Experimental Investigation of Nonlinear Modulation Characteristics of Mode Localization in Electrostatic Coupled Resonators

This paper focuses on the typical configuration of double ended fixed supported electrostatic coupled resonators in mode localization phenomena, and conducts in-depth research on the influence of nonlinear coefficients in electrostatic coupled resonators on their vibration behavior. Firstly, the designed devices are processed based on the standard process of SOI, and the printed circuit board (PCB) of the test circuit is designed using Altium Designer software to connect external circuits with the tested device. Secondly, the working principles of electrostatic drive and capacitance detection are elucidated, and the mechanical model in the system is transformed into an electrical model through equivalent circuit principles, thereby revealing the influence of parasitic feedthrough capacitance on the detection signal. Finally, an open-loop testing circuit was built in a vacuum environment to test the modal localization characteristics of the electrostatic coupling resonator and the performance of the mode localization acceleration sensor.


Introduction
In recent years, coupled array structures have become a development trend in the configuration design of micro mechanical resonant sensors, which can improve the sensitivity of sensors and achieve multifunctional detection.However, as resonator structures tend to miniaturize, micro resonators are prone to exhibit nonlinear vibration behavior under size effects, while electrostatic coupled resonators exhibit more complex nonlinear behavior.Therefore, exploring the nonlinear dynamics of electrostatic coupled resonators has great research value for the practical application of micro resonant sensors.The phenomenon of mode localization arises from coupled array structures.Sensors designed using this phenomenon have high sensitivity and good anti-interference ability, and have been widely used in device design, including micro mass sensors [1], acceleration sensors [2], and electrometers [3].Rabenimanana et al. [4] designed a mode localization micro mass sensor using a coupled cantilever beam structure.By increasing the driving voltage to exhibit nonlinear softening behavior, the amplitude was increased, thereby improving the resolution of the sensor.For modal localization sensors, the vibration amplitude can be increased by increasing the excitation voltage, thereby improving the signal-to-noise ratio.However, it is worth noting that as the excitation voltage increases, the coupled resonator exhibits nonlinear vibration behavior (stiffness softening or stiffness hardening), which can easily cause mutual interference between adjacent mode amplitudes, which is the main obstacle limiting the dynamic working range of such sensors.Therefore, for coupled resonators, there is a mutual constraint relationship between increasing the dynamic operating range and increasing the amplitude.Kacem et al. [5] used 1/2 superharmonic resonance in a single resonator to improve the stability of nonlinear systems, and experimentally verified the results, providing a feasible solution for improving the signal-to-noise ratio of single resonator sensors.At present, there is a lack of understanding of the complex nonlinear phenomena in electrostatically coupled resonators, especially the impact mechanism of nonlinear coefficients on the vibration behavior of electrostatically coupled resonators under different excitation conditions.And exploring the nonlinear dynamic characteristics of electrostatically coupled structures can provide technical support for their design and application in sensors.

Structure of the accelerometer
Taking typical electrostatic coupled resonant structures as the research object, delve into their complex vibration behavior under different excitations.As shown in Fig. 1(a), the model consists of two electrostatically coupled resonators, which are composed of two end fixed beams and driven by fixed electrodes.

Device processing and testing circuits
The structure is all manufactured using the standard SOI process, which has achieved a relatively mature commercialization process in MEMS device processing.Fig. 1 (b) is the structural diagram of the coupled resonators, it can be seen that the perpendicularity of the side walls processed by the device is good, and the etching surface roughness is small.This paper will conduct experimental research on the nonlinear mode localization characteristics of the proposed electrostatic coupled resonator structure by building an experimental platform.
The dynamic current iin generated by the vibration of the resonator is usually at nA~μA, it is difficult to detect directly, therefore it is necessary to amplify the detected signal.Here, a cross impedance amplifier is used to amplify and convert the weak current signal output on the resonator detection electrode into a voltage signal.The schematic diagram of the cross resistance amplification circuit is shown in Fig 2(a), and the amplifier used is a low noise amplifier OPA657, the amplification factor is determined by the resistance Rf.According to the working principle of this module, the circuit board design software Altium Designer was used to design the final PCB physical board, as shown in Fig. 2(b).As shown in Fig. 3, the built experimental equipment mainly includes vacuum chamber, trans impedance amplifier (OPA657), DC power supply and Zurich Instruments HF2LI.And the structure and the fixed PCB board are placed together in the vacuum chamber, and connected with the external DC power supply and the transimpedance amplifier through the BNC adapter on the vacuum chamber wall using the SMA connection line; The Signal Output port of the Lock-in amplifier outputs the AC drive signal, which acts on the drive electrode through the coupling capacitor and the bias voltage.The dynamic current generated by the resonator vibration changes from the detection electrode to the voltage signal through the transimpedance amplifier, and then is input to the Signal Input port of the Lock-in amplifier to obtain the amplitude frequency response of the electrostatic coupled resonator under the open-loop test.Due to the output signal of the resonator device is weak, it is difficult to obtain a high signal to noise ratio in the atmospheric pressure environment test.In order to improve the output signal to noise ratio and make the resonator have a high quality factor, it is usually packaged or tested in a vacuum chamber, a vacuum chamber is used for open-loop testing in here.

Fig. 3 .
Fig. 3.The electrostatic coupled resonator vacuum open loop frequency response platform The amplitude frequency response of the electrostatic coupled resonator under different coupling voltages can be obtained by building a vacuum open-loop test circuit.The open-loop test scheme of the electrostatic coupled resonator is shown in Fig 4.Due to the output signal of the resonator device is weak, it is difficult to obtain a high signal to noise ratio in the atmospheric pressure environment test.In order to improve the output signal to noise ratio and make the resonator have a high quality factor, it is usually packaged or tested in a vacuum chamber, a vacuum chamber is used for open-loop testing in here.

Fig. 4 .
Fig. 4. Open loop test scheme of electrostatic coupled resonator After selecting bias voltage Vdc1=15V, Vdc2=15V, coupling voltage Vc=15V, and AC driving voltage Vac=10 mV, the frequency of the electrostatic coupling resonator is swept using the established open-loop testing scheme.As shown in Fig 5 (a) and (b), the output signals with feedthrough of resonators 1 and 2 under open-loop testing are shown.Through the outputs of the two resonators, it can be observed that under electrostatic coupling, mode 1 is out of phase and mode 2 is in phase.The experimental data after removing the feedthrough capacitor using data post-processing is shown in Fig 5 (c) and (d).By comparing the experimental values with theoretical simulation calculation data, it can be found that the two are in good agreement.

Fig. 5 .
Fig. 5.The response of the electrostatic coupled resonator under open-loop test: (a), (b) is the signals of resonators 1 and 2 under feedthrough; (c), (d) is the amplitudes of resonators 1 and 2 after feedthrough