Design, Fabrication, and Dynamic Modeling of a Printed Circuit Based MEMS Accelerometer Except where reference is made to the work of others, the work described in this thesis is my own or was done in collaboration with my advisory committee. This thesis does not include proprietary or classified information. John E. Rogers Certificate of Approval: Ramesh Ramadoss, Co-Chair Assistant Professor Electrical and Computer Engineering John Y. Hung, Co-Chair Professor Electrical and Computer Engineering Bodgan M. Wilamowski Professor Electrical and Computer Engineering George T. Flowers Interim Dean Graduate School Design, Fabrication, and Dynamic Modeling of a Printed Circuit Based MEMS Accelerometer John E. Rogers A Thesis Submitted to the Graduate Faculty of Auburn University in Partial Fulfillment of the Requirements for the Degree of Master of Science Auburn, Alabama May 10, 2007 Design, Fabrication, and Dynamic Modeling of a Printed Circuit Based MEMS Accelerometer John E. Rogers Permission is granted to Auburn University to make copies of this thesis at its discretion, upon the request of individuals or institutions and at their expense. The author reserves all publication rights. Signature of Author Date of Graduation iii Vita John Elvin Rogers was born in Columbus, Georgia in 1982. He grew up in Smiths, Alabama where he graduated from Smiths Station High School in 2002. John then attended Auburn University where he graduated in December 2005 with a Bachelor of Electrical Engineering degree. He then entered graduate school at Auburn University in Spring 2006. His research interests are in nonlinear systems and control, the design and fabrication of Micro-Electro Mechanical Systems (MEMS), Silicon-on-Insulator (SOI) and Printed Circuit Based (PCB) Fabrication Techniques, and monolithic integration of MEMS with electronics. John?s interests in MEMS has brought him to study various modeling issues associated with these devices that are still unstudied by engineers. iv Thesis Abstract Design, Fabrication, and Dynamic Modeling of a Printed Circuit Based MEMS Accelerometer John E. Rogers Master of Science, May 10, 2007 (B.E.E., Auburn University, 2005) 76 Typed Pages Directed by John Y. Hung & Ramesh Ramadoss A MEMS capacitive-type accelerometer fabricated using printed circuit processing tech- niques is presented. A KaptonR? polymide film is used as the structural layer for fabricat- ing the MEMS accelerometer. The accelerometer proof mass along with four suspension beams are defined in the KaptonR? polyimide film. The proof mass is suspended above a RT/DuroidR? (TeflonR?) substrate using a spacer. The deflection of the proof mass is de- tected using a pair of capacitive sensing electrodes. The top electrode of the accelerometer is defined on the top surface of the KaptonR? film. The bottom electrode is defined in the metallization on the RT/DuroidR? substrate. The initial gap height between the bottom electrode and the KaptonR? film is approximately 41.8 ?m. For an applied external acceler- ation/deceleration (normal to the proof mass), the proof mass deflects towards or away from the fixed bottom electrode due to inertial force. This deflection causes either a decrease or increase in the air gap height thereby either increasing or decreasing the capacitance between the top and the bottom electrodes. v An example PCB MEMS accelerometer with a square proof mass of membrane area 6.4 mm?6.4 mm is reported. The measured resonant frequency of 375 Hz and the Q-factor in air is 1.5. The ability to build MEMS accelerometers using low-cost printed circuit processing techniques allows for integration of electronics, suitability for high-volume manufacturing, and large surface area applications for low-g accelerometers. These are all key advantages for using PCB MEMS accelerometers. vi Acknowledgments I?d like to first and foremost dedicate this to the memory of my father, James Rogers. He has always been a big inspiration in my life and has inspired me to be who I am today, as he was an Electrical Engineer too. I?d also like to give a big thanks to my parents, Elesa and Randy Curenton, for always having been there for me and helping me learn to live life for what it is. I?d also like to thank my grandparents, Mary and John Rogers, for always pushing me to do my best and strive for more in life. I?d also like to give thanks to my grandparents, Elenor and Elvin Nix, and my aunt and uncle, Debbie and Eddie Nix. Last, but not least I?d like to thank my fiancee, Brittany Michelle Camp. She has been there for me through it all and inspires me to be everything I am. A special thanks goes to Dr. John Y. Hung and Dr. Ramesh Ramadoss for their guid- ance and wisdom. Without their kindness, knowledge, and having given me the opportunity to work with them, none of this work would have been made possible. I?d also like to thank my colleague, Phil M. Ozmun, for his help and insight in our research of the analysis and fabrication of various MEMS devices. My final thanks goes to Ms. Madhurima Maddella for her assistance and help with the fabrication of the accelerometer. vii Style manual or journal used Journal of Approximation Theory (together with the style known as ?aums?). Computer software used The document preparation package TEX (specifically LATEX) together with the departmental style-file aums.sty, Microsoft PowerPoint for graphics. viii Table of Contents List of Figures xi 1 PCB MEMS Accelerometer 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Materials, Configuration, & Analysis . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Spring Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Proof Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Natural Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.4 Damping Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.5 Quality Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.1 Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.2 Spacer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.3 KaptonR? Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Detailed Fabrication Process . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Photolithography Masks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.7 Fabrication Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Experimental Characterization 25 2.1 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Conclusions & Future Work 42 3.1 Il Buono . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Il Brutto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Il Cattivo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Bibliography 44 ix A Rigid Body Model for MEMS Accelerometer 46 A.1 Electrical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 A.2 Mechanical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 A.3 Mechanical Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . 49 B Rayleigh-Ritz Method for Determining Equivalent Mass of a Flexible Structure 50 C Linear State Variable Analysis for PCB MEMS Accelerometer 53 C.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 C.2 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 C.3 State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 C.4 Equilibrium State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 C.5 Linearized State Variable Model . . . . . . . . . . . . . . . . . . . . . . . . 55 C.6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 C.7 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 C.8 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 C.9 Stabilizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 C.10 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 C.11 Linear State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 C.12 Observer Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 C.13 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 D MATLAB Code 63 x List of Figures 1.1 Photograph of the fabricated PCB MEMS Accelerometer integrated with capacitive readout chip. Courtesy AMSTC-Auburn University . . . . . . . . 2 1.2 Sequence of Layers for PCB MEMS Accelerometer . . . . . . . . . . . . . . 3 1.3 Deflection of the PCB MEMS Accelerometer . . . . . . . . . . . . . . . . . 4 1.4 Schematic of the PCB MEMS Accelerometer Top view (not to scale) . . . . 6 1.5 Schematic of the PCB MEMS Accelerometer Cross-sectional view (not to scale) 7 1.6 A 2 mil thick KaptonR? film used for fabrication of the PCB MEMS Ac- celerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Kapton Mask used in Photolithography . . . . . . . . . . . . . . . . . . . . 19 1.8 CopperTop Mask used in Photolithography . . . . . . . . . . . . . . . . . . 20 1.9 Subini Mask used in Photolithography . . . . . . . . . . . . . . . . . . . . . 21 1.10 Subcop Mask used in Photolithography . . . . . . . . . . . . . . . . . . . . 22 2.1 A Photograph of an LDS model V408 electromechanical shaker (courtesy Auburn University). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Full photograph of PCB MEMS Accelerometer. Courtesy AMSTC-Auburn University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Laser Interferometric Measurement System . . . . . . . . . . . . . . . . . . 28 2.4 Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer (First Test, Mechanical Response) . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer (Second Test, Electrical Response) . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer (Second Test, Mechanical Response) . . . . . . . . . . . . . . . . . . . . . . 31 xi 2.7 Magnitude Plot of the Experimental and Theoretical Results using Second- Order Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.8 Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer (Third Test, Electrical Response) . . . . . . . . . . . . . . . . . . . . . . . . 34 2.9 Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer (Second Test, Mechanical Response) . . . . . . . . . . . . . . . . . . . . . . 37 2.10 Four pole, three zero, one delay transfer function (MATLAB SYSID) . . . . 38 2.11 Eight pole, six zero, three delay transfer function (MATLAB SYSID) . . . . 39 2.12 Eight pole, seven zero, one delay transfer function (MATLAB SYSID) . . . 40 2.13 Fourth order pole-zero placement transfer function . . . . . . . . . . . . . . 41 A.1 Cross-sectional view of PCB MEMS device . . . . . . . . . . . . . . . . . . 46 B.1 Shape Function for Suspension Beam . . . . . . . . . . . . . . . . . . . . . . 50 C.1 Normalized pull-down voltage curve . . . . . . . . . . . . . . . . . . . . . . 55 C.2 Block diagram for linear state feedback control . . . . . . . . . . . . . . . . 60 C.3 Output response for linear state feedback control . . . . . . . . . . . . . . . 61 C.4 Block diagram for estimated state feedback control . . . . . . . . . . . . . . 61 C.5 Output response for estimated state feedback control . . . . . . . . . . . . . 62 C.6 Estimation error for estimated state feedback control . . . . . . . . . . . . . 62 D.1 PCB MEMS Accelerometer Parameters MATLAB Code . . . . . . . . . . . 63 D.2 PCB MEMS Accelerometer Paramaters MATLAB Code (Cont.) . . . . . . 64 xii Chapter 1 PCB MEMS Accelerometer 1.1 Background Micro-Electro-Mechanical Systems (MEMS) is a technology that integrates many engi- neering fields: electrical, mechanical, materials, computer science, and control systems [1]. MEMS can be classified as either sensors or actuators, and in many applications are a combination of the two. MEMS were first introduced in the early 1960s as discrete open- loop pressure sensors [2]. Typical advantages of MEMS devices include small-size (in the range of 300 nm to 300 ?m), light-weight, low-cost, high performance, and low-power con- sumption [3]. MEMS have found applications in many industries including automotive, biomedical, aerospace, and communications, among many others. MEMS have proven to be a revolutionary technology in many application areas including accelerometers, gyro- scopes, pressure sensors, displays, inkjet nozzles, and fluid pumps. Along with the advances in MEMS technology has been a growing interest in the control of MEMS actuators to achieve extended range of motion [4]. In recent years, there has been a considerable interest in development of meso-scale (on the order of ? 10 mm) MEMS devices fabricated using printed circuit processing techniques known as Printed Circuit Based MEMS or PCB MEMS [5]. In PCB MEMS technology, organic polymer materials are used as substrates, sacrificial layers, and structural layers for fabrication of MEMS devices. PCB MEMS enable monolithic integration of MEMS devices and electronics using low-cost, conventional printed circuit techniques. The advantages of 1 Figure 1.1: Photograph of the fabricated PCB MEMS Accelerometer integrated with ca- pacitive readout chip. Courtesy AMSTC-Auburn University PCB MEMS components include low-cost, ease of integration with electronics, suitabil- ity for high-volume manufacturing, and large surface area applications. Recently, PCB MEMS devices such as flow sensors, tactile sensors [6], pressure sensors [7], salinity sensing system [8], RF MEMS switches [5], and tunable antennas [9] have been demonstrated by various research groups. One type of PCB MEMS device is known as a PCB accelerometer, shown in Figure 1.1. The accelerometer makes use of organic polymers such as KaptonR? film for the structural layer, PolyflonTM bonding film for the spacer layer, and RT/DuroidR? (TeflonR? for the substrate, as shown in Figure 1.2. The proof mass and its four suspension beams, are defined in the KaptonR? polyimide film. A PCB accelerometer is a MEMS-based accelerometer that moves in an out-of-plane motion or a motion orthogonal to the plate mass as shown in 2 Figure 1.3. An accelerometer can be used to measure acceleration of a moving object as well as velocity and position by integrating electronics. The accelerometer may also be used as a pressure sensor. The ability to build accelerometers using low-cost, conventional printed circuit techniques allows for monolithic integration of MEMS with electronics. Materials, configuration, analysis, fabrication, and experimental characterization of an example PCB MEMS accelerometer are discussed in the following sections. Figure 1.2: Sequence of Layers for PCB MEMS Accelerometer 1.2 Materials, Configuration, & Analysis The materials, configuration, and analysis of the PCB MEMS accelerometer are dis- cussed in the following subsections. The materials subsection discusses what materials are 3 used for the PCB MEMS accelerometer. The configuration subsection discusses what the structure is comprised of and how it performs. The analysis subsection discusses the physics behind the structure and how the parameters are derived. Figure 1.3: Deflection of the PCB MEMS Accelerometer 1.2.1 Materials The materials used in fabrication of the PCB MEMS accelerometer include KaptonR?, PolyflonTM, and RT/DuroidR? (TeflonR?). The KaptonR? E polyimide (a polymer of imide monomers) film is available from DuPont and has a permittivity, epsilon1r, of 3.1 at 1 kHz with a 3 ?m copper cladding. KaptonR? E is a premium performance polyimide film for use as a dielectric substrate in flexible printed circuits and high density interconnects. KaptonR? E is a preferred dielectric film for very fine circuitry due to its high modulus and a coeffi- cient of thermal expansion equivalent to that of copper. KaptonR? E also has excellent electrical characteristics and chemical etchability. The PolyflonTM bonding film is available from Daikin Industries and is used as a spacer film. PolyflonTM has minimal deformation under loads, is a good electrical insulator (dielectric breakdown strength), and has good 4 transparency. The RT/DuroidR? 6002 substrate is available from Rogers Corporation and has a permittivity, epsilon1r = 2.94 at 10 GHz, a dissipation factor, tan ? = 0.0012 at 10 GHz with a 1/4 oz (9 ?m) of copper cladding. RT/DuroidR? 6002 is a microwave material with low loss for excellent high frequency performance, extremely low thermal coefficient of dielectric constant, and excellent electrical and mechanical properties. 1.2.2 Configuration The top and cross-sectional views of the PCB MEMS accelerometer are shown in Fig. 1.4 and Fig. 1.5. The proof mass of the accelerometer is defined in the 2 mil (50.8 ?m) thick KaptonR? polyimide film. The proof mass comprises of a square membrane supported by four suspension beams. The proof mass is suspended above a 30 mil (762 ?m) thick RT/DuroidR? substrate using a 2 mil (50.8 ?m) thick PolyflonTM bonding film (spacer). For an applied external acceleration/deceleration (normal to the proof mass), the proof mass deflects towards or away from the substrate due to inertial force. The deflection of the proof mass is detected using a pair of capacitive sensing electrodes as shown in Figure 1.3. The top sensing electrode of the accelerometer is defined in the 3 ?m thick (standard thickness available from DuPont) copper metallization on the top surface of the KaptonR? film. The bottom sensing electrode is defined in the 9 ?m thick (standard thickness available from Rogers Corporation) copper metallization on the RT/DuroidR? substrate. Standard copper metallization materials available from the manufacturers were used in this work. The top and bottom sensing electrodes have different thicknesses due to the standard thicknesses available from their respective manufacturers. The spacer film determines the nominal air gap height between the KaptonR? film and the bottom sensing electrode. The nominal gap 5 Figure 1.4: Schematic of the PCB MEMS Accelerometer Top view (not to scale) height is approximately 41.8 ?m (50.8 ?m spacer - 9 ?m bottom sensing electrode). The deflection of the membrane causes either a decrease or increase in the air gap thereby either increasing or decreasing the capacitance between the top and the bottom electrodes. The sensing leads of width 100 ?m (chosen by design) are used for connecting the electrodes to the capacitance read-out chip. 1.2.3 Analysis To understand what?s happening dynamically with a MEMS device a knowledge of some basic underlying principles is essential. The electrial and mechanical dynamics of lumped element systems have been studied thoroughly by engineers. The various physical 6 Figure 1.5: Schematic of the PCB MEMS Accelerometer Cross-sectional view (not to scale) parameters of the PCB MEMS accelerometer are shown in Fig. 1.4 and Fig. 1.5. The area of the square KaptonR? membrane is d?d. The length and width of the suspension beams are l and w, respectively. The reduction distance between the square area of the KaptonR? membrane and the top electrode is r. In this work, the hypothesis is that the accelerometer may be modeled as a second-order mass-spring-damper system (Appendix A) m?x = ?c?x?kx+ epsilon1AV 2 2 1 (g ?x)2 (1.1) where m is the effective mass, c is the damping constant, k is the spring constant, epsilon1 is the permittivity of the surrounding gas, A is the accelerometer area, V is the applied voltage, and g is the nominal gap distance. 1.3 System Parameters The estimation of parameters in a system is essential in the analysis process. They can be estimated by knowing information about the geometry, material properties, envi- ronmental conditions, etc. If the second-oder model above holds and the parameters can 7 be estimated, a linear state variable control technique can be used to reject disturbrances that would detriment the accelerometer (Appendix C). Some of the system parameters of the example PCB MEMS accelerometer along with their corresponding values are shown in Table 1.1. These parameter values (proof mass, spring consant, damping constant, natural frequency) along with a few others will be derived below. Table 1.1: Designed System Parameters for the Example PCB MEMS Accelerometer Parameter Name Value Units m effective mass 4.499?10?6 kg k spring constant 20.32 N/m c damping constant 18.21?10?2 N-sec/m fo natural frequency 338 Hz g gap distance 41.8 ?m 1.3.1 Spring Constant The mechanical spring constant for a system is defined by the geometry and materials properties of the spring. A system is usually designed around known spring types (i.e. cantilever beams, fixed-fixed beams), load types (i.e. uniform loads, point loads), and materials (i.e. silicon, KaptonR?) for ease of calculations. This allows for ease of estimation when designing. It should be noted that the ability to estimate the spring constant allows for the calculation of other unknown parameters (i.e. damping) in the experimental results. The spring constant for the given accelerometer has four cantilever beams with distributed point loads (Figure 1.4). 8 The spring constant of the bi-material KaptonR?-copper suspension beams of the ac- celerometer is given by k = 48EIl3 (1.2) where EI is the equivalent flexural rigidity [10] given by EI = (wcEct 2c)2 + (wkEkt2 k) 2 + 2wcwkEcEktctk(2t2c + 3tctk + 2t2 k)) 12(wcEctc +wkEktk) (1.3) where Ek and tk are the Young?s modulus and thickness of the KaptonR? film, Ec and tc are the Young?s modulus and thickness of copper, wk and wc are the widths of the KaptonR? and copper in the suspension beams, and l is the length of the suspension beams. 1.3.2 Proof Mass The effective mass, or proof mass, is the total mass that will experience motion as a result of an inertial force. On the accelerometer the plate mass will displace, but there is also spring displacement. For any one spring, a fraction of the spring mass will displace from one end of the pinned or fixed side. The question then arises of how much of the spring will displace? This can be calculated using the Rayleigh-Ritz Method [11] for vibration frequency. The Rayleigh-Ritz Method is based on the principle of energy conservation and is used for calculating the vibration frequency of systems with distributed masses. Experimentally, the distributed mass can be solved for if one knows the vibration frequency and the spring constant. The Rayleigh-Ritz Method for the accelerometer is solved for in 9 Appendix B. The results from the Rayleigh-Ritz Method show the effective proof mass is m = mp + 1335mb (1.4) where mp is the mass of the KaptonR? membrane with the copper top electrode and mb is the total mass of all four suspension beams (along with sense leads). 1.3.3 Natural Frequency The vibration frequency, or natural frequency, is the frequency at which a device natu- rally vibrates or oscillates. The accelerometer should be operated below this frequency (in other words the accelerometer must be shaken below this frequency) in order to operate as an accelerometer. The natural frequency of the accelerometer can be expressed ?o = radicalbigg k m (1.5) where k is the effective spring constant and m is the effective proof mass of the accelerometer. 1.3.4 Damping Constant The accelerometer moves in an out-of-plane motion or a motion orthogonal to the plate mass. Squeeze-film damping is the dominant damping mechanism in this configuration. The squeeze-film damping refers to the energy dissipated in displacing the gas molecules between the moving beam or plate and the substrate. The squeeze-film damping constant 10 for a square membrane [11] is c = 0.42?d 4 g3 (1.6) where ? is the viscosity of air (= 18.27 ?Pa.s at 20oC), d is the side length of the plate mass (KaptonR? film), and g is the nominal gap height between the bottom electrode and the KaptonR? film. 1.3.5 Quality Factor The quality factor is a measure of how well a system dissipates energy. The quality factor is defined as the energy stored over the power loss at resonance. Thus a higher quality factor indicates a lower rate of energy dissipation at resonance. The quality factor, Q, of the accelerometer is given by Q = ?omc (1.7) where ?o is the resonant frequency, m is the proof mass, and c is the damping constant of the accelerometer. In this work, an example PCB MEMS accelerometer with a square membrane of area 6.4 mm?6.4 mm is considered. The length l and width w of the suspension beams are 5.8 mm and 1 mm, respectively. The reduction distance between the square area of the KaptonR? membrane and the top electrode is 270 ?m. The initial gap height g is approxi- mately 41.8 ?m. These parameter values were chosen based on using equations (2.2)-(2.7) to design an accelerometer with a low-frequency resonance for low-g applications. This 11 means that the spring to mass ratio given by Equation (2.5) should be small. The various system parameters can be calculated and are shown in Table 1.1. In the following sections, fabrication and experimental characterization are discussed. 1.4 Fabrication The PCB accelerometer is fabricated using printed circuit techniques. The fabrication of the accelerometer has three layers: a substrate layer, spacer layer, and polyimide layer (Figure 1.5). They will be discussed in further detail below. The substrate layer is made of RT/DuroidR?, a TeflonR? material. The spacer is made of a PolyflonTM bonding layer. The polyimide layer is made of KaptonR? film. 1.4.1 Substrate The substrate chosen is a 30 mils (762 ?m) thick RT/DuroidR? with 9 ?m thick cop- per metallization. The bottom electrode for capacitive sensing is defined in the copper metallization on the RT/DuroidR? substrate. 1.4.2 Spacer The spacer layer provides the required spacing between the substrate and the KaptonR? polyimide layer. Hence, the thickness of the spacer film determines the up-position gap height. A 2 mil (50.8 ?m) thick PolyflonTM bonding film is used as the spacer layer. The bonding film was cut to create openings for the movable membrane with suspension beams. A milling machine could be employed for designs with small characteristic dimensions. 12 1.4.3 KaptonR? Film A 2 mil (50.8 ?m) thick KaptonR? film with 3 ?m thick copper metallization is used as the structural layer for the accelerometer. The KaptonR? film contains copper metallization on both sides as shown in Figure 1.6. A 150-250?A thick nichrome seed layer is present between the KaptonR? film and the copper layer. The film is essentially a sandwich of copper-nichrome-kapton-nichrome-copper. The two major steps in the fabrication of the KaptonR? film are discussed in detail below. Figure 1.6: A 2 mil thick KaptonR? film used for fabrication of the PCB MEMS Accelerom- eter Plasma Etching of KaptonR? Film Fabrication starts with rinsing of the KaptonR? film using Acetone then Methanol followed by cleaning using diluted sulfuric acid to remove any oxide growth on the copper surfaces. The bottom side copper is used to define the mask for DRIE processing. A piece of Dynaflex wafer grip film is attached to a silicon wafer by heating it on a hotplate to 110oC. The Dynaflex wafer grip film is used as an adhesion layer to mount the KaptonR? film to the wafer. When the wafer is sufficiently hot (110oC), the KaptonR? is carefully attached by hand to the wafer in a way as to smooth out all air pockets. The next step is 13 to spin-coat photoresist (PR) onto the KaptonR? film and soft bake. The KaptonR? film is then exposed and developed using the KaptonR? layer mask. The copper is then etched off using an industry standard copper etchant such as CE-200. The photoresist is then removed and the nichrome is fully etched off from the KaptonR? layer pattern. The wafer is then ready for the Deep Reactive Ion Etch (DRIE) process. The wafer is put into the DRIE where the etcher will etch through the KaptonR? and nichrome/copper on the back side. The equipment used for the DRIE process is an STS AOE (Advanced Oxide Etcher). The configuration of the gases used for this etch in Auburn University?s AMSTC facility is 8 sccm carbon tetrafluoride (CF4) and 35 sccm oxygen with 500 Watts of radio frequency (RF) power. Using these processing parameters it takes about 70 minutes to etch a 2 mil thick KaptonR? layer. The bottom copper and nichrome are etched fully using chemical etchants. Then, the wafer grip is dissolved using Amyl Acetate and the KaptonR? film is removed by hand from the wafer. Top Electrode Patterning At this stage, the KaptonR? film contains copper and nichrome layers on only one side. The KaptonR? side is then attached to the dicing tape. The film is then attached to a silicon wafer and photoresist is spin-coated. The top sensing electrode mask is used to define the top electrode in the copper layer on the KaptonR?. Finally, the nichrome is removed. Thermal Compression Bonding Thermo-compression bonding is performed using a Carver Press consisting of two platens. The platens are heated using heaters, whose temperature is sensed by a ther- mocouple. The fixture consists of two steel plates with alignment holes on them. The 14 substrate forms the bottom most layer, the spacer is the middle layer and the KaptonR? film forms the top most layer in this structure. The different layers are aligned by aligning the holes created on the three layers during the fabrication process described earlier. This unit is now placed between the press platens. The bonding is performed at a pressure of 80 psi (a load of 200 lbs.) and a temperature of 130oC. Both pressure and temperature are maintained for 5 min during bonding. Before pressure is released, the assembly is cooled down to the room temperature in a process known as annealing. 1.5 Detailed Fabrication Process Table 1.2: PCB MEMS Accelerometer Process Method Time Kapton Film Cleaning 1. Cleaning Acetone then Methanol DI Rinse 30 Sec N2 Dry 2. Oxide Removal 250mL of H2O + 2 drops of H2SO4 DI Rinse 30 Sec N2 Dry 3. Cleaning Acetone then Methanol DI Rinse 30 Sec N2 Dry Stick Kapton Film on Wafer 1. 4? Wafergrip Application Apply Wafergrip on wafer 2. Soft Bake Hot plate - 110oC 30 Sec 3. Stick Film on Wafer Smooth Kapton Film on wafer 15 Define Kapton Layer 1. Spin Photoresist S1813 Photoresist 30 Sec 2500 RPM 500 r/s ramp 2. Soft Bake Hot plate - 110oC 60 Sec 3. UV Exposure Mask Aligner - expose 30 Sec Kapton Layer mask 4. Develop CD-30 developer 45 Sec DI Rinse 30 Sec N2 Dry 5. Hard Bake Hot plate - 110oC 60 Sec 6. Wet Etch CE-200 etchant 8 Sec DI Rinse 30 Sec N2 Dry 7. Remove Photoresist Acetone then Methanol DI Rinse 30 Sec N2 Dry 8. Wet Etch NaOH + KMNO4 mixture 50 Sec DI Rinse 30 Sec N2 Dry 9. Cleaning Acetone then Methanol DI Rinse 30 Sec N2 Dry Deep Reactive Ion Etch (DRIE) 1. DRIE 4:1 CF4:O2 proces 75 Min 2. Cleaning Acetone then Methanol DI Rinse 30 Sec N2 Dry 3. Remove/Clean Kapton Film Remove Kapton Film from Wafer Clean using Amyl Acetate 4. Cleaning Acetone then Methanol DI Rinse 30 Sec N2 Dry 16 Define CopperTop Layer 1. Spin Photoresist S1813 Photoresist 30 Sec 2500 RPM 500 r/s ramp 2. Soft Bake Hot plate - 110oC 60 Sec 3. UV Exposure Mask Aligner - expose 1:30 Min CopperTop mask 4. Develop CD-30 developer 45 Sec DI Rinse 30 Sec N2 Dry 5. Hard Bake Hot plate - 110oC 60 Sec 6. Wet Etch CE-200 etchant 8 Sec DI Rinse 30 Sec N2 Dry 7. Remove Photoresist Acetone then Methanol DI Rinse 30 Sec N2 Dry 8. Wet Etch NaOH + KMNO4 mixture 50 Sec DI Rinse 30 Sec N2 Dry 9. Cleaning Acetone then Methanol DI Rinse 30 Sec N2 Dry Define Subini Layer 1. Cleaning Acetone then Methanol DI Rinse 30 Sec N2 Dry 2. Oxide Removal 250mL of H2O + 2 drops of H2SO4 DI Rinse 30 Sec N2 Dry 3. Cleaning Acetone then Methanol DI Rinse 30 Sec N2 Dry 4. Tape RT/Duroid Apply blue tape to one side of RT/Duroid 5. Spin Photoresist S1813 Photoresist 30 Sec 2500 RPM 500 r/s ramp 6. Soft Bake Hot plate - 110oC 60 Sec 17 7. UV Exposure Mask Aligner - expose 30 Sec Subini mask 8. Develop CD-30 developer 45 Sec DI Rinse 30 Sec N2 Dry 9. Hard Bake Hot plate - 110oC 60 Sec 10. Wet Etch CE-200 etchant 8 Sec DI Rinse 30 Sec N2 Dry 11. Remove Photoresist Acetone then Methanol DI Rinse 30 Sec N2 Dry Define Subcop Layer 1. Cleaning Acetone then Methanol DI Rinse 30 Sec N2 Dry 2. Spin Photoresist S1813 Photoresist 30 Sec 2500 RPM 500 r/s ramp 3. Soft Bake Hot plate - 110oC 60 Sec 4. UV Exposure Mask Aligner - expose 30 Sec Subcop mask 5. Develop CD-30 developer 45 Sec DI Rinse 30 Sec N2 Dry 6. Hard Bake Hot plate - 110oC 60 Sec 7. Wet Etch CE-200 etchant 8 Sec DI Rinse 30 Sec N2 Dry 8. Remove Photoresist Acetone then Methanol DI Rinse 30 Sec N2 Dry 9. Wet Etch NaOH + KMNO4 mixture 50 Sec DI Rinse 30 Sec N2 Dry 10. Cleaning Acetone then Methanol DI Rinse 30 Sec N2 Dry 18 1.6 Photolithography Masks Figure 1.7: Kapton Mask used in Photolithography 19 Figure 1.8: CopperTop Mask used in Photolithography 20 Figure 1.9: Subini Mask used in Photolithography 21 Figure 1.10: Subcop Mask used in Photolithography 22 1.7 Fabrication Issues Fabrication of a PCB MEMS accelerometer raises many issues. Some of the fabrication issues can be attributed to the fabrication equipment. For example, fine temperature control and pressure control are hard to achieve. If a process calls for a specific temperature and pressure to be maintained over a time period, then undesirable results could be formed and cause further defects in other fabrication steps. There can also be issues with the chemicals used. For example, if a certain chemical is used as an etchant and has been used for several cycles, then the etchant will not etch at the same rate and if the time is kept constant the result will be an unfinished surface. This issue occured in all of the chemical processes due to recycling of the chemicals and was delt with by overetching until it was visibly clear that the process was done. The environmental conditions such as temperature, pressure, moisture can also effect the chemicals, but are usually controlled in a microfabrication clean room. Scattering of the copper layer is another issue in fabrication. Scattering means the copper layer begins to crack and break under the high temperatures present in the DRIE process. This happens due to the large difference in thermal coefficients of expansion be- tween copper and KaptonR?. This issue wasn?t considered before fabrication, thus the fabrication steps were altered to use the backside of the KaptonR? film for the copper layer. To avoid this issue for top and backside copper metallizations aluminum must be deposited before DRIE and removed after DRIE. Residual stress is another possible issue in printed circuit techniques due to the com- bination of KaptonR? with nichrome and copper. Residual stress occurs when a thin film is deposited on a substrate and has in-plane stress. Plane stress is caused by mismatches 23 in thermal expansion of the film and sustrate, which can lead to deformation of the device. This happened when the various layers were pressure bonded in the thermo-compression bonding stage. The gap increased and warped due to these mismatches. These effects could have been supressed by having copper layers on both sides of the KaptonR? to even out the mismatches. Another issue with KaptonR? is the fact that it is a thin polyimide film. This means that every process that requires the film to be moved by hand carries with it the potential to bend the film and make permanent dents thus creating further warpage of the film. This has the potential to happen in the chemical etching stages where the KaptonR? film must be moved back and forth with wafer tweezers in order for the etchant to be effective. There is also the potential for warping in the drying of the KaptonR? film. Every time the KaptonR? film is cleaned or etched it must go through deionized water then dried with nitrogen (N2) gas. The N2 gas has a high pressure rate and when spread across the KaptonR? film has the potential to create dents. These fabrication issues can and do effect the modeling of the device as they will change the various parameters (i.e. mass, spring constant). 24 Chapter 2 Experimental Characterization 2.1 Experimental Set-up A photograph of the fabricated MEMS accelerometer is shown in Fig. 2.2. The MEMS accelerometer was characterized using an LDS model V408 electromechanical shaker shown in Figure 2.1. For testing purposes, the accelerometer substrate was mounted onto a plex- iglass fixture as shown in Figure 2.2 with a mounting screw attached at the center of the bottom surface. The plexiglass is attached to the threaded hole in the shaker head. The shaker vibrates the MEMS accelerometer at a chosen amplitude over a specified frequency range. In response to the applied external acceleration/deceleration, the proof mass vibrates in a direction normal to the substrate. The mechanical displacement of the proof mass was measured by reflecting a laser beam off of the proof mass using a laser interferometric measurement system as shown in Figure 2.3. The experiment yields y(f), the motion of the membrane as a function of frequency. The TeflonR? substrate was used as a reference frame and a second laser interferometric measurement system was used to measure the motion of the reference frame as a function of frequency, x(f), by reflecting a laser beam off of the reference frame. The signals x(f) and y(f) were recorded simultaneously using a signal analyzer. The signal analyzer provides the transmissibility spectrum of the MEMS accelerometer by computing the transfer function T(f) = y(f)/x(f). The transmissibility spectrum is defined as the ratio of the magnitudes of the displacement of the membrane (output) and the reference frame (input) over a range of frequencies otherwise known as the transfer function. 25 Figure 2.1: A Photograph of an LDS model V408 electromechanical shaker (courtesy Auburn University). 2.2 Experimental Results The measured magnitude of the transfer function as a function of frequency for the PCB MEMS accelerometer is shown in Figure 2.4. From this plot, the resonant frequency and the quality factor of the MEMS accelerometer were found to be 375 Hz and 1.5, respectively. The measured resonant frequency is reasonably close to that of the calculated value of 338 Hz (refer Table 1.1). The measured Q is higher than that of the estimated value due to a larger air gap height caused by the thermal expansion of various layers in the accelerometer during fabrication. Using Equation (2.6), the effective air gap height can be estimated to be 125 ?m. 26 Figure 2.2: Full photograph of PCB MEMS Accelerometer. Courtesy AMSTC-Auburn University The first experimental results for the PCB MEMS accelerometer were taken using the set-up described above using two lasers to measure the displacements of both the membrane and the reference frame. The mechanical results for this set-up are shown in Figure 2.4. The second experimental results were taken of the electrical characteristics as well as more mechanical characteristics. The electrical characteristics were taken by having the input signal be the reference frame and the output signal be the electrical output from the capac- itance to voltage or C-V chip. The C-V chip detects a capacitance from the accelerometer and converts it to a voltage in by a linear amount (1 V/pF) and the output is that amount added to a bias voltage of approximately 2.25 V. The change in deflection of the membrane 27 Figure 2.3: Laser Interferometric Measurement System creates a change in capacitance which results in a voltage change. The signal analyzer then creates a transfer function of the output voltage over the reference frame displacement. C = epsilon1Ag (2.1) ?C ?g = ? epsilon1A g2 ?g (2.2) Vout = Vbias +C( 1V1pF ) (2.3) For low frequency there?s a small change in displacement, ?g, which results in a small change in capacitance. The bias voltage, Vbias, falls out in the transfer function and only the voltage 28 Figure 2.4: Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer (First Test, Mechanical Response) due to capacitance change is considered. The electrical results for this set-up are shown in Figure 2.5. The results show a linear increase in voltage of 20dB/decade for low frequency. The plot then rolls off due to the mechanical response. The mechanical characteristics were taken in the same set-up as the first experimental results. There are three plots shown in Figure 2.6 that represent three measurements taken over the membrane at various positions for statistical purposes. These plots were taken by the signal analyzer over a long period of time to insure a sufficiently rich set of data for statistical analysis. The results were taken on a different day and match up with the results 29 Figure 2.5: Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer (Second Test, Electrical Response) taken in the first experiment and thus show the experimental results are replicable. Note however that these results are not ensured for replicated fabrication due to the fabrication issues described in Section 1.5 The results show a resonant frequency around 375 Hz. The results also show a quality factor or Q factor of 1 to 1.5. These results vary from those seen in Table 1.1. By closer inspection of Figure 1.1 it can be seen that the amount of copper left on the beams is much more than that shown in Figure 1.4. This will change the spring constant as well as the effective mass. By using the WYKO Profiler in Auburn University?s CAVE lab it was 30 Figure 2.6: Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer (Second Test, Mechanical Response) observed that the plate was warped and the actual gap distance could not be calculated. A distributed gap then had to be calculated to represent the overall warpage and was calculated to be 125 ?m. This gap is more than three times that of the original gap. This mismatch can be explained by the thermal compression stage of fabrication. PCB devices will experience an expansion in the gap distance due to thermal compression due to mismatches in thermal coefficients of copper and KaptonR? [12]. These changes due to fabrication will change the system parameters as shown in Table 2.1. 31 Table 2.1: Actual System Parameters for the Example PCB MEMS Accelerometer Parameter Name Value Units m effective mass 4.499?10?6 kg k spring constant 24.93 N/m c damping constant 6.8?10?3 N-sec/m fo natural frequency 375 Hz g gap distance 125 ?m There also is seen in the frequency results a second resonant frequency around 1100 Hz in some of the results. In the ideal case the accelerometer would move only in an out-of- plane motion. This resonant frequency may correspond to a torsional motion. Torsional motion has the potential to excite other higher frequency harmonics. There also is a dip before the resonant frequency that may be associated with under-damping. The accelerom- eter doesn?t seem to have a second-order model as described in Section 1.2.3. As to be described in the following section, the accelerometer seems to fit better with a higher-order model. This is most likely due to the accelerometer having more of a flexible structure than a hypothesized rigid structure. Rigid structures can more accurately be described with second-order models, but flexible structures have more of a wave-like nature to them. Flexible structures can thus better be described with distributed parameters rather than lumped-element parameters. The torsional motion notion helps supplement this argument. The quick roll-off at frequencies above the second resonant frequency also seems to support a higher-order model. The accelerometer with the given actual parameters in Table 2.1 should have a second-order response seen in Figure 2.7, but it is obvious that the system has a higher-order response. These issues will be further discussed in the next section. 32 Figure 2.7: Magnitude Plot of the Experimental and Theoretical Results using Second- Order Model A third test was done of the PCB MEMS accelerometer. The test set-up used a refer- ence accelerometer as an input and the output voltage from the PCB MEMS accelerometer as the output. The electromechanical shaker was excited at random frequencies in the range of 10-1600 Hz. The test results are shown in Figure 2.8. The results show a linear ratio be- tween the output voltage and the input acceleration for a given frequency. At the resonant frequency (375 Hz), for an input acceleration of 1 g an output of approximately 4 mV is given. The accelerometer should be operated in the low-frequency range (30-400 Hz) where 33 the curve is mostly flat in order for the accelerometer to give consistant results over the range. Figure 2.8: Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer (Third Test, Electrical Response) 34 2.3 System Identification The experimental data is examined in a control systems analysis technique known as system identification. MATLAB?s System Identification Toolbox was used to develop some higher-order models to fit the experimental data in order to get an insight into the relative order of the system. The System Identification Toolbox is brought up in a GUI window of MATLAB?s command window using the ?Ident? command. The GUI program allows the user to import time domain or frequency domain experimental data. The experimental data used was one of the mechanical response plots from the second test (Figure 2.9). The System Identification Toolbox has three methods for estimating models: parametric estima- tion, process model estimation, and nonparametric estimation. The parametric estimation method was chosen due to its ability to allow the user to select the order of the polynomial for the transfer function. The user then has the ability to select the number of poles, zeros, and time delays. In Figure 2.10 a transfer function with four poles, three zeros, and one delay was estimated to fit the experimental data curve. The curve seems to look like a second-order response, but having a resonant peak at the second resonant point. In Figure 2.11 a transfer function with eight poles, six zeros, and three delays is shown. The curve fits better with the experimental data and even seems to show the first resonant peak, but doesn?t show the dip before the first resonant peak. In Figure 2.12 a transfer function with eight poles, seven zeros, and one delay is shown. The curve fits really well with the experimental data showing the first and second resonant points, and the dip before the first resonant point. Although MATLAB?s System Identification Toolbox can be used to develop higher- order models it is important to have a controls background to interpret the data. All three 35 of the generated transfer functions by MATLAB were unstable in discrete time showing zeros and poles outside the unit circle. It is uncertain how MATLAB does numerically analysis of the experimental data and many of the coefficients are extremely large which could lead to the stability issues. A controls background can allow one to estimate the system order by pole-zero placement using classical control system techniques [13]. A fourth order system was estimated by placing two pairs of complex poles at the resonant points and a pair of complex zeros at the dip before the first resonant point. The placed pole-zero system is shown in Figure 2.13. The system matches the experimental data pretty well for a quick estimate. The conclusion can be made by observing these models that the system is in fact higher-order. A second-order model gives a pretty good estimate, but a higher-order model gives a much better estimate of the actual system. 36 Figure 2.9: Magnitude Plot of the Transfer Function for the PCB MEMS Accelerometer (Second Test, Mechanical Response) 37 Figure 2.10: Four pole, three zero, one delay transfer function (MATLAB SYSID) 38 Figure 2.11: Eight pole, six zero, three delay transfer function (MATLAB SYSID) 39 Figure 2.12: Eight pole, seven zero, one delay transfer function (MATLAB SYSID) 40 Figure 2.13: Fourth order pole-zero placement transfer function 41 Chapter 3 Conclusions & Future Work 3.1 Il Buono A MEMS-based accelerometer was fabricated using printed circuit processing tech- niques. The design, fabrication, and mechanical characterization of the PCB MEMS ac- celerometer were discussed. The resonant frequency of an example accelerometer with a square membrane of area 6.4 mm?6.4 mm was measured to be 375 Hz. The attractive feature of PCB MEMS is that it enables monolithic integration of MEMS devices with electronics using conventional printed circuit techniques. The advantages of the proposed PCB MEMS technology include low-cost, ease of integration with electronics, suitability for high-volume manufacturing, and large area applications. 3.2 Il Brutto The experimental results of the fabricated MEMS accelerometer show that some of the actual parameters didn?t match well with the designed parameters. This is largely due to the two issues: 1) excess copper on the plate that increased the stiffness of the springs, and 2) the large gap increase due to thermal expansion of copper and KaptonR?. More devices should be fabricated using other fabrication techniques to try to eliminate this large gap change. The fabrication of PCB MEMS devices is still at the experimental stage and a single technique hasn?t been developed for various devices thus many tests must be done to refine the process for any one device. 42 3.3 Il Cattivo The experimental results of the fabricated MEMS accelerometer show a higher-order model than the predicted second-order model. This is most likely due to the fact that the structure is flexible. Fabrication of smaller square plates should be done to see if/when the structure will become rigid. Every rigid structure has a point where it becomes flexible due to the large surface area to thickness ratio and material properties. It was not hypothesized that the MEMS accelerometer would be flexible. 3.4 Future Work The new hypothesis now accepts that the structure is flexible and the new question is to what size will the device become rigid. The ability to have a rigid structure will allow for lumped-element modeling of the MEMS accelerometer. The need to conduct tests on multiple devices of the same geometry, but different sizes will help to provide better statistical data on the performance of PCB MEMS accelerometers and how they behave. Lastly, someone should look into applying the science of flexible structures to the modeling of PCB MEMS. This would give further insight into better predicting the behavior of the system. 43 Bibliography [1] B. Borovic, F. L. Lewis, W. McCulley, A. Q. Liu, E. S. Kolesar, and D. O. Popa, ?Control issues for microlectromechanical systems,? IEEE Control Systems Magazine, vol. 26, pp. 18?21, April 2006. [2] E. Bryzek, A. Abbott, D. C. Flannery, and J. Maitan, ?Control issues for mems,? IEEE Conf. Decision and Control, vol. 3, pp. 3039?3047, 2003. [3] C. T.-C. Nguyen, ?Frequency-selective mems for miniaturized low-power communi- cation devices,? IEEE Transactions on Microwave Theory and Techniques, vol. 47, pp. 1486?1503, August 1999. [4] J. Rogers, P. Ozmun, J. Hung, and R. Dean, ?Bi-directional gap closing mems actuator using timing and control techniques,? Proceedings of the 32nd Annual Conference of the IEEE Industrial Electronics Society (IECON?06), pp. 3149?3154, November 2006. [5] R. Ramadoss, S. L. S, Y. Lee, V. Bright, and K. Gupta, ?Rf mems capacitive switches fabricated using printed circuit processing techniques,? IEEE/ASME Journal of Mi- croelectromechanical Systems, vol. 15, pp. 1595?1604, December 2006. [6] X. Wang, J. Engel, and C. Liu, ?Liquid crystal polymer (lcp) for mems: processes and applications,? J. Micromech. Microeng., vol. 13, pp. 628?633, 2003. [7] J. N. Palasagaram and R. Ramadoss, ?Mems capacitive pressure sensor fabricated using printed circuit processing techniques,? IEEE Sensors Journal, vol. 6, pp. 1374?1375, December 2006. [8] D. Fries, G. Steimle, S. Natarajan, S. Ivanov, H. Broadbent, and T. Weller, ?Maskless lithography pcb/laminate mems for a salinity sensing system,? Proc. Int. Microelectron. Packag. Soc. 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Prentice Hall, 10 ed., 2004. 45 Appendix A Rigid Body Model for MEMS Accelerometer substrate plate c x r x o A k Figure A.1: Cross-sectional view of PCB MEMS device Shown above in Figure A.1 is a spring-mass-damper rigid body model for a typical MEMS accelerometer. The system has two plates, a top plate and the substrate. The plate area is defined by A. The spring constant for the system is k, the damping constant for the system is c, and the nominal gap distance between the two plates is xo. When an external force is applied to the substrate, a displacement r is seen and results in a displacement of the top plate given by x. The electrical and mechanical dynamics for the spring-mass-damper are described as follows: A.1 Electrical Dynamics The capacitance for a parallel plate capacitor is given by C = epsilon1Ax o ?x (A.1) 46 Through the charge-voltage relationship q = Cv = epsilon1Avx o ?x (A.2) The work is defined by W = qv2 = epsilon1Av 2 2 parenleftbigg 1 xo ?x parenrightbigg (A.3) The electrostatic force is the change in work Fe = ?W?x = epsilon1A2 parenleftbigg v xo ?x parenrightbigg2 (A.4) where epsilon1 is the permittivity of the surrounding gas, A is the plate area, v is the applied voltage, and xo is the nominal gap distance. A.2 Mechanical Dynamics The inertia force, also known as Newton?s second law Finertia = m?x (A.5) The damping force Fdamper = ?c(?x? ?r) (A.6) 47 The spring force, also known as Hooke?s Law Fspring = ?k(x?r) (A.7) The mechanical force is the sum of the inertia, damper, and spring forces Fm = Finertia +Fdamper +Fspring (A.8) The differential equation for the mechanical force is thus Fm = m?x+c?x+kx (A.9) where m is the proof mass, ?x is the acceleration of the plate mass, c is the damping con- stant, ?x is the velocity of the plate mass, ?r is the velocity of the substrate, k is the spring constant, x is the displacement of the plate mass, and r is the displacement of the substrate. Equating the mechanical and electrical dynamic force equations yields a stable state thus producing an electromechanical model for the MEMS accelerometer m?x = ?c?x?kx+ epsilon1A2 parenleftbigg v xo ?x parenrightbigg2 (A.10) 48 A.3 Mechanical Transfer Function m?x+c(?x? ?r) +k(x?r) = 0 (A.11) ms2X(s) +csX(s)?csR(s) +kX(s)?kR(s) = 0 (A.12) (ms2 +cs+k)X(s) = (cs+k)R(s) (A.13) X(s) R(s) = (cs+k) ms2 +cs+k (A.14) one zero at s = ?kc (A.15) two poles at s = ?c? ?c2 ?4mk 2m (A.16) 49 Appendix B Rayleigh-Ritz Method for Determining Equivalent Mass of a Flexible Structure Figure B.1: Shape Function for Suspension Beam Shown above in Figure B.1 is the shape function for a suspension beam. The length of the beam is given by L. The maximum displacement due to a force is given by xp. The shape function for a suspension beam is given by xb(y) = xp bracketleftBig 3(yL)2 ?2(yL)3 bracketrightBig (B.1) The maximum potential energy Epmax = 12kx2p (B.2) The maximum kinetic energy Ekmax = 12 bracketleftbigg v2pmp + integraldisplay v2bdmb bracketrightbigg (B.3) 50 The change in mass can be re-expressed dmb = mbdyL Ekmax = 12v2pmp + 12 integraldisplay v2bmbdyL (B.4) The velocity can be re-expressed v = ?x Ekmax = 12x2p?2nmp + 12Lmb integraldisplay (?nxb)2dy (B.5) Substituting for xb Ekmax = 12x2p?2nmp + 12Lmb integraldisplay (?nxp bracketleftBig 3(yL)2 ?2(yL)3 bracketrightBig )2dy (B.6) Re-writing the kinetic energy equation Ekmax = 12x2p?2n bracketleftbigg mp + 1335mb bracketrightbigg (B.7) By the principle of energy conservation Epmax = Ekmax (B.8) Equating the kinetic and potential energy 1 2kx 2 p = 1 2x 2 p? 2 n bracketleftbigg mp + 1335mb bracketrightbigg (B.9) 51 The resonant frequency as a function of effective spring constant and proof mass ?2n = bracketleftBigg k mp + 1335mb bracketrightBigg (B.10) The effective proof mass is thus m = mp + 1335mb (B.11) where mp is the plate mass and mb is the beam mass. The plate and beam masses are given by mp = ?Vp (B.12) mb = ?Vb (B.13) 52 Appendix C Linear State Variable Analysis for PCB MEMS Accelerometer C.1 Motivation Electrostatic Micro-Electro Mechanical Systems, or MEMS, are an interesting problem in control systems in that they tend to have second-order nonlinear system dynamics that resemble those of mass-spring-damper systems with a capacitive-like structure. If a structure is rigid the following could be used to design a feedback control system. The objective is to design a control system that stabilizes the output y = x, the actuator position. C.2 Dynamic Model Dynamics of the PCB MEMS Accelerometer are described by the nonlinear ordinary differential equation m?x = ?c?x?kx+ epsilon1oAV 2 2 1 (xo ?x)2 (C.1) Electrostatics of the PCB MEMS Accelerometer are described by the nonlinear equation kx = epsilon1oAV 2 2 1 (xo ?x)2 (C.2) Let m = 4.375?10?6 kg, c = 6.7?10?3 N-sec/m, k = 24.791 N/m, A = 34.34 mm2, and xo = 125 ?m. 53 C.3 State Variables Let the state be z = ? ?? x ?x ? ?? The plant input is u = V, and there is one output. Rewriting dynamic equation C.1 in state variable form ?z = f(z,u) (C.3) y = h(z) (C.4) ?z1 = z2 ?z2 = ? cmz2 ? kmz1 + epsilon1oAV 2 2m 1 (xo ?z1)2 C.4 Equilibrium State The equilibrium state ze, ue can be determined by observing the position versus applied voltage graph (Figure C.1) for the electrostatic equation C.2. The chosen equilibrium point satisfies equation C.2 ze = [10.5?10?6 0], ue = [150]. 54 Figure C.1: Normalized pull-down voltage curve C.5 Linearized State Variable Model Find the linearized state variable model of the form ??z = A?z +B?u (C.5) ?y = C?z (C.6) where ?z = z ?ze and ?u = u?ue. Recall that there is one output variable, y = x. A = ?zf|ze,ue = ? ?? 0 1 ? km + epsilon1oAV 2m 1(xo?z1)3 ? cm ? ?? ze,ue 55 A = ? ?? 0 1 ?4.6?106 ?1532 ? ?? B = ?uf|ze,ue = ? ?? 0 epsilon1oAV m 1 (xo?z1)2 ? ?? ze,ue B = ? ?? 0 0.80 ? ?? C = ?zh|ze,ue = bracketleftbigg 1 0 bracketrightbigg ze,ue = bracketleftbigg 1 0 bracketrightbigg C.6 Stability Analyze stability of the linear state variable model (C.5), (C.6). ? ?z1 = z2 ? ?z2 = ?5.6?106z1 + 119.3?1532z2 + 0.80u Assume a positive-definite Lyapunov function V(z) = z21 + 2z1z2 +z22 d dtV(z) = (?zV)?z ?zV = bracketleftbigg 2z1 + 2z2 2z1 + 2z2 bracketrightbigg 56 The function ?V(z) is negative-definite and thus solutions for ?z are asymptotically stable. C.7 Controllability rank bracketleftbigg B AB bracketrightbigg = 2 = dim(z) The rank of (A, B) is equal to the dimension of z, thus the system is controllable. C.8 Observability rank ? ?? C CA ? ?? = 2 = dim(z) The rank of (A, C) is equal to the dimension of z, thus the system is observable. C.9 Stabilizability |sI ?A| = vextendsinglevextendsingle vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle s ?1 4.6?106 s+ 1532 vextendsinglevextendsingle vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = s2 + 1532s+ 4.6?106 = 0 s = ?766?2003i The system is naturally stable, thus it is stabilizable. 57 C.10 Detectability The system is naturally stable, thus it is stabilizable and detectable, also controllable and observable. C.11 Linear State Feedback A linear state feedback ?u = ?K?z is to be designed to place eigenvalues of (A?BK) at ?800?j800. A?BK = AC = ? ?? 0 1 ?4.6?106 ?0.80K1 ?1532?0.80K2 ? ?? |sI ?(A?BK)| = vextendsinglevextendsingle vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle s ?1 4.6?106 + 0.80K1 s+ 1532 + 0.80K2 vextendsinglevextendsingle vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = s2 + (1532 + 0.80K2)s+ (4.6?106 + 0.80K1) = 0 desire s = ?800?j800 (s+ 800)2 + (800)2 = s2 + 1600s+ 128?104 = 0 K1 = ?4.15?106,K2 = 85 K = bracketleftbigg K1 K2 bracketrightbigg = bracketleftbigg ?4.15?106 85 bracketrightbigg 58 C.12 Observer Gain An observer gain L is to be designed so that the estimation error dynamics are char- acterized by the eigenvalues ?900?j900. A?LC = AO = ? ?? ?L1 1 ?4.6?106 ?L2 ?1532 ? ?? |sI?(A?LC)| = vextendsinglevextendsingle vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle s+L1 ?1 4.6?106 +L2 s+ 1532 vextendsinglevextendsingle vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = s2+(1532+L1)s+(4.6?106+1532L1+L2) = 0 desire s = ?900?j900 (s+ 900)2 + (900)2 = s2 + 1800s+ 162?104 = 0 L1 = 268,L2 = ?3.39?106 L = ? ?? L1 L2 ? ?? = ? ?? 268 ?3.39?106 ? ?? C.13 Simulation Simulate the closed loop behavior of the nonlinear system (C.3) under the linear state feedback control designed in Section C.5. Let the initial condition be z(0) = [0.5?10?6 0]T. Plot the time response of the GCA position x. Simulate the closed loop behavior of the nonlinear system (C.3) under the estimated state feedback control ?u = ?K??z. Use the observer designed in Section C.6. Let the initial 59 Figure C.2: Block diagram for linear state feedback control condition be z(0) = [0.5 ? 10?6 0]T, and the initial value of the estimated state be ??z = [0 0]T. Plot the time response of the GCA position x. Plot the estimation error x? ?x. Since the damping in the problem is so small relative to the mass, increasing the damping constant, c, would allow for better control. This is not usually an easy thing to do, but by isolating the device within the right gas and pressure would do just this. The system is naturally a very stiff system and controlling the damping will thus allow for better control of K and L. 60 Figure C.3: Output response for linear state feedback control Figure C.4: Block diagram for estimated state feedback control 61 Figure C.5: Output response for estimated state feedback control Figure C.6: Estimation error for estimated state feedback control 62 Appendix D MATLAB Code Figure D.1: PCB MEMS Accelerometer Parameters MATLAB Code 63 Figure D.2: PCB MEMS Accelerometer Paramaters MATLAB Code (Cont.) 64