CHAP. IX] ARMATURE REACTION IN D.C. MACHINES 169 



For the sake of simplicity, we replace the actual pole shoe by 

 an equivalent one, without chamfer, of rectangular shape, and 

 with a negligible pole-tip fringing. The ordinates of the curve OX 

 represent to another scale the average flux density on the surface 

 of the pole shoe, because this density is proportional to the total 

 useful flux, or to the induced voltage. Let ab be the flux density 

 corresponding to the required induced e.m.f., and let Oa=Af be 

 the necessary net m.m.f ., without the transverse reaction. Let ac 

 and ad represent the distorting ampere-turns, A/2, at the pole-tips, 

 as given by eq. (95) ; then Oc and Od are the resultant m.m.fs. at 

 the pole-tips. The flux density at the pole-tips of the loaded 

 machine is then equal to eg and dh respectively. 



Since the distorting ampere-turns vary directly as the distance 

 from the center of the pole, and the area of strips of equal width 

 is the same, the abscissae from a represent to scale either distances 

 along the pole face, or areas on the pole face, measured from the 

 center of the pole. Thus, the part gh of the curve OX represents 

 also the distribution of the flux density under the pole, in the 

 loaded machine. Likewise, the line ef represents the distribution 

 of the flux density under the pole at no-load. 



Since the ordinates represent to scale the flux densities and the 

 abscissae the areas of the different parts under the poles, the area 

 under the flux density distribution curve also represents to scale 

 the total flux per pole. The total flux at no load is represented by 

 the area of the reactangle cefd, and to the same scale, the flux in t he 

 loaded machine is represented by the area cghd. If the saturation 

 curve were a straight line, the two areas would be equal, so that 

 the distortion would not modify the value of the total flux per pole. 

 In reality^ the area geb is larger than the area bhf, and the differ- 

 ence between the two represents the reduction in the flux, due to 

 the transverse armature reaction. 



Let now the field excitation be increased by an unknown amount 

 oa' to the value Oa'=M', hi order to compensate for the above 

 explained decrease in the flux. All the points in are shifted 



by the same amount, and are denoted by the same letters wit h t lie 

 sign " prime." The new flux in i he loaded machine is represented 

 by i lie area c'g'h'd', the point a' being the center of t ho line c'd' -erf. 

 If the point a' has been properly selected we must have the con- 

 dition that the 



area cefd -area c'g'h'd', (96) 



