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 increases with temperature. Some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to reduce thermal noise in their circuits. 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.

Thermal noise in an ideal resistor is approximately white, meaning that the power spectral density is nearly constant throughout the frequency spectrum, but does decay to zero at extremely high frequencies (terahertz for room temperature). When limited to a finite bandwidth, thermal noise has a nearly Gaussian amplitude distribution.^[1]

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.^[2]

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.^[2][3]

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.^[2]

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.^[2]

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.^[4][5][2] He described his findings to Harry Nyquist, also at Bell Labs, who was able to explain the results, published in 1928.^[6]

Derivation

As Nyquist stated in his 1928 paper, the sum of the energy in the normal modes of electrical oscillation would determine the amplitude of the noise. Nyquist used the equipartition law of Boltzmann and Maxwell. Using the concept potential energy and harmonic oscillators of the equipartition law,^[7]

${\displaystyle \left\langle H\right\rangle =k_{\rm {B}}T}$

where ${\displaystyle \left\langle H\right\rangle }$ is the noise power density in (W/Hz), ${\displaystyle k_{\rm {B}}}$ is the Boltzmann constant and ${\displaystyle T}$ is the temperature. Multiplying the equation by bandwidth gives the result as noise power:

${\displaystyle N=k_{\rm {B}}T\Delta f}$

where N is the noise power and Δf is the bandwidth.

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 medium (e.g. ions in an electrolyte), not just resistors. 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 power spectral density, or voltage variance (mean square) per hertz of bandwidth, is given by

${\displaystyle {\overline {v_{n}^{2}}}=4k_{\text{B}}TR}$

where k_B is the Boltzmann constant, T is the resistor's absolute temperature, and R is the resistance value. Using this equation for quick calculation, at room temperature:

${\displaystyle {\sqrt {\overline {v_{n}^{2}}}}=0.13{\sqrt {R}}~\mathrm {nV} /{\sqrt {\mathrm {Hz} }}.}$

For example, a 1 kΩ resistor at a temperature of 300 K has

${\displaystyle {\sqrt {\overline {v_{n}^{2}}}}={\sqrt {4\cdot 1.38\cdot 10^{-23}~\mathrm {J} /\mathrm {K} \cdot 300~\mathrm {K} \cdot 1000~\Omega }}=4.07\cdot 10^{-9}~\mathrm {V} /{\sqrt {\mathrm {Hz} }}.}$

For a given bandwidth, the root mean square (RMS) of the voltage, ${\displaystyle v_{n}}$, is given by

${\displaystyle v_{n}={\sqrt {\overline {v_{n}^{2}}}}{\sqrt {\Delta f}}={\sqrt {4k_{\text{B}}TR\Delta f}}}$

where Δf 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.^[8] A useful rule of thumb to remember is that 50 Ω at 1 Hz bandwidth correspond to 1 nV noise at room temperature.

A resistor in a short circuit dissipates a noise power of

${\displaystyle P={v_{n}^{2}}/R=4k_{\text{B}}\,T\Delta f.}$

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

${\displaystyle P=k_{\text{B}}\,T\Delta f}$

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. Taking the Norton equivalent of the voltage source corresponds to dividing the noise voltage by R. This gives the root mean square value of the current source as:

${\displaystyle i_{n}={v_{n} \over R}={\sqrt {{4k_{\text{B}}T\Delta f} \over R}}.}$

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:

${\displaystyle P_{\mathrm {dBm} }=10\ \log _{10}(k_{\text{B}}T\Delta f/1\,{\textrm {mW}})\ {\textrm {dBm}}.}$

At room temperature (300 K) this is approximately

${\displaystyle P_{\mathrm {dBm} }=-173.8\ {\textrm {dBm}}+10\ \log _{10}(\Delta f{\text{ in Hz}})\ {\textrm {dB}}.}$^{[9][10]: 260}

Using this equation, noise power for different bandwidths is simple to calculate:

Bandwidth ${\displaystyle (\Delta f)}$	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

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.^[11] 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:^[12]

${\displaystyle \stackrel{¯<!--\; ¯\; -->}{v_n^2}=\frac{}{}}$Zdroj:https://en.wikipedia.org?pojem=Thermal_noise
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