118 SYNCHRONOUS MOTORS 



Integrating between /o and /, the work done is 



- 9 I sin (aj 2 t-0) sin (atf-O-ftdt. 

 2 



While the second term increases indefinitely, the first is doubly 

 periodical and has the form 



sin co\x sin o^xdx. 



It is known that an integral of this form, taken within limits which 

 comprise a whole number of periods of aj^x and of a} 2 x t is equal to 

 zero, so long as a>\ is different from co 2 . If io\ and oj 2 are very dif- 

 ferent from each other, their zeros are very near each other and the 

 amplitude is very small. On the contrary when w 2 is very near io\ 

 the periodicity of the integral increases in length to the point of becoming 

 infinite. 



Two cases are therefore to be considered: 



i) So long as the speed of rotation is low, the first term is negligible 

 in comparison with the second, provided that the inertia in the motor 

 is sufficient and that only negative work is done. We then have 



T=-- ( E 2 ) / v u /_cos0 



f\ I y.l- / A / O I OO 



v r*2~t 



2 rC/_i 



1 (taz F \ 

 = 1 2 I 



2 \ Wl / 



It is therefore necessary to apply mechanical power to turn the 

 alternator and with Q 2 = the speed corresponding to the period I 2 , and 



] = the speed corresponding to the condition! Q 2 =Q 1 }. the motor 



\ "V 



torque C u will have the following expression: 



