Details

The general first-order transfer function in the Laplace domain is:

,

where is the process gain, is the time constant, is the system dead time or lag and is a Laplace variable. The process gain is the ratio of the output response to the input (unit step for this Demonstration), the time constant determines how quickly the process responds or how rapidly the output changes and the dead time delays the system response by the amount .

This Demonstration converts from the Laplace domain to the time domain for a step-response input. For a first-order transfer function, the time-domain response is:

.

The general second-order transfer function in the Laplace domain is:

,

where is the (dimensionless) damping coefficient.

For a step-response input, the time-domain response for various damping coefficients are:

Overdamped ():

.

Critically damped ():

.

Underdamped ():

.

Reference

[1] D. E. Seborg, D. A. Mellichamp, T. F. Edgar and F. J. Doyle, III, Process Dynamics and Control, 3rd ed., Hoboken: John Wiley & Sons, 2011 pp. 76, 81.