How to apply the Barkhausen criterion in order to know if a system will oscillate?

Question

How to analyze or apply the Barkhausen criterion for oscillation of the astable multivibrator below?

• The criterion talks about the magnitude of the products in a loop must be equal to 1 (ideally)

• The phase must be multiples of 360 starting from zero

I really tried to solve this from my own but I'm not getting anywhere with results that are not meaningful to me in order to understand this.

Also I already obtained the equations for the period, frequency, and time on, for the output waveform taking an initial assumption or state and developing further fulfilling the previous assumptions I've made. Which are correct because I've simulated the circuit on Multisim and I get the same results.

Answer 1

The Barkhausen criteria are usually applied to analyze sine wave type oscillator circuits (Wien bridge, etc.) where a small signal, such as thermal noise, is exponentially amplified around the positive feedback loop to create the output signal. Oscillation is inherently a large signal phenomena and in general can't be analyzed using LTI analysis methods, but the Barkhausen criteria let you predict oscillation from the small signal gain and phase behavior.

It's less clear to me how to directly apply such techniques to this relaxation oscillator circuit, as circuits like this don't have any small signal behavior - there are only 2 stable states. It should be fairly obvious, however, that whatever component values you choose the feedback around the loop will eventually be unity and in phase, i.e. when V+ = V-.