Problem 108-8    Source:   Problem prepared by the site authors.

A split-phase, 127 V, 60 Hz motor with 4 poles has the following impedances:

R₁ = 0.55 Ω   X₁ = 0.83 Ω   R₂ = 1.70 Ω   X₂ = 0.72 Ω   and   X_m = 41.5 Ω

The core losses of this motor are 57 W, while the mechanical losses due to friction and ventilation and supplementary losses total 17.7 W. The motor is operating at nominal voltage and frequency with its starting winding open, and the motor slip is 6.5%. Find the following quantities under these conditions:

a) the stator current;

b) the stator power factor;

c) the machine input power;

d) the machine air gap power;

e) the machine rated power;

f) the machine efficiency.

Solution of the Problem 108-8   

Item a

To calculate the stator current, we need to calculate the total impedance of the motor. To do this, we must calculate Z_P and Z_R. We will use the approximate equations developed in the theoretical section.

For the progressive field impedance, 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 numerical values, we can perform the calculation, finding:

Therefore, the progressive impedance is:

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

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 b

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 c

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

Item d

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

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

Item e

To calculate the machine's nominal power, we must first calculate the copper losses. Then, we will calculate P_Pgap and P_Rgap using equations eq. 108-11 and eq. 108-12.

Note that if we perform the subtraction between P_Pgap and P_Rgap we will find the value of P_gap, as calculated in item d. Then, with this data, we can calculate the losses in the copper of the rotor due to the forward and backward fields given by eq. 108-21, shown below.

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

Therefore, the above value represents the copper losses of the rotor only, as previously stated. To calculate the total losses of the MOTOR, we must add the losses in the motor stator, which are caused by the presence of R₁. So, let's calculate P_R1, or:

With this data, we are able to find the total copper losses of the MOTOR, using eq. 108-22, shown below.

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

Using eq. 108-30 (shown below) and performing an algebraic transformation, it is possible to calculate the rated power of the machine using the values calculated in the problem solution.

Item f

The machine's efficiency is given by eq. 108-31. Therefore: