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DesignCon 2014

How Design of Experiments

Saved my CEI VSR 28G Design

Richard Allred, SiSoft

[rallred@sisoft.com]

Barry Katz, SiSoft

Ishwar Hosagrahar, Inphi Inc.

Chao Xu, Inphi Inc.

Wiley Gillmor, SiSoft

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Abstract

To determine the overall design margin, a CEI 28G VSR/100 Gigabit Ethernet design

required the analysis of 5 million combinations of channel variation, transceiver process

and equalization settings. Brute force simulation would have required 278 days, clearly

outside the available schedule.

Instead, we used Design of Experiments (DOE) and Response Surface Modeling (RSM)

to reduce the number of simulations by a factor of 19,000 and yet produce just as

meaningful results.

This paper will demonstrate the application of DOE and RSM to a CEI 28G VSR design.

We will show the process of creating a DOE, fitting the data to models, determining the

goodness and reliability of the fit and then using the model to perform “what if” analysis,

optimize design factors and quantify the impact of manufacturing variation.

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Authors Biography

Richard Allred is a Senior Member of Technical Staff at SiSoft. Previously, Richard

worked at Inphi where he was responsible for Inphi’s 100G Ethernet PHY (28G per lane)

front plane interface. In the course of that work, Richard contributed to IEEE 802.3 and

OIF-28G-VSR standards discussions on next generation Ethernet. Before that he worked

at Intel, contributing to signal integrity methodology and tool development for

GDDR5/DDR3. He used Design of Experiments and Response Surface Modeling in the

course of this work to predict link performance across high volume manufacturing.

Richard received his MSEE from University of Utah, and has 4 publications.

Barry Katz, President and CTO for SiSoft, founded SiSoft in 1995. As CTO, Barry is

responsible for leading the definition and development of SiSoft’s products. He has

devoted much of his efforts at SiSoft to delivering a comprehensive design methodology,

software tools, and expert consulting to solve the problems faced by designers of leading

edge high-speed systems. He was the founding chairman of the IBIS Quality committee.

Barry received an MSEE degree from Carnegie Mellon and a BSEE degree from the

University of Florida.

Ishwar Hosagrahar is a Senior Staff Engineer at Inphi, with over 17yrs of industry

experience working with Networking & Communication Circuits and Systems. Prior to

Inphi, he worked at Texas Instruments on products ranging from 10/100Mbps Ethernet

PHYs to 15+Gbps SerDes transceivers. Before that, he worked at ArcusTech (later

acquired by Cypress Semiconductor) designing Ethernet and Telecom switch ICs. He

holds a Master's degree in VLSI/Circuit design from University of Texas. In his spare

time, he enjoys flying planes (and anything aviation-related), plays various musical

instruments (albeit somewhat poorly) and dabbles in hi-fi systems.

Chao Xu is a Sr. Director of Platform Engineering at Inphi. He is specialized in

integrated circuits design and architecture related to Computing Architecture, Memory

Architecture, High Speed Digital Communication and Digital Signal Processing in Server

and Data Communication areas. He has extensive experience in system signal integrity in

high speed channel analysis and implementation, high speed mixed signal integrated

circuit designs such as SERDES, PLLs. RF Transceivers, Optical transceivers etc. Chao

received his Ph.D. degree in Electrical Engineering from University of Pennsylvania. He

has more than 10 issued US patents.

Wiley Gillmor is a Principal Engineer engaged in software development at Signal

Integrity Software, Inc. His career in EDA has spanned over 35 years, focusing on tools

for physical design and engineering. Prior to that he had a brief academic career teaching

Mathematical Logic and Computer Science.

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Introduction

With a unit interval of only 35.7 ps, the CEI VSR 28G/100 Gigabit Ethernet link presents

a challenging system level design. In this paper we describe the system design process

which resulted in the industry’s first 100 Gigabit Ethernet CMOS PHY with Inphi’s

GearBox and CDR chips. The jitter budget is tight and with all the variation possible in

the system there are more than 5 million system conditions to check and verify

performance at. Additionally, to ensure that the system is not over-designed or under-

performing, the manufacturing variation impact on performance and manufacturing yield

needs to be estimated. Achieving and optimizing all of these objectives simultaneously is

indeed a serious challenge.

The next generation core routers and data centers require more bandwidth, faster speeds

and lower power. This has forced the industry to implement line speeds as fast as 28

Gbps. An example topology of such interfaces is given below in Figure 1 and shows the

chip to module application, where a pluggable optical transceiver CFP2 module is

connected to a line card host IC. Don’t be fooled by the seemingly clean channel, there

are plenty of impairments here and with a with a bit error ratio (BER) allowance of only

one error in a quadrillion bits, this is a world class challenge.

Figure 1: 28G VSR host to module channel diagram.

The CMOS design allowed Inphi to reduce the power envelope to one third the power

and half the area of competing SiGe and FPGA solutions.

One of the many challenge in the design process was to ensure that the transmitter and

receiver were able to operate in a wide variety of channels, including some that exceeded

the 28G VSR 10 dB host to module channel. Here Design of Experiments (DOE) and

Response Surface Modeling (RSM) was used to determine the maximum allowable trace

lengths, the best layer for PCB routing, the optimum via anti-pad size and the

performance degradation due to the presence of manufacturing variance from IC process,

voltage and temperature (PVT); package impedance; printed circuit board (PCB)

impedance; and via stub length variations. Each of these design objectives will be

addressed and we will illustrate how the DOE/RSM approach provides an informed path

to answering them.

The trouble was, although each simulation of a system configuration took only 4.8

seconds to complete, with 5 million conditions to check it would require 278 days of

compute time to complete! A large compute farm would help but could be easily

overwhelmed with the computation and storage of the results if additional factors were

added to the exploration space or if the simulations required more time to complete. The

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link simulation approached used was a fast analysis approach to estimate the 1e-12 BER

for the link configuration. If a bit by bit time domain analysis was used to estimate

performance then the simulation time could easily approach five minutes each. With this

computational load, it would only require 47.5 years of compute time for the 5 million

cases. Brute force analysis wasn’t going to be a very effective tool in this situation. We

need to find another way.

Ideally, it would be wonderful if there was a magic equation which, given the input

factors such as trace length, impedance and process corner, could reveal exactly what the

resulting system performance would be. This magic equation would enable a multitude of

analyses like optimization, virtual “what if” analysis and the ability to position the design

to minimize the impact of manufacturing variance.

Well of course that isn’t exactly possible, but by using DOE with RSM we can approach

this ideal. DOE is used to sample the factor space and RSM is utilized to create an

equation (or model) which best fits the data. After verifying the correctness of the model

we can utilize it to do all the things mentioned above, to optimize, perform “what if”

analysis and to minimize the impact of manufacturing variation. The DOE/RSM flow is

visualized in Figure 2 below which was made to emphasize the iterative nature of the

analysis. Additionally, each phase of the analysis has certain assumptions which will

need to be revisited and revised before a satisfactory result can be obtained.

Figure 2: Design of Experiments (DOE) and Response Surface Modeling (RSM) methodology flow.

Analysis Approach

The most important step of any type of analysis is to determine the objectives of the

work. Albert Einstein allegedly stated that “If I had only one hour to save the world, I

would spend fifty-five minutes thinking about the problem and only five minutes thinking

about solutions”. Similarly, it is absolutely essential that the questions to be addressed

and the level of accuracy of the answers be clearly defined before starting the actual work

utilizing the DOE/RSM methodology. Clear analysis goals will put you in the best

position to achieve your aim.

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The DOE and RSM flow diagram emphasizes the iterative nature of the analysis as it

allows for the refining of the assumptions at each step of the process. It can also be seen

that without clear analysis goals and exit criteria, such an approach can result in loss of

weekends and never ending reviews.

A statistical mindset to signal integrity, one in which decisions are made in the presence

of uncertainty, is uncomfortable for some. Rather than relying on exhaustively certain

analysis, the DOE/RSM analysis will provide answers couched between confidence

intervals which indicate the accuracy of the results. The only difference between an

exhaustively certain and statistical mindset is that in the former, any uncertainty is

pretended away and in the latter, the uncertainty is quantified, scrutinized and

communicated. There will always be uncertainty, whether we wish it or not and the best

approach is to understand it, reduce it and embrace it. Get comfortable with uncertainty.

Please note that the JMP® statistical discovery software was utilized for the DOE creation

and model fit analysis. Many of the figures in this document are from or derived from

JMP® reports. The system link simulations were performed with SiSoft’s Quantum

Channel Designer ® (QCD).

Assumptions

In addition to a mindset change, the DOE/RSM techniques require an in-depth

knowledge of the related statistical concepts applied, which can be a challenge. The

approach we will use in this paper is to first discuss some of the key concepts and the

assumptions which go into them and then provide solid application examples of the

methodology to the VSR 28G interface design.

In a general sense, to model an object is to utilize a simplified description of some aspect

of interest which allows for an exploration of the object’s characteristics. A physical

example would be the use of a model airplane in a wind tunnel to study its aerodynamic

properties to aid in the design of a full size airplane. For us, the object to be modeled is a

bit more ambiguous but typically is a response of the system performance such as eye

height or width across the study factor ranges.

The modeling objective is to characterize the true response by taking samples of the

factor space (as provided by the DOE table) and then fit a polynomial equation to the

data. In this sense, the model is actually the equation which best fits the data. A visual

example of such a model is shown below in Figure 3 where the eye height of a link is

given versus trace length and impedance.

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Figure 3: Example eye height response surface versus trace length and impedance.

Note that the model is merely an approximation of the actual response and only

represents reality as far as it is accurate. Also, as can be seen from the curvature of the

surface, this model is a two dimensional parabola which best fits the data. The model

form and the coefficient estimates obtained from an ordinary least squares fit is given

below in Equation (1):

𝒚 = 𝜷𝟎 + 𝜷𝟏𝒙𝟏 + 𝜷𝟐𝒙𝟐 + 𝜷𝟑𝒙𝟏𝒙𝟐 + 𝜷𝟒𝒙𝟏𝟐 + 𝛃𝟓𝒙𝟐

𝟐

𝜷𝟎 = −𝟎. 𝟐𝟒𝟕𝟖𝟔𝟐 𝜷𝟏 = 𝟎. 𝟐𝟒𝟑𝟔𝟕 𝜷𝟐 = 𝟎. 𝟎𝟐𝟏𝟎𝟒𝟗𝟐𝜷𝟑 = 𝟎. 𝟎𝟎𝟎𝟏𝟔𝟓𝟓𝟐 𝜷𝟒 = −𝟎. 𝟎𝟎𝟎𝟏𝟐𝟕𝟓 𝜷𝟓 = −𝟎. 𝟎𝟎𝟓𝟗𝟗𝟑𝟖

(1)

Although this surface represents the best fit of the model to the data, it does not address

the question: is this the best model to represent the ‘true response’? In this instance,

adding higher polynomial terms for the length factor would likely capture better the

resonant behavior of the trace. While finding the absolute best model is likely to be an

elusive goal we can certainly approach it with incremental improvements to the model

form. A ‘too simple’ model will result in the smoothing out of important response

characteristics and a ‘too complex’ model can result in over emphasizing certain response

features at the expense of other more important response characteristics.

One might think that to find the best model form one would simply apply a very large

range of polynomial terms to the model fit to see which terms are significant. The

difficulty with this approach is that the model form available for fitting is limited by the

sampling of the response. If there are only two data samples along a factor dimension

then at most a straight line can be fit to the data. If there are three data points then at

most a quadratic line can be fit and further if there are at most four data points then a 3rd

order polynomial can be fit. Thus at some point, given a fixed sampling set, there is a

limit of what model is available to apply to the fit. These limitations lead to the idea that

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the sampling of the factor space must also take into account the model form so that the

model fit can accurately estimate the true response.

Conceptually, with continuous factors one could iterate until the ideal sampling and

model form is obtained to achieve a high quality model fit. In practice though, segments

of the interconnect are often represented by blocks which may only have a discrete

number of levels, such as would be the case from a family of connectors which are

characterized by S-Parameter models. In these situations, the resolutions of the sampling

are fixed and higher order system response characteristics are aliased with lower order

system response characteristics, thus limiting the analysis.

The final aspect of the model assumption is the idea that a polynomial can adequately

describe the response surface. There are other more mathematically sophisticated

models, such as Gaussian Process modeling which have some surprising characteristics

(such as zero residual and spatially cognizant interpolation) but will not be discussed

here. The authors have found that the large majority of the signal integrity applications

of the DOE/RSM methodology are adequately described by polynomial models.

The process of fitting the model to the data is typically performed by Ordinary Least

Squares (OLS) regression analysis. This estimation method assumes that the input

factors are uncorrelated, that the fit error variance does not vary across factors or factor

levels and that the residual error is normally distributed. These assumptions can be

relaxed if needed but require more sophisticated least square methods such as generalized

linear models (allows for other residual error distributions beyond the normal

distribution) and general least squares (allow for correlated factors and non-uniform error

variances). In the analysis here we will make an extensive investigation of the residual to

ensure that the model is adequate and that it meets the OLS assumptions.

Application of methodology to 28G VSR System

Analysis

The whole objective of the study is to answer design questions and quantify

manufacturing variance for the 28G VSR link design. The design questions are, which

layer of the PCB is preferable for the high speed routing, what is the best PCB via anti-

pad size and what is the max PCB trace length that can still pass with adequate system

performance? These questions should be answered in the presence of manufacturing

variation, which will be quantified as well.

The objectives will be obtained by carefully sampling the factor space with the Design of

Experiments approach, evaluating the system performance response at these DOE

conditions, fitting a response surface model to the data to produce a multi-dimensional

equation which can then be used to study the total factor space. Even though this

analysis description hints of a simple progression, in practice it is quite iterative as the

assumptions at each stage are refined to obtain a satisfactory model fit with sufficient

accuracy. It is desired that the uncertainty of any eye height and width predictions be less

than +/- 30 mV and +/- 0.5 ps. Achieving the desired level of accuracy on the first pass

rarely happens (it if does then it should raise your suspicions) but requires subtle

modifications to the model assumption, factor ranges, and DOE creation.

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Table 1 below lists the nine factors which define the space to be explored. The factors

which directly influence design decisions are called design factors and the factors which

in production are not controllable are called manufacturing factors.

Parameter Name Factor Type Min Typ Max

Tx PVT Corner Manufacturing SS TT FF

Tx PKG Manufacturing 90 Ohm 100 Ohm 110 Ohm

Line Card PCB Via Anti-Pad Size

Design 32 mil 36 mil 40 mil

Line Card PCB Via Stub length

Manufacturing 2 mil 10 mil 18 mil

Line Card PCB Routing Layer

Design 3 -- 9

Line Card length Design 1 inch 3 inch 6 inch

Line Card TL Impedance

Manufacturing 90 Ohm 100 Ohm 110 Ohm

Rx PKG Manufacturing 90 Ohm 100 Ohm 110 Ohm

Rx PVT Corner Manufacturing SS TT FF

Table 1: 28G VSR interface factor space definition.

Design the experiment

Traditionally there have been several approaches to sampling a large multi-dimensional

factor space. Some of these approaches include an exhaustive sampling, where every

single condition is evaluated; random sampling, where a number of conditions are chosen

by chance; and one factor at a time sampling, where from a nominal condition, each

factor is swept in isolation. The objective of any type of sampling is to obtain a

representation of the total factor space such that by using statistics of the sample,

inferences about the total factor space can be made. The exhaustive sampling approach is

nice but more than likely an unrealistic approach. The random sampling approach (also

called Monte-Carlo sampling) will likely be unbiased but does not guarantee coverage of

the whole space and requires many sample points to ensure that all regions of the factor

space are considered. Lastly, one factor at a time (sometimes called OFAT) sampling

will miss out on many important factor interactions.

The design of experiments sampling approach attempts to sample the space to provide

good coverage of the whole factor space while minimizing the number of runs. This is

achieved by starting with a random sampling and then modifying each sample point until

the coverage of the DOE set is adequate. The coverage of the sampling is quantified by

the sample prediction variance which can be easily calculated with some straightforward

matrix manipulations. This quantification of the prediction variance allows the

uncertainty of the sampling to be quantified and thus optimized upon. This is why some

call the DOE approach optimal design.

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For our VSR example, the nine factors are sampled with 256 runs and the prediction

variance of the sample is minimized with the D-optimal approach. D-optimal designs

sample the edges of the factor space more than the center and give more accurate model

parameter estimates than other optimality criteria. Since one of the objectives is to find

the worst case conditions of the factor space (these typically occur at an edge of the

space), accurate estimations of the edges and corners of the factor space is important.

The model assumed for the sampling was a 2nd order polynomial with 1st order

interactions between all factors.

Once the simulations are run and the model fit processed, the accuracy of the model fit

predictions is quantified by a confidence interval. A smaller confidence interval will give

more assurance than a larger one. The confidence interval size is dependent on three

things, the desired confidence level (95%, 99%, 99.9% etc.), the model fit error and the

coverage of the DOE sampling. At the DOE creation step of the analysis, knowledge of

the coverage of the DOE sampling allows insight into the relative confidence interval size

and is embodied by the prediction variance. While visualizing a 9 dimensional space is

quite a feat, we can summarize the prediction variance across the whole space with a

fraction of design space plot as shown in Figure 4 below.

Figure 4: Fraction of design space plot which shows the DOE prediction variance over the fraction of the space.

This plot shows that the relative prediction variance for 50% of the factor space is less

than 0.19. While the relative nature of the metric does not lend itself to absolute

guidelines, it allow for the comparison of competing designs. Thus it is recommended

that a few designs be generated and then compared to select the best one.

An ideal design, sometimes known as an orthogonal design, is one where the parameter

estimates are able to be calculated independently. This is only achievable for select

designs and in most situations (given the number of runs and the model form) an

orthogonal design is not possible. What is possible though, is for the design to approach

the orthogonal characteristics. When a design is not orthogonal then two or more

parameter effects are slightly correlated with each other and to that degree

undistinguishable. The degree of correlation can be quantified for all 54 terms of our

model in the color map on correlation plot in Figure 5 below. The names of each of the

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model terms are given for each column of the plot and are the same for each

corresponding row. The color range in the plot goes from blue (un-correlation) to red

(totally correlated). The red diagonal of the plot shows that each model term is perfectly

correlated with itself as expected but most importantly there are no reddish off-diagonal

terms which would be an indication of a poor design.

Figure 5: Color map on correlation plot for all of the terms in the response surface design. The correlation between

two off-diagonal terms is ideally zero and is indicated by blue .

In order to put the upcoming model fit in the best possible position, the design of

experiments sampling has found a sample set which adequately covers the factor space

and allows for near independent estimation of the parameter effects.

Evaluate System Response

The DOE sampling conditions are brought into the EDA link simulation environment for

the 28G VSR topology, simulated and link performance metrics calculated. The metrics

which will be considered here are eye height and eye width at a BER of 1e-12. It is

essential to check the simulation result waveforms for consistency and accuracy. It is

recommended to check the outliers in the performance to ensure that they embody

realistic results. It will be assumed during the model fit that each condition represents the

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actual response thus any discrepancies will propagate errors into the model fit and will

result in poor or even wrong analysis conclusions. Below in Figure 6 is an example of

how to visualize the results in SiSoft’s Quantum Channel Designer ® (QCD).

Figure 6: The simulation results must be carefully evaluated to ensure that all results are reasonable before

proceeding with the model fit.

Response Surface Model Fit

All of the precautions taken up to this point have been done to obtain a good model fit.

Once the fit is complete and its quality measured it will be determined whether those

precautions were sufficient or if more iterations and refinements are necessary. The

model form which we will utilize, called a Response Surface Model, is a multi-

dimensional polynomial with interaction terms as shown in Equation (2) below. Here 𝑦

is the measured response (such as eye height), 𝑥𝑖 is one of the n=9 factors and the 𝛽’s are

the unknown model coefficients which will be estimated by the least squares method.

𝒚 = 𝜷𝟎 + ∑ 𝜷𝒊𝒙𝒊

𝒏

𝒊=𝟏

+ ∑ 𝜷𝒊𝒊𝒙𝒊𝟐

𝒏

𝒊=𝟏

+ ∑ ∑ 𝜷𝒊𝒋𝒙𝒊𝒙𝒋

𝒏

𝒋=𝒊+𝟏

𝒏−𝟏

𝒊=𝟏

(2)

The difference between the simulated eye height and the eye height as predicted by the

response equation is called the error residual. By examining the residual we can obtain

several measures of model quality and validate assumptions.

Goodness of fit

The simplest fit metric is called the coefficient of multiple determination but everyone

just calls it “R-squared” for short and is written as R2. This metric ranges from 0 for a

poor fit to 1 for a good fit. Conceptually, 100*R2 can be thought of as the percentage of

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the variation in the data that can be explained by the model. One interesting fact is that

the R2 metric will always improve if additional model terms are added whether or not

these new terms are actually significant. A modified R2 metric, called R2 adjusted, takes

into account the number of terms used in the model and penalizes for any extra

unnecessary terms. Thus a large difference between the R2 and R2 adjusted is an

indication that there are unnecessary terms in the model.

The error standard deviation of the fit can be estimated by taking the residual for each

point in the data set, squaring it, finding the mean and then taking the square root. This

RMSE metric can also be used as a quick estimate of the prediction confidence interval.

For a 95% confidence interval estimate, simply multiply RMSE by 2. If +/- this value is

larger than the needed accuracy then it will be necessary to go back and revisit earlier

assumptions such as the model form used in the DOE creation and the factor space

definition.

Figure 7: Goodness of fit summaries for eye height (left) and eye width (right).

Shown above in Figure 7, is the JMP statistical software fit summary for the eye height

and eye width fit. We see the R2 and R2adjusted are in the high 90’s and the RMSE is 22

mV for eye height and 0.45 ps for eye width.

Lastly the fit error residual needs to be examined itself to ensure that it is normally

distributed and that it does not contain any “structure”. Figure 8 below shows the

residual distribution of the eye height and width which can be seen as roughly normally

distributed.

Figure 8: Fit error residuals for eye height (left) and eye width (right). These show that the residuals are roughly

normally distributed as required.

Plots of the residual versus the response or other important factors are the best way to

search for “structure” in the residual. Structure, i.e. some systematic relation between the

residual and some explanatory variable, is evidence of a model bias and can provide clues

as to what new terms should be included in the model fit. Often such additional model

terms are not able to be immediately utilized because of insufficient sampling of the DOE

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and require a reformulation of the DOE creation so that the additional model terms can be

added without aliasing other important effects. Below in Figure 9 is the plot of the

residual versus the response. The residual should be normally distributed no matter how

it is viewed but it can be seen from the figure that the lowest predicted eye height cases

have a positive residual as circled in the figure. Further investigation showed that an

additional model term of PCB_LEN*PCB_Z*RX_CORNER improved the fit. Although

there may be more such model terms which may improve the fit, the accuracy was

sufficient for the needs of the study and the model fit was deemed good enough.

Figure 9: Residual versus the predicted eye height (left) and predicted eye width (right). Systematic structure in the

residual is an indication of model bias. Note how the lowest eye height performance cases all have positive residuals,

this observation lead to the inclusion of an additional model term which improved the fit.

While these are very good model fit results, it is interesting to think about the source of

the remaining uncertainty. The residual error can only come from two sources, random

errors and lack of fit. For deterministic signal integrity simulations there is no random

noise so the residual is due solely to lack of fit. In practice, a perfect model fit is not

achieved because the underlying phenomenon is not a perfect polynomial and also

because the true underlying phenomenon factors are garbled and only imperfectly

represented by the study factors.

Explore and Optimize the 28G VSR System

Once everything has been done to ensure a proper fit, we can explore the factor space as

represented by the model with confidence that the uncertainty is roughly understood. In

this application, visualizing a 10 dimensional space is a daunting task, fortunately there

are some tools which facilitate this type of analysis. A plot, called the prediction profiler,

shows what the response would be across each factor if all of the other factors are held

constant. This type of plot is most useful when used interactively, but much can be

gleaned from the static views used below as well.

As an example, consider the prediction profiler plot for the two factor RSM fit as shown

in Figure 10.

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Figure 10: Example prediction profiler plot of the predicted eye height response. It shows the response versus each of

the factors given all of the other factors are held constant.

This prediction profiler plot shows the eye height response across the explanatory factors,

trace length (PCB_LEN) and trace impedance (PCB_Z). Since the slope of the trace

length factor is greatest we can state that this is the most influential factor in this region

of the factor space. If there are any interactions between the factors then the effective

slope of the line could change in other areas of the factor space. Additionally, this plot

indicates that when PCB_LEN=3.5 and PCB_Z = 100 then the predicted eye height is

0.538 V with a 95% confidence interval of [0.514, 0.562] V, which is equivalent to

stating that the predicted eye height is 0.538 +/- 0.0234 V for a 95% confidence interval.

The confidence intervals are represented in the prediction profiler plot by the blue dashed

lines surrounding the solid black predicted response.

A confidence interval (sometimes abbreviated as CI) can be thought of conveying the

following information:

o a CI provides a range of plausible values for the true response with values outside

the range as relatively implausible, or

o a CI gives the precision of the estimation where the upper and lower bounds

provide the likely maximum error of estimation, although there is a possibility of

larger errors.

A confidence level of 95% roughly covers two standard deviations from the predicted

value and a confidence level of 99.5% roughly covers three standard deviations from the

predicted value.

The prediction profiler plot for the middle of the example factor space is given in Figure

11.

Figure 11: Eye height (top row) and eye width (bottom row) Prediction Profiler for the middle of the factor space.

Important factors can be identfied by the slope of the curves

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As was noted before, the trace length is the most influential factor in this region of the

space. The two rows of plots are for the eye height and eye width, respectively. If it was

desired to understand the impact of a factor in this region of the space, this plot easily

provides this information.

Because the fitted response surface is represented by a well-defined function, the space

can be searched for the worst case conditions as shown in the prediction profiler plot in

Figure 12.

Figure 12: Prediction profile plot for eye height (top row) and eye width (bottom row) at the predicted worst case

condition.

It will be noted that while the trace length is still the most influential factor in this region

of the space, the trace impedance and receiver PVT corner ($RX_CORNER) have also

become somewhat influential as compared to their influence at the middle of the factor

space.

Determining Routing Layer for High Speed Signals

The design optimization strategy used here is to first identify the worst case

manufacturing condition and then put the design factors in their best case conditions to

minimize the impact of the worst case performance.

To quantify the impact of the PCB routing layer at the worst case corner, the predicted

response is calculated for PCB_LAYER=3 and PCB_LAYER=9 and compared. As

shown in Figure 13 below (note the vertical axis has been scaled to provide a better view)

the eye height and width difference between the two PCB layers is only 0.012 mV and

0.35 ps. Also note that the 95% confidence interval for eye height is +/-24 mV and for

eye width, +/- 0.5 ps. When the predicted difference is less than the confidence interval

then it can be stated that the model is unable to confidently identify the effect as

significant and provides no actionable information. This could lead to the following

conclusions:

o the PCB routing layer is a weak predictor of the system performance and layers 3

and 9 are equivalent, or

o PCB routing layer 9 has a slight but statistically insignificant advantage over

layer 3.

If either of these conclusions is insufficient then additional information on the impact of

the two layers will be required to make a definitive decision. A more focused DOE/RSM

around this factor could be defined where some of the other insignificant factors are

removed from the study to improve the accuracy of the analysis. Additionally, other

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influences on the routing layer decision, such as cost in PCB space or cost in money,

should be considered when making any decisions.

For the VSR study, it will be concluded that PCB routing layer 9 has a slight advantage

(although statistically insignificant) over layer 3.

Figure 13: Example of how to evaluate "what if" scenarios across the PCB layer factor. The plot on the left shows the

predicted responses for when the PCB routing layer is on layer 3 and the plot on the right shows the predicted

responses for when the PCB routing layer is layer 9.

Determining the best Via Anti-pad size

A similar approach will be taken to understand the impact of the PCB via anti-pad size.

The link performance difference between an anti-pad diameter of 32 and 40 mil for eye

height is 45 mV and 0.47 ps where the eye height confidence interval is +/- 25 mV and

the eye width confidence interval is +/- 0.5 ps for a 95% confidence level. This can

visually be seen from the prediction profiler plots in Figure 14 as the dashed red

horizontal lines for the eye height plots are not contained within the dashed blue

confidence interval lines whereas the dashed red lines are contained by the dashed blue

confidence interval lines for eye width. Therefore we can conclude that an anti-pad size

of 32 mils is a better solution for eye height and is possible to be a better solution for eye

width performance but is statistically insignificant. Additional constraints, such as

increased manufacturing problems with fabricating a given anti-pad diameter should be

carefully weighed with the performance benefits before making a final decision.

For this study, an anti-pad size of 32 mils will be used in future analysis.

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Figure 14: "What If" analysis across the via anti-pad factor showing the performance difference between different via

anti-pad sizes.

One important point to clarify is that while the DOE/RSM methodology can provide a

quantification of the impact of a given factor, it cannot give any indication as to why.

The reasons why a factor is impactful in a given situation must come from subject matter

expertise and engineering judgment. If no satisfactory physical explanation is

forthcoming, then a statistical indication of importance may be the impetus for further

analysis and it can be useful to make hypotheses about competing physical explanations

of the data.

Manufacturing Variation

One approach to quantify the manufacturing variation is to assign probability distribution

functions (PDFs) to each of the factors and then randomly generate millions of cases.

Utilizing the response surface equation, the system performance can be quantified for

each of the random cases and used to give an indication of the probability of yielding a

certain system performance level. Much care must be taken to obtain accurate PDFs as

the weights of the tails can make a large difference in such analysis. Figure 15 shows the

prediction profiler plot and the PDFs assigned to each of the manufacturing factors.

Figure 15: Prediction profiler plot with distributions assigned to each of the manufacturing factors. Randomly

sampling the factor space according to these distributions will give manufacturing Yield information.

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It was desired to understand the manufacturing variation as a function of the line card

trace length. Therefore, for each of the line card lengths of 1, 2, 3, 4, 5 and 6 inches, one

million cases were randomly generated and eye height and width performance calculated

using the response surface equation. The average and standard deviation of the one

million cases for each of the lengths are given below in Figure 16. This analysis clearly

shows that for eye height the impact of manufacturing variation increases with higher line

card trace lengths. A possible explanation for the increase isn’t that more manufacturing

variation happens when the line card trace length equals 6 inches but that the system is

much more susceptible to said variations. It should also be noted that the eye width

variation changes very little with increased trace length.

Figure 16: Manufacturing Yield predictions versus maximum allowed trace length. The vertical bars indicate the

standard deviation of each yield analysis and shows increased eye height variability with increased trace length.

Defects per Million Analysis

The final piece of analysis is an estimation of the defects per million. Given that the

factor PDFs and response surface equations are accurate, the manufacturing yield of the

system can be calculated. The pass/fail criteria for this analysis depends on which link

uncertainties are included in the simulation, and which are budgeted. For this example,

the spec limits for a passing system are an eye height of 200 mV and eye width of 19 ps.

For a system with line card lengths of 6 inches, the yield distribution plots for eye height

and width are given in Figure 17 below.

Figure 17: Defects Per Million (DPM) analysis showing a predicted 225 and 170 DPM according to the eye height and

eye width requirement, respectively.

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Here the lower spec limit (LSL) is plotted on the distributions and the percentage of the

distribution below this limit and the corresponding parts per million (PPM) values of 224

and 170 defects per million are shown.

Conclusion

We have shown the methodology which assisted in the first CMOS 28G VSR / 100G

Ethernet PHY design. The interface design questions: what layer of the PCB to route on,

what size of via anti-pad to use and what maximum trace length to allow were addressed.

The worst case conditions were identified and impact of manufacturing variations on

performance was quantified. All of which was accomplished with the DOE/RSM

methodology. Instead of simulating millions of conditions requiring months of compute

time, the DOE intelligently sampled the factor space with only 256 runs. A least squares

model fit found a response surface which best fit the data and after the model was

validated it was used to predict system performance throughout the factor space. It must

be emphasized that several iterations of the methodology were required as the model

assumption changed. Overall, the DOE/RSM methodology has been shown to be a

powerful approach to comprehend and optimize a dizzyingly large factor space and

contributed to Inphi’s success with the world’s first production ready 100G CMOS

PHY/SerDes Gearbox.

References

Hall, S. & Heck H. (2009). Advanced Signal Integrity for High-Speed Digital Designs.

Montgomery, D. (2009). Design and Analysis of Experiments, 7th Edition.

Goos, P. & Jones, B. (2011). Optimal Design of Experiments: A Case Study Approach.

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