Lecture 4

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In this lecture we investigate limit cycles using the principle of harmonic balance. Like last time, we consider our special feedback class of nonlinear systems, but this time we look for periodic solutions

LaTeX: y(t)\approx{}A\sin\omega{}t.

This solution will be used to approximate a limit cycle with amplitude LaTeX: A and frequency LaTeX: \omega. By the end of this lecture you should be able to:

  • Characterise the output of nonlinear components with sinusoidal inputs using describing functions.
  • Predict the emergence of limit cycles using the principle of harmonic balance, and determine whether such limit cycles are stable or unstable.

Lecture notes that support the video material can be downloaded here: lec06.pdf
The notes we took in class are here: 

Software to help you test out some examples of describing function analysis can be found here: Describing function analysis

Harmonic balance and the describing function

We introduce the principle of harmonic balance, and the role of the describing function in predicting limit cycles.

Calculating the describing function

We explain how to calculate the describing function.

Another example

We calculate the describing function of a saturation. Note, you don't have to use the specific trig identities in the video to complete the calculation (we do things slightly differently in the lecture notes). The key is to be resourceful when tackling the required integrals.

Harmonic balance and stability

We revisit the principle of harmonic balance, and explain how to guess whether any predicted limit cycles are stable or unstable.

Examples

We illustrate the principle of harmonic balance on some examples.

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