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

21.11 Transfer functions 663
R(s) +
–
X 2 (s)
G(s)
Y(s)
X 1 (s)
H(s)
R(s)
G(s)
1 + G(s)H(s)
Y(s)
Figure21.7
Block diagram foranegative feedback loop.
Figure21.8
Simplifiedblock diagram foranegative
feedback loop.
The simplified block diagram for a negative feedback loop isshown inFigure 21.8.
A complicated engineering system may be represented by many differential equations.
The output from one part of the system may form the input to another part. Consider
the following example.
Example21.29 Asystemismodelled bythe differential equations
x ′ +2x=f(t) (21.10)
2y ′ −y =x(t) (21.11)
In Equation (21.10) the input is f (t) and the output is x(t). In Equation (21.11), x(t)
is the input andy(t) is the final output of the system. Find the overall system transfer
functionassumingzero initial conditions.
Solution The output from Equation (21.10) isx(t); this forms the input to Equation (21.11). The
blockdiagramsforEquations(21.10)and(21.11)arecombinedintoasingleblockdiagramasshown
inFigure 21.9.
UsingRule 1,the overall systemtransferfunction can thenbefound:
G(s) = Y(s)
F(s) = 1
(s +2)(2s −1)
This transferfunction relatesY(s)andF(s)(see Figure 21.10).
F(s)
1
X(s)
1
Y(s)
s + 2
2s – 1
F(s)
1
(s + 2)(2s – 1)
Y(s)
Figure21.9
Combinedblock diagram for
Equations(21.10)and (21.11).
Figure21.10
Theoverall systemtransfer function.
Example21.30 Asystemisrepresentedby the differential equations
2x ′ −x=f(t)
y ′ +3y =x(t)
z ′ +z=y(t)
The initial input is f (t) and the final output is z(t). Find the overall system transfer
function, assumingzeroinitial conditions.