An Unconstrained Architecture for Systematic Design of Higher Order $\Sigma\Delta$ Force-Feedback Loops

Abstract

Nowadays, 61-modulation is a widely used technique for analog-to-digital (A/D) conversion, especially when aiming for high resolutions. While being applied initially for purely electrical A/D converters, its application has been expanded to mixed mechanical-electrical systems. This has led to the use of 61 force-feedback for digital readout of high-performance inertial sensors. However, compared with their electrical counterpoint, 61 force-feedback loops often have to deal with three additional issues: 1) an increased stability problem due to phase-lag occurring in the sensor; 2) the injection of relatively high levels of readout noise in the loop; and 3) the lack of degrees-of-freedom of many 61 force-feedback architectures for implementing an arbitrary noise transfer function. As a result, 61 force-feedback loops found in literature are designed in a much less systematic way as compared with electrical 61 modulators. In this paper, we address these issues and propose a new unconstrained architecture. Based on this architecture, we are able to present a systematic approach for designing 61 force-feedback loops. Additionally, the main strengths and weaknesses of different 61 force-feedback architectures are discussed. Index Terms-A/D-conversion, force-feedback, MEMS inertialsensors, sigma-delta (61) modulation, systematic design.

A Closed-Loop Digitally Controlled MEMS Gyroscope With Unconstrained Sigma-Delta Force-Feedback

“…Therefore, the transfer from quantization noise to the output of the Σ∆ forcefeedback loop -commonly called the noise transfer function (NTF) -should be carefully designed. In [8], a (theoretically) nearly optimal unconstrained architecture for such Σ∆ forcefeedback loops was presented, which is also employed here. A diagram of the used structure is shown in Fig.…”

“…Now the configuration consists of a cascade of the sensor with a plain electrical Σ∆ ADC. Note that this can not easily be done with alternative Σ∆ force-feedback configurations which have only mechanical feedback [8]. Response to pseudo-noise actuation of the secondary mode: measured (top) and calculated from the identified system (bottom).…”

“…This way, temperature effects such as drift of the resonant frequency and the quality factor of the primary mode are neutralized. For the secondary (sense) mode the readout is based on Σ∆ force feedback [1][2][3][4][5][6][7][8]. This nearly inherently "digital" solution is fully compatible with our philosophy.…”

A virtually floating dual-mass accelerometer

“…For this, a quantizer is added after the loop filter, before applying the digital feedback. With a proper designed loop filter [16][17][18], the added quantization noise can be made negligible and the performance will be further enhanced.…”

“…Due to the 180 • phase shift in the mechanical part, the force-feedback will only be stable for a dominant pole sufficiently lower in frequency than the mechanical resonance frequency. The system bandwidth could be increased by using a more advanced filter with frequency compensation [17,18], but this is beyond the scope of this prototype. The complete system level diagram for the differentialmode is then shown in Fig.…”