Johnson–Nyquist noise

Johnson–Nyquist noise(thermal noise,Johnson noise,orNyquist noise) is theelectronic noisegenerated by thethermal agitationof thecharge carriers(usually theelectrons) inside anelectrical conductorat equilibrium, which happens regardless of any appliedvoltage.Thermal noise is present in allelectrical circuits,and in sensitive electronic equipment (such asradio receivers) can drown out weak signals, and can be the limiting factor on sensitivity of electrical measuring instruments. Thermal noise is proportional toabsolute temperature,so some sensitive electronic equipment such asradio telescopereceivers are cooled tocryogenictemperatures to improve theirsignal-to-noise ratio.The generic, statistical physical derivation of this noise is called thefluctuation-dissipation theorem,where generalizedimpedanceor generalizedsusceptibilityis used to characterize the medium.

Figure 1.Johnson's 1927 experiment showed that if thermal noise from aresistanceof${\text{R}}$withtemperature${\text{T}}$isbandlimitedtobandwidth$\Delta f$,then itsroot mean squaredvoltage$(V_{\text{rms}})$is${\sqrt {4k_{\text{B}}{\text{T}}\,{\text{R}}\,\Delta f}}$in general, where$k_{\text{B}}$is theBoltzmann constant.

Figure 2. Johnson–Nyquist noise has a nearly a constant4k_BTRpower spectral density per unit offrequency,but does decay to zerodue to quantum effectsat high frequencies (terahertzfor room temperature). This plot's horizontal axis uses alog scalesuch that every vertical line corresponds to apower of tenof frequency inhertz.

Thermal noise in anideal resistoris approximatelywhite,meaning that its powerspectral densityis nearly constant throughout thefrequency spectrum(Figure 2). When limited to a finite bandwidth and viewed in thetime domain(as sketched in Figure 1), thermal noise has a nearlyGaussian amplitude distribution.

For the general case, this definition applies tocharge carriersin any type of conductingmedium(e.g.ionsin anelectrolyte), not justresistors.Thermal noise is distinct fromshot 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 ofAlbert Einstein'sAnnus mirabilispapersthe theory ofBrownian motionwas 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.

Geertruida de Haas-Lorentz,daughter ofHendrik 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.

Walter H. Schottkystudied the problem in 1918, while studying thermal noise using Einstein's theories, experimentally discovered another kind of noise, theshot noise.

Frits Zernikeworking in electrical metrology, found unusual random deflections while working with high-sensitivegalvanometers.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.

The same year, working independently without any knowledge of Zernike's work,John B. Johnsonworking inBell Labsfound the same kind of noise in communication systems, but described it in terms of frequencies.He described his findings toHarry Nyquist,also at Bell Labs, who used principles ofthermodynamicsandstatistical mechanicsto explain the results, published in 1928.

Noise of ideal resistors for moderate frequencies

Figure 3. While thermal noise has an almost constant power spectral density of$4k_{\text{B}}TR$,aband-pass filterwith bandwidth$\Delta f{=}f_{\text{upper}}{-}f_{\text{lower}}$passes only the shaded area of height$4k_{\text{B}}TR$and width$\Delta f$.Note: practicalfiltersdon't havebrickwallcutoffs,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$R$atkelvin temperature$T$andbandlimitedto afrequency bandofbandwidth$\Delta f$(Figure 3) has amean squarevoltage of:

${\overline {V_{n}^{2}}}=4k_{\text{B}}TR\,\Delta f$

where$k_{\rm {B}}$is theBoltzmann constant(1.380649×10⁻²³joulesperkelvin). While this equation applies toideal resistors(i.e. pure resistances without any frequency-dependence) at non-extreme frequency and temperatures, a more accurategeneral formaccounts forcomplex impedancesand quantum effects. Conventional electronics generally operate over a more limitedbandwidth,so Johnson's equation is often satisfactory.

Power spectral density

The mean square voltage perhertzofbandwidthis$4k_{\text{B}}TR$and may be called thepower spectral density(Figure 2).Its square root at room temperature (around 300 K) approximates to 0.13${\sqrt {R}}$in units of⁠nanovolts/√hertz⁠.A 10 kΩ resistor, for example, would have approximately 13⁠nanovolts/√hertz⁠at room temperature.

RMS noise voltage

Figure 4. These circuits are equivalent:

(A)A resistor at nonzero temperature with internal thermal noise;

(B)ItsThévenin equivalentcircuit: a noiseless resistorin serieswith a noisevoltage source;

(C)ItsNorton equivalentcircuit: a noiseless resistancein parallelwith a noisecurrent source.

The square root of the mean square voltage yields theroot mean square(RMS) voltage observed over the bandwidth$\Delta f$:

$V_{\text{rms}}={\sqrt {\overline {V_{n}^{2}}}}={\sqrt {4k_{\text{B}}TR\,\Delta f}}\,.$

A resistor with thermal noise can be represented by itsThévenin equivalentcircuit (Figure 4B) consisting of a noiseless resistor in series with a gaussian noisevoltage sourcewith the above RMS voltage.

Around room temperature, 3 kΩ provides almost one microvolt of RMS noise over 20 kHz (thehuman hearing range) and 60 Ω·Hz for$R\,\Delta f$corresponds to almost one nanovolt of RMS noise.

RMS noise current

A resistor with thermal noise can also beconvertedinto itsNorton equivalentcircuit (Figure 4C) consisting of a noise-free resistor in parallel with a gaussian noisecurrent sourcewith the following RMS current:

$I_{\text{rms}}={V_{\text{rms}} \over R}={\sqrt {{4k_{\text{B}}T\Delta f} \over R}}.$

Thermal noise on capacitors

Idealcapacitors,as lossless devices, do not have thermal noise. However, the combination of a resistor and a capacitor (anRC circuit,a commonlow-pass filter) has what is calledkTCnoise. The noise bandwidth of an RC circuit is$\Delta f{=}{\tfrac {1}{4RC}}.$When this is substituted into the thermal noise equation, the result has an unusually simple form as the value of theresistance(R) drops out of the equation. This is because higherRdecreases the bandwidth as much as it increases the noise.

The mean-square and RMS noise voltage generated in such a filter are:

${\overline {V_{n}^{2}}}={4k_{\text{B}}TR \over 4RC}={k_{\text{B}}T \over C}$

$V_{\text{rms}}={\sqrt {4k_{\text{B}}TR \over 4RC}}={\sqrt {k_{\text{B}}T \over C}}.$

The noisecharge$Q_{n}$is thecapacitancetimes the voltage:

$Q_{n}=C\,V_{n}=C{\sqrt {k_{\text{B}}T \over C}}={\sqrt {k_{\text{B}}TC}}$

${\overline {Q_{n}^{2}}}=C^{2}\,{\overline {V_{n}^{2}}}=C^{2}{k_{\text{B}}T \over C}=k_{\text{B}}TC$

This charge noise is the origin of the term "kTCnoise ". Although independent of the resistor's value, 100% of thekTCnoise arises in the resistor. Therefore, it would incorrect to double-count both a resistor's thermal noise and its associated kTC noise,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:

Thermal noise on capacitors at 300 K

Capacitance	$V_{\text{rms}}{=}{\sqrt {k_{\text{B}}T \over C}}$	Charge noise$Q_{n}{=}{\sqrt {k_{\text{B}}TC}}$
		ascoulombs	aselectrons
1 fF	2 mV	2 aC	12.5 e−
10 fF	640 μV	6.4 aC	40 e−
100 fF	200 μV	20 aC	125 e−
1 pF	64 μV	64 aC	400 e−
10 pF	20 μV	200 aC	1250 e−
100 pF	6.4 μV	640 aC	4000 e−
1 nF	2 μV	2 fC	12500 e−

Reset noise

An extreme case is the zero bandwidth limit called thereset noiseleft on a capacitor by opening an idealswitch.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 isfrozenat a random value withstandard deviationas given above. The reset noise of capacitive sensors is often a limiting noise source, for example inimage sensors.

Any system inthermal equilibriumhasstate variableswith a meanenergyof⁠kT/2⁠perdegree of freedom.Using the formula for energy on a capacitor (E=⁠1/2⁠CV²), mean noise energy on a capacitor can be seen to also be⁠1/2⁠C⁠kT/C⁠=⁠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".

For example, theNISTin 2017 used the Johnson noise thermometry to measure theBoltzmann constantwith uncertainty less than 3ppm.It accomplished this by usingJosephson voltage standardand aquantum Hall resistor,held at thetriple-point temperature of water.The voltage is measured over a period of 100 days and integrated.

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 the2019 redefinition,the kelvin was defined so that the Boltzmann constant is 1.380649×10⁻²³J⋅K⁻¹,and the triple point of water became experimentally measurable.

Thermal noise on inductors

Inductors are thedualof capacitors. Analogous to kTC noise, a resistor with an inductor$L$results in a noisecurrentthat is independent of resistance:

${\overline {I_{n}^{2}}}={k_{\text{B}}T \over L}\,.$

Maximum transfer of noise power

The noise generated at a resistor$R_{\text{S}}$can transfer to the remaining circuit. Themaximum power transfer happens whentheThévenin equivalentresistance$R_{\rm {L}}$of the remaining circuitmatches$R_{\text{S}}$.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:

$P_{\text{max}}=k_{\text{B}}\,T\Delta f\,.$

This maximum is independent of the resistance and is called theavailable noise powerfrom a resistor.

Available noise power in decibel-milliwatts

Signal power is often measured indBm(decibelsrelative to 1milliwatt). Available noise power would thus be$10\ \log _{10}({\tfrac {k_{\text{B}}T\Delta f}{\text{1 mW}}})$in dBm. At room temperature (300 K), the available noise power can be easily approximated as$10\ \log _{10}(\Delta f)-173.8$in dBm for a bandwidth in hertz.Some example available noise power in dBm are tabulated below:

Bandwidth$(\Delta f)$	Available thermal noise power at 300 K (dBm)	Notes
1 Hz	−174
10 Hz	−164
100 Hz	−154
1 kHz	−144
10 kHz	−134	FMchannel of2-way radio
100 kHz	−124
180 kHz	−121.45	OneLTEresource block
200 kHz	−121	GSMchannel
1 MHz	−114	Bluetooth channel
2 MHz	−111	CommercialGPSchannel
3.84 MHz	−108	UMTSchannel
6 MHz	−106	Analog televisionchannel
20 MHz	−101	WLAN 802.11channel
40 MHz	−98	WLAN 802.11n40 MHz channel
80 MHz	−95	WLAN 802.11ac80 MHz channel
160 MHz	−92	WLAN 802.11ac160 MHz channel
1 GHz	−84	UWB channel

Nyquist's derivation of ideal resistor noise

Figure 5. Schematic ofNyquist's1928thought experimentusing two noisy resistors (each represented here by a noise-free resistor in series with a noise voltage source) connected via a long losslesstransmission lineof length$l$.Each resistor's noisesignalpropagates across the line at velocity${\text{v}}$.All impedances are identical, so both signals are absorbed by the opposite resistor instead of beingreflected.

Nyquist then imaginedshortingboth 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 sinusoidalstanding waves.For a band of frequencies$\Delta f$,there are$2l\,\Delta f/{\text{v}}$modes of oscillation.Each mode provides$k_{\rm {B}}T$of energyon average, of which$k_{\rm {B}}T/2$is electric and$k_{\rm {B}}T/2$is magnetic, so the total energy in that bandwidth on average is$k_{\rm {B}}T\cdot 2l\,\Delta f/{\text{v}}.$Each resistor contributed$k_{\rm {B}}T\cdot l\,\Delta f/{\text{v}}$(half of that total energy).

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$l/{\text{v}}$.Dividing the averageenergytransferred fromeachresistor to the line by the transittimeinterval results in a totalpowerof$k_{\rm {B}}T\,\Delta f$transferred over bandwidth$\Delta f$on average from each resistor.

Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors"used concepts aboutpotential energy and harmonic oscillators from the equipartition lawofBoltzmannandMaxwellto explain Johnson's experimental result. Nyquist'sthought experimentsummed the energy contribution of eachstanding wave mode of oscillationon a long losslesstransmission linebetween two equal resistors ($R_{1}{=}R_{2}$). According to the conclusion of Figure 5, the total average power transferred over bandwidth$\Delta f$from$R_{1}$and absorbed by$R_{2}$was determined to be:

${\overline {P_{1}}}=k_{\rm {B}}T\,\Delta f\,.$

Simple application ofOhm's lawsays the current from$V_{1}$(the thermal voltage noise of only$R_{1}$) through the combined resistance is${\textstyle I_{1}{=}{\tfrac {V_{1}}{R_{1}+R_{2}}}{=}{\tfrac {V_{1}}{2R_{1}}}}$,so the power transferred from$R_{1}$to$R_{2}$is the square of this current multiplied by$R_{2}$,which simplifies to:

$P_{\text{1}}=I_{1}^{2}R_{2}=I_{1}^{2}R_{1}=\left({\frac {V_{1}}{2R_{1}}}\right)^{2}R_{1}={\frac {V_{1}^{2}}{4R_{1}}}\,.$

Setting this${\textstyle P_{\text{1}}}$equal to the earlier average power expression${\textstyle {\overline {P_{1}}}}$allows solving for the average of${\textstyle V_{1}^{2}}$over that bandwidth:

${\overline {V_{1}^{2}}}=4k_{\text{B}}T{R_{1}}\,\Delta f\,.$

Nyquist used similar reasoning to provide a generalized expression that applies to non-equal andcomplex impedancestoo. And while Nyquist above used$k_{\rm {B}}T$according toclassicaltheory, Nyquist concluded his paper by attempting to use a more involved expression that incorporated thePlanck constant$h$(from the new theory ofquantum mechanics).

Generalized forms

The$4k_{\text{B}}TR$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 thefluctuation-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 purelypassiveand linear.

Complex impedances

Nyquist's original paper also provided the generalized noise for components having partlyreactiveresponse, e.g., sources that contain capacitors or inductors.Such a component can be described by a frequency-dependent complexelectrical impedance$Z(f)$.The formula for thepower spectral densityof the series noise voltage is

$S_{v_{n}v_{n}}(f)=4k_{\text{B}}T\eta (f)\operatorname {Re} [Z(f)].$

The function$\eta (f)$is approximately 1, except at very high frequencies or near absolute zero (see below).

The real part of impedance,$\operatorname {Re} [Z(f)]$,is in general frequency dependent and so the Johnson–Nyquist noise is not white noise. The RMS noise voltage over a span of frequencies$f_{1}$to$f_{2}$can be found by taking the square root of integration of the power spectral density:

$V_{\text{rms}}={\sqrt {\int _{f_{1}}^{f_{2}}S_{v_{n}v_{n}}(f)df}}$.

Alternatively, a parallel noise current can be used to describe Johnson noise, itspower spectral densitybeing

$S_{i_{n}i_{n}}(f)=4k_{\text{B}}T\eta (f)\operatorname {Re} [Y(f)].$

where$Y(f){=}{\tfrac {1}{Z(f)}}$is theelectrical admittance;note that$\operatorname {Re} [Y(f)]{=}{\tfrac {\operatorname {Re} [Z(f)]}{|Z(f)|^{2}}}\,.$

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 nearabsolute zero), the multiplying factor$\eta (f)$mentioned earlier is in general given by:

$\eta (f)={\frac {hf/k_{\text{B}}T}{e^{hf/k_{\text{B}}T}-1}}+{\frac {1}{2}}{\frac {hf}{k_{\text{B}}T}}\,.$

At very high frequencies ($f\gtrsim {\tfrac {k_{\text{B}}T}{h}}$), the function$\eta (f)$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$\eta (f)=1$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 ofPlanck's law of blackbody radiation.In other words, a hot resistor will create electromagnetic waves on atransmission linejust as a hot object will create electromagnetic waves in free space.

In 1946,Robert H. Dickeelaborated on the relationship,and further connected it to properties of antennas, particularly the fact that the averageantenna apertureover all different directions cannot be larger than${\tfrac {\lambda ^{2}}{4\pi }}$,where λ is wavelength. This comes from the different frequency dependence of 3D versus 1D Planck's law.

Multiport electrical networks

Richard Q. Twissextended Nyquist's formulas to multi-portpassive electrical networks, including non-reciprocal devices such ascirculatorsandisolators. 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 ofcross-spectral densityfunctions relating the different noise voltages,

$S_{v_{m}v_{n}}(f)=2k_{\text{B}}T\eta (f)(Z_{mn}(f)+Z_{nm}(f)^{*})$

where the$Z_{mn}$are the elements of theimpedance matrix$\mathbf {Z}$. 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

$S_{i_{m}i_{n}}(f)=2k_{\text{B}}T\eta (f)(Y_{mn}(f)+Y_{nm}(f)^{*})$

where$\mathbf {Y} =\mathbf {Z} ^{-1}$is theadmittance matrix.

See also

• Fluctuation-dissipation theorem
• Shot noise
• Pink noise
• Langevin equation
• Rise over thermal

References

This article incorporatespublic domain materialfromFederal Standard 1037C.General Services Administration.Archived fromthe originalon 2022-01-22.(in support ofMIL-STD-188).

External links

• Amplifier noise in RF systems
• Thermal noise (undergraduate) with detailed math