Get documents about "Voltage-Controlled Oscillator (VCO)
Shared by: HC120208035354
-
Stats
- views:
- 25
- posted:
- 2/7/2012
- language:
- English
- pages:
- 80
Document Sample
Voltage-Controlled Oscillator (VCO)
fosc
Desirable characteristics: fmax
• Monotonic fosc vs. VC characteristic
with adequate frequency range
• Well-defined Kvco slope = Kvco
fmin
VC
VC
in
VD VC ^
Kv c o out
KPD F(s)
s
^
Kvco
out ^ VC
s KPDKvcoF(s) / N
Noise coupling from VC into PLL
N output is directly proportional to Kvco.
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
1
Oscillator Design
Vin 0
Vout Vout A(s)
A(s) HCL (s)
Vin 1 f A(s)
loop gain
Barkhausen’s Criterion:
f
If a negative-feedback loop satisfies:
f A jo 1
Aj 180 o
then the circuit will oscillate at frequency 0.
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
2
Inverters with Feedback (1)
1 inverter: V2 1 inverter
V1 V2
1 stable
feedback
equilibrium
point
V1
V2
2 inverters:
feedback 3 equilibrium
V1 V2
points: 2 stable,
1 unstable
(latch)
2 inverters
V1
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
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 ( j)
1 j p
A3
Loop gain: Hloop ( j ) inv ( j)
3
0
H
3
1 j p
Applying Barkhausen’s criterion: Hloop ( j) 3 tan1 180 osc 3 p
p
A3
Hloop ( josc) 0
1 A0 2
1 3
3
2
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
4
Ring Oscillator Operation
tp tp tp
VA VB VC
Total phase shift in loop:
Total delay in loop: 3tp
VA 3t
p Tosc 6 t p
tp 2 Tosc
VB
tp
VC
tp
VA 1
Tosc
2
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
5
Variable Delay Inverters (1)
Inverter with variable load capacitance: Current-starved inverter:
Vin Vout
VC
Vin Vout
VC
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
6
Variable Delay Inverters (2)
Interpolating inverter:
ISS R R
Vout-
Vout+
+
VC Vin+ Vin- Vin+ Vin-
_
RG RG
Ifast
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 2009 Prof. M. Green / U.C. Irvine
7
Differential Ring Oscillator
+ + + + −
VA VB VC VD VA
− − − − +
additional inversion
VA (zero-delay)
tp
VB tp Total phase shift in loop:
Total delay in loop: 4tp
4t
VC tp p Tosc 8 t p
2 Tosc
VD tp
Use of 4 inverters makes
quadrature signals available.
VA
1
Tosc
EECS 270C / Spring 2009 2 Prof. M. Green / U.C. Irvine
8
Resonance in Oscillation Loop
Hr ( j)
Hr (s)
1
r
Hr ( j)
Hr (s)
2
r
2
At dc: At resonance:
Since Hr(0) < 1, latch-up does not occur. Hr ( jr ) 1
osc r
Hr ( jr ) 0
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
9
LC VCO
L C
Vin Vout 1
Hr (s) r
Vout LC
Vin
2L
C C
realizes negative
resistance
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
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 inversion
region region
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
11
LC VCO Variations
IS IS
2L 2L
C C C C
2L 2L
C C C C
ISS
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
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
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
13
CML Buffer Input Admittance (1)
1 j / z
Yin jCgs jCgd A0
1 j / p
A0 1 gm R
where: 1/ p CL Cgd R (note p < z)
CL R
1/ z
A0
1 p 1 z
Re Yin A0Cgd 2
2
1 p
Substantial parallel loss at high
frequencies weakens VCO’s
tendency to oscillate
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
14
CML Buffer Input Admittance (2)
Yin magnitude/phase: Yin real part/imaginary part:
magnitude
imaginary
phase
real
Contributes 2k additional parallel resistance
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
15
CML Buffer Input Admittance (3)
3. CML tuned buffer load
Cg = 108 fF
imaginary
1 nH 3.8
3.8 nH
400 fF 400 fF
real
Contributes negative parallel resistance
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
16
CML Buffer Input Admittance (4)
ideal capacitor load
Cg = 108 fF
1 nH 3.8
3.8 nH
400 fF 400 fF
CML buffer load
Loading VCO with tuned CML buffer
allows negative real part at high
frequencies more robust oscillation!
CML tuned buffer load
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
17
Differential Control of LC VCO
Differential VCO control is preferred to reduce VC noise coupling into PLL output.
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
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 – Control voltage applied single-ended
differentially
+ Easier to design; behavior more – Inductors & varactors make design
predictable more difficult and behavior less
predictable
+ Less chip area – More chip area (inductor)
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
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 [ ]
, 0.5
x
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
20
Random Processes (2)
Probability of X within a range is straightforward:
PX(x)
1
0.5
P X [x 1, x 2 ] P(x 2 ) P(x 1)
x1 x2 x
If we let x2-x1 become very small …
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
21
Random Processes (3)
Probability density function pX(x):
Probability that random variable X lies within the range of x and x+dx.
P X x 1, x 2
x2
pX (x) dx PX (x dx) PX (x) pX (x) dx
x1
dPX (x)
p X (x)
dx
PX(x)
pX(x)
1
0.5
dx x x
EECS 270C / Spring 2009 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.
E[X ] X x p 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 ] x 2
p X (x)dx
x X p
2
Variance: E (X X ) 2 2
X (x)dx
Standard deviation: E (X X )2
EECS 270C / Spring 2009 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 , X .
3. Combinations of Gaussian random variables are themselves
Gaussian.
1 f (x)
2
0.607
1 (x X )2 2
f (x) exp 2
2
2 2
f (x)dx 1
X x
X X
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
24
Joint Probability (1)
Consider 2 random variables:
P(x, y) P X x and Y y
If X and Y are statistically independent (i.e., uncorrelated):
P X x, x dx and Y y, y dy p X (x) pY (y) dx dy
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
25
Joint Probability (2)
Consider sum of 2 random variables:
Z X Y
y
P Z z0, z0 dz strip
pX (x)pY (y) dx dy
p X (x)pY (z0 x) dx dz
x y z0 dz
pZ (z0 )
x y z0 dy = dz
determined by convolution
of pX and pY.
x
dx
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
26
Joint Probability (3)
Example: Consider the sum of 2 non-Gaussian random processes:
*
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
27
Joint Probability (4)
3 sources combined:
*
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
28
Joint Probability (5)
4 sources combined:
*
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
29
Joint Probability (6)
Noise sources
Central Limit Theorem:
Superposition of random variables tends toward normality.
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
30
Fourier transform of Gaussians:
1 (x X )2 1 2 2
pX (x) exp F PX () exp X
X 2 2 2
X 2
Recall:
P Z z0 , z0 dz p X (x)pY (z0 x) dx dz
F
pZ (z0 ) pX (x)pY (z0 x) dx PZ () PX () PY ()
1 1 2
exp 2 2 exp Y 2
X
2 2
(z Z)2 F -1 1
exp
1
pZ (z) exp ( 2 2 ) 2
2 2 Y
X
2
2 2 2
X Y
2
X X
Variances of sum of random normal processes add.
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
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 t1 t2
RX ( ) E X (t) X (t ) for any t
Note RX (0)
2
Power spectral density SX():
2
SX () E X (t) e jt dt SX() given in units of [dBm/Hz]
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
32
Relationship between spectral density & autocorrelation function:
1
RX ( ) SX ( ) e j d
2
1
RX (0) 2
SX ( )d
2
infinite variance
Example 1: white noise (non-physical)
SX () RX ( )
SX K RX ( )
K
2
t
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
33
Example 2: band-limited white noise
RX ( ) 1
SX () 2 K p
2
K
p p p
RX ( ) 2e
SX K
2
1 2
p
pX (x)
For parallel RC circuit
capacitor
voltage noise:
2
in
K R 2 2kBTR
f kBT
VC
2
1 C
p
RC x
Prof. M. Green / U.C. Irvine
EECS 270C / Spring 2009
34
Random Jitter (Time Domain)
Experiment:
CLK
data
source RCK
DATA CDR
(DUT) analyzer
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
35
Noise Spectral Density (Frequency Domain)
Power spectral density Single-sideband
of oscillation waveform: spectral density:
Sv()
Ltotal dBc Hz
dBm Hz 1/3 region (-30dBc/Hz/decade)
1/2 region (-20dBc/Hz/decade)
osc (log scale)
osc+
P
Ltotal ( ) 10 log
1Hz osc
Ltotal() given in units of [dBc/Hz]
Ptotal
Ltotal includes both amplitude and phase noise
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
36
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
1
Tosc 1 2 3 4 Histogram plots
fosc
1 2 3 4
trigger
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
37
Jitter Accumulation (2)
2
proportional proportional
to to 2
Observation:
As increases, rms jitter increases.
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
38
Noise Analysis of LC VCO (1)
noise from
resistor
+
vc C L
C L R -R _ inR
active
circuitry jL
Z( j )
1 R
2
r Q 1
LC r L r
Consider frequencies near resonance:
j r L r2
Z j r jL
2r 2
2
2
r
1
r2
R r
r L
R
Q
Z j r j
2Q
r
r
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine 39
Noise Analysis of LC VCO (2)
+
Noise current from resistor:
vc C L 4kT
_ inR i nR
2
f
R
v c i nR | Z( j ) |2
2 2
2
4kT
f r
R
R 2Q
2
4kTR r f
2Q
Leeson’s formula (taken from measurements):
spot noise relative to carrier power
2
kT r 1/ f 3
L 10 log
F
1
1
dBc/Hz
Psig 2Q
Where F and1/f3 are empirical parameters.
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
40
Oscillator Phase Disturbance
ip(t) ip(t)
Current impulse q/t
ip(t)
_ 1 t 2 t
Vosc +
Vosc(t) Vosc(t)
0
0
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
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
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): ( ) qmax
q
change in phase
C Vmax
(normalized to
charge in impulse
signal amplitude)
Example 1: sine wave Example 2: square wave
Vosc (t) Vosc (t)
Vmax
t t
( ) ( )
Note has same period as Vosc.
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
42
Impulse Sensitivity Function (2)
Recall from network theory:
out (s)
i in H(s) out LaPlace transform: H(s)
Iin (s)
t
h(t)
Impulse response: out (t) h(t, ) i in ( ) d
0
time-variant
( ) ( )
impulse response
Recall: ( ) qmax ( ) q
q qmax
ISF convolution integral:
( )
q ) i ( ) d
t t
( can be expressed in terms of
(t)
q max
u(t ) ( ) d
i
max
Fourier coefficients:
0 0
c
from q
1 for (0,t) ( ) k
cos kos c k
k0
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
43
Impulse Sensitivity Function (3)
Case 1: Disturbance is sinusoidal:
i (t) I0 cos mos c t , m = 0, 1, 2, …
(Any frequency can be expressed in terms of m and .)
I
c k
k cos os c d
t
(t) 0 cosk os c m t
q max k0
0
( )
sin (k m) t
k sin (k m)osc t k
I
osc
0 ck
2q max (k m)osc (k m)osc
k0
negligible significant only for
m=k
I0
cm
sin t k
2q max
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
44
Impulse Sensitivity Function (4)
For i (t) I0 cos mos c t
(t)
I0
cm
sin t k 2 2
I0
2
2
cm
2
2qmax 8qmax
Current-to-phase frequency response:
I
osc 2osc
osc osc 2osc 2osc
I0 c0 I0 c1 I0 c2
2qmax 1 2qmax 1 2qmax 1
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
45
Impulse Sensitivity Function (5)
Case 2: Disturbance is stochastic:
2
MOSFET current noise: i n (f ) 4kTgm gm Kf
2 A2/Hz
f Cg f in
i n f
2 2
cm
f 2
2
8q max 2 thermal 1/f
noise noise
2 2
in in
1/f noise
f f
2 Kf
2
gm thermal noise
Cg
c0 c0 c1 c2
gm
4kT
osc 2osc osc 2osc
S
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
46
Impulse Sensitivity Function (6)
1
2
ck 2
gm Kf 2
c0
Total phase noise: S ( ) 2 4kTgm
0
2 3
2
8q max Cg
due to due to 1/f
2 thermal noise noise
in
f
c0 c1 c2
n osc 2osc
S
2
c
2
0
2
ck rms
2
k0
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
47
Impulse Sensitivity Function (7)
2
2
1 rms g Kf
2
S () 4kTgm
2 m
3
2 2
8q max Cg
gm
2 2 2
rms
2
g Kf n,phase
4kTgm 2 m
2kT Cg rms
2 3
Cg
noise corner
frequency n
S (dBc/Hz)
1/(3
region:
−30 dBc/Hz/decade
1/(2 region:
−20 dBc/Hz/decade
n,phase
(log scale)
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
48
Impulse Sensitivity Function (8)
Example 1: sine wave Example 2: square wave
Vosc (t) Vosc (t)
t t
( ) ( )
rms is higher will generate more
1/(2 phase noise
Example 3: asymmetric square wave
Vosc (t)
t
( )
0 will generate more 1/(3 phase noise
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
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 _ Noise from current source will contribute to
phase noise of differential waveform.
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
50
Impulse Sensitivity Function (10)
ID varies over
oscillation waveform Same period as
oscillation
2
in
4kTgm (t)
f
(4kT ) Cox
W
L
VGS (t) Vt
2
Let
i n0
f
(4kT ) Cox
W
L
VGS(DC ) Vt
2 2 VGS (t) Vt
Then i n i n0
(t) where (t)
f f VGS(DC ) Vt
We can use eff ( ) ( ) ( )
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
51
ISF Example: 3-Stage Ring Oscillator
R1A R1B R2A R2B R3A R3B
+
Vout
−
M1A M1B M2A M2B M3A M3B
MS1 MS2 MS3
fosc = 1.08 GHz
PD = 11 mW
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
52
ISF of Diff. Pairs
M1A ISF by tx1 for 3stage differential ring osc
M2A ISF by tx3 for differential ring osc M3A ISF by tx5 for differential ring osc
3 3 3
2 2 2
1
1
1
0 0 0
ISF by tx3
ISF by tx1
ISF by tx5
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
-1 -1 -1
-2 -2 -2
-3 -3 -3
-4 -4 -4
-5 -5 -5
Radian Radian Radian
ISF by tx2 for differential ring osc ISF by tx4 for differential ring osc ISF by tx6 for differential ring osc
M1B M2B M3B
3 3 3
2 2 2
1 1 1
0
0
0
ISF by tx4
ISF by tx2
ISF by tx6
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
-1 -1 -1
-2 -2 -2
-3 -3 -3
-4 -4 -4
-5 -5 -5
Radian Radian Radian
rms 1.86
for each diff. pair transistor
0.26
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
53
ISF of Resistors
R1A R2A R3A
rms 1.72
for each resistor
0.16
EECS 270C / Spring 2009 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 ISF by tail tx3 for differential ring osc
MS1 MS2 MS3
2 2 2
1.5 1.5 1.5
1 1 1
ISF by tail tx1
ISF by tail tx3
ISF by tail tx2
0.5 0.5 0.5
0 0 0
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
-0.5 -0.5 -0.5
-1 -1 -1
-1.5 -1.5 -1.5
-2 -2 -2
Radian Radian Radian
rms 1.00
for each current source transistor
0.12
ISF shows double frequency due to source-coupled node connection.
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
55
Phase Noise Calculation
Using: Cout = 1.13 pF
Vout = 601 mV p-p
qmax = 679 fC
rms (dp ) 4kT gm(dp )
2
rms (res) 4kT R
2
rms (cs) 4kT gm(cs)
2
L 6 2 2
f 6 2 2 2 3 2 2
8 f 2
qmax 8 f qmax 8 f 2
qmax
322 122 70
f 2 f 2 f 2
L
514
f = −112 dBc/Hz @ f = 10 MHz
2
f
EECS 270C / Spring 2009 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?
V
Vos c (t) c v (t) exp j os ct (t)
v Spectrum of Vosc would
v include effects of both
amplitude noise v(t) and
phase noise (t).
osct
EECS 270C / Spring 2009 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 t
Vc(t) Vc(t)
q
t 0 t
osc Amplitude impulse response
exhibits an exponential decay
due to the natural amplitude
limiting of an oscillator ...
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
58
Phase Noise vs. Amplitude Noise (3)
Lamp
L
+
c
Q
Ltotal
Phase noise dominates
at low offset frequencies.
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
59
Phase Noise vs. Amplitude Noise (4)
Sv()
Vosc(t) Vc v (t) cos osct (t)
V c v (t) cos( t) (t) sin(osct)
osc
phase amplitude
noise noise
Vc cos(osct) (t) Vc sin(osct) v (t) cos(osct)
noiseless phase amplitude
oscillation noise noise osc
waveform component component
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 2009 Prof. M. Green / U.C. Irvine
60
Sideband Noise/Phase Spectral Density
Vosc (t) Vc cos osct (t)
Vc cos(osct) (t) sin(osct)
Vc cos(osct) Vc (t) sin(osct)
noiseless phase
oscillation noise
waveform component
1 2 2
Vc
Pphase noise 2 1
2 Lphase S
Psignal 1 2 2
Vc
2
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
61
Jitter/Phase Noise Relationship (1)
NT
1 1 2 3 4 2
E (t ) (t)
Tosc 1
fosc 1 2 3 4 2
osc
2
osc
2
1
E 2 (t ) E 2 (t) 2E (t) (t )
autocorrelation functions R (0) R (0) 2R ( )
(0) R ( )
2
2 R
2
osc
Recall R and S() are a Fourier transform pair:
1
R ( ) S (e j ( ) d( )
)
2
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
62
Jitter/Phase Noise Relationship (2)
R (0)
1
2
S ()d()
R ( )
1
2
S () e
j ( )
d()
2
1
2
S () 1 e
j ( )
d()
osc
1
osc
2
S () 1 cos( ) j sin( )d()
4
2
osc
S ( ) sin
2
d( )
2
0
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
63
Jitter/Phase Noise Relationship (3)
2 3
Jitter from 1/( noise: Jitter from 1/( noise:
^
a b
Let S () Let S ()
()2 ()3
2 ( )
2 ( )
4 ^
a 4 b
2
sin d( ) 2
sin d( )
2
osc () 2
2 2
() 3
2
0 osc
4 a
^
2
osc
2
4
^
a a
2 w here a (2 )2 a
^
osc
2
fosc
Consistent with jitter accumulation measurements!
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
64
Jitter/Phase Noise Relationship (4)
(dBc/Hz)
S f
• Let fosc = 10 GHz
• Assume phase noise dominated by 1/()2
-20dBc/Hz
per decade
-100 a
S f
(f )2
Setting f = 2 X 106 and S =10-10:
f
S 2 106 a
1010 a 400
2 10
2 MHz 2
6
Accumulated jitter:
2
a
400
4 1018
2 109
fc2
2
10 109
Let = 100 ps (cycle-to-cycle jitter):
= 0.02ps rms (0.2 mUI rms)
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
65
Jitter/Phase Noise Relationship (5)
More generally:
(fm )2 10Nm 10
S f
a
(dBc/Hz)
S f (f )2 (f )2
a f 2
2 m 10Nm 10
-20 dBc/Hz 2
fos c fos c
per decade
Nm
f
m 10Nm 20
fosc
ps
fm 10Nm 20 UI
fm f Tosc
Let phase noise increase by 10 dBc/Hz:
N 10 20 10Nm 20 100.5
m
fm 10 m f
Tosc
rms jitter increases by a factor of 3.2
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
66
Jitter Accumulation (1)
in vco
phase loop
out
VCO
detector filter
Kpd F(s) Kvco
fb
N
out K 1
Open-loop characteristic:
G(s) K pd F(s) vco
2s N
NG(s) 1
Closed-loop characteristic: out in vco
1G(s) 1G(s)
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
67
Jitter Accumulation (2)
IchKvco 1 1 sCR
Recall from Type-2 PLL: G(s) 2
N s (C Cp ) 1 sCeq R
-40 dB/decade
S (dBc/Hz)
|1 + G| 1/(3 region:
−30 dBc/Hz/decade
|G|
1/(2 region:
z p
−20 dBc/Hz/decade
2
out
vco
j n,phase
1
As a result, the phase noise at low offset frequencies is
determined by input noise...
80 dB/decade
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
68
Jitter Accumulation (3)
• fosc = 10 GHz
• Assume 1-pole closed-loop PLL characteristic
(dBc/Hz)
S f
a a
, f f0
2 2
f0 f0
-20dBc/Hz
S f
f a , f f
2
-100 1
0
per decade 2
f0 f
R ( ) S (f ) e j (2 f ) d(f )
a
2 f
e 2 f0
0
f0 = 2 MHz f
(0) R ( )
2
2 R
2 fos c
2
a 1 e 2 f0
2
fos c 2 f0
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
69
Jitter Accumulation (4)
a 1
,
a 1 e 2 f0 fosc2
2 (f0 )
2
2
fosc 2 f0 a 1 1
,
fosc 2 (f0 )
2
2 (f0 )
a 4102 For small :
= 0.02 ps rms
f0 = 2 MHz cycle-to-cycle jitter
fosc = 10 GHz
For large :
= 1.4 ps rms
2 (log scale) Total accumulated jitter
a
slope 2
fosc
1
(2 ) (2 MHz)
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
70
Jitter Accumulation (5)
2 (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:
2 (log scale)
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
71
Closed-Loop PLL Phase Noise Measurement
L() for OC-192 SONET transmitter
EECS 270C / Spring 2009 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 2009 Prof. M. Green / U.C. Irvine
73
Jitter/Bit Error Rate (1)
Eye diagram from
sampling oscilloscope
Histogram showing
Gaussian distribution
near sampling point
2 L 2 R
L
R
1UI
Bit error rate (BER) determined by and UI …
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
74
Jitter/Bit Error Rate (2)
2
1 t
2
pR (t)
1
exp
T t
pL (t) 2
exp
2 2 2 2 2
2 2
t0 T T t0
R
0 T
2
x 2
1
Probability of sample at t > t0 from left- PL 2 dx
exp
hand transition: 2 t0
2
2
T x
Probability of sample at t < t0 from right- 1
hand transition: PR
exp dx
2 t0 2 2
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
75
Jitter/Bit Error Rate (3)
x 2
1
PL 2 dx
exp
2 t0
2
2
T x x 2
1 1
PR
exp dx 2
exp
2 t0 2 2 2 Tt 0
2
Total Bit Error Rate (BER) given by:
x 2 x 2
1 1
BER PL PU 2 dx
exp 2 dx
exp
2 t0
2 2 Tt 0
2
1 t0 T 0
t
erfc erfc
2 2
2
2
w here erfc(t) exp x 2 dx
t
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
76
Jitter/Bit Error Rate (4)
Example: T = 100ps
log(0.5)
log BER
2.5 ps
5 ps
t0 (ps)
2.5 ps:
BER 12 for t 18ps, 82ps (64 ps eye opening)
10 0
5 ps:
BER 1012 for t0 36ps, 74ps (38 ps eye opening)
EECS 270C / Spring 2009 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.
2 2 2
RJ
Jrms Jrms
RJ
Jrms
RJ
measured actual trigger
EECS 270C / Spring 2009 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 2009 Prof. M. Green / U.C. Irvine
79
Equivalent Peak-to-Peak Total Jitter
p(t)
RJ
BER JPP Areas sum
10-10 12.7 to BER
10-11 13.4
10-12 14.1
10-13 14.7
10-14 15.3
1 1
n n
, T determine BER 2 2
RJ
BER determines effective JPP
Total jitter given by:
J TJ n J PP
DJ
EECS 270C / Spring 2009 Prof. M. Green / U.C. Irvine
80
