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Generalized Analytical Equations for Injected Ring Oscillator With $RC$ -Load.

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Title: Generalized Analytical Equations for Injected Ring Oscillator With $RC$ -Load.
Authors: Hazeri, Ali Reza1, Miar-Naimi, Hossein1
Source: IEEE Transactions on Circuits & Systems. Part I: Regular Papers. Jan2018, Vol. 65 Issue 1, p223-234. 12p.
Subjects: Injection locked oscillators, Load forecasting (Electric power systems), Differential equations
Abstract: In this paper, a new efficient circuit level analysis is introduced considering the effect of all harmonics and multi-injection for locking and pulling phenomena in $N$ -stage differential and single-ended ring oscillators with the $RC$ -load. Analytical equations are derived for the locking range in both single- and multi-injection cases. At the beginning, using a nonlinear approach, the voltage amplitude and frequency of $RC$ -load free running ring oscillators are derived as simple closed-form equations. Then, to analyze the mentioned phenomena, differential equations describing the circuit behavior are derived by circuit level equations. Furthermore, analyses are extended for general multi-injections with arbitrary amplitude and excess phase for different output nodes. The simplest and most accurate analytical equation is achieved for the locking range. In contrary to few previous works, the proposed analysis is derived directly from equations governing ring oscillators so that we can expect results with more accuracy without any tedious and time-consuming pre-processing like perturbation projection vector. To evaluate the analysis, many simulations have been performed. Simulation results validate presented analytical equations, which provide useful design insights. Absolute percent error of all proposed equations in comparison with simulation results is better than 20%. [ABSTRACT FROM PUBLISHER]
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Database: Engineering Source
Description
Abstract:In this paper, a new efficient circuit level analysis is introduced considering the effect of all harmonics and multi-injection for locking and pulling phenomena in $N$ -stage differential and single-ended ring oscillators with the $RC$ -load. Analytical equations are derived for the locking range in both single- and multi-injection cases. At the beginning, using a nonlinear approach, the voltage amplitude and frequency of $RC$ -load free running ring oscillators are derived as simple closed-form equations. Then, to analyze the mentioned phenomena, differential equations describing the circuit behavior are derived by circuit level equations. Furthermore, analyses are extended for general multi-injections with arbitrary amplitude and excess phase for different output nodes. The simplest and most accurate analytical equation is achieved for the locking range. In contrary to few previous works, the proposed analysis is derived directly from equations governing ring oscillators so that we can expect results with more accuracy without any tedious and time-consuming pre-processing like perturbation projection vector. To evaluate the analysis, many simulations have been performed. Simulation results validate presented analytical equations, which provide useful design insights. Absolute percent error of all proposed equations in comparison with simulation results is better than 20%. [ABSTRACT FROM PUBLISHER]
ISSN:15498328
DOI:10.1109/TCSI.2017.2726100