Barkhausen's criteria states that if A is the gain of the amplifying element in the circuit and β(jω) is the transfer function of the feedback path, so βA is the loop gain around the feedback of the circuit, the circuit will sustain steady-state oscillations only at frequencies for which:
The loop gain is equal to unity in absolute magnitude, that is, |βA|=1|βA|=1 and The phase shift around the loop is zero or an integer multiple of 2π: ∠βA=2πn,n∈0,1,2,….∠βA=2πn,n∈0,1,2,….
Barkhausen's criterion is a necessary condition for oscillation but not a sufficient condition: some circuits satisfy the criterion but do not oscillate
The above criterion is not actually derived from anywhere. It is plain logic.
If net loop gain exceeds unity in any one direction, it will be less than unity in the opposite direction.This will prove that the oscillatory system overall has a net gain of energy in one direction and loss in another direction. By law of conservation of energy, as energy can neither be created, nor destroyed, such oscillators will not have sustained stable oscillations without an external energy source pumping in some energy through the process.
Similarly, if loop does not have a net zero phase shift, the loop gets biased in one direction. This bias causes unstable oscillations.