# Automatic Mode-Matching Method for MEMS Disk Resonator Gyroscopes Based on Virtual Coriolis Force

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## Abstract

**:**

## 1. Introduction

## 2. Architecture and Motion Model of DRG

#### 2.1. Basic Structure of DRG

#### 2.2. Dynamic Model of DRG

#### 2.3. Electrostatic Force and Electrostatic Negative Stiffness Effect

## 3. Automatic Mode-Matching method based on Virtual Coriolis Force

#### 3.1. Electrostatic Stiffness Tuning Theory

#### 3.2. Mode-Matching Control System

#### 3.2.1. System Features in Mode-Matching

#### 3.2.2. Control System Framework

#### 3.2.3. Analysis of Mode-Matching Loop

## 4. Simulation Analysis for Automatic Mode Matching

## 5. Experimental Analysis Results

#### 5.1. Experimental Equipment and Circuit

#### 5.2. Mode-Matching Implementation Process

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Microscope image, mode shapes and microstructure schematic of DRG: (

**a**) microscope images of DRG; (

**b**) mode shapes of the $n=2$ wine-glass modes of DRG; (

**c**) the 3D schematic of the microstructure of the DRG.

**Figure 2.**The fabrication process of the DRG: (

**a**) photo-etching; (

**b**) bonding area etching; (

**c**) metal deposition; (

**d**) anodic bonding; (

**e**) thinning and polishing; (

**f**) dry etching and structure release.

**Figure 6.**Automatic mode-matching process: (

**a**) output response amplitudes (${V}_{d}$ and ${V}_{s}$) of two modes; (

**b**) tuning voltage ${V}_{dcy}$; (

**c**) demodulation Result ${V}_{cq}$; (

**d**) resonant frequency of sense mode ${f}_{yy}$ $({f}_{yy}\triangleq {\omega}_{yy}/\left(2\pi \right))$.

**Figure 7.**Influence of different input angular rates on tuning voltage and resonant frequency: (

**a**) influence of different input angular rates on tuning voltage ${V}_{dcy}$; (

**b**) influence of different input angular rates on resonant frequency ${f}_{yy}$ of sensing mode.

**Figure 10.**Relationship between two mode resonant frequencies and tuning voltage after quadrature stiffness nulling.

**Figure 13.**Scale factor and Allan variance curves under mode-matching and mode mismatch conditions: (

**a**) the scale factor; (

**b**) the Allan variance curves.

Parameter | Value | Unit |
---|---|---|

Radius of disk r | 3000 | $\mathsf{\mu}$m |

Thickness of nested ring d | 120 | $\mathsf{\mu}$m |

Width of nested ring l | 80 | $\mathsf{\mu}$m |

Single electrode width $\vartheta $ | 20 | ${}^{\xb0}$ |

Initial electrode gap ${h}_{0}$ | 5.2 | $\mathsf{\mu}$m |

Radius of anchor R | 750 | $\mathsf{\mu}$m |

Thickness of spoke $st$ | 8 | $\mathsf{\mu}$m |

Parameter | Value | Unit |
---|---|---|

Drive mode resonance frequency ${f}_{1}={\omega}_{1}/\left(2\pi \right)$ | 9546.06 | Hz |

Driving mode quality factor ${Q}_{1}$ | 19,133 | |

Sense mode resonance frequency ${f}_{2}={\omega}_{2}/\left(2\pi \right)$ | 9552.33 | Hz |

Sense mode quality factor ${Q}_{2}$ | 19,265 | |

Quadrature stiffness correction voltage ${V}_{QB}$ | 9.08 | V |

Mode matching loop preset voltage ${V}_{T0}$ | 15.0 | V |

Vacuum permittivity $\u03f5$ | $8.85\times {10}^{-12}$ | F/m |

Mode effective mass ${m}_{0}$ | $6.0\times {10}^{-7}$ | kg |

Parameter | Mode Mismatch | Mode Matching |
---|---|---|

Frequency Split (Hz) | 6.2700 | <0.1000 |

Scale Factor (mV/${}^{\xb0}$/s) | 0.2263 | 4.1277 |

Measurable Range (${}^{\xb0}$/s) | $\pm 10$ | $\pm 10$ |

Bias Instability (${}^{\xb0}$/h) | 30.7575 | 2.8331 |

ARW (${}^{\xb0}/\sqrt{h}$) | 1.0208 | 0.0524 |

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## Share and Cite

**MDPI and ACS Style**

Ruan, Z.; Ding, X.; Qin, Z.; Jia, J.; Li, H.
Automatic Mode-Matching Method for MEMS Disk Resonator Gyroscopes Based on Virtual Coriolis Force. *Micromachines* **2020**, *11*, 210.
https://doi.org/10.3390/mi11020210

**AMA Style**

Ruan Z, Ding X, Qin Z, Jia J, Li H.
Automatic Mode-Matching Method for MEMS Disk Resonator Gyroscopes Based on Virtual Coriolis Force. *Micromachines*. 2020; 11(2):210.
https://doi.org/10.3390/mi11020210

**Chicago/Turabian Style**

Ruan, Zhihu, Xukai Ding, Zhengcheng Qin, Jia Jia, and Hongsheng Li.
2020. "Automatic Mode-Matching Method for MEMS Disk Resonator Gyroscopes Based on Virtual Coriolis Force" *Micromachines* 11, no. 2: 210.
https://doi.org/10.3390/mi11020210