A $480V,45hp,0\mathrm{.}8PF$ leading, Y$-$connection, $60$ Hz synchronous machines has a synchronous reactance of $2\mathrm{.}0$ and a negligible armature resistance. Its friction and windage losses are $1\mathrm{.}5$ kW $,$ and its core losses are $1\mathrm{.}0$ kW $.$ Initially, the shaft is supplying a $25$ hp load, and the motor's power factor is $0\mathrm{.}8$ lagging. The field current $I_F$ at these conditions in $4\mathrm{.}0$A$.$

$(1)$ Sketch the initial phasor diagram of this motor, and find the values of $I_A,$ and $E_A.$

$(2)$ What is the converted power $P_{\text{c}\text{o}\text{n}\text{v}}?$

$(3)$ What is the efficiency $$ of the machine?

$(3)$ If the motor's flux is increased by $35$ percent, sketch the new phasor diagram of the motor. What are $E_A,I_A,$ and the power factor of the motor now?