Problema 105-1    Source:   Adapted from Example 3-2 - page 177 - CHAPMAN, Sthephen J. -    Book: Fundamentos de Máquinas Elétricas - 5ª Edição - Ed. McGraw-Hill - 2013.

Let's assume a simple two-pole generator. The magnetic induction field of the rotor, B_R, is worth 0.2 T and the mechanical rotation speed of the shaft is 3,600 rpm. The diameter of the machine stator is 0.5 m, the length of its coil is 0.3 m and there are 15 turns per coil. The machine is connected to Y. Determine:

a) What are the three phase voltages of the generator as a function of time?

b) What is the effective phase voltage of this generator??

c) What is the effective voltage at the terminals of this generator??

Solution of the Problema 105-1   

Item a

To solve this problem we must remember eq. 75-14   see here!. As the value of the diameter and length of the rotor has been provided, then it is possible to calculate its area, because we know that in a cylinder the area is A = D x L. Therefore, we can write:

To find the operating frequency of the generator we can use eq. 105-01, shown below.

By substituting the respective numerical values ​​and carrying out the calculation, we find

With this data and using eq. 105-2a, repeated below, we can calculate the value of peak or maximum of the induced voltage, because in the calculation of the magnetic flux we use the maximum area of ​​the loop in the cylinder. That way, the eq. 105-2a will give the maximum value of the induced voltage.

By substituting the respective numerical values ​​and carrying out the calculation, we find

Let's write in trigonometric form the three phase voltages produced by the generator.

Where the value 377 is the value of ω = 2 π f_e.

Item b

To find the effective phase voltage we must divide the peak value by √2, or:

Item c

As the generator is connected to Y, the line voltage is equal to the phase voltage multiplied by √3, that is: