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One of the simplest of the op-amp circuits that contains capacitor in the differentiating amplifier.

**Differentiator:**

One of
the simplest of the op-amp circuits that contains capacitor in the
differentiating amplifier.

**Differentiator:**

As the
name implies, 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}
.

The
expression for the output voltage can be obtained KCL eqn written at node V_{2}
as follows,

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 V_{in} with time.

The –sign
=> indicates a 180^{0} 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 (R_{F} /XC_{1} )R with R in frequency at a rate of
20dB/decade. This makes the circuit unstable.

Also
input impedance XC_{1} S with R_{in}
frequency which makes the circuit very susceptible to high frequency noise.

**Basic Differetntiator**

From the
above fig, f_{a} = 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 2 components. R_{1} and C_{F} . This circuit is a
practical differentiator.

From
Frequency f to feedback the gain R_{s} 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} =>
helps 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.

Generally,
the value of Feedback and in turn R1 C1 and RF CF values should be selected
such that

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

**Practical Differentiator**

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 R_{F}
.

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

**Uses:**

Its used
in waveshaping circuits to detect high frequency components in an input signal
and also as a rate of change and detector in FM modulators.

This o/p
for practical differentiator

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