Problem 107-6    Source:   Adapted from example 6-3 - page 326 - CHAPMAN, Stephen J. -    Book: Fundamentals of Electric Machines - 5th edition - Ed. McGrawHill - 2013.

A three-phase induction motor of 460 V, 4 poles, 60 Hz and 25 HP, connected in the Y configuration with 2.2% slip, it has the following impedances per phase referred to the stator circuit:

R₁ = 0.641 Ω    R₂ = 0.332 Ω

X₁ = 1.106 Ω    X₂ = 0.464 Ω    X_m = 26.3 Ω

Also, the total rotational losses are 1,100 W and are assumed to be constant. Core losses are included in rotational losses. Find the following quantities:

a) the speed of the rotor ;

b) the electric current in the stator, I₁ ;

c) motor power factor, PF.

Solution of the Problem 107-6   

Item a

The synchronous speed of this motor is given by eq. 107-02. Soon:

The synchronous angular velocity is:

The mechanical speed of the rotor is given by:

And the angular speed of the rotor is given by:

Item b

In order to calculate the electrical current of the stator, I₁, firstly, we must obtain the equivalent impedance of the entire equivalent circuit of the induction motor, as shows Figure 107-06 in the theoretical part. In this case, we will disregard the value of R_C. So, starting with the impedance of the rotor, we have:

Carrying out the calculation, we get:

Now, we see from the circuit that we must find the parallel of Z_r with X_m. We will name this impedance resulting from the parallel by Z_p. Using the basic rule of two impedances in parallel, we have:

Expanding the above relation, we have:

Carrying out the calculation, we get:

Note from the circuit that this impedance is in series with the stator impedance, represented in the circuit by R₁ and X₁. So by calculating this series we will find the equivalent impedance of the entire circuit. Like this:

Carrying out the calculation, we get:

Before we calculate the current in the stator, we must remember that the calculations carried out up to this moment were based on values ​​of a single phase. As the voltage provided by the problem is a line voltage, then we must find the phase voltage value. As the value of the voltage phase was not provided, then we make this phase our reference, that is, angle equal to zero.

Now we are able to find the value of the stator current and for that we will apply Ohm's law. Then:

Performing the calculate, we have:

Item c

The motor power factor is given by the cos of the equivalent impedance angle. Then: