Signal Processing and Linear Systems-B.P.Lathi copy

E xercise e 21 f ind t he z ero input r esponse o f a

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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...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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