Library / Circuits, electronics and noise / ELN-47
Circuits, electronics and noise
Shot noise current of a junction
RMS shot-noise current from independent charge carriers crossing a junction carrying average current I, in a one-sided bandwidth B, with q = 1.602176634 × 10⁻¹⁹ C.
- Ref
- ELN-47
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- in [A]
Current is made of discrete charges that cross a junction at random times, giving shot noise with an RMS current of in = √(2 q |I| B) in a one-sided bandwidth B. It grows only with the square root of the average current, so the signal-to-noise ratio improves with more current. For 1 mA it is about 18 pA per √Hz, or 1.8 nA RMS in a 10 kHz bandwidth.
RMS shot-noise current from independent charge carriers crossing a junction carrying average current I, in a one-sided bandwidth B, with q = 1.602176634 × 10⁻¹⁹ C. Excess or suppressed noise needs a device model.
Calculator
- Noise density, √(2q|I|)1.79×10−11 A/√Hz
- Signal-to-noise ratio of the DC current, I/in558,600
- Noise as a fraction of the current1.79×10−6
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.
- 0.001 A in 10,000 Hz carries 1.79×10−9 A RMS of shot noise, a signal-to-noise ratio of 558,600.
- Four times the current only doubles the noise, so the ratio improves by a factor of 2; this is why a bright signal is easier to measure.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (A) | Local step | Change in result |
|---|---|---|---|---|---|---|
| B Noise bandwidth | 1 to 1e8 Hz | 1.79e-11 | 1.79e-7 | 1.79e-7 | +10% | +4.88 % |
| I Average junction current | 1e-9 to 1 A | 1.79e-12 | 5.661e-8 | 5.661e-8 | +10% | +4.88 % |
Sweep of Average junction current
| Average junction current (A) | RMS noise current (A) |
|---|---|
| 1e-9 | 1.79e-12 |
| 1.334e-8 | 6.537e-12 |
| 1.778e-7 | 2.387e-11 |
| 2.371e-6 | 8.717e-11 |
| 3.162e-5 | 3.183e-10 |
| 4.217e-4 | 1.162e-9 |
| 0.005623 | 4.245e-9 |
| 0.07499 | 1.55e-8 |
| 1 | 5.661e-8 |
Sweep one variable
Variables and how each one contributes
RMS noise current
What the equation gives you. Everything else feeds it.
Average junction current
The average current through the junction, in A. More current raises the noise current with the square root. Raising it by 10 % (to 0.0011) moves the result by +4.88 %, from 1.79×10−9 to 1.877×10−9 A.
Noise bandwidth
The noise bandwidth, in Hz. A wider bandwidth raises the noise current with the square root. Raising it by 10 % (to 11,000) moves the result by +4.88 %, from 1.79×10−9 to 1.877×10−9 A.
Worked example with the values above
- Product: 2 q I B = 2 × 1.6022×10−19 × 0.001 × 10,000 = 3.2044×10−18 A².
- Take the square root.
- in = 1.7901×10−9 A.
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
- Adding noise voltages directly instead of adding their squares.
- Using a two-sided bandwidth where a one-sided one is needed.
- Forgetting that real filters pass more noise than their −3 dB bandwidth: use the noise bandwidth.
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