THE ARMATURE REACTIONS OF ALTERNATORS 261 



law formerly announced and which is alluded to above. 1 Hence it 

 follows that the fluxes produced by multiple -coil windings differ but 

 little from the mean value of the theoretical fiuxes, and approach the 

 more nearly as the sections are the more numerous, and also as the 

 local variations or fluctuations of the curves between the extreme 

 forms i and 2 of the appended figures are damped out by the Foucault 

 currents of the neighboring pole pieces. The energy expended in 

 these Foucault currents, being supplied by the armature, is represented 

 by an augmentation of its apparent resistance r'. 



The reactions of two-phase armatures would be found in a similar 

 manner, and it will not be necessary to reproduce them more in detail. 

 Moreover, two-phase machines are more and more becoming sup- 

 planted by three-phase, and the latter present reactions of much 

 smaller fluctuations and a better utilization of materials, just as three- 

 phase motors are superior, from this standpoint, to two-phase motors. 



Comparison with Theoretical Coefficients. The theoretical coeffi- 

 cients are easy to establish in the case where a sinusoidal flux is 

 assumed and the harmonics are suppressed. 2 It is then demonstrated 

 that the magnetic potential produced by a polyphase winding of q 

 phases is independent of the number of phases and depends only upon 

 the total number of wires N per double field, and that it is repre- 

 sented by a sinusoid whose amplitude is 2NI . The mean potential 



2 



in the air-gap is therefore 27V/o and the equivalent mean magneto- 



7T 



motive force producing the reactive flux on closed circuit 



that is to say, (4/?r) 2 of the magnetomotive force which will give the 

 same turns if they coincide in position and phase. In this case the 

 sinusoid of potential (Fig. 17) is entirely used and the direct reaction 

 is proportional to the mean ordinate of the area AT,B, and the trans- 

 verse reaction to that of the area -$HBH'io. If, on the contrary, 



1 In alternators with distributed windings (Figs. 12 and 15), the diagrams 

 of windings 10 and 13 respectively may be employed by assuming that each coil 

 is replaced by a zone of wires occupying along the air-gap a breadth of -^ of 

 the field, of of the polar pitch, and having as median line the old outline of 

 the single bobbin which it replaces. 



2 See my above-mentioned memoir of 1899 upon "Rotating Magnetic Fields," 

 see also Arnold and la Cour's " Vorausberechnung der Ein und Mehrphasen- 

 stromgeneratoren." Stuttgart, 1901. 



