Op-Amp Integrator Gain

Formula

Description

An op-amp integrator uses a capacitor in the feedback path instead of a resistor. For a sinusoidal input, the output magnitude decreases with frequency at 20 dB/decade (like a first-order low-pass filter), with a gain of 1/(2πfRC). At DC (f = 0), the ideal gain is infinite, causing the output to drift to the supply rail. In practice, a large feedback resistor is added in parallel with the capacitor to limit DC gain and prevent saturation. Integrators are fundamental building blocks in analog computers, PID controllers (the I term), active filters, and waveform generators (triangle from square wave).

Variables

• V_out — Output voltage magnitude (V)
• V_in — Input voltage magnitude (V)
• f — Signal frequency (Hz)
• R — Input resistor (Ω)
• C — Feedback capacitor (F)

Practical Notes

The unity-gain frequency (where |Vout| = |Vin|) occurs at f = 1/(2πRC). Below this frequency, the integrator amplifies; above it, the integrator attenuates. For a triangle wave generator, apply a square wave to the input: the output triangle amplitude is Vsq × T/(4RC) where T is the square wave period. The feedback resistor Rf for DC stability should satisfy Rf >> 1/(2πf_min×C) where f_min is the lowest signal frequency of interest.