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Automatic Control System
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- 45 -CHAPTER – 4
Transfer Function, Block Diagram and Signal Flow Graph
4.1. TRANSFER FUNCTIONS:
The transfer function of a linear time invariant continuous system is defined to be the ratio of
Laplace transform of the output variable to the Laplace transform of the input variable with all the initial
conditions taken to be zero. The complete transfer function of a system is obtained by determining the
transfer function of the various components of the system and then combining them according to their
connections. Since the characteristics of linear systems depend only on the properties of elements in the
system, the transfer function describes the same.
Consider the linear time invariant system defined by the following differential equations:
n
d y (t ) d n 1 y (t ) dy (t )
a0 n
a1 n 1
….. + a n 1 a n y (t )
dt dt dt
d m x(t ) d m1 x(t ) dx(t )
= b0 m
b1 m 1
........ bm1 bm x(t ) ( n m)
dt dt dt
Where y is the output of the system and x is the input. The transfer function of this system is the ratio of
the Laplace transformed output to the Laplace transformed input when all initial conditions are zero, or
L[output ]
Transfer function = G(s) = zero initial conditions
L[input ]
Y ( s ) b0 S m b1 S m1 .......... bm1 S bm
=
X ( s ) a 0 S n a1 S n 1 ........... a n 1 S a n
By using the concept of transfer function, it is possible to represent system dynamics by algebraic
equations in s. If the highest power of s in the denominator of the transfer function is equal to n, the
system is called an nth order system.
4.1.1. HOW TO DETERMINE TRANSFER FUNCTIONS?
1. Write a set of linearly independent integral-differential equations in (n+1) variables the input
( independent variable), the output and (n-1) other independent variables.
2. Convert the equations in frequency domain by taking Laplace transform assuming initial
conditions as zero.
3. Solve the resulting set of n linearly independent algebraic equations in the n dependent
variables for the Laplace transform of the output in terms of the Laplace transform of the input.
4. Find the relation between Laplace transform of output and input eliminating intermediate
variables. The resultant equation is the transfer function.
4.1.2. THE PROPERTIES OF THE TRANSFER FUNCTIONS:
1. The transfer function is defined only for a linear time invariant system. It is not defined for
non-linear systems.
2. The transfer function between an input variable and an output variable of a system is defined
as the Laplace transform of the impulse response. Alternately, the transfer function between a
pair of input and output variables is the ratio of the Laplace transform of the output to the
Laplace transform of the input.
3. All initial conditions of the system are set to zero.
4. The transfer function is independent of the input of the system.
, - 46 -
5. The transfer function of a continuous data system is expressed only as a function of the complex
variable s. It is not a function of the real variable, time, or any other variable that is used as the
independent variable. For discrete data systems modeled by difference equations, the transfer
functions of z when the z-transform is used.
4.1.3. ADVANTAGES OF TRANSFER FUNCTION:
Transfer function is a very strong tool for analyzing a control system and offers the following
advantages.
1. We get mathematical models all system components hence, of the overall system. It allows us to
analyze the individual components of the system.
2. Stability analysis of the system can easily be carried out. We get information about poles, zeros,
characteristic equations, etc.
3. The use of Laplace transform approach allows converting integral differential time domain
equations to simple algebraic equations.
4. The transfer function is expressed in terms of complex variable ‘s’ which is not a function of the
real variable time or any other variable, which is used as the independent variable.
5. Transfer function is the property and unique equation of the system and its value depends on
the value of parameters of the system and is independent of the input.
4.1.4. DISADVANTAGES OF TRANSFER FUNCTION:
However, transfer function suffers from the following disadvantages.
1. It can be applied only to linear and time invariant systems.
2. We get no information from transfer function regarding the physical structure of the system.
3. Initial conditions do affect the system performance but these are neglected while determining
transfer function and hence, loose their importance.
4.2. BLOCK DIAGRAMS:
A block diagram of a control system is a simplified pictorial representation of various system
components alternatively called system elements. The block diagram is drawn to establish a
mathematical relationship between the input, error and output. A general block diagram consists of the
individual blocks that represent the transfer function of the individual elements.
Let the input output behaviour of a linear system or an element be given by its transfer
C ( s)
function G ( s ) ,
R(s)
Where, R(s) is the Laplace transform of the input variable.
C(s) is the Laplace transform of the output variable.
This can be represented as a block diagram as shown in fig 4.1.
R(s) G(s) C(s)
Input Output
Fig. 4.1.
From the above block diagram, it can be observed that, C(s) = R(s)G(s).
A typical block diagram consists of one or more summing junctions, forward and feedback paths,
take off points. In a block diagram representation three ingredients are commonly present. They are ,
1. Functional Block: This is a symbol that represents the transfer function between the input
R(s) to an element and the output C(s) of the element. The block contains a transfer function.
In figure 4.2 (a).
, - 47 -
2. Summing Point : This has a symbol as shown in fig.4.1(b), the output of summing point is
the algebraic sum of the signals entering to it. Next to each input signal is a plus or minus symbol
indicating the sign associated with the variable. The output of the given summing point is E(s) = R(s) –
C(s). In figure 4.2 (b).
3. Take off Point: A take off point on a branch is a physical point in the system where the
desired signal is tapped off to utilize elsewhere. In figure 4.2 (c).
+
R(s) G(s) C(s) R(s) E(s) C(s) C(s)
--
C(s)
C(s)
Fig. 4.2(a) Fig. 4.2(b) Fig. 4.2(c)
4.2.1. HOW TO DRAW THE BLOCK DIAGRAM:
Consider a simple R-L circuit shown in fig.4.3
R
Vi L Vo
i
Fig. 4.3
Apply KVL,
di
Vi Ri L …………………………………………………………………………. (4.1)
dt
di
V0 L …………………………………………………………………………. (4.2)
dt
Laplace transform of equations (4.1) & (4.2) with initial condition zero
Vi ( s ) I ( s ) R SLI ( s )
Vi ( s ) I ( s )( R SL) …………………………………………………………………... (4.3)
V0 ( s ) SLI ( s ) ……………………………………….................................................... (4.4)
V0 ( s ) sL
From equations (4.3) & (4.4), ………………………………………………….. (4.5)
Vi ( s ) R sL
Vi V0
From fig 4.3 i ……………………………………………………………………………. (4.6)
R
di
V0 L ………………………………………………………..……………………. (4.7)
dt
Laplace transform of equations (4.6) & (4.7)
1
I ( s ) [Vi ( s ) V0 ( s )] ………………………………………………………………. (4.8)
R
V0 ( s ) sLI ( s ) ……………………………………………………………………..… (4.9)
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