A novel multi-degree-of-freedom cascaded hybrid electro-mechanical resonator system with ultra-high sensitivity

Abstract

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.

Introduction

Background and motivation

Due to their small size, low cost and mass production properties, microelectromechanical systems (MEMS) play important roles in many fields¹, including gas sensing², inertial sensing³, force sensing⁴ and mass identification⁵, among many others. Resonant sensing devices are widely used among all miniature sensors due to their high accuracy and quasi-digital output signals⁶. 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 exists⁷. 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 0.5⁴. To satisfy the requirement of sensing a smaller mass/force change, coupled resonators utilizing the mode localization effect have been investigated⁸. 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^9,10,11. Various research studies have been carried out for this type of resonators to find better mass/force sensors^4,5,12,13,14, including 3/4-DoF weakly coupled resonators (WCR) that couple 3 or 4 identical resonators together⁴. 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 reported¹¹.

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. ¹⁵. 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. ⁴ and¹⁵ 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 scheme¹⁶. 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 noise¹⁷. Besides, the sensitivity of the WCR system heavily depends on the stiffness of the coupler^18,19, 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 effect⁴. 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 system²⁰.

While several challenges mentioned above are still under investigation in purely mechanical WCR systems, research on hybrid coupled resonator systems appeared^{21,22,23,24,25}. 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 tunability²¹. 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 14000^22,23,24. 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.

Fig. 1: System-level structure of the proposed multi-DoF cascaded hybrid electro-mechanical resonator system.

a Mass-stiffness-damper model of mechanically coupled WCRs and hybrid coupled WCRs. b Schematic of the proposed hybrid electro-mechanical cascaded resonator system: Res₁ is a mechanical resonator, while Res₂ to Res_n are electrical resonators implemented as IIR filters. c The electrical equivalent of the proposed cascaded hybrid electro-mechanical resonator system; here, all resonators are described as RLC resonators for the theoretical analysis

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 Res₁ and the electrical resonator Res_n 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 ${K}_{{ci}}$), 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 Res₁ and the last electrical resonator Res_n. 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. ^23,24. Several methods to improve the performance of weakly-coupled resonator systems, as mentioned in refs. ^16,17 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 Res₁ is described as an RLC resonator²⁶, with transfer function:

$$H\left(s\right)=\frac{1}{{L}_{1}{s}^{2}+{R}_{1}s+\frac{1}{{C}_{1}}}$$

(1)

where

$$\left\{\begin{array}{l}{L}_{1}={\mu }_{c}{M}_{1}\\ {C}_{1}=\frac{1}{{\mu }_{c}{K}_{1}}\\ {R}_{1}={\mu }_{C}{\eta }_{1}\end{array}\right.$$

(2)

${M}_{1}$ is the proof mass, ${K}_{1}$ is the stiffness, and ${{\rm{\eta }}}_{1}$ is the damping factor of the mechanical resonator, while ${{\rm{\mu }}}_{c}$ 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)²⁷:

$${H}_{i}\left(z\right)=\frac{{a}_{2,i}{z}^{2}+{a}_{1,i}z+{a}_{0,i}}{{b}_{2,i}{z}^{2}+{b}_{1,i}z+{b}_{0,i}}$$

(3)

where

$$\left\{\begin{array}{c}{a}_{2,i}=\frac{1}{{\mu }_{c}\left({M}_{1}{p}^{2}+{\eta }_{1}p+{K}_{1}\right)}\\ {a}_{1,i}=\frac{2}{{\mu }_{c}\left({M}_{1}{p}^{2}+{\eta }_{1}p+{K}_{1}\right)}\\ {a}_{0,i}=\frac{1}{{\mu }_{c}\left({M}_{1}{p}^{2}+{\eta }_{1}p+{K}_{1}\right)}\\ {b}_{2,i}=1\\ {b}_{1,i}=\frac{-2{M}_{1}{p}^{2}+2{K}_{1}}{{M}_{1}{p}^{2}+{\eta }_{1}p+{K}_{1}}\\ {b}_{0,i}=\frac{{M}_{1}{p}^{2}-{\eta }_{1}p+{K}_{1}}{{M}_{1}{p}^{2}+{\eta }_{1}p+{K}_{1}}\end{array}\right.$$

(4)

Here, $p$ represents the complex frequency, which includes the resonant frequencies ${{\rm{\omega }}}_{0,i}=\frac{1}{\sqrt{{L}_{i}{C}_{i}}}$ and the clock period of the digital system ${T}_{s}=\frac{1}{{f}_{s}}$ according to

$$p=\frac{{\omega }_{0,i}}{\tan \left(\frac{{\omega }_{0,i}\cdot {T}_{s}}{2}\right)}=\frac{1}{\tan \left(\frac{\pi \cdot {f}_{0,i}}{{f}_{s}}\right)}$$

(5)

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.

Theoretical analysis

Output metrics and sensitivity analysis

Since the system is used for mass/stiffness perturbation, the sensing input is applied to the mechanical resonator $Re{s}_{1}$, and the relationship between the input $\Delta u$ ($u={M\; or\; K}$) and the output amplitude ratio $A{R}_{1/n}$ between the amplitude of the output signal of $Re{s}_{1}$ and $Re{s}_{n}$ is analyzed as the output metric.

Amplitude ratio derivation

As described in section 1, the output metric for the measurement is the amplitude ratio between the output signal of the mechanical resonator $Re{s}_{1}$ and that of the last electrical resonator $Re{s}_{n}$, which is

$$A{R}_{1/n}=\frac{\left|{Am}{p}_{1}\right|}{\left|{Am}{p}_{n}\right|}$$

(6)

All signals are described in the Laplace domain (s-domain), where $s=j{\rm{\omega }}$. Assuming that an input excitation signal ${x}_{{in}}\left(s\right)$ is applied by an external source, as shown in Fig. 1c, the amplitudes ${Am}{p}_{1}$ and ${Am}{p}_{n}$ can be described as

$$\left\{\begin{array}{l}{Am}{p}_{1}\left(s\right)={H}_{1}\left(s\right)\cdot {G}_{e}\cdot {x}_{{in}}\left(s\right)\\ {Am}{p}_{n}\left(s\right)={K}_{c}^{n-1}\cdot {\prod }_{i=1}^{n}{H}_{i}\left(s\right)\cdot {G}_{e}\cdot {x}_{{in}}\left(s\right)\end{array}\right.$$

(7)

where ${H}_{i}\left(s\right)$ is the transfer function of the i-th resonator described in Eq. (1), and ${K}_{c}$ is the coupling factor mentioned in section 1 (assuming that all ${K}_{{ci}}$ are the same to make the calculation easier), ${G}_{e}$ is the gain of the electrical interface. As a result, the amplitude ratio $A{R}_{1/n}$ is determined as

$$A{R}_{1/n}=\frac{\left|{Am}{p}_{1}\right|}{\left|{Am}{p}_{n}\right|}=\frac{1}{\left|{K}_{c}^{n-1}\cdot {\prod }_{i=2}^{n}{H}_{i}\left(s\right)\right|}$$

(8)

From Eq. (7), it can be concluded that the output signal of $Re{s}_{1}$ has only one resonant frequency ${{\rm{\omega }}}_{1}=\sqrt{\frac{{K}_{1}}{{M}_{1}}}=\frac{1}{\sqrt{{L}_{1}{C}_{1}}}$, which is called the ${1}^{{st}}$ mode, while the n-th resonator can have n resonant frequencies: ${{\rm{\omega }}}_{i}=\frac{1}{\sqrt{{L}_{i}{C}_{i}}}$ (i = 1, 2,…, n). Each possible resonating frequency corresponds to one resonator’s resonant frequency, so there are n modes for the output signal of $Re{s}_{n}$. Assuming that all resonators follow the relationship (note that the electrical resonators’ parameters can be chosen or tuned by the user, such that this assumption holds):

$$\left\{\begin{array}{c}{L}_{1}{C}_{1}={{\rm{\alpha }}}_{2}\cdot {L}_{2}{C}_{2}={{\rm{\alpha }}}_{3}\cdot {L}_{3}{C}_{3}=\ldots ={L}_{n}{C}_{n}=\frac{{M}_{1}}{{K}_{1}}\\ {R}_{1}={R}_{2}={R}_{3}=\ldots ={R}_{n}={{\rm{\mu }}}_{c}{{\rm{\eta }}}_{1}\end{array}\right.$$

(9)

where ${{\rm{\alpha }}}_{i} > 1$ ($i=2,\,3,\,...,{n}-1$) is a factor to make the ${2}^{{nd}}$ to ${\left(n-1\right)}^{{th}}$ modes be located away from the ${1}^{{st}}$ mode and the ${n}^{{th}}$ mode, while the ${1}^{{st}}$ and the ${n}^{{th}}$ mode are overlapping at ${{\rm{\omega }}}_{1}=\frac{1}{\sqrt{{L}_{1}{C}_{1}}}$. Usually, the perturbation applied to the mechanical resonator is only within a few percent of the nominal mass/stiffness of $Re{s}_{1}$. Therefore, the frequency range of interest is around ${{\rm{\omega }}}_{1}$, which doesn’t include the ${2}^{{nd}}$ mode to the ${\left(n-1\right)}^{{th}}$ mode. When a small mass/stiffness perturbation is added to $Re{s}_{1}$, the ${1}^{{st}}$ mode moves away from ${{\rm{\omega }}}_{1}$. The other resonators keep their parameters, thus the overlapping ${1}^{{st}}$ mode and ${n}^{{th}}$ mode are split, one following the change of $Re{s}_{{1}}$ and the other one staying at ${{\rm{\omega }}}_{1}$. Assuming that a mass perturbation $\Delta {M}_{1}$ ($\Delta {L}_{1}$ in the RLC electrical equivalent) is present on the mechanical resonator $Re{s}_{1}$, then the resonating frequencies of the two modes of interest are

$$\left\{\begin{array}{c}{{\rm{\omega }}}_{1{st}}=\frac{1}{\sqrt{\left({L}_{1}+\Delta {L}_{1}\right){C}_{1}}}\\ {{\rm{\omega }}}_{{nth}}=\frac{1}{\sqrt{{L}_{1}{C}_{1}}}\end{array}\right.$$

(10)

Since $Re{s}_{1}$ has only the ${1}^{{st}}$ mode, the output metric to be measured is therefore the amplitude ratio of the ${1}^{{st}}$ mode between $Re{s}_{1}$ and $Re{s}_{n}$. Inserting Eq. (10) into Eq. (8), the amplitude ratio of the ${1}^{{st}}$ mode is

$$A{R}_{1/n,1{st}}=\left|\frac{1}{{K}_{c}^{n-1}}\cdot \left(\mathop{\prod }\limits_{i=2}^{n-1}\left(\frac{1}{{{\rm{\alpha }}}_{i}}\cdot \frac{{{\rm{\alpha }}}_{i}\left(1+\frac{\Delta {L}_{1}}{{L}_{1}}\right)-1}{1+\frac{\Delta {L}_{1}}{{L}_{1}}}+\frac{j}{Q}\right)\right)\cdot \left(\frac{\frac{\Delta {L}_{1}}{{L}_{1}}}{1+\frac{\Delta {L}_{1}}{{L}_{1}}}+\frac{j}{Q}\right)\right|$$

(11)

where $Q=\frac{1}{{R}_{1}}\cdot \sqrt{\frac{{L}_{1}+\Delta L}{{C}_{1}}}$ is the perturbed quality factor of $Re{s}_{1}$. Assuming that only a small perturbation is present and that the quality factor of $Re{s}_{1}$ is large enough, which means that $\frac{\Delta {L}_{1}}{{L}_{1}}\ll 1$ and ${Q}_{1}\gg 1$, Eq. (11) can be approximated as

$$A{R}_{1/n,1{st}}\approx \left|\frac{1}{{K}_{c}^{n-1}}\cdot \left(\mathop{\prod }\limits_{i=2}^{n-1}\left(\frac{{{\rm{\alpha }}}_{i}-1}{{{\rm{\alpha }}}_{i}}\right)\right)\cdot \frac{\Delta {L}_{1}}{{L}_{1}}\right|$$

(12)

Normalized sensitivity analysis

It can be seen from Eq. (12) that the amplitude ratio of the ${1}^{{st}}$ mode is approximately a linear function of the normalized mass perturbation $\frac{\Delta {L}_{1}}{{L}_{1}}$ (equivalent to $\frac{\Delta {M}_{1}}{{M}_{1}}$). If a stiffness perturbation $\frac{\Delta {K}_{1}}{{K}_{1}}$ were applied to $Re{s}_{1}$, the amplitude ratio of the first mode has the same linear function of this normalized stiffness perturbation $\frac{\Delta {K}_{1}}{{K}_{1}}$. Hence, the normalized sensitivity of the n-DoF system is

$${Sensitivit}{y}_{{normalized},\frac{\Delta u}{u}}\approx \frac{1}{{K}_{c}^{n-1}}\cdot \left(\mathop{\prod }\limits_{i=2}^{n-1}\left(\frac{{{\rm{\alpha }}}_{i}-1}{{{\rm{\alpha }}}_{i}}\right)\right)$$

(13)

where $u$ can either be ${M}_{1}$ or ${K}_{1}$. It can be concluded that the normalized sensitivity of the cascaded hybrid resonating sensor depends on the coupling factor ${K}_{c}$, the factors ${{\rm{\alpha }}}_{i}$ ($i=2,\,3,\,...,{n}-1$) and the number of resonators n, which determines the DoF. To get a higher sensitivity, a weaker coupling (smaller ${K}_{c}$) is needed, and the ${{\rm{\alpha }}}_{i}$ should be much larger than 1, which means that the modes $2$ to$\,(n-1)$ should be far away from the ${1}^{{st}}$ and ${n}^{{th}}$ modes. Moreover, a higher DoF order (i.e. a larger n) has a higher sensitivity if the other parameters are fixed, which again illustrates why a multi-DoF system is preferred.

System-level simulation

An electrical equivalent simulation has been carried out to show the behavior of the proposed sensing system. Mass perturbations are applied to the mechanical resonator $Re{s}_{1}$ of an illustrative proposed 3-DoF hybrid resonator system with ω₁≈5 MHz, ${{\rm{\alpha }}}_{2}=2$, ${Q}_{1}\,=\,3000$, respectively. Figure 2a and Fig. 2b show the output frequency response of $Re{s}_{1}$ and $Re{s}_{3}$ when different mass changes are applied to $Re{s}_{1}$. With increasing perturbation, the electrical resonator $Re{s}_{3}$ follows the frequency change of the mechanical resonator $Re{s}_{1}$ on the ${1}^{{st}}$ mode, while the frequency difference between the ${1}^{{st}}$ mode and the ${3}^{{rd}}$ mode increases, and the amplitude ratio between $Re{s}_{1}$ and $Re{s}_{3}$ becomes larger.

Fig. 2: Simulations of the proposed 3-DoF hybrid electro-mechanical resonator system.

Frequency response of the RLC equivalent circuit of the proposed system with coupling factor ${K}_{C}=0.01$ for different mass perturbations: (a) $\Delta {M}_{1}/{M}_{1}=0.1 \%$ and (b) $\Delta {M}_{1}/{M}_{1}=0.5 \%$ (Here only two nearby modes, the ${1}^{{st}}$ and ${3}^{{rd}}$ modes, are plotted). c Amplitude ratio at the 1st mode as a function of the relative mass perturbation for different Kc values. d Relationship between the normalized sensitivity and ${K}_{c}$

A simulation of the amplitude ratio between the output signal of $Re{s}_{1}$ and $Re{s}_{n}$ at the ${1}^{{st}}$ mode as a function of the mass perturbation is now performed for ${K}_{c}=1/100,\,1/150,\,1/200$. The results in Fig. 2c show that the amplitude ratio is linearly proportional to the positive/negative mass perturbation. The relationship between the normalized sensitivity and the coupling factor ${K}_{c}$ is shown in Fig. 2d: the normalized sensitivity is inversely proportional to ${K}_{c}$, which is consistent with the calculation in Eq. (13). For a digital implementation with sufficient bit-width and running speed, the sensitivity can achieve arbitrarily large value by choosing a very small ${K}_{c}$, which is not practical in purely MEMS coupled resonators due to the finite Q factor (the coupling ratio ${{K}}_{c}$ should be larger than $4/Q$ in purely MEMS coupled resonators)²³.

When the perturbation is too small, the split modes are still merged in each other due to the finite quality factor Q caused by the damping factor ${\rm{\eta }}$. This is called mode aliasing (see Fig. 2a), and the relationship between the amplitude ratio and input perturbation becomes nonlinear in this region. To clearly distinguish the two modes and to avoid nonlinear behavior during measuring, a minimum frequency difference $\Delta f$ is needed⁴:

$$\Delta f=2\times \Delta {f}_{3{dB}}$$

(14)

where $\Delta {f}_{3{dB}}=\frac{{f}_{i}}{2{Q}_{i}}$ is the 3-dB bandwidth of the i-th mode, and ${{\rm{\omega }}}_{i}$ and ${Q}_{i}$ are the resonant frequency and the quality factor of the i-th mode, respectively (here we focus on the ${1}^{{st}}$ mode). In order to measure the amplitude ratio as a linear function of the input perturbation, this minimum frequency difference condition should always be satisfied. To achieve this, an intentional frequency bias (difference) larger than $\Delta f$ is to be applied to $Re{s}_{n}$ such that the measurement can guarantee that the system works linearly. The frequency bias is applied by tuning the parameters of the digital resonators to make the ${n}^{{th}}$ mode be out of the 3-dB bandwidth of the ${1}^{{st}}$ mode.

Noise analysis

Another important specification of a sensor is the noise performance, which limitss the smallest input that can be detected. The signal-to-noise ratio (SNR) is used to evaluate the noise performance of the system, which is given as

$${SNR}=\frac{{\left|{S}_{i}\right|}^{2}}{{\left|{N}_{i}\right|}^{2}}$$

(15)

where ${S}_{i}$ and ${N}_{i}$ stand for the desired signal and the integrated noise in the relevant frequency range, respectively. For the amplitude ratio of Eq. (8) discussed in this paper, the useful signal power is ${\left|\frac{{X}_{1}}{{X}_{n}}\right|}^{2}$, where ${X}_{1}$ and ${X}_{n}$ are the output signals of $Re{s}_{1}$ and $Re{s}_{n}$ in the digital domain. For resonators working at several kHz or MHz, the thermal noise dominates the noise power. As derived in ref.⁴, if the noise power at the output of $Re{s}_{1}$ and $Re{s}_{n}$ is Gaussian and uncorrelated, the noise power of the amplitude ratio $\frac{{X}_{1}}{{X}_{n}}$ can be calculated as the variance of the amplitude ratio $\left|\frac{{X}_{1}}{{X}_{n}}\right|$. Hence, the SNR of the amplitude ratio $\frac{{X}_{1}}{{X}_{n}}$ is derived as

$${SN}{R}_{{X}_{1}/{X}_{n}}={\left|\frac{{X}_{1}}{{X}_{n}}\right|}^{2}/{\left|\frac{{X}_{1}}{{X}_{n}}\right|}_{{noise}}^{2}={\left|\frac{{X}_{1}}{{X}_{n}}\right|}^{2}/\left[{\left|\frac{{X}_{1}}{{X}_{n}}\right|}^{2}\left(\frac{{X}_{1,{noise}}^{2}}{{X}_{1}^{2}}+\frac{{X}_{n,{noise}}^{2}}{{X}_{n}^{2}}\right)\right]={\left(\frac{{X}_{1,{noise}}^{2}}{{X}_{1}^{2}}+\frac{{X}_{n,{noise}}^{2}}{{X}_{n}^{2}}\right)}^{-1}=\frac{{SN}{R}_{1}\times {SN}{R}_{n}}{{SN}{R}_{1}+{SN}{R}_{n}}$$

(16)

where ${SN}{R}_{i}\left(i=1,n\right)$ is the SNR of $Re{s}_{1}$ and $Re{s}_{n}$, respectively.

To find the analytic expression for the SNR of the amplitude ratio $\frac{{X}_{1}}{{X}_{n}}$, the expressions for the signal and noise power of $Re{s}_{1}$ and $Re{s}_{n}$ must be derived, and it must be analyzed which noise components are correlated/uncorrelated.

As shown in Fig. 3a, there are three noise sources that need to be considered: (ⅰ) the mechanical noise of $Re{s}_{1}$, which is dominated by mechanical-thermal noise (${F}_{1,n}\left(s\right)$), (ⅱ) the electrical noise ${i}_{n}$ of the transimpedance amplifier in the readout circuit, (ⅲ) the quantization noise ${N}_{q}$ of the ADC. These noise sources will now be described in more detail.

Fig. 3: Noise and measurement range analysis of the proposed hybrid electro-mechanical cascaded resonator system.

a Block diagram of the proposed structure including all noise sources. All noise is introduced before the DSP blocks, since the digital circuitry does not introduce any extra noise into the system. b Noise transfer functions at the output of $Re{s}_{1}$ and $Re{s}_{3}$ without electrical noise in the illustrated 3-DoF hybrid resonator system. c Noise transfer functions at the output of $Re{s}_{1}$ and $Re{s}_{3}$ with electrical noise in the illustrated 3-DoF hybrid resonator system. d Diagram of how useful data (the ADC converted data) moves in the full-length output data of all 3 resonators in the illustrated 3-DoF system. e Frequency response of all 3 resonators of the illustrative 3-DoF system. $Re{s}_{2}$ has the smallest amplitude around the ${1}^{{st}}$ mode

Mechanical noise

For a mechanical resonator with a high Q factor, the noise power in terms of displacement in the range $\pm \Delta {\rm{\omega }}$ near the ${1}^{{st}}$ mode is dominated by the mechanical-thermal noise. Assuming that this mechanical-thermal noise can be described by an input-referred noise power spectral density ${F}_{n}$, the total noise power in terms of displacement ${X}_{1,n}$ of $Re{s}_{1}$ near the ${1}^{{st}}$ mode can be derived as²⁸

$${X}_{1,n}^{2}\approx \frac{1}{2{\rm{\pi }}}{\int }_{{{\rm{\omega }}}_{1{st}}-\Delta {\rm{\omega }}}^{{{\rm{\omega }}}_{1{st}}+\Delta {\rm{\omega }}}{F}_{n}^{2}{H}_{1}^{2}d\omega$$

(17)

where H₁ is the transfer function of $Re{s}_{1}$, as described in Eq. (1). The power spectral density of the driving noise force is²⁹

$${F}_{n}^{2}=4{k}_{B}T{\eta }_{1}$$

(18)

where ${k}_{B}$, $T$ and ${{\rm{\eta }}}_{1}$ are the Boltzmann constant, the ambient temperature and the damping coefficient of $Re{s}_{1}$, respectively.

For a mechanical resonator with a high Q factor, the noise power within the 3-dB bandwidth around the resonance peak dominates the total noise power²⁸. Therefore, for $\Delta {\rm{\omega }}={{\rm{\omega }}}_{1{st}}/\left(2Q\right)\ll {{\rm{\omega }}}_{1{st}}$, the integral in Eq. (17) in the 3-dB bandwidth near the ${1}^{{st}}$ mode can be approximated as³⁰

$${X}_{1,n}^{2}\left(j{\omega }_{1{st}}\right)\approx \frac{4{k}_{B}T{\eta }_{1}Q{\omega }_{1{st}}}{{K}_{1}^{2}}$$

(19)

where ${K}_{1}$ is the stiffness of the mechanical resonator $Re{s}_{1}$.

Electrical noise of the readout circuit

For the readout circuit, and assuming that a standard transimpedance amplifier is used, the current noise power spectral density is²⁸

$${i}_{n}^{2}={i}_{{nop}}^{2}+{\left(\frac{{R}_{m}+{R}_{f}}{{R}_{m}{R}_{f}}\right)}^{2}{e}_{{nop}}^{2}+{\left(\frac{4{k}_{B}T}{{Rf}}\right)}^{2}$$

(20)

where ${i}_{{nop}}$, ${e}_{{nop}}$, ${R}_{m}$ and ${R}_{f}$ are the current and voltage root noise power spectral densities of the opamp used, the equivalent motional resistance of $Re{s}_{1}$ and the feedback resistance, respectively. As above, we only focus on the noise power in the 3-dB bandwidth of the ${1}^{{st}}$ mode, since when the system works normally it resonates at a frequency in this range, and therefore the noise power in this frequency range dominates the noise power of the readout circuit due to the high quality factor of the mechanical resonator.

Quantization noise of the ADC

When an analog signal is converted into the digital domain, the finite resolution of the conversion is limited by the resolution(the number of bits) of the ADC: the quantization error ${e}_{q}$ can take a value between $-\frac{{LSB}}{2}$ and $+\frac{{LSB}}{2}$, where ${LSB}=\frac{{V}_{{ref}}}{{2}^{k}}$ is the least significant bit of the k-bit ADC³¹. Since the quantization error can be an arbitrary value within this range, the probability density function can be assumed to be constant $\frac{1}{{LSB}}$ in this range. The noise power of the quantization noise is then expressed as

$${N}_{q}^{2}={\int }_{-\frac{{LSB}}{2}}^{\frac{{LSB}}{2}}{e}_{q}^{2}p\left({e}_{q}\right)d{e}_{q}={\int }_{-\frac{{LSB}}{2}}^{\frac{{LSB}}{2}}{e}_{q}^{2}\frac{1}{{LSB}}d{e}_{q}=\frac{{LS}{B}^{2}}{12}$$

(21)

The quantization noise is uniformly distributed within half of the Nyquist sampling frequency ${f}_{s}/2$. The output signal of $Re{s}_{1}$ is quantized by the ADC, thus it has a quantization noise ${N}_{q,1}^{2}=\frac{{LS}{B}^{2}}{12}$. For the output signal of the digital resonator $Re{s}_{n}$, it doesn’t introduce any additional noise as it is already digital. However, the output signal of $Re{s}_{n}$ has the same resolution, so it has a quantization noise ${N}_{q,n}^{2}=\frac{{LS}{B}^{2}}{12}$, which is the same as $Re{s}_{1}$.

Noise transfer function simulation

The simulations of the noise transfer functions are performed for the same illustrative 3-DoF hybrid resonator system mentioned in section 2.1.3. As shown in Fig. 3b and Fig. 3c, if only mechanical noise is considered, as in the conventional purely MEMS coupled resonator system, the noise transfer function is the same as the signal transfer function. When the electrical noise (both the noise from the readout circuit and the ADC’s quantization noise) is considered, the noise at the ${3}^{{rd}}$ mode is larger. This is because the electrical noise is amplified by the resonant peak at the ${3}^{{rd}}$ mode and superposed on the mechanical noise already there. Therefore, another reason to leave an intentional frequency difference between the ${1}^{{st}}$ and the ${3}^{{rd}}$ mode is to minimize the influence of the large noise present around the ${3}^{{rd}}$ mode on the ${1}^{{st}}$ mode.

Noise floor of the amplitude ratio $\frac{{X}_{1}}{{X}_{n}}$

The total noise power of $Re{s}_{1}$ consists of three parts: the mechanical noise and the electrical noise of the readout circuit introduced before the ADC in the analog domain and the quantization noise of the ADC. If only the noise sources from the analog domain would be considered, the SNR of $Re{s}_{1}$ within the 3-dB bandwidth of the ${1}^{{st}}$ mode is derived as

$${SN}{R}_{1,{analog}}=\frac{{X}_{1}^{2}\left(j{\omega }_{1{st}}\right){\mu }_{e}^{2}}{\frac{4{k}_{B}T{\eta }_{1}Q{\omega }_{1{st}}{\mu }_{e}}{{K}_{1}^{2}}+{i}_{n}^{2}{R}_{f}^{2}{f}_{3{dB}}}$$

(22)

where ${X}_{1}\left(j{{\rm{\omega }}}_{1{st}}\right)={F}_{{in}}\left(j{{\rm{\omega }}}_{1{st}}\right){H}_{1}$ is the displacement of $Re{s}_{1}$ when applying a driving force ${F}_{{in}}\left(j{{\rm{\omega }}}_{1{st}}\right)$ to the system, ${\mu }_{e}$ is the conversion ratio from mechanical displacement to electrical voltage, ${f}_{3{dB}}=\frac{{{\rm{\omega }}}_{1{st}}}{Q{\rm{\pi }}}$ is the 3-dB bandwidth of the ${1}^{{st}}$ mode.

For the last resonator $Re{s}_{n}$, if we only consider the noise introduced in the analog domain within the 3-dB bandwidth of the ${1}^{{st}}$ mode:

$${SN}{R}_{n,{analog}}=\frac{{X}_{1}^{2}\left(j{\omega }_{1{st}}\right){\mu }_{e}^{2}{\left({\prod }_{i=2}^{i=n}{H}_{i}\left(j{\omega }_{1{st}}\right)\right)}^{2}}{\left(\frac{4{k}_{B}T{\eta }_{1}Q{\omega }_{1{st}}{\mu }_{e}}{{K}_{1}^{2}}+{i}_{n}^{2}{R}_{f}^{2}{f}_{3{dB}}\right){\left({\prod }_{i=2}^{i=n}{H}_{i}\left(j{\omega }_{1{st}}\right)\right)}^{2}}\approx \frac{{X}_{1}^{2}\left(j{\omega }_{1{st}}\right){\mu }_{e}^{2}}{\frac{4{k}_{B}T{\eta }_{1}Q{\omega }_{1{st}}{\mu }_{e}}{{K}_{1}^{2}}+{i}_{n}^{2}{R}_{f}^{2}{f}_{3{dB}}}$$

(23)

where ${H}_{i}\left(j{{\rm{\omega }}}_{1{st}}\right)$ is the transfer function of the ith resonator at the ${1}^{{st}}$ mode in the Laplace domain (the actual circuit is a digital circuit). Equation (23) is nearly equal to Eq. (22) within a narrow band in which the transfer function is nearly constant, hence ${SN}{R}_{1,{analog}}={SN}{R}_{n,{analog}}$.

The quantization noise of $Re{s}_{1}$ and $Re{s}_{n}$ are independent and uncorrelated. As the SNR of the amplitude ratio $\frac{{X}_{1}}{{X}_{n}}$ in the proposed structure is determined by the analog domain noise and the quantization noises of both resonators, according to Eqs. (15) and (16), the SNR of the amplitude ratio $\frac{{X}_{1}}{{X}_{n}}$ is then derived as

$${SN}{R}_{{X}_{1}/{X}_{n}}=\frac{{SN}{R}_{1}\times {SN}{R}_{n}}{{SN}{R}_{1}+{SN}{R}_{n}}=\frac{\frac{{X}_{1}^{2}}{{N}_{1}^{2}}\times \frac{{X}_{n}^{2}}{{N}_{n}^{2}}}{\frac{{X}_{1}^{2}}{{N}_{1}^{2}}+\frac{{X}_{n}^{2}}{{N}_{n}^{2}}}$$

(24)

where ${X}_{1}$, ${N}_{1}$, ${X}_{n}$, ${N}_{n}$ are the amplitude of the output signal and the root mean square noise of $Re{s}_{1}$ and $Re{s}_{n}$, respectively.

The noise floor of the proposed system ${\left\langle \frac{\varDelta u}{u}\right\rangle }_{\min }$ is then given by

$${\left\langle \frac{\varDelta u}{u}\right\rangle }_{\min }=\frac{{\left\langle \frac{{X}_{1}}{{X}_{n}}\right\rangle }_{\min }}{{Sensitivit}{y}_{{normalized},\frac{\varDelta u}{u}}}$$

(25)

where ${\left\langle \frac{{X}_{1}}{{X}_{n}}\right\rangle }_{\min }$ is the minimum detectable amplitude ratio change, which is equal to the noise floor of the amplitude ratio. It can be derived as

$${\left\langle \frac{{X}_{1}}{{X}_{n}}\right\rangle }_{\min }={\left|{X}_{1}/{X}_{n}\right|}_{{noise}}=\sqrt{\frac{{\left({X}_{1}/{X}_{n}\right)}^{2}}{{SN}{R}_{{X}_{1}/{X}_{n}}}}=\sqrt{{\left|\frac{{X}_{1}}{{X}_{n}}\right|}^{2}\left(\frac{{N}_{1}^{2}}{{X}_{1}^{2}}+\frac{{N}_{n}^{2}}{{X}_{n}^{2}}\right)}$$

(26)

The parameters in Eqs. (25) and (26) can be obtained from the measurement results, and Eq. (25) can be used to evaluate the normalized input mechanical resolution of the proposed system.

Measurement range

The measurement range is bounded by the maximum value that the input mass/stiffness perturbation can achieve. For the proposed system as a high-sensitivity sensor, a small perturbation causes a significant amplitude ratio change. As the mechanical resonator $Re{s}_{1}$ remains almost constant in amplitude like an individually driven resonator with a small mass/stiffness perturbation, as discussed in section 2.1.2, this means that the significant amplitude ratio increase is due to a significant amplitude decrease of $Re{s}_{n}$. The maximum detectable mass/stiffness perturbation that the system can detect is therefore given by

$$\left\langle {Maximum\; perturbation}\right\rangle =\frac{{\left\langle {Amplitude}{Ratio}\right\rangle }_{\max }}{{Sensitivit}{y}_{{normalized},\frac{\Delta u}{u}}}=\frac{\frac{{Am}{p}_{1,\max }}{{Am}{p}_{n,\min }}}{{Sensitivit}{y}_{{normalized},\frac{\Delta u}{u}}}$$

(27)

where ${Am}{p}_{1,\max }$ and ${Am}{p}_{n,\min }$ are the maximum amplitude of $Re{s}_{1}$ (the peak amplitude of $Re{s}_{1}$ that the readout circuit and the ADC can handle) and the minimum amplitude of $Re{s}_{n}$ (the smallest detectable amplitude change of $Re{s}_{n}$), respectively.

Assuming that $Re{s}_{1}$ and $Re{s}_{n}$ can have the same maximum amplitude in the digital domain (the digitized data of both resonators have the same word length), then the numerator of Eq. (27) is the dynamic range of $Re{s}_{n}$. From Eq. (27) it can be seen that the largest detectable perturbation is determined by the dynamic range of $Re{s}_{n}$ and the normalized sensitivity of the system. Due to the high sensitivity, a larger dynamic range of $Re{s}_{n}$ is needed to get a reasonably large measurement range. For conventional purely mechanical weakly-coupled resonators, the dynamic range is usually limited by the noise floor of the resonator that has a large amplitude change¹³: when the amplitude of that resonator is too small, its waveform is distorted by the mechanical/electrical noise so that it becomes undetectable.

For the proposed system, when the maximum perturbation is applied, the amplitude of the output of $Re{s}_{1}$ can still achieve the peak amplitude ${Am}{p}_{1,\max }$. Compared to this large signal amplitude, the mechanical and the electrical noise discussed above are negligible; only the quantization noise of $Re{s}_{i}$ ($i=\mathrm{2,3},\ldots ,n$) sets a limitation to the minimum amplitude ${Am}{p}_{n,\min }$ when the maximum input perturbation is applied. For all electrical N-bit resonators used in the proposed system, the useful signal (the measured signal of the mechanical resonator) should always be higher than the noise floor; otherwise, the output of $Re{s}_{n}$ merges with the noise. In other words, the minimum amplitude of $Re{s}_{n}$ is achieved when the smallest output signal among all digital resonators reaches the quantization noise floor. For the proposed system, according to Eqs. (7) and (9), $Re{s}_{n-1}$ has the smallest output signal around the ${1}^{{st}}$ mode. As shown in Fig. 3d, in the illustrated 3-DoF system mentioned in section 2.1.3 and section 2.2.4, the useful output signal of ${{Res}}_{2}$ hits the LSB (digital noise floor) first when a large input perturbation is applied. Figure 3e shows the frequency sweep of the illustrative 3-DoF hybrid resonator system. We can clearly see that for a 3-DoF system, $Re{s}_{2}$ has the smallest amplitude around the ${1}^{{st}}$ mode.

Assuming that the maximum amplitude takes up all the useful bits, and that the smallest amplitude should be ${N}_{0}$ bits higher than the noise floor so that no useful information converted by the ${N}_{0}$-bit ADC is distorted to ensure that output response is linear, then the ideal dynamic range of $Re{s}_{n}$ (in dB) can be calculated as

$$D{R}_{n}=20\cdot {lo}{g}_{10}\left({2}^{B-1-{B}_{\min }}\right)-6.02\times {N}_{0}-1.76[{dB}]$$

(28)

where ${B}_{\min }=$ ${lo}{g}_{2}\left(\frac{{Am}{p}_{\min ,n}}{{Am}{p}_{\min ,n-1}}\right)$ is the number of bits that represents the ratio between the amplitude of $Re{s}_{n}$ and the amplitude of $Re{s}_{n-1}$. This ratio can be estimated by the equation:

$$\frac{{Am}{p}_{\min ,n}}{{Am}{p}_{\min ,n-1}}=\frac{{K}_{c}^{n-1}\cdot {\prod }_{i=1}^{n}{H}_{i}\left(s\right)\cdot {x}_{{in},\min }\left(s\right)}{{K}_{c}^{n-2}\cdot {\prod }_{i=1}^{n-1}{H}_{i}\left(s\right)\cdot {x}_{{in},\min }\left(s\right)}={K}_{c}\cdot {H}_{n}\left(s\right)$$

(29)

As shown in Eq. (28), by increasing the number of bits B in the digital circuitry, a larger dynamic range of $Re{s}_{n}$ can be achieved. In other words, the measurement range of the proposed hybrid resonator system can be larger than that of purely mechanical weakly-coupled resonators by choosing a reasonably large B, while getting a high sensitivity at the same time.

Results

To verify the functionality of the proposed hybrid resonator system, two types of mechanical resonators are used for validation: (1) a quartz crystal microbalance (QCM) resonator is used for the mass perturbation test with the laser bombard method, and (2) a capacitive double-end tuning fork (DEFT) resonator is used for the stiffness perturbation test by tuning its stiffness via the bias voltage.

The electrical part of the resonator system is implemented on an FPGA platform that also integrates the ADC/DAC blocks and a CPU for control, which is needed for the verifications in this paper. Some important specifications and parameters of the FPGA board used are listed in Table 1. More detailed information about the FPGA platform can be found in ref.³².

Table 1 Main specifications of the FPGA board

The schematics of the measurement setup for the QCM and DETF tests are shown in Fig. 4a and Fig. 4b, respectively. A lock-in amplifier (Zurich Instruments HF2LI) is used for the analysis of the proposed system. Since the ADC and DAC on the FPGA board have the same resolution and sampling rate, no extra noise is introduced when the output signals are returned to the lock-in amplifier. A laptop is used for the real-time control of all parameters in the digital blocks. To make the verifications more convincing, both 3-DoF and 5-DoF setups are tested in combination with both mechanical resonators. And for all the measurements, the factors ${{\rm{\alpha }}}_{2}$, ${{\rm{\alpha }}}_{3}$ and ${{\rm{\alpha }}}_{4}$ are chosen to be 2, 4, and 2, respectively. With these factor values, the resonance peaks of all digital resonators are several times smaller than the digital clock frequency, which ensures that all digital resonators are in normal operation.

Fig. 4: System implementation and experimental setup.

a Experimental setup for the proposed system with the QCM. b Experimental setup for the proposed system with the DETF resonator. c Normalized sensitivity error ${\mu }_{c}$ versus the number of processed data in one resonance period of $Re{s}_{1}$ for both the 3-DoF system and the 5-DoF system. d Minimum number of extra bits needed to avoid resolution loss versus the normalized sensitivity for both the 3-DoF system and the 5-DoF system

The bit-width of the coefficients of the digital filters and the processed data is chosen as 32 bits and 40 bits respectively, which is more than sufficient for the resolution of the input perturbations applied and which satisfies the speed limit of the FPGA board, for the precise emulation of mechanical resonators. According to Eqs. (27) and (28), the linear range of $Re{s}_{n}$ can vary from 60 dB to 110 dB with $B=40$ for a reasonably high-Q $Re{s}_{n}$ ($1K < {Q}_{n} < 100K$) and coupling factor ${K}_{c}$ ($0 < {K}_{c} < 0.5$), which indicates a very large maximum detectable input perturbation.

Limitations of the digital implementation

Due to the limitations of the ADC/DAC resolution, the limited FPGA resources and running frequency, the digital implementations are not perfectly the same as the theoretical calculations. There are two main issues to be noted: 1) the difference in sensitivity value between the implemented system and the theoretical models, and 2) the scale factors to make the data in the digital domain compatible with the ADC/DAC resolution.

For an n-DoF system that contains n modes, the transfer function is different from the system-level analysis in section 2.1.1. This is because the transfer functions of the digital resonators only match the transfer function of the mechanical resonators perfectly around their resonating frequencies due to the bilinear transformation, which depends on the digital clock frequency. Due to the limited FPGA clock frequency (125 MHz), there is a difference between the practical sensitivity and the theoretical sensitivity. Fortunately, this difference only appears in the digital part, and since the digital part doesn’t influence the working of the mechanical resonator $Re{s}_{1}$, the difference introduced in the implementation can be represented as a correction factor ${{\rm{\mu }}}_{c}$ (the normalized sensitivity error). Hence, the corrected expression for the sensitivity is

$${Sensitivit}{y}_{{imp},\frac{\Delta u}{u}}\approx \frac{1}{{K}_{c}^{n-1}}\cdot \left(\mathop{\prod }\limits_{i=2}^{n-1}\left(\frac{{{\rm{\alpha }}}_{i}-1}{{{\rm{\alpha }}}_{i}}\right)\right)\cdot \left(1+{{\rm{\mu }}}_{c}\right)$$

(30)

Figure 4c shows the simulation results of how the digital clock affects the normalized sensitivity error ${{\rm{\mu }}}_{c}$ for both the 3-DoF system and the 5-DoF system. The figure indicates that when the digital clock is fast enough, the difference between the practical value and the theoretically calculated value is small, as there are more data processed in one resonance period of $Re{s}_{1}$. In contrast, when the digital clock is not fast enough compared to the resonance frequency, the error to be corrected is larger. Compared to the 5-DoF setup, the proposed 3-DoF system is more robust against lower values of the digital clocks. Since the factor ${{\rm{\mu }}}_{c}$ doesn’t influence the linear behavior of the system, we can obtain the factor ${{\rm{\mu }}}_{c}$ from measurements.

Another issue is to make the digital data compatible with the ADC/DAC resolution. The converted data are only 14 bits; hence, they need to be scaled up before multiplying with the coupling factor ${K}_{c}$ to avoid resolution loss, as ${K}_{c}$ can be a very small value (which implies logic right shifts in binary numbers). Figure 4d shows the minimum number of extra bits needed for scaling up in order to avoid resolution loss for different sensitivity values. A scale factor ${K}_{{scale},{in}}$ is used to describe the scaling up of the input data. The same happens when 40-bit processed data need to be converted again into an analog signal for analysis in the lock-in amplifier by the 14-bit DAC. Thus, scale factors ${K}_{{scale},{out}1}$ and ${K}_{{scale},{outn}}$ are applied to the 40-bit output data of $Re{s}_{1}$ and $Re{s}_{n}$ to avoid overflow for 14-bit resolution. These factors must be included in the calculation of the sensitivity, leading to the sensitivity implemented in the practical setup:

$$\begin{array}{l}Sensitivit{y}_{imp,\frac{\Delta u}{u}}\approx \\ \,\frac{1}{{K}_{c}^{n-1}}\cdot \frac{{K}_{scale,out1}}{{K}_{scale,outn}}\cdot \left(\mathop{\prod }\limits_{i=2}^{n-1}\left(\frac{{{\rm{\alpha }}}_{i}-1}{{{\rm{\alpha }}}_{i}}\right)\right)\cdot \left(1+{{\rm{\mu }}}_{c}\right)\end{array}$$

(31)

Verification with the QCM resonator

QCM resonators experience the piezoelectric effect, which gives a mechanical deformation when a voltage is applied to the two sides of the crystal. It allows resolutions down to several hertz with a resonant frequency within 4-6 MHz due to the high-quality factor, which can range from 10 K to 100 K in air. By applying mass perturbations on the surface of the resonator through the laser bombardment method, its fundamental resonant state changes. This QCM resonator is chosen for the mass perturbation verification of the proposed resonator system. The RLC equivalent electrical parameters of the QCM in air, as measured by an impedance analyzer and lock-in amplifier, are shown in Table 2; these are used to calculate the parameters of the digital filters.

Table 2 RLC equivalent electrical parameters of the QCM used

The resonating frequency of the QCM used is 4.9978 MHz and the calculated Q factor is about 30940 in air. A frequency divider of 2 is used to get a digital clock of 62.5 MHz to avoid setup/hold time violations. The setup of the mass perturbation verification is shown in Fig. 4a.

Verification of the mass perturbation

For the 3-DoF setup, the coupling factor is chosen as ${K}_{c}=1/1024$, while the scale factors ${K}_{{scare},{in}}$, ${K}_{{scare},{out}1}$ and ${K}_{{scare},{out}2}$ are determined as ${2}^{14}$, ${2}^{15}$ and ${2}^{6}$, respectively. These factors ensure no resolution loss in data processing and the output data to the analyzer still have 14 bits resolution. According to Eq. (13) and (29), the sensitivity of the 3-DoF structure is then 524288. An intentional frequency difference of about 5.8 kHz is given to the digital resonators to avoid the mode-aliasing effect.

To apply a reasonably accurate mass perturbation to the QCM as $Re{s}_{1}$, the laser bombardment method is used. Each time, about 0.00126% of the total mass ${M}_{0}$ is removed. The frequency responses of $Re{s}_{1}$ and $Re{s}_{3}$ after several laser bombardments are shown in Fig. 5a. Due to the reduction of the mass, the frequency of the ${1}^{{st}}$ mode (the fundamental resonance frequency of $Re{s}_{1}$) becomes higher after each laser bombard, while the ${3}^{{rd}}$ mode stays fixed. As a result, the frequency difference between the ${1}^{{st}}$ mode and the ${3}^{{rd}}$ mode becomes larger, as mentioned in section 2.1.3. This leads to the expected amplitude ratio reduction in the ${1}^{{st}}$ mode.

Fig. 5: Measurement results of the proposed system with the QCM.

a Frequency sweep of Res1 and Res3 in the proposed 3-DoF structure after several times of laser bombardment. The 1st mode shifts after each laser bombard, while the 3rd mode stays at the same frequency. b Measured amplitude ratio response and the frequency shift with respect to the applied mass perturbation in the 3-DoF implementation. c Measured amplitude ratio response and the frequency shift with respect to the applied mass perturbation in the 5-DoF implementation. d Measured output noise spectral density of the two output signals versus frequency around the ${1}^{{st}}$ and ${3}^{{rd}}$ mode for the 3-DoF implementation in open loop. e Measured output noise spectral density of the two output signals for both 3-DoF and 5-DoF implementation in closed loop. f Allan variance of the output amplitude ratio for both the 3-DoF and 5-DoF implementation

The amplitude ratio between $Re{s}_{1}$ and $Re{s}_{3}$ and the frequency shift of the ${1}^{{st}}$ mode as a function of the input perturbation is shown in Fig. 5b. Each laser bombardment leads to a frequency shift of about 31.4 Hz. The calculated normalized frequency shift sensitivity is

$${Sensitivt}{y}_{\left\{f,{normalized}\right\}}=\frac{\Delta {f}_{1{st}}}{{f}_{1{st}}}/\frac{\Delta {M}_{1}}{{M}_{1}}\approx 0.5$$

(32)

This follows the calculation in ref.³³ that the normalized sensitivity of the frequency shift of a resonator is 0.5; this confirms that the QCM under test is working normally. The measured normalized amplitude ratio sensitivity is 505638. Compared to the theoretical calculation result, there is a difference of only 3.5%. As mentioned above, this difference comes from the error introduced by the implementation of the digital filters. This does not influence the normal operation of $Re{s}_{1}$ and can be modeled as a correction factor ${{\rm{\mu }}}_{{\rm{c}}}$. The amplitude ratio response shows a good linearity: the nonlinearity error is 0.26% across the full range under test.

Then the electrical part is modified and tuned to form a 5-DoF hybrid resonator system, while the coupling factor ${K}_{c}$ is changed to 1/64. The scale factors ${K}_{{scare},{in}}$, ${K}_{{scare},{out}1}$ and ${K}_{{scare},{out}2}$ are determined as ${2}^{20}$, ${2}^{21}$ and ${2}^{12}$, making the amplitude of the output signals from the DACs to the lock-in amplifier at the hundred millivolt level. The calculated normalized amplitude ratio sensitivity is 3145728 in this case. The same laser bombards are applied to $Re{s}_{1}$ to introduce mass perturbations. The measured amplitude ratio response and frequency shift with respect to the mass perturbations are shown in Fig. 5c. The normalized measured amplitude ratio sensitivity derived from this is 3238894, indicating a correction factor ${{\rm{\mu }}}_{c}$ of 2.9%. The nonlinearity error is 0.21% in the range under test. Thus, once the coefficients and parameters are properly set before the test, we can ensure that the amplitude ratio is proportional to the mass perturbation with good linearity, even though the sensitivity derives rightly from the theoretically calculated value.

Noise performance of the proposed system with the QCM

The open-loop noise power spectral density of the 3-DoF system with the QCM is measured with the noise analysis function of the lock-in amplifier without any input drives applied, as shown in Fig. 5d. Here, $Re{s}_{1}$ has only 1 peak, which comes from the resonant frequency of the ${1}^{{st}}$ mode. For $Re{s}_{3}$, the quantization noise and all other detectable noise introduced during readout signal transmission are amplified by the resonant peak at the ${3}^{{rd}}$ mode, so 2 peaks can be observed, which follows the simulations in section 2.2.4. If these two modes are very close to each other, the noise amplified by the ${3}^{{rd}}$ mode starts to dominate the noise power, and the SNR of $Re{s}_{3}$ decreases. Since the quantization noise is uniformly distributed in the 3-dB bandwidth of the ${1}^{{st}}$ mode, and the peak at the ${1}^{{st}}$ mode is higher than the noise floor, the noise of the proposed 3-DoF system is still dominated by the mechanical noise and the noise of the readout circuit.

Then the system is tested in a closed loop with the PLL function of the lock-in amplifier. Figure 5e shows the noise spectral density of the output signals of both resonators in the 3-DoF and 5-DoF system. The normalized noise floor (resolution) of the proposed system with the QCM can be calculated as

$${\left\langle \frac{\varDelta {M}_{1}}{{M}_{1}}\right\rangle }_{\min }=\sqrt{{\left|\frac{{X}_{1}}{{X}_{n}}\right|}^{2}\cdot {\left(\frac{{K}_{{scale},{out}1}}{{K}_{{scale},{out}2}}\right)}^{2}\cdot \left(\frac{{N}_{1}^{2}}{{X}_{1}^{2}}+\frac{{N}_{n}^{2}}{{X}_{n}^{2}}\right)}/{Sensitivit}{y}_{{normalized}}$$

(33)

The variables used in Eqs. (33) and (34) are listed in Table 3. Based on Eq. (33) and Table 3, the normalized noise floor for the 3-DoF and 5-DoF proposed systems with QCM are $9.470\times {10}^{-8}/\sqrt{{Hz}}$ and 10.243$\,\times {10}^{-8}/\sqrt{{Hz}}$, respectively.

Table 3 Variables for noise floor calculation in the proposed system with QCM

Bias instability of the proposed system with the QCM

To evaluate the random drift in the output signals of the proposed system with the QCM, the Allan variance method is employed to calculate the bias instability. In the experiment, the amplitude ratios of the 3-DoF and 5-DoF proposed systems with the QCM have been collected at room temperature with a sampling frequency of about 56 Hz for 30 minutes. The calculated Allan variance as a function of the integration time is shown in Fig. 5f. The normalized bias instability is obtained as $2.098\times {10}^{-8}$ for the 3-DoF implementation and $2.032\times {10}^{-8}$ for the 5-DoF implementation, respectively. These two bias instabilities are approximately the same because the instability is mainly from the QCM resonator, as the digital parts of these two implementations can be considered as deterministic functions during the experiments.

Verification with the DETF resonator

The stiffness perturbation test is performed using one of two identical coupled DETF resonators with electrostatic coupling. The micrograph of the device and the measurement setup are shown in Fig. 4b. Since only 1 mechanical resonator is needed for the cascaded hybrid resonator system, the bias voltage ${V}_{{bias}}$ is applied to tune the stiffness of the mechanical resonator on the $+{V}_{{couple}}$ port. The differential DETF output signals are converted into electrical signals by two transimpedance amplifiers ${TI}{A}_{1}$ and ${TI}{A}_{2}$. Then the instrumentation amplifier INA tunes the differential outputs of the TIAs into a single-ended signal, which is sent to the ADC for processing in the electrical part of the proposed systems.

As described in³⁴, by tuning the bias voltage in the range from 25 V to 35 V, the motional currents are enhanced, which can improve the SNR of the mechanical resonator. In this test, the bias voltage is initially set to 32 V, and negative stiffness perturbations are applied by increasing the bias voltage with a step size of 20 mV. The initial resonance frequency of the mechanical resonator is 74354.4 Hz, while the calculated quality factor is 22840. The digital clock for the electrical setup is chosen as 1.25 MHz.

Verification of the stiffness perturbation

As done in the mass perturbation test above, the stiffness perturbation test also includes the 3-DoF and 5-DoF systems test. The parameters of the digital circuitry of the two implementations are listed in Table 4, which gives the same sensitivity for 3/5-DoF systems as in the QCM test, and makes the output signals compatible with the 14-bit DACs. An intentional frequency difference of about 15 Hz is set to avoid the mode-aliasing effect.

Table 4 Parameters of the digital circuits of the 3/5-DoF systems for the stiffness perturbation test with the DETF

For the 3-DoF system, the frequency response of $Re{s}_{1}$ and $Re{s}_{3}$ for different bias voltages is shown in Fig. 6a. The system shows the same behavior as in the mass perturbation test: the ${1}^{{st}}$ mode of $Re{s}_{3}$ follows the shift of the resonant peak of $Re{s}_{1}$, while the ${3}^{{rd}}$ mode remains unchanged. The evaluated amplitude ratio response and frequency shift with respect to the normalized stiffness perturbation are shown in Fig. 6b. The measured normalized sensitivity of the frequency shift is about 0.5, which indicates that the DETF resonator is in normal operation. The measured normalized amplitude ratio sensitivity is 537288. Then the same test is applied to the 5-DoF system. The result is shown in Fig. 6c: the measured sensitivity of the normalized amplitude ratio is an impressive 3308454.

Fig. 6: Measurement results of the proposed system with the DETF resonator.

a Frequency sweep of $Re{s}_{1}$ and $Re{s}_{3}$ in the proposed 3-DoF structure with the DETF resonator for different bias voltages. The frequency response shows the same behavior as in the QCM mass perturbation test. b Measured amplitude ratio response and the frequency shift with respect to the applied stiffness perturbation in the 3-DoF implementation. c Measured amplitude ratio response and the frequency shift with respect to the applied stiffness perturbation in the 5-DoF implementation. d Measured output noise spectral density of the two output signals versus frequency around the ${1}^{{st}}$ and ${3}^{{rd}}$ mode for the 3-DoF implementation in open loop. e Measured output noise spectral density of the two output signals for both the 3-DoF and 5-DoF implementations in closed loop. f Allan variance of the output amplitude ratio for both the 3-DoF and 5-DoF implementations

Both the 3-DoF and 5-DoF implementations show a good linearity with nonlinearity errors of 0.98% and 0.70%, respectively. Since the coupling factor ${K}_{c}$ for the stiffness perturbation test is identical to the coupling factor used for the mass perturbation test, the theoretical normalized sensitivities for both tests are the same. The differences between the measured and the calculated normalized sensitivity values for the 3-DoF and the 5-DoF systems are 2.4% and 5.2%, respectively.

Noise performance of the proposed system with the DETF resonator

The open-loop output noise power spectral density of the 3-DoF system with the DETF resonator is shown in Fig. 6d, as obtained from the lock-in amplifier. Here again, a strong amplified noise appears at the ${3}^{{rd}}$ mode, similarly as in the mass perturbation test. The peak is even higher since the DETF resonator needs a readout circuit, which the QCM doesn’t need. Here again the mechanical noise and the electrical noise from the actuation and the readout circuit dominate the noise power within the 3-dB bandwidth of the ${1}^{{st}}$ mode.

As in the test with the QCM, the system is tested in closed loop for both the 3-DoF and 5-DoF implementations, the measured results are shown in Fig. 6e. The normalized noise floor (resolution) can be calculated as

$${\left\langle \frac{\varDelta {K}_{1}}{{K}_{1}}\right\rangle }_{\min }=\sqrt{{\left|\frac{{X}_{1}}{{X}_{n}}\right|}^{2}\cdot {\left(\frac{{K}_{{scale},{out}1}}{{K}_{{scale},{out}2}}\right)}^{2}\cdot \left(\frac{{N}_{1}^{2}}{{X}_{1}^{2}}+\frac{{N}_{n}^{2}}{{X}_{n}^{2}}\right)}/{Sensitivit}{y}_{{normalized}}$$

(34)

The variables used in Eq. (34) are listed in Table 5. Based on Eq. (34) and Table 5, the normalized noise floor for the 3-DoF and 5-DoF proposed systems with the DETF resonator are $1.864\times {10}^{-7}/\sqrt{{Hz}}$ and $1.877\,\times {10}^{-7}\,/\sqrt{{Hz}}$, respectively.

Table 5 Variables for noise floor calculation in the proposed system with DETF

Bias instability of the proposed system with the QCM

Like the test with the QCM, the bias instability of the proposed system with the DETF resonator is also calculated based on the recorded data with the same sampling frequency of about 56 Hz for 30 minutes. The calculated Allan variance as a function of the integration time is shown in Fig. 6f. The normalized bias instability is obtained as $3.057\times {10}^{-7}$ for the 3-DoF implementation and $3.706\times {10}^{-7}$ for the 5-DoF implementation, respectively.

Discussion

Table 6 is a summary of all relevant measured and derived results for both resonator types (QCM and DETF) and both the 3-DoF and 5-DoF systems listed above.

Table 6 Measured and evaluated specifications for all tests

The proposed system clearly shows the potential to have an ultra-large amplitude ratio sensitivity independently of the type of mechanical resonator used and the type of perturbation given. According to Eqs. (25) and (26), the minimum detectable input perturbation of the proposed system depends on the amplitude ratio $\frac{{X}_{1}}{{X}_{n}}$, the SNR of the amplitude ratio $\frac{{X}_{1}}{{X}_{n}}$, and the sensitivity of the system. If we insert Eq. (26) into Eq. (25), and ensure that the system is in its linear range, then the minimum detectable input perturbation is derived as

$${\left\langle \frac{\varDelta u}{u}\right\rangle }_{\min }=\frac{\left|\frac{{X}_{1}}{{X}_{n}}\right|\cdot \sqrt{\left(\frac{1}{{SN}{R}_{1}}+\frac{1}{{SN}{R}_{n}}\right)}}{{Sensitivit}{y}_{{normalized},\frac{\varDelta u}{u}}}={\left|\frac{\varDelta u}{u}\right|}_{0}\cdot \sqrt{\left(\frac{1}{{SN}{R}_{1}}+\frac{1}{{SN}{R}_{n}}\right)}$$

(35)

where ${\left|\frac{\varDelta u}{u}\right|}_{0}$ is the input perturbation already applied to the mechanical resonator.

From this Eq. (35) we can conclude that the minimum detectable input perturbation of the proposed system is determined by the input perturbation already applied to the mechanical resonator and the noise of the both output signals. In order to obtain the best noise performance, we should find the optimal operating point of the proposed system. As suggested in ref. ³⁵, the optimal operating point of the 2-DoF mechanical WCR is reported at $\left|\frac{{X}_{1}}{{X}_{2}}\right|\approx 1.22$. However, for the proposed system with ultra-high sensitivity, the linear relationship between the amplitude ratio and the input perturbation is valid only if no mode aliasing occurs, which requires a minimum normalized input perturbation of about $\frac{1}{{Q}_{1}}$, where ${Q}_{1}$ is the quality factor of the mechanical resonator. With this input perturbation, the amplitude ratio is already larger than 1.22 in the proposed system. As discussed in refs. ^16,17, the noise performance is better when the operating point is closer to the optimal operating point, thus, the best noise floor (resolution) the proposed system can get can be calculated as

$$\left\{\begin{array}{c}{\left\langle \frac{\varDelta {M}_{1}}{{M}_{1}}\right\rangle }_{\min ,{QCM}}\approx \frac{1}{{Q}_{{QCM}}}\cdot \sqrt{\left(\frac{1}{{SN}{R}_{1,{QCM}}}+\frac{1}{{SN}{R}_{n,{QCM}}}\right)}=1.589\times {10}^{-8}/\sqrt{{Hz}}\\ {\left\langle \frac{\varDelta {K}_{1}}{{K}_{1}}\right\rangle }_{\min ,{DETF}}\approx \frac{1}{{Q}_{{DETF}}}\cdot \sqrt{\left(\frac{1}{{SN}{R}_{1,{DETF}}}+\frac{1}{{SN}{R}_{n,{DETF}}}\right)}=3.477\times {10}^{-8}/\sqrt{{Hz}}\end{array}\right.$$

(36)

The coupling factor of the conventional MEMS weakly-coupled resonators is limited by the mode-aliasing effect¹⁸: the frequency difference between different modes is related to the coupling stiffness. Taking a 2-DoF weakly-coupled MEMS resonator as example, then the resonance frequencies of the two modes are¹⁹

$$\left\{\begin{array}{c}{f}_{1}=\frac{1}{2\pi }\sqrt{\frac{K}{M}}\\ {f}_{2}=\frac{1}{2\pi }\sqrt{\frac{K+2{K}_{c}}{M}}\end{array}\right.$$

(36)

where $M$, $K$ and ${K}_{c}$ are the proof mass, the stiffness of one resonator and the stiffness of the coupling beam. Since the frequency difference between the two modes should be larger than the 3-dB bandwidth of one resonator to avoid mode-aliasing, the lower value of the stiffness of the coupling beam is limited. And since the sensitivity of the amplitude ratio is inversely proportional to ${K}_{c}$, the maximum sensitivity is thus limited. However, in the proposed cascaded resonator system, the resonators are single-direction coupled, which means that the frequency difference of two modes doesn’t follow the relationships in Eq. (36). Therefore, there is no limitation for the coupling factor. In other words, both the 3-DoF and the 5-DoF system can achieve the same sensitivity since the coupling factor can be arbitrarily small. So the 3-DoF system and the 5-DoF system are theoretically similar in performance when they have the same sensitivity with properly chosen coupling factors.

However, the 3-DoF and 5-DoF systems are different in practical implementations, as discussed in section 3.1. The 3-DoF structure is more robust against changes in the digital clock frequency than higher-DoF structures (as shown in Fig. 4c). For the same digital clock, the normalized sensitivity error of the 3-DoF system is lower than that of the higher-DoF system. This is because for the simpler 3-DoF structure, the influence of the bilinear transformation and the digital clock frequency is smaller, and the number of extra bits needed to avoid resolution loss is also smaller than for the higher-DoF structures if no time violation occurs in the implementation (as shown in Fig. 4d). In addition, the 3-DoF structure requires less resources to implement compared to higher-DoF structures as it has the lowest number of digital resonators among this type of cascaded hybrid resonator systems. Therefore, we can conclude that for a real implementation, the 3-DoF structure is the best choice because of its excellent performance, its stability and simple structure and its low digital resource cost.

Table 7 compares the sensitivity of the proposed cascaded hybrid resonator system with several state-of-the-art works. The proposed multi-DoF hybrid cascaded resonator system shows the highest sensitivity. Compared to the other works in Table 7, besides its ultra-high sensitivity, the proposed hybrid resonator system also has other advantages: (ⅰ) the maximum detectable input perturbation of the proposed system can be much larger than for conventional MEMS WCR systems; (ⅱ) the open-loop structure of the proposed system is simpler than the published electro-mechanical hybrid resonator as reported in²⁴; hence the delays introduced by the data conversion and the digital processing do not influence the performance of the system; (ⅲ) the digital part of the proposed system does not feed back to the mechanical resonator, so the excitation of the system is like the excitation of a single-DoF mechanical resonator, and the entire digital part as well as the data conversion part can be integrated on an ASIC chip; (ⅳ) the digital part of the proposed system is compatible with many different types of mechanical resonators for different applications.

Table 7 Sensitivity comparison with state-of-the-art works

Materials and methods

The specifications of the QCM resonator can be found in ref.³⁶. The setup for the mass perturbation test based on the schematic in Fig. 4a is shown in Fig. 7. The QCM located on the carrier board is put on the laser bombardment platform. Through optical microscope observation, it can be ensured that all pits created by the laser bombardment are uniformly distributed in the central area of the front-side electrode surface. This ensures that each mass removal has approximately the same effect on the QCM resonator.

Fig. 7: Verification of the mass perturbation with the QCM resonator.

a Experimental setup. b The QCM sample under test. c Microscope photo of the laser-bombarded surface

The 2-DoF weakly coupled DETF resonant sensor used for the stiffness perturbation has been fabricated on the basis of a silicon-on-insulator (SOI) wafer process. Table 8 gives the dimensions of the device. The setup for the stiffness perturbation test based on Fig. 4b is shown in Fig. 8. The bonded MEMS device as well as its actuation and readout circuit are located in the vacuum chamber.

Fig. 8: Verification of the stiffness perturbation test with the DETF resonator.

a Experimental setup. b The PCB containing the bonded sample and the actuation/readout circuits. c Microscope photo of the DETF resonator

Table 8 Main dimension parameters of the DETF device used

For both verification tests, the data are recorded by the lock-in amplifier and then post-processed in Matlab.

Conclusion

A new type of multi-DoF cascaded hybrid electro-mechanical resonator system has been introduced and validated. This system presents a new way to link the mechanical resonator and multiple electrical resonators: the open-loop cascaded structure shows WCR-like behavior while it can be driven like a single-DoF MEMS resonator. The concept, structure, and working mechanism have been described in detail, while the theoretical sensitivity, resolution, and measurement range have been calculated. Two types of mechanical resonators (a QCM resonator and a DETF resonator) have been used to validate the functioning of the system as a mass/stiffness change sensor. Both 3-DoF and 5-DoF implementations have been measured. The results are consistent with the theoretical calculations. The proposed sensing system shows the highest normalized sensitivity ever reported. It also has several other advantages over other reported electro-mechanical hybrid WCRs: it is a simple structure, easier to implement as a closed-loop system, and integrated on a chip.

Future work will focus on further reduction of the noise floor, which is the limit to the resolution, and on reducing the power consumption.