162 THE MAGNETIC CIRCUIT [ART. 51 



In order to determine M if the usual method is to mulitply both 

 sides of this equation by sin x and integrate between and TT } 

 because then all upper harmonics give terms equal to zero. In 

 this particular case the limits of integration can be narrowed down 

 to and JTT, because the symmetry of the curve is such that the 

 segment between and JT: is similar to all the rest. Thus, we 

 get 



M r t *sm 2 xf(x)dx=M t f^sin xf(x + fr)dx. . (92) 



From this equation the ratio M t /M can be calculated by the 

 methods shown in Art. 50, i.e., by assuming reasonable forms of 

 the function f(x). Taking again f(x) =cos 2 x and integrating eq. 

 (92) we get &KM=$M t , from which M t /M =0.295. Taking the 

 other extreme case, viz., f(x) =1 from x=Q to x=0, and f(x) =0 

 from x =6 to x =$n, gives, after integration 



(93) 



For 0=0.6(4;:), M,/M=0.29; for d*=Q.7(fr), M,/M=0.39. 1 It 

 will be noted that the cross-magnetizing action of the armature 

 increases considerably with the increasing ratio of pole-arc, to pole- 

 pitch, while the direct reaction slowly diminishes with the increase 

 of this ratio. In machines intended primarily for lighting pur- 

 poses it is advisable to use a rather small ratio of pole-arc to pole- 

 pitch, in order to reduce transverse reaction which affects the volt- 

 age regulation at high values of power-factor in particular. 



Prob. 27. What is the value of Mt/M for the form of f(x) given in 

 problem 24; namely, for /(a;) = 2 cos 2 a; -cos 4 z? Ans. 0.368. 



Prob. 28. Determine the numerical value of the coefficient in formula 

 (81) for the machine used in problem 26. 



1 These values are higher than those derived by E. Arnold. The fact 

 that Arnold's values for the coefficient of transversal reaction are low has 

 been pointed out by Sumec in Elektrotechnik und Maschineribau, 1906, p. 

 67; also by J. A. Schouten, in his article " Ueber den Spannungsabfall mehr- 

 phasiger synchroner Maschinen, " Elektrotechnische Zeitschrift, Vol. 31 (1910), 

 p. 877. 



