Johnson–Nyquist noise

Johnson–Nyquist noise is the voltage or current noise generated by the thermal agitation of the charge carriers 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 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.
File:Johnson-nyquist-noise-power-spectral-density-of-ideal-resistor.svg|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. 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. When limited to a finite bandwidth and viewed in the time domain, thermal noise has a nearly Gaussian amplitude distribution.
For the general case, this definition applies to charge carriers in any type of conducting medium, 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.
Frits Zernike working in electrical metrology, found unusual random 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's 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 for moderate frequencies

Johnson's experiment found that the thermal noise from a resistance at kelvin temperature and bandlimited to a frequency band of bandwidth has a mean square voltage of:
where is the Boltzmann constant. While this equation applies to ideal resistors at non-extreme frequency and temperatures, a more accurate general form accounts for complex impedances and quantum effects. Conventional electronics generally operate over a more limited bandwidth, so Johnson's equation is often satisfactory.

Power spectral density

The mean square voltage per hertz of bandwidth is and may be called the power spectral density. Its square root at room temperature approximates to 0.13, which has the unit. A 10 kΩ resistor, for example, would have approximately 13 nV/ at room temperature.

RMS noise voltage

The square root of the mean square voltage yields the root mean square voltage observed over the bandwidth :
A resistor with thermal noise can be represented by its Thévenin equivalent circuit consisting of a noiseless resistor in series with a gaussian noise voltage source with the above RMS voltage.
Around room temperature, 3 kΩ provides almost one microvolt of RMS noise over 20 kHz and 60 Ω·Hz for corresponds to almost one nanovolt of RMS noise.

RMS noise current

A resistor with thermal noise can also be converted into its Norton equivalent circuit consisting of a noise-free resistor in parallel with a gaussian noise current source with the following RMS current:

Thermal noise on capacitors

Ideal capacitors, as lossless devices, do not have thermal noise. However, the combination of a resistor and a capacitor has what is called kTC noise. The equivalent noise bandwidth of an RC circuit is When this is substituted into the thermal noise equation, the result has an unusually simple form as the value of the resistance 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:
The noise charge is the capacitance times the voltage:
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, 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:

Reset noise

An extreme case is the zero bandwidth limit called the reset noise left on a capacitor by opening an ideal switch. 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 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 per degree of freedom. Using the formula for energy on a capacitor, mean noise energy on a capacitor can be seen to also be C =. 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, 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 results in a noise current that is independent of resistance:

Maximum transfer of noise power

The noise generated at a resistor ' can transfer to the remaining circuit. The maximum power transfer happens when the Thévenin equivalent resistance of the remaining circuit matches '. 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:
This maximum is independent of the resistance and is called the available noise power from a resistor.

Available noise power level

Signal power is often measured as a level with the unit dBm. Available noise power level would thus be in dBm. At room temperature, the available noise power can be easily approximated as in dBm for a bandwidth in hertz. Some example available noise power levels are tabulated below:

Bandwidth	Available thermal noise 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