Quality of S-parameter models
Asian IBIS Summit, Yokohama, November 18, 2011
Yuriy Shlepnev
shlepnev@simberian.com

Copyright © 2011 by Simberian Inc. Reuse by written permission only. All rights reserved.

Outline Introduction
 S-parameters in frequency and time domains
 Constrains on S-parameters in frequency domain
 Quality metrics for reciprocity, passivity, causality
 Rational approximation and final quality metric Conclusion
 Contacts and resources© 2011 Simberian Inc.

2S-parameter models
 S-parameter models are becoming ubiquitous in designof multi-gigabit interconnects
Connectors, cables, PCBs, packages, backplanes, … ,any LTIsystem in general can be characterized with S-parameters fromDC to daylight Electromagnetic analysis or measurements are used tobuild S-parameter Touchstone models
 Very often such models have quality issues:
Reciprocity violations
 Passivity and causality violations
 Common sense violations And produce different time-domain and even frequency-

domain responses in different solvers!
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System response computation requires frequencycontinuous S-parameters from DC to infinity
S ( iω =)∞N ×N− iωtSt⋅e⋅dt,
Siω∈C()()∫−∞FDa ( iω )stimulusb=( iω ) S ( iω ) ⋅ a ( iω )S ( iω )scattering matrix

system response – frequency domain (FD)

Fourier Transforms
TDstimulus

impulse response matrix

S (t )a (t )

system response – time domain (TD)b (=t)∞∫ S ( t − τ ) ⋅ a (τ ) ⋅ dτ

−∞

∞

1iωt
N ×N,=S (t )
Siω⋅e
⋅dω
St∈R()()∫2π −∞

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Possible approximations for discrete models
 Discrete Fourier Transform (DFT) and convolution

Slow and may require interpolation and extrapolation of
tabulated S-parameters (uncontrollable error)

 Approximate discrete S-parameters with rational

functions (RMS error)
Accuracy is under control over the defined frequency band Frequency-continuous causal functions defined from DC to infinity withanalytical impulse response Fast recursive convolution algorithm to compute TD response Results consistent in time and frequency domains Not all Touchstone models are suitable for either

approach

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Common S-parameter model defects
 Model bandwidth deficiency

S-parameter models are band-limited due to limited capabilities ofsolvers and measurement equipment
 Model should include DC point or allow extrapolation, and highfrequencies defined by the signal spectrum Model discretenessTouchstone models are matrix elements at a set of frequencies
 Interpolation or approximation of tabulated matrix elements may
be necessary both for time and frequency domain analyses
 Model distortions due toMeasurement or simulation artifacts
 Passivity violations and local “enforcements”
 Causality violations and “enforcements”
 Human mistakes of model developers and users
 How to estimate and rate quality of the models?
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Causality of LTI system (TD & FD)
The system is causal if and only if all elements of the time-domain
0 at t < 0
impulse response matrix are Si=, j (t )
δ j ( 0)Si , j ( t )delayed causality (for interconnects):t
Si , j ( t ) =0 at t < Ti , j , Ti , j > 0Ti , jThis lead to Kramers-Kronig relations in frequency-domain∞+∞
S ( iω ' ) ω −ε
1'S ( iω=PV ∫⋅ d ω , PV= limε →0  ∫ + ∫ )'iπ−ωω
ω +ε −∞ −∞Sr (ω=)

1π∞

PV ∫−∞Si ( ω ' )ω −ω'⋅ d ω ' , Si (ω=)−1π∞PV ∫−∞

Kramers, H.A., Nature, v 117, 1926 p. 775..
Kronig, R. de L., J. Opt. Soc. Am. N12, 1926, p 547.

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S r (ω ' )ω −ω'⋅ dω 'derivation=S ( t ) sign ( t ) ⋅ S ( t ) ,sign ( t ) = −1, t < 01, t > 0=S ( iω ) F=
{S ( t )}1=
F {sign ( t )} ∗ F {S ( t )}2π2
F {sign ( t )} =iω

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Causality estimation - difficult way
 Kramers-Kronig relations cannot be directly used toverify causality for the frequency-domain response
known over the limited bandwidth at some points
 Causality boundaries can be introduced to estimate
causality of the tabulated and band-limited data sets
Milton, G.W., Eyre, D.J. and Mantese, J.V, Finite Frequency Range
Kramers Kronig Relations: Bounds on the Dispersion, Phys. Rev. Lett.
79, 1997, p. 3062-3064
 Triverio, P. Grivet-Talocia S., Robust Causality Characterization via
Generalized Dispersion Relations, IEEE Trans. on Adv. Packaging, N 3,2008, p. 579-593.
Even if test passes – a lot of uncertainties due
to band limitedness

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Causality estimation - easy way

“Heuristic” causality measure based on the observation that polar plot of acausal system rotates mostly clockwise (suggested by V. Dmitriev-Zdorov)

Plot of Re(S[i,j]) as function ofIm(S[i,j]), or polar plot
End frequency

ReCausality measure (CM) can becomputed as the ratio ofclockwise rotation measure tototal rotation measure in %.

Start frequency
Im

Rotation in complex plane ismostly clockwise around local
centers
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If this value is below 80%, theparameters are reported assuspect for possible violation ofcausality.
Algorithm is good for numericalmodels (to find under-sampling),
but no so good for measured data
due to noise!
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Stability and passivity in time-domainThe system is stable if output is bounded for all bounded inputsa ( t ) < K ⇒ b ( t ) < M , ∀t(BIBO)
 A multiport network is passive if energy absorbed by multiport
E=(t )t∫ a (τ ) ⋅ a (τ ) − b (τ ) ⋅ b (τ ) ⋅ dτ ≥ 0, ∀ttt(does not generate energy)−∞for all possible incident and reflected wavesIf the system is passive according to the above definition, it is alsotcausalta ( t )= 0, ∀t < t0 ⇒ ∫ b (τ ) ⋅ b (τ )  ⋅ dτ ≤ 0 ⇒ b ( t )= 0, ∀t < t0−∞

Thus, we need to check only the passivity of interconnect system!

P. Triverio S. Grivet-Talocia, M.S. Nakhla, F.G. Canavero, R. Achar, Stability, Causality, and Passivity
in Electrical Interconnect Models, IEEE Trans. on Advanced Packaging, vol. 30. 2007, N4, p. 795-808.
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Passivity in frequency domainPower transmitted to multiport is a difference ofpower transmitted by incident andN
reflected waves: Pin= ∑ | an |2 − | bn |2=  a * ⋅ a − b * ⋅ b 

n =1*eigenvals U − S * ⋅ S  ≥ 0**-+or Pin = a ⋅ a − a ⋅ S S ⋅ a = a ⋅ U − S S  ⋅ a
 Quadratic form is non-negative if eigenvaluesof the matrix are non-negative:*Z 01 a1b1 V1 + I Port 11Z 02 a2-+

*eigenvals  S * ⋅ S  ≤ 1I1b2 V2.
.
.

Z0 N aI2+-bN VNPort 2[S ]INN-+I2+-IN

Port N(U is unit matrix)

Sufficient condition only if verified from DC to infinity(impossible for discrete Touchstone models)

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Good Touchstone models of interconnects
 Must have sufficient bandwidth matching signalspectrum
 Must be appropriately sampled to resolve all resonances Must be reciprocal (linear reciprocal materials used inPCBs)t=
Si , j S=Sj ,i or S

 Must be passive (do not generate energy)

Pin = a * ⋅ U − S * S  ⋅ a ≥ 0eigenvals  S * ⋅ S  ≤ 1 from DC to infinity!

 Have causal step or impulse response (response onlyafter the excitation)

Si , j =( t ) 0, t < Tij

Si , j ( t )Ti , j
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Quality metrics (0-100%) to define goodness
First introduced at IBIS forum at DesignCon 2010
PQM

Passivity Quality Measure:
N total 100  −max NPWn  ,0  % total ∑Nn=1  total 

PM n − 1.00001
0 if PM n < 1.00001; otherwise PWn =PWn =
0.1

=
PM n

should be >99%
RQM

Reciprocity Quality Measure:N total
 100  −max NRW∑n  ,0  % totalNn=1 total
should be >99%
max eigenvals ( S * ( f n ) ⋅ S ( f n ) ) RM n − 10−6RWn =0 if RM n < 10 ; otherwise RWn =0.1=
RM n−61∑ S i , j ( f n ) − S j ,i ( f n )N s i, j

Causality Quality Measure: Minimal ratio of clockwise rotation
measure to total rotation measure in % (should be >80% fornumerical models)
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Preliminary quality estimation metrics
 Preliminary Touchstone model quality can be estimatedwith Passivity, Reciprocity and Causality quality metrics(PQM, RQM, CQM)
Metric/Model Icon
Passivity
Reciprocity
Causality- good[100, 99.9][100, 99.9][100, 80]

Color code
Green – goodBlue – acceptable
Yellow – inconclusive
Red - bad

10/23/2011- acceptable(99.9, 99](99.9, 99](80, 50]

Passivity (PQM)[99.9, 100]
[99, 99.9)[80, 99)[0, 80)- inconclusive- bad(99, 80](80, 0](99, 80](80, 0](50, 0]-----Reciprocity (RQM)[99.9, 100]
[99, 99.9)[80, 99)[0, 80)Causality (CQM)[80, 100][50, 80)[20, 50)[0, 20)© 2011 Simberian Inc.

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Example of preliminary quality estimation
Small passivity & reciprocity violations in most of the models
Low causality in some measured data due to noise at high frequencies

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Rational approximation of S-parameters as
the frequency-continuous modelNij
 rij ,nrij*,n   − s⋅Tij Continuous functions⇒ Si , j ( iω ) =+ dij + ∑  ⋅e* ωωipip−−of frequency definedn =1 
ij , nij , n  
a=k 0 k≠ jbib=
S ⋅ a , Si , j =aj

from DC to infinitys= iω , dij − values at ∞, N ij − number of poles,
rij ,n − residues, pij ,n − poles (real or complex), Tij − optional delay Pulse response is analytical, real and delay-causal:
Si , j =( t ) 0, t < Tij

Nij()()Si , j (=t ) dijδ ( t − Tij ) + ∑  rij ,n ⋅ exp pij ,n ⋅ ( t − Tij ) + rij*,n ⋅ exp pij* ,n ⋅ ( t − Tij )  , t ≥ Tij
n =1()

 Stable Re pij , n < 0 Passive if

eigenvals  S (ω ) ⋅ S * (ω )  ≤ 1 ∀ω , from 0 to ∞

 Reciprocal if

S i , j ( ω ) = S j ,i ( ω )10/23/2011© 2011 Simberian Inc.

May requireenforcement

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Rational approximation can be used to Compute time-domain response of a channel with a fastrecursive convolution algorithm (exact solution for PWLsignals)
 Improve quality of tabulated Touchstone models
Fix minor passivity and causality violations
 Interpolate and extrapolate with guarantied passivity Produce broad-band SPICE macro-models

Smaller model size, stable analysis
 Consistent frequency and time domain analyses in any solver Measure the original model quality

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Final quality estimation
 Accuracy of discrete S-parameters approximation withfrequency-continuous macro-model, passive from DC toinfinity
 1 N2=−ωRMSE max SnS
∑ ij ( ) ij ( n ) i, j N n =1



 Can be used to estimate quality of the original dataQ=
100 ⋅ max (1 − RMSE ,0 ) %Model Icon/Quality- good- acceptable- inconclusive- bad- uncertain
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Quality Metric[99, 100][90, 99)[50, 90)[0, 50)[0,100], not passive or not reciprocal

RMSE[0, 0.01](0.01, 0.1]
(0.1, 0.5]
> 0.5

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Example of final quality estimation
All rational macro-models are passive, reciprocal, causal andhave acceptable accuracy (acceptable quality of original models)

Improved tabulatedmodels (re-sampled)

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Conclusion: How to avoid problems with
S-parameter models?
Use reciprocity, passivity and causality metrics for preliminary
analysis
RQM and PQM metrics should be > 99% (acceptable level)
 CQM should be > 80% for all causal numerical models


Use the rational model accuracy as the final quality measureDiscard the model with low RQM, PQM and QM metrics!


The main reason is we do not know what it should beModels that pass the quality metrics may still be not usable or
mishandled by a system simulator

QM should be > 90% (acceptable level)

Due to band-limitedness, discreteness and brut force model fixing

Use rational or BB SPICE macro-models instead of Touchstone
models for consistent time and frequency domain analyses

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Contact and resources Yuriy Shlepnev, Simberian Inc.

shlepnev@simberian.com
Tel: 206-409-2368
 To learn more on S-parameters quality see the following

presentations (also available on request):

Y. Shlepnev, Quality Metrics for S-parameter Models, DesignCon 2010 IBIS Summit, SantaClara, February 4, 2010
H. Barnes, Y. Shlepnev, J. Nadolny, T. Dagostino, S. McMorrow, Quality of High Frequency
Measurements: Practical Examples, Theoretical Foundations, and Successful Techniques thatWork Past the 40GHz Realm, DesignCon 2010, Santa Clara, February 1, 2010.
E. Bogatin, B. Kirk, M. Jenkins, Y. Shlepnev, M. Steinberger, How to Avoid Butchering SParameters, DesignCon 2011
Y. Shlepnev, Reflections on S-parameter quality, DesignCon 2011 IBIS Summit, Santa Clara,February 3, 2011

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