Download Lecture28

EEL5225: Principles of MEMS Transducers (Fall 2003)
Instructor: Dr. Hui-Kai Xie
Lumped Modeling in Thermal Domain
;
Last lecture
z
z
Lumped modeling
Self-heating resistor
z Today:
Self-heating resistor
z Other dissipation mechanisms
z
z
z
Friction, dielectric, magnetic, damping
Coupled flows
Reading: Senturia, Chapter 11, p.278-296
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
Lecture 28 by
1
H.K. Xie 10/31/2003
Resistor Self-Heating
z
Temperature dependent
resistance
z
Voltage-Current relation, V=IR(T)
(Ohm’s Law)
z Now we consider resistor selfheating and temperature
dependence of resistance, R(T)
z
z
fI
+
e
-
V
R
Joule Heating, Pdissipation=VI=I2R >0
For small to moderate temperature
variation, approximate R(T) using
first order Taylor series expansion.
R (T ) ≅ R (T0 )[1 + α R (TR − T0 )]
where α R = temperature coefficient of resistance [1/K]
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
2
Resistor Self-Heating
z
Circuit model for resistor self-heating-Current Source
Thermal reservoir at temperature, T0
Governing equation is:
dTR
TR 2
=−
+ i R0 (1 + α RTR )
CT
dt
RT
i 2 R0
1
dTR
2
=−
1 − α R R0 RT i ) TR +
(
dt
RT CT
CT
10/31/2003
Ref. Senturia, Microsystem Design, p. 231.
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
3
Resistor Self-Heating
z
Circuit model for resistor self-heating
2
i
R0
dTR
1
2
R
R
i
T
=−
1
−
α
+
( R 0T )R C
dt
RT CT
T
i 2 R0
Linear first-order system with input,
.
CT
The time constant is:
RT CT
τi =
1 − α R R0 RT i 2
Steady-state temperature rise is given by
TSS ,i
R0 RT i 2
=
1 − α R R0 RT i 2
Example: Metal fuse
10/31/2003
i < (α R R0 RT )
−1/ 2
Ref. Senturia, Microsystem Design, p. 231.
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
4
Resistor Self-Heating
z
Circuit model for resistor self-heating-Voltage Source
Governing equation is:
dTR
TR
V2
=−
+
CT
dt
RT R0 (1 + α RTR )
Ref. Senturia, Microsystem Design, p. 232.
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
5
Resistor Self-Heating
z
Circuit model for resistor self-heating-Voltage Source
V2
V2
Since the T.C.R., α R , is small,
≅
(1 − α RTR )
R0 (1 + α RTR ) R0
 α RTRV 2  V 2
1 +
+
R0  R0

This linear first-order system has a time constant,
RT CT
and steady-state temperature rise,
τv =
2
α RV
1+ R T
R0
dTR
T
=− R
dt
RT CT
TSS ,V
10/31/2003
RT V 2 / R0
Note that this is stable with no singularities
=
2
α RV
for positive TCR.
1+ R T
R0
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
6
Resistor Self-Heating
z
Implanted resistor embedded in a thermally conducting
silicon substrate
z
Need to estimate lumped elements for this distributed system
Ref. Senturia,
Microsystem
Design, p. 234.
ρL
R=
Thermal capacitance:
CT = ρ mVCˆ p = ρ m ( aWL ) Cˆ p
A
=
L
aW σ e
Electrical resistance:
Cˆ p : specific heat per unit mass (J/(kg-Kelvin))
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
7
Resistor Self-Heating
z
Heat conduction to substrate
Begin with Fourier's law, J Q = −κ∇T .
Consider a semi-cylindrical boundary.
∂T
Q = J Q A(r ) = −κ (π rL )
neglecting end conduction effects.
∂r
TS
r0
Q dr
∫T dT = − ∫a κπ L r
R
Q
 r0 
TR − TS =
ln   = RT Q
κπ L  a 
where RThermal =
r 
ln  0  [Kelvin/Watts]
κπ L  a 
1
T=TS
Ref. Senturia, Microsystem Design, p. 234.
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
8
Resistor Self-Heating
z
Heat conduction to substrate
Geometry: L=300µ m, W=4µ m, a=2µ m, r0 = 10a
Silicon: n-type, N DD =1017 cm −3 , σ e = 15 S / cm,
α R = 2.5 ×10−3 K −1 , ρ m = 2330 kg / m3 ,
Cˆ p = 712 J / kg − K , κ = 148W / m − K
Lumped Elements:
ρL
L
=
= 2.5 × 104 Ω
R0 =
A aW σ e
CT = ρ mVCˆ p = ρ m ( aWL ) Cˆ p = 4 × 109 J / K
RThermal =
10/31/2003
r 
ln  0  = 16.5 K / W
κπ L  a 
1
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
Ref. Senturia, Microsystem
Design, p. 234.
9
Resistor Self-Heating
z
Heat conduction to substrate
ρL
L
= 2.5 ×104 Ω
A aW σ e
CT = ρ mVCˆ p = ρ m ( aWL ) Cˆ p = 4 × 10−9 J / K
R0 =
=
 r0 
RThermal =
ln   = 16.5 K / W
κπ L  a 
For current drive at low current,
1
τi =
RT CT
≅ RT CT
2
1 − α R R0 RT i
τ i = 16.5 K / W × 4 × 10−9 J / K = 66ns
TSS ,i
R0 RT i 2
=
0.04K if i = 0.3mA
2
1 − α R R0 RT i
Ref. Senturia, Microsystem Design, p. 236.
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
10
Resistor Self-Heating
z
Effect of Self-Heating in Resistive Transducers
z Example: Piezoresistor
z Two sources of resistance change
resistance change due to self-heating
resistance change due to piezoresistance
z
z
Maximum
∆R
≅ 1%. If we measure this to 1% accuracy,
R piezoresistance
we need to discern 1% of 1% or 1 part per 10,000 change in resistance.
⇒ Need
∆R
< 1 part per 10,000=10-4 (or 100 ppm).
R self-heating
From R(TR ) ≅ R0 [1 + α R (TR − T0 )],
∆R
10-4
10-4
-4
= α R (TR − T0 ) <10 . Therefore, TR − T0 <
=
= 0.04 K
−6
−1
R self-heating
α R 2500 ×10 K
This can be satisfied if i < 0.3mA (which corresponds to 7.5V across R(=25kΩ))
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
11
Other Dissipation Mechanisms
z Contact
Friction
z Dielectric Losses
z Viscoelastic Losses
z Magnetic Losses
z Fluid Viscosity Losses
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
12
Other Dissipation Mechanisms
z
Contact Friction
Retarding force, Fr that
opposes motion in response
to opposite tangential driving
force, F
z Related to normal force
pressing down
z Approximate model only,
evaluated via experiment
z
Ref. Senturia, Microsystem Design, p. 237.
Linear model of contact friction → Fr ∝ Fn
Fr = µ f Fn where 0 ≤ µ f ≤ 1
Irreversible process, power
dissipated as heat energy
The retarding force is present only when there is relative motion, x.
Fr = bx. Modeled as resistor with value, b.
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
13
Other Dissipation Mechanisms
z
Contact Friction
Depends on surface roughness,
velocity, and initial condition
z Static friction > Sliding friction
z Non-linear friction (Coulomb friction)
z
z
z
z
Threshold tangential force before motion
begins
One key originating force is Coulomb
attraction between charged surfaces
Ref. Senturia,
Microsystem
Design, p. 238.
Internal Friction
z
Viscoelastic effects
z
z
10/31/2003
Internal retarding forces that oppose deformation
Internal friction produces heat, heat flow, entropy
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
14
Other Dissipation Mechanisms
z
Dielectric Losses
z
z
z
Joule heating due to small
conductivity
Internal friction that retards
orientation of dipoles
Since polarization requires
displacement of charged
particles with an inertial mass,
the polarization is frequency
dependent. So, a phase delay
occurs.
Relevant forces: applied electric field,
retarding Coulomb attraction in dipole,
internal friction
G
G
D = (ε ′ + jε ′′) E
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
Ref. Jordan and Balmain,
Electromagnetic Waves and
Radiating Systems, p. 307.
15
Other Dissipation Mechanisms
z
Dielectric Losses
Consider Ampere's Law for a dielectric with finite conductivity, σ e .
K
K
'
"
K K
K ∂ ( ε r + jε r ) ε 0 E
∂D
∇× H = J +
= σeE +
∂t
∂t
Assuming sinusoidal steady-state, e − jωt :
K K
K
K
K
 '
σ e  K
 "
'
"
∇ × H = σ e E − jωε r E + ωε r E = − jω ε r ε 0 + j  ε r ε 0 +   E
ω 


Lumped element equivalent circuit for a parallel plate capacitor:
C=
ε r' ε 0 A
g
R=
(σ
g
"
ωε
+
rε 0 ) A
e
ε = ε r ε 0 where ε 0 = vacuum permitivity=8.854 ×10-14 F / cm
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
16
Other Dissipation Mechanisms
z
Magnetic Losses
I2R Joule heating in finite coil
resistance (inductor or
transformer winding)
z Eddy currents induced in
conducting magnetic core (and
also winding conductors)
K
K K
∂B
Consider Faraday's Law, ∇ × E = −
∂t
The time-varying magnetic flux gives rise to
a voltage (opposing flux change) that induces
electrical currents in the magnetic core.
z
Ref. Erickson, Fundamentals of
Power Electronics, p. 472.
These 'eddy currents' cause i 2 R heating of the core.
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
17
Other Dissipation Mechanisms
z
Magnetic Hysteresis
z
Internal friction due to magnetic domain wall motion when magnetic
flux density attempts to align with magnetic field, H.
z
z
z
z
z
Coercive force, HC, before alignment occurs similar to Coulomb friction.
Remanent flux density, BR, exists at H = 0.
Magnetically hard materials: large HC, such as Sm-Co alloy.
Magnetically soft materials: small HC, such nickel-iron alloy.
The internal area of a hysteresis loop is a measure of the energy lost due
to magnetic hysteresis.
10/31/2003
Ref. Senturia, Microsystem Design, p. 241.
18
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
Coupled Flows
z
Coupled flows:
z
z
z
Temperature gradient and concentration gradient
Simultaneous heat flow and diffusion
Seebeck effect and thermocouples
G
αs: Seeback coefficient
E = α s ∇T
αs,1
TH
TV
αs,2
TC
TH
TV
TV
TC
TH
V
TC
VTC = ∫ α s ,1 dT + ∫ α s ,2 dT + ∫ α s ,1 dT
=
TH
∫ (α
s ,1
− α s ,2 ) dT = (α s ,1 − α s ,2 ) (TH − TC )
TC
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
19
Coupled Flows
z
Peltier effect and Refrigerators
Q = ∏ It
cold
hot
Q = I ( ∏ n − ∏ p ) N
Peltier effect is the converse of Seebeck effect.
The two coefficients have the following relation,
∏ = Tα s
10/31/2003
EEL5225: Principles of MEMS Transducers (Fall 2003) Dr. Xie
20