Modeling I/O Links With X Parameters

José E. Schutt‐Ainé and Pavle Milosevic

Department of Electrical and Computer Engineering,

University of Illinois at Urbana-Champaign Urbana, IL 61801

Wendemagegnehu T. Beyene

Research & Technology DevelopmentRambus Inc.

Los Altos, CA 94022

1

Outline• Motivation• S Parameters• PHD Framework and X Parameters

a. Definitionsb. Propertiesc. Matrix Formulationd. Time‐Domain Simulation

• Application to CMOS Inverter• High-Speed Link Simulations• Conclusions

2

1 11 1 12 2b S a S a

2 21 1 22 2b S a S a

B SA1

N

i ij jj

B S A

0

1,...,

k

iij

Aj k jk N

BSA

For a general N-port

For a two-port

2 22

o

a bIZ

1 11

o

a bIZ

1 1 1 V a b 2 2 2V a b

“ …most successful behavioral models…”

Scattering Parameters

3

– Need to accurately handle very high data rates– Simulate large number of bits to achieve low BER– Non‐linear blocks with time variant systems– Model TX/RX equalization– All types of jitter: (random, deterministic, etc.)– Crosstalk, loss, dispersion, attenuation, etc…– Handle and manage vendor specific device settings– Clock data recovery (CDR) circuits

High speed Serial channels are pushing the current limits of simulation. Models/Simulator need to handle current challenges

These cannot be accurately modeled with S parameters

Challenges in HS Links

4

Adopt X parameters as the framework for high-speed channel design modeling and simulation.

- Mathematically robust framework- Can handle nonlinearities- Instrument exists (NVNA)- Blackbox format vendor IP protection- Matrix format easy incorporation in CAD tools- X Parameters are a superset of S parameters

Advantages

Objective

X Parameters for SI

See Refs [1] & [2] by Verspecht and Root

5

Cascading X Parameters

X-parameters of individual devices can be accurately cascaded within a harmonic balance simulator environment.

X

Vendor A Vendor CVendor B

GOAL: Simulate complete channel by combining X-parameter blocks from different sources into a single composite X matrix.

Foundry In-House FoundryIn-House

6

NVNA instruments will gradually replace all VNAs

Nonlinear Vector Network Analyzer (NVNA)

7

• Polyharmonic distortion (PHD) modeling is a frequency-domain modeling technique

• PHD model defines X parameters which form a superset of S parameters

• To construct PHD model, DUT is stimulated by a set of harmonically related discrete tones

• In stimulus, fundamental tone is dominant and higher-order harmonics are smaller

PHD Modeling

8

• Signal is represented by a fundamental with harmonics

• Signals are periodic or narrowband modulated versions of a fundamental with harmonics

• Harmonic index: 0 for dc contribution, 1 for fundamental and 2 for second harmonic

• Power level, fundamental frequency can be varied to generate complete data for DUT

PHD Framework

9

10

Excitation DesignExcitation 1 Excitation 2

Excitation 3 Excitation 4

Each excitation will generate response with fundamental and all harmonics

PHD Framework

11

Harmonic superposition principle is key to PHD model

In many situations, there is only one dominant large-signal input component present. The harmonic frequency components are relatively small harmonic components can be superposed

Harmonic Superposition

12

! Created Fri Jul 30 07:44:48 2010

! Version = 2.0! HB_MaxOrder = 25! XParamMaxOrder = 12! NumExtractedPorts = 3

! IDC_1=0 NumPts=1! IDC_2=0 NumPts=1! VDC_3=12 NumPts=1! ZM_2_1=50 NumPts=1! ZP_2_1=0 NumPts=1! AN_1_1=100e-03(20.000000dBm) NumPts=1! fund_1=[100 Hz->1 GHz] NumPts=4

TOP: FILE DESCRIPTION

X-Parameter Data File

13

BEGIN XParamData% fund_1(real) FV_1(real) FV_2(real) FI_3(real) FB_1_1(complex) % FB_1_2(complex) FB_1_3(complex) FB_1_4(complex)% FB_1_7(complex) FB_1_8(complex) FB_1_9(complex)% FB_1_12(complex) FB_2_1(complex) FB_2_2(complex) % FB_2_5(complex) FB_2_6(complex) FB_2_7(complex)% FB_2_10(complex) FB_2_11(complex) FB_2_12(complex)% T_1_1_1_1(complex) S_1_2_1_1(complex) T_1_2_1_1(complex)% S_1_4_1_1(complex) T_1_4_1_1(complex) S_1_5_1_1(complex)% T_1_6_1_1(complex) S_1_7_1_1(complex) T_1_7_1_1(complex)% S_1_9_1_1(complex) T_1_9_1_1(complex) S_1_10_1_1(complex)) % T_1_11_1_1(complex) S_1_12_1_1(complex) T_1_12_1_1(complex)% T_2_1_1_1(complex) S_2_2_1_1(complex) T_2_2_1_1(complex)% S_2_4_1_1(complex) T_2_4_1_1(complex) S_2_5_1_1(complex% T_2_6_1_1(complex) S_2_7_1_1(complex) T_2_7_1_1(complex)

% S_2_9_1_1(complex) T_2_9_1_1(complex) S_2_10_1_1(complex)

MIDDLE: FORMAT DESCRIPTION

X-Parameter Data File

14

100 0 0.903921 0.0263984 0.316228 -5.41159e-09 -5.8503e-16 -4.19864e-10 -6.37642e-16 -1.6748e-10 -4.62314e-16-1.25093e-15 -3.79264e-10 -7.91128e-16 -1.51261e-10 1.93535e-17-1.38032e-16 -2.09262e-10 0.107122 -5.52212e-08 0.0739648-0.0081633 -2.40901e-08 -0.00739395 -1.21199e-08 -0.0005307680.000921039 -4.82427e-09 -0.00230559 1.07836e-08 -0.00288533-1.20792e-15 -5.09916e-10 -6.95799e-15 -2.56672e-09 -3.25033e-15-1.2948e-14 3.97284e-10 -7.08201e-15 -2.17127e-09 -1.43757e-143.39598e-15 3.66098e-10 -1.08395e-14 -4.05911e-09 1.67366e-142.76565e-14 5.60242e-09 2.69755e-14 -6.60802e-10 3.99868e-14

BOTTOM: DATA LISTING

X-Parameter Data File

RemarksData is measured or generated from a harmonic

balance simulatorData file can be very large

15

11:P Phase of a

, :ik jlS S type X parameter

, :ik jlT T type X parameter

:ikD B type X parameter

*11 , 11 , 11

( , ) (1,1)

k k l k lik ik ik jl jl ik jl jl

j lb D a P S a P a T a P a

X-Parameter Relationship

16

Sik,jloutport

inport

harmonicout

harmonicin

aikport harmonic

bikport harmonic

Tik,jloutport

inport

harmonicout

harmonicin

Index Convention

17

-5 0 5 10 15 20 25 30-7

-6

-5

-4

-3

-2

-1x 10-3

|A11| (dBm)

S11

,11

- Am

plitu

de (d

B)

0.5 GHz1 GHz

-5 0 5 10 15 20 25 30-100

-90

-80

-70

-60

-50

|A11| (dBm)

T11,

11 -

Am

plitu

de (d

B)

0.5 GHz1 GHz

-5 0 5 10 15 20 25 30-20

-15

-10

-5

0

|A11| (dBm)

S21

,11

- Am

plitu

de (d

B)

0.5 GHz1 GHz

-5 0 5 10 15 20 25 30-60

-50

-40

-30

-20

-10

|A11| (dBm)

T21,

11 -

Am

plitu

de (d

B)

0.5 GHz1 GHz

X Parameters of CMOS

18

-5 0 5 10 15 20 25 30-65

-60

-55

-50

-45

-40

-35

|A11| (dBm)

S12

,11

- Am

plitu

de (d

B)

0.5 GHz1 GHz

-5 0 5 10 15 20 25 30-100

-90

-80

-70

-60

-50

-40

|A11| (dBm)

T12,

11 -

Am

plitu

de (d

B)

0.5 GHz1 GHz

-5 0 5 10 15 20 25 30-50

-45

-40

-35

-30

-25

-20

|A11| (dBm)

S22

,11

- Am

plitu

de (d

B)

0.5 GHz1 GHz

-5 0 5 10 15 20 25 30-100

-80

-60

-40

-20

|A11| (dBm)

T22,

11 -

Am

plitu

de (d

B)

0.5 GHz1 GHz

X Parameters of CMOS

19

• T-Type X ParameterSpectral mapping is non-analyticalReal and imaginary parts in FD are treated differentlyEven and odd parts in TD are treated differentlyT involves non-causal component of signal

• Phase Term PP is phase of large-signal excitation (a11)Contributions to B waves will depend on P In measurements, system must be calibrated for phase

Special Terms

20

*11 , 11 , 11

( , ) (1,1)

k k l k lik ik ik jl jl ik jl jl

j lb D a P S a P a T a P a

*11 , 11 , 11

( , ) (1,1)

k l lik ik ik jl jl ik jl jl

j lb P D a S a P a T a P a

11jP e 11 11where is the phase of a

*11 , 11 , 11

( , ) (1,1)ik ik ik jl jl ik jl jl

j lb D a S a a T a a

where andk kik ik ik ikb b P a a P

we can always express the relationship in terms of modified power wave variables

Multiply through by kP

Handling Phase Term

21

r rr ri r

i ir ii i

b X X ab X X a

,rr r r ri i iX S T X S T

,ir i i ii r rX S T X S T

Because of non-analytical nature of spectral mapping, real and imaginary component interactions must be accounted for separately.

where

we have

Handling R&I Components

22

' '

' '

cos sin cos sinsin cos sin cos

b b rr ri a ar r

b b ir ii a ai i

X Xb aX Xb a

'

'

cos sinsin cos

r b b r

i b b i

b bb b

'

'

cos sinsin cos

r a a r

i a a i

a aa a

Handling Phase TermPhase term can be accounted for by applying following transformations

r rr ri r

i ir ii i

b X X ab X X a

in which

23

Separate real and imaginary componentsAccount for real-imaginary interactionsAccount for harmonic-to-harmonic contributionsAccount for harmonic-to-DC contributions

Matrix size is 2 2mn mnm: number of harmonicsn: number of ports

X Matrix Construction

24

1

2

p

n

aa

aaa

1

2

p

n

bb

bbb

(1)

(1)

(2)

(2)

( )

( )

pr

pi

pr

pi

mprm

pi

aaaa

aa

pa

(1)

(1)

(2)

(2)

( )

( )

pr

pi

pr

pi

mprm

pi

bbbb

bb

pb

b = XaWe wish to use:

*DC term not included

vector size is 2mm: number of harmonicsn: number of ports

(real vectors)

size:2mn

size:2mn

Matrix Formulation*

25

(11) (11) (12) (12) (1 ) (1 )

(11) (11) (12) (12)

(21) (21) (22) (22)

(21) (21) (22) (22)

m mpqrr pqri pqrr pqri pqrr pqri

pqir pqii pqir pqii

pqrr pqri pqrr pqri

pqir pqii pqir pqii

X X X X X XX X X XX X X XX X X X

pqX

( 1) ( 1) ( ) ( )m m mm mmpqir pqii pqir pqiiX X X X

11 12 1n

21 22

pq

n1 nn

X X XX X

X =X

X X

*DC term not included

matrix size is 2mn � 2mnm: number of harmonicsn: number of ports

(real matrix)

size: 2m � 2m

Matrix Formulation*

26

(11) (11) (12) (12) (11) (11) (12) (12)11 11 11 11 12 12 12 12(11) (11) (12) (12) (11) (11) (12) (12)11 11 11 11 12 12 12 12(21) (21) (22) (22) (21) (21)11 11 11 11 12 12

rr ri rr ri rr ri rr ri

ir ii ir ii ir ii ir ii

rr ri rr ri rr ri

X X X X X X X XX X X X X X X XX X X X X X

X

(21) (21)12 12

(21) (21) (22) (22) (21) (21) (22) (22)11 11 11 11 12 12 12 12(11) (11) (12) (12) (11) (11) (12) (12)21 21 21 21 22 22 22 22(11) (11) (12) (12)21 21 21 21 2

rr ri

ir ii ir ii ir ii ir ii

rr ri rr ri rr ri rr ri

ir ii ir ii

X XX X X X X X X XX X X X X X X XX X X X X (11) (11) (12) (12)

2 22 22 22(21) (21) (22) (22) (21) (21) (22) (22)21 21 21 21 22 22 22 22(21) (21) (22) (22) (21) (21) (22) (22)21 21 21 21 22 22 22 22

ir ii ir ii

rr ri rr ri rr ri rr ri

ir ii ir ii ir ii ir ii

X X XX X X X X X X XX X X X X X X X

For instance, X(12)21ri is the contribution to the real part

of the 1st harmonic of the wave scattered at port 2 due to the imaginary part of the 2nd harmonic of the wave incident port in port 1. *DC term not included

(real matrix)

(2 harmonics)X Matrix for 2-Port System*

27

LinearImpedance

PolyharmonicImpedance

NonlinearImpedance

- Time invariant- Linear- Scalar

- Time invariant- Linear- Matrix

- Time variant- Nonlinear- Function

[ ( )] [ ( )][ ( )]V f Z f I f ( ) ( ( ))V t Z I tV ZI

FD & TD FD only

Model assumes that nonlinear effects are mild and are captured via harmonic superposition.

Polyharmonic Impedance

28

(1) (11) (12) (13) (14) (1)

(2) (21) (22) (23) (24) (2)

(3) (31) (32) (33) (34) (3)

(4) (41) (42) (43) (44) (4)

V Z Z Z Z IV Z Z Z Z IV Z Z Z Z IV Z Z Z Z I

Polyharmonic Impedance4-harmonic system

(1) (2) (3) (4)( ) ( ) ( ) ( ) ( )v t v t v t v t v t

(1) (2) (3) (4)( ) ( ) ( ) ( ) ( )i t i t i t i t i t

in frequency domain:

in time domain:

29

Polyharmonic Impedance

-1oZ = 1 + X 1- X Z

V = ZI

ZV

I

oZ : Reference impedance matrix

: Polyharmonic impedance matrix

: Voltage vector

: Current vector Describes interactions between harmonic components of voltage and current.

30

ga = Dv +Γb

-1 ga = 1 -ΓX Dv

b = XaScattered waves

Termination equations

Wave Solution

Voltage Solution

v = 1 + X a

Network Formulation

31

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-100

-80

-60

-40

-20

0

20

40

60

80

100

time(ns)

Volts

Time-Domain Response

VinVout

X Parameter

ADS

cubic term

Steady-State Simulations

32

Generate X parameters for composite systemPower level: 20 dBm, frequency: 1 GHzConstruct X matrixCombine with terminations for simulation

CMOS Driver/Receiver Channel

33

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8

time(ns)

Vol

ts

DC+Fundamental

VinVout

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8

time(ns)

Vol

ts

3 Harmonics

VinVout

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8

time(ns)

Vol

ts

8 Harmonics

VinVout

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8

time(ns)

Vol

ts

12 Harmonics

VinVout

CMOS Driver/Receiver - Harmonics

34

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8

time(ns)

Volts

Time-Domain Response

VinVout

25.2 25.4 25.6 25.8 26.0 26.2 26.4 26.6 26.8 27.0 27.2 27.4 27.6 27.8 28.0 28.2 28.4 28.6 28.8 29.0 29.2 29.4 29.6 29.825.0 30.0

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-7

7

time, nsec

Vin

, VV

out,

V

X Parameter

ADS

Validation

35

• ADS model of Tx (non‐linear) + backplane channel (linear)• Rx is passive termination• Uses a typical BSIM3 model of a 0.25um 2.5V CMOS process, provided in 

ADS– Note: modified nfet and pfet to remove all parasitic caps, in order to run at 

higher speed.

• System Block Diagram:

36

Tx

Main branch

FIR tap 1Channel

Vsrc

Vnear Vfar

Passive termination

Equalized Channel

Impulse Response, BR=5Gbps, tr=20ps

• Unequalized impulse response– Reveals 1‐tap FIR at Tx

will cancel most of ISI (m7)

• Equalized impulse response– FIR tap coefficient set 

to ‐1/3 (ratio of m6 and m7)

– DC shift due to equalizer structure

37

Channel: 40-inch FR4, Z0=50Ohm; terminated with ZL=50 Ohm and Ci=2pF

Channel Analysis

Delay of 1UI = 200ps

• Input signal Vsrc expected: – Single‐ended 2.5V NRZ, 5Gbps, tr=20ps

• FIR filter: modified single‐ended push‐pull– Output signal obtained by voltage dividers– Resistor sizing sets tap coefficients and DC levels

38

Vsrc

R1

R2

Vnear

Main branch

FIR tap 1

[1] Heidar et al., “Comparison of output drivers for high-speed serial links”, ICM 2007.

Transmitter Structure

Transmitter Structure

39

1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

2

Time (ns)

Vol

ts

VinVout

1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

2

Time (ns)

Vol

ts

VinVout

Transient ResponseUnequalized Equalized

X Parameter

ADS

40

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10-10

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6In-phase Signal

Time (s)

Am

plitu

de (A

U)

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10-10

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65In-phase Signal

Time (s)

Am

plitu

de (A

U)

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10-10

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6In-phase Signal

Time (s)

Am

plitu

de (A

U)

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10-10

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65In-phase Signal

Time (s)

Am

plitu

de (A

U)

14 dBm

12 dBm

Unequalized Equalized

Far-End Eye Diagrams

41

Challenges AheadStandardization from different levels of

approximationDefine protocols for X-parameter exchange

Conclusions

X Parameters represent a powerful format for the exchange of nonlinear behavioral models for use in the analysis and design of high-speed links

42

[1] J Verspecht and D. E. Root, ʺPolyharmonic Distortion Modeling,ʺ IEEE Magazine, June 2006, pp. 44‐57.

[2] D.E. Root, J. Verspecht, D. Sharrit, J. Wood, and A.Cognata,“Broad‐band poly‐harmonic distortion (PHD) behavioral models from fast automated simulations and large‐signal vectorial network measurements,”IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp. 3656–3664, Nov. 2005.

[3] “Agilent Nonlinear Vector Network Analyzer (NVNA),” Agilent Technologies, Inc., March 2009.

References

43

The authors thank Agilent Technologies Inc., for encouraging this work and providing the ADS X-parameter generation platform, especially Loren Betts, Steve Fulwider and Bill Wallace for fruitful discussions, insightful comments and helpful suggestions.

X‐parameters is a registered trademark of Agilent Technologies, Inc.

Acknowledgement

44