Integrators, Differentiators, and Filters
Integrator
VOUT(t) = -(1/RC)∫VIN(t)dt
A constant input creates a ramp. A square wave can create an approximately triangular output. A practical integrator adds a resistor across the feedback capacitor so tiny DC errors do not drive the output permanently into saturation.
Differentiator
VOUT = -RC dVIN/dt
Differentiation emphasizes rapid changes, which means it also emphasizes high-frequency noise. Practical circuits deliberately limit their high-frequency gain.
Active low-pass and high-pass
fC = 1/(2πRC)
Adding an op-amp provides buffering, gain, and higher-order responses without needing inductors.
Second-order filters
Sallen-Key and multiple-feedback arrangements can create low-pass, high-pass, band-pass, and notch responses. Q controls peaking and damping.
| Butterworth | Flat magnitude response. |
| Bessel | Good phase and step response. |
| Chebyshev | Sharper transition for a given order, with ripple. |
The op-amp must have enough gain-bandwidth and slew rate for the intended filter. A correct RC calculation alone does not guarantee the real response.