Editing Johnson–Nyquist noise

{{Short description|Electronic noise due to thermal vibration within a conductor}}
[[File:Oscilloscope-setup-johnson-noise.svg|thumb|327x327px|Figure 1. [[John Bertrand Johnson|Johnson]]'s 1927 experiment showed that if thermal noise from a [[Resistance (electricity)|resistance]] of <math>\text{R}</math> with [[Absolute Temperature|temperature]] <math>\text{T}</math> is [[bandlimited]] to [[Bandwidth (signal processing)|bandwidth]] <math>\Delta f </math>, then its [[root mean squared]] [[voltage]] <math>(V_\text{rms})</math> is <math>\sqrt{ 4 k_\text{B} \text{T} \, \text{R} \, \Delta f }</math> in general, where <math>k_\text{B}</math> is the [[Boltzmann constant]].]]
'''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 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 [[Electrical impedance|impedance]] or generalized [[Electric susceptibility|susceptibility]] is used to characterize the medium.
[[File:Johnson-nyquist-noise-power-spectral-density-of-ideal-resistor.svg|thumb|400x400px|Figure 2. Johnson–Nyquist noise has a nearly a constant {{nowrap|{{math|4}}[[Boltzmann constant|{{math|''k''<sub>B</sub>}}]][[Absolute temperature|{{math|T}}]][[Electrical resistance and conductance|{{math|R}}]]}} power spectral density per unit of [[frequency]], but does decay to zero [[Johnson–Nyquist noise#Quantum effects at high frequencies or low temperatures|due to quantum effects]] at high frequencies ([[Terahertz (unit)|terahertz]] for room temperature). This plot's horizontal axis uses a [[log scale]] such that every vertical line corresponds to a [[power of ten]] of frequency in [[hertz]].]]
Thermal noise in an [[ideal resistor]] is approximately [[white noise|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 [[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> 

For the general case, this definition applies to [[charge carriers]] in any type of conducting [[Transmission medium|medium]] (e.g. [[ion]]s in an [[electrolyte]]), not just [[resistor]]s. 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.

== History of thermal noise ==
In 1905, in one of [[Albert Einstein]]'s [[Annus mirabilis papers|''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.<ref name=":0">{{Cite journal |last=Dörfel |first=G. |date=2012-08-15 |title=The early history of thermal noise: The long way to paradigm change |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.201200736 |journal=Annalen der Physik |language=en |volume=524 |issue=8 |pages=117–121 |doi=10.1002/andp.201200736 |issn=0003-3804}}</ref>

[[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.<ref name=":0" /><ref>{{Citation |last=Van Der Ziel |first=A. |title=Advances in Electronics and Electron Physics Volume 50 |chapter=History of Noise Research |date=1980-01-01 |volume=50 |pages=351–409 |editor-last=Marton |editor-first=L. |chapter-url=https://www.sciencedirect.com/science/article/pii/S0065253908610665 |access-date=2024-03-16 |publisher=Academic Press |doi=10.1016/s0065-2539(08)61066-5 |isbn=978-0-12-014650-5 |editor2-last=Marton |editor2-first=C.}}</ref>

[[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]].<ref name=":0" />

[[Frits Zernike]] working in electrical metrology, found unusual random deflections while working with high-sensitive [[Galvanometer|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.<ref name=":0" />

The same year, working independently without any knowledge of Zernike's work, [[John Bertrand Johnson|John B. Johnson]] working in [[Bell Labs]] found the same kind of noise in communication systems, but described it in terms of frequencies.<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 name=":2">{{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><ref name=":0" /> 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.<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>

== Noise of ideal resistors for moderate frequencies ==
[[File:Effect-of-bandwidth-on-selecting-noise.svg|thumb|248x248px|Figure 3. While thermal noise has an almost constant power spectral density of <math>4 k_\text{B} T R</math>, a [[band-pass filter]] with bandwidth <math>\Delta f {=} f_\text{upper} {-} f_\text{lower} </math> passes only the shaded area of height <math>4 k_\text{B} T R</math> and width <math>\Delta f</math>. Note: practical [[Electronic filter|filters]] don't have [[Brickwall filter|brickwall]] [[Cutoff frequencies|cutoffs]], so the left and right edges of this area are not perfectly vertical.]]
Johnson's experiment (Figure 1) found that the thermal noise from a resistance <math>R</math> at [[kelvin temperature]] <math>T</math> and [[bandlimited]] to a [[frequency band]] of [[Bandwidth (signal processing)|bandwidth]] <math>\Delta f </math> (Figure 3) has a [[mean square]] voltage of:<ref name=":2" />

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

where <math>k_{\rm B}</math> is the [[Boltzmann constant]] ({{val|1.380649|e=-23}} [[joules]] per [[kelvin]]). While this equation applies to ''ideal resistors'' (i.e. pure resistances without any frequency-dependence) at non-extreme frequency and temperatures, a more accurate [[Johnson–Nyquist noise#general form|general form]] accounts for [[Complex Impedance|complex impedances]] and quantum effects. Conventional electronics generally operate over a more limited [[Bandwidth (signal processing)|bandwidth]], so Johnson's equation is often satisfactory.

=== Power spectral density ===
The mean square voltage per [[hertz]] of [[Bandwidth (signal processing)|bandwidth]] is <math>4 k_\text{B} T R</math> and may be called the [[power spectral density]] (Figure 2).{{NoteTag|This article is using [[Spectral density#One-sided vs two-sided|"one-sided" (positive-only frequency)]] not "two-sided" frequency.}} Its square root at room temperature (around 300&nbsp;K) approximates to 0.13 <math>\sqrt{R}</math> in units of {{Sfrac|nanovolts|{{sqrt|hertz}}}}. A 10&nbsp;kΩ resistor, for example, would have approximately 13&nbsp;{{Sfrac|nanovolts|{{sqrt|hertz}}}} at room temperature.

=== RMS noise voltage ===
[[File:JohnsonNoiseEquivalentCircuits.svg|thumb|Figure 4. These circuits are equivalent:<br><br>'''(A)''' A resistor at nonzero temperature with internal thermal noise;<br><br>'''(B)''' Its [[Thévenin equivalent]] circuit: a noiseless resistor [[Series and parallel circuits|in series]] with a noise [[voltage source]];<br><br>'''(C)''' Its [[Norton equivalent]] circuit: a noiseless resistance [[Series and parallel circuits|in parallel]] with a noise [[current source]].|303x303px]]The square root of the mean square voltage yields the [[root mean square]] (RMS) voltage observed over the bandwidth <math>\Delta f </math>:

: <math>V_\text{rms} = \sqrt{\overline {V_n^2}} = \sqrt{ 4 k_\text{B} T R \, \Delta f } \, .</math>

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&nbsp;kΩ provides almost one microvolt of RMS noise over 20&nbsp;kHz (the [[human hearing range]]) and 60&nbsp;Ω·Hz for <math>R \, \Delta f</math> corresponds to almost one nanovolt of RMS noise.

=== RMS noise current ===
A resistor with thermal noise can also be [[Thévenin's theorem#Conversion to a Norton equivalent|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:

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

== Thermal noise on capacitors ==
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 <math>\Delta f {=} \tfrac{1}{4RC}.</math><ref name=":3">{{cite web |last=Lundberg |first=Kent H. |title=Noise Sources in Bulk CMOS |url=http://web.mit.edu/klund/www/papers/UNP_noise.pdf |page=10}}</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 |last1=Sarpeshkar |first1=R. |last2=Delbruck |first2=T. |last3=Mead |first3=C. A. |date=November 1993 |title=White noise in MOS transistors and resistors |url=http://users.ece.gatech.edu/~phasler/Courses/ECE6414/Unit1/Rahul_noise01.pdf |journal=IEEE Circuits and Devices Magazine |volume=9 |issue=6 |pages=23–29 |doi=10.1109/101.261888 |s2cid=11974773}}</ref>
: <math>
\overline {V_n^2} = {4 k_\text{B} T R \over 4 R C} = {k_\text{B} T \over C}
</math>
: <math>
V_\text{rms} = \sqrt{4 k_\text{B} T R \over 4 R C} = \sqrt{ k_\text{B} T \over C }.
</math>

The noise [[Electric charge|charge]] <math>Q_n</math> is the [[capacitance]] times the voltage:
: <math>Q_n = C \, V_n = C \sqrt{ k_\text{B} T \over 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 \over 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, it would incorrect to double-count both a resistor's thermal noise and its associated kTC noise,<ref name=":3" /> and the temperature of the resistor alone should be used, even if the resistor and the capacitor are at different temperatures. Some values are tabulated below:

{| class="wikitable" style="text-align:right"
|+ Thermal noise on capacitors at 300 K
! rowspan="2" |Capacitance
! rowspan="2" |<math>
V_\text{rms} {=} \sqrt{ k_\text{B} T \over C }
</math>
! colspan="2" |Charge noise <math>Q_n {=} \sqrt{ k_\text{B} T C }</math>
|-
! as [[Coulomb|coulombs]]!! as [[electrons]]{{NoteTag|The charge of a single electron is e− (the negative of the [[elementary charge]]). So each number to the left of e− represents the total number of electrons that make up the noise charge.}}
|-
|   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>
|}

=== Reset noise ===
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 sensor]]s.

Any system in [[thermal equilibrium]] has [[state variable]]s with a mean [[energy]] of {{Sfrac|kT|2}} per [[degrees of freedom (physics and chemistry)|degree of freedom]]. Using the formula for energy on a capacitor (''E''&nbsp;= {{sfrac|1|2}}''CV''<sup>2</sup>), mean noise energy on a capacitor can be seen to also be {{sfrac|1|2}}''C''{{Sfrac|kT|C}} = {{Sfrac|kT|2}}. Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance.

== Thermometry ==
The Johnson–Nyquist noise has applications in precision measurements, in which it is typically called "Johnson noise thermometry".<ref>{{Cite journal |last1=White |first1=D R |last2=Galleano |first2=R |last3=Actis |first3=A |last4=Brixy |first4=H |last5=Groot |first5=M De |last6=Dubbeldam |first6=J |last7=Reesink |first7=A L |last8=Edler |first8=F |last9=Sakurai |first9=H |last10=Shepard |first10=R L |last11=Gallop |first11=J C |date=August 1996 |title=The status of Johnson noise thermometry |url=https://iopscience.iop.org/article/10.1088/0026-1394/33/4/6 |journal=Metrologia |volume=33 |issue=4 |pages=325–335 |doi=10.1088/0026-1394/33/4/6 |issn=0026-1394|url-access=subscription }}</ref> 

For example, the [[National Institute of Standards and Technology|NIST]] in 2017 used the Johnson noise thermometry to measure the [[Boltzmann constant]] with uncertainty less than 3 [[Parts per million|ppm]]. It accomplished this by using [[Josephson voltage standard]] and a [[Quantum Hall effect|quantum Hall resistor]], held at the [[Triple point of water|triple-point temperature of water]]. The voltage is measured over a period of 100 days and integrated.<ref>{{Cite journal |last1=Qu |first1=Jifeng |last2=Benz |first2=Samuel P |last3=Coakley |first3=Kevin |last4=Rogalla |first4=Horst |last5=Tew |first5=Weston L |last6=White |first6=Rod |last7=Zhou |first7=Kunli |last8=Zhou |first8=Zhenyu |date=2017-08-01 |title=An improved electronic determination of the Boltzmann constant by Johnson noise thermometry |journal=Metrologia |volume=54 |issue=4 |pages=549–558 |doi=10.1088/1681-7575/aa781e |issn=0026-1394 |pmc=5621608 |pmid=28970638}}</ref>

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 [[2019 revision of the SI|2019 redefinition]], the kelvin was defined so that the Boltzmann constant is 1.380649×10<sup>−23</sup> J⋅K<sup>−1</sup>, and the triple point of water became experimentally measurable.<ref>{{Cite press release |date=2016-11-15 |title=Noise, Temperature, and the New SI |url=https://www.nist.gov/news-events/news/2016/11/noise-temperature-and-new-si |website=[[NIST]] |language=en}}</ref><ref>{{Cite press release |date=2017-06-29 |title=NIST 'Noise Thermometry' Yields Accurate New Measurements of Boltzmann Constant |url=https://www.nist.gov/news-events/news/2017/06/nist-noise-thermometry-yields-accurate-new-measurements-boltzmann-constant |website=[[NIST]] |language=en}}</ref><ref>{{Cite journal |last1=Fischer |first1=J |last2=Fellmuth |first2=B |last3=Gaiser |first3=C |last4=Zandt |first4=T |last5=Pitre |first5=L |last6=Sparasci |first6=F |last7=Plimmer |first7=M D |last8=de Podesta |first8=M |last9=Underwood |first9=R |last10=Sutton |first10=G |last11=Machin |first11=G |last12=Gavioso |first12=R M |last13=Ripa |first13=D Madonna |last14=Steur |first14=P P M |last15=Qu |first15=J |date=2018 |title=The Boltzmann project |journal=Metrologia |volume=55 |issue=2 |pages=10.1088/1681–7575/aaa790 |doi=10.1088/1681-7575/aaa790 |issn=0026-1394 |pmc=6508687 |pmid=31080297}}</ref>

== Thermal noise on inductors ==
Inductors are the [[Duality (electrical circuits)|dual]] of capacitors. Analogous to kTC noise, a resistor with an inductor <math>L</math> results in a noise ''current'' that is independent of resistance:<ref name=":1">{{cite journal |last=Pierce |first=J. R. |year=1956 |title=Physical Sources of Noise |url=https://ieeexplore.ieee.org/document/4052064 |journal=Proceedings of the IRE |volume=44 |issue=5 |pages=601–608 |doi=10.1109/JRPROC.1956.275123 |s2cid=51667159|url-access=subscription }}</ref>

: <math>
\overline {I_n^2} = {k_\text{B} T \over L} \, .
</math>

== Maximum transfer of noise power ==
The noise generated at a resistor ''<math>R_\text{S}</math>'' can transfer to the remaining circuit. The [[Maximum power transfer theorem|maximum power transfer happens when]] the [[Thévenin equivalent]] resistance <math>R_{\rm L}</math> of the remaining circuit [[Impedance matching|matches]] ''<math>R_\text{S}</math>''.<ref name=":1" /> In this case, each of the two 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, this maximum noise power transfer is:
: <math>P_\text{max} = k_\text{B} \,T \Delta f \, .</math>
This maximum is independent of the resistance and is called the ''available noise power'' from a resistor.<ref name=":1" />

=== Available noise power in decibel-milliwatts ===
Signal power is often measured in [[dBm]] ([[decibels]] relative to 1 [[milliwatt]]). Available noise power would thus be <math>10\ \log_{10}(\tfrac{k_\text{B} T \Delta f}{\text{1 mW}})</math> in dBm. At room temperature (300 K), the available noise power can be easily approximated as <math>10\ \log_{10}(\Delta f) - 173.8</math> in dBm for a bandwidth in hertz.<ref name=":1" /><ref>{{Citation |last=Vizmuller |first=Peter |title=RF Design Guide |year=1995 |publisher=Artech House |isbn=0-89006-754-6}}</ref>{{rp|260}} Some example available noise power in dBm are tabulated below:
{| class="wikitable"
! Bandwidth <math> (\Delta f )</math>!! Available 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
|}

== Nyquist's derivation of ideal resistor noise ==
[[File:Nyquist-transmission-line-derivation-of-johnson-noise-with-voltage-sources.svg|thumb|396x396px|Figure 5. Schematic of [[Harry Nyquist|Nyquist's]] 1928 [[thought experiment]]<ref name="Nyquist" /><ref name=":0" /> using two noisy resistors (each represented here by a noise-free resistor in series with a noise voltage source) connected via a long lossless [[transmission line]] of length <math>l </math>. Each resistor's noise [[Analog signal|signal]] propagates across the line at velocity <math>\text{v} </math>. All impedances are identical, so both signals are absorbed by the opposite resistor instead of being [[signal reflections|reflected]].<br><br>Nyquist then imagined [[Short circuit|shorting]] both ends of the line, thereby trapping in-flight energy on the line. Because all in-flight energy is now completely reflected (due to the now-mismatched impedance), the in-flight energy can be represented as a summation of sinusoidal [[standing waves]]. For a band of frequencies <math>\Delta f </math>, there are <math>2l \, \Delta f / \text{v}</math> [[Normal mode#Standing waves|modes of oscillation]].{{NoteTag|A standing wave occurs with frequency equal to every integer multiple of <math>\tfrac{\text{v}}{2l}</math>. The line is sufficiently long to make the number of modes within the bandwidth very large, such that the modes will be close enough in frequency to approximate a continuous frequency spectrum.}} Each mode provides <math>k_{\rm B} T</math> [[KT (energy)|of energy]] on average, of which <math>k_{\rm B} T / 2</math> is electric and <math>k_{\rm B} T / 2</math> is magnetic, so the total energy in that bandwidth on average is <math>k_{\rm B} T \cdot 2l \, \Delta f / \text{v} . </math> Each resistor contributed <math>k_{\rm B} T \cdot l \, \Delta f / \text{v} </math> (half of that total energy).<br><br>But since before the shorting there were originally no reflections, the value of that total in-flight energy also equals the combined energy that was transferred from both resistors to the line during the transit time interval of <math>l / \text{v} </math>. Dividing the average '''energy''' transferred from ''each'' resistor to the line by the transit '''time''' interval results in a total '''power''' of <math>k_{\rm B} T \, \Delta f </math> transferred over bandwidth <math>\Delta f </math> on average from each resistor.]]
Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors"<ref name="Nyquist" /> used concepts about [[Equipartition theorem#Potential energy and harmonic oscillators|potential energy and harmonic oscillators from the equipartition law]] of [[Ludwig Boltzmann|Boltzmann]] and [[James Clerk Maxwell|Maxwell]]<ref>{{Cite book |last=Tomasi |first=Wayne |url=https://books.google.com/books?id=LXPWxmakFVgC |title=Electronic Communication |date=1994 |publisher=Prentice Hall PTR |isbn=9780132200622 |language=en}}</ref> to explain Johnson's experimental result. Nyquist's [[thought experiment]] summed the energy contribution of each [[Normal mode#Standing waves|standing wave mode of oscillation]] on a long lossless [[transmission line]] between two equal resistors (<math>R_1 {=} R_2</math>). According to the conclusion of Figure 5, the total average power transferred over bandwidth <math>\Delta f </math> from <math>R_1</math> and absorbed by <math>R_2</math> was determined to be:

: <math>\overline {P_1} = k_{\rm B} T \, \Delta f \, . </math>

Simple application of [[Ohm's law]] says the current from <math>V_1</math> (the thermal voltage noise of only <math>R_1</math>) through the combined resistance is <math display="inline">I_1 {=} \tfrac{V_1}{R_1 + R_2} {=} \tfrac{V_1}{2R_1}</math>, so the power transferred from <math>R_1</math> to <math>R_2</math> is the square of this current multiplied by <math>R_2</math>, which simplifies to:<ref name="Nyquist" />

: <math>P_\text{1} = I_1^2 R_2 = I_1^2 R_1 = \left( \frac{V_1}{2R_1} \right)^2 R_1 = \frac{V_1^2}{4R_1}  \, .</math>

Setting this <math display="inline">P_\text{1}</math> equal to the earlier average power expression <math display="inline">\overline {P_1}</math> allows solving for the average of <math display="inline">V_1^2</math> over that bandwidth:

: <math>\overline{V_1^2} = 4 k_\text{B} T {R_1} \, \Delta f \, .</math>

Nyquist used similar reasoning to provide a generalized expression that applies to non-equal and [[Johnson–Nyquist noise#Complex impedances|complex impedances]] too. And while Nyquist above used <math>k_{\rm B} T</math> according to [[Classical physics|classical]] theory, Nyquist concluded his paper by attempting to use a more involved expression that incorporated the [[Planck constant]] <math>h</math> (from the new theory of [[quantum mechanics]]).<ref name="Nyquist" />

== 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 to moderate 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.

=== Complex 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 approximately 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 taking the square root of integration of the power spectral density:
:<math> V_\text{rms} = \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) {=} \tfrac{1}{Z(f)}</math> is the [[electrical admittance]]; note that <math>\operatorname{Re}[Y(f)] {=} \tfrac{\operatorname{Re}[Z(f)]}{|Z(f)|^2} \, .</math>

=== Quantum effects at high frequencies or low temperatures ===
With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near [[absolute zero]]), the multiplying factor <math>\eta(f)</math> mentioned earlier is in general given by:<ref>{{Cite journal |last1=Callen |first1=Herbert B. |last2=Welton |first2=Theodore A. |date=1951-07-01 |title=Irreversibility and Generalized Noise |url=https://link.aps.org/doi/10.1103/PhysRev.83.34 |journal=Physical Review |volume=83 |issue=1 |pages=34–40 |doi=10.1103/PhysRev.83.34|url-access=subscription }}</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>
At very high frequencies (<math>f \gtrsim \tfrac{k_\text{B} T}{h}</math>), the spectral density <math>S_{v_n v_n}(f)</math> now 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>\tfrac{\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]].

== Notes ==
{{NoteFoot}}

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

== References ==
{{reflist}}
{{FS1037C MS188}}

== External links ==
* [http://www4.tpgi.com.au/users/ldbutler/AmpNoise.htm Amplifier noise in RF systems] {{Webarchive|url=https://web.archive.org/web/20080704180832/http://www4.tpgi.com.au/users/ldbutler/AmpNoise.htm |date=2008-07-04 }}
* [http://www.physics.utoronto.ca/~phy225h/experiments/thermal-noise/Thermal-Noise.pdf Thermal noise (undergraduate) with detailed math]

{{Noise}}

{{DEFAULTSORT:Johnson-Nyquist noise}}
[[Category:Noise (electronics)]]
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[[Category:Electrical parameters]]
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