Question 6.4.3: Design an op-amp circuit that differentiates the input volta......

System Dynamics [1255330]

Design an op-amp circuit that differentiates the input voltage.

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In theory, a differentiator can be created by interchanging the resistance and capacitance in the integrator circuit. The result is shown in Figure 6.4.7, where $Z_{i}(s)=1/C s\;\mathrm{and}\;Z_{f}(s)=R.$ The input-output relation for this ideal differentiator is

\frac{V_{o}(s)}{V_{i}(s)}=-\frac{Z_{f}(s)}{Z_{i}(s)}=-R C s

The model in the time domain is

v_{o}(t)=-R C{\frac{d v_{i}(t)}{d t}}

The difﬁculty with this design is that no electrical signal is “pure.” Contamination always exists as a result of voltage spikes, ripple, and other transients generally categorized as “noise.” These high-frequency signals have large slopes compared with the more slowly varying primary signal, and thus they will dominate the output of the differentiator. Example 6.4.4 shows an improved differentiator design that does not have this limitation.

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