CHAP. V] EXCITING AMPERE-TURNS 93 



mined theoretically by Mr. F. W. Carter. 1 Only the numerical 

 results are given here, in a somewhat simplified practical form; 

 the solution itself presupposing a knowledge of the properties of 

 conjugate functions. 2 



Consider the permeance of two tooth fringes, such as opqr and 

 o'p'ffr 1 (Fig. 24), perpendicular to the plane of the paper. This 

 permeance depends only upon the ratio of the slot width s to the 

 length a of the air-gap, for let both s and a be increased say twice : 

 The length and the cross-section of each elementary tube of force 

 is also increased twice, hence its permeance remains the same. 



The permeance of each fringe can be replaced by the permeance 

 of an equivalent rectangular path of the length a and of a width 

 AJJ (Fig. 26). This is the same as increasing the width of the 

 tooth by the amount Jt and assuming all the lines of force to be 

 parallel to each other in the air-gap. The permeance of the path 

 which replaces the two fringes is equal to fiJt/a. From what has 

 been said^above follows that the ratio Jt/a depends only upon the 

 ratio of s/a; the relationship between the two ratios is plotted in 

 Fig. 26, from Carter's calculations. For the sake of convenience 

 and accuracy, the curve is drawn to two different scales, one for 

 large the other for small values of s/a. 



The curve in Fig. 26 may be interpreted in two ways : It may 

 be said to represent the "geometric permeance" of the fringe (for 

 //=!); or else it may be said to give the correspondings sets of 

 values of s and At, measured in the lengths of the air-gap as the 

 unit. With a given a, Jt increases with s, because the maximum 

 width of the actual fringe is Js. With a given s the width J/ 

 increases toward the pole-tip (if the air-gap is variable), because 

 with a longer air-gap the fringing lines of flux fill a larger part of 

 the air-gap under the slot. 



The corrected width of the tooth ixt' = t + Jt and the permeance 

 of the air-gap, in perms per tooth pitch, is 



(P al -1.25(J/a,+/oJU (48) 



Note on Air-gap In. 1 union, ./num. In*t. Electr. Eng. (British), Vol. 29, 

 (1899-1900), p. 929; Air-gap Imlurti..n. l.lcctrical World, Vol. 38, (1901) 

 p. 884; See also Hawkins ami \V..!lis, The Dynamo (1909), Vol. 1, p. 440; 

 I \rnoM, Die Glcich*tromma*chin* (1906), V< I 1 ; 266. 



'J.C. Maxwell, Electricity and Magnetism, V..I 1 ,. JM: .1 . J Thomson, 

 Recent Researches in Electricity and Magnetism, Chapter III; Horace Lamb, 

 Hydrodynamics (1895), Chapter IV. 



