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EE 247B/ME 218: Introduction to MEMSModule 11: Equivalent Circuits I

CTN 3/16/17

Copyright © 2017 Regents of the University of California

EE C245: Introduction to MEMS Design LecM 11 C. Nguyen 11/6/08 1

EE C247B – ME C218Introduction to MEMS Design

Spring 2017

Prof. Clark T.-C. Nguyen

Dept. of Electrical Engineering & Computer SciencesUniversity of California at Berkeley

Berkeley, CA 94720

Lecture Module 11: Equivalent Circuits I


Lecture Outline

• Reading: Senturia, Chpt. 5

• Lecture Topics:Lumped MassLumped StiffnessLumped DampingLumped Mechanical Equivalent CircuitsElectromechanical Analogies

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Lumped Parameter Mechanical Equivalent Circuit


Equivalent Dynamic Mass

•Once the mode shape is known, the lumped parameter equivalent circuit can then be specified

• Determine the equivalent mass at a specific location x using knowledge of kinetic energy and velocity

Maximum Kinetic Energy

Maximum Velocity @ location x

Equivalent Mass =

Location x

Density

Maximum Velocity Function

z

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• For the folded-beam structure, we’ve already determined the maximum kinetic energy

• And in our resonance frequency analysis, we’ve already determined expressions for velocity


Equivalent Dynamic Stiffness & Damping

• Stiffness then follows directly from knowledge of mass and resonance frequency

• And damping also follows readily from knowledge of Q or other loss measurands

•With mass, stiffness, and damping lumped parameter equivalent circuit

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Get Potential Energy & Frequency

Folded-beam suspension

Anchor

Shuttle w/ mass Ms

h = thickness

Folding truss w/

mass Mt\2

Meq(shuttle) = 2.16x10-11 kg

Keq(shuttle) = 4.8 N/m

Ceq(shuttle) = 1.02x10-10 kg/s

Meq(truss) = 8.64x10-11 kg

Keq(truss) = 19.2 N/m

Ceq(truss) = 4.08x10-10 kg/s


Electromechanical Analogies

meq

keq

ceq

lx rxcx

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Electromechanical Analogies (cont)

•Mechanical-to-electrical correspondence in the current analogy:


Bandpass Biquad Transfer Function

meq

keq

ceq

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-60

-50

-40

-30

-20

-10

0

8.7 8.9 9.1 9.3Frequency [MHz]

Tra

ns

mis

sio

n [

dB

]

Pin=-20dBm

In Out

VP

Sharper roll-off

Sharper roll-off

Loss PoleLoss Pole

Performance:fo=9MHz, BW=20kHz, PBW=0.2%

I.L.=2.79dB, Stop. Rej.=51dB20dB S.F.=1.95, 40dB S.F.=6.45



Design:Lr=40m

Wr=6.5m hr=2m

Lc=3.5mLb=1.6m VP=10.47VP=-5dBm

RQi=RQo=12k

[S.-S. Li, Nguyen, FCS’05]

3CC 3/4 Bridged Mechanical Filter

[Li, et al., UFFCS’04]


Micromechanical Filter Circuit

1/krmr cr 1/krmr cr 1/krmr cr-1/ks -1/ks

1/ks

-1/ks -1/ks

1/ks

1/kb 1/kb

-1/kb

Co Co

1:e e:1

1:c 1:cc:1 c:1

1:b b:1

/4

/4

/4Input

Outputvi

RQ

RQ

vo

VP

Bridging Beam

Coupling Beam

Resonator

vovi

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1/ks

-1/ks -1/ks

1/ks

1/kb 1/kb

-1/kb

Co Co

1:e e:1

1:c 1:cc:1 c:1

1:b b:1

/4

/4

/4Input

Outputvi

RQ

RQ

vo

VP

Bridging Beam

Coupling Beam

Resonator

vovi




1/ks

-1/ks -1/ks

1/ks

1/kb 1/kb

-1/kb

Co Co

1:e e:1

1:c 1:cc:1 c:1

1:b b:1

/4

/4

/4Input

Outputvi

RQ

RQ

vo

VP

Bridging Beam

Coupling Beam

Resonator

vovi

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vovi



1/ks

-1/ks -1/ks

1/ks

1/kb 1/kb

-1/kb

Co Co

1:e e:1

1:c 1:cc:1 c:1

1:b b:1

/4

/4

/4Input

Outputvi

RQ

RQ

vo

VP

Bridging Beam

Coupling Beam

Resonator

All circuit element values determined

by CAD layout

All circuit element values determined

by CAD layout

Amenable to automated circuit

generation

Amenable to automated circuit

generation


-60

-50

-40

-30

-20

-10

0

8.7 8.9 9.1 9.3Frequency [MHz]

Tra

ns

mis

sio

n [

dB

]

Pin=-20dBm

In Out

VP

Sharper roll-off

Sharper roll-off

Loss PoleLoss Pole





Design:Lr=40m

Wr=6.5m hr=2m

Lc=3.5mLb=1.6m VP=10.47VP=-5dBm

RQi=RQo=12k

[S.-S. Li, Nguyen, FCS’05]

3CC 3/4 Bridged Mechanical Filter

[Li, et al., UFFCS’04]

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CTN 3/16/17



Beam Resonator Equivalent Circuits(Pretty Much the Same Stuff)



•Once the mode shape is known, the lumped parameter equivalent circuit can then be specified

• Determine the equivalent mass at a specific location x using knowledge of kinetic energy and velocity

h

W

Maximum Kinetic Energy

Maximum Velocity @ location x

Equivalent Mass =

Location x

Density

Maximum Velocity Function

z

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•We know the mode shape, so we can write expressions for displacement and velocity at resonance

h

W

Location xz


Equivalent Dynamic Stiffness & Damping

• Stiffness then follows directly from knowledge of mass and resonance frequency

• And damping also follows readily

h

W

Location xz

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Equivalent Lumped Mechanical Circuit

h

W

Location x

Meq(x)

Keq(x)

Ceq(x)

2

2

)]([

)]([)(

xu

xdxuAxM

l

oeq

)()( 2 xMxK eqoeq

Q

xMxC

eqoeq

)()(

z


Equivalent Lumped Mechanical Circuit

h

W

Meq(0) = 1.03x10-13 kg

Keq(0) = 19,927 N/m

Ceq(0) = 5.66x10-9 kg/s

Meq(l/2) = 2.78x10-13 kg

Keq(l/2) = 53,938 N/m

Ceq(l/2) = 1.53x10-8 kg/s

Meq(node) = ∞

Keq(node) = ∞

Ceq(node) = ∞

Example: Polysilicon w/ l=14.9m, W=6m, h=2m 70 MHz

z

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