EE 212
Lab 10: OpAmp Differentiators and Integrators
Prelab 10
Opamp circuits are often designed and implemented for signal differentiation
and integration. Until recently (before computerbased control), control
algorithms (such as PID) containing differentials and integrals were implemented
in discrete circuit components. Differentiation is also useful for obtaining
velocity measurements from a signal representing a position or determining
a signal's frequency (recall the amplitude of the time derivative of a sinusoid
is scaled by its frequency). Figure 1 below shows an ideal opamp integrator
and differentiator with inputoutput relationships that are theoretically
correct, but have practical implementation issues discussed below. In this
lab, practically realizable differentiators and integrators will be built
using opamps, resistors and capacitors.
1. The Differentiator

The ideal differentiator is inherently unstable in practice due to the presence
of some high frequency noise in every electronic system. An ideal differentiator
would amplify this small noise. For instance,
if v_{noise}_{ }=
Asin(wt) is differentiated, the output would
be v_{out }=
Awcos(wt). Even
if A = 1mV, when
w = 2p(10MHz)
v_{out} would have an amplitude of 63V! To circumvent this
problem, it is traditional to include a series resistor at the input and
a parallel capacitor across the feedback resistor as shown in figure 2,
converting the differentiator to an integrator at high frequencies for filtering.

Wire up the practical opamp differentiator shown in figure 2 using your
741opamp.

Drive it (via v_{in}(t)) with a 1kHz sine wave, a 1kHz square
wave, and a 1kHz triangle wave. For each input signal, sketch the input and
output waveforms.

Are the output waveforms and their amplitudes what you would expect, i.e.,
does the circuit differentiate the input signal?
2. The Integrator

Opamps allow you to make nearly perfect integrators such as the practical
integrator shown in figure 3. The circuit incorporates a large resistor
in parallel with the feedback capacitor. This is necessary because real opamps
have a small current flowing at their input terminals called the "bias current".
This current is typically a few nanoamps, and is neglected in many circuits
where the currents of interest are in the microamp to milliamp range. However,
if you apply a nanoamp current to a 0.1mF capacitor,
it won't take long until it charges and becomes effectively an open circuit
not allowing any current to flow! The feedback resistor gives a path for
the bias current to flow. The effect of the resistor on the response is
negligible at all but the lowest frequencies.

Wire up the practical opamp integrator shown in figure 3 using your 741
opamp.

Drive the input (via v_{in}(t)) with a 500Hz square wave of
2 V pp amplitude. Sketch the input and output waveforms.

Has the input been integrated?

Calculate the expected output waveform via integration using the circuit
component values and compare to the experimental waveform.

Does the amplitude of the output waveform agree with what it should be from
the circuit values?

Repeat using a sine wave and a triangle wave.
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