3.3 LAPLACE TRANSFORM 151

EXAMPLE 3.3.2

Obtainv(t) in the circuit of Figure E3.3.2(a) by using the Laplace transform method.

+

− +

−

v(t)

i(t) i(t)

Vs(t) = 10 u(t)

(a)

0.5 Ω

3

2

0.25 Ω 1 Ω 0.5 H

2 F 0.5 Ω

+

−

− + +

−

V(s)

I(s) I(s)

(b)

0.5 Ω

3

2

0.25 Ω 1 Ω s/2

0.5 Ω

+

−

10

s^1

2 s

I(s) s/2

(c)

1 Ω

+

+

−

−

I(s)

B

0 (Reference node)

V(s)

3

2

(^1) Ω
4
(^1) Ω
2
(^1) Ω
2
40
s
1
2 s
A
Figure E3.3.2
Solution
All initial values of inductor current and capacitor voltage are zero fort<0 since no excitation
exists. The transformed network is shown in Figure E3.3.2(b). Let us convert all voltage sources
to current sources and use nodal analysis [Figure E3.3.2(c)].
The KCL equations are given by
VA( 4 + 2 s+ 1 )−VB( 1 )=
40
s
−
3
2
Is
−VA( 1 )+VB
(
1 + 2 +
1
s/ 2 + 1 / 2
)

3
2
I(s)
I(s)= 2 sVA