Download Lecture 15 RF MEMS

EEL6935 Advanced MEMS (Spring 2005)
Instructor: Dr. Huikai Xie
Lecture 15
RF MEMS (3)
„
Agenda:
Ê
Ê
S-Parameters
Mechanical Modeling of RF MEMS Devices
– Dynamic Analysis
Ê
Electromagnetic Modeling
Most figures and data in this lecture, unless cited otherwise, were taken from RF MEMS Theory, Design and Technology by G. Rebeiz.
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S-Parameters
„
Y and Z parameter models use
input and output voltage and
current signals.
„
Scattering Parameters, or Sparameters have inputs and
outputs expressed in power.
„
a is assigned to incident values
while b indicates reflected values.
„
S-parameters are transmission
and reflection coefficients.
„
Transmission coefficients are
referred to as gains and
attenuations
„
Reflection coefficients are related
to voltage standing wave ratios
(VSWRs) and Impedances.
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S-Parameters
S21
a1
S11
b1
S22
b2
b1 = S11a1 + S12 a2
a2
b2 = S 21a1 + S 22 a2
S12
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S-Parameters
S21
a1
b1
S11
S22
b2
a2
S12
reflection coefficient Γ = S11
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Smith Chart
4
Dynamic Analysis of MEMS Switches
‰ Damping
‰ Switching time
‰ Release time
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Damping
‰ Gas Fundamentals
λ =
Mean-free path
1
2π N σ
2
N is the number density of gas. σ: gas molecule’s diameter.
λ0=0.07-0.09µm for most gases at STP (standard temperature and
pressure, i.e., P0=101 kPa at room temperature).
For a gas at a pressure Pa, the mean free path can be obtained by
λa =
P0
λ0
Pa
For example, if Pa = 1 mtorr = 7.5 Pa, then λa = 1mm, which is much
greater than the gaps of MEMS devices.
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Damping
‰ Gas Fundamentals
Kn =
Knudsen Number
λ
where g: air gap.
g
Knudsen number is a measure of the viscosity of the gas. The smaller
the Knudsen number, the more viscous the gas or fluid is.
Coefficient of viscosity
µ = 1.2566 × 10
−6
110.33 

T 1 +

T


−1
kg/m ⋅ s
T is in Kelvin and this equation is for ideal and quasi-ideal gases.
At STP, µ = 1.845 ×10-5 kg/(m·s).
Viscosity changes with Knudsen number. Veijola et al derived the
dependence:
µ
µe =
EEL6935 Advanced MEMS
1 + 9.638 K n1.159
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Damping
‰ Gas Fundamentals
Squeeze Number
σ (ω ) =
12 µel 2
ω
Pa g 2
where ω is the applied mechanical frequency.
l: the characteristic length. For a circular
membrane, l is the radius. For a rectangle, l is
the shortest dimension.
Squeeze-film damping Æ b
Gas compression Æ spring force Æ ka
At low frequencies ka ≈ 0. b =
3 µ A2
2π g 03
b
ka
At high frequencies (i.e., high squeeze number),
ka Æ PaA/g
High squeeze number will increase the effective
stiffness and thus the resonance frequency.
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Damping
‰ Quality Factor at Atmospheric Pressure
Fixed-fixed beam
Cantilever Beam
Qcant =
Eρt2
µ ( wl )
2
Q ff =
g 03
where w and l are the width
and length of the cantilever.
Eρt2
µ ( wl / 2 )
2
g 03
Given l=300um, w=60um, t=1um
and g0=3um Æ Q=1.0
At very low pressures, µÆ0. The damping coefficient is limited by anchor
loss, internal friction loss and thermoelastic dissipation (TED).
‰ Damping Variation Versus Gap Height
2

 x  

Qe = Q 1.1 −   

 g 0  

3/ 2
(1 + 9.638K )
1.159
n
Q is small-displacement quality factor at g=g0. Derived by Sadd and Siffler
(Trans. ASME, pp.1366-1370, Nov. 1975)
2005 H. Xie
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Nonlinear Dynamic Analysis
‰ Switching time
Equation of Motion
mx + bx + kx + k s x3 = Fe + Fc
where Fe =
ε 0 AV 2
1
2 ( g 0 + td / ε r − x ) 2
dC 
 dV
V = Vs −  C
+V
R
dt
dt 

Fc =
C1 A
( g0 − x )
3
Attractive van der
Waals force
−
C2 A
( g0 − x )
10
C1 and C2 strongly depends
on surface materials and
conditions. In this analysis,
C1=10-80Nm; C2=10-75Nm8
Repulsive nuclear
contact force
Chan et al, IEEE MTT-S Digest, 1997
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Nonlinear Dynamic Analysis
Parameters of the MEMS Beam for this analysis
Table 3.1
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Switching Time
‰ Switching time of fixed-fixed beams
Matlab/Simulink can be used to obtain the transient response
Resonance frequencies:
• Al beam: 106 kHz
• Au beam: 39.5 kHz
Applied voltage:
Vs = 1.4 Vp
• Constant voltage
• Displacement-dependent damping factor
• Higher Q makes switching faster, but little effect above Q=2
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Switching Time
‰ Switching time varying with applied voltage
• Au beam; Resonance frequency: f0 = 39.5 kHz
• Each cycle: 25 µs; Q = 1
• For Q > 2 (Acceleration-limited), switch time ts ≈ 0.58 (Vp/Vs)/f0
• For Q < 0.5 (Damping-limited), ts ≈ 0.36~1.0 (Vp/Vs)2 /(Qf0)
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Release Time
• Release response can be obtained by setting Fe=0 in the
nonlinear dynamic equation.
• The bridge oscillates if Q>1
• For best release response, Q≈1
• Variable damping has greater effect
• Au beam
• Resonance frequency:
f0 = 39.5 kHz
• Each cycle: 25 µs
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Electromagnetic Modeling
‰ Capacitive Shunt Switches
Ê Circuit model
Ê Current distribution
Ê Series Resistance
Ê Loss
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Capacitive Shunt Switches
‰ Top view
‰ Cross-sectional View
Ground
Signal
l
G: 50-100 µm
W: 50-100 µm
L: 250-400 µm
w: 25-180 µm
g: 1.5-5 µm
Ground
Coplanar Waveguide
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Capacitive Shunt Switches
‰ Current Distribution
ƒ Up-state Position
ƒ Down-state Position
• No RF current on the middle portion of the bridge
• RF current is carried only at the edge of the t-line
• RF current is concentrated on the bridge edge over the CPW gap.
• Changing the bridge width does not affect the current distribution.
• For down-state position, only one edge of the bridge carries the
current. This edge is a short circuit to the incoming wave.
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Capacitive Shunt Switches
‰ Circuit Model
Shunt Impedance
1/ jωC for f f 0
1

=  Rs
Z s = Rs + jω L +
for f = f 0
jω C 
 jω L for f f 0
•
•
•
•
LC Resonant Frequency
f0 =
1
2π LC
C = Cu or Cd depending on the switch position.
Typically, f0,u ~300GHz and f0,d ~50GHz.
For up-state position, the switch can be modeled as a shunt capacitance to ground.
The inductance plays important role for the down-state position.
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Capacitive Shunt Switches
‰ Circuit Model
Series Resistance
Rs1: 0.01~0.1Ω
α: t-line loss, 0.2-2dB/cm (or 0.17-1.7 Np/m)
• If the bridge thickness is smaller than two skin depths,
the switch resistance is constant with frequency.
• For thick MEMS bridges, the switch resistance changes
as (f)1/2 with frequency.
Inductance
L: 1-100 pH
Switches with meander suspensions for small spring
constants have large inductance.
Simulators: IE3D, Sonnet, HFSS
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Capacitive Shunt Switches
‰ Loss
2
Loss = 1 − S11 − S 21
Loss =
2
I s2 Rs
Power loss in MEMS Bridge
=
Power incident on MEMS switch V + 2 / Z
0
Loss ( dB ) = 10 log (1 − Loss )
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Capacitive Shunt Switches
‰ Loss
Up-state Position: Zs >> Z0
Loss =
Rs Z 0
Zs
Total Loss
2
= ω 2 Cu2 Rs Z 0
Lossu ( dB ) = α l (dB ) + ω 2 Cu2 Rs Z 0 (dB )
(~0.05 dB)
Down-state Position: Zs << Z0
4 Rs Z 0
Loss =
Z0
Total Loss
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≈
4 Rs
Z0
Lossd ( dB ) = α l (dB ) +
4 Rs
(dB )
Z0
(~0.15 dB)
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Summary
‰ Introduction to S-Parameters
‰ Mean-free path at STP: 0.07 µm
‰ Knudsen number < 0.1 Æ Viscous
‰ Switching time
‰ Nonlinear equation of motion
‰ Matlab/Simulink can be used to obtain transient response
‰ Increasing drive voltages decreases switching time
‰ Higher Q Æ faster switching
‰ Release time
‰ High Q leads to oscillation
‰ Optimal Q ≅ 1
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