Modelling of Coupled Electrostatic Microsystems - NPTel

Modelling of Coupled Electrostatic Microsystems G. K. Ananthasuresh Professor, Mechanical Engineering Indian Institute of Science Bangalore, India [email protected]

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Pull‐in phenomenon (review) Condition for critical stability

V1 < V2 < V3

Potential energy

g0 / 3

g0

3 (  )  0 AV 2 k g x  2 ( PE ) 2 0    0   V k 0 A x 2 ( g 0  x) 3 k ( g 0  x) g0 1 0 A 2 kx  V  x 2 2 g 0  x  2 3

x

V pull in 

8 k g 30

27  0 A

x

2g0 / 3

Vpull in Vpull in G.K. Ananthasuresh, Indian Institute of Science

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With a dielectric layer: pull‐up and hysteresis (review) Vpull in

td  8k   g 0    27 0 A  r 

3

Dielectric layer x

Vpull up

g0 td

g0

 td 2k  g 0  0 A  r

  

2

x

Pull‐up voltage is found by equating the forces of spring and electrostatics at .x  g 0 Gilbert, J. R., Ananthasuresh, G. K., and Senturia, S. D., “3‐D Modeling and Simulation of Contact Problems and Hysteresis in Coupled Electromechanics,” presented at the IEEE‐MEMS‐96 Workshop, San Diego, CA, Feb. 11‐15, 1996.

Vpull up Vpull in V G.K. Ananthasuresh, Indian Institute of Science

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Pull‐in phenomenon used in a display device www.iridigm.com (a QualComm acquisition) Interference‐modulation by electrostatic actuation of vertically moving membranes.

G.K. Ananthasuresh, Indian Institute of Science

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What about dynamic behavior? Vdynamic pull -in = dynamic pull‐in voltage

Frequency = 

Potential energy

g0 V

t

Lumped 1‐dof model

x

mx  kx 

 0 AV 2

2( g 0  x ) 2

Beam model

 0 wV 2 d 4u 0  wt u  EI 4  2 dx 2( g 0  u )

V 2  (Vdc  Vac sin t ) 2  Vdc2  2VdcVac sin t  Vac2 sin 2 t Will contain a term! 2

So, the response will show two resonance at two frequencies.


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Damping

V


mx  bx  kx 

 0 AV

Beam model

2

2( g 0  x ) 2

 0 wV 2 d 4u  wt u  bu  EI 4  0 2 dx 2( g 0  u )

b How do you obtain ?


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Squeezed film effect

Squeezed‐film damping

V


mx  bx  kx 

 0 AV

Beam model

2

2( g 0  x ) 2

 0 wV 2 d 4u  wt u  bu  EI 4  0 2 dx 2( g 0  u )

b How do you obtain ? Use isothermal, compressible, narrow gap Reynolds equation to model the film of air beneath the beam/plate/membrane. It is widely used in lubrication theory. By analyzing this equation, we can extract the essence of damping as a lumped parameter – the so called “macromodeling”. G.K. Ananthasuresh, Indian Institute of Science

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Modeling squeezed film effects: isothermal Reynolds equation Pressure distribution in the 2‐D x‐y plane

Gap varies in the x‐y plane for a deformable structure (beam,plate, membrane)

1   p ( x, y ) g ( x, y )     p ( x , y ) g ( x , y ) 3 p ( x , y ) t 12





Viscosity of air

For lumped 1‐dof modeling, we have a rigid plate. So,

g

does not depend on . ( x, y )

g3   p ( x, y ) g  g 3      p ( x , y ) p ( x , y )   t 12 12

1 2 2  p x y  ( , )    2

Assume further that pressure distribution is the same along the length of the plate so that it becomes a one dimensional problem.

 p( y) g  g   t 12 3

 1 2 2   p ( y)   2

Assumed pressure distribution

x

y

S. D. Senturia, Microsystems Design, Kluwer, 2001. G.K. Ananthasuresh, Indian Institute of Science

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Behavior with small displacements  p( y) g  g 3  Linearize t 12 p  p0  p

1 2 2  ( p0 , g 0 )   p ( y )  around : 2 

g  g 0  g

y p g , gˆ  Also, use non‐dimensional variables:   , pˆ  w p0 g0 g 02 p0  2 pˆ gˆ g 02 p0  2 pˆ g pˆ width      2 2 2 2 t 12 w  t 12 w  g0 pˆ ( , t )  ~ p ( ) e t Separation of spatial and temporal components: g 02 p0  2 ~ p g ~   p   2 2 12 w  g 0 e t Assume a sudden velocity impulse to the plate. Then, for t > 0, this term is zero.

(with displacement ) x  x0


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Behavior with small displacements (contd.)









2 g 02 p0  2 ~ p 12  w n ~ ~   p  0  p  An sin  n   Bn cos  n   n  2 2 12 w  g 02 p0

Boundary conditions and velocity‐impulse assumption give:

g 02 p0 n 2 2  n  n ;  n  ; n  1,3,5,... 2 12 w  x0 4  n t An   sin( n  ) e  g 0 odd n n 1

Force on the plate =

x0 8  nt f sq (t )  p0 wl  pˆ (t ,  ) d   p0 wl  2 2e g 0 odd n n  0

Take the Laplace transform (continued on the next slide). G.K. Ananthasuresh, Indian Institute of Science

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Finally, getting to lumped approximation…    3  96 l w  1 1  Fsq ( s )  4 3    4  g 0  odd n n 1  s  n  

96 l w3 1 Fsq ( s )   4 g 03 1  s

sX ( s ) 

c

96 l w3 b  4 g 03

  3  96 l w   x  4 3  0  g0   

b 1

s

sX ( s )

c

 2 g 02 p0 c  12 w2

    1 1    4  odd n  n 1  s n  

For only. n 1

Transfer function for general displacement input!

Rb

f sq (t ), Fsq ( s )

x(t ), X ( s ) Damping coefficient

Cut‐off frequency

  sX ( s )   

C 1

bc


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What does it mean mechanically? 96 l w3 b  4 g 03

k

m x

8wp0 k sq  bc  2  g0

x

Thus, squeezed film effect creates two effects: Viscous damping + “air‐spring” Further analysis indicates that at low frequencies, damping dominates, and air‐spring at high frequencies. See S. D. Senturia, Microsystems Design, Kluwer, 2001, for details. G.K. Ananthasuresh, Indian Institute of Science

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Move up to beam modeling…   p ( x, y, t ) {g 0  u ( x, t )} 1    p ( x, y, t ) {g 0  u ( x, t )}3 p ( x, y, t ) t 12





 0 wV 2 (t ) d 4 u ( x, t ) u ( x, t ) w / 2  wt   p ( x, y, t ) dy  EI  0 2 2 4 dx t 2{g 0  u ( x, t )} w / 2 Solve these two coupled equations.

Note that this is still a parallel‐plate approxmation!

An approach Use FDM for pressure equation and FEM or FDM for discretizing the dynamic equation, and integrate in time using the Runge‐Kutta method. G.K. Ananthasuresh, Indian Institute of Science

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FD solution of Reynolds equation  ( Pg )  .( g 3 PP) 12 t 2 2 P g 3  P   P   Pg 2       t 12  x   y   12  

  2 P  2 P  3Pg  2  2  x  12  x

 P g P g  P P  x . x  y . y   g t  


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A typical beam’s response with squeezed film effect The transverse deflection of the mid‐point of a fixed‐fixed beam under (Vdc+Vac) voltage input under the squeezed film effect:

u mid  point

t


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What about this problem now? V

Diaphragm

Flow rate

Passive inlet valve

V sin t

Passive outlet valve

Frequency 

A problem involving three energy domains that are strongly coupled. Furthermore, the fluids part is non‐trivial. G.K. Ananthasuresh, Indian Institute of Science

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Main points 





MEMS are systems that tightly integrate many energetic phenomena, which makes their modeling non‐trivial. Coupled multi‐physics equations need to be solved. Reduced order lumped “macro” models are useful for design and system‐level simulation
