0195136047.pdf

1.2 LUMPED-CIRCUIT ELEMENTS 29

= 2 tmV, 1 ≤t≤ 3 μs

=−t^2 + 8 t−9mV, 3 ≤t≤ 4 μs

=7mV, 4 ≤tμs

which is sketched in the center of Figure E1.2.3(b).

Since the energy stored at any instant is

w(t)=

1

2

Cv^2 (t)=

1

2

( 5 × 10 −^6 )v^2 (t)

it follows that

w(t)= 0 ,t≤− 1 μs

= 2. 5 (

t^2

2

+t+

1

2

)^2 pJ, − 1 ≤t≤ 1 μs

= 10 t^2 pJ, 1 ≤t≤ 3 μs

= 2. 5 (−t^2 + 8 t− 9 )^2 pJ, 3 ≤t≤ 4 μs

= 122 .5pJ, 4 ≤tμs

which is sketched at the bottom of Figure E1.2.3(b).

(c) (i)

1

Ceq

=

1

C 1

+

1

C 2

+

1

C 3

=

3

5 × 10 −^6

, or Ceq=

5

3

× 10 −^6 F=

5

3

μF,

with an initial voltagev(0)=3mV[Figure E1.2.3(c)].

(ii)

Ceq=C 1 +C 2 +C 3 = 3 × 5 × 10 −^6 F= 15 μF

with an initial voltagev(0)=1mV[Figure E1.2.3(d)].

Inductance

Anideal inductoris also an energy-storage circuit element (with no loss associated with it) like a
capacitor, but representing the magnetic-field effect. The inductance in henrys (H) is defined by

L=

λ

i

=

Nψ

i

(1.2.26)

whereλis the magnetic-flux linkage in weber-turns (Wb·t),Nis the number of turns of the coil,
andNψis the magnetic flux in webers (Wb) produced by the currentiin amperes (A). Figure
1.2.7(a) illustrates a single inductive coil or an inductor ofNturns carrying a currentithat is
linked by its own flux.
The general circuit symbol for an inductor is shown in Figure 1.2.7(b). According to Faraday’s
law of induction, one can write

v(t)=

dλ

dt

=

d(Nψ)

dt

=N

dψ

dt

=

d(Li)

dt

=L

di

dt

(1.2.27)