Problem 108-6    Source:   Adapted from Example 9-1 - page 595 - CHAPMAN, Stephen J. -    Book: Fundamentals of Electrical Machines - 5th edition - Ed. McGrawHill - 2013.

A split-phase, 1/3 HP, 110 V, 60 Hz motor with 6 poles has the following impedances:

R₁ = 1.52 Ω   X₁ = 2.10 Ω   R₂ = 3.13 Ω   X₂ = 1.56 Ω   and   X_m = 58.2 Ω

The core losses of this motor are 35 W, while the mechanical losses due to friction and ventilation and supplementary losses total 16 W. The motor is operating at nominal voltage and frequency with its starting winding open, and the motor slip is 5%. Find the following quantities under these conditions:

a) the speed in rpm;

b) the stator current;

c) the stator power factor;

d) the machine input power;

e) the machine air gap power;

f) the machine mechanical power.

Solution of the Problem 108-6   

Item a

To calculate the synchronous speed of the motor we will use eq. 107-02, repeated below.

Substituting the respective numerical values and performing the calculation, we obtain:

Now we can calculate the speed of the machine taking into account the motor slip of 5%, using eq. 107-05.

Item b

To calculate the stator current, we must first find the total impedance that the circuit offers to the voltage source. To do this, we must find the impedances of the forward and backward fields. We will use the approximate equations developed in the theoretical section. To find the impedance of the forward field, we use equations eq. 108-7a and eq. 108-7b, shown below.

Using the data provided in the problem and replacing the variables with their respective values, we can perform the calculation, finding:

Therefore, the progressive impedance is:

Now we must calculate the backward impedance. To do this, we will use equations eq.108-8a and eq.108-8b shown below.

Using the problem data and replacing the variables with their respective numerical values and performing the calculation, we find:

Therefore, the backward impedance is:

After these calculations, it is possible to calculate the value of the total impedance of the circuit, that is:

Using the calculated values, substituting them into the equation above and performing the calculation, we find:

With this data, it is possible to calculate the current flowing through the stator winding.

Performing the calculation, we obtain:

Item c

The stator power factor is given by the phase shift between the applied voltage and the stator current. Since we consider the voltage angle to be zero, the cos of the motor current angle is the power factor. Since the motor impedance is predominantly inductive, the power factor will be lagging. Therefore:

Item d

To find the motor input power, P_ele, we will use eq. 108-29.

Item e

The power in the machine air gap, P_gap, is given by eq. 108-13, repeated below.

Replacing the variables with their respective numerical values and performing the calculation, we obtain:

Item f

To calculate the machine's mechanical power, we use eq. 108-27, shown below.

Observation 😟

When solving this example in Chapman's book, there was a printing error in the progressive impedance value Z_P. This caused a propagation of errors in the calculations of some quantities requested in the problem statement. The correct values are those presented in the solution above. 👆   More details  "Click here"