This paper introduces a new type of multi-degree-of-freedom cascaded hybrid electro-mechanical resonator system with ultra-high sensitivity, enhanced measurement range, and high signal-to-noise ratio, which can be used for many diverse applications. The structure is introduced in detail, and a full theoretical analysis of its performance in terms of sensitivity, noise floor, and measurement range is provided. To validate the theoretical analysis, FPGA implementations of 3- and 5-DoF prototype systems are verified with a QCM resonator and a DETF resonator to detect both mass and stiffness change, showing a high consistency between measurement results and theoretical calculations. By cascading multiple digital IIR resonators with a mechanical resonator and applying small coupling factors (1/1024 for the 3-DoF system and 1/64 for the 5-DoF system) in the digital domain, the normalized sensitivity of the proposed system is 10 times higher than the state-of-the-art weakly-coupled resonator systems. The measurement results confirm that the proposed system achieves an ultra-high normalized sensitivity of about 524000 (3-DoF) and 3145000 (5-DoF), respectively. Due to the high compatibility with different types of mechanical resonators and its high tunability, the proposed system has the potential to be a general solution towards applications requiring ultra-high sensitivity. And because of its cascaded structure, the proposed system can also be implemented as a closed-loop system, like a single mechanical resonator does.
Due to their small size, low cost and mass production properties, microelectromechanical systems (MEMS) play important roles in many fields, including gas sensing, inertial sensing, force sensing and mass identification, among many others. Resonant sensing devices are widely used among all miniature sensors due to their high accuracy and quasi-digital output signals. The most commonly used MEMS resonator sensor measures mass/stiffness changes on a single resonator: when an external mass perturbation on the resonator's proof mass, with the stiffness remaining the same, leads to a detectable frequency shift. The same happens when the stiffness changes and the mass remains the same. This is a single-degree-of-freedom (1-DoF) system since only one mode exists. A major problem of the single-DoF resonator is its limited sensitivity since only the frequency shift can be used as an output metric, which has a normalized sensitivity of only 0.5. To satisfy the requirement of sensing a smaller mass/force change, coupled resonators utilizing the mode localization effect have been investigated. By coupling two identical resonators with a spring with a stiffness that is much weaker than the suspension stiffness of the resonators, a 2-DoF system is formed. This shows an order of magnitude improvement in sensitivity by measuring a new output metric: the displacement amplitude ratio of the two resonators. Various research studies have been carried out for this type of resonators to find better mass/force sensors, including 3/4-DoF weakly coupled resonators (WCR) that couple 3 or 4 identical resonators together. These published works managed to improve the sensitivity to more than 10000 times higher than the conventional single-DoF approach, while other advantages such as an intrinsic common-mode rejection to ambient effects (e.g. the stiffness changes due to temperature, pressure drifts) are also reported.
However, several problems still exist in the conventional mechanically coupled resonator systems: One known challenge is the intrinsic limitations of the fabrication process used for MEMS devices. Mismatches of the MEMS during the fabrication process give inherent mass/stiffness perturbations to the system, which are difficult or nearly impossible to tune after fabrication, resulting in a performance degradation of the sensor. To mitigate the influence of the fabrication mismatch on the performance of the mechanically coupled resonator system, a novel micro-lever coupler structure is introduced in ref. . The energy dissipated on the anchor is substantially attenuated by applying the novel coupler structure, which makes the error value of the coupling stiffness stay within 1%. Another issue for high-sensitivity weakly coupled resonators is that the conventional mechanically coupled resonators have a limited signal-to-noise ratio (SNR) due to their mechanical noise and the electrical noise. Both refs. and reported that the maximum amplitude ratio value is restricted by the SNR of the system, which limits the dynamic range of the system. Several studies have investigated how to increase the dynamic range and improve the noise performance of the WCR systems. Wang et al. reported a force rebalanced closed-loop control scheme for a mode-localized MEMS accelerometer. By introducing an electrostatic force as the compensation for the input acceleration, the amplitude ratio between two resonators remains a constant value, the measurement range is reported to be expanded several times, and the minimum input-referred acceleration noise is significantly reduced compared to the conventional closed-loop scheme. Wang et al. suggested a decouple-decomposition (DD) noise analysis model to help analyze the amplitude noise performance in a closed-loop WCR system, and reported a low input-referred noise. Besides, the sensitivity of the WCR system heavily depends on the stiffness of the coupler, weak coupling is needed for higher sensitivity. However, if the coupling between resonators is very weak, different modes overlap, which is called mode aliasing, resulting in nonlinear behavior of the amplitude ratio output. Zhao et al. introduced an intentional stiffness perturbation to their ultra-high-sensitivity 3-DoF coupled resonator system to shift the initial condition of their system in the desired linear region, so that the dynamic range and linearity of the system are improved compared to the case where the initial condition makes the system suffer from the mode aliasing effect. Zhu et al. established a linear model for a multi-degree-of-freedom WCR system, which points out that, instead of making a weaker coupler, the sensitivity of the amplitude ratio with respect to the input perturbation can be exponentially amplified by coupling multiple resonators in one system.
While several challenges mentioned above are still under investigation in purely mechanical WCR systems, research on hybrid coupled resonator systems appeared. In a hybrid coupled resonator, or virtually coupled resonator system, one or more mechanical resonators from the WCR system are coupled with electrical resonators, and the physical coupler is also replaced with electrical coupling, as shown in Fig. 1a. The electrical resonators, either analog or digital resonators, can be tuned easily. Thus, the fabrication mismatches and defects of the mechanical resonators can precisely be compensated for by tuning the electrical resonator. The electrical coupler can also avoid the influence of the fabrication defects and can have a very small equivalent coupling stiffness, which results in a very high sensitivity. Kasai et al. couple a real microcantilever with an analog virtual cantilever, which shows an enhanced amplitude ratio sensitivity of about 10 times higher than the eigenfrequency shift. The virtual cantilever shows good tunability. The problems of Kasai et al.'s system are that the quality factor (Q factor) of the analog cantilever is low due to the limitations of electrical components and the optical measurement used for the displacement of the microcantilever, in which case, a compact high-sensitivity WCR is hard to build. Humbert et al. present a 2-DoF hybrid WCR system that couples a QCM resonator with a tunable virtual resonator simulated by digital circuits on an FPGA. The 2-DoF hybrid WCR system has an ultra-high normalized sensitivity of 14000. However, this 2-DoF hybrid WCR system needs to be balanced before every mass perturbation test, and the analog-to-digital converters (ADC), digital-to-analog converters (DAC) and digital signal processing (DSP) introduce extra delay into the system, which needs to be compensated carefully. Hence, only open-loop measurements are contained in Humbert et al.'s paper. Moreover, several methods to improve the performance of the mode-localized resonator system, such as closed-loop control, closed-loop noise optimization, and multi-DoF coupled structures, have not been investigated in a hybrid weakly-coupled resonator system yet, allowing for more improvements in the performance of the hybrid weakly-coupled resonator system.
The proposed hybrid system and its electrical equivalent
This work introduces a new type of hybrid electro-mechanical cascaded resonator system with a normalized sensitivity that is at least 10 times higher than all published works. The system-level schematic of the proposed new hybrid resonator system is shown in Fig. 1b. A mechanical resonator is driven by an external source through an actuation circuit; the mechanical resonator's output is converted into electrical signals by the analog readout circuit. This analog electrical signal is then digitized by an ADC to generate a digital input for the remaining part of the proposed system in the digital domain. The quantized output signals of the mechanical resonator Res and the electrical resonator Res are again converted into analog signals through a DAC to send them into the signal analyzer for measurement. As shown in the figure, the mechanical resonator is connected to an arbitrary number of digital resonators in cascade via so-called coupling blocks (with digitally tunable amplification factor ), making the system a hybrid cascaded structure without feedback to the mechanical domain. The mass/stiffness perturbation to be measured is applied to the mechanical resonator, while the output metric is the amplitude ratio between the output signal of the mechanical resonator Res and the last electrical resonator Res This structure guarantees that the mechanical resonator works as an individual resonator; thus any delay introduced by the data conversion and digital blocks does not influence the normal operation of the mechanical resonator, which avoids the problem of the delay in the closed-loop structure as mentioned in refs. . Several methods to improve the performance of weakly-coupled resonator systems, as mentioned in refs. can also be applied to the proposed hybrid electro-mechanical resonator system in future work.
For a detailed analysis of the proposed structure, an electrical equivalent model of the proposed system is shown in Fig. 1c. The mechanical resonator Res is described as an RLC resonator, with transfer function:
where
is the proof mass, is the stiffness, and is the damping factor of the mechanical resonator, while is the conversion ratio from the mechanical displacement into an electrical current.
The electrical resonators are designed to have the same style of transfer function as the mechanical resonator, as given in Eq. (1). However, the signal is converted into the digital domain for easy signal processing. Hence, digital filters with an infinite impulse response (IIR) are used to equivalently and precisely simulate the behavior of the mechanical resonator. A second-order IIR filter as the i-th electrical resonator is built in the z-domain by applying the pre-warp bilinear transform to Eq. (1):
where
Here, represents the complex frequency, which includes the resonant frequencies and the clock period of the digital system according to
Since in general n resonators are located in cascade in the proposed system, this hybrid resonator system is an n-DoF system. All coefficients of the IIR filter shown in Fig. 1b can be tuned or selected by the user according to the actual parameters of the mechanical resonator. Without a feedback signal going back to the mechanical resonator, which influences its working condition, the proposed structure is compatible with many different types of mechanical resonators.
The paper is organized as follows. This "Introduction" section has given general information about the proposed system. Next, the sections "Theoretical analysis" and "Experimental results" will give a detailed analysis of the performance of the system as well as the experimental results. The section "Discussion" will present the advantages of the proposed structure over other published weakly coupled resonators, both purely mechanical and hybrid electro-mechanical designs. The section "Materials and methods" will describe the devices and setup used for the experiments. The section "Conclusion" will summarize the contributions of this work and briefly introduce future work.