CR · Operational amplifier

The Op-Amp Differentiator

Swap the places of just two components in the integrator and a circuit that accumulated becomes one that measures the rate of change. Learn how the differentiator — with a capacitor in place of the input resistor — outputs the slope of its input.

A sloping input makes a flat output

Slide the slope of the input ramp. The input rises at a slant, but the output is flat — its level set only by how fast the input climbs. A steeper input drives the output further down. Set the slope so the output sits on the target of −4 V.

Input slope dVin/dtdVin/dt = +1.5 V/s

Output level and input slope

Vout = -1.5 V  ·  slope = +1.5

Far from −4

Take the input through a capacitor

The integrator made a current Vin/R through the input resistor and poured it into the feedback capacitor. The differentiator swaps the two parts: a capacitor at the input, a resistor in the feedback. A capacitor passes current only when the input changes. That current is proportional to the rate of change, i = C·dVin/dt, and the virtual ground sends all of it through the feedback resistor to set the output.

The output is proportional to the rate of change

The current through the feedback resistor is C·dVin/dt, so the output is that current times R: Vout = −RC·dVin/dt. A constant input has zero rate of change, so the output is zero; the faster the input changes, the larger the output. Feed in a ramp of slope m and the rate of change stays m throughout, so the output is flat at −RC·m. The minus sign is inherited from the inverting amplifier.

Differentiation is reading the slope

In general the output is the time derivative of the input, Vout = −RC·dVin/dt. Differentiation reads the slope of the input curve at every instant. So a triangle wave gives an output that flips sign between the rising and falling parts — a square wave. In the s-domain the input capacitor’s 1/sC produces the s of differentiation, making the gain −sRC, and the integrator is the mirror image with R and C swapped.

ObserveVout = −RC dVin/dt

The output is the derivative of the input.

Chooseinput = ?

A capacitor in the input slot.

Fill inVout = ?

The output is proportional to the rate.

On your owngain = ?

Differentiation is s in the s-domain.

Back to the first screen

The steeper the input ramp, the further down the output sat, and it landed on −4 V when the input slope was +4 (RC = 1). A sloping input became a flat output because the capacitor passes a current proportional to the input’s rate of change, and the ramp’s slope held steady throughout. The integrator that accumulated became, by swapping two components, a differentiator — letting the circuit read the slope on your behalf.

A differentiator replaces the inverting amplifier’s input resistor with a capacitor. The capacitor passes a current i = C·dVin/dt only when the input changes, and that current through the feedback resistor makes the output the derivative of the input, Vout = −RC·dVin/dt. For a ramp input of slope m the output is flat at −RC·m, and a constant input gives zero. In the s-domain the input capacitor’s 1/sC produces the s of differentiation, a gain of −sRC, and swapping R and C gives an integrator.