Notes Integrators& Differentiators

8/13/2019 Notes Integrators& Differentiators

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Integrators & Differentiators


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Integrators

Op Amp RC circuits now

Opens door to a wide range of useful andexciting applications

Integrators and Differentiators are twobasic applications


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Inverting Integrator

Replacing resistors with generalimpedances


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Z 2 (s)= 1/Y 2 (s )

or

Finite dc Gain is given as

Which is the T.F of a low passSTC network

3-dB frequency is


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The capacitor behaves as open circuit at

dc , so by inspection the dc gain is simply-R 2 /R 1

Due to the virtual ground at the invertingterminal the resistance seen by thecapacitor is simply R 2 , hence the timeconstant of the low pass STC network isC 2 R 2


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=

Output voltage is proportional tothe time integral of input voltage

and CR is the integrator TimeConstant.

Negative sign indicates that it is anInverting Integrator (also known asMiller Integrator).


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Bode Plot For The Inverting Integrator

As doubles, the gain is halved (decreases by 6 dB) Intercepts the 0 dB line at =1/RC making gainequal to unity

This is known as IntegratorFrequency

Behaves as a low pass STCnetwork with a corner frequency ofzero

At =0, the T.F becomes infiniteas the capacitor is opened and opamp is operating with open loop

Any tiny dc component in theinput will saturate the output whichis a serious problem with integratorcircuit


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Effect of Input Offset DC voltage and current

R added in the positive lead tokeep the input bias current I B fromflowing through C

Nevertheless I OS flows through C

and causes the op amp output tosaturate.


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Alleviation of dc problem The dc problem can be alleviated by connecting a

resistor R F across the integrator capacitor C It provides a dc path for the dc currents V OS /R and I OS

Now v 0 will have a dc component

instead of rising linearly To keep the dc offset low, one would lower R F

Lower the value of R F , less ideal the integrator circuit

becomes


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Inclusion of R F moves the pole at =0 to =1/CR F and the new T.F becomes

Lower the value of R F , higher the cornerfrequency will be and more non-ideal the

integrator becomes Selection of R F , a trade off between dc performance and signal performance


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The Op Amp Differentiator

Interchanging the positions of R and C inan ideal Integrator results in Differentiatorcircuit

Performs the mathematical operation ofsignal differentiation


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Bode Plot shows that the magnitude doublesfor an octave increase in frequency

Plot is a straight line of slope 6dB/octaveintersecting the 0 dB line at =1/CR (theDifferentiator Time Constant)


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The Frequency response of Differentiator can bethought of a High pass STC network with cornerfrequency at infinity

Differentiator circuit can act as a Noise Magnifieras a spike appears at the output every timethere is a sharp change at the input

This causes stability problems, so they are

generally avoided Sometime a small-valued resistor is used inseries with the capacitor, but it introduces non-ideal behavior to the circuit

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Notes Integrators& Differentiators