I'm starting to learn about phase-shift oscillators and I'm a little confused at the outset. Suppose you have an amplifier with a feedback circuit with transfer functions A and β respectively. My understanding is that the Barkhausen Criterion is a necessary condition for oscillation. In fact a simple algebraic calculation indicates that in order for there to be oscillation, you must have Aβ = 1 (A and β are complex). But this condition isn't sufficient is it?
So in the case where Aβ = 1, my questions are:
- What additional conditions would guarantee oscillation?
- Why is the oscillation sinusoidal?
I'm assuming there must be some 2nd order linear ODE describing the system when Aβ =1, but I don't know how to produce it.
I've been searching on the internet and all the things I've found so far are either naive explanations, which seem to suppose Barkhausen implies oscillation, which it sounds like it doesn't, or research articles in engineering journals which deal with very advanced aspects of topic.
Can someone point me to a reference that might offer some sort of "middle of the road" answer?