Relevance - Chaos can exist in a wide variety of power electronics circuits.
Solvability - As a nonlinear phenomenon, there is no general solution.
Solution - However, there are some approaches to solving chaos problems.
Personal - A personal anecdote.
On the Web - Additional information on the Web.
References - One to a few key papers.
When you look at the output of a dc-dc converter powered by clean (ripple free) inputs and a static load, you expect to see clean output ripple (stable frequency and amplitude), with some broad-band noise associated with the switching moments. If instead you see subharmonics, frequency doubling, quasi-periodicity, unequal amplitudes, or random-like behavior, then chaos or a bifurcation route to chaos may be occurring.
Definitions of Chaos
There is no universally agreed definition of chaos. However, most people would accept the following working definition: Chaos is aperiodic time-asymptotic behaviour in a deterministic system which exhibits sensitive dependence on initial conditions. -- Richard Fitzpatrick ( http://farside.ph.utexas.edu/ )
Chaos appears as a noise-like bounded oscillation with an infinite period and a broadband spectrum where there is no broadband source. -- Brockett, Roger W., and Jonathan R. Wood
Bifurcation is a sudden change in the qualitative dynamics of a function as a parameter is varied. Various types of bifurcations have been identified and named. Period-doubling or subharmonic bifurcations are often seen in power supplies where a frequency component less than the switching frequency appears in the output ripple. Hopf bifurcations are transition from a single point to a limit cycle. Other names seen in the literature, each with its own mathematical definition, are flip bifurcations, fold bifurcations, also called saddle-node or tangent bifurcations, pitchfork bifurcation, transcritical bifurcation, etc.
Usually chaos has the following features: [BROC84A]
- a power spectrum with a continuous part,
- an infinite number of periodic solutions to the associated differential equation, each solution being unstable,
- extreme sensitivity of the trajectories with respect to initial conditions, and
- extreme sensitivities of the trajectories with respect to a parameter.
In a continuous dynamic system, the usual cause of chaos is where a high gain locally initiates instability, but a low gain globally leads to a limitation on the amplitude. [BROC84A].
In switching regulators, chaos often starts with the appearance of subharmonics and progresses to chaos. The progression can often be observed by increasing what caused the subharmonic. For example, if increasing the input voltage brought on subharmonics, increasing it more may cause chaos. (The input voltage is called the bifurcation parameter in this bifurcation-route to chaos.)
Signs of Chaos
Chaos is often observed on the output ripple of converters, by jitter in the duty-cycle control, and by sound.
With no ripple on input sources, static output ripple is equal in period and magnitude. Uneven magnitude or period may be a sign of subharmonics or chaos.
- Duty Cycle
Subharmonics and chaos appear as jitter and worse in the duty-cycle waveforms.
Chaos is often detected by the sound of vibrating magnetics described as a "raucous whine" [DEIS78A] or "frying bacon".[BROC84A]
Hard to tell from random noise, chaos has some structure where random noise does not. Mappings from waveforms can be used to reveal structure. [DEAN90A] Bifurcation
Chaos can exist in a wide variety of power electronics circuits including:
- PWM regulators,
- ripple regulators,
- current-mode controllers,
- overcurrent protection circuits,
- series resonant circuits,
- rectifier-filter circuits,
- ferro-resonant circuits,
- snubbers with nonlinear inductors,
- magnetic amplifiers,
- circuits with a combination high-gain and low-gain loop, etc.
Chaos occurs in nonlinear circuits of second order or higher.
In autonomous circuits (no external driving function) such as a ripple regulator, the circuit must be a third order (three or more energy storage elements) or higher. Energy storage elements that increase the circuit order can include parasitic capacitance and stored charge in diodes and transistors. Autonomous systems whose only periodic solutions are limit cycles do not exhibit chaos. Single and second order autonomous systems are not chaotic. [BROC84A]
In driven circuits, such as the buck converter with PWM voltage-mode control, chaos can occur in second order systems. The driving element in this circuit is the external PWM sawtooth ramp. [DEAN90B]
Chaos is usually bounded and non-destructive and therefore may be useful, although presently it is usually considered undesirable. As it is understood and controlled, it may provide useful in design, for example, smearing discrete harmonics to broadband noise.
Chaotic circuits are always nonlinear, which means there is no general analytical methods for solving them. There is no a priori criteria for determining if a nonlinear circuit will be chaotic.
Although understanding chaos is still in its early stages, there are useful methods for exploring chaos, such as mapping, and for solving specific chaos problems. The phenomena is under active investigation and we will probably better understand it in the future.
Only the simplest problems yield to analysis. Numerical methods and simulation are widely use to explore chaos.
The following are some considerations for solving chaos problems.
Investigators wanting to create chaos often start with nearly unstable circuits. This suggests avoiding chaos by making sure the circuit is robustly stable.
The parasitics of transistors and diodes often provide energy states that contribute to chaos. One of the simplest chaotic circuits is a series R-L-Diode, a prototype for many rectifier circuits showing chaos. These circuit investigations require a slow diode. This suggests that substituting faster diodes or transistors may solve chaos problems. [DEAN89A]
Ripple (also called bang-bang or hysteretic) regulators are deceptively simple, but prone to chaos. Current-mode control circuits and overcurrent protection circuits are also prone to chaos, the latter partly because they are often poorly stabilized. One should push these circuits beyond the worse-case limits while looking for the signs of chaos. If found they may not be detrimental to the design. The timeline of key papers and the bibliography should give additional clues to solution approaches - or lack of them.
I remember when chaos entered my life in the '60s (a chaotic decade). I reported to my manager that, as far as I could tell, the circuit behavior of my 20kHz "bang-bang" regulator was controlled by the phases of the moon. He did not take kindly to blaming the moon for my circuit's misbehavior. I changed to a PWM control and the problem went away. Later, when working with a discontinuous mode boost converter similar to that reported by Tse, I saw similar waveforms. In hindsight I feel vindicated in being perplexed. It was 20 years later, in 1984, when the problem was first discussed at a power electronics conference. We now know a lot more about it and more is becoming available.
On the Web
Chaos has been defined as
- "Ancient God of the shapeless void that preceded the creation of the Earth,"
- "Extreme confusion or disorder," and
- "Stochastic behavior occurring in a deterministic system."
It has been used to explain many things, such as why we can't make long range predictions of the weather or the economy.
If you are interested in learning more about chaos, a good starting point is The Chaos Hypertext book. More mundane is a FAQ sheet on chaos related terms. A sonnet by Edna St. Vincent Millay (1892-1950), I will put Chaos into fourteen lines, Sonnet X, gives it a poetic meaning. Millay died before the resurgence of mathematical chaos and could have known nothing of it, but the poem could describe the beauty of strange attractors as traced by computer graphics. You can see the beauty of some of these strange attractors in the Chaos Gallery at the Chaos at Maryland site, which is a good jumping off place for other links. A search on Chaos on Yahoo yields many other websites. But of recent interest are the new academic tagging websites where researchers can share their reference citations. For example, try CiteULike and search for chaos in title, author last name, abstract, journal name, or tag search types. You will get recent citations on the subject from a wide variety of scientific disciplines.The mathematically inclined may want to review the topic of chaos in Eric Weinstein's World of Mathematics on the Wolfram Research website.
Timeline of key papers: Historical development of problem and solutions
Bibliography: Abstracts of source material.
The definitive book on chaos in power electronics