First- and Second-Order Transfer Functions

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Contributed by: Rachael L. Baumann and Adam J. Johnston (September 2017)
Additional contributions by: John L. Falconer
(University of Colorado Boulder, Department of Chemical and Biological Engineering)
Open content licensed under CC BY-NC-SA



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

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:

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


[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.

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