92 THE MAGNETIC CIRCUIT [ART. 37 



the air-pap at a distance x from the center is a=a +4a(x/w) 2 . The per- 

 meance ol'an infinitesimal path of the width dx is proportional to dx/a x . 

 have the relation 



dx/[a +Aa(x/w) 2 ]= w/a eq , 

 o 



from which 



a eq = Va^a/tan" 1 (X/Ja/a ) . 



Prob. 17. What is the length of the equivalent air-gup in the preced- 

 ing problem if the clearance at the pole-tips is twice the clearance at 

 the center of the pole? Ans. 1.273a - 



Prob. 18, Show that, when the air-gap is non-uniform, the length 

 of the equivalent uniform gap can be determined approximately, accord- 

 ing to Simpson's Rule, from the equation 



where a , fli, and dm are the lengths of the gap at the center, at the tip of 

 the pole, and midway respectively. If the air-gap is uniform under the 

 major portion of the pole, but the pole shoe is chamfered, more terms 

 must be taken in Simpson's formula in order to obtain a eq with a 

 sufficient accuracy. 



Prob. 19. What is the length of the air-gap required in problems 

 16 and 17, according to the formula given in problem 18? 



Ans. 1.276a . 



37. The Method of Equivalent Permeances for the Calculation 

 of Air-gap Ampere-turns. An inspection of Fig. 24 will show that 

 the total permeance of the air-gap is made up of a number of per- 

 meances in parallel. It is equal therefore to the sum of these 

 permeances. For the purpose of calculation two kinds of per- 

 meances are considered separately: those from the teeth to 

 the pole surface proper, and those from the teeth to the pole-tips. 

 The former can be calculated quite accurately, the latter are to 

 some extent estimated. 



The permeance per tooth pitch in the part of the air-gap near 

 the center of the pole can be divided into two parts, that under the 

 tooth-tip, and the fringe from the sides of the slots and in the ven- 

 tilating ducts. The permeance of the paths which proceed from 

 the tooth-tip constitutes the larger portion and is made up of 

 nearly parallel lines; this permeance is therefore easily computed. 

 The values of the permeance of the fringe from the flank of the 

 tooth to the perpendicular surface of the pole have been deter- 



