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[[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 agitation of the charge carriers (usually the [[electron]]s) inside an [[electrical conductor]] at equilibrium, which happens regardless of any applied [[voltage]]. 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 idealistic resistor is approximately [[white noise|white]], meaning that the power [[spectral density]] is nearly constant throughout the [[frequency spectrum]] (however see the section below on extremely high frequencies).  When limited to a finite bandwidth, thermal noise has a nearly Gaussian amplitude distribution.<ref>{{cite book|author=John R. Barry, Edward A. Lee, and David G. Messerschmitt|title=Digital Communications|year=2004|publisher=Sprinter|isbn=9780792375487|page=69|url=http://books.google.com/books?id=hPx70ozDJlwC&pg=PA69&dq=thermal+johnson+noise+gaussian+filtered+bandwidth&hl=en&sa=X&ei=zRLnUbHEDe-0igLT5oCADA&ved=0CEMQ6AEwAw#v=onepage&q=thermal%20johnson%20noise%20gaussian%20filtered%20bandwidth&f=false}}</ref>


== History ==
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This type of noise was first measured by [[John B. Johnson]] at [[Bell Labs]] in 1926.<ref>[http://prola.aps.org/pdf/PR/v29/i2/p350_1 "Proceedings of the American Physical Society: Minutes of the Philadelphia Meeting December 28, 29, 30, 1926"], Phys. Rev. '''29,''' pp. 367-368 (1927) – a February 1927 publication of an abstract for a paper - entitled "Thermal agitation of electricity in conductors" - presented by Johnson during the December 1926 APS Annual Meeting</ref><ref>J. Johnson, [http://link.aps.org/abstract/PR/v32/p97 "Thermal Agitation of Electricity in Conductors"], Phys. Rev. '''32,''' 97 (1928) – details of the experiment</ref> He described his findings to [[Harry Nyquist]], also at Bell Labs, who was able to explain the results.<ref>H. Nyquist, [http://link.aps.org/abstract/PR/v32/p110 "Thermal Agitation of Electric Charge in Conductors"], Phys. Rev. '''32,''' 110 (1928) – the theory</ref>
 
== 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>
\bar {v_{n}^2} = 4 k_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).
Use this equation for quick calculation, at room temperature:
:<math>
\sqrt{\bar {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{\bar {v_{n}^2}} = \sqrt{4 \cdot 1.38 \cdot 10^{-23}~\mathrm{J}/\mathrm{K} \cdot 300~\mathrm{K} \cdot 1~\mathrm{k}\Omega} = 4.07  ~\mathrm{nV}/\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{\bar {v_{n}^2}}\sqrt{\Delta f } = \sqrt{ 4 k_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_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_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 by taking the [[Norton equivalent]] that corresponds simply to divide by ''R''. This gives the [[root mean square]] value of the current source as:
 
:<math>
i_n = \sqrt {{4 k_B T \Delta f } \over R}.
</math>
 
Thermal noise is intrinsic to all resistors and is not a sign of poor design or manufacture, although resistors may also have excess noise.
 
== 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_B T \Delta f \times 1000)</math>
where the factor of 1000 is present because the power is given in milliwatts, rather than watts. This equation can be simplified by separating the constant parts from the bandwidth:
:<math>P_\mathrm{dBm} = 10\ \log_{10}(k_B T \times 1000) + 10\ \log_{10}(\Delta f)</math>
which is more commonly seen approximated for room temperature (T = 300 K) as:
:<math>P_\mathrm{dBm} = -174 + 10\ \log_{10}(\Delta f)</math>
where <math>\Delta f</math> is given in Hz; e.g., for a noise bandwidth of 40&nbsp;MHz, <math>\Delta f</math> is 40,000,000.
 
Using this equation, noise power for different bandwidths is simple to calculate:
 
::{| class="wikitable"
! Bandwidth <math> (\Delta f )</math>!! Thermal noise power !! Notes
|-
| 1&nbsp;Hz    || −174 dBm  ||
|-
| 10&nbsp;Hz  || −164 dBm
|-
| 100&nbsp;Hz  || −154 dBm
|-
| 1&nbsp;kHz  || −144 dBm
|-
| 10&nbsp;kHz  || −134 dBm    || [[Frequency modulation|FM]] channel of [[Walkie-talkie|2-way radio]]
|-
| 15&nbsp;kHz  || −132.24 dBm  || One [[3GPP Long Term Evolution|LTE]] subcarrier
|-
| 100&nbsp;kHz || −124 dBm
|-
| 180&nbsp;kHz || −121.45 dBm || 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 dBm || [[GSM]] channel
|-
| 1&nbsp;MHz  || −114 dBm || Bluetooth channel
|-
| 2&nbsp;MHz  || −111 dBm    || Commercial [[Global Positioning System|GPS]] channel
|-
| 3.84&nbsp;MHz || −108 dBm    || [[UMTS]] channel
|-
| 6&nbsp;MHz  || −106 dBm    || [[Analog television]] channel
|-
| 20&nbsp;MHz  || −101 dBm    || [[IEEE 802.11|WLAN 802.11]] channel
|-
| 40&nbsp;MHz || −98 dBm || [[IEEE 802.11n|WLAN 802.11n]] 40&nbsp;MHz channel
|-
| 80&nbsp;MHz || −95 dBm || [[IEEE 802.11ac|WLAN 802.11ac]] 80&nbsp;MHz channel
|-
| 160&nbsp;MHz || −92 dBm || [[IEEE 802.11ac|WLAN 802.11ac]] 160&nbsp;MHz channel
|-
| 1&nbsp;GHz || −84 dBm || UWB channel
|}
 
== Thermal noise on capacitors ==
Thermal noise on capacitors is referred to as ''kTC'' noise. Thermal noise in an [[RC circuit]] has an unusually simple expression, as the value of the [[electrical resistance|resistance]] (''R'') drops out of the equation. This is because higher ''R'' contributes to more filtering as well as to more noise. The noise bandwidth of the RC circuit is 1/(4''RC''),<ref>Kent H. Lundberg, See pdf, page 10: http://web.mit.edu/klund/www/papers/UNP_noise.pdf</ref> which can substituted into the above formula to eliminate ''R''. The mean-square and RMS noise voltage generated in such a filter are:<ref>R. Sarpeshkar, T. Delbruck, and C. A. Mead, [http://www.rle.mit.edu/avbs/publications/journal_papers/journal_16.pdf "White noise in MOS transistors and resistors"], ''IEEE Circuits Devices Mag.'', pp. 23–29, Nov. 1993.</ref>
 
:<math>
\bar {v_{n}^2} = k_B T / C
</math>
 
:<math>
v_{n} = \sqrt{ k_B T / C }.
</math>
 
Thermal noise accounts for 100% of ''kTC'' noise, whether it is attributed to the resistance or to the [[capacitance]].
 
In the extreme case of 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 equilibrium]] 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.  As an alternative to the voltage noise, the reset noise on the capacitor can also be quantified as the [[electrical charge]] standard deviation, as
 
:<math>
Q_{n} = \sqrt{ k_B T C }.
</math>
 
Since the charge variance is <math>k_B T C</math>, this noise is often called ''kTC noise''.
 
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''), or also ''kT''/2.  Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance.
 
The ''kTC'' noise is the dominant noise source at small capacitors.
 
::{| class="wikitable"
|+ Noise of capacitors at 300 K
! Capacitance !! <math> \sqrt{ k_B T / C } </math> !! Electrons
|-
|      1 fF || 2 mV  || 12.5 e<sup>–</sup>
|-
|    10 fF || 640 µV  || 40 e<sup>–</sup>
|-
|  100 fF  || 200 µV  || 125 e<sup>–</sup>
|-
|  1 pF    ||  64 µV  || 400 e<sup>–</sup>
|-
|  10 pF    ||  20 µV  || 1250 e<sup>–</sup>
|-
| 100 pF    ||  6.4 µV || 4000 e<sup>–</sup>
|-
|  1 nF    ||  2 µV || 12500 e<sup>–</sup>
|}
 
== Noise at very high frequencies ==
{{main|Planck's law}}
The above equations are good approximations at frequencies below about 80 [[gigahertz]] ([[Extremely high frequency|EHF]]). In the most general case, which includes up to optical frequencies, the power [[spectral density]] of the voltage across the resistor ''R'', in V<sup>2</sup>/Hz is given by:<ref>[[Laszlo B. Kish|L.B. Kish]], "Stealth communication: Zero-power classical communication, zero-quantum quantum communication and environmental-noise communication", ''Applied Physics Lett.'' '''87''' (2005), Art. No. 234109; http://arxiv.org/abs/physics/0508135</ref>
 
:<math>
\Phi (f) = \frac{2 R h f}{e^{\frac{h f}{k_B T}} - 1}
</math>
 
where ''f'' is the frequency, ''h'' [[Planck constant|Planck's constant]], ''k<sub>B</sub>'' [[Boltzmann constant]] and ''T'' the temperature in kelvins.
If the frequency is low enough, that means:
 
:<math>
f \ll \frac{k_B T}{h}
</math>
 
(this assumption is valid until few terahertz at room temperature) then the exponential can be expressed in terms of its [[Taylor series]]. The relationship then becomes:
 
:<math>
\Phi (f) \approx 2 R k_B T.
</math>
 
In general, both ''R'' and ''T'' depend on frequency. In order to know the total noise it is enough to integrate over all the bandwidth. Since the signal is real, it is possible to integrate over only the positive frequencies, then multiply by 2.
Assuming that ''R'' and ''T'' are constants over all the bandwidth <math>\Delta f</math>, then the [[root mean square]] (RMS) value of the voltage across a resistor due to thermal noise is given by
 
:<math>
v_n = \sqrt { 4 k_B T R \Delta f },
</math>
 
that is, the same formula as above.
 
==See also==
* [[Fluctuation-dissipation theorem]]
* [[Shot noise]]
* [[1/f noise]]
* [[Langevin equation]]
 
==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]
*[http://www.sengpielaudio.com/calculator-noise.htm Johnson–Nyquist noise or thermal noise calculator{{spaced ndash}}volts and dB]
*[http://www.phys.sci.kobe-u.ac.jp/~sonoda/notes/nyquist_random.ps Derivation of the Nyquist relation using a random electric field, H. Sonoda]
*[http://xformulas.net/applets/thermal_noise.html Applet of the thermal noise.]
 
{{Noise}}
 
{{DEFAULTSORT:Johnson-Nyquist noise}}
[[Category:Noise]]
[[Category:Electrical engineering]]
[[Category:Electronic engineering]]
[[Category:Electrical parameters]]
[[Category:Radar signal processing]]

Latest revision as of 08:54, 26 November 2014


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