phase-domain macromodeling of oscillators for the analysis of noise, interferences and
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Phase-domain Macromodeling of Oscillators for the analysis of
Noise, Interferences and Synchronization effects
Paolo Maffezzoni
Dipartimento di Elettronica, Informazione e BioingegneriaPolitecnico di Milano, Milan, Italy
MIT, Cambridge, MA, 23-27 Sep. 20131
• Mathematical/Theoretical formalization
• Computational issues
• Pulling effects due to interferences
• Phase-noise analysis
Presentation Outline
2
Phase-domain Macromodeling of Oscillators
Presentation Outline
Phase-domain Macromodeling of Oscillators
• Mathematical/Theoretical formalization
• Computational issues
• Pulling effects due to interferences
• Phase-noise analysis
3
Free-Running Oscillator
Ntx )(
Ntxf ))((
State variables
Vector-valued nonlinear function
))(()( txftx
)(txs Vector solutionLimit cycle
Scalar output response
)cos()( 1010 tXtx
)()(0 txtx s
00 2 T
4
Perturbed Oscillator
Transversal variation
Amplitude modulation (AM)
Tangential variation,
Phase modulation (PM)
Franz Kaertner, “Analysis of white and f noise in oscillators”,
International Journal of Circuit Theory and Applications, vol. 18, 1990.
-
5
s(t) small-amplitude
perturbation
)())(()( tsBtxftx
(t) is the time-shift of the perturbed response with respect to
free-running one
))(())(()( ttxttxtx s
)()()())(( ttxtxttx sss
))(( ttx
Pulse Perturbation
6
))(( 11 ttxs
)( 1txs
))(())(( 1111 ttxttxs
Small-amplitude
pulse perturbation
at
1t
Floquet theory of linear time-periodic ODEs
7
Linearization around the limit cycle )(
)()(
txx sx
xftA
)()()( tytAty
)()()( twtAtw T
)()exp()( tutty kk
)()exp()( tvttw kk
Floquet exponent
Left eigenvector
Direct ODE
Adjoint ODE
Right eigenvector N Solutions
)()( 0 tATtA
Phase and Amplitude Modulations
)()(1 txtu s01
t Tkk
N
kk dnBvttutx
02
)()()](exp[)()(
)(1 tv
Tangential variation
is governed by:
)())(()(
tsttdt
td Btvt T )()( 1
Transversal variation
is governed by
Nkk ,,2
0Re k
8
Perturbation-Projection
Vector (PPV)
Small-Amplitude Perturbations
))(()( 0 ttxtxp
))(cos()( 101 ttXtxp
• 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
Excess Phase
Scalar output response
)()( 0 tt 9
Pulse Perturbation Response (1)
10
1tAt time
))(( 11 ttxs
)( 1txs
))(())(( 1111 ttxttxs
Pulse Perturbed Response (2)
11
))(( ttxs
)(txs
))(())(( ttxttxs
At time at
1tt
Pulse Perturbed Response (3)
12
))(( ttxs
)(txs
))(())(( ttxttxs At time at
1tt
)())(()(
tsttdt
td
• Relation between α(t) and s(t) is described by the
periodic scalar function Γ(t)
ttstttttt )())(()()()(
Scalar Differential Equation
13
Phase-Sensitivity Response (PSR) (intuitive viewpoint)
Presentation Outline
• Mathematical/Theoretical formalization
• Computational issues
• Pulling effects due to interferences
• Phase-noise analysis
14 Phase-domain Macromodeling of Oscillators
(i) Franz Kaertner, “Analysis of white and f noise in oscillators,”
International Journal of Circuit Theory and Applications, vol. 18, 1990.
15
-
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
How the PPV and PSR can be computed
State Transition Matrix:
(i) Monodromy Matrix
16
)(
)( 1,1
k
kkk tx
tx
1 kk tt
0,12,11,0
00, )(
)(
MMMMT tx
TtxMonodromy matrix:
N
n
TnnnT tvtuT
1000, )()()exp(
Eigenvalue/eigenvector Expansion:
)()exp()( ,11 knkkknkn tuhtu
kkkTnknk
Tn tvhtv ,11)()exp()(
Integration of direct and adjoint ODE :
Dtx )(
Dttxf )),((
Dtxq ))((
MNA variables
Charges and Fluxes
Resistive term
0))(())(( txftxqdt
d
(ii) With the PSS in a simulator
0)())(())(( tsBtxftxqdt
d
Perturbed Equations
17
Mkkhttk ,...,00
0)()()(),,( 1111 kkkkkk xfM
TxqxqTxxF
M
Th
• The (initial) period T is discretized into a grid of M+1 points
• Integration (BE) at tk gives the equation (dimension D):
)( ktx Initial guess supplied by Transient/Envelope
(very close to PSS final solution) T
)( 0tx
1,...,0 Mk
Periodic Steady State (PSS)
18
0kwhere for )( Mtxis replaced by
0)(),( MM txdt
dtx
0)(000
)(()()()(00
)(()()()(
)(()(00)()(
1
2221
111
M
MMMM
M
txdt
dM
txfthGtCtC
M
txfthGtCtC
M
txftCthGtC
• Jacobian of the system
• DxM+1 unknowns and DxM equations, thus we add an extra constraint
Periodic Steady State (PSS)
19
0
),,(
),,(
),,(
)(
)(
)(
0)(000
)(()()()(00
)(()()()(
)(()(00)()(
1
122
11
2
1
1
2221
111
TxxF
TxxF
TxxF
T
tx
tx
tx
txdt
dM
txfthGtCtC
M
txfthGtCtC
M
txftCthGtC
MMN
M
M
M
MMMM
M
• At convergence, we find a linearization around the PSS response
Ttxtxtx M ;)(),(),( 21 Variables update
Newton-Raphson Iteration
20
Transient ProblemA)
Periodic Steady State Problem
Controllably Periodically Perturbed Problem: Miklos Farkas, Periodic Motion, Springer-Verlag 1994.
B)
TTTpulse
Computing Γ(t)
21
IF:
k
M
M
MMMM
M
tB
h
T
tx
tx
tx
txdt
dM
txfthGtCtC
M
txfthGtCtC
M
txftCthGtC
}
0
0
0
1
)(
)(
)(
0)(000
)(()()()(00
)(()()()(
)(()(00)()(
2
1
1
2221
111
Ttk )(
Computing Γ(t)
22
• Mathematical/Theoretical formalization
• Computational issues
• Pulling effects due to interferences
• Phase-noise analysis
Presentation Outline
23 Phase-domain Macromodeling of Oscillators
• 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
Analysis of Interferences
A) Injection from the Power Amplifier
B) Mutual Injection between Two Oscillators
25
Examples
INPUT OUTPUT
frequency shift
26
Synchronization Effect: Injection Pulling
frequency detuning
s f
Synchronization effect: Injection LockingQuasi-Lock
Injection Locking
27
Synchronization Effect: Injection Locking
s
• Phase Sensitivity Response (PPV component) is To-Periodic:
• For a perturbation with
the Scalar Differential Equation
transforms to:
28
Studying interference with PPV/PSR
0
0 )cos()(n
nn tnt
)cos()( tAts e
0
00 ))(cos(2
)(
nen
n tttnA
dt
td
)())(()(
tsttdt
td
0 e
• The time derivative of (t) is dominated by the “slowly-varying” term:
• Similar to Adler’s equation but generally applicable
29
Averaging Method
))(cos(2
)(00
1 tttA
dt
tde
210 A
k
e 0
))(cos()(
ttkdt
td
)()( 0 tt Notation: , ,
• We make the following assumption:
• Substituting in
30
Approximate Solution (1)
)2sin()sin()( tFtEtt s
where: are unknown parametersFEs ,,,
))(cos()(
ttkdt
td
• Expanding …
31
Approximate Solution (2)
0
32
Closed-Form Expressions: Frequency Shift
))2sin()sin()cos(())(cos()( 0101 tFtEtXttXtx sp
• For a Free-running response
• The perturbed response becomes
)cos()( 010 tXtx
33
Closed-Form Expressions: Amplitude Tones
)cos()()( tAtits einj
srado /1021.387 6
AA 100Current injection:
PPV component
34
Example: Colpitts Oscillator (1)
• Excess Phase• Variable Detuning
• Numerical integration of the Scalar-Differential-Equation
• The average slope of excess phase waveform gives the frequency shift
)()( 0 tt
35
Example: Colpitts Oscillator (2)
• Broken line:
Closed-form estimation
• Square marker:
Numerical solutions of
the Scalar Equation
-11 A0.14
36
Frequency shift vs. Detuning
For detuning
Injection Pulling
2 For detuning
Quasi-Locking
3
s
37
Comparison to Simulations with Spice
• Current injection into nodes E, D
• PPV components:
-11 A9.610D
-11 A0.0E
Injection in E causes no pulling !
38
Example: Relaxation Oscillator
• Injection in D: Ain=25 A
= -1.8 rad/s
• Injection in E: Ain=25 A
= -1.8 rad/s
Spice simulations versus Closed-form prediction
39
Mutual pulling (1)
40
))(())(()( 2221111 ttXgttt
))(())(()( 1112222 ttXgttt
When decoupled:
01221 gg
)(1 t)(1 tX
)(2 tX )(2 t
When coupled: ))(( 22 ttX ))(( 11 ttX
Mutual pulling (2)
41
141221 10 gg Case A: 14
1221 10 gg Case B:
Mutual pulling (3)
42
Case A Case B
Output Spectra
Presentation Outline
• Mathematical/Theoretical formalization
• Computational issues
• Pulling effects due to interferences
• Phase-noise analysis
43 Phase-domain Macromodeling of Oscillators
Phase-Noise Analysis
44
)()()( 11 tntnERn Autocorrelation function
Noise source
Stationary zero-mean Gaussian:
White/Colored
mean value variance0 tDt )(2
• Asymptotically is a non-stationary Gaussian process )(t
)( fSn Power Spectral Density (PSD)
Alper Demir, “Phase Noise and Timing Jitter in Oscillators With Colored-Noise Sources,” IEEE Trans. on Circuits and Syst. I, vol. 49, no. 12, pp. 1782-1791, Dec. 2002.
)())(()(
tnttdt
td
Averaged Stochastic Model
45
(2) (2) Averaged Stochastic
Equation21
0
2
0
0
)(1
T
Wn dT
cc 0
00
)(1 T
Fn dT
cc
White noise source Flicker noise source
(1) Nonlinear Stochastic Equation
tDt )(2Solutions to (1) and (2) have the same
)())(()(
tnttdt
td
)()(
tncdt
tdn
Phase-Noise Spectrum
46
)()(
0 tncdt
tdn )(2)(2 0 fNfjcffj n
)()(2
0 fSf
fcfS n
n
Frequency Domain
Power Spectral Density
Time Domain
)( fS
f
)( fSn
21 f
f
)( fSn
31 f
)( fS
)()( 0 tt
Noise Macro-model
47
)()()()(
tntntndt
tdFWeq
fjfNfNff WW 2)()(2)( 0
20
320
2)(
Tf
A
Tf
AfS FW
Effect of All Noise Sources
)( fS
f
21 f
WW AfS )(
fAfS FF /)(
31 f
cf
Equivalent Noise Sources
• Phase- Noise parameters are derived
by fitting DCO Power Spectrum
Application: Frequency Synthesis in Communication Systems
48
PD Filter VCOref
N
out
• Phase-locked loop (PLL):
• Evolution from Analog towards Digital PLLs
refout N
Bang-Bang PLL (BBPLL)
49
• BPD: single bit quantizer
• DLF: Digital Loop Filter
• DCO: Digitally-Controlled Oscillator
][][][ ktktkt dr
])[sgn(][ ktk
r d
rd 1
1
dt
rt
Digitally-Controlled Oscillator (DC0)
50
Analog Section: Ring Oscillator
Digital-to-Analog Converter (DAC)
wKTT Tv 0
Free-running Period
Period Gain Constant
BBPLL: Design Issues
51
• Harsh nonlinear dynamics: different working regimes
BBPLL: Design Issues
52
• Harsh nonlinear dynamics: different working regimes
• Prone to the generation of spur tones in the output spectrum
Out
put S
pect
rum
[dB
c/H
z]
BBPLL: Design Issues
53
• Harsh nonlinear dynamics: different working regimes
• Prone to the generation of spur tones in the output spectrum: limit-cycle regime .
Out
put S
pect
rum
[dB
c/H
z]
Quantization and Random Noise
54
NTT refv
][kTv vv TkT ][
k
NTT refv
][kTv vv TkT ][
k
How to eliminate spurs
55
(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
Noise-Aware Discrete-Time Model
56
][][][ ktktkt dr ])[sgn(][ ktk
][]1[][ kkk
][][][ DkDkkw
refrr Tktkt ][]1[
][])[(][]1[ 0 kTkwKTNktkt accTdd
Nk
kNiiacc TkT
)1(
1
][
BFD
DLF
REF
DCO
k Index of divider cycle
DCO Model
57
TwKTT Tv 0
][])[(][]1[ 0 kTkwKTNktkt accTdd
Nk
kNiiacc TkT
)1(
1
][
wKTT Tv 0 Period of the Noiseless DCO
Period of the Noisy DCO
T Stochastic variable: fluctuation of DCO period over ONE cycle
Fluctuations accumulated over one reference cycle = N oscillator cycles
Period Variation
58
)]()([2
1)(
02
fNfNfj
efT WW
fTj
Period fluctuation: )()(lim 0 tTtTt
t
eq
Tt
eqt
dndnT )()(lim0
f
AATfS FWT
20)(
• From Phase-Noise parameters,
find PSD of period variation
• Reproduce Noise in the
Discrete-Time-Model
Simulation Results (1)
59
Output Jitter Noise Spectrum
(a)
(c)
(b)
• An optimal parameter setting exists
• Limit-cycle regime and Random-noise regime
Simulation Results (2)
60
Distribution of variable at the BPD input
(a) (b) (c)
Limit-cycle:
uniform distribution
Deep Random-noise:
Gaussian-Laplacian distribution
Intermediate Regime:
Gaussian distribution
Linear behavior !
t
PD
F [
1/s]
Intermediate Regime
61
Linear Analysis, the BPD is replaced by a linear block with gain:
T
FT
Wtrn K
TAK
TAJ
)log()(8 20
20
ttbpd pK
12
)0(2
)( fS T
Closed-form expression of jitter due to DCO random noise only
Limit-cycle regime
62
Nonlinear analysis with the hypothesis of uniform distributed : closed-form expression of jitter due to quantization error only
Ttlc KND
J 3
)1(
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. 21-31, Jan. 2005.
t
Optimal Design: Closed-Form
63
Minimum Total Jitter occurs for:
rnlc JJ
)log()1(
20
0.
TADN
TAK F
WOptT
)log()1(3
)1(2 20
0.
TADN
TADNJ F
WOpttot
Simulations versus Measurements
64
Hardware Implementation: 65-nm CMOS process
Frequency offset [Hz]
Frequency offset [Hz]
Frequency offset [Hz]
Stochastic Resonance
65
(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)
Dithering or Intrinsic DCO Noise ?
66
(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
67
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
MANY THANKS !
MIT, Cambridge, MA, 23-27 Sep. 201368
Phase-domain Macromodeling of Oscillators for the analysis of
Noise, Interferences and Synchronization effects
Presentation Outline
• Mathematical/Theoretical formalization
• Computational issues
• Pulling effects due to interferences
• Synchronization/Frequency division
• Noise analysis
69 Phase-domain Macromodeling of Oscillators
• Frequency shift equates frequency detuning when the term under the square-root becomes zero
• Locking Range:
Closed-Form
estimation under
weak injection
Order 1:1 Injection Locking
70
bea mm 00
me
R
Frequency of the forced oscillator
Frequency of the injected signal
Free-running frequency
0 me
Locking Range
R
Order 1:m, Super-harmonic Injection Locking
71
Improving the LR for small-forcing amplitudes
Improving the LR for moderate-forcing amplitudes
Synchronization Region
72
• 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
Computing the Synchronization Region
oe m
me
R
Harmonic perturbation:
)cos()( tAtb ein
tm
mt e
0
0)(
m
ttt e )(00
Locking condition:
74
Order 1:m Injection Locking
))(cos())(( 0010 ttXttx
Resonant term for k=m:
0
0 )cos()(k
kk tkt
dbtt
)())(()(
dkkA
dAt
k
t
ke
kin
t
ein
))()cos((2
)cos()(
100
0
)cos()( tAtb ein
)cos(2
))(cos(2 00 m
inmme
inm Amm
A
tm
mt e
0
0)(
75
Locking Condition
• For weak sinusoidal injections, 1:m locking condition:
• Multiple-Input Injection at, P1, P2,…, PI
Sensitivity Responses
00
2
mAm
einm
0
)(0
)()( )cos()(k
Pk
Pk
P iii tkt
I
i
pm
pmm
ii j1
)()( )exp(
76
Locking Range
• We study current injection iin(t) at points E1, E2, and at D1, D2.
Free response
77
Example: Relaxation ILFD
• Suitable for odd-number freq. division
• LR is maximized by injecting +iin(t) in E1 and
-iin(t) in E2
Spectrum of )(1 tE Spectrum of )(2 tE
78
Injection at E1, E2
Spectrum of )(2 tD Spectrum of )(1 tD
• Suitable for even-number freq. division
• LR is maximized by injecting +iin(t) with the same sign into both D1 and D2
79
Injection at D1, D2
Divide-by-three LR
multiple input injection
(+)E1, (-)E2
Divide-by-four LR
Multiple input injection
(+)D1, (+)D2
80
Synchronization Regions: comparison with Spice simulations
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