Piezoelectric MEMS Resonator Characterization and Filter Design
by
Joung-Mo Kang
Submitted to the Department of Electrical Engineering and Computer Science
in Partial Fulfillment of the Requirements for the Degrees of
Bachelor of Science in Electrical Engineering and
MASSACHUSETTS INS
OF TECHNOLOGY
of Engineering in Electrical Engineering and Computer Science
JUL 2 0 2004
at the Massachusetts Institute of Technology
June 2004 L F
LIBRARIES
Copyright 0 2003 Joung-Mo Kang. All rights reserved.
The author hereby grants to M.I.T. permission to reproduce and distribute publicly paper
and electronic copies of this thesis and to grant others the right to do so.
Signature of Author
Department of Electrical Enginieering and Computer Science
Jan. 9, 2004
Certified by
Amy E. Duwel, Ph.D.
Charles Stark Draper Laboratory
Thesis Supervisor
Certified by
Charles G. Sodini
Professor of Electrical Engineering
-Thesis Advisor
Accepted by
Arthur C. Smiih
Theses
on
Graduate
Committee
Chairman, Department
BARKER
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2
Piezoelectric MEMS Resonator Characterization and Filter Design
by
Joung-Mo Kang
Submitted to the Department of Electrical Engineering and Computer Science
on Jan. 16, 2004, in partial fulfillment of the requirements for the degrees of
Bachelor of Science in Electrical Engineering and
Master of Engineering in Electrical Engineering and Computer Science
Abstract
This thesis presents modeling and first measurements of a new piezoelectric MEMS
resonator developed at Draper Laboratory. In addition, some simple filter designs
incorporating the resonator with predicted performance parameters were analyzed, with a
special focus on the suitability of using the Draper resonator to implement these filters.
The four-element Butterworth Van-Dyke model, the traditional circuit model used to
describe crystal resonators, was predicted to match the theoretically derived electrical
behavior of the fundamental-mode resonance. A three-element "pi" network model was
used to describe the overall test structure. Transformations and algorithms to convert
measured s-parameter data into best-fit model parameters were developed and
successfully tested on commercial thin film resonators. Measurement of the first Draper
resonators was complicated by fabrication difficulties and a resulting large parasitic
which only allowed low frequency longitudinal resonances to be observed. However, the
observed resonances at 125.3 MHz and 148.3 MHz were found to vary with geometrical
parameters as expected, providing evidence that the design is viable. Initial resonator Q
was estimated to be 542.
Filters were designed with estimated resonator parameters after process optimization.
Three topologies, simple (coupled) ladder, dual resonator ladder, and full lattice, are
described and the limits and tradeoffs among them are discussed given the Draper
resonator properties. Numerical examples and an example filter-plus-resonator design
process are provided. Manufacturing tolerances and their effect on resonator and filter
parameters are discussed. Finally, some considerations when implementing an integrated
filter bank are outlined. The filter analyses bring to light two major goals for the next
stage of resonator development. First, an accurate tuning method must be devised as the
resonator bar's small size makes manufacturing errors on the order of tens of nanometers
significantly affect filter characteristics. Second, a lower impedance level for the
resonator is desirable to allow robust interaction with integrated RF circuitry.
Technical Supervisor: Amy E. Duwel, Ph.D.
Title: MEMS Group Leader
Thesis Supervisor: Charles G. Sodini
Title: Professor of Electrical Engineering
3
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4
Acknowledgements
Dec. 3, 2003
I have many people to thank for encouraging and guiding me to the point where I can
publish this thesis and finally graduate, after almost eight years, from MIT. My gratitude
goes first to my parents, for supporting me in every respect of the word through my
decision to pursue two additional degrees halfway through my (first) senior year. I am
extraordinarily indebted to Dr. Amy Duwel for offering me a suitable thesis project with
time running out in my appointment, and leading me through to its completion. She has
been a wise mentor, gracious supervisor, and encouraging friend and I feel very lucky to
have been one of her first advisees. I am very grateful for Professor Sodini's kind and
supportive advising as well, which reassured me when confusion and doubt arose about
whether I could complete this thesis. I want to thank Professor Steven Senturia for his
advice on choosing a thesis topic and aid in finding an advisor.
I would like to thank all the staff at Draper I worked with during my time here.
Specifically, I thank Angela Zapata and Chris Dub6 for their invitation to work on a
fascinating and exciting project with them, and for teaching me so much. I thank Doug
White for many answers and insights about RF circuits and filters. I thank Dave Carter
for always being incredibly willing to answer my boring fabrication questions and for
working many hours of overtime to fabricate the first resonators, and also congratulate
him for his new daughter. I thank Cristina Davis and Brian Johansson for cheerfully
satisfying my curiosity about various subjects. I thank the people in the Draper Education
Office for their essential role in making my stay at Draper possible: George Schmidt,
Loretta Mitrano, and Joseph Sarcia.
Finally, I would like to thank my friends for all their emotional support and motivation
this past summer and fall. I especially thank Joseph Richards for his technical advice,
Metallica CDs, late-night coffee delivery, and actually reading some of my thesis.
This thesis was prepared at The Charles Stark Draper Laboratory, Inc., under contract
DAAHOl-01-C-R204, sponsored by the U.S. Army Aviation and Missile Command /
DARPA MTO office.
Publication of this thesis does not constitute approval by Draper or the sponsoring agency
of any opinions, findings, conclusions, or recommendations of the author and do not
necessarily reflect the views of the Government. It is published for the exchange and
stimulation of ideas.
James Joung-Mgo Kang
5
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6
Contents
1
Introduction
1.1 Thesis A im s ...............................................................................................
1.2 Project Origins and Goals.........................................................................
1.3 Chapter Summaries....................................................................................
2
Background
13
2.1 . Current Applications of Miniature Resonators in RF Communications
T echnology .............................................................................................
.. 13
2.2 Typical Communications Front-End ........................................................
13
2.3 Requirements for an Integrated Resonator ...............................................
14
11
11
11
12
3 Miniature Resonators
17
3.1 Current Miniature Resonator Technologies ............................................
17
3 .1.1 S A W ............................................................................................
. 17
3.1.2 T FB A R ..........................................................................................
18
3.1.3 Mechanical Resonators..................................................................
19
3.2 Draper Resonator Bar ...............................................................................
20
3.2.1 Device Overview ..........................................................................
21
3.2.2 Analytic Model of Longitudinal Resonance.................................
22
3.3 Butterworth Van-Dyke Model..................................................................
26
3.3.1 BVD Impedance ..........................................................................
27
3.3.2 Draper Resonator Equivalent Circuit Parameters......................... 29
4
Resonator Measurement and Characterization
4.1 S-parameter Measurements ......................................................................
4.1.1 Materials and Setup ......................................................................
4.1.2 Data Collection Procedure.............................................................
4.2 Data Analysis...........................................................................................
4.2.1 Data Transformation....................................................................
4.2.2 Network Impedance Model ..........................................................
4.2.3 Parameter Extraction ...................................................................
4.3 Measurements Results ...............................................................................
4.3.1 TFR Devices .................................................................................
4.3.2 Draper Resonator Measurements..................................................
31
31
31
33
33
34
36
37
38
38
40
5
Filter Design
5.1 B andp ass F ilters.........................................................................................
5.1.1 Bandpass Filter FOMs ..................................................................
5.1.2 Filter Design Goals and Motivations...........................................
5.2 Bandpass Filter Topologies ......................................................................
49
49
49
52
53
7
5.3
5.4
5.5
5.6
6
5.2.1
5.2.2
Simple Ladder..............................................................................
Dual-Resonator Ladder................................................................
54
59
5.2.3
Lattice Filter ..................................................................................
62
Numerical Examples................................................................................
M anufacturing Limits and Tolerances.......................................................
5.4.1 Resonator Bar Dimensional and FOM Limits...............
5.4.2 Manufacturing Tolerances and FOM Sensitivities............
Filter Design .............................................................................................
5.5.1 Filter-level Resonator Requirements .............................................
5.5.2 Resonator Fabrication Tradeoffs .................................................
5.5.3 Simple Ladder Filter Design Process ..........................................
Filter Bank Considerations ......................................................................
5.6.1 Filter Bank Layout.........................................................................
5.6.2 Impedance Matching and Parasitic Loading ................................
5.6.3 Harmonic Mode Cancellation.......................................................
Summary and Conclusions
6.1 Summary of Thesis W ork.........................................................................
6.2 Future Investigations ...............................................................................
8
67
73
73
80
81
81
83
83
86
86
87
88
91
91
92
List of Figures
2.1
3.1
3.2
3.3
3.4
3.5
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
Block diagram of typical RF circuitry for a wireless comm. device............
R esonator bar schem atic .............................................................................
Illustration of fundamental longitudinal vibration.......................................
Finite elements simulation of 0.78 GHz resonator bar I-V characteristic .....
The Butterworth Van-Dyke model for a crystal resonator ..........................
Ideal Butterworth Van-Dyke impedance function......................................
Schematic view of a Draper resonator test structure ...................................
Incident and reflected waves of a two-port system .....................................
Typical TFR S-param eter data ...................................................................
Port voltage and current conventions for a two-port system........................
Tw o-port im pedance it-m odel ......................................................................
Fitting extracted parameters to resonator impedance data ..........................
Fitting extracted parameters to parasitic port impedances ..........................
Overview of Draper resonator test structure fabrication .............................
Masking effect of a large parallel capacitance on resonator response ..........
S21 magnitude data for 25 [Lm and 30 gm long devices ..............................
Transformed impedance data for two 25 ptm long resonators.....................
Transformed impedance data for two 30 ptm long resonators .....................
Fitting extracted parameters to 25 pm resonator impedance data........
Transform ed port impedance data...............................................................
Bandpass filter FOM definitions ...................................................................
Insertion loss definition ...............................................................................
Simple ladder topology................................................................................
Maximally flat response vs. 1 dB passband ripple for a two-pole filter........
Simple ladder filter wideband response......................................................
Dual-resonator ladder topology ....................................................................
Full lattice filter topology ...........................................................................
Location of individual resonance peaks relative to lattice filter passband . .
Bode plot of simple ladder filter transfer function ......................................
Bode plot of dual resonator ladder filter transfer function ..........................
Bode plot of lattice filter transfer function .................................................
Cantilever beam deflection under an applied force .....................................
Deflection vs. bar length under a force of 1000mg for various thicknesses..
Elastic linear torsion under and applied force .............................................
Deflection vs. bar length under a torque of 250mgl for various thicknesses.
One possible switched filter bank implementation.......................................
The Butterworth Van-Dyke model with an additional resonance ...............
Filter transmission characteristic with spurious resonance .........................
Filter transmission characteristic with cancelled spurious resonance ...........
9
14
21
21
22
26
28
32
34
34
35
36
38
39
41
42
43
43
44
45
47
50
50
54
55
59
59
62
63
68
69
70
75
75
76
77
87
88
90
90
List of Tables
3.1
3.2
3.3
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
Example SAW resonators listed with important device characteristics........
Example TFBAR resonators listed with important device characteristics....
Example mechanical resonators listed with important device
characteristics ...............................................................................................
Simple ladder filter simulation parameters..................................................
Sim ple ladder filter FO M s...........................................................................
Dual resonator ladder filter simulation parameters .....................................
Dual resonator ladder filter FOM s................................................................
Lattice filter sim ulation param eters .............................................................
Lattice filter FO M s ......................................................................................
Filter FOM comparison for three filter topologies ...................
Maximum cantilever deflection-limited bar lengths ...................................
Maximum torsion deflection-limited bar lengths ........................................
Minimum resonant frequency achievable at three thicknesses ...................
Maximum resonant frequency achievable at various tether widths ............
Impedance level at resonance achievable for minimum frequencies ..........
Impedance level at resonance achievable for maximum frequencies..........
Simple ladder filter with harmonic mode cancellation simulation
param eters....................................................................................................
10
18
19
20
68
68
69
69
70
70
73
75
77
78
78
79
79
90
Chapter 1
Introduction
1.1
Thesis Aims
The purpose of this thesis is to characterize a new resonator being developed at The
Charles Stark Draper Laboratory. First, expected theoretical behavior of the resonator
will be summarized, along with a model based on these calculations suitable for use in
filter design. Next, radio frequency (RF) measurement and parameter extraction
procedures will be described, applied to prototype resonator devices, and the results
analyzed. Finally, the possibilities of implementing this resonator in high-performance
bandpass filters with be discussed using several design examples.
1.2
Project Origins and Goals
The Draper resonator is the chief element of a project which seeks to develop an array of
RF channel-select filters made from MEMS resonators and integrated with an RF low
noise amplifier, funded by the Defense Advanced Research Projects Agency (DARPA).
To date, the available integrated filters have performance limits in very narrowband, low
insertion loss applications; the large size of current non-integrated RF resonators and
filters makes a compact, portable filter bank infeasible. Integrated MEMS resonators
offer a promising solution for this difficulty, because of their high
Q, mechanical
tuning
techniques (e.g. laser trimming), compatibility with conventional silicon active device
processes, and small size at GHz frequencies (on the order of 10 pim).
I1
The goal of this project was to create resonators with center frequencies ranging from 200
MHz ~ 1.5 GHz, selectable by device geometry, with high enough
Q to allow channel
spacings on the order of 100 kHz to 10 MHz. Approximately twenty of these resonators
would be fabricated to form a filter bank onto a 1-10 mm2 chip with a low noise
amplifier.
1.3
Chapter Summaries
The remainder of this thesis is organized into five chapters. Chapter 2 expounds upon the
problem of integration between RF filters and the other circuitry required for
communications devices. Chapter 3 describes three major miniature resonator
technologies and their capabilities, and then presents the theoretical characteristics of the
Draper resonator. Chapter 4 summarizes in detail the measurement of prototype Draper
devices and the characterization procedures, which were first validated on known TFR
resonators. Chapter 5 begins with an overview of bandpass filter design, and discusses in
detail three filter topologies: simple ladder, dual-resonator ladder, and full lattice, with
numerical examples using the theoretical Draper resonator parameters given. Next the
manufacturing limits and tolerances of the resonator are discussed, and their effect on
filter characteristics. Given the achieveable limits of the Draper resonator, the filter
design process is described, focusing on the suitability of using the Draper resonator for a
given set of requirements. Finally, several topics concerning filter bank design are
discussed. Chapter 6 provides final conclusions for the thesis work and suggestions for
future efforts.
12
Chapter 2
Background
2.1
Current Applications of Miniature Resonators in RF
Communication Technology
As the number of wireless applications have increased, allocations of the frequency
spectrum have grown increasingly crowded. Newer technologies have therefore pushed
their carrier frequencies higher and higher, which complements the demand for smaller
and more portable devices. Greater portability also gives rise to a demand for lower
power consumption. In general, reducing the total size of a piece of circuitry enables
higher frequency operation while reducing power consumption. In current RF devices, all
the major active components can be integrated monolithically, resulting in a small size
and the best performance due to minimal parasitics. However, the passive resonators and
filters necessary for frequency selection and duplex function have no viable integrated
option. The filters used in commercial wireless products today are on the order of
millimeters in dimension [1, 2], which is still by far the largest single component of
modem RF circuitry.
2.2
Typical Communications Front-End
A typical RF receiving circuit begins with an antenna to receive wideband transmissions,
producing an analog electronic signal. The next major step is to bandpass filter this signal
to extract the appropriate frequency band, with the possibility of an intermediate gain
stage depending on the expected signal strength of the antenna signal and the insertion
loss of the filter. After filtering, the signal is amplified and sent to a downmixing element,
13
Gain
to downconverter
BPF
to
antenna
4n
,
.from
.4modulator
BPF
Figure 2.1: Block diagram of typical RF circuitry for a wireless communications device.
and from there it is commonly sampled and processed digitally. In a duplex
communications device, there will also be a transmit circuit which takes a signal
modulated to the carrier frequency, amplifies it, bandpass filters it to prevent interference
from being generated in other frequency bands, then broadcasts it via the antenna (Figure
2.1). The various amplifiers can be integrated with standard silicon fabrication
technologies, but the filters to date have not been integrated in commercial devices. They
are frequently built with ceramic transmission-line resonators, or surface acoustic wave
(SAW) resonators, neither of which is compatible with silicon IC technologies [1, 3]. An
integrated filter solution is highly desirable for its reduced size and consequently smaller
parasitics, and to eliminate the packaging parasitics and longer interconnects due to
switching between on-chip and off-chip circuitry, particularly in the cases where an
additional amplifier stage is required between the antenna and filter.
2.3
Requirements for an Integrated Resonator
In order for a resonator to be integrated with silicon active devices, its production must be
compatible with silicon IC fabrication processes. However, once this requirement is
14
satisfied, the resonator must also be able to compete with the best non-integrated
technologies in performance, reliability, and cost before it can be considered a
worthwhile alternative to them. In addition to process compatibility, the resonator
fabrication must either maintain a certain degree of consistency or a suitable tuning
method must be available for it. Since integrated devices will be much smaller than
discrete elements, fabricating them to the same relative tolerances becomes that much
more difficult.
The other major improvement required of integrated resonators is higher device
Q. First,
as frequencies grow higher, the transition from passband to stopband of the bandpass
filters must grow sharper to allow the same channel spacing, and the sharpness of this
transition is proportional to Q. For example, if a filter allows a 50 kHz channel spacing at
1 MHz center frequency, then with the same
Q at
1 GHz the minimum channel spacing
would be 50 MHz. Given that most communications channels are at most a few MHz
wide, this would be adequate to select a band of channels, but not to do individual
channel selection. This is why most systems rely on tuned down-conversion or IF /
baseband processing to perform channel selection. However, a wide-band input path
creates dynamic-range and adjacent-channel interference constraints that would be
eliminated or reduced if higher-Q, individual channel-select filters were practical.
Second, as resonator dimensions shrink and operational frequencies rise, in general the
impedance level of the resonator increases. At GHz+ frequencies, impedance matching
becomes very important when connecting different circuit elements. In order for highfrequency filters to effectively couple to other active elements without losing too much
signal strength to parasitics, the resonators should have high
at resonance is inversely proportional to
Q, which in turn
Q, since
the impedance level
affects the matching
impedances required. Unfortunately as resonator dimensions decrease, relative loss
mechanisms tend to worsen, making higher
Q more difficult to achieve.
Some of the requirements just listed may be relaxed for the resonator by adding
additional processing steps after conversion to the digital domain. Filtering becomes
much more precise and straightforward this way, but there are various tradeoffs in power,
15
latency, and size, to name a few. The goal of the DARPA project is to accomplish as
much as possible through the resonator component alone.
16
Chapter 3
Miniature Resonators
3.1
Current Miniature Resonator Technologies
Several different types of piezoelectric resonators are under investigation to produce a
viable integratable solution. This section presents a brief overview of the various
competing technologies.
3.1.1
SAW
Surface Acoustic Wave (SAW) resonators are currently one of those most common filter
elements used in commercial RF telecommunications products today, along with ceramic
resonators. They generally are produced by depositing a pair of transducers onto a
smooth piezoelectric substrate. Each transducer consists of thin metal multi-fingered
electrodes interdigitated in a pair; thus there is an input port and an output port. When an
oscillation is applied across the input terminal, the input transducer converts the electrical
signal into acoustic vibrations. A surface wave vibration is generated which propagates in
a direction perpendicular to the electrode fingers, strongly favoring the frequency whose
half-wavelength equals the interdigital spacing. The output transducer converts this
vibration back to an electrical signal which is detected at the output.
The frequency response is determined exclusively by the structure of the electrodes. They
are suitable for use in many applications, but they have one major drawback: high
insertion loss. Each transducer emits energy in both directions, resulting in 6 dB of total
loss. The electrodes are very thin and thus have high resistance. Finally, intentional RF
17
mismatches must be introduced to avoid a rippling phenomenon known as triple-transit
echo. Total insertion losses of SAW filters range from 7 to 30 dB [4].
Conventional SAW resonators are the inline type, where the wave propagates in a straight
line from input to output. These have a typical packaged area of about 15 mm x 6.5 mm
[4]. A smaller type, the Z-path SAW, uses reflectors to bounce the acoustic wave twice to
make a Z-shaped pattern, allowing package dimensions on the order of 5 mm x 5 mm [5].
Author
Type
Q
Size
Frequency
Year
King,
inline SAW
-
15.3 x 6.45
210 MHz
1999
210 MHz
1999
Gopani 4
Franz5
mm2 *
.Z-path SAW
-
5x5mm 2 *
Table 3.1: Example SAW resonators listed with important device characteristics.
3.1.2
TFBAR
Thin Film Bulk Acoustic Resonators (TFBARs) are under heavy development by a
number of research groups due to their low-temperature processing methods, which make
integration with a silicon IC process much more likely, though other obstacles have
barred successful integration to date. TFBARs require two basic ingredients to function: a
transduction method to convert between electrical oscillations and acoustic vibrations,
and an acoustic cavity to trap these vibrations. The former is provided by a piezoelectric
membrane, and the latter by a large acoustic impedance mismatch at each interface of the
membrane, so that most acoustic energy hitting the interfaces is reflected. The method of
generating this impedance mismatch defines the type of TFBAR. The more traditional
method creates an air/crystal interface by either etching away the substrate from the
bottom, or using a sacrificial substrate layer right beneath the resonator which is etched
away after the membrane is deposited, leaving a small gap. Alternatively, the resonator
membrane may be deposited onto a stack of quarter-wavelength thick layers of acoustic
materials forming a Bragg reflector [6]. These solidly mounted resonators (SMRs) enjoy
18
greater structural stability than the air/crystal type since the resonating membrane and
electrodes are fully supported from below.
The acoustic cavity contains the piezoelectric membrane sandwiched between two
electrodes. An electrical signal of the right frequency across the electrodes excites a
standing longitudinal wave in the membrane. The major loss mechanism, and limiting
factor for
Q, is the coupling
at the cavity interfaces with the adjacent air and supporting
substrate. The film is usually about 1-5 pm thick, with lateral dimensions anywhere from
100 to 1000 times the thickness [7].
In general, TFBARs surpass SAW resonators in every way, except for process simplicity
and cost to manufacture [8]. Thus, due to this fact and the desire to invent a process fully
compatible with silicon active devices, experimentation on these devices centers on novel
processing methods.
Author
Type
Q
Size
Frequency
Year
Lakin 9
SM-TFBAR
717
-
644 MHz
1999
Lakin' 0
TFBAR
1090
-
1.6 GHz
2001
Plessky"
SM-TFBAR
641
0.033 mm 2
2 GHz
1998
Ruby"
TFBAR
500-1300
2-4GHz
1994
-0.001-0.1
mm 2
Table 3.2: Example TFBAR resonators listed with important device characteristics.
3.1.3
Mechanical Resonators
Mechanically resonant structures such as deflecting cantilevers and flexural beams show
promise as filter elements due to their high
Q and
possible silicon IC process
compatibility. Their principle of operation is very simple: a simple symmetric structure
with mechanically resonant modes is built out of a piezoelectric material. If an oscillating
force at some frequency is applied to the structure, actuated either mechanically or
piezoelectrically, the structure will vibrate and produce a charge separation proportional
to the amplitude of its vibration. Frequencies near resonance will produce larger
19
amplitude responses, which can be viewed as electrical signals if electrodes are attached
at appropriate points on the structure. When used as a filter element, these types of
resonators are usually one-port devices acting as variable impedances: a voltage signal is
applied to the port and a current waveform is drawn from the source with amplitude
varying with the frequency of the input signal.
A major limiting factor to date with this type of resonator is the loss associated with the
anchor between the resonating element and the substrate. In addition, interfacing
electrically to the resonating element also introduces loss in general. Thus a mechanical
element which could achieve a
Q of perhaps
100,000 in isolation may only get one-tenth
that when modified as an electrical resonator. Another drawback is poor linearity with
amplitude of vibration in many cases.
Author
Type
Q
Cleland12
Flexural
21000 @
beam
4.2 K
Flexural
7450
Nguyen' 3
Frequency
Year
2
82 MHz
2001
-30 x 10 jim 2
90 MHz
2000
Size
3.3 x 2.4 pim
beam
Table 3.3: Example mechanical resonators listed with important device characteristics.
3.2
Draper Resonator Bar
The Draper resonator bar was designed to allow production of arrays of RF
communications filters integrated with silicon active circuitry on a single die. It is a
mechanical resonator designed to operate at the fundamental longitudinal mode. Its range
of dimensions should allow center frequencies from 200 MHz to 1.5 GHz while being
compact enough to fit an entire filter bank into an area on the order of 10 mm 2 .
Preliminary loss analyses predict a device
Q of 104 should be attainable
20
[27].
3.2.1
Device Overview
Figure 3.1 provides a schematic of the resonator bar with the major geometrical
parameters labeled. The bar is suspended over a 1 pim-deep well in the substrate by two
tethers attached to the midpoints of its sides. The bar and the tethers are a single
continuous film of aluminum nitride (AlN). Electrodes cover the top and bottom of the
bar, producing an off-resonance characteristic of a parallel-plate capacitor.
2a
W
Figure 3.1: Resonator bar schematic. The bar is suspended over an empty well while the
tethers rest on the lower electrode and the substrate.
Figure 3.2: Top view of the fundamental longitudinal vibration. One cycle is pictured,
and the dashed outline shows the undeformed bar shape. The amplitude of
motion is exaggerated to illustrate the motion.
21
Longitudinal Mode Resonance
The primary engineered resonance is a longitudinal vibration where the bar expands and
constricts lengthwise (Figure 3.2). The midpoint of the bar, where the tethers attach, is a
node, while the two ends of the bar experience the greatest amplitude of displacement.
The magnitude of vibration will be on the order of nanometers.
The bar has many different natural modes of resonance with several at frequencies below
that of the intended longitudinal mode of operation. However, due to symmetry and the
placement of the electrodes, the charge contribution of these lower order modes at the
device's port should cancel to nearly zero. This calculation was confirmed by a finite
elements simulation of the device's I-V transfer characteristic (Figure 3.3).
1.10 9
0
1 -io1
A
p-1 - 10-110
_
1 -10-12
--
1. 10-13
_
0.3
-
0.4
0.5
0.6
0.7
0.8
0.9
1
frmq GH
Figure 3.3: Finite elements model of the transfer characteristic of a resonator bar with
0.78 GHz longitudinal resonance [14].
3.2.2
Analytic Model of Longitudinal Resonance
By applying the piezoelectric equations of state and standard electrostatics relations to the
resonator bar, an analytic solution for the primary resonance may be obtained. This
solution only models a single resonance, but is immensely helpful during filter design.
A Brief Introduction to Piezoelectric Materials Properties
Piezoelectric properties arise in certain crystals with asymmetric structure. Mechanical
stressing causes a polarization of charge in these materials, with the converse occurring
22
as well: an applied electrical field induces physical deformation. A right-handed
Cartesian set of axes is traditionally introduced to facilitate mathematical analysis of the
various piezoelectric interactions, in accordance with the IEEE Standard on
Piezoelectricity [15]. An electric field in one direction will elicit, in general, a mechanical
response along each of the three axes, and a mechanical stress in one direction will
induce, in general, an electric polarization in each axis as well.
Linear Theory of Piezoelectricity
The piezoelectric equations of state relate the major mechanical variables of interest (e.g.
stress, strain, displacement) to the electric field and charge polarization. These equations
may be linearized by assuming all displacements and vibrational amplitudes are very
small. The various partial derivatives appearing in these equations may now be
considered constants, and define the various material parameters such as stiffness,
permittivity, and the piezoelectric stress and strain constants [16]. In addition, because
the phase velocities of acoustic waves are several orders of magnitude less than the
velocities of electromagnetic waves, quasi-electrostatic conditions are assumed [15]. A
more detailed treatment of the formulation of these equations and the definition of the
mechanical and electrical field variables may be found in [15],[16], and [17].
Longitudinal Mode Analytic Transfer Function
The following analysis was originally performed by Dr. Amy Duwel, and is summarized
here with her permission. Applying the piezoelectric equations of state to AIN results in
the following constitutive equations:
23
T
T2
T3
_
c 11
c12
c 13
0
0
0
S1
0
0
e 31
c 21
c 22
c 23
0
0
0
S2
0
c
c 33
0
0
0
S3
0
0
0
e32
c 31
c44
0
0
c55
0
0
S4
S5
0
e, 5
e 24
0
0
0
0
0
S
0
0
0_
32
T4
0
0
0
T5
0
0
0
T
00
c
E
e 33
E3
eT= piezoelectric coupling: C/M2
c= stiffness matrix: N/m2
S1
DI
(3.2)
D
=
0
0
0
0
e,5
0
0
0
e24
0
0
E,,1
S2
3
+
0E
Ej-
0
E2]
0
22
S5
S6 _
E= dielectric constants: F/M
where T is the stress tensor, S is the strain tensor, E is the electric field vector, and D is
the polarization charge vector. The axis along the bar's length is subscript 1 (2,= x),
across its width is subscript 2 (2 = y), and vertically through its thickness is subscript 3
(X3 = z). Due to symmetry, T and S only have 6 independent elements, with the reduced
subscript mapping as follows:
(3.3)
T1 =J Tj for i = j,T = T_jj for i
j
(3.4)
S = Si for i = j,2Si = S9-i- for i # j
The next step is to apply force balance to Eqn. (3.1), substituting in displacement u and
electric potential O.
T
c]1
c1 2
c 13
0
0
0
u
0
0
e3 ,
T2
c
c
c
23
0
0
0
U2
0
0
e 32
U
0
0
e 33
21
22
cT3
c 32
c 33
0
0
0
(4
0
0
0
c 44
0
0
u 2 3 +u 3 2
0
e24
0
T5
0
0
0
0
c55
0
u 13 +u 3 1
e, 5
0
0
T6
0
0
0
0
0
C66
U 12 +U2. 1
0
0
0
T.
(.)
=+
24
-
-
#2I
-
In this notation, any subscripts after the comma refer to derivatives with respect to that
variable. A number of approximations are now made. First, for a longitudinal vibration in
the x-direction, no y-dependence will be assumed, so u2 and all its derivatives are set to
zero. Second, for the lowest-order longitudinal mode, set stress in the z-direction and x-z
shear to be zero for all z, meaning T, = T, = 0. Third, the inertia in the z direction is
considered negligible. This gives the following acoustic wave equation for the
fundamental longitudinal mode in the x direction, coupled to O:
(3.6)
pau
-1= 3, Le3
a -
_
C33 _
-- C3e
33
C33
_
Solution of the electrostatics equations begins by assuming AlN is non-conducting and
thus its free charge density is 0:
(3.7)
3D
3D,
V-D=0= a 1 + a + 3 3
xi
ax2
aX3
Making the same approximations as above and some algebra results in this equation:
22
(3.8)
.1 Ell+ ee5
+033
55 _
_
C33
_
+
e33
= uI 3,[ei +
e3
1
33
e, 5
e 3 3c 31
C55
C33
_
which has the form of Laplace's equation for an anisotropic material. The system is
described by the coupled equations (3.6) and (3.8), which are difficult to solve due to the
x-z coupling. Eqn. (3.8) may be split into two parts, since the system is linear. If we let
#=
BC + 0
,
4
is the solution to (3.8) with the RHS set to zero and the voltage
BC
boundary conditions applied, and , is the solution to (3.8) with zero voltage boundary
conditions. For
(3.9)
5
BC
, the solution can be approximated as:
#(x, z) ~ f (x) Z
2a
under the constraint that the bar is much longer than it is thick. The boundary conditions
are set as follows: at z = 0 (bottom electrode),
# = 0,
at z = 2a (top electrode), 0 =f(x). As
a first order approximation, Eqn. (3.6) is solved using only #BC on the RHS [14].
Plugging this solution into Eqn. (3.6) allows calculation of the total equation of motion of
the fundamental longitudinal resonance due to a driving voltage function V(s):
25
2e
(3.10) u,(x,s)= 2Cos
)7X
apL
-
V( s)
S2+'8
(2
s
where the following intermediate variables have been defined:
2
(3.11)
c=cH
"13
-
C3
(3.12) e = e 3 e 3
3
1
3
C
33
(3.13)
LLv, p
(3.14) fin =),/Q
2
(3.15)
E =633+
C3
3
Finally, with the complete solution we can integrate D3 over the electrode area, and then
differentiate with respect to time to express the current as a function of voltage:
(3.16) 1-- 4we2
sV(s)
apL s 2 +s3
3.3
+
2+a4
wLV(s)
2a
Butterworth Van-Dyke Model
R
C
L
Co
Figure 3.4: The Butterworth Van-Dyke model for a crystal resonator.
The Butterworth Van-Dyke (BVD) model is a common lumped element circuit model
used by crystal filter designers to simplify the transcendental functions that completely
characterize the resonators used as filter elements [19]. It generally provides an accurate
26
fit for a single resonance plus the other regions of a resonator's transfer function not near
another resonance, modeling those parts as a capacitance. Qualitatively, the R-L-C
branch determines the "series" resonance, where the impedance drops sharply to a
minimum value of R at the frequency where the series inductance and capacitance cancel
each other out. At some higher frequency, the loop reactance hits zero and causes a
"parallel" resonance where most current will travel around the loop instead of past it.
3.3.1
BVD Impedance
The exact transfer function representing the BVD impedance is:
(3.17)
1
s2 LC+sRC+l
I
sLCsRC+I
sC s2 LC+sRC+1+C
Co
ZBVD
This function can also be expressed directly in terms of the major resonator figures of
merit, series resonance ws, parallel resonance wp, and Q:
(3.18)
w
(3.19)
w
= -LC
p
(3.20) Q
=w
1+ C
Co
1I
-= wvL
WSRC
R
S2
(3.21)
ZBVD
1
sCS2
2
+S Ws
+I +2+Wp+
Q
Q
S
1 s 2 +s#+w2
++
+ W2
P
The first term in this impedance is the impedance of the static capacitance Co. If C << Co,
as is generally true of high-Q resonators, the second fraction is very close to unity except
near the resonant frequencies. At those frequencies, there is a complex zero pair followed
closely by a complex pole pair, which creates a very low impedance peak followed by a
very high impedance peak, as expected.
27
The only thing preventing the series and parallel resonant impedances from going to zero
and infinity, respectively, is the resistance R. R is a direct result of finite device
Q and
represents all the losses in the device. R is related to the reactance of the inductor and
capacitor at resonance by a factor of Q. For high-Q resonators, R is frequently ignored
during filter design, and only causes a slightly larger insertion loss in the passband than
calculated otherwise.
160
I
I
I
I
I
I
I
I
140
-c 120
Z
CD
W,
100
80
60
40
7E
0
100
I
I
I
I
I
I
I
I
I
I
785
790
795
800
805
810
815
820
825
830
835
I
I
I
I
I
I
I
I
I
I
830
835
50
Co
a-
0
-50
-1 A
780
I
I
I
785
790
795
800
810
805
Frequency (MHz)
I
I
I
815
820
825
Figure 3.5: Typical Butterworth Van-dyke impedance function.
28
3.3.2
Draper Resonator Equivalent Circuit Parameters
The form of Eqn. (3.16) matches exactly with the BVD impedance. By comparing the
two expressions, the equivalent circuit parameters for the Draper Resonator are found to
be:
(3.22) CO = c-wl
2a
(3.23)
L = apl
4
we2
2
(3.24) C = 4we 2
ac iv
(3.25) R =
Q
Co
4we 2
where the bar dimensions 1, w, and 2a are as defined in Figure 3.1, and the other
parameters are material constants for AlN which may be found in the appendix. There are
a couple noteworthy aspects to these relations. First, the resonant frequency depends only
on the length of the bar:
(3.26) w. =F-
= -
LC
c
i p
Second, the ratio of the motional to the static capacitance (termed r) is a constant with
respect to geometry:
(3.27)
r=
C
C0
8e 2
2
C2e,
1
= 3.22%
31.04
29
30
Chapter 4
Resonator Measurement and
Characterization
In this chapter, the experimental methods used to characterize the Draper resonator
prototype devices and their measurement results are presented. It begins with a summary
of the measurement of packaged resonators manufactured by TFR Technologies, Inc.
(63140 Britta St. C-106, Bend OR 97701), which was performed to validate the testing
and analysis procedures before using them on new experimental devices.
4.1
S-parameter Measurements
S-parameters are usually the measurement variables of choice when characterizing
multiport devices at RF frequencies. At these high frequencies, not only do the lumped
element approximations of circuit elements break down, true opens and shorts become
infeasible to implement, which precludes direct impedance measurement. Through
calibration and mathematical manipulations, impedance data may be acquired from sparameters. Once this data was obtained for each device, it was matched to a circuit
model and equivalent parameters were extracted.
4.1.1
Materials and Setup
The TFR devices were FBAR-type resonators packaged in alumina with two external
ground-signal-ground (GSG) probing pads. The total packaged dimensions were about
31
Network
Analyzer
round
Signal
Ground
Wafer
probes
Figure 4.1: Schematic view of a single Draper resonator test structure showing
placement of testing probe tips. The background gray represents the substrate
silicon. Nickel was used for the top electrode metal and is shown in black.
The bottom electrode metal, Molybdenum, is shown in white, and as shown
is only accessible from above through one set of probe pads.
3.8mm x 2.6mm x 0.8mm. The expected resonant frequency was about 2.1 GHz.
The Draper resonators were fabricated onto a wafer and measured directly from it. Many
devices with a variety of bar dimensions and tether lengths were placed onto one wafer.
Each device was connected to a pair of GSG probe pads for measurement (Figure 4.1).
All s-parameter measurements were performed in air at room temperature. Both the TFR
and Draper devices were designed to be tested in a two-port configuration. When
measuring the TFR devices, a Hewlett-Packard 85 1OC Network Analyzer, Cascade
Microtech probe station, 1000 m pitch GSG probe tips from GGB Industries, Inc., and a
6 GHz GGB calibration substrate were used. For the Draper devices, measurements were
taken with a Hewlett Packard 8753E Network Analyzer and 150 pm pitch GSG probe
tips on the same probe station, with a smaller calibration substrate from the same
company.
32
4.1.2
Data Collection Procedure
Before measurements can be taken, the testing apparatus needs to be calibrated on open,
load, short, and through structures with known parameters. The calibration substrates
provide these structures and the necessary parameter values, which are loaded into the
network analyzer before calibration. During calibration, the network analyzer is swept
through a given frequency range on each test structure multiple times. The data for each
structure are averaged, and with the preloaded parameter values the analyzer calculates
calibration coefficients that remove the effects of the probe tips, transmission cables
linking the probe tips to the analyzer, and the analyzer measurement channels. Each
calibration is only good for a short period of time as environmental variables such as
temperature affect the test apparatus. For these setups it was found that calibrating once
per working day was sufficient for a given frequency range. For each set of devices, data
were initially taken in a wide frequency band to locate the resonance of interest, and then
a very high resolution measurement was taken about the resonant frequency. Sparameters were measured on several consecutive sweeps then averaged and saved to
disk as complex values, to be later imported into MATLAB for analysis.
4.2
Data Analysis
The goal of the data analysis was to produce parameters characterizing the resonators and
closely associated parasitics that would be useful when employing these resonators for a
given application, like a filter. The analysis proceeded as follows: first, the s-parameters
were transformed to z-parameters. Then, a discrete element two-port network model was
applied, and the z-parameters were transformed to discrete impedance values for these
elements. Finally, this data was fitted to a lumped element circuit model for each
impedance element, and circuit element parameters were extracted. From equivalent
circuit parameters, a resonator may be implemented easily into most filter designs using
established techniques.
33
4.2.1
Data Transformation
Four S-parameters were measured for each TFR device: SI i, S12. S21 , S22. Sj is measured
by inputting an electrical wave at port j and measuring the output at port i while setting
the inputs at all other ports to zero and terminating them with matched loads to cancel
reflections. Sj is the ratio of the complex amplitudes of the output wave at port i and the
input at portj [20] (Figure 4.2):
(4.1)
V+
=,
V
k
p
Vi*__+
O
0
2-port
_~
0w
V2
V2p->
network
0-
N
- -
Figure 4.2: Incident and reflected waves of a two-port system
S12 Data
0.8
Cl)
0.6
0.4
Cl)
0.2
CZ
0
0.5
0
CZ
CO)
-0.51-1
-1 51
1.5
2
2.5
Frequency (GHz)
Figure 4.3: Typical S-parameter data
34
3
In theory, any network of passive elements should be reciprocal with S12
= S21,
and
inspection of the data confirmed that these values differed by less than 1% for all devices,
so they were averaged into a composite parameter S 12 (Figure 4.3).
A two-port network can also be fully described by four z-parameters, defined as in Eqns.
4.2 and 4.3 and Figure 4.4:
(4.2)
V1 = Z
(4.3)
V 2 = Z 2 111 + Z212
II + Z 12 12
Once again, for a passive network the cross parameters should be equal, so Z 12 is set
equal to Z21. Z-parameters may be transformed from s-parameters using the following
equations, assuming a reciprocal network where S12
(4.4)
(4.5)
ZI2 = Zo
Z
2
12
(1 - S
-Z
Z22 = Z0
and Z 12 = Z2 1 [20]:
(1+Sa )1-S)+S2
)(1 - S22) +2
t
1
(4.6)
= S21
2in
1-)(1- S22) - 1S2
(1-S ,(+a+S2
S 12_
_1)'S2
(I-
12
SI )(1 -S2)
-
1S
2
For all measurements, the line impedance Zo was 50Q.
+
~
2-port+
0,
network
Figure 4.4: Port voltage and current conventions for a two-port system
35
4.2.2
Network Impedance Model
Zb
Ze
Za
Figure 4.5: Two-port impedance i-model
A two-port "n" network was used to model each device as a collection of discrete
impedances (Figure 4.5). Applying Kirchoff's Laws to this model and combining with
Eqns. 4.2 and 4.3 results in the following expressions for each impedance:
(4.7)
Z
=
Za
z+
Za+Zb +Zc
(4.8)
Z12
(4.9)
Z
ZaZc
Za +Zb +Zc
Zc(Za +Zb)
Za +Zb +ZC
This model is over-simplified; its advantage is that the fitting procedure is greatly
facilitated since this model only has as many independent impedance values as the
number of independent two-port S-parameter values. However, its use can only be
justified if the resulting error is not too significant. The fitting with this model proved
acceptable for the TFR devices, and should be sufficiently accurate for the Draper
devices as well because test structures will be included on-wafer to allow direct
measurement of the major unmodeled parasitic elements. Now, the equations giving the
port characteristics in terms of the internal impedances can be inverted to yield explicit
expressions for each impedance, where IZI
(4.10) Z
1 Z 1 Z 2 2 -Z 122
=zz
SZ22 -Z12
(4.11)
Zb =
Z 12
36
(4.12) Z
-
Z11
4.2.3
12
_
Parameter Extraction
Now from the original s-parameter data, impedance data has been obtained explicitly for
the three model elements Za, Zb, and Zc as a function of frequency. All transformations up
to this point have been exact; the only approximation and source of error so far is
modeling the full device using the three-element pi network, which ignores more
complex parasitic topologies. The magnitude of this error depends on the size of the
unmodeled parasitics and will limit the matching achievable in the parameter fitting step
that follows.
The model element of greatest interest is Zb, which models the resonator. The impedance
data for Zb was compared to the equivalent impedance generated by the Butterworth VanDyke model (Section 3.3). In other words, the four circuit element BVD model was
applied to the Zb data. The MATLAB routine lsqcurvefit was used to vary a set of circuit
element values to get the best fit. The initial values for the fit routine were calculated as
follows: the maximum and minimum impedance magnitudes ("IZpI" and "IZsI") and the
frequencies at which they occur (J, andf,) were extracted from the data for Zb (Figure
4.6). From analysis of the BVD transfer function, the following approximate relations are
obtained [19]:
(4.13)
w = 2nf, =
(4.14) w, =2f,, =
(4.15)
-
Z
w,
C
1+ Co
R
R
1 + jwRC
S
(4.16) Z
LC
=
S
0jwRC
~
1
RC
37
These equations are close approximations to the actual maximum and minimum
impedance points resulting from a BVD resonance, for large
Q (e.g. Q >
100). The exact
equations are so difficult to invert that when measuring real devices it is simpler to fit
parameter values algorithmically to the measured data. However, when fitting to the
sharp resonance peak, an accurate initial guess was necessary to allow the algorithm to
converge, so obtaining one by solving these approximate equations proved necessary. By
plugging the data-extracted values for |41, IZI,f,, andf, into these equations, a very good
approximation for the desired parameters R, L, C, and CO can be obtained.
4.3
Measurement Results
4.3.1
TFR Devices
Measurement and analysis of ten TFR devices was carried out while awaiting fabrication
of the Draper resonator prototypes. Figure 4.6 plots Zb for one measured device alongside
the equivalent BVD impedance calculated from fitted parameters. The matching is
Zb Magnitude and Phase
-060
(D
40
C
-20
-
-- data
-model
01
.
0
20
C
05
E)
-2 1
1. 5
I
2
IL
b.
2.5
Frequency (GHz)
Figure 4.6: Measured and modeled Zb. Simulation parameters:
R = 2.76 K, L = 91.6 nH, C = 0.061 pF, CO = 1.54 pF
38
3
Za Magnitude and Phase
50
E(0 45
CO
= 35
o40
CIS
CL
-
30
ata
mode-I
d--
25
'
CU
-2
E
-2.5'1.5
2
2.5
3
Frequency (GHz)
Zc Magnitude and Phase
:S 50
a)
.~45
0)
CU
S40
Cc 35
,
-data
-- model
-0
a)
CL
E 30
S25
CO
U0
a)
C
a)
E
-3
1.5
2
2.5
3
Frequency (GHz)
Figure 4.7: Measured and modeled impedance data for Za and Z,
excellent, with an average per-point error less than 2% for frequencies near or below the
resonance. The error is about 15% for higher frequencies as the lines begin to deviate,
partly due to unmodeled parasitics whose effects grow more pronounced at higher
frequencies, and partly because the BVD model only contains one resonance while the
actual resonators exhibit higher-order modes.
39
The dominating behavior expected of the shunt impedances Za and Z, is of a capacitor,
from consideration of the physical makeup of these devices. They were fit to a series
RLC circuit instead of only one capacitor to allow for a slightly different slope in
magnitude, and to account for the real component of impedance present in the data.
However the fitted parameters did result in a dominating capacitance. Pictured in Figure
4.7 are the measured impedances Za and Zc of the same device as in Figure 4.6, plotted
with their fitted RLC impedances - these plots were typical for all ten measured devices.
There is some coupling to the resonance evident, which is due to unmodeled factors. The
fitting routine converged for all ten sets of data.
4.3.2
Draper Resonator Measurements
Fabrication of the first Draper resonators was delayed due to difficulty in obtaining high
quality Aluminum Nitride (AlN) films. Since the goal of the first fabrication run was to
assess basic device functionality and frequency scaling behavior, two fabrication steps
were identified that could be postponed till later runs. Their removal had a significant
impact on the frequency response of these initial resonators, so the nature of the
imperfections will be briefly explained before presenting the results.
Impact of Incomplete Fabrication
The major fabrication steps are described by Figure 4.8. The steps skipped were d and e
in the figure. Their purpose is to reduce the extra capacitance that would otherwise be
placed in parallel with the resonator, which is a very significant amount as the probe pad
is on the order of 1000 times the area of the resonator bar. As it turns out, an additional
parallel capacitance of this magnitude mostly overwhelms any resonant peaks, even with
Q as high as
104,
according to simulation (Figure 4.9). Exacerbating this difficulty, the
Q
of these initial resonators is not expected to be over 1000 as the processing has yet to be
optimized. Thus, the resistance at resonance is competing with a very small impedance
shorting out the resonator due to this large capacitance. The shunting is more severe at
higher frequencies because the capacitor impedance is inversely proportional to
frequency, and for the majority of the devices on the wafer no resonance could be
40
(a)
-C
-4-c
(b)
(c)
(d)
(e)
. I
Figure 4.8: Overview of Draper resonator test structure fabrication. The material
depicted in white is Molybdenum, gray is AIN, hatched is Nickel, and black
is a conductive weld. (a) Overhead view of resonator with ground probe
strips. (b) Sideview of initial stack as deposited on silicon substrate. This
stack pattern represents all three GSG strips. (c) Ni and AIN layers are
removed from the left probe pads of all three strips to allow contact with the
bottom electrode. (d) The tethers on the resonator strip only have their
electrodes selectively etched to reduce capacitance. (e) All three strips have
the top and bottom electrodes shorted together to eliminate stray capacitance.
41
0
-20--
Cthru=2pF
Cthru=0
-40
0
-60
C
-80
-100
-120
780
790
800
810
820
830
(a)
-15
-16
co
-17-
S-18
Cl)
-19
-20
188
190
192
194
196
Frequency (MHz)
198
200
(b)
Figure 4.9: Shunting effect of a large parallel capacitance "Cthru" on resonator response.
(a) 6 pm bar length, fo = 800 MHz, Q = 104
(b) 25 pm bar length, fo = 193 MHz, Q = 1000
detected at the expected frequency. Fortunately, a few bars made with extra-large
dimensions had resonant frequencies low enough to be detectable over the capacitor
shunting. Also, the increased bar cross-sectional area meant reduced impedance level
overall, which also helped these larger resonators to have detectable resonances.
Measurement and Analysis of Longitudinal Mode Resonances
Longitudinal-mode resonances were detected for devices of two bar lengths: 25 pm and
30 pm. Typical S2 1 data for both lengths are plotted in Figure 4.10. Note that on this
scale, the resonant peaks are hardly discernible. However it is clear that the wideband
characteristics are very similar, although quite a bit different than that of a single
capacitor. High resolution data was taken for four devices around their resonances, two of
42
- _n
10
CI)
15
20
-
I
F
25
-5
10
C\j
15
U)
20
"F-
I
100
150
200
250
300
350
400
Frequency (MHz)
450
500
550
600
Figure 4.10: S21 magnitude data for 25 pm (top) and 30 prm (bottom) long devices.
Longitudinal length mode resonances are circled.
595
590
585
580
575
580
-i,
570
560
2
550
-81
570
565
-83
-82
-84
-83
-85
-84
-86
)
a
-85
146
7
149
148
147
Frequency (MHz)
148
149
148.5
Frequency (MHz)
Figure 4.11: Transformed impedance Zb for two 25 jm long resonators.
43
149.5
900
960
SG880-
940
920
900
840
-
880
820
(
-71
-76.5
-72
775
-7-
-7
-78.5
-79
a -74
-79.5
-on
-8U
CL
-75
121.5 122
122.5 123 123.
Frequency (MI-z)
124
121.5
122
122.5 123 123.5
Frequency (MHz)
124
Figure 4.12: Transformed impedance Zb for two 30 gm long resonators.
each length. Figures 4.11 and 4.12 show Zb for all these devices after data transformation.
The average series resonant frequencies of 148.3 MHz and 125.3 MHz for each bar
length are about as expected when electrode loading is taken into account (which was
ignored in the prior derivation for the resonant frequency). Inspection of the plots shows
a discrepancy between the data and the ideal BVD impedance, most evidently in the
phase behavior. A BVD resonator has an almost entirely imaginary impedance offresonance, so its impedance phase should be about 90 degrees plus some multiple of 180.
The deviation from one of these values indicates an additional unmodeled component in
the through impedance of these devices which adds a real component to the impedance,
possibly additional parasitic resistance in series with Zb. In the case of the 30 gm bars,
there is clearly a more complex unmodeled behavior affecting the data as the phase
shows a significant slope as well. The net result of these deviations is that algorithmic
fitting of the BVD model to this data is nearly impossible, and could only be attempted
with a model modified to better fit this data. Since these prototypes were fabricated with
a large masking parasitic that hindered measurement, which should be removed from
44
59
0
58
1
simulated
M',---data
_0 57
0
55
0
-80
J-2-
-82-
-84r
_D
F
- -
-
simulated
data
y) (
-86
(0
CD)
10C -88
145.5
146
146.5
147
147.5
148
148.5
14
149.5
150
Frequency (MHz)9
Figure 4.13: Fitted and measured Zb impedance of a 25 gm resonator. R = 11500 W, L =
6.7 mH, C = 0.1718 fF, CO = 1.89 pF
future runs, it was decided premature to develop a more complex model at this point.
Instead, parameters were fitted by hand to the BVD model for the prototype devices. The
results and equivalent BVD parameters are shown in Figure 4.13. The additional
capacitance basically replaces the Co of the resonator, since parallel capacitances add and
the parasitic capacitance is about 1000 times the natural value of Co. The fitted value for
this capacitance is about what was expected from the area of the probe pads. The value of
Q estimated
from these values (which is not affected by CO except through the difficulty
in measurement and fitting presented by the shunting effect) is 542, once again
approximately what was expected from this first round of fabricated devices. The
Q
values for future devices should be much improved as various aspects of the fabrication
process are optimized.
45
As mentioned above, the primary goal of the prototype fabrication round was to confirm
resonator functionality. By finding longitudinal resonances that varied as expected with
length, proper function of these devices was verified.
Probe Pad Parasitics
The transformed data for the probe pad parasitic impedances Za and Z, exhibited some
interesting behavior. One representative set of data is shown in Figure 4.14. The first
important detail is that the port impedances are not symmetric, which makes sense since
the pads themselves are not symmetric: on one side the probe touched down on a single
layer of metal on the substrate, while on the other side the probe touched down on top of
the three-layer stack. Unfortunately, the pad orientation of the measurement ports was not
recorded, so it is not certain which probe pad corresponds to which port impedance.
Some general observations about the impedance data can still be made. First, the origin of
the impedance should primarily be due to fringe capacitance between the probe signal
pads and the two neighboring ground pads in each G-S-G probe pad set. One impedance
looks very much like a capacitor while the other looks like a much smaller capacitive
response with a good deal of complex behavior added in. A possible explanation for this
is that the fringe capacitance for one side is much higher than the other, so the stronger
capacitance dominates its modeled impedance response while the weaker capacitance on
the other side allows more subtle, complex behaviors to become noticeable. The presence
of the piezoelectric stack on only one set of probe pads could change the effective
dielectric constant of the fringe capacitance and explain the different capacitance values.
46
15000
-
a
-0
=3
10000-
cJ)
a(
CZ,
5000
E
160
100
150
200
250
300
350
400
Frequency (MHz)
(a)
450
500
550
600
250
300
350
450
500
550
600
1600Ti
100
1400
a,
S1200
C,)
0)
a,
1000
C)
a)
0-
E
800
600
20
4001'
00 150
20
400
Frequency (MHz)
(a)
Figure 4.14: Transformed impedances Za (a) and Z, (b).
47
48
Chapter 5
Filter Design
5.1
Bandpass Filters
A primary goal for this thesis is to analyze the Draper resonator bar's suitability for its
intended application in high-performance miniaturized communications filters. This
chapter presents the target filter specification goals, analyzes three potential filter
topologies, and discusses the practical concerns and tradeoffs in implementing these
designs.
5.1.1
Bandpass Filter FOMs
Figure 5.1 illustrates most of the major bandpass filter characteristics as traditionally
defined in the literature. These magnitude-based FOMs are usually referenced to the
minimum loss point within the passband, the frequency range where power is transmitted
from the source to the load. The other two regions of interest are the stopband, at which
frequencies most of the input signal is severely attenuated at the output, and the
transitionband, which resides between the passband and stopband. The filter magnitude
characteristics each describe an aspect of one of these regions.
The passband characteristics are bandwidth, center frequency, insertion loss, and
passband ripple. Bandwidth is evaluated for a given corner frequency, for example, the
3dB bandwidth is the distance in frequency between the two points at which the filter
transmission falls below 3dB relative to the minimum loss transmission. Center
frequency is self-explanatory, but there are sometimes subtleties with its exact definition.
49
0 dB
IL
Passband
ripple
3B
Out of band
rejection
3dB Ba dwidth
Stopband
Stopband
fc
Passband
Figure 5.1: Bandpass filter definitions. Plot is in units of dB magnitude vs. frequency.
Rs
U
+
Vs
RL
V
RL
VL
a
~
(a)
Vs
Filter
0
~
0(b)
Figure 5.2: Measuring insertion loss. (a) Reference test circuit. (b) Filter test circuit.
50
It can usually be defined as either the point of minimum insertion loss or the midpoint
between the two bandwidth edges, though the latter definition is usually more accurate. A
common issue that arises is that the center frequency for a given filter topology differs
from the individual resonator frequencies according to a complicated relationship, so
after an initial design is carried out, the filter must be simulated and the resonator
frequencies adjusted to achieve the desired center frequency exactly. Insertion loss is
defined as the relative loss of a filter compared to a short circuit in the test circuit [21].
Figure 5.2 demonstrates a test setup for measuring insertion loss. VL and Vo are measured
as shown, and then the loss is given by:
(5.1)
IL=20log,
0
VLJ
assuming the source and load impedances are matched. If they are not, the insertion loss
may be negative. In such cases it may be preferable to reference VL to the output voltage
generated by an ideal matching network. This voltage for the test circuit of Figure 5.2b is:
(5.2)
VR
S
L
This is known as power loss or flat loss, and is never negative for a passive filter.
Insertion loss is frequently defined as the single lowest loss at any frequency given by the
filter, and thus the other filter characteristics are referenced to it. Passband ripple is the
difference in dB between the maximum transmission in the passband and the minimum,
not including the rolloff into the transitionband.
The stopband characteristics are referenced as dBs of attenuation relative to the minimum
insertion loss, also referred to as out-of-band rejection. The nature of these requirements
vary more widely with application than the passband characteristics, but frequently
include a maximum bandwidth before attaining a certain level of attenuation, and a
minimum attenuation everywhere in the stopband. Additional spurious passbands caused
by higher order modes are usually acceptable in the stopband if they are far enough away
from the carrier frequency.
51
The transitionband characteristics usually deal with the shape or steepness of the rolloff
between the passband and the stopband. A common metric used is called the shape factor,
which is defined for two levels of attenuation, and calculated as the ratio of the
frequencies at which those two levels of attenuation are achieved, such that the answer is
always greater than one. For example, to calculate the 3 dB to 60 dB shape factor for a
filter response, the frequencies at which the attenuation equals 3 dB and 60 dB are
measured (excluding those caused by passband ripple), and then
frequency side and
W3dB/W60dB on
W6OdB/W3dB on
the high
the low frequency side of the passband give the
respective shape factors. They will be roughly equal for symmetric filters, though for
some applications an asymmetric response will be acceptable or even desirable. In all
cases, a value closer to unity is better, as that means a sharper transition from passband to
stopband, permitting less power from unwanted frequencies to leak through. Shape factor
usually only applies to filters that have a monotonic transition from passband to
stopband; there are many filters designed with a transmission zero in the transitionband
which results in an exceedingly sharp passband rolloff and some ambiguity in the
definition of shape factor. For these filters, simply noting the presence of the transmission
zeros is usually sufficient.
The last major frequency-domain filter characteristic of note concerns the filter's phase
response. At times it has no impact for a given application, but when it does, it is usually
required to maintain a certain degree of linearity. A linear phase response through a given
frequency range results in a constant group delay for signals being passed at those
frequencies, which prevents the signal distortion caused by a variable group delay.
5.1.2
Filter Design Goals and Motivations
The goal of the filter design in this chapter is to choose a filter topology that could
theoretically satisfy modern communications needs, and then analyze the Draper
resonator's suitability for implementing that design, along with predicting any potential
obstacles or limitations in doing so. A good example of a current application which could
gain much from the successful development of micromachined resonator filters is the
52
cellular phone industry. In order to be useful in this field, filters would need to be
designed that could conform to the cellular phone band allocations currently in use. An
overview of these can be found in [22]: carrier frequencies range from 800 MHz to just
over 2 GHz, and bandwidths range from 1 % up to 3.1% for most bands.
These specifications will motivate the topology and manufacturing analyses that follow.
The target achievable frequency range will be 200 MHz through 1.5 GHz: this range was
chosen during the resonator's initial design and simulation stages, and would be suitable
for many of the example wireless phone protocols as well as other RF communications
applications. Filter relative bandwidths of 1-3% are desirable. A specification not directly
attached to most consumer-level protocols, but essential to successful product fabrication
is the impedance level at resonance, which translates into insertion loss as a filter FOM.
This needs to be minimized to attain high performance and reduce the effects of noise.
Many current piezoelectric resonators have a 50 Q impedance at resonance (also known
as the Butterworth Van Dyke resistance), so that is a good target value.
5.2
Bandpass Filter Topologies
Three different filter topologies are presented and analyzed in this section. Each filter
transfer function is analyzed assuming no impedance at resonance (e.g., resonator
though the case study simulations which follow include a resistance
Q = 00),
corresponding to Q =
104 to allow a calculation of insertion loss. It is important to note that because all the
filter transmissions were calculated directly as voltages across a load impedance due to a
driving source, the magnitudes plotted do not take into account the insertion loss
reference. This was done so that the plots match directly with the listed equations. Since
the maximum output in each case is half the input, simply shifting each magnitude plot
up by 6 dB will allow direct assessment of insertion loss. Actual devices would be
measured using a RF probe analyzer and such data is usually reported as S21 (see Chapter
4), from which insertion loss may be directly read.
53
5.2.1
Simple Ladder
Z=sL+1/sC
Z=sL+l/sC
C12
Vin
RL
Vout
Figure 5.3: Simple ladder topology. Each resonator is modeled as an L-C pair. Resonator
CO and R are ignored during the design process.
The coupled ladder filter is an old, very well-characterized design widely used to
implement filters with discrete inductors and capacitors. Crystal resonators can be
employed in this topology by modeling each resonator only as an L-C pair. The
consequences of this assumption will be examined later. This topology can only be used
to implement narrowband bandpass filters, with fractional bandwidth less than -0.5%.
The major design variables and exact transfer function are:
wOL
Qs
(5.4)
K -
(5.5)
-
HSIPELDE
HSJMPLE LADDER (S)
Rs
wOL
QL =0
(5.3)
,loaded Q
RL
C 12
RL
K and loaded
=LSZ
2Z+R
Q are mathematical
L
+ Rs +sR LR sC 2 + sC 2 (RL +R S)Z+sZZ2 C
2
constructs that can be used to conveniently express the
more directly applicable figures of merit of the filter characteristics, like fractional
bandwidth and passband ripple. They should not be confused with resonator K 2 and
(unloaded
Q). For example, it is desirable to have
infinite unloaded
Q
Q, but the loaded Q
should be within a range of finite values bounded by other design parameters to meet
desired specifications; an infinite loaded
Q would result in two peaks of arbitrarily
width and no passband.
54
short
-R
Effect of KQ Product on Ladder Filter Response
-10-
A-_-15c,)
-20-
-25-
-30
Frequency
Figure 5.4: Maximally flat response vs. 1 dB passband ripple
The bandwidth is primarily determined by K. The KQ product affects the passband ripple
and shape factor. For a symmetric design (QI = Q2), KQ = 1 results in a maximally flat
or Butterworth response, where the global maximum of the transfer function is at the
center of the passband. As KQ increases, the transmission at the center of the passband
decreases such that the maxima are at the edges of the passband, introducing passband
ripple. At the same time, the shape factor improves (Figure 5.4), resulting in a Chebyshev
response. The phase linearity degrades with increasing ripple [21].
Brief Description of Operation
An ideal L-C pair has zero impedance at resonance, where the positive reactance from the
inductor and the negative reactance from the capacitor cancel out. In either direction in
frequency, the magnitude of the impedance increases linearly without limit. The resonant
peak is very sharp and narrow, so to make a bandpass filter this peak needs to be widened
55
and flattened somehow. This is accomplished by the coupling capacitor. Near the
resonant frequency, the L-C impedances will drop to low magnitudes such that they
become insignificant compared to the coupling impedance. Thus for a band of
frequencies where this inequality holds, the filter will effectively look like the source and
load resistances with just the coupling impedance. If the resistors are sized correctly
relative to the coupling impedance near resonance, the transfer function will be relatively
flat through this band. The parameter K relates the coupling impedance to the resonator
impedance and thus affects bandwidth, while loaded
Q relates
the source and load
resistances to the other impedances and thus affects passband ripple.
Analytical Approximation of Bandwidth
Assuming a symmetrical design (Qs = QL, Rs = RL) and with KQ = 1, an expression for
the fractional bandwidth can be obtained by using a narrow band approximation and
ignoring terms of lower orders of magnitude than KQ. The narrow bandwidth
approximation is:
(5.6)
Z
2
-w+s
s2vC
W _ 2
jww 0 C
2x
jw C
2(w, -w)
jw C
Z is the impedance of the L-C pair representing each resonator - R and CO are ignored
during design (their contributions are discussed below). The variable x is the fractional
distance from the center frequency, and wo is the resonant frequency, defined as V
(5.7)
V0
V
1
_
1+ Z
(1+
jWZC12
.
1
- j2Qx)(2+2Qx+ j)
RL)
1
2+4Qx + j(1 -4Q2X2
1
2
(5.8)
=16Q 4x
4
-4Qx)
+32Q 3 x 3 + 24Q 2x 2 +8Qx+5
Vout
V
This equation may be solved for x when the transfer function equals 1/2 (the maximum
possible transmission), or by differentiating and solving for the maximum. Both methods
56
give the filter's true center frequency at x
=
-1/(2Q). By transforming variables to y
=x
+
1/(2Q), the equation becomes:
(5.9)
2
=16Q 4 y 4 +4
The 3dB bandwidth can be found by solving this equation when
V
out
1
equals
Vill
2,f
which is 3 dB less than the maximum value of one-half. This gives:
(5.10)
y=
1
I
Thus the approximate half-power bandwidth is
(5.11)
BWadB
=
QWO
Q
These results were all expressed in terms of Q because 1/K was set equal to
Q. In general,
for any set value for KQ, all results may be expressed in either K or Q.
Resonator R and Co
Resonator resistance traditionally has had a small effect on filter performance for
piezoelectric types. However, it is a serious issue for the Draper resonator due to its
especially small size and high impedance level. This resistance directly increases
insertion loss at all frequencies, though this is usually acceptable to some extent for most
filter applications. Of greater concern is its effect on the filter shape: as the unloaded
Q
due to the resonator resistance decreases, it no longer becomes insignificant compared to
Q expression defined above. This effectively lowers the KQ product (by
the effective Q) which degrades the shape factor. Practically, this limits the
the loaded
lowering
minimum load and source resistance possible, which can present difficulties in
interfacing to other RF components. The resonator resistance may be decreased by using
multiple identical resonators in parallel and adjusting other values appropriately.
57
The static capacitance Co of each resonator does not affect the passband characteristics
much for a narrow bandwidth, which is why it may be ignored during design. However it
has very significant effects on the out-of-band attenuation. An L-C pair is supposed to
have a low impedance near resonance and high impedance at all other frequencies. At
frequencies lower than resonance, the static capacitance is a large impedance in parallel
with the L-C path and may be ignored. However at higher frequencies it shorts across the
L-C pair. This may lead to a limited attenuation at high frequencies, but using capacitive
coupling avoids this since C 12 shorts as well, resulting in a capacitive voltage divider
which will provide sufficient attenuation for small enough values of K. Another effect is
the parallel resonance added, which makes the resonator impedance peak at infinity at a
frequency slightly higher than the series impedance. This adds a transmission zero to one
side of the passband, which improves the shape factor on that side. However this also
means the resulting filter characteristic is asymmetric. Finally, if the circuit is analyzed
by replacing each resonator with only its static capacitance, which is a very good
approximation outside of the passband, a gradual peak with a one pole rolloff to either
side is obtained (Figure 5.5). For this topology, it is usually desirable to place the
passband far off this peak so that the attenuation immediately next to the passband is
greater. However the attenuation will decrease in the stopband towards this gradual peak,
so this must be taken into account to ensure that stopband requirements are met.
58
Ladder Filter Out-of-Band Attenuation
-20-
-40-
)
-60-
CO
-80-
00 -
1
120 -
-140
102
103
104
Frequency (MHz)
105
106
Figure 5.5: Simple ladder filter wideband response.
5.2.2
Dual-Resonator Ladder
ZR
_L
+
ZP
Vin
h
F
C
RL
Vout
T
I
Figure 5.6: Dual-resonator ladder topology. The series resonance of Z, is matched to the
same frequency as the parallel resonance of Z,. Both resonators have the same
capacitance values and differ only in inductance, though that is not a requirement of the
design; it is assumed here to clarify the analysis. The static capacitance CO must be
included in the resonator model to analyze this topology.
59
(5.12)
ZRESONATOR
(5.13) r =
wo +S 2
sC 0 (+r)w2+s 2
C
Co
(5.14) w, = wo 1+ r
(5.15)
frequency of the parallel resonance
RLZ
HDUALLADDER (S)
(R L + Zs)Z + R s (R L+ Zs + Z )
Brief Description of Operation
There are three important frequencies affecting the passband. Starting from the lowest,
they are the shunt zero (series resonance of Zp), center frequency (parallel resonance of
ZP and series resonance of Z,), and the series zero (parallel resonance of Zs). In the ideal
model with no resonator resistance, both zeros result in a filter transmission of exactly
zero, while the transmission at the center frequency will be exactly one-half, the ideal
maximum. Thus this topology results in an excellent shape factor and very symmetric
response if the source and load resistors are properly sized.
Filter Characteristics
The main drawback to this topology is that the bandwidth is fixed by a physical property
of the resonators, the ratio r of the dynamic and static capacitances. This ratio does not
change with geometry for the Draper resonator. It determines the difference in frequency
between the series and parallel resonances, which is the distance between the center
frequency and each zero in this filter transfer function. The 3dB bandwidth is about twothirds of the bandwidth separating the two zeros according to simulation. The bandwidth
may be adjusted a small amount by moving the two resonant frequencies either closer or
further apart at the expense of passband ripple and decreased phase linearity.
Additional drawbacks to this topology arise from its strong dependence on the static
capacitance of the resonators, Co. Once again, the off-resonance transfer function is well60
approximated by replacing each resonator with its static capacitance, forming a gradual
peak as with the previous topology. However, if the bandwidth is not very narrow, the
shape of this peak affects the passband shape significantly, and the passband must be
placed in the middle of the peak where this function is relatively flat. Thus the attenuation
directly off-resonance is very low. Also, the source and load resistances must be fixed at
about 1/(woCo), which is very large for micrometer-sized devices. These drawbacks are
particularly problematic for the Draper resonator. However, this filter topology is
reasonably common when using other types of resonators such as thin-film surface
acoustic wave or bulk acoustic waves types [9, 10, 23]. These devices are generally much
larger than the Draper device, which is one factor in determining a resonator's impedance
level. The high impedance level of the small Draper resonator will make it difficult to
interface to other components without excessive losses to parasitics. Also, the Draper
resonator cannot vary its static capacitance independently of its motional capacitance,
which determines its impedance level. If the series and shunt devices have impedance
levels that are too different, the shape of the passband is severely degraded, thus if this
topology is implemented with Draper devices the static capacitances must be very similar.
Since the off-peak attenuation is determined by a capacitive divider consisting of the
static capacitances, much additional attenuation can be gained by increasing the shunt
static capacitance relative to that of the series devices, especially if multiple resonator
stages are used, as is the common practice.
In summary, the dual resonator ladder topology nearly achieves the highest bandwidth
theoretically possible with crystal-type resonators, meaning any type for which the
Butterworth Van-Dyke model is applicable, at the expense of flexibility in choosing the
bandwidth. It also has a very good shape factor due to the presence of transmission zeros
in the transitionband. These qualities have made it a common filter implementation for
other thin-film resonators to date, but the Draper resonator's high impedance level and
fixed r make this topology impractical.
61
5.2.3
Lattice Filter
Za
+0
+
R
R
Zb
Vin
Vout
Zb
0l
Za
Figure 5.7: Full lattice filter topology. Two pairs of resonators comprise the filter: one
pair goes straight across each edge of the lattice section, and the other connects
diagonally across, forming an X (although they do not connect in the middle). The
capacitances are kept equal to simplify algebra, so that the pairs differ only in their
inductance and resonant frequencies. The source and load resistances are assumed to be
equal and will be referred to as R in the following analysis.
2
(5.16)
Za=
W
2
S2
W +S
s
sCO (1+r)w +s
(5.17)
V
VIN
= H LATITICE (S
sC 0 ((1+r)w +s 2
2 (b
~
b -a+R)
+R)(Za+R
HM H cE(s)-=RRC
LATTICE
2
(W2_
[W
w2 + S2+sRco((1+r)w
+ S2)W
3b
S2
+ sRc((+r)
+S2)
Brief Description of Operation
The lattice filter attenuates based on the impedance match between the two different
resonators. The filter can be viewed as having two arms, each arm having one of each
resonator Za and Zb, forming a path from the input signal to ground. The arms differ in
the order of resonator placement. The load resistance connects the two arms at the elbows.
When
Za(w)
equals
Zb(w),
the two arms are identical and the voltage at their elbows will
62
be equal, so no signal will be transmitted to the load. If the two impedances are not equal,
the voltages at the elbows will differ so some current will flow through the load.
If resonators with identical static capacitances are used, their impedances will match very
well off-resonance and thus the out-of-band attenuation for the filter will be very high.
Near resonance, the two series resonant frequencies wa and Wb will outline the passband
of the filter (Figure 5.8). Approaching the passband from lower frequencies, the lowerfrequency resonator Za hits its series resonance and effectively becomes a short. While
this does short the load resistor to the source resistor and to the input signal's ground, Zb
steals some current from the load so the transmission is not at the maximum possible. The
maximum output voltage of Vi,/2 occurs at the unique point where Za(W) = -Zb(w) with
matched source and load resistors (the derivation follows in the next discussion). When
Impedance of Za and Zb
-eZa
-- Zb
CD
Wa
Wb
-0
C
Full filter response
Frequency
Figure 5.8: Location of individual resonator frequency peaks relative to lattice filter
passband.
63
the two impedances are of opposite sign, they tend to cancel each other out, and if they
are also of equal magnitude, they effectively disappear. The sign of the Butterworth VanDyke impedance changes when it passes the series resonance, so the impedances will
only cancel in this way between the two series resonance frequencies. This is why a
passband is not also formed between the parallel resonances.
Resistor Sizing
Assuming the source and load resistances are equal, for a maximally flat passband, the
resistors should be sized so that the output voltage reaches its maximum possible value in
the center of the passband. One way to determine this value is to use the point at which
If we define the magnitude of this impedance to be X, then:
Za(W) = -Zb(w).
j2X
R
2 (R + jX)(R - jX)
(5.18) H
ATCE
jRX
R2
+ X
2
We wish to find the maximum value the magnitude of this function may yield. With no
loss of generality, R may be expressed as a ratio on X and thus X may be set to 1. We find
that the maximum possible value is one-half at the point where R = X. Thus one could
measure or simulate the point within the passband where the two resonators have the
same magnitude of impedance and set the resistance to that value to make a maximally
flat filter. As with the two-pole simple ladder, decreasing R improves the shape factor
while adding passband ripple and degrading phase linearity.
Analytical Approximation of Source and Load Resistance
A simple expression for the value of R needed for a maximally flat response may be
obtained if the fractional bandwidth is somewhat smaller than r and the approximations
f1\
(5.19)
--r
1 ~ r
BW
(5.20) We
BW
W
wh
+
2
are used, with BW defined as the bandwidth expressed as a fraction of the center
frequency. The second approximation is saying the filter center frequency is at the
64
geometric mean of the two series resonance frequencies, which is about the same as the
arithmetic mean if they differ by a small percentage. Finally, we define
(5.21) BW =
Wa
Wb
wa Wb
to obtain a simple result, even though this expression is likely to underestimate the true
3dB bandwidth.
Taking the lattice transfer function and evaluating its magnitude at the center frequency:
(5.22)
H
2[wa -
JRC
W + jw,
(5.24)
H - jRC
wbRC 0 ((1
+ r)w,
L1+
wWb
2(w -W)
(5.23)
-W
jRC(wb
Waw,=
rw"
+
a
-Wa
+ kwwhRCO((l + r)wh - W
jw
w, RCO 1+
rw
Wb
h-w _
a _
Wa + Wb
WWb
Wb)
)lwb
-w
wa + wb
j.wwRCO (+
2(wa --
wW
wawbR2CjI
r2
+ r
j~w, wbRCO (2 +r)
B2+
(5.25)
1 |2
_R 2c
2
(w(, +
4BW2
1+1vw
R 2C
2+2r+r2 + 2r 2+jW
Wb 2
WR2C2
r
(5.26)
|H1
2
2R 2 C2
2
~
a4BW2 d ribl
4
tc
((+
w, Wb
2r 2
BW 2
Here we define a temporary variable t:
65
( l+ bww
BR2C
j W4
r)2
2(l+ r)r 2 _
BW 2
I
BW
4
R 2 C 2wVb
(5.27) t=
W2
BW 2
Setting 1H12 = 1/4 and substituting C = Cor yields:
(52)11-=
(5.28)
4
t
1+2t+t
2
(5.29) t = R 2 c 2 wa
BW 2
BW
(5.30) R
C
WaWb
, t2 - 2t +I= 0
b
1
Wb
CWa
Wa
b
This expression should be accurate to well within a factor of 2, and is useful to use both
during the design process and to discuss the size of the load and source resistances
needed when comparing this topology to others for a given set of filter specifications.
Discussion of Bandwidth and Other Filter Characteristics
A considerable advantage of the lattice filter is the wide range of bandwidths it can
achieve. The maximum possible bandwidth is limited by the first parallel resonance, but
any bandwidth less than that is attainable. For the Draper resonators, r is about 3.2%,
which places the parallel resonance about 1.6% higher than the series resonant frequency,
allowing filter responses considered wideband. Filters with passbands higher than the
series-parallel frequency difference are still possible, but the inclusion of the parallel
resonance within the passband causes asymmetry and warping of the response, and
makes the passband phase increasingly nonlinear. It should be noted that the net 3dB
bandwidth possible with this topology can actually be greater than that achieved by the
dual resonator ladder topology. In both cases, two different resonators are employed such
that the parallel resonance of one falls on the series resonance of the other, which more or
less defines the maximum possible bandwidth possible. However, due to the zeros in the
ladder transitionband, its transmission is pulled down at the edges of the passband such
that the 3dB point is reached about two-thirds of the way to the frequency of the zero
itself. The lattice function has no such zeros, and in fact has the worst shape factor of the
three topologies examined, which allows its passband to expand generously up to and
even somewhat past its limits before beginning its gentler rolloff.
66
Bandwidth and out-of-band attenuation are inversely proportional to each other. This is
expected since attenuation increases the better matched the two resonator impedances are,
and if the resonant frequencies are closer they will match better at all frequencies outside
the passband. For most applications, this consideration is unlikely to limit the bandwidth.
One important practical consideration of this full-lattice design is the output and input do
not share the same ground, which presents some complications when interfacing to
unbalanced components. Such a transformation would require a baluns or active buffer
circuitry with differential inputs [24]. The latter choice suggests that this topology is
particularly suited for the possibility of integration with prevailing IC technologies into
complete packaged devices. Differential topologies are especially advantageous for filter
elements of very small dimension due to their excellent common-mode noise rejection.
5.3
Numerical Examples
Here one example of each filter design is presented using typical Draper resonator
parameters. All resonators were simulated with the same dynamic capacitance C = 0.1132
fF and static capacitance 3.514 fF with slightly varying inductance values to realize the
different resonant frequencies required by each design. Such freedom is available when
designing the equivalent circuit parameters of these resonators, as the inductance or
capacitance may be varied independently of the resonant frequency. Each filter was
designed to have a center frequency of approximately 800 MHz. The bar dimensions and
equivalent circuit parameters are listed for each filter example, along with the other filter
element parameters as defined for each topology in Section 5.2. The procedure used was
to first choose appropriate bar dimensions to generate the desired BVD circuit parameters
using the relationships found in Chapter 2. These electrical parameters were then input
into MATLAB as the transfer functions in equations 5.5, 5.15, 5.17 and simulated with
the bode function.
67
Bar
Value
Parameter
1
w
2a
6.04 m
3.22 m
0.5 gm
BVD
Filter
Element
Rs, RL
Loaded Q
Value
Parameter
L
C
R
wo
350.0 gH
0.1132 fF
175.8 Q
799.58 MHz
Value
1758 Q
1000
113.2 fF
C 12
K
10-3
Table 5.1: Simple ladder filter simulation parameters.
Simple Ladder Filter Response
-5
-
-I
-
-I
-
-
-10
Co'
-15
-20
-25
-30
-35
901
U,
0
-90
al)
W~
-180
-270
798
799
800
Frequency (MHz)
801
802
Figure 5.9: Bode plot of simple ladder filter transfer function.
Filter
FOM
fc
BW3dB
IL
Value
Shape
Left
Right
1.0019
1.0121
2.1785
1.0013
1.0042
1.0085
Factor
799.94 MHz
0.1334%
0.9202 dB
3dB-2OdB
3dB-4OdB
3dB-6OdB
Table 5.2: Simple ladder filter FOMs.
68
Out of Band
Attenuation
At w =0
Value
0
At w = O0
Minimum
29.51 dB
@ 26.6 GHz
Bar
Parameter
1
w
2a
6.04 gm
3.22 gm
0.5 pm
6.14 gm
3.17 pm
0.5 gm
BVD
Parameter
L
C
R
350.0 pH
0.1132 fF
175.8 Q
361.3 pH
0.1132 fF
178.7 Q
wo
799.58 MHz
786.98 MHz
Table 5.3: Dual resonator ladder filter simulation parameters. RL = 56649 Q.
Dual Resonator Ladder Filter Response
-5
-10
Co
~0 -15
-
-n
-20
2 -25
-30
-35
90
(D)
(D
a)
0
-90
I
740
760
780
I
I
800
820
Frequency (MHz)
I
840
860
Figure 5.10: Bode plot of dual resonator ladder filter transfer function.
Filter
Value
FOM
f,
799.83 MHz
BW3dB
2.1397%
IL
0.285 dB
Shape
Factor
3dB-2OdB
3dB-4OdB
3dB-6OdB
Left
Right
N/A
N/A
N/A
N/A
N/A
N/A
Out of Band
Attenuation
At w =0
At w =0
Minimum
Value
0
00
3.56 dB near
passband
Table 5.4: Dual resonator ladder filter FOMs.
69
Za
Zb
BVD
Parameter
Za
Zb
6.09 gm
3.195 pm
0.5 pm
6.01 gm
3.24 pm
0.5 gm
L
C
R
355.6 gH
0.1132 fF
177.2 Q
346.0 H
0.1133 fF
174.8 Q
wo
793.26 MHz
803.84 MHz
Bar
Parameter
1
w
2a
Table 5.5: Lattice filter simulation parameters. RL = 28 kM.
Lattice Filter Response
-b
-10
-15
C,
-20
-25
-30
90
-
I
I
I
I
I
790
795
I
I
I
I
I
I
I
I
-
0
(I0
CZ
-90
-180
-270
775
780
785
I
810
805
800
Frequency (MHz)
I
815
820
825
830
Figure 5.11: Bode plot of lattice filter transfer function.
Filter
Value
Shape
Left
Right
Factor
FOM
f,
801.13 MHz
BW3dB
1.8795%
IL
0.0676 dB
3dB-2OdB
3dB-4OdB
3dB-6OdB
Out of Band
Value
Attenuation
1.0198
1.0822
1.2514
Table 5.6: Lattice filter FOMs.
70
1.0203
1.0877
1.2874
=0
00
At w =C00
00
Minimum
none
At w
Discussion of Simulation Results
Shape Factor: In order from best to worst, the shape factors of the three filter designs are
dual ladder, simple ladder, and lattice. The simple ladder function is asymmetrical due to
the parallel resonance caused by the static capacitance Co off the high frequency side of
the passband. Thus the shape factor is excellent to arbitrary levels of attenuation on the
high frequency side, while it drops off much more slowly on the low frequency side,
asymptotically approaching a 20 dB/decade rolloff. However, near the resonance peak
the shape factor is very steep on both sides due to the high loaded
Q of this
topology
allowed by its very narrow bandwidth. To take advantage of the resonant peak's
steepness to greater levels of attenuation, another identical filter section may be added in
series. When shape factor is measured to an attenuation level beyond this initial steep
section, very large values such as the left value for 3 dB to 60 dB shape factor in Table
5.2 will result. The dual resonator ladder filter has the best shape factor due to the
transmission zeros on both sides of its passband. The lattice filter has no transmission
zeros and thus has the most gradual transitionband slope. As with the simple ladder
however, shape factor could be improved if needed by adding additional filter stages in
series - the fact that this design had twice as many resonators as the other two already
helped it compare better with them than it otherwise would have.
Out of Band Attenuation: The lattice filter has the best attenuation characteristic, as it
increases monotonically in both directions infinitely, with a slope no less than 60
dB/decade everywhere, which means in both directions the transmission falls off by at
least a factor of 1000 for every factor in 10 of frequency. A close second is the simple
ladder filter, which has similar (though not as steep) behavior except for the large gradual
minimum in attenuation which occurs far above the passband. This peak can be moved
further away if necessary, or the minimum attenuation increased by adding additional
stages; in any case, it is unlikely to be as large a problem as the additional modes of
resonance exhibited by all resonators which cause spurious passbands throughout the
stopband. Finally, the worst attenuation characteristic is that of the dual resonator ladder
filter. An attenuation of 3-4 dB extending for many filter bandwidths on either side of the
71
passband is almost certainly insufficient for most filter applications. This limitation is
difficult and costly for the Draper resonator to overcome due to its fixed r.
Load Impedance: The smallest load impedance is required by the simple ladder topology,
followed by the lattice, and the largest matching impedance is required by the dual
resonator ladder design. The simple ladder requires a small load impedance because its
narrow bandwidth limitation allows its loaded
Q to be very high. However the impedance
varies directly with the bandwidth, and can only be lowered independently of it to some
degree by allowing passband ripple. The lower limit for RL with this topology is set by
the impedance at resonance of the resonators: as RL approaches this value the shape of
the passband degrades considerably. The load impedance for the dual ladder is the
impedance of Co at resonance, which is very high for the small Draper resonators. If Co =
3 fF (in the middle of the range for these devices), the range for RL given the target
frequency range of 200 MHz - 1.5 GHz would be 35 kQ - 265 kQ. The impedance
required by the lattice design is about half that for the dual ladder at the same bandwidth,
but as with the simple ladder, the load impedance required is lowered with a smaller
bandwidth.
Bandwidth: The three filter topologies have very different bandwidth characteristics. The
dual ladder bandwidth is essentially fixed by the physical properties of the resonator used,
though as it is the maximum bandwidth generally possible for that resonator this topology
is useful for many applications. The simple ladder can only be used for very narrow
bandwidths while maintaining a good passband shape, less than about 0.5% of the center
frequency. The lattice topology has the most flexible bandwidth options by far, as it
essentially allows all bandwidths up to that achieved by the dual ladder.
Insertion Loss: The insertion loss of each filter essentially depends on how the
resonator's BVD resistance compares with Rs and RL. Thus it is subject to vary
depending on the parameters chosen for each filter. In general, the simple ladder topology
will always exhibit more insertion loss than the dual ladder because the dual ladder load
resistance will always be a good deal larger, which helps make the effects of a small
72
additional resistance less significant. The lattice topology is particularly insensitive to this
parasitic resistance in terms of insertion loss because unlike the other two topologies, the
center frequency of the lattice does not coincide with the series resonance of any of its
resonators. Since by design the resonators are not at zero reactance at this point, their
existing reactance helps reduce the impact of the nonideal resistance, which will be a
very small percentage of the reactance if the resonator
Simple ladder
Dual ladder
Full lattice
Achievable
Bandwidths
< 0.5% of fc
2.2% of fc
< 2.2% of fc
Shape
Factor
Good
Excellent
Fair
Q is high.
Out of Band
Attenuation
Good
Poor
Excellent
Load
Impedance
WcL/QL
1I/wcCo
%BW/wcC
Insertion
Loss
Good
Very Good
Excellent
Table 5.7: Filter FOM comparison for three filter topologies.
5.4
Manufacturing Limits and Tolerances
5.4.1
Resonator Bar Dimensional and FOM Limits
As shown in Eqns. (2.22)-(2.25), the equivalent circuit parameters of the resonator
depend on the bar dimensions and material parameters. Resonator FOMs such as the
resonant frequency wo and the impedance level at resonance may also be expressed
directly in terms of the bar dimensions. Therefore, manufacturing and design limits on
the bar determine the filter characteristics this resonator may attain. "Manufacturing"
limits are imposed by what is physically or practically realizable, for example, very thick
films can be prohibitively time-consuming and expensive, and it is hard to control their
stress. "Design" limits help optimize the final resonator characteristics, for example,
setting a lower bound on the length/width ratio helps maintain good modal isolation. All
the limits in the following discussion are not permanent and are subject to further
optimization and improvement as technology advances.
73
Thickness
There is a limit on the minimum total thickness (2a) that can be sputtered by a given
facility, and it becomes increasingly costly to make it very thick. An preliminary limit of
0.2-1.5 gm will be employed throughout this section, though it should be noted that
commercial fabrication technologies could likely achieve a range of at least 0.05-3.0 gm.
Length and Resonant Frequency
The minimum length depends on the width of the tethers holding the bar. Since they were
ignored in the original analysis, their contribution to bar flexing should be minimized,
which can be accomplished by restricting their width relative to the bar's length. This
suggests a design limit of at least five times the tether width. Current technology can
make the tethers as little as 0.5 jim wide, therefore the minimum bar length is 2.5 Pm.
The maximum length will be restricted by the bar's likelihood of hitting the substrate
upon a sharp acceleration. As the bar length increases, it becomes easier to bend and twist
up and down. If it should touch the substrate 1 [tm below, it may stick or break. Applying
the force-deflection relation for a simple cantilever beam:
Ewt 3
(5.31) F = 41t d
41'
where F is the applied force, E is the Young's modulus of AlN, d is the deflection, and 1,
w, and t are the beam dimensions (Figure 5.12), the approximate deflection for a given
force may be calculated. This calculation ignores the electrical contacts on the bar and
displacement due to twisting of the tethers, and assumes the deflection is relatively small.
As an example impact force, 1000 g times the mass of the bar will be used for F. With
thickness 2a and length 0.51:
(5.32) 1000g(p2avv(0.51)) = Ew(2a)3 d
4(0.5l)'
(5.33) d
=
62.5 gl 4 p
Ea2
74
K'
I
I
I
....
.........................
Id
It
'I
F
Figure 5.12: Cantilever beam deflection under an applied force. Under no deflection
the beam is a rectangular prism. The width w goes into the page and is
not shown.
10
9
8
7
E
0
6
2a=0.2 pm
5
4
2a=0.5 gm
3
2
2a=1.0 gm
1
0
0
200
300
Bar Length (pim)
100
400
500
Figure 5.13: Deflection vs. bar length under a force of 1000mg for various thicknesses.
2a (pm)
imax (Rm)
0.2
198
0.5
313
1.0
442
Table 5.8: Total bar length allowing a maximum 1 gm deflection under a 1000mg force.
75
Another means by which the bar may hit the substrate is through twisting of the tethers.
Modeling this mechanism is somewhat complicated. To obtain a simple estimate for the
limiting lengths in this case, the tethers will be modeled as elastic cylinders with diameter
halfway between the width and thickness of the tethers. The torque-twist relation for
cylinders is:
(5.34) T = GK d,40
32h
where G is the shear modulus of AlN and d, is the diameter of the cylinder. The other
variables are defined by two clamp points along the cylinder, for example, if one end is
attached to a fixed surface while the other is attached to a lever applying torque. The
applied torque T and twist 0 are measured between these two points, and h is the distance
between them (Figure 5.14). Assuming no deflection, w is two-thirds 1, tether width is 0.5
gm, tether length is 30 gm (a conservative maximum), and a 1000 mg force is applied at
the beam's midpoint:
(5.35) 1000g (p2aw(O.51))(0.251) = G d,4arctan
32h
~53
l
(5.36) d = tan
2
1280an(__
g_________
(
G(2a+0.5x10-6)4
J
-
2.56ga13 p
1
2
)
tan
Gz(2a + 0.5x 10-6 )4
Clearly the maximum bar length is constrained most by twisting of the tethers, although
this estimate is extremely conservative in using a tether length of 30 pm. A shorter tether
0.51
...................................................................
. ..........................................................
.......................
d
Id
Figure 5.14: Elastic linear torsion. The twisting cylinder has diameter d, and length h
which goes into the page. The near end is assumed fixed while the far end is
attached to one end of a beam with length 0.51. A force F on the beam
creates a torque on the cylinder, and displacement d of the beam's end.
76
10
2jim
9
8
2a=0.2 pm
7
2a=0.5 gm
E
6
0
5
a)
C)
4
3
2a=1.0 pm
2
1
0
I
0
20
10
30
40
I
I
50
60
70
Bar Length (jim)
I
80
I
90
100
Figure 5.15: Deflection vs. bar length under a torque of 250mgl for various thicknesses.
2a (gm)
imax (im)
0.2
35
0.5
40
1.0
50
Table 5.9: Total bar length allowing a maximum 1 gm deflection under a 250mngl torque.
length can almost double the maximum length, since to first order e 0C h-', but these
conservative estimates will serve as acceptable limits for this discussion. Bounding the
values for bar length allows calculation of the resonant frequencies achievable by this bar
design, since wo depends only on 1:
(5.37) fo -
2)r
21
Now the full range of possible resonator frequencies can be calculated and compared to
the target range of 200 MHz to 1.5 GHz. By these calculations, the specifications are
achievable.
77
2a (gm)
lmax (wm)
(MHz)
Wmin
0.2
35
138
0.5
40
121
1.0
50
96.7
Table 5.10: Minimum resonant frequency achievable at three thicknesses, 30gm tethers
Tether width (gim)
min
(gM)
Wmax
(GHz)
0.2
1.0
4.83
0.5
2.5
1.93
1.0
5.0
0.97
Table 5.11: Maximum resonant frequency achievable at various tether widths.
Width and Impedance Level
The final bar geometric parameter is width. It is essentially only limited on the lower end
by lateral manufacturing capabilities to a few tenths of a micron. As it turns out, it will
usually be desirable to keep width as large as possible to minimize impedance, so the
lower limit is not too significant. As width increases relative to the length, the frequency
of the longitudinal width mode grows closer to that of the length mode. To maintain good
modal isolation and to make sure the desirable longitudinal length mode is the lowest in
frequency, a design limit for maximum width is set to two-thirds the length. Another
approach is to make the width much greater than the length, resulting in additional modes
below the length resonant frequency. The advantage to this approach is it reduces the
impedance level of the resonator, and may be explored later for this application, but it
will be ignored in the present discussion.
It is useful to express these geometric limits in terms of the resonator FOMs used during
filter design. If the resonator bar is modeled as an L-C pair, two relevant parameters are
the resonant frequency w0 and the "impedance level", defined as V/LC and
78
%
respectively. The impedance level is the magnitude of the impedance of either the
inductor or capacitor at resonance (they are equal), and is important for scaling the other
elements of the filter appropriately. In terms of material parameters and the bar
dimensions 1, w, and a:
(5.38)
C= ae
1C 4we2
C
As it turns out, the BVD parameter R is equal to the impedance level divided by Q. This
resistance is also known as the "impedance level at resonance". In general this term is a
pure parasitic and it is desirable to reduce it as much as possible. Since a common RF
matching impedance is 50 Q, that is a good order-of-magnitude target for R.
(5.39) RB D
a
4we 2 Q
The impedance level at resonance is inversely proportional to both
Q and
w.
Q is
determined by quality of the manufacturing process and higher-order loss mechanisms, so
it will be treated as a given parameter for now, and estimated at the target minimum value
of 10000. Now given a bar thickness and resonant frequency, the maximum width and the
minimum R can be calculated as shown in Tables 5.12 and 5.13.
2a (jim)
imax
(m)
wo (MHz)
w (jim)
0.2
35
138
23
0.5
40
121
< 27
1.0
50
96.7
< 33
R (Q)
9.87
2 1.0
34.4
Table 5.12: Impedance level at resonance achievable for minimum frequencies.
Tether width (jim)
imi
(pim)
wo (GHz)
w (gm)
R (Q)
0.2
1.0
4.83
5 0.667
851
0.5
2.5
1.93
1.67
340
1.0
5.0
0.97
5 3.33
170
Table 5.13: Impedance level at resonance achievable for maximum frequencies. Bar
thickness 2a was set to 0.5 ptm.
79
The target impedance at resonance of 50 Q is difficult to achieve without going to lower
frequencies. Advanced commercial technologies can produce smaller thicknesses and
impedances than those discussed here, but real-life deviations from these ideal analyses
must also be expected to reduce such gains. One way to address this issue is to increase
the impedance level of components interfacing to the filter to 200 - 400 K2, as has already
been demonstrated in some integrated designs. Another possible solution is to fabricate
two identical resonators in parallel with each other. The resulting pair has an equivalent
impedance of a single resonator with half the resistances and inductances and twice the
capacitances, using the BVD model. Thus the resonant frequency remains the same while
both the impedance level and the impedance at resonance are halved.
5.4.2
Manufacturing Tolerances and FOM Sensitivities
It is important to consider the effects of the imperfect machining capabilities of whatever
equipment is used to fabricate a device. Each physical parameter is subject to finite
manufacturing tolerances, representing the precision with which the machinery can
achieve a target value for that parameter. In the case of the Draper resonator, it was a
specific design goal to have the various resonator properties defined lithographically,
over a wide range. Thus any error in producing the correct bar dimensions will translate
strongly to error in meeting the desired resonator characteristics. The dependence of a
design parameter q on a fabrication parameterf maybe be expressed as the sensitivity S:
d
d
(5.40) Sdf=
dfl
f
)y"
df q
which compares the relative change in q due to a relative change in p.
A sensitivity of one indicates that a 1% change in the independent parameter causes a 1%
change in the dependent parameter. Since the geometric parameters were originally
intended to define the resonator characteristics, the dependence is strong; thus filter
designers should be aware when using this resonator that they are limited by the
tolerances of the technology used.
80
Typical tolerances with current technology are 1% for thickness (a) and about 2% for
small (-3-5 gim) lateral dimensions across the wafer (1, w). These tolerances translate into
a 2% difference in resonant frequency wo between different devices, and a 3% difference
in impedance level
V7. Various tuning methods are under investigation, such as
trimming some mass from the bar corners after fabrication, applying a DC voltage across
the resonator, and coupling multiple resonators mechanically to force them to lock on to
the same resonant frequency even if they are slightly different in length.
5.5
Filter Design
The preceding sections have outlined the capabilities and limitations of the Draper
resonator, and given some examples of how it may be used in a filter. This section will
summarize the process of designing a filter around this resonator, from an applicationspecific point of view. In this case a set of filter specifications is the starting point, such
as a center frequency, minimum bandwidth, and load impedance, specified by a particular
application. In addition, a set of constraints is provided by the capabilities of the Draper
resonator and the technology being used to fabricate it. In general, both sets of
specifications will allow some leeway in the exact resonator characteristics, but some
back-and-forth iteration will be required to find a solution that meets all the requirements.
5.5.1
Filter-level Resonator Requirements
When designing the filter, a topology is chosen and element values are selected for the
electrical components. At this level the designer uses an electrical model of the resonator
and must take into account whatever restrictions the chosen resonator places on the
possible values for each component. For the Draper resonator, it is important to note that
there are fundamentally only two degrees of freedom in its BVD equivalent circuit. The
ratio of the motional and static capacitances is fixed at 1/31.04 (according to simulation),
and choosing L and C determines R through
Q:
81
R=
/c
. Thus a good way to define
the resonator at this level is with its resonant frequency and impedance level, which
together completely determine its geometry and other characteristics. Usually the
resonant frequency is established by the application, which leaves only the impedance
level free for adjustment. The other common filter-level properties of interest,
Q and
r,
have already been set by the process and technology, so the designer should confirm that
these values would allow the filter specifications to be met. Otherwise this resonator is
not suitable for the application.
Impedance Level
Given a resonator with a certain impedance level, the elements of a filter designed to use
it will have parameter values scaled according to that impedance level. If the resonator's
impedance level were changed, the other filter components could be scaled by the same
factor to maintain the same filter characteristic. Thus an obvious way to choose an
impedance level is to meet a desired source or load impedance requirement.
Unfortunately, there are several other factors that change with impedance level, like area
and cost, and modal isolation.
The decision is usually simplified by consideration of parasitics, especially in a highfrequency integrated system. In such systems an overriding concern is the effect of
parasitic capacitances. At high frequencies, these capacitances have small impedances
and steal signal current away by appearing as shorts. To counteract this effect, RFIC
systems usually try to minimize their operational impedance levels, with the goal of being
very small compared to the parasitic impedances. Using typical numbers, this usually
means for the Draper resonator that the smallest impedance level possible before modal
overlap becomes significant is chosen. In other words, there is a minimum load
impedance possible for each filter topology using this resonator, and it is usually
somewhat high so the minimum possible value is chosen. In summary, though this can
vary with application and technology, the filter designer generally does not have much
flexibility with this resonator, and it becomes a decision of whether this resonator will
satisfy the requirements as is.
82
5.5.2
Resonator Fabrication Tradeoffs
The existence of only two independent electrical parameters for the Draper resonator can
be explained mathematically by inspecting the equations relating bar dimensions to those
parameters. Width and thickness always appear in the form wia, which means their only
contribution to determining the electrical parameters is through their ratio. Since the
resonant frequency is uniquely determined by the length, once again there is only one
degree of freedom for a given frequency. However since there are two parameters that go
into this ratio, the choices are not as limited as during the filter design stage.
The same constraints do apply here, such as the desire to minimize cost and area, and
reduce the impedance level. Both of these can be accomplished by maximizing w/a. The
width can only be increased until it reaches the limit of two-thirds the length to maintain
the chosen modal isolation. So the only other choice is to decrease the thickness. There is
an overall minimum set by the process technology on thickness, as well as the
consideration for multiresonator systems that all the thicknesses must be the same. In
practice this means that the thickness will likely be set wafer-wide as small as the
technology will allow. The relative magnitude of the width of each resonator in any
coupled topology can be calculated based on considerations of impedance-matching the
various stages, and the largest relative width will be set at the design maximum, which
minimizes impedance throughout the circuit while maintaining modal isolation. Thus,
once again for a given application and technology the design is likely to be constrained at
one of its limits.
5.5.3
Simple Ladder Filter Design Process
As an example of the filter design procedure, a step-by-step outline using the simple
ladder topology follows. First, the filter requirements must be checked to make sure they
are realizable with the Draper resonator: a center frequency between about 200 MHz and
2 GHz, and a bandwidth less than about 3% of the center frequency. An additional
requirement of this particular topology is that the percentage bandwidth be 1% at the
highest, and preferably less than 0.5%. Next, the filter requirements should be converted
83
to filter K and loaded
Q values
using a published table or analytic equation. One final
check before component selection can begin is to make sure the loaded
Q is sufficiently
Q. The bigger the difference the better, with a minimum resonator
Q of three times the loaded Q to ensure an acceptable filter characteristic.
less than the resonator
Equations to generate normalized k and q values for any number of filter stages can be
found in the Appendix. These parameters relate the impedance values of elements of
neighboring coupled filter stages, as defined in [25]. There is a k between each pair of
stages, so k1,2 relates the first and second stage, k2 ,3 relates the second and third stage, etc.
while there are usually only two q parameters, for the source and load stages. The
lowercase signifies values normalized to a fractional bandwidth of one, whereas
unnormalized values for specific bandwidths are usually written in uppercase. The
difference is just a multiplication or division by the fractional bandwidth as shown in the
Appendix. Throughout this thesis, the symbols K and
Q are used to represent these values
for the two stage, symmetrical topology of Sec. 5.2.1.
When calculating the coupled filter parameters by hand, k and q are most easily obtained
by calculating an intermediate set of parameters called g-codes, used to design low-pass
filter prototypes (i.e. a low pass filter with the same bandwidth and rolloff). There are n +
1 g-codes for a n-order coupled filter.
To illustrate the process, here are two examples for a two-resonator filter, a Butterworth
response (zero passband ripple) and a 1 dB ripple Chebychev response. With n = 2 there
are 4 g-codes. For the Butterworth response, go and g3 are both 1, and g, and g 2 are:
(5.41) g1 =2sin (2 -1)fl=-2
(22
(5.42)
g
2
= 2sin (2
2 l)zlV-2
With I dB ripple, go is still 1, and substituting values for the other codes gives:
(5.43) g, = 1.8219,g
2
= 0.6850,g
3
= 2.6597
Obtaining normalized q values from the g-codes is straightforward [25]:
84
(5.44) qsOURcE = gog 1
(5.45)
qLOAD =
g2 g 3
For symmetric designs, qsOURCE = LOAD The Butterworth q is
-ii,
and the 1 dB ripple q
is 1.8219. Calculating k in general is slightly more complicated because the number of
coefficients increases with the order of the filter. Specifically, there are n - 1 coefficients,
each corresponding to a coupling impedance between two neighboring resonators, which
are numbered from 1 to n going from source to load. For a two-resonator design there is
only one k, and it is labeled k
(5.46) kI, 2
because it is between the first and second resonators.
--
1
This results in 1
2
for Butterworth and 1.1171 with ripple. K and
Q are obtained by
multiplying k by the fractional bandwidth, and dividing q by the same. Note that the
product kq is constant for all bandwidths given n and the passband ripple.
Continuing with the design process, the motional inductance and capacitance of the
resonator are defined as L and C, the impedance level as ZO=
/,and the loaded
Q as
QL. All the other filter elements may be expressed in terms of wo and Zo:
(5.41)
R s= R L=
QL
(5.42) C12 =- =
K wOZOK
using the topology shown in Figure 3.3. Since wo is usually given as a filter specification,
this leaves the choice of Zo to be made. The possible values are bounded by maximum
bar width and minimum thickness on the lower end, proceeding up to very high values
for smaller widths. The choice within this range is determined by the desired interface
impedance, and possibly also to limit the size of C12. Once the value for ZO is set, bar
dimensions can be calculated from wo and ZO and the design is finished.
85
Some further comments about filter K and QL: they are uniquely determined for a given
number of resonators combined with bandwidth and passband ripple or shape factor
specifications. One thing this means is that if bandwidth, passband ripple, and shape
factor are simultaneously specified, they may be impossible to realize without adding
additional resonators. Second, the equations generating K and
Q ignore the effects
of the
resonator impedance at resonance and its static capacitance. Thus the filter should be
overdesigned to exceed specifications to account for this. Finally, filter specifications are
usually given as a bounded value, for example, a minimum bandwidth, a maximum
passband ripple, or a minimum shape factor. If a filter can be designed that exceeds these
bounds, some flexibility in K and
bandwidth,
Q and
Q is
present. For example, with some leeway in
K may be changed to hit a more favorable value for load impedance
without falling out of specification.
5.6
Filter Bank Considerations
One of the most promising applications for an integrated, high-Q resonator is to make an
integrated filter bank. While it is premature at this point to carry out a full design, some
analysis can be made on the eventual feasibility of such a device, and some potential
difficulties may be enumerated.
5.6.1
Filter Bank Layout
One conceptualization of a filter bank is a set of individual filters connected in parallel
between active buffer stages, selected via electrical or possibly MEMS switches (Figure
5.16). Only one switch would be closed at a time, actuated by frequency-selection
circuitry. The maximum number of filter branches between two individual buffers would
be determined by the effect of parasitic loading of the additional paths and parasitics due
to increasing lengths and areas from laying out many multiresonator filters and switches
between the buffer pair.
86
Figure 5.16: One possible switched filter bank implementation.
5.6.2
Impedance Matching and Parasitic Loading
In such a topology, it is vital that each filter be designed for the same source and load
impedance, unless passive matching elements can be placed in series with each filter,
though it is desirable to eliminate matching elements if possible to save on die area and
cost. An advantage of fully integrated filters is the impedance of the active circuits
interfacing with the filters may be adjusted to best accommodate the filter design, for
example at load values of 1000 Q or more, where a higher impedance might be desirable
to minimize relative losses through parasitic resistances. The limit to this flexibility arises
from the parasitic capacitances loading each stage. Each switch is not perfect, and
presents some finite capacitance while open. Additionally there is some capacitance
linking adjacent resonators. These issues are present with single filter circuits, but are
multipled in severity when several filters are connected in parallel in a filter bank. To
prevent too much loss through these parasitic capacitances, the impedance level of each
stage should be kept low to compete favorably for the signal current. Since the parasitic
capacitances in general will be much higher within a filter bank, higher frequencies will
also be harder to achieve compared to a single filter circuit since the frequency at which
these capacitances annihilate the signal will be that much lower.
87
5.6.3
Harmonic Mode Cancellation
Another problem that is present for single filters, but particularly troublesome in a filter
bank is the existence of spurious resonant modes. Such modes would pass signals that
may be important to attenuate for a given application. One example is the width
longitudinal mode, which approximately is governed by the same equations as the length
mode, except length and width are switched. A possible solution to this additional
passband is to use multiple resonators with the same length mode frequency but differing
width mode frequencies. Wherever one resonator is used in a filter, another is added in
series. In theory, the length resonance of the first resonator will overlap fully the
resonance of the second resonator and pass signal at that frequency as originally intended,
while the two dissimilar width resonances should mostly cancel. In a coupled
multiresonator filter topology such as the simple ladder, this cancellation can be designed
into the two existing resonators to achieve this effect.
Figure 5.17: The Butterworth Van-Dyke model with an additional resonance.
Simulation
A numerical simulation to test this solution was performed using the simple ladder filter
topology described in Section 5.2.1. The resonators were modeled as shown in Figure
5.17. By adding another R-L-C branch in parallel with the BVD model, an additional
resonance/anti-resonance pair (corresponding to the width mode) is added to the
frequency response. One resonator was designed with branches with resonant frequencies
88
at 780 MHz (length = 6.19 pm) and 860 MHz (width = 5.62 pm), while the second had
frequencies of 780 MHz and 944 MHz (width = 5.12 gm), which corresponds to a width
half a micron less than in the first resonator. A reference filter using the wider resonator
twice was also simulated for comparison. The other filter parameters and element values
are listed in Table 5.15.
Figures 5.18 and 5.19 show the wideband response of the two filters. In the reference
filter, the minimum loss at both peaks is about 6.5 dB. On the other hand, the mode
canceling filter maintains a primary insertion loss of about 6.5 dB while attenuating the
spurious peaks by about 19 dB and 21 dB. Doubling the total number of resonators
should also approximately double this attenuation while not increasing the primary
insertion loss by too much, although precise matching would become more difficult.
Overall this method of cancellation appears to be a worthwhile solution to unavoidable
spurious resonances, especially if multiple resonators were already called for to meet
attenuation or shape factor requirements.
89
Filter Parameter
WO
Rs, RL
QL
Value
780 MHz
2016.2 Q
500
Table 5.14: Simple ladder filter simulation parameters.
0
K
C 12
0.002
101.1 fF
-50
.
CO)
E
CO)
c -100
IS
= ml 5
iD.I
C,)
a)
a)
a)
I
-9C
_0
CD)
as
180
-270'
700
750
800
850
Frequency (MHz)
900
950
1000
Figure 5.18: Filter transmission characteristic with spurious resonance
0
0
0
0
Cl)
E
-20
-40
-60
-80
-100
C
a)
-120
9C
~0
-9C
a
-180
-270
700
750
850
800
Frequency (MHz)
900
950
1000
Figure 5.19: Filter transmission characteristic with cancelled spurious resonance.
90
Chapter 6
Summary and Conclusions
6.1
Summary of Thesis Work
Draper Resonator
The design goals behind the Draper resonator were presented. It was intended to satisfy a
strong need in the field of RF communications for a high-performance, GHz frequency
resonator compatible with integrated silicon processes. After a brief review of competing
technologies, it is clear that if the optimized resonator can meet performance
specifications, it can fulfill this purpose. The approximate first-order behavior was
derived and found to be equivalent to the Butterworth Van-dyke circuit model about
resonance. The approximate range of resonant frequencies achievable with current
technologies are 200 MHz to 2 GHz, with a range of impedances at resonance of
50-1000 Q possible if the target
Q of 10,000 is reached.
Measurement and Modeling
A procedure for measuring the frequency response of resonators at RF frequencies and
extracting equivalent circuit parameters from these measurements was devised and tested
on optimized TFR resonators. The models and the fitting routine produced an acceptable
fit to the data. However, measurement of the first Draper devices was hindered by a large
parasitic and data could only collected on a small number of extremely large resonators.
In addition, due to this parasitic and the unoptimized experimental process, fitting was
not possible with the developed methods due to significant additional parasitics which
91
those methods did not account for. Hand extraction of resonant parameters estimated an
achieved
Q of 542 at a resonant
frequency of 148 MHz.
Filter Design
Finally, implementation of the Draper resonator in three filter designs was explored. The
different filter topologies were first analyzed generally and their tradeoffs and capabilities
discussed, and then numerical examples with typical parameters expected of the Draper
resonator calculated and presented. Some of the issues concerning single filter design
were extended to a filter bank in concept and the ramifications discussed.
6.2
Future Investigations
Optimization and refinement of the capabilities of the resonator will proceed, hopefully
resulting in high
Q, temperature
stability, and methods of fine-tuning the resonant
frequency. Additional process investigations will focus on precisely matching multiple
resonators on the same wafer to each other. One method to accomplish this may be direct
mechanical coupling of several resonator bars while keeping the various electrical
connections distinct. This will obviously be very important in the design of
multiresonator filters and filter banks.
Another important goal in the next stage of resonator improvements should be lowering
its impedance level. As it is currently, integration at high frequencies with active devices
will be difficult due to parasitic capacitances competing with the high impedances levels.
A promising avenue of research to resolve this is to make the width much wider than the
frequency-determining length. This would reduce impedance by the same factor that is
used to increase the width at the cost of moving the spurious width resonance modes to
lower frequencies. However, higher-order modes of these longitudal resonances are
attenuated proportionately to the mode number, so by placing the larger modes far
enough below the functional resonance, they may be eliminated through associated
circuitry.
92
Finally, once single resonator characteristics have been optimized, the issue of acoustic
isolation must also be addressed to allow multiple resonators to function on the same
substrate without interfering with each other.
93
94
Appendix A
Coupled k and q Coefficients For Ladder Filter Design
This Appendix is referenced in Section 5.5.3 of the text. Coupling coefficients relating
the impedance values of neighboring ladder filter stages can be generated to aid in filter
design. An excellent explanation and definition of the k and q coefficients is found in
[25]. This section will only summarize the equations necessary to generate these
constants as found in [26].
The calculation begins by finding a series of values known as g-codes. These determine
the element values of a low-pass prototype filter. For a Chebychev filter of order n and
passband ripple LAr (in dB), the g-codes go,
B=lIncoth
LAr 4n10
40
-~
c = sinh
nn+
)
Bjl
2" )
ak =sin L2k
~ 2"n
bk =c 2 +sin2 O,
n
, k= 1, 2,
...
n
k=1, 2,... ,n-I
go =1
2a
C
gk
--
bk-
ak_' ,k=2, ...
n
Igk-I
g,+ 1 =1 forn odd
=
for n even.
coth 2
4
95
are given by the following equations:
For the special case of LA, = 0 (a Butterworth response), the equations are:
go = g, 1 = 1
gk
(2* _
- )Zr
, = 1, 2,... ,n
1Ik
2
= 2sin
In both bases, the codes are normalized such that go = 1.
These equations give normalized k and q values as can be found in many published
tables. The index refers to the ladder filter stages, as explained in the references.
Kj~
%BW
=
QSOURCE=-99
%BW
K
A =LO
%BW
where the fractional bandwidth is defined as the 3dB bandwidth divided by the center
frequency:
%BW =
BW 3 dB
fo
To get normalized k and q values, set the fractional bandwidth to one.
96
Appendix B
Material Parameters of Aluminum Nitride
p = 3.3 -10
e3, =
e32
kg/m 3
3
= -0.58
C/M 2
e3 3 =1.55
C/M 2
e4 =1.13
C/m 2
e,5 = -0.48
C/M 2
c1, = 3.45.10"
1.20 -10"
C=
2
N/m
N/M 2
c33 =3.95.10"
N/m
2
1. 18 -10"
N/m
2
c66 =1.10.108
N/M
2
C44=
co,
£33
=8.85 -j -"'2 F/m
=9.5. e,
97
References
[1] Lutsky, J., "A sealed cavity thin-film acoustic resonator process for RF bandpass
filters." Ph.D. diss., Massachusetts Institute of Technology, 1997.
[2] Meier, H., Baier, T., and Riha, G., "Miniaturization and advanced functionalities of
SAW devices," IEEE Transactionson Microwave Theory and Techniques, vol. 49,
no. 2, pp. 743-748, Apr. 2001.
[3] Morkner, H., Ruby, R., Frank, M., and Figueredo, D., "An integrated FBAR filter
and PHEMT switched-amp for wireless applications," IEEE MTT-S International
Microwave Symposium Digest, vol. 4, pp. 1393-1396, 1999.
[4] King, D., and Gopani, S., "SAW filters in CDMA mobile communication networks,"
Wireless Technologies China '99 Conference Proceedings,Shanghai, China, pp.
104-107, Sept. 1999.
[5] Freisleben, S., Bergmann, A., Bauernschmitt, U., Ruppel, C., and Franz, J., "A
highly miniaturized recursive Z-path SAW filter," UltrasonicsSymposium 1999
Proceedings,Caesars Tahoe, NV, vol. 1, pp. 347-350, Oct. 1999.
[6] Dubois, M. A., Muralt, P., Matsumoto, H., and Plessky, V., "Solidly mounted
resonator based on aluminum nitride thin film," UltrasonicsSymposium 1998
Proceedings,Sendai, Japan, vol. 1, pp. 909-912, Oct. 1998.
[7] Lakin, K. M., Kline, G. R., and McCarron, K. T., "High-Q microwave acoustic
resonators and filters," IEEE Transactionson Microwave Theory and Techniques,
vol. 41, no. 12, pp. 2139-2146, Dec. 1993.
[8] Ruby, R. C., Bradley, P., Oshmyansky, Y., Chien, A., and Larson, J. D., III, "Thin
film bulk wave acoustic resonators (FBAR) for wireless applications," Ultrasonics
Symposium 2001, Atlanta, GA, vol. 1, pp. 813-821, Oct. 2001.
[9] Lakin, K. M., "Thin film resonators and filters," UltrasonicsSymposium 1999
Proceedings,Caesars Tahoe, NV, vol. 1, pp. 895-906, Oct. 1999.
[10] Lakin, K. M., McCarron, K. T., Belsick, J., and McDonald, J. F., "Thin film bulk
acoustic wave resonator and filter technology," Radio and Wireless Conference
(RAWCON) 2001, pp. 89-92, Aug. 2001.
[11] Ruby, R., and Merchant, P., "Micromachined thin film bulk acoustic resonators,"
Proceedingsof the 1994 IEEE InternationalFrequency Control Symposium, Boston,
MA, pp. 135-138, June 1994.
98
[12] Cleland, A. N., "Single-crystal aluminum nitride nanomechanical resonators,"
Applied Physics Letters, vol. 79, no. 13, pp. 2070-2072, Sept. 2001.
[13] Wang, K., Wong, A. C., and Nguyen, C. T. C., "VHF free-free beam high-Q
micromechanical resonators," Journalof MicroelectromechanicalSystems, vol. 9,
no. 3, pp. 347-360, Sept. 2000.
[14] Antkowiak, B., Gorman, J. P., Varghese, M., Carter, D. J. D., and Duwel, A. E.,
"Design of a high-Q, low-impedance, GHz-range piezoelectric MEMS resonator,"
The 12th InternationalConference on Solid-State Sensors, Actuators, and
Microsystems (Transducers),Boston, MA, pp. 841-846, June 2003.
[15] ANSI/IEEE Std 176-1987, IEEE Standard on Piezoelectricity.
[16] Zelenka, J., PiezoelectricResonators and their Applications, New York, NY:
Elsevier Science Publishing Co. Inc., 1986.
[17] Gerber, E. A., and Ballato, A., PrecisionFrequency Control, Volume 1: Acoustic
Resonators and Filters,New York, NY: Academic Press, Inc., 1985.
[18] Auld, B. A., Acoustic Fields and Waves in Solids, Vol. I, Malabar, FL: Robert E.
Krieger Publishing Co, Inc., 1990.
[19] Lakin, K. M., "Modeling of thin film resonators and filters," IEEE MTT-S
InternationalMicrowave Symposium Digest, Albuquerque, NM, pp. 149-152, June
1992.
[20] Pozar, D. M., Microwave Engineering,Reading, MA: Addison-Wesley, 1990.
[21] Kinsman, R. G., Crystal Filters,New York, NY: John Wiley and Sons, 1987.
[22] Lakin, K. M., McCarron, K. T., Belsick, J., and Rose, R., "Filter banks implemented
with integrated thin film resonators," 2000 IEEE UltrasonicsSymposium, San Juan,
Puerto Rico, vol. 1, pp. 851-854, Oct. 2000.
[23] Mang, L., Hickernell, E., Pennell, R., and Hickernell, T., "Thin-film resonator ladder
filter," IEEE MTT-S InternationalMicrowave Symposium Digest, Orlando, FL, vol.
2, pp. 887-890, May 1995.
[24] Mang, L., and Hickernell, F., "ZnO thin film resonator lattice filters," Proceedings
of the 1996 IEEE InternationalFrequency Control Symposium, Honolulu, HI, pp.
363-365, June 1996.
[25] Zverev, A. I., Handbook of FilterSynthesis, New York, NY: John Wiley and Sons,
1967.
99
[26] Matthaei, G. L., Young, L., and Jones, E. M. T., Microwave Filters,ImpedanceMatching Networks, and Coupling Structures, Norwood, MA: Artech House, Inc.,
1980.
[27] Duwel, A. E., Charles Stark Draper Laboratory, Inc., personal communication.
100