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Improving microwave circuit testing and producibility with statistical techniques

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IMPROVING MICROWAVE CIRCUIT TESTING AND
PRODUCIBILITY WITH STATISTICAL TECHNIQUES
A Dissertation
by
JAMES MASON CARROLL
Submitted to the Office o f Graduate Studies of
Texas A&M University
in partial fulfillment o f the requirements for the degree o f
DOCTOR OF PHILOSOPHY
August 1995
M ajor Subject: Electrical Engineering
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OMI Number: 9539175
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IMPROVING MICROWAVE CIRCUIT TESTING AND
PRODUCIBILITY WITH STATISTICAL TECHNIQUES
A Dissertation
by
JAMES MASON CARROLL
Submitted to the Office o f Graduate Studies of
Texas A&M University
in partial fulfillment o f the requirements for the degree o f
DOCTOR OF PHILOSOPHY
August 1995
Major Subject: Electrical Engineering
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
OMI Number: 9539175
UMI Microform 9539175
Copyright 1995, by OMI Company. All rights reserved.
This microform edition is protected against unauthorized
copying under Title 17, United States Code.
UMI
300 North Zeeb Road
Ann Arbor, MI 48103
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IMPROVING MICROWAVE CIRCUIT TESTING AND
PRODUCIBILITY WITH STATISTICAL TECHNIQUES
A Dissertation
by
JAMES MASON CARROLL
Submitted to Texas A&M University
in partial fulfillment o f the requirements
for the degree o f
DOCTOR OF PHILOSOPHY
Approved as to style and content by:
Cmu
Cam1 Nguyen
(Member)
Kai Chang
(Chair o f Committee)
/!
to n
Don Halverson
(Member)
A. I
Fred Dahm
(Member)
a
A.D. Patton
(Head of Department)
August 1995
Major Subject: Electrical Engineering
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ABSTRACT
Improving Microwave Circuit Testing and Producibility
with Statistical Techniques. (August 1995)
James Mason Carroll, B.S., Rose-Hulman Institute o f Technology;
M.S., Texas A&M University
Chair o f Advisory Committee: Dr. Kai Chang
Significant improvements to microwave circuits can be made through the use o f statistical
methods. This dissertation addresses three aspects o f the microwave circuit design process:
Modeling, Design Methodology, and Circuit Verification. Advanced statistical modeling
methodology was developed for improved active device variation modeling. Design of
Experiments techniques were applied to the microwave design process within the Computer
Aided Design environment to systematically achieve circuit design goals and reduce
performance variation. A new technique based on bilinear transformations was developed
which helps quantify potential variation in a S-parameter network. Finally, statistical
calibration was applied to the testing process so that wafer probe equipment could be used
to obtain measurements for a fixtured environment. Each of these design tool
enhancements allow microwave engineer designer to produce circuits that exhibit less
performance variability and greater wafer yield in less amount o f time.
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ACKNOWLEDGMENTS
I would like to thank Texas A&M and Texas Instruments for giving me the academic and
industrial education to achieve my doctorate. The expert opinions, encouragement, and
tutelage o f Dr. Kai Chang and Eli Reese were both key drives to me finishing my degree.
The students and employees at both o f these great institutions were both helpful and
friendly. I could not ask for a better bunch o f friends and colleagues.
My mother, father, sister, and grandparents have been supportive and nurturing o f my work.
I appreciate their unwavering dedication to my educational and professional goals.
I had a lot o f people help edit and refine my dissertation. Among them were Dr. Chang,
Kerri Whelan, Sam Pritchett, John Heston, Dr. Dahm, Robert Flynt, and Shirdar
Kanamaluru. This dissertation would not have been as accurate, or readable, as it is without
their excellent help. I would like to thank them all.
Data supporting Chapter III of this dissertation was provided in part by the Application of
Six-Sigma Design Concepts to Integrated Product/Process Development contract (F3361593-C-4328) awarded by Wright Laboratory Manufacturing Technology Directorate of
Wright Patterson AFB to Texas Instruments. Data supporting Chapter V o f this dissertation
was provided in part by the MIMIC Phase 2 contract (N00019-91-C-0210) awarded by
ARPA and NAVAIR to the Raytheon/TI Joint Venture.
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V
TABLE OF CONTENTS
Page
AB STR A CT..................................................................................................................................... iii
ACKNOW LEDGM ENTS.............................................................................................................. iv
TABLE OF CO N TEN TS.................................................................................................................v
LIST OF FIG U RES........................................................................................................................vii
LIST OF TABLES............................................................................................................................ x
CHAPTER I INTRODUCTION.....................................................................................................1
CHAPTER II ADVANCED FET STATISTICAL M ODELING............................................. 5
1. Introduction.......................................................................................................................5
2. Principal Components Background.............................................................................. 9
3. Statistical Equivalence Testing................................................................................... 11
4. FET Parameter Orthogonalization............................................................................... 14
5. Discussion o f Results and A pplications....................................................................25
6. Conclusions.....................................................................................................................27
CHAPTER III STATISTICAL COMPUTER AIDED DESIGN............................................29
1. Introduction....................................................................................................................29
2. DoE B ackground..........................................................................................................31
3. DoE Application - A Design E xam ple......................................................................35
4. Variability Reduction................................................................................................... 46
5. Discussion of Results................................................................................................... 53
6. Conclusions.....................................................................................................................54
CHAPTER IV BILINEAR VARIABILITY COM PARISONS.............................................. 55
1. Introduction....................................................................................................................55
2. Bilinear theory.............................................................................................................. 56
3. Bilinear Mapping Application.................................................................................... 60
4. Discussion o f Results................................................................................................... 67
5. Conclusions....................................................................................................................67
CHAPTER V STATISTICAL WAFER PROBE CALIBRATION....................................... 69
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vi
Page
1. Introduction................................................................................................................... 69
2. Simple Model Statistical Calibration......................................................................... 71
3. More Complex Statistical M odels............................................................................. 81
4. Application And Discussion of Results.....................................................................85
5. Conclusions................................................................................................................... 87
CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS..........................................88
1. Advanced Statistical M odeling...................................................................................88
2. Statistical Design M ethodologies...............................................................................88
3. Statistical Calibration................................................................................................... 88
4. General Conclusions..................................................................................................... 89
REFERENCES............................................................................................................................... 90
APPENDIX EXAMPLE SAS® PROGRAM S.......................................................................... 95
1. Principal Component Modeling of FET E C P s.........................................................95
2. DoE Surface Response Modeling o f Experimental R esults...................................96
3. Statistical Calibration o f Wafer Probe D a ta ............................................................. 97
VITA ..............................................................................................................................................98
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LIST OF FIGURES
FIGURE
Page
1: Critical Design Process A spects............................................................................................. 2
2: Small Signal Model Including Intrinsic, Extrinsic Resistances, and Noise Elem ents
6
3: (a) Correlated Data Set (b) Rotation of Axis by the Principal Component Technique
to Uncorrelate the D ata....................................................................................................... 9
4: Statistical Modeling Methodology Flow C hart...................................................................14
5: Distribution o f Extracted Gm V alues....................................................................................18
6: Comparison o f Extracted and Simulated Cgs Correlation Coefficient with Other FET
E C P s.................................................................................................................................... 19
7: Percent of FET Variation Explained as Number o f Principal Factors Considered in
Model Increases................................................................................................................. 21
8: Comparison o f Principal Component Model Complexity, Correlation of Gm and R ,.... 22
9: Comparison o f Measured and Simulated Correlation Coefficients; Correlation of
Real Part o f S2| with Other FET Responses.................................................................. 23
10: Comparison o f Measured and Simulated Correlation Coefficients; Correlation of
Imaginary Part o f S2I with Other FET Responses.........................................................24
11: Scatter Plot o f Gm and Tau (Correlation Coefficient = 0 .05)......................................... 26
12: Geometric Representation o f Table 5 Experimental D esign.......................................... 33
13: 2-Stage Low Noise Amplifier Topology........................................................................... 36
14: SCAD Methodology for Circuit Design............................................................................ 38
15: Model Error from Random Factor Settings.......................................................................40 '
16: Pareto o f 2-Stage Amplifier G ain........................................................................................42
17: Pareto o f 2-Stage Amplifier Noise Figure.......................................................................... 42
18: Pareto o f 2-Stage Amplifier S ,, ........................................................................................... 43
19: Pareto of 2-Stage Amplifier S22........................................................................................... 43
20: Inner and Outer Array DoE ¥ n Designable (Outer Array) Factor, 0n Noise (Inner
Array) Factor, and Responses ySj .....................................................................................47
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FIGURE
Page
21: Single Stage Amplifier for Variance Reduction Example............................................... 48
22: Single Stage Amplifier Gain Variance Pareto Chart Accounting for Input
Termination V ariability..................................................................................................... 50
23: Single Stage Amplifier Average Gain Pareto Chart Accounting for Input
Termination V ariability..................................................................................................... 51
24: Single Stage Amplifier Gain Variability Pareto Chart Accounting for Intrinsic FET
Variability............................................................................................................................ 52
25: (a) Two Port Network (b) Bilinear Transformation o f an Arbitrary Load Impedance
(F load) onto ^
InPut Reflection Coefficient (TIN) ...................................................... 56
26: (a) r L0AD Distribution (b) Locus Mapping o f TL0AD to TIN Showing Variability
Increase................................................................................................................................ 58
27: (a) Series Feedback Topology Used for FET Variability Reduction (b) Shunt
Feedback T opology............................................................................................................60
28: Normalized Radius for Different Values o f Series Feedback Reflection Coefficient
Expressed in Magnitude (M) and Angle..........................................................................62
29: Series Feedback Contour Mapping, Log o f Normalized Smith Chart R ad iu s.............. 63
30: Shunt Feedback Contour Mapping, Log o f Normalized Smith Chart Radius................64
31: Locus o f Random Load Impedance for T)N Variability Comparison (500 Sam ples)... 65
32: Input Impedance Distributions o f a FET Using Feedback While TL0AD Randomly
Varies (500 Samples)......................................................................................................... 66
33: Typical MMIC Fixture with Coax-to-Microstrip Launchers........................................... 69
34: Probe System Measuring a Die on a un-Diced W afer....................................................... 70
35: WBPA Fixtured Assembly with Bias Capacitors Shaded.................................................72
36: Magnitude o f Input Match (|SU|) Scatterplot..................................................................... 73
37: Magnitude o f Circuit Gain (|S2il) Scatterplot with Linear Segmented Line
Regression M odeling......................................................................................................... 74
38: Magnitude o f Reverse Isolation (|S I2|) Scatterplot............................................................ 74
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FIGURE
Page
39: Magnitude o f Output Match (|S22|) Scatterplot................................................................. 75
40: WBPA Phase Angle o f S2i over Frequency......................................................................77
41: Scatter Plot o f Phase Angle of S21...................................................................................... 78
42: Scatter Plot o f Phase Angle o f S2! after Transformation................................................. 79
43: 2-dB Compression PAE Scatterplot.................................................................................. 80
44: Simple Model Regression Residual Plot for S21 Phase A n g le ........................................82
45: SAS® Program for Fitting Linear Segmented Regression M odel.................................. 84
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LIST OF TABLES
TABLE
Page
1: Common Microwave Circuit Statistical Tests and Their Associated Cumulative
Error L evels........................................................................................................................ 13
2: Mean and Standard Deviation o f Extracted and Simulated FET Parameters...................15
3: Extracted FET Parameters Correlation Coefficients with Statistically Non­
significant Values Shaded................................................................................................ 16
4: Principal Component Factor Pattern Matrix Explaining 96.6% o f Total FET
Parameter V ariation........................................................................................................... 17
5: 3 Factor, 2 Level Full Factorial Design.................................................................................32
6: Low-Noise Amplifier Nominal Design Values and Coding...............................................37
7: Low-Noise Amplifier Performance Cases *Nominal Values Optimized by
Touchstone.......................................................................................................................... 37
8: Low-Noise Amplifier Modeled Response Values............................................................... 39
9: Single Stage Amplifier Nominal Design Values and C oding............................................49
10: Input Match Terminating Impedances for Taguchi Outer Array..................................... 49
11: S-Parameter Response Regression Equation Values - Simple M odel............................ 76
12: Model Regression Values for Fixtured Measurement Model Given Probe Results
for Power Compression, PAE, and G ain........................................................................ 81
13: S-Parameter Response Complex Regression Model V alues............................................83
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1
CHAPTER I
INTRODUCTION
The microwave industry is transitioning from a traditionally defense oriented business to
playing a larger commercial role in the marketplace. This shift has been fueled by both the
recent defense spending cut-backs and the commercial consumer's increasing need for high
frequency circuits found in wireless communication systems. Companies are finding it
more difficult to succeed due to the large amount of competition for the commercial market
and shrinking defense funding for big-budget technologies. In order for a microwave
company to prosper, especially in a commercial market, their products must be continually
improved by making them smaller, more complex, less expensive, and more reliable with
all these characteristics being achieved in a less amount o f time. These types of
improvements are difficult to achieve because each o f these product characteristics is
usually obtained at the expense o f the others. Therefore, there has been a large movement
in the microwave industry to search for methods to improve the design and manufacturing
process in order to achieve better microwave circuits. A good number o f these methods are
based on statistics [1]. This dissertation discusses the research activities at Texas A&M
University and Texas Instruments to fulfill the microwave industry’s need for new
statistical design techniques and methodologies.
Statistical tools were developed to address the three critical aspects o f a product’s design
process as shown in Figure 1. Those aspects are modeling, design methodology, and
product verification. Dependable circuits can not be achieved without successful and
accurate completion o f each o f these three design aspects.
The journal model is IEEE Transactions on Microwave Theory and Techniques.
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2
Design '
Methodology
Statistical
Modeling
Design
Process
Product
Verification
Figure 1: Critical Design Process Aspects
All three aspects are interrelated. For example, the type of design methodology determines
the kind o f modeling that is needed. The verification process serves to build a statistical
database from which old models are validated and new models are developed. Finally,
accurate models assure that the component variations can be taken into account and the
circuits meet the customer’s specifications.
The use of statistics in the circuit design process is still an young and developing field.
Statistical design and optimization tools have been integrated into the commonly available
Computer-Aided Design (CAD) software since the late ‘80s. Much o f these tools that are
available today focus on using methods to predict circuit yield and then maximize it with an
optimizer. Some work has also been done in statistical modeling o f active microwave
devices. However, the techniques are still relatively new and the general engineering
community does not fully understand or use them to their full potential. Most tools are
currently based on Monte Carlo methods which are advocated by the statistical design
experts in electronics [2] and microwave circuits [3]. However, there are drawbacks to the
Monte Carlo method as well areas of microwave design that it does not or can not address.
This dissertation attempts to fill in the “gaps” left by the current methods as well as to
develop innovative new areas in statistical microwave circuit design.
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3
The research described in this dissertation attempted to improve all three aspects o f the
design process shown in Figure 1. The improvements include development o f new
statistical model for simulations which allow accurate modeling of fabrication variations in
microwave circuits. The use o f Design o f Experiments (DoE) and bilinear transformations
are shown to improve the design methodology process in order to obtain much better circuit
designs. Finally, a new approach for product verification has been developed which allows
a more desirable wafer probe measurement system to all but completely replace the old
method of fixturing devices for testing. Admittedly, these developments cover a broad
spectrum o f concepts in the microwave area. However, these improvements serve to open
the door and enlighten designers so that they can begin to design circuits that exhibit low
performance variation and “first-pass” success. The use o f the methodologies developed in
this research will enable designers to create these improved circuits more quickly,
efficiently, and at a lower cost. The time and cost saving improvements are then passed
onto the consumer either in the commercial or military marketplace. Furthermore, the
methods developed in this dissertation will serve as a basis for others to do more in-depth
research into each technique.
It should be mentioned that the ideas developed in this dissertation attempt to improve what
is known as parametric circuit yield, not catastrophic. Parametric circuit failure is when a
circuit does not perform as expected due to the natural circuit component variations
incurred during the fabrication and construction process [2]. Catastrophic failure is when a
circuit process or construction step is not performed correctly and therefore causes the
circuit not to work properly. Examples of catastrophic failure would include, but not be
limited to, circuit breakage, resist layers that are spun on to the wrong thickness, or residues
on the circuit that where not cleaned properly during fabrication. This research only tries to
model the natural, parameteric variations inherent in all fabrication process, and then take
them into account during the circuit design. This allows the designer to compensate for the
variations and create a circuit that is, hopefully, insensitive to those natural changes. The
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4
term “robust circuit” will be used to refer to a circuit that has been designed to be less
sensitive to its natural component variations.
A chapter has devoted to develop each of the three critical aspects o f the design process.
Chapter II discusses improvements to advanced statistical modeling of active device
variations. Chapter III develops a circuit design methodology using DoE techniques in
order to optimize a circuit’s nominal and variation performance. The application o f bilinear
transformations to quantify and reduce potential S-parameter network variability is shown
in Chapter IV. Finally, Chapter V describes the statistical calibration o f microwave wafer
probe and fixtured measurements. Supplementary information and any software code is
included in the Appendices. The software package that was used to do all of the statistical
analysis was SAS because it is commercially available and considered the standard by
many statisticians. Other comprehensive statistical packages could probably be
programmed to perform the same types o f analysis.
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5
CHAPTER II
ADVANCED FET STATISTICAL MODELING
1. Introduction
Statistical modeling has shown increasing popularity with microwave circuit designers
during the last few years [3]. This is due to the incorporation o f statistical yield analysis
and optimization into commercial CAD programs. Statistical modeling allows the
microwave engineer to evaluate circuits on the basis o f their producibility as well as good
electrical performance. This results in more reliable, higher yielding products which are
more commercially competitive.
The foundation for most CAD yield analysis and optimization tools is the Monte Carlo
method [3]. It is well known that all circuit parameters vary randomly around their
nominal, or "designed", values due to fluctuations inherent to the production o f the circuit.
These fluctuations are the result o f each component's intrinsic tolerance which is governed
by technological and cost considerations. For example, GaAs microstrip may be designed
to be 75 pm wide but may vary ± 2 pm due to gold plating limitations. The random
fluctuations in the circuit components causes a corresponding variation in the circuit
response. Commercial microwave CAD packages use the Monte Carlo technique to model
these processing fluctuations as statistically independent, random variables in order to
predict how the circuit will respond.
A CAD package would run a predetermined number o f simulations when using the Monte
Carlo method. For each simulation, values for the component variables are “randomly”
picked from their user defined, independent distributions. The simulator uses these random
variables to determine the circuit response. The simulated values are recorded and used to
determine the mean and variance o f the circuit response after the Monte Carlo method is
finished. However, many natural circuit component fluctuations cannot be expressed in the
form o f independent distributions when selecting the random values. A common and very
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6
influential example o f interrelated variables in microwave circuit modeling are the small
signal FET model parameters [4].
Figure 2 illustrates a conventional small signal model for FET noise and S-parameter
characteristics over frequency [5]. The model is widely used in industry to model
microwave performance over the 0.5 to 26 GHz frequency range by modeling the FET with
lumped circuit elements which is easy to integrate into a CAD simulator. The small signal
circuit model gives reasonable results for small signal conditions by including the intrinsic
(Cdg, Rgs, Cgs, R|, Gm, tau, Cds, Rds), some extrinsic (Rg, Rs, Rd), and noise parameters (Vn,
In, Re_Corr, Im_Corr). The model's main strength is its compactness and ease o f use in the
CAD modeling environment. The potential for FET parameter scalability is also an
advantage which cannot be ignored [6].
Rs
_
Source
Figure 2: Small Signal Model Including Intrinsic, Extrinsic
Resistances, and Noise Elements
The small signal model’s element values can be determined, commonly termed extracted,
so that the measured FET’s scattering and noise parameters can be simulated in the CAD
package. The set o f small signal circuit values that most accurately model a FET’s
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7
measured scattering and noise parameters over some frequency range is called the
Equivalent Circuit Parameters (ECPs) for that active device.
A statistical representation o f a FET population can determined if the ECPs are extracted
from the measurements o f many active devices. When the relationships between the ECPs
are examined it is seen that they are described by highly correlated multivariate
distributions [4]. That is, the value o f one ECP has some influence on the other ECPs when
the entire population is being modeled. This type o f statistical relationship cannot be easily
implemented in existing commercial CAD software simulators because Monte Carlo
simulations assume statistical independence o f the parameters. Some designers have tried
to model the FET parameters as independent random variables with mixed success [4, 7, 8].
Due to the physical correlations existing between FET parameters this modeling scheme
can often result in physically impossible FET parameter combinations during a Monte
Carlo simulation. This situation is undesirable if truly accurate CAD yield predictions are
required. The statistical model that assumes independence of the ECPs will be referred to
as the plus-minus sigma (± g ) model in this research because it uses only the mean and
standard deviation (a) o f the parameters to simulate the entire population.
The Truth Model has been suggested [7,9] and successfully implemented into commercial
CAD packages in order to remedy the shortcomings o f the correlated FET parameters [10].
This method is simple and inherently creates the orthogonality of random variables that the
Monte Carlo method requires. In fact, the Truth Model can be thought o f as making the
entire FET one random variable picked from an S-parameter database during the Monte
Carlo simulation. However, large S-parameter databases are needed to cover all of the
frequency ranges and bias conditions necessary for accurate statistical modeling. The Truth
Model is not compact and cannot be scaled to different FET sizes as can the small signal
FET ECPs. Also, the randomness o f the Monte Carlo sampling is severely limited by the
size o f the S-parameter database. Campbell et al. have suggested database interpolation in
order to reduce the impact o f this limitation [11]. However, this interpolation results in an
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8
even larger database for random FET selections and does not solve the inherent problem.
Finally, S-parameter database access time must be considered for a large number o f Monte
Carlo simulations. In summary, although the Truth Model is accurate, it limits the
economic feasibility o f statistical design due to complexity, database requirements, and
computational inefficiencies.
Purviance et al. suggested statistically characterizing FETs through the use o f a principal
component analysis o f the S-parameters database [12]. This solution has many o f the
disadvantages o f the Truth Model the most important o f which is the large database needed
for the statistical modeling at different frequencies, FET sizes, and biases. However, it will
be shown herein that the principal component technique can also be applied to the smallsignal FET model parameters to obtain an accurate and compact statistical model for circuit
simulation o f both noise and S-parameters.
This dissertation proposes application o f the principal components statistical technique to a
small signal FET equivalent circuit parameter database. In this chapter, background
information on the principal component method will be presented. Correct statistical
testing o f simulated microwave responses is discussed so that they can be applied to
determine the validity o f a statistical modeling approach. Example data from a population
o f four finger low-noise GaAs FETs with a periphery o f 300 pm is used to incorporate the
principal component technique into a currently available CAD microwave simulator. The
correlated parameters will be shown to be easily expressed in terms of uncorrelated random
variables suitable for Monte Carlo analysis. Then, the measured FET parameter's means,
standard deviations, and correlations will be shown to be preserved during a Monte Carlo
simulation. Statistical tests will be used to verify the improvement o f simulated FET noise
and S-parameters over the traditional ±cr model. Finally, improvement in the scattering and
noise parameter modeling o f the original FET population will be shown to be achieved
through the use of the principal component method over the ±cr model. Finally, the results
and other possible applications o f the modeling methodology will be discussed.
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9
2. Principal Components Background
Principal component analysis is a well known statistical technique by which a sample data
set of n correlated variables are linearly transformed into a new data set o f n uncorrelated,
or orthogonal, variables called principal components [13]. Statistically, correlation is
defined as the linear relationship between two or more variables. The principal component
technique essentially rotates the variable axes in order to obtain data with no linear
relationships. Figure 3a shows a set o f data points which have an obvious strong positive
linear relationship with respect to the X and Y coordinate system. Principal components
rotates the axes to produce a new coordinate system described by FI and F2. The data in
Figure 3b is uncorrelated when referenced to this new coordinate system.
•
(a)
•
(b)
Figure 3: (a) Correlated Data Set (b) Rotation o f Axis by the
Principal Component Technique to Uncorrelate the Data
The same concept can be applied to an ^-dimensional coordinate system o f a sample data
set resulting in a new ^-dimensional data set referenced to the orthogonal principal
component axes. Mathematically, this rotation is achieved by determining the eigenvalues
of the n x n correlation matrix o f a sample data set. Equation 1 below shows the vector E
containing the FET parameter variables from Figure 2, the vector F which contains the
orthogonal principal components, A the diagonal eigenvalue matrix, and U a matrix
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10
determined by the eigenvalues and original data. AU'1 is referred to as the factor pattern
matrix because it contains the coefficients that will be multiplied by the principal
components (factors) to reproduce the original data.
E = AU~‘F
(1)
One o f the interesting aspects o f the principal components technique is that the first
eigenvalue, which corresponds to the first factor, is the largest since it is oriented in the
direction responsible for most variation in the original data set. The next eigenvalue is the
second largest because it is oriented, orthogonal to the first, in the direction responsible for
most o f the remaining variation in the original data set. This continues until the «11'
eigenvalue explains the remaining variation. Using all n factors will describe all of the
variation present in the original data. By using the inverse transform o f Equation 1 on each
o f the extracted FET parameters it is possible to derive a new data set which is completely
orthogonal. Each o f the new uncorrelated variables will be standardized according to
Equation 2 where x is the original data's mean and sx is the sample standard deviation.
x -x
\ )
^Standardized
s.
In other words, the principal component variables have a mean o f zero and standard
deviation of one. By using the standardized uncorrelated data set in Equation 1, the linear
combination o f the principal factors will produce the original data in standardized form. To
restore the original FET parameters from standardized data, x must be solved for in
Equation 2.
Most commercial statistical analysis packages will perform the principal component
analysis on a data set. One such commercial statistical analysis package, SAS®, will
determine the new uncorrelated data set from an original data set, calculate the eigenvalues,
cumulative variation explained by each o f the new orthogonal factor, as well as the
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11
coefficients contained in the factor pattern matrix [14]. SAS® can also be used to compute
the means and standard deviations needed to restore the original FET parameters from the
principal components. Sample SAS® code to do this for the 15 small-signal FET ECPs is
included in Appendix A.
Notice that no assumptions have been made o f the original data's distributions. The new
orthogonal data set could be used as it stands. However, if the original data follows a
normal (Gaussian) distribution, the Principal Components will also have a Gaussian
distribution because the linear combination o f Gaussian distributions will be Gaussian.
Each original FET parameter should be checked for this normality assumption by a
statistical test. If the original data has a Gaussian distribution, or one can be obtained by
data transformation, the derived principal components can be defined as having a
standardized Gaussian distribution with a mean o f zero and standard deviation o f one.
Equations 1 and 2 can then be used to define the original data variables as a function o f the
Principal Components. This produces automatic interpolation o f the original FET
parameter database by simulating combinations o f FET parameters that retain the
correlations determined from the measured data but were never actually measured.
3. Statistical Equivalence Testing
Population equivalence testing needs to be done after performing the principal component
analysis on the FET ECP database in order to confirm the statistical model’s accuracy. In
the past, there has been a serious lack o f statistical rigor where modeling examples were
shown to "agree well with" [15] or have "excellent" comparisons [11]. These types o f data
comparisons are qualitative in nature and subjective at best. This section discusses the
statistical tools that are available to determine a quantitative level of statistical accuracy
when comparing measured and simulated microwave circuit populations.
There are two types o f tests that can be applied to multivariate populations to determine
some level o f statistical equivalence. First, a multivariate distribution test can be used to
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12
determine if two populations are equivalent [16]. This test is the most accurate but
unfortunately not commonly incorporated into many commercial statistical packages. The
second method uses pairwise comparisons o f each variable's marginal density distribution.
This does not provide sufficient conditions for multivariate statistical equivalence [17]
except in the case o f the multivariate Gaussian distribution. However, the pairwise testing
is useful even for non-Gaussian distributions because it can help identify which variables of
two multivariate distributions are not statistically equal. Pairwise testing o f the distribution
parameters can be easily achieved with the help o f commercial statistical packages in the
absence o f a true multivariate distribution equivalence test and most statistical texts cover
pairwise statistical testing [18]. The application of these tests to the more common
microwave populations such as S-parameter data sets or ECP sets will be briefly described
here.
Each statistical equivalency test is performed at a predetermined significance level (a)
which is the probability o f finding a difference between population statistics when there
really is none. This is called a Type I error. The person performing the test usually wants
to keep this probability quite low, typically 0.05 to 0.1. Unfortunately, if a population has
many different statistical variables to test, the probability o f making an error accumulates
according to Equation 3 where m is the number o f variables being pairwise tested.
^cumulative
^ 0
&pairwise)
0)
For example, suppose a comparison o f the means of two sample sets o f S-parameters were
to being made. There are four parameters, S u , S 12, S2i, and S22, each with a real and
imaginary part. There will be eight means that would need to be compared to conclude
statistical equivalence. In order to keep the cumulative error small for the entire statistical
test, each pairwise a level must be very low. In fact, the cumulative error would be 0.57 if
each test is performed at an a = 0.1 level. That is, there would be a 57% chance of making
an error if the S-parameter populations were found to be equivalent. The more acceptable
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13
cumulative error o f 0.1 would result if the pairwise comparisons were made at an a = 0.013
level o f significance.
Table 1 shows a compilation o f suggested pairwise a levels and their corresponding
cumulative a for different types o f equivalence testing which are of special interest to
microwave engineers. Significance levels for smaller or larger FET models or different size
S-parameter networks may be derived in a similar fashion with Equation 3.
Table 1: Common Microwave Circuit Statistical Tests and
Their Associated Cumulative Error Levels
Pairwise a-Level
Experiment-wise
Significance
Intrinsic FET Model
S-parameters (2-Port)
7 means or
standard
deviations
21 correlation
coefficients
8 means or
standard
deviations
28 correlations
coefficients
= 0.05
0.0079
0.0024
0.0064
0.0018
= 0.075
0 .0 1 1 1
0.0037
0.0097
0.0028
^cumulative = 0 . 1 0
0.0149
0.0050
0.0130
0.0038
^"cumulative — 0 . 1 5
0.0229
0.0077
0 .0 2 0 1
0.0058
Level
^cumulative
^"cumulative
It can be seen that very low a-level pairwise comparisons need to be made in order to keep
the cumulative error low on any statistical tests that are being made. Table 1 includes
pairwise comparisons o f correlation coefficients along with means and standard deviations.
The number o f correlation coefficients for n different variables is described by Equation 4
[18]. For the 8 S-parameters, this equation shows that there will be 28 correlation
coefficients.
# Correlation Coefficients = — — —
(4)
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14
4. FE T Parameter Orthogonalization
This section illustrates the application o f the Principal Component technique to statistical
FET modeling. The methodology shown in Figure 4 was applied to FETs produced in 1993
at the Texas Instruments GaAs Foundry in Dallas, TX.
'"Fabrication' 1
of FETs
V___________ /
______ I
r Verify Statistical^
Measurement ^
r Monte Carlo ^
< ------Model
DC and RF Probe < — ►
Simulation
v
J
^
J
r
I
t
r Insert Statistical>
r Apply Principal ^
^Extraction o f
------- ►
------- ►
Model in Circuit
Components
FET Model
V
J
^
J
^
J
.
.
A
^
A n n lti D rinA inn]
A I n p o r f Cf*
Figure 4: Statistical Modeling Methodology Flow Chart
Each FET had four gate fingers and a total periphery of 300 pm. Fifty-four FETs were used
from six 100 pm thick GaAs wafers with low-noise doping profiles. Normally, a sample
size o f only fifty-four FETs would be considered small for characterizing an active device
population. However, the purpose o f the study was to prove the usefulness o f this statistical
modeling methodology. Scattering and noise parameter measurements were obtained over
the 0.5 to 26.5 GHz range at 0.5 GHz step intervals at the drain bias level o f 3V and 30mA.
The ECPs shown in Figure 2 were extracted from each FET’s measured response. The ten
intrinsic and extrinsic parameter values were obtained by analytical extraction o f the FET
parameters for each set o f S-parameters similar to Anholt et al [8] and Golio [19]. The five
noise parameters, including Rgs, shown in the Figure 2 were obtained by analytical
extraction using the Hybrid-Pi noise model [5]. All ECPs were optimized to obtain a better
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15
fit to the individual FET measurements. Table 2 shows the mean and standard deviation
values for all o f the fifteen extracted ECPs. The table also details the means and standard
deviations obtained during the principal component and ±<r model simulations. These
values will be discussed after the principal component model is constructed.
Table 2: Mean and Standard Deviation of Extracted and
Simulated FET Parameters
Mean
Extracted
P.C.
Gm (mS)
92.335
92.152
Ces(fF)
389.909
Ri (Q)
Standard Deviation
Extracted
P.C.
92.195
5.251
5.160
5.153
388.341
388.401
28.007
27.178
27.230
2.594
2.600
2.598
0.312
0.314
0.302
Cds(fF)
79.178
79.219
79.217
2.762
2.696
2.757
Rds(fi)
150.393
149.724
150.302
10.955
10.759
10.719
Ced(fF)
32.207
32.213
32.147
3.169
3.148
3.202
Tau (ps)
2.520
2.513
2.524
0.224
0.220
0.221
R a (£2)
0.391
0.395
0.393
0.057
0.056
0.058
R ,(Q )
2.539
2.538
2.547
0.190
0.181
0.191
Rd (Q)
3.678
3.676
3.682
0.173
0.170
0.174
vn
0.050
0.050
0.050
0.004
0.004
0.004
In
ReCorr
704.704
702.277
708.548
98.364
97.213
96.316
-3.088
-3.083
-3.093
0.167
0.163
0.172
ImCorr
-0.286
-0.290
-0.282
0.173
0.171
0.175
R es(Q)
12388.0
12485.0
12348.2
2875.1
2821.5
2874.6
±
±
All of the FET models were extracted and the commercial statistical analysis package SAS®
was used to determine the mean, standard deviation, and correlation matrix o f the FET
parameters with the program listed in Appendix A. Table 3 shows the extracted database's
correlation matrix with the statistically non-significant values shaded. The un-shaded
values were determined to be statistically non-zero by the tests. Fifty-nine of the 105
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16
correlation coefficients have a non-zero value when each was tested at an a = 0.05
significance level. This strongly suggests the independence assumption inherent to a Monte
Carlo analysis with the Monte Carlo method would be violated.
Table 3: Extracted FET Parameters Correlation Coefficients
with Statistically Non-Significant Values Shaded
Im
Re
Corr
Corr
In
Vn
Rd
Tau
Cgd
Rds
Cds
Gm
-0.46 0 26 -0.39 0.74 -0.67 -0.06 0.53 -0.72 0 0 5
0.54
1111
0 )7
Cgs
-0.40 0.51
-0 14 0.41
0 15
Ri
-0.35 0 23
0.10 -0.48 0.51
Rgs
Rs
Rg
-0 26 -0 08 0.32 -0.76 0.55
-0 01 -0 01
0.41
0.58 -0.55
0 15
0 06
1.00
-0.01 -0.16 -0.38 0.28 -0.38 0.26
0.55
Rds
-0 11 0.51
-0 13 -0.44 0.78 -0.36
Cgd
-0 15 -0 15 -0.65
Tau
-0.31
Rg
0.28 -0.36 0.48 -0.75
0.41
012
-0.29
Rs
-0.33
-0.52 0.43
1.00
0.53
0 23 - 0 1 4
0.13 -0.61
0.90 -0.70 -0.12 0.29 -0.60 -0.59
0.51
Rd
Vn
0 08
In
-0.28 0 03 -0.69
ReCorr 0 20
0 04
ImCorr -0.37
1.00
Rgs
000
0.28 -0.38 0.48 -0.05
0 06 0.00 -0 06 0 04
0 27
0.40
0.60 -0.76
000
0 05 -0 0 9
Cgs
Gm
-0.25 0.79
1.00
0.67 -0 16 - O i l
0 08 -0.49 0.19 -0.43
Cds
Ri
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
SAS® was then used to determine the factor pattern matrix values shown in Table 4. The
coefficients in the factor pattern matrix were multiplied by the principal component vector
as shown in Equation 1 to produce an equation for each of the FET ECPs.
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17
Table 4: Principal Component Factor Pattern Matrix
Explaining 96.6% o f Total FET Parameter Variation
Factor 1 Factor 2 Factor 3 Factor 4 Factor 5 Factor 6 Factor 7 Factor 8
Gm
0.808
0.412
-0.006
-0.050
0.323
C£S
0.046
0.822
-0.154
0.020
0.040
-0.115
0.142
Ri
-0.493
0.309
0.251
-0.009
-0.056
-0.025
0.579
-0.463
-0.030
Gds
0.386
0.158
-0.129
-0.312
-0.425
0.559
0.110
0.266
0.202
0.363
-0.313
-0.183
0.850
-0.177
0.317
-0.054
0.185
-0.079
-0.227
0.854
-0.232
-0.275
-0.181
-0.251
-0.071
0.015
-0.028
-0.383
0.862
0.144
-0.026
0.088
0.103
-0.219
0.068
RS
Rs
-0.729
-0.513
0.323
-0.095
0.179
-0.109
-0.031
0.144
0.639
0.073
0.649
0.169
0.093
0.123
-0.073
0.174
Rd
0.50
-0.082
0.582
0.661
-0.289
-0.297
-0.153
0.004
Vn
-0.853
0.252
0.040
0.012
-0.296
0.023
0.168
-0.176
0.956
0.013
-0.095
-0.025
-0.166
-0.040
0.000
-0.044
ReCorr
-0.741
0.106
-0.245
0.182
0.314
-0.422
0.184
-0.033
ImCorr
0.015
0.735
0.138
0.002
-0.205
0.030
0.560
0.279
Rgs
-0.289
-0.488
-0.410
0.464
0.100
0.466
0.011
0.177
^ds
Ced
Tau
Each o f the FET parameters derived from Equation 1 can be placed in the equation block of
a commercial CAD package such as Touchstone [10]. For example, the resulting
expression for Gm is provided in Equation 5 where x Gm is the mean and sGm is the standard
deviation o f the extracted Gm sample. This expression contains all 15 principal component
factors (FI through FI 5) which explain 100% o f the variation present in the original
dataset. This expression for Gm can then be scaled to the desired FET periphery [6].
Gm = x (lm+S0m* (0.808*F1 + 0.412*F2 - 0.006*F3
(5)
- 0.050*F4 + 0.323*F5 - 0.154*F6 + 0.020*F7
+ 0.040*F8 + 0.149*F9 + 0.028*F10 - 0.134*F11
- 0.003*F12 + 0.036*F13 + 0.0216*F14 + 0.034*F15)
Figure 5 shows the distribution o f Gm from the extracted database. The distribution seems
to follow a Gaussian distribution although with just 54 samples the shape is not clearly
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18
defined. Gm, along with all the other ECP variables, each passed a statistical Shapiro-Wilk
normality test at an a = 0.05 level o f significance [20]. This indicates that the principal
components will also follow a Gaussian distribution.
Gm Values (mS)
Figure 5: Distribution of Extracted Gm Values
Therefore, FI through F I 5 were defined in the Touchstone "VARIABLES" block to have a
normal distribution with mean zero and standard deviation o f one [10]. Notice when the
statistical mode o f the CAD package is not being used, that FI through FI 5 will be at their
nominal value, i.e. zero, and Gm will equal the mean o f the entire FET sample. Also, the
sum of the squares o f the principal factor coefficients is equal to one which forces the
standard deviation o f Gmto be S(;„,during the Monte Carlo simulation.
The entire factor pattern matrix for the sample database was used in Equation 1 to
implement all the FET parameters in terms o f the principal component variables FI through
FI 5. One thousand samples were simulated using the Monte Carlo method on the principal
factors and the simulated FET parameters were statistically analyzed. For comparison, one
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19
thousand samples were also simulated using only the FET parameter's mean and standard
deviation which assumes independence of the FET parameters (± a method). Table 2 shows
the means and standard deviations o f the extracted FET parameters, the principal
component model (P.C.) results, and ±cr model. Both the principal component model and
the ± a model are able to accurately reproduce the mean and standard deviations o f the
extracted FET parameters. In fact, both models produced means and standard deviations
statistically equivalent to the original data with a cumulative a = 0.1 level of error.
A representative example o f correlation recovery for the principal component model, ±o
model, and the extracted data is shown in Figure 6 for the correlation o f Cgs with the other
FET ECPs. Figure 6 demonstrates the principal component method correctly recovered all
o f the measured Cgs correlations, while the ± ct method’s correlation coefficients were all
statistically zero.
0.8
4-*
C
0.6
8
0.2
<D
E 04
o
c
o
~
JS
0
br
- 0.2
<D-0.4
L.
O
O *0-6
Gm
Cgd
Cds
Ri
Rds
Tau
Rs
Vn
ImCorr
ReCorr
Rgs
Correlation with Cgs
jjIPI Measured
P.C.
+/- Sigma
Figure 6: Comparison of Extracted and Simulated Cgs
Correlation Coefficient with Other FET ECPs
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20
Pairwise statistical tests verified that all 105 measured and principal component simulated
correlations were statistically equivalent at a cumulative a = 0.1 level o f error. Figure 6
illustrates the findings o f these statistical tests including the fact that the ± g method is not
capable o f modeling the FET parameter correlations because o f the parameter independence
assumption. Therefore, the ± ct method results in simulation o f impossible FET parameter
combinations during the Monte Carlo method. These examples show that principal
components is a superior technique for accurate modeling o f FET ECP statistical variations.
Also, the equations are straightforward to determine and can easily be implemented into
current commercial CAD programs.
Figure 7 shows that a larger portion o f the total variation found in the original FET database
is explained as the number o f Principal Factors used in the model is increased. One
hundred percent o f the total variation is represented when the number o f Principal Factors
equals the number o f original variables. Notice that the first nine factors explain 97.9% of
the total cumulative variation in the FET parameters. This creates the potential to eliminate
some of the less significant principal factors to produce a more compact model for each of
the FET parameters [12].
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21
100
Q.
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Number of Principal Factors Considered
Figure 7: Percent o f FET Variation Explained as Number o f
Principal Factors Considered in Model Increases
To see how the number o f terms affect the model's correlation coefficients, the number of
factors in the Gm principal component model was varied from all fifteen to just the first
principal component factor for a 1000 Monte Carlo run. Figure 8 illustrates the large error
possible for the simulated correlation coefficient between Gm and Rj when only a few
principal factors are used. As the number o f principal factors is increased the error
decreases until it is statistically negligible. Notice that the error for this particular
correlation coefficient is not strictly monotonic. The graph also shows the correlation can
be adequately preserved with just nine factors instead o f the original fifteen.
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22
300
■£ 250
a>
‘o
§o 200
0
1 150
TO
S>
S 100
o
'I
50
u.
HI
2
3
4
5 ’ 6 ' 7 ' 8 ’ 9 '1 0 '1 1 '1 2 '1 3 '1 4 '1 5
Number of Principal Factors Considered
Figure 8: Comparison of Principal Component Model
Complexity, Correlation o f Gm and R|
Reducing the principal factors in the FET parameter statistical model can greatly decrease
the model’s complexity. The number o f terms in Equation 5 could be decreased by using
only the factors FI through F9. A smaller, less complex statistic model has many benefits
such as easier implementation and faster simulation time.
The fifteen ECP principal component equations were used in the FET model shown in
Figure 2 during a 500 run Monte Carlo simulation. The S-parameters from each run were
stored in a database and then used in comparison with the original measured S-parameters
to verify their statistical equivalence. Pairwise comparisons were made on the real and
imaginary parts o f the noise and S-parameters which included Rn, Fmin, and the real and
imaginary part o f r opt. All the measured and principal component simulated FET response
means and standard deviations tested equivalent with an a = 0.05 cumulative error. Figure
9 depicts the correlation coefficient recovery for the measured and simulated FET real part
o f S2i-
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23
lm(S11)
Re(S12)
Re(S22)
NFmln
Im(Gopt)
Correlation with Real Part of S21
M easured Q P . C .
^ + /-S ig m a
Figure 9: Comparison o f Measured and Simulated
Correlation Coefficients; Correlation o f Real Part of S2! with
Other FET Responses
Figure 10 illustrates the same for the imaginary part o f the FET's S2|. The other FET
responses are similar to Figure 9 in that the principal component model recovers the
correlation coefficients fairly well.
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24
1
lm(S11)
R e(S12)
Re(S22)
NFmin
Im(Gopt)
Correlation with Imaginary Part of S21
g j Measured J ^ j P .C .
^ |+ / - S i g m a
Figure 10: Comparison o f Measured and Simulated
Correlation Coefficients; Correlation o f Imaginary Part o f S2,
with Other FET Responses
Equivalence tests between measured and principal component simulated FET responses
with a pairwise a = 0.05 level of error showed that 54 o f the 66 FET response correlations
coefficients were statistically equal. O f those which failed equivalence, only two produced
simulated correlation coefficients that were opposite in sign as the measured values.
A 500 run Monte Carlo simulation was also done using the traditional ±<r method for a
comparison with the principal component method. The measured and simulated FET
response means and standard deviations tested equivalent with a a = 0.05 cumulative error
level. Figure 9 and 10 show the correlation coefficient recovery for the icr method as
compared to the measured and principal component data. Both graphs illustrate that the icr
model produces more significantly different correlation coefficients than does the principal
component method. In fact, only 22 out o f the 66 correlation coefficients tested equivalent
at an a = 0.05 level o f error for the icr method o f FET response simulation. Therefore, the
icy model produced almost four times as many significantly different correlation
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25
coefficients than the principal component method in the simulated runs. O f the
significantly different correlation coefficients, 21 had a sign opposite to that obtained from
the measured FET response database.
This means the ± a method was over 3 times more
likely to produce an incorrect sign for those correlation coefficients that were significantly
different. Both methods, ± ct and principal components, failed the cumulative pairwise
equivalence tests between measured and simulated FET response correlation coefficients
and are therefore can not be statistically equal to the original measured database. However,
it has been shown the principal component method is much more accurate at simulating the
measured database than the traditional ± cj model.
5. Discussion o f Results and Applications
There are several reasons why the principal component model fell short o f the goal to
produce a statistically equivalent simulated database. First, there may be inadequate
modeling o f the individual FETs. This may have been caused by not including the extrinsic
inductances in the FET model. The model optimization during ECP extraction may have
also introduced inaccuracies during the ECP extraction due to local minima in the error
functions. This extraction error could be indicated by the large percentage variances
exhibited by Rg, Rgs, Rj, In, and Im_Corr all o f which have been found to be difficult to
extract. Anholt et al. make an excellent case suggesting that the quality o f the statistical
modeling o f the FET is limited by the accuracy o f the extraction method [8]. It is also
possible that some o f the ECPs exhibited a non-linear relation which would cause errors
when modeling the relation using a correlation coefficient. Correlation is defined as the
linear relation between two variables and cannot accurately account for non-linear
relationships. This possibility was examined by producing scatter plots for all the FET ECP
like shown in Figure 11 for Gm and Tau. Figure 11 infers that there might be a strong
quadratic relationship present even though the correlation coefficient o f 0.05 is quite low.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
105
100
w
E
O
85
80
2.1
2.3
2.5
2.7
2.9
3.1
Tau Values (ps)
Figure 11: Scatter Plot of Gm and Tau (Correlation
Coefficient = 0.05)
Quantifying these non-linear relations is beyond the scope of this dissertation but is a
problem that will need to be overcome. Finally, non-normal distributions for the FET ECPs
may also be a definite problem because o f the Gaussian assumption during the simulation of
the Principal Factors. Normally, this problem could be diminished through the use o f data
transformations to get a more Gaussian distribution. However, the data presented in this
dissertation covered two different lots of wafers which caused some of the FET ECPs to
have distributions that could have been classified as bi-modal. Bi-modal distributions may
be caused by process shifts that will be hard, if not impossible, to model. Larger number of
FET samples would be needed to accurately test for this possibility.
FET parameter orthogonalization has also been shown to pose a better conditioned model
fitting problem [21]. Known correlations between the extracted FET parameters can be
forced on a FET model optimization by using the principal component equations.
Historical FET data or physics based models could be used for these known correlations.
Principal components can also be applied to Design o f Experiments (DoE) which requires
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27
orthogonal variables. The ability to reduce the model into a fewer number of principal
factors than the original FET parameters will enhance the usefulness o f FET variation
modeling in DoE.
Statistical population modeling could also be used as a criteria to monitor the validity of
active device parameter extraction. Once a population of FETs have been modeled, the
methodology illustrated in Figure 4 could be implemented to model the statistical FET
electrical responses and compare them to the original database. Creating a statistically
equivalent simulated population to a measured database is a much harder modeling problem
than representation o f a single active device. Failure to successfully model the measured
FET population could point out processing shifts, erroneous/non-physical extraction, or an
inadequate electrical model. Finally, this type o f parameter orthogonalization could be used
for large signal models as well as the small signal models demonstrated in this dissertation.
Non-linear relationships between the large signal model parameters might pose a problem
because the correlation coefficients only quantify linear relationships. Data transformations
could be used to correct this problem as well as enable the data to be modeled with
Gaussian distributions.
6. Conclusions
Many o f the prior works in statistical modeling o f FET S-parameters are difficult to
implement into current CAD software or are inaccurate in representing the FET population
during Monte Carlo simulations. A new methodology for statistically modeling the
extracted small signal FET parameters was developed and demonstrated. This method uses
the principal component technique to orthogonalize the extracted FET parameters into a
new set o f variables called Principal Factors. Equations for the extracted FET parameters
can then be written in terms o f a linear combination o f the orthogonal principal components
and easily implemented into current commercial CAD software. The modeling approach
was demonstrated on a small sample o f GaAs FETs with a periphery o f 300 pm and was
statistically tested using techniques discussed in this paper. The principal component
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28
methodology was shown to preserve the extracted FET ECP's mean, standard deviation, and
parameter correlations to a high level o f statistical significance. Using the methodology
significantly improved the ability to statistically model measured S-Parameter and noise
FET populations as compared to the traditional assumption o f ECP independence.
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29
CHAPTER III
STATISTICAL COMPUTER AIDED DESIGN
1. Introduction
It is difficult to meet the rigorous performance requirements that are needed in today's
competitive microwave circuit market. The design process eventually becomes a series of
choices made by evaluating circuit performance trade-offs. Unfortunately, the process of
making these choices is more o f an art than a science due to the complex relationships
driving a circuit’s responses. The response relationships make design trade-offs difficult to
quantify and therefore are seldom used to the designer's advantage. Performance optimizers
compound the problem by being extremely sensitive to the user-weighted performance
objectives. Computer optimization routines can create impossible circuit parameter
combinations, design circuit responses that are too sensitive to parameter variations [22], or
end up getting trapped in a local minimum without reaching the optimization goals.
However, a statistical technique known as Design of Experiments (DoE) can be used in
addition to the current design process in order to make circuit design easier and more
systematic.
Design o f Experiments is a well established area o f statistics that is used to make deliberate
changes to the input variables o f a system in order to identify differences in the system's
output responses. Response changes can be fitted using standard statistical regression
techniques to simple mathematical functions of the system's input variables. The
expressions are approximations in a particular region o f the circuit's designable parameter
values and reveal important response trends. The coefficient estimation o f a regression
model fits a linear equation for a product's response as a function o f the input variables
representing a circuit's designable parameters. Interactions between parameters and non­
linear terms can also included in the response model. The types o f circuit responses that
can be characterized can be anything such as simple amplifier gain, amplifier noise figure,
or circuit input impedance. The power of this methodology is obtaining simple empirical
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30
expressions for the product's response which can be used by a designer to gain insight to the
trends within some region o f parameter space.
G. Taguchi introduced the DoE techniques to engineering for quality improvement [23]. In
the past, enhancements to the DoE technique have been used on a production line or
laboratory to derive empirical models and optimize a given process. This approach is called
Response Surface Methodology (RSM) and is becoming increasingly popular in American
industry. However, the DoE technique may be incorporated within Computer-Aided
Design (CAD) packages to give engineers a powerful, yet simple, design tool [24]. A
computer can perform "virtual" experiments using the DoE's systematic methodology and
produce a simple expression which will almost always be less complex than the true
physical relationships that govern circuits. The empirical expression can then be used to
better understand the effect o f design variables, either alone or in combination, on a circuit's
response. This approach to empirical modeling will be called Statistical Computer-Aided
Design (SCAD) in this paper. SCAD is useful in the design environment because it can be
used to quantify performance trade-offs, perform goal optimization, and minimize circuit
variability. This combination of statistical techniques and CAD can enable bad circuit
designs to become good and good designs to become even better.
This paper presents the use o f the SCAD methodology for microwave circuit design. The
basic DoE concepts and terms are presented to give an overview o f the methodology and to
supplement other papers on the subject [25,26]. The intent of this research is to introduce
the microwave circuit designer to a new and beneficial way to design circuits for not only
nominal performance but for manufacturability. It should be emphasized that this statistical
design methodology is a useful alternative, not a replacement, to the current design tools
already available to the engineer. The SCAD methodology will be demonstrated in this
chapter on two microwave amplifier design examples.
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31
2. DoE Background
Many of the DoE concepts were popularized by Taguchi's contributions to the methodology
o f off-line quality design. The basis for his approach is to minimizing the "loss to society"
that occurs when a product's performance varies from a customer-specified target [27].
Taguchi's ideas for parameter and tolerance design have evolved into what industry labels
Design of Experiments for robust product design. The DoE technique describe parameter
settings that maximize the amount of extractable information in the minimum number of
experimental runs, or computer simulations, for fitting a regression model to a system
response. This aspect is a benefit to both physical and virtual experiments because it uses
the minimum amount o f resources (time, money, or computer) to achieve accurate
modeling.
What makes the DoE approach so powerful is that all of the significant controlling
parameters are changed simultaneously according to predetermined levels. This is much
more effective than other non-statistical methods used by engineers and scientists [24]
because important factor interactions can be missed when just changing one variable at a
time. In the DoE methodology, all o f the variables that can affect a product's performance,
such as lengths, doping densities, temperatures, or capacitance, are called factors. The
values that the factors are assigned are levels. There are designable factors which an
engineer can control to make the product perform in a desirable way. Examples o f these are
a circuit's capacitance, transmission line length, and doping levels. Noise factors, or
sometimes called environmental factors, are those which the designer can not control such
as aging effects, temperature, or natural processing variations in the designable component
values. One can only minimize the effect o f the environmental factors on circuit response
by favorable design choices. It should be mentioned that in the DoE approach both types of
factors must be independent, or orthogonal, to each other such that changing one variables
value does not affect any o f the others when using the DoE techniques. An experiment is
when all o f the factors are assigned a particular value, or level. In a DoE, each variable
would be assigned its value for the experiment, and the outputs or responses would be
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32
recorded for statistical analysis and model fitting. Examples o f the most useful
experimental designs for SCAD modeling are Full and Fractional Factorial, Central
Composite, and Box-Behnken [28].
Full Factorial experimental designs are those in which all possible combinations o f factor
levels are used in the analysis. Two factor levels is the most common number o f factor
settings in DoE designs because you only need two points to fit a line in a linear regression.
However, larger number of factor levels may be used depending on the type o f DoE design.
If n factors have 2 different level settings there will be 2n total possible combinations of
experiments. Table 5 shows an example o f a Full Factorial experiment with the 3
designable variables (X l5 X2, X3) each run at 2 level settings.
Table 5: 3 Factor, 2 Level Full Factorial Design
Experiment
Number
Factor X]
Level
Factor X2
Level
Factor X3
Level
1
-1
-1
-1
2
-1
-1
+1
3
-1
+1
-1
4
-1
+1
+1
5
+1
-1
-1
6
+1
-1
+1
7
+1
+1
-1
8
+1
+1
. +1
The values o f the variable levels are coded so that the high and low level experimental
settings are denoted by +1 and -1 respectively. The coding normalizes all o f the parameters
to unitless values which has some beneficial statistical properties [29]. Figure 12 shows the
geometric representation o f the experimental design in Table 5. The nominal point o f the
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33
factors, corresponding to zero for the coded factor level settings, is in the center and is
surrounded by the experimental level settings at each o f the cube's corners.
(+ 1 , + 1 , + 1)
(-1 . +1 ,- 1 ) *
r
^ (+ 1 , +i,i-i)
(0 ,
Variables
Nominal Point
0 , 0)
4
( + 1 , - 1 , + 1)
Exploritory
Point
Figure 12: Geometric Representation of Table 5
Experimental Design
Hopefully, the empirical response model will allow interpolation inside, and perhaps
extrapolate a bit outside, the exploration region defined by the cube in Figure 12 for the
example in Table 5. Normally, high and low level settings for each factor are chosen with
the parameter values that need to be interpolated in-between after the creation o f the
response model. Therefore, it is extremely important for non-linear responses that the
experimental high and low values are not too far apart as to prevent accurate interpolation
o f the user defined model. Experiments w ith more than 3 factors are difficult to visualize
geometrically but follow the same concept presented in Figure 12.
In the Table 5 example, the empirical model is built by setting the 3 factors to the
appropriate levels for each experiment and recording the responses for statistical analysis.
Regression techniques are then used to fit the recorded response values to a user-defined
linear model such as the one shown in Equation 6.
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34
y = P,+
/=1
+ E E m * / + £J
/=l >1
i*j
(6)
where n is total number o f designable factors, P's are the regression coefficients, and Sj
represents the total error in the regression model. The first term, /?„, in the model is the
regression equation's intercept. The second term represents the main factor response effect
o f the
factor. The third term in Equation 6 is the joint effect caused by the first order
interaction o f the / by j main factors. The benefit to this type o f simple model is that it
easily shows the larger response trends. Simple linear equations do not have any local
minima that cause problems for gradient optimizers when finding the best parameter
settings. The limitation o f this type o f modeling is there may be some difficulty using these
equations for non-continuous or quickly changing responses.
Other types o f models are available but the most popular for RSM is the quadratic [30].
The quadratic model is the same as Equation 6 except that the tej restriction is removed
which requires the factors to have more than just two levels settings. Unfortunately, large
number of factor levels dramatically increase the total number o f experiments needed to fit
the model. Certain DoE's have been designed such as the Central-Composite and BoxBehnken which work well with the quadratic model [28].
The designer must keep in mind
that the regression fit is only a simple mathematical model and may not have much physical
significance. The model should only be used in the small "exploration region" of parameter
space that the DoE was performed. The entire equation would probably change
significantly, particularly the interaction terms, when the DoE was performed in another
area of parameter space. However, if the empirical model is good then it should enable a
designer to optimize their process or circuit within the exploration region even though the
model may not hold much physical significance.
As the number o f factors increase, Full-Factorial Experimental designs create prohibitively
large experimental runs. It would not be unusual to have 10 factors in a DoE which would
need 2 1°=1024 total experiments. This many virtual experiments can take a large amount of
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35
processing time even with a powerful computer. Furthermore, by using an ^-factor fullfactorial experiment one obtains information about all possible factor interactions up to and
including the interaction term containing all «-factors. Typically, statisticians do not
include higher than first order interaction terms,
•x ;., because the effects due to higher
order interactions are difficult to interpret. Therefore, one can reduce the total number of
experiments by sacrificing some o f the information about the higher order interactions
which would have typically been dropped from the empirical model anyway. The type of
experimental designs that do not run all o f the possible combinations o f level settings are
called "Fractional Factorial" designs. These types o f designs exhibit confounding which
means that two or more factor effects can not be separated due to the lack of information.
The factors which are confounded can be selected by the user if the experiments are
carefully designed. As mentioned before, the second and higher order interactions are
usually intentionally confounded so as to obtain a smaller number o f experimental runs.
Therefore, knowing what effects are confounded is very important. A person can determine
which interactions are confounded by examining the resolution of the experimental design.
"Resolution V" experiments are needed for all model factors in the quadratic form of
Equation 6 to be unconfounded with each other. This is the type o f resolution that is
recommended for response characterization in RSM. Both Central-Composite and BoxBehnken experimental designs are Resolution V [30].
3. DoE Application - A Design Example
The previous concepts can be applied to statistical modeling of microwave circuits. An
example of the methodology has been developed for the 2-stage low-noise microwave
amplifier shown in Figure 13. The amplifier was designed to operate in the 4.5 to 5.0 GHz
frequency band with over 23 dB o f gain and a noise figure less than 1 dB. The input and
output match performance goals were both to have a return loss less than -8 dB. The two
FETs used in the circuit were arbitrarily picked to be the NEC4583 from an S-parameter
database.
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36
l=x 8
c=x5
l=x 2
L=Xi
IN
f
l=x 2
l=x 3
l=x 4
„
L=1000
l =x 7
l=x 6
*
I— II- •-<— - ■
1
OUT
l=x 8
Figure 13: 2-Stage Low Noise Amplifier Topology
Figure 13 shows the low-noise amplifier had eight designable parameters, variables Xj
through Xg, that were used to adjust the performance o f the amplifier. All o f the factors
were 50Q transmission lines lengths except for X5 which was the value o f the DC blocking
capacitor. Touchstone was used to optimize the design parameters with the user-defined
design goals. The optimized parameter values just met the design specifications and were
accepted as being a valid design. These optimized values are listed in Table 6 and were
coded so that they were the nominal (zero) values in a DoE analysis. Table 7 shows the
performance o f the amplifier using the Touchstone optimized "Nominal Values". To
achieve a better performance, a SCAD DoE methodology was implemented on the circuit to
quantify the design trade-offs and design the better performing circuits listed below the
“Nominal Values” case in Table 7. The other cases will be discussed later.
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37
Table 6: Low-Noise Amplifier Nominal Design Values and
Coding
Variable
Nominal Value
15% (H igh,+1)
-15% (L ow ,-1)
x,
973 pm
1119.0 pm
827.1 pm
X2
2139 pm
1818.2 pm
2459.9 pm
X3
4890 pm
5623.5 pm
4156.5 pm
X4
6879 pm
5847.2 pm
7910.8 pm
X5
0.9609 pF
1.105 pF
0.817 pF
X6
6498 pm
7472.7 pm
5523.3 pm
x7
6099 pm
7013.89 pm
5184.2 pm
X8
1391 pm
1599.7 pm
1182.4 pm
Table 7: Low-Noise Amplifier Performance Cases
*Nominal Values Optimized by Touchstone
R esponse Case
Nominal Values*
M inimize N oise Figure
M aximize Gain
Met All Specifications
Good LN Amp
Coded Values
Gain
NF
s„
$22
(dB)
(dB)
x,
X2
X6
x7 x8
(dB)
(dB)
0
0
0
0 0 0 0 0
0 0.2 1 0
1
0 0 -1 -1 -1
0 -0.1 0 -1 0
1.5 0.1 -0.5 -1 0.5
0 0
-0.4 -1
-0.3 1
-0.2 -0.1
-0.3 -1.4
22.96
24.92
25.05
1.02
0.65
1.32
-8.11 -10.50
-3.94 -6.24
-17.94 -19.81
23.29
23.27
0.92
0.80
-8.25 -10.35
-11.44 -8.58
0
0
X3
X4
X5
The SCAD methodology depicted by the flow chart in Figure 14 was applied to nominal
value circuit design. All o f the eight designable variables listed in Table 6 were identified
as the designable factors in a DoE plan.
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38
Implement Circuit
in CAD Program
Finished ]
Identify Designable
and Noise Factors
Design an
Experimental Plan
Perform Virtual
Experiments
Statistical
Analysis
Yes
No
Meet
Specifications
Implement New
Nominal Values
Optimize Circuit /
Minimize Variability
Figure 14: SCAD Methodology for Circuit Design
A full factorial experimental plan, like the one shown in Table 5, would require 28=256
"virtual" experiments runs. A full-factorial design was impractical both because only the
first order interactions were desired in the response model and this Designed Experiment
was being performed by hand. Therefore, higher interaction confounding was intentionally
•
8 2
introduced by running a Resolution V DoE which consisted of 2 ’ = 6 4 experimental runs.
Modeling of the quadratic response model was desired so center points were added to make
the design Central Composite Fractional Factorial. The center points were chosen using the
commercial statistical software package SAS® interactive DoE Designer which suggested a
total o f 81 experimental runs [28]. Table 6 shows the DoE used high and low values that
were ±15% o f the nominal values which were coded +1 and -1 respectively. The responses
o f interest for each virtual experimental run were the Gain, Noise Factor, S) j, and S22 o f the
amplifier from 4.4 to 5.1 GHz at 0.125 GHz steps. These values were recorded in a
database for statistical analysis.
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39
SAS was used to fit the quadratic form of the Equation 6 model for each o f the responses
at the 4.75 GHz mid-frequency point [14]. The code used to perform the analysis is
included in Appendix A. A statistical measure o f the regression model's "goodness-of-fit"
•
2
2
is called the R-Square (R ) value. In this application, R is the proportion o f observed
variability in the simulated response that is explained by the regression model. This value
is calculated from the total amount o f error in the model as identified by Equation 6 and is
between the values of zero and one with one indicating the statistical model fits the data
with no errors [31]. Table 8 lists each response's R2 value and shows that all o f the
responses are being modeled reasonably well at the experimental design points.
Table 8: Low-Noise Amplifier Modeled Response Values
Response
R2
Gain
0.886
Noise Figure
0.803
s„
s22
0.695
0.684
The R2 is an indication o f "goodness-of-fit" only at the actual factor level settings o f the
experimental runs. The optimal parameter settings for the factors are most likely not to lie
at the high, low, or zero factor level settings. Therefore, one is interested in the accuracy of
the statistical models in-between our high (+1) and low (-1) level settings. To do this,
twenty sets of random parameter values were picked within the limits of each of the
experimental factors and the circuit responses for each set were recorded for both the CAD
and the statistical model results. The difference between these two values was used to
determine the amount o f error in the statistical model for the random parameter sets in what
could be termed a "reality check". Figure 15 shows the experimental error for all four
responses when the 20 random parameter value sets were used.
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40
50
45
40
35
2 30
lu 25
§, 20
|
15
S
10
fc
5
0
-5
-10
-15
18 19 20
Random Trial Number
-a- Gain (dB)-«- NF (dB)
+ S11 (dB) x S22(dB)
Figure 15: Model Error from Random Factor Settings
It can be seen that the models with the largest errors, the input (S) ,) and output (S22)
matches, also had the lowest R2 values shown in Table 8. However, Figure 15 shows that
"reasonable" results can be achieved through the rather simple quadratic models used for
the responses. If the error bounds had been unacceptable then this would have indicated
that the DoE was performed with low and high levels that were set too far away from the
nominal values. The CAD experiments would then need to be repeated with less variations
in the factor level settings. However, in this demonstration one is looking for a trend
analysis and very accurate predictive models are not needed.
Once the empirical model is fit using the regression techniques, one can determine which
factors, or combination o f factors, explain very little o f the variation seen in the response
data. Those factor terms can be dropped from the model for simplification purposes
without sacrificing any significant modeling accuracy. A statistical significance test is
performed on each term to determine which can be dropped. In our example, the
significance test o f each model term was performed using a standard two-tailed t-test at an
a = 0.05 error level o f significance [32]. If the experimental design is orthogonal, then
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41
dropping the non-significant factor does not change the coefficient values o f the other
significant factors. However, the quadratic model causes the experimental design to be
non-orthogonal and the model's significant terms have to be refit after the non-significant
factors are dropped. This was the case for the responses in this particular DoE and the
regression coefficients were re-estimated with the reduced model. An example o f the
equation for S tl containing only significant factors and interactions is shown below in
Equation 7.
S „ = -7.92 + 2.75 X3-X3 + 2.64 X4-X4 + 1.72 X6
+ 1.04 X4 + 0.62 X2-X3 - 0.45 X3
(7)
Equation 7 originally had 45 terms, including the intercept, before the non-significant
factors were dropped leaving only 7 terms in the regression equation. O f course, the
number o f significant terms varies with the actual response characteristics and number of
experimental runs but these examples show that only a few significant factors need to be
included in the response model. Deleting non-significant terms is usually only done to
facilitate writing the equations or to present a reasonable amount o f information to the
designer.
The most significant factor coefficients in the model for each o f the gain, noise figure, input
and output match responses were ranked from largest magnitude to smallest and displayed
on the Pareto Charts in Figures 16, 17, 18, and 19, respectively. A Pareto Chart is a
graphical ranking o f the importance o f response model effects. Typically, only the
statistically significant effects are presented in the Pareto Chart which allow the designer to
easily see what influences the response in question. Pareto Charts are commonly used by
statisticians and industrial engineers.
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42
1.2
% = Negative Coefficient
c 0.8
<D
O
E
g 0.6
O
C
ro
r\ a
o 04
0.2
X4*X6 X3*X3 X7*X7
X5
X2*X3 X5*X6 X4*X5
X4
X7*XB- X3
Factors
Figure 16: Pareto o f 2-Stage Amplifier Gain
0.8
• = Negative Coefficient
gj 0.6
'o
E
d)
o
o
0) 0.4
3
O)
Ll
0)
CO
'o 0.2
0 J—
L+J— L+J— 1-h—H-l—H-l—H-l— H-l 11 I L
X7*X7 X7 X7*X8 X8 X4*X7 X3*X7 X6*X7 X4*X6 X4
Factors
Figure 17: Pareto of 2-Stage Amplifier Noise Figure
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43
• = Negative Coefficient
1.5
X3*X3
X4*X4
X6
X4
X2*X3
'
X3
Factors
Figure 18: Pareto of 2-Stage Amplifier Si
3.5
• = Negative Coefficient
3
2.5
c
0)
ea>
2
o
O
1.5
CM
CM
CO
1
0.5
X 7*xr
X4
X6
X7*X8
X8
X3*X7'
■T~U»L-SL
X5 X3*X6' X3*X4
Factors
Figure 19: Pareto of 2-Stage Amplifier S22
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44
Referring to Equation 7, one can see each o f the significant factor coefficients for the
amplifier's input match Sj | are shown in the Pareto Chart in Figure 18. Negative
coefficients are shown with a dot in the factor's graph bar. The Pareto Charts visually
present the circuit's design trade-offs by showing the relative magnitudes of the most
significant factors. Ranking the model coefficients from largest to smallest lets the designer
see which factors, or combination o f factors, account for the most variation in that response.
For example, Figure 16 shows that the interaction between the two transmission line lengths
in the matching interstage (X4-X6) has the greatest affect on the amplifier gain. One can
also see that the input matching network's transmission line length X3 has the smallest
significant effect on the amplifier gain. It can easily be seen from the Pareto charts that the
transmission line length X! does not significantly influence any o f the responses as would
be expected. Therefore, X! can be totally ignored in the subsequent analysis and
optimization within this volume o f design space.
The Figure 16 Pareto chart lets the designer visualize that if both the Factor X4 and X6 line
lengths were increased to the +1 and +1 factor level settings, then gain o f the entire
amplifier will also increase because the X4-X6 coefficient is positive. If one o f the factors
was increased while the other was decreased (that is, one at +1 the other at -1 causing X4-X6
to be -1) then the amplifier gain would tend to decrease due to the positive interaction
coefficient. Comparing each of the Pareto Charts to each other leads directly to trade-off
conclusions. For example, X4 is shown to be the only main effect factor that affects all the
responses significantly. Referring to Figure 17 , decreasing the noise figure (desirable) of
circuit in Figure 13 by making the X4 factor lower will also decrease gain (undesirable), S(j
and S22 (desirable). The designer can optimize the amplifier by hand choosing the
designable parameter values which give the most desirable trade-offs. Once the designer
changes the designable parameters, the design can be re-simulated and the performance
evaluated. Often, this process would have to be iterated until the design meets the response
specifications as shown in the Figure 4 SCAD methodology flowchart.
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45
The amplifier in Figure 13 was optimized by hand using the Pareto Charts for several
different single frequency performance goals to show the versatility o f the SCAD
methodology. Each o f these performance cases are detailed in Table 7 along with the coded
parameter and performance values. The first two optimization cases, "Minimize Noise
Figure" and "Maximize Gain", used only their respective Pareto Chart to optimize their
response regardless o f the expense o f the other circuit responses. The coded values shown
in Table 7 indicate the designer has tremendous flexibility in optimizing the design using
the SCAD methodology. Minimizing or maximizing a response is aided by the fact that the
designer has an second order equation for a response which can be easily minimized with
respect to a particular variable. Figure 17 shows an example of this for the amplifier's noise
figure which is minimized when X7 is set to -0.4 level because both X7 and X7 X7 factors
are significant.
Both "Met All Specifications" and "Good LN Amp" in Table 7 were optimized by looking
at all of the response Pareto Charts in order to evaluate the performance trade-offs. Both
cases were obtained only through hand tuning with only the Pareto Charts supplying the
needed "roadmap" to find optimal parameter configurations. "Met Specifications" in Table
7 used parameter values which were kept within the ±15 % bounds (-1 to +1) that the DoE
had been performed. However, the "Good LN Amp" case shows that values far outside this
range, such as X2 and Xg, may provide useful optimization points because the general
response trends may continue even when the regression models lose accuracy [33]. Finding
optimum parameter points may be helped by using linear and non-linear programming
techniques for this type o f multiple constrained optimization problem [32]. Taguchi
advocated a "pick the winner" scheme o f optimization by looking for the experimental run
that gives most desirable responses [23]. However, it is not probable that one o f the
planned experiments would happen to set the circuit's designable parameters at their
globally optimum values and this method should not be normally used for optimization.
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46
4. Variability Reduction
Perhaps the most exciting application o f the SCAD methodology is for reduction o f circuit
response variability. All circuits that are produced have some inherent variability in them.
Large circuit variation tends to cause high yield losses. Response variability is due to two
types of factors: designable fa cto r variation and environmental noise. The effect of
designable parameter variation is easily seen by the Pareto Charts. Figure 19 shows that
low noise amplifier output match is much more sensitive to the X4 designable parameter
than the X5 factor. Therefore, if the length X4 was to vary a small percentage while the
circuit was being produced, that would effect the S22 response more than if the length factor
X5 were to vary that same percentage. One can see it is in the designer's best interest to
reduce the fabrication length variations o f the X4 factor more so than the X5 factor’s length
variations. A designer can minimize this propagation o f production variation to the circuit
responses by using the DoE approach to identify, or screen, the most sensitive parameters
and focus effort on controlling their variability [34].
The second type o f circuit variability is due to environmental, or “noise”, variables such as
changes in bias voltages, small signal FET parameters, temperature, or aging. CAD
packages can simulate the effect o f these variables. The variability due to these parameters
can then be minimized by choosing designable parameter settings that cause the circuit to
be least sensitive to these noise variables. Often, the designer cannot totally minimize the
response variability without mis-centering the design. Taguchi described a method o f
achieving this type o f robust circuit design through the use o f inner and outer array DoEs
[27]. The inner array is the designable factor DoE plan such as the one discussed in the
previous section and shown in Table 5. The outer array is a separate designed experiment
using only the noise variables. When replications are made in actual measurements for a
system’s particular level settings, the response will not be the same due to measurement
error and slightly changing environmental conditions. In CAD virtual experiments, re­
running the same level settings always will give the same response. Therefore, circuit
variation must be introduced by using the outer array experiment set as replications o f each
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47
of the designable parameter array's experiments. Figure 20 shows an example o f an
experiment with two designable factor array variables (¥,•) being replicated with a full
factorial array o f two noise factors (0y). The replications across each row will force
variability o f the response variables for each set of designable factor settings (T,). The
variability can be modeled and then minimized by using the designable parameter factors.
Outer Noise Array
Inner Designable Array
-1
-l
+i
+i
©i
©2
*1
^2
-1
+i
-l
+i
-1
-1
yn
yi2
yi3
yi4
-1
+1
y2i
y22
y23
y24
+1
-1
y3i
y32
y33
y34
+1
+1
y4i
y42
y43
y44
Figure 20: Inner and Outer Array DoE
Designable (Outer
Array) Factor, 9n Noise (Inner Array) Factor, and Responses
yy
Any number o f noise factors can be used in the outer array. It should be mentioned that the
outer array design can be a fractional factorial because the noise variables are not used as a
predictor in the regression model equations. However, large numbers o f noise factors can
create prohibitively large outer arrays even when highly fractionated factorial designs are
used. In these cases, using random permutations to obtain the noise array is suggested.
This is equivalent to using the Monte Carlo method to induce variations due to random
noise factors in the design. This approach models the true environmental noise more
accurately than the method o f selecting the noise factor levels. However, to guarantee that
the entire noise parameter space is covered, a large number o f Monte Carlo level
combinations must be made [35]. A SCAD methodology user can use specific noise level
settings in order to run fewer total number o f virtual experiment simulations when only a
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48
small number o f noise parameters are being studied. Either approach should give
equivalent comparisons o f the response variance.
An example o f variance reduction will be performed on the single stage amplifier shown in
Figure 21. A circuit with a small number o f designable factors was chosen to keep the
number o f required simulations low. The environmental noise factor chosen was the
amplifier's input impedance termination. The amplifier's gain will be affected as the
terminating input impedance is changed from the 50Q source impedance that was used
during the nominal design. It would be desirable to make the amplifier performance
insensitive to these variations in the source impedance. DoE provides an easy way to
characterize and then minimize this sensitivity
L4
^ Y Y Y \
OUT
L3
Figure 21: Single Stage Amplifier for Variance Reduction
Example
The single-stage amplifier was designed using the microwave CAD simulator Touchstone®
with lumped inductors to achieve 50Q terminations on both the input and output ports.
Afterward, a Taguchi inner noise array was constructed using the four inductor values in a
Resolution V Box-Behnken DoE. Table 9 shows the inductance values and their high and
low values set at 8% o f nominal values in order to get high modeling accuracy. The noise
factor array consisted in the termination o f the amplifier's input with the 5 different types o f
impedances shown in Table 10.
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49
Table 9: Single Stage Amplifier Nominal Design Values and
Coding
Variable
Nominal Value
8% (High, +1)
-8% (Low, -1)
LI
5.48 nH
5.92 nH
5.04 nH
L2
11.46 nH
12.38 nH
10.54 nH
L3
14.94 nH
16.14 nH
13.75 nH
L4
0.22 nH
0.24 nH
0.20 nH
Table 10: Input Match Terminating Impedances for Taguchi
Outer Array
Resistance (Coding)
Reactance (Coding)
Mag(T)
Ang(T)
50 (Nominal,0)
0 (Nominal,0)
0
0
25 (Low ,-l)
-25 (Low,-l)
0.447
-111.7°
25 (Low,-l)
25 (High,+1)
0.447
111.7°
75 (High,+1)
-25 (Low,-l)
0.277
-33.7°
75 (High,+1)
25 (High,+1)
0.277
33.7°
0
( )
Touchstone defines terminating impedances matches in terms o f magnitude and phase of
the termination's reflection coefficient, with respect to a characteristic impedance of 50Q
[36]. The terminating impedances were picked for perfect match and 4 different quadrants
o f the Smith chart. The outer noise array consisted o f placing each termination on the
amplifier's input and calculating the amplifier's gain for each experiment level setting o f the
inner array.
After the DoE was run, a regression model was used on the gain variance introduced by the
different terminations. Equation 8 shows the resulting expression for the gain's variation
(dB) with significance o f the coefficients determined at an a = 0.05 level.
again = 0.96 + 0.328 LI + 0.081 L2 + 0.232 L3 + 0.024 L4
(8)
- 0.083 LI-LI - 0.062 L1-L2 + 0.022 L1-L3 + 0.024 L1-L4
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50
The Pareto Chart o f the variance equation's ranked coefficients is shown in Figure 22 while
a Pareto Chart o f the amplifier's average gain is in Figure 23. The regressions models for
the gain and gain variance had an R2 o f 0.99 and 0.96 respectively. At the nominal design
point (all inductance codings set to zero), the gain was 16.9 dB with a standard deviation o f
0.96 dB.
0.35
c
<D
'o
G
0)
• = Negative Coefficient
0.3
o 0.25
o
c
o 0.2
ra
> 0.15
o>
Q
•E 0.1
co
TJ
**
c
CO
4-<
0.05
CO
0
L1
L1*L1
L2
L1*L2 L1*L4
L4
L3
JZL
L1*L3
Factor
Figure 22: Single Stage Amplifier Gain Variance Pareto
Chart Accounting for Input Termination Variability
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51
0.4
• = Negative Coefficient
c
0.3
<D
O
t
g 0.2
O
c
'r a
0
L3
L2
L1*L1
L1*L2
m
L1*L3
i m
L1*L4
Factor
Figure 23: Single Stage Amplifier Average Gain Pareto
Chart Accounting for Input Termination Variability
Equation 8 and the Pareto chart in Figure 22 show the total gain variance can be reduced by
picking all of the designable parameters at their coded low (-1) values. The amplifier's
average gain Pareto Chart in Figure 23 indicates that picking all o f the factors at their low (1) will also tend to increase the gain which is a favorable trade-off. With the new all low
designable level settings, the amplifier gain was 17.4 dB, a 0.5 dB increase, while the
standard deviation o f the gain with respect to the noise variables was reduced 58% to 0.41
dB. This shows that the “by-hand” optimization can both increase nominal value while
decreasing the variance o f the circuit gain in a straightforward manner.
Another noise factor which affects the amplifier response is active device variations. DoE
factors must be orthogonal so the Principal Component method was chosen to introduce
physically realizable FET variations into the DoE noise array [37]. The FET model
parameters were varied by changing the first two principal components in the FET model
methodology which corresponded to 86% o f the total variation in the small signal FET
parameters. Picking only some o f the principle components enables the outer array to have
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52
fewer factors and require a smaller amount o f experimental runs. The same inner array DoE
as the previous variation model was run for this set o f virtual experiments. This resulted in
the standard deviation model (in dB) in the following Equation 9 and illustrated in the
Pareto Chart shown in Figure 24 for an a = 0.05 level o f significance. The R2 o f Equation 9
was 0.99.
again = 1.283 + 0.489 LI + 0.162 L2 + 0.022 L4
(9)
- 0.063 LI -LI -0.019 L1-L3
0.7
• = Negative Coefficient
c
0)
'o
0.6
IE
nc
d) 0.5
O
O
|
0.4
ra
5
Q
•o
X(6J
c
(0
0.3
0.2
CO 0.1
0
L1*L3
Factors
Figure 24: Single Stage Amplifier Gain Variability Pareto
Chart Accounting for Intrinsic FET Variability
The model and Pareto chart indicate that setting all o f the factors at their coded low level (1) will reduce the variations in the circuit's gain response. Coincidentally, these are the
same results as were shown when the input termination was used as the noise parameter.
Apparently, those settings for the designable parameters create a circuit robust to a variety
of environmental factor variations. Also, this example shows that designing a more robust
circuit does not always mean sacrificing performance. The principal factor outer array
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53
could have been combined with the input termination array, and others, to form one
standard deviation model for the amplifier gain but this was not done for example clarity.
5. Discussion o f Results
It can be seen that the SCAD modeling methodology gives a reasonably straightforward and
systematic way to optimize circuits. There is an effort to integrate these tools directly into
the microwave CAD packages so the experimental design, response modeling, and Pareto
Charts do not have to be done by hand [38, 25]. Currently, these tools use a goal oriented
approach with Taguchi loss functions which is slightly different from the approach
discussed here. This dissertation shows that minimizing variance in microwave circuit
responses is a very exciting area o f the SCAD methodology. Robust circuits can be
produced in a straightforward manner through the use o f the new type o f variation
introduction and quantification. Other types o f responses such as gain ripple, efficiency, or
third-order intercept could be modeled and optimized using the SCAD modeling
methodology. However, these types o f circuit responses are more complex than circuit gain
or input match. Our own research into yield modeling has shown that yield response
surface is too complex to be modeled with the simple linear regression models advocated in
the paper. Modeling over frequency also seems to be SCAD modeling issue that needs to
be addressed by future research.
DoE response modeling can also be easily combined with more complex statistical models
for predictive circuit response models in a particular region o f parameter space. This
approach is similar to Macro-Modeling which has been used to model certain circuits which
have slow simulation times [39, 40]. Macro-modeling with SCAD could be implemented
for circuits requiring harmonic balance simulations or electromagnetic field solvers. The
required experimental simulations could be done overnight when the computer time is not
normally used. The statistical macro-models then could be used, within some bounds o f the
parameter values, to achieve much faster optimization or design tuning o f a circuit. This is
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54
especially useful to numerical methods requiring meshing because the empirical response
equations will give results that can be interpolated in between mesh points.
6. Conclusions
A new methodology has been demonstrated for microwave circuit design. The approach
uses a combination of Statistical Experimental Design and CAD, called SCAD, and
enables a designer to statistically characterize circuit response in a very systematic way.
Design trade-offs can be quantified with the simple surface response models from the
"virtual" experiments performed by the CAD package. Variation minimization o f a circuit's
response due to noise parameters inherent in circuits can be achieved with this type of
methodology which enables a designer to create robust circuits. The SCAD methodology
has been demonstrated on two different amplifier designs each with varying types o f
designable parameters and optimization goals. Different types o f designed experiments
were used to show the flexibility o f the approach. The introduction o f two types o f
variation noise parameters that are well suited for SCAD, circuit terminations and FET
principal components, were also discussed. The SCAD methodology will prove to be an
invaluable design tool for a designer making better, more robust circuits that exhibit higher
yields.
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55
CHAPTER IV
BILINEAR VARIABILITY COMPARISONS
1. Introduction
It is becoming increasingly important for microwave circuits to exhibit low performance
variability especially as microwave systems continue to become more complex.
Determination o f network variability has usually been achieved by using numerical
techniques such as Monte Carlo analysis [35]. This type o f analysis depends on assignment
o f circuit element distributions which may not be known. This chapter details a method for
determining the variability o f different S-parameter networks in closed-form without
knowledge o f the network’s parameter distributions. The technique allows the designer to
easily compare many different multi-port networks in order to determine the one that has
the lowest variability. Circuits evaluated in this way can be designed to achieve greater
performance predictability and higher manufacturing yield.
The variability comparison has its roots in the bilinear transformation. The complex
impedance o f any circuit element can be uniquely mapped into the complex plane
representing any circuit port’s Y, Z, or S-parameters. This bilinear transformation results in
a second, rotated Smith chart scaled and superimposed onto the network's response of
interest. The second Smith chart represents the extent of changes possible in the circuit’s
response due to changing the value o f the circuit element. Figure 25a represents a 2-port
network terminated in a load impedance with reflection coefficient T, 0AD. A possible
mapping o f TL0AD onto the network’s input reflection coefficient, TIN, is shown in Figure
25b [41].
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56
2-Port
Network
LOAD
F in
Locate
1 LOAD
Here(a)
(b)
Figure 25: (a) Two Port Network (b) Bilinear Transformation
o f an Arbitrary Load Impedance ( r LO a d ) onto the Input
Reflection Coefficient (TIN)
Any load impedance o f interest can be located on the scaled, rotated Smith chart which
determines where T,N will lie on the larger Smith chart. The location, size, and rotation of
the mapped Smith chart is determined by which response is being examined, the bilinear
transformation, and the network’s S-parameters. Bilinear mapping has been used in the
past for filter [41], amplifier, and oscillator design [42]. A novel application o f the bilinear
transformation is the use o f the information about the mapped Smith chart radius to make
network variability comparisons.
2. Bilinear theory
Functions o f linear networks using complex variables are bilinear in nature. That is, those
functions are linear for both the dependent and independent complex variables. Equation
10 shows the generalized bilinear function between a dependent response function w and
the independent function Z, both o f which can be the impedance, admittance, or scattering
parameters o f a multi-port network.
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57
a ,Z + a,
W = ^Q$Z
T . 7+1
V
( 10>
Any bilinear function can be expressed in the form o f Equation 10 such that the function
mapping coefficients a u a2, and a3 can be determined. These coefficients can then be used
to determine the mapping parameters o f the independent variable Z into a response w. O f
particular interest is the radius o f the mapping function which is expressed in Equation 11
in terms o f the mapping coefficients [41].
a2a3 + a x
(11)
R,„ =
a 3 + ° i
For example, the well-known bilinear function for the input impedance S',, o f the 2-port
network shown in Figure 25a is represented by Equation 12. S',, represents the input
reflection coefficient, T,N, which varies as the load impedance is changed.
Ci
_
C
,
/ i o \
"7 T c ~ r
1
22
LOAD
'
)
T,N can be considered dependent on the independent complex variable TL0AD and the
network’s 2-port S-parameters. Therefore, Equation 12 can be manipulated into Equation
10’s form in order to determine the mapping coefficients. Doing so yields the bilinear
coefficients described in Equations 13, 14, and 15 [41].
*^11 “
O ^ l 1*^22
i t s !22
*^ll + (^11 ^22
(13)
^ I2 ^ 2 l)
(14)
1+ ^22
l-S n
«3=77T^
1+ $ 22
05)
These coefficients can be used in Equation 11 to calculate the mapped Smith chart radius
such as was used to obtain Figure 25b. It is important to realize this can be done for any
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58
bilinear function although the coefficients would have a different form than represented in
Equations 13,14, and 15.
One o f the important concepts o f bilinear functions is that points and circles always map
into points and circles due to the linear nature o f the transformation. Suppose there was an
interest in minimizing the changes o f TIN due to the TL0AD variability. An example H
oad
distribution is shown as the locus o f points, roughly a circle, in Figure 26a. When the locus
of r
load' s
bilinearly mapped to the TIN input match Smith chart, the size o f the mapped
distribution depends on the bilinear transform radius Rm The TIN distribution will have a
greater spread, hence more variance, when the mapped Smith chart is large as shown in
Figure 26b.
rL0AD
Mapped
Distribution
T load
Distribution
0
1 LOAD
(a)
(b)
Figure 26: (a) T L 0A d Distribution (b) Locus Mapping of
T l o a d t 0 T in Showing Variability Increase
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59
For this example,
r,Nvariability would be less for the bilinear transformation depicted in
Figure 25b than for the mapping shown in Figure 26b. Thus, in order to make
r,Nmore
robust to variation in TL0AD, the radius o f the bilinear transformation must be minimized.
The bilinear mapping technique only characterizes the potential variability in a multi-port
network due to the fluctuations in a system’s independent variables. These variables can be
any complex value as long as the bilinear mapping transform o f the response can be
expressed or calculated. A response does not have to be expressed in a closed form solution
for application o f the mapping technique [41]. By applying the method, an entire system’s
potential variability can be minimized by designing the circuit’s S-parameters so as to
reduce Rm. The mapping radius can be shrunk or enlarged as well as moving its location
just by changing the network parameters as shown in the example Equations 11,13,14, and
15. Also, the mapping radius value for several different networks can be used to determine
which one has the least potential variability. The decision on what type o f network should
be used can be made by a straightforward comparison of mapping radii.
Minimizing the variability in one type of response can help other system responses. For
example, minimizing the variability in the input and output matches has the side benefit of
reducing the fluctuations in a network’s power gain. Equation 16 is the transducer power
gain of a 2-Port network. A simpler form is shown in Equation 17 which depicts the
transducer power gain o f a network as the inherent active device S2| gain (G0) multiplied by
the input (Gs) and output (GL) match gains [43]. Equation 16 can be applied to both the
gain o f an amplifier or for a single active device such as a FET.
G7.= '
H r ,f
'
|i - r wrs.|
or
Gy = Gs • G0 • G,
H r,
(16)
(17)
Equation 17 indicates that the variance in the transducer gain can be lowered by decreasing
the variance in any o f its components. For a single active device, inherent variability in
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60
active devices S-parameters can not be changed except through modifications in the
fabrication process, which the designer has little control, or possible addition of feedback
circuitry. However, by minimizing the variability of the FET’s input match with additional
circuitry, the inherent fluctuations in FET transducer gain will also be lowered. Similar
work could be done to minimize the variability o f a 2-port network’s
fluctuations in seen in the circuit’s
r0UTresponse due to
rSOURCE.
3. Bilinear Mapping Application
An example application of bilinear mapping radii will be shown in this section to illustrate
the method’s potential. Active devices are known to be a large component o f microwave
circuit performance variability. One o f the popular methods o f reducing this variability is
to incorporate either series or shunt feedback topologies, shown in Figure 27, into a circuit.
However, knowing which feedback topology to use and in what amount has been largely
left to engineering judgment.
H=USeries
f 's h u n t
(a)
(b)
Figure 27: (a) Series Feedback Topology Used for FET
Variability Reduction (b) Shunt Feedback Topology
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61
The bilinear mapping technique allows the designer to easily compare the advantages of
shunt and series feedback in order to select the best circuit topology in terms o f variability.
The radius Rm o f the load Smith chart mapped onto TIN can be calculated with Equation 11
for each topology over all possible feedback values. Equation 18 shows how Rm can then
be normalized to the network having no feedback. RN allows a comparative analysis o f the
feedback’s affect on performance variability. The no-feedback condition is shown in
Equation 19 for both feedback topologies in Figure 27.
n
Feedback
**=7fesar
m
No Feedback:
08)
= 1 Z 1 8 0 ° ,T ^ , = 1Z0°
(19)
Values o f R n that are less than one indicate the response variability has been reduced
compared to a no-feedback case. The variability comparison starts with “adding” feedback
to the active device S-Parameters shown in Figure 27 through the use o f the Z and Yparameters in Equations 20 and 21 [44].
% Serial ~
YSliinil
=
Z \1+ Zl-ealhack
Z 1X + Z Fmlhllck
Y\ 1 + ^Feedback
Y
- Y
_ 21
1 Feedback
Z\2
+ % Feedback
ZjJ + Z Feedback .
Y12 - YFeedback
^22 +
(20 )
(21)
YFeedback .
A particular value o f feedback can be added with Equations 20 and 21 and then each Y and
Z matrix converted back to S-parameters . These new S-parameters are used to calculate
the RmFecdback using the a responses’ bilinear radius such as is shown in Equation 10 for a 2port network’s
r IN. The normalized radius can then be calculated using RmI ccdback and the
original, no feedback, mapping radius with Equation 18. If this is done for different values
o f feedback reflection coefficients, the calculated normalized radius can be graphed in order
to show the variability trends for various feedback values. An example 2-port S-parameters
set for a 2-port network taken from [41] is shown in Equation 22.
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62
S=
0.22 - j0.32
0.00 + j0.05
.0.56 + j2.64
0.80 + j0.39_
(22)
The normalized radius was calculated for the various values o f series feedback that were
added to the example S-parameter set in Equation 20. The feedback impedance (ZFccdback)
was varied over the entire Smith chart by using the series feedback reflection coefficient
magnitudes o f 0, 0.25, 0.5, 0.75,1.0 and changing the angle from -180° to 180°. RN was
calculated for Figure 28 which indicates that when no feedback is added, as in the condition
shown in Equation 19, the normalized mapped radius is one.
£ 3.5
2.5
TJ
0.5
-180
-135
-90
-45
0
45
90
135
180
Reflection Coefficient Angle (Degrees)
M=0
M =0.25____ M=0.5
M=0.75 - e - M=1.0
Figure 28: Normalized Radius for Different Values of Series
Feedback Reflection Coefficient Expressed in Magnitude (M)
and Angle
Also notice the smallest value of RNoccurs when
rScrics= 1Z0° or when the source of the
active device is open circuited. Obviously, this feedback value would not be used in a
design although it exhibits low input match variability. Figure
values o f feedback inductance
28 also shows that small
(-1Z-1200) can dramatically increase the mapped Smith
chart radius causing the affect o f TL0AD fluctuations to be magnified by the bilinear
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63
transformation at the input port. Two different series feedback values can be easily be
compared by looking up their normalized radius value in Figure 28 to see which has the
least potential TIN variability.
Figure 28 can be displayed as a contour map for visualization and easier comparison
between series and shunt feedback topologies. The example S-parameter network in
Equation 22 was used with each feedback topology to determine the potential T,N variability
for different feedback values due to TL0AD fluctuations. The results are displayed in Figures
29 and 30 in logarithmic form in order to plot the wide range of values produced by the
normalized radius equation. The negative values on the contour plots indicate that the
normalized radius was less than one. This means that the transformed TIN distribution
would be smaller on the Smith chart than the TL0AD distribution and tends to reduce the
variability in a circuit’s input match response.
180"
1 .0 1 1
07445
0 7 2 8 -
' ....
.
0 .162
1.294 \
90“
0.445
1,294
0.728
1.011
0:162
0.162
angle(
0:404
0.121
-
0.162
0.162
i8tr
0
Series
0:121
0.445
0.445
0.25
0.5
0.75
I Seriesl
Figure 29: Series Feedback Contour Mapping, Log of
Normalized Smith Chart Radius
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64
0.686
'
1.04 J P4
0.331
a n 9 l e ( S h u n t)
0.686
Shunt
I Shunt I
Figure 30: Shunt Feedback Contour Mapping, Log of
Normalized Smith Chart Radius
A designer could determine a desired feedback reflection coefficient for both the series and
shunt topologies. This decision could be based on such criteria as circuit gain and noise
figure after the addition o f feedback. Then, by locating the feedback value on Figures 29
and 30 contour maps the normalized radii can be determine and the better topology with
respect to the input match variability could be picked. For example, the magnitude of the
original gain (S2)) that the network exhibited in Equation 22 is 8.6 dB. When the feedback
reflection coefficients are
r s.LT(V
,v= 1Z1400 and rv/W
H, = 1Z - 31.5° the gain of the networks
can both be lowered to 6.0 dB. It can be concluded that the shunt feedback would be a
better topology because it gives less input match variability by comparing the contour plots
and/or calculating the normalized radius values o f 1.06 and 1.83 for the shunt and series
feedback topologies respectively.
Five hundred Monte Carlo simulations were done in order to verify that the shunt feedback
was truly the superior circuit topology. The 50 Q load attached to the 2-port network was
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65
given a Gaussian distribution in the reflection coefficient magnitude and uniform in the
angle which represents a two dimensional normal distribution centered in the middle o f the
Smith chart. Figure 31 shows an example distribution of 500 load impedances randomly
picked for the variability comparison.
Figure 31: Locus o f Random Load Impedance for TIN
Variability Comparison (500 Samples)
The Si, input impedance from Equation 12 was used with the FET 2-port network both
with and without each type o f feedback. The locus o f input impedance for each case is
shown in Figure 32. As theorized by the RN values, the tighter distribution resulted from
the shunt feedback network where the larger spread was produced by FET using the series
topology. The distribution o f the FET without feedback can be seen to be the smallest locus
because its normalized radius was one which was a smaller value than the other two
networks. Careful inspection o f the distributions show that they have the same shape
although different rotation, size, and location. Figures 29 and 30 show that other values o f
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66
feedback could have actually achieved a smaller locus o f
riNvalues than the no feedback
case.
Series
Locus
Figure 32: Input Impedance Distributions o f a FET Using
Feedback While TL0AD Randomly Varies (500 Samples)
The example shown above for finding the best feedback topology for the input impedance
was used with Equation 16 to determine potential power gain variability. The same TL0AD
random distribution was used with the addition o f a similar T s o u r c e distribution. The series
feedback network exhibited a mean power gain (magnitude) o f 3.85 and standard deviation
o f 1.22 (31.6% o f mean). The shunt feedback topology had a mean power gain o f 3.71 and
standard deviation of 0.57 (15.4% o f mean). Without feedback, the FET had a mean power
gain of 7.00 and a standard deviation o f 2.02 (28.8% o f mean). The shunt feedback’s gain
performance had the lowest percentage standard deviation which was almost half o f what
was exhibited by the FET without feedback. It can be seen that the series feedback had the
most percentage o f the mean standard deviation and is not be a good choice o f FET
feedback topologies. The shunt feedback topology would be the best choice for the
absolute lowest T[N variability at the desired FET gain levels.
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67
4. Discussion o f Results
This method o f quantifying potential variability in a network has very promising
applications in network synthesis. Multi-port networks can be quickly compared on the
basis of which produces the least variability from an independent element’s fluctuations. In
the feedback example shown in this chapter, both feedback topology value’s power gain
standard deviations were calculated and the shunt feedback was shown to decrease total
percentage standard deviation. With the bilinear mapping method, the design topology that
exhibited the least amount o f input match sensitivity increase and lowest percentage gain
standard deviation was successfully chosen. However, the bilinear technique should not be
the only determinant o f a good network or circuit. Figure 32 shows that while the shunt
topology is more desirable from a variability stand-point, the two networks produce
different locations for the mean input impedance. Circuit responses as well as feedback
topologies should be judged on characteristics other than variability such as noise match,
third-order intercept, and the ability to match the port with a realizable network.
There are limitations to this new technique of characterizing circuit variability. It has been
observed that the radius values are very sensitive to the S-parameters involved in the
transformation. Small changes in S-parameters can cause bilinear mapping applications
such as the feedback variability contours to change radically. These small changes in Sparameters can be due to frequency dependency, FET bias, or even fabrication variation.
As o f yet, there is no way to incorporate these S-parameter changes into the bilinear
mapping method. Therefore, this method is limited to a single point, fixed system analysis.
However this method is simple to use and can be easily programmed into software packages
like MathCAD to give the designer another powerful tool with which to characterize
circuits.
5. Conclusions
A novel application o f bilinear mapping has been developed. It quantifies the potential
variability a multi-port network response due to the fluctuations in an independent response.
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68
This technique has been demonstrated for the input match o f a 2-port network with respect
to the variations in a load impedance. Conclusions on which type o f network topology to
use and in what amount was determined in the example. Using this technique with other
types o f bilinear transformations can be done with equivalent success. This method allows
the designer to easily compare networks in order to obtain the least potential variability and
increase circuit predictability and yield.
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69
CHAPTER V
STATISTICAL WAFER PROBE CALIBRATION
1. Introduction
Industry routinely performs high frequency electrical characterization during and after
MMIC fabrication. Measurements are used to monitor the natural component variations
and verify that the circuits perform to a customer’s specifications. Comparison between
modeled and measured performance is also used to verify existing circuit models as well as
to develop new modeling methodologies. For these reasons, the accuracy, cost, and cycle
time o f the high frequency measurements significantly affect the design process.
In the past, microwave measurements have been made on a representative sample of circuits
mounted in test fixtures similar to the one shown in Figure 33. This process is very time
consuming due to the fact that the circuits are seldom designed to standard sizes and custom
assemblies need to be built.
MMIC Assembly
MMIC
die
Fixture
Base
Launchers (coax)
Figure 33: Typical MMIC Fixture with Coax-to-Microstrip
Launchers
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70
Fixturing can also be very costly in terms o f MMIC wafer yield because an assembled die is
not usually marketable after measurement because it is soldered to the test assembly. Wafer
probing and optical sampling are two alternate measurement methods that have been
developed which have many benefits over traditional fixtured measurements [45]. Wafer
probing is the more common alternative method o f measurement and is quicker, much less
costly, more repeatable, and less destructive to the MMIC die than fixtured testing. Figure
34 displays an example o f an un-diced wafer being measured on a wafer probe chuck with
coax-to-die transition probes.
Wafer
Probes
To
ANA
Un-Diced
W afer
To
ANA
Probe
Chuck
MMIC
die
Figure 34: Probe System Measuring a Die on a un-Diced
Wafer
Both the fixtured and alternative types o f measurement methods do not give exactly the
same result because they use different techniques and equipment. For example, RF wafer
probe measurements performed in the production environment often compromise accuracy
for increased speed. Probed power measurements sometimes use peak voltage detectors
instead o f a more accurate power head. Circuit conditions can also contribute to differences
in the measurements. High power measurements are known to be significantly affected by
the fact that fixtured devices have better heat dissipation due to the solder bonding process
over the vacuum attachment o f the MMIC die to the wafer probe chuck. Fixtured
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71
measurements also have greater gain ripple due to the addition o f bond-wire interconnect
inductance [46,47]. Finally, measurements o f probed circuits can exhibit resonant coupling
to other structures in close proximity to the Device Under Test (DUT) [48]. These types of
conditions can contribute to the measurable difference between the two measurement
techniques.
Wafer probing is quickly becoming the standard measurement method due to its low cost
per die, speed, and ease o f measurement [49]. However, until single function MMICs
become common [50] the circuits will continue to be housed in a module with bond-wire
and Thin Film Network (TFN) interconnects. Because of this fact, measurements done in
test fixtures are considered more representative and will always be desired for use during
the module design process. So far, only relative comparisons between fixtured and probed
measurements have appeared in the literature [51]. This chapter describes a statistical
methodology that can be used to determine an equation that describes the systematic effects
that cause fixtured and probed measurements to be different. Establishing this relationship
enables MMIC measurement with the more desirable wafer probe method and then
mapping these results into the fixtured environment in this way. Even large size RF probe
databases [49] could be converted to a fixtured environment. The predictive mapping
therefore reduces total measurement time and expenses for fixtured environment
characterization as well as gives valuable insight to both measurement techniques [52], A
test example o f this statistical calibration methodology will be given for the Wide Band
Power Amplifier (WBPA) manufactured for the MIMIC Phase 2 program.
2. Simple Model Statistical Calibration
The MIMIC Phase 2 program WBPA produced in 1994 at the Texas Instruments GaAs
fabrication facilities in Dallas, TX was selected for the example statistical calibration. Five
die were selected randomly from 16 wafers representing 5 fabrication lots. The MMIC die
were RF probed for S-Parameters between the frequencies o f 6.5 to 20 GHz at 0.5 GHz
steps before the final wafer separation. Power compression was also measured from 6 to 18
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72
GHz at 2 GHz increments. After processing, the same die were assembled onto the carrier
plates shown in Figure 35 with microstrip thin film networks, bond wires, and bias
capacitors. The assemblies were then measured in fixtured form for the same responses,
frequency ranges, and bias conditions as was done for the wafer probing. A total o f 68 die
had data collected in both probe and fixtured form with most o f the 12 losses being incurred
during the assembly process.
Bond Wires
TFN
Microstrip
0.0 1 u F
Bias Pad
150pF
WBPA
MMIC die
TFN
Microstrip
150pF
0.01 uF
Bias Pad
Figure 35: WBPA Fixtured Assembly with Bias Capacitors
Shaded
The statistical calibration procedure is fairly common in statistics and uses regression
techniques to determine a linear relationship between two measurement databases [53]. A
simple regression model is shown in Equation 23.
Y=
„+
,X
(23)
The dependent variable Y will be predicted with the use o f an independent variable X, an
intercept term 0, and a slope | in Equation 23. The probed data in this example will be
the independent variable and is called the regressor o f the model. The fixtured data is
considered the dependent variable because it will be predicted given the wafer probe results.
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73
The regression modeling software will calculate an intercept and slope that best fits the data
in a Least Squares sense.
The first step in this type o f regression is to produce scatterplots to determine the type of
relationship between the two measurement techniques. For each circuit, there will be
measurements done with the wafer probe and one done in fixtured form. This represents a
data pair. The data pair is graphed on a scatterplot by assigning one measurement technique
to the x-axis and the other measurement technique to they-axis. By plotting each data pair
in this way relationships between the two measurement techniques can be seen.
Scatterplots o f each o f the four S-Parameter magnitudes are shown in Figures 36, 37, 38,
and 39. They display a representative sample o f the 1,904 datapoint pairs for the
measurements done in the 6.5 to 20 GHz frequency range. As expected, a highly linear
relationship is exhibited between the S-parameters o f the two measurement techniques.
Figure 36 shows that a line, albeit “fuzzy” due to measurement uncertainty and error,
describes the relationship between the two measurement techniques. The line has a slope
near one and x-axis intercept o f about 0.1. It is the slope and intercept that need to be
determined for each measurement responses type.
1
-
0.9
w 0.8
a>
3 07
% 0.6
c? 0.5
5 0.4
T3
£ 0.3
I
0.2
^
0.1
'x '
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wafer Probed Magnitude S11
Figure 36: Magnitude o f Input Match (|S n |) Scatterplot
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74
■a 6
o>
Line for Probe
Data> 2.18
Data Kink
Line for Probe
Data< 2.18
0
1
2
3
4
5
6
7
8
9
Wafer Probed Magnitude S21
Figure 37: Magnitude o f Circuit Gain (|S2I|) Scatterplot with
Linear Segmented Line Regression Modeling
0.02
£ 0.018
w 0.016
a)
=j 0.014
1 0.012
ro 0.01
J 0.008
£ 0.006
■| 0.004
0.002
0
0
0.002
0.004
0.006
0.008
0.0
Wafer Probed Magnitude S12
Figure 38: Magnitude o f Reverse Isolation (|SI2|) Scatterplot
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75
1
CM n g
CM u a
w 0.8
(U
■ 3 07
•1 0.6
ra 0.5
5 04
2 0.3
|
0.2
^
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wafer Probed Magnitude S22
Figure 39: Magnitude of Output Match (|S22 |) Scatterplot
The Magnitude of Reverse Isolation (|S I2|) in Figure 38 has almost no scatterplot definition
and will result in a poor regression fit. This is attributed to the fact that the measurement
values are close to the isolation “noise floor” o f both measurement set-ups. The true
relationship between two measurements can not be accurately determined when the
responses are obscured by this type o f measurement inaccuracy. Figure 37 has two
regression lines plotted on it displaying that the amplifier gain may have a “kink” in its
fixtured to wafer probed relationship occurring at about a probed ]S2i| o f 2.5. This is caused
by the larger gain ripple commonly exhibited in fixtured measurements due to the extra
bond wire inductance and large transition reflections [46,47]. Exactly how the two lines
can be modeled so as to preserve the “kink” will be shown later.
Each fixtured S-parameter magnitude was estimated by least squares using the simplest
regression model which included one regressor, the probed magnitude, and an offset
(intercept) term. Least squares estimation is routinely available in commercial statistical
and spreadsheet packages. Sample SAS code to do this type o f analysis is included in
Appendix A. The resulting regression coefficient values are included in Table 11 where the
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76
superscript indicates the fixtured (F) or probed (P) response and the subscript index
indicates the type o f response measurement. All model terms were tested to be statistically
significant when their p-values were compared to a Type I error a-value o f 0.05 [32].
Table 11: S-Parameter Response Regression Equation Values
- Simple Model
Dependent
Offset
Variable
Value
Sn
-0.046
Regressor
Regressor
R2
Coefficient
M
-0.777
1.095
0.957
1.180
0.864
-0.120
0.015
0.004
o /»
0.085
s';
0.785
0.719
A 'i
-71.984
z s /;
0.965
0.928
zs^
-74.190
zsll
1.0084
0.955
z5';
-79.030
zs£
0.980
0.987
S.2
12
Equation 24 shows an example expression for the fixtured gain |S2il from the values in
Table 11.
1^; | = -0.777+1.180-Is/; |
(24)
Intuitively, it is known that the coefficient that is being multiplied by the fixtured data
should be close to one. The simple model regressions, except for S 12, had excellent results
as is indicated by the high R2 values in Table 11. The R2 is the proportion o f observed
variability in the measured fixtured response that is explained by the model. R can range
from zero, no explanation of the data, to one when there is explanation o f all the variation
present in the response being measured. For S12, there is no discernible linear relation
between the fixtured and wafer probed measurements as was seen in Figure 38.
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77
The S-parameter phase measurements require a more involved analysis. Angle information
is an example o f “circular data” [54]. Figure 40 shows a typical S21 phase angle response
for the wafer probed and fixtured WBPA plotted against frequency. The phase data repeats
itself over the range o f -180° to 180°, causing the “Barber’s Pole” banding shown in the
Figure 41 scatter plot.
180
135
o
O)
-45
-90
-135
-180
6
8
10
12
14
16
Frequency (GHz)
18
20
-x - Fixtured Data-e- Probed Data
Figure 40: WBPA Phase Angle of S2i over Frequency
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-180-150-120 -90 -60 -30
0
30 60 90 120 150 180
Wafer Probed Angle S21 (Degrees)
Figure 41: Scatter Plot of Phase Angle o f S2|
This type o f data will be poorly modeled by a simple linear regression. However, the
banded data can be transformed so that the data can be accurately modeled by a simple
linear equation. The easiest transformation is to identify the cluster in the top left-hand area
o f Figure 41 as a continuation o f the lower band as it moves from right to left. Subtracting
360° from that cluster’s fixtured phase angle provides the scatterplot in Figure 42.
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79
w 180
2> 150
S’ 120
O)
0)
Q
90
60
30
0
CM
W -30
0) -60
-90
e -120
< -150
t j -180
S’ -210
3 -240
* -270
L*- -300
-180-150-120-90 -60 -30
0
30 60 90 120 150 180
Wafer Probed Angle S21 (Degrees)
Figure 42: Scatter Plot o f Phase Angle o f S2I after
Transformation
The transformed data can then be regressed to give a good probe to fixtured relationship
which was done for each o f the WBPA S-parameter phase angles. The results for the
simple linear model, a offset and single regressor, are listed in Table 11. The phase angle of
S 12 was not fitted because o f the difficulties with resolving the magnitude values. For the
phase angle o f S2), the R o f the original data was 0.002 in comparison to the transformed
data set which gave 0.955. Obviously, the circular data transformation is important to the
success o f the regression o f phase angle data.
The WBPA compression data included 2-dB compression output power and Power Added
Efficiency (PAE) at that compression point. No satisfactory regression models were
obtained for the data across the entire frequency range. Figure 43 shows a scatterplot of the
PAE for the 6 and 18 GHz data. It can be seen the 6 GHz data has a very linear relationship
while the 18 GHz data has no definition at all. The relationship between the fixtured and
probed power data got progressively worse as the frequency increased. The 2-dB
compression data showed the same trend.
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80
32
g
Data taken at
2-dB Compression point
30
W 28
aj 26
g>24
i- 'ii
%
.v
5 22
I 20
I
18
16
24
26
28
30
32
34
36
38
40
Wafer Probed PAE (%)
+ 6 GHz Data
• 18 GHz Data
Figure 43: 2-dB Compression PAE Scatterplot
Therefore, regressions models were fit at each frequency point using the simple linear
model for the 2-dB compression point, PAE, and small signal gain measured during the
power measurement. The results are shown in Table 12 for each o f the 7 frequency points.
As the frequency increases the fit o f the simple linear regression models get steadily worse.
This indicates either one or both power measurement set-ups have significant error at
frequencies above 12 GHz. The large error at the high frequencies preclude the regression
from being successful over the entire frequency range. For this reason, a model must be
constructed at each frequency point as was done in Table 12.
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81
Table 12: Model Regression Values for Fixtured
Measurement Model Given Probe Results for Power
Compression, PAE, and Gain
2-dB Compression
Freq.
R2
(GHz)
Offset
Value
PAE (% )
Regressor
Coefficient
R2
Offset
Small-Signal Gain
R2
Value
Regressor
Coefficient
O ffset
Value
Regressor
Coefficient
0.946
6
0.848
-5.727
1.158
0.824
-2.863
0.897
0.518
1.623
8
0.871
5.577
0.816
0.771
2.501
0.810
0.558
2.412
0.724
10
0.809
11.095
0.656
0.670
10.317
0.602
0.677
3.275
0.713
0.315
16.450
0.285
0.825
1.257
0.963
12
0.496
19.321
0.366
14
0.127
22.764
0.245
0.020
25.457
-0.067
0.758
-0.248
0.953
16
0.286
19.962
0.337
0.030
24.045
-0.080
0.784
-1.388
0.938
18
0.318
19.733
0.313
0.019
21.479
0.049
0.670
1.182
1.006
3. More Complex Statistical Models
The results presented in the previous section were for the simple regression equation
containing only a single regressor and an intercept term. More complex models can explain
more o f the response variable variation. In fact, while it is always possible to expand a
regression model to explain all o f the variation present in a data set although this model
may not be useful. There is a trade off between model complexity and accuracy. In fact,
common sense dictates that there should be some measurement uncertainty just due to the
inaccuracy o f the test procedures. The goal o f the statistical calibration is to explain just the
“systematic” effects that cause differences between two measurement techniques. The
residual o f a regression is the difference between the actual and predicted values at each
data point. Examination o f regression residuals can reveal systematic lack o f fit in the
regression model [52]. Figure 44 displays the residuals o f the fixtured S2) phase angle and
exhibits a definite systematic error over frequency. This trend is confirmed by careful
examination o f Figure 40 for the difference in probed and fixtured phase angles.
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82
O)
-10
-20
-30
-40
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Wafer Probed Angle S21
Figure 44: Simple Model Regression Residual Plot for S2)
Phase Angle
Use o f more than one regressor is called multiple regression. Least squares estimation of
multiple regression models is available in most commercial statistics packages. When there
are multiple independent variables, it is possible that their effects on the response interact.
Indeed, an interaction effect o f two independent variables can be as large, or larger, than the
individual effect o f either variable. Therefore, using multiple variable regression models
can give significantly different coefficient estimations that was seen I the simple regression
model.
A multiple regression model of the fixtured phase of S2) using both the probed phase angle
and frequency (FREQ) was used to increase the accuracy o f the model. The new relation
for the fixtured S21 phase angle is shown in Equation 25 where the effect FREQ is in GHz.
Z 5 2', = 1.911 +1.032 •Z S ' - 5.745 • FREQ - 0.002 • ZSi[ ■FREQ
(25)
The estimated regression model in Equation 25 had a R2 o f 0.998 indicating a somewhat
better fit than the simple model regression (R2 = 0.955). Table 13 displays the estimated
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83
regression coefficients for the responses that showed a significant improvement in accuracy
as a results o f using the multiple regression model.
Table 13: S-Parameter Response Complex Regression Model
Values
Dependent
Offset
Variable
Value
FREQ
s !;- f re q
Coefficient
Coefficient
Coefficient
-0.213
1.600
0.019
-0.056
0.801
zs(\
18.629
1.049
-6.650
-0.002
0.977
zs^
1.911
1.032
-5.745
-0.002
0.998
z s ';
-1.433
0.924
-5.988
0.010
0.987
R2
Regressing |s'j| on a frequency co-regressor did not significantly improve the model fit.
Therefore, a better model o f |S2)| was attempted by fitting a linear segmented regression
model to the data using non-linear regression techniques. One linear segmented model is
used for the data before the “kink” and then another for the data after the “kink” indicated in
Figure 37. Continuity o f the lines is enforced during the optimization. A more
sophisticated model could be used to permit continuity o f the first derivative at the point of
intersection although this was not done in this example for simplicity. The iterative
estimation procedure starts by guessing a value for the location o f the change in line slope
and for each line’s regression coefficients. Figure 45 shows SAS® code that was used to
simultaneously fit the two regression lines.
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84
P RO C NLIN;
PARMS a l = 0 .2 a2=l.l b l = 0 b2=1.2;
kink =
(Al-Bl)/ ( B 2 -A2 );
IF p r o b e < kink T HEN DO;
M O D E L fixture = al + a2*probe;
DER.al
DER.bl
=1
=0
; DER.a2
; DER.b2
= probe;
= 0;
END;
E LS E DO;
M O D E L fixutre = bl + b2*probe;
DER.al
DER.bl
=1
=0
; DER.a2
; DER.b2
= probe;
= 0;
E ND;
Figure 45: SAS Program for Fitting Linear Segmented
Regression Model
The variables a ( and a2 are the simple model intercept and regressor coefficient for the line
before the kink in the data. The variables b, and b2 are the coefficients for the regression
line after the kink. The kink value location is dependent on these optimization variables
and forces the two regression lines to intersect at the place where the change in slope
occurs. Equation 26 was obtained when this technique was applied to the magnitude of S2I
data. The estimated point o f intersection is 1^1=2.18.
For | | < 2.18: |S211= -0.006 + 0.787 •\s£
For K I > 2.18: IS’*' I = -0.922 +1.207 ■\s£ I
The estimated regression linear segmented line is displayed in Figure 37 with the valid
wafer probe data range indicated. R2 comparisons are inappropriate for non-linear
regression. Instead the likelihood ratio for adequacy o f the model in Equation 26 relative to
the model in Equation 25 [56].
For a non-linear regression, the quality o f the model can determined from a similar test
done by calculating the likelihood ratio. The likelihood ratio test essentially compares the
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85
difference in amount o f variation explained by the non-linear and linear models [56]. If
significantly more variation is explained by the non-linear model then it is considered to be
better. The calculated value o f the likelihood ratio test statistic for the linear segmented
regression model was 6.63 which is significant at an alpha level o f 0.05. Therefore, the
linear segmented regression model was considered to be an improvement over the linear
model. Because o f the large amount o f data, this test procedure has large statistical power.
That is, even slight improvement in the model will be deemed significant.
4. Application And Discussion of Results
Application o f the procedure discussed in this chapter is intended to reduce the amount of
fixtured testing required for the design verification process. Initially, a representative
sample of circuits needs to be measured using both the probe and fixtured methods. The
relationship between the two measurement environments can be established using the
techniques discussed in this chapter. Once thus calibrated, the more desirable wafer
probing method can then be used exclusively to determine the fixtured environment circuit
performance through the performance mapping equations. Periodically, the calibration
equations must be updated, or maintained, to preserve the model integrity against changing
circuit conditions. Response control charts can be used to indicate any fabrication
fluctuations that may cause the statistical models to be unreliable. Many texts contain
further information on statistical model construction, estimation, and other considerations
that must be taken into account during the calibration [52].
The sampling o f the data used for the regression model coefficient estimation needs to be
considered carefully. Coefficient and variance estimates can be affected if any correlation
exists between the presumably independent datapoint pairs. The 1,904 data samples used
for the regressions in this chapter had correlations due to the fact that “blocks” o f samples
were taken over frequency for the same circuit. The gain value o f a particular circuit at 6
GHz is related to that same circuit’s gain at 7 GHz. The coefficient estimations are usually
not effected greatly by this type o f correlation. However, the estimates o f the coefficient
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86
variation are affected causing incorrect conclusions about the significance o f each term in
the regression model. There are ways to adjust for this type o f error but they are beyond the
scope o f this discussion and can be found in common statistical texts.
The regression equations can give valuable insight into the probe and fixtured measurement
techniques. First, the inability to fit a good regression line to the S 12 S-parameter data
indicates that the measurement isolation for either, or both, test set-ups is insufficient.
Better test-set isolation would have to be attained for the relationship between fixtured and
probed measurements o f the low amplifier S)2 values to be accurately estimated. The
analysis on the phase angles can also give valuable test information. Table 13 shows the
more complex regression models used to improve the data fit for the S-parameter phase
angles and magnitude o f S22. For the phase angle regressions, the complex model intercept
terms are much closer to zero than for the simple models displayed in Table 11. Also, the
simple model’s offset terms were all around -74 degrees. The linearly increasing difference
in phase angles with frequency indicates that the error could have been caused by one o f
two mechanisms. First, there may be some type o f reference plane error in either o f the
measurements. Both measurements were supposedly calibrated to the WBPA input and
output planes. The bond wire inductances could also have caused the difference in the two
measurements by adding a complex impedance jcoL in series with the input and output
ports. However, adding a series inductance would cause a varying phase shift for different
input and output match values. This would be indicated by a large interaction between
FREQ and the magnitude or angle o f that measurement. For the phase angle measurements,
it is believed the systematic phase shift was caused by a reference plane error due to the fact
that the FREQ coefficient dominates the ZSjj»FREQ interaction. For the |S22| model, the
bond wire inductances are considered the dominating factor because the \Sjj\»FREQ
interaction is larger than the frequency main effect as shown on Table 13.
The regression can also give insight to the greater gain ripple effect caused by the fixtured
bondwires and transition reflections. The estimated joint point for the linear segmented
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87
regression model showed that the ripple was particularly significant when the gain o f the
amplifier became greater than 2.18.
The power compression regressions enable understanding of the frequency dependency of
the test equipment. The regressions’ R values point to the fact that one, or both, o f the
measurement set-ups have significant error in the compression and PAE above 10 GHz.
Also, the regression equations deviate from having a slope near one as the test frequency
increases. The small-signal gain measured by the power compression test equipment was
analyzed with the simple model regression at each frequency and is displayed on Table 12.
2
2
The R for each analysis were all larger than 0.512 and greatly exceeded the R values for
the compression or PAE measurements. This demonstrates that the small-signal gain
measured before the amplifier was driven into compression had relatively good regression
fits. Therefore, the errors in power compression measurements appear to be an artifact of
the power measurement process itself rather than causes such as repeatability o f the
amplifier connections. Also, the best PAE and 2-dB compression relations where obtained
when the small-signal gain regressions where at their worst. Overall, because the smallsignal gain and the compression measurements are interrelated both types o f measurements
need to be improved.
5. Conclusions
Least squares regression methods were applied to calibrate wafer probe and fixtured Sparameter and compression measurements. Simple linear relationships were estimated for
each of the magnitude and phase angles o f those measurements. Additional accuracy was
obtained for certain calibration equations by developing more complex regression models.
It was shown that many different cause and effect relationships can be determined through
the use o f statistical calibration and analysis. The methodology was successfully
demonstrated on a MMIC WBPA over the 6.5 to 20 GHz range. The use o f this method for
MMIC characterization can help reduce test time and cost while still producing the fixtured
environment measurement results needed for accurate module design.
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88
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
1. Advanced Statistical Modeling
The principal component method presented in this dissertation is a significant advancement
in the area o f advanced statistical modeling. With the methodology, designers can
immediately integrate the simple equations relating the model parameters into their CAD
software to provide accurate variation simulation of their active devices. However,
continuing research must be done on characterizing non-Gaussian parameter distributions
and non-linear relationships between model parameters. These will be issues that must be
resolved if the principal component method is to be used for a wide variety o f models
including those o f the large signal class.
2. Statistical Design Methodologies
The use o f Design o f Experiments methodology can greatly benefit the design of circuits for
nominal and variability performance. The methodology shown in this research has been
shown to work well for simultaneously optimizing different amplifier performance
objectives. The response trends are visually presented to the designer so that performance
trade-off and trends can be easily seen. There is no reason why the techniques wouldn’t
work as effectively for other types of circuits responses such as those found in oscillators or
mixers. However, further work must be done to model circuit responses over frequency.
Quickly changing responses within the design space can also pose some serious modeling
issues. Higher order response models may be used to solve these types o f problems.
Finally, these DoE tools need to be seemlessly integrated into the CAD simulators for ease
o f use and quick presentation of results.
3. Statistical Calibration
The statistical calibration methodology has never been applied to fixtured versus probed
results before. Previously, the two techniques were shown to agree well with each other
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89
rather than actually quantifying the amount o f agreement between the two. Also, statistical
characterization o f the probe and fixtured measurements showed how much insight can be
gained into the two measurement technique’s relationship. The measurement model for
mapping the probe results into the fixture environment has been determined and is often
very simple even over a large frequency range. Two additional topics need to be developed
in this area. First, another power amplifier test case needs to be analyzed in the same way
as was done for the MIMIC Phase IIW BPA in this dissertation. This will prove if the
relationships that were determined in this dissertation can only apply to the WPBA or to all
power amplifiers in general. It is believed that the calibration must be done for each
different design that is being characterized. Finally, research into other types o f responses
must be performed. Noise, Intermodulation, and Load Pull are all examples o f microwave
measurements that could be used with the methods discussed in this dissertation.
4. General Conclusions
The research detailed in this dissertation makes contributions to the three critical aspects of
microwave circuit design. The methodologies and techniques developed in this research
can be used by designers to create circuits while keeping manufacturability and high yield
in mind. This leads directly to more robust circuits that are designed in a less amount of
time. This research is by no means finished and needs to built upon by others in the area of
microwave circuits. Hopefully, the methods in this dissertation will be further developed
and integrated in to CAD packages so that the large amount o f information needed to
perform statistical design can be handled more efficiently. With this, engineers will use the
tools more fluently and then better, more robust designs will result. These designs will be
fabricated and the resulting circuits will can be less expensive and more commercially
competitive.
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90
REFERENCES
[1] M.J.Harry, J.R.Lawson, Six Sigma Producibility Analysis and Process
Characterization,.Reading, MA: Addison-Wesley, 1992, pp. 1.1-1.6.
[2] R.Spence, R.S.Soin, Tolerance Design o f Electronic Circuits, Wokingham, England:
Addison-Wesley, 1988, pp.108-113.
[3] M. Meehan and J. Purviance, Yield and Reliability in Microwave Circuit and System
Design, Boston: Artech, 1993, pg. xv.
[4] J. Purviance, D. Criss, and D. Monteith, "FET Model Statistics and Their Effects on
Design Centering and Yield Prediction for Microwave Amplifiers", in Proc. IEEE
M TTSymp. Digest, 1990, pp. 315-318.
[5] P. Ikalainen, "Extraction o f Device Noise Sources from Measured Data Using Circuit
Simulator Software," IEEE Trans. Microwave Theory Tech., vol. 41, no. 2, pp. 340343, Feb. 1993.
[6] J.M. Golio, Microwave MESFETs and HEMTs, Boston: Artech, 1991, pp. 79-80.
[7] J. Purviance, M.D. Meehan, and D.M. Collins, "Properties o f FET Statistical Data
Bases", in IEEE M TTSymp. Digest, 1990, pp. 567-570.
[8] R. Anholt, R. Worley, and R. Neidhard, "Statistical Analysis o f GaAs MESFET SParameter Equivalent-Circuit Models", International Journal o f Microwave and
Millimeter-Wave Computer-Aided Engineering, vol. 1, no. 3, pp. 263-270, March
1991.
[9] M.D. Meehan, "Accurate Design Centering and Yield Prediction Using the 'Truth
Model'", in IEEE M TTSymp. Digest, 1991, pp. 1201-1204.
[10] EEsof Manual Version 3.5, EEsof Inc., Westlake, CA, 1993.
[11] L. Campbell, J. Purviance, C. Potratz, "Statistical Interpolation o f FET Data Base
Measurements", in IEEE M TTSymp. Digest, 1991, pp. 201-204.
[12] J. Purviance, M Petzold, and C. Potratz, "A Linear Statistical FET Model Using
Principal Component Analysis", IEEE Trans, on Microwave Theory and Tech., vol.
MTT-37, no. 9, pp. 1389-1394, Sept. 1989.
[13] N.R. Draper, H. Smith, Applied Regression Analysis, 2nded., New York: John Wiley
& Sons, 1981, pg. 327.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
91
[14] SAS/STAT Software Version 6: STAT User's Guide, SAS Institute Inc., Cary, NC,
1992, pp. 773-821.
[15] J.W. Bandler, R.M. Biernacki, S.H. Chen, J.F. Loman, M.L. Renault, Q.J. Zhang,
"Combined Discrete/Normal Statistical Modeling of Microwave Devices", Proc. o f
the 19th European Microwave Conference, 1989, pp.205-210.
[16] M.Meehan, L.Campbell, "Statistical Techniques for Objective Characterization of
Microwave Device Statistical Data", in IEEE M TTSymp. Digest, 1991, pp. 12091212 .
[17] L.Devroye, Non-Uniform Random Variate Generation, New York: Springer-Verlag
Publishing Co., 1986.
[18] J.S. Milton, J.C. Arnold, Introduction to Probability and Statistics, 2nd ed., New
York: McGraw-Hill Inc., 1990, pg. 381.
[19] J.M. Golio, Microwave MESFETs and HEMTs, Boston: Artech House, 1991, pp. 207236.
[20] SAS Software Version 6: Procedures Guide, SAS Institute Inc., Cary, NC, 1993, pp.
617-634.
[21] A.D. Patterson, V.F. Fusco, J.J. McKeown, J.A.C. Stewart, "A Systematic
Optimization Strategy for Microwave Device Modeling", IEEE Trans, on
Microwave Theory and Tech., vol. MTT-41, no. 3, pp. 395-405, March 1993.
[22] J.Purviance, M.Meehan, “CAD for Statistical Analysis and Design o f Microwave
Circuits”, International Journal o f Microwave and Millimeter-Wave ComputerAided Engineering, vo l.l, no. 1, pp. 59-76, Jan. 1991.
[23] J.J. Pignatiello and J.S. Ramberg, "Top Ten Triumphs and Tragedies o f Genichi
Taguchi", Quality Engineering, pp. 211-225, April 1991.
[24] T.A. Donnelly, "Response-Surface Experimental Design", IEEE Potentials, pp. 19-21,
February 1992.
[25] A.Howard, “Higher Manufacturing Yields Using DoE”, Microwave Journal, pp. 92110, July 1994.
[26] D.Pleasant, "Using Design o f Experiments to Optimize Filter Tuning Steps", RF
Design, pp. 58-64, June 1994.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
92
[27] J.S. Ramberg, S.M. Sanchez, P.J. Sanchez, and L.J. Hollick, "Designing Simulation
Experiments: Taguchi Methods and Response Surface Metamodels", in Proc.
Winter Simulation Conference, 1991, pp. 167-176.
[28] S.R. Schmidt and R.G. Launsby, Understanding Industrial Designed Experiments,
Colorado Springs, CO: Air Academy Press, 1992, pp. 3.1-3.50.
[29] S.R. Schmidt and R.G. Launsby, Understanding Industrial Designed Experiments,
Colorado Springs, CO: Air Academy Press, 1992, pp. 4.42-4.45.
[30] J.M. Donohue, E.C. Houck, and R.H. Myers, "Some Optimal Simulation Designs for
Estimating Quadratic Response Surface Functions", in Proc. Winter Simulation
Conference, 1990, pp. 337-343.
[31] S.R. Schmidt and R.G. Launsby, Understanding Industrial Designed Experiments,
Colorado Springs, CO: Air Academy Press, 1992, pp. 4.25-4.28.
[32] K.K. Low, S.W. Director, "An Efficient Macromodeling Approach to Statistical IC
Process Design", in Proc. IEEE Int. Conf. on CAD, 1988, pp. 16-19.
[33] J.J.Hanrahan and T.A. Baltus, "Efficient Engineering through Computer-Aided Design
of Experiments", IEEE Trans, on Industry Applications, vol. 28, no. 2, pp. 293-296,
March/April 1992.
[34] K.L. Virga and R.J. Engelhardt Jr., "Efficient Statistical Analysis of Microwave
Circuit Performance Using Design o f Experiments", in IEEE M TTSymp. Digest,
1993, pp. 123-126.
[35] M. M eehan and J. Purviance, Yield and Reliability in Microwave Circuit and System
Design, Boston: Artech, 1993, pp. 65-68.
[36] EEsof Manual, EEsof Inc., Westlake, CA, 1993, pg. 2.25.
[37] J.M. Carroll, K.A. Whelan, S. Pritchett, and D. Bridges, "FET Parameter
Orthogonalization with Principal Components", in IEEE M TTSymp. Digest, 1994,
pp. 409-412.
[38] EEsof Manual Series IV v5.0: Circuit User's Guide, EEsof Inc., Westlake, CA, 1994,
pp. 7.54-7.84.
[39] N. Salamina and M.R. Rencher, "Statistical Bipolar Circuit Design Using MSTAT", in
Proc. IEEE Computer-Aided Design C onf, 1989, pp. 198-201.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
93
[40] R.M. Biernacki, J.W. Bandler, J. Song, and Q.J. Zhang, "Efficient Quadratic
Approximation for Statistical Design", IEEE Trans, on Circuits and Systems, vol
CAS-36, no. 11, pp. 1449-1454, Nov. 1989.
[41] T.R.Cuthbert, Jr., Circuit Design Using Personal Computers, Malabar, FL: Krieger
Publishing, 1994, pp. 230-246.
[42] R.M.Dougherty, “Feedback Analysis and Design Techniques”, Microwave Journal,
pp. 133-150, April 1985.
[43] G.Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, Englewood
Cliffs, NJ: Prentice-Hall, 1984, pp. 92-94.
[44] D.M.Pozar, Microwave Engineering, Reading, MA: Addison-Wesley, 1990, pp. 237240.
[45] S.Lucyszyn, C.Stewart, I.D.Robertson, A.H.Aghvami, “Measurement Techniques for
Monolithic Microwave Integrated Circuits”, Electronics & Communication
Engineering Journal, pp. 69-76, April 1994.
[46] S.Nelson, M.Youngblood, J.Pavio, B.Larson, R.Kottman, “Optimum Microstrip
Interconnects”, in Proc. IEEE M TTSym p., 1991, pp. 1071-1074.
[47] HP Application Note 8510-8, “Network Analysis: Applying the HP 8510B TRL
Calibration for Non-Coaxial Measurements”, Palo Alto, CA, 1987.
[48] T.H.Miers, A.Cangellaris, D.Williams, R.Marks, “Anomalies Observed in Wafer
Level Microwave Testing”, ”, in IEEE MTTSymp. Digest, 1991, pp. 1121-1124.
[49] A.Lum, “Production Worthiness o f a GaAs Wafer Fab as Demonstrated Through
Automated RF Probe Measurements”, 77 Internal Document, Dallas, TX, 1993.
[50] W.R.Wisseman, L.C.Witkowski, G.E.Brehm, R.P.Coats, D.D.Heston, R.D.Hudgens,
R.E.Lehmann, H.M.Macksey, H.Q.Tserng, “X-Band GaAs Single-Chip T/R Radar
Module”, Microwave Journal, vol. 30, pp. 167-173, Sept. 1987.
[51] A.Lum, C.Dale, D.Ragle, M.Vernon, “High Power CW RF Probe Measurements”, in
Proc. GaAs IC Symposium, 1992, pp. 191-193.
[52] N.R.Draper and H.Smith, Applied Regression Analysis, New York: John Wiley &
Sons, 1966, pp. 234-242.
[53] P.J.Brown, “Multivariate Calibration (with Discussion)”, Journal o f the Royal
Statistical Society Series B, Vol. 44, No.3, pp. 287-321, 1982.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
94
[54] N.I.Fisher, Statistical Analysis o f Circular Data, Cambridge, Great Britain:
Cambridge University Press, 1993, pp. xv-xviii.
[55] N.I.Fisher, Statistical Analysis o f Circular Data, Cambridge, Great Britain:
Cambridge University Press, 1993, pp. 168-197.
[56] A.R.Gallant, Nonlinear Statistical Models, New York: John Wiley & Sons, 1987, pp.
47-59.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
95
APPENDIX
EXAMPLE SAS® PROGRAMS
1. Principal Component Modeling o f FET ECPs
O P T I O N PS=55 ls=150;
^ i n c l u d e '/ h o m e / c a r r o l l / s a s p r o g / o u t l i e r _ m a c r o ' ;
F I L E N A M E f i l e d a t a '/ h o m e / c a r r o l l / s a s p r o g / B E - C O / T E S T ';
D A T A devices;
INFI LE filedata;
INPUT fl 1-7 f 2 8-9 fa $ f 3 f4 f5 Vds Ids Vgs Vg2 Rg Rs Rd G m Cgs Ri
Cds Rds C g d T au Ls Lg Ld Cos t Vn In R e _ c o r r I m_ c o r r Rgs;
RUN = _N_;
**Use O u t l i e r m a c r o (Ma halanobis d i s t a n c e deviate) to s c r e e n d ata
for m u l t i v a r i a t e o u t l i e r s this a s t a n d a r d SAS m a c r o **;
* S c r e e n t w i c e to m a k e su r e b a d ou tli e r s a r e t a k e n out ;
% o u t l i e r (v a r = G m Cgs Ri Cds Rds C g d Ta u Vn In R e _ c o r r
pv a lue =.0 5, passes= 2)
dat a new;
set chiplot;
if d s q > 3 0 t h e n delete;
Im_corr,
id=RUN,
% o u t l i e r (v a r = G m Cgs Ri Cds Rds C g d Ta u V n In R e _ c o r r
pval u e = . 0 5 , passes= 2)
dat a new;
set chiplot;
if d s q > 3 0 t h e n delete;
p r o c sort; b y run;
Im_corr,
id=RUN,
p r o c corr;
var G m Cgs Ri Cds Rds C g d Tau Rg Rs Rd V n
*
In R e _ c o r r
Print out m e a n s a n d s t a n d a r d d e v i a t i o n s ;
pr o c u n i v a r i a t e normal;
v ar G m Cg s Ri Cds Rds C g d Tau Rg Rs R d Vn In R e _ c o r r
I m _ c o r r Rgs Cost;
I m _ c o r r Rgs Cost;
*
Print out fac t o r p a t t e r n m a t r i x ;
pr o c fa ct o r s i m p l e m i n e i g e n = 0 n f a c t o r s = 1 4 c o r r ou t=f a c t d a t ;
va r G m C gs Ri Cds Rds C g d Tau Rg Rs R d V n In R e _ c o r r I m _ c o r r Rgs;
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96
2. DoE Surface Response Modeling of Experimental Results
O P T I O N L S= 7 5
PS=4 0;
* L O A D IN E X P E R I M E N T A L F A C T O R S E T T I N G S (X1-X8) A N D THE R E S P O N S E S ;
FI LE N A M E D O E _ D A T A ' e : \ d o c \ p a p e r s \ d o e \ l a m p r s l t . t x t ' ;
D A T A W;
INF I L E DOE _DATA;
INPUT e xp e r XI X2 X 3 X4 X5 X6 X7 X8 F r e q Gai n Sta b N F Sll S22 m a g G a i n
m a g N F m a g S l l m a g S 2 2 magStab;
RU N = _N_;
* M O D E L I N G O N L Y A T O N E F R E Q U E N C Y POINT
D A T A new;
set W;
if freq ne 4.75 t h e n delete;
;
*
USE T H E R E S P O N S E S U R F A C E M O D E L I N G P R O C E D U R E TO FIT A Q U A D R A T I C WI T H
*
THE 8 F A C T O R S F O R G A I N (dB) ;
PROC R SR E G D A T A = NEW;
M O D E L G A I N = XI X2 X3 X4 X5 X6 X7 X8 ;
;
*
USE T H E R E S P O N S E S U R F A C E M O D E L I N G P R O C E D U R E TO FIT A Q U A D R A T I C WI T H
*
THE 8 F A C T O R S F OR N O I S E FIGURE;
PROC RSR E G D A T A = NEW;
M O D E L N F = XI X2 X3 X4 X5 X6 X7 X8 ;
;
*
USE TH E R E S P O N S E S U R F A C E M O D E L I N G P R O C E D U R E
*
TH E 8 F A C T O R S F O R M A G N I T U D E O F GAIN;
PROC R SR E G DATA=N EW;
M O D E L M A G S 11 = XI X2 X3 X4 X5 X 6 X7 X8 ;
TO FIT A Q U A D R A T I C
W IT H ;
*
USE T H E R E S P O N S E S U R F A C E M O D E L I N G P R O C E D U R E
*
THE 8 F A C T O R S F O R Sll;
PROC R S R E G D A T A = NEW;
M O D E L Sll = XI X2 X3 X4 X5 X6 X7 X8 ;
TO FIT A Q U A D R A T I C
WI T H ;
*
USE THE R E S P O N S E S U R F A C E M O D E L I N G P R O C E D U R E TO FIT A Q U A D R A T I C WI T H
*
T H E 8 F A C T O R S F O R S22;
PROC RSR E G D A T A = NEW;
M O D E L S22 = XI X2 X3 X4 X5 X6 X7 X8 ;
;
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97
3. Statistical Calibration of Wafer Probe Data
OP T I O N S
P A G E S I Z E = 6 0 L I N E S I Z E = 9 0 N O C E N T E R NODATE;
F I LE NAM E F ILE1
'C:\data\doc\papers\probe\eg6353.dat';
DATA M easured;
INFILE FILE1 LRECL=4 50;
INPUT LOT R_C M EAS $ F REQ MS11 A S H MS21 AS21 MS12 A S1 2 M S 22 AS22 ;
if FREQ A= FLOOR(FREQ) then DELETE;
* GET o nl y 6.5, 7, 7.5 ... data;
if F REQ < 6 . 5 t hen DELETE;
if FREQ > 20 t he n DELETE;
if LOT = 941 8 1 0 1 0 3 t h e n DELETE;
* Canno t find this lot's RF p robe
D AT A probed;
SET Measured;
if M EA S = 'F' then DELETE;
pMSll = MS11;
pMS22 = MS22;
pMS21 = M S 2 1 ;
pAS21 = AS21;
DROP M E AS M S I 1 A S 1 1 M S2 1 AS21 MS12 AS12 MS22 AS22;
PROC SORT DATA=pro bed ;
BY LOT R _ C FREQ;
D AT A fixtured;
SET Measured;
if M E AS = ’P' then DELETE;
fMS21 = M S 2 1 ;
fAS21 = A S 2 1 ;
DROP M E AS M S 11 A S11 MS21 AS21 MS12 AS12 MS22 AS22;
PROC S ORT D A T A =f ixt ure d;
BY LOT R _ C FREQ;
D AT A Both;
set fixtured;
set probed;
* Do S21 A n g l e T r a n s f o r m a t i o n a nd put into TfAS21 ;
TfAS21 = fAS21 ;
if (pAS21 < -30) a n d (fAS21 > 30) then TfAS21 = fAS21 - 360
shift ;
;
* Data
PROC G L M DATA=both;
MO D EL fMS21 = p M S21 / SSI ;
PROC G L M DATA=both;
M OD EL fMS21 = p MS 21 | FREQ / SSI;
PROC G L M DATA=both;
M O D E L fAS21 = p AS21 / SS3 ;
PROC G L M DATA=both;
M O D E L T f AS 21 = p A S 2 1 / SS3 ;
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98
VITA
James Mason Carroll was born in Houma, Louisiana in November 1968. He received his
Bachelor of Science Degree in Electrical Engineering from Rose-Hulman Institute of
Technology in 1990. Afterward, Mr. Carroll attended Texas A&M University under the
direction o f Dr. Kai Chang. He received his Master’s Degree in Electrical Engineering in
1992 with the thesis title o f “Accurate Characterization and Improvement o f GaAs
Microstrip Attenuation”. He also has been working at the Texas Instruments Advanced
Microwaves Group as a Summer Engineer since getting the Advanced Microwaves TI
Fellowship in 1991. Mr. Carroll has also worked at Texas A&M University as a Lecturer
for Electrical Engineering fo r Non-Majors and both Lecturer and Lab Assistant roles for
Digital Circuit and System Design (Major’s Class). His interests include microwave theory
and MMIC design. Mr. Carroll can be contacted at Texas Instruments, Mail Stop 245,
13510 North Central Expressway, Dallas, TX. 75265.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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