The Barkhausen criterion for system with positive feedback (being \$B(s)\$ the feedback network transfer function and \$A(s)\$ the gain network transfer function) afirms that for a system to keep oscillating without input signal and without loss, we need the poles of:
\$H(s) = \dfrac{A(s)}{1 – B(s)A(s)}\$
in the imaginary axes at the complex plane.That means there should be a solution to:
\$B(j \omega_o) \cdot A(j \omega_o) = 1\$
and ωo would be the oscillating frequency.
On the other hand, if i have a system with negative feedback (being B(s) the feedback network transfer function and A(s) the gain network transfer function), the overall transfer function will be:
\$H(s) = \dfrac{A(s)}{1+B(s)A(s)}\$
So, in order to have poles in the imaginary axis (no loss), I need a frequency ωd that solves:
\$B(\omega_d).A(\omega_d) = -1\$
That loop transfer function being -1, should mean that the sinusoidal input with frequency ωd wil be shifted by 180º and will keep the same amplitude after passing through the loop \$A(s).B(s)\$, then will be shifted by 180° again because of the negative feedback and will return to its original form to keep oscillating indefinitely.
It will also mean that the transfer function \$H(s)\$ will have infinity gain at jωd which is a requirement to keep a system living without any input signal. Isn't it enough to implement a oscillating system that will keep oscillating any sinusoidal with frequency ωd that enters the loop?
Why do we need positive feedback to implement an oscillating system ?
Best Answer
The answer is relatively simple: Each linear oscillator needs a loop gain of (at least) unity (unity magnitude and phase shift of zero deg) at one frequency only.
That means: We need a frequency-selective circuitry which can meet this condition at the desired frequency.
If we want to use a fixed gain stage (which is not always the case, we also can use integrators) we have two choices: inverting or non-inverting.
1.) Non-inverting (phase shift zero): The passive network must produce a phase shift of zero at the desired frequency (example: bandpass).
2.) Inverting (180 deg phase shift): The passive network must produce -180 deg at the desired frequency (example: three RC lowpass stages).
Answer to your final question: Why do we need positive feedback to implement an oscillating system ?
In order to avoid misinterpretations, I think that we should say: We need always a positive loop gain (of unity or - in practice - slightly larger) at the desired frequency. More than that - at the same time we need negative loop gain (negative feedback) for DC (stable bias point).