71 THE MAGNETIC CIRCUIT [ART. 31 



quontly, the vectors of the fifth harmonic are combined at an angle 

 of 120X5=600 degrees, or what is the same, 120 degrees. Therefore 

 the proportion of the fifth harmonic in the line voltage is the same as 

 that in the phase voltage. 



Prob. 32. Solve the foregoing problem when the winding pitch is 7/9. 

 Ans. 0.33 per cent. This shows that by properly selecting the 

 winding pitch an objectionable higher harmonic can be 

 reduced to a negligible amount. 



Prob. 33. Show that the line voltage of a Y-connected machine can 

 have no 3d, 9th, 15th, etc. harmonics, that is to say, harmonics the num- 

 bers of which are multiples of 3, no matter to what extent such harmonics 

 are present in the induced e.m.fs. in each phase. 



Prob. 34. Prove that in order to have even harmonics in the induced 

 e.m.f. of an alternator two conditions are necessary: (a) the flux distribu- 

 tion under the alternate poles must be different; (fe) the distribution of 

 the armature conductors under the alternate poles must also be different 

 from one another. Indicate pole shapes and an arrangement of the arma- 

 ture winding particularly favorable for the production of the second har- 

 monic. Note: In spite of a different distribution of flux densities the 

 total flux is the same under all the poles. Therefore, the average voltages 

 for both half cycles are equal (see Art. 24), though the shape of the two 

 halves of the curve may be different, due to the presence of even har- 

 monics. This shows that there is no " continuous- voltage component " 

 in the wave, or rather that the voltage is in no sense unidirectional, and 

 that a direct-current machine cannot be built with alternate poles without 

 the use of some kind of a commutating device. 



31. The Induced E.M.F. in a Direct-current Machine. The 



e.m.f. induced in the armature coils of a direct-current machine 

 (Fig. 20) is alternating, but due to the commutator, the voltage 

 between the brushes of opposite polarity remains constant. 

 This voltage is equal at any instant to the sum of the instantaneous 

 e.m.fs. induced in the coils which are connected in series between 

 the brushes. When a coil is transferred from one circuit to 

 another, a new coil in the same electromagnetic position is intro- 

 duced into the first circuit, and in this wise the voltage between 

 the brushes is maintained practically constant, except for the small 

 variations which occur while the armature is coming back to a 

 symmetrical position. These variations are due to the coils short- 

 circuited by the brushes and to the fact that the number of 

 commutator segments is finite. 



Thus, to obtain the value of the voltage between the brushes, 

 it is necessary to find the sum of the e.m.fs. induced at some 

 instant in the individual armature coils which are connected in 



