Heterodyned Electrostatic Transduction Oscillators Evade Low Frequency Noise Aliasing

We recently demonstrated a polarization technique that eliminates the frequency drift due to dielectric charge in MEMS oscillators [1]. In this work we show that these oscillators are also firstorder insensitive to low-frequency voltage noise, with potential for significant positive impact on predicted close-to-carrier phase noise performance. Close-to-carrier noise also determines the limits of some resonant sensor applications [2, 3]. INTRODUCTION Electrostatic MEMS resonators are typically polarized with dc voltages. By simply changing resonator polarization vp to ac (at frequency ωB, Fig. 1), it has been shown that frequency drift due to dielectric charging disappears [1]. Under this scheme, the system electrically oscillates at two frequencies simultaneously (ωR±ωB), an effect caused by heterodyning, i.e. frequency mixing, at both the input and output transducers of the device. The mixing of frequencies within the oscillator is summarized in Fig. 2. The polarization vp(t) can be generated by frequency division of the oscillator’s output as discussed previously in [1]. Kaajakari [4] identifies the inherent force nonlinearity and current nonlinearity to be the primary sources of the aliasing of low frequency noise (Fig. 3(a)) into close-to-carrier current noise in conventional electrostatic MEMS oscillators. This aliased lowfrequency noise determines the phase noise performance of the oscillator. In this work we show that although these sources of aliasing still exist under the ac actuation scheme, they cause low-frequency noise (such as 1/ f noise) to appear far from the carrier frequencies. Instead, it is the higher frequency noise near the polarization signal at ωB (Fig. 3(b)) that aliases to the carrier sidebands. Figure 1: General oscillator representation and test setup with artificial ‘noise’ injection. The polarization source is generalized to frequency ωB and voltage amplitude vB. For DC polarization we set ωB = 0. Figure 2: A simplified model of circulating signals in a heterodyned electrostatic transduction oscillator. Here, ωR is the device resonance frequency, and ωB is the polarization frequency. Although the device continues to mechanically resonate at frequency ωR, we can only observe ωR±ωB in the electrical domain. Figure 3: Dominant noise contributions from vp(t). (a) With conventional (DC) polarization, the dominant close-to-carrier noise source in the oscillator output is the aliased 1/f noise near 0 Hz (marked in blue). (b) With AC polarization, the dominant noise source in the oscillator is adjacent to the polarization frequency (marked in blue), and not low frequency noise. THEORY We propose an extended version of Leeson’s phase noise model [5] for ac polarized MEMS oscillators.