Transfer Function is the ratio of Laplace transform of the output to the Laplace transform of the input, assuming all initial conditions zero.

- In general

- Where
*x*is the input of the system and*y*is the output of the system.

- When order of the denominator polynomial is greater than the numerator polynomial the transfer function is said to be ‘proper’. Otherwise it is ‘improper’.

## Properties of Transfer Function

Transfer function can be used to check

- The stability of the system from characteristic equation.
- Time domain and frequency domain characteristics of the system.
- Output of the system for any given input.
- Poles/zeros of the system.

## Advantages of Transfer Function

- It gives the gain of the given system.
- Integral and differential equations can be converted into simple algebraic equations.
- Once the transfer function is known the output of the system can be determined for any input.
- Poles and zeros can be found out by transfer function and they play important role in response of the system.

## Disadvantages of Transfer Function

- Transfer function is valid only for Linear Time Invariant (LTI) systems.
- In transfer function initial conditions are neglected hence initial conditions loose their importance.
- It does not give any idea how the present output is progressing.