Presentation Of Derivation of transfer function of electrical system

1. GYANMANJARI INSTITUTE OF TECHNOLOGY
Control System Engineering
(21509009)

2. Derivationoftransferfunctionofelectrical system
Prepared By,
Asit Bhavsar (151290109007)
Vishal Chauhan (151290109009)
Hiren Chovatiya (151290109010)
Priyank Dave (151290109012)
Faculty Guide
Prof. Utsav Yagnik

3. Outline
What Is Transfer Function..?
Definition of Transfer Function
Laplace Transform
What is Laplace transform…?
Why input, output and other signals are represented in Laplace form
in a control system…?
 Properties Of Transfer Function
 Advantages of T.F.
 disadvantages of T.F.
 Procedure and Methods of Transfer Function
Concept of Transfer Function
 Practice Example

4.  In engineering, a transfer function also known as system function or network
function. In control engineering and control theory the transfer function is
derived using the Laplace transform.
 A control system consists of an output as well as an input signal. The output
is related to the input through a function call transfer function. This function
is represented by a block and the complete diagram of control system using
these blocks which represent transfer function and arrows which represent
various signals, is collectively known as block diagram of a control system.
What Is Transfer Function..?

5. The transfer function of a control system is defined as, ‘The ratio of the
Laplace transform of the output variable to Laplace transform of the input
variable assuming all initial conditions to be zero’. And also We can define
Transfer Functions as , ‘The way to represent system dynamics, via the S-
representation gotten from Laplace transforms’. Now Let’s see some theory
of Laplace Transform…
Definition of Transfer Function

6. LAPLACE TRANSFORM
 Laplace is remembered as one of the greatest scientists
of all time. Known as ‘French Newton’ or ’Newton of
France’.
 Began work in calculus which led to the Laplace-
Transform
 Developed mathematics in Astronomy, Physics, and
Statistics
 One of the first Scientists to suggest the existence of
Black holes
 Focused later on Celestial Mechanics
Pierre Simon Laplace
(French mathematician and
astronomer)

7. What is Laplace transform…?
 Laplace transform will convert a function in some domain into a function in
another domain, without changing the value of the function.
 Generally, Laplace Transforms is a method for solving differential
equations, converts differential equations in time ‘t’ into algebraic equations
in complex variable ‘s’.
• Let f(t) be defined for t ≥ 0. The Laplace transform of f(t), denoted by F(s) or
L{f(t)}, is an integral transform given by the Laplace integral:
0
s t
e dt

 
 

8. But for mathematical analysis, of a system all kinds of signals should be represented in a similar form. This is done
by transforming all kinds of signal to their Laplace form. Hence a basic block diagram of a control system can be
represented as,
Why input, output and other signals are represented in Laplace form in a control system…?
It is not necessary that output and input of a control system are of same category. For example,
Where, r(t) and c(t) are time domain function of input and output signal respectively.

9. 1. The transfer function of a system is the Laplace transform of its impulse
response for zero initial conditions.
2. The transfer function can be determined from system input-output pair by
taking ratio Laplace of output to Laplace of input.
3. The transfer function is independent of the inputs to the system.
4. The system poles/zeros can be found out from transfer function.
5. The transfer function is defined only for linear time invariant systems. It is
not defined for non-linear systems.
Properties Of Transfer Function :

10. 1. It is a mathematical model that gives the gain of the given block/system.
2. Integral and differential equations are converted to simple algebraic
equations.
3. Once the transfer function is known, any output for any given input, can be
known.
4. System differential equation can be obtained by replacement of variable ‘s’
by ‘d/dt’.
5. The value of transfer function is dependent on the parameters of the system
and independent of the input applied.
Advantages of t.f. :

11. disadvantages of t.f. :
1. Transfer function is valid only for Linear Time Invariant
systems.
2. It does not take into account the initial conditions. Initial
conditions loose their significance.
3. It does not give any idea about how the present output is
progressing.

12. Procedure for determining the transfer function of a control system are as follows :
1. We form the equations for the given system.
2. Now we take Laplace transform of the system equations, assuming initial conditions as
zero.
3. Specify system output and input.
4. Lastly we take the ratio of the Laplace transform of the output and the Laplace transform
of the input which is the required transfer function.
There are major two ways of obtaining a transfer function for the control system. The ways are,
 Block Diagram Method : It is not convenient to derive a complete transfer function for a complex control
system. Therefore the transfer function of each element of a control system is represented by a block
diagram. Block diagram reduction techniques are applied to obtain the desired transfer function.
 Signal Flow Graphs : The modified form of a block diagram is a signal flow graph. Block diagram gives a
pictorial representation of a control system. Signal flow graph further shortens the representation of a
control system.
Methods of Obtaining a Transfer Function :

13. The transfer function is generally expressed in Laplace Transform and it is nothing but the relation between input
and output of a system. Let us consider a system consists of a series connected resistance (R) and inductance (L)
across a voltage source (V).
From the circuit, we get,
Now applying Laplace Transform, we get,
The transfer function of the system, G(s) = I(s)/V(s), the ratio of
output to input.
Concept of Transfer Function
Here, Initially Inductor behaves Open, So i(0+) = 0
So,

14. Practice Example :
Find the Transfer Function of Given Electrical Network.