Johnson–Nyquist noise

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 electrons) inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. Thermal noise is present in all electrical circuits, and in sensitive electronic equipment (such as radio receivers) 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 impedance or generalized susceptibility is used to characterize the medium.

Johnson-Nyquist noise has a nearly a constant 4k_BTR power spectral density per unit of frequency, but does decay to zero due to quantum effects at high frequencies (terahertz for room temperature).

Thermal noise in an ideal resistor is approximately white, meaning that its power spectral density is nearly constant throughout the frequency spectrum (see plot). When limited to a finite bandwidth, thermal noise has a nearly Gaussian amplitude distribution.

For the general case, this definition applies to charge carriers in any type of conducting medium (e.g. ions in an electrolyte), not just resistors. 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 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.

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.

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.

Frits Zernike working in electrical metrology, found unusual aleatory deflections while working with high-sensitive 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.

The same year, working independently without any knowledge of Zernike's work, John B. Johnson working in Bell Labs found the same kind of noise in communication systems, but described it in terms of frequencies. 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.

Noise of ideal resistors

Although a resistor produces significant noise across all but the most extreme frequencies, measuring devices (e.g. oscilloscopes) and electronic applications generally operate over a more limited bandwidth. Johnson's experiment used a narrow resonant circuit or wider band-pass filter to limit observation of the resistor's noise to a definite frequency band (from frequency $f_{1}$ to $f_{2}$) to accurately measure its intensity. Johnson observed that over a $\Delta f{=}f_{2}{-}f_{1}$ bandwidth, a resistance with value of $R$ at a kelvin temperature of $T$ provided a mean square voltage variance of:

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

where $k_{\rm {B}}$ is the Boltzmann constant (1.380649×10⁻²³ joules per kelvin). This equation applies to ideal resistors (i.e. pure resistances without any frequency-dependence) at non-extreme frequency and temperatures. A more accurate general form accounts for complex impedances and quantum effects.

Power spectral density

This voltage variance per hertz of bandwidth is $4k_{\text{B}}TR$ and may be called the power spectral density. Its square root at room temperature (around 300 K) approximates to 0.13 ${\sqrt {R}}$ in units of nanovolts/√hertz. A 10 kΩ resistor, for example, would have approximately 13 nanovolts/√hertz at room temperature.

RMS noise voltage

These circuits are equivalent: (A) A noisy resistor at nonzero temperature; (B) A noiseless resistor in series with a noise-creating voltage source (i.e. the Thévenin equivalent circuit); (C) A noiseless resistance in parallel with a noise-creating current source $I_{\text{rms}}{=}V_{\text{rms}}/R$ (i.e. the Norton equivalent circuit).

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

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

Rough calculations may consider that a nanovolt of RMS noise at room temperature corresponds to a little over 50 Ω·Hz for $R\,\Delta f$, so values of $R\,\Delta f$ that are x times larger have almost √x nV of RMS noise at room temperature. For example, 5 kΩ provides almost 1 μV of RMS noise over 10 kHz (half of the human hearing range) at room temperature.

RMS noise current

The earlier representation of the resistor can be converted into its Norton equivalent circuit as a noise-free resistor in parallel with the following current noise source:

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

Thermal noise on capacitors

Ideal capacitors, as lossless devices, do not have thermal noise, but as commonly used with resistors in an RC circuit, the combination has what is called kTC noise. The noise bandwidth of an RC circuit is Δf = 1/4RC. When this is substituted into the thermal noise equation, the result has an unusually simple form as the value of the 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:

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

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

The noise charge is the capacitance times 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 "kTC noise".

Although independent of the resistor's value, 100% of the kTC noise arises in the resistor. Therefore, if the resistor and the capacitor are at different temperatures, the temperature of the resistor alone should be used in the above calculation.

An extreme case is the zero bandwidth limit called 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 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 sensors.

Any system in thermal equilibrium has state variables with a mean energy of kT/2 per degree of freedom. Using the formula for energy on a capacitor (E = 1/2CV²), mean noise energy on a capacitor can be seen to also be 1/2CkT/C = kT/2. Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance.

Thermal noise on capacitors at 300 K

Capacitance	${\sqrt {k_{\text{B}}T/C}}$	${\sqrt {k_{\text{B}}TC}}$	Electrons
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−

Thermal noise on inductors

Inductors are the dual of capacitors. Just like kTC noise is independent of resistance, a resistor with an inductor $L$ also results in a noise current that 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 $R_{\text{source}}$ can transfer to the remaining circuit. The maximum noise power transfer happens when the Thévenin equivalent resistance $R_{\rm {th}}$ of the remaining circuit matches $R_{\text{source}}$. 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 the available noise power from a resistor.

Available noise power in decibel-milliwatts

Signal power is often measured in dBm (decibels relative to 1 milliwatt). 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	FM channel of 2-way radio
100 kHz	−124
180 kHz	−121.45	One LTE resource block
200 kHz	−121	GSM channel
1 MHz	−114	Bluetooth channel
2 MHz	−111	Commercial GPS channel
3.84 MHz	−108	UMTS channel
6 MHz	−106	Analog television channel
20 MHz	−101	WLAN 802.11 channel
40 MHz	−98	WLAN 802.11n 40 MHz channel
80 MHz	−95	WLAN 802.11ac 80 MHz channel
160 MHz	−92	WLAN 802.11ac 160 MHz channel
1 GHz	−84	UWB channel

Nyquist's classical derivation of ideal resistor noise

Schematic of Nyquist's 1928 thought experiment using two noisy resistors (each represented by a noise-free resistor in series with a noise voltage source) connected via a long lossless transmission line. Each resistor's noise signal propagates across the line. All impedances are identical, so both signals are absorbed by the opposite resistor instead of being reflected. Nyquist then imagined 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 from $f_{1}$ to $f_{1}{+}\Delta f$, there are $2\cdot {\text{length}}\cdot \Delta f/f_{1}$ modes of oscillation. Because each mode provides one degree of freedom and $k_{\rm {B}}T$ of energy on average, the total energy in that bandwidth on average is thus $k_{\rm {B}}T$ times that number of modes. But since before the shorting there were originally no reflections, the value of that total energy also equals the combined energy that was transferred from both resistors to the line during the transit time interval of ${\text{length}}/f_{1}$. Dividing the average energy transferred from each resistor to the line by the transit time interval results in $2\cdot {\text{length}}/f_{1}$ being cancelled out. So the total power transferred over bandwidth $\Delta f$ on average from each resistor is $k_{\rm {B}}T\,\Delta f$.

Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors" used concepts about potential energy and harmonic oscillators from the equipartition law of Boltzmann and Maxwell to explain Johnson's experimental result. Nyquist's thought experiment (key details in figure) summed the kT energy contribution of each standing wave mode of oscillation on a long lossless transmission line between two equal resistors ($R_{1}{=}R_{2}$). The electromotive force of each resistor's thermal noise is represented as a voltage noise source in series with a noise-free resistor. The total average power per bandwidth $\Delta f$ transferred from $R_{1}$ and absorbed by $R_{2}$ was determined to be:

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

Simple application of Ohm's law says 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:

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

Setting this general expression for ${\textstyle P_{\text{1}}}$ to the earlier average power ${\textstyle {\overline {P_{1}}}}$ 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's generalization of this noise expression showed that that $R_{1}$ didn't have to equal $R_{2}$ (as assumed above), but rather only depended on the value of the resistor producing the noise.

Generalized forms

The $4k_{\text{B}}TR$ voltage noise described above is a special case for a purely resistive component for low 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 passive and linear.

Complex impedances

Nyquist's original paper also provided the generalized noise for components having partly reactive response, e.g., sources that contain capacitors or inductors. Such a component can be described by a frequency-dependent complex electrical impedance $Z(f)$. The formula for the power spectral density of 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, its power spectral density being

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

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

Quantum effects at high frequencies or low temperatures

While Nyquist's classical derivation used $k_{\rm {B}}T$ as the energy per mode of vibration (of which half the energy is electric and half is magnetic), Nyquist concluded his 1928 paper by attempting to use a more involved expression that incorporated the Planck constant $h$ from the new theory of quantum mechanics. With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near absolute 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 of Planck's law of blackbody radiation. 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, and further connected it to properties of antennas, particularly the fact that the average antenna aperture over 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. Twiss extended Nyquist's formulas to multi-port passive electrical networks, including non-reciprocal devices such as circulators and isolators. 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,

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

where the $Z_{mn}$ are the elements of the impedance 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 the admittance matrix.

See also

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

References

 This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22. (in support of MIL-STD-188).

External links

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