Page 138 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS 127

3. Inductance L:
di(t )
~(t) L- t, V(s) = sLI(s) - Li(0-) (3.50)
=
dt
The second model of the inductance L in Fig. 3-10 is obtained by rewriting Eq. (3.50) as
1 1
i(t) t, I(s) = -V(s) + -i(O-) (3.51)
sL S
4. Capacitance C:

dm
i(t) = C- t* I(s) = sCV(s) - Cu(0-) (3.52)
dt
The second model of the capacitance C in Fig. 3-10 is obtained by rewriting Eq. (3.52) as

1 1
u(t) t* V(s) = -I(s) + -u(O-) (3.53)
sc S

Solved Problems

LAPLACE TRANSFORM

3.1. Find the Laplace transform of
(a) x(t) = -e-atu( -t)
(b) x(t)=ea'u(-t)
(a) From Eq. (3.3)

m
( ) = - ( e-afu(-r)e-fldt = - (O-e-("")' df

Thus, we obtain
1
-e-"'u( -I) H -
s+a
(b) Similarly,