Page 170 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS

Then, using the initial value theorem (3.971, we have
s+l
+
1
1
v,,(O+) = lim sVcl(s) = lim --7 + 1 = 2 V
=
s4m s-rm S + a
S-7 I
+
+
s-+m S + ,
1
ucJO+) = lim sVC-(s) = lim ---7 2 = 3 V
2
=
s-+m
Note that ucl(O+)# u,1(0-) and ucz(O+)# ~~$0-). This is due to the existence of a
capacitor loop in the circuit resulting in a sudden change in voltage across the capacitors.
This step change in voltages will result in impulses in i,(t) and i2(t). Circuits having a
capacitor loop or an inductor star connection are known as degener~ti~e circuits.

Supplementary Problems

3.43. Find the Laplace transform of the following x(t 1:

2 s
(c) If a > 0, X(s) = - -a < Re(s) < a. If a < 0, X(s) does not exist since X(s) does
sz - ,2 '
not have an ROC.
(d) Hint: x(t) = u(t) + u(-t)
X(s) does not exist since X(s) does not have an ROC.
(el Hint: x(t)=u(t) -u(-t)
X(s) does not exist since X(s) does not have an ROC.

3.44. Find the Laplace transform of x(t) given by

t, _< t s t,
x(t) =
0 otherwise

1
Am. X(s) = -[e-"I - e-"21, all s
S