Editing Johnson–Nyquist noise (section)

== 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
|}