Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared to the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinise the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points and it pinpoints the crucial need of considering this effect when investigating critical transitions

Nonlinear dynamics of hysteretic oscillators

The dynamic response and bifurcations of a harmonic oscillator with a hysteretic restoring force and sinusoidal excitation are investigated. A multilinear model of hysteresis is presented. A hybrid system approach is used to formulate and study the problem. A novel method for obtaining exact transient and steady state response of the system is discussed. Simple periodic orbits of the system are analyzed using the KBM method and an analytic criterion for existence of bound and unbound resonance is derived. Results of KBM analysis are compared with those from numerical simulations. Stability and bifurcations of higher period orbits are studied using Poincar´e maps. The Poincar´e map for the system is constructed by composing the corresponding maps for the individual subsystems of the hybrid system. The novelty of this work lies in a.) the study of a multilinear model of hysteresis, and, b.) developing a methodology for obtaining the exact transient and steady state response of the system

Analysis of output loading effects in autonomous circuits

A methodology is presented to analyze the impact of the termination load on the oscillation frequency and output power of autonomous circuits. Variations of this load can also lead to an extinction of the oscillation signal, due to their effect on the impedance seen by the active device(s). The new methodology enables an efficient analysis and mitigation of the pulling effects, in the case of undesired output mismatch, as well as an efficient oscillator synthesis in large-signal conditions, for specified values of oscillation frequency and output power. The method is based on the calculation of constant-amplitude and constant-frequency contours, traced in the Smith chart. Oscillation extinctions and some forms of hysteresis can be predicted through the inspection of these contours. However, the stability properties will generally depend on the frequency characteristic of the termination impedance. In an oscillator synthesis, the selected impedance, providing the specified values of oscillation frequency and output power, must be implemented in order to guarantee a stable solution. The dependence of the phase-noise spectral density on the particular implementation is predicted, combining an analysis based on the variance of the phase deviation with the conversion-matrix approach.This work was supported by the Spanish Ministry of Economy and Competitiveness under the research project TEC2014-60283-C3-1-R, the European Regional Development Fund (ERDF/FEDER) and Juan de la Cierva Research Program IJCI-2014-19141 and by the Parliament of Cantabria under the project Cantabria Explora 12.JP02.64069

Systematic methodology for the global stability analysis of nonlinear circuits

A new methodology for the detection of Hopf, flip, and turning-point bifurcations in nonlinear circuits analyzed with harmonic balance (HB) is presented. It enables a systematic determination of bifurcation loci in terms of relevant parameters, such as input power, input frequency, and bias voltages, for instance. It does not rely on the use of continuation techniques and is able to globally provide the entire loci, often containing multivalued sections and/or disconnected curves, in a single simulation. The calculation of Hopf and flip bifurcations is based on the extraction of a small-signal admittance/impedance function from HB and the calculation of its zeros through a geometrical procedure. The method is ideally suited for the investigation of the global stability properties of power amplifiers and other nonlinear circuits. The turning-point locus, associated with either jump phenomena or synchronization, is obtained by taking into account the annihilation/generation of steady-state solutions that is inherent to this type of bifurcation. A technique is also presented for the exhaustive calculation of oscillation modes in multidevice oscillators and oscillators loaded with multiresonance networks. The new methodologies are illustrated through their application to a power amplifier and a multimode oscillator.This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under research projects TEC2014-60283-C3-1-R and TEC2017-88242-C3-1-R

Analysis and elimination of hysteresis and noisy precursors in power amplifiers

Power amplifiers (PAs) often exhibit instabilities leading to frequency division by two or oscillations at incommensurate frequencies. This undesired behavior can be detected through a large-signal stability analysis of the solution. However, other commonly observed phenomena are still difficult to predict and eliminate. In this paper, the anomalous behavior observed in a Class-E PA is analyzed in detail. It involves hysteresis in the power-transfer curve, oscillation, and noisy precursors. The precursors are pronounced bumps in the power spectrum due to noise amplification under a small stability margin. The correction of the amplifier performance has required the development of a new technique for the elimination of the hysteresis. Instead of a trial-and-error procedure, this technique, of general application to circuit design, makes use of bifurcation concepts to suppress the hysteresis phenomenon through a single simulation on harmonic-balance software. Another objective has been the investigation of the circuit characteristics that make the noisy precursors observable in practical circuits and a technique has been derived for their elimination from the amplifier output spectrum. All the different techniques have been experimentally validated

Hysteresis and oscillation in high-efficiency power amplifiers

Hysteresis in power amplifiers (PAs) is investigated in detail with the aid of an efficient analysis method, compatible with commercial harmonic balance. Suppressing the input source and using, instead, an outer-tier auxiliary generator, together with the Norton equivalent of the input network, analysis difficulties associated with turning points are avoided. The turning-point locus in the plane defined by any two relevant analysis parameters is obtained in a straightforward manner using a geometrical condition. The hysteresis phenomenon is demonstrated to be due to a nonlinear resonance of the device input capacitance under near optimum matching conditions. When increasing the drain bias voltage, some points of the locus degenerate into a large-signal oscillation that cannot be detected with a stability analysis of the dc solution. In driven conditions, the oscillation will be extinguished either through synchronization or inverse Hopf bifurcations in the upper section of the multivalued curves. For an efficient stability analysis, the outer-tier method will be applied in combination with pole-zero identification and Hopf-bifurcation detection. Departing from the detected oscillation, a slight variation of the input network will be carried out so as to obtain a high-efficiency oscillator able to start up from the noise level. All the tests have been carried out in a Class-E GaN PA with measured 86.8% power-added efficiency and 12.4-W output power at 0.9 GHz.This work was supported by the Spanish Ministry of Economy and Competitiveness (MINECO) under Project TEC2014-60283-C3-1-R and Project TEC2014-58341-C4-1-R, with FEDER co-funding, the Parliament of Cantabria (12.JP02.64069) and by the Predoctoral Fellowship for Researchers in Training of the University of Cantabria and the Regional Ministry of Education of the Government of Cantabria

Multiphysics modelling and experimental validation of microelectromechanical resonator dynamics

The modelling of microelectromechanical systems provides a very challenging task in microsystems engineering. This field of research is inherently multiphysics of nature, since different physical phenomena are tightly intertwined at microscale. Typically, up to four different physical domains are usually considered in the analysis of microsystems: mechanical, electrical, thermal and fluidic. For each of these separate domains, well-established modelling and analysis techniques are available. However, one of the main challenges in the field of microsystems engineering is to connect models for the behavior of the device in each of these domains to equivalent lumped or reduced-order models without making unacceptably inaccurate assumptions and simplifications and to couple these domains correctly and efficiently. Such a so-called multiphysics modelling framework is very important for simulation of microdevices, since fast and accurate computational prototyping may greatly shorten the design cycle and thus the time-to-market of new products. This research will focus on a specific class of microsystems: microelectromechanical resonators. MEMS resonators provide a promising alternative for quartz crystals in time reference oscillators, due to their small size and on-chip integrability. However, because of their small size, they have to be driven into nonlinear regimes in order to store enough energy for obtaining an acceptable signal-to-noise ratio in the oscillator. Since these resonators are to be used as a frequency reference in the oscillator circuits, their steady-state (nonlinear) dynamic vibration behaviour is of special interest. A heuristic modelling approach is investigated for two different MEMS resonators, a clamped-clamped beam resonator and a dog-bone resonator. For the clamped-clamped beam resonator, the simulations with the proposed model shows a good agreement with experimental results, but the model is limited in its predictive capabilities. For the dogbone resonator, the proposed heuristic modelling approach does not lead to a match between simulations and experiments. Shortcomings of the heuristic modelling approach serve as a motivation for a first-principles based approach. The main objective of this research is to derive a multiphysics modelling framework for MEMS resonators that is based on first-principles formulations. The framework is intended for fast and accurate simulation of the steady-state nonlinear dynamic behaviour of MEMS resonators. Moreover, the proposed approach is validated by means of experiments. Although the multiphysics modelling framework is proposed for MEMS resonators, it is not restricted to this application field within microsystems engineering. Other fields, such as (resonant) sensors, switches and variable capacitors, allow for a similar modelling approach. In the proposed framework, themechanical, electrical and thermal domains are included. Since the resonators considered are operated in vacuum, the fluidic domain (squeeze film damping) is not included. Starting from a first-principles description, founded on partial differential equations (PDEs), characteristic nonlinear effects from each of the included domains are incorporated. Both flexural and bulk resonators can be considered. Next, Galerkin discretization of the coupled PDEs takes place, to construct reduced-order models while retaining the nonlinear effects. The multiphysics model consists of the combined reduced-order models from the different domains. Designated numerical tools are used to solve for the steady-state nonlinear dynamic behaviour of the combined model. The proposed semi-analytical (i.e. analytical-numerical) multiphysics modeling framework is illustrated for a full case study of an electrostatically actuated single-crystal silicon clamped-clamped beam MEMS resonator. By means of the modelling framework, multiphysics models of varying complexity have been derived for this resonator, including effects like electrostatic actuation, fringing fields, shear deformation, rotary inertia, thermoelastic damping and nonlinear material behaviour. The first-principles based approach allows for addressing the relevance of individual effects in a straightforward way, such that the models can be used as a (pre-)design tool for dynamic response analysis. The method can be considered complementary to conventional finite element simulations. The multiphysics model for the clamped-clamped beam resonator is validated by means of experiments. A good match between the simulations and experiments is obtained, thereby giving confidence in the proposed modelling framework. Finally, next to themodelling approach for MEMS resonators, a technique for using these nonlinear resonators in an oscillator circuit setting is presented. This approach, called phase feedback, allows for operation of the resonator in its nonlinear regime. The closedloop technique enables control of both the frequency of oscillation and the output power of the signal. Additionally, optimal operation points for oscillator circuits incorporating a nonlinear resonator can be defined

Efficient simulation of solution curves and bifurcation loci in injection-locked oscillators

A new method is presented for the two-level harmonic-balance analysis of multivalued synchronized solution curves in injection-locked oscillators. The method is based on the extraction of a nonlinear admittance function, which describes the circuit response from the input source terminals. It does not require any optimization or parameter switching procedures, this constituting a significant advantage compared with previous analysis techniques. With additional mathematical conditions, it enables a straightforward determination of the turning point and Hopf bifurcation loci that delimit the stable injection-locked operation bands. The codimension two bifurcation point at which the turning point and Hopf bifurcation loci merge is analyzed in detail, as well as the saddle-connection locus. As it is shown, a second intersection of the saddle-connection locus with the turning point locus acts as a boundary between synchronization points and points associated with jumps and hysteresis. The likely observation of chaotic solutions in the neighborhood of the saddle-connection locus is discussed too. The techniques have been validated by application to several injection-locked oscillators, obtaining good agreement with the experimental results.This work was supported by the Spanish Ministry of Economy and competitiveness under contract TEC2011-29264-C03-01 and the predoctoral fellowship for researchers in training of the University of Cantabria and the Regional Ministry of Education of the Government of Cantabria