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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 ContinuousTime 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) •
6.42 lim [sF(s)  f(O)]
sO 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 ighthand 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+) + sloo
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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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