40 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



permeability may be expressed in terms of B, giving the equation: 



m"(5" - B) 



M = ^, . 



(12) 



If this relation is applied to a part of a magnetic circuit, such as the 



core of a relay, the reluctance (Re of this part may be written as f/(ij.a), 



where / is the length and a is the cross-sectional area of the part. Then 



from (10), (Re is given by: 



// 



(Re = cs\c -^ , (13) 



(f — (f 



where (Re = (/{ix"a), ip = Ba, the flux through the part, and tp" is the 

 saturation value of <p. 



For increasing magnetization, this expression is applicable only for 

 values of B, or (p/a, beyond the knee of the magnetization curve, the 

 point of maximum permeabihty. Thus (13) is applicable for vahies of 

 B above B', the density at which ix has its maximum value jj.' . Writing 

 (Re for f/{fx'a) and tp' for B'a, (13) may be written in the alternative 

 form : 



(Re = (R'c %^ . (13A) 



^ - V 



For values of B below B' the permeability of initially demagnetized 

 material varies greatly with B, as shown in Fig. 5. In normal use, how- 

 ever, an electromagnet is rarely operated from a fully demagnetized 

 state. Fig. 7(a) shows the magnetization relations for an electromagnet 

 in repeated operation, and Fig. 7(b) shows the corresponding relations 

 for its core. The solid line corresponds to initial operation from a demag- 

 netized condition, the dotted line to decreasing magnetization in release, 

 and the dashed line to subsequent remagnetization. The latter corre- 

 sponds to higher permeability and a lower core reluctance than those for 

 initial magnetization. As a convenient approximation, linear magnetiza- 

 tion may be taken as a representative condition for B less than B' , in 

 which case the core reluctance is constant for v^.less than <p' . In this low 

 density region, then, the magnetic circuit constants may be considered 

 to be independent of the flux. This, of course, is never strictly true, but 

 it is a satisfactory approximation for most ordinary electromagnets. 



Jlijperholic Approximation to Decreasing Alagnetization Curves 



As the hyperbolic approximation is a purely empirical relation, it may 

 be applied to the (^ — J relation for an electromagnet as well as to that 



