Modified Nodal Analysis for MEMS Design Using SUGAR

1. 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

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

3. Netlist simulator SPICE SUGAR Introduction Current simulation approaches for MEMS devices: • FEM, BEMMEMCAD, 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 simulationNODAS, SUGAR

4. 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.

5. SUGAR(cont.) • Four versions released free on the web since June 1998. http://www-bsac.eecs.berkeley.edu/~cfm • 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.

6. SUGAR Release History

7. Process Files Netlist Input ODE Element Models SPICE–like Environment Simulation Engine (Static, Transient, Steady State)

8. 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 ……

9. g1 n1 b1 n2 n3 a2 a3 a1 v1 n4 n5 g 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] []

10. Y-axis Accelerometer

11. 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]

12. 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 ……

13. 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 x0 , iterates Until Static Analysis (DC) (tolerance)

14. Lb 6 + O Experimental results Simulation results V - Pull-in Voltages (V) Gap distance at node 6 (um) Length of the beam L (um) Voltage V (v) Static Simulation Example • 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;

15. 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.

16. -11 -5 -12 -6 -7 -13 log10(magnitude) log10(magnitude) -8 -14 -9 -15 -10 2 3 4 5 6 10 10 10 10 10 2 3 4 5 6 10 10 10 10 10 Frequency (Hz) Frequency (Hz) 100 200 50 100 0 phase(degree) 0 phase(degree) -50 -100 -100 -200 2 3 4 5 6 10 10 10 10 10 2 3 4 5 6 10 10 10 10 10 Frequency (Hz) Frequency (Hz) 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 mass The response of induced current in lower comb

17. Mode 1 Mode 2 at 15454 Hz at 26983 Hz Mode 3 Mode 6 at 123010 Hz at 31112 Hz Modal Simulation Example