THE ARMATURE REACTIONS OF ALTERNATORS 265 



in the opposite direction with the same speed and is therefore 

 displaced with respect to the field magnets with a speed double that 

 of synchronism ; this gives rise to a pulsating flux in the field magnets 

 of frequency double that of the e.m.f. induced in the armature, as 

 I have also shown experimentally. 1 This reaction is somewhat weakened 

 by the currents which it produces in the closed circuit of excitation, 

 but it is not completely extinguished and produces therefore an e.m.f. 

 of normal frequency in the armature which should be taken into con- 

 sideration. But, while the fixed reaction may be analyzed, exactly 

 as in polyphase alternators, into direct and transverse reactions, the 

 parasitical rotating reaction is effected directly through the field mag- 

 nets, as well as transversely through the pole pieces, and may be 

 represented, consequently, by a mean coefficient of self-induction 

 similar to the self-induction of the stray field cos, to which it is added. 

 It may be directly demonstrated by calculations that this analysis 

 readily lends itself to the interpretation of the facts, 2 and that a 

 single-phase alternator whose armature at rest presents a self-induction 



1 Blondel, "Photographic Record of Periodic Curves," Lumiere Electrique, 

 August, 1891. 



2 In fact, the seff-induction of the armature L, varying between the two 

 values X and X' according to position, may be represented by an expression 



L=A + B cos 2<ut + ti)S 

 and abbreviating, 



indicating always by s the inductance of the stray fields in the slots. Let 

 e = E s'mii}t be the internal E.M.F., and z = 7 sin (oi/ <p) the strength of the 

 current, and there is immediately obtained as the difference of potential at the 

 terminals of the machine the following expression: 



r di dX di 



u = e-ri-L-iws 



at tit dt 



Replacing e, L, i by their values, and neglecting a term cos ($wt</>) which 



produces an upper harmonic, there remains 



M=,)Sin aitrI sin (wt ) to I A -- j 7 cos SF cos utt 



w (A H I 7 sin $? sin uit wsl^ cos (wt ^) 

 = E sin wt rlo sin (wt ) -7, cos (wt ) 7 cos * cos wt 



-- 7 sin SF sin att ws7 cos (a)^f). 



I X' 



The values of and representing that which we have uniformly denoted 



by / and /' for all the machines in the construction of the diagrams. 



