CHAP. V] EXCITING AMPERE-TURNS 97 



gap (Fig. 23) : At each instant some stator teeth are opposite rotor 

 teeth, others bridge over some rotor slots, and vice versa. The 

 amount of overlap varies from instant to instant, causing periodic 

 fluctuations in the air-gap reluctance. 



Assume first that both the stator and the rotor have smooth 

 surfaces facing the air-gap. Let the permeance of such a machine 

 be (P 9 . If now the armature be slotted, the cross-section of 

 the paths in the air-gap (neglecting the fringe) is reduced in the 

 ratio ti/Ai where t\ and A x are the stator tooth width and tooth 

 pitch respectively. The permeance (P, is also reduced in the same 

 ratio. Let the rotor be also provided with slots; the average 

 cross-section of the path is thereby further reduced in the ratio 

 fe/^2), where 2 and ^2 are the tooth width and the tooth pitch on 

 the surface of the rotor. Thus, disregarding the spread of the 

 flux, the average air-gap permeance of an induction motor is 



the symbol (P a being put in parentheses to indicate that a further 

 correction for the tooth fringe is necessary. 1 



In order to take the fringe into consideration, an empirical cor- 

 rection is made hi this formula. Namely, it is assumed that the 

 actual permeance of the fringes of the stator teeth is the same as 

 if the rotor had a smooth core, and vice versa. Accordingly, in tin 1 

 preceding formula, the values t\ and li of the tooth widths are 

 corrected for the fringe, using Carter's curve (Fig. 26). The 

 formula becomes then 



^=(*i'A)(' 2 'A)<p., ..... 



where l\ and t 2 ' are the corrected widths of the stator and rotor 

 teeth respectively. This formula has been found to U- in :i satis- 

 factory agreement with experimental results. 2 



r a more rigorous proof of this formula see C. A. Adams, "A Study 

 in th<- Design of Induction Motors/' Trans. Amer. Inst. Elcctr. Engs., Vol. 24 

 (190:. 



2 T. I . Wall. Tin- Ilcluctance of tin- Air-^ip in Dynamo-machines, Joum. 

 In*t. Electr. Engrs. (K.in-l, . Vol. 40 (1907-8), p. 568. E. Arnold in his 

 WechseUtromtechnik, Vol. 5, Part 1, pp. U. i:t. calculates the value of * 

 for :m induction machine in a somewhat different way. With open slots 

 'Id's met!. ;.>\ver values of k a than they an- in reality. 



Sec Hoock and ilellmund, Bcitrag sur Berechnung des Magnetiiierungs- 



