Op-Amp Series – Part 6: The Integrator Amplifier

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The Integrator Amplifier

The integrator amplifier is an operational amplifier circuit that performs a mathematical integration of the input signal. 

Unlike the previous amplifier circuits, the output of an integrator doesn’t simply scale or invert the input. Instead, it accumulates the input over time, producing an output voltage proportional to the area under the input waveform.

Integrator circuits are widely used in:

  • Signal processing
  • Waveform generation
  • Analog computing
  • Control systems
  • Audio and synth circuits

We’ll start with the theory and maths, then build a real, working integrator on the bench using an LM358.

What Is an Integrator Amplifier?

An integrator is a modified inverting op-amp circuit where the feedback resistor is replaced with a capacitor. This can be seen in the circuit below.

Integrator Amplifier Circuit

The basic circuit consists of the following: -
  • Input resistor: R
  • Feedback capacitor: C
  • Non-inverting input tied to ground
  • Output fed back through the capacitor

The capacitor forces the op-amp to continuously adjust its output to maintain the virtual ground at the inverting input.

How the Integrator Works

Capacitors resist changes in voltage by charging and discharging over time.

  • A constant input voltage causes the capacitor to charge linearly
  • A square wave becomes a triangle wave
  • A DC input causes the output to ramp up or down continuously

The op-amp drives the output just enough to keep the inverting input at 0 V, which means the capacitor must absorb the input current.

The Maths Behind the Integrator

Let’s derive the integrator equation step by step.

Step 1: Virtual Ground Assumption

For an ideal op-amp:

V- = V+ = 0 V

No current flows into the op-amp input.

Step 2: Input Current

The input current through resistor R is:

Iin = Vin / R

This same current must flow into the capacitor.

Step 3: Capacitor Current Equation

For a capacitor:

Ic = C × (dV / dt)

Here, the voltage across the capacitor is the output voltage Vout.

So:

Ic = C × (dVout / dt)

Step 4: Equate the Currents

Vin / R = C × (dVout / dt)

Rearranging:

dVout / dt = Vin / (R × C)

Step 5: Integrate Both Sides

Vout = - (1 / (R × C)) × ∫ Vin dt

The negative sign comes from the inverting configuration.

Final Integrator Equation

Vout(t) = - (1 / (R × C)) × integral of Vin(t) over time

This is the defining equation of an op-amp integrator.

What the Integrator Does to Signals

The table provided states the output for an integrator op-amp for a given input signal.

This makes integrators extremely useful as low-pass filters and waveform shapers.

Practical Integrator Circuit

Now let’s build a real integrator and see it working.

Final circuit working

Components

  • LM358 op-amp (DIP package)
  • 1 × 10 kΩ resistor
  • 1 × 10 µF electrolytic capacitor
  • Breadboard
  • Jumper wires
  • Bench power supply or 5 V source
  • Oscilloscope
  • Signal generator

The integration rate depends on RC:

RC = 10,000 × 10 µF
RC = 0.1 seconds

This gives a slow, visible ramp that’s easy to measure with an oscilloscope.

Step 1: Power the Op-Amp

The LM358 pinout (for one amplifier):

  • Pin 8 → +5 V
  • Pin 4 → Ground (0 V)

Important:

Double-check polarity before powering up. Incorrect power connections can damage the chip.

Step 2: Ground the Non-Inverting Input

To create the virtual ground required for integration connect pin 3 (non-inverting input) directly to ground This fixes the reference point at 0 V.

Step 3: Add the Input Resistor

Now build the input path:

  • Connect one end of the 10 kΩ resistor to your input signal
  • Connect the other end of the resistor to pin 2 (inverting input)

This resistor sets the input current into the integrator.

Step 4: Add the Feedback Capacitor

This is the key part that makes the circuit an integrator.

  • Connect the 10 µF capacitor between:
    • Pin 1 (output)
    • Pin 2 (inverting input)

Electrolytic capacitor polarity matters:

  • Positive lead → pin 2
  • Negative lead → pin 1

This completes the integrator feedback loop.



Testing the Integrator

First power the bench power supply to 6V
Set the function generator to output a square wave of 1KHz at an amplitude of 2 Volts
Connect the oscilloscope, bench power supply and function generator to the op-amp appropriate pins.
Result:

  • Output becomes a triangle wave

  • The slope depends on input amplitude and RC value


YouTube Video


What’s Next?

In the next part of the series, we’ll look at the Differentiator Amplifier — the mathematical opposite of the integrator — and see how it reacts to fast-changing signals.

👉 Op-Amp Series – Part 7: The Differentiator Amplifier

Hope you enjoyed this post.
Thanks for reading
Matty

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