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Design and analysis of microwave oscillators

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D esig n and A nalysis o f M icrowave
O scillators
A lek san der M . D e c
Submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
in the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
1998
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©1998
A leksander M. Dec
All R ig h ts Reserved
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A b stract
D esig n an d A n alysis o f M icrow ave O scillators
A leksander M . D ec
T his dissertatio n investigates the application of m icrom achined electro-m echanically
tu n a b le capacitors to design of microwave low-noise voltage-controlled oscillators.
T he
problem of phase noise in microwave m icrom achincd-based voltage controlled oscillators
d u e to electrical and m echanical noise is also addressed.
M icrom achined electro-m echanically tunable capacitors w ith two and three parallel
plates have been developed for microwave applications in the 1-2 GHz range. Experim ental
devices were fab ricated using a MUMPs polysilicon surface m icrom achining process. The
tw o-plate tu n ab le cap acito r has a measured nom inal capacitance o f 2.05 pF, a Q-factor of
20 a t 1 GHz, an d achieves a tuning range of 1.5:1. T he th ree-p late version has a nominal
capacitance of 4.0 p F , a Q-factor of 15.4 a t 1 GHz, and a tu n in g range of 1.87:1. Effects
due to various physical phenom ena such as tem perature, gravity, and shock have been
investigated in d etail.
In order to d em o n strate the feasibility of the electro-m echanically tunable capacitors,
voltage controlled oscillators w ith 1.9 GHz and 2.4 GHz nom inal oscillatiou frequencies
have been developed. A 0.5 ftm CMOS process was used to fabricate nou-microinachined
com ponents. T h e oscillators were assembled by bonding tog eth er th e CMOS and MUMPs
dice. Bonding w ire has been employed to form a high Q -factor inductor. The 1.9 GHz
oscillator has a tu n in g range of 9 percent and a phase noise o f —126.3 dB c/H z at a 600 kHz
offset from th e carrier. T h e 2.4 GHz oscillator has a tu n in g range o f 3.4 percent and a phase
noise of —122.5 d B c/H z a t a 1 MHz offset from the carrier.
The analysis o f phase noise due to electrical noise has been carried out where a
sm all-signal linear periodically tim e-varying oscillator model is used to predict the oscillator
o u tp u t power sp ectra l density.
An analytical expression for th e o u tp u t power spectral
with p erm ission o f th e copyright ow ner. Further reproduction prohibited without perm ission.
density has been o b tained for a class of oscillators th a t can be modeled by a positive
feedback system w ith an M £/l-order filter, a com parator-type nonlinearity, and a white noise
in p u t. For a special case o f a 2nrf-order bandpass filter, the expression can be simplified and
show n to reduce to a well-known result under a high Q-factor assum ption. T he theoretical
resu lts m atch well w ith th e experim ental results. Phase noise due to mechanical noise of a
m icrom echanical tu n ab le capacitor has also been analyzed.
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T ab le o f C on ten ts
1 In tr o d u ctio n
1
1.1
M otivation
......................................................................................................................
I
1.2
W ireless S ta n d a r d s .........................................................................................................
4
1.3
T hesis O r g a n iz a tio n ......................................................................................................
6
7
2 S in u so id a l O scillators
2.1
I n tr o d u c tio n ......................................................................................................................
7
2.2
P ositive Feedback M o d e l...............................................................................................
8
2.3
N egative Resistance Model
.....................................................................................
9
2.4
Series to Parallel T ra n s fo rm a tio n ..............................................................................
11
2.5
C o lp itts O s c illa to rs ..........................................................................................................
13
2.6
H artley O s c illa to r s ..........................................................................................................
16
2.7
D ifferential Pair O s c ill a to r s .......................................................................................
18
2.8
S u m m a r y .........................................................................................................................
21
3 N o is e A n alysis o f Sinusoidal O scillators
22
I n tro d u c tio n ......................................................................................................................
22
3.2 L inear Tim e Invariant A n a ly s is ..................................................................................
23
3.3 L inear Periodically Tim e-V arying A n a ly s is ...............................................................
25
3.1
3.3.1
G eneral A p p r o a c h .............................................................................................
25
3.3.2
O scillator w ith C om parator N o n lin e arity ......................................................
27
3.3.3
Second-O rder B andpass F ilter E x a m p l e .....................................................
30
3.3.4
Experim ental V e rific a tio n ...............................................................................
31
3.4
A pplication of T heory to P ractical O s c illa to rs ......................................................
31
3.5
M easurem ent Techniques
...........................................................................................
37
Spectrum A n a ly z e r.............................................................................................
37
3.5.2 P hase Lock Loop M e t h o d ................................................................................
38
3.5.1
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3.5.3
3.6
4
39
S u m m a r y ...........................................................................................................................
40
In tegrated In d u ctors an d C apacitors
41
4.1
In tro d u c tio n ......................................................................................................................
41
4.2
Integrated I n d u c t o r s .....................................................................................................
42
4.2.1 P lan ar S piral Inductors and T ra n s fo rm e rs.................................................
42
4.2.2 B onding W ire I n d u c to r s ..................................................................................
45
Integrated C apacitors
..................................................................................................
48
4.3.1 M etal-M etal C a p a c ito r s ...................................................................................
48
4.3.2 P-N Ju n ctio n V a r a c t o r s ...................................................................................
49
4.3.3
MOS C a p a c i t o r s ...............................................................................................
51
4.3.4
M icrom achined Tunable C a p a c ito rs..............................................................
54
S u m m a r y ..........................................................................................................................
55
4.3
4.4
5
Delay-Line M e th o d ...........................................................................................
M icrom achined T u n ab le C apacitors
56
5.1
In tro d u c tio n .......................................................................................................................
56
5.2
Principle of O p e r a t i o n ..................................................................................................
57
5.2.1
T w o-P late Elcctro-M echanically Tunable C a p a c ito r ...............................
57
5.2.2
T h ree-P late Electro-M echanically Tunable C a p a c ito r ............................
60
5.3
Device Models
................................................................................................................
62
5.3.1
Linear Equivalent Circuit M o d e l .................................................................
62
5.3.2
N onlinear Equivalent C ircuit M o d e l ...........................................................
64
5.3.3
M odeling o f th e M echanical Loss and N o i s e .............................................
65
.......................................................................................................................
67
5.5 M icromachined T unable C apacitor D e s ig n ..............................................................
68
5.4 Technology
5.5.1
T unable C apacitor w ith Tw o Parallel P la t e s .............................................
68
5.5.2
T unable C apacitor w ith T hree Parallel P l a t e s ..........................................
70
5.5.3
P ad D e s i g n ..........................................................................................................
70
5.5.4 Suspension Beam Design
...............................................................................
71
5.6 Experim ental R e s u lts ......................................................................................................
73
5.6.1
M icrom achined Tw o-Plate Tunable C a p a c ito r ..........................................
73
5.6.2 M icrom achined T hree-P late Tunable C a p a c ito r .......................................
78
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5.7
5.8
6
D iscu ssio n ...........................................................................................................................
80
5.7.1
Effect of R esidual S tr e s s ..................................................................................
80
5.7.2
Effect o f Various Loss Mechanisms on Q - F a c to r.....................................
83
5.7.3
Effect o f A ir as a D ie le c tr ic ...........................................................................
85
5.7.4
Effect o f T e m p e r a tu r e .....................................................................................
86
5.7.5
Effect o f Gravity, Acceleration, and S h o c k ................................................
86
5.7.6
Effect o f E tch H o l e s .........................................................................................
87
5.7.7
Effect o f Process Variations on Spring C o n s ta n t......................................
87
S u m m a r y ..........................................................................................................................
88
M icrow ave M icrom achin ed -B ased V C O s
89
6.1
In tro d u c tio n ......................................................................................................................
89
6.2
Phase Noise due to Mechanical and E lectrical Noise
........................................
90
6.3
Microwave CM O S Inverter VCO - 1.9 GHz
.........................................................
92
6.4
6.5
6.3.1
VCO Design
......................................................................................................
92
6.3.2
Noise A n a l y s i s ...................................................................................................
95
6.3.3
E x p erim en tal Results
.....................................................................................
97
Microwave CM OS Differential P air VCO - 2.4 G H z ............................................
103
6.4.1
VCO Design
.......................................................................................................
104
6.4.2
Noise A n a l y s i s ...................................................................................................
106
6.4.3
E x perim ental Results
.....................................................................................
109
S u m m a r y ..........................................................................................................................
115
7
C onclusions an d F uture D irections
117
A
In d u ctan ce C o m p u ta tio n via G reenhouse
134
A .l
Inductance F o r m u la s ......................................................................................................
134
A.2
Inductance C alculation P r o g r a m ..............................................................................
136
A.3 Sam ple In p u t File
B
.........................................................................................................
M axim um T u n in g R ange
B .l
153
154
Voltage Source as Means of C apacitance C o n t r o l ................................................
B.2 C harge Source as Means of C apacitance C ontrol
................................................
C P h a se N o ise d u e to M echanical N o ise
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154
156
157
L ist o f F igu res
1.1
Block diagram of a frequency synthesizer.................................................................
2
2.1
Positive feedback m odel o f an LC oscillator.............................................................
8
2.2
Negative resistance m odel of an LC oscillator.........................................................
10
2.3
Scries to parallel tran sfo rm atio n ..................................................................................
11
2.4
C om m on-drain C o lpitts oscillator: (a) single-ended and (b) differential.
. .
14
2.5
Com m on-gate C o lpitts oscillator: (a) singled-ended and (b) differential.
. .
14
2.6
Comm on-source C olpitts, also known as Pierce, oscillator: (a) singled-ended
and (b) differential.............................................................................................................
14
2.7
C om m on-drain H artley oscillator: (a) single-ended and (b) differential.
. .
17
2.8
C om m on-gate H artley oscillator: (a) singled-ended and (b) differential.
. .
17
2.9
Differential pair oscillator w ith capacitive voltage divider....................................
20
2.10 Differential pair oscillator w ith inductive voltage divider......................................
20
3.1
Simplified diagram of a positive feedback oscillator...............................................
23
3.2 Large-signal model of th e oscillator..............................................................................
26
3.3 Sm all-signal model of th e oscillator..............................................................................
27
3.4
Schem atic of th e circuit used in m easurem ents........................................................
32
3.5
O scillator in p u t S x an d o u tp u t S y(uj) when Q — 1: (a)com puted and (b)
m easured...............................................................................................................................
3.6
33
O scillator in p u t S x an d o u tp u t S y(uj) when Q = 5: (a)com puted and (b)
m easured...............................................................................................................................
33
3.7
A n LC oscillator circuit an d its equivalent block level d iag ram ..........................
34
3.8
Modified block level d iag ram of th e LC oscillator...................................................
35
3.9
Simplified circuit schem atic of a practical oscillator...............................................
36
3.10 D iagram o f phase noise m easurem ent using a spectrum analyzer......................
38
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3.11 D iagram of phase noise m easurem ent using a narrow -bantl PL L .......................
38
3.12 D iagram of phase noise m easurem ent using a delay-line frequency discrimi­
n a to r
39
4.1
D iagram o f a planar spiral inductor..........................................................................
43
4.2
Equivalent circuit model of a planar sp iral inductor............................................
43
4.3
Diagram o f integrated planar transform ers: (a) 1:1 transform er and (b) 2:1
transform er.........................................................................................................................
4.4
46
D iagram o f a differential bonding wire inductor: (a) bonding diagram and
(b) device model................................................................................................................
47
4.5
Equivalent circuit model of the differential bonding wire inductor..................
47
4.6
D iagram o f a parallel plate capacitor........................................................................
49
4.7
Cross-section of a pn-junction varactor in a CMOS te c h n o lo g y ....................
50
4.8
Layout o f a pn-junction varactor in a CM OS technology.
.............................
50
4.9
Equivalent circuit of a pn-junction varactor............................................................
50
4.10 Cross-section of a MOS capacitor in a CM OS technology.
.............................
53
........................................
53
4.12 Equivalent circuit of a MOS capacitor......................................................................
53
4.13 D iagram o f a micromachined tunable cap acito r.....................................................
54
5.1
Functional model of a two-plate electro-m echanically tunable capacitor.
58
5.2
C onceptual model of a three-plate electro-m echanically tunable capacitor.
61
5.3
Linear equivalent circuit model of th e tw o-plate tunable capacitor..................
64
5.4
N onlinear equivalent circuit model of th e tw o-plate tunable capacitor.
65
5.5
Simplified top view (a) and cross-section (b) of the micromachined two-plate
4.11
Layout o f a MOS capacitor in a CM OS technology.
. .
...
tu n ab le capacitor (0.6 pF design value)....................................................................
5.6
Simplified top and cross-section views of th e micromachiued three-plate tun­
able capacitor (1.9 pF design value)...........................................................................
5.7
69
71
Simplified top view (a) and cross-section (b) of a conventional (1.5 pF) and
a low p arasitic (0.25 pF) pad in M U M Ps process..................................................
72
5.8
C onceptual diagram and an equivalent spring model of the suspension.
73
5.9
M icrophotograph of th e tw o-plate tu n ab le capacitor (0.6 pF design value).
74
5.10 M icrophotograph of th e tw o-plate tu n a b le capacitor (1.0 pF design value).
74
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. .
5.11 M easured S u of the two-plate tunable capacitor (0.6 pF design value). . .
5.12 M easured tu n in g characteristics of the two-plate tunable capacitor (0.6
76
pF
design value)......................................................................................................................
5.13 M easured 5 u of the two-plate tunable capacitor (1.0 pF design value). . .
76
77
5.14 M easured tu n in g characteristics of the tw o-plate tunable capacitor (1.0 pF
design value)......................................................................................................................
77
5.15 M icrophotograph of the three-plate tunable capacitor (1.9 pF design value).
79
5.16 M easured 5 n of the three-plate tunable capacitor (1.9 pF design value).
79
5.17 M easured tu n in g characteristics of the three-plate tunable capacitor (1.9
pF
design value): (a) V) swept while V? = 0 V and (b) V2 sw ept while V\ = 0 V.
80
5.18 SEM photograph of th e two-plate tunable capacitor (1.0 pF design value).
82
5.19 SEM photograph of the three-plate tunable capacitor (1.9 pF design value).
82
5.20 High-frequency model of the polysilicon m icromachiued tu n ab le capacitor.
84
6.1
Schem atic of th e CMOS inverter oscillator and its equivalent parallel RLC
circu it...................................................................................................................................
6.2
93
T heoretical phase noise due to mechanical and electrical noise (1.9 GHz
V C O )....................................................................................................................................
96
6.3
C om plete circuit schem atic of the 1.9 GHz VCO...................................................
99
6.4
M icrophotograph of th e 1.9 GHz V CO ......................................................................
99
6.5
M easurem ent test setup for the experim ental 1.9 GHz V C O ..............................
100
6.6
M easured o u tp u t spectrum of the 1.9 GHz V CO ...................................................
100
6.7
P hase noise m easurem ent of the 1.9 GHz VCO......................................................
102
6.8
M easured tu ning characteristics of th e 1.9 GHz V CO ...........................................
102
6.9
Schem atic of th e CMOS differential pair oscillator its equivalent parallel RLC
circu it...................................................................................................................................
104
6.10 T heoretical phase noLse due to mechanical and electrical noise (2.4 GHz
V C O )....................................................................................................................................
108
6.11 C om plete circuit schem atic of the 2.4 GHz VCO...................................................
110
6.12 M icrophotograph of the 2.4 GHz VCO......................................................................
110
6.13 M easurem ent test setup for the experim ental 2.4 GHz V CO ..............................
112
6.14 M easured o u tp u t spectrum of the 2.4 GHz V CO ...................................................
112
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6.15 P hase noise m easurem ent of th e 2.4 GHz V C O ......................................................
113
6.16 M easured tuning characteristics of th e s 2.4 GHz VCO.......................................
113
7.1
Top an d cross section views o f a differential tunable capacitor..........................
119
7.2
C onceptual diagram of a m icrom achined sw itch.....................................................
121
A .l
Tw o parallel conductors w ith rectan g u lar cross-scction........................................
135
B .l
S pring and electrostatic forces as a function of th e displacem ent......................
154
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L ist o f T ables
1.1
Perform ance of the sta te of the a rt V CO s..................................................................
4
1.2
C om parison of mobile stan d ard s....................................................................................
5
5.1
S um m ary of selected M UM Ps process param eters..................................................
68
5.2
S um m ary of inicrom achincd tu n ab le capacitor m easurem ents.............................
81
6.1
S um m ary of the MOS transistors in th e 1.9 GHz V C O .........................................
97
6.2
C om parison of sim ulated and m easured results (1.9 GHz V C O ).........................
103
6.3
Sum m ary of the m easurem ent results (1.9 GHz V C O )...........................................
103
6.4 Sum m ary of the MOS transistors sizes in th e 2.4 GHz VCO ...............................
109
6.5 C om parison of sim ulated and m easured results (2.4 GHz V C O ).........................
114
6.6 S um m ary of th e m easurem ent results (2.4 GHz V C O )...........................................
115
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A ck n ow led gm en ts
I would like to express my sincere g ratitu d e to my advisor, Prof. Ken Suyarna, for his
guidance, enthusiasm , an d encouragem ent during the course of this research. I also want
express my th anks to Prof. Yoshihiko Horio for his guidance, help, an d all the invaluable
knowledge th a t he passed on to me during th e first two years of my graduate work.
am equally grateful to Prof.
I
Laszlo T o th for his encouragem ent, support, and all the
discussions on noise analysis of nonlinear system s. In addition, I would like to thank Prof.
Jo h n K houry an d Prof. Yannis Tsividis for th e ir many helpful suggestions over th e course
of my g rad u ate studies. I would especially like to thank members of th e defense com mittee:
Prof. K en Suyarna, Prof. Jo h n Khoury, Prof. Charles Zukowski, Prof. Karl Sigman, and
Prof. R ichard Longm an for th eir helpful com ments.
I am indebted to my colleagues at the M icroelectronic C ircuits and Systems Labo­
ratory, S hanthi Pavan, N agendra K rishnapura, Hai Tao, M aurice Tarsia, D andan Li, Ilya
Yusim, R u b en H errera, an d Gregory lonis for many enjoyable discussions. I equally wish
to th a n k Peicheng J u who from the beginning of my graduate studies has been a valuable
friend an d a great m entor. I also would like to thank C hristophe T retz for his guidance,
su p p o rt, and all th e educational discussions on digital integrated circuit design.
T h is work would not be possible w ithout the help of Susanne Arney of Lucent Tech­
nologies who a t th e early stages of this research provided many invaluable pointers on mi­
crom achined device fabrication. Moreover, I wish to acknowledge th e help of John Shaffer,
C arl Peterson, David V allancourt, and Ali Eshraghi of Lucent Technologies for their help
w ith th e packaging of th e chips. T he interaction w ith Hongmo W ang of th e Communications
System Research L aboratory a t Bell Laboratories, M urray Hill, during th e summers of 1995
and 1996 was an extrem ely educational experience. In addition, I benefited greatly from
my interactio n w ith M ihai Banu of the Silicon Research L aboratory a t Bell Laboratories,
M urray Hill, during th e sum m er of 1997.
Furtherm ore, I would also like to acknowledge th e generous financial support provided
ix
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by th e A nalog D evelopm ent Funding C orporation and National Science Foundation.
A nd finally, I would like to dedicate this thesis to my parents who su p p o rted and
encouraged me through all these years as well as my brother and sister-in-law who provided
s u p p o rt an d u n d erstan d in g in difficult times.
x
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1
C h ap ter 1
In tro d u ctio n
1.1
M o tivation
Recent developm ents in integrated circuit technologies have m ade possible the re­
alization of com plete system s on a single integrated circuit [1], M odern system s such as
disk drive read channels or wireless transceivers often require a frequency synthesizer which
generates a program m able set of reference frequencies [2]-[3]. Frequency synthesizers for
wireless applications present many special design challenges not only because analog and
d ig ital circuits m ust op erate a t very high frequencies b u t also because sensitive analog cir­
cu its m ust coexist w ith noisy digital logic. W ireless cellular telephones th a t conform to the
GSM an d DCS1800 stan d ard s, for exam ple, o p erate at frequencies as high as 900 MHz and
1800 MHz, respectively [4]. W ireless local area networks (LANs), on the o th er hand, have
been allocated th e so-called industrial, scientific, and m edical (ISM) bands a t 900 MHz,
2.4 GHz, and 5.8 GHz [4]. C ellular telephone applications are particularly difficult since
these devices m ust be p ortable, and hence to extend b attery life, power consum ption m ust
be as low as possible. T h e power consum ption of digital logic, for instance, is proportional
to th e switching frequency and th e p ro d u ct of th e voltage swing and th e supply voltage.
Due to high frequency of o peration, th e power consum ption can only be m inim ized by re­
ducing voltage swing and power su p p ly voltage, however, a t a cost of d eteriorated noise
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2
m argins. Moreover, p o rtab le applications require small size, light weight, and of course
low cost. Therefore, th e in tegration of such wireless transceiver com ponents as frequency
synthesizers onto a single in teg rated circuit is desirable.
A block diagram o f a m odern frequency synthesizer is shown in Fig. 1.1. T h e phase
d etecto r performs a com parison between the reference frequency and the divided-down
oscillation frequency of th e voltage controlled oscillator (V CO). T he o u tp u t of the phase
detecto r is lowpass filtered an d th e o u tp u t signal of the filter is used as the control voltage
of th e VCO. T he phase-lock loop (PLL) ensures th a t the frequency a t th e o u tp u t of the
digital divider is equal to th e in p u t reference frequency. T he reference frequency is norm ally
generated using a cry stal oscillator whose frequency is extrem ely stable.
f ref
oul
Phase
Detector
VCO
Divide by
F igure 1.1: Block diagram o f a frequency synthesizer.
In many frequency synthesizer im plem entations, th e loop filter and th e VCO often
rem ain external to th e chip. T he loop filter is difficult to in teg rate in p articular because it
m ust realize a low frequency pole, which in tu rn requires large capacitance. Unfortunately,
on-chip capacitors w ith sufficiently high density are simply n o t available in m odern inte­
g rated circuit technologies. T h e VCO or its LC resonant circu it has rem ained off-chip in
m any instances m ainly because, w ith an integrated VCO, it is difficult to satisfy th e phase
noise specifications o f m any cellular telephone system s [5]-[8]. Even though th e VCO is used
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3
in a P L L an d is phase-locked to a very low noise crystal oscillator, the PLL has high loop
gain only a t frequencies th a t fall w ithin the loop ban d w id th of th e PLL. Hence, the phase
noise o f th e synthesizer is low only a t offset frequencies less th an th e loop bandw idth, b u t
outside th e loop ban d w id th , the synthesizer phase noise is dom inated by the noise of th e
voltage controlled oscillator. W ith loop bandw idths of many cellular telephone synthesizers
on th e o rd er of 10-20 kHz, a VCO w ith sufficiently low noise m ust be designed.
In general, VCOs or sim ply oscillators can be classified into two categories: relaxation
oscillators and sinusoidal oscillators. Most commonly used oscillators today in integrated
circuit applications arc relaxation oscillators prim arily because only a capacitor is required
to set th e frequency of oscillation. Although these types of oscillators can achieve very
wide tu n in g range, they often exhibit poor phase noise perform ance [9]-[12]. Sinusoidal
oscillators use an LC resonant circuit as the frequency selective elem ent and have not been
widely used in in tegrated circuit applications m ainly because an on-chip inductor is required.
However, inductors, such as bonding wires and p lan ar spiral inductors, have been shown to
be practical in th e 1-2 GHz range, and integrated LC VCOs have become possible [13]-[22],
T he p hase noise of th is type of oscillators is inversely proportional to the square of th e
q u ality factor (Q -factor) of the LC tank. Since it is difficult to realize integrated tu n ab le
capacitors an d inductors w ith sufficiently high quality factors, high quality off-chip elem ents
are often used in com m ercial products. T he perform ance of the state of the a rt integrated
VCOs is sum m arized in Table 1.1. The best phase noise perform ance is achieved by th e
VCO d em o n strated by J. C raninckx [15] where bonding wires arc used as a high Q-factor
inductor.
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4
VCO
Frequency
Offset
Phase Noise
Tuning Range
N. Nguyen [13]
1.68 GHz
100 kHz
-88 dB c/H z
12.5%
R. D uncan [14]
1.2 GHz
100 kHz
-87 dB c/H z
20.0%
J. C raninckx [15]
1.8 GHz
200 kHz
-115 d B c/H z
7.0%
A. Rofougaran [16]
0.9 GHz
100 kHz
-85 dB c/H z
15.3%
J. C raninckx [17]
1.81 GHz
600 kHz
-116 d B c/H z
14.0%
B. Razavi [18]
1.77 GHz
500 kHz
-100 d B c/H z
7.3%
L. D auphinee [19]
1.5 GHz
100 kHz
-105 d B c/H z
10.0%
B. Jansen [20]
2.2 GHz
100 kHz
-99 dB c/H z
12.8%
M. Zannoth [21]
2.0 GHz
4.7 MHz
-136 d B c/H z
8.5%
P. Kingct [22]
4.7 GHz
1.0 MHz
-110 d B c/H z
4.3%
Table 1.1: Perform ance of the state of th e a rt VCOs.
1.2
W ireless S tan d ard s
Table 1.2 shows a set of specifications for the GSM an d DCS1800 digital cellular
telephone stan d ard s and th e D E C T digital cordless telephone sta n d a rd [23]. T he GSM and
DCS1800 stan d ard have a low reference sensitivity of ab o u t —100 dBm while the D ECT
system has a reference sensitivity of only —83 dBm . Since these systems employ phase
m odulation, the linearity requirem ents are not very stringent, an d hence the input referred
third-order intercept point I I P 3 requirem ents are on the order of —20 dBm for all three
system s.
T h e GSM and DCS 1800 phase noise specifications reflect th e problem of reciprocal
m ixing where th e —23 dB m in-band blocker a t a 3 MHz offset from the desired channel and
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Wireless Standard
GSM
DSC1800
DECT
Transm it B and [MHz]
880-915
1710-1785
1880-1900
Receive B and [MHz]
925-960
1805-1880
1880-1900
C hannel B andw idth [kHz]
200
200
1728
N um ber o f C arriers
175
375
10
Bit R ate [Kbps]
13
13
32
M axim um T ran sm it Power [dBm]
30
30
24
Reference S ensitivity [dBm]
-102
-100
-83
Max In p u t [dBm]
-15
-23
N /A
Required SN R a t D etecto r [dB]
9
9
12
M inim um / / P 3 [dBm]
-18
-19
-23
Phase Noise [dBc/Hz]
-139 @ 3 MHz
-138 @ 3 MHz
-132 @ 4.684 MHz
T ab le 1.2: C om parison of mobile stan d ard s.
the noise o f th e local oscillator (LO) signal mix together down to an interm ediate frequency
(IF) an d fall on to p o f th e dow n-converted desired channel a t th e IF frequency. T h e power
of the desired signal a t IF frequency in a single channel b an d w id th m ust be must be 9 dB
(i.e. the required signal to noise ratio) higher th a n th e power d u e to th e product of th e local
oscillator phase noise a n d th e interfering signal [23]. T he D E C T system requires th a t the
emission o f th e tra n s m itte r due to th e intcrm odulation in th e th ird adjacent channel be less
—47 dB m , from which th e required phase noise of —132 d B c/H z a t a 4.684 MHz offset from
the carrier can b e derived w hen 25 dB m o u tp u t power is taken into account [21]. N ote th a t
the DCS 1800 phase noise requirem ent is m ore stringent th a n th e D E C T requirem ent which
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at a 3 MHz offset from the carrier equivalently requires a phase noise of —128 dB c/H z. It
should be also noted th a t only th e C raninckx [15] VCO from Table 1.1 meets the stringent
phase noise requirem ents o f th e DCS1800 system .
T h is thesis attem p ts to address th e problem of low noise VCO design by focusing on
phase noise in oscillators, th e design an d fabrication of high quality m icrom achined tunable
capacitors, and finally th e realization o f low-noise VCOs based on m icrom achincd capacitors
and b onding w ire inductors o p eratin g a t 1.9 GHz an d 2.4 GHz.
1.3
T h e sis O rgan ization
In C h ap ter 2, negative resistance and positive feedback oscillator models arc defined,
and different practical oscillator topologies are reviewed. C h ap ter 3 presents the analyses
of noise in oscillators. F irst, a linear tim e invariant model of the oscillator is used to obtain
an expression for the o u tp u t noise power spectral density. Second, a linear periodically tim e
varying m odel of the oscillator is developed an d an analytical expression for th e o u tp u t noise
power sp ectra l density is derived. Finally, techniques for the m easurem ent of phase noise
are discussed an d com pared. Different m ethods o f im plem enting tunable and non-tunable
reactive elem ents in integrated circu it technologies are presented in C h ap ter 4. C hapter 5
dem o n strates th e application of surface polysilicon m icrom achining technology to the design
and fabrication of m icrom achined tu n ab le capacitors. An application o f micromachiued
tu n ab le capacitors to microwave VCOs is d em onstrated in C h ap ter 6. C h ap ter 7 concludes
w ith a research sum m ary an d provides directions for further research.
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I
C h ap ter 2
Sinu soid al O scillators
2.1
In trod u ction
An oscillator is an autonom ous circuit. It produces a periodic o u tp u t w ithout any
periodic in p u t present. In essence, it converts th e dc power into an ac power. O scillators
are often classified based on the shape o f th eir steady-state o u tp u t waveform. Square-wave
oscillators (i.e. oscillators w ith a square-wave o u tp u t), for exam ple, are generally used as
clocks in b o th analog and digital discrete-tim e systems. Sinusoidal o r harm onic oscillators
(i.e. oscillators with a nearly sinusoidal o u tp u t), on th e other hand, often find use in m odern
wireless com m unication systems.
O ne of th e simplest sinusoidal oscillators is an ideal parallel LC circuit, which oscillates
indefinitely provided th a t initial conditions are not zero. However, if loss is present in the LC
circuit, th e oscillations will decay exponentially and eventually d isap p ear. Since in practice
it is not possible to realize a lossless parallel LC circuit, an active circu it th a t com pensates
or cancels th e loss is generally required.
In this chapter, positive feedback an d negative resistance m odels of a n LC oscillator
are introduced, and conditions for su stain ed oscillations are presented. In addition, a scries
to parallel transform ation is developed. Finally, LC oscillators such as C olpitts, Hartley,
an d differential pair configurations are reviewed.
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8
2.2
P o sitiv e Feedback M od el
Fig. 2.1 shows a positive feedback model of am LC oscillator where ft, L, an d C are
the resistance, inductance, and capacitance of the resonant circuit, respectively, and G m
is the tran sco n d u ctan cc of th e active device. Barkhausen criteria states th a t the positive
O
)o
o
o
out
JO
Figure 2.1: Positive feedback model of an LC oscillator.
feedback system will oscillate provided th a t th e loop gain L(.s) is unity [24], In other words,
a t the frequency o f oscillation, th e to ta l phase shift around the loop m ust be 360 degrees
and th e gain aro u n d th e loop m ust be unity [24], It can be shown th a t the loop gain L(s)
of the LC oscillator m odel shown in Fig. 2.1 is given by:
5
L(s) = GmR x o2 _j RC
5_ _i L_
b ^ RC ^ LC
(2 . 1)
A ccording to B arkhausen criteria th e loop gain L(jcu) m ust be equal to unity, and equation
(2.1) can be arran g ed as follows:
L(juj) = GmR x
U
RC
uf
_ i( * _ u,2) + RC
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(2.2 )
Since th e to tal phase shift around th e loop m ust be equal to 360 degrees, th e im aginary p art
in the den o m in ato r o f (2.2) m ust disappear, which occurs a t a single frequency uiq given by:
=
ic
(23)
At th e frequency ujq, th e loop gain L(jui) must be equal to unity as follows:
L(juo) = G mR = 1
(2.4)
an d hence oscillations occur provided th a t Gm = 1/ R . In practice, however, the resistance
R represents
th e loss o f th e inductor and the capacitor an d usually th e transconductance
Gm m ust be chosen two or three tim es larger th an strictly necessary to
ensure oscillations
over process a n d tem p eratu re variations.
2.3
N e g a tiv e R esista n ce M od el
Fig. 2.2 shows a negative resistance model of th e LC oscillator where Gneg represents
th e negative con d u ctan ce of the active network and Y {juj) represents th e lossy parallel R.LC
network. T h e loss in th e capacitor Ca is indicated w ith th e resistor R a while resistor Rb
represents th e loss of th e inductor Lb- T he resistor R c represents th e load resistance and
o u tp u t resistance of th e active network.
If th e negative ad m ittan ce is zero, the oscillations will decay exponentially given th a t
th e initial conditions are not zero. In order to ensure oscillations, G neg must be sufficiently
large so th a t it cancels or exceeds th e loss in the LC circu it [24].
In other words, the
ad m ittan ce of th e negative conductor and th e parallel R LC network m ust be equal to zero
as shown:
Gneg + Y i j w ) = 0
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(2.5)
10
Figure 2.2: Negative resistance model of an LC oscillator.
T h e ad m itta n c e of th e RLC network Y ( j u ) is given by:
V’(jw )
=
7T +
1
1
r> +
JlZc H +
ju iC a + uJ2C%Ra
Rb - j u L b
(n
1 + « 2C 2K 2
R-l + uj2 Lj;
K
Rc
~
+ Rb
.
T h e real an d im aginary p arts of eq uation (2.6) can be shown to be as follows:
Rc(Y(ju)) =
— + L+ J f j i p * +
I m {Y(juj)) =
l + w-2 C2H2 “
R2
+ jz
L2
+ ~2 L 2
(2'7)
(2-8)
To satisfy eq uation (2.5), two conditions m ust be satisfied. F irst, th e im aginary p a rt of
Y ( j u ) m u st equal to zero, which occurs a t a single frequency ujq given by:
2_ i n -(%)r>ca
°
UC.
Second, th e su m o f th e negative conductance Gncg and the real p a rt of Y(juj) a t the fre­
quency u/o m ust be equal also to zero, an d hence:
G neg = - R e ( Y { j o j 0))
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(2.10)
11
Again, th e negative conductance m ust b e chosen to be two or three tim es larger th a n strictly
necessary in order to ensure oscillations over process and tem p eratu re variations.
2.4
Series to P arallel T ransform ation
It is often convenient to transform th e parallel R L C network w ith a lossy inductor
an d capacitor to an ideal parallel R ' L ' C ' circuit w ith lossless inductors an d capacitors, as
shown in Fig. 2.3.
Y(j(0)
C
O
L
)o
<—
'
-O
Rc
R:
C a =t=
R
Ri
Figure 2.3: Scries to parallel transform ation.
T h e Q-factor of an inductor or a capacitor is often defined as th e ratio of imaginary
to real p art of the corresponding elem ent im pedance as shown:
~
1
Qa = —r—~
ClJLnR'a
,
^
ujLb
and Qb =
Rb
(2 . 11)
Using th e definitions of Q a an d Qb, th e real p art o f Y(jui), given by equation (2.7), can be
rew ritten as follows:
1
1
tT +
R e ( Y ( j u ) ) = -5 " +
Rc
R a( l + Q l )
R b(l + Q'i)
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( 2 . 12)
12
Similarly, th e im aginary p a rt of Y(ju;), given by equation (2.8), can be simplified to:
uj3L l C a(l + 4 j ) —wLb( 1 + jjr)
(213)
N ote th a t a parallel R L C network resonates when th e imaginary p a rt of its adm ittance
becom es zero. T h e relationship between th e elem ents in the R L C an d R ' L ' C 1 networks
can be found, by setting th e im aginary p a rt of the adm ittance Y(juj) to zero, solving for
the resonant frequency u>0, and equating th e resonant frequencies o f th e R L C and R ' L ' C '
networks, as follows:
“ u c a x T T J =
(214)
Also, note th a t, a t resonance, th e ad m ittan ces Y ( j u ) and Y'(juj) are purely real. Finally,
separatin g term s in (2.14) and equating th e adm ittances Y ( j u 0) an d Y'{jui0), th e elem ent
values of th e R ' L ' C ' network can be as:
L'b
= Lb(l + -^2)
V6
=
R'c
(rfVy
= R c\\(Ra( l + Q l ) m R b ( l + Q l ) )
(2.15)
(2' 161
(2-17)
T h e above scries to parallel transform ation is valid only for frequencies sufficiently close to
th e resonant frequency cjq. Clearly, th e adm ittances Y(juj) and Y'(ju>) arc not equivalent
a t dc where 1 /T ( j0 ) = Rb\\Rc an d l / Y '{ j Q ) = 0. Similarly, a t frequencies well above
o;0, th e ad m ittan ces Y(juj) an d Y'(juj) are not equivalent since l / Y ( j o o ) = /?a ||7r!c and
l / y '( j o o ) = 0 .
Nevertheless, the series to parallel transform ation is extrem ely useful when analyzing
p ractical integrated circuits where low Q-factor elements are often used.
For exam ple,
w hen a positive feedback oscillator model is used, fairly com plicated expressions for loop
gain can be obtained if low Q-factor reactive elements are present. However, the simple
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13
analysis, shown in Sec. 2.2, can be carried o ut once th e series to parallel transform ation
has been obtained. Note also th a t, from th e above equations, an inductor w ith a low Qfactor produces a reduction in resonant frequency. Interestingly, th e use of a low Q-factor
capacitors results in an increase o f th e resonant frequency.
2.5
C o lp itts O scillators
Many different circuit configurations can be used to realize an LC oscillator, b u t prob­
ably th e m ost com m on an d well-known group of LC oscillators is C olpitts oscillators [25].
T h e distin ct feature of C o lp itts oscillators is its frequency selective network, which consists
of two capacitors an d a single inductor. T h e single-ended an d differential sm all-signal cir­
cuits of com m on-drain, com m ou-gate, and coramon-source C o lp itts oscillators are shown
in Fig. 2.4, Fig. 2.5, an d Fig. 2.6, respectively. Differential circuits have been obtained
by observing th a t the m idpoints of balanced circuits are a t a sm all signal ground. Hence,
single-ended circuits can be com bined together where th e grounds of the corresponding
single-ended circuits are connected together and converted to floating connections. N ote
th a t only differential oscillations are possible, since th e negative resistance exists ouly for
differential signals. T he m ain advantage of the C olpitts oscillators lies prim arily in th e fact
th a t all the parasitic capacitances can be lum ped together w ith th e capacitors of th e m ain
LC network, an d hence a unique oscillation frequency is defined. T he C olpitts oscillator,
shown in Fig. 2.4, is p articu larly popular, since th e oscillator o u tp u t can be conveniently
taken from th e d ra in of th e M OS tran sisto r w ithout loading th e resonant circuit.
T he single-ended com m on-drain C olpitts oscillator, shown in Fig. 2.4(a), is analyzed,
as an exam ple, using th e negative resistance approach. It can b e shown th a t the in p u t ad ­
m ittan ce Yin ( j u ) looking into th e MOS transistor w ith capacitive voltage divider consisting
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14
2L
.0000.
2R
S7
^7
V7
(a )
(b)
Figure 2.4: C om m on-drain C olpitts oscillator: (a) single-ended and (b) differential.
2L
_QQQQ_
—V^Vv—1
T
2R
<7
<7
Ci
V
(a)
/
N7
12
(b)
F igure 2.5: C om m on-gatc C olpitts oscillator: (a) singled-cnded and (b) differential.
C i/2
~ \\-
uooa
L
, 000Q
L
2R
11
C ,/2
C,
__QQQQ_^,
It
\7
(a)
(b)
Figure 2.6: Com m on-source C olpitts, also known as Pierce, oscillator: (a) singled-ended
a n d (b) differential.
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of C i and Ci is given by:
Y i n ( j " ) = -------------------------------------------------------------------------- ( 2 1 8 )
7m +
+ C2)
where gm is the transconductance of the MOS transistor. Since th e parasitic capacitances
can be lum ped together w ith capacitances Ci and C 2, th e p arasitic capacitances associ­
ated with th e MOS tran sisto r have been neglected for simplicity. S eparating the real and
imaginary p arts o f (2.18), th e expression for Yin(jui) reduces to:
„
'
gm ^2C t C2
9 1
. V C , C 2(C t + C 2)
\2
9 m + ^ \ C \ + C 2)2
3 2
a tn
9m + V * { C i
1 \2
+ C 2r
(2.19)
From the above equation, it follows th a t the MOS tra n sisto r and capacitive voltage divider
realize the negative resistance. T h e oscillation criteria requires th a t the im aginary p art
of the overall ad m ittan ce be zero a t the oscillation frequency u>o and th a t the negative
resistance cancels th e loss of the LC tank, as shown:
Re(Yin( j u 0)) + 2 itp
=
0
(2.20)
I m ( Y tn(jcj0)) + - I —
JOJqL
=
0
(2.21)
While the oscillation frequency u>o is defined by equation (2.21), th e necessary value of the
negative resistance to ensure sustained oscillations is given by (2.20). S u b stitu ting (2.19)
into (2.20) and (2.21),
ujq
an d assum ing th a t oj2 (Ci + C 2)2 > > 7™,th e oscillation frequency
and the required transconductance of the MOS tra n sisto r are given as follows:
u>l
=
9m
M C , + C 2)2
= ----5------p r p ;----tip
U 1O2
0
— ^ ----L ( cxca ^
u V.C1+C2/
(2.22)
(2-23)
It should be noted th a t the analysis for th e differential circuit, shown in Fig. 2.4(b), is
identical. T he C o lp itts oscillators in Fig. 2.5 an d Fig. 2.6 can be analyzed in sim ilar fashion.
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16
2.6
H a rtley O scillators
A lthough not as widely used as C o lp itts oscillators, H artley oscillators form another
com m on an d well-known class of LC oscillators [25]. T he unique characteristic of H artley
oscillators, as in Colpitts oscillators, is its frequency selective network, which is com posed of
a single capacitor and two either coupled o r un-coupled inductors. Fig. 2.7 and Fig. 2.8 show
th e single-ended and differential sm all-signal circuits of the com m on-drain and com m on-base
H artley oscillators, respectively. Differential H artley oscillators have been o b tained in the
sam e principle as the differential C olpitts oscillators in Sec. 2.5.
T h e m ain disadvantage of H artley oscillators is the fact th a t two inductors are used
an d hence m ultiple parallel resonances may be present.
Normally, the cap acitor of the
H artley LC network sets th e resonant frequency, b u t the parasitic capacitances o f MOS
tran sisto rs can also resonate w ith one o f th e inductors and hence oscillations a t m ultiple
frequencies may occur. It is also ap p aren t from Fig. 2.7 and Fig. 2.8 th a t th e biasing of the
active circuit is difficult due to inductors. Large coupling capacitors are required, which
is undesirable in IC technologies, since bypass capacitors occupy large chip area and have
large b o tto m plate parasitic capacitance.
Using th e negative resistance approach, the single-ended H artley oscillator, shown in
Fig. 2.8(a), is analyzed as an exam ple. T h e in p u t adm ittance Yin(ju>) looking into the drain
of th e MOS transistor w ith inductive divider formed by inductors L\ and L i can be shown
to be given as follows:
Y ,M * ) = r - L - - {am*
JuLi
gm +
w here gm is th e transconductance of th e MOS transistor.
(2.24)
For sim plicity, the parasitic
capacitances of the MOS transistors are neglected. Assuming th a t
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>>
17
(a)
(b)
Figure 2.7: C om m on-drain H artley oscillator: (a) single-ended an d (b) differential.
(a)
(b)
F igure 2.8: C om m on-gate H artley oscillator: (a) singled-ended an d (b) differential.
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18
th e real an d imaginary p a rts of Yin {jui) simplify to:
Re{Yxn( ju ) )
=
- 7T T T %
(Li + L i Y
(2-25)
Irn(Ym {ju>))
=
-
(2.26)
- 1 —
u { L i 4- L2)
N ote th a t the MOS tran sisto r along w ith the inductive voltage d iv id er realize a negative
resistance. To ensure oscillations, th e im aginary p art of the to tal ad m itta n c e m ust become
zero a t th e
oscillation frequency ujq an d th e negative conductance m u st cancel th e parallel
resistan ce of the LC circu it, as follows:
Rc(Yin{j<jJo)) + -jr-
=
0
(2.27)
I m (Yin {jwo)) -+- juo C
=
0
(2.28)
tip
N ote th a t the frequency o f oscillation is determ ined by (2.28), while th e necessary transcon­
d u ctan ce of the MOS tran sisto r is determ ined by (2.27). S u b stitu tin g (2.25) into (2.27) and
(2.26) into (2.28), the oscillation frequency u 0 and the required tran sco n d u ctan ce of the
M OS tran sisto r can be show n as:
=
(Z T T Z S c
(2'29)
=
jtipr (- 't->l
r L/2
i- 2
(2'30)
T h e analysis for the differential circuit, shown in Fig. 2.8(a), proceeds in identical fashion.
It should be also noted th a t th e analysis of common-gate H artley oscillator is sim ilar to the
analysis o f the com m on-gate C o lp itts oscillator presented in Sec. 2.5.
2 .7
D ifferential P air O scillators
A lthough differential C o lpitts an d H artley oscillators have been exam ined in the previ­
ous sections, the original circu its involved only a single tu b e or a tran sisto r, an d differential
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19
p air oscillators arc often classified as a separate class of oscillators [26]-[27]. Fig. 2.9 and
Fig. 2.10 show two positive feedback differential pair oscillators w ith capacitive and in­
ductive voltage dividers, respectively. Differential oscillators offer m any advantages over
singled-ended circuits. For example, differential oscillators generally have a 3 dB better
carrier-to-noise ratio or phase noise. Note th a t, when using a differential oscillator, the
voltage swing increases by 6 dB while the noise increases only by 3 dB. In addition, differ­
ential oscillators like many differential circuits have good power supply rejection ratio and
are less sensitive to fluctuations on the power supply th an th e single-ended circuits.
As an exam ple, th e differential pair oscillator w ith th e capacitive voltage divider,
shown in Fig. 2.9, is analyzed using th e positive feedback approach. It can be shown th a t
the loop gain L(s) is given by:
\
s
Vc i +C’2 )
L(s) =
(2.31)
9 m R ( c i + C2 ) s 2 +
+ <•(§% )
\
S u b stitu tin g s = juj an d rearranging equation (2.31), the loop gain L(ju>) becomes:
\
L{ju>) -
9mR{ci +c2)
(2.32)
.
|
y
^ |
,( iZiga.') ~ U>
I+
R( c i f i . )
/
B arkhausen criteria m ust be satisfied a t th e oscillation frequency in th a t th e phase shift
around th e loop is 360 degrees and the m agnitude of loop gain is unity. T he phase shift of
L(jui) is 360 degrees a t a unique frequency u>o a t which th e im aginary p a rt of th e denomi­
nator of (2.32) becom es zero, as shown:
1
(2.33)
To ensure oscillations, th e loop gain L(juj) m ust become unity a t ujq- T he required transcon-
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20
a
L
JO
Figure 2.9: D ifferential p air oscillator with capacitive voltage divider.
Figure 2.10: D ifferential p air oscillator with inductive voltage divider.
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d uctan ce o f th e M OS transistors to achieve sustained oscillations is given by:
3m
1
— T5Rv
(2 .3 4 )
Identical analysis can be carried o u t for the circuit shown in Pig. 2.10.
2.8
S u m m a ry
In th is ch ap ter, positive feedback and negative resistance models of a n oscillator have
been introduced an d th e conditions for oscillations have been derived. A scries to parallel
transfo rm atio n has been discussed.
Comm on oscillator configurations such as C olpitts,
H artley, an d differential p air oscillators have also been exam ined.
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22
C h a p ter 3
N o ise A n a ly sis o f S in u soid al O scillators
3.1
In tro d u ctio n
T h e noise in an oscillator m anifests itself in random am plitude and phase fluctuations
of its stead y -state signal. In d a ta com m unications, where th e tim ing of d a ta is crucial, the
behavior o f noise in th e tim e-dom ain m ust be exam ined [28]-[30]. On the o th er hand, in radio
com m unications, th e location an d spacing of radio channels are im p o rtan t a n d a com plete
knowledge of th e oscillator o u tp u t sp ectru m is necessary. A com mon approach of predicting
the o u tp u t sp ectru m is to model an oscillator as a linear tim e invariant (LTI) circuit [31][33] such th a t th e sta n d a rd phasor analysis can be used to find the o u tp u t power spectral
density (PSD ). T h is approach, however, fails to predict the effects of the nonlincarity such
as m ixing of noise w ith th e steady-state signal. T he nonlinear effects are typically accounted
for by solving for th e conversion gain from each noise source to the o u tp u t. T his approach,
com bined w ith th e LTI analysis, can th en be used to estim ate th e o u tp u t PSD of the
oscillator [34]. A nother approach is to perform the linear analysis assum ing th a t am plitudes
an d phases of voltage an d current phasors are slowly tim e-varying. T he sp ectru m of phase
an d am p litu d e fluctuations can then be com puted [35]. This approach has been modified
la ter to account for nonlinear effects [36]-[37]. A lthough several techniques dealing w ith
th e general noise analysis in nonlinear circuits have recently been reported [38]-[44], these
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23
m etho d s rely on numerical techniques an d are best suited for com puter im plem entation.
T his chapter begins w ith a linear tim e-invariant analysis of the oscillator noise where
th e oscillator is modeled as an LTI system w ith positive feedback. Next, a linear periodically
tim e-varying (LPTV) model of th e oscillator is developed, and an analytical expression for
th e o u tp u t PSD is developed for a special b u t practically im portant class of oscillators
consisting of an M th-ordcr filter, a com p arato r nonlinearity, and a w hite noise in p u t [45].
T heoretical results are then applied to practical oscillators and a useful expression for the
oscillator phase noise is developed. T h e chapter concludes with an overview of different
oscillator phase noise m easurem ent techniques and sum m ary of the theoretical results.
3 .2
Linear T im e Invariant A n alysis
Fig. 3.1 shows a simplified d iagram of an oscillator consisting of an am plifier w ith
gain characteristics >l(s) and a frequency selective feedback network w ith tran sfer function
B ( s ) w here Vtn is the input voltage an d Vout is the o u tp u t voltage.
A (s)
B (s)
Figure 3.1: Simplified diagram of a positive feedback oscillator.
T h e oscillator model in Fig. 3.1 is a linear time invariant model an d hence the transfer
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24
function from in p u t to o u tp u t can be simply given as follows:
Vq u t , x _
Ms)
Vi n
1 +A{s)D(s)
n
T h e am plifier gain is assum ed to be frequency independent and th e frequency selective
network is assum ed to have a second-order bandpass filter characteristic as shown:
A(.s)
=
A
(3-2)
(*&)s
<33>
+
By replacing s w ith joj, th e transfer function can be simplified to:
Voirr.. x
V' "
A
1 I .1
A(u§ - OJ2 + j u f )
,
u,o~“2+&{■#)
u $ - u ? + M % ) ( A + 1)
’
T h e oscillations a t u = wo occur provided th a t A = —1, in which case th e transfer function
simplifies fu rth er to:
"V/.v
X L iM
Let u; = a/o +
- ~ 1(“°u>
'Q/— uV)-1 “ V
- (-D (»
- q£ —
* w2
5 I
y +W
and the transfer function becomes:
V 'ot/r , ■
(j(o * + <M) = ( - 1 ) 1 + J V}yy
I
—2o;Q<ia; + 8 u}^
Assum ing Suj «
(« )
(3-6)
a>0, and evaluating (3.6), we obtain:
Vo u t .
VlN
Using (3.7), th e m agnitude of th e transfer function squared can be shown to be:
___________
V° UT [j
(oj0 + <M)
Vw
”
"
, i + - L
4Q2
, W
V<W
! ,
_L x f ^ ) 2
4Q 1
\SojJ
as)
’
Given noise in p u t w ith th e w hite noise power spectral density S x , the o u tp u t power spectral
density S y{8 (jj) can be given by:
*(*-> -
><(£ )’
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(3'9)
25
Sy(Su>) in (3.9) includes b o th the effects o f random am plitude and phase fluctuations of
th e stead y -sta te signal. Assuming th e o u tp u t power spectral density due to th e random
a m p litu d e fluctuations is equal th e o u tp u t PSD due to the phase flu ctu atio n s [32], the
o u tp u t PSD due to phase fluctuations o r sim ply phase noise only is as follows:
<310)
N o te th a t th e o u tp u t noise Ce( 6 u)) d u e to th e phase fluctuations decreases a t a rate of
—20 d B /d c c from the carrier and can be m inim ized by increasing th e Q -factor of the band­
pass filter.
3 .3
Linear P eriod ically T im e-V aryin g A n alysis
In this section, a linear periodically tim e-varying model is developed for general os­
c illato r, an d analytical expressions are derived for the o u tp u t power sp ectra l density of an
oscillato r w ith an M tfl-order filter and th resh o ld nonlinearity which we define as com parator
n o n lin carity [46].
3 .3 .1
G e n e r a l A p p ro a c h
M any practical oscillators can be m odeled by the system in Fig. 3.2 where h{t) is an
im pulse response of an M th-order frequency selective filter, /(•) is a m cm oryless nonlinearity,
y(£) is the o u tp u t, and x(t) is th e in p u t. T h e stead y -state o u tp u t of th e oscillator is denoted
as yo(t) when x(t) = 0 and is assum ed to be periodic w ith a unique p erio d Tosc as follows:
Vo{t) = y0(t -I- Tosc)-
(3-11)
From Fig. 3.2, it follows th a t y(£) = h(t) * (x(£) + /(y (£ ))) and yo(£) = h(t) * /(yo(£))
w here * is the convolution operator. L et x(£) be a small input, then a n expression for the
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26
m
Figure 3.2: Large-signal model of the oscillator.
sm all-signal o u tp u t y(t) can be obtained:
y(o =
m - yo(t) = mo * (x(t)+n m ) - n vo m -
(3-12)
A pplying a first-order Taylor expansion around yo(0> / ( y ( 0 ) —/ ( 2/o(0) — f ' i y o (0 )y (O i ar»d
th e sm all-signal behavior o f th e oscillator can be modeled by:
y(t) = h(t) * (x(t) + g(t)y(t)) where g(t) =
(3.13)
au
u —yo(i)
Fig. 3.3 shows th e sm all-signal model of the oscillator given by (3.13). Since yo(£) is periodic,
g(t) is also periodic, an d th e sm all-signal system is a linear periodically tim e-varying (L PT V )
system . N ote th a t th e period of g{t), denoted here as T , is generally different from Tosc
given in (3.11).
Let h(t,
t )
be th e response of an LPTV system to a delayed delta function S(t —r ) .
T hen, th e o u tp u t PSD Sy (uj) of the LPTV system w ith a statio n ary w hite noise in p u t x(t)
a n d th e in p u t PSD S x is given by
S y{u;) = ^
JQr \H{u;,T)? dT
(3.14)
w here H ( uj, t ) is th e Fourier transform of /i( t,r ) [47]. Similarly, a non-stationary w hite
noise, th a t is a bias dependent w hite noise, can be modeled as w(t) — f{ t ) x {t ) w here f ( t )
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27
X(t)
Figure 3.3: Small-signal model of th e oscillator.
reflects th e bias dependence. T h e o u tp u t PSD S y (u>) due to the bias dependent noise w(t),
then can be similarly given by:
Syicj) =
^
m r ^ H ^ r ^ d r
(3.15)
l o se JO
An expression for H(u>, r ) can be obtained for the LPT V system in Fig. 3.3 by applying
the Fourier transform to b o th sides of /i(£, r ) = h{t) * (<5(£ — t ) + g(t)h( t,r) ):
= H ( u ) ( e - i “T + f ;
\
GnH ( u j - n ^ , T ) \
n = —oo
(3.16)
/
where th e transfer function of th e filter H(u;) is the Fourier transform o f h(t) and G n is the
n-th Fourier coefficientin th e com plex Fourier series expansion of g{t) =
g(t 4- T).
Generally, the o u tp u t PSD S y(uj) cannot be expressed in an analytical form because
num erical techniques are typically required to obtain th e stead y -state solution. However,
when a com parator nonlinearity (i.e. f ( y ) = A sgn(y)) is assum ed, a n analytical expression
for th e o u tp u t PSD can b e obtained, as shown next.
3 .3 .2
O sc illa to r w it h C o m p a r a to r N o n lin e a r ity
It is assumed here th a t th e period of oscillation is TOJC = 2T a n d th a t the tim e between
each consecutive zero crossings of yo (£) is T . Furtherm ore, it is assum ed th a t poles Pk of
the M 1*1-order filter arc u n ique an d Re[pk) < 0 for any k, and a p a rtia l fraction expansion
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28
o f the filter transfer function H ( uj) exists where Htt ’s are the partial fraction coefficients.
Then, H ( u ) an d th e corresponding h(t) are given by:
M
ft
= y
k
t \3 U -P k
M
an d h(t) = y H kcPktu(t).
fx
(3.17)
Note th a t u(t) is th e unit step function where u(t) = 1 if t > 0 and u{t) = 0 if t < 0. It
follows th a t th e steady-state o u tp u t of the co m p arato r is given by:
f(y0(t))= 2A
y
( —l ) " u ( f - n T ).
(3.18)
n ——oo
Using (3.18) an d observing th a t by chain rule df{ya{t))/dt = g(t)y'Q(t) where
<3' w ,
D enoting th e unit-step response of the filter by p(t) an d using linearity and tim e invariance,
OO
y0 ( t ) = 2 A
y
( —1)kp ( t ~ k T )
(3.20)
{ - l ) kh ( t - h T )
(3.21)
k ——oo
and
OO
y'0 ( t ) = 2 A
y
k ——oo
since h(t) = p'(t). Using (3.17) and (3.21), an expression for y'0 (nT) can be obtained:
y'0 (nT) = 2j4( —l)"C /(—1,0)
(3.22)
where for convenience U (z, r ) is defined as:
M
U{z, t )
=
y
H ke~PkT [ze~PkT - l)
.
(3.23)
Jfc=i
Using (3.19) an d (3.22), g(t) can be w ritten as:
1
9<i) = t T ( ^ 0 )
00
?
H t-n T ).
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(3.24)
29
FYom (3.24), it follows th a t G n = (U( —1 ,0 )T ) _1 for any n [48]. S u b stitu tin g for G n into
(3.16),
H W .r) = h M (« -* " + j
A lthough (3.25) expresses
t
'
'
in im plicit form, an explicit expression for
(:,-25)
can
be found in a sim ilar fashion as in [49]:
* ( „ . r ) = H (U ) ( e - > - +
.
(3 .2 6 )
Finally, su b stitu tin g (3.26) into (3.14), sep aratin g r-d ep en d en t term s, an d evaluating cor­
responding integrals, an an aly tical expression for th e o u tp u t PSD is obtained:
S y(v)
=
a(w )
=
^ \ H ( u j ) \ 2 ( T + 2Re(H(uJ)a (u)) +
( c / ( - l ,0 ) - C / ( e > u/r,0 ) ) " 1
“
Now,
+Jui)'r ) ■
(P* + Pn)(1 -
(3.27)
(3.28)
( ■ }
consider a bias dependent noise source w(t) = f ( t ) x ( t ) where f ( t ) is equal to
zero over th e first half an d is unity over th e second h alf of the period T osc as follows:
1
/(< ) = 2
rp
OO
£
u( t - n T - - ) - u ( t - ( n + l ) T )
(3.30)
n = —oo
A pplying (3.15), an d su b stitu tin g (3.30), th e o u tp u t power sp ectral density S y (u) can be
shown to sim plify to:
S y(uj) = ^
\ H ( ^ r ) \ 2d r
(3.31)
It can be easily verified by inspecting equations (3.14) and (3.31) th a t given th e bias depen­
d en t noise source
w(t) th e o u tp u t power spectral density S y(u>) is sim ply one h alf of th a t
given by (3.27) as follows:
SyM
=
^ \ H { u > ) \ 2 {T + 2R e{ H { u, ) a {u )) + 0 ( u, ))
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(3.32)
30
«M
=
( { / ( - l , 0 ) - t f ( e > wT,0 ))
bm
=
f
{
1
(3.33)
f3 34.
^
(P k
+ p ;)(l -
e < P > -i» Y T ){
1 - e(P S + ^ ) ‘
1 ^
In order to gain some insight into these general results, an oscillator w ith a second-order
bandpass filter is considered as an example.
3 .3 .3
S e c o n d -O r d e r B a n d p a ss F ilte r E x a m p le
A second-order bandpass filter w ith a unity gain, a center frequency uiq and a quality
factor Q has a tran sfer function given by:
= (j
c V)v ia+. J^-Q
+ VO2 -
<3-35)
It can be shown [50] th a t, if Q > 1/2, th e frequency of oscillation is:
v osc - v 0J i
V - —^
10(1 T = '-^r- = — -■
4Q z 2
a)osc
(3-36)
R earranging (3.35) into th e form in (3.17), the poles and p artial fraction coefficients are:
Pi
-
P2 = - ^ + M > *
c
(3.37)
<M8>
T h e o u tp u t PSD is given by (3.27), where a(cu) and /3(u>) sim plify to:
^
( Siuh^ ^ ’^+ 27 sinh2( ^ - ) tan( ^ ) )
P{v) = — cosh ( ^ ) sec2 ( ^ ) sinh3 ( ^ - ) UIo
4Q
2
4Q
It
is of
p ractical in terest to
find the o u tp u t PSD in th e vicinity of
(3-39)
(3.40)
ujosc.For frequencies
close to ujosci H{uj) « 1, sec2 (o»T/2) a 4 / ( T 2 (u> — ljosc)2), and (3.27) simplifies further to:
cosh (t£ c ) sinh3 ( t£ ? )
(u; - wosc)-4T-*u/o
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31
If Q »
1, then a>osc ~ u>o, and (3.41) reduces to th e well-known result obtained by the LTI
approach [31]-[32]:
5 •<"> - s
3 .3 .4
-
r
a
-
(X 42)
E x p e r im e n ta l V e r ific a tio n
An oscillator consisting of a Tow-Thomas second-order bandpass filter and a com­
p arato r was b u ilt using discrete com ponents to verify th e theoretical results. An external
w hite noise source, set a t a level o f -44 dBm, was used as the input x{t). T he noise level was
chosen to be much larger th a n the internal noise of th e circuit. T he filter com ponents were
arb itrarily chosen so th a t tJo — 48.4 krad/sec, Q = 1, and uj0sc = 41.9 krad/sec. Fig. 3.5
shows the calculated an d m easured PSDs of th e oscillator. To verify the quality factor
dependence on the o u tp u t PSD, the filter com ponents were changed so th a t Q = 5 and
u OSc = 48.9 k rad /sec (Fig- 3.6). T he general results, (3.27), (3.39), and (3.40), were used to
com pute th e o u tp u t PSD s. A lthough Fig. 3.5 and 3.6 indicate a good m atch between the­
ory and experim ent, one should note th a t S y(nuJoxc) —> oo for odd n (Fig. 3.5(a) & 3.6(a)).
In practice, S v (lj) does n o t go to infinity a t n u oxc (Fig. 3.5(b) & 3.6(b)). Nonlinear anal­
ysis shows however th a t th e proposed LPTV approach is valid for frequencies very close
to n ljosc [38]. It should also be noted th a t th e LTI analysis, unlike the LPTV approach,
predicts only th a t S y {u>osc) —> oo and fails to predict peaking of noise a t odd harmonics
of UJo s c .
3.4
A p p lic a tio n o f T heory to P ra ctica l O scillators
Fig. 3.7 shows th e an LC oscillator circuit an d its equivalent block level diagram .
T h e noise cu rren t /„ represents th e noise current of th e transconductor G m as well as the
equivalent parallel resisto r R e q -
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32
R
QR
QR
I
Figure 3.4: Schem atic o f the circuit used in m easurem ents.
T h e impedance Z( s ) in the block level diagram is sim ply th e parallel com bination of
th e L e q , R e q , and C e q elem ents as shown:
. “‘ ‘£ ‘2----- ,------ | b b q
ReqCeQ LeqCeq /
ZM = (
\S
A m odified block diagram
resistan ce R eq
of th e LC oscillator is shown in Fig. 3.8
(3.43)
where th e equivalent
has been shifted to th e input and th e feedback elem ent.
T he transfer
functio n H( s ) = Z ( s ) / R e q is unity a t th e center frequency u>o = 1/ \ / L e q C e q 2
T h e equivalent in p u t noise voltage V lN due to th e bias independent noise current of
th e equivalent resistor R e q is given by:
V ) N = 4 kbT R EQ
(3.44)
w here kb is the B oltzm an constant an d T is the tem p eratu re [51].
A ssum ing th a t the
n onlinearity /(•) is a com parator-type nonlinearity w ith sa tu ratio n levels o f 0 and I max,
a n d using the theoretical results developed in th e previous section (i.e. equation (3.42)) the
2
o u tp u t noise voltage V o u t (5uj) can be shown to be as follows:
(S )2
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»•«>
33
at
(a)
(b)
Figure 3.5: O scillator in p u t S x a n d o u tp u t S y (ui) w hen Q = 1: (a) com puted and (b)
m easured.
(a)
(b)
Figure 3.6: O scillator in p u t S x an d o u tp u t S y{u) w hen Q = 5: (a) com puted and (b)
m easured.
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34
EQ
I
In
Z(s)
rout
!(•)
Figure 3.7: An LC oscillator circuit and its equivalent block level diagram.
T h e bias dependent noise source of the transconductor is m odeled as a product of stationary
w h ite noise w ith equivalent in p u t power spectral density v ] N given by
Til
VIN ~ 4kbT OtGmR e q
(3.46)
an d a d eterm inistic signal /(£ ) given by equation (3.30) w here G m is the transconductance
w hen th e o u tp u t cu rren t is Imax and a is a noise factor specific to the im plem entation of
th e transconductor. Using th e results from the previous section, th e o u tp u t noise voltage
d u e to th e bias d ep en d en t noise source can be shown to be as follows:
?2
,2
—2
,r ,
1 4kbT a G m R EQ f UJo \ '
V outM = j
W l --------
(3.47)
C om bining equations (3.45) and (3.47) together, th e o u tp u t noise voltage V OUT(Su}) is
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35
H(s)
rm
out
F igure 3.8: Modified block level diagram of th e LC oscillator,
sim ply given by:
(3.48)
w here F c m has been defined as Fc,;m = a G mR.EQ- T he noise factor Fam can be more
conveniently viewed as sim ply being the ratio of th e transconductor noise current and the
equivalent resistor noise current as follows:
4 kbT a G m
r, D
Fc.n = ,, — t— = a G mREQ
4 kbT H e q
(3.49)
It should be noted th a t th e factor of 1/2 in equation (3.48) does not come up when using
LTI analysis since all bias dependent noise sources arc trea ted as stationary noise sources.
Fig. 3.9 shows a simplified circuit schem atic of a practical voltage controlled oscillator
where series resistors R c and R e have been added to indicate lossy reactive elements and
capacitor C p has been added to illustrate th e parasitic capacitance. T he practical oscillator
circuit can be tran sfo rm ed into simple parallel LC oscillator circuit using series to parallel
transform ation (see Sec. 2.4). T he equivalent parallel resistance R eq and th e Q-factor of
th e LC tan k can be show n to be as follows:
R e q = /ZplKl + Q I R l M I + Q c R c )
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(3.50)
36
Figure 3.9: Simplified circuit schem atic of a practical oscillator.
Q — uq R e q C e q ~
+ i )C d
(3.51)
where C p — j C p and C p / ( l -t- 1/ Q c ) ~ C p has been assum ed. T h e o u tp u t uoisc voltage
V q (IT(Su>) hence is given by:
n „ ,( M
= ^ U
+ i F &, ) ^ ( £ ) 2
<3.52,
A pplying definitions of Q c and Q t , and using ljqLe q = 1 / ( u o C e q ) , th e R e q / Q 2 term can
be reduced to:
R EQ
Q2
r eq
ulR?EQ( l + y ) 2 C't)
= _______ 1
uj$(1 + 'y)2 C l , R p
-Rr +
^
( 1 + t )2
(3 531
Using (3.52), the expression for th e o u tp u t noise voltage can be shown to finally reduce to:
v l VT(S„) = i £ ( l + I f c J ( - I ir _ L 5 p - + RL + - 5 ^ ) ( £ ) '
(3.54)
T he phase noise or carrier to noise ratio a t a frequency offset Soj from th e carrier wq, denoted
as C{ 6 uj), is th e ratio of (3.54) a n d th e rins voltage squared of th e stead y -state waveform,
as shown:
C(Su)
* £ ( i + \ F Gm) f ^
+ & l + (T T ^ j ( ^ ) 2
-----------------------V -°----------°------------------------- L--------ottt.pcqfe
2
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(3-55)
Design o f a low phase noise oscillator requires reactive com ponents w ith high Q-factors and
transconductors w ith low noise factor F c m ■ In addition, a tran sco n d u cto r w ith high o u tp u t
im pedance and large voltage swing oscillations across th e LC tan k are desirable.
Note
th a t th e Q-factor p arasitic capacitance C p reduces the effect o f th e lossy capacitor Co- In
practice, however, th e lossy capacitor may represent a tunable cap acito r and hence th e high
Q-factor parasitic capacitance improves phase noise b u t only a t th e cost of reducing the
tu n in g range.
3.5
M ea su rem en t T ech n iqu es
In this section, th ree different m ethods of phase noise m easurem ent arc presented:
the sp ectru m analyzer, phase lock loop, an d delay line m ethod.
T h eir advantages and
disadvantages are discussed.
3 .5 .1
S p e c tr u m A n a ly z e r
T h e phase noise o f th e oscillator can be measured d irectly using a spectrum analyzer,
as shown in Fig. 3.10.
M any problem s can arise, however, w hen one attem p ts to make
accurate m easurem ents using th is m ethod [52].
First, th e in tern al signal source of the
sp ectru m analyzer m u st have a much lower phase noise th a n th e signal source under test.
Second, there are m any m easurem ent inaccuracies w ith th e sp ectru m analyzer m ethod due
to inaccuracies in gain, IF filter ban d w id th , and am plitude-log linearity. For exam ple, a
2.5 dB erro r is often induced due to th e fact th a t th e video sm oothing is done after the log
am plification. In ad d itio n , th e m easurem ent of signal sources w ith large o u tp u t power and
low noise is often difficult since th e signal takes up most o f th e available dynam ic range of
th e sp ectru m analyzer.
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38
A n a ly z e r
<s>S ig n a l Sou rce
U n d er T est
Figure 3.10: Diagram o f phase noise m easurem ent using a spectrum analyzer.
3 .5 .2
P h a s e L ock L o o p M e t h o d
T h e phase fluctuations of th e signal source under test can be dem odulated directly
using a phase lock loop, as shown in Fig. 3.11 [53]. T he phase lock loop bandw idth is usually
chosen to be very narrow so th a t the dynam ics of the loop do not affect th e measurement.
Some of th e advantages of th e PLL m easurem ent m ethod include the suppression of the
Baseband
Analyzer
Signal Source
Under Test
Reference Source
Figure 3.11: D iagram o f phase noise m easurem ent using a narrow -band PLL.
carrier signal which enables a wide m easurem ent dynam ic range. Provided th a t a low phase
noise reference source is used, a very good sensitivity can be obtained w ith this method.
Finally, since phase lock loop m aintains lock, th e m easurem ents are not sensitive to the
drift of th e source under te st a n d hence m easurem ents close to th e carrier can be performed
using th is m ethod.
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39
3 .5 .3
D e la y -L in e M e th o d
Frequency fluctuations of th e signal source can also be dem odulated using a frequency
d iscrim inator [53]. Fig. 3.12 shows a block diagram of a delay-line frequency discrim inator.
T h e delay line has a delay r while th e phase shifter 4> is ad ju sted so th a t the inputs to the
m ixer are in perfect q u ad ratu re. T h e o u tp u t of this frequency dem odulator is given by:
y(t)
Signal Source
Under Test
Figure 3.12: D iagram of phase noise measurem ent using a delay-line frequency discrim ina­
tor.
y(t) =
'
(3.56)
7T /T
w here K $ is th e gain of th e m ixer an d S f is the frequency deviation of the signal source
under test. For m odulation frequencies / < < l/(27rr), th e o u tp u t becomes:
y(t) = K ^ n r S f
(3.57)
T h e sensitivity of th is m easurem ent m ethod can be im proved by increasing the am ount
o f delay r , b u t th a t can be done only a t the cost of th e m easurem ent bandw idth. The
m ain advantage of this m easurem ent m ethod is the speed w ith which the measurement
can be conducted.
Moreover, a reference signal source is not required an d hence very
high sensitivity can be achieved w ith this m ethod. In addition, m easurem ents a t offset
frequencies close to th e carrier can be also carried out.
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40
3.6
S u m m ary
A linear tim e-invariant model of an oscillator has been used to obtain an expression
for th e oscillator PSD o u tp u t. Furtherm ore, a linear tim e-varying model of an oscillator has
been developed an d an analytical expression for th e o u tp u t PSD of an oscillator consisting
of a n M th-order frequency selective filter, a w hite noise in put, an d a com parator has been
developed. T h e theoretical results have been verified by experim ent. These th eoretical
results have been applied to a practical oscillator an d a useful design expression for the
oscillator phase noise has been developed. Various phase noise m easurem ent techniques
have been reviewed a n d compared.
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41
C h a p ter 4
In te g r a te d In d u cto rs and C ap acitors
4.1
In tr o d u c tio n
Passive devices such as capacitors, inductors, an d transform ers arc essential elements
o f m any radio-frequency an d microwave circuits [54]. An im p o rtan t design consideration for
m icrowave oscillators is th e Q-factor o f th e resonant circuit. In th e previous chapter, it was
show n th a t th e p h ase noise of an oscillator is inversely p roportional to th e Q-factor of the
resonant circu it, an d hence, in low noise oscillator applications, capacitors and inductors
w ith high (^-factors are required. O ften the frequency of oscillations m ust be electronically
tunab le, especially in an ap p licatio n such as frequency synthesizers, an d hence, a t least
one of these reactive elem ents m ust also be tunable. In practice, it is difficult to realize a
high quality electronically tu n ab le inductor, and thus an electronically tu n ab le capacitor is
generally used.
A lthough p la n a r sp iral inductors and transform ers have been used in GaAs technolo­
gies for several decades [55], p lan ar spiral inductors have been shown to be practical in
BiCM O S an d CM O S IC technologies only recently [56]. T h e m ain lim itatio n of spiral in­
ducto rs in silicon processes th u s far has been th eir low Q-factor. T he Q-factor of these
inducto rs is lim ited by conductor an d eddy current losses in th e silicon su b strate. A high
(^-factor in d u cto r, however, can be realized using bonding wire inductance [57].
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42
In CM OS IC technologies, the cap acito r w ith the highest Q-factor is a m etal-m etal
capacitor, b u t such capacitor is not tunable. O ften, a p-n ju n c tio n varactor, formed between
p + diffusion an d the n-well, is used as a n electronically tu n ab le capacitor [58]. An MOS
cap acito r can also be used as a tunable elem ent and a 900 MHz VCO based on an MOS
cap acito r has ju s t been dem onstrated [59]. Recently, m icrom achincd electro-mechanically
tu n ab le capacitors have been proposed as tu n in g elements for microwave VCOs [60]-[61].
In this chapter, th e analysis and design of planar sp iral inductors as well as bonding
wire inductors are presented. Different cap acito r im plem entation options are discussed and
finally a m icrom achined electro-m echanically tunable cap acito r is proposed.
4.2
In tegrated In d u ctors
In th is section, p lan ar spiral an d bonding wire inductors are introduced and their
characteristics are discussed.
4 .2 .1
P la n a r S p ira l I n d u c to r s a n d T ra n s fo rm e rs
Fig. 4.1 shows a layout diagram o f a p lan ar spiral inductor. T h e turns of the inductor
are usually realized w ith th e top m etal layer in order to m inim ize th e parasitic capacitance
to s u b stra te where th e inner tu rn is usually connected using a lower level m etal layer.
T h e in d u cto r turns can also be realized using m ultiple m etal layers stacked together to
reduce th e series resistance and to enhance th e inductor (^-factor [62]. Recently, physicallybased equivalent circuit model for th e p lan ar spiral inductors has been developed [63].
Fig. 4.2 shows an equivalent circuit m odel o f a planar spiral inductor. The inductance in
m icrohenries is approxim ately given by:
45^nV
22r - 14a
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v
’
43
Figure 4.1: D iagram of a planar spiral inductor.
o-
L
IXKKL
Rs
c o x /2
Ri
-o
AAAA*
C OX / 2
Ri
X 7
\ /
Figure 4.2: Equivalent circuit model of a planar sp iral inductor.
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44
w here no is th e perm eability constant (no = 4 n x 10“ ' H /m ), n is the num ber of tu rn s, r and
a are th e radius and the m ean radius o f th e sp iral in m eters, respectively. T he in d u ctance of
th e sp iral can be calculated exactly by using an electro-m agnetic field solver or G reenhouse
m ethod .
T he im plem entation of th e inductance com putation algorithm via G reenhouse
m etho d is shown in the A ppendix A [64]. N ote th a t Greenhouse m ethod takes into account
parallel m etal segments and hence only square an d rectangular spiral inductors can be
analyzed using this m ethod. G eneral expressions for arb itrarily positioned conductors have
been developed by Grover [65] an d several general inductance com putation program s have
been recently w ritten [66]-[68]. T he series resistance R s of th e spiral is given by:
« • “ wi ( i
(4-2>
w here p is th e resistivity o f th e m etal, I is th e to tal length of the spiral inductor w inding, w
an d t are the w idth and thickness of th e m etal interconnect. Note th a t th e above expression
reflects th e effect of skin effect w here a t high frequencies most of th e current flows on the
surface o f th e conductor. T he skin d e p th S is given by:
*= J —
Vujpo
(4-3)
w here u is the frequency. T h e sh u n t capacitance Cp between the inner tu rn co n tact wire
an d th e n tu rn s is given by:
Cp = nw 2 ■
tox
(4 .4 )
w here tOI is the thickness of th e oxide betw een th e underpass and the spiral tu rn s an d eox is
th e dielectric constant of th e oxide. T h e capacitance between the spiral an d the s u b stra te
is given by:
Cox = wz •
eftox-
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(4 .5 )
45
T h e resistance R \ is given by:
~ - 777—
wlG sub
(4.6)
w here G sub is a fitting p aram eter which has a typical value of 10 7 S / f i m 2. T he capacitance
Ci is given by:
C, =
(4.7)
w here C xub is a fitting p aram eter w ith a typical range o f 10~3 to 10-2 fF/[xm2.
In b ip o lar IC technologies, w here high resistivity su b strates are often used, integrated
p lan ar sp iral inductors can
achieve Q-factors as high as 12 at 4 GHz, for example, for a
2.8 nH in d u cto r [69].However, CM O S IC technologies generally use lightly doped su b strates
which have an undesirable effect on th e (^-factor of th e spiral inductor. For example, a 3 nH
spiral in d u cto r fabricated in a CM OS IC technology typically has a Q -factor of only 3 a t
2 GHz [70]. In d u cto rs w ith su p erio r characteristics can be obtained in new technologies
th a t use low-loss su b strates an d copper as interconnect [71].
In a d d itio n to p lan ar sp iral inductors, planar transform ers can also be realized in a
typical IC technology. Fig. 4.3 shows a diagram o f I : 1 and 2 : 1 p lan ar transform ers.
In teg rated transform ers have coupling coefficients in th e 0.6 to 0.8 range [72].
4 .2 .2
B o n d in g W ir e I n d u c to r s
Fig. 4.4 shows a d iag ram o f a floating inductor where two bonding wires are used to
realize an in d u cto r [73]. T h e self-inductance in nanohenries of a straig h t conductor w ith a
circular cross-section is given by:
L = -l x ( M 7 ) - 0-75 + 0
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(4.8)
46
(a)
(b)
F igure 4.3: D iagram of integrated p lan ar transform ers: (a) 1:1 transform er and (b) 2:1
transform er.
w here I a n d r are th e length and th e radius of the wire in m illim eters [73]. Similarly, the
m u tu al inductance in nanohenries betw een two parallel wires can be given by:
w here d is th e separation between th e two parallel wires in m illim eters [73]. Since th e current
th ro u g h th e bonding wires is th e sam e b u t opposite direction, th e overall inductance of the
differential in d u cto r in Fig. 4.4 is sim ply given by:
L d iff = 2 L - 2 M
(4.10)
N ote th a t for sim plicity the inductance an d resistance due to th e m etal segment th a t con­
nects th e two bonding wires has been neglected. T he to tal dc resistance of th e differential
bonding w ire in d u cto r is simply given by:
Rdc = ^
7rr z
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(4.U )
47
MI
2r
M
Bonding Wire
'ME
(b)
(a)
Figure 4.4: D iagram of a differential bonding wire inductor: (a) bonding diagram and (b)
device model.
Rs /2
Figure 4.5: Equivalent circuit m odel of th e differential bonding wire inductor.
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48
Since th e skin d ep th is sm all com pared w ith the diam eter o f the bonding wire, th e resistance
can b e readily com puted as:
*- = s s
(412)
w here p = 2.35 x 10~8 Sim for gold.
A n equivalent circuit m odel of th e bonding wire in d u cto r is shown in Fig. 4.5. As an
exam ple, consider a differential bonding wire inductor w ith length I — 3 mm, an d radius
r = 12.5 um (i.e. I mil bonding wire), and separation d = 500 p m . T h e self inductance
and m u tu al inductance are approxim ately 3.25 nH an d 1 nH, respectively. T h e coupling
coefficient k betw een th e two bonding wires is 0.3 and th e overall differential inductance
L d iff is approxim ately 4.5 nH. T h e dc resistance R,ir is a b o u t 0.14 SI while th e ac resistance
Rac a t 1.9 GHz is approxim ately 0.5 S2. At 1.9 GHz, the in d u cto r has a Q-factor o f 107. In a
CM OS process, th e p arasitic capacitance to substrate o f a pad is typically 200 fF. A lthough
th e geom etry of a bonding wire is not as well defined as th e geom etry of a spiral inductor,
only a 6 percent variation in bonding wire inductance has been observed [73].
4.3
In teg ra ted C ap acitors
In this section, m etal-m etal capacitors, pn-junction varactors, an d MOS capacitors
are exam ined. Finally, a m icrom achined electro-m echanically tu n ab le cap acito r is proposed
as a high quality microwave com ponent.
4 .3 .1
M e t a l- M e t a l C a p a c ito r s
An integrated capacitor th a t has the highest Q- factor is n atu rally a m etal-m etal
capacitor [74]. A diagram of a parallel-plate capacitor is shown in Fig. 4.6 where w and I
is th e w idth an d length o f th e capacitor plate, respectively. T h e height of th e dielectric is
denoted by tox an d th e dielectric eox is the dielectric co n stan t of th e oxide.
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49
►
w
Figure 4.6: D iagram o f a parallel plate capacitor.
T h e capacitance of a parallel-platc capacitor, shown in Fig. 4.6, is given by:
0 = * -^
tox
(4.13)
M etal-m etal capacitors can have £?-factors as high as 80 a t 2.5 GHz [62j. The m ain dis­
advan tag e of th e m etal-m etal capacitor, however, is its low capacitance per area an d its
large bo tto m p late parasitic capacitance. A lthough it is not tunable, a fixed capacitor such
as m etal-m etal capacitor can be used to create a tunable capacitor using active com po­
nents [75]-[76]. However, these techniques usually introduce noise, produce d istortion, and
usually significantly reduce the (^-factor of th e capacitor.
4 .3 .2
P - N J u n c t io n V a r a c to r s
In most microwave applications, a pn-junction varactor is used as a tu n ab le capaci­
to r [77]. In CMOS IC technology, th e pn-junction varactor can be formed using p-F diffusion
a n d n-well as shown in Fig. 4.7. A practical layout of the pn-junction varactor, suitable
for high frequency application is show n in Fig. 4.8. In order to minimize losses, b o th ends
of p-F an d n-F diffusion stripes sh o u ld be contacted. T he capacitance of th e pn-junction
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50
•o
—
■
-c>
n - well
p - s u b s tra te
Figure 4.7: Cross-section of a pn-junction varactor in a CMOS technology.
p-difTasion contacts
n-wdl
Figure 4.8: Layout of a pn-junction varactor in a CMOS technology.
a
G
v A /W ----------
Rnw
F igure 4.9: Equivalent circuit of a pn-junction varactor.
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51
varactor shown in Fig. 4.7 is given by [78]:
c
C]Wl
+
=
^
+
(4 14)
(1 + Y% J-)Mjsw
where Vfo05 is th e bias voltage, <f>o is th e bulk junction potential, C} is the zero-bias ju n ctio n
capacitance per area, CJSW is th e zero-bias sidewall capacitance, M j is the bulk ju n ctio u
grading coefficient an d M js x v is th e bulk-sourcc/drain sidewall grading coefficient, w and /
are th e w idth an d th e length of th e p-F diffusion strip e. A simple equivalent circuit model
of th e pn-junction varactor is shown in Fig. 4.12. T h e loss due to series resistance of the
n-well is represented by th e RnW resistor, which can approxim ately be given by:
^ nw = 12 * JV ' ^ nu,/ D ’ w
(4 1 5 )
where Rnw/a is th e sheet resistance of n-wcll per square, and N is the num ber o f varactor
stripes. Note th a t th e above expression reflects d istrib u ted RC effects and assum es double
contact of the p-F an d n-F diffusions. It can be shown th a t the Q-factor of th e pn-junction
varactor is given by:
I
Q pn =
where
C a r ea
~
b jC p n R n w
_ _______ 12_
=
— -------- 5 ------------ H
^ C a r e a l ^ n w/O ' ^
<4 - 1 6 )
is th e pn-junction capacitance per area. It follows from the above equation,
th a t m inim um dim ensions should be used in the im plem entation of the p-F diffusion stripe.
In CMOS IC technology, a pn-junction varactor with for exam ple 0.5 pF capacitance
typically has a Q-factor of 5 a t 4 GHz [79]. An integrated pn-junction varactor has norm ally
a tun in g range of 2:1 given the supply voltage is used as the maximum bias voltage.
4 .3 .3
M O S C a p a c ito r s
An MOS cap acito r is a highest density capacitor available in CM OS and BiCMOS
IC technologies. I t is rarely used since it is nonlinear and has a high voltage coefficient.
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52
In addition, it is often poorly characterized and in ad equately modeled. Fig. 4.10 shows
a cross-section o f a n MOS capacitor realized in n-well process. A practical layout of an
M OS capacitor, which is suitable for high frequency applications, is shown in Fig. 4.11.
T h e polysilicon g ate an d th e n-i- diffusion should be co ntacted on b o th sides of the ca­
pacito r stripes to m inim ize loss. T he MOS capacitor shown in Fig. 4.10, when biased in
accum ulation, has a capacitance Cmo* given by:
\Vg b - V f b \ + 2 4 > )
I
(417)
w here COT is th e g ate oxide capacitance, Vg b >s the g ate-to -b o d y voltage, <pt is th e therm al
voltage, and Vf b is th e flat-band voltage [80]. Fig. 4.12 shows a sim ple equivalent circuit
m odel of an MOS capacitor, where Rpoly represents th e loss due to series resistance of
polysilicon gate an d Rnw represents th e loss due to series resistance of th e n-well. T h e
resistances Rpoly a n d Rnw are given by [81]:
Rpoly
=
Rnw
~
To 'T r Rpoly/o ' ~r
12 N
n,oiy/
I
1
1
/
n
'
m ' Rnw/a 12 AT i2nw/a ‘ wr
(4-18)
(4.19)
w here Rp 0 iy/a is th e sheet resistance of polysilicon p er square, Rnw/a is th e sheet resistance
o f n-well p er square, and N is th e num ber of MOS cap acito r stripes. T h e above expressions
take into account d istrib u ted RC effects and assum e double contact of polysilicon gate and
n + diffusion. T h e Q-factor of the MOS capacitor can b e shown to be [81]:
Qmos
uCmos(Rpoly + Rnw)
uCarea{Rnw/a
'^ +
Rpoly/a
' w^)
From th e above equation, it follows th a t minimum length MOS stru c tu re m ust be used while
th e w idth is th e o p tim ization param eter. However, it is im practical to use a n MOS capac­
itor w ith th e m inim um w idth since th e interconnect capacitance will introduce significant
capacitance an d reduce tu n in g range.
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n - w e ll
F ig u re 4.10: C ross-section of a MOS capacitor in a CM OS technology.
n-wcllcontact
n-wc(l
F igure 4.11: Layout o f a MOS capacitor in a CM OS technology.
Rpoly
O------- vV W ^ -
mos
G
vW
A
Rnw
Figure 4.12: Equivalent circuit o f a MOS capacitor.
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54
An M OS capacitor, in a typical CM OS process, has a tuning range of approxim ately
2:1 and capacitance density o f approxim ately 2 fF /fim 2. Recently, devices w ith (^-factors
as high as 100 a t 900 MHz have been dem onstrated [81].
4 .3 .4
M ic r o m a c h in e d T u n a b le C a p a c ito r s
M icrom achining techniques have been widely used in the realization of p lan ar induc­
tors where th e su b strate and oxide are etched away to remove the parasitics [82]. How­
ever, tu n ab ie capacitors based on m icrom achining techniques have only been proposed and
d em o n strated recently [60]-[61]. Fig. 4.13 shows a functional diagram of an in teg rated mi­
crom achined tu n ab le capacitor. T h e tunable capacitor consists of a mechanically secured
p late and a suspended plate, which is secured to the substrate via some spring arrange­
m ent w ith equivalent spring constant k. T h e capacitance can be tuned by applying dc bias
voltage Vbias, as will be explained in detail in C hapter 5.
Spring
k/2
Spring
k/2
Figure 4.13: D iagram of a micromachined tunable capacitor.
A dvantages of such a tunable capacitor include high electrical Q-factor a t microwave
frequencies, which makes th e realization of low noise microwave oscillators possible. In ad­
dition, th e m icrom echanical capacitor does not respond to microwave frequencies, which are
well above th e mechanical resonant frequencies of these devices, and hence these devices are
not expected to produce much harm onic content. Furthermore, these devices can accom-
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55
m odate large voltage swings, which is advantageous since th e phase noise can be improved
by allowing large voltage swings across the LC tan k . T his is normally not possible with
pn-junction varactors where the designer must ensure th a t the pn-junction varactor does
not enter the forw ard bias region over the tuning voltage range.
Aluminum m icrom achined tunable capacitors can achieve electrical Q-factors as high
as 60 a t I GHz for a 2 p F capacitor and tuning ranges of 1.16:1 within 5.5 V control volt­
age [60]. In th e next chapter, micromachincd tun ab le capacitors with two an d three parallel
plates im plem ented in a polysilicon surface m icrom achining process will be dem onstrated.
4 .4
S u m m ary
P lanar spiral inductors and transformers in BiCM O S and CMOS have been discussed
an d the im plem entation of a differential inductor using bonding wire has also been intro­
duced. Finally, m etal-m etal capacitors, pn-junction varactors, MOS capacitors have been
exam ined and a m icrom achined electro-mechanically tun ab le capacitor has been proposed
for microwave applications.
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56
C h ap ter 5
M icrom ach in ed T un able C apacitors
5.1
In tro d u ctio n
Recent developm ents in m icrom achining suggest th a t technologies w here microrria-
chined devices and electronics can b o th reside on a single chip will be widely available in
the future. Com m ercial p roducts based on a BiCM OS technology w ith an integ rated surface
polysilicon process are already available today [83]. Such a technology not only makes possi­
ble th e realization o f integrated mechanical resonant devices b u t also of in tegrated electrical
resonant circuits where one o f th e reactive elem ents is m ade tu n ab le by electro-m echanical
means.
F ilters an d oscillators based on a m echanical resonant devices w ith a high ^-factor,
which o p erate in the 10s o f MHz, have already been d em onstrated [84]-[86]. A lthough these
devices are not suited for applications in th e 1-2 GHz range, these devices m ay well find use
in the interm ediate frequency (IF) section o f a radio transceiver. T he m echanical resonant
frequency, however, is not electronically tu nable, since the resonant frequency is set by the
proof m ass an d th e equivalent spring co n stan t of th e suspension.
A tu n ab le high Q-factor resonator, o p eratin g in the 1-2 GHz range, m ay be possible
w ith an electrical resonant circuit w here one of th e reactive elem ents is electro-m echanically
tunable. M icrom achined electro-m echanically tunable capacitors are not expected to re­
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57
spond to R F frequencies in th e 1-2 GHz range especially since their mechanical resonant
frequencies norm ally lie in th e 10-100 kHz. Therefore, w ith R F frequencies 10,000 times
th e m echanical bandw idth, these devices are unlikely to produce significant am ount of
harm onic content. Moreover, m icrom achincd tunable capacitors can accom m odate large
voltage swings, unlike p-n ju n c tio n varactors where large voltage swings m ust be avoided in
order not to forward bias th e pn-junction. As discussed in Sec. 4.3.4, micromachincd tu n ­
able capacitors have been show n to exhibit an adequate ^-facto r when they are fabricated
in eith er an alum inum [60] or polysilicon [61] surface m icrom achining technology. T he m ain
lim itatio n of these m icromechanical devices, however, has been th e fact th a t their tuning
ranges th u s far have been less th a n th e theoretical calculations suggest [60]-[61].
T h is ch ap ter dem onstrates m icrom achincd electro-m echanically tunable capacitors
fabricated in a stan d ard surface polysilicon m icrom achining technology th a t achieve ade­
q u ate Q-factors and the w idest tu ning range reported to d ate [87]-[88]. In this chapter,
principle o f operation, design, experim ental results, an d discussion on micromachincd tu n ­
able capacitors w ith two and th ree parallel plates are presented.
5.2
P rin cip le o f O p eration
In th is section, the o p eratio n of two- and th ree-plate electro-m echanically tunable
capacito rs is explained and an aly tical expressions for the capacitor tu ning characteristics
are derived.
5 .2 .1
T w o - P la t e E le c tr o -M e c h a n ic a lly T u n a b le C a p a c ito r
Fig. 5.1 shows a functional model of a n electro-m echanically tunable capacitor th a t
consists o f two parallel plates.
T h e to p p la te of th e tu n ab le capacitor is suspended by
a sprin g w ith spring constant k, while th e bo tto m p late of th e capacitor is mechanically
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58
secured. W hen a bias voltage V\ is applied across the capacitor plates, the suspended plate
is a ttra c te d toward the b o tto m p late due to th e resultant electrostatic force. T h e suspended
p la te moves toward the fixed plate until an equilibrium between the electro static and the
spring forces is reached.
Jim.
Suspended
plate
4
Fc
d. + x
Fixed plate
F igure 5.1: Functional m odel o f a tw o-plate electro-mechanically tu n a b le capacitor.
T h e capacitance of such a parallel plate capacitor is simply given by the well-known expres­
sion below:
CD =
d \+ x
(5.1)
w here A is the area of th e capacitor plates, ej is the dielectric constant o f air (ej = eatreo
w here eair = 1.00054 an d e0 = 8.85415 x 10~ 12F /m ), and d\ is th e separation of the
capacito r plates when th e spring is in its relaxed state.
T h e spring force Fs follows Hooke’s law [89], and hence is pro p o rtio n al to th e spring
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59
constan t an d to the displacem ent o f the spring from its relaxed state as given by:
Fs = k x
(5.2)
T h e electrostatic force Fe is equal to the gradient of energy th at is stored on the capaci­
to r [90], which simplifies to:
I d C n .,2
Fe "
2
I
edA V ?
~ d T V' ~ ~ 2 (d, + x)*
A t th e equilibrium , Fs = Fe, an d hence:
1 edA V f
kx= R earranging
2
iiT W
‘
(5'4)
(5.4)and moving all the term s to the left han d side of th e equation, a cubic
polynom ial in x is obtained:
x 3 + (2d)x2 + (d2)x + C -± ^ -) = 0
(5.5)
By solving th e cubic polynom ial, an explicit expression for th e desired capacitance C o(V i)
can be o b tain ed [91], as follows:
C n lV i)
=
---- — -----
*{Vx)
=
( a i( V ,) + s 2( V i ) ) - ^ -
si(Vi)
=
^ ( V i ) + \J u 3 + u;2 ( V i ) j
S2(^i)
=
^iu(Vi) - yju3 + tn2(Vi)^
m
U
di+ i(V i)
(5.6)
- - -f
It should be noted th a t an equilibrium between the spring and the electrostatic forces exists
only for displacem ents 0 > x > —d i/3 (see Appendix B). T he suspended plate will make
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60
contact w ith th e bo tto m plate if the electrostatic force is greater th an th e spring force,
which occurs when x < —d i/3 . The m axim um capacitance th a t such a tu n a b le capacitor
can be tu n e d to is 3C d / 2, and the m axim um theoretical tu n in g range is 1.5:1. A lthough the
tu n ab le capacitor is biased here using a voltage source, in principle a charge source could be
used to control th e am ount of charge on the capacitor plates in which case th e equilibrium
between th e forces would exist for 0 > x > —d\ (see A ppendix B).
5 .2 .2
T h r e e -P la te E le c tr o -M e c h a n ic a lly T u n a b le C a p a c ito r
Fig. 5.2 shows a conceptual model of a three-plate electro-inechanically tunable ca­
pacitor where under zero bias condition th e distances between parallel plates are d\ and
d-2 , respectively. T he top and bottom plates of the capacitor are fixed m echanically while
th e m iddle p late is suspended by two springs w ith a spring constant k / 2 each. If a bias
voltage V\ is applied a n d Vi = 0 V, th e electrostatic force causes th e suspended plate to
move tow ard th e top plate. Similarly, if a bias voltage Vi is applied an d V\ — 0 V. the
suspended plate moves tow ard the b o tto m plate.
T h e desired capacitance Co an d th e parasitic capacitance Cp are given by the wellknow n expressions below:
CD =
di + x
an d C P =
di —x
(5.7)
Using (5.7), th e electrostatic forces Fe\ and Fei can be found to be as follows:
F"
~
Fe2 =
Ld C D 2
1 e jA V ?
2 ~ d T V' ~ ~ 2 { d l + x ) i
2 dx
\2 (a*
(ddA
Vxl\)i1
2—
(5'8)
(59)
A t th e equilibrium , Fs = Fe\ + Fe2, an d thus:
1 edA V ?
1 edA V j
2 (d i + x )2
2 (^2 — xY
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(5.10)
61
m
J M
Spring
k/2
Spring
Fixed plate
k/2
*
d, + x
“ F,
i>
Suspended
d2 - x
plate
T
£
1
I Fe l + F c2
V2
i
+
Fixed plate
Figure 5.2: C onceptual m odel of a three-plate electro-m echanically tunable capacitor.
Tw o cubic equations can b e derived from (5.10) by setting eith er V\ or V2 to zero, as follows:
x 3 + ( 2 d ) x 2 + (d2)x +
x
3 +
( - 2
V
(5.11)
edAV.?
) = 0 if Vi = 0 V
2k
(5.12)
= 0
2k
d ) x 2 + (d2)x + ( -
if V2 = 0
W hen V2 is set to zero, th e desired capacitance C d {V i) is given by (5.7). T he expression
for th e desired capacitance C d (V 2), when V\ is set to zero, is sim ilarly given by:
C d (V2)
=
x{V i)
=
si{ V 2)
=
edA
(5.13)
d\ + x ( V 2)
(w {V 2) + yju 3 + w 2 [V2)^
1
S2(V2)
=
XL —
w (V 2)
=
^iu(V2) — \J v ? - 1- u;2 (V2)^
A
9
A
27
edA V i
4k
'
T h e m axim um capacitance th a t this capacitor can be tu n e d to is still 3C d / 2 (see Ap-
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62
pendix B).
However, the m inim um capacitance th at this capacitor can be tuned to is
3 C d /4 , if distances di and di are equal. Hence, the maximum theoretical tu n in g range is
2:1. Note th a t in a practical circuit application, the middle plate o f th e tun ab le capacitor
m ust be connected to a sm all-signal ground so th at only the desired capacitance C o plays
a role in th e ac tu a l circuit.
5.3
D e v ic e M od els
Design o f high perform ance integrated circuits requires accurate device modeling.
In this section, linear and nonlinear equivalent circuit models for th e tw o-plate tunable
capacitor are developed. T he mechanical loss and noise models are also introduced.
5 .3 .1
L in e a r E q u iv a le n t C ir c u it M o d e l
T he cu rren t i(t) flowing through the capacitor C q is simply th e derivative of charge
w ith respect to tim e as shown:
By su b stitu tin g (5.1) into (5.14), an expression for current flowing th ro u g h the capacitor
can be obtained as shown in (5.15). T he second term in (5.14) accounts for th e current th a t
is generated w hen th e suspended plate moves through the electric field.
t dA
dVx
,(1> = d ^ - j T
edAVx
dx
-
,r
( J ' 15)
T h e dynam ic behavior of th e electro-mechanically tunable capacitor can be modeled m ath ­
em atically by:
d?X
A*
,
1 d C p ^ r l, ^
m d f i + r d i + kX = 2
/r-
(!U 6)
where m is th e m ass of th e suspended plate, r is the mechanical resistance, k is the spring
constant of th e suspension system .
Since equations (5.15) an d (5.16) are coupled and
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63
nonlinear, an analytical solution may be obtained by carrying out the p ertu rb atio n analysis.
S u b stitu tin g Xdc+xar for x and V ^i + u acl for V\ into (5-16), assuming th a t x ac < < x dc
a n d vaci << Vdcl, applying first-order Taylor expansion to the right hands side of (5.16),
an d finally separating the DC and AC term s, th e DC operating point can be obtained as
follows:
(5 1 7 )
w here Vdc\ is th e dc bias voltage across the capacitor and x dc is the displacem ent o f the
suspended plate. T he small-signal relationship is given by:
(i^xac
d x ac
m-diT + r- d r +kl~ =
1 ( —2)e^>lV^cj
2 l d , + ^ x“
.
I. 2edAVdrl
+
.
1
.
’
Sim ilarly, su b stitu tin g x dc -f- x ac for x and Vdd -f vacy for V\ into (5.15), assum ing th a t
x ac << x dc an d varl «
Vdcy, and neglecting higher order term s, (5.15) simplifies to:
edA
dvacy
dt -
' (1) =
edAVdcy d xac
(rf, + * * )* ~dT
^
Since (5.18) and (5.19) compose a linear system of differential equations, Laplace transform s
can now be used to obtain an equivalent circuit model of the electro-mechanically tunable
capacitor. T h e equivalent adm ittance Yij(s) = I ( s ) / V ( s ) of the tunable capacitor can be
show n to be:
Y „ M = sC D +
s "b \ Rm/ LJm) s "b l/( L x n C m )
(5.20)
w here
CD =
(5-21)
d\ +•
( 5 -2 2 >
/
Rm
-
\fa i
(
C
G m
—
“
\ ^
t-n n V-i~t
1-X C
to A Kiel
V
IU(di
+TdcP)
gl+
I <*cJ~ )
< o A V 2 .,
">
AVZi
(di+Xdc)3
/r 24\
•
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64
Fig. 5.3 shows th e sm all-signal equivalent circuit model of th e two-plate tunable capacitor.
N ote th a t th e second term in (5.15) is typically neglected, an d hence the denom inator in
(5.24) differs from m ost analyses [92]. O m ission of the second term in (5.15) can result
in errors in th e resonant frequency as high as 25 percent. T h e model developed here is
very useful for sm all signal analysis, b u t for large excitations num erical techniques m ust be
em ployed. It should be also noted th a t the above model is valid only a t low frequencies. At
frequencies above th e mechanical resonance frequency, th e Rm LmCm series resonant circuit
becom es open and V//(s) « sC p . In practice, some loss will be present in the 1-2 GHz
range, an d th u s an o th er approach is needed to characterize th e tunable capacitor a t high
frequencies.
GY l f (s )
v W A
R.
ix m
m
F igure 5.3: Linear equivalent circuit m odel of th e tw o-plate tunable capacitor.
5 .3 .2
N o n lin e a r E q u iv a le n t C ir c u it M o d e l
T he above equations (5.15) and (5.16) have been im plem ented in HSPICE, a sta n d a rd
circuit sim u lato r [93]. T h e equivalent nonlinear circuit m odel of the electro-m echanical
tu n a b le cap acito r w ith two parallel plates is shown in Fig. 5.4.
Although in a typical
circuit sim u lato r it is n o t possible to im plem ent signal sources w ith arb itrary uonlinearities,
th e nonlinear an d signal dependent cu rren t and voltage sources can be im plem ented in
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65
H S P IC E w ith relative ease.
Note th a t th e m odel in Fig. 5.4 preserves noise properties
o f th e mechanical system . In addition, th e m agnitudes of th e force an d the displacement
have to be carefully scaled in order to avoid num erical problem s. T h e model finds use in
sim ulation of circuits which employ the electro-m cchanically tu n ab le capacitors.
Electrical Domain
cA
Mechanical Domain
,w ( 0 = l
2
(xo)+j,r
O', =k
Gv =r
m
G2 = C2
i(t)
m
x(t)
V(t)
F igure 5.4: N onlinear equivalent circuit m odel of the tw o-plate tunable capacitor.
5 .3 .3
M o d e lin g o f t h e M e c h a n ic a l L o ss a n d N o is e
Modeling of th e m echanical loss or m echanical resistance r in micromachined devices is
non trivial. In many m icrom echanical devices such as capacitive sensors or micromcchanical
tu n a b le capacitors, however, squeczc-film d am ping often is th e d o m in an t source of loss [94].
T h e squeeze-film d am p in g is th e result of loss associated w ith squeezing the air out from
betw een the moving surfaces.
For a m echanical system th a t consists o f two parallel p lates w ith an area A and
sep arated by a distan ce d i, th e mechanical resistance r ju m is given by [94]:
3/iajr A2
r / t ,m "
2 n d :l
w here fxair is the fluid viscosity of air (p 0ir = 18 x 10~6 kg-s- l -m- 1 ).
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(
)
66
T h e squeeze-film dam ping can be reduced by perfo ratin g one of the plates, since the
air can escape
th ro u g h the perforations instead of being squeezed out the
plate edges. For
a m echanical sy stem th a t consists of two parallel plates w here only one plate is perforated,
the mechanical resistance rpeTj is given by [94]:
r~ '
=
a(p)
=
( 5 -2 6 )
2
8
4
8
(5.27)
w here N is the to ta l num ber of holes and p is the fraction of open area in the plate.
T he presence o f loss or dissipation in a m echanical system also requires th a t a random
Q uctating force b e present as well [94], where the m ean squared force F 2ext has a w hite noise
power spectral d en sity and is given by:
F 2ext = 4kbT r
(5.28)
w here th e B o ltzm an n ‘s constant kb = 1.38 x 10-23 J-K ~ 1 is and T is the tem p erature in
2
K elvin. T he flu ctu atin g force F cxt produces random fluctuations in displacement x which
can be described as follows:
m<~ j^ +
+ k x = Fext
(5.29)
w here m is th e m ass, r is the mechanical resistance, k is th e spring constant, and Frxt is
the random ex citatio n force.
By dividing (5.29) by m, taking the Laplace transform of both sides, an expression
for th e transfer function X / F exl(s) can be obtained:
*
M = ^ • ( , . ,L — A
(5.30)
T h e mechanical resonant frequency and th e Q-factor of th e system can now be directly
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67
obtained from (5.30) as follows:
wjj
=
m
(5.31)
Q
=
^
(5.32)
Finally, by evaluating (5.30) a t s = jo/ and squaring th e m agnitude response, an expression
for the mean squared displacem ent as a function frequency can be found as follows:
X (u j ) 2 = — ■ ( —t
™2 \ ( s “ a;2)2 + (m w) /
(5.33)
Note th a t the mechanical system , given by (5.29), has a Iowpass response. T h e m agnitude of
th e displacem ent X (lj ) 2 is constant for frequencies below the mechanical resonant frequency
an d decays a t a ra te of —40 d B /d cc for frequencies above th e mechanical resonant frequency.
T h e m agnitude o f the random displacem ent can be minimized in three ways. First, a rigid
suspension w ith a large spring constant can be used. Second, th e mass of th e suspended plate
can be maximized. Finally, th e mechanical loss itself can be minim ized by use of vacuum
as dielectric (w hich has sm all fluid viscosity) or use of a large num ber of perforations.
5.4
T ech n ology
Despite th e superior electrical properties of alum inum , polysilicon was chosen as the
structu ral m aterial for tu n ab le capacitors due to its good mechanical properties [95]. A
standard polysilicon surface m icrom achining process (M UM Ps) [96] has been selected for
th e fabrication o f p rototype devices. T h e process features three layers of polysilicon (polyO,
polyl, and poly2) and one layer o f gold (gold can only be deposited on poly2). Selected
M UMPs process param eters are sum m arized in Table 5.1.
MUM Ps devices are released using an HF sacrificial layer etch followed by a supercrit­
ical carbon dioxide drying process [96]. M icromachincd tunable capacitors th a t are released
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68
Layer
Thickness
Sheet Resistance
PolyO
0.5 /im
30 « / □
Polyl
2.0 ;im
10 fi/D
Poly2
1.5 /im
20 fl/D
Gold
0.5 /im
0.06 fi/D
T able 5.1: Sum m ary of selected MUMPs process param eters.
using an HF etch an d conventional drying m ethods may suffer from a stiction problem
w hich can resu lt in an unsatisfactory device yield. During th e drying process, attractiv e
capillary forces can bring the capacitor plates into perm anent contact (often referred to
as th e stictio n effect) effectively shorting th e tunable capacitor perm anently [97]. T he su­
p ercritical carb o n dioxide drying process alleviates the stiction problem an d dram atically
im proves device yield [98]-
5 .5
M icro m a ch in ed T u n ab le C apacitor D esign
Several m icrom achined electro-m echanically tunable capacitors w ith two an d three
parallel plates have been designed using th e MUM Ps process. A new pad w ith low parasitic
capacitance to su b stra te has been also developed.
5 .5 .1
T u n a b le C a p a c ito r w it h T w o P a r a lle l P la te s
Fig. 5.5 shows th e simplified top view an d cross-section of a 0.6 pF tunable capacitor
w ith two parallel plates.
P olyl an d poly2/gold layers were selected as capacitor plates,
since in th e M U M Ps process p o ly l an d poly2/gold layers arc the most conductive layers
(see T able 5.1). Given th e air gap o f 0.75 /im (after sacrificial layer release), the desired
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69
capacitance can bo achieved w ith a 210 p m by 230 p m p late area. A spring constant of
37.2 N /m is necessary in order to achieve a m axim um capacitance of 0.9 pF under the
m axim um bias voltage o f V\ = 3.3 V. T he mass of th e suspended plate, which is composed
of the poly2/gold layers, is 0.6 pgrains, an d the mechanical resonant frequency is estim ated
to be 39 kHz.
A nchors
10 p m
(a)
S u sp en sio n
V iew
100 p m
210 p m
A n ch o r
A ir G ap
S uspen d ed P late
0.7 5 p m
C ross
(b )
A ir G ap
1.5 p m
=p CP
Fixed Plate
N itride
A n ch o r
P o ly l
2.0 p m
P o ly2
1.5 p m
G old
0.5 p m
Figure 5.5: Simplified top view (a) and cross-section (b) of th e m icromachined two-plate
tunable capacitor (0.6 pF design value).
A 1.0 pF tu n ab le capacitor has also been designed. T h e required capacitance can
be obtained w ith 295 p m by 295 p m plates, given th e a ir gap of 0.75 pm . In order to
achieve a m axim um capacitance of 1.5 pF when V\ = 3.3 V, a spring co n stan t of 65.3 N /m
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is required. T h e m echanical resonant frequency is estim ated a t 38.8 kHz. while the mass of
the poly2/gold suspended plate is 1.1 /xgrams.
5 .5 .2
T u n a b le C a p a c ito r w ith T h r e e P a r a lle l P la te s
Fig. 5.6 shows the top view and the cross-section of a 1.9 pF tunable capacitor, which
consists of th ree parallel plates. A capacitor w ith an air gap of 0.75 {im and a 400 /xm
by 400 n m p late area achieves the desired capacitance. A spring constant of 122 S / m is
necessary in o rder to obtain a maximum capacitance of 2.85 pF when V\ = 3.3 V and
V2 = 0 V. T h e mass of the suspended plate, which is composed of polyl layer, is then
0.7 /igram s an d th e mechanical resonant frequency is estim ated at 65.8 kHz. Dimples are
used to prevent th e middle plate (polyl) from touching and sticking to the b ottom plate
(polyO) in th e presence of excessive bias voltage V2.
5 .5 .3
P a d D e s ig n
The p arasitic capacitance of the pads is extrem ely critical, especially since die bonding
and flip-chip technology are often used to integrate active circuits w ith m icromachined
devices. T h e sta n d a rd pad in the MUMPS process has a parasitic capacitance of 1.5 pF
which significantly lim its the tuning range. Therefore, a low parasitic pad, which has a
parasitic capacitance of only 0.25 pF (Fig. 5.7), has been developed. A sm aller capacitance
per area is achieved by the use of only poly2 and gold layers, which are deposited on 2.25 /im
thick oxide p rio r to sacrificial layer release. To p rotect the oxide underneath the pad from
HF etch, anchors are placed around the edges of th e pad. In addition, a sm aller pad (86 ,um
by 86 fj.m) is used.
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Nitride
i m ^°iy'
A nchor
im
2.0 fim
m u P°ly2
^
1.5 |Am
G old
NSnI 0.5 (im
Figure 5.6: Simplified top and cross-section views of th e m icrom achined three-plate tunable
capacitor (1.9 pF design value).
5.5.4
Suspension B eam D esign
Fig. 5.8 shows a conceptual diagram of the suspension which is used in the design of
tunable capacitors. T he equivalent spring constant of this suspension can be approxim ately
modeled as a parallel an d series com bination of springs, as follows:
k \ 2 ki
keq ~ k : + 2fc2 '
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(5.34)
72
(anchor not shown)
Top
V ie w
Standard
pad
118 jam
Low parasitic
pad
86 p m
T
Cross
Section
Oxide
N itride
Anchor
P oiyl
2.0 fim
PolyO
0.5 p m
Poly2
1.5 fim
Gold
0.5 jim
Figure 5.7: Simplified top view (a) an d cross-section (b) of a conventional (1.5 pF ) and a
low p arasitic (0.25 pF) pad in M UM Ps process.
T he stiffness constant for a double clam ped suspension beam may be used, if th e capacitor
p late is assum ed to be rigid. T he stiffness constant is given by:
kl =
EpW tT f
L?
(5.35)
w here W t. Tt . L, are the w idth, thickness, an d length of the the beam, respectively, and
E p is th e Young’s m odulus (approxim ately 160 G Pa) of polysilicon [99]. Since the overall
suspension consists of four such beam arrangem ents, the total spring constant is k tot = 4k eq.
Since th e R F signal passes through th e suspension beam s, gold is deposited on top of the
suspension beam s in order to m inim ize series losses. Deposition of gold on suspension beam s
is not expected to have significant effect on the overall spring constant because the gold
trace has a sm aller w idth and thickness relative to the polysilicon beam.
Furtherm ore,
the Young’s m odulus of gold is 50 percent lower th an th a t of polysilicon (approxim ately
80 G Pa).
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73
MIL
w.
Figure 5.8: C onceptual diagram and an equivalent spring model of the suspension.
5.6
E x p erim en ta l R esu lts
All m easurem ents were done using an HP 8753D network analyzer, a Cascade probe
statio n , and W inC al software. The measurements include the parasitic capacitances of the
pads. An open p ad m easurem ent indicates a pad to su b strate capacitance of approxim ately
0.26 pF.
5.6.1
M icrom achined T w o-P late Tunable Capacitor
M icrophotographs of the 0.6 pF (210
by 230 p m capacitor) and 1.0 pF (295 p m
by 295 p m capacitor) tw o-plate m icrom achined tunable capacitors are shown Fig. 5.9 and
Fig. 5.10. respectively.
T he 0.6 p F tun ab le capacitor has a Q-factor of 20 a t 1 GHz and Q-factor of 11.6 at
2 GHz (Fig. 5.11). T h e self-resonant frequency is beyond 6 GHz. the maximum operating
frequency of th e H P 8753D network analyzer. The tu ning characteristics of the tunable
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Figure 5.9: M icrophotograph of the two-plate tunable capacitor (0.6 pF design value).
Figure 5.10: M icrophotograph of the two-plate tunable capacitor (1.0 p F design value)
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capacitor are shown in Fig. 5.12. W hen a zero bias voltage {V\ = 0 V) is applied, the
m easured capacitance is 2.05 pF. The measured capacitance is 3.1 pF when V\ = 4 V is
applied. The tuning range of the 0.6 pF micromachined tu n ab le capacitor is hence 1.5:1.
S tatistics show th a t th e average measured nominal capacitance is 1.98 pF and the standard
deviation is 0.14 pF using the d ata from 14 functional devices (16 devices were fabricated).
Fig. 5.13 shows th e measured S u of the 1.0 pF tu n ab le capacitor. The 1.0 pF tunable
capacitor has a Q-factor of 13.6 at I GHz and Q-factor of 5.5 a t 2 GHz. T he self-resonant
frequency is also beyond 6 GHz. Fig. 5.14 shows the tu n in g characteristics of the 1.0 pF
tunable capacitor. T he capacitor can be tuned from 3.3 pF (when the bias voltage is set to
zero volts) to 5.0 pF (w hen the bias voltage is set to 3.3 V) and hence the tuning range is
1.5 : I. Sixteen devices were fabricated and only one device was found to be non-functional.
T he average m easured nominal capacitance was found to be 3.39 pF and the standard
deviation was 0.28 pF.
M icrom achined tw o-plate tunable capacitors have been tested from —10° C to 100° C.
T he devices were functional over the tem perature range, b u t their tuning characteristics
changed as th e polysilicon and gold characteristics changed w ith tem perature. The tunable
capacitor w ith two parallel plates (0.6 pF design value) has a tem p eratu re coefficient of
2050 p p m /°C . At high tem peratures, the stiffness coefficient of the suspension decreases,
and consequently a lower control voltage is required to achieve the 1.5:1 tuning range.
A 100 Hz. 3 V square-wave was applied across the tu n ab le capacitor with two parallel
plates (295 /xm x 295 £tm capacitor) in order to test th e reliability of a micromachined
tunab le capacitor. No change in tunable capacitor characteristics has been observed even
after 120 million cycles (i.e. 2 weeks).
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V- -
Figure 5.11: M easured S n of the tw o-plate tunable capacitor (0.6 pF design value).
3 2
3 0
_rT
2 8
“
2. 6
2. 4
2. 2
2. 0
00
0 5
10
1 5
2 0
2 5
3 0
3 5
4. 0
V, [V]
Figure 5.12: M easured tu n in g characteristics of the two-plate tu n a b le capacitor (0.6 pF
design value).
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77
2 GHz
4.8 - j 49.7 £2 I
Figure 5.13: M easured S n of th e tw o-plate tunable capacitor (1.0 pF design value).
55
50
u.
a
45
o
35
3.0
0 0
05
1.0
15
2 0
25
30
3.3
V, [V]
Figure 5.14: M easured tuning characteristics of the tw o-plate tunable capacitor (1.0 pF
design value).
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78
5.6.2
M icromachined T h ree-P late Tunable Capacitor
The fabricated micromachined tunable capacitor with three parallel plates (400 pm
by 400 pm capacitor) is shown in Fig. 5.15.
The tunable capacitor has a m easured Q-
factor of 15.4 at I GHz an d 7.1 a t 2 GHz (Fig. 5.16).
The self-resonant frequency is
approxim ately 6 GHz. Fig.
5.17 shows the tuning characteristics of th e tu n ab le capacitor.
U nder zero bias conditions
(i.e. V\ = 0 V and Vi — 0
V).the m easured capacitance (i.e.
the desired capacitance C p ) is 4.0 pF. T he measured capacitance is approxim ately 6.4 pF
when V \ = 1.8 V and V i = 0 V are applied. W hen V \ = 0 V and V i = 4.4 V are set. the
m easured capacitance is 3.4pF. The tu n ab le capacitor has hence a tu n in g range of 1.87:1.
If th e Vi is more than 4.4 V while V\ = 0 V. bistability and discontinuity in tuning are
observed. The capacitance suddenly drops to approxim ately 2.2 pF an d retu rn s to 2.3 pF
when the bias voltages are set back to zero volts. T he capacitor is still tunable, but the
tuning range in this mode is only 1.12:1. The device returns to th e previous mode (i.e.
4.0 pF nominal capacitance), provided V\ = 5.0 V and Vi = 0 V are applied before the bias
voltages are reset to zero volts. O ut of th e 96 fabricated devices. 7 tu n a b le capacitors were
not functional. The average m easured nom inal capacitance was 3.63 pF an d the standard
deviation was 0.52 pF.
The measured results for 210 p m x 230 /im and 295 pm x 295 p m two-plate capaci­
tors. and a 400 pm x 400 pm three-plate capacitor are sum m arized in Table. 5.2. A lthough
further statistical d a ta is necessary, this prelim inary d a ta seems to indicate that tuning
range of micromachined tunable capacitors is wide enough to cover process variations.
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79
Figure 5.15: M icrophotograph of the three-plate tun ab le capacitor (1.9 pF design value).
' 2.6 - j 18.442
<
2 GHz
. 1 6 -j 40 42
Figure 5.16: M easured S n of the three-plate tun ab le capacitor (1.9 pF design value).
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80
6.4
-i
5.9
—
5.4 -
/
i
4.9
4.4
3.9
3.3
0.0
0.4
O.a
1.2
1.6
0.0
0 .8
1.6
2.4
V, [VI
V, [V]
fa)
(b)
3.2
4.0
F igure 5.17: M easured tuning characteristics of the three-plate tun ab le capacitor (1.9 pF
design value): (a) Vi swept while Vo = 0 V and (b) \ri swept while V\ = 0 V.
5 .7
D iscu ssio n
Since m icrom achined tunable capacitors are mechanical devices, effects of various
physical phenom enon such as residual stress, loss, tem perature, hum idity, pressure, gravity,
acceleration, and shock on the capacitor characteristics m ust be exam ined.
5.7.1
Effect o f R esidual Stress
T he m easured capacitances of th e experim ental devices are significantly larger th an
th e designed values. Parasitic capacitance to the su b strate adds approxim ately 0.37 pF.
0.4 pF . an d 0.6 pF to th e overall designed capacitance of th e 0.57 pF . 1.0 pF. and 2.0 pF
tu n ab le capacitors, respectively. Furtherm ore, residual stress in the polysilicon plates pro­
duces w arping of th e capacitor plates, which effectively results in a different capacitor plate
separatio n an d thus a different parallel-plate capacitance [100].
Fig. 5.18 shows an SEM photograph of a two-plate tu n ab le capacitor (1.0 pF design
value). A sm all upw ard displacem ent of suspension beam s is clearly visible. A lthough the
top plate does not ap p ear to be w arped, the low-magnificat ion m icrophotographs indicate
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81
Three-plate
Varactor Type
Two-plate
Two-plate
P la te A rea
210 jim x 230 p m
295 pm x 295 pm
D esigned C apacitance
!---------------- -—
j M easured C apacitance
0.6 pF
1.0 pF
2.05 pF
3.3 p F
4.0 pF
! M easured Q- factor
i
20.0
13.6
15.4
! T u n in g Range
1
| T u n in g Voltages
1.5:1
1.5:1
1.87:1
4 V
3.5 V
16
16
96
2
1
7
A verage C apacitance
1.98 pF
3.39 pF
3.63 pF
S ta n d a rd D eviation
0.14 pF
0.28 pF
0.52 pF
j F ab ricated Devices
|
| N onfunctional Devices
i
------------------------------ j
400 pm x 400 pm
.
1.9 pF
j
!
j
------------------ 1
1.8 V & 4.4 V
----------------------------- 1
Table 5.2: Sum m ary of m icrom achined tunable capacitor measurements.
some dow nw ard w arping. T he shape of the b ottom plate cannot be observed.
T h e SEM photograph of a tu n ab le capacitor w ith three parallel plates (1.9 pF design
value) is shown in Fig. 5.19. From th e photograph, it is clear th a t the top plate is w arped
upw ard. It is difficult to evaluate th e shape of the m iddle plate from the photograph. The
sh ap e o f the b o tto m plate does not change upon sacrificial layer release since it is directly
dep o sited on the nitride.
I t is believed th a t th e bistability observed in the three-plate tunable capacitor is due to
the w arping of th e top or th e m iddle plate, since the direction of plate warping (i.e. upw ard
or dow nw ard) can be changed w ith an application of ap p ro p riate dc voltages. S tructures
w ith low residual stress m ust be investigated to minimize w arping and bistability so th a t
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82
Figure 5.18: SEM photograph of the tw o-plate tunable capacitor (1.0 pF design value)
Figure 5.19: SEM photograph of th e three-plate tunable capacitor (1.9 pF design value).
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83
the tuning range of three-plate tunable capacitors can be further improved.
5 .7 .2
E ffect o f V a rio u s L oss M e c h a n ism s on Q -F actor
In
many RF applications, the electrical Q-factor of the tunable capacitor is an ex­
trem ely im portant characteristic. The top plate of tunable capacitors isim plem ented using
poly2/gold layers, which provide a low resistance. However, the bo tto m plate of the twop late and the m iddle p late of the three-plate tunable capacitors are composed of polyl
which has rath er high sheet resistance. In order to evaluate the effect of polyl plate on the
Q-factor. the tunable capacitor is modeled as a d istributed RC line (Fig. 5.20). where the
poly2/gold plate is assum ed to be equipotential. T he input adm ittance to such a d istrib u ted
RC line is given by:
,
yhf(s) =
y/ar s in h y / s r
7—7=
Rp 1 -+- c o s h y / S T
n ^
and " = R-pCd-
where R p is the resistance of the polyl p late and C q is the desired capacitance.
, c 0<M
(0.36)
From
(5.36). an expression for the Q-factor as a function of frequency can be obtained:
Q(co)
4-
1
s in { y / 2 u r ) 4- 2cosh(yJ ^ ) s i n { s^ )
s i n h ( y / 2 u j r ) 4 2 sin h (< J ~ ^ -)c o s
T he expression derived above represents th e maximum theoretically achievable Q-factor.
since losses due to the interconnect are not considered. At 1 GHz. the calculated Q-factors
for 0.6 pF. 1.0 pF. an d 1.9 pF tunable capacitors are 318. 191. and 95. respectively, which
suggests th a t a high Q-factor can be achieved in a MUM Ps process. T he Q-factor estim ate
is lower, however, when the series interconnect resistance is considered, in particular the
resistance of th e polyl-poly2 via and the resistance of the deposited gold on the suspension
beam . W ith interconnect resistance considered, the Q-factors are estim ated at 84.8. 55.2.
and 26.6. a t 1 GHz, for 0.6 pF, 1.0 pF. and 1.9 pF tunable capacitors, respectively. However,
the (J-factor values for the experim ental devices fall short of their theoretical estim ates.
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84
Suspended Plate i Poly 1)
4 —A / W V
R p/2n
V
s'
- V /V 'v — O
R p/n
-V /V ,
— V /V V
Rp/n
R p/2n
C D/n
C D/n
Top Plate <Poly 2 * Gold)
F igure 5.20: High-frequency model of th e polysilicon m icrom achined tu n a b le capacitor.
W hile th e {^-factor of a micromachined tu n a b le capacitor is limited by th e series resistance
of th e interconnect, the m easured capacitance is higher th a n the designed value and hence
a lower <5-factor is measured. Taking th e m easured nom inal capacitances into account, the
Q -factors are estim ated at 25.5. 17.2. an d 16.5 for 2.05 pF. 3.2 pF. an d 4.0 pF (i.e. 0.6 pF.
1.0 pF . an d 1.9 pF design values) tu n ab le capacitors, respectively.
T h e skin-effect is not a problem since th e skin d ep th of gold is 2.4 |zm a t 1 GHz and
1.7 fjm at 2 GHz (given 0.5 ^ m thick gold). However, since M UM Ps process is not a pla­
narized process, steps in gold interconnect can result in a high interconnect resistance [96].
T h o u g h M UM Ps is not a planarized process, surface m icrom achining technologies th a t use
chem ical-m echanical polishing process to achieve layer planarization are available [101].
Using d a ta from 20 prior M UM Ps fabrication runs, the variation in the resistance of
polysilicon is estim ated to be approxim ately 20 percent. Consequently, sim ilar variation in
the (^-factor of the m icrom achined cap acito r can be expected.
Since the Q-factor of the m icrom achined tu n ab le capacitor is lim ited by the series
interconnect, alternative suspension designs th a t can provide th e desired spring constant
an d yet have a low series resistance m u st be investigated in the future. In addition, wider
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85
interconnect m ust be used in order to increase th e Q-factor even further.
5.7.3
Effect o f A ir as a D ielectric
Since air is used as a dielectric for the tunable capacitor, its properties as a function
of pressure, te m p eratu re, an d hum idity must be exam ined. An em pirical expression for the
dielectric co n stan t of air as a function of pressure and tem perature is shown below:
eatr(P-T) = 1 + 0 .1 5 7 ^ .
(5.38)
where P is the pressure in atm an d T is the absolute tem perature in Kelvin [102]. Under
the norm al atm ospheric conditions (i.e. 293 K and 1 atm ), the dielectric constant of air
has a te m p eratu re coefficient of —2 ppm /K w hen hum idity is zero percent and 10 p p m /K
when hum idity is 100 percent [103]. It is apparent from (5.38) th a t practical variations in
atm ospheric conditions have a sm all effect on th e dielectric constant of air and hence on
the capacitance.
U nder norm al atm ospheric conditions, dielectric stren g th of air is 0.8 kV /'m m ’103].
which suggests th a t arcing betw een capacitor plates m ay be present. F urther exam inations,
however, show th a t th e dielectric breakdown field of air is a nonlinear function of electrode
spacing [104]. For instance, under normal atm ospheric conditions, th e breakdown field of
air is 4.5 k V /m m for a 1 m m air gap (breakdown voltage of 4.5 kV) and 12.5 k V /m m for
a 0.06 mm air gap (breakdow n voltage of 750 V ).
The tu n a b le cap acito r may be operated in vacuum , which resolves many issues th a t
can arise w hen air is used as a dielectric. For exam ple, the mechanical resistance r can
be dram atically reduced an d hence the mechanical Q-factor can be significantly improved
when the tu n a b le capacitor is placed in vacuum. In ad d ition, the mechanical noise due to
the th erm al ag itatio n of air molecules can be reduced as mechanical resistance r is reduced.
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Furtherm ore, there is no dielectric breakdow n in vacuum. However, specialized packaging
is needed to ensure vacuum.
5 .7 .4
E ffe c t o f T e m p e r a tu r e
Although tem perature has a small effect on the dielectric constant of air. the difference
in therm al expansion of polysiiicon and gold layers results in the deformation of capacitor
plates and hence significant capacitance change.
T h e therm al expansion is often given
by [89]:
A L = aL A T .
(5.39)
where a is the linear expansion coefficient. A T is th e tem perature change in Kelvin. L
is th e length in meters, an d A L is th e increase in length in m eters due to the therm al
expansion. T he therm al coefficient of expansion a is 2.6 p p m /K for silicon and 14 p p m /K
for gold [105]. For example, a 295 /j.m x 295 ^ m p o ly l plate expands by 0.07 ^m . given a
100 Kelvin change in tem perature, while a gold plate w ith the same dimensions expands by
0.38 (im.
5 .7 .5
E ffe c t o f G r a v ity , A c c e le r a tio n , a n d S h o c k
In order to evaluate th e effect o f gravity on th e m icrom achined tunable capacitor, the
displacem ent x g should be calculated, as follows:
x9
=
(5.40)
where g is the gravity (g = 9.80665 m / s 2). For exam ple, the 0.6 pF capacitance would
change only by approxim ately 0.02 percent due to th e gravitational pull on the suspended
plate. Conversely, an acceleration of approxim ately 432 g is required in order to observe a
10 percent change in th e capacitance value.
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87
An im p o rtan t consideration when dealing with m icromachined devices is the ability
of these devices to survive shock. For example, a common criterion is th e ability of the
m icrom achined device to survive a shock of approxim ately 5.000 g [106].
5.7.6
Effect of Etch H oles
In o rd er to ensure the proper release of the micromachined tu n ab le capacitor, a suf­
ficient num ber of etch holes must be placed across the capacitor plates. A lthough the me­
chanical stre n g th of th e polysilicon plate is reduced, the reduction of the effective Young's
m odulus is only 18 percent [107].
5.7.7
E ffect o f P rocess Variations on Spring C onstant
It is equally im p o rtan t to investigate the effect of process variations on the spring
constant o f th e double clam ped beam , given by (5.35). T he effect of process variations on
the spring co n stan t can be exam ined using sensitivity analysis, as shown:
dkx
dEp
difct
dW,
dkx
&TX
d kt
dLx
kt
Ep
kx
^
^
3Art
T\ ^
3ki
Li
Akt _ AEP
kt
EP
t^kx
X
= wt
AA;,
Tx
X
=
A £.
Akt
‘1 * '
O
ki
Lt
—
—
(5.41)
(5.42)
(5.43)
(5.44)
Like in any IC process, in a surface polysilicon m icrom achining technology, variations in
w idth an d length are fairly small, since these dimensions are set by th e lithography. Thick­
ness of polysilicon layers, however, cannot be controlled as well. In th e MUMPs process,
for exam ple, th e variations in thickness for polyO, polyl, and poly2 layers have been com­
puted (calculated over th e last 20 MUMPs fabrication runs) as 3.0. 1.3. and 3.5 percent,
respectively. U nfortunately, many vendors do not provide d a ta w ith respect to variations in
Young's m odulus of polysilicon over process and tem perature. Since th e suspension beams
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are usually designed w ith rath er large dim ensions, the small variations in th e spring con­
s ta n t d u e to im perfections in the lithography are normally negligible. T h e spring constant
of th e suspension beam is most sensitive to variations in the thickness Tt where a variation
in th e spring constant of 10.5 percent can be expected due to th e thickness changes of the
poly2 beam s.
5.8
S um m ary
M icrom achined electro-m echanically tunable capacitors, fab ricated in a polysilicon
surface m icrom achining process, have been dem onstrated.
Even th o u g h polysilicon was
used as th e stru ctu ra l m aterial, theoretical calculations show th a t tu n a b le capacitors w ith
relatively high Q-factors are possible.
In addition, experim ental devices achieve tuning
ranges which are near theoretical lim its. T he two-plate tu n ab le cap acito r has a nom inal
capacitan ce of 2.05 pF. 1.5:1 tu n in g range w ith 4.0 V bias voltage, an d a Q-factor of 20 at
1 GHz. T he three-plate device has a nom inal capacitance of 4.0 p F . 1.87:1 tuning range
w ith 4.4 V bias voltage, and Q-factor of 15.4 a t 1 GHz. No changes in th e tw o-plate device
characteristics due to wear have been observed even after 120 m illion cycles. T he twop late tu n ab le capacitor was also found to be functional over —10° C to 100° C tem perature
range. Moreover, dependence of various physical phenom ena such as tem p eratu re, pressure,
hum idity, and shock on capacitor characteristics has been exam ined.
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89
C h a p ter 6
M icrow ave M icro m a ch in ed -B a sed V C Os
6.1
In trod u ction
M icrom achined electro-m echanically tunable capacitors may provide means of real­
izing integrated low phase noise VCOs especially since commercial CM OS IC technologies
w ith integrated polysilicon surface m icrom achining are already in existence today.
Al­
though to d ay 's CMOS technologies w ith integrated m icrom achining processes do not feature
fast enough transistors to realize microwave circuits, feasibility of m icrom achined electrom echanically tunable capacitors in microwave VCOs should nevertheless be investigated. A
low phase noise 714 MHz CMOS VCO based on a high Q-factor alum inum micromachined
tu n ab le capacitor and an off-chip in d u cto r has been d em onstrated already [108]. However,
the micro m achined-based VCOs have thus far exhibited tuning ranges as low as 2 per­
cent [108], which is insufficient to even cover a single frequency band of a typical m odern
com m unication sytem.
As discussed in C h ap ter 3. th e phase noise of an oscillator is inversely proportional
to th e overall Q-factor of the resonant circuit, and hence inductors and capacitors with
high ^-factors are required. In C h a p te r 5, m icrom achined electro-m echanically tunable
capacitors with relatively wide tu n in g range and high ^-factors have been presented. In
this chapter, the application of m icrom achined tunable capacitors, developed in C hapter 5.
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90
to microwave VCOs is investigated through two prototype L.9 GHz and 2.4 GHz CMOS
VCOs. The 1.9 GHz VCO satisfies th e phase noise specifications of th e DCS1800 system and
the 2.4 GHz VCO meets the requirem ents of the IEEE 802.11 direct sequence wireless LAN
standard. The prototype VCOs are the widest tuning range microwave micromachinedbased VCOs presented to date.
6.2
P h a se N o ise d u e to M echanical and E lectrical N oise
T he phase noise of an oscillator due to the mechanical noise in a micromachined
tunable capacitor is derived in A ppendix C. In this chapter, only differential oscillators
are considered and hence the phase noise reflects two m icrom echanical tunable capacitors
connected in series. At a frequency offset Slj. the phase noise due to mechanical noise is
given as follows:
(6.
1)
where the random displacem ent fluctuations of the capacitor plate are given by:
It should be noted th a t, for offset frequencies Suj below the m echanical resonant frequency,
the phase noise due to m echanical noise decreases at a rate o f —20 dB /dec.
At offset
frequencies well above th e m echanical resonant frequency, the phase noise due to mechanical
noise reduces at a rate of - 6 0 d B /d e c from the carrier. This is m ainly because the spectrum
of th e random displacem ent fluctuations is white for frequencies well w ithin the mechanical
resonant frequency, while the displacem ent fluctuations decay at —40 d B /d e c at frequencies
above the resonant frequency. T he phase noise due to mechanical noise can be minimized in
four ways. F irst, th e m echanical resistance should be minimized. T his can be accomplished
either by increasing the num ber of etch holes or by using vacuum as the dielectric. Second,
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91
the mass of the suspended plate could in principle be increased, but it does not ap p ear to
be a practical solution given th a t the mechanical layers (i.e. polysilicon, gold, alum inum ) in
m any m icromachining technologies are 1-2 (itn thick. T hird, a suspension w ith a very large
spring constant could theoretically be employed to minim ize the mechanical noise, b u t only
a t a cost of larger tuning voltages. Finally, the separation between the capacitor plates d\
could be increased. However, it is not a practical solution since a much larger capacitor
area would be required to achieve the same capacitance.
T he phase noise of an oscillator due to electrical noise has been exam ined in C h ap ter 3.
T he phase noise at a 6 u offset from the u>o carrier is given below:
+ 5Fc J (u,»(i+7)ac£Rp
£ e( M = ----------------------
+ r l
+ (T^yr) (s£)2
(6-3)
out,peak
2
w here the noise factor FGm is sim ply defined as the ratio of noise currents from th e transcon­
ductor and from the equivalent parallel conductance, as shown:
FGm =
(6-4)
d l GEQ
Note th a t the phase noise due to the electrical statio n ary and non-stationary w hite noise
decreases a t a rate of —20 d B /d e c from the carrier. T he phase noise caused by the electrical
noise can be minimized in three ways. F irst, the loss in th e LC tank could be minimized. In
other words, the series resistances of the inductor and capacitor. R i and R c - respectively,
should be minimized while the parallel resistance R p should bemaximized.
transconductor w ith a low noise factor F c m should be used.Finally,
Second, a
th e am plitude of
oscillations or rath er the voltage swing across the LC ta n k must be maximized.
T he active devices such as MOS transistors often exhibit therm al noise behavior. T he
therm al drain current noise of MOS devices is given as follows:
i2np = AkTctgmv
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(6.5)
_2
*nn =
f6.6)
where gmn and gmp are th e transconductances of the NMOS an d PM O S tran sisto rs, respec­
tively. T he scaling factor a is equal to 2/3 for long channel devices in satu ratio n . However.
noise behavior of short channel devices is not yet w ell-understood an d a scaling factor a of
1.5 to 3 has been often used in the literature [109j-[110j. T h e u n certain ty of the MOS noise
m odel makes it difficult to evaluate the phase noise of oscillators precisely an d differences in
phase noise calculations of ab o u t 3 dB can be expected d ep en d in g on which scaling factor
a is used.
6.3
M icrow ave C M O S Inverter VCO - 1.9 G H z
In this section, th e design of a 1.9 GHz CMOS VCO based on cross-coupled inverters,
its noise analysis, and experim ental results are presented.
6.3.1
VCO D esign
Fig. 6.1 shows th e circuit schematic of th e CMOS in verter oscillator an d its equivalent
parallel RLC circuit. T he two cross-coupled NMOS tran sisto rs
an d the two cross­
coupled PM OS tran sisto rs A/3-M 4 realize th e negative resistance, while the LC resonator
is formed w ith th e tu n ab le capacitors C q and the fixed value inductors L f . Resistors R c
and R l are shown to illu strate the finite Q-factor of these reactive elem ents. The CMOS
inverter oscillator can be represented by an equivalent p arallel RLC circuit using series to
parallel transform ation, as shown in Fig. 6.1 (see Sec. 2.4). T h e values o f th e elements in
th e equivalent parallel RLC circuit can be shown to be as follows:
G M ' S EG
9mn ■+■ gmp
2
(6.7)
( 6.8 )
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93
M,
M,
G M.NHG
—
-D
-V A A ------- W A
•H
Lp
• LOAD
Rl
EQ
I
H
r-
■me— vwv— ,yWV
WA—
G
VA
'EeQ
■V.'Vv-
mT
Rl
L
I
C
load
Leq
ja m .
i
I
<7
M,
Figure 6.1: Schem atic of the CMOS inverter oscillator and its equivalent parallel RLC
circuit.
G eq
L eq
2(1 + Q2l ) R l
2(1 + Q 2c ) R c
(6.9)
2r 0
r op
(6.
2 £ f (1
10 )
and
Cg
Cp
=
2 (C g d .V
s x
+ C
qsp
C d .v tC d p
C
load
+ Cg d p)
(
6 . 11 )
where C q d n an d C g d p are the gate to drain overlap capacitances. C'g s .v and C g s p are
the gate to source capacitances. C o s and C q p are th e drain diffusion capacitances. ron and
are th e o u tp u t resistances, and gmn and gmp are th e transconductances of the NMOS
and PM O S tran sisto rs, respectively. The load capacitance C l o a d represents the parasitic
capacitance of th e interconnect and the input capacitance of the circuits th a t the oscillator
is intended to drive.
From th e equivalent RLC network of the oscillator (see C hapter 2). it is apparent
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94
th a t the oscillation frequency u/q is given by:
L
( 6 . 12 )
UJQ
s/
L
eq
C
eq
where at cuo- th e conductances of the inductor and capacitor are equal and opposite in sign,
and hence for oscillations to begin, the following condition m ust be met:
G m .x e g > G e q
(6.13)
T he m icrom achined tunable capacitor employed in this design, has a nominal capacitance
of 2 pF. a series resistance R c of 4.3 fi. and a ^ -facto r of 9.7 a t 1.9 GHz. For ease of design,
it is assum ed th a t Q-factors of tunable capacitors an d fixed value inductors are sufficiently
large so th a t 1/ Q c ~ 0 am * ^ / Q \ ~ 0- *n practice, th e parasitic capacitance Cp is often as
large as th e desired tu ning capacitance Co- Denoting C p = ~/ C d where 7 = 1. and setting
the desired oscillation frequency to 1.9 GHz. the required equivalent inductance L e q can
be now calcu lated as follows:
LeQ
=
u j 's C e q
= (2tt
x
1.9
x
109)2
x
2
x
1 0 "12
~ 3 '°
nH
(6' U )
T he 3.5 nH in d u cto r can be realized w ith a 0.7 mil bonding wire (r = 8.75 pm ) where
I = 2.2 m m a n d d = 0.48 mm (see chapter 4). The bonding wire inductor has an estim ated
resistance of 0.5 fl and a Q-factor of 80 at 1.9 GHz.
For convenience, the transistor o u tp u t resistances r on and
enough so th a t th e ir contribution to G
eq
are assumed to be large
is negligible. Using (6.9) and (6.13). the required
negative conductance G.vr.,VEG can be found to be:
C*t \y \j c’/'
™'NEG
7
■
2 ( 1 + Q \)R l
4"
^
——
4“
" s: 1.4 m S
2(1 + Q2
c )R c
6401
817.7
(6.1 o )
Hence, to en su re oscillations, the CMOS inverter m ust have a transconductance G \ ( j y v as
follows:
G.Vf./.W = Jm n 4” ffmp '■> 2.8 TTlS
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(6.16)
95
In order to sustain oscillations over process and tem perature variations. G u j x v has been
conservatively selected to be approxim ately 18 mS which is approxim ately 6 tim es higher
th an th e required transconductance.
Normally, a 3 times larger tran sconductance than
strictly necessary is chosen. However, in this design, the oscillator is designed to oscillate
even if th e m icromachined tunable capacitor has a 50 percent lower Q-factor th a n expected.
T he CMOS inverter VCO has been sim ulated using H SPICE [93]. Sim ulations show
th a t the circuit oscillates at a nominal oscillation frequency of 1.88 GHz where the differential am plitude of oscillations across th e LC tank is 2.3 V or 17.2 dBm. The oscillator draws
•5.3 mA from a 2.7 V power supply. Given th a t the micromachined tunable capacitor has a
m axim um tuning range of 1.5:1. the sim ulated tuning range of the VCO has been found to
be 13.6 percent.
6.3.2
N oise A nalysis
From Fig. 6.1. it follows th a t th e equivalent noise current th a t appears in parallel
w ith the RLC resonant circuit is sim ply given by:
(6.17)
where th e scaling factor a = 2/3 has been assumed. The noise factor F c m of the transcon­
d ucto r can be calculated simply as follows:
Fc
■IkT
X
,(g m n +
(Jm p )
AkTGp
(6.18)
T he mass of the suspended plate m and the equivalent spring constant of the suspension arrangem ent k for the 2 pF m icrom achined tunable capacitor have been estim ated
a t 0.8 /j grams and 85 N /m . respectively. T he mechanical resistance has been estim ated at
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96
1.4 x 10_1N s /m using equation (5.26) in Sec. 5.3.3 where th e 2 pF micromachined capacitor
has a to tal of 144 etch holes w ith an area of 9 fzm2 each. Phase noise equations sum m a­
rized in Sec. 6.2 have been used to plot th e theoretical phase noise due to b oth mechanical
an d electrical noise (see Fig. 6.2) where 1.9 GHz carrier has been assumed. Note th a t the
20 dB/dec
-20
-4 0
-60
O
T3
GO
*80
20 dB/dec
<0
O
Z
<
D
cn
-100
•C
-120
a.
Electrical
-1 4 0
-1 6 0
M echanical
-1 8 0
-200
1000
10000
100000
1e+06
la + 0 7
Frequency [Hz]
Figure 6.2: T h eo retical phase noise due to m echanical and electrical noise (1.9 GHz VCO).
phase noise of th e VCO is dom inated by th e m echanical noise at offset frequencies below
400 kHz. while th e electrical noise dom inates th e overall phase noise a t offset frequencies
above 400 kHz. At offset offset frequencies below approxim ately 10 kHz. the phase noise
decays a t a ra te of —20 dB /dec. Similarly, a t offset frequencies above 400 kHz. the phase
noise decreases a t —20 dB /dec.
As an exam ple, th e phase noise is evaluated using equations (6.1) and (6.3) a t 100 kHz
and 600 kHz offset frequencies. Since th e m echanical noise dominates the phase noise a t
a t offset frequencies below 400 kHz. th e phase noise a t a 100 kHz offset from th e 1.9 GHz
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97
carrier is given by:
1
1
2.7 x 10-29
/1 9 0 0 M H z y
16 X 4 X (0.75 x lO' 6)2 X \ 0.1 M H z )
£ m(100 kHi
=
—95.6 d B c / H z
(6.19)
At 600 kHz offset frequency from th e 1.9 GHz carrier, the phase noise of the VCO is
dom inated by the electrical circu it noise and is given by:
.XI2121,! + 8.551(0.50
£*(600 k Hz )
=
if E ) ( f g > $ ^ ) 2
(2.3 V)2
2
=
—127.2 d B c / H z
( 6 . 20 )
where, for simplicity, the cu rren t through the MOS transistors was assumed to be a 50 per­
cent d uty cycle square wave.
6 .3 .3
E x p e r im e n ta l R e s u lt s
Fig. 6.3 shows the com plete schem atic of the CMOS inverter VCO. The cross coupled
inverters, transistors M \- M \ form th e core of the oscillator. T he o u tp u t buffers consist of
transisto rs
A f5- A f 12
where th e resistors R 1-R 4 (500 Q. each) are used to stabilize the o u tp u t
com mon mode. The sizes of th e MOS transistors in Fig. 6.3 are sum m arized in Table 6.1.
Transistors
Size
M 1-M 4
24 x 4.5 p m / 0.6 p m
m 5- m 8
10 x 4.5 p m /0 .6 pm
A/g-A/12
24 x 4.5 p m /0 .6 p m
Table 6.1: S um m ary o f th e MOS transistors in the 1.9 GHz VCO.
T he prototype VCO has been fabricated in a 0.5 p m CMOS process while the mi­
crom achined tunable capacitors have been fabricated in a MUM Ps surface polysilicon mi-
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98
crom achining process. T he CMOS and M UM Ps dice have been packaged in a stan d ard
quad flat-pack ceram ic package (MS I 5P20M) and th e two dice have been bonded together.
In order to o b tain low noise performance, a high (^-factor bonding wire inductor has been
employed for the LC resonator. The m icrophotograph of the fully assembled integrated
VCO is show n in Fig. 6.4.
All m easurem ents were performed using an H P 4352B V C O /PLL Analyzer.
The
diagram o f th e m easurem ent test setup is shown in Fig. 6.5. The measurements include also
the d eg rad atio n effect of a M /A -C O M (ETC l.6-4-2-3) 4 : 1 transm ission line transform er
which is used to perform differential to single-ended conversion.
T he core of the VCO consumes 15 mW of power and the ou tp u t buffers consum e
30 m W o f power, while operating from a 2.7 V power supply. Using the internal PLL of the
V C O /P L L A nalyzer, the frequency of the prototype VCO has been set to 1.9 GHz. T he
o u tp u t sp ectru m of th e VCO is shown in Fig. 6.6. T h e VCO delivers 3.4 dBm into the 50 fi
load.
Fig. 6.7 shows the m easurem ent of phase noise obtained using the HP 4352B ana­
lyzer w hen the VCO is locked to a low noise 1.9 GHz frequency reference. The HP 4352B
analyzer uses th e PLL phase noise measurement m ethod. The VCO achieves a phase noise
of —97.5 d B c/H z a t a 100 kHz offset from the carrier and a phase noise of —126.3 dB c/H z
at a 600 kHz offset from the carrier. Assuming th a t th e phase noise decreases a t a rate
of —20 d B /d e c from th e carrier, the VCO has an estim ated phase noise of —140 dB c/H z
at a 3 M Hz offset from the carrier, which satisfies th e requirem ents of the DCS 1800 cel­
lular telephone sta n d a rd (see C hapter 1). T he m easured phase noise decays at a rate of
- 2 0 d B /d e c for offset frequencies below 20 kHz and a t a rate of approxim ately - 3 0 d B /d ec
for offset frequencies between approxim ately 20 kHz an d 400 kHz. At offset frequencies
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99
■o V
o V,
TUNE
o VF
« Vf
M UM PS
die
CM OS
L
die
Figure 6.3: Com plete circuit schem atic o f the 1.9 GHz VCO.
Figure 6.4: M icrophotograph of th e 1.9 GHz VCO.
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« Vr
100
100 p F
50
a
HP 4352B
VCO/PLL
Analyzer
OP
VCO
ON
100 pF
cc
C ontrol V oltage
TUNE
Supply V oltage
Figure 6.5: M easurem ent test setup for th e experim ental 1.9 GHz VCO.
10 d B / REF 10 dBm
ExR
A fC
ATM 10 dB
CENTER
1 . 9 GHz
L0
VBU 30 0 Hz
1 . 8 7 3 9 GHz
SUP
3 .7 9 3 sec
SPAN 2 HHz
Figure 6.6: M easured o u tp u t sp ectru m of the 1.9 GHz VCO.
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above 400 kHz. th e m easured phase noise decays again at a rate of —20 dB /dec. which
corresponds reasonably well w ith the theoretical predictions shown in Sec. 6.3.2.
Tuning characteristics of the VCO are shown in Fig. 6.8. T h e oscillation frequency is
tunable from 1925 MHz to 1765 MHz with the tuning voltage of 1.25 V to 5.25 V. which
results in a tuning range o f 9.0 percent. Note th at the oscillation frequency and the gain
of the VCO vary in a nonlinear fashion with respect to the control voltage. The use of
such a VCO in a frequency synthesizer results generally in undesirable settling behavior
where the settling tim e varies based on which channel the synthesizer is switched to and
from. Since variations in th e control voltage of the VCO are slow in practice, a nonlinear
preprocessing circuit in principle can be inserted in front of the VCO to produce an overall
linear frequency versus voltage characteristic. The sharp changes in tuning characteristics
are attrib u te d to th e n o nlinear behavior of micromachined tu n ab le capacitors.
Sim ulation an d m easurem ent results are com pared in Table 6.2. T he sim ulated and
measured results agree relatively well. The difference in the o u tp u t power of the VCO is
a ttrib u ted to th e in sertio n loss in the 4:1 transm ission line transform er, which is specified
to be less th a n 3 dB over its usable frequency range of 500 MHz-2500 MHz. W hile the
predicted and m easured phase noise match surprisingly well, th e m easured tuning range
falls short of the sim u lated value.
The m easurem ent results of the 1.9 GHz CMOS inverter VCO are summarized in
Table 6.3. As can be seen from the table the VCO can o perate a t a 3.3 V power supply and
can deliver 7.5 dB m into a 50 ft load. In addition, th e VCO can operate at a power supply
as low as 2.0 V and can deliver -16.9 dBm of power into a 50 ft load. While operating
from the 2.0 V power supply, the VCO consumes 7.0 mW of power, where about 2.3 mW
is consumed in th e core of th e VCO and 4.7 mW in th e o u tp u t buffers. Despite the low
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L02
P hase N oise
20 d B / REF 0 dBc
-1 2 6 .3
dBc
ExR
A fc
Avg
ATN 15 dB
START 1 kHz
N8U 1 Hz
LO 1 . 8 9 6 GHz
SUP
2 .3 0 8 sec
STOP
I riHz
Figure 6.7: Phase noise measurement of the L.9 GHz VCO.
F requency
ATM 10 dB
START
1 .2 5 V
50 HHz/ REF 1 . 8 4 GHz
FRES
1 kHz
SUP 3 6 0 s e c
STOP
5 .2 5 V
Figure 6.8: M easured tuning characteristics of the 1.9 GHz VCO.
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103
Sum m ary
Simulated
Measured
Nom inal Frequency
1.88 GHz
1.92 GHz
Tuning Range
13.5%
9.1 %
P hase Noise @ 600 kHz
-127.2 dB c/H z
-126.3 dB c/H z
O u tp u t Power
5.8 dBm
3.4 dBm
Power Supply
2.7 V
2.7 V
S upply C urrent
16.0 mA
17.0 mA
i
•4
Table 6.2: Com parison of sim ulated an d measured results (1.9 GHz VCO).
power supply voltage, the VCO achieves a phase noise of -120.4 dB c/H z at 600 kHz offset.
Power Supply
2.0 V
2.1 V
2.4 V
2.1 V
O u tp u t Power [dBm]
-16.9
-8.0
-0.2
3.4
3.0 V
3.3 V !
i
5.7
7.5
..
Supply C u rren t [mA]
3.5
6.0
11.9
17.0
22.1
Phase Noise @ 100 kHz [dBc/Hz]
-95.1
-96.5
-97
-97.5
-97.6
Phase Noise @ 600 kHz [dBc/Hz]
-120.4
-122.7
-123.5
-126.3
-126.3
.
I
i
i
27.1
j
!
-97.1 j
1
-126.4 !
Table 6.3: Sum m ary of th e m easurem ent results (1.9 GHz VCO).
6.4
M icrow ave C M O S D ifferential Pair V C O - 2.4 GHz
This section discusses the design of a 2.4 GHz CMOS differential pair oscillator. Its
noise perform ance is analyzed and th e experim ental results are presented.
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104
6.4.1
V C O D esign
T h e circu it schem atic of th e CM OS differential pair oscillator an d its equivalent par­
allel RLC circuit are shown in Fig. 6.9. T he oscillator consists of a cross-coupled differential
pair which realizes th e negative resistance necessary to overcome th e loss in the LC tank.
T he LC ta n k consists of the tu n ab le capacitors C p and fixed value inductors L f where the
resistors R c an d R i are shown to illu strate the finite Q-factor of these com ponents. Series
1M.NEG
—
LOAD
LOAD
>
Figure 6.9: Schem atic of th e CM OS differential pair oscillator its equivalent parallel RLC
circuit.
to parallel tran sfo rm atio n can be applied to transform the CMOS differential pair oscillator
to a sim plified equivalent RLC circuit, as shown in Fig. 6.9 (see C h a p te r 2). T he com ponent
values o f th e equivalent parallel RLC circuit are:
ffmn
( 6 .2 1 )
Cd
i T
Cp
”
2(1 + Q \)R l + 2(1 + Q2
c )Rc + 2rm
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( 6 . 22 )
(6.23)
105
L eq
=
2 L e (1 -‘t —j-)
('6.24)
and
C r
= 2C cd v ^
C r . s s * C p _ v +_ C L o *
d
(„ M )
where C o d s is th e overlap capacitance between the gate to drain. Cgs.v is the gate to source
capacitance. C o s is the diffusion capacitance at the drain,
is the o u tp u t resistance. gmn
is the tran sco n d u ctan ce of the NMOS transistors M \ and M 2. The parasitic capacitance of
the interconnect and th e input capacitance of the following stages is lum ped together into
the load capacitance C l o a d It is clear from th e equivalent parallel RLC network th at the oscillation frequency uo
is given by:
U* =
-1 X—
V L £ q Cs EQ
(6-26)
At frequency uiq. th e ad m ittan ces of the capacitor and the inductor are equal and opposite
in sign, an d oscillations s ta rt provided that:
G \ j,s e g > G eq
(6.27)
In this design, an in d u cto r is im plem ented using a 1.0 mil bonding wire where I = 2.2 mm.
d = 0.48 m m (see chap ter 4). which results in an inductance of L eq = 3.3 nH. The resistance
of the bonding wire in d u cto r is estim ated a t 0.4 fl and the Q-factor of the inductor is
calculated to be 117 a t 2.4 GHz. The t?-factors of th e tunable capacitors and the fixedvalue inductors are assum ed to be sufficiently large so th at 1/Q ^ ^ 0 and 1j Q \ ~ 0. In
order to achieve an oscillation frequency of 2.4 GHz, th e equivalent capacitance must be as
follows:
CeQ = uj$LP = (2tr x 2.4 x 109)2 x 3.3 x 10“ 9 “ L3 ? F
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(6 28)
106
Note th a t if parasitics are assum ed to contribute to one half of the capacitance to the
equivalent capacitance (i.e.
7 = 1). a L.3 pF tunable capacitance will be required.
A
1.39 pF micromachined tunable capacitor has been fabricated and has a Q-factor of 11 at
2.4 GHz and a series resistance of 4.1 H. Neglecting the ou tp u t resistance
of th e NMOS
transisto rs and using (6.27) and (6.23). the required negative transconductance m ust meet
the following criteria:
G.Vf.<V EC >
2(1 + QI ) R l ’’’ 2(1 + Q 2
C )RC
10952 + 1000
= 1.1 m S
(6.29)
T he transistors in the differential pair hence m ust have a transconductance gmn as follows:
9mn > 2.2 TTlS
(6.30)
T he transconductance of the differential pair has been chosen to be approxim ately 7 times
larger th a n the m inim um value necessary so th a t the circuit oscillates over process and
tem p eratu re variations even if th e micromachined tunable capacitor had a <5- factor °f 5.
T he VCO circuit has been sim ulated using H SPICE [93]. Sim ulation results indicate
th a t th e circuit has a nom inal oscillation frequency of 2.35 GHz and a differential am plitude
of oscillations of 1.9 V or 15.6 dBm.
T he circuit operates from a 2.7 V power supply
and consumes 13.5 mW of power. T he sim ulated tuning range of the VCO is 225 MHz
or 9.3 percent, given the m axim um tuning range of 1.5:1 for the m icromachined tunable
capacitor.
6.4.2
N oise A nalysis
T h e equivalent noise current due to th e transistors in the differential pair and the
noise due to the bias circuitry in Fig. 6.9 is given by:
(6.31)
t t T x ? ( l + N )g m 4
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(6.32)
107
where gm4 = *Vgm3. Note th a t if perfect switching o f th e differential p air is assum ed, the
noise cu rren t
will appear directly at the o u tp u t of the differential pair. T h a t is to
say. when tran sisto r M \ is on and M 2 is off. the noise current will end up a t the d rain of
M i. Similarly, the noise current will end up a t the drain of M2, if tran sisto r M \ is off and
M 2 is on. Since the w hite G aussian noise is not correlated in time, the bias noise current
bnas ap p ears directly a t the o u tp u t of the differential pair. However, this noise appears as
am p litu d e noise: th a t is. it does not affect the zero crossings of the stead y sta te waveform.
In oth er words, in practice the noise current due to the bias appears as com m on mode when
the differential pair is in its linear region and hence this noise source does not affect the zero
crossings. Therefore, it does not co n trib u te to phase noise. Consequently, the ij^as term is
not included in the subsequent noise calculations. T he noise factor Fq m can be calculated
as follows:
P
X \d m n
—
Gm
~
K P ~
A kT G >
- i - (rra) -
-
T he m icrom achined tu n ab le cap acito r has an estim ated mass of the suspended plate of
0.77 digrams and an equivalent sp rin g constant of the suspension of 220 N /m . Using (5.26)
in Sec. 5.3.3, the mechanical resistance of th e m icromachined tunable cap acito r is estim ated
a t 1.8 x 10_ 3N -s/m where the 1.4 pF m icrom achined capacitor has a to ta l of 121 etch holes
w ith an area of 9 /jm 2 each. Fig. 6.10 shows the theoretical phase noise (2.4 GHz carrier) due
to th e electrical and mechanical noise where equations sum m arized in Sec. 6.2 have been used
to o b tain th e plot. It should be noted th a t th a t the mechanical noise dom inates for offset
frequencies less th a n approxim ately 400 kHz. For offset frequencies below approxim ately
20 kHz. th e phase noise decays a t a rate of —20 d B /dec. while at offset frequencies between
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108
20 dB/dec
-20
-40
-60
-80
20 dB/dec
-100
-1 2 0
Electrical
-1 4 0
-1 6 0
Mechanical
-1 8 0
-200
—
100000
F re q u e n c y [Hz)
10000
1000
10+07
F igure 6.10: T h eo retical phase noise due to m echanical and electrical noise (2.4 GHz VCO).
approxim ately 20 kHz an d 400 kHz. the phase noise decreases a t a rate of -3 0 d B /d ec. T he
phase noise decays a t a rate of - 2 0 d B /d ec a t offset frequencies beyond 400 kHz.
As an exam ple, th e phase noise is com puted using equations (6.1) and (6.3) a t 100 kHz
an d 1 M Hz offset frequencies. At offset frequencies below 400 kHz. the mechanical noise
dom inates th e phase noise, and hence a t a LOO kHz offset from the 2.4 GHz carrier, the
phase noise is given by:
£ m (100
, rr x
H z) =
1
1
— x - x
2.2 x 10~29
/2400 M H z \ ~
^ x iq _ ^ 2 x ^ Q t
J
-9 4 .5 d B c / H z
(6.34)
T h e phase noise of th e VCO at a 1 MHz offset from the 2.4 GHz carrier is dom inated
by the electrical circuit noise and is given as shown:
4 x l 0 ~ 21
-(1 -i- 9.7)(0.5fi + 5 f ! ) ( 2\ ° V £ f ) 2
Ce( 1 M H z )
=
— -
(1 .9 V ) l
2
=
-1 2 7 .7 d B c / H z
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(6.35)
109
w here, for simplicity, the differential p air was assumed to be fully switching w ith 50 percent
du ty cycle.
6 .4 .3
E x p e r im e n ta l R e s u lts
T h e complete circuit schem atic of the 2.4 GHz CMOS VCO is shown in Fig. 6.11.
T ran sisto rs iVfg-.Wl3 provide th e bias for the oscillator core and the o u tp u t buffers. The
oscillator consists of the negative resistance formed by the NMOS tran sisto rs M \-M i and
the LC tan k formed by th e m icrom achined tunable capacitors denoted as C D and the
bo n d in g wire inductors denoted as L b - T he output buffer consists of source followers XI-\.Vf4 an d th e cascoded differential p air am plifier formed with the NMOS tran sisto rs XU-XI*.
R esistors R \-R .2 are on-chip 50
resistors. Table 6.4 shows the sizes of th e MOS transistors
show n in Fig. 6.11.
1
Transistors
Size
Transistors
X I i - A‘1'2
16 x 10.5 n m / 0.6 /rm
m9
Size
|
_____ _______________ jI
2 x 10.5 ii m i 0.6 ^im i
. .1
XU-XL
8 x 10.5 jum /0.6 fj.m
XI 10
1
16 x 10.5 \j.m l0.6 /xm |
Xfr>-XI$
16 x 10.5 \ i m f 0.6 fim
XI12
1
24 x 10.5 /x m /0.6 p m j
XI--XL
8 x 10.5 (j.m j0.6 fjm
XI[[ and XI 1 3
12 x 10.5 /x m /0.6 /xm j
i
Table 6.4: Sum m ary of th e MOS transistors sizes in the 2.4 GHz VCO.
T h e 2.4 GHz VCO has been fabricated in a 0.5 /xm CMOS process an d micromachined
tu n a b le capacitors have been fab ricated in a MUMPs surface polysilicon micromachining
process. T he VCO has been assem bled by bonding together the M UM Ps an d CMOS dice
to g eth er in an (M S I 5P20M) ceram ic quad flat-pack package. The m icrophotograph of the
packaged VCO is shown in Fig. 6.12.
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110
-r
______ O r
i
V
’TUNE
M U M PS
die
CD
O
l
C
L
g
“
R,
!
CD
R,
U op v°n
M,
'rrjX'—
CM OS
die \
- Vcc
* -
O
'— l [ M7
^
'--------------
j
r
i BIAS»-
H
m
J M-.
M,
i-
4
Ms
<F;
r
<r M„
;1>M 10
ill
12
.r
^M
13
- V ,EE
Figure 6.11: C om plete circuit schem atic of th e 2.4 GHz VCO.
mmmmmmmma
Figure 6.12: M icrophotograph of the 2.4 GHz VCO.
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.*•
I ll
All m easurem ents have been perform ed using an HP 4352B V C O /P L L analyzer. The
m easurem ents also include th e effects of the M /A -C O M (ETC1.6-4-2-3) 4:1 transm ission
line transform er, which is used to perform differential to single-ended conversion. Fig. 6.13
shows the m easurem ent test setup for the 2.4 GHz VCO. T h e VCO draws an average
quiescent current of 20 mA while operating from a 2.7 V power supply where the VCO core
and th e o u tp u t buffers consum e 5 mA and 15 mA of current, respectively. The oscillation
frequency o f th e pro to ty p e VCO has been set to 2.4 GHz by phase locking the VCO to a
low phase noise frequency reference w ith the internal narrow band PLL of the HP 4352B
analyzer. T he o u tp u t sp ectru m of the VCO is shown in Fig. 6.14. T he power at th e o u tp u t
of the VCO test setu p is —14.3 dBm.
T he results of th e phase noise measurement are shown in Fig. 6.15. At a 100 kHz
offset from th e 2.4 GHz carrier, the measured phase noise is —93.1 dB c/H z while a t a
1 MHz offset from th e carrier th e m easured phase noise is —122.4 dB c/H z. T he phase noise
decreases at ab o u t —30 dB c/d ec a t offset frequencies above 10 kHz and below 400 kHz.
while a t offset frequencies below 10 kHz and above 400 kHz. th e phase noise decreases at
a ra te of approxim ately - 2 0 dB /dec. T he 2.4 GHz VCO m eets the requirem ents of the
IE E E 802.11 direct sequence wireless LAN standard, which requires th a t the phase noise
a t a 100 kHz offset from a 2.4 GHz carrier be b etter th a n —90 dB c/H z [23].
T he tu n in g characteristics o f the 2.4 GHz CMOS VCO axe shown in Fig. 6.16. T he
oscillation frequency is tu n ab le from 2404 MHz to 2324 MHz. w ithin a control voltage range
of 6 V. T he tu n in g range is hence approxim ately 3.4 percent.
T he sim ulated and m easured results are shown in T able 6.5.
The sim ulated and
m easured results such as th e oscillation frequency and power consum ption agree fairly well.
T he difference in th e o u tp u t power of the experim ental VCO is partially a ttrib u ted to the
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112
100 pF
HP 4352B
VCO/PLL
Analyzer
OP
VCO
ON
100 pF
CC
BIAS
C ontrol V oltage
TUNE
S up p ly V oltage
R BIAS
F igure 6.13: M easurem ent test setup for the experim ental 2.4 GHz VCO.
S p e c tru *
10 d8 / REF 0 d8 »
flfc
CENTER
2 . A GHZ
RBU 3 kHz
VBU
LO
2 .3 7 6
GHz
300 Hz
SUP
1 8 .0 9 s e c
SPAN
10 MHz
Figure 6.14: M easured o u tp u t spectrum of the 2.4 GHz VCO.
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L13
P h a s e H o is e
- 1 2 2 .4 5 dBc
20 d 8 / REF 0 dBc
ExR
START
1 kHz
LO
NBU 1 Hz
2 .3 7 6 GHz
SUP
2 .3 0 8 s e c
STOP
1 nHz
Figure 6.15: Phase noise measurement of the 2.4 GHz VCO.
F re q ue n cy
2 .-1 0 4 3 7 5 GHZ
20 H H z/ REF 2 .3 6 GHz
ExR
Id
ATM 10 dB
START 2 . 7 V
FRES
1 kHz
SUP
120 s e c
Figure 6.16: M easured tuning characteristics of the s 2.4 GHz VCO.
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loss in the 4:1 transform er, which has an operating frequency range of 500 MHz-2500 MHz
and has a specified maxim um insertion loss of 3 dB. The assembled VCO achieves a tuning
range o f only 3.4 percent within 6 V control voltage well below the sim ulated 9.3 percent.
The 1.4 pF micromachined tunable capacitor has a measured tu n in g range of 1.5:1 w ithin a
5 V control voltage. The difference in the sim ulated and m easured tuning range of the VCO
is a ttrib u te d to th e change in tuning characteristics of the 1.4 pF m icrom achined tunable
capacitor as a result of wire bonding of the MUMPs and CM OS dice. The com puted and
m easured phase noise numbers differ by approxim ately 5 dB. T h e disagreem ent is a ttrib u ted
to the fact th a t p a rt of the noise generated in the bias circuit is in practice converted to
phase noise because of non-ideal switching of the differential pair.
Sum m ary
Simulated
M easured
Nominal Frequency
2.35 GHz
2.4 GHz
T uning Range
9.3%
3.4%
P hase Noise @ 1 MHz
-127.7 dB c/H z
-122.5 dB c/H z
O u tp u t Power
-7.8 dBm
-14.3 dBm
Power Supply
2.7 V
2.7 V
Supply C urrent
21 mA
20 mA
j
j
1
1
Table 6.5: Com parison of sim ulated and measured results (2.4 GHz VCO).
T he m easurem ent results are sum m arized in Table 6.6 for different bias conditions.
T h e VCO operates a t supply currents below 17 mA, but the o u tp u t level of the VCO test
setup is then insufficient to drive the HP 4352B analyzer directly.
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115
! Supply Current
17 mA
20 mA
23 mA
26 m A
j O u tp u t Power [dBm]
-15.7
-14.3
-13.2
-12.2
Phase Noise @ 100 kHz [dBc/Hz]
-93.3
-93.1
-92.7
-92.2
P hase Noise
-123.70
-122.5
-125.2
-123.8
j
|
1 MHz [dBc/Hz]
Table 6.6: Sum m ary of the m easurem ent results (2.-1 GHz VCO).
6.5
S u m m ary
Microwave voltage controlled oscillators, based on m icrom achined tu n ab le capacitors
and bonding wire inductors, have been dem onstrated. The m icrom achined tun able capac­
itors have been fabricated in a M UM Ps surface polysilicon m icrom achining process while
the experim ental oscillators have been fabricated in a 0.5 p m CMOS process. T he CMOS
and M UM Ps dice were assembled using wire bonding and were packaged in a stan d ard quad
flat-pack ceram ic package.
T h e 1.9 GHz VCO achieves a tun in g range of 9.0 percent an d a phase noise of
-1 2 6 d B c/H z a t a 600 kHz offset from th e carrier. The VCO meets th e requirem ents of
the DCS1800 system th a t requires a phase noise of —125 dB c/H z a t a 600 kHz offset from
the carrier. W hile op eratin g from a 2.7 V power supply, the VCO and th e o u tp u t buffers
consume 15 m W an d 30 mW of power, respectively. Moreover, the VCO can operate at
power supply voltages as low as 2 V. however, at the expense of phase noise performance.
T he VCO an d th e o u tp u t buffers consum e 2.3 mW and 4.7 mW of power, respectively, when
operating from 2 V power supply. At 2 V. the VCO achieves a phase noise of —120.4 dB c/H z
a t 600 kHz offset.
T he 2.4 GHz oscillator has a tu ning range of 3.4 percent a n d a phase noise of
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—122.5 d B c/H z a t a 1 MHz offset from the carrier. The VCO satisfies the specifications of
the IE E E 802.11 direct sequence wireless LAN standard. T he VCO operates from a 2.7 V
power su p p ly an d consum es an average quiescent current of 20 mA where 5 mA is consum ed
in the core o f th e VCO and 15 mA is consumed in the o u tp u t buffers.
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117
C h a p ter 7
C o n clu sio n s and F u tu re D irection s
In this thesis, the design and fabrication of m icromachined electro-mechanically tu n ­
able capacitors in surface poiysilicon m icrom achining technology has been dem onstrated.
T h e ap p licatio n of micromachined electro-m echanically tunable capacitors to microwave LC
V C O s in CM OS IC technology has also been dem onstrated. The analysis of oscillator noise
due to electrical and mechanical noise sources has been performed and analytical theoretical
resu lts have been developed.
T h e m icrom achined electro-m echanically tu n ab le capacitors have been fabricated us­
ing surface polysilicon surface m icrom achining process.
Two- and three-plate m icrom a­
chined tu n ab le capacitors have been investigated. T he two-plate micromachined tunable
cap a cito r w ith 2.05 pF nominal capacitance achieves a tuning range of 1.5:1 given a 4.0 V
m axim um bias voltage and a Q-factor of 20 a t 1 GHz. T he three-plate tunable capacitor
features a nom inal capacitance of 4.0 pF. a tu n in g range of 1.87:1 w ithin a 4.4 V maxim um
control voltage, and a Q-factor of 15.4 a t 1 GHz. D espite the fact th a t polysilicon is used
as th e electrical m aterial, these experim ental devices achieve sufficient ^-factor for VCOs
of such wireless system standards as GSM , DCS1800, D ECT, and IEEE 802.11.
Microwave VCOs at 1.9 GHz an d 2.4 GHz based on the micromachined tunable
cap acito rs have been also investigated.
T h e 1.9 GHz VCO achieves a tuning range of
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118
9.0 percent and a phase noise of —126 dB c/H z at a 600 kHz offset from the carrier. The
VCO and o u tp u t buffers consum e 15 mW and 30 mW of power, respectively, from a 2.7 V
power supply.
The 1.9 GHz VCO meets the phase noise requirem ents of the DCS1800
cellular telephone stan d ard discussed in C hapter. 1. The 2.4 GHz oscillator has a tuning
range of 3.4 percent and a phase noise of —122.5 dB c/H z at a 1 MHz offset from the carrier.
T he VCO operates from a 2.7 V power supply and consumes an average quiescent current
o f 20 mA. T he 2.4 GHz VCO satisfies the phase noise specifications of the IEEE 802.11
direct sequence wireless LAN stan d ard .
The effect of electrical circuit noise on the oscillator phase noise has been exam ined
using linear tim e-invariant an d linear periodically time-varying models of an oscillator. An
analytical expression has been developed for the output PSD of an oscillator consisting of
an .Vff/l-order frequency selective filter, a w hite noise source input, and a com parator. The
effect of mechanical noise in m icromachined tunable capacitors on the oscillator phase noise
has also been investigated. T heoretical predictions agree relatively well w ith the phase noise
m easurem ents of 1.9 GHz an d 2.4 GHz CMOS VCOs.
M icromachined electro-m echanically tunable capacitors may in the future find use in
such microwave applications as tunable m atching networks, electronically tunable filters,
and voltage controlled oscillators. These devices are especially im p o rtan t since both a high
Q-factor an d a wide tu n in g range are achievable. Wide tuning range is necessary not only
to provide sufficient tu n in g to cover the desired frequency band b u t also to accommodate
process variations. High Q-factor is required, on the other hand, to realize tunable filters
w ith high dynam ic range and voltage controlled oscillators w ith low phase noise. F urther­
more. unlike the integrated p-n ju n ctio n varactors. micromachined tunable capacitors can
sustain large voltage swings and are not expected to produce much harm onic content. Since
with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission
119
m icrom achined tunable capacitors do not respond to frequencies beyond their resonant fre­
quencies. the capacitance of these micromechanical devices will not change w ith respect
to microwave signals. T he lack of harm onic content w ith respect to microwave signals is
p articu larly advantageous in microwave filter applications. A lthough these devices are sen­
sitiv e to excitations w ithin th eir mechanical resonant frequencies, when used in a voltage
controlled oscillator application, a wide band phase lock loop can be employed to correct
for such low frequency excitations. In the microwave filter applications, an au tom atic filter
tu n in g loop could in principle accomplish the same task.
Anchors
Suspension
Anchor
Cross
Section
Suspended Plate
Nitnde
H
Anchor
Figure 7.1: Top and cross section views of a differential tunable capacitor.
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F uture research on micromachined tun ab le capacitors may include the optim ization of
the suspension design as to minimize the effects of residual stress. The effect of tem p eratu re
on th e Young‘s m odulus of polysilicon could also be investigated, especially since the existing
surface polysilicon micromachining foundries currently do not provide such data.
Since a significant am ount of loss in micromachined tunable capacitors comes from
the series resistance of the suspension beam s, different suspension designs th a t can achieve
an equivalent spring constant but a much lower series resistance could be investigated in the
future. Furtherm ore, a com pletely different stru ctu re for micromachined tu n ab le capacitor
design could be employed. For exam ple, th e stru ctu re in Fig. 7.1 avoids entirely the flow of
current th ro u g h through the suspension beam s [111]. Although in general th e bo ttom plate
parasitic is qu ite undesirable, the effect of parasitics can be nullified if such a m icromachined
tunab le capacitor was to be used in an application where the bottom p la te parasitic would
be at a sm all-signal ground.
A ltern ativ e means of tuning the micromechanical capacitor could b e investigated as
well. For exam ple, a capacitive comb drive [106] could be used to change eith er the overlap
area of th e capacitor plates or the distance between capacitor plates.
M oreover, micromachined electro-m echanically tunable capacitors can be used in mi­
crowave tu n ab le filters. Such research could be of particular im portance if a filter w ith a
high dynam ic range by taking advantage of linearity could be d em onstrated.
A lthough integrated technologies where b oth CMOS and polysilicon surface m icrom a­
chined devices can reside on the same die are in existence today, such technologies feature
very few m echanical layers and often do not provide fast enough tran sisto rs for microwave
applications. However, as CMOS technologies scale down, sub-m icron CM OS technologies
w ith in teg rated polysilicon surface m icrom achining processes may possibly be available in
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m
th e future, and fully monolithic microwave integrated circuits th a t can take advantage of
m icrom echanical com ponents m ay become feasible in the future. For example, a fully mono­
lithic microwave VCO th a t utilizes a m icrom achined electro-m echanically tunable capacitor
as the tu n in g element has yet to be dem onstrated. Since currently only wire bonding or
flip chip technologies can be used to integrate these microwave devices, a fully m onolithic
solution would remove the necessity for bonding pads and hence a much larger tuning range
due to sm aller parasitics could be achieved. In addition, the unpredictable mechanical ef­
fects of wire bonding on the behavior of these sensitive micromechanical devices could be
avoided.
T he application of polysilicon surface m icromachining technology to the design and
fabrication of micromachined microwave switches could also be investigated in the future.
Since polysilicon surface m icrom achining technology such as MUM Ps has one gold layer, a
Figure 7.2: C onceptual diagram of a m icromachined switch.
m icrom achined electrostatically controlled microwave switch th a t achieves low insertion loss
and high isolation can possibly be realized. A simplified diagram of such a micromachined
electrostatically controlled sw itch is shown in Fig. 7.2.
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122
B ib liograp h y
:1] D. Bursky. "M egagate integration is creating systems on a chip." Electronic Design.
vol. 43. no.22. pp. 48. 50. 52-4. Oct. 1995.
[2] R. Alini. G. B etti. R. Castello. F. Heydari. G. M aguire. L. Fredrickson. L. Voltz.
and D. Stone. "A 200 M Sam ples/s trellis-coded PRM L read /w rite channel w ith dig­
ital serv o /’ in IE E E ISSC C Dig. Tech. Papers, pp. 318-319. Feb. 1998.
[3] T. S tetzler. I. Post. J. Havens. M. Koyama. '’A 2.7-4.5 V single chip GSM transceiver
R p integrated circuit." IE E E J. Solid-State Circuits, vol. 30. no. 12. pp. 1421-1429.
Dec. 1995.
r4j L. Larson. R F and Microwave Circuit Design fo r Wireless Communications. Artech
House. Boston-London, pp. 20-21, 1996.
[5] M. T h am sirian u n t and T. Kwasniewski, ’A 1.2 p.m CM OS im plem entation of a lowpower 900-MHz mobile radio frequency synthesizer.-’ in IE E E Proc. CICC, pp. 383-386.
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13 4
A p p e n d ix A
In d u c ta n c e C o m p u ta tio n v ia G reen h ou se
A .l
In d u cta n ce Form ulas
T h e in ductance of a planar spiral inductor can be calculated by adding the self in­
ductances of each section of the inductor and adding all m utual inductances between all
parallel sections of the inductor as shown:
L = Y . L' + H Mi*
i
(A -1}
i.k
where L, is th e self-inductance of an i-th conductor and Af,,* is the m utual inductance be­
tween th e i-th and k-th conductor. Fig. A .l shows two parallel conductors w ith rectangular
cross-sections. T he self-inductance in microhenries of a single conductor is given as follows:
L = 0 002 • I ( l n ( ——— ) + 0.50049 +
\
W -r t
Oi
/
(A.2)
where I. w. a n d t are the length, w idth, and thickness of the conductor in centim eters [64].
T he m u tu al inductance in microhenries between two parallel conductors w ith equal length
I which are sep arated by distance d is given by [64]:
M = 2IQi
(A.3)
where
Ql
~
I
ln{G M D ] *
I2
+ GMD*] ~
V1
I2
GMD
+ GMD2 +
I
V1
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
.
(
x
}
135
o.
T
o
o
W
F ig u re A .l: Two parallel conductors w ith rectangular cross-section.
and
I n ( G M D ) = ln(d) -
- 6 0 (£ )4 ~ 168(A)6 " 3 60(£)8 ~ 6 6 0 (£ )10
(A '0)
T he m utual in d u ctan ce M ( j . m ) between two parallel conductors of unequal lengths j and
m. as shown in Fig. A .l. when p ^ q. p = q. and p = 0. is given as follows [64]:
M (j.m)
= ( m + p ) Q m^ p - ( m + q ) Q m+<i ~ p QP ~ <lQq i t p £ <1
(A.6)
M (j, m )
= 2(m -f- p ) Q m~ P - 2p Q p if p = q
(A .7)
M{j. m)
= j Q j 4- m Q m — q Q q if p = 0
(A .8)
Since in d u ctan ce form ulas for arb itrarily positioned conductors of arb itrary length have
been developed [65], inductors of various geom etries can be analyzed in sim ilar fashion [66][68]. However, th e program th a t follows takes into account only parallel conductors, uses
(A .1)-(A .8). a n d hence accom m odates only square and rectangular spiral inductors.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
136
A .2
In d u cta n ce C alcu lation P rogram
T h is p ro gram c a l c u l a t e s t h e i n d u c t a n c e , p a r a s i t i c r e s i s t a n c e ,
p a r a s i t i c c a p a c i t a n c e , s e l f - r e s o n a n t f r e q u e n c y , maximum Q, and t h e
f r e q u e n c y o f maximum Q. The i n d u c t o r s t r u c t u r e c o n s i s t s o f s q u a r e
s p i r a l m e t a l s e g m e n ts . T h is program was v e r i f i e d w ith examples from
G r e e n h o u s e , and Nguyen.
w r i t t e n by, A l e k s a n d e r Dec
Columbia U n i v e r s i t y , New Y ork, NY
March 4 t h , 1995
=====================================================================
^ in clu d e
# in clu d e
# in clu d e
^in clu d e
« /
< std io .h >
< m ath. h>
< std lib .h >
< m a llo c .h >
# d e f i n e MAX_NUM 500
# d e f i n e MAX.P 10
# d e f in e PI 3 .1 4
d o u b l e CM, RM, RP, LAMBDA, L I, L2, W, T, S, N;
d o u b l e BEGIN [MAX.P] , STOP[MAX.P], STEP[MAX.P];
d o u b l e GOAL, TOL;
i n t FLAG1, XGRAPH, LAYOUT, VARY.INDEX, VARY[MAX.P];
T h i s f u n c t i o n r e t u r n s e r r o r m e s s a g e s . Type o f t h e e r r o r message
d e p e n d s on f l a g s t h a t a r e t o be p a s s e d when t h e f u n c t i o n i s c a l l e d .
I f t h e r e i s an e r r o r , t h i s f u n c t i o n w i l l e x i t program .
======================================================================
void e r r o r _ c o d e ( x ,y ,z )
in t x ,y ,z ;
{
i f (x == 1) {
p r i n t f (" USAGE: s p i r a l < f i l e n a m e > \ n \ n " ) ;
p r i n t f (" The i n p u t f i l e must c o n t a i n t h e f o l l o w i n g p a r a m e t e r s : \ n \ n " ) ;
< m etal 1 p a r a s i t i c c a p a c i t a n c e i n fF/um *2>\n")
p r i n t f (" CM1
< m etal 2 p a r a s i t i c c a p a c i t a n c e i n fF/um *2>\n")
p r i n t f (" CM2
< m etal 3 p a r a s i t i c c a p a c i t a n c e i n fF/um *2>\n")
p r i n t f (" CM3
< m etal 1 r e s i s t a n c e i n O h m s/sq u are > \n ")
p r i n t f (" RM1
< m etal 2 r e s i s t a n c e i n Q hm s/sq u are > \n ")
p r i n t f (" RM2
<m etal 3 r e s i s t a n c e i n O hm s/sq u are > \n ")
p r i n t f (" RM3
p r i n t f (" LAYER < m etal l a y e r > \ n " ) ;
< len g th in m ic r o n s > \n " ) ;
p r i n t f (" LI
< l e n g t h in m i c r o n s > \ n " ) ;
p r i n t f (" L2
< t r a c k w id th i n m i c r o n s > \ n " ) ;
p r i n t f (" W
< m etal t h i c k n e s s i n m i c r o n s > \ n " ) ;
p r i n t f (" T
< s p a c in g betw een t r a c k s i n m i c r o n s > \ n " ) ;
p r i n t f (" S
<number of t u m s > \ n " ) ;
p r i n t f (" N
with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission
« /
137
p r i n t f (" RP
p r i n t f (" FLOAT
< su b strate re s is ta n c e > \n ”) ;
<f l o a t i n g / g r o u n d e d i n d u c t o r > \ n \ n " ) ;
>
e x i t Cl) ;
}
/«
=======================================================================
T his f u n c tio n p r i n t s out th e le n g h ts of a l l lin e seg m en ts.
=======================================================================
. /
v o id p r in t_ o u tC lin e )
d ouble * l i n e ;
{
in t
2
,i ;
z = C in t) l i n e [ 0 ] ;
f o r C i = l ; i< = z ;i+ + )
p r i n t f C"Line [’/,d] = 7,f \ n " , i , l i n e [ i ] ) ;
>
/*
=======================================================================
T h is f u n c t i o n p r i n t s a h e a d e r .
======================================================================
void p rin t_ h e a d C fo o l,fo o 2 ,d )
char f o o l [ 1 0 ] ,foo 2 [1 0 ];
i n t d;
{
if
CXGRAPH == 0)
i f Cd == 1) {
p r i n t f C " '/ . s \ t L \ t R \ t C \ t f _ r e s \ t f _ q m a x \ t O m a x \ n " , f o o l ) ;
p r i n t f C" 7 ,s \ t [nH] \ t [ohm] \ t [pF] \ t [GHz] \ t [GHz] \ t \ n " , f o o 2 ) ;
>
e lse {
p r i n t f C "L \tR \tC \tf_ re s\tf_ q m a x \tQ m a x \n " );
p r i n t f C” [ n H ] \ t [ o h m ] \ t [ p F ] \ t [ G H z ] \ t [ G H z ] \ t \ n " ) ;
>
>
e lse {
i f CXGRAPH == 1) {
p r i n t f C ' t i t l e t e x t : I n d u c t a n c e v s '/,s\n" , f o o l ) ;
p r i n t f C x u n i t t e x t : )Cs, [ % s ] \ n " , f o o l , f o o 2 ) ;
p r i n t f C ’y u n i t t e x t : L, [ n H ] \ n " ) ;
>
if
CXGRAPH == 2) {
p r i n t f C’t i t l e t e x t : R e s i s t a n c e v s V.sNn" , f o o l ) ;
p r i n t f C x u n i t t e x t : 7.s, [*/.s]\n" , f o o l , foo2) ;
p r i n t f C ’y u n i t t e x t : R, [ o h m s ] \ n " ) ;
}
if
CXGRAPH == 3) {
p r i n t f C’t i t l e t e x t : C a p a c i t a n c e vs '/,s\n" , f o o l ) ;
p r i n t f C x u n i t t e x t : */.s , [7.s] \ n " , f o o l , f o o 2 ) ;
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
« /
138
p r i n t f C y u n i t t e x t : C, [ p F ] \ n " ) ;
>
if
(XGRAPH == 4) {
p r i n t f ( " t i t l e t e x t : S e l f R eso nant F req u e n cy vs '/,s\n" , f oo 1) ;
p r i n t f C x u n i t t e x t : 7.s, [7.s] \ n " , f o o l , fo o2 ) ;
p r i n t f C y u n i t t e x t : f _ r e s , [GHz]\nM) ;
>
if
(XGRAPH == 5) {
p r i n t f ( " t i t l e t e x t : Maximum Q F requency vs 7.s\n" , f o o l ) ;
p r i n t f C x u n i t t e x t : '/.s, C’/ .s ] \ n " , f o o l , fo o 2) ;
p r i n t f C y u n i t t e x t : f_qmax, [G H z]\n");
>
if
(XGRAPH == 6) {
p r i n t f ( " t i t l e t e x t : Maximum Q vs 7.s\n" , f o o l ) ;
p r i n t f C x u n i t t e x t : V.s, [ ' X s ] \ n " , f o o l , f o o 2 ) ;
p r i n t f C y u n i t t e x t : Q \n") ;
>
>
>
/*
======================================================================
T h is fu n c tio n in d u cto r sp ec s.
=======================================================================
. /
v o id p r i n t _ s p e c s ( a , b , c , d , e , f , g , f o o )
d o u b le a , b , c , d , e , f , g ;
in t foo;
if
((XGRAPH = = 0 )
II (VARY_INDEX == 0 ))
{
i f ( f o o == 1)
p r i n t f ( "7.. 3 f \ t 7 . . 3 f \ t 7 . . 3 f \ t 7 . . 3 f \ t 7 . . 3 f \ t 7 . . 3 f \ t 7 . . 3 f \ n " , a , b , c , d , e , f , g ) ;
else
p r i n t f ("7. - 3 f \t7 . - 3f \ t 7 . . 3 f \ t 7 . . 3 f \ t 7 . . 3 f \ t 7 . . 3 f \ n " , b , c , d , e , f , g) ;
>
else
i f (XGRAPH == 1) p r i n t f ("*/..3 f \ t 7 . . 3 f \ n " , a , b ) ;
i f (XGRAPH == 2) p r i n t f ("*/..3 f \ t 7 . . 3 f \ n " , a , c ) ;
if
(XGRAPH== 3) p r i n t f ("7 ..3 f \ t 7 . . 3 f \ n " , a , d ) ;
if
(XGRAPH== 4) p r i n t f C 7 . - 3 f \ t 7 . . 3 f \ n " , a , e ) ;
if
(XGRAPH== 5) p r i n t f ( " 7 . - 3 f \ t 7 . . 3 f \ n " , a , f ) ;
if
(XGRAPH== 6) p r i n t f ( " 7 . . 3 f \ t 7 . . 3 f \ n " , a , g ) ;
}
>
/ * ======================================================================
T h i s f u n c t i o n r e t u r n s t h e minimum of two num bers.
=======================================================================
d o u b l e minimum(x, y)
d ouble x , y ;
{
d ou b le r e s u l t ;
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
, /
139
if
(x >= y)
r e s u l t = y;
else
r e s u l t = x;
re tu rn ( r e s u l t) ;
>
/* =======================================================================
T h i s f v i n c t i o n f i g u r e s o u t maximum p o s s i b l e number o f t u r n s .
====================================================================== , /
d o u b le m ax _ tu rn ()
d o u b le m in , r e s u l t ;
m in = minimum(L1 , L2) ;
r e s u l t = (min - Vf)/(2 . 0 * (W + S ) ) ;
re tu rn ( r e s u l t ) ;
>
/ * =======================================================================
T his f u n c tio n p r i n t s ou t th e g l o b a l v a r i a b l e s .
=======================================================================
«/
v o id p r i n t _ s t a t ( )
if
(XGRAPH == 0)
{
p rin tf("\n ");
p r i n t f (" L I = ,/ , . 3 f \ t L 2 = 7..3f\tW = 7 . . 3 f \ t S = 7 . . 3 f \ n ” , L l * l e 4 , L 2 * l e 4 , V * l e 4 , S * l e 4 ) ;
p r i n t f ("T = 7 ..3 f\tN = ’/ i.3f\tWmax = 7..3 f \ n " , T * l e 4 ,N , m a x _ t u m ( ) ) ;
p rin tf("\n ");
>
>
/«. =======================================================================
T h i s f u n c t i o n f i g u r e s o u t a l l t h e segm ent l e n g t h s and s t o r e s t h e
r e s u l t i n an a r r a y .
======================================================================= «/
v o i d i n i t i a l i z e ( l i n e , z)
do u b le “l i n e ;
i n t z;
■C
i n t y;
l i n e [ 0 ] = (d o u b le ) z;
l i n e [1] = LI - W;
l i n e [2] = L2 - W;
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
140
f o r (y=2; y<=z; y++) {
l i n e [ 2 * y ] = l i n e [ l ] - (y - 1) * (W + S) ;
l i n e [ 2 * y - 1] = l i n e [2] - (y - 2) * (W +■ S ) ;
}
>
/ * =======================================================================
T h i s f u n c t i o n c a l c u l a t e s and r e t u r n s t h e s e l f i n d u c t a n c e o f t h e
s i n g l e l i n e s e g m e n t.
======================================================================= «/
d ouble s e l f _ i n d ( l )
d o u b l e 1;
{
d o u b le f o o l , f o o 2 , f o o 3 , r e s u l t ;
f o o l = (1 * 2) / (W + T) ;
f oo2 = (W + T) / (3 * 1) ;
fo o 3 = 0 .0 0 2 « 1;
r e s u l t = fo o 3 * ( l o g ( f o o l ) + 0 .5 0 0 4 9 + foo 2 )
re tu rn ( r e s u l t ) ;
>
/ * ======
T his f u n c tio n c a lc u l a t e s th e t o t a l s e l f in d u c ta n c e .
o f e a c h l i n e segm ent i s c a l c u l a t e d and t h e n a d d e d .
The c o n t r i b u t i o n
======================================================================= «/
doub le s u m _ s e lf(lin e )
d o u b le * l i n e ;
{
double r e s u l t ;
i n t y ,z;
z = (in t) lin e [ 0 ] ;
r e s u l t = 0;
f o r ( y = l;y < = z ;y + + )
r e s u l t = r e s u l t +■ s e l f _ i n d ( l i n e [y] ) ;
r e s u l t = 1000 « r e s u l t ;
re tu rn ( r e s u l t ) ;
>
/ . ======================================================================
T h i s f u n c t i o n c a l c u l a t e s t h e mean g e o m e t r i c d i s t a n c e .
======================================================================= . /
d o u b l e g m d (d ,v )
d o u b le d , w;
{
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm issio n :
141
d o u b le f o o , r e s u l t ;
foo = d/w;
r e s u lt = lo g (d )
re su lt = r e s u lt
re su lt = re s u lt
re su lt = re s u lt
re su lt = r e s u lt
re su lt = re s u lt
;
-
(1 .0 /1 2 .0 )
(1 .0 /6 0 .0 )
(1 .0 /1 6 8 .0 )
(1 .0 /3 6 0 .0 )
(1 .0 /6 6 0 .0 )
* p o w (fo o , - 2 . 0 ) ;
* p o w (fo o , - 4 . 0 ) ;
* p o w (fo o , - 6 . 0 ) ;
* p o w (fo o ,-8 .0 );
* p o w (fo o , - 1 0 . 0 ) ;
re su lt = e x p ( re s u lt);
re tu rn ( r e s u l t ) ;
>
/* ========■========================================================
T h is f u n c t i o n c a l c u l a t e s t h e Q, a q u a n t i t y w hich i s needed when
c a l c u l a t i n g m u tu a l i n d u c t a n c e .
d o u b le q _ f u n c ( l , g )
do u b le 1 , g;
{
d o u b le f o o l , f o o 2 , f o o 3 , fo o 4 , r e s u l t ;
fo o l
foo2
foo3
foo4
=
=
=
=
1 /g ;
s q r t (1 + p o w ( ( 1 / g ) , 2 . 0 ) ) ;
s q r t ( l + p o w ((g /1 ),2 .0 )) ;
g /1 ;
r e s u l t = l o g ( f o o l + fo o 2) - foo3 + fo o 4 ;
re tu rn ( r e s u lt) ;
>
/« =======================================================================
T h is f u n c t i o n c a l c u l a t e s t h e m utu al i n d u c t a n c e b etw een two w i r e s .
======================================================================= «/
d o u b le m u t u a l ( m , p , q , d i s t , w )
d o u b le m , p , q , d i s t , w ;
d o u b le m _p lus_ p= 0, m _plus_q=0, m_p=0, m_q=0;
d o u b le f o o , r e s u l t ;
foo = g m d ( d i s t . w ) ;
m_plus_p = (m + p) * q _ f u n c ( ( m + p ) , f o o ) ;
m _plus_q = (m + q) * q _ f u n c ((m + q ), f o o ) ;
i f (p != 0) m_p = p * q _ f u n c ( p , f o o ) ;
i f (q != 0) m_q = q * q _ f u n c ( q , f o o ) ;
r e s u l t = m _plus_p + m _plus_q - m_p - m_q;
re tu rn ( r e s u l t ) ;
>
/* =======================================================================
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
T h i s f u n c t i o n adds a l l p o s i t i v e m u tu a l i n d u c t a n c e s and r e t u r n s
th e r e s u lt .
C u r r e n t i s fl o w i n g i n o p p o s i t e d i r e c t i o n .
d o u b le su m _ m u t_ p o s (lin e )
double * lin e ;
d o u b le r e s u l t ;
d o u b le m , p , q , d i s t ;
i n t y , n , z , f o o l , fo o 2 ,fo o 3 ;
z = (in t) l in e [ 0 ] ;
fo o 2 = ( i n t ) (z / 4 ) ;
r e s u l t = 0;
fo r(n = l;n< = foo 2;n + + ) {
f o o l = z - 4 * n;
f o r ( y = l ; y < = f o o l; y + + ) {
m = l i n e [ y + 4*n] ;
p = (W
S) » n;
q = (W + S) * n;
d i s t = (W + S) * n;
foo3 = W;
r e s u lt = r e s u lt + m u tu a l(m ,p ,q ,d is t,fo o 3 );
>
>
re s u lt = 2 * re s u lt;
re tu rn ( r e s u l t ) ;
>
/« ===============================================================
T h i s f u n c t i o n adds a l l p o s s i b l e n e g a t i v e m u tu a l i n d u c t a n c e s and
r e t u r n s t h e r e s u l t . C u r r e n t i s f l o w i n g i n t h e same d i r e c t i o n .
d o u b l e su m _ m u t_ n e g (lin e )
do u b le * lin e ;
d o u b le r e s u l t ;
do u b le m , p , q , d i s t ;
in t y ,n ,z ,fo o l,fo o 2 ,fo o 3 ;
z = (in t) lin e [0 ];
fo o 2 = ( i n t ) (z / 4 ) ;
r e s u l t = 0;
fo r(n = l;n< = fo o2;n+ + ) {
f o o l = z - 4 * n + 2 ;
f o r ( y = l ; y < = f o o l; y + + ) {
m =l i n e C y + 4*n
- 2];
p =
(W+ S) « n;
q =
(V + S) - (n - 1) ;
d i s t = l i n e [ y +4*n - 3] +■ (n - 1) * (W + S) ;
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited without p erm issio n .
143
fo o 3 = W;
r e s u l t = r e s u l t +■ m u t u a l ( m , p , q , d i s t , f o o 3 ) ;
>
>
re s u lt = 2 * re s u lt;
re tu rn ( r e s u l t) ;
/* =======================================================================
T h i s f u n c t i o n c a l c u l a t e s t h e r e s i s t a n c e t h e i n d u c t o r . T h is l i m i t s
i n d u c t o r Q. The t o t a l l e n g t h i s f i r s t o b t a i n e d and t h e n r e s i s t a n c e
i s c a l c u l a t e d a c c o r d i n g t o t h e f o l l o w i n g e q u a t i o n : R = R sh e e t x L/W
=====================================================================
,/
d o u b le c a l c _ r e s ( l i n e )
d o u b le * l i n e ;
d o u b le r e s u l t , le n g th ;
in t z , i ;
z = (in t) lin e [0 ];
l e n g t h = 0;
f o r ( i = l ; i < = z ; i++)
le n g th = len g th + l i n e [ i ] ;
r e s u l t = ( le n g th /W ) * RM;
re tu rn ( r e s u l t) ;
>
/ . =====================================================================
T h i s f u n c t i o n c a l c u l a t e s p a r a s i t i c c a p a c i t a n c e o f t h e i n d u c t o r . T h is
l i m i t s t h e s e l f r e s o n a n t f r e q u e n c y o f t h e i n d u c t o r . Beyond t h a t
f r e q u e n c y i n d u c t o r i s c a p a c i t i v e . A rea of t h e m e t a l i s f i r s t
c a l c u l a t e d and t h e n c a p a c i t a n c e i s c a l c u l a t e d a c c o r d i n g t o t h e
f o l l o w i n g f o r m u l a : C = C s h e e t x A rea
=======================================================================
d o u b le c a lc _ c a p ( l in e )
d o u b le « l i n e ;
{
d o u b le r e s u l t , le n g th ;
in t z , i ;
z = (in t) lin e [ 0 ] ;
l e n g t h = 0;
f o r ( i = l ; i< = z ; i + + )
le n g th = len g th + l i n e [ i ] ;
r e s u l t = l e n g t h * W * CM * l e 5 ;
re tu rn ( r e s u l t) ;
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited without perm ission.
«/
144
T h i s f u n c t i o n c a l c u l a t e s t h e s e l f r e s o n a n t fr e q u e n c y o f t h e i n d u c t o r .
T h ere a r e two c a s e s h e r e :
(1 )
Grounded I n d u c t o r -> f l a g = 1
(2 )
F l o a t i n g I n d u c t o r => f l a g = 2
The e q u a t i o n u s e d h e r e i s t a k e n from N.Nguyen’s p a p e r on i n d u c t o r s i n
silic o n .
======================================================================= . /
dou ble s e l f _ r e s _ f r e q ( i n d ,c a p ,r e s _ s )
doub le in d ,c a p ,r e s _ s ;
{
do u b le r e s u l t , f o o l , f o o 2 , f o o 3 ;
double c p , l s , r s , r p ;
if
(FLAG1 == 1) {
cp = ( c a p / 2 ) * l e - 1 2 ;
I s = in d * l e - 9 ;
r s = res_ s;
r p = RP;
>
if
(FLAG1 == 2) {
cp = ( c a p / 4 ) * l e - 1 2 ;
I s = in d * l e - 9 ;
rs = re s .s ;
r p = RP * 2;
>
fo o l = l / s q r t( ls * c p );
f o o 2 =1 - p o w ( r s , 2 . 0 ) * ( c p / l s ) ;
fo o 3 =1 - p o w ( r p , 2 . 0 ) * ( c p / l s ) ;
if
( ( f o o 2 / f o o 3 ) > 0)
r e s u l t = fo o l * s q rt( f o o 2 /f o o 3 ) * (1 /(2 » P I)) * le-9 ;
e lse
r e s u l t « 0;
re tu rn ( r e s u lt) ;
>
/« ======================================================================
T h is f u n c t i o n c a l c u l a t e s t h e f r e q u e n c y a t which t h e i n d u c t o r h a s t h e
maximum Q. T h ere a r e two c a s e s h e r e :
(1 )
Grounded I n d u c t o r => f l a g = 1
(2 )
F l o a t i n g I n d u c t o r => f l a g = 2
The e q u a t i o n u s e d h e r e i s t a k e n from N.Nguyen’s p a p e r on i n d u c t o r s in
silic o n .
======================================================================= «/
dou ble m a x _ q _ fre q (in d ,c a p ,re s _ s )
d o u b le i n d , c a p , r e s _ s ;
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
d o u b le r e s u l t , f o o l , f o o 2 , f o o 3 ;
d o u b le c p , l s , r s , r p ;
i f (FLAG1 == 1) {
cp = ( c a p / 2 ) * l e - 1 2 ;
I s = in d * l e - 9 ;
rs = res_s;
rp = RP;
}
if
(FLAG1 == 2) {
cp = ( c a p / 4 ) « l e - 1 2 ;
I s = in d * l e - 9 ;
rs = res_ s;
r p = RP * 2;
>
fool = l / s q r t ( l s * c p );
fo o 2 = s q r t (1 + (4 /3 ) « ( r p / r s ) ) - 1;
fo o 3 = r s / (2 * r p ) ;
r e s u l t = f o o l * s q r t ( f o o 2 * fo o 3 ) * ( 1 / ( 2 * P I ) ) * l e - 9 ;
re tu rn ( r e s u l t) ;
/* ====================================================================
T h is f u n c t i o n c a l c u l a t e s t h e Q of t h e i n d u c t o r a t a g i v e n f r e q u e n c y .
d ouble i n d _ q ( i n d ,c a p ,r e s _ s ,f r e q )
d ouble i n d , c a p , r e s _ s , f r e q ;
{
d o u b le r e s u l t , imag, r e a l , f o o l , f o o 2 , omega;
d o u b le c p , l s , r s , r p ;
if
(FLAG1 == 1) {
cp = ( c a p / 2 ) * l e - 1 2 ;
I s = in d « le - 9 ;
rs = res_ s;
r p = RP;
>
if
(FLAG1 == 2) {
cp = ( c a p / 4 ) * l e - 1 2 ;
I s = in d * le -9 ;
rs = res_s;
r p = RP * 2;
>
omega = 2 * PI « ( f r e q « l e + 9 ) ;
f o o l = p o w (r p , 2 .0 )
fo o 2 = p o w ( r s , 2 .0 )
+ pow (( 1 / (om ega* cp)) , 2 . 0 ) ;
+ pow ((om ega*Is), 2 .0 ) ;
imag = ( 1 / (om ega«cp)) * fo o2 - (om eg a » ls) * f o o l ;
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission
r e a l = rp * foo2 + r s * f o o l ;
r e s u l t = - im a g /re a l;
re tu rn ( r e s u lt) ;
>
/ at
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
T h i s f u n c t i o n r e a d s i n a l l t h e d a t a from t h e i n p u t f i l e .
v o id f i l e . r e a d ( x . y )
i n t x;
c h a r « y [] ;
{
FILE « in ;
c h a r dummy[1 0 ];
i n t fo o ;
d o u b l e cml, cm2, cm3, r m l , rm2, rm3;
/ * *****
I n i t i a l i z e th e g lo b al v a ria b le s
***** * /
CM = 0 .0 ;
RM = 0 .0 ;
RP = 0 .0 ;
LAMBDA = 0 . 0 ;
LI = 0 .0 ;
L2 = 0 .0 ;
V = 0 .0 ;
T = 0 .0 ;
S = 0 .0 ;
T = 0 .0 ;
N = 0 .0 ;
VARY.INDEX = 0;
FLAG1 = 0;
XGRAPH = 0;
LAYOUT = 0;
/ * *****
P arse th ro u g h th e in p u t f i l e
***** * /
i f ( ( i n = f o p e n ( y [ l ] , " r t " ) ) == NULL) e r r o r _ c o d e ( l , 0 , 0 ) ;
w h ile ( ! f e o f ( i n ) )
•C
f s c a n f ( in ,'" /.s " , dummy);
i f (strcasecm p(dum m y, "cm l") == 0) {
f s c a n f ( i n , "’/ .s" .dummy) ;
cml = ato f(d u m m y );
>
if
(strcasecm p(dum m y,"cm 2") == 0) {
f s c a n f ( i n , "7aS" .dummy) ;
cm2 = atof(dum m y);
>
if
(strcasecm p(dum m y, "cm3") == 0) {
f s c a n f ( i n , "’/ .s" .dummy) ;
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
cm3 = a t o f ( d u m m y ) ;
>
i f (strc a s e c m p (d u m m y , " r m l" ) == 0) {
f s c a n f ( i n , "7,s" .dummy) ;
rm l = a t o f ( d u m m y ) ;
>
i f (strc a s e c m p (d u m m y , "rm2") == 0) {
f s c a n f ( i n , "V.s" .dummy) ;
rm2 = a t o f (dummy);
>
i f (strc a s e c m p (d u m m y , "rm3") == 0) {
f s c a n f ( i n , "7.s” .dummy) ;
rm3 = a t o f ( d u m m y ) ;
>
if
(strc asec m p (d u m m y , "lambda") == 0) {
f s c a n f ( i n , "V.s" .dummy);
LAMBDA = a to f ( d u m m y ) ;
>
i f (s trc a s e c m p (d u m m y ," 1 1 " ) == 0) {
f s c a n f ( i n , "7,s" .dummy);
LI = atof(du m m y ) * l e - 4 ;
>
if
(strc a se c m p (d u m m y ," 1 2 " ) == 0) {
f s c a n f ( i n , '"/.s" .dummy) ;
L2 = atof(dum m y) * l e - 4 ;
>
if
( strca secm p (d u m m y , " v ") = = 0 )
f s c a n f ( i n , "V.s" .dummy) ;
W = atof(dum m y) * l e - 4 ;
{
(strc a s e c m p (d u m m y , " t " ) = = 0 )
f s c a n f ( i n , "7,s" .dummy);
T = atof(dum m y) * l e - 4 ;
{
(strc a s e c m p (d u m m y , " s " ) = = 0 )
f s c a n f ( i n , "7,s" .dummy) ;
S = atof(dum m y) * l e - 4 ;
{
>
if
>
if
>
i f (s trc a s e c m p (d u m m y , " n " ) == 0) {
f s c a n f ( i n , "7,s" .dummy);
N = ato f(d u m m y );
>
i f (s tr c a s e c m p ( d u m m y ," r p " ) == 0) {
f s c a n f ( i n , "7.s " , dummy) ;
RP = ato f(d u m m y );
>
if
(s trc a s e c m p (d u m m y , " f l o a t " ) == 0) {
f s c a n f ( i n , "7.s" .dummy) ;
i f (s trc a s e c m p ( d u m m y , " t r u e " ) == 0) FLAGl = 1;
e l s e FLAGl = 2 ;
>
i f ( s tr c a s e c m p C d u m a y ," l a y e r " ) == 0) {
f s c a n f ( i n , "7,s " .dummy);
fo o = a t o i ( d u m m y ) ;
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
148
if
(fo o == L) {
RM = rm l;
CM = cml;
>
if
(fo o == 2) {
RM = rm2;
CM = cm2;
>
if
(fo o == 3) {
RM = rm3;
CM = cm3;
>
>
if
(strcasecm p(du m m y, " v a r y " ) == 0) {
f s c a n f ( i n , "V.s" .dummy) ;
i f (s trc a s e c m p (d u m m y ,"1 1 ") == 0) {
VARY[VARY.INDEX] = 1;
f s c a n f ( i n , " V.s " , dummy) ;
BEGIN[VARY.INDEX] = atof(dum m y) * l e - 4 ;
f s c a n f ( i n , '"/.s" .dummy) ;
STOP [VARY.INDEX] = atof(dum m y) * l e - 4 ;
fsc a n f(in ,"V s" .d u m m y );
STEP[VARY.INDEX] = atof(dum m y) * l e - 4 ;
>
if
(strcasec m p (d u m m y , " 1 2 " ) == 0) {
VARY[VARY.INDEX] = 2;
f s c a n f ( i n , "7,s" .dummy) ;
BEGIN[VARY.INDEX] = atof(dumm y) * l e - 4 ;
f s c a n f ( i n , "7.s" .dummy) ;
STOP[VARY.INDEX] = atof(dum m y) « l e - 4 ;
f s c a n f ( i n , "7.s" .dummy) ;
STEP[VARY.INDEX] = atof(dum m y) « l e - 4 ;
>
i f (s trc asec m p (d u m m y , "w") == 0) {
VARY[VARY.INDEX] = 3;
f s c a n f ( i n , '"/.s" .dummy) ;
BEGIN[VARY.INDEX] = atof(dumm y) * l e - 4 ;
f s c a n f ( i n , '"/.s" .dummy) ;
STOP[VARY.INDEX] = atof(dum m y) * l e - 4 ;
f s c a n f ( i n , '"/.s" .dummy) ;
STEP[VARY.INDEX] = atof(dum m y) * l e - 4 ;
>
i f (s trc asec m p (d u m m y , " s " ) == 0) {
VARY[VARY.INDEX] = 4;
f s c a n f ( i n , "V.s" .dummy) ;
BEGIN[VARY.INDEX] = atof(dum m y) * l e - 4 ;
f s c a n f ( i n , ”’/ ,s ” .dummy) ;
STOP[VARY.INDEX] = atof(dum m y) * l e - 4 ;
f s c a n f ( i n , "V.s" .dummy) ;
STEP[VARY.INDEX] = atof(dum m y) * l e - 4 ;
>
i f (strc asec m p (d u m m y , " t " ) == 0) {
VARY[VARY.INDEX] = 5;
f s c a n f ( i n , "V.s" .dummy) ;
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission
BEGIN [VARY.INDEX] = a t o f (dummy) « l e - 4 ;
f s c a n f ( i n , "V.s" .dummy);
STOP [VARY.INDEX] = a t o f (dummy) * l e - 4 ;
f s c a n f ( i n , '"/.s" .dummy) ;
STEP [VARY.INDEX] = a t o f (dummy) * l e - 4 ;
>
if
( strca secm p(du m m y,"n") == 0) {
VARY[VARY.INDEX] = 6;
f s c a n f ( i n , "/(s" .dummy) ;
BEGIN[VARY.INDEX] = atof(dum m y);
f s c a n f ( i n , "V.s" .dummy);
STOP[VARY.INDEX] = atof(dum m y);
f s c a n f ( i n , "‘/(s" .dummy);
STEP[VARY.INDEX] = ato f(du m m y );
}
if
(strcasecm p(dum m y , "llfe2") == 0) {
VARY[VARY.INDEX] = 7;
f s c a n f ( i n , "*/(s" .dummy) ;
BEGIN [VARY.INDEX] = a t o f (dummy) * l e - 4 ;
f s c a n f (in,'"/.s",dummy) ;
STOP[VARY.INDEX] = atof(dummy) * l e - 4 ;
f s c a n f ( i n , '"/.s" .dummy);
STEP[VARY.INDEX] = atof(dummy) « l e - 4 ;
>
VARY.INDEX = VARY.INDEX + 1;
>
if
(strc asec m p (d u m m y , " x g rap h ") == 0) {
f s c a n f ( i n , '" / , s " .dummy);
if
(strcasecm p (d um m y, "L") == 0) XGRAPH = 1;
if
(strcasecm p (d um m y , "R") == 0) XGRAPH = 2;
if
(strcasecm p(dum m y,"C ") == 0) XGRAPH = 3;
i f ( s tr c a s e c m p ( d u m m y , " f .r e s " ) == 0) XGRAPH = 4;
i f (strcasecm p(dum m y,"f_qm ax") == 0) XGRAPH = 5;
i f (strc asec m p(du m m y,"q.m ax") == 0) XGRAPH = 6;
>
>
fc lo se (in );
/ ,
===================================
T h i s f u n c t i o n a n a ly z e s t h e i n d u c t o r .
v o id i n d _ a ( a ,b , c , d ,e ,h )
double * a ,* b ,* c ,* d ,* e ,* h ;
{
d o u b l e s e l f , m u t .p o s , m u t.n e g , t o t a l , c a p . i n d , r e s . i n d ;
d o uble s . r . f , m . q . f , i . q ;
d o uble * lin e _ s e g ;
i n t Z;
Z = (in t)
(4 * N);
l i n e . s e g = (d o u b le *) malloc(MAX_NUM * s i z e o f ( d o u b l e ) ) ;
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission
150
in itia liz e (lin e _ s e g ,Z );
/********
I n d u c t a n c e C a l c u l a t i o n s *«**«<****/
s e l f = s u jn _ s e lf ( l i n e . s e g ) ;
m ut_pos = s u m _ m u t_ p o s (lin e _ s e g ) ;
m ut_neg = s u m _ m u t_ n e g ( l in e _ s e g ) ;
t o t a l = s e l f + mut_pos - m u t.n eg ;
/********
I n d u c t o r Sp ecs C a l c u l a t i o n s **»*«**«*/
re s_ in d = c a lc _ r e s ( lin e _ s e g ) ;
c a p _ in d = c a l c _ c a p ( l i n e _ s e g ) ;
s_r_f = s e l f _ r e s _ f r e q ( t o t a l , c a p .in d ,re s _ in d );
m_q_f = m a x _ q _ f r e q ( t o t a l , c a p _ i n d , r e s _ i n d ) ;
i _ q = i n d _ q ( t o t a l , c a p _ in d ,r e s _ i n d ,m _ q _ f ) ;
«a
*b
*c
«d
*e
«h
=
=
=
=
=
=
to ta l;
re s .in d ;
c a p _ in d ;
s_ r_ f;
m _ q _ f;
i_q;
fre e (lin e .se g );
>
/« =======================================================================
T h is f u n c t i o n s t o r e s a l l t h e v a r i a b l e s
====================================================================== «/
v o i d s t o r e ( l l _ o l d , 1 2 _ o ld , w _old, t _ o l d , s _ o l d , n _ o ld )
d o u b le « l l _ o l d , * 1 2 _ o ld , * v _ o ld , » t _ o l d , » s _ o l d , «n_ o ld ;
{
* l l _ o l d = L I;
* 1 2 _ o ld = L2;
*w_old = W;
* t _ o l d = T;
* s _ o l d = S;
* n _ o ld = N;
/* ===========================================================-==-=-==T his f u n c tio n r e s t o r e s a l l th e v a r ia b l e s
=================================================================== «/
v o i d r e s t o r e ( l l _ o l d , 1 2 _ o ld , w_old, t _ o l d , s _ o l d , n _ o ld )
d o u b le l l _ o l d , 1 2 _ o l d , w_old, t _ o l d , s _ o l d , n _ o ld ;
{
LI = l l _ o l d ;
L2 = 1 2 _ o ld ;
W = w_old;
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission
151
T = t_ o ld ;
S = s_ o ld ;
N = n_old;
/ . =======================================================================
T his f u n c ti o n v a r ie s in d u c to r p a ra m e te rs .
=======================================================================
v o i d s w e e p ()
•c
d o u b l e max_n;
d o u b l e a , b , c , d , e , f , i n c [ M A X _ P ] ,f o o 3 ;
d o u b l e l l _ o l d , 1 2 _ o ld , w _old, t _ o l d , s _ o l d , n _ o l d ;
c h a r fo o lC lO ] , f o o 2 [ l C ] ;
in t i ;
s t o r e ( 4 1 1 _ o l d , t l 2 _ o l d , few_old, f c t . o l d , 4 s _ o l d , & n _ o ld );
if
(VARY.INDEX > 0)
•c
f o r (i=0;i<VARY_INDEX;i++)
{
if
(VARYCi] == 1) {
strc p y C fo o l, "L I");
s trc p y (fo o 2 ," u m " );
>
if
(VARYCi] == 2) {
s t r c p y ( f o o l , "L 2");
strc p y (fo o 2 ," u m " );
>
if
(VARYCi] == 3) {
s t r c p y ( f o o l , "W");
s trc p y (fo o 2 ," u m " );
}
if
(VARYCi] == 4) {
s trc p y (fo o l," S " );
s trc p y (fo o 2 ," u m " );
>
if
(VARYCi] == 5) {
s t r c p y ( f o o l, "T ");
s t r c p y ( f o o 2 , "u m ");
>
if
(VARYCi] == 6) {
s t r c p y ( f o o l , "N ");
s trc p y (fo o 2 ," tu rn s " );
}
if
(VARYCi] == 7) {
s t r c p y ( f o o l , "L1=L2");
s trc p y (fo o 2 ," u m " );
>
p rin t_ sta t();
p rin t_ h e a d (fo o l,fo o 2 ,l) ;
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission
./
152
in c C i] = BEGIN [ i ] ;
w h i l e ( i n c C i ] <= STO P[i])
{
i f (VARY[i] == 1) {
LI = i n c [ i ] ;
foo3 = LI *> l e 4 ;
>
i f (VARYCi] == 2) {
L2 = i n c C i ] ;
foo3 = L2 * l e 4 ;
>
i f (VARYCi] == 3) -C
W = in cC i];
foo3 = W « l e 4 ;
>
i f (VARYCi] == 4) {
S = in c C i];
fo o3 = S * l e 4 ;
>
i f (VARYCi] == 5) {
T = in c C i];
foo3 = T * l e 4 ;
>
i f (VARYCi] == 6) {
N = in c C i];
foo3 = N;
>
i f (VARYCi] == 7) {
LI = i n c Ci] ;
L2 = i n c Ci] ;
fo o3 = LI * l e 4 ;
}
m ax.n = a a x . t u m O ;
i f (max_n > N)
{
in d _ a ( & a ,t b ,& c ,& d ,& e ,& f ) ;
p rin t.s p e c s (fo o 3 .a .b .c .d .e .f.l);
>
i n c C i ] = i n c C i ] + STEPCi];
>
r e s t o r e ( l l _ o l d , 1 2 _ o ld , w . o l d , t _ o l d , s _ o l d , n . o l d ) ;
>
>
else {
p r i n t _ s t a t ();
p r i n t _ h e a d ( " " , ' " , ,0) ;
in d _ a (& a ,& b ,& c ,& d ,& e ,& f);
p r i n t _ s p e c s ( O . 0 , a , b , c , d , e , f ,0) ;
>
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
This is the main - TOP function of the program
i n t m a i n ( a r g c , a rg v )
in t argc;
c h a r - a r g v [] ;
{
file _ re a d (a rg c ,argv);
sweep 0 ;
r e t u r n 0;
>
A .3
«
S am p le In p u t File
T echn olo gy D e f i n i t i o n s **
CM1 0.C36 fF /u m ‘ 2
CM2 0 .0 1 1 fF /u m -2
CM3 0 .0 0 7 fF/um*2
RM1 0 .0 8 O hm s/square
RM2 0 .0 7 O hm s/square
RM3 0 .0 3 O hm s/square
LAMBDA 0 . 5 m ic ro n s
LI 300 m ic ro n s
L2 300 m ic ro n s
W 9 m ic ro n s
T 1 .2 m ic ro n s
S 3 .0 m ic ro n s
V 3 tu rn
RP 1 ohms
FLOAT f a l s e
LAYER 3
«» C o n t r o l S t a t e m e n t s -*
-VARY -W
10
1
300
100
VARY L1&2
0 .5
10
5
0 .2 5
-VARY -N
1
-VARY -S
1 .5
15
0 .5
-VARY *T
0 .2
2
0 .1
XGRAPH
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
154
A p p e n d ix B
M a x im u m T uning R an ge
B .l
V oltage Source as M eans o f C apacitance C ontrol
The m axim um tu ning of the electro-mechanically tunable capacitor can be best il­
lu stra ted graphically.
Fig. B .l shows th e spring and electrostatic forces for the 0.6 pF
tw o-plate tunable capacitor where the electrostatic force is plotted for three control volt­
ages
= 1.75 V. V\ = 3.3 V. and V\ = 4.5 V. As Fig. B .l illustrates for small control
1.75
-20
3.3 V
4.5 V
O -60
-80
-100
-0.75
-0.625
-0.5
-0.375
-0.25
-0.125
0
Displacement [jim]
Figure B .l: Spring and electrostatic forces as a function of th e displacement.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
voltages (i.e. V'! < 3.3 V). there are two real solutions. T h e physically m eaningful solution
corresponds to th e intersection which goes through zero when Vi = 0 V. For large control
voltages (i.e. V: > 3.3 V). there is no real solution and th e m agnitude of the electrostatic
force is always larger th a n the spring force.
The suspended plate will hence come into
contact w ith the m echanically secured plate. W hen V\ = 3.3 V. however, there is only one
real solution, which is at approxim ately 0.25 ^xm (i.e. a b o u t a th ird of 0.75 u m air gap).
T h e m axim um displacem ent can be derived analytically [112]. b u t first some obser­
vations m ust be m ade. At a point of m axim um displacem ent. F , = Fe and
dFs __ dF,
dx
dx
(B .l)
Taking th e corresponding derivatives:
d Fs
dx
dFe
dx
=
k
(B.2)
( n
V 2/
( edA V l2
y c ^ + x ) 2/
—9
Vdi-rx
(B.3)
S u b stitu tin g (B.2) and (B.3) into (B .l):
Taking advantage of Fs — Fe:
k = k x ■( - — -— )
Kd\ - r X ,
(B.5)
R earranging (B.5) and finally solving for the m axim um displacem ent x m ai:
dx
3
(B .6 )
The m inim um capacitance C min th a t cam be achieved is given by (B.7) where x = 0 under
zero bias. T h e m axim um capacitance Cmax is given by (B.8) w here x max = —di/3.
C mtn
=
fdA
^
dx
~
dx-dx/3 “
(B.7)
2 Cmm
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
( B ' S)
156
Hence, the m axim um tu n in g range C mai/C mm is 1.5 : 1.
B .2
C h a r g e S o u r c e a s M e a n s o f C a p a c i t a n c e C o n tr o l
In the previous section, it has be shown th a t th e suspended plate of micromechani­
cal tunable cap acito r can be displaced by only a th ird of the original plate separation d\
when voltage source is used as a control variable.
In this section, the use of charge as
means of controlling the capacitance of the m icrom echanical capacitor is investigated. The
electrostatic force is given by:
V 2 ) {d.1 -r x ) 2
T he charge <71 sto red on th e capacitor plates is given by:
q i = C DV!
IB .10)
Using (B.9) an d (B.10), the electrostatic force in term s o f charge can be obtained as follows:
Fe =
2edA
f B .l 1)
At the equilibrium of th e forces. Fs = Fe. and hence the displacem ent in terms of fixed
am ount charge can be given:
,
a
i
2
>
From (B.12). it follows th a t a unique solution exists when charge source is used as means
of controlling th e capacitance. Hence, by depositing an app ro p riate am ount of charge on
the capacitor p lates, th e displacem ent can take on any physically meaningful displacem ent
value from —d\ to 0.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
157
A p p e n d ix C
P h a se N o ise d u e to M ech an ical N oise
E lectro-m echanicallv tunable capacitors exhibit m echanical noise which produces ran­
dom displacem ent fluctuations around the operating point. The power spectral density of
the displacem ent is given by:
1
AkbT r
= ~
— ,T - ~ ~
^ ( £ - * 2)2 - M £ w )2
as discussed in chapter 5.These random displacem ent
fc.i)
fluctuations m odulate
capacitor Cp- Hence, w hen an electro-mechanicallv tu n ab le capacitor
the tunable
isemployed in a
LC VCO application, th e mechanical noise will be d irectly frequency m odulated by the
oscillator. T he frequency of oscillations ^o(x) is given by:
*o(z) =
l T i n 1 n - ; t?
\/ L ( C p ■+■C p ( x ) )
(C-2 )
where L is the inductance. Cp is a parasitic capacitance, and Cp{ x ) is the tunable capac­
itance. T h e oscillation frequency is nonlinear with respect to the displacem ent. Since the
displacem ent due to m echanical noise is small. (C.2) can be linearized as follows:
^o(^) —^d(Xdc)
(^*3)
where Kf m(xdc) is th e FM m odulation gain and can b e easily shown to simplify to:
Kj mi Xd c) = ^ 5
= - \ ( L { C p -r C D( x dc) ) ) - l L
d x \x=Iic
2
dx !r=Xdc
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(C.4)
158
The tunable capacitance and its derivative can be found to be:
^
dCo
C D(x) = — — = > — !
d \ -r- X
i
C Di x dc)
= - - ----------
(IX
fC.o)
aj
S ub stitu tin g th e result of (C.5) into (C.4). we obtain:
K f m( xdc) = ^ ( L,C p- X--) ) ( ------------7 )
2 V d x ^ x dc J \ ( L ( C p ^ C D( x dc)))> J
(C .6)
Let Cp = ~' Co(xdc). th en (C .6 ) reduces to:
f m. Xdc
1 f L C D(xdc) \ (
1
2 W ! - r x * J \( (1 -r j ) L C D( Xdc)) ■
I i^Jo(xdc)
1
2 d x -+- x dc 1 -+- 7
(C.7)
In m any cases, noise can be represented as a sum m ation of sinusoids w ith random am plitude
and phase I471. Here, only single sinusoid is considered with frequency equal to
8u
and
am plitude equal to square root of noise power in 1 Hz b andw idth around frequency 6u j . as
shown:
x = \J 2 X 2 (ui)cos(Sujt)
fC .8)
T he fundam ental frequency com ponent of th e oscillator is assum ed to be in the form:
Kmf(t) = Aocos( 6 (t))
(C.9)
where .4ois th e am plitude of the fundam ental and o(t) isthe phase. T h e expression for the
phase o(t). which is the integral of frequency uo(x), can be easily o b tain ed by using (C.3)
and (C .8) as follows:
0( t )
C
= /
Uj0 (x)dt =
J-Jc
jjQ{xdc)t
~
J 2 X 2 {uj)
_ x
—I
Sin(du)t)
6 uj
K f m ( x dc) -
(C.10)
By su b stitu tin g (C.10) into (C.9). equation (C.9) simplifies to:
Vmit(t)
=
A qK f m i xdc)\J2X~ {Su>)
Aocos(ujo(xdc)t) 1----------------—---------------- cos((cjq
2 ou)
-
A o K f r n i x ^ y / tt
:
^
— ---------------cos((ujQ - du)t)
Zdu)
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6 u})t)
(C. l l )
L59
From ( C. l l ) . th e ratio of th e sideband noise to the carrier power £ m(<5u;) can be found as
follows:
Cm(Suj) = \ ( - /r^ C— ■) X 2 (Suj)
2 \
)
l,C. 12)
Finally, su b stitu tin g (C.7) into (C.12). an d making th e necessary sim plifications we obtain:
r
tz
\
^ ( K f m { x dc)\
= 2I
iZ
"v2 ^r
>
' 'j A
I
1
t * 0( j Je) y
8 (1 -r ~f)2 (d\ -r x dc)2 V du )
Hence, to m inim ize th e noise of th e oscillator due to th e m echanical noise, the ratio of the
random displacem ent X 2 (6 lj) to (dt 4- xdc)2 must be kept small. Increasing the separation
d\ -t-Xdc betw een the capacitor plates plates in not practical since to achieve the same capac­
itance a much larger area would be required. On the o th er hand, the random displacement
fluctuations X 2 (Su;) can be kept small by reducing th e m echanical resistance r. which can
be achieved by th e use of a large num ber of etch holes or use of vacuum instead of air.
N ote th a t in differential circuit applications th e tu n ab le cap acito r C d often consists
of two equal m icrom achined tu n ab le capacitors Cox an d C p 2 connected in series. Since two
m echanical noise sources are involved, superposition can be applied. F irst, considering C o i
as a fixed capacitor, th e m echanical noise due to C o 2 can be first accounted for. Given th a t
C o \ = C 0 2 - th e m od u latio n gain K f m is simply half of (C.7) as shown:
rs /_ \ _ ^ uJo('*‘dc)
1
Kfm(^dc) — . i
,
4 di -r Xdc 1 * 7
//-t ,
(C.14)
Using (C.12) an d (C.14). th e phase noise due to single m icrom achined tu n ab le capacitor is
hence given by:
/. ,S ,
1
1
X 2 (6 u>) fujQ( x d c ) \ 2
Cm (OUJ) = — — — —2 7^—
I—r
32 (1 -f- 7 ) (d\ -f- xdc)~ \
du> J
( C-l o l
However, since two electro-m echanical capacitors are used and th e ir m echanical noise sources
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160
are uncorrelated, the noise sidebands sim ply ad d and hence the phase noise due to mechan­
ical noise of two identical m icrom achined capacitors connected in series is given by:
= l
1 j
16 (1 — -v)^ (rfi
-Trfc) V
J
'C .16)
Hence, differential oscillator w ith two m icrom achined tunable capacitors connected in series
across the inductor has the phase noise given by (C.16) which is sim ply 50 percent sm aller
th a n th e phase noise given by the expresion (C.13)
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IMAGE EVALUATION
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