A non-destructive electro-acoustic method to characterize the pull-in voltage of electrostatic actuators

For electrostatic actuators, the pull-in marks an upper limit for the operation range. Once reached, the electrodes come into contact and are shorted without further protection. A non-destructive measurement technique to predict this failure mode is of high interest to allow, e.g. fabrication monitoring or reliability studies. To this end, we develop a surprisingly simple nonlinear lumped parameter model (LPM) for a rather complex electrostatic actuator, designed for an in-ear loudspeaker application. It turns out that a single degree-of-freedom model with only one parameter is sufficient. Our key approach is to experimentally determine this free model parameter by analysing harmonic distortions at low frequencies. Harmonic distortions are a very sensitive tool for nonlinearities. Our method is suggested by simulations with a 2D stationary finite element method (FEM), demonstrating how the analysis of harmonic distortions for voltages far below the pull-in can predict not only the DC pull-in but also the quasi-static AC pull-in voltages at different working points. The distortion analysis of electrostatic actuator ensembles therefore seems a viable route for their non-destructive characterization in the nonlinear domain.


Introduction
The pull-in phenomenon is a critical effect for electrostatic solutions limiting their operation range [23]. The elastic force is counteracted by the electrostatic force. Due to the strong nonlinear character of the latter, a critical voltage, the pull-in voltage, at a critical displacement, the pull-in displacement leads to an acceleration of the movable electrode towards the counter electrode. Once they are in contact the device without mechanical stop or isolation layer is short-circuited. Pull-in predictions within the literature are mainly based on the comparison of a model prediction for a given geometry with the force-deflection characteristic (e.g. [13]) or the resonance frequency shift by the applied voltage due to an electrostatic softening (e.g. [1]). Both methods utilize a single, nearly monotonic curve. Those fit methods are highly affected by the chosen voltage range. For low voltages, the response change is small. This prevents a non-destructive pull-in measurement. Typically, destructive measurements are therefore the only viable route to quantify the pull-in. Recently, actuators based on the nanoscopic electrostatic drive (NED) principle with an air motion setting have been used for the development of micro-loudspeakers based on microelectromechanical systems (MEMS) [4,9]. Motivated by electrostatic balanced systems, Schenk et al. proposed a NED equivalent [19], suppressing the otherwise dominant even harmonic distortions [8]. A detailed description of this loudspeaker can be found in [10]. Since reaching the pull-in often causes terminal device failure, a non-destructive prediction of the pull-in voltage is highly desirable, enabling, e.g. a more detailed device evaluation and advanced reliability studies. Experimentally, it is difficult to use the resonance frequency shift as a non-destructive characterization method for MEMS loudspeakers, due to high damping. The balanced NED actuators show, e.g. a resonator quality of only Q = 3.9 [16]. Likewise, for the NED-loudspeaker, it is not practical to directly measure the force-displacement characteristic optically, as it is done, e.g. for diaphragm-based loudspeakers [14]. This is because the actuator should be characterized under its actual working conditions within a closed cavity. This requires the actuators in a NED based MEMS loudspeaker to be concealed between a top and a bottom cover wafer. The pressure signal on the other hand is the loudspeakers intended output signal and thus, a suitable probe for the device performance. In practice, FEM modelling alone is not well suited to relate measured harmonic distortions to experimental data. This is because such a simulation requires a broad range of input parameters, all subject to process variations when it comes to a silicon chip under test.
In contrast, lumped parameter models can serve as a quite suitable link between the actuator design and its measured performance [18], focusing only on the relevant aspects of the analysis. As shown already for, e.g. comb-drives, simplified spring-plate capacitor models can be utilized to investigate the pull-in [6]. Although the balanced NED actuator has a complex geometry, it is still possible to describe its behaviour with a one degree-of-freedom (1-DOF) spring plate capacitor model [10]. Thereby, the dimensionless equation for a voltage-response characteristic reduces the task of finding the pull-in voltage to finding a voltage scaling factor. Our approach here is to extract this voltage scaling from distortion analysis. Harmonic distortion coefficients offer the advantage to be dependent on the voltage scale, characterizing the device and its nonlineari-ties; however, they are independent of the length scale, measuring the beam deflection (response). For the distortion analysis, the deflection amplitude is decomposed as a Fourier series into its harmonic components. The distortion coefficients K i are then defined by the ratio of the i-th higher harmonic A i to the response amplitude A 1 of the fundamental frequency, Harmonic distortions are in fact a powerful base for audio system characterization, as well as the detection of defects in loudspeakers [11,12]. Within the MEMS context, only a few publications refer to distortion and intermodulation measurements for the system's modelling. Girbau et al. described how intermodulation products provide a base for modelling high-Q RF-MEMS switches [7]. Bounouh et al. make use of the electrical distortion products of MEMS-based energy harvester for estimating mechanical resonance frequencies and damping factors of MEMS devices [3].
Here, we outline how harmonic distortions, generated for a single frequency signal at voltages far below the critical voltage, can be used to predict the pull-in voltage. To this end, we present and motivate in Sect. 2.1 our lumped parameter model for the NED actuator described above. Next, we explain in Sect. 2.2 how harmonic distortions, recorded at operation voltages far below the critical voltage, offer a non-destructive method to estimate the relevant voltage scale. In Sect. 2.3, we demonstrate that our LPM is capable of accurately matching the FEM results regarding the AC and the DC pull-in voltage, after fixing the voltage scale. In Sect. 2.4, we present the respective experimental results, estimating the voltage scale using the third harmonic. Sound pressure level (SPL) measurements and the respective distortion analysis can only be done for loudspeakers, i.e. for ensembles of actuators. This poses the question of parameter variance within this statistical ensemble. To assess the homogeneity of the actuator ensemble used and to further scrutinize our model, we introduce in Sect. 2.5 the method of intentionally asymmetric voltage detuning. We discuss the results and their perspectives in Sect. 3. Our main conclusions are summarized in Sect. 4. A brief summary of applied methods regarding the loudspeaker device, the acoustic measurement setup and the Fig. 1 Schematic clamped-free balanced NED actuator with two NED cells. The applied voltage U at time τ leads to a deflection a. The shaded area represents the actuator's clamped boundary, and the dashed line marks the symmetry of the actuator's geometry. Black rectangles mark the isolation between the three voltage potentials. The geometrical dimensions are altered in this image evaluation of harmonic distortions is given in appendix A, B and C.

Lumped parameter model
The lumped parameter approach of Spitz et al. inspires a similar approach for a balanced NED actuator [10,22]. Figure 1 illustrates the basic appearance of the actuator. A detailed motivation for the following mathematical model structure based on a spring-plate capacitor model with three elastically coupled, movable electrodes as well as further FEM simulations is given online in the supplementary information (Online Resource 4, sec. 5 and 6, respectively), The dynamic variable a represents the dimensionless actuator deflection. For the quasi-static limit at low frequencies, the damping is neglected. Due to the clamped-free configuration, the elastic spring is set linear. Further, the beam consists of two electrostatically driven layers. They result in opposite deflection for the same applied voltage. The upper layer leads to a positive deflection. The actuator is built symmetrically, and bias voltages with opposing signs are applied at the outer electrodes. A signal voltage at the middle electrode leads to a voltage difference in phase with the initial signal within one layer, while the difference in the other one is 180 • phase shifted. This differential drive is referred to as balanced driving here. Voltages are transferred by Eq. (3) into the dimensionless bias voltage u dc as well as the dimensionless time-dependent signal u s = u ac cos(ωτ ) with the forcing frequency ω and the dimensionless time τ . The DC pull-in voltage is utilized as scaling factor U PI and translates the applied voltages U into their dimensionless counterparts u. For a more in depth device characterization, the balanced driving is intentionally detuned towards an asymmetric force distribution to evaluate the impact of small asymmetries. Experimentally, this can be reached by applying bias voltages U dc,1 and U dc,2 which differ in their amplitude. The asymmetry is thereby expressed in the parameter δ. It represents the deviation from the mean bias voltage U dc . Further, a symmetric drive with U 1 = U 2 and the single-sided actuation with U 1 = U and U 2 = 0 are utilized for the development of the LPM. An overview of the actuation conditions and the expected response is given in the supplementary information (Online Resource 4, sec. 5.1). In accordance with the NED design logic, for each layer, multiple electrodes are attached to the beam. The electrostatic force is modelled by two effective capacitors in series representing the two layers. Each capacitor is char-acterized by the applied voltage u i and the effective gap γ i ± a. The outer electrodes have a finite stiffness. Thus, an applied bias voltage will lead to a gap reduction for both capacitors, parameterized by g u . By the electrostatic actuation of any one of the layers, a bending moment is induced due to the electrodes geometry, leading to a deflection of the beam. A stationary FEM simulation reveals that the leverage factor , linking the middle electrode motion and the actuator's entire deflection, is constant with respect to the voltage. The beam bending leads to a compressive stress within the layer in deflection direction and a tensile stress within the opposing layer. The beam deflection deforms the cell electrodes and changes thereby the capacitors gap. The parameter g a captures this effect. A LPM including g u and g a can qualitatively explain the gap variability, but the effect seems to be overestimated in the LPM: As seen by inspecting the resonance frequency shift, as simulated by FEM, a model with and without those two parameters is capable of producing similar results regarding the pull-in voltage (Online Resource 4, sec. 6.1). Further, the gap reduction by the outer electrodes in FEM simulations is lower than 1% of the initial gap, while the LPM suggests a reduction of maximal 15% (Online Resource 4, sec. 6.2 and 6.3). From an experimental point of view, the simpler model version with less variables is preferable. This simpler model with only the DC pull-in voltage as free parameter is shown in the following to be well suited to reproduce all experimental data on harmonic distortions and to allow predicting the pull-in voltages with high precision.

2D-FEM versus LPM: distortion analysis
For a balanced driving, the actuator's harmonic distortion is dominated by the third harmonic K 3 . Figure 2 displays a comparison between the LPM predictions and FEM generated data. K 3 is computed based on Eq. (2) for a variety of bias voltages at fixed signal to bias ratios (sec. C). Higher driving voltages generate higher harmonic distortions. Within the investigated voltage range, these distortions were all below 3%. The maximum bias voltage of 20 V used in this simulation amounts to 49% of the pull-in voltage and is compatible with typical operating conditions for an electrostatic MEMS speaker. Harmonic distortions are independent of the deflection scaling (Eq. 1). Thus, fixing the scale parameter of the LPM only requires the re-scaling of the x-axis by a variation of U 0 , yielding a DC pull-in voltage of U PI = 43.03 V (Eq. 2). For this parameter estimation, the sum absolute residuals were used as the figure of merit, instead of using the sum of squared residuals. This approach intends to avoid underestimating deviations significantly smaller than 1. All curves shown were fitted simultaneously. The resulting maximal absolute deviation for a single working point is 0.04%. At U dc = 20 V, this is equivalent to a relative error of 1.6% and 0.15% for a signal to bias ratio of 0.2 and 0.4, respectively.

2D-FEM versus LPM: prediction of the pull-in voltage
The DC pull-in was extracted by means of a 2D modal FEM as voltage with zero effective stiffness (Online Resource 4, sec. 6.1). This procedure yields a FEM pull-in voltage of U PI,FEM = 43.6 V.
Compared to the LPM result of U PI = 43.03 V, we find a match between LPM and FEM within a relative margin of 1.3%.
Furthermore, the maximal applicable bias for an applied signal voltage is evaluated. A stationary 2D-FEM model is solved with an arc-length solver for dif- ferent signal to bias ratios (similar to [15,17]). The maximal reached voltage is interpreted as pull-in voltage. The LPM-based prediction matches the FEM results within an error margin below 0.26 V and a relative error below 0.9%, Fig. 3. An overview about the methods used to determine the pull-in as well as representative bifurcation diagrams can be found in Online Resource 4, sec. 5.6. Consequently, not only the stationary DC pull-in voltage but also the quasi-static AC pull-in can be accurately predicted by our LPM model.

Experimental distortion measurements for an actuator ensemble
In practical acoustic measurements, the sound pressure reflects the performance of an ensemble of fabricated actuators rather than a single idealized actuator, as assumed in the previous simulations. Additionally, each actuator is now connected to an acoustic load. Despite this, we find a good match between theory and experiment for the balanced NED loudspeaker (Figs. 4 and 5).
The device was measured with a fixed sweep for the signal amplitude at each bias level from low to high voltages (Fig. 4). Further information about the experimental raw data can be found in Online Resource 4, section 7. In comparison, Fig. 5 shows the simulated K 3 by the LPM parameterized with U PI = 34.6 V found by the fit procedure of Sect. 2.2. Figure 6 displays the working points extracted from Fig. 4 utilized for this.  Note that the extracted pull-in voltage U PI,exp = 34.6 V is lower compared to the FEM simulation. The actuator's harmonic distortions are experimentally confirmed to be dominated by the third harmonic over the selected voltage range indicating a highly symmetric device.
The absolute deviation between the model's and experimental K 3 is maximal around ±0.12% and averaged over all curves around ±0.033%. As a reference, the experimental error of K 3 is overestimated by the highest and lowest neighbouring measurement point within the raw measurement for each voltage combination. This leads to a mean error of ±0.07%, suggesting a high compatibility between experimental data and the LPM.

Asymmetric driving
An asymmetric driving scheme is examined in order to test the LPM's robustness as well as to evaluate the homogeneity of the actuator ensemble. With this driving, the system generates even harmonics as dominant distortion. The respective distortion coefficient K 2 as computed by FEM simulations is well reproduced by the LPM with the voltage scale found in Sect. 2.2, emphasizing the fit robustness and the suitability of our LPM, Fig. 7.
As can be seen in Fig. 8, the same holds true for the LPM match with the respective measurements for U dc = 8 V and a variation of the signal amplitude. Again, the experimental and FEM-based pull-in voltages differ. Measurements, FEM and LPM modelling all confirm that K 2 increases linearly for small detuning δ. Small actuator geometry variations, causing an actuator asymmetry, can be compensated by a suitable detuning δ. The minimum of K 2 denotes ideally the point of perfect symmetry.
A manufacturing process variation affecting the symmetry of all actuators in the same way would simply shift the value of this optimal δ. If on the other hand, each actuator within the loudspeaker ensemble was affected differently, the common minimum of all K 2 curves would stop to exist and the occurrence of even harmonics could not be avoided.
Here, neither case is observed. Even with high signal voltages, the distortions for a balanced driving (δ ≈ 0) are below the background level, indicating a good homogeneity of the actuator ensemble and therefore a high loudspeaker quality (Online Resource 4, sec. 7.1 and 7.2). The voltage polarity impacts the amount of even harmonics only slightly. The experimental data match towards the LPM with a pull-in voltage U PI = 34.6 V below ±0.04% depending on δ. On the other hand, the maximum difference between negative and positive bias supply is 0.07%. Together this underlines the high compatibility between the 1-DOF model and the experimental data. In summary, the 1-DOF model in Eq. (2) reproduces not only the sound pressure level [10] but simultaneously the harmonic distortions for a balanced as well as asymmetric driving within a wide voltage range also for experimental measurements with an actuator ensemble.  Figure 9 summarizes the main results. The designed pull-in voltages are higher compared to the experimentally predicted ones. The real pull-in voltage is estimated by destructive measurements at different chips from the same wafer. Optical measurements with a digital holographic microscope (DHM) as in [10] are performed without the cover waver. At stable positions, quasi-static oscillations with 50 Hz are measured. At 15 V bias and signal voltage, the electrodes are shorted by a contact. The destructive acoustic measurement was performed with an ear-simulator (GRAS RA0045) like in [19] at 1 kHz by a signal amplitude sweep at U dc = 19 V. At the unstable position, a sudden SPL drop with a higher current was observed. These measurements indicate a compatibility towards the predicted pull-in by distortion measurements. In general, experimentally pull-in voltages at higher signal to bias voltages should be used as reference rather than an investigation of the balanced case. For the latter, the device is extremely sensitive towards asymmetries. For this device, the pull-in is destructive. Thus, a detailed pull-in measurement at different AC levels was not possible. The distortion level as well as the pressure level depend on the acoustic enclosure. The hydrostatic pressure leads approximately to a linear hardening of the actuators' stiffness. This reduces the distortion level as well as increases the pull-in voltage. Thus, for applying the 1-DOF model to another acoustic setup as with the ear simulator, a scaling factor for the effective linear stiffness might be necessary. The designed pullin voltage of U PI = 43.6 V, is higher compared to the effective pull-in voltage determined experimentally with a model parameter U PI = 34.6 V. In-plane stray fields leading to a higher electrostatic force at the middle electrode are not included in the FEM simulation. These lower the pull-in voltage. The pull-in voltage by the LPM is reduced, e.g. by a smaller effective gap or stiffness. The difference may be addressed to fabrication tolerances, e.g. thinner outer side walls as well as 3D effects like non-straight side walls. The actuator geometry of the FEM simulation is not adapted to fit the experimental results because it refers to the simulation of a single, idealized actuator without the acoustic domain. In contrast, the 1-DOF model is parameterized with effective variables for the loudspeaker device consisting of multiple actuators. Further, the predicted pullin voltage by the LPM is related to a measurement with multiple actuators and can be understood as effective parameter. In general, the possibility of predicting the pull-in voltage for this rather complex actuator geometry suggests the applicability of harmonic distortion fits for the pull-in estimation likewise for less complex structures. Applications could be, e.g. fabrication monitoring tests or reliability studies. Harmonic distortions capture the nonlinearity of the actuator movement with high precision, already at voltages where the actuator deflection is small. Moreover, the distortions are not limited to the acoustic domain. Precise measurements of a deflection or capacity variation would allow the same strategy based on other experimental observables. In addition, the fit strategy is even applicable to static measurements. The deflection per load, measured directly or indirect by, e.g. the capacity, can be translated into a series of distortion curves by the same strategy as the FEM distortions are analysed here (Online Resource 4, sec. 8).

Conclusion
Despite the rather complex actuator geometry, a 1-DOF model is found to describe the actuator's static behaviour. Effects like the voltage-induced gap reduction and the impact of the beam deflection on the electrodes cancel each other. Thus, a model with only one free parameter, namely the DC pull-in voltage, is sufficient. Stationary 2D-FEM simulations are reproduced with a high accuracy. Overall, the main finding is the method to determine the free parameter. The pull-in voltage is predicted by a non-destructive method based on harmonic distortions with a driving far below this critical point. This offers the perspective of a new experimental method for the characterization of electrostatic actuators. Funding Open Access funding enabled and organized by Projekt DEAL. Open Access enabled and organized by Project DEAL.

Data availability
The datasets generated during and/or analysed during the current study are available in the supplementary Online Resource 1, 2 (FEM Data) and 3 (Measurement Data).

Conflict of interest
The authors declare no competing interests. The authors have no relevant financial or non-financial interests to disclose.
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Appendix A: Loudspeaker device
The test chip consists of two electrically independent parts with the same bias voltages at the outer electrodes for each half. Chip rows are selectively activated by the AC signal. Each chip half is divided into 12 rows with 10 actuators per row. The outer rows 1 and 12 are independently connected to the signal voltage, while the other rows are driven pairwise. In total, 240 actuators are placed on the chip. Here, 70 actuators on one chip half are used for the characterization. Bond wires connect the chip to a printed circuit board (PCB). A detailed description of the actuator and the fabrication process can be found in [10]. Figure 10 shows the experimental setup and the NED actuator as well as the MEMS chip. The bias voltage at the outer electrodes is generated independently by a three-channel voltage source (HAMEG instruments, Power Supply HM8142). The audio analyser system, NTI Audio Flexus FX100, generates as well as analyses the signal voltage. All AC voltages are mentioned as peak values. Moreover, the signal's amplitude is adjusted by an amplifier (Krohn-Hite 7602 M). The signal form is fixed to a cosine wave. For the acoustic characterization, the test device on the PCB is connected through its rear side by a 3D printed adapter with a half inch microphone attached to a pre-amplifier (Microtech Gefell MK301 and MV210). The acoustic resonance of the measurement volume is identified with a modal FEM analysis around 19 kHz which is high compared to the actuators resonance at 4.8 kHz. Further, for all distortion measurements the IEEE-standard (Eq. 1) is used to minimize the impact of measurement noise [20]. In order to evaluate the low frequency device behaviour, a signal frequency of 250 Hz (≈ 15% f res /3) is chosen which is far below the actuators resonance frequency of 4.8 kHz. The sound pressure level as well as the distortion products show no resonances near this frequency. This measurement as well as the raw-measurements can be found in Online Resource 4, section 7.

Appendix C: Numerical evaluation method for low frequency distortions
Low frequency distortions can be extracted based on the static voltage deflection curve (u ac − a curve). These curves can be generated by a reduced order model, a FEM simulation or even experimental data making this analysis method widely applicable. The timedependent voltage signal is translated point-by-point by the static voltage deflection curve into a corresponding deflection response. The time scale of the signal and response is only introduced to define the Fourier transformation, which is used secondly to analyse the harmonic distortions of the deflection response. The Moore-Penrose pseudo-inverse is one efficient way of calculating harmonics [2,5,21]. Only necessary frequencies are computed, while a higher number of sampling points in the time domain is taken into account. Reference data for a single 2D actuator are generated with stationary FEM simulations in ANSYS like in [10]. Chosen simulation settings ensure a numerical error below 10 −2 for the distortions. A detailed description of the method, mesh images and an analysis of error sources is placed online (Online Resource 4, sec. 8).