Phase Noise in CMOS

Ali Hajimiri

Stanford University, Stanford, CA 94305

Thomas H. Lee

Differential LC Oscillators

Outline

Introduction and Definitions

Tank Voltage

Measurement Results

Conclusion

Noise Sources

Effect of Tail Current Source

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)

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( )

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 spectral density of

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

Γrms is the rms value of the ISF.

,

where

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=

∞

∑+=

1/f 3 Corner of Phase Noise Spectrum

ω1 f

3⁄ω1 f⁄

ΓdcΓrms--------------

2

=

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

L/2

C

Vdd

A Symmetric LC Oscillator

Uses the same current twice for high transconductance.

WN/L

WP/L

WN/L

WP/L

Adjust ratiosto fine tune

[Also appears in: J.Craninckx, et al, Proceedings of CICC 97.]

symmetry

L/2

Outline

Introduction and Definitions

Tank Voltage

Measurement Results

Conclusion

Noise Sources

Effect of Tail Current Source

L

C

VDD

Itail

Itail

Itail

Tank Voltage Amplitude

Itail

-Itail

t

i(t)

Assuming fast switching of the

differential pair, the current can

be approximated as:

ReqLCi(t)

Itail

-Itail

t

i(t)

Tank Voltage Amplitude

Assuming rectangular waveform:

Vmax4π--- I tailReq=

Effectively, the current waveform

is closer to sinusoidal, therefore:

Vmax I tailReq≈

“Current limited” mode .

2 4 6 8 10 12 14 16 18 20

Tail current (mA)

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Tan

k vo

ltage

sw

ing

(vol

t)

Vdd=1.5V

Vdd=2.5V

Vdd=3.0V

Modes of Amplitude Limiting

Voltage Limited

Current Limited

Complementary cross-coupled LC oscillator

Outline

Introduction and Definitions

Tank Voltage

Measurement Results

Conclusion

Noise Sources

Effect of Tail Current Source

i p12 i p2

2

in12

in22

i tail2

L

C

VDD

bias

vrs2 rs

Major Noise Sources

Different noise sources affect

phase noise differently.

in2

∆f------ 4kTγµCox

WL----- VGS VT–( )=

Valid in both long and short

channel regimes.

Inductor Noise:

vn2

∆f------ 4kT rs=

C

L

i1 t( )

i2 t( )r r

R

C

L

i2 t( ) i1– t( )

2---------------------------

2r

Equivalent Circuit for Differential Sources

DifferentialEquivalent

in2

∆f------

diff pair–

14---

in12

∆f--------

in22

∆f--------

i p12

∆f--------

i p22

∆f--------+ + +

12---

in2

∆f------

i p2

∆f------+

= =

Outline

Introduction and Definitions

Tank Voltage

Measurement Results

Conclusion

Noise Sources

Effect of Tail Current Source

0.5

1.0

1.5

2.0

2.5

Nod

e V

olta

ge (

V)

0.0 1.0 2.0 3.0 4.0 5.0 6.0x (radians)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

ISF

V+V-

Vtail

NMOS ISF

PMOS ISF

Tail ISF

Waveform and ISF

Effect of Tail Current Source

L

C

VDD

2ω0ω0

ω

i n2

∆f------- ω( )

1f--- Noise

i tail2

For the tail current source, only

in the vicinity of even harmonics

of the tail current source affect

phase noise.

low frequency noise and noise

Outline

Introduction and Definitions

Tank Voltage

Measurement Results

Conclusion

Noise Sources

Effect of Tail Current Source

Die Photo of the Complementary Oscillator

700µm x 800µm

L L

C

Active

Bypass Bypass

Driver

Pad limited

0.25µm process

103

104

105

106

107

Offset from carrier (Hz)

-140

-130

-120

-110

-100

-90

-80

-70

-60

-50P

hase

noi

se b

elow

car

rier

(dB

c/H

z)

Complementary cross coupled LC oscillatorf0=1.8GHz, Pdiss=6mWatt

106 107103 104 105

Phase Noise vs. Offset from Carrier

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

Conclusion

● The effect of tail current source on the amplitude is discussed.

● Major noise sources are identified.

● Phase noise of differential MOS oscillators is analyzed.

● The effect of tail current source noise is shown.

● Good agreement between theory and measurements is observed.