A $208$V$,30$hp$,0\mathrm{.}85$PF leading, Y$-$connection, $60$ Hz synchronous machines has a synchronous reactance of $2\mathrm{.}2$ and a negligible armature resistance. Its friction and windage losses are $1\mathrm{.}2$ kW $,$ and its core losses are $1\mathrm{.}0$ kW $.$ Initially, the shaft is supplying a $10$ hp load, and the motor's power factor is $0\mathrm{.}80$ leading.

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

$(2)$ Assume that the shaft load is now increased to $25$ hp $.$ Sketch the behavior of the phasor diagram in response to this change.

$(3)$ Find $I_A,I_L,V_T$ and $E_A$ after the load change. What is the new motor power factor?