In nstitu ution nen f

In nstitu ution nen f
In
nstitu
ution
nen för
f ssysteemteknikk
Departm
D
ment of Electrica
E
al Engineeering
Exam
mensarbeete
A Stud
dy and Design of High Pe
erforma
ance
Voltage-C
Contro
olled Oscillat
O
tors in 65nm CMOS
S
Tech
hnolog
gy
Maste
er thesis perform
med in Ellectroniic Devic
ces
by
Kamra
an Afgh
hari
LiTH-ISYL
-EX--12/46646--SE
Deceember 2012
TEKNISK
KA HÖGSK
KOLAN
LIINKÖPING
GS UNIVE
ERSITET
Departmeent of Electrical Engineeering
Linköping University
y
S-581 83 Linköping, Sweden
L
Linköpings ttekniska höggskola
IInstitutionenn för systemtteknik
5581 83 Linkööping
ii A Study and Design of High Performance
Voltage-Controlled Oscillators in 65nm CMOS
Technology
Master thesis
Performed in Electronic Devices
Department of Electrical Engineering
Linköping Institute of Technology
by
Kamran Afghari
LiTH‐ISY‐EX‐‐12/4646‐‐SE Supervisor: Assoc. Professor Jerzy Dabrowski Fahad Qazi Linköping University Examiner: Assoc. Professor Jerzy Dabrowski Linköping University Linköping, December 2012 iii iv Presentation Date
2012-12-17
Department and Division
Department of Electrical Engineering
Publishing Date (Electronic version)
Language
English
Other (specify below)
Number of Pages
74
Type of Publication
ISBN (Licentiate thesis)
Licentiate thesis
Degree thesis
Thesis C-level
Thesis D-level
Report
Other (specify below)
ISRN: LiTH-ISY-EX--12/4646--SE
Title of series (Licentiate thesis)
Series number/ISSN (Licentiate thesis)
URL, Electronic Version
http://www.ep.liu.se
Publication Title
A Study and Design of High Performance Voltage-Controlled Oscillators in 65nm CMOS Technology
Author(s)
Kamran Afghari
Abstract
In recent years, oscillators are considered as inevitable blocks in many electronic systems. They are commonly used in
digital circuits to provide clocking and in analog/RF circuits of communication transceivers to support frequency
conversion. Nowadays, CMOS technology is the most applicable solution for VLSI and especially for modern integrated
circuits used in wireless communications. The main purpose of this project is to design a high performance voltagecontrolled oscillator (LC VCO) using 65nm CMOS technology. To meet the state-of-the-art requirements, several circuit
solutions have been explored and the design work ended-up with a Quadrature VCO. The circuit operates at center
frequency of 2.4 GHz. The phase noise of QVCO obtained by simulation is -140 dBc/Hz at 1MHz offset frequency which is
6 dB less compared to conventional LC VCOs. The power consumption is 3.6mW and the tuning voltage can be swept from
0.2 V to 1.2 V resulting in 2.25 GHz - 2.55 GHz frequency range.
Keywords
QVCO, Power Consumption, Phase Noise, Tuning Voltage, Communication Transceivers
v vi Abstract In recent years, oscillators are considered as inevitable blocks in many electronic systems. They are commonly used in digital circuits to provide clocking and in analog/RF circuits of communication transceivers to support frequency conversion. Nowadays, CMOS technology is the most applicable solution for VLSI and especially for modern integrated circuits used in wireless communications. Additionally, the trend towards single chip implementation makes the circuit design increasingly challenging. The main purpose of this project is to design a high performance voltage‐
controlled oscillator (LC VCO) using 65nm CMOS technology. In the beginning, a brief study of different VCO architectures is carried out. Next, a wide comparison between different VCO topologies is performed in terms of phase noise and power consumption. The effect of VCO phase noise on RF transceivers is also analyzed. In the following, all the phase noise contributors in a typical LC VCO are identified to enable design optimization. To meet the state‐of‐the‐art requirements, several circuit solutions have been explored and the design work ended‐up with a Quadrature VCO. The design is verified for the intended tuning range and process, temperature, and supply voltage (PTV) variations. The circuit operates at center frequency of 2.4 GHz. The phase noise of QVCO obtained by simulation is ‐140 dBc/Hz at 1MHz offset frequency which is 6 dB less compared to conventional LC VCOs. The power consumption is 3.6mW and the tuning voltage can be swept from 0.2 V to 1.2 V resulting in 2.25 GHz ‐ 2.55 GHz frequency range. vii viii Acknowledgments First, I should show my gratitude to Professor Jerzy Dabrowski for his support and supervision during the project. He showed patience for solving problems encountered throughout my thesis. The smart suggestions provided by Professor Jerzy Dabrowski gave me a new inspiration for designing a state of the art circuit. Furthermore, his experience and knowledge helped me to tackle the problems quicker. I should appreciate the PhD students in Electronic Devices for giving me some advice during my thesis. Also, I am really grateful to all of my friends for sharing happy moments in Linköping University. Here, I would like to thank my parents for supporting me during my studies. They always encouraged me to do my best during the stressful moments in my life. Last but not least, I should thank my brother, Erfan, for his support and persuasion during my project. ix x Contents Abstract………………………………………………………................................................. vii Acknowledgments……………………………………….................................................. ix Abbreviations…………………………………………………………………………………………….. xix 1. INTRODUCTION…………………………………………................................................ 1 1.1 Objective and Scope……………………………………………………………………………….. 1 1.2 VCO Design Specifications………………………………………………………………………. 2 1.3 VCOs in Phase‐Locked Loops………………………………………………………………….. 3 1.4 Thesis Organization……………………………………………………………………………….. 3 2. OSCILLATOR BASICS………………………………………………………………………………… 5 2.1 Introduction………………………………………………………………………………………….. 5 2.2 Oscillator Transfer Function………………………………………………………………….. 5 2.3 Ring Oscillators…………………………………………………………………………………….. 7 2.4 LC Oscillators……………………………………………………………………………………... 10 2.4.1 LC Tank…………………………………………………………………………………………. 10 2.4.2 One‐port Model of LC VCO…………………………………………………………… 12 2.4.3 Cross‐Coupled LC Oscillator……………………………………………………….…. 13 2.4.3.1 NMOS Cross‐Coupled LC Oscillator………………………………………… 14 2.4.3.2 CMOS Cross‐Coupled LC Oscillator…………………………………….…… 17 xi 3. OSCILLATOR NOISE………………………………………………………………………….. 20 3.1 General Idea……………………………………………………………………………….. 20 3.2 Phase Noise Effect on RF Transceiver Topologies………………………… 22 3.3 Analysis of Oscillator Phase Noise……………………………………………….. 24 3.3.1 Leeson’s Equation of Phase Noise…………………………………………. 24 3.3.2 Hajimiri and Lee’s Phase Noise Equation………………………………. 29 3.3.3 F‐Parameter…………………………………………………………………………. 35 3.4 Analysis of Phase Noise Sources in CMOS LC VCOs……………………… 36 3.4.1 Transistor Noise Contribution………………………………………………. 36 3.4.2 Noise Contributors in MOSFET……………………………………………… 37 3.4.2.1 Thermal Noise………………………………………………………………… 37 3.4.2.2 Flicker Noise…………………………………………………………………… 39 3.4.2.3 Bias Noise………………………………………………………………………. 39 3.4.2.4 Switching Pair Noise……………………………………………………… 40 3.4.2.5 Resonator Noise…………………………………………………………... 40 xii 4. STATE‐OF‐THE‐ART DESIGN……………………………………………………………. 41 4.1 Introduction………………………………………………………………………………… 41 4.2 Low Phase Noise QVCO……………………………………………………………….. 41 4.2.1 Circuit Design………………………………………………………………………… 42 4.2.2 Design Specifications…………………………………………………………….. 44 4.2.3 Simulation Results…………………………………………………………………. 45 4.3 Low Noise Low Power CMOS LC VCO………………………………………….. 45 4.3.1 CMOS LC VCO………………………………………………………………………. 46 4.3.2 Power Analysis…………………………………………………………………….. 48 5. LC VCO DESIGN……………………………………………………………………………… 52 5.1 CMOS Cross‐Coupled Models……………………………………………….……. 52 5.1.1 Analysis and Comparison………………………………………………….…. 52 5.1.2 Design Procedure……………………………………………………………..…. 57 5.1.2.1 Cross‐Coupled LC VCO…………………………………………………… 57 5.1.2.2 NMOS LC VCO………………………………………………………….……. 60 5.1.2.3 PMOS LC VCO…………………………………………………………….…. 62 5.1.3 Comparison of Different VCO Topologies………………………….…. 63 5.2 Bias Circuitry Design………………………………………………………………….. 65 xiii 6. COMPLEMENTARY SIMULATIONS AND RESULTS…………………..….…… 66 6.1 Introduction…………………………………………………………………………….… 66 6.2 Phase Noise and Frequency vs. Control Voltage……………………….… 66 6.3 Reference Current Source Variation…………………………………………… 67 6.4 Phase Noise and Frequency vs. Temperature…………………….………. 68 7. CONCLUSION………………………………………………………………………………. 70 7.1 Summary……………………………………………………………………….………... 70 7.2 State‐of‐the‐Art Comparison…………………………….......................... 71 References………………………………………………………………………………….…… 72 xiv List of Figures Figure 2.1: Oscillator’s feedback system……………………………………………………….…… 6 Figure 2.2: Ring oscillator architecture……………………………………………………………... 7 Figure 2.3: Current starved inverter………………………………………………………………..… 8 Figure 2.4: Differential symmetric pair…………………………………………………………….… 9 Figure 2.5: Series resistance conversion into equivalent parallel resistance……… 11 Figure 2.6: Phase behavior and magnitude of a resonator………………………………...12 Figure 2.7: One‐port model of an LC VCO………………………………………………….……... 13 Figure 2.8: Bottom biased LC oscillator with equivalent losses……………………………14 Figure 2.9: NMOS LC oscillator analysis……………………………………………………..……… 15 Figure 2.10: Waveforms of a cross‐coupled LC oscillator…………………………………… 16 Figure 2.11: Top‐biased complementary LC oscillator……………………………………….. 18 Figure 3.1: Comparison of phase and jitter noise………………………………………………. 21 Figure 3.2: Output waveforms comparison of an ideal and a real oscillator………. 22 Figure 3.3: Phase noise impact on receivers……………………………………………………… 23 Figure 3.4: Phase noise impact on transmitters………………………………………………… 24 Figure 3.5: LC oscillator model with noise generators………………………………………. 25 Figure 3.6: Phase noise spectrum of a real oscillator………………………………………… 28 Figure 3.7: Impulse response of an ideal oscillator…………………………………………… 29 Figure 3.8: Impulse impact on a time‐varying system………………………………………. 30 Figure 3.9: Amplitude noise of an oscillator in one period……………………..……….. 30 xv Figure 3.10: Current impulse behavior on a time‐varying system…………………...... 30 Figure 3.11: Impulse sensitivity function of a real oscillator………………………….…… 31 Figure 3.12: Current impulse effect on the output signal……………………………….…..32 Figure 3.13: ISF block diagram……………………………………………………………………..…….33 Figure 3.14: Procedure of current noise conversion into phase noise………….……..34 Figure 3.15: Noise sources in LC cross‐coupled VCO…………………………….…………….36 Figure 3.16: Lumped noise model……………………………………………………………………….37 Figure 4.1: Circuit schematic of Quadrature VCO………………………………………….…….42 Figure 4.2: Conventional Quadrature VCO……………………………………………………..…. 43 Figure 4.3: CMOS LC oscillator circuit schematic…………………………………………….…. 47 Figure 5.1: Different designs of CMOS cross‐coupled VCO…………………………….….. 53 Figure 5.2: NMOS phase noise at 100 kHz offset frequency…………………………….… 54 Figure 5.3: PMOS phase noise at 100 kHz offset frequency………………………………..55 Figure 5.4: CMOS phase noise at 100 kHz offset frequency……………………….……….55 Figure 5.5: NMOS phase noise at 1MHz offset frequency…………………………………..56 Figure 5.6: PMOS phase noise at 1MHz offset frequency……………………………….…. 56 Figure 5.7: CMOS phase noise at 1MHz offset frequency……………………………….…. 57 Figure 5.8: CMOS cross‐coupled LC oscillator with NMOS bias circuit……………….. 58 Figure 5.9: Phase noise of CMOS cross‐coupled VCO with NMOS bias circuit……. 59 Figure 5.10: Phase noise of CMOS cross‐coupled VCO with PMOS bias circuit..... 60 Figure 5.11: Phase noise of self‐biased CMOS cross‐coupled VCO…………………….. 60 Figure 5.12: Phase noise of NMOS VCO with NMOS bias circuit……………………….. 61 xvi Figure 5.13: Phase noise of NMOS VCO with PMOS bias circuit………………………… 61 Figure 5.14: Phase noise of PMOS VCO with NMOS bias circuit………………………… 62 Figure 5.15: Phase noise of PMOS VCO with PMOS bias circuit…………………………. 63 Figure 6.1: QVCO phase noise (red) and frequency (blue) versus control voltage……………………………………………………………………………………………………………… 67 Figure 6.2: Impact of reference current source variation on QVCO center frequency…………………………………………………………………………………………………………. 68 Figure 6.3: Impact of temperature variation on QVCO phase noise (red) and frequency (blue)……………………………………………………………………………………………….. 69 xvii List of Tables Table 1.1: Ultimate VCO design specifications…………………………………………………….. 2 Table 5.1: CMOS cross‐coupled LC VCO dimensions………………………………………….. 58 Table 5.2: NMOS cross‐coupled LC VCO dimensions…...........................................61 Table 5.3: PMOS cross‐coupled LC VCO dimensions……………………………………………62 Table 5.4: Comparison of different VCO topologies regarding the phase noise and power consumption………………………………………………………………………………………….. 64 Table 7.1: State‐of‐the‐art comparison................................................................. 71 xviii Abbreviations VCO Voltage‐Controlled Oscillator QVCO Quadrature Voltage‐Controlled Oscillator Q Quality factor FOM Figure of Merit PLL Phase‐Locked Loop MOSFET Metal Oxide Semiconductor Field Effect Transistor IF Intermediate Frequency RF Radio Frequency OFDM Orthogonal Frequency‐Division Multiplexing QAM Quadrature Amplitude Modulation DC Direct Current ISF Impulse Sensitivity Function AM Amplitude Modulation xix xx 1. INTRODUCTION Nowadays, oscillators are considered as inevitable blocks in many electronic devices. They are mainly applicable in communication systems such as radio transceivers. Also, other applications take advantage of high performance oscillators. For instance, clock generation in microprocessors can be achieved by some types of VCOs. Today, CMOS technology is the most popular solution for designing a high performance voltage‐controlled oscillator. It should be pointed out that in most cases a VCO is not a standalone block but rather a part of a larger system like PLL‐based frequency synthesizer. 1.1 Objective and Scope The first objective of this project was a broad analysis of different VCO topologies used in the contemporary applications. A specific objective was to design a high performance VCO based on a design study of different LC VCO circuits. By practical comparison of critical specifications like phase noise and power consumption the choice of the right circuit could be done. To meet the state‐of‐
the‐art specifications, a quadrature VCO (QVCO) was chosen as the ultimate design goal. The QVCO was assumed to drive a balanced mixer for frequency conversion in a data OFDM transceiver. Since the OFDM‐QAM signal is extremely sensitive to the local oscillator phase noise, the phase noise was the primary specification that the QVCO was optimized for. Additionally, to check for the application in a PLL, the process, temperature, and supply voltage (PTV) variations and tuning range were taken carefully into account. 1 1.2 VCO Design Specifications To design a VCO, different requirements should be fulfilled. In this section, we define the VCO metrics individually. In particular, we should meet the oscillation frequency, power consumption, tuning range and phase noise requirements which are the most important in a VCO design. The oscillation frequency may vary from one design to another due to different applications and architectures. The tuning voltage range is determined by required frequency variations in different applications. The other major issues that should be considered especially in a high performance VCO design are phase noise and power consumption. Generally, it is difficult to fulfill all of the requirements at the same time. For instance, there is usually a tradeoff between power consumption and phase noise. On the other hand, some VCO topologies can improve the phase noise performance while other architectures can dissipate less power. Consequently, regarding the design specifications and their priorities, the designers have to choose the appropriate VCO topology but still are exposed to design tradeoffs. The specifications for our ultimate VCO design aimed at 65‐nm CMOS technology are as shown in Table 1.1. QVCO specifications Center frequency Supply voltage Phase noise at 1MHz offset frequency
Power consumption Tuning voltage Frequency range Value 2.4 GHz 1.2 V < ‐130 dBc/Hz < 5 mW 0.2 ‐ 1.2 V 2.25 ‐ 2.55 GHz Table 1.1: Ultimate VCO design specifications 2 1.3 VCOs in Phase‐Locked Loops Voltage‐controlled oscillators are mostly implemented as a component of phase‐
locked loops (PLLs). PLLs can be used in different areas such as clocking of microprocessors, providing carriers for wireless transceivers or other transmission systems. Usually, in communication applications PLLs require VCOs with a wide tuning range to serve up‐ or down‐conversion over the system bandwidth. Interestingly, in PLLs the VCO phase noise requirements can be relaxed. In other words, the noise produced by a voltage‐controlled oscillator at the oscillation frequency will be to some extent filtered out by the system. Therefore, VCO topologies with wide tuning range are usually preferred. In high performance applications where a low phase noise or jitter is required, VCOs using LC tanks are preferred for their high Q‐factor. Therefore, LC‐based VCOs will be in focus of the presented designs. 1.4 Thesis Organization In this project, firstly, various VCO architectures are analyzed in detail. In particular, regarding phase noise and power consumption, a wide comparison is carried out. As a result, we have come up to the state‐of‐the‐art design in which the VCO specifications can be fulfilled. In the following chapter, the foundations of the oscillator theory are discussed. Moreover, different VCO architectures such as the ring oscillator, negative gm oscillator, and CMOS LC VCOs are analyzed. At the end of this chapter the advantages and drawbacks regarding different topologies are compared. In Chapter 3, we study the oscillator noise using different models acknowledged in literature. Afterwards, the phase noise is defined and its effect on an RF transceiver is analyzed. Finally, the phase noise contributors in a VCO are discussed. 3 In chapter 4, some state of the art topologies are presented and their specification requirements are defined. In Chapter 5, seven variants of low‐noise low‐power LC VCO are designed for the same power consumption and then they are compared in terms of phase noise. In Chapter 6, complementary simulations of the QVCO regarding PTV variations are presented showing it is in line with the state‐of‐the‐art designs. In Chapter 7, conclusions from the presented design are provided. At the end of this chapter, the performance of the designed LC QVCO circuit is compared with other reported LC VCOs. 4 2. OSCILLATOR BASICS 2.1 Introduction In the second chapter, we describe the basic oscillation theory. This theory refers to the feedback system model or the negative resistance one‐port model. Different types of oscillators including the ring, negative gm oscillators, CMOS LC VCOs and some other topologies are discussed. Nowadays, among different types of oscillators, mostly the LC cross‐coupled VCOs are preferred. Hence, their advantages and drawbacks are mainly emphasized in this project. 2.2 Oscillator Transfer Function A VCO can be modeled as a feedback system as shown in Fig. 2.1. The equivalent transfer function of this system can be formulated as: G jω
Y
H
H
H
(2.1) According to the above equation, at the desired frequency of ω0, if H(jω0)=1, the closed loop gain will be driven to infinity. It causes the system noise to increase and produces a periodic signal. From another point of view, the oscillation will be stable if H(jω0) is precisely on the imaginary axis [11]. Practically, the small signal loop gain should be at least two to fulfill the initial oscillation conditions. As we know, this value decreases gradually and goes to unity as the amplitude improves due to the nonlinearities. The following equations are two conditions that should be satisfied for the oscillation startup at ω0: |H (jω0)| = 1 , arg (H (jω0)) = 0⁰ (2.2) 5 The phaase in (2.2)) should be
e changed to a value of 180⁰ w
when the lo
oopback gaain is negative. These equations aare known as Barkhausen’s critteria. Howeever, in special situations, the tw
wo above
e conditions can no
ot satisfy the oscilllation staartup complettely. In other word
ds, the Baarkhausen’’s criteria are neceessary but not sufficien
nt. For exaample, in ssome pracctical casess, the outp
put remain
ns saturateed or grounde
ed, even th
hough the startup co
onditions aare satisfied
d. e 2.1: Oscillator’s feeedback system Figure
he Barkhaausen’s critteria ners use various v
top
pologies to
o satisfy th
Nowadaays, design
mention
ned in the
e above se
ection. Mo
oreover, LLC and rin
ng oscillato
ors are maainly preferre
ed in high ffrequency oscillator design. 6 2.3 Rin
ng Oscillators A ring o
oscillator iss a device ccomposed of an oddd number o
of inverters connecteed in a feedb
back loop. IIn other w
words, the N
NOT gatess are placed in a chaiin in which
h the output of last inverter is fed f back into i
the first one. TThe total delay of eeach inverterr cell dettermines the oscillation freqquency w
which is fformulated
d as followin
ng: f
N
((2.3) Conside
ering the power conssumption aand phase noise perfformance, the numbeer of stages is chosen. There are different topologiess regardingg the ring oscillatorss but the diffe
erential an
nd the singgle‐ended m
models aree mainly diiscussed in
n this projeect. Figure
e 1.2: Ringg oscillator architectu
ure a) Diffferential m
model b) Singgle‐ended model 7 t
invertters have high outtput gain, the In single‐ended model, ass far as the Barkhau
usen’s criteria is alw
ways satisfiied. Since each delaay cell has a large siignal phase shift s
of 180
0⁰, the sin
ngle‐ended
d model is designed with an o
odd numbeer of cells. Th
he current is only use
ed during ttransitions of the NO
OT gates. The inttrinsic cap
pacitors arre charged
d and disccharged by this con
nstant currrent generatted by the transistorss. Thereforre, the delaay time of each cell iis defined. In order to have aa quicker ttransitions and higheer oscillatio
on frequen
ncy, we should apply a higher currrent. d type of d
delay cell is current‐‐starved in
nverter. Ass shown in
n the Another improved
Fig. 2.3,, the frequ
uency can be controlled when two extra transistors are addeed to the cell. Figure 2.2: Currrent starvved inverteer 8 As show
wn in the FFig. 2.4, th
he differential structuure consistts of an NMOS or PM
MOS differen
ntial transiistors and an outputt load. The cell delaay is deterrmined byy the charge stored in tthe circuit and the cu
urrent driv ing the loaad. Resisto
ors can be u
used for a fixed frequ
uency. If PMOS P
tran
nsistors arre applied
d as our lloads, we can improve
e the oscillator to be b tunable with diffeerent rangge of voltaages. PMO
OS or NMOS ttransistors can be ap
pplied as crross‐coupleed or symm
metric mod
dels. Figure
e 2.3: Diffe
erential sym
mmetric paair ption, frequ
uency and number o
of stages, tthe differential For a given power consump
e performance thann its singlee‐ended counterparrt. In ring osccillator haas a worse
other words, w
the
e differenttial ring osscillator’s phase noise is N(1+Vchar/(RL**Itail)) times higher than
n the single
e‐ended sttructure [220]. On thee other hand, the sin
ngle‐
ended architecture consumes lesss power than th
he differeential mo
odel. Additionally, it haas a better phase no
oise for sppecific pow
wer consum
mption du
ue to the inverse propo
ortionalityy of the phase noisee to poweer dissipattion. Howeever, differen
ntial ring osscillators aare often p
preferred inn digital cirrcuits due to their beetter commo
on noise rejection r
[20]. [
More
eover, lesss noise iss injected to the o
other compon
nents on the chip. N
Nowadays, this archittecture is widely useed for statte of 9 the art designs. For instance, a quadratic signal is generated by differential architecture since even number of inverters should be maintained. Passive devices are not used in ring oscillators. Regarding this issue, there are benefits and drawbacks which can be discussed. The main advantage of using active elements is easy integration on chip. Additionally, inductors occupy a considerable amount of area on chip. However, using an inductor with a capacitor on the chip makes a band pass filter that leads to better phase noise performance. Overall, ring oscillators show worse phase noise performance than the LC VCOs. The main usage of ring oscillators is for data transmission as clock recovery or clock propagation on the chip. 2.4 LC Oscillators An LC oscillator consists of a resonator tank in which the inductor and the capacitor are connected in parallel. There are various LC VCO architectures but Colpitts and Negative gm oscillators are the most popular structures. In the following sections, these two models are widely compared regarding their benefits and drawbacks. 2.4.1 LC Tank In VCO implementation, LC network is mainly in focus due to its filtering benefits. Since the LC tank consists of an inductor and a capacitor connected in parallel, it is also called a parallel resonator circuit. In an ideal condition, if a current impulse is applied to the LC tank, the energy transfers back and forth between the inductance and the capacitance. This leads the circuit to oscillate for an infinite period time. However, this occurs in an ideal case where both of the elements are lossless. In other words, some dissipation is inevitable for the inductor and the capacitor. Otherwise, they should have an infinite quality factor that is not possible in practice. Actually, the resonator’s loss is due to the series resistances of the inductor and the capacitor. As far as RF VCOs oscillate over a small range of 10 frequen
ncies, the p
parasitic re
esistances can be rem
modeled to
o their parrallel structture. The ove
erall resistaance is sho
own by Rp iin the follo
owing figurres. It dissip
pates the energy e
and
d thus leads the osccillation to
o fade awaay. In ordeer to compen
nsate this lossy circu
uit, an activve circuit sshould be added in parallel to
o the tank. Th
his active ccircuit prod
duces a neggative resistance to ccancel the Rp. Thereffore, it helps the oscillaation to surrvive. ure 2.4: Se
eries resistaance conve
ersion into
o equivalen
nt parallel resistancee Figu
The main drawback is that integrated
d inductorss have veryy low quality factorss and occupy a larger area on the t
chip than t
the ring oscilllators. Neverthelesss, LC oscillato
ors have a better phaase noise p
performance comparred to ringg oscillatorss. The magnitude an
nd phase b
behavior o
of an LC tank is show
wn in the FFig. 2.6. Att the resonan
nce freque
ency, the n
network be
ehaves likee a resistan
nce. In oth
her words,, it is purely rreal. The re
esonance ffrequency is formulated as follo
owing: f
(2.4) √LC
nce and capacitance respectiveely. Where, L and C arre inductan
11 Figure 2.5: Phaase behavio
or and maggnitude off a resonator 2.4.2 O
One‐port Model off LC VCO
Negativve gm or one‐port model m
represents thee oscillator as the combinatio
on of two individual one
e‐port mod
dels. This aapproach m
makes the analysis o
of LC oscillaators much easier. The procedure
e is depicte
ed in the Fiig. 2.7. ntioned before, the resonance
e circuit with the serries loss o
of the indu
uctor As men
and the
e capacitorr can be modeled m
as its paralllel equivaalent. At each period
d, Rp preventts oscillatio
on to be stable s
by consuming
c
g some am
mount of sstored LC tank energy. Therefore
e, an activve negativve resistannce which is equal to –Rp should compen
nsate the loss of the tank. In otther words, we shou
uld create a lossless tank with the
e infinite p
parallel ressistance att the resonnance freq
quency. Th
his conditio
on is satisfied
d by the fo
ollowing eq
quation: G
R ((2.5) 12 Figure 2.7: One‐p
port modell of an LC V
VCO As mentioned earrlier, Gm is called as the large ssignal tran
nsconductaance when
n the VCO is in stable co
ondition. 2.4.3 C
Cross‐Cou
upled LC O
Oscillatorr As mentioned in aabove secttions, in ne
egative Gm procedurre, the losss of the tank is neutraliized by acctive devices for a sttable oscilllation. In CMOS cro
oss‐coupled LC VCO, th
he transisstors act similar s
to an activee negativee resistance. To saatisfy oscillation startup
p condition, negative resistance usuallyy is defined
d at least two times larger than
n the parrallel loss of the taank. The m
main advaantage of this topologgy is the sim
mple desiggn and imp
plementatiion on thee chip. The cross‐coupled LC oscillator is categorized
c
d into three types: PMOS, N
NMOS and CMOS crross‐
coupled
d. Each of them can be implemented ussing top‐b
biased, botttom‐biaseed or self‐biassing curren
nt source. 13 2.4.3.1
1 NMOS C
Cross‐Cou
upled LC O
Oscillator As show
wn in the Fig. 2.8, a a bottom‐biased NM
MOS cross‐‐coupled LLC oscillato
or is presentted. The lo
osses of th
he tanks are depicteed as well. It consistts of an NM
MOS differen
ntial pair and a two resonators in which their series and paarallel resisstive losses aare analyze
ed in detaill. Figure 2.6: Bottom
m biased LLC oscillato
or with equ
uivalent lossses OS cross‐co
oupled LC
C oscillato
or is analyyzed. For the In this section, the NMO
umed that the comm
mon‐
simplicity, the cirrcuit is divided into two parts.. It is assu
mode noise n
is grrounded so
o that the
e Vsource is biased to
o ground. However, this assump
ption is nott complete
ely true be
ecause the current so
ource has a finite ou
utput resistan
nce. As we
e know, th
he output nodes havve differen
ntial swingg with a phase differen
nce of 180
0⁰. Therefo
ore, the gaates of thee transisto
ors are exp
pressed as Vout and –Vout
sistance seen from ou
utput nodee to ground is formulated as: o . The res
V
I
V
G V
G
(2.6) 14 Figure
e 2.7: NMO
OS LC oscilllator analyysis ore, if the ttwo tanks aare same, we can convert them
m to an equ
uivalent circuit Therefo
in which
h: Lp=2Lp1 ,, Cp=Cp1/2, Rp=2Rp1 aand the neggative resiistance wo
ould be ‐2/Gm1. For a cross‐coup
c
t be in stable conddition, thee small siggnal loop gain led VCO to should be greaterr than unitty. Therefo
ore, the gaain value sshould be chosen as two in the m
minimum case. There
efore, the sstartup conndition is aas following: α
R ((2.7) In the above equaation, αmin satisfies th
he startup condition which sho
ould be at lleast two. we should ffind out th
he relation
nship betw
ween the ssmall signaal gain and
d the Next, w
large signal gain. The small signal gaain is αmin times largger than th
he large siignal gain. Ass the oscillation ente
ers the steady state, the small signal gain
n of the circuit will deccrease. Fin
nally, it will be equal to the large sign
nal gain because off the nonlinearities. Th
he nonlinear charaacteristic iis figured out by analyzing the transistors functio
on in triod
de and cu
ut‐off regio
on. In cut‐‐off region
n, the possitive output swing is clipped un
nder the bias voltaage. Additiionally, th
he large siignal transconductance
e of an NMOS transistor is eequal to µnCox (W/LL) Vds in linear region. Actually, in linear region, we w have tthe directt proportio
onality of the transconductance
e to drain vvoltage. On
n the other hand, byy applying aa small voltage 15 of ∆V att Vg1, the V
Vg2 or Vd1 w
will decreaase corresppondingly. If ∆V is co
onsidered large enough, the transsistor will e
enter the liinear regio
on. Here, th
he gm1 starrts to decrease and the
e loop gain will be staable at one
e. As obse
erved in th
he results, symmetrical wavefforms are obtained at the outtput. is on in haalf of the period so the curreent behavior is Therefo
ore, each transistor t
depicted as a squ
uare waveform for each e
transsistor [21]. The mean value off the output would be the same as for the
e circuit operating in
n DC mode. The currrent obtaine
ed at the center freq
quency is e
equal to (11/π)Ibias. Th
he LC tank filters outt the current harmonics so the ou
utput amplitude is fo
ormulated as: V
I
R ((2.8) ure 2.8: Waaveforms o
of a cross‐ccoupled LC
C oscillatorr Figu
16 2.4.3.2 CMOS Cross‐Coupled LC Oscillator A CMOS cross‐coupled LC oscillator consists of two NMOS and PMOS transistor pairs. In this topology, usually one inductor is used. A differential voltage is applied on the inductor which leads to better output waveforms than the previous topology. To obtain symmetrical waveforms, we should generate equal parallel resistance on the both sides of the resonator. If not, the phase noise would increase. Therefore, if the inductance is not designed symmetrical, two inductors in series should be implemented alternatively. In this structure, as shown in Fig. 2.11, the PMOS transistors reuse the bias current. Therefore, a larger output voltage is generated. As the output voltage increases, one of the NMOS transistors is switched on while the other one goes off. This is the same case for the PMOS pair but in the other way. For instance, in the half period, the current is driven into the M4 and then the resonator and finally it is conducted by the M1. One of the advantages of this structure over other topologies is that the current goes through the resonator’s resistance in each half period. Consequently, the output would be twice larger than the PMOS or NMOS cross‐coupled architecture. Regarding the Leeson’s equation, the output voltage improvement leads to better phase noise performance. Additionally, the PMOS transistors produce an extra negative resistance which can increase the overall negative resistance when added to NMOS pair. Consequently, the complementary architecture obtains an overall negative resistance of ‐2/Gm,NMOS‐2/Gm,PMOS. To produce symmetrical waveforms, the NMOS and PMOS transconductance should be the same. So, the overall negative resistance would be ‐4/Gm. Additionally, when using the complementary architecture, the startup condition would be satisfied easily for a specific bias current. Moreover, when equal transconductance is considered for the NMOS and PMOS pair, the most symmetric output can be obtained comparing to the NMOS or PMOS cross‐coupled counterparts. Therefore, the rise and fall times will boost and hence the flicker noise upconversion will improve [20]. In our design, we 17 obtain a a more reasonable phase noisse in 1/f3 region com
mpared to
o NMOS crross‐
2
coupled
d architectture. The 1/f phase
e noise is improved
d as well. However, the PMOS transistors are the main contrib
butors of thhermal noise in this region. There are a also so
ome drawbacks whe
en using C
CMOS crosss‐coupled LC VCOs. We encounter some o
output swing constraaints in this architeccture. In an
n NMOS crross‐
coupled
d architectture, conssidering th
he voltagee headroo
om, the o
output voltage swing iss set at Vddd‐Vdiff while
e in a com
mplementary architeccture the o
output voltage degrade
es to (Vdd‐V
Vdiff)/2. Ho
owever, wh
hen the ouutput is biaased at Vddd, the VCO
O will be more
e influence
ed by the ssupply volttage distorrtions. Figgure 2.9: TTop‐biased complementary LC oscillator In a complementtary LC arcchitecture, adding PPMOS pairr to the circuit lead
ds to extra paarasitic cap
pacitances seen in th
he resonator. Regard
ding the biaas current,, the PMOS and a NMOSS sizing maay vary correspondinngly. In otther wordss, the sizin
ng of PMOS ttransistors should be
e chosen qu
uite largerr than NMO
OS pair. Th
his will balaance the ovverall traansconducttance of the deesign. Theerefore, the requ
uired 18 transconductance is met in the complementary LC architecture. Since the overall intrinsic parasitic capacitance enhances in this topology, the resonator value should be calculated carefully to meet the desired oscillation frequency. Consequently, decreasing the value of the capacitor leads to tuning range limitations. 19 3. OSCILLATOR NOISE In this chapter, we mainly describe the oscillator phase noise. In the beginning, a brief definition of the phase noise is given. Afterwards, its effect on RF transceivers is analyzed. In the following sections, the Leeson’s phase noise model is discussed and followed by two other models using the impulse sensitivity function. Finally, a detailed analysis of the phase noise sources in a complementary LC oscillator is carried out. 3.1 General Idea There are some types of noise sources which influence the functionality of an LC VCO. The external noise is produced by the other components that are interacting with the oscillator. The filter generating the tuning voltage of the oscillator and the frequency divider in a Phase‐Locked Loop are the blocks which can produce extrinsic noise in our circuit. The internal noise is generated by the circuit layout of the components designed in the VCO. As a matter of fact, both noise sources can inject some disturbances into our design. Therefore, the output amplitude and frequency are influenced by this issue. Since the transistors nonlinearities compensate the amplitude noise, its effect is commonly neglected during the VCO design. Actually, a small deviation of frequency is regarded as phase noise in analog circuits. Nevertheless, if we consider the output in a time‐dependent scale, it is considered as a deviation in zero crossing points. The phase noise plays an important role in VCO functionality. In the digital circuits, phase noise can be defined as jitter. In the Fig. 3.1, the difference can be identified clearly. 20 Figure 3.1:
F
Comparison of phasse and jitteer noise n formulatte the circuit outputt as x (t) == Acos [ω
ω0t + φn (t)]. The ou
utput We can
waveform is sinussoidal whe
ere A is the
e amplitudde of VCO, ω0 is the desired ceenter frequen
ncy and φn (t) is the ccorrespond
ding phasee noise. Fo
or a detaileed analysis,, the output signal can be rewrittten as follo
owing: x(t)= A
A[cos(ω0t )cos(φn(t))−sin(ω0t)sin(φn(tt))] ((3.1) For smaall values o
of phase no
oise, we ge
et: sin(φn(t)) ≈ φ
(
t) ((3.2) n(t
cos(φn(t)) ≈ cos(0) = 1 ((3.3) Therefo
ore, the output is app
proximated
d as follow
wing: x(t) ≈ A
Acos(ω0t) − Aφn(t)ssin(ω0t) ((3.4) As show
wn in the FFig. 3.2, th
he output w
waveform of an ideaal and a reeal oscillattor is is responssible for w
comparred. Moreo
over, the resonator r
wiping out the unwanted signals to some extent. As A we mo
ove away from the center frrequency, less undesired signals are obserrved due to
o the filterring techniique. Therrefore, the real oscillato
or spectrum
m differs ffrom what is expecteed in theorry. In practtice, the phase noise is similar to a skirt in ffrequency domain. 21 Figgure 3.2: O
Output wavveform com
mparison o
of an ideal and a real oscillator
we propose
e a formulaa for calculating the pphase noisse over a 1Hz bandwidth. Here, w
Therefo
ore, we sho
ould consid
der a specific distancce from ou
ur carrier frequency. This distance
e is called an offsett frequency which iss depicted
d by ∆ω in
n the equaation (3.5). To obtain the t phase noise at this t
offset frequencyy, we takee advantagge of logarith
hmic scale in which the t noise power is ddivided byy the carrier power. The corresponding forrmula is as following: L ∆ω
10 log
g
P
∆ , H
P
((3.5) Ptone sho
ows the no
oise power at the de
esired offseet frequen
ncy from th
he carrier over a 1Hz b
bandwidth and Pcarrierr representts for the carrier power. L(Δω) is defined by dBc/Hz which rep
presents th
he phase noise valuee. In other words, it iillustrates how much th
he Ptone is b
below the Pcarrier in dB
B scale. 3.2 Pha
ase Noise
e Effect o
on RF Tran
nsceiver TTopologiies As we kknow, the ssignal reaching to an antenna sshould be sset at a low
wer frequeency. This fre
equency is called IF o
or interme
ediate freqquency. Ho
owever, the signal sh
hape should not change. On the other han
nd, we enccounter some disturb
bances staaying near ou
ur signal. TTherefore, after downconversio
on, we gett both desiired signal and unwantted interferer. The ph
hase noise
e of the loccal oscillato
or covers tthe main siignal and the
e interferer and hence the sign
nals overlaap each otther at thee intermed
diate 22 frequen
ncy. Overaall, the siggnal to noise ratio w
will decreaase at thee intermed
diate frequen
ncy. This ph
henomeno
on is called reciprocal mixing. Figure 3.3: Phase n
noise impaact on receeivers The Fig.. 3.3 corresponds to the receivvers. Howeever, it is th
he same fo
or transmittting the sign
nals. In a trransceiver,, the powe
er amplifie r strengthens the un
nwanted siignal which iss modulate
ed by the oscillator. On the otther hand, the interfferer specttrum gets larger as the offset frequency moves farther. This isssue corrup
pts the dessired signal to
o some exttent. 23 Figure 3.4
4: Phase no
oise impactt on transm
mitters 3.3 Analysis of Oscillator Phase N
Noise In this section, several the
eories are analyzed in detail regarding the oscilllator noise and iits impact on the cirrcuit perfo
ormance. A
Additionally, two fam
mous phase n
models are prese
ented in th
he followin
ng sections. At the eend of thiis chapter,, the noise contrributors are
e identified
d one by o
one. phase n
3.3.1 LLeeson’s EEquation of Phase
e Noise In this part, the VCO is asssumed as a one‐po
ort model. We divid
de the inteernal noises of o an osciillator into
o two grou
ups. The ffirst noise contributtor is the tank circuit w
while the ssecond one
e is because of the aactive devices. In oth
her words,, the lossy re
esonator and a
the trransistor pairs p
are tthe main noise con
ntributors that should be taken into account. The active components maainly produ
uce flickerr and thermal noise in the circuit. The corresponding m
model is giiven in thee Fig. 3.5. 24 Figgure 3.5: LC
C oscillatorr model wiith noise generators As depiicted in th
he model, the noise
e sources of an LC oscillator are identtified clearly and this would w
help
p us to specify how these eleements inffluence on
n the noise. To simplify the
e calculatio
ons, we deefine an id
deal resonaator. Howeever, phase n
if a non‐ideal tankk is assume
ed, its loss will be covvered by th
he active d
devices. equency of o ω, the im
mpedance seen at tthe resonaator input is as At our desired fre
ng: followin
L
Z ω
L C
((3.6) If we co
onsider an offset freq
quency of ∆
∆ω from thhe oscillation frequency, we geet: ω
ω
∆ω ((3.7) On the other hand
d, the oscillation freq
quency is:
ω
L C
((3.8) By replaacing (3.7) in (3.6), w
we get: ∆
∆
((3.9) ∆
Now, w
we expand tthe freque
ency terms in the dennominator. So, we geet: ∆
∆
∆
∆
25 (33.10) As it is clear from (3.8), 1
is equal to zero. On the other hand, the term is so small that can be ignored. So, we get: ∆
∆
(3.11) ∆
Since ω0 is quite larger than ∆ω, we can ignore ∆ω in the numerator. So the equation (3.11) will be refined as following: Z ω
(3.12) ∆
As explained in the previous chapter, the tank generates a parallel resistance which is illustrated by Rp. By obtaining the value of equivalent parallel resistance, we can easily calculate the quality factor of the tank. The equation is as following: Q
ω R C (3.13) By replacing (3.13) in (3.12), we obtain the following formula which can show the tank impedance as a function of the quality factor and the equivalent parallel resistance. R
Q
(3.14) ∆
Here, we square both sides of the equation and hence we obtain the absolute value of the tank impedance as a function of the offset frequency. |
R
|
(3.15) Q∆
Next, a new equation for calculating the overall output noise is presented. By using (3.15), we obtain: R
∆
∆
R
∆
|Z
ω | (3.16) 26 As observed in (3.16), the overall noise of the oscillator is determined. The equation is rewritten as following: R
∆
1
∆
R
R
∆
ω | (3.17) |Z
∆
The term in the parenthesis can be defined as following: F ∆ω
1
R
R
∆
∆
(3.18) On the other hand, for the parallel resistance noise generated by the tank, we get: T
R
R
∆
(3.19) The output noise spectral density is formulated as below: R
∆
∆
ω |
F ∆ω |Z
4KTF ∆ω R
Q∆
(3.20) To obtain the phase noise formula, we apply the equipartition theory to the (3.20). It proves that if a sinusoidal wave is obtained at the output, the noise is equally divided into two parts. One equation is dedicated to the phase noise while the other one points to the amplitude noise [22]. ∆
P
2KTF ∆ω R
Q∆
(3.21) The following formula is presented for calculating the phase noise at ∆ω offset frequency. L ω
10 log
S
∆
P
(3.22) The final formula can be rewritten as: L ω
10 log
KTF ∆
P
Q∆
(3.23) 27 The value of F(∆ω) is conssidered as two and hence thee overall p
phase noisse is formulaated as: 10 log
KT
P
Q∆
(3
3.24) As it is u
understood from the
e phase no
oise formulla, we get ‐20dBc/Hzz transition
n per decade. Howeverr, in practtice, the phase noiise does n
not follow
w this concept her words,, the noise of active ddevices neeeds a morre complicated complettely. In oth
analysiss. In this model, since the trransistors are the m
main contrributors off the thermal noise, th
hey are co
onsidered as resistorrs. Nevertheless, the flicker n
noise ed by the ttransistor p
pairs should be takenn into acco
ount. On th
he other hand, produce
the biass circuitry generatess some no
oise which influences the outp
put wavefo
orm. Moreovver, the acttive device
es show low
w impedannce when functionin
ng in the trriode region aand hence this cause
es the resonator’s quality factor to degrad
de. Figgure 3.6: P
Phase noise
e spectrum
m of a real oscillator As obse
erved in Fig. 3.6, the
e phase no
oise spectrrum differss from wh
hat is expeected for an ideal oscillator. For a a non‐ideaal oscillato
or, the phaase noise cchanges ‐3
30dB per deccade for lo
ow offset frequencie
f
es. Howeveer, for som
me offset ffrequenciees as shown in Fig. 3.6
6, the ram
mp is set at a ‐20dB/ddecade. M
Moreover, at high offset ncies, the n
noise floor causes the
e ramp to go to zero. frequen
ented an experiment
e
tal model which fullfills the equation (3
3.24) Leeson has prese
e phase noiise behavio
or of a real oscillatorr simultaneeously [12]]. The Leesson’s and the
experim
mental form
mula is as ffollowing:
28 10 log
1
1
∆
∆
/
|∆ |
(3
3.25) 5), ∆ / is the point p
wherre 1/ aand 1/ regions m
meet. In equaation (3.25
Leeson’’s formula depicts the inverse
e proportio
onality of the phasee noise to
o the resonattor’s powe
er dissipation. Based on this isssue, we o
optimize th
he phase n
noise value byy improvin
ng the oscilllation amp
plitude. Addditionally,, for our deesign, we h
have chosen efficient in
nductors w
with high quality facto
or. 3.3.2 H
Hajimiri and Lee’s Phase No
oise Equaation Since Leeson’s L
model m
is empirical, e
it cannott identify the effect of all n
noise generattors on th
he phase noise. On the otheer hand, ∆
∆ / and
d F terms are experim
mental in (3.25). Fo
or instance
e, if theree are seveeral activee compon
nents produciing flicker noise in th
he circuit, Leeson’s fo
ormula do
oes not rulee any morre. In other w
words, therre is no datta to identtify whose flicker noise should be chosen
n for calculatting the ∆ / . As understoo
od from (33.25), increeasing the quality faactor improve
es the overall phase noise perfformance. However, the F paraameter risees as well wh
hich degrades the ph
hase noise
e performaance [14]. Thereforee, the Leesson’s model cannot co
ompletely interpret the phasee noise beehavior off an oscillaator. uently, Lee
e and Hajimiri prese
ented ano
other model to stud
dy the imp
pulse Subsequ
responsse of an ide
eal oscillattor [13]. Figure 3.7: Impulse re
esponse off an ideal o
oscillator In this p
part, the im
mpact of current imp
pulse on thhe tank is iinvestigateed as show
wn in the Fig. 3.8. Forr instance,, if we apply an im
mpulse att the outp
put peak, the ude change
es while th
he phase re
emains thee same. Th
he amplitu
ude variatio
on is amplitu
estimatted as ∆v ∆q⁄c . In
n this form
mula, ∆q shhows the charge acro
oss the tan
nk. In other case, if we apply a pulse p
at the time axiis where the signal is crossingg the 29 zero po
oint, the ou
utput phasse will be aaffected. H
However, d
during anyy other tim
me of oscillation, the im
mpulse leads to phase and am
mplitude vvariation siimultaneously. on of the ssystem is id
dentified bby the timee of impulse occurreence. The phaase variatio
Consequently, an oscillator is considered as a time‐varying system. Moreover, the n is directlyy proportio
onal to thee pulse amplitude. value off amplitude variation
Figgure 3.8: Im
mpulse imp
pact on a ttime‐varyin
ng system As explained in previous p
se
ections, th
he designeers usuallyy do not ccare aboutt the ude noise due to th
he nonline
earities off active deevices in the oscillaator. amplitu
Figure 3
3.9 briefly d
describe th
his phenom
menon. Figurre 3.9: Amp
plitude noise of an oscillator in
n one perio
od Therefo
ore, to op
ptimize the
e phase noise n
performance, the noisee disturbances should o
occur at th
he peak of output siggnal. Figure 3.10: Curren
nt impulse
e behavior on a time‐‐varying syystem 30 The imp
pulse respo
onse of a ttime‐varying system is depicteed in Fig. 3
3.10. As sh
hown in the fiigure, a ph
hase variatiion is intro
oduced in tthe system
m. Overall, tthe oscillator’s phase vvariation du
ue to the ccurrent imp
pulse is forrmulated aas followin
ng: ,
(3
3.26) presents for f impulsse sensitivvity functio
on (ISF). It illustrattes the phase Γ(x) rep
variatio
on when a ccurrent impulse is ap
pplied at a specific tim
me. Figure
e 3.11: Imp
pulse sensiitivity funcction of a reeal oscillattor Obvioussly, when tthe oscillator is morre susceptiible to currrent impulse driven into the reso
onator, the
e amplitud
de of ISF fu
unction risees. This leaads to phaase variatio
on of the output signal. w take ad
dvantage of o the sup
perpositionn principlee to calculate the phase Here, we noise ge
enerated b
by the currrent impulsse. Γ
,
(33.27) understand
ding of equ
uation (3.227), we preesent the ffollowing b
block To obtaain a clear u
diagram
m. It showss how a currrent impu
ulse influennces the ou
utput signaal. 31 Figure 3.12: Cu
urrent imp
pulse effectt on the ou
utput signaal Since ISSF function
n is periodic, we take
e advantage of Fourieer series to
o remodel it as the follo
owing equation: Γ
∑
cos
c
(3
3.28) All of th
he parameters are re
eal in equation (3.28)). The harm
monics phaase is depicted by in
n the ISF fu
unction equ
uation. The
erefore, thhe overall p
phase of th
he ISF funcction is as following: ∑
cos
(3
3.29) dea of th
he equatio
on (3.29), the equivvalent blo
ock diagram
m is To get a clear id
own in Fig. 3.13. In this mo
odel, the coefficien
nts of imp
pulse presentted as sho
sensitivvity functio
on should be calculaated to esttimate thee overall p
phase noisee. In other words, w
sevveral noise
e compone
ents are ggenerated due to cu
urrent imp
pulse injection to the tank. They are locate
ed at diffeerent frequ
uencies regarding to
o the nts of impu
ulse sensittivity functtion. This b
behavior is illustrateed in Fourier coefficien
3. The ISF block diagram and th
he proceduure of convverting thee current n
noise Fig. 3.13
to phase noise are
e illustrate
ed in Fig. 3..13 and 3.114 respectively. 32 Fiigure 3.13:: ISF block diagram ncy of ω0 iis upconveerted The 1/f noise of tthe transisttor pairs aat the centter frequen
Actually, th
he flicker n
noise stayss close to tthe carrierr. Addition
nally, to 1/ region. A
the c1 coefficient
c
t in Fourie
er series represents r
for the n
noise nearr to oscillaation frequen
ncy. On the
e other haand, the th
hermal noiise of the componen
nts is the m
main contribu
utor of phaase noise iin 1/ reggion. The cc2 coefficieent determ
mines the n
noise of 2ω0 w
while c3 represents th
he noise co
orrespond ing to 3ω0 and so forth. The equ
uivalent ph
hase noise
e of ISF blo
ock diagram
m is calcullated as eq
quation (3.30). The relaative phase
e noise is cconsidered
d in 1/ reegion. ∆
10 log
g
∆
.
∆
(3
3.30) 33 Figgure 3.14: Procedure
e of currentt noise connversion in
nto phase n
noise [14]
he magnituude of currrent noise, ∆f is regarded In equation (3.30)), in represents for th
bandwidth
h of noise impulse, flicker noise frequency of the ccomponen
nts is as the b
illustratted by ω1/ff, the desirred offset frequencyy for calculating the phase noise is shown b
by ∆ω, the
e maximum
m charge to
olerated by capacito
ors in the taank is giveen by qmax and finally the Γrms shows the root mean square off the impu
ulse sensittivity function
n. Here, we w have presented a formula to determ
mine the phase noiise of the 1/f3 region. To obtain a precise value of p
phase noisse in 1/f3 rregion, we should deefine the currrent noise
e generate
ed by the relative fflicker noisse. The exxpression iis as followin
ng: ı
, /
ı
∆
/
(3
3.31) 34 The c0 coefficient in Fourier series indicates the DC value of the impulse sensitivity function. Therefore, we combine the equations (3.30) and (3.31) to determine the phase noise produced in 1/f3 region. ∆
10 log
∆
/
∆
∆
(3.32) To calculate the relative corner frequency of 1/f3, we set the equations (3.30) and (3.32) equal to each other. Therefore, we get: Δω
/
/
/
(3.33) As we know, ∑
is equal to 2Γ so we can improve the phase noise performance at all offset frequencies by reducing the coefficients of impulse sensitivity function. It is understood from equation (3.33) that we can reduce the noise spectrum of 1/f3 region by reducing the constant value of impulse sensitivity function. In practice, this idea is achieved by reducing the rise and fall times of the oscillator to generate more symmetrical outputs. Additionally, this formula reveals the interaction between the phase noise and the resonator dissipation. In other words, if the resonator dissipation is compensated by transistor pairs; the overall phase noise will be improved. This idea is verified when the output is at its peak value. 3.3.3 F‐Parameter Rael and Abidi dedicated their research to identify the effect of noise generators on overall phase noise performance of a VCO [15]. They presented a formula in which all noise sources and their effects are taken into account. Their experimental equation is as following: F
1
,
(3.34) Now, we consider that the oscillator is in current‐limited region. Therefore, the second term in equation (3.34) will be equal to 2γ. On the other hand, by 35 increasiing the bias b
curre
ent, the phase no
oise perfo
ormance will imprrove. Neverth
heless, the
e output amplitude a
is restrictted by the Vdd in vvoltage‐lim
mited region. Therefore
e, V0 doess not chan
nge while the bias current increases. This causes the F‐paarameter to increaase and consequeently the phase n
noise perform
mance will be corrup
pted. Overrall, the beest phase noise is aachieved w
when the traansistors are a biased
d between
n the currrent‐limitted and vvoltage‐lim
mited regions. 3.4 Analysis of Phase No
oise Sources in CM
MOS LC VCOs In this section, the phase noise contributo
c
rs are ideentified in
n our design. Therefo
ore, it is be
etter to anaalyze each of the phaase noise ccontributo
ors individu
ually. Howeve
er, unlike previous chapter, the resonnator noisse is explained briiefly. Moreovver, a comp
plementary mechaniism for achhieving a b
better noisee performaance is introd
duced. A simple sche
ematic of LC VCO is shown in Fig. 3.15. A
As we see,, the phase n
noise contrributors are
e illustrate
ed individuually. Figgure 3.15: Noise sources in LC cross‐coup
pled VCO 3.4.1 T
Transistorr Noise Contributiion As we d
described b
briefly in Section 3.3,, when thee circuit is oscillatingg in very low or high fre
equencies,, it sufferss from the
e phase nnoise correesponding to the acctive devices. Therefore, it is more logical tto investiggate all of tthe noise ccontributo
ors in an oscillator regarrdless of th
he operatio
on frequenncy. 36 3.4.2 N
Noise Con
ntributorss in MOSFET The noise contributors in a MOSFET device are ddivided intto two groups of inteernal and extternal noises. As shown in Fig. 3.16, a lum
mped mod
del is preseented in w
which all of th
he noise ge
enerators aare illustraated. The eexternal no
oises show
wn in the figure are produced by tterminal p
parasitic capacitors. A
Additionallly, the resiistances off the metal to semicon
nductor connection can c add so
ome extrin
nsic noise to our design. On the other han
nd, the inte
ernal noise
e is also divided into
o two grou
ups of thermal and flickker noise. Figu
ure 3.16: LLumped no
oise model 3.4.2.1
1 Thermal Noise There are a some sources s
which are the main ccontributors of the tthermal no
oise. The noise produced by the resistance
e of the gate is one o
of the main sources. The gate ressistance is expressed
d as followiing: R
W R
N L
(3
3.35) In equaation (3.35
5), Rg gene
erates the
ermal noise in which R is deffined as sheet resistan
nce and N representss for numb
ber of finggers. Howeever, the reesistance sseen at the ggate of the transistors can be optimized w
when perfo
orming thee circuit layyout. If we consider c
th
he gate re
esistance as a an indiividual eleement, its value willl be optimized as follo
owing: 37 R
R
,
(3.36) Nevertheless, in the layout procedure, the gate resistance can be even less than above and hence this leads to better phase noise performance. As a result, the overall noise seen from the gate resistance is formulated as following: v
,
4kTR , ∆f (3.37) The Boltzmann’s constant is shown by k while T represents the relative temperature. As understood from equation (3.35), the gate resistance is proportional to width, length and number of fingers. Therefore, we should keep these issues in mind while performing the final layout of the circuit. The channel noise is considered as the other contributor of thermal noise in the circuit. The second source of thermal noise in a MOSFET is the thermal channel noise. The carriers in the channel produce a relatively large drain noise which is regarded as a current noise. It is the most considerable noise source in a MOSFET. If the transistor is saturated, it corresponding noise is estimated as following: ı
4kTγg ∆f (3.38) In equation (3.38), the gamma changes proportionally to the bias current while gd0 represents the channel transconductance. However, we can replace gm instead of gd0 since the oscillator is functioning in the saturation region. The gamma parameter is proportional to the drain to source voltage. In other words, increasing Vds produces a larger γ. For simplicity, gamma is estimated as 2/3 in our design. However, for short channel devices, the value differs. On the other hand, a parasitic capacitance is seen between the gate and source of the transistor which generates a negligible noise called the induced gate noise. The corresponding expression is as following: ı
4kTβ
C
∆f (3.39) 38 In equation (3.39), β is a parameter depending on the bias current while kgs is a coefficient for long channel devices. It is usually set to 4 in our design. Moreover, the induced gate noise will change due to the frequency variations. Obviously, we cannot use the above equation when the circuit is oscillating at very high frequencies. 3.4.2.2 Flicker Noise As a matter of fact, for the lower oscillation frequencies, the noise spectral density improves. The flicker noise behaves differently from what we discussed about thermal noise. Actually, the flicker noise range is estimated between 100 kHz to some MHz. Since it is inversely proportional to frequency, we call it 1/f noise. Here, we discuss about a theory to clarify the flicker noise behavior. It shows that the charges stored near the silicon interface under the transistor’s gate contribute to flicker noise. This theory is called the carrier density fluctuation. On the other hand, it illustrates that the gate bias voltage cannot affect the flicker noise while the noise power interacts closely with the charges stored beneath the gate of MOSFET. The flicker noise is formulated as following for the saturation region. ı
K
∆
C
WL
(3.40) 3.4.2.3 Bias Noise In the related bias circuit, there are also several noise contributors that should be taken into account. As mentioned before, if the circuit is oscillating at low frequencies, the flicker noise is mostly in focus while at high frequencies the circuit suffers from the thermal noise. In the design, the transistor pairs behave like mixers due to their switching capability. Therefore, the fundamental harmonic at ωn is upconverted to ω0‐ωn and ω0+ωn. The frequencies are located symmetrically from the center frequency. Since the phase variations compensate each other, the overall noise is generated as AM. 39 3.4.2.4 Switching Pair Noise In this section, we discuss the switching pair noise. It can be analyzed easily compared to previous models. As mentioned earlier, we focus on flicker noise at low frequencies due to its dominant effect on the circuit performance. In this region, a small impedance is measured from the transistor drain. However, we neglect it and hence it will be considered as short. Consequently, the flicker noise can be modeled as parallel with the bias noise [15]. Moreover, the flicker noise will be upconverted. On the other hand, the active devices linearity plays an important role in the overall phase noise performance. To reduce the flicker noise, we can decrease the width of transistors. 3.4.2.5 Resonator Noise Finally, we should consider the noise corresponding to our tank circuit. As discussed in previous chapter, we can assume all of the noise contributors in the VCO as a negative resistance thermal noise. On the other hand, the F‐parameter connects all of the noise sources to each other. For simplicity, we ignore the noise of active devices and hence the F‐parameter will be equal to unity. In other words, we are analyzing the equation (3.34) that is presented by Rael and Abidi for the tank noise. The following formula expresses the noise produced by the loss of passive elements in the resonator. 10 log
∆
(3.41) 40 4. STATE‐OF‐THE‐ART DESIGN 4.1 Introduction In this chapter, we present some low phase noise LC architectures. One of the most recent topologies is a quadrature VCO with integrated back gate coupling. Nowadays, quadrature VCOs are widely applicable in transceivers. One of the drawbacks of this topology is higher power consumption than the previous LC VCO architectures. To get an optimum result regarding the power and the phase noise simultaneously, a much simpler design is presented. At the end of this chapter, we compare all of the benefits and drawbacks of different VCO architectures. Finally, a low noise low power CMOS LC oscillator is designed. 4.2 Low Phase Noise QVCO Nowadays, CMOS technology is the most applicable solution for modern wireless communication devices. The challenge of being implemented on a single chip makes the design much complicated. In this topology, high efficient transmission is performed by the help of quadrature‐amplitude modulation and the frequency division technique. Recently, some transceivers use quadrature oscillators to drive mixers for performing frequency conversion. However, the signal is susceptible to the phase noise disturbances. There are different methods to produce the quadrature signal. The differential voltage controlled oscillator, the quadrature coupling of two simple LC VCOs, the ring oscillators and the frequency division technique are the most common procedures for producing a low phase noise signal. The quadrature topology is popular among designers due to its high performance regarding the phase noise. The quadrature topology can be done in different ways. Back‐gate coupling or adding some transistors to the VCO core are some common procedures. One of these approaches is called source resistive degeneration which has a noticeable impact on phase noise improvement in quadrature VCOs. In this state‐of‐the‐art design, we take advantage of source resistive degeneration and back gate coupling simultaneously. In other words, we put the both procedures into one single model to achieve a significant output with low power dissipation and low phase noise at the same time. 41 4.2.1 C
Circuit De
esign In this ssection, a 6
65nm CMO
OS quadratture VCO w
with a signiificant low phase noiise is designe
ed. As men
ntioned in introductory sectioon, we mixx two diffeerent meth
hods into one
e model to
o achieve tthe best phase noisee performaance. As observed in
n the final simulation results, th
he phase noise and the power consumption h
have improve
ed significantly in a well‐desiggned QVCO
O. Howeveer, a largeer area on
n the chip sh
hould be dedicated for the design. O
Overall, wee achieve an optim
mum perform
mance regaarding the phase noisse and pow
wer consumption. Th
he phase n
noise is ‐140 dBc/Hz at 1 MHz offfset from 2.4 GHz. TThe QVCO consumess 3 mA fro
om a 1.2 V po
ower supply. The QV
VCO circuitt schematiic is shown in the FFig. 4.1. Th
he two CM
MOS VCOss are coupled
d back to back by th
heir gates. In other words, thee circuit consists of two CMOS LLC VCOs w
with eight transistors.. In our deesign, the PPMOS bulkks of each VCO are connected by coupling ccapacitors as shown in the circu
uit schemaatic. Quadraturee VCO [18] Figgure 4.1: Ciircuit schematic of Q
ure VCO hhas an advvantage co
omparing with This tecchnique ussed in ourr quadratu
conventtional VCO
Os. As we n
notice in o
old QVCO designs, fo
our more ttransistorss are 42 used ass coupling elements. A simple conventional quadraature VCO
O is depicteed in Fig. 4.2.. Figure 4
4.2: Conventional Quuadrature V
VCO proposed d
design, the
ese extra ttransistors are omittted due to the back‐‐gate In our p
couplingg techniqu
ue. Conseq
quently, th
he circuit pperforman
nce improvves due to
o the reductio
on of noise sources. As described in previous chapters, adding m
more transistors leads to extra noise n
of th
he circuit. Flicker noise of NMOS and PM
MOS transistors are the t
main factors th
hat shouldd be con
nsidered in
n our design. Therefo
ore, the correspondin
ng phase noise can bbe formulatted as follo
owing: L ∆
10 log
g
,
.∆
∆
.
⁄ ,
∆
,
.∆
∆
.
⁄ ,
∆
(4.1) C0 show
ws the sym
mmetry of the outpu
ut signal. A
Actually, it is the coeefficient off the Fourier series for the impulse function. The , ∆ reprresents thee overall n
noise d by NMOS pairs an
nd its correspondingg frequenccy is show
wn as density produced
. . For the PMOS P
tran
nsistors, the terms , ∆ and ⁄ , rrepresentss the overall noise denssity and the corner frequency rrespectiveely. NMOS transistorss are the maiin noise co
ontributorss in our de
esign due tto the flickker noise u
up‐converssion. Therefo
ore, it is usseful to takke care of this issue by applyin
ng some reesistors on
n the source of NMOS transisttors. This is calledd source resistive degeneraation technique. There
efore, by applying a resistannce at eaach NMOSS source, the e will deccrease by a coefficient of 1⁄ 1
. transconductance
,
As far as ⁄ , is propo
ortional to
o , the an
ngular corn
ner frequeency ,
decreasses as well. On the other hand,, the transsconductan
nce is set aat a fixed p
point ⁄ ,
43 by the help of Rsource. This technique gives more symmetry to the drain current. In this case, more improvement in phase noise is achieved. A designer might think that this technique can be considered for PMOS transistors as well. However, in our circuit, the PMOS transistors should have quite large gm for phase locking. Consequently, this procedure might not be suitable for PMOS transistors. On the other hand, we encounter some limitations when designing the Rsource. High Rsource value can ruin the oscillation initial condition and produce some additional disturbances. The circuit is designed in CMOS 65nm technology. There is a big challenge for selecting a proper inductance. The quality factor of the inductor should be considered as well. We use a spiral structure for designing the inductor. Its value is 2.1 nH and its corresponding quality factor is 15. As observed in the final results, the value of Rsource is 26 ohm. 4.2.2 Design Specifications The oscillation frequency is functioning between 2.25 to 2.55 GHz when the Vtune is tuned from 0.2 to 1.2 V. Our designed output power varies from ‐0.5 to ‐1.6 dBm. In the frequency range of 2.25 to 2.55 GHz, the best result is achieved at 2.4 GHz. The varactor used in our design has a hyperbolic capacitance versus voltage curve. This makes the middle of tuning range a critical point. At 2.4 GHz, the phase noise is ‐140dBc/Hz at 1 MHz offset frequency. Our designed quadrature VCO consumes 3mA from a 1.2V supply voltage. The equation for calculating the figure of merit for VCO is as following: FOM
L∆
20 log
∆
10 log
44 DC
(4.2) 4.2.3 Simulation Results As described in the introductory part, we have designed a 2.4 GHz quadrature VCO. Its corresponding phase noise is ‐140 dBc/Hz at 1 MHz offset frequency. In the design procedure, two simple CMOS LC VCO are coupled together to satisfy the oscillation condition at the desired frequency. The body terminal of PMOS transistors are connected together via coupling capacitors. Additionally, four resistances are added to the source of NMOS transistors to reduce the transconductance as much as possible. Therefore, we have less gm variation at the output. On the other hand, phase noise is decreased as well. This is called source resistive degeneration technique. 4.3 Low Noise Low Power CMOS LC VCO Oscillators are inevitable blocks in designing communication systems. There are different LC VCO topologies in communication electronics. LC VCOs are mainly applicable in highly efficient transmitters and receivers. VCOs are used as inputs for the mixers to produce desired outputs. Therefore, they are quite noticeable in highly integrated transceivers. Low noise and high signal amplitude should be achieved for obtaining a reasonable performance in a VCO design. To obtain a state‐of‐the‐art design, two specifications should be met at the same time. LC VCOs are mostly popular due to this issue. They achieve an ultra‐low noise with low power dissipation simultaneously. Therefore, the designers are encouraged to design efficient LC VCO topologies. Nowadays, lots of investments have been focused on designing CMOS LC VCOs using on‐chip resonators. The drawback is that fully integrated LC VCOs consume lots of power. Therefore, external LC VCO topologies are still used in recent cell phones. In this design, we aim for an optimal circuit using fully integrated VCOs. Our goal is to produce outputs with lower phase noise and power dissipation comparing with conventional off‐chip LC VCO topologies. This work mainly concentrates on design of fully integrated VCOs with optimized power consumption and phase noise lower than VCOs with external resonators. For this design, we mainly discuss the complementary LC VCO structure. Then, we compare its performance, regarding phase noise and power consumption, with conventional VCOs. 45 4.3.1 CMOS LC VCO There are some noticeable advantages that make CMOS LC VCO an identical topology. Complementary VCOs are more economical than their conventional counterparts. Old topologies use only NMOS or PMOS transistors. In our design, we use both type of transistors at the same time. By adding PMOS transistors to conventional NMOS only VCOs, much larger transconductance is achieved. As we know, the tank used in the circuit is lossy. Since we generate a noticeably large transconductance, less current is needed to compensate the resonator loss. Therefore, much power is saved in this topology. On the other hand, using the PMOS and NMOS pairs simultaneously, we produce symmetrical waveforms at the output. Consequently, the flicker noise upconversion to the 1/f3 region is reduced. The CMOS LC VCO is illustrated in the Fig. 4.3. If the VCO requirements are fulfilled, the circuit oscillates properly. In the theory, the amplitude increases gradually and stops in a point. Actually, when the negative resistance cannot compensate the resonator loss any further, the output will be stable. However, it is the case when the Vdd is not putting constraints on the output swing at the oscillation startup point. When operating at the current limited regime, the CMOS cross‐coupled VCO is the best choice for the state‐of‐the‐art design. Applying the same voltage and bias current, it generates a better phase noise comparing with its NMOS or PMOS counterparts. The phase noise can be analyzed in different aspects. First, the CMOS cross‐coupled VCO tolerates a larger charge for the output swing. This maximum swing is illustrated as qmax in the Hajimiri’s model. As the phase noise is inversely proportional to the maximum charge, the CMOS cross‐coupled VCO generates a better phase noise comparing with its conventional counterparts. On the other hand, we can improve it to the Without Tail (WT) structure. The WT structure shows even a better performance than the fixed biasing topology. In the WT topology, the number of transistors is reduced. In another words, we decrease the number of noise sources. Therefore, the flicker noise sources are just of the cross‐coupled pairs. Since switched biasing is applied in the design, the cross‐coupled pairs do not affect the phase noise performance that much due to their low flicker noise. Overall, in this design, we improve the phase noise performance by omitting the tail transistor. On the other hand, since 46 the tail transistor is neglecte
ed, we do not have vvoltage headroom lim
mitation in
n our design. However,, the tail transistorr in the ffixed biasiing structu
ure put some constraints on the voltage headroom
m which caan be prob
blematic fo
or low voltage topologgies. In fixe
ed biasingg topology, the outpput signal has a low
w power. SSince phase n
noise is de
efined as signal to no
oise ratio, low signaal power caan worsen
n the phase noise n
perfformance. Therefore
e, in the ffixed biasin
ng model,, an indiviidual circuit should s
be designed to bias the extra traansistor. C
Consequen
ntly, the circuit dissipattes more power and on the other hannd we sho
ould modeel more n
noise generattors for the
e fixed biasing design
n. In the FB topologyy, the noise produceed by the extrra biasing ccircuit is trransferred to the taill transistorr. In our deesign, we h
have solved tthis proble
em as well. hematic Figure 4.3: CMOS LC oscillator circuit sch
In our p
presented structure, since the amplitudee changes all the tim
me, the currrent varies ccorrespond
dingly. To m
maintain a proper osscillation condition, tthe NMOSS and PMOS pairs use
ed in ourr design should prroduce an
n approprriate negaative 47 resistance. The transconductance produced by cross‐coupled pairs should be inversely proportional to the overall resistance of the resonator. As obtained in our experimental simulations, we can optimize the power dissipation by improving the quality factor of our resonator. As a matter of fact, this will optimize the required transconductance as well. However, using fully integrated inductors generates some obstacles for the design. Fully integrated VCOs have low quality factors. On the other hand, there are some boundaries for increasing the quality factor of the inductors. 4.3.2 Power Analysis If we apply Vdd as our supply voltage to the circuit, the voltage measured at the output could be estimated as Vdd/2. Vm represents the output amplitude by which the gate to source voltages can be formulated as following: Vgs1=Vdd‐Vgs3=Vdd/2+Vmsin(ωt) (4.3) Vgs2=Vdd‐Vgs4=Vdd/2‐Vmsin(ωt) (4.4) The NMOS transistors switch on when Vgs≥Vth,NMOS and the PMOS transistors switch on when Vsg ≥ |Vth,PMOS|. As observed in the theory, if we subtract the NMOS and PMOS current from each other the resonator current will be identified. Since one of the PMOS or NMOS transistors is switched on at each cycle, a larger current is conducted into the resonator. Consequently, when the current is driven by one of the M1 or M3 at each cycle, the power dissipation is reduced. On the other hand, since we reduce the number of transistors in each cycle, less noise is driven into the resonator. As observed in above equations, we should consider some limitations for choosing the right supply voltage. If we apply a supply voltage which is larger than the overall threshold voltage of the M1 and M3 transistors(Vth,NMOS+Vth,PMOS), they will be switched on at the same time. Consequently, the circuit dissipates more power and extra noise will be conducted into the resonator. Increasing the overall noise in the circuit has an inverse impact on the phase noise performance. Overall, to dissipate less power and improve the phase noise performance, we should present a state‐of‐the‐art structure. In this topology, M1 and M3 are not allowed to conduct at the same 48 time in each cycle. This is the same case for M2 and M4 transistors. Now, it is understood the reason to minimize the supply voltage to overall threshold of PMOS and NMOS transistors. Applying the Vdd equal to Vth,NMOS+Vth,PMOS , the output voltage will be estimated as NMOS threshold voltage(Vth,NMOS). This ensures that M1 and M3 or M2 and M4 transistors would not be switched on simultaneously. Therefore, it guarantees that each of the NMOS or PMOS transistors is switched on for half of the oscillation cycle. The other issue that should be analyzed in details is choosing a right inductor with a suitable resistance. The noise produced by the inductor has a power equal to 4
. Optimizing the inductor’s value has several impacts on the performance of the whole circuit. By reducing the inductor’s value, its overall resistance will be decreased as well. Consequently, the phase noise performance will be improved because less noise is produced by the resistance. On the other hand, when the inductor generates less resistance, the corresponding transconductance for the transistors will reduce as well. This leads to less current and hence the power consumption will be optimized. Another advantage of choosing a small inductor is to decrease the reciprocal effect of inductors designed on our chip. On the other hand, when the inductor’s size is minimized, a larger capacitor should be chosen to keep the oscillation frequency at the desired value. Larger capacitors will increase the maximum charge that can be tolerated. Based on Hajimiri’s formula, increasing the qmax will improve the phase noise performance. If we consider a defined unit area on the chip, the capacitance value that can be allocated to that area is much larger than the inductance that can be specified to that space. Therefore, by reducing the inductor size and increasing the capacitor’s value to fix the oscillation frequency at the desired value, the needed area on the chip will be minimized. However, there are some constraints for decreasing the inductor’s value. The tank amplitude can be modeled using a current source which turns on and off very fast from one transistor pair to the other. Since the voltage direction on the resonator changes in every moment, the current direction reverses dynamically through the resonator. Therefore, we can model the whole circuit as current source switching in two directions of Ibias and –Ibias. The current source is feeding the parallel RLC 49 tank all the time. Req is defined as the equivalent resistance of the resonator. At the resonance frequency, the inductor and the capacitor cancel each other due to their admittances. At the end, what remains is the equivalent resistance of the tank (Req). Since the LC tank mainly weakens the effect of individual harmonics of the input current, the fundamental harmonic can produce a noticeable output swing. Its corresponding amplitude can be estimated as (4/π)IbiasReq. However, the output can be estimated as a sinusoidal waveform at higher frequencies. In sinusoidal waveforms, the output can be estimated as IbiasReq. Therefore, without considering these limitations, reducing the inductor value can be problematic. When decreasing the inductor value, the equivalent parallel resistance decreases as well. Consequently, the tank amplitude decreases noticeably. In our design, the overall resistance in parallel is estimated as R
r
Q which is around 320Ω. To fulfill the startup condition, the transconductance should fit in the following formula: g
,
1/R (4.5) To minimize the flicker noise upconversion effect, equal transconductance for the NMOS and PMOS transistors should be chosen. The width and length of transistors is decided by following equations: W
L NMOS
W
L PMOS
,NMOS
IB µ C
,PMOS
IB µ C
(4.6) (4.7) To minimize the short channel noise, proper length and width should be chosen. Regarding phase noise calculation, Hajimiri presents a model as following: ∆
10 log
∆
∑
∆
(4.8) 50 Cn represents the coefficients of the VCO Fourier series. As mentioned earlier, qmax shows the maximum charge that can be stored in the capacitor. The noise power spectrum is shown by the term ∆
. As discussed earlier, we can reduce the inductor noise by decreasing its value. Regarding this issue, the current needed for the compensation of lossy resonator will minimize. Consequently, the VCO suffers from less noise which is one of our goals. To meet the desired center frequency, if we reduce the inductor size the capacitor value should be increased. Applying the proper Vdd equal to the overall threshold voltage of an NMOS and a PMOS transistor, only two transistors will conduct in each of the half oscillation periods. This issue saves the power and decreases the overall noise. Overall, the phase noise performance will be improved. 51 5. LC VCO DESIGN In this part, some basic cross‐coupled LC oscillators are compared regarding low phase noise and power consumption. Selecting the most appropriate model, the oscillator is improved by reducing the effect of the phase noise to achieve a high FOM. Additionally, some LC voltage‐controlled oscillators are implemented in order to decrease the power consumption as well as satisfying the phase noise. Usually, the bias current is maintained the same. 5.1 CMOS Cross‐Coupled Models The most common cross‐coupled LC model is implemented. Additionally, NMOS, PMOS and CMOS cross‐coupled topologies are analyzed in detail. In the following sections their benefits and drawbacks are compared as well. 5.1.1 Analysis and Comparison The aim of this chapter is to do a deep analysis between VCO models by applying equal power and using elements in 65nm library. We choose an inductor with a value of 3nH. Its quality factor is almost 10 at 2.4GHz. The supply voltage is fixed at 1.2V. The current source is set to 2.7mA. As we know, the CMOS model has larger swing output. The output of the CMOS topology is proportion to the bias current and the loss of tank. The active devices are the standard 1.2V PMOS and NMOS transistors in the cadence tool. It indicates that Vds, Vbs and Vgs voltages should not go beyond 1.2V. An appropriate capacitor is chosen and its value is set regarding the oscillation frequency at 2.4GHz. An initial condition of ∆V is applied to the capacitors. We keep the transconductance of NMOS and PMOS transistors the same. By selecting the same gm for PMOS and NMOS transistors, a better phase noise is obtained since a symmetrical output is accomplished. The comparison is made by sweeping the transistor parameters in PSS analysis. 52 Figurre 5.1: Diffe
erent desiggns of CMO
OS cross‐coupled VC
CO er, we havve not take
en some isssues into considerattion. In ou
ur compariison, Howeve
the currrent source is ideal. Half of the
e bias currrent is drivven by the transistorrs on each sid
de. It can llead to a n
negative vo
oltage dro
op on bias current. For example, in CMOS ttopology iff the W/L ratio is low
w, the Vgs will be able to have a larger vvalue than Vdd/2. However, the de
esigner mu
ust take thhe voltage headroom
m into acco
ount. Additionally, the noise of the t bias circuit shouuld be con
nsidered ass well. On
n the other hand, h
the oscillation
n initial situation is not the same in ttheory an
nd in practice
e. It starts at a lower loop gain
n compareed to a praactical circcuit. The o
other issue is considerin
ng the ideaal capacitaance in our comparisson. The ccapacitor h
has a very larrge quality factor. When using a varactorr in the dessign, the q
quality facttor is conside
erably less.. In the reaal design tthe dimenssions of th
he transistors are sw
wept. Therefo
ore, the total parasitic capacitaance variess leading tto shifts off frequenccy. In cross‐co
oupled CM
MOS, the frrequency change c
co
onsiderablyy due to lo
ow mobilitty of PMOS ccouples. Re
egarding th
he Leeson’’s formula,, as the freequency riises, the phase noise off the circuiit will incre
ease. 53 Howeve
er, the de
esigner ge
ets into an a ambiguuous undeerstanding through this comparrison. It is aalmost com
mplicated to interpreet the phaase noise p
performancce in detail. In I PSS anaalysis, the results are shown bbriefly in tthe following figuress for s. As men
differen
nt offset frequencie
f
ntioned beefore, it iss considerrably tough to interpre
et the grap
phs. Hence
e, the designer can hardly givve precise commentts on phase n
noise behavvior. In this section, s
we compare
e differentt topologiees regardin
ng the Wp//Wn ratio. The sizing of o NMOS transistors t
providingg an appro
opriate gaiin is 18/0.15. For PM
MOS design, the dime
ension of the transistors is 448/0.15. In
n theory, ffor low offset frequen
ncies, PMO
OS satisfiess the phase noise reequirements but NM
MOS transisstors may no
ot fulfill the
e phase noise perfo
ormance. A
As we know, high trransistor w
width causes lower flickker noise. A
At 1MHz o
offset, crosss‐coupled
d structuree accomplishes the mo
ost desirab
ble phase noise perrformance while thee phase n
noise of o
other structurres is app
proximately 6dBc/H
Hz greaterr than th
he CMOS cross‐coupled topologgy. As men
ntioned beffore, at higgh offset ffrequenciees the flicker noise efffect decreasses and the most no
oticeable region r
is 11/f2 region. Overall, the discusssion shows tthat a soph
histicated ccircuit shou
uld be des igned for low offset frequenciees. Figure
e 5.2: NMO
OS phase n
noise at 100 kHz offseet frequen
ncy 54 Figure
e 5.3: PMO
OS phase n
noise at 1000 kHz offseet frequency Figure
e 5.4: CMO
OS phase n
noise at 1000 kHz offseet frequency 55 Figurre 5.5: NM
MOS phase noise at 1M
MHz offsett frequenccy Figure 5.6: PM
MOS phase noise at 1M
MHz offsett frequenccy 56 Figure 5.7: CM
MOS phase noise at 1M
MHz offsett frequenccy 5.1.2 D
Design Pro
ocedure Various types of ccross‐coup
pled LC VCO
O are analyzed in this section. In this design, a 500m
mV output swing is obtained with a bbias curren
nt set at 2.7mA. W
When choosin
ng the pro
oper transsistor dime
ension, thhe phase noise specifications are fulfilled for all de
esigned VC
CO topologgies. The lowest len
ngth shoulld be used
d for CMOS ttransistors to decreaase parasitic capacito
ors. The traansistor w
which is useed in the bias circuit should havve a low overdrive o
vvoltage du
ue to the limited su
upply voltage. 5.1.2.1
1 Cross‐Co
oupled LC
C VCO At this point, the cross‐coup
pled LC oscillator witth NMOS bias circuitt shown in
n the Fig. 5.8 is implemented. App
plying a supply voltagge of 1.2V,, with prop
per W/L raatios, the Vgs will be so high that tthe headro
oom voltagge of the b
bias transisstor decreaases. Therefo
ore, it avoids the de
esired currrent to flo
ow in the circuit. As describeed in previou
us chapter,, we can upgrade u
th
he circuit to the WTT topologyy to solve this problem
m. 57 Figure 5.8:: CMOS cro
oss‐coupled LC oscillaator with N
NMOS biass circuit e some connditions th
hat should be taken into When aanalyzing tthe circuit,, there are
the account durin
ng our dessign. When increasinng the W//L ratio of f the switcching pair, the designerr encounte
ers some issues. By increasingg the W/L ratio, the bias transistor copes with w the voltage v
drop so the current fflows easily through
h the circuit. However, the parasitic elements reveal more than before. Consequently, the tuning range w
would be limited. CMO
OS LC Oscillator NMOS
Refference Capacitaance
Current W[µm] W
L[µm] W[µ
µm] L[µm
m] W[µm
m] L[µm] [pF]]
PMOSS‐Bias 60 0.15
0
34
3
0.15
25
0.15 0.62
2
NMOSS‐Bias 60 0.15
0
34
3
0.15
50
0.15 1.3
Self‐‐Bias 27 0.15
0
15
1
0.15
‐
‐ 0.9
Taable 5.1: CMOS crosss‐coupled LLC VCO dim
mensions PMOSS 58 Below, we prese
ent a sim
mulation of the CM
MOS cross‐‐coupled ttopology. The importaant challen
nge is the b
bias transisstor in thee design. A
As mention
ned in prevvious sectionss, it should
d operate iin saturation region.. On the otther hand,, there is aa low voltage drop on V
Vds. In the ssimulation
n, the tail ccurrent is 22.4mA whiich is less tthan the the
eoretical assumptio
on. The sizing s
of ttransistorss is largeer than o
other topologgies. The parasitic p
capacitors c
play an im
mportant role for eestimating the overall capacitancce. Therefo
ore, a 0.62
2pF capacittor is need
ded to set the oscillaation at the d
desired freq
quency wh
hen using tthe PMOS ttransistor for the biaas circuitryy. As we observe o
in the follow
wing simullation resuults, the seelf‐biased architecture is more preferable.
p
. In this design, we w minimize the tu
uning rangge limitatiions. Therefo
ore, by cho
oosing the suitable w
width and length forr the transsistors, wee can easily m
meet the de
esign specifications.
ure 5.9: Phaase noise o
of CMOS cross‐couplled VCO with NMOS bias circuit Figu
59 Figure 5.10: Ph
hase noise of CMOS ccross‐couppled VCO w
with PMOS bias circuit Figure 5.11: Phase noise of se
elf‐biased CMOS cross‐coupled
d VCO 5.1.2.2
2 NMOS LLC VCO Two diffferent dessigns are analyzed in this sectio
on. In this topology, we can saatisfy biasing condition easily. The
erefore, th
he sizing o
of the transsistors is d
decreased by a coefficie
ent. We increase the
e value of ccapacitancce so the circuit will o
oscillate att the desired frequencyy. 60 Figure 5
5.12: Phase
e noise of NMOS VCO
O with NM
MOS bias circuit Figure 5
5.13: Phase
e noise of NMOS VCO
O with PM
MOS bias cirrcuit ollowing taable, we sp
pecify the ttransistor ddimension
ns of an NM
MOS LC VCO. In the fo
NMOSS LC Oscillaator NMO
OS
Refference Capacitaance
Cu
urrent W[µm]
W
L[µm]
W[µm] L[µm] [pF]]
NMOS‐Bias 18
0.15
15
0.06 2.37
7
PMOS‐Bias 18
0.15
30
0.06 2.37
7
Taable 5.2: NMOS crosss‐coupled LLC VCO dim
mensions 61 5.1.2.3
3 PMOS LLC VCO As men
ntioned in
n the pre
evious chaapter, the cross‐cou
upled top
pology can
n be optimized by usin
ng proper ttransistors in the biaas circuit. In the follo
owing part,, the sizing off the transsistors and phase noise analysiss are show
wn in detaill. PMO
OS
PMOSS LC Oscillaator Reference Capacitaance
urrent Cu
W[µm]
W
L[µm]
W[µm]] L[µm] [pF]]
NMOS‐Bias 48
0.15
10
0.06 1.97
7
PMOS‐Bias 48
0.15
20
0.06 2.1
Taable 5.3: PMOS crosss‐coupled LLC VCO dim
mensions Figure 5
5.14: Phase
e noise of PMOS VCO
O with NM
MOS bias cirrcuit 62 Figure 5
5.15: Phase noise of PMOS VCO
O with PMOS bias cirrcuit 5.1.3 C
Comparison of Diffferent VC
CO Topologies Generally, the cro
oss‐couple
ed topologgy using NMOS and PMOS pair is prefeerred mostly among de
esigners du
ue to its phase noisee performance. How
wever, it iss not always the case. In some designs, d
the CMOS to
opology puts some constraintts on the sup
pply voltage. Therefo
ore, the op
ptimum phhase noise performance canno
ot be met by CMOS crross‐couple
ed topologgy all the time. To solve this problem, the sizing of o the transistors and the W/LL ratio are increased
d. Consequ
uently, wee will have a large tran
nsconductaance. Therrefore, it will worseen the theermal noisse of switchin
ng transisttors. Addiitionally, the t
output suffers from a laarge harm
monic distortio
on due to the nonlin
nearity of the transiistors. On the otherr hand, low
w Vgs causes tthe oscillattor to funcction impro
operly. As it is o
observed iin simulation results,, the PMOS bias circuit exhibitts better phase noise performanc
p
ce than itts NMOS counterpaart. Howeever, at high frequeency offsets, this imprrovement does not persist. Thhis differeence can b
be analyzeed in differen
nt aspectss. PMOS bias transsistor has a larger width th
han its NM
MOS counterrpart. On the otherr hand, in the 65nm
m technolo
ogy, it is observed that PMOS ttransistor h
has the leaast flicker n
noise. Therrefore, it iss much loggical to usee the PMOS b
bias circuit in the dessign. 63 Generally, the PMOS topology is more practical in LC VCO design. However, it has some drawbacks that should be taken into account. The main disadvantage is that the output transistor is biased at Vss so it is prone to noises on ground. In practice, this can degrade the function of PLL in some cases. For instance, when accompanying with a digital circuit on the same Vss, the functionality of the whole circuit will be affected. In the following table, we have compared the different LC VCO topologies regarding the phase noise and power consumption. Architecture NMOS‐Nbias NMOS‐Pbias PMOS‐Nbias PMOS‐Pbias CMOS‐Nbias CMOS‐Pbias CMOS‐Self‐
bias Phase Noise at Phase Noise at Supply Voltage 100kHz 3MHz [V] [dBc/Hz] [dBc/Hz] ‐103 ‐129
1.2
‐107 ‐132
1.2
‐100 ‐130
1.2
‐109 ‐131
1.2
‐106 ‐127
1.2
‐111 ‐128
1.2
‐107 ‐133
1.2
Bias Current [mA] 2.65
2.68
2.68
2.7
2.4
2.47
2.6
Table 5.4: Comparison of different VCO topologies regarding the phase noise and power consumption 64 5.2 Bias Circuitry Design In LC VCO design, the sizing of the current mirror has an effect on selecting the proper value for supply voltage. The current is related to Vds in saturation region. The drain to source voltage should be considerably large to pave the way for the desired value of current. Simply, Vds should be equal to Vgs in 65nm technology. Theoretically, Vsupply is equal to Vds+(Vgs‐Vth). Here, the transistors transconductance is larger than the inductor loss. The sizing of the transistors should be chosen properly to maintain the desired Vgs. In our simulations, we have increased the size of bias transistors to observe their effect on the phase noise performance. As described in previous chapters, one of the main sources of the phase noise in 1/f3 region is the flicker noise upconversion of the bias transistors. 65 6. COMPLEMENTARY SIMULATIONS AND RESULTS 6.1 Introduction In this chapter, we verify our previous results by a more detailed analysis. Here, we do the final integration of VCO sub‐blocks to approach the state‐of‐the‐art phase noise and power consumption. To meet the design specifications precisely, we have tested VCO performance for varying temperature, supply voltage and tuning range. Additionally, a wider comparison is made between different VCO topologies regarding the phase noise and power consumption. As a result, we have obtained a deeper understanding of different VCO topologies regarding their applications and frequency range of oscillation. 6.2 Phase Noise and Frequency vs. Control Voltage In this section, to verify the frequency range of our designed QVCO, the tuning voltage is swept from 0.2 V to 1.2 V. From the simulation results, the frequency range is observed from 2.25 GHz to 2.55 GHz. The center frequency is 2.4 GHz which matches our specification requirements. Moreover, due to control voltage and frequency variations, the phase noise changes correspondingly. In Fig. 6.1, the blue plot demonstrates the frequency variation controlled by tuning voltage while the red curve shows the phase noise variation versus frequency and tuning voltage. As understood from the red plot, the phase noise is considerably high at low tuning ranges. At low tuning voltages, the quality factor of the varactor is quite small. Therefore, the overall quality factor of the resonator reduces and this leads to higher phase noise. However, as the tuning range increases, the phase noise improves and finally it reaches to our desired value at the center frequency. 66 Figure
e 6.1: QVCO
O phase no
oise (red) aand frequeency (blue)) versus co
ontrol voltaage 6.3 Refference C
Current Source Vaariation In this ssection, we
e have anaalyzed the impact of reference current so
ource variaation on the center fre
equency. The T supply voltage iss set at 1.2 V while the refereence current is swept from 150µA
A to 210µA
A. As obse
erved from
m Fig. 6.2, there is a small devviation fro
om the cen
nter frequeency when th
he reference current is swept ffrom 150µA
A to 210µA
A. 67 Figgure 6.2: Im
mpact of re
eference current souurce variatiion on QVC
CO center frrequency 6.4 Pha
ase Noise
e and Fre
equency vvs. Tempeerature In this section, we have analyzed our propposed QVC
CO perforrmance veersus temperature variaation. The tail curren
nt is set att 3 mA wh
hile the supply voltagge is 1.2 V. A
Applying a tuning volltage of 0.6
6 V, the teemperature is sweptt from ‐50°°C to 175°C. e noise dependency on tempeerature. Ass we The Fig. 6.3 demonstrates the phase
nd this lead
ds to know, tthe thermaal noise is directly prroportionaal to tempeerature an
poor ph
hase noise performan
nce as the temperatuure increasses. 68 Figu
ure 6.3: Imp
pact of tem
mperature variation o
on QVCO p
phase noise (red) and
d frequ
uency (bluee) 69 7. CONCLUSION 7.1 Summary At the beginning of the project work, we studied the basic theory of an oscillator. In the following sections, the noise contributors affecting the VCO performance are identified. Furthermore, different models interpreting the noise impact on a voltage‐controlled oscillator are presented. The main purpose of this project is to implement a state‐of‐the‐art design considering optimal phase noise and power consumption. Initially, to achieve a high performance VCO, we have designed different high performance LC VCO architectures. Moreover, a wide comparison is carried out regarding the VCO specifications such as phase noise at different offset frequencies, power consumption, FOM and so forth. As a result, a suitable LC VCO topology is chosen for further analysis. Afterwards, we have improved our design to a Quadrature VCO with back‐gate coupling and source resistive degeneration. The designed QVCO oscillates at the center frequency of 2.4 GHz. The phase noise estimated by simulation at 1MHz offset frequency is ‐140dBc/Hz. The circuit consumes a power of 3.6mW which is less than conventional QVCO architectures. Finally, to verify our design, process, temperature, and reference current variations were tested. As a result, the specification requirements have been met in our design. 70 7.2 State‐of‐the‐Art Comparison Here, the performance of the designed LC QVCO circuit is compared with other reported LC VCOs. As we see, the implemented QVCO performance is in line with the state‐of‐the‐art designs in terms of power consumption and phase noise. Topology Technology [1] [2] [3] [4] [5] [6] [7] [8] [9] This work 90 nm 0.25 µm 0.18 µm 0.25 µm 0.18 µm 0.18 µm 0.18 µm 0.18 µm 0.18 µm 65 nm Center Phase Power Supply FOM frequency Noise at consumption Voltage (dBc/Hz) (GHz) 1MHz (mW) (V) (dBc/Hz) 2.42
2.45
5.47
2.07
1.87
3.5
5.37
3.2
2.4
2.4
‐119.7
‐113.3
‐122.4
‐124.4
‐111
‐116.5
‐123
‐133
‐134
‐140
0.515
3.1
5
3
0.5
4.7
18
4.4
4.6
3.6
Table 7.1: State‐of‐the‐art comparison 71 1.8 1.8 1.2 1.5 0.9 1.8 1.8 1 1 1.2 ‐190.26
‐‐‐
‐190.2
‐186
‐180
‐‐‐
‐185
‐196.6
‐195
‐197
References [1] P.K. Rout, U.K. Nanda, D.P. Acharya, G. Panda, “Design of LC VCO for optimal figure of merit performance using CMODE”, 1st International Conference on Recent Advances in Information Technology (RAIT), pp. 761 – 764, March 2012. [2] S. Zihir, F. Tasdemir, T. Dinc, Y. Gurbuz, “A new resonant circuit for 2.45 GHz LC VCO with linear frequency tuning”, Microwave Integrated Circuits Conference (EuMIC), pp. 390 – 393, Oct. 2011. [3] P. Liu, P. Upadhyaya, J. Jung, D. Heo, J.‐H. Kim, B.‐S. Kim, “Dynamically switched low‐phase‐noise LC VCO with harmonic filtering”, Electronics Letters, vol. 47, pp. 792 – 793, July 2011. [4] B. Soltanian, P. Kinget, “A Low Phase Noise Quadrature LC VCO Using Capacitive Common‐Source Coupling”, 32nd European Solid‐State Circuits Conference (ESSCIRC), pp. 436 – 439, Sept. 2006. [5] Han‐il Lee, Tae‐Young Choi, S. Mohammadi, L.P.B. Katehi, “An extremely low power 2 GHz CMOS LC VCO for wireless communication applications”, The European Conference on Wireless Technology, pp. 31 – 34, Oct. 2005. [6] Harikrishnan Ramiah, Tun Zainal Azni Zulkifli, “A Low‐Phase Noise and High Output Swing RF LC‐VCO in CMOS Technology”, RF and Microwave Conference, pp. 197 – 200, Sept. 2006. [7] Yang Zhang, Peng Liu, Tang‐Nian Luo, Yi‐Jan Emery Chen, Deukhyoun Heo, “A low‐voltage low‐phase‐noise bottom‐series LC QVCO using capacitor tapping technique”, Microwave Symposium Digest, pp. 237 – 240, June 2008. [8] M D Wei, S Chang, Y Yang, "A CMOS Backgate‐Coupled QVCO Based on Back‐
to‐Back Series Varactor Configuration for Minimal AM‐to‐PM Noise Conversion ", Microwave and Wireless Components Letters, IEEE , pp. 320 ‐ 322, (2009). 72 [9] Zhenbiao Li, Kenneth K. O, "A low‐phase‐noise and low‐power multiband CMOS voltage‐controlled oscillator", IEEE Journal of Solid‐State Circuits, pp. 1296 – 1302, June 2005. [10] A. Hajimiri and T. H. Lee, “Design issues in CMOS differential LC oscillators”, IEEE J. Solid‐State Circuits, vol. 34, pp. 717–724, May 1999. [11] Behzad Razavi, "Design of analog CMOS integrated circuits", McGraw‐Hill, (2001). [12] D. Leeson, “A simple model of feedback oscillator noise spectrum,” Proceedings of the IEEE, vol. 54, pp. 329 – 330, Feb. 1966. [13] A. Hajimiri, “A general theory of phase noise in electrical oscillators,” IEEE Journal of Solid‐State Circuits, vol. 33, no. 2, pp. 179–194, Feb. 1998. [14] T. Lee and A. Hajimiri, “Oscillator phase noise: a tutorial,” IEEE Journal of Solid‐State Circuits, vol. 35, no. 3, pp. 326–336, Mar. 2000. [15] Rael, J. J. and Abidi, A. A., “Physical Processes of Phase Noise Differential LC Oscillators”, IEEE Custom Integrated Circuits Conference, Orlando, Florida, May 21‐24, 569‐572, (2000). [16] E. Hegazi, H. Sjoland, A. A. Abidi, "A filtering technique to lower LC oscillator phase noise", IEEE Journal of Solid‐State Circuits, pp. 1921 ‐ 1930, (2001). [17] M. Ebrahimzadeh, “Design of an ultra‐low power low phase noise CMOS LC oscillator”, World Academy of Science, (2011). [18] Muh‐Dey Wei, Sheng‐Fuh Chang, Chi‐Sheng Chen, “A Low Phase‐Noise QVCO With Integrated Back‐Gate Coupling and Source Resistive Degeneration Technique”, IEEE Microwave and Wireless Components Letters, vol. 19, pp. 398 – 400, June 2009. [19] Behzad Razavi, "RF Microelectronics”, Second Edition, Prentice Hall, September 2011. 73 [20] Ali Hajimiri, Thomas H. Lee, "The Design of Low Noise Oscillators”, Kluwer Academic Publishers, (1999). [21] M. Perrott, “High speed communication circuits and systems”, MIT open courseware, Lecture 11, (2003). [22] T. H. Lee, "The Design of CMOS Radio‐Frequency Integrated Circuits”, Cambridge University Press, (1998). 74 
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