A Design Method for $\Delta\sum$ Force-Feedback Accelerometer Interface Systems

Delta-Sigma $(\Delta\Sigma)$ technique represents an optimum way for realizing force-feedback electromechanical systems, especially for capacitive sensors. However, when operating the sensor in feedback, the stability of the system becomes a concern, particularly, in $\Delta\Sigma$ -based systems, and the higher the order of the system, the harder it becomes to achieve stability. Hence, following a systematic design flow for these systems is essential. While the design of stable electrical $\Delta\Sigma$ loops is well established, the design of electromechanical $\Delta\Sigma$ loops presents a challenge due to the nature of the capacitive sensor resonator. In this work, a way to stabilize high-order $\Delta\Sigma$-based interface systems for inertial capacitive sensors is introduced and a systematic design approach is proposed. The design approach is based on noise transfer function (NTF) matching which translates the system design problem to an NTF design problem as in electrical $\Delta\Sigma$ loops. The design approach is applied to the design of a fifth-order $\Delta\Sigma$ based interface for a capacitive accelerometer. The sensor has a $0.12 \ \mu \mathrm{g}$ proof-mass, a resonance frequency of 1.8 kHz, a displacement-to-capacitance factor of $3.22 \ \text{pF}/ \mu \mathrm{m}$ and a feedback factor of $0.7 \ \mu \mathrm{N/V}^{2}$. The designed system achieves a signal-to-quantization noise ratio (SQNR) of 181 dB.