264 METHODS OF CALCULATION 



mean flux-density is then known which must be developed in the 

 air-gap, the teeth, and the pole pieces, and, moreover, one has from 

 the curves of potential the value of the magnetomotive forces acting 

 at all points of the air-gap. From this may be deduced the real 

 flux-density at every point, and consequently the true variation of 

 the total flux produced by the reaction. These expressions of self- 

 inductions given above should be consequently replaced by the integrals 

 of the form 



N c( 

 = ^= I I o.^ 



O2\' 2/oJ \ 



indicating by x the abscissa, and by y the ordinate of the curve, and 

 R the reluctance per unit of surface at this point, b represents the 

 length of the armature, q the number of phases, N the number of 

 peripheral wires per field. 



Case of Single-phase Alternators. The problem of the reactions 

 of single-phase alternators is more complex than that of polyphase 

 alternators, as will be seen. It does not appear to have been fully 

 understood by the authors who have previously treated it. It may 

 be analyzed by the same method as that which I have formerly developed 

 for asynchronous motors, 1 but taking into account this important 

 difference, that, in general, the rotating reactions (rotating magnetic 

 field with respect to the armature, which we suppose fixed), are sup- 

 pressed in motors by the short-circuited windings on the rotor, but 

 not. in alternators, except in the case where they are furnished with 

 massive poles, or especially with the damping windings of Leblanc 

 (which, however, only give a partial suppression). 



I have shown that each coil of a single-phase armature produces 

 a reactive flux capable of being decomposed in space into sinusoidal 

 harmonics, of which the first, the only one which need be considered 



in practice, has for amplitude - , denoting by nl, the ampere- 



K 



turns of the coil, and by R, the reluctance of the magnetic circuit 

 traversed by the flux which it produces. In accordance with the 

 theorem of Leblanc, I decompose this pulsating sinusoid into two 

 rotating sinusoids of one-half amplitude: one turning synchronously 

 with the rotor, and not displaced therefore with respect to the 

 field-poles; it provides only a fixed reaction; the other rotate 



1 Blondel, "Properties of Rotating Magnetic Fields," Eclairage Eledrique, 

 May, 1908. 



