How Does Overvoltage Affect Transformer Core Saturation and Current Flow?

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I'm trying to understand what happens to a distribution transformer in relation to overvoltage conditions (in terms of saturation and current flow; not insulation failure).
I understand that once the voltage has risen high enough to drive the magnetic core past the saturation point, that the magnetizing current will rise dramatically in relation to the increase in the flux field.

1. Is the current increase (on the primary) at this point analogous to a motor at locked rotor, where the resistance of the windings themselves are the limiting factor?

2. Is there a substantial difference in the amount of this current in relation to full load vs. no load on the secondary (other than the load itself)?
 
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I don't think you will get any saturation with no load on the secondary, regardless of how high the primary voltage goes. Saturation comes from too high an impressed H field, which comes from the coil currents. And even at full load, hopefully an overvoltage in the primary won't cause saturation. That's not much design margin, IMO.

When a coil saturates, then yes, the final limiting factor is the coil windings. Look at the B -- H curve for the magnetic material to see what the effective mu does as you get farther and farther into saturation.