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

a 22 e xample 2 2 a voltage i tt l oe 3t ut is

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Unformatted text preview: ause the voltage across the inductor is L (dyo/dt) or yo(t), this equation can be written as follows: + 3yo(t) + v ert) = 0 yo(O) + 3yo(0) + ve(O) Setting t = 0, we o btain =0 (2.14) But yo(O) = 0 a nd vc(O) = 5. Consequently, Yo(O) = -5 Therefore, the desired initial conditions are Yo(O) = 0 and Yo(O) = -5 Thus, the problem reduces to finding yo(t), the zero-input component of y(t) o fthe system specified by the equation (D2 + 3 D + 2)y(t) = D f(t), when the initial conditions are yo(O) = 0 and yo(O) = - 5. We have already solved this problem in Example 2.1a, where we found yo(t) = - 5e- t + 5 e- 2t (2.15) This is the zero-input component of the loop current y (t). I t will be interesting to find the initial conditions a t t = 0 - and 0+ for the total response y (t). Let us compare y(O-) and y(O-) with y(O+) and Y(O+). The two pairs can be compared by writing the loop equation for the circuit in Fig. 2.1a a t t = 0 - and t = 0+. The only difference between the two situations is t hat a t t = 0 -, the input f (t) = 0, whereas a t t = 0 +, the input f (t) = 10 (because f (t) = l Oe- 3t ). Hence, the two loop equations are + 3 y(0-) + v e(O-) = y(O+) + 3y(0+) + vc(O+) = y (O-) 0 10 The loop current y(O+) = y(O-) = 0 because it cannot change instantaneously in the absence of impulsive voltage. The same is true of the capacitor voltage. Hence, ve(O+) = v c(O-) = 5. S ubstituting these values in the above equations, we obtain y (O-) = - 5 and y(O+) = 5. Thus (2.16) 2, E xercise E 2.3 • In the circuit in Fig. 2.1a, the inductance L = 0 and the initial capacitor voltage vc(O) = 30 volts. Show that the zero-input component of the loop current is given by yo(t) = _10e- 2t / 3 for t ~ O. V' 113 I n t his example we c omputed t he z ero-input component without using t he i nput f (t). T he z ero-state component c an b e c omputed from t he knowledge of t he i nput f (t) alone; t he i nitial conditions are assumed t o b e zero (system in zero s tate). T he two components of t he s ystem response (the zero-input a nd z ero-state components) are independent of each other. The two worlds o f z ero-input response a nd z ero-state response coexist s ide by side, n either o f t hem knowing o r c aring what the other is doing. For each component, the other is t otally irrelevant. Role of Auxiliary Conditions in Solution o f Differential Equations F ig. 2 .1 yo(t) S ystem Response t o I nternal Conditions: Zero-Input Response Independence of Zero-Input and Z ero-State Response 30 IH 2.2 S olution of a differential equation requires additional pieces of information (the a uxiliary c onditions). W hy? We now show t hat a differential equation does not, i n general, have a unique solution unless some additional constraints (or conditions) o n t he solution a re known. T he r eason is t hat, as explained in t he discussion on invertibility (Sec. 1.7), t he d ifferentiation o peration is n ot invertible unless one piece of information a bout y (t) is given. Thus, differentiation is a n irreversible (noninvertible) o p...
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