Johnson–Nyquist noise
182745
225030300
2008-07-11T15:27:31Z
PaulLowrance
5012437
/* Thermal noise on capacitors */ Adding further clarification.
'''Johnson–Nyquist noise''' ('''thermal noise''', '''Johnson noise''', or '''Nyquist noise''') is the [[electronic noise|electronic]] [[noise (physics)|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]].
Thermal noise is approximately [[white noise|white]], meaning that the power [[spectral density]] is nearly equal throughout the [[frequency spectrum]] (however see the section below on extremely high frequencies). Additionally, the amplitude of the signal has very nearly a [[Normal distribution|Gaussian]] [[probability density function]].<ref>{{cite web | url = http://focus.ti.com/lit/an/slod006b/slod006b.pdf | title = Op Amps For Everyone | accessdate = 2006-12-06 | last = Mancini | first = Ron | coauthors = others | year = 2002 | month = August | format = [[Portable Document Format|PDF]] | work = Application Notes | publisher = [[Texas Instruments]] | pages = [http://focus.ti.com/lit/an/slod006b/slod006b.pdf#page=148 p. 148]
| quote = Thermal noise and shot noise (see below) have Gaussian probability density functions. The other forms of noise do not.}}</ref>
== History ==
This type of noise was first measured by [[John B. Johnson]] at [[Bell Labs]] in [[1928]].<ref>J. Johnson, [http://link.aps.org/abstract/PR/v32/p97 "Thermal Agitation of Electricity in Conductors"], Phys. Rev. 32, 97 (1928) – 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 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:
:<math>
\bar v_{n} = 0.13 \sqrt{R} ~\mathrm{nV}/\sqrt{\mathrm{Hz}}</math>.
For example, a 1 kΩ resistor at a temperature of 300 K has
:<math>
\bar v_{n} = \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} = \bar v_{n}\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 kΩ resistor at room temperature and a 10 kHz bandwidth, the RMS noise voltage is 400 nV.<ref>[http://www.google.com/search?q=sqrt%284*k*295+K*1+kiloohm*%2810+kHz%29%29+in+microvolt Google Calculator result] for 1 kΩ room temperature 10 kHz bandwidth</ref>
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 the noise power transfer to the circuit 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 in decibels ==
In [[Telecommunication|communications]], power is often measured in [[decibels]] relative to 1 milliwatt ([[dBm]]), assuming a 50 ohm load resistance. With these conventions, thermal noise for a resistor at [[room temperature]] can be estimated as:
:<math>
P_\mathrm{dBm} = -174 + 10\ \log(\Delta f)
</math>
where P is measured in [[dBm]]. For example:
::{| class="wikitable"
! Bandwidth !! Power !! Notes
|-
| 1 Hz || −174 dBm ||
|-
| 10 Hz || −164 dBm
|-
| 1000 Hz || −144 dBm
|-
| 10 kHz || −134 dBm || FM channel of 2-way radio
|-
| 1 MHz || −114 dBm
|-
| 2 MHz || −111 dBm || Commercial GPS channel
|-
| 6 MHz || −106 dBm || Analog television channel
|-
| 20 MHz || −101 dBm || WLAN 802.11 channel
|}
The actual amount of thermal noise received by a radio receiver having a 50 Ω input [[Electrical impedance|impedance]], connected to an [[Antenna (radio)|antenna]] with a 50 Ω [[radiation resistance]] would be scaled by the [[noise figure]] (NF), shown as follows:
:<math>P_\mathrm{receiver noise} = P_\mathrm{resistor noise}+10\ \log_{10}(10^{NF/10}-1)</math>
Here P<sub>receivernoise</sub> is the noise generated in the receiver itself. P<sub>resistornoise</sub> is the value from the table above for thermal noise from a resistor. NF is in [[Decibel | dB]]. Ten raised to the power of NF/10 is called the [[noise factor]] and is simply the linear version of noise figure. We subtract one from the noise factor here because a noise factor of 1 is a perfect receiver (which contributes no noise to the signal). Noise factor is defined this way because in the [[laboratory]], it is measured by connecting a resistor to the receiver's input and comparing the output noise to what one would expect if the noise of the resistor were simply amplified by the [[gain]] of the receiver. A ratio of 1 between the actual output noise in the [[numerator]] and the gain times the resistor thermal noise in the [[denominator]] means that the receiver added no noise of its own to the input.
Note that the radiation resistance of the antenna does not convert power to heat, and so is not a source of thermal noise. Likewise, the load impedance of the input of the receiver does not contribute directly to received noise. Therefore, it is indeed possible, and even common, for a receiver to have a noise factor of less than 2X (or equivalently, a noise figure of less than 3 dB).
For example a 6 MHz wide channel such as a television channel received signal would compete with the tiny amount of power generated by room temperature in the input stages of the receiver, which, for a TV receiver with a noise figure of 3 dB would be −106 dBm, or one fortieth of a picowatt. For a TV with a noise figure of 1 dB, the noise power would be −112 dBm. The actual source of this noise is a combination of thermal noise in physical resistances of wires and [[semiconductors]], thermal noise in other lossy devices such as [[Transformer | transformers]], as well as [[shot noise]].
The 6 MHz [[Bandwidth (signal processing)|bandwidth]] could be the 6 MHz between 54 and 60 MHz (corresponding to TV channel 2) or the 6 MHz between 470 MHz and 476 MHz (corresponding to TV channel UHF 14) or any other 6 MHz in the spectrum for that matter. The bandwidth of any channel should never be confused with the transmitting frequency of a channel. For example, a channel transmit frequency may be as high as 2450 MHz for a WIFI signal, but the actual width of the channel may be only 20 MHz, and that 20 MHz would be the correct value to use in computing the Johnson–Nyquist noise.
Note that it is quite possible to detect a signal whose amplitude is less than the noise contained within its bandwidth. The [[Global Positioning System]] (GPS) and [[Glonass]] system both have signal amplitudes that are less than the received noise in a typical receiver at ground level. In the case of GPS, the received signal has a power of −133 dBm. The newer batch of satellites have a more powerful transmitter. To achieve this feat, GPS uses [[spread spectrum]] techniques, while some other communication systems use [[Coding theory | error control coding]]. There is still a fundamental limit to the ability to discern the meaning of a signal in the midst of noise, given by the [[Shannon–Hartley theorem]].
== 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.
== 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. Specifically the RMS noise voltage generated in such a filter is:<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>
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]]. Consider a capacitor of C farads with parallel resistance of R ohms. The thermal noise voltage generated by a resistor is
:<math>
v_{n} = \sqrt{ 4 k_B T R \Delta f }
</math>
where ''k<sub>B</sub>'' is the [[Boltzmann constant]], ''T'' is temperature in Kelvin, ''R'' is resistance, and <math>\Delta f</math> is bandwidth. The bandwidth of the RC circuit is 1 / (4 ''R C'').<ref>Kent H. Lundberg, See pdf, page 10: http://web.mit.edu/klund/www/papers/UNP_noise.pdf</ref> Thus, the thermal noise caused by the parallel resistance across a capacitor is
:<math>
v_{n} = \sqrt{ 4 k_B T R \Delta f } = \sqrt{ 4 k_B T R (1 / (4 R C)) } = \sqrt{ k_B T / C) }
</math>
which accounts for all kTC noise. kTC noise is caused by parallel resistance since all capacitors have parallel resistance.
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=1/2*C*V<sup>2</sup>), mean noise energy on a capacitor can be seen to also be 1/2*C*(k*T/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.
{| border="1"
|+ 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 µF || 2 µV || 12500 e<sup>–</sup>
|-
|}
== Noise at very high frequencies ==
The above equations are good approximations at any practical radio frequency in use (i.e. frequencies below about 80 [[terahertz]]). In the most general case,which includes up to optical frequencies, the power [[spectral density]] of the voltage across the resistor ''R'', in <math>\mathrm{V^2/Hz}</math> is given by:
:<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) 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==
* [[Shot noise]]
* [[1/f noise]]
* [[Harry Nyquist]]
* [[John B. Johnson]]
==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 — volts and dB]
*[http://www.licha.de/astro_article_ccd_bias_dark.php Thoughts about Image Calibration for low dark current and Amateur CCD Cameras to increase Signal-To-Noise Ratio]
*[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]
[[Category:Noise]]
[[Category:Electrical parameters]]
[[ca:Soroll tèrmic]]
[[de:Wärmerauschen]]
[[es:Ruido térmico]]
[[fr:Bruit thermique]]
[[it:Rumore termico]]
[[nl:Thermische ruis]]
[[ja:熱雑音]]
[[pt:Ruido térmico]]
[[ru:Тепловой шум]]
[[sr:Термални шум]]