0195136047.pdf

32 CIRCUIT CONCEPTS

EXAMPLE 1.2.4

(a) Consider a 5-μH inductor to which a current sourcei(t) is attached, as shown in Figure

E1.2.3(b). Sketch the inductor voltage and stored energy as a function of time.

(b) Let a voltage sourcev(t) shown in Figure E1.2.3(a) be applied to the 5-μH inductor

instead of the current source in part (a). Sketch the inductor current and energy stored as

a function of time.

(c) If three identical 5-μH inductors with initial current of 1 mA are connected (i) in series

and (ii) in parallel, find the equivalent inductance for each case.

Solution

(a) From the principle of duality and for the given values, it follows that the inductor-voltage

waveform is the same as the capacitor-current waveform of Example 1.2.3(a), in which

i(t) is to be replaced byv(t), and 25 and−50 mA are to be replaced by 25 and−50 mV.

The stored energyw(t) is the same as in the solution of Example 1.2.3(a).

(b) The solution is the same as that of Example 1.2.3(b), except thatv(t)inmVistobe

replaced byi(t)inmA.

(c) (i) Looking at the solution of Example 1.2.3(c), part (ii),

Leq=L 1 +L 2 +L 3 = 3 × 5 × 10 −^6 H= 15 μH

with an initial currenti(0)=1 mA.

(ii) Following the solution of Example 1.2.3(c), part (i),

1

Leq

=

1

L 1

+

1

L 2

+

1

L 3

=

3

5 × 10 −^6

or Leq=

5

3

× 10 −^6 H=

5

3

μH

with an initial currenti(0)=3 mA.

When more than one loop or circuit is present, the flux produced by the current in one

loop may link another loop, thereby inducing a current in that loop. Such loops are said to

be mutually coupled, and there exists amutual inductancebetween such loops. The mutual

inductance between two circuits is defined as the flux linkage produced in one circuit by a current

of 1 ampere in the other circuit. Let us now consider a pair of mutually coupled inductors, as

shown in Figure 1.2.9. The self-inductancesL 11 andL 22 of inductors 1 and 2, respectively, are

given by

L 11 =

λ 11

i 1

(1.2.36)

and

L 22 =

λ 22

i 2

(1.2.37)

whereλ 11 is the flux linkage of inductor 1 produced by its own currenti 1 , andλ 22 is the flux

linkage of inductor 2 produced by its own currenti 2. The mutual inductancesL 12 andL 21 are

given by