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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