Phase Macromodels Research Articles

Analysis of Steady-State Oscillations in an Autonomous Oscillator under Multi-Frequency Excitation

The analysis of the behavior of an oscillator under multi-frequency excitation is considered in the paper. The investigation is based on the phase macromodel. The paper shows that three steady-state modes can exist in oscillator under multi-frequency excitation. The synchronized (locked) mode can be defined as the coincidence of the oscillator fundamentals with the excitation fundamentals in the region of sufficiently large excitation magnitude. The unsynchronized (unlocked) mode exists outside the synchronized region and its spectrum contains additional intrinsic fundamental besides the excitation ones. Singular points mode in some isolated points outside the synchronized region is characterized by the equality of the number of the oscillator fundamentals with the number of the excitation fundamentals. Performed numerical experiments confirmed the appearance of bifurcation points while transition of oscillator into the synchronization mode. The existence of singular points outside the synchronization region and their isolated character was also experimentally demonstrated. The problems of finding a steady-state solution of the phase equation of an excited oscillator by the Harmonic Balance (HB) method are considered. It is shown that main difficulties are connected with the presence of linear term in the steady-state solution. A transformation is proposed to provide the formation of HB equations for the phase micromodel in a standard form. Additional difficulties of HB simulations of synchronized oscillator phase equations are discussed.

Fast and Accurate Time-Domain Simulations of Integer-N PLLs

We present a methodology to simulate industrial integer- $N$ phase-locked loops (PLLs) at a verification level, as accurate as and faster than transistor-level simulation. The accuracy is measured on the PLL factors of interest, i.e., locking time, power consumption, phase noise and jitter (period and long-term). The speedup factor tends to the division ratio $N$ for device-noise simulations. We develop a unifying technique which is able to deal with both noise-free and device-noise analyses, taking into account nonlinear and second-order effects visible at transistor-level simulation only, whereas previous works focused on one of the two analyses, separately. The procedure is based on oscillator’s sensitivity analysis and on the creation of a phase macromodel for the voltage-controlled oscillator (VCO) together with the loop divider (the phase model is called VCODIV), whilst the other PLL’s blocks remain at transistor level. The macromodel’s phase law is characterized by a piecewise linear curve, representing the sensitivity of the VCODIV output’s phase deviation with respect to the voltage variation of the VCO’s control pin, and by the effects of all the VCO’s and divider’s noise sources on the model’s output. We show two experiments on industrial PLLs, and provide guidelines for designers which highlight the steps needed to implement the methodology by using well-known analyses in circuit simulation and Verilog-A for the creation of the macromodel.

Influence of Amplitude Fluctuations on the Noise-Induced Frequency Shift of Noisy Oscillators

We discuss the effect of the estimation of the amplitude variations on the noise-induced frequency shift in nonlinear oscillators. The analysis is based on a general reduction approach to a phase macromodel making use of the Ito interpretation of the stochastic differential equations representing noisy autonomous systems. We show that a significant improvement of the accuracy can be obtained by using the average value of the amplitude component determined from a linearization of the amplitude equation.

Hierarchical abstraction of weakly coupled synchronized oscillator networks

SummaryWe provide a constructive and numerically implementable proof that synchronized groups of coupled, self‐sustaining oscillators can be represented as a single effective Perturbation Projection Vector (PPV) (or Phase Response Curve) phase macromodel – in other words, that a group of synchronized oscillators behaves as a single effective oscillator with respect to external influences. This result constitutes a foundation for understanding and predicting synchronization/timing hierarchically in large, complex systems that arise in nature and engineering. We apply this result hierarchically to networks of synchronized oscillators, thereby enabling their efficient and scalable analysis. We illustrate our theory and numerical methods with examples from electronics (networks of three‐stage ring oscillators), biology (Fitzhugh–Nagumo neurons) and mechanics (pendulum clocks). Our experiments demonstrate that effective PPVs extracted hierarchically can capture emergent phenomena, such as pattern formation, in coupled oscillator networks. Copyright © 2015 John Wiley & Sons, Ltd.

Influence of Noise on the Phase and Amplitude of Second-Order Oscillators

We investigate noise effects in second-order oscillators described in terms of amplitude and phase. We show that phase noise in an oscillator is a convection-diffusion problem, i.e., noise is responsible for both phase diffusion and frequency shift. The magnitude of a noise-induced frequency shift depends on both the noise intensity and on the curvature of iscohrons, i.e., manifolds describing the phase of a nonlinear oscillator. We discuss this dependence and to what extent amplitude fluctuations can be neglected. In the case of additive noise, ignoring amplitude fluctuations gives a trivial phase macromodel that can be integrated directly, and the resulting phase variable is a Gaussian-distributed random process.

Quadratic Approximations for the Isochrons of Oscillators: A General Theory, Advanced Numerical Methods, and Accurate Phase Computations

The notion of isochrons for oscillators, introduced by Winfree and thereon heavily utilized in mathematical biology, were instrumental in introducing a notion of generalized phase and form the basis for oscillator perturbation analyses. Computing isochrons is a hard problem, existing brute-force methods incurring exponential complexity. In this paper, we present a precise and carefully developed theory and numerical techniques for computing local but quadratic approximations for isochrons. Previous work offers the techniques needed for computing only local linear approximations. Our treatment is general and applicable to oscillators with large dimension. We present examples for isochron computations, verify our results against exact calculations in a simple analytically calculable case, test our methods on complex oscillators, and show how quadratic approximations of isochrons can be used in formulating accurate, novel phase computation schemes and finally allude to second-order accurate compact phase macromodels. Oscillator studies seem to have progressed independently in electronics and biology. Even though analyses in electronics did not make use of the notion of isochrons, similar models and methods, expressed in totally different terminologies, have been developed in both disciplines. In this paper, we also reveal the connection between oscillator analysis work in these two seemingly disparate disciplines.

Simulation of Mutually Coupled Oscillators Using Nonlinear Phase Macromodels

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Design of integrated RF circuits requires detailed insight in the behavior of the used components. Unintended coupling and perturbation effects need to be accounted for before production, but full simulation of these effects can be expensive or infeasible. In this paper, we present a method to build nonlinear phase macromodels of voltage-controlled oscillators. These models can be used to accurately predict the behavior of individual and mutually coupled oscillators under perturbation at a lower cost than full circuit simulations. The approach is illustrated by numerical experiments with realistic designs. </para>

Capturing Oscillator Injection Locking via Nonlinear Phase-Domain Macromodels

Injection locking is a nonlinear dynamical phenomenon that is often exploited in electronic and optical oscillator design. Behavioral modeling techniques for oscillators that predict this phenomenon accurately are of significant scientific and practical importance. In this paper, we propose a nonlinear approach for generating small phase-domain oscillator/voltage-controlled oscillator (VCO) macromodels that capture injection locking well. Our nonlinear phase-domain macromodels are closely related to recent oscillator phase noise and jitter theories, and can be extracted efficiently by algorithm from SPICE-level descriptions of any oscillator or VCO. Using LC and ring oscillators as test cases, we confirm the ability of nonlinear phase macromodels to capture injection locking, and also obtain significant computational speedups over full SPICE-level circuit simulation. Furthermore, we show that our approach is equally effective for capturing the dynamics of transition to locking, including unlocked tones and phase jump phenomena.

3D Simulation of a Shape Memory Microactuator

A three-dimensional (3D) model has been developed for simulation of the physical properties of an electrically driven shape memory alloy (SMA) microactuator used for control of a microvalve. The mechanical behavior is decribed by a two phase macromodel taking into account material and geometrical nonlinearity. The simulation makes use of mechanical, electrical and thermal finite element programs, which are coupled by a program for management of the simulation sequence as well as the exchange of data between the different finite element programs. Mechanical tests confirm the simulated forces and displacements. From the pressure-dependent gas flow in the valve an effective heat transfer coefficient is determined to simulate the convective heat exchange. The simulated heat transfer times are in quantitative agreement with experimentally determined time constants of the microvalve.