0195136047.pdf

30 CIRCUIT CONCEPTS

i(t)

i(t)

(a)

L

+

−

+

−

v(t) N turns v(t)

L = λ/i = Nψ/i

λ = Nψ

ψ

(b)

v(t) = Ldidt;i(t) =L^1 ∫^ v(τ) dτ

t

L

=^1

∫ 0 v(τ) dτ + i(0)

t

Figure 1.2.7An inductor.(a)A single inductive coil ofNturns.(b)Circuit symbol.

whereLis assumed to be a constant and not a function of time (which it could be if the physical

shape of the coil changed with time). Mathematically, by looking at Equations (1.2.17) and

(1.2.27), the inductor is thedualof the capacitor. That is to say, the terminal relationship for

one circuit element can be obtained from that of the other by interchangingvandi, and also by

interchangingLandC.

The terminali–vrelationship of an inductor can be obtained by integrating both sides of

Equation (1.2.27),

i(t)=

1

L

∫t

−∞

v(τ) dτ (1.2.28)

which may be rewritten as

i(t)=

1

L

∫t

0

v(τ) dτ+

1

L

∫ 0

−∞

v(τ) dτ=

1

L

∫t

0

v(τ) dτ+i( 0 ) (1.2.29)

wherei(0) is the initial inductor current att=0.

The instantaneous power delivered to the inductor is given by

p(t)=v(t)i(t)=Li(t)

di(t)

dt

(1.2.30)

whose average value can be shown (see Problem 1.2.13) to be zero for sinusoidally varying current

and voltage as a function of time. The energy stored in an inductor at a particular time is found

by integrating,

w(t)=

∫t

−∞

p(τ) dτ=

1

2

Li^2 (t)−

1

2

Li^2 (−∞) (1.2.31)

Assuming the inductor current to be zero att=−∞, the stored energy in the inductor at some

timetis given by

w(t)=

1

2

Li^2 (t) (1.2.32)

which depends only on the inductor current at that time, and represents the stored energy in the

magnetic field produced by the current carried by the coil.

If the current flowing through the coil does not change with time, no voltage across the coil

exists, as seen from Equation (1.2.27). The following relations hold:

L=

λ

I

; V= 0 ; W=

1

2

LI^2 (1.2.33)