Op-Amp Integrator and Differentiator Calculator

Find the corner frequency and gain magnitude of an op-amp integrator or differentiator circuit from its resistor and capacitor values.

🔌 Op-Amp Integrator and Differentiator Calculator
Ω
nF
Evaluation frequency (f)1000 Hz
Hz
10 Hz20 kHz
Ω
nF
Evaluation frequency (f)1000 Hz
Hz
10 Hz20 kHz
Gain magnitude |Av(f)|
—
Corner frequency (f0)
—
Step-by-step working
Gain magnitude |Av(f)|
—
Corner frequency (f0)
—
Step-by-step working

🔌 What is the Op-Amp Integrator and Differentiator Calculator?

The Op-Amp Integrator and Differentiator Calculator finds the corner frequency f0 and gain magnitude of the two classic reactive op-amp circuits, the integrator and the differentiator, from their resistor and capacitor values, at any frequency you choose to evaluate.

Analog designers use these circuits to build waveform generators (a triangle wave from an integrated square wave), analog PID controllers, edge detectors, and active filter building blocks. Understanding how gain varies with frequency for each topology is essential to keeping both circuits stable and predictable in real designs.

A common point of confusion is that both circuits share the exact same corner frequency formula, f0 = 1/(2 pi R C), only the direction of their frequency response differs, the integrator's gain falls with frequency while the differentiator's gain rises with frequency, meeting at a gain of 1 at f0.

This calculator computes the corner frequency and the gain magnitude at your chosen evaluation frequency for whichever topology you select, and plots the full gain-versus-frequency curve so you can see where your operating point sits relative to f0.

📐 Formula

f0 = 1 / (2πRC)      |Av(f)|int = 1 / (2πfRC)      |Av(f)|diff = 2πfRC
f0 = corner frequency where gain magnitude = 1 (Hz)
R = resistance (Ω), C = capacitance (F), f = evaluation frequency (Hz)
Integrator example: R = 10,000 Ω, C = 10 nF, f = 1,000 Hz → f0 ≈ 1591.55 Hz, |Av| ≈ 1.5915.
Differentiator example: R = 10,000 Ω, C = 10 nF, f = 1,000 Hz → f0 ≈ 1591.55 Hz, |Av| ≈ 0.6283.

📖 How to Use This Calculator

Steps

1
Choose Integrator or Differentiator mode. Select the circuit topology you are analyzing.
2
Enter R, C, and the evaluation frequency. Type the resistance in ohms, the capacitance in nanofarads, and the frequency f at which to evaluate the gain.
3
Read the corner frequency and gain magnitude. The calculator reports f0 and the gain magnitude |Av(f)| at your chosen frequency.

💡 Example Calculations

Example 1 — Integrator at 1 kHz

Integrator mode: R = 10,000 Ω, C = 10 nF, f = 1,000 Hz

1
f0 = 1 / (2π × 10,000 × 10×10⁻⁹) = 1591.55 Hz
2
|Av| = 1 / (2π × 1,000 × 10,000 × 10×10⁻⁹) = 1.5915
f0 = 1591.55 Hz, |Av| = 1.5915
Try this example →

Example 2 — Integrator with Larger Capacitor

Integrator mode: R = 1,000 Ω, C = 100 nF, f = 500 Hz

1
f0 = 1 / (2π × 1,000 × 100×10⁻⁹) = 1591.55 Hz
2
|Av| = 1 / (2π × 500 × 1,000 × 100×10⁻⁹) = 3.1831
f0 = 1591.55 Hz, |Av| = 3.1831
Try this example →

Example 3 — Differentiator at 1 kHz

Differentiator mode: R = 10,000 Ω, C = 10 nF, f = 1,000 Hz

1
f0 = 1 / (2π × 10,000 × 10×10⁻⁹) = 1591.55 Hz
2
|Av| = 2π × 1,000 × 10,000 × 10×10⁻⁹ = 0.6283
f0 = 1591.55 Hz, |Av| = 0.6283
Try this example →

Example 4 — Differentiator Below Its Corner Frequency

Differentiator mode: R = 1,000 Ω, C = 1,000 nF, f = 200 Hz

1
f0 = 1 / (2π × 1,000 × 1,000×10⁻⁹) = 159.15 Hz
2
|Av| = 2π × 200 × 1,000 × 1,000×10⁻⁹ = 1.2566
f0 = 159.15 Hz, |Av| = 1.2566
Try this example →

❓ Frequently Asked Questions

What is an op-amp integrator?+
An op-amp integrator is a circuit whose output voltage is proportional to the time integral of its input voltage, built from an op-amp with a feedback capacitor C and input resistor R. Its gain magnitude falls as frequency rises: |Av(f)| = 1/(2 pi f R C).
What is an op-amp differentiator?+
An op-amp differentiator is a circuit whose output voltage is proportional to the rate of change (time derivative) of its input voltage, built with a feedback resistor R and input capacitor C. Its gain magnitude rises as frequency rises: |Av(f)| = 2 pi f R C.
What is the corner frequency f0 in these circuits?+
The corner frequency f0 = 1/(2 pi R C) is the frequency at which the gain magnitude of either circuit equals exactly 1 (0 dB). Below f0 the integrator has gain above 1, above f0 the differentiator has gain above 1, this shared reference point makes it easy to compare the two.
Why does the integrator's gain fall with frequency?+
A capacitor's impedance falls as frequency rises (Xc = 1/(2 pi f C)), and since the integrator's feedback impedance is that capacitor, the closed-loop gain magnitude 1/(2 pi f R C) falls proportionally, this is the defining -20 dB/decade rolloff of a true integrator.
Why does the differentiator's gain rise with frequency?+
In the differentiator, the capacitor is at the input rather than in the feedback path, so as frequency rises and the capacitor's impedance falls, more current (and thus gain) is delivered to the feedback resistor, giving the +20 dB/decade rise in gain 2 pi f R C.
Why do real differentiator circuits need extra stabilization?+
An ideal differentiator's gain grows without bound at high frequency, which amplifies high-frequency noise and can make the circuit unstable. Practical designs add a small resistor in series with the input capacitor to cap the high-frequency gain and improve stability.
Why do real integrator circuits need a bleed resistor?+
An ideal integrator has infinite gain at DC (0 Hz), so any small input offset voltage integrates over time and eventually saturates the output. Practical designs add a large resistor in parallel with the feedback capacitor to bleed off this DC buildup while barely affecting the intended AC integration.
Can I use the same R and C values for both modes?+
Yes, the corner frequency formula is identical for both topologies, f0 = 1/(2 pi R C). What differs is the circuit topology (which element is in the feedback path) and therefore whether the gain rises or falls away from f0.
What units does this calculator use?+
Resistance is entered in ohms, capacitance in nanofarads, and frequency in Hertz. The corner frequency f0 is reported in Hertz, and gain magnitude |Av| is dimensionless (a ratio).
Does this calculator show the output waveform shape?+
No, this calculator reports the frequency-domain gain magnitude and corner frequency, not a time-domain waveform. For a sine-wave input, the output is also a sine wave (scaled and phase-shifted by 90 degrees) at the gain magnitude shown here.
Tags: Op Amp Integrator Calculator Op Amp Differentiator Calculator Integrator Circuit Calculator Differentiator Circuit Calculator Operational Amplifier Calculator Calculator Formula Free Online

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What is an op-amp integrator?

An op-amp integrator is a circuit whose output voltage is proportional to the time integral of its input voltage, built from an op-amp with a feedback capacitor C and input resistor R. Its gain magnitude falls as frequency rises: |Av(f)| = 1/(2 pi f R C).

What is an op-amp differentiator?

An op-amp differentiator is a circuit whose output voltage is proportional to the rate of change (time derivative) of its input voltage, built with a feedback resistor R and input capacitor C. Its gain magnitude rises as frequency rises: |Av(f)| = 2 pi f R C.

What is the corner frequency f0 in these circuits?

The corner frequency f0 = 1/(2 pi R C) is the frequency at which the gain magnitude of either circuit equals exactly 1 (0 dB). Below f0 the integrator has gain above 1, above f0 the differentiator has gain above 1, this shared reference point makes it easy to compare the two.

Why does the integrator's gain fall with frequency?

A capacitor's impedance falls as frequency rises (Xc = 1/(2 pi f C)), and since the integrator's feedback impedance is that capacitor, the closed-loop gain magnitude 1/(2 pi f R C) falls proportionally, this is the defining -20 dB/decade rolloff of a true integrator.

Why does the differentiator's gain rise with frequency?

In the differentiator, the capacitor is at the input rather than in the feedback path, so as frequency rises and the capacitor's impedance falls, more current (and thus gain) is delivered to the feedback resistor, giving the +20 dB/decade rise in gain 2 pi f R C.

Why do real differentiator circuits need extra stabilization?

An ideal differentiator's gain grows without bound at high frequency, which amplifies high-frequency noise and can make the circuit unstable. Practical designs add a small resistor in series with the input capacitor to cap the high-frequency gain and improve stability.

Why do real integrator circuits need a bleed resistor?

An ideal integrator has infinite gain at DC (0 Hz), so any small input offset voltage integrates over time and eventually saturates the output. Practical designs add a large resistor in parallel with the feedback capacitor to bleed off this DC buildup while barely affecting the intended AC integration.

Can I use the same R and C values for both modes?

Yes, the corner frequency formula is identical for both topologies, f0 = 1/(2 pi R C). What differs is the circuit topology (which element is in the feedback path) and therefore whether the gain rises or falls away from f0.

What units does this calculator use?

Resistance is entered in ohms, capacitance in nanofarads, and frequency in Hertz. The corner frequency f0 is reported in Hertz, and gain magnitude |Av| is dimensionless (a ratio).

Does this calculator show the output waveform shape?

No, this calculator reports the frequency-domain gain magnitude and corner frequency, not a time-domain waveform. For a sine-wave input, the output is also a sine wave (scaled and phase-shifted by 90 degrees) at the gain magnitude shown here.

📌 Quick Tips

💡Both the integrator and differentiator share the same corner frequency formula f0 = 1/(2*pi*R*C), only the direction of the gain response differs, integrator gain falls with frequency, differentiator gain rises with frequency.
💡A practical differentiator circuit usually needs a small series resistor added at the input to limit high-frequency noise gain, an ideal differentiator amplifies noise without bound as frequency increases.
💡A practical integrator circuit usually needs a large feedback resistor in parallel with the capacitor to prevent DC offset from saturating the output over time, an ideal integrator has no DC gain limit.
💡At the corner frequency f0, both circuits have a gain magnitude of exactly 1 (0 dB), this is a useful reference point when relating either circuit to standard filter theory.
💡The gain formulas assume an ideal op-amp and single-pole RC response, real circuits roll off further at very high frequency due to the op-amp's own finite gain-bandwidth product.