Question 6.4.2: Determine the transfer function Vo(s)/Vi(s) of the circuit s......

System Dynamics [1255330]

Determine the transfer function $V_{o}(s)/V_{i}(s)$ of the circuit shown in Figure 6.4.6.

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The impedance of a capacitor is 1/Cs. Thus, the transfer function of the circuit is found from (6.4.2) with $Z_{i}(s)=R\ {\mathrm{and}}\ Z_{f}(s)=1/C s.$ It is

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

Thus in the time domain, the circuit model is

$v_{o}=-{\frac{1}{R C}}\int_{\mathbf{0}}^{t}v_{i}\,d t$ (1)

Thus the circuit integrates the input voltage, reverses its sign, and divides it by RC. It is called an op-amp integrator, and is used in many devices for computing, signal generation, and control, some of which will be analyzed in later chapters.

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