80 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1054 



In Fig. 4, (Rn , (Rl , and .1 are constants, so that (R is a function of x 

 only, independent of (p. Equation (3) can then apply only to the low 

 density region, (p < (p'. For most ordinary electromagnets, the magnetiza- 

 tion relations for (p < (p' conform to (3), and hence to the schematic of 

 Fig. 4. 



For ip > (f', the expression for (R must provide for the variation with 

 (p. It usually suffices to use an expression in which the only term varying 

 with tp is that corresponding to the path in which saturation first occurs. 

 The most common case is that in which saturation first occurs in the 

 core. The reluctance can then be taken as conforming to the schematic 

 of Fig. 5, in which (R02 , (Rl2 and A^ are constants, and (Re is a function of 

 (P only. The total reluctance is given by: 



(R = (Re -\- (Re, (4) 



where : 



(Rl2((R02 + ~ 



(Re = ^ ^ . (5) 



(RV2 + (R02 + J 



If the variation with (p is taken as conforming to the empirical Frohlich- 

 Kennelly equation, (Re is given by: 



«c = m^^^=(R:.C^^', (0) 



(f — cp (p — (p 



where <p" is the saturation flux and ip' is the flux for maximum perme- 

 ability or minimum reluctance (Re . If these assvimptions apply, the 

 minimum values of (R for all \'alues of x must lie on (p', as for the results 

 of Fig. 3. This common minimum is thus evidence that the core is con- 

 trolling with respect to the \-ariation with <p, and that (4) and (6) are 

 applicable. As ((>) applies only for ^ > tp', (Re is the value of (Re not 

 only at (p = (p', l)ut throughout the low density region tp < tp'. In the 

 alternati\'e form given by (G), (R" is defined by this ec[uation, and repre- 

 sents merely the intercept of a plot of (Re extended below the region to 

 which (6) applies. 



In some electromagnets saturation occurs in the armature rather than 

 in the core. This is the case, for example, in high speed relays in which 

 the armature section is minimized to reduce its mass. In such cases, the 

 reluctance conforms to the schematic of Fig. Ofa), in which (R .4 represents 

 the reluctance of the armature, which varies with (Pa ■ As the ratio (Pa/<p 



