A $3-$phase, Y$-$connected, $60$ Hz generator is rated for a line voltage magnitude of $208V_{RMS}.$ For purposes of analysis, treat the generator's line$-$to$-$neutral voltage phasor $V_{an}$ as the reference phasor. The generator has an armature resistance of $0\mathrm{.}5$ and a synchronous reactance of $1\mathrm{.}5.$ The generator has a mutual inductance $L_{sf}=1\mathrm{.}2H.$ The generator powers a $3-$phase load that draws $40$ kW at a power factor of $0\mathrm{.}809$ lagging at its rated terminal voltage level.

a$.$ Calculate the $3-$phase complex power $S_{3-\text{p}\text{h}\text{a}\text{s}\text{e}}$ for the load. $(3$ points$)$

b$.$ Calculate the line current phasor $I_a$ at the generator's terminals. Treat $V_{an}($line$-$to$-$neutral voltage at generator terminals$)$ as the reference phasor. $(5$ points$)$

c$.$ Calculate the excitation voltage $E_a$ for the generator. $(5$ points$)$

d$.$ Calculate the field current $I_f$ required to operate the generator under these conditions. $(2$ points$)$