Johnson-Nyquist noise
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Johnson-Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the noise generated by the equilibrium fluctuations of the electric current inside an electrical conductor, which happens regardless of any applied voltage, due to the random thermal motion of the charge carriers (the electrons).
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History
This type of noise was first measured by J.B. Johnson at Bell Labs in 1928. He described his findings to H. Nyquist, also at Bell Labs, who was able to explain the results by deriving a fluctuation-dissipation relationship.
Explanation
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).
The thermal noise power, P, in watts, is given by <math>P = { 4 k_B T \Delta f }</math>, where kB is Boltzmann's constant in joules per kelvin, T is the conductor temperature in kelvins, and Δf is the bandwidth in hertz. Thermal noise power, per hertz, is equal throughout the frequency spectrum, depending only on kB and T. It is white noise, in other words.
In communications, noise power is often used. Thermal noise at room temperature can be estimated in decibels as:
- <math>
P = -174 + 10\ \log(\Delta f) </math>
Where P is measured in dBm (0 dBm = 1 mW) and Δf is bandwidth in Hz. For example:
| Bandwidth | Power |
|---|---|
| 1 Hz | -174 dBm |
| 10 Hz | -164 dBm |
| 1000 Hz | -144 dBm |
| 5 kHz | -137 dBm |
| 1 MHz | -114 dBm |
| 6 MHz | -106 dBm |
Electronics engineers often prefer to work in terms of noise voltage across the resistor (vn) and noise current (in) going through the resistor. These also depend on the electrical resistance, R, of the conductor. In general, the spectral density of the voltage across the resistor R 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, kB Boltzmann constant and T the temperature in kelvins. If the frequency is low enough, that means:
- <math>
f << \frac{k_B T}{h} </math>
(this assumption is valid until few gigahertz) 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 (r.m.s.) value of the voltage across a resistor due to thermal noise is given by the square root of the total noise, that means:
- <math>
v_n = \sqrt { 4 k_B T R \Delta f } </math>
and, dividing it by R we obtain the r.m.s. current:
- <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.
See also
- Shot noise
- 1/f noise
- Harry Nyquist, J. Johnson
References
- 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
- adapted in part from Federal Standard 1037C and from MIL-STD-188
External links
- Amplifier noise in RF systems
- Introduction to low-noise measurements
- Thermal noise (undergraduate) with detailed math
- Johnson-Nyquist noise or thermal noise calculator - volts and dB
- Thoughts about Image Calibration for low dark current and Amateur CCD Cameras to increase Signal-To-Noise Ratio
- Derivation of the Nyquist relation using a random electric field, H. Sonodade:Johnson-Rauschen
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