CHAP. VI] EXCITING AMPERE-TURNS 101 



parts, where n is an even number. Then, we have that 



on 



+2(H 2 + H 4 +... + H n - 2 )]. . (53) 



When the flux density in the teeth is considerable, say between 

 18 and 24 kilomax wells per square centimeter, an appreciable part 

 of the total flux passes through the slots between the teeth, also 

 through the air-ducts, and in the insulation between the lamina- 

 tions. Dividing, therefore, the flux per tooth pitch by the net 

 cross-section of the tooth, one gets only the so-called apparent flux 

 density in the tooth, which density is higher than the true density. 

 With highly saturated teeth, a small difference hi the estimated 

 flux density makes an appreciable difference in the required number 

 of ampere-turns; it is therefore of importance to know how to 

 determine the true density in a tooth, knowing the apparent 

 density. 



Consider first the case of a machine with a large diameter, in 

 which the taper of the teeth can be neglected. Assume the con- 

 centric cylindrical surfaces at the tips and at the roots of the teeth 

 to be equipotential surfaces, and the lines of force to be all parallel 

 to each other, hi the slots as well as in the iron. In reality, some 

 lines of force enter the teeth on the sides of the slots (Fig. 24), so 

 that the foregoing assumptions are not quite correct; but they are 

 the simplest ones that can be made. Any other assumptions 

 would lead to calculations too complicated for practical use. 



Let B real be the true flux density in the iron of the tooth, and 

 let B npp be the apparent flux density hi the tooth under the assump- 

 tion that no flux passes through the slots, air-ducts, or insulation 

 between the laminations. Then, denoting the actual flux density 

 in the air by B a , we have the following expression for the total 

 flux per tooth pitch : 



A iB a pp A iB^i + A a B a , 



where Ai and A a arc the cross-sort inns in square rent imeters of tho 

 paths per tooth pitch, in the iron and air respectively. Sinee tin- 

 iron and tin- air paths are of equal length, and an- in parallel, the 

 in. in. f. gradient is the same in Ix.th. l.<t // U this gradient . in 



