# Method of Measuring the Mismatch of Parasitic Capacitance in MEMS Accelerometer Based on Regulating Electrostatic Stiffness

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method of Measuring the Mismatch of Parasitic Capacitance

#### 2.1. Influence of Parasitic Capacitance

_{m1}between C

_{p1}and C

_{p2}—including in the sensor, package, and circuit—will confuse the differential effective capacitances ΔC between C

_{top}and C

_{bottom}that would produce an offset. The mismatch ΔC

_{m2}between C

_{p3}and C

_{p4}will also have an influence on the output. Besides, the parasitic capacitances, C

_{p5}and C

_{p6}, can affect the influence of ΔC

_{m1}and ΔC

_{m2}on the output.

#### 2.2. Theory of Measuring the Mismatch

## 3. Measurement Results and Discussion

#### 3.1. Measurement Results

#### 3.1.1. Verification Experiment and Results

^{2}of the linear fitting is 0.9999 which shows highly linear correlation between ${V}_{ref}^{2}$ and ${F}_{e}^{\prime}$. The strong linear relationship validates the theory of formula deduction. From the linear fitting formula, the linear coefficient can be obtained which is −1.98205 × 10

^{−8}. Through calculation according to this number, the bending value ${x}_{2}$ of the beam owing to the mismatch of parasitic capacitance which is also the deviation from the geometrical center is −13.48 nm. It should be noted that the bending value of the beam is a vector. That is to say it can be positive or negative. The bending direction of the beam depends on the sum of ${x}_{1}$ and ${x}_{2}$, and the minus sign of this ${x}_{2}$ indicates that the beam bends to the bottom plate, owing to the mismatch of parasitic capacitance. Correspondingly, the mismatch of parasitic capacitance is −69.372 fF and the offset caused by the mismatch is 219 mg.

#### 3.1.2. Applications and Results

_{p3}and C

_{p4}was made in MEMS accelerometer using diode ring detecting circuit and a change of 0.5 g on output was observed, so it is necessary to study the influence of the parasitic capacitance between the fixed plate and GND. It should be noted that the effect of this equivalent mismatch on output is not equal to the effective differential capacitance, so its equivalent mismatch cannot be measured using the direct measuring method. The experiment for measuring the equivalent mismatch of the parasitic capacitance between the fixed plate and GND is carried out.

#### 3.2. Discussion

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Dong, Y.; Zwahlen, P.; Nguyen, A.M.; Frosio, R.; Rudolf, F. Ultra-high precision MEMS accelerometer. In Proceedings of the 16th International Conference on Solid-State Sensors, Actuators and Microsystems, Beijing, China, 5–9 June 2011. [Google Scholar]
- Ullah, P.; Ragot, V.; Zwahlen, P.; Rudolf, F. A new high performance sigma-delta MEMS accelerometer for inertial navigation. In Proceedings of the 2015 Dgon Inertial Sensors and Systems Symposium, Karlsruhe, Germany, 22–23 September 2015. [Google Scholar]
- Yazdi, N.; Kulah, H.; Najafi, K. Precision readout circuits for capacitive microaccelerometers. In Proceedings of the IEEE Sensors 2004, Vienna, Austria, 24–27 October 2004. [Google Scholar]
- Zhu, Z.Y.; Liu, Y.D.; Jin, Z.H. The parasitic capacitance’s influence on noise in a MEMS accelerometer sensor. Chin. J. Sens. Actuators
**2013**, 26, 17–20. [Google Scholar] - He, J.B.; Xie, J.; He, X.P.; Du, L.M.; Zhou, W. Research on nonlinear error of micro-accelerometers considering the fringe and parasitic capacitance. Mech. Sci. Technol. Aerosp. Eng.
**2016**, 35, 752–757. [Google Scholar] - Jeong, Y.; Ayazi, F. A novel offset calibration method to suppress capacitive mismatch in MEMS accelerometer. In Proceedings of the Samsung Electro-Mechanics Best Paper Award, Seoul, Korea, 13 November 2013. [Google Scholar]
- Lajevardi, P.; Petkov, V.P.; Murmann, B. A delta sigma interface for MEMS accelerometers using electrostatic spring constant modulation for cancellation of bondwire capacitance drift. IEEE J. Solid-State Circ.
**2016**, 48, 265–275. [Google Scholar] [CrossRef] - Xiong, X.X.; Wu, Y.L.; Jone, W.B. Control circuitry for self-repairable MEMS accelerometers. In Technological Developments in Education and Automation, 1st ed.; Iskander, M., Kapila, V., Karim, M.A., Eds.; Springer: Berlin, Germany, 2010; pp. 265–270. [Google Scholar]
- Ko, H. Highly configurable capacitive interface circuit for tri-axial MEMS microaccelerometer. Int. J. Electron.
**2012**, 99, 945–955. [Google Scholar] [CrossRef] - Liu, M.J.; Dong, J.X. Compensation for bias in capacitive micro accelerometer. J. Chin. Inert. Technol.
**2008**, 16, 86–89. [Google Scholar] - Zhou, H.Y.; Shen, T.Y.; Li, L.R.; Xu, F.Y.; Hu, J.W.; Xie, Y.H. Study on the measurement of micro-capacitance based on RLC series resonance and increment. In Proceedings of the International Conference on Mechanical, Electronic and Information Technology Engineering, Chongqing, China, 21–22 May 2016. [Google Scholar]
- Dascher, D.J. Measuring parasitic capacitance and inductance using TDR. Hewlett-Packard J.
**1996**, 47, 83–96. [Google Scholar] - Li, J.; Dong, J.X.; Liu, Y.F.; Wu, T.Z. Effects of preload voltage on performance of force-rebalance micro silicon accelerometer. J. Transducer Technol.
**2004**, 23, 35–40. [Google Scholar]

**Figure 4.**Measuring the equivalent mismatch between fixed plate and GND: (

**a**) the initial state; (

**b**) state of adding a capacitance of 1 pF.

**Figure 5.**Mismatch of different circuit design: (

**a**) result of before-optimization circuit; (

**b**) result of after-optimization circuit.

${V}_{ref}\text{\hspace{0.17em}}(\mathrm{V})$ | ${U}_{out}\text{\hspace{0.17em}}(\mathrm{LSB})$ | ${K}_{1}\text{\hspace{0.17em}}(\mathrm{LSB}/\mathrm{g})$ | ${V}_{ref}^{2}\text{\hspace{0.17em}}({\mathrm{V}}^{2})$ | ${F}_{e}^{\prime}\text{\hspace{0.17em}}(\mathrm{N})$ |
---|---|---|---|---|

1.00 | 5058 | 137,837 | 1.00 | 7.33 × 10^{−8} |

2.00 | 1526 | 68,051 | 4.00 | 1.96 × 10^{−8} |

3.00 | −49 | 45,092 | 9.00 | −8.48 × 10^{−8} |

4.00 | −1079 | 33,768 | 16.00 | −2.29 × 10^{−7} |

5.00 | −1946 | 26,993 | 25.00 | −4.09 × 10^{−7} |

6.00 | −2651 | 22,491 | 36.00 | −6.22 × 10^{−7} |

7.00 | −3329 | 19,253 | 49.00 | −8.73 × 10^{−7} |

8.00 | −4043 | 16,811 | 64.00 | −1.18 × 10^{−6} |

9.00 | −4682 | 14,942 | 81.00 | −1.51 × 10^{−6} |

Sensor | X_{2} (m) | Mismatch/fF |
---|---|---|

1 | −1.12 × 10^{−8} | −57.64 |

2 | −0.99 × 10^{−8} | −50.95 |

3 | −0.98 × 10^{−8} | −50.43 |

4 | −1.14 × 10^{−8} | −58.67 |

5 | −1.17 × 10^{−8} | −60.21 |

6 | −1.18 × 10^{−8} | −60.73 |

average | −1.10 × 10^{−8} | −56.44 |

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**MDPI and ACS Style**

Dong, X.; Yang, S.; Zhu, J.; En, Y.; Huang, Q.
Method of Measuring the Mismatch of Parasitic Capacitance in MEMS Accelerometer Based on Regulating Electrostatic Stiffness. *Micromachines* **2018**, *9*, 128.
https://doi.org/10.3390/mi9030128

**AMA Style**

Dong X, Yang S, Zhu J, En Y, Huang Q.
Method of Measuring the Mismatch of Parasitic Capacitance in MEMS Accelerometer Based on Regulating Electrostatic Stiffness. *Micromachines*. 2018; 9(3):128.
https://doi.org/10.3390/mi9030128

**Chicago/Turabian Style**

Dong, Xianshan, Shaohua Yang, Junhua Zhu, Yunfei En, and Qinwen Huang.
2018. "Method of Measuring the Mismatch of Parasitic Capacitance in MEMS Accelerometer Based on Regulating Electrostatic Stiffness" *Micromachines* 9, no. 3: 128.
https://doi.org/10.3390/mi9030128