An analytical formulation for phase noise in MEMS oscillators.

In recent years, there has been much interest in the design of low-noise MEMS oscillators. This paper presents a new analytical formulation for noise in a MEMS oscillator encompassing essential resonator and amplifier nonlinearities. The analytical expression for oscillator noise is derived by solving a second-order nonlinear stochastic differential equation. This approach is applied to noise modeling of an electrostatically addressed MEMS resonator-based square-wave oscillator in which the resonator and oscillator circuit nonlinearities are integrated into a single modeling framework. By considering the resulting amplitude and phase relations, we derive additional noise terms resulting from resonator nonlinearities. The phase diffusion of an oscillator is studied and the phase diffusion coefficient is proposed as a metric for noise optimization. The proposed nonlinear phase noise model provides analytical insight into the underlying physics and a pathway toward the design optimization for low-noise MEMS oscillators.