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