Library / Circuits, electronics and noise / ELN-48
Circuits, electronics and noise
Input-referred amplifier and source noise density
Total input-referred noise voltage density of an amplifier and its real source resistance Rs, from the voltage noise, the current noise flowing through Rs and the resistor’s own thermal noise.
- Ref
- ELN-48
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- et [V/√Hz]
Noise sources that are not correlated add as powers. The total input-referred noise density is et = √(en² + (in Rs)² + 4 kB T Rs): the amplifier’s voltage noise, its current noise flowing in the source resistance, and the thermal noise of Rs itself. For a low Rs the voltage noise dominates, for a high Rs the current noise, and the best amplifier depends on the source.
Total input-referred noise voltage density of an amplifier and its real source resistance Rs, from the voltage noise, the current noise flowing through Rs and the resistor’s own thermal noise. The three are uncorrelated at one frequency.
Calculator
- Amplifier voltage-noise term, en5×10−9 V/√Hz
- Current-noise term, in Rs1×10−9 V/√Hz
- Thermal term of Rs, √(4kBTRs)4.07×10−9 V/√Hz
- Source resistance where the two amplifier terms are equal, en/in5,000 Ω
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.
- The total input noise density is 6.524×10−9 V/√Hz; the three terms are 5×10−9, 1×10−9 and 4.07×10−9 V/√Hz.
- The two amplifier terms are equal at a source resistance of 5,000 Ω, which is the best match between this amplifier and the source.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (V/√Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Rs Source resistance | 10 to 1,000,000 Ω | 5.017e-9 | 1.008e-6 | 1.003e-6 | +10% | +2.17 % |
| en Voltage noise density | 1e-9 to 1e-7 V/√Hz | 4.309e-9 | 1.001e-7 | 9.578e-8 | +10% | +5.99 % |
| in Current noise density | 1e-14 to 1e-11 A/√Hz | 6.447e-9 | 1.19e-8 | 5.451e-9 | +10% | +0.246 % |
| T Absolute temperature | 4 to 1,000 K | 5.121e-9 | 9.013e-9 | 3.892e-9 | +10% | +1.93 % |
Sweep of Source resistance
| Source resistance (Ω) | Total input noise density (V/√Hz) |
|---|---|
| 10 | 5.017e-9 |
| 42.17 | 5.07e-9 |
| 177.8 | 5.289e-9 |
| 749.9 | 6.163e-9 |
| 3,162 | 9.348e-9 |
| 13,340 | 2.059e-8 |
| 56,230 | 6.418e-8 |
| 237,100 | 2.453e-7 |
| 1,000,000 | 1.008e-6 |
Sweep one variable
Variables and how each one contributes
Total input noise density
What the equation gives you. Everything else feeds it.
Voltage noise density
The amplifier’s voltage noise density, in V/√Hz. Adds in power, so it matters most for a small source resistance. Raising it by 10 % (to 5.5×10−9) moves the result by +5.99 %, from 6.524×10−9 to 6.915×10−9 V/√Hz.
Current noise density
The amplifier’s current noise density, in A/√Hz. Adds in power through Rs, so it matters most for a large source resistance. Raising it by 10 % (to 1.1×10−12) moves the result by +0.246 %, from 6.524×10−9 to 6.54×10−9 V/√Hz.
Source resistance
The source resistance, in Ω. Raises both the current-noise and thermal terms, so the total rises. Raising it by 10 % (to 1,100) moves the result by +2.17 %, from 6.524×10−9 to 6.666×10−9 V/√Hz.
Absolute temperature
The absolute temperature, in K. Raises the thermal noise of the source resistance with the square root. Raising it by 10 % (to 330) moves the result by +1.93 %, from 6.524×10−9 to 6.65×10−9 V/√Hz.
Worked example with the values above
- Voltage-noise term squared: en² = 2.5×10−17.
- Current-noise term squared: (in Rs)² = 1×10−18; thermal term squared: 1.6568×10−17.
- et = √(4.2568×10−17) = 6.5244×10−9 V/√Hz.
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