An Efficient Method to Analyze Lock Range in Ring Oscillators With Multiple Injections

In this brief, a simple method is proposed to analyze the lock range (LR) of ring oscillators with multiple injection signals by utilizing the generalized Adler equation. An exact formula is introduced for the LR in the case of a single injection. The proposed method is employed to study the asymmetry in high and low LRs. To describe the dependence of the oscillator LR to injection phases in the case of multiple injections, a simple closed-form formula and an empirical formula are derived in weak injection and strong injection, respectively. All of the proposed mathematical findings comprise the results of recent works that have been done in this area and are verified by circuit simulation.