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Modified Nodal Analysis for MEMS Design Using SUGAR Ningning Zhou, Jason Clark, Kristofer Pister, Sunil Bhave, BSAC David Bindel, James Demmel, Depart. of CS, UC Berkeley Sanjay Govindjee, Depart. of CEE, UC Berkeley Zhaojun Bai, Depart. of CS, UC Davis Ming Gu, Jianlin Xia, Depart. of Mathematics, UC Berkeley January, 2001

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Outline Background SUGAR introduction Netlist input Algorithms with examples Element models More examples Conclusion

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Introduction Current simulation approaches for MEMS devices: FEM, BEM MEMCAD, AutoBEM, ANSYS etc. –Device/Process oriented; –Not well integrated with other domains such as circuits; –Poorly suited to do higher level design and optimization. System level simulation NODAS, SUGAR SUGAR SPICE Netlist simulator

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SUGAR Simulation package for MEMS devices implemented in MATLAB. Using Modified Nodal Analysis method modeled on SPICE. Ability to perform simulation in multi-energy domains such as electrical circuits, mechanical, thermal etc. Implemented static(DC), steady state (SS), modal frequency, transient and sensitivity analysis in different versions of SUGAR.

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SUGAR(cont.) Four versions released free on the web since June Hundreds of downloads from all over the world. For example, in the period of 02/2000 ~ 04/2000, 121 downloads from universities(~40%), industries(10~20%), research labs(5~10%) etc.. Active interaction with users.

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SUGAR Release History V0.5V1.01V1.1V2.0 Release time06/9811/9907/00Now 2D(DC, SS, Modal, TA) 3D(DC, SS, Modal, TA) Mechanical (beams, anchors, gaps) Simple electrical elements Open framework for new models, New netlist input allowing subnets Sensitivity analysis

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SPICE–like Environment Netlist InputProcess Files ODE Element Models Simulation Engine (Static, Transient, Steady State)

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Elements and Models Elements: Beams Anchors Plate mass Electrostatic gaps Circuits elements (resistor, voltage source) …… Models: Beam Linear mechanical model Nonlinear mechanical model Mechanical-electrical model etc. Gap Nonlinear electro-mechanical model Anchor Mechanical model Electro-mechanical model ……

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Input Netlist uses mumps.net v1 Vsrc * [n1 g] [V=10] e1 eground * [g] [] a1 anchor p1 [n1] [l=5e-6 w=10e-6 oz=180 R=100] b1 beam2de p1 [n1 n2] [l=1e-4 w=2e-6 oz=0 R=1000] g1 gap2de p1 [n2 n3 n4 n5] [l=1e-4 w1=1e-5 w2=2e-6 … gap=2e-6 R1=100 R2=100 oz=0] a2 anchor p1 [n4] [l=5e-6 w=1e-5 oz=-90 R=100] e2 eground * [n4] [] a3 anchor p1 [n5] [l=5e-6 w=1e-5 oz=-90 R=100] e3 eground * [n5] [] v1 a1 b1 g1 a2a3 g n1 n2n3 n4 n5

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Y-axis Accelerometer

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Netlist of Y-axis Accelerometer uses mumps.net subnet XSusp [B] [susp_len=* angle=*][ a1 anchor parent [A] [l=10u w=10u h=6u oz=90+angle] b1 beam3d parent [A a1] [l=susp_len w=2u h=6u oz=0+angle] b2 beam3d parent [a1 a2] [l=10u w=2u h=6u oz=-90+angle] b3 beam3d parent [a2 B] [l=susp_len w=2u h=6u oz=180+angle] b4 beam3d parent [A a3] [l=susp_len w=2u h=6u oz=180+angle] b5 beam3d parent [a3 a4] [l=10u w=2u h=6u oz=-90+angle] b6 beam3d parent [a4 B] [l=susp_len w=2u h=6u oz=0+angle] ] subnet XMass [A B] [finger_len=*][ b1 beam3d parent [A b1] [l=25u w=50u h=6u oz=-90] b2 beam3d parent [b1 B] [l=25u w=50u h=6u oz=-90] b3 beam3d parent [b1 b2] [l=finger_len w=2u h=6u oz=0] b4 beam3d parent [b1 b3] [l=finger_len w=2u h=6u oz=180] ] XSusp p1 [c(1)] [susp_len=200u angle=0] for k=1:10 [ mass(k) XMass p1 [c(k) c(k+1)] [finger_len=100u] ] XSusp p1 [c(11)] [susp_len=200u angle=180]

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Modified Nodal Analysis Finding nodal variables (unknowns) by formulating and solving nodal equations at each node. Nodal variables: mechanical displacements electrical potentials thermal temperatures…… Nodal equations at each node: sum of forces = 0 sum of currents = 0 sum of heat flux = 0 ……

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Static Analysis (DC) Finding the equilibrium point of the system SUGAR uses Newton-Raphson method solving nonlinear equation system x is the equilibrium nodal variables Starting from an initial guess x 0, iterates Until (tolerance)

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Static Simulation Example Pull-in Voltages (V) Length of the beam L (um) O Experimental results Simulation results Test structures are fabricated by MCNC; Beam: Nominal Lb=100um, w=2um, h=2um. Measured : L=100um, w=1.74um, h=2.003um Gap plate: Lg=100um, w=10um, h=2.003um. Young’s Modulus: assume 165GPa. Simulation was done by considering fringing-field effects; Contact force model was used to get pull-in voltage; Voltage V (v ) Gap distance at node 6 (um) - + V Lb 6

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Steady State and Modal Analysis Finding the sinusoidal response of the system Linearizing the system at a DC equilibrium point, solving linear ODE system where u = sinusoidal excitation y = system output response C = output matrix D = feed forward matrix Modal frequencies and modal shapes can be found by solving for system eigenvalues and eigenvectors.

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Steady State Simulation Examples Simulation of a linear multiple mode resonator by Reid Brennen. Sugar results matches his measurements within 5%. The response of vertical displacement of massThe response of induced current in lower comb

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Modal Simulation Example Mode 3 at Hz Mode 2 at Hz Mode 1 at Hz Mode 6 at Hz

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