Phase-domain Macromodeling of
Oscillators for the analysis of
Noise, Interferences and
Synchronization effects
Paolo Maffezzoni
Dipartimento di Elettronica, Informazione e Bioingegneria
Politecnico di Milano, Milan, Italy
MIT, Cambridge, MA, 23-27 Sep. 2013
1
Presentation Outline
•
Mathematical/Theoretical formalization
•
Computational issues
•
Pulling effects due to interferences
•
Phase-noise analysis
2
Phase-domain Macromodeling of Oscillators
Presentation Outline
•
Mathematical/Theoretical formalization
• Computational issues
•
Pulling effects due to interferences
•
Phase-noise analysis
Phase-domain Macromodeling of Oscillators
3
Free-Running Oscillator
x (t )  f ( x(t ))
x(t )  N
f ( x(t ))  N
xs (t )
State variables
Vector-valued nonlinear function
Vector solution
Limit cycle
0  2 T0
x0 (t )  xs (t )
Scalar output
response
x0 (t ) X1 cos(0t  1 )  
4
Perturbed Oscillator
x (t )  f ( x(t ))  B  s(t )
s(t) small-amplitude
perturbation
Franz Kaertner, “Analysis of white and f -a noise in oscillators”,
International Journal of Circuit Theory and Applications, vol. 18, 1990.
x(t )  xs (t  a (t ))  x(t  a (t ))
a(t)
is the time-shift of the perturbed response with respect to
free-running one
Tangential variation,
xs (t  a (t ))  xs (t )  xs (t ) a (t ) Phase modulation (PM)
x(t  a (t ))
Transversal variation
Amplitude modulation (AM)
5
Pulse Perturbation
Small-amplitude
pulse perturbation

at t1
xs (t1  a (t1 ))
xs (t1  a (t1 ))  x(t1  a (t1 ))
xs (t1 )
6
Floquet theory of linear
time-periodic ODEs
Linearization around the limit cycle
Direct ODE
y (t )A(t ) y(t )
N Solutions
f ( x)
A(t ) 
x x xs (t )
A(t  T0 )  A(t )
y(t )  exp(k t )uk (t )
Right eigenvector
Floquet exponent
Adjoint ODE
 (t ) A(t )T w(t )
w
w(t )  exp(k t )vk (t )
Left eigenvector
7
Phase and Amplitude Modulations
Tangential variation
is governed by:
1  0
u1 (t )  xs (t )
v1 (t )
da (t )
 (t  a (t ))  s (t )
dt
k
Transversal variation
is governed by
Perturbation-Projection
Vector (PPV)
(t )  v1 (t )T  B
k  2,, N
Rek   0
N
t
k 2
0
x(t )   uk (t )  exp[k (t   )]vkT ( ) B  n( )d
8
Small-Amplitude Perturbations
• Limit cycle is stable: small-amplitude signals give
negligible transversal deviations from the orbit
• Phase is a neutrally stable variable: weak signals
induce large phase deviations that dominate the
oscillator dynamics
x p (t )  x0 (t  a (t ))
Scalar output response
x p (t )  X1 cos(0t   (t ) 1 )  
 (t )  0a (t )
Excess Phase
9
Pulse Perturbation Response (1)

t
At time 1
xs (t1  a (t1 ))
xs (t1  a (t1 ))  x(t1  a (t1 ))
xs (t1 )
10
Pulse Perturbed Response (2)
xs (t  a (t ))  x(t  a (t ))
xs (t  a (t ))
xs (t )
At time at
t  t1
11
Pulse Perturbed Response (3)
At time at t  t1
xs (t  a (t ))  x(t  a (t ))
xs (t )
xs (t  a (t ))
12
Phase-Sensitivity Response (PSR)
(intuitive viewpoint)
• Relation between α(t) and s(t) is described by the
periodic scalar function Γ(t)


a (t )  a (t  t )  a (t )  (t  a (t ))  s(t )  t
da (t )
 (t  a (t ))  s (t )
dt
Scalar Differential
Equation
13
Presentation Outline
•
Mathematical/Theoretical formalization
•
Computational issues
•
Pulling effects due to interferences
• Phase-noise analysis
Phase-domain Macromodeling of Oscillators
14
How the PPV and PSR can be
computed
(i)
Franz Kaertner, “Analysis of white and f -a noise in oscillators,”
International Journal of Circuit Theory and Applications, vol. 18, 1990.
Eigenvalue/eigenvector expansion of the Monodromy matrix
(ii) A. Demir, J. Roychowdhury, “A reliable and efficient procedure for
oscillator PPV computation, with phase noise macromodeling
applications ,” IEEE Trans. CAD, vol. 22, 2003.
Exploits the Jacobian matrix of PSS within a simulator
15
(i) Monodromy Matrix
tk 
tk 1
State Transition Matrix:
Monodromy matrix:
T ,0 
Eigenvalue/eigenvector
Expansion:
N
T ,0   exp(nT )un (t0 )  vnT (t0 )
n 1
k 1,k
x(t k 1 )

x(t k )
x(t0  T )
 M ,M 1  M 1,M 2 1,0
x(t0 )
Integration of direct
and adjoint ODE :
un (tk 1 ) exp(n hk )k 1,k  un (tk )
vnT (tk ) exp(n hk )vnT (tk 1 )k 1,k
16
(ii) With the PSS in a simulator
d
q( x(t ))  f ( x(t ))  0
dt
q( x(t ))  D
f ( x(t ),t )  D
x(t )  D
MNA variables
Charges and Fluxes
Resistive term
d
q ( x(t ))  f ( x(t ))  B  s (t )  0
dt
Perturbed Equations
17
Periodic Steady State (PSS)


x (tk ) T
Initial guess supplied by Transient/Envelope
(very close to PSS final solution)
• The (initial) period T is discretized into a grid of M+1 points
tk  t0  kh
k  0,...,M
T
h
M
• Integration (BE) at tk gives the equation (dimension D):
T
Fk 1 ( xk 1 , xk , T )  q( xk 1 )  q( xk ) 
f ( xk 1 )  0
M
where for k  0
x(t0 )
is replaced by
k  0,...,M  1
x(tM )
18
Periodic Steady State (PSS)
• DxM+1 unknowns and
DxM equations, thus we add
an extra constraint
x(t M ),
d 
x (t M )  0
dt
• Jacobian of the system

0
0
 C (t M )
C (t1 )  hG(t1 )

  C (t1 )
C (t 2 )  hG(t 2 )







0
0
 C (t M 1 ) C (t M )  hG(t M )


d 

0
0
0
x (t M )
dt

f ( x(t1 ) 
M 
f ( x(t 2 ) 

M 

f ( x(t M ) 
M 

0 
19
Newton-Raphson Iteration

0
0
 C (t M )
C (t1 )  hG(t1 )

  C (t1 )
C (t 2 )  hG(t 2 )







0
0
 C (t M 1 ) C (t M )  hG(t M )


d 
0
0
0
x (t M )

dt

x(t1 ),x(t2 ),x(tM ) ; T
f ( x(t1 ) 
M   x(t1 ) 
 F1 ( x1 , xM , T ) 

f ( x(t 2 ) 
 F (x , x ,T ) 
 x(t 2 ) 

 2 2 1

M  



      


f ( x(t M )  x(t )
F
(
x
,
x
,
T
)
M 

 N M M 1 
M  


0
  T 
0 

Variables update
• At convergence, we find a linearization around the
PSS response
20
Computing Γ(t)
A)
Transient Problem
IF: Tpulse  T  T
B)
Periodic Steady State
Problem
Controllably Periodically
Perturbed Problem:
Miklos Farkas, Periodic Motion,
Springer-Verlag 1994.
21
Computing Γ(t)

0
0
 C (t M )
C (t1 )  hG(t1 )

  C (t1 )
C (t 2 )  hG(t 2 )







0
0
 C (t M 1 ) C (t M )  hG(t M )


d 
0
0
0
x (t M )

dt

f ( x(t1 ) 
M   x(t1 ) 
0
f ( x(t 2 )  
 B  }t
 x(t 2 ) 
k

M  
1 

       h 0
 
f ( x(t M )  x(t )
M 



M
 0 
  T 
0 

(tk )  T
22
Presentation Outline
• Mathematical/Theoretical formalization
•
Computational issues
•
Pulling effects due to interferences
• Phase-noise analysis
Phase-domain Macromodeling of Oscillators
23
Analysis of Interferences
• Signal leakage through the packaging and
the substrate in ICs
• Weak interferences (-60/-40 dB) may have
tremendous effect on the oscillator response
• This depends on the injection point and the
frequency detuning
• Purely numerical simulation is not suitable
to explore all the potential injection points
24
Examples
A) Injection from the
Power Amplifier
B) Mutual Injection between
Two Oscillators
25
Synchronization Effect: Injection Pulling
INPUT
OUTPUT
  frequency detuning

f
s
frequency shift
26
Synchronization Effect: Injection
Locking

Synchronization
effect: Injection
Locking
Quasi-Lock

Injection Locking
s  
27
Studying interference with PPV/PSR
• Phase Sensitivity Response (PPV component) is To-Periodic:

(t )   n cos(n0t   n )
n 0
• For a perturbation
s(t ) A cos(et ) with
the Scalar Differential Equation
transforms to:
e  0
da (t )
 (t  a (t ))  s (t )
dt
da (t )  n A

cos(n0t  0a (t )   n  et )
dt
n 0 2
28
Averaging Method
• The time derivative of a(t) is dominated by the
“slowly-varying” term:
da (t ) 1 A

cos( 0t  0a (t )  et )
dt
2
0 1 A ,
Notation: k 
2
  0  e ,
 (t )  0a (t )
d (t )
 k cos( t   (t ))
dt
• Similar to Adler’s equation but generally applicable
29
Approximate Solution (1)
• We make the following assumption:
 (t )  st  E sin(t )  F sin(2t )
where:
s ,  , E, F are unknown parameters
d (t )
 k cos( t   (t ))
• Substituting in
dt
30
Approximate Solution (2)
• Expanding …
0
31
Closed-Form Expressions:
Frequency Shift
32
Closed-Form Expressions:
Amplitude Tones
• For a Free-running response
x0 (t )  X1 cos(0t )
• The perturbed response becomes
x p (t )  X1 cos(0t   (t ))  X1 cos((0  s )t  E sin(t )  F sin(2t ))
33
Example: Colpitts Oscillator (1)
o  387.21106 rad / s
PPV component
Current injection: s(t )  iinj (t )  A cos(et )
A  100A
34
Example: Colpitts Oscillator (2)
• Variable Detuning
• Excess Phase
 (t )  0a (t )
• Numerical integration of the
Scalar-Differential-Equation
• The average slope of excess
phase waveform gives the
frequency shift
35
Frequency shift vs. Detuning
• Square marker:
Numerical solutions of
the Scalar Equation
• Broken line:
Closed-form estimation
1  14.0 A-1
36
Comparison to Simulations with Spice
s
For detuning 2
Injection Pulling
For detuning 
3
Quasi-Locking
37
Example: Relaxation Oscillator
• Current injection into
nodes E, D
Injection in E causes no pulling !
• PPV components:
1D  610.9 A-1
1E  0.0A-1
38
Spice simulations versus Closed-form
prediction
•
Injection in D: Ain=25 A
 = -1.8 rad/s
•
Injection in E: Ain=25 A
 = -1.8 rad/s
39
Mutual pulling (1)
When decoupled:
g 21  g12  0
X1 (t ) 1 (t )
X 2 (t )
When coupled:
2 (t )
X 1 (t  a1 (t )) X 2 (t  a 2 (t ))
a1 (t ) 1 (t  a1 (t ))  g21  X 2 (t  a 2 (t ))
a 2 (t ) 2 (t  a 2 (t ))  g12  X1 (t  a1 (t ))
40
Mutual pulling (2)
4
1
Case A: g21  g12  10 
Case B: g21   g12  104 1
41
Mutual pulling (3)
Output Spectra
Case A
Case B
42
Presentation Outline
•
Mathematical/Theoretical formalization
• Computational issues
•
Pulling effects due to interferences
• Phase-noise analysis
Phase-domain Macromodeling of Oscillators
43
Phase-Noise Analysis
da (t )
 (t  a (t ))  n(t )
dt
Noise source
Stationary zero-mean Gaussian:
White/Colored
Rn ( )  En(t1 )  n(t1   )
Autocorrelation function
Sn ( f )
Power Spectral Density (PSD)
• Asymptotically a (t ) is a non-stationary Gaussian process
mean value a  0
variance a2 (t )  Da t
Alper Demir, “Phase Noise and Timing Jitter in Oscillators With ColoredNoise Sources,” IEEE Trans. on Circuits and Syst. I, vol. 49, no. 12, pp. 17821791, Dec. 2002.
44
Averaged Stochastic Model
da (t )
 (t  a (t ))  n(t )
dt
(1) Nonlinear Stochastic
Equation
da (t )
 cn  n(t )
dt
(2) Averaged Stochastic
Equation
1
cn  cW  
 T0
12

T0
0

 ( )d 

2
White noise source
1
cn  cF 
T0
T0

0
( )d
Flicker noise source
Solutions to (1) and (2) have the same
a2 (t )  Da t
45
Phase-Noise Spectrum
Time Domain  (t )  0a (t )
d (t )
 cn  0 n(t )
dt
Frequency Domain
j 2f( f )  cn j 2f 0 N ( f )
2
Power Spectral Density
 cn f 0 
  S n ( f )
S ( f )
 f 
S ( f )
S ( f )
1 f2
1 f3
Sn ( f )
f
Sn ( f )
f
46
Noise Macro-model
Effect of All Noise Sources
da (t )
 neq (t )  nW (t )  nF (t )
dt
( f )  2f 0 NW ( f )  NW ( f ) j 2f
Equivalent Noise Sources
SW ( f ) AW
SF ( f ) AF / f
AW
AF
S ( f ) 2 2  3 2
f T0
f T0
S ( f )
1 f3
1 f2
• Phase- Noise parameters are derived
f
fc
by fitting DCO Power Spectrum
47
Application: Frequency Synthesis in
Communication Systems
• Phase-locked loop (PLL): out  N  ref
ref
PD
Filter
VCO
out
N
• Evolution from Analog towards Digital PLLs
48
Bang-Bang PLL (BBPLL)
• BPD: single bit quantizer
t[k ]  tr [k ]  td [k ]
 [k ]  sgn(t[k ])
r
td d
  1
• DLF: Digital Loop Filter
tr
r
d
  1
• DCO: Digitally-Controlled
Oscillator
49
Digitally-Controlled Oscillator (DC0)
Digital-to-Analog Converter (DAC)
Free-running Period
Tv  T0  KT w
Period
Gain
Constant
Analog Section: Ring Oscillator
50
BBPLL: Design Issues
• Harsh nonlinear dynamics: different working regimes
51
BBPLL: Design Issues
• Harsh nonlinear dynamics: different working regimes
Output Spectrum [dBc/Hz]
• Prone to the generation of spur tones in the output
spectrum
52
BBPLL: Design Issues
• Harsh nonlinear dynamics: different working regimes
Output Spectrum [dBc/Hz]
• Prone to the generation of spur tones in the output
spectrum: limit-cycle regime .

53
Quantization and Random Noise
Tv [k ]  Tv
Tv [k ]
k
Tv  Tref N
Tv [k ]  Tv
Tv [k ]
k
Tv  Tref N
54
How to eliminate spurs
(i) Dithering: addition of extra noise
- extra hardware, higher power dissipation
- eliminate cycles but increase noise floor and total jitter
(ii) Exploiting VCO intrinsic noise sources
- accurate knowledge and control of VCO noise
55
Noise-Aware Discrete-Time Model
BFD t[k ]  tr [k ]  td [k ]
k Index of divider cycle
 [k ]  sgn(t[k ])
DLF  [k ]   [k  1]   [k ]
w[k ]  [k  D]  a [k  D]
REF
tr [k 1]  tr [k ]  Tref
DCO
td [k  1]  td [k ]  N  (T0  KT w[k ])  Tacc[k ]
Tacc[k ] 
( k 1) N
 T
i  kN 1
i
56
DCO Model
Tv  T0  KT w
Period of the Noiseless DCO
Tv  T0  KT w  T
T
Period of the Noisy DCO
Stochastic variable: fluctuation of DCO
period over ONE cycle
Tacc[k ] 
( k 1) N
 T
i  kN 1
i
Fluctuations accumulated over one
reference cycle = N oscillator cycles
td [k  1]  td [k ]  N  (T0  KT w[k ])  Tacc[k ]
57
Period Variation
Period fluctuation: T  lima (t  T0 )  a (t )
t 
t T0
t

T  lim  neq ( )d   neq ( )d 

t    

e j 2fT0  1
T ( f ) 
[ NW ( f )  NW ( f )]
j 2f

AF 

ST ( f )T  AW 
f 

2
0
• From Phase-Noise parameters,
find PSD of period variation
• Reproduce Noise in the
Discrete-Time-Model
58
Simulation Results (1)
Output Jitter
Noise Spectrum
(b)
(a)
(c)
• An optimal parameter setting exists
• Limit-cycle regime and Random-noise regime
59
Simulation Results (2)
Distribution of t variable at the BPD input
(b)
(c)
PDF [1/s]
(a)
Limit-cycle:
uniform distribution
Deep Random-noise:
Gaussian-Laplacian
distribution
Intermediate Regime:
Gaussian distribution
Linear behavior !
60
Intermediate Regime
Linear Analysis, the BPD is replaced
by a linear block with gain:
Kbpd  2 pt (0) 
2 1
  t
ST ( f )
Closed-form expression
AW T0

of jitter due to DCO
J rn   t 
KT
2
2
8 ( KT )  AF T0 log( )
random noise only
61
Limit-cycle regime
Nonlinear analysis with the hypothesis of uniform
distributed t : closed-form expression of jitter due to
quantization error only
J lc   t
(1  D)

NKT
3
Nicola Da Dalt, “ A design-oriented study of the nonlinear dynamics of digital
bang-bang PLLs” IEEE Trans. on Circuits and Syst. I, vol. 52, no. 1, pp. 2131, Jan. 2005.
62
Optimal Design: Closed-Form
Minimum Total Jitter
occurs for:
J lc  J rn
KT
Opt .

AW T0
2
 AF T0 log( )
N (1  D)
AW T0
2 N (1  D)
2
J tot Opt . 
 AF T0 log( )
N (1  D)
3
63
Simulations versus Measurements
Hardware Implementation: 65-nm CMOS process
Frequency offset [Hz]
Frequency offset [Hz]
Frequency offset [Hz]
64
Stochastic Resonance
(i) System contains a threshold device, i.e. the BPD
(ii) Unintentional noise (i.e. DCO noise) is modulated by a
loop parameter
(iii) Noise enhances quality
(iv) System performance
shows a peculiar dependence
on noise (i.e. on loop-parameter)
65
Dithering or Intrinsic DCO Noise ?
(i) With dithering added to DCO noise
(ii) Only DCO noise
• With dithering spur reduction is more robust
• Optimal design achieves no spurs and minimum jitter
66
Conclusions and future work
• Oscillator macro-modeling works (reliability/synchronization)
• Amplitude modulation effects
• Large-amplitude Pulse Injection Locked oscillators
• Pulling in VCO closed in PLLs
67
Phase-domain Macromodeling of
Oscillators for the analysis of
Noise, Interferences and
Synchronization effects
MANY THANKS !
MIT, Cambridge, MA, 23-27 Sep. 2013
68
Presentation Outline
•
Mathematical/Theoretical formalization
• Computational issues
•
Pulling effects due to interferences
• Synchronization/Frequency division
• Noise analysis
Phase-domain Macromodeling of Oscillators
69
Order 1:1 Injection Locking
• Frequency shift equates frequency detuning
when the term under the square-root
becomes zero
• Locking Range:
Closed-Form
estimation under
weak injection
70
Order 1:m, Super-harmonic Injection
Locking
Free-running frequency
e  m0
Frequency of the injected signal
Frequency of the forced oscillator
R
R 
e
m
Locking Range
m0  a  e  m0  b
71
Synchronization Region
Improving the LR for smallforcing amplitudes
Improving the LR for
moderate-forcing amplitudes
72
Computing the Synchronization Region
• Extensive detailed simulations
- generally applicable
- time-consuming
- no synthesis information
• Behavioral macro-models
- small forcing amplitudes
- explore many possible injection strategy
- explore many possible parameters settings
73
Order 1:m Injection Locking
Harmonic perturbation:
b(t )  Ain cos(et )
e  mo
Locking condition: x0 (t  a (t )) X1 cos(0t  0a (t ))  
R 
e
m
0t  0a (t ) 
e t
m
e  m0
a (t ) 
t
m0
74
Locking Condition
t
a (t )   ( a ( ))  b( )d

t
a (t )   Ain0 cos(e )d
b(t )  Ain cos(et )

(t )   k cos(k0t   k )
k 0

Aink
cos((k0  e )  k0a ( )  k )d


2
k 1

a (t ) 
t

e  m0
t
m0
Resonant term for k=m:
m Ain
m Ain
cos( m0  e  m0a ( )   m ) 
cos(  m )
2
2
75
Locking Range
• For weak sinusoidal injections, 1:m locking condition:
m0 m Ain
  e  m 0
2
•
Multiple-Input Injection at, P1, P2,…, PI
Sensitivity Responses 

( Pi )
(t )   k( Pi ) cos(k0t   k( Pi ) )
k 0
m 
I
( pi )
( pi )


exp(
j

 m
m )
i 1
76
Example: Relaxation ILFD
Free response
• We study current injection iin(t) at points E1, E2,
and at D1, D2.
77
Injection at E1, E2
Spectrum of  E1 (t )
Spectrum of  E 2 (t )
• Suitable for odd-number freq. division
• LR is maximized by injecting +iin(t) in E1 and
-iin(t) in E2
78
Injection at D1, D2
Spectrum of
 (t )
D1
Spectrum of
 D 2 (t )
• Suitable for even-number freq. division
• LR is maximized by injecting +iin(t) with the same sign
into both D1 and D2
79
Synchronization Regions: comparison
with Spice simulations
Divide-by-three LR
multiple input injection
(+)E1, (-)E2
Divide-by-four LR
Multiple input injection
(+)D1, (+)D2
80