Transcript Jitter in PLLs - Henry Samueli School of Engineering

Voltage-Controlled Oscillator (VCO)
fosc
Desirable characteristics:
• Monotonic fosc vs. VC characteristic
with adequate frequency range
• Well-defined Kvco
KPD
VD
F(s)
+
VC
slope = Kvco
fmin
fVC
fin
fmax
VC
^
Kvco
s
fout
fout
Noise coupling from VC into PLL
output is directly proportional to Kvco.
¸N
EECS 270C / Spring 2014
K^ vco
=
× fVC
s + KPD K^ vco F (s) / N
Prof. M. Green / U.C. Irvine
1
Oscillator Design
Vin Þ 0
A(s)
Vout
Vout
A(s)
º HCL (s) =
Vin
1+ f × A(s)
loop gain
f
Barkhausen’s Criterion:
If a negative-feedback loop satisfies:
then the circuit will oscillate at frequency 0.
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
2
Inverters with Feedback (1)
1 inverter:
V1
V2
1 inverter
V2
feedback
1 stable
equilibrium
point
V1
V2
2 inverters:
V1
feedback
V2
3 equilibrium
points: 2 stable,
1 unstable
(latch)
2 inverters
V1
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3
Inverters with Feedback (2)
3 inverters forming an oscillator:
V2
V1
V2
1 unstable
equilibrium point
due to phase shift
from 3 capacitors
V1
A0
Let each inverter have transfer function Hinv ( jw ) =
1+ jw p
A30
3
Loop gain: Hloop ( jw ) = [Hinv ( jw )] =
3
1+ jw p
(
)
Applying Barkhausen’s criterion:
Hloop ( jw o ) =
EECS 270C / Spring 2014
A30
[1+ 3]
Prof. M. Green / U.C. Irvine
3
> 1 Þ A0 > 2
2
4
Ring Oscillator Operation
tp
VA
tp
tp
VB
VC
VA
tp
VB
1
Tosc = 3t p
2
Þ Tosc = 6t p
tp
VC
tp
VA
1
Tosc
2
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5
Variable Delay Inverters (1)
Inverter with variable load capacitance:
Vin
Current-starved inverter:
Vout
VC
Vin
Vout
VC
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6
Variable Delay Inverters (2)
Interpolating inverter:
ISS
+
VC
_
R
Vout+
R
Vout-
Vin+
Vin- Vin+
VinRG
Ifast
RG
Islow
• tp is varied by selecting weighted sum of fast and slow inverter.
• Differential inverter operation and differential control voltage
• Voltage swing maintained at ISSR independent of VC.
EECS 270C / Spring 2014
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7
Differential Ring Oscillator
+
−
+
−
VA
+
−
VB
VA
VB
VC
VC
+
−
VD
−
+
VA
additional inversion
(zero-delay)
tp
tp
1
Tosc = 4t p
2
Þ Tosc = 8t p
tp
tp
VD
Use of 4 inverters makes
quadrature signals available.
VA
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1
Tosc
2
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8
Resonance in Oscillation Loop
Hr ( jw)
Hr (s)
1
Hr (s)
+
p
ÐHr ( jw)
r
2
r
-
At dc:
Since Hr(0) < 1, latch-up does not occur.

p
2
At resonance:
Hr ( jwr ) > 1
ÐHr ( jwr ) = 0
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
Prof. M. Green / U.C. Irvine
Þ wo = wr
9
LC VCO
L
Vin
Hr (s)
C
Vout
wr =
Vout
Vin
1
LC
Hr (s)
Þ
Þ
2L
C
C
Hr (s)
realizes negative
resistance
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10
Variable Capacitance
varactor = variable reactance
Cj
A. Reverse-biased p-n junction
+
VR
–
VR
B. MOSFET accumulation capacitance
Cg
p-channel
–
VBG
+
n diffusion in n-well
VBG
accumulation
region
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
inversion
region
11
LC VCO Variations
IS
2L
C
C
C
C
2L
2L
C
IS
2L
C
C
C
ISS
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12
Effect of CML Loading
1.
1. ideal capacitor load
1 nH
3.8 
400 fF
400 fF
108 fF
108 fF
2.
Cg = 108fF
1 nH
3.8 
400 fF
400 fF
2. CML buffer load
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13
CML Buffer Input Admittance (1)
Yin = jwCgs + jwCgd A0 ×
A0 = 1+ gm R
(
1+ jw / z
1+ jw / p
)
where: 1/ p = CL +Cgd R
1/ z =
( )
(note p < z)
CL R
A0
Re Yin = A0Cgd w 2 ×
1 p -1 z
(
1+ w p
)
2
Substantial parallel loss at high
frequencies  weakens VCO’s
tendency to oscillate
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14
CML Buffer Input Admittance (2)
Yin magnitude/phase:
Yin real part/imaginary part:
magnitude
imaginary
phase
real
Contributes 2k additional parallel resistance
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15
CML Buffer Input Admittance (3)
3. CML tuned buffer load
Cg = 108 fF
1 nH
imaginary
3.8 
400 fF
400 fF
3.8 nH
real
Contributes negative parallel resistance
EECS 270C / Spring 2014
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16
CML Buffer Input Admittance (4)
ideal capacitor load
Cg = 108 fF
1 nH
3.8 
400 fF
400 fF
3.8 nH
CML buffer load
Loading VCO with tuned CML buffer
allows negative real part at high
frequencies  more robust oscillation!
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
CML tuned buffer load
17
Differential Control of LC VCO
Differential VCO control is preferred to reduce VC noise coupling into PLL output.
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18
Oscillator Type Comparison
Ring Oscillator
LC Oscillator
– slower
+ faster
– low Q  more jitter generation
+ high Q  less jitter generation
+ Control voltage can be applied
differentially
– Control voltage applied single-ended
+ Easier to design; behavior more
predictable
– Inductors & varactors make design
more difficult and behavior less
predictable
+ Less chip area
– More chip area (inductor)
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19
Random Processes (1)
Random variable: A quantity X whose value is not exactly known.
Probability distribution function PX(x): The probability that a random variable
X is less than or equal to a value x.
PX(x)
1
Example 1:
Random variable
X Î [-¥,+¥]
0.5
x
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20
Random Processes (2)
Probability of X within a range is straightforward:
PX(x)
1
(
0.5
)
P X Î [x1, x2 ] = P(x 2 ) - P(x1)
x1 x2
x
If we let x2-x1 become very small …
EECS 270C / Spring 2014
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21
Random Processes (3)
Probability density function pX(x):
Probability that random variable X lies within the range of x and x+dx.
pX (x) ×dx = PX (x + dx) - PX (x)
Þ pX (x) =
(
) ò
P X Î [ x 1, x 2 ] =
dPX (x)
dx
PX(x)
x2
x1
pX (x) dx
pX(x)
1
0.5
dx
EECS 270C / Spring 2014
x
x
Prof. M. Green / U.C. Irvine
22
Random Processes (4)
Expectation value E[X]: Expected (mean) value of random variable X
over a large number of samples.
+¥
ò x ×p
E[X ] º X =
X
(x)dx
-¥
Mean square value E[X2]: Mean value of the square of a random
variable X2 over a large number of samples.
+¥
E[X 2 ] =
òx
2
× p X (x)dx
-¥
Variance:
[
+¥
]
E (X - X ) º s =
2
2
ò (x - X ) p
X
(x)dx
-¥
[
Standard deviation: s = E (X - X )2
EECS 270C / Spring 2014
2
]
Prof. M. Green / U.C. Irvine
23
Gaussian Function
1. Provides a good model for the probability density functions of many
random phenomena.
2. Can be easily characterized mathematically s , X .
3. Combinations of Gaussian random variables are themselves
Gaussian.
(
)
f (x)
1
s 2p
é -(x - X )2 ù
ú
f (x) =
expê
2
êë 2s
úû
s 2p
1
0.607
s 2p
2
+¥
ò f (x)dx = 1
X -s
-¥
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
X
X +s
x
24
Joint Probability (1)
Consider 2 random variables:
(
P(x, y) º P X £ x and Y £ y
)
If X and Y are statistically independent (i.e., uncorrelated):
(
)
P X Î [ x, x + dx ] and Y Î [ y, y + dy ] = pX (x) × pY (y) ×dx dy
EECS 270C / Spring 2014
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25
Joint Probability (2)
Consider sum of 2 random variables:
Z = X +Y
(
) òò
é
=ê ò
ë
P Z Î [ z0, z0 + dz] =
y
strip
pX (x)pY (y) dx dy
ù
p X (x)pY (z0 - x) dx ú dz
-¥
û
¥
x + y = z0 + dz
pZ (z0 )
dy = dz
x + y = z0
dx
EECS 270C / Spring 2014
determined by convolution
of pX and pY.
x
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26
Joint Probability (3)
Example: Consider the sum of 2 non-Gaussian random processes:
*
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27
Joint Probability (4)
3 sources combined:
*
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28
Joint Probability (5)
4 sources combined:
*
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29
Joint Probability (6)
Noise sources
Central Limit Theorem:
Superposition of random variables tends toward normality.
EECS 270C / Spring 2014
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30
Fourier transform of Gaussians:
pX (x) =
é
2ù
-(x
X
)
ú
expê
ê 2s 2
ú
2p
ë
û
X
1
sX
æ 1 2 2ö
PX (w) = expç- s X w ÷
è 2
ø
F
Recall:
é
P Z Î [ z0, z0 + dz] = ê
ë
(
ù
p X (x)pY (z0 - x) dx ú dz
-¥
û
) ò
pZ (z0 ) =
ò
¥
-¥
¥
pX (x)pY (z0 - x) dx
F
PZ (w) = PX (w) ×PY (w)
æ 1 2 2ö
æ 1 2 2ö
= expç- s X w ÷ ×expç- sY w ÷
è 2
ø
è 2
ø
pZ (z) =
1
(
2p s 2X + s Y2
)
é
-(z - Z)2
expê
ê2 s 2 +s 2
X
Y
ë
(
)
ù
ú
ú
û
F -1
æ 1
ö
= expç- (s 2X + s 2X )w 2 ÷
è 2
ø
Variances of sum of random normal processes add.
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31
Autocorrelation function RX(t1,t2): Expected value of the product of 2
samples of a random variable at times t1 & t2.
RX (t1,t2 ) = E [ X (t1) × X (t2 )]
For a stationary random process, RX depends only on the time
difference t = t1 - t 2
RX (t ) = E [ X (t) × X (t + t )] for any t
2
Note RX (0) = s
Power spectral density SX():
2ù
é +¥
ê
ú
SX (w ) = Eê X (t) ×e - jwt dt ú
êë -¥
úû
ò
EECS 270C / Spring 2014
SX() given in units of [dBm/Hz]
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32
Relationship between spectral density & autocorrelation function:
1
RX (t ) =
2p
ò
¥
-¥
SX (w) ×e jwt dw
ò
1
Þ RX (0) = s =
2p
2
¥
-¥
SX (w)dw
infinite variance
(non-physical)
Example 1: white noise
SX (w)
( )
SX w = K
EECS 270C / Spring 2014
RX (t )


RX (t ) =
Prof. M. Green / U.C. Irvine
K
×d t
2p
()
33
Example 2: band-limited white noise
RX (t )
SX (w)
1
2
s 2 = Kw p
K
-w p
( )
SX w =
wp

K
RX (t ) = s 2e
-w p t

w2
1+ 2
wp
pX (x)
For parallel RC circuit
capacitor voltage noise:
wp =
1
RC
EECS 270C / Spring 2014
s V2C =
kBT
C
-s
Prof. M. Green / U.C. Irvine
+s
x
34
Random Jitter (Time Domain)
Experiment:
CLK
data
source
EECS 270C / Spring 2014
DATA
CDR
(DUT)
RCK
Prof. M. Green / U.C. Irvine
analyzer
35
Jitter Accumulation (1)
Experiment:
Observe N cycles of a free-running VCO on an oscilloscope over a long
measurement interval using infinite persistence.
NT
Free-running
oscillator output
Tosc =
3
2
1
4
1
fosc
s1
s2
s3
s4
Histogram plots
trigger
EECS 270C / Spring 2014
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36
Jitter Accumulation (2)
s t2
t
proportional
to 
proportional
to 2
Observation:
As  increases, rms jitter increases.
EECS 270C / Spring 2014
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37
Noise Spectral Density (Frequency Domain)
Single-sideband
spectral density:
Power spectral density
of oscillation waveform:
( ) [dBc Hz]
Ltotal Df
Sv(f)
[dBm Hz]
1/f3 region (-30dBc/Hz/decade)
1/f2 region (-20dBc/Hz/decade)
fosc
(
fosc+f
é P f + Df
1Hz osc
Ltotal (Df ) = 10 logê
êë
Ptotal
EECS 270C / Spring 2014
f (log scale)
) ùú
úû
Ltotal(f) given in units of [dBc/Hz]
Ltotal includes both amplitude and phase noise
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38
Noise Analysis of LC VCO (1)
noise from
resistor
+
C
L
R
-R
vc
_
C
L
active
circuitry
wr =
1
LC
inR
Z( jw ) =
R
Q=
wr L
jwL
æ w ö2
1- ç ÷
è wr ø
Consider frequencies near resonance:
wr
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
39
Noise Analysis of LC VCO (2)
+
vc
_
Spot noise current from resistor:
C
L
inR
Leeson’s formula (taken from measurements):
spot noise relative to carrier power
dBc/Hz
Where F and1/f3 are empirical parameters.
EECS 270C / Spring 2014
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40
Oscillator Phase Disturbance
ip(t)
ip(t)
Current impulse q/t
ip(t)
_
Vosc +
t1
t2
t
Vosc(t)
t
Vosc(t)
Vosc jumps by q/C
• Effect of electrical noise on oscillator phase noise is time-variant.
• Current impulse results in step phase change (i.e., an integration).
 current-to-phase transfer function is proportional to 1/s
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41
Impulse Sensitivity Function (1)
The phase response for a particular noise source can be determined at each point 
over the oscillation waveform.
Impulse sensitivity function (ISF):
= C ×Vmax
change in phase
charge in impulse
(normalized to
signal amplitude)
Example 1: sine wave
Example 2: square wave
Vosc (t)
Vosc (t)
Vmax
t
t
G(t )
G(t )


Note  has same period as Vosc.
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42
Impulse Sensitivity Function (2)
Recall from network theory:
fout
H(s)
i in
LaPlace transform:
Fout (s)
= H(s)
Iin (s)
t
h(t)
Impulse response: fout (t) =
ò h(t,t ) ×i
in
(t ) dt
0
time-variant
impulse response
Recall:
ISF convolution integral:
t
f (t) =
ò
0
G(t )
×u(t - t ) × [i (t ) ×dt ] =
q max
t
ò qG(t ) ×i(t ) ×dt
0
max
 can be expressed in terms of
Fourier coefficients:
¥
from q
G(t ) =
= 1 for t Î (0,t)
åc
k
(
cos kwosct + q k
k=0
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43
)
Impulse Sensitivity Function (3)
Case 1: Disturbance is sinusoidal:
, m = 0, 1, 2, …
(Any frequency can be expressed in terms of m and .)
G(t )
negligible
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significant only for
m=k
44
Impulse Sensitivity Function (4)
For
Current-to-phase frequency response:
I
osc

´
I0 c0
2q max w1
2osc
osc osc+
´
I0 c1
2q max w1
2osc 2osc+
´

I0 c2
2q max w1


EECS 270C / Spring 2014

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45
Impulse Sensitivity Function (5)
Case 2: Disturbance is stochastic:
A2/Hz
MOSFET current noise:
thermal
noise
in
1/f
noise
1/f noise
gm2
´c0
2p ×Kf
Cg w
thermal noise
4kTggm
osc
2osc
´c0

´c2
´c1
osc
2osc

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46

Impulse Sensitivity Function (6)
Total phase noise:
due to
thermal noise
´c0
due to 1/f
noise
´c1
n
osc
´c2
2osc

( )
c = G
2
0
2
¥
åc = (G )
2
k

EECS 270C / Spring 2014
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k=0
2
rms
47
Impulse Sensitivity Function (7)
Þ
noise corner
frequency n
(dBc/Hz)
1/(3 region:
−30 dBc/Hz/decade
1/(2 region:
−20 dBc/Hz/decade
(log scale)
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48
Impulse Sensitivity Function (8)
Example 1: sine wave
Example 2: square wave
Vosc (t)
Vosc (t)
t
t
G(t )
G(t )


Grms is higher  will generate more
1/(2 phase noise
Example 3: asymmetric square wave
Vosc (t)
t
G(t )

EECS 270C / Spring 2014
G > 0  will generate more 1/(3 phase noise
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49
Impulse Sensitivity Function (9)
Effect of current source in LC VCO:
Due to symmetry, ISF of this noise source
contains only even-order coefficients − c0 and c2
are dominant.
+
Vosc
EECS 270C / Spring 2014
_
 Noise from current source will contribute to
phase noise of differential waveform.
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50
Impulse Sensitivity Function (10)
ID varies over
oscillation waveform
Same period as
oscillation
Let
where a (t) =
Then
We can use
EECS 270C / Spring 2014
VGS (t) -Vt
VGS(DC ) -Vt
Geff (t ) = G(t ) × a(t )
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51
ISF Example: 3-Stage Ring Oscillator
R1A
R1B
M1A
M1B
MS1
R2A
R2B
M2A
M2B
MS2
R3A
R3B
M3A
+
Vout
−
M3B
MS3
fosc = 1.08 GHz
PD = 11 mW
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52
ISF of Diff. Pairs
GM2A
ISF by tx1 for 3stage differential ring osc
3
2
2
2
1
1
1
0
0
-1
0
1
2
3
4
5
6
-2
7
-1
0
1
2
3
4
5
6
-2
ISF by tx5
3
7
-3
-3
-4
-4
-5
-5
ISF by tx2 for differential ring osc
GM2B
1
1
5
-2
6
7
0
0
1
2
3
4
5
-1
-2
6
-1
-3
-4
-4
-4
-5
Grms = 1.86
G = -0.26
7
0
1
2
3
4
5
6
7
-2
-3
-5
Radian
Radian
6
0
7
-3
-5
5
3
ISF by tx6
-1
4
ISF by tx4
2
1
3
4
ISF by tx6 for differential ring osc
GM3B
2
2
3
Radian
ISF by tx4 for differential ring osc
2
0
2
-5
3
1
1
Radian
3
0
0
-2
-4
GM1B
ISF by tx5 for differential ring osc
0
-1
-3
Radian
ISF by tx2
GM3A
ISF by tx3 for differential ring osc
3
ISF by tx3
ISF by tx1
GM1A
Radian
for each diff. pair transistor
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53
ISF of Resistors
GR1A
GR2A
Grms = 1.72
G = -0.16
GR3A
for each resistor
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
54
ISF of Current Sources
ISF by tail tx1 for differential ring osc
ISF by tail tx2 for differential ring osc
GMS2
2
1.5
1.5
1.5
1
1
1
0
-0.5
0
1
2
3
4
5
6
7
0.5
0
-0.5
0
1
2
3
4
5
6
ISF by tail tx3
2
0.5
7
0.5
0
-0.5
-1
-1
-1
-1.5
-1.5
-1.5
-2
-2
Radian
Grms = 1.00
G = -0.12
ISF by tail tx3 for differential ring osc
GMS3
2
ISF by tail tx2
ISF by tail tx1
GMS1
0
1
2
3
4
5
6
-2
Radian
Radian
for each current source transistor
ISF shows double frequency due to source-coupled node connection.
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
55
7
Phase Noise Calculation
Using: Cout = 1.13 pF
Vout = 601 mV p-p
qmax = 679 fC
= −112 dBc/Hz @ f = 10 MHz
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
56
Phase Noise vs. Amplitude Noise (1)
How are the single-sideband noise spectrum Ltotal() and phase
spectral density S() related?
[(
)]
Vosc (t) = [Vc +v(t)] ×exp j wosc t + f (t)
v

osct
EECS 270C / Spring 2014
v
Spectrum of Vosc would
include effects of both
amplitude noise v(t) and
phase noise (t).
Prof. M. Green / U.C. Irvine
57
Phase Noise vs. Amplitude Noise (2)
Recall that an input current impulse causes an enduring phase
perturbation and a momentary change in amplitude:
i(t)
i(t)
t
Vc(t)
t
Vc(t)
Dt = 0
EECS 270C / Spring 2014
Dt =
Dq
w osc
Prof. M. Green / U.C. Irvine
Amplitude impulse response
exhibits an exponential decay
due to the natural amplitude
limiting of an oscillator ...
58
Phase Noise vs. Amplitude Noise (3)
+
wc
Q



Phase noise dominates
at low offset frequencies.

EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
59
Phase Noise vs. Amplitude Noise (4)
(
» (V
) (
+v (t)) × [cos(w
Vosc (t) = Vc +v (t) ×cos wosc t + f (t)
c
osc
Sv()
)
t) - f (t) ×sin(wosc t)]
= Vc cos(wosc t) - f (t) ×Vc sin(wosc t) +v (t) ×cos(wosc t)
noiseless
oscillation
waveform
phase
noise
component
amplitude
noise
component
phase
noise
amplitude
noise
osc

Phase & amplitude noise
can’t be distinguished in a
signal.
Amplitude limiting will decrease amplitude noise
but will not affect phase noise.
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
60
Sideband Noise/Phase Spectral Density
(
)
Vosc (t) = Vc ×cos wosc t + f (t)
» Vc × [cos(wosc t) - f (t) ×sin(wosc t)]
Vc ×cos(wosc t) -Vc × f(t) ×sin(wosct)
noiseless
oscillation
waveform
Pphase noise
Psignal
EECS 270C / Spring 2014
phase
noise
component
1 2 2
Vc × f
=2
= f2
1 2
Vc
2
Prof. M. Green / U.C. Irvine
61
Jitter/Phase Noise Relationship (1)
NT
s t2 º
=
1
2
wosc
1
2
wosc
{[
] [
Rf (0)
2
w
}
]
× E f 2 (t + t ) + E f 2 (t) - 2E [f (t) × f (t + t )]
autocorrelation functions
Þ s t2 =
2ü
ì
×E í[f (t + t ) - f (t)] ý
î
þ
2
osc
Rf (0)
2Rf (t )
× [Rf (0) - Rf (t )]
Recall R and S() are a Fourier transform pair:
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
62
Jitter/Phase Noise Relationship (2)
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
63
Jitter/Phase Noise Relationship (3)
3
2
Jitter from 1/( noise:
Jitter from 1/( noise:
^
Let
Let
^
4 a^ pt
=
×
2
pwosc 4
a^
= 2 ×t
wosc
=
=z ×t2
a
2
fosc
× t where a^ º (2p )2 ×a
Consistent with jitter accumulation measurements!
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
64
Jitter/Phase Noise Relationship (4)
(dBc/Hz)
• Let fosc = 10 GHz
• Assume phase noise dominated by 1/()2
-20dBc/Hz
per decade
-100
Setting f = 2 X 106 and S =10-10:
(
f
2 MHz
)
Sf 2 ×106 =
a
(2 ×10 )
6
2
= 10-10 Þ a = 400
Accumulated jitter:
s t2 =
a
400
×
t
=
fc2
10 ×109
(
EECS 270C / Spring 2014
)
2
[
]
× t = 4 ×10-18 × t
[
]
st = 2 ×10-9 × t
Let  = 100 ps (cycle-to-cycle jitter):
  = 0.02ps rms (0.2 mUI rms)
Prof. M. Green / U.C. Irvine
65
Jitter/Phase Noise Relationship (5)
More generally:
(dBc/Hz)
-20 dBc/Hz
per decade
2
æ
ö
a
f
s t2 = 2 × t = ç m ÷ ×10Nm 10 × t
fosc
è fosc ø
Nm
æ f ö N 20
m
÷ ×10 m × t
è fosc ø
st = ç
st
fm
f
Tosc
= fm ×10Nm 20 × t
[ps]
[UI]
Let phase noise increase by 10 dBc/Hz:
 rms jitter increases by a factor of 3.2
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
66
Jitter Accumulation (1)
in
fb
vco
phase
detector
loop
filter
VCO
Kpd
F(s)
Kvco
+
out
¸N
Open-loop characteristic:
fout
K
1
= G(s) = K pd ×F (s) × vco ×
fe
2ps N
Closed-loop characteristic: fout =
EECS 270C / Spring 2014
NG(s)
1
× fin +
× fvco
1+G(s)
1+G(s)
Prof. M. Green / U.C. Irvine
67
Jitter Accumulation (2)
G(s) =
Recall from Type-2 PLL:
Ich Kvco
1
1+ sCR
× 2
×
N
s (C +Cp ) 1+ sCeq R
-40 dB/decade
(dBc/Hz)
|1 + G|
1/(3 region:
−30 dBc/Hz/decade
|G|
z

p

1/(2 region:
−20 dBc/Hz/decade

1
As a result, the phase noise at low offset frequencies is
determined by input noise...
80 dB/decade

EECS 270C / Spring 2014

Prof. M. Green / U.C. Irvine
68
Jitter Accumulation (3)
• fosc = 10 GHz
• Assume 1-pole closed-loop PLL characteristic
(dBc/Hz)
-100
-20dBc/Hz
per decade
f0 = 2 MHz
EECS 270C / Spring 2014
f
Prof. M. Green / U.C. Irvine
69
Jitter Accumulation (4)
a = 4 ´102
f0 = 2 MHz
fosc = 10 GHz
s t2 (log scale)
slope =
For small :
 = 0.02 ps rms
cycle-to-cycle jitter
For large :
 = 1.4 ps rms
Total accumulated jitter
a
2
fosc

1
(2p ) ×(2 MHz)
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
70
Jitter Accumulation (5)
s t2 (log scale)
proportional to 
(due to 1/f noise)
proportional to 
(due to thermal noise)

The primary function of a PLL is to place a bound on cumulative jitter:
s t2 (log scale)

EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
71
Closed-Loop PLL Phase Noise Measurement
L() for OC-192 SONET transmitter
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
72
Other Sources of Jitter in PLL
• Clock divider
• Phase detector
Ripple on phase detector output can cause high-frequency jitter. This
affects primarily the jitter tolerance of CDR.
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
73
Jitter/Bit Error Rate (1)
Eye diagram from
sampling oscilloscope
Histogram showing
Gaussian distribution
near sampling point
2s L
2s R
L
R
1UI
Bit error rate (BER) determined by  and UI …
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
74
Jitter/Bit Error Rate (2)
2ù
é
T
t
1
ê
ú
pR (t) =
×expê2s 2 ú
s 2p
êë
úû
(
é t2 ù
1
pL (t) =
×expê- 2 ú
ë 2s û
s 2p
2s
0
2s
t0
T
2
T - t0
R
T
é x2 ù
PL =
× expê- 2 ú dx
Probability of sample at t > t0 from leftë 2s û
s 2p t 0
hand transition:
2ù
é
T
x
¥
Probability of sample at t < t0 from right1
ê
ú
P
=
×
exp
dx
R
2
hand transition:
ê
ú
t0
2s
s 2p
êë
úû
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
1
ò
ò
¥
(
)
75
)
Jitter/Bit Error Rate (3)
é x2 ù
PL =
× expê- 2 ú dx
ë 2s û
s 2p t 0
1
ò
¥
é
1
ê T -x
PR =
× expê2s 2
s 2p t 0
êë
ò
¥
(
)
ù
1
ú
dx
=
×
ú
s 2p
úû
2
é x2 ù
expê- 2 ú
T-t 0
ë 2s û
ò
¥
Total Bit Error Rate (BER) given by:
é x2 ù
1
BER = PL + PU =
× expê- 2 ú dx +
×
t0
ë 2s û
s 2p
s 2p
ò
1
¥
é x2 ù
expê- 2 ú dx
T-t 0
ë 2s û
ò
¥
æ t ö
æT - t öù
1é
0
0
= êerfcçç
÷÷ + erfcçç
÷÷ú
2 êë
è 2s ø
è 2s øúû
where erfc(t) º
EECS 270C / Spring 2014
2
p
×
ò
t
¥
( )
exp -x 2 dx
Prof. M. Green / U.C. Irvine
76
Jitter/Bit Error Rate (4)
Example: T = 100ps
log(0.5)
log BER
s = 2.5 ps
s = 5 ps
·
·
·
·
t0 (ps)
s = 2.5 ps :
BER £ 10-12 for t0 Î [18ps, 82ps] (64 ps eye opening)
s = 5 ps :
BER £ 10-12 for t0 Î [36ps, 74ps] (38 ps eye opening)
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
77
Bathtub Curves (1)
The bit error-rate vs. sampling time can be measured directly using a bit
error-rate tester (BERT) at various sampling points.
Note: The inherent jitter of the analyzer trigger should be considered.
( )
RJ
Jrms
2
measured
( )
RJ
= Jrms
EECS 270C / Spring 2014
2
actual
( )
RJ
+ Jrms
2
trigger
Prof. M. Green / U.C. Irvine
78
Bathtub Curves (2)
Bathtub curve can easily be numerically extrapolated to very low BERs
(corresponding to random jitter), allowing much lower measurement times.
Example:
10-12 BER with T = 100ps is
equivalent to an average of 1 error
per 100s. To verify this over a
sample of 100 errors would require
almost 3 hours!
·
·
·
·
t0 (ps)
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
79
Equivalent Peak-to-Peak Total Jitter
p(t)
BER
RJ
JPP
10-10
12.7 × s
10-11
13.4 × s
10-12
14.1×s
10-13
14.7 × s
10-14
15.3 × s
Areas sum
to BER
1
ns
2
, T determine BER
1
ns
2
RJ
BER determines effective JPP
Total jitter given by:
(
)
DJ
J TJ = n × s + JPP
EECS 270C / Spring 2014
Prof. M. Green / U.C. Irvine
80