Exercise 9.1 AN RC-phase shift oscillator

The op-amps in the phase-shift oscillator below are all ideal: symmetrical, $A_0\to \infty$, $R_{\mathit{in}}\to \infty$ and $R_{\mathit{out}}\to 0\mathrm{\Omega }$. The circuit is designed to create a harmonic signal.

a)

Derive an expression for the signal transfer of each of the 3 (identical) stages at the right hand side of the oscillator.

b)

Derive an expression for the total (combined) signal transfer of the 3 (identical) stages.

c)

Draw a Bode plot (magnitude and phase) of the transfer function of one of the 3 stages AND of the combined 3 stages. For these Bode plots you may use e.g. $R=1\mathrm{\Omega }$ and $C=1F$.

d)

Node $z$ will be connected to either $x$ or $y$. The node that is NOT connected to $z$ will be connected to ground. Determine which of the nodes $x$ and $y$ of OA1 has to be the inverting one to obtain oscillation.

e)

Derive an equation for the (radian) frequencies/frequency for which this transfer is purely real

f)

Determine the value of $\beta$ such that the oscillation condition is satisfied.