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We have considered the case of a restricted three-body problem (material points), when the masses of the two main attracting bodies are equal. Their orbits have been assumed to be ellipses. The problem regarding the third body's motion of negligible mass under the influence of the main body's gravitational attraction (the Sitnikov problem) allows particular solutions for the following: the third body moves along a straight line passing through the center of mass of the main bodies and perpendicular to the plane of their orbits. We have assumed that the eccentricity of the main body orbits is small and we investigated the nonlinear problem of the existence of the third body's periodic motion with a period multiple of the revolution period of the main bodies in their orbits. The problem of stability (in Lyapunov's sense) of these periodic motions under both perturbations keeping the trajectory of the third body rectilinear and arbitrary spatial perturbations has also been solved. [ABSTRACT FROM AUTHOR]

Fast transient response, simple control scheme and low cost are features that distinguish constant‐on‐time buck converters from others. Simplicity of the control scheme is reflected in a single requirement for power electronics engineers: a design that ensures enough low amplitude resistive ripple component at the feedback to precisely starts the on‐time pulse. There are design rules that guide the selection of the components. A well known one reported in the literature predicts if the converter suffers of the sub‐harmonic oscillations undesirable phenomenon. The authors present a better simple design expression to avoid the onset of this unwanted drawback. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear dynamics of a PV-fed high-voltage-gain single-switch quadratic boost converter loaded by a grid-interlinked DC-AC inverter is explored in its parameter space. The control of the input port of the converter is designed using a resistive control approach ensuring stability at the slow time-scale. However, time-domain simulations, performed on a full-order circuit-level switched model implemented in PSIM© software, show that at relatively high irradiance levels, the system may exhibit undesired subharmonic instabilities at the fast time-scale. A model of the system is derived, and a closed-form expression is used for locating the subharmonic instability boundary in terms of parameters of different nature. The theoretical results are in remarkable agreement with the numerical simulations and experimental measurements using a laboratory prototype. The modeling method proposed and the results obtained can help in guiding the design of power conditioning converters for solar PV systems, as well as other similar structures for energy conversion systems.

This paper deals with controllers of fast-scale instabilities in DC-DC switching power converters from a frequency domain standpoint with the aim of understanding their working principle and hence simplifying their design. Some approaches for controlling fast-scale instabilities and their limitations are revisited. Considering the frequency domain transfer function of already existing controllers, a simple and extended notch filter centered at half of the switching frequency is proposed to avoid these instabilities. However, a switching converter under this controller may still exhibit the undesired slow-scale instability. Accordingly, the paper explores an alternative approach based on amplifying the harmonic at the switching frequency. Numerical simulations show that the new proposed controller can concurrently improve both fast-scale and slow-scale stability margins. The results from the different controllers are contrasted in terms of stability boundaries, indicating that the last one presents a wider stability range. [ABSTRACT FROM AUTHOR]

V2Ic control provides very fast dynamic performance to the Buck converter both under load steps and under voltage reference steps. However, the design of this control is complex since it is prone to subharmonic oscillations and several parameters affect the stability of the system. This paper derives and validates a very accurate modeling and stability analysis of a closed-loop V2Ic control using the Floquet theory. This allows the derivation of sensitivity analysis to design a robust converter. The proposed methodology is validated on a 5-MHz Buck converter. The work is also extended to V2 control using the same methodology, showing high accuracy and robustness. The paper also demonstrates, on the V2 control, that even a low bandwidth-linear controller can affect the stability of a ripple-based control. [ABSTRACT FROM AUTHOR]

This paper presents measurements, simulations, and a theoretical basis to support the observation of spurious peaks on the output spectrum of a microwave phase-locked loop (PLL) exposed to a continuous wave electromagnetic interference (EMI). It is shown that high-power EMI combined to nonlinear characteristics of the PLL may induce subharmonic oscillation phenomena, which provoke the appearance of spurious peaks on the output spectrum. Measurements show that in addition to their dependence on the EMI power and frequency, these spurious peaks depend also on the natural frequency of the PLL and on its stability. Transient circuit simulations of the PLL allowing the prediction of these spurious peaks are presented. Finally, a theoretical explanation based on an analysis of a second-order PLL is given. [ABSTRACT FROM PUBLISHER]

Microcontroller-based peak current mode control of a buck converter is investigated. The new solution uses a discrete time controller with digital slope compensation. This is implemented using only a single-chip microcontroller to achieve desirable cycle-by-cycle peak current limiting. The digital controller is implemented as a two-pole, two-zero linear difference equation designed using a continuous time model of the buck converter and a discrete time transform. Subharmonic oscillations are removed with digital slope compensation using a discrete staircase ramp. A 16 W hardware implementation directly compares analog and digital control. Frequency response measurements are taken and it is shown that the crossover frequency and expected phase margin of the digital control system match that of its analog counterpart. [ABSTRACT FROM AUTHOR]

Active-clamp dc–dc converters are pulsewidth-modulated converters having two switches featuring zero-voltage switching at frequencies beyond 100 kHz. Generalized equivalent circuits valid for steady-state and dynamic performance have been proposed for the family of active-clamp converters. The active-clamp converter is analyzed for its dynamic behavior under current control in this paper. The steady-state stability analysis is presented. On account of the lossless damping inherent in the active-clamp converters, it appears that the stability region in the current-controlled active-clamp converters get extended for duty ratios, a little greater than 0.5 unlike in conventional hard-switched converters. The conventional graphical approach fails to assess the stability of current-controlled active-clamp converters, due to the coupling between the filter inductor current and resonant inductor current. An analysis that takes into account the presence of the resonant elements is presented to establish the condition for stability. This method correctly predicts the stability of the current-controlled active-clamp converters. A simple expression for the maximum duty cycle for subharmonic-free operation is obtained. The results are verified experimentally. [ABSTRACT FROM AUTHOR]

In this work, through applying a comprehensive bifurcation analysis method, we study the nonlinear radial oscillations of the bubble oscillator. The frequency of the driving force is chosen as the linear resonance frequency (f r) and linear subharmonic (SH) resonance frequency (f s h = 2 f r) of the bubble. Results show that, when the bubble is sonicated with 2 f r , period doubling is more likely to result in non-destructive oscillations. The evolution of the period-2 (P2) oscillations of the bubble makes the shape of a bowtie for bubbles with an initial diameter of 0.74 μm and above. When f = 2 f r , the phase portrait of the P2 attractor is distinctly different from a P2 attractor when f = f r , and subharmonic component of the backscattered pressure is maximum. When sonicated by 2 f r , due to lower oscillation amplitude and gentler bubble collapse, the bubble can sustain stable P2 oscillations for a longer duration and over a broader range of applied acoustic pressure. • Period two oscillation regimes of a bubble are investigated. • Period two occurs over a wider pressure range when the driving frequency is twice the resonance. • Period doubling may result in bubble destruction when sonication frequency is the bubble resonance. • When driving frequency is twice the bubble resonance, period doubling occurs at gentle oscillation regimes. [ABSTRACT FROM AUTHOR]

Purpose. Development of mathematical model for calculation of starting modes of asynchronous motor connected in series with capacitors. Method. Mathematical modeling of dynamic modes of asynchronous motors with lateral capacitor compensation of reactive power. Results. The calculation algorithm and results of mathematic modeling of processes during starting modes of asynchronous motor feeding from the network through capacitors connected in series are presented. It is shown that for some values of capacitance the self-excitation processes and subharmonic oscillations can appear. Scientific novelty. Mathematic modeling and research of processes in asynchronous motor under its feeding through capacitors is carried out for the first time. The calculation algorithm is based on the mathematical model of asynchronous motor with high level of adequacy, which takes into account the magnetic core saturation and the current displacement in limbs of the rotor. Practical implication. Developed mathematical model makes it possible to investigate the possibility of self-excitation modes appearing in condition of their feeding from line with lateral compensation of reactance in order to avoid the negative effects typical for them. [ABSTRACT FROM AUTHOR]