Exercise 9.17 An harmonic oscillator with opampe?

(a)

Explain why $A\cdot \beta (j{\omega }_{\mathit{osc}})=1$ needs to be satisfied in order for a circuit using positive feedback to function as a harmonic oscillator. You may derive and make use of the closed-loop transfer of the following closed-loop system in your explanation:

(b)

Given is the following second-order Sallen-Key low-pass filter:

Show that the transfer function of this filter is $H(\mathit{j\omega })=\frac{1}{1+2\mathit{j\omega RC}+j^2{\omega }^2{R}^2{C}^2}$ for $A\to \infty$.

(c)

Given is the following second-order filter:

Show that the transfer function of this filter is $H(\mathit{j\omega })=\mathit{j\omega }\frac{L}{R}\cdot \frac{1-\mathit{j\omega RC}}{1+\mathit{j\omega }\frac{L}{R}}$.

(d)

Now we want to see if it is possible to make a harmonic oscillator if we connect the low-pass filter from above in series with a voltage buffer and a couple of other opamp circuits in a closed-loop, see the circuit diagram below. The opamps are ideal, with a real valued $A\to \infty$. The voltage gain of the voltage buffer at the top of the schematic is a real valued negative factor $A_{\mathit{buf}}$.

Note that the value of all resistors is the same (R), that the capacitors have identical values (C), and that all inductors have identical values (L).

Derive the transfer function for every stage.

(e)

Draw the corresponding Bode plots (both magnitude and phase plots). You may assume that the corner frequencies ${\omega }_c$ are the same for all stages.

(f)

Calculate the total loop gain and draw the Bode plot (magnitude and phase) of this total loop gain (assume $A_{\mathit{buf}}=1$ and assume the same corner frequencies ${\omega }_c$).

(g)

Can the circuit operate as a harmonic oscillator for $A_{\mathit{buf}}>0$? If so, at what frequency and for what $A_{\mathit{buf}}$?

(h)

Can the circuit operate as a harmonic oscillator for $A_{\mathit{buf}}<0$? If so, at what frequency and for what $A_{\mathit{buf}}$?