A $\frac{220}{380}V,4-$pole,$50-Hz,$ Y$-$connected induction motor has the following parameters:

$R_1=0\mathrm{.}08,x_1=0\mathrm{.}4,R_2^{\text{'}}=0\mathrm{.}05,x_2^{\text{'}}=0\mathrm{.}6,$ and $x_m=15$

Neglect core and rotational losses. Rotor frequency

The motor runs at the speed of $1440$ rpm $,$

Calculate,

a$.$ Motor line current.

b$.$ Motor power factor.

c$.$ Rotor copper losses.

d$.$ Developed torque.

e$.$ Motor efficiency.

f$.$ Maximum torque.

State whether the speed $1050$ rpm is stable or not and explain why.

Determine the slip of maximum torque and maximum torque when the rotor resistance is doubled.

Determine the motor efficiency at starting.

Determine the developed starting torque.

Determine the motor developed torque at synchronous speed.

The motor is braked by plugging, determine immediately after plugging:

a$.$ The rotor frequency

b$.$ The braking developed torque

c$.$ Motor efficiency

The motor is braked by regenerative method by reducing the motor phase voltage and frequency to $132$ V and $30$ Hz $.$ Calculate the developed torque immediately after regenerative braking.