16 THE MAGNETIC CIRCUIT [ART. 9 



In some cases two or more magnetic paths are in parallel, for 

 instance, when there is magnetic leakage (see below). In most 

 cases the engineer has to consider complicated magnetic circuits 

 which consist partly of paths in series, partly of paths in parallel. 

 Thus, in the same machine, the m.m.f . or the difference of mag- 

 netic potential between the pole-tips is 6500 ampere-turns. This 

 m.m.f. maintains a useful flux of say 2.5 megalines through the 

 armature, and say 0.5 megaline of leakage or stray flux between 

 the pole-tips. Thus the total flux in the field frame is 3 mega- 

 lines. 



The fundamental law of the magnetic circuit, as expressed by 

 eq. (1), is analogous to Ohm's law for the simple electric circuit. 

 Therefore magnetic paths hi series and in parallel are combined 

 according to the same rule that ele'ctrical conductors are combined 

 in series and in parallel. Namely, when two or more magnetic 

 paths are in series, their reluctances are added ; when two or more 

 magnetic paths are in parallel their permeances are added. Or, 

 for a series combination, 



........ (17) 



and for a parallel Combination 



(P eg = 2(P ........ (18) 



It will be remembered that similar relations hold also for impe- 

 dances and admittances in the alternating current circuit, and 

 for elastances and permittances in the electrostatic circuit. 



The proof of formulae (17) and (18) is similar to that usually 

 given for the combination of electric resistances in series and in 

 parallel. Namely, when reluctances are in series the total mag- 

 netomotive force is equal to the sum of component m.m.f .s., or 



(I7o) 



Dividing both sides of this equation by the common flux eq. 

 (17) is obtained. When permeances are in parallel, the total flux 

 is the sum of the component fluxes, or 



(18a) 



Dividing both sides of this equation by the common M, eq. (18) is 

 obtained. 



