Analysis of the Σ-Δ pulsed digital oscillator for MEMS

The aim of this paper is the analysis, simulation, and experimental verification of the Σ-Δ pulsed digital oscillator (PDO) topology. As it has been shown in previous works, the oscillation frequency and output spectrum in the PDO depend on the sampling frequency, the natural frequency of the microelectromechanical systems (MEMS) resonator and its damping factor. Here, extensive discrete-time simulations have been carried out which show that the normalized oscillation frequency as a function of the normalized natural frequency of the resonator is very similar to a distorted Devil\´s Staircase fractal. This nonlinear behavior is a direct consequence of the damping losses of the MEMS resonator. Analytical conditions for a perfect oscillation at the natural frequency of the resonator are also calculated. For this set of what we call "perfect" frequencies, it is also shown that the energy transfer from the electrical to the mechanical domain is maximum. Then a more generalized structure of the oscillator is considered and the drawn conclusions are tested against experimental results obtained from an oscillator prototype which uses a MEMS resonator with thermoelectric actuation and piezoresistive position sensing.