Johnson–Nyquist noise

Norton equivalent

circuit).

Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the

is used to characterize the medium.

Thermal noise in an ideal resistor is approximately

History of thermal noise

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.^[2][3]

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.^[2]

Frits Zernike working in electrical metrology, found unusual aleatory 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.^[2]

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.^[4][5][2] He described his findings to Harry Nyquist, also at Bell Labs, who was able to explain the results, published in 1928.^[6]

Derivation

As Nyquist stated in his 1928 paper, the sum of the energy in the normal modes of electrical oscillation would determine the amplitude of the noise. Nyquist used the

${\displaystyle \left\langle H\right\rangle =k_{\rm {B}}T}$

where ${\displaystyle \left\langle H\right\rangle }$ is the noise power density in (W/Hz), ${\displaystyle k_{\rm {B}}}$ is the Boltzmann constant and ${\displaystyle T}$ is the temperature. Multiplying the equation by bandwidth gives the result as noise power:

${\displaystyle N=k_{\rm {B}}T\Delta f}$

where N is the noise power and Δf is the bandwidth.

Noise voltage and power

Thermal noise is distinct from

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

${\displaystyle {\overline {v_{n}^{2}}}=4k_{\text{B}}TR}$

where k_B is the Boltzmann constant, T is the resistor's absolute temperature, and R is the resistance value. Using this equation for quick calculation, at room temperature:

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

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

${\displaystyle {\sqrt {\overline {v_{n}^{2}}}}={\sqrt {4\cdot 1.38\cdot 10^{-23}~\mathrm {J} /\mathrm {K} \cdot 300~\mathrm {K} \cdot 1000~\Omega }}=4.07\cdot 10^{-9}~\mathrm {V} /{\sqrt {\mathrm {Hz} }}.}$

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

${\displaystyle v_{n}={\sqrt {\overline {v_{n}^{2}}}}{\sqrt {\Delta f}}={\sqrt {4k_{\text{B}}TR\Delta f}}}$

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.^[8] 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

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

The noise generated at the resistor can transfer to the remaining circuit; the maximum noise power transfer happens with

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

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

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. Taking the

value of the current source as:

${\displaystyle i_{n}={v_{n} \over R}={\sqrt {{4k_{\text{B}}T\Delta f} \over R}}.}$

Noise power in decibels

Signal power is often measured in

, in dBm, is then:

${\displaystyle P_{\mathrm {dBm} }=10\ \log _{10}(k_{\text{B}}T\Delta f/1\,{\textrm {mW}})\ {\textrm {dBm}}.}$

At room temperature (300 K) this is approximately

${\displaystyle P_{\mathrm {dBm} }=-173.8\ {\textrm {dBm}}+10\ \log _{10}(\Delta f{\text{ in Hz}})\ {\textrm {dB}}.}$[9]^[10]: 260

Using this equation, noise power for different bandwidths is simple to calculate:

Bandwidth ${\displaystyle (\Delta f)}$	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

Thermal noise on capacitors

Ideal capacitors, as lossless devices, do not have thermal noise, but as commonly used with resistors in an

(R) drops out of the equation. This is because higher R decreases the bandwidth as much as it increases the noise.

The mean-square and RMS noise voltage generated in such a filter are:^[12]

${\displaystyle {\overline {v_{n}^{2}}}={4k_{\text{B}}TR \over 4RC}=k_{\text{B}}T/C}$

${\displaystyle v_{n}={\sqrt {4k_{\text{B}}TR \over 4RC}}={\sqrt {k_{\text{B}}T/C}}.}$

The noise charge is the capacitance times the voltage:

${\displaystyle Q_{n}=Cv_{n}=C{\sqrt {k_{\text{B}}T/C}}={\sqrt {k_{\text{B}}TC}}}$

${\displaystyle {\overline {Q_{n}^{2}}}=C^{2}{\overline {v_{n}^{2}}}=C^{2}k_{\text{B}}T/C=k_{\text{B}}TC}$

This charge noise is the origin of the term "kTC noise".

Although independent of the resistor's value, 100% of the kTC noise arises in the resistor. Therefore, if the resistor and the capacitor are at different temperatures, the temperature of the resistor alone should be used in the above calculation.

An extreme case is the zero bandwidth limit called the reset noise left on a capacitor by opening an ideal switch. The resistance is infinite, yet 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 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 kT/2 per degree of freedom. Using the formula for energy on a capacitor (E = 1/2CV²), mean noise energy on a capacitor can be seen to also be 1/2C(kT/C) = kT/2. Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance.

Noise of capacitors at 300 K

Capacitance	${\displaystyle {\sqrt {k_{\text{B}}T/C}}}$	${\displaystyle {\sqrt {k_{\text{B}}TC}}}$	Electrons
1 fF	2 mV	2 aC	12.5 e−
10 fF	640 µV	6.4 aC	40 e−
100 fF	200 µV	20 aC	125 e−
1 pF	64 µV	64 aC	400 e−
10 pF	20 µV	200 aC	1250 e−
100 pF	6.4 µV	640 aC	4000 e−
1 nF	2 µV	2 fC	12500 e−

Generalized forms

The ${\displaystyle 4k_{\text{B}}TR}$ 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

Nyquist's original paper also provided the generalized noise for components having partly reactive response, e.g., sources that contain capacitors or inductors.^[6] Such a component can be described by a frequency-dependent complex electrical impedance ${\displaystyle Z(f)}$. The formula for the

of the series noise voltage is

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

The function ${\displaystyle \eta (f)}$ is simply equal to 1 except at very high frequencies, or near absolute zero (see below).

The real part of impedance, ${\displaystyle \operatorname {Re} [Z(f)]}$, is in general frequency dependent and so the Johnson–Nyquist noise is not white noise. The rms noise voltage over a span of frequencies ${\displaystyle f_{1}}$ to ${\displaystyle f_{2}}$ can be found by integration of the power spectral density:

${\displaystyle {\sqrt {\langle v_{n}^{2}\rangle }}={\sqrt {\int _{f_{1}}^{f_{2}}S_{v_{n}v_{n}}(f)df}}}$.

Alternatively, a parallel noise current can be used to describe Johnson noise, its

being

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

where ${\displaystyle Y(f)=1/Z(f)}$ is the

; note that ${\displaystyle \operatorname {Re} [Y(f)]=\operatorname {Re} [Z(f)]/|Z(f)|^{2}}$

Quantum effects at high frequencies or low temperatures

Nyquist also pointed out that quantum effects occur for very high frequencies or very low temperatures near absolute zero.^[6] The function ${\displaystyle \eta (f)}$ is in general given by^[13]

${\displaystyle \eta (f)={\frac {hf/k_{\text{B}}T}{e^{hf/k_{\text{B}}T}-1}}+{\frac {1}{2}}{\frac {hf}{k_{\text{B}}T}},}$

where ${\displaystyle h}$ is the Planck constant and ${\displaystyle \eta (f)}$ is a multiplying factor.

At very high frequencies ${\displaystyle f\gtrsim k_{\text{B}}T/h}$, the function ${\displaystyle \eta (f)}$ 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 ${\displaystyle \eta (f)=1}$ for conventional electronics work.

Relation to Planck's law

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.^[14] 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,

over all different directions cannot be larger than ${\displaystyle \lambda ^{2}/(4\pi )}$, where λ is wavelength. This comes from the different frequency dependence of 3D versus 1D Planck's law.

Multiport electrical networks

Richard Q. Twiss extended Nyquist's formulas to multi-port passive electrical networks, including non-reciprocal devices such as circulators and isolators.^[16] 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

functions relating the different noise voltages,

${\displaystyle S_{v_{m}v_{n}}(f)=2k_{\text{B}}T\eta (f)(Z_{mn}(f)+Z_{nm}(f)^{*})}$

where the ${\displaystyle Z_{mn}}$ are the elements of the

${\displaystyle \mathbf {Z} }$. 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

${\displaystyle S_{i_{m}i_{n}}(f)=2k_{\text{B}}T\eta (f)(Y_{mn}(f)+Y_{nm}(f)^{*})}$

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

Continuous electrodynamic media

The full generalization of Nyquist noise is found in

. The equations of fluctuation electrodynamics provide a common framework for describing both Johnson–Nyquist noise and free-space

See also

• Fluctuation-dissipation theorem
• Shot noise
• Pink noise
• Langevin equation
• Rise over thermal

References

 This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22. (in support of MIL-STD-188).

External links

• Amplifier noise in RF systems
• Thermal noise (undergraduate) with detailed math