Why is the short circuit test on a transformer done at the high voltage side ? Is this not to damage with transformer by having lots of current flowing through it ?

 

On a short circuit test one doesn't apply the rated terminal voltage to the other winding. Normally a variable voltage controller (such as an induction regulator) is interposed between the excited winding and the AC supply with the initial setting at zero volts. The excitation voltage is then carefully increased while the short circuit current is monitored up to the rated current, at which point the various parameters are measured.

 

Why is the short circuit test on a transformer done at the high voltage side ? Is this not to damage with transformer by having lots of current flowing through it ?

The SCT is preformed on the high voltage side because the rated current on the HV side is lower. (The SCT must be preformed at rated current)

Since we have to apply a source to provide this excitation current, wouldn't you rather it be a smaller current as opposed to a larger one?

It is much easier to find a source that can supply a smaller current, as opposed to a larger one.