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Relaxation Oscillators:

 

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Vibe:Relaxation Oscillators:van Der Pol Oscillator

The van Der Pol Oscillator is the first relaxation oscillator. van Der Pol described this oscillator in a paper in 1928 in a model of human heartbeat,
  • The Heartbeat considered as a Relaxation oscillation, and an Electrical Model of the Heart, Balth. van der Pol and J van der Mark, (1928) Phil. Mag. Suppl. #6 pp 763--775.

Here is the second-order differential equation for a van Der Pol oscillator:

Omega controls how much voltage is injected into the system. Alpha controls the way in which voltage flows through the system. Think of the oscillator as having two distinct phases: a slow recovery phase and a fast relase phase (that is in fact where the term relaxation comes from: vaccuum tubes quickly release or relax their voltage after slowly building up tension.) The variable alpha controls the rate at which the slow build up occurs. As alpha is decreased, the voltage collects more slowly, making the slow phase take longer and hence slowing down the oscillator.

Also notice that the amplitude of the oscillator remains constant as alpha and/or omega vary. What changes is the shape of the voltage curve and the frequency of the oscillator.

Vibe Hints and Experiments
  • First, click on the Launch Simulator! button in the upper left. If your browser supports Java, you will get the Vibe simulator in a separate window. If you don't see a Launch Simulator button, you are either browsing Vibe with Java turned off or, alternately, Java is not working on your browser.
  • Now run the oscillator with default settings for a few timesteps.
  • Click the Layer Button to run a second simulation
  • This time try varying either coupling or voltage and then click Start to run the new simulation.
  • Repeat as desired.
  • You should see that peak voltage is the same whatever settings you use, but that frequency and curve-shape change drastically.

Vibe Oscillator Simulator © 2000-2003 Douglas S. Eck.
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Thu Jul 24 09:12:34 EDT 2003