The Differentiator 

To obtain an op amp integrator, we replaced the feedback resistor with a capacitor. What if we keep the feedback resistor but use an input capacitor instead? Will we get a differentiator?
The circuit to the right shows an op amp connected as a differentiator. Since the input circuit element is a capacitor, this circuit will only experience input current in response to changes in input voltage — the faster and larger the change in input voltage, the greater the input current, therefore the greater the output voltage in response.
Since the output voltage will reflect the rate of change of the input, this circuit will indeed perform differentiation. The general equation for the output voltage is:
V_{out} = RC  dV_{in} 
dt 
The "d/dt" notation indicates differentiation with respect to time. If you're not familiar with differential calculus, don't worry about it here; you won't need it for these pages.
The op amp differentiator is not used in any analog computer application, and indeed not generally. The basic reason for this is that highfrequency noise signals will not be suppressed by this circuit; rather they will be amplified far beyond the amplification of the desired signal.
In some applications, it may be possible to add a series input resistor, as shown in the schematic diagram to the right. This limits the high frequency gain of the circuit to the ratio R_{f}/R_{in}. The low frequency gain is still set by R_{f} and C, as before. The cutoff frequency, where these two effects meet, is determined by R_{in} and C, according to the expression: f_{co} = 1/2R_{in}C.
Higherfrequency signals are still amplified more than lowfrequency signals, so any noise present in the circuit will still be amplified more than the desired signal. If an application can suppress such noise and doesn't require higherfrequency components, this modified circuit may serve the need. In other cases, if differentiation is absolutely required, a passive RC circuit is generally used instead, and the inevitable signal losses compensated later.


 
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