0195136047.pdf

3.3 LAPLACE TRANSFORM 149

v(t)

v(t) = Ri(t)

i(t)^12 R

Resistance

+− V(s)

V(s) = RI(s)

I(s)^1 R

Time − Domain Network Transformed Network in Frequency Domain

2

+−

I(s)

I(s)

I(s)

v(t)

v(t) = Ldtdi

i(t)

12 L

Inductance

+−

V(s) = LsI(s) − Li(0+)

V(s)

or

Ls Li(0+)

(^12) −+

• −
  I(s) =+VLs(s) i(0s
  +)
  V(s)
  i(0+)/s
  12 Ls


• −
  I(s)
  v(t)
  v(t) =C^1 ∫i(t)dt
  i(t)^12 C
  Capacitance
  +− V(s)
  or
  1/Csv(0+)/s
  1/Cs
  (^12) +−


• −
  V(s) =Cs^1 I(s) +v(0s
  +)
  V(s)
  I(s) = Cs V(s) − Cv(0+)
  Cv(0+)
  12


• −
  (a)
  (b)
  (c)
  Figure 3.3.2Time-domain networks and transformed network equivalents in frequency domain for
  R, L,andC.

4. By taking the inverse Laplace transform of the frequency-domain response, obtain the
   time-domain response.

Let us illustrate the use of this procedure with some examples.

EXAMPLE 3.3.1

Consider the circuit shown in Figure E3.3.1(a) in which the switchShas been in position 1 for a
long time. Let the switch be changed instantaneously to position 2 att=0. Obtainv(t) fort≥ 0
with the use of the Laplace transform method.