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