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

21.11 Transfer functions 661
R(s)
G(s)
Y(s)
Figure21.3
Therelationship
Y(s) =G(s)R(s)holds
forasingle block.
by which these signals are modified. The word signal is used in a very general sense
and is not restricted to, say, voltage. On this basis each of the elements of the system
can be modelled by a transfer function. A transfer function defines the relationship between
an input signal and an output signal. The relationship is defined in terms of the
Laplace transforms of the signals. The advantage of this is that the rules governing the
manipulation of transfer functions are then of a purely algebraic nature. Consider
Figure 21.3. If
then
R(s) = L{r(t)} = Laplace transform of the input signal
Y(s) = L{y(t)} = Laplace transformof the output signal
G(s) = transferfunction
Y(s) =G(s)R(s)
Transfer functions are represented schematically by rectangular blocks, while signals
are represented as arrows. Engineers often speak of the time domain and the s domain
in order to distinguish between the two mathematical representations of an engineering
system. However, it is important to emphasize the equivalence between the
two domains.
Oftenwhenconstructingamathematicalmodelofasystemusingtransferfunctions,
it is convenient first to obtain transfer functions of the elements of the system and then
combinethem.Beforetheoveralltransferfunctioniscalculatedablockdiagramisdrawn
which shows the relationship between the various transfer functions. Block diagrams
consistof threebasic components. These are shown inFigure 21.4.
R(s)
G(s)
Y(s)
R(s)
+
–
R(s) – X(s)
Y(s)
Y(s)
X(s)
(a) (b) (c)
Figure21.4
Thethree components ofblock diagrams. (a)Abasic block; the block contains a
transferfunction which relatesthe input and outputsignals. (b)Asummingpoint.
(c) Atake-off point.
We have already examined the basic block which is governed by the relationship
Y(s) =G(s)R(s).Asummingpointaddstogethertheincomingsignalstothesumming
point and produces an outgoing signal. The polarity of the incoming signals is denoted
bymeansofapositiveornegative sign.Therecanbeseveral incomingsignalsbutonly
one outgoing signal. A take-off point is a point where a signal is tapped. This process
of tapping the signal has no effect on the signal value; that is, the tap does not load the
original signal. There are several rules governing the manipulation of block diagrams.
Only two will beconsidered here.
21.11.1 Rule1.Combiningtwotransferfunctionsinseries
Consider Figure 21.5. The following relationships hold:
X(s) =G 1
(s)R(s)
Y(s) =G 2
(s)X(s)
Y(s)