View source for Johnson–Nyquist noise

{{Short description|Electronic noise due to thermal vibration within a conductor}}
[[File:JohnsonNoiseEquivalentCircuits.svg|thumb|These three circuits are all equivalent: '''(A)''' A resistor at nonzero temperature, which has Johnson noise; '''(B)''' A noiseless resistor [[Series and parallel circuits|in series]] with a noise-creating [[voltage source]] (i.e. the [[Thévenin equivalent]] circuit); '''(C)''' A noiseless resistance [[Series and parallel circuits|in parallel]] with a noise-creating [[current source]] (i.e. the [[Norton equivalent]] circuit).]]

'''Johnson–Nyquist noise''' ('''thermal noise''', '''Johnson noise''', or '''Nyquist noise''') is the [[electronic noise]] generated by the [[Thermal energy|thermal agitation]] of the [[charge carrier]]s (usually the [[electron]]s) inside an [[electrical conductor]] at equilibrium, which happens regardless of any applied [[voltage]].  Thermal noise is present in all [[electrical circuit]]s, and in sensitive electronic equipment (such as [[radio receiver]]s) can drown out weak signals, and can be the limiting factor on sensitivity of electrical measuring instruments.  Thermal noise increases with temperature.  Some sensitive electronic equipment such as [[radio telescope]] receivers are cooled to [[cryogenic]] temperatures to reduce thermal noise in their circuits.  The generic, statistical physical derivation of this noise is called the [[fluctuation-dissipation theorem]], where generalized [[Electrical impedance|impedance]] or generalized [[Electric susceptibility|susceptibility]] is used to characterize the medium.

Thermal noise in an ideal resistor is approximately [[white noise|white]], meaning that the power [[spectral density]] is nearly constant throughout the [[frequency spectrum]], but does decay to zero at extremely high frequencies ([[Terahertz (unit)|terahertz]] for [[room temperature]]).  When limited to a finite bandwidth, thermal noise has a nearly [[Normal distribution|Gaussian amplitude distribution]].<ref>{{cite book|author1=John R. Barry |author2=Edward A. Lee |author3=David G. Messerschmitt |title=Digital Communications|year=2004|publisher=Sprinter|isbn=9780792375487|page=69|url=https://books.google.com/books?id=hPx70ozDJlwC&q=thermal+johnson+noise+gaussian+filtered+bandwidth&pg=PA69}}</ref>

== History ==
This type of noise was discovered and first measured by [[John Bertrand Johnson|John B. Johnson]] at [[Bell Labs]] in 1926.<ref>{{Cite journal |doi = 10.1103/PhysRev.29.350|title = Minutes of the Philadelphia Meeting December 28, 29, 30, 1926|journal = Physical Review|volume = 29|issue = 2|pages = 350–373|year = 1927|last1 = Anonymous|bibcode = 1927PhRv...29..350.}}</ref><ref>{{cite journal|first=J.|last=Johnson|title=Thermal Agitation of Electricity in Conductors|journal= Physical Review|volume=32|pages=97–109|number=97|date=1928|doi=10.1103/physrev.32.97|bibcode=1928PhRv...32...97J}}</ref>  He described his findings to [[Harry Nyquist]], also at Bell Labs, who was able to explain the results.<ref name=Nyquist>{{cite journal|first=H.|last=Nyquist|title=Thermal Agitation of Electric Charge in Conductors|journal= Physical Review|volume=32|pages=110–113|number=110|date=1928|doi=10.1103/physrev.32.110|bibcode=1928PhRv...32..110N}}</ref>

== 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 [[Equipartition theorem|equipartition law]] of Boltzmann and Maxwell. Using the concept [[Equipartion theorem#Potential energy and harmonic oscillators|potential energy and harmonic oscillators]] of the equipartition law,<ref>{{Cite book|url=https://books.google.com/books?id=LXPWxmakFVgC|title=Electronic Communication|last=Tomasi|first=Wayne|date=1994|publisher=Prentice Hall PTR|isbn=9780132200622|language=en}}</ref>

<math>\left \langle H \right \rangle=k_{\rm B} T</math>

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

<math>N=k_{\rm B} T  \Delta f </math>

where ''N'' is the noise power and ''Δf'' is the [[Bandwidth (signal processing)|bandwidth]].

== 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 [[Transmission medium|medium]] (e.g. [[ion]]s in an [[electrolyte]]), not just [[resistor]]s. 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 [[Spectral density|power spectral density]], or voltage variance (mean square) per [[hertz]] of [[Bandwidth (signal processing)|bandwidth]], is given by

:<math>
\overline {v_{n}^2} = 4 k_\text{B} T R
</math>

where ''k''<sub>B</sub> is [[Boltzmann's constant]] in [[joule]]s per [[kelvin]], ''T'' is the resistor's absolute [[temperature]] in kelvins, and ''R'' is the resistor value in [[ohm]]s (Ω).
Using this equation for quick calculation, at room temperature:
:<math>
\sqrt{\overline {v_{n}^2}} = 0.13 \sqrt{R} ~\mathrm{nV}/\sqrt{\mathrm{Hz}}.</math>

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

:<math>
\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}}.</math>

For a given bandwidth,  the [[root mean square]] (RMS) of the voltage, <math>v_{n}</math>, is given by

:<math>
v_{n}  = \sqrt{\overline {v_{n}^2}}\sqrt{\Delta f } = \sqrt{ 4 k_\text{B} T R \Delta f }
</math>

where Δ''f'' is the bandwidth in hertz over which the noise is measured. For a 1&nbsp;kΩ resistor at room temperature and a 10&nbsp;kHz bandwidth, the RMS noise voltage is 400&nbsp;nV.<ref>[https://www.google.com/search?q=sqrt(4*k*295+Kelvin*1+kiloOhm*(10+kHz))+in+nanovolt Google Calculator result] for 1&nbsp;kΩ room temperature 10&nbsp;kHz bandwidth</ref> A useful rule of thumb to remember is that 50&nbsp;Ω at 1&nbsp;Hz bandwidth correspond to 1&nbsp;nV noise at room temperature.

A resistor in a short circuit dissipates a noise power of
:<math>
P = {v_{n}^2}/R = 4 k_\text{B} \,T \Delta f.
</math>

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

:<math>
P = k_\text{B} \,T \Delta f
</math>

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 [[Norton equivalent]] of the voltage source corresponds to dividing the noise voltage by ''R''. This gives the [[root mean square]] value of the current source as:

:<math>
i_n = {v_n \over R} = \sqrt {{4 k_\text{B} T \Delta f } \over R}.
</math>

== Noise power in decibels ==
Signal power is often measured in [[dBm]] ([[decibels]] relative to 1 [[milliwatt]]). From the equation above, noise power in a resistor at [[room temperature]], in dBm, is then:
:<math>P_\mathrm{dBm} = 10\ \log_{10}(k_\text{B} T \Delta f / 1\,\textrm{mW})\ \textrm{dBm}.</math>
At room temperature (300 K) this is approximately
:<math>P_\mathrm{dBm} = -173.8\ \textrm{dBm} + 10\ \log_{10}(\Delta f \text{ in Hz})\ \textrm{dB}.</math><ref>{{cite journal|first=J. R.|last=Pierce|url=https://ieeexplore.ieee.org/document/4052064|title=Physical Sources of Noise|journal=Proceedings of the IRE|year=1956|volume=44|issue=5|pages=601–608|doi=10.1109/JRPROC.1956.275123|s2cid=51667159}}</ref><ref>{{Citation |last=Vizmuller|first= Peter |year= 1995 |title= RF Design Guide  |publisher= Artech House |isbn=0-89006-754-6  }}</ref>{{rp|260}}

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

{| class="wikitable"
! Bandwidth <math> (\Delta f )</math>!! Thermal noise power<br/>at 300 K ([[dBm]]) !! Notes
|-
| 1&nbsp;Hz    || −174 ||
|-
| 10&nbsp;Hz   || −164 ||
|-
| 100&nbsp;Hz  || −154 ||
|-
| 1&nbsp;kHz   || −144 ||
|-
| 10&nbsp;kHz  || −134 || [[Frequency modulation|FM]] channel of [[Walkie-talkie|2-way radio]]
|-
| 100&nbsp;kHz || −124 ||
|-
| 180&nbsp;kHz || −121.45 || One [[3GPP Long Term Evolution|LTE]] resource block<!-- Twelve of these subcarriers together (per slot) is called a resource block so one resource block is 180 kHz -->
|-
| 200&nbsp;kHz || −121 || [[GSM]] channel
|-
| 1&nbsp;MHz   || −114 || Bluetooth channel
|-
| 2&nbsp;MHz   || −111 || Commercial [[Global Positioning System|GPS]] channel
|-
| 3.84&nbsp;MHz || −108 || [[UMTS]] channel
|-
| 6&nbsp;MHz   || −106 || [[Analog television]] channel
|-
| 20&nbsp;MHz  || −101 || [[IEEE 802.11|WLAN 802.11]] channel
|-
| 40&nbsp;MHz || −98 || [[IEEE 802.11n|WLAN 802.11n]] 40&nbsp;MHz channel
|-
| 80&nbsp;MHz || −95 || [[IEEE 802.11ac|WLAN 802.11ac]] 80&nbsp;MHz channel
|-
| 160&nbsp;MHz || −92 || [[IEEE 802.11ac|WLAN 802.11ac]] 160&nbsp;MHz channel
|-
| 1&nbsp;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 [[RC circuit]], the combination has what is called ''kTC'' noise.  The noise bandwidth of an RC circuit is &Delta;''f'' = 1/(4''RC'').<ref>{{cite web|first=Kent H.|last=Lundberg|page=10|url=http://web.mit.edu/klund/www/papers/UNP_noise.pdf|title=Noise Sources in Bulk CMOS}}</ref>  When this is substituted into the thermal noise equation, the result has an unusually simple form as the value of the [[electrical resistance|resistance]] (''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:<ref>{{cite journal
 |first1=R. |last1=Sarpeshkar |first2=T. |last2=Delbruck |first3=C. A. |last3=Mead
 |title=White noise in MOS transistors and resistors
 |journal= IEEE Circuits and Devices Magazine |volume=9 |issue=6 |pages=23–29
 |date=November 1993 |doi=10.1109/101.261888
 |s2cid=11974773 |url=http://users.ece.gatech.edu/~phasler/Courses/ECE6414/Unit1/Rahul_noise01.pdf
}}</ref>

:<math>
\overline {v_n^2} = {4 k_\text{B} T R \over 4 R C} = k_\text{B} T / C
</math>
:<math>
v_n = \sqrt{4 k_\text{B} T R \over 4 R C} = \sqrt{ k_\text{B} T / C }.
</math>

The noise ''charge'' is the capacitance times the voltage:
: <math>
Q_n = C v_n = C \sqrt{ k_\text{B} T / C } = \sqrt{ k_\text{B} T C }
</math>
: <math>
\overline{Q_n^2} = C^2 \overline{v_n^2} = C^2 k_\text{B} T / C = k_\text{B} T C
</math>
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 sensor]]s.

Any system in [[thermal equilibrium]] has [[state variable]]s with a mean [[energy]] of ''kT''/2 per [[degrees of freedom (physics and chemistry)|degree of freedom]]. Using the formula for energy on a capacitor (''E''&nbsp;= ½''CV''<sup>2</sup>), mean noise energy on a capacitor can be seen to also be ½''C''(''kT''/''C'') = ''kT''/2.  Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance.

{| class="wikitable" style="text-align:right"
|+ Noise of capacitors at 300 K
! Capacitance !! <math>\sqrt{ k_\text{B} T / C }</math> !! <math>\sqrt{ k_\text{B} T C }</math> !! Electrons
|-
|   1 fF ||   2 mV   || {{#expr:sqrt(1.380649e-23 * 300 * 1e-15) * 1e18 round  1 }} aC ||    12.5 e<sup>−</sup>
|-
|  10 fF || 640 µV   || {{#expr:sqrt(1.380649e-23 * 300 * 1e-14) * 1e18 round  1 }} aC ||    40 e<sup>−</sup>
|-
| 100 fF || 200 µV   || {{#expr:sqrt(1.380649e-23 * 300 * 1e-13) * 1e18 round  0 }} aC ||   125 e<sup>−</sup>
|-
|   1 pF ||  64 µV   || {{#expr:sqrt(1.380649e-23 * 300 * 1e-12) * 1e18 round  0 }} aC ||   400 e<sup>−</sup>
|-
|  10 pF ||  20 µV   || {{#expr:sqrt(1.380649e-23 * 300 * 1e-11) * 1e18 round -1 }} aC ||  1250 e<sup>−</sup>
|-
| 100 pF ||   6.4 µV || {{#expr:sqrt(1.380649e-23 * 300 * 1e-10) * 1e18 round -1 }} aC ||  4000 e<sup>−</sup>
|-
|   1 nF ||   2 µV   || {{#expr:sqrt(1.380649e-23 * 300 * 1e-9)  * 1e15 round  1 }} fC || 12500 e<sup>−</sup>
|}

== Generalized forms ==
The <math>4 k_\text{B} T R</math> 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 [[Passivity (engineering)|passive]] and linear.

=== Reactive impedances ===
Nyquist's original paper also provided the generalized noise for components having partly [[electrical reactance|reactive]] response, e.g., sources that contain capacitors or inductors.<ref name=Nyquist/> Such a component can be described by a frequency-dependent complex [[electrical impedance]] <math>Z(f)</math>. The formula for the [[power spectral density]] of the series noise voltage is
:<math>
S_{v_n v_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Z(f)].
</math>
The function <math>\eta(f)</math> is simply equal to 1 except at very high frequencies, or near absolute zero (see below).

The real part of impedance, <math>\operatorname{Re}[Z(f)]</math>, is in general frequency dependent and so the Johnson–Nyquist noise is not white noise.
The rms noise voltage over a span of frequencies <math>f_1</math> to <math>f_2</math> can be found by integration of the power spectral density:
:<math> \sqrt{\langle v_n^2 \rangle} = \sqrt{\int_{f_1}^{f_2} S_{v_n v_n}(f) df}</math>.

Alternatively, a parallel noise current can be used to describe Johnson noise, its [[power spectral density]] being
:<math>
S_{i_n i_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Y(f)].
</math>
where <math>Y(f) = 1/Z(f)</math> is the [[electrical admittance]]; note that <math>\operatorname{Re}[Y(f)] = \operatorname{Re}[Z(f)]/|Z(f)|^2</math>

=== 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.<ref name=Nyquist/> The function <math>\eta(f)</math> is in general given by<ref>{{cite journal |last1=Callen |first1=Herbert |title=Irreversibility and Generalized Noise |journal=Physical Review |volume=83 |issue=1 |page=34 }}</ref>
:<math>\eta(f) = \frac{hf/k_\text{B} T}{e^{hf/k_\text{B} T} - 1}+\frac{1}{2}
\frac{h f}{k_\text{B} T},</math>
where <math>h</math> is [[Planck's constant]] and <math>\eta(f)</math> is a multiplying factor.

At very high frequencies <math>f \gtrsim k_\text{B} T/h</math>, the function <math>\eta(f)</math> 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 <math>\eta(f)=1</math> 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|Planck's law of blackbody radiation]].<ref>{{cite book |title=Fundamentals of Microwave Photonics |page=63 |url=https://books.google.com/books?id=mg91BgAAQBAJ&pg=PA63 |first1=V. J.|last1=Urick|first2=Keith J.|last2=Williams|first3=Jason D.|last3=McKinney|isbn=9781119029786 |date=2015-01-30 |publisher=John Wiley & Sons }}</ref> 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,<ref>{{Cite journal| doi = 10.1063/1.1770483| volume = 17| issue = 7| pages = 268–275| last = Dicke| first = R. H.| title = The Measurement of Thermal Radiation at Microwave Frequencies| journal = Review of Scientific Instruments| date = 1946-07-01| pmid=20991753| bibcode = 1946RScI...17..268D| s2cid = 26658623| doi-access = free}}</ref> and further connected it to properties of antennas, particularly the fact that the average [[antenna aperture]] over all different directions cannot be larger than <math>\lambda^2/(4\pi)</math>, 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 (circuit theory)|port]] passive electrical networks, including non-reciprocal devices such as [[circulator]]s and [[Isolator (microwave)|isolator]]s.<ref>{{Cite journal | doi = 10.1063/1.1722048| title = Nyquist's and Thevenin's Theorems Generalized for Nonreciprocal Linear Networks| journal = Journal of Applied Physics| volume = 26| issue = 5| pages = 599–602| year = 1955| last1 = Twiss | first1 = R. Q.| bibcode = 1955JAP....26..599T}}</ref> 
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,
:<math>
S_{v_m v_n}(f) = 2 k_\text{B} T \eta(f) (Z_{mn}(f) + Z_{nm}(f)^*)
</math>
where the <math>Z_{mn}</math> are the elements of the [[impedance matrix]] <math>\mathbf{Z}</math>.
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
:<math>
S_{i_m i_n}(f) = 2 k_\text{B} T \eta(f) (Y_{mn}(f) + Y_{nm}(f)^*)
</math>
where <math>\mathbf{Y} = \mathbf{Z}^{-1}</math> is the [[Admittance parameters|admittance matrix]].

=== Continuous electrodynamic media ===
<!--
TODO in this section:
Mention foundational work by Kubo, Rytov
Mention applications such as generalized electrodynamic thermal transfer (nearfield, etc.)
Someone should write [[fluctuation electrodynamics]].
-->
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 [[permeability (electromagnetism)|magnetic permeability]].
The equations of fluctuation electrodynamics provide a common framework for describing both Johnson–Nyquist noise and free-space [[blackbody radiation]].<ref>{{cite book
 |author1-link=Lev Pitaevskii
 |first1=L. P.
 |last1=Pitaevskii
 |author2-link=Evgeny Lifshitz
 |first2=E. M.
 |last2=Lifshitz
 |year=1980
 |title=Statistical Physics, Part 2: Theory of the Condensed State
 |edition=1st |volume=9
 |chapter=Chapter VIII. Electromagnetic Fluctuations
 |publisher=[[Butterworth-Heinemann]]
 |isbn=978-0-7506-2636-1
}}</ref>

==See also==
* [[Fluctuation-dissipation theorem]]
* [[Shot noise]]
* [[Pink noise]]
* [[Langevin equation]]
* [[Rise over thermal]]

==References==
<references/>
{{FS1037C MS188}}

== External links ==
*[http://www4.tpgi.com.au/users/ldbutler/AmpNoise.htm Amplifier noise in RF systems]
*[http://www.physics.utoronto.ca/~phy225h/experiments/thermal-noise/Thermal-Noise.pdf Thermal noise (undergraduate) with detailed math]

{{Noise}}

{{DEFAULTSORT:Johnson-Nyquist noise}}
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