# Voltage-Controlled Oscillator (VCO)

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```							                       Voltage-Controlled Oscillator (VCO)

fosc
Desirable characteristics:                                               fmax
• Monotonic fosc vs. VC characteristic
• 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 jo  1
                                               Aj  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 tan1  180  osc  3  p
p 

A3
Hloop ( josc)        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
−                          −                   −                                  −             +

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
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 ( jr )  1
 osc  r
Hr ( jr )  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

1.
1 nH    3.8 

400 fF                             400 fF   108 fF   108 fF

2.
Cg = 108fF

1 nH      3.8 

400 fF                             400 fF

EECS 270C / Spring 2009                              Prof. M. Green / U.C. Irvine
13

1 j / z
Yin  jCgs  jCgd 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

Yin magnitude/phase:                                          Yin real part/imaginary part:

magnitude

imaginary

phase

real

EECS 270C / Spring 2009               Prof. M. Green / U.C. Irvine
15

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

Cg = 108 fF

1 nH    3.8 

3.8 nH
400 fF                                 400 fF

allows negative real part at high
frequencies  more robust oscillation!

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 jt 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)

               

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                                                              jL
Z( j ) 
1            R                                                                                                
2

r                 Q                                                                                              1  
LC          r L                                                                                            r 

 Consider frequencies near resonance:
                                                                                              


j r   L                      r2

Z j r                                                    jL
  2r                           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 and1/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 kos c   k         
                                                                                                       k0




EECS 270C / Spring 2009                      Prof. M. Green / U.C. Irvine
43
Impulse Sensitivity Function (3)
Case 1: Disturbance is sinusoidal:

                
i (t)  I0 cos mos 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     k0
0

( )

     
sin (k  m)   t  
       
 k  sin (k  m)osc  t   k                        

                I

osc
 0                ck                                                                                  
2q max                     (k  m)osc             (k  m)osc                                  
k0                                                                                     

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 mos c   t               
 (t) 
I0
cm 
sin  t   k   2  2
I0
  2 
2
cm  

 
2
2qmax                          8qmax 

Current-to-phase frequency response:

                                    I

osc                               2osc           
                 osc osc                   2osc 2osc

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 )  4kTgm  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                     2osc                                              osc                   2osc         
                                                                                                                        

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              4kTgm 
0
 2                       3 
                   
2
8q max                              Cg     
                                            

                                            

due to                due to 1/f
2                     thermal noise               noise
in
      f

c0                          c1                        c2


n           osc                        2osc         
                                                 

 
S 

2
c  
2
0



          
2
                                                                                                   ck  rms
2

                                                          k0
EECS 270C / Spring 2009                    Prof. M. Green / U.C. Irvine
47
Impulse Sensitivity Function (7)

                                      2 
                              
2
1              rms                g Kf     
2
S ()              4kTgm 
                        2         m
3 
                              
2                          2
8q max                                Cg     

                                         

                                                                          gm         
2                         2                                                          2
rms                         

2
g Kf                                        n,phase               
4kTgm                     2    m

                                                                                                     2kT Cg rms   
                                                                                     
2                              3
Cg    
noise corner
frequency n

                                                   
S  (dBc/Hz)

1/(3

region:


1/(2 region:

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
 4kTgm (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
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

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

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

 
-100                                                                    a
S f 
(f )2
Setting f = 2 X 106 and S =10-10:


f              
S 2 106                      a
 1010  a  400
2 10 
2 MHz                                                                         2
6

Accumulated jitter:

 2 
a
 
400
  4 1018  
                                            
  2 109         
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 
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 
                       2s N

NG(s)            1
Closed-loop characteristic: out                              in          vco
                   1G(s)         1G(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


 
S  (dBc/Hz)

|1 + G|                             1/(3 region:

|G|
1/(2 region:
z        p       
2
out
vco
j                                                                            n,phase           
1

                                                                                   
As a result, the phase noise at low offset frequencies is
determined by input noise...

                 
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
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  4102                       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                       Tt 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                                               Tt 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  1012 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

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