First-order lag, linear lag, exponential transfer lag, First order system transfer function,steady state gain or static gain or simply the gain of the process,time constant

A first-order system is one whose output, y(t), is modeled by a first order,linear differential equation


where f(t) is the input (forcing function). If a0 # 0, then above eqation yields,



Define,



𝜏p is known as the time constant of the process

and KP is called the steady state gain or static gain or simply the gain of the process.

From eqn, it is easily found that the transfer function of a firstorder process is given by;


A first-order process with a transfer function given by above eqn. is also known as: first-order lag, linear lag, exponential transfer lag.


If on the other hand, a0 = 0, then from eqn. we take


which gives a transfer function



In such case the process is called purely capacitive or pure integrator