Lumped-Element System Dynamics
Joel Voldman*
Massachusetts Institute of Technology
*(with thanks to SDS)
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 1
Outline
> Our progress so far
> Formulating state equations
> Quasistatic analysis
> Large-signal analysis
> Small-signal analysis
> Addendum: Review of 2nd-order system dynamics
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 2
Our progress so far…
> Our goal has been to model multi-domain systems
> We first learned to create lumped models for each
domain
> Then we figured out how to move energy between
domains
> Now we want to see how the multi-domain system
behaves over time (or frequency)
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 3
Our progress so far…
> The Northeastern/ADI RF Switch
> We first lumped the mechanical domain
Fixed support
Images removed due to copyright restrictions.
Figure 11 on p. 342 in: Zavracky, P. M., N. E. McGruer, R. H. Morrison,
and D. Potter. "Microswitches and Microrelays with a View Toward Microwave
Applications." International Journal of RF and Microwave Comput-Aided
Engineering 9, no. 4 (1999): 338-347.
I
Dashpot b
+
V
1 µm
Silicon
0.5 µm
-
Cantilever
Pull-down
electrode
Anchor
Spring k
Mass m
g
z
Fixed plate
Image by MIT OpenCourseWare.
Adapted from Figure 6.9 in Senturia, Stephen D. Microsystem
Design. Boston, MA: Kluwer Academic Publishers, 2001, p.
138. ISBN: 9780792372462.
Image by MIT OpenCourseWare.
Adapted from Rebeiz, Gabriel M. RF MEMS: Theory, Design, and
Technology. Hoboken, NJ: John Wiley, 2003. ISBN: 9780471201694.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 4
Our progress so far…
> Then we introduced a two-port capacitor to convert
energy between domains
• Capacitor because it stores potential energy
• Two ports because there are two ways to store energy
» Mechanical: Move plates (with charge on plates)
» Electrical: Add charge (with plates apart)
• The system is conservative: system energy only depends on
state variables
.
g
I
2
Q g
W (Q, g ) =
2εA
+
V
-
1/k
+
C
m
F
b
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 5
Our progress so far…
∂W (Q, g )
Q2
=
F=
∂g
2εA
Q
> We first analyzed system
quasistatically
> Saw that there is VERY different
V=
∂W (Q, g )
Qg
=
∂Q
εA
g
behavior depending on whether
• Charge is controlled
Î stable behavior at all gaps
• Voltage is controlled
dW = VdQ + Fdg
dW * = QdV − Fdg
Î pull-in at g=2/3g0
> Use of energy or co-energy
depends on what is controlled
• Simplifies math
∂W *
Q=
∂V
=
g
εA
g
Vin
εAVin2
∂W *
=
F=
∂g V
2g 2
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 6
Today’s goal
> How to move from quasi-static to dynamic analysis
> Specific questions:
• How fast will RF switch close?
> General questions:
• How do we model the dynamics of non-linear systems?
• How are mechanical dynamics affected by electrical domain?
> What are the different ways to get from model to
answer?
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 7
Outline
> Our progress so far
> Formulating state equations
> Quasistatic analysis
> Large-signal analysis
> Small-signal analysis
> Addendum: Review of 2nd-order system dynamics
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 8
Adding dynamics
Fixed support
> Add components to complete
Resistor
R
the system:
Dashpot b
Mass m
Spring k
+
• Source resistor for the voltage
•
I
Vin
source
Inertial mass, dashpot
+
V
-
g
z
-
Fixed plate
> This is now our RF switch!
> System is nonlinear, so we
can’t use Laplace to get
transfer functions
> Instead, model with state
g
I
+
Vin
-
R
+
V
-
C
1/k
z
+
m
F
W(Q,g)
b
equations
Image by MIT OpenCourseWare.
Adapted from Figure 6.9 in Senturia, Stephen D. Microsystem Design. Boston,
MA: Kluwer Academic Publishers, 2001, p. 138. ISBN: 9780792372462.
Electrical domain
Mechanical domain
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 9
State Equations
> Dynamic equations for general
system (linear or nonlinear) can be
formulated by solving equivalent
circuit
> In general, there is one state variable
for each independent energy-storage
element (port)
> Good choices for state variables:
the charge on a capacitor
(displacement) and the current in an
inductor (momentum)
Goal:
⎡Q ⎤
⎞
d ⎢ ⎥ ⎛ functions of
⎟⎟
g ⎥ = ⎜⎜
⎢
Q,g,g or constants ⎠
dt
⎝
⎢⎣ g ⎥⎦
> For electrostatic transducer, need
three state variables
• Two for transducer (Q,g)
• One for mass (dg/dt)
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 10
Formulating state equations
> Start with Q
+ eR - I
> We know that dQ/dt=I
> Find relation between I and
state variables and constants
KVL : Vin − eR − V = 0
Vin − IR − V = 0
dQ
1
= I = (Vin − V )
dt
R
dQ 1 ⎛
Qg ⎞
= ⎜ Vin −
dt R ⎝
ε A ⎟⎠
Vin
+
-
R
+
C
V
W(Q,g)
-
eR = IR
Qg
V=
εA
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 11
Formulating state equations
> Now we’ll do
g
> We know that dg
.
g
dt
KVL :
F − ek − em − eb = 0
F − kz − mz − bz = 0
= g
ek = kz
em = mz
eb = bz
z = g 0 − g ⇒ z = − g , z = − g
.
z
1/k
C
+
+ ek - +
W(Q,g)
F
em
-
m
- eb + b
F − k ( g 0 − g ) + mg + bg = 0
1
[F − k ( g 0 − g ) + bg ]
m
⎤
dg
1 ⎡ Q2
=− ⎢
− k ( g 0 − g ) + bg ⎥
dt
m ⎣ 2εA
⎦
g = −
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 12
Formulating state equations
> State equation for g is easy:
dg
= g
dt
> Collect all three nonlinear state equations
⎡
⎤
1⎛
Qg ⎞
Vin −
⎢
⎥
⎜
⎟
R⎝
εA⎠
⎡Q ⎤ ⎢
⎥
d ⎢ ⎥ ⎢
⎥
g
g
=
⎥
dt ⎢ ⎥ ⎢
⎢⎣ g ⎥⎦ ⎢ 1 ⎡ Q 2
⎤⎥
⎢ − m ⎢ 2ε A − k ( g 0 − g ) + bg ⎥ ⎥
⎣
⎦⎦
⎣
> Now we are ready to simulate dynamics
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 13
Outline
> Our progress so far
> Formulating state equations
> Quasistatic analysis
> Large-signal analysis
> Small-signal analysis
> Addendum: Review of 2nd-order system dynamics
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 14
Quasistatic analysis
⎡1 ⎛
⎤
Qg ⎞
⎟
⎢ ⎜Vin −
⎥
Q
ε
A
R
⎡
⎤
⎝
⎠
⎢
⎥
d
⎢ g ⎥ = ⎢ g
⎥
dt ⎢⎣ g ⎥⎦ ⎢ 1 ⎡ Q 2
⎤⎥
k
g
g
b
g
(
)
−
−
−
+
0
⎥⎥
⎢ m ⎢ 2εA
⎦⎦
⎣
⎣
Fixed-point
analysis
Given static Vin, etc.
State eqns
What is deflection,
charge, etc.?
(Just now)
.
g
I
Vin
+
-
R
+
C
V
W(Q,g)
-
.
1/k
z
+
m
F
-
Follow
causal path
(Wed)
b
Equivalent circuit
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 15
State equations
For transverse
electrostatic actuator:
State variables
Inputs
x = f (x, u)
y = g (x, u)
Outputs
State variables:
⎡Q ⎤
x = ⎢g ⎥
⎢⎣ g ⎥⎦
Inputs:
u = [Vin ]
⎡1 ⎛
⎤
Qg ⎞
⎢ ⎜ Vin −
⎥
⎟
εA⎠
⎢R ⎝
⎥
⎥
f (x,u) = ⎢ g
⎢
⎥
2
⎤⎥
⎢ 1⎡Q
⎢ − m ⎢ 2ε A − k ( g 0 − g ) + bg ⎥ ⎥
⎣
⎦⎦
⎣
Outputs: y = [g ] = g (x, u)
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 16
Fixed points
> Definition of a fixed point
• Solution of f(x,u) = 0
• time derivatives Æ 0
> Global fixed point
• A fixed point when u = 0
• Systems can have multiple global fixed points
• Some might be stable, others unstable (consider a
pendulum)
> Operating point
• Fixed point when u is a non-zero constant
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 17
Fixed points of the electrostatic actuator
> This analysis is analogous to what we did last time…
Qg ⎞
1⎛
0 = ⎜Vin −
⎟
R⎝
εA ⎠
0 = g
⎤
1 ⎡ Q2
0=− ⎢
− k ( g 0 − g ) + bg ⎥
m ⎣ 2εA
⎦
Qg
εA
Q 2 Vin 2ε A
=
= k ( g0 − g )
2
2ε A
2g
Vin =
normalized gap
Operating point
stable
1
0.5
0
0
0.5
1
normalized voltage
Last time…
g = g0 −
εAVin2
2kg 2
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 18
Outline
> Our progress so far
> Formulating state equations
> Quasistatic analysis
> Large-signal analysis
> Small-signal analysis
> Addendum: Review of 2nd-order system dynamics
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 19
Large-signal analysis
⎡1 ⎛
⎤
Qg ⎞
⎟
⎢ ⎜Vin −
⎥
Q
ε
A
R
⎡
⎤
⎝
⎠
⎢
⎥
d
⎢ g ⎥ = ⎢ g
⎥
dt ⎢⎣ g ⎥⎦ ⎢ 1 ⎡ Q 2
⎤⎥
k
g
g
b
g
(
)
−
−
−
+
0
⎥⎥
⎢ m ⎢ 2εA
⎦⎦
⎣
⎣
Integrate
state eqns
Given a step input
Vin(t)u(t)
State eqns
(Earlier)
.
g
I
Vin
+
-
R
+
C
V
W(Q,g)
-
What is g(t), Q(t), etc.?
.
1/k
z
SPICE
+
m
F
b
Equivalent circuit
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 20
Direct Integration
> This is a brute force approach: integrate the state
equations
• Via MATLAB® (ODExx)
• Via Simulink®
> We show the SIMULINK® version here
• Matlab® version later
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 21
Electrostatic actuator in Simulink®
⎡Q ⎤
x = ⎢⎢ g ⎥⎥, u = [Vin ]
⎢⎣ g ⎥⎦
⎡1 ⎛
⎤
Qg ⎞
V
−
⎜
⎟
⎢
⎥
in
εA ⎠
⎢R ⎝
⎥
⎥
f(x, u) = ⎢ g
⎢
⎥
2
⎡
⎤
⎢ 1 Q
⎥
⎢− m ⎢ 2εA − k ( g 0 − g ) + bg ⎥ ⎥
⎣
⎦⎦
⎣
(u[1]^2)/(2*e*A)
+
+
+
Electrostatic force
Spring
Sum2
k
_
b
-1/m
Inertia
Damping
1
s
3
Velocity
gdot
1
s
+
2
Position
At-rest gap g _ 0
1/(e*A)
g
*
Qg
_
1
V_in
+
1/R
1
s
1
1/R
Charge
Q
Image by MIT OpenCourseWare.
Adapted from Figure 7.8 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer
Academic Publishers, 2001, p. 174. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 22
Electrostatic actuator with contact
<
Accel > 0 ?
+ +
g_min
>
g > g_min?
+
+
+
(u[1]^2)/(2*e*A)
Electrostatic force
-1/m
1
s
Inertia
Switch
Spring
Sum2
k
_
b
Velocity
0
Damping
gdot
Zero
1
s
+
2
Position
At-rest gap g _ 0
1/(e*A)
3
g
*
Qg
_
1
V_in
+
1/R
1/R
1
s
Charge
1
Q
Image by MIT OpenCourseWare.
Adapted from Figure 7.9 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2001,
p. 175. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 23
Behavior through pull-in
7 10 8
1.6
Drive (scaled)
Charge
6 10 8
1.4
1.2
5 10 8
Position
4 10 8
3 10 8
2 10 8
1.0
Pull-in
0.8
0.6
0.4
1 10 8
0 10 0
Release
0.2
0
50
100
150
Time
200
250
300
0.0
0
50
100
150
200
250
300
Time
Image by MIT OpenCourseWare.
Adapted from Figure 7.10 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2001,
p. 176. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 24
Behavior through pull-in
1.0
1.4
1.2
Release
0.0
Drive
1.0
Position
Velocity
0.5
Pull-in
-0.5
Release
0.8
0.6
Discharge
0.4
-1.0
0.2
-1.5
0
50
100
150
Time
200
250
300
0.0
Pull-in
0
50
100
150
200
250
300
Time
Image by MIT OpenCourseWare.
Adapted from Figure 7.11 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2001, p. 177. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 25
Outline
> Our progress so far
> Formulating state equations
> Quasistatic analysis
> Large-signal analysis
> Small-signal analysis
> Addendum: Review of 2nd-order system dynamics
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 26
Small-signal analysis
Given O.P.
State eqns
Linearize
Linearized
state eqns
(Jacobians)
Time
response
Integrate
Take LT
n
ct
nf s
g e si
Ei aly
An
form TFs
Equivalent Linearize Linearized
circuit
circuit
What is g(t)
due to small
changes in
Vin(t)?
Form TFs
Transfer SSS Frequency
functions
response
poles &
zeros
Natural
system
dynamics
Given O.P.
How fast
can I wiggle
tip?
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 27
Small-signal analysis
Given O.P.
State eqns
Linearize
Linearized
state eqns
(Jacobians)
Integrate
Time
response
What is g(t)
due to small
changes in
Vin(t)?
Given O.P.
Equivalent
circuit
How fast
can I wiggle
tip?
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 28
Linearization about a fixed point
> This is EXTREMELY common in MEMS literature
> This is also done in many other fields, with different
names
• Small-signal analysis
• Incremental analysis
• Etc.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 29
Linearization About an Operating Point
> Using Taylor’s theorem, a system can
x = f (x, u)
x (t ) = X 0 + δx (t )
u(t ) = U0 + δu(t )
be linearized about any fixed point
> We can do this in one dimension or
Operating point
many
f(x)
df/dx
~ f(X0+δ x)
f( X0)=Y0
δx
⎛ ∂f
(
)
d
δ
x
= f (X 0 , U0 ) + ⎜⎜ i
X 0 +
∂x
dt
⎝ j
x
X0
f ( X 0 + δx ) ≈ f ( X 0 ) +
df
dx
d (δx )
X 0 +
= f (X 0 + δx, U0 + δu)
dt
Multi-dimensional Taylor
δx
X0
⎞
⎛
⎟δx (t ) + ⎜ ∂f i
⎟
⎜ ∂u
X 0 ,U 0 ⎠
⎝ j
⎞
⎟δu(t )
⎟
X 0 ,U 0 ⎠
Cancel
d (δxi (t ) ) ⎛⎜ ∂f i
=⎜
∂x
dt
⎝ j
⎞
⎛
⎟δx (t ) + ⎜ ∂f i
⎟ i
⎜ ∂u
X 0 ,U 0 ⎠
⎝ j
J1
⎞
⎟δu (t )
⎟ i
X 0 ,U 0 ⎠
J2
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 30
Linearization About an Operating Point
δx (t ) = J1δx + J2δu(t)
> The resulting set of
equations are linear, and
have dynamics described
by the Jacobians of f(x,u)
evaluted at the fixed point.
> These describe how much
a small change in one state
variable affects itself or
another state variable
>
>
⎡ ∂f1
⎢
⎢ ∂Q O.P.
⎢ ∂f
J1 = ⎢ 2
⎢ ∂Q O.P.
⎢ ∂f 3
⎢
⎢⎣ ∂Q O.P.
⎛
⎜ gˆ 0
⎜−
⎜ RεA
⎛ δQ ⎞ ⎜
The O.P. must be evaluated d ⎜⎜ δg ⎟⎟ = ⎜ 0
dt ⎜ ⎟ ⎜
to use the Jacobian
⎝ δg ⎠ ⎜ Q0
⎜−
Example – linearization of
⎜ mεA
⎜
the voltage-controlled
⎝
electrostatic actuator
−
Q0
RεA
0
−
k
m
J1
∂f1
∂g
∂f 2
∂g
∂f 3
∂g
O. P.
O. P.
O.P.
∂f1
∂g
∂f 2
∂g
∂f 3
∂g
⎤
⎥
O. P. ⎥
⎥
⎥
O. P. ⎥
⎥
⎥
O. P. ⎥
⎦
⎞
⎟
0 ⎟
⎟
⎛1⎞
δ
Q
⎜ ⎟
⎞
⎛
⎟
1 ⎟⎜ ⎟ ⎜ R ⎟
(δV )
δg +
⎟⎜ ⎟ ⎜ 0 ⎟ in
⎜ ⎟ ⎜ ⎟
b ⎟⎝ δg ⎠ ⎜ 0 ⎟
⎝ ⎠
− ⎟
m⎟
⎟
J2
⎠
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OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 31
State Equations for Linear Systems
> Normally expressed with:
•
•
•
•
x: a vector of state variables
u: a vector of inputs
y: a vector of outputs
Four matrices, A,B,C, D
x = Ax + Bu
y = Cx + Du
> For us, Jacobian matrices take the place of A and B
> C and D depend on what outputs are desired
• Often C is identity and D is zero
> Can use to simulate time responses to arbitrary
SMALL inputs
• Remember, this is only valid for small deviations from O.P.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 32
Direct Integration in Time
Charge (Q)
Charge
> Can integrate via Simulink®
model (as before) or
MATLAB®
500
> First define system in
0
2
> Can use MATLAB®
commands step, initial,
impulse etc.
> Response of electrostatic
actuator to impulse of voltage
• Parameters from text (pg
167)
Velocity
alternate method
Displacement
MATLAB®
• using ss(J1,J2,C,D) or
Impulse Response
1000
0
-2
-4
4
2
0
-2
-4
-6
0
5
10
15
20
25
Time (sec)
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 33
Small-signal analysis
Given O.P.
State eqns
Linearize
Linearized
state eqns
(Jacobians)
Integrate
Time
response
Take LT
What is g(t)
due to small
changes in
Vin(t)?
Form TFs
Transfer SSS Frequency
functions
response
poles &
zeros
Equivalent
circuit
Given O.P.
How fast
can I wiggle
tip?
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 34
Solve via Laplace transform
> Use Laplace Transforms
to solve in frequency
domain
• Transform DE to algebraic
equations
• Use unilateral Laplace to
allow for non-zero IC’s
x = Ax + Bu
⇓ Unilateral Laplace
sX ( s ) − x (0) = AX( s ) + BU( s )
sIX ( s ) − x (0) = AX( s ) + BU( s )
(sI − A )X( s) = x(0) + BU( s)
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 35
Transfer Functions
> Transfer functions H(s) are useful for obtaining
compact expression of input-output relation
• What is the tip displacement as a function of voltage
> Most easily obtained from equivalent circuit
> But can also be obtained from linearized state eqns
• Depends on A, B, C (or J1, J2, C) matrices
• Can do this for fun analytically (see attachment at
end)
• Matlab can automatically convert from s.s to t.f.
formulations
> For our actuator, we would get three transfer
functions
⎡ Q( s ) ⎤
⎢
⎥
V
(
s
)
in
⎢
⎥
⎢ g( s) ⎥
H(s) = ⎢
⎥
(
s
)
V
⎢ in ⎥
⎢ g ( s ) ⎥
⎢
⎥
(
s
)
V
⎣ in ⎦
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 36
Sinusoidal Steady State
> When a LTI system is
driven with a sinusoid, the
steady-state response is a
sinusoid at the same
frequency
> The amplitude of the
response is |H(jω)|
u (t ) = U 0 cos(ωt )
Y ( jω ) = H ( jω )U ( jω )
y sss (t ) = Y0 cos(ωt + θ )
> The phase of the response
relative to the drive is the
angle of H(jω)
> A plot of log magnitude vs
log frequency and angle
vs log frequency is called
a Bode plot
Y0 = H ( jω ) U 0
Im{H ( jω )}
tanθ =
Re{H ( jω )}
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 37
Bode plot of electrostatic actuator
command
bode with previously
defined system sys
> Evaluate only one of TFs
40
20
0
Magnitude (dB)
> Use
Bode Diagram
Matlab®
-40
-60
-80
-100
g( s)
H(s) =
Vin ( s )
-120
-140
180
> This tells us how quickly
135
Phase (deg)
we can wiggle tip!
• At a certain OP!
-20
90
45
0
-45
-90
-2
10
-1
10
0
1
10
10
Frequency (rad/sec)
2
10
3
10
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 38
Small-signal analysis
Given O.P.
State eqns
Linearize
Linearized
state eqns
(Jacobians)
Time
response
Integrate
Take LT
What is g(t)
due to small
changes in
Vin(t)?
Form TFs
Transfer SSS Frequency
functions
response
poles &
zeros
Equivalent
circuit
Natural
system
dynamics
Given O.P.
How fast
can I wiggle
tip?
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 39
Poles and Zeros
> For our models, system function is a ratio of polynomials in s
> Roots of denominator are called poles
• They describe the natural (unforced) response of the system
> Roots of the numerator are called zeros
• They describe particular frequencies that fail to excite any output
> System functions with the same poles and zeros have the
same dynamics
−Q0
g( s)
ε ARm
H(s) =
=
Vin ( s )
⎛ 1
Q02 1 ⎞
b⎞ 2 ⎛ 1 b k⎞ ⎛ 1 k
3
+ ⎟s +⎜
+ ⎟s+⎜
− 2 2
s +⎜
⎟
ε
RC
m
RC
m
m
RC
m
A
Rm
0
0
0
⎝
⎠
⎝
⎠ ⎝
⎠
where C0 =
εA
gˆ 0
> MATLAB solution for poles is VERY long
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 40
Pole-zero diagram
> Displays information
tip velocity
Imaginary Axis
about dynamics of
system function
• Matlab command
Pole-Zero Map
1
pole
zero
0.5
ωd
0
-0.5
pzmap
-1
-9
-8
-7
-6
> Useful for examining
-4
-3
-2
-1
0
-3
-2
-1
0
R l Aaxis
i
Real
Pole-Zero Map
1
gap
Imaginary Axis
dynamics, stability,
etc.
-5
0.5
0
-0.5
-1
-9
-8
-7
-6
-5
-4
Real Axis
Real axis
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 41
Small-signal analysis
Given O.P.
State eqns
Linearize
Linearized
state eqns
(Jacobians)
Time
response
Integrate
Take LT
n
ct
nf s
g e si
Ei aly
An
Form TFs
Transfer SSS Frequency
functions
response
poles &
zeros
Equivalent
circuit
What is g(t)
due to small
changes in
Vin(t)?
Natural
system
dynamics
Given O.P.
How fast
can I wiggle
tip?
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 42
Eigenfunction Analysis
> For an LTI system, we can find the eigenvalues and
eigenvectors of the J1 (or A) matrix describing the internal
dynamics
For scalar 1st-order system:
dx
= λx
dt
⇒ x(t ) = K 0 e λt + K1
If we try solution:
x (t ) = Ke λt
Plug into DE:
Our linear (or linearized)
homogeneous systems look like:
δx (t ) = J1δx + J2δu(t)
dx
= Ax
dt
d (δx )
= J1δx
dt
λx = Ax
•This is an eigenvalue
equation
•If we find λ we can find
natural frequencies of
system
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 43
Eigenfunction Analysis
> These λ are the same as the poles si of the system
> Can solve analytically
• Find λ from det(A-λI)=0
> Or numerically eig(sys)
-8.9904
-0.2627 + 0.8455i
-0.2627 - 0.8455i
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 44
Linearized system poles
> We can use either λi or si to
determine natural frequencies
of system
> As we increase applied
voltage
• Stable damped resonant
Increasing voltage
> Plotting poles as system
jω
frequency decreases
changes is a root-locus plot
σ
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 45
Spring softening
> Plot damped resonant frequency versus applied voltage
> Resonant frequency is changing because net spring constant
k changes with frequency
k'=k−
ε AV 2
g
3
This is called
spring softening
Damped resonant frequency
> This is an electrically tuned mechanical resonator
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0
0.01 0.02 0.03 0.04 0.05 0.06
Voltage
Image by MIT OpenCourseWare.
Adapted from Figure 7.5 in Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 169. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 46
Small-signal analysis
Given O.P.
State eqns
Linearize
Linearized
state eqns
(Jacobians)
Time
response
Integrate
Take LT
n
ct
nf s
g e si
Ei aly
An
form TFs
Equivalent Linearize Linearized
circuit
circuit
What is g(t)
due to small
changes in
Vin(t)?
Form TFs
Transfer SSS Frequency
functions
response
poles &
zeros
Natural
system
dynamics
Given O.P.
How fast
can I wiggle
tip?
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 47
Linearized Transducers
> Can we directly linearize
our equivalent circuit?
YES!
> This is perhaps the most
R
+
Vin(t) +
-
C
V
-
common analysis in the
literature
> First, choose what is load
1/k
I
W
U
+
+
F
Fout
-
-
m
b
Source
Transducer
Load
Find OP
and what is transducer
Linearize
• Here we include spring
with transducer
R
δVin(t)
+
-
δU
δI
+
δV
-
Linearized
Transducer
+
δFout
m
b
Source
Load
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 48
Linearized Transducer Model
> First, find O.P.
V0 , gˆ 0 , Q0
> Next, generate matrix to
relate incremental port
variables to each other
• Start from energy and force
relations
Qg
εA
Q2
=
− k ( g0 − g )
2ε A
V=
Fout
• Linearize (take partials…)
⎡ gˆ 0
⎡δ V ⎤ ⎢ ε A
⎢δ F ⎥ = ⎢ Q
⎣ out ⎦ ⎢ 0
⎢⎣ ε A
Q0 ⎤
ε A ⎥ ⎡δ Q ⎤
⎥⎢ ⎥
δg
k ⎥⎣ ⎦
⎥⎦
> Recast in terms of port variables
⎡δ I ⎤
δ
Q
⎡ ⎤ ⎢ s ⎥
⎢δ g ⎥ = ⎢δ U ⎥
⎣ ⎦
⎣
s⎦
> Define intermediate variables
Q0
εA
C0 =
, V0 =
gˆ 0
C0
> Final expression
⎡ 1
⎡δ V ⎤ ⎢⎢ sC0
⎢δ F ⎥ = ⎢ V
⎣ out ⎦
0
⎢ sgˆ
⎣ 0
V0 ⎤
sgˆ 0 ⎥ ⎡δ I ⎤
⎥⎢ ⎥
k ⎥ ⎣δ U ⎦
s ⎥⎦
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 49
Linearized Transducers
> Now we want to convert this relation into a circuit
> Many circuit topologies are consistent with this matrix relation
> THIS IS NOT UNIQUE!
i'
i
+
+
-1/k'
1/k
+
+
F
υ
{
+
i
u
u'
1:Γ
υ'
υ
_
Co
F'
_
_
i
_
Γ2/κ∗
Co
+
Co∗
u
F
1/k
_
_
1:Γ
+
υ
1/k*
u
1:Γ/κ2
_
u
i
+
+
F
υ
_
_
Co
1_Γ
1
k_Γ/C
Co
Γ
+
o
F
_
Image by MIT OpenCourseWare.
Adapted from Figure 5 on p. 163 in Tilmans, Harrie A. C. "Equivalent Circuit Representations of Electromechanical
Transducers: I. Lumped-parameter Systems." Journal of Micromechanics and Microengineering 6, no. 1 (1996): 157-176.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 50
Linearized Transducers
> This is the one used in the text
⎡ δ V ⎤ ⎡ Z EB ϕ Z EB ⎤ ⎡ δ I ⎤
⎢δ F ⎥ = ⎢ϕ Z
⎥ ⎢δ U ⎥
Z
⎦
MO ⎦ ⎣
⎣ out ⎦ ⎣ EB
Z MS
⎛ ϕ 2 Z EB ⎞
= Z MO ⎜1 −
⎟
Z MO ⎠
⎝
> Uses a transformer
• Transforms port variables
• Doesn’t store energy
> What we want to do now is
identify ZEB, ZMS and ϕ, and
figure out what they mean…
⎛ e2 ⎞ ⎛⎜ ϕ
⎜⎜ ⎟⎟ =
⎝ f 2 ⎠ ⎜⎝ 0
0 ⎞⎛ e1 ⎞
⎟
− 1 ⎟⎜⎜ f ⎟⎟
ϕ ⎠⎝ 1 ⎠
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 51
Linearized Transducers
⎡ δ V ⎤ ⎡ Z EB ϕ Z EB ⎤ ⎡ δ I ⎤
⎢δ F ⎥ = ⎢ϕ Z
⎥ ⎢δ U ⎥
Z
⎦
MO ⎦ ⎣
⎣ out ⎦ ⎣ EB
⎛ ϕ 2 Z EB ⎞
Z MS = Z MO ⎜1 −
⎟
Z MO ⎠
⎝
⎡ 1
⎡δV ⎤ ⎢ sC0
⎢δF ⎥ = ⎢ V
⎣ ⎦ ⎢ 0
⎢⎣ sgˆ 0
Z MS
V0 ⎤
sgˆ 0 ⎥ ⎡δI ⎤
⎥
k ⎥ ⎢⎣δU ⎥⎦
s ⎥⎦
2 1
2
⎞
⎛
k ⎜ ⎛ Q0 ⎞ sC0 ⎟ k ⎛⎜ ⎛ Q0 ⎞ 1 ⎞⎟
= 1 − ⎜⎜ ⎟⎟
= ⎜1 − ⎜⎜ ⎟⎟
⎟
k
s ⎜ ⎝ gˆ 0 ⎠ C0 k ⎟
s ⎜ ⎝ gˆ 0 ⎠
⎟
⎝
⎠
s ⎠
⎝
2
Q
k⎛
Q02 ⎞
0
⎟⎟
= ⎜⎜1 −
k
k
=
−
'
s ⎝ εAkgˆ 0 ⎠
εAgˆ
ϕ=
C0V0
gˆ 0
0
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 52
Linearized Transducers
> C0 represents the
capacitance of the
structure seen from the
electrical port
> It is simply the capacitance
at the gap given by the
operating point
C0 =
εA
gˆ 0
> As Vin increases, C0 will
increase until the structure
pulls in
> This is a tunable capacitor
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 53
Linearized Transducers
> k’ represents the effective
spring
> A combination of the
mechanical spring k and
the electrical spring
> This is an electrically
tunable spring!
• Spring softening shows up
Q02
k'= k −
εAgˆ 0
in k’
> As Vin increases, k’ will
decrease from k (at Vin=0)
to 0 (at Vin=Vpi)
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 6.777J/2.372J Spring 2007, Lecture 10 - 54
Linearized Transducers
> ϕ represents the
electromechanical
coupling
> Represents how much the
capacitance changes with
gap
> A measure of sensitivity
ϕ=
ϕ = −V0
= V0
=
∂C
∂g
C0V0 Q0
=
gˆ 0
gˆ 0
= −V0
O. P.
∂ εA
∂g g O. P.
εA
gˆ 02
C0V0
gˆ 0
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 55
Transfer Functions
> Can use linearized
>
circuit to construct H(s)
using complex
impedances
Usually helpful to
“eliminate” transformer
> Transformer changes
impedances
Z2 =
Z1
1:ϕ
Z1
ϕ2
Z2
1:ϕ
R
δVin
δU
m
+
-
C0
b
ϕ2/k’
R
δVin
1/k’
+
-
δU
m/ϕ2
C0
2
b/ϕ
Can now get any transfer
function using standard
circuit analysis
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 56
Linearized Transducer Models
> Now we can understand Nguyen’s filter!
Image removed due to copyright restrictions.
Image removed due to copyright restrictions.
Figure 9 on p. 17 in Nguyen, C. T.-C. "Vibrating RF MEMS
Figure 12 on p. 62 in: Nguyen, C. T.-C. "Micromechanical
Overview: Applications to Wireless Communications."
Filters for Miniaturized Low-power Communications."
Proceedings of SPIE Int Soc Opt Eng 5715 (January 2005): 11-25.
Proceedings of SPIE Int Soc Opt Eng 3673 (July 1999): 55-66.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 57
Small-signal analysis summary
Given O.P.
State eqns
Linearize
Linearized
state eqns
(Jacobians)
Time
response
Integrate
Take LT
n
ct
nf s
g e si
Ei aly
An
form TFs
Equivalent Linearize Linearized
circuit
circuit
What is g(t)
due to small
changes in
Vin(t)?
Form TFs
Transfer SSS Frequency
functions
response
poles &
zeros
Natural
system
dynamics
Given O.P.
How fast
can I wiggle
tip?
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 58
Conclusions
> We can now analyze and design both quasistatic and
dynamic behavior of our multi-domain MEMS
> We have much more powerful tools to analyze linear
systems than nonlinear systems
> But most systems we encounter are nonlinear
> Linearization permits the study of small-signal inputs
> Next up: special topics in structures, heat transfer,
fluids
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 59
Review: analysis of a 2nd-order linear system
> Spring-mass-dashpot
.
x
F
+
-
1/k
+ ek - +
em
- eb + -
m
b
⎤
d ⎡ x ⎤ ⎡ x
= ⎢1
⎥
⎢
⎥
dt ⎣ x ⎦ ⎢⎣ m (F − kx − bx )⎥⎦
State
eqns
x = Ax + Bu
y = Cx + Du
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 60
Direct Integration in Time
> Example: Spring-mass-dashpot step response
• k=m=1;b=0.5;
Step Response
Amplitude
>> A=[0 1;-1 -0.5]; B=[0;1];
>> C=[1 0;0 1]; D=[0;0];
>> sys=ss(A,B,C,D);
>> step(sys)
Position
1.5
1
0.5
Velocity
0
1
0.5
0
-0.5
0
5
10
15
Tim e (sec)
20
25
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 61
Transfer Functions
> Can get TFs from A,B,C matrices
[
Y ( s ) = CX ( s ) + DU( s )
]
Y ( s ) = C (sI − A ) x (0) + (sI − A ) BU( s ) + DU( s )
−1
−1
Assume transient has died out (XZIR=0)
No feed-through (D=0)
−1
⎡
Y( s ) = C ( sI − A ) B ⎤ U( s )
⎣
⎦
Y( s ) = H ( s )U( s )
−1
H ( s ) = ⎡C ( sI − A ) B ⎤
⎣
⎦
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 62
Transfer Functions
> Let’s do analytically & via MATLAB
1 ⎤
⎡ s 0⎤ ⎡ 0
−⎢ k
sI − A = ⎢
⎥
b ⎥
−
−
s
0
⎣
⎦ ⎢⎣ m
m⎥⎦
−1 ⎤
⎡ s
= ⎢k
b ⎥
+
s
⎢⎣ m
m⎥⎦
1 ⎡ s + b m 1⎤
−1
⎥
(sI − A ) = ⎢ k
Δ⎢−
s⎥
m
⎣
⎦
Δ = s ( s + b m) + k m = s 2 + s b m + k m
⎡1 0⎤ 1 ⎡ s + b m 1⎤ ⎡0 ⎤
−1
⎥⎢ ⎥
⎢
C(sI − A ) B = ⎢
⎥
1
k
⎣0 1⎦ Δ ⎢⎣ − m s ⎥⎦ ⎢⎣ m⎥⎦
⎡1 ⎤
1 ⎢ m⎥
=
Δ ⎢s ⎥
⎣ m⎦
⎡ X( s ) ⎤ ⎡
1
⎤
⎢ F( s) ⎥ ⎢ 2
ms + sb + k ⎥
⎥
⎢
H( s) =
=⎢
⎥
⎢ X( s ) ⎥ ⎢
s
⎥
⎥ ⎢ ms 2 + sb + k ⎥
⎢
⎦
⎣ F( s) ⎦ ⎣
1
⎡
⎤
⎢ s 2 + 0.5s + 1⎥
H( s) = ⎢
⎥
s
⎢
⎥
2
⎢⎣ s + 0.5s + 1⎥⎦
>> [n,d]=ss2tf(A,B,C,D)
n=
s1
s0
s2
0 -0.0000 1.0000
0 1.0000 -0.0000
d=
1.0000
0.5000
1.0000
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 63
Transfer Functions
> Can also construct H(s) directly
using complex impedances and
circuit model
.
1/k
x
F
+
-
+ ek - +
em
- eb + b
F − ek − em − eb = 0
k
ek = kx = x
s
eb = bx
em = mx = msx
m
(s)
X
1
= H2 ( s ) =
F( s )
Z( s )
1
=
b + ms + k s
s
=
ms 2 + bs + k
( s ) sX( s )
X
=
F( s ) F( s )
1
X( s )
= H1 ( s ) =
ms 2 + bs + k
F( s )
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 64
Poles and Zeros
> For 2nd-order system, easy to get
poles and zeros from TFs
1
⎤
⎡
2
1 ⎢⎢ s + s b m + k m ⎥⎥
H(s) =
s
m⎢
⎥
⎢ s2 + s b m + k m ⎥
⎦
⎣
1
⎤
⎡
⎥
⎢
1 ⎢ ( s − s1 )( s − s2 ) ⎥
=
s
m⎢
⎥
⎢ ( s − s )( s − s ) ⎥
1
2 ⎦
⎣
where
2
b
k
⎛ b ⎞
± ⎜
s1,2 = −
−
⎟
2m
2
m
⎝
⎠ m
these are the poles
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 65
Spring-mass-dashpot system
> It is a second order system, with
two poles
> We conventionally define
•
•
•
•
Undamped resonant frequency
Damping constant
Damped resonant frequency
Quality factor
s2 + b
m
s+k
m
= s 2 + 2αs + ω 02
k
m
b
α=
2m
ω0 =
s1, 2 = −α ± α 2 − ω 02
For underdamped systems (α < ω 0 )
s1, 2 = −α ± jω d
where
ω d = ω 02−α 2
Quality factor :
Q=
ω 0 mω 0
=
2α
b
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 66
Pole-zero diagram
> Displays information about dynamics of system
function
s
H2 ( s ) = 2
s + 0.5s + 1
s1, 2 = −0.25 ± j 1 − 1 = −0.25 ± j 0.97
16
Pole-Zero Map
2
1.5
pole
Imaginary Axis
1
0.5
zero
ωd
0
-0.5
-1
-1.5
-2
-2
-1
0
Real Axis
1
2
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 67
SMD-position frequency response
1
1
H( jω ) =
m (ω 02 − ω 2 ) 2 + 4α 2ω 2
Bode Diagram
0
-10
-20
-30
-40
360
Phase (deg)
⎛ 2αω ⎞
⎟
∠H( jω ) = − atan⎜⎜ 2
2 ⎟
⎝ ω0 − ω ⎠
Magnitude (dB)
1
1
H( jω ) =
m − ω 2 + 2αjω + ω 02
10
315
270
225
180
-1
10
0
10
Frequency (rad/sec)
10
1
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 68
Eigenfunction Analysis
> Find eigenvalues numerically using MATLAB and A matrix
1 ⎤
⎡0
A=⎢
⎥
0
0
.
5
−
⎣
⎦
[ V , Λ ] = eig( A)
0
⎡λ1 0 ⎤ ⎡− 0.25 + 0.97 j
⎤
=
Λ=⎢
⎥ ⎢
⎥
λ
0
−
−
0
0
.
25
0
.
97
j
⎦
⎣
2⎦
⎣
0.707
0.707
⎡
⎤
V = [v1 v2 ] = ⎢
⎥
⎣− 0.18 − 0.68 j − 0.18 − 0.68 j ⎦
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 69