Exercise 9.6 Something like a Wien bridge oscillator

a)

An answer:

$$\begin{array}{rcll}& H_{\mathit{loop}}(\mathit{j\omega })& =A\cdot H_1(\mathit{j\omega })\cdot H_2(\mathit{j\omega })& \\ & & H_1(\mathit{j\omega })=\frac{1∕\mathit{j\omega }{C}_1}{R_1+1∕\mathit{j\omega }{C}_1}=\frac{1}{1+\mathit{j\omega }{C}_1{R}_1}& \\ & & H_2(\mathit{j\omega })=\frac{R_2}{R_2+1∕\mathit{j\omega }{C}_1}=\frac{\mathit{j\omega }{R}_2{C}_2}{1+\mathit{j\omega }{C}_2{R}_2}& \\ & H_{\mathit{loop}}(\mathit{j\omega })& =A\cdot \frac{1}{1+\mathit{j\omega }{C}_1{R}_1}\cdot \frac{\mathit{j\omega }{R}_2{C}_2}{1+\mathit{j\omega }{C}_2{R}_2}& \\ & & =A\cdot \frac{\mathit{j\omega }{R}_2{C}_2}{1+\mathit{j\omega }{R}_2{C}_2+\mathit{j\omega }{R}_1{C}_1-{\omega }^2{R}_1{C}_1{R}_2{C}_2}& \\ & & =A\cdot \frac{1}{\frac{1}{\mathit{j\omega }{R}_2{C}_2}+1+\frac{R_1{C}_1}{R_2{C}_2}+\mathit{j\omega }{R}_1{C}_1}& \end{array}$$

yielding:

$$\begin{array}{rcll}& {\omega }_{\mathit{osc}}& =\frac{1}{\sqrt{R_1{C}_1{R}_2{C}_2}}& \end{array}$$

b)

An answer:

$$\begin{array}{rcll}& H_{\mathit{loop}}({\omega }_{\mathit{osc}})& =A\cdot \frac{1}{1+\frac{R_1{C}_1}{R_2{C}_2}}& \\ & A& =1+\frac{R_1{C}_1}{R_2{C}_2}& \end{array}$$