Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the
electronic noise generated by the
thermal agitation of the
charge carrier
In physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and holes. The term i ...
s (usually the
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
s) inside an
electrical conductor at equilibrium, which happens regardless of any applied
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
. Thermal noise is present in all
electrical circuit
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage source ...
s, and in sensitive electronic equipment (such as
radio receiver
In radio communications, a radio receiver, also known as a receiver, a wireless, or simply a radio, is an electronic device that receives radio waves and converts the information carried by them to a usable form. It is used with an antenna. Th ...
s) 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
A radio telescope is a specialized antenna and radio receiver used to detect radio waves from astronomical radio sources in the sky. Radio telescopes are the main observing instrument used in radio astronomy, which studies the radio frequency ...
receivers are cooled to
cryogenic
In physics, cryogenics is the production and behaviour of materials at very low temperatures.
The 13th IIR International Congress of Refrigeration (held in Washington DC in 1971) endorsed a universal definition of “cryogenics” and “cr ...
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
White is the lightest color and is achromatic (having no hue). It is the color of objects such as snow, chalk, and milk, and is the opposite of black. White objects fully reflect and scatter all the visible wavelengths of light. White ...
, meaning that the power
spectral density
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies ...
is nearly constant throughout the
frequency spectrum, but does decay to zero at extremely high frequencies (
terahertz for
room temperature
Colloquially, "room temperature" is a range of air temperatures that most people prefer for indoor settings. It feels comfortable to a person when they are wearing typical indoor clothing. Human comfort can extend beyond this range depending on ...
). When limited to a finite bandwidth, thermal noise has a nearly
Gaussian amplitude distribution.
History
This type of noise was discovered and first measured by
John B. Johnson at
Bell Labs
Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984),
then AT&T Bell Laboratories (1984–1996)
and Bell Labs Innovations (1996–2007),
is an American industrial research and scientific development company owned by mul ...
in 1926. He described his findings to
Harry Nyquist
Harry Nyquist (, ; February 7, 1889 – April 4, 1976) was a Swedish-American physicist and electronic engineer who made important contributions to communication theory.
Personal life
Nyquist was born in the village Nilsby of the parish Stora ...
, also at Bell Labs, who was able to explain the results.
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,
where
is the noise power density in (W/Hz),
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
and
is the
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
. Multiplying the equation by bandwidth gives the result as noise power.
where ''N'' is the noise power and ''Δf'' is the
bandwidth.
Noise voltage and power
Thermal noise is distinct from
shot noise
Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process.
In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where sh ...
, 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
Medium may refer to:
Science and technology
Aviation
* Medium bomber, a class of war plane
* Tecma Medium, a French hang glider design
Communication
* Media (communication), tools used to store and deliver information or data
* Medium ...
(e.g.
ions in an
electrolyte
An electrolyte is a medium containing ions that is electrically conducting through the movement of those ions, but not conducting electrons. This includes most soluble salts, acids, and bases dissolved in a polar solvent, such as water. Upon ...
), not just
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...
s. 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
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
of
bandwidth, is given by
:
where ''k''
B is
Boltzmann's constant in
joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
s per
kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
, ''T'' is the resistor's absolute
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
in kelvins, and ''R'' is the resistor value in
ohm
Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm.
Ohm or OHM may also refer to:
People
* Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm''
* Germán Ohm (born 1936), Mexican boxer
* Jörg Ohm (bor ...
s (Ω).
Using this equation for quick calculation, at room temperature:
:
For example, a 1 kΩ resistor at a temperature of 300 K has
:
For a given bandwidth, the
root mean square
In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
(RMS) of the voltage,
, is given by
:
where Δ''f'' 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. 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
:
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
:
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 by taking the
Norton equivalent
In direct-current circuit theory, Norton's theorem, also called the Mayer–Norton theorem, is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of te ...
that corresponds simply to dividing by ''R''. This gives the
root mean square
In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
value of the current source as:
:
Noise power in decibels
Signal power is often measured in
dBm (
decibels
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a ...
relative to 1
milliwatt
The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wat ...
). From the equation above, noise power in a resistor at
room temperature
Colloquially, "room temperature" is a range of air temperatures that most people prefer for indoor settings. It feels comfortable to a person when they are wearing typical indoor clothing. Human comfort can extend beyond this range depending on ...
, in dBm, is then:
:
At room temperature (300 K) this is approximately
:
Using this equation, noise power for different bandwidths is simple to calculate:
Thermal noise on capacitors
Ideal capacitors, as lossless devices, do not have thermal noise, but as commonly used with resistors in an
RC circuit
A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC ...
, the combination has what is called ''kTC'' noise. The noise bandwidth of an RC circuit is Δ''f'' = 1/(4''RC''). 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:
:
:
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, 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
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
as given above. The reset noise of capacitive sensors is often a limiting noise source, for example in
image sensor
An image sensor or imager is a sensor that detects and conveys information used to make an image. It does so by converting the variable attenuation of light waves (as they pass through or reflect off objects) into signals, small bursts of c ...
s.
Any system in
thermal equilibrium
Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in ...
has
state variables with a mean
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
of ''kT''/2 per
degree of freedom. Using the formula for energy on a capacitor (''E'' = ½''CV''
2), mean noise energy on a capacitor can be seen to also be ½''C''(''kT''/''C'') = ''kT''/2. Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance.
Generalized forms
The
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.
Reactive 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
In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.
Quantitatively, the impedance of a two-terminal circuit element is the ratio of the c ...
. The formula for the power spectral density of the series noise voltage is
:
The function is simply equal to 1 except at very high frequencies, or near absolute zero (see below).
The real part of impedance,