23S THE MAGNETIC CIRCUIT [ART. 68 



outside the armature windings, and beside there are two short - 

 circuitin<r paths through the coils undergoing commutation: a 

 long path and a short path. Thus, the problem becomes 

 indefinite, because it is not possible to tell the relative amounts of 

 the current through these different paths. Disregarding the short 

 path, the criterion becomes the same as in the case of a machine 

 with two sets of brushes only. 1 This is on the safe side, and 

 the commutation may be expected to be better than that cal- 

 culated, or at the worst, as good. 



With multiplex windings, the expression for e ave is the 

 same as that given above, provided that proper values are selected 

 for /!, s, and /. With regard to the latter quantity it must 

 be remembered that b' is much larger than the actual thickness 

 of mica. Namely, with respect to the component winding under 

 consideration the metal of the commutator segments belonging 

 to the other component windings is equivalent to insulation. 

 This fact must not be lost sight of in choosing the correct value 

 for b f to be used in the expression (158). 



With fractional-pitch windings the reactance voltage is smaller 

 than with the corresponding full-pitch winding, because the 

 conductors short-circuited under the adjacent sets of brushes 

 (Fig. 57) are situated in part or totally in different slots, and 

 have a smaller common magnetic flux, or none at all. When 

 the winding pitch is reduced considerably, s instead of 2s must 

 be used in the preceding formulae; otherwise a value between 

 s and 2s must be chosen, according to one's judgment. 2 



Prob. 22. The armature of a 6-pole, 600-r.p.m., multiple-wound, 

 direct-current machine has the following dimensions: Diameter, 85 cm.; 

 gross length, 22 cm.; three air-ducts, 1 cm. each; 1008 face conductors. 



1 For an analysis of this case see C. A. Adams, Reactance E.M.F. and 

 the Design of Commutating Machines, Electrical World and Engineer, Vol. 

 46 (1905), p. 346. 



2 For an advanced and more scientific theory of commutation, see Arnold, 

 Die Gleichstromaschine, Vol. 1 (1906), pp. 354 to 513; in particular the 

 approximate formula (170) on bottom of p. 498; also Vol. 2 (1907), chapter 

 14. A simpler and more concise treatment will be found in Tomiilen's 

 Electrical Engineering. A good practical treatment will also be found in 

 Pichelmayer's Dynamobau, pp. 86-118; it is considerably simplified as 

 compared to Arnold's treatment, and is accurate enough for practical pur- 

 poses, because the numerical values of unit permeances are known only 

 approximately. 



