GENERAL PRINCIPLES OF SYNCHRONOUS MOTORS 23 



But even for easy questions the use of this diagram is too com- 

 plicated; the same thing is true of the diagrams devised by Kapp, 1 after 

 Blakesley, and based on the same principles. The author has also 

 given 2 a more convenient method of representation of power by 

 circles which cut off segments directly from the lines OA and OB. 



In all that follows, we will give preference to another graphical 

 method, still more simple, which will be explained in Chapter II. 



Equation of the Synchronous Motor by the Method of Complex 

 Variables. Finally, to complete the review of the different methods 

 of study proposed or used for the representation and study of the 

 phenomena, attention should be given to the equations which translate 

 the preceding diagram by the method of complex variables, already 

 applied to alternating current problems, in the exponential form, by 

 Oberbeck, Cornu, Chaperon, etc., and in the linear form by Kennelly 

 Steinmetz, Guilbert, etc. It is merely an application of the ordinary 

 geometrical representation of imaginary quantities. 



We will employ the notation of Steinmetz modified by M. Guil- 

 bert (La Lumiire Electrique, Vol. L, p. 451, and VEdairage Electrique, 

 Vol. XIV, p. 69), i.e., by expressing the impedance as (r+si) instead 

 of (rsi), the latter expression being scarcely logical, since the reactance 

 acts in the same direction as a resistance to reduce the current. It 

 thus becomes possible to retain the ordinary axes. 3 



(i) Let OX, OY (Fig. 14) represent two rectangular co-ordinate 

 axes. Let OA be a vector repre- 

 senting the sinusoidal function whose 

 projections are x and y. Geomet- 

 rically, the vector A is defined by the 

 two projections, and analytically it 

 is represented by a single imaginary 

 value 



x+yj=(A cos 6) + (A sin 



in which j is the imaginary sym- FIG. 14. 



bol V^T. 



All the harmonic functions may be also represented by imaginary 

 quantities. 



1 Electrical Transmission of Energy. London, Whittaker, 1892. 



2 La Lumiere Electrique, Vol. XLV, 1892, p. 415, and Bulletin de la Societe des 

 Electriciens, 1893. 



3 Compare with C. P. .Steinmetz, Theory of the Synchronous Motor (Transactions 

 of the American Institute of Electrical Engineers, Oct. 17, 1894) 



