GENERAL PRINCIPLES OF SYNCHRONOUS MOTORS 15 



single period. The fields turning in the same direction as the 

 inducing poles will have the same angular velocity , as the latter, 

 and will drag them in very much the same way as in polyphase 

 motors. On the other hand, the fields which turn in the opposite 

 direction will have a relative velocity, 2a, which is contrary to and 

 double that of the field-poles, so that their attracting or repelling 

 actions, since they succeed each other in inverse directions, will 

 produce no resisting torque. These reversed revolving fields will 

 give rise only to supplemental losses by hysteresis and by eddy 

 currents. 



By this simple analysis (which is, in reality, only approximate) 

 the operation of single-phase motors can, it is seen, be discussed and 

 explained in the same way as that of polyphase motors. 



It has been supposed, in what precedes, that the armature is station- 

 ary and the field movable. In the contrary case the explanation is 

 the same if we consider the relative velocities of the two portions, but 

 the fields displace themselves only with respect to the armature and 

 therefore remain stationary in space the same as the field-poles. 



Equations of Synchronous Motors. Analytical Theory. We have 

 just examined the phenomena of synchronous motors from a physical 

 point of view. We shall now represent them analytically, according 

 to the theory first expounded by Dr. J. Hopkinson, but with a few 

 modifications in form. We shall suppose with him that the E.M.F.'s 

 and currents follow the sinusoidal law, and that the reactances of the 

 machine are constant. 



Let us suppose, then, a single-phase A.C. generator and motor, 

 defined by their induced E.M.F.'s, their resistances, and their mean 

 inductances, which are all supposed constant. 



Let r=the duration of the period; 



27T 



a> = = the speed of pulsation of the currents; 



ei and 2 = the instantaneous values of the generator and motor 



E.M.F.'s respectively, at the instant t; 



E\ and 2 = the effective values equal to the amplitudes of the sine- 

 functions, i.e., the maximum value, divided by Va; 



-^ = the phase-difference between e\ and e^\ 

 # = the angle of lag (phase-difference) corresponding to 



