Johnson-Nyquist Noise

Introduction

Johnson-Nyquist Noise, also known as thermal noise or Johnson noise, is the electronic noise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium. It’s an essential consideration in electronics and signal processing and was first measured accurately by John B. Johnson and then theoretically analyzed by Harry Nyquist, both at Bell Labs, in the 1920s.

Fundamentals of Thermal Noise

The root cause of thermal noise is the random motion of charge carriers in a conductor due to thermal agitation. These random movements of carriers create fluctuations in the voltage and current, which we interpret as noise. The power of the Johnson-Nyquist noise is given by the formula:

where is Boltzmann’s constant, is the absolute temperature in Kelvin, and is the bandwidth in Hertz over which the noise is measured.

Noise Voltage and Current

The root-mean-square (rms) noise voltage across a resistor is given by:

where is the resistance. For noise current, the rms noise current through a resistor is given by:

White Noise

Johnson-Nyquist Noise is often referred to as “white noise” because it has equal intensity at different frequencies, much like white light in optics. This frequency-independent nature makes it an important type of noise in many fields, from electrical engineering to acoustics and even finance.

Nyquist’s Theorem

Harry Nyquist developed a detailed and insightful theory explaining the origin and behavior of Johnson noise. Nyquist’s theorem, using the principles of statistical mechanics and the equipartition theorem, showed that the mean square noise in a resistor is directly proportional to the absolute temperature and the bandwidth over which it is measured. His theory was crucial for the correct understanding and utilization of noise in electronics.

Noise Measurement and Practical Implications

In practical applications, Johnson-Nyquist noise is a fundamental limit to the sensitivity of electronic equipment. It sets a lower bound to the signal-to-noise ratio (SNR) achievable within a given bandwidth at a given temperature. This noise can only be reduced by lowering the temperature or narrowing the bandwidth.

Noise in Quantum Limit

In the realm of quantum mechanics, Johnson-Nyquist noise maintains its importance. At very low temperatures, the quantum nature of energy becomes significant, and we transition from the classical Johnson-Nyquist noise to the so-called “shot noise”. This transition and the underlying physics are active areas of research.

Conclusion

Johnson-Nyquist noise is a fundamental aspect of electronics and signal processing, representing the interaction between thermodynamics and electrical phenomena. Understanding this type of noise is critical for designing and interpreting electronic systems, particularly those requiring high sensitivity. Current research aims to better understand thermal noise at very low temperatures and high frequencies, where quantum effects become significant.

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