# Self-Test and Self-Calibration of Digital Closed-Loop Accelerometers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Architecture

_{out}. Since the output is a 1-Bit Σ∆ modulated bit stream, the linearity is guaranteed, and the need for high precision D/A converter is alleviated. Although there is a large amplitude of quantization noise introduced by1-Bit quantizer, it is shaped to high frequency both by the electrical loop filter and the MEMS sensing element. Thus, the in-band noise is kept unaffected, and a digitalized high-performance closed-loop accelerometer is obtained. The working principle of digital Σ∆ closed-loop accelerometer is well illustrated in many other papers, we will not reiterate here and concentrate on the digital self-test and self-calibration loop [13,14,15].

_{T}will inversely transfer from the MEMS sensing element to the system output since there exists high in-band gain provided by the integral section of PID loop compensator. Thus, the inner property of the MEMS sensing element is excited by the digital self-test excitation. Ideally, the MEMS sensing element will be servo-controlled at its balancing position. The square-law relationship could be linearized by using 1-Bit feedback, as shown in Figure 2. The same linearization effect can be obtained by taking either three points in 1.5-Bit feedback or two points in 1-Bit feedback from the square-law relationship of electrostatic force and feedback voltage. However, the 1.5-Bit feedback curve will be a broken line when there is a DC displacement deviation $x$, whereas 1-Bit force feedback curve will still be a straight line. This is because whenever you take a two-point approximation of whatever curve, the approximation curve will always be a straight line, whereas there are inevitably parasitic and residue stress in the MEMS structure which induces the deviation of the closed-loop servo position to the central balancing position. The DC bias of the closed-loop accelerometer will drift, and the linearity performance is destructed in this circumstance which will be discussed in detail. Thus, the nonlinearity or, in other words, the harmonic distortion is chosen to be a flag to identify the output DC bias drift. An on-chip harmonic distortion analysis unit is realized by the digital synchronous orthogonal demodulation circuit, which could extract the magnitude and phase information of the second-order harmonic distortion in the output signal under self-test mode. According to the magnitude and phase information extracted, a digital self-calibration algorithm is established, which could automatically tune the value and direction of the compensation capacitance array. Therefore, the imbalance of the front-end capacitance bridge is compensated, and the output DC bias is calibrated.

## 3. Harmonic Distortion-Based Self-Test Technique

_{in}to the displacement variation x, C

_{0}is the static value of sensing capacitance, d

_{0}is static gap clearance, V

_{S}is the reference voltage applied onto the sensing capacitance at the capacitance detection stage, C

_{f}is the feedback capacitance of capacitance detection circuit, ${H}_{C}\left(z\right)$ is the transfer function of the PID loop compensator, STF

_{Σ∆}is the signal transfer function of 3-order electrical Σ∆ modulator. In ideal circumstances, there is no servo deviation, x

_{drift}is equal to zero. The proof mass is located in the middle of fixed plates, and the clearance on either side is equal to d

_{0}. There will be a corresponding displacement variation x when there is an input acceleration a

_{in}. This displacement is close to zero when the in-band loop gain is large enough, whereas, when there is a stress or parasitic mismatch in the MEMS sensing element, the proof mass will be servo-controlled at the imbalance position. Thus, there is a deviation x

_{drift}between the closed-loop servo-position and the central balancing position. Under this condition, the DC operating point of the displacement variation will change to ${x}^{\prime}$, which equals the sum of x and x

_{drift}(e.g., ${x}^{\prime}=x+{x}_{drift}$).

_{elec}and the applied electrostatic voltage V

_{f}represent a square-law relationship. Although using fully differential feedback can linearize it, the residue displacement also contributes to the nonlinearity, especially when there is a DC deviation. The existence of nonlinearity is detrimental to the feedback control system: (1) The linearity of the system will be degraded, and harmonic distortion will be introduced in the output. (2) Out of band quantization noise can be demodulated into the band of interest through this nonlinearity relationship, elevate the in-band noise floor, reduce SNR. (3) If the above problem is severe enough, the stability condition may not be sustained, and the system may be pushed into malfunction.

_{T}to the digital output D

_{out}is mainly determined by the MEMS sensing element and front-end charge amplifier. In other words, the following relationships can be held.

_{elec}(x,D

_{out}) is the expression of digital electrostatic feedback force, which is not only related to the digital output D

_{out}but also to the residue displacement variation x when taking the displacement modulation effect into consideration. The relationship can be written as:

_{T}:

_{i}can be obtained by substituting Equation (8) into Equation (7) and comparing the coefficients of corresponding terms ${V}_{T}^{i}$.

_{drift}≠ 0:

- (1)
- The even-order harmonic distortion terms will occur in the self-test response.
- (2)
- There is a near-linear relationship between the amplitude of even order harmonic distortion and the servo deviation x
_{drift}.

_{drift}<< d

_{0}. However, in the actual sensitive structure of the accelerometer, the conversion from displacement to capacitance is nonlinear, and the drift position cannot fully satisfy the condition, which will introduce nonlinearity part in the second-order harmonic distortion. Inevitably, parasitic mismatch capacitance is introduced by routing, bonding wire, and packaging. When the MEMS sensing element is incorporated in a closed servo loop, the mismatch in sensing capacitance will cause the proof mass to be servo-controlled in the wrong position, and there is a displacement deviation x

_{drift}. As shown in Equation (6), it will introduce a DC bias term η and contribute to the final output DC bias.

## 4. Digital Automatic Self-Calibration

_{n}(t) is the quantization noise introduced by the 1-Bit Σ∆ modulation. The second-order harmonic distortion is extracted from the output bitstream of the digital Σ∆ closed-loop accelerometer. The synchronous orthogonal demodulation is used in order to suppress the accompanying quantization noise in the output bitstream. Besides, it also extracts the phase information of the detected second-order harmonic so as to determine the calibrating direction.

- (1)
- ADC component contains the magnitude and phase information of the second-order harmonic of self-test response.
- (2)
- The other harmonic distortion component is modulated to nω
_{T}and will be filtered out by a succeeding low-pass filter. - (3)
- Since the noise varies randomly, only the component which has both the same frequency and phase relationship with the second-order harmonic is demodulated to baseband. Thus, most of the noise gets suppressed.

_{I}and Y

_{Q}, respectively, which can be expressed as:

^{N}× ∆C. In each step, the current control bit is set to ‘1’ firstly, and the capacitance with the value 2

^{i}× ∆C is set to parallel with the sensing capacitance.

## 5. Test Results and Discussion

^{2}of 0.997, whereas the third-order harmonic distortion is unaffected. There exists a nonlinearity part in the relationship between second-order harmonic distortion and mismatch of sensing capacitance which is caused by non-ideal drift and displacement.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Shkel, A.M. Precision navigation and timing enabled by microtechnology: Are we there yet? In Proceedings of the ION 2013 Pacific PNT Meeting, Honolulu, HI, USA, 23–25 April 2013; pp. 1049–1053. [Google Scholar]
- Deb, N.; Blanton, R.D. Built-in self test of CMOS-MEMS accelerometers. In Proceedings of the International Test Conference, Baltimore, MD, USA, 10–10 October 2002; pp. 1075–1084. [Google Scholar]
- Trusov, A.A.; Zotov, S.A.; Simon, B.R.; Shkel, A.M. Silicon accelerometer with differential frequency modulation and continuous self-calibration. In Proceedings of the 2013 IEEE 26th International Conference on Micro Electro Mechanical Systems (MEMS), Taipei, Taiwan, 20–24 January 2013; pp. 29–32. [Google Scholar]
- Balachandran, G.K.; Petkov, V.P.; Mayer, T.; Baislink, T. 27.1 a 3-axis gyroscope for electronic stability control with continuous self-test. In Proceedings of the 2015 IEEE International Solid-State Circuits Conference—(ISSCC) Digest of Technical Papers, San Francisco, CA, USA, 22–26 February 2015; pp. 1–3. [Google Scholar]
- Basith, I.I.; Kandalaft, N.; Rashidzadeh, R. Built-in self-test for capacitive MEMS using a charge control technique. In Proceedings of the 2010 19th IEEE Asian Test Symposium, Shanghai, China, 1–4 December 2010; pp. 135–140. [Google Scholar]
- Dhayni, A.; Mir, S.; Rufer, L. MEMS built-in-self-test using MLS. In Proceedings of the Ninth IEEE European Test Symposium, ETS 2004, Corsica, France, 26–26 May 2004; pp. 66–71. [Google Scholar]
- Xiong, X.; Wu, Y.; Jone, W. A dual-mode built-in self-test technique for capacitive MEMS devices. IEEE Trans. Instrum. Meas.
**2005**, 54, 1739–1750. [Google Scholar] [CrossRef] - Chu, Y.; Liu, Y.; Dong, J.; Chi, B. Elimination of nonlinearity in Σ∆ MEMS accelerometer. In Proceedings of the 2015 IEEE SENSORS (2015), Busan, Republic of Korea, 1–4 November 2015; pp. 1–4. [Google Scholar]
- de Bruin, D.; Allen, H.; Terry, S. Second-order effects in self-testable accelerometers. In Proceedings of the IEEE 4th Technical Digest on Solid-State Sensor and Actuator Workshop (1990), Hilton Head, SC, USA, 4–7 June 1990; pp. 149–152. [Google Scholar]
- Frosio, I.; Stuani, S.; Borghese, N.A. Auto calibration of MEMS accelerometer. In Proceedings of the 2006 IEEE Instrumentation and Measurement Technology Conference, Sorrento, Italy, 24–27 April 2006; pp. 519–523. [Google Scholar]
- Yu, H.; Ye, L.; Guo, Y.; Su, S. An Innovative 9-Parameter Magnetic Calibration Method Using Local Magnetic Inclination and Calibrated Acceleration Value. IEEE Sens. J.
**2020**, 20, 11275–11282. [Google Scholar] [CrossRef] - Painter, C.; Shkel, A. Active structural error suppression in MEMS vibratory rate integrating gyroscopes. IEEE Sens. J.
**2003**, 3, 595–606. [Google Scholar] [CrossRef] [Green Version] - Petkov, V.; Boser, B. A fourth-order/spl Sigma//spl Delta/interface for micromachined inertial sensors. IEEE J. Solid-State Circuits
**2005**, 40, 1602–1609. [Google Scholar] [CrossRef] - Ismail, A.H.; Elsayed, A. Σ − ∆ based force-feedback capacitive micro-machined sensors: Extending the input signal range. In Proceedings of the 2017 IFIP/IEEE International Conference on Very Large Scale Integration (VLSI-SoC), Abu Dhabi, United Arab Emirates, 23–25 October 2017; pp. 1–6. [Google Scholar]
- Grinberg, B.; Feingold, A.; Koenigsberg, L.; Furman, L. Closed-loop MEMS accelerometer: From design to production. In Proceedings of the 2016 DGON Intertial Sensors and Systems(ISS), Karlsruhe, Germany, 20–21 September 2016; pp. 1–16. [Google Scholar]

**Figure 1.**The system architecture of the proposed digital closed-loop readout interface circuit and digital self-test and self-calibration circuit.

**Figure 8.**The harmonic distortion with respect to the mismatch in sensing capacitances. (

**a**) Amplitude of harmonic distortions with different sensing capacitance mismatch. (

**b**) Measurement RMSE results for the second-order harmonic distortion.

Physical Quality | Value |
---|---|

Sensitivity | 10 pF/g |

Quality | 62 × 10^{−6} kg |

Static capacitance | 180 pF |

Coefficient of damping | 0.01 N/m/s |

Spacing of comb | 2 μm |

Coefficient of stiffness | 2760 N/m |

Resonant frequency | 1000 Hz |

Quality Factor | >30 |

Brown Noise Equivalent Acceleration | <60 ng/Hz^{1/2} |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sun, Z.; Wang, M.
Self-Test and Self-Calibration of Digital Closed-Loop Accelerometers. *Sensors* **2022**, *22*, 9933.
https://doi.org/10.3390/s22249933

**AMA Style**

Sun Z, Wang M.
Self-Test and Self-Calibration of Digital Closed-Loop Accelerometers. *Sensors*. 2022; 22(24):9933.
https://doi.org/10.3390/s22249933

**Chicago/Turabian Style**

Sun, Zhiyuan, and Miao Wang.
2022. "Self-Test and Self-Calibration of Digital Closed-Loop Accelerometers" *Sensors* 22, no. 24: 9933.
https://doi.org/10.3390/s22249933