Library / Circuits, electronics and noise / ELN-31
Circuits, electronics and noise
Closed-loop gain with finite open-loop gain
Closed-loop gain of a negative-feedback amplifier whose open-loop gain A₀ is finite and whose feedback fraction is β.
- Ref
- ELN-31
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- Acl [V/V]
With open-loop gain A₀ and a feedback fraction β (the fraction of the output fed back), the closed-loop gain is A₀/(1 + A₀β). For a large loop gain A₀β the gain tends to 1/β, set by the resistors. The error is about 1/(A₀β) as a fraction, so a gain of 10 (β = 0.1) with A₀ = 1000 is 1 % low. Raising the loop gain improves the accuracy.
Closed-loop gain of a negative-feedback amplifier whose open-loop gain A₀ is finite and whose feedback fraction is β. When A₀β is large it approaches 1/β.
Calculator
- Ideal closed-loop gain, 1/β10 V/V
- Loop gain, A₀β100
- Gain error compared with 1/β0.9901 %
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 ideal gain is 10, but with a loop gain of 100 the actual gain is 9.901, 0.99 % lower.
- For 0.1 % accuracy the loop gain must be at least 1000, which is why high-gain stages need op amps with a gain well above A₀ = 1000.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (V/V) | Local step | Change in result |
|---|---|---|---|---|---|---|
| β Feedback fraction | 0.001 to 1 | 0.999 | 500 | 499 | +10% | −9.01 % |
| A0 Open-loop gain at low frequency | 10 to 1,000,000 V/V | 5 | 10 | 5 | +10% | +0.0901 % |
Sweep of Open-loop gain at low frequency
| Open-loop gain at low frequency (V/V) | Closed-loop gain (V/V) |
|---|---|
| 10 | 5 |
| 42.17 | 8.083 |
| 177.8 | 9.468 |
| 749.9 | 9.868 |
| 3,162 | 9.968 |
| 13,340 | 9.993 |
| 56,230 | 9.998 |
| 237,100 | 10 |
| 1,000,000 | 10 |
Sweep one variable
Variables and how each one contributes
Closed-loop gain
What the equation gives you. Everything else feeds it.
Open-loop gain at low frequency
The open-loop gain of the amplifier at the frequency of interest, no units. A larger A₀ brings the closed-loop gain closer to 1/β. Raising it by 10 % (to 1,100) moves the result by +0.0901 %, from 9.901 to 9.91 V/V.
Feedback fraction
The feedback fraction β, between 0 and 1. A larger β lowers the closed-loop gain, roughly in inverse proportion. Raising it by 10 % (to 0.11) moves the result by −9.01 %, from 9.901 to 9.009 V/V.
Worked example with the values above
- Loop gain: A₀β = 1,000 × 0.1 = 100.
- Denominator: 1 + A₀β = 101.
- Acl = 1,000 / 101 = 9.901.
Assumptions and limits
- A voltage-feedback op amp with a single dominant pole and a constant feedback network.
- Small-signal formulas used below the slew limit and with the output in its linear range.
Common mistakes
- Using the closed-loop gain where the noise gain is needed: they differ, for instance, in an inverting stage.
- Entering a peak-to-peak voltage in the slew-rate formula, which needs the peak.
- Forgetting that input offset adds to the output through the noise gain.
Statistics and uncertainty
Open-loop gain, offset voltage, bias current and gain-bandwidth product all vary a lot between parts and with temperature, and data sheets give typical and maximum values. Use the worst case for error budgets.
Where it is used
- Choosing an op amp for gain, bandwidth, output swing and DC accuracy.
- Estimating the errors of a precision amplifier circuit.
Maths you need
- Noise gain: 1 + Rf/Rg for a non-inverting stage; the gain seen by the amplifier’s own errors
- Gain-bandwidth product: gain times bandwidth is constant
- Slew rate: the largest rate of change of the output