Linearity, Time-Variation, Phase Modulation and Oscillator ...class.ece.iastate.edu/djchen/ee507/PhaseNoiseTutorialLee.pdf · Linearity, Time-Variation, Phase Modulation and ... Time-Variation,

T. Lee, Stanford University Center for Integrated Systems

Linearity, Time-Variation, Phase Modulation and Oscillator Phase

Linearity, Time-Variation, Phase Modulation and Oscillator Phase Noise

Prof. Thomas H. LeeStanford University

[email protected]://www-smirc.stanford.edu

Tom Lee, Stanford University Center for Integrated Systems

A Phase Noise Tutorial

Preliminaries (to refresh dormant neurons...)

A system is linear as long as superposition holds.

Scaling of a single input is included, since scaling may al-ways be viewed as the result of summing.

The response to an impulse then yields sufficient informa-tion to deduce the response to any arbitrary input.

All real systems may be made to act nonlinearly for some in-puts (e.g., the response to 1mV may differ in shape and odor from the response to a gigavolt).

Linearity thus holds only over some restricted range of excita-tions, in practice.

A system is time-invariant if the only result of time-shifting any input is to shift the response by precisely the same amount.

If a system is LTI, it may be shown that excitation at

f

produces a response only at

f

.



Preliminaries

If a system is LTV, it is no longer generally true that an excitation at

f

produces a response at the same frequen-cy.

Superposition still holds, however, so the response to the sum of two inputs may be deduced from the response to each.

If a system is nonlinear, the response may also contain spectral components not present in the excitation.

Dependency of output on combination of inputs not neces-sarily linear; this difference can be used as a basis for deter-mining whether spectral shaping is due to time-variation or nonlinearity.

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

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.


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


Isubstrate

Vsupply

Isupply

Substrate and Supply Noise



Phase Noise: General Considerations

Consider simple oscillator:

RLC

+

noiseless

negative

R

:

Can show that noise-to-signal power ratio is

Negative-

R

must cancel the tank loss in steady state.

Noise current sees pure

LC

impedance.

G C LNoiselessV

4kTG

EnergyRestorer

NS

Vn

2

Vsig

2

kTE

stored

ωkTQP

diss

= = =



General Considerations

Practical oscillators operate in one of two regimes:

Current-limited

, in which the oscillation amplitude is linear-ly proportional to

I

bias

R

tank

.

Voltage-limited

, in which the oscillation amplitude is large-ly independent of bias current.

Ibias

Vosc

Current-limited

Voltage-limited




In the voltage-limited regime, increases in bias current do not increase carrier power.

Additional dissipation only increases noise power, so CNR degrades (decreases).

In the current-limited regime, increases in bias current increase signal power faster than noise power.

CNR increases until boundary with voltage-limited region is approached.

Best oscillator performance is typically achieved near the transition point between current- and voltage-limit-ed modes of operation.


Short Course: CMOS RF IC Design

Oscillator Phase Noise

An expression for noise-to-carrier ratio reveals important optimization objectives:

Vcarrier∝ IbiasRtank==> Pcarrier ∝ (Ibias)2Rtank in the current-lim-ited regime of oscillation.

Pnoise = kT/C = kTω2L, if dominated by tank loss.

So, N/C ∝ kTω2L/(Ibias)2Rtank to an approximation.

Generally want to minimize L/R to optimize oscillator for a given oscillation frequency and power consumption. This result contradicts much published advice, which advocates

maximizing tank inductance.

N/C is important, but also need to know noise spectrum.




Assuming all noise comes from tank loss, PSD of tank voltage is given approximately by

.

Noise power splits evenly between phase and ampli-tude domains. Then, we finally have

.

Funny units: Some dBc/hertz at a certain offset frequency. Example: –110dBc/Hz @ 600kHz offset, at 1.8GHz.

vn

2

∆f

in

2

∆fZ

2⋅ =4kTG 1

G

ω0

2Q∆ω⋅

2

=4kTRω

02Q∆ω

2

=

L ∆ω =10 logv

n

2∆f⁄

vsig

2⋅ =10 log 2kT

Psig

ω0

2Q∆ω

2

⋅⋅



Simple LTI Model vs. Reality

Previous expression doesn’t quite describe real phase noise spectra:

log ∆ω

L(∆ω)

2−

3−

∆ω1 f

3⁄

ωc

RealityOur LTI eqn.



Leeson Model

Leeson provided empirical

fix to remove discrepancies:

.

Factor

F

accounts for excess noise in all regions.

∆ω

1/

f

3

accounts for 1/

f

3

region close to carrier.

First additive factor of 1 accounts for noise floor.

Problem: Can’t compute these fudge factors

a priori

; they are basically

post hoc

fitting parameters.

Need to revisit unstated assumptions.

Is an oscillator truly a linear, time-invariant system?

L ∆ω =10 log 2FkTP

sig

1ω

02Q∆ω

+

2

1

∆ω1 f

3⁄

∆ω+

⋅ ⋅⋅


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( )

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=

∞

∑+=


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.


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

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

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


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


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


Short Course: Phase Noise in Oscillators

Plus ça change...

To exploit cyclostationary effects, arrange to supply energy to the tank impulsively, where the ISF is a minimum.

This idea is actually very old; mechanical clocks use an es-capement to deliver energy from a spring, to a pendulum in impulses.

Coupled topendulum

Drivenby spring



Phase Noise

In the best implementations, impulses are delivered at or near the pendulum’s velocity maxima. The escape-ment thus restores energy without disturbing the period of oscillation. [Airy, 1826]

See: www.database.com/~lemur/dmh-airy-1826.html (my thanks to Byron Blanchard for finding this reference.)

Similarly, the optimal moments for an

LC

oscillator are near the voltage maxima.

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

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

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------+

= =

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

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


Another example (Margarit et al.)

Gnd

Bias

VCC

Bias

L2 L1

R2 R1C2

C1

C4 C3

1.6 mA1.2 mA0.8 mA0.4 mA

0

227ns 228ns 229ns 230ns 231ns

227ns 228ns 229ns 230ns 231ns

Col

lect

or c

urre

nth φφ

(°/p

C)

20

10

0

–10

–20h φφef

f (°/

pC) (

dash

ed)

227ns 228ns 229ns 230ns 231ns

227ns 228ns 229ns 230ns 231ns

VCO

outp

ut vo

ltage 500 mV

–500 mV

h φφ (°

/pC)

4

2

0

–2

–4


Predicted vs. measured

LTV model

Spectre

Offset frequency (Hz)102 103 104 105 106

0

–10

–20

–30

–40

–50

–60

–70

–80

–90

–100

–110

–120

–130

–140

–150

dBc/Hz


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:


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


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)


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


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



Amplitude Noise

Phase noise generally dominates close-in spectrum. Amplitude noise typically dominates far-out spectrum.

Effect of amplitude noise may be accommodated with the same general approach: Investigate impulse re-sponse.

If amplitude control mechanism acts as a first-order system (e.g., if it is well damped), amplitude impulse response will die out with a time constant equal to the inverse bandwidth of the control loop.

For an

LC

tank, this bandwidth is the tank bandwidth,

ω

0

/Q.

Corresponding contribution to noise spectrum is flat to frequency offset equal to that bandwidth, then rolls off; produces pedestal in overall response.

If amplitude control is underdamped (e.g., behaves as 2nd order), can get peaking in the spectrum.



Amplitude Response

Possible responses corresponding to these control dy-namics look roughly as follows:

log ∆ω

L(∆ω)

2−

From amplitude noise

From phase noise

Underdamped

Well-damped



Summary and Conclusions

LTI theories say:

Maximize signal power and resonator

Q

and operate at edge of current-limited regime, with minimum ratio

L

/

R

consis-tent with oscillation.

Can’t do anything about 1/

f

3

corner frequency.

Corner frequency is strictly technology-limited.

LTV theory says:

Continue to maximize signal power, resonator

Q

, and

R

/

L.

Use tapped tanks (à la Clapp, e.g.).

Maximize symmetry (in the ISF sense) to reduce 1/

f

3

corner frequency.

Choose topologies and bias conditions so that energy is re-turned to tank impulsively.



Acknowledgments

Prof. Ali Hajimiri of Caltech, who developed this theory while a Ph.D. student at Stanford.

David Leeson, for graciously encouraging us to build on his theory.

Web URL is, again: http://www-smirc.stanford.edu

Documents

Linearity, Time-Variation, Phase Modulation and Oscillator ...class.ece.iastate.edu/djchen/ee507/PhaseNoiseTutorialLee.pdf · Linearity, Time-Variation, Phase Modulation and ... Time-Variation,