Problem $1$: A $4-$pole, $60$ Hz $,$ Y$-$connected, $3-$phase generator has a regulated line voltage of $V_{AB}=\frac{450}{{?}_{30}}{V}_{rms},$ a synchronous reactance of $0\mathrm{.}05-,$ a stator resistance of $0\mathrm{.}003,$ and $L_{sf}=0\mathrm{.}018H.$ The balanced $3-$phase load draws $3$ MW at a lagging $0\mathrm{.}75$ power factor.

a$.$ Compute the $3-$phase complex power of the load $($magnitude and angle$).$

$S_{3\text{p}\text{h}\text{a}\text{s}\text{e}}=\frac{4}{{?}_{41\mathrm{.}4}}$MVA

b$.$ Compute the generator phase current $($magnitude and angle; hint: the angle of the current should be negative for the load to have a lagging power factor$).$

$I_a=\frac{5132}{{?}_{-}}41\mathrm{.}4A_{rms}$

$E_a=\frac{476}{{?}_{22\mathrm{.}4}}{V}_{rms}$

d$.$ Calculate the required field current.

$I_f=99\mathrm{.}2A_{DC}$

e$.$ State the required speed of the prime mover $($ rpm $)$

$N_m=1800RPM$