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. Thermal noise is present in all electrical circuits, and in sensitive electronic equipment (such as radio receivers) can drown out weak signals, and can be the limiting factor on sensitivity of electrical measuring instruments. Thermal noise is proportional to absolute temperature, so some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to improve their signal-to-noise ratio. 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 ideal resistor is approximately white, meaning that its power spectral density is nearly constant throughout the frequency spectrum (Figure 2). When limited to a finite bandwidth and viewed in the time domain (as sketched in Figure 1), thermal noise has a nearly Gaussian amplitude distribution.[1]
For the general case, this definition applies to charge carriers in any type of conducting medium (e.g. ions in an electrolyte), not just resistors. 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.
In 1905, in one of Albert Einstein's Annus mirabilis papers the theory of Brownian motion was first solved in terms of thermal fluctuations. The following year, in a second paper about Brownian motion, Einstein suggested that the same phenomena could be applied to derive thermally-agitated currents, but did not carry out the calculation as he considered it to be untestable.[2]
Geertruida de Haas-Lorentz, daughter of Hendrik Lorentz, in her doctoral thesis of 1912, expanded on Einstein stochastic theory and first applied it to the study of electrons, deriving a formula for the mean-squared value of the thermal current.
Walter H. Schottky studied the problem in 1918, while studying thermal noise using Einstein's theories, experimentally discovered another kind of noise, the shot noise.
Frits Zernike working in electrical metrology, found unusual random deflections while working with high-sensitive galvanometers. He rejected the idea that the noise was mechanical, and concluded that it was of thermal nature. In 1927, he introduced the idea of autocorrelations to electrical measurements and calculated the time detection limit. His work coincided with de Haas-Lorentz' prediction.
The same year, working independently without any knowledge of Zernike's work, John B. Johnson working in Bell Labs found the same kind of noise in communication systems, but described it in terms of frequencies.[3] [4] He described his findings to Harry Nyquist, also at Bell Labs, who used principles of thermodynamics and statistical mechanics to explain the results, published in 1928.[5]
Johnson's experiment (Figure 1) found that the thermal noise from a resistance
R
T
\Deltaf
\overline
2} | |
{V | |
n |
=4kBTR\Deltaf
where
k\rm
The mean square voltage per hertz of bandwidth is
4kBTR
\sqrt{R}
The square root of the mean square voltage yields the root mean square (RMS) voltage observed over the bandwidth
\Deltaf
Vrms=\sqrt{\overline
2}} | |
{V | |
n |
=\sqrt{4kBTR\Deltaf}.
A resistor with thermal noise can be represented by its Thévenin equivalent circuit (Figure 4B) consisting of a noiseless resistor in series with a gaussian noise voltage source with the above RMS voltage.
Around room temperature, 3 kΩ provides almost one microvolt of RMS noise over 20 kHz (the human hearing range) and 60 Ω·Hz for
R\Deltaf
A resistor with thermal noise can also converted into its Norton equivalent circuit (Figure 4C) consisting of a noise-free resistor in parallel with a gaussian noise current source with the following RMS current:
Irms={Vrms\overR}=\sqrt{{4kBT\Deltaf}\overR}.
Ideal capacitors, as lossless devices, do not have thermal noise. However, the combination of a resistor and a capacitor (an RC circuit, a common low-pass filter) has what is called kTC noise. The noise bandwidth of an RC circuit is
\Deltaf{=}\tfrac{1}{4RC}.
The mean-square and RMS noise voltage generated in such a filter are:[7]
\overline
2} | |
{V | |
n |
={4kBTR\over4RC}={kBT\overC}
Vrms=\sqrt{4kBTR\over4RC}=\sqrt{kBT\overC}.
Qn
Qn=CVn=C\sqrt{kBT\overC}=\sqrt{kBTC}
2} | |
\overline{Q | |
n |
=C2
2} | |
\overline{V | |
n |
=C2{kBT\overC}=kBTC
Vrms{=}\sqrt{kBT\overC} | Charge noise Qn{=}\sqrt{kBTC} | |||
---|---|---|---|---|
as coulombs | as electrons | |||
1 fF | 2 mV | aC | 12.5 e− | |
10 fF | 640 μV | aC | 40 e− | |
100 fF | 200 μV | aC | 125 e− | |
1 pF | 64 μV | aC | 400 e− | |
10 pF | 20 μV | aC | 1250 e− | |
100 pF | 6.4 μV | aC | 4000 e− | |
1 nF | 2 μV | fC | 12500 e− |
An extreme case is the zero bandwidth limit called the reset noise left on a capacitor by opening an ideal switch. Though an ideal switch's open resistance is infinite, the formula still applies. However, now the RMS voltage 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 fluctuations 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.
Any system in thermal equilibrium has state variables with a mean energy of per degree of freedom. Using the formula for energy on a capacitor (E = CV2), mean noise energy on a capacitor can be seen to also be C = . Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance.
The Johnson–Nyquist noise has applications in precision measurements, in which it is typically called "Johnson noise thermometry".[8]
For example, the NIST in 2017 used the Johnson noise thermometry to measure the Boltzmann constant with uncertainty less than 3 ppm. It accomplished this by using Josephson voltage standard and a quantum Hall resistor, held at the triple-point temperature of water. The voltage is measured over a period of 100 days and integrated.[9]
This was done in 2017, when the triple point of water's temperature was 273.16 K by definition, and the Boltzmann constant was experimentally measurable. Because the acoustic gas thermometry reached 0.2 ppm in uncertainty, and Johnson noise 2.8 ppm, this fulfilled the preconditions for a redefinition. After the redefinition, the kelvin was defined so that the Boltzmann constant is 1.380649×10−23 J⋅K−1, and the triple point of water became experimentally measurable.[10] [11] [12]
Inductors are the dual of capacitors. Analogous to kTC noise, a resistor with an inductor
L
\overline
2} | |
{I | |
n |
={kBT\overL}.
The noise generated at a resistor
RS
R\rm
RS
Pmax=kBT\Deltaf.
Signal power is often measured in dBm (decibels relative to 1 milliwatt). Available noise power would thus be
10 log10(\tfrac{kBT\Deltaf}{1mW
10 log10(\Deltaf)-173.8
Bandwidth (\Deltaf) | Available thermal noise power at 300 K (dBm) | Notes | |
---|---|---|---|
1 Hz | −174 | ||
10 Hz | −164 | ||
100 Hz | −154 | ||
1 kHz | −144 | ||
10 kHz | −134 | FM channel of 2-way radio | |
100 kHz | −124 | ||
180 kHz | −121.45 | One LTE resource block | |
200 kHz | −121 | GSM channel | |
1 MHz | −114 | Bluetooth channel | |
2 MHz | −111 | Commercial GPS channel | |
3.84 MHz | −108 | UMTS channel | |
6 MHz | −106 | Analog television channel | |
20 MHz | −101 | WLAN 802.11 channel | |
40 MHz | −98 | WLAN 802.11n 40 MHz channel | |
80 MHz | −95 | WLAN 802.11ac 80 MHz channel | |
160 MHz | −92 | WLAN 802.11ac 160 MHz channel | |
1 GHz | −84 | UWB channel |
Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors" used concepts about potential energy and harmonic oscillators from the equipartition law of Boltzmann and Maxwell[14] to explain Johnson's experimental result. Nyquist's thought experiment summed the energy contribution of each standing wave mode of oscillation on a long lossless transmission line between two equal resistors (
R1{=}R2
\Deltaf
R1
R2
\overline{P1}=k\rmT\Deltaf.
Simple application of Ohm's law says the current from
V1
R1
R1
R2
R2
P1=
2 | |
I | |
1 |
R2=
2 | |
I | |
1 |
R1=(\tfrac{V1}{2R
2 | |
1}) |
R1=
2}{4R | |
\tfrac{V | |
1} |
.
Setting this equal to the earlier average power expression allows solving for the average of over that bandwidth:
2} | |
\overline{V | |
1 |
=4kBT{R1}\Deltaf.
Nyquist used similar reasoning to provide a generalized expression that applies to non-equal and complex impedances too. And while Nyquist above used
k\rmT
h
The
4kBTR
Z(f)
S | |
vnvn |
(f)=4kBTη(f)\operatorname{Re}[Z(f)].
η(f)
The real part of impedance,
\operatorname{Re}[Z(f)]
f1
f2
Vrms=
f2 | |
\sqrt{\int | |
f1 |
S | |
vnvn |
(f)df}
Alternatively, a parallel noise current can be used to describe Johnson noise, its power spectral density being
S | |
inin |
(f)=4kBTη(f)\operatorname{Re}[Y(f)].
Y(f){=}\tfrac{1}{Z(f)}
\operatorname{Re}[Y(f)]{=}\tfrac{\operatorname{Re}[Z(f)]}{|Z(f)|2}.
With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near absolute zero), the multiplying factor
η(f)
η(f)=
hf/kBT | + | |||||
|
1 | |
2 |
hf | |
kBT |
.
f\gtrsim\tfrac{kBT}{h}
η(f)
η(f)=1
Nyquist's formula is essentially the same as that derived by Planck in 1901 for electromagnetic radiation of a blackbody in one dimension—i.e., it is the one-dimensional version of Planck's law of blackbody radiation.[16] In other words, a hot resistor will create electromagnetic waves on a transmission line just as a hot object will create electromagnetic waves in free space.
In 1946, Robert H. Dicke elaborated on the relationship,[17] and further connected it to properties of antennas, particularly the fact that the average antenna aperture over all different directions cannot be larger than
\tfrac{λ2}{4\pi}
Richard Q. Twiss extended Nyquist's formulas to multi-port passive electrical networks, including non-reciprocal devices such as circulators and isolators.[18] 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 | |
vmvn |
(f)=2kBTη(f)(Zmn(f)+Znm(f)*)
Zmn
Z
S | |
imin |
(f)=2kBTη(f)(Ymn(f)+Ynm(f)*)
Y=Z-1