Phase Noise in Multi-Gigahertz CMOS Ring Oscillatorssmirc.stanford.edu/papers/CICC98s-ali.pdf · Stanford University, Stanford, CA 94305 Phase Noise in Multi-Gigahertz Ali Hajimiri

Stanford University, Stanford, CA 94305

Phase Noise in Multi-Gigahertz

Ali Hajimiri

CMOS Ring Oscillators

Thomas H. LeeSotirios Limotyrakis

http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: [email protected]

Outline

Review of Time-Variant Phase Noise Model

Measurement Results

Conclusion

Calculation of the ISF for Ring Oscillators

Expression for Phase Noise of Ring Oscillators


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

Spectrum Analyzer

Oscillator


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)

i t( )

1 2 3 4 5δ t τ–( )

tτ

i(t)

Oscillators Are Time-Variant Systems

Phase error

Amplitude change

persists indefinitely.

is quenched.

WN/L

WP/L


∆V∆q

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

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

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

∆q qswing«

2ns 3ns 4nsTime

1.0

3.0

5.0

Nod

e V

olta

ge (

Vol

t)

∆φ

∆V

Impulse Response of a Ring Oscillator


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.


2.0 4.0 6.0

1.0

3.0

5.0

2.0 4.0 6.0x (radians)

-0.5

0.0

0.5

1.0

-1.0

Waveform

ISF

V 1 t( )

Γ x( )

Impulse Sensitivity Function (ISF)

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


φ t( ) hφ t τ,( )i τ( )dτ∞–

∞

∫i τ( )

qmax------------------Γ ω0τ( )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 NΓrms

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.

[A. Hajimiri and T.H. Lee, “A general model of phase noise in electrical oscillators,”

IEEE Journal of Solid-State Circuits, vol. 33, no. 2, Feb. 1998.]


1/f 3 Corner of Phase Noise Spectrum

ω1 f

3⁄ω1 f⁄

c0Γrms--------------

2

=

The 1/f 3 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.


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.


t

V out t( ) V out t( )

t

i t( )

t

i t( ) i t( )

A low frequency current induces a frequency change for the asymmetric waveform.

C

Effect of Symmetry


Outline


Measurement Results

Conclusion




0.0 2.0 4.0 6.0x (Radian)

-1.5

-0.5

0.5

1.5

ISF for Various Length, Equal Frequency Ring Oscillators

N=3

N=5N=7

9111315

ISF

Effect of Number of Stages on the ISF

The ISF for different number of stages can be approximated by similar triangles.


Approximate ISF for Ring Oscillators

2π

2f 'max------------

1f 'max------------

Γ x( )

x

2π x

f 'max

f x( )

f– 'max

1

1f 'max------------

2f 'max------------

The peak of the ISF is inversely proportional to the maximum slope of the normalized waveform.

Γrms2 1

2π------ Γ2 x( ) xd0

2π∫

42π------ x 2 xd

0

1 f 'max⁄

∫2

3π------1

f 'max-------------

3= = =


Γrms2 2π2

3η3--------- 1

N3

------=

t Dη

f 'max-------------=

Risetime and Delay Relationship

2π x

f 'max

f x( )

f– 'max

1

1f 'max------------

ηf 'max------------

ηf 'max------------

2π 2Nt D2Nηf 'max-------------= =Period:

Stage Delay:

1f 'max-------------

πNη--------=⇒

ISF RMS:


10Number of Stages, N

0.1

1.0

RM

S V

alue

of I

SF

ISF RMS vs. Number of Stages

2 3

+: Fixed Frequency, 5V Supplyo: Fixed Drive, Similar Invertersx: Fixed Frequency, 3V Supply

Inverter Chain CMOS Ring Oscillator

__: 4/N1.5 (η=0.75)


10Number of Stages, N

0.1

RM

S v

alue

of I

SF

3 4

0.2

0.3

+: Fixed Power, Fixed Swingo: Fixed Power, Fixed Load Resistancex: Fixed Tail Current, Fixed Load Resistance

ISF RMS vs. Number of Stages

Differential MOS Ring Oscillator

__: 3/N1.5 (η=0.9)


Outline


Measurement Results

Conclusion




in2

∆f------ 4kTγ gd0 4kTγµCox

WL----- VGS VT–( )= =

I D

µCox

2------------W

L----- VGS VT–( )2

=

in2

∆f------ 8kT

γ I D

VGS VT–-----------------------=

in2

∆f------ 8kT

I D

Vchar-------------=

Vchar VGS VT–( ) γ⁄=

I D

µCox

2------------WEc VGS VT–( )=

in2

∆f------ 8kT

γ I D

ECL----------=

Vchar EcL γ⁄=

Long Channel Short Channel

Valid for both short and long channel regime.

Channel Noise of MOS Transistors


VDD

bias

Gnd

RL RL

Itail

W/L W/Lin2

∆f------

N

in2

∆f------

Load

P NItailVDD=

f 01

2NtD------------- 1

2ηNtr---------------

I tail

2ηNqmax-----------------------≈ ≈=

in2

∆f------

in2

∆f------

N

in2

∆f------

Load

+ 4kT Itail1

Vchar------------- 1

RLI tail----------------+

= =

L min ∆f 83η------ N⋅ kT

P------

VDD

Vchar-------------

VDD

RLI tail----------------+

f 0

∆f------

2

⋅ ⋅ ⋅≈

Vchar VGS VT–( ) γ⁄=

Vchar EcL γ⁄=

N Stages

Power Dissipation:

Frequency:

Noise:

Short channel:

Long channel:

Phase Noise in Differential Ring Oscillator


L min ∆f 83η------ N⋅ kT

P------

VDD

Vchar-------------

VDD

RLI tail----------------+

f 02

∆ f2

---------⋅ ⋅ ⋅≈

For a given power and frequency, phase noise degrades

with number of stages, N, in differential ring oscillators.

Doubling the number of stages:

VDD

bias

Gnd

RL/2 RL/2

Itail /2

W/L W/L

RL is divided by 2 to keep the frequency constant,

Itail is divided by 2 to keep the power constant,

Therefore:Maximum charge swing, qmax, is 4 times smaller.

This is NOT the case for single-ended ring, since the swing is constant.

Effect of Number of Stages on Phase Noise


Outline


Measurement Results

Conclusion




VDD

bias

Gnd

RL RL

Itail

Control

W/L W/L

Differential Ring Oscillators

Effective channel length: Leff=0.25µm.

Unsilicided poly load resistors.

N Stages

Maximum frequency of 5.4GHz.


0 10 20 30 40 50 60 70 80NMOS width (µm)

-104

-102

-100

-98

-96

-94

-92

-90

Pha

se N

oise

at 1

MH

z of

fset

(dB

c/H

z)

Predicted and Measured Phase Noise

Differential Ring Oscillators; RL=1kΩ, Leff=0.25µm

4.47GHz

3.39GHz

2.24GHz1.19GHz

Measured

Predicted


Die Photo of 12-Stage Differential Ring Osc.


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


Conclusion

An expression for the phase noise of ring oscillators is derived.

The effect of number of stages on the phase noise is discussed.

A closed form approximate expression for the ISF is obtained.

Reduction in the upconversion of 1/f noise due to symmetry is shown.

Good agreement between theory and measurements is observed.

The new time-variant phase noise model is applied to ring oscillators.

Documents

Phase Noise in Multi-Gigahertz CMOS Ring Oscillatorssmirc.stanford.edu/papers/CICC98s-ali.pdf · Stanford University, Stanford, CA 94305 Phase Noise in Multi-Gigahertz Ali Hajimiri