Why do you have to use the nominal voltage of the transformer for the no-load or open-circuit test?
I think I understand your question so first here's a representation of the equivalent circuit of a transformer: -



Under no load conditions the only current flowing into the primary is the current taken by the "parallel components" Xm and Rc. For a normal power transformer that current will be small compared to the current normally taken by the primary when the secondary is driving a load. For that reason you can ignore (short circuit) Xp and Rp and of course the secondary is only producing an open circuit voltage so Rs and Xs are of no consequence.

The "thing" in the middle that looks like a transformer is a perfect power transformer and because no current is being delivered to the secondary that perfect transformer takes no current.

So, it boils down to Rc and Xm being connected to the incoming power and no further components need to be analysed.


  I was wondering why exactly you have to use the nominal voltage of the
  transformer for the open-circuit or no-load test.


There is one very important reason for this and that is core saturation - if you don't use the normal applied voltage you'll either have too much saturation or too little and you won't have a representive measurement. Saturation of the core is non-linear with voltage so it's important to use the right applied voltage. Look at the BH curve to see why: -



You can see that it is very non-linear once you start approaching saturation and, most transformers will be designed to run at a magnetic field strength (H) in the early to mid areas of saturation. This of course means a smaller transformer size and less iron. Commercial reasons prevail.

So, to do the test justice you need to run at nominal levels.

If you were doing a test to find out the values of the series components then you'd run the primary from a variac and short circuit the output. The sort of voltage that is now applied is a fraction of the nominal voltage so core losses are low (very linear BH curve) and eddy current losses (Rc) are also quite small.