Full-scale linear sensing of mode-localized sensors enabled by optimized readout circuit

This paper proposes a novel single-port readout circuit to realize the full-scale linear sensing of mode-localized sensors. In our proposal, in addition to the conventional frequency closed-loop control, the amplitude closed-loop based on automatic gain control (AGC) is adopted to stabilize the amplitude product of resonant units at a normalized value so that the difference of squared amplitudes can be regarded as the sensor readout to characterize the variations of external input signals. A micro-electro-mechanical system (MEMS) mode-localized accelerometer prototype is tested based on our proposed readout circuit. The experiment results demonstrate that the tested accelerometer prototype realizes the linear sensing in the full measurement range and shows a mechanical sensitivity of 0.1182 /g with a maximum non-linearity of -6.54 %, which are basically consistent with the output metric based on the subtraction of reciprocal amplitude ratios (SRAR).


Introduction
Different from the traditional resonant sensors by detecting the frequencies (eigenvalues) of single-degree-of-freedom resonant (1DoF) systems, the mode-localized sensors are sensitive to the change of the inherent characteristics of the sub-resonant system by detecting the amplitude ratios (eigenstates) of the multi-degree-of-freedom (mDoF) weakly coupled resonant systems [1].The amplitude ratios-based readouts of mode-localized sensors exhibit significant advantages in relative mechanical sensitivity and ambient robustness to temperature and atmospheric pressure fluctuations [2][3][4][5].
However, the output metric based on amplitude ratios of the weakly coupled resonant system has strong non-linearity near the zero point, that is, the loci veering phenomenon [6].The linear working range is severely limited, thus reducing the measurement range of the mode-localized sensors.To deal with this issue, our group proposed a novel output metric based on the subtraction of reciprocal amplitude ratios (SRAR) [7].Although the SRAR-based output metric effectively enhances the output linearity of the mode-localized sensors, the convenient and accurate measurement of SRAR remains a challenge.This paper proposes an analogue-digital hybrid readout circuit scheme, which can be utilized to realize the convenient and accurate extraction of SRAR.

Mode Localization of Weakly Coupled Resonators
The simplified mass-damping-spring model of two-degree-of-freedom (2DoF) weakly coupled resonators (WCRs) with double-side stiffness perturbation is established as shown in Figure 1.By solving the motion equations, the eigenvalues and eigenstates (i.e., natural frequencies and amplitude ratios) of the coupled second-order system can be obtained as: where m, c, k are the equivalent mass, damping coefficient and equivalent elastic stiffness of resonant unit, respectively; kc is the coupling stiffness; Δk represents the external stiffness perturbation; The variable quantity i = 1,2 represents the sequence of vibration modes of the dual-mass resonant system.Based on the parametric values listed in Table 1, the variations of the natural frequencies and amplitude ratios versus the external stiffness perturbation are plotted in Figure 2 accordingly.Obviously, no matter the output metrics based on frequencies or amplitude ratios, there is always an inherent veering zone with strong non-linearity near the zero point, which limits the measurement range of mode-localized sensors [6].Amplitudes of sinusoidal excitation forces 5×10 -7 N Figure 2. Numerically calculated natural frequencies and amplitude ratios of a 2DoF weakly coupled resonant system.

Full-scale Linear Sensing
The SRAR-based readout can be defined as the difference between the amplitude ratio and the reciprocal of the amplitude ratio of resonant units whether the in-phase mode or anti-phase mode of the weakly coupled resonant system is stimulated, as expressed by: 2 where Ai,j represents the vibration displacement amplitude of the j-th resonant unit in the i-th mode.It can be seen that the theoretical full-scale linear sensing of mode-localized sensors can be realized by simple algebraic operations in the signal processing process without changing device architecture.Subsequently, the accurate extraction of SRAR is critical for the linear sensing of mode-localized sensors.
The SRAR of a weakly coupled resonant system is usually extracted by accurately demodulating the amplitudes of the resonant units and then performing further algebraic operations.The system output of the mode localized sensors in this scheme is not intuitive, and the division operation consumes tremendous computing resources and inevitably introduces truncation errors [8].Therefore, we are trying to convert the readout circuit into a single port output system through optimization design, thereby simplifying the extraction process of SRAR.
The SRAR expression (3) can be further rewritten as follows: From equation (4), we can infer that when the amplitude product of resonant units is stabilized at a normalized value, the difference between the squared amplitudes can be utilized to characterize the changing trend of SRAR, namely: Therefore, by manipulating the amplitude product of resonant units, the relatively complex measurement of SRAR can be converted into the relatively simple measurement of the difference between the squared amplitudes, which can be realized by multipliers and amplitude closed-loop circuits.The readout circuit is finally converted into a single-port output system.

Optimized Modular Hybrid Readout Circuit
The circuit implementation scheme of the proposed single-port readout system is shown in Figure 3.The driving circuit is designed to quickly and accurately track the resonant frequencies of the operating modes of WCRs, and to stabilize the amplitude product of the resonant units, which can be realized by combining a phase-locked (PLL) and automatic gain control (AGC).In addition to the analogue circuit components such as the capacitance detection circuit based on carrier modulation, the drive interface circuit based on low-noise operational amplifiers, and the algebraic operation module for further processing the amplitude signals, the proportion-integral (PI) control, phase-locked closed-loop control, digital filtering and demodulation of the vibration signals are all implemented in a field programmable gate array (FPGA) chip.The digital PLL consists of a phase-frequency detector (PFD) for the signal phase recognition, a low-pass filter (LPF) based on the infinite impulse response (IIR) topology, a PI controller, and a digital controlled oscillator (DCO) implemented by the coordinate rotation digital computer (CORDIC) algorithm.The drive displacement signals of the resonant units are rectified and filtered, and then multiplied.The obtained amplitude product signal is converted by a high-precision analogue-digital converter (ADC) and sent to the FPGA chip for comparison with the set reference value to obtain a deviation signal.Based on the calculated deviation signal, the controllable gain of the AGC module is extracted after the PI controller.The AGC module continuously adjusts the controllable gain through the PI controller and multiplies it with the output signal of the DCO, thereby generating a sinusoidal excitation signal and feeding it back to the WCRs to stabilize the amplitude product of the resonant units.

Results
In this work, a mode-localized accelerometer fabricated based on the standard deep dry  After fixing the tested mode-localized accelerometer on a precision goniometer, gradually adjust the rotation angle of the goniometer from -90°to 90°to obtain the input acceleration of -1g to 1g.According to the measured data, the steady-state response curve of the difference between the squared amplitudes is plotted as shown in Figure 5.Likewise, we plot the response curve of SRAR-based readout without activating the AGC module for comparison.It can be seen that when the AGC module is activated, the tested accelerometer shows a scale factor of 0.1182 /g and a maximum non-linearity of -6.54 %.Meanwhile, the scale factor of the tested accelerometer without activating the AGC module is 0.1105 /g, and the maximum non-linearity is 4.46 %.Overall, the two output metrics exhibit substantially similar steady-state output characteristics, and the error between the measured scale factors is 6.5 %, which can be attributed to the instability of the test environment.

Conclusions
In this paper, a novel single-port readout scheme is proposed to realize the full-scale linear sensing of mode localized sensors.By stabilizing the amplitude product of resonant units at a normalized value, the difference of squared amplitudes can be regarded as the sensor readout to characterize the variations of external input signals so as to convert the conventional dual-port readout system of mode-localized sensors to the single-port readout system without affecting output characteristics.

Figure 3 .
Figure 3. Circuit implementation of the proposed single-port readout system.
of Physics: Conference Series 2740 (2024) 012022 silicon etching on glass (DDSOG) is utilized to evaluate the proposed single-port readout circuit.Figure4illustrates the device architecture schematic diagram and the detailed microscope photographs of the fabricated prototype.

Figure 4 .
Figure 4.The mode-localized accelerometer for experimental verification.(a) Schematic diagram of device architecture.(b) Microscope photographs of the fabricated prototype.

Figure 5 .
Figure 5. Steady-state responses of the tested mode-localized accelerometer in terms of different output metrics.

Table 1 .
Parametric values adopted in numerical calculation