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By checking the output V while operational. There is also a static test for resistance.
If the rectifier is wye connected, ripple will increase. If it is delta connected, ripple will substantially increase. Whether or not the rest of the rectifier, filter, and regulator are damaged will depend on how much current is being pulled by whats left, and by how deep the ripple actually becomes.
A: actually a full wave rectifier does not regulate. It will however follow the input minus diodes voltage drops. depending on current it is assumed to be .6 to .7 volts for silicon diodes.
bridge rectifier is the best rectifier.
Filter capacitors smooth out the pulsating DC. Without the capacitors, the pulsating DC drops to zero at the AC crossing point, unless it is a three phase rectifier. Sometimes, that is sufficient. Sometimes, they are followed up with a regulator circuit that further stabilizes the DC.