overviewewh.ieee.org/r2/lehigh_valley/sscs/events/06feb2018/slides.pdfcurrent harmonic 1 o lc w= 1...

RF Harmonic OscillatorsIntegrated in Silicon Technologies

Pietro AndreaniDept. of Electrical and Information Technology (EIT)

Lund University, Sweden

SSCS Distinguished Lecture

Lehigh University, PATuesday, Feb. 6th, 2018

Overview

o Popular harmonic oscillatorsn Phase noise

o Architectures for low 1/f2 and/or 1/f3 phasenoise

o Series-resonance oscillator

o Design techniques for very wide frequencytuning range RF CMOS VCOs

P. Andreani, SSCS DL, Lehigh Univ., PA, 2/6/2018 2 of 78

LC resonator

We begin with an inductor-capacitor resonator

L CL C

RL RC

01w =LC

RLC

( )0w =Z R0

1w »LC


Building a harmonic oscillator

LC

R ‒Ractive

Tank losses are compensated by an activenegative resistance in parallel to the tank

activeR R< 00

RQ RCL

ww

= =


Colpitts oscillator

P. Andreani, SSCS DL, Lehigh Univ., PA, 2/6/2018

Amplifier + resonator

Feedback network

Noise

Oscillator as positive-feedback system

IB

Vout

VBC1

C2

L R

Amplifier

Resonator

FB

5 of 78

Colpitts oscillator

IB

Vout

VB C1

C2

L R

Vout

t

Itank

VDD

Itank

VDD


class-C operation

6 of 78

Classical embodiment of active negativeresistance with only one active device

Analysis with Describing Function


( )( )

01

2 1out

B

A I R n

I R nw= -

» -

C1

C2

Vout

L R

Fundamentalcurrent harmonic

1o LC

w =

1

1 2

CnC C

=+

1 2

1 2

C CCC C

=+

2 BI»0Iw

DSI

t

7 of 78

Cross-coupled differential-pair oscillatorExtremely popular oscillator family

V-

IB

N1 N2

V+

VDDIB IB

t

t

V-

V+

VDD

Operation in class-B


22BRA A I

p+ -= =2

diff BA I Rp

=

8 of 78

Class-B with double switch pair

Compared to single-switch-pair:double amplitude, but also doubleVDD (no swing above VDD)

4diff BA I R

p=

V-V+

IB

t

VDDV+ V-

IB

-IBt

VDD


Overview






Real oscillations

o Phase uncertainty grows with time à jittern Caused by various noise sources

o Jitter increases without bound in a free-runningoscillator

o In the frequency domain, the oscillator displaysphase noise

Phase noise,in dBc/Hz

f0 f

dBc


Why bother?

Phase noise in transceiver is important for at least threereasons:q In a receiver, it can downconvert large nearby

signals on top of the desired signalq In a transmitter, it can increase the noise floor in

the receive bandq In both, it can directly corrupt the phase

information in the signal– Not seldom, the phase noise of the VCO is the

bottleneck for the whole radio performance


Reciprocal mixing


Noise floor raised by mixing of LO phase noise with blocker

0 f1fo

fo+ f1

LO

Desired signal + Blocker

IFfo

13 of 78

LO

Phase noise and SNR (EVM)


02

BW BW

BWSNR PN PN

-= =ò ò

2010SNR

EVM-

=

Example: 64QAM inWCDMA RX; EVM = 2.5%

f

PN

BW

PLL phase noise

Error vector magnitude

14 of 78

Phase noise – LTI approach

( ) ( )211 ww w

w w

-æ ö= + + - =ç ÷

è øact

LCY G j C G

j L j L

( ) 0 0 00

0 0

122 2

w w ww w

w w ww w

+ D = = × × = ×D D D

L LZ R RR Q

( ) ( )( )( )

0 00 2

0

0

1 2

w w ww w

ww ww

+ D+ D » » -

D- + D

j L LZ jLC


LC

R ‒Ract

LC tank

Active device

15 of 78

Phase noise from tank losses

LC2

ni

( )2 22 2

2 0 00

14 42 2

n nB B

v iZ k TG R k TR

f f Q Qw ww w

w wæ ö æ ö

= × + D = × × = ×ç ÷ ç ÷D D D Dè ø è ø

q Both amplitude and phase noise, but amplitude noise is rejectedq Thus, phase noise is defined as half the above expression, normalized

to the output signal power (in dB below the carrier per Hertz, dBc/Hz):

( )22

010 102 2

2 2 110log 10log2 2 2

n B

pk pk

v k TRLA A Q

ww

w

æ ö æ öæ öç ÷ ç ÷D = = ×ç ÷ç ÷ç ÷ Dè øè øè ø


Leeson’s equation

( )2

010 2

2 110log2 2

www

æ öé ùæ öç ÷D = ×ê úç ÷ç ÷Dê úè øë ûè ø

B

pk

k TRLA Q

( )L wD

( )log wD

Ideal ( )wDL

Thermal noise (20 dB/dec, 1/f2 region)

F

0

2Qw

Noise floor

1+

31 fwD

Up-converted 1/f noise (30dB/dec, 1/f3 region)

311w

w

æ ö× +ç ÷ç ÷è

D

øDf


Hajimiri and Lee’s theory of phase noise

( )td t- ( )outV t

Conversion of noise intophase noise is time-dependent – LTV phasenoise analysis needed!

Hajimiri and Lee, JSSC Feb. ‘98

LC

outV

( )td t-

DV

outV

( )td t-

DV

No phase noise Maximum phase noise


Impulse sensitivity function (ISF, G)

q Current noise source is weighed by associated

à effective current noise

q ISF is dimensionless, frequency- and amplitude independent,with period 2p:

( )ni f ( )ni

fG

( ) ( ) ( ), nn eff n ii if f f= ×G

( ) ( )0

1cos

2 n nn

c c nf f f¥

=

G = + +å

( )0tf w=


Phase noise expression

( )( ) ( )

2, ,

2 210log

2n eff rms

pk

iL

CAw

w

æ öç ÷D =ç ÷Dè ø

If is a (cyclo)stationary white current noise source,its contribution to 1/f2 phase noise is

( )ni f

( )( )cos w f+pkA t t+

-( )ni f


Graphical interpretation


Example of ISF – LC oscillators

( )sin fG = -ni

( )cos f=out pkV A

f

f

Hajimiri and Lee, JSSC Feb. ’98; Andreani and Wang, JSSC Nov. ‘04

+

-( )ni f outV


A particularly simple caseParallel RLC resonator again

( )( ) ( ) ( ) ( )

( ) ( )

22,

2,,

2 22 2

2 2

10log 10log2 2

4 1 210log2

ww w

w

æ ö æ öç ÷ ç ÷D = =ç ÷ ç ÷D Dè ø è ø

æ ö×ç ÷=ç ÷Dè ø

Gnnn eff rms

pk pk

B

p

rm

k

i siiL

CA CA

k TG

CA2

02

2 1 10 log2 2

ww

æ öæ öç ÷= ç ÷ç ÷Dè øè ø

B

pk

k TRA Q

LC

2n

i

Leeson with F=1recovered without anyad hoc assumptions!

– phase noise from tank losses:


Phase noise from MOS pair

V-

IB

N1 N2

V+

Two commutations in one oscillation period

( ) ( )21 , 14N B n m Ni k T gf g f= ( ) ( ) ( )2 2 2

1, 1 1N eff N Ni if f f= ×G

F-

1NI2NI

F

( )1 1N fG »


Total phase noise

( )( )

( )22 2tank

10log 1Bn

k TLRA C

w gw

é ùD = +ê ú

Dê úë û

q Transistors appear only through channel noise factor gn

q Transistor phase noise always proportional to tank noise(60% from tank, 40% from MOS pair, if gn = 2/3)

q This is because: 1) transistor noise is proportional tocommutation time, 2) which is inversely proportional to theoscillation amplitude, 3) which is proportional to the tankparallel resistance

q A simple-minded LTI analysis would yield very wrongpredictions (i.e., MOS phase noise increases with MOS gm)

Tank MOS pair

1 nF g= +


MOS phase noise – invariance

( ) ( )21 , 14N B n m Ni k T gf g f=

tank

21 Bn

n

IA b

F =

Two effects balance each other:1) Larger MOS produces more

noise during currentcommutation, and

2) Larger MOS allows a fastercommutation

Result: the two areas areidentical

nF,2n bFn-F ,2n b-F

2 nb´

Andreani et al., JSSC May ‘05

1 nb´


Single-ended Colpitts oscillator


Hajimiri and Lee, JSSC Feb. ‘98; Andreani et al., JSSC May ‘05

IB

VoutVB

C1

C2

L R( )MOS fG

( )2MOSi f

( )2,MOS effi f

Noise injected into tank when ISF is near zero à excellent!

27 of 78

Phase noise in Colpitts oscillator


( )( ) ( )22 3 22

1101

log 14

w gw

é ùæ öD » +ê úç ÷è øD

-

-ê úë û

Bn

B

k TLI R C

nnn

However, contrary to what was once (justifiably) believed,Colpitts is more noisy than the differential-pair LC oscillator!

0.30 for 2 31 3 for 10.36 for 1.5

n

opt n

n

nggg

=ìï= =íï =î

Minimum for:

Andreani et al., JSSC May ‘05

28 of 78

Harmonic oscillators – a general result


1) G sinusoidal and in quadrature with tank voltage2) Active devices work as transistors3) Transistor current noise proportional to gm

Transistor effective noise depends onlyon tank loss and topology

J. Bank, ”A harmonic oscillator design methodology based on describing functions”, PhD thesis, Gothenburg, Sweden, 2006Mazzanti and Andreani, JSSC Dec. ’08; Murphy et al, TCAS-I June ‘10

29 of 78

More on inductive vs capacitive losses

o Asymmetry between phase noise fromrC vs rL – depends on (large) gm×rC

o Lost with ”equivalent” parallel tank losseso PN from GM always proportional to PN from

rC and rL together (here, g = 1)Pepe and Andreani, TCAS-II June 2016

rLgm

rCGM

rL = rC


Matrix-based Fourier-series LTV approach,starting from

All quantities are functions of w0, 2w0, … , nw0

Alternative phase noise analysis

MGY

+

-ResonatorTransconductor

VI

I V= Yr ur

dI dV= MGuur uuur

and


Pepe and Andreani, TCAS-I Feb. 2017

31 of 78

Results of new phase noise analysis

o Rigorous analysis under very broad hypothesesn GM pure transconductance; Y linear; GM noise

proportional to GM via g

o Phase noise from GM always in proportion of g:1to phase noise from Y, independently ofresonator and transconductor nature

o Phase noise expressions as functions of V and Yo Closed-form, explicit phase noise expressions if

Q is highn General case of Y resonating at multiples of w0


What do these expressions look like?

General equation

( ) ( )( )

( )( )20

10 1 12 2

1 1

1 110log 2 Reg w

ww

*

é ù+ê ú

D = × ×ê ú× D ×ê úë û

uuuuruuur

uuur uuuur uuur

T

BT

L k T LV LVDV LV RV

YΔY


MGY

+

-

33 of 78

Matrix algebra


1 10 00; ; 0T T

M DV RV DV DV LV M× = = × =uuur r uuur uuur uuur uuur r

MGY

+

-* * * * *

2 0 1 2 3 4* * * *

1 1 0 1 2 3* *

0 2 1 0 0 1 2*

3 2 1 1 0 1

4 3 2 1 2 0

é ùê ú- - - - -ê úê ú- - - - -ê ú= - - - - -ê úê ú- - - - -ê ú

- - - - -ê úê úë û

O M M M M M N

K K

K K

K K

K K

K K

N M M M M M O

Y g g g g gg Y g g g g

M g g Y g g gg g g Y g gg g g g Y g

( ) ( )( )

( )( )20

10 1 12 2

1 1

1 110log 2 Reg w

ww

*

é ù+ê úD = × ×ê ú

× D ×ê úë û

uuuuruuur

uuur uuuur uuur

T

BT

L k T LV LVDV LV RV

YΔY

34 of 78

Tank with multiple resonances

( ) ( )

( )2 2

110 22

2 21

110log

gw

w

é ù+ê úD = ê úDê úë û

åå

Nn nB

Nn n

n A Rk TL

n A C

n high-Q resonances


( ) ( ) 2

010 102 2 2 2

1 21 110log 10log2 2

g ww

w w

æ öé ù+ æ öç ÷D = = ×ê ú ç ÷ç ÷D Dê ú è øë û è ø

B B

pk pk

k T k TRLA C R A Q

Single resonance à we recover the well-known equation

35 of 78

Tank resonating at w0 and 3w0

o Steeper V-waveform, but now also R3 contributeso Advantageous only if Q3 (much) larger than Q1

o Difficult to enforce in practicePepe and Andreani, TCAS-I Feb. 2017; Garampazzi et al., JSSC Mar. 2014

( ) ( )( )

2 21 1 3 3

10 22 2 21 1 3 3

1 910log9

Bk T A R A RLA C A C

gw

w

é ù+ +ê úD =ê úD +ë û


Double-switch pair vs. single-switch pair

IB

V+ V-

IB

V+ V-

Double-switch (DS) pair oscillator Single-switch (SS)pair oscillator

2SS BA I R

p=

4DS BA I R

p=

What phasenoise difference

should weexpect?


DS pair vs. SS pair – phase noise

( )2

02

2 110log 12 2 2

n pBDS

DS

k TRLA Q

g gww

w

æ ö+æ öæ öç ÷D = +ç ÷ç ÷ç ÷Dè ø è øè ø

( ) ( )2

02

2 110log 12 2

BSS n

SS

k TRLA Q

ww g

w

æ öæ öç ÷D = +ç ÷ç ÷Dè øè ø

2 6DS SS DS SSA A L L dB= ® = -

o 60% from tank, 40% from transistorsn If gn = gp = 2/3

o If IB,DS = IB,SS and gn = gp à

Andreani and Fard, JSSC Dec. 2006

(!)


DS vs. SS – MOS noise

,n SSF,n DSF,n SS-F ,n DS-F

Area (SS) = 2 · Area (DS)

4 DS transistors makeas much noise as 2 SStransistors!

DS

SS


Phase noise vs power consumption


• If IB,SS = 2IB,DS à ASS = ADS à same phase noise

• In this case, VDD,SS = ½ VDD,DS à same powerconsumption

• Thus, same maximum achievable “figure-of-merit”(FoM, phase noise for a given power consumption)

• If VDD,DS = VDD,SS and IB,DS = ¼ IB,SS à ASS,max =2ADS,max à LSS,min = LDS,min ‒ 6dB

• Again, FoMSS,max = FoMDS,max

40 of 78

power efficiency

Figure of merit (FoM)

( )( ) [ ]

2 20 3 210

BDC mW

QFoML P k T F

w w hw

-D= = × ×

D ×

noise factor

High tank Q crucial for high FoM

Phase noise normalized to power consumption,oscillation frequency, and frequency offset

Andreani et al., JSSC Dec. 2006; Garampazzi et al., JSSC Jul. 2015; Murphy et al., ISSCC 2015


SS vs DS – PN and FoM with fixed Vdd

SS

SSmax FoM

DSmax FoM

DS

SS oscillation limitDS oscillation limit

IB 4IBLiscidini et al., ISSCC 2012, JSSC Mar. 2014

PhaseNoise

FoM


Current bias – resistive source

V-

IB

N1 N2

V+

Resistor

Very simple biascircuitry

No 1/f noisegeneration

Low-Z source

Significantupconversion of1/f noise fromN1-N2

Abidi and Ismail, ISSCC 2003


Current bias – MOS source

High-Z source

Less upconversionof 1/f noise fromN1-N2

Own 1/f noisegeneration

“Tail” current source

VB

V-

IB

N1 N2

V+


Impact of parasitic tail capacitance

o Ctail,par + cross-coupledMOS entering linear regionà MOS contribution tophase noise increases,even by a large amount

o Ctail,par good for filtering HFnoise from bias

o But, increase of 1/f noiseupconversion

Ctail,par


Overview






Possible solution – noise filter

o Noise filter: Ctail,par resonateswith Ltail at à MOSswitches see high-Z at

o CBIG filters tail noise and ac-grounds Ltail

o CBIG includes CDB of MOS tailà long and large MOS, low1/f noise

o Drawbacks: narrow-band,Ctail,par must be known withsome precision, extra Ltail

Ctail,par

CBIG

Ltail

02w

Hegazi, Sjöland, Abidi, JSSC Dec. 2001

02w


Dramatic performance improvemento 0.35μm CMOSo 1.2GHz, 3.5mA, 2.5Vo Ltail = 10nH, CBIG = 40pFo FoM = 196dBc/Hzo TR?


More on tail filtero Many variations on the

same basic themeo Extremely popular!

CBIG

LtailCtail,par

Hegazi, Sjöland, Abidi, JSSC Dec. 2001


A recent variation

o Tail/top resonance with Xrfmo Very low PN and great FoM

(up to 195.6dBc/Hz)o 200-400kHz 1/f3 cornerGarampazzi et al., JSSC July 2015

@20MHz

/2

7mW


Alternative to tail resonance

Babaie et al., RFIC 2013, JSSC Mar. 2015; Shahmohammadi et al., ISSCC 2015; Murphy et al., ISSCC 2015

o Design tank for differential resonance at andcommon-mode resonance atn Also here, the resonance must track the

resonance – two capacitor banks

02w0w

02w 0w


Class-F2 (or, here, F2,3) oscillator

Shahmohammadi et al., ISSCC 2015

o 5.4-7.0GHz; 1V, 10-12mWo Low 1/f3 corner (60-130kHz)o Very good FoM (~191dBc/Hz) at

very low PN (-124dBc/Hz @ 1MHz)o Very low VDD pushing (12-23MHz/V)


Implicit common-mode resonance

Murphy et al., ISSCC 2016

o 4.7-5.4GHz; 0.7V, 0.5mWo Low 1/f3 corner (200kHz)o Great FoM (194-196dBc/Hz)


A totally different approach – class-C

Mazzanti and Andreani, JSSC Dec. 2008

IB

V+ V-

Ctail

N1N2

bias2 I» ×p0

Iw

biasI»0

IwN 2IN 1I

o Ctail turns class-B into class-C:optimal differential “Colpitts”oscillator

o Ideally, 3.9dB lower phasenoise for the same bias current

o Also here, Ctail filters off high-frequency noise from tail, andincludes tail CDB à long andlarge MOS, low 1/f noise


Original prototypeo 4.90GHz < fc< 5.65GHzo 1V, 1.4mWo 193.5dBc/Hz<FoM<196dBc/Hz

fosc = 4.90GHz

fosc= 5.52GHz


Design issues in class-C CMOS VCO

o Diff-pair must not enter thelinear region (otherwise, largePN penalty) à shift of MOS DCgate voltage VB (which may begenerated with feedback loop)

o RC bias should not load tank(transformer feedback possible)

o Nevertheless, higher maximumoscillation amplitude in theideal class-B CMOS oscillator

o Class-C very attractive for BJTVCOs

Mazzanti and Andreani, JSSC Dec. 2008; Fanori and Andreani, JSSC July 2013

Ctail

V+ V-

VB


Colpitts VCO in SiGe BiCMOS process

Boscolo et al., ESSCIRC 2017

o 18.8-23.1GHz; 4.0V, 17.5mAo PN = -119dBc/Hz @ 1MHz (best)o FoM =188dBc/Hz


Overview






Oscillation with series resonance


( )0sin wpV t0sin

2pQ V t pwæ ö× -ç ÷

è ø

o Voltage driven

o Gain equal to quality factor à internaloscillation may be much higher than VDD

n Attractive for ultra-low phase noise

o p/2 phase shift between input and output

sr L

C

P. Andreani, L. Fanori, and T. Mattsson, “Series-resonance oscillator,” U.S. Patent 2015 0381 157, 2015

59 of 78

Phase shift by quadrature


90V 180V

o We disregard the (important) issue of start-up

o Square wave between VDD and GND at LC input

o MOS devices work almost exclusively as switches àchannel resistance in series with the tank’s

Tohidian et al., MWCL Aug. 2015; Pepe, Bevilacqua, Andreani, TCAS-I Feb. 2018

60 of 78

Phase noise


( )( )

( )( )

( )2

010 102 2

4 4 110log 1 10log 14 4

ww

ww w

é ù é ùæ öê ú ê úD = + = +ç ÷ê ú ê úDè øD Dë û ë û

B s B

pk pk s

k Tr k TL F FQI L I r

o MOS work as switches à previous phase noisetheorems do not apply

o F accounts for 1) MOS are non-ideal switches, and2) they do work as transconductors for a (tiny)fraction of the oscillation period

o Ideally, F is negligible!

2p

= DDpk

s

VIr

Pepe, Bevilacqua, Andreani, TCAS-I Feb. 2018

61 of 78

Figure of merit


o Usual dependence on Q2

o Very large power consumption, ultra-low phasenoise (plus quadrature phases for free)

o However:n MOS resistance is critical (current-based architectures such

as class-B and class-C are much more robust)n Stray resistances of GND and power supply distribution are

also criticaln Very large internal voltages make frequency tuning difficult

23 210

1h-= × ×+B

QFoMk T F

Ideally, close to 1

Ideally, close to 0

62 of 78

Overview






VCOs in modern radios – I

o Carrier aggregation requires several harmonicVCOsn Active at the same timen Should not pull one another

o Band proliferation favors VCOs with a very widetuning range (TR)n Wider than 1 octave is particularly attractive


VCOs in modern radios – II

o VCO with 8-shaped tank inductorn Much less sensitive to external magnetic fieldsn Generates itself a vanishing magnetic fieldn Slightly lower Q acceptablen Often used

M. Nilsson et al., ISSCC 2011


Very-Wide-TR VCOs – I

o Two or more VCOs with overlapping TRsn Saves power, costs area

n Very popular choice in real-life products

Hadjichristos et al., ISSCC 2009


Very-Wide-TR VCOs – II

o Large switchable C in parallel to small Ln floating switchesn power wasted at low frequencies, compared to reasonable

phase-noise specsn power cannot be decreased without killing the oscillation

Sjöland, TCAS-II May 2002


Very-Wide-TR VCOs – III

o Switchable Ln Ultra-wide TR possiblen Difficult to obtain low PN at high FoMn Additional issue: switchable 8-shaped

inductor

Sadhu et al., CICC 2009


Very-Wide-TR VCOs – IVo Transformer-based VCOs

n Two resonances with overlapping TRsn TR > 1 octaven Difficult to design an 8-shaped transformer

Bevilacqua et al., TCAS-II Apr. 2007; Li et al., JSSC June 2012


Very-Wide-TR VCOs – V

o Mode-switching VCOn 4 inductors, two oscillation modesn Rejects external magnetic fieldsn TR > 1 octaven Excellent PM and FoMn Large area

Taghivand et al., ISSCC 2014


Very-Wide-TR VCOs – VIo Double-core VCO

n Two concentric 8-shaped coils – do not interfere (much)with each other

n TR > 1 octave; saves inductor area, sub-optimal Q

Fanori et al., ISSCC 2014


Very-Wide-TR VCOs – VII

o Reconfigurable active coren Standard LC tank design (i.e., with very large capacitance)n Negative resistance: either single-switch (nMOS) pair –

SS moden or, double (complementary nMOS-pMOS) switch pair –

DS moden DS mode avoids power waste at lower frequencies

Liscidini et al., ISSCC 2012, JSSC Mar. 2014


SS pair vs. DS pair, again

SS

SSmax FoM

DSmax FoM

DS

SS oscillation limitDS oscillation limit

IB 4IB


Very-Wide-TR reconfigurable VCO

SS

SSDS

SS

SSDS

Fanori et al., RFIC 2015

o STM 28nm UTBB FD-SOI CMOSo 2.8-5.8GHzo -154<PN (dBc/Hz @20MHz)<-142o 186 < FoM (dBc/Hz) < 189o 300kHz < 1/f3 corner < 3MHz

Phase noise DS-mode, IDC = 4 mAPhase noise SS-mode, IDC = 8 mA


Conclusions

o Rigorous phase noise resultsn For transconductor-based oscillators

o Class-B VCOs à simple, robust, ubiquitousn Tail filter improves phase noise, even largelyn Recent proposals: common-mode tank resonance at

o Class-C VCOs à better efficiency than standardclass-B, but more complicatedn Class-C must be enforced for all working conditionsn Excellent for BJT VCOs

o Series-resonance oscillator à great potential, butimportant issues to be solved

o Several techniques for very wide frequency tuningrangen None is a clear winner

02w


References – I1. A. Hajimiri and T. H. Lee, A general theory of phase noise in electrical oscillators,” IEEE J. Solid-

State Circuits, vol. 33, no. 2, pp. 179–194, Feb. 1998.2. P. Andreani and X. Wang, “On the Phase-Noise and Phase-Error Performances of Multiphase LC

CMOS VCOs,” IEEE J. Solid-State Circuits, vol. 39, no. 11, pp. 1883–1893, Nov. 2004.3. P. Andreani et al., “A study of phase noise in Colpitts and LC-tank CMOS oscillators,” IEEE J. Solid-

State Circuits, vol. 40, no. 5, pp. 1107–1118, May 2005.4. A. Mazzanti and P. Andreani, “Class-C harmonic CMOS VCOs, with a general result on phase noise,”

IEEE J. Solid-State Circuits, vol. 43, no. 12, pp. 2716–2729, Dec. 2008.5. J. Bank, ”A harmonic oscillator design methodology based on describing functions”, PhD thesis,

Gothenburg, Sweden, 2006.6. D. Murphy et al., “Phase noise in LC oscillators: “A phasor-based analysis of a general result and of

loaded”, IEEE Trans. Circuits Syst. I, vol. 57, no. 6, pp. 1187–1203, Jun. 2010.7. F. Pepe and P. Andreani, “Still More on the 1/f2 Phase Noise Performance of Harmonic Oscillators,”

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