I'm getting interested in high voltage circuits, and I already have drawn out a couple of circuits (I have not yet built anything). However, I'm just wondering as to what would happen if a pole transformer rated at 120V and 50-60 Hz were run at 120V and 50-60 kilohertz? Would the transformer overheat? Explode?
A transformer that's designed for 50/60Hz will not work as a transformer at frequencies much above that.
The core will be designed with laminations 'just thin enough' for reasonable eddy current losses at 60Hz. As the frequency rises, losses in the laminations increase. 400Hz aircraft transformers have much thinner laminations than 60Hz transformers.
Fortunately, as you raise the frequency, the peak flux in the core drops for any given terminal voltage, so the hysteresis losses per cycle in the core drop. However, as there are more cycles per second, the core losses tend to rise anyway as the frequency rises.
By the time you reach kHz, the transformer behaves more as a heater, with the windings coupling the input power into core heating. Pump in enough power and it will get hot, with all the normal bad things that that entails.
If you want to operate a transformer at kHz or 10s of kHz, then you need to switch to a ferrite core material, with higher resistivity (to defeat eddy current losses) and lower hysteresis losses.
Yes, it would likely overheat or explode. The core material in a low frequency transformer is not designed for kHz operation, and the core losses would cause overheating to destruction. It would not make a very good high frequency transformer anyway, because (aside from the horrendous losses) the leakage inductances which are manageable at 50/60 Hz would become a problem at 50-60 kHz.
Other answers mention inductance, but transformers also have interwinding capacitance. On such a large transformer it won't be negligible.
Depending on the wiring and construction, the HF impedance of the primary might be capacitive enough to draw more current than expected, or the primary waveform might couple to the secondary through the interwinding capacitance.
This is hand-waving as it depends a lot on how the transformer is built, but please keep capacitance it in mind.
The inductive reactance of 50Hz / 60Hz transformer will be too high at KHz. That alone will drastically reduce the amount of power you can get out of the transformer.