$($c$)$ The field excitation current is maintained at $72$ A $.$ The prime$-$mover power is increased up to the stability limit of the generator in the steady state, with the output line voltage remaining at $22$ kV $.$ Neglecting the stator resistance, determine the corresponding real power output and electromagnetic torque of the generator.

$[5$ marks$]$

$($d$)$ The generator is discornected from the infinste bus and is run at its rated speed. If the output terminals are short$-$circuited, determine the field current required so as to limit the stator current to the rated value.

$[3$ marks$]$

A $3-$phase, $4-$pole, Y$-$connected synchronous generator is rated $1\mathrm{.}5$ MVA, $22$ kV $($line$,rms$ and has a non$-$salient type rotor. The machine has a per$-$phuse synchronous reactance of $25,$ armature resistance of $1,$ and is connected to an infinite bus with a line voltage of $22$ kV $($ mms $).$ The generator runs at synchronous speed. The effects of magnetic saturation can be neglected.

$($a$)$ When the field excitation current is adjusted to $72$ A $,$ the generator supplies its rated current at a power factor of $0\mathrm{.}9($lagging$).$

Determine:

$($i$)$ The armature phase voltage generated.

$[7$ marks$]$

$($ii$)$ The armature to field mutual inductance.

$[2$ marks$]$

$($b$)$ Sketch the phasor diagram for Part $($a$),$ showing all currents and volage phasors.