0195136047.pdf

34 CIRCUIT CONCEPTS

wherek 1 =ψ 21 /ψ 11 andk 2 =ψ 12 /ψ 22. Whenkapproaches unity, the two inductors are said

to be tightly coupled; and whenkis much less than unity, they are said to be loosely coupled.

While the coefficient of coupling can never exceed unity, it may be as high as 0.998 for iron-core

transformers; it may be smaller than 0.5 for air-core transformers.

When there are only two inductively coupled circuits, the symbolMis frequently used to

represent the mutual inductance. It can be shown that the mutual inductance between two electric

circuits coupled by a homogeneous medium of constant permeability is reciprocal,

M=L 12 =L 21 =k

√

L 11 L 22 (1.2.45)

The energy considerations that lead to such a conclusion are taken up in Problem 1.2.30 as an

exercise for the student.

Let us next consider the energy stored in a pair of mutually coupled inductors,

Wm=

i 1 λ 1

2

+

i 2 λ 2

2

(1.2.46)

whereλ 1 andλ 2 are the total flux linkages of inductors 1 and 2, respectively, and subscriptm

denotes association with the magnetic field. Equation (1.2.46) may be rewritten as

Wm=

i 1

2

(λ 11 +λ 12 )+

i 2

2

(λ 22 +λ 21 )

=

1

2

L 11 i 12 +

1

2

L 12 i 1 i 2 +

1

2

L 22 i 22 +

1

2

L 21 i 1 i 2

or

Wm=

1

2

L 11 i 12 +Mi 1 i 2 +

1

2

L 22 i 22 (1.2.47)

Equation (1.2.47) is valid whether the inductances are constant or variable, so long as the

magnetic field is confined to a uniform medium of constant permeability.

Where there arencoupled circuits, the energy stored in the magnetic field can be expressed as

Wm=

∑n

j= 1

∑n

k= 1

1

2

Ljkijik (1.2.48)

Going back to the pair of mutually coupled inductors shown in Figure 1.2.9, the flux-linkage

relations and the voltage equations for circuits 1 and 2 are given by the following equations, while

the resistances associated with the coils are neglected:

λ 1 =λ 11 +λ 12 =L 11 i 1 +L 12 i 2 =L 11 i 1 +Mi 2 (1.2.49)

λ 2 =λ 21 +λ 22 =L 21 i 1 +L 22 i 2 =Mi 1 +L 22 i 2 (1.2.50)

υ 1 =

dλ 1

dt

=L 11

di 1

dt

+M

di 2

dt

(1.2.51)

υ 2 =

dλ 2

dt

=M

di 1

dt

+L 22

di 2

dt

(1.2.52)

For the terminal voltage and current assignments shown in Figure 1.2.9, the coil windings

are such that the fluxes produced by currentsi 1 andi 2 are additive in nature, and in such a case

the algebraic sign of the mutual voltage term is positive, as in Equations (1.2.51) and (1.2.52).

In order to avoid drawing detailed sketches of windings showing the sense in which each

coil is wound, adot conventionis developed, according to which the pair of mutually coupled

inductors of Figure 1.2.9 are represented by the system shown in Figure 1.2.10. The notation