10/21/2003 1EEL5225: Principles of MEMS Transducers (Fall 2003)
Nonlinear DynamicsPackaging
� Today:� Nonlinear dynamics
� Linearization� Example: Electrostatic actuator
� Packaging
� Reading: Senturia, pp. 137-138, pp. 164-178, Chapter 17, pp. 453-468
Lecture 24 by H.K. Xie 10/20/2003
EEL5225: Principles of MEMS Transducers (Fall 2003)Instructor: Dr. Hui-Kai Xie
� Last lecture� Linear dynamics� Example: mass-spring
10/21/2003 2EEL5225: Principles of MEMS Transducers (Fall 2003)
Nonlinear Dynamics
�x = f(x,u)
y = g(x,u)
A nonlinear system can be expressed in the following general form:
where f and g are nonlinear functions of the state and the input.
� Two practical analysis methods:� Linearization about an operating point� Numerical integration of state equations
(SIMULINK)
10/21/2003 3EEL5225: Principles of MEMS Transducers (Fall 2003)
Linearization
� Linearization about an operating point
Operating point (X0, U0): a fixed state that is established by constant inputs. And f(X0, U0)=0.
� Consider a small fluctuation near the operating point, i.e., ( ) ( )
( ) ( )0
0
x t X x t
u t U u t
δδ
= +
= +
� Substitute x(t) and u(t) into the state equations
( ) ( ) ( ) ( )
( )( ) ( ) ( )( )0 0 0 0 0 0
2 2
0 0 0 0 0 0
( , ) , , ,
1, 2 , ,
2!
x u
xx xu uu
f x u f X U f X U x f X U u
f X U x f X U x u f X U u
δ δ δ
δ δ δ δ
= = + +
+ + + +
�
�
�
x t
0
Taylor series0 0 0
10/21/2003 4EEL5225: Principles of MEMS Transducers (Fall 2003)
Linearization
� Linearization about an operating point
� Drop the higher-order terms,
� Write in matrix form,
( )0 0 0 0
( ) ( ), ,
f fx t u t
X U X Ux uδ δ δ
∂ ∂= + ∂ ∂ �x t
1 1 1 1
1 11 1 1
1 10 0 0 0, ,
n n
n n n n n n n
n n
f f f f
x x u ux x u
x f f x f f u
x x u uX U X U
δ δ δ
δ δ δ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
= + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
� ��
� � � � � � � � �
�� �
Jacobian J
10/21/2003 5EEL5225: Principles of MEMS Transducers (Fall 2003)
Linearization
� Example: Electrostatic actuator
� For electrical domain,1
in
QgI V Q
R Aε = − =
�
Ref. Senturia, p.138
� For mechanical domain,2
2
QF
Aε=
� The governing equation,2
0( ) 02
Qmg bg k g g
Aε+ + + − =�� �
10/21/2003 6EEL5225: Principles of MEMS Transducers (Fall 2003)
Linearization
� Example: Electrostatic actuator
� Select state variables,
1
2
3
x Q
x g
x g
=== �
� The state equations become
( )
1 21
2 3
21
3 2 0 3
1
1
2
in
x xx V
R A
x x
xx k x g bx
m A
ε
ε
= −
=
= − + − +
�
�
�
Nonlinear
10/21/2003 7EEL5225: Principles of MEMS Transducers (Fall 2003)
Linearization
� Example: Electrostatic actuator
� Jacobian matrix:
� Find the operating point by setting f(x,u)=0:
2 1
1
0
0 0 1
X X
R A R AJ
X k b
m A m m
ε ε
ε
− −
= − − −
Nonlinear
where X1 and X2 are the operating point
( )
1 2
3
21
2 0 3
0
0
02
in
X XV
AX
Xk X g bX
A
ε
ε
− =
= + − + =
10/21/2003 8EEL5225: Principles of MEMS Transducers (Fall 2003)
Linearization
� Example: Transducer Model for Electrostatic actuator
� Characteristic Equations:
� Linearization:
2
out 0
QgV =
εA
QF = - k(g - g)
2εA
Senturia, p.170� Operating point:� At gap , which corresponds to Vin,0
� Charge:
g0: gap at V=0
0g
0 ,00ˆ in
AQ V
g
ε=
0 0
0
g QQA A
Q gk
A
δ δε εδ δ
ε
=
V
F
10/21/2003 9EEL5225: Principles of MEMS Transducers (Fall 2003)
Linearization
� Example: Transducer Model for Electrostatic actuator
� Linearization
Senturia, p.171
S ince
Laplace transform ,
S im ilarly,
Q Idt
IQ
sU
gs
δ δ
δδ
δδ
=
=
=
∫ 0 0
0
g QIs A s A
Q Uk
s A s
δ δε εδ δ
ε
=
V
F
( )
0
0
0
0
2
220
0
ˆ
ˆ
1
ˆ
E B M O
E M M E
E M
E B
M S e
E Me
E B M O
g kZ Z
s A sQ
T Ts A
QT
Z g
kZ k
s
QTk
Z Z A kg
ε
ε
φ
ε
= =
= =
= =
= −
= =
10/21/2003 10EEL5225: Principles of MEMS Transducers (Fall 2003)
Direct Integration of State Equations
� MATLAB/SIMULINK
http://www.rpi.edu/dept/chem-eng/WWW/faculty/bequette/lou/simtut/simtut_html.html
http://www.messiah.edu/acdept/depthome/engineer/Resources/tutorial/matlab/simu.html
� Simulink is an interactive tool for modeling, simulating, and analyzing dynamic, multidomain systems.
� It lets you accurately describe, simulate, evaluate, and refine a system's behavior through standard and custom block libraries.
� Simulink integrates seamlessly with MATLAB, providing you with immediate access to an extensive range of analysis and design tools.
� These benefits make Simulink the tool of choice for control system design, signal processing system design, communications system design, and other simulation applications.
� www.mathworks.com
Simulink Tutorial:http://www.mathworks.com/products/simulink/demos.jsp#
10/21/2003 11EEL5225: Principles of MEMS Transducers (Fall 2003)
Direct Integration of State Equations
mx bx kx f
or
f b kx x x
m m m
′′ ′+ + =
′′ ′= − −
� Example: mass-spring-damper system
10/21/2003 12EEL5225: Principles of MEMS Transducers (Fall 2003)
Direct Integration of State Equations
� Example: Electrostatic Actuator
Senturia, p.174
( )
1 21
2 3
21
3 2 0 3
1
1
2
in
x xx V
R A
x x
xx k x g bx
m A
ε
ε
= −
=
= − + − +
�
�
�
10/21/2003 13EEL5225: Principles of MEMS Transducers (Fall 2003)
“Back-end Processing”� Processing done after silicon chip is completed� Packaging� Test� Calibration
� Example of cost breakdown for integrated pressure transducer� Silicon chip (35%)� Package (45%)� Calibration and test (20%)
MEMS Packaging
10/21/2003 14EEL5225: Principles of MEMS Transducers (Fall 2003)
“Back-end Processing”
� Packaging� Conventional ICs and MEMS have different requirements
� IC package issues� (hermetic “moisture-tight” package, thermal dissipation,
parasitic lead capacitances and inductances, mechanical reliability)
� May test at the chip or “die” level� MEMS package issues
� Unique to each MEMS transducer (some MEMS require coupling to environment, others do not; mechanical properties of package affect mechanical performance of MEMS transducer)
� Difficult to test until after packaged
� Conclusion: Packaging must be designed from beginning with the MEMS device!
10/21/2003 15EEL5225: Principles of MEMS Transducers (Fall 2003)
� Functions of MEMS packages� Mechanical support� Protection from environment� Electrical connection to other system components� Coupling (optical, wireless, etc.)� Thermal considerations
� Types of MEMS packages� Metal packages� Ceramic packages� Thin-film multilayer packages� Plastic packages
MEMS Packaging
Reading: Gerke, MEMS Packaging
10/21/2003 16EEL5225: Principles of MEMS Transducers (Fall 2003)
MEMS Packaging
� Pressure sensors� Inertial sensors� RF-MEMS� Microfluidics� BioMEMS� Optical MEMS
10/21/2003 17EEL5225: Principles of MEMS Transducers (Fall 2003)
Flow-Chart for Packaging and Device Design
Ref. Senturia, Microsystem Design, p. 456.
10/21/2003 18EEL5225: Principles of MEMS Transducers (Fall 2003)
System Partitioning
� Key System Partitioning Decision� To integrate or not to integrate system
electronics with MEMS device� Cost issues
� Final product cost� Development cost
� Recommendation: MEMS and electronic integration kept to minimum unless absolutely necessary
� Alternative: Hybrid packaging
10/21/2003 19EEL5225: Principles of MEMS Transducers (Fall 2003)
Packaging: Hybrid Approach
Ref. Senturia, Microsystem Design, p. 457.
•Example: Hybrid Accelerometer
10/21/2003 20EEL5225: Principles of MEMS Transducers (Fall 2003)
MEMS Packaging
Ref. Senturia, Microsystem Design, p. 461.
• Example: Manifold Absolute Pressure (MAP)
Pre-molded plastic case Silicone die bond
10/21/2003 21EEL5225: Principles of MEMS Transducers (Fall 2003)
MAP Packaging
Ref. Senturia, Microsystem Design, p. 462.
10/21/2003 22EEL5225: Principles of MEMS Transducers (Fall 2003)
MAP Packaging
Ref. Senturia, Microsystem Design, p. 466.