Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the noise generated by the thermal agitation of the charge carriers (the electrons) inside an electrical conductor in equilibrium, which happens regardless of any applied voltage.

History

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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.

Explanation

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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 conductor medium (e.g. ions in an electrolyte). Thermal noise is generated in resistors. It can be modeled by voltage source in series with the noise generating resistor. The root mean square (rms) of the voltage, $v\_\{n\}$, can be calculated from,

$$

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

where k_B is Boltzmann's constant in joules per kelvin, T is the conductor temperature in kelvins, R is the resister value in Ohm,and Δf is the bandwidth in hertz.

The noise generated at the resistor can transfer to the remaining circuit, the maximum noise power transfer happens with impedance matching when the thevenin 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,

$$

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. Also the noise is white noise, equal throughout the frequency spectrum.

In communications, decibels(dBm) are often used. Thermal noise at room temperature can be estimated as:

$$

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

Where P is measured in dBm. 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

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:

$$

i_n = \sqrt - 1}

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:

$$

f << \frac{k_B T}{h}

(this assumption is valid until few gigahertz) then the exponential can be expressed in terms of its Taylor series. 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 $\backslash Delta\; f$, 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.

See also

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• Shot noise
• 1/f noise
• Harry Nyquist, J. Johnson

References

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• 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

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• Amplifier noise in RF systems
• 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. Sonoda

Noise | Electronics

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