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According to Barkhausen's criteria, for oscillation the feedback should be made after phase difference of 2pi and gain has to be >=1.

Now when I consider a ring oscillator if 2 inverters are connected as in:

Inverter Schematic

Here the phase change after 2 inverters is 2pi. But this would actually not oscillate since output will not change. This is a contradiction right?

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That's because Barkhausen only applies to linear circuits. Ring oscillators are made from inverter stages that have so much gain that they have to be treated as nonlinear (saturating) elements.

Two inverters (or any even number of inverters) form a bistable system.

A single inverter generally does not have enough delay to oscillate; instead, it drives itself to an intermediate analog state.

Larger odd numbers of inverters have enough delay to oscillate, since the output of each gate reaches saturation before its input changes again.

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The most important condition for such a ring oscillator is an uneven number of inverter stages. Only in this case you have negative feedback for DC. This is necessary in order not to lock the circuit at one of the two possible states.

Although it is correct that the original Barkhausen condition (unity loop gain) applies to LINEAR circuits only, the modified Barkhausen condition - which always is applied for practical circuits - requiring a loop phase of 360 deg (0 deg) and a loop gain magnitude larger than unity does apply here. Thus, the oscillation frequency will occur at that frequency, for which the DELAY is identical to a 360deg phase shift.

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