Op-Amp Integrator Calculator (Ramp Rate & Unity-Gain Frequency)
By Santaji GadepatilinUpdated November 6, 2026
An op-amp integrator charges a feedback capacitor with a current set by the input resistor, so a constant input produces a linear ramp and a sine input is attenuated by 20 dB per decade. This op-amp integrator calculator gives the ramp rate, the output after a chosen time and the unity-gain frequency — see the op-amp integrator and differentiator explained.
How to use the op-amp integrator calculator
Enter the input voltage, the input resistor and the feedback capacitor. Add an integration time to see where the ramp has reached, and an initial output voltage if the capacitor is not discharged. The signal frequency gives the AC gain for sine-wave inputs.
Op-amp integrator formula
Vout(t) = V₀ − (1/RC) ∫ Vin dt · constant input: Vout = V₀ − Vin t / RC
|G| = 1 / (2π f R C) · f₀ = 1 / (2π R C)
Worked example
1 V input, R = 10 kΩ, C = 100 nF (RC = 1 ms):
- Ramp rate = −1 ÷ 0.001 = -1,000 V/s
- After 1 ms the output has moved to -1.00 V
- Unity-gain frequency f₀ = 159.15 Hz; at 1 kHz the gain is 0.1592
Gain vs frequency (R = 10 kΩ, C = 100 nF)
| Frequency | Gain | Gain (dB) | Phase |
|---|---|---|---|
| 10.0 Hz | 15.92 | 24.0 dB | 90° lead (inverted: −90° + 180°) |
| 100.0 Hz | 1.592 | 4.0 dB | 90° lead (inverted: −90° + 180°) |
| 159.2 Hz | 1 | 0.0 dB | 90° lead (inverted: −90° + 180°) |
| 1,000.0 Hz | 0.1592 | -16.0 dB | 90° lead (inverted: −90° + 180°) |
| 10,000.0 Hz | 0.01592 | -36.0 dB | 90° lead (inverted: −90° + 180°) |
Design notes
- The output can only ramp until it reaches the op-amp’s output swing limit; reset the capacitor with a switch for repeating ramps.
- Use a low-leakage film (C0G, polypropylene) capacitor — electrolytics leak and drift.
- For a square-wave input the integrator produces a triangle wave of peak-to-peak amplitude Vin T/(2RC) for a symmetric ±Vin square of period T.
Frequently asked questions
What is the output of an op-amp integrator?
Vout(t) = V₀ − (1/RC)∫Vin dt. For a constant input that is a straight ramp of slope −Vin/(RC) volts per second.
What is the unity-gain frequency of an integrator?
f₀ = 1/(2πRC). Below f₀ the gain is greater than one, above it less, falling at 20 dB per decade.
Why does a practical integrator need a resistor across the capacitor?
Without it the DC gain is the op-amp's open-loop gain, so input offset voltage and bias current slowly drive the output into the rail. A resistor R_f in parallel with C limits DC gain to −R_f/R, at the cost of integrating only above 1/(2πR_fC).
Where are integrators used?
In the I term of analogue PID controllers, triangle-wave and ramp generators, dual-slope ADCs, active filters and charge amplifiers.