First Order Systems - Transfer Function

First Order System Descriptions

A first order system has a differential equation - given below.

tdx/dt + x(t) = G_dcu(t)

With this differential equation, the terms are:

x(t) = Response of the System,
u(t) = Input to the System,
t = The System Time Constant,
G_dc = The DC Gain of the System.

If the differential equation is Laplace transformed we get:

tX(s) + X(s) = G_dcU(s), or, solving for the transfer function, we get:
Transfer Function = X(s)/U(s) = G_dc/(st + 1)