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General closed-form subharmonic oscillation conditions are obtained for ${\rm V}^2$ -controlled buck converters. Both constant switching period and constant on-time operations are analyzed with the outer voltage loop closed or open. Once an arbitrary linear feedback is given, the associated closed-form stability condition of the converter is readily obtained. Past research results based on the describing function technique and the Floquet theory become special cases. This paper provides an alternative and easier method to analyze subharmonic instability.

This paper is an extension of the author's recent research in which only buck converters were analyzed. Similar analysis can be equally applied to other types of converters. In this paper, a unified model is proposed for buck, boost, and buck-boost converters under peak or average current mode control to predict the occurrence of subharmonic oscillation. Based on the unified model, the associated stability conditions are derived in closed forms. The same stability condition can be applied to buck, boost, and buck-boost converters. Based on the closed-form conditions, the effects of various converter parameters including the compensator poles and zeros on the stability can be clearly seen, and these parameters can be consolidated into a few ones. High-order compensators such as type-II and PI compensators are considered. Some new plots are also proposed for design purpose to avoid the instability. The instability is found to be associated with large crossover frequency. A conservative stability condition, agreed with the past research, is derived. The effect of the voltage loop ripple on the instability is also analyzed.

Sampled-data analysis and harmonic balance analysis are applied to analyze switching DC-DC converters under constant on-time control. Design-oriented boundary conditions for the period-doubling bifurcation and the saddle-node bifurcation are derived. The required ramp slope to avoid the bifurcations and the assigned pole locations associated with the ramp are also derived. The derived boundary conditions are more general and accurate than those recently obtained. Those recently obtained boundary conditions become special cases under the general modeling approach presented in this paper. Different analyses give different perspectives on the system dynamics and complement each other. Under the sampled-data analysis, the boundary conditions are expressed in terms of signal slopes and the ramp slope. Under the harmonic balance analysis, the boundary conditions are expressed in terms of signal harmonics. The derived boundary conditions are useful for a designer to design a converter to avoid the occurrence of the period-doubling bifurcation and the saddle-node bifurcation.

Although existence of multiple periodic orbits in some DC–DC converters have been known for decades, linking the multiple periodic orbits with the saddle-node bifurcation (SNB) is rarely reported. The SNB occurs in popular DC–DC converters, but it is generally reported as a strange instability. Recently, design-oriented instability critical conditions are of great interest. In this article, average, sampled-data and harmonic balance analyses are applied and they lead to equivalent results. Many new critical conditions are derived. They facilitate future research on the instability associated with multiple periodic orbits, sudden voltage jumps or disappearances of periodic orbits observed in DC–DC converters. The effects of various converter parameters on the instability can be readily seen from the derived critical conditions. New Nyquist-like plots are also proposed to predict or prevent the occurrence of the instability.

A general instability critical condition in terms of the loop gain is derived for the constant on-time controlled buck converter. The instability is associated with a discrete-time pole at -1. Given an arbitrary control scheme, a systematic procedure is proposed to derive the critical condition, which also shows the required ramp slope to stabilize the converter. Different control schemes are shown to have similar forms of critical conditions. Some previously known critical conditions become special cases in this generalized framework, and their inaccuracies are identified.

Critical conditions of period-doubling bifurcation and saddle-node bifurcation are derived for a class of switched linear systems. They can be applied to other similar hybrid, Filippov, or piecewise-smooth systems. Such switched linear systems are common in DC-DC converters. Those previously known critical conditions in the past research become special cases in the generalized framework. Given an arbitrary control scheme, a systematic procedure is proposed to derive the critical condition, which shows the stable range of a system parameter and the minimum stabilizing ramp slope. Two different systems with similar loop gains at high frequency have similar forms of critical conditions. The effect of switching delay is also analyzed.

Local feedback stabilization of limit cycles in PWM DC-DC converters isconsidered, using recently developed general sampled-data models. Thepaper focuses on converters for which the nominal operating conditionhas lost stability due to off-design operation. The results apply tostabilization of the nominal periodic operating condition. In addition,the same approach can be used to stabilize other limit cycles such asthose embedded in a possible chaotic trajectory. Two feedbackstabilization schemes are proposed and studied in detail. The firststabilization technique uses voltage reference compensation and thesecond uses dynamic ramp compensation. Both employ discrete-time washoutfilters to ensure preservation of the size and shape of the limit cycle.Washout filters ensure that the nominal operating branch is unaffectedby the control, without the need for accurate knowledge of the limitcycle.

Previous models of quasi-resonant converters generally use averaging and assume infinite filter inductance and capacitance to reduce circuit complexity, but at the expense of accuracy. In this paper, exact sampled-data modelling is used. A general block diagram model applicable to various topologies of quasi-resonant converters is proposed. Large-signal analysis, steady-state analysis and small-signal analysis are all studied. They agree closely with the experimental results in the literature. Compared with the averaging approach, the sampled-data approach is more systematic and accurate. Copyright © 2001 John Wiley & Sons, Ltd.

A general critical condition of subharmonic oscillation in terms of the loop gain is derived. Many closed-form critical conditions for various control schemes in terms of converter parameters are also derived. Some previously known critical conditions become special cases in the generalized framework. Given an arbitrary control scheme, a systematic procedure is proposed to derive the critical condition for that control scheme. Different control schemes share similar forms of critical conditions. For example, both V2 control and voltage mode control have the same form of critical condition. A peculiar phenomenon in average current mode control where subharmonic oscillation occurs in a window value of pole can be explained by the derived critical condition. A ripple amplitude index to predict subharmonic oscillation proposed in the past research has limited application and is shown invalid for a converter with a large pole., Submitted to an IEEE Journal on Dec. 23, 2011, and resubmitted to IEEE Transactions on Circuits and Systems-I on Feb. 14, 2012. My current six papers in arXiv have a common reviewer

By transforming an exact stability condition, a new Nyquist-like plot is proposed to predict occurrences of three typical instabilities in DC-DC converters. The three instabilities are saddle-node bifurcation (coexistence of multiple solutions), period-doubling bifurcation (subharmonic oscillation), and Neimark bifurcation (quasi-periodic oscillation). In a single plot, it accurately predicts whether an instability occurs and what type the instability is. The plot is equivalent to the Nyquist plot, and it is a useful design tool to avoid these instabilities. Nine examples are used to illustrate the accuracy of this new plot to predict instabilities in the buck or boost converter with fixed or variable switching frequency., Comment: Submitted to an IEEE journal in 2011

Feedback stabilization of the nominal periodic operating condition of PWM DC-DC converters is considered, using recently developed general sampled-data models. Two types of discrete-time washout filter aided feedback stabilization scheme are proposed and studied in detail. These are voltage reference compensation and dynamic ramp compensation. The stabilization schemes preserve the nominal periodic operating condition through washout filters as the system parameters vary.

A general sampled-data model of PWM DC-DC converters 1, 2 is employed to study types of loss of stability of the nominal (periodic) operating condition and their connection with local bifurcations. In this work, the nominal solution's periodic nature is accounted for via the sampled-data model. This results in more accurate predictions of instability and bifurcation than can be obtained using the averaging approach. The local bifurcations of the nominal operating condition studied here are period-doubling bifurcation, saddle-node bifurcation, and Neimark bifurcation. Examples of bifurcations associated with instabilities in PWM DC-DC converters are given. In particular, input filter instability is shown to be closely related to the Neimark bifurcation.