Define thermal noise (Johnson-Nyquist noise) in electronic systems.

The Johnson-Nyquist noise was first discovered and mathematically described by John B. Johnson and Harry Nyquist in the early 20th century. According to their theory, the thermal noise voltage (V_n) across a resistor at temperature T (in Kelvin) is given by:

V_n = √(4 * k * T * R * Δf)

Where:

k is Boltzmann's constant (approximately 1.380649 × 10^-23 J/K)

T is the absolute temperature in Kelvin

R is the resistance of the conductor or resistor in ohms

Δf is the bandwidth of the measurement in hertz

The key characteristics of thermal noise are as follows:

Randomness: Thermal noise is random and exhibits a Gaussian (normal) distribution of voltage fluctuations over time. This means that its amplitude varies unpredictably, with equal probabilities of positive and negative fluctuations.

White noise: It is called "white" noise because it has a flat frequency spectrum, meaning it has equal power density across all frequencies within its bandwidth. Hence, thermal noise contains a wide range of frequencies, covering both high and low frequencies.

Temperature dependence: The magnitude of thermal noise is directly proportional to the square root of temperature (T). As the temperature increases, so does the thermal noise level.

Proportional to resistance: The magnitude of thermal noise is proportional to the square root of the resistance (R). Higher resistance values result in higher thermal noise levels.

Thermal noise is an unavoidable phenomenon in electronic circuits and systems, and its impact is especially significant in low-noise applications or high-impedance circuits where even small voltage fluctuations can be problematic. Engineers and designers must carefully consider and mitigate thermal noise effects to ensure proper performance and accuracy in electronic devices and communication systems.