## Differentiator and differentiator circuit

Table of Contents

**Differentiator**– One of the Simplest of the op-amp Circuits that Contains Capacitor in the Differentiating Amplifier.

**Differentiator:**

The circuit, as its name suggests, performs differentiation in mathematics,. Where the output waveform is the derivative of the input waveform. If an input resistor R1 is swapped out for a capacitor C1, a basic inverting amplifier can be used to create the differentiator.

Since the differentiator performs the integrator function in reverse,. The following KCL Equation written at node V2 can be used to obtain the expression for the output voltage.

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 â€“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}Â SÂ withÂ R_{in}Â frequency which makes the circuit very susceptible to high frequency noise.

**Basic Differentiator**

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}Â C_{1}Â (i.e) T > R_{F}Â C_{1}

**Practical Differentiator**

A Workable Differentiator can be designed by Implementing the following steps.

1.Â Select fa equal to the highest Frequency of the input signal to be Differentiated then assuming a value of C_{1}< 1Î¼f. Calculate the value of R_{F}Â .

2.Â Choose fb = 20fa and Calculate the values of R_{1}Â and C_{F}.Â So that R_{1}Â C_{1}Â = R_{F}C_{F}Â .

**Uses:**

It serves as a rate of change and Detector in FM Modulators as well as a high Frequency Component Detector in wave shaping Circuits.

This o/p for practical Differentiator