Numerical simulation of electromagnetic radiation using high-order discontinu...
07lecture10
1. 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
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 1
2. 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].
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3. 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
4. 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,
I Dashpot b
and D. Potter. "Microswitches and Microrelays with a View Toward Microwave
Applications." International Journal of RF and Microwave Comput-Aided Spring k
Engineering 9, no. 4 (1999): 338-347. + Mass m
V g
z
-
1 µm Cantilever Fixed plate
0.5 µm Pull-down Image by MIT OpenCourseWare.
Silicon electrode Anchor 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
5. 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
. 1/k
I g
2 + +
Q g
W (Q, g ) = V F m
2εA
C
- -
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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6. Our progress so far…
∂W (Q, g ) Q2
> We first analyzed system F= =
∂g Q
2εA
quasistatically ∂W (Q, g ) Qg
V= =
> Saw that there is VERY different ∂Q g
εA
behavior depending on whether
• Charge is controlled dW = VdQ + Fdg
stable behavior at all gaps
• Voltage is controlled dW * = QdV − Fdg
pull-in at g=2/3g0
> Use of energy or co-energy ∂W * εA
depends on what is controlled Q= = Vin
∂V g
g
• Simplifies math
∂W * εAVin 2
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
7. 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
8. 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
9. Adding dynamics
Fixed support
> Add components to complete Resistor
R I
the system: Dashpot b Spring k
Mass m
+
• Source resistor for the voltage +
source Vin V
- g
-
z
• Inertial mass, dashpot Fixed plate
> This is now our RF switch!
g z 1/k
> System is nonlinear, so we I
R + +
can’t use Laplace to get +
C
F m
Vin V
transfer functions - W(Q,g)
- -
> Instead, model with state 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
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 9
10. 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) Goal:
> Good choices for state variables: ⎡Q ⎤
d ⎢ ⎥ ⎛ functions of ⎞
⎜ ⎟
the charge on a capacitor ⎢ g ⎥ = ⎜ Q,g,g or constants ⎟
⎢g ⎥ ⎝ ⎠
(displacement) and the current in an dt
inductor (momentum) ⎣ ⎦
> 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
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 10
11. Formulating state equations
> Start with Q
+ eR - I
> We know that dQ/dt=I R + C
+
> Find relation between I and Vin V
- W(Q,g)
state variables and constants -
KVL : Vin − eR − V = 0
eR = IR
Vin − IR − V = 0
dQ 1
= I = (Vin − V )
dt R Qg
V=
εA
dQ 1 ⎛ Qg ⎞
= ⎜ Vin −
dt R ⎝ εA⎟⎠
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 - 11
12. Formulating state equations
> Now we’ll do .
g . 1/k
g z
> We know that dg
=g C + + ek - +
dt
F em m
W(Q,g)
- - eb + -
b
KVL : ek = kz
F − ek − em − eb = 0 F − k ( g 0 − g ) + mg + bg = 0
em = mz
F − kz − mz − bz = 0 eb = bz 1
g = − [F − k ( g 0 − g ) + bg ]
m
dg 1 ⎡ Q2 ⎤
z = g0 − g ⇒ z = − g, z = − g =− ⎢ − k ( g 0 − g ) + bg ⎥
dt m ⎣ 2εA ⎦
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
13. Formulating state equations
> State equation for g is easy: dg
=g
dt
> Collect all three nonlinear state equations
⎡ 1⎛ Qg ⎞ ⎤
⎢ ⎜ Vin − ⎟ ⎥
⎡Q ⎤ ⎢ R⎝ εA⎠ ⎥
d ⎢ ⎥ ⎢ ⎥
g⎥= g
dt ⎢ ⎢ ⎥
⎢ g ⎥ ⎢ 1 ⎡ Q2
⎣ ⎦ ⎤⎥
⎢ − 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
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 13
14. 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
15. Quasistatic analysis
⎡1 ⎛ Qg ⎞ ⎤
⎢ ⎜V − ⎟ ⎥
d ⎡Q ⎤ ⎢ R ⎝ in εA ⎠ ⎥ Fixed-point
⎢g ⎥ = ⎢g ⎥ analysis
dt ⎢ g ⎥ ⎢ 1 ⎡ Q 2
⎣ ⎦ ⎤⎥
− ⎢
⎢ m 2εA − k ( g 0 − g ) + bg ⎥ ⎥
⎣ ⎣ ⎦⎦ Given static Vin, etc.
State eqns
What is deflection,
(Just now) charge, etc.?
. . 1/k
I g z
R + C + Follow
+ F m causal path
Vin V
- W(Q,g)
(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
16. State equations
For transverse
electrostatic actuator:
State variables ⎡Q ⎤
State variables: x = ⎢g ⎥
Inputs ⎢g ⎥
⎣ ⎦
x = f (x, u) Inputs: u = [Vin ]
y = g (x, u)
⎡1 ⎛ Qg ⎞ ⎤
⎢ ⎜ Vin − ⎟ ⎥
Outputs ⎢R ⎝ εA⎠ ⎥
f (x,u) = ⎢ g ⎥
⎢ ⎥
⎢ 1⎡Q 2
⎤⎥
⎢ − 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
17. Fixed points
> Definition of a fixed point
• Solution of f(x,u) = 0
• time derivatives 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
18. Fixed points of the electrostatic actuator
> This analysis is analogous to what we did last time…
Operating point stable
1⎛ Qg ⎞
normalized gap
0 = ⎜Vin − ⎟ 1
R⎝ εA ⎠
0=g
1 ⎡ Q2 ⎤ 0.5
0=− ⎢ − k ( g 0 − g ) + bg ⎥
m ⎣ 2εA ⎦
0
0 0.5 1
Qg normalized voltage
Vin =
εA Last time…
Q 2 Vin 2ε A εAVin
2
= = k ( g0 − g ) g = g0 −
2ε A 2g 2
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
19. 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
20. Large-signal analysis
⎡1 ⎛ Qg ⎞ ⎤
⎢ ⎜V − ⎟ ⎥
d ⎡Q ⎤ ⎢ R ⎝ in εA ⎠ ⎥ Integrate
⎢g ⎥ = ⎢g ⎥ state eqns
dt ⎢ g ⎥ ⎢ 1 ⎡ Q 2
⎣ ⎦ ⎤⎥
− ⎢
⎢ m 2εA − k ( g 0 − g ) + bg ⎥ ⎥
⎣ ⎣ ⎦⎦ Given a step input
State eqns Vin(t)u(t)
(Earlier) What is g(t), Q(t), etc.?
. . 1/k
I g z
R + C + SPICE
+ F m
Vin V
- W(Q,g)
- -
b
Equivalent circuit
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 - 20
21. 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
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 21
22. Electrostatic actuator in Simulink®
⎡Q ⎤
x = ⎢ g ⎥, u = [Vin ]
(u[1]^2)/(2*e*A) +
⎢ ⎥ + -1/m
Electrostatic force
⎢g ⎥
⎣ ⎦
+
Inertia
⎡1 ⎛ Qg ⎞ ⎤ 1
⎢ ⎜Vin − ⎟ ⎥ s 3
⎢R ⎝ εA ⎠ ⎥ Spring k b Damping Velocity gdot
f(x, u) = ⎢ g ⎥
⎢ ⎥
⎢ 1 ⎡ Q2 ⎤⎥ Sum2
1
s 2
⎢− m ⎢ 2εA − k ( g 0 − g ) + bg ⎥ ⎥ _
+ g
⎣ ⎣ ⎦⎦ Position
At-rest gap g _ 0 1/(e*A) *
Qg
_
1/R 1
1 + s 1
V_in 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
23. Electrostatic actuator with contact
<
g_min
Accel > 0 ? >
+ +
g > g_min?
(u[1]^2)/(2*e*A) +
+ -1/m
Electrostatic force + Inertia 1
s
Switch Velocity
3
0
Spring k b Damping gdot
Zero
1
Sum2 s 2
_
+ g
Position
At-rest gap g _ 0 1/(e*A) *
Qg
_
1/R 1 1
1 + s
1/R Q
V_in Charge
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
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 23
24. Behavior through pull-in
7 10 8 1.6
6 10 8 Drive (scaled) 1.4
Charge Release
1.2
5 10 8
1.0
4 10 8
Position
Pull-in
0.8
3 10 8
0.6
2 10 8
0.4
1 10 8 0.2
0 10 0 0.0
0 50 100 150 200 250 300 0 50 100 150 200 250 300
Time 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
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 24
25. Behavior through pull-in
1.0
1.4
0.5 1.2
Release Drive
1.0
0.0 Release
Velocity
Position
Pull-in 0.8
-0.5 0.6 Discharge
0.4
-1.0
0.2
Pull-in
-1.5 0.0
0 50 100 150 200 250 300 0 50 100 150 200 250 300
Time 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
26. 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
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 26
27. Small-signal analysis
Given O.P.
Linearized Time What is g(t)
Linearize Integrate
State eqns state eqns response due to small
(Jacobians) changes in
Take LT Vin(t)?
Ei aly Form TFs
An
g e si
nf s
ct Transfer SSS Frequency
n
functions response
poles & Given O.P.
form TFs zeros
How fast
Natural
Equivalent Linearize Linearized can I wiggle
system
circuit circuit tip?
dynamics
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 27
28. Small-signal analysis
Given O.P.
Linearized Time What is g(t)
Linearize Integrate
State eqns state eqns response due to small
(Jacobians) changes in
Vin(t)?
Given O.P.
How fast
Equivalent can I wiggle
circuit tip?
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 - 28
29. 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
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 29
30. Linearization About an Operating Point
> Using Taylor’s theorem, a system can x = f (x, u)
be linearized about any fixed point x (t ) = X 0 + δx (t )
u(t ) = U0 + δu(t )
> We can do this in one dimension or
many Operating point
d (δx )
f(x) X0 + = f (X 0 + δx, U0 + δu)
df/dx dt
~ f(X0+δ x) Multi-dimensional Taylor
f( X0)=Y0 ⎛ ∂f ⎞ ⎛ ⎞
d (δx ) ⎟δx (t ) + ⎜ ∂f i
X0 + = f (X 0 , U0 ) + ⎜ i
⎜ ∂x ⎟ ⎜ ∂u
⎟δu(t )
⎟
dt
⎝ j X 0 ,U 0 ⎠ ⎝ j X 0 ,U 0 ⎠
δx
x Cancel
X0
d (δxi (t ) ) ⎛ ∂f i ⎞ ⎛
⎟δx (t ) + ⎜ ∂f i
⎞
=⎜
⎜ ∂x ⎟ i ⎜ ∂u
⎟δu (t )
⎟ i
dt
df ⎝ j X 0 ,U 0 ⎠ ⎝ j X 0 ,U 0 ⎠
f ( X 0 + δx ) ≈ f ( X 0 ) + δx J2
dx X0
J1
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 30
31. Linearization About an Operating Point
> The resulting set of δx (t ) = J1δx + J2δu(t)
equations are linear, and ⎡ ∂f1 ⎤
∂f1 ∂f1
have dynamics described ⎢ ⎥
⎢ ∂Q O.P. ∂g O. P.
∂g O. P. ⎥
by the Jacobians of f(x,u) ⎢ ∂f ∂f 2 ∂f 2 ⎥
J1 = ⎢ 2 ⎥
evaluted at the fixed point. ⎢ ∂Q O.P. ∂g O. P.
∂g O. P. ⎥
⎢ ∂f 3 ∂f 3 ∂f 3 ⎥
> These describe how much ⎢ ⎥
⎢ ∂Q O.P.
⎣ ∂g O.P.
∂g O. P. ⎥
⎦
a small change in one state
⎛ ⎞
variable affects itself or ⎜ g0ˆ Q0 ⎟
⎜ − − 0 ⎟
another state variable ⎜ RεA RεA ⎟ ⎛1⎞
⎛ δQ ⎞ ⎜ ⎟⎛ δQ ⎞ ⎜ ⎟
The O.P. must be evaluated d ⎜ δg ⎟ = ⎜ 0 0 1 ⎟⎜ ⎟ ⎜ R ⎟
> ⎜ ⎟ δg +
⎟⎜ ⎟ ⎜ 0 ⎟ in
(δV )
to use the Jacobian dt ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟
⎝ δg ⎠ ⎜ Q0 k b ⎟⎝ δg ⎠ ⎜ 0 ⎟
⎜− − − ⎟ ⎝ ⎠
> Example – linearization of ⎜ mεA m m⎟
the voltage-controlled ⎜ ⎟ J2
⎝ J1 ⎠
electrostatic actuator
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 31
32. State Equations for Linear Systems
> Normally expressed with:
• x: a vector of state variables
• u: a vector of inputs x = Ax + Bu
• y: a vector of outputs
y = Cx + Du
• Four matrices, A,B,C, D
> 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.
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 32
33. Direct Integration in Time
Impulse Response
> Can integrate via Simulink® 1000
model (as before) or
Charge
Charge (Q)
500
MATLAB®
> First define system in 0
MATLAB® 2
• using ss(J1,J2,C,D) or
Displacement
alternate method 0
> Can use MATLAB® -2
commands step, initial,
impulse etc. -4
4
2
> Response of electrostatic
0
Velocity
actuator to impulse of voltage
-2
• Parameters from text (pg -4
167)
-6
0 5 10 15 20 25
Time (sec)
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 33
34. Small-signal analysis
Given O.P.
Linearized Time What is g(t)
Linearize Integrate
State eqns state eqns response due to small
(Jacobians) changes in
Take LT Vin(t)?
Form TFs
Transfer SSS Frequency
functions response
poles & Given O.P.
zeros
How fast
Equivalent can I wiggle
circuit tip?
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 34
35. Solve via Laplace transform
> Use Laplace Transforms
to solve in frequency x = Ax + Bu
domain ⇓ Unilateral Laplace
• Transform DE to algebraic sX ( s ) − x (0) = AX( s ) + BU( s )
equations
sIX ( s ) − x (0) = AX( s ) + BU( s )
• Use unilateral Laplace to
allow for non-zero IC’s (sI − A )X( s) = x(0) + BU( s)
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 35
36. 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. ⎡ Q( s ) ⎤
formulations ⎢ ⎥
⎢ Vin ( s ) ⎥
> For our actuator, we would get three transfer ⎢ g( s) ⎥
H(s) = ⎢ ⎥
functions ⎢ Vin ( s ) ⎥
⎢ g( s) ⎥
⎢ ⎥
⎣ Vin ( s ) ⎦
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 36
37. Sinusoidal Steady State
> When a LTI system is
driven with a sinusoid, the u (t ) = U 0 cos(ωt )
steady-state response is a
sinusoid at the same Y ( jω ) = H ( jω )U ( jω )
frequency
> The amplitude of the y sss (t ) = Y0 cos(ωt + θ )
response is |H(jω)|
> The phase of the response
relative to the drive is the
angle of H(jω) Y0 = H ( jω ) U 0
> A plot of log magnitude vs Im{H ( jω )}
tanθ =
Re{H ( jω )}
log frequency and angle
vs log frequency is called
a Bode plot
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 37
38. Bode plot of electrostatic actuator
Bode Diagram
> Use Matlab®
command 40
20
bode with previously 0
Magnitude (dB)
-20
defined system sys
-40
-60
> Evaluate only one of TFs -80
-100
g( s)
H(s) = -120
Vin ( s ) -140
180
> This tells us how quickly 135
we can wiggle tip! 90
Phase (deg)
• At a certain OP! 45
0
-45
-90
-2 -1 0 1 2 3
10 10 10 10 10 10
Frequency (rad/sec)
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 38
39. Small-signal analysis
Given O.P.
Linearized Time What is g(t)
Linearize Integrate
State eqns state eqns response due to small
(Jacobians) changes in
Take LT Vin(t)?
Form TFs
Transfer SSS Frequency
functions response
poles & Given O.P.
zeros
How fast
Natural
Equivalent can I wiggle
system
circuit tip?
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 - 39
40. 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 ) 3 ⎛ 1 b⎞ 2 ⎛ 1 b k⎞ ⎛ 1 k Q02 1 ⎞
s +⎜ + ⎟s +⎜ + ⎟s+⎜ − 2 2 ⎟
⎝ RC0 m ⎠ ⎝ RC0 m m ⎠ ⎝ RC0 m ε A Rm ⎠
εA
where C0 =
ˆ
g0
> MATLAB solution for poles is VERY long
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 40
41. Pole-zero diagram
> Displays information 1
Pole-Zero Map
tip velocity
about dynamics of 0.5
pole
ωd
Imaginary Axis
zero
system function 0
• Matlab command -0.5
pzmap
-1
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
> Useful for examining Real A i
R l axis
dynamics, stability, 1
Pole-Zero Map
gap
etc. 0.5
Imaginary Axis
0
-0.5
-1
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
Real Axis
Real axis
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 41
42. Small-signal analysis
Given O.P.
Linearized Time What is g(t)
Linearize Integrate
State eqns state eqns response due to small
(Jacobians) changes in
Take LT Vin(t)?
Ei aly Form TFs
An
g e si
nf s
ct Transfer SSS Frequency
n
functions response
poles & Given O.P.
zeros
How fast
Natural
Equivalent can I wiggle
system
circuit tip?
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 - 42
43. 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: If we try solution:
dx
= λx ⇒ x(t ) = K 0 e λt + K1 x (t ) = Ke λt
dt
Plug into DE:
λx = Ax
Our linear (or linearized)
homogeneous systems look like:
•This is an eigenvalue
δx (t ) = J1δx + J2δu(t) equation
dx d (δx ) •If we find λ we can find
= Ax = J1δx natural frequencies of
dt dt system
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 43
44. 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
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 44
45. Linearized system poles
> We can use either λi or si to
determine natural frequencies
of system
> As we increase applied
voltage
• Stable damped resonant
frequency decreases Increasing voltage
jω
> Plotting poles as system
changes is a root-locus plot
σ
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 45
46. Spring softening
> Plot damped resonant frequency versus applied voltage
> Resonant frequency is changing because net spring constant
k changes with frequency
> This is an electrically tuned mechanical resonator
1.00
Damped resonant frequency
0.90
0.80
ε AV 2 0.70
k'=k− 3
0.60
g 0.50
0.40
This is called 0.30
0 0.01 0.02 0.03 0.04 0.05 0.06
spring softening 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.
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 46
47. Small-signal analysis
Given O.P.
Linearized Time What is g(t)
Linearize Integrate
State eqns state eqns response due to small
(Jacobians) changes in
Take LT Vin(t)?
Ei aly Form TFs
An
g e si
nf s
ct Transfer SSS Frequency
n
functions response
poles & Given O.P.
form TFs zeros
How fast
Natural
Equivalent Linearize Linearized can I wiggle
system
circuit circuit tip?
dynamics
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 - 47
48. Linearized Transducers
> Can we directly linearize
our equivalent circuit? I
1/k U
R
YES! +
C
+ +
> This is perhaps the most Vin(t) + F Fout m
- V
common analysis in the -
W - -
b
literature
Source Transducer Load
> First, choose what is load Find OP
and what is transducer
Linearize
• Here we include spring
δI δU
with transducer R
+ +
+ Linearized
δVin(t) δV δFout m
- Transducer
- -
b
Source Load
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 48
49. Linearized Transducer Model
> First, find O.P.
ˆ
V0 , g 0 , Q0 > Recast in terms of port variables
δI
> Next, generate matrix to ⎡δ Q⎤ ⎡ s ⎤
⎢ ⎥
relate incremental port ⎢δ g ⎥ = ⎢δ U ⎥
⎣ ⎦
variables to each other ⎣ s⎦
• Start from energy and force > Define intermediate variables
relations εA Q0
C0 = , V0 =
V=
Qg
ˆ
g0 C0
εA
Q2
Fout = − k ( g0 − g ) > Final expression
2ε A
• Linearize (take partials…) ⎡ 1 V0 ⎤
ˆ
⎡ g0 Q0 ⎤ ⎡δ V ⎤ ⎢ sC0 sg 0 ⎥ ⎡δ I ⎤
ˆ
⎡δ V ⎤ ⎢ ε A ε A ⎥ ⎡δ Q ⎤ ⎢δ F ⎥ = ⎢ V ⎥⎢ ⎥
k ⎥ ⎣δ U ⎦
⎢δ F ⎥ = ⎢ Q ⎥⎢ ⎥ ⎣ out ⎦ ⎢ 0
⎣ out ⎦ ⎢ 0 k ⎥⎣ ⎦
δg ⎢ sg s ⎥
⎢ε A
⎣ ⎥
⎦ ⎣ ˆ0 ⎦
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 49
50. 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' u' u i 1:Γ/κ2 u
1:Γ
+ + + + + +
-1/k' 1/k ∗
Co
{
1/k*
υ υ' F' F υ F
Co 1/k
_ _ _ _ _ _
i u i u
1:Γ
+ + + +
Γ2/κ∗ Co 1
1_Γ k_Γ/C o
υ F υ Co F
Co
_ Γ
_ _ _
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.
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JV: 6.777J/2.372J Spring 2007, Lecture 10 - 50
51. Linearized Transducers
> This is the one used in the text
⎡ δ V ⎤ ⎡ Z EB ϕ Z EB ⎤ ⎡ δ I ⎤
⎢δ F ⎥ = ⎢ϕ Z Z MO ⎥ ⎢δ U ⎥
⎣ out ⎦ ⎣ EB ⎦⎣ ⎦
⎛ ϕ 2 Z EB ⎞
Z MS = 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 0 ⎞⎛ e1 ⎞
⎜ ⎟=⎜
⎜ f ⎟ ⎜0
⎟
− 1 ⎟⎜ f ⎟
⎜ ⎟
⎝ 2⎠ ⎝ ϕ ⎠⎝ 1 ⎠
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