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.