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

50 d yt d2 y 2 d 2yt dt 151 a nd so on

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Unformatted text preview: Moreover, C onsequently, t he l oop e quation (1.48) c an b e e xpressed as (D (1.58) f (t) Multiplying b oth s ides o f t he a bove e quation b y D ( that is, differentiating t he a bove e quation), we o btain (1.59a) (15D + 5)x(t) = D f(t) y (t) J&lt;t) 91 S ystem M odel: I nput-Output D escription + l )y(t) (1.60) = f (t) ( 1.55) or w hich i s i dentical t o E q. ( 1.52). R ecall t hat E q. ( 1.55) is n ot a n a lgebraic e quation, a nd D2 + 3 D + 2 i s n ot an a lgebraic t erm t hat m ultiplies y (t)j i t i s a n o perator t hat o perates o n y (t). I t m eans t hat w e m ust p erform t he f ollowing o perations o n y (t): t ake t he s econd d erivative o f y (t) a nd a dd t o i t 3 t imes t he f irst d erivative o f y (t) a nd 2 t imes y (t). C learly, a p olynomial i n D m ultiplied b y y (t) r epresents a c ertain d ifferential o peration o n 3 dy dt • E xample 1 .11 F ind t he e quation r elating t he i nput t o o utput for t he series R C c ircuit o f Fig. 1.32 if t he i nput is t he v oltage f (t) a nd o utput is ( a) t he loop c urrent x (t) ( b) t he c apacitor v oltage y (t). T he loop e quation for t he c ircuit is R x(t) + 11' c -00 x(T)dT = f(t) 5['00 X(T) dT = f(t) • E xercise E 1.lS For the R LC circuit in Fig. 1.31, find the input-output relationship if the output is t he inductor voltage VL(t). Answer: (D2 + 3 D + 2) VL(t) = D2 f(t) 'V Hint: VL(t) = L Dy(t) = D y(t). For the R LC circuit in Fig. 1.31, find the input-output relationship if the o utput is the capacitor voltage v c(t). Hint: v c{t) = d riy(t) = t&gt;y(t). Answer: (D2 + 3 D + 2) v c{t) = 2 f(t) 'V ' T(t) (1.56) ) ~i1=Jr or 15x(t) + (1.61) E xercise E 1.l7 !:!. y (t). !:!. + y(t) = f (t) B 8(t) (1.57) F ig. 1 .33 B (b) A rmature c ontrolled dc m otor. W ith o perational n otation, t his e quation c an b e e xpressed as numerator as well as in the denominator. This happens, for instance, in circuits with all-inductor loops or all-capacitor cutsets. To eliminate this problem, avoid the integral operation in system equations so t hat t he resulting equations are differential rather than integra-differential. In electrical circuits, t his can be done by using charge (instead of current) variables in loops containing capacitors and using current variables for loops without capacitors. In the literature this problem of commutativity of D and 1 / D is largely ignored. As mentioned earlier, such procedure gives erroneous results only in special systems, such as the circuits with all-inductor loops or all-capacitor cutsets. Fortunately such systems constitute a very small fraction of the systems we deal with. For further discussion of this topic and a correct method of handling problems involving integrals, see Ref. 4 • E xample 1 .12 I n r otational s ystems, t he e quations o f m otion a re s imilar t o t hose in t ranslational s ystems. I n p lace o f force F , we have t orque T . I n p lace o f m ass M , we have m oment o f i nertia J ( the r otational m ass), a nd in place o f l inear accel...
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