Modelling and simulation of the capacitive acceleration sensor : Luận văn ThS. Kỹ thuật điện tử - viễn thông : 2.07.00

VIETNAM N A TIO N A L U N IV ERSITY , HANOI COLLEGE OF TECH NO LO G Y np Ằ_ _ 'TI A T ran Đ ứ c Tan M O D E L IN G A N D S IM U L A T IO N O F TH E C A P A C IT IV E A C C E L E R A T IO N S E N S O R Field: Electronics and Telecommunication Code: 2.07.00 MASTER THESIS Scientific Advisor: P ro f D r N g u y e n P h u T h u y Hanoi, 2004 Đ Ạ I H Ọ C Q U Ố C G IA H À N Ộ I TRƯỜNG ĐẠ • • I H Ọ•C C Ô N G N G H Ệ • TP Ậ_ T\ w _ np £ _ _ Tran Đ ứ c lâ n M Ô HÌNH HĨA VÀ MƠ PHỎNG SENSO R GIA TỐC KIỂU TỤ C h u y ê n n g n h : K ỹ th u ậ t vô tu y ế n đ iện tử v th ô n g tin liên lạc M ã số : 0 L U Ậ N V Ă N T H Ạ C SỸ C án h n g dẫn: G S T S K H N g u y ễ n P h ú T hu ỳ H N ộ i, 0 TABLE OF CONTENTS ASSURAïvCE ACKNOWLEDGMENTS LI ST OF TABLES LI ST OF FIGURES FOREWORD CHAPTER INTRODUCTION 1.1 Overview of MEMS 1 -> 1.2 Silicon Micro Accelerometers 1.2.1 Electromechanical Accelerometers 1.2.2 Piezoelectric Accelerometers 1.2.3 Piezoresistive Accelerometers 1.2.4 Electrostatic Accelerometers 1.2.5 Resonant Accelerometer 1.3 MEMS Modeling and Simulation CHAPTER ACCELEROMETER: FROM THEORY TO DESIGN 2.1 Operational Principles 2.1.1 Open-Loop Design 2.1.2 Force-Balance Design 20 2.1.3 Comparisons 2.2 Capacitive Accelerometer 2.2.1 Position Measurement with Capacitance 2.2.2 Noise Analysis CHAFTER MODELING AND SIMULATION OF THE ACCELEROMETER 3.1 Overview of SUGAR 3.2 Nodal Analysis Approach 3.3 Simulation Program based on SUGAR 24 26 27 32 35 35 36 45 14 14 15 3.4 Simulation Result 3.4.1 Single Capacitive Accelerometer 3.4.2 Differential Capacitive Accelerometer with a Single Beam 3.4.3 Differential Capacitive Accelerometer with Two Symmetric Beams 3.4.4 Two Parallel Beams Accelerometer 47 47 52 54 56 3.4.5 Four Symmetric Beam Accelerometers 3.5 Experimental Calibration Set-up and Experimental Results 57 55 3.6 Comparison o f the simulation and experimental results 63 CHAPTER CONCLUSIONS 4.1 Concluding Remarks 4.2 Future Work 4.3 List o f Publications REFERENCE APPENDIX 67 67 68 70 71 74 Abbreviation ASIC Application-Specific Integrated Circuit CAD Computer-Aided Design CTS Clear To Send Ctf Capacitance to Mtage Converter FEA Finite Element Analysis FHSS Freqency H oping Spread Spectrum ICs Integrated Circuits ITIMS International Training Institute for Materials Science LIGA Lithography Galvanoforming Moulding Processes MEMS Microelectromechanical Systems MOEMS M icroO ptoE lectroM echanical Systems ODEs Ordinary Differential Eqations PDM Pulse D ensity M odulation P-P Peak to Peak PWM Pulse W idth M odulation RTS R eqest To Send List of Tables T able 1.1 A vailable M EM S sim ulation tools, by level and view 13 T able 3.1 Physical param eters o f the sim ple capacitive accelerom eter 47 T able 3.2 G eom etry param eters o f the single capacitive accelerom eter 48 T able 3.3 R elation betw een b eam ’s thickness and resonant frequency 50 T able 3.4 R elation betw een b eam ’s length and resonant frequency 51 T able 3.5 R elation betw een b eam ’s thickness and resonant frequency 55 T able 3.6 R elation betw een b eam ’s thickness and resonant frequency 58 List of Figures Figure 1.1 A compression type piezoelectric accelerometer arrangement Figure 1.2: Piezoresistive acceleration sensor Figure 1.3 Capacitive measurement of acceleration Figure 1.4 Resonant accelerometer Figure 1.5 Cantilever beam and beam - capacitor options 11 Figure 1.6 Nodal analysis and finite elements analysis 12 Figure 1.7 Ideal and actual cantilever beams (side view) 12 Figure 2.1 Open loop accelerometer 15 Figure 2.2 Frequency response and phase response with various damping 19 Figure 2.3 Transient responses with various damping 20 Figure 2.4 Force - balance accelerometer 21 Figure 2.5 Variety o f capacitor structures used for position sensing 27 Figure 2.6 Variety o f differential capacitor structures 27 Figure 2.7 Typical circuit use o f a differential capacitor 28 Figure 2.8 Transimpedance amplifier capture the capacitor current 29 Figure 2.9 Feedback capacitor is added to circuit of Fig 2.8 30 Figure 2.10 Measurement the output voltage of a differential capacitor 31 Figure 3.1 A simple MEMS structure 37 Figure 3.2 A bent beam showing nodal forces, moments, and coordinates 39 Figure 3.3 the level-2 model o f an electrostatic actuator 44 Figure 3.4 Flow chart o f the simulation program 46 Figure 3.5 Single capacitive accelerometer 48 Figure 3.6 Steady responses of the single capacitive accelerometers with different beam thicknesses 49 Figure 3.7 Relation between beam’s thickness and resonant frequency 50 Figure 3.8 Relation between beam’s width and resonant frequency 51 Figure 3.9 Differential capacitive accelerometer with a single beam 53 Figure 3.10 Relation between the voltage and the proof mass’s displacement of the differential capacitive accelerometer with single beam 53 Figure 3.11 A differential capacitive accelerometer with two symmetric beams 54 Figure 3.12 Relation between the voltage and proof mass’sdisplacement of the differential capacitive accelerometer with two symmetric beams Figure 3.13 Two Parallel Beams Accelerometer 55 56 Figure 3.14 Relation between voltage and acceleration of two parallel beams accelerometers 57 Figure 3.15 Four symmetric beam accelerometer 57 Figure 3.16 Relation between voltage and acceleration of the four symmetric beams accelerometer with different beam’s thickness Figure 3.17 The response frequency o f system under dam ping 58 59 Figure 3.18 Differential capacitive acceleration sensors with: double beamsand (b) four symmetrical beams 60 Figure 3.19 Calibration set-up consisting of the rotating disk, circuits and the capacitive acceleration sensor under test Figure 3.20 The CVC circuit and interface of the calibration set-up the CVC, wireless 61 61 Figure 3.21 Relation between voltage and acceleration of the different sensors with the configurations noted in the figures 62 Figure 3.22 Relation betw een the voltage and the acceleration: com parison betw een sim ulation and experim ental results 65 Figure 4.1 A newly suggested structure with four symmetric beams 68 Figure 4.2 A comb structure proposed for future work 69 Foreword MEMS technology has been developed since 1960 and MEMS products have been commercialized and widely used around the world since 1980 In Vietnam, however, this new field of technology has only been studied several years ago Following this trend, the College of Technology o f VNUH started research on MEMS devices and their applications in 2003 This thesis is a continuation o f this effort and is the first attempt to investigate and design MEMS sensor by modeling and simulation My thesis includes four chapters The first chapter introduces an overview of MEMS and discusses some types of accelerometers Capacitive accelerometer has been chosen to be the object of my thesis because of its high sensitivity, good dc response, noise performance, low drift, low temperature sensitivity, low-power dissipation, and simple structure Chapter discusses operational principles of openloop and force-balance accelerometers In addition, results of position measurement and noise analysis of the capacitive accelerometer are given Chapter focuses on modeling and simulation of different structures using SUGAR language in MATLAB environment In particular, the simulation results are compared to experimental results Finally, the conclusions of this research and proposal for future study are presented in chapter of this thesis Chapter I CHAPTER INTRODUCTION 1.1 Overview of MEMS M icroelectrom echanical systems (M EM S) are collection o f m icrosensors and actuators that have the ability to sense its environm ent and react to changes in that environm ent w ith the use o f a m icrocircuit control They also include the conventional m icroelectronics packaging, integrating antenna structures for com m and signals into m icroelectrom echanical structures for desired sensing and actuating functions The system may also need m icropow er supply, microrelay, and m icrosignal processing units M icrocom ponents m ake the system faster, more reliable, cheaper, and capable o f incorporating m ore com plex functions In the beginning o f 1990s, M EM S appeared w ith the aid o f the developm ent o f integrated circuit fabrication processes, in w hich sensors, actuators, and control functions are co-fabricated in silicon [1] Since then, rem arkable research progresses have been achieved in M EM S under the strong prom otions from both governm ent and industries In addition to the com m ercialization o f some less integrated M EM S devices, such as m icroaccelerom eters, inkjet printer head, m icrom irrors for projection, etc., the concepts and feasibility o f m ore complex M EM S devices have been proposed and dem onstrated for the applications in such varied fields as m icrofluidics, aerospace, biom edical, chem ical analysis, wireless com m unications, data storage, display, optics, etc Som e branches o f M EM S appearing as m icrooptoelectrom echanical system s (M O E M S), m icro total analysis system s, etc., have attracted a great research since their potential applications' m arket At the end o f 1990s, m ost o f M EM S devices w ith various sensing or actuating m echanism s w ere fabricated using silicon bulk m icrom achining, surface Modeling and simulation o f the capacitive accelerometer Chapter I m icrom achining, and lithography, galvanoform ing, m oulding (L1GA) processes [2 Three-dim ensional m icrofabrication processes incorporating m ore m aterials were presented for M EM S recently because o f specific application requirem ents (e.g biom edical devices) and higher output pow er m icroactuators M icrom achining has becom e the fundamental technology for the fabrication o f M EM S devices and, in particular, m iniaturized sensors and actuators Silicon m icrom achining is the m ost advanced o f the m icrom achining technologies, and it allow s for the fabrication o f M EM S that have dim ensions in the subm illim eter range It refers to fashioning m icroscopic m echanical parts out o f silicon substrate or on a silicon substrate, m aking the structures three dim ensional and bringing new principles to the designers Em ploying m aterials such as crystalline silicon, polycrystalline silicon, silicon nitride, etc., a variety o f m echanical m icrostructures including beam s, diaphragm s, grooves, orifices, springs, gears, suspensions, and a great diversity o f other com plex m echanical structures have been conceived In som e applications, stresses and strains to which the structure is subjected to may pose a problem for conventional cabling In others, environm ental effects may affect system perform ance A dvances in ultra flat antenna technology coupled w ith M EM S sensors and actuators seem to be an efficient solution The integration o f m icrom achining and m icroelectronics on one chip results in so-called sm art sensors [3], In sm art sensors, small sensor signals are am plified, conditioned, and transform ed into a standard output format They may include m icrocontroller, digital signal processor, application-specific integrated circuit (ASIC), self-test, self-calibration, and bus interface circuits sim plifying their use and m aking them m ore accurate and reliable Silicon m icrom achining has been a key factor for the vast progress o f M EM S in the last decade This refers to the fashioning o f m icroscopic m echanical parts out o f silicon substrates and, more recently, other m aterials It is used to fabricate such features as clam ped beam s, m em branes, cantilevers, grooves, orifices, springs, gears, suspensions, etc These can be assem bled to create a variety o f sensors Bulk m icrom achining is the com m only used m ethod, but it is being replaced by surface m icrom achining that offers the attractive possibility o f integrating the m achined Modeling and simulation o f the capacitive accelerometer Chapter Figure 3.22 Relation betw een the voltage and the acceleration: com parison betw een sim ulation and experim ental results A lthough the sim ulation results are relied on sim ple m odels, they seem to be conformable to the experim ent and they can be used to explain logically experimental results A t low acceleration, the difference betw een the sim ulation and experim ental results are very small But, there occurs a non-neglected difference when applying high acceleration to the structure This draw back can be explained by following reasons: ■ The electronic noise and dissipation in m easuring system: noise and dissipation are certainly non-neglected effects in every circuit (Sec 2.2) We can reduce these effects by optim izing and packing the circuit in one chip * U nsuitable conditions applying for sim ulation: As w e m entioned above about the stiffness param eter, w e used energy m ethods to sim ulate the behavior o f the structure w hen undergone a large deform ation (a high acceleration) The sim ulation results could reflect this effect but not very well It is quite difficult to sim ulate an elastic body undergone a large deform ation ■ The gravity field: In sim ulation program s, w e could totally neglect the effect o f the gravity field but in the real condition, w e could not eliminate it perfectly ■ Errors in fabrication and packaging: in order to perform anodic bonding betw een a glass and a silicon w afer either for defining the capacitor or for packaging reasons, high voltage exceeding by Modeling and simulation o f the capacitive accelerometer far critical and high 65 Chapter tem perature are applied The presence o f very strong electrostatic forces w ould causes errors on the sensor O therw ise, im perfect packaging could not prevent the device from the environm ent effects Rem inding, how ever, that m ost o f these sensors are designed for low acceleration (up to 2-g applications), we conclude that the sim ulation results can reflect correctly the behavior o f the real accelerom eters Modeling and simulation o f the capacitive accelerometer 66 Chapter CHAPTER C O N C L U S IO N S 4.1 C oncluding R em arks This thesis aims at m odeling and sim ulation o f different vertical structures o f the capacitive accelerom eter W e have developed different m odels, such as: ■ Single capacitive accelerom eter ■ D ifferential capacitive accelerom eter w ith single beam ■ D ifferential capacitive accelerom eter w ith two sym m etric beams ■ Tw o parallel beam accelerom eter ■ Four sym m etric beam accelerom eter These M EM S-based sensors are m odeled by SUGAR language in M ATLAB environm ent W e have used SU G A R to com pute the following characteristics o f these sensors: ■ The resonant frequency (the w orking range) ■ The relation betw een force and displacem ent (m echanical sensitivity) ■ The relation betw een acceleration and voltage (electronic sensitivity) In particular, we com pared the sim ulation results and the experim ental ones, which w ere obtained by using a specific m easurem ent system on M EMS accelerom eters fabricated in ITIM S The sim ulation results are conform able to explain logically experim ental ones B oth open-loop sensor and force-balance (closed-loop) sensor are discussed in chapter O pen loop accelerom eter was chosen to be m odeled and sim ulated because o f its sim plicity and low cost A lthough the sim ulation results in chapter are relied on sim ple m odels, they are conform able to explain logically experimental Modeling and simulation o f the capacitive accelerometer 67 Chapter results These sim ulation programs can also be used to design the structures that m eet the desire Though the results derived from sim ulation and experim entation are not very w ell fitness at high acceleration region, but it is quite acceptable in general 4.2 F uture W ork This thesis provides different m odels o f M EM s-based accelerom eter but m uch m ore w ork is necessary to model m ore advance structures To reduce the size and im prove the sensitivity and working range o f the sensor, we will continue m odeling the accelerom eters in three directions: To obtain large sensitivity and low noise, a large p ro o f mass is needed, which suggests the use o f bulk m icrom achined techniques Figure 4.1 shows a newly suggested structure w ith four symmetric beams Figure 4.1 A newly suggested structure with four symmetric beams O ne other way to increase the sensitivity is producing the sensing capacitance in the com b structure so that their capacitance can be drastically increased Figure 4.2 shows a comb structure that is under developing This kind o f accelerom eter can be fabricated by using surface m icrom achined technique Modeling and simulation o f the capacitive accelerometer 68 C hapter Figure 4.2 A comb structure proposed for future work A s discussed in chapter 2, force-balance accelerom eters can provide high sensitivity and large linear working range Therefore I will certainly concentrate also on m odeling o f this type o f acceleration sensor Modeling and simulation o f the capacitive accelerometer 69 Chapter 4.3 L ist of P ub lication s [1] Bùi Thanh rùng, Chủ' Đức Trình, Nguyễn Phú Thuỳ, Vũ Ngọc Hùng, Trần Đức l ân Vũ Việt Hùng (2004), “ủ n g dụng cảm biến áp suất M EM S thiết bị điện tử V tế” Tuyển tập hộ/ nghị điện tử toàn quốc lần thứ Thành phố Hồ Chí M inh 52003, tr 3 - [2] Trần Đức Tân, Chử Đức Trình, Nguyễn Phú Thuỳ, Vũ Ngọc Hùng (2004), “Cảm biến gia tốc kiểu tụ”, H ội nghị khoa học Khoa Công N ghệ, 10-11 -2004, in Journal o f Science, Đại học Q uốc Gia Hà Nội, 2005 [3] Trần Đ ức Tân, N guyễn Phú Thuỳ, Chử Đức Trình, N guyễn Thăng Long, Vù N gọc H ùng “Cảm biến gia tốc kiểu tụ: So sánh kết m ô thực nghiệm ” Sẽ trình bày H ộ i nghị Tự động hố tồn quốc VICA6, 2005 M odeling and simulation o f the capacitive accelerometer 70 REFERENCE Petersen, K E “Silicon as a m echanical m aterial”, Proc IE E E , 70(982) pp.42('-457 M ohamed G ad-el-H ak (2002), The M E M S handbook, CRC Press, New York Bhara: Brushan (1999), H andbook o f M icro/N ano Tribology, CRC Press Londcn L.C.Spangler, C.J.Kem p (1996), “ISAAC: Integrated silicon automotive accelerom eter”, Sens Actuators, A54 pp 523-529 Bernhard E Boser, R oger T Howe, (1996), “ Surface M icrom achined A ccelerom eter”, IEE E Journal o f solid - state circuits, Vol 31(3), pp 179191 Harvey W einberg (2002) “M EM S Sensors Are D riving the Autom otive Industry”, http://w ww.sensorsm ag.com /articles/articlesZ0202/36/m ain.shtm / S Senturia (2001), M icrosystem D esign, K luw er A cadem ic Publishers M ukherjee and G K Fedder (1997), “ Structured m icroelectrom echanical system s”, Proc o f 34th D esign Conference (D A C ’97), A naheim , CA, pp 680-685 design o f Automation Varadan, V K and Varadan (1995), “3D M EM S Structures and their A pplications”, Invited Paper p resen ted at the International Symposium on M icrosystems, Intelligent M aterials, a n d Robots, Tohoku University, Japan 10 J V Clark, N Zhou, D Bindel, L Schenato, W W u, J Dem m el, K S J Pister (2002), “3D M EM S Sim ulation M odeling U sing M odified Nodal A nalysis”, Tech Digest, Solid-State Sensor and A ctuator W orkshop, pp 191196 11 G Li and N R Aluru (2001), “Linear, nonlinear and m ixed-regim e analysis o f electrostatic M E M S”, Sensors a n d Actuators, A91(3) pp 2782 ,2 0 M o d e l i n g and s i m u l a t i o n o f t h e c a p a c i t i v e a c c e l e r o m e t e r 71 12 J H K ane (1994), Boundary Elem ent Analysis in Engineering Continuum M echanics, Prentice-H all 13.G K Fedder (1999) “ Structured design for integrated M EM S”, Proc IEEE M E M S '99, Orlando, FL, pp 1-8, 1999 14 L C Spangler, C.J Kemp (1996) ‘in terg rated accelerom eter”, Sens Actuators, A54 pp 523-529 silicon autom otive 15.Bernstein, Jonathan (2003), “A n O verview o f M EM S Inertial Sensing Technology Sensors” http.VAvww.sensorsmag com /articles/0203/14/main, shtm l 16 Harvey W einberg (2003) “Building a Tiny A ccelerom eter to D etect Very Small Signals”, http://w \v\v.sensorsmag com /articles/articles/0201/20/m ain, shtml 17.N R Swart, S F Bart, M H Zam an, M M ariappan, J Gilbert, and D M urphy (1998) “AutoM M : A utom aticG eneration of Dynamic M acrom odels for M EM S D evices”, Heidelberg, G erm any, pp 178-183 18 R Puers, S Reyntjens (1998) “Design and processing experim ents o f a new m iniatured capacitive triaxial accelerom eters”, Sens A ctuators, A68 pp.324328 19 R obert Puers, D aniel Lapadatu (1996), “Electrostatic forces and their effects on capacitive m echanical sensors”, Sens A ctuator, A56 pp 203-210 20 W olfgang K uehnel (1994), “M odeling o f the m echanical behavior o f a differential capacitor accelerom eter sensor”, Sens Actuator, A48, pp 101108 21 B ernhard, E.Boser (2002), “C apacitive Sensor Interfaces”, Berkeley Sensor a n d A ctu a to r C enter, http:// bernhard.bsac.berkeley.edu 22 J C hae, H Kulah and K Najafi (2004), “An In-plane High Sensitivity, Low N oise M icro-g Silicon A ccelerom eter w ith CM OS Readout C ircuitry”, IEEE Jo u rn a l o f M icroelectrom echanical System s, Vol 13(4), pp 628-635 D avid B indel, Jason Clark, N ingning Zhou (2002), “ SUGAR 3.0: A MEMS S im ulation Program ”, http://bsac.berkeley.edu M odeling and s i m u l a t i o n o f the c a p a c i t i v e a c c e le r o m e te r 72 24 J V Clark, D Bindel, N Zhou, S Bhave, Z Bai, J D em m el, K S J Pister (2001), “A dvancem ents in 3D M ulti D om ain Sim ulation Package for M E M S’', P roceeding o f the M icro scale System s, pp 40-45 25 Vu N goc Hung, N guyen Phu Thuy, N guyen Due Chien, Chu Due Trinh N guyen Thi M inh Hang (2004), “A sim ple approach in fabrication o f capacitive acceleration sensor based on bulk m icrom achining”, Proceeding o flC M T pp 355-340 26.T M ineta, S.K obayashi, Y W atanabe, S.K anauchi, I.N agakaw a, E.Suganuma M Esashi (1995) “Three-axis capacitive accelerom eter with uniform axial sensitivities”, Transducer 95, Stokholm, Sweden, p p 544-577 M o d e l i n g and s i m u l a t i o n o f the c a p a c i t i v e a c c e l e r o m e t e r 73 APPENDIX The follow ing listing codes can run if SUGAR is installed in M ATLAB environment SU G A R can be accessible on the W orld W ide W eb (W W W ) http://bsac.berkeley.edu/cadtools/sugar SU G A R ’S m odel subnet beam2de [a b] [1=* w=* h=* R=? G=? resistivity=? density=* fluid=* viscosity=* Youngsmodulus=*] [ call assert(isdef(R) | isdef(G) | isdef(resistivity), 'R, G, or resistivity must be defined in beam2de') if isdef(R) [ R parent [a b] [R=R] ] else if isdef(G) [ R parent [a b] [G=G] ] else if isdef(resistivity) & isdef(l) & isdef(w) & isdef(h) f R parent [a b] [R=resistivity*l/(w*h)] ] else [ call assert(0, 'R, G, or resistivity must be defined') ] beam2d parent [a bj [1=1 w=w h=h density=density viscosity=viscosity Youngsniodulus=Youngsmodulus] ] subnet beam3de [a b] [1=* w=* h=* R=? G=? resistivity=? density=* fluid=* viscosity=* Youngsmodulus^] [ if isdef(R) [ R parent [a b] [R=R] ] else if isdef(G) [ R parent [a b] [G=G] ] else if isdef(resistivity) & isdef(l) & isdef(w) & isdef(h) [ R parent [a b] [R=resistivity*l/(w*h)] ] else [ call assert(0, 'R, G, or resistivity must be defined') beam3d parent [a b] [1=1 w=w h=h density=density viscosity=viscosity M o d e l i n g and s i m u l a t i o n o f t h e c a p a c i t i v e a c c e l e r o m e t e r 74 Youngsmodulus^ Youngsmodulus] subnet gap2de [a b e d ] [I-* w=? w l= ? w2=? h=* R=? R -? R -? G=? G -? G -? resistivity-? density-* fluid-* viscosity-* Youngsmodulus-* gap-* permittivity-*] call assert(isdef(w) | (isdef(w l) & isdef(w2)), 'Must define beam widths') call assert(isdef(resistivity) | isdef(R) | isdef(G) | ((isdef(R 1) | isdef(G 1)) & isdef(R2) | isdef(G2)), 'Must define beam resistivities') wl w2 R1 R2 G1 G2 - cond(isdef(wl), w l, w) cond(isdef(w2), w2, w) cond(isdef(Rl), R l, R) cond(isdef(R2), R2, R) cond(isdef(G 1), G 1, G) cond(isdef(G2), G2, G) R l - cond(isdef(Rl) | isdef(G l), R l, resistivity *l/(w 1*h)) R2 - cond(isdef(R2) | isdef(G2), R2, resistivity*I/(w2*h)) beam2de parent [a b] [1=1 w -w l h-h density=density fluid-fluid viscosity-viscosity Y oungsm odulus^Y oungsm odulus R -R G -G ] beam2de parent [c d] [1-1 w=w2 h-h density-density fluid-fluid viscosity-viscosity Youngsmodulus-Youngsmodulus R - R l G=G1] gap2dforce parent [a b e d ] [1-1 w l- w l w2-w2 gap-gap permittivity-permittivity] ] subnet gap3de [a b e d ] [1=* w=? w l- ? w2=? h=* R -? R -? R -? G -? G = ? G -? resistivity-? density-* fluid-* viscosity-* Youngsmodulus-* gap-* permittivity-*] t call assert(isdef(w) | (isdef(w l) & isdef(w2)), 'Must define beam widths') M o d e l i n g and s i m u l a t i o n o f t h e c a p a c i t i v e a c c e l e r o m e t e r 75 call asser(isdef(resistivity) | isdef(R) | isdef(G) | ((isd ef(R l) | isdef(G l)) & isdef(R2) | isdef(G2)), 'Must define beam resistivities') w l = eord(isdef(wl), w l, w) w2 = cord(isdef(w2), w2, w) R1 = cord(isdef(Rl), R l, R) R2 = cord(isdef(R2), R2, R) G1 = cord(isdef(G 1), G 1, G) G2 = cord(isdef(G2), G2, G) R l = cord(isdef(Rl) | isdef(Gl), R l, resistivity*l/(w 1*h)) R2 = cord(isdef(R2) | isdef(G2), R2, resistivity*l/(w2*h)) beam3de parent [a b] [1=1 w =w l h=h density=density fluid=fluid viscosity=viseosity Youngsmodulus=Youngsmodulus R=R1 G=G1] beam3de parent [c d] [1=1 w=w2 h=h density=density fluid=fluid viscosity=viscosity Youngsmodulus=Youngsmodulus R=R1 G=G1] gap3dforce parent [a b e d ] [1=1 w l= w l w2=w2 gap=gap permittivity=permittivity] subnet gap2dV [a b e d ] [I-* w~? w l= ? w2=? R=? R 1=? R2=? G=? G 1=? G2=? resistivity=? density=* fluid=* viscosity=5,t Youngsmodulus=* gap=* permittivity=* V=*] [ gap2dV parent [a b e d ] [1=1 w=w w l= w l w2=w2 R=R R1=R1 R2=R2 G=G G1=G1 G2=G2 resistivity=resistivity density=density fluid=fluid viscosity=viscosity Youngsmodulus=Youngsmodulus gap=gap permittivity=permittivity] Vsrc * [d b] [V=V] eground * [b] [] ] subnet gap3dV [a b e d ] [1=* w=? w l= ? w2=? R=? R 1=? R2=? G=? G 1=? G2=? resistivity=? density=* fluid=* viseosity=* Youngsmodulus=* gap=* permittivity=* V=*] [ M o d e l i n g and s i m u l a t i o n o f t h e c a p a c i t i v e a c c e l e r o m e t e r 76 gap3dV parent [a b c d| [1=1 w=w w l= w l w2=w2 R=R R1=R1 R2=R2 G=G G1=G1 G2=G2 resistivity=resistivity density=density fluid=fluid viscosity=viscosity Youngsmodulus=Youngsmodulus gap=gap permittivity=permittivity] Vsrc * [d b] [V =V ] eground * [b] [] Comb’s netlist: useC'std2cvt.net") pi = 3.1415926535897932385 poly = material { Poisson = 0.3, thermcond = 2.33, viscosity = 1.78e-5, fluid = 2e-6, density = 2300, Youngsmodulus = 165e9, permittivity = 8.854e-12, sheetresistance = 20, stress = 0, straingradient = 0, thermalexpansion = 0, ambienttemperature = } pi = material { parent = poly, h = 2e-6, Poisson = 0.3, thermcond = 2.33, viscosity = 1.78e-5, fluid = 2e-6, density = 2300, Youngsmodulus = 165e9, permittivity = 8.854e-12, sheetresistance = 20, stress = 0, straingradient = 0, thermalexpansion = 0, ambienttemperature = } p2 = material { parent = poly, h = 1.5e-6, Modeling and s i m u l a t i o n o f the c a p a c i t i v e a c c e le r o m e te r Poisson = 0.3, thermcond = 2.33, viscosity = 1.78e-5, fluid = 2e-6, density = 2300, Youngsmodulus - 165e9, permittivity = 8.854e-12, sheetresistance = 20, stress = 0, straingradient - 0, thermalexpansion = 0, ambienttemperature = } d2 = material { parent = poly, fluid = 0.75e-6, Poisson = 0.3, thermcond = 2.33, viscosity = 1.78e-5, fluid = 2e-6, density = 2300, Youngsmodulus - 165e9, permittivity = 8.854e-12, sheetresistance = 20, stress = 0, straingradient = 0, thermalexpansion = 0, ambienttemperature = } subnet mfXSusp (B, material, su spjen) local parent = material _currnodes["B"] = B su sp jen = su sp je n or (material and material.suspjen) mfanchor {_n("A "); material = parent, = lOu, w = lOu, h - lOu, oz = deg(90)} mfbeam3d {_n("A "), _ n (" a r ); material = parent, = suspjen, w = 2u, h = lOu, oz = deg(0)} mfbeam3d {_ n ("a r'), _n("a2"); material = parent, = lOu, w = 2u, h = lOu, oz = -(deg(90))} mfbeam3d {_n("a2"), _n("B"); material = parent, = suspjen, w = 2u, h = lOu, oz = deg( 180)} mfbeam3d {_n("A "), _n(Ma3M); material = parent, = suspjen, w = 2u, h = lOu, oz = deg(l 80)} mfbeam3d {_n("a3"), _n("a4"); material = parent, = lOu, w = 2u, h = lOu, oz - -(deg(90))} mfbeam3d {_n("a4M), _n("B"); material = parent, = suspjen, w = 2u, h = lOu, oz = deg(0)} end subnet mfXMass (A, B, material, fingerjen) local parent = material _currnodes["AM] = A _currnodes[”B"] = B fin g erjen = fin g erje n or (material and material.fingerjen) mfbeam3d {_n("A "), _ n ("b lM); material = parent, = 25u, w = 50u, h = lOu, oz = -(deg(90))} mfbeam3d {_ n ("b r')5_n("B"); material = parent, = 25u, w = 50u, h = lOu, oz = -(deg(90))} M odeling and s i m u l a t i o n o f the c a p a c i t i v e a c c e le r o m e te r mfbeam3d {_n(Mb 1M), _n("b2"); material = parent, = fingerjen, w = 2u, h = lOu, oz deg(O)} mfbeam3d {_n ("b l"), _n("b3M); material = parent, I = finger_len, w = 2u, h = lOu, oz deg( 180)} end mfXSusp {_n("c" (1)); material = p i, suspjen = 200u} for k = , 10 mfXMass { n("c" (k)), _n(McM ((k )+ (l))); material = p i, fingerjen = 100u} mfXSusp {_n("c" relpos treewalkO (11)); material = p i, suspjen = 200u, oz = pi} MATLAB program: % for k = l : % param V=k- l ; % net = cho_load('I v 1.net',param); % dq = cho_dc(net,dq); % titIe('Deflected structure'); % dy(k) = cho_dq_view(dq, net, 'x3', 'y'); %end %plot((0.00001+dy),'LineWidth',3); clc;clear; dq= [ ]; for k=25:40 param.thick=k/l 000000; net = cho Jo ad ('lvl net',param); [f, egv, dq] = cho_mode(net); cho_modeshape(net, f, egv, dq, 1); y(k)=f(l)/(2*pi); end stem((25:40)/l 000000,y(25:40),'.'); clc; for k = l:39 param vv=k-l; net = cho_load('diffl net',param); dq = cho_dc(net); dy(k)= cho_dq_view(dq, net, 'x3', 'y'); %stem(0:k-l,dy,V); %hold on; plot(0:k-l,dy); hold on; plot(0:k-l ,dy,'p'); M odeling and s i m u l a t io n o f the c a p a c i t i v e a c c e le r o m e te r 79