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Johnson–Nyquist noise

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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 approximately white, meaning that the power spectral density is equal throughout the frequency spectrum. Additionally, the amplitude of the signal has very nearly a Gaussian probability density function.^[1]

1 History
2 Noise voltage and power
3 Noise in decibels
4 Noise current
5 Thermal noise on capacitors
6 Noise at very high frequencies
7 See also
8 References
9 External links

History

This type of noise was first measured by John B. Johnson at Bell Labs in 1928^[2]. He described his findings to Harry Nyquist, also at Bell Labs, who was able to explain the results.^[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. 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 power spectral density, or voltage variance (mean square) per hertz of bandwidth, is given by

<math>

\bar v_{n}^2 = 4 k_B T R </math> where k_B 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. For example, a resistor of 1 kΩ at an average temperature (300 K) has

<math>

\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}}</math>. For a given bandwidth, the root mean square (rms) of the voltage, <math>v_{n}</math>, is given by

<math>

v_{n} = \bar v_{n}\sqrt{\Delta f } = \sqrt{ 4 k_B T R \Delta f } </math> where Δf is the bandwidth in hertz over which the noise is measured. 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.[1] 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 the noise power transfer to the circuit is given by

<math>

P = k_B \,T \Delta f </math> 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 resistance. With these conventions, thermal noise at room temperature can be estimated as:

<math>

P_\mathrm{dBm} = -174 + 10\ \log(\Delta f) </math> where P is measured in dBm. For example:

Bandwidth	Power	Notes
1 Hz	-174 dBm
10 Hz	-164 dBm
1000 Hz	-144 dBm
10 kHz	-134 dBm	FM channel of 2-way radio
1 MHz	-114 dBm
2 MHz	-111 dBm	Commercial GPS channel
6 MHz	-106 dBm	Analog television channel
2.4 GHz	-80 dBm

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 load of receiver, which would be -106 dBm, or one fortieth of a picowatt. The 6 MHz could be the 6 MHz between spectrum at 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 2.4 GHz in the chart should not be confused with the Johnson-Nyquist noise generated in a 6 MHz wide channel at that starting frequency, which would be -106 dBm. 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 at ground level. In the case of GPS, the received signal has a power of -133 dBm. The newer batch of satellites have a more powerful transmitter.

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. This gives the root mean square value of the current source as:

<math>

i_n = \sqrt {{ 4 k_B T \Delta f } \over R} </math> Thermal noise is intrinsic to all resistors and is not a sign of poor design or manufacture, although resistors may also have excess noise.

Thermal noise on capacitors

Johnson noise in an RC circuit can be expressed more simply by using the capacitance value, rather than the resistance and bandwidth values. The rms voltage noise on a capacitance C is

<math>

v_{n} = \sqrt{ k_B T / C } </math> independent of the resistor value, since bandwidth varies reciprocally with resistance in an RC circuit.^[4] In the case of the reset noise left on a capacitor by opening an ideal switch, the resistance is infinite, and 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. As an alternative to the voltage noise, the reset noise on the capacitor can also be quantified as the charge standard deviation, as

<math>

Q_{n} = \sqrt{ k_B T C } </math> Since the charge variance is <math>k_B T C</math>, this noise is often called kTC noise. Any system in thermal equilibrium has particles with an energy of half of kT per degree of freedom. Using the formula for energy on a capacitor (E=1/2*C*V^2), the energy on a capacitor can be seen to also be 1/2*C*(k*T/C), or also kT/2. The kTC noise is the dominant noise source at small capacitors.

Noise of capacitors at 300°K
Capacitor size	<math> \sqrt{ k_B T / C } </math>	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 low frequencies. In general, the power spectral density of the voltage across the resistor R, in <math>\mathrm{V^2/Hz}</math> is given by:

<math>

\Phi (f) = \frac{2 R h f}{e^{\frac{h f}{k_B T}} - 1} </math> where f is the frequency, h Planck's constant, k_B Boltzmann constant and T the temperature in kelvins. If the frequency is low enough, that means:

<math>

f \ll \frac{k_B T}{h} </math> (this assumption is valid until few terahertz) then the exponential can be expressed in terms of its Taylor series. The relationship then becomes:

<math>

\Phi (f) \approx 2 R k_B T </math> 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 <math>\Delta f</math>, then the root mean square (rms) value of the voltage across a resistor due to thermal noise is given by

<math>

v_n = \sqrt { 4 k_B T R \Delta f } </math>, that is, the same formula as above.

References

^ Mancini, Ron; others (August 2002). Op Amps For Everyone (PDF). Application Notes p. 148. Texas Instruments. Retrieved on 2006-12-06. “Thermal noise and shot noise (see below) have Gaussian probability density functions. The other forms of noise do not.”
^ J. Johnson, "Thermal Agitation of Electricity in Conductors", Phys. Rev. 32, 97 (1928) – the experiment
^ H. Nyquist, "Thermal Agitation of Electric Charge in Conductors", Phys. Rev. 32, 110 (1928) – the theory
^ R. Sarpeshkar, T. Delbruck, and C. A. Mead, "White noise in MOS transistors and resistors", IEEE Circuits Devices Mag., pp. 23–29, Nov. 1993.