How it works
For the ideal inverting integrator, Vout = −(1/RC)∫Vin dt. With R = 10 kΩ and C = 10 nF, RC = 100 μs. A 1 V DC input ramps the output at −100 V/s; in real circuits, input offset voltage and bias current cause the output to saturate unless a large feedback resistor (typically 10·R or more) is placed across C to reset the DC operating point. The ideal differentiator gives Vout = −RC·(dVin/dt). Practical differentiator adds a series resistor R1 (typically R1 = R/10) and sometimes a feedback capacitor CF to limit gain at high frequencies above f = 1/(2πR1C).
Key points to remember
The RC time constant is the single most important design parameter: too small and the integrator output is tiny, too large and it saturates. For the integrator, the 3 dB frequency is f = 1/(2πRC) — below this frequency it integrates faithfully, above it the gain is less than −20 dB/decade. Practical integrators used in analog PID controllers use RC values in the 10 kΩ–100 kΩ, 100 nF–10 μF range. Differentiators are inherently noisy because gain increases at 20 dB/decade with frequency — the practical fix, adding R1 in series with C, creates a high-pass filter with a gain ceiling of R/R1. DC offsets cause integrator drift — the most common fault question in lab exams.
Exam tip
The examiner always asks you to draw both the ideal and practical differentiator circuits side by side and explain why the practical version adds R1 — state that it limits high-frequency gain and prevents noise amplification, then give the limiting frequency formula.