Phase Noise in Oscillators - Stanford University

Phase Noise in Oscillators

Ali Hajimiri

Stanford University, Stanford, CA 94305

http://smirc.stanford.edu/papers/Orals98s-ali.pdf Email: [email protected]

Outline

Introduction and Definitions

Time-Variant Phase Noise Model

Upconversion of 1/f Noise

Cyclostationary Noise Sources

Measurement Results

Conclusion

Substrate and Supply Noise


Frequency Instability: Time Domain

σ2

∆T

Known As Clock Jitter.

Oscilloscope

Oscillator


D Q

CLK

Flip-Flop

tsetup thold

Oscillator

Data

Clock

Timing Jitter in Digital Applications


Ideal

Actual

Frequency Instability: Frequency Domain

f

f

f0

Spectrum Analyzer

Oscillator

Known As Phase Noise.


Desired SignalStrong Adjacent Channel

Desired Signal

RF

LO

IF

The desired signal is buried under the phase noise of an adjacent strong channel.

Phase Noise in RF Applications

f

f

f

fRF

fLO

fIF

RF

LO

IF


Units of Phase Noise

log(∆ω)=log(ω-ω0)

L ∆ω 1

f 3------

1

f 2------

Measured in dB below carrier per unit bandwidth.

(-20dB/dec)

(-30dB/dec)

Sv ω( )

ω

∆ω

ω0

dBc Hz⁄[ ]

ω1 f 3⁄

dBc


Internal noise sources set a fundamental limit for phase noise.

i n2

∆f------

MOSBJT

1f---

Low frequency noise can be an important contributor to the system noise.

Thermal and 1/f Noise

log(f)


Isubstrate

Vsupply

Isupply



Outline





Measurement Results

Conclusion



Oscillator with Input Noise Sources

i n1 t( )

i n2 t( )

i nM t( )

V out t( ) A t( ) f ω0t φ t( )+[ ]⋅=

t

Vout

Oscillator

Non-ideal waveform

Ideal waveform

V out t( ) A ω0t φ0+( )cos⋅=

Noise current

v n1 t( )

v n2 t( )

v nN t( )

sources.

Noise voltagesources.


C Li(t)δ t τ–( )

t

i(t)

t

Vout

t

Vout

Oscillators Are Time-Variant Systems

τ

Impulse injected at the peak of amplitude.

∆V

∆V

Even for an ideal LC oscillator, the phase response is Time Variant.

Impulse injected at zero crossing.τ

τ


Once Introduced, phase error persists indefinitely.

Non-linearity quenches amplitude changes over time.

θ

∆θ

V

dVdt--------

LimitCycle

a

b

∆V

Active Device

C Lδ t τ–( )

t

i(t)

τ

G -G(A)

Amplitude Restoring Mechanism

i(t)


∆V∆q

Cnode---------------=

∆φ Γ ω0t( ) ∆VVswing--------------- Γ ω0t( ) ∆q

qswing--------------= =

CR

i(t)

V(t)

V(t)

∆φ

∆V

Vin

Vout

V+V-

+V1

-V1

Impulse Response of a Relaxation Oscillator

∆q qswing«


2ns 3ns 4ns

1.0

3.0

5.0

2ns 3ns 4nsTime

1.0

3.0

5.0

Nod

e V

olta

ge (

Vol

t)N

ode

Vol

tage

(V

olt)

∆V∆q

Cnode---------------=

∆φ Γ ω0t( ) ∆VVswing--------------- Γ ω0t( ) ∆q

qswing--------------= =

i t( )

1 2 3 4 5δ t τ–( )

tτ

i(t)

Impulse Response of a Ring Oscillator

∆q qswing«


Phase Impulse Response

φ t( )hφ t τ,( )

0 t

i(t)

τ 0 τ

hφ t τ,( )Γ ωoτ( )

qmax-------------------u t τ–( )=

t

i t( )

The unit impulse response is:

Γ x( ) is a dimensionless function periodic in 2π, describing how much

phase change results from applying an impulse at time: t Tx

2π------=

The phase impulse response of an arbitrary oscillator is a time varying step.


t

t

t

t

V out t( ) V out t( )

Γ ω0t( ) Γ ω0t( )

LC Oscillator Ring Oscillator

Impulse Sensitivity Function (ISF)

The ISF quantifies the sensitivity of every point in the waveform to perturbations.

Waveform

ISF


φ t( ) hφ t τ,( )i τ( )dτ∞–

∞

∫1

qmax---------------- Γ ω0τ( )i τ( )dτ

∞–

t

∫= =

φ t( )i t( )hφ t τ,( )

Γ ω0τ( )

qmax-------------------u t τ–( )=

Superposition Integral:

Phase Response to an Arbitrary Source

Γ ω0t( )

∞–t

∫ ω0t φ t( )+[ ]cosi t( )

qmax----------------

φ t( )ψ t( ) V t( )

IdealIntegration

PhaseModulation

Equivalent representation:


i t( ) φ t( ) V t( )hφ t τ,( ) ω0t φ t( )+[ ]cos

LTV system Nonlinear system

Phase Noise Due to White Noise

i n2

∆f---------

L ∆ω Γrms

2

qmax2

----------------in

2 ∆f⁄

2∆ω2-------------------⋅=

For a white input noise current with the spectral density of

The phase noise sideband power below carrier at an offset of ∆ω is:

Γrms is the rms value of the ISF.


ISF is a periodic function:

cn nω0t θn+( )cos

c0

c1 ω0t θ1+( )cos

Σ ω0t φ t( )+[ ]cos

i t( )qmax-------------

φ t( )

∞–t

∫

∞–t

∫

∞–t

∫

V t( )

ISF Decomposition

Phase can be written as:

φ t( ) 1qmax------------- c0 i τ( )dτ

∞–

t

∫ cn i τ( ) nω0τ( )cos dτ∞–

t

∫n 1=

∞

∑+=

Γ ωot( ) c0 cn nω0τ θn+( )cosn 1=

∞

∑+=


c0c1 c2 c3

2ω0ω0 3ω0ω

Sφ ω( )

i n2

∆f------- ω( )

1f--- Noise

Sv ω( )

2ω0ω0 3ω0ω

ω

PM∆ω

∆ω ∆ω ∆ω

∆ω

Noise Contributions from nωo

φ t( ) 1qmax------------- c0 i τ( )dτ

∞–

t

∫ cn i τ( ) nωτ( )cos dτ∞–

t

∫n 1=

∞

∑+=


log(ω)

Sφ f( )

1f 2-----

L ∆ω( )1f 3-----

1f 2-----

Amplifier Noise Floor

PSD of φ(t) PSD of V(t)

Power Spectrum of Phase Noise

c0

c1

c2

c3

ω1f---

ω 1

f 3----

Noise components around integer multiples of the oscillation frequency havethe strongest effect on phase noise, and their effect is weighted by the Fouriercoefficients of the ISF, cn.

log(ω−ω0)


Outline





Measurement Results

Conclusion



c012π------ Γ x( )dx

0

2π

∫=

t

t

V out t( )

Γ ωt( )

Symmetric rise and fall time

t

t

V out t( )

Γ ωt( )

Asymmetric rise and fall time

Effect of Symmetry

The dc value of the ISF is governed by rise and fall time symmetry, and

controls the contribution of low frequency noise to the phase noise.


1/f 3 Corner of Phase Noise Spectrum

ω1 f

3⁄ω1 f⁄

c0Γrms--------------

2

=

The 1/f3 corner of phase noise is NOT the same as 1/f corner of device noise

log(ω-ωo)

L ∆ω( )

1f 2-----

c0

c1

c2

c3

ω1f---

ω 1

f 3----

By designing for a symmetric waveform, the performancedegradation due to low frequency noise can be minimized.


6.4µm0.8µm----------------

6.4µm0.8µm----------------

3.2µm0.8µm----------------

3.2µm0.8µm----------------

i1 t( ) i2 t( )

Ring Oscillator with an Asymmetric Stage

The effect of asymmetry can be seen by comparing the effect of

low frequency injection into symmetric and asymmetric nodes.


0.8GHz 0.9GHz 1.0GHz 1.1GHzFrequency

-100

-50

0

Pow

er b

elow

Car

rier

(dB

c)

Injection into Asymmetric Node

0.8GHz 0.9GHz 1.0GHz 1.1GHzFrequency

-100

-50

0

Injection into Symmetric Node

-40dBc -52dBc

Low Frequency Upconversion


0.5 1.0 1.5 2.0 2.5Wp to Wn ratio

-80.0

-70.0

-60.0

-50.0

-40.0

-30.0

Sid

eban

d po

wer

bel

ow c

arrie

r (d

Bc)

Sidebands Due to Low Frequency Injection

Effect of Rise and Fall Time Symmetry

WN/L

WP/L

fo=1GHzfoff=50MHz

i t( )

1 2 3 4 5Analytical ExpressionSimulation


25µm1µm

---------------25µm1µm

---------------

150µm1µm

------------------150µm

1µm------------------

1mA

i injection t( )

Gnd

Effect of Differential Symmetry

The noise sources on each of the differential nodes are not fully correlated.

It is the symmetry of the half circuit that matters.


600MHz 700MHz 800MHz 900MHzFrequency (Hz)

-80

-60

-40

-20

0

Pow

er (

dB)

Low Frequency Current Injection into Differential Ring

-46dBc

Effect of Differential Symmetry

Differential symmetry does not automatically eliminate the low frequency upconversion.


L

C

Vdd

A Symmetric LC Oscillator

Possible to Adjust Symmetry Properties of the Waveform

WN/L

WP/L

WN/L

WP/L

Adjust ratiosfor symmetry


Outline





Measurement Results

Conclusion



0

5

10

15

20

Col

lect

or V

olta

ge (

Vol

ts)

0 5ns 10ns 15nsTime

0

1mA

2mA

3mA

Col

lect

or C

urre

nt

Time Varying Current in Colpitts Oscillator

10kΩ

40pF

200pF500µA

200nH

Gnd

Gnd

Vcc


i n t( ) i n0 t( ) α ω0t( )=

i n0 t( )

α ω0t( )

Cyclostationary Properties, Time Domain

φ t( ) in τ( )Γ ω0τ( )

qmax------------------dτ

∞–

t

∫=

in0 τ( )α ω0τ( )Γ ω0τ( )

qmax-------------------------------------dτ

∞–

t

∫=

Γeff x( ) Γ x( ) α x( )⋅=

Effective ISF:

A cyclostationary source can be modeled as stationary with a new ISF.

Noise Modulating Function (NMF)


0.0 2.0 4.0 6.0x (radians)

-1.5

-0.5

0.5

1.5

Colpitts Oscillator

Γeff x( ) Γ x( )α x( )=

Γ x( )

α x( )

Effective ISF

ISFNMF α x( )

Γ x( )

Γeff x( )

Gnd

Vcc


0.0 2.0 4.0 6.0x (radians)

-1.0

-0.5

0.0

0.5

1.0

5 Stage Ring Oscillator

Γeff x( ) Γ x( )α x( )=

Γ x( )

α x( )

Effective ISF

ISFNMF

1 2 3 4 5


Outline





Measurement Results

Conclusion



i t( )i t( ) i t( ) i t( )i t( )

Fully Correlated Sources

1 2 3 4 5

Γk ωot( ) Γ ωot2πkN

---------+ =Similar stages:

ej2π5

------

ej8π5

------

ej4π5

------

ej6π5

------Γk ω0τ( )k 1=

N

∑ Nc0≈

Superposition: φtotal t( ) φ t( )k 1=

N

∑ i τ( )qmax----------- Γk ω0τ( )

k 1=

N

∑ dτ∞–

t

∫Nc0

qmax----------- i τ( )dτ

∞–

t

∫≈= =


φtotal t( )Nc0

qmax----------- i τ( )dτ

∞–

t

∫≈

i t( )i t( ) i t( ) i t( )i t( )

1 2 3 4 5

Fully Correlated Sources

Only the low frequency portion of substrate and supply noise is important, provided:

1. Stages and loadings are the same

2. The noise sources are identical

This is good news since c0 can be significantly reduced by adjusting the symmetry.


Outline





Measurement Results

Conclusion



c0c1 c2 c3

2ω0ω0 3ω0ω

Sφ ω( )

i n2

∆f------- ω( )

1f--- Noise

Sv ω( )

2ω0ω0 3ω0ω

ω

PM∆ω

∆ω ∆ω ∆ω

∆ω

Noise Contributions from nωo

φ t( ) 1qmax------------- c0 i τ( )dτ

∞–

t

∫ cn i τ( ) nωτ( )cos dτ∞–

t

∫n 1=

∞

∑+=

∆ω


10kHz 100kHz 1MHzOffset from Multiple Integer of Carrier (fm)

-50.0

-40.0

-30.0

-20.0

-10.0

Sid

eban

d P

ower

bel

ow C

arrie

r (d

Bc)

Injection at Integer Multiples of f0

f=fmf=f0+fmf=2f0+fmf=3f0+fmf=4f0+fm

1 2 3 4 5

i t( ) I n f 0 f m+( )sin=

fm dBc

f0


1µA 10µΑ 100µΑ 1mAInjected Current (amperes)

-80.0

-60.0

-40.0

-20.0

Sid

eban

d P

ower

bel

ow C

arrie

r (d

Bc)

f=fmf=f0+fmf=2f0+fmf=3f0+fm

Sideband Power vs. Injection Current

i t( ) I n f 0 f m+( )sin=

1 2 3 4 5

fm dBc

f0


100kHz 1MHzFrequency of Injection (Hz)

-55

-45

-35

Sid

eban

d P

ower

bel

ow C

arrie

r (d

Bc)

Symmetric, n1

Symmetric, n4

Asymmetric, n1

Asymmetric, n4

Symmetric vs. Asymmetric Ring Oscillator

i1 t( )

1 2 3 4 5

i4 t( )

fm dBc

f0


102 103 104 105 106

Offset from the Carrier (Hz)

-130

-80

-30

Sid

eban

d be

low

Car

rier

per

Hz

(dB

c/H

z)

f0=232MHz, 2µm Technology

L ∆f 10 0.84 ∆f2⁄( )log=

L 500kHz 114.7dBcHz----------–=

ω 1

f 3-----

75kHz=

L 500kHz 114.5dBcHz----------–=

ω 1

f 3-----

80kHz=

Predicted:

Measured:

5-Stage Single-Ended Ring Oscillator


102 103 104 105 106-130

-80

-30

Sid

eban

d P

ower

bel

ow C

arrie

r pe

r H

z (d

Bc/

Hz)

f0=115MHz, 2µm Process

Offset from the Carrier (Hz)

L ∆f 10 0.152 ∆f2⁄( )log=

L 500kHz 122.1dBcHz----------–=

ω 1

f 3-----

43kHz=

L 500kHz 122.5dBcHz----------–=

ω 1

f 3-----

45kHz=

Predicted:

Measured:

11-Stage Single-Ended Ring Oscillator


W/LNtail

W/LNinv

W/LPtail

W/LPinv

Nbias

Pbias

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Symmetry Voltage (VPbias+VNbias) [V]

200kHz

400kHz

600kHz

800kHz

1.0MHz

1.2MHz

1/f3

Cor

ner

Fre

quen

cy

9-Stage Current Starved Single-Ended VCO

Vsym=VPbias+VNbias

f0=600MHz, 0.25µm Process


2 4 6 8 10 12 14 16Tail Current (mA)

-126

-124

-122

-120

-118

-116

-114

-112

Pha

se N

oise

at 6

00K

Hz

offs

et (

dBc/

Hz)

f0=1.8GHz, 0.25µm Process (VDD =3V)

Measurement

Complementary Cross-Coupled LC Oscillator

C

L

Vdd

bias

Gnd

Itail

Γ2rms=0.5

Simulated ISF


Complementary Cross-Coupled VCO

1.5

2

2.5

3

24

68

1012

1416

x 10−3

−126

−124

−122

−120

−118

−116

−114

−112

f0=1.8GHz, 0.25µm Process

Pha

se n

oise

bel

ow c

arrie

r at

600

kHz

offs

et

Vdd Itail (mA)

-121dBc/Hz@600kHz

f0=1.8GHz

P=6mW

CL

Vdd

bias

Gnd

Itail


Complementary vs. NMOS-Only VCO

Vdd

C

bias

Gnd

L/2 L/2

Itail

CL

Vdd

bias

Gnd

Itail

11.5

22.5

3

2 4 6 8 10 12 14 16

x 10−3

−126

−124

−122

−120

−118

−116

−114

−112

f0=1.8GHz, 0.25µm Process

VddItail (mA)

Pha

se n

oise

bel

ow c

arrie

r at

600

kHz

offs

et

NMOS-Only

Complementary


Die Photo of the Complementary Oscillator

700µm x 800µm

L L

C

Active

Bypass Bypass

Driver

Pad limited

0.25µm Process


Conclusion and Contributions

A new general model for phase noise is introduced, which:

is useful both as an analysis and a design tool,

is valid for arbitrary sources of noise and interference,

is independent of the topology of the oscillator,

predicts the effect of symmetry on the upconversion of 1/f noise,

incorporates cyclostationary noise sources naturally,

shows agreement among theory, simulation and measurements.

reduces to previously existing models as special cases,

predicts the effect of correlation on phase noise,


AcknowledgmentsAdvisor: Prof. Thomas H. Lee.

Stanford faculty: Prof. K. Saraswat and Prof. J. McVittie.

Tom Lee group: Arvin, Dave, Derek, Hamid, Hirad, Kevin, Mar, Mohan, Raf, Ramin and Tam.

Committee: Prof. Bruce Lusignan, Prof. Bruce Wooley and Prof. Mark Horowitz.

Brains : Ken, Masoud, Mehrdad, Sotirios and Stefanos, Tom Lee group, H. Swain, D. Leeson.

Other groups: Wooley group, Horowitz group, SNF.

Companies: Rockwell Semiconductors, Texas Instrument and Lucent Technologies.

DEDICATED TO ALL MY TEACHERS.

Family: Tabassom, Mom, Dad and Emad.

Friends: Adrian, Ali (x3), Amir (x2), Amirmasoud, Arash, Ardavan, Arvin, Azam, Azita,Babak(x2), Bijan(x2), David(x2), Derek, Eric, Faranak, Farid, Fati, Hamid (x3),Hirad, Hossein, Jalil, Joe, Kambiz, Kati, Kevin, Kiarash, Koohyar, Manohar,Mar, Maryam, Masoud, Matt, Mehdi(x3), Mehrdad, Mina, Mohan, Nogol(x2),Patrick, Philip, Raf, Ramin(x2), Rasoul, Reza(x2), Sha, Shahram, Shideh,

Real power: Ann Guerra.

Sotirios, Stefanos, Tamara, Tayebe, Tina, Yasmin, Younes.

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Phase Noise in Oscillators - Stanford University

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