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Automated Matching Control System Using Load Estimation and Microwave Characterization
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Title:
Automated Matching Control System Using Load Estimation and Microwave Characterization
Creator:
Kim, Jaeseok
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (133 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Eisenstadt, William R.
Committee Members:
Harris, John G.
O, Kenneth K.
Wiens, Gloria J.
Graduation Date:
12/19/2008

Subjects

Subjects / Keywords:
Analyzers ( jstor )
Circles ( jstor )
Electric potential ( jstor )
Estimation methods ( jstor )
Geometric centers ( jstor )
Impedance matching ( jstor )
Least squares ( jstor )
Microwaves ( jstor )
Neural networks ( jstor )
Varactor diodes ( jstor )
Electrical and Computer Engineering -- Dissertations, Academic -- UF
characterization, estimation, impedance, matching, microwave, reflectometer, rftest
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Electrical and Computer Engineering thesis, Ph.D.

Notes

Abstract:
The automation of the impedance matching of radio frequency (RF) ports enables the test engineer to compensate the undesired effects, which are not uncommon in RF systems and make the impedance matching iterative, time-consuming, and an empirical process. Numerous reconfigurable 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 reflectometer 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 filter, a load estimation technique using the microstrip filter, a new lumped matching network for the automatic matching in embedded RF testing, and a new matching control algorithm using the load estimation. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2008.
Local:
Adviser: Eisenstadt, William R.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-06-30
Statement of Responsibility:
by Jaeseok Kim.

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Source Institution:
University of Florida
Holding Location:
University of Florida
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Copyright Kim, Jaeseok. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
6/30/2009
Resource Identifier:
430115874 ( OCLC )
Classification:
LD1780 2008 ( lcc )
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AUTOMATEDMATCHINGCONTROLSYSTEMUSINGLOADESTIMATIONANDMICROWAVECHARACTERIZATIONByJAESEOKKIMADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2008 1

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c2008JaeseokKim 2

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Tomyfamily 3

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ACKNOWLEDGMENTS Iwouldliketoexpressmysinceregratitudetomyadvisor,ProfessorWilliamR.Eisenstadt,forhisinvaluableadvice,encouragement,andsupport.Thisdissertationwouldnothavebeenpossiblewithouthisguidanceandsupport.MydeeprecognitiongoestoProfessorKennethO,ProfessorJohnG.Harris,andProfessorGloriaJ.Wiensforservingonmysupervisorycommitteeandfortheirvaluablesuggestions.ManythanksgotoMr.LarryLucefromFreescaleSemiconductorfortheirvaluableinputandgenerousfundingforthisresearch.ThanksalsogotomycolleaguesintheElectronicCircuitsLaboratory(ECL)fortheirdiscussionofideasandyearsoffriendship.Lastbutnotleast,Ioweaspecialdebtofgratitudetomyfamily.Withouttheirselessloveandsupport,IcannotimaginewhatIwouldhaveachieved. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTION .................................. 13 1.1WhyAutomaticImpedanceMatching? .................... 13 1.2MaximumPowerTransferonImpedanceMatching ............. 14 1.3ChallengesofAutomaticImpedanceMatching ................ 15 1.4ProposedSolution:AutomaticMatchingControlwithLoadEstimation .. 16 2BACKGROUND ................................... 17 2.1ImpedanceMatchingNetworkswithLumpedElements ........... 17 2.1.1L-typeMatchingNetwork ........................ 18 2.1.2-typeMatchingNetwork ....................... 19 2.1.3T-typeMatchingNetwork ....................... 20 2.1.4BandwidthofMatchingNetwork .................... 21 2.2PatternRecognition .............................. 22 2.2.1Thek-NearestNeighborClassier ................... 23 2.2.2ArticialNeuralNetwork(ANN) .................... 24 2.2.3PrincipalComponentAnalysis(PCA) ................. 26 2.3Multi-PortReectometers ........................... 28 2.3.1Six-PortReectometer ......................... 28 2.3.2Four-PortMultistateReectometer .................. 30 3AUTOMATICMATCHINGCONTROL ...................... 34 3.1Overview .................................... 34 3.2SystemOverview ................................ 35 3.2.1ImpedanceMatchingTuner ....................... 36 3.2.2Controller ................................ 36 3.2.3Searchalgorithm ............................ 39 3.3ExperimentalResults .............................. 39 3.3.1CharacterizationofAutomaticTunerSystem ............. 39 3.3.2MeasurementResults .......................... 41 3.4ConclusionandDiscussion ........................... 47 5

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4LOADIMPEDANCEESTIMATION ........................ 52 4.1Overview .................................... 52 4.2SystemOverview ................................ 53 4.2.1AutomaticMatchingControl(AMC) ................. 53 4.2.2LoadEstimationMethod ........................ 54 4.3ExperimentalResults .............................. 57 4.4Conclusion .................................... 58 5COUPLER-FREELOADESTIMATIONUSINGTHREE-PORTEFLECTOMETER .................................. 59 5.1Overview .................................... 59 5.2SystemOverview ................................ 60 5.3HighImpedanceProbe ............................. 63 5.3.1LeastSquareFitting .......................... 64 5.3.2ArticialNeuralNetwork ........................ 65 5.4LoadEstimationMethods ........................... 65 5.4.1RadicalCenter .............................. 66 5.4.2LeastSquareFitting .......................... 67 5.5ExperimentalResults .............................. 68 5.5.1HighImpedanceProbeEstimation ................... 68 5.5.2LoadEstimation ............................. 69 5.5.3ComparisonofCouplerandCoupler-FreeLoadEstimation ..... 73 5.6Conclusion .................................... 79 6THREEPORTANDFOURPORTREFLECTOMETERS ............ 80 6.1Overview .................................... 80 6.2MultistateReectometers ........................... 81 6.3TunableMatchingNetwork .......................... 84 6.4LoadEstimationforMultistateReectometer ................ 87 6.5ExperimentalResults .............................. 88 6.6Conclusion .................................... 90 7AUTOMATICMATCHINGCONTROLUSINGLOADESTIMATION ..... 94 7.1Overview .................................... 94 7.2MatchingControlProcedures ......................... 95 7.3CharacterizationofMatchingNetwork .................... 97 7.4BiasSearchforImpedanceMatching ..................... 100 7.5CharacterizationResults ............................ 101 7.6ImpedanceMatchingResults .......................... 106 7.7Conclusion .................................... 110 8CONCLUSION .................................... 114 APPENDIX 6

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ADERIVATIONOFLOADIMPEDANCECIRCLEEQUATIONUSINGINPUTIMPEDANCEMAGNITUDE ............................ 117 BTHREE-PORTANDFOUR-PORTMULTISTATEREFLECTOMETERS ... 121 CSTANDARDANDMIXED-MODES-PARAMETERTRANSFORMATION .. 124 DDERIVATIONOFDIFFERENTIALINPUTREFLECTIONCOEFFICIENT 126 REFERENCES ....................................... 129 BIOGRAPHICALSKETCH ................................ 133 7

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LISTOFTABLES Table page 3-1Specicationofthematchingtuner ......................... 37 3-2Mismatchedloadspecication ............................ 40 3-3Comparisonofbrute-force,greedy,andsingle-stepalgorithms .......... 46 3-4Percentageofcatastrophiccaseforbrute-force,greedy,andsingle-stepalgorithms 47 3-5Comparisonofbrute-force,greedy,andsingle-stepalgorithmsavoidingcatastrophiccases .......................................... 47 4-1Summaryofmismatchedloadestimationresults .................. 57 5-1Summaryofheavilyandslightlymismatchedloads ................ 73 5-2Comparisonofcouplerandcoupler-freeloadestimationintermofmeansquareerror(MSE) ...................................... 74 6-1Summaryofmismatchedloads ............................ 89 7-1Meansquareerror(MSE)ofneuralnetworkttingmodelsusingtrainingandtestingdata ...................................... 106 7-2Averageerrorofclosed-formmodels ......................... 106 7-3Mappingtablebetweenbiasvoltageandamismatchedloadtobematched ... 107 7-4Inversemappingtablebetweenamismatchedloadtobematchedandbiasvoltage 107 7-5Impedancematchingresultsusingcoupler-freeloadestimationandS-parametermeasurementdata .................................. 109 7-6Impedancematchingresultsusingthree-portreectometerloadestimationandS-parametermeasurementdata ........................... 109 7-7Impedancematchingresultsusingcoupler-freeloadestimationandS-parametersestimatedbyneuralnetworkmodels ........................ 109 7-8Impedancematchingresultsusingthree-portreectometerloadestimationandS-parametersestimatedbyneuralnetworkmodels ................. 110 8

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LISTOFFIGURES Figure page 1-1Networkmatchinganarbitraryloadimpedancetoatransmissionline ...... 14 1-2Maximumpowerdeliveredwhensourceandloadimpedancesarematched. ... 14 1-3Automaticmatchingcontrol(AMC) ........................ 16 2-1TheLsectionmatchingnetworks .......................... 19 2-2Thesectionmatchingnetwork .......................... 20 2-3TheTsectionmatchingnetwork .......................... 20 2-4Typicalpatternrecognitionssystemdiagram .................... 22 2-5Nearest-neighborruleleadstoapartitioningoftheinputspaceintoVoronoicells 23 2-6Multilayerperceptron ................................ 24 2-7Articialneuron ................................... 25 2-8Sigmoidalnonlinearity ................................ 25 2-9Principalcomponentanalysis ............................ 28 2-10Six-portreectometer ................................ 29 2-11Determinationof)]TJ /F4 7.97 Tf 125.44 -1.79 Td[(Lfromtheradicalcenterofthreecircles ............ 31 2-12Four-portreectometer ................................ 31 3-1Automaticmatchingcontrol(AMC)systemdiagram ............... 35 3-2Recongurableve-stubmatchingtuner ...................... 37 3-3Comparisonofsimulationandmeasurementofmatchingtuner.Thematchingtuneristunedwithtypicalbias(2.562.562.56)andmatchedload ........ 38 3-4Mismatchedload1measurement(S11=0.11+j0.09at3.5GHz).Motorpositionsare(16725,2262,5000).ReectioncoecientjS11j=0.14. ............. 40 3-5Mismatchedload2measurement(S11=0.19+j0.40at3.5GHz).Motorpositionsare(17105,1424,5000).ReectioncoecientjS11j=0.45. ............. 41 3-6Mismatchedload3measurement(S11=-0.11+j0.22at3.5GHz).Motorpositionsare(17722,2005,5000).ReectioncoecientjS11j=0.24. ............. 41 3-7Matchingcapabilityofve-stubmatchingtunerat3.5GHz ........... 42 9

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3-8Matchingtunermeasurementwithmatchedload.Typicalandautomatedbiasesare(2.56V2.56V2.56V)and(4.8V3.84V4.64V) ................. 44 3-9Matchingtunermeasurementwithmismatchedload1.Typicalandautomatedbiasesare(2.56V2.56V2.56V)and(3.2V3.86V3.52V) .............. 45 3-10Matchingtunermeasurementwithmismatchedload2.Typicalandautomatedbiasesare(2.56V2.56V2.56V)and(4.8V3.84V4.64V) .............. 49 3-11Matchingtunermeasurementwithmismatchedload3.Typicalandautomatedbiasesare(2.56V2.56V2.56V)and(4.8V3.84V4.64V) .............. 50 3-12Optimizationsurfaceofgreedyalgorithm ...................... 51 4-1Systemdiagramofloadimpedanceestimationforautomaticimpedancematching 53 4-2Radicalcenterofthreecircles,)]TJ /F4 7.97 Tf 193.14 -1.79 Td[(L1,)]TJ /F4 7.97 Tf 14.46 -1.79 Td[(L2,and)]TJ /F4 7.97 Tf 37.22 -1.79 Td[(L3. ................. 56 4-3Impedancematchingprocedureusingestimatedloadimpedance. ......... 56 5-1Automaticmatchingcontrol(AMC) ........................ 60 5-2ThreeportreectometerintegratedwithahighimpedanceprobeA)SystemdiagramwithschematicB)FabricationonFR4board ............... 61 5-3VaractorSMV1405capacitanceversusreversevoltage ............... 62 5-4VaractorSPICEmodel ................................ 62 5-5Radicalcenterofthreecircles,)]TJ /F4 7.97 Tf 193.14 -1.79 Td[(L1,)]TJ /F4 7.97 Tf 14.46 -1.79 Td[(L2,and)]TJ /F4 7.97 Tf 37.22 -1.79 Td[(L3. ................. 67 5-6Leastsquarenonlinearttingofhighimpedanceprobemodel .......... 70 5-7Articialneuralnetworkofhighimpedanceprobemodel ............. 71 5-8Highimpedanceprobeestimationerrordistribution ................ 72 5-9LoadestimationusingestimatedS11fromhighimpedanceprobe ......... 74 5-10Couplerandcoupler-freeloadestimationwithmismatched#1 .......... 75 5-11Couplerandcoupler-freeloadestimationwithmismatched#2 .......... 76 5-12Couplerandcoupler-freeloadestimationwithmismatched#3 .......... 77 5-13Couplerandcoupler-freeloadestimationwithmismatched#4 .......... 78 6-1ReectometersA)Three-portB)Four-port ..................... 81 6-2Schematicofalumpedpowerdivider.Port3isacoupledport,whichcanbeusedasareferenceport. ............................... 83 10

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6-3Recongurablethree-portmatchingnetworkA)SchematicB)ImplementationonFR4board .................................... 85 6-4Tunableelementimpedancewithabiasfrom0Vto10V.Avaractorisinparallelandinserieswithaninductor.anddenoteS11andS21,respectively.A)Inserieswith3.3nHB)Inparallelwith1.8nH .................... 86 6-5Matchingcapabilityofthree-portreectometerat2.4GHz ............ 91 6-6Theq-pointdistributionofthree-portreectometerat2.4GHz .......... 92 6-7MultistatereectometerestimationusingestimatedS11fromhighimpedanceprobeA)Smallmismatch(MSE=0.09)B)Largemismatch(MSE=0.80) .... 93 7-1Automaticmatchingcontrolsupportsloadestimation. .............. 97 7-2Neuralnetworkmodelsfora2-portmatchingnetworkweretrainedbybackpropagation.A)S11B)S21C)S12D)S22 ............................. 103 7-3Neuralnetworkmodelsfora2-portmatchingnetworkweretestedby10%ofmeasurementdata.A)S11B)S21C)S12D)S22 .................. 104 7-4Closedformmodelsfora2-portmatchingnetworkweretrainedbynonlinearleastsquaretting.A)S11B)S21C)S12D)S22 .................. 105 7-5S-parametersfromvectormeasurementandneuralnetworkmodelwerecomparedintermsofcoupler-freeloadestimationassistedmatchingcontrol.A)VectormeasurementbylargemismatchB)NeuralnetworkmodelbylargemismatchC)VectormeasurementbysmallmismatchD)Neuralnetworkmodelbysmallmismatch ....................................... 111 7-6S-parametersfromvectormeasurementandneuralnetworkmodelwerecomparedintermsofreectometerloadestimationassistedmatchingcontrol.A)VectormeasurementbylargemismatchB)NeuralnetworkmodelbylargemismatchC)VectormeasurementbysmallmismatchD)Neuralnetworkmodelbysmallmismatch ....................................... 112 11

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyAUTOMATEDMATCHINGCONTROLSYSTEMUSINGLOADESTIMATIONANDMICROWAVECHARACTERIZATIONByJaeseokKimDecember2008Chair:WilliamR.EisenstadtMajor:ElectricalandComputerEngineering Theautomationoftheimpedancematchingofradiofrequency(RF)portsenablesthetestengineertocompensatetheundesiredeects,whicharenotuncommoninRFsystemsandmaketheimpedancematchingiterative,time-consuming,andanempiricalprocess.Numerousrecongurablematchingnetworkshavebeenpresentedforautomatedmatching.However,theautomationstillreliesonaniterativecontroltoachieveamatchinggoal,becauseitlackstheknowledgeoftheRFtargetandthematchingnetwork.Ourgoalsweretodevelopanautomaticmatchingcontrolsystemthatusesthisknowledgetosettheimpedancematchinginanon-iterativefashionandtodevelopamethodtoextractcircuitparameterssystematicallywhilekeepingtheadditionalnecessarypartstoaminimum.Toachievethisgoal,weusetheprinciplesofareectometertoextractknowledgeoftheRFtargetandvariousmicrowavemodelingmethodstocharacterizethematchingnetwork.Ourresultsdemonstratetheproposedideasandincludeanautomaticmatchingcontrolusingatunablemicrostripbandpasslter,aloadestimationtechniqueusingthemicrostriplter,anewlumpedmatchingnetworkfortheautomaticmatchinginembeddedRFtesting,andanewmatchingcontrolalgorithmusingtheloadestimation. 12

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CHAPTER1INTRODUCTION 1.1WhyAutomaticImpedanceMatching? Thedesignprocessforradiofrequency(RF)systemsoftenstartswiththedenitionoftheinterfacebetweenthesmallercomponents,wheretheimpedancematchingiscriticalforthefollowingreasons. Delivermaximumpowerbetweencomponents Improvethesignal-to-noiseratio(SNR) Reduceamplitudeandphaseerrors Provideisolationbetweencircuitstages However,theactualimplementationoftheimpedancematchingisimpairedeasilybyadeviationfromthetypicalcomponentvalueusedduringthedesignprocess.Itiscriticaltobeabletoevaluatethisdeviationwhensellingacomponentcommercially,sotechniquestocreateandevaluatematchingareimportantfortestingRFproductionICs. Forexample,theimpedanceofradiofrequency(RF)portsinadeviceundertest(DUT)toaloadboardispoorlydenedduetounwantedeects,suchaspogopinconnections,socketparasitics,andmanufacturingvariation.Inaddition,theinputimpedanceofahandhelddeviceantenna,especiallythereactancepartoftheimpedance,isvaryingastheenvironmentaroundthedeviceischanging.TheunwantedeectsandvaryingenvironmentarenotuncommoninRFsystemsandthematchingofapoorlydenedimpedanceisdicult,slow,andexpensive,duringparttest. ArecongurablematchingnetworkcanbeusedtosettheimpedancematchingofRFmatchtargetsagainstthedeviationcausedbytheunwantedeects.Therecongurationisconductedbyaclosed-loopfeedbackcontrol,aso-calledautomaticmatchingcontrol,whichsensestheimpedancemismatchanddeterminespropercomponentvaluesfortheimpedancematching. 13

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Figure1-1. Networkmatchinganarbitraryloadimpedancetoatransmissionline Figure1-2. Maximumpowerdeliveredwhensourceandloadimpedancesarematched. Inthisthesis,wewilldeveloparecongurablematchingnetworkandamatchingcontrolalgorithmforanautomaticmatchingcontrol(AMC)system,whichautomaticallysetstheimpedancematchingofRFmatchtargets. 1.2MaximumPowerTransferonImpedanceMatching ThebasicideaoftheimpedancematchingisillustratedinFigure 1-1 .Atwo-portnetworkisplacedbetweenanarbitraryload(ZL)andatransmissionline(Z0).TheloadimpedanceZLisconvertedtobematchedwiththesourceimpedanceZ0.Impedancematchingenablesthedeliveryofmaximumpowertotheload,althoughtherearemultiplereectionsbetweenthematchingnetworkandtheload. Maximumpowerdeliveryisexplainedasfollows.InFigure 1-2 ,ACpowerisbeingtransferredfromthesource,withphasormagnitudevoltagejVSj(peakvoltage)andxedsourceimpedanceZS,toaloadwithimpedanceZL,resultinginaphasormagnitudecurrentjIj.jIjissimplythesourcevoltagedividedbythetotalcircuitimpedance. jIj=jVSj jZS+ZLj(1{1) TheaveragepowerPLdissipatedintheloadisthesquareofthecurrentmultipliedbytheresistiveportion(therealpart)RLoftheloadimpedance. PL=I2rmsRL=1 2jIj2RL=1 2jVSj jZS+ZLj2RL=1 2jVSj2RL (RS+RL)2+(XS+XL)2(1{2) 14

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Thedenominatorisminimizedbymaking XL=)]TJ /F5 11.955 Tf 9.3 0 Td[(XS(1{3) Thepowerequationisreducedto PL=1 2jVSj2RL (RS+RL)2(1{4) Similarly,thepowerismaximizedbymaking RL=RS(1{5) Therefore,twoconditions,Equation 1{3 and 1{5 ,arecombinedandwrittenascomplexconjugatematchingcondition. ZL=ZS(1{6) where*denotesacomplexconjugate.Thecomplexconjugatematchingisthematchinggoaloftheproposedautomaticmatchingcontrol. 1.3ChallengesofAutomaticImpedanceMatching Recongurablematchingnetworkshavebeenproposedfortheautomaticmatchingcontrol(AMC).Theproposednetworksweredesignedusingvarioustuningelementssuchasvaractors[ 1 ][ 2 ][ 3 ][ 4 ],CMOSswitches[ 5 ],p-i-ndiodes[ 6 ][ 7 ],orMEMSswitches[ 8 ]. Therecongurablematchingnetworkisadaptedbyamatchingcontrolalgorithmwhichdeterminescomponentvaluesbytheiterationofaclosed-loopfeedback.WhenRFmatchtargetsarenotpreciselydened,ndingnecessarymatchingcomponentsbecomeshighlyiterative,timeconsuming,andanempiricalprocess.EveniftheyarefoundforacertainRFmatchtarget,theyseldomprovidetheimpedancematchingtoothertargets. Animprovementontheimpedancematchingcontrolalgorithmhasbeenachievedbyusingheuristicapproaches[ 6 ].Althoughthepresentedalgorithmscanconvergefasterthanabrute-forceapproach,theymaystillgettrappedinalocalminimumwhichisoneoftypicaloptimizationproblems.Further,theiterativenatureofthealgorithmsmakes 15

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Figure1-3. Automaticmatchingcontrol(AMC) convergencetimetoachievetheimpedancematchingincreasingproportionallytothecomplexityofthematchingnetwork. ForembeddedRFtesting,lumpedelementsarepreferredtodistributedelementsduetothecompactsizeatthefrequencyequaltoorlessthan2.4GHz.Mostmatchingcontrolassumesthatadegreeofmismatchismeasuredthroughadirectionalcoupler,whichistoolargetobeembedded. Sofarthechallengingproblemsoftheautomaticmatchingcontrolhavebeendescribed.Nextsectionwewillproposesolutionstothem. 1.4ProposedSolution:AutomaticMatchingControlwithLoadEstimation Aproposedsolutiontothechallengesoftheimpedancematchingisanautomaticmatchingcontrol(AMC)supportedbyaloadimpedanceestimation.AsshowninFigure 1-3 ,theautomaticmatchingcontrolreconguresthematchingnetworkthroughtheclosed-loopfeedbackconsistingofacoupler,areectioncoecientdetector,andamatchdecisioncircuit.Anestimateofaloadreectioncoecienthelpsthematchingcontroltoachieveanimpedancematchwithouttryingdierentbiasesiteratively.ThisthesisfocusesondevelopingrecongurablematchingnetworksandmatchingcontrolalgorithmswithloadestimationespeciallyforembeddedRFtesting. 16

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CHAPTER2BACKGROUND 2.1ImpedanceMatchingNetworkswithLumpedElements Whencircuitdimensionsarenotsmallrelativetothewavelength,thephasechangefromonepointtoanotherinthecircuitisnotnegligible.Inthiscase,theequivalentvoltageandcurrentwavesalongatransmissionlineareexpressedasthesumoftheincidentandreectivewaves,givenby V(z)=V+e)]TJ /F4 7.97 Tf 6.58 0 Td[(|z+V)]TJ /F5 11.955 Tf 7.09 -4.94 Td[(e+|z(2{1) I(z)=V+ Z0e)]TJ /F4 7.97 Tf 6.59 0 Td[(|z)]TJ /F5 11.955 Tf 13.15 8.09 Td[(V)]TJ ET q 0.478 w 260.59 -215.19 m 277.1 -215.19 l S Q BT /F5 11.955 Tf 262.47 -226.38 Td[(Z0e+|z(2{2) whereZ0andarethecharacteristicimpedanceandthepropagationconstantofthetransmissionline,respectively. IfthetransmissionlineisterminatedwithotherthanitscharacteristicimpedanceZ0,thereectionhappensasaresultofdiscontinuities.Thereectioncoecient)-327(isdenedastheratiooftheincidenttothereectedwavealongatransmissionline,givenby \(z)=V)]TJ /F5 11.955 Tf 7.09 -4.34 Td[(e+|z V+e)]TJ /F4 7.97 Tf 6.59 0 Td[(|z=V)]TJ ET q 0.478 w 259.64 -377.07 m 276.15 -377.07 l S Q BT /F5 11.955 Tf 259.64 -388.26 Td[(V+e+|2z(2{3) IfthetransmissionlineisterminatedwithaloadimpedanceZL,theloadreectioncoecientandtheloadimpedancearewrittenas )]TJ /F4 7.97 Tf 7.31 -1.8 Td[(L=\(z)z=0=V)]TJ ET q 0.478 w 268.89 -472.7 m 285.4 -472.7 l S Q BT /F5 11.955 Tf 268.89 -483.89 Td[(V+(2{4) ZL=V(z) I(z)z=0=Z0V++V)]TJ ET q 0.478 w 229.84 -514.85 m 277.47 -514.85 l S Q BT /F5 11.955 Tf 229.84 -526.04 Td[(V+)]TJ /F5 11.955 Tf 11.96 0 Td[(V)]TJ /F1 11.955 Tf 11.6 4.75 Td[(=Z01+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(L 1)]TJ /F1 11.955 Tf 11.96 0 Td[()]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L(2{5) TheloadreectioncoecientcanbealsoexpressedintermsofZLandZ0as )]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L=ZL)]TJ /F5 11.955 Tf 11.95 0 Td[(Z0 ZL+Z0(2{6) 17

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Thereectioncoecient)-326(isacomplexnumberthatdescribesboththemagnitudeandthephaseshiftofthereection.Thesimplestcases,whentheimaginarypartof)-326(iszero,are )-326(=)]TJ /F1 11.955 Tf 9.3 0 Td[(1Maximumnegativereection,whenshort-circuited. )-326(=0Noreection,whenperfectlymatched. )-326(=+1Maximumpositivereection,whenopen-circuited. Thevoltagestandingwaveratio(VSWR),whichrepresentsthedegreeofreectioninanotherway,isdenedastheratioofthemaximumtotheminimummagnitudeofthevoltagewave.Thevoltagestandingwaveratiocanbeexpressedintermsofthereectioncoecientgivenby VSWR=jV(z)jmax jV(z)jmin=jV+jje)]TJ /F4 7.97 Tf 6.59 0 Td[(|z+)]TJ /F4 7.97 Tf 19.07 -1.8 Td[(Le+|zjmax jV+jje)]TJ /F4 7.97 Tf 6.59 0 Td[(|z+)]TJ /F4 7.97 Tf 19.08 -1.79 Td[(Le+|zjmin=1+j)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(Lj 1)-222(j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(Lj(2{7) Conversely,thereectioncoecientcanbeobtainedfromthemeasurementofvoltagestandingwaveratioalongthetransmissionline. j)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(Lj=VSWR)]TJ /F1 11.955 Tf 11.95 0 Td[(1 VSWR+1(2{8) Inatraditionalvectornetworkanalyzer,acomplexreectioncoecientiscalculatedfromthemeasurementofincidentandreectedwavepowers.Ifthephaseofareectioncoecientisnotnecessary,e.g.,thedegreeofmismatch,themagnitudeofareectioncoecientcanbemeasuredbyaVSWRdetector. 2.1.1L-typeMatchingNetwork ThesimplestmatchingnetworkisaL-typenetworkwithtwolumpedelements.Thevaluesofthelumpedelementscanbefoundthroughtheanalyticsolutionasfollows.TheimpedanceseenlookingintothematchingnetworkshowninFigure 2-1A shouldbeequaltoZ0forimpedancematching: Z0=|X+1 |B+1=(RL+|XL)(2{9) 18

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A B Figure2-1. TheLsectionmatchingnetworks wheretheloadimpedanceZL=RL+|XL.Rearrangingandseparatingintorealandimaginarypartsgivestwoequationsgivenby B(XRL)]TJ /F5 11.955 Tf 11.96 0 Td[(XLZ0)=RL)]TJ /F5 11.955 Tf 11.95 0 Td[(Z0(2{10) X(1)]TJ /F5 11.955 Tf 11.95 0 Td[(BXL)=BZ0RL)]TJ /F5 11.955 Tf 11.96 0 Td[(XL(2{11) ThesolutionforBandXaregivenby B=XLp RL=Z0p R2L+X2L)]TJ /F5 11.955 Tf 11.96 0 Td[(Z0RL R2L+X2L(2{12) X=1 B+XLZ0 RL)]TJ /F5 11.955 Tf 19.11 8.09 Td[(Z0 BRL(2{13) NotethatthesolutionsexistonlywhenRL>Z0. Similarly,thesolutionforFigure 2-1B isgivenby B=p (Z0)]TJ /F5 11.955 Tf 11.96 0 Td[(RL)=RL Z0(2{14) X=p RL(Z0)]TJ /F5 11.955 Tf 11.96 0 Td[(RL))]TJ /F5 11.955 Tf 11.96 0 Td[(XL(2{15) NotethatthesolutionsexistonlywhenRL
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Figure2-2. Thesectionmatchingnetwork Figure2-3. TheTsectionmatchingnetwork TheimpedanceseenlookingintothematchingnetworkshowninFigure 2-2 shouldbeZ0. 1 Z0=|B1+1 |X+1 |B2+1=(RL+|XL)(2{16) Rearrangingandseparatingintorealandimaginarypartsgivestwoequationsgivenby B1Z0(X+XL))]TJ /F5 11.955 Tf 11.96 0 Td[(B1Z0B2XXL)]TJ /F5 11.955 Tf 11.95 0 Td[(B2(XRL)]TJ /F5 11.955 Tf 11.95 0 Td[(XLZ0)=Z0)]TJ /F5 11.955 Tf 11.96 0 Td[(RL(2{17) X(1)]TJ /F5 11.955 Tf 11.95 0 Td[(B2XL)=(B1+B2)Z0RL)]TJ /F5 11.955 Tf 11.96 0 Td[(XL)]TJ /F5 11.955 Tf 11.96 0 Td[(B1B2XZ0RL(2{18) Therearethreevariables,X,B1,andB2,denedbytwoequations.Hence,thereexistmultiplesolutionssatisfyingtheimpedancematching.Forauniquelydeterminedsolution,thequalityfactor,whichdeterminesthebandwidthofmatchingnetworks,canbeusedasoneofdesignspecications. 2.1.3T-typeMatchingNetwork Similartothe-typenetwork,aT-typenetworkisanalyzedasfollows.TheimpedanceseenlookingintothematchingnetworkshowninFigure 2-3 shouldbeZ0. Z0=|X1+1 |B+1=(RL+|(X2+XL))(2{19) 20

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Rearrangingandseparatingintorealandimaginarypartsgivestwoequationsgivenby BZ0(X2+XL))]TJ /F5 11.955 Tf 11.95 0 Td[(BX1RL=Z0)]TJ /F5 11.955 Tf 11.95 0 Td[(RL(2{20) X1(1)]TJ /F5 11.955 Tf 11.96 0 Td[(B(X2+XL))=BZ0RL)]TJ /F1 11.955 Tf 11.95 0 Td[((X2+XL)(2{21) 2.1.4BandwidthofMatchingNetwork Inaresonantcircuit,theratioofitsresonantfrequencyf0toitsbandwidthisknownastheloadedQofthecircuit. QL=!0 Bandwidthinradian=f0 BandwidthinHertz(2{22) Thematchingnetworksareusedforanimpedancematchatacertainfrequency.Thefrequencyresponseofthenetworkscanbeclassiedaseitheralow-passorahigh-passlter.Ateachnodeofthenetworks,thereisanequivalentseriesinputimpedance,denotedbyRs+|Xs.Hence,acircuitnodeQ,denotedbyQn,canbedenedateachnodeas Qn=jXsj Rs(2{23) IftheequivalentparallelinputadmittanceatthenodeisGp+|Bp,thecircuitnodeQcanbeexpressedintheform Qn=jBpj Gp(2{24) Foranarrowbandrangeoffrequenciesaroundf0,amatchingnetworkcanbeviewedasabandpasslter.Hence,theloadedQofthebandpasslterisgivenby QL=!0 BW=!0RTCT=jBTj GT=RT jXTj(2{25) Ifthematchingnetworkholdsthecomplexconjugatematchingateachnode,Rs1=Rs2andXs1=)]TJ /F5 11.955 Tf 9.3 0 Td[(Xs2.Rs1andRs2aretheresistanceseenlookingintoasourceandaloadatthenode.Xs1andXs2arethereactanceseenlookingintoasourceandaloadatthe 21

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node.TheloadedQcanbewrittenastheratiooftheresistancetothereactance. QL=Rs1jjRs2 jXs1j=Rs1jjRs2 jXs2j=Qn 2(2{26) IfmultipleinternalnodesexistsuchasTornetworks,theQLofthematchingcircuitsisthehalfvalueofthehighestQninthecircuits.Ingeneral,highervaluesofQLcanbeobtainedusingmatchingcircuitswithmoreelements.Forexample,addingoneelementtoaLnetworkresultsineitheraTnetworkoranetwork,whichhashighervaluesofQLthanaLnetwork. Whenabandwidthofmatchingnetworksisadesignconsideration,LnetworksarenotsuitablebecausetheQLofthenetworksisxedwiththecomplexconjugatematchingcondition.Instead,thehigherorderladdernetworkscanhaveeitherhigherorlowerQLdependingonthehighestQnofthenetworks. 2.2PatternRecognition Mostpatternrecognitionsystemsarepartitionedintofourcomponentssuchaspreprocessing,featureextraction,classication,andpostprocessing,asshowninFigure 2-4 .Preprocessingsimpliessubsequentoperationswithoutlosingrelevantinformation,featureextractionmeasuresusefulpropertiesforclassication,classicationassignstheextractfeatureintoacategory,andpostprocessingusestheclassicationresultstodecideontherecommendedaction. Figure2-4. Typicalpatternrecognitionssystemdiagram 22

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Figure2-5. Nearest-neighborruleleadstoapartitioningoftheinputspaceintoVoronoicells Themicrowavecomponentshavebeensuccessfullymodeledbythepatternrecognitiontechniquesthankstotheiradaptabilitytocomplexphenomena[ 9 ].ApplyingthepatternrecognitiontechniquestotheproposedmatchingcontrolbecomesanewresearchopportunityfortheadvancementoftheautomatedRFsystem.Inthissection,twoclassiers,anearestneighborclassierandaneuralnetwork,andprincipalcomponentanalysis(PCA),oneofpopularpreprocessingmethods,areintroducedanddescribed. 2.2.1Thek-NearestNeighborClassier Thetaskoftheclassieristousethefeaturevectorprovidedbythefeatureextractortoassigntheobjecttoacategory.Thenearest-neighborruleforclassifyingatestpointxistoassignitthelabelassociatedwiththetrainingpointx0nearesttoit.Thisruleallowsustopartitionthefeaturespaceintocellsconsistingofallpointsclosertoagiventrainingpointx0{aso-calledVoronoitessellationofthespace,asshowninFigure 2-5 Anobviousextensiontoofthenearest-neighborruleisthek-nearest-neighborrule.Adecisionismadebyexaminingthelabelsontheknearestneighborsandtakingavote. 23

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Figure2-6. Multilayerperceptron 2.2.2ArticialNeuralNetwork(ANN) Articialneuralnetworks(ANNs)aredistributed,adaptive,generallynonlinearlearningmachinesbuiltfrommanydierentprocessingelements(PEs)[ 10 ].Asanonlinearstatisticaldatamodelingtool,neuralnetworksareusedtomodelcomplexrelationshipsbetweeninputsandoutputsortondpatternsindata. Oneofthemostpopularandpowerfulneuralnetworkdesignsisknownasamulti-layerperceptron.Amultilayerperceptronconsistsofasetofneuronsinterconnectedbyweightedconnections,asillustratedinFigure 2-6 .Thereareaninputlayer,oneormorehiddenlayers,andanoutputlayer.AsshowninFigure 2-7 ,eachneuronhasweightedcoecientswl;j;mthatareadjustedtotrainthealgorithm,whicharelinearlycombined,thenpassedthroughanonlinearactivationfunctionf.Themultilayerperceptroncanberepresentedas yk=f Xjw2;j;kf Xiw1;i;jxi!!(2{27) 24

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Figure2-7. Articialneuron Figure2-8. Sigmoidalnonlinearity wherefistheactivationfunction(typically,nonlinearsigmoidalfunction),wl;j;kisaweightfromthekthnodeofthe(l-1)thlayertothejthnodeofthelthlayer,xiandykdenotetheithinputandthekthoutputnode,respectively. Theactivationfunctionf()providesthenonlinearitynecessaryfortheneuralnetwork.Ingeneral,anymonotonicallyincreasingfunctioncanbeused.Onecommonlyusedfunctionisthesigmoidalnonlinearity,denedby f(x)=1 1+e)]TJ /F4 7.97 Tf 6.59 0 Td[(x(2{28) 25

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andplottedinFigure 2-8 .Thisfunctionhasthepropertythatthederivativeiseasytocompute. f0(x)=f(x)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(f(x))(2{29) Intrainingthemultilayerperceptron,asupervisedtrainingalgorithmisused,inwhichasetofknowninput/outputdatacombinationsarepresentedtothenetwork.Usingabackpropagationalgorithm,thenetworkistrainedsothatthenetworkoutputmatchesascloselyaspossiblethedesiredoutput,foreachinputdatapoint. 2.2.3PrincipalComponentAnalysis(PCA) Tightcorrelationamongfeaturesmakesdicultthetrainingofneuralnetworks.Principalcomponentanalysisisusedtodecorrelatefeaturesbyrepresentingdataalongthedirectionwiththelargestvarianceandisoftenusedtogetherwithneuralnetworks.Howfeaturevectorscanberepresentedintermsofprincipalcomponentsisdescribedinthissection. Theproblembeginswithaideaofhowasinglevectorx0representsallvectorsinasetofnd-dimensionalsamplesx1;x2;;xn.Oneofsolutionsistondavectorx0suchthatthesumofthesquareddistancesbetweenx0andthevariousxkisassmallaspossible.Thesquared-errorcriterionfunctionJ0(x0)isdenedas J0(x0)=nXk=1jjx0)]TJ /F7 11.955 Tf 11.95 0 Td[(xkjj2(2{30) andisusedtoseekthevalueofx0tominimizeJ0.Thesolutionisthesamplemeanm,denedas m=1 nnXk=1xk(2{31) However,thesamplemeaniszero-dimensionalrepresentationofthedataset.Itismoreinterestingtogetone-dimensionalrepresentationbyprojectingthedataontoalinerunningthroughthesamplemean.Thelineequationiswrittenas x=m+ae(2{32) 26

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whereeisaunitvectorinthedirectionoflineandthescalaracantakeanyrealvalue.Ifxkisrepresentedbym+ake,thenanoptimalsetofcoecientakcanbeobtainedbyminimizingthesquared-errorcriterionfunction. Thesquared-errorcriterionfunctioniswrittenas J1(x)=J1(a1;a2;;an;e)=nXk=1jj(m+ake))]TJ /F7 11.955 Tf 11.95 0 Td[(xkjj2 =nXk=1a2kjjejj2)]TJ /F1 11.955 Tf 11.96 0 Td[(2nXk=1akeT(xk)]TJ /F7 11.955 Tf 11.95 0 Td[(m)+nXk=1jjxk)]TJ /F7 11.955 Tf 11.95 0 Td[(mjj2(2{33) ThederivativeofJ1withrespecttoakissettozeroandtheoptimalsetofcoecientakisgivenby @J1 @ak=2nXk=1(akjjejj2)]TJ /F7 11.955 Tf 11.96 0 Td[(eT(xk)]TJ /F7 11.955 Tf 11.95 0 Td[(m))=0(2{34) ak=eT(xk)]TJ /F7 11.955 Tf 11.95 0 Td[(m)(2{35) wheretheunityvectorjjejj2=1. BysubstitutingthesolutionakintoJ1,J1iswrittenas J1(x)=J1(e)=)]TJ /F4 7.97 Tf 17.36 14.95 Td[(nXk=1[eT(xk)]TJ /F7 11.955 Tf 11.95 0 Td[(m)]2+nXk=1jjxk)]TJ /F7 11.955 Tf 11.96 0 Td[(mjj2(2{36) =)]TJ /F4 7.97 Tf 17.36 14.94 Td[(nXk=1eT(xk)]TJ /F7 11.955 Tf 11.96 0 Td[(m)(xk)]TJ /F7 11.955 Tf 11.96 0 Td[(m)Te+nXk=1jjxk)]TJ /F7 11.955 Tf 11.95 0 Td[(mjj2(2{37) =)]TJ /F7 11.955 Tf 9.29 0 Td[(eTSe+nXk=1jjxk)]TJ /F7 11.955 Tf 11.96 0 Td[(mjj2| {z }independentofe(2{38) wherethescattermatrixisdenedas S=nXk=1(xk)]TJ /F7 11.955 Tf 11.95 0 Td[(m)(xk)]TJ /F7 11.955 Tf 11.96 0 Td[(m)T(2{39) UsingthemethodofLagrangemultipliers,thevectoreminimizingJ1becomesaneigenvectorofthescattermatrixasfollows. Se=e(2{40) 27

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Figure2-9. Principalcomponentanalysis )]TJ /F7 11.955 Tf 11.95 0 Td[(eTSe=)]TJ /F5 11.955 Tf 9.3 0 Td[(eeT=)]TJ /F5 11.955 Tf 9.3 0 Td[((2{41) Insummary,ndingthebestone-dimensionalprojectionofthedata(bestinleast-sum-of-squared-errorsense)isequivalenttoprojectingthedataontoalinethroughthesamplemeaninthedirectionoftheeigenvectorofthescattermatrixhavingthelargesteigenvalue.Thecoecientsakarecalledtheprincipalcomponents.Principalcomponentanalysisreducesthedimensionalityoffeaturespacebylimitingdirectionsalongwhichthescattering(variance)isthegreatest.ThosedirectionsaregeometricallyillustratedinFigure 2-9 2.3Multi-PortReectometers 2.3.1Six-PortReectometer Asix-portreectometerwasproposedasanalternativenetworkanalyzer[ 11 ].Thebasicstructureofthesix-portreectometerisillustratedinFigure 2-10 .Theport1(P1)isconnectedtoasignalsource,theport2(P2)isterminatedwithadeviceundertest 28

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Figure2-10. Six-portreectometer (DUT)havinganimpedanceZL,andtheport3,4,5,and6(P3,P4,P5,andP6)areconnectedtopowerdetectors.Thesix-portjunctionischaracterizedby12complexwavesaiandbi,i=1;;6andthescatteringequationsdenedas 0BBBBBBB@b1b2...b61CCCCCCCA=266666664S11S12S16S21S22S26............S61S62S663777777750BBBBBBB@a1a2...a61CCCCCCCA(2{42) or bi=6Xj=1Sijaj;i=1;;6(2{43) Moreover,thedetectorsareterminatedwithdenedloads.Theterminationsaredescribedbyadditionalequationsgivenby ai=bi)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(ii=3;;6(2{44) where)]TJ /F4 7.97 Tf 41.13 -1.8 Td[(iistheinputreectioncoecientoftheithpowerdetector.Equation 2{43 and 2{44 canbesolvedintermsofa2andb2.Then,theincidentwavesonthedetectorsarewrittenas bi=Aia2+Bib2=b2Aia2 b2+Bi Ai=b2Ai()]TJ /F6 7.97 Tf 11.87 -1.8 Td[(2)]TJ /F5 11.955 Tf 11.95 0 Td[(qi)i=3;;6(2{45) 29

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whereAiandBiarefunctionsofSijand)]TJ /F4 7.97 Tf 30.07 -1.79 Td[(j Thepowerdetectoryieldsonlyamplitudeorpowerresponse,thereforetheoutputoftheithdetectoriswrittenas Pi=jbij2=jb2j2jAij2j)]TJ /F6 7.97 Tf 7.32 -1.8 Td[(2)]TJ /F5 11.955 Tf 11.95 0 Td[(qij2(2{46) Withananalogywithasix-portreectometeranalysis,oneofportisareferenceport(P4),whereA4isideallyzero.Thepowerdetectedinthereferenceport4iswrittenas P4=jb4j2=jA4a2+B4b2j2=jB4j2jb2j2(2{47) Thenormalizedpowerbythereferenceportiswrittenas Pi P4=jAij2 jB4j2j)]TJ /F6 7.97 Tf 7.31 -1.8 Td[(2)]TJ /F5 11.955 Tf 11.95 0 Td[(qij2i=3;5;6(2{48) ThereectioncoecientoftheDUTcanbeexpressedintermsofthedetectorpoweras j)]TJ /F6 7.97 Tf 7.31 -1.79 Td[(2)]TJ /F5 11.955 Tf 11.96 0 Td[(qij2=jB4j2 jAij2Pi P4i=3;5;6(2{49) TheequationrepresentscirclesontheSmithchartandthereectioncoecientofaDUT)]TJ /F6 7.97 Tf 7.31 -1.79 Td[(2isthecrosspointofthecircles. Althoughthesecirclesshouldbeintersectedinasinglepointideally,itmaynothappeninrealityduetomeasurementerrors.Inthiscase,thereectioncoecientcanbedeterminedbytheradicalcenterofthreecircles,asshowninFigure 2-11 2.3.2Four-PortMultistateReectometer Asix-portreectometerhasbeenpaidattentionbymanyresearcherduetoitssimplestructure.Thefundamentalsofthesix-portreectometerwereextendedtothedesignofamultistatereectometerandafour-portmultistatereectometerwithonlytwodetectorswasproposed[ 12 ][ 13 ][ 14 ].ThediagramofthemultistatereectometerisillustratedinFigure 2-12 30

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Figure2-11. Determinationof)]TJ /F4 7.97 Tf 98.52 -1.79 Td[(Lfromtheradicalcenterofthreecircles Figure2-12. Four-portreectometer 31

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Withananalogywiththesix-portreectometer,theoperationofafour-portreectometerisdescribedbythefollowingequations, bi=Aia2+Bib2=b2Aia2 b2+Bi Ai=b2Ai()]TJ /F4 7.97 Tf 11.86 -1.79 Td[(L)]TJ /F5 11.955 Tf 11.95 0 Td[(qi)i=3;4(2{50) )]TJ /F6 7.97 Tf 7.31 -1.8 Td[(2)]TJ /F5 11.955 Tf 11.96 0 Td[(q3 )]TJ /F6 7.97 Tf 7.31 -1.8 Td[(2)]TJ /F5 11.955 Tf 11.96 0 Td[(q42=A4 A32P3 P4(2{51) wherethesystemparametersA3andA4aregiven(RefertoAppendix B forderivation)by A3=S21S32)]TJ /F5 11.955 Tf 11.95 0 Td[(S22S31 S21q3=S31 S22S31)]TJ /F5 11.955 Tf 11.95 0 Td[(S21S32(2{52) and A4=S21S42)]TJ /F5 11.955 Tf 11.95 0 Td[(S22S41 S21q4=S41 S22S41)]TJ /F5 11.955 Tf 11.95 0 Td[(S21S42(2{53) Themultistatereectometerisequippedwiththefacilitytooperateindierentstatesk(k=1,2,3)inordertoprovideenoughinformationforthedeterminationofboththemagnitudeandthephaseofthereectioncoecientofadevice-under-test(DUT). Theequationsarewrittenwithsuperscriptk,whichdenotesastate. )]TJ /F6 7.97 Tf 7.32 -1.79 Td[(2)]TJ /F5 11.955 Tf 11.96 0 Td[(q(k)3 )]TJ /F6 7.97 Tf 7.32 -1.79 Td[(2)]TJ /F5 11.955 Tf 11.96 0 Td[(q(k)42=A(k)4 A(k)32P(k)3 P(k)4(2{54) Theoptimum-performancecriteriaforthemultistatereectometerareasfollows. Thecentersq(k)shouldbeequallyspacedaroundtheorigin(i.e.,120angularseparation). Thecentersq(k)shouldbeequidistantfromtheorigin(i.e.,equalmagnitudes). ThesystemparametersB(k)shouldhavezeromagnitudes. Similartothesix-portreectometer,thereferenceport,whichdependsonlyonb2tosetA4tozero,isassignedtoport4.Theequationcanbewrittenas )]TJ /F6 7.97 Tf 7.31 -1.8 Td[(2)]TJ /F5 11.955 Tf 11.96 0 Td[(q(k)32=B(k)4 A(k)32P(k)3 P(k)4(2{55) 32

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Thepowermeasurementwiththreedierentnetworkstatesresultsinthreecirclesdenedbytheequation.Thecomplexreectioncoecientofadevice-under-test(DUT)canbedeterminedinthesamewayasthesix-portreectometer. 33

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CHAPTER3AUTOMATICMATCHINGCONTROL 3.1Overview TheabilitytotestRFdevicesontheloadboardrequiresagoodbroadbandmatch.Moreover,thebandwidthofmanywirelessdevicesisgovernedbytheinputimpedanceofanantenna.However,theinputimpedanceisoneoftheparametersthatvariesmostindierentenvironmentsontheloadboardandnexttoanantennaandtheinputreactancevarieswithfrequencymorethandoestheinputresistanceoftheantenna[ 15 ][ 16 ]. Variousautomaticmatchingtunersandcontrolalgorithmshavebeenproposed.Aphasedetectorwasusedtocorrectthereactivepartoftheantennamismatchin[ 17 ].Generalmatchingnetworkdesignandtuningstrategieswerestudiedin[ 18 ].Agenericalgorithmhasbeenwidelyusedasatuningalgorithm[ 18 ][ 19 ][ 20 ]andheuristicsearchalgorithmswerestudiedin[ 6 ].Inaddition,variousnarrowbandtechniquesatdierentfrequencieshavebeenstudiedforCMOSswitchedcapacitorsat2.4GHz[ 5 ]andap-i-ndiodeswitchedcapacitorsat390MHz[ 6 ]. Priortothiswork,abroadbandrecongurablematchingnetworkwasproposed,butasupportingmatchingcontrolsystemandalgorithmswerenotdeveloped[ 1 ].Inthiswork,anovelautomaticbroadbandmatchingcontrolwasdevelopedforabroadbandrecongurablematchingnetworkfrom2.5to4.5GHz.Thematchingnetworkconsistsofave-stubmicrostriplterandthreevaractors.Amatchingalgorithmwasdevelopedusingagreedysearchalgorithmtodeterminethevaractorbiasforimpedancematchoverthelargefractionalbandwidth(>50%).Theworkdemonstratesthefeasibilityoftheautomaticmatchingcontrolcircuitover2.5GHzto4.5GHz. Therestofthischapterisorganizedasfollows:Section 3.2 givesanoverviewoftheproposedsystem.Section 3.3 presentstheexperimentalresults.Section 3.4 concludesanddescribesthefuturedirection. 34

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Figure3-1. Automaticmatchingcontrol(AMC)systemdiagram 3.2SystemOverview AsshowninFigure 3-1 ,theproposedautomaticmatchingcontrolsystemconsistsofarecongurabletuner,anetworkanalyzer,ahostcomputer,andamicrocontroller.Inthecurrentimplementation,thetunerisdirectlyconnectedtothenetworkanalyzer.Variousantennamismatchesweresimulatedbyanautomaticload-pullsystem. Theautomaticmatchingisperformedbyaclosed-loopfeedbackoperationasfollows.First,thenetworkanalyzermeasuresthereectedwavepower(S11parameter).Second,thehostcomputercalculatestheavailablebandwidth.Last,thealgorithmsearchesforthevaractorbiasforbroadbandimpedancematchandthemicrocontrollersetsthevaractorbias.Theaboveproceduresarerepeateduntilthealgorithmsearchconvergestotheoptimalbias,whichminimizestheimpedancemismatch.Thistechniquecanbeappliedtoanindustrialautomatictestequipment(ATE)systemandaloadboardforconductingRFparttest. 35

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3.2.1ImpedanceMatchingTuner AsshowninFigure 3-2 ,thetuner,manufacturedonRogersCorporationDuroid6006boardbytheUniversityofArizona,hasvestubsalongamicrostripline.Threestubsareconnectedtoareverse-biasedvaractordiodeMPV1965,whereastwostubshavenotuningelements.Theeectivelengthofthetunablestubiscontrolledbythecapacitanceofthevaractor.Thevaractorreversebiasandcapacitancerangefrom0to5.12Vand4.5to1pF,respectively. Thetunerisdesignedtoprovidewidebandwidth(upto2.5GHz)atthefrequencyof3.5GHz,asshowninFigure 3-3 .Thefractionalbandwidthis71%=2.5/3.5.AgilentE8358APNAnetworkanalyzerwasusedtomeasureS-parametersandthebandwidthwasmeasuredbelow10dBforthereturnloss.Theinsertionlossisaslowas2dB.ThedimensionsofthetunerareshowninTable 3-1 Thematchingcapabilityofthematchingtunercanberepresentedbytheloadreectioncoecient)]TJ /F4 7.97 Tf 112.67 -1.79 Td[(Ltobematchedbythetuner.Theinputreectioncoecient)]TJ /F4 7.97 Tf 317.52 -1.79 Td[(iniswrittenas )]TJ /F4 7.97 Tf 7.31 -1.79 Td[(in=S11+S12S21)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L(3{1) Thematchingcapabilityisderivedfromtheloadreectioncoecientbysettingtheinputreectioncoecienttozero. )]TJ /F4 7.97 Tf 7.32 -1.79 Td[(L)]TJ /F13 5.978 Tf 5.29 -1.22 Td[(in=0=S11 S11S22)]TJ /F5 11.955 Tf 11.96 0 Td[(S12S21(3{2) Thedistributionofthematchingcapabilitywiththefullrangeofvaractorbiasvoltageillustratesthecoverageofmismatchedloadsthatcanbecompensatedbythematchingtuner. 3.2.2Controller Thecontrollerconsistsofahostcomputer,amicrocontroller,andadigital-to-analogconverter(DAC).Thecomputerprovidesinterfaceswithmeasurementinstrumentsandexecutessearchalgorithm.Themicrocontrollerconsistsofa12-channel12-bit 36

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Figure3-2. Recongurableve-stubmatchingtuner Table3-1. Specicationofthematchingtuner Stub/InterconnectWidth(mils)Length(mils) Stub#1򀰸 Between#1and#239342 Stub#2񧈝 Between#2and#336481 Stub#334384 analog-to-digitalconverter(ADC),a4-channel12-bitDAC,auniversalasynchronousreceiver/transmitter(UART),anddigitalinput/outputs.Toprovideasucientnumberofvoltagebiases,anadditionaloctal12-bitDAC(AD5328)wassolderedonthemicrocontrollerevaluationboard.Microcontrollerrmwarewasdevelopedintheembedded-Clanguageandstoredintheinternalashmemory.Thermwaresupportsthecontroller'scommunicationwithboththehostcomputerandtheDACthroughRS-232CandI2C.ItcaninterpretandexecuteGPIB-likecommands,suchas*IDN?,fromthehostcomputer. 37

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ASimulation BMeasurement Figure3-3. Comparisonofsimulationandmeasurementofmatchingtuner.Thematchingtuneristunedwithtypicalbias(2.562.562.56)andmatchedload 38

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ThehostcomputerandotherinstrumentswerelogicallyconnectedthroughtheNI-VISAstandardprovidedbyMATLAB.Adedicatedinterfaceprogramwasdevelopedtoautomatethecontroloftheautomaticload-pullsystem.Theprogramsupportsthecommunicationbetweenthehostandtheload-pullsystemandisnormallycalledbyMATLAB. 3.2.3Searchalgorithm Agreedyalgorithmwasusedtondthevaractorbiasforimpedancematch.Thealgorithmsearchesforthelocallyoptimalbiaspersinglevaractoratatimeandcanbeexpressedas ^vi=argmaxviBW(vi);i=1;;d;vi=0;vmax 2r)]TJ /F1 11.955 Tf 11.96 0 Td[(1;;vmax| {z }2r;(3{3) whereBW()isthemeasuredbandwidth,vmaxisthemaximumreversebiasvoltageappliedtovaractors,disthenumberofvaractors,risthenumberofDACbits. ThealgorithmhasacomputationalcomplexityofO(n),comparedwithO(n3)ofbrute-forceapproach(dimension=#ofvaractors=3).Althoughitmayfailtoconvergetothegloballyoptimalbias,itcanreducethesearchingtimesignicantly.Forexample,supposethataDACresolutionis6bits.Twoalgorithms'searchingtimearedimensionresolution=326=192andresolutiondimension=263=262144,respectively.Thegreedyalgorithmisfasterbythefactorof2rd)]TJ /F16 5.978 Tf 5.75 0 Td[(1=d=263)]TJ /F16 5.978 Tf 5.75 0 Td[(1=3'1365. 3.3ExperimentalResults AcontrolprogramwasdevelopedusingMATLABtoperformalltestingproceduresincludinginstrumentinitialization,mismatchloadsetup,S-parametermeasurement,andsearchforthevaractorbias. 3.3.1CharacterizationofAutomaticTunerSystem WeusedMauryMicrowave'sautomatictunersystem(ATS)andmechanicalload-pulltunersforvariousmismatchedloads.TheS-parametersoftheload-pulltunerare 39

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Table3-2. Mismatchedloadspecication TypeS11jS11jMotorposition Matched0+j00(100,5000,5000) Mismatched10:11+|0:090.14(16725,2262,5000) Mismatched20:19+|0:400.45(17105,1424,5000) Mismatched3)]TJ /F1 11.955 Tf 9.3 0 Td[(0:11+|0:220.24(17722,2005,5000) Mismatched40:14)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:010.14(20464,2228,5000) Mismatched50:35+|00.35Simulated Mismatched60:55+|00.55Simulated Mismatched70:65+|00.65Simulated Mismatched80:75+|00.75Simulated Figure3-4. Mismatchedload1measurement(S11=0.11+j0.09at3.5GHz).Motorpositionsare(16725,2262,5000).ReectioncoecientjS11j=0.14. determinedbythreemotorsandhavetobemeasuredoverthefrequencyrangeofinterestbecausetheyareunknownandfrequency-dependent. ThecharacterizationwasperformedbyMauryMicrowave'sATSsoftwareMT993andthemappingle,so-calledtunerle,wascreated.Eachlineofthetunerlecontainsmotorpositions,S-parameters,andoperatingfrequency. Fromthetunerle,onematchedloadandthreemismatchedloadswereselected.ThemeasurementsofthreemismatchedloadsareshowninFigure 3-4 to 3-6 .TheS11at3.5GHzandmotorpositionsaresummarizedinTable 3-2 40

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Figure3-5. Mismatchedload2measurement(S11=0.19+j0.40at3.5GHz).Motorpositionsare(17105,1424,5000).ReectioncoecientjS11j=0.45. Figure3-6. Mismatchedload3measurement(S11=-0.11+j0.22at3.5GHz).Motorpositionsare(17722,2005,5000).ReectioncoecientjS11j=0.24. Threemismatchedloadsusedinthisworkshowthealmost-constantreturnandinsertionlossfrom2.5to4.5GHz.AsdescribedinTable 3-2 ,threemismatchedloadshavejS11jat3.5GHz=0.14,0.45,and0.24,respectively. 3.3.2MeasurementResults ThematchingcapabilityS11=(S11S22)]TJ /F5 11.955 Tf 11.97 0 Td[(S12S21)atthecenterfrequencyof3.5GHzwascalculatedfromtheS-parametermeasurementresults,wherethevaractorbiasrangesfrom0Vto5.12Vby0.16Vstepin32voltagelevels.AsshowninFigure 3-7 ,thematchingcapabilitycancoverasmallportionontheSmithchart,becausethetunerwasdesigned 41

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Figure3-7. Matchingcapabilityofve-stubmatchingtunerat3.5GHz forbroadbandmatching.Generally,theoverallperformanceofthebroadbandmatchingcanbeevaluatedbytheavailablebandwidth,whichhasbeenusedastheperformancemetricthroughthiswork. TheS-parametersofthematchingtunerweremeasuredwithtypicalandoptimalbias.2.56Vwassetasthetypicalbiasforthevaractor.Theoptimalbiaswasfoundbythesearchalgorithmasdescribedbefore. Thesearchalgorithmfoundtheoptimalbiasforthe50matchedloadandthemismatchedload1,whereasitfailedtondforthemoreseverelymismatchedload2and3.Forthemismatchedload2and3,theoptimalbiasfoundforthe50casewasused. 42

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AsshowninFigure 3-8 andFigure 3-9 ,theoptimalbiasforthe50matchedloadincreasedtheavailablebandwidthfrom1GHzto2GHz.However,theoptimalbiasdoesnotincreasethebandwidthforothermismatchedloads,becausethematchingtunertogetherwiththelargermismatch()-327(=0.24,0.45)didnotrespondtothechangeinvaractorcapacitance. Thecomparisonofbrute-force,single-stepproposedin[ 6 ],andthegreedyalgorithmsispresentedinTable 3-3 ,wheretheperformancemetricforthecostfunctionofthreealgorithmsisthe10dBavailablebandwidthandthenumberoftrialsrepresentshowmanybiaseseachalgorithmhasevaluateduntilitconverges.100dierentinitialstateswererandomlygeneratedandusedforeachexperimentandtheresultswereaveragedover100experiments.ThemismatchedloadsusedinthisexperimentaresummarizedinTable 3-2 .Thedegreeofmismatchcoversfrom0to0.75.AsshowninTable 3-3 ,thegreedyalgorithmoutperformsbrute-forceandsingle-stepalgorithmsintermsofthenumberoftrialsandtheavailablebandwidth. Forseverelymismatchedloads,thesingle-stepalgorithmoftenfailedtohaveavailablebandwidth.Inthiscatastrophiccase,itisnotfairtocomparetheperformancebytakingtheaverageofexperimentalresults.Instead,thecatastrophiccasewasanalyzedandexcludedfromtheresult.First,wedene10%ofthecenterfrequency(0.35GHz)asthebandwidthlimitforthecatastrophiccase.Duringtheexperiment,thenumberofthecatastrophiccaseswerecountedtoshowhowoftenthecatastrophiccasehappensforeachalgorithm.AssummarizedinTable 3-4 ,thesingle-stepalgorithmsueredfromthecatastrophiccaseevenformodestlymismatchedloads,e.g.)]TJ /F4 7.97 Tf 334.11 -1.79 Td[(L=0:24andallthreealgorithmsfailedtooptimizefortheseverelymismatchedload,e.g.)]TJ /F4 7.97 Tf 356.74 -1.79 Td[(L=0:75.Thisresultshowedthesimilartrendasthebroadbandmatchingbandwidthfordierentloadimpedancepresentedby[ 1 ],whichlimitedtherealandimaginarypartofmismatchedloadimpedanceto25to100and)]TJ /F1 11.955 Tf 9.29 0 Td[(50to50,respectively.Second,thecatastrophiccasewasexcludedfromTable 3-3 .Theexperimentalresultavoidingthecatastrophiccaseis 43

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ATypicalbias BAutomatedbias Figure3-8. Matchingtunermeasurementwithmatchedload.Typicalandautomatedbiasesare(2.56V2.56V2.56V)and(4.8V3.84V4.64V) 44

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ATypicalbias BAutomatedbias Figure3-9. Matchingtunermeasurementwithmismatchedload1.Typicalandautomatedbiasesare(2.56V2.56V2.56V)and(3.2V3.86V3.52V) 45

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Table3-3. Comparisonofbrute-force,greedy,andsingle-stepalgorithms Type#ofTrialsAvailableBW(GHz) brutegreedysinglebrutegreedysingle Matched(Measured)32768273.60358.291.281.221.17 Mismatched1(Measured)32768272.64414.991.241.211.09 Mismatched2(Measured)32768340.80454.950.760.680.25 Mismatched3(Measured)32768332.16420.391.241.160.63 Mismatched4(Measured)32768281.28353.701.241.201.04 Mismatched5(Simulated)32768323.52473.851.161.100.40 Mismatched6(Simulated)32768312.00550.530.720.510.06 Mismatched7(Simulated)32768281.28642.870.360.360.00 Mismatched8(Simulated)32768284.16642.870.320.320.00 Average32768300.16479.160.920.860.52 summarizedinTable 3-5 .Providedthatthecatastrophiccasecanbealwaysavoidedbysometechnique,theperformancedierencebetweenthegreedyandothertwoalgorithmsisnotaslargeaswhenthecatastrophiccasehappens.However,thetechniquetoavoidthecatastrophiccasecanbeadditionaloverheadaddedtomatchingcontrolalgorithms. Therearetworeasonstomakethegreedyalgorithmoutperformotheralgorithms.First,thegreedyalgorithmislessdependentonaninitialstatecomparedwiththesingle-step,becausethegreedyalgorithmsearchesforthesuboptimalsolutionforeachvaractor.Second,thesuboptimalsolutionofthegreedyalgorithmturnedouttobeclosetotheglobaloptimalsolutionofthebrute-force.Intheotherword,thegreedyalgorithmmanagedtoreachclosetotheglobaloptimalsolution,whereasthesingle-stepmaygettrappedinlocalminima. TheoptimizationsurfaceduringanexperimentonthegreedyalgorithmisillustratedinFigure 3-12 .Thez-axisrepresentstheavailablebandwidthinGHzandthex-andy-axesrepresentthe32varactorvoltagelevelsfrom0Vto5.12Vby0.16Vstep.Theothervaractorbiasissetto16thvoltagelevel.Thesurfaceshowedthesteepthreshold,whichcanbemodeledbyeithersigmoidorroundfunctions.Theatsurfaceabovethe 46

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Table3-4. Percentageofcatastrophiccaseforbrute-force,greedy,andsingle-stepalgorithms Typej)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(LjPercentageofcatastrophiccases(%) brute-forcegreedysingle-step Matched(Measured)0.01000 Mismatched1(Measured)0.14003 Mismatched2(Measured)0.450065 Mismatched3(Measured)0.240040 Mismatched4(Measured)0.14001 Mismatched5(Simulated)0.350057 Mismatched6(Simulated)0.5503993 Mismatched7(Simulated)0.6500100 Mismatched8(Simulated)0.75100100100 Table3-5. Comparisonofbrute-force,greedy,andsingle-stepalgorithmsavoidingcatastrophiccases Type#ofTrialsAvailableBW(GHz) brutegreedysinglebrutegreedysingle Matched(Measured)32768273.60358.291.281.221.17 Mismatched1(Measured)32768272.64424.481.241.211.12 Mismatched2(Measured)32768340.80348.690.760.680.59 Mismatched3(Measured)32768332.16287.551.241.161.05 Mismatched4(Measured)32768281.28356.451.241.201.05 Mismatched5(Simulated)32768323.52345.981.161.100.91 Mismatched6(Simulated)32768388.72366.430.720.720.48 Mismatched7(Simulated)32768281.28N/A0.360.36N/A Mismatched8(Simulated)N/AN/AN/AN/AN/AN/A Average32768311.75355.411.000.960.91 thresholdexplainswhythesuboptimalsolutionofgreedyalgorithmisprettyclosetotheglobaloptimalsolution. 3.4ConclusionandDiscussion Thenovelautomaticbroadbandmatchingcontrolwasdevelopedforthebroadbandmicrostripmatchingtuner,whichwasprovidedbytheUniversityofArizona.Theproposedmatchingcontroldemonstratedthefeasibilityoftheautomaticbroadbandmatchingcontrolusingtheclosed-loopfeedbackandthegreedysearchalgorithm.The 47

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measurementresultsshowedthatthegreedysearchalgorithmcouldndtheoptimalbiasforthematchingtuner,butthematchingtunerdidnotshowthegoodtunabilityagainstthelargemismatches,j)]TJ /F2 11.955 Tf 7.32 0 Td[(j>0:14.Currentlyweareworkingonimprovingthetuner'stunabilityagainstthelargemismatch. 48

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ATypicalbias BAutomatedbias Figure3-10. Matchingtunermeasurementwithmismatchedload2.Typicalandautomatedbiasesare(2.56V2.56V2.56V)and(4.8V3.84V4.64V) 49

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ATypicalbias BAutomatedbias Figure3-11. Matchingtunermeasurementwithmismatchedload3.Typicalandautomatedbiasesare(2.56V2.56V2.56V)and(4.8V3.84V4.64V) 50

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Figure3-12. Optimizationsurfaceofgreedyalgorithm 51

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CHAPTER4LOADIMPEDANCEESTIMATION 4.1Overview Theoverallgoaloftheproposedresearchistodevelopacompactautomaticmatchingcontrol(AMC)circuitthatsetstheimpedancematchforradiofrequency(RF)portsofasocketeddeviceundertest(DUT)underautomatictestequipment(ATE)test[ 21 ].AnAMCprovidesmatchsettingandexaminationcapabilityforRFtestfrequenciestoassistinavarietyofRFtestprotocols.Inthischapter,anewloadimpedanceestimationtechniqueispresentedthatfacilitatestheAMCsystem.AloadimpedancewasnotabletobedeterminedinanAMCsystempriortothiswork. Varioussearchmethodsfortheoptimalstateofatunablematchingnetworkhavebeenproposed[ 6 ][ 17 ][ 22 ].However,theexhaustivesearchmethodsrequiremillionsofmeasurementsdependingonthenumberofthematchingnetworkstatesandbecomealimitingfactortothedesignofthematchingnetwork. Inpriorwork,mostsearchmethodstrytoreducesearchingtimeusingheuristicapproachbasedonexhaustivesearch,buttheirsearchingtimeincreasesproportionallytothematchingnetworkcomplexity.IftheloadimpedanceZLisknowntotheAMCsystem,thenthesearchisnolongernecessary.Instead,theAMCsystemcansetamatchingnetworktopologyandcomponentvaluesthroughdirectanalysis.Thus,theAMCsystemneedstosearchonlyafewtimestoachievetheimpedancematch. Theproposedimpedanceestimationtechniqueisbasedontheprincipleofamulti-statereectometer.Asanalternativenetworkanalyzer,measurementmethodsbasedonamulti-statereectometerwerepresentedusingvariousapproaches,suchasascalarnetworkanalyzer[ 23 ],afour-portjunction[ 13 ][ 14 ],andatunablemicrostriplter[ 24 ].Similartothepriorwork,theproposedmethodmeasuresasingleportmultipletimesinsteadofthemeasurementofmultipleportsinasix-portreectometer[ 11 ],thenacomplexreectioncoecientisdeterminedbysomemathematicalmanipulation.The 52

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Figure4-1. Systemdiagramofloadimpedanceestimationforautomaticimpedancematching measuredpowerrepresentsthemagnitudeofareectioncoecientandisdepictedasacircleontheSmithchart.Becausethemeasuredpowerreadingresultsfromthedierentnetworksandthesameload,theintersectpointofthesecirclesbecomestheloadreectioncoecient.Inthiswork,wedemonstratethatthetunablematchingnetworkdesignedforimpedancematchcanperformloadestimation. Therestofthischapterisorganizedasfollows:Section 4.2 givesanoverviewoftheAMCandtheproposedmethod.Section 4.3 presentstheexperimentalresults.Section 4.4 concludesanddescribesthefuturedirection. 4.2SystemOverview 4.2.1AutomaticMatchingControl(AMC) AsshowninFigure 4-1 ,anautomaticmatchingcontrolsystemconsistsofatunablematchingnetwork,apowerdetector,andacontroller.Theautomaticmatchingisperformedbyaclosed-loopfeedbackoperationasfollows.First,thepowerdetector 53

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measuresthereectedwavepower.Second,thecontrollercalculatesthedegreeofmismatch(S11parameter).Last,thecontrolalgorithmsearchesforthevaractorbiasforimpedancematchandthecontrollersetsthevaractorbias.Theaboveproceduresarerepeateduntilthecontrolalgorithmachievesaspeciedmatchinggoal.Thematchingcontrolsystempresentedin[ 25 ],whichwasnotabletoestimateaload,wasusedinthiswork. 4.2.2LoadEstimationMethod Aninputreectioncoecientisexpressedintermsofaloadreectioncoecientandtwo-portS-parametersas )]TJ /F4 7.97 Tf 7.31 -1.79 Td[(in=S11+S12S21)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(L 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L(4{1) Acomplexreectioncoecient,suchas)]TJ /F4 7.97 Tf 215.22 -1.8 Td[(inand)]TJ /F4 7.97 Tf 30.08 -1.8 Td[(L,isdeterminedbybothmagnitudeandphaseofareectedwave.However,aphasedetectorisnotascompactandaccurateasapowerdetector.Forthisreason,anembeddedtestcircuitoftenmeasuresonlywavemagnitudeusingpowerdetectorsandthephaseinformationofthereectioncoecientbecomesunavailable.Inthiscase,acomplexinputreectioncoecientsisalsomeasuredonlyinmagnitudeandEquation 4{1 canbewrittenas j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj=S11+S12S21)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(L 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(L(4{2) TheloadreectioncoecientderivedfromthemagnitudeoftheinputreectioncoecientisdepictedasacircleontheSmithchart.BymanipulatingEquation 4{2 inthesamewayasstabilitycircleequationderivation[ 26 ],thecircleequationisgivenby )]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L)]TJ /F1 11.955 Tf 13.15 6.7 Td[((S22j)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(inj2)]TJ /F1 11.955 Tf 11.96 0 Td[(S11) jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)-222(jj2=S12S21 jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)-222(jj2j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj (4{3) where=S11S22)]TJ /F5 11.955 Tf 11.95 0 Td[(S12S21and*denotesacomplexconjugate. 54

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Theloadreectioncoecient)]TJ /F4 7.97 Tf 162.1 -1.79 Td[(Lisexpressedasacirclewhosecenterandradiusare CL=(S22j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[(S11) jS22j2j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2)-221(jj2(center) (4{4) RL=S12S21 jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)-222(jj2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj(radius) (4{5) (RefertoAppendix A forthederivation.)Therelationshipbetweeninputandloadreectioncoecientswasalsoanalyzedasacircleequationin[ 8 ].Theirresultlooksdierentbutismerelyaspecialcaseofthegeneralequationpresentedinthiswork. Basedonthederivedequations,anestimationmethodmeasuresareectedwavepowerj)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(injofaninputportmultipletimes.ThemeasuredpowerrepresentsonlythemagnitudeofareectioncoecientandisdepictedasacircleontheSmithchart.Becausethemeasuredpowerreadingresultsfromthedierentnetworkandthesameload,theintersectpointofthesecirclesbecomestheloadreectioncoecient.Weperformedtheloadimpedanceestimationasfollows. 1. Measurepowerintheinputportofamatchingnetworkwiththreedierentsetsofbiases 2. Calculatethecenterandradiusofthecircleoftheloadreectioncoecient 3. Calculatetheradicalcenterofthreecircles Theradicalcenteristheapproximationofthecenterofthreecircles'overlappedregion,asillustratedinFigure 4-2 .Thecoordinatesoftheradicalcenteraregivenby x=R2L1)]TJ /F5 11.955 Tf 11.96 0 Td[(R2L2+x22 2x2 (4{6) y=R2L1)]TJ /F5 11.955 Tf 11.96 0 Td[(R2L3+x23+y23)]TJ /F1 11.955 Tf 11.96 0 Td[(2xx3 2y3 (4{7) Notethattheleast-squaresmethoddevelopedforthesix-portmeasurementcanbeusedforhigheraccuracy[ 27 ]. Assumingtheloadisfoundthroughtheproposedestimationmethod,thecontrolalgorithmcansetimpedancematchingusingS-parametersofthematchingnetworkand 55

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Figure4-2. Radicalcenterofthreecircles,)]TJ /F4 7.97 Tf 166.22 -1.79 Td[(L1,)]TJ /F4 7.97 Tf 14.47 -1.79 Td[(L2,and)]TJ /F4 7.97 Tf 37.23 -1.79 Td[(L3. Figure4-3. Impedancematchingprocedureusingestimatedloadimpedance. 56

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Table4-1. Summaryofmismatchedloadestimationresults TypejS11jMeansquarederror(MSE)MSEexcludingjc)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(Lj>1 Mismatched#10.140.5963000.011048 Mismatched#20.457.9827290.080232 Mismatched#30.240.0663340.066334 Mismatched#40.140.1483510.031391 allowquickmatchingcontroloftheAMC.TherevisedmatchingcontrolprocedureisillustratedinFigure 4-3 4.3ExperimentalResults Inourexperiments,fourdierentloadswereusedtoevaluatetheproposedestimationmethods.MauryMicrowave'sAutomaticTunerSystem(ATS)generatedthespeciedloadspreciselyeverymeasurementandAgilentE8538PNAnetworkanalyzermeasuredtheinputreectioncoecientj)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj.Thereectioncoecientmagnitudeoftheloadsarej)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(L1j0:14,j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(L2j0:45,j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L3j0:24,j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L4j0:14overtherangeoffrequencyfrom2.5GHzto4.5GHz.Threesetsofvaractorbiaseswerearbitrarilyselectedtosetthematchingnetworktodierentstates.Thesevaluesare(2.564.84.8),(4.83.844.64),and(4.84.82.56). ThecenterandradiusoftheloadreectioncoecientwerecalculatedusingEquation 4{4 and 4{5 andtheloadreectioncoecientc)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(LwasestimatedusingtheradicalcenterequationsEquation 4{6 and 4{7 Whenthecentersoftwoormoreofthreecirclesareclosetoeachother,theestimationisnotaccuratebecausetheradicalcenterisoftennotinsidetheoverlappedareaofthreecircles.Sometimes,theestimatedloadreectioncoecientgoesoutofaunitcircleontheSmithchart,whichisdenitelyincorrectforapassiveload.Theincorrectestimateswereexcludedfromtheaccuracystatistics. Theestimationaccuracyisevaluatedbythemeansquarederror(MSE)betweenmeasuredandestimatedloadreectioncoecients,Efj)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L)]TJ /F8 11.955 Tf 12.89 3.02 Td[(c)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(Lj2g.Theproposedmethod 57

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achievedthemeansquarederrors(MSE),0.011,0.080,0.066,and0.031,respectively.TheexperimentalresultissummarizedinTable 4-1 4.4Conclusion Theautomaticmatchingcontrol(AMC)systemprovidesimpedancematchandexaminationcapabilityforradiofrequency(RF)portsbytheclosed-loopfeedback.However,theAMCsystempriortothisworkwasnotabletoestimatealoadimpedance.Inthiswork,wedemonstratedthatthemeasuredpowerprovidesnotonlythedegreeofmismatchbutalsotheestimateofanunknownloadthroughthemultistatereectometermeasurement.TheproposedloadestimationallowsthequickcontroloftheAMCsystembyachievingamatchinggoalwithoutanexhaustiveiteration. Althoughthevaractorbiasesforestimationwerearbitrarilyselected,selectingvaractorbiasesshouldbebasedonthecarefularrangementofq-pointsaccordingtothesix-portreectometertheory.Theidealarrangementofq-pointsisknownasthesamemagnitudeand120phasedierence[ 11 ].Asfuturework,weareworkingonnewmatchingnetworkdesignandbiasselectiontoaccountfortheeectofq-points. 58

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CHAPTER5COUPLER-FREELOADESTIMATIONUSINGTHREE-PORTEFLECTOMETER 5.1Overview Mobiledevicesequippedwithradiofrequency(RF)subsystemsarewidelyusedanddriveemergingtechnologiesforthebetterperformanceatlowpowerconsumption.ImpedancematchingbetweentheRFsubsystemsplaysacriticalroleinimprovingthelow-powersystemeciencybymaximizingpowertransferandsignal-to-noiseratio(SNR).However,RFimpedancematchingisoftenhighlyiterativeandtime-consuming,whentheRFportbetweensubsystemsispoorlydened. AsaneorttoautomatetheRFimpedancematching,anautomaticmatchingcontrol(AMC)circuithasbeendevelopedforcellularphoneantenna[ 6 ]andloadboardtesting[ 21 ].Thepriorresearchworkdemonstratedthatanunknownimpedancemismatchcanberesolvedbyatunablematchingnetworkandaniterativesearchmethod. Thedisadvantageoftheautomaticmatchingcontrolisthattheiterativesearchmayslowdowntheautomationprocessandmakeimpedancematchinginreal-timeverychallenging.Toimproveslowresponsetime,aloadimpedanceestimationmethodwasdevelopedtodiscoveranunknownloadandtoenableimmediateimpedancematching[ 25 ].However,theloadestimationmethod,aswellastheautomaticmatchingcontrol,measuresthedegreeofmismatchthroughadistributeddirectionalcoupler.Whenthesystemareaorvolumematters,forexample,on-chipembeddedRFtestingorcompactloadboardtesting,thecouplerispreferredtoberemovedorreplacedwithotherequivalentparts.Asalternativestothedirectionalcoupler,alumped-elementcoupler[ 28 ][ 29 ]andanactivecoupler[ 30 ]havebeenproposed. Weproposeanovelcoupler-freeloadestimationmethodforanautomaticmatchingcontrol(AMC)circuitusingalumped-elementthree-portreectometer.Theproposedthree-portreectometerconsistsofatwo-portlumped-element-typematchingnetworkandahighimpedanceprobingportattachedtotheinputportofthematchingnetwork. 59

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Figure5-1. Automaticmatchingcontrol(AMC) Thereectometercanbereconguredbythreevaractor-basedtunablecapacitorsandusedtosetimpedancematchingwithintheunitcircleontheSmithchartat2.4GHz. Thecoupler-freeloadestimationcanestimateanunknownloadfromthecombinedpowerofincidentandreectedwavesthroughahigh-impedanceprobe.Theuseofthehigh-impedanceprobeandcoupler-freeestimationcaneliminatetheneedofthedirectionalcouplerandmakecompactfabricationofimpedancemeasurementonachipfeasible. Therestofthischapterisorganizedasfollows:Section 5.2 givesanoverviewoftheautomaticmatchingcontrolandthecoupler-freereectometer.Section 5.3 showsthehighimpedanceprobemodelandttingmethods.Section 5.4 describesradicalcenterandleast-squareloadestimationmethods.Section 5.5 presentstheexperimentalresults.Section 5.6 concludesanddescribesthefuturedirection. 5.2SystemOverview Figure 5-1 illustratesthesystemdiagramofanautomaticmatchingcontrol(AMC).Theautomaticmatchingcontrolreconguresthematchingnetworkthroughtheclosed-loopfeedbackconsistingofacoupler,areectioncoecientdetector,andamatchdecisioncircuit.Theclosed-loopfeedbackenablesthematchdecisioncircuittosearchforvaractorbiasesforminimizingtheRFsignalreectioninatrial-and-errorprocess.Inthiswork,thecouplerhasbeenreplacedwithahighimpedanceprobe,whichenablestheautomaticmatchingcontroltoperformthecoupler-freeloadestimation. 60

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A B Figure5-2. ThreeportreectometerintegratedwithahighimpedanceprobeA)SystemdiagramwithschematicB)FabricationonFR4board 61

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Figure5-3. VaractorSMV1405capacitanceversusreversevoltage Figure5-4. VaractorSPICEmodel Theproposedtunablematchingnetworkisbasedonthebandpasstopology,whichisknowntoprovideimpedancematchingwithanypointontheSmithchart[ 3 ][ 31 ].AsshowninFigure 5-2 ,thematchingnetworkisa-typeband-passlterwithlumpedinductorsandtunablecapacitorsandwasfabricatedon6:34:8mmFR4printedcircuitboard(PCB).Thenetworkwasdesignedtosetimpedancematchingat2.4GHzandhasthreeportsforinput,output,andprobe. SkyworksSMV1405-074LFwasusedasthetunablecapacitor.Itcontainstwohyper-junctionvaractordiodesinasinglepackage,wheretwovaractorsareconnectedincommoncathode.Thecapacitanceofasinglevaractorrangesfrom1pF(bias=-8V)to2.7pF(bias=0V),asshowninFigure 5-3 .Thecommoncathodepinisconnected 62

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toabiasvoltagesupplythroughaDCblockingchokeinductor.TheSPICEmodelofSMV1405,asshowninFigure 5-4 ,wasusedforpreliminarysimulation. Ahighimpedanceprobewasimplementedwitha250(=5Z0)chipresistorandatransmissionline.Thehighimpedanceprobewasusedtomeasuretheinputreectioncoecientofthematchingnetwork.TherelationshipmodelbetweenthehighimpedanceprobeandtheinputreectioncoecientwillbepresentedinnextSection. 5.3HighImpedanceProbe AsshowninFigure 5-2 ,ahighimpedanceprobeconsistsofa250chipresistoratthemeasurementnodeandatransmissionlineterminatedwithamatchedport,whichisconnectedtoanetworkanalyzer.Foranembeddedtestsystem,thehighimpedanceprobeandthenetworkanalyzercanbereplacedbyanon-chipbipolarpowerdetectorpresentedin[ 32 ]. ThefollowingareworthytonoteinthisdemonstrationasshowninFigure 5-2 1. Serieschipresistor(250)isattachedtoavectornetworkanalyzerthroughatransmissionlinetoemulatetheoperationofahigh-impedanceprobe. 2. UsedavectornetworkanalyzertomeasureandmanipulatefullS-parameterdata 3. Derivedj)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(injfromS-parametersmeasuredbyavectornetworkanalyzertoemulatetheoperationofapowerdetector. 4. Usedj1+)]TJ /F4 7.97 Tf 29.02 -1.79 Td[(injfromavectornetworkanalyzertoseethesystemresponsewithapowerdetectorandtoverifyEquation 5{1 Thehighimpedanceprobehasnodirectivityandmeasuresthesummationofbothincidentandreectedwaves.Themeasurementatthehighimpedanceprobecanbewrittenas an+bn=an+)]TJ /F4 7.97 Tf 19.07 -1.8 Td[(nan=an(1+)]TJ /F4 7.97 Tf 32.14 -1.8 Td[(n)(5{1) wherean,bn,)]TJ /F4 7.97 Tf 14.46 -1.79 Td[(narenormalizedincidentandreectedwavesandareectioncoecientofportn.ThenetworkanalyzerportismatchedwithimpedanceZ0andconnectedinserieswiththechipresistor5Z0.ThenetworkanalyzermeasuresZ0=(Z0+5Z0)ofa 63

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probenodeandactsasavoltagedivider.Inotherwords,theoutputresponseofthehighimpedanceprobeisscaleddownbyavoltagedividerratioZ0=(Z0+5Z0)andfollowedbythephase-delay(e|)ofatransmissionline.Assumingthattheinputportofthematchingnetworkisport1andthehighimpedanceprobeisport3,themeasuredpowerinthehighimpedanceprobeiswrittenas P3=jb3j2=jS31a1j2=Z0 5Z0+Z0e|S11a1(1+)]TJ /F4 7.97 Tf 32.14 -1.79 Td[(in)2;(5{2) whereanandbnarenormalizedincidentandreectedwavesofportn.TheidealrelationshipdescribedbyEquation 5{2 doesnotexactlytthemeasurementresultduetonon-idealcircuiteects,suchasniteimpedanceoftheprobeandtransmissionlineloss.Insteadoftheidealmodel,arelationshipmodelwithparasiticsandaneuralnetworkapproximationmodelwereusedtotakethenon-idealeectsintoaccount. Thenodemeasuredbythehighimpedanceprobeisconnectedtotheport1throughatransmissionline.Themeasurementofthenodeisdeterioratedduetothenon-idealeects,whichareincludedinamodiedrelationshipmodel.TherelationshipbetweenthemeasuredS-parametersfromtheinputport1andthehighimpedanceprobeismodiedandwrittenas S11=(S31r1e|1)]TJ /F1 11.955 Tf 11.95 0 Td[(1)r2e|2(5{3) wherer1;r2;1;2aremagnitudeandphasettingparameters.Therelationshipmodelcanbetrainedbyaleastsquarenonlineartting.Anarticialneuralnetwork,awell-knownnonlinearapproximationmodel,wasalsousedtorepresenttherelationship.Thetrainingprocedurefortwomodelswillbedescribedinmoredetail. 5.3.1LeastSquareFitting Leastsquarenonlinearttingwasappliedtotraintherelationshipmodel.Thettingparameters,r1;r2;1;2,areobtainedbyminimizingtheleastsquareequationwrittenas ^r1;^r2;^1;^2=argminXVS11(V))]TJ /F1 11.955 Tf 11.96 0 Td[((S31(V)r1e|1)]TJ /F1 11.955 Tf 11.96 0 Td[(1)r2e|22(5{4) 64

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whereV=(V1;V2;V3)Tisabiasvoltagevectorforthreevaractors.Thettingalgorithmisthelarge-scalealgorithm,asubspacetrustregionmethodbasedontheinterior-reectiveNewtonmethoddescribedin[ 33 ][ 34 ]. 5.3.2ArticialNeuralNetwork Anarticialneuralnetworkiswellknownfortheapproximationofanonlinearmodel.ThenonlinearrelationshipbetweenS11andthehighimpedanceprobe,describedbyEquation 5{3 ,isapproximatedbyanarticialneuralnetwork.TheinputandoutputoftheneuralnetworkarethemeasuredS31andtheestimateofS11.Theusedneuralnetworkhas15perceptronsinthehiddenlayerandwastrainedbytraditionalbackpropagationalgorithm. 5.4LoadEstimationMethods Theequationforcoupler-freeloadestimationisbasedonthestabilitycirclederivationaspresentedin[ 26 ,Chapter11]and[ 25 ].Accordingtothestabilitycirclederivation,thecenterandradiusofthecircleweregivenas CL=(S22j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[(S11) jS22j2j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2)-221(jj2(center) (5{5) RL=S12S21 jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)-222(jj2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj(radius) (5{6) Whentheincidentandreectedwavesarecombinedundertheabsenceofacoupler,themeasuredpoweroftheinputportisexpressedas pin=ja1+b1j2=ja1j2j1+)]TJ /F4 7.97 Tf 27.59 -1.8 Td[(inj2(5{7) Therefore,j1+)]TJ /F4 7.97 Tf 29.79 -1.79 Td[(injinsteadofj)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(injcanbeusedtoestimatealoadimpedance.ThemanipulatedS-parameterdatafromanetworkanalyzer,j1+)]TJ /F4 7.97 Tf 28.02 -1.8 Td[(inj,wereusedtoverifythepowerdetectorequationrewrittenas j1+)]TJ /F4 7.97 Tf 27.58 -1.8 Td[(inj=1+S11+S12S21)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(L 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(L(5{8) 65

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j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj=1+S11)]TJ /F1 11.955 Tf 11.95 0 Td[((1+S11)S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L+S12S21)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L 1)]TJ /F5 11.955 Tf 11.95 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L(5{9) Bytakingthesquareofbothsides,theequationiswrittenas j1)]TJ /F5 11.955 Tf 11.95 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(Lj2j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2=j1+S11)]TJ /F1 11.955 Tf 11.96 0 Td[((S22+))]TJ /F4 7.97 Tf 33.38 -1.79 Td[(Lj2(5{10) where =S11S22)]TJ /F5 11.955 Tf 11.95 0 Td[(S12S21(5{11) Thenextderivationfollowsthestabilitycirclederivationpresentedin[ 25 ].Then,thecenterandradiusofthecircleforcoupler-freeloadestimationaregivenas CL=(S22j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.96 0 Td[((S22+)(1+S11)) jS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)-222(jS22+j2(center) (5{12) RL=S12S21 jS22j2j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2)-222(jS22+j2j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj(radius) (5{13) where=S11S22)]TJ /F5 11.955 Tf 12.04 0 Td[(S12S21and*denotesacomplexconjugate.ThedetailinderivationisgiveninAppendix A Thecenterandradiusofthreecircleswereusedtoestimateanunknownload.Theestimationisbasedonradicalcenterandleastsquarettingwidelyusedforasix-portreectometer.Theradicalcenterofthreecirclesisthesimplestgeometricmethodandtheleastsquarettingismoreaccurateandastatisticalapproachbyminimizingthesumofsquareddistancefromthreecircles. 5.4.1RadicalCenter Theradicalcenteristheapproximationofthecenterofthreecircles'overlappedregion,asillustratedinFigure 5-5 .Thecoordinatesoftheradicalcenteraregivenas x=R2L1)]TJ /F5 11.955 Tf 11.96 0 Td[(R2L2+x22 2x2 (5{14) y=R2L1)]TJ /F5 11.955 Tf 11.96 0 Td[(R2L3+x23+y23)]TJ /F1 11.955 Tf 11.96 0 Td[(2xx3 2y3 (5{15) Notethatallmismatchedloadsarepassiveandshouldbewithinaunitcircle.Ifthemagnitudeoftheestimatedreectioncoecientislargerthanone,thelargeerrorcan 66

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Figure5-5. Radicalcenterofthreecircles,)]TJ /F4 7.97 Tf 166.22 -1.8 Td[(L1,)]TJ /F4 7.97 Tf 14.47 -1.8 Td[(L2,and)]TJ /F4 7.97 Tf 37.23 -1.8 Td[(L3. dominatetheoverallmeansquareerrortoleadtowrongperformancestatistics.Inthiscase,themagnitudeissettoone(ontheunitcircle)whilekeepingthephaseinformation. 5.4.2LeastSquareFitting Leastsquarenonlinearttingwasappliedtoestimatealoadandtheradicalcenterwasusedasaninitialparameter,asdescribedin[ 27 ].Theleastsquarettingcanprovidemoreaccurateresultsinsacriceofhighcomputationcost.Thesamettingalgorithmusedinthehighimpedanceprobewasusedhere.Thettingparameteristhereectioncoecientoftheunknownload,acomplexnumberontheSmithchart,andisobtainedbyminimizingtheleastsquareequationwrittenas ^)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(L=argminXVjCircle(V))]TJ /F1 11.955 Tf 11.96 0 Td[()]TJ /F4 7.97 Tf 7.31 -1.79 Td[(Lj2(5{16) 67

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whereV=(V1;V2;V3)Tisabiasvoltagevectorforthreevaractors.NotethatCircle(V)isdescribedbyEquation 5{12 and 5{13 andthedistancebetweenacircleandapointisdenedasadistancebetweenatangentiallinetothecircleandthepoint. 5.5ExperimentalResults Avectornetworkanalyzerwasusedtomeasurethethree-portS-parametersofthenetworkat2.4GHz.Thevaractorbiasvoltagerangesfrom0to5.12Vby0.32Vstep,16voltagelevelsandthenumberofstatesofthethree-varactormatchingnetworkis163=4096.Therefore,thetotalnumberofthree-portS-parameteris16332=36;864.Thefollowingexperimentalresultscomefromthe36,864S-parameterdataof4096statesofthematchingnetwork. First,ttingthemeasurementdatafromthehighimpedanceprobetotheinputportS-parameterwillbepresented.Thettingwereperformedbytheleastsquarettingoftherelationshipmodelwithparasiticsandthebackpropagationtrainingoftheneuralnetworkapproximationmodel.Theerrordistributionofbothmethodsisalsopresented. Second,acoupler-freeloadestimationisperformedbyradicalcenterandleastsquareandtheresultsusingtwomethodsarecomparedintermsofmeansquareerror(MSE).Theinputdatatotheloadestimationistheestimateddatafromthehighimpedanceprobe. Last,forafaircomparisonofcouplerandcoupler-freeloadestimationmethods,theestimationperformanceisevaluatedwiththesamemeasurementdatausedin[ 25 ]. 5.5.1HighImpedanceProbeEstimation Twomodelsbasedonleastsquarettingandarticialneuralnetworkwereusedforthehighimpedanceprobeestimation.Thettingparametersofleastsquarearer1;r2;1,and2,magnitudeandphaseoftwocomplexnumbersrepresentingnon-idealeects.Leastsquarettingwastrainedbyalarge-scalealgorithm,asubspacetrustregionmethodbasedontheinterior-reectiveNewtonmethod.ThehighimpedanceprobeS-parameters 68

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S31wereconvertedbytherelationshipmodelandfourtrainedttingparametersandcomparedwiththeinputreectioncoecientS11. TheneuralnetworkusedinthisworkhastwoinputnodesrepresentingtherealandimaginarypartsofthehighimpedanceprobeS-parameterS31,15perceptronsinthehiddenlayer,andtwooutputnodesrepresentingtherealandimaginarypartsoftheinputreectioncoecientS11.Thetraditionalbackpropagationalgorithmwasusedtotraintheneuralnetwork. Figure 5-6 and 5-7 showestimationresultsoftheinputreectioncoecientS11fromthehighimpedanceprobeS-parameterS31usingleastsquarettingandarticialneuralnetwork,respectively.AsexplainedinSection 5.3 ,thedistributionofthehighimpedanceprobeS-parametersisthedownscaleoftheinputreectioncoecientthroughvoltagedivider.Theexperimentresultsshowedthattheinputreectioncoecientcanbesuccessfullyestimatedthroughthehighimpedanceprobewithoutdisturbingthematchingnetwork. Meansquareerror(MSE)forleastsquarettingandthetrainedneuralnetworkare0.000732and0.000366.Althoughtheneuralnetworkachievedlowerestimationerror,leastsquarettingalsohasadvantageoffastertrainingandsimplermodelrepresentationovertheneuralnetwork.ThemagnitudeoferrordistributionofbothmethodsareshowninFigure 5-8 .Asexpected,theneuralnetworkshoweditspeakoferrordistributionclosertozerothanleastsquaretting. 5.5.2LoadEstimation Theinputreectioncoecientsestimatedfromthehighimpedanceprobewereusedtoestimateanunknownloadusingthecoupler-freeloadestimation.Onematchedload,threeslightlymismatchedloads,andthreeheavilymismatchedloadswereusedasmismatchedloadsfortheexperiments.Theywereclassiedintotwogroups,slightly-mismatchedandheavily-mismatchedloadsandbothgroupsincludethematchedloadas 69

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AInputreectioncoecientS11 BHighimpedanceprobeS-parameterS31 CFittingS31toS11 Figure5-6. Leastsquarenonlinearttingofhighimpedanceprobemodel 70

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AInputreectioncoecientS11 BHighimpedanceprobeS-parameterS31 CFittingS31toS11 Figure5-7. Articialneuralnetworkofhighimpedanceprobemodel 71

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ALeastsquare BNeuralnetwork Figure5-8. Highimpedanceprobeestimationerrordistribution 72

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Table5-1. Summaryofheavilyandslightlymismatchedloads MismatchedloadjS11jat2.4GHzTunermotorposition Matched0.01(100,5000,5000) Slightlymismatched#10.13(16725,2262,5000) Slightlymismatched#20.38(17105,1424,5000) Slightlymismatched#30.20(20464,2228,5000) Heavilymismatched#10.78(15781,526,5000) Heavilymismatched#20.76(17835,624,5000) Heavilymismatched#30.69(20572,804,5000) areference.AllusedloadsarecarefullygeneratedbyMauryMicrowave'sload-pullsystem(MT986AandMT982B01)andtheirspecicationissummarizedinTable 5-1 Theloadestimationwasperformedseparatelyforslightlymismatchedandheavilymismatchedgroups.Duringtheexperiment,adierentsetofthreestatesofmatchingnetworkweremanuallychosentogivebetterestimationresults.Thechosenvoltagebiasesforthreedierentstatesforslightlymismatchedandheavilymismatchedgroupsare((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. AsshowninFigure 5-9 ,theestimationonslightlymismatchedloadsshowedsmallererrorthanheavilymismatchedloads.Althoughtheaccuracyisnotgoodenoughtoreplaceanhigh-precisioninstrument,theroughestimateofmagnitudeandphaseoftheloadreectioncoecientisveryusefulfortheautomaticmatchingcontroltosetimmediateimpedancematching. 5.5.3ComparisonofCouplerandCoupler-FreeLoadEstimation Forthecomparisonofcouplerandcoupler-freeloadestimationmethods,themeasurementdatausedinthepriorresearchwork[ 25 ]wereusedtoevaluatetheestimationperformance. TwoloadestimationmethodswereevaluatedwithradicalcenterandleastsquarettingdescribedinSection 5.4 .Thefourmismatchedloadsusedinthepriorresearch 73

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ASmallmismatch(MSE=0.21) BLargemismatch(MSE=0.86) Figure5-9. LoadestimationusingestimatedS11fromhighimpedanceprobe Table5-2. Comparisonofcouplerandcoupler-freeloadestimationintermofmeansquareerror(MSE) MismatchedjS11jRadicalcenterLeastsquare loadCouplerCoupler-freeCouplerCoupler-free Mismatched#10.140.0568400.0508650.0135760.003665 Mismatched#20.450.1209090.1345960.0349830.061899 Mismatched#30.240.0356070.0692650.0099300.009934 Mismatched#40.140.0540440.0563330.0181380.035334 workwerealsousedforfaircomparison.TheexperimentalresultsaresummarizedinTable 5-2 intermsofmeansquareerror(MSE). Althoughmeansquareerror(MSE)ofthecoupler-freeloadestimationisslightlyhigherthanMSEofcouplerloadestimation,theestimationaccuracyoftwomethodsarecomparableforbothradicalcenterandleastsquarettingalgorithms,asshowninFigure 5-10 to 5-13 .Whenthecoupler-freeestimationisemployedtoanautomaticmatchingcontrolsystem,thesystemdimensionwillbedramaticallyreducedwithoutcompromisingthesystemperformance. 74

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ARadicalcenterwithcoupler BRadicalcenterwithoutcoupler CLeastsquarewithcoupler DLeastsquarewithoutcoupler Figure5-10. Couplerandcoupler-freeloadestimationwithmismatched#1 75

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ARadicalcenterwithcoupler BRadicalcenterwithoutcoupler CLeastsquarewithcoupler DLeastsquarewithoutcoupler Figure5-11. Couplerandcoupler-freeloadestimationwithmismatched#2 76

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ARadicalcenterwithcoupler BRadicalcenterwithoutcoupler CLeastsquarewithcoupler DLeastsquarewithoutcoupler Figure5-12. Couplerandcoupler-freeloadestimationwithmismatched#3 77

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ARadicalcenterwithcoupler BRadicalcenterwithoutcoupler CLeastsquarewithcoupler DLeastsquarewithoutcoupler Figure5-13. Couplerandcoupler-freeloadestimationwithmismatched#4 78

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5.6Conclusion Wepresentedanovelcoupler-freeloadestimationandalumped-elementreectometerintegratedwithahighimpedanceprobe.Theestimateofanunknownmismatchedloadcanhelpanautomaticmatchingcontrol(AMC)tosetimpedancematchingontheloadwithoutiterativesearch. Ahighimpedanceprobewaspresentedfortheintegratedon-chipRFtesting.Tworelationshipmodelstogetherwithestimationmethods,leastsquareandarticialneuralnetwork,achievedmeansquareerror0.000732and0.000366,respectively.Theleasesquarettinghasonlyfourttingparameterscomparedwithabout20parametersoftheneuralnetworkmodelandenablesfastertrainingwithoutcompromisingtheaccuracy.Theestimatedresultswerealsousedasaninputtothecoupler-freeloadestimation,whichdiscoveredanunknownloadfortheautomaticmatchingcontrol. Anovelcoupler-freeloadestimationwasproposedwithalumped-elementreectometer.Theproposedmethoddemonstratedthatitcoulddiscoveranunknownmismatchedloadandprovidewiththehighaccuracyenoughforanautomaticmatchingcontroltoexploit.Thecoupler-freeloadestimationremovedtheneedofanydistributedcomponentsandachievedcompactsystemdimensionwithoutcompromisingtheestimationaccuracy.Theexperimentalresultsshowedthattheproposedcoupler-freeestimationachievedthecomparableaccuracyasthepriorresearchworkadoptingacoupler.Theproposedloadestimationcanbeintegratedwithanon-iterativeautomaticmatchingcontrolsystemwithoutextracomponentsorperformancedegradation. 79

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CHAPTER6THREEPORTANDFOURPORTREFLECTOMETERS 6.1Overview Automaticorself-recongurableradiofrequency(RF)systemshavereceivedattentionduetotheirpotentialtoovercometheuncertaintyofRFsystemssuchasparasiticsandmanufacturingvariation.Recently,anautomaticmatchingcontrol(AMC)wasproposedtoprovidetheimpedancematchingoftheradiofrequency(RF)portsofadeviceundertest(DUT)oracellularphoneantenna[ 21 ][ 6 ].Also,loadestimationwasproposedtoovercometheiterativenatureoftheautomaticmatchingcontrolsystem[ 25 ]. Recongurablematchingnetworksinthepriorworkwerenotdesignedfortheloadestimation[ 1 ][ 5 ][ 6 ].Eventheloadestimationproposedin[ 25 ]usedthearbitrarilyreconguredmatchingnetworks,whosedesignwasnottargetedforloadestimationpurpose.Inthiswork,three-portandfour-portreectometersweredesignedtosupportboththeautomaticmatchingcontrolandtheloadestimationusinglumpedelementsandcompactpowerdetectorsforsizereductionandembeddedRFtesting. Inthenetworkanalyzerresearchcommunity,anewmethodusingdetectorpowerreadingsandsomemathematicalmanipulationwasintroducedtomeasureacomplexreectioncoecientofadeviceundertest(DUT).Thenewmethod,so-calledasix-portreectometer,wasrstproposedasanalternativeofaconventionalnetworkanalyzer[ 11 ].Later,four-portmultistatereectometerswerepresentedtoreducetherequirednumberofportsandparts[ 12 ][ 13 ][ 14 ].Recently,alumped-elementstructurewasalsopresentedfortheintegrationonachip[ 35 ]. Weproposethree-portandfour-portlumped-elementmultistatereectometersforbotharecongurablematchingnetworkandloadestimation.Carefullychosenq-pointscanimprovetheaccuracyoftheloadestimationanditcanstillrecongureamatchingnetworkfortheautomaticmatchingcontrol. 80

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A B Figure6-1. ReectometersA)Three-portB)Four-port 6.2MultistateReectometers AsshowninFigure 6-1 ,port1isfedwithasignalsourcewithimpedanceZ0,port2isconnectedwithadeviceundertest(DUT)havingacomplexreectioncoecient)]TJ /F6 7.97 Tf 449.18 -1.79 Td[(2(sometimesloadreectioncoecientdenotedby)]TJ /F4 7.97 Tf 259.77 -1.79 Td[(L)tobemeasured,andport3and4arepowerdetectorports. Similartoasix-portreectometeranalysis,powerreadingscanbeexpressedintermsofincidentandreectedwavepowersofport2.Thepowerreadingsofann-portreectometeriswrittenas bi=Aia2+Bib2;i=3;4;;n(6{1) 81

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Pi=jbij2=jb2j2jAij2j)]TJ /F6 7.97 Tf 7.31 -1.79 Td[(2)]TJ /F5 11.955 Tf 11.96 0 Td[(qij2;i=3;4;;n(6{2) where )]TJ /F6 7.97 Tf 7.32 -1.79 Td[(2=a2 b2;qi=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(Bi Ai(6{3) ThecomplexreectioncoecientofaDUTcanbecalculatedbypowerdetectorreadingsandsomemathematicalmanipulation,whichisexplainedinmoredetailin[ 11 ]. Byaddinganetworkstatek,theoperationofmultistatereectometerscanbedescribedbythefollowingequation, j)]TJ /F6 7.97 Tf 7.32 -1.79 Td[(2)]TJ /F5 11.955 Tf 11.95 0 Td[(q(k)ij2=P(k)i jb(k)2j2jA(k)ij2;i=3;4;;n;k=1;2;3(6{4) wherekisanetworkstate(typically,setbydierentbiases)andthreenetworkstatesareneededinordertodeterminethecomplexreectioncoecientofaDUT. Especiallyforafour-portcase,port3isassignedasareferenceportwhichdependsonlyonb2(inotherwords,b3=B3b2orA3=0).Theequationcanbewrittenas j)]TJ /F6 7.97 Tf 7.32 -1.79 Td[(2)]TJ /F5 11.955 Tf 11.95 0 Td[(q(k)4j2=B(k)3 A(k)42P(k)4 P(k)3;k=1;2;3(6{5) IfallpowerdetectorsareperfectlymatchedtoZ0,thenthecalibrationconstants,Ai,Bi,andqi,areexpressedintermsofS-parametersandgivenby(refertoAppendix B forthederivation) A(k)i=S(k)21S(k)i2)]TJ /F5 11.955 Tf 11.95 0 Td[(S(k)22S(k)i1 S(k)21;B(k)i=S(k)i1 S(k)21;q(k)i=S(k)i1 S(k)22S(k)i1)]TJ /F5 11.955 Tf 11.96 0 Td[(S(k)21S(k)i2(6{6) Theabovecalibrationconstantscanbeobtainedbyeitheradirectmeasurementormodeling. Thereferenceportcanberealizedbyalumpedpowerdivider,whichwaspresentedfortheintegrationonachipin[ 35 ],asshowninFigure 6-2 .Inpriorwork,adirectionalcouplerwaswidelyusedforareferenceport,butthecompactlumpedpowerdividerispreferredforsizereductioninembeddedtest.Thescatteringmatrixofthedividerwith 82

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Figure6-2. Schematicofalumpedpowerdivider.Port3isacoupledport,whichcanbeusedasareferenceport. respecttoZ0isgivenby 0BBBB@b1b2b31CCCCA=26666666401 1+Za Z01 1+Zc Z01 1+Za Z0001 1+Zc Z0003777777750BBBB@a1a2b31CCCCA(6{7) Athree-portreectometerhasthesamebasicstructureasafour-portoneexceptthatareferenceportandapowerdividerdonotexist.Becauseareferenceportdoesnotexist,aninputsignalpowerhastobeknowntoareectometerforthenormalizationofwavepowers.Inotherwords,areferenceportcanbereplacedbyapredenedinputsignalpowerinsomecases,e.g.,testingRFpartsunderanautomatictestequipment(ATE).Hence,thewavepowerjb2jcanbeestimatedthroughtherelationshipbetweenjb2jandja1jifaninputpowerja1jispreciselydened.Therelationshipisdescribedbytheequationgivenby b2=S21a1+S22a2=S21a1+S22)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(Lb2(6{8) b2=S21 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(La1(6{9) Insummary,thethree-portreectometercanbeusedifthesignalsourceispreciselydenedandthefour-portreectometerisapplicabletoothergeneralcases. 83

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6.3TunableMatchingNetwork Thesimplestlumped-elementmatchingnetworkisL-type,butcannotsetimpedancematchingofallpointsontheSmithchart.T-and-typesareknownformatchingcapabilitywithanypointonaSmithchart[ 31 ].AlthoughT-and-typetopologiesareequivalentintermsofmatchingcapability,the-typeispreferredforimpedancematchingpurposebecauseithasasmallernumberoftunableelementsalongwithasignalpath,whichresultinlowerinsertionlossthanT-type. Eachbranchofthematchingnetworkhasaninductorconnectedwithavaractoreitherinparallelorseries.AsshowninFigure 6-3 ,thebasicstructureis-typebandpassltertunedatthecenterfrequencyof2.4GHz,wherecapacitorsarereplacedwithtunablevaractors. ThetunablecapacitorusedinthematchingnetworkisSkyworksSMV1405-074LF,whichcontainstwocommoncathodediodevaractorsinasinglepackage.Thecommoncathodeportisconnectedtoabiasvoltagesupplythroughachokecoil.Thetypicalcapacitanceofeachvaractorwithabias1Vis1.21(min)to1.45(max)pFandthecapacitancerangesfrom2.1pF(0.5V)to0.95pF(10V).ThecapacitanceversusreversevoltageisshowninFigure 5-3 .TheSPICEmodelusedinsimulationsisshowninFigure 5-4 AgilentADSwasusedtoperformS-parametersimulations.Accordingtothesimulationresults,thevaractorinserieswithaninductance3.3nHandinparallelwith1.8nHshowsthelargesttuningrangeat2.4GHzasshowninFigure 6-4 .Theinductancevalueof3.3nHand1.8nHwasusedforthedesignofthematchingnetwork. Anoninvasivemeasurementismandatorynottodisturbtheoriginaldesignofamatchingnetwork.Similartothepriorwork[ 35 ],ahighimpedancepowerdetectormeasuresaninternalnode.Thenoninvasivepowerdetectorisemulatedbyinsertinghighresistanceinserieswithameasurementportanddeembeddingtheeectoftheresistance.Asexplainedearlier,thefour-portreectometerneedsapowerreadingthatdependsonly 84

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A B Figure6-3. Recongurablethree-portmatchingnetworkA)SchematicB)ImplementationonFR4board 85

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A B Figure6-4. Tunableelementimpedancewithabiasfrom0Vto10V.Avaractorisinparallelandinserieswithaninductor.anddenoteS11andS21,respectively.A)Inserieswith3.3nHB)Inparallelwith1.8nH 86

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onthereectedwavepowerfromaDUT.Thereferenceportcanbeeasilyrealizedusingalumpedpowerdivider. 6.4LoadEstimationforMultistateReectometer Thematchingcapabilityofatunablematchingnetworkcanberepresentedbytheloadreectioncoecient)]TJ /F4 7.97 Tf 138.03 -1.79 Td[(Ltobematchedbythematchingnetwork.Thematchingcapabilityisderivedasfollows.Theinputreectioncoecient)]TJ /F4 7.97 Tf 332.03 -1.79 Td[(iniswrittenas )]TJ /F4 7.97 Tf 7.31 -1.8 Td[(in=S11+S12S21)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(L 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(L(6{10) Theequationcanberewrittenintermsoftheloadreectioncoecientas )]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L=S11)]TJ /F1 11.955 Tf 11.95 0 Td[()]TJ /F4 7.97 Tf 7.32 -1.79 Td[(in S11S22)]TJ /F5 11.955 Tf 11.95 0 Td[(S12S21)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(in(6{11) Thematchingcapabilityisderivedfromtheloadreectioncoecientbysettingtheinputreectioncoecienttozero. )]TJ /F4 7.97 Tf 7.32 -1.8 Td[(L)]TJ /F13 5.978 Tf 5.29 -1.22 Td[(in=0=S11 S11S22)]TJ /F5 11.955 Tf 11.96 0 Td[(S12S21(6{12) ThecoverageontheSmithchartspeciedthematchingcapabilityillustratesthedistributionoftheloadreectioncoecienttobematched. Accordingtothesix-portreectometerprinciple,anunknownloadisthesameasthepointintersectedbythreecircles,speciedbyacenter,so-calledq-point,andaradius.ThecircleisrepresentedbycalibrationconstantsdenedbyEquation 6{6 .Notethattheq-pointdoesnotchangeevenifamismatchedloadvaries,whereastheloadestimationmethodproposedin[ 25 ]haschangedthecirclecenterasamismatchedloadvaries.Theconstantq-pointenablestokeeptheoptimum-performancecriteriaforthemultistatereectometerovervariousmismatchedloads. Inreality,thethreecirclesrepresentedbycalibrationconstantsseldomintersectatapointduetothenon-idealeects.Thegeometriccenteroftheoverlapofthecirclesis 87

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estimatedbytheradicalcenterandleastsquareestimationmethodswidelyusedforthesix-portreectometer[ 27 ]. Theradicalcenteristheapproximationofthecenterofthreecircles'overlappedregion.Thecoordinatesoftheradicalcenteraregivenas x=R2L1)]TJ /F5 11.955 Tf 11.96 0 Td[(R2L2+x22 2x2 (6{13) y=R2L1)]TJ /F5 11.955 Tf 11.96 0 Td[(R2L3+x23+y23)]TJ /F1 11.955 Tf 11.96 0 Td[(2xx3 2y3 (6{14) Leastsquarettingcanenhancetheaccuracyofloadestimationespeciallywhenthreecirclesfailedtomeettheoptimum-performancecriteria.Theloadreectioncoecientcanbeobtainedbytheleastsquareequationas ^)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(L=argminXVjCircle(V))]TJ /F1 11.955 Tf 11.96 0 Td[()]TJ /F4 7.97 Tf 7.31 -1.8 Td[(Lj2(6{15) whereV=(V1;V2;V3)Tisabiasvoltagevectorforthreevaractors.NotethatCircle(V)isrepresentedbycalibrationconstantsdenedbyEquation 6{6 andthedistancebetweenacircleandapointisdenedasadistancebetweenatangentiallinetothecircleandthepoint. 6.5ExperimentalResults Thematchingcapabilityoftheproposedthree-portreectometer,denedasS11=(S11S22)]TJ /F5 11.955 Tf 12.28 0 Td[(S12S21),isshowninFigure 6-5 .Thematchingcapabilitywasmeasuredbychangingeachvaractorbiasfrom0Vto5.12Vin16levelsby0.32Vstep.ThematchingcapabilitycoversaunitcircleontheSmithchartcompletely,showingitscapabilityonanypassivemismatchedload. Oneofimportantcalibrationconstantsisq-pointgivenbytheequation qi=Si1 S22Si1)]TJ /F5 11.955 Tf 11.96 0 Td[(S21Si2(6{16) Theq-pointsweremeasuredwithrespecttothesamebiasrangeasusedforthematchingcapability.Themeasuredq-pointsaredistributedalongtheunitcircleasshowninFigure 88

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Table6-1. Summaryofmismatchedloads MismatchedloadjS11jat2.4GHzTunermotorposition Matched0.01(100,5000,5000) Slightlymismatched#10.13(16725,2262,5000) Slightlymismatched#20.38(17105,1424,5000) Slightlymismatched#30.20(20464,2228,5000) Heavilymismatched#10.78(15781,526,5000) Heavilymismatched#20.76(17835,624,5000) Heavilymismatched#30.69(20572,804,5000) 6-6 .Somesetofq-pointscanbeselectedfromthedistributioninordertosatisfytheoptimum-performancecriteriaforthemultistatereectometer.Asetofq-pointswascarefullyselectedforbetterestimationperformance. Theloadestimationusingthree-portreectometerwasperformedseparatelyonslightlyandheavilymismatchedloads.ThespecicationofmismatchedloadsisgiveninTable 6-1 .First,theS-parameteroftheinputportwasconvertedfromthehighimpedanceportemulatinghighimpedancepowerdetector.ThecalibrationconstantswereobtainedfromtheS-parametersthroughdirectmeasurementofthethree-portreectometerandconversionusingthehighimpedancepowerdetector.Theq-pointswerechosentoachievehigherestimationaccuracyfortwoseparateexperiments.Then,radicalcenterestimationwasappliedtoestimateanunknownloadreectioncoecient.Whenthemagnitudeoftheestimatedreectioncoecientislargerthanone,itisincorrectforpassivemismatchedloads.Inthiscase,themagnitudewassettoonewithkeepingthephase. AsshowninFigure 6-7 ,estimationofslightlymismatchedloadsshowedmuchsmallerestimationerrorthanheavilymismatchedloads.Themeansquareerrorforslightlyandheavilymismatchedloadsare0.09and0.80,respectively.Duetothelargerestimationerror,theestimatedloadreectioncoecientoftheheavilymismatchedloadsoftengobeyondaunitcircle.Asdescribed,themagnitudewassetto1andonlythephasewaskeptforimpedancematching.However,theestimatedphaseisstillquiteusefulbecause 89

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theestimatedmagnitudelargerthanoneoftenresultsfromheavilymismatchedload,whosemagnitudeisclosetoone.Thenextchapterwilldemonstratethatanautomaticmatchingcontrolcanachieveimpedancematchingusingtheestimatedphaseinformation. 6.6Conclusion Weproposedathree-portlumped-elementreectometerforbothloadestimationandimpedancematching.Theproposedreectometercanbeeasilyextendedtoafour-portreectometerbyaddingthesuggestedpowerdivider.Theloadestimationmethoddemonstratedthatthetunablemultistatereectometercanhelptheautomaticmatchingcontrol(AMC)toestimatealoadreectioncoecientaswellastosetimpedancematching.Thehighimpedancepowerdetectorreplacedthedistributedcouplerandrealizedthedramaticsizereductionofanautomaticmatchingcontrolsystemwithoutcompromisingtheloadestimationandmatchingcapability.ThematchingcapabilitycoveredcompletelytheunitcircleontheSmithchart.Althoughtheloadestimationresultisnotaccuratetobeusedasanhigh-precisioninstrument,theestimatedphaseinformationcanstillenabletheautomaticmatchingcontroltoachievefasterimpedancematchingonheavilymismatchedloads.Weareworkingtowardtheintegrationoftheproposedloadestimationandanovelautomaticmatchingcontrolsystemcapableofanimmediateimpedancematching. 90

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Figure6-5. Matchingcapabilityofthree-portreectometerat2.4GHz 91

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Figure6-6. Theq-pointdistributionofthree-portreectometerat2.4GHz 92

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A B Figure6-7. MultistatereectometerestimationusingestimatedS11fromhighimpedanceprobeA)Smallmismatch(MSE=0.09)B)Largemismatch(MSE=0.80) 93

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CHAPTER7AUTOMATICMATCHINGCONTROLUSINGLOADESTIMATION 7.1Overview Anautomaticmatchingcontrol(AMC)systemhasbeendevelopedtoautomatetime-consumingimpedancematchingprocedure[ 5 ][ 6 ][ 21 ].Theimpedancematchingoftheautomaticmatchingcontrolwasperformedbyreconguringatunablematchingnetworkuntilthelowestmismatchisachieved.Therecongurationwascontrolledbyheuristiciterativemethods,whichshowedagoodtrade-obetweensystemresponseandimpedancematchingaccuracy.Also,loadestimationreusingtheexistingtunablematchingnetworkoftheautomaticmatchingcontrolsystemwasproposedtofacilitatetheautomationofimpedancematching[ 25 ].Inthiswork,theloadestimationtechniquewasintegratedwiththeexistingautomaticmatchingcontrolsystemtoachieveimmediateimpedancematchingwithoutcompromisingmatchingaccuracy. Traditionalautomaticmatchingcontrolsystemsachievedimpedancematchingofunknownorevenvaryingmismatchedloadsbythefeedbackloopofatunablematchingnetwork,amismatchdetector,andmatchcontrolcircuit[ 21 ].Thefeedbackloopiscontrolledbyiterativemethodsofamatchcontrolcircuit,whichsearchesforthevalueoftuningelementsinatrial-and-errorprocess.However,thetrial-and-errorapproachsloweddownthesystemresponseandvariousheuristicapproacheshavebeendevelopedtoimprovethesystemresponsewithoutcompromisingmatchingcapability.Nevertheless,thesystemresponseoftheheuristicapproachesisstillproportionaltothecomplexityofthematchingnetworkandgetsslowerasmoretuningelementsandlevelsareadded. Wewilldemonstratethatanestimatedloadcanbeusedforamatchingcontrolcircuittoachieveimmediateimpedancematchingwithoutusingheuristicapproaches.Theproposedmatchingcontrolcanndthevalueoftuningelementsbyexaminingthecharacterizationtableofamatchingnetwork.Therefore,theprecisecharacterizationas 94

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wellastheloadestimationplayanimportantroleinthisimmediateimpedancematchingapproach. Variouscharacterizationmethodsforamicrowavedevicehavebeenreportedtoreducethemicrowavedesigncomplexity.Aneuralnetworkhasbeenwidelyusedtocharacterizemicrowavedevices,suchastheapproximationofS-parametersofBJTs[ 36 ]andmodelingparametersofmicrowavecomponents[ 9 ].Also,closedformequationwasalsopresentedforS-parametersofBJTs[ 37 ].Inthiswork,neuralnetworkmodelsandtheclosedformequationwereusedtoapproximatemeasuredS-parametersandtheaccuracyofthecharacterizationmethodswasevaluatedintermsofmeansquarederror(MSE)betweentrueandestimatedvalues. Theproposedmatchingcontrolconsistsoftwotasks.First,acharacterizationtableintermsoftuningelementswasbuiltfromthedirectmeasurementofthematchingnetworkorapproximationmodelssuchasaneuralnetworkandclosedformequations.Next,thevalueofthetuningelementswasfoundbyminimizingthedegreeofmismatch.Thedegreeofmismatchwascalculatedfromthemagnitudeoftheinputreectioncoecient.Theexperimentalresultsoftheimmediateimpedancematchingapproachwillbepresented. 7.2MatchingControlProcedures Thesamelumped-elementtunablematchingnetworkthatisusedforloadestimationwasusedtodevelopmatchingcontrolproceduressupportingloadestimationpresentedinChapter 5 and 6 .Thematchingnetworkhasa-typebandpassltertopologyandthreevaractordiodesastuningelements.Therecongurationofthematchingnetworkwasperformedbychangingthevaractorbiasvoltages.ItsmatchingcapabilitycoversallreectioncoecientswithintheunitcircleontheSmithchartat2.4GHz. Theloadreectioncoecient)]TJ /F4 7.97 Tf 162.1 -1.79 Td[(Lofadeviceundertest(DUT)isassumedtobeestimatedbyloadestimationtechniquespresentedinChapter 5 and 6 .WhentheDUTisconnectedtotheport2ofatunablematchingnetwork,theinputreectioncoecient)]TJ /F4 7.97 Tf 452.79 -1.79 Td[(in 95

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lookingintotheport1ofthematchingnetworkiswrittenasfollows. )]TJ /F4 7.97 Tf 7.31 -1.79 Td[(in=S11+S12S21)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L(7{1) whereSijistheS-parameterfromportjtoportiofthematchingnetwork.NotethattheS-parametersarethefunctionofabiasvoltagevector,denotedbyv.Therefore,theinputreectioncoecientcanbeexplicitlywrittenasthefunctionofv. )]TJ /F4 7.97 Tf 7.31 -1.79 Td[(in(v)=S11(v)+S12(v)S21(v))]TJ /F4 7.97 Tf 11.87 -1.8 Td[(L 1)]TJ /F5 11.955 Tf 11.95 0 Td[(S22(v))]TJ /F4 7.97 Tf 11.86 -1.79 Td[(L;v=(v1;v2;;vn)T(7{2) whereTdenotesatransposeandvnisthenthbiasvoltage.TheloadreectioncoecientoftheDUTcanbederivedfromtheinputreectioncoecient. )]TJ /F4 7.97 Tf 7.31 -1.8 Td[(L=S11(v))]TJ /F1 11.955 Tf 11.95 0 Td[()]TJ /F4 7.97 Tf 7.32 -1.79 Td[(in(v) S11(v)S22(v))]TJ /F5 11.955 Tf 11.95 0 Td[(S12(v)S21(v))]TJ /F5 11.955 Tf 11.96 0 Td[(S22(v))]TJ /F4 7.97 Tf 11.87 -1.8 Td[(in(v)(7{3) Themismatchedloadtobematchedbythematchingnetworksetbyabiasvoltagev,denotedby)]TJ /F4 7.97 Tf 67.8 -1.8 Td[(M,istheloadreectioncoecientthatmakestheinputreectioncoecientzero. )]TJ /F4 7.97 Tf 7.31 -1.79 Td[(M(v)=)]TJ /F4 7.97 Tf 27.61 -1.79 Td[(L)]TJ /F13 5.978 Tf 5.28 -1.21 Td[(in=0=S11(v) S11(v)S22(v))]TJ /F5 11.955 Tf 11.95 0 Td[(S12(v)S21(v)(7{4) Now,letusdeneanoptimalbiasvoltagevector,denotedbybv,asabiasvoltagevectorthatminimizesthemagnitudeoftheinputreectioncoecient)]TJ /F4 7.97 Tf 370.59 -1.79 Td[(in,thedegreeofmismatch. bv=argminvj)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(in(v)j(7{5) Ifthebiasvoltage,thatminimizestheinputreectioncoecienttozero,canbefoundforallpossibleloadreectioncoecients,theoptimalbiasvoltagevectorbvcanbeexpressedusing)]TJ /F4 7.97 Tf 37.94 -1.79 Td[(Masfollows. bv=nv)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(in(v)=0o=nv)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(M(v)=)]TJ /F4 7.97 Tf 27.61 -1.8 Td[(Lo(7{6) Findingtheoptimalbiasvoltagecanbeexpressedasndingabiasvoltagewhose)]TJ /F4 7.97 Tf 430.26 -1.79 Td[(Misequalto)]TJ /F4 7.97 Tf 64.28 -1.79 Td[(L.Therefore,themappingtablebetween)]TJ /F4 7.97 Tf 214.77 -1.79 Td[(Mandvshouldbecalculated 96

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Figure7-1. Automaticmatchingcontrolsupportsloadestimation. toperformthebiassearchandthemappingtablecanbeconvertedfromthematchingnetworkcharacterizedbytheS-parameters.ThisprocedureisbasedonEquation 7{4 .TheS-parameterscanbeobtainedfromdirectmeasurementusingavectornetworkanalyzeroraneuralnetworkttingmodel.ThecharacterizationmethodswillbeintroducedinthenextSection. 7.3CharacterizationofMatchingNetwork ThecharacterizationofamatchingnetworkisaproceduretodiscovertheS-para-meterfunctionstobeusedtocalculateaninputreectioncoecientoramismatchedloadtobematched.TheS-parametersofthematchingnetworkweremeasuredusingavectornetworkanalyzerwhilechangingthebiasvoltage.Themeasurementpointsweredeterminedbythenumberofvaractorsandbiasvoltagelevels.Althoughthemorevoltagelevelscanproducemoreaccuratecharacterizationresults,16voltagelevelswerechosenasgoodtrade-obetweenmeasurementtimeandcharacterizationaccuracy.The 97

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characterizationresultswereconvertedtoaformofamappingtable,)]TJ /F4 7.97 Tf 365.45 -1.79 Td[(M(v),foreasyaccessandsearches. UnknownS-parametersbetweenmeasurementpointswereapproximatedbyamultivariatelinearinterpolation.Themultivariatelinearinterpolationisinterpolatingafunctionofmultiplevariablesonaregulargrid,asanextensionofalinearinterpolation.Itperformslinearinterpolationrstononedirection,thenagainintheotherdirection.Supposewewanttointerpolateavalueofanunknownfunctionfatthepoint(x;y).Thevalueofthefunctionfatfourneighborpointsonaregulargrid,f(x1;y1),f(x1;y2),f(x2;y1),andf(x2;y2),areassumedtobeknown,thentheinterpolationofthefunctionfatthepoint(x;y)canbewrittenasfollows. f(x;y)f(x1;y1) (x2)]TJ /F5 11.955 Tf 11.95 0 Td[(x1)(y2)]TJ /F5 11.955 Tf 11.95 0 Td[(y1)(x2)]TJ /F5 11.955 Tf 11.96 0 Td[(x)(y2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)+f(x2;y1) (x2)]TJ /F5 11.955 Tf 11.95 0 Td[(x1)(y2)]TJ /F5 11.955 Tf 11.95 0 Td[(y1)(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x1)(y2)]TJ /F5 11.955 Tf 11.96 0 Td[(y)+f(x1;y2) (x2)]TJ /F5 11.955 Tf 11.95 0 Td[(x1)(y2)]TJ /F5 11.955 Tf 11.95 0 Td[(y1)(x2)]TJ /F5 11.955 Tf 11.96 0 Td[(x)(y)]TJ /F5 11.955 Tf 11.95 0 Td[(y1)+f(x2;y2) (x2)]TJ /F5 11.955 Tf 11.95 0 Td[(x1)(y2)]TJ /F5 11.955 Tf 11.95 0 Td[(y1)(x)]TJ /F5 11.955 Tf 11.96 0 Td[(x1)(y)]TJ /F5 11.955 Tf 11.95 0 Td[(y1) (7{7) Iftheunknownfunctiontointerpolatehasasmoothsurfaceoverneighborpoints,thelinearinterpolationreducesthenumberofmeasurementpointssignicantlywithoutthelossofthecharacterizationdetail. TheothermethodtoestimateunknownS-parametersbetweenmeasurementpointsisanapproximationttingfunctiontothemeasuredS-parameters.Asdescribed,themeasuredS-parametersarethefunctionsofthebiasvoltagevectorv,givenasfollows. S(v)=264S11(v)S12(v)S21(v)S22(v)375(7{8) TheS-parameterfunctionswereapproximatedbyacurvettingmodel,suchasaclosedformandaneuralnetwork.First,anarticialneuralnetworkwasusedtoapproximate 98

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theS-parameterfunctions.Fourindependentneuralnetworkmodelsapproximatetwo-portS-parameterfunctions,S11(v),S12(v),S21(v),andS22(v).TheinputandoutputoftheneuralnetworkarebiasvoltagevectorandrealandimaginarypartsoftheS-parameter,respectively.Thefeed-forwardtopologywasusedwith15perceptronsinthehiddenlayer.Thewell-knownbackpropagationalgorithmwasusedtotraintheneuralnetworks. ThecascadednetworkiseasilyrepresentedbyABCD-parametersandthematchingnetworkwasdecomposedintobasiccomponents,suchasatransmissionline,seriesimpedance,andshuntadmittance.TherepresentationoftheABCD-parameterswasusedasclosed-formequations.TheABCD-parametersforthebasiccomponentsaregivenasfollows. 264cos(2)|Z0sin(2)|Y0sin(2)cos(2)375(transmissionline) (7{9) 2641Z01375(seriesimpedance) (7{10) 26410Y1375(shuntadmittance) (7{11) Theclosed-formABCD-parameterswerederivedfromthecascadednetworkofatransmissionline(1),anetwork(Y1;Z3;andY2),andatransmissionline(2).ThevaractorcapacitancewascalculatedfromthevaractorSPICEmodelusedinChapter 5 and 6 ABCD11=(cos(21)(1+Y2Z3)+|Z0sin(21)(Y1+Y2+Y1Y2Z3))cos(22)+|(cos(21)Z3+|Z0sin(21)(1+Y1Z3))Y0sin(22) (7{12) ABCD12=|(cos(21)(1+Y2Z3)+|Z0sin(21)(Y1+Y2+Y1Y2Z3))Z0sin(22)+(cos(21)Z3+|Z0sin(21)(1+Y1Z3))cos(22) (7{13) 99

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ABCD21=(|Y0sin(21)(1+Y2Z3)+cos(21)(Y1+Y2+Y1Y2Z3))cos(22)+|(|Y0sin(21)Z3+cos(21)(1+Y1Z3))Y0sin(22) (7{14) ABCD22=|(|Y0sin(21)(1+Y2Z3)+cos(21)(Y1+Y2+Y1Y2Z3))Z0sin(22)+(|Y0sin(21)Z3+cos(21)(1+Y1Z3))cos(22) (7{15) TheadmittanceY1;Y2andimpedanceZ3arethefunctionofbiasvoltagewhichdeterminesthevaractorcapacitance.Thetransmissionlinedelay1;2,parasiticparameters,ttingparametersofthevaractorSPICEmodelaretrainedbyanonlinearleastsquarettingalgorithm.TheABCD-parametersbasedttingmodelwasconvertedtoS-parametersforfaircomparisonwithdirectmeasurementandneuralnetworkmodel. 7.4BiasSearchforImpedanceMatching Thegoalofbiassearchistondtheoptimalbiasvoltagevectorgivenby bv=nv)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(in(v)=0o=nv)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(M(v)=)]TJ /F4 7.97 Tf 27.61 -1.8 Td[(Lo(7{16) Duetothediscretemeasurementdataofamappingtable)]TJ /F4 7.97 Tf 307.73 -1.79 Td[(M(v),itisnotalwayspossibletondtheoptimalbiasvoltagevector.Instead,wechoosethebiasvoltagevectorclosesttotheoptimalbiasvoltagevectorandthisprocedurecanbewrittenasfollows. bv=argminvj)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(M(v))]TJ /F1 11.955 Tf 11.96 0 Td[()]TJ /F4 7.97 Tf 7.31 -1.79 Td[(Lj(7{17) Fromnowon,theoptimalbiasvoltagevectorisredenedasthebiasminimizingthemagnitudebetween)]TJ /F4 7.97 Tf 110.39 -1.79 Td[(Mand)]TJ /F4 7.97 Tf 30.08 -1.79 Td[(L. Thebiassearchconsistsoftwosteps,coarseandnesearch.ThecoarsesearchwasperformedonthemappingtableconvertedfromthedirectmeasurementofS-parameters. 100

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Then,thenesearchwasperformedagainonthelinearinterpolationofavoxel 1 ofthebiasvoltagevectorfoundinthecoarsesearch. Thecostfunctionforbothcoarseandnesearchismeansquareerror(MSE)ofmagnitudebetween)]TJ /F4 7.97 Tf 110.39 -1.79 Td[(Mofeightbiasvoltagevectorsofavoxelandanestimatedload)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L.Notethatavoxelfor3Ddatahaseightvertexes,similartoacubic.Forexample,aunitvoxelconsistsofeightpoints,(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).Eightpointsofthevoxelforthecoarsesearchwereselectedfromthevoltagebiasusedforthedirectmeasurement.Oncethevoxelisfoundbythecoarsesearch,themappingtableofthevoxelisinterpolatedbymultivariatelinearinterpolationwithhigherresolution.Then,thenesearchisperformedontheinterpolateddata.Theproposedtwo-stepsearchreducestherequirednumberofmeasurementpointsandthememoryusagebythepartialinterpolation.Inaddition,thebiassearchcanbemadefasterusingbinarysearchalgorithmonsorteddata. 7.5CharacterizationResults TheS-parametersofatunablematchingnetworkarethefunctionofbiasvoltage.TheS-parametersandthecorrespondingbiasvoltagesshouldbemeasuredtogether.Duringthemeasurement,thebiasvoltageofeachvaractorwassetbya12bitdigital-to-analogconverter(DAC)from0to5Vby0.32Vstepin16levels.TheS-parametersweremeasuredusingavectornetworkanalyzerat2.4GHz.AllmeasurementprocedureswereautomatedbyaninstrumentcontrolprogramwritteninMATLAB.TheprogramrunninginahostcomputercommunicatedwithamicrocontrollerandaDACtosetbiasvoltage,thensentaGPIBcommandforthenetworkanalyzertomeasureS-parameters.Notethatthemicrocontrollerwilleventuallyreplacethehostcomputerandimplementallcontrolprograms.Duetotheslowresponseofthenetworkanalyzer,thenetworkanalyzerfailedtomeasurecorrectS-parametersimmediatelyafterchangingbiasvoltage. 1 Avoxelisabasicunitcellfor3Ddata,similartoapixelfor2Dpicture. 101

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ThecontrolprogrampausedforafewsecondsafterbiassettingsuchthatthenetworkanalyzercouldgetintosteadystateandmeasurecorrectS-parameters.Thebiasvoltageofthreevaractorswaschangedin16levels,thereforethetotalnumberofmeasurementsis161616=4096. FourneuralnetworkmodelsapproximatedS11,S12,S21,andS22,respectivelyandweretrainedbybackpropagationalgorithm.Theinput,hidden,andoutputlayersconsistof3biasvoltages,15hiddenperceptrons,and2(realandimaginary)partsofS-parameters,respectively.Becausethematchingnetworkisareciprocalpassivenetwork,theneuralnetworkmodelsforS12andS21shouldgeneratesimilaroutputsandcanbemergedintoasinglemodel.Duringtrainingtheneuralnetworks,10%oftheS-parameterdatawereusedforvalidationand10%fortestingpurposes. Theestimationresultsusingthetrainedneuralnetworkshowedgoodagreementwiththemeasurementdata,asshowninFigure 7-2 .Asexpected,theresultsforS12andS21werecloseenoughtomergeintoasinglemodel.Tochecktheoverttingproblemofneuralnetworks,thetrainednetworkswerecomparedusingtrainingandtestingdata.Thecomparedestimationresults,asshowninFigure 7-3 ,demonstratedthecomparableerrorforbothdata,thereforethetrainednetworkshavenooverttingproblemandcanapproximateunknownS-parameterdata. Fourclosed-formmodelsrepresentingS11,S12,S21,andS22,weretrainedbyanonlinearleastsquarealgorithm.Theestimationresultsusingthetrainedclosed-formmodelswerecomparedwiththeS-parametermeasurementdata.Theestimationerrorwashigherthantheneuralnetworkmodelsbyafewordersofmagnitude,asshowninFigure 7-4 .Toimprovetheclosed-formmodels,moreparasiticeectsandttingparameterscanbeadded,butmoreparametersmaycauseoptimizationproblemssuchasinitialparametersettingandlocalminima.Inthiswork,onlytheneuralnetworkmodelswereusedtoevaluatetheautomaticmatchingcontrol.Theestimationstatisticsofneuralnetworkandclosed-formmodelsaresummarizedinTable 7-1 and 7-2 102

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A B C D Figure7-2. Neuralnetworkmodelsfora2-portmatchingnetworkweretrainedbybackpropagation.A)S11B)S21C)S12D)S22 103

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A B C D Figure7-3. Neuralnetworkmodelsfora2-portmatchingnetworkweretestedby10%ofmeasurementdata.A)S11B)S21C)S12D)S22 104

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A B C D Figure7-4. Closedformmodelsfora2-portmatchingnetworkweretrainedbynonlinearleastsquaretting.A)S11B)S21C)S12D)S22 105

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Table7-1. Meansquareerror(MSE)ofneuralnetworkttingmodelsusingtrainingandtestingdata TypeTrainingdataTestingdata S111:9010)]TJ /F6 7.97 Tf 6.59 0 Td[(42:2510)]TJ /F6 7.97 Tf 6.59 0 Td[(4 S211:5010)]TJ /F6 7.97 Tf 6.59 0 Td[(41:6910)]TJ /F6 7.97 Tf 6.59 0 Td[(4 S121:7910)]TJ /F6 7.97 Tf 6.59 0 Td[(42:0610)]TJ /F6 7.97 Tf 6.59 0 Td[(4 S222:4910)]TJ /F6 7.97 Tf 6.59 0 Td[(42:8010)]TJ /F6 7.97 Tf 6.59 0 Td[(4 Table7-2. Averageerrorofclosed-formmodels TypeMeansquareerrorAverageerror S111:2010)]TJ /F6 7.97 Tf 6.59 0 Td[(21:1010)]TJ /F6 7.97 Tf 6.59 0 Td[(1 S215:9910)]TJ /F6 7.97 Tf 6.59 0 Td[(37:7410)]TJ /F6 7.97 Tf 6.59 0 Td[(2 S126:0010)]TJ /F6 7.97 Tf 6.59 0 Td[(37:7510)]TJ /F6 7.97 Tf 6.59 0 Td[(2 S227:9510)]TJ /F6 7.97 Tf 6.59 0 Td[(38:9210)]TJ /F6 7.97 Tf 6.59 0 Td[(2 TheS-parameterswereconvertedtoamappingtable,whichisusedforbiassearchandmatchingcontrol,andtheconversionprocedureisdescribedasfollows.Twodierentformsofthemappingtablewereusedinthiswork.Therstformofthemappingtable,asshowninTable 7-3 ,consistsofabiasvoltagevectorandthecorrespondingmismatchedloadtobematchedbyamatchingnetwork.ThemappingtablewasobtainedbyEquation 7{4 andS-parameterdata.However,themagnitudeofthemismatchedloadtobematchedmaybeoutoftheunitcircleontheSmithchartandthevalueisincorrectforapassivenetwork.Inthiscase,thecorrespondingbiasvoltagecannotbechosenforimpedancematchingofanymismatchedloadandcanberemovedfromthemappingtable.Thesecondformofthemappingtable,asshowninTable 7-4 ,isaninversemappingtable,wheremismatchedloadstobematchedareevenlydistributedontheSmithchartandthecorrespondingoptimalbiasvoltagevectorswerecalculatedthroughthetwo-stepsearch. 7.6ImpedanceMatchingResults Impedancematchingwasperformedbymatchingcontrolusingbiassearchonthemappingtable.Mismatchedloadswasestimatedbycoupler-freeandreectometerload 106

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Table7-3. Mappingtablebetweenbiasvoltageandamismatchedloadtobematched Biasvoltage(V)Mismatchedloadreectioncoecienttobematchedbymatchingnetwork 0.00,0.00,0.00+0:54)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:99 0.00,0.00,0.31+0:36)]TJ /F5 11.955 Tf 11.95 0 Td[(|1:08 0.00,0.00,0.63+0:13)]TJ /F5 11.955 Tf 11.95 0 Td[(|1:16 0.00,0.00,0.94)]TJ /F1 11.955 Tf 9.3 0 Td[(0:18)]TJ /F5 11.955 Tf 11.95 0 Td[(|1:22 0.00,0.00,1.26)]TJ /F1 11.955 Tf 9.3 0 Td[(0:60)]TJ /F5 11.955 Tf 11.95 0 Td[(|1:18 0.00,0.00,1.57)]TJ /F1 11.955 Tf 9.3 0 Td[(1:11)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:91 0.00,0.00,1.89)]TJ /F1 11.955 Tf 9.3 0 Td[(1:54)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:32 0.00,0.00,2.20)]TJ /F1 11.955 Tf 9.3 0 Td[(1:58+|0:55 0.00,0.00,2.52)]TJ /F1 11.955 Tf 9.3 0 Td[(1:13+|1:24 0.00,0.00,2.83)]TJ /F1 11.955 Tf 9.3 0 Td[(0:50+|1:53 ...... Table7-4. Inversemappingtablebetweenamismatchedloadtobematchedandbiasvoltage Mismatchedloadreectioncoecient tobematchedbymatchingnetworkBiasvoltage(V) +0:00(4.22,4.12,2.83) +0:50(2.61,2.01,4.72) +0:25+|0:43(4.72,3.46,2.99) )]TJ /F1 11.955 Tf 9.3 0 Td[(0:25+|0:43(2.30,2.20,2.01) )]TJ /F1 11.955 Tf 9.3 0 Td[(0:50+|0:00(1.89,1.67,1.64) )]TJ /F1 11.955 Tf 9.3 0 Td[(0:25)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:43(1.57,0.00,0.31) +0:25)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:43(0.31,0.85,4.63) +1:00(1.35,0.13,0.06) +0:87+|0:50(0.00,0.00,4.72) +0:50+|0:87(2.64,1.98,0.82) +0:00+|1:00(0.50,0.09,0.35) )]TJ /F1 11.955 Tf 9.3 0 Td[(0:50+|0:87(0.09,0.09,0.00) )]TJ /F1 11.955 Tf 9.3 0 Td[(0:87+|0:50(3.15,3.78,4.60) )]TJ /F1 11.955 Tf 9.3 0 Td[(1:00(4.38,3.18,4.12) )]TJ /F1 11.955 Tf 9.3 0 Td[(0:87)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:50(2.36,4.72,3.15) )]TJ /F1 11.955 Tf 9.3 0 Td[(0:50)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:87(3.59,2.86,3.49) )]TJ /F1 11.955 Tf 9.3 0 Td[(0:00)]TJ /F5 11.955 Tf 11.95 0 Td[(|1:00(4.72,3.27,2.99) +0:50)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:87(2.30,2.11,1.79) +0:87)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:50(0.72,0.00,4.72) 107

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estimationmethods.Themappingtablewasconvertedfromthecharacterizationresultsandexaminedbythetwo-stepsearchforanoptimalbiasvoltage.Theoptimalbiasvoltagewasappliedtothematchingnetworkandthedegreeofmismatch,themagnitudeoftheinputreectioncoecient,wasmeasuredtoevaluatethecapabilityofimpedancematching.TheslightlyandheavilymismatchedloadsusedforloadestimationwereconguredasthesamewayusedinChapter 5 and 6 Theexperimentalresultsdemonstratedthatthematchingcontrolcouldachieveimmediateimpedancematchingthroughtheloadestimationandthemappingtable.TheoptimalbiasvoltageandthedegreeofmismatchweresummarizedinTable 7-5 7-6 7-7 ,and 7-8 Becausethereectometerloadestimationmethodachievedlowerestimationerrorthanthecoupler-freeloadestimation,theimpedancematchingbythereectometerresultedinthelowerdegreeofmismatch.Inaddition,S-parameterdataestimatedbytheneuralnetworkmodelsweresoclosetomeasurementdatathatthebiassearchandthecorrespondingimpedancematchingwerealmostidenticalconsideringthemeasurementerror. Theimpedancematchingresultsdemonstratedthattheneuralnetworkmodelcanbeusedfortheautomaticmatchingcontrol.Thenumberofcoecientsofeachneuralnetworkmodelsisaslowas20,becausetheinput,hidden,andoutputlayershave3,15,and2nodes.ComparedwiththelargesizeofS-parametermeasurementdata,163=4096,theneuralnetworkmodelismoreappropriateforcompactmicrocontrollersystemwithlimitedavailablememory. TheestimateofthemismatchedloadsandtheloadreectioncoecientmatchedbythematchingnetworkwereplottedinFigure 7-5 and 7-6 ,respectively.Duetothesmallestimationerror,afewhundredth,ofneuralnetworkmodels,themappingtableconvertedfromtheneuralnetworkmodelswasalmostidenticaltothatofmeasurement 108

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Table7-5. Impedancematchingresultsusingcoupler-freeloadestimationandS-parametermeasurementdata Mismatchedload)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(LOptimalbiasvoltage(V) Inputreectioncoecient)]TJ /F4 7.97 Tf 61.95 -1.79 Td[(inDegreeofmismatchj)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj +0:14)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:77(2.03,0.17,4.39)+0:04+|0:160.16 +0:64+|0:40(4.71,0.02,4.71)+0:22+|0:090.24 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:69+|0:04(0.02,4.71,2.38)+0:05)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:060.08 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:01+|0:01(4.49,4.64,4.45)+0:03+|0:030.04 +0:12)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:01(4.05,1.05,1.59)+0:02)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:010.02 +0:38+|0:06(3.70,0.08,4.45)+0:03+|0:010.03 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:20)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:01(4.45,4.30,4.64)+0:01+|0:030.03 Average0.09 Table7-6. Impedancematchingresultsusingthree-portreectometerloadestimationandS-parametermeasurementdata Mismatchedload)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(LOptimalbiasvoltage(V) Inputreectioncoecient)]TJ /F4 7.97 Tf 61.95 -1.79 Td[(inDegreeofmismatchj)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj +0:14)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:77(1.68,0.87,4.42)+0:05+|0:170.18 +0:64+|0:40(2.44,0.14,4.68))]TJ /F1 11.955 Tf 9.3 0 Td[(0:05)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:050.07 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:69+|0:04(0.02,4.01,2.28))]TJ /F1 11.955 Tf 9.3 0 Td[(0:13+|0:030.13 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:01+|0:01(4.23,4.30,4.20)+0:01+|0:000.01 +0:12)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:01(4.11,0.87,1.68))]TJ /F1 11.955 Tf 9.3 0 Td[(0:02)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:010.02 +0:38+|0:06(3.20,0.02,4.64)+0:01)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:010.01 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:20)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:01(4.49,4.20,4.71)+0:01+|0:020.02 Average0.06 Table7-7. Impedancematchingresultsusingcoupler-freeloadestimationandS-parametersestimatedbyneuralnetworkmodels Mismatchedload)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(LOptimalbiasvoltage(V) Inputreectioncoecient)]TJ /F4 7.97 Tf 61.95 -1.79 Td[(inDegreeofmismatchj)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj +0:14)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:77(1.28,4.30,0.02)+0:01+|0:150.15 +0:64+|0:40(4.71,4.71,4.14)+0:21+|0:080.22 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:69+|0:04(1.94,0.11,2.28)+0:01)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:060.06 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:01+|0:01(4.61,1.34,4.49)+0:02+|0:050.05 +0:12)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:01(4.61,1.46,4.23)+0:02)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:000.02 +0:38+|0:06(4.52,3.60,4.23)+0:03+|0:010.03 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:20)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:01(4.11,0.08,4.71)+0:01+|0:030.03 Average0.08 109

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Table7-8. Impedancematchingresultsusingthree-portreectometerloadestimationandS-parametersestimatedbyneuralnetworkmodels Mismatchedload)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(LOptimalbiasvoltage(V) Inputreectioncoecient)]TJ /F4 7.97 Tf 61.95 -1.8 Td[(inDegreeofmismatchj)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(inj +0:14)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:77(1.28,4.42,0.02)+0:05+|0:170.18 +0:64+|0:40(2.69,2.41,3.89))]TJ /F1 11.955 Tf 9.3 0 Td[(0:04)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:050.06 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:69+|0:04(1.68,0.05,2.35))]TJ /F1 11.955 Tf 9.3 0 Td[(0:15+|0:030.15 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:01+|0:01(4.20,0.74,4.45)+0:01+|0:000.01 +0:12)]TJ /F5 11.955 Tf 11.96 0 Td[(|0:01(4.42,1.90,4.55))]TJ /F1 11.955 Tf 9.3 0 Td[(0:01)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:000.01 +0:38+|0:06(4.33,3.07,4.11)+0:02)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:020.03 )]TJ /F1 11.955 Tf 9.3 0 Td[(0:20)]TJ /F5 11.955 Tf 11.95 0 Td[(|0:01(4.17,0.02,4.71)+0:02+|0:020.03 Average0.07 data.Therefore,theimpedancematchingresultsfrombothmappingtableswerealsoalmostidentical. Thematchingcontrolachievedthelowdegreeofmismatch,-26dBorless,fortheslightlymismatchedloads.Thelargerestimationerrorfortheheavilymismatchedloadsresultsinthelargerdegreeofmismatch,around-12dBorless,dependingontheaccuracyofloadestimation.Thematchingcontrolcanbeintegratedwithoneoftheloadestimationmethods.Thecoupler-freeloadestimationcanbeimplementedinacompactsize,whereasthereectometerloadestimationcanachievebetterimpedancematchingforheavilymismatchedloads. 7.7Conclusion Wedemonstratedthatanautomaticmatchingcontrolsystemcouldachieveanimmediateimpedancematchingbyutilizingloadestimationtechniquesandthecharacterizationofamatchingnetwork.Thematchingnetworkwascharacterizedbyneuralnetworkmodelsandclosed-formequationsaswellasS-parametermeasurementdata.Theneuralnetworkmodelsachievedthecomparablecharacterizationaccuracywithmuchfewercoecientsthanmeasurementdata.Thesmallernumberofcoecientscanbeeasilystoredandimplementedbyacompactmicrocontrollerwithbuilt-inmemory. 110

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A B C D Figure7-5. S-parametersfromvectormeasurementandneuralnetworkmodelwerecomparedintermsofcoupler-freeloadestimationassistedmatchingcontrol.A)VectormeasurementbylargemismatchB)NeuralnetworkmodelbylargemismatchC)VectormeasurementbysmallmismatchD)Neuralnetworkmodelbysmallmismatch 111

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A B C D Figure7-6. S-parametersfromvectormeasurementandneuralnetworkmodelwerecomparedintermsofreectometerloadestimationassistedmatchingcontrol.A)VectormeasurementbylargemismatchB)NeuralnetworkmodelbylargemismatchC)VectormeasurementbysmallmismatchD)Neuralnetworkmodelbysmallmismatch 112

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Thebiassearchwasperformedonamappingtablebetweenthemismatchedloadandbiasvoltage,whichapproximatedtheinversefunctionofthemismatchedloadtobematched.Thetwo-stepcoarseandnesearchalgorithmcouldachievethesamedegreeofmismatchwithafewernumberofmeasurementdatapointsthanone-stepfullsearch.Theexperimentalresultsshowedthatthedegreeofmismatchcouldbeaslowas-12dBforheavilymismatchedloadsand-26dBforslightlymismatchedloads.Notethatthebiassearchcanbemadefasterbyusingbinarysearchandthesortedmappingtable. Althoughthematchingcontrolandmatchingnetworkinthisworkweredevelopedforon-chipembeddedtestsystems,theycanbeappliedtodierentmatchingnetworks,suchasdistributedorbroadband.Inaddition,loadestimationandmicrowavecharacterizationwillgetmoreattentionwithariseofautomatedRFsystems,becausetheautomaticmatchingcontrolsystemutilizingloadestimationneedstoknowthecharacteristicsofamatchingnetwork. 113

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CHAPTER8CONCLUSION Theimpedancematchingofradiofrequency(RF)portsofasocketeddeviceundertest(DUT),whichusuallysuersfromapogopinconnection,poorlydenedRFtolerance,andmanufacturingvariation,isdicult,time-consuming,andempiricalprocess.Inthiswork,arecongurablematchingnetworkworkswithadaptivematchingcontroltoachievetheimpedancematchingandtoreducetheundesiredeects.Inaddition,thematchingcontrolisassistedbythecharacterizationandmicrowavemodelingofthematchingnetwork. BackgroundtheorypresentedinChapter 2 providesthebasicofalumpedimpedancematching,variousnonlinearmicrowavemodelingtechniques,andthefundamentalsofsix-portandfour-portreectometers. Anautomaticmatchingcontrol(AMC)presentedinChapter 3 facilitatestheimpedancematchingoverabroadbandfrequencyusingamicrostripbandpasslter.Themicrostriplterconsistsofvestubsandthreevaractorsconnectedtotheendofthestub.Thecenterfrequencyandthebandwidthare3.5GHzand2GHz,respectivelyandtheinsertionlossoverthepassbandisaslowas2dB.Theproposedsystememploysagreedysearchalgorithmtodeterminethevaractorbiasesforimpedancematchoveralargefractionalbandwidth71%=2.5/3.5.Thegreedyalgorithmoutperformsbrute-forceandsingle-stepalgorithmsintermsofthenumberoftrialsandtheavailablebandwidth,respectively.Theworkdemonstratesthefeasibilityoftheautomaticmatchingcontrolcircuitoverthebroadbandfrequencies. LoadestimationtechniquespresentedinChapter 4 exploitstheprincipleofasix-portreectometertocalculatethecomplexreectioncoecientofaDUTfromthreewavepowerreadings.Threewavepowerdetectorsarerealizedbychangingvaractorbiasesandmeasuringreectedwavepowerusingascalarnetworkanalyzer.Theperformanceoftheestimationisevaluatedbythemeansquareerror(MSE)betweentrueandestimated 114

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reectioncoecients.ItdemonstratesthattunablematchingnetworksandcompactpowerdetectorscanbeusedtoestimatetheloadreectioncoecientofaDUTfortheautomaticmatchingcontrol(AMC)system. Acoupler-freeloadestimationpresentedinChapter 5 demonstratesthedramaticreductionoftheautomaticmatchingcontrolsystemsizebyreplacingadistributedcouplerwithahighimpedanceprobe.Thecoupler-freeloadestimationdiscoversanunknownloadfromthepowerofthecombinedincidentandreectedwavesmeasuredbyahighimpedanceprobe.Theexperimentalresultsshowthatthecouplerandcoupler-freeloadestimationmethodsarecomparableintermsofestimationperformance. Thelumped-elementreectometerpresentedinChapter 6 isdesignednotonlyforanautomaticmatchingcontrol,butalsousedasamultistatereectometerforloadestimation.Alosslesssectionnetworkisusedasanetworktopologyfortheproposedthree-port.Thefour-portreectometercanbeextendedbyaddingareferenceporttothethree-portreectometerandthereferenceportcanberealizedwitharesistivepowerdivider.Thethree-portmatchingnetworkwasfabricatedonaFR4printedcircuitboard(PCB)usinglumpedchipinductorsandvaractors.ThereectometeranalysisandexperimentalresultshowedthatthematchingcapabilitycouldcoveraunitcircleontheSmithchartandprovidetheoptimumperformancecriteriaofthematchingnetwork. ThenovelmatchingcontrolandcharacterizationmethodspresentedinChapter 7 aredevelopedtosupportloadestimationtechniques.AtunablematchingnetworkischaracterizedbyvectormeasurementorneuralnetworkmodelsofS-parameters.TheS-parameterdataareconvertedtoamappingtableofaloadreectiontobematchedbythematchingcontrol.Theequationofthemappingtableisderivedfromtheanalysisofabiasvoltagevector,aninputreectioncoecient,andaloadreectioncoecient.Theoptimalbiasminimizingthedegreeofmismatchisdiscoveredbytwo-stepbiassearch.Thetwo-stepsearchperformscoarsesearchonthemappingtablefollowedbynesearchonthemultivariatelinearinterpolationofthemappingtable.Theexperimentalresults 115

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showthattheneuralnetworkttingmodelachievescomparableaccuracyasvectormeasurementandthattheimpedancematchingbytheproposedmatchingcontrolisaslowas-12and-26dBforheavilyandslightlymismatchedloads,respectively. Oneofthemostchallengingtasksinimplementingtheautomaticmatchingcontrolsystemwithloadestimationisthefullvectormeasurementofthematchingnetwork.IncaseoftheembeddedRFtestsystemwheretheembeddedmatchingnetworkhasnoexternalnode,itisoftenverydicult,evenifnotimpossible,tomeasureusingaregularvectornetworkanalyzer.Thedevelopmentofameasurementprocedureusingonlyexistingpowerdetectorswillbevaluable.Someresearchershaveproposedacalibrationmethodusingonlypowerdetectors[ 38 ]andthecalibrationofmultistateperturbation-two-port(PTP)[ 39 ].Themeasurementoftheembeddedmatchingnetworkcanbeimplementedinsimilarway.Inaddition,althoughsingle-endedRFsystemwasassumedinthiswork,theproposedworkcanbeextendedtoadierentialRFsystem.ThepreliminaryanalysisofaninputreectioncoecientofdierentialRFsystemisprovided(Appendix C and D ). 116

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APPENDIXADERIVATIONOFLOADIMPEDANCECIRCLEEQUATIONUSINGINPUTIMPEDANCEMAGNITUDE Acomplexinputreectioncoecientiswrittenintermsoftwo-portS-parametersSijandaloadreectioncoecient)]TJ /F4 7.97 Tf 170.55 -1.8 Td[(Las )]TJ /F4 7.97 Tf 7.31 -1.79 Td[(in=S11+S12S21)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L(A{1) Themagnitudeoftheinputreectioncoecientbecomes j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj=S11+S12S21)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(L 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(L(A{2) j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj=S11)]TJ /F5 11.955 Tf 11.96 0 Td[(S11S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L+S12S21)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L 1)]TJ /F5 11.955 Tf 11.95 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L(A{3) Bytakingthesquareofbothsides,theequationiswrittenas j1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(Lj2j)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(inj2=jS11)]TJ /F1 11.955 Tf 11.96 0 Td[()]TJ /F4 7.97 Tf 17.07 -1.8 Td[(Lj2(A{4) where =S11S22)]TJ /F5 11.955 Tf 11.95 0 Td[(S12S21(A{5) Theequationismanipulatedasfollows. j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2+jS22j2j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(Lj2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(Lj)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F9 7.97 Tf 7.32 4.34 Td[(Lj)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2=jS11j2+jj2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(Lj2)]TJ /F1 11.955 Tf 11.95 0 Td[()]TJ /F4 7.97 Tf 17.07 -1.79 Td[(LS11)]TJ /F1 11.955 Tf 11.96 0 Td[()]TJ /F9 7.97 Tf 7.31 4.34 Td[(LS11 (A{6) where*denotesacomplexconjugate. (jS22j2j)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(inj2)-221(jj2))]TJ /F4 7.97 Tf 11.87 -1.8 Td[(L)]TJ /F9 7.97 Tf 7.32 4.34 Td[(L)]TJ /F1 11.955 Tf 11.96 0 Td[((S22j)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[(S11))]TJ /F4 7.97 Tf 11.86 -1.8 Td[(L)]TJ /F1 11.955 Tf 9.3 0 Td[((S22j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[(S11))]TJ /F9 7.97 Tf 11.87 4.34 Td[(L=jS11j2)-222(j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2 (A{7) )]TJ /F4 7.97 Tf 7.32 -1.79 Td[(L)]TJ /F9 7.97 Tf 7.31 4.34 Td[(L)]TJ /F1 11.955 Tf 13.15 8.09 Td[((S22j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[(S11))]TJ /F4 7.97 Tf 11.87 -1.79 Td[(L+(S22j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[(S11))]TJ /F9 7.97 Tf 11.87 4.34 Td[(L jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)-222(jj2=jS11j2)-222(j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2 jS22j2j)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(inj2)-221(jj2 (A{8) 117

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)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(L)]TJ /F1 11.955 Tf 13.15 8.08 Td[((S22j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.96 0 Td[(S11) jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(inj2)-222(jj22=jS11j2)-222(j)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(inj2 jS22j2j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2)-221(jj2+S22j)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[(S11 jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)-222(jj22 (A{9) =jS11j2jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)-222(jj2jS11j2)-222(jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj4+jj2j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2 jjS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)-222(jj2j2+jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj4+jj2jS11j2)]TJ /F1 11.955 Tf 11.95 0 Td[(S11S22j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[(S11S22j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2 jjS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)-222(jj2j2 (A{10) =jS11j2jS22j2+jj2)]TJ /F1 11.955 Tf 11.95 0 Td[(S11S22)]TJ /F1 11.955 Tf 11.95 0 Td[(S11S22 jjS22j2j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2)-221(jj2j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2 (A{11) =S11S22)]TJ /F1 11.955 Tf 11.95 0 Td[( jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(inj2)-222(jj22j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj2 (A{12) =S12S21 jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(inj2)-222(jj22j)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(inj2 (A{13) Now,theequationrepresentsacircleontheSmithchart. )]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L)]TJ /F1 11.955 Tf 13.15 8.09 Td[((S22j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.96 0 Td[(S11) jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)-222(jj2=S12S21 jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj2)-222(jj2j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(inj (A{14) Thecenterandradiusofthecirclearegivenby CL=(S22j)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[(S11) jS22j2j)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(inj2)-221(jj2(center) (A{15) RL=S12S21 jS22j2j)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(inj2)-222(jj2j)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(inj(radius) (A{16) Whentheincidentandreectedwavesarecombinedundertheabsenceofacoupler,themeasuredpoweroftheinputport(P1)isexpressedas pin=ja1+b1j2=ja1j2j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2(A{17) 118

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Therefore,j1+)]TJ /F4 7.97 Tf 29.79 -1.79 Td[(injinsteadofj)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(injcanbeusedtoestimatealoadimpedance.Thecomplexinputreectioncoecientisrewrittenas 1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(in=1+S11+S12S21)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L 1)]TJ /F5 11.955 Tf 11.95 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L(A{18) Themagnitudeoftheinputreectioncoecientbecomes j1+)]TJ /F4 7.97 Tf 27.58 -1.8 Td[(inj=1+S11+S12S21)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(L 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.32 -1.8 Td[(L(A{19) j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj=1+S11)]TJ /F1 11.955 Tf 11.95 0 Td[((1+S11)S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L+S12S21)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L 1)]TJ /F5 11.955 Tf 11.95 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L(A{20) Bytakingthesquareofbothsides,theequationiswrittenas j1)]TJ /F5 11.955 Tf 11.95 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(Lj2j1+)]TJ /F4 7.97 Tf 27.59 -1.8 Td[(inj2=j1+S11)]TJ /F1 11.955 Tf 11.96 0 Td[((S22+))]TJ /F4 7.97 Tf 33.38 -1.8 Td[(Lj2(A{21) where =S11S22)]TJ /F5 11.955 Tf 11.95 0 Td[(S12S21(A{22) Theequationismanipulatedasfollows. j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2+jS22j2j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(Lj2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(Lj1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F9 7.97 Tf 7.32 4.34 Td[(Lj1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2=j1+S11j2+jS22+j2j)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(Lj2)]TJ /F1 11.955 Tf 11.96 0 Td[((S22+))]TJ /F4 7.97 Tf 33.39 -1.79 Td[(L(1+S11))]TJ /F1 11.955 Tf 11.96 0 Td[((S22+))]TJ /F9 7.97 Tf 11.87 4.34 Td[(L(1+S11) (A{23) where*denotesacomplexconjugate. (jS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.8 Td[(inj2)-222(jS22+j2))]TJ /F4 7.97 Tf 11.87 -1.8 Td[(L)]TJ /F9 7.97 Tf 7.32 4.34 Td[(L)]TJ /F1 11.955 Tf 11.96 0 Td[((S22j1+)]TJ /F4 7.97 Tf 27.58 -1.8 Td[(inj2)]TJ /F1 11.955 Tf 11.96 0 Td[((S22+)(1+S11)))]TJ /F4 7.97 Tf 16.42 -1.8 Td[(L)]TJ /F1 11.955 Tf 9.29 0 Td[((S22j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[((S22+)(1+S11)))]TJ /F9 7.97 Tf 16.42 4.34 Td[(L=j1+S11j2)-222(j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2 (A{24) )]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L)]TJ /F9 7.97 Tf 7.31 4.34 Td[(L)]TJ /F1 11.955 Tf 13.15 8.09 Td[((S22j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.96 0 Td[((S22+)(1+S11)))]TJ /F4 7.97 Tf 16.42 -1.79 Td[(L+(S22j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[((S22+)(1+S11)))]TJ /F9 7.97 Tf 16.42 4.34 Td[(L jS22j2j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2)-222(jS22+j2=j1+S11j2)-222(j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2 jS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)-222(jS22+j2 (A{25) 119

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)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(L)]TJ /F1 11.955 Tf 13.15 8.08 Td[((S22j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[((S22+)(1+S11)) jS22j2j1+)]TJ /F4 7.97 Tf 27.59 -1.8 Td[(inj2)-221(jS22+j22=j1+S11j2)-221(j1+)]TJ /F4 7.97 Tf 27.59 -1.8 Td[(inj2 jS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)-222(jS22+j2+S22j1+)]TJ /F4 7.97 Tf 27.59 -1.8 Td[(inj2)]TJ /F1 11.955 Tf 11.95 0 Td[((S22+)(1+S11) jS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)-222(jS22+j22 (A{26) =j1+S11j2jS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)-222(jS22+j2j1+S11j2)-221(jS22j2j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj4+jS22+j2j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2 jjS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)-222(jS22+j2j2+jS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj4+jS22+j2j1+S11j2 jjS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)-222(jS22+j2j2)]TJ /F1 11.955 Tf 10.5 8.09 Td[((S22+)(1+S11)S22j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2+(S22+)(1+S11)S22j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2 jjS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)-222(jS22+j2j2 (A{27) =j1+S11j2jS22j2+jS22+j2 jjS22j2j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2)-222(jS22+j2j2j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 10.5 8.09 Td[((S22+)(1+S11)S22+(S22+)(1+S11)S22 jjS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)-222(jS22+j2j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2 (A{28) =(1+S11)S22)]TJ /F1 11.955 Tf 11.95 0 Td[((S22+) jS22j2j1+)]TJ /F4 7.97 Tf 27.59 -1.8 Td[(inj2)-222(jS22+j22j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj2 (A{29) =S12S21 jS22j2j1+)]TJ /F4 7.97 Tf 27.59 -1.8 Td[(inj2)-222(jS22+j22j1+)]TJ /F4 7.97 Tf 27.59 -1.8 Td[(inj2 (A{30) Now,theequationrepresentsacircleontheSmithchart. )]TJ /F4 7.97 Tf 7.32 -1.79 Td[(L)]TJ /F1 11.955 Tf 13.15 8.09 Td[((S22j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)]TJ /F1 11.955 Tf 11.96 0 Td[((S22+)(1+S11)) jS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)-222(jS22+j2=S12S21 jS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.79 Td[(inj2)-222(jS22+j2j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj (A{31) Thecenterandradiusofthecirclearegivenby CL=(S22j1+)]TJ /F4 7.97 Tf 27.58 -1.8 Td[(inj2)]TJ /F1 11.955 Tf 11.96 0 Td[((S22+)(1+S11)) jS22j2j1+)]TJ /F4 7.97 Tf 27.58 -1.8 Td[(inj2)-222(jS22+j2(center) (A{32) RL=S12S21 jS22j2j1+)]TJ /F4 7.97 Tf 27.59 -1.8 Td[(inj2)-222(jS22+j2j1+)]TJ /F4 7.97 Tf 27.59 -1.79 Td[(inj(radius) (A{33) 120

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APPENDIXBTHREE-PORTANDFOUR-PORTMULTISTATEREFLECTOMETERS Port3isterminatedwithadetectorwhoseloadreectioncoecientis)]TJ /F6 7.97 Tf 374.13 -1.79 Td[(3.Theincidentwavepowera1iswrittenas a3=)]TJ /F6 7.97 Tf 19.74 -1.8 Td[(3b3(B{1) Therefore,three-portS-parameterequationscanbeexpressedinamatrixformas 0BBBB@b1b2b31CCCCA=266664S11S12S13)]TJ /F6 7.97 Tf 7.31 -1.8 Td[(3S21S22S23)]TJ /F6 7.97 Tf 7.31 -1.79 Td[(3S31S32S33)]TJ /F6 7.97 Tf 7.31 -1.79 Td[(33777750BBBB@a1a2b31CCCCA(B{2) Theincidentpowerwaveintoaloadterminatingtheport3,b3iswrittenas b3=S31a1+S32a2+S33)]TJ /F6 7.97 Tf 7.31 -1.8 Td[(3b3(B{3) b3=1 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S33)]TJ /F6 7.97 Tf 7.32 -1.79 Td[(3(S31a1+S32a2)(B{4) Also,theincidentpowerwaveintotheport1isgivenby a1=1 S21()]TJ /F5 11.955 Tf 9.3 0 Td[(S22a2+b2)]TJ /F5 11.955 Tf 11.96 0 Td[(S23)]TJ /F6 7.97 Tf 7.31 -1.8 Td[(3b3)(B{5) BypluggingEquation B{5 intoEquation B{4 ,theequationbecomes b3=1 1)]TJ /F5 11.955 Tf 11.95 0 Td[(S33)]TJ /F6 7.97 Tf 7.31 -1.79 Td[(3S31)]TJ /F5 11.955 Tf 10.49 8.09 Td[(S22 S21a2+1 S21b2)]TJ /F5 11.955 Tf 13.15 8.09 Td[(S23)]TJ /F6 7.97 Tf 7.31 -1.79 Td[(3 S21b3+S32a2 (B{6) 1+S23S31)]TJ /F6 7.97 Tf 7.31 -1.79 Td[(3 S21(1)]TJ /F5 11.955 Tf 11.96 0 Td[(S33)]TJ /F6 7.97 Tf 7.32 -1.79 Td[(3)b3=1 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S33)]TJ /F6 7.97 Tf 7.32 -1.79 Td[(3S32)]TJ /F5 11.955 Tf 13.15 8.09 Td[(S22S31 S21a2+S31 S21b2 (B{7) Now,thereectedpowerwaveb3iswrittenas b3=S21(1)]TJ /F5 11.955 Tf 11.95 0 Td[(S33)]TJ /F6 7.97 Tf 7.31 -1.79 Td[(3) S21(1)]TJ /F5 11.955 Tf 11.96 0 Td[(S33)]TJ /F6 7.97 Tf 7.32 -1.8 Td[(3)+S23S31)]TJ /F6 7.97 Tf 7.31 -1.8 Td[(3S21S32)]TJ /F5 11.955 Tf 11.95 0 Td[(S22S31 S21(1)]TJ /F5 11.955 Tf 11.96 0 Td[(S33)]TJ /F6 7.97 Tf 7.31 -1.8 Td[(3)b2a2 b2)]TJ /F5 11.955 Tf 44.7 8.08 Td[(S31 S22S31)]TJ /F5 11.955 Tf 11.96 0 Td[(S21S32=S21S32)]TJ /F5 11.955 Tf 11.96 0 Td[(S22S31 S21(1)]TJ /F5 11.955 Tf 11.96 0 Td[(S33)]TJ /F6 7.97 Tf 7.32 -1.8 Td[(3)+S23S31)]TJ /F6 7.97 Tf 7.32 -1.8 Td[(3b2()]TJ /F4 7.97 Tf 11.86 -1.8 Td[(L)]TJ /F5 11.955 Tf 11.95 0 Td[(q3) (B{8) 121

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where )]TJ /F4 7.97 Tf 7.31 -1.8 Td[(L=a2 b2q3=S31 S22S31)]TJ /F5 11.955 Tf 11.95 0 Td[(S21S32(B{9) Tosimplifytheaboveequations,supposethatthedetectorportismatched()]TJ /F6 7.97 Tf 402.07 -1.79 Td[(3=0).Theequationsissimpliedas b3)]TJ /F16 5.978 Tf 5.29 -1.1 Td[(3=0=S21S32)]TJ /F5 11.955 Tf 11.95 0 Td[(S22S31 S21b2()]TJ /F4 7.97 Tf 11.87 -1.79 Td[(L)]TJ /F5 11.955 Tf 11.96 0 Td[(q3)(B{10) andalsoexpressedintermsofA3andB3as b3=A3a2+B3b2(B{11) where A3=S21S32)]TJ /F5 11.955 Tf 11.96 0 Td[(S22S31 S21B3=S31 S21(B{12) Also,thereectedwavepowerb2issimpliedas b2)]TJ /F16 5.978 Tf 5.29 -1.11 Td[(3=0=S21a1+S22)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(Lb2(B{13) b2=S21 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(La1(B{14) BypluggingEquation B{14 intoEquation B{10 ,b3canbesimpliedas b3=S21S32)]TJ /F5 11.955 Tf 11.96 0 Td[(S22S31 S21b2()]TJ /F4 7.97 Tf 11.87 -1.8 Td[(L)]TJ /F5 11.955 Tf 11.96 0 Td[(q3)=S21S32)]TJ /F5 11.955 Tf 11.96 0 Td[(S22S31 S21S21 1)]TJ /F5 11.955 Tf 11.96 0 Td[(S22)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(La1()]TJ /F4 7.97 Tf 11.87 -1.79 Td[(L)]TJ /F5 11.955 Tf 11.96 0 Td[(q3)=S21S32)]TJ /F5 11.955 Tf 11.95 0 Td[(S22S31 1)]TJ /F5 11.955 Tf 11.95 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(La1()]TJ /F4 7.97 Tf 11.87 -1.8 Td[(L)]TJ /F5 11.955 Tf 11.95 0 Td[(q3) (B{15) Similartothethree-portreectometer,afour-portreectometerwithmatcheddetectorports()]TJ /F6 7.97 Tf 87.7 -1.8 Td[(3=)]TJ /F6 7.97 Tf 19.74 -1.8 Td[(4=0)isgovernedbythefollowingequationas b4=S21S42)]TJ /F5 11.955 Tf 11.96 0 Td[(S22S41 S21b2()]TJ /F4 7.97 Tf 11.87 -1.79 Td[(L)]TJ /F5 11.955 Tf 11.96 0 Td[(q4)=S21S42)]TJ /F5 11.955 Tf 11.95 0 Td[(S22S41 1)]TJ /F5 11.955 Tf 11.95 0 Td[(S22)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(La1()]TJ /F4 7.97 Tf 11.87 -1.8 Td[(L)]TJ /F5 11.955 Tf 11.95 0 Td[(q4) (B{16) 122

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where q4=S41 S22S41)]TJ /F5 11.955 Tf 11.95 0 Td[(S21S42(B{17) Theequationisalsoexpressedintermsofa2andb2as b4=A4a2+B4b2(B{18) where A4=S21S42)]TJ /F5 11.955 Tf 11.96 0 Td[(S22S41 S21B4=S41 S21(B{19) 123

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APPENDIXCSTANDARDANDMIXED-MODES-PARAMETERTRANSFORMATION Common-anddierential-modenormalizedpowerwavesaredenedby ac1=1 p 2(a1+a2)ad1=1 p 2(a1)]TJ /F5 11.955 Tf 11.96 0 Td[(a2)(C{1) bc1=1 p 2(b1+b2)bd1=1 p 2(b1)]TJ /F5 11.955 Tf 11.96 0 Td[(b2)(C{2) Thecommon-anddierential-modepowerwavesarewritteninacompactmatrixformas bmm=0B@~b1~b21CA=0BBBBBBB@bc1bd1bc2bd21CCCCCCCA=1 p 226666666411001)]TJ /F1 11.955 Tf 9.3 0 Td[(1000011001)]TJ /F1 11.955 Tf 9.3 0 Td[(13777777750BBBBBBB@b1b2b3b41CCCCCCCA=Mbstd(C{3) Themixed-andstandard-modenormalizedpowerwavesareconvertedtoeachotherbyusingthefollowingrelationshipas bmm=Mbstd(C{4) M=M)]TJ /F6 7.97 Tf 6.58 0 Td[(1=1 p 226666666411001)]TJ /F1 11.955 Tf 9.3 0 Td[(1000011001)]TJ /F1 11.955 Tf 9.3 0 Td[(1377777775(C{5) Now,mixed-modeS-parameterequationsarewritteninamatrixformas bmm=0BBBBBBB@bc1bd1bc2bd21CCCCCCCA=266666664Scc11Scd11 Scc12Scd12Sdc11Sdd11 Sdc12Sdd12 Scc21Scd21 Scc22Scd22Sdc21Sdd21 Sdc22Sdd223777777750BBBBBBB@ac1ad1ac2ad21CCCCCCCA=264S11S12S21S223750B@~a1~a21CA=Smmamm(C{6) 124

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Theequationismanipulatedasfollows. bmm=Smmamm(C{7) Mbstd=SmmMastd(C{8) bstd=M)]TJ /F6 7.97 Tf 6.59 0 Td[(1SmmMastd(C{9) Therefore,thetransformationbetweenmixed-andstandard-modeS-parametersaregivenby Sstd=M)]TJ /F6 7.97 Tf 6.59 0 Td[(1SmmM(C{10) and Smm=MSstdM)]TJ /F6 7.97 Tf 6.59 0 Td[(1(C{11) 125

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APPENDIXDDERIVATIONOFDIFFERENTIALINPUTREFLECTIONCOEFFICIENT Anetworkwithonedierentialportcanbewrittenas ~b1=Smm11~a1(D{1) wheremmdenotesmixed-modeand ~a1=0B@ac1ad11CA;~b1=0B@bc1bd11CA;Smm11=264Scc11Scd11Sdc11Sdd11375(D{2) Itcanbeeasilyextendedtotwodierentialportnetworkasfollows. 0B@~b1~b21CA=264Smm11Smm12Smm21Smm223750B@~a1~a21CA(D{3) Supposethatadierentialloadisconnectedtothedierentialport2.Then,thereisthefollowingrelationshipbetweenincidentandreectedwaves. ~a2=)]TJ /F4 7.97 Tf 8.08 4.93 Td[(mmL~b2(D{4) where )]TJ /F4 7.97 Tf 8.08 4.94 Td[(mmL=264)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(Lcc)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(Lcd)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(Ldc)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(Ldd375(D{5) Therefore,thes-parameterequationcanbewrittenas 0B@~b1~b21CA=264Smm11Smm12Smm21Smm223750B@~a1)]TJ /F4 7.97 Tf 8.08 4.34 Td[(mmL~b21CA(D{6) ~b2canbewrittenas ~b2=Smm21~a1+Smm22)]TJ /F4 7.97 Tf 8.08 4.93 Td[(mmL~b2(D{7) Itcanbesolvedas (I)]TJ /F7 11.955 Tf 11.96 0 Td[(Smm22)]TJ /F4 7.97 Tf 8.08 4.94 Td[(mmL)~b2=Smm21~a1(D{8) 126

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~b2=(I)]TJ /F7 11.955 Tf 11.95 0 Td[(Smm22)]TJ /F4 7.97 Tf 8.08 4.93 Td[(mmL))]TJ /F6 7.97 Tf 6.58 0 Td[(1Smm21~a1(D{9) FromtheS-parameterequationandtheaboveequation,~b1canbewrittenas ~b1=Smm11~a1+Smm12)]TJ /F4 7.97 Tf 8.08 4.94 Td[(mmL~b2=Smm11~a1+Smm12)]TJ /F4 7.97 Tf 8.08 4.94 Td[(mmL(I)]TJ /F7 11.955 Tf 11.96 0 Td[(Smm22)]TJ /F4 7.97 Tf 8.08 4.94 Td[(mmL))]TJ /F6 7.97 Tf 6.59 0 Td[(1Smm21~a1(D{10) Theinputreectionmatrixisdenedas ~b1=)]TJ /F4 7.97 Tf 8.08 4.94 Td[(mmin~a1(D{11) where )]TJ /F4 7.97 Tf 8.08 4.94 Td[(mmin=264)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(incc)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(incd)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(indc)]TJ /F4 7.97 Tf 7.32 -1.79 Td[(indd375(D{12) BythecomparisonofEquation D{10 and D{11 ,theinputreectionmatrixis )]TJ /F4 7.97 Tf 8.08 4.93 Td[(mmin=Smm11+Smm12)]TJ /F4 7.97 Tf 8.08 4.93 Td[(mmL(I)]TJ /F7 11.955 Tf 11.96 0 Td[(Smm22)]TJ /F4 7.97 Tf 8.08 4.93 Td[(mmL))]TJ /F6 7.97 Tf 6.59 0 Td[(1Smm21(D{13) Themixed-modereectionmatrixcanbeconvertedtostandard-modeasfollows. )]TJ /F4 7.97 Tf 8.08 4.94 Td[(stdin=M)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F4 7.97 Tf 8.08 4.94 Td[(mminM(D{14) =M)]TJ /F6 7.97 Tf 6.59 0 Td[(1Smm11M+M)]TJ /F6 7.97 Tf 6.58 0 Td[(1Smm12MM)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F4 7.97 Tf 8.08 4.93 Td[(mmLMM)]TJ /F6 7.97 Tf 6.58 0 Td[(1(I)]TJ /F7 11.955 Tf 11.96 0 Td[(Smm22)]TJ /F4 7.97 Tf 8.08 4.93 Td[(mmL))]TJ /F6 7.97 Tf 6.59 0 Td[(1MM)]TJ /F6 7.97 Tf 6.58 0 Td[(1Smm21M(D{15) Thestandard-modeinputreectioncoecientmatrixiswrittenas )]TJ /F4 7.97 Tf 8.08 4.94 Td[(stdin=Sstd11+Sstd12)]TJ /F4 7.97 Tf 8.08 4.94 Td[(stdL(I)]TJ /F7 11.955 Tf 11.95 0 Td[(Sstd22)]TJ /F4 7.97 Tf 8.08 4.94 Td[(stdL))]TJ /F6 7.97 Tf 6.58 0 Td[(1Sstd21(D{16) where )]TJ /F4 7.97 Tf 8.08 4.93 Td[(stdin=264)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(in11)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(in12)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(in21)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(in22375;)]TJ /F4 7.97 Tf 8.08 4.93 Td[(stdL=264)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(L55)]TJ /F4 7.97 Tf 7.31 -1.8 Td[(L56)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L65)]TJ /F4 7.97 Tf 7.31 -1.79 Td[(L66375(D{17) Sstd11=264S11S12S21S22375;Sstd22=264S33S34S43S44375(D{18) 127

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Sstd12=264S13S14S23S24375;Sstd21=264S31S32S41S42375(D{19) 128

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REFERENCES [1] R.B.Whatley,Z.Zhou,andK.L.Melde,\RecongurableRFimpedancetunerformatchcontrolinbroadbandwirelessdevices,"IEEETrans.AntennasPropag.,vol.54,no.2,pp.470{478,Feb.2006. [2] W.C.E.Neo,Y.Lin,X.D.Liu,L.C.N.d.Vreede,L.E.Larson,M.Spirito,M.J.Pelk,K.Buisman,A.Akhnoukh,A.d.Graauw,andL.K.Nanver,\Adaptivemulti-bandmulti-modepoweramplierusingintegratedvaractor-basedtunablematchingnetworks,"IEEEJ.Solid-StateCircuits,vol.41,no.9,pp.2166{2176,Sep.2006. [3] C.Hoarau,P.E.Bailly,J.D.Arnould,P.Ferrari,andP.Xavier,\ARFtunableimpedancematchingnetworkwithacompletedesignandmeasurementmethodology,"inMicrowaveConference,2007.European,Oct.9{12,2007,pp.751{754. [4] J.H.SinskyandC.R.Westgate,\Designofanelectronicallytunablemicrowaveimpedancetransformer,"inMicrowaveSymposiumDigest,1997.,IEEEMTT-SInternational,vol.2,Jun.8{13,1997,pp.647{650. [5] P.SjoblomandH.Sjoland,\AnadaptiveimpedancetuningCMOScircuitforISM2.4-GHzband,"CircuitsandSystemsI:RegularPapers,IEEETransactionson,vol.52,no.6,pp.1115{1124,Jun.2005. [6] J.d.Mingo,A.Valdovinos,A.Crespo,D.Navarro,andP.Garca,\AnRFelectronicallycontrolledimpedancetuningnetworkdesignanditsapplicationtoanantennainputimpedanceautomaticmatchingsystem,"IEEETrans.Microw.TheoryTech.,vol.52,no.2,pp.489{497,Feb.2004. [7] H.Zhang,H.Gao,andG.-P.Li,\Broad-bandpoweramplierwithanoveltunableoutputmatchingnetwork,"IEEETrans.Microw.TheoryTech.,vol.53,no.11,pp.3606{3614,Nov.2005. [8] K.BritoandR.N.d.Lima,\Tunableimpedancematchingnetwork,"inMicrowaveandOptoelectronicsConference,2007.IMOC2007.SBMO/IEEEMTT-SInterna-tional,Brazil,Oct.29{Nov.2007,2007,pp.117{121. [9] F.Wang,V.K.Devabhaktuni,andQ.-J.Zhang,\Ahierarchicalneuralnetworkapproachtothedevelopmentofalibraryofneuralmodelsformicrowavedesign,"IEEETrans.Microw.TheoryTech.,vol.46,no.12,pp.2391{2403,Dec.1998. [10] J.C.Principe,N.R.Euliano,andW.C.Lefebvre,NeuralandAdaptiveSystems:FundamentalsthroughSimulationswithCD-ROM.NewYork,NY,USA:JohnWiley&Sons,Inc.,1999. [11] G.F.Engen,\Thesix-portreectometer:Analternativenetworkanalyzer,"IEEETrans.Microw.TheoryTech.,vol.25,no.12,pp.1075{1080,Dec.1977. 129

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[12] L.C.Oldeld,J.P.Ide,andE.J.Grin,\Amultistatereectometer,"IEEETrans.Instrum.Meas.,vol.34,no.2,pp.198{201,Jun.1985. [13] S.P.YeoandM.Cheng,\Improvedfour-portinstrumentusingtwopowerdetectorstomeasurecomplexreectioncoecientsofmicrowavedevices,"ElectronicsLetters,vol.32,pp.565{566,Mar.14,1996. [14] S.P.YeoandS.T.Tay,\Improveddesignformultistatereectometer(withtwopowerdetectors)formeasuringreectioncoecientsofmicrowavedevices,"IEEETrans.Instrum.Meas.,vol.49,no.1,pp.61{65,Feb.2000. [15] K.R.Boyle,\TheperformanceofGSM900antennasinthepresenceofpeopleandphantoms,"inAntennasandPropagation,2003.(ICAP2003).TwelfthInternationalConferenceon(Conf.Publ.No.491),vol.1,Mar.31{Apr.3,2003,pp.35{38. [16] K.L.VirgaandY.Rahmat-Samii,\Low-proleenhanced-bandwidthPIFAantennasforwirelesscommunicationspackaging,"IEEETrans.Microw.TheoryTech.,vol.45,no.10,pp.1879{1888,Oct.1997. [17] F.Meng,A.vanBezooijen,andR.Mahmoudi,\Amismatchdetectorforadaptiveantennaimpedancematching,"inMicrowaveConference,2006.36thEuropean,Sep.2006,pp.1457{1460. [18] Y.SunandJ.K.Fidler,\High-speedautomaticantennatuningunits,"inAntennasandPropagation,1995.ICAP'95.NinthInternationalConferenceon(Conf.Publ.No.407),Eindhoven,Apr.4{7,1995,pp.218{222. [19] D.Qiao,R.Molno,S.M.Lardizabal,B.Pillans,P.M.Asbeck,andG.Jerinic,\AnintelligentlycontrolledRFpoweramplierwitharecongurableMEMS-varactortuner,"IEEETrans.Microw.TheoryTech.,vol.53,no.3,pp.1089{1095,Mar.2005. [20] F.PourmohammadiandM.Hakkak,\Adaptivematchingofantennainputimpedance,"inAntennaTechnologySmallAntennasandNovelMetamaterials,2006IEEEInternationalWorkshopon,Mar.6{8,2006,pp.373{376. [21] J.Kim,W.Eisenstadt,H.-H.Yeh,andK.Melde,\Automaticmatchingcontrolsystemforloadboardtest,"inTestofWirelessCircuitsandSystems,2008.WTW2008.7thWorkshopon,Apr.27,2008. [22] I.Ida,J.Takada,T.Toda,andY.Oishi,\Anadaptiveimpedancematchingsystemandconsiderationsforabetterperformance,"inCommunications,2004andthe5thInternationalSymposiumonMulti-DimensionalMobileCommunicationsProceedings.The2004JointConferenceofthe10thAsia-PacicConferenceon,vol.2,Aug.29{Sep.1,2004,pp.563{567. [23] K.HomannandZ.Skvor,\Anovelvectornetworkanalyzer,"IEEETrans.Microw.TheoryTech.,vol.46,no.12,pp.2520{2523,Dec.1998. 130

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[24] J.Vian,\Double-stubimpedancematchingalgorithm,"inAntennasandPropagationInternationalSymposium,2007IEEE,Jun.9{15,2007,pp.4477{4480. [25] J.KimandW.Eisenstadt,\Loadimpedanceestimationforautomaticimpedancematchcontrolcircuits,"inTechnologyandTalentforthe21stCentury,2008.TECHCON2008.10thTechnicalConferenceon,Sep.15{16,2008. [26] D.M.Pozar,Microwaveengineering,2nded.NewYork,NY:JohnWiley&Sons,Inc.,1998. [27] G.F.Engen,\Aleastsquaressolutionforuseinthesix-portmeasurementtechnique,"IEEETrans.Microw.TheoryTech.,vol.28,no.12,pp.1473{1477,Dec.1980. [28] Y.-C.ChiangandC.-Y.Chen,\Designofawide-bandlumped-element3-dbquadraturecoupler,"IEEETrans.Microw.TheoryTech.,vol.49,no.3,pp.476{479,Mar.2001. [29] T.-Y.Song,J.-H.Kim,S.-H.Kim,J.-B.Lim,andJ.-S.Park,\Designofanovellumpedelementbackwarddirectionalcouplerbasedonparallelcoupled-linetheory,"inMicrowaveSymposiumDigest,2002IEEEMTT-SInternational,vol.1,Jun.2{7,2002,pp.213{216. [30] K.Jung,\Broadbandbalunembeddedmeasurementfordierentialcircuits,"Ph.D.dissertation,UniversityofFlorida,Dec.2007. [31] C.Hoarau,P.E.Bailly,J.D.Arnould,P.Ferrari,andP.Xavier,\AccuratemeasurementmethodforcharacterisationofRFimpedancetuners,"ElectronicsLetters,vol.43,pp.1434{1436,Dec.6,2007. [32] T.Zhang,W.R.Eisenstadt,R.M.Fox,andQ.Yin,\BipolarmicrowaveRMSpowerdetectors,"IEEEJ.Solid-StateCircuits,vol.41,no.9,pp.2188{2192,Sep.2006. [33] T.F.ColemanandY.Li,\Aninteriortrustregionapproachfornonlinearminimizationsubjecttobounds,"SIAMJournalonOptimization,vol.6,no.2,pp.418{445,May1996. [34] ||,\Ontheconvergenceofreectivenewtonmethodsforlarge-scalenonlinearminimizationsubjecttobounds,"MathematicalProgramming,vol.67,no.2,pp.189{224,1994. [35] F.Wiedmann,B.Huyart,E.Bergeault,andL.Jallet,\Newstructureforasix-portreectometerinmonolithicmicrowaveintegrated-circuittechnology,"IEEETrans.Instrum.Meas.,vol.46,no.2,pp.527{530,Apr.1997. [36] I.Majid,A.E.Nadeem,andF.eAzam,\Smallsignals-parameterestimationofBJTsusingarticialneuralnetworks,"inMultitopicConference,2004.ProceedingsofINMIC2004.8thInternational,Dec.24{26,2004,pp.669{673. 131

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[37] A.E.NadeemandW.R.Eisenstadt,\Improvedclosed-formexpressionsfors-parametersofBJTsusingmodiedgummel-poonmodel,"inMultiTopicCon-ference,2003.INMIC2003.7thInternational,Dec.8{9,2003,pp.202{207. [38] G.Koers,J.Stiens,andR.Vounckx,\Scalarcalibrationofquasi-opticalreectionmeasurements,"IEEETrans.Microw.TheoryTech.,vol.54,no.7,pp.3121{3126,Jul.2006. [39] K.HomannandZ.Skvor,\CalibrationofthePTPvectornetworkanalyzer,"inMicrowavesandRadar,1998.MIKON'98.,12thInternationalConferenceon,vol.3,May20{22,1998,pp.710{714. 132

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BIOGRAPHICALSKETCH JaeseokKimwasborninSeoul,Korea.HereceivedtheB.S.degreein1994,fromInhaUniversity,majoringinelectronicengineering.HereceivedtheM.S.degreein2002fromtheUniversityofFlorida,majoringinelectricalandcomputerengineering.Since2006,hehasbeenwiththeElectronicCircuitLaboratory(ECL)attheUniversityofFlorida,pursuinghisPh.D.degree.HisresearchinterestsincludeRFimpedancematchingcontrolalgorithmandsystem,machineintelligenceofRFsystems,andembeddedRFon-chiptesting. 133