Transfer Function

Definition of transfer function:

The transfer function of a liner time invariant system is defined as the ratio of Laplace transform of the output to the Laplace transform of input of the system assuming that all the initial conditions are zero.

The Transfer functions are used in control systems for analysis of the systems. 

If y(t) is output variable of a system given by y’(t) + y(t) = x(t). Where x(t) is the input variable of the system. The transfer function of this system is given as below,

Taking Laplace transform,

L{y’(t) + y(t)}=L{x(t)}

⇒ sY(S) + Y(S) = X(S)

⇒ Y(S)(s + 1) = X(S) ⇒ Y(s)/X(s) =1/(1+s)

Therefor the transfer function is equal to \[\frac{L(output)}{L(input)}=\frac{Y(S)}{X(S)}=\frac{1}{S + 1}\].

In transfer function s is a complex variable and its highest power in the denominator indicates the order of the system. The highest power of s in the denominator of above system is unity and hence it is a first order system.