Transcript Modified Nodal Analysis for MEMS Design Using SUGAR Ningning Zhou, Jason Clark, Kristofer Pister, Sunil Bhave, BSAC David Bindel, James Demmel, Depart.

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
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Background
SUGAR introduction
Netlist input
Algorithms with examples
Element models
More examples
Conclusion
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
Netlist simulator
SUGAR
SPICE
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 V2.0
Release time
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
06/98
11/99
07/00
Now
SPICE–like Environment
Netlist Input
Process Files
Simulation
Engine
(Static, Transient,
Steady State)
ODE Element Models
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
n1
a1
v1
b1
n2
a2 n4
g1
n3
n5
a3
g
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]
[]
Y-axis Accelerometer
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]
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 ……
Static Analysis (DC)
• Finding the equilibrium point of the system
• SUGAR uses Newton-Raphson method solving nonlinear
equation system
f x   0
x is the equilibrium nodal variables
Starting from an initial guess x0 , iterates
1
 f 
xn 1  xn  
 f ( xn )
 xn 
Until
xn1  xn  
(tolerance)
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
Lb
6
• Gap plate: Lg=100um, w=10um, h=2.003um.
• Young’s Modulus: assume 165GPa.
• Simulation was done by considering fringing-field effects;
V +
-
• Contact force model was used to get pull-in voltage;
1.8
20
1.6
18
1.4
16
Pull-in Voltages (V)
Gap distance at node 6 (um)
22
1.2
1
0.8
0.6
0.4
Simulation results
14
12
10
8
6
4
0.2
0
O Experimental results
6
6.5
7
7.5
8
Voltage V (v)
8.5
9
9.5
10
2
40
60
80
100
120
140
160
180
Length of the beam L (um)
200
220
240
Steady State and Modal Analysis
• Finding the sinusoidal response of the system
• Linearizing the system at a DC equilibrium point, solving
linear ODE system
x  Ax  Bu
y  Cx  Du
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.
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
-11
-7
-8
-9
-10
2
10
phase(degree)
log10(magnitude)
-6
10
3
4
10
Frequency (Hz)
10
5
10
-12
-13
-14
-15
2
10
6
200
100
100
50
phase(degree)
log10(magnitude)
-5
0
-100
-200
2
10
10
3
4
10
Frequency (Hz)
10
5
10
6
10
3
4
10
Frequency (Hz)
10
5
10
6
0
-50
-100
2
10
10
3
4
10
Frequency (Hz)
10
5
10
6
Modal Simulation Example
Mode 1
at 15454 Hz
Mode 2
at 26983 Hz
Mode 3
at 31112 Hz
Mode 6
at 123010 Hz