INF5490 RF MEMS
L3: Modeling, design and analysis
S2008, Oddvar Søråsen
Department of Informatics, UiO
1
Today’s lecture
• MEMS - function
– Transducer principles
– Sensor principles
• Methods for RF MEMS modeling
– 1. Simple mathematical models
– 2. Convert to electrical equivalents
– (3. Analyzing using Finite Element Methods)
• Æ L4
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Transducers for (RF) MEMS
• Electromechanical transducers
– Transforming
electrical energy ÅÆ mechanical energy
• Transducer principles
– Electrostatic
– Electromagnetic
– Electro thermal
– Piezoelectric
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Transducer principles
• Electrostatic transducers
– Principle: Forces between electric charges
• ”Coulombs law”
– Stored energy when mechanical or electrical work is performed
on the unit can be converted to the other form of energy
– The most used form of electromechanical energy conversion
– Fabrication is simple
– Often implemented using a capacitor with movable plates
• Vertical movement: parallel plates
• Horizontal movement: Comb structures
4
Electrostatic transducers
• + Beneficial due to simplicity
• + Actuation controlled by voltage
• voltage Æ charge Æ attractive force Æ movement
• + Movement gives current
• movement Æ variable capacitor Æ current when voltage is
constant
• ÷ Need environmental protection (dust)
• Packaging required (vacuum)
• ÷ Transduction mechanism is non-linear
• Gives distortions (force is not proportional to voltage)
• Solution: small signal variations around a DC voltage
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Transducer principles, contd.
• Electromagnetic transducers
– Magnetic winding pulls the element
• Electro thermal actuator
– Different thermal expansion on different
locations due to temperature gradients
• Large deflections can be obtained
• Slow
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Transducer principles, contd.
• Piezoelectric transducers
– In some anisotropic crystalline materials the charges
will be displaced when stressed Æ electric field
• stress = ”mechanical stress”
– Similarly, strain results when an electric field is
applied
• strain = ”mechanical strain”
– Ex. PZT (lead zirconate titanates) – ceramic materials
•
(Electrostrictive transducers
–
•
Mechanical deformation by electric field
Magnetostrictive transducers
–
Deformation by magnetic field)
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Comparing different principles
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Sensor principles
• Piezoresistive detection
• Capacitive detection
• Piezoelectric detection
• Resonance detection
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Sensor principles
• Piezoresistive detection
– Resistance varies due to external
pressure/stress
– Used in pressure sensors
• deflection of membrane
– Piezo-resistors placed on membrane where
strain is maximum
– Resistor value is proportional to strain
– Performance of piezoresistive micro sensors
are temperature dependent
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Pressure sensor
Senturia
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Sensor principles, contd.
• Capacitive detection
– Exploiting capacitance variations
– Pressure Æ electric signal
• Detected by change in oscillation
frequency, charge, voltage (V)
– Potentially higher performance
than piezoresistive detection
• + Better sensitivity
• + Can detect small pressure
variations
• + High stability
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Sensor principles, contd.
• Piezoelectric detection
– Electric charge distribution changed due to
external force Æ electric field Æ current
• Resonance detection
– Analogy: stress variation on a string gives
strain and is changing the “natural”
resonance frequency
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Methods for modeling RF MEMS
• 1. Simple mathematical models
– Ex. parallel plate capacitor
• 2. Converting to electrical equivalents
• 3. Analysis using Finite Element
Methods
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1. Simple mathematical models
• Equations, formulas describing physical
phenomena
– Simplification, approximations
– Explicit solutions for simple problems
• linearization around a bias point
– Numerical solution of the set of equations
• Typical differential equations
• + Gives the designer insight/ understanding
– How the performance changes by parameter
variations
– May be used for initial estimates
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Ex. On mathematical modeling
• Important equatins for many RF MEMS
components:
– Æ Parallel plate capacitor!
– Electrostatic actuation of the capacitor with
one spring-suspended plate
– Calculating ”pull-in”
• Formulas and figures Æ
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Electrostatics
Electric force between charges: Coulombs law
+q
-q
F=
F
1
4πε 0
⋅
q1q2
r2
r
Electric field = force pr. unit charge
E=
F
q0
b
Work done by a force = change in potential energy
Wa →b = ∫ F ⋅ d l = U a − U b
a
Potential, V = potential energy pr. unit charge
Voltage = potential difference
V=
U
q0
b
Va − Vb = ∫ E ⋅ d l
a
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Capacitance
+Q
Definition of capacitance
a
E
b
A
C=
Vab
d
Q
Vab
-Q
Surface charge density = σ
σ Q 1
E= = ⋅
ε0 A ε0
C=
Voltage
Vab = E ⋅ d =
Q
A
= ε0
Vab
d
Energy stored in a capacitor, C,
that is charged to a voltage V0 at a current
Q
⋅d
Aε 0
dV
i = Q& = C
dt
0
ε0 A 2
dv
1
2
U = ∫ v ⋅ i ⋅ dt = ∫ v ⋅ C ⋅dt = C ∫ v ⋅ dv = CV0 =
V0
dt
2
2d
0
V
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Parallel plate capacitor
Attractive force between plates
εAV 2
∂ εA 2
∂U
=− ( V )=
F =−
2d 2
∂d 2d
∂d
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Movable capacitor plate
• Assumptions for calculations:
– Suppose air between plates
– Spring attached to upper plate
• Spring constant: k
– Voltage is turned on
• Electrostatic attraction
– At equilibrium
• Forces up and forces down are in balance Æ
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Force balance
k = spring constant
deflection from start position
d0 = gap at 0V and zero spring strain
d = d0 – z
z=d0 – d
Force on upper plate at V and d:
= 0 at equilibrium
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Two equilibrium positions
ς = 1 – d/d0
Senturia
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Stability
• How the forces develop when d decreases
– Suppose a small perturbation in the gap at constant voltage
Suppose the gap decreases
If the upward force also deceases,
the system is UNSTABLE!
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Stability, contd.
Stability condition:
Pull-in when:
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Pull-in
Pull-in when:
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Pull-in
Senturia
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2. Converting to electrical
equivalents
• Mechanical behavior can be modeled using
electrical circuit elements
– Mechanical structure Æ simplifications Æ equivalent
electrical circuit
• ex. spring/mass Æ R, C, L
– Possible to “interconnect” electrical and mechanical
energy domains
• Simplified modeling and co-simulation of electronic and
mechanical parts of the system
– Proper analysis-tools can be used
• Ex. SPICE
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Converting to electrical equivalents,
contd.
• We will discuss:
– Needed circuit theory
– Conversion principles
• effort - flow
– Example of conversion
• Mechanical resonator
– In a future lecture:
• Co-existence and coupling between various
energy domains
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Circuit theory
• Basic circuit elements: R, C, L
• Current and voltage equations for basic
elements (low frequency)
– Ohms law, C and L-equations
• V = RI, I = C dV/dt, V = L dI/dt
– Laplace transformation
• From differential equations to algebraic (s-polynomial)
• Æ Complex impedances: R, 1/sC, sL
• Kirchhoffs equations
– Σ current into nodes = 0, Σ voltage in a loop = 0
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Effort - flow
•
Electrical circuits are described by a set of variables:
conjugate power variables
– Voltage V: across or effort variable
– Current I: through or flow variable
– An effort variable drives a flow variable through an impedance, Z
•
Circuit element is modeled as a
1-port with terminals
– Same current (f = flow) in
and out and through the element
• Positive flow into a terminal defining a positive effort
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Energy-domains, analogies
• Various energy domains exist
– Electric, elastic, thermal, for liquids etc.
• For every energy domain it is possible to
define a set of conjugate power variables
that may be used as basis for lumped
component modeling using equivalent
circuits elements
• Table 5.1 Senturia ->
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Ex. of conjugate power variables
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Conjugate power variables: e,f
• Assume conversion between energy domains
were the energy is conserved!
• Properties
– e * f = power
– e / f = impedance
• Generalized displacement represents the
state, f. ex. position or charge
f (t ) = q& (t )
– e * q = energy
t
q(t ) = ∫ f (t )dt + q(t0 )
t0
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Generalized momentum
t
p(t ) = ∫ e(t )dt + p(t0 )
t0
– Mechanics: impulse
• F*dt = mv – mv0
– General: p * f = energy
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Ex.: Mechanical energy domain
force
velocity
position
momentum
force x time
work/time = power
force*distance = work =
energy
energy
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Ex.: Electrical energy domain
voltage
current
charge
power
energy
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e Æ V - convention
• Senturia and Tilmans use the
eÆV –convention
• Ex. electrical and mechanical circuits
–
–
–
–
e Æ V (voltage)
equivalent to F (force)
f Æ I (current)
equivalent to v (velocity)
qÆ Q (charge)
equivalent to x (position)
e * f = ”power” injected into the element
H. Tilmans, Equivalent circuit representation of electromagnetical transducers:
I. Lumped-parameter systems, J. Micromech. Microeng., Vol. 6, pp 157-176, 1996
37
Other conventions
• Different conventions exist for defining throughor across-variables
*
alternativt
38
Generalized circuit elements
• One-port circuit elements
– R, dissipating element
– C, L, energy-storing elements
– Elements can have a general function!
• Can be used in various energy domains
39
Generalized capacitance
Compare with a simplified case:
- a linear capacitor
definisjon av C
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Generalized capacitance, contd.
Capacitance is associated with stored potential energy
Co-energy:
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Energy stored in parallel plate capacitor
Energy:
Co-energy:
W * (V ) = W (Q )
Q
Q
q
Q2
W (Q ) = ∫ e ⋅ dq = ∫ ⋅ dq =
C
2C
0
0
V
V
CV 2
W (V ) = ∫ q ⋅ de = ∫ C ⋅ v ⋅ dv =
2
0
0
*
for linear capacitance
42
Mechanical spring
Hook´s law:
F =k⋅x
Stored energy
W (Q ) =
Compare with capacitor
Q
x1
Æ 1/C equivalent to k
1 1 2
⋅ ⋅Q
2 C
displacement
displacement
43
”Compliance”
• ”Compliance” = ”inverse stiffness”
Cspring
1
=
k
• Stiff spring Æ small capacitor
• Soft spring Æ large capacitor
44
Generalized inductance
Energy also defined as:
Energy = stored kinetic energy
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Ex.: Electrical inductor
Co-energy:
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Analogy between mass
(mechanical inertance) and inductance L
A mechanical system has linear momentum: p = mv
Flow:
Co-energy:
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Analogy between m and L
Compare with:
L is equivalent to m
m=L
inertance
Mechanical inertance = mass m
is analog to inductance L
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Interconnecting elements
• e Æ V follows two basic principles
– Elements that share a common flow , and
hence a common variation of displacement,
are connected in series
– Elements that share a common effort are
connected in parallel
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Ex. of interconnection:
”Direct transformation”
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System without damping (b=0, R=0)
ω0 =
k
1
, ω0 =
LC
m
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System without damping, contd.
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With damping
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Damped system, contd.
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What is the meaning of ”damping time”?
Power
initial conditions
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Q-factor and damping time
General equation
mechanical
electrical
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Amplitude at resonance for forced vibrations
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