##  [Johnson-Nyquist Noise](/johnson-nyquist-noise-0) 

 Definition

Random electrical voltage or current fluctuations generated by the thermal agitation of charge carriers in a resistive element at thermal equilibrium; for a linear resistor at temperature T the equilibrium one‑sided voltage power spectral density is S_v(f) = 4 k_B T R (classically frequency‑independent, with quantum corrections when hf ≳ k_B T).

 

 

 

 

 

 





## Principle

Principle

Thermodynamic equilibrium of a passive resistor imposes a fundamental, temperature‑proportional noise power that is determined by Boltzmann’s constant and the resistor value; it cannot be eliminated by linear, passive circuit design and sets a lower bound on signal detectable in a bandwidth-limited measurement.

 

 

 

 

 





## Demonstration

Demonstration

Illustrative scenario — Situation: a 50 Ω termination at room temperature connected to a low‑noise amplifier and a spectrum analyzer. Recognition: the measured flat noise floor corresponds to S_v ≈ 4 k_B T R across audio to radio frequencies. Action: averaging spectra and subtracting amplifier contribution isolates resistor noise. Consequence: measured noise power within bandwidth B equals k_B T B, limiting achievable signal‑to‑noise ratio for small signals.

 

 

 

 

## Misapplication

Misapplication

Treating Johnson‑Nyquist noise as identical to shot noise or 1/f noise; the semantic error is conflating a thermal equilibrium, broadband fluctuation (proportional to T and R) with noise that arises from discrete carrier statistics (shot) or device/technology dependent low‑frequency processes (flicker).

 

 

 

 

 





## Consequence

Consequence

It defines a thermodynamic floor for passive sensing and communication: for a given temperature, resistance and measurement bandwidth the minimum root‑mean‑square noise is fixed, driving requirements for cooling, impedance matching, bandwidth reduction, or active amplification to achieve desired sensitivity.

 

 

 

 

## Reversal

Reversal

At frequencies or temperatures where quantum effects are non‑negligible (hf comparable to or greater than k_B T), the classical flat spectral formula is replaced by the quantum expression derived from Planck’s law; in non‑equilibrium resistive elements additional excess noise sources (e.g., current‑driven shot noise, generation‑recombination noise) can dominate.

 

 

 

 

 





## Boundary

Boundary

Clearly within: passive linear resistors in thermal equilibrium measured across terminals. Boundary case: a biased resistor carrying dc current — thermal noise still exists but is accompanied by shot and excess noise that require separate accounting. Clearly outside: noise produced by active sources, switching transients, or ionic leakage and moisture‑induced conductance on surfaces.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Sensitivity Versus Practicality — lowering temperature or resistance reduces thermal noise but increases cost, complexity, or other tradeoffs (e.g., cooling systems, altered impedance matching), forcing a design trade between thermodynamic limits and system constraints.

 

 

 

 

 





## Synthesis

Synthesis

Johnson‑Nyquist noise is a universal, equilibrium‑thermodynamic noise source tied to temperature and resistance; understanding it as an irreducible baseline focuses engineering choices on changing temperature, bandwidth, impedance, or adding controlled amplification rather than attempting impossible elimination.