Question 6.4.4: The differentiator analyzed in Example 6.4.3 is susceptible ......

System Dynamics [1255330]

The differentiator analyzed in Example 6.4.3 is susceptible to high-frequency noise. In practice, this problem is often solved by using a redesigned differentiator, such as the one shown in Figure 6.4.8. We will analyze its performance in Chapter 9 when we study the response of systems to sinusoidal inputs. Derive its transfer function.

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Using the op-amp equation (6.4.2), we have

$\frac{V_{o}(s)}{V_{i}(s)}\approx-\frac{Z_{f}(s)}{Z_{i}(s)}$ (6.4.2)

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

where $Z_{i}(s)=R_{1}+1/C s\;\mathrm{and}\;Z_{f}(s)=R.$ The circuit’s transfer function is

${\frac{V_{o}(s)}{V_{i}(s)}}=-{\frac{Z_{f}(s)}{Z_{i}(s)}}=-{\frac{R C s}{R_{1}C s+1}}$ (1)

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