Johnson-Nyquist Noise

Prerequisites

• Central Limit Theorem

Short Topical Videos

• The Origin of Noise in Electronics (Aaron Parsons)
• Introduction to Thermal Noise (Darryl Morrell)

Reference Material

• Johnson-Nyquist Noise (wikipedia)
• Maximum Power Transfer Theorem (wikipedia)
• Derivation of Johnson Noise (Romero, MIT)
• Electronic Noise (Pellett, UC Davis)

Johnson-Nyquist Noise

Johnson-Nyquist noise is the thermal noise generated by the random motions of electrons within a resistor. The random thermal motion of electrons corresponds a current that has zero mean but non-zero variance. This current sets up a fluctuating voltage across a resistor. The frequency power spectrum of this noise is approximately white (flat versus frequency), although it does roll over at higher frequencies.

The variance in the voltage across a resistor of a given temperature is given by

$V^{2}=4k_{B}TBR,\,\!$

where $V$ is voltage, $k_{B}$ is Boltzmann’s constant, $T$ is the temperature of the resistor, $B$ is the frequency bandwidth of the measurement, and $R$ is the resistance of the resistor. Using that $P=IV$, and Ohm’s Law, we also have:

$P_{sig}=4k_{B}TB,\,\!$

where $P_{sig}$ is the power of the signal generated by the resistor.

A model of Johnson-Nyquist resistor noise driving a load

However, when we talk about the noise generated by a resistor, we are usually interested because this noise is being propagated through a circuit. According to the Maximum Power Transfer theorem, not all of the signal power generated by the resistor can be passed on through the circuit. In fact, the maximum amount of power is transferred only if the impedance of the downstream circuit ($R_{load}$ above) matches the resistor generating the noise ($R_{src}$ above). In this case, we have a simple voltage divider where the voltage output is halved. As a result, if you go to measure the signal power from the resistor that dissipated in a downstream circuit (say, into the input of an amplifier), you will find:

${\frac {P}{B}}=k_{B}T.\,\!$

As you will see later in the context of receiver/amplifier temperatures, this jives with our understanding that resistors add a temperature $T$ to the noise temperature of a system.