Exercise 9.4 A 3-stage RC-phase shift oscillator

The circuit schematic below shows the circuit of an RC phase shift oscillator. In this circuit you can assume that the RC sections hardly influence each other because $C_1=10\cdot C_2=100\cdot C_3$ and $R_1=\frac{R_2}{10}=\frac{R_3}{100}$. You may assume that there is no influence whatsoever as simplification.

a)

Derive an expression for the voltage transfer of the feedback network $\beta$ consisting of the three RC stages (that is from the output of the amplifier to the input of the amplifier).

b)

Draw a Bode plot (magnitude and phase) of the feedback network.

c)

Derive an expression for the oscillation frequency $f_{\mathit{osc}}$ or the radian oscillation frequency ${\omega }_{\mathit{osc}}$.

d)

Derive an expression for the amplifier gain A needed for harmonic oscillation.

e)

Draw a polar plot of A$\beta$ and indicate relevant angular frequencies $\omega =0$, $\omega \to \infty$, and $\omega ={\omega }_{\mathit{osc}}$.