POWERELECTRONIC SYSTEMDESIGN
LinkingDifferential Equations,LinearAlgebra, andImplicitFunctions
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1.Capacitorandinductor1
1.1Capacitorequationindifferentialform1
1.2Capacitorequationinintegralform2
1.3Inductorequationindifferentialform3
1.4Inductorequationinintegralform4
1.5DefinitionofinductanceandFaraday’slaw4
1.6Magneticcouplingandmutualinductance5
1.7Transformerequation7
1.8Nonidealcapacitor,nonidealinductor,andequivalentcircuit10
1.9Transformerequivalentcircuits11
1.10Physicalsizeofcapacitorandinductor13
1.11Specificationsforcapacitorandinductor15
2First-ordercircuits19
2.1RCnetworkwithperiodicdrivesource19
2.2Sawtooth(triangleramp)generator30
2.3Full-waverectifierwithRCload33
2.4AbrushlessDCMotorwithpermanentmagnetsrotor38
2.5ABLDCmotorspeeddetector45 References47
3Currentsource49
3.1Semiconductordiodeequation49
3.2Simplecurrentsource50
3.3BobWidlarcurrentsource54
3.4Improvedcurrentsource58
3.5Sourceimpedance60
3.6555timer64
3.7Precisioncurrentloop70
3.8Current-modelaserdriver74
3.9LEDarraydriver76
3.10JFETcurrentsource77
3.11MOSFETcurrentsource78
4Secondorder81
4.1Form81 4.2Root83
4.3Timedomain85
4.4Frequencydomain89
4.5Parallelandserialresonance92
4.6Eigenvalueapproach103
4.7RCfiltersandSallen–Keyfilters104
4.8Powerfilters111
4.9Oscillator113
4.10Implicitfunction120
5Gainblocks123
5.1Class-Adirect-coupledbipolartransistoramplifiers123
5.2Class-AB,B,Cbipolartransistoramplifiers129
5.3Transformer-coupledtransistoramplifiers133
5.4Class-Dswitch-modepoweramplifiers135
5.5Pulsewidthmodulator139
5.6Digital(clocked)windowcomparator140
5.7Linearoperationalamplifiers142
5.8Tunedamplifiersandimplicitfunction147
5.9Compositenonlinearoperationalamplifiers150
5.10Unity-gainbandwidthofop-amp153
5.11Largesignalgainofop-amp156
6Feedbackapproaches167
6.1Voltagefeedback167
6.2Currentfeedback170
6.3PIDfeedback175
6.4Statefeedback178
6.5Feedbackisolation180
7Controlpractices189
7.1Levelcontrol189
7.2Modecontrol190
7.3Zonecontrol192
7.4Variablestructures193
7.5Sensor196
7.6Openloop198
7.7Closeloop200
7.8Loopcontention203
7.9Timecontrol204
7.10Sequentialtimecontrol206
8Linearregulator213
8.1Bipolarseriesvoltageregulator213
8.2MOSFETseriesvoltageregulator223
8.3Multipleimplicitfunctionapproach227
8.4Designprocedureforloopstability228
8.5Designprocedureforerroramplifiers230
8.6Current-modelaserdriverdesignprocedure236
8.7Shuntregulators238
9Switch-modeDC/DCconverters241
9.1Powerfilter,inductor,andcapacitor243
9.2Fundamentaltopologies249
9.3Operationaldynamicsofbasicbucktopology254
9.4Operationaldynamicsofbasicboosttopology257
9.5Operationaldynamicsofbasicflybackconverter259
9.6Cascadedconverter—nonisolated261
9.7Isolatedconverter—forwardconverter264
9.8Isolatedconverter—half-bridgeconverter269
9.9Isolatedconverter—push–pullconverter272
9.10Isolatedconverter—full-bridgeconverter272
9.11Isolatedconverter—quasi-resonantconverter273
9.12Analogfeedback275
9.13Closeloop—analog288
9.14Closeloop—digital296
10ACdrives,rectification,andinductiveloads299
10.1ReexamineRC-loadedrectifier299
10.2ACdrivewithunidirectionalRLload301
10.3Half-waveACdrivewithnonpulsatingcurrentfeedingRLload304
10.4Full-waveACdrivewithnonpulsatingcurrentfeedingRLload305
10.5Phase-controlledACdrivewithRLload307
10.6Phase-controlledACdrivewithfree-wheeldiodeandRLload309
10.7Phase-controlledfull-waveACdrivewithRLload311
10.8Three-phasecircuits313
11Rotation,three-phasesynthesis,andspacevectorconcepts319 11.1Magneticfield(flux)319 11.2Synthesisofthree-phasesourcesandinverters323 11.3Vectorconcept331
AppendixAAcceleratedsteady-stateanalysisforaparallelresonant networkfedbynonsinusoidal,half-waverectifiedcurrent347
AppendixBMatrixexponential349
AppendixCExample4.7MATLABm-file351
AppendixDExample8.1353
AppendixEAgeneralmass-spring-dashpotsecond-ordersystem; firstalternative359
AppendixFAgeneralmass-spring-dashpotsecond-ordersystem; secondalternative363
AppendixGAgeneralmass-spring-dashpotsecond-ordersystem; thirdalternative365
AppendixHMatrixexponential—Jordanform367
AppendixIAstep-by-stepprimerondigitalpower-supplydesign369
AbouttheAuthor
KengC.Wu,anativeofChiayi( ),Dalin( ),Taiwan,receivedtheB.S. degreefromChiaotungUniversity,Taiwan,in1969andtheM.S.degree fromNorthwesternUniversity,Evanston,Illinoisin1973.
Hewasaleadmember,technicalstaff,ofLockheedMartin,Moorestown, NJ.Hehaspublishedfivebooks: PulseWidthModulatedDC-DCConverters Chapman&Hall,January1997; TransistorCircuitsforSpacecraftPowerSystem KluwerAcademicPublishers,November2002; Switch-modePowerConverters: DesignandAnalysis AcademicPress,Elsevier,November2005; PowerRectifiers,Inverters,andConverter Lulu.com November2008.; PowerConverterswith DigitalFilterFeedbackControl,Elsevier,AcademicPress,2016.Heholdsa dozenUSpatents,wasawardedAuthoroftheYeartwice(2003and2006 atLockheedMartin),andpresenteda3-houreducationalseminaratIEEE APEC-2007S17.
Preface
Yearsago,Prof.EmeritusChi-TsongChen,theauthorof LinearSystem TheoryandDesign,averysuccessfultextbook(OxfordUniversityPress),met theauthorathisFlushing,NewYorkresidence.Inthemeeting,andinthe prefaceof SignalsandSystems–AFreshLook hislastpublication(PDFform freetoallglobally),Prof.Chenlamentedthat“Feedbacksfromgraduates thatwhattheylearnedinuniversityisnotusedinindustrypromptedmeto ponderwhattoteachinsignalsandsystems.”
Sadly,andbasedonlongprofessionalcareerservingRCA/GE/Lockheed Martinspacesector,theauthorcandefinitivelyconfirmthefactProf. Chenwassadabout.Theless-than-desirablestatehadexisted,andisstill present,intheformthatmanydegree-holdingengineersincludingelectrical, electronic,mechanical,andotherspecialtiesarefallingshortinapplying mathematicaltoolstheyweretaughtincollege.Givenelectricalschematic drawings,theywereunabletoformulateandexpresssystems’dynamicsin statevariablesandstatetransitionusingthefirst-orderdifferentialequations andlinearalgebratechnique.Asaresult,theywereunabletoboosttheir productivityusingsoftwaresuchasMATLAB.
Thisbookintendstobridgethegap—whatistaughtincollegeandhow itisbeingappliedinindustry.Inessence,thiswritingshallbeconsidered didactic.
ItbeginswithChapteronegivingcapacitorsandinductors,twoindispensableenergystoragecomponents,anin-depthexaminationfromthe viewpointofthefirst-orderderivative,itscorrespondingintegralform, anditsphysicalimplications.ChaptertwocoversRC-andRL-typenetworksgovernedbyasingledifferentialequation.Keystepsmovingsystem differentialequationstoLaplacetransforminafrequencydomainandto astate-spacetransitionformareintroduced.Alongtheway,unconventionalapproachesderivingFourierseries,explainingorthogonalproperty, ortreatingboundaryvalueproblemsarealsoexplored.Chapterthreecovers currentsourcingcircuitsincludingcurrentmirror,theworkhorseofanalog integratedcircuits,andprecisioncurrentgeneratorloopscriticaltoinstrumentation.ChapterfourextendsChaptertwotonetworksofsecondorder governedbytwofirst-orderdifferentialequations.Procedurestransforming multipledifferentialequationstoLaplaceform,tostate-transitionform,and tostate-transitionsolutionareshown.Chapterfiveexaminescircuitblocks
andmodulesperformingamplification,voltage-to-timewindow,dutycycle modulation,etc.Chaptersixcoversfeedbackpracticesincludingvoltage, current,isolation,summativecurrent,subtractivecurrent,andstatefeedback. Chaptersevendiscussesconfigurationsofcontrolloopsincludingsingleloop, multipleloop,openloop,closedloop,nestedloop,loopcontention,etc. Chaptereightdealswithlinearregulatorsincludingseriesvoltageregulator andcurrentshuntinparallel.Chapternineexploresswitch-modepower processing.Chaptertenpresentscomplexitiesarisingfrominductiveloadfed byrectifiedACsourcesofsinglephase,multiplephases,andphasecontrol. Employingtheconceptofelectromagneticvectorsinspace,Chaptereleven focusesontheformationofmagneticfluxvectorplacedintentionallyalong selectedorientation,time-varyingfluxintensity,androtationalfluxvector thatmakesmotorspin.
Consideringthewriter’sgoalistobridgematerialstaughtincollegeand applicationsofthematerialinactualindustrialsettings,thetopicsoutlined aboveandorganizedinthatparticularorderaresuitableforcollegeseniors andnoviceprofessionalsintheindustry.Followingthematerial,andwhen facingareal-worlddesignschematic,readerswillbeableto(1)assignstate variables(circuitnodevoltage,inductorbranchcurrent),(2)writedown multipledifferentialequations,(3)placeequationsetinastate-transition form,(4)selecttheapproachoneismorecomfortableandconfident,forthe timebeing,(5)obtainsystemresponsesolutionscorrespondingtovarious drivesindifferenttimeframes,(6)stitchtogetherasteady-stateresponse solutioninclosed-formanalyticalexpressions.
Giventimeandpractice,andwhenfacingsystemorderexceedingthree, mostreaderswillquicklyrealizethatstate-transitionequationandsolution invokingmatrixoperationdelineatedinlinearalgebraaremoreeffective, evenelegant,inhandlinghigh-ordersystems.
Thiswriterhaddefinitelyexperiencedthatawareness,andexpectsall readertodothesame.
Asindicatedinthesubtitleofthiswriting,alongthepresentation, mathematicalnotesareinsertedwhereappropriatenessisnotviolated.Quite afewmaybeconsideredunconventional.Thisisdoneinthespiritofnever takingauthoritydogmatically—atrueopenmindrespectingtheunlimited possibilitiesofviewingnaturefrommultipleanglesandabeliefthatwhat
Preface xv wassaidtrueinthepastmaynotbetrueinthefuturewhennewdiscoveries seethedaylight.
Onthebackdropoftheaboveconviction,thisauthortookadditional effortstomakethiswritingalsoavailableinChineselanguage;thankstopublisherElsevierforgrantingsuchtranslationright.Thanksarealsoextended toMr.,atITRI(IndustrialTechnologyResearchInstitute,HsinchuTaiwan), whohadperformedthetranslation,averydemandingtaskconsideringthe limitationsofChineselanguageinhandlingtechnicalsubjects.
Withtheadvanceofminiaturizedelectronichardwareandsupercomputerequippedwithmathematicalco-processors,engineeringdesigntasks arenowmostlycarriedoutbythesimulationandcomputation.The implementationofbothalwaysrequiresdesignformulationintheform ofanalyticalexpressionsbasedon,inmostcases,systemsofdifferential equationswithcoefficientsdependingoncomponents/partsvalues.
Inthecourseofalmostfourdecades‘Ł‘™professionalcareerinaerospace industries,theauthorhaddefinitelyderivedsignificantbenefitsfromfollowingthepathoutlinedabove.
You,readers,cancertainlydothesame.
KengC.Wu Princeton,NJ. Dec.2020
Capacitorandinductor
Twocomponents,capacitorsandinductors,playirreplaceablerolesinelectricalpowerprocessingfortheirenergy-storageproperties.Bypresentingthe analyticalequationsgoverningbothineitherdifferentialorintegralforms, thischapterilluminatestheelectromagneticbehaviorsofthosedevicesand elucidatesitsphysicalsignificancewhenworkingwithdrivingsources.
MATH.NOTE: Inmostcalculustextbooks,derivativesandintegralsare introducedintheformsof f´(x) = dy/dx = df(x)/dx and ∫ydx = ∫f(x)dx, given y = f(x)atwo-dimensionalplanecurveand x istheindependentvariable,withlittlephysicalmeaningattachedexcepttheconceptof“tangential slope,”associatedwiththederivative,and“geometricalarea,”associatedwith theintegral,employingtheapproachoflimit.Theindependentvariable x is bynomeansrestrictedtosignifyingonlyspacequantity.Itcertainlycanstand fortime,andmanyothervariablesaswell.Thesimpleactofreplacing dx with dt,aninfinitesimaltimeincrement,introducesinteresting,andimportant, physicalmeaningtoderivative f´(t) = dy/dt = df(t)/dt.As dt appearsinthe numerator(inverseoftime),derivativeagainsttimeyieldsthedimensionof speed,velocity,and/orfrequency;thetemporalchangesofatime-dependent variable.
1.1Capacitorequationindifferentialform
Almostwithoutexception,theactionofcapacitorsisintroducedintextbooksinadifferentialform;whichlinkscurrentthroughthedeviceand timerateofvoltagechangeacrossitwithapositivesignasshown.
MATH.NOTE: Atamorefundamentallevel,thecurrentisexpressedas therateofchargecarriers’changes, i(t) = dQ(t)/dt,inwhich Q(t) = Cv(t)and C,thecapacitanceandaconstantwithinreason,isafunctionofgeometry andmaterialproperty.
Powerelectronicsystemdesign. Copyright©2021ElsevierInc. DOI:10.1016/B978-0-32-388542-3.00004-2Allrightsreserved. 1
Fig.1.1 Terminalcurrentandvoltageofacapacitor.
Whatdoesthisformtellusaboutacapacitiveelement?
1.When dv/dt = 0,thatiswhenthedevicevoltagereachesanextreme, amaximumoraminimum,thecorrespondingdevicecurrentcrosseszero value.Stateddifferently,thedevice’stime-domaincurrentwaveformmakes azero-crossingatthetimeitscorrespondingvoltagewaveformpeaks,or bottomsout.Inotherwords,andinagraphicalform,terminalvoltageand throughcurrentforacapacitormustholdarelationasshownin Fig.1.1
2.Asthevoltagevariablein Eq.(1.1) appearsasaderivative,thecurrent variabledoesnotchangeitsvalueif Eq.(1.1) isrewrittenas
Inthisform,oneimportantpropertyofcapacitorstandsout.Thatis,the devicesustainsaDC(directcurrent)voltage, VDC ,whichhoweverdoesnot contributetoitscurrent.Thesignificanceofthisattributeisthatacapacitor blocksDCcurrent.Or,DCcurrentdoesnotflowthroughacapacitor.Only AC(alternating)currentdoes.
3.CapacitorallowstheapplicationofaDCvoltagewithinlimit;the breakdownvoltage.
1.2Capacitorequationinintegralform
Eq.(1.1) canofcourseberewrittenas
MATH.NOTE: ThisisactuallyarewordingofthepreviousMATH NOTE,thatis,chargeisequaltothetimeintegralofcurrent.
Incontrasttothederivativeform,theintegralform (Eq.1.3),inparticular theright-handside,conveysanextremelyimportanteffectofthecapacitive statevariable:voltage.
4.Inaverystraightforwardmanner,itdeclaresthecontinuousnatureof capacitorvoltage.
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