In a differentiator circuit, the output voltage is the differentiation of the input voltage. There are two types of differentiator called passive differentiator and active differentiator. The active differentiator using active components like op-amp.

The output voltage is given by

                                                Vout = - 1/ (RfCf) [dVin / dt]
                              Time constant    = - RfCf

The negative sign indicates that there is a phase shift of 180 degree between input and output. The main advantage of such an active differentiator is the small time constant which gives perfect differentiation.

Sometimes a compensation resistance is needed to connect to the non-inverting terminal to provide the bias compensation. The compensation resistance values is given by Rcomp = (Rf  parallel with R1 ).


Integrator design

The output voltage is given by

                                                Vout = - 1/ (RfCf)   Vin (t) + Vo (0)
                              Time constant    = Rf Cf

1)    To find Cf

             The gain value is given by A          = (Rf / R1) / (1 + jω RfCf) ------------------- (1)
            The corner frequency is           fc    = 1 / 2RfCf    -------------------------------- - (2)
Choose,           fc    = 100Hz and
                        Rf    = 10KΩ
By substituting all in equation (2), calculate the value of Cf .

2)    To find R1  

                    Let   Gain (A) =   1 and substitute all remaining values in equation (1), then               find the value of R1.



Differentiator design

The gain value is given by A          = - jω RfC1 / (1 + jω R1C1)2                ------------------- (1)

The lower corner frequency is           fa    = 1 / 2R1C1      ----------------------------------- (2)

The upper corner frequency is           fb    = 1 / 2RfC1      ----------------------------------- (3)

Always assume fa < fb < fc   and Rf C1 < T. Where T is time constant.

Design procedure

1.    Choose fa as the highest frequency of the input signal. i.e. fa    =   100Hz

2.    Choose C1 to be less than 1 micro Farad and calculate the value of R1.

Choose C = 1micro Farad and    from equation (2) and Calculate R1.

3.    Choose fb as 10 times fa which ensures that fa < fb. That is fb =10 fa. Now find Rf.

4.    To find Cf ,  use   RfC1 =  R1C1 and Rcomp  = R1 parallel with Rf .


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