INDUCTANCES AND RESISTANCES 



e, M 

 so =7" 



Hence 



rwr«5 ra//o — If a tightly coupled mutual inductance be fed from some 

 kind of generator as shown in Figure 4.15, we have 



and ^2 = K^N.2. d(f)ldt 



because of the tight coupling <f)^ = ^2 

 so eje^ = A^i/^2 



Loose coupled windings — the coupling factor k 



If the coils of a mutual inductance are loosely coupled, M is less than 

 (LiLg)^'^ ^"^ ^'^^ introduce the factor k, such that M = k{L-J^^^i". k then varies 

 from zero — when the windings are indefinitely far apart, or are perpendicu- 

 larly oriented — to nearly unity, when the coils are linked by an iron core and 

 are close together. 



Self inductances in series, but possessing some mutual coupling 



If two coils, which considered separately have self inductances L^ and L^ 

 are connected in series, their effective inductance is no longer merely L^ + L^ 

 if there is mutual coupling between them (as when they share the same core); 

 for let them be connected to a generator of any varying current /; there is 

 a back e.m.f. —L^ dildt across L^ due to the current in L^, and a back e.m.f. 

 — M dijdt across L^ due to the current in Lg. Similarly, there is a back e.m.f. 



Effective inductance A-B = Ly*L2*2M 



(-^vtnnnrinp-| ^^rffYvyvy^ 



-o 



aO ' "B 



Effective inductance /4- B =/., +/.2"2M 

 Figure 4.16 



— L2 dijdt across L^ due to the current in L^, and a back e.m.f. —M difdt 

 across L^ due to the current in L^. Thus the total back e.m.f. is (L^ + Z-a + 

 2M) dijdt, hence the effective self inductance for the two windings is L1 + L2 + 

 2M. This assumes that the sense of the windings is such that their m.m.f.'s 

 are acting in the same direction in the core. If the m.m.f.'s oppose one 

 another, the effective self inductance is L^ + -^2 ~~ 2Af {Figure 4.16). 



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