# Automated matching control system using load estimation and microwave characterization

код для вставкиСкачатьAUTOMATED MATCHING CONTROL SYSTEM USING LOAD ESTIMATION AND MICROWAVE CHARACTERIZATION By JAESEOK KIM A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 1 UMI Number: 3370928 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ______________________________________________________________ UMI Microform 3370928 Copyright 2009 by ProQuest LLC All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. _______________________________________________________________ ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106-1346 c 2008 Jaeseok Kim ? 2 To my family 3 ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisor, Professor William R. Eisenstadt, for his invaluable advice, encouragement, and support. This dissertation would not have been possible without his guidance and support. My deep recognition goes to Professor Kenneth O, Professor John G. Harris, and Professor Gloria J. Wiens for serving on my supervisory committee and for their valuable suggestions. Many thanks go to Mr. Larry Luce from Freescale Semiconductor for their valuable input and generous funding for this research. Thanks also go to my colleagues in the Electronic Circuits Laboratory (ECL) for their discussion of ideas and years of friendship. Last but not least, I owe a special debt of gratitude to my family. Without their sel?ess love and support, I cannot imagine what I would have achieved. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 CHAPTER 1 2 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1 1.2 1.3 1.4 . . . . 13 14 15 16 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 . . . . . . . . . . . . 17 18 19 20 21 22 23 24 26 28 28 30 AUTOMATIC MATCHING CONTROL . . . . . . . . . . . . . . . . . . . . . . 34 3.1 3.2 34 35 36 36 39 39 39 41 47 2.2 2.3 3 3.3 3.4 Why Automatic Impedance Matching? . . . . . . . . Maximum Power Transfer on Impedance Matching . Challenges of Automatic Impedance Matching . . . . Proposed Solution: Automatic Matching Control with Impedance Matching Networks with Lumped 2.1.1 L-type Matching Network . . . . . . . 2.1.2 ?-type Matching Network . . . . . . 2.1.3 T-type Matching Network . . . . . . 2.1.4 Bandwidth of Matching Network . . . Pattern Recognition . . . . . . . . . . . . . 2.2.1 The k -Nearest Neighbor Classi?er . . 2.2.2 Arti?cial Neural Network (ANN) . . . 2.2.3 Principal Component Analysis (PCA) Multi-Port Re?ectometers . . . . . . . . . . 2.3.1 Six-Port Re?ectometer . . . . . . . . 2.3.2 Four-Port Multistate Re?ectometer . Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview . . . . . . . . . . . . . . . . . . . . . . . System Overview . . . . . . . . . . . . . . . . . . . 3.2.1 Impedance Matching Tuner . . . . . . . . . . 3.2.2 Controller . . . . . . . . . . . . . . . . . . . 3.2.3 Search algorithm . . . . . . . . . . . . . . . Experimental Results . . . . . . . . . . . . . . . . . 3.3.1 Characterization of Automatic Tuner System 3.3.2 Measurement Results . . . . . . . . . . . . . Conclusion and Discussion . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Load Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 LOAD IMPEDANCE ESTIMATION . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1 4.2 . . . . . . 52 53 53 54 57 58 COUPLER-FREE LOAD ESTIMATION USING THREE-PORT EFLECTOMETER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 4.4 5 5.1 5.2 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AUTOMATIC MATCHING CONTROL USING LOAD ESTIMATION . . . . . 94 Overview . . . . . . . . . . . . . . . . Matching Control Procedures . . . . . Characterization of Matching Network Bias Search for Impedance Matching . Characterization Results . . . . . . . . Impedance Matching Results . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 81 84 87 88 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 6.2 6.3 6.4 6.5 6.6 Overview . . . . . . . . . . . . . . . . . . . . Multistate Re?ectometers . . . . . . . . . . . Tunable Matching Network . . . . . . . . . . Load Estimation for Multistate Re?ectometer Experimental Results . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation . . . . . . . . . . . . . 80 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Overview . . . . . . . . . . . . . . . . . . . . . . . . System Overview . . . . . . . . . . . . . . . . . . . . High Impedance Probe . . . . . . . . . . . . . . . . . 5.3.1 Least Square Fitting . . . . . . . . . . . . . . 5.3.2 Arti?cial Neural Network . . . . . . . . . . . . Load Estimation Methods . . . . . . . . . . . . . . . 5.4.1 Radical Center . . . . . . . . . . . . . . . . . . 5.4.2 Least Square Fitting . . . . . . . . . . . . . . Experimental Results . . . . . . . . . . . . . . . . . . 5.5.1 High Impedance Probe Estimation . . . . . . . 5.5.2 Load Estimation . . . . . . . . . . . . . . . . . 5.5.3 Comparison of Coupler and Coupler-Free Load Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . THREE PORT AND FOUR PORT REFLECTOMETERS . . . . . . . . . . . . 5.6 8 . . . . . . 59 60 63 64 65 65 66 67 68 68 69 73 79 5.5 7 . . . . . . . . . . (AMC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 6 Overview . . . . . . . . . . . . . . System Overview . . . . . . . . . . 4.2.1 Automatic Matching Control 4.2.2 Load Estimation Method . . Experimental Results . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 95 97 100 101 106 110 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 APPENDIX 6 A DERIVATION OF LOAD IMPEDANCE CIRCLE EQUATION USING INPUT IMPEDANCE MAGNITUDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 B THREE-PORT AND FOUR-PORT MULTISTATE REFLECTOMETERS . . . 121 C STANDARD AND MIXED-MODE S-PARAMETER TRANSFORMATION . . 124 D DERIVATION OF DIFFERENTIAL INPUT REFLECTION COEFFICIENT . 126 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7 LIST OF TABLES Table page 3-1 Speci?cation of the matching tuner . . . . . . . . . . . . . . . . . . . . . . . . . 37 3-2 Mismatched load speci?cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3-3 Comparison of brute-force, greedy, and single-step algorithms . . . . . . . . . . 46 3-4 Percentage of catastrophic case for brute-force, greedy, and single-step algorithms 47 3-5 Comparison of brute-force, greedy, and single-step algorithms avoiding catastrophic cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4-1 Summary of mismatched load estimation results . . . . . . . . . . . . . . . . . . 57 5-1 Summary of heavily and slightly mismatched loads . . . . . . . . . . . . . . . . 73 5-2 Comparison of coupler and coupler-free load estimation in term of mean square error (MSE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6-1 Summary of mismatched loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7-1 Mean square error (MSE) of neural network ?tting models using training and testing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7-2 Average error of closed-form models . . . . . . . . . . . . . . . . . . . . . . . . . 106 7-3 Mapping table between bias voltage and a mismatched load to be matched . . . 107 7-4 Inverse mapping table between a mismatched load to be matched and bias voltage107 7-5 Impedance matching results using coupler-free load estimation and S-parameter measurement data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7-6 Impedance matching results using three-port re?ectometer load estimation and S-parameter measurement data . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7-7 Impedance matching results using coupler-free load estimation and S-parameters estimated by neural network models . . . . . . . . . . . . . . . . . . . . . . . . 109 7-8 Impedance matching results using three-port re?ectometer load estimation and S-parameters estimated by neural network models . . . . . . . . . . . . . . . . . 110 8 LIST OF FIGURES Figure page 1-1 Network matching an arbitrary load impedance to a transmission line . . . . . . 14 1-2 Maximum power delivered when source and load impedances are matched. . . . 14 1-3 Automatic matching control (AMC) . . . . . . . . . . . . . . . . . . . . . . . . 16 2-1 The L section matching networks . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2-2 The ? section matching network . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2-3 The T section matching network . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2-4 Typical pattern recognitions system diagram . . . . . . . . . . . . . . . . . . . . 22 2-5 Nearest-neighbor rule leads to a partitioning of the input space into Voronoi cells 23 2-6 Multilayer perceptron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2-7 Arti?cial neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2-8 Sigmoidal nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2-9 Principal component analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2-10 Six-port re?ectometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2-11 Determination of ?L from the radical center of three circles . . . . . . . . . . . . 31 2-12 Four-port re?ectometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3-1 Automatic matching control (AMC) system diagram . . . . . . . . . . . . . . . 35 3-2 Recon?gurable ?ve-stub matching tuner . . . . . . . . . . . . . . . . . . . . . . 37 3-3 Comparison of simulation and measurement of matching tuner. The matching tuner is tuned with typical bias (2.56 2.56 2.56) and matched load . . . . . . . . 38 3-4 Mismatched load 1 measurement (S11 = 0.11 + j0.09 at 3.5 GHz). Motor positions are (16725, 2262, 5000). Re?ection coe?cient |S11 |=0.14. . . . . . . . . . . . . . 40 3-5 Mismatched load 2 measurement (S11 = 0.19 + j0.40 at 3.5 GHz). Motor positions are (17105, 1424, 5000). Re?ection coe?cient |S11 |=0.45. . . . . . . . . . . . . . 41 3-6 Mismatched load 3 measurement (S11 = -0.11 + j0.22 at 3.5 GHz). Motor positions are (17722, 2005, 5000). Re?ection coe?cient |S11 |=0.24. . . . . . . . . . . . . . 41 3-7 Matching capability of ?ve-stub matching tuner at 3.5 GHz 9 . . . . . . . . . . . 42 3-8 Matching tuner measurement with matched load. Typical and automated biases are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V) . . . . . . . . . . . . . . . . . 44 3-9 Matching tuner measurement with mismatched load 1. Typical and automated biases are (2.56V 2.56V 2.56V) and (3.2V 3.86V 3.52V) . . . . . . . . . . . . . . 45 3-10 Matching tuner measurement with mismatched load 2. Typical and automated biases are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V) . . . . . . . . . . . . . . 49 3-11 Matching tuner measurement with mismatched load 3. Typical and automated biases are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V) . . . . . . . . . . . . . . 50 3-12 Optimization surface of greedy algorithm . . . . . . . . . . . . . . . . . . . . . . 51 4-1 System diagram of load impedance estimation for automatic impedance matching 53 4-2 Radical center of three circles, ?L1 , ?L2 , and ?L3 . . . . . . . . . . . . . . . . . . 56 4-3 Impedance matching procedure using estimated load impedance. . . . . . . . . . 56 5-1 Automatic matching control (AMC) . . . . . . . . . . . . . . . . . . . . . . . . 60 5-2 Three port re?ectometer integrated with a high impedance probe A) System diagram with schematic B) Fabrication on FR4 board . . . . . . . . . . . . . . . 61 5-3 Varactor SMV1405 capacitance versus reverse voltage . . . . . . . . . . . . . . . 62 5-4 Varactor SPICE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5-5 Radical center of three circles, ?L1 , ?L2 , and ?L3 . . . . . . . . . . . . . . . . . . 67 5-6 Least square nonlinear ?tting of high impedance probe model . . . . . . . . . . 70 5-7 Arti?cial neural network of high impedance probe model . . . . . . . . . . . . . 71 5-8 High impedance probe estimation error distribution . . . . . . . . . . . . . . . . 72 5-9 Load estimation using estimated S11 from high impedance probe . . . . . . . . . 74 5-10 Coupler and coupler-free load estimation with mismatched #1 . . . . . . . . . . 75 5-11 Coupler and coupler-free load estimation with mismatched #2 . . . . . . . . . . 76 5-12 Coupler and coupler-free load estimation with mismatched #3 . . . . . . . . . . 77 5-13 Coupler and coupler-free load estimation with mismatched #4 . . . . . . . . . . 78 6-1 Re?ectometers A) Three-port B) Four-port . . . . . . . . . . . . . . . . . . . . . 81 6-2 Schematic of a lumped power divider. Port 3 is a coupled port, which can be used as a reference port. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 10 6-3 Recon?gurable three-port matching network A) Schematic B) Implementation on FR4 board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6-4 Tunable element impedance with a bias from 0 V to 10 V. A varactor is in parallel and in series with an inductor. и and О denote S11 and S21 , respectively. A) In series with 3.3 nH B) In parallel with 1.8 nH . . . . . . . . . . . . . . . . . . . . 86 6-5 Matching capability of three-port re?ectometer at 2.4 GHz . . . . . . . . . . . . 91 6-6 The q-point distribution of three-port re?ectometer at 2.4 GHz . . . . . . . . . . 92 6-7 Multistate re?ectometer estimation using estimated S11 from high impedance probe A) Small mismatch (MSE=0.09) B) Large mismatch (MSE=0.80) . . . . 93 7-1 Automatic matching control supports load estimation. . . . . . . . . . . . . . . 97 7-2 Neural network models for a 2-port matching network were trained by backpropagation. A) S11 B) S21 C) S12 D) S22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7-3 Neural network models for a 2-port matching network were tested by 10% of measurement data. A) S11 B) S21 C) S12 D) S22 . . . . . . . . . . . . . . . . . . 104 7-4 Closed form models for a 2-port matching network were trained by nonlinear least square ?tting. A) S11 B) S21 C) S12 D) S22 . . . . . . . . . . . . . . . . . . 105 7-5 S-parameters from vector measurement and neural network model were compared in terms of coupler-free load estimation assisted matching control. A) Vector measurement by large mismatch B) Neural network model by large mismatch C) Vector measurement by small mismatch D) Neural network model by small mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7-6 S-parameters from vector measurement and neural network model were compared in terms of re?ectometer load estimation assisted matching control. A) Vector measurement by large mismatch B) Neural network model by large mismatch C) Vector measurement by small mismatch D) Neural network model by small mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 11 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Ful?llment of the Requirements for the Degree of Doctor of Philosophy AUTOMATED MATCHING CONTROL SYSTEM USING LOAD ESTIMATION AND MICROWAVE CHARACTERIZATION By Jaeseok Kim December 2008 Chair: William R. Eisenstadt Major: Electrical and Computer Engineering The automation of the impedance matching of radio frequency (RF) ports enables the test engineer to compensate the undesired e?ects, which are not uncommon in RF systems and make the impedance matching iterative, time-consuming, and an empirical process. Numerous recon?gurable matching networks have been presented for automated matching. However, the automation still relies on an iterative control to achieve a matching goal, because it lacks the knowledge of the RF target and the matching network. Our goals were to develop an automatic matching control system that uses this knowledge to set the impedance matching in a non-iterative fashion and to develop a method to extract circuit parameters systematically while keeping the additional necessary parts to a minimum. To achieve this goal, we use the principles of a re?ectometer to extract knowledge of the RF target and various microwave modeling methods to characterize the matching network. Our results demonstrate the proposed ideas and include an automatic matching control using a tunable microstrip bandpass ?lter, a load estimation technique using the microstrip ?lter, a new lumped matching network for the automatic matching in embedded RF testing, and a new matching control algorithm using the load estimation. 12 CHAPTER 1 INTRODUCTION 1.1 Why Automatic Impedance Matching? The design process for radio frequency (RF) systems often starts with the de?nition of the interface between the smaller components, where the impedance matching is critical for the following reasons. ? ? ? ? Deliver maximum power between components Improve the signal-to-noise ratio (SNR) Reduce amplitude and phase errors Provide isolation between circuit stages However, the actual implementation of the impedance matching is impaired easily by a deviation from the typical component value used during the design process. It is critical to be able to evaluate this deviation when selling a component commercially, so techniques to create and evaluate matching are important for testing RF production ICs. For example, the impedance of radio frequency (RF) ports in a device under test (DUT) to a loadboard is poorly de?ned due to unwanted e?ects, such as pogo pin connections, socket parasitics, and manufacturing variation. In addition, the input impedance of a handheld device antenna, especially the reactance part of the impedance, is varying as the environment around the device is changing. The unwanted e?ects and varying environment are not uncommon in RF systems and the matching of a poorly de?ned impedance is di?cult, slow, and expensive, during part test. A recon?gurable matching network can be used to set the impedance matching of RF match targets against the deviation caused by the unwanted e?ects. The recon?guration is conducted by a closed-loop feedback control, a so-called automatic matching control, which senses the impedance mismatch and determines proper component values for the impedance matching. 13 Z0 Matching network Load ZL Figure 1-1. Network matching an arbitrary load impedance to a transmission line ZS VS I ZL VL Figure 1-2. Maximum power delivered when source and load impedances are matched. In this thesis, we will develop a recon?gurable matching network and a matching control algorithm for an automatic matching control (AMC) system, which automatically sets the impedance matching of RF match targets. 1.2 Maximum Power Transfer on Impedance Matching The basic idea of the impedance matching is illustrated in Figure 1-1. A two-port network is placed between an arbitrary load (ZL ) and a transmission line (Z0 ). The load impedance ZL is converted to be matched with the source impedance Z0 . Impedance matching enables the delivery of maximum power to the load, although there are multiple re?ections between the matching network and the load. Maximum power delivery is explained as follows. In Figure 1-2, AC power is being transferred from the source, with phasor magnitude voltage |VS | (peak voltage) and ?xed source impedance ZS , to a load with impedance ZL , resulting in a phasor magnitude current |I|. |I| is simply the source voltage divided by the total circuit impedance. |I| = |VS | |ZS + ZL | (1?1) The average power PL dissipated in the load is the square of the current multiplied by the resistive portion (the real part) RL of the load impedance. PL = 2 Irms RL 1 1 = |I|2 RL = 2 2 ( |VS | |ZS + ZL | )2 14 RL = 1 |VS |2 RL 2 (RS + RL )2 + (XS + XL )2 (1?2) The denominator is minimized by making XL = ?XS (1?3) The power equation is reduced to PL = 1 |VS |2 RL 2 (RS + RL )2 (1?4) Similarly, the power is maximized by making RL = RS (1?5) Therefore, two conditions, Equation 1?3 and 1?5, are combined and written as complex conjugate matching condition. ZL = ZS? (1?6) where * denotes a complex conjugate. The complex conjugate matching is the matching goal of the proposed automatic matching control. 1.3 Challenges of Automatic Impedance Matching Recon?gurable matching networks have been proposed for the automatic matching control (AMC). The proposed networks were designed using various tuning elements such as varactors [1][2][3][4], CMOS switches [5], p-i-n diodes [6][7], or MEMS switches [8]. The recon?gurable matching network is adapted by a matching control algorithm which determines component values by the iteration of a closed-loop feedback. When RF match targets are not precisely de?ned, ?nding necessary matching components becomes highly iterative, time consuming, and an empirical process. Even if they are found for a certain RF match target, they seldom provide the impedance matching to other targets. An improvement on the impedance matching control algorithm has been achieved by using heuristic approaches [6]. Although the presented algorithms can converge faster than a brute-force approach, they may still get trapped in a local minimum which is one of typical optimization problems. Further, the iterative nature of the algorithms makes 15 Source Z0 Coupler Reconfigurable matching network Reflection coefficient detector Match decision and biasing circuit Load ZL Figure 1-3. Automatic matching control (AMC) convergence time to achieve the impedance matching increasing proportionally to the complexity of the matching network. For embedded RF testing, lumped elements are preferred to distributed elements due to the compact size at the frequency equal to or less than 2.4 GHz. Most matching control assumes that a degree of mismatch is measured through a directional coupler, which is too large to be embedded. So far the challenging problems of the automatic matching control have been described. Next section we will propose solutions to them. 1.4 Proposed Solution: Automatic Matching Control with Load Estimation A proposed solution to the challenges of the impedance matching is an automatic matching control (AMC) supported by a load impedance estimation. As shown in Figure 1-3, the automatic matching control recon?gures the matching network through the closed-loop feedback consisting of a coupler, a re?ection coe?cient detector, and a match decision circuit. An estimate of a load re?ection coe?cient helps the matching control to achieve an impedance match without trying di?erent biases iteratively. This thesis focuses on developing recon?gurable matching networks and matching control algorithms with load estimation especially for embedded RF testing. 16 CHAPTER 2 BACKGROUND 2.1 Impedance Matching Networks with Lumped Elements When circuit dimensions are not small relative to the wavelength, the phase change from one point to another in the circuit is not negligible. In this case, the equivalent voltage and current waves along a transmission line are expressed as the sum of the incident and re?ective waves, given by V (z) = V + e???z + V ? e+??z (2?1) V + ???z V ? +??z ? e e Z0 Z0 (2?2) I(z) = where Z0 and ? are the characteristic impedance and the propagation constant of the transmission line, respectively. If the transmission line is terminated with other than its characteristic impedance Z0 , the re?ection happens as a result of discontinuities. The re?ection coe?cient ? is de?ned as the ratio of the incident to the re?ected wave along a transmission line, given by V ? +?2?z V ? e+??z ?(z) = + ???z = + e V e V (2?3) If the transmission line is terminated with a load impedance ZL , the load re?ection coe?cient and the load impedance are written as ?L = ?(z) V (z) ZL = I(z) = Z0 z=0 = z=0 V? V+ V++V? 1 + ?L = Z0 + ? V ?V 1 ? ?L (2?4) (2?5) The load re?ection coe?cient can be also expressed in terms of ZL and Z0 as ?L = ZL ? Z0 ZL + Z 0 17 (2?6) The re?ection coe?cient ? is a complex number that describes both the magnitude and the phase shift of the re?ection. The simplest cases, when the imaginary part of ? is zero, are ? ? ? ? = ?1 ?= 0 ? = +1 Maximum negative re?ection, when short-circuited. No re?ection, when perfectly matched. Maximum positive re?ection, when open-circuited. The voltage standing wave ratio (VSWR), which represents the degree of re?ection in another way, is de?ned as the ratio of the maximum to the minimum magnitude of the voltage wave. The voltage standing wave ratio can be expressed in terms of the re?ection coe?cient given by VSWR = |V (z)|max 1 + |?L | |V + ||e???z + ?L e+??z |max = = + ???z +??z + ?L e |min |V (z)|min |V ||e 1 ? |?L | (2?7) Conversely, the re?ection coe?cient can be obtained from the measurement of voltage standing wave ratio along the transmission line. |?L | = VSWR ? 1 VSWR + 1 (2?8) In a traditional vector network analyzer, a complex re?ection coe?cient is calculated from the measurement of incident and re?ected wave powers. If the phase of a re?ection coe?cient is not necessary, e.g., the degree of mismatch, the magnitude of a re?ection coe?cient can be measured by a VSWR detector. 2.1.1 L-type Matching Network The simplest matching network is a L-type network with two lumped elements. The values of the lumped elements can be found through the analytic solution as follows. The impedance seen looking into the matching network shown in Figure 2-1A should be equal to Z0 for impedance matching: Z0 = ?X + 1 ?B + 1/(RL + ?XL ) 18 (2?9) ?X ?X Z0 ZL ?B Z0 A ?B ZL B Figure 2-1. The L section matching networks where the load impedance ZL = RL + ?XL . Rearranging and separating into real and imaginary parts gives two equations given by B(XRL ? XL Z0 ) = RL ? Z0 (2?10) X(1 ? BXL ) = BZ0 RL ? XL (2?11) The solution for B and X are given by ? ? XL ▒ RL /Z0 RL2 + XL2 ? Z0 RL B= RL2 + XL2 X= 1 XL Z0 Z0 + ? B RL BRL (2?12) (2?13) Note that the solutions exist only when RL > Z0 . Similarly, the solution for Figure 2-1B is given by ? (Z0 ? RL )/RL B=▒ Z0 ? X = ▒ RL (Z0 ? RL ) ? XL (2?14) (2?15) Note that the solutions exist only when RL < Z0 . 2.1.2 ?-type Matching Network The L-type network cannot provide an impedance match of every point on the Smith chart. By adding the third element, the whole Smith chart can be covered. The third element added to the L-type networks results in either ?- or T-type network. In this section, the analysis of the ?-type network is presented as follows. 19 ?X Z0 ?B1 ?B2 ZL Figure 2-2. The ? section matching network ?X1 ?X2 ?B Z0 ZL Figure 2-3. The T section matching network The impedance seen looking into the matching network shown in Figure 2-2 should be Z0 . 1 = ?B1 + Z0 1 1 ?X + ?B2 + 1/(RL + ?XL ) (2?16) Rearranging and separating into real and imaginary parts gives two equations given by B1 Z0 (X + XL ) ? B1 Z0 B2 XXL ? B2 (XRL ? XL Z0 ) = Z0 ? RL (2?17) X(1 ? B2 XL ) = (B1 + B2 )Z0 RL ? XL ? B1 B2 XZ0 RL (2?18) There are three variables, X, B1 , and B2 , de?ned by two equations. Hence, there exist multiple solutions satisfying the impedance matching. For a uniquely determined solution, the quality factor, which determines the bandwidth of matching networks, can be used as one of design speci?cations. 2.1.3 T-type Matching Network Similar to the ?-type network, a T-type network is analyzed as follows. The impedance seen looking into the matching network shown in Figure 2-3 should be Z0 . Z0 = ?X1 + 1 ?B + 1/(RL + ?(X2 + XL )) 20 (2?19) Rearranging and separating into real and imaginary parts gives two equations given by 2.1.4 BZ0 (X2 + XL ) ? BX1 RL = Z0 ? RL (2?20) X1 (1 ? B(X2 + XL )) = BZ0 RL ? (X2 + XL ) (2?21) Bandwidth of Matching Network In a resonant circuit, the ratio of its resonant frequency f0 to its bandwidth is known as the loaded Q of the circuit. QL = ?0 f0 = Bandwidth in radian Bandwidth in Hertz (2?22) The matching networks are used for an impedance match at a certain frequency. The frequency response of the networks can be classi?ed as either a low-pass or a high-pass ?lter. At each node of the networks, there is an equivalent series input impedance, denoted by Rs + ?Xs . Hence, a circuit node Q, denoted by Qn , can be de?ned at each node as Qn = |Xs | Rs (2?23) If the equivalent parallel input admittance at the node is Gp + ?Bp , the circuit node Q can be expressed in the form Qn = |Bp | Gp (2?24) For a narrowband range of frequencies around f0 , a matching network can be viewed as a bandpass ?lter. Hence, the loaded Q of the bandpass ?lter is given by QL = ?0 |BT | RT = ?0 RT CT = = BW GT |XT | (2?25) If the matching network holds the complex conjugate matching at each node, Rs1 = Rs2 and Xs1 = ?Xs2 . Rs1 and Rs2 are the resistance seen looking into a source and a load at the node. Xs1 and Xs2 are the reactance seen looking into a source and a load at the 21 node. The loaded Q can be written as the ratio of the resistance to the reactance. QL = Rs1 ||Rs2 Qn Rs1 ||Rs2 = = |Xs1 | |Xs2 | 2 (2?26) If multiple internal nodes exist such as T or ? networks, the QL of the matching circuits is the half value of the highest Qn in the circuits. In general, higher values of QL can be obtained using matching circuits with more elements. For example, adding one element to a L network results in either a T network or a ? network, which has higher values of QL than a L network. When a bandwidth of matching networks is a design consideration, L networks are not suitable because the QL of the networks is ?xed with the complex conjugate matching condition. Instead, the higher order ladder networks can have either higher or lower QL depending on the highest Qn of the networks. 2.2 Pattern Recognition Most pattern recognition systems are partitioned into four components such as preprocessing, feature extraction, classi?cation, and postprocessing, as shown in Figure 2-4. Preprocessing simpli?es subsequent operations without losing relevant information, feature extraction measures useful properties for classi?cation, classi?cation assigns the extract feature into a category, and postprocessing uses the classi?cation results to decide on the recommended action. Preprocessing Feature Extraction Classification Postprocessing Figure 2-4. Typical pattern recognitions system diagram 22 Voronoi edge Voronoi vertex v Figure 2-5. Nearest-neighbor rule leads to a partitioning of the input space into Voronoi cells The microwave components have been successfully modeled by the pattern recognition techniques thanks to their adaptability to complex phenomena [9]. Applying the pattern recognition techniques to the proposed matching control becomes a new research opportunity for the advancement of the automated RF system. In this section, two classi?ers, a nearest neighbor classi?er and a neural network, and principal component analysis (PCA), one of popular preprocessing methods, are introduced and described. 2.2.1 The k -Nearest Neighbor Classifier The task of the classi?er is to use the feature vector provided by the feature extractor to assign the object to a category. The nearest-neighbor rule for classifying a test point x is to assign it the label associated with the training point x? nearest to it. This rule allows us to partition the feature space into cells consisting of all points closer to a given training point x? ? a so-called Voronoi tessellation of the space, as shown in Figure 2-5 An obvious extension to of the nearest-neighbor rule is the k-nearest-neighbor rule. A decision is made by examining the labels on the k nearest neighbors and taking a vote. 23 P P Input Layer P f () P f () P f () f () f () Hidden Layer Output Layer Figure 2-6. Multilayer perceptron 2.2.2 Artificial Neural Network (ANN) Arti?cial neural networks (ANNs) are distributed, adaptive, generally nonlinear learning machines built from many di?erent processing elements (PEs) [10]. As a nonlinear statistical data modeling tool, neural networks are used to model complex relationships between inputs and outputs or to ?nd patterns in data. One of the most popular and powerful neural network designs is known as a multilayer perceptron. A multilayer perceptron consists of a set of neurons interconnected by weighted connections, as illustrated in Figure 2-6. There are an input layer, one or more hidden layers, and an output layer. As shown in Figure 2-7, each neuron has weighted coe?cients wl,j,m that are adjusted to train the algorithm, which are linearly combined, then passed through a nonlinear activation function f . The multilayer perceptron can be represented as yk = f ( ? ( w2,j,k f j ? i 24 )) w1,i,j xi (2?27) wl,j,0 wl,j,1 wl,j,2 P wl,j,3 vl,j yl,j f (и) wl,j,ml?1 Layer l, Neuron j Figure 2-7. Arti?cial neuron 1 0.8 f(x) 0.6 0.4 0.2 0 -10 -5 0 x 5 10 Figure 2-8. Sigmoidal nonlinearity where f is the activation function (typically, nonlinear sigmoidal function), wl,j,k is a weight from the kth node of the (l-1)th layer to the jth node of the lth layer, xi and yk denote the ith input and the kth output node, respectively. The activation function f () provides the nonlinearity necessary for the neural network. In general, any monotonically increasing function can be used. One commonly used function is the sigmoidal nonlinearity, de?ned by f (x) = 1 1 + e?x 25 (2?28) and plotted in Figure 2-8. This function has the property that the derivative is easy to compute. f ? (x) = f (x)(1 ? f (x)) (2?29) In training the multilayer perceptron, a supervised training algorithm is used, in which a set of known input/output data combinations are presented to the network. Using a backpropagation algorithm, the network is trained so that the network output matches as closely as possible the desired output, for each input data point. 2.2.3 Principal Component Analysis (PCA) Tight correlation among features makes di?cult the training of neural networks. Principal component analysis is used to decorrelate features by representing data along the direction with the largest variance and is often used together with neural networks. How feature vectors can be represented in terms of principal components is described in this section. The problem begins with a idea of how a single vector x0 represents all vectors in a set of n d-dimensional samples x1 , x2 , и и и , xn . One of solutions is to ?nd a vector x0 such that the sum of the squared distances between x0 and the various xk is as small as possible. The squared-error criterion function J0 (x0 ) is de?ned as J0 (x0 ) = n ? ||x0 ? xk ||2 (2?30) k=1 and is used to seek the value of x0 to minimize J0 . The solution is the sample mean m, de?ned as 1? xk m= n k=1 n (2?31) However, the sample mean is zero-dimensional representation of the data set. It is more interesting to get one-dimensional representation by projecting the data onto a line running through the sample mean. The line equation is written as x = m + ae 26 (2?32) where e is a unit vector in the direction of line and the scalar a can take any real value. If xk is represented by m + ak e, then an optimal set of coe?cient ak can be obtained by minimizing the squared-error criterion function. The squared-error criterion function is written as J1 (x) = J1 (a1 , a2 , и и и , an , e) = n ? ||(m + ak e) ? xk ||2 k=1 = n ? a2k ||e||2 ? 2 n ? k=1 ak eT (xk ? m) + k=1 n ? ||xk ? m||2 (2?33) k=1 The derivative of J1 with respect to ak is set to zero and the optimal set of coe?cient ak is given by n ? ?J1 =2 (ak ||e||2 ? eT (xk ? m)) = 0 ?ak k=1 (2?34) ak = eT (xk ? m) (2?35) where the unity vector ||e||2 = 1. By substituting the solution ak into J1 , J1 is written as J1 (x) = J1 (e) = ? n ? [e (xk ? m)] + T 2 =? ||xk ? m||2 (2?36) k=1 k=1 n ? n ? e (xk ? m)(xk ? m) e + T T n ? ||xk ? m||2 (2?37) k=1 k=1 = ?eT Se + n ? ||xk ? m||2 |k=1 {z (2?38) } independent of e where the scatter matrix is de?ned as S= n ? (xk ? m)(xk ? m)T (2?39) k=1 Using the method of Lagrange multipliers, the vector e minimizing J1 becomes an eigenvector of the scatter matrix as follows. Se = ?e 27 (2?40) 15 10 Principal Component 1 Dimension 2 5 0 ?5 Principal Component 2 ?10 ?15 ?15 ?10 ?5 0 Dimension 1 5 10 15 Figure 2-9. Principal component analysis ? eT Se = ??eeT = ?? (2?41) In summary, ?nding the best one-dimensional projection of the data (best in least-sum-of-squared-error sense) is equivalent to projecting the data onto a line through the sample mean in the direction of the eigenvector of the scatter matrix having the largest eigenvalue. The coe?cients ak are called the principal components. Principal component analysis reduces the dimensionality of feature space by limiting directions along which the scattering (variance) is the greatest. Those directions are geometrically illustrated in Figure 2-9. 2.3 2.3.1 Multi-Port Reflectometers Six-Port Reflectometer A six-port re?ectometer was proposed as an alternative network analyzer [11]. The basic structure of the six-port re?ectometer is illustrated in Figure 2-10. The port 1 (P1) is connected to a signal source, the port 2 (P2) is terminated with a device under test 28 a1 a2 Six-port Reflectometer ZL P5 a6 a4 b6 P4 P3 b5 b4 b3 a5 b2 a3 b1 ?2 P6 Figure 2-10. Six-port re?ectometer (DUT) having an impedance ZL , and the port 3, 4, 5, and 6 (P3, P4, P5, and P6) are connected to power detectors. The six-port junction is characterized by 12 complex waves ai and bi , i = 1, и и и , 6 and the scattering equations ? ? ? b S S иии ? 1 ? ? 11 12 ? ? ? ?b2 ? ?S21 S22 и и и ? ? ? ?.?=? . .. . . ?.? ? . . . ?.? ? . ? ? ? b6 S61 S62 и и и or bi = 6 ? de?ned as ?? ? S16 a ? ? 1? ?? ? ? ? S26 ? ? ?a2 ? ? ? .. ? ?? . ? . ? ? .. ? ?? ? S66 a6 (2?42) i = 1, и и и , 6 Sij aj , (2?43) j=1 Moreover, the detectors are terminated with de?ned loads. The terminations are described by additional equations given by i = 3, и и и , 6 a i = b i ?i (2?44) where ?i is the input re?ection coe?cient of the ith power detector. Equation 2?43 and 2?44 can be solved in terms of a2 and b2 . Then, the incident waves on the detectors are written as ( bi = Ai a2 + Bi b2 = b2 Ai a2 Bi + b2 Ai ) = b2 Ai (?2 ? qi ) 29 i = 3, и и и , 6 (2?45) where Ai and Bi are functions of Sij and ?j The power detector yields only amplitude or power response, therefore the output of the ith detector is written as Pi = |bi |2 = |b2 |2 |Ai |2 |?2 ? qi |2 (2?46) With an analogy with a six-port re?ectometer analysis, one of port is a reference port (P4), where A4 is ideally zero. The power detected in the reference port 4 is written as P4 = |b4 |2 = |A4 a2 + B4 b2 |2 = |B4 |2 |b2 |2 (2?47) The normalized power by the reference port is written as Pi |Ai |2 = |?2 ? qi |2 P4 |B4 |2 i = 3, 5, 6 (2?48) The re?ection coe?cient of the DUT can be expressed in terms of the detector power as |?2 ? qi |2 = |B4 |2 Pi |Ai |2 P4 i = 3, 5, 6 (2?49) The equation represents circles on the Smith chart and the re?ection coe?cient of a DUT ?2 is the cross point of the circles. Although these circles should be intersected in a single point ideally, it may not happen in reality due to measurement errors. In this case, the re?ection coe?cient can be determined by the radical center of three circles, as shown in Figure 2-11. 2.3.2 Four-Port Multistate Reflectometer A six-port re?ectometer has been paid attention by many researcher due to its simple structure. The fundamentals of the six-port re?ectometer were extended to the design of a multistate re?ectometer and a four-port multistate re?ectometer with only two detectors was proposed [12][13][14]. The diagram of the multistate re?ectometer is illustrated in Figure 2-12 30 ?L6 6 RL q6 ?L3 Radical center 3 RL R L5 q3 q5 ?L5 Figure 2-11. Determination of ?L from the radical center of three circles a1 a2 Four-port Reflectometer ZL P3 ?2 b4 b3 a4 b2 a3 b1 P4 Figure 2-12. Four-port re?ectometer 31 With an analogy with the six-port re?ectometer, the operation of a four-port re?ectometer is described by the following equations, ( bi = Ai a2 + Bi b2 = b2 Ai a2 Bi + b2 Ai ) = b2 Ai (?L ? qi ) i = 3, 4 ?2 ? q3 2 A4 2 P3 ?2 ? q4 = A3 P4 (2?50) (2?51) where the system parameters A3 and A4 are given (Refer to Appendix B for derivation) by A3 = S21 S32 ? S22 S31 S21 q3 = S31 S22 S31 ? S21 S32 (2?52) A4 = S21 S42 ? S22 S41 S21 q4 = S41 S22 S41 ? S21 S42 (2?53) and The multistate re?ectometer is equipped with the facility to operate in di?erent states k (k = 1, 2, 3) in order to provide enough information for the determination of both the magnitude and the phase of the re?ection coe?cient of a device-under-test (DUT). The equations are written with superscript k, which denotes a state. ? ? q (k) 2 A(k) 2 P (k) 2 4 3 = (k) 3 (k) A3 P4(k) ? 2 ? q4 (2?54) The optimum-performance criteria for the multistate re?ectometer are as follows. ? ? ? The centers q (k) should be equally spaced around the origin (i.e., 120? angular separation). The centers q (k) should be equidistant from the origin (i.e., equal magnitudes). The system parameters B (k) should have zero magnitudes. Similar to the six-port re?ectometer, the reference port, which depends only on b2 to set A4 to zero, is assigned to port 4. The equation can be written as 2 B (k) 2 P (k) (k) ?2 ? q3 = 4(k) 3(k) A3 P4 32 (2?55) The power measurement with three di?erent network states results in three circles de?ned by the equation. The complex re?ection coe?cient of a device-under-test (DUT) can be determined in the same way as the six-port re?ectometer. 33 CHAPTER 3 AUTOMATIC MATCHING CONTROL 3.1 Overview The ability to test RF devices on the loadboard requires a good broadband match. Moreover, the bandwidth of many wireless devices is governed by the input impedance of an antenna. However, the input impedance is one of the parameters that varies most in di?erent environments on the loadboard and next to an antenna and the input reactance varies with frequency more than does the input resistance of the antenna [15][16]. Various automatic matching tuners and control algorithms have been proposed. A phase detector was used to correct the reactive part of the antenna mismatch in [17]. General matching network design and tuning strategies were studied in [18]. A generic algorithm has been widely used as a tuning algorithm [18][19][20] and heuristic search algorithms were studied in [6]. In addition, various narrowband techniques at di?erent frequencies have been studied for CMOS switched capacitors at 2.4 GHz [5] and a p-i-n diode switched capacitors at 390 MHz [6]. Prior to this work, a broadband recon?gurable matching network was proposed, but a supporting matching control system and algorithms were not developed [1]. In this work, a novel automatic broadband matching control was developed for a broadband recon?gurable matching network from 2.5 to 4.5 GHz. The matching network consists of a ?ve-stub microstrip ?lter and three varactors. A matching algorithm was developed using a greedy search algorithm to determine the varactor bias for impedance match over the large fractional bandwidth (> 50%). The work demonstrates the feasibility of the automatic matching control circuit over 2.5 GHz to 4.5 GHz. The rest of this chapter is organized as follows: Section 3.2 gives an overview of the proposed system. Section 3.3 presents the experimental results. Section 3.4 concludes and describes the future direction. 34 Microcontroller (ADuC7026) Magnitude/Phase Measurement Chip Coupler 12-Channel 12-Bit ADC Microprocessor Core Serial Interface 4-Channel 12-Bit DAC Lookup Table for Varactor Bias R S 232C Reconfigurable Tuner Magnitude Phase Network Analyzer (Agilent E8358A) 300k ~ 9GHz GPIB Computer (LabVIEW) Choke Coil Figure 3-1. Automatic matching control (AMC) system diagram 3.2 System Overview As shown in Figure 3-1, the proposed automatic matching control system consists of a recon?gurable tuner, a network analyzer, a host computer, and a microcontroller. In the current implementation, the tuner is directly connected to the network analyzer. Various antenna mismatches were simulated by an automatic load-pull system. The automatic matching is performed by a closed-loop feedback operation as follows. First, the network analyzer measures the re?ected wave power (S11 parameter). Second, the host computer calculates the available bandwidth. Last, the algorithm searches for the varactor bias for broadband impedance match and the microcontroller sets the varactor bias. The above procedures are repeated until the algorithm search converges to the optimal bias, which minimizes the impedance mismatch. This technique can be applied to an industrial automatic test equipment (ATE) system and a loadboard for conducting RF part test. 35 3.2.1 Impedance Matching Tuner As shown in Figure 3-2, the tuner, manufactured on Rogers Corporation Duroid 6006 board by the University of Arizona, has ?ve stubs along a microstrip line. Three stubs are connected to a reverse-biased varactor diode MPV1965, whereas two stubs have no tuning elements. The e?ective length of the tunable stub is controlled by the capacitance of the varactor. The varactor reverse bias and capacitance range from 0 to 5.12 V and 4.5 to 1 pF, respectively. The tuner is designed to provide wide bandwidth (up to 2.5 GHz) at the frequency of 3.5 GHz, as shown in Figure 3-3. The fractional bandwidth is 71% = 2.5 / 3.5. Agilent E8358A PNA network analyzer was used to measure S-parameters and the bandwidth was measured below 10 dB for the return loss. The insertion loss is as low as 2 dB. The dimensions of the tuner are shown in Table 3-1. The matching capability of the matching tuner can be represented by the load re?ection coe?cient ?L to be matched by the tuner. The input re?ection coe?cient ?in is written as ?in = S11 + S12 S21 ?L 1 ? S22 ?L (3?1) The matching capability is derived from the load re?ection coe?cient by setting the input re?ection coe?cient to zero. ?L ?in =0 = S11 S11 S22 ? S12 S21 (3?2) The distribution of the matching capability with the full range of varactor bias voltage illustrates the coverage of mismatched loads that can be compensated by the matching tuner. 3.2.2 Controller The controller consists of a host computer, a microcontroller, and a digital-to-analog converter (DAC). The computer provides interfaces with measurement instruments and executes search algorithm. The microcontroller consists of a 12-channel 12-bit 36 Choke Inductor DC Block Capacitor Stub #2 Stub #1 Varactor 1 Stub #4 Stub #3 Varactor 2 Stub #5 Varactor 3 Figure 3-2. Recon?gurable ?ve-stub matching tuner Table 3-1. Speci?cation of the matching tuner Stub / Interconnect Width (mils) Length (mils) Stub #1 & #5 Between #1 and #2 Stub #2 & #4 Between #2 and #3 Stub #3 27 39 22 36 34 416 342 429 481 384 analog-to-digital converter (ADC), a 4-channel 12-bit DAC, a universal asynchronous receiver/transmitter (UART), and digital input/outputs. To provide a su?cient number of voltage biases, an additional octal 12-bit DAC (AD5328) was soldered on the microcontroller evaluation board. Microcontroller ?rmware was developed in the embedded-C language and stored in the internal ?ash memory. The ?rmware supports the controller?s communication with both the host computer and the DAC through RS-232C and I2 C. It can interpret and execute GPIB-like commands, such as *IDN?, from the host computer. 37 0 Magnitude (decibels) -10 -20 -30 -40 S11 S21 S12 S22 -50 -60 0 1 2 3 4 5 Freq [GHz] 6 7 8 9 6 7 8 9 A Simulation 0 -10 Magnitude (decibels) -20 -30 -40 -50 S11 S21 S12 S22 -60 -70 0 1 2 3 4 5 Freq [GHz] B Measurement Figure 3-3. Comparison of simulation and measurement of matching tuner. The matching tuner is tuned with typical bias (2.56 2.56 2.56) and matched load 38 The host computer and other instruments were logically connected through the NI-VISA standard provided by MATLAB. A dedicated interface program was developed to automate the control of the automatic load-pull system. The program supports the communication between the host and the load-pull system and is normally called by MATLAB. 3.2.3 Search algorithm A greedy algorithm was used to ?nd the varactor bias for impedance match. The algorithm searches for the locally optimal bias per single varactor at a time and can be expressed as BW(vi ), v?i = argmax vi i = 1, и и и , d, vmax vi = 0, r , и и и , vmax , 2 ? 1 {z } | (3?3) 2r where BW() is the measured bandwidth, vmax is the maximum reverse bias voltage applied to varactors, d is the number of varactors, r is the number of DAC bits. The algorithm has a computational complexity of O(n), compared with O(n3 ) of brute-force approach (dimension = # of varactors = 3). Although it may fail to converge to the globally optimal bias, it can reduce the searching time signi?cantly. For example, suppose that a DAC resolution is 6 bits. Two algorithms? searching time are 3 dimension О resolution = 3 О 26 = 192 and resolutiondimension = 26 = 262144, respectively. The greedy algorithm is faster by the factor of 2r 3.3 d?1 3?1 /d = 26 /3 ? 1365. Experimental Results A control program was developed using MATLAB to perform all testing procedures including instrument initialization, mismatch load setup, S-parameter measurement, and search for the varactor bias. 3.3.1 Characterization of Automatic Tuner System We used Maury Microwave?s automatic tuner system (ATS) and mechanical load-pull tuners for various mismatched loads. The S-parameters of the load-pull tuner are 39 Table 3-2. Mismatched load speci?cation Type S11 |S11 | Motor position Matched Mismatched Mismatched Mismatched Mismatched 1 2 3 4 0 + j0 0.11 + ?0.09 0.19 + ?0.40 ?0.11 + ?0.22 0.14 ? ?0.01 0 0.14 0.45 0.24 0.14 (100, 5000, 5000) (16725, 2262, 5000) (17105, 1424, 5000) (17722, 2005, 5000) (20464, 2228, 5000) Mismatched Mismatched Mismatched Mismatched 5 6 7 8 0.35 + ?0 0.55 + ?0 0.65 + ?0 0.75 + ?0 0.35 0.55 0.65 0.75 Simulated Simulated Simulated Simulated +j1.0 0 +j2.0 S11 S21 S12 S22 +j0.5 -5 +j0.2 5.0 2.0 0.0 1.0 -20 +j5.0 0.5 -15 0.2 Magnitude (decibels) -10 \infty -25 -30 -j0.2 -35 S11 S21 S12 S22 -40 -45 0 1 -j5.0 -j0.5 2 3 4 5 Freq [GHz] 6 7 8 9 -j2.0 -j1.0 Figure 3-4. Mismatched load 1 measurement (S11 = 0.11 + j0.09 at 3.5 GHz). Motor positions are (16725, 2262, 5000). Re?ection coe?cient |S11 |=0.14. determined by three motors and have to be measured over the frequency range of interest because they are unknown and frequency-dependent. The characterization was performed by Maury Microwave?s ATS software MT993 and the mapping ?le, so-called tuner file, was created. Each line of the tuner ?le contains motor positions, S-parameters, and operating frequency. From the tuner ?le, one matched load and three mismatched loads were selected. The measurements of three mismatched loads are shown in Figure 3-4 to 3-6. The S11 at 3.5 GHz and motor positions are summarized in Table 3-2. 40 +j1.0 0 +j2.0 S11 S21 S12 S22 +j0.5 -5 +j0.2 5.0 2.0 1.0 0.0 0.5 -15 +j5.0 0.2 Magnitude (decibels) -10 \infty -20 -j0.2 -25 S11 S21 S12 S22 -30 -35 0 1 -j5.0 -j0.5 2 3 4 5 Freq [GHz] 6 7 8 -j2.0 -j1.0 9 Figure 3-5. Mismatched load 2 measurement (S11 = 0.19 + j0.40 at 3.5 GHz). Motor positions are (17105, 1424, 5000). Re?ection coe?cient |S11 |=0.45. +j1.0 0 +j2.0 S11 S21 S12 S22 +j0.5 -5 +j0.2 +j5.0 5.0 2.0 0.0 1.0 -20 0.5 -15 0.2 Magnitude (decibels) -10 \infty -25 -j0.2 -j5.0 -30 S11 S21 S12 S22 -35 -40 0 1 -j0.5 2 3 4 5 Freq [GHz] 6 7 8 9 -j2.0 -j1.0 Figure 3-6. Mismatched load 3 measurement (S11 = -0.11 + j0.22 at 3.5 GHz). Motor positions are (17722, 2005, 5000). Re?ection coe?cient |S11 |=0.24. Three mismatched loads used in this work show the almost-constant return and insertion loss from 2.5 to 4.5 GHz. As described in Table 3-2, three mismatched loads have |S11 | at 3.5 GHz = 0.14, 0.45, and 0.24, respectively. 3.3.2 Measurement Results The matching capability S11 /(S11 S22 ? S12 S21 ) at the center frequency of 3.5 GHz was calculated from the S-parameter measurement results, where the varactor bias ranges from 0 V to 5.12 V by 0.16 V step in 32 voltage levels. As shown in Figure 3-7, the matching capability can cover a small portion on the Smith chart, because the tuner was designed 41 +j1.0 +j2.0 +j0.5 +j5.0 5.0 2.0 1.0 0.5 0.0 0.2 +j0.2 \infty ?j5.0 ?j0.2 ?j2.0 ?j0.5 ?j1.0 Figure 3-7. Matching capability of ?ve-stub matching tuner at 3.5 GHz for broadband matching. Generally, the overall performance of the broadband matching can be evaluated by the available bandwidth, which has been used as the performance metric through this work. The S-parameters of the matching tuner were measured with typical and optimal bias. 2.56 V was set as the typical bias for the varactor. The optimal bias was found by the search algorithm as described before. The search algorithm found the optimal bias for the 50 ? matched load and the mismatched load 1, whereas it failed to ?nd for the more severely mismatched load 2 and 3. For the mismatched load 2 and 3, the optimal bias found for the 50 ? case was used. 42 As shown in Figure 3-8 and Figure 3-9, the optimal bias for the 50 ? matched load increased the available bandwidth from 1 GHz to 2 GHz. However, the optimal bias does not increase the bandwidth for other mismatched loads, because the matching tuner together with the larger mismatch (? = 0.24, 0.45) did not respond to the change in varactor capacitance. The comparison of brute-force, single-step proposed in [6], and the greedy algorithms is presented in Table 3-3, where the performance metric for the cost function of three algorithms is the 10 dB available bandwidth and the number of trials represents how many biases each algorithm has evaluated until it converges. 100 di?erent initial states were randomly generated and used for each experiment and the results were averaged over 100 experiments. The mismatched loads used in this experiment are summarized in Table 3-2. The degree of mismatch covers from 0 to 0.75. As shown in Table 3-3, the greedy algorithm outperforms brute-force and single-step algorithms in terms of the number of trials and the available bandwidth. For severely mismatched loads, the single-step algorithm often failed to have available bandwidth. In this catastrophic case, it is not fair to compare the performance by taking the average of experimental results. Instead, the catastrophic case was analyzed and excluded from the result. First, we de?ne 10% of the center frequency (0.35 GHz) as the bandwidth limit for the catastrophic case. During the experiment, the number of the catastrophic cases were counted to show how often the catastrophic case happens for each algorithm. As summarized in Table 3-4, the single-step algorithm su?ered from the catastrophic case even for modestly mismatched loads, e.g. ?L = 0.24 and all three algorithms failed to optimize for the severely mismatched load, e.g. ?L = 0.75. This result showed the similar trend as the broadband matching bandwidth for di?erent load impedance presented by [1], which limited the real and imaginary part of mismatched load impedance to 25 to 100 ? and ?50 to 50 ?, respectively. Second, the catastrophic case was excluded from Table 3-3. The experimental result avoiding the catastrophic case is 43 0 -10 Magnitude (decibels) -20 -30 -40 -50 S11 S21 S12 S22 -60 -70 0 1 2 3 4 5 Freq [GHz] 6 7 8 9 6 7 8 9 A Typical bias 0 -10 Magnitude (decibels) -20 -30 -40 -50 S11 S21 S12 S22 -60 -70 0 1 2 3 4 5 Freq [GHz] B Automated bias Figure 3-8. Matching tuner measurement with matched load. Typical and automated biases are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V) 44 0 -10 Magnitude (decibels) -20 -30 -40 -50 S11 S21 S12 S22 -60 -70 0 1 2 3 4 5 Freq [GHz] 6 7 8 9 6 7 8 9 A Typical bias 0 -10 Magnitude (decibels) -20 -30 -40 -50 S11 S21 S12 S22 -60 -70 0 1 2 3 4 5 Freq [GHz] B Automated bias Figure 3-9. Matching tuner measurement with mismatched load 1. Typical and automated biases are (2.56V 2.56V 2.56V) and (3.2V 3.86V 3.52V) 45 Table 3-3. Comparison of brute-force, greedy, and single-step algorithms Type # of Trials Available BW (GHz) brute greedy single brute greedy single Matched (Measured) Mismatched 1 (Measured) Mismatched 2 (Measured) Mismatched 3 (Measured) Mismatched 4 (Measured) 32768 32768 32768 32768 32768 273.60 272.64 340.80 332.16 281.28 358.29 414.99 454.95 420.39 353.70 1.28 1.24 0.76 1.24 1.24 1.22 1.21 0.68 1.16 1.20 1.17 1.09 0.25 0.63 1.04 Mismatched Mismatched Mismatched Mismatched 32768 32768 32768 32768 323.52 312.00 281.28 284.16 473.85 550.53 642.87 642.87 1.16 0.72 0.36 0.32 1.10 0.51 0.36 0.32 0.40 0.06 0.00 0.00 32768 300.16 479.16 0.92 0.86 0.52 5 6 7 8 (Simulated) (Simulated) (Simulated) (Simulated) Average summarized in Table 3-5. Provided that the catastrophic case can be always avoided by some technique, the performance di?erence between the greedy and other two algorithms is not as large as when the catastrophic case happens. However, the technique to avoid the catastrophic case can be additional overhead added to matching control algorithms. There are two reasons to make the greedy algorithm outperform other algorithms. First, the greedy algorithm is less dependent on an initial state compared with the single-step, because the greedy algorithm searches for the suboptimal solution for each varactor. Second, the suboptimal solution of the greedy algorithm turned out to be close to the global optimal solution of the brute-force. In the other word, the greedy algorithm managed to reach close to the global optimal solution, whereas the single-step may get trapped in local minima. The optimization surface during an experiment on the greedy algorithm is illustrated in Figure 3-12. The z-axis represents the available bandwidth in GHz and the x- and y-axes represent the 32 varactor voltage levels from 0 V to 5.12 V by 0.16 V step. The other varactor bias is set to 16th voltage level. The surface showed the steep threshold, which can be modeled by either sigmoid or round functions. The ?at surface above the 46 Table 3-4. Percentage of catastrophic case for brute-force, greedy, and single-step algorithms |?L | Type Percentage of catastrophic cases (%) brute-force greedy single-step Matched (Measured) Mismatched 1 (Measured) Mismatched 2 (Measured) Mismatched 3 (Measured) Mismatched 4 (Measured) 0.01 0.14 0.45 0.24 0.14 0 0 0 0 0 0 0 0 0 0 0 3 65 40 1 Mismatched Mismatched Mismatched Mismatched 0.35 0.55 0.65 0.75 0 0 0 100 0 39 0 100 57 93 100 100 5 6 7 8 (Simulated) (Simulated) (Simulated) (Simulated) Table 3-5. Comparison of brute-force, greedy, and single-step algorithms avoiding catastrophic cases Type # of Trials Available BW (GHz) brute greedy single brute greedy single Matched (Measured) Mismatched 1 (Measured) Mismatched 2 (Measured) Mismatched 3 (Measured) Mismatched 4 (Measured) 32768 32768 32768 32768 32768 273.60 272.64 340.80 332.16 281.28 358.29 424.48 348.69 287.55 356.45 1.28 1.24 0.76 1.24 1.24 1.22 1.21 0.68 1.16 1.20 1.17 1.12 0.59 1.05 1.05 Mismatched Mismatched Mismatched Mismatched 32768 32768 32768 N/A 323.52 388.72 281.28 N/A 345.98 366.43 N/A N/A 1.16 0.72 0.36 N/A 1.10 0.72 0.36 N/A 0.91 0.48 N/A N/A 32768 311.75 355.41 1.00 0.96 0.91 5 6 7 8 (Simulated) (Simulated) (Simulated) (Simulated) Average threshold explains why the suboptimal solution of greedy algorithm is pretty close to the global optimal solution. 3.4 Conclusion and Discussion The novel automatic broadband matching control was developed for the broadband microstrip matching tuner, which was provided by the University of Arizona. The proposed matching control demonstrated the feasibility of the automatic broadband matching control using the closed-loop feedback and the greedy search algorithm. The 47 measurement results showed that the greedy search algorithm could ?nd the optimal bias for the matching tuner, but the matching tuner did not show the good tunability against the large mismatches, |?| > 0.14. Currently we are working on improving the tuner?s tunability against the large mismatch. 48 0 -10 Magnitude (decibels) -20 -30 -40 -50 S11 S21 S12 S22 -60 -70 0 1 2 3 4 5 Freq [GHz] 6 7 8 9 6 7 8 9 A Typical bias 0 -10 Magnitude (decibels) -20 -30 -40 -50 S11 S21 S12 S22 -60 -70 0 1 2 3 4 5 Freq [GHz] B Automated bias Figure 3-10. Matching tuner measurement with mismatched load 2. Typical and automated biases are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V) 49 0 -10 Magnitude (decibels) -20 -30 -40 -50 S11 S21 S12 S22 -60 -70 0 1 2 3 4 5 Freq [GHz] 6 7 8 9 6 7 8 9 A Typical bias 0 -10 Magnitude (decibels) -20 -30 -40 -50 S11 S21 S12 S22 -60 -70 0 1 2 3 4 5 Freq [GHz] B Automated bias Figure 3-11. Matching tuner measurement with mismatched load 3. Typical and automated biases are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V) 50 1.4 1.2 1 0.8 0.6 0.4 0.2 0 40 35 30 30 25 20 20 15 10 10 0 5 0 Figure 3-12. Optimization surface of greedy algorithm 51 CHAPTER 4 LOAD IMPEDANCE ESTIMATION 4.1 Overview The overall goal of the proposed research is to develop a compact automatic matching control (AMC) circuit that sets the impedance match for radio frequency (RF) ports of a socketed device under test (DUT) under automatic test equipment (ATE) test [21]. An AMC provides match setting and examination capability for RF test frequencies to assist in a variety of RF test protocols. In this chapter, a new load impedance estimation technique is presented that facilitates the AMC system. A load impedance was not able to be determined in an AMC system prior to this work. Various search methods for the optimal state of a tunable matching network have been proposed [6][17][22]. However, the exhaustive search methods require millions of measurements depending on the number of the matching network states and become a limiting factor to the design of the matching network. In prior work, most search methods try to reduce searching time using heuristic approach based on exhaustive search, but their searching time increases proportionally to the matching network complexity. If the load impedance ZL is known to the AMC system, then the search is no longer necessary. Instead, the AMC system can set a matching network topology and component values through direct analysis. Thus, the AMC system needs to search only a few times to achieve the impedance match. The proposed impedance estimation technique is based on the principle of a multi-state re?ectometer. As an alternative network analyzer, measurement methods based on a multi-state re?ectometer were presented using various approaches, such as a scalar network analyzer [23], a four-port junction [13][14], and a tunable microstrip ?lter [24]. Similar to the prior work, the proposed method measures a single port multiple times instead of the measurement of multiple ports in a six-port re?ectometer [11], then a complex re?ection coe?cient is determined by some mathematical manipulation. The 52 Zin Tunable Impedance Matching Network Z0 ZL Power Detector Control varactors Automatic Matching Controller Figure 4-1. System diagram of load impedance estimation for automatic impedance matching measured power represents the magnitude of a re?ection coe?cient and is depicted as a circle on the Smith chart. Because the measured power reading results from the di?erent networks and the same load, the intersect point of these circles becomes the load re?ection coe?cient. In this work, we demonstrate that the tunable matching network designed for impedance match can perform load estimation. The rest of this chapter is organized as follows: Section 4.2 gives an overview of the AMC and the proposed method. Section 4.3 presents the experimental results. Section 4.4 concludes and describes the future direction. 4.2 4.2.1 System Overview Automatic Matching Control (AMC) As shown in Figure 4-1, an automatic matching control system consists of a tunable matching network, a power detector, and a controller. The automatic matching is performed by a closed-loop feedback operation as follows. First, the power detector 53 measures the re?ected wave power. Second, the controller calculates the degree of mismatch (S11 parameter). Last, the control algorithm searches for the varactor bias for impedance match and the controller sets the varactor bias. The above procedures are repeated until the control algorithm achieves a speci?ed matching goal. The matching control system presented in [25], which was not able to estimate a load, was used in this work. 4.2.2 Load Estimation Method An input re?ection coe?cient is expressed in terms of a load re?ection coe?cient and two-port S-parameters as ?in = S11 + S12 S21 ?L 1 ? S22 ?L (4?1) A complex re?ection coe?cient, such as ?in and ?L , is determined by both magnitude and phase of a re?ected wave. However, a phase detector is not as compact and accurate as a power detector. For this reason, an embedded test circuit often measures only wave magnitude using power detectors and the phase information of the re?ection coe?cient becomes unavailable. In this case, a complex input re?ection coe?cients is also measured only in magnitude and Equation 4?1 can be written as S12 S21 ?L |?in | = S11 + 1 ? S22 ?L (4?2) The load re?ection coe?cient derived from the magnitude of the input re?ection coe?cient is depicted as a circle on the Smith chart. By manipulating Equation 4?2 in the same way as stability circle equation derivation [26], the circle equation is given by 2 ? ? (S |? | ? ?S ) 11 ?L ? 22 2 in 2 |S22 | |?in | ? |?|2 S S 12 21 |?in | = |S22 |2 |?in |2 ? |?|2 where ? = S11 S22 ? S12 S21 and * denotes a complex conjugate. 54 (4?3) The load re?ection coe?cient ?L is expressed as a circle whose center and radius are ? ? (S22 |?in |2 ? ?S11 ) CL = |S22 |2 |?in |2 ? |?|2 S S 12 21 |?in | RL = |S22 |2 |?in |2 ? |?|2 (center) (4?4) (radius) (4?5) (Refer to Appendix A for the derivation.) The relationship between input and load re?ection coe?cients was also analyzed as a circle equation in [8]. Their result looks di?erent but is merely a special case of the general equation presented in this work. Based on the derived equations, an estimation method measures a re?ected wave power |?in | of an input port multiple times. The measured power represents only the magnitude of a re?ection coe?cient and is depicted as a circle on the Smith chart. Because the measured power reading results from the di?erent network and the same load, the intersect point of these circles becomes the load re?ection coe?cient. We performed the load impedance estimation as follows. 1. 2. 3. Measure power in the input port of a matching network with three di?erent sets of biases Calculate the center and radius of the circle of the load re?ection coe?cient Calculate the radical center of three circles The radical center is the approximation of the center of three circles? overlapped region, as illustrated in Figure 4-2. The coordinates of the radical center are given by 2 2 RL1 ? RL2 + x22 2x2 2 2 R ? RL3 + x23 + y32 ? 2xx3 y = L1 2y3 x= (4?6) (4?7) Note that the least-squares method developed for the six-port measurement can be used for higher accuracy [27]. Assuming the load is found through the proposed estimation method, the control algorithm can set impedance matching using S-parameters of the matching network and 55 ?L3 3 RL (x3 , y3 ) ?L1 Radical center 1 RL R L2 (0,0) (x2 , 0) ?L2 Figure 4-2. Radical center of three circles, ?L1 , ?L2 , and ?L3 . Start Measure reflected power Estimate load impedance Calculate bias for matching Reflected power = 0 No Yes End Figure 4-3. Impedance matching procedure using estimated load impedance. 56 Table 4-1. Summary of mismatched load estimation results Type Mismatched Mismatched Mismatched Mismatched #1 #2 #3 #4 |S11 | Mean squared error (MSE) 0.14 0.45 0.24 0.14 0.596300 7.982729 0.066334 0.148351 cL | > 1 MSE excluding |? 0.011048 0.080232 0.066334 0.031391 allow quick matching control of the AMC. The revised matching control procedure is illustrated in Figure 4-3. 4.3 Experimental Results In our experiments, four di?erent loads were used to evaluate the proposed estimation methods. Maury Microwave?s Automatic Tuner System (ATS) generated the speci?ed loads precisely every measurement and Agilent E8538 PNA network analyzer measured the input re?ection coe?cient |?in |. The re?ection coe?cient magnitude of the loads are |?L1 | ? 0.14, |?L2 | ? 0.45, |?L3 | ? 0.24, |?L4 | ? 0.14 over the range of frequency from 2.5 GHz to 4.5 GHz. Three sets of varactor biases were arbitrarily selected to set the matching network to di?erent states. These values are (2.56 4.8 4.8), (4.8 3.84 4.64), and (4.8 4.8 2.56). The center and radius of the load re?ection coe?cient were calculated using Equation cL was estimated using the radical center 4?4 and 4?5 and the load re?ection coe?cient ? equations Equation 4?6 and 4?7. When the centers of two or more of three circles are close to each other, the estimation is not accurate because the radical center is often not inside the overlapped area of three circles. Sometimes, the estimated load re?ection coe?cient goes out of a unit circle on the Smith chart, which is de?nitely incorrect for a passive load. The incorrect estimates were excluded from the accuracy statistics. The estimation accuracy is evaluated by the mean squared error (MSE) between cL |2 }. The proposed method measured and estimated load re?ection coe?cients, E{|?L ? ? 57 achieved the mean squared errors (MSE), 0.011, 0.080, 0.066, and 0.031, respectively. The experimental result is summarized in Table 4-1. 4.4 Conclusion The automatic matching control (AMC) system provides impedance match and examination capability for radio frequency (RF) ports by the closed-loop feedback. However, the AMC system prior to this work was not able to estimate a load impedance. In this work, we demonstrated that the measured power provides not only the degree of mismatch but also the estimate of an unknown load through the multistate re?ectometer measurement. The proposed load estimation allows the quick control of the AMC system by achieving a matching goal without an exhaustive iteration. Although the varactor biases for estimation were arbitrarily selected, selecting varactor biases should be based on the careful arrangement of q-points according to the six-port re?ectometer theory. The ideal arrangement of q-points is known as the same magnitude and 120? phase di?erence [11]. As future work, we are working on new matching network design and bias selection to account for the e?ect of q-points. 58 CHAPTER 5 COUPLER-FREE LOAD ESTIMATION USING THREE-PORT EFLECTOMETER 5.1 Overview Mobile devices equipped with radio frequency (RF) subsystems are widely used and drive emerging technologies for the better performance at low power consumption. Impedance matching between the RF subsystems plays a critical role in improving the low-power system e?ciency by maximizing power transfer and signal-to-noise ratio (SNR). However, RF impedance matching is often highly iterative and time-consuming, when the RF port between subsystems is poorly de?ned. As an e?ort to automate the RF impedance matching, an automatic matching control (AMC) circuit has been developed for cellular phone antenna [6] and loadboard testing [21]. The prior research work demonstrated that an unknown impedance mismatch can be resolved by a tunable matching network and an iterative search method. The disadvantage of the automatic matching control is that the iterative search may slow down the automation process and make impedance matching in real-time very challenging. To improve slow response time, a load impedance estimation method was developed to discover an unknown load and to enable immediate impedance matching [25]. However, the load estimation method, as well as the automatic matching control, measures the degree of mismatch through a distributed directional coupler. When the system area or volume matters, for example, on-chip embedded RF testing or compact loadboard testing, the coupler is preferred to be removed or replaced with other equivalent parts. As alternatives to the directional coupler, a lumped-element coupler [28][29] and an active coupler [30] have been proposed. We propose a novel coupler-free load estimation method for an automatic matching control (AMC) circuit using a lumped-element three-port re?ectometer. The proposed three-port re?ectometer consists of a two-port lumped-element ?-type matching network and a high impedance probing port attached to the input port of the matching network. 59 Source Z0 Coupler Reconfigurable matching network Reflection coefficient detector Match decision and biasing circuit Load ZL Figure 5-1. Automatic matching control (AMC) The re?ectometer can be recon?gured by three varactor-based tunable capacitors and used to set impedance matching within the unit circle on the Smith chart at 2.4 GHz. The coupler-free load estimation can estimate an unknown load from the combined power of incident and re?ected waves through a high-impedance probe. The use of the high-impedance probe and coupler-free estimation can eliminate the need of the directional coupler and make compact fabrication of impedance measurement on a chip feasible. The rest of this chapter is organized as follows: Section 5.2 gives an overview of the automatic matching control and the coupler-free re?ectometer. Section 5.3 shows the high impedance probe model and ?tting methods. Section 5.4 describes radical center and least-square load estimation methods. Section 5.5 presents the experimental results. Section 5.6 concludes and describes the future direction. 5.2 System Overview Figure 5-1 illustrates the system diagram of an automatic matching control (AMC). The automatic matching control recon?gures the matching network through the closed-loop feedback consisting of a coupler, a re?ection coe?cient detector, and a match decision circuit. The closed-loop feedback enables the match decision circuit to search for varactor biases for minimizing the RF signal re?ection in a trial-and-error process. In this work, the coupler has been replaced with a high impedance probe, which enables the automatic matching control to perform the coupler-free load estimation. 60 Tunable Impedance Matching Network Zin 3.3nH Z0 5Z0 Power Detector 1.8nH 1.8nH ZL Control varactors Automatic Matching Controller A B Figure 5-2. Three port re?ectometer integrated with a high impedance probe A) System diagram with schematic B) Fabrication on FR4 board 61 Capacitance (pF) 100.0 SMV1413 10.0 1.0 SMV1405 SMV1408 0.1 0 5 10 15 20 25 30 Reverse Voltage (V) Figure 5-3. Varactor SMV1405 capacitance versus reverse voltage Cathode Anode Figure 5-4. Varactor SPICE model The proposed tunable matching network is based on the bandpass ? topology, which is known to provide impedance matching with any point on the Smith chart [3][31]. As shown in Figure 5-2, the matching network is a ?-type band-pass ?lter with lumped inductors and tunable capacitors and was fabricated on 6.3 О 4.8 mm FR4 printed circuit board (PCB). The network was designed to set impedance matching at 2.4 GHz and has three ports for input, output, and probe. Skyworks SMV1405-074LF was used as the tunable capacitor. It contains two hyper-junction varactor diodes in a single package, where two varactors are connected in common cathode. The capacitance of a single varactor ranges from 1 pF (bias = -8 V) to 2.7 pF (bias = 0 V), as shown in Figure 5-3. The common cathode pin is connected 62 to a bias voltage supply through a DC blocking choke inductor. The SPICE model of SMV1405, as shown in Figure 5-4, was used for preliminary simulation. A high impedance probe was implemented with a 250?(= 5Z0 ) chip resistor and a transmission line. The high impedance probe was used to measure the input re?ection coe?cient of the matching network. The relationship model between the high impedance probe and the input re?ection coe?cient will be presented in next Section. 5.3 High Impedance Probe As shown in Figure 5-2, a high impedance probe consists of a 250? chip resistor at the measurement node and a transmission line terminated with a matched port, which is connected to a network analyzer. For an embedded test system, the high impedance probe and the network analyzer can be replaced by an on-chip bipolar power detector presented in [32]. The following are worthy to note in this demonstration as shown in Figure 5-2. 1. Series chip resistor (250?) is attached to a vector network analyzer through a transmission line to emulate the operation of a high-impedance probe. 2. Used a vector network analyzer to measure and manipulate full S-parameter data 3. Derived |?in | from S-parameters measured by a vector network analyzer to emulate the operation of a power detector. 4. Used |1 + ?in | from a vector network analyzer to see the system response with a power detector and to verify Equation 5?1. The high impedance probe has no directivity and measures the summation of both incident and re?ected waves. The measurement at the high impedance probe can be written as an + bn = an + ?n an = an (1 + ?n ) (5?1) where an , bn , ?n are normalized incident and re?ected waves and a re?ection coe?cient of port n. The network analyzer port is matched with impedance Z0 and connected in series with the chip resistor 5Z0 . The network analyzer measures Z0 /(Z0 + 5Z0 ) of a 63 probe node and acts as a voltage divider. In other words, the output response of the high impedance probe is scaled down by a voltage divider ratio Z0 /(Z0 + 5Z0 ) and followed by the phase-delay (e?? ) of a transmission line. Assuming that the input port of the matching network is port 1 and the high impedance probe is port 3, the measured power in the high impedance probe is written as 2 Z0 ?? P3 = |b3 | = |S31 a1 | = e S11 a1 (1 + ?in ) , 5Z0 + Z0 2 2 (5?2) where an and bn are normalized incident and re?ected waves of port n. The ideal relationship described by Equation 5?2 does not exactly ?t the measurement result due to non-ideal circuit e?ects, such as ?nite impedance of the probe and transmission line loss. Instead of the ideal model, a relationship model with parasitics and a neural network approximation model were used to take the non-ideal e?ects into account. The node measured by the high impedance probe is connected to the port 1 through a transmission line. The measurement of the node is deteriorated due to the non-ideal e?ects, which are included in a modi?ed relationship model. The relationship between the measured S-parameters from the input port 1 and the high impedance probe is modi?ed and written as S11 = (S31 r1 e??1 ? 1)r2 e??2 (5?3) where r1 , r2 , ?1 , ?2 are magnitude and phase ?tting parameters. The relationship model can be trained by a least square nonlinear ?tting. An arti?cial neural network, a well-known nonlinear approximation model, was also used to represent the relationship. The training procedure for two models will be described in more detail. 5.3.1 Least Square Fitting Least square nonlinear ?tting was applied to train the relationship model. The ?tting parameters, r1 , r2 , ?1 , ?2 , are obtained by minimizing the least square equation written as ( ) ? ? ? S11 (V) ? (S31 (V)r1 e??1 ? 1)r2 e??2 2 r?1 , r?2 , ?1 , ?2 = arg min V 64 (5?4) where V = (V1 , V2 , V3 )T is a bias voltage vector for three varactors. The ?tting algorithm is the large-scale algorithm, a subspace trust region method based on the interior-re?ective Newton method described in [33][34]. 5.3.2 Artificial Neural Network An arti?cial neural network is well known for the approximation of a nonlinear model. The nonlinear relationship between S11 and the high impedance probe, described by Equation 5?3, is approximated by an arti?cial neural network. The input and output of the neural network are the measured S31 and the estimate of S11 . The used neural network has 15 perceptrons in the hidden layer and was trained by traditional backpropagation algorithm. 5.4 Load Estimation Methods The equation for coupler-free load estimation is based on the stability circle derivation as presented in [26, Chapter 11] and [25]. According to the stability circle derivation, the center and radius of the circle were given as ? ? (S22 |?in |2 ? ?S11 ) CL = |S22 |2 |?in |2 ? |?|2 S S 12 21 |?in | RL = |S22 |2 |?in |2 ? |?|2 (center) (5?5) (radius) (5?6) When the incident and re?ected waves are combined under the absence of a coupler, the measured power of the input port is expressed as pin = |a1 + b1 |2 = |a1 |2 |1 + ?in |2 (5?7) Therefore, |1 + ?in | instead of |?in | can be used to estimate a load impedance. The manipulated S-parameter data from a network analyzer, |1 + ?in |, were used to verify the power detector equation rewritten as S S ? 12 21 L |1 + ?in | = 1 + S11 + 1 ? S22 ?L 65 (5?8) 1 + S11 ? (1 + S11 )S22 ?L + S12 S21 ?L |1 + ?in | = 1 ? S22 ?L (5?9) By taking the square of both sides, the equation is written as |1 ? S22 ?L |2 |1 + ?in |2 = |1 + S11 ? (S22 + ?)?L |2 (5?10) ? = S11 S22 ? S12 S21 (5?11) where The next derivation follows the stability circle derivation presented in [25]. Then, the center and radius of the circle for coupler-free load estimation are given as ? (S22 |1 + ?in |2 ? (S22 + ?)(1 + S11 ))? |S22 |2 |1 + ?in |2 ? |S22 + ?|2 S12 S21 |1 + ?in | RL = |S22 |2 |1 + ?in |2 ? |S22 + ?|2 CL = (center) (5?12) (radius) (5?13) where ? = S11 S22 ? S12 S21 and * denotes a complex conjugate. The detail in derivation is given in Appendix A. The center and radius of three circles were used to estimate an unknown load. The estimation is based on radical center and least square ?tting widely used for a six-port re?ectometer. The radical center of three circles is the simplest geometric method and the least square ?tting is more accurate and a statistical approach by minimizing the sum of squared distance from three circles. 5.4.1 Radical Center The radical center is the approximation of the center of three circles? overlapped region, as illustrated in Figure 5-5. The coordinates of the radical center are given as 2 2 RL1 ? RL2 + x22 2x2 2 2 R ? RL3 + x23 + y32 ? 2xx3 y = L1 2y3 x= (5?14) (5?15) Note that all mismatched loads are passive and should be within a unit circle. If the magnitude of the estimated re?ection coe?cient is larger than one, the large error can 66 ?L3 3 RL (x3 , y3 ) ?L1 Radical center 1 RL R L2 (0,0) (x2 , 0) ?L2 Figure 5-5. Radical center of three circles, ?L1 , ?L2 , and ?L3 . dominate the overall mean square error to lead to wrong performance statistics. In this case, the magnitude is set to one (on the unit circle) while keeping the phase information. 5.4.2 Least Square Fitting Least square nonlinear ?tting was applied to estimate a load and the radical center was used as an initial parameter, as described in [27]. The least square ?tting can provide more accurate results in sacri?ce of high computation cost. The same ?tting algorithm used in the high impedance probe was used here. The ?tting parameter is the re?ection coe?cient of the unknown load, a complex number on the Smith chart, and is obtained by minimizing the least square equation written as ??L = arg min ? |Circle(V) ? ?L |2 V 67 (5?16) where V = (V1 , V2 , V3 )T is a bias voltage vector for three varactors. Note that Circle(V) is described by Equation 5?12 and 5?13 and the distance between a circle and a point is de?ned as a distance between a tangential line to the circle and the point. 5.5 Experimental Results A vector network analyzer was used to measure the three-port S-parameters of the network at 2.4 GHz. The varactor bias voltage ranges from 0 to 5.12 V by 0.32 V step, 16 voltage levels and the number of states of the three-varactor matching network is 163 = 4096. Therefore, the total number of three-port S-parameter is 163 О 32 = 36, 864. The following experimental results come from the 36,864 S-parameter data of 4096 states of the matching network. First, ?tting the measurement data from the high impedance probe to the input port S-parameter will be presented. The ?tting were performed by the least square ?tting of the relationship model with parasitics and the backpropagation training of the neural network approximation model. The error distribution of both methods is also presented. Second, a coupler-free load estimation is performed by radical center and least square and the results using two methods are compared in terms of mean square error (MSE). The input data to the load estimation is the estimated data from the high impedance probe. Last, for a fair comparison of coupler and coupler-free load estimation methods, the estimation performance is evaluated with the same measurement data used in [25]. 5.5.1 High Impedance Probe Estimation Two models based on least square ?tting and arti?cial neural network were used for the high impedance probe estimation. The ?tting parameters of least square are r1 , r2 , ?1 , and ?2 , magnitude and phase of two complex numbers representing non-ideal e?ects. Least square ?tting was trained by a large-scale algorithm, a subspace trust region method based on the interior-re?ective Newton method. The high impedance probe S-parameters 68 S31 were converted by the relationship model and four trained ?tting parameters and compared with the input re?ection coe?cient S11 . The neural network used in this work has two input nodes representing the real and imaginary parts of the high impedance probe S-parameter S31 , 15 perceptrons in the hidden layer, and two output nodes representing the real and imaginary parts of the input re?ection coe?cient S11 . The traditional backpropagation algorithm was used to train the neural network. Figure 5-6 and 5-7 show estimation results of the input re?ection coe?cient S11 from the high impedance probe S-parameter S31 using least square ?tting and arti?cial neural network, respectively. As explained in Section 5.3, the distribution of the high impedance probe S-parameters is the downscale of the input re?ection coe?cient through voltage divider. The experiment results showed that the input re?ection coe?cient can be successfully estimated through the high impedance probe without disturbing the matching network. Mean square error (MSE) for least square ?tting and the trained neural network are 0.000732 and 0.000366. Although the neural network achieved lower estimation error, least square ?tting also has advantage of faster training and simpler model representation over the neural network. The magnitude of error distribution of both methods are shown in Figure 5-8. As expected, the neural network showed its peak of error distribution closer to zero than least square ?tting. 5.5.2 Load Estimation The input re?ection coe?cients estimated from the high impedance probe were used to estimate an unknown load using the coupler-free load estimation. One matched load, three slightly mismatched loads, and three heavily mismatched loads were used as mismatched loads for the experiments. They were classi?ed into two groups, slightlymismatched and heavily-mismatched loads and both groups include the matched load as 69 +j1.0 +j1.0 +j2.0 5.0 ?j0.2 ?j5.0 ?j0.5 2.0 0.0 +j5.0 1.0 +j0.2 Ц ?j0.2 +j2.0 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 +j0.5 0.2 +j0.5 ?j2.0 ?j5.0 ?j0.5 ?j2.0 ?j1.0 ?j1.0 A Input re?ection coe?cient S11 B High impedance probe S-parameter S31 +j1.0 Fitting from S31 +j2.0 S11 +j0.5 5.0 2.0 1.0 0.5 +j5.0 0.2 +j0.2 0.0 ?j0.2 Ц ?j5.0 ?j0.5 ?j2.0 ?j1.0 C Fitting S31 to S11 Figure 5-6. Least square nonlinear ?tting of high impedance probe model 70 Ц +j1.0 +j1.0 +j2.0 5.0 ?j0.2 ?j5.0 ?j0.5 2.0 0.0 +j5.0 1.0 +j0.2 Ц ?j0.2 +j2.0 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 +j0.5 0.2 +j0.5 ?j2.0 ?j5.0 ?j0.5 ?j2.0 ?j1.0 ?j1.0 A Input re?ection coe?cient S11 B High impedance probe S-parameter S31 +j1.0 Fitting from S31 +j2.0 S11 +j0.5 5.0 2.0 1.0 0.5 +j5.0 0.2 +j0.2 0.0 ?j0.2 Ц ?j5.0 ?j0.5 ?j2.0 ?j1.0 C Fitting S31 to S11 Figure 5-7. Arti?cial neural network of high impedance probe model 71 Ц 1400 1200 1000 800 600 400 200 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 A Least square 1400 1200 1000 800 600 400 200 0 0 0.01 0.02 0.03 0.04 0.05 0.06 B Neural network Figure 5-8. High impedance probe estimation error distribution 72 0.07 0.08 Table 5-1. Summary of heavily and slightly mismatched loads Mismatched load |S11 | at 2.4 GHz Tuner motor position Matched 0.01 (100, 5000, 5000) Slightly mismatched #1 Slightly mismatched #2 Slightly mismatched #3 0.13 0.38 0.20 (16725, 2262, 5000) (17105, 1424, 5000) (20464, 2228, 5000) Heavily mismatched #1 Heavily mismatched #2 Heavily mismatched #3 0.78 0.76 0.69 (15781, 526, 5000) (17835, 624, 5000) (20572, 804, 5000) a reference. All used loads are carefully generated by Maury Microwave?s load-pull system (MT986A and MT982B01) and their speci?cation is summarized in Table 5-1. The load estimation was performed separately for slightly mismatched and heavily mismatched groups. During the experiment, a di?erent set of three states of matching network were manually chosen to give better estimation results. The chosen voltage biases for three di?erent states for slightly mismatched and heavily mismatched groups are ((0.32, 2.56, 4.48), (4.48, 0, 2.88), (4.8, 0, 0)) and ((0.32, 4.16, 3.52), (1.92, 0, 3.84), (3.2, 0.32, 0)), respectively. As shown in Figure 5-9, the estimation on slightly mismatched loads showed smaller error than heavily mismatched loads. Although the accuracy is not good enough to replace an high-precision instrument, the rough estimate of magnitude and phase of the load re?ection coe?cient is very useful for the automatic matching control to set immediate impedance matching. 5.5.3 Comparison of Coupler and Coupler-Free Load Estimation For the comparison of coupler and coupler-free load estimation methods, the measurement data used in the prior research work [25] were used to evaluate the estimation performance. Two load estimation methods were evaluated with radical center and least square ?tting described in Section 5.4. The four mismatched loads used in the prior research 73 +j1.0 +j1.0 True +j2.0 Estimate ?j0.2 ?j0.2 ?j5.0 ?j0.5 5.0 0.0 2.0 Ц +j5.0 1.0 +j0.2 0.5 5.0 2.0 1.0 0.0 0.5 +j5.0 0.2 +j0.2 True +j2.0 Estimate +j0.5 0.2 +j0.5 ?j2.0 Ц ?j5.0 ?j0.5 ?j2.0 ?j1.0 ?j1.0 A Small mismatch (MSE=0.21) B Large mismatch (MSE=0.86) Figure 5-9. Load estimation using estimated S11 from high impedance probe Table 5-2. Comparison of coupler and coupler-free load estimation in term of mean square error (MSE) |S11 | Mismatched load Mismatched Mismatched Mismatched Mismatched #1 #2 #3 #4 0.14 0.45 0.24 0.14 Radical center Least square Coupler Coupler-free Coupler Coupler-free 0.056840 0.120909 0.035607 0.054044 0.050865 0.134596 0.069265 0.056333 0.003665 0.061899 0.009934 0.035334 0.013576 0.034983 0.009930 0.018138 work were also used for fair comparison. The experimental results are summarized in Table 5-2 in terms of mean square error (MSE). Although mean square error (MSE) of the coupler-free load estimation is slightly higher than MSE of coupler load estimation, the estimation accuracy of two methods are comparable for both radical center and least square ?tting algorithms, as shown in Figure 5-10 to 5-13. When the coupler-free estimation is employed to an automatic matching control system, the system dimension will be dramatically reduced without compromising the system performance. 74 +j1.0 +j1.0 True +j2.0 Estimate ?j0.2 5.0 2.0 0.0 +j5.0 1.0 Ц ?j0.2 ?j5.0 ?j0.5 ?j2.0 ?j0.5 ?j2.0 ?j1.0 A Radical center with coupler B Radical center without coupler +j1.0 +j1.0 True +j2.0 Estimate ?j5.0 ?j0.5 5.0 2.0 0.0 1.0 Ц +j5.0 0.5 +j0.2 0.2 5.0 2.0 1.0 0.5 ?j0.2 True +j2.0 Estimate +j0.5 +j5.0 0.2 +j0.2 ?j0.2 ?j2.0 ?j1.0 ?j1.0 C Least square with coupler D Least square without coupler Figure 5-10. Coupler and coupler-free load estimation with mismatched #1 75 Ц ?j5.0 ?j0.5 ?j2.0 Ц ?j5.0 ?j1.0 +j0.5 0.0 +j0.2 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 True +j2.0 Estimate +j0.5 0.2 +j0.5 +j1.0 +j1.0 True +j2.0 Estimate ?j0.2 5.0 2.0 0.0 +j5.0 1.0 Ц ?j0.2 ?j5.0 ?j0.5 ?j2.0 ?j0.5 ?j2.0 ?j1.0 A Radical center with coupler B Radical center without coupler +j1.0 +j1.0 True +j2.0 Estimate ?j5.0 ?j0.5 5.0 2.0 0.0 1.0 Ц +j5.0 0.5 +j0.2 0.2 5.0 2.0 1.0 0.5 ?j0.2 True +j2.0 Estimate +j0.5 +j5.0 0.2 +j0.2 ?j0.2 ?j2.0 ?j1.0 ?j1.0 C Least square with coupler D Least square without coupler Figure 5-11. Coupler and coupler-free load estimation with mismatched #2 76 Ц ?j5.0 ?j0.5 ?j2.0 Ц ?j5.0 ?j1.0 +j0.5 0.0 +j0.2 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 True +j2.0 Estimate +j0.5 0.2 +j0.5 +j1.0 +j1.0 True +j2.0 Estimate ?j0.2 5.0 2.0 0.0 +j5.0 1.0 Ц ?j0.2 ?j5.0 ?j0.5 ?j2.0 ?j0.5 ?j2.0 ?j1.0 A Radical center with coupler B Radical center without coupler +j1.0 +j1.0 True +j2.0 Estimate ?j5.0 ?j0.5 5.0 2.0 0.0 1.0 Ц +j5.0 0.5 +j0.2 0.2 5.0 2.0 1.0 0.5 ?j0.2 True +j2.0 Estimate +j0.5 +j5.0 0.2 +j0.2 ?j0.2 ?j2.0 ?j1.0 ?j1.0 C Least square with coupler D Least square without coupler Figure 5-12. Coupler and coupler-free load estimation with mismatched #3 77 Ц ?j5.0 ?j0.5 ?j2.0 Ц ?j5.0 ?j1.0 +j0.5 0.0 +j0.2 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 True +j2.0 Estimate +j0.5 0.2 +j0.5 +j1.0 +j1.0 True +j2.0 Estimate ?j0.2 5.0 2.0 0.0 +j5.0 1.0 Ц ?j0.2 ?j5.0 ?j0.5 ?j2.0 ?j0.5 ?j2.0 ?j1.0 A Radical center with coupler B Radical center without coupler +j1.0 +j1.0 True +j2.0 Estimate ?j5.0 ?j0.5 5.0 2.0 0.0 1.0 Ц +j5.0 0.5 +j0.2 0.2 5.0 2.0 1.0 0.5 ?j0.2 True +j2.0 Estimate +j0.5 +j5.0 0.2 +j0.2 ?j0.2 ?j2.0 ?j1.0 ?j1.0 C Least square with coupler D Least square without coupler Figure 5-13. Coupler and coupler-free load estimation with mismatched #4 78 Ц ?j5.0 ?j0.5 ?j2.0 Ц ?j5.0 ?j1.0 +j0.5 0.0 +j0.2 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 True +j2.0 Estimate +j0.5 0.2 +j0.5 5.6 Conclusion We presented a novel coupler-free load estimation and a lumped-element re?ectometer integrated with a high impedance probe. The estimate of an unknown mismatched load can help an automatic matching control (AMC) to set impedance matching on the load without iterative search. A high impedance probe was presented for the integrated on-chip RF testing. Two relationship models together with estimation methods, least square and arti?cial neural network, achieved mean square error 0.000732 and 0.000366, respectively. The lease square ?tting has only four ?tting parameters compared with about 20 parameters of the neural network model and enables faster training without compromising the accuracy. The estimated results were also used as an input to the coupler-free load estimation, which discovered an unknown load for the automatic matching control. A novel coupler-free load estimation was proposed with a lumped-element re?ectometer. The proposed method demonstrated that it could discover an unknown mismatched load and provide with the high accuracy enough for an automatic matching control to exploit. The coupler-free load estimation removed the need of any distributed components and achieved compact system dimension without compromising the estimation accuracy. The experimental results showed that the proposed coupler-free estimation achieved the comparable accuracy as the prior research work adopting a coupler. The proposed load estimation can be integrated with a non-iterative automatic matching control system without extra components or performance degradation. 79 CHAPTER 6 THREE PORT AND FOUR PORT REFLECTOMETERS 6.1 Overview Automatic or self-recon?gurable radio frequency (RF) systems have received attention due to their potential to overcome the uncertainty of RF systems such as parasitics and manufacturing variation. Recently, an automatic matching control (AMC) was proposed to provide the impedance matching of the radio frequency (RF) ports of a device under test (DUT) or a cellular phone antenna [21][6]. Also, load estimation was proposed to overcome the iterative nature of the automatic matching control system [25]. Recon?gurable matching networks in the prior work were not designed for the load estimation [1][5][6]. Even the load estimation proposed in [25] used the arbitrarily recon?gured matching networks, whose design was not targeted for load estimation purpose. In this work, three-port and four-port re?ectometers were designed to support both the automatic matching control and the load estimation using lumped elements and compact power detectors for size reduction and embedded RF testing. In the network analyzer research community, a new method using detector power readings and some mathematical manipulation was introduced to measure a complex re?ection coe?cient of a device under test (DUT). The new method, so-called a six-port re?ectometer, was ?rst proposed as an alternative of a conventional network analyzer [11]. Later, four-port multistate re?ectometers were presented to reduce the required number of ports and parts [12][13][14]. Recently, a lumped-element structure was also presented for the integration on a chip [35]. We propose three-port and four-port lumped-element multistate re?ectometers for both a recon?gurable matching network and load estimation. Carefully chosen q-points can improve the accuracy of the load estimation and it can still recon?gure a matching network for the automatic matching control. 80 a1 a2 Three-port Reflectometer ZL b2 ?2 b3 a3 b1 P3 A a1 a2 Four-port Reflectometer ZL ?2 b4 b3 P3 a4 b2 a3 b1 P4 B Figure 6-1. Re?ectometers A) Three-port B) Four-port 6.2 Multistate Reflectometers As shown in Figure 6-1, port 1 is fed with a signal source with impedance Z0 , port 2 is connected with a device under test (DUT) having a complex re?ection coe?cient ?2 (sometimes load re?ection coe?cient denoted by ?L ) to be measured, and port 3 and 4 are power detector ports. Similar to a six-port re?ectometer analysis, power readings can be expressed in terms of incident and re?ected wave powers of port 2. The power readings of an n-port re?ectometer is written as bi = Ai a2 + Bi b2 , 81 i = 3, 4, и и и , n (6?1) Pi = |bi |2 = |b2 |2 |Ai |2 |?2 ? qi |2 , i = 3, 4, и и и , n (6?2) where ?2 = a2 , b2 qi = ? Bi Ai (6?3) The complex re?ection coe?cient of a DUT can be calculated by power detector readings and some mathematical manipulation, which is explained in more detail in [11]. By adding a network state k, the operation of multistate re?ectometers can be described by the following equation, (k) |?2 ? (k) qi |2 = Pi (k) (k) |b2 |2 |Ai |2 i = 3, 4, и и и , n, , k = 1, 2, 3 (6?4) where k is a network state (typically, set by di?erent biases) and three network states are needed in order to determine the complex re?ection coe?cient of a DUT. Especially for a four-port case, port 3 is assigned as a reference port which depends only on b2 (in other words, b3 = B3 b2 or A3 = 0). The equation can be written as B (k) 2 P (k) (k) |?2 ? q4 |2 = 3(k) 4(k) , k = 1, 2, 3 A4 P3 (6?5) If all power detectors are perfectly matched to Z0 , then the calibration constants, Ai , Bi , and qi , are expressed in terms of S-parameters and given by (refer to Appendix B for the derivation) (k) (k) Ai = (k) (k) (k) S21 (k) (k) S21 Si2 ? S22 Si1 , (k) Bi = Si1 (k) S21 (k) , (k) qi = Si1 (k) (k) (k) (k) S22 Si1 ? S21 Si2 (6?6) The above calibration constants can be obtained by either a direct measurement or modeling. The reference port can be realized by a lumped power divider, which was presented for the integration on a chip in [35], as shown in Figure 6-2. In prior work, a directional coupler was widely used for a reference port, but the compact lumped power divider is preferred for size reduction in embedded test. The scattering matrix of the divider with 82 2 Za Z0 1 2 0 20 1 1 3 30 Zc 10 20 3 30 Figure 6-2. Schematic of a lumped power divider. Port 3 is a coupled port, which can be used as a reference port. respect to Z0 is given by ? ? 1 1 ? ? ? ? Za Zc ? ? 0 1 + Z0 1 + Z0 ? a 1 ?b1 ? ? ? ?? ? ? ? ? 1 ?? ?b ? = ? ? ? 0 0 ? ? a ? 2 ? ? 1 + Za ? 2? ? Z0 ? ? ? ?? ? ? 1 ? b3 b3 0 0 1 + ZZ0c (6?7) A three-port re?ectometer has the same basic structure as a four-port one except that a reference port and a power divider do not exist. Because a reference port does not exist, an input signal power has to be known to a re?ectometer for the normalization of wave powers. In other words, a reference port can be replaced by a prede?ned input signal power in some cases, e.g., testing RF parts under an automatic test equipment (ATE). Hence, the wave power |b2 | can be estimated through the relationship between |b2 | and |a1 | if an input power |a1 | is precisely de?ned. The relationship is described by the equation given by b2 = S21 a1 + S22 a2 = S21 a1 + S22 ?L b2 b2 = S21 a1 1 ? S22 ?L In summary, the three-port re?ectometer can be used if the signal source is precisely de?ned and the four-port re?ectometer is applicable to other general cases. 83 (6?8) (6?9) 6.3 Tunable Matching Network The simplest lumped-element matching network is L-type, but cannot set impedance matching of all points on the Smith chart. T - and ?-types are known for matching capability with any point on a Smith chart [31]. Although T - and ?-type topologies are equivalent in terms of matching capability, the ?-type is preferred for impedance matching purpose because it has a smaller number of tunable elements along with a signal path, which result in lower insertion loss than T -type. Each branch of the matching network has an inductor connected with a varactor either in parallel or series. As shown in Figure 6-3, the basic structure is ?-type band pass ?lter tuned at the center frequency of 2.4 GHz, where capacitors are replaced with tunable varactors. The tunable capacitor used in the matching network is Skyworks SMV1405-074LF, which contains two common cathode diode varactors in a single package. The common cathode port is connected to a bias voltage supply through a choke coil. The typical capacitance of each varactor with a bias 1 V is 1.21(min) to 1.45(max) pF and the capacitance ranges from 2.1 pF (0.5 V) to 0.95 pF (10 V). The capacitance versus reverse voltage is shown in Figure 5-3. The SPICE model used in simulations is shown in Figure 5-4. Agilent ADS was used to perform S-parameter simulations. According to the simulation results, the varactor in series with an inductance 3.3 nH and in parallel with 1.8 nH shows the largest tuning range at 2.4 GHz as shown in Figure 6-4. The inductance value of 3.3 nH and 1.8 nH was used for the design of the matching network. A noninvasive measurement is mandatory not to disturb the original design of a matching network. Similar to the prior work [35], a high impedance power detector measures an internal node. The noninvasive power detector is emulated by inserting high resistance in series with a measurement port and deembedding the e?ect of the resistance. As explained earlier, the four-port re?ectometer needs a power reading that depends only 84 P3 High input impedance power detector Z0 ZL A B Figure 6-3. Recon?gurable three-port matching network A) Schematic B) Implementation on FR4 board 85 S(2,1) S(1,1) freq (2.400GHz to 2.400GHz) S(2,1) S(1,1) A freq (2.400GHz to 2.400GHz) B Figure 6-4. Tunable element impedance with a bias from 0 V to 10 V. A varactor is in parallel and in series with an inductor. и and О denote S11 and S21 , respectively. A) In series with 3.3 nH B) In parallel with 1.8 nH 86 on the re?ected wave power from a DUT. The reference port can be easily realized using a lumped power divider. 6.4 Load Estimation for Multistate Reflectometer The matching capability of a tunable matching network can be represented by the load re?ection coe?cient ?L to be matched by the matching network. The matching capability is derived as follows. The input re?ection coe?cient ?in is written as ?in = S11 + S12 S21 ?L 1 ? S22 ?L (6?10) The equation can be rewritten in terms of the load re?ection coe?cient as S11 ? ?in S11 S22 ? S12 S21 ? S22 ?in ?L = (6?11) The matching capability is derived from the load re?ection coe?cient by setting the input re?ection coe?cient to zero. ?L ?in =0 = S11 S11 S22 ? S12 S21 (6?12) The coverage on the Smith chart speci?ed the matching capability illustrates the distribution of the load re?ection coe?cient to be matched. According to the six-port re?ectometer principle, an unknown load is the same as the point intersected by three circles, speci?ed by a center, so-called q-point, and a radius. The circle is represented by calibration constants de?ned by Equation 6?6. Note that the q-point does not change even if a mismatched load varies, whereas the load estimation method proposed in [25] has changed the circle center as a mismatched load varies. The constant q-point enables to keep the optimum-performance criteria for the multistate re?ectometer over various mismatched loads. In reality, the three circles represented by calibration constants seldom intersect at a point due to the non-ideal e?ects. The geometric center of the overlap of the circles is 87 estimated by the radical center and least square estimation methods widely used for the six-port re?ectometer [27]. The radical center is the approximation of the center of three circles? overlapped region. The coordinates of the radical center are given as 2 2 RL1 ? RL2 + x22 2x2 2 2 R ? RL3 + x23 + y32 ? 2xx3 y = L1 2y3 x= (6?13) (6?14) Least square ?tting can enhance the accuracy of load estimation especially when three circles failed to meet the optimum-performance criteria. The load re?ection coe?cient can be obtained by the least square equation as ??L = arg min ? |Circle(V) ? ?L |2 (6?15) V where V = (V1 , V2 , V3 )T is a bias voltage vector for three varactors. Note that Circle(V) is represented by calibration constants de?ned by Equation 6?6 and the distance between a circle and a point is de?ned as a distance between a tangential line to the circle and the point. 6.5 Experimental Results The matching capability of the proposed three-port re?ectometer, de?ned as S11 /(S11 S22 ? S12 S21 ), is shown in Figure 6-5. The matching capability was measured by changing each varactor bias from 0 V to 5.12 V in 16 levels by 0.32 V step. The matching capability covers a unit circle on the Smith chart completely, showing its capability on any passive mismatched load. One of important calibration constants is q-point given by the equation qi = Si1 S22 Si1 ? S21 Si2 (6?16) The q-points were measured with respect to the same bias range as used for the matching capability. The measured q-points are distributed along the unit circle as shown in Figure 88 Table 6-1. Summary of mismatched loads Mismatched load |S11 | at 2.4 GHz Tuner motor position Matched 0.01 (100, 5000, 5000) Slightly mismatched #1 Slightly mismatched #2 Slightly mismatched #3 0.13 0.38 0.20 (16725, 2262, 5000) (17105, 1424, 5000) (20464, 2228, 5000) Heavily mismatched #1 Heavily mismatched #2 Heavily mismatched #3 0.78 0.76 0.69 (15781, 526, 5000) (17835, 624, 5000) (20572, 804, 5000) 6-6. Some set of q-points can be selected from the distribution in order to satisfy the optimum-performance criteria for the multistate re?ectometer. A set of q-points was carefully selected for better estimation performance. The load estimation using three-port re?ectometer was performed separately on slightly and heavily mismatched loads. The speci?cation of mismatched loads is given in Table 6-1. First, the S-parameter of the input port was converted from the high impedance port emulating high impedance power detector. The calibration constants were obtained from the S-parameters through direct measurement of the three-port re?ectometer and conversion using the high impedance power detector. The q-points were chosen to achieve higher estimation accuracy for two separate experiments. Then, radical center estimation was applied to estimate an unknown load re?ection coe?cient. When the magnitude of the estimated re?ection coe?cient is larger than one, it is incorrect for passive mismatched loads. In this case, the magnitude was set to one with keeping the phase. As shown in Figure 6-7, estimation of slightly mismatched loads showed much smaller estimation error than heavily mismatched loads. The mean square error for slightly and heavily mismatched loads are 0.09 and 0.80, respectively. Due to the larger estimation error, the estimated load re?ection coe?cient of the heavily mismatched loads often go beyond a unit circle. As described, the magnitude was set to 1 and only the phase was kept for impedance matching. However, the estimated phase is still quite useful because 89 the estimated magnitude larger than one often results from heavily mismatched load, whose magnitude is close to one. The next chapter will demonstrate that an automatic matching control can achieve impedance matching using the estimated phase information. 6.6 Conclusion We proposed a three-port lumped-element re?ectometer for both load estimation and impedance matching. The proposed re?ectometer can be easily extended to a four-port re?ectometer by adding the suggested power divider. The load estimation method demonstrated that the tunable multistate re?ectometer can help the automatic matching control (AMC) to estimate a load re?ection coe?cient as well as to set impedance matching. The high impedance power detector replaced the distributed coupler and realized the dramatic size reduction of an automatic matching control system without compromising the load estimation and matching capability. The matching capability covered completely the unit circle on the Smith chart. Although the load estimation result is not accurate to be used as an high-precision instrument, the estimated phase information can still enable the automatic matching control to achieve faster impedance matching on heavily mismatched loads. We are working toward the integration of the proposed load estimation and a novel automatic matching control system capable of an immediate impedance matching. 90 +j1.0 +j0.5 +j2.0 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 ?j0.2 \infty ?j5.0 ?j0.5 ?j2.0 ?j1.0 Figure 6-5. Matching capability of three-port re?ectometer at 2.4 GHz 91 +j1.0 +j2.0 +j0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 ?j0.2 \infty ?j5.0 ?j2.0 ?j0.5 ?j1.0 Figure 6-6. The q-point distribution of three-port re?ectometer at 2.4 GHz 92 +j1.0 +j1.0 True +j2.0 Estimate ?j0.2 ?j5.0 ?j0.5 5.0 0.0 2.0 Ц +j5.0 1.0 +j0.2 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 True +j2.0 Estimate +j0.5 0.2 +j0.5 ?j0.2 ?j5.0 ?j0.5 ?j2.0 Ц ?j2.0 ?j1.0 ?j1.0 A B Figure 6-7. Multistate re?ectometer estimation using estimated S11 from high impedance probe A) Small mismatch (MSE=0.09) B) Large mismatch (MSE=0.80) 93 CHAPTER 7 AUTOMATIC MATCHING CONTROL USING LOAD ESTIMATION 7.1 Overview An automatic matching control (AMC) system has been developed to automate time-consuming impedance matching procedure [5][6][21]. The impedance matching of the automatic matching control was performed by recon?guring a tunable matching network until the lowest mismatch is achieved. The recon?guration was controlled by heuristic iterative methods, which showed a good trade-o? between system response and impedance matching accuracy. Also, load estimation reusing the existing tunable matching network of the automatic matching control system was proposed to facilitate the automation of impedance matching [25]. In this work, the load estimation technique was integrated with the existing automatic matching control system to achieve immediate impedance matching without compromising matching accuracy. Traditional automatic matching control systems achieved impedance matching of unknown or even varying mismatched loads by the feedback loop of a tunable matching network, a mismatch detector, and match control circuit [21]. The feedback loop is controlled by iterative methods of a match control circuit, which searches for the value of tuning elements in a trial-and-error process. However, the trial-and-error approach slowed down the system response and various heuristic approaches have been developed to improve the system response without compromising matching capability. Nevertheless, the system response of the heuristic approaches is still proportional to the complexity of the matching network and gets slower as more tuning elements and levels are added. We will demonstrate that an estimated load can be used for a matching control circuit to achieve immediate impedance matching without using heuristic approaches. The proposed matching control can ?nd the value of tuning elements by examining the characterization table of a matching network. Therefore, the precise characterization as 94 well as the load estimation play an important role in this immediate impedance matching approach. Various characterization methods for a microwave device have been reported to reduce the microwave design complexity. A neural network has been widely used to characterize microwave devices, such as the approximation of S-parameters of BJTs [36] and modeling parameters of microwave components [9]. Also, closed form equation was also presented for S-parameters of BJTs [37]. In this work, neural network models and the closed form equation were used to approximate measured S-parameters and the accuracy of the characterization methods was evaluated in terms of mean squared error (MSE) between true and estimated values. The proposed matching control consists of two tasks. First, a characterization table in terms of tuning elements was built from the direct measurement of the matching network or approximation models such as a neural network and closed form equations. Next, the value of the tuning elements was found by minimizing the degree of mismatch. The degree of mismatch was calculated from the magnitude of the input re?ection coe?cient. The experimental results of the immediate impedance matching approach will be presented. 7.2 Matching Control Procedures The same lumped-element tunable matching network that is used for load estimation was used to develop matching control procedures supporting load estimation presented in Chapter 5 and 6. The matching network has a ?-type bandpass ?lter topology and three varactor diodes as tuning elements. The recon?guration of the matching network was performed by changing the varactor bias voltages. Its matching capability covers all re?ection coe?cients within the unit circle on the Smith chart at 2.4 GHz. The load re?ection coe?cient ?L of a device under test (DUT) is assumed to be estimated by load estimation techniques presented in Chapter 5 and 6. When the DUT is connected to the port 2 of a tunable matching network, the input re?ection coe?cient ?in 95 looking into the port 1 of the matching network is written as follows. ?in = S11 + S12 S21 ?L 1 ? S22 ?L (7?1) where Sij is the S-parameter from port j to port i of the matching network. Note that the S-parameters are the function of a bias voltage vector, denoted by v. Therefore, the input re?ection coe?cient can be explicitly written as the function of v. ?in (v) = S11 (v) + S12 (v)S21 (v)?L , 1 ? S22 (v)?L v = (v1 , v2 , и и и , vn )T (7?2) where T denotes a transpose and vn is the nth bias voltage. The load re?ection coe?cient of the DUT can be derived from the input re?ection coe?cient. ?L = S11 (v) ? ?in (v) S11 (v)S22 (v) ? S12 (v)S21 (v) ? S22 (v)?in (v) (7?3) The mismatched load to be matched by the matching network set by a bias voltage v, denoted by ?M , is the load re?ection coe?cient that makes the input re?ection coe?cient zero. ?M (v) = ?L = ?in =0 S11 (v) S11 (v)S22 (v) ? S12 (v)S21 (v) (7?4) b, as a bias voltage Now, let us de?ne an optimal bias voltage vector, denoted by v vector that minimizes the magnitude of the input re?ection coe?cient ?in , the degree of mismatch. b = arg min |?in (v)| v v (7?5) If the bias voltage, that minimizes the input re?ection coe?cient to zero, can be found for b can be expressed all possible load re?ection coe?cients, the optimal bias voltage vector v using ?M as follows. } } { { b = v?in (v) = 0 = v?M (v) = ?L v Finding the optimal bias voltage can be expressed as ?nding a bias voltage whose ?M is equal to ?L . Therefore, the mapping table between ?M and v should be calculated 96 (7?6) Start Characterization Measure power three-times Estimate load impedance Calculate bias for matching End Figure 7-1. Automatic matching control supports load estimation. to perform the bias search and the mapping table can be converted from the matching network characterized by the S-parameters. This procedure is based on Equation 7?4. The S-parameters can be obtained from direct measurement using a vector network analyzer or a neural network ?tting model. The characterization methods will be introduced in the next Section. 7.3 Characterization of Matching Network The characterization of a matching network is a procedure to discover the S-parameter functions to be used to calculate an input re?ection coe?cient or a mismatched load to be matched. The S-parameters of the matching network were measured using a vector network analyzer while changing the bias voltage. The measurement points were determined by the number of varactors and bias voltage levels. Although the more voltage levels can produce more accurate characterization results, 16 voltage levels were chosen as good trade-o? between measurement time and characterization accuracy. The 97 characterization results were converted to a form of a mapping table, ?M (v), for easy access and searches. Unknown S-parameters between measurement points were approximated by a multivariate linear interpolation. The multivariate linear interpolation is interpolating a function of multiple variables on a regular grid, as an extension of a linear interpolation. It performs linear interpolation ?rst on one direction, then again in the other direction. Suppose we want to interpolate a value of an unknown function f at the point (x, y). The value of the function f at four neighbor points on a regular grid, f (x1 , y1 ), f (x1 , y2 ), f (x2 , y1 ), and f (x2 , y2 ), are assumed to be known, then the interpolation of the function f at the point (x, y) can be written as follows. f (x1 , y1 ) (x2 ? x)(y2 ? y) (x2 ? x1 )(y2 ? y1 ) f (x2 , y1 ) + (x ? x1 )(y2 ? y) (x2 ? x1 )(y2 ? y1 ) f (x1 , y2 ) (x2 ? x)(y ? y1 ) + (x2 ? x1 )(y2 ? y1 ) f (x2 , y2 ) + (x ? x1 )(y ? y1 ) (x2 ? x1 )(y2 ? y1 ) f (x, y) ? (7?7) If the unknown function to interpolate has a smooth surface over neighbor points, the linear interpolation reduces the number of measurement points signi?cantly without the loss of the characterization detail. The other method to estimate unknown S-parameters between measurement points is an approximation ?tting function to the measured S-parameters. As described, the measured S-parameters are the functions of the bias voltage vector v, given as follows. ? ? ?S11 (v) S12 (v)? (7?8) S(v) = ? ? S21 (v) S22 (v) The S-parameter functions were approximated by a curve ?tting model, such as a closed form and a neural network. First, an arti?cial neural network was used to approximate 98 the S-parameter functions. Four independent neural network models approximate two-port S-parameter functions, S11 (v), S12 (v), S21 (v), and S22 (v). The input and output of the neural network are bias voltage vector and real and imaginary parts of the S-parameter, respectively. The feed-forward topology was used with 15 perceptrons in the hidden layer. The well-known backpropagation algorithm was used to train the neural networks. The cascaded network is easily represented by ABCD-parameters and the matching network was decomposed into basic components, such as a transmission line, series impedance, and shunt admittance. The representation of the ABCD-parameters was used as closed-form equations. The ABCD-parameters for the basic components are given as follows. ? ? ? cos(2??) ?Z0 sin(2??)? ? ? ?Y0 sin(2??) cos(2??) ? ? ?1 Z ? ? ? 0 1 ? ? ? 1 0? ? ? Y 1 (transmission line) (7?9) (series impedance) (7?10) (shunt admittance) (7?11) The closed-form ABCD-parameters were derived from the cascaded network of a transmission line (?1 ), a ? network (Y1 , Z3 , and Y2 ), and a transmission line (?2 ). The varactor capacitance was calculated from the varactor SPICE model used in Chapter 5 and 6. ABCD11 =(cos(2??1 )(1 + Y2 Z3 ) + ?Z0 sin(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) cos(2??2 ) +? (cos(2??1 )Z3 + ?Z0 sin(2??1 )(1 + Y1 Z3 )) Y0 sin(2??2 ) (7?12) ABCD12 =? (cos(2??1 )(1 + Y2 Z3 ) + ?Z0 sin(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) Z0 sin(2??2 ) + (cos(2??1 )Z3 + ?Z0 sin(2??1 )(1 + Y1 Z3 )) cos(2??2 ) 99 (7?13) ABCD21 =(?Y0 sin(2??1 )(1 + Y2 Z3 ) + cos(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) cos(2??2 ) +? (?Y0 sin(2??1 )Z3 + cos(2??1 )(1 + Y1 Z3 )) Y0 sin(2??2 ) (7?14) ABCD22 =? (?Y0 sin(2??1 )(1 + Y2 Z3 ) + cos(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) Z0 sin(2??2 ) + (?Y0 sin(2??1 )Z3 + cos(2??1 )(1 + Y1 Z3 )) cos(2??2 ) (7?15) The admittance Y1 , Y2 and impedance Z3 are the function of bias voltage which determines the varactor capacitance. The transmission line delay ?1 , ?2 , parasitic parameters, ?tting parameters of the varactor SPICE model are trained by a nonlinear least square ?tting algorithm. The ABCD-parameters based ?tting model was converted to S-parameters for fair comparison with direct measurement and neural network model. 7.4 Bias Search for Impedance Matching The goal of bias search is to ?nd the optimal bias voltage vector given by { } { } b = v?in (v) = 0 = v?M (v) = ?L v (7?16) Due to the discrete measurement data of a mapping table ?M (v), it is not always possible to ?nd the optimal bias voltage vector. Instead, we choose the bias voltage vector closest to the optimal bias voltage vector and this procedure can be written as follows. b = arg min |?M (v) ? ?L | v v (7?17) From now on, the optimal bias voltage vector is rede?ned as the bias minimizing the magnitude between ?M and ?L . The bias search consists of two steps, coarse and ?ne search. The coarse search was performed on the mapping table converted from the direct measurement of S-parameters. 100 Then, the ?ne search was performed again on the linear interpolation of a voxel1 of the bias voltage vector found in the coarse search. The cost function for both coarse and ?ne search is mean square error (MSE) of magnitude between ?M of eight bias voltage vectors of a voxel and an estimated load ?L . Note that a voxel for 3D data has eight vertexes, similar to a cubic. For example, a unit voxel consists of eight points, (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), and (1,1,1). Eight points of the voxel for the coarse search were selected from the voltage bias used for the direct measurement. Once the voxel is found by the coarse search, the mapping table of the voxel is interpolated by multivariate linear interpolation with higher resolution. Then, the ?ne search is performed on the interpolated data. The proposed two-step search reduces the required number of measurement points and the memory usage by the partial interpolation. In addition, the bias search can be made faster using binary search algorithm on sorted data. 7.5 Characterization Results The S-parameters of a tunable matching network are the function of bias voltage. The S-parameters and the corresponding bias voltages should be measured together. During the measurement, the bias voltage of each varactor was set by a 12 bit digitalto-analog converter (DAC) from 0 to 5 V by 0.32 V step in 16 levels. The S-parameters were measured using a vector network analyzer at 2.4 GHz. All measurement procedures were automated by an instrument control program written in MATLAB. The program running in a host computer communicated with a microcontroller and a DAC to set bias voltage, then sent a GPIB command for the network analyzer to measure S-parameters. Note that the microcontroller will eventually replace the host computer and implement all control programs. Due to the slow response of the network analyzer, the network analyzer failed to measure correct S-parameters immediately after changing bias voltage. 1 A voxel is a basic unit cell for 3D data, similar to a pixel for 2D picture. 101 The control program paused for a few seconds after bias setting such that the network analyzer could get into steady state and measure correct S-parameters. The bias voltage of three varactors was changed in 16 levels, therefore the total number of measurements is 16 О 16 О 16 = 4096. Four neural network models approximated S11 , S12 , S21 , and S22 , respectively and were trained by backpropagation algorithm. The input, hidden, and output layers consist of 3 bias voltages, 15 hidden perceptrons, and 2 (real and imaginary) parts of S-parameters, respectively. Because the matching network is a reciprocal passive network, the neural network models for S12 and S21 should generate similar outputs and can be merged into a single model. During training the neural networks, 10% of the S-parameter data were used for validation and 10% for testing purposes. The estimation results using the trained neural network showed good agreement with the measurement data, as shown in Figure 7-2. As expected, the results for S12 and S21 were close enough to merge into a single model. To check the over?tting problem of neural networks, the trained networks were compared using training and testing data. The compared estimation results, as shown in Figure 7-3, demonstrated the comparable error for both data, therefore the trained networks have no over?tting problem and can approximate unknown S-parameter data. Four closed-form models representing S11 , S12 , S21 , and S22 , were trained by a nonlinear least square algorithm. The estimation results using the trained closed-form models were compared with the S-parameter measurement data. The estimation error was higher than the neural network models by a few orders of magnitude, as shown in Figure 7-4. To improve the closed-form models, more parasitic e?ects and ?tting parameters can be added, but more parameters may cause optimization problems such as initial parameter setting and local minima. In this work, only the neural network models were used to evaluate the automatic matching control. The estimation statistics of neural network and closed-form models are summarized in Table 7-1 and 7-2. 102 +j1.0 +j1.0 True +j2.0 Estimate ?j0.5 A B +j1.0 +j1.0 ?j5.0 ?j0.5 5.0 5.0 0.0 1.0 Ц +j5.0 0.5 +j0.2 0.2 5.0 2.0 1.0 ?j0.2 True +j2.0 Estimate +j0.5 +j5.0 0.5 ?j2.0 ?j1.0 True +j2.0 Estimate ?j0.2 ?j2.0 Ц ?j5.0 ?j0.5 ?j2.0 ?j1.0 ?j1.0 C D Figure 7-2. Neural network models for a 2-port matching network were trained by backpropagation. A) S11 B) S21 C) S12 D) S22 103 Ц ?j5.0 ?j1.0 +j0.2 2.0 ?j0.2 ?j2.0 +j0.5 0.2 1.0 0.0 ?j5.0 ?j0.5 0.0 Ц +j5.0 2.0 ?j0.2 +j0.2 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 True +j2.0 Estimate +j0.5 0.2 +j0.5 +j1.0 +j1.0 True +j2.0 Estimate ?j0.5 A B +j1.0 +j1.0 ?j5.0 ?j0.5 5.0 5.0 0.0 1.0 Ц +j5.0 0.5 +j0.2 0.2 5.0 2.0 1.0 ?j0.2 True +j2.0 Estimate +j0.5 +j5.0 0.5 ?j2.0 ?j1.0 True +j2.0 Estimate ?j0.2 ?j2.0 Ц ?j5.0 ?j1.0 +j0.2 2.0 ?j0.2 ?j2.0 +j0.5 0.2 1.0 0.0 ?j5.0 ?j0.5 0.0 Ц +j5.0 2.0 ?j0.2 +j0.2 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 True +j2.0 Estimate +j0.5 0.2 +j0.5 Ц ?j5.0 ?j0.5 ?j2.0 ?j1.0 ?j1.0 C D Figure 7-3. Neural network models for a 2-port matching network were tested by 10% of measurement data. A) S11 B) S21 C) S12 D) S22 104 +j1.0 +j1.0 True +j2.0 Estimate ?j0.5 A B +j1.0 +j1.0 ?j5.0 ?j0.5 5.0 5.0 0.0 1.0 Ц +j5.0 0.5 +j0.2 0.2 5.0 2.0 1.0 ?j0.2 True +j2.0 Estimate +j0.5 +j5.0 0.5 ?j2.0 ?j1.0 True +j2.0 Estimate ?j0.2 ?j2.0 Ц ?j5.0 ?j1.0 +j0.2 2.0 ?j0.2 ?j2.0 +j0.5 0.2 1.0 0.0 ?j5.0 ?j0.5 0.0 Ц +j5.0 2.0 ?j0.2 +j0.2 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 True +j2.0 Estimate +j0.5 0.2 +j0.5 Ц ?j5.0 ?j0.5 ?j2.0 ?j1.0 ?j1.0 C D Figure 7-4. Closed form models for a 2-port matching network were trained by nonlinear least square ?tting. A) S11 B) S21 C) S12 D) S22 105 Table 7-1. Mean square error (MSE) of neural network ?tting models using training and testing data Type Training data Testing data S11 S21 S12 S22 1.90 О 10?4 1.50 О 10?4 1.79 О 10?4 2.49 О 10?4 2.25 О 10?4 1.69 О 10?4 2.06 О 10?4 2.80 О 10?4 Table 7-2. Average error of closed-form models Type Mean square error Average error S11 S21 S12 S22 1.20 О 10?2 5.99 О 10?3 6.00 О 10?3 7.95 О 10?3 1.10 О 10?1 7.74 О 10?2 7.75 О 10?2 8.92 О 10?2 The S-parameters were converted to a mapping table, which is used for bias search and matching control, and the conversion procedure is described as follows. Two di?erent forms of the mapping table were used in this work. The ?rst form of the mapping table, as shown in Table 7-3, consists of a bias voltage vector and the corresponding mismatched load to be matched by a matching network. The mapping table was obtained by Equation 7?4 and S-parameter data. However, the magnitude of the mismatched load to be matched may be out of the unit circle on the Smith chart and the value is incorrect for a passive network. In this case, the corresponding bias voltage cannot be chosen for impedance matching of any mismatched load and can be removed from the mapping table. The second form of the mapping table, as shown in Table 7-4, is an inverse mapping table, where mismatched loads to be matched are evenly distributed on the Smith chart and the corresponding optimal bias voltage vectors were calculated through the two-step search. 7.6 Impedance Matching Results Impedance matching was performed by matching control using bias search on the mapping table. Mismatched loads was estimated by coupler-free and re?ectometer load 106 Table 7-3. Mapping table between bias voltage and a mismatched load to be matched Bias voltage (V) Mismatched load re?ection coe?cient to be matched by matching network 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, +0.54 ? ?0.99 +0.36 ? ?1.08 +0.13 ? ?1.16 ?0.18 ? ?1.22 ?0.60 ? ?1.18 ?1.11 ? ?0.91 ?1.54 ? ?0.32 ?1.58 + ?0.55 ?1.13 + ?1.24 ?0.50 + ?1.53 .. . 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, .. . 0.00 0.31 0.63 0.94 1.26 1.57 1.89 2.20 2.52 2.83 Table 7-4. Inverse mapping table between a mismatched load to be matched and bias voltage Mismatched load re?ection coe?cient to be matched by matching network Bias voltage (V) +0.00 +0.50 +0.25 + ?0.43 ?0.25 + ?0.43 ?0.50 + ?0.00 ?0.25 ? ?0.43 +0.25 ? ?0.43 +1.00 +0.87 + ?0.50 +0.50 + ?0.87 +0.00 + ?1.00 ?0.50 + ?0.87 ?0.87 + ?0.50 ?1.00 ?0.87 ? ?0.50 ?0.50 ? ?0.87 ?0.00 ? ?1.00 +0.50 ? ?0.87 +0.87 ? ?0.50 (4.22, (2.61, (4.72, (2.30, (1.89, (1.57, (0.31, (1.35, (0.00, (2.64, (0.50, (0.09, (3.15, (4.38, (2.36, (3.59, (4.72, (2.30, (0.72, 107 4.12, 2.01, 3.46, 2.20, 1.67, 0.00, 0.85, 0.13, 0.00, 1.98, 0.09, 0.09, 3.78, 3.18, 4.72, 2.86, 3.27, 2.11, 0.00, 2.83) 4.72) 2.99) 2.01) 1.64) 0.31) 4.63) 0.06) 4.72) 0.82) 0.35) 0.00) 4.60) 4.12) 3.15) 3.49) 2.99) 1.79) 4.72) estimation methods. The mapping table was converted from the characterization results and examined by the two-step search for an optimal bias voltage. The optimal bias voltage was applied to the matching network and the degree of mismatch, the magnitude of the input re?ection coe?cient, was measured to evaluate the capability of impedance matching. The slightly and heavily mismatched loads used for load estimation were con?gured as the same way used in Chapter 5 and 6. The experimental results demonstrated that the matching control could achieve immediate impedance matching through the load estimation and the mapping table. The optimal bias voltage and the degree of mismatch were summarized in Table 7-5, 7-6, 7-7, and 7-8. Because the re?ectometer load estimation method achieved lower estimation error than the coupler-free load estimation, the impedance matching by the re?ectometer resulted in the lower degree of mismatch. In addition, S-parameter data estimated by the neural network models were so close to measurement data that the bias search and the corresponding impedance matching were almost identical considering the measurement error. The impedance matching results demonstrated that the neural network model can be used for the automatic matching control. The number of coe?cients of each neural network models is as low as 20, because the input, hidden, and output layers have 3, 15, and 2 nodes. Compared with the large size of S-parameter measurement data, 163 = 4096, the neural network model is more appropriate for compact microcontroller system with limited available memory. The estimate of the mismatched loads and the load re?ection coe?cient matched by the matching network were plotted in Figure 7-5 and 7-6, respectively. Due to the small estimation error, a few hundredth, of neural network models, the mapping table converted from the neural network models was almost identical to that of measurement 108 Table 7-5. Impedance matching results using coupler-free load estimation and S-parameter measurement data Mismatched load ?L Optimal bias voltage (V) Input re?ection coe?cient ?in Degree of mismatch |?in | +0.14 ? ?0.77 +0.64 + ?0.40 ?0.69 + ?0.04 (2.03, 0.17, 4.39) (4.71, 0.02, 4.71) (0.02, 4.71, 2.38) +0.04 + ?0.16 +0.22 + ?0.09 +0.05 ? ?0.06 0.16 0.24 0.08 ?0.01 + ?0.01 +0.12 ? ?0.01 +0.38 + ?0.06 ?0.20 ? ?0.01 (4.49, (4.05, (3.70, (4.45, +0.03 + ?0.03 +0.02 ? ?0.01 +0.03 + ?0.01 +0.01 + ?0.03 0.04 0.02 0.03 0.03 4.64, 1.05, 0.08, 4.30, 4.45) 1.59) 4.45) 4.64) 0.09 Average Table 7-6. Impedance matching results using three-port re?ectometer load estimation and S-parameter measurement data Mismatched load ?L Optimal bias voltage (V) Input re?ection coe?cient ?in Degree of mismatch |?in | +0.14 ? ?0.77 +0.64 + ?0.40 ?0.69 + ?0.04 (1.68, 0.87, 4.42) (2.44, 0.14, 4.68) (0.02, 4.01, 2.28) +0.05 + ?0.17 ?0.05 ? ?0.05 ?0.13 + ?0.03 0.18 0.07 0.13 ?0.01 + ?0.01 +0.12 ? ?0.01 +0.38 + ?0.06 ?0.20 ? ?0.01 (4.23, (4.11, (3.20, (4.49, +0.01 + ?0.00 ?0.02 ? ?0.01 +0.01 ? ?0.01 +0.01 + ?0.02 0.01 0.02 0.01 0.02 4.30, 0.87, 0.02, 4.20, 4.20) 1.68) 4.64) 4.71) Average 0.06 Table 7-7. Impedance matching results using coupler-free load estimation and S-parameters estimated by neural network models Mismatched load ?L Optimal bias voltage (V) Input re?ection coe?cient ?in Degree of mismatch |?in | +0.14 ? ?0.77 +0.64 + ?0.40 ?0.69 + ?0.04 (1.28, 4.30, 0.02) (4.71, 4.71, 4.14) (1.94, 0.11, 2.28) +0.01 + ?0.15 +0.21 + ?0.08 +0.01 ? ?0.06 0.15 0.22 0.06 ?0.01 + ?0.01 +0.12 ? ?0.01 +0.38 + ?0.06 ?0.20 ? ?0.01 (4.61, (4.61, (4.52, (4.11, +0.02 + ?0.05 +0.02 ? ?0.00 +0.03 + ?0.01 +0.01 + ?0.03 0.05 0.02 0.03 0.03 1.34, 1.46, 3.60, 0.08, 4.49) 4.23) 4.23) 4.71) Average 0.08 109 Table 7-8. Impedance matching results using three-port re?ectometer load estimation and S-parameters estimated by neural network models Mismatched load ?L Optimal bias voltage (V) Input re?ection coe?cient ?in Degree of mismatch |?in | +0.14 ? ?0.77 +0.64 + ?0.40 ?0.69 + ?0.04 (1.28, 4.42, 0.02) (2.69, 2.41, 3.89) (1.68, 0.05, 2.35) +0.05 + ?0.17 ?0.04 ? ?0.05 ?0.15 + ?0.03 0.18 0.06 0.15 ?0.01 + ?0.01 +0.12 ? ?0.01 +0.38 + ?0.06 ?0.20 ? ?0.01 (4.20, (4.42, (4.33, (4.17, +0.01 + ?0.00 ?0.01 ? ?0.00 +0.02 ? ?0.02 +0.02 + ?0.02 0.01 0.01 0.03 0.03 0.74, 1.90, 3.07, 0.02, 4.45) 4.55) 4.11) 4.71) Average 0.07 data. Therefore, the impedance matching results from both mapping tables were also almost identical. The matching control achieved the low degree of mismatch, -26 dB or less, for the slightly mismatched loads. The larger estimation error for the heavily mismatched loads results in the larger degree of mismatch, around -12 dB or less, depending on the accuracy of load estimation. The matching control can be integrated with one of the load estimation methods. The coupler-free load estimation can be implemented in a compact size, whereas the re?ectometer load estimation can achieve better impedance matching for heavily mismatched loads. 7.7 Conclusion We demonstrated that an automatic matching control system could achieve an immediate impedance matching by utilizing load estimation techniques and the characterization of a matching network. The matching network was characterized by neural network models and closed-form equations as well as S-parameter measurement data. The neural network models achieved the comparable characterization accuracy with much fewer coe?cients than measurement data. The smaller number of coe?cients can be easily stored and implemented by a compact microcontroller with built-in memory. 110 +j1.0 +j1.0 True +j2.0 Estimate Matched by AMC ?j0.2 ?j5.0 ?j0.5 ?j0.2 ?j0.5 ?j2.0 ?j2.0 ?j1.0 ?j1.0 A B +j1.0 True +j2.0 Estimate Matched by AMC ?j0.2 ?j5.0 5.0 0.0 2.0 Ц +j5.0 1.0 +j0.2 0.5 5.0 2.0 1.0 0.5 +j5.0 0.2 +j0.2 True +j2.0 Estimate Matched by AMC +j0.5 0.2 +j0.5 ?j0.5 Ц ?j5.0 +j1.0 0.0 5.0 0.0 2.0 Ц +j5.0 1.0 +j0.2 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 True +j2.0 Estimate Matched by AMC +j0.5 0.2 +j0.5 ?j0.2 ?j2.0 Ц ?j5.0 ?j0.5 ?j2.0 ?j1.0 ?j1.0 C D Figure 7-5. S-parameters from vector measurement and neural network model were compared in terms of coupler-free load estimation assisted matching control. A) Vector measurement by large mismatch B) Neural network model by large mismatch C) Vector measurement by small mismatch D) Neural network model by small mismatch 111 +j1.0 +j1.0 True +j2.0 Estimate Matched by AMC ?j0.2 ?j5.0 ?j0.5 ?j0.2 ?j0.5 ?j2.0 ?j2.0 ?j1.0 ?j1.0 A B +j1.0 True +j2.0 Estimate Matched by AMC ?j0.2 ?j5.0 5.0 0.0 2.0 Ц +j5.0 1.0 +j0.2 0.5 5.0 2.0 1.0 0.5 +j5.0 0.2 +j0.2 True +j2.0 Estimate Matched by AMC +j0.5 0.2 +j0.5 ?j0.5 Ц ?j5.0 +j1.0 0.0 5.0 0.0 2.0 Ц +j5.0 1.0 +j0.2 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 True +j2.0 Estimate Matched by AMC +j0.5 0.2 +j0.5 ?j0.2 ?j2.0 Ц ?j5.0 ?j0.5 ?j2.0 ?j1.0 ?j1.0 C D Figure 7-6. S-parameters from vector measurement and neural network model were compared in terms of re?ectometer load estimation assisted matching control. A) Vector measurement by large mismatch B) Neural network model by large mismatch C) Vector measurement by small mismatch D) Neural network model by small mismatch 112 The bias search was performed on a mapping table between the mismatched load and bias voltage, which approximated the inverse function of the mismatched load to be matched. The two-step coarse and ?ne search algorithm could achieve the same degree of mismatch with a fewer number of measurement data points than one-step full search. The experimental results showed that the degree of mismatch could be as low as -12 dB for heavily mismatched loads and -26 dB for slightly mismatched loads. Note that the bias search can be made faster by using binary search and the sorted mapping table. Although the matching control and matching network in this work were developed for on-chip embedded test systems, they can be applied to di?erent matching networks, such as distributed or broadband. In addition, load estimation and microwave characterization will get more attention with a rise of automated RF systems, because the automatic matching control system utilizing load estimation needs to know the characteristics of a matching network. 113 CHAPTER 8 CONCLUSION The impedance matching of radio frequency (RF) ports of a socketed device under test (DUT), which usually su?ers from a pogo pin connection, poorly de?ned RF tolerance, and manufacturing variation, is di?cult, time-consuming, and empirical process. In this work, a recon?gurable matching network works with adaptive matching control to achieve the impedance matching and to reduce the undesired e?ects. In addition, the matching control is assisted by the characterization and microwave modeling of the matching network. Background theory presented in Chapter 2 provides the basic of a lumped impedance matching, various nonlinear microwave modeling techniques, and the fundamentals of six-port and four-port re?ectometers. An automatic matching control (AMC) presented in Chapter 3 facilitates the impedance matching over a broadband frequency using a microstrip bandpass ?lter. The microstrip ?lter consists of ?ve stubs and three varactors connected to the end of the stub. The center frequency and the bandwidth are 3.5 GHz and 2 GHz, respectively and the insertion loss over the passband is as low as 2 dB. The proposed system employs a greedy search algorithm to determine the varactor biases for impedance match over a large fractional bandwidth 71% = 2.5 / 3.5. The greedy algorithm outperforms brute-force and single-step algorithms in terms of the number of trials and the available bandwidth, respectively. The work demonstrates the feasibility of the automatic matching control circuit over the broadband frequencies. Load estimation techniques presented in Chapter 4 exploits the principle of a six-port re?ectometer to calculate the complex re?ection coe?cient of a DUT from three wave power readings. Three wave power detectors are realized by changing varactor biases and measuring re?ected wave power using a scalar network analyzer. The performance of the estimation is evaluated by the mean square error (MSE) between true and estimated 114 re?ection coe?cients. It demonstrates that tunable matching networks and compact power detectors can be used to estimate the load re?ection coe?cient of a DUT for the automatic matching control (AMC) system. A coupler-free load estimation presented in Chapter 5 demonstrates the dramatic reduction of the automatic matching control system size by replacing a distributed coupler with a high impedance probe. The coupler-free load estimation discovers an unknown load from the power of the combined incident and re?ected waves measured by a high impedance probe. The experimental results show that the coupler and coupler-free load estimation methods are comparable in terms of estimation performance. The lumped-element re?ectometer presented in Chapter 6 is designed not only for an automatic matching control, but also used as a multistate re?ectometer for load estimation. A lossless ? section network is used as a network topology for the proposed three-port. The four-port re?ectometer can be extended by adding a reference port to the three-port re?ectometer and the reference port can be realized with a resistive power divider. The three-port matching network was fabricated on a FR4 printed circuit board (PCB) using lumped chip inductors and varactors. The re?ectometer analysis and experimental result showed that the matching capability could cover a unit circle on the Smith chart and provide the optimum performance criteria of the matching network. The novel matching control and characterization methods presented in Chapter 7 are developed to support load estimation techniques. A tunable matching network is characterized by vector measurement or neural network models of S-parameters. The S-parameter data are converted to a mapping table of a load re?ection to be matched by the matching control. The equation of the mapping table is derived from the analysis of a bias voltage vector, an input re?ection coe?cient, and a load re?ection coe?cient. The optimal bias minimizing the degree of mismatch is discovered by two-step bias search. The two-step search performs coarse search on the mapping table followed by ?ne search on the multivariate linear interpolation of the mapping table. The experimental results 115 show that the neural network ?tting model achieves comparable accuracy as vector measurement and that the impedance matching by the proposed matching control is as low as -12 and -26 dB for heavily and slightly mismatched loads, respectively. One of the most challenging tasks in implementing the automatic matching control system with load estimation is the full vector measurement of the matching network. In case of the embedded RF test system where the embedded matching network has no external node, it is often very di?cult, even if not impossible, to measure using a regular vector network analyzer. The development of a measurement procedure using only existing power detectors will be valuable. Some researchers have proposed a calibration method using only power detectors [38] and the calibration of multistate perturbation-two-port (PTP) [39]. The measurement of the embedded matching network can be implemented in similar way. In addition, although single-ended RF system was assumed in this work, the proposed work can be extended to a di?erential RF system. The preliminary analysis of an input re?ection coe?cient of di?erential RF system is provided (Appendix C and D). 116 APPENDIX A DERIVATION OF LOAD IMPEDANCE CIRCLE EQUATION USING INPUT IMPEDANCE MAGNITUDE A complex input re?ection coe?cient is written in terms of two-port S-parameters Sij and a load re?ection coe?cient ?L as ?in = S11 + S12 S21 ?L 1 ? S22 ?L (A?1) The magnitude of the input re?ection coe?cient becomes S S ? 12 21 L |?in | = S11 + 1 ? S22 ?L S11 ? S11 S22 ?L + S12 S21 ?L |?in | = 1 ? S22 ?L (A?2) (A?3) By taking the square of both sides, the equation is written as |1 ? S22 ?L |2 |?in |2 = |S11 ? ??L |2 (A?4) ? = S11 S22 ? S12 S21 (A?5) where The equation is manipulated as follows. ? ? |?in |2 + |S22 |2 |?L |2 |?in |2 ? S22 ?L |?in |2 ? S22 ?L |?in |2 ? = |S11 |2 + |?|2 |?L |2 ? ??L S11 ? ?? ??L S11 (A?6) where * denotes a complex conjugate. ? )?L (|S22 |2 |?in |2 ? |?|2 )?L ??L ? (S22 |?in |2 ? ?S11 ? |?in |2 ? ?? S11 )??L = |S11 |2 ? |?in |2 ?(S22 ?L ??L ? ? ? |?in |2 ? ?? S11 )??L )?L + (S22 (S22 |?in |2 ? ?S11 |S22 |2 |?in |2 ? |?|2 |S11 |2 ? |?in |2 = |S22 |2 |?in |2 ? |?|2 117 (A?7) (A?8) 2 ? ? 2 ) (S |? | ? ?S 22 in 11 ?L ? |S22 |2 |?in |2 ? |?|2 ? 2 S22 |?in |2 ? ?S11 |S11 |2 ? |?in |2 + = |S22 |2 |?in |2 ? |?|2 |S22 |2 |?in |2 ? |?|2 |S11 |2 |S22 |2 |?in |2 ? |?|2 |S11 |2 ? |S22 |2 |?in |4 + |?|2 |?in |2 = ||S22 |2 |?in |2 ? |?|2 |2 ? ? |S22 |2 |?in |4 + |?|2 |S11 |2 ? ?? S11 S22 |?in |2 ? ?S11 S22 |?in |2 + ||S22 |2 |?in |2 ? |?|2 |2 = ? ? |S11 |2 |S22 |2 + |?|2 ? ?? S11 S22 ? ?S11 S22 |?in |2 2 2 2 2 ||S22 | |?in | ? |?| | (A?9) (A?10) (A?11) 2 S11 S22 ? ? |?in |2 = 2 2 2 |S22 | |?in | ? |?| (A?12) 2 S12 S21 |?in |2 = |S22 |2 |?in |2 ? |?|2 (A?13) Now, the equation represents a circle on the Smith chart. 2 ? ? ?L ? (S22 |?in | ? ?S11 ) |S22 |2 |?in |2 ? |?|2 S12 S21 |?in | = 2 2 2 |S22 | |?in | ? |?| (A?14) The center and radius of the circle are given by ? ? (S22 |?in |2 ? ?S11 ) CL = 2 2 |S22 | |?in | ? |?|2 S12 S21 |?in | RL = 2 2 2 |S22 | |?in | ? |?| (center) (A?15) (radius) (A?16) When the incident and re?ected waves are combined under the absence of a coupler, the measured power of the input port (P1) is expressed as pin = |a1 + b1 |2 = |a1 |2 |1 + ?in |2 118 (A?17) Therefore, |1 + ?in | instead of |?in | can be used to estimate a load impedance. The complex input re?ection coe?cient is rewritten as 1 + ?in = 1 + S11 + S12 S21 ?L 1 ? S22 ?L (A?18) The magnitude of the input re?ection coe?cient becomes S S ? 12 21 L |1 + ?in | = 1 + S11 + 1 ? S22 ?L 1 + S11 ? (1 + S11 )S22 ?L + S12 S21 ?L |1 + ?in | = 1 ? S22 ?L (A?19) (A?20) By taking the square of both sides, the equation is written as |1 ? S22 ?L |2 |1 + ?in |2 = |1 + S11 ? (S22 + ?)?L |2 (A?21) ? = S11 S22 ? S12 S21 (A?22) where The equation is manipulated as follows. ? ? |1 + ?in |2 + |S22 |2 |?L |2 |1 + ?in |2 ? S22 ?L |1 + ?in |2 ? S22 ?L |1 + ?in |2 ? ? = |1 + S11 |2 + |S22 + ?|2 |?L |2 ? (S22 + ?)?L (1 + S11 ) ? (S22 + ?? )??L (1 + S11 )(A?23) where * denotes a complex conjugate. ? (|S22 |2 |1 + ?in |2 ? |S22 + ?|2 )?L ??L ? (S22 |1 + ?in |2 ? (S22 + ?)(1 + S11 ))?L ? ? ?(S22 |1 + ?in |2 ? (S22 + ?? )(1 + S11 ))??L = |1 + S11 |2 ? |1 + ?in |2 ?L ??L ? (A?24) ? ? ? + ?? )(1 + S11 ))??L |1 + ?in |2 ? (S22 ))?L + (S22 (S22 |1 + ?in |2 ? (S22 + ?)(1 + S11 |S22 |2 |1 + ?in |2 ? |S22 + ?|2 |1 + S11 |2 ? |1 + ?in |2 = (A?25) |S22 |2 |1 + ?in |2 ? |S22 + ?|2 119 2 ? ? 2 (S |1 + ? | ? (S + ?)(1 + S )) 22 in 22 11 ?L ? |S22 |2 |1 + ?in |2 ? |S22 + ?|2 ? 2 S22 |1 + ?in |2 ? (S22 + ?)(1 + S11 |1 + S11 |2 ? |1 + ?in |2 ) + = |S22 |2 |1 + ?in |2 ? |S22 + ?|2 |S22 |2 |1 + ?in |2 ? |S22 + ?|2 (A?26) |1 + S11 |2 |S22 |2 |1 + ?in |2 ? |S22 + ?|2 |1 + S11 |2 ? |S22 |2 |1 + ?in |4 + |S22 + ?|2 |1 + ?in |2 = ||S22 |2 |1 + ?in |2 ? |S22 + ?|2 |2 |S22 |2 |1 + ?in |4 + |S22 + ?|2 |1 + S11 |2 + ||S22 |2 |1 + ?in |2 ? |S22 + ?|2 |2 ? ? (S ? + ?? )(1 + S11 )S22 |1 + ?in |2 + (S22 + ?)(1 + S11 )S22 |1 + ?in |2 ? 22 (A?27) ||S22 |2 |1 + ?in |2 ? |S22 + ?|2 |2 |1 + S11 |2 |S22 |2 + |S22 + ?|2 2 2 |1 + ?in | 2 2 2 ||S22 | |1 + ?in | ? |S22 + ?| | ? ? ? ? (S + ? )(1 + S11 )S22 + (S22 + ?)(1 + S11 )S22 |1 + ?in |2 ? 22 ||S22 |2 |1 + ?in |2 ? |S22 + ?|2 |2 = (A?28) (1 + S11 )S22 ? (S22 + ?) 2 |1 + ?in |2 = |S22 |2 |1 + ?in |2 ? |S22 + ?|2 (A?29) 2 S12 S21 |1 + ?in |2 = |S22 |2 |1 + ?in |2 ? |S22 + ?|2 (A?30) Now, the equation represents a circle on the Smith chart. 2 ? ? ?L ? (S22 |1 + ?in | ? (S22 + ?)(1 + S11 )) |S22 |2 |1 + ?in |2 ? |S22 + ?|2 S12 S21 |1 + ?in | = 2 2 2 |S22 | |1 + ?in | ? |S22 + ?| (A?31) The center and radius of the circle are given by ? ))? (S22 |1 + ?in |2 ? (S22 + ?)(1 + S11 |S22 |2 |1 + ?in |2 ? |S22 + ?|2 S12 S21 |1 + ?in | RL = 2 2 2 |S22 | |1 + ?in | ? |S22 + ?| CL = 120 (center) (A?32) (radius) (A?33) APPENDIX B THREE-PORT AND FOUR-PORT MULTISTATE REFLECTOMETERS Port 3 is terminated with a detector whose load re?ection coe?cient is ?3 . The incident wave power a1 is written as a 3 = ?3 b 3 (B?1) Therefore, three-port S-parameter equations can be expressed in a matrix form as ? ? ? ?? ? ?b1 ? ?S11 S12 S13 ?3 ? ?a1 ? ? ? ? ?? ? ?b ? = ?S S S ? ? ?a ? ? 2 ? ? 21 22 23 3 ? ? 2 ? ? ? ? ?? ? S31 S32 S33 ?3 b3 b3 (B?2) The incident power wave into a load terminating the port 3, b3 is written as b3 = S31 a1 + S32 a2 + S33 ?3 b3 b3 = (B?3) 1 (S31 a1 + S32 a2 ) 1 ? S33 ?3 (B?4) Also, the incident power wave into the port 1 is given by a1 = 1 (?S22 a2 + b2 ? S23 ?3 b3 ) S21 (B?5) By plugging Equation B?5 into Equation B?4, the equation becomes 1 b3 = 1 ? S33 ?3 ( ( S23 S31 ?3 1+ S21 (1 ? S33 ?3 ) ( S31 ) 1 S23 ?3 S22 ? a2 + b2 ? b3 S21 S21 S21 1 b3 = 1 ? S33 ?3 (( ) S22 S31 S32 ? S21 ) + S32 a2 ) S31 a2 + b2 S21 (B?6) ) (B?7) Now, the re?ected power wave b3 is written as ( ) S21 (1 ? S33 ?3 ) S21 S32 ? S22 S31 a2 S31 b3 = b2 ? S21 (1 ? S33 ?3 ) + S23 S31 ?3 S21 (1 ? S33 ?3 ) b2 S22 S31 ? S21 S32 S21 S32 ? S22 S31 = b2 (?L ? q3 ) S21 (1 ? S33 ?3 ) + S23 S31 ?3 121 (B?8) where ?L = a2 b2 q3 = S31 S22 S31 ? S21 S32 (B?9) To simplify the above equations, suppose that the detector port is matched (?3 = 0). The equations is simpli?ed as b3 = ?3 =0 S21 S32 ? S22 S31 b2 (?L ? q3 ) S21 (B?10) and also expressed in terms of A3 and B3 as b3 = A3 a2 + B3 b2 (B?11) where A3 = S21 S32 ? S22 S31 S21 B3 = S31 S21 (B?12) Also, the re?ected wave power b2 is simpli?ed as b2 ?3 =0 = S21 a1 + S22 ?L b2 b2 = S21 a1 1 ? S22 ?L (B?13) (B?14) By plugging Equation B?14 into Equation B?10, b3 can be simpli?ed as S21 S32 ? S22 S31 b2 (?L ? q3 ) S21 S21 S32 ? S22 S31 S21 a1 (?L ? q3 ) = S21 1 ? S22 ?L S21 S32 ? S22 S31 = a1 (?L ? q3 ) 1 ? S22 ?L b3 = (B?15) Similar to the three-port re?ectometer, a four-port re?ectometer with matched detector ports (?3 = ?4 = 0) is governed by the following equation as S21 S42 ? S22 S41 b2 (?L ? q4 ) S21 S21 S42 ? S22 S41 = a1 (?L ? q4 ) 1 ? S22 ?L b4 = 122 (B?16) where q4 = S41 S22 S41 ? S21 S42 (B?17) The equation is also expressed in terms of a2 and b2 as b4 = A4 a2 + B4 b2 (B?18) where A4 = S21 S42 ? S22 S41 S21 123 B4 = S41 S21 (B?19) APPENDIX C STANDARD AND MIXED-MODE S-PARAMETER TRANSFORMATION Common- and di?erential-mode normalized power waves are de?ned by 1 ac1 = ? (a1 + a2 ) 2 1 ad1 = ? (a1 ? a2 ) 2 (C?1) 1 bc1 = ? (b1 + b2 ) 2 1 bd1 = ? (b1 ? b2 ) 2 (C?2) The common- and di?erential-mode power waves are written in a compact matrix ? form as bmm ? b ?1 ? ? ? c1 ? ? ? ? ?bd1 ? ?b1 1 ? ? ? ? ? ?1 ? =? ?=? ?= ? ?b ? 2? ?b2 ? c2 ? ?0 ? ? ? 0 bd2 ? 1 0 ?1 0 0 1 0 1 ?? ? 0 ? ?b1 ? ?? ? ? ? 0? ? ?b2 ? ? ? ? = Mbstd ? ? 1? ? ?b3 ? ?? ? b4 ?1 (C?3) The mixed- and standard-mode normalized power waves are converted to each other by using the following relationship as bmm = Mbstd M = M?1 ? 1 ? ? 1 ? ?1 ? = ? 2? ?0 ? 0 (C?4) ? 1 0 0 ? ? ?1 0 0 ? ? ? ? 0 1 1? ? 0 1 ?1 (C?5) Now, mixed-mode S-parameter equations are written in a matrix form as ? ? ? ?? ? S S S S b a ? c1 ? ? cc11 cd11 cc12 cd12 ? ? c1 ? ? ?? ? ? ? ? ?? ? ?bd1 ? ?Sdc11 Sdd11 Sdc12 Sdd12 ? ?ad1 ? ? ? ? ? ? ? ?S11 S12 ? ??a1 ? mm mm bmm = ? ? = ? ?? ? = ? ?? ? = S a ? ? ? ?? ? S21 S22 ?a2 ?bc2 ? ?Scc21 Scd21 Scc22 Scd22 ? ?ac2 ? ? ? ? ?? ? bd2 ad2 Sdc21 Sdd21 Sdc22 Sdd22 (C?6) 124 The equation is manipulated as follows. bmm = Smm amm (C?7) Mbstd = Smm Mastd (C?8) bstd = M?1 Smm Mastd (C?9) Therefore, the transformation between mixed- and standard-mode S-parameters are given by Sstd = M?1 Smm M (C?10) Smm = MSstd M?1 (C?11) and 125 APPENDIX D DERIVATION OF DIFFERENTIAL INPUT REFLECTION COEFFICIENT A network with one di?erential port can be written as ?b1 = Smm?a1 11 where mm denotes mixed-mode and ? ? ? ? bc1 ? ?ac1 ? ?b1 = ? ?a1 = ? ? , ? ?, ad1 bd1 (D?1) ? ? ?Scc11 Scd11 ? Smm ? 11 = ? Sdc11 Sdd11 It can be easily extended to two di?erential port network as follows. ? ? ? ?? ? mm mm ? ?b1 ? ?S11 S12 ? ??a1 ? ? ?=? ?? ? mm ?b2 Smm S ?a2 21 22 (D?2) (D?3) Suppose that a di?erential load is connected to the di?erential port 2. Then, there is the following relationship between incident and re?ected waves. ? ?a2 = ?mm L b2 where ? (D?4) ? ??Lcc ?Lcd ? ?mm = ? ? L ?Ldc ?Ldd Therefore, the s-parameter equation can be written as ? ? ? ? ?? mm mm ? ?b1 ? ?S11 S12 ? ? ?a1 ? = ? ? ? ? ?? ? ?b2 Smm Smm ?mm 21 22 L b2 (D?5) (D?6) ?b2 can be written as ?b2 = Smm?a1 + Smm ?mm?b2 21 22 L (D?7) mm mm ? a1 (I ? Smm 22 ?L )b2 = S21 ? (D?8) It can be solved as 126 ?b2 = (I ? Smm ?mm )?1 Smm?a1 22 L 21 (D?9) From the S-parameter equation and the above equation, ?b1 can be written as ?b1 = Smm?a1 + Smm ?mm?b2 = Smm?a1 + Smm ?mm (I ? Smm ?mm )?1 Smm?a1 11 12 L 11 12 L 22 L 21 (D?10) The input re?ection matrix is de?ned as ?b1 = ?mm?a1 in ? where (D?11) ? ??incc ?incd ? ?mm ? in = ? ?indc ?indd (D?12) By the comparison of Equation D?10 and D?11, the input re?ection matrix is mm mm mm mm mm ?1 mm ?mm in = S11 + S12 ?L (I ? S22 ?L ) S21 (D?13) The mixed-mode re?ection matrix can be converted to standard-mode as follows. ?1 mm ?std in = M ?in M ?1 mm ?1 mm ?1 mm mm ?1 ?1 mm = M?1 Smm 11 M + M S12 MM ?L MM (I ? 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Xavier, ?Accurate measurement method for characterisation of RF impedance tuners,? Electronics Letters, vol. 43, pp. 1434?1436, Dec. 6, 2007. [32] T. Zhang, W. R. Eisenstadt, R. M. Fox, and Q. Yin, ?Bipolar microwave RMS power detectors,? IEEE J. Solid-State Circuits, vol. 41, no. 9, pp. 2188?2192, Sep. 2006. [33] T. F. Coleman and Y. Li, ?An interior trust region approach for nonlinear minimization subject to bounds,? SIAM Journal on Optimization, vol. 6, no. 2, pp. 418?445, May 1996. [34] ??, ?On the convergence of re?ective newton methods for large-scale nonlinear minimization subject to bounds,? Mathematical Programming, vol. 67, no. 2, pp. 189?224, 1994. [35] F. Wiedmann, B. Huyart, E. Bergeault, and L. Jallet, ?New structure for a six-port re?ectometer in monolithic microwave integrated-circuit technology,? IEEE Trans. Instrum. Meas., vol. 46, no. 2, pp. 527?530, Apr. 1997. [36] I. Majid, A. E. Nadeem, and F. e Azam, ?Small signal s-parameter estimation of BJTs using arti?cial neural networks,? in Multitopic Conference, 2004. Proceedings of INMIC 2004. 8th International, Dec. 24?26, 2004, pp. 669?673. 131 [37] A. E. Nadeem and W. R. Eisenstadt, ?Improved closed-form expressions for s-parameters of BJTs using modi?ed gummel-poon model,? in Multi Topic Conference, 2003. INMIC 2003. 7th International, Dec. 8?9, 2003, pp. 202?207. [38] G. Koers, J. Stiens, and R. Vounckx, ?Scalar calibration of quasi-optical re?ection measurements,? IEEE Trans. Microw. Theory Tech., vol. 54, no. 7, pp. 3121?3126, Jul. 2006. [39] K. Ho?mann and Z. Skvor, ?Calibration of the PTP vector network analyzer,? in Microwaves and Radar, 1998. MIKON ?98., 12th International Conference on, vol. 3, May 20?22, 1998, pp. 710?714. 132 BIOGRAPHICAL SKETCH Jaeseok Kim was born in Seoul, Korea. He received the B.S. degree in 1994, from Inha University, majoring in electronic engineering. He received the M.S. degree in 2002 from the University of Florida, majoring in electrical and computer engineering. Since 2006, he has been with the Electronic Circuit Laboratory (ECL) at the University of Florida, pursuing his Ph.D. degree. His research interests include RF impedance matching control algorithm and system, machine intelligence of RF systems, and embedded RF on-chip testing. 133 e 6-4. The inductance value of 3.3 nH and 1.8 nH was used for the design of the matching network. A noninvasive measurement is mandatory not to disturb the original design of a matching network. Similar to the prior work [35], a high impedance power detector measures an internal node. The noninvasive power detector is emulated by inserting high resistance in series with a measurement port and deembedding the e?ect of the resistance. As explained earlier, the four-port re?ectometer needs a power reading that depends only 84 P3 High input impedance power detector Z0 ZL A B Figure 6-3. Recon?gurable three-port matching network A) Schematic B) Implementation on FR4 board 85 S(2,1) S(1,1) freq (2.400GHz to 2.400GHz) S(2,1) S(1,1) A freq (2.400GHz to 2.400GHz) B Figure 6-4. Tunable element impedance with a bias from 0 V to 10 V. A varactor is in parallel and in series with an inductor. и and О denote S11 and S21 , respectively. A) In series with 3.3 nH B) In parallel with 1.8 nH 86 on the re?ected wave power from a DUT. The reference port can be easily realized using a lumped power divider. 6.4 Load Estimation for Multistate Reflectometer The matching capability of a tunable matching network can be represented by the load re?ection coe?cient ?L to be matched by the matching network. The matching capability is derived as follows. The input re?ection coe?cient ?in is written as ?in = S11 + S12 S21 ?L 1 ? S22 ?L (6?10) The equation can be rewritten in terms of the load re?ection coe?cient as S11 ? ?in S11 S22 ? S12 S21 ? S22 ?in ?L = (6?11) The matching capability is derived from the load re?ection coe?cient by setting the input re?ection coe?cient to zero. ?L ?in =0 = S11 S11 S22 ? S12 S21 (6?12) The coverage on the Smith chart speci?ed the matching capability illustrates the distribution of the load re?ection coe?cient to be matched. According to the six-port re?ectometer principle, an unknown load is the same as the point intersected by three circles, speci?ed by a center, so-called q-point, and a radius. The circle is represented by calibration constants de?ned by Equation 6?6. Note that the q-point does not change even if a mismatched load varies, whereas the load estimation method proposed in [25] has changed the circle center as a mismatched load varies. The constant q-point enables to keep the optimum-performance criteria for the multistate re?ectometer over various mismatched loads. In reality, the three circles represented by calibration constants seldom intersect at a point due to the non-ideal e?ects. The geometric center of the overlap of the circles is 87 estimated by the radical center and least square estimation methods widely used for the six-port re?ectometer [27]. The radical center is the approximation of the center of three circles? overlapped region. The coordinates of the radical center are given as 2 2 RL1 ? RL2 + x22 2x2 2 2 R ? RL3 + x23 + y32 ? 2xx3 y = L1 2y3 x= (6?13) (6?14) Least square ?tting can enhance the accuracy of load estimation especially when three circles failed to meet the optimum-performance criteria. The load re?ection coe?cient can be obtained by the least square equation as ??L = arg min ? |Circle(V) ? ?L |2 (6?15) V where V = (V1 , V2 , V3 )T is a bias voltage vector for three varactors. Note that Circle(V) is represented by calibration constants de?ned by Equation 6?6 and the distance between a circle and a point is de?ned as a distance between a tangential line to the circle and the point. 6.5 Experimental Results The matching capability of the proposed three-port re?ectometer, de?ned as S11 /(S11 S22 ? S12 S21 ), is shown in Figure 6-5. The matching capability was measured by changing each varactor bias from 0 V to 5.12 V in 16 levels by 0.32 V step. The matching capability covers a unit circle on the Smith chart completely, showing its capability on any passive mismatched load. One of important calibration constants is q-point given by the equation qi = Si1 S22 Si1 ? S21 Si2 (6?16) The q-points were measured with respect to the same bias range as used for the matching capability. The measured q-points are distributed along the unit circle as shown in Figure 88 Table 6-1. Summary of mismatched loads Mismatched load |S11 | at 2.4 GHz Tuner motor position Matched 0.01 (100, 5000, 5000) Slightly mismatched #1 Slightly mismatched #2 Slightly mismatched #3 0.13 0.38 0.20 (16725, 2262, 5000) (17105, 1424, 5000) (20464, 2228, 5000) Heavily mismatched #1 Heavily mismatched #2 Heavily mismatched #3 0.78 0.76 0.69 (15781, 526, 5000) (17835, 624, 5000) (20572, 804, 5000) 6-6. Some set of q-points can be selected from the distribution in order to satisfy the optimum-performance criteria for the multistate re?ectometer. A set of q-points was carefully selected for better estimation performance. The load estimation using three-port re?ectometer was performed separately on slightly and heavily mismatched loads. The speci?cation of mismatched loads is given in Table 6-1. First, the S-parameter of the input port was converted from the high impedance port emulating high impedance power detector. The calibration constants were obtained from the S-parameters through direct measurement of the three-port re?ectometer and conversion using the high impedance power detector. The q-points were chosen to achieve higher estimation accuracy for two separate experiments. Then, radical center estimation was applied to estimate an unknown load re?ection coe?cient. When the magnitude of the estimated re?ection coe?cient is larger than one, it is incorrect for passive mismatched loads. In this case, the magnitude was set to one with keeping the phase. As shown in Figure 6-7, estimation of slightly mismatched loads showed much smaller estimation error than heavily mismatched loads. The mean square error for slightly and heavily mismatched loads are 0.09 and 0.80, respectively. Due to the larger estimation error, the estimated load re?ection coe?cient of the heavily mismatched loads often go beyond a unit circle. As described, the magnitude was set to 1 and only the phase was kept for impedance matching. However, the estimated phase is still quite useful because 89 the estimated magnitude larger than one often results from heavily mismatched load, whose magnitude is close to one. The next chapter will demonstrate that an automatic matching control can achieve impedance matching using the estimated phase information. 6.6 Conclusion We proposed a three-port lumped-element re?ectometer for both load estimation and impedance matching. The proposed re?ectometer can be easily extended to a four-port re?ectometer by adding the suggested power divider. The load estimation method demonstrated that the tunable multistate re?ectometer can help the automatic matching control (AMC) to estimate a load re?ection coe?cient as well as to set impedance matching. The high impedance power detector replaced the distributed coupler and realized the dramatic size reduction of an automatic matching control system without compromising the load estimation and matching capability. The matching capability covered completely the unit circle on the Smith chart. Although the load estimation result is not accurate to be used as an high-precision instrument, the estimated phase information can still enable the automatic matching control to achieve faster impedance matching on heavily mismatched loads. We are working toward the integration of the proposed load estimation and a novel automatic matching control system capable of an immediate impedance matching. 90 +j1.0 +j0.5 +j2.0 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 ?j0.2 \infty ?j5.0 ?j0.5 ?j2.0 ?j1.0 Figure 6-5. Matching capability of three-port re?ectometer at 2.4 GHz 91 +j1.0 +j2.0 +j0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 ?j0.2 \infty ?j5.0 ?j2.0 ?j0.5 ?j1.0 Figure 6-6. The q-point distribution of three-port re?ectometer at 2.4 GHz 92 +j1.0 +j1.0 True +j2.0 Estimate ?j0.2 ?j5.0 ?j0.5 5.0 0.0 2.0 Ц +j5.0 1.0 +j0.2 0.5 5.0 2.0 1.0 0.5 0.0 +j5.0 0.2 +j0.2 True +j2.0 Estimate +j0.5 0.2 +j0.5 ?j0.2 ?j5.0 ?j0.5 ?j2.0 Ц ?j2.0 ?j1.0 ?j1.0 A B Figure 6-7. Multistate re?ectometer estimation using estimated S11 from high impedance probe A) Small mismatch (MSE=0.09) B) Large mismatch (MSE=0.80) 93 CHAPTER 7 AUTOMATIC MATCHING CONTROL USING LOAD ESTIMATION 7.1 Overview An automatic matching control (AMC) system has been developed to automate time-consuming impedance matching procedure [5][6][21]. The impedance matching of the automatic matching control was performed by recon?guring a tunable matching network until the lowest mismatch is achieved. The recon?guration was controlled by heuristic iterative methods, which showed a good trade-o? between system response and impedance matching accuracy. Also, load estimation reusing the existing tunable matching network of the automatic matching control system was proposed to facilitate the automation of impedance matching [25]. In this work, the load estimation technique was integrated with the existing automatic matching control system to achieve immediate impedance matching without compromising matching accuracy. Traditional automatic matching control systems achieved impedance matching of unknown or even varying mismatched loads by the feedback loop of a tunable matching network, a mismatch detector, and match control circuit [21]. The feedback loop is controlled by iterative methods of a match control circuit, which searches for the value of tuning elements in a trial-and-error process. However, the trial-and-error approach slowed down the system response and various heuristic approaches have been developed to improve the system response without compromising matching capability. Nevertheless, the system response of the heuristic approaches is still proportional to the complexity of the matching network and gets slower as more tuning elements and levels are added. We will demonstrate that an estimated load can be used for a matching control circuit to achieve immediate impedance matching without using heuristic approaches. The proposed matching control can ?nd the value of tuning elements by examining the characterization table of a matching network. Therefore, the precise characterization as 94 well as the load estimation play an important role in this immediate impedance matching approach. Various characterization methods for a microwave device have been reported to reduce the microwave design complexity. A neural network has been widely used to characterize microwave devices, such as the approximation of S-parameters of BJTs [36] and modeling parameters of microwave components [9]. Also, closed form equation was also presented for S-parameters of BJTs [37]. In this work, neural network models and the closed form equation were used to approximate measured S-parameters and the accuracy of the characterization methods was evaluated in terms of mean squared error (MSE) between true and estimated values. The proposed matching control consists of two tasks. First, a characterization table in terms of tuning elements was built from the direct measurement of the matching network or approximation models such as a neural network and closed form equations. Next, the value of the tuning elements was found by minimizing the degree of mismatch. The degree of mismatch was calculated from the magnitude of the input re?ection coe?cient. The experimental results of the immediate impedance matching approach will be presented. 7.2 Matching Control Procedures The same lumped-element tunable matching network that is used for load estimation was used to develop matching control procedures supporting load estimation presented in Chapter 5 and 6. The matching network has a ?-type bandpass ?lter topology and three varactor diodes as tuning elements. The recon?guration of the matching network was performed by changing the varactor bias voltages. Its matching capability covers all re?ection coe?cients within the unit circle on the Smith chart at 2.4 GHz. The load re?ection coe?cient ?L of a device under test (DUT) is assumed to be estimated by load estimation techniques presented in Chapter 5 and 6. When the DUT is connected to the port 2 of a tunable matching network, the input re?ection coe?cient ?in 95 looking into the port 1 of the matching network is written as follows. ?in = S11 + S12 S21 ?L 1 ? S22 ?L (7?1) where Sij is the S-parameter from port j to port i of the matching network. Note that the S-parameters are the function of a bias voltage vector, denoted by v. Therefore, the input re?ection coe?cient can be explicitly written as the function of v. ?in (v) = S11 (v) + S12 (v)S21 (v)?L , 1 ? S22 (v)?L v = (v1 , v2 , и и и , vn )T (7?2) where T denotes a transpose and vn is the nth bias voltage. The load re?ection coe?cient of the DUT can be derived from the input re?ection coe?cient. ?L = S11 (v) ? ?in (v) S11 (v)S22 (v) ? S12 (v)S21 (v) ? S22 (v)?in (v) (7?3) The mismatched load to be matched by the matching network set by a bias voltage v, denoted by ?M , is the load re?ection coe?cient that makes the input re?ection coe?cient zero. ?M (v) = ?L = ?in =0 S11 (v) S11 (v)S22 (v) ? S12 (v)S21 (v) (7?4) b, as a bias voltage Now, let us de?ne an optimal bias voltage vector, denoted by v vector that minimizes the magnitude of the input re?ection coe?cient ?in , the degree of mismatch. b = arg min |?in (v)| v v (7?5) If the bias voltage, that minimizes the input re?ection coe?cient to zero, can be found for b can be expressed all possible load re?ection coe?cients, the optimal bias voltage vector v using ?M as follows. } } { { b = v?in (v) = 0 = v?M (v) = ?L v Finding the optimal bias voltage can be expressed as ?nding a bias voltage whose ?M is equal to ?L . Therefore, the mapping table between ?M and v should be calculated 96 (7?6) Start Characterization Measure power three-times Estimate load impedance Calculate bias for matching End Figure 7-1. Automatic matching control supports load estimation. to perform the bias search and the mapping table can be converted from the matching network characterized by the S-parameters. This procedure is based on Equation 7?4. The S-parameters can be obtained from direct measurement using a vector network analyzer or a neural network ?tting model. The characterization methods will be introduced in the next Section. 7.3 Characterization of Matching Network The characterization of a matching network is a procedure to discover the S-parameter functions to be used to calculate an input re?ection coe?cient or a mismatched load to be matched. The S-parameters of the matching network were measured using a vector network analyzer while changing the bias voltage. The measurement points were determined by the number of varactors and bias voltage levels. Although the more voltage levels can produce more accurate characterization results, 16 voltage levels were chosen as good trade-o? between measurement time and characterization accuracy. The 97 characterization results were converted to a form of a mapping table, ?M (v), for easy access and searches. Unknown S-parameters between measurement points were approximated by a multivariate linear interpolation. The multivariate linear interpolation is interpolating a function of multiple variables on a regular grid, as an extension of a linear interpolation. It performs linear interpolation ?rst on one direction, then again in the other direction. Suppose we want to interpolate a value of an unknown function f at the point (x, y). The value of the function f at four neighbor points on a regular grid, f (x1 , y1 ), f (x1 , y2 ), f (x2 , y1 ), and f (x2 , y2 ), are assumed to be known, then the interpolation of the function f at the point (x, y) can be written as follows. f (x1 , y1 ) (x2 ? x)(y2 ? y) (x2 ? x1 )(y2 ? y1 ) f (x2 , y1 ) + (x ? x1 )(y2 ? y) (x2 ? x1 )(y2 ? y1 ) f (x1 , y2 ) (x2 ? x)(y ? y1 ) + (x2 ? x1 )(y2 ? y1 ) f (x2 , y2 ) + (x ? x1 )(y ? y1 ) (x2 ? x1 )(y2 ? y1 ) f (x, y) ? (7?7) If the unknown function to interpolate has a smooth surface over neighbor points, the linear interpolation reduces the number of measurement points signi?cantly without the loss of the characterization detail. The other method to estimate unknown S-parameters between measurement points is an approximation ?tting function to the measured S-parameters. As described, the measured S-parameters are the functions of the bias voltage vector v, given as follows. ? ? ?S11 (v) S12 (v)? (7?8) S(v) = ? ? S21 (v) S22 (v) The S-parameter functions were approximated by a curve ?tting model, such as a closed form and a neural network. First, an arti?cial neural network was used to approximate 98 the S-parameter functions. Four independent neural network models approximate two-port S-parameter functions, S11 (v), S12 (v), S21 (v), and S22 (v). The input and output of the neural network are bias voltage vector and real and imaginary parts of the S-parameter, respectively. The feed-forward topology was used with 15 perceptrons in the hidden layer. The well-known backpropagation algorithm was used to train the neural networks. The cascaded network is easily represented by ABCD-parameters and the matching network was decomposed into basic components, such as a transmission line, series impedance, and shunt admittance. The representation of the ABCD-parameters was used as closed-form equations. The ABCD-parameters for the basic components are given as follows. ? ? ? cos(2??) ?Z0 sin(2??)? ? ? ?Y0 sin(2??) cos(2??) ? ? ?1 Z ? ? ? 0 1 ? ? ? 1 0? ? ? Y 1 (transmission line) (7?9) (series impedance) (7?10) (shunt admittance) (7?11) The closed-form ABCD-parameters were derived from the cascaded network of a transmission line (?1 ), a ? network (Y1 , Z3 , and Y2 ), and a transmission line (?2 ). The varactor capacitance was calculated from the varactor SPICE model used in Chapter 5 and 6. ABCD11 =(cos(2??1 )(1 + Y2 Z3 ) + ?Z0 sin(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) cos(2??2 ) +? (cos(2??1 )Z3 + ?Z0 sin(2??1 )(1 + Y1 Z3 )) Y0 sin(2??2 ) (7?12) ABCD12 =? (cos(2??1 )(1 + Y2 Z3 ) + ?Z0 sin(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) Z0 sin(2??2 ) + (cos(2??1 )Z3 + ?Z0 sin(2??1 )(1 + Y1 Z3 )) cos(2??2 ) 99 (7?13) ABCD21 =(?Y0 sin(2??1 )(1 + Y2 Z3 ) + cos(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) cos(2??2 ) +? (?Y0 sin(2??1 )Z3 + cos(2??1 )(1 + Y1 Z3 )) Y0 sin(2??2 ) (7?14) ABCD22 =? (?Y0 sin(2??1 )(1 + Y2 Z3 ) + cos(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) Z0 sin(2??2 ) + (?Y0 sin(2??1 )Z3 + cos(2??1 )(1 + Y1 Z3 )) cos(2??2 ) (7?15) The admittance Y1 , Y2 and impedance Z3 are the function of bias voltage which determines the varactor capacitance. The transmission line delay ?1 , ?2 , parasitic parameters, ?tting parameters of the varactor SPICE model are trained by a nonlinear least square ?tting algorithm. The ABCD-parameters based ?tting model was converted to S-parameters for fair comparison with direct measurement and neural network model. 7.4 Bias Search for Impedance Matching The goal of bias search is to ?nd the optimal bias voltage vector given by { } { } b = v?in (v) = 0 = v?M (v) = ?L v (7?16) Due to the discrete measurement data of a mapping table ?M (v), it is not always possible to ?nd the optimal bias voltage vector. Instead, we choose the bias voltage vector closest to the optimal bias voltage vector and this procedure can be written as follows. b = arg min |?M (v) ? ?L | v v (7?17) From now on, the optimal bias voltage vector is rede?ned as the bias minimizing the magnitude between ?M and ?L . The bias search consists of two steps, coarse and ?ne search. The coarse search was performed on the mapping table converted from the direct measurement of S-parameters. 100 Then, the ?ne search was performed again on the linear interpolation of a voxel1 of the bias voltage vector found in the coarse search. The cost function for both coarse and ?ne search is mean square error (MSE) of magnitude between ?M of eight bias voltage vectors of a voxel and an estimated load ?L . Note that a voxel for 3D data has eight vertexes, similar to a cubic. For example, a unit voxel consists of eight points, (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), and (1,1,1). Eight points of the voxel for the coarse search were selected from the voltage bias used for the direct measurement. Once the voxel is found by the coarse search, the mapping table of the voxel is interpolated by multivariate linear interpolation with higher resolution. Then, the ?ne search is performed on the interpolated data. The proposed two-step search reduces the required number of measurement points and the memory usage by the partial interpolation. In addition, the bias search can be made faster using binary search algorithm

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