Colored noise in oscillators. Phase-amplitude analysis and a method to avoid the Ito-Stratonovich dilemma

We investigate the effect of time-correlated noise on the phase fluctuations of nonlinear oscillators. The analysis is based on a methodology that transforms a system subject to colored noise, modeled as an Ornstein-Uhlenbeck process, into an equivalent system subject to white Gaussian noise. A description in terms of phase and amplitude deviation is given for the transformed system. Using stochastic averaging technique, the equations are reduced to a phase model that can be analyzed to characterize phase noise. We find that phase noise is a drift-diffusion process, with a noise-induced frequency shift related to the variance and to the correlation time of colored noise. The proposed approach improves the accuracy of previous phase reduced models

A cascaded two-port network model for the analysis of harmonic chain-based energy harvesters

This paper analyzes a recently proposed energy harvester model, based on a harmonic chain of coupled oscillators connected to a piezoelectric transducer. We discuss first how random mechanical vibrations can be modelled as a superposition of sinusoidal signals with random amplitude and phase. Using a mechanical-electrical analogy, we derive equivalent circuits in the form of two-port networks, for both the single mass and the N-mass energy harvester. The transfer functions for both the single mass and the N-mass (valid for any number of masses in the harmonic chain) devices are then obtained cascading the two-port representations of the individual sections, and we give expressions for computing the output power and the power efficiency. Pros and cons of the harmonic chain model are also briefly discussed

An Impedance Matching Solution to Increase the Harvested Power and Efficiency of Nonlinear Piezoelectric Energy Harvesters

Circuit theory and nonlinear dynamics are instrumental to design efficient energy harvesters for ambient mechanical vibrations. In this work, we show that an impedance matching networks can be designed that maximizes the harvested power, and improves the power efficiency. The proposed matching network achieves impedance matching at a single frequency, that can be chosen at will by the designer, and does not need to coincide with the resonant frequency of the harvester. Moreover, the matching network also increases the harvested power over a wide frequency bandwidth. According to our numerical simulations, the matching network increases the maximum harvested power by a factor greater than 3, and the power harvested over the whole frequency spectrum by a factor of 6. The frequency bandwidth can be further extended considering nonlinear energy harvesters. Even using the matching network designed for the linear case, performance is significantly nonetheless improved for the nonlinear harvester

On the application of circuit theory and nonlinear dynamics to the design of highly efficient energy harvesting systems

Ambient dispersed mechanical vibrations are a viable energy source, that can be converted into usable electric power. Ambient vibrations are random process, that can be modeled by superposition of periodic signals. When most of the energy is concentrated in a narrow frequency band, a single periodic function may be a reasonable approximation. This work shows that circuit theory, complemented with nonlinear dynamics methods, are instrumental in designing efficient energy harvesters for ambient mechanical vibrations. It is also shown that the average extracted power can be maximized by a proper load matching, and that the introduction of nonlinearities results in a larger frequency bandwidth, increasing the efficiency of the harvester at frequencies close to the resonance. Even for the nonlinear harvester, the matched load boosts the performance by a large amount

The Complex World of Oscillator Noise: Modern Approaches to Oscillator (Phase and Amplitude) Noise Analysis

The study of fluctuations in oscillators has been a classical research topic in mathematics, physics, and engineering since the first half of the 20th century [1]-[4]. Besides the intellectual fascination for mathematically difficult problems, the importance of the topic is deeply rooted in practical applications, mainly in the fields of RF and microwave electronics and also telecommunications. In fact, defining a precise frequency reference is fundamental for many applications, both electrical (e.g., transmitters and receivers) and optical (e.g., lasers). The broadening of the generated spectral line is mainly due to the phase-noise component of oscillator fluctuations, which consequently is the most commonly studied feature of oscillator noise (see [5] for a recent and exhaustive review). In a dual perspective, the definition of a precise time reference is also extremely important for digital applications, thus implying the necessity to keep under control the time jitter in clocked and in sampled systems. From a theoretical standpoint, phase noise and time jitter are simply two sides of the same coin, a manifestation of the oscillator's noisy behavior. As the microwave engineer is more often interested in the phase-noise characterization, we discuss only the latter. The time-jitter estimation is discussed, for instance, in [6]

Colored Noise in Oscillators. Phase-Amplitude Analysis and a Method to Avoid the Itô-Stratonovich Dilemma

We investigate the effect of time-correlated noise on the phase fluctuations of nonlinear oscillators. The analysis is based on a methodology that transforms a system subject to colored noise, modeled as an Ornstein-Uhlenbeck process, into an equivalent system subject to white Gaussian noise. A description in terms of phase and amplitude deviation is given for the transformed system. Using stochastic averaging technique, the equations are reduced to a phase model that can be analyzed to characterize phase noise. We find that phase noise is a drift-diffusion process, with a noise-induced frequency shift related to the variance and to the correlation time of colored noise. The proposed approach improves the accuracy of the previous phase reduced models

Leveraging circuit theory and nonlinear dynamics for the efficiency improvement of energy harvesting

We study the performance of vibrational energy harvesting systems with piezoelectric and magnetic inductive transducers, assuming the power of external disturbance concentrated around a specific frequency. Both linear and nonlinear harvester models are considered. We use circuit theory and equivalent circuits to show that a large improvement in both the harvested energy and the power efficiency is obtained, for linear systems, through a proper reactive modification of the load. For nonlinear systems, we use methods of nonlinear dynamics to derive analytical formulae for the output voltage, the harvested energy and the power efficiency.We show that also for the nonlinear case, the modified load significantly boosts the performance

Logic gates based on nonlinear oscillators

Networks of coupled nonlinear oscillators are among the recently proposed computation structures that can possibly overcome bottlenecks and limitations of current designs. It has been shown that coupled oscillator networks are capable of solving complex combinatorial optimization problems, such as the MAX-CUT problem and the Boolean Satisfiability (SAT) problem. The goal of this work is to provide a theoretical framework for designing logic gates based on coupled nonlinear oscillators. We show how a simplified model for the network can be derived using the phase reduction technique. The phase deviation equations obtained are then used to design simple networks that achieve the desired phase patterns implementing the corresponding logic gates

Moment-Based Stochastic Analysis of a Bistable Energy Harvester with Matching Network

We discuss the analysis of a piezoelectric energy harvester for random mechanical vibrations, and we assess the performance improvement guaranteed by interposing a matching network between the transducer and the electrical load, in terms of average output power and power efficiency. The mathematical model describing the harvester is a system of stochastic differential equations, where both cases of linear and nonlinear devices are considered. In the linear case, the power delivered to the load is increased by a factor of about 20 with respect to the direct connection, with a similar increase in the conversion efficiency. In the nonlinear case, we use a moment closure technique to calculate the first- and second-order moments of the electro-mechanical variables in the weak noise limit. Moment calculation is used to determine the optimal values of the matching network components that maximize the performance. In the strong noise limit, the state equations are integrated numerically to determine the same performance metrics. Our analysis shows that a properly designed matching network improves the performance by a significant amount, especially at low noise intensity

Model order reduction and stochastic averaging for the analysis and design of micro-electro-mechanical systems

Electro-mechanical systems are key elements in engineering. They are designed to convert electrical signals and power into mechanical motion and vice-versa. As the number of networked systems grows, the corresponding mathematical models become more and more complex, and novel sophisticated techniques for their analysis and design are required. We present a novel methodology for the analysis and design of electro-mechanical systems subject to random external inputs. The method is based on the joint application of a model order reduction technique, by which the original electro-mechanical variables are projected onto a lower dimensional space, and of a stochastic averaging technique, which allows the determination of the stationary probability distribution of the system mechanical energy. The probability distribution can be exploited to assess the system performance and for system optimization and design. As examples of application, we apply the method to power factor correction for the optimization of a vibration energy harvester, and to analyse a system composed by two coupled electro-mechanical resonators for sensing applications