80 ALTERNATING CURRENTS 



and is increasing in a positive direction when the time t is 0. 

 Obviously the voltage induced in coil 6 will be 120 electrical tiino- 

 degrees behind E ca and that induced in coil c will be 240 elec- 

 trical time-degrees behind E oa , as shown in Figs. 77 (a) and 77 

 (6). These three voltages constitute the elementary voltages 

 generated in a three-phase system. 



An examination of Fig. 77 (a) shows that for any particular 

 instant of time, the algebraic sum of these three voltages is zero. 

 When one voltage is zero, the other two are 86.6 per cent, of their 

 maximum values and have opposite signs. When any one 

 voltage wave is at its maximum, each of the others has the 

 opposite sign to this maximum and each is 50 per cent, of its 

 maximum value. 



Figure 77 (6) shows the vectors representing these three volt- 

 ages, the vectors being 120 apart. 



Each of the coils of Fig. 76 (a) can be connected through its 

 two slip-rings to a single-phase circuit. This gives six slip-rings 

 and three independent single-phase circuits. With a rotating 

 field and stationary armature type of generator, which is the 

 most common type met in practice, the six slip-rings would 

 not be necessary, but six leads would be taken directly from the 

 armature. 



In practice, however, a machine seldom supplies three inde- 

 pendent circuits by the use of six wires. 



45. Y-connection. The three coils of Fig. 76 are shown in 

 simple diagrammatic form in Fig. 78. The three corresponding 

 ends, one for each coil, are tied together at the common point o. 

 This is called the Y-connection of the coils. Ordinarily only 

 three wires, aa', W and cc f , lead to the external circuit, although 

 the neutral wire oo' is sometimes carried along, making a three- 

 phase, four-wire system. 



Figure 79 (a) again shows the three coils and Fig. 79 (6) the 

 three corresponding voltage vectors, E oa , Bob, and E oc . These 

 three voltages are called the coil or Y-voltages. Let it be 

 required to find the three line voltages E a b, E bc and E ca . The line 

 voltage f ab = E ao + #<* (Par. 43). E ao is not on the original 

 diagram but is obtained by reversing E oa . E ao is then added 

 vectorially to E^ giving E a b- 



From geometry, E^ lags the coil voltage E b by 30 and is 



