 to f_2 can be found by integration of the power spectral density:

 \sqrt{\langle v_n^2 \rangle} = \sqrt{\int_{f_1}^{f_2} S_{v_n v_n}(f) df}.

Alternatively, a parallel noise current can be used to describe Johnson noise, its power spectral density being

S_{i_n i_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Y(f)].

where Y(f) = 1/Z(f) is the electrical admittance; note that \operatorname{Re}[Y(f)] = \operatorname{Re}[Z(f)]/|Z(f)|^2

Quantum effects at high frequencies[edit]

Nyquist also pointed out that quantum effects analogous to Planck's law occur for very high frequencies.[4] The function \eta(f) is in general given by

\eta(f) = \frac{hf/k_\text{B} T}{e^{hf/k_\text{B} T} - 1},

where h is Planck's constant. At very high frequencies f \gtrsim k_\text{B} T/h, the function \eta(f) starts to exponentially decrease to zero. At room temperature this transition occurs in the terahertz, far beyond the capabilities of conventional electronics, and so it is valid to set \eta(f)=1 for conventional electronics work.

Multiport electrical networks[edit]

Richard Q. Twiss extended Nyquist's formulas to multi-port passive electrical networks, including non-reciprocal devices such as circulators and isolators.[8] Thermal noise appears at every port, and can be described as random series voltage sources in series with each port. The random voltages at different ports may be correlated, and their amplitudes and correlations are fully described by a set of cross-spectral density functions relating the different noise voltages,

S_{v_m v_n}(f) = 2 k_\text{B} T \eta(f) (Z_{mn}(f) + Z_{nm}(f)^*)

where the Z_{mn} are the elements of the impedance matrix \mathbf{Z}. Again, an alternative description of the noise is instead in terms of parallel current sources applied at each port. Their cross-spectral density is given by

S_{i_m i_n}(f) = 2 k_\text{B} T \eta(f) (Y_{mn}(f) + Y_{nm}(f)^*)

where \mathbf{Y} = \mathbf{Z}^{-1} is the admittance matrix.

Continuous electrodynamic media[edit]

The full generalization of Nyquist noise is found in fluctuation electrodynamics, which describes the noise current density inside continuous media with dissipative response in a continuous response function such as dielectric permittivity or magnetic permeability. The equations of fluctuation electrodynamics provide a common framework for describing both Johnson–Nyquist noise and free space blackbody radiation.[9]

See also[edit]


  4. ^ a b c H. Nyquist, "Thermal Agitation of Electric Charge in Conductors", Phys. Rev. 32, 110 (1928) – the theory
  8. ^ Twiss, R. Q. (1955). "Nyquist's and Thevenin's Theorems Generalized for Nonreciprocal Linear Networks". Journal of Applied Physics 26 (5): 599. doi:10.1063/1.1722048.  edit
  9. ^ L.P. Pitaevskii, E.M. Lifshitz (1980). "Chapter VIII. Electromagnetic Fluctuations". Statistical Physics, Part 2: Theory of the Condensed State. Vol. 9 (1st ed.). Butterworth-Heinemann. ISBN 978-0-7506-2636-1. 


External links[edit]