Thermal Noise

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. Ion s in an Electrolyte ).

The thermal Noise Power , ''P'', in watts, is given by

P = { 4 k_B T \Delta f }

, where ''k_B'' is Boltzmann's Constant in joules per Kelvin , ''T'' is the conductor Temperature in kelvins, and Δ''f'' is the Bandwidth in Hertz .
Thermal noise power, per hertz, is equal throughout the Frequency spectrum, depending only on ''k_B'' 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:

:

P = -174 + 10\ \log(\Delta f)

Where P is measured in DBm (0 dBm = 1 MW ) and Δ''f'' is Bandwidth in Hz . For example:

Electronics engineers often prefer to work in terms of noise Voltage across the resistor (''v_n'') and noise Current (''i_n'') 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:

:

\Phi (f) = rac{2 R h f}{e^{rac{h f}{k_B T}} - 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 << rac{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

\Delta f

, 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:

:

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

and, dividing it by ''R'' we obtain the r.m.s. current:

:

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Thermal Noise