How this instrument works
A resistor generates a tiny AC voltage across its own terminals purely from being warm, with zero current required. Free electrons inside any conductor are in constant, random thermal motion — the same jostling behind Brownian motion — and at any instant slightly more of them happen to drift toward one terminal than the other, producing a fluctuating open-circuit voltage. Johnson-Nyquist noise names that fluctuation: Vₙ = √(4kBTRΔf), an RMS figure because the underlying signal averages to zero and only its power is well defined.
The square root traces back to how Harry Nyquist derived the formula in 1928: not from circuit theory but from thermodynamics, by treating a resistor connected to a matched transmission line as a one-dimensional blackbody. Equipartition assigns kBT of energy per mode, the line carries kBTΔf watts of available noise power into a matched load, and because power divides as V²⁄4R for a source driving its own resistance, solving for V pulls a resistance and a square root into the answer. Every factor in the formula earns its place from that derivation; none of it is a curve fit to measured data.
The formula is a classical approximation and knows its own limits. It assumes the noise power spectral density is flat across Δf, which holds from near-DC up to frequencies where the photon energy hf becomes comparable to kBT — around 6 terahertz at room temperature — beyond which the full Planck-corrected expression rolls the spectrum off rather than letting it climb forever. It also only describes a genuinely resistive part; an ideal capacitor or inductor produces no Johnson noise of its own, though it will filter whatever noise a real resistance nearby contributes to the same network.
- Enter the resistor's value in Resistance — ohms or kilohms.
- Enter the resistor's actual operating temperature in Temperature — °C or °F; the calculator converts it to kelvin internally before using it.
- Set Bandwidth to the measurement or circuit bandwidth in question — Hz, kHz, or MHz — not the resistor's own effectively unlimited physical bandwidth.
- Read RMS thermal noise voltage — the noise floor that this resistance, temperature, and bandwidth together impose, in volts or millivolts.
Worked example — a 1 kΩ resistor across 1 MHz of bandwidth
Take a plain 1 kΩ resistor sitting at room temperature, 27°C, feeding a measurement chain with 1 MHz of bandwidth — a fairly ordinary span for an audio-to-RF test setup. Converting to kelvin first, T = 27 + 273.15 = 300.15 K. The formula then gives Vₙ = √(4 × 1.380649×10⁻²³ × 300.15 × 1000 × 1,000,000) = 4.07137223722×10⁻⁶ V, about 4.07 microvolts RMS.
That 4.07 µV is not measurement error or a flaw in the part; it is the thermal noise floor no amount of precision manufacturing can design around, present the instant the resistor reaches thermal equilibrium with its surroundings, current or no current flowing through it. An engineer sizing the first stage of a sensitive preamplifier compares this figure directly against the amplifier's own input-referred noise to see which one actually limits the design; only cooling the resistor or narrowing the downstream bandwidth will bring it down.
Questions
Why does thermal noise appear even with no current flowing through the resistor?
Because Johnson-Nyquist noise comes from the random thermal motion of charge carriers, not from current flow. Every resistor above absolute zero has electrons jostling under thermal agitation, and that motion alone creates a tiny fluctuating voltage across the terminals whether or not any current is driven through it. That separates it from shot noise, which does require current and vanishes at zero bias.
Why is temperature entered in Celsius when the formula needs kelvin?
Because engineers usually think and measure in Celsius, so the field accepts °C or °F for convenience, then the calculator adds 273.15 internally to reach the absolute, kelvin temperature the physics requires. Skipping that step — plugging a bare Celsius number straight into 4kBTRΔf — is a common hand-calculation error, and it understates the noise for any resistor near everyday room temperature.
Does doubling the resistance double the noise voltage?
No — it multiplies the noise voltage by √2, about 1.41 times, not 2. Noise power scales linearly with resistance, but voltage is the square root of power for a fixed reference, so that exponent carries through the whole expression under the radical. Ten times the resistance raises Vₙ by √10, roughly 3.16 times, never tenfold.
What bandwidth, Δf, should I actually enter?
The bandwidth of whatever measures or uses the noise downstream — typically the −3 dB bandwidth of an amplifier, filter, or ADC front end — not the resistor's own essentially unlimited physical bandwidth. A wideband RF front end and a narrowband audio channel see very different noise voltages from the identical resistor because each integrates that flat spectral density over a different Δf.
Can a resistor's Johnson noise be reduced without changing R or Δf?
Only by lowering its temperature. Cryogenic receiver front ends in radio telescopes and quantum measurement systems cool the input resistance to a few kelvin specifically because Vₙ scales with the square root of T, and that is the only remaining lever once resistance and bandwidth are fixed by the rest of the design.
Is Johnson-Nyquist noise the only noise source in a real circuit?
No. Real circuits also carry shot noise from discrete charge crossing a biased junction, flicker or 1/f noise that dominates at low frequencies in semiconductors, and noise contributed by active devices themselves. Johnson-Nyquist noise is unique in being present in every resistive element from temperature alone, setting an unavoidable floor beneath all the others.