Phase Noise in Oscillators - Stanford University

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Phase Noise in Oscillators

Ali Hajimiri

Stanford University, Stanford, CA 94305

http://smirc.stanford.edu/papers/Orals98s-ali.pdf Email: hajimiri@smirc.stanford.edu

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

Known As Clock Jitter.

Oscilloscope

Oscillator

Flip-Flop

tsetup thold

Oscillator

Timing Jitter in Digital Applications

Actual

Frequency Instability: Frequency Domain

Spectrum Analyzer

Oscillator

Known As Phase Noise.

Desired SignalStrong Adjacent Channel

Desired Signal

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

Phase Noise in RF Applications

Units of Phase Noise

log(∆ω)=log(ω-ω0)

L ∆ω 1

f 3------

f 2------

Measured in dB below carrier per unit bandwidth.

(-20dB/dec)

(-30dB/dec)

Sv ω( )

dBc Hz⁄[ ]

ω1 f 3⁄

Internal noise sources set a fundamental limit for phase noise.

∆f------

MOSBJT

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( )+[ ]⋅=

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–( )

Oscillators Are Time-Variant Systems

Impulse injected at the peak of amplitude.

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

Impulse injected at zero crossing.τ

Once Introduced, phase error persists indefinitely.

Non-linearity quenches amplitude changes over time.

dVdt--------

LimitCycle

Active Device

C Lδ t τ–( )

G -G(A)

Amplitude Restoring Mechanism

∆V∆q

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

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

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

Impulse Response of a Relaxation Oscillator

∆q qswing«

2ns 3ns 4ns

2ns 3ns 4nsTime

∆V∆q

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

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

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

i t( )

1 2 3 4 5δ t τ–( )

Impulse Response of a Ring Oscillator

∆q qswing«

Phase Impulse Response

φ t( )hφ t τ,( )

τ 0 τ

hφ t τ,( )Γ ωoτ( )

qmax-------------------u t τ–( )=

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.

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

φ t( ) hφ t τ,( )i τ( )dτ∞–

qmax---------------- Γ ω0τ( )i τ( )dτ

∞–

∫= =

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

∆f---------

L ∆ω Γrms

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

c1 ω0t θ1+( )cos

Σ ω0t φ t( )+[ ]cos

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

φ t( )

∞–t

V t( )

ISF Decomposition

Phase can be written as:

φ t( ) 1qmax------------- c0 i τ( )dτ

∞–

∫ cn i τ( ) nω0τ( )cos dτ∞–

∫n 1=

Γ ωot( ) c0 cn nω0τ θn+( )cosn 1=

c0c1 c2 c3

2ω0ω0 3ω0ω

Sφ ω( )

∆f------- ω( )

1f--- Noise

Sv ω( )

2ω0ω0 3ω0ω

PM∆ω

∆ω ∆ω ∆ω

Noise Contributions from nωo

φ t( ) 1qmax------------- c0 i τ( )dτ

∞–

∫ cn i τ( ) nωτ( )cos dτ∞–

∫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

ω1f---

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

V out t( )

Γ ωt( )

Symmetric rise and fall time

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

3⁄ω1 f⁄

c0Γrms--------------

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-----

ω1f---

f 3----

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

6.4µm0.8µm----------------

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

Injection into Asymmetric Node

0.8GHz 0.9GHz 1.0GHz 1.1GHzFrequency

Injection into Symmetric Node

-40dBc -52dBc

Low Frequency Upconversion

0.5 1.0 1.5 2.0 2.5Wp to Wn ratio

Sidebands Due to Low Frequency Injection

Effect of Rise and Fall Time Symmetry

fo=1GHzfoff=50MHz

i t( )

1 2 3 4 5Analytical ExpressionSimulation

25µm1µm

---------------25µm1µm

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

150µm1µm

------------------150µm

1µm------------------

i injection t( )

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)

Low Frequency Current Injection into Differential Ring

-46dBc

Effect of Differential Symmetry

Differential symmetry does not automatically eliminate the low frequency upconversion.

A Symmetric LC Oscillator

Possible to Adjust Symmetry Properties of the Waveform

Adjust ratiosfor symmetry

Outline

Measurement Results

Conclusion

0 5ns 10ns 15nsTime

Time Varying Current in Colpitts Oscillator

200pF500µA

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

∞–

in0 τ( )α ω0τ( )Γ ω0τ( )

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

∞–

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)

Colpitts Oscillator

Γeff x( ) Γ x( )α x( )=

Γ x( )

α x( )

Effective ISF

ISFNMF α x( )

Γ x( )

Γeff x( )

0.0 2.0 4.0 6.0x (radians)

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=

∑ Nc0≈

Superposition: φtotal t( ) φ t( )k 1=

∑ i τ( )qmax----------- Γk ω0τ( )

∑ dτ∞–

∫Nc0

qmax----------- i τ( )dτ

∞–

∫≈= =

φtotal t( )Nc0

qmax----------- i τ( )dτ

∞–

∫≈

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( )

∆f------- ω( )

1f--- Noise

Sv ω( )

2ω0ω0 3ω0ω

PM∆ω

∆ω ∆ω ∆ω

Noise Contributions from nωo

φ t( ) 1qmax------------- c0 i τ( )dτ

∞–

∫ cn i τ( ) nωτ( )cos dτ∞–

∫n 1=

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

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

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

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

100kHz 1MHzFrequency of Injection (Hz)

Symmetric, n1

Symmetric, n4

Asymmetric, n1

Asymmetric, n4

Symmetric vs. Asymmetric Ring Oscillator

i1 t( )

1 2 3 4 5

i4 t( )

fm dBc

102 103 104 105 106

Offset from the Carrier (Hz)

f0=232MHz, 2µm Technology

L ∆f 10 0.84 ∆f2⁄( )log=

L 500kHz 114.7dBcHz----------–=

f 3-----

75kHz=

L 500kHz 114.5dBcHz----------–=

f 3-----

80kHz=

Predicted:

Measured:

5-Stage Single-Ended Ring Oscillator

102 103 104 105 106-130

f0=115MHz, 2µm Process

Offset from the Carrier (Hz)

L ∆f 10 0.152 ∆f2⁄( )log=

L 500kHz 122.1dBcHz----------–=

f 3-----

43kHz=

L 500kHz 122.5dBcHz----------–=

f 3-----

45kHz=

Predicted:

Measured:

11-Stage Single-Ended Ring Oscillator

W/LNtail

W/LNinv

W/LPtail

W/LPinv

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

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)

f0=1.8GHz, 0.25µm Process (VDD =3V)

Measurement

Complementary Cross-Coupled LC Oscillator

Γ2rms=0.5

Simulated ISF

Complementary Cross-Coupled VCO

x 10−3

−126

−124

−122

−120

−118

−116

−114

−112

f0=1.8GHz, 0.25µm Process

Vdd Itail (mA)

-121dBc/Hz@600kHz

f0=1.8GHz

Complementary vs. NMOS-Only VCO

L/2 L/2

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)

NMOS-Only

Complementary

Die Photo of the Complementary Oscillator

700µm x 800µm

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.

Phase Noise in Oscillators - Stanford University

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