Johnson (Thermal) Noise

Johnson noise (also called Nyquist or thermal noise) is the fundamental noise floor below which no resistor can be quieter — it arises from random Brownian motion of charge carriers and depends only on absolute temperature T (kelvin), resistance R (ohms) and Boltzmann's constant k (1.38 × 10⁻²³ J/K). The open-circuit RMS noise voltage in bandwidth B is V_n = √(4kTRB), which evaluates to about 4 nV/√Hz for 1 kΩ at room temperature, 13 nV/√Hz for 10 kΩ, and 130 nV/√Hz for 1 MΩ.

Because Johnson noise is white (flat in frequency) and Gaussian, designers integrate it over the system bandwidth. A 10 kΩ resistor in a 1 MHz audio chain contributes 13 nV/√Hz × √(1×10⁶) = 13 μV RMS of noise — sometimes the dominant noise source in low-level amplifiers. Cooling helps: at 77 K (liquid nitrogen) the noise is √(77/293) ≈ 0.51× that at room temperature. Lowering R is the only other practical knob, hence why instrumentation-amplifier input bias networks use the smallest possible feedback resistors.

Real resistors add a small excess 1/f noise on top of Johnson, captured by the noise-index figure (see noise-figure entry). Wirewound and bulk metal foil are essentially pure-Johnson; thick film and carbon composition can add 10–30 dB of 1/f at low frequencies.