This preview shows page 1. Sign up to view the full content.
Unformatted text preview: c omponent yo(t) ( response d ue t o t he i nitial c onditions alone w ith
/ (t) = 0) a nd t he z ero-state c omponent r esulting f rom t he i nput a lone w ith all
i nitial c onditions zero. A t t = 0 -, t he r esponse y(t) c onsists solely o f t he z ero-input
c omponent yo(t) b ecause t he i nput h as n ot s tarted y et. Hence t he i nitial c onditions
o n y(t) a re i dentical t o t hose o f volt). T hus, y(O-) = Yo(O-), y(O-) = Yo(O-), a nd
so on. Moreover, yo(t) is t he r esponse d ue t o i nitial c onditions a lone a nd d oes n ot
d epend o n t he i nput / (t). H ence, a pplication o f t he i nput a t t = 0 d oes n ot affect
volt). T his m eans t he i nitial c onditions o n volt) a t t = 0 - and 0+ a re i dentical; t hat
is Yo(O-), yo(O-), . .. a re i dentical t o Yo(O+), Yo(O+), . .. , r espectively. I t is clear
t hat for yo(t), t here is no d istinction b etween t he i nitial c onditions a t t = 0 -, 0
a nd 0 +. T hey a re a ll t he s ame. B ut t his is n ot t he c ase w ith t he t otal r esponse
y (t), w hich consists of b oth, t he z ero-input a nd t he z ero-state c omponents. T hus,
i n g eneral, y(O-) ¥ y(O+), y(O-) ¥ y(O+), a nd so on.
• + 4D + k )y(t) = (3D + 5 )/(t) determine the zero-input component of the response if the initial conditions are yo(O) = 3,
and yo(O) = - 7 f or two values of k: ( a) 3 ( b) 4 ( c) 40.
( a) 2.2 E xample 2 .2
A voltage I tt) = l Oe- 3t u(t) is applied a t t he input of the R LC circuit illustrated in
Fig. 2.1a. Find the loop current y (t) for t ;:::: 0 if t he initial inductor current is zero; t hat
is, y (O-) = 0 a nd the initial capacitor voltage is 5 volts; t hat is, v c(O-) = 5.
T he differential (loop) equation relating y(t) to / (t) was derived in Eq. (1.55) as
(D2 + 3D + 2) y(t) = D /(t) T he zero-state component of y(t) resulting from the input / (t), assuming t hat all initial
conditions are zero; t hat is, y (O-) = v c(O-) = 0, will be obtained later in Example 2.5. In
this example we shall find the zero-input component yo(t). For this purpose, we need two
initial conditions yo(O) and yo(O). These conditions can be derived from the given initial
conditions, y (O-) = 0 and v c(O-) = 5, as follows. Recall t hat Yo(t) is t he loop current
when the input terminals are shorted a t t = 0, so t hat the input / (t) = 0 (zero-input) as
depicted in Fig. 2.1b. We now compute yo(O) and yo(O), the values of the loop current and
its derivative a t t = 0, from the initial values of the inductor current and the capacitor
voltage. Remember t hat t he inductor current cannot change instantaneously in the absence
of an impulsive voltage. Similarly, the capacitor voltage cannot change instantaneously in
the absence of an impulsive current. Therefore, when the input terminals are shorted a t
t = 0, t he inductor current is still zero and the capacitor voltage is still 5 volts. Thus,
yo(O) =0 112 2 T ime-Domain Analysis of Continuous-Time Systems
H1 IH ~ [ (I) 9 +
V c(l) ~F (a) +
v c(t) ~F ( b) To determine yo(O), we use the loop equation for the circuit in Fig. 2.1b. Bec...
View Full Document