A $3-$phase induction motor, wye connection, $60$ Hz is connected to a $380$V

source. This $380$ V is the line voltage of a $3-$phase supply source. The slip is

$25\%$ and rotor speed is $900$ rpm$.$ The equivalent circuit per phase has the

following parameters:

R$1=$ Stator resistance $=0\mathrm{.}31$

X$1=$ Stator leakage inductive reactance $=1$

R$2=$ Rotor resistance $=0\mathrm{.}82$

X$2=$ Rotor leakage inductive reactance $=3\mathrm{.}5$

Rc $=$ no$-$load loses resistance $=$ Infinity

Xm $=$ magnetizing reactance $=10$

$($a$)$ Prove that the number of poles is $6.$

$($b$)$ Prove that the input impedance of this motor is $2+$ j$4($or $4\mathrm{.}47263\mathrm{.}435$o $)$

$($c$)$ Based on the value givenv$($or calculated$)$ in part $($b$),$ calculate the input

power.

$($d$)$ Calculate the mechanical power

$($e$)$ Find the developed torque

$($f$)$ Find the efficiency