With the continuous miniaturization of electronic devices, microelectromechanical system (MEMS) oscillators that can be combined with integrated circuits have attracted increasing attention. This study reports a MEMS Huygens clock based on the synchronization principle, comprising two synchronized MEMS oscillators and a frequency compensation system. The MEMS Huygens clock improved short-time stability, improving the Allan deviation by a factor of 3.73 from 19.3 to 5.17 ppb at 1 s. A frequency compensation system based on the MEMS oscillator’s temperature-frequency characteristics was developed to compensate for the frequency shift of the MEMS Huygens clock by controlling the resonator current. This effectively improved the long-term stability of the oscillator, with the Allan deviation improving by 1.6343 × 105 times to 30.9 ppt at 6000 s. The power consumption for compensating both oscillators simultaneously is only 2.85 mW∙°C−1. Our comprehensive solution scheme provides a novel and precise engineering solution for achieving high-precision MEMS oscillators and extends synchronization applications in MEMS.
During modernization and digitalization, oscillators have been widely utilized as time-reference sources in electronic devices such as communication [1], radar [2], and global positioning system (GPS) [3]. Owing to difficulties in the miniaturization and customization of frequencies, the quartz oscillator, the most widely utilized time-reference source [4], hinders the development of electronic devices. With miniaturization [5], high integration with complementary metal-oxide semiconductor (COMS) electronics [6], [7], [8], [9], low cost, low power consumption [10], and other advantages, microelectromechanical system (MEMS) resonators have good prospects for applications in timing and frequency control. They are expected to replace conventional quartz crystal oscillators. There are two main challenges in the research on MEMS oscillators. MEMS oscillators are more easily excited into the nonlinear regime based on the size effect [11], which causes performance degradation [12], and the short-term stability of MEMS oscillators cannot match that of quartz oscillators. In addition, owing to the temperature-dependent Young’s modulus of silicon, MEMS oscillators exhibit a strong linear frequency-temperature dependence [13], which weakens their long-term stability.
Improving short-term stability is a crucial technical problem for MEMS oscillators. With years of development, scientists have successfully solved many fundamental problems, such as the optimization of excitation methods [14] and resonant mode exploration [15], [16]. In addition, the nonlinearity that emerges from the size effect and mode coupling is usually considered undesirable for resonators; however, recently, parametric feedback [17], parametric symmetry breaking [18], phase feedback [19], bifurcation topology [20], antisymmetric vibration [21], modal localization [22], and synchronization [23] have been found to be beneficial for the stability of resonators.
Among these nonlinear effects, Huygens first discovered synchronization in coupled pendulums in the 17th century. A Dutch mathematician invented the first pendulum clock in history and improved time measurements by reducing the loss of time by clocks from approximately 15 min to approximately 15 s per day. His observation of synchronization in coupled pendulums inspired many studies on the sympathetic rhythms of interacting nonlinear oscillators in many areas of science [24]. The first inter-synchronization of two MEMS oscillators via electrostatic coupling was achieved by Agrawal et al. [25]. Many studies have demonstrated the advantages of synchronization, such as reduced phase noise and enhanced frequency stability [26]. When one of the coupled oscillators is perturbed by an external signal, the frequency of the other oscillator synchronously changes with the perturbed oscillator frequency. This has been widely utilized in frequency sensing research, such as superharmonic synchronization [27], [28] and subharmonic synchronization [29].
Here, we constructed a MEMS Huygens clock with two micromechanical resonators, in which the oscillation signals of the two resonators were injected into each other, and one of the oscillator frequencies was adjusted to achieve synchronization. The short-term stability of the two resonators is improved after synchronization, improving the Allan deviation by a factor of 3.73 from 19.3 to 5.17 ppb at 1 s. In addition, a Joule heating-based frequency compensation system, which changes the current of the resonator to control the Joule heat, was utilized to compensate for the frequency shift of the MEMS oscillator caused by the induced temperature. Unlike the high-power heat source utilized in the previous active compensation, this compensation system effectively enhances the long-term stability of the oscillator at 2.85 mW∙°C−1, improving the Allan variance at 6000 s by 452 times based on the fuzzy proportion integration differentiation (PID) control. By compensating for one oscillator of the MEMS Huygens clock, the long-term stability of the other oscillator was synchronously enhanced, and the MEMS Huygens clock was much more stable than an independent compensated oscillator based on the synchronization effect, with a 1.6343 × 105 times improvement in Allan deviation at 6000 s. Compared with the micro-oven control method adopted by Ortiz et al. [30], our experimental results provide better long-term stability when tested at 25 °C. This study enriches the synchronization application in MEMS and provides a novel and precise engineering solution for achieving high-precision MEMS oscillators.
2. Working principle of MEMS Huygens clock
The original Huygens clock comprises two pendulums that conduct mechanical vibrations in both directions, coupled via a single beam. Accordingly, the MEMS Huygens clock described here has two MEMS oscillators that conduct two unidirectional electrical signals, as illustrated in Fig. 1(a).
The MEMS Huygens clock has two oscillator systems and two electrical signal injection paths, as illustrated in Fig. 1(b). The two disk resonators achieve closed-loop oscillations via their respective phase-locked loops to form two MEMS oscillator systems. The closed-loop signal from the active oscillator (Sync 1) is injected into the passive oscillator through Comparator 2. The closed-loop signal from the passive oscillator (Sync 2) is injected into the active oscillator in the same manner as in Comparator 1. Owing to processing errors, the frequencies of the two oscillators are different, and the resonance frequency of the passive oscillator is thermally tuned by regulating the direct current (DC) voltage. When the frequency difference is within the synchronous bandwidth, the two frequencies will be close to each other and finally become the same.
Two edge-anchored disk MEMS resonators with identical structural parameters were utilized to ensure frequency matching. The resonator structure is disc-type, supported by four anchors. The resonator's diameter and thickness are 750 and 40 μm, respectively. The resonator was surrounded by 12 capacitive electrodes to drive and detect resonator vibrations. The device was fabricated with standard silicon-on-glass technology. The operating mode of the resonator is illustrated in Fig. 1(c). The microfabricated MEMS resonator after the wire bonding is illustrated in Fig. 1(d).
The MEMS resonator is packaged on a ceramic chip. The entire device is operated in a vacuum environment (0.1 Pa) and at room temperature (25 °C). The bias voltages and regulation DC were generated by low-noise voltage sources (Keithley 2400, Tektronix, USA and Keysight B2961A, Keysight Technologies, USA). The drive and synchronization signals (Syncs 1 and 2) were provided by a two-channel lock-in amplifier (HF2LI, Zürich Instruments, Switzerland).
The resonator motion was detected with a capacitive detection scheme. In the experiments, a passive oscillator with a frequency of 2.6743 MHz and an active oscillator with a frequency of 2.6721 MHz were actuated and sensed with 100-V DC biasing and 1-V alternating current (AC) voltage excitement. In addition, synchronization signals Syncs 1 and 2 were set to a strength of 50 mV.
The synchronization process is affected by synchronization interference and excitation forces [31]. To visually demonstrate the synchronization process of the MEMS Huygens clock, Syncs 1 and 2 were set differently, and the frequency of the passive oscillator was adjusted by regulating the DC to observe the different phenomena when the frequencies of the two oscillators were close, as demonstrated in Fig. 2. Without the loaded synchronization signal, the two oscillators are independent regardless of whether their frequencies are close to or away from each other, as illustrated in Fig. 2(a). When only Sync 2 was loaded, the active oscillator stability was destroyed owing to the Sync 2 disturbance, and the active oscillator frequency tracked the passive oscillator frequency when the frequency difference was less than the synchronous bandwidth, as illustrated in Fig. 2(b). When Syncs 1 and 2 are loaded, the stability of both oscillators is destroyed by signal disturbances, and their frequencies track each other when the frequency difference is less than the synchronous bandwidth, as illustrated in Fig. 2(c).
Owing to the synchronization effect, when one of the coupled oscillators is perturbed by an external signal, the frequency of the other oscillator changes synchronously with the perturbed oscillator frequency. Here, each oscillator of the Huygens clock is driven by both the excitation force of the closed-loop feedback and the synchronous perturbation force. The synchronous perturbation force can be modeled as a linear term in normalized form Ref. [32]. The spectral responses of the active and passive oscillators during a typical synchronization process are presented in Fig. 3(a). Utilizing the electrical coupling method, a sidelobe was observed in the spectra of both resonators, which disappeared when the frequencies of the two oscillators were synchronized.
From the above analysis, we know that the synchronization effect is beneficial for improving the short-term stability of the oscillator but not the long-term stability. Factors affecting the resonator frequency shift include changes in temperature, acceleration, and aging. The combination of these factors causes the oscillator’s stability to decrease as time increases. A Joule-heating-based frequency compensation system was built to adjust the oscillator frequency shift, as illustrated in Fig. 4(a). The system comprises a comparator, a frequency counter (constructed from the field programmable gate array (FPGA) program module), a frequency-voltage calculation system (executed by the STM32 program), a digital to analog converter (DAC), and an oscillator body resistor. When the frequency counter detects a frequency change in the oscillator, the frequency-voltage calculation system accurately calculates the required drain current and adjusts the frequency shift by changing the current to control the Joule heat induced by the DAC. The temperature-voltage relationship of the resonator was simulated, as demonstrated in Fig. 4(b).
The simulation results reveal that the temperature (T) is uniformly distributed throughout the resonator and can be expressed as follows:
where is the ambient temperature; is the drain current passing through the resonator body; is the thermal dissipation coefficient of the thermal diffusivity of material and the structure; is the support beam resistance; is the disc resistance; is the length of the beam; is the resistivity; is the resonator thickness; is the circle radius, and is the width of the connecting beam and that of the connection between the connecting beam and the disc.
Increasing the temperature changed the Young’s modulus of the resonator, resulting in a change in the frequency.
where is the modal constant; is the Young’s modulus at the ambient temperature; is the density; is the Poisson’s ratio; is the temperature variation. is the variation in Young’s modulus with temperature.
The frequency of a resonator is affected by complex environmental factors. Therefore, a frequency compensation system requires fast frequency retuning under shock and high-frequency stability in different external environments. We compared the differences between the PID and fuzzy PID in the experiments.
Therefore, taking frequency error as the input of the system and voltage as the output, the classical transfer function of the PID can be expressed as
where is the proportionality coefficient; is the integration coefficient; is the differentiation coefficient, and is the Laplacian variable.
The fuzzy PID is constructed based on conventional PID control theory and the rule of adaptive fuzzy control, which can tune the controller parameters automatically. , , and are modified by error and the change rate of error , which changes before each calculation, as demonstrated in Fig. 5(a).
where , , and are the fuzzy PID parameters for the previous period. , , and are inference parameters calculated from the and by fuzzy rules. , , and are the revised fuzzy PID parameters.
Triangular membership functions [33] were selected as membership functions of the fuzzy variables. A general fuzzy rule [34] based on engineering design was adopted here. Table 1 lists the parameter ranges for the basic theory domain, fuzzy subset, and scale factor of the fuzzy PID control variables.
The oscillator was set in three cases: without compensation, with PID control, and with fuzzy PID control for comparison experiments at room temperature (25 °C). In the 100 min test, the frequency shift of the uncompensated oscillator was more than 2 ppm, the frequency drift of the oscillator under PID control was more than ±0.15 ppm, and the fuzzy PID-controlled oscillator achieved the best experimental results with a frequency drift of less than ±0.1 ppm, as illustrated in Fig. 5(b). Fig. 5(c) presents the Allan deviation of the oscillation frequency with a sampled frequency of 28 Hz for the unconstrained, PID control, and fuzzy PID control cases. The oscillator under fuzzy PID control exhibits optimal long-term stability, with the Allan deviation improved to 0.752 ppb at 300 s (compared with 9.51 ppb under PID control and 340 ppb without compensation), as illustrated in Fig. 5(c).
4. Frequency compensated MEMS Huygens clock
The synchronization effect enhances the short-term stability of the oscillators. In contrast, the oscillators’ frequencies track each other after synchronization, which implies that when one oscillator is frequency-compensated, the stability of the coupled oscillator will be improved synchronously. Therefore, in the proposed MEMS Huygens clock, the active oscillator is connected to a temperature-compensation system to improve the long-term stability of the overall synchronization system. The passive oscillator is responsible for achieving synchronization by manually adjusting the DC voltage, as illustrated in Fig. 1(b). The system operates in a sequence where oscillators first achieve synchronization before executing frequency compensation. Fig. 6(a) presents a schematic of the experimental setup of the MEMS Huygens clock combined with the Joule heating-based frequency compensation system.
The results of the 2 h test reveal that the two oscillators of the MEMS Huygens clock are strongly synchronized, and their frequency drift is less than ±100 ppb at room temperature, as demonstrated in Fig. 6(b). Fig. 6(c) presents the Allan deviation of the oscillation frequency with a sampled frequency of 28 Hz for the different cases. The stability of synchronized oscillators with a frequency compensation system is improved, and the long-term stability is further improved compared to before synchronization with an Allan deviation of 30.9 ppt at 6000 s (compared with 5.05 ppm under un-synced, 5.10 ppm under synced, and 3.99 ppb under compensated).
5. Conclusion
The oscillator is the timing reference of electronic devices utilized for operation, and the accuracy of the oscillator will directly affect the performance of precision instruments. MEMS oscillators, which are expected to replace quartz crystal oscillators, face two key challenges: short- and long-term stability. This study evaluated a MEMS Huygens clock combined with a frequency compensation system. The MEMS Huygens clock is developed by synchronizing two micromechanical resonators via electrical signal injection, leading to an improvement of its frequency’s Allan deviation by 3.73 times from 19.3 to 5.17 ppb at 1 s. A Joule heating-based frequency compensation system was built to compensate for the frequency shift of one oscillator of the MEMS Huygens clock. The stability of the coupled oscillator was improved based on the synchronization effect, which improved the Allan deviation by 1.6343 × 105 times from 5.05 ppm to 30.9 ppt at 6000 s. This is the best result of MEMS oscillators reported to date. This study provides a novel, precise engineering solution for achieving a high-precision MEMS clock. The development of MEMS-IC technology is promising for creating a full-chip MEMS Huygens clock that incorporates a frequency compensation system into the chip.
Acknowledgments
This work was financially supported by the National Key Research and Development Program of China (2022YFB3203600), the National Natural Science Foundation of China (52075432), and the Program for Innovation Team of Shaanxi Province (2021TD-23).
Compliance with ethics guidelines
Xueyong Wei, Mingke Xu, Qiqi Yang, Liu Xu, Yonghong Qi, Ziming Ren, Juan Ren, Ronghua Huan, and Zhuangde Jiang declare that they have no conflict of interest or financial conflicts to disclose.
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