A $208-$V Y$-$connected synchronous motor is drawing $50$ A at unity power factor from a $208-V$ power system. The field current flowing under these conditions is $2\mathrm{.}7$ A $.$ Its synchronous reactance is $1\mathrm{.}6.$ Assume a linear open$-$circuit characteristic.

$($a$)$ Find $v_q$ and $E_1$ for these conditions.

$($b$)$ Find the torque angle $.$

$($c$)$ What is the static stability power limit under these conditions?

$($d$)$ How much field current would be required to make the motor operate at $0\mathrm{.}80$ PF leading?

$($e$)$ What is the new torque angle in part $($d$)?$