Chaos in Power Electronics & Power Supplies (1 of 5)

Problem

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
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]

1. a power spectrum with a continuous part,
2. an infinite number of periodic solutions to the associated differential equation, each solution being unstable,
3. extreme sensitivity of the trajectories with respect to initial conditions, and
4. extreme sensitivities of the trajectories with respect to a parameter.

Continuous System
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].

Switching Regulators
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.

• Ripple
  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.
• Sound
  Chaos is often detected by the sound of vibrating magnetics described as a "raucous whine" [DEIS78A] or "frying bacon".[BROC84A]

Structure
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

Do not use this information for design without independent verification of the information.