Analyzing Slowly Modulated Oscillator Behavior
نویسندگان
چکیده
This paper presents a novel method for analyzing slowly modulated oscillator behavior. This occurs, for example, when the values of some oscillator components are changed through an external signal that varies much slower than the oscillation time constant. Applications exhibiting this type of behavior are phase-locked loops (PLLs) or frequency-modulation-based transmitters. Both examples involve an oscillator whose frequency is adjusted by a slowvarying input signal, modulating the value of one of the oscillator’s circuit components (typically the capacitor value). By exploiting the structure of the differential equations governing these type of systems and using concepts from dynamical systems theory, it is shown how the original set of (circuit) equations can be reduced to a nearly integrable system. Such a system is characterized by a set of rapidly rotating angular variables together with a set of slow-varying action variables. Extending the ideas discussed in [2], [11], it is then possible to construct an averaging transformation that extracts those components of the system equations, responsible for the (dominant) slow-varying part of its behavior. This resulting set of equations makes up a valuable starting point for efficient simulation, behavioral modeling and stability analysis of the system. The approach is illustrated for a harmonic oscillator subject to slow modulation of its capacitor value. Keywords—modeling, oscillators, averaging, perturbation analysis, dynamical systems
منابع مشابه
Chaotic motion of a weakly nonlinear, modulated oscillator.
A weakly nonlinear oscillator of natural frequency omega(0) and damping ratio delta is driven by an amplitude-modulated force of dimensionless amplitude epsilon, carrier frequency omega, and modulation frequency omega(1). Each of (omega - omega(0))/omega(0), omega(1)/omega(0), and delta is O(epsilon(2/3)) as epsilon --> 0. The response in this resonant neighborhood is a slowly modulated sine wa...
متن کاملThe Transient Behavior of LC and Ring Oscillators under External Frequency Injection
In this work, time domain analysis is used to solve Adler’s equation in order to obtain the required time, for an oscillator under external injection, reaching the steady-state condition. Mathematical approach has been applied to fully describe the transient of frequency acquisition in injection-locked LC and Ring oscillators considering their time-varying nature. Then, the analysis is verifie...
متن کاملHomoclinic bifurcation and Chaos in φ6-Rayleigh oscillator with Three wells Driven by an amplitude Modulated Force
With amplitude modulated excitation, the effect on chaotic behavior of Φ-Rayleigh oscillator with three wells is investigated in this paper. The Melnikov theorem is used to detect the conditions for possible occurrence of chaos. The results show that the domain of the appearance of chaos is enlarged as both amplitudes of modulated and unmodulated forces increase. The effect of these two amplitu...
متن کاملNoise-induced phase bistability via stochastic rocking.
We study the effect of a randomly modulated harmonic driving on the phase behavior of a nonlinear oscillator. A multiple-scale analysis shows that the system is formally equivalent to a rocked oscillator, in which a modulated harmonic driving locks the system at one of two phases, both of which are in quadrature with that of the driving. This theoretically predicted noise-induced bistable phase...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2002