4.1. The Differentiator

A differentiator is a circuit whose output is the first derivative of its input. For a beginner, this means the output voltage is proportional to the rate of change of the input voltage. If the input voltage is changing quickly, the output voltage will be large; if the input is steady, the output will be zero.

The formula for the output voltage is: V₀ = -RC * (dVᵢ/dt)

Key Insight: By choosing the values of the resistor (R) and capacitor (C) such that their product RC = 1, the circuit’s output becomes the direct, though inverted, differentiation of the input: V₀ = -(dVᵢ/dt).

4.2. The Integrator

An integrator is a circuit that produces an output that is the integration of the input signal. In simple terms, the output represents the accumulation of the input voltage over time.

The formula for the output voltage is: V₀ = -(1/RC) * ∫Vᵢ dt

Key Insight: Just as with the differentiator, if we choose the values of R and C such that their product RC = 1, the output becomes the direct, but inverted, integration of the input: V₀ = -∫Vᵢ dt.

These calculus circuits demonstrate the incredible signal-processing power of the simple op-amp.