Johnson–Nyquist noise

thumb|327x327px|Figure 1. [[John Bertrand Johnson|Johnson's 1927 experiment showed that if thermal noise from a resistance of <math>\text{R}</math> with temperature <math>\text{T}</math> is bandlimited to bandwidth <math>\Delta f </math>, then its root mean squared voltage <math>(V_\text{rms})</math> is <math>\sqrt{ 4 k_\text{B} TR \Delta f }</math> in general, where <math>k_\text{B}</math> is the Boltzmann constant.]]

Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the voltage or current 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.

thumb|400x400px|Figure 2. Johnson–Nyquist noise has a nearly a constant power spectral density per unit of [[frequency, but does decay to zero due to quantum effects at high frequencies (terahertz for room temperature). This plot's horizontal axis uses a log scale such that every vertical line corresponds to a power of ten of frequency.]]

Thermal noise in an ideal resistor is approximately white, meaning that its power spectral density is nearly constant throughout the frequency spectrum (Figure 2). When limited to a finite bandwidth and viewed in the time domain (as sketched in Figure 1), 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 discovered shot noise in 1918, while studying Einstein's theories of thermal noise.

Noise of ideal resistors for moderate frequencies

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 filters don't have brickwall 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 <math>\Delta f </math> (Figure 3) has a mean square voltage of: 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 spectral density of the noise in the passband.

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

: <math>

\overline {V_\text{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 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, all 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,

For example, the NIST in 2017 used the Johnson noise thermometry to measure the Boltzmann constant with uncertainty less than 3 ppm. It accomplished this by using Josephson voltage standard and a quantum Hall resistor, held at the triple-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 the 2019 redefinition, the kelvin was defined so that the Boltzmann constant has an exact value (), and the triple point of water became experimentally measurable.

Thermal noise on inductors

Inductors are the 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:

: <math>

\overline {I_\text{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 happens when the Thévenin equivalent resistance <math>R_{\rm L}</math> of the remaining circuit matches <math>R_\text{S}</math>. Some example available noise power levels are tabulated below:

{| class="wikitable"

! Bandwidth <math> (\Delta f )</math>!! Available thermal noise <br />power level at 300 K !! 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 derivation of ideal resistor noise

thumb|396x396px|Figure 5. Schematic of [[Harry Nyquist|Nyquist's 1928 thought experiment to explain Johnson's experimental result. Nyquist's thought experiment summed the energy contribution of each standing wave mode of oscillation on a long lossless transmission line between two equal resistors (<math>R_1 {=} R_2</math>). According to the conclusion of Figure 5, the total average power transferred over bandwidth <math>\Delta f </math> from <math>R_1</math> and absorbed by <math>R_2</math> was determined to be:

: <math>\overline {P_1} = k_{\rm B} T \, \Delta f \, . </math>

Simple application of Ohm's law says the current from <math>V_1</math> (the thermal voltage noise of only <math>R_1</math>) through the combined resistance is <math display="inline">I_1 {=} \tfrac{V_1}{R_1 + R_2} {=} \tfrac{V_1}{2R_1}</math>, so the power transferred from <math>R_1</math> to <math>R_2</math> is the square of this current multiplied by <math>R_2</math>, which simplifies to:

: <math>\eta(f) = \frac{hf/k_\text{B} T}{e^{hf/k_\text{B} T} - 1}+\frac{1}{2}

\frac{h f}{k_\text{B} T} \, .</math>

At very high frequencies (<math>f \gtrsim \tfrac{k_\text{B} T}{h}</math>), the spectral density <math>S_{v_n v_n}(f)</math> now 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 <math>\eta(f)=1</math> 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 <math>\tfrac{\lambda^2}{4\pi}</math>, 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,

: <math>S_{v_m v_n}(f) = 2 k_\text{B} T \eta(f) (Z_{mn}(f) + Z_{nm}(f)^*)</math>

where the <math>Z_{mn}</math> are the elements of the impedance matrix <math>\mathbf{Z}</math>.

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

: <math>S_{i_m i_n}(f) = 2 k_\text{B} T \eta(f) (Y_{mn}(f) + Y_{nm}(f)^*)</math>

Johnson–Nyquist noise

History of thermal noise

Noise of ideal resistors for moderate frequencies

Thermal noise on inductors

Maximum transfer of noise power

Nyquist's derivation of ideal resistor noise

Relation to Planck's law

Multiport electrical networks

Notes

See also

References

External links