Exercise 9.13 An harmonic oscillator?

Given is the circuit below; this circuit is intended to be an harmonic oscillator.

• The voltage gain of the voltage buffer at the left hand side of the schematic is a real valued factor $A_v$ in questions (a)-(d); this value can be positive or negative. The voltage gain of the other voltage buffers is 1; for all voltage buffers $r_{\mathit{in}}\to \infty \mathrm{\Omega }$ and $r_{\mathit{out}}=0\mathrm{\Omega }$.
• The value of all resistors is the same (R), the capacitors have identical values (C), and all inductors have identical values (L) - this is to simplify your derivations.

(a)

Which condition(s) must be satisfied to get harmonic oscillation?

(b)

Derive the voltage transfer for each of the 4 stages in the figure. Combined these form the loop gain.

(c)

Draw a Bode plot (magnitude and phase) of the loop gain; for this question you may assume $R=1\mathrm{\Omega }$, $C=1F$ and $A_v=1$ when making this plot.

(d)

Determine whether the given circuit can function as harmonic oscillator for a specific positive (real) value of $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $A_{\mathit{buf}}$.

(e)

Determine whether the given circuit can function as harmonic oscillator for a specific negative (real) value of $A_{\mathit{buf}}$. If your answer shows that harmonic oscillation can be possible, derive an expression for the (radian) oscillation frequency and for the required $A_{\mathit{buf}}$.