A. El Aroudi

BIFURCATIONS IN DC-DC SWITCHING CONVERTERS: REVIEW OF METHODS AND APPLICATIONS

This paper presents, in a tutorial manner, nonlinear phenomena such as bifurcations and chaotic behavior in DC–DC switching converters. Our purpose is to present the different modeling approaches, the main results found in the last years and some possible practical applications. A comparison of the different models is given and their accuracy in predicting nonlinear behavior is discussed. A general Poincare map is considered to model any multiple configuration of DC–DC switching converters and its Jacobian matrix is derived for stability analysis. More emphasis is done in the discrete-time approach as it gives more accurate prediction of bifurcations. The results are reproduced for different examples of DC–DC switching converters studied in the literature. Some methods of controlling bifurcations are applied to stabilize Unstable Periodic Orbits (UPOs) embedded in the dynamics of the system. Statistical analysis of these systems working in the chaotic regime is discussed. An extensive list of references i...

A Design-Oriented Combined Approach for Bifurcation Prediction in Switched-Mode Power Converters

In this brief, two different approaches are combined for studying the stability of a buck switching power dc-dc converter and for predicting its bifurcations. Instability indexes derived from both approaches are combined to get a complete design-oriented perspective of bifurcation analysis in terms of practical circuit parameter and to show the effect of each parameter on the system behavior. The first approach is based on the conventional Routh-Hurwitz criterion applied to the averaged converter dynamics, and it is suitable for detecting low-frequency oscillations but is unsuitable for predicting fast-scale period-doubling instabilities. Complementarily, the second approach considers the ripple component before the pulsewidth modulator to quantitatively predict the occurrence of subharmonic oscillations. The usefulness of the combined approach is shown by analytically deriving inequalities that compress the complete design space into simple instability indexes, which, used complementarily, allow a division of the design space into the different instability regions (both period-doubling or fast-scale instability, and Hopf or slow-scale instability). Comparison of the design-oriented complete map obtained combining both methods and their related stability indexes with the results obtained both from time-domain numerical simulations of the exact switched state equations, as well as the stability border obtained from discrete recurrent maps, corroborates the approach.

A Ripple-Based Design-Oriented Approach for Predicting Fast-Scale Instability in DC–DC Switching Power Supplies

This paper presents a design-oriented analytical approach for predicting fast-scale instability in power electronics converters under voltage-mode control strategy. This approach is based on the use of the ripple amplitude of the feedback control voltage as an index for predicting subharmonic oscillations in these systems. First, the work revisits the stability analysis technique based on the nonlinear discrete-time model, demonstrating that the ripple amplitude can be included within the expression of the Jacobian matrix of this model, hence giving a mathematical support to extend the ripple index to more complex topologies. A simple but representative buck converter under voltage-mode control is used to illustrate the approach. Using the ripple-based index, closed-form expressions of stability boundaries are derived. Unlike other available results obtained from existing methods, the stability boundary, in this work is expressed analytically in terms of both power stage and controller design parameters. Moreover, one can determine how these parameters are involved in the closed form expressions and, furthermore, how each parameter affects the stability of the system. The approach is validated by numerical simulations from the state equations and also experimentally within a wide range of the design parameter space.

Dynamics and Stability Issues of a Single-Inductor Dual-Switching DC–DC Converter

A single-inductor two-input two-output power electronic dc-dc converter can be used to regulate two generally nonsymmetric positive and negative outputs by means of a pulsewidth modulation with a double voltage feedback. This paper studies the dynamic behavior of this system. First, the operation modes and the steady-state properties of the converter are addressed, and, then, a stability analysis that includes both the power stage and control parameters is carried out. Different bifurcations are determined from the averaged model and from the discrete-time model. The Routh-Hurwitz criterion is used to obtain the stability regions of the averaged (slow-scale) dynamics in the design parameter space, and a discrete-time approach is used to obtain more accurate results and to detect possible (fast-scale) subharmonic oscillations. Experimental measurements were taken from a system prototype to confirm the analytical results and numerical simulations. Some possible nonsmooth bifurcations due to the change in the switching patterns are also illustrated.

TWO-DIMENSIONAL BIFURCATION DIAGRAMS: BACKGROUND PATTERN OF FUNDAMENTAL DC–DC CONVERTERS WITH PWM CONTROL

One of the usual ways to build up mathematical models corresponding to a wide class of DC–DC converters is by means of piecewise linear differential equations. These models belong to a class of dynamical systems called Variable Structure Systems (VSS). From a classical design point of view, it is of interest to know the dynamical behavior of the system when some parameters are varied. Usually, Pulse Width Modulation (PWM) is adopted to control a DC–DC converter. When this kind of control is used, the resulting mathematical model is nonautonomous and periodic. In this case, the global Poincare map (stroboscopic map) gives all the information about the system. The classical design in these electronic circuits is based on a stable periodic orbit which has some desired characteristics. In this paper, the main bifurcations which may undergo this orbit, when the parameters of the circuit change, are described. Moreover, it will be shown that in the three basic power electronic converters Buck, Boost and Buck–Boost, very similar scenarios are obtained. Also, some kinds of secondary bifurcations which are of interest for the global dynamical behavior are presented. From a dynamical systems point of view, VSS analyzed in this work present some kinds of bifurcations which are typical in nonsmooth systems and it is impossible to find them in smooth systems.

Stabilizing Technique for AC–DC Boost PFC Converter Based on Time Delay Feedback

It is well known that the ac-dc power factor correction (PFC) boost preregulator can present instability at the line frequency. This nonlinear phenomenon can jeopardize the system performances by increasing the total harmonic distortion and decreasing the power factor. In this brief, a new stabilizing technique is applied using time delay feedback when the system exhibits slow-scale instability under a traditional controller. First, the technique is applied to the averaged model. The results are then validated by numerical simulations using power simulator package. An analytical expression for the stability domain is also provided. It is proven that the proposed technique introduces many advantages to the most and widely used average current mode control through widening the stability region of the PFC converter. Moreover, this technique can also bring the same advantages to other existing commercial control methods for boost and other PFC topologies.

Suppression of Line Frequency Instabilities in PFC AC-DC Power Supplies by Feedback Notch Filtering the Pre-Regulator Output Voltage

Due to nonlinear effects, Power Factor Correction (PFC) ac-dc converters working in Continuous Conduction Mode (CCM) can exhibit nonlinear phenomena such as subharmonic oscillations and chaotic regimes at the line frequency. These undesirable behaviors increase the Total Harmonic Distortion (THD) and shift the dc input line current to non null values. This results in a malfunctioning of the system or even damages caused by the increase of the temperature which can jeopardize enormously the performances and shorten the lifetime of the system. In order to avoid these problems, subharmonic line frequency instabilities must be suppressed. In this paper, a selective notch filter based controller is combined with the output voltage compensator to suppress these instabilities and to stabilize a two-stage PFC ac-dc converter. A frequency domain analysis is used to explain the control mechanism of directing subharmonic and chaotic oscillations into stable periodic motion. A detailed study of the effect of this selective filter on the dynamical behavior of the system is presented. A simple harmonic model is used to obtain the control stability domain. This control technique is simple and allows widening the stability domain and improving the performances of the system being its experimental implementation possible using standard devices such as OAs and passive elements. The correctness of the proposed technique is verified with numerical simulations and experimental measurements.

Modelling and analysis of multicell converters using discrete time models

The main drawback of the discrete time models reported in the literature for predicting nonlinear phenomena in power electronic circuits is their complexity which make their use in system design very minted. The availability of approximated discrete time models that retain the accuracy of the exact model and at the same time makes the system design simple would give new perspectives in the control design of such systems. In this paper we give a detailed analytical study of a two-cell DC-DC buck converter for high voltage applications by using discrete time formulation. Different operating modes are possible and they can be modeled by a unified discrete time model. A digital controller is considered for the system. This controller includes a dynamic compensator in the form of digital integrator for the output variable regulation. An approximated discrete time model in the form of current recurrence equation which accurately describes the dynamical behavior of the system is derived. This model is use to predict instabilities when some design parameters are varied. The Jury test is applied to the characteristic polynomial in order to obtain boundary of stability in the design parameter space. Numerical simulations confirm the theoretical predictions

Analysis of Bifurcation Behavior of a Piecewise Linear Vibrator with Electromagnetic Coupling for Energy Harvesting Applications

Recently, nonlinearities have been shown to play an important role in increasing the extracted energy of vibration-based energy harvesting systems. In this paper, we study the dynamical behavior of a piecewise linear (PWL) spring-mass-damper system for vibration-based energy harvesting applications. First, we present a continuous time single degree of freedom PWL dynamical model of the system. Different configurations of the PWL model and their corresponding state-space regions are derived. Then, from this PWL model, extensive numerical simulations are carried out by computing time-domain waveforms, state-space trajectories and frequency responses under a deterministic harmonic excitation for different sets of system parameter values. Stability analysis is performed using Floquet theory combined with Filippov method, Poincare map modeling and finite difference method (FDM). The Floquet multipliers are calculated using these three approaches and a good concordance is obtained among them. The performance of the system in terms of the harvested energy is studied by considering both purely harmonic excitation and a noisy vibrational source. A frequency-domain analysis shows that the harvested energy could be larger at low frequencies as compared to an equivalent linear system, in particular, for relatively low excitation intensities. This could be an advantage for potential use of this system in low frequency ambient vibrational-based energy harvesting applications.

Stability analysis and bifurcations of SEPIC DC-DC converter using a discrete-time model

In this work, stability analysis and bifurcations of a SEPIC dc-dc converter, operating in the continuous conduction mode (CCM) and under current-programmed control is carried out by means of a discrete-time model. Large-signal model is built, and its corresponding linearised version is performed in order to study the stability of the nominal (1-periodic orbit) operating regime. Varying the reference current, it is obtained that the system functioning changes from a stable buck-like converter to an unstable boost-like converter. The 1-periodic orbit loses its stability via flip bifurcation (FB) and the resulting attractor is a 2-periodic orbit. This latter, bifurcates to chaos via border collision bilification (BCB). It will be shown that allows in order to get a boost-like converter with a stable periodic behavior, the control polarity should be inverted.