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24437POWERELECTRONICSSINUSOIDALPULSEWIDTHMODULATIONTHESWITCHESINTHEVOLTAGESOURCEINVERTERSEEFIG1CANBETURNEDONANDOFFASREQUIREDINTHESIMPLESTAPPROACH,THETOPSWITCHISTURNEDONIFTURNEDONANDOFFONLYONCEINEACHCYCLE,ASQUAREWAVEWAVEFORMRESULTSHOWEVER,IFTURNEDONSEVERALTIMESINACYCLEANIMPROVEDHARMONICPROFILEMAYBEACHIEVEDINTHEMOSTSTRAIGHTFORWARDIMPLEMENTATION,GENERATIONOFTHEDESIREDOUTPUTVOLTAGEISACHIEVEDBYCOMPARINGTHEDESIREDREFERENCEWAVEFORMMODULATINGSIGNALWITHAHIGHFREQUENCYTRIANGULARCARRIERWAVEASDEPICTEDSCHEMATICALLYINFIG2DEPENDINGONWHETHERTHESIGNALVOLTAGEISLARGERORSMALLERTHANTHECARRIERWAVEFORM,EITHERTHEPOSITIVEORNEGATIVEDCBUSVOLTAGEISAPPLIEDATTHEOUTPUTNOTETHATOVERTHEPERIODOFONETRIANGLEWAVE,THEAVERAGEVOLTAGEAPPLIEDTOTHELOADISPROPORTIONALTOTHEAMPLITUDEOFTHESIGNALASSUMEDCONSTANTDURINGTHISPERIODTHERESULTINGCHOPPEDSQUAREWAVEFORMCONTAINSAREPLICAOFTHEDESIREDWAVEFORMINITSLOWFREQUENCYCOMPONENTS,WITHTHEHIGHERFREQUENCYCOMPONENTSBEINGATFREQUENCIESOFANCLOSETOTHECARRIERFREQUENCYNOTICETHATTHEROOTMEANSQUAREVALUEOFTHEACVOLTAGEWAVEFORMISSTILLEQUALTOTHEDCBUSVOLTAGE,ANDHENCETHETOTALHARMONICDISTORTIONISNOTAFFECTEDBYTHEPWMPROCESSTHEHARMONICCOMPONENTSAREMERELYSHIFTEDINTOTHEHIGHERFREQUENCYRANGEANDAREAUTOMATICALLYFILTEREDDUETOINDUCTANCESINTHEACSYSTEMWHENTHEMODULATINGSIGNALISASINUSOIDOFAMPLITUDEAM,ANDTHEAMPLITUDEOFTHETRIANGULARCARRIERISAC,THERATIOMAM/ACISKNOWNASTHEMODULATIONINDEXNOTETHATCONTROLLINGTHEMODULATIONINDEXTHEREFORCONTROLSTHEAMPLITUDEOFTHEAPPLIEDOUTPUTVOLTAGEWITHASUFFICIENTLYHIGHCARRIERFREQUENCYSEEFIG3DRAWNFORFC/FM21ANDTL/RT/3TPERIODOFFUNDAMENTAL,THEHIGHFREQUENCYCOMPONENTSDONOTPROPAGATESIGNIFICANTLYINTHEACNETWORKORLOADE/2RLE/2D1D2T1T2ILVLFIG1SIMPLEVOLTAGESOURCEDINVERTER1AMGOLE,200024437POWERELECTRONICSDUETHEPRESENCEOFTHEINDUCTIVEELEMENTSHOWEVER,AHIGHERCARRIERFREQUENCYDOESRESULTINALARGERNUMBEROFSWITCHINGSPERCYCLEANDHENCEINANINCREASEDPOWERLOSSTYPICALLYSWITCHINGFREQUENCIESINTHE215KHZRANGEARECONSIDEREDADEQUATEFORPOWERSYSTEMSAPPLICATIONSALSOINTHREEPHASESYSTEMSITISADVISABLETOUSESOTHATALLTHREEWAVEFORMSARESYMMETRICFCFM3KKN,06012018024030036011060120180240300360CARRIERDESIREDWAVEFORMFIG2PRINCIPALOFPULSEWIDTHMODULATIONEAVD/2VD/2ACAM060120180240300360105510IA2AMGOLE,200024437POWERELECTRONICSNOTETHATTHEPROCESSWORKSWELLFORFOR,THEREAREPERIODSOFTHETRIANGLEWAVEINWHICHTHEREISNOINTERSECTIONOFTHECARRIERANDTHESIGNALASINFIG4HOWEVER,ACERTAINAMOUNTOFTHIS“OVERMODULATION”ISOFTENALLOWEDINTHEINTERESTOFOBTAININGALARGERACVOLTAGEMAGNITUDEEVENTHOUGHTHESPECTRALCONTENTOFTHEVOLTAGEISRENDEREDSOMEWHATPOORERNOTETHATWITHANODDRATIOFORFC/FM,THEWAVEFORMISANTISYMMETRICOVERA360DEGREECYCLEWITHANEVENNUMBER,THEREAREHARMONICSOFEVENORDER,BUTINPARTICULARALSOASMALLDCCOMPO0601201802403003601106012018024030036006012018024030036055100V100VIAAFIG3SPWMWITHFC/FM48,L/RT/3M1M13AMGOLE,200024437POWERELECTRONICSNENTHENCEANEVENNUMBERISNOTRECOMMENDEDFORSINGLEPHASEINVERTERS,PARTICULARLYFORSMALLRATIOSOFFC/FMSPWMSPECTRAALTHOUGHTHESPWMWAVEFORMHASHARMONICSOFSEVERALORDERSINTHEPHASEVOLTAGEWAVEFORM,THEDOMINANTONESOTHERTHANTHEFUNDAMENTALAREOFORDERNANDN2WHERENFC/FMTHISISEVIDENTFORTHESPECTRUMFORN15ANDM08SHOWNINFIG5NOTETHATIFTHEOTHERTWOPHASESAREIDENTICALLYGENERATEDBUT120OAPARTINPHASE,THELINELINEVOLTAGEWILLNOTHAVEANYTRIPLENHARMONICSHENCEITISADVISABLETOCHOOSE,ASTHENTHEDOMINANTHARMONICWILLBEELIMINATEDITISEVIDENTFROMFIG5B,THATTHEDOMINANT15THHARMONICINFIG5AISEFFECTIVELYELIMINATEDINTHELINEVOLTAGECHOOSINGAMULTIPLEOF3ISALSOCONVENIENTASTHENTHESAMETRIANGULARWAVEFORMCANBEUSEDASTHECARRIERINALLTHREEPHASES,LEADINGTOSOMESIMPLIFICATIONINHARDWAREITISREADILYSEENTHATASTHEWHEREEISTHEDCBUSVOLTAGE,THATTHERMSVALUEOFTHEOUTPUTVOLTAGESIGNALISUNAFFECTEDBYTHEPWMPROCESSTHISISSTRICTLYTRUEFORTHEPHASEVOLTAGEASTRIPLENHARMONICORDERSARECANCELLEDINTHELINEVOLTAGEHOWEVER,THEPROBLEMATICHARMONICSARESHIFTEDTOHIGHERORDERS,THEREBYMAKINGFILTERINGMUCHEASIEROFTEN,THEFILTERINGISCARRIEDOUTVIATHENATURALHIGHIMPEDANCECHARACTERISTICOFTHELOAD06012018024030036022060120180240300360FIG4OVERMODULATIONM13NOINTERSECTIONFCFM3KKN,PWM2E24AMGOLE,200024437POWERELECTRONICSSELECTIVEHARMONICELIMINATIONALSOCALLEDOPTIMALPWMNOTICETHATINTHESPWMSTRATEGYDEVELOPEDABOVE,ALARGENUMBEROFSWITCHINGSAREREQUIRED,WITHTHECONSEQUENTASSOCIATEDSWITCHINGLOSSESWITHTHEMETHODOFSELECTIVEHARMONICELIMINATION,ONLYSELECTEDHARMONICSAREELIMINATEDWITHTHESMALLESTNUMBEROFSWITCHINGSTHISMETHODHOWEVERCANBEDIFFICULTTOIMPLEMENTONLINEDUETOCOMPUTATIONANDMEMORYREQUIREMENTSFORATWOLEVELPWMWAVEFORMWITHODDANDHALFWAVESYMMETRIESANDNCHOPSPERQUARTERCYCLEASSHOWNINFIG4,THEPEAKMAGNITUDEOFTHEHARMONICCOMPONENTSINCLUDINGTHEFUNDAMENTAL,AREGIVENBYEQN1024681012141618202224262830323436384050100HARMONICNUMBERMAGNITUDEV0246810121416182022242628303234363840100200HARMONICNUMBERMAGBITUDEFIG5SPWMHARMONICSPECTRAN15,M08APHASEVOLTAGEBLINEVOLTAGE5AMGOLE,200024437POWERELECTRONICS1HEREHIISTHEMAGNITUDEOFTHEITHHARMONICANDJISTHEJTHPRIMARYSWITCHINGANGLEEVENHARMONICSDONOTSHOWUPBECAUSEOFTHEHALFWAVESYMMETRYTHENCHOPSINTHEWAVEFORMAFFORDNDEGREESOFFREEDOMSEVERALCONTROLOPTIONSARETHUSPOSSIBLEFOREXAMPLENSELECTEDHARMONICSCANBEELIMINATEDANOTHEROPTIONWHICHISUSEDHEREISTOELIMINATEN1SELECTEDHARMONICSANDUSETHEREMAININGDEGREEOFFREEDOMTOCONTROLTHEFUNDAMENTALFREQUENCYACVOLTAGETOFINDTHESREQUIREDTOACHIEVETHISOBJECTIVE,ITISSUFFICIENTTOSETTHECORRESPONDINGHSINTHEABOVEEQUATIONSTOTHEDESIREDVALUES0FORTHEN1HARMONICSTOBEELIMINATEDANDTHEDESIREDPERUNITACMAGNITUDEFORTHEFUNDAMENTALANDSOLVEFORTHESEQUATION1CANBEREADILYPROVEDBYFINDINGTHEFOURIERCOEFFICIENTSOFTHEWAVEFORMSHOWNINFIG4INGENERAL,FORAPERIODICWAVEFORMWITHPERIOD,THEFOURIERCOSINEANDSINECOEFFICIENTSAREGIVENBYH14E121COS22COS23COS2NCOSH34E31231COS232COS23323NCOSCOSHK4EK12K1COS2K2COS2K32KNCOSCOS/21EETN1N2FIG4ATWOLEVELPWMWAVEFORMWITHODDANDHALFWAVESYMM26AMGOLE,200024437POWERELECTRONICS2BECAUSEOFTHEHALFCYCLESYMMETRYOFTHEWAVEFORMOFFIG4,ONLYODDORDERHARMONICSEXISTALSO,ITISEASYTOSEETHATTHEFOURIERCOSINECOEFFICIENTSDISAPPEARWITHTHECHOICEOFCOORDINATEAXESUSEDUTILIZINGTHEQUARTERCYCLESYMMETRY,THEFOURIERSINECOEFFICIENTSBECOME3SUBSTITUTINGTHETWOVALUEDPWMWAVEFORMFOR,ONEOBTAINSSEEFIG44THEFOLLOWINGEXAMPLEILLUSTRATESTHEUSEOFTHREECHOPSPERQUARTERCYCLEWHICHALLOWFORTHREEDEGREESOFFREEDOMWEMAYUSETHESETOELIMINATETWOHARMONICSANDCONTROLTHEMAGNITUDEOFTHEFUNDAMENTALTOANYDESIREDVALUEEXAMPLESELECTIVEHARMON
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