Parametric Analysis of Electrostatic Comb Drive for Resonant Sensors Operating under Atmospheric Pressure

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Introduction
With the rapid development of microelectromechanical systems (MEMS), MEMS resonators, as a basic MEMS device, are widely used in many domains, including sensing [1][2][3][4], energy harvesting [5], timing [6], radio frequency communication [7], and others.Electrostatic driving, thermal driving, piezoelectric driving, and electromagnetic driving are the most common types of driving for MEMS resonators.Due to its low power consumption, compatibility with complementary metal oxide semiconductor processes, ease of downsizing, and electronic control, the electrostatic comb drive structure has become one of the most widely used drive mechanisms in MEMS.Due to the issues of high fabrication and maintenance costs, high technical level requirements, low reliability, and short life of vacuum packaging [8][9][10], electrostatic comb drive for resonant sensors like gyroscopes [11][12][13], magnetic field strength sensors [14], resonant micromirrors [15,16], and electric field sensors [2,3] are operating at atmospheric pressure.In addition, some special microelectrostatic resonators, such as ultrasonic transducers [17], can only function under normal pressure.However, the microcomb resonator under atmospheric pressure is limited in its practical applicability by its high driving voltage and severe feed-through coupling noise.Therefore, improving the displacement and reducing the driving voltage is the research focus of electrostatic comb structure design.
When the resonator is operating, a periodic electrostatic excitation force is formed between the moveable and fixed comb fingers, and the moving comb finger generates horizontal vibration under the influence of the electrostatic force and the elastic restoring force of the spring.The electromechanical coupling between the vibrating and fixed comb finger serves as the electromechanical energy conversion module for the entire resonator construction, converting the input voltage signal into a mechanical vibration signal.In the prior study, raising the electrostatic force and decreasing the spring's mechanical equivalent stiffness were proposed as means to increase the displacement and decrease the driving voltage.The study of Legtenberg et al. [18] and Gupta et al. [19], respectively, compare the modeling and simulation of three different types of suspensions: clamped-clamped beam, crab-leg flexure, and folded flexure.According to the analysis, the spring stiffness in the direction of actuation decreases as the length of the cantilever rises, which causes the driving voltage to be minimized and the displacement to be maximized.However, these designs have a few severe flaws: reducing the mechanical stiffness of the beam reduces the sensitivity of the system, increasing the resolution and response time of the measurement.In addition, the system will get more complex, and the necessary size will rise if the subactuators and the back-end signal modulation system are added.
To avoid a reduction in the microresonator's resonant frequency, domestic and international research focuses primarily on enhancing the electrostatic force density.Ahmed et al. [20] used a direct coupled finite element method to investigate the influence of the overlapping length of the comb finger on the driving performance, and the computed displacement of the comb finger exhibits a nonlinear proportionality with the overlapping length.Tang [21] and Priyadarshini and Mahapatra [22] calculated displacement and optimized the design of the traditional electrostatic comb drive using finite element analysis software.Simulation results indicate that to generate a comb drive with a high electrostatic force density, it is possible to achieve a high comb capacitance change rate by designing densely thin comb fingers with thick polysilicon layers.Consequently, the electrostatic force output is strongly dependent on the smallest gap that can be manufactured.To improve the generated electrostatic force without decreasing the minimum etched trench width, tapered [23][24][25] and stepped [26] comb finger designs were created.
The aforementioned research on the best design of the comb shape and parameters can only guarantee that the electrostatic displacement of the microresonator is maximized at the minimum driving voltage or that the dynamic displacement is maximized while the microresonator is resonating in a vacuum.For the MEMS comb resonator that performs high-speed resonant motion at atmospheric pressure, the ultimate optimization objective is the dynamic resonant amplitude.The calculation of the resonant amplitude of MEMS comb resonators integrates theories from the fields of the electrostatic field, structural mechanics, and fluid mechanics.Due to the scale effect of the microstructure, the air friction between the comb finger is a significant influence in determining the resonator's dynamic characteristics, such as the quality factor and displacement amplitude.Therefore, to improve the microresonator's structural characteristics, both the changes in electrostatic force output and air damping must be considered.
In this paper, the structural parameter dependencies of electrostatic force, air damping, and resonant displacements of the microcomb resonator are investigated exhaustively using the analytical models and the coupled finite element models.Based on the results of the research, general design guidelines for the electrostatic comb finger of resonant sensors under atmospheric pressure are provided.The structure of this paper is as follows: The second section introduces the electrostatic comb-drive resonator for electric field sensor, which is used as an illustration, and focuses on the influence of structural parameters of the microresonator on the electrostatic force and air damping, respectively.The third section describes the 3D electrostatic-structure-fluid coupling model of the resonator and presents simulation results of resonance amplitudes.A general design guideline for the electrostatic comb drive for resonant sensors at atmospheric pressure is presented based on simulation results.Section 4 compares simulations and measurements of the fabricated MEMS device with shapes optimized from numerical simulations.Section 5 concludes the document.

Models of 3D Electrostatic Comb-Drive Resonant Sensors
2.1.The Structure of the Resonator.Figure 1 depicts the 3D model of the microelectrostatic resonance electric field sensor.The device is manufactured with the silicon-on-insulator process.To limit the feedthrough effect, the microresonant sensor is a differentially driven and differentially sensing structure, as seen in the top view of Figure 1(a).The microresonant sensor is comprised of two folding beams, a bilateral electrostatic comb drive, and the moving mass in the center.One end of the folding beam is attached to the vibrating comb fingers, while the other end is anchored.Beams that fold enable elastic recovery for system motion.The moving comb fingers and the fixed comb fingers are staggered to create a plurality of parallel plate capacitor designs.When the resonator is operational, the moving comb fingers generate lateral vibration in response to the electrostatic force and elastic restoring force of the folded beam.In addition, the micromechanism accomplishes electrical isolation through device layer trenching.Figure 1(c) depicts the back aspect of the suspended structure, which is achieved by hollowing out the handle layer and releasing the buried oxygen layer.
Figure 2 depicts the specification of the comb finger unit structure and capacitance, while Table 1 provides the variable names and typical size reference values for every parameter.It is assumed that the construction dimensions of the moving and fixed comb fingers are identical.The length of the comb finger is L, the width is w, the thickness of the film is h, the gap between the comb finger is d, and the distance between the end face of the moveable comb finger and the bottom surface of the fixed comb finger is g 0 , and the overlapping length between the comb fingers is l 0 .C L and C V represent the lateral and vertical capacitances, respectively, between the vibrating and fixed comb fingers.At any given time, l is the length of the overlapping part between the moving and fixed comb fingers, g is the distance between the end surface of the moving comb fingers and the bottom 2 Journal of Sensors surface of the fixed comb fingers, and x is the displacement of the moving comb fingers.Then, g ¼ g 0 − x and l ¼ l 0 − x.
The parameters are provided for the following modeling and analysis of electrostatic forces and damping.
2.2.Electrostatic Force 2.2.1.Analytical Model.3D effects such as fringing fields are ignored in the analytical model since they lead to an underestimated share of the transverse electrostatic force below 5% [18].Let C represent the capacitance between the moveable and fixed comb fingers.When the driving voltage U is put between them, the capacitor's stored electrostatic energy is as follows: Then, the electrostatic force on the comb finger is as follows: First, the capacitance and force of the single-sided comb drive are analyzed.Assuming that the single-sided comb drive has n comb fingers, the capacitance between the moving and fixed comb fingers can be separated into the lateral capacitance (C L ) and the vertical capacitance (C V ).The comb structure of the entire resonator can then be viewed as being formed by 2n parallel plate capacitors.Since the thickness of the comb structure in the selected typical size is significantly greater than the width of the comb finger, only the fringing effect along the thickness direction is taken into account when calculating the lateral capacitance, while the fringing   electric field along the width direction is disregarded.Using the capacitance calculation formula presented in the study of Zhao et al. [27] in consideration of the electric field fringe effect, the single lateral capacitance can be calculated as follows: where ε is the dielectric constant of air.The change rate of the lateral capacitance on a single side can be obtained as follows: where g ¼ g 0 − x.
Similarly, the vertical capacitance and its rate of change can be determined as follows: To reduce interference, the bilateral electrostatic drive excitation method is employed, and the DC bias and opposite-phase AC voltage are applied on both sides, i.e., the applied voltage on the left and right sides of the microcomb resonator is set as follows: where V dc represents the DC bias voltage, V ac represents the AC voltage, and ω a is the circular frequency of the excitation.
Assuming that the rate of change of the lateral capacitance and vertical capacitance along the direction of motion remains fixed [20] and substituting the value of the equilibrium position, i.e., taking the rate of change of capacitance when x ¼ 0, the total electrostatic force can be calculated as follows: Substituting Equations ( 4), ( 6), (7), and ( 8) into (9) to get the following: Table 1 illustrates the structural parameters and reference values of the microresonator.According to Equation (10), the capacitance change rate (or the electrostatic force) is proportional to the film thickness h.Additionally, the capacitance change rate is influenced by the comb finger width, the comb finger gap, the lateral distance between the comb fingers, and the length of the overlapping section of the comb fingers.Consequently, while computing the capacitance change rate, these five parameters are swept from 67% to 167% of their reference values; the resulting calculation is depicted in Figure 3.As anticipated [20], the rate of capacitance change (electrostatic force) is proportional to the comb structure's thickness h, and inversely proportional to the finger gap, d.The remaining three parameters have a minor effect, which means that the vertical capacitance change rate ∂C V =∂x is significantly greater than the lateral capacitance change rate ∂C L =∂x.The comb finger spacing g 0 and the comb finger width w are associated with the lateral capacitance C L .Therefore, the electrostatic force increases slightly upon reducing the comb finger spacing g 0 and increasing the comb finger width w.Besides, the overlap length of the comb fingers l 0 is the parameter with the least influence.Therefore, to achieve the high ∂C=∂x, it is possible to construct narrow comb fingers packed using thick polysilicon films.2) is built using the electromechanics multiphysics interface in COMSOL Multiphysics.Similar to the process of theoretical analysis, the five parameters (h, d, l 0 ; g 0 , and w) are swept and changed to achieve a desirable design that minimizes damping coefficient c 0 and maximizes capacitance change rate ∂C=∂x.The simulation parameters are consistent with those in Table 1 and Figure 3 from the analytical calculations.The electric field absolute value distribution of the comb fingers in the electromechanical coupling model is depicted in Figure 4 as a top view.It can be observed that the electric field in the overlapping portion of the comb fingers is relatively strong, particularly in the sharp corners of the ends of the comb fingers.In accordance with the analytical calculation, the electrostatic force corresponding to the lateral capacitance (the overlapping portion of the comb fingers) is significantly greater than the electrostatic force corresponding to the vertical capacitance (the end of the comb fingers).
Figure 5 depicts the results of the influence of structural parameters on the rate of capacitance change.These results are highly consistent with the analytical calculations except the overlap length of the comb finger l 0 .The analytical model ignores the edge effect of the lateral capacitance (C L ) along the l 0 direction, resulting in inconsistent trends of the curve.However, it has been verified from the analysis and simulation results that l 0 hardly affects the capacitance change rate, so the simplification of the model is reasonable.

Air Damping
2.3.1.Analytical Model.Air damping is the primary cause of energy loss for microresonators oscillating in the air.The moveable component of the resonator consists of the folding beam, the moving comb finger, and the center moving mass.As the electrostatically driven structure is dense interdigitated combs, the air damping of the microresonator under atmospheric pressure is mainly composed of the sliding film damping and the squeezing film damping of the comb fingers.The distribution of air damping on the comb finger is depicted in Figure 6.(1) Sliding film damping between moving and fixed comb fingers The moving comb fingers vibrate laterally to generate sliding film damping by driving the air damping between the comb fingers.Because the microresonator's vibration frequency is relatively high, the flow velocity field created by the surrounding fluid movement driven by viscosity has a short effective penetration depth; hence, the sliding film damping calculation utilizes the more accurate Stokes flow model [28]: where μ is the viscosity of the airflow, δ is the penetration depth, which is defined as the distance when the fluid velocity drops to 1% of its maximum velocity, δ ¼ ffiffiffiffiffiffiffiffiffiffi ffi 2v=ω p ; v is the kinematic viscosity of the airflow, w is the vibration circular frequency of the resonator, b d 1 ¼ d=δ, and A 1 is the total area of the overlapped region between the moving and the fixed comb fingers, which is as follows: (2) Sliding film damping between the lower surface of the moving comb fingers and the substrate Similarly, the sliding film damping coefficient between the moving part of the resonator and the substrate can be calculated as follows: where b d 2 ¼ H=δ; A 2 is the lower surface of the moving comb finger, and (3) Sliding film damping on the upper surface of the moving comb fingers Assuming the microresonator is far from any object above it, the gap approaches infinity, or b d approaches zero.Therefore, the sliding film damping coefficient over the upper surface of the moveable section of the resonator can be calculated as follows: (4) Squeeze film damping between the ends of the moving comb finger Squeeze film damping refers to the damping caused by the squeezing action of the end (or back) of the moveable comb finger on the air film between the end (or back) of the matching fixed comb finger.Air can be regarded as an incompressible fluid for a MEMS resonator operating at atmospheric pressure because its density changes very little.The moving comb finger (the end) is considered to be an ordinary rectangular plate, and its film-damping coefficient is calculated as follows [29]: (5) The total damping coefficient of the comb finger unit In summary, the total damping coefficient of the system at the equilibrium position can be obtained as follows: where Similarly, when evaluating the damping coefficient of comb teeth, the five parameters (h, d, l 0 ; g 0 , and w) vary 6 Journal of Sensors between 67% and 167% of their respective reference values.Figure 7 displays the calculated outcomes.Because the damping coefficient increases as the area of the air film increases and as the thickness of the air film decreases, the calculation result of Figure 7 is plausible.The damping coefficient has a superlinear relationship with the comb finger's thickness h and is inversely related to the finger gap d and finger spacing g 0 .In addition, Figure 7 demonstrates that the damping coefficient rises as the comb finger overlap length l 0 and comb finger width w grow.Journal of Sensors 77.5, respectively.Figure 9 demonstrates that in the equilibrium position, the damping coefficient has a superlinear relationship with the thickness of the comb finger h, which is the most significant structural parameter.The relationship between the absolute value of the curve slopes is as follows: structure thickness h > comb finger spacing g 0 > comb finger gap d > comb finger width w > comb finger overlap length l 0 .

Fluid-Solid Coupling Calculation
Comparing Figures 7 and 9, it can be observed that the variation trends of the analytical curves of all parameters and the simulation results are in good agreement.However, the absolute values of the two are inconsistent.The reason for the difference is that the damping caused by the relative motion of the interdigitated comb fingers is not special but more general in the fluid feature size and boundary conditions.The limitation of the analytical method is that the simple approximation of the aforementioned squeezing film damping and sliding film damping will result in inaccuracies.When the overlapping area of the comb fingers rises gradually, the thickness of the air film reduces progressively, and the results of the finite element simulation will approach those of the analytical calculation.

Parametric Simulation Analysis of Displacement Amplitude
Comparing the results of the finite element simulation in Figures 5 and 9, it can be shown that the changing trend of capacitance change rate under the sweep of five structural parameters is the same as that of air damping, that is the electrostatic force, damping coefficient rise or decrease in pair.
The general optimization law cannot be derived directly.In addition, since the objective of the optimization is to maximize the resonance displacement under the same driving voltage and chip area, an electrostatic-fluid-structure multiphysics coupling 3D finite element model is established.

Electrostatic-Structure-Fluid Coupling Model.
To more accurately quantify the dynamic electrostatic force and damping coefficient, and output displacement of the electrostatic comb drive resonator, the electrostatic-structure-fluid three-field indirect coupling model was developed using the COMSOL Multiphysics.The detailed model construction procedure is described in our published research [30].Figure 10 illustrates the three-field coupling approach.As depicted in Figure 10(a), the primary idea is to separate the electrostatic-fluid-structure three-field coupling into the fluid-structure coupling and the electromechanical coupling.In the fluid-structure coupling transient analysis, a block modeling method for the viscous damping of a microresonator is proposed.As shown in Figure 10(b), for the modeling of the air damping of the plate structure, sliding film damping and squeezing film damping models are used to realize direct coupling.In detail, the sliding damping of the proof mass and the squeeze-film damping of the parallel sensing electrode part are simulated with the thin film interface in COMSOL.For incompressible flow damping of irregular structures (comb elements and folded-flexure suspension) with a general fluid characteristic structure and boundary conditions, a fluid-structure coupling analysis model is established to extract the fitting curve of the damping coefficient with the displacement (x) of the motion.The function of the damping coefficient with displacement (x) is used as the boundary condition of the electromechanical coupling frequency domain analysis for repeated iterative solutions, thereby re-establishing the three-field indirect coupling.Briefly, we extracted the damping coefficient, which is related to the operating point (vibration amplitude), and then substituted it into the electromechanical coupling model to obtain the displacement amplitude.It should be noted that the specific study type of electromechanical forces coupling is the prestressed analysis, frequency domain, which could take into consideration the shift of resonate frequency due to the stress-softening effect by DC load [31].Compared to the measured data, the microresonator displacement computed by this model is quite close to the experimental value, and the accuracy of the resonance amplitude approaches 84.5% [20].

Simulation Results
. The resonant displacement is calculated by using the above finite element model to simultaneously consider the changes in the effective mass and resonant frequency of the system caused by changes in the structural parameters.Similarly, the calculation results of the normalized resonance amplitude X with five structural parameters (h, d, l 0 ; g 0 , and w) swept from 67% to 167% of their reference values are obtained, as shown in Figure 11.
(1) The structural layer thickness h The resonance displacement decreases fast as the structural layer thickness of the resonator h rises, as seen in Figure 11.This is because air damping increases more rapidly than electrostatic force.Electrostatic force and air damping rise linearly and superlinearly, respectively, as the thickness of the comb finger grows.Therefore, it is advisable to keep the structural layer's thickness as thin as feasible.Journal of Sensors (2) The finer spacing g 0 As the spacing between the end faces of the comb fingers g 0 decreases, the damping force on the end faces of the comb fingers (yz plane), which is the principal contribution of air damping (Figure 8), increases rapidly, whereas the electrostatic force does not change appreciably.Therefore, the resonance displacement grows rapidly as the comb finger spacing g 0 increases.As the spacing between the comb fingers grows, so does the lateral breadth of the resonator.To achieve the optimal value, a compromise must be made in the actual design.
(3) The comb finger width w When the comb finger width w grows, so does the fluid load on the end faces of the comb fingers, the displacement reduces, and the silicon area of the chip increases.Therefore, the minimum comb finger width to avoid side pull-in of individual comb fingers [32] should be chosen under conditions permitted by the manufacturing process, that is, as follows: is the effective elastic modulus in bending, E and ν are Young's modulus and Poisson's ratio of the structural material, and Ṽ PI is the normalized pull-in voltage, and V max is the maximum value of the driving voltage.
(4) The overlapping length of the comb finger l 0 Changes in l 0 have little effect on the resonance displacement.However, to lower the lateral width of the chip, the overlapping length of the comb finger l 0 should be minimized as much as feasible until it equals the maximum output displacement necessary.
(5) The vertical gap of the comb finger d When the vertical gap of the comb finger d decreases, the electrostatic force grows faster than the air damping, the displacement increases, and the number of comb fingers inside the same chip area increases.Therefore, the electrostatic force comb also depends on the minimum gap size that the process can manufacture.If the minimal feature and minimal trench are equal (i.e., w ¼ d), then the minimal gap d and finger width w can be obtained from Equation (22): In conclusion, except for the comb finger gap d, the changing trends of the resonance displacement and electrostatic force are completely different when the other four parameters change.For electrostatic comb drive resonators that oscillate under atmospheric pressure or for resonators with air damping as the primary damping contribution, to obtain high voltage driving efficiency, a thin polysilicon film can be used to design narrow comb fingers that are dense in the vertical direction and loose in the lateral direction.

Experiments and Discussion
To verify the accuracy of the calculation results and the parametric analysis, an electrostatic comb-drive resonator with shapes optimized from numerical simulations was fabricated.Figure 12 shows the frequency response curves comparing both simulation and experimental results under the applied voltage of V dc ¼ 9 V and V ac ¼ 4:5 V.It can be seen that the resonant frequencies of the two are close (a difference of 4.35%), but the quality factor of the simulation curve is higher, and its damping coefficient is lower, as will be discussed later.
Table 2. shows the definition of the parametric optimization for a microelectrostatic comb-driven resonator.Here, X z ð Þ is the resonance amplitude function, A z ð Þ is the function which describes the total area of comb drive, and f z ð Þ is the function of drive efficiency per unit comb drive area.z is the vector of optimization variables.Lower bounds (lb) and upper bounds (ub) limit the variables to reasonable physical values considering the manufacturing process and practical application.Therefore, the optimization criteria for the structural parameters of the comb drive is to maximize the resonance amplitude under the same chip area.Table 3 shows the dimension parameter values of the initial and optimized comb drive structure, whose SEM images at different positions are shown in Figure 13.The detailed fabrication process flow and vibration test procedures are illustrated in our published paper [30].Journal of Sensors Figure 14 shows the comparison curves of the measured vibration amplitudes and the simulated results of the initial MEMS device [30] and the optimized one, respectively.It can be seen in Figure 14(b) that the resonance frequency error and the vibration amplitude average error of the optimized device are 0.52% and 10.26%, respectively, which indicating the simulation is in good agreement with the experimental results.The reasons for the differences mainly include the neglected damping terms such as material internal losses, thermoelastic damping and friction in joints, fabrication process error, measurement error, and the error brought by the equivalent linearization of the three-field indirect coupling model [30].Besides, comparing the measurement results of the devices before and after optimization, it can be seen that the optimized device area is only increased by 29%, but the driving efficiency is enhanced by 102%, which validates the above parametric analysis and shows the superiority of the optimized shapes.

Conclusions
In this paper, an electrostatic force analytical model that accounts for edge effects and a simplified damping analytical model based on the thin-film damping model is used to

Journal of Sensors
analyze the electrostatic force and all damping terms, as well as their dependence on the structural parameters (h, d, l 0 ; g 0 , and w) of a microelectrostatic comb drive resonator operating at atmospheric pressure.The results show that the structural parameters of the comb fingers influence both the electrostatic force output and the air damping of the resonator.The finite element simulations further validated this result.Among them, the electrostatic force is linear with the thickness (h) of the structure and inversely proportional to the comb finger gap (d).The remaining three parameters have little impact.The damping coefficient at the equilibrium position is superlinear with the structure's thickness (h) and is inversely proportional to the comb finger spacing (g 0 ).As the comb finger gap (d) lowers and the comb finger width (w) increases, the damping coefficient increases slightly.
Changing the comb finger's overlapping length (l 0 ) has almost no effect on the damping.
The resonant displacement is also directly calculated by using the electrostatic-fluid-structure multiphysics coupling 3D finite element model to simultaneously consider the changes in the effective mass and resonant frequency of the system caused by changes in the structural parameters.The results show that, for electrostatic comb drive resonators that oscillate under atmospheric pressure or for electrostatic comb drive resonators with air damping as the primary damping contribution to obtain high voltage driving efficiency, a thin polysilicon film can be used to design narrow comb fingers that are dense in the vertical direction and loose in the lateral direction.To validate the parametric analysis, an electrostatic comb-drive resonator with shapes optimized from numerical simulations was fabricated.The results show that the resonance frequency error and the vibration amplitude average error of the optimized device are 0.52% and 10.26%, respectively, which confirms the agreement between simulations and measurements; the driving efficiency is enhanced by 102% with the chip area increased by 29%, which validates the above parametric analysis and shows the superiority of the optimized shapes.

FIGURE 2 :
FIGURE 2: The definition of the structural parameters of the electrostatic comb drive unit.

FIGURE 1 :
FIGURE 1: The 3D model of the electrostatic comb-drive resonator for electric field sensor: (a) front view; (b) detail of the electrostatic comb drive; (c) backside view.
(nF/m) Length ratio (%) Film thickness h Finger gap d Finger overlap l 0 Finger spacing g 0 Finger width w

FIGURE 3 :
FIGURE 3: Theoretical calculation results of the capacitance change rate with film thickness, finger length, finger overlap, finger spacing, and finger width swept from 67% to 167% of their reference values.

FIGURE 4 :Finger overlap l 0 Finger spacing g 0 Finger width w ðbÞ FIGURE 5 :
FIGURE 4: Distribution map of the absolute value of an electric field of comb finger.Note.The voltage difference between the moving comb and the fixed comb is 1 V. Filleting the geometry does not affect the capacitance results of the finite element calculation.

FIGURE 6 :
FIGURE 6: Schematic diagram of viscous damping distribution of a single comb finger unit.

ðbÞFIGURE 8 :
FIGURE 8: Simulation results of fluid-solid coupling calculation model: (a) the stress distribution on the surface of comb finger; (b) the x component of fluid load distribution on the surface of comb finger.Note.At this time, the x component of the comb finger speed is 3.5468 × 10 −4 m/s.

FIGURE 7 :
FIGURE 7: Theoretical calculation results of the damping coefficient with film thickness, finger length, finger overlap, finger spacing, and finger width swept from 67% to 167% of their reference values.
Finger gap d Finger overlap l 0 Finger spacing g 0 Finger width w

FIGURE 9 :
FIGURE 9: Finite element simulation results of the effect of the structural parameters on the damping coefficient.

Finger overlap l 0 Finger spacing g 0 FIGURE 11 :
FIGURE 11: Calculation results of the displacement amplitude swept with structural parameters in the electrostatic-fluid-structure multiphysics coupling 3D finite element model.TABLE 2: Definition of the parametric optimization for a combdriven resonator.

FIGURE 12 :
FIGURE 12:  The frequency response curves comparing both simulation and experimental results under the applied voltage of V dc ¼ 9 V and V ac ¼ 4:5 V.

TABLE 1 :
Typical dimension parameter values of comb drive structure.Symbol Parameter Typical value L Comb finger length 15 μm w Comb finger width 3 μm d Comb finger gap 3 μm h Film thickness 30 μm g 0 Comb finger spacing 11 μm l 0 Comb finger overlapping length 4 μm s Main drive arm width 18 μm D Back spacing of comb finger 29 μm H Cavity depth 400 μm n Unilateral comb finger number 720

TABLE 2 :
Definition of the parametric optimization for a combdriven resonator.

TABLE 3 :
Dimension parameter values of initial and optimal comb drive structure.
FIGURE 13: SEM images of the optimized comb drive structure at different positions.