phase noise in oscillators - stanford university
TRANSCRIPT
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
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Frequency Instability: Time Domain
σ2
∆T
Known As Clock Jitter.
Oscilloscope
Oscillator
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D Q
CLK
Flip-Flop
tsetup thold
Oscillator
Data
Clock
Timing Jitter in Digital Applications
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Ideal
Actual
Frequency Instability: Frequency Domain
f
f
f0
Spectrum Analyzer
Oscillator
Known As Phase Noise.
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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
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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
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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)
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Isubstrate
Vsupply
Isupply
Substrate and Supply Noise
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Outline
Introduction and Definitions
Time-Variant Phase Noise Model
Upconversion of 1/f Noise
Cyclostationary Noise Sources
Measurement Results
Conclusion
Substrate and Supply Noise
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.
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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
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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)
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∆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«
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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«
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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.
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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
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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:
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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.
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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=
∞
∑+=
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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=
∞
∑+=
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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)
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Outline
Introduction and Definitions
Time-Variant Phase Noise Model
Upconversion of 1/f Noise
Cyclostationary Noise Sources
Measurement Results
Conclusion
Substrate and Supply Noise
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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.
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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.
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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.
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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
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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
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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.
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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.
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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
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
http://smirc.stanford.edu/papers/Orals98s-ali.pdf Email: [email protected]
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
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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)
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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
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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
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Outline
Introduction and Definitions
Time-Variant Phase Noise Model
Upconversion of 1/f Noise
Cyclostationary Noise Sources
Measurement Results
Conclusion
Substrate and Supply Noise
http://smirc.stanford.edu/papers/Orals98s-ali.pdf Email: [email protected]
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
∫≈= =
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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.
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
http://smirc.stanford.edu/papers/Orals98s-ali.pdf Email: [email protected]
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Die Photo of the Complementary Oscillator
700µm x 800µm
L L
C
Active
Bypass Bypass
Driver
Pad limited
0.25µm Process
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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,
http://smirc.stanford.edu/papers/Orals98s-ali.pdf Email: [email protected]
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.