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This is to certify that the
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PARAMETRICALLY-EXCITED
MICROELECTROMECHANICAL OSCILLATORS WITH
FILTERING CAPABILITES

presented by

JEFFREY FREDERICK RHOADS

has been accepted towards fulﬁllment
of the requirements for the

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PARAMETRICALLY-EXCITED MICROELECTROMECHANICAL
OSCILLATORS WITH FILTERING CAPABILITIES

By

Jeffrey Frederick Rhoads

A THESIS
Submitted to
Michigan State University

in partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2004

ABSTRACT

PARAMETRICALLY-EXCITED MICROELECTROMECHANICAL
OSCILLATORS WITH FILTERING CAPABILITIES

By

Jeffrey Frederick Rhoads

This thesis investigates a class of tunable microelectromechanical (MEM)
oscillators that can be implemented in systems that act as signal ﬁlters. Unlike
most mechanical and electrical ﬁlters that rely on direct linear resonance for
ﬁltering, the MEM ﬁlters presented in this work employ parametric resonance. As
a result, these ﬁlters feature, in addition to the general beneﬁts associated with
MEM devices, nearly ideal stopband rejection and extremely sharp response roll-
off. Unfortunately, the introduction of parametric instabilities into the system
does present some complications with regard to ﬁltering. These issues are
addressed by means of novel tuning schemes and logic implementation that
make use of a pair of MEM oscillators and associated circuitry. This thesis
includes a brief introduction to bandpass ﬁlters, an analysis of a dynamic model
for electrostatically driven MEM oscillators, descriptions of methods by which the
aforementioned deﬁciencies can be overcome by parameter selection and
tuning, and a method through which two MEM oscillators can be combined to

yield a highly effective MEM bandpass ﬁlter.

To my parents, for their continual support.

iii

ACKNOWLEDGEMENTS

This work would not have been possible without the assistance and guidance of
my advisor, Dr. Steven Shaw. I would also like to thank Barry Demartini, Dr.
Kimberly Turner, and the other members of the MEMS Characterization
Laboratory at the University of California, Santa Barbara for the integral role they
played in this work, and especially for contributing preliminary design information
and MEM oscillator images to this endeavor. Additional thanks go to the Air
Force Ofﬁce of Scientiﬁc Research for providing the necessary funding for the
project, Drs. Cevat Gokcek and Alan Haddow for providing valuable insight and
sitting on my committee, and to my colleague, Brian Olson, for acting as a

sounding board throughout this research.

iv

Table of Contents

LIST OF TABLES ......................................................................... vi

LIST OF FIGURES ....................................................................... vii

CHAPTER 1

Introduction and Background ............................................................ 1

CHAPTER 2

Bandpass Filter Speciﬁcations .......................................................... 6

CHAPTER 3

Employing Parametric Resonance for Filtering ................................... 10

CHAPTER 4

THE Dynamics of a Single Parametrically-Excited MEM Oscillator .......... 17

CHAPTER 5

Linear Tuning: Manipulating the Instability Zone ................................. 28

CHAPTER 6

Nonlinear Tuning: Conditioning the Nontrivial Response ...................... 33
6.1 Averaged Equations of Motion .............................................. 33
6.2 Analysis of the Steady-State Responses ................................. 35
6.3 Tuning the MEM Oscillator’s Nonlinearity ................................ 55
6.4 Summary ......................................................................... 60

CHAPTER 7

Creating a Bandpass Filter ............................................................ 62

CHAPTER 8

Design Robustness Issues ............................................................ 68
8.1 Parameter Sensitivity .......................................................... 69
8.2 Effects of Noise ................................................................. 73

CHAPTER 9

Conclusions and Directions for Future Work ...................................... 75

REFERENCES ........................................................................... 77

LIST OF TABLES

Table 1. Nondimensional parameter deﬁnitions [1, 2].

Table 2. A comprehensive list of the design parameters used throughout this
work [1].

Table 3. Redeﬁned nondimensional parameters [1, 2].

Table 4. Summary of tuning requirements for bandpass ﬁltering [1]. The
“Nonlinearity Region” refers to Figure 9.

vi

LIST OF FIGURES

Figure 1. A sample frequency response function for a bandpass ﬁlter (Adapted
from [3]).

Figure 2. An approximate Strutt-lnce stability chart for the linear Mathieu
equation (,u = 0) near principle parametric resonance. “U” denotes those regions
wherein the equation’s trivial solution is unstable. Likewise, “S” denotes those
regions wherein the trivial solution is stable. Note that with the addition of small
damping the transition curve emanating from .0 = 2 shifts upward in the
parameter space, most notably near the vertex.

Figure 3. The dominant instability zone in the .(2 - ,1 parameter space with a
superimposed nonlinear frequency response for a given value of 2. (Adapted from
[4]); (a) with hardening nonlinearity (p > O) (b) with softening nonlinearity (p < 0).
Note that solid lines indictate a stable response and dashed lines indicate an
unstable response. Also shown is the effect of damping on the instability region.

Figure 4. Schematic of both interdigitated and non-interdigitated electrostatic
comb drives (Adapted from [5]). The arrows designate the dominant direction of
motion. (a) An interdigitated comb drive. (b) A non-interdigitated comb drive in
an ‘aligned’ position. (c) A non-interdigitated comb drive in a ‘misaligned’
position.

Figure 5. (a) A representative parametrically excited MEM oscillator [1]. (b) A
close up view of the non-interdigitated comb drives [1]. The oscillator’s backbone
“B” is the primary mass, the springs “S” provide attachment to ground as well as
the mechanical restoring force, and the non-interdigitated comb drives “N” are
used to provide parametric excitation. “AC” and “DC” indicate voltage sources.
[Pictures courtesy of Barry Demartini and Dr. Kimberly Turner, University of
California, Santa Barbara]

Figure 6. A CAD drawing of a representative MEM oscillator [1].

Figure 7. The regions of parametric instability in the VA - .(2 parameter space
produced using both the system’s averaged equations and simulations of the
system's full equation of motion with the linear parameters deﬁned in Table 2.
(a) The case of pure AC voltage excitation. (b) A nominal case which results in
a symmetric wedge of instability [1].

vii

Figure 8. Parametric instability zones in the VA - .(2 parameter space (for rm > 0)
created using both analytical techniques and simulations. (3) p = O and p = 1/2.
(b) p= 0 and p = -1/2.

Figure 9. The 73 - ’13 parameter space [6]. The roman numerals are used to
designate the various response regions.

Figure 10. Representative frequency response plots showing response
amplitude versus detuning frequency for each of the eight regions delineated in
Figure 8 [6]. The frequency regions designated by the letters correspond to the
phase portraits of Figure 11.

Figure 11. Representative phase plots corresponding to the frequency response
regions deﬁned in Figure 9 [6]. Solid dots indicate equilibrium points.

Figure 12. Sample parameter spaces showing how the oscillator's nonlinearity
can transition between response regimes under a varying input excitation
amplitude; (a) for the p = -1/2 design presented in Table 2, (b) for the p = 1/2
design presented in Table 2 [6].

Figure 13. Sample response curves — amplitude vs. frequency: (a) p = 1/2 and
(b) p = -1/2.

Figure 14. One proposed logic implementation scheme, which is capable of
producing a bandpass ﬁlter [1, 2].

Figure 15. Simulation results from the ﬁlter system of Figure 14 [1]. Data points
are frequency thresholds for the passband boundaries at various AC amplitude
levels. The insets show the system outputs at the points indicated.

Figure 16. Sensitivity plots, which show the sensitivity of p with respect to each

of its components. The black squares designate the nominal design parameters
used for the p = 1/2 oscillator under consideration here.

viii

Chapter 1

Introduction and Background

As the demand for commercial and tactical wireless communications
devices continues to increase, so too does the demand for effective and efﬁcient
bandpass signal ﬁlters. These devices, used to attenuate signals with frequency
components outside of a speciﬁc bandwidth while passing those within, are an
integral part of such technology. This is especially true in the case of cellular

phones, the driving force in the wireless communications industry.

Due to its importance in a number of applications, bandpass ﬁlter design
has been considered a viable research topic since the late 19th century.
However, it was not until the early 20th century that the full potential of bandpass
ﬁlters was realized. At that time engineers working at Bell Laboratories showed
the potential of electrical, and later mechanical, bandpass ﬁlters in practical

engineering applications, such as phonographs [7-10].

Since then, filter research has undergone a steady progression, with

numerous design alternatives being proposed. Amongst the mechanical and

electromechanical ﬁlter designs brought forth, one speciﬁcally stands out, namely
the design proposed by Adler in the late 19403 [10, 11]. This design, which
utilized weakly-coupled mechanical resonators to produce a narrow banded
frequency response function, exhibited high quality factors (deﬁned in the case of
bandpass ﬁlters as the ratio between the ﬁlter’s resonant frequency and its
bandwidth) and much improved operational stability compared to its
predecessors. Following Adler, numerous ﬁlter designs appeared that utilized
ceramic and quartz crystal resonators [12-14]. While these designs advanced
the capabilities of mechanical ﬁlters, they tended to be bulky in comparison with

other designs and thus were utilized in a limited number of applications.

Despite differing in form and appearance, each of the mechanical and
electromechanical ﬁlter designs proposed in the early to mid 20th century
exhibited a number of common characteristics. Most signiﬁcantly, each featured
a relatively high quality (Q) factor and robust operational stability. However, they
proved to be costly and generally incompatible with integrated circuitry and thus
had limited potential following the emergence of such technology in the late 20th

century.

In the electronic rich era of the late 20th century, ﬁlter design focused
primarily on electrical and electromechanical bandpass ﬁlters (see, for example,
[14-181). The proposed designs consisted of chains of coupled resonators, such

as RLC circuits, and also more modern technologies, such as surface acoustic

waves (SAWs) [16-18] and even discrete-time digital systems [14, 19-22]. In
comparison to their predecessors, these designs offer ease of integration with IC
technology. However, they too exhibit somewhat limited potential in some
ﬁltering applications due to limitations in their size, frequency operating ranges,

and quality factors [18].

The introduction of microelectromechanical (MEM) devices into
mainstream engineering has opened a promising new path of study regarding the
development of ﬁlters [1-3, 23-28]. This promise is founded upon the distinct
advantages MEM ﬁlters hold over more conventional designs. Namely, MEM
ﬁlters are typically smaller, consume less power to operate, and have the
potential to operate with signiﬁcantly higher Q factors. MEM ﬁlters also offer

seamless integration with IC technology and are highly tunable in nature.

To date, a number of MEM ﬁlter designs have been proposed, with the
three most prevalent designs being the electrostatically-driven coupled resonator
ﬁlter, the electrostatically-driven clamped-clamped beam ﬁlter, and the
piezoelectrically-driven clamped-clamped beam ﬁlter. The coupled resonator
ﬁlter, popularized primarily by Nguyen and Howe [3, 23-25, 29-31], utilizes a
design similar to that proposed by Adler nearly sixty years ago, but developed at
microscale. This design has gained popularity largely due to the exceedingly
high Q factors it produces, which, have been reported to be as high as 80,000 in

vacuum [3]. An alternative design proposed Nguyen [25, 29, 32, 33] utilizes an

electrostatically-driven clamped-clamped beam ﬁlter. While the Q factors
reported for this ﬁlter are signiﬁcantly lower (about 8000), the clamped-clamped
beam design is both smaller and easier to fabricate. An analogous design
proposed by Piekarski, et al [26] also utilizes a single clamped-clamped beam
resonator, but excited through piezoelectric means. While the Q factors reported
for this ﬁlter are signiﬁcantly lower than those reported for both the coupled
resonator and electrostatically-driven clamped-clamped beam ﬁlter designs
(about 1000), the piezoelectrically-driven clamped-clamped beam design does
not require the specially tailored input (no bias voltage is required) and drive

electrodes of its counterparts and is easy to fabricate.

Whereas the three ﬁlter designs discussed above utilize coupled
oscillators and/or linear phenomena for ﬁltering, the present work describes a
ﬁlter design based on the nonlinear response of parametrically-excited MEM
oscillators. With further development, this design may have signiﬁcant potential
in many communications applications. The thesis is outlined as follows. Chapter
2 reviews some basic information with regards to ﬁlter design and performance.
Chapter 3 reviews parametric resonance and discusses its relevance to MEMS
and its potential use in ﬁltering applications. In Chapter 4, a single MEM
oscillator is modeled and its dynamic response is analyzed. Chapters 5 and 6
present tuning schemes, linear and nonlinear, respectively, that are capable of
improving oscillator performance, speciﬁcally for ﬁltering applications. Chapter 7

describes one possible ﬁlter design which utilizes two tuned MEM oscillators.

The robustness of the proposed ﬁlter design is examined in Chapter 8, and the
thesis concludes in Chapter 9 with a brief discussion and some concluding
remarks. Note that a considerable portion of this work has been submitted for

publication in three papers that were co-authored by the author [1, 2, 6].

Chapter 2

Bandpass Filter Specifications

The subject of this thesis is bandpass ﬁlters, which are systems that pass
the components of signals, ideally unaltered, with frequencies inside a speciﬁed
passband, while attenuating those components outside of the passband. Since
Adler’s development in the 19403, most bandpass ﬁlter designs have been based
on a one-dimensional chain of identical resonators. In such an approach, the
natural frequency of the system determines the ﬁlter’s center frequency (see
Figure 1). Weak coupling (generally of the “nearest neighbor’ type) is then used
to produce a system with multiple natural frequencies in close proximity to the
ﬁlter’s center frequency. An input signal is provided at one end of the resonator
chain and the system response, a ﬁltered version of the input, is measured at the
opposite end. The result, when the system is operated in a lightly damped
environment (vacuum or partial vacuum in the case of MEMS), is a highly
desirable ﬁlter frequency response function (FRF), which features closely

packed, lightly damped resonances, as shown in Figure 1.

0 ﬁr
Insertion Loss

 

       
 
 

I Ripple

 

40 db

r Roll-off l
Stopband

Rejection

I

 

 

H

40 db
Bandwidth

 

Magnitude (db)

 

 

1‘

Center Passband Frequency a)

Figure 1. A sample frequency response function for a bandpass
ﬁlter (Adapted from [3]).

Figure 1 highlights some of the key features which should be considered

when designing a bandpass ﬁlter or assessing a ﬁlter’s performance. The

characteristics of greatest interest are [1, 2, 14]:

The center passband frequency — the nominal operating frequency of
the ﬁlter.

The bandwidth — the range of frequencies that will pass through the
ﬁlter with minimal loss in signal strength.

The stopband rejection — the amount by which the signal is attenuated
outside of the passband.

The insertion loss — the reduction in signal amplitude as the signal
passes through the ﬁlter.

The sharpness of the roll-off — the width of the frequency range

between the edges of the passband and the stopband.

. The ﬂatness of the passband response - the degree to which ripples

are present in the ﬁlter’s passband frequency response.

Note that the aforementioned linear ﬁlters can be designed to exhibit near-
optimal characteristics. In fact, a number of well established design techniques
exist whose sole purpose is optimizing the performance of certain ﬁlter properties
[14, 15, 34], though often to the detriment of others. Chebyshev design, for
example, results in a ﬁlter with an equiripple passband and monotonic stopband,
but only average roll-off characteristics. Similarly, Butterworth ﬁlter design
generates both a monotonic stopband and a monotonic passband, but at the cost
of poor roll-off characteristics. Other design approaches include elliptical design,
which yields a highly selective ﬁlter with extremely sharp response roll-off, but
also distinct ripples in both the passband and stopband, and Bessel design,
which produces excellent time response characteristics and a linearly-varying
phase, but also poor rejection characteristics. Independent off the design
method is the fact that the shape of the respective FRF is altered by increasing
the ﬁlter’s order or, in ﬁlters based on Adler’s design, adding additional oscillators
to the chain. Thus, while certain ﬁlter characteristics can be optimized, it is
usually done at the expense of the ﬁlter’s other response characteristics, its
complexity, and its insertion loss. In comparison, it is believed that the
performance characteristics of the proposed parametrically-excited MEM ﬁlter

are largely independent of one another and can be optimized individually with

minimal change in overall performance [1, 2]. However, this can be achieved

only by exploiting special parameter tuning, which is readily achievable in MEMS.

Chapter 3

Employing Parametric Resonance
for Filtering

Parametric excitation is a form of excitation that affects a system’s
response only when the states of the system are non-zero. Mathematically this
means one or more of the coefﬁcients of the dynamic states in a system’s
equation of motion are time varying. Originally discovered by Faraday in the
18303 during experimentation with a vertically excited column of water [35, 36],
parametric excitation is mathematically exempliﬁed today by a special case of the

Hill equation, the Mathieu equation, which can be expressed as
z”+(1+/icosQr)z+,u23 =0. (3.1)

While this equation can be used to describe a variety of electrical and
mechanical systems, the most prevalent physical model is that of a single

pendulum with purely vertical base excitation. In this system, periodic excitation

10

at the pendulum’s base produces an oscillating effective gravity, which, in turn,

leads to the time varying effective stiffness in the system’s equation of motion.

 

stability boundaries
without damping

      
  
 

stability boundaries
with damping

 

2 22

Figure 2. An approximate Strutt-lnce stability chart for the linear
Mathieu equation (a = 0) near principle parametric resonance. "U”
denotes those regions wherein the equation’s trivial solution is
unstable. Likewise, “S” denotes those regions wherein the trivial
solution is stable. Note that with the addition of small damping the
transition curve emanating from .0 = 2 shifts upward in the
parameter space, most notably near the vertex.

The response characteristics of the Mathieu system are well understood,

in particular, the effects of variations in the system’s damping (not included in the

model presented above), excitation amplitude, forcing frequency, natural

frequency, and nonlinearities (also not shown) are known (see, for example, [35,

37-441). Of direct relevance to this work are the stability and resonance

characteristics of a parametrically-excited system (such as Equation 3.1). To

explore this further, ﬁrst consider the Strutt-lnce stability chart shown in Figure 2,

11

developed by van der Pol and Strutt in 1928 [38], which delineates the stability of
the trivial response of the Mathieu equation (Equation 3.1) in terms of the
parameters {2 and 11. Of direct relevance here is that within certain parameter
regions, hereafter referred to as “instability zones” or “wedges of instability”, the
trivial response becomes unstable and, in fact, unbounded in the absence of
nonlinearities. While the addition of damping has no impact with regard to
limiting the unstable response, it does produce a fundamental change in the
Strutt-lnce stability chart. Namely, it shifts the transition curves, such as those
emanating from .(2 = 2, upward in the parameter space. Figure 2 shows this for
the .(2 = 2 case. The addition of nonlinearities in the system can limit the
response inside the instability zones. In particular, this addition leads to
nontrivial steady-state responses within the unstable domain. These solutions
may or may not persist outside of the unstable domains, depending on the types
and magnitudes of both the nonlinearity and the damping, but when they do
persist, the solutions coexist with the stable trivial solution, which results in
hysteresis in the system’s frequency response. Examples of this situation are
shown in Figures 3(a) and 3(b) for hardening and softening nonlinearities,
respectively. The amplitude dependent nature of the bandwidth of the instability
zone should also be noted. In particular, it is observed that as the parametric
excitation amplitude, A, is increased the bandwidth of the instability zone

increases.

12

non-zero responses
7» ll / /

stability boundaries

 

 

2 22

non-zero responses

S U S /zero response

stability boundaries

 

 

(b)

Figure 3. The dominant instability zone in the {2 - It parameter
space with a superimposed nonlinear frequency response for a
given value of It (Adapted from [4]); (a) with hardening nonlinearity
(,u > 0) (b) with softening nonlinearity (,u < 0). Note that solid lines
indictate a stable response and dashed lines indicate an unstable
response. Also shown is the effect of damping on the instability
region.

13

The resonant nature of a parametrically-excited system is also noteworthy.
Unlike a directly-excited system, where small excitations produce resonant
responses only when the forcing frequency is near the system’s linear natural
frequency, small excitations in a parametrically-excited system (including a
Mathieu system) can produce resonant responses for other frequency conditions
as well [35]. For example, if the excitation frequency is taken to be 260,
parametric resonances can occur when the excitation frequency, a), approaches
coo / n (n = 1, 2, 3...), where 020 is the linear natural frequency of the system.
While the n = 1 resonance (primary parametric resonance) is by far the most
common one encountered, higher order resonances do occur, although their
appearance is heavily dependent on the amount of damping present in the
system, due to the upward shifts mentioned above. These resonances appear
only in lightly damped environments [45]. It should also be noted that due to the
unique nature of parametric resonance, parametrically-excited systems exhibit

quite abrupt transitions between trivial and nontrivial responses.

Due to the unique actuation methods and lightly damped environments
inherent to MEM devices, parametric excitation and its associated resonances
arise quite naturally in certain MEM oscillators. In particular, such phenomena
have been observed in oscillators electrostatically driven by non-interdigitated
comb drives (see Figure 4) [4549]. Similar to their interdigitated counterparts,
non-interdigitated comb drives utilize a voltage source to induce motion through

electrostatic effects [5, 50, 51]. However, unlike interdigitated drives where the

14

motion induced is parallel to the ‘tines’ or ‘ﬁngers’ of the comb, the motion
induced by non-interdigitated drives is perpendicular to the ‘tines’; see Figure 4.
More relevantly, under an alternating voltage input the non-interdigitated drives,
through fringing electrical ﬁelds, produce a ﬂuctuating electrostatic stiffness,
which in turn results in a parametric excitation upon the oscillator [5]. Note that
the excitation is purely parametric only if the tines are symmetrically aligned so

that there is zero net force in the mechanical equilibrium state.

 

(a) (b) (C)
Figure 4. Schematic of both interdigitated and non-interdigitated
electrostatic comb drives (Adapted from [5]). The arrows designate
the dominant direction of motion. (a) An interdigitated comb drive.

(b) A non-interdigitated comb drive in an ‘aligned’ position. (c) A
non-interdigitated comb drive in a perfectly ‘misaligned’ position.

From a ﬁltering point of view, the existence of parametric resonance in
MEM oscillators is highly desirable for a few reasons. Namely, if the oscillator’s
instability zones are exploited as passbands and the input of its AC comb drive is
exploited as a system input, because of the abrupt on/off nature of parametric
resonance, such oscillators, acting as ﬁlters, would exhibit nearly ideal stopband

rejection and an extremely sharp response roll-off [1, 2]. However, these

15

beneﬁts come at a cost, as the addition of parametric excitation (and the
associated parametric instabilities) introduces a number of complications.
Among the most obvious complications are [1, 2]:

o The center frequency and bandwidth of the passband, or the center
and width of the instability zone, depend on the amplitude of the
excitation.

o Nontrivial responses can exist outside of the passband due to
hysteresis.

. There is a nonlinear relationship between the system’s input and
output.

. Higher order resonances can occur.

Fortunately, these complications can be largely overcome through careful tuning

and implementation of the oscillators, as presented in the forthcoming chapters.

16

Chapter 4

The Dynamics of a Single
Parametrically-Excited MEM
Oscillator

In an attempt to gain a better understanding of the proposed bandpass
ﬁlter design, its beneﬁts, and how it is achieved in light of the shortcomings of the
features of parametric resonance, consider a model for a single degree of
freedom parametrically-excited MEM oscillator, such as that shown in Figures 5
and 6. This oscillator, similar to those oscillators examined in [1, 2, 4, 5, 52], but
designed for the speciﬁc tuning characteristics required for a ﬁlter, consists of a I
shuttle mass, essentially the oscillator’s backbone “B”, anchored to a substrate
by four folded-beam springs “S” and excited by two sets of non—interdigitated
comb drives “N”, which are individually powered by AC and DC voltage sources

(for tuning purposes, as described subsequently).

17

 

(b)

Figure 5. (a) A representative parametrically-excited MEM
oscillator [1]. The oscillator’s backbone “B" is the primary mass, the
springs “S" provide attachment to ground as well as the mechanical
restoring force, and the non-interdigitated comb drives “N” are used
to provide parametric excitation. “AC” and “DC" indicate voltage
sources. (b) A close up view of the non-interdigitated comb drives
[1]. [Pictures courtesy of Barry Demartini and Dr. Kimberly Turner,
University of California, Santa Barbara]

 

 

Figure 6. A CAD drawing of a representative MEM oscillator [1].

The equation of motion for this oscillator has been previously developed,

and can be expressed as [1, 2,4]:

mx+CX+F,(x)+Fes(x,t)=0, (4.1)

where the elastic restoring force from the springs, F,(x), is accurately modeled by

a cubic function of displacement of the form,

F,(x)= k1x+k3x3, (4.2)

which is, in general, mechanically hardening, that is, k3 > O. The electrostatic
driving and restoring forces, Fes(x,t), are generated by two independent non-
interdigitated comb drives, as shown in Figures 5(a) and 5(b). One comb drive
utilizes a DC input voltage of amplitude V0 and the other a square-rooted voltage

signal of the form

V(t) = VA(/1+cos(cot), (4.3)

which is used to ensure the isolation of harmonic and parametric effects [45]. As
with the elastic restoring force, cubic functions of displacement provide an
accurate model of the resulting force, which is also proportional to the square of
the respective voltage signals [45]. Thus, the combined force from the two sets

of comb drives can be expressed as

Fes(x,t)=(r1ox + r3(,x:’)/02 +(r1Ax + rmx3 W} (1 + cos(wt)), (4.4)
where r10, r30, rm, and rm are electrostatic coefﬁcients that depend on the
geometry of the comb drives. Substituting these forces into Equation 4.1 results
in a net equation of motion for the shuttle mass, or the oscillator’s backbone, of

the form [1, 2]:

mx + C)? + k1x + k3x3 +(r10x + r30x3)/02 +(r1Ax + ran3 )l,f(1 + cos(a)t)) = 0.

(4.5)

To facilitate analysis this equation is nondimensionalized by rescaling time

according to

r=w0t, (4.6)

20

where coo is the purely elastic natural frequency deﬁned as,

(00 = —, (4.7)

51’22 = i, (4.8)

where x0 is a characteristic length of the system (eg. the length of the oscillator’s
backbone) and a is a scaling parameter introduced to aid the analysis. This

results in a nondimensional equation of motion of the form

2" + 2542' + z(1+ 13v1 + 5211 cos(Qr))+ 523(1 + v3 + 2.3 COS(QZ')) = 0, (4.9)

where the new derivative operator and the nondimensional parameters are
deﬁned as in Table 1 [1, 2]. Note that by introducing the scaling parameter, a,
small damping and small parametric excitation have been assumed, which are
consistent with the operation of MEM oscillators, especially near resonance [1].
It is also important to note that due to the tunable nature of MEM oscillators,
many of the parameters present in Equation 4.9 can be speciﬁed through the
design of the oscillator and the associated comb drives, or even tuned “on the ﬂy”

[1]. However, it should be noted that VA and V0 appear in many terms and this

21

imposes constraints on how the parameters are selected (In fact, this
complication is the focus of the tuning in Chapters 5 and 6). For the present
study, two sets of parameter values are used, as listed in Table 2 (their p = i 1/2
designations will become evident subsequently). Note that these parameters are

taken from preliminary designs of a device similar to that shown in Figure 5.

 

 

 

 

 

 

 

 

 

 

 

 

Deﬁnition Nondimensional Parameter
(,)._ d(-) Scaled Time Derivative
dr
8; _ C Scaled Damping Ratio
2mw0
5v _ rmvo2 + r1 AVA? Linear Electrostatic Stiffness Coefﬁcient
1 k1
r1 AVA? Linear Electrostatic Excitation Amplitude
5,1, = ——
k.
9 _ g Nondimensional Excitation Frequency
(00
kaxci Nonlinear Mechanical Stiffness Coefﬁcient
_ k,
= x3 (raovo2 + GAVE) Nonlinear Electrostatic Stiffness Coefﬁcient
k.
13 _ xgraAv} Nonlinear Electrostatic Excitation Amplitude
k.

 

 

 

 

Table 1. Nondimensional parameter deﬁnitions [1, 2].

22

 

p = 1/2 Oscillator

p = -1I2 Oscillator

 

 

 

k1= 10 uN/um k1= 10 uN/um
k3 = 0.05 uN/um3 k3 = 0.05 pN/pm3
X0 = 1 11m X0 = 1 pm

 

m = 4.0528E-10 kg

m = 4.0528E-10 kg

 

§= 0.01

§= 0.01

 

r1. = 25-3 uN/umV2

rm = 25-3 (111/..th2

 

r10 = -55-4 uN/umV2

r10 = -5E-4 pN/umvz

 

r3. = 15-3 pN/um3V2

r3. = 15-3 pN/um3V2

 

 

r30 = -2554 (1N/11m3v2

 

r30 = -7554 uN/um3V2

 

 

Table 2. A comprehensive list of the design parameter used
throughout this work [1].

Once nondimensionalized, the equation of motion presented in Equation
4.9 provides a foundation for the analysis of the MEM oscillator’s response,
especially with regard to varying physical parameters and system inputs. It is
important to note that Equation 4.9 is not the typical nonlinear Mathieu equation,
due to the presence of parametric excitation acting on the cubic term. While this
signiﬁcantly complicates the oscillator’s response, as described in Chapter 6, it
has no effect on the linear stability of the system’s trivial response. As such, the
stability characteristics of the trivial response are compatible with those outlined
in Chapter 3. To conﬁrm this, the averaged equations (derived in Chapter 6) can
be used to analytically approximate the ﬁrst instability zone for the oscillator’s
trivial response. This is shown in Figures 7(a) and 7(b) in terms of the VA - I2

parameter space, that is, the physical excitation amplitude (the alternating

voltage amplitude) versus nondimensional excitation frequency parameter space,

23

for both pure AC voltage excitation (Figure 7(a)) and a reference case which
results in a symmetric wedge of instability (Figure 7(b)) achieved by a means
described in Chapter 5. The amplitude-dependent critical frequency values are
denoted .001 and .002, such that the zero response is unstable in the range .002 <
.0 < .0c1 (for 21 > 0). Using perturbation methods (see Chapters 5 and 6) these

can be shown to be

co, = 2 + {144.121} (4.10)
and
002 =2+8(V1-521], (4.11)

respectively. Also note that the accuracy of the analytical predictions has been
veriﬁed with results from simulations of the full equation of motion, from which the
stability boundary is determined by examining the behavior (growth or decay) of
small perturbations from the oscillator’s trivial response. The following section

examines the critical features of the instability region.

24

Alternatlng Voltage Amplltude vs. Nondlmenelonal Frequency

 

  
 

 

 

 

25 — — ——
20 » * 4
>‘
g, 15 4— — —— —
8» .
g 10 .4 ”A w -4 m* ---_ “_-_ a
> ——»»._-——— .,
g e rho = 1 Simulation
8 E:rhci1_Tl19<>r>'._.
g 5 _. _ -_ E _ _._- __ _ _
2
o i-’_ "_‘—_'—‘ —“““Y‘_' ' v r v _—T' T‘_..' 1'“- “" T'ﬁ
1.980 2.000 2.020 2.040 2.060 2.080 2.100 2.120 2.140
Nondimensional Frequency - 0
(a)
Alternating Voltage Amplltude ve. Nondimensional Frequency
25.000 em -— —~

   
 

Sm; 8188;19—
._ ”31311191131011
5.000 F 77* A A , , ,. 5,- -- — 7- h V p _ _v_ _.__ k

Alternating Voltage Amplitude - V. (V)
g
l
l
l
l
l
l
l
l
l
l
l
l
l
l

 

 

0.0001 ~-—-e ~ ,L__. ~»«-- 1 . . .
1.950 1.960 1.970 1.980 1.990 2.000 2.010 2.020 2.030 2.040 2.050

Nondlmenelonel Frequency - n

(b)

Figure 7. The regions of parametric instability in the VA - .0
parameter space produced using both the system’s averaged
equations and simulations of the system’s full equation of motion
with the linear parameters deﬁned in Table 2. (a) The case of pure
AC voltage excitation. (b) A nominal case which results in a
symmetric wedge of instability [1].

 

—-—~1 —— — 1 —~- ~—y"_ —— —~,

25

Due to the nature of the ﬁltering approach proposed in this work, the
instability zones depicted in Figure 7 are of particular interest. Analogous to the
instability zone emanating from .0 = 2 in Figure 2, these “wedges of instability”
exhibit stability characteristics very similar to those presented in Chapter 3. In
fact, the only noticeable difference is that, unlike the primary instability zone
shown in Figure 2, which features a constant center frequency and nearly linear
transition curves, the center frequency and the transition curves in Figure 7 have
a distinct curvature since the excitation amplitude is VA2 and the oscillator’s

natural frequency depends on the square of the input voltage amplitudes, namely

 

2 2
(02 = k1+r10v0 + rlAVA
" m

(4.12)
As explained in Chapter 3, inside the instability zone the trivial response is
unstable, which results in a non-zero response that is dictated by the
nonlinearities present in the system. Outside of the boundary, the trivial
response is stable. This, however, does not ensure a zero response outside of
the wedge, since the system can have multiple possible steady states, and
nontrivial responses can occur outside of the instability zone, as thoroughly
investigate in Chapter 6. This leads to potential hysteresis in the response,
which is highly undesirable in a ﬁlter. Another characteristic to note is that the
base of the instability wedge originates at the nondimensional frequency of 2,
and therefore ﬁltering takes place at twice the oscillator’s natural frequency, that

is, at twice the frequency of ﬁlters which utilize direct excitation [1, 2]. The effect

26

of damping on this instability zone is also similar to the results presented in
Chapter 3. Namely, the height of the instability zone’s origin is dictated in a
sensitive manner by the amount of aerodynamic and structural damping present
in the system [1, 2]. As such, there is a critical AC voltage input required for the
system to oscillate. This, however, is of minimal concern since the oscillators
typically act in an environment with extremely low damping (near vacuum for
testing) where Q factors can range into the thousands. In addition, the input
(excitation) signal can be ampliﬁed, if needed, to attain the critical excitation
amplitude. Finally, the amplitude-dependent nature of the stability boundary
should be noted. A direct result of these facts is that the both bandwidth and
center frequency of the system depend on the excitation amplitude, one of the
primary drawbacks discussed in Chapter 3. The bandwidth deﬁciency is
considered in the following chapter, while the hysteresis mentioned above is

covered in Chapter 6 and the center frequency shift is discussed in Chapter 7.

27

Chapter 5

Linear Tuning: Manipulating the
Instability Zone

As Figure 6 shows, the oscillator’s activation frequencies (where the trivial
solution undergoes a stability change) are strongly dependent on VA, the
amplitude of the AC voltage input. As such, when an untuned oscillator is
employed as a ﬁlter, its bandwidth will be dependent on the excitation amplitude.
However, it is possible to partially negate this effect through the implementation
of a speciﬁc linear tuning scheme, namely, one in which the natural frequency of
the oscillator, can, is made to be dependent on VA through a tuning of the linear
electrostatic stiffness coefﬁcients [1, 2]. This is accomplished by selecting the
DC voltage in the DC comb drive to be dependent on VA, which is the amplitude
of the AC voltage that acts on the AC comb drive. A description of this tuning, as
well as the net results it produces are presented here. Note that the theoretical
basis for this tuning is largely founded on the averaged equations derived in

Chapter 6.

28

To begin, it should be recognized that the signal to be ﬁltered is the AC
signal, and that the DC signal has been introduced in this analysis solely for this
linear tuning. The DC voltage is to be dictated by the amplitude of the AC signal
as follows. A designer-speciﬁed constant of proportionality a is introduced that

relates the amplitude of the DC and AC voltage amplitudes according to
V0 = aVA, (5.1)

which results in the redeﬁned parameters given in Table 3. Substituting these

parameters into Equation 4.9 results in a revised equation of motion given by
z" + z = —g(2;z' + 221(p + cos(Qr))+ 23(1 + v3 + 2., cos(Qr))), (5.2)

wherein a new tuning parameter, p, is introduced that relates the linear

electrostatic stiffness coefﬁcient to the linear excitation amplitude, according to

V1 ’10"
p=—==1+—————. (5.3)
21.1

The parameter p represents the net effect that the AC amplitude, expressed in

terms of 2.1, has on the natural frequency of the oscillator. Speciﬁcally,

(on =,/1+£p2., =,/1+.ev1 . (5.4)

29

 

 

 

 

Definition Nondimensional mrameter
r 2 + r Linear Electrostatic Stiffness Coefﬁcient
_ L11 V2
5V1 ‘ k A
1
3 = [ x§(r30a2 + r3“)]V} Nonlinear Electrostatic Stiffness Coefﬁcient
k.

 

 

 

 

Table 3. Redeﬁned nondimensional parameters [1, 2].

The instability zone can thus be distorted through variation of p, since this
produces a change in the linear natural frequency of the oscillator in a manner
that is dependent on the input amplitude, VA, through 111 [1, 2]. This results in a
rotation of the wedge of instability away from the reference conﬁguration, here
taken as the symmetric case corresponding to p = 0, as shown in Figure 7(b). In
particular, for rm > 0, by selecting p > 0 the wedge will rotate clockwise, and by
selecting p < 0 the wedge will rotate counterclockwise. Note that one has the
ability to set p by designing the comb drives with the desired electrostatic
characteristics and then specifying a accordingly. Also note that if one does not
activate the set of DC combs, and simply drives the system in the common

manner, this corresponds to p = 1, since a = 0, as shown in Figure 7(a).

Using perturbation calculations (outlined in Chapter 6), it can be shown

that by selecting p = 1/2 the left stability boundary of the wedge becomes vertical

(to leading order in the perturbation calculations), as shown in Figure 8(a), and,

30

similarly, by selecting p = -1/2 the right stability boundary becomes vertical, as
shown in Figure 8(b) [1, 2]. Higher order perturbation methods can be used to
improve this verticality, however, the resulting improvement in performance is
minimal, so nominal values of p = :1/2 are used in this work. Note that the
results described here correspond to rm > 0, the opposite trends in term of

rotation occur for rm < 0.

The aforementioned verticality, achieved by selecting p = _+_1/2, has the
distinct advantage that it renders one of the oscillator's activation frequencies to
be amplitude independent and, as such, makes it act essentially like a high or
low pass switch. In particular, for rm > 0, by selecting p = 1/2, the oscillator will
act as a high pass switch and by selecting p = -1/2 a low pass switch is achieved

[1, 2].

While this tuning provides a solution to one difﬁculty, others remain. The

possible existence of nontrivial responses occurring outside of the instability zone

is addressed in the next chapter.

31

Alternating Voltage Amplitude vs. Nondimensional Frequency

 

I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I

 

 

Lrho = 0 Theory T]

_ _ _ 5 55‘ A rho=OSimulation I‘m
—rho = 1/2 Theory I

I a rho =1/2 Simulatithj

01

I
I

Alternating Voltage Amplitude - V. (V)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
l

 

I

o +—__ ﬁ ' .T —_‘— _"~'—'—T “‘~—'—_‘ 1* ﬁ' T

1940 1.960 1.980 2.000 2.020 2.040 2.060 2.080 2.100
Nondimensional Frequency - Q

(a)

_ "_——“—T—' .. —._—_.‘

 

Alternating Voltage Amplitude vs. Nondimensional Frequency

 

I
I
I
I
I
I
I
I
I
I
I
I
I
I
I

151*: ___v_.__.A __..—_. —_—_———

IJZEB:0?R£6‘T'I ,
A rho = 0 Simulation F _-_ _~__ -7 LL - 5 __. _

8
I
I
I
|
I
I
I
I
I
I
I
I
I
I
I
I
I

I
i
I
I
I
I
I
I
I
I
I
I

Alternating VoItage Amplitude - V‘ (V)

 

 

5 « ~~«4 ——
—-rho = -1/2 Theory
I._-.th= -1/2 Simulation,
0 +> '_——T '_'T‘ *1”. _ "T *7 “*F MO—T“ m—T—w—‘fﬁq—‘H—TW" '"“‘—‘ '_ _ ”'TIF ~~~~~~~~~~~~
1.900 1.920 1.940 1.960 1.980 2.000 2.020 2.040 2.060

Nondimensional Frequency - n

(b)
Figure 8. Parametric instability zones in the VA - .0 parameter

space (for rm > 0) created using both analytical techniques and
simulations. (a) p = 0 and p = 1/2. (b) p = 0 and p = -1/2.

32

Chapter 6

Nonlinear Tuning: Conditioning the
Nontrivial Response

One deﬁciency of the tuned oscillators presented in Chapter 5 is that they
have the propensity to feature non-zero response amplitudes outside of the
wedge of instability, due to the presence of hysteresis in the system. However,
the ﬂexibility of the comb drives in these MEM devices allows one to adjust the
system nonlinearities through electrostatic forces. Speciﬁcally, using well
established techniques, the cubic nonlinearity produced by electrostatic effects in
the comb drives can be tuned such that the oscillator’s overall nonlinearity
exhibits hardening, softening, or mixed hardening/softening characteristics [1, 2,
4—6, 52]. This can be veriﬁed by considering the effective nonlinearities of the

MEM oscillator, as determined through a perturbation analysis, as follows.
6.1 Averaged Equations of Motion

To begin, Equation 5.2 is converted to the correct form for the application
of the method of averaging by employing the standard coordinate transformation

to amplitude and phase coordinates, speciﬁcally,

33

z(r) = a(r)cos(9§’-+ 110)]. (5.1)

z'(r) = —a(r)%sin[%5+w(r)]. (6.2)

In order to capture the response near the primary parametric resonance, an

excitation frequency detuning parameter, a, is also introduced, deﬁned by,

Q = 2 + 50', (6.3)
in which 0' measures the closeness of the excitation frequency to the principle
parametric resonance condition. Equation 5.2 is then transformed to the
amplitude/phase coordinates, a(r) and (0(7), and the resulting equations that

govern the dynamics of these variables are averaged over 4n / .0 in the rdomain

[1, 2, 6]. This results in the averaged equations

a’ = g-aaI— 8; + (2211 + a223)sin(21//)]+ C(82) (6.4)

(11' = %a[3a2(1 + v3)+ 411p — 40' + 2(211 + azﬂg)cos(2w)]+ 0(62 I (65)

34

Note that the presence of the nonlinear electrostatic parametric excitation (13)
leads to a complicating term in the averaged equations, as compared to the
usual nonlinear Mathieu equation. Consequently, the nontrivial steady-state
solutions of the system take on a more complicated form, and the system has

much more interesting response characteristics.

6.2 Analysis of the Steady-State Responses

Since the characteristic form of the system nonlinearity (e.g. hardening,
softening, etc.) is the critical feature of the response for current purposes, and
this is unaffected by damping, zero damping (; = O) is assumed in order to
simplify the analysis. Examining the steady-state responses in this case, reveals
that the response has three sets of nontrivial branches. The ﬁrst two sets, which
can be derived by setting the left hand sides of Equations 6.4 and 6.5 equal to
zero, solving for the cosine and sine terms, and evoking the ﬁrst Pythagorean

Identity, appear in pairs with amplitudes given by

 

 

 

 

51 = ifa—ZMZp—ﬂ’ (6.6)
3(z+va)-243

.52 : i_\/40'—2/1.,(2p+1) . (6.7)
3(,1/+v3)+2/13

35

(Note that each +/- pair represents the same physical response; they appear this
way due to the phase relation of this subharmonic response to the excitation [35])
For each of these solutions, the sign of the term under the square root
determines the frequency ranges over which these branches are real, and thus
physically meaningful. It should be noted that the stability of the a = 0 solution
cannot be determined directly from the averaged equations in polar form, due to
the fact that the phase is undeﬁned at a = 0. However, one can determine the
stability of the trivial response using Cartesian coordinates, or it can be inferred
from the nontrivial responses in Equations 6.6 and 6.7, since the frequency (0')
values at which these branches become zero correspond to the stability limits of
a = 0. Furthermore, it is found that the trivial response is unstable within the
frequency domain bounded by the critical values given in Equations 4.10 and
4.11. It is also easily seen from Equations 6.6 and 6.7 that by selecting p = :1/2
one can render one of the boundary frequencies (that is, the 0' values) where a
branch appears independent of the input amplitude 21, leading to the desired

rotation of wedges described in the previous chapter.

Of direct interest here is the role played by the nonlinearities, which differ
for each of these frequency-dependent branches. To describe these effects,
effective nonlinear coefﬁcients 71 and 72 are deﬁned that dictate the hardening or
softening behavior of the branches. These coefﬁcients are taken from the

denominator of the expressions above and are given by

36

71 = 3(2 + V3 ) - 243 (6-8)

and

72:3(g+v3)+223. (6.9)

Selecting 71 < 0 and 72 < 0 locally results in the usual overall system
softening nonlinearity. Likewise, selecting 71 > 0 and )9 > 0 locally results in an
overall system hardening nonlinearity. However, due to the unusual nature of
this oscillator, two mixed cases also exist, namely, 71 > 0 and y; < 0, and y1 < 0
and y; > 0, which correspond to the two branches bending toward or away from
each other, as determined by the sign of ’11 and the magnitude of p [6]. This
result is summarized in the chart presented in Figure 9 which delineates the

various response regions within the 73 - I13 parameter space, where,

n=z+e me

is the component of the cubic nonlinearity that does not multiply the time varying

term cos(.0r) while 23 is the component that does multiply it. Note that the

transitions (bifurcations) between response types occur when one or both of the

effective nonlinearities equal zero.

37

The solutions in the third set of nontrivial responses are determined by
examining the g = 0 version of Equation 6.4. They have constant amplitude (for

zero damping) and are given by

53:: “:21. (5.11)

 

These solutions are, however, of comparatively less importance from a ﬁltering
point of view due to the nature of their stability and range of existence (though

from a nonlinear systems point of view they are quite interesting) [6].

The nonlinear coefﬁcients 1.3 and 73 are related to the input voltage
amplitudes (see Tables 1 and 2), and thus the nonlinear characteristics of the
response can change as the input amplitude varies. This must be accounted for
in the design of the oscillators and comb drives, and may limit the range of
allowable AC input voltages for the resulting ﬁlter. The following discussion
addresses the intricacies of the oscillator’s overall frequency response, as well as
the effective nonlinearities’ dependence on the system’s input voltage, and how

these vary with system design parameters.

38

-l
x 10 1‘3.

 

 

 

(a: M "*7 9
‘I T; ; 4' ’l
1 ,._. '1 , "94*“ II A“
‘1‘“: , .d‘ . V
I , "Hugh‘s
05 IV 1.532.; i
011‘:
a” 0
-0‘5 VIII
}’
1 VI 9 \
4° 0
~\\
“:1 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 O 8 1
Y3 x 10‘1

Figure 9. The 73 - 23 parameter space [6]. The roman numerals

are used to designate the various response regions.

To facilitate further analysis of the oscillator’s response and stability, the
linear electrostatic stiffness coefﬁcient is assumed to be zero (v1= O, and thus p
= 0), the linear electrostatic excitation amplitude, ’11, is taken to be positive, and
damping is neglected. These assumptions introduce additional symmetries into
the problem, which simplify the analysis of the system’s various responses
without compromising the applicability of the results to a more general system,
i.e., the results provide an accurate “road map" for the overall response that is
still qualitatively valid when the assumptions are relaxed. With these
assumptions, frequency response curves (in terms of the detuning) are easily
produced for parameter values within each of the eight response regions
delineated in Figure 9. In order to obtain a complete picture, the stability of the

response branches must be determined; this is considered next.

39

Though the local stability of steady-state responses is generally
addressed by considering the local linear behavior of the averaged equations
near the steady-state responses in the (a, W) polar coordinate space, singularities
at the origin require that a coordinate change be introduced. As such, the
following coordinate change is evoked, which converts the polar coordinates (a,

(a) into Cartesian coordinates (x, y),
X(r) = a(r)cos[i//(r)], (6.12)
ym = a(r)sin[w(r)]- (6.13)

This coordinate change results in new averaged equations of the form [6]

x' = %g[(22, + 40‘)y — 3y3x2 y + (— 373 + 22,)y3 — 8;x]+ C(52), (6.14)

y' = %£[(2/i1 — 48-)x + 373xy2 + (3), + 22,)x3 — 8gyj+ o(.92). (8.15)

Using these equations, with g = 0, local stability results are inferred by
considering the response of the system when linearized about the steady states.

Letting

40

_ X17)
X(r)_[ym] (8.18)
and
X' = [x] (8.17)
y

where the stars designate steady-state values, the local linearized equation of

motion for the system can be written as

Y'(r) = J|x. Y(r), (6.18)
where

Y(r) = X(z')— X“ (6.19)
and J is the Jacobian matrix of the averaged equations presented above,
Equations 6.14 and 6.15, evaluated at one of the system’s steady states, X'. The
stability of the steady-state response is governed by the eigenvalues of the

Jacobian matrix evaluated at the ﬁxed points that correspond to the steady-state

responses.

41

To simplify the analysis, we will consider the stability in terms of the trace
and determinant of the Jacobian matrix [53]. This method proves to be sufﬁcient
due to the direct correlation between the eigenvalues of the Jacobian and its
trace and determinant. In particular, the eigenvalues can be expressed in terms

of the trace, T, and the determinant, A, as

11,, = %(T i- 12 — 4A). (8.20)

For the undamped case under consideration here, the trace of the
Jacobian for each of the steady-state responses is zero, thus only two
equilibrium types are possible for nontrivial eigenvalues: saddles (unstable) and
centers (marginally stable). Which of these two equilibrium types exists depends
solely on the sign of the Jacobian’s determinant. In particular, when A > 0, the
equilibrium will appear as a center, and when A < 0, the equilibrium will appear
as a saddle. The remaining case, A = 0, corresponds to the case of two
identically zero eigenvalues, and as such is used here only to calculate where

stability changes occur.

The trivial solution features a determinant of the form

42

A ”1188203 —402). (8.21)

By ﬁnding the roots of A = 0, the critical frequency values (detuning values) at

which stability changes occur can be shown to be

a, = -5 (6.22)

and

02 2 g (8.23)

respectively. The stability in each frequency (detuning) region is then determined
by evaluating the determinant at a single point within the domain or by calculating
the derivative of the determinant at the critical frequency value. As a quick
computation will show, the trivial solution appears as a center for 0' < 0'1 and a >
02, and as a saddle for 01 < a < 02. Using similar techniques the constant
solutions of Equation 6.11 can be shown to be saddles for all frequency values

where the solutions exists (determined by considering their phases), namely,

min{—2%(— 373 i 2,) } < a < max{?’:—e-(— 373 i 2,) }. (8.24)

43

The two pair of nontrivial response branches are slightly more complicated
in that their stability is heavily dependent on which region of the 73 - 2.3 parameter
space (see Figure 9) the system falls within. Regardless, the stability for each of
the branches can be found by evoking the fact that the response presented in

Equation 6.6 has a determinant given by

A z .20, + 2011311. - 2.2. + 24.0)

 

41373-24.) ' (“25’

which features critical frequency values of

a, = 121 (8.28)
and

a, = —‘—1—(— 373 + 2,) , (8.27)

245

and the response presented in Equation 6.7 has a determinant given by

A z 62(4 — 20X34y. + M. + 24.0) (6.28)

 

4(373 +243) ’

44

which features critical frequency values of

a. = {12(— 32. — 1,) (6.29)
and
02 = 521 (8.30)

Combining each of these results with the steady-state solutions derived above
yields a complete portrait of the oscillator’s possible frequency responses, as

shown in Figure 10.

45

Response Amplitude vs. Detunlng In Region I (y, = 0.025, A, = 0.025)

 

 

 

 

 

 

 

 

0'

(a)

Response Amplitude vs. Detunlng In Region ll (1, I 0.025, 1., = 0.050)

 

 

 

 

 

 

 

8 +_——._ —,v *4 v - ~——. ——M __——_w_——

 

 

 

 

 

~2.5 -2 -1.5 -1 05 0 0.5 1 1.5 2 2.5

46

Response Amplitude vs. Detunlng In Region III (73 a -0.025, A, = 0.050)

 

 

 

 

 

6 L_‘LA________E_ ___-_. - .___ u __.‘r,_ ..._.__L__.____

\
.I_.._ .__-_. 55‘ _.__ _ 55---.
\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

47

Response Amplitude vs. Detunlng In Region V (13 = -0.025, A; = -0.025)

 

 

 

 

I———K L t

 

 

 

 

 

 

 

 

 

-2.5 -2 -1.5 -1 -O.5 0 0.5 1 1.5 2 2.5

0

(6)

Response Amplitude vs. Detunlng In Region VI (1, = 41.025, 1., I -0.050)

 

 

 

 

 

 

 

 

 

25, 2““.
I
I: M .i——~ 0 ~— P o s
| ’/
20I~ 4 4 4 ——~— e—-—~— —I_4 —~—eIe——4—‘—————,—’—-—w
/
I /
. /
. /
I ,’
15r‘_‘————- ~—— ————~—e —~~—~~7— --———~ -A__._,
I //
j /
/
10 T———_——_—— 4_—-- ~34“ -_,2__..5,__~_ ____‘
~-_ //
~‘
I ~~1 ———.P V
5’ ....... ._1: ”w-‘d--EALELAih_~_LALA
I
OI T r —T_— T 1 m
2.5 .2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
U

48

Response Amplitude vs. Detunlng In Region VII (73 =- 0.025, A, = -0.050)

 

 

 

 

25r—-— __
I
f i o——~R+———s o 4.
204—.\V_WW-**-WA__¥#*_‘_
I \\
I \
\
I \\
15I*7**~-W7——**~~—v AAAAAA
f \
I! \I\
\
10I__ _’_______‘_ _A.__-__ _____ __ e____ _ _ _
I X
I U JI-—— '9’,’
5IﬁiiiiiﬁiEvh- 745-.'+**
I

 

 

 

 

 

20 1* 4~ *_______i-_.___-_‘ﬂ_,_,p~_ ~—~~ » ~ - --

 

 

 

 

 

 

 

 

 

(0)

Figure 10. Representative frequency response plots showing
response amplitude versus detuning frequency for each of the eight
regions delineated in Figure 9 [6]. The frequency regions
designated by the letters correspond to the phase portraits of
Figure 11 .

49

Due to the unique nature of many of the frequency response curves
presented in Figure 10, each are brieﬂy discussed here. To begin, the response
curves depicted in Figures 10(a) and 10(d). corresponding to regions I and IV in
Figure 9, are considered. As quick examination reveals, these regions appear
consistent with normal softening and hardening nonlinear behavior. That is, the
nontrivial solutions branch off in two distinct pitchfork bifurcations, one subcritical
and the other supercritical, and all solutions remain globally bounded. The
frequency responses shown in Figures 10(b) and 10(c), corresponding to
topologically equivalent regions labeled II and III in Figure 9, are slightly more
complicated. Here the nontrivial solutions also branch off, but in each instance a
subcritical pitchfork bifurcation occurs. The net result is that the two response
branches actually bend away from each other and some solutions are globally

unbounded.

In the lower half-plane of the parameter space depicted in Figure 9, the
existence of the additional nontrivial (constant amplitude) solution complicates
matters. First, consider Figures 10(e) and 10(h), corresponding to regions V and
VIII in Figure 9. Here the response locally resembles that of Figures 10(a) and
10(d). However, as the frequency is swept further away from zero detuning (0' =
0), the system undergoes two additional bifurcations, corresponding to the
creation and annihilation of the constant amplitude response, which connects the

other two nontrivial branches. Though this still results in a globally stable system

50

with quasi softening or hardening characteristics, it is distinct from the usual
hardening and softening response curves of the upper half-plane, due to the fact
that the stability in the two non-constant branches switches via the constant
amplitude branch that connects them. The responses shown in Figures 10(f) and
10(g), corresponding to the regions labeled VI and VII in Figure 9, are also quite
unique. Again, the responses feature four distinct bifurcations, two of the
pitchfork variety where the nontrivial solution branches off from zero and two
corresponding to the creation and annihilation of the constant amplitude solution,
again connecting the other two nontrivial branches. The net result is a globally
unstable system with nontrivial response branches which initially bend towards
one another, are both initially stable, and lose stability via bifurcations with the

constant amplitude branch.

To provide additional insight into the bifurcations described above,
representative phase planes corresponding to each of the frequency regimes
dictated in Figure 10 are depicted in Figure 11. Note that these phase planes are
only the topological equivalent of those in the speciﬁed regimes, as many of the

topologies occur in more than one set of response curves.

51

 

 

 

 

(b)

(a)

 

 

(d)

 

 

 

 

 

 

 

(e)

52

 

 

.1I1.Ii..la... I1, . II 1 a .1 IA
5 r . . . .
ﬂ 1_ , Ii . ._ . w
. .. _ _. _
L. x t _ 2
__ \\ ,. L f.
,. A / \x A
T ,, ,. II . . t_ W
L . .. .
L __ x. ..
A . .5. _
. _, A {\t ,._
IIt ..4 ,_ 5
.. a

—— H ’ih—r~r 'ﬁ‘iY
/

 

 

~10

. s
(I
.
.
L

.EI. I..- \\..I/I I I...
A
_

 

_

r i,

5 V

A -

5 / \ J

51.1.... _.. 5v - .....

 

 

 

 

 

53

 

a
'0

>4
\
/
52

\

I~ I
I
I II
/
A A T i A LI—IFLI
, I I)
\\ I
I I
/
/ /

/
/

I

 

 

‘ I
FTTTI
8.
I

I

II... 5
II I III I I I I I I I I I I I L
5 0 5 O 6 0 5 5 O 5 D 6 O 5
1 1 1 4 1 1 1 1

 

 

 

(r)

54

(o)
(q)

 

 

 

 

 

5 \\ , / . . . 7,.. f a
I \x‘ \ 4 , ,, y ‘ c
II \ fr _— w\\ i“ If
>~ OI I I O \ I I I 4
‘ I \
I I 1' I\ \‘ T 4 J ‘
/ ./i I, i- .5 _. ‘7‘"‘5 ‘ I‘\ _‘ 14
62/ T‘ \‘ \.;
I-~- —-~~-V - 1
~10) “
1 RT“ _ 7 I
-15 . ~ '
' 15 10 -5 T 75 7 10 15
X x
(S) (t)

Figure 11. Representative phase plots corresponding to the
frequency response regions deﬁned in Figure 9 [6]. Solid dots
indicate equilibrium points.

6.3 Tuning the MEM Oscillator’s Nonlinearity

From a ﬁltering point of view, the ﬂexibility in selecting the nature of the
nonlinearity is very useful for limiting the existence of non-zero solutions outside
of, or at least on one side of, the instability zone. In particular, by specifying a
hardening nonlinearity (Region I) or a quasi-hardening nonlinearity (Region Vlll),
that is 71 > 0 and 72 > 0, for a high pass switch (p = 1/2, for rm > 0), nontrivial
responses below the activation frequency can be avoided [1]. Similarly, by
specifying a softening nonlinearity (Region IV) or a quasi-softening nonlinearity
(Region V), that is 111 < 0 and 72 < 0, for a low pass switch (p = -1/2, for rm > 0),

nontrivial responses above the activation frequency can be avoided [1].

55

 

As alluded to earlier, this nonlinear tuning can be achieved through careful
design of the comb drives [1]. Precise values of the effective nonlinear
coefﬁcients are not required, since one simply needs to ensure that hardening or
softening (or even quasi-hardening or quasi-softening) persists, even in the face
of the mechanical (hardening) nonlinearity and the AC input. Here r30 and r3A,
two parameters determined by the geometry of the ﬁngers of the comb drives,
must be selected such that the desired response characteristics are achieved
over a given range of voltages. If this cannot be achieved in conjunction with the
linear tuning constraints on rm and rm, it would be possible to add an additional
comb drive to the oscillator, whose sole purpose is to tune the nonlinearity. It
should also be noted that since the mechanical nonlinearity is generally
hardening, one will need signiﬁcant electrostatic softening in the low pass switch
in order to achieve overall softening over a reasonable range of voltages (this
has recently been accomplished experimentally at U083 [54]). Of course,
maintaining the form of the nonlinearity is complicated signiﬁcantly by the
excitation amplitude (VA) dependent nature of the system’s nonlinearities, as

examined next.

As shown in Tables 1 and 3, the nonlinear electrostatic terms that
comprise the effective nonlinearities of the system, V3 and 2.3, depend on the

amplitude of the excitation voltage of the system, VA. Accordingly, the system

can actually transition from one response regime to another (in Figure 9) as the

56

amplitude of the alternating voltage changes. This can be conﬁrmed by
examining the form of the nonlinearity in the 73 - ’13 parameter space. Taking the

deﬁnitions of ’13 and )9,

2 2
1:, ___ XOr3AvA (6.31)
k.
and
X3 2 2
73 =Z+V3 =Z+T(r3A+a r30)vA (6-32)

1

it is seen that at zero voltage the system starts at (73. 13) = (Z. 0) (on the
boundary between Regions | and VIII) and moves along a straight line in the 73 -

’13 parameter space given by

13 = [-—r3‘——](73 - 1) (633)

as VA is increased. To validate this consider the oscillator design designated ,0 =
-1/2 in Table 2. As Figure 12(a) shows, this oscillator’s nonlinearity will move
through regions I, II, III, and IV as the input voltage is varied from 0 to
approximately 5 V. Similarly, the oscillator designated ,0 = 1/2 in Table 2 will

move through regions I and II as the input voltage is varied from O to

57

 

approximately 18 V, as shown in Figure 12(b). To avoid such transitions within a
given oscillator’s operating voltage range, consideration must be given to these
effects during oscillator design. The following relationships are useful in
determining the transition voltages, the critical voltage values corresponding to

qualitative changes in the systems nonlinearity, for a given oscillator:

 

‘ k3 (6.34)

 

VA,Ct : 2 5

 

_k3

 

(6.35)

VA.c2 =

2 1

Note that to achieve an overall hardening behavior it is beneﬁcial to have a
hardening mechanical nonlinearity, as it ensures that a minimum voltage
threshold will not be required to realize the desired nonlinearity. Likewise, it
would be beneﬁcial to have a softening mechanical nonlinearity in systems where
an overall softening is desired. Unfortunately, this is not realizable in current
designs, though it is currently being considered by the author. However, these
systems will, due to the nature of the parametric resonance, have a threshold
operating voltage in any case, and it has been experimentally observed that the

transition from hardening to softening takes place at relatively low voltages [54].

58

 

0.04 ,
0.03.

0.02 I

 

IV

 

0.02]

0.03 _ —

 

vA=4.2v

0.0:?

 

' 7101" '“ Wo’ "

 

6.0T _

73

(3)

IV

___ _— _ﬁ.__

vA $17.3v
II

0.02

 

' 0.03

 

”#0072 7

 

i007 ‘ 'A o. ‘

vA=ov

3.0T ‘—

Ya

(b)

tioT

 

_003

Figure 12. Sample parameter spaces showing how the oscillator’s
nonlinearity transitions between response regimes under a varying
input excitation amplitude; (a) for the p = -1/2 design presented in
Table 2, (b) for the p = 1/2 design presented in Table 2 [6].

59

6.4 Summary

The above analysis gives a quite complete picture of the response
characteristics of the MEM oscillator of interest. For general parameter values
(including the inclusion of damping) approximate response curves can be
numerically produced from the averaged equations. Figure 13 highlights sets of
curves for both a p = 1/2 oscillator with a hardening nonlinearity and a p = -1/2
oscillator with a softening nonlinearity. As predicted, each response curve
originates at the chosen activation frequency (0 = 2) regardless of the AC
excitation amplitude, VA. This conﬁrms that the instability wedges have been
rotated as desired (see Figure 8). In addition, the response quickly increases
from zero to a ﬁnite amplitude after activation. To conﬁrm the validity of the
perturbation calculations, results from numerical simulation of the full nonlinear
equation of motion have also been included in Figures 8 and 13 [1]. Note that
the averaged solutions provide excellent accuracy at low AC voltage amplitudes,

and are reasonable at larger excitation voltages.

6O

Response Amplitude vs. Nondimensional Frequency
p I 1I2

 

   

Response Amplitude
l
l
l
l
l
l

   

I— - 10V- Analytical
_. 7— -12V— Analytical
l—15V-Analyticai

0 10V- Simulation
I I 12V- Simulation
{___ 0 15V- Simulation

 

 

 

2.000 2.005 2.010 2.015 2.020
Nondimensional Frequency - n

(a)

Response Amplitude vs. Nondimensional Frequency
pBJR

 

 

   

 

1.5I—~- H~~ ___w.
- - 1OV- Analytical”

. — -12V- Analytical
— 15V - Analytical
0 10V - Simulation L
I 12V- Simulation _ - #

II__:‘ 0 15V- Simulati_o_n_J
00 - ~~———-~-~~-———

1 980 1.985 1.990 1.995 2.000
Nondimensional Frequency - a

(b)

Figure 13. Sample response curves — amplitude vs. frequency: (a)
p =1/2 and (b) p = -1/2 [1].

0.5 l

 

 

_..__.___ ___—v- .___ _-.-.__-‘___.

61

Chapter 7

Creating a Bandpass Filter

While the tuning schemes presented in Chapters 5 and 6 effectively
condition the response of an individual oscillator, they result in efﬁcient
frequency-activated switches, but not a bandpass ﬁlter [1, 2]. Furthermore,
these switches are good at only one frequency, as the other instability boundary
is still amplitude-dependent, and the responses will likely exhibit hysteresis
beyond the switch point. However, two such oscillators, tuned to act as
amplitude-independent switches at nearby frequency thresholds, have great
potential for such use [1, 2]. In particular, it may be possible to create a

bandpass ﬁlter through the implementation scheme shown in Figure 14.

The idea is to generate a bandpass ﬁlter which features a center
passband frequency of {20 and a bandwidth of Ana, where A is deﬁned to be a
small parameter which describes the bandwidth as a percentage of the center
frequency (see Figure 1). This parameter may also be used to deﬁne an
“effective quality (Q) factor”, where Qeﬂ’ is independent of system damping, as

follows,

62

(7.1)

DI—s

To achieve this design, two oscillators are required. One is a high pass
switch (with p = 1/2 for rm > O) that has been nonlinearly tuned such that it
exhibits a hardening nonlinearity, which will henceforth be designated as ‘H’.
The other is a low pass switch (with p = -1/2 for rm > 0) that has been nonlinearly
tuned such that it exhibits a softening nonlinearity, which will henceforth be
designated as ‘L’. In addition, both the L and H oscillators must have their linear
mechanical frequencies tuned such that the base points of their respective
wedges of instability are slightly shifted from .00, so that the passband is created.
This can be achieved by designing the oscillators so that their zero-voltage (i.e.,
purely mechanical) linear instability threshold frequencies are as follows: for the L
oscillator the threshold is selected to be ADO/2 above .00, and for the H oscillator
the threshold is selected to be Ana/2 below {20. With this, the oscillators’ tuning
is complete. A summary of the required tuning conditions for the two oscillators

is given in Table 4.

63

 

 

 

 

 

Oscillator Tuning Condition
H Oscillator Zero-Voltage Linear Frequency km 9o A
”I" ‘ m? ' 7i "5)
Amplitude Dependent Linear Tuning [r1003] 1
(p = 1/2) rm H 2
Hardening Nonlinearity Nonlinearity Region I or VIII
L Oscillator Zero-Voltage Linear Frequency

kit 90 A
= -—'-=— 1 _
w“ \lm, 2 [ +2)

 

(p= -1/2)

Amplitude Dependent Linear Tuning

l—‘w’l --’
r1 A L 2

 

 

 

Softening Nonlinearity

 

Nonlinearity Region IV or V

 

 

Table 4. Summary of tuning requirements for bandpass ﬁltering
[1]. The “Nonlinearity Region” refers to Figure 9.

 

 

 

 

 

SC

 

 

 

 

 

 

 

 

 

 

——>

 

 

 

 

 

 

 

 

 

0+
—>

 

 

 

 

 

AND

 

 

 

 

L,

 

 

 

 

 

 

 

 

 

 

1 enable
\

 

R=VAcos(0)t)

64

 

/P output

Figure 14. One proposed logic implementation scheme, which is
capable of producing a bandpass ﬁlter [1, 2].

 

Once the two oscillators have been tuned in accordance with the
conditions set forth in Table 4, they are ready for implementation in the ﬁlter
system presented in Figure 14. This system is designed to function as follows.
A harmonic input signal, R, of the form R = V, cos(a)t) is supplied to the system.
This signal travels to a signal conditioner (SC) that produces an excitation signal
appropriate for the oscillators’ comb drives, namely the square-rooted input
described in Chapter 4. This signal is then used to drive both the H and L
oscillators, via the AC comb drives, and is also provided to the block designated
F1. This block represents an AC to DC converter that produces the amplitude of
R, namely V, (which is monotonically related to VA), or some proportion thereof.
This DC signal is sent to the oscillators, where it is used to drive the two
resonators, via the DC comb drives, and to tune each oscillator through their
linear tuning parameters, p = i1/2 (recall that the p tuning is set by a, which sets
V0 in relation to VA.) Each oscillator, acting as described in previous sections,
ﬁlters the provided input signal and acts as a switch, producing a zero (in
practice, the noise ﬂoor) or ﬁnite amplitude oscillatory response, depending on
the frequency of the excitation signal. The respective signal from each oscillator
is then sent to another block designated F2 which converts the signal into a
constant voltage, or into a digital signal. For example, F2 may produce a 0 when
the oscillator’s output is zero (noise ﬂoor) and 1 when the output is oscillating or,
this is simply and AC to DC converter. The signal from each F2 block then
proceeds to an AND junction, which provides a non-zero signal to an enabling

input of block P only when the frequency of the input signal falls within the

65

desired bandwidth. If the enabling input of device P receives a non-zero signal, it
allows the original ﬁlter input to pass unimpeded; otherwise, it blocks the signal.
The result is a bandpass ﬁlter with ideal stopband rejection and optimal roll-off in

its frequency response.

To verify the operation of the ﬁltering scheme presented in Figure 14,
numerical simulations of the system were carried out using SimulinkTM [55]. As
Figure 15 shows, the results for a bandpass ﬁlter designed with an effective
quality factor of 500 are essentially as expected. The ﬁlter’s bandwidth and
center frequency are nearly amplitude independent (note the horizontal scale),
and could be made even more so by reﬁning the tuning parameters, the p’s. In
addition, the attenuation outside of the passband is absolute, which veriﬁes that
the ﬁlter’s stopband rejection is ideal. Finally, the ﬁltering takes place at twice the
natural frequency of the system, or twice the frequency of those ﬁlters which

utilize linear resonance.

66

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67

Chapter 8

Design Robustness Issues

So far, the discussion of ﬁlters within this work has been largely based on
the assumption that the ﬁlters’ input is both known and deterministic and that
target values for design parameters are readily achievable. However, in practice
uncertainties arising in both the system’s input and design parameters must be
characterized and addressed. in particular, the effect these uncertainties have
on ﬁlter performance must be noted. A complete robustness analysis is beyond
the scope of this thesis, but we provide here an overview and some preliminary
results of such an analysis. Of particular importance are uncertainties arising in
the following:

. mechanical properties, including the mass and stiffness,

o damping,

o electrostatic properties, including the forces generated by the comb

dnves,

. the input voltage signal, including any noise that may be present, and

. background system noise.

68

The importance of these uncertainties is largely due to the direct effect they have
on ﬁlter characteristics, including,
. the linear natural frequency of the ﬁlter,
0 the stability thresholds and the ability to manipulate them through
linear tuning, and

o the system nonlinearities and the ability to manipulate them through

nonlinear tuning.

8.1 Parameter Sensitivity

The robustness issues arising from design parameter uncertainty are
perhaps best addressed through sensitivity analysis, as it offers a means through
which the effects of parameter uncertainty can be estimated. For current

purposes, the standard deﬁnition of sensitivity is used, namely,

S(A,B)= —— (8.1)

which represents in the percent change in A due to a percent change in 8.

Due to the nature of the ﬁltering method described within this work, the
sensitivity of only two ﬁlter parameters are of utmost importance, namely, the
sensitivity of p, which dictates the orientation of the instability region, and the

sensitivity of (00, which dictates the location of the apex of the instability region.

69

All other parameters, once designed within acceptable ranges, can deviate from

their speciﬁed values with minimal loss of performance.

A brief analysis shows that calculating the sensitivity of 000 is trivial. In

particular, one ﬁnds that

(8.2)

8(a20,m) = g. (8.3)

As such, for every 1% change in m or k1, mo will exhibit 0.5% change. As both
the linear stiffness and the mass are highly predictable, this indicates that

uncertainties associated with the purely elastic linear natural frequency will likely

have minimal effect on the system.

The sensitivities associated the linear tuning parameter, p, however, are

slightly more interesting. Recall from Equation 5.3 that

p=—=1+——, (8-4)

70

where no and rm are electrostatic coefﬁcients dependent on the geometry of the
ﬁngers of the comb drives and a is a tuning parameter which speciﬁes the
relationship between AC and DC input voltages. Evaluating the sensitivities

reveals that

2ra
S , =——‘-°——, 8.5
(pa) WW. ( )
r a
s . =——‘°—, d 8.6
(Prro) r1A+r10a2 an ( )
r a
8(p.r )=—-—‘°——. (8.7)
“ annoaz

which, when plotted for the p = 1/2 oscillator design presented in Table 2 results
in the sensitivity curves of Figure 16. From these plots it is clear that the linear
electrostatic coefﬁcients and the voltage tuning parameter must be selected with
high precision to ensure the verticality of the stability boundary, and thus its
amplitude independence. This is not difﬁcult for the user-speciﬁed voltage ratio
a. However, for no and rm this is easier said than done, as the electrostatic
coefﬁcients are determined largely through approximate methods, including
electrostatic ﬁnite element techniques (typically a two-dimensional analysis) and

curve ﬁtting.

71

s(ps r10)

 

 

 

l

-10.00 H — 7 77* ﬂ *7 iiiiiiiiiiiiiiiiiii L .L_...____z___

 

.1200 + — _+r - w e — — f _‘.+___*-‘_-_._ _.-__L._ _. ______ ﬂ

-6.0E-04 -5 8E-04 -.5 6E -04 -.5 4E-04 5. 2E-04 -.5 0E-04 -4 8E-04 -4 6E-04 4. 4E-04 -4. 26-04 -4 0E-04

'10

(b)

72

 

 

 

 

 

 

0.00 +-— .—, -——- —- —e ‘ -—-- -——- —— +- --—— ._ —.

1.2E-03 1.4E~03 1.6E-03 1.8E-03 2.0E-03 2.2E-03 2.4E-03 2.6E—03 2.8E-03

VIII

(6)
Figure 16. Sensitivity plots, which show the sensitivity of p with
respect to each of its components. The black squares designate

the nominal design parameters used for the p = 1/2 oscillator under
consideration here.

8.2 Effects of Noise

While the sensitivity methods presented above are extremely useful for
determining the required accuracy of system design parameters and the net
effect parameter uncertainty will have on the system, another fundamental issue
remains. In particular, the ﬁlter’s capability to handle uncertain inputs must be
considered. Whereas the analysis considered to this point has assumed
perfectly harmonic input signals, in reality ﬁlters will also see both pure noise
inputs and harmonic inputs with superimposed noise. While the effects such

inputs have on this ﬁlter are beyond the scope of this work, it is worth noting that

73

a bevy of literature exists which examines the response of both single-degree-of-
freedom (SDOF) and multi-degree-of-freedom (MDOF), parametrically-excited
systems with stochastic inputs [56-65]. The complication here is that a full
analysis of the system’s response to stochastic inputs would require one to
assume both random amplitude and phase, which would affect both the linear

and nonlinear behavior of the oscillator, thus signiﬁcantly complicating matters.

74

 

 

Chapter 9

Conclusions and Directions for
Future Work

Filtering based on parametric excitation has some very attractive features,
as summarized in Figure 15. For the implementation considered here, the most
obvious drawbacks are:

. a damping-dependent critical AC excitation amplitude is required for
operation (which can be addressed by restricting the input voltage to a
speciﬁed range);

. design robustness issues may arise, for example, the required
accuracies of the linear tuning strategy that rotates the wedge,
temperature sensitivity, effects of noise, etc;

. the insertion loss of the physical system cannot yet be quantiﬁed, since
this will depend on the hardware implemented for the ﬁlter;

0 higher order resonances may appear in the system, which will lead to
non-trivial responses well away from the passband; and

. one must be able to measure, or at least estimate, the amplitude of the

harmonic input of interest.

75

These issues will be addressed in forthcoming studies, including planned

experimentation.

In addition to the methods described within this work, a number of
alternative ﬁltering methods based on parametric excitation are currently under
consideration. Amongst the most promising alternatives are:

. coupled oscillator ﬁltering systems, designed such that logic circuitry, like
that shown in Chapter 7, is unnecessary;

. parametrically-excited ﬁlters with specially tailored inputs, such as
constant amplitude square waves, which may be used to avoid the
amplitude dependent nature of a single oscillator’s passband and center
frequency; and

. parametrically-excited ﬁlters tailored to handle noise.

While this work focused on bandpass ﬁlters, it is worth noting that an
equally ideal band gap ﬁlter can theoretically be produced by simply changing
block P such that it enables with a zero amplitude signal instead of a non-zero
amplitude signal. It is also expected that these parametric-based frequency

switches can be used to develop high and low pass ﬁlters.

The ultimate goal of this line of work is to achieve fully functional ﬁlters

wherein the parametrically excited MEM oscillators and the associated circuitry

are integrated into a single chip.

76

 

10.

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