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Unformatted text preview: n general. However, in t his case, b1 = 0,
a nd t here is o nly o ne feedforward connection w ith coefficient bo = 5 f rom t he o utput o f
t he i ntegrator t o t he o utput s ummer. O bserve t hat b ecause t here is o nly o ne s ignal t o
b e s ummed a t t he o utput s ummer, a s ummer is n ot n eeded. For t his r eason, t he o utput
s ummer is o mitted i n Fig. 6.23. . -L
8 +7 o ne. sig~al ~o ~e s ummed a t t he o utput s ummer, t he o utput s ummer is o mitted. T he
realizatIOn I II Fig. 6.25a is r edrawn i n a more convenient form, as i llustrated i n Fig. 6.25b.
H (s) = 4 s + 28
s2 + 6s + 5
T his is a second-order s ystem w ith bo = 28, b l = 4, b2 = 0, ao = 5, a l = 6.
F igure 6.26 shows a realization w ith t wo feedback connections a nd two feedforward
connections. = s +5
s +7 T he r ealization a ppears in Fig. 6.24. Here H (s) is a first order t ransfer f unction w ith
ao = 7 a nd bo = 5 , bl = 1. T here is a single feedback connection ( with coefficient 7) from
t he i ntegrator o utput t o t he i nput s ummer. T here a re t wo feedforward connections from
t he o utputs o f t he i ntegrator a nd t he i nput s ummer t o t he o utput s ummer.t F ig. 6 .24 R ealization of m F ig. 6 .26 R ealization o f .i~t;!5
6 ( c)
H (s) = _ s_ 8 +7 E xercise E 6.9
Realize the transfer function H(s) = T his f irst-order t ransfer f unction is similar t o t hat i n ( b), e xcept t hat bo = 0. T herefore,
t he r ealization i s s imilar t o t hat in Fig. 6.24 w ith t he feedforward connection from t he
o utput o f t he i ntegrator missing, as d epicted i n Fig. 6.25a. Also, because t here is only
tWhen m = n (as in this case), H (s) can also be realized in another way by recognizing t hat H(s) = 1 - _2_
8 +7 We now realize H ( s) as a parallel combination of two transfer functions, as indicated by t he above
equation. 6 .6-2 82 28 + 6 . + 25 Cascade and Parallel Realizations A n n th-order t ransfer f unction H (s) c an b e e xpressed a s a p roduct o r a s um o f
n fi~st-order t ransfer f unctions. A ccordingly, w e c an a lso r ealize H (s) a s a c ascade ~senes) o r p arallel f orm o f t hese n f irst-order t ransfer f unctions.
I nstance, t he t ransfer f unction i n p art ( d) o f t he l ast e xample: C onsider, f or 420 6 C ontinuous-Time System Analysis Using t he Laplace Transform 6.6 T he complex poles in H (s) s hould be realized as a second-order (quadratic)
factor because we c annot implement multiplication by complex numbers. Consider,·
for example, We can express H (s) as = 4s + 28
(s + l )(s + 5) (48 + 28) ( 1 )
= ~ 8+ 5 H (s) _
lOs + 50
- (s + 3)(s2 + 4s + 13) (6.76a) ' -v-"---"""-H1(8) lOs + 50
(s + 3)(s + 2 - j3)(s + 2 + j 3) H 2(s) We can also express H (8) as a sum of p artial fractions as 48 + 28 l )(s + 5) 2 1 + j2
1 - j2
s + 2 - j3
8 + 2 + j3
We cannot realize first-order transfer functions individually with t he poles - 2 ± j 3
because they require multiplication by complex numbers in t he feedback a nd t he
feed forward path...
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