Second-order Oscillator Research Articles

Design and Analysis of CCII-Based Oscillator with Amplitude Stabilization Employing Optocouplers for Linear Voltage Control of the Output Frequency

This paper shows the topology design of a simple second-order oscillator based on two three-port current conveyors, two resistors, and two grounded capacitors, as well as its modification to a voltage-controlled oscillator (VCO). In comparison with many previous works, the following useful conceptual novelties and improvements were made in this study. Both resistors presented in the topology can be employed to tune of the oscillation frequency by the simultaneous driving of two optocouplers with resistive output stage. The current gain of the current conveyor ensures the control of the oscillation condition. The proposed solution offers advantages (in comparison with many standard so-called single-resistance-controllable types) of improved dependence of the frequency of oscillation (FO) on a driving force (extended tuning of the FO), constant ratio of amplitudes of generated waveforms when the FO is tuned, low complexity (taking into account auxiliary circuitry for optocouplers), and comfortable tuning of the FO by a single control voltage. The oscillator produces waveforms with tunable frequency having a constant 45-degree phase shift between them. The relative sensitivities of the proposed solution achieve typical values for these second-order systems (−0.5). Experimental verification confirmed the expected behavior in the operational band between 1 and 10 MHz tuned by a DC voltage from 1.7 to 5 V. This indicates a significant reduction of the driving force ratio (3:1 in our case) in comparison with standard tuning approaches required for a ratio of 10:1 for FO adjustment. Output amplitudes reached 100 and 150 mV in the observed tunability range with distortion ranging between 0.7 and 3.3%.

Synchronization of stochastic complex networks with time delay via feedback control based on discrete-time state observations

Abstract In this paper, synchronization of stochastic complex networks (SCNs) with time delay is researched. A feedback control method is introduced to drive such SCNs to achieve inner synchronization in mean square. Different from previous researches, the feedback control we designed is based on discrete-time state observations. Moreover, by exploiting Lyapunov method and Kirchhoff’s Matrix Tree Theorem in graph theory, some sufficient criteria are obtained to guarantee synchronization in mean square and asymptotical synchronization in mean square of SCNs. What is more, we make use of the theoretical results to analyze asymptotical synchronization in mean square of second-order Kuramoto oscillators with time delay and stochastic disturbances and get a sufficient criterion. At the end of this paper, a numerical example is provided to validate the effectiveness and feasibility of our theoretical results.

Adaptive Fixed-Time Fast Terminal Sliding Mode Control for Chaotic Oscillation in Power System

The second-order chaotic oscillation system model is used to analyze the dynamic behavior of chaotic oscillations in power system. To suppress chaos and stabilize voltage within bounded time independent of initial condition, an adaptive fixed-time fast terminal sliding mode chaos control strategy is proposed. Compared with the conventional fast terminal sliding mode control strategy and finite-time control strategy, the proposed scheme has advantages in terms of convergence time and maximum deviation. Finally, simulation results are given to demonstrate the effectiveness of the proposed control scheme and the superior performance.

Methods for Extended Tunability in Quadrature Oscillators Based on Enhanced Electronic Control of Time Constants

New methods for extending the tunability range in second-order voltage-controlled oscillators (VCOs) with quadrature outputs are presented and compared in this paper. Linear dependence is the most common type of relation between frequency of oscillations (FO) and control voltage (or current) in communication subsystems (VCOs). However, there are other nonlinear types of dependencies that are advantageous (quadratic and cubic) and should be further studied. Positions of controllable blocks within the structure of the oscillator (providing parameters influencing time constants of integrators) in two scenarios are studied together. The impact of controllable parameters of these additional elements on FO as well as on produced output signal levels is evaluated. The simulation and experimental results are shown to confirm expected behavior of solutions in the case of the quadratic type of control of FO. Nowadays, standard linear control offers a tunability ratio 10:1 (for the same ratio of parameters that control the FO) only. However, in this paper, the experimental tests yield a tunability ratio 80:1 (50 kHz–4 MHz) for dc control voltage within the range 0.175–1.75 V (10:1). The presented approach is very beneficial for low supply voltage systems requiring a wide range of tunability and also for utilizing active devices having limited range of adjustability of their controllable parameters while still requiring wide tunability.

Proximate Time-Optimal Control of a Harmonic Oscillator

This paper considers near-time-optimal setpoint tracking of a second-order oscillator. First, the time-optimal setpoint tracking feedback law is derived by recasting the problem as a regulator problem in the error coordinates with shifted control limits. This result is used to construct a derated approximation of the optimal control. Initially, the control law is derived as a regulator controller. This result is then extended to include setpoint tracking. Global stability is proven for the setpoint tracking controller in the zero-damping case. A scaling method is developed, which allows an efficient implementation of the control laws using a single lookup table. Both the damped and undamped controllers are validated experimentally.

Synchronization analysis of stochastic coupled systems with time delay on networks by periodically intermittent control and graph-theoretic method

In this paper, the issue of inner exponential synchronization (IES) for a class of stochastic coupled systems with time-varying delay on networks (SCSTVDNs) is investigated. A periodically intermittent control strategy is introduced to drive such SCSTVDNs to achieve IES. By exploiting Lyapunov method and Kirchhoffs matrix tree theorem in graph theory, two types of sufficient IES criteria are respectively established in the form of Lyapunov-type theorem and coefficients-type theorem. Furthermore, to show the practicability of the theoretical results, the analysis of IES for the second-order Kuramoto oscillators with stochastic perturbations and time-varying delay is performed. A numerical example is finally provided to demonstrate the validity and feasibility of our analytical results.

Semi-analytical solutions of the nonlinear oscillator with a matrix Lagrange multiplier

In this paper, a semi-analytical method based on variational iteration formulae is proposed. Second-order nonlinear oscillator equations are numerically investigated by this method. A matrix Lagran...

Chaos in a second-order non-autonomous Wien-bridge oscillator without extra nonlinearity

PurposeThe purpose of this paper is to develop a simple chaotic circuit. The circuit can be fabricated by less discrete electronic components, within which complex dynamical behaviors can be generated.Design/methodology/approachA second-order non-autonomous inductor-free chaotic circuit is presented, which is obtained by introducing a sinusoidal voltage stimulus into the classical Wien-bridge oscillator. The proposed circuit only has two dynamic elements, and its nonlinearity is realized by the saturation characteristic of the operational amplifier in the classical Wien-bridge oscillator. After that, its dynamical behaviors are revealed by means of bifurcation diagram, Lyapunov exponent and phase portrait and further confirmed using the 0-1 test method. Moreover, an analog circuit using less discrete electronic components is implemented, and its experimental results are measured to verify the numerical simulations.FindingsThe equilibrium point located in a line segment varies with time evolution, which leads to the occurrence of periodic, quasi-periodic and chaotic behaviors in the proposed circuit.Originality/valueUnlike the previously published works, the significant values of the proposed circuit with simple topology are inductor-free realization and without extra nonlinearity, which make the circuit can be used as a paradigm for academic teaching and experimental illustraction for chaos.

Simple current control method for three‐phase VSC under unbalanced grid condition

Injection of both positive and negative sequence currents is desired for three-phase voltage source converters (VSC) under unbalanced grid fault condition. However, the second-order oscillation exists in the positive and negative sequence synchronous reference frame. Several approaches have been reported to cope with this second-order oscillation. Here, a simple direct proportional plus integral current control method in double synchronous reference frame is proposed. The proposed method does not require the additional filter or decoupling circuit, and does not require signal reconstruction for the current reference. Theoretical analysis and experimental study show that the proposed current control not only achieves zero steady-state error, but also exhibits good transient response.

Influence of stochastic perturbations on the cluster explosive synchronization of second-order Kuramoto oscillators on networks.

Explosive synchronization in networked second-order Kuramoto oscillators has been well studied recently and it is revealed that the synchronization process is featured by cluster explosive synchronization. However, little attention has been paid to the influence of noise or perturbation. We here study this problem and discuss the influences of noise and perturbation. For the former, we interestingly find that noise has significant influence on the cluster explosive synchronization of those nodes with smaller degrees, i.e., their synchronization will change from the first-order to second-order transition and the critical points for both the forward and backward synchronization depend on the strength of noise. Especially, when the strength of noise is in an optimal range, a synchronization of the nodes with smaller degrees will be induced in the region of coupling strength where they do not display synchronization in the absence of noise. For the latter, we find that the effect of perturbation is similar to that of noise when its duration W is small. However, the perturbation will induce a change from cluster explosive synchronization to explosive synchronization when W is large. Furthermore, a brief theory is provided to explain the influence of perturbations on the critical points.

Dynamical vs. geometric anisotropy in relativistic heavy-ion collisions: Which one prevails?

We study the influence of geometric and dynamical anisotropies on the development of flow harmonics and, simultaneously, on the second- and third-order oscillations of femtoscopy radii. The analysis is done within the Monte Carlo event generator HYDJET++, which was extended to dynamical triangular deformations. It is shown that the merely geometric anisotropy provides the results which anticorrelate with the experimental observations of either $v_2$ (or $v_3$) or second-order (or third-order) oscillations of the femtoscopy radii. Decays of resonances significantly increase the emitting areas but do not change the phases of the radii oscillations. In contrast to the spatial deformations, the dynamical anisotropy alone provides the correct qualitative description of the flow and the femtoscopy observables simultaneously. However, one needs both types of the anisotropy to match quantitatively the experimental data.

Multiplexing induced explosive synchronization in Kuramoto oscillators with inertia

Explosive synchronization (ES) of coupled oscillators on networks is shown to be originated from the existence of correlation between natural frequencies of oscillators and degrees of corresponding nodes. Here, we demonstrate that ES is a generic feature of multiplex network of second-order Kuramoto oscillators and can exist in the absence of a frequency-degree correlation. A monoplex network of second-order Kuramoto oscillators bearing homogeneous (heterogeneous) degree distribution is known to display the first-order (second-order) transition to synchronization. We report that multiplexing of two such networks having homogeneous degree distribution support the first-order transition in both the layers thereby facilitating ES. More interesting is the multiplexing of a layer bearing heterogeneous degree distribution with another layer bearing homogeneous degree distribution, which induces a first-order (ES) transition in the heterogeneous layer which was incapable of showing the same in isolation. Further, we report that such induced ES transition in the heterogeneous layer of multiplex networks can be controlled by varying inter- and intra-layer coupling strengths. Our findings emphasize the importance of multiplexing or the impact of one layer on the dynamical evolution of other layers of systems having inherent multiplex or multilevel architecture.

Synchronization of stochastic coupled systems via feedback control based on discrete-time state observations

In this paper, we formulate and investigate the synchronization of stochastic coupled systems via feedback control based on discrete-time state observations (SCSFD). The discrete-time state feedback control is used in the drift parts of response system. Combining Lyapunov method with graph theory, the upper bound of duration between two consecutive state observations is provided. And a global Lyapunov function of SCSFD is presented, which derives some sufficient criteria to guarantee the synchronization of drive–response systems in the sense of mean-square asymptotical synchronization. In addition, the theoretical results are applied to stochastic coupled oscillators and second-order Kuramoto oscillators. Finally, two numerical examples are given to verify the effectiveness of the theoretical results.

Cascadable and Tunable Analog Building Blocks Using EX-CCCII

A new cascadable voltage-input, current-output first-order all-pass filter and its applications in second-order filter and oscillator are presented. The proposed circuit employs a single active element namely extra-X current-controlled current conveyor (EX-CCCII) and only a single grounded capacitor. The circuit exhibits high input and high output impedances, so that the filter can be cascaded without additional buffers. The pole frequency is electronically tunable and the circuit requires no component matching constraints. Effects of nonidealities and parasitics are also discussed. As applications, a second-order transadmittance (TA)-mode all-pass filter and a quadrature oscillator are also realized using the proposed voltage-input, current-output first-order all-pass filter. These examples validate easy cascading feature of the proposed circuit. The validity of the proposed circuit is verified through PSPICE simulations using 0.25[Formula: see text][Formula: see text]m parameters with a supply voltage of [Formula: see text][Formula: see text]V.

Stochastic Stability Criteria for Second-Order Oscillator Parametrically Excited by Colored Noise

A second-order oscillator is considered having a random perturbation in its stiffness. This is given by a colored Gaussian or non-Gaussian process. In this way, the oscillator may be stochastically stable or unstable according to the intensity of the excitation. The almost sure (sample) stochastic stability and the stability in the first three response statistical moments are compared for different excitation processes: process with exponential autocorrelation, second-order Gaussian process, bounded noise process. Notable differences in the stability boundaries are found either according to the stability criteria or to the type of excitation. These comparisons are lacking in literature.

Broadband Microwave Photonic Sub Harmonic Downconverter with Phase Shifting Ability

A microwave photonic signal processor that is capable of simultaneously realizing sub-harmonic frequency down conversion and IF signal phase shifting operation is presented. The IF signal generated by the frequency down conversion process is obtained by the beating of the second-order local oscillator (LO) sideband with the fundamental RF signal sideband at the photodetector. The phase of the IF signal can be tuned continuously from 0° to 360° by controlling the optical phase of the RF signal sideband. The optical carrier is eliminated at the signal processor output to achieve high conversion efficiency mixing operation. The signal processor can be designed to have a very wide bandwidth as it only involves optical components. Experimental results are presented that demonstrate frequency down conversion with > 8.8 dB conversion efficiency over 6 to 40 GHz RF signal bandwidth and 0.05 to 10 GHz IF signal bandwidth, and a continuous 0° to 360° phase shift on the output IF signal.

Improved Power Decoupling Scheme for a Single-Phase Grid-Connected Differential Inverter With Realistic Mismatch in Storage Capacitances

A single-phase differential inverter consists of two elementary dc–dc converters, sharing a common dc source and a common ac output terminal. The other ac terminals of the two converters are connected to the grid in the case of grid-connected applications. The differential inverter has subsequently been shown to have a differential flow path for power transfer and a common-mode path for shifting the usual second-order power oscillation away from the dc source. This capability is referred to as power decoupling, which when implemented properly, may prolong the lifespan of the dc source. Existing studies related to power decoupling using a differential inverter have however focused on developing control schemes with equal storage capacitances assumed for the two elementary converters. This is unquestionably not realistic since the two capacitances will vary in practice. It is therefore the intention of this paper to quantify ac and dc imperfections experienced by the differential inverter when storage mismatch occurs. A simple improved scheme is then proposed for raising performance of the differential inverter (or the differential rectifier where desired). Simulation and experimental results provided have verified the computation and control scheme developed.

Response Characteristics of Looseness-Rubbing Coupling Fault in Rotor-Sliding Bearing System

For the diagnosis of looseness-rubbing coupling fault of rotor-sliding bearing system caused by bolt looseness fault, the mechanical model and finite element model of dual-disc rotor system with looseness-rubbing coupling fault are established based on the nonlinear finite element method, nonlinear oil film force, looseness stiffness model, and Hertz contact theory. With the augmented Lagrange method, contact constraint conditions are dealt with to ensure that the rotary disk and casing contact with each other meeting boundary penetrating depth within the prescribed tolerance range. And then the dynamics characteristics of the health rotor system supported by sliding bearing are studied. Combined with experimental study and simulation analysis, it is found that the looseness-rubbing coupling fault is often characterized by rubbing fault, the lower part of the time-domain fluctuated shape is denser, while the upper part is relatively loose, and multiple nested half ellipse is shown in orbit diagram. Because of the loosing stiffness and rubbing force, the phenomenon of unstable oil film is depressed. The appearance of the first- and second-order oil film oscillation phenomenon is delayed. It could be used as a theoretical basis for diagnosing looseness-rubbing coupling fault of rotor-sliding bearing system.

A Hybrid Modular Multilevel Converter with Partial Embedded Energy Storage

Modular and cascaded multilevel converters provide a functional solution for the integration of energy storage systems (ESSs). This paper develops a hybrid multilevel converter based on the modular multilevel converter (MMC) that can be functionally extended with partial embedded ESS as a fraction of the overall converter power rating. The configuration, which can operate as a typical DC-AC converter, enables multi-directional power flow between the DC- and AC-side of the converter, as well as the embedded energy storage elements. The use of a three-phase flying-capacitor submodule eliminates the second-order harmonic oscillations present in modular cascaded multilevel converters. Current, voltage and power control are discussed in the paper while simulation results illustrate the operation of the hybrid MMC as a DC-AC converter in a typical inverter application and the additional functions and control of the embedded ESS.

Many-Objective Optimal and Robust Design of Proportional-Integral-Derivative Controls With a State Observer

This paper presents a many-objective optimization (MaOO) approach to design control and observer gains simultaneously. The many-objective optimization (MaOO) approach finds trade-offs between different nonagreeable design goals. We report the MaOO results of a proportional-integral-derivative (PID) control with a state estimator applied to a second-order oscillator. The overall system is optimized to minimize the peak time, overshoot, tracking error, and control energy, and maximize the rejection of external disturbance and measurement noise, while the relative stability, which is defined by maximum real part of eigenvalues of the closed-loop system, is constrained to be less than or equal to −1. The numerical simulations show promising findings of the proposed method.