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A Mechanistic Approach to Tuning of MEMS Resonators

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Abhyudai Singh

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A Mechanistic Approach to Tuning of MEMS Resonators
By
Abhyudai Singh

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2006

ABSTRACT
A Mechanistic Approach to Tuning of MEMS Resonators
By
Abhyudai Singh

A novel method of bidirectional frequency tuning of a beam-type MEMS resonator is
presented. This method is based on the use of follower and axial end loads that are shown to
decrease and increase the resonant frequency of the beam. The end loads were implemented
in the resonator through electrostatic forces that act near the free end of the beam through
specially designed comb fingers. Preliminary experimental results on fabricated resonators
confirm some of the theoretical results and illustrate the change in resonant frequency with
varying end load. A control scheme that uses these end loads to tune the resonant frequency
of the resonator to any desired frequency is provided. For a particular design, we show
that fast and effective double sided tuning, i.e. both increase and decrease of the natural

frequency, can be achieved with a range of upto 20% of the original resonant frequency.

To my family

iii

ACKNOWLEDGMENTS

I would like to express my deepest appreciation to my advisor Prof. Ranjan Mukherjee
for his guidance and insight. I would also like to thank Prof. Kimberly Turner, Maria
N apoli and Weibin Zhang for fabrication and testing of the resonator devices. Last but not
the least, I thank my parents (Yatindra and Neeta Singh) and my fiancee (Swati Singh).

They have been a great source of strength all through this work.

iv

TABLE OF CONTENTS

LIST OF FIGURES vii
1 Introduction 1
2 Mechanics of a beam with end forces 5
3 Implementation of Follower and Axial End Force 10
3.1 Design I .................................... 10

3.2 Design 11 ................................... 16

4 MEMS Resonator Design and Testing 21
4.1 Resonator Designs .............................. 21
4.2 Preliminary Test Results ........................... 27

5 Frequency 'Ihning 32
5.1 Initial Setup .................................. 32

5.2 Controller design and analysis ........................ 36
5.2.1 Continuous-time controller ...................... 36

5.2.2 Discrete-time controller ....................... 39

6 Conclusion 45

BIBLIOGRAPHY

vi

48

1.1

2.1

3.1

3.2

4.1

4.2

4.3

4.4

4.5

4.6

4.7
4.8

5.1
5.2

5.3

LIST OF FIGURES

A cantilever beam with axial and follower end loads ............ 2

Variation in the first two natural frequencies of the cantilever beam with
a) variation in the axial force for zero follower force, and b) variation in
follower force for zero axial force. ...................... 7

Design I : A cantilever beam with two arc-shaped comb fingers in its (a)

unperturbed state (b) perturbed state. .................... 11
Design II : A cantilever beam with two arc-shaped comb fingers and spe-
cially designed fixed elements A and B. ................... l6
MEMS resonator design based on Design I. It uses tensile follower and
axial end forces to tune resonant frequency. ................. 22
MEMS resonator design based on Design I. It uses compressive follower
and axial end forces to tune resonant frequency. ............... 24
MEMS resonator design based on Design II. It uses tensile follower and
axial end forces to tune resonant frequency. ................. 25
MEMS resonator design based on Design 11. It uses compressive follower
and axial end forces to tune resonant frequency. ............... 26
The layout of the four designs shown in Figures 4.1—4.4 with the voltage
pads. ..................................... 28
Fabricated MEMS resonators based on design shown in Figure 4.1. . . . . 29
Fabricated MEMS resonators based on design shown in Figure 4.1. . . . . 30
Resonant frequency of the MEMS resonator presented in Figure 4.4 for
voltages VA == V3 = V. ............................ 31
Setup for frequency tuning of the MEMS resonator. ............ 35

Tuned resonant frequency of the resonator for the controller presented in
Section 5.2.1 and setup as in Figure 5.1 .................... 38
The peaking effect in the function g(e). ................... 39

vii

5.4

5.5

5.6
5.7

Tuned resonant frequency of the resonator for the controller presented in
Section 5.2.1. The setup is as per Figure 5.1 and g is passed through an
extra saturation element of maximum and minimum amplitude of i1. . . . 40
Tuned resonant frequency of the resonator (dashed line) for the controller
presented in Section 5.2.1. The setup is as per Figure 5.1 with no low pass
filter, i.e, y(t) is directly fed to the controller. ................ 40
Voltage V[n] used for generating the follower force after different iterations. 43
Tuned frequency of the MEMS resonator after different iterations. The
desired frequency is shown in the solid line. ................. 44

viii

CHAPTER 1

Introduction

Micro resonators have been used in a number of applications, including filters [4], sensors
[6, 2], accelerometers [12] and gyroscopes [13]. A common problem in the use of
resonators for such applications is the variability in the resonant frequency, which can
arise from a number of sources, such as manufacturing tolerances, thermal effects, etc.
Various frequency tuning methods, which maintain the natural frequency at a fixed value,
have been proposed and include electrostatically adjusting the stiffness of the structure
[14, 15], changing internal stress to induce change in the natural frequency [1 l], and using
deposition processes [10]. This paper presents another method for tuning the frequency of

a micro-scale cantilever beam resonator by applying axial and follower end forces.

The transverse vibrations of slender beams subjected to constant amplitude follower
and axial end forces is a classical, well-understood subject. It is known that axial and
follower end loads affect transverse vibration natural frequencies of a beam [1]. In fact a
compressive (tensile) axial end load, which always remain aligned with the undefonned

neutral axis of the beam, decreases (increases) the first natural frequency of the beam.

RIM
=l\

 

 

Cantilever Beam

.-_ \\\\\

i
=0 x=l

XL.

Figure 1.1. A cantilever beam with axial and follower end loads

A compressive axial end load, denoted by n, acting on a cantilever beam is shown in
Figure 1.1. In this figure, 0' denotes a compressive follower end load, which always remain
parallel to the slope at the free end of the beam. A compressive (tensile) follower end force
increases (decreases) the frequency of the fundamental mode of the beam [1]. If both axial
and follower end loads can be realized on a microscale cantilever beam, then, double-sided
tuning, i.e. both increase and decrease of natural frequency, can be achieved. Double-sided
tuning can be achieved with a compressive axial end load and a compressive follower end
load, or a tensile axial end load and a tensile follower end load. Since compressive axial
and follower forces can lead to buckling and flutter instability, respectively, tensile end

loads are preferred for double-sided tuning.

We introduce two novel structural designs to realize axial/follower end loads. These end
loads are implemented using electrostatic forces that act near the free end of the beam
through specially designed comb fingers. The combs are arranged so that the orientation

of the applied electrostatic force can remain parallel to the undeformed neutral axis of the

beam, or can move with the beam so as to always act tangentially to the beam, and hence
generate axial and follower end loads, respectively. Based on this, we present various
designs of MEMS resonators. For one particular design we show that bi-directional
frequency tuning in the range of 20% of the original resonant frequency could be achieved.
For a given desired frequency cup in this range, different control schemes which use
follower/axial end loads to tune the first resonant frequency of the resonator a). to this
desired frequency are explored. Similar to what has been proposed in [8, 9], we first obtain
the error e = ((01 — (Up) by implementing a multiplication type phase detector which
consists of an analog multiplier and a low pass filter. It can be shown that the steady-state
output of this phase detector is a function g(e) where g(0) = 0, g(e) > 0 if e > 0 and
g(e) < 0 if e < 0. Thus, depending on the sign of g, the appropriate end load is applied to

increase or decrease an and converge it to the desired frequency cop.

Using the output from the phase detector, a continuos time controller is implemented,
which using integral action, applies the appropriate magnitude of follower or axial force
and ensures robust asymptotic regulation of the error to zero. Implementation of the above
controller in Matlab results in limit cycles with the resonant frequency of the resonator
oscillating about the desired frequency. As was noted in [8], this typically occurs due to the
transient response of the resonator and the peaking effect in g(e) near e = 0. Introducing
saturation in g lowered the amplitude of the limit cycles but did not eliminate them. The
above problems were circumvented by introducing a new controller which only used
information about the sign of g(e). Unlike the previous controller, this controller was

implemented in discrete time with time between each step corresponding to the time taken

by the resonator to reach steady state. By implementing a binary search algorithm [16], the
controller finds the appropriate magnitude of the end load for which e = 0. Simulations for

this controller show fast and effective tuning of the resonant frequency.

We begin with a brief review of the dynamics of a slender cantilever beam with ax—
ial and follower forces applied to the free end, emphasizing the effects that these loads
have on the fundamental natural frequency of the beam. The proposed means of realizing
these forces in a micro-scale beam resonator are then presented. This is followed by pre-
liminary experimental results on fabricated resonators illustrating the change in resonant
frequency with varying end load. Towards the end, we present control schemes to tune the

resonant frequency followed by conclusions and directions for future work

CHAPTER 2

Mechanics of a beam with end forces

Consider a uniform cantilever beam that has both a follower force F and an axial end load
P applied to its free end. A dimensionless version of this system is shown in Figure 1.1. If
E denotes the Young’s modulus, I the second area moment, p the mass density, A the cross
sectional area, and L the length of the beam, then, the dimensionless end load parameters
are defined as 0' = F L2 /EI and n 2 PL2 / E1. While the axial force 7) always remains
parallel to the axis of the beam, the follower force 0' changes direction continuously so as
to always act tangentially to the beam. With the usual assumption of the elementary theory
of bending, the normalized equation of small oscillations of the beam is

84 .t 82 ,t 82 ,t
+3: 25:: >+ 2::

 

where z(x,t) is the transverse deflection of the beam normalized by the beam length L and

measured such that x = 0 is the fixed end. Time has been rescaled by the characteristic

frequency \/ E1 /ApL4. The boundary conditions can be written as

 

z(x,t) lx=0 = 0 (2a)
8z(x,t) _
Tito—0 <2b>
822(x,t)
-8—2x_ix=l : O (26)
832(x,t) 82(x,t)
Vix=l ax ix=l : 0' (2d)

Considering a separation of variable solution z(x,t) = Z (x)T(t) for (1) yields

—+(0'+7l)—=*—=wn (3)

where prime and dot denote derivatives with respect to the spatial variable x, and the tem-

poral variable t, respectively. From equations (2) and (3), we have

z””+(o+n)z”—w32=0 (4)
and the boundary conditions
Z (O) = 0 (5a)
z’(0) = 0 (519)
z”( 1) = 0 (56)
Z”’(l)+nZ’(l)=0. (5d)

\ 30

. l
\ ._'
- 30.05
\

20;
!

  
 

 

     
 

 

 

i i
1' (I)
”’l 2.48 3 “10/:
t , (”I
l 1 .
,z,_ . .zx _ . .. .. . _-p "I ,_.. 1 - - ‘b 0
-IS -10 -5 5 1() IS -IS -10 -5 5 It) 15

Figure 2.1. Variation in the first two natural frequencies of the cantilever beam with a)
variation in the axial force for zero follower force, and b) variation in follower force for
zero axial force.

If we assume a spatial solution of the type Z (x) = exp(lx), we get

24+(o+n)22—w3=0 (6)

which yields

Z(x) = A cosh(llx) + Bsinh(llx) + Ccos(/12x) + Dsin(2.2x) (7)

where

 

 

«earl/{aaiw
*2=\ (3?“M/ {1392M-

We now use the boundary conditions (5) to numerically solve for the natural frequencies

 

 

 

a)" of the beam for different values of the follower force 0', and axial force, 1]. The first

three equalities in (5) give us

C=—A
8/11
D — —_A—2_

1112 sinh()..) - A] A; sinh(lz)

A = B .
ficoshflz) — 212 cosh(ll)

 

Substituting back in (5d) with A] , 22 and eliminating B we have

G(o,n,co,,)=0

for some function G. Natural frequencies of the beam are then given by the roots of this
function for given 0' and 17. Of particular interest to us, is the variation in the natural fre-
quencies of the beam with variation in the magnitude of the follower force when the axial
force is zero, and variation in the natural frequencies of the beam with variation in the mag-
nitude of the axial force when the follower force is zero. Figure 2.1(a) shows the first two
natural frequencies of the beam as a function of the axial end load for zero follower force.
This indicates the decreasing nature of the natural frequencies as 17 increases, and shows
the end load at which the beam buckles to be 116, = 2.48. Figure 2.1(b) shows the first two
natural frequencies as a function of the follower force for zero axial force. Note that as the
follower force increases the first natural frequency increases while the second natural fre-
quency decreases. These two frequencies merge, resulting in flutter instability, at a critical
value of the follower force given by CC, = 20.05. To avoid buckling and flutter instability
one typically prefers to work with tensile axial and follower forces which correspond to

negative values of 0' and 77. From the plots in Figure 2.1, its clear that a tensile axial force

will increases the first resonant frequency of the beam where as a tensile follower force will
decrease the first resonant frequency. Hence if both types of forces can be realized on a
resonator, double sided tuning, i.e. both increase and decrease of the first natural frequency,

can be achieved with a large range of frequency variation.

CHAPTER 3

Implementation of Follower and Axial

End Force

In this chapter we present two different structural designs to realize tensile follower and

axial forces end loads using electrostatic forces.

3.1 Design I

Consider the cantilever beam in Figure 3.1(a) with two arc-shaped comb fingers. Each of
the two fingers subtend angle a at the base of the beam and remain parallel to the fixed
elements A and B, respectively. The fixed elements are located symmetrically with respect
to the beam in the unperturbed state, and [3 denotes the angle of no overlap between the
fingers and the fixed elements. The perturbed state is shown in Figure 3.1(b) for small
angular deflection of the beam, 0. In this state, the angular overlap between the fingers
and the fixed elements A and B are (a — [3 + 6) and (a — B — 9), respectively. If FA and

F3 denote the electrostatic forces applied on the fingers by the fixed elements A and B,

10

fixed element A

 

fixed element B
(a)

fixed element A

\
(a-BCB)

rs ‘ 5 i
' (a-B-B)
/ ._

 

fixed element B

(b)

Figure 3.1. Design I : A cantilever beam with two arc-shaped comb fingers in its (a) unper-
turbed state (b) perturbed state.

11

respectively, then the resultant of these forces have the following x and 2: components

2
[72 : FA sin (Eigfl) _FBsin (w) .

Ft : FA COS (W) +FBCOS (W)

2

The resultant force subtends angle W with the x axis which is given by

F.
tan at = F° (8)

For some constant K, the following choice of FA and F3
FAszin[(a+[3+9)/2], F3=Ksin[(a+,B-9)/2] (9)
leadsto

Fx 2 §[sin(a+[3+9)+sin(a+l3—9)]
= Ksin(a+B)cos(6)
F. = §[cos(a+fi—9)—cos(a+fi+9)]

= Ksin(a +13) sin(0)

VF} +Fz2 = Ksin((x+[3).

Substituting the force components back in (8) we have V = 9. Thus, we have generated a

12

tensile follower force of constant magnitude

0: —Ksin(a+B). (10)

Similarly one can show that

FA=Ksin[(a+B—9)/2], F3=Ksin[(a+l3+9)/2] (11)

leads to W = O, and hence, generates a tensile axial force with

n = —Ksin(a +I3).

From electrostatic principles, the forces FA and F3 are also given by the relations

80

Iq=zgmfiua—B+m,fi=

80

ngfiua—p—m on

where 80 is the electrostatic permittivity of vacuum, g is the radial gap between the comb
fingers and fixed elements, h is the height of the resonator, L is the length of the beam, and
VA and V3 are the voltages applied between the comb fingers and the fixed elements A and
B, respectively. By choosing voltages VA and V3 appropriately (as functions of 6) in (12),
one can apply forces FA and F3 as given by (9) and (11), and hence realize follower and

axial end loads, respectively. In particular, substituting the voltages VA and V3

 

 

_ (a—B)sin[(a+fi+9)/2] A
V" ‘ V\/(a—B+9)sin[(a+B)/2] ‘ V7149)

l3

 

 

_ (a—B)sinl(a+fi-9)/Zl A
V” ‘ Vl/(a—r —9)sinl(a+B)/2] ‘ ”5”")

in (12) we have FA and F3 of the form as in (9) with

_ sohv2L(a—B)
_ 2828in[(a+l3)/21

 

and a follower force of 0 equal to

_ _£0hV2L(a —B)sin(a+l3)
_ 2g2 sinl(a+B)/21

 

Similarly, the choice of voltages

 

 

_ (a—B)sin[(a+B—6)/2] A
V" ‘ V \/(a —p + 6)sin[(a+B)/2] ‘ ”5(9)

 

_ (oc-—,/3)sin[(0£+l3+9)/2]A
VB‘ V\/(a—fi—9)sinl(a+fi)/21 ”Tim

generates an axial force. In particular, substitution of the above voltage expressions in (12)

results in FA and F3 of the form as in (11) with

_ £0hV2L(a —[3)sin(a +13)
_— 2gzsin[(a+B)/2] '

 

In the sequel we refer to the functions TA’f(0), T15 f(9), TAU?) and T549) as follower and
axial force generators. Using them tensile axial and follower forces of any appropriate

magnitude can be realized by simply adjusting the voltage V. As one can see, these fol-

l4

lower/axial force generators are functions of the angle of deflection 9, which can be simply
calculated by measuring the transverse deflection at some point along the beam using a laser
vibrometer. Also, as the deflection angle is small, the above expression can be expanded
using Taylor series to yield much simpler expressions which will be easier to implement.

After a Taylor expansion, we have the following approximation

33(9) = 1+O.5a16+0.125a202
rgfw) = 1—0.5a19+0.125a292
7“],(9) = l—O.5a36+0.125a492

T349) 1+0.5a36 +0.1250t492

for TA’f(9), Téf(6), TA’x(9) and Téx(9) respectively, where

c = 06+[3
d = a—fi
06 — 05cot(£)—l
' _ ' 2 d
_ 1 cot(c/2) 1 2
a2 — 4 [d—z -O.125——2-2——] — [O.5COI(C/2)— 2]

a3 = 0.5cot(§)+%

cot(c/2)

1 1 2
a4 = 4[6-[—2-—0.125+ 2d J—[0'5C0t(6/2)+c—1] .

Note that al, which is a difference of two terms, can be made small or even zero by pr0per

designing. Then, ignoring the quadratic term in 6 for small angles of deflection, we will

15

Fixed
Element A

  
   

 

Fixed
Element B

Figure 3.2. Design II : A cantilever beam with two arc-shaped comb fingers and specially
designed fixed elements A and B.

have T/(f(6) z T5146) 3 1 which implies that follower forces can be realized by applying
voltages VA 2 V3 = V independent of 6. As 0.5 cot (c / 2) z 1 /c for small angles of deflec-
tion, we need c m d for a1 = O which implies B 2 0. This is physically unrealistic when

you have many comb fingers one after another.

3.2 Design 11

We now present a second structural design to realize follower and axial forces as has been
shown in Figure 3.2. Here, unlike Design I in Figure 3.1, fixed elements A and B have an
protruded structure at their end. The angular position of the beginning and end of the pro—

trusion, from the undeformed axis of the beam is denoted by angles a and B, respectively.

16

In this design the angle of overlap between the protrusion and the comb finger of the beam
remains fixed at a — B, for small transverse deflections of the beam. Thus the electrostatic

forces from the fixed elements are given by

80 8(
a = @hviua—B), 3,: ~2—g’5hv5ua—rs) (14)

and make a fixed angle of (a + B) / 2 with the x axis. The x and 2 components of the

resultant force is given by

F, = FAcos(a;—B)+F3cos(a:‘6)
F: = FAsin(a:B)—F3sin(a:fi).

 

 

 

 

If one chooses
FA 2 K sin [(a+B +26)/2], F3 = Ksin [(a+B —26)/2] (15)
then we get

F, = Kcos (a +B)cos(6)

F. = Ksin(a +B)sin(6)

(/F,(2+Fz2 = Ksin(a+B).

l7

This implies from (8) that I]! = 6, and hence, a tensile follower force of constant magnitude

0' = —Ksin(a+B).

Similarly, the choice

 

FA=Ksin(9%P—), F3=Ksin(a:'8) (16)

leads to

Fx = Ksin(a+B)

F: = 0
which implies w = 0 and a tensile axial force of magnitude
1] = —Ksin(a +B).

For the following choice of voltages VA and V3

 

 

_ sin[(oz+13+2e)/2]A
VA“V\/ sinl(a+B)/21 ”77‘9“”

 

 

_ sin[(a+B—29)/2] A
VB‘Vl/ sin[(a+l3)/2] ””55“”

18

the force FA and F3 are of the form as in (15) with

__ 80hV2L(a—l3)
_ 2gzsin[(a+B)/2]'

 

This results in a follower force of magnitude

_ _£0hV2L(a—B)sin(a+B)
— 2g2sin[(a+B)/2]

 

Similarly, the choice of

VA = V3 = V
leads to an axial force of the same magnitude, i.e

_ __80hV2L(a —B)sin(a+B)
— 2gzsinl(a+l3)/21

 

Note that these voltages are independent of the angle 9.

For small angles of deflection the follower and axial force generators TA’HB) and

Té}(9) can be binomially expanded to give

 

r1119) = v [1+cot(a:l3) 9 —O.562]

 

Trifle) = V[1—cot(a:fi)9—O.592].

l9

Note that similar magnitude compressive axial and follower forces can be generated in
both the designs by putting the fixed elements A and B behind the comb fingers of the beam

instead of having them in front as in Figures 3.1 and 3.2.

20

CHAPTER 4

MEMS Resonator Design and Testing

4.1 Resonator Designs

Based on the structural design presented in the previous section, we now provide a
schematic of a resonator whose first resonant frequency can be tuned up or down through
the application of axial and follower end forces. The resonator, shown in Figure 4.1, has
a set of anterior combs for the application of tensile end force (axial or follower) and an
additional set of lateral combs for actuation and sensing. The actuator combs will be used
to provide sinusoidal excitation for resonance. Although the deflection of the beam will re-
main small, all curved elements are designed by taking the radius of curvature of the beam
(corresponding to the first mode shape) into consideration. More specifically, with the first
normalized mode shape of the cantilever beam given by (7), radius of curvature (R) for a

comb finger located at x = W, is given by

217 x=W

21

 
  
   

pad for fixed elements A

   
  

comb actuator

anchor

 

 

 

 

 

comb sensor

pad for fixed elements B

 

Lav : 430“ —’

a ~— 77 330 VA—s 25
2 “ 55011 M

 

 

Figure 4.1. MEMS resonator design based on Design I. It uses tensile follower and axial
end forces to tune resonant frequency.

22

and its center of curvature is located at distance C = W —— R from the fixed end. This
ensures that the gaps between the fixed and moving elements remain constant and prevents

potential malfunctioning due to varying gap or mechanical contact.

The MEMS resonator in Figure 4.1 has the following specific dimensions and mate-

rial property
a: 16°, B=8°, 11:30“, 3:211, E: 170x 10% (17)

Using a lumped-mass model of the fingers, the first natural frequency was found to be
approximately 7.5 KHz. For any voltage V, end loads 0' and n can be calculated form (13)
with an, in place of L. Here n = 9 is the number of anterior fingers for application of end
force, and Lav = 43011 is the average distance of the anterior fingers from the anchor. For

these values one can calculate the magnitude of the end load to be

v 2A 2
22(6) (.8)

Hence with voltage V upto 70 volts one could realize non-dimensional follower and axial
forces upto 0 = —1 and n = —1. From the analysis in Section 2, it can be seen that
while 0' = —1 decreases the first natural frequency by 425 Hz, 17 = —1 increases the
natural frequency by 1.1 KHz. Thus, bi-directional frequency tuning with a range of

approximately 1.5 KHz (20%) can be achieved through voltage adjustments.

23

    
 

Fixed
Element

Comb
Actuator

     
    
 

 

Anchor

Comb
Sensor

Fixe
Element B

  

Figure 4.2. MEMS resonator design based on Design I. It uses compressive follower and
axial end forces to tune resonant frequency.

Note that this resonator uses Design I presented in the previous chapter to realize
tensile follower and tensile axial end loads for decreasing and increasing the first resonant
frequency, respectively. Variations of this resonator where frequency tuning is done using
Design II are also possible. Moreover by moving the fixed elements behind the comb
fingers one could work with compressive follower and axial end loads. Figures 4.2, 4.3
and 4.4 illustrate some of the obvious options. Table 4.1 provides the voltages VA and V3
needed to apply a follower and axial force for Design I and II based on parameters given
in (17). These voltages were based on the calculations on page 15 and 19 after ignoring

the quadratic 92 terms.

24

 
   
  
 
 

Fixed
Element A

Comb
Actuator

  
   

Anchor

Sensor

Fixed
Element B

Figure 4.3. MEMS resonator design based on Design II. It uses tensile follower and axial
end forces to tune resonant frequency.

Table 4.1. Voltages VA and V3 needed to apply a follower and axial force for Design I and
II based on parameters given in (17).

 

 

 

Design description Voltages for follower forces Voltages for axial forces
Design I VA = V(l — 2.49) VA = V(l —4.766)
V3=V(l+2.49) V3=V(l—4.760)
Design II VA = V(l +4716) VA = V
V3=V(l—4.710) V3=V

 

 

 

25

     
  
     
 

Comb

Actuator

Anchor

Comb
Sensor

Figure 4.4. MEMS resonator design based on Design II. It uses compressive follower and

axial end forces to tune resonant frequency.

4.2 Preliminary Test Results

All the above designs shown in Figures 4.1—4.4 were fabricated and tested at the MEMS
facilities at the University of California, Santa Barbara. The layout of the four designs is
shown in Figure 4.5 with voltage pads, which will be used for actuation of the resonators
and applying voltages VA and V3. Using this layout, the devices were fabricated from a
801 (Silicon On Insulator) wafer by first etching it by DRIE (Deep Reactive Ion Etching)
process. The insulator was then etched using HF which released the resonators and was
followed by drying and wirebonding. Figure 4.6 and 4.7 shows the fabricated resonator

corresponding to the design in Figure 4.1.

With the present setup we were unable to measure the angle 6 and hence apply an-
gle depend voltages VA and V3. Thus we were only able to verify realization of axial force
using Design II which as can be seen from Table 4.1 requires angle independent voltages
VA 2 V3 = V. Figure 4.8 plots the resonant frequency of the MEMS resonator presented in
Figure 4.4 for different voltages V. As one would expects from Figure 2.1 (a), the resonant
frequency decreases with increasing voltage V, i.e increasing compressive axial loads. In
conclusion, as of now, these experimental results are inconclusive and more work will be
needed to experimentally verify the frequency increase/decrease for axial and follower end

loads.

27

 

Figure 4.5. The layout of the four designs shown in Figures 4.1—4.4 with the voltage pads

28

 

 

Figure 4.6. Fabricated MEMS resonators based on design shown in Figure 4.1.

29

 

Figure 4.7. Fabricated MEMS resonators based on design shown in Figure 4.1.

30

7750 . fl

 

 

 

 

Ts?
.C
E. at
>
8 7700- *
Q)
8- ate
9
LL
ale
7650 ‘
o 5 10 15 20
V

Figure 4.8. Resonant frequency of the MEMS resonator presented in Figure 4.4 for voltages
VA = VB = V.

31

CHAPTER 5

Frequency Tuning

In this chapter we present a procedure for frequency tuning of the MEMS resonator pre-

sented in the previous chapter to any prescribed frequency.

5.1 Initial Setup

We begin by presenting an idealized model for the MEMS resonator. If z(t) is the transverse
deflections at some point on the resonator and u(t) the actuation signal applied on the
resonator, then the dynamics can be described by a lightly damped second order system

given by
210+ ZC (012(1) + wiar) = Qu(t)

where C is the damping factor, Q is some positive constant, w] = f (0', n) is the first natural
frequency of the beam and is a function of the applied follower and axial end loads. This

constant Q can easily be determined from experiments by applying a constant voltage and

32

measuring the output in the static configuration. The transfer function from u(t) to z(t) is

then given by

Q

— 32+2Cwls+wiT

 

T(s)

Our goal is to place the resonant frequency of the resonator at a prescribed frequency (03.
As has been proposed in [8, 9], we obtain the error e 2 (an — (03) by implementing a
multiplication type phase detector. Taking the actuation input as u(t) = U sin(th), we get

the output

z(t) = |T(ja)D)|Usin(a)Dt +®(ja)3)) +transients

where |T(ja)3)| and @(jwp) denotes the magnitude and phase angle of T(ja)3), respec-
tively. After allowing, the transients to decay, we implement a phase detector by first
multiplying the steady state sinusoidal output by U sin(a)3t) to get the signal y(t). Thus we

have

y(t) = |T(jw3)|Uzsin(th+®(jc03))sin(wpt)

- 2
= WU— [cos(®(ja)p)) — COs(2th + @0090)”-

One can then get rid of the time-varying part in y(t) by either doing time averaging of the

signal, i.e integrate and then divide by time, or by implementing a low pass filter with cutoff

33

frequency well below (03. After this, we have the constant output

where

C(e)

 

 

 

U2 . .
7|T(JwD)ICOS(®(JwD))
(12 ,
7 Re (70010))
112 Q
— Re 2 2 .
2 (DI—(DD‘i-chlij
UZQ (00 + (101
a) — a)
2 (mg—mf)2+4§2wgmlz( l D)
UZQ 2w3+e e
2 [(2w3+e)2(e)2+4§2wg(w3+e)2]
G(e)e
(1)1 - (DD
UgQ 2wp+e

 

2 [(ZwD + e)2(e)2 -l- 4C2w12)(a)1) + e)2]

>0.

Note that if g > 0, then a); > (1)3 and we need to apply a tensile follower end force to reduce

a), down and similarly if g < 0 we need to apply a tensile axial end force to increase (01 up.

We will now use the output g from the phase detector to design controllers for frequency

tuning. In particular, the goal of the control design is to tune voltage V in (18) to give

the appropriate magnitude of end loads for which e = 0. Figure 5.1 summarizes the above

discussion and shows the setup for frequency tuning of the MEMS resonator.

34

 

 

 

 

 

 

 

 

 

 

 

Product y Lomtgeriss
Angle

 
   
 

 

 

Generator

 

 

 

 

 

 

 

 

 

 

 

 

_ Follower/Axial V
Force Controller
Generator m

 

 

Figure 5.1. Setup for frequency tuning of the MEMS resonator.

35

5.2 Controller design and analysis

In this section we investigate two different control designs to tune the frequency of the

resonator.

5.2.1 Continuous-time controller

As the first natural frequency (1)1 = f (0', n) we have

=w,=§§a+-§%x (19)

Note form Figure 2 that 8 f / 80' is positive while a f / an is negative. Assume that with no
end forces, g is positive, which implies that the error e is positive and we need to implement

a tensile follower force. We do this by

VA = VTAlf(6)

where

36

 

and K; > 0 is an appropriately chosen gain. Using (18) and (19) we have

0' 2: —pW
7? = 0
3f _ 3f
e —%pW —a pKlG(e)e

Note that n = 0 as we are only applying a tensile follower force. With 8 f / 80pK 1 G(e) > 0,

the error e asymptotically converges to zero.

If initially, with no end forces, g < 0, then we can similarly implement an axial

force with

VA = VTXAQ)
V3 = mm)
v = t/W

W = —K18

and similar analysis as above shows the asymptotic convergence of the error to zero.

The above controller is implemented on Matlab by assuming that the natural fre-
quency of the beam without end forces is a). = 47,007 rad/sec or 7.48 KHz, the desired
frequency is 46,000 rad/sec, C = .001, Q = 1, U = 104 and K 1 = 1. As the desired resonant

frequency is lower than the actual resonant frequency we will be using a tensile follower

37

 

4.72 I I I I I I I I I

P
‘1
T

1

.e
B
I
J

.e

82
I

/
l

4.62 e \ . _

4 6 _ \\ 'I/m /\ \ / \\ /\ /\ I 1‘ / 1“ / A
. \J/ / \ J/ \ / \/ \l/ \ I/i \ I l/

4.58 — ‘ ~

Frequency (rad/sec)
&
9

 

 

 

4.56 1 l 1 l l l 1 1 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.2. Tuned resonant frequency of the resonator for the controller presented in Sec-
tion 5.2.1 and setup as in Figure 5.1.

force to decrease it. The low pass filter was taken to be

1
013+]

to remove the time varying part from y(t). As shown in Figure 5.2, this lead to limit cycles
with the natural frequency of the resonator oscillating about the desired natural frequency.
As was noted in [8], this typically occurs due to the transient response of the resonator and
the peaking effect in g(e) near e = 0. This effect can be seen in Figure 5.3 which plots
g as a function of e for the above parameters. To remove the peaking effect we passed
g through a saturation element of maximum and minimum amplitude of :1:1. As shown

in Figure 5.4 this lowered the amplitude of the limit cycle. Lowering the maximum and

38

 

r T \ r r 1 r

&
I
l

 

 

 

F\ ..
a) 0 - *
ERR ‘Kr
. \\\ .
-2 .- \\ _
-4 — -
-6 — 1 1 1 1 1 1 1 1 1 —
-1 000 —800 -600 -400 -200 0 200 400 600 800 1 000

Figure 5.3. The peaking effect in the function g(e).

minimum amplitude of the saturation element further reduced the amplitude of the limit
cycle but did not eliminate it. Directly feeding y(t) to the controller, i.e removing both the
low pass filter and the saturation element, lead to the elimination of these large amplitude
limit cycles but small, high frequency oscillations appeared as can be seen from the insert
of Figure 5.5. As this performance is not adequate for our application we turn to a different

controller.

5.2.2 Discrete-time controller

Here we present a very simple discrete-time controller V[n] to tune the natural frequency of
the resonator. Unlike the above controller here we do not explicitly use the product G(e)e

in the control design but only care for its sign. Also this controller waits for the beam to

39

 

 

 

 

x 10
4.72 I I I I I I I
4.7 b r
A 4.58 * \ \ ‘
§ .
B _
g 4.66 r \\ —
E 4 64 ~ \‘ .
3 ' \.
E \
LL 4.62 ~ \ -
\
\,\
’\ f‘ /\
4 5 ~ \ / \ 11/ \ /\ -
\ / \/ \/ \/ \
4.58 I I I I g l l
0 0 1 O 2 0 3 0.4 0 5 0 6 O 7 0 8
t

Figure 5.4. Tuned resonant frequency of the resonator for the controller presented in Sec-

tion 5.2.1. The setup is as per Figure 5.1 and g is passed through an extra saturation element
of maximum and minimum amplitude of i1.

 

 

 

 

 

 

 

 

x104
4.72 I I 4 I I T I 1
x10
468 4.5995~_‘ ., ' p . .
3‘ I 1 ' ~ -' llilllllllllllllllllllll
llllillllllllllllllllllllllllilllll. ,
3 4.5995] 1 . j ‘ .
'8 4.66- 0.78 0.7805 0.781 0.7815 0.782 0.7825 0.783 0.7835 0.784.
‘6‘
(134.64
3
U'
9
"-4.62~ —
‘41
4.6 e1
4'58 1 ' I l ' l ‘ I L
o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.5. Tuned resonant frequency of the resonator (dashed line) for the controller

presented in Section 5.2.1. The setup is as per Figure 5.1 with no low pass filter, i.e, y(t) is
directly fed to the controller.

40

reach steady-state as was assumed earlier in the analysis. To further simplify the controller
we use a simple time averaging scheme instead of a low pass filter to extract the constant g
from the time varying signal y(t). We begin at t = 0 with the resonator being actuated with
u(t) = U sin(a)3t) and no end forces, i.e. V[l] = 0. After waiting for .1 seconds, in which
time transients die out“ the corresponding g[l] is calculated as outlined above. Suppose
g[1] > 0, then we need to apply a follower force by choosing an appropriate 0' E (—l,0)
(similarly if g[l] < 0 one would apply an axial force instead of a follower force). More

precisely this 0' needs to be the solution 6* to the equation
g(f(0*,0) - (DD) = 0,

which we find using a binary search algorithm [16]. Hence, we start an interval, divide
it into two halves and picks the subinterval in which the root exists and then divides this

subinterval into two halves and so on.

We start with the interval (—l,0) and divide the interval into two halves (—1,—0.5)

and (—0.5,0). In order to find where the root of (5.1) is, we apply 0' = —0.5 to the

 

“Assuming C = .001 and a). z 47,100 rad/sec = 7.5 KHz, the time constant l/wIC is given by .02
second. Four times the time constant is .08 second which is less that .1 second

41

resonator by taking

VA = V[ZITAf(9)
VB = V[2]T3f(9)

v12] 2 @.

After again waiting for .1 seconds the corresponding g[2] is obtained. If g[2] is still positive
then we need to apply a larger follower force, and hence the root exists in (— 1, —0.5). This

interval in then further subdivided and checked for roots by taking 0' = —0.75 and thus

.75
V[3] = —0—.
19

But if g[2] is negative, then we need to apply a lesser follower force then previously applied
and hence the root exists in (——O.5, 0). In this case we proceed further by taking 0' = —0.25

for which

0.25
v13] = —.
P

As both functions g, f are continuous, this binary search or bisection method, guarantees

the convergence of V[n] to the value

*

 

lim V[n] =

"—700

and lim g[n] = 0
n—mo

42

 

 

 

 

60 I I I I I I I
50~ . g ‘ ~
1
l r_____
l W.”
40- i ! 1___ ....J -
l l
L. I .~
2?
a 30F _.
Z
w
01
£9
a 20_ ..
>
10~ a
o a
-10 1 I I 1 I I .l
2 3 4 5 6 9

 

 

 

Figure 5.6. Voltage V[n] used for generating the follower force after different iterations.

even though the precise forms of functions f and g are uncertain. Figures 5.6 and 5.7
show a simulation of the above iteration where as before, the natural frequency of the
beam without end forces is 47,007 rad/sec or 7.48 KHz, the desired frequency is 46,000
rad/sec, C = .001, Q = l and U = 3 :1: 104. In this simulation the above iteration was stopped
once |g[n]| became less than a pre-determined constant. Here this constant was chosen to
correspond to an error of approximately :El Hz. As one can see from the figures it took
about 9 steps, which would be less than a second, to tune the frequency of the resonator.
This is very good considering the fact that the frequency of the MEMS resonator changes

slowly due to temperature and other imperfection.

43

 

4.72 l I I F I I

3?

'2

9

(D

N
1

Frequency (rad/sec)

 

."
o:

4.58 r

4.56 — ————— ' «

 

 

)—
)—
h-
)—

 

4.54 ' ' 1
4

Figure 5.7. Tuned frequency of the MEMS resonator after different iterations. The desired
frequency is shown in the solid line.

44

CHAPTER 6

Conclusion

A method to tune the resonant frequency of a micro-scale cantilever beam MEMS
resonator using axial and follower end loads was presented. These end loads were realized
by using electrostatic forces that act through specially designed comb fingers at the free
end of the beam. The magnitude of these electrostatic forces were appropriately changed
based on the angle of deflection of the beam, such that the resultant applied force can
remain parallel to the undeformed neutral axis of the beam, or can move with the beam
so as to always act tangentially to the beam, and hence generate axial and follower end
loads, respectively. Two control schemes were explored for frequency tuning. In these
scheme, a multiplicative phase detector is implemented to provided information about the
error between the desired and the actual resonant frequency of the resonator. In the first
control scheme, a continuous-time controller was implements which led to limit cycle and
oscillation of the actual resonant frequency about the desired resonant frequency. In the
second scheme, a discrete-time controller was implemented which finds the appropriate

magnitude of the axial/follower end load for which the error is zero through a binary

45

 

search algorithm. We demonstrated through simulation that using this control scheme,
fast and effective double sided tuning with a range of upto 20% of the original resonant

frequency can be done.

As has been mentioned before, experimental verification of implementation of fol-
lower and axial end forces is still inconclusive and is a direction of future work. This can
only be done if we can estimate the angle of deflection of 6 and apply angle dependent
voltages VA and V3. At present the resonant frequency is estimated using a laser vibrometer.
A direction of future work would be to build a microchip which directly uses the comb
sensor as shown in Figure 4.1 to estimate the resonant frequency, angle of deflection (6)
and apply axial/follower end forces to tune frequency. Another direction of future work
includes exploring other types of resonator designs that use such end forces for frequency

tuning.

46

 

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47

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[41

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49

 

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