Influence of period‐doubling bifurcations in the appearance of border collisions for a ZAD‐strategy‐controlled buck converter

The first period‐doubling bifurcation of a dc–dc buck converter controlled by a zero‐average dynamic strategy is studied in detail. Owing to the saturation of the duty cycle, this bifurcation is followed by a border‐collision bifurcation, which is the main mechanism to introduce instability and chaos in the circuit. The multiparameter analysis presented here leads to a complete knowledge of the relatioship between these two bifurcations. The results are obtained by using a frequency‐domain approach for the study of period‐two oscillations in maps. Copyright © 2010 John Wiley & Sons, Ltd.     

Design of high‐gain wide‐band harmonic self‐oscillating mixers

In this work, a new design method is presented for the design of wide‐band harmonic self‐oscillating mixers (HSOM) with high conversion gain. The optimum design for the HSOM circuit is obtained using bifurcation analysis‐ and control‐techniques in combination with nonlinear optimization techniques based on the use of an auxiliary generator. The design method is illustrated through the design of an 11.25–1.5 GHz third harmonic self‐oscillating mixer (3HSOM) with a 6.5 dB down‐conversion gain over a 1.6 GHz bandwidth. It is also demonstrated how the frequency variation of the conversion gain can be shaped by means of the use of bifurcation control techniques. A good agreement between the simulated and experimental results has been found. Copyright © 2009 John Wiley & Sons, Ltd.     

Bifurcation analysis of a piecewise smooth system with non‐linear characteristics

In previous works, there are no results about the bifurcation analysis for a piecewise smooth system with non‐linear characteristics. The main purpose of this study is to calculate the bifurcation sets for a piecewise smooth system with non‐linear characteristics. We first propose a new method to track the bifurcation sets in the system. This method derives the composite discrete mapping, Poincaré mapping. As a result, it is possible to obtain the local bifurcation values in the parameter plane. As an illustrated example, we then apply this general methodology to the Rayleigh‐type oscillator containing a state‐ period‐dependent switch. In the circuit, we can find many subharmonic bifurcation sets including global bifurcations. We also show the bifurcation sets for the border‐collision bifurcations. Some theoretical results are verified by laboratory experiments. Copyright © 2005 John Wiley & Sons, Ltd.     

Fast‐scale bifurcation in power‐factor‐correction buck‐boost converters and effects of incompatible periodicities

In this paper, we derive the discrete‐time model for the power‐factor‐correction (PFC) buck‐boost converter in terms of a stroboscopic switching map. Fast‐scale instability is analysed through a fold diagram, which exposes the periodicity of the operation as well as the locations of the critical phase angles of the line voltage at which instability begins to occur along a half‐line cycle. The asymmetrical locations of the critical phase angles along a half‐line cycle is explained in terms of ‘under‐developed’ bifurcation. Border collision bifurcations are observed and analysed in detail. Copyright © 2006 John Wiley & Sons, Ltd.     

Fast‐scale instability in a PFC boost converter under average current‐mode control

This paper investigates the fast‐scale instability in a power‐factor‐correction (PFC) boost converter under a conventional average current‐mode control. The converter is operated in continuous mode. Computer simulations and theoretical analysis are performed to study the effects of the time‐varying input voltage under the variation of some chosen parameters on the qualitative behaviour of the system. It is found that fast‐scale instability may occur during a line cycle, which can cause distortion to the line current and degrade the practical power factor. The results provide useful information for the design of PFC boost converters to avoid distortion due to fast‐scale bifurcation. Copyright © 2003 John Wiley & Sons, Ltd.     

Bifurcation behaviour in parallel‐connected boost converters

This paper describes the bifurcation phenomena of a system of parallel‐connected d.c./d.c. boost converters. The results provide important information for the design of stable current sharing in a master–slave configuration. Computer simulations and experiments are performed to capture the effects of variation of some chosen parameters on the qualitative behaviour of the system. In particular, it is found that variation of some parameters leads to Neimark–Sacker bifurcation. Analysis is presented to establish the possibility of the bifurcation phenomena. Copyright © 2001 John Wiley & Sons, Ltd.     

Relationship of fast‐scale and slow‐scale instabilities in switching circuit with multiple inputs

Power converter circuits, such as current‐controlled or voltage‐controlled converters and inverters often have multiple inputs in the controller. The multiple inputs cause high‐frequency and low‐frequency oscillations. In earlier studies, the characteristics of circuits in fast‐scale and slow‐scale dynamics have been investigated. However, in many cases, circuits with multiple inputs have three or more dimensional topology which makes detailed analysis difficult. In this paper, we analyze a simple interrupted electric circuit in order to understand essential characteristics of fast‐scale and slow‐scale dynamics. The advantage of this simple interrupted circuit is that it is possible to derive a 1‐dimensional map, which facilitates rigorous studies. Based on the structure of the return map and the characteristic multiplier, we explain the characteristics of the system. We report the occurrence of pitchfork, period doubling, and border collision bifurcations in slow scale, and period doubling bifurcation in fast scale. We found that local bifurcation, which appears in fast‐scale dynamics, does not significantly affect the global behavior of the system while instabilities in the slow‐scale dynamics strongly affect the system behavior. Copyright © 2016 John Wiley & Sons, Ltd.     

Fast‐scale stability limits of a two‐stage boost power converter

There are many applications in power electronics that demand high step‐up conversion ratio between the source and the load. A simple way of achieving such a high voltage ratio is by cascading DC–DC boost converters in a few stages. The individual converters in such a cascaded system are usually designed separately applying classical design criteria. This paper investigates the stability of the overall system of a cascade connection of two boost converters under current mode control. We first demonstrate the bifurcation behavior of the system, and it is shown that the desired periodic orbit can undergo fast‐scale period doubling bifurcation leading to subharmonic oscillations and chaotic regimes under parameter variation. The value of the intermediate capacitor is taken as a design parameter, and we determine the minimum ramp slope in the first stage required to maintain stability. It is shown that smaller capacitance values give rise to wider stability range. We explain the bifurcation phenomena using a full‐order model. Then, in order to simplify the analysis and to obtain a closed‐form expression to explain the previous observation, we develop a reduced‐order model by treating the second stage as a current sink. This allows us to obtain design‐oriented stability boundaries in the parameter space by taking into account slope interactions between the state variables in the two stages. Copyright © 2015 John Wiley & Sons, Ltd.     

Circuit implementation and model of a new multi‐scroll chaotic system

In this paper, a multi‐scroll chaotic system from the improved Chua's system is proposed. Moreover, non‐linear dynamics are analyzed including phase‐space trajectories, bifurcation diagrams, Poincaré maps and so on. The most important thing is that we discovered phase‐space trajectories, bifurcation diagrams and Poincaré maps are unified and closely related, which can describe different aspects of the multi‐scroll chaotic system. Furthermore, the corresponding improved module‐based circuits are designed for realizing two to four‐scroll chaotic attractors, and the experimental results are also obtained, which are consistent with the numerical simulations. Copyright © 2012 John Wiley & Sons, Ltd.     

The design and implementation of a multi‐wing chaotic attractor based on a five‐term three‐dimension system

The last two decades have seen great progress about the generation and circuit realization of multi‐wing chaotic attractor. In this paper, several multi‐scroll chaotic attractors are generated from a five‐term system by adding a piecewise linear function. Moreover, some basic properties in terms of symmetry and dissipation, equilibrium points, eigenvalues of the Jacobian matrices, Lyapunov exponent spectrum, bifurcation diagram, and Poincaré map are studied. In particular, an analog circuit is designed to implement the proposed multi‐scroll attractors, which are different from the traditional attractors. Furthermore, an integrated circuit diagram is designed to realize the fractional‐order multi‐scroll attractors. Finally, the performed experimental results confirm the theoretical analysis, and our work lends itself to many potential applications in engineering. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical modeling of the route‐to‐chaos of semiconductor lasers under optical feedback and its dependence on the external‐cavity length

This paper investigates numerically influence of the external‐cavity length on the type of the route‐to‐chaos of semiconductor lasers under external optical feedback. The study is based on numerical solution of a time‐delay model of rate equations, and the solutions are employed to construct bifurcation diagrams and to examine the Fourier frequency spectrum of the laser output. The ratio of the relaxation frequency to the external‐cavity resonance frequency is employed to measure the influence of the length of the external cavity. The route‐to‐chaos is period doubling when this frequency ratio is less than unity. The route is sub‐harmonic when the frequency ratio increases up to 2.25. When the frequency ratio increases further, the transition to chaos becomes quasi‐periodic characterized by the compound‐cavity frequency and the relaxation frequency as well as their difference. Copyright © 2009 John Wiley & Sons, Ltd.     

Second order mem‐circuits

This paper presents a comprehensive taxonomy of so‐called second‐order memory devices, which include charge‐controlled memcapacitors and flux‐controlled meminductors, among other novel circuit elements. These devices, which are classified according to their differential and state orders, are necessary to get a complete extension of the family of classical nonlinear circuit elements (resistors, capacitors, and inductors) for all possible controlling variables. Using a fully nonlinear formalism, we obtain nondegeneracy conditions for a broad class of second‐order mem‐circuits. This class of circuits is expected to yield a rich dynamic behavior; in this regard, we explore certain bifurcation phenomena exhibited by a family of circuits including a charge‐controlled memcapacitor and a flux‐controlled meminductor, providing some directions for future research. Copyright © 2014 John Wiley & Sons, Ltd.     

Theoretical and experimental analysis of a simple PWM‐1 controlled interrupted electric circuit

In this paper, we analyze a simple PWM‐1 controlled interrupted electric circuit in order to essentially understand the circuit fundamental characteristics. First, we explain the circuit dynamics, and then we define the return map by using the exact solution. Next, we focus on the existence region of the solution (invariant interval) and bifurcation phenomena in the circuit. In particular, we find the circuit has three types of the invariant interval depending on the parameter. We also clarify that the period‐doubling bifurcation and the border‐collision bifurcation effect in the existence region of the periodic solution in a wide parameter plane. Finally, the mathematical results are verified by the laboratory experiment. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical characterization of nonlinear oscillatory waves in a composite right‐ and left‐handed traveling‐wave field‐effect transistor

A specialized type of traveling‐wave field‐effect transistor, the gate and drain lines of which have composite right‐ and left‐handed structures, is considered as the platform to support nonlinear oscillatory waves. The cubic–quintic complex Ginzburg–Landau equation is obtained by application of the reductive perturbation method, by which we quantify the homogeneous oscillations including the property of the Andronov–Hopf bifurcation point, oscillation frequency, and amplitude. Several numerical calculations follow to validate the Ginzburg–Landau equation‐based analysis. Finally, the dynamics of numerically obtained stationary flat‐top pulses are discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

Modeling of switching frequency instabilities in buck‐based DC–AC H‐bridge inverters

In this paper, the dynamical behavior of a full bridge DC–AC buck inverter controlled by fixed frequency and PWM is studied. After showing that the system can undergo both period‐doubling and Neimark–Sacker bifurcation at the fast scale (switching period) by using the exact switching model, an exact solution discrete‐time model able to predict both instability phenomena is derived. The model is obtained without making the quasi‐static approximation and it can be used to obtain the useful operation region in the multi‐dimensional design parameter space from time domain simulations in a very fast and accurate manner. Based on the study of the system, some design guidelines are provided. Copyright © 2010 John Wiley & Sons, Ltd.     

Fast‐scale bifurcation in single‐stage PFC power supplies operating with DCM boost stage and CCM forward stage

This paper describes the fast‐scale bifurcation phenomena of a single‐stage single‐switch power‐factor‐correction (PFC) regulator comprising a boost stage operating in discontinuous conduction mode (DCM) and a forward stage operating in continuous conduction mode (CCM). The two stages combine into a single stage by sharing one main switch and one control loop. Using ‘exact’ cycle‐by‐cycle computer simulations, the effects of various circuit parameters on fast‐scale instabilities are studied. The results are qualitatively verified by experimental measurements. This work provides a clear picture of how the variation of certain practical parameters can render such a circuit fast‐scale unstable. Copyright © 2006 John Wiley & Sons, Ltd.     

Improved static and dynamic performances of a two‐cell DC–DC buck converter using a digital dynamic time‐delayed control

Multi‐cell converters have been developed to overcome shortcomings in usual switching devices. The control system in these circuits is twofold: first, to balance voltages of the switches and second to regulate the load current to a desired value. However, with a purely proportional controller, the system presents a static error. With a PI controller the static error is annihilated, but at the expense of shortening the stability region and increasing settling time. In this work, a zero static error dynamic controller for a two‐cell DC–DC buck converter is designed. To achieve zero current error, we propose a generalized scheme of a dynamic controller. Then, using nonlinear analysis and Lyapunov stability theory and bifurcation prediction tools, we prove that zero static error is achieved. The proposed controller outperforms the PI controller in terms of settling time in the presence of saturating effect during the start‐up transients. Numerical simulations in the form of time domain waveforms and bifurcation diagrams from switched circuit‐based model are presented to confirm our theoretical results. Copyright © 2010 John Wiley & Sons, Ltd.     

Crisis‐induced coexisting multiple attractors in a second‐order nonautonomous memristive diode bridge‐based circuit

This paper reports a second‐order nonautonomous memristive diode bridge‐based circuit, upon which a system model is established. The AC and DC equilibrium points and their stability evolutions are theoretically analyzed, and the mechanisms of complex dynamical behaviors are explored in detail. Furthermore, the stimulus‐dependent dynamical behaviors are numerically performed by the single‐parameter bifurcation diagrams, Lyapunov exponents, and phase portraits. Of particular concern, it should be highly emphasized that multiple kinds of crisis scenarios associated with the initial conditions are found in a specified parameter region, resulting in that coexisting multiple attractors under different initial conditions are discovered for the fixed system parameters. Additionally, hardware experiments and PSpice circuit simulations are used to confirm the numerically simulated results.     

Analysis and experimental verification of the linear correlation of the operation condition among the bifurcation points of current‐mode‐controlled buck–boost converter

The operation parameter of the buck–boost converter is examined for its boundary of the first flip bifurcation. It is found that the parameters exhibit a linear relation for the same parameter at another bifurcation point. The bifurcation parameters also have a linear correlation between them. Theoretical analysis is presented to explain how this can occur. Simulation and experimental results at 20 kHz are used to present these novel results. Copyright © 2005 John Wiley & Sons, Ltd.     

Spiral periodic structure inside chaotic region in parameter‐space of a Chua circuit

In this letter we investigate, via numerical simulations, the parameter‐space of the set of autonomous first‐order differential equations of a Chua circuit. We show that this parameter‐space presents self‐organized periodic structures immersed in a chaotic region, forming a single spiral structure that coils up around a focal point. Additionally, bifurcation diagrams are used to show that those periodic structures also organize themselves in period‐adding cascades, along specific directions that point towards this same focal point. Copyright © 2010 John Wiley & Sons, Ltd.