Johnson noise calculator

Johnson–Nyquist noise is the unavoidable voltage fluctuation that appears across any resistive element because charge carriers jiggle with thermal energy. The closed-form RMS value is the square root of four times Boltzmann’s constant times absolute temperature times resistance times noise bandwidth, written Vn equals square root of four k T R Δf. Every RF front end, ADC chain, sensor bridge, and low-noise amplifier design begins with that estimate so the engineer can compare thermal noise to quantization noise, shot noise, and amplifier noise figures. This Johnson noise calculator accepts resistance in ohms, absolute temperature in kelvin, and noise bandwidth in hertz, evaluates the deterministic square-root expression, and returns the RMS voltage together with the open-circuit spectral density in volts per square-root hertz and a short printable expression. Negative bandwidth is rejected with a clear validation error; negative resistance or temperature fails the same way as non-physical for this model.

How Johnson–Nyquist thermal noise voltage works

In thermal equilibrium a resistor behaves as a noise voltage source in series with its ideal resistance. The mean-square open-circuit voltage in a frequency interval Δf is four k T R Δf, so the root-mean-square amplitude is the square root of that product.

Boltzmann’s constant k is fixed by the SI definition at one point three eight zero six four nine times ten to the minus twenty-three joules per kelvin, temperature must be absolute kelvin, resistance is in ohms, and Δf is the equivalent noise bandwidth of the measurement or signal path in hertz.

For a brick-wall filter Δf is simply the passband width; for a single-pole RC low-pass the equivalent noise bandwidth is pi over two times the three-decibel corner, which is larger than the half-power frequency because noise power continues past the corner. Doubling resistance or temperature multiplies Vn by the square root of two; quadrupling bandwidth doubles Vn.

At room temperature near two hundred ninety kelvin a fifty-ohm source over one megahertz of noise bandwidth produces on the order of nine hundred nanovolts RMS, a number RF engineers memorize as a sanity check. The formula assumes white thermal noise over the band of interest, neglects quantum corrections that matter only at very high frequencies or cryogenic extremes for most circuit work, and does not include amplifier excess noise or 1/f flicker. Negative bandwidth is explicitly invalid; zero bandwidth correctly yields zero noise voltage.

Where Johnson noise calculators show up in real work

Introductory electronics and RF courses assign thermal-noise drills so students can convert resistor values, room temperature, and filter bandwidths into microvolts or nanovolts before they compare a scope floor or spectrum-analyzer marker to theory. Sensor and instrumentation teams estimate the thermal limit of bridge and shunt resistors that sit in front of precision ADCs, and a trusted closed form keeps notebooks and data-acquisition scripts aligned. Receiver and LNA designers budget source thermal noise against noise figure and system temperature when writing link budgets for radios, radars, and satellite downlinks.

Keep shot-noise, flicker-noise, noise-figure, and SNR calculator landings separate so each search intent has one clear page rather than an overloaded multi-formula tool that confuses both students and search engines. If you only need spectral density without integrating bandwidth, the response already returns that intermediate quantity so you can scale later by the square root of any other Δf without a second formula lookup.