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The transfer function of a linear time invariant differential equation system is defined to be the ratio of the Laplace transform of the output variable (response function) to the Laplace transform of the input variable (driving function) under the assumption that all the initial condition are zero.

Consider the linear time invariant system defined by the following differential equation:

**A**

_{0}y^{n}+ A_{1}y^{n-1}+ A_{2}y^{n-2}+....................+ A_{n}y = B_{0}x^{m}+_{ }B_{1}x^{m-1}+ B_{2}x^{m-2}+......................+ B_{m}x
Where y is the system’s output and x is the input. As we know that transfer function is the Laplace transform of the output response to the Laplace transform of input variable when all initial conditions are zero.

**Transfer Function G(s) =**

^{Y(s)}**/**

_{X(s)}=^{B0xm + B1xm-1 + B2xm-2 +......................+ Bmx}/_{A0yn + A1yn-1 + A2yn-2 +....................+ Any}

_{ }^{th}order system.

"When the transfer function of a physical system is determined, the system can be represented by a block, which is a short hand pictorial representation of the cause and effect relationship between input and output of the system. The signal flowing into the block (called input) flows out it called (output) after being processed by the transfer function characterizing the block. Functions operation of a system can be more readily visualised by examination of a block diagram rather than by the examination of the equations describing the physical system. Therefore when working with a linear time invariant system, one can think of a system or its sub system simply as interconnected blocks with each block described by its own transfer function.

#### Procedure for deriving transfer functions

The following assumptions are made in deriving the transfer function of physical system.

- It is assumed that there is no loading that is no power drawn at the output of the system. Considering the system has more than one non loading elements in random then the transfer function of each element should be determined first independently and then overall transfer function of the physical system is obtained by multiplying the individual transfer functions.
- The system should be approximated by a linear lumped constant parameters modelled by making suitable assumptions.

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