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Case Study : How to do integration and differentiation beforedigital electronic computers Choice #1: analytically

BUT: for calculation of e.g. the times of high tide and low tides this was notfeasible

Solution: mechanical Integrators and Differentiators large machines

High cost, not always too accurate (http://www.usscod.org/tdc-restore.html)

From the 1950s 1960s: analogue electronic integrators with the help of OPAMPs

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Op Amp as an integrator: Why does it integrate the input voltage?

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Op Amp as an integrator:

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Op Amp as an integrator:

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Improvement of plot?

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Improvement of plot?

Log log plot

straight line slope -1

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Op Amp as differentiator:

2 O d ff d h l

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Op Amp as differentiator:

21 O A i t t diff d th li ti

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Op Amp as integrator / differentiator:

Up to now, we have done our calculations in the timedomain

The magnitude and the phase shift of the output signal

can

of course - also be worked out using the frequency representation, i.e. the complex reactance of the

capacitor

including the negative sign of the gain

Xc = -j/CGain = - Xc/RI for the integrator (i.e. Inverting amp

and an additional phase shift of -90 degrees)

21 O A i t t diff d th li ti

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Op amp as a square wave oscillator:

2 feedback

paths

21 O A i t t diff d th li ti

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Op amp as a square wave oscillator:- How does this work out in detail?

- NI = non inverting input, II= inverting input

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Op amp as a square wave oscillator:- How does this work out in detail?

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Op amp as a square wave oscillator:- Result

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