0195136047.pdf

1.2 LUMPED-CIRCUIT ELEMENTS 33

L 12 =

λ 12

i 2

(1.2.38)

and

L 21 =

λ 21

i 1

(1.2.39)

whereλ 12 is the flux linkage of inductor 1 produced by the currenti 2 in inductor 2, andλ 21 is the
flux linkage of inductor 2 produced by the currenti 1 in inductor 1.
If a current ofi 1 flows in inductor 1 while the current in inductor 2 is zero, the equivalent
fluxes are given by

ψ 11 =

λ 11

N 1

(1.2.40)

and

ψ 21 =

λ 21

N 2

(1.2.41)

whereN 1 andN 2 are the number of turns of inductors 1 and 2, respectively. That part of the flux
of inductor 1 that does not link any turn of inductor 2 is known as the equivalentleakage fluxof
inductor 1,

ψl 1 =ψ 11 −ψ 21 (1.2.42)

Similarly,

ψl 2 =ψ 22 −ψ 12 (1.2.43)

Thecoefficient of couplingis given by

k=

√

k 1 k 2 (1.2.44)

i (^1) i 2 = 0 i 1 = 0 i 2
v 2
open
v 1
open
(a) (b)

• −


• 
  

  −

  
  


• 
  

  −

  
  


• 
  

  −
  v 1 v 2
  N 1
  turns
  N 2
  turns
  N 1
  turns
  N 2
  turns
  ψ 21
  ψl 1 ψl 2
  ψ 12
  i 1 i 2
  v 2
  (c)

  
  


• 
  

  −

  
  


• 
  

  −
  v 1
  N 1 turns N 2 turns
  ψ 21
  ψ 12
  ψl 1
  ψl 2
  Figure 1.2.9Mutually coupled inductors.