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## Q. 2.14

Transfer Function—Inverting Operational Amplifier Circuit

PROBLEM: Find the transfer function, $V_{0}(s)/V_{i}(s)$, for the circuit given in Figure 2.11. ## Verified Solution

The transfer function of the operational amplifier circuit is given by
Eq. (2.97).

$\frac{V_{0}(s) }{V_{i}(s)} = -\frac{Z_{2}(s)}{Z_{1}(s)}$

Since the admittances of parallel components add,$Z_{1}(s)$ is the reciprocal of the sum of the admittances, or

$Z_{1} (s)=\frac{1}{C_{1}s+\frac{1}{R_{1} } } =\frac{1}{5.6+10^{-6}S+\frac{1}{360\times 10^{3} } }= \frac{360\times 10^{3} }{2.016S+1}$

For $Z_{2}(s)$ the impedances add, or

$Z_{2}(s)=R_{2} +\frac{1}{C_{2}s }=220\times 10^{3}+\frac {10^{7} }{s}$

Substituting Eqs. (2.98) and (2.99) into Eq. (2.97) and simplifying, we get

$\frac{V_{0}(s) }{V_{i}(s)}=-1.232\frac{s^{2}+45.95s+22.55 }{s}$

The resulting circuit is called a PID controller and can be used to improve the performance of a control system. We explore this possibility further in Chapter 9.