Phase-domain Macromodeling of Oscillators for the analysis of Noise, Interferences and...

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


Phase-domain Macromodeling of Oscillators





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)






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

• 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

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


52


• Prone to the generation of spur tones in the output spectrum

Out

put S

pect

rum

[dB

c/H

z]


53


• 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]



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





• Synchronization/Frequency division

• Noise analysis


• 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

Phase-domain Macromodeling of Oscillators for the analysis of Noise, Interferences and...

Documents

Transcript of Phase-domain Macromodeling of Oscillators for the analysis of Noise, Interferences and...