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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?

An op amp differentiator circuit

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:

Vout = -RC  dVin


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 high-frequency noise signals will not be suppressed by this circuit; rather they will be amplified far beyond the amplification of the desired signal.

A differentiator with limited high-frequency gain.

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 Rf/Rin. The low frequency gain is still set by Rf and C, as before. The cutoff frequency, where these two effects meet, is determined by Rin and C, according to the expression: fco = 1/2RinC.

Higher-frequency signals are still amplified more than low-frequency 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 higher-frequency 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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