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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The micromechanical silicon resonant accelerometer has attracted considerable attention in the research and development of high-precision MEMS accelerometers because of its output of quasi-digital signals, high sensitivity, high resolution, wide dynamic range, anti-interference capacity and good stability. Because of the mismatching thermal expansion coefficients of silicon and glass, the micromechanical silicon resonant accelerometer based on the Silicon on Glass (SOG) technique is deeply affected by the temperature during the fabrication, packaging and use processes. The thermal stress caused by temperature changes directly affects the frequency output of the accelerometer. Based on the working principle of the micromechanical resonant accelerometer, a special accelerometer structure that reduces the temperature influence on the accelerometer is designed. The accelerometer can greatly reduce the thermal stress caused by high temperatures in the process of fabrication and packaging. Currently, the closed-loop drive circuit is devised based on a phase-locked loop. The unloaded resonant frequencies of the prototype of the micromechanical silicon resonant accelerometer are approximately 31.4 kHz and 31.5 kHz. The scale factor is 66.24003 Hz/g. The scale factor stability is 14.886 ppm, the scale factor repeatability is 23 ppm, the bias stability is 23 μg, the bias repeatability is 170 μg, and the bias temperature coefficient is 0.0734 Hz/°C.

A micromechanical silicon resonant accelerometer converts the acceleration signals to be tested into the frequency variation of the resonator. Thus, the output is a quasi-digital signal. Moreover, the micromechanical silicon resonant accelerometer has the advantages of a wide dynamic range, strong anti-interference capacity and high stability. The output signal need not experience A/D conversion before entering the digital system, which greatly facilitates the signal processing. Thus, this type of sensor can easily achieve high-precision measurements. In addition, it possesses the numerous other advantages of silicon micro-inertia devices. It is one of several new-generation, high-precision MEMS accelerometers.

In recent years, the micromechanical silicon resonant accelerometer has invoked great interest worldwide. Some famous companies and research institutions have thoroughly studied this type of accelerometer [

A prototype device that was developed by the University of California, Berkeley, has base resonator frequencies of 145 kHz and a scale factor of 17 Hz/g [

Kim from Seoul National University designed inertial-grade vertical-type and lateral-type differential accelerometers. They consist of an out-of-plane (for the z-axis) accelerometer and in-plane (for the x- and y-axes) accelerometers. The sensing principle of the accelerometer is based on the gap-sensitive electrostatic stiffness changing effect. The out-of-plane resonant accelerometer shows a bias stability of 2.5 μg, a sensitivity of 70 Hz/g and a bandwidth of 100 Hz at a resonant frequency of 12 kHz. The in-plane resonant accelerometer shows a bias stability of 5.2 μg, a sensitivity of 128 Hz/g and a bandwidth of 110 Hz at the resonant frequency [

Draper Laboratory was one of the pioneers in the study of micromechanical accelerometers, and their results remain at the cutting edge of international research. The Draper studies show that a 0.01 °C temperature control will be maintained if the scale factor stability is better than 1 ppm. The principle prototype they developed provides the best overall performance, with a scale factor stability of better than 1 ppm and a bias stability superior to 1 μg [

China's research on micromechanical silicon resonant accelerometers started recently. At most institutions, the research remains at the simulation stage of the micromechanical structure. Laboratory prototypes have rarely been developed. In 2004, the Institute of Microelectronics at Peking University, and the Department of Precision Instruments at Tsinghua University, collaboratively developed a prototype of a micromechanical silicon resonant accelerometer with a sensitivity of 27.3 Hz/g and a resolution of 167.8 μg [^{−3}. The temperature coefficient of the resonator is 42 Hz/°C [^{−6}/°C. The bias stability approaches 42.5 μg within 1.5 h [

The domestic SOG technique is currently adopted by most research institutions for fabricating micromechanical silicon resonant accelerometers. In this technique, the anodic bonding process is used to form a tight silicon-oxygen bond to adhere the silicon wafer and the glass wafer together. Because of the mismatching thermal expansion coefficients of silicon and glass, thermal stress will be produced during the fabrication, packaging and use of the accelerometer. This thermal stress will seriously affect the accelerometer performance. In this study, the structure of the micromechanical silicon resonant accelerometer is optimized to reduce the temperature influence on the accelerometer. Thus, the closed-loop drive circuit is designed based on the phase-locked loop. A performance test is also performed on the developed prototype.

A structural diagram of the micromechanical silicon resonant accelerometer is shown in

Two identical double-ended tuning forks (DETFs) serve as the stress-sensitive resonators. The two DETFs are symmetrically arranged and connected by the proof mass, which converts the acceleration into an inertial force, which is later magnified by leverage before being transmitted to the resonators. The resonant frequency of one resonator will decrease under the compressive force, and the resonant frequency of the other resonator will increase under the tensile force. The magnitude of the input acceleration will be calculated from the difference between the resonant frequencies of the two resonators.

By simplifying the relevant theoretical formula [_{0}

The differential output of the accelerometer is given as follows:

Taylor expansion is performed on

Because the beat frequency is far below the unloaded resonant frequency, the accelerometer bias is greatly reduced.

The scale factor is two times that of a single resonator.

The ^{2}

The effect of the common-mode errors, such as temperature and stress, on the output is weakened.

The DETFs serve as resonators in the micromechanical silicon resonant accelerometer. When there is an acceleration input, the axial force on the resonant beam will induce changes in the resonant frequency. In addition, the thermal stress caused by variations in the ambient temperature results in the variation of the resonant frequency. Thus, the additional stress induced by variations in the ambient temperature should be minimized in the structural design.

A thermal analysis simulation is performed for the two DETF structures using the ANSYS software. The thermal stress in the DETF structure can be obtained by simulation.

As shown in

To amplify the resonant frequency variation, which is caused by acceleration, and to increase the scale factor of the entire device, a single-stage microleverage mechanism is used to magnify the inertial force. The microleverage mechanism is directly connected to the proof mass and the resonators. The major consideration in the structural design is whether the connection mode releases the axial thermal stress on the resonant beam, which is related to the resulting size variation because of the temperature variation.

A thermal analysis simulation is performed for the two types of accelerometers. The thermal stress caused by the change in the ambient temperature is imposed on the accelerometer as the pre-stress, and a structural dynamic analysis is performed on the accelerometer. The resonant frequency of the reverse motion between the two beams can be obtained at different temperatures, as shown in

Ideally, the frequency shift of the two resonators, which is caused by the change in the ambient temperature, can be eliminated using a differential structure. However, during fabrication and packaging, the highest temperature required by the technology is 400 °C. Because of the mismatching thermal expansion coefficients of silicon and glass, a large thermal stress will be generated in the resonant beams. The large thermal stress may deform or damage the structure.

After the accelerometer is cooled from a high temperature to room temperature, there will be a large residual stress in the resonant beams. The residual stress results in a large frequency deviation and may lead to structural deformation or damage. Some accelerometer structures with common DETF have produced deformation after packaging.

A structural dynamics simulation is performed on the improved accelerometer, and the working modes are shown in

The working frequencies of the upper resonator and the lower resonator are 31,279.8 Hz and 31,285.6 Hz, respectively. There is a small difference between the two resonators because of the calculation error. When a different acceleration is applied to the accelerometer, a different frequency shift is obtained. The calculated scale factor from the process is 62.6 Hz/g.

The resonant beam in the micromechanical silicon resonant accelerometer is sensitive to the acceleration variation. In the measuring range, the resonant frequency is allowed to vary over a wide range. In this frequency range, the drive circuit shows an equivalent stable phase shift and accurately tracks the frequency variation of the resonant beam. The phase-locked loop (PLL) is essentially an automatic control closed-loop system that synchronizes the phases of two electric signals. The frequency of the input signals is constantly tracked in a certain range. In addition, this loop strongly inhibits the input noise. When applied to the micromechanical silicon resonant accelerometer, the phase-locked loop has distinctly superior performance [

The closed-loop drive circuit diagram of the micromechanical silicon resonant accelerometer is shown in

The dynamics equation of the resonant beam of the micromechanical silicon resonant accelerometer is expressed as follows:
_{n}_{n}_{n}_{n}_{0} is the free oscillation frequency of the VOC, _{o}_{e}(t)

The open-loop transfer function of the phase-locked loop with respect to the phase is expressed as follows:
_{v}_{v}_{d}K_{a}K_{o}_{d}_{a}

The closed-loop transfer function of the system is expressed as follows:
_{i}_{o}

The driving amplitude control circuit is composed of a full-wave rectifier and an integrator. Suppose that the vibration displacement of the resonant beam is written as follows:
_{S}

The integrator is a low-pass filter. When the cut-off angular frequency is much smaller than _{REF}_{2}_{I}_{REF}_{2} is considered the AC drive amplitude, the drive amplitude control constitutes the negative feedback loop of the vibration amplitude, which stabilizes the vibration amplitude of the resonant beam.

The normalized transfer function of the low-pass loop filter is expressed as follows [

When

The continuous acceleration that slowly varies in a finite time can be approximated as a superimposition of several step-change accelerations as follows:

In general, the passive low-pass filter has no pole. Therefore,

If _{v}_{n}

The active proportion-integration filter in

The transfer function of this procedure is expressed as:
_{1}=1/_{1}_{1}, _{2}=1/_{2}_{1}, and _{3}=1/_{2}_{2}. Then, the steady-state error is calculated as:

Thus, the no-error phase control is achieved in the resonant circuit.

The steady-state phase error can be decreased by changing the gain _{v}_{v}

The micromechanical silicon resonant accelerometer is fabricated using the SOG technique. Silicon and glass are the structural layer and the substrate of the MEMS device, respectively. The standard SOG process is illustrated in

First, the accelerometer is electrically pre-heated at room temperature. There is an input of ±1 _{1} is calculated according to _{+}_{1g}_{-}_{1g}_{+}_{1g}_{-}_{1g}

After the pre-heating at room temperature, 7 scale factors were repeatedly measured for one start-up. The time of each measurement is 10 min. The scale factor stability is calculated using _{1stab}_{1m}_{1} is the mean of the scale factor, and is the total number of measurements.

The accelerometer is powered off for 30 min before the next round of measurements. The scale factor of the accelerometer is measured using the detailed rules in Section 5.1. The accelerometer is powered off six times, _{1r}_{1m}

A clamp is installed to ensure that the input shaft of the accelerometer is in the horizontal position, which is nearly 0 g. After the accelerometer is electrically pre-heated for 20 min at room temperature, the prototype is tested for 60 min at a sampling rate of 1 Hz. The average is taken for every 10 groups of 3,600 groups of data. The standard deviation (1 σ) is calculated as the stability index.

After pre-heating at room temperature, the accelerometer is rolled over at four positions, namely +0 g, +1 g, −0 g and −1 g. The output velocity is recorded at a sampling frequency of 1 Hz. The time of each measurement should not exceed 30 s. The measurements are averaged. The bias is calculated using _{+}_{0g}_{+0g}

The accelerometer is powered off for 30 min at room temperature. Then, it is pre-heated again, and the accelerometer bias is measured. The accelerometer is powered off six times, and seven measurements are conducted. The bias repeatability is calculated using _{0}_{r}_{0}_{m}_{0} is the bias mean.

The accelerometer is placed in a temperature-controlled oven, where the temperature is maintained at −40 °C, −20 °C, 0 °C, +20 °C, +40 °C and +60 °C, each for 1 h. Then, the resonant frequency of the DETF at each temperature is measured. The output data of the DETF is recorded at a sampling frequency of 1 Hz. The measurement time at each temperature is 30 s. The average value of 30 samples is considered the output at each temperature.

The test results of the two kinds of accelerometers are listed in

The accelerometer structure is improved to reduce the temperature effect on the accelerometer performance. In practical applications, a different structure can eliminate most of the common-mode errors that are caused by the temperature effects. The improved DETF structure can greatly reduce the thermal stress caused by the temperature change. A complete set of micromechanical silicon resonant accelerometers are designed and fabricated. The closed-loop driving mechanism based on the phase-locked loop is analyzed in detail, and the corresponding circuit diagram is realized. A prototype of the micromechanical silicon resonant accelerometer is developed. According to the test results, the unloaded resonant frequencies of the prototype are approximately 31.4 kHz and 31.5 kHz, respectively; the scale factor is 66.24003 Hz/g; the scale factor stability is 14.886 ppm; the scale factor repeatability is 23 ppm; the bias stability is 23 μg; and the bias repeatability is 170 μg. The bias temperature coefficient is 0.0734 Hz/°C. The test provides the basis for the subsequent development of micromechanical silicon resonant accelerometer prototypes. There is a gap between the prototype and the existing state-of-the-art micromechanical silicon resonant accelerometer. The study on temperature compensation and temperature control is in progress, which can further reduce the temperature effect on the accelerometer and improve the accelerometer performance.

This work was supported by the National Natural Science Foundation of China (No. 61101021), the Jiangsu Provincial Natural Science Foundation of China (No. BK2010401), the Foundation (No. KL201103) of Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology, Ministry of Education, China and the Fundamental Research Funds for the Central Universities (3222003102).

The authors declare no conflict of interest.

^{1/2}resolution

Structural diagram of the micromechanical silicon resonant accelerometer.

(

(

The stress distributions of the two DETF structures when the ambient temperature decreases from room temperature to −40 °C. (

The stress distributions of the two DETF structures when the ambient temperature increases from room temperature to +60 °C; (

(

Simulation models of the two types of accelerometers. (

The deformation of the accelerometer with common DETF after packaging.

The working modes of the improved accelerometer. (

The closed-loop drive circuit diagram of the micromechanical silicon resonant accelerometer.

The active proportion-integration filter.

Curves of the variation of the steady-state phase error.

The standard SOG process. (

The local structure of the improved micromechanical silicon resonant accelerometer under the 3D video microscope.

(

(

The bias stability measurement curve.

Resonant frequency of the DETF at different temperatures (reverse motion).

− |
|||
---|---|---|---|

Frequency of the common DETF structure (Hz) | 28,288.2 | 31,287.5 | 33,130.0 |

Frequency of the improved DETF structure (Hz) | 29,439.3 | 31,282.0 | 32,449.7 |

The parameters of the microleverage mechanism.

Microleverage | 800 | 50 | 60 |

Input beam | 120 | 8 | 60 |

Output beam | 50 | 8 | 60 |

Pivot beam | 50 | 8 | 60 |

Working frequencies of the resonators in the two types of accelerometers at different temperatures.

− |
|||
---|---|---|---|

Working frequency of the common DETF (Hz) | 31,080.9 | 31,280.5 | 31,412.7 |

Working frequency of the improved DETF (Hz) | 31,236.2 | 31,279.8 | 31,308.8 |

The scale factors of the seven measurements for one start-up.

Scale factor (Hz/g) | 66.24005 | 66.24165 | 66.23948 | 66.24175 | 66.24052 | 66.24018 | 66.24201 |

The scale factors of the seven measurements for seven start-ups.

Scale factor (Hz/g) | 66.24730 | 66.24791 | 66.25065 | 66.25112 | 66.25086 | 66.25064 | 66.25027 |

The bias values of seven measurements for seven start-ups.

Bias (g) | 1.589646 | 1.589737 | 1.589785 | 1.589854 | 1.589968 | 1.589993 | 1.590142 |

The full temperature test result of the improved accelerometer.

− |
− |
+ |
+ |
+ | ||
---|---|---|---|---|---|---|

Resonant frequency of the upper resonator (Hz) | 31,331.12 | 31,379.3 | 31,426.16 | 31,462.37 | 31,496.56 | 31,514.02 |

Resonant frequency of the lower resonator (Hz) | 31,439.55 | 31,486.05 | 31,531.33 | 31,565.55 | 31,598.49 | 31,615.11 |

Frequency difference (Hz) | 108.43 | 106.75 | 105.17 | 103.18 | 101.93 | 101.09 |

The full temperature test result of the old accelerometer.

− |
− |
+ |
+ |
+ | ||
---|---|---|---|---|---|---|

Resonant frequency of the upper resonator (Hz) | 31,327.09 | 31,402.31 | 31,471.82 | 31,544.49 | 31,617.52 | 31,686.85 |

Resonant frequency of the lower resonator (Hz) | 31,402.41 | 31,479.45 | 31,551.78 | 31,627.08 | 31,702.98 | 31,775.43 |

Frequency difference (Hz) | 75.32 | 77.14 | 79.96 | 82.59 | 85.46 | 88.58 |