Impact of the closed-loop phase shift on the frequency stability of capacitive MEMS oscillators

Transcription

1 Impact of the cloed-loop phae hift on the frequency tability of capacitive MEMS ocillator Alexi Brene, Jérôme Juillard, Filipe Vinci do Santo To cite thi verion: Alexi Brene, Jérôme Juillard, Filipe Vinci do Santo. Impact of the cloed-loop phae hift on the frequency tability of capacitive MEMS ocillator. 016 Sympoium on Deign, Tet, Integration and Packaging of MEMS/MOEMS (DTIP), May 016, Budapet, Hungary. Deign, Tet, Integration and Packaging of MEMS/MOEMS (DTIP), 016 Sympoium on, pp , 016, 016 Sympoium on Deign, Tet, Integration and Packaging of MEMS/MOEMS (DTIP) < /DTIP >. <hal > HAL Id: hal Submitted on 30 Aug 016 HAL i a multi-diciplinary open acce archive for the depoit and diemination of cientific reearch document, whether they are publihed or not. The document may come from teaching and reearch intitution in France or abroad, or from public or private reearch center. L archive ouverte pluridiciplinaire HAL, et detinée au dépôt et à la diffuion de document cientifique de niveau recherche, publié ou non, émanant de établiement d eneignement et de recherche françai ou étranger, de laboratoire public ou privé.

2 Impact of the cloed-loop phae hift on the frequency tability of capacitive MEMS ocillator A. Brene 1,,3, J. Juillard and F. Vinci do Santo 3 1 THALES Avionic Valence, France alexi.brene@centraleupelec.fr GeeP Group of electrical engineering Pari, UMR CNRS 8507, CentraleSupélec, Univ. Pari-Sud, Sorbonne Univerité, UPMC Univ. Pari 06, 3,11 rue Joliot-Curie, Plateau de Moulon F-9119 Gif-ur-Yvette CEDEX. 3 CentraleSupélec Advanced Analog Deign Chair, 3 rue Joliot-Curie, 9119 Gif-ur-Yvette, France Abtract Phae noie i a clue performance indicator for MEMS-baed reonant enor. The optimal reolution achievable with thee enor i limited by the cloe-to-the-carrier phae noie reulting from the modulation of noie ource by the mechanical reonator. In thi paper, we focu on the effect of a white noie input ource on a one-ided capacitive MEMS reonator. We tudy how the cloedloop phae hift affect it frequency tability. Our tudy reveal the exitence of optimal point, with phae noie minimization and SNR maximization, which cannot be predicted by a traditional third-order Taylor-Serie approach. We reveal that, due to the modulation of the actuation force by the electrotatic nonlinearity, tuning the cloed-loop phae hift to improve the ocillator frequency tability i epecially relevant. Keyword Electrotatic nonlinearity, cloed-loop ytem, ocillator, frequency tability, capacitive MEMS. I. INTRODUCTION MEMS reonant enor are known for their high reolution achievable with mall and relatively cheap ytem [1] []. In thee enor, the meaurement proce i baed on the dependency of the natural frequency of the reonator on the meaured quantity. One olution to track thi natural frequency i to place the reonator in a elf-ocillating loop. However, the reolution of the enor baed on thi principle depend on the frequency tability of the cloed loop in the preence of noie ource. Several work have already addreed theorie of phae noie in ocillator [3] [4] [5]. The phae noie of MEMS ocillator reult from noie injection at different tage, uch a the input of the reonator or the feedback electronic [3]. The pectrum of an ocillator i expected to be made of two principal component. One component reult from additive white noie at frequencie cloe to the carrier and eventually the effect of nonlinearitie which mix the noie at other frequencie into the pa band of interet. The other main component i due to the long-term parameter variation linked to everal reliability iue uch a dielectric charging, creep or fatigue [6]. Thee latter effect are outide the cope of thi paper. In thi tudy, we will only focu on the effect of additive white noie introduced by random procee uch a thermal fluctuation. The maximal reolution achievable with a enor i fundamentally limited by the converion of amplitude noie into phae noie through the electromechanical tranduction [7]. To etimate thi optimal performance, a precie model of the reonator i required. Among the tudie of phae noie, ome have already addreed phae noie model for cloed-loop electronic MEMS ocillator [8] [9] [10]. One recent tudy ha even focued on the impact of the phae introduced in the feedback loop on the frequency tability of ocillator ubject to cubic hardening retoring force [11]. However, the effect of the electrotatic nonlinearity on capacitive MEMS reonator induce effect which cannot be explained by the preence of a cubic nonlinear force, neither in open-loop [1], nor in cloed-loop ytem [13]. None of the aforementioned article have tudied the impact of the feedback phae hift on the frequency tability of ocillator baed on capacitive MEMS reonator. Thi i the main purpoe of thi paper. In thi tudy, we etimate the performance of a nonlinear MEMS-baed ocillator in term of frequency tability by computing the variance of the frequency variation like in the work of Sancho et al. [14]. Compared to them, we derive a imple cloed-form expreion of thi variance valid for high-q MEMS-baed ocillator. Finally, we adapt thee expreion to capacitive MEMS reonator, which are ubject to electrotatic oftening. Thi paper i organized a follow: in Section II, we adapt the theory given by Sancho et al. [14] to a reonator actuated by a linear force but ubject to a nonlinear retoring force. We illutrate thi approach on a MEMS reonator affected by cubic hardening and tudy the impact of the operating point on the frequency tability. Finally, in Section III, we generalize thi method to a reonator ubject to a nonlinear actuation force and take the example of a capacitive MEMS reonator. II. FREQUENCY STABILITY ESTIMATION In thi ection, we tudy the cae of a MEMS-baed ine-wave ocillator whoe reonator i ubject to a nonlinear retoring force. The actuation force i firtly aumed to be diplacement-independent (i.e. linear).

3 A. General expreion Let u conider a reonator ubject to tatic memoryle nonlinear retoring force placed in a ine-wave ocillation loop. A noie ource b i placed at the input of the reonator. The ytem i depicted in Fig. 1. b( + + Nonlinear reonator Force Diplacement x Feedback electronic f Fig. 1 Cloed-loop circuit of the ocillator. The feedback loop introduce a phae hift f correponding to a pecific operating point. The condition for elfocillation i written f=- The behavior of thi ocillator may be decribed by: d x 1 dx x(1 g( x)) F in t b( (1) dt Q dt where x i the normalized diplacement of the reonator, t=where i the natural frequency of the reonator and i the time variable, Q the Q-factor of the reonator, the actuation frequency normalized with repect to 0 and b a tochatic noie. The function g correpond to a nonlinear retoring force. In (1), b will be conidered a a mall perturbation to the teady-tate nonperturbed regime: x( A in t () where i the phae between the actuation force and the diplacement. Since we focu on high-q reonator, the diplacement of the reonator will be retricted to it firt harmonic. Let u write: x( A( in( ( ) (3) The perturbation of the teady-tate regime i uppoedly mall o that we will aume: d A dt A and da d QA (with dt dt ) (4) The method of harmonic balance applied to (1) yield to: A1 g1( A) F co b da d A c A F in b dt dt Q where 1 t g1( A) g( Ain ( t')) in ( t') d t t 1 b b( t')in ( t') d t t c 1 b b( t')co ( t') d t Since b i a mall perturbation, we write: (5) (6)

4 A A A and ( A A and ). Neglecting the econd order term in A and lead to: db b c hb d h dt Q 1 1 h (7) dt Q QF in where dg1 F co A da A h (8) Q F in Auming that b c and b are not correlated, the frequency noie pectrum i given by: S TF S TF S c (9) b c b b b where S (repectively S c ) i the pectrum of b (repectively b c). b b One can notice from (9) that the pectrum of the ocillator i the um of two component repectively related to the pectrum of the in-phae and quadrature component of the injected noie. Thi i coherent with the interpretation of the LTV approach developed by Hajimiri and Lee [5]. To calculate S() in the cae of white noie, we may write: b S c S (10) b b where b i the tandard deviation of b. Finally, the optimal performance of the ocillator can be derived by tudying the variance of, which i calculated through: S d (11) 0 The actuation force of the reonator in a MEMS ocillator alway ha a component in quadrature with the diplacement (otherwie it cannot ocillate). Hence, we will uppoe that i far from 0 or. Since ocillator are uually made of high-q reonator we will then aume h 1, which give: b 1 Qh (1) 4Q F in To operate the reonator in cloed-loop, the phae f introduced in the feedback loop ha to verify: f. (13) One can etimate from (1) the optimal cloed-loop phae hift f to minimize the frequency noie. Thi i illutrated in the Subection II.B. B. Cae-tudy of a reonator ubject to cubic hardening Among the varietie of nonlinearitie affecting MEMS reonator [11], cubic hardening model accurately the increaing tiffne of mechanical reonator at large diplacement amplitude [15] [16]. In Fig., we tudy the behavior of a MEMS reonator ubject to a cubic nonlinear retoring force etimated from (14). Thi ituation correpond to: (14) g( x) x The illutration i given for =0.8, Q=100 and b=10-4. In order to validate our model, we have performed tranient imulation of (1) for different actuation amplitude F and changed the operating point by chooing the cloed-loop phae hift. One can notice, in Fig., that the variance of obtained in tranient imulation correpond to our etimation given by (1).

5 Fig. Variance of the frequency intability (lef, ocillation amplitude (center) and ocillation frequency (righ with repect to the cloed-loop phae hift for increaing actuation amplitude (F=10-3, F= and F= ). The olid line correpond to the olution given by (13) and the circle to tranient imulation of (1). A long a the actuation force and the diplacement of the reonator are mall, the cubic nonlinearity may be neglected. In thi cae, the optimal phae hift with repect to, on the one hand, the maximal amplitude and, on the other hand, the frequency noie i located at quadrature (blue line in Fig. 1). However, at larger amplitude, the optimal operating point with repect to the frequency tability tart to hift toward phae far from quadrature. The level at which thi phenomenon arie trongly depend on the value of and Q but it hould generally not be mijudged. The ame concluion ha been reached by Villanueva et al. [17] where it ha been interpreted a the combination of the frequency enitivity to thermal noie and to the reonator parameter variation (like the quality factor or the Duffing coefficien. However, our analyi how that the preence of white noie at the input of the MEMS reonator i ufficient to generate a imilar effect. In Section III, we will generalize our method to a nonlinear actuation force and compare the reult to the previouly tudied cubic nonlinear retoring force. III. PERFORMANCE OF CAPACITIVE MEMS-BASED OSCILLATORS In thi ection, we etimate the effect of the cloed-loop phae hift f on the frequency tability of a capacitive MEMS-baed ocillator. We focu on a one-ided parallel-plate capacitive MEMS reonator. The behavior of uch a reonator i known to be affected by a nonlinear actuation force which can be written [15]: F e 1 V0 int 1 x( ( x) (15) where x i the diplacement normalized with repect to the gap ditance between the electrode, the electromechanical tranduction coefficient and V 0 the amplitude of the actuation voltage normalized with repect to the bia voltage V b applied on the reonator. Since the actuation force i uually mall compared to the bia voltage, we will aume V 0<<1. The exiting tudie relating to the influence of the cloed-loop phae hift on the frequency tability of MEMS ocillator have addreed a cubic nonlinear retoring force and a linear force [11] [17]. In the cae of a capacitive MEMS, thi would correpond to a third-order Taylor erie expanion of F e for x<<1, neglecting all the term in V 0x: 3 1 x 3x 4x V int Fe ( x) FTS ( x) 0 (16) In thi cae, the approximate behavior of the reonator would be given by: d x 1 dx x F ( x) b( TS (17) dt Q dt One can notice that the econd-order term of the Taylor-erie expanion ha no impact on the firt harmonic of the diplacement x. Thu, the behavior of the electrotatic force at mall amplitude i imilar to a cubic nonlinear retoring force decribed in Section II auming: 4 and F V0 (18) However, at larger amplitude (i.e. when x 1 doe not hold), we have to model the ytem decribed by:

6 1 d x 1 dx V 0 int x b( (19) dt Q dt 1 x( In thi more complex cae, the method of harmonic balance detailed in Section II lead to the reult preented in Fig. 3 where we compare the reult obtained with the Taylor-Serie approximation (17) to the one obtained from the complete expreion of the electrotatic force (19). The reult are plotted for =-4=-0.04 and Q which highlight the difference. Fig. 3 Variance of the frequency intability (lef, ocillation amplitude (center) and ocillation frequency (righ with repect to the feedback phae for increaing actuation force amplitude (F= (blue), F= 10-6 (red) and F= (dark)). The thick line correpond to the model decribed by (0) wherea the thin line relate to the third-order erie expanion (17). A hown in Fig. 3, a imilar phenomenon a in Fig. appear, where the bet operating point with repect to the frequency tability i not alway located at quadrature. However, Fig. 3 highlight that a cubic nonlinear behavior largely underetimate the pecific behavior of a capacitive MEMS. Surpriingly, thi phenomenon may arie at relatively low amplitude (e.g. 0% of the gap ditance here). The poition of the optimal operating point depend on the reonator characteritic uch a the electromechanical tranduction coefficient and the quality factor Q. If the phae hift introduced in the cloed loop i not deigned properly, improving the ignalto-noie ratio (SNR) by increaing the actuation voltage would lead to poor phae noie performance (ee local maximum on the olid dark line in Fig. 3). However, a good etimation of the characteritic of the ytem make the operation in large amplitude and low phae-noie regime poible by preciely tuning the cloed-loop phae hift. IV. CONCLUSION In thi paper, we obtained a cloed-form expreion to etimate the performance of MEMS ocillator in term of frequency tability. We found out that, with adequate precaution, driving a capacitive high-q reonator in cloed loop into large amplitude regime may improve the phae noie performance of the ocillator. The only noie ource which ha been conidered in thi work i a white noie (typically thermal noie) placed at the input of the MEMS ocillator. Obviouly, the exitence of other noie ource, uch a flicker or white electronic noie coming from other component placed inide the loop would deteriorate the performance of the enor. Hence, our tudy model a bet-cae cenario which could be interpreted a a fundamental limit. Studying the impact of the noie coming from detection and/or amplifier tage on the global performance of the whole ytem i the ubject of ongoing work.

7 V. REFERENCES [1] X. Zou and A. Sehia, "A high-reolution reonant MEMS accelerometer," 18th International Conference on Solid-State Senor, Actuator and Microytem (TRANSDUCERS), pp , 015. [] U. Krihnamoorthya, R. Olon, G. Bogartb, M. Bakera, D. Carrb, T. Swilera and P. Clewa, "In-plane MEMS-baed nano-g accelerometer with ubwavelength optical reonant enor," Senor and Actuator A, Vol , pp , 008. [3] P. Ward and A. Duwel, "Ocillator Phae Noie: Sytematic Contruction of an Analytical Model Encompaing Nonlinearity," IEEE Tranaction on Ultraonic, Ferroelectric, and Frequency Control, vol. 58, no. 1, pp , 011. [4] D. Leeon, "A Simple Model of Feedback Ocillator Noie Spectrum," RFIC Virtual Journal, IEEE, vol. 54, no., pp , [5] A. Hajimiri and T. H. Lee, "A General Theory of Phae Noie in Electrical Ocillator," IEEE Journal of Solid-State Circuit, vol. 33, no., pp , [6] X. Sun, R. Horowitz and K. Komvopoulo, "Stability and Reolution Analyi of a Phae-Locked Loop Natural Frequency Tracking Sytem for MEMS Fatigue Teting," Journal of Dynamic Sytem, Meaurement and Control, vol. 14, pp , 00. [7] K. L. Ekinci, Y. T. Yang and M. L. Rouke, "Ultimate limit to inertial ma ening baed upon nanoelectromechanical ytem," Journal of Applied Phyic, vol. 95, no. 5, pp , 004. [8] V. Kaajakari, J. K. Kokinen and T. Mattila, "Phae Noie in Capacitively Coupled Micromechanical Ocillator," IEEE Tranaction on Ultraonic, Ferroelectric, and Frequency Control, vol. 5, no. 1, pp , 005. [9] D. K. Agrawal and A. A. Sehia, "An Analytical Formulation for Phae Noie in MEMS Ocillator," IEEE Tranaction on Ultraonic, Ferroelectric and Frequency Control, vol. 61, no. 1, pp , 014. [10] M. Pardo, L. Sorenon, W. Pan and F. Ayazi, "Phae Noie Shaping via Forced Nonlinearity in Piezoelectrically Actuated Silicon Micromechanical Ocilaltor," in MEMS, Cancun, 011. [11] E. Kenig, M. C. Cro, L. G. Villanueva, R. B. Karabalin, M. H. Matheny, R. Lifhitz and M. L. Rouke, "Optimal operating point of ocillator uing nonlinear reonator," Phyical Review E, vol. 86, no. 5, p , 01. [1] J. Juillard, "A comparative tudy of reduced-order modeling technique for nonlinear MEMS beam," in Sympoium on Deign, Tet, Integration & Packaging of MEMS and MOEMS, Montpellier, France, 015. [13] J. Juillard, A. Bonnoit, E. Avignon, S. Hentz and E. Colinet, "Large amplitude dynamic of micro-/nanomechanical reonator actuated with electrotatic pule," Journal of Applied Phyic, vol. 107, p , 010. [14] S. Sancho, F. Ramírez and A. Suárez, "Analyi and reduction of the ocillator phae noie from the variance of the phae deviation, determined with harmonic balance," in Microwave Sympoium Diget, 008 IEEE MTT-S International, Atlanta, 008. [15] M.-H. Bao, Micro Mechanical Tranducer, Elevier, 000. [16] R. Lifhitz and M. C. Cro, "Nonlinear Dynamic of Nanomechanical and Micromechanical Reonator," in Review of Nonlinear Dynamic and Complexity, Wiley, 008. [17] L. G. Villanueva, E. Kenig, R. B. Karabalin, M. H. Matheny, R. Lifhitz, M. C. Cro and M. L. Rouke, "Surpaing Fundamental Limit of Ocillator Uing Nonlinear Reonator," Phyical Review Letter, vol. 110, no. 17, p , 013.