082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

662 Chapter 21 The Laplace transform
R(s)
G 1 (s)
X(s)
G 2 (s)
Y(s)
R(s)
G 1 (s) G 2 (s)
Y(s)
Figure21.5
Two blocksin series.
Figure21.6
Figure21.5issimplifiedtoasingleblock.
EliminatingX(s) from these equations yields
Y(s) =G 1
(s)G 2
(s)R(s)
Y(s)
R(s) =G 1 (s)G 2 (s)
Finally the overall transfer function, G(s), is given by G 1
(s)G 2
(s) as shown in
Figure 21.6.
Fortwo transferfunctions inseries the overall transferfunction isgiven by
G(s) =G 1
(s)G 2
(s)
21.11.2 Rule2.Eliminatinganegativefeedbackloop
Consider Figure 21.7 which shows a negative feedback loop. It is so called because the
outputsignalis‘fedback’andsubtractedfromtheinputsignal.Suchloopsarecommon
inavarietyofengineeringsystems.ThequantitiesX 1
(s)andX 2
(s)representintermediate
signals in the system. We wish to obtain an overall transfer function for this system
relatingY(s)andR(s). For the two transferfunctions the following hold:
Y(s) =G(s)X 2
(s) (21.6)
X 1
(s) =H(s)Y(s) (21.7)
For the summingpoint
X 2
(s) =R(s) −X 1
(s) (21.8)
Combining Equations(21.7)and (21.8) gives
X 2
(s) =R(s) −H(s)Y(s) (21.9)
Combining Equations(21.6)and (21.9) gives
Y(s) =G(s)(R(s) −H(s)Y(s)) =G(s)R(s) −G(s)H(s)Y(s)
Y(s)(1 +G(s)H(s)) =G(s)R(s)
Y(s)=
G(s)R(s)
1 +G(s)H(s)
Theoverall transferfunctionforanegative feedback loop isgiven by
Y(s)
R(s) = G(s)
1 +G(s)H(s)