Question 19.16: What is the behavior of a ring with only n = 1 inverter, as ......

Based upon (19.80)

$f_{0} = \frac{1}{T_{0}} = \frac{1}{2nT_{d}}$      (19.80)

and the preceding discussion, one might think that the circuit will oscillate at a frequency $f_{0} = 1 / (2T_{d})$. But the first-order small-signal model of an inverting amplifier in Fig. 19.19(b) is clearly an undriven single-time constant circuit whose solution is an exponential decay,

$v_{1}(t) = v_{1}(0)exp\left\lgroup-\frac{t}{R_{o~1}C_{1}} \right\rgroup$

When the inverting amplifier is a simple digital CMOS inverter, this analysis is accurate and there are no oscillations, as one may readily verify in the lab. The voltage quickly settles to a dc value between ground and the supply. In fact, this can be a useful bias circuit for establishing a dc voltage precisely equal to the trip point of the inverter, where $v_{out} = v_{in}$.
If the inverting amplifier has other internal nodes not represented by the small-signal schematic in Fig. 19.19(b), it will have a higher-order response and the solution is more complex.