A simpler system or element maybe governed by first order or second order differential equation.

**Transfer Function**

A simpler system or element maybe governed by first order or second order differential equation. When several elements are connected in sequence, say "n" elements, each one with first order, the total order of the system will be nth order

In general, a collection of components or system shall be represented by nth order differential equation.

In control systems, transfer function characterizes the input output relationship of components or systems that can be described by Liner Time Invariant Differential Equation

In the earlier period, the input output relationship of a device was represented graphically.

In a system having two or more components in sequence, it is very difficult to find graphical relation between the input of the first element and the output of the last element. This problem is solved by transfer function

Transfer function of a LTIV system is defined as the ratio of the Laplace Transform of the output variable to the Laplace Transform of the input variable assuming all the initial condition as zero.

The transfer function of a system is the mathematical model expressing the differential equation that relates the output to input of the system.

The transfer function is the property of a system independent of magnitude and the nature of the input.

The transfer function includes the transfer functions of the individual elements. But at the same time, it does not provide any information regarding physical structure of the system.

The transfer functions of many physically different systems shall be identical.

If the transfer function of the system is known, the output response can be studied for various types of inputs to understand the nature of the system.

If the transfer function is unknown, it may be found out experimentally by applying known inputs to the device and studying the output of the system.

Write the differential equation of the system.

Take the L. T. of the differential equation, assuming all initial condition to be zero. Take the ratio of the output to the input. This ratio is the T. F.

A control system is a collection of physical object connected together to serve an objective. The mathematical model of a control system constitutes a set of differential equation.

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