Page 166 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS

1 I
Now (t) = - i(r) dr
C -,

and

Hence, Eq. (3.107) reduces to

Solving for I(s), we obtain
v-0, 1 v-u, 1
I(s) = - =-
s R + 1/Cs R s + l/RC

Taking the inverse Laplace transform of I(s), we get

(b) When u,(r) is the output and the input is u,(t), the differential equation governing the
circuit is

Taking the unilateral Laplace transform of Eq. (3.108) and using Eq. (3.441, we obtain

Solving for V,(s), we have
v 1 uo
Vc(s) = - +
RCs(s+l/RC) s+l/RC

Taking the inverse Laplace transform of I/,(s), we obtain

uc(t) = V[1 - e-t/RC]u(t) + ~,e-'/~~u(t)
Note that uc(O+) = u, = u,(O-). Thus, we write uc(t) as

uc(t) = V(1 -e-'IRC) + ~~e-'/~~
trO

3.40. Using the transform network technique, redo Prob. 3.39.

(a) Using Fig. 3-10, the transform network corresponding to Fig. 3-14 is constructed as shown
in Fig. 3-15.