The circuit performs the mathematical operation of differentiation (i.e.) the output waveform is the derivative of the input waveform.

**Differentiator:**

The
circuit performs the mathematical operation of differentiation (i.e.) the
output waveform is the derivative of the input waveform. The differentiator may
be constructed from a basic inverting amplifier if an input resistor R_{1}
is replaced by a capacitor C_{1}. Since the differentiator performs the
reverse of the integrator function. Thus the output V_{0} is equal to R_{F}
C_{1} times the negative rate of change of the input voltage Vin with
time. The –sign indicates a 180º phase shift of the output waveform V_{0}
with respect to the input signal. The below circuit will not do this because it
has some practical problems.

The
gain of the circuit (RF /XC1) R with R in frequency at a rate of 20dB/decade.
This makes the circuit unstable. Also input impedance XC1s with R in frequency
which makes the circuit very susceptible to high frequency noise.

From the above fig. fa = frequency at
which the gain is 0dB and is given by

Both stability and high frequency noise
problems can be corrected by the addition of two components. R_{1} and
C_{F}. This circuit is a practical differentiator.

From Frequency fa to feedback the gain
Rs at 20dB/decade after feedback the gain S at 20dB/

decade. This 40dB/ decade change in gain
is caused by the R_{1}C_{1} and R_{F}C_{F}
combinations.

The gain limiting frequency fb is given by,

Where
R_{1}C_{1} = R_{F}C_{F}

R_{1}C_{1}
and R_{F}C_{F} help to reduce the effect of high frequency
input, amplifier noise and offsets.

All
R_{1}C_{1} and R_{F} C_{F} make the circuit more
stable by preventing the R in gain with frequency.

The
input signal will be differentiated properly, if the time period T of the input
signal is larger than or equal to RFC_{1} (i.e) T > RFC_{1}
generally, the value of Feedback and in turn R_{1}C_{1} and R_{F}

C_{F}
values should be selected such that

R_{F}
C_{1}>> R_{1} C_{1}

A
workable differentiator can be designed by implementing the following steps.

1.
Select
fa equal to the highest frequency of the input signal to be differentiated then
assuming a value of C_{1} < 1μf. Calculate the value of RF.

2. Choose fb = 20fa and calculate the
values of R_{1} and C_{F} so that R_{1} C_{1} =
R_{F} C_{F}.

It
is used in wave shaping circuits to detect high frequency components in an
input signal and also as a rate of change and detector in FM modulators.

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