CHAP. VIII] REACTION IN SYNCHRONOUS MACHINES 159 



electrical radian is (Pf(x). The function/(z) must be periodic and 

 such that /(O) = 1, and/(i*) =0, /(*) = 1, etc., because the perme- 

 ance reaches its maximum value under the centers of the poles 

 and is practically nil midway between the poles. 



The direct armature m.m.f., acting alone, without any excita- 

 tion on the poles, would produce in each half of a pole a flux 



The magnetomotive force M d placed on the real poles, acting 

 alone, must produce the same total flux, so that 



Equating the two preceding expressions we get 



MC**cosxf(x)dx=M d C**f(x)dx. . . . (89) 



The ratio of M d to M can be calculated from this equation, l>y 

 assuming a proper law/(x) according to which the permean 

 the active layer varies with x, in poles of the usual shapes. Hav- 

 ing a drawing of the armature and of a pole, the magnetic field can 

 be mapped out by the judgment of the eye, assisted if necessary 

 by Lehmann's method (Art. 41 above). A curve can then U> 

 plotted, giving the relative permeances per unit peripheral length, 

 against x as abscissae. Thus, the function /(.n is -i\ vn irraphirally, 

 and the two integrals which enter into eq. (89) can be determined 

 graphically or be calculated by Simpson's Rule. Or else, 

 f(x) can be expanded into a Fourier series and the integration 

 performed analytically. Such calculations performed on poles of 

 the usual proportions give values of Mj/M of between 0.81 and 

 0.85. 



It is also possible to assume for/(x) a few simple analytical 

 expressions, and integrate eq, < s ') directly. Take for in 

 /(x)=cos 2 ar. By plotting this function against x as abscissas the 

 reader will see that the function heroines /.em midway between 

 the poles, is equal to unity opposite th 9 of the polos, and 



has a reasonnhle .irenend shape at intermediate points. Suhsti- 

 tuting cos 2 x for /(j) int. ') and inte-ratii j.U - 



from which 



