For a 3 phase induction motor, say rated 110 kW, that uses to drive a high inertia centrifugal fan. If we stop the motor, the induction motor still rotates due to load inertia. So is it harmful to run the induction motor again while it's running by load inertia?
This will lead to induction generation. Voltage will be induced in stator. Now there are many possibilities.
Next starting: if motor was running due to flywheel effect and again mains voltage applied then depending on phase difference, either very high voltage will be applied to motor or lower voltage would be applied momentarily. Both the conditions are harmful. Heavy inrush current, very high toque generation (leading to mechanical failures) is even possible. Also, frequency of mains and motor generated voltage would never be same. Motor will be applied distorted voltage in totality.
Now if you had directly connected capacitor across motor terminal then your motor stop time would increase drastically. This will be on account of motor generated voltage charging capacitor and charged capacitors working as source to motor.
The motor stator (after disconnection) has an easily measured voltage that decays in magnitude at a rate set by rotor/magnetizing circuit time constant, and in frequency (hence phase) at a rate set by load and inertia. If you wait before reconnecting until the residual voltage is small enough then reclosure is safe. At the other extreme if you restart immediately after stopping, the voltage will still be nearly 100% but could be 180 degrees out of phase (relative to system voltage) depending on the instant of restart. Inrush current will then be twice normal, and stator winding forces (proportional to current squared) will be four times those for a normal start.
Most small motors have no explicit protection against this, but rely on operator skill to not push "start" immediately after "stop". Large and important motors may have deliberate provision for fast re-energisation that checks synchronism.
The rotor is energized by transformer action from the stator. However it is an inductor, and when the stator current is interrupted, the rotor current continues (as a decaying dc transient from the initial value). You can get a sense for this from the conventional Stienmetz equivalent circuit - there is a closed path through the rotor inductance and the magnetizing inductance. The usual model is for steady-state (except slip) conditions, but a variation on this model (with transient sources) is how you predict the behavior for transient conditions. The rotor current induces a voltage on the stator, decaying in voltage and frequency.