I have an unpleasant understanding problem about the Barkhausen criterion for oscillators.

It states that the amplitude of the closed-loop transfer function has to be 1 and the phase of it an interger multiple of 2*pi.

This means that the resulting complex number has to be real-valued with an amplitude of 1, hence it is on the real axis on the right side (the circle with a absolute value of 1 cuts the real axis on the right side).

But now comes the confusion for me: The poles-zero plot tells us that a system is stable (the amplitude decreases with time) on the left side.

Exactly on the imaginary axis the system would be stable without damping and on the right side of the imaginary axis the system´s amplitude would rise to infinity.

So the Barkhausen criterion has a pole at 1 on the real axis and therefore should not be stable, but with a transfer function amplitude of 1 it doesnt grow......Im confused about this and would appreciate any help to understand this better.

Thank you very much!