# Improving microwave circuit testing and producibility with statistical techniques

код для вставкиСкачатьReproduced 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 M ajor 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. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. 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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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. 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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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 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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; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 ; ; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 ; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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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