
[MARO78A] Marotto shows that the existence of a snapback repeller is sufficient for chaos (as reported in Baillieul below and Mees below).

[DEIS78A] Deisch, in a paper on currentmode control, implies chaos (raucous whine from the transformer unrelated to the switching frequency) in a fixed frequency currentmode controlled buck derived converter, even though chaos is not mentioned.

[BAIL80A] Baillieul, Brockett, and Washburn, in a highly mathematical paper, extend the criteria for chaos beyond the presence of a snapback repeller to a wider variety of feedback systems and illustrate the technique with a firstorder currentmode controlled PWM regulator. They identify other power converters as potentially chaotic systems.

[BROC84A] Brockett and Wood provide an early introduction to chaotic behavior in power electronic circuits. Chaos does not occur in autonomous circuits described by first and second order differential equations or those systems whose only periodic solutions are stable limit cycles. Chaotic circuits described include a peakcurrentlimit overcurrent protection circuit in a PWM forward converter, and a ripple regulator.

[MEES86A] Mees, in a book edited by Holden, looks at chaos from the viewpoint of feedback theory. He uses the Marotta, and the Baillieul, et. al. papers above as a starting point for discussing chaos in a currentmode controlled pulsewidth modulated (PWM) feedback system.

[HAMI88A] Hamill and Jefferies use difference equations and a zigzag return map to show chaos in a first order PWM voltagemode controlled converter.

[WOOD89A] Wood updates the 1984 Brockett and Wood paper. Describes the same circuits but with an improved introduction and additional lab results from the earlier ripple regulator circuit. A good first introduction to chaos in power electronics.

[DEAN89A] Deane and Hamill classify a wide variety of power electronics and support circuits as potentially chaotic. They use mappings to distinguish chaotic from random circuits and bifurcation diagrams to show the sequence from subharmonic to quasiperiodicity to chaotic behavior with bands of intermittency. A short glossary of terms is included. Circuits discussed in detail are a simple driven RLDiode (slow diode) circuit and a series resonant converter modification; a series RLC circuit with a nonlinear inductance (similar to some snubbers); a controlled rectifier circuit; and a buck converter driven by a MC34060 currentmode controller. A good introduction to chaos. Reprinted in PELS'90.

[DEAN90A] Deane and Hamill republish their PESC'89 paper in PELS'90. A good introduction to chaos.

[DEAN90B] Deane and Hamill examine the first order (no output capacitor) and second order voltagemode controlled PWM Buck converter. The first order converter can exhibit multiple pulsing but not chaos. With input voltage the bifurcation parameter, the second order converter is shown to bifurcate from stable to 1/2, 1/4, 1/8 ... subharmonics to chaos. Chaos is shown by determining boundary conditions on the circuit differential equations, by SPICE simulation, and by an experimental circuit, with good agreement. Data is displayed in bifurcation diagrams, the iv phase plane, timeamplitude plots, and Poincare' sections.

[KREI90B] Krein and Bass discuss three types of instability: unbounded (destructive); chatter (destructive); and multiple operating modes with bifurcations to chaos (nondestructive). Chaotic circuits discussed are a buck converter with boundary control (ripple regulator), and a boost converter with currentmode control. Chatter in sliding mode control is also discussed.

[HAMI92A] Hamill, Deane, and Jefferies describe iterated mappings, a nonlinear discrete modeling technique and its advantages over averaging techniques (which do not predict subharmonics or chaos), discrete time modeling (predicts subharmonics but not chaos) and iterative mappings (predicts stability, subharmonics, and chaos). They apply the results to a voltagemode controlled PWM buck converter. The paper provides a good introduction to chaos in power converters.

[TSE94A] Tse publishes the first paper reporting chaos in discontinuousmode switching regulators, containing theory, simulations, and experimental results. A companion paper reports similar results for the buck converter.

[TSE95A] Tse and Chan report Chaos in a fourthorder Cuk converter under currentmode control. Experimental confirmation is reported in a separate paper.

[HAMI95A] Hamill reviews the state of power electronics including history, solid state switches, power converters, the buck converter, the shortcomings of the averaged models, especially in predicting chaos, models that predict chaos, and the pitfalls of computer simulation of these circuits. An excellent review paper that contains an extensive bibliography of key power electronics papers (72) and its own timeline of key chaos papers related to power electronics.

[PODD95A] Poddar, Chakrabarty, and Banerjee propose a chaos prevention circuit for a single order currentmode controlled PWM buck converter that does not require complex online computations or an additional periodic reference voltage. The prevention circuit is far more complex than the buck converter.

[HAMI97A] Hamill, Deane, and Aston review the nonlinear dynamics of power electronics circuits and suggest some uses for operating them in a chaotic mode. The peakcurrent controlled boost converter is used as the example. Data is provided on how chaotic operation can reduce electromagnetic interference by spreading the spectrum. Other suggestions include using chaos to for targeting and control, and for increasing the agility of converters operating on the edge of stability.