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

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%.