In the Short circuit test of a synchronous machine, the armature terminals are short-circuited through ammeters. Then the machine is rotated at rated speed while gradually increasing the field current.
So, in case of short-circuiting, V=0 and hence the rate of change of total flux must also be zero. So the short-circuit current have to rise only to oppose the main field flux (i.e. demagnetisation effect of the armature reaction). As the effect is demagnetising, the machine works at low flux density. Therefore, the chance of occurrence of saturation is negligible. Thus, the armature current increases in proportional to the field current and Short-circuit characteristic plot (armature current vs. field current) is linear.