Op-Amp Differentiator Gain

Formula

Description

An op-amp differentiator uses a capacitor at the input and a resistor in the feedback path. The output magnitude increases linearly with frequency at 20 dB/decade (like a first-order high-pass filter). At high frequencies, the gain can become very large, amplifying noise and potentially causing oscillation. Therefore, practical differentiators always include a small series resistor with the input capacitor or a small feedback capacitor to limit the high-frequency gain. Differentiators are used in rate-of-change detection, edge detection, frequency demodulation, and the D term of PID controllers.

Variables

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

Practical Notes

The unity-gain frequency occurs at f = 1/(2πRC), the same as the integrator but with gain increasing instead of decreasing. To stabilize the circuit, add a series resistor Rs with the input capacitor: Rs ≈ R/(10-100) limits the maximum gain to R/Rs. Alternatively, add a feedback capacitor Cf in parallel with R, creating a rolloff at f = 1/(2πRCf). The noise gain of a differentiator increases without bound, making it inherently noisier than an integrator.