= Solution
Interchange the input and feedback components: put the <capacitor> $C$ between the input and the inverting terminal, and the <resistor> $R$ between the output and the inverting terminal. Keep the non-inverting terminal grounded, as in the lower circuit above. For an <ideal operational amplifier> under unsaturated <negative feedback>, the inverting node remains at <virtual ground>.
The input <capacitor current> is $I_C=C\,dV_{\rm in}/dt$. It flows through the feedback <resistor>, giving
$$
C\frac{dV_{\rm in}}{dt}=\frac{0-V_{\rm out}}R,\qquad\boxed{V_{\rm out}=-RC\frac{dV_{\rm in}}{dt}}.
$$
This is an inverting <op amp differentiator>. Its ideal gain magnitude increases with <frequency>, so it strongly amplifies high-frequency noise. Practical differentiators include components that restrict the useful band and stabilize feedback; that qualification does not change the displayed ideal low-amplitude relation within the operating band.
Back to article page
