Analysis of 3 Stage CS Ring Oscillator
For a stable DC bias point we need at zero Hertz a phase shift of -180deg. This is ensured because we have three inverting stages in a closed loop (number n of stages is odd).

According to the Barkhausen criterion we need in total -360 deg (identical to zero deg) for a certain frequency. For this purpose, we have three 1st order lowpass stages - each with a cut-off frequency at app. \$wc=1/R_DC_L\$. 

Hence, the oscillation criterion is fullfilled for another additional phase shift of -180deg at the desired frequency wo.

That means: Each stage must contribute -60 deg. - and the oscillation frequency is somewhat larger than the cut-off (which gives only -45 deg).   

Comment: The mentioned -90deg per stage is achieved at infinite frequencies only.