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

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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

Ask a homework question - tutors are online