The transfer function of a system is a mathematical representation of the relationship between the input and output of a linear system. It is typically expressed as a ratio of the Laplace transform of the output to the Laplace transform of the input, with all initial conditions set to zero. It is typically used in the field of control engineering to determine how a system should respond to a given input.

The formula T(s) for a transfer function is,

**T(s) = Y(s) / U(s)**

Where Y(s) is the Laplacian Transform of the output signal, and U(s) is the Laplacian Transform of the input signal.

The input can be anything from a simple switch to a complex set of sensor readings. The output can be anything from a single light bulb to the movement of a complex mechanical device. In between the input and output is a controller, which is the brain of the control system. The controller takes in the input and uses it to calculate the appropriate output.

## What is the order of a transfer function?

The order of a transfer function is the highest order of the derivative in the numerator and denominator of the transfer function.

Between the order of the denominator and numerator, the transfer function order is determined by the numerator. It doesn’t matter what is in the denominator, the order is simply the highest exponent on the s in the numerator. So for example, if you have a transfer function like:

The order would be 2 because that is the highest exponent on s in the numerator.