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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: hajimiri@smirc.stanford.edu
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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
∆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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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.
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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.
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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:
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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.]
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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.
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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.
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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.
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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= = =
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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:
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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)
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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)
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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.
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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
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Die Photo of 12-Stage Differential Ring Osc.
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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
http://smirc.stanford.edu/papers/CICC98s-ali.pdf Email: hajimiri@smirc.stanford.edu
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.
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