Synchronization of oscillators coupled through narrow-band networks

The ability of two coupled oscillators to synchronize depends critically on the coupling network. Previous analyses have accurately predicted the performance of quasi-optical microwave oscillator arrays for both weak and strong coupling, but have been limited to coupling networks with bandwidths considerably larger than the locking bandwidths of the oscillators. In this paper, the authors develop a method for deriving a suitable system of nonlinear differential equations describing the oscillator amplitude and phase dynamics using a generalization of Kurokawa's method. The method is applied to the case of two Van der Pol oscillators coupled through a resonant network for a wide range of coupling strengths and bandwidths. Simple approximate formulas are developed for the size of the frequency locking region as functions of the basic circuit parameters     

Chaos Synchronization in Complex Oscillators Networks with Time Delay via Adaptive Complex Feedback Control

This paper investigates the problem of chaos synchronization in complex oscillators networks. An adaptive complex feedback controller is designed to realize chaos synchronization in complex oscillators networks. The adaptive complex feedback controller consisting of real part controller and imaginary part controller. The real part controller and the imaginary part controller control the real part system and the imaginary part system of complex oscillators system of complex dynamical networks, respectively. Chaos synchronization in complex oscillators networks with constant delay and chaos synchronization in complex oscillators networks with time-varying coupling delay are researched. Finally, the numerical simulations show the effectiveness of theoretical analysis.     

On the steady-state phase distribution in a hexagonally coupled oscillator array

The distribution of the phase of the oscillations of a set of mutually injection locked electronic oscillators is investigated for the case of a hexagonal nearest neighbor coupling topology. It is shown that planar distributions are not, in general, steady-state solutions of the system of nonlinear differential equations governing the system. An exact steady-state solution of the equations is obtained and shown to be planar only for six discrete azimuth angles. The degree of deviation from a planar distribution is determined and its impact on the directivity of a phased array excited by the oscillator output signals is estimated.     

Pulse Synchronization of Intestinal Myoelectrical Models

Electrical stimulation of the stomach and small intestine in animals has been shown to give a "pacing" effect, whereby the normal frequency of the myoelectrical slow-waves can be altered to that of the pulse stimulus if it has a frequency close to that of the natural rhythm. In this paper the effect of pulse stimulation on three different types of models used for gastrointestinal studies is investigated. Each of the models is an electronic implementation comprising coupled oscillators, where the unit oscillators are either based on van der Pol's equation, Hodgkin-Huxley type equations, or a relaxation switching circuit. The synchronization range is investigated for each model for variations in stimulus pulse height and width. The effect of the number of oscillators in a chain, the waveshape of the individual oscillators, and the coupling between oscillators are also studied. It is shown that the relaxation model has different synchronization characteristics than the other two models. These differences are that a weakly coupled system is easier to synchronize than a strongly coupled system, that increasing pulsewidth does not always increase the pacing band, phase lead cannot be induced, and the synchronization-band can be entirely above the unpaced system frequency.     

多涡卷混沌振荡电路同步的实验研究

以5涡卷混沌振荡器为例进行了单向、双向耦合同步实验研究，实验结果显示耦合同步法同步性能良好，具体表现为：实现简单方便，鲁棒性较好，且双向耦合比单向耦合同步性能更好。以5涡卷混沌振荡电路为例用脉冲同步法进行同步实验研究，结果表明：脉冲同步法在合理选择脉冲的频率（周期）、占空比及幅度参数的情况下，可使收、发端的混沌振荡电路保持同步，且抗干扰性能良好，RC乘积越小，同步性能越好，实验结果与仿真结果一致。     

Analysis of interinjection-locked oscillators for integrated phased arrays

The behavior of a system of coupled oscillators is shown to have potential applications in the generation of power for integrated phased antenna arrays. Nonlinear differential equations are derived to describe a system of oscillators coupled by an arbitrary frequency-dependent network. State-variable analysis of the linearized equations leads to closed-form solutions for one- and two-dimensional phased array systems. Experimental data for a VHF prototype system is presented, and practical considerations in system design are discussed.     

Active antenna array lumped ring configuration

Coupled active antenna oscillator arrays are used for power-combining at microwave and millimeter-wave frequencies. It is known that the relative phase determined by the element separation distance ultimately determines the array operational mode and, hence, far-field radiation characteristics. Separately, it is known that coupled oscillator modal stability is achieved by coupling oscillators through lumped capacitive elements. In this paper, an arrangement whereby lumped capacitive elements (placed across the oscillator loads) and radiative coupling are employed concurrently is investigated. The arrangement takes the form of a ring of coupled oscillators used to excite a 2×2 antenna array. The effect that these couplings have on array behavior are evaluated using time-domain analysis and analytically derived equations. Experimental results for far-field radiation patterns are discussed in relation to coupled oscillator dynamical behavior. These suggest that the theoretical predictions are valid, offering a robust design tool for studies of larger power-combining arrays     

Estimation of the time delay of coupling between oscillators from time realizations of oscillation phases for different properties of phase dynamics

The problem of interval estimation of the time delay of the coupling between oscillatory systems from observed time series is considered. It is shown that the known asymptotic estimates based on the empirical model in the form of a system of first-order phase oscillators and the maximum likelihood formalism can lead to false inferences of the value of the time delay in two typical situations: 1) nonlinear low-dimensional systems whose phases are well-determined but, as a result of significant amplitude fluctuations, the phase approximation is insufficient for describing the dynamics and 2) systems whose phases are defined not quite well because of too large amplitude fluctuations. A method for empirical diagnostics of problematic situations and its modification (coarse estimation) providing a low probability of false inferences in these situation are proposed. The efficiency of the diagnostic criterion and coarse estimation suggested is demonstrated on reference systems with different dynamic properties (linear stochastic oscillators, van der Pol oscillators, and Ressler and Lorenz chaotic systems).     

Coupled oscillators at a distance: applications to coupledsemiconductor lasers

A theory for coupled oscillators is developed and applied to data for coupled semiconductor lasers. The theory generalizes much of the early work on this subject, as it includes the effects of time delays along the coupling path. By allowing for these time delays, additional degrees of freedom, or longitudinal modes are introduced in the coupled oscillator system. It is shown that quality phase locking requires that only one coupled-cavity mode be excited in the system, and that this limits the allowable levels of coupling to a range that extends from the minimum at which locking first occurs (the Adler condition) to a maximum coupling at which the lock begins to degrade; this latter level occurs only in oscillators coupled at a distance     

Efficient analysis of microstrip-coupled nonradiative dielectric(NRD) resonators for hybrid integrated circuits

Nonradiative dielectric (NRD) resonators have been recently known as excellent stabilizing components for the design of a new class of oscillators based on hybrid integration of planar circuits and NRD-guides. However, an accurate characterization of such a component is needed in order to develop efficient computer-aided-design programs. In this paper, a hybrid planar integrated circuit comprising an NRD resonator is accurately modeled and is coupled to input and output planar microstrip circuits by means of long slots located on the top ground plane of the NRD structure. A reciprocity approach is used to solve the coupling problem between the resonator and microstrip lines. A modal expansion of the field equations has been applied to the NRD resonator in obtaining its rigorous field expressions. Finally, a transmission-line analysis is proposed for the structure, and scattering parameters have been calculated for different slot positions. Results are discussed for various parametric effects on coupling and Q-factor characteristics     

N个振荡器相互注入锁定同步振荡系统

本文对Ｎ个振荡器相互注入锁定同步振荡系统提出了一种通用的分析方法。建立了该系统的完整非线性等效模型，导出了系统的状态方程。对于各种不同电路形式的振荡系统，只要将具体的电路参数代入，就可计算其输出功率，功率合成效率及工作频率等参数，从而使这类系统的计算机辅助分析和设计成为可能。     

基于耦合振荡器阵列的非线性有源天线阵

利用耦合振荡器阵列实现有源集成天线阵电扫描,可以实现相控阵天线高效率、低成本、小型化。文中利用Y-参数对利用任意耦合网络实现的耦合振荡器阵列进行了分析,得到了系统幅度及相位非线性动力系统的微分方程组,然后针对利用耦合振荡器阵列实现非线性有源天线阵列这一实际应用,提出了一种不用移相器实现有源天线阵列波束扫描的方法,并进行了稳定性分析。最后利用计算机仿真验证了该方法的可行性。     

Nonlinear analysis of a cross-coupled quadrature harmonic oscillator

The dynamic equations governing the cross-coupled quadrature harmonic oscillator are derived assuming quasi-sinusoidal operation. This allows for an investigation of the previously reported tradeoff between close-to-carrier phase noise and quadrature precision. The results explain how nonlinearity in the coupling transconductances, in conjunction with a finite amplitude relaxation time and de-tuning of the individual oscillators, cause close-to-carrier AM-to-PM noise conversion. A discussion is presented of how the theoretic results translate into design rules for quadrature oscillator ICs. SPECTRE RF simulations verify the developed theory.     

The Single-Cavity Multiple-Device Oscillator

The output power from 12 IMPATT diodes has been combined in a single-cavity multiple-device oscillator. The oscillator is free from the well-known moding problem of multiple-device oscillators. The objective of this paper is to present the oscillator-circuit theory, which clearly indicates why this particular circuit configuration can give a stable operation free from moding problems. To handle the formidable equations necessary for the analysis, the eigenfunction approach is extensively used. The condition for stable operation, the noise performance, and the locking behavior of the oscillator are all discussed. The noise performance is similar and the locking behavior identical to those of single-device oscillators.     

Generalized Analysis of Parallel Two-Post Mounting Structures in Waveguide

An analytical expression is obtained for the reactance of parallel two-post mounting structures having unsymmetrical strip and gap positions and different strip and gap widths. An equivalent circuit is derived and analytical expressions for its components established giving a physical insight into the problem of coupling between the two gaps. The analysis is based on deriving a variational expression for the structure reactance from the boundary conditions at the structure position. The theoretical results are experimentally verified for a wide range of coupling conditions. Exploitation of this model for the design of wide-band varactor-tuned negative resistance oscillators and multidiode parallel mounts in general is discussed.     

Phase-noise measurement using two inter-injection-locked microwave oscillators

Phase noise in two mutually coupled oscillators is analyzed by the describing function method, and the after-lock phase noise of the oscillators is calculated in terms of their free-running phase noise. A new phase-noise measurement technique based on inter-injection locking of two similar oscillators is proposed. Experimental results are presented, which confirm the theory. It is shown that in the case of zero phase of coupling coefficient, the system is in the optimum state where the only required parameter for the measurement is the locking bandwidth. In this optimum state, as far as the locking bandwidth is measured correctly, imperfections such as the frequency drift, parameters discrepancy, and nonlinear susceptance of the oscillators have no serious effect on the measurement accuracy. The proposed method is compared to the conventional ones.     

CMOS second-harmonic quadrature voltage controlled oscillator using substrate for coupling

In this work, a new circuit configuration for second-harmonic quadrature voltage controlled oscillator (QVCO) with CMOS technology is proposed. The proposed QVCO is made by coupling two identical cross-connected VCOs. The coupling elements (two resistors and two capacitors) do not increase the power consumption of the core VCOs and do not disturb the resonant frequency of the tank circuit in the core VCOs and also, according to the simulations the coupling elements do not adversely affect the phase noise. The role of the substrate terminal of cross-connected MOSFETs of the core oscillators as common mode nodes is demonstrated. Using this node for coupling makes it possible to eliminate the tail transistors in the core oscillators and therefore the circuit can operate with supply voltages as low as 0.5 V. Using the same method to couple more than two core oscillators, results in a multiphase VCO.     

Phase and Amplitude Noise Analysis in Microwave Oscillators Using Nodal Harmonic Balance

In this paper, a nodal harmonic balance (HB) formulation is presented for the phase and amplitude noise analysis of free-running oscillators. The implications of using different constraints in the resolution of the perturbed-oscillator equations are studied. The obtained formulation allows the prediction of the possible spectrum resonances without ill conditioning at low frequency offset from the carrier. The noise spectrum is meaningfully expressed in terms of the eigenvalues of a newly defined matrix, obtained from the linearization of the nodal HB system about the steady-state solution. The cases of real or complex-conjugate dominant eigenvalues are distinguished. The developed phase-noise formulation is extended to a system of two coupled oscillators. The phase and amplitude noise analyses have been applied to a push-push oscillator at 18 GHz, a bipolar oscillator at 1 GHz, and a coupled system of two field-effect transistor oscillators at 6 GHz.     

Operating principles of reactance-less memristor-based oscillators

The aspects of design of reactance-less oscillators based on memristors are considered. The requirements on load elements in the circuits of such oscillators are formulated. The equations describing the dynamical behavioral of the memristor oscillators with the use of linear-drift models are presented. Examples of the analysis and synthesis of oscillators using such equations are presented. The possibility of generation in the circuits under consideration of periodic signals with a waveform close to harmonic oscillations is shown.