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Electromagnetic System Design for Wireless Power

Permanent Link: http://ufdc.ufl.edu/UFE0041449/00001

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Title: Electromagnetic System Design for Wireless Power
Physical Description: 1 online resource (149 p.)
Language: english
Creator: Casanova, Joaquin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Wireless communications technology has freed electronics from communication cables. The natural next step is to cut the last wire of portable wireless devices, the power cable. Wireless power systems would permit charging many different devices equipped with receiving coils, in addition to delivering power through rooftops and through the atmosphere. The approaches to wireless power transfer can be categorized as near-field, midrange, and far-field. To date, the latter is still impractical for consumer applications due to the high power and large antenna requirement necessary to achieve levels of power comparable to a wall supply. On the other hand, near-field inductive coupling has more promise as a wireless power technology for charging battery-operated devices. Midrange power transfer has the most potential for applications such as vehicle charging and power transmission through walls and rooftops. Far-field applications include radiofrequency (RF) energy harvesting and transmission of power from space. This dissertation presents several aspects of the design and testing of wireless power systems. Circuit topology and electromagnetic design of a near-field system is considered, as well as the extension of the system to multiple coils. In addition, the use of ferrite shielding and detection and estimation algorithms are considered for the near-field system. The near-field architecture is extended and modified for a midrange system. Finally, far-field power transfer through the atmosphere and the environment are considered through the numerical solution of the radiative transfer equation.
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.
Statement of Responsibility: by Joaquin Casanova.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Lin, Jenshan.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041449:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041449/00001

Material Information

Title: Electromagnetic System Design for Wireless Power
Physical Description: 1 online resource (149 p.)
Language: english
Creator: Casanova, Joaquin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Wireless communications technology has freed electronics from communication cables. The natural next step is to cut the last wire of portable wireless devices, the power cable. Wireless power systems would permit charging many different devices equipped with receiving coils, in addition to delivering power through rooftops and through the atmosphere. The approaches to wireless power transfer can be categorized as near-field, midrange, and far-field. To date, the latter is still impractical for consumer applications due to the high power and large antenna requirement necessary to achieve levels of power comparable to a wall supply. On the other hand, near-field inductive coupling has more promise as a wireless power technology for charging battery-operated devices. Midrange power transfer has the most potential for applications such as vehicle charging and power transmission through walls and rooftops. Far-field applications include radiofrequency (RF) energy harvesting and transmission of power from space. This dissertation presents several aspects of the design and testing of wireless power systems. Circuit topology and electromagnetic design of a near-field system is considered, as well as the extension of the system to multiple coils. In addition, the use of ferrite shielding and detection and estimation algorithms are considered for the near-field system. The near-field architecture is extended and modified for a midrange system. Finally, far-field power transfer through the atmosphere and the environment are considered through the numerical solution of the radiative transfer equation.
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.
Statement of Responsibility: by Joaquin Casanova.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Lin, Jenshan.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041449:00001


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ELECTR OMAGNETICSYSTEMDESIGNFORWIRELESSPOWER By JOAQUINJESUSCASANOVA ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010

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c 2010 JoaquinJesusCasanova 2

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T omyfamily,fortheirsupportandencouragement 3

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A CKNOWLEDGMENTS Firstandforemost,I'dliketothankDr.JenshanLinforbeingthemosthelpful, understanding,andencouragingadvisorastudentcouldaskfor.Heisoneoftherare professorswhowillgivehisstudentstoexploretheirresearchontheirown,andin doingso,allowsthemtotrulylearn.Thanksarealsoduetomycommitee,Dr.Henry Zmuda,Dr.RobertMoore,andDr.SubrataRoy,fortheirencouragementandinsightful questions.Theyputmeateasewithoutgoingeasyonme.Ioweadebtofgratitude toZhenNingLowfortakingtherststepsonthisproject,forhishelpunderstanding powerampliers,andforhisfriendshipandconversation.Hekeptmesane.Ithank JasonTaylor,AshleyTrowell,andRaulChingafortheirtechnicalsupport,guidance,and friendshipinworkingonthisproject.Thanksalsogoouttomyparentsandmybrother, whoalwayssupportedme,evenifitdidseemlikemylifewasnothingbutmyresearchto theexclusionofallelse.Finally,I'dliketothankFloridaHighTechCorridorandFlorida DepartmentofEnvironmentalProtectionforfundingandsupport. 4

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T ABLEOFCONTENTS page A CKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................8 LISTOFFIGURES .....................................9 ABSTRACT .........................................14 CHAPTER 1INTRODUCTIONTOWIRELESSPOWERTRANSFER .............16 2LOOSELY-COUPLEDNEARFIELDWIRELESSPOWER ............18 2.1Introduction ...................................18 2.2Analysis .....................................18 2.2.1DesignEquationfor C rx ........................20 2.2.2DesignEquationfor L out ........................21 2.2.3DesignEquationfor C out ........................21 2.2.4DesignEquationfor C t .........................22 2.3Tests .......................................23 2.4Conclusion ...................................28 3NEAR-FIELDELECTROMAGNETICANALYSIS .................30 3.1Introduction ...................................30 3.2CoilFields ....................................30 3.3CoilInductance .................................31 3.4CoilParasitics ..................................33 3.4.1RoundConductor ............................33 3.4.1.1Skineffect ..........................33 3.4.1.2Proximityeffect ........................35 3.4.2RectangularConductor .........................36 3.4.2.1Skineffect ..........................37 3.4.2.2Proximityeffect ........................40 3.5LitzWire .....................................42 3.6Regulations ...................................42 3.7Conclusion ...................................43 4OPTIMALPRIMARYCOILDESIGN ........................45 4.1Introduction ...................................45 4.2PlanarWirelessPowerSystem ........................45 4.3CoilDesign ...................................46 4.4Testing ......................................47 5

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4.5 Results .....................................48 4.6Conclusion ...................................49 5M:NANALYSIS ....................................52 5.1Introduction ...................................52 5.2Analysis .....................................52 5.3TestsResults ..................................55 5.3.1Verication ...............................57 5.3.2ReceiverDecoupling ..........................58 5.3.3ImpactonEfciencyandTotalReceivedPower ...........63 5.4Conclusion ...................................65 6OPTIMALPRIMARYCOILDESIGNFORMULTIPLECOILS ..........67 6.1Introduction ...................................67 6.2CoilDesign ...................................67 6.3System .....................................68 6.4Testing ......................................71 6.5Results .....................................71 6.6Conclusion ...................................73 7INCLUSIONOFFERRITES .............................74 7.1Introduction ...................................74 7.2InductanceEstimation .............................75 7.3LossEstimation .................................78 7.4ThicknessandWidthEffects .........................78 7.5ExperimentalEvaluation ............................80 7.6Conclusion ...................................83 8BAYESIANLOAD/FAULTTRACKING .......................84 8.1Introduction ...................................84 8.2Technology/Data ................................86 8.3Theory/Methods ................................86 8.3.1State/MeasurementModel .......................86 8.3.2ParticleFilterAlgorithm ........................87 8.3.2.1Datasetgeneration .....................87 8.3.2.2Initialization ..........................88 8.3.2.3State .............................88 8.3.2.4Measurement ........................88 8.3.2.5Update ............................89 8.3.2.6Estimate ...........................89 8.3.3Tests ...................................90 8.3.4Implementation .............................90 8.4SimulationResults ...............................91 6

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8.5 MeasuredResults ...............................95 8.6Conclusion ...................................96 9MIDRANGEWIRELESSPOWERTRANSFER ..................103 9.1Introduction ...................................103 9.2Analysis .....................................104 9.2.1CoilDesign ...............................105 9.2.2ComponentSelection .........................106 9.2.2.1Series-parallel ........................107 9.2.2.2Series-series .........................108 9.2.2.3T-network ...........................108 9.3PreliminaryTests ................................109 9.3.1RectifyingDiodeEffects ........................109 9.3.2FrequencyandInductanceEffects ..................111 9.3.3TopologyEffects ............................113 9.3.4Sensitivity ................................113 9.4Synthesis ....................................118 9.4.150cmSeparation ............................118 9.4.21mSeparation .............................119 9.5Conclusion ...................................123 10FAR-FIELDWIRELESSPOWERTRANSFER ...................125 10.1Introduction ...................................125 10.2Theory ......................................127 10.3SolutionDetails .................................128 10.4PhysicalProperties ...............................129 10.4.1Soil ....................................129 10.4.2Atmosphere ...............................130 10.4.2.1Gaseouswatervapor ....................130 10.4.2.2Waterdroplets ........................131 10.4.2.3Icecrystals ..........................133 10.4.3Vegetation ................................134 10.5ResultsandDiscussion ............................135 10.5.1AtmosphericLossEstimationforSolarPowerSatellite .......136 10.5.2LossEstimationforRadiofrequency-HarvestingSensorUnder VegetationCanopy ...........................138 10.5.3Flux ...................................139 10.6Conclusions ...................................139 11CONCLUSIONS ...................................142 REFERENCES .......................................143 BIOGRAPHICALSKETCH ................................150 7

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LIST OFTABLES T able page 2-1 Designparameters. .................................23 2-2Componentvalues. ..................................23 4-1Summaryofsystemperformance. .........................49 5-1Componentvaluesfor1and2transmittersystems. ................57 5-2Maximum P rx andmaximum c fordifferentM:Narrangements. .........63 6-1Designparameters. .................................68 6-2Componentvalues. ..................................70 6-3Summaryofsystemperformance. .........................71 7-1Ferriteproperties. ..................................81 7-2Ferriteexperimentalevaluationwithsolenoidcoil. .................82 9-1Componentvalues. ..................................110 9-2Componentvalues. ..................................111 9-3Componentvalues. ..................................112 9-4Componentvalues. ..................................118 9-5Summaryof1mtests. ................................120 10-1ParametervaluesforRTE. ..............................136 8

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LIST OFFIGURES Figure page 2-1 One-to-onewirelesspowersystemblockdiagram. ................18 2-2ClassEdrivingcircuitforawirelesspowersystem. ...............19 2-3Testsetup. ......................................24 2-4Fromtopleft,clockwise: R in \Z tx ,drainvoltagewaveformat R L =10 4 and Q ........................................25 2-5Receivedpowerandtotalefciencyasafunctionof R L .............26 2-6Receivedpowerandtotalefciencyasafunctionof R L ,andtheir95%condence intervals. .......................................27 2-7Efciencyasafunctionof R L ...........................28 3-1CurrentstickforMQSanalysis. ..........................31 3-2MagneticeldcomponentsusingMQSandMoMtechniquesofa1mby1m squarecoil. .....................................32 3-3MagneticeldmagnitudeusingMQSandMoMtechniquesofa1mby1m squarecoil. .....................................32 3-4Conductorcrosssectionshowingeldandcurrentinroundconductorunder skineffect. ......................................34 3-5Conductorcrosssectionshowingeldandcurrentinroundconductorunder proximityeffect. ...................................35 3-6Conductorcrosssectionshowingeldandcurrentinrectangularconductor underskineffect. ..................................37 3-7Conductorcrosssectionshowingeldandcurrentinrectangularconductor underproximityeffect. ...............................40 4-1Transmittertestsetup. ...............................44 4-2Coillayout. ......................................45 4-3Calculatedz-directedmagneticeld,assuming1Acurrent(A/m). .......46 4-4Fieldprobemeasurement(mV). ..........................47 4-5Receivedpower(W)asafunctionofthelocationofthecenterofthereceiving coil. ..........................................49 4-6Power(W)andefciency(%)atloadsfrom10 to2k ............49 9

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5-1 M:Nblockdiagram. .................................52 5-2Coilarrangements. .................................55 5-3Measuredvs.predicted P rx ............................57 5-4Powerspaceplotsfortwo-receivertestswithsmallreceivers. .........58 5-5Powerspaceplotsfortwo-receivertestswithlargereceivers. ..........58 5-6Powerspaceplotforthree-receivertest. .....................60 5-7Powervs.efciencyplotfortwo-receivertestswithsmallreceivers. ......60 5-8Powervs.efciencyplotfortwo-receivertestswithlargereceivers. ......61 5-9Powervs.efciencyplotforthree-receivertestswithsmallreceivers. .....61 5-10Totalreceivedpowerasafunctionof R L ,andits95%condenceintervals ..63 5-11Totalefciencyasafunctionof R L ,andits95%condenceintervals. .....64 6-1Coillayout. ......................................68 6-2Calculatedz-directedmagneticeld,assuming1Acurrent(A/m). .......68 6-3Transmittertestsetup. ...............................69 6-4Overlapofdualtransmittercoils. ..........................69 6-5Receivedpower(W)asafunctionofthelocationofthecenterofthereceiving coil. ..........................................71 6-6Power(W)andefciency(%)atloadsfrom75 to4k ............71 7-1Diagramofferriteshielding. ............................73 7-2Empirical e predictions(redx)andobservations(bluecircle). ........76 7-3Flux-eldhyteresisloop ..............................76 7-4Effectsofthicknessandrelativewidthofferriteoninductance. .........78 7-5Effectsofthicknessandrelativewidthofferriteonresistance. .........79 8-1Generatedmeasurementsin( V in ,I DC )space. ...................90 8-2True(blue)andestimated(red)modeforN=10,withresampling. .......91 8-3True(blue)andestimated(red)modeforN=100,withresampling. .......92 8-4True(blue)andestimated(red)modeforN=1000,withresampling. ......93 10

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8-5 True(blue)andestimated(red)modeforN=10,withoutresampling. ......94 8-6True(blue)andestimated(red)modeforN=100,withoutresampling. .....95 8-7True(blue)andestimated(red)modeforN=1000,withoutresampling. ....96 8-8RMSEofmodeandstates. .............................97 8-9Modeandchargeestimatevariance,withresampling. ..............97 8-10Modeandchargeestimatevariance,withoutresampling. ............98 8-11Testofdifferentmodesin( V in ,I DC )space,inrealsystem. ............98 8-12True(blue)andestimated(red)modeforN=100,withoutresampling,inreal system. .......................................99 8-13Predictedandobservedpower,resistance,andinputvoltage,andDCinput currentforN=100,withoutresampling,inrealsystem. .............99 8-14True(blue)andestimated(red)modeforN=1000,withoutresampling,inreal system. .......................................100 8-15Predictedandobservedpower,resistance,andinputvoltage,andDCinput currentforN=1000,withoutresampling,inrealsystem. .............100 8-16True(blue)andestimated(red)modeforN=10000,withoutresampling,in realsystem. .....................................101 8-17Predictedandobservedpower,resistance,andinputvoltage,andDCinput currentforN=1000,withoutresampling,inrealsystem. .............101 9-1MidrangeclassEseries-parallelarchitecture. ..................106 9-2MidrangeclassEseries-seriesarchitecture. ...................107 9-3MidrangeclassETnetworkarchitecture. .....................108 9-4Diodeeffectsonsystemperformance. ......................109 9-5Frequencyandinductanceeffectsonsystemperformance. ...........110 9-6Topologyeffectsonsystemperformance. .....................111 9-7Effectof D d ,and f ontotalefciencyat N =8. .................113 9-8Effectof D d ,and f onreceivedpowerat N =8. .................113 9-9Effectof N f ,and D ontotalefciencywhere D = d ..............114 9-10Effectof N f ,and D onreceivedpowerwhere D = d .............114 11

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9-11 Coiloffseteffectsonsystemperformance. ....................116 9-12Efciencyatnominalcomponentvalues(blackline)and95%condenceintervals. 117 9-13Poweratnominalcomponentvalues(blackline)and95%condenceintervals. 118 9-1450cmsystemperformance. ............................120 9-151msystemsetup. ..................................121 9-161msystemperformance. .............................122 10-1Anexampleofaradiofrequency(RF)harvestingwirelesssensornode[3]. ..124 10-2AnillustrationoftheSolarPowerSatellite(SPS)concept[61]. .........124 10-3Atmospheremodelschematicusedinthischapter[63]. .............125 10-4Canopymodelschematicusedinthischapter[64]. ...............125 10-5Clouddropletdistributionandabsorptioncrosssection. .............131 10-6Raindropletdistributionandabsorptioncrosssection. .............132 10-7Clouddropletdistributionandscatteringcrosssection. .............133 10-8Raindropletdistributionandscatteringcrosssection. ..............134 10-9Icespheredistributionandabsorptioncrosssection. ..............135 10-10Icespheredistributionandscatteringcrosssection. ...............136 10-11Logintensitydistributionthroughcloud. ......................137 10-12Logintensitydistributionthroughrain. .......................138 10-13Logintensitydistributionthroughice. .......................139 10-14Logintensitydistributionthroughvegetationwithdiffuseandspecularlower boundary. ......................................140 10-15Fluxprolethroughdifferentmedia. ........................140 12

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Abstr actofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ELECTROMAGNETICSYSTEMDESIGNFORWIRELESSPOWER By JoaquinJesusCasanova May2010 Chair:JenshanLin Major:ElectricalandComputerEngineering Wirelesscommunicationstechnologyhasfreedelectronicsfromcommunication cables.Thenaturalnextstepistocutthelastwireofportablewirelessdevices, thepowercable.Wirelesspowersystemswouldpermitchargingmanydifferent devicesequippedwithreceivingcoils,inadditiontodeliveringpowerthroughrooftops andthroughtheatmosphere.Theapproachestowirelesspowertransfercanbe categorizedasnear-eld,midrange,andfar-eld.Todate,thelatterisstillimpractical forconsumerapplicationsduetothehighpowerandlargeantennarequirement necessarytoachievelevelsofpowercomparabletoawallsupply.Ontheotherhand, near-eldinductivecouplinghasmorepromiseasawirelesspowertechnologyfor chargingbattery-operateddevices.Midrangepowertransferhasthemostpotential forapplicationssuchasvehiclechargingandpowertransmissionthroughwallsand rooftops.Far-eldapplicationsincluderadiofrequency(RF)energyharvestingand transmissionofpowerfromspace. Thisdissertationpresentsseveralapsectsofthedesignandtestingofwireless powersystems.Circuittopologyandelectromagneticdesignofanear-eldsystemis considered,aswellastheextensionofthesystemtomultiplecoils.Inaddition,the useofferriteshieldinganddetectionandestimationalgorithmsareconsideredforthe near-eldsystem.Thenear-eldarchitectureisextendedandmodiedforamidrange 13

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system. Finally,far-eldpowertransferthroughtheatmosphereandtheenvironmentare consideredthroughthenumericalsolutionoftheradiativetransferequation. 14

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CHAPTER 1 INTRODUCTIONTOWIRELESSPOWERTRANSFER Thelargenumberofbatteryoperatedconsumerelectronicsandtheassociated tangleofwall-wartchargershasgeneratedinterestindesigningasingle,convenient chargingplatform[ 1].Wirelessbatterychargingsystemswouldpermitchargingmany differentdevicesequippedwithreceivingcoilsandcutthelastwireofportablewireless devices.Severaltechniquesexistfortransmittingpowerbyelectromagneticelds. Theydifferprimarilybythedistancebetweenreceiverandtransmitter( D )andthe characteristicdimensionofthetransmitter( d ),relativetowavelength( ),andbytheir typicalpowerlevelsandapplications[ 2].Far-eld,orradiative,powertransferoccurs whenthedistancebetweenthetransmitterandthereceiverexceedstheRayleigh distance, D > 2d 2 = and d >.Atnearerdistances,( D d )isconsideredmidrange. Where D << d electromagneticcouplingisconsiderednear-eld. Far-eldpowertransferisimpracticalforconsumerapplicationsduetothehigh powerandlargeantennarequirementnecessarytoachievelevelsofpowercomparable toawallsupply[ 2].Twomajorapplicationsofradiativewirelesspowertransfer(WPT) areinambientradiofrequency(RF)harvestingandtheSolarPowerSatellite(SPS). Theideabehindthersttechniqueistoconverttheradiowavesfromcommunications intopower[3 4].TheSPSisanideawhichcameaboutinthelate1960s[ 5 ],wherethe principleistocollectsolarenergyinspaceusingasatelliteandbeamittoareceiving stationonearth. Todate,midrangehasshownpromiseintheorybutpracticaltestsathighpower levelsarelackingintheliterature.Thisisanappropriaterangeforapplicationssuch aswirelesschargingofelectricvehicles[ 6].Evanescentcouplngemploysresonant structurestoensureastronglinkbetweentransmitterandreceiver[ 7]inthisregime. Ontheotherhand,near-eldinductivecouplinghasmorepromiseasaconsumer-level wirelesspowertechnology.ThisisthefamiliarprinicpleusedintransformersandAC 15

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and DCmachinery.ItcanalsobeexploitedforWPTtochargebattery-operateddevices [811]. Becauseofthis,thisdissertationwillfocusonthedesignofanear-eldsystem: Chapters 2 -7 considerthenear-eldsystem;Chapters 9-10 considerthemidrange andfar-eldsystems,respectively.Chapter 2 discussesanear-eldwirelesspower systemcircuitarchitectureanddesignrulesforthiscircuit.Chapter 3 discusses theelectromagnetictheorybehindcalculationofcoilpropertiesofinductanceand resistance.AtechniquefortheoptimaldesignoftheprimarycoilisdiscussedinChapter 4.Thearchitectureisextendedtoasystemwithanarbitrarynumberoftransmittersand receiversinChapter 5 .Coildesignformultipletransmittingcoilsinparallelisdiscussed inChapter 6.Chapter 7 discussestheuseandevaluationofferriteshielding.Chapter 8 describesthedevelopmentandtestingofaBayesiantrackingalgorithmforreceiver discriminationandchargestatusdeterminationinthenear-eldsystem.Theextension ofthesystemandcoildesigntomidrangeispresentedinChapter 9.Chapter 10 describestheuseofradiativetransfermodelingforestimatinglossesoffar-eldwireless powertransmission.Chapter 11 presentssomeconcludingremarks. 16

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CHAPTER 2 LOOSELY-COUPLEDNEARFIELDWIRELESSPOWER 2.1Introduction Fig. 2-1 showsablockdiagramforageneralizedwirelesspowersystemandthe circuitdiagramisshowninFig. 2-2.TheinverterisaclassEamplier[ 12]drivenby alow-powerclockat240kHz,followedbyaseries-parallelimpedancetransformation network[ 13 ].Selectingthevaluesof C rx L out C out ,and C t foroptimumperformanceof thewirelesspowersystempresentsachallenge.[ 14]showsthedesignmethodolgyfor asimilararchitecturewithclosedloopcontrol.[ 15 ]demonstrateshowtochoosedesign valuesforaclassEwithoutrelyingonRaab'swaveformequations;however,itinvolves numericalrootnding.[ 16]presentsaselectiontechniqueforanopen-loopsystemthat reliesonsweepingcomponentvaluesnumericallyuntiltheimpedanceanddrainvoltage satisfycertainconstraints.Whilethismethodsuccessfullyndsappropriatecomponent values,itistimeconsuming.Thischapterderivessimpleformulasfortheoptimum componentvaluesbyapplyingthesameconstraints. 2.2Analysis Theoptimumvaluesfor C rx L out C out ,and C t canbederivedbyapplyingseveral constraintsonthesystemsresponsetothevariableloadresistance R L .Inthisanalysis, itisassumedthatthecomponentsarelossless.Inaddition,assumptionsaremade abouttheclassEtoallowuseofRaab'sequations[ 17 ],namelythatthetransistoris Figure 2-1.One-to-onewirelesspowersystemblockdiagram. 17

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Figure 2-2.ClassEdrivingcircuitforawirelesspowersystem. aperfectswitchandthatthechokeinductanceisinnite.Beforethesederivationsitis necessarytohaveexpressionsforreceiverimpendance, Z rx ,inputimpedancelooking intothetransmittercoil, Z in ,andimpedancelookinginto L out Z tx Z rx canbeexpressed asfollows: Z rx = R rx + jX rx = R L jj C rx (2) = R L j C rx R 2 L 1 + 2 R 2 L C 2 rx (2) Z in canbefoundbyexaminingthecouplingequations: V 1 = j L 1 I 1 + j MI 2 V 2 = j MI 1 + j L 2 I 2 (2) where L 1 ,L 2 ,and M ,aretransmittercoil,receivercoil,andmutualinductance, respectively. Z in is V 1 /I 1 18

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Z in = R in + jX in = 2 M 2 R rx R 2 rx + ( L 2 + X rx ) 2 + j L 1 2 M 2 (! L 2 + X rx ) R 2 rx + ( L 2 + X rx ) 2 (2) Z tx isjust Z in withadditionalseriesreactancefrom C out and L out : Z tx = R tx + jX tx = R in + j L out 1 C out + X in (2) 2.2.1 DesignEquationfor C rx Selectionof C rx isdeterminedonthebasisofefciencyandqualityfactor Q .Ifthe realpartof Z in istoolowcomparedtothecoilparasitics,thesystemwillbeinefcient.If itistoolarge,itisdifculttoget Q highenoughforclassEoperation(about1.78[ 18]). Byforcingthepeakrealpartof Z in tobeaspeciedvalue( R 0 ),acompromisebetween efciencyand Q canbereached.Toderivewhich C rx forcesthemaximumrealpartof Z in tobe R 0 ,the R L correspondingtothepeakvalueisfoundbysetting @ R in @ R L to zero.This yieldsapolynomialofdegreesix,wherefouroftherootsarecomprisedbyadouble conjugatepairandcanthusbeignored R L = j C rx (2) There aretworealroots R L = L 2 1 2 L 2 C rx (2) which canbesubstitutedbackintoEq.( 2);thentherealpartissetto R 0 19

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R in = M 2L 2 M 2 C rx L 2 1 = R 0 (2) Solving for C rx yieldstworoots C rx = R 0 L 2 1 2 M 2 R 0 2 L 2 2 (2) The negativerootgivesa C rx whichensuresthat Z in phasewillincreasewith increasing R L ,whichhasthedesirableeffectofloweringpowerdeliveryathighload resistance.Thisisdesirablebecauseinthecaseofadevicebeingcharged,highload resistance(thousandsof )correspondstofully-chargedconditionandthuslowpower requirement. 2.2.2DesignEquationfor L out Thepurposeof L out istoensurethecircuithasaminimum Q highenoughfor properfunctioningoftheclassE. Q issmallestwhentherealpartof Z in ishighest,at R 0 Since L out contributesthelargestpartofthereactanceof Z tx Q L out R 0 (2) L out is found: L out = 1 QR 0 (2) 2.2.3DesignEquationfor C out C out bringstherangeofthephaseof Z tx toarangewhichallowsZVSoperation oftheclassEandmaximumefciency.From[ 19],thisphaserangeis40 o to70 o .By settingtheminimumphasetoaspeciedvalue, ,efcientoperationcanbeachieved. Thelocationoftheminimumphaseiswhere @ \ (Z tx ) @ R L = 0 (2) whichyieldsaquadraticequationin R L withtheroots 20

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R L = 2 R 0 L 2 M 2 s C out (! L 1 (k 2 1) R 0 Q ) 1 C out (! L 1 (k 2 1) R 0 (Q + 2)) 1 (2) where k = q M 2 L 1 L 2 SubstitutingEq.( 2)intoEq.( 2)andsetting tan ()= X tx R tx and squaringtoeliminatetheradicalinthenumeratoryieldsaquadraticequationin C out ,withtworealroots. C out = 1 f! L 1 (1 k 2 )+ R 0 (Q +1 sec ( )) g fR 2 0 (Q 2 +2 Q tan () 2 )+2 QR 0 L 1 (1 k 2 ) + 2 L 2 1 (1 k 2 ) 2 +2 L 1 R 0 (1 k 2 )g 1 (2) Thegreaterroot,correspondingtopositive ,yields C out = 1 L 1 (1 k 2 ) + R 0 (Q +1 sec ()) (2) aftersimplication. 2.2.4DesignEquationfor C t Finally, C t isselectedtoguaranteezerovoltageswitching(ZVS)operationofthe classE.From[ 17],thisoptimumvalueof C t ,givenaloadresistance R is C t = 2 1 (1 + 2 4 )R (2) Since theloadresistanceinthewirelesspowersystemisvariable, C t isselected basedonthemaximum R ,whichforthecircuitunderconsiderationisthemagnitudeof Z tx as R L increasestoinnity.Takingthislimit, 21

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T able2-1.Designparameters. P arameter V alue L 1 34.58 H L 2 4.05 H M 1.65 H R 0 7.5 Q 2 65 o T able2-2.Componentvalues. Component Calculated Measured C rx 100.55 nF 100.00 nF C out 11.90 nF 11.57 nF C t 15.15 nF 14.55 nF L out 9.95 H 9.52 H R = L out + 1 C out 2 M 2 C rx 2 C rx L 2 1 (2) and substitutinginthederivedvaluesfortheothercomponents(Eqs.( 2),( 2), (2)),theoptimum C t isfound: C t = 2 1 (1 + 2 4 )(1 + sec ())R 0 (2) 2.3Tests Havingderivedtheoptimumcomponentvaluesforagiven L 1 L 2 M R 0 Q ,and thissectiondemonstratestheperformanceofthesystem. Atestsystemwasbuilt,consistingofa16cmby18cm,13turn,spiraltransmitting coil,designedbythetechniquedescribedin[ 20 ],andarectangular4cmby5cm, 6turn,receivingcoil.Bothcoilswereconstructedof100strand,40AWGLitzwire tominimizecoilparasitics.Fig. 2-3 showsapictureofthetestsystem.Table 2-1 givesvaluesoftheinductances L 1 L 2 M ,anddesignparameters R 0 Q ,and R 0 waschosenas7.5 basedonthetotaltransmittingandreceivingcoilparasitics whichamountedtoabout0.5 .Ingeneral,selectionof R 0 issystem-dependentbutit 22

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Figure 2-3.Testsetup.Topisadiagramshowingthecoils,whereredisthereceiverand blueisthetransmitter.Bottomisaphotographofthecoils. 23

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Figure 2-4.Fromtopleft,clockwise: R in \Z tx ,drainvoltagewaveformat R L =10 4 ,and Q shouldbeatleastanorderofmagnitudehigherthantheparasitics.Howlargean R 0 is acceptabledependsontheavailabilityofinductorsofsufcientsizetoachieveminimum Q .Minimum Q and valueswerechosenbasedonallowablevaluesgivenin[ 18 19 ], withadditionalbuffertotoleratesomedeviationsofrealcomponents.Basedonthe proposedmethodandequationsinSection 2.2,theoptimumcomponentvalueswere calculated.Table 2-2 givesthecalculatedcomponentvaluesandalsoliststhemeasured valuesoftheactualcomponentsbeingused.Fig. 2-4 showsthecalculated R in \ Z tx drainvoltagewaveform,and Q ,demonstratingthatthedesiredconstraintsaremetusing thecomponentselectionformulas. Oneofthekeychallengesforawirelesspowersystemistohavedesirable performancerespondingtovariableload.Toevaluatethesystemperformancewith regardstopowerandefciency, R L wassweptfrom60to4000 bymeansofan electronicload.TheDCvoltageandcurrentweremeasuredattheloadandatthe supply.Supplyvoltagewas12V. 24

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Figure 2-5.Receivedpowerandtotalefciencyasafunctionof R L .Circlesare measuredvaluesandthesolidlineissimulated,usingthecalculated componentvalues. Fig. 2-5 showsthereceivedpowerandtotalefciencyversusloadresistance,and thesimulatedvaluesofpowerandefciency,usingtheideal,calculatedcomponent values.Thesystemhaspowerdeliveryofover3.7W,andpeakefciencyover66%. TheimportantfeatureinFig. 2-5 isthetrendofdecreasingreceivedpowerandtotal efciencywithincreasing R L ,whichisguaranteedbythecomponentselection.Theideal performanceisclosetotherealsystem.Powerandefciencyarelowerby5%-10%in theactualsystem,primarilyduetothedeviationof L out and C out intherealcomponents. Deviationsfromtheiridealvaluescausethe and Q toshiftwhichhasasubstantial effectontheclassEefciency. Tofurtherinvestigatethesensitivitytocomponentselection,aMonteCarlo simulationwasrun,assumingthecomponentsarenormallydistributed,withmeans givenbythederivedcomponentformulasandwithstandarddeviations, ,suchthat 3 isthecomponenttolerance.Thesesimulationswerecarriedoutattolerancelevelsof 25

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Figure 2-6.Receivedpowerandtotalefciencyasafunctionof R L ,andtheir95% condenceintervalswith5%componenttolerance(circles),10%component tolerances(pluses),and20%componenttolerances(squares). 5%,10%,and20%.Fig. 2-6 showsthe95%condenceintervalsforreceivedpower andtotalefciencyatthethreetolerancelevels.Ascanbeseen,thepowerisskewed low,withtolerancesofabout+35/-25%at5%tolerance,+115/-45%at10%tolerance, and+425/-70%at20%tolerance.Theefciencyisskewedveryhigh,withtolerances ofabout+3.4/-18%at5%tolerance,+3.6/-38%at10%tolerance,and+4.3/-70%at 20%tolerance.Thisskewlowinthepowercondenceintervalsandskewhighinthe efciencycondenceintervalsshowsthatthesystemisnotoptimizedformaximum powerdeliverybutratherefciency.Thismakessense,asalloftheconstraints( R 0 Q ,andZVS)usedforcomponentselectionarechosentomaximizetheefciencyofthe classE.Attoleranceof5%orless,thecondenceintervalshowsthattheperformance isstilldecent.Forgreatertolerances,thepotentialvariabilityofperformanceisprobably unacceptableformostapplications. 26

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Figure 2-7.Efciencyasafunctionof R L .SquaresareclassEefciency,plusesare couplingefciency,andcirclesaretotalefciency. Fig. 2-7 showstheamplierefciency,couplingefciency,andtotalefciency. AmplierefciencyisACtransmittingpoweroverDCinputpower;couplingefciency isDCpowerattheloadoverACtransmittedpower;andtotalefciencyistheproduct ofthesetwoefciencies.FromFig. 2-7,thecouplingefciencycomprisesmostofthe losses.Thecouplingefciencydecreaseswithincreasingloadresistancebecauseas R L increases, R in decreases,whiletheparasiticsremainthesame,somorepoweris dissipatedthroughtheparasitics.Theamplierefciencypeakswhere \ Z tx isinthe 40 o 70 o range,whichbydesignisintheneighborhoodofthephaseminimum. 2.4Conclusion Thischapterhaspresentedasetofdesignequationsforoptimizingtheperformance ofaclassEamplierusedininductivelycoupledwirelesspowersystem.Byapplying constraintsontherealpartoftheinputimpedancetotheprimarycoil,thephaseofthe inputimpedance,theminimum Q ,andthedrainvoltagewaveform,componentscan beselectedtoguaranteedesirableoperatingcharacteristicsofthesystem,namely,the 27

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ZVS operationoftheclassEandthetrendofdecreasingpowerreceivedwithincreasing loadresistance.Theproposedoptimizationmethodwastestedinasystemcomposed ofa16cmby18cmprimarycoilanda4cmby5cmsecondarycoilwithavariable load.Thesystemshowspowerdeliveryofover3.7W,andpeakefciencyover66%, inadditiontothedesirabletrendofdecreasingpowerandefciencywithrespectto increasingloadresistance. 28

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CHAPTER 3 NEAR-FIELDELECTROMAGNETICANALYSIS 3.1Introduction Theperformanceofanear-eldwirelesspowersystemdependsheavilyonthe electromagneticpropertiesoftheprimaryandsecondarycoils.Inparticular,theelds producedbythecoil,thecoilinductance,andthecoilparasiticsareimportanttoknow. Thischapterderivesanalyticallythesequantitiesforroundandrectangularconductors foruseinsystemdesign. 3.2CoilFields Atfrequencieslessthan500kHz,insteadofsolvingMaxwell'scoupledequations, itisstillsufcientlyaccuratetocalculatethemagneticeldusingthemagnetostatic solution,thatis,theBiot-SavartLaw.Thisisknownasthemagnetoquasistatic(MQS) solution[ 21 ].TheanalyticalMQSsolutionforalineofcurrentispresentedhere(see Fig. 3-1).Theeldsproducedbyapolygonalcoilcanbeconstructedbysuperposition. H = 1 4 Z V 0 ~ J ( ~ r 0 ) ^ i r 0 r j ~ r ~ r 0 j 2 dV 0 (3) = I 4 Z c b ~ c ~ a d ja j( 2 + r 2 0 ) 3 =2 (3) = I 4 ~ c ~ a ja j r 2 0 ( 2 + r 2 0 ) 1 =2 c b (3) = I 4 ~ c ~ a j ~ c ~ a j 2 ~ a ~ c jc j ~ a ~ b jb j (3) T otesttheMQSanalyticalsolution,itwascomparedtoamethodofmoments (MoM)solution[ 22 ]calculatedusingtheNumericalElectromagneticsCode(NEC)[ 23]. Inparticular,asingleturn,1mby1msquarecoilwastestedusingbothtechniques. Theeldswerecalculatedata2mby2mplane5cmabovethecoil.Whilefrequencyis notusedinMQS,itisinMoM,andthefrequencyusedinthiscasewas240kHz. 29

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Figure 3-1.CurrentstickforMQSanalysis. Fig. 3-2 showsthemagnitudeofthetheeldcomponents( H x ,H y H Z )forbothMQS andMoMtechniques.Fromtheplots,itisevidentthatthedistributionsaresimilar.The magnitudeofthedifference,intermsofrootmeansquaredifferenceofthex-component is6.3239 10 5 A/m,ofthey-componentis9.4997 10 5 A/m,andofthez-component is8.4331 10 5 .Thepeakeldmagnitudeinbothcasesis1A/m. 3.3CoilInductance Theinductancematrix M ij relatestheuxfromaprimarycoil i throughasecondary coil j totheprimary'scurrent.[ 24]presentsmanyinductanceformulasfordifferent geometries.ThemostgeneralforalamentarycoilistheNeumannformula: M ij = 0 4 I c i I c j ~ ds i ~ ds j j ~ R ij j (3) The selfinductanceissimilar: M ii = 0 4 I c i I c i ~ ds i ~ ds i j ~ R ii j jR j a =2 + L p ,ii (3) where L p ,ii is theinternalinductanceoftheconductorand a istheradius. 30

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Figure 3-2.MagneticeldcomponentsusingMQSandMoMtechniquesofa1mby1m squarecoil. Figure 3-3.MagneticeldmagnitudeusingMQSandMoMtechniquesofa1mby1m squarecoil. 31

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3.4 CoilParasitics Theeffectofthedistributionofcurrentwithintheconductorismagniedathigher frequency.Internalresistanceandinductancedependonthedistributionofcurrent. Therearetwomainmechanismsforhighfrequencylossesandinternalinductance: theskineffectandproximityeffect.Theskineffectiswhencurrentdistributionis mostlytowardsthesurfaceoftheconductor,raisingtheresistance.Proximityeffectis whenmagneticeldsfromnearbyconductorsinduceeddycurrentsontheconductor surface,increasingthepowerdissipation.Thesetwoeffectsarediscussedinthe contextoftransformerdesignintheliterature;thereareanalytical[ 25 26 ]andempirical [27 28 ]treatmentsoftheseeffects.However,theresultspublishedintheliterature areincompleteinthattheexactcurrentdistributionsarenotgiven,andneitheristhe effectonconductorinternalinductance.Inaddition,nopaperderivestheseeffectsfor aconductorofarbitraryrectangularcrosssection,suchasaPCBtrace.Thissection derivesthecurrentdistributionandresistanceandinductanceunderskinandproximity effectsforbothroundandrectangularconductors. 3.4.1RoundConductor [ 29]presentsapartialdescriptionoftheskineffectinroundconductor.Thissection presentsacompletedescriptionofskinandproximityeffects.Bothskinandproximity effectparasiticsforaroundconductorarederivedthroughapplicationofMaxwell's equations. 3.4.1.1Skineffect Fig. 3-4 showsthewirecrosssectionandboundaryconditions.Usingthecurrent distributionequation,separationofvariables,andthesurfaceelectriceldandsymmetry 32

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Figure 3-4.Conductorcrosssectionshowingeldandcurrentinroundconductorunder skineffect. asboundaryconditions,thetotalcurrentcanberelatedtothesurfaceelectriceld E 0 : r 2 J z = j J z (3) 2 = j (3) @ 2 J z @ r 2 + 1 r @ J z @ r + 2 J z = 0 (3) J z = C 1 J 0 ( r ) (3) J z = E 0 J 0 ( r ) J 0 ( r 0 ) (3) E z = E 0 J 0 ( r ) J 0 ( r 0 ) (3) H = 1 j @ E z @ r (3) = E 0 J 0 0 ( r ) J 0 ( r 0 ) (3) I ~ H d ~ l = I = 2 r 0 H ( r 0 ) (3) I = E 0 J 0 0 ( r ) J 0 ( r 0 ) (3) where J z is current, H is -directedmagneticeld, E z isz-directedelectriceld,and I is totalcurrent. 33

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Figure 3-5.Conductorcrosssectionshowingeldandcurrentinroundconductorunder proximityeffect. Thenextstepsrelatethetotalcurrenttotheparasitics.Withthedenitionofinternal impedance, Z = E z (r 0 ) I (3) = R + j L (3) R s = 1 (3) = 1 p f (3) = p 2 r 0 (3) R p = R s p 2 r 0 Ber ( )Bei 0 ( ) Bei ( ) Ber 0 ( ) Ber 0 2 ( ) + Bei 0 2 ( ) (3) L p = R s p 2 r 0 Ber ( )Ber 0 ( ) Bei ( )Bei 0 ( ) Ber 0 2 ( ) + Bei 0 2 ( ) (3) where Bei and Ber aretheimaginaryandrealKelvinfunctions. 3.4.1.2Proximityeffect Similarly,proximityeffectcanbehandledbyapplyingtheboundaryconditionofan imposedtangentialsurfacemagneticeld.Fig. 3-5 showsthewirecrosssectionand boundaryconditions.Beginningwiththecurrentdistributionequation,theeldscanbe relatedtotherealandreactivepower.Thepowertermsmayberelatedtotheparasitics 34

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through thetotalcurrent: 1 r 2 @ 2 J z @ 2 + 1 r @ @ r r @ J z @ r + 2 J z = 0 (3) J z = C 1 J 1 ( r )cos( ) (3) J z = H 0 J 1 ( r ) J 1 ( r 0 ) cos( ) (3) E z = H 0 J 1 ( r ) J 1 ( r 0 ) cos( ) (3) j ~ H = r E z (3) ~ H = H 0 2 ^ r r J 1 ( r ) J 1 ( r 0 ) sin( ) + ^ J 0 1 ( r ) J 1 ( r 0 ) cos( ) (3) P real = 2 Z 2 0 Z r 0 0 jE j 2 rdrd (3) P imag = 2 Z 2 0 Z r 0 0 jH j 2 rdrd (3) P real = 2 H 2 0 Z r 0 0 J 1 ( r ) J 1 ( r 0 ) 2 rdr (3) P imag = 2 H 2 0 2 Z r 0 0 1 ( r ) 2 J 1 ( r ) J 1 ( r 0 ) 2 + J 0 1 ( r ) J 1 ( r 0 ) 2 rdr (3) R p = 2P real I 2 p (3) L p = 2P imag I 2 p (3) 3.4.2 RectangularConductor Arectangularconductorofwidth w andthickness t ,suchasaprintedcircuitboard (PCB)tracecanbehandledwiththecurrentdistributionPDEinCartesiancoordinates. Similarderivationsasintheprevioussectioncanbecarriedoutinordertoderivethe parasiticsforarectangularcrosssectionconductor.Thefollowingsectionsderivethe parasiticsthroughapplicationofMaxwell'sequations. 35

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Figure 3-6.Conductorcrosssectionshowingeldandcurrentinrectangularconductor underskineffect. 3.4.2.1Skineffect Fig. 3-6 showstherectangularcrosssectionandboundaryconditionsusedinthis section.Theboundaryconditionisoneofaconstanttangentialelectriceld.Usingthe reactiveandrealpowerswithinthecrosssection,andthetotalcurrent,theparasitics maybederived.ApplyingthePDEandseparationofvariables,wherethetotalcurrent distributionmustbehandledbysuperpositionoftwosolutions,oneassociatedwitha homogeneousx-direction( J z 1 )andanotherassociatedwithahomogeneousy-direction (J z 2 ): 36

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r 2 J z = 2 J z (3) @ 2 J z @ x 2 + @ 2 J z @ y 2 + 2 J z = 0 (3) J z = J z 1 + J z 2 (3) J z 1 = X (x ) Y ( y ) (3) X (x )= A cos( x )+ B sin( x ) (3) X 0 (0)=0 (3) X (w =2)=0 (3) X = A cos( n x ) (3) n =(2 n +1) w (3) Y (y ) = C cosh( y )+ D sinh( y ) (3) 2 n = 2 n 2 (3) Y 0 (0)=0 (3) Y (t =2)X (x )= E 0 (3) Y n = a n cosh( n y ) (3) a n = R w = 2 0 E 0 cos( n x )dx cosh( n t = 2) R w =2 0 cos 2 ( n x ) (3) a n = 4( 1) n (2 n + 1)cosh( n t = 2) (3) J z 1 = E 0 1 0 a n cosh( n y )cos( n x ) (3) 37

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J z 2 = X (x )Y (y ) (3) Y (y ) = A cos( y )+ B sin( y ) (3) Y 0 (0)=0 (3) Y (t = 2)=0 (3) Y = A cos( n y ) (3) n =(2 n +1) t (3) X (x ) = C cosh( x )+ D sinh( x ) (3) 2 n = 2 n 2 (3) X 0 (0)=0 (3) X (w = 2) Y (x )= E 0 (3) X n = b n cosh( n x ) (3) b n = R t = 2 0 E 0 cos( n y )dy cosh( n w =2) R t = 2 0 cos 2 ( n y ) (3) b n = 4( 1) n (2 n + 1)cosh( n w =2) (3) J z 2 = E 0 1 0 b n cosh( n x )cos( n y ) (3) 38

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J z = E 0 1 0 4( 1) n (2 n + 1) (3) cosh( n y )cos( n x ) cosh( n t = 2) + cosh( n x ) cos( n y ) cosh( n w = 2) (3) j H x = @ J z @ y (3) j H y = @ J z @ x (3) H x = E 0 2 1 0 4( 1) n (2 n + 1) (3) n sinh( n y )cos( n x ) cosh( n t = 2) n cosh( n x ) sin( n y ) cosh( n w = 2) (3) H y = E 0 2 1 0 4( 1) n (2 n + 1) (3) n cosh( n y )sin( n x ) cosh( n t = 2) n sinh( n x ) cos( n y ) cosh( n w =2) (3) I 4 = Z t = 2 0 Z t = 2 0 J z dxdy (3) I p = E 0 1 0 16(1) n (2 n + 1) tanh( n t = 2) w +tanh( n w = 2) t (3) P real = 2 Z 2 0 Z r 0 0 jE j 2 rdrd (3) P imag = 2 Z 2 0 Z r 0 0 jH j 2 rdrd (3) R p = 2P real I 2 p (3) L p = 2P imag I 2 p (3) 3.4.2.2 Proximityeffect Fig. 3-7 showstherectangularcrosssectionandboundaryconditionsusedinthis section.Theboundaryconditionisoneofanarbitrary,constant,tangential,magnetic eld.Usingthereactiveandrealpowerswithinthecrosssection,andthetotalcurrent, theparasiticsmaybederived.Proximityeffectcanbehandledwiththeimpositionof 39

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Figure 3-7.Conductorcrosssectionshowingeldandcurrentinrectangularconductor underproximityeffect. tangentialmagneticeldontheverticalsides,andoddsymmetryaboutthey-axis: r 2 J z = 2 J z (3) @ 2 J z @ x 2 + @ 2 J z @ y 2 + 2 J z = 0 (3) J z = X (x )Y (y ) (3) Y (y )= A cos( y )+ B sin( y ) (3) Y 0 (0)=0 (3) Y (t =2)=0 (3) Y = A cos( n y ) (3) n =(2 n +1) t (3) X (x ) = C cosh( x )+ D sinh( x ) (3) 2 n = 2 n 2 (3) X (w = 2) Y (x )= E 0 (3) X n = a n sinh( n x ) (3) 40

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2 H o y = @ J z @ x x = w = 2 (3) a n = 2 H 0 y R t = 2 0 E 0 cos( n y )dy n cosh( n w = 2) R t = 2 0 cos 2 ( n y ) (3) a n = 2 H 0y 4( 1) n +1 (2 n + 1) n cosh( n w =2) (3) J z = 1 0 a n sinh( n x )cos( n y ) (3) I 4 = Z t =2 0 Z t =2 0 J z dxdy (3) I p = E 0 1 0 16( 1) n (2 n + 1) tanh( n t = 2) w +tanh( n w =2)t (3) P real = 2 Z 2 0 Z r 0 0 jE j 2 rdrd (3) P imag = j 2 Z 2 0 Z r 0 0 jH j 2 rdrd (3) R p = 2 P real I 2 p (3) L p = 2 P imag I 2 p (3) 3.5 LitzWire Tomitigatethehighfrequencyparasiticsdescribedintheprevioussections,Litz wirecouldbeusedasthecoilconductor[ 30].Litzwireisstrandedwirewherethe strandsareinsulatedfromoneanother.Sincethesizeoftheindividualconductorsis muchlessthantheskindepth,theskinandproximityeffectsareminimized.Typically, Litzwireisspeciedintermsofthenumberofstrandsandthegaugeoftheindividual wires.Forinstance,/40Litzwireis100strandsof40AWGwire. 3.6Regulations Astheapplicationofwirelesspowerconsideredinthisdissertationislargely consumerelectronics,somediscussionofthehealthandhumansafetyaspects,aswell asappropriatefederalregulations,isinorder.Thissectionsummarizessomeregulatory constraints. 41

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FCC Part18[ 31 ]concernsunlicensedintential,unintential,orincidentalradiators forindustrial,medical,orscitic(ISM)andnon-ISMequipment.AccordingtoPart 18,operationwithinspecicsearchandrescuebands(490-510kHz,2170-2194 kHz,8354-8374kHz,121.4-121.6MHz,156.7-156.9MHz,and242.8-243.2MHz). Additionallythereareeldstrengthlimitsfordifferentfrequencybandsandapplications andconductedemissionlimits.IfawirelssdeviceisdesignedtoworkinnonISMbands thetheeldstrengthlimitis15 V/mat300mandtheconductedemissionlimitis66-56 dBV(quasi-peak)and56-46dB V(average) FCCPart15[ 32 ]isconcernedwithradiofrequencydevices.Specically,subpartB concernsunintentionalradiators.Thereareradiatedandconductedemissionlimits.For conductedemissions,thelimitis66-56dB V(quasi-peak)and56-46dB V(average). Radiatedemissionlimitsare(in V/m),forfrequencybetween9and490kHZ,2400 dividedbythefrequencyinkHz,measuredat300m.Forfrequencybetween490and 1705kHZ,24000dividedbythefrequencyinkHz,measuredat30m. IEEEC95.1describeslimitsoneldstrength,currentandspecicabsorption rate(SAR)forhealthandhumansafety[ 33 ].Exposurelimitsaredenedfortwo cases,controlledanduncontrolledenvironments(thegeneralpublic).Forconsumer electronics,andnear-eldinduction,theprimaryconcernwouldbemagneticeld strengthlimitsinuncontrolledenvironments.Maximumpermissibleexposures(MPEs), forheadandtorso,inuncontrolledenvironments,between3.35-5000kHz,areanrms uxof0.205mTandeldof163A/m.Inthelimbs,MPEsareanrmsuxof1.13mT andeldof900A/m.Additionally,therearespecicabsorptionrate(SAR)limits:forthe wholebody,0.08W/kg;foranylocalized10goftissue,2W/kg;andforextremities,4 W/kg. 3.7Conclusion Thischapteranalyticallyderivedusefulrelationshipsfortheeldsproducedbythe coil,thecoilinductance,andthecoilparasiticsforroundandrectangularconductors. 42

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These relationshipscanbeusedtohelpestimateperformanceandguidethedesign ofthecoilsbeforeconstructionofasystem.Combinedwiththedesignequationsfrom Chapter 2 ,thisprovidesabasisforanelectronicdesignautomation(EDA)codeused throughouttherestofthisdissertation. 43

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CHAPTER 4 OPTIMALPRIMARYCOILDESIGN 4.1Introduction Ifmultipledevicesaretobechargedsimultaneouslyonthesamesystem,the transmittingcoilmustbelargeenoughtoaccomodatethem.Thisposesachallenge,as toensureuniformpowerdeliverytodevices,regardlessofposition,theelectromagnetic elddistributionmustbeeven.Inparticular,thedistributionofthez-componentof themagneticeldintheplaneofthereceivingcoilsmustbeasuniformaspossible. Transmittingcoilsmaybedesignedtoproducesuchelds;oneapproachistheoptimal hybridcoildesign[ 34],whichisdemonstratedforaslargeacoilas15cmby15cm. Thischapterdescribesadifferenttechniqueforcoildesign(theprimarydifferencebeing theparameterizationofthecoilshape),whichisdemonstratedfora20cmby20cmcoil. 4.2PlanarWirelessPowerSystem Inthischapter,thesystemwasconguredwithseries-parallelcompensationas describedin[ 13].Thetransmittingcoilfollows,whichisinturninductivelycoupledto thereceivingcoil,arectangularcoilof6cmby8cmand6turns.Bothtransmitterand receivercoilswereconstructedbyhandusingLitzwiretoreduceresistivelossesfrom proximityandskineffects.Thereceivingcoilisconnectedtothesecondhalfofthe Figure 4-1.Transmittertestsetup. 44

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Figure 4-2.Coillayout. transformationnetwork,aparallelcapacitor,andfollowedbyarectierandareceiver load.ApictureofthetestsetupisshowninFig. 4-1. 4.3CoilDesign Thetransmittingcoilisarectangularspiralwithbluntedcorners,wheretheratioof thewidthofaturntotheoverallwidth, f ,isdenedby: f =1 (1 (N n +1) = N ) k (4) where n istheturnnumber,countingfromtheoutside,and N isthenumberofturns, k isaparameter,and givesthefractionofofeachcornertoberemovedtobluntthe corners.Thespiralgeometryisentirelydescribedbyfreeparameters N k ,and ;and thelengthandwidth,whicharexedat20cmby20cmforthisexample.Bysweeping theparametervalues,evaluatingtheelds,andcalculatinganobjectivefunction,the 45

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Figure 4-3.Calculatedz-directedmagneticeld,assuming1Acurrent(A/m). coildesignwhichproducedmaximallyateldsforacoilofthespeciedsizewas determined.Theanalyticalmagnetoquasistatic(MQS)solutionforalineofcurrent(see Chapter 3 )wasusedtobuildtheeldsfortheentirecurrentinaplane1mmabovethe coil.Theobjectivefunctionwaschosenasthecoefcientofvariation(COV,thestandard deviationdividedbythemean)ofthez-componentofthemagneticeld.Minimizing theCOVminimizestherelativevariationsintheeld,ensuringasmoothdistribution. ThenaloptimalcoillayoutisshowninFig. 4-2,andthecorrespondingMQSeldsare showninFig. 4-3. 4.4Testing Thetransmittingcoilwastestedinthreeways.First,thez-directedmagneticeld wasmeasuredwitha6cmdiametereldprobe.Second,thereceiverpositionwas variedovertheentiretransmittercoilareatogaugetheuniformityofwirelesspower transfer.Finally,thereceiverwasxedatthecenterofthetransmitterandtheload resistancewassweptfrom10to2000 46

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Figure 4-4.Fieldprobemeasurement(mV). ThechiefguresofmeritconsideredinthetestweretheDCpowersuppliedtothe amplier,theACpowertransmitted,andtheDCpowerdeliveredtotheresistiveload, inadditiontotheamplier,thecoupling,andthetotalefciency.Amplierefciency isdenedastheratiooftransmittedpowertosuppliedpower;couplingefciency isdenedastheratioofpowerreceivedbytheloadtotransmittedpower;andtotal efciencyisdenedastheproductoftheprevioustwo. 4.5Results Table 4-1 summarizestheperfomancecharcteristicsofthecoil.Theeldmeasurement resultsareshowninFig. 4-4,intermsofthevoltagemeasuredontheeldprobe.The peakinthelowerleftcornercorrespondstothelocationoftheinputleads,andthe peaksinothercornersareduetothesuperpositionofeldsatcornersofthespiral. Theapparentdrop-offattheedgesisduetothespatialaveragingeffectoftheprobe. Asmallportionoftheprobewasoutsideofthecoil,wheretheeldreversesand 47

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T able4-1.Summaryofsystemperformance. Siz e 20 cmby20cm P eakdeliveredpower 11.8 W P eaktotalefciency 80.9% P eakcouplingefciency 88.4% CO V 2.2% Self-inductance 45.00 H Resistance 0.37 becomes negative,pullingtheaveragedown.Asidefromthisartifactoftheeldprobe measurement,thegeneraltrendoftheeldmatchestheMQScalculations. FromFig. 4-5,thespatialuniformityofthereceivedpowerisshown.Thetwo notablepeaksmatchtheeldpeaksatthecornersandneartheleads,whichcanbe seenintheeldplots.Thesepeaksarerelativelysmall,however,asthemaximum variationshownintheplotis0.8W,lessthan10%ofthemean.TheCOVislikewise small,at2.2%. Fig. 4-6 showstheresultsfromthevariableloadingtest.Ascanbeseen,the receivedpowerismaximumat25 andavalueof11.8W.Themaximumtotal efciencyis80.9%at100 andthemaximumcouplingefciencyis88.4%at250 .Theefcienciesarehighunderawiderangeofloads(itshouldbenotedthatthe efciencyislessenedslightlyinthepresenceofmultipleloads).Thisdemonstrates thesystem'srobustness,notonlywithrespecttoreceiverplacement,butalsoloading conditions. 4.6Conclusion Thischapterhasdemonstratedthefeasibilityoflargetransmittingcoilsforopenair inductivelycoupledpowertransfer.Largetransmittingcoilssuchasthismaybeused forwirelesschargingofmultiplebattery-powereddevicesequippedwithreceiving coils,suchascellphonesandPDAs.Theprimarychallengeindesigningsucha coilisachievinganevenpowerdeliveryregardlessofreceiverposition,inorderto accommodatemultipledevices.Suchacoildesignwasachievedthroughoptimization, andthe20cmby20cmcoilwasbuiltandtestedwithaswitchmodepoweramplier 48

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Figure 4-5.Receivedpower(W)asafunctionofthelocationofthecenterofthe receivingcoil. Figure 4-6.Power(W)andefciency(%)atloadsfrom10 to2k 49

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and a6cmby8cmreceivingcoil.Itwasfoundtohaveamaximumefciencyof80.9% andamaximumpowerdeliveryof11.8W.Ataxedload,thepowerdeliveryhasa coefcientofvariationof2.2%asthereceivingcoil'spositionisvariedonthetransmitter, andthepeakspatialvariationislessthan10%ofthemeanpowerdelivery.Ingeneral, thesystemisrobustandefciencyishigh,irrespectiveofreceiverplacementand loadingconditions.Thisdemonstratedthefeasibilityofeliminatingthelastwireof wirelessportabledevicestoachieveacompletelywirelesssolution. 50

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CHAPTER 5 M:NANALYSIS 5.1Introduction Regardlessoftechnique,largertransmittingcoilsrequiremoreturnstoachievean evenelddistribution,raisingtheinductance.Thisisaproblembecausetheamplier operationissensitivetocomponentvariationinthetransformationnetworkfollowing thedrivingcircuit.Astheinductanceoftheprimarycoilincreases,theseriescapacitor inthenetworkneedstobesmaller,andtheclassEbecomesincreasinglysensitive tosmallvariationsinthecomponentvalues,sometimesseverelyhinderingsystem performance.Tocircumventthisproblem,theinductancecouldbeloweredbyusing twoormoreprimarycoilsinparallel.Thisreducestheinductancewhilestillallowing alargechargingarea.Inaddition,havingmultipletansmittingcoilsinparallelreduces theinuenceofoneload'spowerconsumptiononthatofanyotherload.Thischapter derivesandveriesthemathematicaldescriptionofthecouplingbetweenMtransmitters andNreceiversanddemonstratestheadvantagesofsuchasystemexperimentally. 5.2Analysis ThemathematicalanalysisofpowertransferintheM:Ncasecanbeperformedby applyingKirchoffvoltageandcurrentlawstothecircuitshowninFig. 5-1.Theprimary coilsarenumbered 1 through M ,andthereceivingcoilsarenumbered M +1 through M + N .Thevoltage-currentmatrixequationis: ZI = V (5) b = M + N X b =1 Z ab I b = V a (5) where I b isthecurrentonthe b th coiland V a isthevoltageonthe a th coil,and Z ab is the (a b ) th elementoftheimpedancematrix,denedas 51

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Figure 5-1.M:Nblockdiagram. Z ab = 8 > < > : j L a + R a for a = b j M ab otherwise (5) where isangularfrequency, L a and R a aretheself-inductanceandparasitic resistanceofthe a th coil,and M ab isthemutualinductancebetweenthe a th and b th coils.Relatingcurrentandvoltageineachofthecoils, V b canbefound.Fortheprimary coils(inparallel),thevoltageisthesameforall,theinputvoltage( V b = V in ).Forcoils M +1 through M + N V b = I b Z Lb ,where Z Lb istheimpedanceoftheloadandany transformationnetworkattachedtothe b th coil.Thenalconstraintisthatthesumofthe currentsintheprimarycoilsmustbeequaltotheinputcurrent, I in = Z in V in .Applyingthis toEq. 5, ( Z Z L )I =0 (5) where Z isdenedasbefore, I isavectorofthecurrents,and Z L isdenedas 52

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Z L = 8 > > > > < > > > > : Z in f or 1 a M and 1 b M Z Lb for a = b and b > M 0 otherwise (5) Eq. 5 canbesolvedfor Z in bysplittingitintoseveralsubmatricesasfollows: Z Z L = 2 6 4 Z III (Z II ) T Z II Z I 3 7 5 (5) I = 2 6 4 I II I I 3 7 5 (5) where Z III hasdimensions M M ; Z II hasdimensions N M ; Z I hasdimensions N N ; I II hasdimensions M 1 ;and I I hasdimensions N 1 .Dening Z IV = Z III + Z in 1 MM (where 1 MM isan M M matrixofones),andwithsomemanipulations, Z I I I = Z II I II (5) (Z IV Z in 1 MM )I II = (Z II ) T I I (5) Inputcurrent I in isthesumofcurrentsinthetransmittingcoils,statedmathematically as(where 1 1 M isa1by M vectorofones): I in =1 1M I I (5) UsingEq. 5 through 5 [Z IV (Z II ) T (Z I ) 1 Z II ]I II = Z in 1 MM I II (5) Substituting V in = Z in 1 MM I II andusing Z in = V in = I in : 53

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Z in = 1= f1 1M [Z IV (Z II ) T (Z I ) 1 Z II ] 1 1 M 1 g (5) Havingaclosed-formexpressionfortheinputimpedanceallowsderivationofthe currentsintheindividualcoils.Bysubtracting Z in 1 MM I II frombothsidesofEq. 5 : I II = null (X) (5) X = Z IV (Z II ) T ( Z I ) 1 Z II Z in 1 MM (5) I I = (Z I ) 1 (Z II I II ) (5) Nowknowingthecurrentsinthetransmitterandreceivercoils,thepowerreceived byload b maybecomputedsimplyas (I I b ) 2 Re ( Z Lb ).Theseequationsareextensibleto differentreceivertopologies,suchasparallelorseriescapacitors,andnonlinearities (suchasrectiers,orproximityandskineffectsonresistanceandinductance)maybe consideredaswell,throughtheuseofxed-pointiteration. 5.3TestsResults Toverifythecorrectnessoftheprecedingequationsaswellastodemonstratethe benetofusingmultipleprimarycoilsinparallel,simulationsandtestswerecarriedout forthe1:1,1:2,1:3,2:2,and2:3cases.Forallexceptthethree-receivercases,two receiversizeswereconsidered.Inaddition,thetwo-transmittertestswereperformed withthetransmittingcoilsadjacentandseparated.Fig. 5-2 showstheelevendifferent congurationsforthetestsetup. Theprimarycoilinductanceis34.44 H,reducedbyhalfwhenthetwo-coilcase isconsidered.ComponentselectionprocedurefortheclassEwasdescribedin[ 16 ], andcomponentvaluesarespeciedinTable 5-1 (forallcases L dc was500 Hand L out was9.5 H).Notablythevaluesfor C out arehigherwiththe2transmittersystem. 54

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Figure 5-2.Startingtoprow,left-to-right:coilarrangements(thickredlineisreceiver, thinbluelineistransmitter),for(a)1:1small-rx,(b)1:2small-rx,(c)1:3 small-rx,(d)2:2small-rx,(e)2:3small-rx,(f)1:1big-rx,(g)1:2big-rx,(h)2:2 big-rx,(i)2:2split-txsmall-rx,(j)2:3split-txsmall-rx,and(k)2:2split-tx big-rx. 55

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T able5-1.Componentvaluesfor1and2transmittersystems. M Rx size C rx (nF) C out (nF) C t (nF) 1 Small 100 11.5 14.7 1 Large 53.8 15.3 6.8 2 Small 100 22.3 27.3 2 Large 53.8 22.3 18.3 Higher capacitancemeanstheimpedancewillbelesssensitivetocomponentvariations becauseoftheinverserelationshipbetweencapacitanceandreactance.Thederivative ofreactancewithrespecttocapacitancegoesastheinversesquareofcapacitance,so highercapacitancevaluesmeansamuchlowersensitivity.Tomitigateproximityand skineffects,weusedLitzwireforcoilwindings.Thesmallreceiverswereall4cmby 5cmrectangularcoilsof6turns,thelargereceiverswere7cmby8cmwith6turns, andthetransmitterswere16cmby18cmwith13turns,designedbythetechnique describedin[ 20]. Foreachtransmitter/receiverpairing,theresistiveloadattachedtoeachreceiver wassweptfrom60 to4000 bymeansofprogrammableelectronicloads.The resistiveloadisanapproximationofthechargestatusofabattery;afullycharged deviceappearsasalargeresistiveload(thousandsof )andanunchargeddevice appearsasalowresistiveload(ahandfulof ).DCreceivedpower( P rx )owwas measuredattheelectronicloads. 5.3.1Verication ToverifytheaccuracyoftheequationsdevelopedinSection 5.2,simulationswere performedusingMATLABcodeimplementingtheanalyticaltreatmentoftheclassE amplierbyRaab[ 17]foraloadwithimpedancedenedasinEq. 5. L a and M ab are caluculatedusinganumericalintegrationoftheNeumannformula[ 24].Themeasured andpredicted P rx foreachoftheM:Ncasesconsideredinthischapterareshownin Fig. 5-3.Thepredictedvs.observedplotsshowaone-to-onecorrespondence,aside fromsomespreadduetouncertaintyinsecondaryandprimarycoilpositions.For1:3 thereisapartiularlylargeamountofspread.Withthreereceiversincloseproximityto 56

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Figure 5-3.Measuredvs.predicted P rx ,for:(a)1:1small-rx,(b)1:2small-rx,(c)1:3 small-rx,(d)2:2small-rx,(e)2:3small-rx,(f)1:1big-rx,(g)1:2big-rx,(h)2:2 big-rx,(i)2:2split-txsmall-rx,(j)2:3split-txsmall-rx,and(k)2:2split-tx big-rx.Scaleisasindicatedin(i)forallsubplots. eachother,uncertaintiesintheirrelativepositionshaveamorepronouncedeffecton predictedpower. 5.3.2ReceiverDecoupling Toshowthathavingmultipleprimarycoilsreducestheinuenceofonereceiveron theothers,wemaptheloadingcondition ( R l 1 R l 2 ,..., R lN ) toacorrespondingreceived powerdeliverycondition ( P rx 1 P rx 2 ,..., P rxN ),usingthedatafromtheelectronicload sweeps.Thoughitisimpossibletofullyexplorethepowerdeliveryspaceduetothe discretenatureofthetests,lookingatthisdiscretesetofloadingconditionsallowsusto outlinethephysicallyrealizablepowervaluesthatcanbereceivedbymultipleloadson thesameprimarycoilorcoils. Figs. 5-4 and 5-5 showsthisforthetworeceivercondition.InFig. 5-4,1:2and 2:2showsimilarpowerspacesbecausethereceiversaresmallandfurtherapartso theyareweaklycoupled.Fig. 5-5 demonstratesthatwhenthereceiversizeislarge,for 57

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Figure 5-4.Powerspaceplotsfortwo-receivertestswithsmallreceivers. Figure 5-5.Powerspaceplotsfortwo-receivertestswithlargereceivers. 58

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1:2, thepowerspaceissqueezedintoamuchnarrowerarea,whilefor2:2thepower spaceisclosetoasquare10Woneachside.Theconstrictedpowerspacefor1:2 occursbecausewhenoneloadislarge(eg,afullychargeddevice),itchokespower deliverytotheother,lowload(eg,anunchargeddevice).Thisphenomenoncanbe seeninthebluedots(1:2)inFig. 5-5:whenreceiver1hashighloadresistanceand receiveslowpower(lessthan0.2W),receiver2islimitedtolessthan0.2W.This amountstothepinchedshapeofthepowerspace.Suchpowerdeliverylimitationsare unacceptable.Thesameplotdemonstratesthatfor2:2,thepowerdeliveredtoreceiver 2canstillreachabout10Wwhenreceiver1haslowpower,highresistanceconditions. Thoughasimplication,itcanbesaidthatwithmultipletransmitters,thereceiversare essentiallyinparallelwhilewithonetransmittertheyareessentiallyinseries.Witha constantvoltagesource,powerdeliverytoresistiveloadsinseriesisgovernedbythe totalresistance,whileloadsinparallelreceiveindependentpowerdelivery.Multiple primarycoilsparallelizespowerdelivery. Inthesameplots,theeffectofsplittransmitterisalsodemonstrated.Thekey differenceforthesplittransmitterisareductioninreceivedpower,seenasashiftingof thepowerspacetowardstheorigin.Thisisbecausethefringingeldsoftheprimary coilsdissipateintothenearbyenvironmentinsteadofintoaneighboringcoil. Fig. 5-6 showsthepowerspacewithsmallreceiversfor1:3andfor2:3(large receiverscouldnotbeconsideredfor1:3becauseofinsufcientroomonthetransmitter). Thoughthedifferenceislesspronouncedthanthatofthe N =2 condition,itisapparent thatthe1:3powerspaceismorecurved,withanupwardsweep,whilethe2:3power spaceisadistinctrectangularprism.Whenonereceiverisinahighresistance,low powercondition,thepowerreceivedbytheotherreceiversislessin1:3thanin2:3.Fig. 5-6 similarlydemonstratesthedecouplingeffect,onlywithasplittransmitter.Theeffect isthesameasdiscussedintheprecedingparagraph,andforsimilarreasons. 59

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Figure 5-6.Powerspaceplotforthree-receivertest. Figure 5-7.Powervs.efciencyplotfortwo-receivertestswithsmallreceivers. 60

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Figure 5-8.Powervs.efciencyplotfortwo-receivertestswithlargereceivers. Figure 5-9.Powervs.efciencyplotforthree-receivertestswithsmallreceivers. 61

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T able5-2.Maximum P rx andmaximum c fordifferentM:Narrangements. Arrangement Max P rx (W) Max c (fr action) 1:2, small-rx 3.44 0.75 2:2, small-rx 3.88 0.75 2:2, small-rx,split-tx 2.60 0.68 1:2, large-rx 1.82 0.82 2:2, large-rx 9.45 0.88 2:2, large-rx,split-tx 7.86 0.87 1:3, small-rx 1.91 0.74 2:3, small-rx 3.08 0.74 2:3, small-rx,split-tx 2.40 0.67 5.3.3 ImpactonEfciencyandTotalReceivedPower Transmittedpowerwasmeasuredusingacurrentprobe(AgilentN2783A),avoltage probe(AgilentN2863A),andanoscilloscope(AgilentDSO5034A),withameasurement accuracyof1%and0.5%,respectively.Thiscorrespondstoanaccuracyofpower measurementof1.5%.Duetotemperatureeffectsandtheeffectoftransmissiondelay onthephaseofmeasurement,theactualaccuracyisestimatedtobearound5%. ReceivedpowerwasmeasuredusingtheDCelectronicloads(BK8500),whichhavea (worst-case)accuracyof0.4%forcurrentand0.38%forvoltage,givingameasurement accuracyforpowerofabout0.8%. Fig. 5-7 showstotalreceivedpower, P rx ,andcouplingefciency( c ,denedasthe totalreceivedpoweroverthetransmittedpower)forthe2smallreceivertests.It'sclear fromtheplotthattheimpactonefciencyisminimal;themaximum c for1:2and2:2is 0.75anddropsto0.68withsplittransmitters.Withlargereceivers(Fig. 5-8),theeffect ofchangingfrom1:2to2:2isseenasanincreaseinreceivedpower,asthemaximum P rx isincreasedfrom1.82to9.45.Likewise,with3receivers,Fig. 5-9 demonstrates thatthereisalsoanincreaseinreceivedpower,whilethemaximumefciencyremains aboutthesame.Usingthesplittransmitterdecreases c to0.67.Itseemsthatusing multipletransmittersthatarespatiallyseparatedfromeachotherreducesefciencyand receivedpowerasthefringingeldsaredissipatedintothenearbyenvironmentinstead 62

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Figure 5-10.Totalreceivedpowerasafunctionof R L ,andits95%condenceintervals with5%componenttolerance(redlines),10%componenttolerances(blue dashes),and20%componenttolerances(blackdash-dot),forboth1:2 (left)and2:2(right)cases. ofcouplingintoaneighboringcoil.Table 5-2 givesthemaximum P rx and c foreach test. Toinvestigatethesensitivitytocomponentvariation,aMonteCarlosimulation wasrun,assumingthecomponentsarenormallydistributed,withmeansgivenby thederivedcomponentformulasandwithstandarddeviations, ,suchthat 3 isthe componenttolerance.Thesesimulationswerecarriedoutattolerancelevelsof5%, 10%,and20%,forthe1:2and2:2congurations,usingthelargereceivers.One receiverwasxedat500 andtheotherwassweptfrom60to4000 .Figure 5-10 showsthe95%condenceintervalsfortotalreceivedpoweratthethreetolerance levels.Figure 5-11 showsthe95%condenceintervalsfortotalefciencyatthethree tolerancelevels.Ascanbeseen,thepowerisskewedlow,andwithtightertolerances for1:2thanfor2:2.Efciencyisskewedhigh,withtightertolerancesforthe2:2system thanforthe2:1.Thisskewlowinthepowercondenceintervalsandskewhighinthe 63

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Figure 5-11.Totalefciencyasafunctionof R L ,andits95%condenceintervalswith 5%componenttolerance(redlines),10%componenttolerances(blue dashes),and20%componenttolerances(blackdash-dot),forboth1:2 (left)and2:2(right)cases. efciencycondenceintervalsshowsthatthesystemisnotoptimizedformaximum powerdeliverybutratherefciency.Thismakessense,asallofthecomponentselection forthesystemisdoneonthebasisofefcientoperationoftheclassE.The2:2system's efciencyislesssensitivetocomponentvariationprimarilybecauseof C out which governsthephaserangeseenbytheclassEandthusitsefciency. C out islargerin the2:2system,thereforeitsreactanceislesssensitivetovariations.Fortotalreceived power,the1:2systemislesssensitivethanthe2:2systemtocomponentvariations, becausethetworeceiversinthe2:2systemcanvarymoreindependentlyduetothe decouplingeffect. 5.4Conclusion InductivewirelesspowertransferbetweenMprimarycoilscoupledtoNsecondary coilsisderivedanalyticallyanddemonstratedexperimentallyfor M =1,2 and N = 64

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1, 2,3 .Usingmultipleprimarycoilsinparallelhasadvantagesoverasingleprimarycoil. First,thereducedinductanceofthetransmittingcoilsmakestheamplierlesssensitive tocomponentvariations.Second,withmultiplereceivingcoils,thepowerdeliveryto anindividualreceiverislesssensitivetochangesintheloadsattachedtoothercoils, decouplingreceiversfromeachother.Inaddition,usingmultipletransmittersisshown toincreasereceivedpowerwithlimitedimpactoncouplingefciency.Themultiple transmittingcoilarchitectureincreasesthefeasibilityandeffectivenessofsimultaneous multipledevicechargingaswellasmakingtheampliermorerobusttocomponent variation. 65

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CHAPTER 6 OPTIMALPRIMARYCOILDESIGNFORMULTIPLECOILS 6.1Introduction Chapter 4 presentedatechniqueforcoildesignthatensuresaneveneld. However,tomaintaineveneldsovergreaterareasrequiresacoilofhigherinductance. FromChapters 2 and 5 ,thisleadstoincreasedsensitivitytocomponentvariation. Chapter 5 developsthetheoryformultipletransmittingcoils,which,amongotherthings, reducessensitivitytocomponentvariations.Naturally,thenextstepistocombinethe multiplecoilideawiththecoildesigntechnique.Thischapterpresentsacoildesign techniqueformultipletransmittingcoils.Specically,asystemwithtwotransmittingcoils inparallelisdesigned. 6.2CoilDesign ThecoildesignfortwotransmittersdiffersfromthesysteminChapter 5 inthatthe dual-coilsystemthereusedtwoidenticalcoilsdesignedtoworkindividually.Here,they areconsideredtobeworkingtogethertoestablishalargerareaofevenelds.The basicprinciplebehinddesignformultiplecoilsisthesameasforsinglecoils.Thatis,the geometryshouldbeparameterized,andthentheparameterscanbeoptimizedtogive minimumeldvariationsasmeasuredbythecoefcientofvariation.Followingthesame basedesignfromChapter 4 ,wheresuccessiveturns'widthsarerelatedtotheoverall widthby f : f =1 (1 (N n +1) = N ) k (6) andwiththecornersbluntedbyafraction .Consideratwo-coilsystemwithoverall y dimension W and x dimension L.Thetwocoilsoverlapinthe y directionbysome amount b .Alistofcoordinates (x y ) isgeneratedbythemethodinChapter 4 forthecoil associatedwithparameters (N k ,) withdimensions L and w : 66

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T able6-1.Designparameters. N 15 k 2.64 0.10 a 0.40 b 9.00 cm W 35.00 cm w = (W 2b )= 2 (6) The y coordinatesarethenskewedbyanexponent a andoffsetby b : y 1 = w ( y = w ) a + b (6) y 2 = w (y = w ) a b (6) Thisstretchesoutthecoils,sothattheturnsarespacedfurtherapartintheregion wherethetwocoilsoverlap.Soessentially,themultiplecoildesignisthesameassingle coildesignwithtwoadditionalparameters,skew a andoverlap b Foratwocoilsystemwith W =35cmand L=25cm,theoptimumparametersare givenTable 6-1.Thesedimensionswerechosensuchthatthecoilcouldaccommodate alaptopequippedwithareceivingcoil. Fig. 6-1 showsthecorrespondingcoilsandFig. 6-2 showstheMQSestimateofthe elddistribution. 6.3System Thedualcoilsystemwastested,usingtheparametersinTable 6-1 andareceiver of8cmby10cmwith6turns.ThecomponentvaluesfortheclassEwereselected accordingtothegeneralprinciplesdescribedinChapter 2 bysweepingthecomponent valuesandhandtuning.ThevaluesusedaregiveninTable 6-2.Figs. 6-3 and 6-4 show thecoilsandcircuits. 67

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Figure 6-1.Coillayout. Figure 6-2.Calculatedz-directedmagneticeld,assuming1Acurrent(A/m). 68

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Figure 6-3.Transmittertestsetup. Figure 6-4.Overlapofdualtransmittercoils. Table6-2.Componentvalues. C rx 53.80 nF C out 13.50 nF L out 9.12 H C t 18.00 nF 69

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T able6-3.Summaryofsystemperformance. Siz e 25 cmby35cm P eakdeliveredpower 5.62 W P eaktotalefciency 59.3% P eakcouplingefciency 75.6% CO V 17.8% Coil 1self-inductance 50.41 H Coil 1resistance 0.44 Coil 2self-inductance 51.27 H Coil 2resistance 0.44 Coil 1and2mutualinductance 4.87 H T otalself-inductance 27.86 H T otalresistance 0.22 6.4 Testing Totestthesystem,thereceivedpower,transmittedpower,andinputpower,were measuredataloadresistanceof100 at5cmgridpointsoverthecoilstoevaluatethe spatialvariability.Inaddition,thereceivedpower,transmittedpower,andinputpower, weremeasuredatloadsof75,100,250,500,750,1000,and4000 andthereceiver centeredonthetransmitter. 6.5Results Fig. 6-5 isaplotofthereceivedpowerovertheareaofthecoils.Thisshowsthe spatia1uniformityofthepowerdistributionovertheareaofthecoil( 17.5 < y < 17.5 and 12.5 < x < 12.5 ).Thereismorevariabilityinthe y directionthaninthe x direction. Thecoefcientofvariationis17.79%.Theratioofreceiverareatotransmitterareais about0.09,whereasisChapter2iswas0.12,whichexplainsthegreatervariability. Fig. 6-6 showstheplotoftheloadresistanceresponse.Theresponsetrendis similartothoseseenforsystemsinChapters 2, 4,and 5.Ingeneral,theefcienciesare lowerthanthosepresentedinChapter 4 duetothelowermutualinductancebetween thereceiverandtransmitter. Table 6-3 givesasummaryoftheperformancecharacteristicsofthesystem. 70

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Figure 6-5.Receivedpower(W)asafunctionofthelocationofthecenterofthe receivingcoil. Figure 6-6.Power(W)andefciency(%)atloadsfrom75 to4k 71

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6.6 Conclusion Thischapterhasdemonstratedtheapplicationofthedesignstrategyforasingle coiltomultiplecoils.Thisconferstheadvantagesoflargecoilsizeandreducedprimary inductance.Theprimarychallengeindesigningsuchacoilisachievinganevenpower deliveryregardlessofreceiverposition,inordertoaccommodatemultipledevices.Such acoildesignwasachievedthroughoptimization,andthe25cmby35cmcoilwasbuilt andtestedwithaclassEpoweramplieranda8cmby10cmreceivingcoil.Itwas foundtohaveamaximumefciencyof59.3%andamaximumpowerdeliveryof5.62 W.Ataxedload,thepowerdeliveryhasacoefcientofvariationof17.78%asthe receivingcoil'spositionisvariedonthetransmitter. 72

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CHAPTER 7 INCLUSIONOFFERRITES 7.1Introduction Inanear-eldwirelesspowersystem,ifthedeviceunderchargeiselectrically conductive,orifthereareelectricallyconductiveobjectsunderneaththetransmitting coil,theeldsgeneratedbytheprimarywillbedissipatedinsteadofcontributingto powertransfer.ItwouldbedesireabletohaveanEMshieldthatwouldalloworeven enhancepowertransferwithoutlettingeldsdissipate.Forinstance,acellphone equippedwithareceivingcoilwouldneedashieldbetweenthecoilandthebatteryin orderfortheretobeeffectivewirelesspowertransfer. Onewaytodothisisthroughuseofhighmagneticpermeabilitymaterials,such asferrite,backedbycopper([ 35, 36]).Thecoilisplacedontopofferrite,whichison topofthecopper.Theprincipleofoperationisbasedonthemagneticeldboundary conditionsonthenormalandtangentialcomponentsoftheeld: H t 1 = H t 2 (7) B n 1 = B n 2 (7) 0 H n 1 = r 0 H n 2 (7) wheresubscript t denotesthetangentialcomponent, n denotesthenormal component,material1isair,andmaterial2isferrite.Sincethetangentialeldis Figure 7-1.Diagramofferriteshielding. 73

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contin uous,andthenormalelddropsinmagnitudeduetothelargeincreasein permeabilityacrosstheboundary,theeldisguidedthroughtheferriteandreducedin magnitude.Theeldsreachingthecopperarethusreducedsolittlepowerisdissipated inthecopper.Soinsteadofuselesslygeneratingcurrentsinconductiveobjectsbehind thecoil,theferriteguidestheeldthroughthecoil.Intermsofthewirelesspowercircuit parameters,theinductanceisincreasedandtheparasiticresistanceisdecreased. Thischapterpresentsseveralaspectsoftheuseofferriteforshielding.Sections 7.2 and 7.3 discusstheoreticaleffectsofferritepropertiesonthecoilinductanceandlosses. Section 7.4 usesnumericalsimulationtoestablishwidthandthicknesseffectsofaferrite shieldontheinductanceandresistance.Finally,Section 7.5,presentsexperimental evaluationofseveralcommercialferrites. 7.2InductanceEstimation [ 37]presentsaderivationoftheeffectofasemi-innite,losslessferritesubstrateon theinductanceofaradiallysymmetriccoilusingimagetheory.Theendresultis L = 2 + 1 L 0 (7) where L 0 isthefree-spaceinductanceofthecoil.So,inthelimit,theferrite inductanceistwicethefree-spaceinductance.Clearlythissortofimpactonthe inductancematrixwillrequirecarefulconsiderationofcircuittuning.Forferritesof nitethickness,butinniteplanarextent,theeffectsontheinductanceofradialcoils isexploredanalyticallyin[ 3840].However,therealcaseofaniteferriteshield overanitecoppershieldismorecomplicated.Thiscouldpotentiallybehandled bySchwarz-Christoffelmapping[ 41 ].Bymappingthesemi-innitesolutionforthe vectorpotential A tothenitegeometrytheinductanceinthenitegeometrycouldbe estimated.Ifthevectorpotentialinthesemi-innitegeometryisdenoted A( z ),and theSchwarz-Christoffeltransformbetweentheoriginalcoordinates z andthenew coordinates w is w = g (z ),themappedvectorpotentialis A: 74

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A(w ) = A(g 1 (w ) g 0 (g 1 ( w )) (7) and thenewinductanceis L = H C Ads I (7) where C is thepathofthecoil. Asimplewaytoevaluatetheeffectofaferriteshieldoninductancewouldbeto useanempiricalfunctionofthecoilpropertiesandtheferritepropertiestocalculatean effectivepermeability,similartotheeffectivedielectricconstantusedinmicrostripdesign [42 ].Deningthisconstantas e = L L 0 (7) where L is theinductancewithferrite,and L 0 istheinductanceinfreespace.A simplegeometricalparameterfordescribingthecoilisthecoil'slength l .Theferritecan bedescribedbyitsrelativepermeability r andbyitsthickness h .Usingafunctional relationshipfunctionalrelationshipsimilartothatoftheeffectivedielectric, e = 2 r r + 1 C (7) C = A(1+ B h l ) n (7) where A, B and n arettingparameters.Todeterminethevaluestouseforthese parameters,thecoilinductanceofcircularcoilsofvaryingdiameterandnumberofturns wasmeasuredinfreespaceandoverthreedifferentferrites,of r 125,2000,and2100. Usingthesemeasurements,thebest-t A is0.8128, B is97.7237,and n is-0.09. 75

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Figure 7-2.Empirical e predictions(redx)andobservations(bluecircle). Figure 7-3.Flux-eldhyteresisloop Fig. 7-2 showsthemeasureddataandvaluesestimatedbythebesttfunction.Of course,thisfunctionisagrosssimplication,asitignoresmanygeometricaleffects,but itprovidesaquickestimateofaparticularferrite'seffect. 76

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7.3 LossEstimation Besidestheeffectontheinductancematrix,aferriteshieldintroduceslossesdueto twoeffects.Therstishysteresislossesfromthealternatingelds.Duetosaturation, thepeakuxandeldis B =min(B s 0 H p ) (7) H = H c + B 0 (7) Using these,thehysteresislosscanbeestimatedoneway,approximatingthe B H curveasaparallelogram(Fig. 7-3) P hyst = fH c B (7) where H c isthecoercivity. Or,usingthecomplexpermeability 0 + j 00 P hyst = f 00 H 2 (7) Conductivelossesarethesecondlossmechanismandareusuallylessthanthe hysteresislosses.Thesecanbeestimatedaccordingto[ 43]as: P cond = ( hfB ) 2 6 (7) where is resistivity. 7.4ThicknessandWidthEffects Apracticalconsiderationforferriteshieldingisthenecessarydimensionand necessarypermeabilityoftheshieldmaterial.Todeterminewhatsizeandpermeability shieldwouldbesufcient,nite-elementMQSsimulationswereruninAnsoftMaxwell ofasquarecoilrestingonaferriteofthickness h ,permeability ,andrelativewidth F F isdenedastheratiooftheferritewidthtothecoilwidth.Thesethreevariableswere sweptandateach( h F )pointtheratiooftheinductancetothefree-spaceinductance 77

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Figure 7-4.Effectsofthicknessandrelativewidthofferriteoninductance. wascalculatedandplottedinFig. 7-4.Thisratioshowstheexpectedincreasingtrend withincreasing h ,and F .Thisaloneisinsufcienttodeterminearequirementon shieldsize. Thelossesmustalsobeconsidered.Fig. 7-5 showstheratiooftheresistanceto freespaceresistance.As F increases,thisratiodecreases,approaching1because fewerfringingeldsarebeingdissipatedinthecopperbacking.Theeffectof F attensoffaround1.2sotheshieldwidthshouldbeabout1.2timesthecoilsize. Moreconcretely,the 1 =e calculatedfoldinglengthfortheaverageofthecurvesinFig. 7-4 is1.08andinFig. 7-5 is1.09.Using1.2asaguideprovidessomedegreeofsafety marginfordesignpurposes. For =600,apracticalvalue,theeffectof h variationissmall,soitshouldbe possibletouseashieldasthinas0.2mmor0.3mm. 78

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Figure 7-5.Effectsofthicknessandrelativewidthofferriteonresistance. 7.5ExperimentalEvaluation Toevaluatethelossesandshieldingeffectivenessofpotentialcommercially availableferrites,currentwasrunthroughasolenoidtestcoilplacedovertheferriteor theferrite/coppercombination.Thecurrentandvoltagewaveformswerecapturedfor oneperiodtocalculatetheinputimpedanceasfollows: I = I p cos (! t ) (7) V = V p cos ( t + ) (7) jZ j = V p I p (7) = arccos( IV ) (7) Thepowerdissipatedis: 79

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T able7-1.Ferriteproperties. Ferrite 0 00 B s (T) H c (A/m) ( m) F airRite42 2530 118 0.40 5 5 10 4 3M 600 710 125 0.273 68 10 8 F errox3C96 2000 10 0.50 13 5 P = 1 2 I 2 p <(Z ) (7) F orasolenoidcoil,theeldstrengthproducedis, H p = NI p h coil (7) where N =18 turnsand h coil ,theheightofthesolenoid,is20mmwithadiameter ofabout20mmfortheparticulartestcoil.Thetestresultsforseveralferritesand thicknesseswithandwithoutcoppershieldingarepresentedinTable 7-2. Theresistance( R ),reactance( X ),powerdissipation( P ),andinductance( L)were calculatedusingthevoltageandcurrentwaveforms. L 0 and R 0 arethefree-space inductanceandresistanceofthesolenoidtestcoil.Themostdesireablematerialwould havean R = R 0 ascloseto1.00anda L = L 0 greaterthan1.00,forasthinaspossible ferrite,overacopperbacking. SamplesofFairRite42wereonlyavailableinonethickness,1.00mm.Itshows similarperformancewiththecopperbackingandwithout.Thisindicatesitisaneffective shieldingmaterial.However,1.00mmmaybetoothickandheavytobeusedasashield inaportabledevice. 3M600sampleswereobtainedatthreethicknesses,0.4mm,0.3mm,and0.2mm. IthasgreaterhysteresislossesthantheFairRite42,asits 00 and H c arehigher.Its conductivelossesarelower,withitsgreaterresistivity.Fromthetestdata,the0.3mm thick3M600shouldprovideadequateshielding. 80

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T able7-2.Ferriteexperimentalevaluationwithsolenoidcoil. Ferrite h (mm) I p (A) H p (A/m) P (W) X ( ) R ( ) L = L 0 R = R 0 No ferrite 2.88 2592 0.51 13.19 0.11 1.00 1.00 F airRite42 1.00 2.80 2520 0.46 15.71 0.12 1.19 1.09 F airRite42w/Cu 1.00 2.84 2556 0.49 15.77 0.12 1.20 1.09 3M 600 0.40 2.64 2376 0.52 15.76 0.15 1.19 1.36 3M 600w/Cu 0.40 3.16 2844 1.01 14.90 0.16 1.13 1.45 3M 600 0.30 2.72 2448 0.45 15.88 0.12 1.20 1.09 3M 600w/Cu 0.30 2.32 2088 0.39 15.00 0.14 1.14 1.27 3M 600 0.20 2.72 2484 0.41 15.94 0.11 1.21 1.00 3M 600w/Cu 0.20 3.24 2916 0.97 15.06 0.19 1.14 1.73 F errox3C96 4.76 2.56 2304 0.48 15.94 0.15 1.21 1.36 F errox3C96w/Cu 4.76 2.40 2160 0.40 15.17 0.14 1.15 1.27 F errox3C96 3.18 2.84 2556 0.58 16.06 0.14 1.22 1.27 F errox3C96w/Cu 3.18 3.20 2880 1.09 15.25 0.21 1.16 1.91 F errox3C96 1.59 3.04 2736 0.68 15.00 0.15 1.14 1.36 F errox3C96w/Cu 1.59 3.08 2772 0.59 13.68 0.25 1.04 2.27 81

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Of thethreetestferrites,theFerrox3C96hasphysicalpropertiesindicatingthe lowesthysteresisandhighestconductivelosses.Intermsof L = L 0 ,and R =R 0 ,itisthe worstperformingatthethicknessesavailableassamples.Ofcourse,thickerferrites, allotherthingsbeingequal,shouldhavegreaterpowerdissipationduetothegreater volumeofmaterial.Inaddition,thecopperbackinghastheeffectoflowering L = L 0 and raising R = R 0 butitseffectonpowerdissipationintestsappearsferriteandthickness dependent.Overall,thebestpracticalshieldingwouldbethe3M600atathicknessof 0.3mm. 7.6Conclusion Thischapterpresentedseveralaspectsoftheuseofferriteforshielding:theoretical effectsofferritepropertiesonthecoilinductanceandlosses,numericalsimulationto establishwidthandthicknesseffectsofaferriteshieldonthecoilelectricalproperties, andempiricalformulastoestimatetheseeffects.Inaddition,thechapterpresented experimentalevaluationofseveralcommercialferrites.3Mferriteof = 710and thickness0.3mmand20%widerthanthecoilwouldbeagoodshieldforwirelesspower applications. 82

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CHAPTER 8 BAYESIANLOAD/FAULTTRACKING 8.1Introduction Awirelesspowersystemcouldchargemultipledevicesofdifferenttypes,simultaneously. Ofcourse,oneconcernismetalobjectsnearoronthetransmitter.Theseeffectively shortoutthetransmittedelds.Anotherconcernisdevicesunderchargeghtingeach otherforpowerasdescribedinChapter 5,whichcouldleadtoinstabilitiesorperhaps excessivechargetimes.Naturally,then,itwouldbedesirabletodetectthepresenceof faults,thenumberofloads,andtheirbatterychargestatusorsomesimilarmeasure. Theproblemofloadorfaultdetectionisessentiallyaproblemofstateestimation [44 ].Thestateestimationproblemposedinthischapterisacombinationofdiscreteand continuousstates.Theproblemofdetectionandestimationinthewirelesssystemis compoundedbythefactthatonlytwomeasurementsareavailableonthetransmitter: DCinputcurrentandvoltageonthetransmittercoil.Therearemultiplepossiblestates thatareacombinationofcontinuousanddiscreterandomvariables:differentnumbers ofloadsandtheirloadcurrentsorcharges,orpossiblefaultconditions.Inaddition, whenmultipleloadsarepresent,theyinuenceeachothers'receivedpower.So,a detection/estimationschemeforthissystemshouldbeabletohandlediscreteand continuousvariables.Itshouldalsoberobust,becauseafaileddetectionofafault conditioncouldbedangerous. Therearemanyqualitativeorquantititativemethodsforthiskindofestimationin industrialprocessesandothercomplexsystems[ 4547 ].Threefamiliesoftechniques areoutlinedbelow,allofwhicharesomeformofBayesiantracking. TheKalmanlterisapowerfultoolforestimatingthestateofaprocessinthe presenceofprocessandmeasurementnoise.Thebasicideaistousetheknown systemandmeasurementdynamicsandatimeseriesofmeasurementstoestimate thecurrentstateofthesystem[ 48 ].Itisusedforcontinuousvariables.Ateachtime 83

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step thereisapredictionandupdatestep.Thepredictionstepusestheknownsystem dynamicsandthepreviousestimateofthestatetoestimatethecurrentstate.In theupdatestep,aKalmangainisappliedwhichessentiallyweightstheestimate andthemeasurementtoupdatethecurrentestimate.TheclassicKalmanFilterhas requirementsoflinearityandnormalitybutthesecanberelaxedsomewhatinalternate versions,suchastheExtendedKalmanFilterandtheUnscentedKalmanFilter. HiddenMarkovModelsareusefulinstateestimationwhenstatesarefroma discreteset.AnHiddenMarkovModelconsistsofasetofnitestates,transition probabilitiesbetweenthesesstates,andprobabilitydistributionsofobservationsymbols conditionedonthesestates[ 49].TheHiddenMarkovModelproblemis,givena sequenceofobservationsymbols,whatisthesequenceofstatesthatcausedthese symbols?ThefamousViterbialgorithmisoneapproach.Theimportantrestrictionisthat thestatespaceisdiscrete. ParticleltersareaexiblemethodofstateestimationusingMarkovChainMonte Carlotechniques.Theessentialideabehindaparticlelteristouserandomsamples torepresenttheprobabilitydistributionofpossiblestates,updatingthesamplesas thesystemevolveswithtimeandwithnewmeasurements[ 50].Itcombineselements oftheKalmanFilterandHiddenMarkovModel.Thefactthatitusessamplesrather thanadistributionliftsanyrestrictiononnormalityorevenlinearity.Inaddition,itcan beextendedtoincludecombinationsofdiscreteandcontinuousstates,hierarchiesof states,andrisk-weightedstates[ 51, 52].Thismeansthatimprobablebutdangerous stateswillnotbemissed.Anotherkeyfeatureisitscomputationalsimplicity,makingit easiertoimplementinalesspowerfulmicrocontrollerasmightbeusedinaprocess monitoringsituation.Forthesereasons,thischapterwillfocusontheparticlelterina wirelesspowersystem. 84

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8.2 Technology/Data Inthischapter,thetransmittingcoilisa16by18cmcoilwith13turnsandthe receiversareall4by5cmcoilswith6turns.Thetransmittingcoilisdesignedaccording tothetechniquedescribedinChapter 4. Ratherthanusetestdata,thischapterusessimulationsofthesystemwithamodel alreadydevelopedinMatlab,whichmakesuseofclassicanalyticalsolutionsforthe classEamplier[ 17, 19]andnumericalintegrationforthecalculationofcoilinductances andmutualinductances[ 24 ].Inthesystem,thereismeasurementnoise(thermal) andprocessnoise(duetoloaductuations).Componentsareassumedtohavezero toleranceandthereceivingcoils'positionsareassumedxedonceplacedonthe transmittingcoil. 8.3Theory/Methods 8.3.1State/MeasurementModel Fivepossiblediscretestatesareconsidered:zerothroughthreeloads,orfault mode(metalobjectonthetransmitter).Fornoloadorfaultmode,thereisonlythe discretestatetobeestimated.Forthe1-3loadcases,thediscretestate(numberof loads)aswellasthecontinuousstate(chargestatusoftheloads'batteries)mustbe estimated.Thegoverningequationisforthechargeinthebatteryanditstimerateof change,wherevectorsareusedtoindicatethepossibilityofamultiplicityofloads.Given theDCreceivedpower( P DC ),andthexedregulatoroutputvoltage( V reg ), @ ~ Q @ t = ~ I = ~ P DC V reg (8) Relating chargetoequivalentresistance,thendiscretizing: @ f ( ~ R ) @ t = ~ P DC V reg (8) @ f @ R @ ~ R @ t = ~ P DC V reg (8) 85

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~ R k = ~ R k 1 + @ f @ R 1 ~ P DC ( ~ R k 1 ) V reg + n k 1 (8) where k is timeindexand n isprocessnoise.Themeasurementequationis (V in I DC )= h ( ~ R k )+ k (8) where h isthemeasurementfunctionand ismeasurementnoise.Thefunctional relationships ~ P DC ( ~ R k 1 ) and h ( ~ R k ) areknownfromderivationsinpreviouschaptersand alreadycodedinMatlab.Itsufcestosaytheyarenonlinear. f relateschargetoload resistanceandisdenedasfollowsforthepurposesofthischapter: Q = Q 0 R R 0 (8) f or R R 0 Q =( Q 1 Q 0 ) R R 0 R 1 R 0 + Q 0 (8) f or R 0 < R R 1 Q = Q 1 (8) for R > R 1 Q Q 1 ,and Q 0 areinCoulombs. R R 0 ,and R 1 arein .Forthis instance, Q 0 =36, Q 1 =3600 R 0 =1,and R 1 =100. 8.3.2ParticleFilterAlgorithm Theparticlelteralgorithmiscomposedofseveralsteps,detailedinthefollowing sections. 8.3.2.1Datasetgeneration Thisrststepforloadtrackingisnottechnicallyaparticlelterstepbutisnecessary fortests.AtruthdatasetisgeneratedusingtheMonteCarloMarkovChainmethod andstate/measurementmodelpreviouslydescribed.Thisprovidesthemeasurements 86

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V in and I DC to beusedandmode( m k )statevalues( R k )tobeestimated.Thischapter usesaone-hoursequenceofmeasurementswithsampleseveryonesecond. 8.3.2.2Initialization Therststepistogenerateaninitialgroupof N particles.Thisisdonewitha uniformpriordistributionforthediscretemodes ^ m i 0 andGaussiandistributionforeachof theloads ^ R i 0 (i isparticleindex).Attheendofthisstep,eachparticlehasbeenassigned adiscretemode,andifthemodehasloadsasociatedwithit,theloadshaveresistance values. 8.3.2.3State Thenextstepistoupdatetheparticlesaccordingtothestatemodel.TheMarkov transitionmatrix a ij isusedatthispointtodetermineifanyparticlewillmovetoanew mode.A(0,1)uniformrandomvariable u isgeneratedandtheCDFforeachcurrent modeiscalculated(thecurrentmodeis i ): F n = j = n j =1 a ij (8) Thisisusedtodeterminethenextmode.Foreach u ,when F n 1 < u < F n (8) thenextmodeismode n .Ifthereisamodechangefromtime k 1 totime k ,the statesofanyloadsarere-initialized.Forexample,ifthetransitionisfromno-loadto two-load,thetwoloadsareassignedinitialstatevaluesasdescribedinSection 8.3.2.2. 8.3.2.4Measurement Usingthemeasurementequations,estimatesof ^ V in and ^ I DC areobtained.Then, usingthe(assumedknown)distributionsofmeasurementnoise,andmeasured V in and I DC ,theconditionalprobabilitiesofeachparticle i attime k (w i k )areobtained.The collectionof Nw i k 'sarethenscaledsothattheysumtoone. 87

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w i k = w i k 1 p (V in I DC j ^ V in ^ I DC ) (8) = w i k 1 P ( V in ^ V in I DC ^ I DC ) j ~ (8) w i k = w i k i w i k (8) where P is thenoisepdfand ~ isitsparametervector. 8.3.2.5Update These w areusedtoupdatetheparticles.Twomethodsareconsideredinthis chapter.Therstisasdescribedabove,keepingthesameparticlesbutrescalingtheir weights.ThesecondincorporatesresamplingtheparticlesusingtheCDF C j k fromtheir weights C j k = i =j i =1 w i k (8) Thisisusedtoresamplebygenerating N instancesa(0,1)uniformrandomvariable u .Foreach u i ,when C j 1 k < u i < C j k (8) thenewresampledparticle i istheoldparticle j ,andalltheresampledparticles weightsareconsidereduniform.Thisissupposedtoallieviateproblemsofthe distributiondegeneratingtoasingleparticlewithweight1. Thesemethodswillbereferredtoasparticlelterwithandwithoutresampling. 8.3.2.6Estimate Theestimateateachtimestepistheweighted(by w )sumoftheparticles'values. 88

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^ m k = w i k m i k (8) ^ Q k = w i k Q i k (8) = w i k f (R i k ) (8) Themodesaregivennumericalvaluesasfollows: m =1isoneload,2istwoloads, 3isthreeloads,4isnoload,and5isfaultmode. 8.3.3Tests Thestatemodel,measurementmodel,andparticlelters(withandwithout resampling)describedabovewereusedwithagenerateddataset. N wassettobe 10,100,or1000.Thepriordistributionofmodeswasassumeduniform( p =1= 5 for eachmode),andtransitionprobabilitiesbetweenmodesweredened a ij =0.996 if i = j and 0.001 otherwise.ProcessnoiseandmeasurementnoisewereassumedGaussian, thoughitshouldbenotedthatthisisnotarequirementoftheparticleltertechnique. Noisecancomefromanyarbitrarydistributionwithaparticlelter.Thestandard deviationandmeanofthepriordistributionoftheloadresistances(forinitialization)are 0.1and1 .Thestandarddeviationandmeanoftheprocessnoiseare0.1and0.0 For V in ,themeanofthemeasurementnoiseis0.0Vandthestandarddeviationis0.1V; for I DC ,themeanofthemeasurementnoiseis0.0Aandthestandarddeviationis0.01 A. 8.3.4Implementation Theparticlelterloadtrackingschemewasimplementedinaphysicalsystem, usingthetestsetupasinChapter 5 ,withthe4cmby5cmreceivers.Thecomponent selectionisthesame.TheMATLABparticleltercodewasimplementedinC++ andcombinedwiththecodeusedforcontrollingtheelectronicloads,DCsource, oscilloscope,andfunctiongenerator.The V in and I DC measurementswereobtainedfrom theoscilloscopeandDCsource,andtheelectronicloadswereprogrammedtobehave 89

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according tothepiecewise-linearmodelestablishedearlier.Processnoisewas0.1 ;voltagenoisewas0.5V;andcurrentnoisewas0.03A.Thetransitionprobabilities betweenmodesweredened a ij =0.9 if i = j ,and 0.025 otherwise.Thesystemwas testedindifferentmodesindividually,andinasequenceofnoload,1load,2loads,3 loads,2loads,1load,0load,andfault.Forthesequencetests,theparticlelteruseda hierarchicalimplementation,inwhichrstthenoloadandfaultmodeareruledout.If I dc wasgreaterthan1Athenfaultmodewasdetermined.If I dc waslessthan0.4A,and V in wasgreaterthan70V,thesystemwasdeterminedtobein0loadcondition.Otherwise, theparticlelterwasasintheMATLABsimulations. 8.4SimulationResults Figure 8-1.Generatedmeasurementsin( V in ,I DC )space. Fig. 8-1 showsthegeneratedtruthdatasetinthe (V in I DC ) space.Thepoints arecolorcodedtoindicatethediscretemode.Noticehowclosethedifferentmodes 90

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are inthefeaturespace.Actually,inrealwirelesspowersystemstestedinthelab,this separationisgreater,sodiscriminationshouldbeeasier. Figure 8-2.True(blue)andestimated(red)modeforN=10,withresampling. Figs. 8-2, 8-3,and 8-4 showtheestimatedandtruemodetimeseries( ^ m k and m k ) for N as10,100,and1000.Whatisthephysicalinterpretationofthetruthtimeseries? Thesystemstartsinfaultmode(sayametalsheetisplacedonthetransmittingcoil), thenaloadisplacedonthetransmittingcoil.Thisloadcharges,thenthesystemgoes intofaultmodeagain.Themetalsheetisremovedandthesystemisinnoloadstate. Oneloadisplacedonthetransmittingcoil;itcharges,thenanotherloadisplaced;it charges,thenit'sremoved.Thesystemissomehowplacedinfaultmodeagain,followed byalongstretchoftimewiththreeloads.Thisisfollowedbyabriefunloading,then oneload,thentwoloads,thenashortunloading,thenfaultmode.Finally,twoloadsare placedonthetransmittingcoiltocharge. 91

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Figure 8-3.True(blue)andestimated(red)modeforN=100,withresampling. Theestimatedmodematchesthetruemodemorecloselyas N isincreased. Transitionsbetweenmodesaremissedbecausenoneoftheparticles,afterresampling, areinthenewmode,soitisimpossibletodetectthatchange.Essentially,with resampling,theparticle'sdistributionlacksenoughvariabilitytodetectsuddenchanges. Figs. 8-5,8-6,and 8-7 showtheestimatedandtruemodetimeseriesfor N as10, 100,and1000.Ingeneral,withoutresampling,theestimateismorevariablebecause alloftheparticlesarethere,insteadofbeingresampled.Ononehand,thisisbad, becausetheestimateisnoisy.Ontheotherhand,thisisgoodbecausetheincreased variablityallowstheltertocapturetransitionsthatthelterwithresamplingmisses. Plotsofthechargestatetimeseriesarenotincludedastheychangenotjustin valuebutalsoindimension(1,2,or3loads)overtime. Fig. 8-8 showsthemodeandstaterootmeansquareerror(RMSE)asafunction of N ,forbothwithandwithoutresampling.StateRMSEiscalculatedintermsofthe 92

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Figure 8-4.True(blue)andestimated(red)modeforN=1000,withresampling. chargestatusoftheloads,nottheloadresistances.RMSEsdecreasewithincreasing N aswouldbeexpected.ThemodeRMSEsarelowerfortheparticlelterwithout resamplingthanwithresampling;thestateRMSEsarelowerwithresamplingthan without.Withresampling,theparticleshavenergranularity(lowervarianceinmode andcharge;seeFig. 8-9);withoutresampling,theparticleshavecoarsergranularity (greatervarianceinmodeandcharge;seeFig. 8-10).Theimplicationofthisis thatthelterwithoutresamplingwillbemoreabletocatchthesuddentransitions betweenmodes,butnotthegradualchargingofthedeviceordevices.Withresampling, transitionsaremorelikelytobemissedbecauseitcouldbethatafterresamplingthatall theparticleshavethesamemode. 93

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Figure 8-5.True(blue)andestimated(red)modeforN=10,withoutresampling. 8.5MeasuredResults Fig. 8-11 showsthepredictedandobserved( V in I DC )spaceforall5modes,tested individually,withN=10000.Fromthisitisevidentthatthestate/measurementmodelis accurate.Theprimarydifcultyisindetectingtransitionsbetweenmodes. Figs. 8-12 through 8-17 showsthepredictedandobservedmodetimeseries,and thepredictedandobservedstateandmeasurementvariablesforN=100,N=1000,and N=10000.Duetothehierarchicaltechnique,noloadandfault(modes0and4)are detectedreadilyforallNvalues.However,resultsaregenerallypoor. ForN=100andN=1000(Figs. 8-12 and 8-14)theestimatedmodeisusuallywithin 2.ForN=10000(Fig. 8-16 )theestimateiswithin1.Thepooraccuracycouldbedueto thenumerousuncertaintiesinthesystemthatareunknowninthetrackingmodel.For instance,thetruetransitionprobabilitymatrix,processandmeasurementnoiseareall unknown.Amorerenedtechniquemightnotrelyonsuchpreciseknowledge. 94

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Figure 8-6.True(blue)andestimated(red)modeforN=100,withoutresampling. TheloadresistanceandpowerdeliveryestimatesforN=100and1000(Figs. 8-13 and 8-15 )showpoorperformance,dueinnosmallparttotheincorrectlyestimated mode.ThisperformanceimprovesforN=1000(Fig. 8-17)asthemodeismoreclosely estimated.WithoutahigherN,thecontinuousanddiscretestatetrackingaccuraciesare probablyinsufcient.Thisnumberofparticles(morethan10000)isprobablyimpractical foranon-boardmicroprocessor. 8.6Conclusion Inconclusion,theparticlelterdoesworkforfaultdetection/loadtracking.However, thenumberofparticlestoachievesatisfactoryresultsisimpracticalforanon-board microprocessor.Twochangescouldbeimplemented.Oneistoonlydoselective resampling:iftheprobabilitymassfunctionoftheparticleshasinsufciententropy(or someothermetric)onlythenisresamplingconducted.Thiswouldmaintainvariability whileavoidingdegeneracyofthedistribution.Thestatemodelcouldbesimplied 95

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Figure 8-7.True(blue)andestimated(red)modeforN=1000,withoutresampling. whichwouldreducecomputationalcomplexityforimplementationonanARMorsimilar microprocessor. Inadditiontochangesforperformance,modelchangescouldbeimplemented. Thecharge-resistancemodelusedinthischapterwasfairlyarbitrary;ameasured charge-resistancecurveforaparticulardeviceshouldbeused.Differentreceivertypes andrelativepositionscouldbeincluded.Ingeneral,theparticlelteralgorithmisan effectivetoolforchallengingnonlineardiscriminationproblemssuchaswirelessload tracking. 96

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Figure 8-8.RMSEofmodeandstates. Figure 8-9.Modeandchargeestimatevariance,withresampling. 97

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Figure 8-10.Modeandchargeestimatevariance,withoutresampling. Figure 8-11.Testofdifferentmodesin( V in ,I DC )space,inrealsystem. 98

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Figure 8-12.True(blue)andestimated(red)modeforN=100,withoutresampling,inreal system. Figure 8-13.Predictedandobservedpower,resistance,andinputvoltage,andDCinput currentforN=100,withoutresampling,inrealsystem. 99

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Figure 8-14.True(blue)andestimated(red)modeforN=1000,withoutresampling,in realsystem. Figure 8-15.Predictedandobservedpower,resistance,andinputvoltage,andDCinput currentforN=1000,withoutresampling,inrealsystem. 100

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Figure 8-16.True(blue)andestimated(red)modeforN=10000,withoutresampling,in realsystem. Figure 8-17.Predictedandobservedpower,resistance,andinputvoltage,andDCinput currentforN=1000,withoutresampling,inrealsystem. 101

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CHAPTER 9 MIDRANGEWIRELESSPOWERTRANSFER 9.1Introduction Inpreviouschapters,thereceiver-transmitterdistanceconsideredwasless than5mm.Thisisappropriatefortheprimaryapplicationconsidered,charging battery-operatedelectronics.However,forsomeapplicationsitmaybedesirableto extendthereceiver/transmitterdistancetoaboutthesizeofthecoilsthemselves. Thisisconsideredmidrangecoupling.Theapproachtakenin[ 7 53]istousehigh-Q, electromagneticallyresonantstructurestoformastrongcoupling.Thefrequenciesused areinexcessof10MHzandcouplingefcienciesof90%areachievedatdistancesof 75cm.Inaddition,ratherthantheconventionalinductivecouplingequationsconsidered earlierinthisdissertation,[ 7 ]usescoupledmodetheory[ 5456].[53 ]usesahigher frequency(10MHz)andcoupledmodetheory,relyingontheself-capacitanceofthe coilstoachieveresonance,thoughtotalefciencyislow( 40%),dueinparttotheir selectionofaColpittsoscillatorasthedrivingcircuit.[ 57 ]usesthisresonanttechnique butwithlumpedcapacitorstopoweranLEDatadistanceofafewcentimeters.The papersusingresonanceallrelyonatotaloffourcoilsforeveryreceiver-transmitterpair: atransmittingcoil,tworesonantlycoupledintermediarycoils,andareceivingcoil.This chaptertriestoextendthearchitectureconsideredinpreviouschapterstomidrange distanceswhilemaintaininghightotalefciency.First,thecoupledmodetheoryanalysis iscomparedtotheinductivecouplinganalysis.Next,coildesignisreconsideredfor midrange.Then,designrulesaredevelopedfortheseries-parallelarchitecture(and others)toextendtheclassE'sutilitytomidrangecoupling.Inductance,frequency, andcircuittopologyeffectsaretestedonanactualsystemandevaluatedintermsof powerandefciency.Theidealtopologyformidrangeisfoundandtestedwithregards toitssensitivitytocomponenttolerancesandrelativereceiver-transmitterpositioning. Ultimately,onesystemisdesignedwithpeaktotalefciencyof69.2%andpeakreceived 102

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po werof0.94Wandat25cm,andanotherisdesignedwithpeaktotalefciencyof 57.9%andpeakreceivedpowerof3.78Wat1m. 9.2Analysis Coupledmodetheoryfortworesonantobjectscanbedescribed[ 53]by da 1 dt = j 1 a 1 1 a 1 + j a 2 (9) da 2 dt = j 2 a 2 2 a 2 + j a 1 (9) where a i is astatevariableofobject i ,suchthatitssquarehasunitsofenergy; i istheresonantfrequency; i istheloss,withunitsoffrequency;and isthecoupling coefcient,withunitsoffrequency. Tocomparethistotheinductivesystem,statevariables,losses,andcouplingare denedasfollows: a i = r L i 2 I i (9) i = R pi L i (9) = M p L 1 L 2 (9) = k (9) Q i = i L i R pi (9) i = 1 p L i C i (9) where L i s thecoilinductance, I i isthecoilcurrent, R pi istheparasiticresistance, M is themutualinductance, Q i isthequalityfactor,and C i istheresonantcapacitancevalue. Strongcoupling,necessaryformidrangetransfer,occurswhen p 1 2 1 (9) 103

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which isequivalentto M p R p 1 R p 2 1 (9) 9.2.1 CoilDesign Coildesignforthemidrangesystemhasdifferentconsiderationsthanforthe near-eldsystem.Whereasbefore,theprimaryconcernwasevenelddistribution, nowtheobjectiveismaximalefciencyatalargeseparationdistance.Todothis, theremustbestrongcouplingasdescribedabove,andthe Q mustbeashighas possible.Thismeanstheinductanceshouldbeashighaspossiblewiththeminimum parasiticresistance.So,bothreceiverandtransmittercoilsshouldhavethemaximum inductanceforagivenlengthofcoil.Sinceweareconsideringaseparationdistance( d ) approximatelyequaltothedimensionofthecoil( D ), D isconstrainedbythedesired distance.Usingtheinductanceformulasfrom[ 58 ],andforsimplicity,assumingDC resistance,the Q ofaplanarcircularcoilis: Q = L R (9) L = (D = 2) 2 N 2 8( D = 2) +11N (2 a ) (9) R = DN 2 a 2 (9) L = D 2 N 2 16 D + 88 Na (9) R = DN 2 a 2 (9) Q = 2! a 2 DN 16D + 88Na (9) where isresistivity, N isnumberofturns,and a iswireradius.Forasquarecoil, 104

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Q = L R (9) L = N 2 2D (ln ( D = a ) 0.77401) (9) R = 4DN a 2 (9) Q = 2! N a 2 ( ln (D = a ) 0.77401) (9) Since D is xed, Q canbeincreasedbyincreasing a or N .Foragivenwireradius, forbothcoils, Q isanincreasingfunctionof N .Intuitively,inductancegoesas N 2 and parasiticresistancegoesas N ,thereshouldbenooptimal N .Theconstraintthen becomestheimpactofthecoilinductanceoncomponentselection,ie,thecapacitor sensitivitydiscussedinChapter 5. 9.2.2ComponentSelection Inthischapter,thereceiverandtransmittercoilsareconsideredindenticalin geometryandthusinductancetosimplifycomponentselection.Theyaremaderesonant throughpropercapacitorselection: L 1 = L 2 (9) C 1 = C 2 (9) C =( 2 L ) 1 (9) where C 1 istheresonanttransmittercapacitorand C 2 isthereceiverresonantcapacitor. Becausethecapacitorselectionisdoneforresonance,thedesignrulesare differentfromthosederivedinChapter 2.Inaddition,thedesireableperformanceof theseries-paralleltopologyfornear-eldinduction(decreasingpowerdeliverywith increasingload)isnotpresentwhenthereceivercapacitorischosenforresonance. Thus,threecircuittopologieswillbeconsideredforthemidrangesystem:theseries-parallel, 105

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Figure 9-1.MidrangeclassEseries-parallelarchitecture. theseries-series,andtheTnetwork.Thefollowingsectionsdetailthedesignrulesfor each. 9.2.2.1Series-parallel Designrulesfortheseries-paralleltopologyinresonance,showninFig. 9-1,are basedonsimilarconstraintstothosedevelopedinChapter 2 butwiththereceiver capacitorchosenforresonanceinsteadofforobtainingaspecic R 0 atthemaximum realpartof Z in Z in = 4 M 2 C 2 R L j 3 M 2 C (9) Z tx = 1 j C out + j L out + 4 M 2 C 2 R L j 3 M 2 C (9) As before, L out and C out arechosentomeet Q andphaserequirements,and C t id chosentoobtainZVS: L out = Q 4 M 2 C 2 R L + 3 M 2 C (9) C out =( L out 3 M 2 C (1+ tan ( ) CR L )) 1 (9) C t =2((1+ 2 = 4) 4 M 2 C 2 R L ) 1 (9) where R L istheloadresistancewherethepahseis 106

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Figure 9-2.MidrangeclassEseries-seriesarchitecture. 9.2.2.2Series-series Similarly,designrulesfortheseries-seriesarchitecture,showninFig. 9-2,are developed: Z in = 2 M 2 R L (9) Z tx = 1 j C out + j L out + 2 M 2 R L (9) L out = Q 2 M 2 R L (9) C out = L out 2 M 2 R L tan() 1 (9) C t = 2 (1+ 2 = 4) 2 M 2 R L 1 (9) 9.2.2.3 T-network TheideabehindtheTnetwork,showninFig. 9-3,istoaddanadditionaldegreeof freedomtothedesignofthemidrangeseries-parallelarchitecturesothatadesireable impedanceresponsemaybeobtained. X 2 and X 3 aregeneralreactances. 107

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Figure 9-3.MidrangeclassETnetworkarchitecture. Z in = 4 M 2 C 2 R L j 3 M 2 C (9) Z tx = jX 3 + 1 jX 2 + 1 Z in 1 (9) Z tx = jX 3 + 3 M 2 CX 2 + j 4 M 2 R L C 2 X 2 4 M 2 R L C 2 + j (X 2 3 M 2 C ) (9) X 3 = k 2 R L tan()X 2 2 + k MX 2 (X 2 k M ) k 4 R 2 L X 2 (X 2 k M ) 2 + k 4 R 2 L (9) 9.3 PreliminaryTests Toevaluatehowtobestmaximizeefciencyatmidrangedistances,theeffectsof coilinductance,operatingfrequency,andcircuittopologywereinvestigated.Inaddition, theimpactofthediodeparasiticcapacitancewasinvestigated. FortestsinSections 9.3.1, 9.3.2,and 9.3.3,theDCsupplyvoltagewas6Vandthe transistorusedwastheIRLR3410NMOS. 9.3.1RectifyingDiodeEffects Therectifyingdiodeinthehalf-waverectiercontributessomecapacitance(10-30 pF).Astheoperatingfrequencyincreases,thisbecomesincreasinglyimportantinterms ofachievingresonance.Todemonstratethediode'seffect,thesystemwascongured withtheseries-parallelarchitectureat761.79kHz,withreceiverandtransmittercoils30 108

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T able9-1.Componentvalues. Component w/ diodecompensation w/o diodecompensation C 2 0.614 nF 0.808 nF C 1 0.808 nF 0.808 nF C out 10.07 nF 10.07 nF C t 6.63 nF 6.63 nF L out 5.7 H 5.7 H Figure 9-4.Diodeeffectsonsystemperformance. cmsquare,with8turns,constructedof420/42Litzwire.Theseparationdistancewas 25cm.ThecomponentselectionsfortwosystemsareshowninTable 9-1;onesystem hascompensationforthediodecapacitance,andtheotherdoesnot.Thevaluesshown aremeasuredvalues. Fig. 9-4 showstheperformancecurvesforthetwosystems.Thenon-compensated systemhasworseperformance.Peakefciencyisabout20%lower,andthepeak receivedpowerisabout0.05Wlower,comparedtowhenthediodeistakeninto account. 109

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T able9-2.Componentvalues. Component 240 kHz,5turns 757 kHz,5turns 761.79 kHz,8turns C 2 18.45 nF 1.856 nF 0.614 nF C 1 18.45 nF 1.858 nF 0.808 nF C out 70.66 nF 10.06 nF 10.07 nF C t 26.3 nF 10.01 nF 6.63 nF L out 9.68 H 5.7 H 5.7 H Figure 9-5.Frequencyandinductanceeffectsonsystemperformance. 9.3.2FrequencyandInductanceEffects Toevaluatetheeffectsofcoilinductanceandfrequencyonpowerdelivery,three systemsweretested.A240kHzsystem,with30cmsquarecoilsof5turns;a757kHz system,with30cmcoilsof5turns;anda761.79kHzsystem,with30cmcoilsof8 turns.Theseparationdistancewas25cm.Table 9-2 showstherelevantcomponent values. Fig. 9-5 showstheperformancecurvesofthethreesystems.Ingeneral,as inductanceandfrequencygoup(anincreasein Q ofthecoils),thepeakefcienygoes 110

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T able9-3.Componentvalues. Component Ser ies-parallel Ser ies-series T -network C 2 0.614 nF 0.808 nF 0.614 nF C 1 0.808 nF 0.808 nF 0.808 nF C out 10.07 nF 10.81 nF 8.96 nF L p n/a n/a 1.05 H C t 6.63 nF 5.71 nF 11.1 nF L out 5.7 H 5.7 H 5.7 H Figure 9-6.Topologyeffectsonsystemperformance. upandpeakpowerdeliverygoesdown.Thisisbecausetherealpartof Z in increases withincreasing M and .Thehigherrealpartisincreasinglygreaterthantheparasitics, leadingtohigherefciency,andthehigherrealpartresultsinlowercurrentandthus lowerpowerdelivery. Thestrengthofcouplingasdescribedbycoupledmodetheorymentionedin Equation 9 isgreaterthan1foronlythe761.79kHzsystem. 111

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9.3.3 TopologyEffects Threetopologieswereevaluated:series-parallel,series-series,andtheT-network, allat761.79kHzwiththe8turncoilsasbefore.Thecomponentselectionsareshownin Table 9-3. Fig. 9-6 showstheperformancecurvesforthethreetopologies.Theseries-series hasthehighestpowerandefciency,inadditiontopreservingthedesireabletrend ofdecreasingpowerwithincreasingpowerdelivery.Thisisbecausefortheresonant series-series, Z in isapurelyreal,decreasingfunctionof R L ,andthephaseof Z tx is monotonicallyincreasing.Forseries-parallelandT-network,theimpedanceresponseis asinpreviouschapters,excepttheminimumphaseisatamuchlargerloadresistance. Additionally,thetopologieswhichhaveaparallel C rx haveahighervoltageontheload, potentiallybeyondthevoltageratingsoftherectierandcapacitor., 9.3.4Sensitivity Havingestablishedtheseries-seriesasthebesttopologyformidrangetransfer, thissectionperformsasensitivityanalysiswithregardsto:frequency,size,separation distance,andnumberofturns;andforthe761.79kHzsystemdescribedintheprevious sectionthepositionandcomponenttolerances. M = N 2 D log p d 2 + D 2 + D d 2 p d 2 + 2D 2 + D p d 2 + 2D 2 D + d + p d 2 + D 2 2 p d 2 + 2D 2 (9) (9) [ 59] P tx = 2 V 2 cc 1 + 2 = 4 R L + R p 2 M 2 + R p (R L + R p ) (9) P rx = 2 V 2 cc R L 1 + 2 = 4 (R L + R p ) 2 =k 2 + 2 M 2 (! 2 M 2 + R p (R L + R p )) 2 (9) 112

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Figure 9-7.Effectof D d ,and f ontotalefciencyat N =8. Figure 9-8.Effectof D d ,and f onreceivedpowerat N =8. 113

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Figure 9-9.Effectof N f ,and D ontotalefciencywhere D = d Figure 9-10.Effectof N f ,and D onreceivedpowerwhere D = d 114

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E = 1 (9) c = R L R L + Rp (R L + R p ) 2 = k 2 + 2 M 2 2 M 2 + R p (R L + R p ) (9) Using theseries-seriestopology,thecoildimension( D ),theseparationdistance (d ),thefrequency( f ),andthenumberofturns( N ),weresweptsimultaneously.For every D d f N point,themaximumreceivedpowerandtotalefciencywerecalculated. Fig. 9-7 showstheeffectsof D d ,and f ontotalefciencyat N =8.Efciencyis highestwhen d D andincreasesas f increases. Fig. 9-8 showstheeffectsof D d ,and f onreceivedpowerat N =8.Poweris highestwhere d isslightlyhigherthan D .Thisisbecausetheweakercouplingleadsto asmallerrealpartof Z in ,leadingtolowerefciencybuthigherpowerdelivery. Fig. 9-9 showstheeffectsof N f ,and D ontotalefciencywhere D = d .Efciency increaseswithincreasing f ,increasing N ,andincreasing D .Thisisbecausethereal partof Z in increaseswith f andmutualinductance,andmutualinductanceincreases withincreasingnumberofturnsandcoildimension. Fig. 9-10 showstheeffectsof N f ,and D onreceivedpowerwhere D = d Powerdeliveryshowsthetrendofdecreasingwithincreasing N whileincreasingwith increasing f .As D increases,thehighestpowerdeliveryoccursatlower N Thesystems'sensitivitytoreceiverplacementwastestedbymeasuringtheload responseatdifferentvertical( z )andlateral( x y )offsets.Theperformancecurvesare showninFig. 9-11.Theoffsetvectorforeachcurveisindicatedinthelegend.For example, [0,7.5,0] indicatesthereceiver'scenterisdisplaced7.5cminthe y direction fromthetransmitter'scenter.Ingeneral,theefciencyandpowerdeliverydecrease withincreasingreceiveroffset.Theworstperformanceiswhenthereceiverisoffsetby [15,15,0] (thelargestabsoluteoffset):thereceivedpowerisdecreasedby0.5Wand theefciencyisdecreasedby15%.However,thesedecreasesaresmallenoughforall 115

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Figure 9-11.Coiloffseteffectsonsystemperformance. otheroffsets(lessthan0.2Wand9%)toconsiderthesystemrobustwithregardsto receiver/transmitterplacement. AnidenticalMonteCarloanalysisasinChapter 2 wasperformedfortheseries-series system.The95%condenceintervalsofefciencyareshowninFig. 9-12 andofpower areshowninFig. 9-13.Theefciencycondenceintervalsexhibitaskewsimilartothat inChapter 2,indicatingthecomponentselectionisdonetooptimizeefciency.The powercondenceintervalsshowhighupperboundsandapeakaround100-200 .This isbecauseasthecomponentsvary,thesystemgoesoff-resonance.Theoffresonant systemislikethesystemfrompreviouschapters,withapeakpowerdeliveryratherthen amonotonicallydecreasingpowerdelivery.Thepowerdeliverycanbehigher(though theefciencyislower)becauseatoff-resonancetherealpartof Z in issmallerthanat resonance. 116

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Figure 9-12.Efciencyatnominalcomponentvalues(blackline)and95%condence intervalsat5%(redlines),10%(bluedash),and20%(blackdash-dot) componenttolerancesforthemidrangeseries-seriessystem. Table9-4.Componentvalues. Component V alue C 2 0.371 nF C 1 0.372 nF C out 19.3 nF C t 3.02 nF L out 5.7 H 9.4 Synthesis 9.4.150cmSeparation Thesystemwasconguredwiththeseries-seriesarchitectureat758.1kHz,with receiverandtransmittercoils50cmsquare,with8turns,constructedof100/40Litzwire. Theseparationdistancewas50cm.Thecomponentselectionsforthesystemisshown inTable 9-4.Thevaluesshownaremeasuredvalues. 117

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Figure 9-13.Poweratnominalcomponentvalues(blackline)and95%condence intervalsat5%(redlines),10%(bluedash),and20%(blackdash-dot) componenttolerancesforthemidrangeseries-seriessystem. Fig. 9-14 showstheperformancecurves.Thepeakefciencyisabout52.6%and thepeakpowerdeliveryisabout0.13W.Greaterefciencycouldbeachievedwith betterLitzwire,asthecoilparasiticsforthe50cmsystemwererelativelyhigh(2.6 ,comparedtothe30cmsystem's0.6 ).Itshouldbepossibletoincreasethetotal efciencyatgreaterdistancesbyusinglesslossycoils,withLitzofhighergaugeand greaterstrandnumber. 9.4.21mSeparation Asystemwith1mcoilseparationwasconstructedusing1725strand,48AWGLitz wiretobuildcoilsof1msquareandisshowninFig. 9-15 .Thecoilswereattachedto foamposterboardandhungfromtheceiling.Numberofturns,frequency,supplyand gatevoltages,anddutycyclewerevariedinanattempttomaximizeefciency.The 118

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T able9-5.Summaryof1mtests. Maxim umefciency(%) 44.6 47.9 48.8 51.2 55.1 56.3 56.8 57.9 54.3 Maxim umreceivedpower(W) 4.91 4.98 0.82 1.65 5.59 4.57 4.37 3.78 3.84 T urns 4 4 4 6 6 6 6 6 8 M (H) 1.83 1.83 1.83 4.10 4.10 4.10 4.10 4.10 7.23 L 1 (H) 69.13 69.13 69.13 158.74 158.74 158.74 158.74 158.74 274.70 L 2 (H) 68.51 68.51 68.51 160.67 160.67 160.67 160.67 160.67 268.79 f (kHz) 730.00 706.00 701.00 510.34 510.34 511.88 511.88 513.50 438.55 Duty cycle(%) 50 50 50 50 50 50 45 40 50 Vds (V) 12 12 6 12 20 20 20 20 20 Vgs (V) 5 6 5 10 10 10 10 10 10 C 2 (nF) 0.605 0.653 0.653 0.609 0.609 0.609 0.609 0.609 0.500 C 1 (nF) 0.590 0.651 0.607 0.609 0.609 0.609 0.609 0.609 0.490 C out (nF) 1.62 1.65 6.72 7.58 7.58 7.58 7.58 7.58 8.08 C t (nF) 0.991 0.503 3.30 3.90 3.90 3.90 3.90 3.90 2.20 L out ( H) 29.10 29.10 8.62 18.46 18.46 18.46 18.46 18.46 18.46 119

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Figure 9-14.50cmsystemperformance. transistorusedwastheIRF640NMOS,andthereceiverrectierwasafull-bridgeusing IR10MQ060Ndiodes. Table 9-5 summarizestheresultsofmanytestsconductedtondthedesign maximizingefciencyforthe1msystem.Increasingnumberofturnsincreasesthe mutualandselfinductances,raisingthe Q andincreasingefciency;however,sincethe inductanceissohigh,thevoltageonthetransmittingcoil'sleadsishighenoughtolead toarcingbetweenadjacentturns.Thismakestuningandtestingthesystemformore than6turnsdifcult.Increasingfrequencyraisestherealpartoftheimpedanceseen bytheclassEinverter,whichwouldincreaseefciencyifnotforthefactthatthecoil parasiticsincreaseaswell.Increasingsupply(Vds)andgatevoltages(Vgs)ensures theMOSFETswitchescompletelyintosaturationwhenon.Inaddition,lowerVdsis associatedwithstrongernonlinearityintheoutputcapacitanceoftheMOSFET,making 120

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Figure 9-15.1msystemsetup. 121

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Figure 9-16.1msystemperformance. tuning C t forZVSmoredifcult.Finally,changingthedutycyclecanensureZVSand zeroderivativeswitching(ZDS)atloadresistanceswhereotherwisethiswouldnotbe thecase. Fig. 9-16 showstheperformancecurvesofthebestperformingdesign.Thepeak efciencyisabout57.9%andthepeakpowerdeliveryisabout3.78W.Theefciency wasimprovedbyusinglesslossyLitz(forthe6turn1msystemat513.50kHz,the parasiticsareabout2.6 ,whilewith420strand,42AWGLitztheparasiticswereabout twiceashigh)orbyincreasingthenumberofturns. 9.5Conclusion Thenear-eldwirelesspowersystemconsideredin[ 16]and[ 60 ]wasextendedto midrangedistances,wherethecoilsizeiscomparabletotheseparationdistance.The effectsofcoilinductances,frequency,circuittopology,andcoilpositionsaretestedona system.Itisfoundthattheseries-seriestopologyisbestformidrangepowertransfer; 122

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that thereisatradeoffbetweenpowerandefciency,whereefciencyincreasesand powerdecreaseswithincreasingfrequencyandcoilinductance;andthattheresonant tuningmakesthesystemrobustwithrespecttovariationsincoilpositions.Ultimately, onesystemisdesignedwithpeaktotalefciencyof69.2%andpeakreceivedpowerof 0.94Wandat25cm,andanotherisdesignedwithpeaktotalefciencyof57.9%and peakreceivedpowerof3.78Wat1m. 123

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CHAPTER 10 FAR-FIELDWIRELESSPOWERTRANSFER 10.1Introduction Figure 10-1.Anexampleofaradiofrequency(RF)harvestingwirelesssensornode[ 3]. Figure 10-2.Anillustrationofthesolarpowersatellite(SPS)concept[ 61 ]. Far-eld,orradiative,powertransferoccourswhenthedistancebetweenthe transmitterandthereceiverexceedstheRayleighdistance, D > 2d 2 =,where d is thecharacteristicdimensionofthetransmitter,undertheconditionthat d islargerthan .Twomajorapplicationsofradiativewirelesspowertransfer(WPT)areinambient radiofrequency(RF)harvesting(Fig: 10-1)andtheSolarPowerSatellite(SPS)(Fig. 10-2). Theideabehindthersttechniqueistoconverttheradiowavesfromcommunications intopower.Sincethepowerlevelsarelow,typicalapplicationsincludewireless sensorsfreefrombatteriesandRFIDtagsequippedwithsmallcomputationalabilities 124

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Figure 10-3.Atmospheremodelschematicusedinthischapter[ 63 ]. Figure 10-4.Canopymodelschematicusedinthischapter[ 64 ]. [3, 4 ].TheSPSisanideawhichcameaboutinthelate1960s[ 5]andhasseenreceived somerecentrevival[ 62].Theprincipleistocollectsolarenergyinspaceusinga geosynchronoussatellitewithlargesolarpanelsandthenconverttheenergyin microwaveformandbeamittoareceivingstationonearth. BothRFharvestingandSPSinvolveradiativetransferthroughparticipatingmedia. InRFharvesting,theparticipatingmediaisabsorbingandscatteringvegetationand urbanobstacles.InSPS,themediaincludestheionosphereandtheiceandwater 125

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par ticlesintheatmosphere.Thischapterwilldiscussthesimplifyingassumptionsmade andtheresultingsolutionstotheradiativetransferequationaswellasthepractical implications,fortwoexampleframeworks:transmissionthroughtheatmospherefrom spacebyasolarpowersatellite,andtransmissionthroughavegetationcanopyby anRFtransmittingtowerofsomekindtoanRF-harvestingsensornode.Fig. 10-3 illustratesthesimpliedexampleoftheformerconsideredinthischapter;Fig. 10-4 illustratesthesimpliedexampleofthelatter.Theframeworkforanalyzingthepower transferisaniteplane-parallelabsorbingand/orscatteringmedium,withanexternal beamofincidentuxattheupperboundary.Thecoordinatesystemstartsat0and extendsdownward.Thelowerboundaryisreecting. 10.2Theory Eachoftheseframeworksareexaminedwiththeradiativetransferequation(RTE). From[65]: cos( ) @ I @ z = I b I (10) + s 4 Z 2 0 Z 0 p ( 0 0 ) (10) I ( 0 0 z ) sin( 0 )d 0 d 0 (10) where I isintensity, isextinctioncoefcient, isabsorptioncoefcient,and s is scatteringcoefcient. p isthescatteringphasefunction.Thiscanbesimpliedusing axialsymmetry.Inaddition,sinceatmicrowavefrequencies,thermalemissionisvery lowincomparisontotheincidentpower,itisneglectedinthefollowinganalyses. @ I @ = I + $ 2 Z +1 1 p (, 0 )I ( 0 )d 0 (10) = cos( ) (10) = Z z 0 (z 0 )dz 0 (10) 126

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The boundaryconditionat z =0 is I (0, )=0 (10) for 1 0 ,withtheincidentradiation I (0, )= F ( 0 ) (10) where 0 isthedirectioncosineoftheincidentbeam. Thereisalowerboundary,withreectance.Intermsofaboundarycondition,this canbestatedas: I (h )= 1 2 Z 1 0 00 ( 0 )I (h 0 ) 0 d 0 (10) f or 1 < 0. 10.3SolutionDetails Thesolutiontechniqueusedhereisnitedifference(in )withGaussianquadrature (in ).InGaussianquadrature,intensity I isdiscretizedatspecic j ,thequadrature points,and R Id isapproximatedasaweightedsumoverthese mu j j a j I j .Applying nitedifferenceapproximation, i I i ,k +1 I i ,k 1 2 + I i ,k = j a j p ( i j )I j ,k (10) where i and j areangleindicesand k istheopticaldepthdiscretizationindex. Forapurelyabsorbingmedium,anisotropicallyscatteringmedium,andaRayleigh scatteringmedium,respectively,thephasefunctionsare p ( i j )=0 (10) p ( i j )=1 (10) p ( i j )=1+ P 2 ( i )P 2 ( j ) 2 (10) 127

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The upperboundaryconditionbecomes I i ,0 =0 (10) for i < 0 a i I i ,0 = F (10) for i = 0 Thelowerboundarycondition,fordiffusereectance,becomes R j a j I j ,K = I i ,K (10) f or i < 0 and j > 0 .Forspecularreectance,thisbecomes RI j ,K = I i K (10) for i = 0 and j = 0 10.4PhysicalProperties Sincemicrowavepowertransferschemesdescribedin 10.1 arelessthan10GHz, propertieswillbeconsideredinthisrange,specicallyat2.45GHz.Thepropertiesare consideredhomogeneous,andtheatmosphericthicknessistakenas10km. 10.4.1Soil Themicrowavereectanceofsoilisatopicwell-coveredintheremotesensing literature.Twofamouspapersare[66 ]and[ 67].[ 66]discussestheeffectsofroughness, soilmoisture,andangleofincidenceonreectanceand[ 67]formulatesawidely-used modelforthedielectricofsoil.Theirsemiempiricalmodelgivesthesoildielectricas: soil =1+ b s ( s 1) + m v fw m v (10) s =(1.01+0.44 s ) 2 0.062 (10) 128

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where and are ttingparameters, s and b aresolidandbulkdensities, m v is volumewaterfraction, fw isfreewaterdielectric.Atafrequencyof2.45GHz,and0.3 volumemoisturefraction,thedielectricconstantofsandyloamsoilisabout 18+ j 3 Whiletherearemultiplewaysofvaryingdegreesofcomplexitytocharacterizethe soil'sreectance,thetwomoststraightforwardaretotreatitasspecularorasdiffuse. Forspecularreectance,usingtheFresnelformula,thereectanceis0.6112. WhenconsideringtheSPS,thereectancewillbeconsidered 0 becausethe downwellingbeamisstrikingarectennaarrayandnotsoil. 10.4.2Atmosphere TheSPSinvolvesradiationthroughtheatmosphere,includingtheionosphere. Experiments[ 68, 69 ]showthatmicrowavetransmissionathighfrequencynonlinearly excitesvariouselectrostaticplasmawaves;subsequentnumericalsimulation[ 70] ofusingparticle-in-cellmodelingshowsthatthereisthreewavecoupling,where thetransmittedwaveservesasapumpwave,abackscatteredwaveoccurswith aslightlylowerfrequency,andamuchlowerfrequencyelectrostaticwavecauses electronheating.Wherethegeomagneticeldisparalleltothetransmissionbeam,the electrostaticwaveleadstoheatingwhichpreventsformationofthethree-wavecoupling andtransmissionefciencyismaintainedat90%orhigher,whilewheretheeldis perpendicularthethree-wavecouplingcontinuesperiodically,causingmost(80%)of theenergytobeconsumedinthecouplingprocess.[ 71]showsthattheionosphere wouldhaveminimaleffectontransmissionthroughtheatmospheresotheeffectsof atmosphericplasmasarenotconsideredhere. Forthemicrowaveradiativetransfer,therearethreeatmosphericcomponentswhich participatenoticably:gaseouswatervapor,waterdroplets,andicecrystals. 10.4.2.1Gaseouswatervapor [ 72]givesthedielectricofwatervaporasafunctionoffrequencyandconcentration aswellasmeasurements.Fromthistheabsorbtioncoefcientcanbededuced.[ 73 ] 129

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sho wstheimpactofatmosphericwatervapor(amongotherthings)onmicrowave remotesensingofsoilmoistureat19GHzandabove.[ 74 ]givesanRTmodelfor microwavetransferincirrusclouds.AthighGHzfrequencies,watervaporisstrongly absorbingenoughtobeasignicantlossatatmosphericconcentration,butnotatlow GHz.Fromthesestudies,wecanignoretheeffectsofwatervaporforradiativeWPT throughtheatmosphere. 10.4.2.2Waterdroplets Waterdropletscanplayasignicantroleinmicrowavescatteringandabsorption. [63 ]performacomplexmicrowaveradiativesimulationofanevolvingcloudusinga different,timevaryingdropsizedistributionandaHenley-Greensteinscatteringphase function.[ 75]examinesmicrowaveWPTusingatwo-streampolarimetricmodelwith verticalandhorizontalpolarizations.[ 76]describesaradiativetransfermodelusing differentexponentialdropsizedistributionsforcloudsorrain,combinedwithafrequency dependentdielectricfunction.Theygivescatteringandabsorbingcoefcientsatarange offrequencies. Thedropsizedistributionisamodiedgamma(withunitsof m 1 = cm 3 ): p (a )= K 1 a P exp(K 2 a Q ) (10) whereforcloudstheparametersare K 1 =1.26 10 13 K 2 =0.75, P =15,and Q =1 ;forrain K 1 =7.41 10 28 K 2 =0.025, P =10,and Q =1 .Theygivethe dielectricas drop =5.5+ 82.5 1 + j 0.0359= (10) =81.47 j 22.27 (10) 130

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At lowfrequencies 2 a =<< 1 soRayleighscatteringdominates.Thescattering andabssorbingcrosssectionsforRayleighare C sca = a 2 8 3 m 2 1 m 2 + 2 x 4 (10) C abs = 4 a 2 = m 2 1 m 2 + 2 x (10) = Z 1 0 p (a )C abs (a )da (10) s = Z 1 0 p (a )C sca (a )da (10) Numericallyintegratingovertheparticledistributiongivesvaluesof s =1.96 10 15 and =1.10 10 8 forcloudsand 1.39 10 11 and 6.88 10 9 forrain,withunitsin cm 1 Figure 10-5.Clouddropletdistributionandabsorptioncrosssection. Figs. 10-5-10-8 showthecrosssectionsanddistributionfunctionsforcloudsand rain. 131

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Figure 10-6.Raindropletdistributionandabsorptioncrosssection. 10.4.2.3Icecrystals Atmosphericicecrystalsarestrongscatterersofmicrowaves.Theyoccurina widerangeofshapesincludingspheres,plates,needles,anddendriticshapes,so theirscatteringpropertiesvarywidely.[ 77]examinesthesingle-scattering,polarimetric effectsofdifferenticecrystalsusingpolarimetricRayleighscatteringandagammasize distribution.[ 78 ]usesaDiscreteDipoleApproximationforpolarimetric,azimuthally dependenticescatteringformanyshapes.Usingthemodiedgammawith K 1 = 1.99 10 11 K 2 =0.01 P =2,and Q =1,withthedielectricevaluatedat2.45GHz[ 79] ice =3+ 97 1 + j 2 f 57 10 6 (10) =3 j 1.12 10 4 (10) gives s =7.45 10 14 and =2.07 10 13 at2.45GHz. 132

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Figure 10-7.Clouddropletdistributionandscatteringcrosssection. Figs. 10-9-10-10 showthecrosssectionsanddistributionfunctionsforice. 10.4.3Vegetation FortheRF-harvestingsensorexampleconsideredinthischapter,theprimary concernsareabsorbtionandscatteringbyvegetation.Thishasbeenheavilyinvestigated bythemicrowaveremotesensingcommunity. Vegetationscattersandabsorbsmicrowaves.Sincethereisnotypicalvegetation, oftenthisiscalculatedwithempiricalrelationships.[ 80 ]computessingle-scattering albedoandopticaldepthforcornandalfalfaatX(7-12.5GHz)andKabands(20-30GHz) andshowsadependenceofopticaldepthonplantwatercontent.[ 81 ]develops amicrowavedielecricmodelforvegetativetissueswhichplaysanimportantrole insubsequentstudiesofmicrowave-plantinteractions.Torelatethisdielectricto absorption,sometimesthewatercloudmodelisused[ 82].[83 ]takesthesemiempirical 133

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Figure 10-8.Raindropletdistributionandscatteringcrosssection. approachfurtherbyincorporatinganotherbiophysicalparameter,leafareaindex.For thischapter,a2mcorncanopywillbeconsidered,with =1.8 and $ =0.08. Otherpaperstakeamoreelectromagneticallyrigorousapproach.AtL-band,[ 64 ] treatsthetrunksinaforeststandasdihedralreectors,andthecanopyasvolumeof waterdropletsandincorporatesmultiplereections.[ 84 ]incorporatescatteringfrom leaves(disks)andstems(cylinders)orientedrandomlyinafullypolarimetricsimulation. Measurementsandmodelingofscatteringpropertiesofvegetationwithdifferent leaf/stemgeometriesaregiven[ 85]. 10.5ResultsandDiscussion TheRTEissolvedforparametervaluesasdeterminedfromtheliteraturereview. Allsimulationsaredonewith32quadraturepointsand8discretizationpointsinthe direction. AllintensitiesareshownindBW/srtohighlightdifferencesatlowintensitylevels. 134

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Figure 10-9.Icespheredistributionandabsorptioncrosssection. Table10-1.ParametervaluesforRTE. Parameter Cloud Rain Ice V egetation 0 1.00 1.00 1.00 0.71 1.1010 8 6.8810 9 2.0710 13 9.0010 3 s 1.9610 15 1.3910 11 7.4510 14 7.8310 4 $ 1.7810 7 2.0110 3 2.6010 1 8.0010 2 h 1.1010 2 6.9010 3 2.8110 7 1.8010 0 R 0.00 0.00 0.00 0.61 10.5.1 AtmosphericLossEstimationforSolarPowerSatellite Fromtheliteraturereview,thedominantmethodsofmediaparticpationthrough theatmosphereareRayleighscatteringandabsorbtionbyclouds,rain,andice.For allthreescatterers,where < 0 (upwellingradiation)thereisadropinintensitydue tothenon-reectinglowerboundary;comparativelylittleradiationintensityisdirected upwards,andthatwhichisisdueentirelytoRayleighscattering.Atthelowerboundary, 135

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Figure 10-10.Icespheredistributionandscatteringcrosssection. theupwellingradiation( = h and < 0)iszero,whichis inlogscalesoitdoesn't showuponthelogintensityplot. Fig. 10-11 showstheintensitydistributioninacloudyatmosphere.Thesingle-scattering albedoisverylow(seeTable 10-1)sotheatmosphereisprimarilyabsorbing. Fig. 10-12 showstheintensitydistributioninarainyatmosphere.Thesingle-scattering albedoissmallbuthigherthanforcloudsasthemodeparticlesizeislargersothe atmosphereismorestronglyscattering.Thismanifestsitselfintheintensityeldas higherintensitylevelsoffofthemainbeamcomparedtoforclouds. Fig. 10-13 showstheintensitydistributioninanicyatmosphere.Thesingle-scattering albedoismuchhigherthanforcloudsorrain.However,themagnitudeofthe s and arequitesmallduetothefactthatthemassdensityofparticlesis1-2ordersof magnitudelowerthanforcloudsandrain,andthedielectricforiceismuchlowerand lesslossythanforliquidwater.Hencetheopticaldepthismuchlowerforthesame 136

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Figure 10-11.Logintensitydistributionthroughcloud. verticaldistance.Thismanifestsitselfintheintensityeldashigherintensitylevelsoffof themainbeam,duetoscattering,butlessloss,duetothedielectricoficecomparedto water. 10.5.2LossEstimationforRadiofrequency-HarvestingSensorUnderVegetation Canopy Vegetationishandledasanabsorbingandisotropicallyscatteringmedia,with empiricalparametersfromtheliterature. Fig. 10-14 showstheintensityeldinthevegetativemediumforadiffuseanda specularlyreectingboundary.Sincetheopticaldepthismuchhigherforavegetation canopy,theintensitydropismuchhigher.Forthediffuselyreectingboundary,where < 0,theupwellingradiationisspreadout.Forthespecularlyreectingboundary, where < 0,theupwellingradiationisadistinctbeam.Thesinglescatteringalbedohas theeffectofincreasingtheoff-beamintensity. 137

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Figure 10-12.Logintensitydistributionthroughrain. 10.5.3Flux Integratingoverthesolidanglesateachzcoordinateandnormalizingto1W/m 2 incidentgivestheuxateachheightthatwouldbereceived.Theprolesareshownin Fig. 10-15.Usingthesethetransmissionefciencycanbeestimated:98.9%forclouds, 99.3%forrain,99.999%forice,3.49%forvegetationwithspecularsoil,and8.96%for vegetationwithdiffusesoil. 10.6Conclusions Thischapterexaminedwirelesspowertransmissioninthemicrowaveregime.The far-eld,radiativemodesofwirelesspowertransfer(WPT)throughclouds,rain,ice, andvegetationwerestudiedusinganumericalsolutionoftheequationofradiative transfer.Theprimaryconclusionisthattheopticaldepthduetoatmosphericparticlesis muchlowerthanthatforvegetationcanopy.Atmosphericicehaslessattentuationthan 138

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Figure 10-13.Logintensitydistributionthroughice. atmosphericwaterdropletsduetothedifferentdielectricconstantsoffrozenandliquid water.Highersingle-scatteringalbedoincreasesoff-beamintensities. Ultimately,theSPScannotbeconsideredimpracticalduetotransmissionefciency. Othertechnicalchallengessuchasthesatelliteconstructionandlaunch,andthe efciencyofthereceivingarray,maylimittheproject.RFharvesting,relyingonlow powerlevels,ismuchmoretolerantofthelowtransmissionefcienciesassociatedwith transferthroughurbanandruralenvironments.Far-eldWPThasapplicability,justas near-eldandmidrangepowertransfer. 139

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Figure 10-14.Logintensitydistributionthroughvegetationwithdiffuseandspecular lowerboundary. Figure 10-15.Fluxprolethroughdifferentmedia. 140

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CHAPTER 11 CONCLUSIONS Aproliferationofbattery-operateddevices,fromcellphonestoelectriccars,has createdademandforawirelesschargingsystem.Thechallengeofwirelesspower transfer(WPT)canbehandled,broadlyspeakng,inthreeregimes.Innear-eld WPT,thetransmissiondistanceismuchlessthanthecharacteristicdimensionofthe transmitter,andmagneticuxfromonecoilinducescurrentinthereceiver.Mid-range nonradiativeWPToccurswhenthetransmissiondistanceisonetoseveraltimes thecharacteristicdimension,andpoweristransferredbymeansofslowly-decaying evanescentmodesbetweentwohigh-Qresonantstructures.Far-eldWPTisradiativein natureandisattransmissiondistancesgreaterthantheRayleighdistance. ThisdissertationhaspresentedseveralaspectsofthedesignofaWPTsystem. Thepowerelectronics,electromagnetic,detectionandestimation,andradiativetransfer aspectsofwirelesspowersystemwereconsidered.Chapter 2 describedthetheory behind,andderiveddesignequationsfor,aclassEdrivingcircuitandimpedance transformationnetwork.Chapter 3 derivedrelevantelectromagneticquantitiesfora near-eldsystem.InChapter 4,acoildesignprovidingeveneldswasdeveloped. Chapter 5 extendedthesystemtoincludemultipletransmittingcoilsinparallel.Coil designformultipletransmittingcoilsinparallelisdiscussedinChapter 6.Chapter 7 discussedtheuseandevaluationofferriteshieldingandfoundasuitablematerial candidate.Chapter 8 showedthedevelopmentandtestingofaBayesiantracking algorithmforreceiverdiscriminationandchargestatusdetermination.Theextension ofthesystemandcoildesigntomidrangewaspresentedinChapter 9.Chapter 10 describedtheuseofradiativetransfermodelingforestimatinglossesoffar-eldwireless powertransmission. 141

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BIOGRAPHICAL SKETCH JoaquinCasanovawasborninGainesville,Florida.Somestuffhappened.Then, in2006,hegothisbachelor'sinagriculturalandbiologicalengineeringfromUniversity ofFlorida,withafocusonagrisystemsengineeringandaseniorprojectdesigning athermallyregulatedtableforseedgerminationstudies.In2007,hereceivedthe master'sdegreeinthesamesubjectforhisworkonmicrowaveremotesensingof soilandvegetation.AftertransferringtotheUFElectricalandComputerEngineering Department,heearnedanothermaster'sdegreein2008,designingathree-dimensional fractalheatsinkantenna.Shiftingresearchfocustowirelesspowertransfer,withabrief asideintohigh-voltageplasmagenerationelectronics,heobtainedhisdoctoratein2010. 149