Problem 108-9    Source:   Example 9.2 - Electrical Machines - Umans, Stephen & Fitzgerald - 7th edition - McGraw Hill Education - 2014

A 1/4 HP, 110 V, 60 Hz, 4-pole capacitor-start motor has the following equivalent circuit parameters:

R₁ = 2.02 Ω  R₂ = 4.12 Ω   X₁ = 2.79 Ω   X₂ = 2.12 Ω   and   X_m = 66.8 Ω  

The motor operates at nominal voltage and frequency with the auxiliary winding open. Knowing that core losses are 24 W, rotational losses are 13 W, and slip s = 0.05, determine:

a) the stator current;

b) the power factor, PF, of the motor;

c) the nominal power;

d) the synchronous angular velocity;

e) the torque on the motor shaft;

f) the efficiency of the machine.

Solution of the Problem 108-9   

Item a

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 progressive and retrograde fields. We will use the approximate equations developed in the theoretical section. To find the impedance of the progressive field, we use equations eq. 108-7a and eq. 108-7b, shown below.

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

Therefore, the progressive impedance is:

Now we must calculate the retrograde 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 values and performing the calculation, we find:

Therefore, the retrograde 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:

Now we can calculate the current flowing through the stator winding.

Performing the calculation, we find:

Item b

To find the power factor, simply use the stator current angle, since we adopted a zero angle for the voltage. Since the motor impedance is predominantly inductive, the power factor is lagging, or:

Item c

To find the nominal power, we will calculate the air gap power using eq. 108-13, repeated below.

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

With the value of the power in the air gap, we can calculate the value of the mechanical power using eq. 108-27, shown below.

Therefore, replacing the variables with their respective numerical values and performing the calculation, we find:

To find the nominal power, we must subtract the losses mentioned in the problem from the mechanical power, that is:

Item d

To find the synchronous angular velocity we will use eq. 108-14 shown below.

Therefore, replacing the variables with their respective numerical values and performing the calculation, we find:

Item e

To calculate the torque on the motor shaft we must know the angular speed of the rotor, or:

Then, using eq. 108-17 shown below.

Therefore, replacing the variables with their respective numerical values and performing the calculation, we find:

Item f

To find the machine's efficiency, we will calculate the input power, P_ele, using the equation eq. 108-29, shown below.

So, replacing the variables with their respective numerical values and performing the calculation, we find:

Now using the equation eq. 108-31, the yield will be:

Power Balance

Let's check whether the powers involved in the problem are in accordance with energy conservation. The power dissipated in R₁, R_P, and R_R are:

Adding these three powers, we find P_sum = 61.15 W. To this value we must add the losses constant in the problem statement. Therefore:

Também podemos calcular as perdas totais da máquina, fazendo:

Note that this small difference of 2 W is within the tolerance margin. All the results of the calculations performed with our approximated equations, according to the theoretical part, fully agreed with the results obtained in the source book.