Self-Oscillation

October 28, 2011

Oscillators are important devices in modern electronics technology. You're likely reading this article on a computer that's clocked at two gigahertz or higher. The dual-core Linux system on which this article is being written runs at 2.66 GHz. The basic clock rate is established by a quartz crystal resonator that oscillates at a lower frequency that is multiplied by a phase-locked-loop.

Similar crystal oscillators are built into your cellphone, television, cable box, and nearly every other electronic device in your house or automobile. Oscillators can be mechanical, also, such as pendulum clocks. All these examples of oscillators are self-oscillators; that is, the energy source is not tuned to the resonator, but the resonator mechanism itself controls the positive feedback, or restoring force, that keeps the oscillation going.

These self-oscillators are distinct from forced oscillators that are driven by an external energy source that's tuned to their resonant frequency. Bowed and wind musical instruments, and the heartbeat, are all self-oscillators.

Alejandro Jenkins, a physicist at Florida State University (Tallahassee, FL) has just published a nice review of self-oscillation on the arXiv Preprint Server.[1] One thing I found while reading his paper was that my physics education was incorrect in one regard. I was shown a film clip of the Tacoma Narrows Bridge collapse and directed to the mainstay textbook of the era, Resnick and Halliday, that said the cause was forced oscillation.

Jenkins sets the record straight that the bridge collapse was caused by self-oscillation. The bridge had a natural resonant frequency of 0.2 Hz, whereas the Strouhal frequency of turbulent vortex shedding for the 42 mph (68 km/h) winds would be 1 Hz. I can't be too critical of my professors, one of whom was actually Robert Resnick, since half of what I learned has become obsolete in the past forty years.

One fundamental example of a self-oscillator is the Pearson-Anson relaxation oscillator, as shown in the figure. These were fun to build in the days when neon bulbs were still plentiful, although we would use a rectifier attached to the household voltage supply rather than a battery.

A Pearson-Anson relaxation oscillator using a neon bulb. Fig. 8 of Ref. 1, via the arXiv Preprint Server[1].

It can be seen from the circuit that there is no resonant element. There's a capacitor, but no inductor, so its oscillating action is necessarily one of self-oscillation. The "circuitry" for the human heartbeat functions along the same principle.

Neon bulbs may be rare, but operational amplifiers are ubiquitous, so we can give the example of a triangle wave generator, as shown below. As in the relaxation oscillator, there is no resonant element in this circuitry.

Triangle wave generator using two operational amplifiers. The leftmost amplifier is wired as a comparator. The comparator supplies the driving force that enables the self-oscillation. One way to look at this force is as a "negative damping" component. (Circuit diagram by the author)

The equations of damped harmonic motion are well developed. An example of such a damped system is a pendulum with friction at its pivot. Jenkins writes that one simple way to look at self-oscillation is as a system with negative damping.

Negative damping corresponds to an in-phase force acting on an oscillator, which is an ideal way to keep the oscillation going. Nonlinearities in the oscillator limit the amplitude, which would grow exponentially otherwise.

Jenkins also describes the operation of a thermodynamic self-oscillator, the Rijke tube, as shown in the figure. The device is so simple, I'm surprised that I've never seen a science fair project using it.[2] A Rijke tube is a tube that's open at both ends, so air can flow through it. Low in the tube is an electrically heated mesh. The mesh can be heated with a torch, but the heat from the torch complicates matters. When the mesh is hot, an acoustic tone is emitted by the tube,

The tone (an oscillation) is reinforced by the fact that the oscillating air column causes air to pass through the mesh in both directions, thereby enhancing the heat transfer from the mesh to the air.

The Rijke tube thermodynamic self-oscillator. Electrically heating the mesh will cause emission of an acoustic tone. Fig. 12 of Ref. 1, via the arXiv Preprint Server[1].

References:

1. Alejandro Jenkins, "Self-oscillation," arXiv Preprint Server, September 29, 2011
2. Students can be quite imaginative in their science fair projects. I once viewed an elementary school science fair in which one project on sleep apnea had a most unusual display. There were several Barbie Dolls in tiny beds. The dolls had wires affixed to their heads. This was years before Robot Chicken.