Analysis and design of sub-harmonically injection locked oscillators

Abstract

Abstract-Sub-harmonic injection locking (SHIL) is an interesting phenomenon in nonlinear oscillators that is useful in RF applications, e.g., for frequency division. Existing techniques for analysis and design of SHIL are limited to a few specific circuit topologies. We present a general technique for analysing SHIL that applies uniformly to any kind of oscillator, is highly predictive, and offers novel insights into fundamental properties of SHIL that are useful for design. We demonstrate the power of the technique by applying it to ring and LC oscillators and predicting the presence or absence of SHIL, the number of distinct locks and their stability properties, lock range, etc.. We present comparisons with SPICElevel simulations to validate our method's predictions.is a nonlinear phenomenon in which a self-sustaining oscillator's phase becomes precisely locked (i.e., entrained or synchronized) to that of an externally applied signal. The phenomenon, together with the related effect of injection pulling, has often been regarded as an unwanted disturbance, causing, among other things, malfunction in serial clock/data recovery, increased timing jitter and clock skew, increased BER in communications, etc.. Over the years, however, IL has also been put to good use in electronics -e.g., for quadrature signal generation [4]; for microwave generators in laser optics [5]; for fast, low-power frequency dividers [6]; and in PLLs [7] and wireless sensor networks [8]. Moreover, IL is an important enabling mechanism in biology (e.g., [9], [10]). When an oscillator locks to an external signal whose frequency is close to the oscillator's natural frequency, the phenomenon is termed fundamental harmonic IL. It is also possible, however, for oscillators to phase-lock at a frequency that is an exact integral sub-multiple of the frequency of the externally applied signal; this is termed subharmonic IL (or SHIL, described further in Sec. II-B) and is useful in frequency division applications [6], [7]. Design of circuits exploiting SHIL has tended to rely predominantly on trial-and-error based methodologies, using brute-force transient simulations to assess impact on SHIL-based circuit function. Existing analyses of fundamental and sub-harmonic IL (e.g., [11], [2], [12]) have been limited to very specific circuit topologies (e.g., LC oscillators), while more general analyses [13], that apply to any kind of oscillator, do not consider SHIL. The work of Daryoush et. al. [14] presented a computationally complicated method limited to negative feedback oscillators, and provided no insights about multiple lock states for SHIL. To our knowledge, there is no general analysis or theory that provides the correct design intuition and predictive power for SHIL and related phenomena. In this paper, we develop and validate a general method for analysing and understanding sub-harmonic injection locking. The method applies to any self-sustaining, amplitude-stable oscillator, not only from electronics but also from other domains such as biology. ...

OIM: Oscillator-Based Ising Machines for Solving Combinatorial Optimisation Problems

“…We present working hardware prototypes of our oscillator based Ising machines. We first show that the phase dynamics of any network of coupled, self-sustaining, amplitude-stable oscillators can be abstracted using the Generalised Adler model [14,15], a generalisation of the well-known Kuramoto model [16][17][18]. The model's phase dynamics are governed by an associated Lyapunov function, i.e., a scalar function of the oscillators' phases that is always non-increasing and settles to stable local minima as phase dynamics evolve.…”

“…In general, however, oscillator phases do not settle to the discrete values 0/π, but span a continuum of values instead. In order to binarise oscillator phases (i.e., get them to settle to values near 0/π), we inject each oscillator with a second harmonic signal (dubbed SYNC) that induces sub-harmonic injection locking (SHIL), which makes the phase of each oscillator settle to a value near either 0 or π [14,19]. We devise a new Lyapunov function that governs the network's dynamics with SHIL; this Lyapunov function is also essentially identically to the Ising Hamiltonian at phase values of 0/π.…”

Design tools for oscillator-based computing systems

“…By plotting the LHS and RHS of (5) and looking for intersections, designers can predict whether IL or SHIL will happen given a design [1] without running long and expensive transient simulations.…”

A V‐band frequency quadrupler using subharmonic injection‐locked oscillator with push–push technique

“…A fundamental signal with including high power and low phase noise is also a demand if a passive frequency multiplier is used in such systems. Conversely, the injection‐locked oscillators (ILO) obtain a low phase noise and a sufficient output power without any complicated filters and frequency multiplying circuits into a multiplier chain . Nevertheless, it usually has a disadvantage of small locking range especially in a high‐order subharmonic injection‐locked oscillator (SILO).…”