Influence of the driving frequency and equivalent parameters on displacement amplitude of electrostatic linear comb actuator

Vietnam Journal of Mechanics, VAST, Vol. 40, No. 4 (2018), pp. 397 – 406 DOI: https://doi.org/10.15625/0866-7136/13215 INFLUENCE OF THE DRIVING FREQUENCY AND EQUIVALENT PARAMETERS ON DISPLACEMENT AMPLITUDE OF ELECTROSTATIC LINEAR COMB ACTUATOR Hoang Trung Kien1, Vu Cong Ham2, Pham Hong Phuc1,∗ 1Hanoi University of Science and Technology, Vietnam 2Le Quy Don University, Vietnam ∗E-mail: phuc.phamhong@hust.edu.vn Received Octorber 22, 2018 Abstract. A new method determining the equivalent dynamic parameters such as stiff- ness, vibrating mass, and air damping factor in motion direction of shuttle (i.e. in y- direction) is proposed, thence the differential motion equation of shuttle is established and solved to achieve a typical displacement formula. Simulation and experimental results show that the change of ELCA’ displacement is inappreciable while the range of driving frequency up to 27 Hz (error of 10% with driving voltage is a square wave). Moreover, the range of driving frequency for the ELCA can be extended up to 1 kHz with displacement amplitude error of 10% while the shape of driving voltage is a harmonic sine wave. Keywords: Electrostatic Linear Comb Actuator (ELCA), driving frequency, dynamic pa- rameters, displacement amplitude. 1. INTRODUCTION Micro electromechanical system (MEMS) is an advance technology and widely ap- plied in many fields like robotics, transportation, aerospace, medicine, or in electronic industry, etc. Micro actuators are one of the MEMS products and play an important role which generates a power to drive the micro devices as well as micro systems. Electro- static Linear Comb Actuator (ELCA) is a typical actuator with working principle based on the electrostatic force producing linear movement. The advantages of the ELCA sys- tem are fast response, high performance, low energy consumption, and simple config- uration. That is the reason why it is commonly used for driving MEMS devices, such as resonators [1], gyroscopes [2], micro conveyer/transportation systems [3], micro mo- tor [4], or micro gripper [5]. c© 2018 Vietnam Academy of Science and Technology 398 Hoang Trung Kien, Vu Cong Ham, Pham Hong Phuc Recently, there are many researches on the ELCA in various aspects. They cover main topics such as: new structures, working quality, manufacturing process improve- ment, and application of smart materials. Particularly, the fabrication quality as well as the stability of the ELCA’s motion determines the working accuracy of the devices. The shapes of suspension beams such as a crab-leg and fold [6, 7] have been designed in order to reduce the stiffness and improve the displacement. Another research on improv- ing displacement has been published [8], by reducing out-of-plane movement from that establishes in-plane displacement. The authors in [9] have proposed a structure of an ac- tuator by adding a module which combines of a sensor and a sub-actuator for controlling and minimizing instability. The influence of side etching effect on displacement of the ELCA after deep-Reactive ion etching (D-RIE) process has been mentioned particularly by [10]. When calculating the displacement of the ELCA in design task [3, 4, 10–12], the au- thors used the theoretical formula which is established based on static equilibrium equa- tion, aiming simplifies computing and reduces a design time. In fact, these devices work in state of motion though applying AC voltage likes a square or sine wave, etc. There- fore, considering the influence of inertia force and air damping plays an important role in predicting the fact displacement of the ELCA. The dynamic of the ELCA has studied in [13, 14], in which the electrostatic actuators are used for driving resonators and gyro- scopes. However, the parameters of differential equation of motion such as an effective mass and conversion air damping factor have not clearly determined in these publica- tions. In this paper, we propose equivalent dynamic parameters, i.e. the effective mass of the ELCA’s movable part and conversion air damping factor via equivalent transforma- tion kinetic energy and the air drag method. The ELCA’s displacement is examined and discussed corresponding to the change of driving frequency by solving the differential equation of motion. 2. THEORETICAL DISPLACEMENT OF ELCA 2.1. Configuration and displacement of ELCA Fig. 1 shows the configuration of ELCA, which consists of a movable part (shuttle) 1 and fixed part (anchor pads 2 and fixed electrodes 3 ). The movable part (consists of movable comb fingers) which is suspended by four elastic beams system moving in y-direction under the electrostatic force, one end of each beam is fixed on anchor pad 2 . The structure and dimensions of movable and fixed comb fingers are the same. Tab. 1 describes the geometric parameters of ELCA as shown in Fig. 1, where b is the width of a beam, h is the thickness of device layer, and ga is the gap between the shuttle’s bottom and substrate. Table 1. Value of the ELCA geometric parameters Length (µm) Width (µm) Thickness (µm) Gap (µm) Comb finger lc = 40 w = 3 h = 30 g = 2 Suspension beam L = 515 b = 3.6 h = 30 ga = 4 Influence of the driving frequency and equivalent parameters on displacement amplitude of electrostatic linear . . . 399 2 Hoang Trung Kien, Vu Cong Ham, and Pham Hong Phuc tors and gyroscopes. However, the parameters of differential equation of motion such as an effective mass and conversion air damping factor have not clearly determined in these publications. In this paper, we propose equivalent dynamic parameters, i.e. the effective mass of the ELCA’s movable part and conversion air damping factor via equivalent transformation kinetic energy and the air drag method. The ELCA’s displacement is examined and discussed corresponding to the change of driving frequency by solving the differential equation of motion. 2. THEORETICAL DISPLACEMENT OF ELCA 2.1. Configuration and displacement of ELCA Fig. 1 shows the configuration of ELCA, which consists of a movable part (shuttle)  and fixed part (anchor pads  and fixed electrodes ). The movable part (consists of movable comb fingers) which is suspended by four elastic beams system moving in y-direction under the electrostatic force, one end of each beam is fixed on anchor pad . The structure and dimensions of movable and fixed comb fingers are the same. Table 1 describes the geometric parameters of ELCA as shown in Fig. 1, where b is the width of a beam, h is the thickness of device layer, and ga is the gap between the shuttle’s bottom and substrate. Movable part Fixed part Beams y x Shuttle L ② ③ ② ③ Fix electrode ① lc w0 g0 Anchor Fig. 1. Configuration of ELCA Table 1. Value of the ELCA geometric parameters Length (µm) Width (µm) Thickness (µm) Gap (µm) Comb finger lc = 40 w0 = 3 h =30 g0 = 2 Suspension beam L = 515 b = 3.6 h = 30 ga = 4 While applying a voltage V between anchor pads  and fixed electrodes  of the ELCA, the total electrostatic force in y-direction is expressed as follows: 20 0 e nh F V g   (1) Where n is a total number of the movable comb fingers; ε0 = 8.854×10-6 pF/µm and ε = 1 are permit- tivity of vacuum and air, respectively; V is the driving voltage. The total stiffness in y-direction of four suspension beams system can be calculated as [4, 10]: 3 3 4Ehb K L  (2) Fig. 1. Configuration of ELCA While applying a voltag V between anc or pads 2 and fixed electrodes 3 of the ELCA, the total electrostatic force in y-direction is expressed as follows Fe = nhεε0 g0 V2, (1) where n is a total number of the movable comb fingers; ε0 = 8.854×10−6 pF/µm and ε = 1 are permittivity of vacuum and air, respectively; V is the driving voltage. The total stiffness in y-direction of four suspension beams system can be calculated as [4, 10] K = 4Ehb3 L3 , (2) where E = 169 GPa i silicon Young’s modulus. While applying a DC voltage for ELCA, the displacement y0 of ELCA (i.e. displace- ment of the shuttle) is directly inferred from force balance equation Fe − Fel = 0 or nhεε0g0 V 2 = 4Ehb3 L3 y0. Here, Fel is a total elastic force of suspension beams. We have y0 = nεε0 4Eg0 ( L b )3 V2. (3) Eq. (3) indicates the quadratic relation between the displacement of the ELCA and the DC driving voltage. 2.2. Dynamic model of ELCA In case driving voltage is an AC: considering the motion of the ELCA’s shuttle, this motion is similar to the motion of a mass with one-degree of freedom shown in Fig. 2. K, C, M are the conversion stiffness, conversion damping coefficient and conversion ef- fective mass of the ELCA system, respectively. Fe(t) is an external total force (here is an electrostatic driving force). 400 Hoang Trung Kien, Vu Cong Ham, Pham Hong Phuc Influence of the driving frequency and equivalent parameters on displacement amplitude of electrostatic linear comb actuator 3 Where E = 169GPa is silicon Young’s modulus. While applying a DC voltage for ELCA, the displacement y0 of ELCA (i.e. displacement of the shuttle) is directly inferred from force balance equation: 3 20 03 0 4 0 or: .e el nh Ehb F F V y g L     Here, Fel is a total elastic force of suspension beams. We have: 3 20 0 04 n L y V Eg b         (3) Equation (3) indicates the quadratic relation between the displacement of the ELCA and the DC driving voltage. 2.2. Dynamic model of ELCA In case driving voltage is an AC: considering the motion of the ELCA’s shuttle, this motion is similar to the motion of a mass with one-degree of freedom shown in Fig. 2. Where K, C, M are the conversion stiffness, the conversion damping coefficient and the conversion effective mass of the ELCA system, respectively; Fe(t) is an external total force (here is an electrostatic driving force). M K C Fe(t) Fig. 2. 1-DOF dynamic model of the ELCA The differential equation of ELCA’s motion in y-direction (Fig. 1) can be expressed: ( )eMy Cy Ky F t   (4) In order to calculate exactly the displacement of the ELCA through differential equation, the equivalent coefficients K, M, C must be accurately identified. These coefficients depend on various parameters such as geometric dimensions, material properties, working environment, etc. of the sys- tem. The conversion stiffness K in y-direction can be determined by equation (2) above. The effective mass of movable parts (including the elastic beams) converted to shuttle’s motion is based on rule: the kinetic energy of effective mass M is equal to total kinetic energy of all movable parts of the ELCA, thus we have: 2 0 4 ( ) ( ) L b s M y x M M dx L y L          (5) Where Ms, Mb are the mass of the shuttle and of one single beam, respectively; ẏ(L), ẏ(x) are the veloc- ity of the shuttle and the section of a single beam at x-position. We assume that directional movement of the shuttle and beam cross-sections at x-position are the same. Thus, relation between ẏ(x) and ẏ(L) is obtained through the relation of displacements: Fig. 2. 1-DOF dynamic model of the ELC The differential equation of ELCA’s motion in y-direction (Fig. 1) can be expressed My¨ + Cy˙ + Ky = Fe(t). (4) In ord r o calculate exactly the displacement of the ELCA t rough d ff rential quation, the equivalent coefficients K, M, C must b accurat ly identified. These coefficients de- pend on various parameters such as geometric dimensions, material properties, working environment, etc. of the system. The conversion stiffness K in y-direction can be determined by Eq. (2) above. The effective mass of movable parts (including the elastic beams) converted to shut- tle’s motion is based on rule: the kinetic energy of effective mass M is equal to total kinetic energy of all movable parts of the ELCA, thus we have M = Ms + 4Mb L L∫ 0 ( y˙(x) y˙(L) )2 dx, (5) where Ms, Mb are the mass of the shuttle and of one single beam, respectively; y˙(L), y˙(x) are the velocity of the shuttle and the section of a single beam at x-position. We assume that directional movement of the shuttle and beam cross-sections at x- position are the same. Thus, relation between y˙(x) and y˙(L) is obtained through the relation of displacements y(x) = 3Lx2 − 2x3 L3 y(L)⇒ y˙(x) = 3Lx 2 − 2x3 L3 y˙(L). (6) From (5) and (6), we have M = Ms + 4 13Mb 35 = ρh(At + 52bL 35 ), (7) where ρ = 2330 kg/m3 is the mass density of silicon; At is the total top side area of the shuttle. According to [15], the conversion air-damping coefficient C in Eq. (4) can be calcu- lated C = C1 + C2 + C3, C1 is the air slide-film damping coefficient on shuttle’s bottom and movable fingers’ face versus substrate and fixed fingers, respectively C1 = µ ( Abt ga + 2Asc g0 ) , (8) Influence of the driving frequency and equivalent parameters on displacement amplitude of electrostatic linear . . . 401 where µ = 1.85×10−5 Pa·s (at room temperature T0 = 20◦C) is the viscosity coefficient of air; Abt, Asc are the total areas of the shuttle’s bottom and comb fingers’ face, respectively. C2 is the air slide-film damping coefficient on the top and two sides of the shuttle C2 = µ At + Ass δ , (9) where At, Ass are the top and two sides area of the shuttle; δ = √ 2µ ρaω is the effective distance; ρa is the mass density of air, ω is the angle frequency of system. C2 is trivial and can be ignored in some cases. C3 is the air drag coefficient on the vertical surface of the movable part and can be approximated as follows C3 ≈ 323 µ h 2 . (10) Using Eqs. (8), (9) and (10), we have C = µ ( Abt ga + 2Asc g0 + At + Ass δ + 16h 3 ) . (11) The force Fe(t) in Eq. (4) is a total electrostatic force, which is generated by applying voltage on two electrodes of the ELCA. This force depends on amplitude, frequency and kind of driving voltage wave. In case of the driving voltage is the square wave as in Fig. 3. 4 Hoang Trung Kien, Vu Cong Ham, and Pham Hong Phuc 2 3 3 3 2 ( ) ( ) Lx x y x y L L   2 3 3 3 2 ( ) ( ) Lx x y x y L L    (6) From (5) and (6), we have: 13 52 4 ( ) 35 35 b s t M bL M M h A     (7) Where ρ = 2330 kg/m3 is the mass density of silicon; At is the total top side area of the shuttle. According to [15], the conversion air-damping coefficient C in equation (4) can be calculated: 1 2 3C C C C   C1 is the air slide-film damping coefficient on shuttle’s bottom and movable fingers’ face versus substrate and fixed fingers, respectively: 1 0 2bt sc a A A C g g         (8) Where µ = 1.85×10-5 Pa∙s (at room temperature T0 = 200C) is the viscosity coefficient of air; Abt, Asc are the total areas of the shuttle’s bottom and comb fingers’ face, respectively. C2 is the air slide-film damping coefficient on the top and two sides of the shuttle: 2 t ssA AC     (9) Where At, Ass are the top and two sides area of the shuttle; 2 a      is the effective distance; ρa is the mass density of air, ω is the angle frequency of system. C2 is trivial and can be ignored in some cases. C3 is the air drag coefficient on the vertical surface of the movable part and can be approximat- ed as follows: 3 32 3 2 h C   (10) Combination of equations (8), (9) and (10), we have: 0 2 16 3 bt t ss a A A A A h C g          (11) The force Fe(t) in the equation (4) is a total electrostatic force, which is generated by applying voltage on two electrodes of the ELCA. This force depends on amplitude, frequency and kind of driv- ing voltage wave. In case of the driving voltage is the square wave as in Fig. 3 T/2 T/2 V t 0 Fig. 3. Square wave of driving voltage The Fe (t) force function can be expressed as: Fig. 3. Square wave of driving voltage The Fe(t) force function can be expressed as Fe(t) =  nhεε0 g0 V2, 0 ≤ t ≤ T/2 0, T/2 < t ≤ T (12) where T is one cycle of the applying voltage. Accordingly, the displacement of differential equation (4) has the form y1 = e−βt (A1 cos(ωt) + A2 sin(ωt)) + Fe K . (13) Here β = C 2M ; ω = √ ω2n − β2 is the damped angle frequency; ωn = √ K M is the natural angle frequency. 402 Hoang Trung Kien, Vu Cong Ham, Pham Hong Phuc According to the initial condition: y(0) = 0 and y˙(0) = 0, we have: A1 = −FeK ; A2 = − β ω Fe K . The general displacement formula of the shuttle in this case can be expressed y1 = nεε0 4Eg0 ( L b )3 V2 [ 1− ( cos(ωt) + β ω sin(ωt) ) e−βt ] . (14) In case of the driving voltage is the sine wave function as shown in Fig. 4. Influence of the driving frequency and equivalent parameters on displacement amplitude of electrostatic linear comb actuator 5 20 0 0 / 2 ( ) 0 / 2 e nh V t T gF t T t T         (12) Where T is one cycle of the applying voltage. Accordingly, the displacement result of differential equation (4) has formation:  1 1 2cos( ) sin( ) t eFy e A t A t K      (13) Here: 2 C M   ; 2 2n   is the angle frequency while damping; n K M   is the nature angle frequency. According to the initial condition: (0) 0y  and (0) 0y  , we have: 1 eFA K   ; 2 eFA K     The general displacement formula of the shuttle in this case can be expressed: 3 20 1 0 1 cos( ) sin( ) 4 tn Ly V t t e Eg b                   (14) In case of the driving voltage is the sine wave function as shown in Fig. 4. V t V0 Fig. 4. Sine wave of driving voltage V = V0(1 + sin(Ωt)) By setting: 200 0 0 nh F V g   , the driving force can be expressed as: 0 3 1 2sin( ) os(2 ) 2 2 esF F t c t          (15) In this case, the displacement result of differential equation (4) is calculated:           20 0 1 2 2 2 2 20 2 2 2 2 1 2 3 2 sin( ) os( ) 2 4 os(2 ) 2 sin(2 ) 2 4 8 cos( ) sin( ) s t F F y K M t C c t K K M C F K M c t C t K M C e B t B t                                (16) According to initial condition: 1 (0) 0sy  and 1 (0) 0sy  , we have: Fig. 4. Sine wave of driving voltage V = V0(1+ sin(Ωt)) By setting: F0 = nhεε0 g0 V20 , the driving force can be expressed as Fes = F0 [ 3 2 + 2 sin(Ωt)− 1 2 cos(2Ωt) ] (15) In this case, the displacement result of differential equation (4) is calculated y1s = 3F0 2K + 2F0 (K−MΩ2)2 + C2Ω2 [( K−MΩ2) sin(Ωt)− CΩ cos(Ωt)] − F0 2(K− 4MΩ2)2 + 8C2Ω2 [( K− 4MΩ2) cos(2Ωt) + 2CΩ sin(2Ωt)] + e−βt [B1 cos(ωt) + B2 sin(ωt)] . (16) According to initial condition: y1s(0) = 0 and y˙1s(0) = 0, we have B1 = F0 { 2CΩ (K−MΩ2)2 + C2Ω2 + K− 4MΩ2 2 (K− 4MΩ2)2 + 8C2Ω2 − 3 2K } , B2 = F0 ω { 2CΩ2 (K− 4MΩ2)2 + 4C2Ω2 − 2Ω(K−MΩ 2) (K−MΩ2)2 + C2Ω2 } + β ω B1. 3. ANALYSIS, SIMULATION AND MEASUREMENT RESULTS Material properties of the silicon [4, 10] and air are listed in Tab. 2. The conversion coefficients in differential equation (4) calculated by Eqs. (2), (7) and (11) are shown in Tab. 3. The ELCA’s simulation is implemented by using a finite element method (FEM) in ANSYS. At driving voltage of 100 V, the maximum stress on suspension beams is 28.2 MPa and the obtained displacement y of the shuttle is 38.42 µm, as shown in Fig. 5. Influence of the driving frequency and equivalent parameters on displacement amplitude of electrostatic linear . . . 403 Table 2. Material properties of silicon and air Young’s modulus of silicon E (Pa) 1.69E+11 Mass density of silicon ρ (kg/m3) 2330 Mass density of air (20◦C, 1atm) ρa (kg/m3) 1.205 Viscosity coefficient of air (20◦C, 1atm) µ (Pa·s) 1.85E-5 Permittivity constant of vacuum ε (pF/µm) 8.854E-6 Permittivity coefficient of air ε 1 Table 3. Conversion coefficients of the ELCA Conversion stiffness K (µN/µm) 6.927 Conversion effective mass M (kg) 1.008E-08 Conversion damping C (µN·s/µm) 2.322E-06 6 Hoang Trung Kien, Vu Cong Ham, and Pham Hong Phuc         2 1 0 2 2 2 2 2 2 2 2 2 2 0 2 12 2 2 2 2 2 2 2 2 4 3 22 4 8 2 2 ( ) 4 4 C K M B F KK M C K M C F C K M B B K M C K M C                                          3. ANALYSIS, SIMULATION AND MEASUREMENT RESULTS Material properties of the silicon [4, 10] and air are listed in Table 2: Table 2. Material properti s of silicon and air Young’s modulus of silicon E (Pa) 1.69E+11 Mass density of silicon ρ (kg/m3) 2330 Mass density of air (200C, 1atm) ρa (kg/m3) 1.205 Viscosity coe ficient of air (200C, 1atm) µ (Pa∙s) 1.85E-5 Permittivity constant of vacuum ε0 (pF/µm) 8.854E-6 Permittivity coefficient of air ε 1 The conversion coefficients in differential equation (4) calcul ted by equations (2), (7) and (11) are shown in Table 3: Table 3. Conversion coefficients of the ELCA Conversion stiffness K (µN/µm) 6.927 Conversion ffective mass M (kg) 1.008E-08 Conversion damping C (µN∙s/µm) 2.322E-06 The ELCA’s simulation is implemented by using a finite ele ent method (FEM) in ANSYS. At driving voltage of 100V, the maximum stress on suspension beams is 28.2MPa and the obtained dis- placement y of the shuttle is 38.42µm, as shown in Fig. 5. The natural frequency n of the system determined by theoretical analysis is 4172Hz, and by ANSYS is 4168Hz, i.e. deviation is only 0.1%. (a) Displacement b) Natural frequency Fig. 5. Simulation of the ELCA The model of ELCA is fabricated by using standard SOI-MEMS technology with a 30μm thick silicon device layer, 4μm thick buried silicon dioxide SiO2 and 450μm thick silicon substrate of SOI (a) Displacement 6 Hoang Trung Kien, Vu Cong Ham, and Pham Hong Phuc         2 1 0 2 2 2 2 2 2 2 2 2 2 0 2 12 2 2 2 2 2 2 2 4 3 22 4 8 2 2 ( ) 4 4 C K M B F KK M C K M C F C K M B B K M C K M C                                          3. ANALYSIS, SIMULATION AND MEASUREMENT RESULTS Material properties of the silicon [4, 10] and air are listed in Table 2: Table 2. Material properties of silicon and air Young’s modulus of silicon E (Pa) 1.69E+11 Mass density of silicon ρ (kg/m3) 2330 Mass density of air (200C, 1atm) ρa (kg/m3) 1.205 Viscosity coefficient of air (200C, 1atm) µ (Pa∙s) 1.85E-5 Permittivity constant of vacuum ε0 (pF/µm) 8.854E-6 Permittivity coefficient of air ε 1 The conversion coefficients in differential equation (4) calculated by equations (2), (7) and (11) are shown in Table 3: Table 3. Conversion coefficients of the ELCA Conversion stiffness K (µN/µm) 6.927 Conversion effective mass M (kg) 1.008E-08 Conversion damping C (µN∙s/µm) 2.322E-06 The ELCA’s simulation is implemented by using a finite element method (FEM) in ANSYS. At driving voltage of 100V, the maxi um stress on suspension beams is 28.2MPa and the obtained dis- placement y of t e shuttle is 38.42µm, as shown in Fig. 5. The n tural fr quency n of the system determined by theoretic l nalysi is 4172Hz, and by ANSYS is 4168Hz, i. . deviation is only 0.1%. (a) Displacement b) Natural frequency Fig. 5. Simulation of the ELCA The model of ELCA is fabricated by using standard SOI-MEMS technology with a 30μm thick silicon device layer, 4μm thick buried silicon dioxide SiO2 and 450μm thick silicon substrate of SOI (b) atural frequency Fig. 5. Simulation of the ELCA The natural frequencyωn of the system determined by theoretical analysis is 4172 Hz, and by ANSYS is 4168 Hz, i.e. deviation is only 0.1%. The model of ELCA is fabricated by using standard SOI-MEMS technology with a 30 µm thick silicon device layer, 4 µm thick buried silicon dioxide SiO2 and 450 µm thick silicon substrate of SOI (silicon-on-insulator) wafer. Details of fabrication process, such as photolithography, DRIE, and vapor HF etching can be referred in our previous publications [3, 4]. Scanning Electron Microscopy (SEM) images of the structures after fabrication is shown in Fig. 6. Graph on Fig. 7 compares the displacement’s value of the shuttle in four cases: the- oretical calculation via Eq. (3); influence of conversion coefficients K, M, C via Eq. (14); simulation at f = 1 Hz and experimental results referred in [10]. The displacement re- sults calculated by theoretical are matching to the results when considering influence of 404 Hoang Trung Kien, Vu Cong Ham, Pham Hong Phuc Influence of the driving frequency and equivalent parameters on displacement amplitude of electrostatic linear comb actuator 7 (silicon-on-insulator) wafer. Details of fabrication process, such as photolithography, DRIE, and vapor HF etching can be referred in our previous publications [3, 4]. Scanning Electron Microscopy (SEM) images of the structures after fabrication is shown in Fig. 6. Shuttle (movable part) Fixed electrodes Beams Fig. 6. Structure of ELCA after fabrication Fig. 7. Relation between the ELCA displacement and the driving voltage. Graph on Fig. 7 compares the displacement’s value of the shuttle in four cases: theoretical calculation via equation (3); influence of conversion coefficients K, M, C via equation (14); simula- tion at f = 1Hz and experimental results referred in [10]. The displacement results calculated by the- oretical are matching to the results when considering influence of conversion coefficients (K, M, C) and confirms that at low frequencies (a few Hz), the influence of inertial mass and air damping to ELCA’s displacement are not significant. The displacement error between the theoretical and the simulation is about 0.21% (see Fig. 7). Tiny deviation can be explained because of the difference of stiffness value calculated separately by theoretical and simulation. The difference of theoretical, simulation and measurement results are 19.81% and 20.06% at driving voltage of 100V, respectively. The reason of larger deviation is be- cause of the voltage loss in the contact points between the probe tip and the anchor surface, as well as because of the fabricated comb gap is bigger than the design gap g0. In other words, the real elec- trostatic force loaded on the ELCA will be smaller than the value calculated by equation (1). Fig. 6. Structure of ELCA after fabrication Influence of the driving frequency and equivalent parameters on displacement amplitude of electrostatic linear comb actuator 7 (silicon-on-insulator) wafer. Details of fabrication process, such as photolithography, DRIE, and vapor HF etching can be referred in our previous publications [3, 4]. Scanning Electron Microscopy (SEM) images of the structures after fabrication is shown in Fig. 6. Shuttle (movable part) Fixed electrodes Beams Fig. 6. Structure of ELCA after fabrication Fig. 7. Relation between the ELCA displacement and the driving voltage. Graph on Fig. 7 compares the displacement’s value of the shuttle in four cases: theoretical calculation via equation (3); influence of conversion coefficients K, M, C via equation (14); simula- tion at f = 1Hz and experimental results referred in [10]. The displacement results calculated by the- oretical are matching to the results when considering influence of conversion coefficients (K, M, C) and confirms that at low frequencies (a few Hz), the influence of inertial mass and air damping to ELCA’s displacement are not significant. The displacement error between the theoretical and the simulation is about 0.21% (see Fig. 7). Tiny deviation can be explained because of the difference of stiffness value calculated separately by theoretical and simulation. The difference of theoretical, simulation and measurement results are 19.81% and 20.06% at driving voltage of 100V, respectively. The reason of larger deviation is be- cause of the voltage loss in the contact points between the probe tip and the anchor surface, as well as because of the fabricated comb gap is bigger than the design gap g0. In other words, the real elec- trostatic force loaded on the ELCA will be smaller than the value calculated by equation (1). Fig. 7. Relation between the ELCA displace- ment and the driving voltage conversion coefficients (K, M, C) and confirms that at low frequencies (a few Hz), the influence of inertial mass and air damping to ELCA’s displacement are not significant. The displacement error between the th oretical and the simulation is about 0.21% (see Fig. 7). Tiny deviation can be explained because of the difference of stiffness value calculated separately by theoretical and s mulation. The diff rence of th retical, simu a- tion and measurement results are 19.81% and 20.06% at driving voltage of 100 V, respec- tively. The reason of larger deviation is because of the voltage loss in the contact points between the probe tip and the anchor sur ace, as well as becaus of t fabricated comb gap is bigger than the design gap g0. In other words, the real electrostatic force loaded on the ELCA will be smaller than the value calculated by Eq. (1). 8 Hoang Trun Kien, Vu C ng Ham, and Pham Hong Phuc Fig. 8. Relation between maximum amplitude and the driving frequency Fig. 8a shows that the maximum amplitude of the ELCA is nearly constant at low frequencies (less than 10Hz) in case of applying square wave voltage. The relative error of amplitude is 10% matching to working frequency of 27Hz (Fig. 8b). In case of applying sine wave is shown in Fig. 8c, the maximum amplitude is nearly consistently in a larger range of driving frequency; here, the error is also under 10% until ranging frequency up to 1kHz (Fig. 8d). Consequently, for the displacement of comb-drive structure, the sine wave is better than the square wave driving voltage with the range of working frequency is also larger. The quality factor (Q factor) of the ELCA system can be calculated by [16]: KM Q C  (17) 108 109 110 111 112 113 114 115 0 200 400 600 800 1000 Q f (Hz) Fig. 9. Q factor depends on a sine wave frequency When frequency changing from 1Hz to 1000Hz, Q factor value decreases about 4.6% from 114 to under 109 (see Fig. 9). It shows that the Q factor of ELCA reduces insignificantly, even thought the larger range of sine wave frequency value. 4. CONCLUSION This paper studied and simulated an influence of vibration at different frequencies on the accu- racy of an ELCA’s displacement. Theoretical, simulation results and measurement data were com- pared and evaluated. Comments can be summarized as follows: - The conversion coefficients K, M, and C of the ELCA’s differential motion equations are more exactly determined via the equivalent kinetic energy and air drag methods. Fig. 8. Relation between maximum amplitude and the driving frequency Fig. 8(a) show that the maxim m amplitude of the ELCA is nearly constant at low frequencies (less than 10 Hz) in case of applying square wave voltage. The relative er- ror of amplitude is 10% mat hing to working frequency of 27 Hz (Fig. 8(b)). In cas of Influence of the driving frequency and equivalent parameters on displacement amplitude of electrostatic linear . . . 405 applying sine wave is shown in Fig. 8(c), the maximum amplitude is nearly consistently in a larger range of driving frequency; here, the error is also under 10% until ranging frequency up to 1 kHz (Fig. 8(d)). Consequently, for the displacement of comb-drive structure, the sine wave is better than the square wave driving voltage with the range of working frequency is also larger. The quality factor (Q factor) of the ELCA system can be calculated by [16] Q = √ KM C . (17) 8 Hoang Trung Kien, Vu Cong Ham, and Pham Hong Phuc Fig. 8. Relation between maximum amplitude and the driving frequency Fig. 8a shows that the maximum amplitude of the ELCA is nearly constant at low frequencies (less than 10Hz) in case of applying square wave voltage. The relative error of amplitude is 10% matching to working frequency of 27Hz (Fig. 8b). In case of applying sine wave is show in Fig. 8c, the maximum amplitude is nearly consistently in a larger range of driving frequency; here, the error is also under 10% until ranging frequency up to 1kHz (Fig. 8d). Consequently, for the displacement of comb-drive structure, the sine wave is better than the square wave driving voltage with the range of working frequency is also larger. quality factor (Q factor) f the ELCA system can be calculated by [16]: KM Q C  (17) 108 109 110 111 112 113 114 115 0 200 400 600 800 1000 Q f (Hz) Fig. 9. Q factor depends on a sine wave frequency When frequency changing from 1Hz to 1000Hz, Q factor value decreases about 4.6% from 114 to under 109 (see Fig. 9). It shows that the Q factor of ELCA reduces insignificantly, even thought the larger range of sine wave frequency value. 4. CONCLUSION This paper studied and simulated an influence of vibration at different frequencies on the accu- racy of an ELCA’s displacement. Theoretical, simulation results and measurement data were com- pared and evaluated. Comments can be summarized as follows: - The conversion coefficients K, M, and C of the ELCA’s differential motion equations are more exactly determined via the equivalent kinetic energy and air drag methods. Fig. 9. Q factor depen a sine wave frequency When frequency changing from 1 Hz to 1000 Hz, Q factor value decreases about 4.6% from 114 to under 109 (see Fig. 9). It shows that the Q factor of ELCA reduces insignificantly, even thought the larger range of sine wave frequency value. 4. CONCLUSION This paper studied and simulated the influence of vibration at different frequencies on the accuracy of an ELCA’s displacement. Theoretical, simulation results and measure- ment data wer compared and evaluated. Comments can be summarized as follows: - The conversion coefficients K, M, and C of the ELCA’s differential motion equations are more exactly determined via the equivalent kinetic energy and air drag methods. - In medium of air, while driving frequency is lower than 10 Hz, the displacement of ELCA is nearly independent on damping and inertial force of the system. - The displacement amplitude deviation in term of using sine wave voltage is smaller than 10% with driving frequency up to 1 kHz, while the displacement amplitude devi- ation if using square wave is approximately to 10% with driving frequency ranging is only 27 Hz. It is clear that the sine wave frequency gives the displacement amplitude and working frequency range better than the square wave. This is a valuable suggestion for analysis and design of micro devices using the electrostatic comb-drive actuators such as micro motor, micro gripper or capacitive accelerometer. REFERENCES [1] W. C. Tang, T.-C. H. Nguyen, M. W. 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