Transfer Functions and the Laplace Transform
An intro to the mysteries of the frequency domain and Laplace transform and how they're used to understand mechanical and electrical systems.
Course Information
Certificate Track
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About this Course
This course is about the Laplace Transform, a single very powerful tool for understanding the behavior of a wide range of mechanical and electrical systems: from helicopters to skyscrapers, from light bulbs to cell phones. This tool captures the behavior of the system and displays it in highly graphical form that is used every day by engineers to design complex systems.
This course is centered on the concept of the transfer function of a system. Also called the system function, the transfer function completely describes the response of a system to any input signal in a highly conceptual manner. This visualization occurs not in the time domain, where we normally observe behavior of systems, but rather in the “frequency domain.” We need a device for moving from the time domain to the frequency domain; this is the Laplace transform.
We will illustrate these principles using concrete mechanical and electrical systems such as tuned mass dampers and RLC circuits.
This is part of a 5-part series in Differential Equations:
What you'll learn
You’ll learn how to:
- Pass back and forth between the time domain and the frequency domain using the Laplace Transform and its inverse.
- Use a toolbox for computing with the Laplace Transform.
- Describe the behavior of systems using the pole diagram of the transfer function.
- Model for systems that have feedback loops.
- Model sudden changes with delta functions and other generalized functions.
Prerequisites
Meet your instructors
Haynes Miller
Professor of Mathematics, emeritus
Haynes Miller joined the MIT mathematics faculty as professor in 1986. A graduate of Harvard, he received the Ph.D. from Princeton under the direction of John Moore in 1974. Following assistant professorships at Harvard and Northwestern Universities, he joined the faculty of the University of Washington in 1977, and the faculty of Notre Dame as professor in 1984. Professor Miller is an algebraic topologist. In 1992-93, he served as Chair of the Pure Mathematics Committee. From 2004 to 2013 Professor Miller has chaired the Undergraduate Mathematics Committee, which he continued as Associate Department Head from July 2011-June 2013. He has worked on many educational initiatives, such as the Cambridge-MIT Exchange program, for which he received the Cambridge-MIT fellowship in 2003. Professor Miller was selected by MIT to be a MacVicar Faculty Fellow in 2005 for "exemplary and sustained contributions to the teaching and education of undergraduates at MIT." He also received the Graduate Student Council Teaching Award of the School of Science in 2006.
