A simple RC circuit will integrate or differentiate waveforms:
![](/sites/default/files/demomanual/electricity_and_magnetism/ac_circuits/rc_integration_and_differentiation1.gif)
(Of course, the derivative and integral of a sine wave is the leading and lagging cosine wave; these are just the normal 90° phase shifts.) The circuit below integrates.
![](/sites/default/files/demomanual/electricity_and_magnetism/ac_circuits/rc_integration_and_differentiation2.gif)
The resistance R is made large and the capacitative reactance Xc is made small by using a large C and/or a large W. Then the current into the circuit is set by R and proportional to Vin. The capacitor stores and integrates the charge.
![](/sites/default/files/demomanual/electricity_and_magnetism/ac_circuits/rc_integration_and_differentiation.gif)
To differentiate the circuit is wired as below:
![](/sites/default/files/demomanual/electricity_and_magnetism/ac_circuits/rc_integration_and_differentiation4.gif)
We arrange for nearly all the input voltage to drop across the capacitor (Vc >> Vout ) by making R small and Xc large using a small C and/or w Thus the voltage drop across R measures i without disturbing Vc.
![](/sites/default/files/demomanual/electricity_and_magnetism/ac_circuits/rc_integration_and_differentiation5.gif)
The circuit looks essentially like a capacitor to the input. The current is set by C and the small R is placed in series to sense it. RL circuits will perform the same operations.