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Nonlinear feedback system for an inverter-based ring oscillator

In integrated circuits, ring oscillator (RO) has many applications. In these applications, how to obtain the accurate oscillation frequency is an important issue for the design. In this paper, we explore the feedback problem of a N-stage ring oscillator. This paper proposes a more rigorous approach to analyze the ring oscillator. It can be approximated that the feedback system of a ring oscillator can be a nonlinear Lur'e problem. With this Lur'e problem, circle criterion can be used to determine the stability of overall feedback system. With this Lur'e problem, the describing-function method is used to determine the oscillation frequency of the ring oscillator. A new formula will be presented. It can be also observed that if N (number of inverters) is large enough, the proposed formula will approach the conventional formula. Moreover, with describing function method, a "necessary condition" for the existence of fundamental mode and higher order modes of oscillations are presented. Furthermore, it can be shown that as N ≥ 7 and the voltage gain is large enough; the higher harmonic oscillation may exists. Finally, with Tsypkin Function method, a more accurate formula for the oscillation frequency of ring oscillator will be presented. Finally, Simulation examples will illustrate these results.