DETC THE EFFECT OF HIGH ORDER NON-LINEARITIES ON SUB-HARMONIC EXCITATION WITH PARALLEL PLATE CAPACITIVE ACTUATORS

Proceeings of the ASME 27 International Design Engineering Technical Conferences & Computers an Information in Engineering Conference IDETC/CIE 27 September 4-7, 27, Las Vegas, Nevaa, USA DETC27-34996 THE EFFECT OF HIGH ORDER NON-LINEARITIES ON SUB-HARMONIC EXCITATION WITH PARALLEL PLATE CAPACITIVE ACTUATORS Alexaner A. Trusov MicroSystems Laboratory Dept. of Mechanical an Aerospace Engineering University of California, Irvine Irvine, CA 92697-3975 Email: atrusov@uci.eu Anrei M. Shkel MicroSystems Laboratory Dept. of Mechanical an Aerospace Engineering University of California, Irvine Irvine, CA 92697-3975 Email: ashkel@uci.eu ABSTRACT Electrostatic actuation of motion is commonly use in resonant MEMS. Drive signal fee-through is unesirable as it masks the etection signal. This paper reports analysis an emonstration of a resonant motion excitation scheme, which uses a combination of a DC bias with a sinusoial AC voltage, frequency of which is twice of the mechanical resonant frequency. This configuration is experimentally emonstrate to excite a gyroscope with a parallel plate rive capacitor into high amplitue perioic vibrations. The fee-through has a frequency higher than the main motional harmonic an thus can be eliminate by simple low pass filtering. Full nonlinear ynamics along with several higher-orer approximations are consiere. Analysis of the effect of the approximation orer on the frequency response accuracy shows that the complete nonlinear equation shoul be use for moeling of high amplitue actuation. Two istinct types of frequency responses are examine as functions of riving AC an DC voltages an amping. INTRODUCTION Several important classes of MEMS evices, such as resonators [], gyroscopes [2], an chemical sensors [3], rely on resonance phenomenon in their operation. In these evices, resonant motion nees to be actuate, etecte, an controlle. Capacitive phenomena are commonly use for transuction in vi- Aress all corresponence to this author. bratory MEMS evices ue to the ease of fabrication, low sensitivity to temperature changes, an other practical avantages [4], [5]. Conventional electrostatic actuation schemes use a combination of DC an AC voltages to excite mechanical motion at the resonant frequency of the structure. This is typically achieve by matching the frequency of the rive AC voltage to the resonant frequency of the evice. In this case, the amplitue of motion is ictate by the amplitues of the riving voltages an the quality factor of the mechanical resonator. For this type of actuation, the electrical rive fee-through from rive to sense electroes cause by parasitic capacitance has the same frequency content as the mechanical motion. Thus, the motional signal is maske by the fee-through, an intricate etection schemes, such as electromechanical amplitue moulation [6], are require to extract an measure the mechanical response. These factors increase the overall system complexity an limit the performance. Nonlinearity of parallel plate capacitors presents esign opportunities for introucing new electrostatic actuation schemes. Issues of large amplitue parallel plate actuation using a conventional combination of DC an AC voltage were extensively stuie, for example in [7 9]. Parametric excitation phenomenon in MEMS was reporte in []. In its simplest form, parametric excitation using electrostatic parallel plate force is escribe by a classical linear time variant LTV) Mathieu equation. Dynamics of a clampeclampe beam uner parametric excitation was stuie in [] Copyright c 27 by ASME

using small amplitue assumption. The effect of an aitional thir orer term was stuie in [2]. An interesting parametrically enhance harmonic actuation scheme using two separate sets of capacitors an AC voltages was propose in [3]. Most of the relevant literature stuies parallel plate parametric effects using approximate equations of motion. The focus of this paper is actuation of high amplitue vibrations in capacitive MEMS using a combination of a DC bias with a single AC sinusoial voltage, frequency of which is twice of the mechanical resonant frequency. In Section we introuce a general electro-mechanical moel of a MEMS resonator with parallel plate actuation an erive the equations of motion. The phenomenon of high amplitue parametric excitation is emonstrate experimentally in Section 2 using a prototype of a capacitive MEMS gyroscope. In Section 3 we efine two ifferent norms of solutions use for frequency response analysis for the nonlinear system. Section 4 presents a comparative numerical stuy of system ynamics using the complete nonlinear moel an its several finite orer approximations. Section 5 summarize the obtaine results. PARAMETRIC EXCITATION Figure shows a general schematic of a capacitive microresonator, a basic element of various micro-sensors. The electromechanical iagram inclues the mechanical resonator an the electrostatic rive an sense electroes. Mass m of the resonator is suspene by a spring k an is constraine to move only along the horizontal x-axis. The variable sense capacitance is efine as C s x), an the rive capacitance as C x), where x is the isplacement. Typically in MEMS evices, the rive an sense terminals are not completely insulate, but are electrically couple by stray parasitic capacitors an resistors [6,4,5]. In this paper we assume, without loss of generality, that the parasitic circuit consists of a single lumpe capacitor C p. An AC riving voltage V t) = v sinω t) is applie to the rive electroe, an a DC bias V c is applie to the mass voltage values are reference with respect to a common groun). The sense capacitor C s is forme between the mobile mass an the fixe sense electroe. The sense electroe is connecte to the inverting input of an operational amplifier which is configure as a trans-impeance amplifier, [6]. Due to motion the sense capacitance C s x) changes, causing a flow of motional current I s = C sv s ) t, where V s is the sensing voltage across the sense capacitor. The total pick-up current It) = I s t)+i p t) consists of both the motional an the parasitic fee-through currents an is converte to the final output voltage V t) with trans-impeance gain R. Parasitic current is inuce by the rive voltage V an therefore has the same frequency ω. The total sensing voltage is the DC bias V c. Accoring to the laws of electrostatics, the total pick-up current that flows through the fee-back resistor of the trans-impeance amplifier is It) = t [V t)c p + V c C s t)]. Drive ~ Vt) Amplitue, B Vc 2 4 6 8 Figure. Drive Electroe C Parasitic Circuit Mechanical resonance x Moveable Mass Vc Cp Sense Electroe Cs Ist) Sense Itt)=Is+Ip R - V=-RI Feethrough Ipt) Pick-up Voltage motional + parasitic Frequency + sub harmonic resonance no resonance, fee through Transimpeance Amp. Schematic of a capacitive MEMS resonator with parallel plate sub-harmonic excitation. Since the parasitic capacitance C p an the voltage V c are constant, It) = V c C s t) t The motion of the resonator is governe by ẍ + ω n Q ẋ + ω2 nx = 2m V t) +C p. ) t C x) V c +V t)) 2, 2) x k where ω n = m is the unampe natural frequency an Q is the quality factor. Let us enote the initial gap between plates in the parallel plate riving capacitor at rest by g, the overlap length of iniviual parallel plate pairs by L, height of the plates i.e. structural layer thickness) by y, an permittivity of the meia by ε. The total overlap area in the rive capacitor is given by A = NLy, where N is a number of parallel plate pairs in the rive capacitor. Then, the variable rive capacitance is C x) = εa g x = εa g x/g. 3) 2 Copyright c 27 by ASME

We introuce the nominal rive capacitance C n = εa/g an gap-normalize imensionless isplacement χ = x/g <. From Eqn. 3), using Taylor series expansion with respect to χ, the rive capacitance is C = C n χ = C n n= χ n. 4) The electrostatic force prouce by the rive capacitor is proportional to the graient of the capacitance C x = ) Cn = C n x χ g χ) 2 = C n g n= n + )χ n. 5) Finally, from Equations 2) an 5), normalize isplacement χ of the resonator is governe by χt) + ω n Q χt) + ω2 nχt) = C n 2mg 2 V c +V ) 2 χ) 2, 6) or equivalently, by n= χt) + ω n Q χt) + ω2 nχt) = C n 2mg 2 { Vc 2 + 2 v2 ) + 2V cv sinω t) 2 v2 cos2ω t) n + )χ n 7) }. This equation has a nonlinear an time epenent right han sie. The nonlinearity is ue to the parallel plate capacitance graient, Fig. 2; the voltage-square term causes the time epenency of the excitation. Let us consier several special cases. A linear time invariant LTI) approximate moel can be obtaine by neglecting the linear time-epenent an the nonlinear terms on the right han sie. The resulting moel is ientical to the case of classical lateral comb actuation. For this moel, the resonant motion can be achieve in the following two cases: riving AC voltage v t) is applie at the resonance frequency or at its half. In the first case, ω ω n an the forcing term C n V mg 2 c v sinω t) provies resonant excitation. In the secon case, ω 2 ω n an C n v 2 4mg 2 cos2ω t) provies resonant excitation. The rawback of ω ω n is that motion an parasitics occur at the same frequency. The main isavantage of ω ω n /2 is the low actuation gain, cause by the 2 v2 coefficient. The next level of approximation can be obtaine in the following way. First, we assume V c V an neglect terms proportional to v 2. Secon, we linearize the capacitance graient Normalize capacitive graient C χ) /χ, log Figure 2. Parallel plate rive complete an approximate moels complete / χ) 2 orer 5 approx. orer 3 approx. orer 2 approx. linear approx. constant approx. 2 Gap normalize isplacement χ=x/g, log orer approximations. Nonlinear capacitive graient of parallel plates an its finite C n g + 2χ). Finally, Eqn. 7) is simpli- using Eqn. 5) as C x fie to which is equivalent to χt) + ω n Q χt) + ω2 nχt) = C n 2mg 2 { V 2 c + 2V c v sinω t) } + 2χ), 8) χt) + ω n Q χt) + ω 2 n V ) cc n mg 2 [V c + 2v sinω t)] χt) = = V cc n 2mg 2 [V c + 2v sinω t)]. 9) This equation obtaine by neglecting v 2 an linearization with respect to χ) is very similar to the classical Mathieu equation [], which escribes zero-input parametric instability, cause by harmonic moulation of the effective stiffness. However, Eqn. 9) is not homogenous an is simultaneously force with a static force an a sinusoial force at ω frequency. This equation, common in MEMS literature on parallel plate actuation [], qualitatively explains two phenomena inherent to parallel-plate actuation: active resonant frequency tuning an parametric resonance. The effective natural frequency for a parallel-plate actuate motion is given by ω 2 n V cc n ) [V mg 2 c + 2v sinω t)]. The stiffness reuces ) quaratically with applie DC bias as k ω 2 n C n V 2 mg 2 c. This effect is known as frequency tuning an the corresponing mathematical moel is known as negative 3 Copyright c 27 by ASME

CDIP-4 Boning pas 89 Frequency Tuning EFAB DMG) 882.5 Biase Resonant Frequency, Hz 875 867.5 86 852.5 845 837.5 experimantal parabolic fit Unbiase resonant frequency is 888.4 Hz Figure 3. EFAB ie Package prototype of istribute mass gyroscope DMG) fabricate in EFAB process. electrostatic spring. In Section 2 frequency tuning of a MEMS gyroscope prototype is experimentally characterize. Classical parametric resonance is a property of a linear timevarying ynamic system to evelop unstable motion from nontrivial initial conitions uner harmonic moulation of the effective stiffness. Parametric resonance for both non-ampe an ampe systems governe by Mathieu equation is a well stuie problem in both cases motion with infinitely growing amplitue occurs when the stiffness moulation frequency satisfies the conition ω = 2 k ω n, k =,2,...,. The case of k =, ω = 2ω n, is of the greatest practical interest, because it is the only k for which the frequency of the fee-through is above the frequency of motion. Feasibility of this scheme of excitation is experimentally stuie in Section 2. Numerical analysis of ifferent orer approximations of Eqn. 7) is presente in Section 4. 2 DEMONSTRATION OF SUB-HARMONIC EXCITA- TION The simplifie LTV approximation given by Eqn. 9) allows to preict phenomenon of parametric resonance in parallel plate actuate MEMS. Specifically, we can expect the system to evelop significant amplitues of motion, when riving AC voltage is applie at ouble of the resonant frequency taking into account frequency own-tuning ue to the DC voltage). The motion occurs at the tune resonant frequency, which is half of the riving AC voltage frequency. In this section we experimentally characterize this nonlinear actuation metho. Figure 4. Characterization of parallel plate frequency tuning in the evice. 83 5 3 45 6 75 9 5 2 35 5 DC Bias Voltage, V 2. Test-Be A prototype of a Distribute Mass Gyroscope DMG) [5] was use for experimental emonstration of parametric actuation. The evice was fabricate using EFAB TM [7], a commercial process from Microfabrica. The core of the EFAB process is a sequential electroeposition of multiple structural layers. Square ies 5 5) mm were forme on.5 mm thick alumina Al 2 O 3 ) substrate. The first µm thick Ni-Co structural layer was eposite using electroplating an patterne. Then, blanket Cu layer was electroforme an planarize to the thickness of the structural layer. The cycle of these two steps was repeate 2 times, after which the sacrificial Cu layer was chemically etche away. As a result, a 2 µm high 3D metal structures on a nonconuctive substrate were forme. For the fabricate gyroscope, the minimal line-with was 2 µm an minimal spacing was 3 µm. Figure 3 shows a package an wire-bone evice. The rive moe of the gyroscope was use in this paper to stuy high amplitue actuation base on nonlinear parametric resonance phenomenon. 2.2 Structural Characterization In orer to characterize frequency tuning an the unbiase natural frequency of the evice, resonance was visually etecte at ifferent values of V c. Figure 4 shows the collecte ata along with the curve fitting results. Structural properties of the evice were also characterize in air an in vacuum using electrostatic metho of parasitics-free half-frequency sweep [5]. Figure 5 shows the experimental ata an the curve fitting results. The resonant frequency was 878 Hz for a DC bias of 6 V; the quality factor was 275 in air an 86 in 2 mtorr vacuum. 4 Copyright c 27 by ASME

8 Structural Characterization of EFAB DMG in Air fixe plate mobile plate Amplitue Response, Linear Scale 7 6 5 4 3 2 fn =878.4 Hz Q =274.7 experimental analytical fit 25 m 2 m Amplitue Response, Linear Scale 8 82 84 86 88 9 92 9495 Drive frequency, Hz 9 8 7 6 5 4 3 2 a) In air Structural Characterization of EFAB DMG in Vacuum fn =878.8 Hz Q =859.3 experimental analytical fit 8 82 84 86 88 9 92 94 95 Drive Frequency, Hz Figure 5. b) In 2 mtorr vacuum Structural characterization of the evice using electrostatic actuation an capacitive etection. 2.3 High Amplitue Parametric Resonance Nonlinear parametric excitation of motion at resonant frequency by a twice higher frequency AC voltage is experimentally characterize in this section. This actuation metho provies high amplitue of motion an robustly separates motion from the parasitics in frequency omain. The experimental setup consiste of a package gyroscope assemble on a breaboar together with a trans-impeance amplifier. The teste moe of vibration was characterize utilizing parallel plate actuation an sensing, as shown in Fig.. The ca- Figure 6. 3 m mobile plate at rest Superimpose optical photographs of parallel plate evice at rest an at resonant motion parametrically excite by a twice higher frequency AC voltage. The blurre area shows amplitue of motion 8% of the gap). pacitive gap between 2 µm wie parallel plates was 3 µm. The DC bias voltage was applie to the boy of the resonator. The riving AC voltage was impose onto the anchore parallel plate structure. Sensing of the motion was one using anchore parallel plate structure, which was virtually groune by means of trans-impeance amplifier with a MΩ feeback resistor. The sense capacitor between fixe an moveable parallel plates ha a fixe DC voltage ifference across it. Velocity epenent motional current flows through the feeback resistor of the transimpeance amplifier. Also, capacitive coupling between rive an sense electroe contributes to the output current. The total pick-up current was converte to voltage an fe into a signal analyzer HP 35665A for spectral analysis an ata capturing. During the experiments, the ω frequency of the riving AC voltage V t) was manually swept in the vicinity of 2ω n. The amount of motion was etecte visually using a microscope, Fig. 6. At the same time, the Fast Fourier Transform FFT) of the pick-up signal was continuously performe, upate, isplaye an store by the ynamic signal analyzer. We compare results of the FFT at resonance an off-resonance in our experimental setup this means that motional amplitues were smaller than optically resolvable.25µm). From this comparison, shown in Fig. 7, we etermine the frequency content of the fee-through signals an istinct features of the pick-up signal in the presence of parametrically excite motion. 5 Copyright c 27 by ASME

Amplitue, BVrms 2 2 4 6 Excitation of resonant motion by a superharmonic AC voltage motion 2n motion+ st rive at resonance no resonance 4th motion+ 2n rive 6th motion+ 3r motion 3n rive 5th motion 3 FREQUENCY RESPONSE NORMS Small amplitue parametrically excite motion can be moele by classical Mathieu equation []. The focus of this paper, however, is the steay-state large amplitue vibrations excite using parallel plate capacitors. Parametric excitation of large amplitue vibrations was experimentally emonstrate in Section 2. In this section we moel an analyze the ynamics using the complete Eqn. 7) along with several orers of approximations. These moels of the ynamics can be uniformly escribe by 8 Figure 7. 2 3 4 5 Frequency, Hz Experimental pick-up signal PSD for parametrically excite resonator. The riving AC voltage is provie at ouble of the resonant frequency. The strong frequency separation between first main) motional component an parasitics is achieve. We registere the large amplitue of motion when the riving AC voltage was applie at frequency close to twice of the DC tune resonant frequency i.e. ω 2ω n, f 2 85 =.7 khz). The amplitue of motion was several times higher than in the case of conventional LTI excitation i.e. ω ω n ) using the same combination of DC an AC voltages. Due to the capacitive parasitic coupling an non-ieality of the AC voltage generator, the fee-through of the rive occurs at all multiple harmonics of the riving AC frequency i.e. at n.7 =.7,3.4,5.,... khz). In other wors, the rive voltage leakage is present at all even orer multiples of the resonant frequency. However, the motion mainly occurs at the resonant frequency. Thus, the main vibrational moe is sub-harmonic with respect to the riving AC voltage. In aition to the main moe at the resonant frequency, the motion also occurs at the resonant frequency an all of its multiples i.e. at n.85 =.85,.7,2.55,3.4,4.25,5.,... khz). To conclue, nonlinear parametric excitation of resonant motion by a higher frequency voltage was emonstrate using a parallel plate actuate evice. This non-lti actuation approach experimentally exhibits several important avantages over the conventional LTI riving schemes. The avantages inclue actuation of large amplitues of motion 8% of the gap) using relatively small voltages, an robust frequency separation between the motion an the rive AC voltage fee-through. Parametric excitation of perioic resonant motion by a higher frequency AC voltage can potentially be use to provie resonant evices with wier banwith an higher gain while simplifying the etection of motion. N n= χt) + ω n Q χt) + ω2 nχt) = C n 2mg 2 { Vc 2 + 2 v2 ) + 2V cv sinω t) 2 v2 cos2ω t) n + )χ n ) }, where N efines the orer of the Taylor sum approximation for the nonlinear capacitive graient on the right han sie. Orer N = correspons to the often use LTV moel, while infinite summation correspons to the complete nonlinear moel of C χ)/ χ = / χ) 2. Equation ) was solve numerically in the complete nonlinear form along with N =,2,3,5 orer approximations using integration routine oe45 in MATLAB. Base on the sensitivity analysis, the time increment was set to 2 of the resonant vibrations perio to ensure precision of the numerical solutions. The total solution time interval was s. The output of the coe was a multi-imensional array of time omain solutions, corresponing to ifferent orers N an values of the system parameters. Frequency responses were calculate from the steay-state response parts of the time omain solutions by a separate MAT- LAB coe. It is important to explicitly efine the notion of the frequency response, since for the non-lti system escribe by Eqn. ) the motion occurs at an infinite number of frequencies. We efine two alternative norms of the χt) solutions for a given rive AC frequency ω : χt) max an χt) ω. The total amplitue of motion norm χt) max is efine as the maximum span of χt) on a certain time interval. This norm etermines if the solution is boune an physically feasible. This approach to frequency response analysis is typically use to stuy instability conitions of parametrically excite micro-resonators, for example []. Operation of many resonant MEMS sensors, however, is base on a certain moe of vibration. For instance, in vibratory gyroscopes the input angular rate stimulus is amplituemoulate an subsequently etecte at the frequency of the resonant vibrational moe. If parametric phenomenon is use to actuate such evices, the amplitue of only the main resonant motion harmonic efines the scale factor an other performance 6 Copyright c 27 by ASME

characteristics of the evice. Therefore, for micro-evices base on sinusoial motion an Fourier signal processing, a ifferent efinition of norm shoul be use to stuy frequency responses. This norm is the amplitue of the sinusoial component of the motion at the resonant frequency. In other wors, χt) ω is the norm of the orthogonal projection of χt) onto the Fourier basis function e i ω t 2, i.e. χ ω = R t t χt)e i ω 2 t) t R t t e iωt). ) t To summarize, two ifferent types of frequency response analyses were numerically stuie for a parametrically excite micro-evice. It was conclue, that rive frequency ω epenency of χt) max escribes the instability regions, while χ ω efines performance characteristics of sinusoial motion base resonant micro-evices. 4 MODELING OF NON-LINEARITIES In this section we iscuss the results of ynamics moeling. The moel of the resonator is base on the evice, which was characterize in Section 2; the evice has the following nominal parameters: f n = 85 Hz, Q = 27, effective m = 9.3e-7 kg, C n =.5 pf, V c = 2 V, v = 5 V pk. 4. Effect of Approximation Orer Figure 8a) shows χ max frequency responses of the resonator, obtaine using N =, 2, 3 orer approximations of Eqn. ) an a complete nonlinear moel. Mathieu equation type LTV moel shows classical unboune parametric resonance, which is ifferent from experimentally observe high amplitue vibration; the frequency omain instability region given by this moel is also ifferent from the actual. Secon orer moel shown in Fig. 8a) is also quantitatively ifferent from the complete moel, however the preicte resonant region is very close to the actual. Thir orer moel is close to the actual, however the amplitue of the total χ max response is overestimate. In general, finite orer N approximations of Eqn. ) result in overestimate amplitues of motion; amount of the overestimation reuces with higher orer approximation orer N. Figure 8b) shows a comparison between χ max an χt) ω frequency responses for a complete moel an a thir orer approximation. The frequency omain resonant regions coincie, an the profiles of response curves are similar. Most of the motion is prouce by a harmonic at χt) ω frequency. The harmonic frequency responses o not exhibit a character- Normalize amplitue of motion, χ = x /g Normalize amplitue of motion, χ = x /g.8.6.4.2 orer 3 moel Parametric rive ifferent moels orer 2 moel LTV moel complete moel.87.88.89.9.9.92.93.94 Normalize frequency of V ac, f.8.6.4.2 Figure 8. a) Maximum amplitue of motion χt) max. Parametric rive effective frequency responses Nonlin3 total Nonlin3 resonant Complete total Complete resonant.88.89.9.9.92.93.94 Normalize frequency of V ac, f b) Comparison of χt) max an χt) ω frequency responses. Frequency responses of parametrically excite vibrations efine by two ifferent norms of solutions. istic peak, which is observe in the total amplitue frequency response. 4.2 Effect of AC Drive Voltage Amplitue v Figure 9 shows χt) ω frequency responses of the resonator for ifferent AC riving voltages v = 2,3,4,5,8,2 V base on the complete moel. Depening on the amplitue of the riving AC voltage, there are three regimes of the ynamics. For small AC amplitues in this case, approximately below V ), there is no parametric resonance. For a range of voltages in this case, approximately between an 6 V ), there is a well pronounce high amplitue pass-ban with a sharp transi- 7 Copyright c 27 by ASME

Normalize amplitue of motion, χ = x /g.5.4.3.2. Parametric rive effect of Vac 8V 5V 4V 2V.84.86.88.9.92.94.96 Normalize frequency of V ac, f Figure 9. Resonant harmonic frequency response χt) ω for parametric excitation. Effect of the AC riving voltage amplitue. 3V 2V Normalize amplitue of motion, χ = x /g.7.6.5.4.3.2. 35 V Parametric rive effect of Vc V c 3 V 25 V 2 V 5 V V.5.6.7.8.9 2 Normalize frequency of V ac, f Figure. Resonant harmonic frequency response χt) ω for parametric excitation. Effect of the DC bias. tion on the left sie an a smooth slope on the right sie. Finally, for voltages above a certain bifurcation threshol, the pass-ban breaks into two ifferent regions. The left part of the pass-ban shows instability, while the right part allows for stable high amplitue motion. This behavior is specific to the complete nonlinear moel of parallel plate parametric excitation. 4.3 Effect of DC Bias Voltage V c Figure shows the effect of the DC bias on the χt) ω frequency response. As expecte from the negative electrostatic spring effect, the pass-ban shifts to the left with increasing V c. Aitionally, the pass-ban becomes wier an shorter. This is ue to the reuction of the effective parallel plate gap, cause by the static isplacement of the mass in response to the DC force. 4.4 Effect of Quality Factor Q In classical Mathieu equation, amping is known to efine parametric instability regions without saturating the amplitue of motion []. Figure shows the effect of Q-factor, or, equivalently the amping, on the χt) ω frequency response of a complete nonlinear parallel plate moel. Frequency response has a low sensitivity to the quality factor in the wie range of values 2 4. This relaxes packaging requirements for parametrically actuate evices []. Quality factors in the range of 2 3 are typical of vibratory gyroscopes in air an appear favorable for the parametric excitation. Normalize amplitue of motion, χ = x /g.5.4.3.2. Parametric rive effect of Q Q=256 Q=24 Q=248 Q=892.87.88.89.9.9.92.93.94 Normalize frequency of V ac, f Figure. Resonant harmonic frequency response χt) ω for parametric excitation. Effect of the quality factor Q. 5 CONCLUSIONS In this paper we stuie high amplitue actuation of capacitive vibratory evices using parametric resonance. The actuation scheme uses a simple combination of a DC bias with a sinusoial AC voltage, frequency of which is twice of the resonator s resonant frequency. Experiments emonstrate excitation of high amplitue motion in the parallel plate rive moe of a gyroscope. Since motion mostly occurs at half of the riving AC voltage frequency, fee-through signals can be efficiently eliminate by simple low-pass filtering of the pick-up signal. This removes the nee for intricate etection methos an simplifies the overall Q 8 Copyright c 27 by ASME

system esign. In its linear approximation, the phenomenon is represente by a non-homogenous LTV Mathieu equation. This approach can be use to stuy small amplitue motion an instability conitions. However, it is not aequate for stuy of large amplitue responses; higher orer terms of the parallel plate capacitive graient nee to be taken into account. In this paper we presente a etaile analysis of two types of frequency responses of parametrically riven resonators using the complete nonlinear moel. Maximum isplacement χt) max frequency response etermines if the solution is boune an physically feasible. A ifferent type of frequency response, efine as the amplitue of the main sinusoial motion at half of the AC voltage frequency χt) ω, prescribes the scale factor of such resonant evices as vibratory gyroscopes. Finite orer approximations of the complete nonlinear electro-mechanical ynamics result in significant overestimation of the actual steay-state response amplitue. Also, lower orer approximations escribe the resonant region of the frequency response imprecisely. The full nonlinear moel shoul be use in orer to precisely moel the ynamics. The escribe actuation approach is insensitive to amping an can be use to improve performance of vibratory gyroscopes an other resonant MEMS by proviing high amplitues of motion using small voltages an simplifying the etection by isolating parasitics at higher frequencies. ACKNOWLEDGMENT This research was partially supporte by the National Science Founation Grant CMS-49923. We woul like to acknowlege Cenk Acar for esign of the teste evice, Microfabrica for the fabrication of the prototypes, an Aam R. Schofiel an Professor Faryar Jabbari for iscussions. REFERENCES [] Stemme, G., 99. Resonant silicon sensors. IOP Journal of Micromechanics an Microengineering, vol., pp. 3 25. [2] Yazi, N., Ayazi, F., an Najafi, K., 998. Micromachine inertial sensors. Proceeings of the IEEE, vol. 86 issue 8. [3] Turner, K. K., an Zhang, W. Design an analysis of a ynamic mem chemical sensor. Proceeings of the American Control Conference 2, Arlington, VA, USA, June 25-27, 2. [4] Baxter, L. K., 996. Capacitive Sensors: Design an Applications. Wiley-IEEE Press. [5] Kovacs, G. T., 998. Micromachine Transucers Sourcebook. McGraw-Hill. [6] Cao, J., an Nguyen, C.-C. Drive amplitue epenence of micromechanical resonator series motional resistance. Digest of Technical Papers, -th International Conference on Soli-State Sensors an Actuators, Senai, Japan, June 7 -, 999. [7] Lu, M. S.-C., an Feer, G. K. Close-loop control of a parallel-plate microactuator beyon the pull-in limit. Soli-State Sensor, Actuator an Microsystems Workshop, Hilton Hea Islan, South Carolina, USA, June 2-6, 22. [8] Seeger, J., an Boser, B. Parallel-plate riven oscillations an resonant pull-in. Soli-State Sensor, Actuator an Microsystems Workshop, Hilton Hea Islan, South Carolina, USA, June 2-6, 22. [9] Fargas-Marques, A., an Shkel, A. On electrostatic actuation beyon snapping conition. IEEE Sensors Conference, Irvine, California, USA, October 3 - November 3 25. [] Turner, K. L., Miller, S. A., Hartwell, P. G., MacDonal, N. C., Strogatz, S. H., an Aams, S. G., 998. Five parametric resonances in a micromechanical system. Nature, vol. 396, November. [] Nayfeh, A. H., an Younis, M. I., 25. Dynamics of mems reonators uner superharmonic an subharmonic excitations. IOP Journal of Micromechanics an Microengineering, vol. 5, pp. 84 847. [2] Zhang, W., Baskaran, R., an Turner, K. L., 22. Effect of cubic nonlinearity on auto-parametrically amplifie resonant mems mass sensor. Sensors an Actuators A, vol. 3578, pp.. [3] Gallacher, B. J., Buress, J. S., an Harish, K. M., 26. A control scheme for a mems electrostatic resonant gyroscope excite using combine parametric excitation an harmonic forcing. IOP Journal of Micromechanics an Microengineering, vol. 6, pp. 32 33. [4] Cagaser, B., Jog, A., Last, M., Leibowitz, B. S., Zhou, L., Shelton, E., Pister, K. S., an Boser, B. E. Capacitive sense feeback control for mems beam steering mirrors. Soli- State Sensor, Actuator an Microsystems Workshop, Hilton Hea Islan, South Carolina, USA, June 6 -, 24. [5] Trusov, A., Acar, C., an Shkel, A. M., 26. Comparative analysis of istribute mass micromachine gyroscopes fabricate in scs-soi an efab. Proceeings of SPIE Volume: 674, Smart Structures an Materials 26: Sensors an Smart Structures Technologies for Civil, Mechanical, an Aerospace Systems. [6] Horowitz, P., an Hill, W., 989. The Art of Electronics. Cambrige University Press. [7] Cohen, A., Frois, U., Tseng, F.-G., Zhang, G., Mansfel, F., an Will, P., 999. Efab: low-cost automate electrochemical batch fabrication of arbitrary 3- microstructures. SPIE Conference on Micromachining an Microfabrication Process Technology V. 9 Copyright c 27 by ASME