Energy-conserving Transducers - MIT OpenCourseWareJV: 6.777J/2.372J Spring 2007, Lecture 9 - 14...

Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

JV: 6.777J/2.372J Spring 2007, Lecture 9 - 1

Energy-conserving Transducers

Joel Voldman*

Massachusetts Institute of Technology

(*with thanks to SDS)



Outline

> Last time

> The two-port capacitor as a model for energy-conserving transducers

> The transverse electrostatic actuator

> A look at pull-in

> Formulating state equations



Last time: equivalent circuits

> Learned how to describe systems as lumped elements and equivalent circuits

b

m

1/kx.

F +-

+

++

- -

-

eb

em

ek

m

k

b

F

x

Images removed due to copyright restrictions. Figure 11 on p. 342 in: Zavracky, P. M., N. E. McGruer, R. H. Morrison, and D.Potter. "Microswitches and Microrelays with a View Toward MicrowaveApplications." International Journal of RF and Microwave Comput-Aided Engineering 9, no. 4 (1999): 338-347.

Silicon0.5 µm

1 µm

Pull-downelectrode

Cantilever

Anchor

Image by MIT OpenCourseWare.Adapted from Rebeiz, Gabriel M. RF MEMS: Theory, Design, and Technology.Hoboken, NJ: John Wiley, 2003. ISBN: 9780471201694.



Last time: equivalent circuits

> Saw that lumped elements in different domains all had equivalent circuits

> Introduced generalized notation to describe many different domains

∫+=

=

t

o fdtqq

dtdqf

0∫+=

=

t

o edtpp

dtdpe

0

General Electrical Mechanical Fluidic Thermal

Effort (e) Voltage, V Force, F Pressure, P Temp. diff., ΔT

Flow (f) Current, I Velocity, v Vol. flow rate, Q Heat flow,

Displacement (q) Charge, Q Displacement, x Volume, V Heat, QQ

Momentum (p) - Momentum, p Pressure Momentum, Γ

-

Resistance Resistor, R Damper, b Fluidic resistance, R

Thermal resistance, R

Capacitance Capacitor, C Spring, k Fluid capacitance, C

Heat capacity, mcp

Inertance Inductor, L Mass, m Inertance, M -

Node law KCL Continuity of space Mass conservation

Heat energy conservation

Mesh law KVL Newton’s 2nd law Pressure is relative

Temperature is relative



Equivalent circuit elements



Today’s goal

> How do we model an electrical force applied to the cantilever?

> How can we describe converting energy between domains?

> This leads to energy-conserving transducers



Outline

> Last time



> A look at pull-in


Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT penCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

JV: 6.777J/2.372J Spring 2007, Lecture 9 - 8O

General Considerations> In MEMS, we are often interested in sensors and actuators

> We can classify sensors and actuators by the way they handle energy:

• Energy-conserving transducers» Examples: electrostatic, magnetostatic, and piezoelectric

actuators • Transducers that use a dissipative effect

» Examples: resistive or piezoresistive sensors

> There are fundamental reasons why these two classes must be treated differently.

• Energy-conserving transducers depend only on the state variables that control energy storage. Therefore, quasi-static analysis is OK.

• Dissipative transducers depend, in addition, on state variables that determine the rate of energy dissipation, and are more complex as a result.



An Energy-Conserving Transducer

> By definition, it dissipates no energy, hence contains no resistive elements in its representation

> Instead, it can store energy from different domains –this creates the transducer action

> Because the stored energy is potential energy, we use a capacitor to represent the element, but because there are both mechanical and electrical inputs, this must be a new element: a two-port capacitor



Capacitor with moveable plate

> A charged capacitor has a force of attraction between its two plates

> If one of the plates is moveable, one can make an electrostatic actuator.

I

+ Moveable plate

Vg

z

-Fixed plate

Adapted from Figure 6.1 in: Senturia, Stephen D. MicrosystAcademic Publishers, 2001, p. 126. ISBN: 9780792372462.

Image by MIT OpenCourseWare.em Design . Boston, MA: Kluwer



Various ways of charging

> Charging at fixed gap• An external force is required to

prevent plate motion• No movement No mechanical

work

> Charging at zero gap, then lifting

• No electrical energy at zero gap• Must do mechanical work to lift

the plate

> Either method results in stored energy

+Charge at fixed gap

io

-

V g

I

Force

+

-

V g

I

Force

Pull up

+

Charge at zero gap, then....

io

-

V+Q

+Q

-Q

-Q

I

stemDesign.. Boston, MA: Kluwer Academic Publishers, 2001, p. 127.ISBN: 9780792372462.

Image by MIT OpenCourseWare.Adapted from Figure 6.2 in Senturia, Stephen D. Microsy



Charging at Fixed Gap

> The stored energy is obtained directly from the definition for a linear capacitor

> Anticipating that the gap might vary, we now explicitly include the gap as a variable that determines the stored energy

AgQ

CQgQW

dQCQVdQedqW

QQq

ε22),(

22

000

==

=== ∫∫∫

QqVe

→→

CQV =

gAC ε

=



Pulling Up at Fixed Charge> Putting charge at zero gap stores no electrical energy

> Once charge is applied, determining stored energy is a mechanics problem.

> In determining the force, we must avoid double-counting of charge

E-field of bottom plate

∫ ε==

g

AgQFdggQW

0

2

2),(

The final stored energy is same as before!ONLY depends on Q and g, not the path!

AQQEF

AQE

ε

ε

2

22

==

= Q on top plate

02

2

0→=⇒∞→

→ CQWC

g

- - - - - - -

+++++++



Lossless transducers

> The energy in the system ONLY depends on the STATE variables (e.g., Q, g) and NOT how we put the energy in

• The system is lossless/conservative

FdgVdQdWdtdgF

dtdQV

dtdW

gFVI

PPdt

dWmechanicalelectrical

+=

+=

+=

+=

Adapted from Figure 6.3 in Senturia, Stephen D. Microsystem Design.Boston, MA: Kluwer Academic Publishers, 2001, p. 129. ISBN: 9780792372462.

CF

W(Q,g)V

+

I

-

+

-

g

Image by MIT OpenCourseWare.



A Differential Version> Since we can modify the stored energy either by changing the

charge or moving the plate, we can think of the stored energy asdefined differentially

This leads to a pair of differential relations for the force andvoltage

QggQWF

∂∂

=),(

gQgQWV

∂∂

=),(

FdgVdQdW +=



Revisit charging the capacitor

> The energy only depends on Q, g• These are thus the STATE variables for this transducer

gg1

Q1

Q

Move plates, then charge

Charge, then move plates

AgQgQW

ε2),(

2

=



The two-port capacitor> This transducer is what will couple our electrical

domain to our mechanical domain

AQ

ggQWF

Q ε2),( 2

=∂

∂=

AQg

QgQWV

g ε=

∂∂

=),(

AgQgQW

ε2),(

2

=

CF

W(Q,g)V

+

I

-

+

-

g

Adapted from Fig sign.Boston, MA: Kluwer Academic Publishers, 2001, p. 129. ISBN: 9780792372462.

Image by MIT OpenCourseWare.ure 6.3 in Senturia, Stephen D. Microsystem De



A different example

> What if the material in the gap could move?

CQxQW2

),(2

=

( )

( )

( )

0 0

2

0 0

20

20 0

( )

( , ) 12 ( )

2 ( )

Q

lC x x xgW Q x Q gF

x l x x x x

Q gFl x x x

ε ε

ε ε

ε εε ε

= + −

∂ ∂= =

∂ ∂ + −

−=

+ −

x0

x

g

l +Q

-Qε ε0



Outline

> Last time



> A look at pull-in




The Electrostatic Actuator

> If we now add a spring to the upper plate to supply the external mechanical force, a practical actuator results

> We are getting closer to our RF switch…

Fixed support

Fixed plate

Spring k

zgV

I+

-

C

W(Q,g)

+

-

V

+

-

F 1/k

I g.

z.

Adapted from Figure 6.4 in: Boston, MA: Kluwer Academic Publishers, 2001, p. 130. ISBN: 978079237246.


Senturia,Stephen D. Microsystem Design..Image by MIT OpenCourseWare.

Adapted from Figure 6.4 in: Senturia, Stephen D. Microsystem Design. Boston,MA: Kluwer Academic Publishers, 2001, p. 130. ISBN: 9780792372462.



Two methods of electrical control

> Charge control• Capacitor is charged from a current source, specifically

controlling the charge regardless of the motion of the plate• This method is analyzed with the stored energy

> Voltage control• Capacitor is charged from a voltage source, specifically

controlling the voltage regardless of the motion of the plate• This method is analyzed with the stored co-energy



Charge control> Following the causal path

1. Current source determines the charge

2. Charge determines the force (at any gap!)

3. Force determines the extension of the spring

4. Extension of the spring determines the gap

5. Charge and gap together determine the voltage

AkQgg

zggkFz

AQ

gWF

dttiQ

Q

t

in

ε

ε

2

2

)(

2

0

0

20

−=

−=

=

=∂∂

=

= ∫1)

2)

3)

4)

initial displacement

Fixed support

Fixed plate

Spring k

zgV

I+

-

C

W(Q,g)

+

-

Viin(t)

+

-

F 1/k

I g z

Adapted from Figure 6.5 in: Senturia, Stephen D. Microsystem Design. Boston,MA: Kluwer Academic Publishers, 2001, p. 131. ISBN: 9780792372462.


Adapted from Figure 6.4 in: Senturia,Stephen D. Microsystem Design..Boston, MA: Kluwer Academic Publishers, 2001, p. 130. ISBN: 9780792372462.




Charge control

> Let’s get voltage, normalize and plot

> Normalize variables to make easier to plot• First normalize V and Q to some nominal values• Introduce ξ (normalized displacement) that goes from 0 (g=g0)

to 1 (g=0)

• Define Q0 and V0 using expression above

AAk

QgQ

AQg

QWV

g εε

ε

⎟⎟⎠

⎞⎜⎜⎝

⎛−

==∂∂

=2

2

0

( )0

0 0 0 0 g gQV zv qV Q g gξ −= = = =

020

000 2 AkgQ

AgQV ε

ε==



Charge control

> Now, plug in to non-dimensionalize

> Now we get expressions relating voltage and displacement to charge

( ) ( )( )( )

22

0

2200

02

00

20

00

)1(11

)1()1(

2

qqgg

qqvqqAgQV

AgqgqQ

AAk

qQgqQV

=⇒−−=−=

−=⇒−=

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

ξξε

εεε

AAk

QgQV

εε ⎟⎟

⎠

⎞⎜⎜⎝

⎛−

=2

2

0



Charge control> Actuator is stable at all

gaps – the voltage goes to zero at zero gap

> The voltage is multivalued the charge uniquely determines the state and thus the energy

AAk

QgQ

AQg

QWV

g εε

ε

⎟⎟⎠

⎞⎜⎜⎝

⎛−

==∂∂

=2

2

0

0 0.5 10

0.2

0.4

norm

aliz

edvo

ltage

0 0.5 10

0.5

1

normalized charge (q)

norm

aliz

ed

disp

lace

men

t (ζ)



Co-Energy> For voltage control, we cannot use W(Q,g) directly, because we

cannot maintain constant charge. Instead we use the co-energy• So we change variables

( )[ ] [ ]FdgVdQVdQQdVgVdW

gQdWQVdgVdWgQWQVgVW

+−+=

−=

−=

),(),(),(

),(),(

*

*

*

)()( 1111* qWeqeW −=

V

g

ggVWF

VgVWQ

∂∂

−=⇒

∂∂

=⇒

),(

),(

*

*

FdgQdVgVdW −=),(*

Recall:



Voltage control> Following the causal path

1. Voltage and gap (implicitly) determines the force

2. Force determines the spring extension

3. And thus the gap4. Voltage and gap together

determine the charge

* 2 2

2*

2

0

2

0 2

1( , )2 2

2

2

in in in

in

V

in

in in

AW V g CV Vg

AVWFg g

g g zFzk

AVg gkg

AQ V CVg

ε

ε

ε

ε

= =

∂= − =

∂

= −

=

= −

= =

1)

2)

3)

4)

C

W*(V,g)

+

-

VVin(t)

+

-

F 1/k

I g z.

+-

.

Adapted from Figure 6.6 in: Senturia, Stephen D. Microsystem Design . Boston, MA:Kluwer Academic Publishers, 2001, p. 132. ISBN: 9780792372462.




Outline

> Last time



> A look at pull-in




Forces and stability

> Nondimensionalize again

02

)(

0

2

2

0 =−−=

=−=

gAVggk

FFF elecmechNet

ε0

)1(274

0274

0278

2

2

2

2

20

2

30

2

2

0

=−

−

=−

=−=

ξξ

ξ

εεξ

vggv

Akg

gAvgkFNet

( )

AkgV

VVv

ggg

PI

PI

ε

ξ

278

/

302

00

=

=

−=

0 0.2 0.4 0.6 0.8 10

0.20.40.60.8

1

normalized displacement (ζ )

norm

aliz

ed fo

rce

Spring force Electrical force

Increasing v

positiveforce increases gap

> Let’s examine the net force on the actuator



Stability criterion> At low voltage, there are

two intersections• Which is stable?

> At higher voltages, there are none

• What is happening?

The position of the actuator is stable only when there is a net restoring force when the system is disturbed from equilibrium

unstable

g

stableFnet Fnet

g



0 0.5 1-1

-0.5

0

0.5

1

normalized gap (g/g0)

norm

aliz

ed fo

rce

Stability criterion

0

1 gg

ξ− =unstablestable

> We can plot the normalized NET force versus normalized gap and check

Increasing v

2

0 2

2

2

20

2

0

( )2

427(1 )

4127

Net mech elec

net

F F FAVk g gg

v

g vfg g

g

ε

ξξ

= −

= − −

= −−

⎛ ⎞= + −⎜ ⎟

⎛ ⎞⎝ ⎠⎜ ⎟⎝ ⎠



Stability criterion

> So what we want is a negative slope

> In this example, this means that the spring constant must exceed a critical value that varies with voltage

kgAV

gAVk

gF

gAVggkF

Net

Net

<

<⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

∂∂

−−=

3

2

3

2

2

2

0

0

2)(

ε

ε

ε

Stability:



Stability criterion

> If the voltage is too large, the system becomes unstable, and we encounter pull-in

> Right at pull-in, the spring constant is AT the critical value AND static equilibrium is maintained

2

3

2

0 2

0

30

0

( )2

( )2

82723

PI

PI

PIPI

PI

PIPI

PI

PI

AVkg

AVk g gg

kgk g g

kgVA

g g

ε

ε

ε

=

− =

− =

=

=

At pull-in:



Stability analysis of pull-in> Plot normalized gap versus

normalized voltage

> Solve cubic equation

0 0.5 10

0.5

1

normalized voltage

norm

aliz

ed g

apstable

unstable

2

2

0 2kgAVgg inε

−=

In Matlab: g = fzero(@(g)(g - g0 + eps*A*V^2/(2*k*g^2)),g0);



Release voltage after pull-in> After pull-in less voltage is

needed to keep beam down

> Find force when pulled down

> Equate to mechanical force to get hold-down voltage

> Is usually much less than pull-in voltage

( ) 00

2

2

2kggkF

AVF

gmech

ingelec

≈−=

=

=

=

δδ

ε

δ

δ

AkgV

kgAV

HD

HD

εδ

δε

02

2

02

2

22

=

=

14

272

0

2

<⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛gV

VPI

HD δ

g0δ

AkgVPI ε27

8 302 =

Normalizeto VPI



Macro pull-in?

> Can we do a macroscopic pull-in demo?

> Use soft spring k = 1 N/m

> Use • A = 8.5” x 11” plates• g0 = 1 cm

> Not easy… this is why pull-in is a MEMS-specific phenomenon

( ) ( )( )

30

3

212

827

8(1)(0.01)27 8.85 10 8.5 11 0.0254

750 V

PIkgV

Aε

−

=

=× × ×

≈



Outline

> Last time



> A look at pull-in




Adding dynamics

> Add components to complete the system:

• Source resistor for the voltage

Vin

ResistorR I

Dashpot b Spring kMass m

zgV

Fixed plate

Fixed support

+

+

--

CF

W(Q,g)

+ +R

Vin V

- -

I g z

m

1/k

b

+

-

source• Inertial mass, dashpot

> This is now our RF switch!

> System is nonlinear, so we can’t use Laplace to get transfer functions

> Instead, model with state equations

Electrical domain Mechanical domain

Image by MIT OpenCourseWare.Adapted from Figure 6.9 in Senturia, Stephen D. Microsystem Design . Boston,MA: Kluwer Academic Publishers, 2001, p. 138. ISBN: 9780792372462.



The System is Now General

> The addition of the source resistor breaks up the distinction between voltage-controlled and charge-controlled actuation:

• For small R, the system behaves like a voltage-controlled actuator

• For large R, the system behaves like a charge-controlled actuator at short times because the “impedance” of the rest of the circuit is negligible the voltage source delivers a constant current V/R*

*See, for example, Castaner and Senturia, JMEMS, 8, 290 (1999)



State Equations> Dynamic equations for general

system (linear or nonlinear) can be formulated by solving equivalent circuit

> In general, there is one state variable for each independent energy-storage element (port)

> Good choices for state variables: the charge on a capacitor (displacement) and the current in an inductor (momentum)

> For electrostatic transducer, need three state variables

• Two for transducer (Q,g)• One for mass (dg/dt)

Goal:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

constantsor of functions

gQ,g,ggQ

dtd



Formulating state equations

( )1

1

in

in

dQ I V Vdt R

dQ QgVdt R Aε

= = −

⎛ ⎞= −⎜ ⎟⎝ ⎠

> Start with Q

> We know that dQ/dt=I

> Find relation between I and state variables and constants

KVL : 0

0

in R

in

V e V

V IR V

− − =

− − =

IReR =

QgVAε

=

I

Vin

R+-

+

-V

C

W Q( ,g)

+ -eR




0

0:KVL

=−−−

=−−−

zbzmkzF

eeeF bmk

> Now we’ll do

> We know that

k

m

b

e kze mze bz

=

==

g

gdtgd =

gzgzggz −=−=⇒−= ,0

[ ]

⎥⎦

⎤⎢⎣

⎡+−−−=

+−−−=

=++−−

gbggkA

Qmdt

gd

gbggkFm

g

gbgmggkF

)(2

1

)(1

0)(

0

2

0

0

ε

C

W(Q,g)

1/kzg

+

+

-

-

ek

eb

F emm

b

+ +

- -

. .




2

0

1

1 ( )2

inQgV

Q R Ad g gdt

g Q k g g bgm A

ε

ε

⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟⎡ ⎤ ⎝ ⎠⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥⎣ ⎦ − − − +⎢ ⎥⎢ ⎥

⎣ ⎦⎣ ⎦

> State equation for g is easy:

> Collect all three nonlinear state equations

> Now we are ready to simulate dynamics (WED)

gdtdg

=



What have we wrought?

> We have modeled a complex multi-domain 3D structure using

• Equivalent circuits• A two-port nonlinear capacitor

> What can we now get• Actuation voltage: VPI

• Tip dynamics

> What have we lost• Capacitor plates are not really parallel during actuation• Neglected fringing fields• Neglected stiction forces when beam is pulled in

Figure 9 on p. 17 in: Nguyen, C. T.-C."Vibrating RF MEMS Overview: Applic ations to W ireless Communications." Proceedings of SPI E Int Soc Opt Eng 5715 (January 2005): 11-25.

Images removed due to copyright restrictions. Figure 11 on p. 342 in: Za vracky, P. M ., N. E. Mc Grue r, R. H. Mo rris on, and D.Potter. "Microsw itches and Microrelays w ith a View To ward Micro waveApplications." Internati onal Journal of RF and Microw ave Com put-Aided Engineering 9, no. 4 (1999): 338-347.

Images removed due to copyright restrictions.



Conclusions

> We can successfully model nonlinear transducers with a new element: the two-port capacitor

> Know when to use energy or co-energy for forces• At best a sign error• At worst just wrong

> Under charge control, transverse electrostatic actuator is well-behaved

> Under voltage control, exhibits pull-in

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