160 THE MAGNETIC CIRCUIT [A RT . 51 



Another extreme assumption is that of poles without chamfer, 

 with a constant air-gap. Neglecting the fringe at the pole-tips, 

 f(x) = 1 from x =0 to x =0, and f(x) =0 from x =6 to x =$n. Inte- 

 grating eq. (89) between the limits and we obtain 



(90) 



The poles usually cover between 60 and 70 per cent of the periph- 

 ery. For 0=0.6(i7r) the preceding equation gives M d /M =0.86, 

 and for 6 =0.7 fa), M d /M=QM. 



Prob. 22. Let the permeance of the active layer decrease from the 

 center of the poles according to the straight-line law, so that 



/(*)-! -(2/*)s. 



What is the ratio of M d /M? Ans. 0.81 1 . 



Prob. 23. The permeance of the active layer decreases according 

 to a parabolic law, that is, as the square of the distance from the center 

 of the poles. What is the ratio of M d /M? Ans. 0.774. 



Prob. 24. The law f(x) =cos 2 x assumed in the text above presup- 

 poses that the permeance varies according to a sine law of double 

 frequency with a constant term, because cos 2 x = % + % cos 2x. In reality, 

 the permeance varies more slowly under the poles and more rapidly 

 between the poles than this law presupposes (Fig. 39). A correction can 

 be brought in by adding another harmonic of twice the frequency to the 

 foregoing expression, thus making it un symmetrical, and of the form 

 f(x) =a + b cos 2x+c cos 4x. Show that f(x) =2 cos 2 x cos 4 x contains 

 the largest relative amount of the fourth harmonic, consistent with the 

 physical conditions of the problem, and compare graphically this curve 

 with /(z)= cos 2 z. 



Prob. 25. What is the value of M d /M for the form of f(x) given in 

 the preceding problem? Ans. 0.815. 



Prob. 26. Plot the curve f(x) for a given machine, estimating the 

 permeances by Lehmann's method, and determine the value of the coef- 

 ficient in formula (79). 



51. The Calculation of the Value of the Coefficient of Transverse 

 Reaction in Eq. (81). The average value 0.33 of the ratio of the 

 maximum distorting armature m.m.f. to the equivalent number of 

 ampere-turns, M t , on the fictitious poles is given in Art. 48 without 

 proof. The following computations show the reasonable theoret- 

 ical limits of this ratio. The problem is more complicated than 

 that of finding the ratio of M d /M, because there the field ampere- 

 turns, the actual demagnetizing armature-m.m.f., and the equiva- 

 lent ampere-turns M d are all acting on the same permeance of the 



