# Measurement -based modeling of vector network analyzer calibration standards and nonlinear microwave devices using artificial neural networks

код для вставкиСкачатьMEASUREMENT-BASED MODELING OF VECTOR NETWORK ANALYZER CALIBRATION STANDARDS AND NONLINEAR MICROWAVE DEVICES USING ARTIFICIAL NEURAL NETWORKS by JEFFREY ARENDT JARGON B. S., University of Colorado, 1990 M. S., University of Colorado, 1996 A thesis submitted to the Faculty of the Graduate School of the Uni versity of Colorado in partial fulfillment of the requirements of the degree of Doctor of Philosophy Department of Electrical and Computer Engineering 2003 I _______________________ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission UMI N um ber: 3087553 UMI UMI Microform 3087553 Copyright 2003 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This thesis for the Doctor of Philosophy degree by Jeffrey Arendt Jargon has been approved for the Department of Electrical and Computer Engineering by K. C. Gupta c Donald C. DeGroot Date Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I Jargon, Jeffrey Arendt (Ph.D. Electrical Engineering) Measurement-Based Modeling of Vector Network Analyzer Calibration Standards and Nonlinear Microwave Devices Using Artificial Neural Networks Thesis Directed by Professor K. C. Gupta and Dr. Donald C. DeGroot This thesis is comprised of two parts. The first segment covers artificial neural network (ANN) modeling for improved vector network analyzer (VNA) calibrations. Specifically, measurement-based ANNs are applied to model a variety of on-wafer and coaxial vector network analyzer calibrations, including open-short-load-thru (OSLT) and line-reflect-match (LRM). A sensitivity analysis of the ANNs is performed by determining the training error as functions of the number of hidden neurons and the number of training points. The respective accuracies of these calibrations are then assessed using the ANN-modeled standards. As a major research result, this doctoral thesis shows that ANN models offer a number of advantages over using calibrated measurement data files or equivalent circuit models, namely: they do not require the numerous details and parameters of physical models; calibration times can be reduced because only a few training points are required to accurately model the standards; ANN model descriptions are much more compact than large measurement data files; ANN models, trained on only a few measurement points can be much more accurate than direct calibrations when limited calibration data are available; ANNs give an optimized estimate in the presence of noise; and ANN models are able to accurately model loads with measured DC resistances slightly outside of their training range. in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the second part of this thesis, new frequency-domain models and figures of merit for nonlinear microwave circuits are developed for sparse-tone inputs. This section begins with a method for preserving time-invariant phase relationships when ratios are taken between two harmonically related signals by introducing a third signal that is used as a phase reference. Then, as another major research result, this doctoral thesis introduces nonlinear large-signal scattering (§>) parameters, a new type of frequency-domain mapping that relates incident and reflected signals. A general form of nonlinear large-signal ^-parameters is presented. It is shown that they reduce to classic ^-parameters in the absence of nonlinearities. Nonlinear large-signal impedance (%) and admittance (|j) parameters are also introduced, and equations relating the different representations are derived. Next, definitions of power gain, transducer gain, and available gain are expanded by taking harmonic content into account. An example is provided showing how the expanded definitions of gain and nonlinear large-signal ^-parameters allow one to examine the behavior of a nonlinear model by simply performing a harmonic balance simulation. Next, this thesis illustrates how nonlinear large-signal ^-parameters can be used as a tool in the design process of a nonlinear circuit, specifically a single-diode 1-2 GHz frequency-doubler. For the case where a nonlinear model is not readily available, a method of extracting nonlinear large-signal ^-parameters is developed using ANN models trained with multiple measurements made by a nonlinear vector network analyzer equipped with two sources. Finally, nonlinear large-signal ^-parameters are compared to another form of nonlinear mapping, known as nonlinear scattering functions. The nonlinear large-signal ^-parameters are shown to be more general. IV permission of the copyright owner. Further reproduction prohibited without permission. I DEDICATION To my wife Soogy, and our children, Trevor and Valerie. Words can’t possibly describe how much you all mean to me. I love you so much and appreciate your encouragement, patience, and understanding throughout this project. And to Mom and Dad for absolutely everything. You’re the best parents I could ever ask for. I love you both. I also thank the rest of my wonderful family for their love and support throughout the years, including Jangmonim; my in-laws, nieces, and nephews in Korea; Jon and Maria; Julie and Craig; Granddad and Rada; Grandma Boam; Grandpa Jargon and my late Grandma Jargon. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I thank all of the individuals who helped make this dissertation possible. Special thanks to my thesis advisors, K. C. Gupta and Donald C. DeGroot, for their invaluable support, guidance, and encouragement. Our last few years working together have been very educational and enjoyable. I also thank my co-conspirator and best friend, Michael Janezic, for making this journey together a pleasant one. I am grateful for the support of everyone at the National Institute of Standards and Technology, especially Robert Judish, Susie Rivera, Kate Remley, and Dennis Friday. I am indebted to Dominique Schreurs, Alessandro Cidronali, Pete Kirby, Huantong Zhang, Jan Verspecht, and Marc Vanden Bossche for their valuable contributions and suggestions. I also thank Y. C. Lee, James Baker-Jarvis, and Melinda Piket-May for serving on my committee and providing me with valuable feedback. And last, but also least, a tip of the hat to Victor Kushmann for “keepin’ it real.” j vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS CHAPTER 1. 2. I i INTRODUCTION 1 1.1. ANN Modeling for ImprovedVector Network Analyzer Calibrations 1 1.2. Frequency-Domain Models and Figures Nonlinear Circuits 3 1.3. Organization of the Thesis 6 ARTIFICIAL NEURAL NETWORKS 11 of Merit for 2.1. Neural Network Structure 12 2.2. Model Development 16 Model Verification 18 2.3. PART I - ANN MODELING FOR IMPROVED VECTOR NETWORK ANALYZER CALIBRATIONS 3. 4. LINEAR NETWORK ANALYSIS 22 3.1. Scattering Parameters 22 3.2. Vector Network Analyzer Architecture 26 3.3. Vector Network Analyzer Error Models 28 3.4. Vector Network Analyzer Calibration Techniques 30 3.5. ANN Modeling for Improved VNA Calibrations 32 ANN MODELING OF ON-WAFER OSLT STANDARDS 34 4.1. Modeling the Standards 35 4.2. Calibration Comparisons 45 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3. 46 Discussion 5. MODELING LOAD VARIATIONS WITH ANNS TO IMPROVE ON-WAFER OSLT CALIBRATIONS 49 5.1. Modeling the Standards 50 5.2. Advantages of ANN Models 53 5.3. Calibration Comparisons 53 5.4. Discussion 57 6 . ANN MODELING FOR ON-WAFER LRM CALIBRATIONS 58 6.1. Load Modeling 58 6.2. Calibration Comparisons 60 6.3. Discussion 62 7. ANN MODELING FOR COAXIAL LRM CALIBRATIONS 63 7.1. Load Modeling 64 7.2. Calibration Comparisons 67 7.3. Discussion 68 PART II - DEVELOPING FREQUENCY-DOMAIN MODELS FOR NONLINEAR CIRCUITS BASED ON LARGE-SIGNAL MEASUREMENTS 8 . NONLINEAR NETWORK ANALYSIS 70 8.1. Linear V ersus N onlinear Behavior 70 8.2. Nonlinear Vector Network Analysis 71 8.3. Nonlinear Vector Network Analyzers 75 8.3.1. Architecture 75 8.3.2. Calibration 77 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I 8.4. 9. Developing Frequency-Domain Models for Circuits Based on Large-Signal Measurements Nonlinear 79 CALCULATING RATIOS OF HARMONICALLY-RELATED, COMPLEX SIGNALS 81 9.1. Introduction 81 9.2. Method 80 9.3. Example 84 9.4. Discussion 89 10. NONLINEAR LARGE-SIGNAL SCATTERING PARAMETERS 90 10.1. General Form 90 10.2. Nonlinear Large-Signal Impedance Parameters 93 10.3. Relating 94 10.4. Nonlinear Large-Signal Admittance Parameters 97 10.5. Relating &>and $ Parameters 98 10.6. One-Port Network with Single-Tone Excitation 101 10.7. Two-Port Network with Single-Tone Excitation 103 10.8. Discussion 105 and Z Parameters 11. EXPANDING DEFINITIONS OF GAIN HARMONIC CONTENT INTO ACCOUNT BY TAKING 107 11.1. Commonly Used Definitions of Gain 107 11.2. Expanded Definitions of Gain 109 11.3. Expanded Definitions of Gain in Terms of ^-Parameters 111 11.4. Discussion 113 12. USING ^-PARAMETERS AND EXPANDED DEFINITIONS OF GAIN TO EXAMINE THE BEHAVIOR OF A PHEMT MODEL IX Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 12.1. Using ^-Parameters to Examine the Behavior of a PHEMT Model 116 12.2. Using Expanded Definitions of Gain to Examine the Behavior of a PHEMT Model 121 12.3. Discussion 125 13. USING ^-PARAMETERS TO DESIGN A DIODE FREQUENCY DOUBLER WITH A HARMONIC-BALANCE SIMULATOR 126 13.1. Diode Only 128 13.2. With 1&2 GHz Filters 129 13.3. With 1&2 GHz Filters, and Input Matching 130 13.4. With 1&2 GHz Filters, Input and Output Matching 130 13.5. With 1&2 GHz Filters, Input & Output Matching Optimized 131 13.6. With 1&2,4&6 GHz Filters, In & Out Matching Optimized 131 13.7 Discussion 132 14. DETERMINING ^-PARAMETERS FROM ARTIFICIAL NEURAL NETWORK MODELS TRAINED WITH MEASUREMENT DATA 134 14.1. Methodology 134 14.2 Sensitivity Analysis of ANN Models 142 14.3. Results and Comparison 147 14.4. Discussion 150 15. COMPARING ^-PARAMETERS SCATTERING FUNCTIONS WITH NONLINEAR 16. SUMMARY AND FUTURE WORK 16.1. ANN Modeling for Improved Vector Network Analyzer Calibrations Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 \ 58 153 16.1.1. Summary 158 16.1.2. Other Applications 160 16.1.3. Future Work 161 16.2. Frequency-Domain Models and Figures of Merit for Nonlinear Circuits 164 16.2.1. Summary 164 16.2.2. Future Work 167 169 BIBLIOGRAPHY XI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Figure 2.1. MLP3 artificial neural network architecture. 13 Figure 2.2. Sigmoidal activation function used in hidden and output layer neurons. 15 Figure 2.3. Linear scaling of neural network data. 17 Figure 2.4. Three ANN models illustrating (a) over-learning, (b) underlearning, and (c) good-learning. 19 Figure 3.1. Determining two-port scattering parameters. 25 Figure 3.2. Four-sampler vector network analyzer. 27 Figure 3.3. Three-sampler vector network analyzer. 27 Figure 3.4. Twelve-term error model for vector network analyzers. 29 Figure 3.5. Eight-term error model for vector network analyzers. 30 Figure 3.6. Standards measured for the OSLT, LRM, TRL, and multiline TRL calibrations. 32 Figure 4.1. Training error versus the number of neurons in the hidden layer for various OSLT calibration standards. 37 Figure 4.2. Magnitude of the ANN-modeled reflection coefficient errors (|ASn|) for the open standard with varying numbers of training points. 38 Figure 4.3. Magnitude of the ANN-modeled reflection coefficient errors (|ASn|) for the load standard with varying numbers of training points. 38 Figure 4.4. Comparison of magnitude and phase of the reflection coefficients [|Sn| and Arg(Sn)] for the load standard using an ANN model trained with 5 points, linear interpolation with TRL using the same 5 points, and TRL with 192 points as the reference. 40 xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.5. Magnitude and phase of the reflection coefficients [|Sn| and Arg(Sn)] for the open standard measured by multiline TRL and ANN modeling. 41 Figure 4.6. Magnitude and phase of the reflection coefficients [|Sn| and Arg(Sn)] for the short standard measured by multiline TRL and ANN modeling. 42 Figure 4.7. Magnitude and phase of the reflection coefficients [|Sn| and Arg(Sn)] for the load standard measured by multiline TRL and ANN modeling. 43 Figure 4.8. Magnitude and phase of the transmission coefficients [|S2i| and Arg(S2i)] for the thru standard measured by multiline TRL and ANN modeling. 44 Figure 4.9. Magnitude and phase of the scattering parameters of a calibrated 19-mm CPW transmission line. 46 Figure 4.10. Magnitude of the scattering parameter differences (|ASij|) of a calibrated 19-mm CPW transmission line. 47 Figure 5.1. Real and imaginary components of Z\\ for the load standards measured by multiline TRL and modeled by an ANN. 52 Figure 5.2. Magnitude and phase of S21 for a calibrated 1.764-mm CPW transmission line. 55 Figure 5.3. Magnitude and phase of S 11 for a calibrated 1.764-mm CPW transmission line. 56 Figure 5.4. Magnitudes of the scattering parameter differences of a calibrated 1.764-mm CPW transmission line. 57 Figure 6.1. Magnitude and phase of measured and modeled scattering parameters of the on-wafer load for the LRM calibration. 59 Figure 6.2. Worst-case error bounds between measurements of passive devices from on-wafer LRM and TRL calibrations and the multiline TRL calibrations. 61 Figure 7.1. Real and imaginary components of the measured and modeled impedance of the coaxial load for the LRM calibration. 66 Xlll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.2. Worst-case error bounds between measurements of passive devices from coaxial LRM and TRL calibrations and the multiline TRL calibrations. 69 Figure 8.1. Nonlinearities transfer energy from the stimulus frequency to products at new frequencies. 71 Figure 8.2. Complex traveling waves at the ports of a nonlinear device when a set of harmonically related signals is present. 72 Figure 8.3. Flow diagram for a two-port network with a singlefrequency excitation at port 1 and a terminating load at port 2. (a) Linear case, (b) Nonlinear case considering three harmonics. 74 Figure 8.4. Simplified block diagram of a nonlinear vector network analyzer. 76 Figure 9.1. Phasor plot of the fundamental reference x\, the dividend z%, and the divisor >’3 at the first phase reference [all phasors identified by superscript ( 1)]. 84 Figure 9.2. Time-domain plot of the fundamental reference x\, the dividend Z2, and the divisor y 3 at the first phase reference. 88 Figure 9.3. Phasor plot of the fundamental reference x\, the dividend Z2, and the divisor _y3 at the second phase reference [using superscript (2 )]. 88 Figure 9.4. Phasor plot of the fundamental reference xi, the dividend Z2, and the divisor y 3 at the third phase reference [using superscript (3)]. 89 Figure 12.1. Circuit diagram of a pHEMT device operating in a two-port, common-source configuration. 115 Figure 12.2. Magnitude of as a function of input power for a nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GFIz and a bias of Vds - 3V and VGS = 0.0, -0.2, -0.4, -0.6, -0.8, and -1.0 V. 117 Figure 12.3. Magnitude of # 21*1 as a function of input power for a nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of VDS = 3V and Vgs - 0.0, -0.2, -0.4, -0.6, -0.8, and -1.0 V. 119 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I Figure 12.4. Phase of £>m\ and as a function of input power for a 120 nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of VDS - 3V and Vgs - 0.0, -0.4, and -1.0 V. Figure 12.5. Expanded power gain i as a function of input power for a nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of Vds - 3V and VGS = 0.0, -0 .2 , -0.4, -0 .6 , -0 .8 , -1.0, and - 1.2 V. 121 Figure 12.6. Expanded transducer gain C$r as a function of input power for a nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of VDS = 3V and V g s = 0 .0 , -0 .2 , -0.4, -0 .6 , -0 .8 , -1.0, and - 1.2 V. 122 Figure 12.7. The ratio ®i/(S as a function of power for a nonlinear lumped-element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of V d s ~ 3V and V g s = 0.0, -0.2, -0.4, -0.6, -0.8, -1.0, and -1.2 V. 123 Figure 12.8. The 1 dB gain compression point for a nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of Vds - 3V and Vgs - -0.4 V. 125 Figure 13.1. Block diagram of a single-diode resistive doubler. 126 Figure 13.2. Nonlinear large-signal ^-parameters used to characterize a two-port device excited by a single-tone signal at port 1 . 127 Figure 13.3. Final design of the single-diode resistive frequency doubler. Electrical lengths shown are all at 1 GHz. 133 Figure 13.4. Time-domain plots of a\ and 62 for the simulated 1-2 GHz frequency-doubler circuit. 133 Figure 14.1. Block diagram of a nonlinear vector network analyzer equipped with a second source and isolators. 135 Figure 14.2. An ANN model that maps real and imaginary values of o ’s to b's for different real and imaginary values of amn \{m±X)/\(n£V)\. 136 1 xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 Figure 14.3. An ANN model that interpolates b’s from the measured results for nonzero values of amn [(m^l)A(n^l)] to the desired values for amn [(m^l)A(n^l)] equal to zero. Outputs of the ANN model yield values of &iuti- 136 Figure 14.4. Schottky diode in a series configuration located in the middle of a CPW transmission line. (White area is metal.) 138 Figure 14.5. Five hundred measurements of a2\ in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 138 Figure 14.6. Five hundred measurements of an in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 139 Figure 14.7. Five hundred measurements of au in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 139 Figure 14.8. Five hundred measurements of a 22 in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 140 Figure 14.9. Five hundred measurements of a22i in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 140 Figure 14.10. Five hundred measurements of b\\ in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 141 Figure 14.11. Five hundred measurements of b2\ in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 141 Figure 14.12. Average testing errors as functions of the number of hidden neurons for ANN models trained to map aj and a2 to bj and ai and a 2 to b 2. The models were developed using 250 training points and verified using 250 testing points. 146 j j ! xvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 14.13. Average testing errors as functions of the number of training points for ANN models trained to map and &2 to bi and sl\ and a 2 to b 2. The models were developed using 14 hidden neurons and verified using 250 testing points. 146 Figure 14.14. Average testing errors as functions of the number of testing points for ANN models trained to map ai and a 2 to bi and ai and a2 to b 2. The models were developed using 14 hidden neurons and 250 training points. 147 The 250 measurements of bn used for training (circles). Values of B in r^ n were determined from the measurementbased ANN model (square) and the harmonic balance simulation using a compact model (triangle). 149 Figure 14.16. The 250 measurements of bn used for training (circles). Values of B>2in-tfn were determined from the measurementbased ANN model (square) and the harmonic balance simulation using a compact model (triangle). 149 Figure 15.1. Gkpij and Hkpy serve to map ay circles centered at zero to bkP ellipses with variable axes also centered at zero, neglecting FkP for illustrative purposes. 152 Figure 15.2. If B y/ is a complex constant for a given bias and fundamental drive level, it has the limitation that it can only map circles into circles. 157 Figure 14.15. X V ll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I LIST OF TABLES Table 3.1. Error terms for the twelve-term model. 28 Table 5.1. Measured DC resistances of the five load terminations. 51 Table 9.1. Determining the ratios of Z2 to y?, using three methods at three different phase references. 87 Table 13.1 Simulated values for & im - §>n6i, &2111 ~ &2i6i> ®2, and (g2/(& for each of the design stages of the diode frequency doubler. 129 Table 14.1. Average testing errors and correlation coefficients as functions of the number of hidden neurons for ANN models trained to map values from the first five harmonics of ai and a 2 to the first five harmonics of bi. All models were developed using 250 training points and verified using 250 testing points. 144 Table 14.2. Average testing errors and correlation coefficients as functions of the number of training points for ANN models trained to map values from the first five harmonics of aj and a 2 to the first five harmonics of bi. All models were developed using 14 hidden neurons and verified using 250 testing points. 144 Table 14.3. Average testing errors and correlation coefficients as functions of the number of testing points for ANN models trained to map values from the first five harmonics of ai and a 2 to the first five harmonics of bi. All models were developed using 250 training points and 14 hidden neurons. 144 Table 14.4. Average testing errors and correlation coefficients as functions of the number of hidden neurons for ANN models trained to map values from the first five harmonics of ai and a 2 to the first five harmonics of b 2. All models were developed using 250 training points and verified using 250 testing points. 145 xvm Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Table 14.5. Average testing errors and correlation coefficients as functions of the number of training points for ANN models trained to map values from the first five harmonics of ai and a 2 to the first five harmonics of b 2- All models were developed using 14 hidden neurons and verified using 250 testing points. 145 Table 14.6. Average testing errors and correlation coefficients as functions of the number of testing points for ANN models trained to map values from the first five harmonics of ai and a 2 to the first five harmonics of b 2. All models were developed using 250 training points and 14 hidden neurons. 145 Table 14.7. Differences between the measurement-based, ANN-modeled results and the compact model simulated in commercial harmonic-balance software. 148 I 1 xix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I CHAPTER 1 INTRODUCTION This thesis is comprised of two parts. The first segment covers artificial neural network (ANN) modeling for improved vector network analyzer (VNA) calibrations. The second segment covers frequency-domain models and figures of merit for nonlinear circuits based on large-signal measurements. In this chapter, an introduction and motivation is provided for each part of the thesis. Following that, the organization of the thesis is presented along with a brief summary of each chapter. 1.1. ANN Modeling for Improved Vector Network Analyzer Calibrations Vector network analyzers (VNAs) are one of the most versatile instruments in the RF and microwave industry. They can be found in calibration facilities, research laboratories, design facilities, and on production lines. VNAs are used to measure complex scattering parameters of devices and circuits. Engineers use them to verify their designs, confirm proper performance, and diagnose failures. The accuracy of a VNA measurement is highly dependent on its calibration, which accounts for imperfections in the instrument such as impedance mismatch, loss in the cables and connectors, the frequency response of the source and receiver, and directivity and crosstalk due to signal leakage. There are a wide variety of calibration methods available to VNA users, most of which can be classified into one of two categories depending on the type of calibration standards used. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. The first category makes use of transmission lines as standards, and includes such calibration methods as thru-reflect-line (TRL) [1-2] and multiline TRL [3-4], Multiline TRL is the most accurate means of VNA calibration and is especially useful for on-wafer environments, since the characteristic impedance can be calculated from dimensional measurements of the standards, which consist simply of a number of transmission lines of varying line lengths and a reflective termination. In an on-wafer environment, the disadvantages of this method are that it requires a lot of real estate on the wafer, due to the numerous long lines required for an accurate calibration, and the different lengths of line necessitate changing the separation between probes during the calibration process. In a coaxial environment, the disadvantages are that a large number of expensive airline standards and numerous interconnects are required. Consequently, a second category of VNA calibrations, which makes use of compact, lumped-element standards, is often preferred. The most common of these calibrations are the open-short-load-thru (OSLT) and the line-reflect-match (LRM) [5] methods. The trade-off is that these methods tend to be less accurate, since it is more difficult to calculate the reflection coefficients of the standards from independent physical measurements. But, if the compact calibration kits are characterized using a benchmark calibration, such as multiline TRL, it is possible to perform an accurate lumped-element calibration. Once the lumped-element standards for a given calibration kit are characterized, one must decide whether to develop a model for each of the standards or to directly use the measurement data obtained from the benchmark calibration. In the first part of this thesis, ANNs are applied to improve the modeling of lumped- with permission of the copyright owner. Further reproduction prohibited without permission. element standards in both on-wafer and coaxial environments. As a major research result, this doctoral thesis shows that ANN models offer a number of advantages over the use of calibrated-measurement data files and equivalent circuit models, namely, the following: ( 1) they do not require the numerous details and parameters of physical models; (2 ) calibration times can be reduced since only a few training points are required to accurately model the standards; (3) ANN model descriptions are much more compact than large measurement files; (4) ANN models, trained on only a few measurement points, can be much more accurate than direct calibrations when limited data are available; and (5) they give an optimized estimate in the presence of noise. Specifically, this thesis describes how on-wafer OSLT standards have successfully been modeled, in one case assuming the standards can be reproduced from wafer to wafer with little variation, and in another case where the loads exhibit significant difference among the wafers studied. This thesis also describes how load standards have been modeled to improve both on-wafer and coaxial LRM calibrations. 1.2. Frequency-Domain Models and Figures of Merits for Nonlinear Circuits Although the measurement of ^-parameters using a vector network analyzer (VNA) is invaluable to the microwave designer of linear circuits, such measurements are oftentimes inadequate for nonlinear circuits since nonlinearities transfer energy from the stimulus frequency to products at new frequencies. Thus, a different and more sophisticated instrument is required to measure nonlinear devices and circuits. A recently introduced nonlinear vector network analyzer (NVNA) is capable of with permission of the copyright owner. Further reproduction prohibited without permission. providing accurate waveform vectors by acquiring and correcting the magnitude and phase relationships between the fundamental and harmonic components in the periodic signals [6-10]. An NVNA excites a nonlinear device under test (DUT) with one or more sine-wave signals and detects the response of the DUT at its signal ports. Assuming the DUT exhibits neither sub-harmonic nor chaotic behavior, the input and output signals will be combinations of sine-wave signals, due to the nonlinearity of the DUT combined with mismatches between the system and the DUT. Even though ^-parameters cannot adequately represent nonlinear circuits, some type of parameters relating incident and reflected signals are beneficial so that | the designer can “see” application-specific engineering figures of merit that are similar to what he or she is accustomed to. Another major result of this thesis is the proposal of definitions of such ratios that are referred to as nonlinear large-signal scattering (# ) parameters. First, a general form of time-invariant nonlinear large- | signal ^-parameters is presented. Then nonlinear large-signal impedance (Z) and j admittance (fl) parameters are introduced, and equations for relating the different representations are derived. It is important to note that these parameters are developed for sparse tone inputs, rather than wideband modulation. j In addition to nonlinear large-signal ^-parameters, another figure of merit for ] s | nonlinear circuits is introduced by expanding the definitions of power gain, j transducer gain, and available gain by taking harmonic content into account. Under j special conditions, these expanded definitions of gain can be expressed in terms of j \ i ! two-port, nonlinear large-signal ^-parameters. 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For existing nonlinear models, nonlinear large-signal ^-parameters can readily be generated by performing a harmonic balance simulation. For devices, with no model available, these parameters can be extracted from artificial neural network (ANN) models that are trained with multiple frequency-domain measurements made on a nonlinear DUT with an NVNA. To illustrate applications and generation of nonlinear large-signal parameters, three examples are presented. First, nonlinear large-signal ^-parameters and the expanded definitions of gain are shown to be useful in discovering valuable information regarding the behavior of a nonlinear model. Specifically, a lumpedelement model of a pseudomorphic high electron mobility transistor (pHEMT) device operating in a two-port, common-source configuration is examined. Second, this thesis illustrates how nonlinear large-signal ^-parameters can be used as a tool in the design process of a nonlinear circuit, specifically a single-diode 1-2 GHz frequencydoubler. And finally, a method for generating nonlinear large-signal ^-parameters based upon ANN models trained on frequency-domain data measured using a nonlinear vector network analyzer (NVNA) is described. A diode circuit model, generated using this method, is compared to a harmonic balance simulation of a commercial device model. In both parts of this thesis, when measurement-based ANNs are utilized, not only are demonstrations presented to show their usefulness, but specific procedures are outlined for determining optimal models. | 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 1.3. Organization of the Thesis In Chapter 2, an overview of artificial neural networks is presented since they are common to both parts of this thesis. This chapter covers ANN structure, with special emphasis on the multi-layer perceptron architecture, model development, and model verification. The first part of the thesis, which deals with ANN modeling for improved vector network analyzer calibrations, is covered in Chapters 3-7. In Chapter 3, an overview of linear network analysis is presented, which includes an introduction to scattering parameters, VNA architecture and error models, calibration techniques, and motivations for using ANNs to model calibration standards. In Chapter 4, ANNs are applied to improve the modeling of on-wafer OSLT standards. The ANNs are trained with measurement data obtained from a benchmark calibration. The training errors and training times are quantified as functions of both the number of training points and the number of neurons in the hidden layer. The OSLT calibration using the ANN-modeled standards compares favorably (less than a 0.02 difference in magnitude) to the benchmark calibration over a 40 GHz bandwidth. The assumption made in Chapter 4 is that the standards can be reproduced from wafer to wafer with little variation. One study, however, found that while open, short, and thru standards can be reproduced with minimal variance, load standards exhibit a significant difference among wafers. So in Chapter 5, it is demonstrated that on-wafer OSLT calibrations can be further improved by applying ANNs to model the correlation between measured DC resistance and RF variations in load terminations. Furthermore, ANN models are shown to offer a number of advantages over using 6 with permission of the copyright owner. Further reproduction prohibited without permission. calibrated measurement files or equivalent circuit models, including ease of use, reduced calibration times, and compactness. In Chapter 6 , a load is modeled using an ANN to improve an on-wafer LRM calibration. The accuracy of the LRM calibration using the ANN-modeled load compares favorably to a benchmark calibration with an average worst-case scattering parameter error bound of 0.017 over a 40-GHz frequency range. In Chapter 7, a load is modeled using an ANN to improve a coaxial LRM calibration and improve its accuracy over an earlier reported equivalent circuit model. This chapter shows that an ANN model allows one to develop a compact description of the standard without having to formulate a detailed physical model. The second part of the thesis, which deals with frequency-domain models and figures of merit for nonlinear circuits based on large-signal measurements, is covered in Chapters 8-15. Chapter 8 compares linear and nonlinear network analysis. A brief overview of nonlinear vector network analyzers (NVNAs) is presented, which includes instrument architecture, error models, and calibration techniques. Finally, motivations are presented for introducing application-specific figures of merit such as nonlinear large-signal scattering parameters and expanded definitions of gain. In Chapter 9, a method is presented for preserving time-invariant phase relationships when ratios are taken between two harmonically related, complex signals by introducing a third signal that is used as a phase reference. This technique shows that a reference signal must be present at the fundamental frequency in order for time-invariant phase relationships to exist between ratios of any two harmonically related signals. A simple example is provided to illustrate the technique. This method Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. must be implemented when defining the time-invariant, nonlinear large-signal scattering parameters in the following chapter. In Chapter 10, the concept of nonlinear large-signal scattering parameters is introduced. Like commonly used linear 5-parameters, nonlinear large-signal scattering (# ) parameters can also be expressed as ratios of incident and reflected wave variables. However, unlike linear 5-parameters, nonlinear large-signal f¶meters depend upon the signal magnitude and must take into account the harmonic content of the input and output signals since energy can be transferred to other frequencies in a nonlinear device. After presenting the general form of nonlinear large-signal ^-parameters, nonlinear large-signal impedance (Z) and admittance (|f) parameters are also presented, and equations for relating the different representations are derived. Next, two simplifications are made, considering the cases of a one-port network with a single-tone excitation and a two-port network with a single-tone excitation. In Chapter 11, the definitions of power gain, transducer gain, and available gain are expanded by taking harmonic content into account. Under special conditions, these expanded definitions of gain can be expressed in terms of the two-port, nonlinear large-signal scattering parameters defined in the previous chapter. In Chapter 12, an example is provided showing how nonlinear large-signal scattering parameters and the expanded definitions o f gain, introduced in the previous chapters, can be used to discover valuable information regarding the behavior of a nonlinear model. Specifically, a lumped-element model of a pseudomorphic high Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. electron mobility transistor (pHEMT) device operating in a two-port, common-source configuration is examined. Chapter 13 illustrates how nonlinear large-signal scattering parameters can be used to as a tool in the design process of a simple nonlinear circuit, specifically a single-diode 1-2 GHz frequency-doubler circuit. In Chapter 14, an approach is described for generating ANN models for nonlinear devices, based upon frequency-domain data measured using a nonlinear vector network analyzer (NVNA). Specifically, models are developed for a Schottky diode in a series configuration. This chapter demonstrate that the ANN models give accurate descriptions of the input-output relationships of the device over the span of the measurements. A sensitivity analysis is performed to determine how many training points, testing points, and hidden neurons are required to adequately train the ANN models. The models are also used to extract nonlinear large-signal scattering parameters using appropriate ratios of wave variables measured under large-signal conditions. An independent check is obtained by comparing the diode-circuit models, generated by means of this methodology, to a compact model simulated in commercial harmonic-balance software. In Chapter 15, nonlinear large-signal scattering parameters are compared to another form of nonlinear mapping, known as nonlinear scattering functions [11, 12]. The two formulations are shown not to be equivalent. Specifically, nonlinear largesignal scattering parameters are more general than nonlinear scattering functions, which are useful in approximating a specific class of nonlinearity in a more compact form. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Finally, Chapter 16 provides a summary of the thesis with concluding remarks and directions for future research. 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2 ARTIFICIAL NEURAL NETWORKS Artificial neural networks (ANNs) are neuroscience-inspired computational tools that are trained using input-output data to generate a desired mapping from an input stimulus to the targeted output. ANNs consist of multiple layers of processing elements called neurons. Each neuron is linked to other neurons in neighboring layers by varying coefficients that represent the strengths of these connections. Learning is accomplished by adjusting these coefficients until the network provides output results that meet prescribed values. ANNs have been applied to diverse areas such as speech and pattern recognition, financial and economic forecasting, telecommunications, and nuclear power plant diagnosis, and have just recently been introduced into the area of microwave engineering [13]. In particular, researchers have successfully used ANNs to model microstrip vias [14], packaging and interconnects [15], spiral inductors [16], MESFET devices [17], CPW circuit components [18], effective dielectric constant of microstrip lines [19], and HBT amplifiers [20], to name just a few. ANNs offer several advantages over other modeling methods [13], First, they tend to be more accurate than polynomial regression models and allow for more dimensions than look-up tables [21]. Second, prior knowledge about the input-output mapping is not required to develop an ANN model. Relationships between input and output data are developed from the training process. Third, ANNs can generalize, which means that they can provide the correct output for inputs that have not been 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I used to train the model. And fourth, ANNs have the ability to model both linear and nonlinear input-output mappings. 2.1. Neural Network Structure A neural network has two types of components, the processing elements known as neurons and the connections between them known as links. Each neuron receives stimulus from connected neighboring neurons, processes the information, and produces an output. Input neurons receive stimulus from outside of the network, hidden neurons receive stimuli from other neurons and send stimuli to other neurons in the network, and output neurons produce stimuli that are used externally. A variety of neural network architectures have been developed for numerous applications. O f these, multi-layer perceptrons (MLP) are the most popular type in use today. They belong to a class of structures known as feed-forward neural networks and are capable of approximating generic classes of functions [22], In an MLP network, neurons are grouped into layers, namely an input layer, one or more hidden layers, and an output layer. Figure 2.1 illustrates a three-layer network (MLP3). One of the reasons that MLP3 networks are so widely used is due to the proof of the universal approximation theorem for MLP networks, which in summary states that there always exists a three-layer perceptron that can approximate an arbitrary, continuous, multidimensional function with any desired accuracy [23-24]. From this theorem, we can conclude that any errors in an MLP3 neural model can be attributed 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to too many or too few neurons, inadequate training, or the presence of a stochastic relationship between inputs and outputs. In an MLP3 network, each neuron, except those in the input layer, receive and process stimuli from other neurons. Each input is multiplied by a corresponding weight parameter w, and the resulting products are added to produce a weighted sum, which is passed through a neuron activation function g to produce the final output of the neuron. The output can then become a stimulus for neurons in the next layer. z, / Y 1 / Output Layer Hidden Layer Input Layer w \\ W // * —* g(Uj) Zj " 11 WKi\\ Kj \ \ —► v = t g(vK) * k=l,2,...,K Figure 2.1. MLP3 artificial neural network architecture. 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. K Analytical functions corresponding to the network of Figure 2.1 can be written down as follows [25], The input of the y'th hidden neuron is obtained by forming a weighted linear combination of the I input variables x, and the corresponding weights w(l)i to give Uj = X w^Xf (2.1) i=0 where w(1)7o is the additional bias input to the jth neuron. The output zj of the y'th hidden neuron is obtained by transforming the linear sum of eq.(2.1) using the activation function of the neuron g to give zj = g ( U j ) . (2.2) The network outputs are obtained in a similar manner. For each output neuron k, a weighted linear combination of the outputs of the hidden layer neurons is formed to give V* = Z •. ( 2 .3 ) 7=0 The output y>k of the Mi output neuron is obtained by transforming the linear sum of eq. (2.3) using the activation function of the neuron g to give y k = g ( vk )• Combining eqs. (2.1)-(2.4), the kth output of the network is 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2-4) yk=g ^ j =0 (2) / T (1) 1 2 \ w )/X i (2.5) Ki=0 Equation (2.5) can be written in matrix-vector form as Y = g[W2 #g(W 1 *X>] (2.6) where X is the input vector, Y is the output vector, and Wi and W 2 are the weight matrices between the input and hidden layers and between the hidden and output layers, respectively. The most widely used activation function is the sigmoidal function given by g(«) = 1 1+ exp(-w) (2.7) and is shown in Figure 2.2. The advantages of this activation function are that it is bounded between 0 and 1, it is continuous, monotonic, and continuously differentiable. 1 .0 - 1 0 .8 0.6 0.4 > r2 _ -10 Figure 2.2. Sigmoidal activation function used in hidden and output layer neurons. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2. Model Development The first step in developing a neural network is the generation of training and testing data, which may be obtained from either simulation or measurement. Training data are used for model development while testing data are used for model validation. Before an ANN model is developed, the training data, both inputs and outputs, are usually re-scaled. This is especially useful when different variables have values that significantly vary. With linear re-scaling, each variable is assigned the same importance for model development resulting in weights that should not be substantially different. Linear scaling, shown in Figure 2.3 is given by X = im in + - * ~ ^ nn_ ■*max ( ; max < ( 2 .8 ) x min Model training is accomplished by adjusting the weights of the network by comparing the outputs of the network to the target output variables. At the beginning of training, the weights are initialized to small random values. The inputs are passed through the network and the outputs are computed. The output values are compared to target values and the derivatives of the error between them with respect to each of the weights are calculated. Weights are then adjusted based upon the contribution to the error such that the overall network error decreases. This procedure continues in an iterative fashion until the overall error of the network is reduced to a prescribed level. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I X max mm X Xmax Xmm , Figure 2.3. Linear scaling of neural network data. One of the most popular algorithms for neural network training is back propagation [26]. This refers to the propagation of errors backward through the network to find the error derivatives with respect to each weight. During training, the neural network performance is evaluated by computing the difference between outputs of the network y* to the target values tk for all N of the training samples. The difference E, also known as the error, is given by e = \ X Z b 'b . - t h , ? . (2.9) 1 n=\k=1 Derivatives of the error with respect to each weight are computed, and the weights w are updated along the negative direction of the gradient of E as 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I w = w - 77^— , aw (2.10) until E becomes small enough. Here, the parameter T| is called the learning rate. Although back propagation is the most popular training algorithm, it is relatively slow in converging, so second-order, training algorithms such as conjugategradient and Quasi-Newton are oftentimes preferred for their increased efficiency [13]. The Quasi-Newton approach is relatively fast due to its quadratic convergence property, although more computer memory and implementation effort is required since it relies on the Hessian matrix and its inverse to be calculated. The conjugategradient method is a nice compromise both in terms of memory requirements and implementation effort, since the descent direction runs along the conjugate direction, which can be accomplished without matrix computations. It is important to point out that there are potential hazards when training an ANN. One possibility is over-learning, which means that the network matches the training data but cannot generalize well. This can occur when there are insufficient training data or too many hidden neurons. Another possibility is under-learning, which means the network has not learned the input-output relationship well and cannot even match the training data. Reasons for this include insufficient hidden neurons, inadequate training, or the training procedure being stuck on a local minimum. Good-learning is when the network matches the training and testing data well. Figure 2.4 illustrates the three cases. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 -I 6 - 42 ■ □ - 4 6 Training D ata T estin g D ata ANN 10 8 12 (a) lO -i □□ 6- 4- □ □ 2 4 6 ■ □ Training D ata T esting D ata ANN 10 8 12 (b) 6- 42 ■ □ - 0 2 4 6 8 Training D ata T esting D ata ANN 10 12 (c) Figure 2.4. Three ANN models illustrating (a) over-learning, (b) under-learning, and (c) good-leaming. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I 2.3. Model Verification Once a neural network model is trained, its quality should be tested with an independent set of testing data. There are a number of measures that can be used to indicate the quality of a model, a few of which are discussed below. The relative test error S« for the Mi output of a neural model at the /th data sample is defined as S ., = thl....... tk ,m a x k = l t ' " t K . j e Te (2.11) *knmin where Te is the set of test data. The average test error 8aVg is given by ~ I sfel _ k = lle T e avg " Sizc(Te)K ( 2. 12) The worst-case test error 8w-c is calculated from K ? w -c = max max \ou (2.13) k=1 /e L Another measure that indicates the correlation between a neural model and test data is the correlation coefficient p and it is defined as H £ (ykl - Tavg \^kl ~ ^avg ) k=lleT„ K K Y Y {ykl Y Y {fkl t'dvg) T avg) k - \ 1&TP k=\l<aTP (2.14) where 20 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. K (2.15) 7av8= Siz« T e)K and 1 ' a ,6 = Sizt*Te)K {1 u -t (2.16) In the following portions of this thesis, ANNs will be shown to be a powerful tool for developing measurement-based, frequency-domain models of VNA calibration standards and nonlinear devices. ! 21 3 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 LINEAR NETWORK ANALYSIS This chapter reviews the basic concepts of linear network analysis as applied to microwave frequencies, which in its most fundamental form involves the measurement of incident, reflected, and transmitted waves that travel along waveguides. Since it is difficult, if not impossible, to measure total voltage and current at microwave frequencies, scattering parameters are usually measured. These parameters relate to familiar concepts such as reflection coefficient, insertion loss, and gain. Aside from the physical interpretations, scattering parameters are a convenient way to represent linear networks for a number of reasons, namely: they can be cascaded for multiple devices to predict system performance; they can be converted into impedance and admittance parameters; and they are conveniently used with computer-aided engineering circuit simulation software. Finally, scattering parameter ratios enable measurements to be made that are independent of absolute power. In the following sections, we briefly review scattering parameter definitions, VNA architecture and error models, and a discuss some of the most popular types of VNA calibration techniques. 3.1. Scattering Parameters When an electric circuit operates at low frequencies, its size is relatively small compared to the wavelength and thus it may be treated as an interconnection of lumped components with unique voltages and currents defined at any point in the 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. network. This leads to the formulation of impedance and admittance parameters. However, when an electric circuit operates at microwave frequencies, its size can be much larger than the wavelength and thus voltages and currents may not be welldefined. In this case, normalized wave variables (a and b) are more suitable for characterizing circuits at microwave (and higher) frequencies. Relationships between a’s and b’s for a linear circuit can expressed by a scattering matrix whose elements are known as scattering parameters, or ^-parameters [27-30]. The number of ^-parameters for a given device is equal to the square of the number of ports. For example, a two-port device has four 5-parameters. Each Sparameter Sy has two subscripts, i and y, where i refers to the port at which power emerges and j refers to the port number at which power enters. When i and j are identical, Su indicates a reflection coefficient, and when i and j are different, Sy indicates a transmission coefficient. Consider an JV-port network. Normalized wave variables aj and bj at the y'th port are proportional to the incoming and outgoing waves, respectively, and may be defined in terms of the voltages associated with these waves as follows: aJ V1 V7 *J = (3.1) Z 0J where V'j and Vj represent voltages associated with the incoming and outgoing waves in the transmission lines connected to the y'th port, and Z oJ is the characteristic impedance of the line at the y'th port. The scattering matrix S of the network expresses the relationship between a’s and b’s at various ports through the matrix equation 23 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 1 (3.2) b = Sa, where b and a are iV-element column vectors and S is an (/Vx/V)-eleinent square matrix. For a two-port network, eq. (3.2) becomes CN 1 1 i1 i <>T V "A A. A (3.3) The ^-parameters along the diagonal of the scattering matrix are reflection coefficients defined as S jj bJ VJ a J a,= 0 (i* j) V/ ' (3.4) Off-diagonal terms o f the scattering matrix are transmission coefficients defined as VflJZoi auj ai = Q { i * j ) • (3.5) V ;/Jz~ Sy may be determined by finding bt for a value aj (power input to port j) with incident wave variables at all other ports equal to zero. Figure 3.1 illustrates this for a two-port network. I 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Forward Transmitted Incident a . DUT a 2 ~ Reflected 0 Reflected b, . Su =------------ = — a, =0 Incident ax S21 - Transmitted b0 , = a2=0 Incident ax Reverse Reflected b2 , S22 = ------------ = a\ Incident a2 o Transmitted b, . 512 --------------- = — Incident a2 =0 Reflected a1 b 1 DUT Clr Transmitted Incident Figure 3.1. Determining two-port scattering parameters. It should be noted that all of the normalized wave variables, characteristic impedances, and S-parameters are generally complex values. Making use of magnitude and phase is required to fully characterize a linear network. Furthermore, complex impedance must be measured in order to design efficient matching networks. Additionally, the transformation from the frequency-domain to the time-domain requires magnitude and phase information in order to perform an inverse Fourier transform. 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I 3.2. Vector Network Analyzer Architecture A vector network analyzer (VNA) is an instrument that is used to measure complex 5-parameters [31-37], It does this by sampling the incident and reflected waves at both ports of a device under test (DUT), and then forming ratios that are directly related to the reflection and transmission coefficients of the device. Frequency is swept to obtain 5-parameters over a band of interest. Most commercial VNAs make use of either three-sampler or four-sampler architectures. High-end VNAs often use four samplers, while less expensive VNAs employ three samplers [4, 38]. Figure 3.2 is a simple operational schematic of a four-sampler VNA. Directional couplers behind each test port can sample the incident and reflected waves with the source switched to either port while the other port is terminated by a nominally ideal load. Typically, four-sampler VNAs do not make use of all four data channels. Rather, only the transmitted signal, and not the reflected signal, is sampled on the terminating side. This limits the cost of the electronics and increases sweep speed. Neglecting the reflected signal introduces additional error in the measurement. However, this error can be corrected provided that the load reflection coefficients are repeatable and can be measured at least once. Figure 3.3 is a simple operational schematic of a three-sampler VNA. Directional couplers behind each test port can sample the reflected waves with the source switched to either port while the other port is terminated by a nominally ideal load. A third directional coupler samples the incident wave independent of the switch position. 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I DUT Figure 3.2. Four-sampler vector network analyzer. VW DUT Figure 3.3. Three-sampler vector network analyzer. 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3. Vector Network Analyzer Error Models When using a VNA for microwave measurements, calibration is required regardless of the architecture of the instrument. Calibrations account for serious systematic imperfections in the VNA such as impedance mismatch, loss in the cables and connectors, the frequency response of the source and receiver, and directivity and cross talk due to signal leakage. That is not to say that calibrations account for all of the possible sources of error. Calibrations do not account for system drift, repeatability in the switches and connectors, instrument noise, or errors in the calibration standards. The most popular way to describe a VNA’s systematic imperfections is the twelve-term error model, as shown in Figure 3.4 [39-40]. There are 6 error terms for the forward direction (Su and S21) and 6 error terms for the reverse direction (Sn and £ 22) • These 12 terms are defined in Table 3.1. Table 3.1. Error terms for the twelve-term model. Error Term eoo en ^10^01 £30 @22 £10^32 e ’33 e ’n e 23 e 32 £03 e ’22 e 23 e ’01 Description Directivity Port 1 Match Reflection Tracking Leakage Port 2 Match Transmission Tracking Directivity Port 1 Match Reflection Tracking Leakage Port 2 Match Transmission Tracking 28 with permission of the copyright owner. Further reproduction prohibited without permission. F orw ard "30 ------------------------ ► -----------------------e 1 0 e 32 IP p 1 * ^ b 2 ° l 4%P F 1 ^ ^00 *1 ki ^ ^ II S 21 i9 ^ S 22 r 5 U i i ^11 e 10 e 01 W 19 S n | 9 ^ i | a 2 D U T * 1 R everse * iW ^ ^22 b\ a A’, e 23 e 32 J 21 a a - e 23 e 01 U T 22 - b \ D U T 22 ® 33 A 1 — ^ ---- S’, 2 4 ^ « 2 03 Figure 3.4. Twelve-term error model for vector network analyzers. Solving the twelve-term error model signal flow graph gives 4 equations relating the corrected 5-parameters to the uncorrected, measured 5-parameters via the 12 error terms. Another common error model is the eight-term model, shown in Figure 3.5. In this model, all of the errors are accounted for by a pair of “error box” matrices on either side of the DUT, but requires imperfect switch conditions to be satisfied. The eight-term model and twelve-term models can be derived from one another [40], but that discussion is beyond the scope of this chapter. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a« o --------► -------- \ >----------------------------------- ► -----*10 V em eu i L *01 V pP ^ ^ P Sn i a% IP p 1 VI p *23 V e 33 i ^12 a2 *^21 ^22 P^ - DUT ^ r 14 i b2 e22 i { *32 O -------- M — — i i------------------------------------ M -------- Figure 3.5. Eight-term error model for vector network analyzers. 3.4. Network Analyzer Calibration Techniques There are a wide variety of calibration methods available to VNA users, most of which can be classified into one of two groups depending on the type of calibration standards used. Before we discuss the various calibration methods, it is worth mentioning that the basic procedure for calibrating a VNA is the same, regardless of what type of calibration is performed. First, known standards are measured. Next, the measured data is processed to determine the error coefficients. And finally, the measured data for a DUT is corrected using the error coefficients. The first category of VNA calibrations makes use of transmission lines as standards, and includes such calibration methods as thru-reflect-line (TRL) [1-2] and multiline TRL [3-4]. Multiline TRL is the most accurate means of VNA calibration 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and is especially useful for on-wafer environments, since the characteristic impedance can be calculated from dimensional measurements of the standards, which consist simply of a number of transmission lines of varying line lengths and a reflective termination. In an on-wafer environment, the disadvantages of this method are that it requires a lot of real estate on the wafer, due to the numerous long lines required for an accurate calibration, and the different lengths of line necessitate changing the separation between probes during the calibration process. In a coaxial environment, the disadvantages are that a large number of expensive airline standards and numerous interconnects are required. Consequently, a second category of VNA calibrations, which makes use of compact, lumped-element standards, is often preferred. The most common of these calibrations are the open-short-load-thru (OSLT) and the line-reflect-match (LRM) [5, 41] methods. The trade-off is that these methods tend to be less accurate, since it is more difficult to calculate the reflection coefficients of the standards from independent physical measurements. But, if the compact calibration kits are characterized using a benchmark calibration, such as multiline TRL, it is possible to perform an accurate lumped-element calibration. Figure 3.6 illustrates the calibration standards that are measured for the previously mentioned techniques of OSLT, LRM, TRL, and multiline TRL. ! 31 j i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LRM Standards OSLT Standards Open Thru Short Reflect Load Match Thru Multiline TRL Standards Thru TRL Standards |------- j Thru Reflect 1------ , Reflect Line 1 Line Line 2 Figure 3.6. calibrations. ,---- Standards measured for the OSLT, LRM, TRL, and multiline TRL 3.5. ANN Modeling for Improved VNA Calibrations Assuming a compact calibration kit is preferred, once the lumped-element standards are characterized, we must decide whether to develop a model for each of the standards or to directly use the measurement data obtained from the benchmark calibration. Recently, we have applied ANNs to improve the modeling of lumpedelement standards in both on-wafer and coaxial environments [42-45]. We have shown that ANN models offer a number of advantages over the use of calibratedmeasurement data files and equivalent circuit models, namely, the following: (1) they do not require the numerous details and parameters of physical models; (2) calibration 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. times can be reduced since only a few training points are required to accurately model the standards; (3) ANN model descriptions are much more compact than large measurement files; (4) ANN models, trained on only a few measurement points, can be much more accurate than direct calibrations when limited data are available; and (5) they give an optimized estimate in the presence of noise. In the following four chapters, we summarize our work in this area, describing how we have successfully modeled on-wafer OSLT standards, in one case assuming the standards can be reproduced from wafer to wafer with little variation (Chapter 4), and in another case where the loads exhibit significant difference among the wafers studied (Chapter 5). We also describe how we modeled load standards to improve both on-wafer and coaxial LRM calibrations (Chapters 6 and 7). 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A PTE R 4 ANN MODELING OF ON-WAFER OSLT STANDARDS The OSLT calibration [5] is one of the most widely used techniques for calibrating VNAs. It is mainly used with devices that contain coaxial or waveguide interfaces, but is also often applied to on-wafer environments such as microstrip and coplanar waveguide (CPW). The calibration procedure consists of a “thru” connection of the two VNA ports as well as the measurement (on both ports) of three one-port standards, typically a nominal open, a nominal short, and a nominally matched load. None of these standards needs to be ideal, but we must know their reflection coefficients. In practice, our definition of these reflection coefficient values is typically drawn from a model of the standard. Manufacturers of calibration kits typically provide a description of the standards based on equivalent circuit parameters, known as Calibration Kit Parameters [36, 46] or Calibration Component Coefficients [37]. These parameters assume single, real values for both load impedance and characteristic impedance and describe the open and short circuit terminations as frequency-polynomials of capacitance and inductance, respectively. With coaxial and waveguide standards, the equivalent circuit approximations have worked to the satisfaction of most users, but for on-wafer standards, a recent study [47] reported errors in scattering parameters of up to 0.5. Considering the maximum possible value for passive devices is a magnitude of 1, such errors are clearly unacceptable. DeGroot et al. [48] recently documented the OSLT models and developed a general description of transmission 34 permission of the copyright owner. Further reproduction prohibited without permission. lines to express offset reflection standards and finite-length thru standards that accounts for lossy environments with complex impedance. Implementing this more general description, however, still requires physical models or measurement data of each for the individual standards. Here, we apply ANNs to improve the modeling of on-wafer OSLT standards used for calibrating VNAs [42]. The ANNs are trained with measurement data obtained from a benchmark multiline TRL calibration. We assess the accuracy of an OSLT calibration using these ANN-modeled standards and find that it compares favorably (less than a 0.02 difference in magnitude) to the benchmark multiline TRL calibration over a 40 GHz bandwidth. We also quantify the training errors and training times as a function of both the number of training points and the number of neurons in the hidden layer. 4.1. Modeling the Standards In this study, the OSLT and multiline TRL standards and devices were constructed of CPW transmission lines fabricated with 1.5 jam thick gold conductors evaporated on 500 jam thick semi-insulating GaAs [49]; the gold center conductor was 73 |am wide and separated from the ground plane by 49 |am gaps. The five line standards included a 0.55-mm thru line and four additional lines that were 2.135, 3.2, 6.565, and 19.695 mm longer. The load standard was fabricated by terminating a 275jam section of the CPW with a single 73 jam by 73 jam nickel-chromium thin-film resistor. All of the standards were measured using on-wafer probes. The OSLT open circuit was defined by lifting the probe heads off the wafer, as recommended by 35 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. probe manufacturers. For each standard, we measured scattering parameters at 192 frequencies from 0.5 to 40 GHz. Once the OSLT standards had been characterized using a multiline TRL calibration, we determined how many neurons in the hidden layer were required to develop accurate ANN models. Our first experiment was to vary the number of neurons in the hidden layer for each of the standards. We started with 1 hidden neuron, noting the training error reported by the software after training was completed, and repeated this process while incrementing the number of hidden neurons until we reached a total of 10 neurons. We performed this experiment on each of the standards using all 192 measured frequencies as training data. Figure 4.1 illustrates the results for S\ i of the open, short, and load, and for S21 of the thru. Each of the standards had different errors, but no discemable improvements could be seen for more than 5 neurons. We also measured the training time for each standard while varying the number of hidden neurons, and found that the training time varied linearly with the number of hidden neurons. One hidden neuron required approximately 2 seconds of training time on the computer used, while 10 hidden neurons required about 20 seconds of training time. These training times undoubtedly vary depending on the speed of the computer, but the values give a relative idea of how much time is required per hidden neuron. After we decided that 5 hidden neurons were sufficient, we studied how many training points were required to accurately model each standard. We trained each standard using all 192 points, and then tried smaller subsets of the measurement points, namely 3, 5, 9, and 41 points. After the models for each of the standards were 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I trained for the five sets of data, we compared them to the measurement data. To our surprise, we found that we could achieve good accuracy with as few as 9 training points, and that as few as 5 training points were adequate for the open and short. Figures 4.2 and 4.3 show the magnitudes of the vector differences of Sn (|ASn|) between the measured data and the ANN models for various numbers of training points for the open and load standards, respectively. From these two plots, we see that the ANN model of the open standard agrees with measurement data to within 0.015 using as few as 5 training points, and that the ANN model of the load standard agrees with measurement data to within 0.04 for most frequencies using as few as 9 training points. 0 . 001 - —ir - Short S u —▼•••• Thru S2I Open S u •••■»■• Load S u 1 2 3 4 5 6 7 8 9 10 Neurons in Hidden Layer Figure 4.1. Training error versus the number of neurons in the hidden layer for various OSLT calibration standards. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 .16 -, 0 .1 4 - 0.12 0.10 - ------ ANN M odel - 192 pts. ------ANN M odel - 41 pts. ........ ANN M odel - 9 pts. ------ ANN M odel - 5 pts. — ANN M odel - 3 pts. - 0 .0 8 - 0 .0 6 - 0.04- 0 .0 2 - 0 .0 0 - 0 20 30 40 Frequency (GHz) Figure 4.2. Magnitude of the ANN-modeled reflection coefficient errors (|ASn|) for the open standard with varying numbers of training points. SO -, 40- — 0 10 20 ANN M odel - 192 pts. ANN M o d e l-4 1 pts. ANN M odel - 9 p ts . ANN M odel - 5 pts. ANN M odel - 3 p ts . 30 40 Frequency (GHz) Figure 4.3. Magnitude of the ANN-modeled reflection coefficient errors (|ASn|) for the load standard with varying numbers of training points. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Our observation that so few training points are sufficient to model our standards highlights an important advantage in using ANN models over calibrated measurement data files. We found that it is possible to cut down on calibration times by measuring only a few frequency points and developing an ANN model rather than measuring numerous points and carrying around large data files. The ANN model, trained on only a few measurement points, can be much more accurate than linearly interpolating, as is commonly done in practice. For example, if one were to measure the load standard at 5 points and perform linear interpolation between frequencies, as shown in Figure 4.4, the maximum error would be 0.045, as opposed to only 0.016 for the ANN model trained using the same 5 points. Next, we developed ANN models for each of the OSLT standards using 5 hidden neurons and all 192 measured points, since we already had the data on hand. Figures 4.5-4.7 show the magnitude and phase of Su using both measured and ANN model data for the open, short, and load standards, respectively. Figure 4.8 shows the magnitude and phase of S21 using the measured and ANN model data for the thru standard. Notice that the ANN models for each standard follow the trends of the measured data, but avoid the scatter of multiline TRL calibrated measurements. Whether or not this scatter is real, we see that ANNs follow general trends but omit the scatter, which is usually desirable in a model as long as the scatter is less than the repeatability of the measurements. > 39 with permission of the copyright owner. Further reproduction prohibited without permission. I 0.14-, lilik 0 . 12 - 0 . 10 0 .0 8 t/5 — TRL Calibration (192 points) - Reference — TRL Calibration (5 points) ANN Model (5 points) 0.060.0 4 0.0 2 - 0.00 0 10 20 Frequency (GHz) 30 40 10 20 Frequency (GHz) 30 40 -l.O-i -1 .5 - - 2. 0 - -2 .5 - -3.0 0 Figure 4.4. Comparison of magnitude and phase of the reflection coefficients [|Sn| and Arg(Sn)]for the load standard using an ANN model trained with 5 points, linear interpolation with TRL using the same 5 points, and TRL with 192 points as the reference. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.020 1.015- TRL Calibration ANN Model 1. 010 1.0051.0000.995 0.990 0 10 20 Frequency (GHz) 30 40 TRL Calibration ANN Model i •S e 0.8 0.6 0.4 0.2 0.0 0 10 20 Frequency (GHz) 30 40 Figure 4.5. Magnitude and phase of the reflection coefficients [|Sn| and Arg(Si for the open standard measured by multiline TRL and ANN modeling. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.020-, 1.015 TRL Calibration ANN Model 1. 010 1-0051.0000.995 0.990 0 10 20 Frequency (GHz) 30 40 30 40 3.16 3.14 3.12 ^1 3.10 X 3.08 czT « 3 06 % 3.04 TRL Calibration ANN Model 3.02 3.00 0 10 20 Frequency (GHz) Figure 4.6. Magnitude and phase of the reflection coefficients [|Sn| and Arg(Si for the short standard measured by multiline TRL and ANN modeling. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I 0.14-1 0 . 12 0 . 10 0.0 8 0.06TRL Calibration ANN Model 0.0 4 0.0 2 - 0.00 0 10 20 Frequency (GFIz) 30 40 30 40 0 .0 - i -0 .5 -1 .0 - TRL Calibration ANN Model -1 .5 CZ3 | -2 .0 -2 .5 -3.0 0 10 20 Frequency (GHz) Figure 4.7. Magnitude and phase of the reflection coefficients [|Sn| and Arg(Sn)] for the load standard measured by multiline TRL and ANN modeling. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. l.OOlO-i 1.0005- TRL Calibration ANN Model (N 1.0000 0.9995- 0.9990 0 10 20 Frequency (GHz) 30 40 30 40 800 -i 600400- - 200 - 200 - TRL Calibration ANN Model -600-800 0 10 20 Frequency (GHz) Figure 4.8. Magnitude and phase of the transmission coefficients [|S2i| and Arg(S 2i)] for the thru standard measured by multiline TRL and ANN modeling. 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Figures 4.5 and 4.6, the measured magnitudes of the reflection coefficient for the open and short standards are slightly greater than 1, which is not possible for passive devices. This discrepancy can be attributed to random errors in the TRL calibration, which are typically as high as 0.02 at 40 GHz. Our measurements never exceed 1 by more than this repeatability error. A similar argument can be made for the transmission coefficients of the thru standard. 4.2. Calibration Comparisons We performed two OSLT calibrations, one using the calibrated measurement data of the standards and the other using the ANN models of the standards. We calibrated a 19-mm CPW transmission line; using both OSLT calibrations, and compared the results to measurements calibrated directly using the benchmark multiline TRL calibration. Figure 4.9 compares the magnitudes and phases of the scattering parameter data [|Sn|, Arg(S'n), l&il, and ArgfS^i)] for all three calibrations. The agreement is good. To get a more quantitative idea of the differences, we plotted the maximum vector differences of the scattering parameters (|ASij|) for the 19-mm line between the two OSLT calibrations and the multiline TRL calibration. Figure 4.10 illustrates the differences. Not surprisingly, the OSLT calibration, using the calibrated measurement data, compares better to the multiline TRL calibration, since they both make use of the same calibration data. However, the OSLT using the ANN models still compares well with less than a 0.02 difference in magnitude at all frequencies. The difference here does not necessarily mean that the OSLT, using ANN models, is in error. The 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i differences could be due to the presence of noise in the TRL calibration that the ANN models avoided. Regardless of the source of error, a 0.02 difference between two onwafer calibrations spanning 40 GHz is impressive, considering that the repeatability between two multiline TRL calibrations is usually on the order of 0.015. 4 -i 0.95 n OSLT (TRL) OSLT (ANN) TRL 0 .9 0 - Kfi U 20- 0 .8 0 0.75 0 10 20 30 0 40 10 20 30 40 30 40 Frequency (GHz) Frequency (GHz) 1001 6 -1 80”0 60- 200 10 20 30 0 40 10 20 Frequency (GHz) Frequency (GHz) Figure 4.9. Magnitude and phase of the scattering parameters of a calibrated 19-mm CPW transmission line. 4.3 Discussion We have successfully applied ANNs to model on-wafer OSLT standards, and shown that such a calibration compares favorably (less than a 0.02 difference in magnitude) to the benchmark multiline TRL calibration. In modeling these standards, we quantified the training errors and training times as a function of both the number of training points and the number of neurons in the hidden layer. We found that 5 46 with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 -1 OSLT (TRL) vs. TRL OSLT (ANN) vs. TRL 15- o » 10- 0 10 20 30 40 Frequency (GHz) Figure 4.10. Magnitude of the scattering parameter differences (|ASij|) of a calibrated 19-mm CPW transmission line. neurons in the hidden layer of an MLP3 architecture and that fewer than 10 training points were sufficient to accurately model our standards. In practice, ANN-modeled calibration standards can be easily implemented using existing or custom software packages. In our case, we utilized MultiCal [4], a free program developed by the National Institute of Standards and Technology, to perform our benchmark multiline TRL calibration. The internal software on any commercial network analyzer can also be used if the user has confidence in another calibration method such as single-line TRL or LRM (line-reflect-match). Then, once the OSLT standards are measured, one of a number of ANN programs may be used to model the standards. We used software developed by Zhang et al [50] to construct our 47 with permission of the copyright owner. Further reproduction prohibited without permission. ANN models. Finally, a program that can perform OSLT calibrations using exported ANN models is required. We wrote custom software to perform this task, using the equations found in references [5] and [48] to perform the OSLT calibrations. We have shown that ANN models offer a number of advantages over using calibrated measurement data files or equivalent circuit models, namely: (1) They do not depend on the numerous details and parameters of physical models. (2) Calibration times can be reduced since only a few training points are required to accurately model the standards. (3) ANN model descriptions are much more compact than large measurement data files. (4) ANN models, trained on only a few measurement points, can be much more accurate than direct calibrations, when limited calibration data is available. (5) They give an optimized estimate in the presence of noise. 48 I\ \ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5 MODELING LOAD VARIATIONS WITH ANNS TO IMPROVE ON-WAFER OSLT CALIBRATIONS The assumption made in the previous chapter was that the standards can be reproduced from wafer to wafer with little variation. Kirby et al. [51] studied variations in OSLT standards from wafer to wafer on a CPW calibration set designed for GaAs substrates, and found that open, short, and thru standards can be reproduced with minimal variance, but that load standards exhibit a significance difference among the wafers they studied. Furthermore, they discovered that RF variations in the load terminations correlate directly to their measured DC resistances. Here, we demonstrate that on-wafer OSLT calibrations of VNAs can be further improved by applying ANNs to model the correlation between DC resistance and RF variations in load terminations [43]. The ANNs are trained with measurement data obtained from a benchmark multiline TRL calibration. The open, short, and thru standards do not vary significantly from wafer to wafer, so we also model these standards using ANNs trained with calibrated measurement data chosen from an arbitrary wafer. We assess the accuracy of five OSLT calibrations with varying load terminations using the ANN-modeled standards, and find that they compare favorably (a difference of less than 0.04 in magnitude at most frequencies) to the benchmark multiline TRL calibration over a 66 GHz frequency range. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1. Modeling the Standards In this study, the OSLT and multiline TRL standards and devices were constructed of CPW transmission lines fabricated from 4.5 (im plated gold on a 625 urn thick GaAs. The load terminations were composed of TiWN (titanium tungsten nitride) thin film resistive material [51]. The four line standards included a thru line and three additional lines that were 0.9552, 1.239, and 1.764 mm longer. All of the standards were measured using on-wafer probes. For each standard, we measured scattering parameters at 165 frequencies from 1 to 67 GHz. Since the open, short, and thru standards did not vary significantly from wafer to wafer we modeled these standards with ANNs using calibrated measurement data chosen from an arbitrary wafer. The ANN architecture for the open, short, and thru standards consisted of one input (frequency) and two outputs (the real and imaginary components) for each measured scattering parameter. Since we measured reflection coefficients for the two terminations at both ports and all four scattering parameters of the thru connection, we ended up with eight ANN models, excluding the load. From our previous study in [42], we determined that 5 neurons were sufficient for the hidden layer. We trained each model of the standards using all 165 frequencies since we already had the data on hand. The ANN architecture for the load standards consisted of two inputs (frequency and DC resistance) and two outputs (the real and imaginary components) for the impedance parameters at each port. We were unable to generate one model that included both ports due to a systematic difference between the load measurements at port 1 and port 2, so we settled on separate models for each port. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ten neurons were chosen for the hidden layers since the ANN models for the loads included an additional input compared to the other standards. The measured DC resistances for the loads are listed in Table 5.1. For each port, we trained the models using 3 of the 5 loads. We chose loads 1, 4, and 5 since load 1 had the lowest DC resistance, load 5 had the highest, and load 4 had an intermediate value. It is important to train ANNs at the expected boundary values of the input parameter space in order to ensure good performance of the model [13]. By purposely not training the ANN with loads 2 and 3, we could test how effective the model behaved at other DC resistances. Figure 5.1 shows the real and imaginary components of port 1 impedance, Z\\, of both measured and ANN-modeled data for the 5 load standards. Table 5.1. Measured DC resistances of the five load terminations. Load 1 2 3 4 5 DC Resistance (Q) Port 1 44.73 45.85 45.20 45.38 46.45 DC Resistance (Q) Port 2 45.01 46.13 45.27 45.64 46.71 a 51 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 5 2 -i Load 1 Load 2 Load 3 Load 4 Load 5 50- 13. ? 48' Pi 46' 44- _l-------------- !-------------- 1------------ 1------------ 1------------ 1------------ 1 10 20 30 40 50 60 70 50 60 70 Frequency (GHz) 8 -| 6- 53 4 - N 2- 0-1 10 20 30 40 Frequency (GHz) Figure 5.1. Real and imaginary components of Z\\ for the load standards measured by multiline TRL and modeled by an ANN. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2. Advantages of ANN Models One of the advantages of using ANN models as opposed to calibrated measurement files is the compact description possible with an ANN. For example, the ANN model we developed for the load at port 1 required 62 real-valued parameters to generate complex S-parameters as a function of frequency and DC resistance. In contrast, a single measurement file contains 495 real-valued numbers (165 frequency points plus the real and imaginary components at each point). If a measurement database of just 5 loads is utilized, the combined files would contain 2475 real-valued numbers. We also explored the use of ANN models for extrapolation outside the bounds of the training data. (Generally, it is believed that ANN models are good at interpolating but not extrapolating.) We did this by training an ANN model at port 1 using 3 of the 5 loads once again, but this time we chose loads 2, 3, and 4. By purposely not training the ANN model with loads 1 and 5, we could test how effective the model behaved at extrapolating. Surprisingly, both the interpolating and extrapolating ANN models exhibited almost identical deviations between measured and predicted values. This bodes well for the application of ANN models to our loads, since it is conceivable that other wafers may possess DC resistances slightly outside the range of the 5 loads we used to train the models. 5.3. Calibration Comparisons We performed 5 OSLT calibrations, each one making use of the same ANNmodeled open, short, and thru standards as well as the ANN-modeled loads with their 53 1 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. respective DC resistances. We calibrated a 1.764-mm long CPW transmission line using each of the OSLT calibrations and compared the results to measurements calibrated directly using the benchmark multiline TRL calibration. Figures 5.2 and 5.3 compare the magnitudes and phases of S 21 and Sn for all 6 calibrations. The agreement is remarkably good except at a few points where the multiline TRL calibration shows a lot of scattering. To obtain a more quantitative idea of the differences, we plotted the maximum magnitude of the vector differences of the scattering parameters [maxd^l)] for the 1.764-mm line for each of the OSLT calibrations and the multiline TRL calibration. Figure 5.4 illustrates the differences. All of the OSLT calibrations using ANNmodeled standards compare favorably to the benchmark multiline TRL calibration, with a difference of less than 0.04 in magnitude at most of the frequencies over the 66 GHz frequency range. Not surprisingly, the OSLT calibrations for loads 2 and 3 show slightly higher differences since they were not used to train the ANN model. The differences between the 5 OSLT calibrations and the TRL calibration do not necessarily mean the OSLT calibrations are in error. The differences are likely due to the presence of noise in the TRL calibration that the ANN models avoided. Regardless of the source of error, a 0.04 difference between two on-wafer calibrations spanning 66 GHz is impressive, considering that the repeatability between two multiline TRL calibrations is usually on the same order. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 1 .0 4 - 1 1. 0 2 - 1. 0 0 - 0.98- Xfl 0.96- Multiline TRL OSLT - LI ANN OSLT - L2 A N N OSLT - L3 A N N OSLT - L4 A N N OSLT - L5 ANN 0.94- 0.92- 0.90- “I 1 ------------ 1------------ 1------------ 1---10 20 30 40 50 60 70 Frequency (GHz) 2- m S5 -1 -j -2 -3 - 1 ------------ 1------------ 1------------ 1------------ 1------------ 1------------ 1 10 20 30 40 50 60 70 Frequency (GHz) Figure 5.2. Magnitude and phase of S21 for a calibrated 1.764-mm CPW transmission line. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 .1 2 —i 0. 1 0 - 0.08 Multiline TRL OSLT - LI A N N OSLT - L2 ANN OSLT - L3 A NN OSLT - L4 ANN OSLT - L5 ANN 0 .0 4 - 0.02 0.00 10 20 30 40 50 60 70 50 60 70 Frequency (GHz) 2 -| 1 '3 2 0- -2 - 10 20 30 40 Frequency (GHz) Figure 5.3. Magnitude and phase of Sn for a calibrated 1.764-mm CPW transmission line. 56 | Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 -| 60O SLT - L I ■O SLT - L 2 O S L T -L 3 O SLT - L4 O S L T -L 5 ANN ANN ANN ANN ANN 40- 20- ~T~ 10 n --------------------------r 20 30 40 ~ r 50 60 70 Frequency (GHz) Figure 5.4. Magnitudes of the scattering parameter differences of a calibrated 1.764mm CPW transmission line. 5.4 Discussion We have successfully applied ANNs to model the correlation between DC resistance and RF variations in load terminations and the RF performance of open, short, and thru standards used for on-wafer OSLT calibrations of vector network analyzers. We have shown that these modeled standards compare favorably (a difference of less than 0.04 in magnitude at most frequencies) to the benchmark multiline TRL calibration over a 66 GHz bandwidth. In addition, we have shown that ANN models are able to accurately model loads with measured DC resistances slightly outside their training range. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. j CHAPTER 6 | ANN MODELING FOR ON-WAFER LRM CALIBRATIONS ! ! ! | Another popular compact calibration method is LRM (line-reflect-match), | which requires only a short transmission-line connection, a load, and a reflection [41, | 52]. Here, the reference impedance is set to that of the standard load. As shown . above, the impedance of many on-wafer loads, however, is non-ideal, which can lead to significant error in LRM calibrations. Here, we model a load using an ANN to improve an on-wafer LRM ] | calibration [44], The ANN is trained with measurement data obtained from a single- j line TRL calibration. Using a single-line TRL calibration enables us to build an j j effective model of the load using minimal real estate on the wafer. This methodology | results in an LRM calibration with less overall error than by simply applying the j j single-line TRL calibration. The accuracy of the LRM calibration using the ANN- I modeled load compares favorably to a benchmark multiline TRL calibration with an average worst-case scattering parameter error bound of 0.017 over a 40-GHz | frequency range. 6.1. Load Modeling In this study, the LRM and multiline TRL standards and devices were | constructed of coplanar waveguide (CPW) transmission lines fabricated from 1.5 pm gold conductors on a layer of 500 pm thick semi-insulating GaAs [49]; the gold 1 I I1 3 1;l | center conductor was 73 pm wide, and was separated from the ground plane by 49 58 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lim gaps. The five line standards included a 0.55-mm thru line and four additional lines that were 2.135, 3.2, 6.565, and 19.695 mm longer. All of the standards were measured using on-wafer probes. For each standard, we measured scattering parameters at 192 frequencies from 0.5 to 40 GHz. In Figure 6.1, we plot measurements of magnitude and phase of the load’s reflection coefficients. They were determined by a TRL calibration using only the thru connection and the 2.135-mm line, and applying an impedance transformation to the calibration, which yielded the measured 5-parameters referenced to 50 £2 [53]. Use of only a single line explains the inaccuracy at multiples of 26.65 GHz, where the difference in line lengths corresponds to a multiple of half a wavelength. The figure shows that the load deviates significantly from 50 Q. 0.20 i - -40 --6 0 Measured w/ Single-Line TRL ANN Model 0.15 - — 80 --1 0 0 (JQ - -140 0 .0 5 - - -160 0.00 -180 0 10 20 30 40 Frequency (GHz) Figure 6.1. Magnitude and phase of measured and modeled scattering parameters of the on-wafer load for the LRM calibration. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We then used an ANN to model the load. The ANN for the load standard consisted of one input (frequency) and two outputs (the real and imaginary components) for the S-parameters. Based on previous experience [42], five neurons were chosen for the hidden layer. Figure 6.1 shows the magnitude and phase of the Sparameters of both measured and ANN-modeled data for the load standard. Notice that the ANN model for the load standards follows the trends of the measured data, but avoids both the spike near 26.65 GHz as well as scatter of the TRL-calibrated measurements. Whether or not this scatter is real, we see that ANNs follow general trends but omit noise, which is usually desirable in a model. 6.2. Calibration Comparisons First, two consecutive multiline TRL calibrations, using all five lines, were compared to assess the limitations on calibration repeatability caused by contact error and instrument drift. The technique of [54] was used to determine an upper bound on this repeatability error. Briefly, the comparison determines the upper bound for |S'ySij\ for measurements on any passive device, where S'y are the scattering parameters of a device measured with respect to the first calibration and Sy are the scattering parameters measured with respect to the second calibration. The bound is obtained from a linearization, which assumes the two calibrations are similar to the first order. The result, plotted as a solid curve in Figure 6.2, roughly indicates the minimum deviation between any pair of calibrations. The average of the worst-case error bounds for repeatability was 0.013. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I So S-H t/5 1 ffl ti 0.01 Multiline TRL Repeatability Single-line TRL Simple LRM LRM with Measured Load LRM w ih ANN-M odeled Load £ 0.001 0 20 10 30 40 Frequency (GHz) Figure 6.2. Worst-case error bounds between measurements of passive devices from on-wafer LRM and TRL calibrations and the multiline TRL calibrations. We also compared the single-line TRL calibration, which was used to develop the ANN model, to the multiline TRL calibration. The result is plotted in Figure 6.2. Since we used only the 2.135-mm line standard, our calibration accuracy is poor near multiples of 26.65 GHz, where the difference in line lengths corresponds to a multiple of half a wavelength [3]. Otherwise the single-line TRL calibration is nearly as accurate as the multiline TRL calibration at most frequencies. We assessed the accuracy of the LRM calibrations by comparing them to a 50-0 multiline TRL calibration. First, we compared a simple LRM calibration, where the load is assumed to be ideal, to the multiline TRL calibration. Figure 6.2 illustrates a large difference since the reference impedance of the LRM calibration, which is equal to the impedance of the non-ideal load, deviates significantly from 50 Q. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In order to assess the accuracy of the best LRM calibration, we compared the multiline TRL calibration to the LRM with a fully characterized load, which involved calibrating the load with the benchmark multiline TRL calibration and using the calibrated measurement data file to define the load. This comparison is once again shown in Figure 6.2. The average of the worst-case error bounds for this calibration was 0.011. Figure 6.2 also shows the worst-case error bounds for the LRM calibration based on the ANN-modeled load. Here, the average worst-case error bound was 0.017. 6.3. Discussion The use of ANNs to model on-wafer LRM load standards compares favorably to a benchmark multiline TRL calibration, with an average worst-case scatteringparameter error bound of 0.017 over a 40-GHz bandwidth. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7 ANN MODELING FOR COAXIAL LRM CALIBRATIONS Multiline TRL can also be applied in coaxial environments. However, a set of coaxial lines, some relatively long, is required to obtain a wide-band measurement. Coaxial airlines also require considerable care to ensure a good connection without damaging the standard. Furthermore, a set of lines can be costly. In contrast, the LRM calibration, which requires only a thru connection, a coaxial load, and a reflection, overcomes these limitations. Here, the reference impedance is set to that of a standard load. The impedance of many coaxial loads, however, is non-ideal, which can lead to significant error in LRM calibrations. There are two approaches we can take to characterize an imperfect load. One is to characterize it in terms of its reflection coefficient, which requires access to a full multiline TRL calibration set. Alternatively, we can postulate a physical model of the load and apply a minimal calibration sufficient to determine the model coefficient. Jargon et al [55] applied this notion to coaxial lines, employing the measurement of the load after a single-line TRL calibration to fit the parameters of an equivalent circuit model. This provided a means for obtaining an accurate wide-band LRM calibration with a compact coaxial standard set consisting of a reflection, a match standard, and a line of short length. Although the equivalent circuit model used in [55] was effective, there were a number of difficulties with it. First, the model was specifically tailored for the load used, so considerable time was required to develop an adequate model. The load was 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. approximated by the impedance R+qaf+jcoL, preceded by a lossless line of characteristic impedance Z q, length /, and effective permittivity er,eff. The value of R was determined by measuring the dc resistance of the load. Zq was chosen to be 50 £2, and er,eff was assumed to be 1. Then, L, q, and I were determined by optimizing the model. Another disadvantage of this model is that it is not guaranteed to work for other loads. A completely different impedance might be required to model other loads correctly. In an attempt to improve the accuracy of the LRM calibration, we use a single-line TRL calibration to train an ANN model of the load. The following sections describe our implementation of ANNs and assess the accuracy of the LRM calibration using the ANN-modeled load, comparing it to the equivalent circuit model and measured data [45]. 7.1. Load Modeling We used a set of commercially available GPC-7 artifacts for these experiments. The artifacts consisted of 2.25-cm, 10-cm, and 30-cm airlines, a short circuit, and a nominally 50 Q coaxial load. We assumed that our sexless GPC-7 connectors mated perfectly with our line, allowing a direct connection between the two ports to serve as a thru line. In Figure 7.1 we plot measurements of the real and imaginary parts of the load impedance. The impedance was determined by a TRL calibration using only the thru connection and the 2.25-cm line, and applying an impedance transformation to the calibration, which yielded the measured 5-parameters referenced to 50 Q. The 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. characteristic impedance of the line was determined from its capacitance and propagation constants, allowing the reference impedance of the TRL calibration to be accurately set to 50 Q, [53], Use of only a single line explains the inaccuracy at multiples of 6.67 GHz, where the difference in line lengths corresponds to a multiple of half a wavelength. The figures show that the load deviates significantly from 50 Q. To account for the non-idealities, we developed an ANN model for the load using 15 hidden neurons and 181 measured points. We trained our ANN model with data taken from the single-line TRL calibration to illustrate that loads, or for that matter almost any artifact, can be modeled using only a simple set of calibration standards rather a large set of expensive airlines. Figure 7.1 shows that the real and imaginary parts of the ANN-modeled load correspond closely to the measured values, and that the model did avoid the spikes present at multiples of 6.67 GHz. Also plotted in Figure 7.1 are the real and imaginary parts of the load as determined by the equivalent circuit model of [55]. Not only does the ANN model match the measured values closer than the circuit model, but it was also developed in a small fraction of the time needed to develop and optimize the circuit model. 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 -i Measured Single-line TRL ANN Model Circuit Model 60- 55O N <L> 45- 0 5 10 Frequency (GHz) 15 20 10 Frequency (GHz) 15 20 15-i 10 - N 5 -5- -10 0 5 Figure 7.1. Real and imaginary components of the measured and modeled impedance of the coaxial load for the LRM calibration. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.2. Calibration Comparisons First, two consecutive multiline TRL calibrations, using all three airlines, were compared to assess the limitations on calibration repeatability caused by contact error and instrument drift. The technique of [54] was used to determine an upper bound on this repeatability error. Briefly, the comparison determines the upper bound for l^y - Sy\ for measurements on any passive device, where S'y are the scattering parameters of a device measured with respect to the first calibration and Sy are the scattering parameters measured with respect to the second calibration. The bound is obtained from a linearization, which assumes the two calibrations are similar to the first order. The result, plotted as a solid curve in Figure 7.2, roughly indicates the minimum deviation between any pair of calibrations. The average worst-case error bound for repeatability was 0.013. We then compared the single-line TRL calibration, which was used to develop both the ANN and equivalent-circuit models, to the mutliline TRL calibration. The result is plotted in Figure 7.2. Since we only used the 2.25-cm line standard, our calibration accuracy is poor near multiples of 6.67 GHz, where the difference in line lengths corresponds to a multiple of half a wavelength [3]. Otherwise, the single-line TRL calibration is nearly as accurate as the multiline TRL calibration at most frequencies. We assessed the accuracy of the LRM calibrations by comparing them to the 50 Q multiline TRL calibration. Figure 7.2 shows the maximum possible difference |S'y - S'y] where S'y corresponds to the simple LRM calibration (load assumed to be ideal), and Sy corresponds to the multiline TRL calibration. Here, the difference is 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. large since the reference impedance of the LRM calibration, which is equal to the impedance of the non-ideal load, deviates significantly from 50 Q. To see how accurate the best LRM calibration was, we compared the multiline I j TRL calibration to the LRM with a fully characterized load, which involved j calibrating the load with the benchmark multiline TRL calibration and using the j calibrated measurement data file to define the load. This comparison is once again j shown in Figure 7.2. The average of the worst-case error bounds for this calibration | \I j was 0.016. Figure 7.2 also shows the worst-case error bounds for the LRM calibrations based on both the ANN and equivalent circuit models. The average of the worst-case error bounds for the ANN-modeled LRM calibration was 0.024, while the average for the circuit-modeled LRM was 0.034. 7.3. Discussion j 1 The use of ANNs to model coaxial LRM load standards compares favorably ! | to a benchmark multiline TRL calibration, with an average worst-case scattering- | 3 parameter error bound of 0.024. j |j In the case of LRM calibrations,7 we have shown that ANN models offer ] j advantages over equivalent circuit models since they do not require detailed physical models. Our ANN model required far less development time than our equivalent | circuit model and still managed to achieve higher accuracy. ANN model descriptions are also preferred over calibrated measurement data files since they are much more compact in size. Additionally, loads, or for that matter almost any artifact, can be ! 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. modeled using only a simple set of calibration standards rather than being fully characterized with a large set of expensive airlines. St Multiline T R L R epeatability Single-line T R L Sim ple LRM LR M with M easured Load LR M with C ircuit-M odeled Load LR M with A N N -M odeled Load 'M 1 CO 0 . 01 - 0.001 0 2 4 6 10 8 12 14 16 18 Frequency (G H z) Figure 7.2. Worst-case error bounds between measurements of passive devices from coaxial LRM and TRL calibrations and the multiline TRL calibration. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I CHAPTER 8 NONLINEAR VECTOR NETW ORK ANALYSIS In the second portion of this thesis, we focus on frequency-domain models and figures of merit for nonlinear circuits based on large-signal measurements. 8.1. LINEAR VERSUS NONLINEAR BEHAVIOR As mentioned in Chapter 3, vector network analyzers (VNAs) are one of the most versatile instruments in the RF and microwave industry. They are used to measure complex scattering parameters (^-parameters) of devices or circuits. Engineers use them to verify their designs, confirm proper performance, and diagnose failures. A VNA works by exciting a linear device under test (DUT) with a series of sine wave signals, one frequency at a time, and detecting the response of the DUT at its signal ports. Since the DUT is linear, the input and output signals are at the same frequency as the source, and can be described by complex numbers that account for the signals’ amplitudes and phases. The input-output relationships are described by ratios o f complex numbers, known as ^-parameters. For a two-port network, four Sparameters completely describe the behavior of a linear DUT when excited by a sine wave at a particular frequency. Although the measurement of S-parameters by VNAs is invaluable to the microwave designer for modeling and measuring linear circuits, this is oftentimes inadequate for nonlinear circuits operating at large-signal conditions since nonlinearities transfer energy from the stimulus frequency to products at new 70 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. frequencies, as shown in Figure 8.1. Thus, conventional linear network analysis, which relies on the assumption of superposition, must be replaced by a more general type of analysis, which we refer to as nonlinear network analysis. Linear Device Time Time Incident Transmitted Frequency I Frequency Nonlinear Device Time Time Incident Transmitted | Frequency Frequency Figure 8.1. Nonlinearities transfer energy from the stimulus frequency to products at new frequencies. 8.2. Nonlinear Vector Network Analysis Nonlinear network analysis involves characterizing a nonlinear device under realistic, large-signal operating conditions. To do this, complex traveling waves (rather than ratios) are measured at the ports of a DUT not only at the stimulus frequency (or frequencies), but also at other frequencies where energy may be created. Assuming the input signals are sine-waves and the DUT exhibits neither sub harmonic nor chaotic behavior, the input and output signals will be combinations of sine-wave signals, due to the nonlinearity of the DUT in conjunction with mismatches between the system and the DUT. If a single excitation frequency is present, new 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. frequency components will appear at harmonics of the excitation frequency, and if multiple excitation frequencies are present, new frequency components will appear at the intermodulation products as well as at harmonics of each of the excitation frequencies. Figure 8.2 illustrates this concept for a set of harmonically related signals present at the port of a DUT. In practice, there will be a limited number of significant harmonics and intermodulation products. The set of frequencies at which energy is present and must be measured is known as a frequency grid. ... con ©1, ©2, ... ©n L ^ l a, b. Nonlinear a, C0„0)2, ... con ©1, 0)2, ... ©n L _ j_ Figure 8.2. Complex traveling waves at the ports of a nonlinear device when a set of harmonically related signals is present. To get a feel for the added complexity of nonlinear network analysis, as compared to linear network analysis, we examine the flow diagram of a two-port network excited by a source at a single frequency at port 1 and terminated with a load at port 2. For a linear device, shown in Fig. 8.3(a), four scattering parameters relate the incoming and outgoing waves. The outgoing wave b\ can easily be solved in terms of a\ as 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Tin is referred to as the input reflection coefficient of the network. For a nonlinear device, the flow diagram becomes much more complicated. Figure 8.3(b) shows a nonlinear device with only the first three harmonics considered. In this case, we still have a single-source excitation at port 1 (a \\), but now we must account for the fact that power may be transferred to higher harmonic frequencies. Note that the wave variables, a and b, in the nonlinear case contain two positive integer subscripts, the first refers to the port number and the second to the spectral component number. Here, we have three reflection coefficients for the terminating load and 16 nonlinear large-signal scattering parameters (&#*/) that relate the incoming and outgoing waves, assuming a single-source excitation and no downconversion. (Nonlinear large-signal scattering parameters will be discussed in more detail in Chapter 10.) The outgoing wave b\\ can be solved in terms of a\\ in a similar fashion to the linear case, giving j u - » m i + * li l' * i!‘i r ‘ - a ll 1 *2211 1 (8-2) Solving for bn and bu in terms of an , however, gives more involved expressions: ^12 _ g, a \\ + ^1221^2111^1 + ^1222^2121^2 + ^1222^2221^211 l^ l^ 1—^2211^1 1—^2222-^2 0 —^2211^1 ) 0 —&2222^2) /g 3 -v and _^13__^ a \\ ! ^1231^2111^1 ! ^1233^2131^3 | I - ^221 l^ i I - 8*2233^3 ^1233^2231^2111^1^3 0 _ ^2 2 1 1 ^ 1 ) 0 ~ ^ 2 2 3 3 ^ 3 ) 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (8 4) In this simplified case, we have not considered harmonics above 3, but if we did, eqs. (8.4) and (8.5) would become even more involved at frequencies that are multiples of more than one sub-harmonic. For example, the fourth harmonic is a multiple of both the first and second harmonics. (a) «i 'l ' 21*1 *22kl ^ 11*1 12ki Figure 8.3. Flow diagram for a two-port network with a single-frequency excitation at port 1 and a terminating load at port 2. (a) Linear case, (b) Nonlinear case considering three harmonics. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I 8.3. Nonlinear Vector Network Analyzers A class of instruments known nonlinear vector network analyzers (NVNA) are capable of providing accurate waveform vectors by acquiring and correcting the magnitude and phase relationships between the fundamental and harmonic components in the periodic signals [6-8]. An NVNA excites a nonlinear DUT with one or more sine wave signals and detects the response of the DUT at its signal ports. Assuming the DUT does not exhibit any sub-harmonic or chaotic behavior, the input and output signals will be combinations of sine wave signals due to the nonlinearity of the DUT in conjunction with mismatches between the system and the DUT. With these facts in mind, the major difference between a linear VNA and an NVNA is that a VNA measures ratios between input and output waves one frequency at a time while an NVNA measures the actual input and output waves simultaneously over a broad band of frequencies. 8.3.1. Architecture Figure 8.4 illustrates a simplified block diagram of an NVNA with the RF source connected to port 1 and bias tees located at both ports. The incident and reflected waves at both ports of the DUT are measured through directional couplers. The measured RF spectrum are down-converted to an IF spectrum using harmonic mixing, which is based on the same principle as in sampling oscilloscopes. The compressed signals are digitized using precision analog-to-digital (ADC) cards. The RF source, the four down-converters, and the four ADC cards are timed by a common reference clock in order to obtain a fully synchronous, phase-coherent measurement system. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Synchronous Clock ADC ADC ADC ADC Broadband Down Converters M RF Source BiasT DUT BiasT Figure 8.4. Simplified block diagram of a nonlinear vector network analyzer. Despite the disadvantages of using some non-commercially available parts, a high price tag, and a complex infrastructure that requires powerful software, the NVNA offers a number of significant advantages over other types of measurement systems. First, the use of four couplers and four-channel data acquisition allows the entire spectrum of incident and reflected waves to be measured simultaneously at both ports of a DUT, thus circumventing any phase synchronization problems. Second, because of four couplers, all mismatches can be taken into account during calibration without making any assumptions regarding perfect terminating loads. 76 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Third, data acquisition is fast. A power-bias sweep requires only a few seconds. And fourth, the system has good dynamic range, typically around 60 dB. 8.3.2. Calibration The calibration procedure for an NVNA is based on the assumption that the four measured wave quantities (aM\k, aMik, bM\k, and bM2k) are linearly related to the four physical voltage waves at the ports of the device (au, aik, b\k, and bjk) [10], Here, a represents the incident wave, b represents the scattered wave, the first subscript refers to the port number (1 or 2), the second subscript k refers to the frequency index, and the superscript M refers to the measured wave quantities. The relationship between the physical waves and the digitized quantities is characterized by complex square matrices (one matrix for each frequency) with sixteen elements, as given by ~<*\k b\k 0 0 M a \k Pk 0 0 0 p.2k \k a„M 2k 0 Yik P ->2k . 1 P \k a2k Y\k 0 p2k _ 0 = \Ck\emCk) M (8.5) Generally, the calibration procedure solves for the sixteen elements for all of the matrices, but oftentimes as in eq. (8.5), the cross-talk terms are assumed to be negligible and are set to zero, leaving only eight terms to be solved. The calibration of an NVNA consists of three steps: a relative calibration that is identical to that used in a linear VNA, an amplitude calibration that makes use of a power meter, and a phase distortion calibration that makes use of a reference generator. The relative calibration, such as TRL, OSLT, or LRM, accounts for 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. directivity and cross-talk errors due to signal leakage, impedance mismatches and losses in the cables and connectors. This calibration solves for the following seven terms: pa, yik, 5u, 0C2k, P2k, Jik, and 82*. There is also a complex number C* common to the entire matrix that must be solved for in order to achieve an absolute calibration. The amplitude calibration solves for the magnitude of C*, and the phase calibration solves for the phase of Q , The phase calibration is achieved by using a reference generator that generates a fundamental signal and harmonics with a known phase relationship, the main components of which are a power amplifier and a step recovery diode. The reference generator is characterized by a sampling oscilloscope, which in turn is characterized by a nose-to-nose calibration [56, 57], The main principle behind the nose-to-nose calibration is that the “impulse response” of each sampler in an oscilloscope can also be used as a pulse generator. This “kick-out” pulse occurs when an offset voltage is applied to the hold capacitors of the sampler. It turns out that the “kick-out” pulse is proportional to the “impulse response.” The nose-to-nose calibration refers to the process of connecting two oscilloscopes together, and using one scope to measure the other’s “kick-out” pulse. If the scopes are identical, the resulting waveform is the convolution of the scope’s impulse response with the kick-out pulse. The impulse response can then be calculated. Since no two scopes are exactly identical, three oscilloscopes are needed to perform an accurate nose-to-nose calibration. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. j 8.4. Developing Frequency-Domain Models for Nonlinear Circuits Based on Large-Signal Measurements Even though 5-parameters cannot adequately represent nonlinear circuits, some type of parameters relating incident and reflected signals are beneficial so that the designer can “see” application-specific engineering figures of merit that are similar to what he or she is accustomed to. In this portion of the thesis, we propose definitions of such ratios that we refer to as nonlinear large-signal scattering (§>) parameters. After describing a method for preserving time-invariant phase relationships when ratios are taken between any two harmonically related, complex signals, we present a general form of time-invariant nonlinear large-signal ¶meters. Then we introduce nonlinear large-signal impedance (Z) and admittance (H) parameters, and present equations for relating the different representations. In addition to nonlinear large-signal ^-parameters, we introduce another figure of merit for nonlinear circuits by expanding the definitions of power gain, transducer gain, and available gain by taking harmonic content into account. We show that under special conditions, these expanded definitions of gain can be expressed in terms of two-port, nonlinear large-signal ^-parameters. For existing nonlinear models, we can readily generate nonlinear large-signal ^-parameters by performing a harmonic balance simulation. For devices, with no model available, we can extract these parameters from artificial neural network (ANN) models that are trained with multiple frequency-domain measurements made on a nonlinear DUT with an NVNA. 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To illustrate applications and generation of nonlinear large-signal 0 parameters, we present three examples. First, we show how nonlinear large-signal 0 parameters and the expanded definitions of gain can be used to discover valuable information regarding the behavior of a nonlinear model. Specifically, we examine a lumped-element model of a pseudomorphic high electron mobility transistor (pHEMT) device operating in a two-port, common-source configuration. Second, we illustrate how nonlinear large-signal ^-parameters can be used to as a tool in the design process of a simple nonlinear circuit, specifically a single-diode 1-2 GHz frequency-doubler circuit. And finally, we describe a method for generating nonlinear large-signal 0-parameters based upon ANN models trained on frequency-domain data measured using a nonlinear vector network analyzer (NVNA). We compare a diode circuit model, generated using this method, to a harmonic balance simulation of a commercial device model. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 9 CALCULATING RATIOS OF HARMONICALLY RELATED, COMPLEX SIGNALS In this chapter, we describe a method for preserving time-invariant phase relationships when ratios are taken between any two harmonically related, complex signals [58]. We provide a simple example to illustrate our technique. Then in the following chapter, we will show how this method is implemented when defining time-invariant nonlinear large-signal scattering parameters. 9.1. Introduction When two complex signals z and y exist at the same frequency w/27i, the ratio R of the two quantities may be expressed as (9.1) where the phasor notation of z is represented by |z|Zcpz and that of y is represented by [y|Z(py. When two complex signals exist at different frequencies, obtaining a timeinvariant phase of the ratio is more involved. When ratios are taken between two harmonically related signals, we can preserve time-invariant phase relationships by introducing a third signal that acts as a phase reference. We show that this reference signal must have a component at the fundamental frequency in order that the ratios of any two harmonically related signals contain a time-invariant phase relationship. We provide a simple example to illustrate our technique. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.2. Method Consider two complex signals zk and yi that are harmonically related. Here k and I are positive integers representing signals at the Mi and /th harmonic terms, respectively. In phasor form, (9.2) Note that all phases (p considered here are in units of degrees (in terms of their respective frequencies) and have a modulus of 360° (i.e., 0° < cp < 360°). At first glance, a commonly assumed equation for taking the ratio of two harmonically related complex signals zk and yi is nt _ \Zk K ki ~ Z (9.3) The factor kll serves to translate the phase from the /th harmonic of the divisor to the Mi harmonic of the dividend, resulting in the phase of the ratio R \i given in terms of the Mi harmonic. The superscript‘t’ is used because eq. (9.3) gives a timevariant phase. Specifically, if kll is not an integer, there will be a phase ambiguity of 360°//. In order to try to avoid a phase ambiguity, we modify eq. (9.3) by referencing the phases of signals zk and yi to some reference phase of a third signal x„ at the nth harmonic, which gives Ru Zk W (9.4) / where 82 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. In eq. (9.5), the kin factor serves to translate the arbitrary phase (0° < cp* < 360°) from the nth harmonic of x„ to the kth harmonic of z* and the Hn factor serves to translate the phase from the nth harmonic of x„ to the /th harmonic of yi. Combining eqs. (9.4) and (9.5) gives U V <t>zk <t>x, n ' k ( rh - L r h J I $yi n $x. ' (9.6) Here, we still have the problem that if kin or Hn are not integers, eq. (9.6) gives an inconsistent phase. Specifically, if kin or Hn are not integers, there will be up to n (n < I) possible answers with a phase ambiguity of 360°//. If & is a multiple of / or vice versa, there will be fewer than n possibilities, but still more than one in general. In order to avoid any phase ambiguity, kin and lln must be integers. In order for this to be true for all k and /, n must equal one. If the frequency of the reference signal (xn) is set to its fundamental frequency (n - 1), then eq. (9.6) becomes Rkl ~ z (&* ~ k(Pxx) - y fay, (9.7) ) Note that eq. (9.7) is algebraically identical to eq. (9.3). It is important, however, to leave the (p.C| terms in and perform the phase references for cpZi and in order to ensure that eq. (9.7) provides a time-invariant phase. Equation (9.7) can be simplified in the case of Rk\ ( /- l) if >’i serves as both the divisor and the reference signal: 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. **. = ~ f k ~ kK)- <’ -8 > 9.3. Example Here, we provide a simple example illustrating that eq. (9.3) gives a timevariant phase, and that eq. (9.6) gives a time-variant phase for n > 1. However, eq. (9.6) does provide a time-invariant phase if n - 1. In this example (see Figure 9.1), we consider signals with three phase references, the first one being arbitrary, where the reference at the fundamental is xi = 1Z0°, the reference at the second harmonic is x2 = 1Z0°, the dividend (at the second harmonic) is z2 = 0.7Z800, and the divisor (at the third harmonic) is yi = 0.4Z1700. Figure 9.2 shows the time-domain representation of xi, z2, and y^. From the figure, we 1.0n Im z\l) =0.7Z80‘ 0.5- >{l) =0.4Z170( - 1.0 Re -0.5 0.5 .•() -0.5- -1 .0 J Figure 9.1. Phasor plot of the fundamental reference xi, the dividend z2, and the divisor ^3 at the first phase reference [all phasors identified by superscript (1)]. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. can see that the 80° phase delay in z2 corresponds to a time delay of 0.111 fundamental-unit period, and the 170° phase delay in yi corresponds to a time delay of 0.157 fimdamental-unit period. Figure 9.2 illustrates that there is no ambiguity in the time domain if all of the signals are synchronous with the fundamental signal. Equation (9.7) ensures the same in the frequency domain. But if, on the other hand, all of the signals are synchronous with a harmonic signal, portions of the waveforms at lower frequencies will be lost, resulting in possible phase ambiguities. At the first phase reference, we determine definition uses from eq. (9.3), which by as the reference in this case: K - 3^3 (9.9) Next, we calculate 7?(2)23 from eq. (9.6), using x2 as the reference, as n(2) _ lZ2l z "X2 23 (9.10) where the superscript ‘(2)’ denotes the phase reference of x2. Finally, we calculate from eq. (9.6), using xi as the reference, as (1) _ lZ2| z R 23 (9.11) where the superscript ‘(1)’ denotes of phase reference of x\. At this first phase reference, eqs. (9.9) - (9.11) give the same answer, of 1.75Z326.670, as shown in the first column of Table 9.1. Next, we consider a second phase reference, where the phase of the fundamental frequency is shifted by 100°. This means that the phase at the second 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. harmonic is shifted by 2 times 100°, or 200°, and the phase at the third harmonic is shifted by 3 times 100°, or 300°. So now, the reference at the fundamental is x\ = 1Z1000, the reference at the second harmonic is X2 = 1Z2000, the dividend is z2 0.7Z2800, and the divisor is ys = 0.4Z4700 = 0.4Z110°. These values are plotted in Figure 9.3. At this second phase reference, we again determine Rl22, R(1)23, and R(2)23 using eqs. (9.9) - (9.11). Here, R ^ - 1.75Z206.670 is inconsistent with the answer determined at the first phase reference by 120° (360°/3). The ratio R(2)23 = 1.75Z326.670 is consistent with the answer determined at the first phase reference. Likewise, the ratio i ?(1)23 - 1.75Z326.670 is also consistent with the answer determined at the first phase reference. The values of all of the quantities at the second phase reference are shown in the second column of Table 9.1. Finally, we consider a third phase reference, where the phase of the fundamental frequency is shifted by 200°. This means that the phase at the second harmonic is shifted by 2 times 200°, or 400°, and the phase at the third harmonic is shifted by 3 times 200°, or 600°. So now, the reference at the fundamental is x\ 1Z2000, the reference at the second harmonic is x2 ~ 1Z40°, the dividend is z2 = 0.7Z1200, and the divisor is yi = 0.4Z500. These values are plotted in Figure 9.4. At this third phase reference, we again determine /?*23, Z?(1)23, and /?(2)23 using eqs. (9.9) (9.11). Here, R \ 5 = 1.75Z86.670 is inconsistent with the answers determined at the first and second phase references by 120° (360°/3). The ratio R(2)23 = 1.75Z206.670 is also inconsistent with the answers determined at the first and second phase references. The ratio i ?(1)23 = 1.75Z326.670, however, is consistent with the answers 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. *2 K Quantity 1st Phase Reference Xi K 0O 2nd Phase Reference 1Z100° 0O Table 9.1. Determining the ratios of z2 to yj using three methods at three different phase references. 1Z 2000 Z2 0.7Z800 0.7Z2800 0.7Z120 0 T3 0.4Z1700 0.4Z110° 0.4Z500 ^23 [eq. (9)] 1.75Z326.670 1.75Z206.670 3rd Phase Reference 1Z 2000 1Z40° 1.75Z86.670 R(2)23 [eq. (10)] 1.75Z326.670 1.75Z326.670 1.75Z206.670 R(1)23 [eq. ( 11)] 1.75Z326.670 1.75Z326.670 1.75Z326.670 determined at the first and second phase references. The values of all of the quantities at the third phase reference are shown in the third column of Table 9.1. Examining the fifth row of Table 9.1, we see that R l23 does indeed give a timevariant phase. Since kll - 2/3 is not an integer, there is a phase ambiguity of 360° / 3, or 120°. Examining the sixth row of Table 9.1, we see that that R(2)23 also gives a time-variant phase since the reference signal is located at the second harmonic. Since lln = 3/2 is not an integer, there are 2 possible answers with a phase ambiguity of 360° / 3, or 120°. Examining the seventh row of Table 9.1, we see that that R(1)23 gives a time-invariant phase since the reference signal is located at the fundamental frequency. Thus, the only time-invariant ratio is R0)23. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 .0 “ i 0.5 0.5u = 0.157 0.0 0.0 -0.5 -0.5 - 1.0 - - 0.0 0.2 0.4 0.6 Time (fundamental unit period) 1.0 0.8 0.2 Figure 9.2. Time-domain plot of the fundamental reference x\, the dividend Z2 , and the divisor ^3 at the first phase reference. IZKHf + 200‘ +3001 - 1.0 Res ■0.5 0.5 +IW -0.5- 12) =0.7Z280‘ -1.0 J Figure 9.3. Phasor plot of the fundamental reference x\, the dividend Z2, and the d iv iso r^ at the second phase reference [using superscript (2)]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1-O-i Im - +400' Re +600°f 0.5 ♦ 200 -0.5 -1.0 J Figure 9.4. Phasor plot of the fundamental reference x\, the dividend Z2 , and the divisor J 3 at the third phase reference [using superscript (3)]. 9.4. Discussion We described a method for preserving time-invariant phase relationships when ratios are taken between two harmonically related signals by introducing a third signal that is used as a phase reference. We showed that a reference signal must be present at the fundamental frequency in order for time-invariant phase relationships to exist between ratios of any two harmonically related signals. We provided a simple example to illustrate our technique. In the next chapter, we will implement this method when defining time-invariant nonlinear large-signal scattering parameters. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 10 NONLINEAR LARGE-SIGNAL SCATTERING PARAMETERS In this chapter, we introduce the concept of nonlinear large-signal scattering parameters. Like commonly used linear 5-parameters, nonlinear large-signal scattering (# ) parameters can also be expressed as ratios of incident and reflected wave variables. However, unlike linear 5-parameters, nonlinear large-signal U>parameters depend upon the signal magnitude and must take into account the harmonic content of the input and output signals since energy can be transferred to other frequencies in a nonlinear device. After presenting the general form of nonlinear large-signal ^-parameters, we also introduce nonlinear large-signal impedance (2) and admittance ($) parameters, and present equations for relating the different representations. Next, we make two simplifications, considering the cases of a one-port network with a single-tone excitation and a two-port network with a single-tone excitation. 10.1. General Form Consider an iV-port network. Normalized wave variables a# and bji at the y'th port and /th harmonic are proportional to the incoming and outgoing waves, respectively, and may be defined in terms of the voltages associated with these waves as follows: 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a - I L . „ - I I . ( 10. 1) where K1}/ and Vji represent voltages associated with the incoming and outgoing waves in the transmission lines connected to they'th port and at the /th harmonic, and Z0j is the characteristic impedance of the line at the y'th port. The nonlinear large-signal scattering matrix & of the network expresses the relationship between a ’s and b’s at various ports and harmonics through the matrix equation b=$a. (10.2) where b and a are (/VxVlf)-element column vectors. Here N refers to the number of ports and M refers to the number of harmonics being taken into account. & is an (NxM) -element square matrix. We assume all a ’s and V s are phase referenced to a\\ to enforce time invariance, as discussed in the previous chapter. For a two-port network with 3 harmonics, for example, eq. (10.2) becomes [® l2 f *1 -h _ [» 2 2 l. J » 2 ll a x (10.3) a2 where v11 >#21 ' i j 12 •'ij 13 h j2 2 h j2 3 >#31 *ij 32 >,33 (10.4) For each nonlinear large-signal scattering parameter Bgki the index i refers to the port number of the b wave, the index j refers to the port number of the a wave, k is the 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. harmonic index of the b wave, and / is the harmonic index of the a wave. The vectors 1 1 I '— 1 aj and bt are (M=3)-element vectors: 1 & U> 1 II 1-cT II isT aj2 (10.5) bn bn _ Equation (10.3) can be expanded to *1111 ^1112 ^1113 ^1211 ^\2\2 ® 1213 a \\ ^12 ® 1121 ^1122 ® 1123 ^1221 ^1222 ^1223 a \2 ^13 ® 1131 ® 1132 ^1133 ® 1231 ^1232 ^1233 a tt ^21 ^2111 ® 2112 ^2113 ^2211 ^2212 ^2213 a 2\ ^22 ® 2121 ^2122 ^2123 ^2221 ^2222 ^2223 a 22 _^2131 ^2132 ^2133 ^2231 ^2232 ® 2233_ V _^23_ _a 23 (10.6) Note that in each of the four sub-matrices, the diagonal elements contain the samefrequency scattering parameters, the upper right elements contain the frequency down-conversion scattering parameters, and the lower left elements contain the frequency up-conversion scattering parameters. If the device under consideration contains no nonlinearities (i.e. no power is transferred to other frequencies), then eq. (10.6) reduces to 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 0 ^1211 0 0 au ^12 ®hlll 0 ® 1122 0 0 ^1222 0 a \2 ^13 0 0 ® 1133 0 0 ^1233 an ^21 ^2111 0 0 ® 2211 0 0 a 2\ ^22 0 ^2122 0 0 ^2222 0 a 22 0 0 ^2133 0 0 V (10.7) o 1 ^ 2 2 3 3 . _a 23 which is the matrix representation of the well-known linear 5-parameters containing three frequencies. 10.2. Nonlinear Large-Signal Impedance Parameters Rather than expressing the relationship between a ’s and b's in terms of a nonlinear large-signal scattering matrix &, we can alternatively express the relationship between voltages (F s) and currents (Ps) in terms of a nonlinear largesignal impedance matrix Z, as follows V = Z I, (10.8) where V and I are (iVxM)-element column vectors. Once again N refers to the number of ports and M refers to the number of harmonics being taken into account. Z is an (/VxAf)2-element square matrix. For a two-port network with 3 harmonics, for example, eq. (10.8) becomes > r '[2„] [Z12] JZ 2 1 ]J [Z 22] LK 2j 13*21 L*22JJL72j 72. where 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (10.9) 7)'l 1 7/12 -021 7y22 7y23 7/32 7y33 [ * ,] = (10.10) For each nonlinear large-signal impedance parameter Zy«the index i refers to the port number of the voltage V, the index j refers to the port number of the current I, k is the harmonic index of V, and / is the harmonic index of I. The vectors Vt and Ij are 1 ■Jy\ II y,= V t2 7*3. i 1 1 (M=3)-element vectors: (10.11) *J2 J jK Equation (10.9) can be expanded to Vn Vn Vn v21 v22 V23_ 7 lll 7112 7113 7211 7212 7213 ~h1 7121 7122 7123 7221 7222 7223 ^12 7l31 2^1132 7133 7231 7232 7233 ^13 7 lll 7 l l2 7 ll3 ^2211 7212 7213 7l 7l21 7122 7l23 7221 7222 7223 72131 7 l3 2 7l33 7231 7232 7233. 122 J23 (10.12) 10.3. Relating & and Z Matrices The & and 2 matrices can be expressed in terms of one another, if we know how a and b relate to V and I. From eq. (10.1), we can express Iji and Vtk in terms of dji and bik as follows: v ik = v ik + v ik ~ 4 ^ o i (a ik + bik) > where the subscripts refer to the /'th port and at the Mi harmonic, and 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (10.13) (10.14) i j , = i ; + /} = Y f o - v ; ) = - j = = ( a/, - b „ ) , z °j where the subscripts refer to the /th port and at the /th harmonic. For simplicity, we will assume for now the network under consideration consists of two ports. Later, we can easily generalize the equations relating the & and Z matrices for any /V-port network. If we allow the two transmission lines or waveguides connecting the two ports to have different characteristic impedances, Z0\ and Z02, eq. (10.14) can be expressed in matrix form as X \ u y z 01 [0 ] E°] [U]/Zo2 \ \ X ik*i -h_ X (10.15) Yi. where [U] is the identity matrix. Eq. (10.9) can be expressed as X +X Y2Y Yi. "[Z11] [Zi2]“ X _[Z21] [Z 2 2 L X (10.16) Combining eqs. (10.15) and (10.16) gives X +X Y2Y Yi. "[Z11] [Z 12 ] [U]/Zol [0] J * 2 l] [ZaalJ [0] [ u y z o2^ Vx+ vs vs v; (10.17) or r +X Y2Y Y2Y '[Z n ] .[Z 21] / \ X <N X [Z22 ] J V Y2Y X Y2Y / where 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (10.18) [Zi2r ■[Zl'l] [z12r \u y z 01 _[Z21] [z22]. [°] [0] [U]/Zo2_ "[Z11] .[Z2 1 ] \Z 2 2 \_ (10.19) is the normalized impedance matrix. Eq. (10.18) can be rewritten as r r [Z n ] [%2] 1 ~IW M [Z2 2 ]. l] _1_ .[0] [0]~ \ > f" VI ) Y Y f '[Zn] [z«r V] [0 ]' [Z2 2 ]. _[0] V _[Z'21] \ > , +" [U]_ / Yi. (10.20) and eq.(10.3) can be rewritten as [0] V V ^Y i [°] [ [ » ! ,] [*12 ]" [ u v f e ^ _[*2l] [ » 2 2 l [0] > f" [U]/y[Z^_ Y i . [0] [ u v 4 z 7 i_ h+J (10.21) Combining eqs. (10.20) and (10.21) allows us to solve for & in terms of Z: "»n] J * 2 l] [*12 f [*22 ]_ [0] [UVyfZ,o2 [0] T \%\\] [%2] M i ] [Z22L /r[2n] [Z,'2]' + [Z2 1 ] [Z2 2 1 \ '[U] [0 ]“ .[° ] V\ J [U] [0]' [0] [U] -1 m j z 01 [°] (10.22) [0 ] [U]/ J z o2 _ If Z0\=Z02 , eq. (10.22) reduces to hi] ► 21] ► i2] »22] ‘[Zn] [z;2]' + '[£/] [0]T _[Z21] [Z22]_ _[°] ml) "[Z[,] [z;2]' "[tf] [0]T .[Z2 1 ] [Z2 2 ]. _[°] 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (10.23) Alternatively, we can combine eqs. (10.20) and (10.21) to solve for Z in terms of 0 : ' [U] '[Z»] [z;2r [Z2 2 L ~[U] [0]" + JO] [U]_ [0] -1 [U] [0] M L >11] >12] ^21] ^22 ] [0] ■\lZ o2 r [u] [0]' LEO] [U]\ [V] -1 [0] [U] yjZo2 J \ _1 [0] 4 Zo1 4 Z oX 1 I” 1 0 ~[U] [0] 4 Z oX [0] [»22lJ Ll»2l] [0] [U] & o2 (10.24) If Z0\-Z 02 , eq. (10.24) reduces to '[Zn] \%2\ \% x\ [Z 22I / V / ~[U] [0]~ _[°] [U}_ ~[U] \ _[°] I»12l" _j1_ J » 2 ,l \ [» 2 2 l. / \~ [0 ]' " [ » ,,] » 1 2 ]' lul J » 2 ll [»22 ]_ / (10.25) 10.4. Nonlinear Large-Signal Admittance Parameters We can also express the relationship between voltages (T’s) and currents (/’s) in terms of a nonlinear large-signal admittance matrix as follows I (10.26) where $ is an (7VxA/)2-element square matrix. 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For a two-port network with 3 harmonics, for example, eq. (10.26) becomes in] . ^^ IK fcil 1 X l,2] Vi 122] (10.27) where 1,11 1,12 1,13 « , ] = 1,21 1,22 1,23 1,31 1,32 1,33 (10.28) For each nonlinear large-signal impedance parameter %jki the index i refers to the port number of the current 7, the index j refers to the port number of the voltage V, k is the harmonic index of 7, and / is the harmonic index of V. The vectors V/ and 7, are once again (M=3)-element vectors, defined in eq. (11). Equation (10.27) can be expanded as X I\2 In 1 21 1 tos 1 22 lllll I 1121 lll31 I 2 III 12121 12131 llll2 ll 122 ll 132 12112 12122 12132 ll 113 ll 123 ll 133 12113 12123 12133 ll211 ll221 ll231 I 2 2 II 12221 12231 ll212 ll222 ll232 12212 12222 12232 ll213 >11 ll223 V 12 ll233 V ,3 12213 V 2 1 12223 V22 12233 723 (10.29) 10.5. Relating 9 and $ Matrices The & and $ matrices can also be expressed in terms of one another, using eqs. (10.13) and (10.14) which show how a and b relate to V and 7. Once again, for simplicity, we will assume the network under consideration consists of two ports. If we allow the two transmission lines or waveguides 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. connecting the two ports to have different characteristic impedances, Z0\ and Z0i, eq. (10.14) can be expressed in matrix form as V ~[U]/Z01 h. [°] f77 77 \ [U ]/Zo2_ 77 7 C _ V ) [0] (10.30) where [t/] is the identity matrix. Eq. (10.27) can be expressed as \ Vi+ + V f H a l . V 72 . 7C_ ) H i,] ra n .r a n h . (10.31) Combining eqs. (10.30) and (10.31) gives 7 7 7 2 . 7 7 \ u y z 01 7 2\ [0] -i [0] [U]/Zo2_ \ 7 + + vx~ l22] V7 2\ 727 ) (10.32) + (10.33) ll2] In] *21] r - ____ - — or 7+ W l v x~ In] « « ]' 727 In] [«22] where \ u y z ol Itn] JI2 1 ] [f22]_ [°] [0] -1 'ran ran [U]IZo2_ ran ran (10.34) is the normalized admittance matrix. Eq. (10.33) can be rewritten as / [0 ]' V _ [°] 1 '[ i n ] [in f J in ] [ f 2 2 ]. / 7 7 727 V \U ] [0 ]" JO ] [U]_ m ' j flK i] V, [ i l '2 r [ W ill J and eq.(10.3) can be rewritten as 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (10.35) ! ! [U]/Jz4 [0] r^ni [ ^ i2] i ivyJzT, [0] _[®21 ] [^22]_ [0] [ £ /] /,/V Yi. [0] kJ (10.36) Combining eqs. (10.35) and (10.36) allows us to solve for & in terms of '[» „ ) l®12 ]" \u yjz^ ; ® 2 .i [» 2 2 l. [°] / ~[U] [0] ~[U] [0 ]' .[0 ] R . [0]' 11 " [In ] luyjz. 2 . V .[0 ] VI fl& i] \ ' [ « » ] m 2] [ u v - j z ^ flE l] [ I 22 (10.37) -1 [o] l u y j z ,o l [0 ] / [%2] N-1 i%2]_ If Z0\=Z02 , eq. (10.37) reduces to: / ►l2] h i] ►n] hi ] V f V \ "[£/] [O f " t i n ] + .[0] [tf]_ "[C/] [0]' .[0] lu l [in i' [I22 ]_ (10.38) \ '[ i n ] i t u i «2l] [I22 ]_/ Alternatively, we can combine eqs. (10.35) and (10.36) to solve for $ in terms of rm in ] u 122] [t/] [0]' [0] [tf]_ -1 [0] 4 Z oi [0] [U] -A [V] [[Sul [»«]] VZ.l L[»2ll [®22lJ [0] 4 Z o l_ [U] [U] [0]' [0] [U] [0] [ [U] [0] [U\ [U] yjZo2 \^ -a -1 K.] + [0] [0] 4 z oi ^221 [0] o2 [U] 4 Z o2 (10.39) If Z0\=Z02 , eq. (10.39) reduces to: 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -[fl] flE i] [ i : 2r m 2]_ / ~[U] [» ,2 lT [0]' J » 2 ll VJO] // \U] [0]" XJO] [U]_ 4- 'l& lll I » 2 ,l [®22lJJ (10.40) 1*12 i f [^22 ]_y 10.6. One-Port Network with Single-Tone Excitation For a one-port network with a single-tone excitation at the fundamental frequency, we can extract a reflection coefficient as '11*1 _K \ A k ~ kK la11 ) a \m ~ 0 f°r V/n(/n ^ l) (10.41) The limitation imposed on the equation is that all other incident waves other than a\\ equal zero. Instead of simply taking the ratio of b u to an, we reference the phase of b\k to that of a\\. To do this we must subtract k times the phase of a\\ from b\k. This concept is identical to the simplified case presented in eq. (9.8), where a\\ serves as both the reference and the divisor. For a one-port network with a single-tone excitation at the fundamental frequency, we can show that the equation relating £? and 2 reduces to the same wellknown equation as for the linear case if we assume that no energy is created in the form of frequency down-conversion. To illustrate this, we will once take into account M=3 harmonics for the sake of simplicity. Eq. (10.6) reduces to 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. hi ^ n ii ^1112 ^1113 an hi = ® 1121 ^1122 ^1123 0 .^ 1 1 3 1 ^1132 ^1133_ 0 hi_ (10.42) _ for a one-port network with a single-tone excitation a\\. This matrix can be rewritten as a set of three equations: Z>i = 1 n iai ; bn = ^ \ \ 2 \ aw 1 > h i = ^ i u i an - (10.43) Likewise, eq. (10.12) reduces to Vn ^1111 ^*1112 ^1113 hi Vn = ^*1121 ^1122 ^1123 hi .^ 1 1 3 1 ^1132 ^1133 _ ^13. (10.44) Ju where the voltage at the first harmonic V\\ can be expressed as ^11 = ^llllAl + ^1112^12 ^’^'1113^13 • (10.45) From eqs. (10.13) and (10.14), we know ^ii = y[^oiiau + ^ n ) ’ h i = ~T==iau ~ h i ) ’ \ Jo\ 0 h i ~ rz— (ai2 h i ) - ' \ hi ~ o\ r r — (a \l ~ h l ) ~ ~ Jil (10.46) y [ z7 i’ hi ^01 ■yj^oi Combining eqs. (10.45) and (10.46) gives y [z^ (a n +bn ) — i——[%xm (au v bn ) Z1112Z>12 ^ iiiih i\- ol Substituting eq. (10.43) into eq. (10.47) and solving for 2 im gives 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (10.47) Z o l( l + ^ 1 1 1 1 ) + *1111 = % n 12 ^1 1 2 1 (l + (10.48) ^ 1 1 1 3 ^ 1 1 3 1 m) If no energy is created in the form of frequency down-conversion (i.e., Ziii2=Zni3=0), then eq. (10.48) reduces to the same equation as in the linear case: 1049 (1 + S n ) ( . ) Zll=Zol(l-S n )' A similar derivation can be performed to show that M 1050 ( . ) — H i 1 2 ® 1 121 ~ H l l 3 ^ 1 1 3 1 mi Once again, if no energy is created in the form of frequency down-conversion (i.e., %\\\2- § \\n - ^ ) , then eq. (10.50) reduces to the same equation as in the linear case: 1 5 r,,=— = f -15'1?. z„ z„,(i+su) (io. i) 10.7. Two-Port Network with Single-Tone Excitation For a two-port network excited at port 1 by a single-tone excitation at the fundamental frequency, we can extract an input reflection coefficient as 'm i \K \A k \aii ~ kK ) a \ / \T am„ = 0 for \fm\/n[(m ^ l) a (n * l)j (10.52) As with eq. (10.41), instead of simply taking the ratio of bu to a\\, we phase reference to a\\. To do this we must subtract k times the phase o f an from b\k. And the limitation once again imposed on the equation is that all other incident waves other than a\ \ equal zero. 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Another valuable parameter, the forward transmission coefficient, is similarly extracted as _ \ b 2k \ A K ~ k K ) 2141 \a.11 amn ~ ®f°r ± l) a (n * l)]. (10.53) This parameter provides a value of the gain or loss through a device either at the fundamental frequency or converted to a higher harmonic frequency. In addition to the previous two parameters, shown in eqs. (10.52) and (10.53), an output reflection coefficient can also be useful when trying to determine the output matching network. If a nonlinear DUT is operating under its normal drive condition (>a\\ at some constant signal level), and a second source, excited by a small-signal tone at frequency fk, is placed at port 2 of the DUT, one of the equations in the matrix found in eq. (10.6) reduces to ^24 = ^2\k\a\\ + ^22kka2k • (10.54) If we solve eq. (10.54) for 82244, we get v22kk 21kV 11 . a2k (10.55) a2k From eq. (10.55), it is obvious that the output reflection coefficient 82244 cannot be determined by simply taking the ratio of 624 to a^k, since it also depends on a\\ through 8*2i4i- When we have small levels of aik, we can generate another signal A«24 that is offset slightly from the frequency of interest fk by Afk. Doing this, eq. (10.54) becomes 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^2k + ^ b 2k - ^ >2 m a U + ^22kk(a 2k + ^ ° 2 k ) ’ where Aci2k « (10.56) <*ik and S»22m remains constant over this frequency range. Subtracting eq. (10.54) from eq. (10.56) gives ^b>2k — ^22kk^a 2k > (10.57) which does not depend on &2Ui-If we solve eq. (10.57) for ^ 22kk, _ ^b 2k ?22kk ~ A Aa2k (10.58) Large a xj , Small A a 2k Eq. (10.58) is a quasi-linear approximation of the output reflection coefficient under normal operating conditions, and is consistent with the definition of “Hot £ 22,” which has been used to measure the degree of mismatch at the output port of a power amplifier at its excitation frequency. 10.8. Discussion In this chapter, we presented the general form of nonlinear large-signal parameters. Unlike linear A-parameters, nonlinear large-signal ^-parameters depend upon the signal magnitude and must take into account the harmonic content of the input and output signals since energy can be transferred to other frequencies in a nonlinear device. We also introduced nonlinear large-signal impedance (2) and admittance ($) parameters, and presented equations for relating the different representations. Next, we made two simplifications, considering the cases of a oneport network with a single-tone excitation and a two-port network with a single-tone 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. excitation. For the one-port case with a single-tone excitation at the fundamental frequency, we showed that the equation relating % and Z reduces to the same wellknown equation as for the linear case if we assume that no energy is created in the form of frequency down-conversion. For the two-port case excited at port 1 by a single-tone excitation at the fundamental frequency, we extracted an input reflection coefficient a forward transmission coefficient l5>2ui, and a quasi-linear output reflection coefficient: nikk- 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 11 EXPANDING DEFINITIONS OF GAIN BY TAKING HARMONIC CONTENT INTO ACCOUNT In this chapter, we expand the definitions of power gain, transducer gain, and available gain by taking harmonic content into account. Furthermore, we show that under special conditions, these expanded definitions of gain can be expressed in terms of nonlinear large-signal scattering parameters. These expanded forms of gain and nonlinear large-signal scattering parameters can provide us with valuable information regarding the behavior of nonlinear models. 11.1. Commonly Used Definitions of Gain Three types of gain are commonly used as figures of merit for two-port networks - power gain, transducer gain, and available gain [59]. Power gain G is defined as the ratio of the power delivered to the load Pi connected at port 2 of the device to the input power Pm at port 1 of the device (11.1) where Pl and Pm can be described in terms of wave-variables referenced to some real-valued impedance (11.2) and 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. d 1 k I2 = "21 ' *1 I2 • (11.3) Here at and bi refer to the complex incident and reflected power-normalized waves, respectively, where the subscript / denotes the port number. Transducer gain G j is defined as the ratio of the power delivered to the load Pi to the power available from the source P a ys Gt s - A . (11.4) AVS The power available from the source P ays is the maximum power that can be delivered to the network. This occurs when the input impedance of the terminated network is conjugate matched to the source impedance, and can be described in terms of power-normalized waves as Pays ~ Pin r*s 1rin — ~1 z2 (11.5) Available gain GA is defined as the ratio of the power available from the network P avn to the power available from the source P ays G = E d Z iL (11.6) AVS The power available from the network P avn is the maximum power that can be delivered to the load. This occurs when the output impedance of the terminated network is conjugate matched to the load impedance, and can be described in terms of power-normalized waves as 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AVN ~ r (11.7) I h l 2- OUT One limitation of the gain definitions described in eqs. (1), (4), and (6) is that they are restricted to the fundamental operating frequency, and thus do not take into account energy at higher harmonic frequencies. 11.2. Expanded Definitions of Gain In the case of a sinusoidal input to a two-port nonlinear device, power may be transferred to higher harmonic frequencies. Thus, we can modify the definitions of power, described in the previous section by taking into account the harmonic contributions and summing over all K harmonics considered. First, the input power P in given in eq. (11.2) can be modified to d _ 1 v I I2 _ l k=\ 1 v l k Ia (11.8) I2 =i Here, a,* and bik refer to the complex incident and reflected power-normalized waves, respectively, where the subscript i denotes the port number and k denotes the spectral component number. All waves must be taken with respect to some real-valued, reference impedance. Similarly, we can modify the other definitions as follows: (11-9) I k=\ AVS k=\ A-k=\ \k \ ’ and 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (11.10) Pavn —~'E\b2k\ • (H-H) Z- k=\ Equations (11.8-11.11) are valid under the assumption that the power in the network is confined to a grid of frequencies that are harmonically related. Here, the expanded power gain (S is defined as the ratio of Pl to P in and is given by l^ i P " i2 l^ i i2 (11.12) — EM Z k=\ z k=l the expanded transducer gain 0f>7 is defined as the ratio of P l to P a vs and is given by P p" s and the expanded available gain k k f - k k l 2 2 ‘=1 2 i - ‘------- • ^Ekl Zk=\ (11.13) C$a is defined as the ratio of P a v n to P A vs and is given A r AVN _ Z k= 1 P 1 K , Pavs by * v Ia I2 (11.14) ,2 Z k=\ 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11.3. Expanded Definitions of Gain in Terms of ^-Parameters Next, we show that under special conditions, the expanded definitions of gain can be expressed in terms of the two-port, nonlinear large-signal scattering parameters introduced in the previous chapter. When we extract the nonlinear large-signal scattering parameters I$iui and &2ui, described in eqs. (10.52) and (10.53), only aw is present and all other o ’s are forced to zero. In this case, eq. (11.8) is reduced to l k =1 z Likewise, eq. (11.9) becomes 2 k =1 and eq. (11.10) simplifies to Pavs A2 k iif- (n -17> Equation (11.11), however, remains unchanged as Pa w 4 S M Zk=\ 2. (H-18) since it contains no ay terms. In this case, where only a\\ is present and all other a’s are forced to zero, the expanded power gain (6 can be expressed as ill Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 1 £ |, Z*=l ■IN ±.\ a ii |2 (11.19) '*=1 i* Dividing both the numerator and denominator of eq. (11.19) by |tfn|2, and substituting in eqs. (10.52) and (10.53) gives the power gain in terms of the nonlinear large-signal scattering parameters: K 2\k\ k=\ (11.20) K 1*=l itti The power gain confined to the nth harmonic frequency is '2 1 n l ’n k (11.21) *=1 The ratio (S«/(S expresses the fraction of the power gain confined to the nth harmonic frequency compared to the power gain over all of the harmonic frequencies considered. Likewise, we can express transducer gain and available gain in terms of nonlinear large-signal scattering parameters. With only an present and all other a’s forced to zero, (Br and (§u reduce to the same expression: = « * = £ |»21«| *=1 (11.22) Here, the transducer, or available, gain confined to the nth harmonic frequency is 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. |» 2,„ |2 • Once again, ratio (H-23) or C e x p r e s s e s the fraction of the transducer, or available, gain confined to the nth harmonic frequency compared to the transducer, or available, gain over all of the harmonic frequencies considered. Note that (8„/(S = 11.4. Discussion In this chapter, we expanded the definitions of power gain, transducer gain, and available gain by taking harmonic content into account. We also showed that under special conditions, these expanded definitions of gain can be expressed in terms of nonlinear large-signal scattering parameters. In the following three chapters we will present three examples illustrating applications and extraction of nonlinear large-signal ^-parameters and the expanded definitions of gain. First, we will show how nonlinear large-signal ^-parameters and the expanded definitions of gain can be used to discover valuable information regarding the behavior of a nonlinear model. Specifically, we examine a lumpedelement model of a pseudomorphic high electron mobility transistor (pHEMT) device operating in a two-port, common-source configuration. Second, we illustrate how nonlinear large-signal ^-parameters can be used to as a tool in the design process of a simple nonlinear circuit, specifically a single-diode 1-2 GHz frequency-doubler circuit. And finally, we describe a method for generating nonlinear large-signal &>parameters based upon ANN models trained on frequency-domain data measured 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. using a nonlinear vector network analyzer (NVNA). We compare a diode circuit model, generated using this method, to a harmonic balance simulation of a commercial device model. 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 12 USING ^-PA RA M ETERS AND EXPANDED DEFINITIONS OF GAIN TO EXAMINE THE BEHAVIOR OF A PHEM T MODEL In this chapter, we provide an example showing how nonlinear large-signal scattering parameters along with the expanded definitions of gain, introduced in the previous chapter, can be used to discover valuable information regarding the behavior of a nonlinear model. Specifically, we examine a lumped-element model of a 2x90 pm GaAs pseudomorphic high electron mobility transistor (pHEMT) device operating in a two-port, common-source configuration, shown in Figure 12.1. The pHEMT model was developed by Cidronali et al [60, 61] using 5-parameter measurements and electromagnetic simulations of the device layout. GS DS ax Figure 12.1. Circuit diagram of a pHEMT device operating in a two-port, commonsource configuration. 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I j I j ] | 12.1. Using 0-Parameters to Examine the Behavior of a PHEMT Model First, we will look at 0 m i and 02 ui as a function of power for various bias conditions, and then we will reduce the 0-parameter data set by calculating the expanded power gain (E, the expanded transducer gain <&r, and the ratio of the power ! gain confined to the first harmonic to that of the overall power gain (Si/(S. We examine the model operating at a frequency of 5 GHz and powers ranging from a j | small-signal level of -20 dBm to a level of 10 dBm, which is close to the maximum | device rating. Since the model is valid for an ^-channel device, the drain is biased i j | | j i I positively with respect to the common source. In this case, we choose Vus = 3 V. The I I j we vary gate is biased negatively with respect to the source since this condition controls the width of the depletion region and blocks part of the conducting channel region. Here, V gs from 0.0 V to -1.2 V in steps of 0.2 V. We can easily determine the two-port, nonlinear large-signal scattering parameters, described in eqs. (10.52) and (10.53), as a function of power and bias using a commercial harmonic balance simulator with all a’s other than an forced to zero. Figure 12.2 plots the magnitudes of 0 i u i for the first four harmonics (k = 1,2, | 1 ! ! 3, and 4). We see that |0 im | remains relatively flat at all bias conditions with varying input power \au\- The value of |0 n n | decreases as Vgs decreases from 0 V to -0.2 V, reaches a minimum of -2.325 dB at Vgs = -0.2 V, and then increases as Vgs is further decreased from -0.2 V to -1.0 V. The parameters |0i 121(, |0i 1311, and |0n4i| generally increase with input power at all bias conditions. This reveals that as the input power is increased, more energy is converted to higher harmonic frequencies and appears at port 1. ! 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0V as - 20- “ O.OV S 'S' -2 0 •a lin 1121 I 1121 |> -4 0 2 1131 I mi 1141 1131 1141 -80 -20 -10 0 -20 10 -10 0 10 0 10 0 10 a n (dBm) a xj (dBm) V r,, - -0.4 V m C$ -4 0 - s 1111 1111 1121 1121 1131 1131 1141 1141 -6 0 -80 -20 -10 0 -20 10 -10 an (dBm) a, j (dBm) 20 -i 0 - Va , ~ -0.8 V CO -a ^-o -20 3 §> -40 I -6 0 - -20 "O g 1 | -4 0 I -6 0 - 1121 1131 1141 1121 1131 1141 -80 0 -20 10 -20 -10 a, j (dBm) a , . (dBm) Figure 12.2. Magnitude of S i u t as a function of input power for a nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of VDs= 3V and VGS = 0.0, -0.2, -0.4, -0.6, -0.8, and -1.0 V. 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 12.3 plots the magnitudes of S>2«i for the first four harmonics (k = 1, 2, 3, and 4). At small input signals, |02in| increases as VGS decreases from 0 V to -0.4 V, reaches a maximum of 15.481 dB at VGs = -0.4 V, and then decreases as VGS is further decreased from -0.4 V to -1.0 V. The parameter |02m| remains relatively flat and then gradually decreases with increasing |«n| for values of VGs from 0 V to -0.6 V. At VGs less than -0.6 V, |02in| actually increases with increasing \au \ due to self-biasing. We also see that |0 2 i2i|, |0 2 i3i|, and |0 2 i4i| generally increase with input power at all bias conditions. This reveals that as the input power is increased, some of the energy converted to higher harmonic frequencies appears at port 2. To gain some physical insight into the pHEMT model, we can examine the nonlinear large-signal ^-parameter data near pinch-off ( VGS = -1.0 V and Vos - 3 V) and near Ids ~ Idss (V gs - 0 Y and Vos - 3 V). In these two regions, the second harmonic content is maximum (02 121 > 02u-i for k > 2), which can be seen in Figure 12.3. Near the pinch-off region, the device draws current only for the positive part of the gate voltage waveform, which results in a clamped waveform drain current, while near Ids ~ Idss, the device saturates at the positive part of the gate voltage waveform and the device draws current only for the negative portion. Thus, taking the Fourier transform gives even harmonics of the same amplitude in both cases, but they are 180° out of phase. From Figure 12.4, we can see that near pinch-off the phase of 02i2i is approximately 160°, while near I d s = (V gs = -1.0 V), loss ( V g s = 0 V), the phase of 02i2i is approximately -50°. The difference in phase is about 210°, which is close to the expected value of 180°. 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 20 n OH 1o -2 0 - - £ si Si * “ -4 0 - rGS =o.ov %2111 &2121 -6 0 -80-20 .g 1 1 -10 r o V™ = -0.2 V -4 0 - 2 -6 0 - &2131 &2141 "l o“ -2 0 - -80-20 1 10 axj (dBm) 20 0 0 - m T3 -20- 10 -6 0 - -60 H “ i-----1 -20 -20 h M ITO -4 0 - -4 0 - -80 - T3 1 | I \ -10 o a xx (dBm) 20-i e ^ —i— 1----r r -10 0 axj (dBm) -80 n -20 10 r ~i-----1 -10 0 10 axx(dBm) 20- | 20 -i OH CQ TD ^ T3 -20 h ^ ”2 .n I 1 ICg -40' -20- = -1.0V -4 0 - 2111 2121 -6 0 - -60 2131 2141 -80 -80 ~_l----- 1— '----- 1— 1— I -20 -10 0 a xj (dBm) -20 10 -10 0 10 a x! (dBm) Figure 12.3. Magnitude of &21/H as a function of input power for a nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of VDS= 3V and VGS = 0.0, -0.2, -0.4, -0.6, -0.8, and -1.0 V. 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 200 - - 2111 100- 2121 P 03 1 <D 2131 o- 0- 2141 l-i < 1121 -100 1131 - — 100- 1141 -20 -15 5 -10 0 -20 CO 8 If "O -15 -10 5 0 a n (dBm) # ii (dBm) 200- 200- 100- 100 — s i o- 3 < 0- 2121 1131 - 100- -20 2131 -1 0 0 1141 - 10 - 2141 0 -20 10 200- 200 100 100- 0 10 - S3 <u 0- o1121 fgs =-i .ov — » 2111 2 121 2131 131 -100 -10 a n (dBm) a xj (dBm) U < 2111 1121 -100 - - 2141 1141 -20 -10 0 a n (dBm) -20 10 -10 0 a n (dBm) 10 Figure 12.4. Phase of and &2ui as a function of input power for a nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of Vds = 3V and VGS = 0.0, -0.4, and -1.0 V. 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12.2. Using Expanded Definitions of Gain to Examine PHEMT Model the Behavior of a Rather than looking at numerous ^-parameter graphs, we can get a more concise view of the modeled behavior of a device by reducing the nonlinear largesignal ^-parameter data set into the compact expression of power gain (S. Figure 12.5 plots the calculated values of (S using eq. (11.20) for K = 4, as a function of power for Vqs varying from 0.0 V to -1.2 V. At small signals, the power gain increases as Vgs decreases from 0 V to -0.4 V. The power gain reaches a maximum of 19.72 dB at = -0.4 V, and then decreases as V gs V gs is further decreased from -0.4 V to -1.2 V. This is consistent with the fact that the trans-conductance gm, which is proportional to Y2\ and hence £ 21, peaks at a bias of Vqs = -0.4 V. We can also see from Figure 12.5 that the gain stays relatively flat and then gradually decreases with increasing \au\ for values of V gs from 0 V to -0.6 V. At Vg s less than -0.6 V, the gain increases with increasing \an \ due to self-biasing and harmonic production. 2 0 -i -0.4 V 15- -0.2 V -0.8 V "°-6V vGS(S) 10- — 0.0 - 50- -20 -1.0V -1.2 V -15 0.2 — -0.4 — - - — - - -10 5 axj (dBm) 0 5 - 0.6 0.8 1.0 - 1.2 - 10 Figure 12.5. Expanded power gain (S as a function of input power for a nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of VDS= 3V and VGS = 0 .0 , -0 .2 , -0.4, -0 .6 , -0 .8 , - 1 .0 , and - 1.2 V. 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We can also reduce the nonlinear large-signal ^-parameter data set by using the compact expression of transducer gain d r • Figure 12.6 plots the calculated values of d r using eq. (11.22) for K = 4, as a function of power for Vg s varying from 0.0 V to -1.2 V. At small signals, the power gain increases as Vgs decreases from 0 V to -0.4 V. The transducer gain reaches a maximum of 15.48 dB at decreases as V gs Vqs - -0.4 V, and then is further decreased from -0.4 V to -1.2 V. And like power gain, the transducer gain stays relatively flat and then gradually decreases with increasing |an| for values of Vgs from 0 V to -0.6 V. At Vq s less than -0.6 V, the transducer gain increases with increasing \an \ due to self-biasing and harmonic production. 2010 | - -0.4 V -°-2 V 0- -1 0 - -0.6 V -0.8 V 0.0 -1.0 V -1.2 V - 0.2 -0.4 - 0.6 - 1.2 -20 -20 -15 -10 5 a {j (dBm) 0 5 10 Figure 12.6. Expanded transducer gain d r as a function of input power for a nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of Vd s ~ 3V and V q s = 0.0, -0.2, -0.4, -0.6, -0 .8 , -1.0, and - 1.2 V. Comparing the plots of power gain and transducer gain in Figures 12.5 and 12.6, we can see that the values of transducer gain are less than those of power gain, due to the large values of H>iin which inflate the values of power gain. Another difference to 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. note is that the power gain at VGS = -0.4 V is always greater than at Vgs - -0.2 V, but the transducer gain at VGs = -0.4 V is only greater than at VGs - -0.2 V up to |an| = 2.6 dBm. At higher input powers, the transducer gain is slightly higher at Vgs = -0.2 V than at Vgs = -0.4 V. Figure 12.7 plots the ratio of the power gain confined to the first harmonic to that of the overall power gain, (&i/CS. Recall that ($|/(S = (Sri/®r- At small signals, (&i/(S is relatively high for all values of V g s, but is at a maximum of 0.9958 for -0.4 V. As |an| increases, ($i/($ remains relatively high for At Vgs V gs Vgs - at -0.2 V and -0.4 V. less than -0.4 V, (Hu/d decreases dramatically with increasing |an| since much of the energy in the device is converted to higher harmonic frequencies. In fact, at VGs = -1.2 V, (&i/(8 drops to 0.731. 1.00 0.95 -0.4 V -0.2 V V:;s(V) 0 .9 0 - 0.0 - 0.2 — — -0.4 - - 0.6 - - - 0.8 - 1.0 - 1.2 — -1.0 V - 0 .7 5 - -1.2 V 0.70 -20 -15 -10 5 a ]| (dBm) 0 5 10 Figure 12.7. The ratio ®i/(K as a function of input power for a nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of Vds= 3V and VGS = 0 .0 , -0 .2 , -0.4, -0 .6 , -0 .8 , - 1 .0 , and - 1.2 V. 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From Figures 12.5, 12.6, and 12.7, we can clearly see that the optimum gateto-source bias condition is near Vgs - -0.4 V for the device at Vos = 3 V, where d , d r, and d i / d are highest. Furthermore, if we look at the power gain at this particular bias condition, shown in Figure 12.8, we find the 1 dB gain compression point to be didB = 18.716 dB, which occurs at \aw\~ 1.40 dBm. Also shown in Figure 12.8 is the traditional power gain G. For this example, where a\\ is the only incident wave present, G can be expressed in terms of nonlinear large-signal scattering parameters as: O = l°'2l" l — (12.1) H & in if We see from Figure 12.8 that at small input signals d and G are nearly identical, but at higher powers d is greater than G since the traditional power gain does not take into account the power generated at higher harmonic frequencies. Using G, we get the same 1 dB gain compression point of GidB = 18.716 dB, but it occurs at |tfn| = 0.78 dBm. 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 0 .0 -1 19.0- ldB ldB 18.0•g 17.0g 16.015.0- — G 14.013.0 -20 -15 -10 a 5 x, (dBm) 0 5 10 Figure 12.8. The 1 dB gain compression point for a nonlinear lumped-element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of Vos - 3V and VGS = -0.4 V. 12.3. Discussion We provided an example showing how the expanded definitions of gain and nonlinear large-signal scattering parameters can be used to examine the behavior of a nonlinear model by simply performing a harmonic-balance simulation with all a's other than a\\ forced to zero. Looking at the nonlinear large-signal scattering parameters gives us an in-depth view of the modeled behavior by allowing us to separate out the input reflection coefficients and transmission coefficients at each of the frequency components, while reducing the nonlinear large-signal scattering parameter data set into the compact expressions of power gain (S, transducer gain (Sr, and the ratio (&i/(S, gives us a more concise view of the modeled behavior. 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 13 USING NONLINEAR LARGE-SIGNAL ^-PARAM ETERS TO DESIGN A DIODE FREQUENCY-DOUBLER CIRCUIT W ITH A HARMONIC-BALANCE SIMULATOR Resistive frequency doublers operate on the principle that a sinusoidal waveform is distorted by the nonlinear UV characteristic of a Schottky-barrier diode [62]. This distortion causes powers to be generated at higher harmonic frequencies. The design o f such doublers reduces to separating the input and output signals by filters and determining the optimum input and output matching circuits, as illustrated in Figure 13.1. Although single-diode resistive doublers are not very efficient (analysis predicts a conversion loss of at least 9 dB [63]), we choose this circuit because it is simple enough to clearly illustrate how nonlinear large-signal ^-parameters can be used as a design tool. Input Matching Network Output Filtering Network Input Filtering Network Output Matching Network Figure 13.1. Block diagram of a single-diode resistive doubler. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the following sections, we describe the various steps involved in designing a single-diode 1-2 GHz frequency-doubler circuit. Since we are using a simulator, we can force the stimulus to consist of only |a\\\, with all other amn = 0 , where m and n are positive integers such that m * 1 and n ^ 1. (This condition can never be completely realized in a measurement environment.) With only an an component present, we need only consider the parameters &n*i (eq. 10.52), a measure of the large-signal input match at the Mi harmonic, &21« (eq- 10.53), a measure of the largesignal conversion loss or gain at the Mi harmonic, and the quasi-linear B 2222 (eq. 10.58) to determine the output matching network at the second harmonic. Figure 13.2 illustrates the setups required for determining these parameters. Determining B 2222 requires a second source at port 2 at a frequency slightly offset from CO2 • G1 inti Nonlinear DUT 21k \ (a) G2 G1 Nonlinear DUT 2222 (b) Figure 13.2. Nonlinear large-signal ^-parameters used to characterize a two-port device excited by a single-tone signal at port 1. 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the first step, we perform a simulation on the diode alone and use #2121 to determine the optimum bias condition for converting power from the fundamental frequency to the second harmonic. Second, we add filtering networks to separate the input and output signals, and verify their proper performances by looking at &2111 and §>1121. Third, we make use of §>1111 to determine the input matching network. Fourth, with the input matching network in place, we place a second source at port 2 and find the quasi-linear value of ^ 2222 , which allows us to determine the output matching network. Fifth, we use the optimization feature of the simulator to minimize & im b y varying the line lengths of the input and output matching circuits. And finally, sixth, we add 4 GHz and 6 GHz filters at the output (and re-determine the proper input and output matching circuits) in order to reduce the values of #2141 and # 2161, which in turn increases the value of &2121 and cleans up the output waveform. 13.1. Diode Only In this example, we use a compact model to simulate a commercial Schottkybarrier diode. The model includes a series resistance, Rs, of 14 Q, a junction capacitance at zero voltage, C/0, of 0.08 pF, and a reverse saturation current, Is, of 3xlO'10 A. First, we perform a harmonic-balance simulation on the diode, sweeping the bias voltage to determine which condition gives the highest value of &2121 for an 1.0 V. Note that in all simulations we set the generator impedance, Zg, and the load 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 13.1. Simulated values for & im - &ii 6i, &2111 of the design stages of the diode frequency doubler. Quantity 1£ i m | | £ l l 2 l| 1® 1 1 3 l l l& 114l| l& 115ll | £ l l 6 ll l & 211 ll l & 212 l | 1& 2 1 3 l l 1^2141 l® 215l| l& 216ll (dB) ® 2/<S <62 Diode Only 0.464 0.170 3 .2xl0'2 2.4x10‘2 1.7xl0"2 3 .9xl0'3 0.536 0.170 3.2x1 O'2 2.4xl0'2 1.7x1 O'2 3.9x10'3 -14.16 0.091 *2 1 6 1 : ($2 , and (£2/(6 for each 9.4x1 O'2 8.8x1 O'6 4 .0 x l0 '3 Diode w/ 1 ,2 GHz Filters, Input & Output Match 8.7x10‘2 8.0x1 O’6 1.4x1 O'2 Diode w/ 1 ,2 GHz Filters, Input & Output Match Opt. 6.0xl0'3 9.5x1 O’6 l.lx lO '2 Diode w/ 1 ,2 , 4 ,6 GHz Filters, Input & Output Match Opt. 2.1xl0'4 9.9x10'6 2.2x1 O'2 3.7x1 O'2 2.4x1 O'2 1.9x10'3 9.7xl0'7 4.0x10'5 0.328 1.5x10'6 5.1X10'2 2.5x10’3 3.3xl0'5 0.268 3.5x1 O'7 l . l x l O'2 1.0x1 O'6 4.0x10'5 0.326 3.3xl0'7 2.8xl0'2 2.3x10'3 l.lx lO '6 4.0xl0'5 0.331 l.lx lO '6 2.0x1 O'6 5.0x1 O'5 0.332 1.7x1 O'7 3.5xl0'2 4.5x1 O'2 4.1xl0'2 4.0x1 O'2 1.4x1 O'6 7.6x1 O'7 2.0x1 O'2 -9.73 0.978 l.lxlO"6 2.5 x l0 '2 -9.69 0.976 2.5x1 O'6 2.6x1 O'2 -9.65 0.979 2.3X10'6 2.9 x l0 '2 3.0x1 O'6 2.7x1 O'6 -9.56 0.999 Diode w/ 1, 2GHz Filters Diode w/ 1 ,2 GHz Filters, Input Match 0.569 1.3x1 O'5 4,9x10'3 3.5xl0'2 l.lx lO '2 1.0x1 O'6 -9.60 0.978 impedance, Zi, to 50 Q. After sweeping the voltage, we determine that the optimum forward bias is +0.48 V. 13.2. Diode with 1 & 2 GHz Filters With a stimulus of an = 1.0 V and a forward bias of +0.48 V, we add filtering networks to separate the input and output signals. On the input side, we place a 2 GHz, A/4 (A/8 at 1 GHz) open-circuited stub. This creates an RF short at 2 GHz, preventing the output power generated in the diode from traveling backward. On the output side, we place a 1 GHz, A/4 open-circuited stub. This creates an RF short at 1 GHz, preventing any signal at 1 GHz from traveling forward. 129 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Table 13.1 lists the simulated values for 8 »im - 0n6i and ^2111 - ^ 2161, as well as the power gain confined to the second harmonic CS2 and the ratio OS2/®, for each of the design stages. With the 1 GHz and 2 GHz filters in place, we see that the value of IS'iml decreases from 0.170 to 1.3xl0'5, the value of |^ 2in| decreases from 0.536 to 3.3xlO'5, and (§2 increases from -14.16 dB to -9.73 dB, when compared to the diode-only case. 13.3. Diode with 1& 2 GHz Filters and Input Matching Once the filters are placed in the circuit, we make use of the complex-valued &1111 to design the input matching network with the well-known single, open- circuited stub technique. This is possible, assuming that no energy is created in the form of frequency down-conversion, as discussed in Section 10.6. We see in Table 13.1 that m il reduces from 0.569 without the input matching network to 9.4x10 -2 with the input matching network in place. Likewise, (§2 increases from -9.73 dB to -9.69 dB. 13.4. Diode with 1& 2 GHz Filters, Input and Output Matching Whereas our input matching network is designed for 1 GHz, our output matching network must be designed for 2 GHz. While the circuit is operating under its normal drive condition (an = 1.0 V and a forward bias of +0.48 V) we place a second source at port 2, excited by a small-signal tone (A«22 = 0.01 V) at a frequency offset of 10 kHz from the desired 2 GHz, to give us the quasi-linear value of 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. §>2222 , which allows us to determine the output matching network. We make use of ^ 2 2 2 2 to design the output matching network with the well-known single, open-circuited stub technique. We see in Table 13.1 that with the output matching network in place, the value of I&21211is only marginally increased from 0.326 to 0.328. This is because the value of §>2222 is relatively low, which means the output is already almost matched to 50 Q. We also note that (82 increases from -9.69 dB to -9.65 dB. 13.5. Diode with 1& 2 GHz Filters, Input and Output Matching Optimized With the filters and matching networks in place, we use the optimization feature of the simulator to minimize #1111 by varying the lengths of the lines in the input and output matching circuits. Doing this decreases the value of |&mi| from 8.7xl0 '2 to 6.0xl0 '3 while increasing the value of I&2121I from 0.328 to 0.331 and (82 from -9.65 dB to -9.60 dB. 13.6. Diode with 1& 2, 4 & 6 GHz Filters, Input and Output Matching Optimized From Table 13.1, we see that at the output port, |02in|> I&2131I, and I&2151I all have values less than or equal to 4.0xl0'5, but I&2141I and |&2i6i| have noticeably higher values (at least 2.9x10'2). In order to clean up the output waveform, we add 4 GHz and 6 GHz filters, in the form of A/4 open-circuited stubs, at the output. With these filters placed in the circuit, we re-determine the proper input and output matching conditions. After optimizing the circuit once again, the value of I&2141I decreases from 4.0x10 '2 to 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.4x10' and the value of |&2i6i| decreases from 2.9x10' to 2.7x10". The addition of these filters, in turn, slightly increases I&2121I from 0.331 to 0.332 and (S2 from -9.60 dB to -9.56 dB. At this final design stage, the overall power gain is very near -9.56 dB since the ratio CI2/® = 0.999. The semi-empirical analysis of [62] predicts a maximum gain of -9 dB. Figure 13.3 illustrates the final design of the single-diode resistive doubler circuit. And Figure 13.4 shows the time-domain plots of a\ and bi for the final design o f the simulated 1-2 GHz frequency-doubler circuit. 13.7. Discussion We illustrated how nonlinear large-signal ^-parameters can be used as a tool in the design process of a single-diode 1-2 GHz frequency-doubler. Specifically, we used i n n to determine the input matching network, §>2222 to determine the output matching network, and fifiui, €>21*1 (for k = 1 to 6), and ($ 2 to quantify the performance of the circuit at each stage. By the final stage of the design, we had created a doubler with an overall power gain of -9.56 dB, not far from the maximum possible predicted value of -9 dB. 132 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Input Match 4, 6 GHz Filters Diode Doubler Output Match +0.48 V 65.72° -W 56.29° - 50 Q* vd *T) 50 Q 0 fln=1.0V /,=1.0 GHz Figure 13.3. Final design of the single-diode resistive frequency doubler. Electrical lengths shown are all at 1 GHz. 1 1 D D D D •1 D •1 time, rtsec time, nsec Figure 13.4 Time-domain plots of a\ and bi for the simulated 1-2 GHz frequencydoubler circuit. 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 14 DETERMINING NONLINEAR LARGE-SIGNAL ^-PARAMETERS FROM ARTIFICIAL NEURAL NETWORK MODELS TRAINED WITH MEASUREMENT DATA Although the nonlinear large-signal ^-parameters, introduced in Chapter 10, can be easily determined for an existing model in a commercial harmonic balance simulator by forcing all a ’s other than an to zero, they cannot be determined directly from measurements. With currently available nonlinear vector network analyzers (NVNAs), described in Chapter 9, the nonlinear device under test (DUT), in conjunction with the impedance mismatches and harmonics from the system make it impossible to set all a ’s other than an (assuming port 1 excitation) to zero. In order to overcome this obstacle, we propose a method [64, 65] that makes use of multiple measurements of a DUT using a second source and isolators, as shown in Figure 14.1. This measurement set-up is similar to that introduced by Verspecht et al. [9, 10] to generate ‘nonlinear scattering functions.’ As a side note, in the following chapter, we will compare and contrast the ‘nonlinear scattering functions’ with our definitions of nonlinear large-signal scattering parameters. 14.1. Methodology To illustrate our technique of generating nonlinear large-signal ^-parameters, let us consider the case where a DUT is excited at port 1 by a single-tone signal at frequency f\ and signal level \an\. Utilizing a second source, multiple measurements of a nonlinear circuit are taken for different values of amn [(m£\ )A(n^l)]. These data 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. \ are then used to develop an ANN model that maps values of a ’s to b’s, as shown in Figure 14.2. Once the ANN model is trained and verified, the nonlinear large-signal ^-parameters are obtained by interpolating V s from the measured results for nonzero values of amn [(w ^I)a(r^I)] to the desired values for amn [(m^l)A(«^l)] equal to zero, as shown in Figure 14.3. Alternatively, other conditions may be called for where amn ^ 0 depending on the desired application-specific figure of merit. The ANN architecture used for this modeling is the feedforward, three-layer perceptron structure that we discussed in Chapter 2. To briefly review, this architecture consists of an input layer, a hidden layer, and an output layer. The hidden layer allows complex models of input-output relationships. And once again, we utilized software developed by Zhang et al. [42] to construct the ANN models. Precision ADC Broadband Downconverter Bias T r DUT Bias T s Figure 14.1. Block diagram of a nonlinear vector network analyzer equipped with a second source and isolators. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. : \a i l l Re,Im{<2 12} Re,Im{6n } Re,Im Re,Im{a1A:} Re,Im{«21} Re,Im{&12} ANN Model Re,Im {blk} Re,Im{a22} Re,Im{a2;t} Re,Im{617J Figure 14.2. An ANN model that maps real and imaginary values of a ’s to b’s for different real and imaginary values of amn [(m^l)A(n^l)]. \a Re,Im{6n } i l l A 'm i Re,Im{612} aXk{ k * \) - Q ) tf2,(a lU )= (P ANN M odel '1121 Re,Im{6u } A 'm i R e ,I m { V } A 1\ K \ Figure 14.3. An ANN model that interpolates h’s from the measured results for nonzero values of amn [(m*X)A{n£\)] to the desired values for amn [{m^ 1 ) a ( « ^ 1 ) ] equal to zero. Outputs of the ANN model yield values of 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To test our method of generating nonlinear large-signal ^-parameters, we fabricated a wafer-level test circuit using a Schottky diode in a series configuration, as shown in Figure 14.4. The two-port diode circuit was fabricated on an alumina substrate by bonding a beam-lead diode package to the gold metalization layer with silver epoxy. The diode was located in the middle of the coplanar waveguide (CPW) transmission lines, with short lines connecting the diode to probe pads at both ports. We measured the test circuit on an NVNA using an on-wafer line-reflect-reflectmatch (LRRM) VNA calibration, along with signal amplitude and phase calibrations. This process places the reference plane at the tips of the wafer probes used to connect with the CPW leads. For all measurements, the first source, located at port 1, was set to a sine-wave excitation of frequency 900 MFIz and magnitude \aw\ ~ 0.178 V ( - 5 dBm in a 50 Q environment) at the probe tips. The second source was connected to port 2 and was set to a sine-wave excitation of frequency 900 MHz and \ci2 \\ ~ 0.178 V. The diode was forward-biased to +0.2 V through the probe tips. In order to obtain the nonlinear large-signal ^-parameters, |§>im and f$2 \k\, the excitation from source 1 was held constant, while the phase of source 2 was randomly changed for 500 different measurements that varied slightly in magnitude. Figure 14.5 plots the resulting measurements of an in the complex plane. The nonlinearities in the test circuit, along with impedance mismatches, created other input components at higher harmonics, as shown in Figures 14.6 - 14.9 for the second and third harmonics {an, an, an, and an). These variations in ay allowed us to create an ANN model that could be used to interpolate Ifs from the measured results for nonzero values of amn [(m^l ) a ( / i + 1 ) ] , as 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shown in Figures 14.10 and 14.11 for bn and 621, to the desired values for amn [(m^l)A(rt^l)] equal to zero, or alternatively another desired device condition. (i Ci Figure 14.4. Schottky diode in a series configuration located in the middle of a CPW transmission line. (White area is metal.) •• 0.1- - 0.2 - 0.2 0.1 > -0.1- -0 .2 - Figure 14.5. Five hundred measurements of a 2 \ in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 - i Im{a12} ( V ) x 103 ♦y. v ; *• •• s . '; *. A Re{a12} (V) x 103 -2 - r 1 • & - l- -2J Figure 14.6. Five hundred measurements of a n in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. Im{a13} (V) x 103 1- Figure 14.7. Five hundred measurements of ab in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 - i Im {a22} (V) X 103 1- Re{a22} (V) x 103 1 -2 -1 -1 -2Figure 14.8. Five hundred measurements of <222 in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. Re{a23} (V) x 103 1 -2 -1 1 2 -1- -2J Figure 14.9. Five hundred measurements of <223 in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 . 2 - , Im{*u }(V ) 0.1- - 0.1 - -0 .2 J Figure 14.10. Five hundred measurements of bn in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 0.2-, im{&21} (V) - 0.2 - 0.1 - 0.1 - 0.2 - 1 Figure 14.11. Five hundred measurements of bii in the complex plane with the excitation from source 1 held constant and the output from source 2 set to random phases with constant amplitude. 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14.2. Sensitivity Analysis of ANN Models Data from the 500 measurements were used to develop two ANN models, one for mapping values from the first five harmonics of ai and a 2 (a\\, an, ..., flis, a 2i, an, ..., a25 ) to the first five harmonics of bi (bn, bn, ■■■, bn), and the other for mapping values from the first five harmonics of ai and a 2 to the first five harmonics of b2 (£>21, £>22, •••, £>25)- We performed a sensitivity analysis to determine how many training points, testing points, and hidden neurons are required to adequately train the two ANN models. Tables 14.1 - 14.3 summarize the results for the first model, where we map values from the first five harmonics of ai and a 2 to the first five harmonics of bi, and Tables 14.4 - 14.6 summarize the results for the second model, where we map values from the first five harmonics of ai and a 2 to the first five harmonics of b 2. First, we varied the number of hidden neurons from 1 to 20. All other parameters were held constant. Specifically, the 500 measurements points were divided into 250 training points and 250 testing points, and the conjugate gradient method was used for training. Table 14.1 lists the average testing errors and correlation coefficients for the models that map ai and a 2 to bi, and Table 14.4 lists the average testing errors and correlation coefficients for the models that map ai and a 2 to b 2. Both mappings show similar trends. The average testing errors decreased with increasing numbers of hidden neurons until around 14 or 16, where the errors were minimized. For more than 16 hidden neurons, the trend reversed and the errors appeared to start increasing again. Figure 14.12 plots the average testing errors as a function of the number of hidden neurons for both mappings. 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Next, we varied the number of training points from 5 to 250. All other parameters were held constant. The number of hidden neurons was set to 14 since we found that to be an ideal number from the previous analysis, and 250 testing points were used for verification. Table 14.2 lists the average testing errors and correlation coefficients for the models that map ai and a 2 to bi, and Table 14.5 lists the average testing errors and correlation coefficients for the models that map ai and a 2 to b 2. Once again, both mappings showed similar trends. The average testing errors decreased for an increasing number of training points. However, as more and more training points were added, diminishing returns on the testing errors were evident. Figure 14.13 plots the average testing errors as a function of the number of training points for both mappings. Finally, we varied the number of testing points from 5 to 250. All other parameters were held constant. The number of hidden neurons was once again set to 14, and the same 250 training points were used for model development. Table 14.3 lists the average testing errors and correlation coefficients for the models that map ai and a 2 to bi, and Table 14.6 lists the average testing errors and correlation coefficients for the models that map ai and a 2 to b 2. Both mappings showed that the average testing errors varied little with the number of testing points. Figure 14.14 plots the average testing errors as a function of the number of testing points for both mappings. 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 14.1. Average testing errors and correlation coefficients as functions of the number of hidden neurons for ANN models trained to map values from the first five harmonics of ai and si2 to the first five harmonics of bi. All models were developed using 250 training points and verified using 250 testing points. Hidden Neurons 1 2 4 6 8 10 12 14 16 18 20 Average Testing Error (%) 16.86 10.84 4.56 1.66 1.15 1.08 0.80 0.72 0.72 0.84 0.70 Correlation Coefficient 0.94814 0.98896 0.99715 0.99971 0.99989 0.99991 0.99996 0.99997 0.99997 0.99996 0.99997 Table 14.2. Average testing errors and correlation coefficients as functions of the number of training points for ANN models trained to map values from the first five harmonics of ai and a to the first five harmonics of bj. All models were developed using 14 hidden neurons and verified using 250 testing points. 2 Training Points 5 10 25 50 125 250 Average Testing Error (%) 20.10 9.01 3.64 1.91 0.95 0.72 Correlation Coefficient 0.96764 0.99556 0.99891 0.99979 0.99995 0.99997 Table 14.3. Average testing errors and correlation coefficients as functions of the number of testing points for ANN models trained to map values from the first five harmonics of ai and a to the first five harmonics of bi. All models were developed using 250 training points and 14 hidden neurons. 2 Testing Points 5 10 25 50 125 250 Average Testing Error (%) 0.80 0.74 0.68 0.68 0.72 0.72 Correlation Coefficient 0.99998 0.99997 0.99998 0.99998 0.99997 0.99997 144 i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 14.4. Average testing errors and correlation coefficients as functions of the number of hidden neurons for ANN models trained to map values from the first five harmonics of ai and a 2 to the first five harmonics of b 2. All models were developed using 250 training points and verified using 250 testing points. Hidden Neurons 1 2 4 6 8 10 12 14 16 18 20 Average Testing Error (%) 17.88 13.22 6.48 2.04 1.43 0.90 0.82 0.78 0.73 0.78 0.99 Correlation Coefficient 0.74320 0.91161 0.96659 0.99893 0.99951 0.99985 0.99989 0.99989 0.99992 0.99988 0.99983 Table 14.5. Average testing errors and correlation coefficients as functions of the number of training points for ANN models trained to map values from the first five harmonics of ai and a 2 to the first five harmonics of b 2. All models were developed using 14 hidden neurons and verified using 250 testing points. Training Points 5 10 25 50 125 250 Average Testing Error (%) 27.08 12.99 3.72 1.75 1.09 0.78 Correlation Coefficient 0.50237 0.91962 0.99628 0.99940 0.99978 0.99989 Table 14.6. Average testing errors and correlation coefficients as functions of the number of testing points for ANN models trained to map values from the first five harmonics of ai and a 2 to the first five harmonics of b 2. All models were developed using 250 training points and 14 hidden neurons. Testing Points 5 10 25 50 125 250 | I Average Testing Error (%) 0.87 0.84 0.81 0.80 0.81 0.78 Correlation Coefficient 0.99995 0.99993 0.99988 0.99989 0.99988 0.99989 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N 0sO 20 1 15- f 10 ce <D H <u ^ Model (ap a2 -> bj) I- Model (ap a2 -> b2) - .* * *. * 2 o > • 0 I 5 0 "T~ 15 10 20 Number o f Hidden Neurons Figure 14.12. Average testing errors as functions of the number of hidden neurons for ANN models trained to map ai and a2 to bi and ai and a2 to b2. The models were developed using 250 training points and verified using 250 testing points. 0X s g 30 - | 25- Model (ap a2 -> bj) Model (ap a2 -> b2) 20 15- CD H 8<D 10 f.• • k' - 5- V. 0 0 50 100 150 200 250 Number o f Training Points Figure 14.13. Average testing errors as functions of the number of training points for ANN models trained to map ai and a2 to bi and ai and a2 to b2. The models were developed using 14 hidden neurons and verified using 250 testing points. 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 .0 -1 go 0 .6 - .3 Model (ap a2 -> bj) —■* Model (ap a2 -> b2) <i 0 .0 0 50 100 150 200 250 Number o f Testing Points Figure 14.14. Average testing errors as functions of the number of testing points for ANN models trained to map ai and a 2 to bi and ai and a 2 to b 2. The models were developed using 14 hidden neurons and 250 training points. 14.3. Results and Comparison With the sensitivity analysis complete, we decided to use 250 training points and 250 testing points to train and verify the two ANN models. We chose to use 14 hidden neurons for mapping values from the first five harmonics of ai and a 2 to the first five harmonics of bi and 16 hidden neurons for mapping values from the first five harmonics of ai and a 2 to the first five harmonics of b 2. The testing error was 0.72% for the bi model and 0.73% and for the b 2 model, with respective correlation coefficients of 0.99997 and 0.99992. After the ANN models were developed, the nonlinear large-signal g>parameters, and 02Ui (k = 1, 2, ..., 5), were obtained by interpolating b u and 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. from measured results for nonzero values of an, an, ..., a n and # 21, # 22, •••> ^25 to the desired values for <212, <213, ..., <215 and #21, 0 -22 , ■■■, a 25 equal to zero. Figure 14.15 shows the interpolated value of 611 (= Shm -an) when <212, an, ..., an and ai\, 022, ..., <225 were set equal to zero, and Figure 14.16 shows the interpolated value of 621 (= ftm rtfn ) when <212, an, ■■■, an and a2 \, <222, ^25 were set equal to zero. We compared our results to a compact model provided by the manufacturer and simulated in commercial harmonic-balance software to get an independent check on our methodology. Our comparison was accomplished by providing the simulator with the identical biasing conditions on the diode and a stimulus of the same magnitude used in the measurements for a\\ and setting all other a ’s to zero. Providing the simulated circuit with <211 of the same magnitude as the measurement should give the same values of bn and b2k as the interpolated values of bn (= & m r« n ) and 62/t (= &2u rtfn ) determined by the ANN models when an, an, ■■■, an and <221 , a 22, ■■■, <225 are set equal to zero. Figures 14.15 and 14.16 show that the simulated values bn and 621 agree with those determined from the measurementbased ANN models. Quantitatively, the differences between the ANN and equivalentcircuit models are shown in Table 14.7. Table 14.7. Differences between the measurement-based, ANN-modeled results and the compact model simulated in commercial harmonic-balance software. Quantity ^1111 &1121 ^1131 ^1141 ^1151 Difference (%) 3.38 1.23 3.29 0.40 1.67 Quantity Difference (dBV) -44.5 -53.3 -44.8 -63.1 -50.6 &2111 &2121 &2131 &2141 &2151 Difference (%) 3.95 7.15 5.93 0.72 0.85 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Difference (dBV) -43.2 -38.0 -39.6 -57.9 -56.5 ! 0.2-| Im{6n}(V) • Measurements ■ ANN Model ▲ Compact Model 0 .1 - - 0.2 - 0.1 -0 . 1 - - 0 .2 -1 Figure 14.15. The 250 measurements of bn used for training (circles). Values of S iiirflu were determined from the measurement-based ANN model (square) and the harmonic balance simulation using a compact model (triangle). • Measurements ■ ANN Model ▲ Compact Model 0.1 r 2111 - 0.2 0.2 •• -0.2-1 Figure 14.16. The 250 measurements of 621 used for training (circles). Values of &2iirtfn were determined from the measurement-based ANN model (square) and the harmonic balance simulation using a compact model (triangle). 149 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 14.4. Discussion We described a method of extracting nonlinear large-signal ^-parameters, using an NVNA equipped with isolators and a second source. First, we showed how multiple measurements of a nonlinear circuit could be used to train artificial neural networks. Then, we extracted the desired ^-parameters by interpolating the ANN models for all a ’s equal to zero other than a\\. We checked our approach by comparing our results to a compact model simulated in commercial harmonic-balance software, and showed that the two methods agree well. We also performed a sensitivity analysis on the ANN networks, and discovered the following: (1) The average testing error decreases for an increasing number of training points. However, as more and more training points are added, diminishing returns on the testing errors are evident. (2) As the number of hidden neurons are increased, the average testing error decreases until around 14 hidden neurons at which point more hidden neurons have no benefit and can actually lead to increases in testing error. (3) The number of testing points does not drastically affect the testing error. In fact, no more than 25 testing points are needed for the models tested. 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 15 COMPARING NONLINEAR LARGE-SIGNAL 0-PARAMETERS WITH NONLINEAR SCATTERING FUNCTIONS Here, we compare the nonlinear large-signal ^-parameters, introduced previously, to another form of nonlinear mapping, known as nonlinear scattering functions, introduced by Verspecht [11, 12]. For a two-port nonlinear device, excited by a single-tone signal, and assuming all harmonic signals are relatively small compared to the fundamental signals, Verspecht defines nonlinear scattering functions as ^kp where ay F'kp kp X Z-i i= 1,2 ijJ) & kpij kpij "F ' X H kkpij p ij ,) > (15.1) i= \, 2 j= 2 ,...,M and bkp represent the wave variables proportional to the incoming and outgoing waves, respectively, and M refers to the number of harmonics being taken into account. Fkp, Gkpij, and Hkpy are functions of the fundamental components Re(an), Re(a 2i), and Im(tf2i). The imaginary component of an is omitted, with the assumption that the wave variables are phase referenced such that the phase of a n is set to zero. F ^, Gkpij, and H ^y are assumed complex constants for a given bias and fundamental drive condition. Note that these three terms do not depend upon the higher harmonic signal levels. With the ay wave variables split into real and imaginary components, Gkpij and Hkpij serve to map ay circles centered at zero to bkp ellipses with variable axes also centered at zero, as shown in Figure 15.1. The Fkp terms translate the ellipses about the complex plane. 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i 0 .5 - -i 0.5 T— - 1.0 -0.5 Re{«} -h 1.0 -0.5- ••••• b = G • Re(a) + H ■Im(a) 1 l.O-i 0.5- - 1.0 0.5 • -0.5 -0.5- • •••••' -1.0J Figure 15.1. Gkpij and Hkpij serve to map a# circles centered at zero to bkP ellipses with variable axes also centered at zero, neglecting Fkp for illustrative purposes. 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. j For illustrative purposes, let us consider bn, taking into account the first three harmonics. Doing this, eq. (15.1) reduces to b n = F n + I G,ijRe(as) + 2 «ii( Im(%) /= 1,2 7 = 2 ,3 (15-2) ;= 1,2 y = 2 ,3 or *11 = -*11 + ^ 1112 R e ( a 12 ) + **1 112 Im ( a 12 ) + * -h ll3 R e ( a 1 3 ) + *^1113 I m ( a 13) (15.3) + ^1122 Re(«22) + H \ \2 2 Im(<322) + ^1123 Re(a23) + ifn23 Im(a23) . If we now consider the nonlinear large-signal ^-parameter representation for bn, once again assuming a two-port network and taking into account the first three harmonics, we have *11 = Z &ijuaji O 5 -4 ) J = 1.2 1=2,3 or * 11 - ^ 1 1 1 1 ^ 1 1 + ^ 1 1 1 2 a 12 + ® 1 1 1 3 a 13 (15.5) + ^ 1 2 1 1 a 21 + ^ 1 2 1 2 ^ 2 2 + ^ 1 2 1 3 a 23 • I 2 Here, S>ijki are functions o f all of the harmonics, not just the fundamental terms. So for any change in any ap, a new set of &,#/ will need to be determined. Separating the real and imaginary components of the a ’s, we can express eq. (15.5) as 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R e ^ 0 +%uu [Re(^i2) + J lm(an )] + £ n 13[Re(tz13) + j Im<>13)] + £ 121j[Re(a21) + j Im(a21)] '11 - ^ 1 1 1 1 + j '1 2 1 2 (15.6) [Re(fl22) +7M «22 )] +^1213tRe(«23) +7Im0 23)] • The imaginary component of an is omitted once again, with the phase reference such that the phase of an is set to zero. We can now equate the nonlinear large-signal ^-parameters of eq. (15.6) to the nonlinear scattering functions of eq. (15.3), with the understanding that this is only generally valid for the special case when the nonlinear large-signal ^-parameters are constant for a given bias and fundamental drive level, like Fkp, Gkpij, and Hkpy are defined. Normally, however, the nonlinear large-signal ^-parameters are dependent upon the higher harmonics as well as the bias and fundamental drive level. The implication of this special case will be discussed shortly, after eqs. (15.3) and (15.6) are equated. Equating the corresponding real and imaginary components of the a wave variables in eqs. (15.3) and (15.6) gives Fn = &H JRe(an ) + ^ \ 2 \\a 2 \ • (15.7) 1 Additionally, *1112 1113 _ ^ 1 1 1 2 _ < A h 3 > 7 ^ 1 1 1 2 “ # 1 1 1 2 ; 7 ^ 1 1 1 3 - # 1 H 3 ’ (15.8) 5 (15.9) ^1212 - ^1122 » 7 ^ 1 2 1 2 ~ # 1 1 2 2 > and 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (15.10) '1 2 1 3 (15.11) _ ^1123 5 7 ^ 121 3 “ ^1123 • Eqs. (15.8) - ( 1 5 .1 1 ) im ply ^ 1 1 1 2 = ~ j H 1112 5 ^ 1 1 1 3 = ~ j H 1113 > ^ 1 1 2 2 = ~ J H 1122 > ^ 1 1 2 3 = ~ J H 1123 ’ (15.12) which means Re(Gkpij )= lm (H kf)ii) ; Re(Hkinj) = -lm (G kpij) . (15.13) Eq. (15.13) satisfies the conditions of the Cauchy-Riemann equations [66], 9 [ R e ( ^ ) ] _ 3 [Im (^ )] a[Re(ap)] which implies 3[Im (^)] 3[Re(^p)] ’ 3[Im(ap)] must be an analytic function of ay. _ d [Im (^ )] 3 [R e(^ )] ’ (15.14) A complex-valued function is said to be analytic on an open set W if it has a derivative at every point of W. This is only generally true when bkp is a linear function of ay. Thus, equating the nonlinear large-signal ^-parameters with the nonlinear scattering functions is only generally valid in the small-signal, linear case. As we mentioned earlier, eqs. (15.7) - (15.12) are only generally valid in the special case when the nonlinear large-signal ^-parameters are constant for a given bias and fundamental drive level, like Fkp, GkPy, and Hkpy are defined. Since this is not generally true, the formulations for nonlinear large-signal ^-parameters and nonlinear scattering functions are not equivalent. We can make at a few important conclusions, however, after attempting to equate the two formulations. First, if Gkpij and Hkpy are allowed to be functions of higher harmonics, then only one of them, either Gkpy or Hkpy, or equivalently &yu, is required since eq. (15.12) shows that they are not independent. Second, if the 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. nonlinear large-signal ^-parameters are complex constants for a given bias and fundamental drive level and are not functions of the higher harmonics, they have the limitation that they cannot map circles into ellipses, but rather can only map circles into circles, as shown in Figure 15.2. This is because I i s a single, complex constant rather than a pair of independent complex constants like G ^ j and Hkpjj. Thus, if Biju is not dependent upon higher harmonics, it acts like a linear ^-parameter. We have shown above that the two formulations are not equivalent. Nonlinear large-signal ^-parameters are more general than the nonlinear scattering functions, which are useful in approximating a specific class of nonlinearity in a more compact form. Nonlinear large-signal ^-parameters have the advantage of being able to map circles into any arbitrary shape, rather than being limited to ellipses. 156 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 0 .5 - -0.5 0.5 -0 .5 - **•••«!•§•. >•••••••** b = S ■a I 1.0 -, .......... 0 .5 - R e{b} - 1.0 -0.5 0.5 1.0 -0 .5 ••• -l.O-1 Figure 15.2. If &#*/ is a complex constant for a given bias and fundamental drive level, it has the limitation that it can only map circles into circles. 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 16 SUMMARY AND FUTURE WORK 16.1. ANN Modeling for Improved Vector Network Analyzer Calibrations 16.1.1. Summary In the first part of this thesis, we successfully applied ANNs to model onwafer and coaxial lumped-element calibration standards. For the on-wafer OSLT standards, we showed that a calibration using ANN-modeled standards compares favorably (less than a 0.02 difference in magnitude) to the benchmark multiline TRL calibration. In modeling these standards, we quantified the training errors and training times as functions of both the number of training points and the number of neurons in the hidden layer. We found that 5 neurons in the hidden layer of an MLP3 architecture and that fewer than 10 training points were sufficient to accurately model our standards. We then expanded upon our method of modeling on-wafer OSLT standards using ANNs by taking into account the load variations from wafer to wafer. Specifically, we modeled the correlation between measured DC resistance and RF variations in load terminations as well as the RF performances of the open, short, and thru standards. We showed that these modeled standards compare favorably (a difference of less than 0.04 in magnitude at most frequencies) to the benchmark multiline TRL calibration over a 66 GHz frequency range. We also used ANNs to successfully model on-wafer and coaxial LRM load standards. For the on-wafer case, we showed that an LRM calibration using an ANNmodeled load compares favorably to a benchmark multiline TRL calibration, with an average worst-case scattering parameter error bound of 0.017 over a 40 GHz 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. frequency range. For the coaxial case, we found the average worst-case error bound to be 0.024, an improvement over the equivalent circuit-modeled average worst-case bound of 0.034. In practice, ANN-modeled calibration standards can be easily implemented using existing or custom software packages. In our case, we utilized MultiCal [4], a free program developed by the National Institute of Standards and Technology, to perform our multiline TRL and LRM calibrations. The internal software on any commercial network analyzer can also be used if the user has confidence in another calibration method. Then, once the lumped-element standards are measured, one of a number of ANN programs may be used to model the standards. We used software developed by Zhang et al. [50] to construct our ANN models. For the OSLT experiments, we wrote custom software to perform the calibrations with exported ANN models, using the equations found in references [5] and [48]. We have shown that ANN models offer a number of advantages over using calibrated measurement data files or equivalent circuit models, namely: (1) They do not require the numerous details and parameters of physical models. (2) Calibration times can be reduced since only a few training points are required to accurately model the standards. (3) ANN model descriptions are much more compact than large measurement data files. (4) ANN models, trained on only a few measurement points, can be much more accurate than direct calibrations, when limited calibration data are available. (5) They give an optimized estimate in the presence of noise. 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6) ANN models are able to accurately model loads with measured DC resistances slightly outside their training range. 16.1.2. O ther Applications In addition to modeling calibration standards, ANNs have been used as a measurement-based modeling tool in a number of other RF and microwave applications. Recently, they have been developed as a means of transforming the measured reflection coefficients of liquids, using an open-circuited coaxial probe, to their respective permittivity values. Tuck and Coad implemented an ANN using training data from nine different liquids in the frequency range of 200 MHz to 16 GHz [67]. In another study, Bartley et al. implemented an ANN using training data from eleven liquids (different mixtures of water and isopropyl alcohol) in the frequency range of 200 MHz to 6 GHz [68], Both of these studies suggest that ANNs can be successfully applied to determine the dielectric properties of materials from uncalibrated reflection coefficient measurements made on an open-ended coaxial probe. Bartley et al. also applied an ANN to determine the moisture content in wheat from microwave transmission measurements [69], The ANN was trained for moisture contents between 10.6 and 19.2 %, which is referred to as “wet basis,” and bulk densities varying from 0.72 to 0.88 g/cm3. Measurements were made from free-space transmission-coefficient measurements on layers of wheat placed between two antennas connected to a VNA at eight frequencies ranging from 10-18 GHz. The ANN architecture consisted of one hidden layer with fifteen neurons. Sixteen inputs 160 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (the magnitudes and phases of each of the eight measured transmission coefficients) were used for the ANN model. The one output was moisture content. For the 179 sample measurements, Bartley et al. calculated the mean squared error between the ANN model and the measured results to be 0.028. Here, the authors point out that this method has the potential for on-line, nondestructive moisture content measurement for flowing grain. 16.1.3. Future Work ANNs have many potential applications in the area of RF and microwave measurement-based modeling [70]. For example, in the area of materials characterization, the most accurate way of measuring the complex permittivity of a low-loss dielectric material is by using one of a number of resonator methods [71]. Although usually limited to a single frequency, resonators provide the required accuracy that broadband methods lack. The disadvantage of most resonator techniques, however, is the need for accurately machined samples of the material of interest. Recently, Krupka et al. [72] introduced a new type of resonator that allows for nondestructive permittivity measurements, referred to as the split-post resonator. Briefly, the permittivity of a sample is determined from shifts in the quality factor and resonant frequency from measurements taken with and without the sample in place. Krupka et al. utilize the Rayleigh-Ritz method to theoretically determine the shift in resonant frequency and quality factor from the complex permittivity and sample thickness. Using a wide range of permittivities and sample thicknesses, they created a look-up table that is 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. used to calculate the sample permittivity from measured shifts in resonant frequency and quality factor for a given sample thickness. For values that are not explicitly listed in the table, they interpolate between the two closest values. Researchers at the National Institute of Standards and Technology (NIST) are currently developing an independent mode-matching model for the split-post resonator to compare with the method of Krupka et al. In a similar manner as described above, they will develop a model to determine the shift in resonant frequency and quality factor from the complex permittivity and sample thickness. But rather than creating a look-up table, they are interested in developing an ANN model, trained by data from the modematching technique, to provide the complex permittivity for a given sample thickness from measured shifts in resonant frequency and quality factor. The ANN will provide an efficient model for the transformation that would otherwise be time-consuming using the mode-matching computations each time. Another potential application of ANNs comes in the area of power measurement. The measurement of microwave power is a fundamental test requirement necessary for determining output levels of signal generators, transmitters, and radar, just to name a few. Invariably, commercial power sensors are ultimately traceable to measurements made by primary national standards laboratories, most of which utilize a water-bath microcalorimeter and a reference standard for coaxial and waveguide measurements. The reference standards are usually substitution-type bolometric power detectors, which use heat sensitive resistors terminating a transmission line that absorbs microwave power [73]. 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I \ I ! Not all the microwave energy incident on a power detector is absorbed by the bolometer element, however. Some of the power is dissipated in the connector, the transmission line, and the bolometer mounting structure, and some additional power is lost to leakage in the mount. These factors result in a dimensionless measurement error called the mount efficiency, which is always less than one. Additionally, the bolometer elements are not heated identically by the same amounts of RF and DC power. This is referred to as the RF-DC substitution error. The combination of these two errors is defined as the effective efficiency, which is independent of mismatch corrections. Customers who wish their devices to be directly traceable to a national standards laboratory submit their devices for calibration. After the measurements are complete, the customer receives a table of effective efficiencies for the frequencies measured. Since a typical measurement of 360 frequency points requires approximately 48 hours due to the time required for the thermopile to stabilize at each point, ANNs have the potential to lessen measurement times by reducing the number of frequency points. An ANN model, could in principle, be trained to interpolate the effective efficiency between a reduced set of measurements, especially for devices that have been previously measured and are known to behave correctly, without unpredictably large spikes that occur over a short frequency span. In addition to reducing measurement times for microcalorimeter measurements without noticeably degrading accuracy, ANNs can potentially be used in other areas where measurement times are lengthy, such as the determination of noise parameters. 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16.2. Frequency-Domain Models and Figures of Merit for Nonlinear Circuits 16.2.1. Summary In the second part of this thesis, we developed new frequency-domain models and figures of merit for nonlinear microwave circuits. We began by introducing a method for preserving time-invariant phase relationships when ratios are taken between two harmonically related signals by introducing a third signal that is used as a phase reference. We showed that a reference signal must be present at the fundamental frequency in order for time-invariant phase relationships to exist between ratios of any two harmonically related signals. Then, we introduced nonlinear large-signal scattering parameters, a new type of frequency-domain mapping that relates incident and reflected signals. Unlike classical 5-parameters, nonlinear large-signal ^-parameters take harmonic content into account and depend on the signal magnitudes. First, we presented a general form of nonlinear large-signal ^-parameters and showed that they reduce to classic Sparameters in the absence of nonlinearities. We also introduced nonlinear large-signal impedance (2) and admittance (|j) parameters, and presented equations for relating the different representations. Next, we made two simplifications, considering the cases of a one-port network and a two-port network, each with a single-tone excitation. For the one-port network, we showed that the equation relating and 2 reduces to the same well-known equation for the linear case assuming no power is transferred in the form of frequency down-conversion. For the two-port case, we extracted input reflection coefficients and forward transmission coefficients, which can be useful for designing circuits such as amplifiers and frequency multipliers. In 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. addition, we derived a quasi-linear approximation of the output reflection coefficient under normal operating conditions. These three two-port parameters allow a designer to “see” application-specific engineering figures of merit that are similar to what he or she is accustomed to in the linear world. Next, we expanded the definitions of power gain, transducer gain, and available gain by taking harmonic content into account, and showed that under special conditions, this generalized power gain can be expressed in terms of nonlinear large-signal scattering parameters. We provided an example showing how the expanded definitions of gain and nonlinear large-signal scattering parameters allow us to examine the behavior of a nonlinear pHEMT model by simply performing a harmonic balance simulation with all a’s other than a\\ forced to zero. Looking at the nonlinear large-signal scattering parameters gives us an in-depth view of the modeled behavior by letting us separate out the input reflection coefficients and transmission coefficients at each of the frequency components, while reducing the nonlinear large-signal scattering parameter data set into the compact expressions of power gain (S, transducer gain (Br, and the ratio (Si/<8, gives us a more concise view of the modeled behavior. We also illustrated how nonlinear large-signal ^-parameters can be used as a tool in the design process of a single-diode 1-2 GHz frequency-doubler. Specifically, we used 111 to determine the input matching network, #2222 to determine the output matching network, and n iki, (for k = 1 to 6), and ($2 to quantify the performance of the circuit at each stage. By the final stage of the design, we had 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. created a doubler with an overall power gain of -9.56 dB, not far from the maximum possible predicted value of -9 dB. For the case where a nonlinear model is not readily available, we described a method of extracting nonlinear large-signal ^-parameters, using an NVNA equipped with isolators and a second source. First, we showed how multiple measurements of a nonlinear circuit could be used to train artificial neural networks. Then, we extracted the desired ^-parameters by interpolating the ANN models for all o ’s equal to zero other than a\\. We checked our approach by comparing our results to a compact model simulated in commercial harmonic-balance software, and showed that the two methods agree well. We also performed a sensitivity analysis on the ANN networks, and discovered the following: (1) The average testing error decreases for an increasing number of training points. However, as more and more training points are added, diminishing returns on the testing errors are evident. (2) As the number of hidden neurons are increased, the average testing error decreases until around 14 hidden neurons at which point more hidden neurons have no benefit and can actually lead to increases in testing error. (3) The number of testing points does not drastically affect the testing error. In fact, no more than 25 testing points are needed for the models tested. Finally, we compared our nonlinear large-signal ^-parameters to another form of nonlinear mapping, known as nonlinear scattering functions. We showed that the two formulations are not equivalent. Nonlinear large-signal ^-parameters are more general than the nonlinear scattering functions, which are useful in approximating a specific class of nonlinearity in a more compact form. Nonlinear large-signal &- 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. parameters, however, have the advantage of being able to map circles into any arbitrary shape, rather than being limited to ellipses. 16.2.2. Future Work There are a number of future research topics that can be done to extend the work presented in the second half of this thesis. One idea is to examine whether the method of preserving time-invariant phase relationships when ratios are taken between two harmonically related signals can be generalized or modified to preserve consistent phase relationships when ratios are taken between two signals not harmonically related. In this case, the third signal would occur at a frequency that is a common factor of the first two, but may not be readily available for use as a reference. Such a method, if discovered, could be very useful for mixer applications. Another topic for further consideration is to examine how well the quasilinear approximation of the output reflection coefficient, derived in Chapter 10, can be measured using an NVNA equipped with two sources. This concept works very well in harmonic balance simulators, but it would be interesting to see how this parameter varies in the presence of noise and at various offset frequencies and power levels. To show how nonlinear large-signal scattering parameters can be used to design nonlinear circuits, it would be useful to provide more examples. Designing a high-efficiency amplifier, such as a Class E or Class F configuration, where harmonic 167 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I terminations are important, would provide a convincing argument for the use of these parameters. In order to make the ANN models we derived from measurements more useful, it would be beneficial if they could be incorporated into a commercial harmonic balance simulator. Currently, most nonlinear models that are imported into simulators are represented in the time-domain, but if a “hook” could be provided that would also allow frequency-domain models, these ANN models could be placed into the circuit of interest and used in the overall simulation, as well as in the optimization stage of the design process. The measurement set-up we presented for extracting nonlinear large-signal 1S>parameters was shown to be useful in diode circuits, but in the case of a transistor, where there can be more isolation between ports 1 and 2, this may not be adequate. A third source, placed at port 1, and possibly other additional hardware, may be required to simultaneously cover the origins of the a\k waves, while the second source, located at port 2, covers the origins of the au waves. Further investigation will be required to answer this question. These are just a few of the many research topics that can be explored in this area. In the broader sense of measurement-based modeling, uncertainty statements on the measured o ’s and V s will be required for the designer to fully trust and validate his or her models. 168 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY 1. G. F. Engen and C. A. Hoer, Thru-reflect-line: an improved technique for calibrating the dual six-port automatic network analyzer, IEEE Trans Microwave Theory Tech 27 (1979), 987-993. 2. B. Bianco, M. Parodi, S. Ridella, and F. 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