Temperature-drift characterization of a micromachined resonant accelerometer with a low-noise frequency readout

Sensors and Actuators A 300 (2019) 111665

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Temperature-drift characterization of a micromachined resonant accelerometer with a low-noise frequency readout Zhengxiang Fang, Yonggang Yin, Xiaofei He, Fengtian Han ∗ , Yunfeng Liu ∗ Department of Precision Instrument, Tsinghua University, Beijing, 100084, China

a r t i c l e

i n f o

Article history: Received 22 June 2019 Received in revised form 26 August 2019 Accepted 7 October 2019 Available online 14 October 2019 Keywords: Resonant accelerometer Scale factor Frequency readout Temperature drift Temperature control Bias stability

a b s t r a c t This paper describes the design and experimental evaluation of a silicon micromachined resonant accelerometer with high scale factor and low temperature drift. The device was fabricated by bulk micromachining and sealed in a hermetic metallic package to ensure the resonator operates with high Q-factor. By optimizing the structure, the scale factor of the resonant accelerometer was increased to 361 Hz/g, which is close to the simulated result of 336 Hz/g taking into account the measurement range of ±14 g. The bias-temperature drift coefficient fell to 4.4 ␮g/◦ C. A digital frequency readout was implemented using a continuous time stamping method. The measured relative resolution within the operating frequency range of the resonator was better than 1.8 ppb at 1 Hz data update rate. Thus, we have produced a miniaturized resonator accelerator without the bulky commercial frequency counters used in our previous work. A test chamber at a constant temperature varying by only ±0.01◦ C was used to characterize the temperature drift of the accelerometer prototypes by isolating the disturbances due to the ambient temperature. After increasing the scale factor and decreasing the temperature sensitivity, the 3-day bias stability was measured to be 2.19 ␮g and 0.51 ␮g for room-temperature and constant-temperature operations, respectively. Furthermore, the long-term bias stability was 1.77 ␮g in 30-day measurements when the temperature variations were controlled to be within ±0.01 ◦ C. The experimental results indicate that this resonant accelerometer exhibits excellent long-term temperature stability, which offers the promise for high-performance shipborne inertial navigation applications. © 2019 Published by Elsevier B.V.

1. Introduction Micromachined accelerometers are widely used in many fields from consumer electronics for mobile devices [1] and medical instruments [2] to inertial navigation [3]. Their advantages include small size, low power consumption, low cost, and high integration [4]. Micromachined resonant accelerometer (MRA) is a type of high-performance accelerometers that convert an input acceleration into a resonant frequency change. It generally comprises a pair of double-ended tuning fork (DETF) resonators connected to a common proof mass. Since the differential operation of the two resonators produces high common-mode error rejection, MRAs exhibit potential for high-precision applications such as inertial navigation and seismology [5–8]. The temperature drift characterization of an MRA is significant for high-performance navigation and gravimeters with desirable short- or long-term accuracy. It is essential that the thermal sta-

∗ Corresponding authors. E-mail addresses: [email protected] (F. Han), yfl[email protected] (Y. Liu). https://doi.org/10.1016/j.sna.2019.111665 0924-4247/© 2019 Published by Elsevier B.V.

bility is improved by active suppression of the temperature and time drift [9–12]. Though the temperature drift can be considerably suppressed using a fully differential design for the MRA geometry, there is still a residual temperature drift owing to imperfect fabrication tolerances, which degrades the MRA performance. To achieve excellent stability, the careful design of a structure with a low sensitivity to thermal stress is significant, combined with various temperature-drift compensation, electric stiffness modulation [13] or constant temperature control schemes. Draper Laboratory (USA) has reported two high-precision prototypes MRA with excellent stability required for strategic and high-precision navigation applications [14]. One resonant accelerometer for long-term navigation had an excellent bias stability of 0.19 ␮g and 2 ␮g for 3-day and 30-day measurements under 0.01 ◦ C temperature control. A single-anchor resonant accelerometer integrated with an on-chip micro-oven was proposed to suppress the error induced by temperature drift [15]. The measured bias temperature-drift coefficient (TDC) was about -0.117 mg/◦ C from −20 ◦ C to 80 ◦ C and the bias stability was 1.6 ␮g for a measurement duration of 2.2 h at a controlled temperature of 75 ◦ C. For another resonant accelerometer with a temperature self-compensation model, the bias TDC

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decreased from 4.754 mg/◦ C to 0.035 mg/◦ C [16]. Two structures with a low sensitivity to thermal stress using optimized anchor location or single-anchor isolation frame were reported in our previous work [17]. At room temperature, the 15-day bias stability was 1.4 ␮g without temperature drift compensation. However, not much has been published on the experimental characterization of the temperature-dependent drift and stability, which is significant for high-performance MRAs. In addition, the temperature stability of an MRA can be improved by increasing its scale factor, which is applicable to most precise inertial navigation systems with a relatively small dynamic range requirement [18,19]. The scale factor of a resonant accelerometer can be enhanced by increasing the inertial force and the force-to-frequency amplification factor. Moreover, it is desirable to achieve high resolution and low noise by an optimal MRA design with high scale factor and adequate measurement range. Typically, high-performance inertial navigation applications require the accelerometer measurement range to be larger than ±10 g and thus, the scale factor of the MRAs in the literature is limited to about 200 Hz/g [8,14,16,20]. This paper presents the design and test results of a resonant accelerometer with a high scale factor and excellent thermal stability for long-endurance high-performance navigation applications. The scale factor was increased by optimizing the structural parameters of the resonators, micro-levers, proof mass, and suspension beams, which was also helpful in reducing the bias TDC. To replace the bulky commercial frequency counters used in our previous work and thus to miniaturize the prototype inertial sensor, a precise digital frequency readout was realized using the continuous time stamping method. A precise constant-temperature chamber was developed to test and characterize the temperature drift performance under different temperature conditions. The bias stability and Allan deviation of a prototype MRA were measured and compared at room temperature and at a precisely controlled temperature. The experimental results demonstrate the excellent long-term bias stability with precise temperature control.

2. Design of the MRA structure 2.1. Structure and operation of the MRA The device topology is like that of our previous work [17]. Fig. 1 is a schematic representation of half of the fully symmetric MRA structure. Two DETF resonators are arranged so that they are loaded differentially by the proof mass. These elements form a monolithic silicon structure that is supported above and anodically bonded to a glass substrate via a set of anchors. The vibration of the resonators is excited electrostatically and sensed capacitively by the electrodes distributed as comb teeth on the resonating beams. In this work, each resonator pair is driven in an anti-phase vibration mode to improve drive efficiency [21]. When an input acceleration occurs along the sensitive axis (the x-axis in Fig. 1), one resonator is placed in tension while the other one is under compression to generate a differential vibration frequency change under the applied load. This differential frequency change is measured and serves as the acceleration output of the MRA. Clearly, this differential design doubles the scale factor of the MRA and results in the mechanical cancellation of the various error sources that are common to both DETF resonators [13].

2.2. Structure design and optimization To increase the scale factor and reduce the temperature sensitivity of the MRA, the structure parameters were further optimized

Fig. 1. Half of the fully symmetric MRA structure.

based on our previous design. The resonant frequency of the fundamental mode is given by [22] f0 =

1 2



K1 1 = Me q 2



199Etw3 12l3 t(0.396wl

+ Sc )

(1)

where E and  are the density and Young’s modulus of singlecrystal silicon, l, w, and t are the length, width, and thickness of the vibration beam, respectively, and Sc is the area of the comb teeth. Then the scale factor of the resonant accelerometer is given by SF =MP · AF · SFR

(2)

where MP is the mass of the proof mass, AF is the effective fore magnification of the micro-lever, and SFR is the sensitivity of the DEFT resonator. Clearly, optimization of the sensor geometry, microlever, and DETF resonator would be an effective way to increase the scale factor. The scale factor of the DEFT resonator can be expressed as SFR = f0

6l2 2 Etw3

F

(3)

where F is the equivalent load applied to the resonant beam. Next, the effective magnification of the micro-lever is closely related to the stiffness of the micro-lever and suspension beam as AF =

KL AL 2KU + KL

(4)

where KL and KU are the stiffnesses of the micro-lever and suspension beam, and AL is the force amplification factor of the micro-lever. A decrease in the suspension beam stiffness will increase the scale factor of the MRA but at the expense of a degraded overload limit. Generally, the drift of each resonator mainly results from the sensitivity of Young’s modulus to temperature and the thermal stress between the glass substrate and the silicon structure [9,23]. Thus, the bias TDC of the MRA is dependent on the frequency temperature coefficient (FTC, ppm/◦ C) of the two resonators. It can be expressed as Bias TDC =

FTC1 · f1 − FTC2 · f2 SF

(5)

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Table 1 Main structure parameters of the MRA. Parameters

Symbols

Units

Value

Area of the proof mass Thickness of the proof mass Length of the DETF Width of the DETF Area of the comb teeth Length of the suspension beam Width of the suspension beam Lever ratio

S t l w Sc lU wU AL

mm2 ␮m ␮m ␮m ␮m2 ␮m ␮m ␮m:␮m

15.054 60 932 6 19268 500 4 1110:40

Fig. 3. Simulated linearity error vs. input acceleration.

Fig. 2. Simulated differential frequency output vs. input acceleration.

where f1 and f2 are the resonant frequency of two resonators. It is clear that the offset drift is virtually independent of the frequency mismatch between the two resonators. The differential FTC of the two resonators mainly results from the fabrication tolerances and residual stress in the fabrication and packaging process. However, the FTC is generally constant for an MRA device after fabrication and packaging. Accordingly, the bias TDC can be reduced by a welldesigned MRA structure with high scale factor. In our MRA design, the structural parameters were investigated by trial and error via finite-element simulation in ANSYS Workbench to give an optimized higher scale factor. The resulting structural parameters of the MRA are listed in Table 1. The anti-phase resonant mode is demonstrated in the inset of Fig. 2 where the natural resonant frequency of the resonator (f0 ) is 20,067 Hz. The simulated scale factor for a measurement range of ±14 g is 333.9 Hz/g and the FTC of each resonator is −29.00 ppm/◦ C (−0.5819 Hz/◦ C). The 1% linearity error limit of the scale factor is obtained at about ±4.1 g and the full-scale range of ±14 g exhibits a linearity error of 1.38%. It is clear that the nonlinear error of the scale factor as the simulated result shown in Fig. 3 is symmetrical and can be characterized by a nonlinear model. Thus, this linearity error can be compensated largely after careful calibration by experiments. Furthermore, shock survival is one of the sensor design guidelines. During the shock simulation, the sensor structure is loaded at an acceleration of 100 g in three orthogonal directions respectively. The simulated results indicate that the sensitive x-axis input exhibits the weakest shock survival, where the proof mass responds to shock with a maximum displacement of 9.84 ␮m and a stress up to 101.8 MPa. However, the shock responses are smaller than their allowable limits of 11.5 ␮m and 240 MPa, respectively. This indicates the sensor offers an anti-shock limit over 100 g.

Fig. 4. Micrograph of a fabricated MRA sensing structure (inset is a photograph of the vacuum packaged device).

2.3. Fabrication and testing This MRA was fabricated using bulk micromachining based on silicon on glass. It was then packaged under vacuum into a hermetic metallic case and maintained at a vacuum of 0.1 Pa using a getter. A micrograph of the MRA structure and a photograph of the vacuum packaged device are shown in Fig. 4. The die size is 8.7 mm × 8.7 mm × 0.58 mm and the vacuum metallic package size is 15 mm × 15 mm × 5 mm. The scale factor of the prototype MRA was tested using a precise turntable. By using the tilt method in the Earth’s gravitational field, the calibrated scale factor was measured to be 361 Hz/g. Under 0 g input and room temperature, the open-loop sweepingfrequency measurements indicate that the two resonators vibrate at 19 518 Hz and 19 525 Hz at 10 mV drive voltage respectively, as shown in Fig. 5. The measured Q-factor of the two resonators is over 3.81 × 105 , which shows excellent vacuum condition inside the device package. Clearly, the measured resonant frequencies are slightly less than the simulated value while the scale factor is larger than the predic-

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Fig. 5. Open-loop frequency response of the two resonators.

Fig. 7. Block diagram of the interface and electronic frequency readout.

Fig. 6. The effect of the over-etching error on the resonant frequency and scale factor.

tion (as indicated in Fig. 2). This suggests an imperfect over-etching on the beams during fabrication existed. The beamwidth of both the suspension beam and the DETF were adjusted in the ANSYS simulation. The results shown in Fig. 6 illustrate the effect of the over-etching error. Considering the measured scale factor and resonant frequency of the MRA device, the beam width reduction due to the over-etching tolerance was estimated to be 0.16 ␮m. 3. Frequency readout The output signal of the MRA is the two-channel vibrationsensing voltage of the resonators, which is typically in the form of a sine wave. The indicated acceleration is generally resolved from precise measurements of the resonant frequency of the two DETF resonators. In our previous work, two bulky universal frequency counters (Keysight 53230A) were used for the high-resolution digital frequency readout. To miniaturize the MRA and to meet the needs of future applications of this micromachined sensor, an embedded electronic frequency readout was considered to be an ideal solution for enabling precise frequency measurements for this low-noise accelerometer. The DEFT resonator is excited by a closed-loop electronic oscillator interface and operated as a self-sustained oscillation with a constant vibration amplitude. The vibration frequency of each resonator is measured by a precise frequency counter. Fig. 7 shows a block diagram of the interface and the electronic frequency readout. Two frequency readout channels, one for each DETF, was imple-

mented in one PCB. The analog electronic interface is like that in our previous work [24]. The closed-loop electronic interface mainly comprises capacitive vibration sensing, phase and amplitude regulation, and electrostatic excitation. The precise electronic frequency readout for the MRA was developed on a field programmable gate array (FPGA) as schematically shown in Fig. 7. Its input signal is from the output of the capacitive vibration detection, whose frequency lies in the vicinity of the nominal resonant frequency of the resonators, i.e., 20 kHz in this design. Considering the scale factor of 361 Hz/g and the measurement range of ±14 g, the frequency readout should have a dynamic range from 15 kHz to 25 kHz. A measurement of the resonating frequency signal normally has to deal with noise components outside the input range. Therefore, an analog bandpass filter with a center frequency of 19.8 kHz is used to improve the signal-to-noise ratio before the signal is sent to the frequency counter. Then the sinusoidal input signal is converted into a square wave and used as the input to the FPGA by triggering an ultrafast comparator. A precise reference clock running at 200 MHz was used for the control gate and master clock of the FPGA-based counter. An ultra-low temperature-drift clock was based on an oven-controlled crystal oscillator (OCXO) running at a 10 MHz reference frequency, which is part of the frequency readout PCB, as shown in Fig. 8. As the resolution of the acceleration for the MRA is better than 1 ␮g at 1 Hz data update rate, the equivalent resolution of the frequency counter must be much higher than the resolution of the sensor. In this work, the design goal of the counter is a frequency resolution of at least 100 ng for MRA measurements, which is equivalent to a relative resolution of 1.8 ppb. The common methods for frequency counters include conventional counting, reciprocal counting, and interpolating methods for reciprocal counting [25]. At 1 s measured time, the quantization error in reciprocal counting is ±100 ns with a 10 MHz reference clock. To obtain a high resolution, either the frequency of reference clock or measurement duration can be increased, as the measurement error mainly comes from the quantization error of the reference clock. So, some interpolating methods, such as Vernier and time-to-digital conversion, have been developed to give more refined measurements [26,27]. Recently, continuous time stamping

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Fig. 10. Comparison of measured resolution of the three frequency measurement methods.

Clearly, the continuous time stamping method has a much lower frequency noise, over one order of magnitude less, than the reciprocal counting. The relative resolution of the continuous time stamping counter realized by the electronic frequency readout is 25.9 ␮Hz / 20 kHz =1.29 ppb. Furthermore, the noise for a commercial universal frequency counter (Keysight 53230A) is shown in Fig. 10. The bias between the dotted line and the solid line comes from the discrepancy between the reference clocks used in the two counters. The standard deviation (1␴) of frequency measurements using the 53230A is 24.8 ␮Hz, which is very close to the experimental result for the continuous time stamping method. This implies that a commercial counter can be replaced by our miniature electronic frequency readout in a performance test of our high-resolution MRA. The relative uncertainty of the measured frequency in the continuous time stamping method is

Fig. 8. A photograph of the frequency readout PCB.

s(fˆ ) s(Tˆ ) (T ) ≈ =   Tˆ fˆ (xk − x¯ )2 Tˆ

Fig. 9. Schematic diagram of the continuous time stamping.

has been used to enhance the resolution [28,29]. Fig. 9 is a schematic of continuous time stamping. The status of input events is monitored at each rising edge of the output of the reference clock. The input events and the clock cycles are continuously counted without being reset. The values for the event counter and the clock counter are latched into registers when the rising edge of an input event occurs. In this way, there is no dead time for counting. For every measurement of an event period, quantization errors are produced in the phase state since the rising edges of the reference clock and input events are independent and cannot be aligned ideally. With this approach, a series of data {xk , Tk } is recorded where xk is the accumulated input counts and Tk is the accumulated time at every sampling moment. The resolution of the frequency measurements can be improved using least squares fitting. Thus, the estimated frequency of the input events is [30] 1 fˆ = = Tˆ





(xk − x¯ )2

(xk − x¯ )(Tk − T¯ )

 2  2 n xk − ( xk )    = n

xk Tk −

xk

Tk

(6)

where ˆT is the estimated period of the input signal, which is the reciprocal of the frequency, andˆf is the inverse slope of the recorded data. To evaluate the FPGA-based frequency counter, a 20 kHz sinusoidal signal generated by a signal generator (HP 33120A) was used as the input. A comparison of the measured frequency noise between the reciprocal counting and continuous time stamping is shown in Fig. 10. The sampling time, gate time, and measurement duration are, respectively, 1 s, 0.95 s, and 1 h for both methods.

(7)

where (T) is the uncertainty for all accumulated times, and s(ˆf) and s(ˆT) are the variances of the measured frequency and the measured period of the signal. As {xk } is the continuous natural sequence from 1 to N, the relative uncertainty of the measured frequency is finally given as (T ) s(fˆ ) (T ) =  ≈  fˆ Tˆ N(N 2 − 1)/12



12Tˆ 

(8)

where  is the measured time and N is the total sample count in one control gate. The relative resolution of the measured frequency is proportional to the -3/2 power of the measurement time from Eq. (8). Fig. 11 shows the experimental results of the relative resolution of the measured frequency as a function of the measuring time using the continuous time stamping method. The uncertainty (T) mainly consists of three parts: (1) the quantization error q (T) limited by the resolution of the time interval of the reference clock, (2) the trigger error t (T) induced by the comparator and the FPGA, and (3) the inherent noise n (T) of the input signal in this test. The uncertainty for all accumulated times can be expressed as



(T ) =

q2 (T ) + t2 (T ) + n2 (T )

(9)

In this work, the quantization error q (T) is 5 ns while the reference clock ticks at 200 MHz. The trigger error t (T) is about 8.4 ns from the device datasheet, and the inherent noise n (T) of the wave

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4. Performance test 4.1. MRA setup

Fig. 11. Relative resolution of the measured frequency vs. measuring time.

A schematic of the MRA setup is shown in Fig. 13. The MRA and the interface electronics are fixed inside an aluminum case to isolate external interference. The prototype MRA is mounted inside a chamber in which the temperature varies by only ±0.01 ◦ C, which was developed especially for the temperature drift evaluation. A platinum resistor, which is attached to the sidewall of the Al case, is used as a thermometer after careful calibration. A polyimide film, which is attached to the inner wall of the chamber, is used as a heater. The middle of the wall is made of heat-insulated foam, which acts as an insulator around the heater to isolate the impact of external heat. The outer stainless steel case provides a stable base for the MRA. The chamber temperature is regulated by a microcontroller with a proportional-integral-differential (PID) algorithm. The temperature control loop has a sampling rate of 10 Hz and its data update rate to a host computer is set at 1 Hz. The maximum heater power is about 7.5 W and has a 15 V DC power supply. During following measurements, the data readout rate of the accelerometer is also set at 1 Hz.

4.2. Temperature sensitivity

Fig. 12. Relative resolution of the measured frequency vs. input signal frequency.

The temperature-dependent frequency drift is one of the major errors of an MRA. Although the differential design of the MRA structure can cancel most common-mode errors, a residual temperature drift exists due to thermal stress and fabrication tolerances. The measured frequency drifts of the two resonators and the differential frequency versus operating temperature in the range from 20 ◦ C to 60 ◦ C are shown in Fig. 14. The measured FTCs of the two resonators are -28.42 ppm/◦ C (-0.5547 Hz/◦ C) and -28.49 ppm/◦ C (0.5563 Hz/◦ C), respectively, which agree well with the simulation result -29.00 ppm/◦ C (-0.5819 Hz/◦ C). Note that the differential frequency FTC is only 0.0016 Hz/◦ C, which is equivalent to a bias TDC of 4.4 ␮g/◦ C. This shows an ultra-low temperature drift of the zerog output and exhibits an excellent bias stability of 1.1 ppm/◦ C [13], assuming the uncompensated linearity error of 1% and full-scale range of ±4.1 g.

4.3. Short-term bias stability generator is 0.1 ns, which can be ignored. The uncertainty (T) is almost constant at 9.8 ns. Thus, the relative resolution of the estimated frequency can be calculated using Eq. (8), as shown in Fig. 11. If the gate time ranges from 0.095 s to 0.95 s, then the experimental results agree with the calculated results. It is clear that the relative resolution of the measured frequency decreases rapidly when the measuring time is 0.095 s. In addition, the relative frequency resolution was tested in the resonator response range of the MRA, which is from 15 kHz to 25 kHz. Three repeated test results with a frequency interval of 1 kHz are shown in Fig. 12. The measuring time was set at 0.95 s for 1 Hz data update and each point on the figure was calculated from 3600 data points. These relative frequency resolutions are at almost the same level. From Eq. (8), the relative frequency resolution is constant regardless of the input signal frequency. The results were influenced by the ambient temperature fluctuations in the tests, which indicates that the relative frequency resolution of the continuous time stamping method is better than 1.8 ppb. According to the range and scale factor of the MRA, the equivalent resolution of the acceleration for the frequency readout can reach 100 ng, which satisfies the frequency measurement requirement well.

The short-term stability of the accelerometer bias was measured. This is one of the most crucial parameters for inertial navigation applications. A comparison of the bias stability is shown in Fig. 15. The prototype MRA was tested at room temperature and in a temperature-controlled chamber. As the ambient temperature changes over a 24 -h cycle, the short-term temperature drift can be characterized with a 72 -h bias test. During this test of the bias stability, the room temperature fluctuated by about 1 ◦ C over the 72 h. The chamber had a set point of 45 ◦ C, varying by ±0.01 ◦ C. It provides a more stable and uniform environment. Since the device is insensitive to temperature, the measured bias-temperature drift at room temperature is less than 6.4 ␮g peak-to-peak, which is slightly larger than the predicted value of 4.4 ␮g calculated solely from the temperature sensitivity, as presented in Section 4.2. The additional drift is possibly due to the non-uniform distribution of the temperature field and the thermal deformation of the fixture. By contrast, a precise temperature controlled chamber can provide a more stable and uniform thermal environment for better performance. The measured bias stability (1␴) over 3 days was 2.19 ␮g and 0.51 ␮g at room temperature and at the constant temperature, respectively. Clearly, the temperature-controlled chamber is

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Fig. 13. a) Schematic of the MRA temperature control setup; b) cross-section of the temperature control chamber.

Fig. 14. Measured bias-temperature drift of the MRA. Fig. 16. Comparison of Allan deviation results.

data recorded over 72 h in Fig.15 is shown in Fig. 16. The result shows the bias instability at room temperature is 75 ng, which is achieved using an averaging time of about 70 s. When the prototype was operated with precise temperature control, the bias instability reduces to 55 ng due to the large reduction in temperature-induced noise. It is clear that the temperature-controlled operation of the MRA is effective for improving the overall performance including bias stability, noise, etc. 4.5. Long-term bias stability

Fig. 15. Short-term bias stability of the MRA.

an effective solution for realizing excellent bias stability, especially for high-performance navigation and gravimetry applications. 4.4. Allan deviation The Allan deviation is widely used to identify random errors in inertial sensors using a time domain characterization. In an Allan deviation plot, different slopes correspond to different noise mechanisms. The minimum of the Allan deviation indicates the bias instability of the resonant accelerometer, which limits the resolution of the MRA. A comparison of the Allan deviations using the

In high-performance shipborne inertial navigation systems, the accelerometers must operate continuously for months with longterm accuracy. To evaluate the long-term bias stability, a 30-day test using the setup shown in Fig. 13 was performed. The results are shown in Fig. 17. The temperature point was set at 45 ◦ C and the controlled temperature fluctuated within ±0.01 ◦ C over 30 days, as shown in the top panel of Fig. 17. The lower panel is the differential output of the accelerometer. It had a long-term bias stability of 1.77 ␮g over 30 days (1␴). Several jerks can be observed in the bias measurements, which may be due to ambient shocks or vibrations to the MRA case. There is also a slowly changing drift, which is possibly due to the deformation of the Al case, the base of the temperature-controlled chamber, or the vibration-isolated base. These measurements indicate that the MRA presented in this work has excellent long-term stability under precise temperature control. A summary of several resonant accelerometers is listed in Table 2 and compared with the experimental results in this work.

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Table 2 Comparisons between this work and the results in the literature. Parameters

[14]

[19]*

[15]

[16]

This work

Scale factor (Hz/g) Resonant frequency (kHz) Q-factor Range (g) Bias TDC (␮g/◦ C) Bias stability (g) Bias instability (g)

258.8 20 1 × 105 ±2 – 2 (30 d) 0.08

496 140 – ±0.22 −141 – 0.52

427.1 800 6.0 × 104 – −117 1.6 (2.2 h) 0.16

127 20 – ±30 35 11.5 (1 h) 4

361 20 3.0 × 105 ±14 4.4 1.77 (30 d) 0.055

*The MRA was designed for a tilt sensor.

Acknowledgments This work was supported by the National Natural Science Foundation of China (grant 41774189).

References

Fig. 17. Long-term bias stability of the MRA. Top is operating temperature and bottom is output bias.

Clearly, the prototype MRA has an ultra-low temperature sensitivity, excellent long-term bias stability, and low noise. 5. Conclusions An MRA with high sensitivity and excellent temperature stability has been presented. A comprehensive analysis of the sensing structure indicates that the scale factor can be enhanced by increasing the proof mass, the effective magnification of the micro-lever, or the sensitivity of the resonator. It also shows that the bias TDC drops with an increase in the scale factor. After optimizing the structural parameters in finite-element simulation, the scale factor of the MRA reaches up to 361 Hz/g with a ±14 g measurement range. Moreover, an FPGA-based miniaturized frequency counter was developed for the miniaturized prototype MRA using the continuous time stamping method. The relative resolution of the measured frequency is better than 1.8 ppb at 1 Hz data update rate, which is close to that of high-performance commercial frequency counters. With the optimized sensor structure and precise temperature-controlled chamber, the measured bias stability and noise performance of the MRA have been enhanced. Measurements over 30 days with the temperature varying by only ±0.01 ◦ C demonstrate that the MRA has an excellent long-term bias stability of only 1.77 ␮g. The experimental results indicate that the MRA offers the promise for high-performance and long-endurance shipborne inertial navigation applications. This accelerometer prototype exhibits a potential to be an alternative of commercial high-precision products, such as costly quartz flex accelerometers, where the MEMS resonant accelerometers will offer a more competitive cost and size. Future work will focus on miniaturizing the device package and matching an application-specific integrated circuit (ASIC) to significantly reduce the size of the sensor and the power consumption.

[1] M. Grankin, E. Khavkina, A. Ometov, Research of MEMS accelerometers features in mobile phone, 2012 12th Conference of Open Innovations Association (FRUCT) and Seminar on e-Travel (2012) 31–36. [2] S. Roy, S. Mandal, N. Hanumaiah, MEMS accelerometer: from engineering to medicine, IEEE Potentials 35 (2016) 30–33. [3] D.K. Shaeffer, MEMS inertial sensors: a tutorial overview, IEEE Commun. Mag. 51 (2013) 100–109. [4] A. Albarbar, S.H. Teay, Advanced Mechatronics and MEMS Devices II, 2017, pp. 19–40. [5] A.A. Seshia, M. Palaniapan, T.A. Roessig, R.T. Howe, R.W. Gooch, T.R. Schimert, S. Montague, A vacuum packaged surface micromachined resonant accelerometer, J. Microelectromech. Syst. 11 (2002) 784–793. [6] J.K. Saunders, D.E. Goldberg, J.S. Haase, Y. Bock, D.G. Offield, D. Melgar, J. Restrepo, R.B. Fleischman, A. Nema, J. Geng, C. Walls, D. Mann, G.S. Mattioli, Seismogeodesy using GPS and low-cost MEMS accelerometers: perspectives for earthquake early warning and rapid response, Bull. Seismol. Soc. Am. 106 (2016) 2469–2489. [7] A. D’Alessandro, R. D’Anna, L. Greco, G. Passafiume, S. Scudero, S. Speciale, G. Vitale, Monitoring earthquake through MEMS sensors (MEMS project) in the town of Acireale (Italy), 2018 IEEE International Symposium on Inertial Sensors and Systems (INERTIAL) (2018) 1–4. [8] X. Zou, P. Thiruvenkatanathan, A.A. Seshia, A seismic-grade resonant MEMS accelerometer, J. Microelectromech. Syst. 23 (2014) 768–770. [9] R. Melamud, S.A. Chandorkar, B. Kim, H.K. Lee, J.C. Salvia, G. Bahl, M.A. Hopcroft, T.W. Kenny, Temperature-insensitive composite micromechanical resonators, J. Microelectromech. Syst. 18 (2009) 1409–1419. [10] J. Dong, A. Qiu, R. Shi, Temperature influence mechanism of micromechanical silicon oscillating accelerometer, 2011 IEEE Power Engineering and Automation Conference (PEAM 2011) vol. 3 (2011) 385–389. [11] S.A. Zotov, B.R. Simon, A.A. Trusov, A.M. Shkel, High quality factor resonant MEMS accelerometer with continuous thermal compensation, IEEE Sens. J. 15 (2015) 5045–5052. [12] B. Kim, R.N. Candler, M.A. Hopcroft, M. Agarwal, W. Park, T.W. Kenny, Frequency stability of wafer-scale film encapsulated silicon based MEMS resonators, Sens. Actuators A Phys. 136 (2007) 125–131. [13] C.R. Marra, A. Tocchio, F. Rizzini, G. Langfelder, Solving FSR versus offset-drift trade-offs with three-axis time-switched FM MEMS accelerometer, J. Microelectromech. Syst. 27 (2018), 790-199. [14] R. Hopkins, J. Miola, R. Setterlund, B. Dow, W. Sawyer, The silicon oscillating accelerometer: a high-performance MEMS accelerometer for precision navigation and strategic guidance applications, the 61st Annual Meeting of The Institute of Navigation (2005) 1043–1052. [15] D.D. Shin, C.H. Ahn, Y. Chen, D.L. Christensen, I.B. Flader, T.W. Kenny, Environmentally robust differential resonant accelerometer in a wafer-scale encapsulation process, 30th IEEE International Conference on Micro Electro Mechanical Systems (MEMS 2017) (2017) 17–20. [16] Z. Jing, Q. Anping, S. Qin, B. You, X. Guoming, Research on temperature compensation method of silicon resonant accelerometer based on integrated temperature measurement resonator, 12th IEEE International Conference on Electronic Measurement and Instruments (ICEMI 2015) vol. 3 (2015) 1577–1581. [17] Y. Yin, Z. Fang, Y. Liu, F. Han, Temperature-insensitive structure design of micromachined resonant accelerometers, Sensors 19 (2019) 1544. [18] C. Comi, A. Corigliano, G. Langfelder, A. Longoni, A. Tocchio, B. Simoni, A resonant microaccelerometer with high sensitivity operating in an oscillating circuit, J. Microelectromech. Syst. 19 (2010) 1140–1152. [19] X. Zou, P. Thiruvenkatanathan, A.A. Seshia, A high-resolution micro-electro-mechanical resonant tilt sensor, Sens. Actuators A Phys. 220 (2014) 168–177. [20] X. Wang, Y.P. Xu, G.M. Xia, Y. Zhao, J. Zhao, A.P. Qiu, Y. Su, A 27␮W MEMS √ silicon oscillating accelerometer with 4-␮g bias instability and 10-␮g/ Hz

Z. Fang, Y. Yin, X. He et al. / Sensors and Actuators A 300 (2019) 111665

[21]

[22] [23] [24]

[25] [26] [27] [28] [29]

[30]

noise floor, 5th IEEE International Symposium on Inertial Sensors and Systems (INERTIAL 2018) (2018) 1–4. X. Wang, R. Huan, D. Pu, X. Wei, Effect of nonlinearity and axial force on frequency drift of a T-shaped tuning fork micro-resonator, J. Micromech. Microeng. 28 (2018), 125012. C.M. Harris, A.G. Piersol, Harris’ Shock and Vibration Handbook, McGraw-Hill, New York, 2002. R. Tabrizian, G. Casinovi, F. Ayazi, Temperature-stable silicon oxide (SilOx) micromechanical resonators, IEEE Trans. Electron Dev. 60 (2013) 2656–2663. Y. Yin, Z. Fang, F. Han, B. Yan, J. Dong, Q. Wu, Design and test of a micromachined resonant accelerometer with high scale factor and low noise, Sens. Actuators A Phys. 268 (2017) 52–60. S. Johansson, T. Horita, Frequency counter, Rev. Laser Eng. 10 (2011) 775–779. L. Bengtsson, Embedded Vernier TDC with sub-nanosecond resolution using fractional-N PLL, Measurement 108 (2017) 48–54. J. Wu, Y. Shi, D. Zhu, A low-power wave union TDC implemented in FPGA, J. Instrum. 7 (2012). E. Rubiola, On the measurement of frequency and of its sample variance with high-resolution counters, Rev. Sci. Instrum. 5 (2005). S. Johansson, New frequency counting principle improves resolution, Seventh International Conference on Electronic Measurement & Instruments (ICEMI 2005) vol. 2 (2005) 592–599. M.T. Crawley, Statistics: An Introduction Using R, 2nd edition, 2014.

Biographies Zhengxiang Fang received the B.S. degree in mechanical engineering and automation from Tsinghua University, Beijing, China, in 2015. He is currently working toward the Ph.D. degree in instrumentation science and technology in Tsinghua University. His research interests include MEMS resonant accelerometers and interface circuit design.

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Yonggang Yin received the B.S. degree in mechanical engineering and automation from Tsinghua University, Beijing, China, in 2014. He is currently working toward the Ph.D. degree in instrumentation science and technology in Tsinghua University. His research interests include MEMS resonant accelerometers and 3-axis electrostatically suspended accelerometers. Xiaofei He received the B.S. degree in mechanical engineering from Ningxia University, Ningxia, China, in 2017. She is currently working toward the Master’s degree in mechanical design and theory in Beijing Jiaotong University. Her research interests include design and optimization of MEMS resonant accelerometers. Fengtian Han received the B.S. degree in automation instrumentation from Nanjing University of Science and Technology, Nanjing, China, in 1990, the M.S. degree in industrial automation from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1996, and the Ph.D. degree in precision instrumentation from Tsinghua University, Beijing, China, in 2002. From 2002 to 2004, he was a Post-Doctoral Scholar with the Department of Precision Instrument, Tsinghua University, where he is currently a Professor. His research interests include MEMS inertial sensors and inertial navigation. Yunfeng Liu received the B.S. degree in School of aeronautic science and engineering from Beijing Aerospace University, in 1996, the M.S. degree and Ph.D. degree in precision instrumentation from Tsinghua University, Beijing, China, in 2006. He is currently an Associate Professor in the Department of Precision Instrument, Tsinghua University. His research interests include MEMS inertial sensors and inertial technology and gravity detection.