26 SYNCHRONOUS MOTORS 



As for P 2 , it can, according to the rule already stated, be expressed as 

 follows: 



E 2 cos 6[R (Ei - E 2 cos 0) + ajLE 2 sin d] 



R 2 +aj*L 2 -E 2 sin 0[RE 2 sin 6-uL(E l -E 2 cos 0)] 



= (E lC os( r -6)-E 2 cos r ]. 



VR 2 +a> 2 L 2 



This expression could have been written directly, by symmetry. 



Such is the method of imaginaries for establishing the fundamental 

 equations. It will be seen that the calculations thereby made are more 

 simple than with the analytical method of Hopkinson, because they 

 give, immediately, the general solution, and do not require any inte- 

 gration. In reality, however, complex variables only constitute an 

 artifice for writing down the results of the graphical method. In place 

 of detailed reasonings which are rendered more precise by means of 

 diagrams, they substitute algebraical operations, which are effected 

 mechanically, without benefit in helping the mind to understand the 

 physical phenomena. In all that follows we will therefore adopt the 

 graphical method. 



Excitation of Synchronous Motors. Synchronous motors may be 

 excited in three ways: 



(1) As in the case of alternators, it is possible to use the current 

 produced by a small exciter-dynamo mounted on the same shaft. 

 The machine is put in operation by one of the methods indicated later, 

 or else by running the exciter-dynamo as a motor, by means of current 

 from a storage batter}'. This method of excitation is advantageously 

 used for large motors. 



(2) To avoid the complication of the exciter-dynamo, especially for 

 small motors, the machine is often made self-exciting by means of corn- 

 mutated currents, by sending through the field-circuit a portion of the 

 current obtained from the mains. The oldest method of commutating 

 this current consisted in using a simple shell-commutator by which the 

 current was reversed at each change of polarity. Such was the Ganz 

 commutator shown diagrammatically in Fig. 15 for a six-pole motor. 

 The inducing circuit connects with three segments p, p', p" connected 

 in parallel with one pole, and the other end of the circuit connects 

 with three other segments n, /, n", which are connected in parallel 

 with the other pole. The alternating current to be commutated is 



