Theory
A differentiator circuit consists of an operational amplifier, resistors are used at feedback side and capacitors are used at the input side. The circuit is based on the capacitor's current to voltage relationship:
where I is the current through the capacitor, C is the capacitance of the capacitor, and V is the voltage across the capacitor. The current flowing through the capacitor is then proportional to the derivative of the voltage across the capacitor. This current can then be connected to a resistor, which has the current to voltage relationship:
where R is the resistance of the resistor.
Note that the opamp input has a very high input impedance (it also forms a virtual ground) so the entire input current has to flow through R.
If Vout is the voltage across the resistor and Vin is the voltage across the capacitor, we can rearrange these two equations to obtain the following equation:
From the above equation following conclusions can be made:
- Output is proportional to the time derivative of the input. Hence opamp acts as a differentiator.
- The minus sign indicates the phase difference of 180 degrees between the output and the input.
- Above equation is true for any frequency signal.
Thus, it can be shown that in an ideal situation the voltage across the resistor will be proportional to the derivative of the voltage across the capacitor with a gain of RC.
Read more about this topic: Differentiator
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