Limit cycles in a digitally controlled buck converter

European Conference on Circuit Theory and Design (ECCTD), Linkoping, Sweden, 29-31 Aug. 2011We describe the mathematical model of a digitally controlled buck converter. This model is an autonomous discrete-time discontinuous piecewise-linear dynamical system in three dimensions. Investigating this system, we find its equilibrium points, describe the shape and size of possible limit cycles (i.e. stable periodic motions), and derive conditions for their existence and non-existence.Irish Research Council for Science, Engineering and Technologyti, ke, ab, li - TS 26.04.1

Frequency locking of modulated waves

We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs. Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals

Limit cycles in a digitally controlled buck converter

European Conference on Circuit Theory and Design (ECCTD), Linkoping, Sweden, 29-31 Aug. 2011We describe the mathematical model of a digitally controlled buck converter. This model is an autonomous discrete-time discontinuous piecewise-linear dynamical system in three dimensions. Investigating this system, we find its equilibrium points, describe the shape and size of possible limit cycles (i.e. stable periodic motions), and derive conditions for their existence and non-existence.Irish Research Council for Science, Engineering and Technologyti, ke, ab, li - TS 26.04.1

Frequency quantization in first-order digital phase-locked loops with frequency-modulated input

Presented at the International Workshop on Nonlinear Maps and their Applications (NOMA '07), INSA, Toulouse, December 13-14, 2007Frequency granularity in a digital phase-locked loop arises from quantization in the number-controlled oscillator which prevents the loop from locking exactly onto its reference signal and introduces unwanted phase jitter. Based on a nonlinear analysis of trajectories in the phase space, we have recently investigated the effect of frequency quantization in a first-order loop with a frequency-modulated input signal and have derived useful bounds on the steady-state phase jitter excursion. In this paper, we continue that work and derive the maximum modulation amplitude such that loop cycle slipping is avoided. We also examine in detail the loop behavior in acquiring phase-lock.Science Foundation Irelandke, ab, co - TS 16.04.1

Phase jitter dynamics of first-order digital phase-locked loops with frequency-modulated input

IEEE International Symposium on Circuits and Systems (ISCAS), Seattle, USA, 18-21 May 2008Inherent to digital phase-locked loops is frequency quantization in the number-controlled oscillator which prevents the loop from locking exactly onto its reference signal and introduces unwanted phase jitter. This paper investigates the effect of frequency quantization in a first-order loop with a frequency-modulated input signal. Using tools of nonlinear dynamics, we show that, depending on the modulation amplitude, trajectories in the phase space eventually fall into either an invariant region or a trapping region, the boundaries of which give useful bounds on the steady-state phase jitter excursion. We also derive a sufficient condition for the maximum modulation amplitude to prevent loop cycle slipping.Science Foundation Irelandke, ab, co li - TS 17.04.1

Frequency quantization in first-order digital phase-locked loops with frequency-modulated input

Presented at the International Workshop on Nonlinear Maps and their Applications (NOMA \u2707), INSA, Toulouse, December 13-14, 2007Frequency granularity in a digital phase-locked loop arises from quantization in the number-controlled oscillator which prevents the loop from locking exactly onto its reference signal and introduces unwanted phase jitter. Based on a nonlinear analysis of trajectories in the phase space, we have recently investigated the effect of frequency quantization in a first-order loop with a frequency-modulated input signal and have derived useful bounds on the steady-state phase jitter excursion. In this paper, we continue that work and derive the maximum modulation amplitude such that loop cycle slipping is avoided. We also examine in detail the loop behavior in acquiring phase-lock.Science Foundation Irelandke, ab, co - TS 16.04.1

Phase jitter dynamics of first-order digital phase-locked loops with frequency-modulated input

IEEE International Symposium on Circuits and Systems (ISCAS), Seattle, USA, 18-21 May 2008Inherent to digital phase-locked loops is frequency quantization in the number-controlled oscillator which prevents the loop from locking exactly onto its reference signal and introduces unwanted phase jitter. This paper investigates the effect of frequency quantization in a first-order loop with a frequency-modulated input signal. Using tools of nonlinear dynamics, we show that, depending on the modulation amplitude, trajectories in the phase space eventually fall into either an invariant region or a trapping region, the boundaries of which give useful bounds on the steady-state phase jitter excursion. We also derive a sufficient condition for the maximum modulation amplitude to prevent loop cycle slipping.Science Foundation Irelandke, ab, co li - TS 17.04.1