### Transfer Function

1. Definition
Mathematically it is defined as the ratio of Laplace transform of output (response) of the system to the Laplace transform of input (excitation or driving function), under the assumption that all initial conditions are zero.
Symbolically system can be represented as shown in the Fig.1(a). while the transfer function of system can be shown as in the Fig.1(b).
Transfer function of this system is C(s)/R(s) , where C(s) is Laplace of c(t) and R(s) is Laplace of r(t).
2. Advanatages and Features of Transfer Function
The various features of the transfer function are,
i ) It give mathematical models of all system components and hence of the overall system. Individual analysis of various components is also possible by the transfer function approach.
ii) As it uses a Laplace approach, it converts time domain equations to simple algebraic equations.
iii) it suggests operational method of expressing equations which relate output to input.
iv) The transfer function 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.
v) It is the property and characteristics of the system itself. Its value is dependent on the parameters of the system and independent of the values of inputs. In the example1, if the output i.e. focus of interest is selected as voltage across resistance R rather than the voltage across capacitor C, the transfer function will be different. So transfer function is to be obtained for a pair of input and output and then it remains constant for any selection of input as long as output variable is same. It helps in calculating the output for any type of input applied to the system.
vi) Once transfer function is known, output response for any type of reference input can be calculated.
vii) It helps in determining the important information about the system i.e. poles, zeros, characteristics equation etc.
viii) It helps in the stability analysis of the system.
ix) The system differential equation can be easily obtained by replacing variables 's' by d/dt.
x) Finding inverse, the required variable can be easily expressed in the time domain. This is much more easy than to analyse the entire system in the time domain.
The few limitation of the transfer function approach are,
i) Only applicable to linear time invariant systems.
ii) It does not provide any information concerning the physical structure of the system. From transfer function, physical nature of the system whether it is electrical, mechanical, thermal or hydraulic, cannot be judged.
iii) Effect arising due to initial conditions are totally neglected. Hence initial conditions loose their importance.
4. Procedure to Determine the Transfer Function of a Control System
The procedure used in Ex 1 and Ex 2 can be generalized as below :
1) Write down the time domain equations for the system by introducing different variables in the system.
2) Take the Laplace transform of the system equations assuming all initial conditions to be zero.
3) Identify system input and output variables.
4) Eliminating introduced variables, get the resultant equation in terms of input and output variables.
5) Take the ratio of Laplace transform of output variables to Laplace transform of input variables to get the transfer function model of the system.
Note : The network in Ex2 and Ex 3 is same but as focus of interest i,e, output is changed, the transfer function is changed. For a fixed output, transfer function is constant and independent of any type of input applied to the system. If the output variable is changed, the T.F. also changes accordingly.

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