**Module overview.**

*This module defines the describing function (DF) of a nonlinearity and shows how it may be used to study the behavior of a feedback loop with a single nonlinearity. DFs are derived for single and double valued nonlinearities and examples are given to show how DFs can be used for examining the stability of a nonlinear system. Since the method is approximate comments are made on its accuracy.*

**Module objectives.**
*When you have completed this module you should be able to:*

- Define a describing function of a nonlinearity
- Evaluate the describing function of a given nonlinearity
- Use the describing function approach to understand the dynamic behavior of a feedback loop with a single nonlinearity
- The approximations and limitations of the describing function method.

**Module prerequisites.**
*First year mathematics, Fourier series,transfer functions,frequency response, stability*

- Introduction
- Evaluation of the Describing Function
- DFs of some common nonlinearities
- A cubic nonlinearity
- An ideal saturation nonlinearity
- Relay with dead zone and hysteresis
- Further Comments

- Determination of the Stability of a Feedback Loop
- DF Accuracy
- Concluding Remarks
- Problems

D. P. Atherton 2005-01-27