a$)$ For a Y$-$connected three$-$phase synchronous generator with a nonsalient rotor, the line$-$to$-$line terminal voltage under the open$-$circuit test with an excitation current of $3$ A is $220$ V $.$ When the generator supplies power to a Y$-$connected load with $\backslash (5\backslash$Omega $\backslash )$ resistor in each phase, an excitation current of $3\mathrm{.}8$ A is needed to maintain the terminal voltage at $220$ V $.$ Assume that the armature resistance is neglected, calculate the synchronous reactance.

b$)$ A Y$-$connected three$-$phase synchronous generator with a non$-$salient rotor has a synchronous reactance with per unit value as $\backslash ($ X$\_\{$C$\}=1\mathrm{.}8\backslash ).$ The rated power factor is $\backslash (\backslash$cos $\backslash$varphi$\_\{$N$\}=0\mathrm{.}87\backslash )($lagging$).$ If the armature resistance and leakage reactance are neglected, draw the corresponding phasor diagram and calculate the phase angle $\backslash (\backslash$theta $\backslash )$ between the terminal voltage and induced EMF with the rated conditions.

c$)$ With the same scenario of Q$2-$b$,$ the magnetic circuit is assumed to have linear characteristic. When the per unit value of the field current is $\backslash ($ i$\_\{$f$\}=1\backslash ),$ the corresponding per unit value of open circuit electromotive force $($EMF$)$ is $\backslash ($ E$\_\{0\}=1\backslash ).$ If the volume of air gap is doubled, calculate the per unit value of the corresponding synchronous reactance. Furthermore, calculate the per unit value of field current that can produce the rated open$-$circuit voltage.

d$)$ When a stationary synchronous motor is directly connected to a symmetrical three$-$phase power source, can this motor be started up and reach the synchronous speed? If yes, justify your answer; if not, propose a solution for that.