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

T he i nitial condition voltages in t he t hree b

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Unformatted text preview: c apacitor C1 (leaving t he n ode a), is [Va(s) - Va(S)]C1S = [Va(S) - KVb(S)]C1S. T he s um of a ll t he t hree currents is zero. Therefore or C;S Ie, ( s) IT ( ) V + 2S b S = 0 G 1G2 = C 1C 2S 2 + [G 1G2 + G2C2 + G 2C1(1- K)]s + G1G2 wo 2 (6.63b) where T he two node e quations (6.63a) a nd (6.63b) in two unknown node voltages Vats) a nd Vb(S) c an b e expressed i n m atrix form as (6.64) where a nd 2 WO G1G2 = C1C2 = R 1R2C 1C2 (6.65a) (6.65b) 6 410 Continuous-Time System Analysis Using t he Laplace T ransform Now 6.5 Block Diagrams 411 Comment: T he i nitial value theorem should be applied only if F (s) is s trictly proper (m < n ), because for m 2: n , lim s _ s F(s) does not exist, a nd t he t heorem does not apply. To prove t he final value theorem, we let s - + 0 in Eq. (6.34a) t o o btain oo Therefore (6.66) • 6.4-2 lim [sF(s) - f(O-)] s-O Initial and Final Values In certain applications, it is desirable to know t he values. of f (t) as t - > 0 a nd t - > 0 0 [initial a nd final values of f(t)] from t he kn0-:vledge o~ Its Laplace transform F ( s). I nitial a nd final value theorems provide such mformatlOn. T he i nitial v alue t heorem s tates t hat if f (t) and its derivative df / dt a re b oth Laplace transformable, t hen f(O+) = lim s F(s) (6.67) 8 -00 provided t hat t he limit on t he r ight-hand side of Eq. (6.67) exists. (6.6S) lim f (t) = lim s F(s) t -oo • 100 = 0- df dt dt f(t)\OO = lim f (t) - f(O-) t_oo E xample 6 .17 Determine the initial and final values of yet) if its Laplace transform y es) is given by Y es) = 10(2s + 3) S(s2 + 2s + 5) Equations (6.67) and (6.68) yield y(O+) = lim s Y(s) = lim .-00 8 -00 + 3) + 2s + 5) 10(28 (8 2 = ° y(oo) = lim s Y(s) = lim ( 10(2S + 3) ) = 6 • 2 • _0 0 10 - 0- d _fe- 8t dt dt a d eduction which leads to t he desired result (6.6S) Comment: T he final value theorem applies only if t he poles of s F(s) a re in t he LHP. I f t here is a pole on t he i maginary axis, t hen l ims_o s F(s) does not exist. If t here is a pole in t he RHP, limt_oo f (t) does not exist. provided t hat s F(s) h as no poles in t he RHP or on t he i maginary axis. To prove these theorems, we use Eq. (6.34a) s F(s) - f(O-) = 1 00 0- 8 -+0 00 s_O = T he f inal v alue t heorem s tates t hat if b oth f (t) a nd df / dt a re Laplace transformable, t hen = lim . -0 s + 2s + 5 df - st d -e t dt = { 0+ df e -st dt + 10 10- dt = f(t)\o+ 0- 00 0+ + df _ e- st dt dt roo df e -stdt 10+ = f(O+) - f(O-) dt + r df e - st dt 10+ dt Therefore a nd lim s F(s) = f(O+) 8 -+00 = f(O+) + sl-oo im 00 0 10 + + roo df 10+ dt df - st d dt e t ( lim e - st ) dt s_OO 6.5 Block Diagrams Large systems may consist of an enormous number of components or elements. Analyzing such systems all a t once could be next t o impossible. Anyone who has seen t he circuit diagram of a radio or a T V receiver can appreciate this fact. In such cases, i t is convenient t o r epresent a system by suitably interconnected subsystems, each o f which can be readily analyzed. Each subsystem can be characterized...
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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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