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terms of its i nput-output relationships. A linear (sub)system can be characterized
by its transfer function H(8). F igure 6.1Sa shows a block diagram of a system with
a transfer function H (s) a nd its i nput a nd o utput r epresented by their frequency
domain descriptions F (s) a nd Y (s) respectively.
Subsystems may be interconnected by using three elementary types of interconnections (Figs. 6.1Sb, 6.1Sc, 6.1Sd): cascade, parallel, a nd feedback. W hen two
transfer functions appear in cascade, as depicted in Fig. 6.1Sb, t he t ransfer function
of t he overall system is t he p roduct of t he two transfer functions. This conclusion
follows from t he fact t hat in Fig. 6.1Sb y es) _ W (s) y es) _
( ) ()
F (s) - F(s) W (s) - HI S H2 s
T his result can be extended t o a ny number of transfer functions in cascade. 6 412 Continuous-Time System Analysis Using t he Laplace Transform F ( s) ~
! H I ( s) - ------I R(S) L H2 ( 8) r-~--- = I \ F (s)
....... _--I H I ( s)H 2 (s)I-_~~ (b) Y eS) F (s) System Realization so t hat (a) I-_---y(s) H (s) 6.6 = 413 y es)
F(s) = 1 + G (s)H(s) (6.69) • Therefore, t he feedback loop can be replaced by a single block with the transfer
function shown in Eq. (6.69) (see Fig. 6.18d).
In deriving these equations, we implicitly assume t hat when t he o utput of one
subsystem is connected to t he i nput of a nother s ubsystem, t he l atter does not load
the former. For example, t he t ransfer function Hl(S) in Fig. 6.18b is c omputed
by assuming t hat t he second subsystem H 2( s) was not connected. This is t he
s ame as assuming t hat H2(S) does not load Hl(S). I n other words, t he i nputoutput relationship of Hl(S) will remain unchanged regardless of whether H2(S) is
connected or not. Many modern circuits use op amps with high i nput impedances,
so t his a ssumption is justified. When such a n a ssumption is not valid, H I (s) m ust
be computed under operating conditions ( that is, when H2(S) is connected).
T he MATLAB example C6.3 allows us t o d etermine t he t ransfer function of
the feedback system in Fig. 6.18d [Eq. (6.69)J, when t he t ransfer functions G(s) a nd
H (s) a re given. o C omputer E xample C 6.3
Find the transfer function of the feedback system of Fig. 6.1Sd when G(s) = S(~8)'
H(s) = 1, and K = 7, 16, and SO.
For the sake of generalization, we shall split G(s) into two terms Gl(S) = K and
G2(S) = '(.~8)' Such generalization allows us to use this program when G(s) is made up
of two subsystems in cascade. ( e) Y ( s) % (c62.m)
G lnum=[O 0 K );Glden=[O 0 I );
G 2num=[0 0 1 );G2den=[1 8 0];
H num=[O 0 1 );Hden=[0 0 1);
p rintsys(CIGnum,CIGden) (d) F ig. 6.18 Elementary connections of blocks and their equivalents.
Similarly w hen two transfer functions, Hl(S) a nd H2(S), a ppear i n parallel, as
illustrated in F ig. 6.18c, t he overall transfer funct~o~ is giv~n by H l (s)+H2(S), t he
s um of t he two transfer functions. T he p roof is t nvlal. T his r esult can be extended
t o any number of systems in parallel....
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