Library / Circuits, electronics and noise / ELN-46
Circuits, electronics and noise
Johnson noise voltage of a resistor
Open-circuit RMS thermal noise voltage of a resistor at absolute temperature T in a one-sided noise bandwidth B.
- Ref
- ELN-46
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- vn [V]
Thermal agitation of the charge carriers in a resistor produces a random voltage with a flat spectrum, vn = √(4 kB T R B), in a one-sided bandwidth B. The value is RMS, and the peak-to-peak value is roughly six times larger. A 10 kΩ resistor at room temperature gives about 13 nV per √Hz, or 1.3 µV RMS over 10 kHz. It does not depend on the current or the type of resistor.
Open-circuit RMS thermal noise voltage of a resistor at absolute temperature T in a one-sided noise bandwidth B. It is white noise with k_B = 1.380649 × 10⁻²³ J/K.
Calculator
- Noise density, √(4kBTR)1.287×10−8 V/√Hz
- Noise power in the bandwidth, 4kBTB (available)4.142×10−17 W
- Approximate peak-to-peak noise (6 × RMS)7.723×10−6 V
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 10,000 Ω resistor at 300 K in 10,000 Hz has 1.287×10−6 V RMS (1.287 µV) of noise, about 7.723 µV peak-to-peak.
- Cooling to 77 K lowers the noise by only a factor of 1.97, and halving the bandwidth by 1.41, because it depends on the square root.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (V) | Local step | Change in result |
|---|---|---|---|---|---|---|
| B Noise bandwidth | 1 to 1e8 Hz | 1.287e-8 | 1.287e-4 | 1.287e-4 | +10% | +4.88 % |
| R Resistance | 10 to 1e7 Ω | 4.07e-8 | 4.07e-5 | 4.066e-5 | +10% | +4.88 % |
| T Absolute temperature | 4 to 1,000 K | 1.486e-7 | 2.35e-6 | 2.201e-6 | +10% | +4.88 % |
Sweep of Resistance
| Resistance (Ω) | RMS noise voltage (V) |
|---|---|
| 10 | 4.07e-8 |
| 56.23 | 9.652e-8 |
| 316.2 | 2.289e-7 |
| 1,778 | 5.428e-7 |
| 10,000 | 1.287e-6 |
| 56,230 | 3.052e-6 |
| 316,200 | 7.238e-6 |
| 1,778,000 | 1.716e-5 |
| 1e7 | 4.07e-5 |
Sweep one variable
Variables and how each one contributes
RMS noise voltage
What the equation gives you. Everything else feeds it.
Absolute temperature
The absolute temperature, in K. A higher temperature raises the noise voltage with the square root. Raising it by 10 % (to 330) moves the result by +4.88 %, from 1.287×10−6 to 1.35×10−6 V.
Resistance
The resistance, in Ω. A larger resistance raises the noise voltage with the square root. Raising it by 10 % (to 11,000) moves the result by +4.88 %, from 1.287×10−6 to 1.35×10−6 V.
Noise bandwidth
The noise bandwidth, in Hz. A wider bandwidth raises the noise voltage with the square root. Raising it by 10 % (to 11,000) moves the result by +4.88 %, from 1.287×10−6 to 1.35×10−6 V.
Worked example with the values above
- Product: 4 kB T R B = 4 × 1.3806×10−23 × 300 × 10,000 × 10,000 = 1.6568×10−12 V².
- Take the square root.
- vn = 1.2872×10−6 V.
Assumptions and limits
- White noise with a uniform spectral density over the stated one-sided bandwidth.
- Independent, uncorrelated noise sources, so powers (not voltages) add.
Common mistakes
- Using a two-sided bandwidth.
- Using Celsius in place of kelvin.
- Adding the noise of several resistors as voltages instead of as squares.
Statistics and uncertainty
Noise values are RMS statistics: the voltage at any moment is random, and its peak-to-peak value is about six times the RMS for a Gaussian distribution. Measured noise also depends on the bandwidth and on flicker noise at low frequencies.
Where it is used
- Estimating the noise floor, resolution and signal-to-noise ratio of sensor and measurement circuits.
- Choosing amplifiers for a given source resistance.
Maths you need
- Spectral density: power or voltage squared per hertz
- RMS: the root of the mean square
- Boltzmann’s constant: 1.380649 × 10⁻²³ J/K