Exercise 9.5 A 3-stage oscillator with NPNs

The circuit below is an harmonic oscillator, consisting of three identical stages. The transistors can be assumed to have infinite current gain $\mathit{\beta fe}$ and infinite output resistance. Furthermore, you may use the numerical values $\frac{\mathit{kT}}{q}=25\mathit{mV}$ and $\frac{q}{\mathit{kT}}=40V^{-1}$ in your expressions.

a)

Derive an equation for the small signal voltage gain of a single stage.

b)

Show that the small-signal open-loop gain can be written as $A_{\mathrm{OL}}\approx -{\left(\frac{40V_{\mathit{CC}}-24}{1+\mathit{j\omega RC}}\right)}^3$.

c)

Derive an expression for the oscillation frequency $f_{\mathit{osc}}$.

d)

Derive an expression for the supply voltage $V_{\mathit{CC}}$ for stable oscillation.

e)

Explain what happens:

(a)

if $V_{\mathit{CC}}$ is lower than what is derived in c),

(b)

and if $V_{\mathit{CC}}$ is higher than what is derived in c).

f)

If we build the same circuit with 4, 5 or 6 identical stages (BJT+R+C), which of these can also be harmonic oscillators (with $V_{\mathit{CC}}$ adjusted accordingly to enable potential harmonic oscillation) and which cannot? Explain/prove/illustrate your answer for each number of stages.