How to calculate Thermal Noise Power?

There are 3types of noise in electrical circuit- thermal noise, shot noise and flicker noise(see Types of Noise that you should know about). Here the thermal noise is discussed and the thermal noise power equation derived.

Consider a electrical or electronics wire which has some finite resistance R. Let the room temperature be T Kelvin. The electrons in the wire are in constant random motion manifested due to their kinetic energy which is due to the temperature T. These motion will produce fluctuating voltage across the wire. The average value of this voltage is zero but the mean square value is not zero. The mean square value of this voltage is given by Planck's Black Body radiation law as follows:

where h is Planck’s constant (6.546 × 10−34 J · s), k the Boltzmann constant (1.38 × 10−23 J/K), T the temperature in kelvin, B the system bandwidth in hertz, and f the center frequency of the bandwidth in hertz.

In the denominator,  we have two quantities in the exponential term- hf and kT. At temperature T > 10K and frequency f < 100GHz, kT will be larger than hf. Hence the term exp(hf/kT) can be approximated as follows-

and the mean square value of voltage becomes,

This approximation is also called Rayleigh–Jeans approximation.

By using Thevenin theorem we may replace the noisy resistor R with an equivalent circuit which has the mean square voltage and a noiseless resistor R.

The power delivered to a load resistor R is then,

where we have used the Rayleigh–Jeans approximation for mean square voltage. This equation is the equation for Thermal Noise Power.