Resistor Thermal Noise
# Resistor Thermal Noise: A Quiet Circuit Still Has Fluctuations
A resistor at a finite temperature produces thermal electrical noise even when no DC current flows through it. Also called Johnson–Nyquist noise, it comes from random thermal motion of charge carriers. In the classical frequency range it is approximately white: its voltage-noise power per hertz is constant for an ideal resistance at a fixed temperature.
## 1. Represent the noise once
*Conceptual carrier-motion illustration and equivalent two-terminal circuit. The source icon denotes random noise, not a sinusoidal signal. The series resistor is noiseless because the separate source already accounts for the physical resistor’s thermal noise.*
The one-sided open-circuit voltage power spectral density is Sv = 4 kB T R, in V²/Hz. Its amplitude spectral density is en = √(4 kB T R), in V/√Hz. Here kB = 1.380649 × 10⁻²³ J/K, T is absolute temperature in kelvin, and R is resistance in ohms.
The equivalent Norton model uses a noise current source in parallel with a noiseless resistor, with in = √(4 kB T/R) in A/√Hz. These are alternative descriptions of the same noise. Do not add both sources independently for one resistor.
## 2. Integrate over the bandwidth you actually use
*Analytical example, not measured data. The graph assumes a rectangular passband and an unloaded resistor at 300 K. Its frequency axis is logarithmic; its RMS-noise axis is linear.*
For a rectangular noise bandwidth B, vrms = √(4 kB T R B). A 1-kilohm resistor at 300 K has about 4.07 nV/√Hz of open-circuit voltage-noise density. Across 10 kHz this becomes about 0.407 µV RMS; across 1 MHz it becomes about 4.07 µV RMS. Increasing bandwidth by 100 increases RMS noise by 10.
For a real circuit, propagate each source through its transfer function and integrate the resulting output power spectral density: vout,rms² = ∫₀∞ Sv(f) |H(f)|² df. For flat input noise and unity low-frequency gain, equivalent noise bandwidth is Bn = ∫₀∞ |H(f)|² df. A unity-gain single-pole low-pass with −3 dB frequency fc has Bn = (π/2)fc. Its −3 dB bandwidth alone would underestimate the integrated white noise.
## 3. Include loading and other sources correctly
The open-circuit voltage is not automatically the noise delivered to a load. A load forms a divider with R, and reactive loading shapes the spectrum. With a matched noiseless load, the source resistor’s available noise power in a rectangular band is kB T B in the classical model. A physical load resistor also generates its own thermal noise; include that source when predicting the total observed voltage.
Independent noise sources add in mean-square units. If two independent contributions are 3 µV RMS and 4 µV RMS, their combined value is 5 µV RMS. Correlated sources require cross terms and cannot generally be combined by this rule.
| Change | Ideal thermal-noise effect | Circuit-level qualification |
|---|---|---|
| Increase R by 4× | Open-circuit voltage density doubles | Current-noise density halves; loading also changes |
| Increase T by 4× | Density doubles | Device parameters and self-heating may also change |
| Increase bandwidth by 4× | Integrated RMS doubles | Use the actual noise transfer function |
| Remove DC bias | Thermal noise remains | Bias-dependent excess noise may decrease |
## 4. Design and verify a noise budget
Lower resistor values can reduce open-circuit voltage noise, but may increase loading or power consumption. In a feedback network, changing resistance can also alter the noise gain, bandwidth and sensitivity to amplifier current noise. Evaluate the complete circuit rather than ranking resistors by voltage density alone.
Thermal noise is distinct from excess resistance noise, amplifier flicker noise, shot noise, interference and sampling effects. Real resistor technology and bias can affect excess noise; the ideal thermal contribution remains determined by resistance and temperature within the model’s range. At frequencies where photon energy is comparable to kB T, the classical white-noise approximation needs quantum correction.
In simulation, identify each resistor’s temperature, the output quantity and the integration band. Compare spectral-density units carefully: V/√Hz is an amplitude density, while V²/Hz is a power density. In measurement, include instrument noise, gain calibration, grounding and the filter response. Report RMS noise with its bandwidth and loading conditions.
## References
- [Analog Devices: Resistor Noise Can Be Deafening and Hard to Reduce](https://www.analog.com/en/resources/analog-dialogue/raqs/raq-issue-25.html) — physical origin and resistance-dependent noise.
- [Analog Devices: Ask the Applications Engineer — Op-Amp Noise](https://www.analog.com/en/resources/analog-dialogue/articles/what-should-i-know-about-opamp-noise.html) — noise density, integration and circuit noise budgeting.
- [Texas Instruments: Noise Sources in a Circuit](https://www.ti.com/document-viewer/lit/html/SBOA345/GUID-F87CE11A-8998-4FB4-BEA6-8D520E81351E) — resistor thermal noise and RMS calculations.