Introduction
The indication of cause and effect relationship existing between input and output mathematically means to decide the transfer function of the given system. It is commonly used to characterize the input-output relationship of the system.
Mathematically, it is defined as the ratio of Laplace transform of output of the system to Laplace transform of input, under the assumption that all initial conditions are zero.
Theory
Transfer function also known as the system function or network function and, when plotted as a graph, transfer curve is a mathematical representation for fit or to describe inputs and outputs of black box models.
Technically, it is a representation in terms of spatial or temporal frequency, relation between the input and output of a linear time-invariant system with zero initial conditions and zero-point equilibrium. With optical imaging devices, for example, it is the Fourier transform of the point spread function (hence a function of spatial frequency) i.e. the intensity distribution caused by a point object in the field of view.
The closed-loop transfer function is measured at the output. The output signal waveform can be calculated from the closed-loop transfer function and the input signal waveform.
An example of a closed-loop transfer function is shown below:
The summing node and the G(s) and H(s) blocks can all be combined into one block, which would have the following transfer function:
Define an intermediate signal Z, as shown in fig.
Using this figure we write:
Features of Transfer function :
- It gives mathematical modes of all system components and hence of the overall system.
- As it uses Laplace approach, it converts integro-differential time domain equations to simple algebraic equations.
- It helps in determining the important information about the system i.e poles, zeros, characteristic equation etc.
- It helps in stability analysis of the system.
- It is only applicable to linear time invariant system.
- It does not provide any information concerning the physical structure of the system.
Objective
To find the transfer function of a negative feedback system using scilab.
Experiment
Click on Sci Lab Simulator for 64 bit windows:
Download Simulator and download it.
Click on Sci Lab Simulator for 32 bit windows:
Download Simulator and download it.
Launch open source scilab software .
Go to Xcos simulator. Xcos can be launched from the toolbar, via the icon
Manual
Scilab is a programming language associated with a rich collection of numerical algorithms covering many aspects of scientific computing problems.
Whatever your platform is (i.e. Windows, Linux or Mac), Scilab binaries can be downloaded directly from the Scilab homepage
or from the
Download area.
Scilab elementary mathematical operators:
+ addition
- subtraction
* multiplication
/ right division, i.e. x/y =xy-1
\ left division, i.e. x\y = x-1y
^ power, i.e. xy
** power (same as ^)
' transpose conjugate
Here we use syslin command. Syslin defines a linear system as a list and checks consistency of data. dom='c' for a continuous time system, dom='d' for a discrete time system, n for a sampled system with sampling period n (in seconds), dom=[] if the time domain is undefined.
Disp -Display of objects defined by tlist may be overloaded by the definition of a function. This function must
have no output argument a single input argument and it's name is formed as follows:
% < tlist_type >_p where % < tlist_type > stands for the first entry of the tlist type component.
Sys1 * Sys2 gives the multiplication of transfer function of two cascade systems.
(-1)*Sys3 gives the value of transfer function when negative feedback is applied.
The overall transfer function is given by the ratio of Laplace of output by input.
Procedure/Program
- Open the scilab window.
- Write the program mentioned below in the command window and then execute the commands.
Program
s=%s;
sys1=syslin('c',(s+3)/(s+1))
sys2=syslin('c',0.2/(s+2) )
sys3=syslin('c',50/( s+4) )
y=sys1*sys2
z=(-1)*sys3
x=y/.z
disp(x,"C(s)/R(s)=")
- Define the systems by their Laplace equation.
- Execute the program.
- You will get the result as C(s)/R(s).
Result
sys1 =
3 + s
-----
1 + s
sys2 =
0.2
-----
2 + s
sys3 =
50
----
4 + s
y =
0.6 + 0.2s
----------
2 + 3s + s2
z =
- 50
----
4 + s
-->x=y/.z
x =
2.4 + 1.4s + 0.2s2
-----------------
- 22 + 4s + 7s2 + s3
C(s)/R(s)=
2.4 + 1.4s + 0.2s2
-----------------
- 22 + 4s + 7s2 + s 3
Quizzes
Q1. The transfer function is applicable to
(a) Linear and time-invariant system
(b) Linear and time variant system
(c) Linear system
(d) Non-linear system
Answer: a
Q2. The transfer function technique is inadequate under which the following conditions?
(a) System having complexities and non-linearities
(b) System having stability problem
(c) System having multiple input disturbances
(d) All of the above
Answer: d
Q3. In a closed loop system, with appositive value of feedback gain the overall gain of the system will
(a) Decrease
(b) Increase
(c) Be unaffected
(d) Any of the above
Answer: a
Q4. The output of feedback control system must be a function of
(a) Reference and output
(b) Reference and input
(c) Input and feedback signal
(d) Output and feedback signal
Answer: a
Q5. What value in the transfer function defines the order of system?
a) The highest power of ‘s’ in the characteristic equation
b) Number of zeroes at origin
c) The lowest power of ‘s’ in the characteristic equation
d) None of the above
Answer: a
