Op-Amp Integrator & Differentiator Calculator
This calculator estimates ideal inverting op-amp integrator and differentiator behavior from R, C, input voltage, time interval, and input slew rate.
It reports RC time constant and characteristic frequency using ideal transfer-function terminology. The characteristic frequency is not a practical cutoff frequency for an ideal integrator or ideal differentiator.
Engineering tool
Op-Amp Integrator & Differentiator Calculator
Calculate ideal inverting op-amp integrator output, differentiator output, RC time constant, and characteristic frequency.
Ideal inverting op-amp RC network resistor.
Ideal inverting op-amp RC network capacitor.
Constant input voltage integrated over the selected time.
Duration over which the constant input is integrated.
Output voltage at the start of the integration interval.
Final output voltage
-1 mV
Characteristic frequency is not a practical cutoff frequency for the ideal circuit.
Result console
- RC time constant (τ)
- 1,000ms
- Characteristic frequency (fRC)
- 0.159155Hz
- Transfer function
- H(s) = -1 / (sRC)
- Phase
- +90° phase shift, equivalent to -270°
- Output voltage change (ΔVout)
- -1mV
- Final output voltage
- -1mV
- Integration rate
- -1V/s
- Initial output voltage
- 0V
- Resistor R
- 10kΩ
- Capacitor C
- 100µF
Ideal integrators continuously integrate DC offset, input offset voltage, and input bias current. Practical circuits usually add a feedback resistor to limit DC gain.
Formula reference
Integrator and Differentiator Formula
The calculator separates time-domain equations, ideal transfer functions, and the RC characteristic frequency. fRC is the unity-gain frequency of the ideal RC transfer function, not a practical bandwidth limit.
Integrator: Vout(final) = Vout(initial) − Vin × Δt / (RC)Integrator: H(s) = −1 / (sRC)Integrator: |H(jω)| = 1 / (ωRC), phase = +90°Differentiator: Vout = −RC × ΔVin / ΔtDifferentiator: H(s) = −sRCDifferentiator: |H(jω)| = ωRC, phase = −90°Variable definitions
- R
- Op-amp RC network resistor
- C
- Op-amp RC network capacitor
- τ
- RC time constant
- fRC
- Characteristic frequency, 1 / (2πRC)
- s
- Complex frequency variable
- ω
- Angular frequency in rad/s
Variable Description
- R
- Resistor in the ideal inverting op-amp RC network.
- C
- Capacitor in the ideal inverting op-amp RC network.
- Vin
- Constant input voltage used for the integrator time-domain calculation.
- ΔVin
- Input voltage transition amount used for differentiator slew-rate calculation.
- Δt
- Integration time or transition time, depending on mode.
- Vout(initial)
- Integrator output voltage at the start of the time interval.
- Vout(final)
- Integrator output voltage after the selected integration time.
- τ
- RC time constant, equal to R multiplied by C.
- fRC
- Characteristic frequency or unity-gain frequency of the ideal RC transfer function.
- s
- Complex frequency variable used in Laplace-domain transfer functions.
- ω
- Angular frequency in radians per second, equal to 2πf.
Worked Examples
Example 1: Integrator with Positive DC Input
- R
- 10 kΩ
- C
- 100 nF
- Vin
- 1 V
- Δt
- 1 ms
- Vout initial
- 0 V
τ = 1 ms; fRC ≈ 159.1549 Hz; ΔVout = −1 V × 1 ms / 1 ms = −1 V
Vout final = −1 V
A positive DC input drives the ideal inverting integrator output negative.
Example 2: Integrator with Negative Input
- R
- 100 kΩ
- C
- 1 µF
- Vin
- −2 V
- Δt
- 50 ms
- Vout initial
- 0.5 V
τ = 0.1 s; fRC ≈ 1.59155 Hz; ΔVout = +1 V
Vout final = 1.5 V
A negative input produces a positive output ramp because the ideal topology is inverting.
Example 3: Differentiator with Rising Input
- R
- 10 kΩ
- C
- 10 nF
- ΔVin
- 1 V
- Δt
- 1 ms
τ = 100 µs; Input slew rate = 1000 V/s; Vout = −RC × ΔVin/Δt
Vout = −0.1 V
A rising input produces a negative output pulse in the ideal inverting differentiator.
Example 4: Differentiator with Falling Input
- R
- 47 kΩ
- C
- 100 nF
- ΔVin
- −2 V
- Δt
- 10 ms
τ = 4.7 ms; Input slew rate = −200 V/s; Vout = −4.7 ms × (−200 V/s)
Vout = +0.94 V
A falling input produces a positive output because the differentiator is inverting.
Ideal op-amp integrator
An ideal integrator produces output proportional to the accumulated input over time.
Ideal op-amp differentiator
An ideal differentiator produces output proportional to the input voltage rate of change.
Inverting operation
Both V1 models are inverting, so polarity is opposite the applied integral or derivative term.
RC time constant
τ = RC sets the scaling factor for both ideal time-domain equations.
Characteristic frequency
fRC is the unity-gain frequency of the ideal RC transfer function, not a practical cutoff frequency.
Integrator ramp output
A constant Vin produces a linear output ramp until the real op-amp saturates.
Input slew rate
The differentiator output depends on ΔVin/Δt, so faster transitions create larger output magnitude.
Initial conditions
Integrator output depends on Vout(initial), so initial voltage matters.
Output saturation
Ideal output may exceed real supply rails and output swing.
Op-amp gain-bandwidth product
Real circuits must verify GBW against frequency, gain, noise gain, and stability needs.
Slew rate limitation
Real op-amps may not reproduce fast output ramps or pulses predicted by the ideal model.
Input offset voltage
Integrator circuits can accumulate offset into output drift.
Input bias current
Bias current through the RC network can create drift and error.
Noise amplification
Ideal differentiators amplify high-frequency noise unless bandwidth is limited.
DC drift
Ideal integrators have unlimited DC gain and practical versions need drift control.
Practical circuits
Practical integrators add feedback resistance; practical differentiators add compensation to limit high-frequency gain.
Common Mistakes
Treating an integrator as a normal RC low-pass filter
An ideal op-amp integrator has a different transfer function and no ordinary first-order cutoff.
Calling fRC a practical cutoff frequency
Use characteristic frequency or unity-gain frequency for the ideal RC transfer function.
Ignoring the inverting sign
Positive input can create negative output change in both ideal modes.
Forgetting initial output voltage
Integrator final output depends on where the output starts.
Mixing ms, µs, and ns
Time unit mistakes directly scale the calculated output.
Mixing nF and µF
Capacitance unit mistakes can shift output magnitude by orders of magnitude.
Ignoring output saturation
Ideal calculated output can exceed real op-amp supply rails.
Ignoring integrator drift
Input offset and bias current can integrate into long-term output drift.
Building an ideal differentiator without bandwidth limiting
Uncompensated differentiators can amplify noise and become unstable.
Ignoring GBW and slew rate
Real op-amps must support the required speed, gain, and output movement.
Support reference
FAQ
What is an op-amp integrator?
An op-amp integrator is an inverting circuit whose output changes according to the time integral of the input voltage. A constant input produces a ramping output.
What is an op-amp differentiator?
An op-amp differentiator is an inverting circuit whose output is proportional to the input voltage rate of change.
How do I calculate integrator output for a constant input?
Use Vout(final) = Vout(initial) - Vin × Δt / (RC). The minus sign comes from the inverting op-amp topology.
Why is the integrator output inverted?
The ideal circuit is an inverting op-amp configuration, so a positive constant input drives the output in the negative direction.
What does the RC characteristic frequency mean?
fRC = 1 / (2πRC) is the unity-gain frequency of the ideal RC transfer function. It should not be treated as a practical cutoff frequency for the ideal integrator or differentiator.
Why does a practical integrator need a feedback resistor?
A feedback resistor limits DC and very-low-frequency gain so input offset voltage, bias current, and leakage do not integrate indefinitely into saturation.
Why does an ideal differentiator amplify noise?
The differentiator transfer magnitude increases with frequency, so high-frequency noise can be amplified unless the practical circuit limits bandwidth.
Can the calculated output exceed the op-amp supply rails?
Yes. The calculator shows the ideal result and warns when the magnitude is large. Real op-amps are limited by supply rails, output swing, slew rate, and device behavior.
Related Engineering Guides
Dedicated op-amp guides are planned for this topic cluster. These guide topics are reserved for future publication and are shown without links until the pages exist.
Planned Engineering Guide
Op-Amp Integrator Basics
Planned guide covering ideal integration, ramp output, input polarity, and practical feedback leakage paths.
Planned Engineering Guide
Op-Amp Differentiator Basics
Planned guide explaining input slew rate, differentiator polarity, noise gain, and practical bandwidth limits.
Planned Engineering Guide
Practical Integrator Design
Planned guide covering feedback resistors, drift control, saturation, reset behavior, and real op-amp limits.
Planned Engineering Guide
Practical Differentiator Design
Planned guide covering compensation networks, high-frequency gain limiting, noise, and stability.
Planned Engineering Guide
Understanding Op-Amp Slew Rate
Planned guide covering slew-rate limits, large-signal behavior, and waveform distortion.
Planned Engineering Guide
Op-Amp Bandwidth and Stability
Planned guide covering gain-bandwidth product, phase margin, compensation, and stable active circuits.
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AvailableSlew Rate Calculator
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Engineering Disclaimer
This calculator uses ideal inverting op-amp integrator and differentiator equations for estimation and education. Real designs must verify supply rails, output swing, gain-bandwidth product, slew rate, input offset voltage, input bias current, noise, stability, practical compensation components, and device-specific datasheet limits before hardware release.