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. Electronic noise is an unwanted signal characteristic of all electronic circuits. In Science, and especially in Physics and Telecommunication, noise is fluctuations in and the addition of external factors to the stream of target The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Science and engineering, a conductor is a material which contains movable Electric charges. Electrical tension (or voltage after its SI unit, the Volt) is the difference of electrical potential between two points of an electrical

Thermal noise is approximately white, meaning that the power spectral density is nearly equal throughout the frequency spectrum (however see the section below on extremely high frequencies). White noise is a random signal (or process with a flat Power spectral density. In Statistical signal processing and Physics, the spectral density, power spectral density ( PSD) or energy spectral density ( Familiar concepts associated with a Frequency are colors musical notes radio/TV channels and even the regular rotation of the earth Additionally, the amplitude of the signal has very nearly a Gaussian probability density function. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability [1]

## History

This type of noise was first measured by John B. Johnson at Bell Labs in 1928. John Bertrand "Bert" Johnson (1887–1970 ( né Johan Erik Bertrand was a Swedish -born American electrical engineer and physicist Bell Laboratories (also known as Bell Labs and formerly known as AT&T Bell Laboratories and Bell Telephone Laboratories) is the Research organization Year 1928 ( MCMXXVIII) was a Leap year starting on Sunday (link will display full calendar of the Gregorian calendar. [2] He described his findings to Harry Nyquist, also at Bell Labs, who was able to explain the results. Harry Nyquist ( né Harry Theodor Nyqvist pron, not as often pronounced ( February 7, 1889 – April 4, 1976) was an important [3]

## Noise voltage and power

Thermal noise is to be distinguished from shot noise, which consists of additional current fluctuations that occur when a voltage is applied and a macroscopic current starts to flow. Shot noise is a type of Electronic noise that occurs when the finite number of particles that carry energy such as Electrons in an electronic circuit or Photons For the general case, the above definition applies to charge carriers in any type of conducting medium (e. A transmission medium' (plural transmission media) is a material substance ( Solid, Liquid or Gas) which can propagate g. ions in an electrolyte), not just resistors. An ion is an Atom or Molecule which has lost or gained one or more Valence electrons giving it a positive or negative electrical charge An electrolyte is any substance containing free Ions that behaves as an electrically conductive medium |- align = "center"| |width = "25"| | |- align = "center"| || Potentiometer |- align = "center"| | | |- align = "center"| Resistor| | 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 power spectral density, or voltage variance (mean square) per hertz of bandwidth, is given by

$\bar v_{n}^2 = 4 k_B T R$

where kB is Boltzmann's constant in joules per kelvin, T is the resistor's absolute temperature in kelvins, and R is the resistor value in ohms. In Statistical signal processing and Physics, the spectral density, power spectral density ( PSD) or energy spectral density ( Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics The joule (written in lower case ˈdʒuːl or /ˈdʒaʊl/ (symbol J) is the SI unit of Energy measuring heat, Electricity The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature The ohm (symbol Ω) is the SI unit of Electrical impedance or in the Direct current case Electrical resistance, Use this equation for quick calculation:

$\bar v_{n} = 0.13 \sqrt{R} ~\mathrm{nV}/\sqrt{\mathrm{Hz}}$.

For example, a resistor of 1 kΩ at an average temperature (300 K) has

$\bar v_{n} = \sqrt{4 \cdot 1.38 \cdot 10^{-23}~\mathrm{J}/\mathrm{K} \cdot 300~\mathrm{K} \cdot 1~\mathrm{k}\Omega} = 4.07 ~\mathrm{nV}/\sqrt{\mathrm{Hz}}$.

For a given bandwidth, the root mean square (rms) of the voltage, vn, is given by

$v_{n} = \bar v_{n}\sqrt{\Delta f } = \sqrt{ 4 k_B T R \Delta f }$

where Δf is the bandwidth in hertz over which the noise is measured. In Mathematics, the root mean square (abbreviated RMS or rms) also known as the quadratic mean, is a statistical measure of the The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second. For a resistor of 1 kΩ at room temperature and a 10 kHz bandwidth, the RMS noise voltage is 400 nV or 0. 4 microvolt. [4]

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. Impedance matching is the electronics design practice of setting the Output impedance ( Z S of a signal source equal to the Input impedance ( In electrical circuit theory, Thévenin's theorem for linear Electrical networks states that any combination of Voltage sources Current sources In this case the noise power transfer to the circuit is given by

$P = k_B \,T \Delta f$

where P is the thermal noise power in watts. Notice that this is independent of the noise generating resistance

## Noise in decibels

In communications, power is often measured in decibels relative to 1 milliwatt (dBm), assuming a 50 ohm load resistance. The decibel ( dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity relative to For other uses see DBM (disambiguation dbm was the first of a family of simple Database engines originally written by Ken With these conventions, thermal noise for a resistor at room temperature can be estimated as:

$P_\mathrm{dBm} = -174 + 10\ \log(\Delta f)$

where P is measured in dBm. Room temperature (also referred to as ambient temperature) is a common term to denote a certain Temperature within enclosed space at which humans are accustomed For other uses see DBM (disambiguation dbm was the first of a family of simple Database engines originally written by Ken For example:

BandwidthPowerNotes
1 Hz−174 dBm
10 Hz−164 dBm
1000 Hz−144 dBm
10 kHz−134 dBmFM channel of 2-way radio
1 MHz−114 dBm
2 MHz−111 dBmCommercial GPS channel
6 MHz−106 dBmAnalog television channel
20 MHz−101 dBmWLAN 802. 11 channel

The actual amount of thermal noise received by a radio receiver having a 50 ohm input impedance, connected to an antenna with a 50 ohm radiation resistance would be scaled by the noise figure (NF), shown as follows:

$P_\mathrm{receiver noise} = P_\mathrm{resistor noise}+10\ \log_{10}(10^{NF/10}-1)$

Note that the radiation resistance of the antenna does not convert power to heat, and so is not a source of thermal noise. Likewise, the load impedance of the input of the receiver does not contribute directly to received noise. Therefore, it is indeed possible, and even common, for a receiver to have a noise factor of less than 2X (or equivalently, a noise figure of less than 3 dB).

For example a 6 MHz wide channel such as a television channel received signal would compete with the tiny amount of power generated by room temperature in the input stages of the receiver, which, for a TV receiver with a noise figure of 3 dB would be −106 dBm, or one fortieth of a picowatt. For a TV with a noise figure of 1 dB, the noise power would be −112 dBm. The actual source of this noise is a combination of thermal noise in physical resistances of wires and semiconductors, thermal noise in other lossy devices such as transformers, as well as shot noise. A semiconductor' is a Solid material that has Electrical conductivity in between a conductor and an insulator; it can vary over that A transformer is a device that transfers Electrical energy from one circuit to another through inductively coupled Electrical conductors Shot noise is a type of Electronic noise that occurs when the finite number of particles that carry energy such as Electrons in an electronic circuit or Photons

The 6 MHz bandwidth could be the 6 MHz between 54 and 60 MHz (corresponding to TV channel 2) or the 6 MHz between 470 MHz and 476 MHz (corresponding to TV channel UHF 14) or any other 6 MHz in the spectrum for that matter. The bandwidth of any channel should never be confused with the transmitting frequency of a channel. For example, a channel transmit frequency may be as high as 2450 MHz for a WIFI signal, but the actual width of the channel may be only 20 MHz, and that 20 MHz would be the correct value to use in computing the Johnson–Nyquist noise.

Note that it is quite possible to detect a signal whose amplitude is less than the noise contained within its bandwidth. The Global Positioning System (GPS) and Glonass system both have signal amplitudes that are less than the received noise in a typical receiver at ground level. Basic concept of GPS operation A GPS receiver calculates its position by carefully timing the signals sent by the constellation of GPS Satellites high above the Earth In the case of GPS, the received signal has a power of −133 dBm. The newer batch of satellites have a more powerful transmitter. To achieve this feat, GPS uses spread spectrum techniques, while some other communication systems use error control coding. Spread-spectrum techniques are methods by which Energy generated in a particular bandwidth is deliberately spread in the Frequency domain, resulting Coding theory is one of the most important and direct applications of Information theory. There is still a fundamental limit to the ability to discern the meaning of a signal in the midst of noise, given by the Shannon–Hartley theorem. In Information theory, the Shannon–Hartley theorem is an application of the Noisy channel coding theorem to the archetypal case of a continuous-time analog communications

## Noise current

The noise source can also be modeled by a current source in parallel with the resistor by taking the Norton equivalent that corresponds simply to divide by R. Norton's theorem for Electrical networks states that any collection of Voltage sources Current sources and Resistors with two terminals is This gives the root mean square value of the current source as:

$i_n = \sqrt {{ 4 k_B T \Delta f } \over R}$

Thermal noise is intrinsic to all resistors and is not a sign of poor design or manufacture, although resistors may also have excess noise. In Mathematics, the root mean square (abbreviated RMS or rms) also known as the quadratic mean, is a statistical measure of the

## Thermal noise on capacitors

Thermal noise on capacitors is referred to as kTC noise. Thermal noise in an RC circuit has an unusually simple expression as the value of the resistance (R) drops out of the equation. A resistor–capacitor circuit (RC circuit, or RC filter or RC network, is an Electric circuit composed of resistors and capacitors driven by This is because higher R contributes to more filtering as well as to more noise. Specifically the rms noise voltage generated in such a filter is:[5]

$v_{n} = \sqrt{ k_B T / C }$

Thermal noise caused by resistance accounts for 100% of kTC noise. Consider a capacitor of C farads with parallel resistance of R ohms. The known thermal noise voltage generated by a resistor is Vn = sqrt(4 k T R BW), where k is Boltzmann constant, T is temperature in Kelvin, R is resistance, and BW is bandwidth. The known bandwidth of the RC circuit is 1 / (4 R C)[6]. Thus, the known thermal noise caused by the resistor across a capacitor is Vn = sqrt(4 k T R BW) = sqrt(4 k T R (1 / (4 R C))) = sqrt(k T / C), which accounts for all kTC noise.

In the extreme case of 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 equilibrium 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. An image sensor is a device that converts an optical image to an electric signal As an alternative to the voltage noise, the reset noise on the capacitor can also be quantified as the charge standard deviation, as

$Q_{n} = \sqrt{ k_B T C }$

Since the charge variance is kBTC, this noise is often called kTC noise.

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=1/2*C*V2), mean noise energy on a capacitor can be seen to also be 1/2*C*(k*T/C), or also kT/2. Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance.

The kTC noise is the dominant noise source at small capacitors.

 Capacitor size $\sqrt{ k_B T / C }$ Electrons 0. 001 pF 2 mV 12. 5 e- 0. 01 pF 640 µV 40 e- 0. 1 pF 200 µV 125 e- 1 pF 64 µV 400 e- 10 pF 20 µV 1250 e- 100 pF 6. 4 µV 4000 e-

## Noise at very high frequencies

The above equations are good approximations at any practical radio frequency in use (i. e. frequencies below about 80 terahertz). Electromagnetic waves sent at terahertz frequencies, known as terahertz radiation, submillimeter radiation, terahertz waves, terahertz In the most general case,which includes up to optical frequencies, the power spectral density of the voltage across the resistor R, in V2 / Hz is given by:

$\Phi (f) = \frac{2 R h f}{e^{\frac{h f}{k_B T}} - 1}$

where f is the frequency, h Planck's constant, kB Boltzmann constant and T the temperature in kelvins. In Statistical signal processing and Physics, the spectral density, power spectral density ( PSD) or energy spectral density ( The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics If the frequency is low enough, that means:

$f \ll \frac{k_B T}{h}$

(this assumption is valid until few terahertz) then the exponential can be expressed in terms of its Taylor series. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives The relationship then becomes:

$\Phi (f) \approx 2 R k_B T$

In general, both R and T depend on frequency. In order to know the total noise it is enough to integrate over all the bandwidth. Since the signal is real, it is possible to integrate over only the positive frequencies, then multiply by 2. Assuming that R and T are constants over all the bandwidth Δf, then the root mean square (rms) value of the voltage across a resistor due to thermal noise is given by

$v_n = \sqrt { 4 k_B T R \Delta f }$,

that is, the same formula as above. In Mathematics, the root mean square (abbreviated RMS or rms) also known as the quadratic mean, is a statistical measure of the