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

Outline

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

IntroductionCurrent 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 SPICENetlist simulator

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.

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.

SUGAR Release History

V0.5 V1.01 V1.1 V2.0Release time 06/98 11/99 07/00 Now

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

SPICE–like Environment

Netlist Input Process Files

ODE Element Models

Simulation Engine(Static, Transient, Steady State)

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 ……

Input Netlist

uses mumps.netv1 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

a2 a3

g

n1 n2 n3

n4 n5

Y-axis Accelerometer

Netlist of Y-axis Accelerometer uses mumps.netsubnet 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]

Modified Nodal AnalysisFinding 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 ……

Static Analysis (DC)• Finding the equilibrium point of the system• SUGAR uses Newton-Raphson method solving nonlinear

equation system

0xf x is the equilibrium nodal variables

)(1

1 nn

nn xfx

fxx

Starting from an initial guess x0 , iterates

nn xx 1Until (tolerance)

Static Simulation Example

40 60 80 100 120 140 160 180 200 220 2402

4

6

8

10

12

14

16

18

20

22

Pull-

in V

olta

ges

(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;

6 6.5 7 7.5 8 8.5 9 9.5 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Voltage V (v)

Gap

dist

ance

at n

ode

6 (u

m)

-+V

Lb

6

Steady State and Modal Analysis

DuCxy

BuAxx

• Finding the sinusoidal response of the system

• Linearizing the system at a DC equilibrium point, solving linear ODE system

whereu = sinusoidal excitationy = system output responseC = output matrixD = feed forward matrix

Modal frequencies and modal shapes can be found by solvingfor system eigenvalues and eigenvectors.

Steady State Simulation Examples

• Simulation of a linear multiple mode resonator by Reid Brennen. Sugar results matches his measurements within 5%.

102

103

104

105

106

-10

-9

-8

-7

-6

-5

Frequency (Hz)

log1

0(m

agni

tude

)

102

103

104

105

106

-200

-100

0

100

200

Frequency (Hz)

phas

e(de

gree

)

The response of vertical displacement of mass

102

103

104

105

106

-15

-14

-13

-12

-11

Frequency (Hz)

log1

0(m

agni

tude

)

102

103

104

105

106

-100

-50

0

50

100

Frequency (Hz)

phas

e(de

gree

)

The response of induced current in lower comb

Modal Simulation Example

Mode 3at 31112 Hz

Mode 2at 26983 Hz

Mode 1at 15454 Hz

Mode 6 at 123010 Hz