University Press of Florida
Introduction to Physical Electronics
Buy This Book ( Related Link )
CITATION PDF VIEWER
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/AA00011724/00001
 Material Information
Title: Introduction to Physical Electronics
Physical Description: Book
Language: en-US
Creator: Wilson, Bill
 Subjects
Subjects / Keywords: Engineering, Electrical Engineering, Physics, Electronics, Conduction, Semiconductors, P-N Junction, Gauss' Law, Diode Equation, Light Emitting Diode, Solar Cells, Bipolar Transistors, Transistor Equations, MOSFETs, MOS Structure, MOS Transistor, Inverters, CMOS Logic, JFET, Electrostatic Discharge, Transmission Lines, Telegrapher's Equations, Transmission …
Electromechanical Technology, Electronics, Engineering
Science / Engineering
 Notes
Abstract: An introduction to solid state device including field effect and bipolar transistors. Contents: 1) Conductors, Semiconductors and Diodes. 2) Bipolar Transistors. 3) FETs. 4) IC Manufacturing. 5) Introduction to Transmission Lines. 6) AC Steady-State Transmission. 7) Glossary.
General Note: Expositive
General Note: Community College, Higher Education
General Note: http://www.ogtp-cart.com/product.aspx?ISBN=9781616100445
General Note: Adobe PDF Reader
General Note: Bill Wilson, Rice University
General Note: Textbook
General Note: http://cnx.org/content/col10114/latest/
General Note: http://florida.theorangegrove.org/og/file/61c67a61-35f0-83cc-dfd3-1a7ab8713c74/1/IntroPhysElectr.pdf
 Record Information
Source Institution: University of Florida
Rights Management: Copyright © 2008 Bill Wilson. This selection and arrangement of content is licensed under a Creative Commons Attribution License. You are free to copy, distribute and transmit this work and to adapt this work if you attribute authorship and use the work for non-commercial purposes. If you alter, transform, or build upon this work, you may distribute the …
Resource Identifier: isbn - 9781616100445
System ID: AA00011724:00001

Downloads

This item is only available as the following downloads:

( PDF )


Full Text

PAGE 1

IntroductiontoPhysicalElectronics By: BillWilson

PAGE 3

IntroductiontoPhysicalElectronics By: BillWilson Online: < http://cnx.org/content/col10114/1.4/ > CONNEXIONS RiceUniversity,Houston,Texas

PAGE 4

2008BillWilson ThisselectionandarrangementofcontentislicensedundertheCreativeCommonsAttributionLicense: http://creativecommons.org/licenses/by/1.0

PAGE 5

TableofContents 1Conductors,SemiconductorsandDiodes 1.1 SimpleConduction..........................................................................1 1.2 IntroductiontoSemiconductors..............................................................6 1.3 DopedSemiconductors.....................................................................12 1.4 P-NJunction:PartI.......................................................................19 1.5 PN-Junction:PartII.......................................................................21 1.6 Gauss'Law.................................................................................24 1.7 DepletionWidth............................................................................26 1.8 ForwardBiased.............................................................................29 1.9 TheDiodeEquation........................................................................34 1.10 ReverseBiased/Breakdown................................................................38 1.11 Diusion..................................................................................41 1.12 LightEmittingDiode.....................................................................44 1.13 LASER...................................................................................48 1.14 SolarCells................................................................................51 2BipolarTransistors 2.1 IntrotoBipolarTransistors.................................................................55 2.2 TransistorEquations.......................................................................57 2.3 TransistorI-VCharacteristics...............................................................58 2.4 CommonEmitterModels...................................................................61 2.5 SmallSignalModels........................................................................62 2.6 SmallSignalModelforBipolarTransistor..................................................66 3FETs 3.1 IntroductiontoMOSFETs..................................................................71 3.2 BasicMOSStructure.......................................................................72 3.3 ThresholdVoltage..........................................................................77 3.4 MOSTransistor............................................................................82 3.5 MOSRegimes..............................................................................84 3.6 PlottingMOSI-V..........................................................................89 3.7 Models.....................................................................................93 3.8 InvertersandLogic.........................................................................96 3.9 TransistorLoadsforInverters...............................................................99 3.10 CMOSLogic.............................................................................102 3.11 JFET....................................................................................107 3.12 ElectrostaticDischargeandLatch-Up....................................................109 4ICManufacturing 4.1 IntroductiontoICManufacturingTechnology.............................................115 4.2 SiliconGrowth............................................................................117 4.3 Doping....................................................................................118 4.4 Fick'sFirstLaw...........................................................................120 4.5 Fick'sSecondLaw.........................................................................121 4.6 Photolithography..........................................................................123 4.7 IntegratedCircuitWellandGateCreation.................................................127 4.8 ApplyingMetal/Sputtering................................................................132 4.9 IntegratedCircuitManufacturing:Bird'sEyeView........................................134 4.10 DiusedResistor.........................................................................137 4.11 Yield.....................................................................................138

PAGE 6

iv Solutions.......................................................................................141 5IntroductiontoTransmissionLines 5.1 IntroductiontoTransmissionLines:DistributedParameters...............................143 5.2 Telegrapher'sEquations...................................................................146 5.3 TransmissionLineEquation...............................................................148 5.4 TransmissionLineExamples...............................................................151 5.5 ExcitingaLine............................................................................153 5.6 TerminatedLines..........................................................................156 5.7 BounceDiagrams..........................................................................159 5.8 CascadedLines............................................................................169 Solutions.......................................................................................179 6ACSteady-StateTransmission 6.1 IntroductiontoPhasors...................................................................181 6.2 A/CLineBehavior........................................................................183 6.3 TerminatedLines..........................................................................186 6.4 LineImpedance...........................................................................189 6.5 CrankDiagram............................................................................189 6.6 StandingWaves/VSWR...................................................................193 6.7 BilinearTransform........................................................................196 6.8 TheSmithChart..........................................................................200 6.9 IntroductiontoUsingtheSmithChart....................................................211 6.10 SimpleCalculationswiththeSmithChart................................................212 6.11 Power....................................................................................216 6.12 FindingZL...............................................................................218 6.13 Matching................................................................................222 6.14 IntroductiontoParallelMatching........................................................225 6.15 SingleStubMatching....................................................................226 6.16 DoubleStubMatching...................................................................230 6.17 OddsandEnds..........................................................................237 Glossary ............................................................................................242 Index ...............................................................................................243 Attributions ........................................................................................247

PAGE 7

Chapter1 Conductors,SemiconductorsandDiodes 1.1SimpleConduction 1 Ourinitialstudieswillmoreorlessbeareviewoftopicsinelectricitythatyoumayhaveseenbeforein physics.However,ifexperienceisanyguide,thereisnogreatharmingoingbackoverthismaterial,forit seemsthatformanystudents,thewholeconceptofjusthowelectricityactuallyworksisjustalittlehazy. Consideringthatyouhopetobecalledanelectricalengineeroneofthesedays,thismightevenbeagood thingtoknow! Mostofthe"laws"ofhowelectricitybehavesarereallyjustmathematicalrepresentationsofanumber ofempiricalobservations,basedonsomeassumptionsandguesseswhichweremadeinattempttobringthe "laws"intoacoherentwhole.EarlyinvestigatorsFaraday,Gauss,Coulomb,Henryetc....allthoseguys determinedcertainthingsaboutthisstrange"invisible"thingcalledelectricity.Infact,theelectronitself wasonlydiscoveredalittleover100yearsago.Evenbeforetheelectronitselfwasobserved,peopleknew thatthereweretwokindsofelectriccharge,whichwerecalled positive and negative .Likechargesexhibit arepulsiveforcebetweenthemandoppositechargesattractoneanother.Thisforceisproportionaltothe productoftheabsolutevalueofpositiveandnegativecharge,andvariesinverselywiththesquareofthe distancebetweenthem.Dierentchargecarriershavedierentmass,someareverylight,andothersare signicantlyheavier.Electricalchargescanexperienceforces,andcanmoveabout.Sinceforcetimesdistance equalswork,awholesystemofenergy potential aswellas kinetic andenergylosshadtobedescribed. Thishasleadtoourcurrentsystemofelectrostaticsandelectrodynamics,whichwewillnotreviewnowbut bringupalongthewayasthingsareneeded. Justtomakesureeveryoneisonthesamefootinghowever,let'sdeneafewquantitiesnow,andthen wewillseehowtheyinteractwithoneanotheraswegoalong. Thetotalchargeinsomeregionisdenedbythesymbol Q andithasunitsofCoulombs.Thefundamental unitofchargethatofanelectronoraprotonissymbolizedeitherbyalittle q orby e .Sincewe'lluse e forotherthings,inthiscoursewewilltrytostickwith q .The chargeofanelectron q ,hasavalueof 1 : 6 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(19 Coulombs. Sincechargecanbedistributedthroughoutaregionwithvaryingconcentrations,wewillalsotalkabout the chargedensity ,whichhasunitsof Coulombs cm 3 .Inthisbook,wewilluseamodiedMKSsystem ofunits.Inkeepingwithmostworkersinthesolid-statedeviceeld,volumewillusuallybeexpressedasa cubiccentimeter,ratherthanacubicmeter-acubicmeterofsiliconisjustfartoomuch!Inmostcases, thechargedensityisnotuniformbutisafunctionofwhereweareinspace.Thus,whenwehave distributedthroughoutsomevolume, V Q = Z V d .1 1 Thiscontentisavailableonlineat. 1

PAGE 8

2 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES describesthetotalchargeinthatvolume. Weknowthatwhenweapplyanelectriceldtoachargethatthereisaforceexertedonit,andthatif thechargeisabletomoveitwilldoso.Themotionofchargegivesrisetoan electriccurrent ,whichwe call I .Thecurrentisameasureofhowmuchchargeispassingagivenpointperunittime Coulombs second Itwillbehelpfulifwehavesomekindofmodelofhowelectricityowsinaconductor.Thereareseveral approacheswhichonecantake,somemoreintuitivethanothers.Theonewewilllookat,whilenotcorrect inthestrictestsense,stillgivesaverygoodpictureofhowelectricalconductionworks,andisperfectlyne touseinavarietyofsituations.Inthe Drudetheory ofconduction,theinitialhypothesisconsistsofa solid,whichcontainsmobilechargeswhicharefreetomoveaboutundertheinuenceofanappliedelectric eld.Therearealsoxedchargesofpolarityoppositethatofthemobilecharges,sothateverywherewithin thesolid,thenetchargedensityiszero.Thishypothesisisbasedonthemodeloftheatom,withapositively chargednucleusandnegativelychargedelectronssurroundingit.Inasolid,theatomsarexedinposition inthelattice,butitisassumedthatsomeoftheelectronscanbreakfreeoftheir"host"atomandmove abouttootherplaceswithinthesolid.Inourmodel,letuschoosethepolarityofthemobilechargesto bepositive;thisisnotusuallythecase,butwecanavoidalotof"minusones"thisway,andhaveabetter chanceofendingupwiththerightanswerintheend. Figure1.1: Modelofaconductor. AsshowninFigure1.1,themodeloftheconductorconsistsofanumberofmobilepositivechargesrepresentedbytheballswiththe"+"signinthemandanequalnumberofxednegativechargesrepresented bythebare"-"sign.Insubsequentgures,wewillleaveoutthexedcharge,sinceitcannotcontributein anywaytotheconductionprocess,butkeepinmindthatitisthere,andthatthetotalnetchargeiszero withinthematerial.Eachofthemobilechargecarriershasamass, m ,andanamountofcharge, q .

PAGE 9

3 Figure1.2: Applyingapotentialtoaconductor Inordertohavesomeconduction,wehavetoapplyapotentialorvoltageacrossthesampleFigure1.2. Wedothiswithabattery,whichcreatesapotentialdierence, V ,betweenoneendofthesampleandthe other.Wewillmakethesimplestassumptionthatwecan,andsaythatthevoltage, V ,givesrisetoauniform electriceldwithinthesample.Themagnitudeoftheelectriceldisgivensimplyby E = V L .2 where L isthelengthofthesample,and V isthevoltagewhichisplacedacrossit.Intruth,weshould beshowing E aswellassubsequentforcesetc.asvectorsinourequations,butsincetheirdirectionwillbe obvious,andunambiguous,let'skeepthingssimple,andjustwritethemasscalers. Electricpotential ,or voltage,isjustameasureofthechangeinpotentialenergyperunitchargegoingfromoneplacetoanother. Sinceenergy,orworkissimplyforcetimesdistance,ifwedividetheenergyperunitchargebythedistance overwhichthatpotentialexists,wewillendupwithforceperunitcharge,orelectriceld, E .Ifyouare notsureaboutwhatyoujustread,writeitoutasequations,andseethatitisso. TheelectriceldwillexertaforceonthemovablechargesAndthexedonestooforthatmatter,but sincetheycannotgoanywhere,nothinghappenstothem.Theforceisgivensimplyastheproductofthe electriceldstrengthtimesthecharge F = qE .3 TheforceactsonthechargesandcausesthemtoaccelerateaccordingtoNewton'sequationsofmotion F = m d dt v t = qE .4 or d dt v t = qE m .5 Thus,thevelocityofaparticlewithnoinitialvelocitywillincreaselinearlywithtimeas: v t = qE m t .6 Therateofaccelerationisproportionaltothestrengthoftheelectriceld,andinverselyproportionaltothe massoftheparticle.Theparticlecannotcontinuetoaccelerateforeverhowever.Sinceitislocatedwithin asolid,soonerorlateritwillcollidewitheitheranothercarrier,orperhapsoneofthexedatomswithin thesolid.Wewillassumethatthecollisioniscompletelyinelastic,andthatafteracollision,theparticle

PAGE 10

4 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES comestoastop,onlytobeacceleratedagainbytheelectriceld.Ifweweretomakeaplotoftheparticles velocityasafunctionoftime,itmightlooksomethinglikeFigure1.3. Figure1.3: Velocityasafunctionoftimeforchargecarrier Althoughtheparticleachievesvariousvelocities,dependinguponhowmuchtimethereisbetweencollisions,therewillbesomeaveragevelocity, v ,whichwilldependuponthedetailsofthecollisionprocess. Letusdeneascatteringtime s whichwillgiveusthataveragevelocitywhenwemultiplyittimesthe accelerationoftheparticle.Thatis: v = qE s m .7 or s m v qE .8 Nowlet'stakealookatjustasmallsectionoftheconductorFigure1.4.Itwillhavethecrosssectionof thesample, A ,butwillonlybe v t long,where t isjustsomearbitrarytimeinterval. Figure1.4: Sectionoftheconductor Afteratime t haspassed,allofthechargeswithintheboxwillhaveleftit,astheyareallmovingwith thesameaveragevelocity, v .Ifthedensityofchargecarriersintheconductoris n perunitvolume,then thenumberofcarriers N withinourlittleboxisjust n timesthevolumeofthebox v tA N = n v tA .9 Thusthetotalcharge, Q ,whichleavestheboxintime t isjust qN .Thecurrentow, I ,isjusttheamount

PAGE 11

5 ofchargewhichowsoutoftheboxperunittime I = qn v tA t = qn vA = q 2 n s EA m = Q t .10 Wenowhavetwochoices,wecanlookatourresultfromaeldquantitypointofview,inwhichcasewewill beinterestedinthe currentdensity J ,whichisjustthecurrent, I ,dividedbythecross-sectionalarea J = I A = q 2 n s m E = E .11 where iscalledthe conductivity ofthematerial.Ifwelookattheconductorfromamacroscopicpoint ofview,thenweareinterestedintherelationshipbetweenthevoltageandthecurrent.Thevoltageisjust theelectriceldtimesthelengthofthesample,andthecurrentisjustthecurrentdensitytimesiscross sectionalarea.Thuswehave I = AJ = AE = A V L .12 or V = L A I = RI .13 where R istheresistanceofthesample.Wehavediscovered Ohm'slaw Notethat.13tellsusthattheresistanceofthesampleisproportionaltoitslengththelongerthe sample,thehighertheresistanceandinverselyproportionaltoitscrosssectionalareathefatterthesample, thelowertheresistance.Thesampleresistanceisalsoinverselyproportionaltotheconductivity ofthe sample.Sometimes,insteadofconductivity,the resistivity ,isspeciedforaresistivematerial.The resistivityissimplytheinverseoftheconductivity = 1 .14 andthus: R = L A .15 And,inaneorttowardscompleteness,thereisoneotherquantitywhichyoumightruninto,andthatis thecarrier mobility .Themobilityisjusttheproportionalityfactorbetweentheaveragevelocityofthe particleandtheelectriceld.Thatis: v = E .16 Youshouldcheckthatthefollowingtworelationshipsarecorrect: = nq .17 = q s m .18 Ifwetakeanordinaryconductorandwewillhavetodenelaterwhatwemeanbythatandheatitup, theatomswithinthematerialstarttovibratefasterduetotheelevatedtemperature,andthecarrierssuer signicantlymorecollisions.Themeancollisiontime s decreases,andhencetheconductivitygoesdown, andtheresistanceofthesamplegoesup.

PAGE 12

6 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES 1.2IntroductiontoSemiconductors 2 Ifweonlyhadtoworryaboutsimpleconductors,lifewouldnotbeverycomplicated,butontheother handwewouldn'tbeabletomakecomputers,CDplayers,cellphones,i-Podsandalotofotherthings whichwehavefoundtobeuseful.Wewillnowmoveon,andtalkaboutanotherclassofconductorscalled semiconductors. Inordertounderstandsemiconductorsandinfacttogetamoreaccuratepictureofhowmetals,or normalconductorsactuallywork,wereallyhavetoresorttoquantummechanics.Electronsinasolidare verytinyobjects,anditturnsoutthatwhenthingsgetsmallenough,theynolongerexactlyfollowingthe classical"Newtonian"lawsofphysicsthatweareallfamiliarwithfromeverydayexperience.Itisnotthe purposeofthiscoursetoteachyouquantummechanics,sowhatwearegoingtodoinsteadisdescribethe resultswhichcomefromlookingatthebehaviorofelectronsinasolidfromaquantummechanicalpointof view. Solidsatleasttheoneswewillbetalkingabout,andespeciallysemiconductorsarecrystallinematerials, whichmeansthattheyhavetheiratomsarrangedinaorderedfashion.Wecantakesiliconthemost importantsemiconductorasanexample.SiliconisagroupIVelement,whichmeansithasfourelectrons initsouterorvalenceshell.Siliconcrystallizesinastructurecalledthe diamond crystallattice.Thisis showninFigure1.5.Eachsiliconatomhasfourcovalentbonds,arrangedinatetrahedralformationabout theatomcenter. Figure1.5: Crystalstructureofsilicon Intwodimensions,wecanschematicallyrepresentapieceofsingle-crystalsiliconasshowninFigure1.6. Eachsiliconatomsharesitsfourvalenceelectronswithvalenceelectronsfromfournearestneighbors,lling theshellto8electrons,andformingastable,periodicstructure.Oncetheatomshavebeenarrangedlike this,theoutervalenceelectronsarenolongerstronglyboundtothehostatom.Theoutershellsofallof theatomsblendtogetherandformwhatiscalleda band .Theelectronsarenowfreetomoveaboutwithin thisband,andthiscanleadtoelectricalconductivityaswediscussedearlier. 2 Thiscontentisavailableonlineat.

PAGE 13

7 Figure1.6: A2-Drepresentationofasiliconcrystal Thisisnotthecompletestoryhowever,foritturnsoutthatduetoquantummechanicaleects,thereis notjustonebandwhichholdselectrons,butseveralofthem.Whatwillfollowisaveryqualitativepicture ofhowtheelectronsaredistributedwhentheyareinaperiodicsolid,andtherearenecessarilysomedetails whichwewillbeforcedtoglossover.Ontheotherhandthiswillgiveyouaprettygoodpictureofwhatis goingon,andmayenableyoutohavesomeunderstandingofhowasemiconductorreallyworks.Electrons arenotonlydistributedthroughoutthesolidcrystalspatially,buttheyalsohaveadistributioninenergy aswell.Thepotentialenergyfunctionwithinthesolidisperiodicinnature.Thispotentialfunctioncomes fromthepositivelychargedatomicnucleiwhicharearrangedinthecrystalinaregulararray.Adetailed analysisofhowelectron wavefunctions ,themathematicalabstractionwhichonemustusetodescribe howsmallquantummechanicalobjectsbehavewhentheyareinaperiodicpotential,givesrisetoanenergy distributionsomewhatlikethatshowninFigure1.7. Figure1.7: Schematicofthersttwobandsinaperiodicsolidshowingenergylevelsandbands Firstly,unlikethecaseforfreeelectrons,inaperiodicsolid,electronsarenotfreetotakeonanyenergy valuetheywish.Theyareforcedintospecicenergylevelscalled allowedstates whicharerepresentedby thecupsinthegure.Theallowedstatesarenotdistributeduniformlyinenergyeither.Theyaregrouped

PAGE 14

8 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES intospeciccongurationscalled energybands .Therearenoallowedlevelsatzeroenergyandforsome distanceabovethat.Movingupfromzeroenergy,wethenencountertherstenergyband.Atthebottom ofthebandthereareveryfewallowedstates,butaswemoveupinenergy,thenumberofallowedstates rstincreases,andthenfallsoagain.Wethencometoaregionwithnoallowedstates,calledanenergy bandgap .Abovethebandgap,anotherbandofallowedstatesexists.Thisgoesonandon,withanygiven materialhavingmanysuchbandsandbandgaps.ThissituationisshownschematicallyinFigure1.7,where thesmallcupsrepresentallowedenergylevels,andtheverticalaxisrepresentselectronenergy. Itturnsoutthateachbandhasexactly 2 N allowedstatesinit,where N isthetotalnumberofatoms intheparticularcrystalsamplewearetalkingabout.Sincethereare10cupsineachbandinthegure,it mustrepresentacrystalwithjust5atomsinit.Notaverybigcrystalatall!Intothesebandswemustnow distributeallofthevalenceelectronsassociatedwiththeatoms,withtherestrictionthatwecan onlyput oneelectronintoeachallowedstate .Thisistheresultofsomethingcalledthe Pauliexclusionprinciple Sinceinthecaseofsiliconthereare4valenceelectronsperatom,wewould just llupthersttwobands, andthenextwouldbeempty.Ifwemakethelogicalassumptionthattheelectronswillllinthelevels withthelowestenergyrst,andonlygointohigherlyinglevelsiftheonesbelowarealreadylled.This situationisshowninFigure1.8. Here,wehaverepresentedelectronsassmallblackballswitha"-"signonthem.Indeed,thersttwo bandsarecompletelyfull,andthenextisempty.Whatwillhappenifweapplyanelectriceldtothesample ofsilicon?Rememberthediagramwehaveathandrightnowisan energy basedone,weareshowinghow theelectronsaredistributedinenergy,nothowtheyarearrangedspatially.Onthisdiagramwecannot showhowtheywillmoveabout,butonlyhowtheywillchangetheirenergyasaresultoftheappliedeld. Theelectriceldwillexertaforceontheelectronsandattempttoacceleratethem.Iftheelectronsare accelerated,thentheymustincreasetheirkineticenergy.Unfortunately,therearenoemptyallowedstates ineitherofthelledbands.Anelectronwouldhavetojumpallthewayupintothenextemptybandin ordertotakeonmoreenergy.Insilicon,thegapbetweenthetopofthehighestmostoccupiedbandandthe lowestunoccupiedbandis1.1eV.OneeVisthepotentialenergygainedbyanelectronmovingacrossan electricalpotentialofonevolt.The meanfreepath ordistanceoverwhichanelectronwouldnormally movebeforeitsuersacollisionisonlyafewhundredangstroms )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(300 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(8 cmandsoyouwould needaverylargeelectriceldseveralhundredthousand volts cm inorderfortheelectrontopickupenough energyto"jumpthegap".Thismakesitappearthatsiliconwouldbeaverybadconductorofelectricity, andinfact,verypuresiliconisverypoorelectricalconductor.

PAGE 15

9 Figure1.8: Silicon,withrsttwobandsfullandthenextempty Ametalisanelementwithan odd numberofvalenceelectronssothatametalendsupwithanupper bandwhichisjusthalffullofelectrons.ThisisillustratedinFigure1.9.Hereweseethatonebandisfull, andthenextisjusthalffull.ThiswouldbethesituationfortheGroupIIIelementaluminumforinstance. Ifweapplyanelectriceldtothesecarriers,thosenearthetopofthedistributioncanindeedmoveinto higherenergylevelsbyacquiringsomekineticenergyofmotion,andeasilymovefromoneplacetothenext. Inreality,thewholesituationisabitmorecomplexthanwehaveshownhere,butthisisnottoofarfrom howitactuallyworks.

PAGE 16

10 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Figure1.9: Electrondistributionforametalorgoodconductor So,backtooursiliconsample.Iftherearenoplacesforelectronsto"move"into,thenhowdoessilicon workasa"semiconductor"?Well,intherstplace,itturnsoutthatnotalloftheelectronsareinthe bottomtwobands.Insilicon,unlikesayquartzordiamond,thebandgapbetweenthetop-mostfullband, thenextemptyoneisnotsolarge.Aswementionedaboveitisonlyabout1.1eV.Solongasthesiliconis notatabsolutezerotemperature,someelectronsnearthetopofthefullbandcanacquireenoughthermal energythattheycan"hop"thegap,andendupintheupperband,calledthe conductionband .This situationisshowninFigure1.10.

PAGE 17

11 Figure1.10: Thermalexcitationofelectronsacrossthebandgap Insiliconatroomtemperature,roughly 10 10 electronspercubiccentimeterarethermallyexcitedacross theband-gapatanyonetime.Itshouldbenotedthattheexcitationprocessisacontinuousone.Electrons arebeingexcitedacrosstheband,butthentheyfallbackdownintoemptyspotsinthelowerband.On averagehowever,the 10 10 ineach cm 3 ofsiliconiswhatyouwillndatanygiveninstant.Now10billion electronspercubiccentimeter seems likealotofelectrons,butletsdoasimplecalculation.Themobilityof electronsinsiliconisabout1000 cm 2 volt-sec .Remember,mobilitytimeselectriceldyieldstheaveragevelocity ofthecarriers.Electriceldhasunitsof volts cm ,sowiththeseunitswegetvelocityin cm sec asweshould.The chargeonanelectronis 1 : 6 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(19 coulombs.Thusfromthisequation.17: = nq =10 10 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1 : 6 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(19 1000 =1 : 6 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 mhos cm .19 Ifwehaveasampleofsilicon1cmlongby mm mm square,itwouldhavearesistanceof R = L A = 1 : 6 10 )]TJ/F6 4.9813 Tf 5.396 0 Td [(6 : 1 2 =62 : 5 M .20 whichdoesnotmakeitmuchofa"conductor".Infact,ifthiswerealltherewastothesiliconstory,we couldpackupandmoveon,becauseat any reasonabletemperature,siliconwouldconductelectricityvery poorly.

PAGE 18

12 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES 1.3DopedSemiconductors 3 Toseehowwecanmakesiliconausefulelectronicmaterial,wewillhavetogobacktoitscrystalstructure. Supposesomehowandwewilltalkabouthowthisisdonelaterwecouldsubstituteafewatomsof phosphorusforsomeofthesiliconatoms. Figure1.11: Asiliconcrystal"doped"withphosphorus Ifyousneakalookattheperiodictable,youwillseethatphosphorusisagroupVelement,ascompared withsiliconwhichisagroupIVelement.Whatthismeansisthephosphorusatomhas ve outeror valence electrons,insteadofthefourwhichsiliconhas.Inalatticecomposedmainlyofsilicon,theextraelectron associatedwiththephosphorusatomhasno"mating"electronwithwhichitcancompleteashell,andsois leftlooselydanglingtothephosphorusatom,withrelativelylowbindingenergy.Infact,withtheaddition ofjustalittlethermalenergyfromthenaturalorlatentheatofthecrystallatticethiselectroncanbreak freeandbelefttowanderaroundthesiliconatomfreely.Inour"energyband"picture,wehavesomething likewhatweseeinFigure1.12.ThephosphorusatomsarerepresentedbytheaddedcupswithP'sonthem. Theyarenewallowedenergylevelswhichareformedwithinthe"bandgap"nearthebottomoftherst emptyband.Theyarelocatedcloseenoughtotheemptyor"conduction"band,sothattheelectrons whichtheycontainareeasilyexcitedupintotheconductionband.There,theyarefreetomoveaboutand contributetotheelectricalconductivityofthesample.Notealso,however,thatsincetheelectronhasleft thevicinityofthephosphorusatom,thereisnowonemoreprotonthanthereareelectronsattheatom, andhenceithasanetpositivechargeof1 q .Wehaverepresentedthisbyputtingalittle"+"signineach P-cup.Notethatthispositivechargeisxedatthesiteofthephosphorousatomcalleda donor sinceit "donates"anelectronupintotheconductionband,andisnotfreetomoveaboutinthecrystal. 3 Thiscontentisavailableonlineat.

PAGE 19

13 Figure1.12: Silicondopedwithphosphorus Howmanyphosphorusatomsdoweneedtosignicantlychangetheresistanceofoursilicon?Suppose wewantedour1mmby1mmsquaresampletohavearesistanceofoneohmasopposedtomorethan60 M .Turningtheresistanceequationaroundweget = L RA = 1 1 0 : 1 2 =100 mho cm .21 AndhenceIfwecontinuetoassumeanelectronmobilityof 1000 cm 2 voltsec n = q = 100 : 6 10 )]TJ/F6 4.9813 Tf 5.396 0 Td [(19 =6 : 25 10 17 cm 3 .22 Nowaddingmorethan 6 10 17 phosphorusatomspercubiccentimetermightseemlikealotofphosphorus, untilyourealizethattherearealmost 10 24 siliconatomsinacubiccentimeterandhenceonlyoneinevery 1.6millionsiliconatomshastobechangedintoaphosphorusonetoreducetheresistanceofthesample fromseveral10sofM downtoonlyone .Thisistherealpowerofsemiconductors.Youcanmake dramaticchangesintheirelectricalpropertiesbytheadditionofonlyminuteamountsofimpurities.This processiscalled" doping "thesemiconductor.Itisalsooneofthegreatchallengesofthesemiconductor manufacturingindustry,foritisnecessarytomaintainfantasticlevelsofcontroloftheimpuritiesinthe materialinordertopredictandcontroltheirelectricalproperties. Again,ifthisweretheendofthestory,westillwouldnothaveanycalculators,stereosor"Agentof Doom"videogamesOratleasttheywouldbeverybigandcumbersomeandunreliable,becausetheywould havetoworkusingvacuumtubes!.Wenowhavetofocusonthefew"empty"spotsinthelower,almost fullbandCalledthe valenceband .Wewilltakeanotherviewofthisband,fromasomewhatdierent perspective.ImustconfessatthispointthatwhatIamgivingyouisevenfurtherfromthewaythingsreally work,thenthe"cupsatdierentenergies"picturewehavebeenusingsofar.Theproblemis,thatinorder todothingsright,wehavetogetinvolvedinmomentumphase-space,alotmorequantummechanics,and

PAGE 20

14 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES generallyabunchofmathandconceptswedon'treallyneedinordertohavesomeideaofhowsemiconductor deviceswork.Whatfollowbelowisreallyintendedasamotivation,sothatyouwillhavesomefeelingthat whatwestateasresults,isactuallyreasonable. ConsiderFigure1.13.Hereweshowalloftheelectronsinthevalence,oralmostfullband,andfor simplicityshowonemissingelectron.Let'sapplyanelectriceld,asshownbythearrowinthegure.The eldwilltrytomovethenegativelychargedelectronstotheleft,butsincethebandisalmostcompletely full,theonlyonethatcanmoveistheonerightnexttotheemptyspot,or hole asitiscalled. Figure1.13: Bandfullofelectrons,withonemissing Onethingyoumaybeworryingaboutiswhathappenstotheelectronsattheendsofthesample.This isoneoftheplaceswherewearegettingasomewhatdistortedviewofthings,becauseweshouldreallybe lookinginmomentum,orwave-vectorspaceratherthan"real"space.Inthatpicture,theymagicallydrop oonesideand"reappear"ontheother.Thisdoesn'thappeninrealspaceofcourse,sothereisnoeasy waywecandealwithit. AshorttimeafterweapplytheelectriceldwehavethesituationshowninFigure1.14,andalittle whileafterthatwehaveFigure1.15.Wecaninterpretthismotionintwoways.Oneisthatwehaveanet owofnegativechargetotheleft,orifweconsidertheeectoftheaggregateofalltheelectronsinthe bandwhichwehavetodobecauseofquantummechanicalconsiderationsbeyondthescopeofthisbookwe couldpicturewhatisgoingonasasinglepositivecharge,movingtotheright.ThisisshowninFigure1.16. Notethatineitherviewwehavethesameneteectinthewaythetotal net chargeistransportedthrough thesample.Inthemostlynegativechargepicture,wehaveanetowofnegativechargetotheleft.Inthe singlepositivechargepicture,wehaveanetowofpositivechargetotheright.Bothgivethesamesignfor thecurrent!

PAGE 21

15 Figure1.14: Motionofthe"missing"electronwithanelectriceld Figure1.15: Furthermotionofthe"missingelectron"spot Figure1.16: Motionofa"hole"duetoanappliedelectriceld Thus,itturnsout,wecanconsidertheconsequencesoftheemptyspacesmovingthroughtheco-ordinated motionofelectronsinanalmostfullbandasbeingthemotionofpositivecharges,movingwhereverthese emptyspaceshappentobe.Wecallthesechargecarriers"holes"andtheytoocanaddtothetotalconduction ofelectricityinasemiconductor.Using torepresentthedensityin cm )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 ofspacesinthevalenceband

PAGE 22

16 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES and e and h torepresentthemobilityofelectronsandholesrespectivelytheyareusuallynotthesame wecanmodifythisequation.17togivetheconductivity ,whenbothelectrons' holes arepresent. = nq e + q h .23 Howcanwegetasampleofsemiconductorwitha lot ofholesinit?Whatif,insteadofphosphorus,we dopeoursiliconsamplewithagroupIIIelement,sayboron?ThisisshowninFigure1.17.Nowwehave some missing orbitals,orplaceswhereelectronscouldgoiftheywerearound.Thismodiesourenergy pictureasfollowsinFigure1.18.Nowweseeasetofnewlevelsintroducedbytheboronatoms.Theyare locatedwithinthebandgap,justalittlewayabovethetopofthealmostfull,orvalenceband.Electrons inthevalencebandcanbethermallyexcitedupintothesenewallowedlevels,creatingemptystates,or holes,inthevalenceband.Theexcitedelectronsarestuckattheboronatomsitescalled acceptors ,since they"accept"anelectronfromthevalenceband,andhenceactas xed negativecharges,localizedthere. Asemiconductorwhichisdopedpredominantlywithacceptorsiscalled p-type ,andmostoftheelectrical conductiontakesplacethroughthemotionofholes.Asemiconductorwhichisdopedwithdonorsiscalled n-type ,andconductiontakesplacemainlythroughthemotionofelectrons. Figure1.17: SilicondopedwithBoron Figure1.18: P-typesilicon,duetoboronacceptors Inn-typematerial,wecanassumethatallofthephosphorousatoms,or donors ,arefullyionizedwhen theyarepresentinthesiliconstructure.Sincethenumberofdonorsisusuallymuchgreaterthanthenative,

PAGE 23

17 orintrinsicelectronconcentration, )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(10 10 cm )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 ,if N d isthedensityofdonorsinthematerial,then n theelectronconcentration, N d Ifanelectrondecientmaterialsuchasboronispresent,thenthematerialiscalled p-type silicon,and theholeconcentrationisjust p N a theconcentrationof acceptors ,sincetheseatoms"accept"electrons fromthevalenceband. Ifbothdonorsandacceptorsareinthematerial,thenwhicheveronehasthehigherconcentrationwins out.Thisiscalled compensation .Iftherearemoredonorsthanacceptorsthenthematerialisn-type and n N d )]TJ/F11 9.9626 Tf 10.21 0 Td [(N a .Iftherearemoreacceptorsthandonorsthenthematerialisp-typeand p N a )]TJ/F11 9.9626 Tf 10.211 0 Td [(N d Itshouldbenotedthatinmostcompensatedmaterial,onetypeofimpurityusuallyhasamuchgreater severalorderofmagnitudeconcentrationthantheother,andsothesubtractionprocessdescribedabove usuallydoesnotchangethingsverymuch. 10 18 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 16 10 18 Oneotherfactwhichyoumightndusefulisthat,again,becauseofquantummechanics,itturnsout thatthe product oftheelectronandholeconcentrationinamaterialmustremainaconstant.Insiliconat roomtemperature: np n i 2 10 20 cm )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 .24 Thus,ifwehaveann-typesampleofsilicondopedwith 10 17 donorspercubiccentimeter,then n ,the electronconcentrationisjustand p ,theholeconcentration,is 10 20 10 17 =10 3 cm )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 .Thecarrierswhichdominate amaterialarecalled majoritycarriers ,whichwouldbetheelectronsintheaboveexample.Theother carriersarecalled minoritycarriers theholesintheexampleandwhile 10 3 mightnotseemlikemuch comparedto 10 17 thepresenceofminoritycarriersisstillquiteimportantandcannotbeignored.Notethat ifthematerialisundoped,thenitmustbethat n = p and n = p =10 10 Thepictureof"cups"ofdierentallowedenergylevelsisusefulforgainingapictorialunderstandingof whatisgoingoninasemiconductor,butbecomessomewhatawkwardwhenyouwanttostartlookingat deviceswhicharemadeupofbothnandptypesilicon.Thus,wewillintroduceonemorewayofdescribing whatisgoingoninourmaterial.ThepictureshowninFigure1.19iscalledabanddiagram.A band diagram isjustarepresentationoftheenergyasafunctionofpositionwithasemiconductordevice.Ina banddiagram,positiveenergyforelectronsisupward,whileforholes,positiveenergyisdownwards.That is,ifanelectronmoves upward ,itspotentialenergy increases justasawithanormalmassinagravitational eld.Also,justasamasswill"falldown"ifgivenachance,anelectronwillmovedownaslopeshownina banddiagram.Ontheotherhand,holesgainenergybymoving downward andsotheyhaveatendancyto "oat"upwardifgiventhechance-muchlikeabubbleinaliquid.Thelinelabeled E e inFigure1.19shows theedgeoftheconductionband,orthebottomofthelowestunoccupiedallowedband,while E v isthetop edgeofthevalence,orhighestoccupiedband.Thebandgap, E g forthematerialisobviously E c )]TJ/F11 9.9626 Tf 9.453 0 Td [(E v .The dottedlinelabeled E f iscalledthe Fermilevel andittellsussomethingaboutthechemicalequilibrium energyofthematerial,andalsosomethingaboutthetypeandnumberofcarriersinthematerial.Moreon thislater.Notethatthereisnozeroenergylevelonadiagramsuchasthis.WeoftenuseeithertheFermi leveloroneorotherofthebandedgesasareferencelevelonlieuofknowingexactlywhere"zeroenergy" islocated. Figure1.19: Anelectronband-diagramforasemiconductor

PAGE 24

18 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES ThedistanceinenergybetweentheFermilevelandeither E c and E v givesusinformationconcerning thedensityofelectronsandholesinthatregionofthesemiconductormaterial.Thedetails,onceagain,will havetobebeggedoongroundsofmathematicalcomplexity.TakeSemiconductorDevicesELEC462in yoursenioryearandndouthowisreallyworks!Itturnsoutthatyoucansay: n = N c e )]TJ/F27 6.9738 Tf 6.226 7.682 Td [( E c )]TJ/F9 4.9813 Tf 5.397 0 Td [(E f kT .25 p = N v e )]TJ/F27 6.9738 Tf 6.226 7.682 Td [( E f )]TJ/F9 4.9813 Tf 5.396 0 Td [(E v kT .26 Both N c and N v areconstantsthatdependonthematerialyouaretalkingabout,butaretypicallyon theorderof 10 19 cm )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 .TheexpressioninthedenominatoroftheexponentialisjustBoltzman'sconstant, k ,timesthetemperature T ofthematerialinabsolutetemperatureorKelvin. Boltzman'sconstant k = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(8 : 63 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 eV K .Atroomtemperature kT =1 = 40 ofanelectronvolt.Lookcarefullyatthenumerators intheexponential.Noterstthatthereisaminussigninfront,whichmeansthebiggerthenumberinthe exponent,thefewercarrierswehave.Thus,thetopexpressionsaysthatifwehaven-typematerial,then E f mustnotbetoofarawayfromtheconductionband,whileifwehavep-typematerial,thentheFermi level, E f mustbedownclosetothevalenceband.Thecloser E f getsto E c themoreelectronswehave.The closer E f getsto E v ,themoreholeswehave.Figure1.19thereforemustbeforasampleofn-typematerial. NotealsothatifweknowhowheavilyasampleisdopedThatis,weknowwhat N d isforexampleand fromthefactthat n N d wecanuse.25tondouthowfarawaytheFermilevelisfromtheconduction band E c )]TJ/F11 9.9626 Tf 9.963 0 Td [(E f = kTln N c N d .27 Tohelpfurtherinourabilitytopicturewhatisgoingon,wewilloftenaddtothisbanddiagram,somesmall signedcirclestoindicatethepresenceofmobileelectronsandholesinthematerial.Notethattheelectrons arespreadoutinenergy.Fromour"cups"pictureweknowtheyliketostayinthelowerenergystatesif possible,butsomewillbedistributedintothehigherlevelsaswell.Whatisdistortedhereisthescale.The band-gapforsiliconis1.1eV,whilethe actual spreadoftheelectronswouldprobablyonlybeafewtenths ofaneV,notnearlyasmuchasisshowninFigure1.20.Letslookatasampleofp-typematerial,justfor comparison.Notethatforholes,increasingenergygoes down notup,sotheirdistributionisinvertedfrom thatoftheelectrons.Youcankindofthinkofholesasbubblesinaglassofsodaorbeer,theywanttooat tothetopiftheycan.Notealsoforbothnandp-typematerialtherearealsoafew"minority"carriers,or carriersoftheoppositetype,whicharisefromthermalgenerationacrosstheband-gap. Figure1.20: Banddiagramforann-typesemiconductor

PAGE 25

19 1.4P-NJunction:PartI 4 Figure1.21: Banddiagramforap-typesemiconductor Figure1.22: Anon-equilibriump-njunction Wearenowreadytomakeanactualusefuldevice!Let'stakeapieceofn-typematerial,andapieceofp-type material,andstickthemtogether,asshowninFigure1.22.Thiswaywewillbemakinga pn-junction ,or diode ,whichwillbeourrstrealelectricdeviceotherthanasimpleresistor. ThereareacoupleofthingswrongwithFigure1.22.Firstofall,oneoftherulesregardingtheFermi levelisthatwhenyouhaveasystemat equilibrium thatis,whenitisarest,andisnotbeinginuenced byexternalforcessuchasthermalgradients,electricalpotentialsetc.,theFermilevelmustbethesame everywhere.Secondly,wehaveabigbunchofholesontherightandabigbunchofelectronsontheleft,and sowewouldexpect,thatintheabsenceofsomeforcetokeepthemthisway,theywillstarttospreadout untiltheirdistributionismoreorlessequaleverywhere.Finally,werememberthataholeisjustanabsence ofanelectron,andsinceanelectronintheconductionbandcanlowerthesystemenergybyfallingdowninto oneoftheemptyholestates,itseemslikelythatthiswillhappen.Thisprocessiscalled recombination Theplacewherethisismostlikelytooccur,ofcourse,wouldberightatthejunctionbetweenthenandp regions.ThisisshowninFigure1.23. 4 Thiscontentisavailableonlineat.

PAGE 26

20 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Figure1.23: Recombinationofholesandelectrons Nowismightseemthatthisrecombinationeectmightjustgoonandon,untiltherearenocarriersleft inthesample.Thisisnotthecasehowever.Inordertoseewhatbringseverythingtoahalt,weneedyet anotherdiagram.Figure1.24ismorephysicalthanwhatwehavebeenlookingatsofar.Itisapictureof theactualp-njunction,showingboththeholesandtheelectrons.Wealsoneedtoputinthedonorsand acceptorshowever,ifwewanttoseewhatgoeson.Thexedcan'tmovearoundchargesofthedonorsand acceptorsarerepresentedbysimple"+"and"-"signs.Theyarearrangedinanicelattice-likearrangement toremindusthattheyarestucktothecrystallattice.Inrealityhowever,eventhoughtheyarestuckin thecrystallattice,therearesofewofthemcomparedtothesiliconatomsthattheirdistributionwouldbe quiterandom.Forthemobileholesandelectrons,wewillstaywiththelittlecircleswithchargesignsin them.Thesearerandomlydistributed,toremindusthattheyarefreetomoveaboutthecrystal. Figure1.24: Spatialschematicofap-njunction Wewillnowhavetoallowsomeoftheholesandelectronsagainnearthejunctiontorecombine. Remember,whenanelectronandaholerecombine,theybothareannihilatedanddisappear.Notethatthis processconserveschargeandifwecouldcalculateitmomentumaswell.Thereisobviouslysomeenergy lost,butthiswillsimplyshowupasvibrations,orheat,withinthecrystallattice.Or,inthecaseofan LED,aslightemittedfromthedevice.See,alreadyweknowenoughaboutsemiconductorstounderstand somewhathowanactualdeviceworks.LightcommingfromanLEDissimplytheenergywhichisrealeased whenanelectronandholerecombine.Wewilltakealookatthisinmoredetaillater.Let'sallowsome recombinationtooccur,asshowninFigure1.25.

PAGE 27

21 Figure1.25: Thejunctionaftersomerecombinationhasoccurred AndtheninFigure1.26somemore..... Figure1.26: Afterfurtherrecombination 1.5PN-Junction:PartII 5 Ifyoulookcloselyatthesepictures,youwillnoticesomething.Asweremovemoreandmoreelectrons andholes,wearestartingto"uncover"thexedchargesassociatedwiththedonorsandacceptors.Weare makingwhatisknownasa depletionregion ,sonamedbecauseitis depleted ofmobilecarriersholes andelectrons.Theuncoverednetchargeinthedepletionregionisseparated,withnegativechargeinthe p-region,andpositivechargeinthen-region.Whatwillsuchachargeseparationgiveriseto?Why,an electriceld!Ofcourse!Whichwaywilltheeldpoint?Theelectriceldwhicharisesfromaseparationof chargesalwaysgoesfromthepositivecharge,towardsthenegativecharge.ThisisshowninFigure1.27. Figure1.27: Thepn-junctionwiththeresultantbuilt-inelectriceld 5 Thiscontentisavailableonlineat.

PAGE 28

22 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Whateectwillthiseldhaveonourdevice?Itwillhavethetendencytopushtheholesbackinto thep-regionandtheelectronsintothen-region.Thisisjustwhatweneedtocounteracttherecombination whichhasbeengoingon,andhopefullybringittoastop. Nowtrytothinkthroughwhateectthiseldcouldhaveonourenergybanddiagram.Theband diagramisforelectrons,soifanelectronmovesfromtherighthandsideofthedevicethen-regiontowards thelefthandsidethep-region,itwillhavetomovethroughanelectriceldwhichisopposingitsmotion. Thismeansithasdosomework,orinotherwords,thepotentialenergyfortheelectronmustgoup.We canshowthisonthebanddiagrambysimplyshiftingthebandsonthelefthandsideupward,toindicate thatthereisashiftinpotentialenergyaselectronsmovefromrighttoleftacrossthejunction. Figure1.28: Energybanddiagramforap-njunctionatequilibrium Theshiftofthebands,whichisjustthedierencebetweenthelocationoftheFermilevelinthen-region andtheFermilevelisthep-region,iscalledthe built-inpotential V bi .Thisbuilt-inpotentialkeepsthe majorityofholesinthep-region,andtheelectronsinthen-region.Itprovidesapotentialbarrier,which preventscurrentowacrossthejunction.Onthebanddiagramwehavetomultiplythebuilt-inpotential V bi bythechargeofanelectron, q ,sothatwecanrepresenttheshiftinenergyintermsof electronvolts theunitofpotentialenergyusedinbanddiagrams. Howbigis V bi ?Thisisnottoohardtogureout.Let'slookatFigure1.28alittlemorecarefully. Remember,weknowfromthisequation.25andthisequation.26thatsince n = N d inthen-region and p = N a inthep-region,wecanrelatethedistanceoftheFermilevelfrom E c and E f by E c )]TJ/F11 9.9626 Tf 9.963 0 Td [(E f = kTln N c N d .28 and E f )]TJ/F11 9.9626 Tf 9.962 0 Td [(E v = kTln N v N a .29 LookatFigure1.28andseeifyoucanagreethat qV BI = E g )]TJ/F8 9.9626 Tf 9.962 0 Td [( E c )]TJ/F11 9.9626 Tf 9.963 0 Td [(E f )]TJ/F8 9.9626 Tf 9.963 0 Td [( E f )]TJ/F11 9.9626 Tf 9.963 0 Td [(E v = E g )]TJ/F11 9.9626 Tf 9.962 0 Td [(kTln N c N d )]TJ/F11 9.9626 Tf 9.963 0 Td [(kTln N c N d = E g )]TJ/F11 9.9626 Tf 9.962 0 Td [(kTln N c N v N d N a .30 Where N d and N a arethedopingdensitiesinthenandpsincrespectively.Remember, kT =1 = 40 eV = 0 : 025 eV E g =1 : 1 eV and N c and N v areboth )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(10 19 .Thus, qV BI =1 : 1 eV )]TJ/F8 9.9626 Tf 9.962 0 Td [(0 : 025 eVln 10 38 N d N a

PAGE 29

23 Herethe q infrontofthe V BI andthe e in eV areboththechargeof1electronandtheycanceloutmaking V BI = 1 : 1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(0 : 025 ln 10 38 N d N a volts Supposeboth N d and N a arebothabout[10tothe15th]-notuncommonvalues.Howbigwouldthebuilt-in potentialbeinthiscase? Itturnsoutthatwecanactuallyderivesomespecicdetailsaboutthedepletionregionifwemake onlyacoupledofsimplifyingandoftenjustiedassumptions.Inordertomakethematheasier,andalso becausemanyp-njunctionsarebuiltthisway,wewillconsiderwhatisknownasa onesidedjunction Figure1.29isapictureofsuchabeast:Inthisdiode,onesideismuchmoreheavilydopedthantheother. Inthisparticularexample,thep-sideisheavilydoped,andthen-sideisrelativelylightlydoped.Wecan notshowthetruepicturehere,becausetypically,themoreheavilydopedsidewillbedoped severalorders ofmagnitude greaterthanthelightlydopedside.Typicalvaluesmightbe N a =10 19 and N d =10 16 Regardlessofhowbigthedierenceishowever,theremustbeexactlythesameamountof"uncovered" chargeonbothsideofthejunction.Why?Becauseeachtimeaholeandelectronrecombinetoformthe depletionregion,theyeachleavebehindeitheradonororanacceptor.Acarefulcountoftheexposedcharge inFigure1.29showsthatIwascarefulenoughtodrawmygureaccuratelyforyou.Wedonotneedto haveaone-sideddiodetodotheanalysisthatwillfollow,buttheequationsareeasiertosolveifwedo. Figure1.29: Anexampleofaone-sideddiode Inordertoproceedfromhere,therstthingwedoismakeaplotofthechargedensity x aswemove throughthejunction.Naturally,inthebulk,sincetheholesandtheacceptorsinthep-side,ortheelectrons andthedonorsinthen-sidejustequaloneanother,thenetchargedensityiszero.Inthedepletionregion, thechargedensityis)]TJ/F11 9.9626 Tf 7.749 0 Td [(q N a onthep-sideand + q N d onthedonorside.Allthemobilecarriersare gone,andweareleftwithjustthechargedacceptorsordonors.Wewillmaketheassumptionthatonthe n-side,thedepletionextendsadistance )]TJ/F11 9.9626 Tf 7.749 0 Td [(x n fromthejunction.Onthep-side,theacceptorchargedensityis solarge,thatwewilltreatitisa -function,withessentiallynowidth.Theareasofthetwoboxesmustbe thesameequalamountofpositiveandnegativechargeandhence,thetallthinboxactuallyhasawidth of N d N a x n ,which,since N a isseveralordersofmagnitudegreaterthan N d ,meansthatthetallboxhasavery verysmallwidthcomparedtothelower,widerone,whichis qN d tall,andhasawidthof x n .

PAGE 30

24 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Figure1.30: Chargedensityasafunctionofposition 1.6Gauss'Law 6 Nowwehavetoreviewsomeeldtheory.Wewillbeusingeldsfromtimetotimeinthiscourse,andwhen weneedsomeaspectofeldtheory,wewillintroducewhatweneedatthatpoint.Thisseemstomakemore sensethanspendingseveralweekstalkingaboutalotofabstracttheorywithoutseeinghoworwhyitcan beuseful. Therstthingweneedtorememberis Gauss'Law .Gauss'Law,likemostofthefundamentallaws ofelectromagnetismcomesnotfromrstprinciple,butratherfromempiricalobservationandattemptsto matchexperimentswithsomekindofself-consistentmathematicalframework.Gauss'Lawstatesthat: H s DdS = Q encl = H v v dV .31 where D isthe electricdisplacementvector ,whichisrelatedtothe electriceldvector E ,bythe relationship D = E iscalledthe dielectricconstant .Insiliconithasavalueof 1 : 1 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 F cm .Note that D musthaveunitsof Coulombs cm 2 tohaveeverythingworkoutOK. Q encl isthetotalamountofcharge enclosedinthevolume V ,whichisobtainedbydoingavolumeintegralofthechargedensity v 6 Thiscontentisavailableonlineat.

PAGE 31

25 Figure1.31: PictorialrepresentationofGauss'Law. .31justsaysthatifyouaddupthesurfaceintegralofthedisplacementvector D overaclosedsurface S ,whatyougetisthesumofthetotalchargeenclosedbythatsurface.Usefulasitis,theintegralform ofGauss'Law,whichiswhat1.31iswillnothelpusmuchinunderstandingthedetailsofthedepletion region.Wewillhavetoconvertthisequationtoitsdierentialform.Wedothisbyrstshrinkingdownthe volume V untilwecantreatthechargedensity v asaconstant ,andreplacethevolumeintegralwitha simpleproduct.Sincewearemaking V small,let'scallit V toremindusthatwearetalkingaboutjust asmallquantity. I v v dV v .32 Andthus,Gauss'Lawbecomes: H s DdS = H s EdS = V .33 or 1 V 0 @ I s EdS 1 A = .34 Now,by denition thelimitoftheLHSof.34as V 0 isknownasthedivergenceofthevector E div E .Thuswehave lim V 0 1 V H s EdS = div E = .35 Notewhatthissaysaboutthedivergence.Thedivergenceofthevector E isthelimitofthesurfaceintegral of E overavolume V ,normalizedbythevolumeitself,asthevolumeshrinkstozero.Iliketothinkofasa kindof"pointsurfaceintegral"ofthevector E .

PAGE 32

26 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Figure1.32: Smallvolumefordivergence If E onlyvariesinonedimension,whichiswhatweareworkingwithrightnow,theexpressionfor thedivergenceisparticularlysimple.Itiseasytoworkoutwhatitisfromasimplepicture.Lookingat Figure1.32weseethatif E isonlypointedalongonedirectionlet'ssay x andisonlyafunctionof x ,then thesurfaceintegralof E overthevolume V = x y z isparticularlyeasytocalculate. I s EdS = E x + x y z )]TJ/F81 9.9626 Tf 9.962 0 Td [(E x y z .36 Wherewerememberthatthesurfaceintegralisdenedasbeingpositiveforanoutwardpointingvector andnegativeforonewhichpointsintothevolumeenclosedbythesurface.Nowweusethedenitionofthe divergence div E = lim V 0 1 V H s EdS = lim V 0 E x + x )]TJ/F82 6.9738 Tf 6.227 0 Td [(E x y z x y z = lim V 0 E x + x )]TJ/F82 6.9738 Tf 6.226 0 Td [(E x x = @ @x E x .37 So,wehaveforthedierentialformofGauss'law: @ @x E x = x .38 Thus,inourcase,therateofchangeof E with x d dx E ,orthe slopeof E x isjustequaltothecharge density, x ,dividedby 1.7DepletionWidth 7 WecannowgobacktothechargedensityasafunctionofpositiongraphFigure1.30andeasilyndthe electriceldinthedepletionregionasafunctionofposition.IfweintegrateGauss'Law.38,wegetfor theelectriceld: E x = 1 Z x dx .39 7 Thiscontentisavailableonlineat.

PAGE 33

27 We could writedownanexpressionfor x andthenformallyintegrateittoget E x butwecanalso justdoitgraphically,whichisaloteasier,andgivesusamuchmoreintuitivefeelingforwhatisgoingon. Let'sstartdoingourintegralat[xequals-innity]Wheneverweperformanintegralsuchas.39,we've gottoremembertoaddaconstanttoouranswer.Sincewecannothaveanelectriceldwhichextendsto innityeitherplusorminushowever,wecansafelyassume E )]TJ/F11 9.9626 Tf 7.749 0 Td [(infinity =0 andremainsatthatvalue untilwegettotheedgeofthedepletionregionatessentiallyxequalszero.Sincethechargedensityis zeroallthewayuptotheedgeofdepletionregion,Gausstellsusthattheelectriceldcannotchangehere either.Whenwegettox=0weencounterthelargenegativedelta-functionofnegativechargeattheedgeof thedepletionregion.Ifyoucanrememberbacktoyourcalculus,whenyouintegrateadeltafunction,you getastep.Sincethechargeinthep-sidedeltafunctionisnegative,whenweintegrateit,wegetanegative step.Sincewedon'tknowyethowbigthestepwillbe,let'sjustcallit-|Emax|. Figure1.33: Findingtheelectriceldinthep-typeregion

PAGE 34

28 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Figure1.34: Finishingtheintegral Inthen-sideofthedepletionregion x =+ q N d = @ @x E .40 andsoweplot E x withapositiveslopeof qN d ,startingat E x = )]TJ/F11 9.9626 Tf 7.748 0 Td [(Emax atx=0.Thislinecontinues withthispositiveslopeuntilitreachesavalueof0atx= x n .WeknowthatExmustequal0atx= x n becausethereisnofurtherchargeoutsideofthedepletionregionandEmustbe0outsidethisregion. Wearenowdonedoingtheintegral.Wewouldknoweverythingaboutthisproblem,ifwejustknew what x n was.Notethatsinceweknowtheslopeofthetrianglenow,wecannd )]TJ/F11 9.9626 Tf 7.748 0 Td [(E max intermsofthe slopeand x n .Wecanderiveanexpressionfor x n ,ifwerememberthattheintegraloftheelectriceldover adistanceisthepotentialdropacrossthatdistance.Whatisthepotentialdropingoingfromthep-sideto then-sideofthediode? Asareminder,Figure1.35showsthejunctionbanddiagramagain.Thepotentialdropmustjustbe V bi the"built-in"potentialofthejunction.Obviously V bi cannotbegreaterthan1.1V,theband-gappotential. Ontheotherhand,bylookingatFigure1.35,andrememberingthatthebandgapinsiliconis1.1eV,itwill notbesomevaluelike0.2or0.4voltseither.Let'smakelifeeasyforourselves,andsay V bi =1 Volt .This willnotbetoofaro,andasyouwillseeshortly,theanswerisnotverysensitivetothe exact valueof V bi anyway.

PAGE 35

29 Figure1.35: Banddiagramforap-njunction Theintegralof E x isnowjusttheareaofthetriangleinFigure1.34.Gettingtheareaiseasy: area = 1 2 base height =1 = 2 x n qN d x n = qN d x n 2 2 = V bi .41 Wecansimplyturn.41aroundandsolvefor x n x n = s 2 V bi qN d .42 Aswesaid,forsilicon, Si =1 : 1 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 .Let'slet N d =10 16 cm )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 donors.Aswealreadyknowfrombefore, q =1 : 6 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(19 Coulombs.Thismakes x n =3 : 7 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(5 cm or0.37 mlong.Notaverywidedepletion region!Howbigis j E max j ?Pluggingin E max = qN d x n .43 Wend j E max j =53 ; 000 V cm !Whysuchabigelectriceld?Well,we'vegottoshiftthepotentialbyabout avolt,andwedonothavemuchdistancetodoitinlessthanamicron,andsotheremustbe,bydefault, afairlylargeeldinthedepletionregion.Remember,potentialiselectriceld times distance. Enoughp-njunctionelectrostatics.Thepointofthisexercisewastwo-fold; a: soyouwouldknow somethingaboutthedetailsofwhatisreallygoingoninap-njunction; b: toshowyouthatwithjust someverysimpleelectrostaticsandalittlethinking,itisnotsohardtogurethesethingsout! 1.8ForwardBiased 8 Nowlet'stakealookatwhathappenswhenweapplyanexternalvoltagetothisjunction.Firstweneed someconventions.Wemakeconnectionstothedeviceusing contacts ,whichweshowascross-hatched blocks.Thesecontactsallowthefreepassageofcurrentintoandoutofthedevice.Currentusuallyows throughwiresintheformofelectrons,soitiseasytoimagineelectronsowingintooroutofthen-region. Inthep-region,whenelectronsow out ofthedevice into thewire,holeswillowintothep-regionso astomaintaincontinuityofcurrentthroughthecontact.Whenelectronsowintothep-region,theywill recombinewithholes,andsowehavetheneteectofholesowingoutofthep-region. 8 Thiscontentisavailableonlineat.

PAGE 36

30 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Figure1.36: Ap-ndiodewithcontactsandexternalbias Withtheconventionthata positiveappliedvoltage meansthattheterminalconnectedtothep-region ispositivewithrespecttotheterminalconnectedtothen-region.Thisiseasytoremember;"pispositive, nisnegative".Letustrytogureoutwhatwillhappenwhenweapplyapositiveappliedvoltage V a .If V a ispositive,thenthatmeansthatthepotentialenergyforelectronsonthep-sidemustbe lower thanitwas undertheequilibriumcondition.Wereectthisonthebanddiagramby lowering thebandsonthep-side fromwheretheywereoriginally.ThisisshowninFigure1.37.

PAGE 37

31 Figure1.37: Ap-njunctionunderforwardbias AswecanseefromFigure1.37,whenthep-regionisloweredacoupleofthingshappen.Firstofall,the Fermilevelthedottedlineisnolongeraatline,butratheritbendsupwardingoingfromthep-region tothen-region.Theamountitbendsandhencetheamountofshiftofthebandsisjustgivenby qV a wheretheenergyscaleweareusingforthebanddiagramisin electron-volts which,aswesaidbefore, isacommonmeasureofpotentialenergywhenwearetalkingaboutelectronicmaterials.Theotherthing wecannoticeisthattheelectronsonthen-sideandtheholesonthep-sidenow"see"alowerpotential energybarrierthantheysawwhennovoltagewasapplied.Infact,itlooksasifalotofelectronsnowhave sucientenergysuchthattheycouldmoveacrossfromthen-regionandowintothep-region.Likewise, wewouldexpecttoseeholesmovingacrossfromthep-regionintothen-region. Thisowofcarriersacrossthejunctionwillresultinacurrentowacrossthejunction.Inordertosee howthiscurrentwillbehavewithappliedvoltage,wehavetousearesultfromstatisticalthermodynamics concerningthedistributionofelectronsintheconductionband,andholesinthevalenceband.Wesaw fromour"cups"analogy,thattheelectronstendtollintheloweststatesrst,withfewerandfewerof themaswegoupinenergy.Formostsituations,averygooddescriptionofjusthowtheelectronsare distributedinenergyisgivenbyasimpleexponentialdecay.Thiscomesaboutfromastatisticalanalysis ofelectrons,whichbelongtoaclassofparticlescalled Fermions .Fermionshavethepropertiesthatthey are: a: indistinguishablefromoneanother; b: obeythe PauliExclusionPrinciple whichsaysthattwo Fermionscannotoccupythesameexact state energyandspin; c: remainatsomexedtotalnumber N If n E tellsushowmanyelectronstherearewithanenergygreaterthansomevalue E c then n E is givensimplyas: n E = N d e )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( E )]TJ/F9 4.9813 Tf 5.396 0 Td [(E c kT .44 TheexpressioninthedenominatorisjustBoltzman'sconstanttimesthetemperatureinKelvins.Atroom temperature kT hasavalueofabout 1 = 40 ofaneVor25meV.Thisnumberissometimescalledthe thermal voltage V T ,butit'sokforyoutojustthinkofitasaconstantwhichcomesfromthethermodynamicsof

PAGE 38

32 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES theproblem.Because kT 1 = 40 ,youwillsometimessee.44andsimilarequationswrittenas n E = N d e )]TJ/F7 6.9738 Tf 6.226 0 Td [(40 E )]TJ/F10 6.9738 Tf 6.227 0 Td [(E c .45 Whichlooksalittlestrangeifyouforgetwherethe40camefrom,andjustseeitsittingthere. Iftheenergy E is E c theenergyleveloftheconductionband,then n E c = N d ,thedensityofelectrons inthen-typematerial.As E increasesabove E c ,thedensityofelectronsfallsoexponentially,asdepicted schematicallyinFigure1.38:Nowlet'sgobacktotheunbiasedjunction. Figure1.38: Distributionofelectronsintheconductionbandwithenergy Remember,aswesaidbefore,therearecurrentsowingacrossthejunction,evenifthereisnobias. Thecurrentwehaveshownas I f isduetothoseelectronswhichhaveanenergygreaterthanthebuilt-in potential.Theyareowingfromrighttoleft,asshownbytheopenarrow,which,ofcourse,givesacurrent owingfromlefttoright,asshownbythesolidarrows.Basedon.44thecurrentshouldbeproportional to: I f / N d e )]TJ/F27 6.9738 Tf 6.227 7.681 Td [( qV bi kT .46

PAGE 39

33 Figure1.39: Balancedowacrossajunction Theprincipleofdetailedbalancesaysthatatzerobias, I f = )]TJ/F11 9.9626 Tf 7.749 0 Td [(I r andso I R /)]TJ/F1 9.9626 Tf 19.925 14.048 Td [( N d e )]TJ/F27 6.9738 Tf 6.227 7.681 Td [( qV bi kT .47 I R = )]TJ/F8 9.9626 Tf 9.41 0 Td [( I f )]TJ/F11 9.9626 Tf 9.963 0 Td [(N d e )]TJ/F27 6.9738 Tf 6.227 7.682 Td [( qV BI kT .48 Now,whathappenswhenweapplythebias?Fortheelectronsoveronthen-side,thebarrierhasbeen reducedfromaheightof qV bi to q V bi )]TJ/F11 9.9626 Tf 9.963 0 Td [(V a andhencetheforwardcurrentwillbesignicantlyincreased. I f / N d e )]TJ/F27 6.9738 Tf 6.227 9.773 Td [( q V bi )]TJ/F9 4.9813 Tf 5.397 0 Td [(V a kT .49 Thereversecurrenthowever,willremainjustthesameasitwasbefore.47.

PAGE 40

34 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Figure1.40: Currentwhenthejunctionisforwardbiased Thetotalcurrentacrossthejunctionisjust I f + I r N d e qV a kT )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .50 wherewehavefactoredoutthe N d e )]TJ/F27 6.9738 Tf 6.227 7.682 Td [( qV bi kT termoutofbothexpressions.Wearenotprepared,withwhat weknowatthispoint,togettheothertermsintheproportionalitythatareinvolvedhere.Also,theastute readerwillnotethatwehavenotsaidanythingabouttheholes,butitshouldbeobviousthattheywillalso contributetothecurrent,andtheargumentswehavemadeforelectronswillholdfortheholesjustaswell. Wecantaketheeectoftheholes,andtheotherunknownsabouttheproportionality,andbindthem allintooneconstantcalled I sat ,sothatwewrite: I = I sat e qV a kT )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .51 Thisisthefamous diodeequation andisaveryimportantresult. 1.9TheDiodeEquation 9 Thereasonforcallingtheproportionalityconstant I sat willbecomeobviouswhenweconsiderreversebias. Letusnowmake V a negative insteadofpositive.Theappliedelectriceldnow addsinthesamedirection tothebuilt-ineld.Thismeansthebarrierwill increase insteadofdecrease,andsowehavewhatisshown inFigure1.41.Notethatwehavemarkedthebarrierheightas q V bi )]TJ/F11 9.9626 Tf 9.962 0 Td [(V a asbefore.Itisjustthatnow, V a isnegative,andsothebarrierisbigger. 9 Thiscontentisavailableonlineat.

PAGE 41

35 Figure1.41: P-Njunctionunderreversebias V a < 0 Remember,theelectronsfalloexponentiallyaswemoveupinenergy,soitdoesnottakemuchofa shiftofthebandsbeforethereareessentially no electronsonthen-sidewithenoughenergytogetoverthe barrier.Thisisreectedinthediodeequation.51where,ifwelet V a beanegativenumber, e qV a kT very quicklygoestozeroandweareleftwith I = )]TJ/F11 9.9626 Tf 7.748 0 Td [(I sat .52 Thus,whileintheforwardbiasdirection,thecurrentincreasesexponentiallywithvoltage,inthereverse directionitsimplysaturatesat )]TJ/F11 9.9626 Tf 7.749 0 Td [(I sat .Aplotof I asafunctionofvoltageoran I-Vcharacteristiccurve mightlooksomethinglikeFigure1.42. Figure1.42: IdealizedI-Vcurveforap-ndiode Infact,for realdiodes onesmadefromsilicon I sat issuchasmallvalueontheorderof 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(10 amps thatyoucannotevenseeitonmostcommonmeasuringdevicesoscilloscope,digitalvoltmeteretc.andif youweretolookonadevicecalleda curvetracer whichyouwilllearnmoreaboutinElectronicCircuits [ELEC342]whatyouwouldreallyseewouldbesomethinglikeFigure1.43.

PAGE 42

36 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Figure1.43: RealisticI-Vcurve Weseewhatlookslikezerocurrentinthereversedirection,andinfact,whatappearstobenocurrent untilwegetacertainamountofvoltageacrossthediode,afterwhichitveryquickly"turnson"withavery rapidlyincreasingforwardcurrent.Forsilicon,this"turnon"voltageisabout0.6to0.7volts. DigitalvoltmetersDVM'susethischaracteristicfortheir"diodecheck"function.Whattheydois, whenthe"red"orpositiveleadisconnectedtothep-sideanode,orarrowinthediagramandthe"black" ornegativeleadisconnectedtothen-sidecathode,orbarinthediagramofadiode,themeterattempts topassusually1mAofcurrentthroughthediode.Ifthe1mAofcurrentisallowedtoow,themeter thenindicatestheamountofforwardvoltagedevelopedacrossthediode.Ifitreadssomethinglike0.673 volts,thenyoucanbeprettysurethediodeisOK.Reversetheleads,andthediodeisreversebiased,and themetershouldread"OL"overloadorsomethinglikethattoindicatethatnocurrentisowing. Thediodeequation.51isusuallyapproximatedbytwosomewhatsimplerequations,dependingupon whetherthediodeisforwardorreversebiased: I 8 < : 0 if V a < 0 I sat e qV a kT if V a > 0 .53 Forreversebias,aswesaid,thecurrentisessentiallynil.Intheforwardbiascase,theexponentialterm quicklygetsmuchlargerthanunity,andsowecanforgetthe"-1"terminthediodeequation.51. Remember,wesaidthat kT atroomtemperaturehadavalueofabout1/40ofaneV,so q kT 40 V )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ,this meanswecanalsosayforforwardbiasthat I = I sat e 40 V a .54 Fromthisequationitiseasytoseethatonlyasmallpositivevaluefor V a isneededinordertomakethe exponentialmuchgreaterthanunity. Nowlet'sconnectthis"idealdiodeequation"totherealworld.Onethingyoumightaskyourselfis"How couldIchecktoseeifanactualdiodefollowstheequationgivenhere.49?"Aswesaid, I sat isavery smallcurrent,andsotryingtodothereversetestisprobablynotgoingtobesuccessful.Whatisusually doneistomeasurethediodecurrentandforwardvoltageoverseveralordersofmagnitudeofcurrent. Note: Whilethecurrentcanvarybymanyordersofmagnitude,thevoltageismoreorless limitedtovaluesbetween0and0.6to0.7volts,notbyanyfundamentalprocess,butrathersimply bythefactthattoomuchforwardcurrentwillburnupthediode. Ifwetakethenaturallogofbothsidesofthesecondpieceof.53,wend: lnI = lnI sat + qV a kT .55 Thus,aplotof lnI asafunctionof V a shouldyieldastraightlinewithaslopeof q kT ,or40.

PAGE 43

37 Well,Iwentintothelab,grabbedarealdiodeandmadesomemeasurements.Figure1.44isaplotofthe naturallogofthecurrentasafunctionofvoltagefrom0.05to0.70volts.Includedwiththisplot,isalinear curvettothedatawhichisplottedasadottedline.Thelineartgoesthroughthedatapointsquitenicely, sothecurrentissurelyanexponentialfunctionoftheappliedvoltage!Fromtheexpressionforthebestt, whichisprintedabovethegraph,weseethat lnI sat = )]TJ/F8 9.9626 Tf 7.749 0 Td [(19 : 68 .Thatmeansthat I sat = e )]TJ/F7 6.9738 Tf 6.226 0 Td [(19 : 68 =2 : 89 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(9 amps,whichisindeedaverysmallcurrent.Lookattheslopehowever.Itssupposedtobe40,andyetitturns outtobeslightlymorethan20!Thiscomesaboutbecauseofsomecomplexdetailsofexactlywhathappensto theelectronsandholeswhentheycrossthejunction.Inwhatiscalledthe diusiondominatedsituation electronsandholesareinjectedacrossthejunction,afterwhichtheydiuseawayfromthejunction,andalso recombine,untileventuallytheyareallgone.ThisisshownschematicallyinFigure1.45.Theotherregime iscalled recombinationdominated andhere,themajorityofthecurrentismadeupoftheelectronsand holesrecombiningdirectlywitheachotheratthejunction.ThisisshowninFigure1.46.Forrecombination dominateddiodebehavior,itturnsoutthatthecurrentisgivenby I = I sat e qV a 2 kT .56 Figure1.44: Plotshowing lnI asafunctionof V a fora1N4123silicondiode Figure1.45: Diusiondominateddiodebehavior

PAGE 44

38 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Figure1.46: Recombinationdominateddiodebehavior Ingeneral,aparticulardiodemighthaveacombinationofthesetwoeectsgoingon,andsopeopleoften useamoregeneralformforthediodeequation: I = I sat e qV a nkT .57 where n iscalledthe idealityfactor andisanumbersomewherebetween1and2.Forthediodewhichgave thedataforourexample n =1 : 92 andsomostofthecurrentisdominatedbyrecombinationofelectrons andholesinthedepletionregion. 1.10ReverseBiased/Breakdown 10 Beforeweleavediodes,itwouldbeworthwhileexploringsomeothermodesofoperation,aswellassomespecicapplicationswhichwillbeofinterest.Wesaidthatwhenthediodewasreverse-biasedp-regionnegative withrespecttothen-regionthattheonlycurrentwhichowsisthereversesaturationcurrent,resulting fromthefewthermallygeneratedminoritycarrierswhichcanfalldownorupthebarrierFigure1.47. Figure1.47: Reversesaturationcurrent Ifwemakethereversebiasevengreater,thesamecurrentows,butthecarrierspickupmoreenergy astheyfalldownthenowlargerjunctionpotential.Astheydothis,itispossibleforthemtopickupso 10 Thiscontentisavailableonlineat.

PAGE 45

39 muchenergy,thatwhentheycollidewithalatticesite,theycreateanadditionalelectron-holepairthrough aprocesscalled impactionization Figure1.48.Whenthisoccurs,wenowhavecurrentconsistingof twoelectronsandonehole.Theseadditionalcarrierscanthemselvescollideandgenerateadditionalelectron holepairsaswell.Thecurrentnowconsistsofveelectronsandtwoholes.Thisprocessiscalled avalanche multiplication Figure1.49,becausewestartwithonecarrier,andthroughasuccessionofimpactscreate moreandmorecurrent.Thisprocesscaninfactrunaway,muchlikeanavalancheonasnowymountain side,inaprocesscalled avalanchebreakdown Figure1.48: ImpactIonization Theneteectistochangethereversecharacteristicsofthediodesomewhat.Ifweincludetheeectof breakdownintheI-Vcurveforthediode,wewouldseesomethinglikethatinFigure1.50.

PAGE 46

40 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Figure1.49: Avalanchemultiplication Figure1.50: DiodeI-VCurveshowingboththeforwardcharacteristicsandreversebreakdown Thereisnowasuddenonsetofcurrentaftertheavalanchebreakdownvoltagehasbeenexceeded.Do notbeconfusedintothinkingthatthis"breakdown"meansthatthediodehasbeendamaged.Theprocess ofavalanchingitselfisnotdestructive.ButasyoucanseefromFigure1.48,thediodecurrentincreasesvery rapidlyoncethebreakdownthresholdhasbeenexceeded.Thus,ifthereisnotsomethinginserieswiththe diodetolimitthemaximumcurrentthroughit,itcouldbedamagedbyoverheating.Diodesinbreakdown areusedasvoltagereferencesthevoltageacrossthemismoreorlessindependentofthecurrentrunning throughthembutyouwillalwaysndaseriescurrentlimitingresistorusedalongwiththem.Suchdiodes arecalled ZenerDiodes namedafterthegrandfatherofWillRice'sGeorgeZenerwhograduatedafew yearsago...thatisGeorgedid,nothisgrandfatherbutthenameiskindofamisnomer.The ZenerEect isalsoareversebreakdownphenomena,butcomesfromdirecteldgenerationofextracarriers,ratherthan

PAGE 47

41 asaresultofimpactionization.Intruth,youcannottelltheoneeectfromtheotherbylookingatthe diodeI-Vcurve,andsoalldiodesusedinreversebreakdownarecalledZenerDiodes.AcircuitusingaZener diodeasavoltagereferenceisshowninFigure1.51. Figure1.51: Voltageregulatorcircuit 1.11Diusion 11 1.11.1Introduction Letusturnourattentiontowhathappenstotheelectronsandholes,oncetheyhavebeeninjectedacrossa forward-biasedjunction.Wewillconcentratejustontheelectronswhichareinjectedintothep-sideofthe junction,butkeepinmindthatsimilarthingsarealsohappeningtotheholeswhichenterthen-side. Aswesawawhileback,whenelectronsareinjectedacrossajunction,theymoveawayfromthejunction regionbyadiusionprocess,whileatthesametime,someofthemaredisappearingbecausetheyare minoritycarrierselectronsinbasicallyp-typematerialandsotherearelotsofholesaroundforthemto recombinewith.ThisisallshownschematicallyinFigure1.52DiusionacrossaP-NJunction. DiusionacrossaP-NJunction Figure1.52: Processesinvolvedinelectrontransportacrossap-njunction 1.11.2DiusionProcessQuantied Itisactuallyfairlyeasytoquantifythis,andcomeupwithanexpressionfortheelectrondistributionwithin thep-region.Firstwehavetolookalittlebitatthediusionprocesshowever.Imaginethatwehavea 11 Thiscontentisavailableonlineat.

PAGE 48

42 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES seriesofbins,eachwithadierentnumberofelectronsinthem.Inagiventime,wecouldimaginethatall oftheelectronswouldowoutoftheirbinsintotheneighboringones.Sincethereisnoreasontoexpect theelectronstofavoronesideovertheother,wewillassumethatexactlyhalfleavebyeachside.Thisis allshowninFigure1.53Firstexampleofadiusionproblem.Wewillkeepthingssimpleandonlylookat threebins.ImagineIhave4,6,and8electronsrespectivelyineachofthebins.Aftertherequired"emptying time,"wewillhaveanetuxofexactlyoneelectronacrosseachboundaryasshown. Firstexampleofadiusionproblem Figure1.53 Diusionfrombins Figure1.54 Nowlet'sraisethenumberofelectronsto8,12and16respectivelytheelectronsmayoverlapsomenow inthepicture.Wendthatthenetuxacrosseachboundaryisnow2electronsperemptyingtime,rather thanone.Notethatthegradientslopeoftheconcentrationintheboxeshasalsodoubledfromoneper boxtotwoperbox.Thisleadsustoaratherobviousstatementthattheuxofcarriersisproportionalto thegradientoftheirdensity.Thisisstatedformallyinwhatisknownas Fick'sFirstLawofDiusion : Flux = )]TJ/F11 9.9626 Tf 7.749 0 Td [(D e d dx n x .58 Where D e issimplyaproportionalityconstantcalledthe diusioncoecient .Sincewearetalkingabout themotionofelectrons,thisdiusionuxmustgiverisetoacurrentdensity J e diff .Sinceanelectronhasa charge )]TJ/F11 9.9626 Tf 7.748 0 Td [(q associatedwithit, J e diff = qD e d dx n .59 Nowwehavetoinvokesomethingcalledthe continuityequation .Imaginewehaveavolume V which islledwithsomecharge Q .Itisfairlyobviousthatifweaddupallofthecurrentdensitywhichisowing

PAGE 49

43 outofthevolumethatitmustbeequaltothetimerateofdecreaseofthechargewithinthatvolume.This ideasisexpressedintheformulabelowwhichusesa closed-surfaceintegral ,alongwiththealltheother integralstofollow: I S JdS = )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( d dt Q .60 Wecanwrite Q as Q = I V v dV .61 wherewearedoingavolumeintegralofthechargedensity overthevolume V .NowwecanuseGauss' theoremwhichsayswecanreplaceasurfaceintegralofaquantitywithavolumeintegralofitsdivergence: I S JdS = Z V divJdV .62 So,combining.60,.61and.62,wehavenotewearestilldealingwithsurfaceandvolumeintegrals: Z V divJdV = )]TJ/F1 9.9626 Tf 9.409 20.025 Td [(0 @ Z V d dt dV 1 A .63 Finally,weletthevolume V shrinkdowntoapoint,whichmeansthequantitiesinsidetheintegralmust beequal,andwehavethedierentialformofthecontinuityequationinonedimension divJ = @ @x J = )]TJ/F1 9.9626 Tf 9.409 8.07 Td [()]TJ/F10 6.9738 Tf 7.266 -4.147 Td [(d dt x .64 1.11.3WhatabouttheElectrons? Nowlet'sgobacktotheelectronsinthediode.Theelectronswhichhavebeeninjectedacrossthejunction arecalled excessminoritycarriers ,becausetheyareelectronsinap-regionhenceminoritybuttheir concentrationisgreaterthanwhattheywouldbeiftheywereinasampleofp-typematerialatequilibrium. Wewilldesignatethemas n 0 ,andsincetheycouldchangewithbothtimeandpositionweshallwritethem as n 0 x;t .Nowtherearetwowaysinwhich n 0 x;t canchangewithtime.Onewouldbeifwewereto stopinjectingelectronsinfromthen-sideofthejunction.Areasonablewaytoaccountforthedecaywhich wouldoccurifwewerenotsupplyingelectronswouldbetowrite: d dt n 0 x;t = )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( n 0 x;t r .65 Where r calledthe minoritycarrierrecombinationlifetime .Itisprettyeasytoshowthatifwestart outwithanexcessminoritycarrierconcentration n o 0 at t =0 ,then n 0 x;t willgoesas n 0 x;t = n 0 0 e )]TJ/F9 4.9813 Tf 5.396 0 Td [(t r .66 But,theelectronconcentrationcanalsochangebecauseofelectronsowingintooroutoftheregion x .The electronconcentration n 0 x;t isjust x;t q .Thus,duetoelectronowwehave: d dt n 0 x;t = 1 q d dt x;t = 1 q div J x;t .67

PAGE 50

44 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES But,wecangetanexpressionfor J x;t from.59.Reducingthedivergencein.67toonedimension wejusthavea @ @x J wenallyendupwith d dt n 0 x;t = D e d 2 dx 2 n 0 x;t .68 Combining.68and.65electronswill,afterall,suerfrombothrecombinationanddiusionandwe endupwith: d dt n 0 x;t = D e d 2 dx 2 n 0 x;t )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(n 0 x;t r .69 Thisisasomewhatspecializedformofanequationcalledthe ambipolardiusionequation .Itseems kindofcomplicatedbutwecangetsomeniceresultsfromitifwemakesomesimplyboundarycondition assumptions.Let'sseewhatwecandowiththis. 1.11.3.1UsingtheAmbipolarDiusionEquation Foranythingwewillbeinterestedin,wewillonlylookat steadystatesolutions .Thismeansthatthe timederivativeontheLHSof.69iszero,andsowehaveletting n 0 x;t becomesimply n 0 x sincewe nolongerhaveanytimevariationtoworryabout d 2 dt 2 n 0 x )]TJ/F8 9.9626 Tf 19.314 6.74 Td [(1 D e r n 0 x =0 .70 Let'spickthenotunreasonableboundaryconditionsthat n 0 = n 0 theconcentrationofexcesselectrons justatthestartofthediusionregionand n 0 x 0 as x !1 theexcesscarriersgotozerowhenwe getfarfromthejunctionthen n x = n 0 e )]TJ/F27 6.9738 Tf 6.227 7.682 Td [( x p D e r .71 Theexpressionintheradical p D e r iscalledthe electrondiusionlength L e ,andgivesussomeideaas tohowfarawayfromthejunctiontheexcesselectronswillexistbeforetheyhavemoreorlessallrecombined. Thiswillbeimportantforuswhenwemoveontobipolartransistors. Justsoyoucangetafeelforsomenumbers,atypicalvalueforthediusioncoecientforelectronsin siliconwouldbe D e =25 cm 2 sec andtheminoritycarrierlifetimeisusuallyaroundamicrosecond.Thus L e = p D e r = p 25 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 =5 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 cm .72 whichisnotveryfaratall! 1.12LightEmittingDiode 12 Let'stalkabouttherecombiningelectronsforaminute.Whentheelectronfallsdownfromtheconduction bandandllsinaholeinthevalenceband,thereisanobviouslossofenergy.Thequestionis;where doesthatenergygo?Insilicon,theanswerisnotveryinteresting.Siliconiswhatisknownasan indirect band-gapmaterial .Whatthismeansisthatasanelectrongoesfromthebottomoftheconductionband tothetopofthevalenceband,itmustalsoundergoasignicantchangeinmomentum.Thisallcomesabout fromthedetailsofthebandstructureforthematerial,whichwewillnotconcernourselveswithhere.Aswe allknow,wheneversomethingchangesstate,wemuststillconservenotonlyenergy,butalsomomentum.In thecaseofanelectrongoingfromtheconductionbandtothevalencebandinsilicon,bothofthesethings 12 Thiscontentisavailableonlineat.

PAGE 51

45 canonlybeconservedifthetransitionalsocreatesaquantizedsetoflatticevibrations,called phonons or"heat".Phononsposses both energyandmomentum,andtheircreationupontherecombinationofan electronandholeallowsforcompleteconservationofbothenergyandmomentum.Alloftheenergywhich theelectrongivesupingoingfromtheconductionbandtothevalenceband.1eVendsupinphonons, whichisanotherwayofsayingthattheelectronheatsupthecrystal. Insomeothersemiconductors,somethingelseoccurs.Inaclassofmaterialscalled directband-gap semiconductors ,thetransitionfromconductionbandtovalencebandinvolvesessentiallynochangein momentum.Photons,itturnsout,possessafairamountofenergyseveraleV/photoninsomecasesbut theyhaveverylittlemomentumassociatedwiththem.Thus,foradirectbandgapmaterial,theexcess energyoftheelectron-holerecombinationcaneitherbetakenawayasheat,ormorelikely,asaphotonof light.This radiativetransition thenconservesenergyandmomentumbygivingolightwheneveran electronandholerecombine.Thisgivesrisetoforusanewtypeofdevice,thelightemittingdiodeLED. EmissionofaphotoninanLEDisshownschematicallyinFigure1.55Radiativerecombinationinadirect band-gapsemiconductor. Radiativerecombinationinadirectband-gapsemiconductor Figure1.55 ItwasPlanckwhopostulatedthattheenergyofaphotonwasrelatedtoitsfrequencybyaconstant, whichwaslaternamedafterhim.IfthefrequencyofoscillationisgivenbytheGreekletter"nu" ,then theenergyofthephotonisjust h ,where h isPlanck'sconstant,whichhasavalueof 4 : 14 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(15 eVseconds E = h .73 Whenwetalkaboutlightitisconventionaltospecifyitswavelength, ,insteadofitsfrequency.Visible lighthasawavelengthontheorderofnanometersRedisabout600nm,greenabout500nmandblueisin the450nmregion.Ahandy"ruleofthumb"canbederivedfromthefactthat = c v ,where c isthespeed oflight.Since c =3 10 8 m sec or c =3 10 17 nm sec nm = hc E eV = 1242 E eV .74 Thus,asemiconductorwitha2eVband-gapshouldgiveolightatabout620nminthered.A3eV band-gapmaterialwouldemitat414nm,intheviolet.Thehumaneye,ofcourse,isnotequallyresponsive toallcolors.WeshowthisinFigure1.56Relativeresponseofthehumaneyetovariouscolors,wherewe havealsoincludedthematerialswhichareusedforimportantlightemittingdiodesLEDsforeachofthe dierentspectralregions.

PAGE 52

46 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Relativeresponseofthehumaneyetovariouscolors Figure1.56 Asyounodoubtnotice,anumberoftheimportantLEDsarebasedontheGaAsPsystem.GaAsisa directband-gapsemiconductorwithabandgapof1.42eVintheinfrared.GaPisanindirectband-gap materialwithabandgapof2.26eVnm,orgreen.BothAsandParegroupVelements.Hencethe nomenclatureofthematerialsas III-Vcompoundsemiconductors .WecanreplacesomeoftheAswith PinGaAsandmakeamixedcompoundsemiconductor GaAs 1-x P x .Whenthemolefractionofphosphorous islessthanabout0.45thebandgapisdirect,andsowecan"engineer"thedesiredcolorofLEDthatwe wantbysimplygrowingacrystalwiththeproperphosphorusconcentration!ThepropertiesoftheGaAsP systemareshowninFigure1.57BandgapfortheGaAsPsystem.Itturnsoutthatforthissystem,there areactually two dierentbandgaps,asshownintheinsetFigure1.57:BandgapfortheGaAsPsystem. Oneisadirectgapnochangeinmomentumandtheotherisindirect.InGaAs,thedirectgaphaslower energythantheindirectonelikeintheinsetandsothetransitionisaradiativeone.Aswestartadding phosphoroustothesystem,boththedirectandindirectbandgapsincreaseinenergy.However,thedirect gapenergyincreasesfasterwithphosphorousfractionthandoestheindirectone.Atamolefraction x of about0.45,thegapenergiescrossoverandthematerialgoesfrombeingadirectgapsemiconductorto anindirectgapsemiconductor.At x =0 : 35 thebandgapisabout1.97eVnm,andsowewould onlyexpecttogetlightuptotheredusingtheGaAsPsystemformakingLED's.Fortunately,people discoveredthatyoucouldaddanimpuritynitrogentotheGaAsPsystem,whichintroducedanewlevel inthesystem.Anelectroncouldgofromtheindirectconductionbandforamixturewithamolefraction greaterthan0.45tothenitrogensite,changingitsmomentum,butnotitsenergy.Itcouldthenmakea directtransitiontothevalenceband,andlightwithcolorsallthewaytothegreenbecamepossible.Theuse ofanitrogen recombinationcenter isdepictedintheFigure1.58Additionofanitrogenrecombination centertoindirectGaAsP.

PAGE 53

47 BandgapfortheGaAsPsystem Figure1.57 AdditionofanitrogenrecombinationcentertoindirectGaAsP Figure1.58

PAGE 54

48 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Ifwewantcolorswithwavelengthsshorterthanthegreen,wemustabandontheGaAsPsystemandlook formoresuitablematerials.AcompoundsemiconductormadefromtheII-VIelementsZnandSemakeup onepromisingsystem,andseveralresearchgroupshavesuccessfullymadeblueandblue-greenLEDsfrom ZnSe.SiCisanotherweakblueemitterwhichiscommerciallyavailableonthemarket.Recently,workers atatiny,unknownchemicalcompanystunnedthe"displayworld"byannouncingthattheyhadsuccessfully fabricatedablueLEDusingtheII-VmaterialGaN.AgoodblueLEDhasbeenthe"holygrail"ofthe displayandCDROMresearchcommunityforanumberofyearsnow.Obviously,addingbluetothealready workinggreenandredLED'scompletesthesetof3primarycolorsnecessaryforafull-coloratpaneldisplay HangaTVscreenonyourwalllikeapicture?.UsingablueLEDorlaserinaCDROMwouldmorethan quadrupleitsdatacapacity,asbitdiameterscalesas ,andhencetheareaas 2 1.13LASER 13 Speakingoflasers,whatisthedierencebetweenanLEDandasolidstatelaser?Therearesomedierences, butbothdevicesoperateonthesameprincipleofhavingexcesselectronsintheconductionbandofa semiconductor,andarrangingitsothattheelectronsrecombinewithholesinaradiativefashion,givingo lightintheprocess.Whatisdierentaboutalaser?InanLED,theelectronsrecombineinarandomand unorganizedmanner.Theygiveolightbywhatisknownas spontaneousemission ,whichsimplymeans thattheexacttimeandplacewhereaphotoncomesoutofthedeviceisuptoeachindividualelectron,and thingshappeninarandomway. Thereisanotherwayinwhichanexcitedelectroncanemitaphotonhowever.Ifaeldoflightorasetof photonshappenstobepassingbyanelectroninahighenergystate,thatlighteldcaninducetheelectron toemitanadditionalphotonthroughaprocesscalled stimulatedemission .Thephotoneld stimulates theelectrontoemititsenergyasanadditionalphoton,whichcomesout inphasewiththestimulatingeld Thisisthebigdierencebetween incoherentlight whatcomesfromanLEDoraashlightand coherent light whichcomesfromalaser.Withcoherentlight,alloftheelectriceldsassociatedwitheachphonon areallexactlyinphase.Thiscoherenceiswhatenablesustokeepalaserbeamintightfocus,andtoallow ittotravelalargedistancewithoutmuchdivergenceorspreadingout. SohowdowerestructureanLEDsothatthelightisgeneratedbystimulatedemissionratherthan spontaneousemission?Firstly,webuildwhatiscalleda heterostructure .Allthismeansisthatwebuild upasandwichofsomewhatdierentmaterials,withdierentcharacteristics.Inthiscase,weputtwowide band-gapregionsaroundaregionwithanarrowerbandgap.Themostimportantsystemwherethisis doneistheAlGaAs/GaAssystem.AbanddiagramforsuchasetupisshowninFigure1.59Double HeterostructureGaAs/AlGaAslaser.AlGaAspronounced"Al-Gas"hasalargerband-gapthendoes GaAs.Thepotential"well"formedbytheGaAsmeansthattheelectronsandholeswillbeconnedthere, andalloftherecombinationwilloccurinaverynarrowstrip.Thisgreatlyincreasesthechancesthatthe carrierscaninteract,butwestillneedsomewayforthephotonstobehaveinthepropermanner.Figure1.60 LaserDiodeisapictureofwhatarealdiodemightlooklike.WehavetheactiveGaAslayersandwich in-betweenthetwoheterostructureconnementlayers,withacontactontopandbottom.Oneitherend ofthedevice,thecrystalhasbeen"cleaved"orbrokenalongacrystallatticeplane.Thisresultsinashiny "mirror-like"surface,whichwillreectphotons.Thebacksurfacewhichwecannotseehereisalsocleaved tomakeamirrorsurface.Theothersurfacesarepurposelyroughenedsothattheydonotreectlight.Now letuslookatthedevicefromtheside,anddrawjustthebanddiagramfortheGaAsregionFigure1.61 Buildupofaphotoneldinalaserdiode.Westartthingsowithanelectronandholerecombining spontaneously.Thisemitsaphotonwhichheadstowardsoneofthemirrors.Asthephotongoesbyother electrons,however,itmaycauseoneofthemtodecaybystimulatedemission.Thetwoinphasephotons hitthemirrorandarereectedandstartbacktheotherway.Astheypassadditionalelectrons,they stimulatethemintoatransitionaswell,andtheopticaleldwithinthelaserstartstobuildup.Afterabit, thephotonsgetdowntotheotherendofthecavity.Thecleavedfacet,whileitactslikeamirror,isnota 13 Thiscontentisavailableonlineat.

PAGE 55

49 perfectone.Somelightisnotreected,butrather"leaks";though,andsobecomestheoutputbeamfrom thelaser.Thedetailsofndingwhattheratioofreectedtotransmittedlightiswillhavetowaituntillater inthecoursewhenwetalkaboutdielectricinterfaces.Therestofthephotonsarereectedbackintothe cavityandcontinuetostimulateemissionfromtheelectronswhichcontinuetoenterthegainregionbecause oftheforwardbiasonthediode. DoubleHeterostructureGaAs/AlGaAslaser Figure1.59 LaserDiode Figure1.60

PAGE 56

50 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Buildupofaphotoneldinalaserdiode Figure1.61 OutputCoupling Figure1.62 Inreality,thephotonsdonotmovebackandforthinabig"clump"aswehavedescribedhere,rather theyaredistributeduniformlyalongthegainregion.Theeldwithinthecavitywillbuilduptothepoint wherethelossofenergybylightleakingoutofthemirrorsjustequalstherateatwhichenergyisreplaced bytherecombiningelectrons.

PAGE 57

51 1.14SolarCells 14 Nowletuslookattheoppositeprocessoflightgenerationforamoment.Considerthefollowingsituation. P-Ndiodeunderilllumination Figure1.63 Wehavejustaplainoldnormalp-njunction,onlynow,insteadofapplyinganexternalvoltage,we imaginethatthejunctionisbeingilluminatedwithlightwhosephotonenergyisgreaterthantheband-gap. Inthissituation,insteadofrecombination,wewillgetphoto-generationofelectronholepairs.Thephotons simplyexciteelectronsfromthefullstatesinthevalenceband,and"kick"themupintotheconduction band,leavingaholebehind 15 .AsyoucanseefromFigure1.63P-Ndiodeunderilllumination,thiscreates excesselectronsintheconductionbandinthep-sideofthediode,andexcessholesinthevalencebandof then-side.Thesecarrierscandiuseovertothejunction,wheretheywillbesweptacrossbythebuilt-in electriceldinthedepletionregion.Ifweweretoconnectthetwosidesofthediodetogetherwithawire,a currentwouldowthroughthatwireasaresultoftheelectronsandholeswhichmoveacrossthejunction. Whichwaywouldthecurrentow?AquicklookatFigure1.63P-Ndiodeunderillluminationshows thatholespositivechargecarriersaregeneratedonthen-sideandtheyoatuptothep-sideastheygo acrossthejunction.Hencepositivecurrentmustbecomingoutoftheanode,orp-sideofthejunction. Likewise,electronsgeneratedonthep-sidefalldownthejunctionpotential,andcomeoutthen-side,but sincetheyhavenegativecharge,thisowrepresentscurrentgoing into thecathode.Wehaveconstructed a photovoltaicdiode ,or solarcell !Figure1.64Schematicrepresentationofaphotovoltaiccellisa pictureofwhatthiswouldlooklikeschematically.Wemightliketoconsiderthepossibilityofusingthis deviceasasourceofenergy,butthewaywehavethingssetupnow,sincethevoltageacrossthediodeis zero,andsincepowerequalscurrenttimesvoltage,weseethatwearegettingnadafromthecell.Whatwe need,obviously,isaloadresistor,solet'sputonein.ItshouldbeclearfromFigure1.65Photovoltaiccell withaloadresistorthatthephotocurrentowingthroughtheloadresistorwilldevelopavoltagewhich itbiasesthediodeinthe forward direction,which,ofcoursewillcausecurrenttoowbackintotheanode. Thiscomplicatesthings,itseemswehavecurrentcoming out ofthediodeandcurrentgoing into thediode allatthesametime!Howarewegoingtogureoutwhatisgoingon? 14 Thiscontentisavailableonlineat. 15 Thisissimiliartothethermalexcitationprocesswetalkedaboutearlier.

PAGE 58

52 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES Schematicrepresentationofaphotovoltaiccell Figure1.64 Photovoltaiccellwithaloadresistor Figure1.65 Theansweristomakeamodel.Thecurrentwhicharisesduetothephotonuxcanbeconveniently representedasacurrentsource.Wecanleavethediodeasadiode,andwehavethecircuitshownin Figure1.66ModelofPVcell.Eventhoughweshow I out comingoutofthedevice,weknowbytheusual polarityconventionthatwhenwedene V out asbeingpositiveatthetop,thenweshouldshowthecurrent forthephotovoltaic, I pv ascurrentgoingintothetop,whichiswhatwasdoneinFigure1.66Modelof PVcell.Notethat I pv = I diode )]TJ/F11 9.9626 Tf 10.289 0 Td [(I photo ,soallweneedtodoistosubtractthetwocurrents;wedothis graphicallyinFigure1.67Combiningthediodeandthecurrentsource.Notethatwehavenumberedthe fourquadrantsintheI-VplotofthetotalPVcurrent.InquadrantIandIII,theproductof I and V isa positivenumber,meaningthatpowerisbeing dissipated inthecell.ForquadrantIIandIV,theproductof I and V isnegative,andsowearegettingpower from thedevice.Clearlywewanttooperateinquadrant IV.Infact,withouttheadditionofanexternalbatteryorcurrentsource,thecircuit,will only runinthe IV'thquadrant.Consideradjusting R L ,theloadresistorfrom0ashortto 1 anopen.With R L =0 ,we wouldbeatpointAonFigure1.67Combiningthediodeandthecurrentsource.As R L startstoincrease fromzero,thevoltageacrossboththediodeandtheresistorwillstarttoincreasealso,andwewillmoveto

PAGE 59

53 pointB,say.As R L getsbiggerandbigger,wekeepmovingalongthecurveuntil,atpointC,where R L is anopenandwehavethemaximumvoltageacrossthedevice,but,ofcourse,nocurrentcomingout! ModelofPVcell Figure1.66 Combiningthediodeandthecurrentsource Figure1.67 Poweris VI soatBforinstance,thepowercomingoutwouldberepresentedbytheareaenclosedbythe twodottedlinesandthecoordinateaxes.SomeplaceaboutwhereIhavepointBwouldbewherewewould begettingthemostpoweroutofoutsolarcell. Figure1.68Arealsolarcellshowsyouwhatarealsolarcellwouldlooklike.Theyareusuallymade fromacompletewaferofsilicon,tomaximizetheusablearea.Ashallow.25 mjunctionismadeonthe

PAGE 60

54 CHAPTER1.CONDUCTORS,SEMICONDUCTORSANDDIODES top,andtopcontactsareappliedasstripesofmetalconductorasshown.Ananti-reectionARcoatingis appliedontopofthat,whichaccountsforthebluishcolorwhichatypicalsolarcellhas. Arealsolarcell Figure1.68 Thesolarpoweruxontheearth'ssurfaceisconvenientlyabout 1 kW m 2 or 100 mW cm 2 .Soifwemadeasolar cellfroma4inchdiameterwafertypicalitwouldhaveanareaofabout 81 cm 2 andsowouldbereceiving auxofabout8.1Watts.Typicalcellecienciesrunfromabout10%tomaybe15%unlessspecialand costlytricksaremade.Thismeansthatwewillgetabout1.2Wattsoutfromasinglewafer.LookingatB on2.59wecouldguessthat V out willbeabout0.5to0.6volts,thuswecouldexpecttogetmaybearound 2.5ampsfroma4inchwaferat0.5voltswith15%eciencyundertheilluminationofonesun.

PAGE 61

Chapter2 BipolarTransistors 2.1IntrotoBipolarTransistors 1 Let'sleavetheworldoftwoterminaldeviceswhichareallcalleddiodesbytheway;diodejustmeans two-terminalsandventureintothemuchmoreinterestingworldofthreeterminals.Therstdevicewewill lookatiscalledthe bipolartransistor .ConsiderthestructureshowninFigure2.1BipolarTransistor Structure: BipolarTransistorStructure Figure2.1: StructureofanNPNbipolartransistor Thedeviceconsistsofthreelayersofsilicon,aheavilydopedn-typelayercalledtheemitter,amoderately dopedp-typelayercalledthebase,andthird,morelightlydopedlayercalledthecollector.Inabiasing appliedDCpotentialcongurationcalled forwardactivebiasing ,theemitter-basejunctionisforward biased,andthebase-collectorjunctionisreversebiased.Figure2.2forward_active_biasingshowsthe biasingconventionswewilluse.Bothbiasvoltagesarereferencedtothebaseterminal.Sincethebaseemitterjunctionisforwardbiased,andsincethebaseismadeofp-typematerial, V EB mustbenegative.On theotherhand,inordertoreversebiasthebase-collectorjunction V CB willbeapositivevoltage. 1 Thiscontentisavailableonlineat. 55

PAGE 62

56 CHAPTER2.BIPOLARTRANSISTORS forward_active_biasing Figure2.2: Forwardactivebiasingofannpnbipolartransistor Figure2.3: Banddiagramandcarrieruxesinabipolartransistor Now,let'sdrawtheband-diagramforthisdevice.Atrstthismightseemhardtodo,butweknow whatforwardandreversebiasedbanddiagramslooklike,sowe'lljuststickoneofeachtogether.Weshow thisinFigure2.3.Figure2.3isaverybusygure,butitisalsoveryimportant,becauseitshowsallof theimportantfeaturesintheoperationthetransistor.Sincethebase-emitterjunctionisforwardbiased, electronswillgofromthen-typeemitterintothebase.Likewise,someholesfromthebasewillbeinjected intotheemitter. InFigure2.3,wehavetwodierentkindsofarrows.Theopenarrowswhichareattachedtothecarriers, showuswhichwaythecarrierismoving.Thesolidarrowswhicharelabeledwithsomekindofsubscripted I ,representcurrentow.Weneedtodothisbecauseforholes,motionandcurrentowareinthesame direction,whileforelectrons,carriermotionandcurrentowareinoppositedirections. Justaswesawinthelastchapter,theelectronswhichareinjectedintothebasediuseawayfromthe emitter-basejunctiontowardsthereversebiasedbase-collectorjunction.Astheymovethroughthebase, someoftheelectronsencounterholesandrecombinewiththem.Thoseelectronswhich do gettothebase-

PAGE 63

57 collectorjunctionrunintoalargeelectriceldwhichsweepsthemoutofthebaseandintothecollector. They"fall"downthelargepotentialdropatthejunction. TheseeectsareallseeninFigure2.3,witharrowsrepresentingthevariouscurrentswhichareassociated witheachofthecarriersuxes. I Ee representsthecurrentassociatedwiththeelectroninjectionintothebase. Itpointsintheoppositedirectionfromthemotionoftheelectrons,sinceelectronshaveanegativecharge. I Eh representsthecurrentassociatedwithholesinjectionintotheemitterfromthebase. I Br represents recombinationcurrentinthebase,while I Ce representstheelectroncurrentgoingintothecollector.It shouldbeeasyforyoutoseethat: I E = I Ee + I Eh .1 I B = I Eh + I Br .2 I C = I Ce .3 Ina"good"transistor,almostallofthecurrentacrossthebase-emitterjunctionconsistsofelectrons beinginjectedintothebase.Thetransistorengineerworkshardtodesignthedevicesothatverylittle emittercurrentismadeupofholescomingfromthebaseintotheemitter.Thetransistorisalsodesigned sothatalmostallofthoseelectronswhichareinjectedintothebasemakeitacrosstothebase-collector reverse-biasedjunction.Somerecombinationisunavoidable,butthingsarearrangedsoastominimizethis eect. 2.2TransistorEquations 2 Thereareseveral"guresofmerit"fortheoperationofthetransistor.Therstoftheseiscalledthe emitter injectioneciency .Theemitterinjectioneciencyisjusttheratiooftheelectroncurrentowingin theemittertothetotalcurrentacrosstheemitterbasejunction: = I e I Ee + I Eh .4 Ifyougobackandlookatthediodeequation.51youwillnotethattheelectronforwardcurrent acrossajunctionisproportionalto N d thedopingonthen-sideofthejunction.Clearlytheholecurrent willbeproportionalto N a ,theacceptordopingonthep-sideofthejunction.Thus,atleasttorstorder = N d E N d E + N a B .5 Therearesomeotherconsiderationswhichweareignoringinobtainingthisexpression,buttorstorder, andformost"real"transistors,.5isaverygoodapproximation. Thesecond"gureofmerit"isthebasetransportfactor, T .Thebasetransportfactortellsuswhat fractionoftheelectroncurrentwhichisinjectedintothebaseactuallymakesittocollectorjunction.This turnsouttobegiven,toaverygoodapproximation,bytheexpression T =1 )]TJ/F8 9.9626 Tf 11.158 6.739 Td [(1 2 W B L e 2 .6 Where W B isthephysicalwidthofthebaseregion,and L e istheelectrondiusionlength,denedin theelectrondiusionlengthequation.72. L e = p D e r .7 2 Thiscontentisavailableonlineat.

PAGE 64

58 CHAPTER2.BIPOLARTRANSISTORS Clearly,ifthebaseisverynarrowcomparedtothediusionlength,andsincetheelectronconcentration isfallingolike e )]TJ/F9 4.9813 Tf 5.397 0 Td [(x L e theshorterthebaseiscomparedto L e thegreaterthefractionofelectronswhowill actuallymakeitacross.Wesawbeforethatatypicalvaluefor L e mightbeontheorderof0.005cmor50 m.Inatypicalbipolartransistor,thebasewidth, W B isusuallyonlyafew mandso canbequiteclose tounityaswell. LookingbackatthisgureFigure2.3,itshouldbeclearthat,solongasthecollector-basejunction remainsreverse-biased,thecollectorcurrent I Ce ,willonlydependonhowmuchofthetotalemittercurrent actuallygetscollectedbythereverse-biasedbase-collectorjunction.Thatis,thecollectorcurrentICisjust somefractionofthetotalemittercurrent I E .Weintroduceyetonemoreconstantwhichreectstheratio betweenthesetwocurrents,andcallitsimply" ."Thuswesay I C = I E .8 Sincethe electron currentintothebaseisjust I E and T ofthatcurrentreachesthecollector,wecan write: I C = I E = T I E .9 LookingbackatthestructureofannpnbipolartransistorFigure2.1:BipolarTransistorStructure,we canuseKircho'scurrentlawforthetransistorandsay: I C + I B = I E .10 or I B = I E )]TJ/F11 9.9626 Tf 9.963 0 Td [(I C = I C )]TJ/F11 9.9626 Tf 9.963 0 Td [(I C .11 Thiscanbere-writtentoexpress I C intermsof I B as: I C = 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( I B I B .12 Thisisthefundamentaloperationalequationforthebipolarequation.Itsaysthatthecollectorcurrent isdependentonlyonthebasecurrent.Notethatif isanumberclosetobutstillslightlylessthanunity, then whichisjustgivenby = 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( .13 willbeafairlylargenumber.Typicalvaluesforawillbeontheorderof0.99orgreater,whichputs ,the currentgain,ataround100ormore!Thismeansthatwecancontrol,oramplifythecurrentgoingintothe collectorofthetransistorwithacurrent100timessmallergoingintothebase.Thisalloccursbecausethe ratioofthecollectorcurrenttothebasecurrentisxedbytheconditionsacrosstheemitter-basejunction, andtheratioofthetwo, I C to I B isalwaysthesame. 2.3TransistorI-VCharacteristics 3 Let'snowtakealookatsomecurrentvoltagerelationshipsforthebipolartransistor.Intheabsenceofany voltageorcurrentontheemitter-basejunction,ifweweretomakeaplotof I C asafunctionof V CB it wouldlooksomethinglikeFigure2.4.Checkbackwiththevoltageconventionintheguresonthestructure 3 Thiscontentisavailableonlineat.

PAGE 65

59 Figure2.1:BipolarTransistorStructureandforwardactivebiasingFigure2.2:forward_active_biasing ofabipolartransistortomakesureyouagreewithwhatIdrew.Allwe'vegothereisapnjunctionordiode. Itjusthappenstobebiasedinareversedirection,soitconductswhen V CB isnegativeandnotwhen V CB ispositive.Thus,allweneedtodoisdrawadiodecurve,butupsidedown! Figure2.4: I-Vforthecollector-baseterminalsofthebipolartransistor Whathappensifwenow also havesomebiasappliedtotheemitter-basejunction?Aswesaw,solong asthebase-collectorjunctionisreversebiased,almostallofthecollectorcurrentconsistsofelectronswhich havebeeninjectedintothebasebytheemitter,diuseacrossthebaseregion,andthenfalldownthebasecollectorjunction.Therateatwhichelectronsfalldownthejunctiondoesnotdependonhowlargeadrop thereise.g.howbig V CB is.Theonlythingthatmatters,insofarasthecollectorcurrentisconcerned, ishowfastelectronsarebeinginjectedintothebaseregion,whichis,ofcourse,determinedbytheemitter current I E Thusforseveraldierentvaluesofemittercurrent, I E 1 I E 2 ,and I E 1 ,wemightseesomethinglike Figure2.5.Intherstquadrant,whichisinthe"forwardactivebiasmode,"theoutputfromthecollector terminallooksmoreorlesslikeacurrentsource;thatis I C isaconstant,regardlessofwhat V CB is.Note however,thatwemustusea controlledsource ,inthiscase,acurrent-controlledcurrentsource,since I C dependsonwhat I E happenstobe.Obviously,lookingintheforwardbiasedemitter-baseterminal,we seetheusualp-njunction.Thus,ifwewereinterestedinbuildinga"model"ofthisdevice,wemightcome upwithsomethinglikeFigure2.6.Notethatthebaseterminaliscommontobothinputs.Sincewewould actuallyliketothinkofthetransistorasatwo-portdevicewithaninputandanoutputthemodelforthe transistorisoftendrawnasshowninFigure2.7.

PAGE 66

60 CHAPTER2.BIPOLARTRANSISTORS Figure2.5: Commonbasecharacteristicsofthebipolartransistor Figure2.6: Modelforthecommonbasetransistor Figure2.7: Re-drawncommonbasetransistor Theonlydrawbackwithwhatwehavesofaristhatexceptinsomespecializedhigh-frequencycircuits, thebipolartransistorisveryrarelyusedinthecommonbaseconguration.Mostofthetime,youwill seeitineitherthecommonemittercongurationFigure2.8,orthecommoncollectorconguration.The commonemitterisprobablythewaythetransistorismostoftenused.

PAGE 67

61 Figure2.8: Congurationforthecommonemittercircuit Notethatwehaveacurrentsourcedrivingthebase,andwehaveappliedjustonebatteryalltheway fromthecollectortotheemitter.Thebatterynowhastodotwothing:aIthastoprovidereversebias forthebase-collectorjunctionandbithastoprovideforwardbiasforthebaseemitterjunction.Forthis reason,the I C asafunctionof V CE curveslookalittledierentnow.Itisnownecessaryfor V CE tobecome slightlypositiveinordertogetthetransistorintoitsactivemode.Theotherdierence,ofcourse,isthat thecollectorcurrentisnowshownasbeing I B thebasecurrentinsteadof I E theemittercurrent. Figure2.9: Commonemittercharacteristiccurvesforthetransistor 2.4CommonEmitterModels 4 Let'sgoaheadanddrawamodelforthetransistorinthecommonemittercongurationFigure2.10.We againhaveadiodeconnectedbetweenthebaseandtheemitter,andanewcurrentcontrolledcurrentsource betweencollectorandemitter.Thereisonesmallcaveatwhichweneedtokeepinmindhoweverwhen drawingthecommonemittercircuit.Thediodeweseeinthebasecircuitis not thesameoneaswehadin thecommonbasemodel.Inthecommonbasemodel,itwastruethat I E = I sat e qV BE kT )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 .14 4 Thiscontentisavailableonlineat.

PAGE 68

62 CHAPTER2.BIPOLARTRANSISTORS Figure2.10: Discretemodelforthecommonemitterconguration Forthebasehowever,onlyasmallfractionofthecurrentthatgoesthroughthis"diode"actuallygoes inthroughthebase,therestiscominginthroughthecollector.Thuswehavetomakeacoupleofchanges I C = I E = I sat e qV BE kT )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .15 I B = I C = I sat e qV BE kT )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 .16 Sotheoperationalequationforthediodeinthebasecircuitstillistheusualexponentialfunctionof V BE exceptthatitnowhasasaturationcurrentof I sat insteadofjust I sat Inprincipleyoucouldputthismodelintoacircuit,andanalyzeittondallofthenecessaryvoltages andcurrents.However,thiswouldnotbeveryconvenient.Thebase-emitterjunctionisconnectedbya diode,whichasweknow,hasaverynon-linearI-Vrelationship.Itwouldbeniceifwecouldcomeupwith a linear modelwhich,atleastoversomelimitedrangeofinputs,wecouldusewithcondence. 2.5SmallSignalModels 5 Inordertodothisweneedtointroducetheconceptof bias ,and largesignal and smallsignaldevice behavior.Considerthefollowingcircuit,showninFigure2.11.Weareapplyingthesumoftwovoltagesto thediode, V B ,the biasvoltage whichisassumedtobeaDCvoltageandvsthe signalvoltage whichis assumedtobeAC,orsinusoidal.Bydenition,wewillassumethat j v s j ismuchlessthan j V B j Asaresult ofthesevoltages,therewillbeacurrent I B owingthroughthediodewhichwillconsistoftwocurrents, I B theso-called biascurrent ,and i s ,whichwillbethe signalcurrent .Again,weassumethat i s ismuch smallerthan I B 5 Thiscontentisavailableonlineat.

PAGE 69

63 Figure2.11: Puttingtogetheralargesignalbias,andasmallsignalACexcitation Whatwewouldliketodoistoseeifwecanndalinearrelationshipbetween v s and i s whichwecould useinoursignalanalysis.Therearetwowayswecanattacktheproblem;agraphicalapproach,anda purelymathematicalapproach.Letstrythegraphicalapproachrst,asitismoreintuitive,andthenwe willconrmwhatwendoutwithamathematicalone. Let'sremindourselvesabouttheI-Vcharacteristicsofadiode.Inthepresentsituation, V D isthesum oftwovoltages,aDCbiasvoltage V B andanACsignal, v s Let'splot V D t onthe V D axisasshownin Figure2.13.Howarewegoingtogureoutwhatthecurrentis?Whatweneedtodoistoprojectthe voltageupontothecharacteristicI-Vcurve,andthenprojectovertotheverticalcurrentaxisWedothisin Figure2.14.Notethattheoutputcurrentsignalissomewhatdistorted,whichmeanswedonothavelinear behavioryet.Let'sreducetheamplitudeofthesignalvoltage,asshowninFigure2.15.Nowweseetwo things:atheoutputismuchlessdistorted,sowemustgettingamorelinearbehavior,andbwecouldget theamplitudeoftheoutputsignal i s simplybymultiplyingtheinputsignal v s bytheslopeoftheI-Vcurve atthepointwherethedeviceisbiased. Wehavereplacedthenon-linearI-Vcurveofthediodebyalinear one,whichisapplicableovertherangeofthesignalvoltage. i s = d dV D I D j I D = I B .17 Figure2.12: DiodeI-Vbehavior

PAGE 70

64 CHAPTER2.BIPOLARTRANSISTORS Figure2.13: BiasandsignalexcitationofadiodeI-Vcurve Figure2.14: GraphicallyndingtheACresponse

PAGE 71

65 Figure2.15: Withasmallersignal,theresponseismorelinear Togettheslope,weneedafewsimpleequations: I D = I sat e qV D kT )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 I sat e qV D kT .18 d dV D I D = q kT I sat e qV D kT .19 Whenweevaluatethepartialderivativeatvoltage V D ,wenotethat I sat e qV D kT = I B .20 andhence,theslopeofthecurveisjust q kT I B or 40 I B ,since q kT justhasavalueof 40 V )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 atroomtemperatures.Notethatcurrentdividedbyvoltageisjustconductance,whichisjusttheinverseofresistance andsowehavefoundthe smallsignallinearconductanceforthediode AsfarastheACsignalgeneratorisconcerned,wecouldreplacethediodewitharesistorwhosevalueis theinverseoftheconductance,or r = 1 40 I B ,where I B istheDCbiascurrentthroughthediode. Studentsaresometimesconfusedabouthowwecanreplaceadiode,whichonlyconductsinonedirection, witharesistor,whichconductsbothways.TheansweristolookcarefullyatFigure2.15.AstheACsignal voltagerisesandfalls,theACoutputcurrentalsoincreasesanddecreasesinthesamemanner.Overthe limitedrangeoftheACsignalparameters,thediodeisindeedalinearsignalelement,notarectifyingone, asitisforlargesignalapplications. Nowlet'sgetthesameanswerfromapurelymathematicalapproach. I D = I B + i s = I sat e qV D kT )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 e q V B + v s kT .21 Inthelastexpression,wedroppedthe )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 asitisverysmallcomparedtotheexponentialtermandcanbe neglected. Nowwenotethat: e q V B + v s kT = e qV B kT e qv s kT .22 And,forthesecondexponential,if qV B ismuchlessthan kT e qv s kT 1+ qv s kT + ::: .23

PAGE 72

66 CHAPTER2.BIPOLARTRANSISTORS wherewehaveusedthepowerseriesexpansionfortheexponential,buthaveonlytakenthersttwoterms. Thus I B + i s I sat e qV B kT 1+ qv s kT .24 Obviously I B = I sat e qV B kT .25 and i s = I sat e qV B kT )]TJ/F10 6.9738 Tf 8.902 -3.625 Td [(q kT v s = q kT I B v s .26 whichgivesusthesameresultasbefore i s v s = q kT I B .27 2.6SmallSignalModelforBipolarTransistor 6 ThusifwegobacktothecircuitmodelFigure2.10forthecommonemittertransistor,andre-drawitas a smallsignalmodel itwouldlooksomethinglikeFigure2.16.Herewehavereplacedthediodewitha linearelementaresistor,called r andwehavechangedthenotationforthecurrentsfrom I B and I C to i b and i c respectively,toremindusthatwearenowtalkingaboutsmallsignalacquantities,notlargesignal ones.Thebiascurrents I B and I C arestillowingthroughthedeviceandwewillleaveittoELEC342to discusshowthesearegeneratedandsetupbuttheydonotappearinthesmallsignalmodel.Thismodel isonlyusedtogureouthowthetransistorbehavesfortheacsignalgoingthroughit,nothaveitresponds tolargeDCvalues. Figure2.16: Smallsignallinearmodelforthecommonemittertransistor Now r theequivalentsmallsignalresistanceofthebase-emitterdiodeisgivensimplybytheinverseof theconductanceoftheequivalentdiode.Remember,wefound r = 1 q kT I B = 1 q kT I C = 40 I C .28 6 Thiscontentisavailableonlineat.

PAGE 73

67 wherewehaveusedthefactthat I C = I B and q kT =40 V )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 .Aswesaidearlier,typicalvaluesfor ina standardbipolartransistorwillbearound100.Thus,foratypicalcollectorbiascurrentof I C =1 mA r willbeabout2.5k Thereisonemoreitemweshouldconsiderinputtingtogetherourmodelforthebipolartransistor.We didnotgetthingscompletelyrightwhenwedrewthecommonemittercharacteristiccurvesFigure2.9 forthetransistor.Thereisasomewhatsubtleeectgoingonwhen V CE isincreased.Remember,wesaid thatthecurrentcomingoutofthecollectorisnoteectedbyhowbigthedropwasinthereversebiased base-collectorjunction.Thecollectorcurrentjustdependsonhowmanyelectronsareinjectedintothe basebytheemitter,andhowmanyofthemmakeitacrossthebasetothebase-collectorjunction.Asthe base-collectorreversebiasisincreasedbyincreasing V CE thedepletionwidthofthebase-collectorjunction increasesaswell.Thishastheeectofmakingthebaseregionsomewhatshorter.Thismeansthatafew moreelectronsareabletomakeitacrossthebaseregionwithoutrecombiningandasaresult andhence increasesomewhat.Thisthenmeansthat I C goesupslightlywithincreasing V CE .Theeectiscalled basewidthmodulation .Letusnowincludethateectinthecommonemittercharacteristiccurves.As youcanseeinFigure2.18,thereisnowaslopetothe I C V CE curve,with I C increasingsomewhatas V CE increases.TheeecthasbeensomewhatexaggeratedinFigure2.17,andIwillnowmaketheslopeeven biggersothatwemaydeneanewquantity,calledthe EarlyVoltage Figure2.17: Commonemitterresponsewithbase-widthmodulationeect Figure2.18: FindingtheEarlyVoltage

PAGE 74

68 CHAPTER2.BIPOLARTRANSISTORS Backintheverybeginningofthetransistorera,anengineeratBellLabs,JimEarly,predictedthatthere wouldbeaslopetothe I C curves,andthattheywouldallprojectbacktothesameintersectionpointon thehorizontalaxis.Havingmadethatprediction,Jimwentdownintothelab,madethemeasurement,and conrmedhisprediction,thusshowingthatthetheoryoftransistorbehaviorwasbeingproperlyunderstood. Thepointofintersectionofthe V CE axisisknownasthe EarlyVoltage .Sincethesymbol V E ,forthe emittervoltagewasalreadytaken,theyhadtolabeltheEarlyVoltage V A instead.Eventhoughthe intersectionpointinonthenegativehalfofthe V CE axis, V A isuniversallyquotedasapositivenumber. HowcanwemodeltheslopingI-Vcurve?Wecandoalmostthesamethingaswedidwiththesolarcell. Thehorizontalpartofthecurveisstillacurrentsource,andtheslopedpartissimplyaresistorinparallel withit.HereisagraphicalexplanationinFigure2.19. Figure2.19: Combiningacurrentcourseandaresistorinparallel Usually,theslopeismuchlessthanwehaveshownhere,andsoforanygivenvalueof I C ,wecanjust taketheslopeofthelineas I C V A ,andhencetheresistance,whichisusuallycalled r o isjust V A I c .Thus,we add r o tothesmallsignalmodelforthebipolartransistor.ThisisshowninFigure2.20.Inagoodquality moderntransistor,theEarlyVoltage, V A willbeontheorderof150-250Volts.Soifwelet V A =200 ,and weimaginethatwehaveourtransistorbiasedat1mA,then r o = 200 V 1 mA =200 k .29 whichisusuallymuchlargerthanmostoftheotherresistorsyouwillencounterinatypicalcircuit.Inmost instances, r o canbeignoredwithnoproblem.Ifyougetintohighimpedancecircuitshowever,asyoumight ndinainstrumentationamplier,then v be hastobetakenintoaccount.

PAGE 75

69 Figure2.20: Includingrointhesmallsignallinearmodel Sometimesitisadvantageoustouseamutualtransconductancemodelinsteadofacurrentgainmodel forthetransistor.Ifwecalltheinputsmallsignalvoltage v be ,thenobviously i b = v be r = v be 40 I C .30 But i c = i b = v be 40 I C =40 I C v be g m v be .31 Where g m iscalledthemutualtransconductanceofthetransistor.Noticethat hascompletelycancelled outintheexpressionfor g m andthat g m dependsonlyuponthebiascurrent, I C ,owingthroughthecollector andnotonanyofthephysicalpropertiesofthetransistoritself! Figure2.21: Transconductancesmallsignallinearmodel Finally,thereisonelastphysicalconsiderationweshouldmakeconcerningtheoperationofthebipolar transistor.Thebase-collectorjunctionisreversebiased.Weknowthatifweapplytoomuchreversebiasto apnjunction,itcanbreakdownthroughavalanchemultiplication.Breakdowninatransistorissomewhat "softer"thanforasimplediode,becauseonceasmallamountofavalanchemultiplicationstarts,extra holesaregeneratedwithinthebase-collectorjunction.Theseholesfallup,intothebase,wheretheyactas additionalbasecurrent,which,inturn,causes I C toincrease.ThisisshowninFigure2.22.

PAGE 76

70 CHAPTER2.BIPOLARTRANSISTORS Figure2.22: Ionizationatthebase-collectorjunctioncausesadditionalbasecurrent AsetofcharacteristiccurvesforatransistorgoingintobreakdownisalsoshowninFigure2.23. Figure2.23: BipolarTransistorgoingintobreakdown Well,wehavelearnedquiteabitaboutbipolartransistorsinaveryshortspace.Gobackoverthis chapterandseeifyoucanpickoutthetwoorthreemostimportantideasofequationswhichwouldmake upasetof"facts"thatyoucouldstickawayinyouheadsomeplace.Dothissoyouwillalwayshavethem torefertowhenthesubjectofbipolarscomesupInsay,ajobintervieworsomething!.

PAGE 77

Chapter3 FETs 3.1IntroductiontoMOSFETs 1 Wenowmoveontoanotherthreeterminaldevice-alsocalleda transistor .Intruththisdevicereallyhasat leastfour,andprobablyve,terminals,butwewillleavethesubtledetailsforalatertime.Thistransistor, however,worksonmuchdierentprinciplesthandoesthebipolarjunctiontransistorofthelastchapter.We willnowfocusonadevicecalledthe FieldEectTransistor ,or Metal-Oxide-SemiconductorField EectTransistor orsimply,the MOSFET .Considerthefollowing: Figure3.1: Thestartofaeldeecttransistor Herewehaveablockofsilicon,dopedp-type.Intoitwehavemadetworegionswhicharedopedn-type. Toeachofthosen-typeregionsweattachawire,andconnectabatterybetweenthem.Ifwetrytoget somecurrent, I ,toowthroughthisstructure,nothingwillhappen,becausethen-pjunctionontheRHS isreversebiasedWehavethepositiveleadfromthebatterygoingtothen-sideofthep-njunction.Ifwe attempttoremedythisbyturningthebatteryaround,wewillnowhavetheLHSjunctionreversebiased, andagain,nocurrentwillow.If,forwhateverreason,wewantcurrenttoow,wewillneedtocomeup withsomewayofformingalayerofn-typematerialbetweenonen-regionandtheother.Thiswillthen connectthemtogether,andwecanruncurrentinoneterminalandouttheother. 1 Thiscontentisavailableonlineat. 71

PAGE 78

72 CHAPTER3.FETS Toseehowwewilldothis,let'sdotwothings.Firstwewillgrowalayerof SiO 2 silicondioxide,or justplain"oxide"ontopofthesilicon.Thisturnsouttoberelativelyeasy,wejuststickthewaferin anovenwithsomeoxygenowingthroughit,andheateverythinguptoabout 1100 C foranhourorso, andweendupwithanice,high-qualityinsulating SiO 2 layerontopofthesilicon.Ontopoftheoxide layerwethendepositaconductor,whichwecallthegate.Inthe"olddays"thegatewouldhavebeena layerofaluminumHencethe"metal-oxide-silicon"orMOSname.Today,itismuchmorelikelythata heavilydopedlayerofpolycrystallinesiliconpolysilicon,ormoreoftenjust"poly"wouldbedepositedto formthegatestructure.Iguess"POS"soundedfunnytopeopleintheeld,becauseitnevercaughtonas anameforthesedevices.Polysiliconismadefromthereductionofagas,suchassilane SiH 4 through thereaction SiH 4 g Si s +2 H 2 g .1 Thesiliconispolycrystallinecomposedoflotsofsmallsiliconcrystallitesbecauseitisdepositedontop oftheoxide,whichisamorphous,andsoitdoesnotprovideasinglecrystal"matrix"whichwouldallowthe silicontoorganizeitselfintoonesinglecrystal.Ifwehaddepositedthesiliconontopofasinglecrystalsilicon wafer,wewouldhaveformedasinglecrystallayerofsiliconcalledan epitaxiallayer Epitaxy comes fromtheGreek,anditjustmeans"orderedupon".Thusanepitaxiallayerisonewhichfollowstheorderof thesubstrateonwhichitisgrown.Thisissometimesdonetomakestructuresforparticularapplications. Forinstance,growingan-typeepitaxiallayerontopofap-typesubstratepermitsthefabricationofavery abruptp-njunction. 3.2BasicMOSStructure 2 Figure3.2: FormationoftheMOSstructure Figure3.2showsthestepsnecessarytomaketheMOSstructure.Itwillhelpusinourunderstandingifwe nowrotateourpicturesothatitispointingsidewaysinournextfewdrawings.Also,wewillforgetabout thetwon-regionsforawhile,andpickthembackuplaterwhenwerotatethestructurerightsideupagain. Figure3.3showstherotatedstructure.Notethatinthep-siliconwehavepositivelychargedmobileholes, andnegativelycharged,xedacceptors.Becausewewillneeditlater,wehavealsoshownthebanddiagram forthesemiconductorbelowthesketchofthedevice.Notethatsincethesubstrateisp-type,theFermilevel islocateddownclosetothevalanceband. 2 Thiscontentisavailableonlineat.

PAGE 79

73 Figure3.3: BasicMOSstructure Letusnowplaceapotentialbetweenthegateandthesiliconsubstrate.Supposewemakethegate negativewithrespecttothesubstrate.Sincethesubstrateisp-type,ithasalotofmobile,positively chargedholesinit.Someofthemwillbeattractedtothenegativechargeonthegate,andmoveoverto thesurfaceofthesubstrate.Thisisalsoreectedinthebanddiagrambelowthesketchofthestructure Figure3.4.RememberthatthedensityofholesisexponentiallyproportionaltohowclosetheFermilevel istothevalencebandedge.Weseethatthebanddiagramhasbeenbentupslightlynearthesurfaceto reecttheextraholeswhichhaveaccumulatedthere. Figure3.4: Applyinganegativegatevoltage

PAGE 80

74 CHAPTER3.FETS Anelectriceldwilldevelopbetweenthepositiveholesandthenegativegatecharge.Notethatthe gateandthesubstrateformakindofparallelplatecapacitor,withtheoxideactingastheinsulatinglayer in-betweenthem.Theoxideisquitethincomparedtotheareaofthedevice,andsoitisquiteappropriate toassumethattheelectriceldinsidetheoxideisauniformone.Wewillignorefringingattheedges. Theintegraloftheelectriceldisjusttheappliedgatevoltage V g .Iftheoxidehasathickness x ox then since E ox isuniform,itisgivenby E ox = V g x ox .2 Ifwefocusinonasmallpartofthegate,wecanmakealittle"pill"boxwhichextendsfromsomewhere intheoxide,acrosstheoxide/gateinterfaceandendsupinsidethegatematerialsomeplace.Thepill-box willhaveanarea s .NowwewillinvokeGauss'lawwhichwereviewedearlier.Gauss'lawsimplysaysthat thesurfaceintegraloveraclosedsurfaceofthedisplacementvector D whichis,ofcourse,just times E isequaltothetotalchargeenclosedbythatsurface.Wewillassumethatthereisasurfacechargedensity )]TJ/F11 9.9626 Tf 7.749 0 Td [(Q g Coulombs cm 2 onthesurfaceofthegateelectrodeFigure3.5.TheintegralformofGauss'Lawisjust: I ox E d S = Q encl .3 Figure3.5: Findingthesurfacechargedensity Notethatwehaveused ox E inplaceof D .Inthisparticularset-uptheintegraliseasytoperform,since theelectriceldisuniform,andonlypointinginthroughonesurface-itterminatesonthenegativesurface chargeinsidethepill-box.Thechargeenclosedinthepillboxisjust )]TJ/F8 9.9626 Tf 9.409 0 Td [( Q g s ,andsowehavekeepingin mindthatthesurfaceintegralofavectorpointingintothesurfaceisnegative H ox E d S = )]TJ/F8 9.9626 Tf 9.409 0 Td [( ox E ox s = )]TJ/F8 9.9626 Tf 9.409 0 Td [( Q g s .4 or ox E ox = Q g .5 Now,wecanuse.2toget ox V g x ox = Q g .6

PAGE 81

75 or Q g V g = ox x ox c ox .7 Thequantity c ox iscalledthe oxidecapacitance .Ithasunitsof Farads cm 2 ,soitisreallyacapacitance per unitarea oftheoxide.Thedielectricconstantofsilicondioxide, ox ,isabout 3 : 3 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(13 F=cm .Atypical oxidethicknessmightbe250or 2 : 5 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(6 cm .Inthiscase, c ox wouldbeabout 1 : 30 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 F cm 2 .The unitsweareusinghere,whiletheymightseemalittlearbitraryandconfusing,aretheonesmostcommonly usedinthesemiconductorbusiness.Youwillgetusedtotheminashortwhile. Themostusefulformof.7iswhenitisturnedaround: Q g = c ox V g .8 asitgivesusawaytondthechargeonthegateintermsofthegatepotential.Wewillusethisequation laterinourdevelopmentofhowtheMOStransistorreallyworks. Itturnsoutwehavenotdoneanythingveryusefulbyapplyanegativevoltagetothegate.Wehave drawnmoreholesthereinwhatiscalledan accumulationlayer ,butthatisnothelpingusinoureort tocreatealayerofelectronsintheMOSFETwhichcouldelectricallyconnectthetwon-regionstogether. Let'sturnthebatteryaroundandapplya positive voltagetothegate.Actually,let'stakethebattery outofthesketchFigure3.6fornow,andjustlet V g beapositivevalue,relativetothesubstratewhich willtietoground.Making V g positiveputspositive Q g onthegate.Thepositivechargepushestheholes awayfromtheregionunderthegateanduncoverssomeofthenegatively-chargedxedacceptors.Now theelectriceldpointstheotherway,andgoesfromthepositivegatecharge,terminatingonthenegative acceptorchargewithinthesilicon. Figure3.6: Increasingthevoltageextendsthedepletionregionfurtherintothedevice Theelectriceldnowextends into thesemiconductor.Weknowfromourexperiencewiththep-njunction thatwhenthereisanelectriceld,thereisashiftinpotential,whichisrepresentedinthebanddiagram bybendingthebands.Bendingthebandsdownasweshouldmovingtowardspositivechargecausesthe valencebandtopullawayfromtheFermilevelnearthesurfaceofthesemiconductor.Ifyourememberthe expressionwehadforthedensityofholesintermsof E v and E f electronandholedensityequations.25 itiseasytoseethatindeed p = N v e )]TJ/F27 6.9738 Tf 6.226 7.682 Td [( E f )]TJ/F9 4.9813 Tf 5.396 0 Td [(E v kT .9

PAGE 82

76 CHAPTER3.FETS thereisadepletionregionregionwithalmostnoholesneartheregionunderthegate.Once E f )]TJ/F11 9.9626 Tf 10.286 0 Td [(E v getslargewithrespectto kT ,thenegativeexponentcauses p 0 Figure3.7: Threshold, E f isgettingcloseto E c Theelectriceldextendsfurtherintothesemiconductor,asmorenegativechargeisuncoveredandthe bandsbendfurtherdown.Butnowwehavetorecalltheelectrondensityequation.25,whichtellsushow manyelectronswehave n = N c e )]TJ/F27 6.9738 Tf 6.226 7.681 Td [( E c )]TJ/F9 4.9813 Tf 5.397 0 Td [(E f kT .10 AglanceatFigure3.7aboverevealsthatwiththismuchbandbending, E c theconductionbandedge, and E f theFermilevelarestartingtogetclosetooneanotheratleastcomparedto kT ,whichmeans that n ,theelectronconcentration,shouldsoonstarttobecomesignicant.Inthesituationrepresentedby Figure3.7,wesayweareat threshold ,andthegatevoltageatthispointiscalledthe thresholdvoltage V T Now,let'sincrease V g above V T .Here'sthesketchinFigure3.8.

PAGE 83

77 Figure3.8: Inversion-Electronsformaninversionlayerunderthegate Eventhoughwehaveincreased V g beyondthethresholdvoltage, V T ,andmorepositivechargeappears onthegate,thedepletionregionnolongermovesbackintothesubstrate.Insteadelectronsstarttoappear underthegateregion,andtheadditionalelectriceldlinesterminateonthesenewelectrons,insteadofon additionalacceptors.Wehavecreatedan inversionlayer ofelectronsunderthegate,anditisthislayer ofelectronswhichwecanusetoconnectthetwon-typeregionsinourinitialdevice. Wheredidtheseelectronscomefrom?Wedonothaveanydonorsinthismaterial,sotheycannot comefromthere.Theonlyplacefromwhichelectronscouldbefoundwouldbethroughthermalgeneration. Remember,inasemiconductor,therearealwaysafewelectronholepairsbeinggeneratedbythermal excitationatanygiventime.Electronsthatgetcreatedinthedepletionregionarecaughtbytheelectric eldandaresweptovertotheedgebythegate.Ihavetriedtosuggestthiswiththeelectrongeneration eventshowninthebanddiagraminthegure.Ina real MOSdevice,wehavethetwon-regions,anditis easyforelectronsfromoneorbothto"fall"intothepotentialwellunderthegate,andcreatetheinversion layerofelectrons. 3.3ThresholdVoltage 3 Ourtasknowistogureouthowmuchvoltageweneedtoget V g upto V T andthentogureouthowmuch negativechargethereisunderthegate,once V T hasbeenexceeded.Therstpartisactuallyprettyeasy.It isalotliketheproblemwelookedat,withtheone-sideddiode,butwithjustalittleaddedcomplication.To startout,letsmakeasketchofthechargedensitydistributionundertheconditionsofthisimageFigure3.7, justwhenwegettothreshold.Wellincludethesketchofthestructuretoo,soitwillbeclearwhatcharge wearetalkingabout.ThisisshowninFigure3.9.Now,wejustusetheequationwedevelopedbeforefor theelectriceld.39,whichcamefromintegratingthedierentialformofGauss'Law. E x = Z x dx .11 3 Thiscontentisavailableonlineat.

PAGE 84

78 CHAPTER3.FETS Figure3.9: Chargedistributionatthreshold Asbefore,wewilldotheintegralgraphically,startingattheLHSofthepicture.Theeldoutsidethe structuremustbezero,sowehavenoelectricelduntilwegettothedeltafunctionofchargeonthegate, atwhichtimeitjumpsuptosomevaluewewillcall E ox .Thereisnochargeinsidetheoxide,so d dx E is zero,andthus E x mustremainconstantat E ox untilwereachtheoxide/siliconinterface. Figure3.10: Electriceldintheoxide Ifweweretoputourlittle"pillbox"ontheoxide-siliconinterface,theintegralof D overthefacein thesiliconwouldbe Si E Si S ,where E Si isthestrengthoftheelectriceldinsidethesilicon.Ontheface insidetheoxideitwouldbe )]TJ/F8 9.9626 Tf 9.41 0 Td [( ox E ox S ,where E ox isthestrengthoftheelectriceldintheoxide.The

PAGE 85

79 minussigncomesfromthefactthattheeldontheoxidesideisgoingintothepillboxinsteadofoutofit. Thereisnonetchargecontainedwithinthepillbox,sothesumofthesetwointegralsmustbezero.The integraloverthe entire surfaceequalstheenclosedcharge,whichiszero. Si E max S )]TJ/F11 9.9626 Tf 9.963 0 Td [( ox E ox S =0 .12 or Si E max = ox E ox .13 Figure3.11: UsingGauss'Lawatthesilicon/oxideinterface Thisisjustastatementthatitisthenormalcomponentofdisplacementvector, D ,whichmustbe continuousacrossadielectricinterface,nottheelectriceld, E .Solving.12fortheelectriceldinthe silicon: E Si = ox Si E ox .14 Thedielectricconstantofoxidesaboutonethirdthatofthedielectricconstantofsilicondioxide,sowe seea"jump"downinthemagnitudeoftheelectriceldaswegofromoxidetosilicon.Thechargedensity inthedepletionregionofthesiliconisjust )]TJ/F8 9.9626 Tf 9.409 0 Td [( qN a andsotheelectriceldnowstartsdecreasingatarate )]TJ/F7 6.9738 Tf 6.227 0 Td [( qN a Si andreacheszeroattheendofthedepletionregion, x p .

PAGE 86

80 CHAPTER3.FETS Figure3.12: electriceldandvoltagedropsacrosstheentirestructure Clearly,wehavetwodierentregions,eachwiththeirownvoltagedrop.Remembertheintegralof electriceldisvoltage,sotheareaundereachregionof E x representsavoltagedrop.Thedropinthe littletriangularregionwewillcall V Si anditrepresentsthepotentialdropingoingfromthebulk,down tothebottomofthedroopingconductionbandatthesilicon-oxideinterface.Lookingbackattheearlier gureFigure3.7onthreshold,youshouldbeabletoseethatthisisnearlyonewholeband-gapsworthof potential,andsowecansafelysaythat V Si 0 : 8 1 : 0 V Justaswiththesingle-sideddiode,thewidthofthedepletionregion x p ,iswhichwesawinaprevious equation.42: x p = s 2 Si V Si qN a .15 fromwhichwecangetanexpressionfor E Si E Si = qN a Si x p = q 2 qN a V Si Si .16 bymultiplyingtheslopeofthe E x linebythewidthofthedepletionregion, x p Wecannowuse.14tondtheelectriceldintheoxide E ox = Si ox E Si = 1 ox p 2 q Si N a V Si .17 Finally, V ox issimplytheproductof E ox andtheoxidethickness, x ox V ox = x ox E ox = x ox ox p 2 q Si N a V Si .18 Notethat x ox dividedby ox issimplyoneover c ox ,theoxidecapacitance,whichwedescribedearlier. Thus V ox = 1 c ox p 2 q Si N a V Si .19

PAGE 87

81 Andthethresholdvoltage V T isthengivenas V T = V Si + V ox = V Si + 1 c ox p 2 q Si N a V Si .20 whichisnotthathardtocalculate!.20isoneofthemostimportantequationsinthisdiscussionofeld eecttransistors,asittellsuswhentheMOSdeviceisturnedon. .20hasseveral"handles"availabletothedeviceengineertobuildadevicewithagiventhreshold voltage.Weknowthatasweincrease N a ,theacceptordensity,thattheFermilevelgetsclosertothe valanceband,andhence V si willchangesome.Butaswesaid,itwillalwaysbearound0.8to1Volt,so itwillnotbethedrivingtermwhichdominates V T .Let'sseewhatwegetwithanacceptorconcentration of 10 17 .Justforcompleteness,let'scalculate E f )]TJ/F11 9.9626 Tf 9.963 0 Td [(E v p = N a = N v e E f )]TJ/F9 4.9813 Tf 5.397 0 Td [(E v kT .21 thus E f )]TJ/F11 9.9626 Tf 9.963 0 Td [(E v = kTln N v N a Insilicon, N v is 1 : 08 10 19 andthismakes E f )]TJ/F11 9.9626 Tf 9.4 0 Td [(E v =0 : 117 eV whichwewillcall E .Itisconventional tosaythatasurfaceisinvertedif,atthesiliconsurface, E c )]TJ/F11 9.9626 Tf 9.252 0 Td [(E f ,thedistancebetweentheconductionband andtheFermilevelisthesameasthedistancebetweentheFermilevelandthevalancebandinthebulk. Withalittletimespentlookingat.14,youshouldbeabletoconvinceyourselfthatthetotalenergy changeingoingfromthebulktothesurfaceinthiscasewouldbe q V Si = E g )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 E =1 : 1 eV )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 : 117 eV =0 : 866 eV .22 Figure3.13: Exampleofnding V Si Using N A =10 17 Si =1 : 1 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(12 F cm and q =1 : 6 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(19 C ,wendthat p 2 q si N a V Si =1 : 74 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 .23

PAGE 88

82 CHAPTER3.FETS Wesawearlierthatifwehaveanoxidethicknessof250,wegetavaluefor c ox of 1 : 3 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 F cm 2 Coulombs Vcm 2 ,andso V ox = 1 c ox p 2 q Si N a V Si = 1 1 : 3 10 )]TJ/F6 4.9813 Tf 5.397 0 Td [(7 1 : 74 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 =1 : 32 V .24 and V T = V Si + V ox =0 : 866+1 : 32 =2 : 18 V .25 3.4MOSTransistor 4 Nowwecangobacknowtoourinitialstructure,shownintheintroductiontoMOSFETsFigure3.1, onlythistimewewilladdanoxidelayer,agatestructure,andanotherbatterysothatwecaninvertthe regionunderthegateandconnectthetwon-regionstogether.Wellalsoidentifysomenamesforpartsof thestructure,sowewillknowwhatwearetalkingabout.Forreasonswhichwillbeclearlater,wecall then-regionconnectedtothenegativesideofthebatterythe source ,andtheotheronethe drain .We willgroundthesource,andalsothep-typesubstrate.Weaddtwobatteries, V gs betweenthegateandthe source,and V ds betweenthedrainandthesource. Figure3.14: BiasingaMOSFETtransistor Itwillbehelpfulifwealsomakeanothersketch,whichgivesusaperspectiveviewofthedevice.For thiswestripothegateandoxide,butwewillimaginethatwehaveappliedavoltagegreaterthan V T to thegate,sothereisan-typeregion,calledthe channel whichconnectsthetwo.Wewillassumethatthe channelregionis L longand W wide,asshowninFigure3.15. 4 Thiscontentisavailableonlineat.

PAGE 89

83 Figure3.15: Theinversionchannelanditsresistance Nextwewanttotakealookatalittlesectionofchannel,andnditsresistance dR ,whenthelittle sectionis dx long. dR = dx s W .26 Wehaveintroduceda slightly dierentformforourresistanceformulahere.Normally,wewouldhavea simple inthedenominator,andanarea A ,forthecross-sectionalareaofthechannel.Itturnsouttobe veryhardtogureoutwhatthatcrosssectionalareaofthechannelishowever.Theelectronswhichform theinversionlayercrowdintoaverythinsheetof surfacecharge whichreallyhaslittleornothickness, orpenetrationintothesubstrate. If,ontheotherhandweconsiderasurfaceconductivityunits:simplymhos, s ,where s = s Q chan .27 thenwewillhaveanexpressionwhichwecanevaluate.Here, s isasurfacemobility,withunitsof cm 2 Vsec Weraninto inearlierchaptersSection1.1,whenwewerebuildingoursimpleconductionmodel.Itwas thequantitywhichrepresentedtheproportionalitybetweentheaveragecarriervelocityandtheelectriceld. v = E .28 = q m .29 Thesurfacemobilityisaquantitywhichhastobemeasuredforagivensystem,andisusuallyjustanumber whichisgiventoyou.Somethingaround300 cm 2 Vsec isaboutrightforsilicon. Q chan iscalledthesurface chargedensityorchannelchargedensityandithasunitsof Coulombs cm 2 .Thisislikeasheetofcharge,which isdierentfromthebulkchargedensity,whichhasunitsof Coulombs cm 2 .Notethat: cm 2 Voltsec Coulombs cm 2 = Coul sec Volt = I V = mhos .30 Itturnsoutthatitisprettysimpletogetanexpressionfor Q chan ,thesurfacechargedensityinthe channel.Foranygivengatevoltage V gs ,weknow 5 thatthechargedensityonthegateisgivensimplyas: Q g = c ox V gs .31 5 "BasicMOSStructure",

PAGE 90

84 CHAPTER3.FETS However,untilthegatevoltage V gs getslargerthan V T wearenotcreatinganymobileelectronsunder thegate,wearejustbuildingupadepletionregion.We'lldene Q T asthechargeonthegatenecessaryto gettothreshold. Q T = c ox V T .Anychargeaddedtothegateabove Q T ismatchedbycharge Q chan inthe channel.Thus,itiseasytosay: Q channel = Q g )]TJ/F11 9.9626 Tf 9.962 0 Td [(Q T .32 or Q chan = c ox V g )]TJ/F11 9.9626 Tf 9.963 0 Td [(V T .33 Thus,putting.32and.27into.26,weget: dR = dx s c ox V gs )]TJ/F11 9.9626 Tf 9.962 0 Td [(V T W .34 IfyoulookbackatFigure3.14,youwillseethatwehavedenedacurrent I d owingintothedrain. Thatcurrentowsthroughthechannel,andhencethroughourlittleincrementalresistance dR ,creatinga voltagedrop dV c acrossit,where V c isthechannelvoltage. dV c x = I d dR = I d dx s c ox V gs )]TJ/F10 6.9738 Tf 6.227 0 Td [(V T W .35 Let'smovethedenominatortotheleft,andintegrate.Wewanttodoourintegralcompletelyalongthe channel.Thevoltageonthechannel V c x goesfrom0ontheleftto V ds ontheright.Atthesametime, x isgoingfrom0to L .Thusourlimitsofintegrationwillbe0and V ds forthevoltageintegral dV c x and from0to L forthe x integral dx Z V ds 0 s c ox V gs )]TJ/F11 9.9626 Tf 9.962 0 Td [(V T WdV c = Z L 0 I d dx .36 Bothintegralsareprettytrivial.Let'sswaptheequationorder,sinceweusuallywant I d asafunction ofappliedvoltages. I d L = s c ox W V gs )]TJ/F11 9.9626 Tf 9.962 0 Td [(V T V ds .37 Wenowsimplydividebothsidesby L ,andweendupwithanexpressionforthedraincurrent I d ,interms ofthedrain-sourcevoltage, V ds ,thegatevoltage V gs andsomephysicalattributesoftheMOStransistor. I d = s c ox W L V gs )]TJ/F11 9.9626 Tf 9.963 0 Td [(V T V ds .38 3.5MOSRegimes 6 ThisequationlooksalotliketheI-Vcharacteristicsofaresistor! I d issimplyproportionaltothedrain voltage V ds .Theproportionalityconstantdependsonthedimensionsofthedevice,WandLasthey intuitivelyshould.Thecurrentincreasesasthetransistorgetswider,itdecreasesasitgetslonger.Italso dependson c ox and s ,andonthedierencebetweenthegatevoltageandthethresholdvoltage V T .Note thatifweadjust V gs wecanchangetheslopeoftheI-Vcurve.Wehavemadeavoltage-controlledresistor! 6 Thiscontentisavailableonlineat.

PAGE 91

85 Figure3.16: TheMOSFETI-Vinthelinearregime Cautionisadvisedwiththisresulthowever,becausewehaveoverlookedsomethingquiteimportant.Lets gobacktoourpictureofthegateandthebatteriesinvolvedintheoperationoftheMOStransistor.Here wehaveexplicitlyshownthechannelasablackbandandwehaveintroducedanewquantity, V c x ,the voltagealongthechannel,andacoordinate x ,whichtellsuswhereweareonthechannelrelativetothe sourceanddrain.Notethatonceweapplyadrainsourcepotential, V ds ,thepotentialinthechannel V c x changeswithdistancealongthechannel.Atthesourceend, V c =0 ,asthesourceisgrounded.Atthe drainend, V c L = V ds .Wewilldeneavoltage V gc whichisthepotentialdierencebetweenthegate voltageandthevoltageinthechannel. V gc x V gs )]TJ/F11 9.9626 Tf 9.963 0 Td [(V c x .39 Thus, V gc goesfrom V gs atthesourceendto V gs )]TJ/F11 9.9626 Tf 9.963 0 Td [(V ds atthedrainend. Figure3.17: Eectof V ds onchannelpotential Thenetchargedensityinthechanneldependsuponthepotentialdierencebetweenthe gateandthe channelateachpointalongthechannel ,notjust V gs )]TJ/F11 9.9626 Tf 9.438 0 Td [(V T .Thuswehavetomodifytheequationofanother module.32totakethisintoaccount Q chan = c ox V gc x )]TJ/F11 9.9626 Tf 9.963 0 Td [(V T = c ox V gs )]TJ/F11 9.9626 Tf 9.963 0 Td [(V c x )]TJ/F11 9.9626 Tf 9.962 0 Td [(V T .40 This,inturn,modiestheintegralrelation.36between I d and V gs Z V ds 0 s c ox V gs )]TJ/F11 9.9626 Tf 9.963 0 Td [(V T )]TJ/F11 9.9626 Tf 9.962 0 Td [(V c x WdV c x = Z L 0 I d dx .41

PAGE 92

86 CHAPTER3.FETS .41isonlyslightlyhardertointegratethantheonebeforeNowwhat is theintegralofxdx,andso wegetforthedraincurrent I d = s c ox W L V gs )]TJ/F11 9.9626 Tf 9.963 0 Td [(V T V ds )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(V 2 ds 2 .42 Thisequationiscalledthe SahEquation afterC.T.Sah,whorstdescribedtheMOStransistor operationthiswaybackin1964.ItisveryimportantbecauseitdescribesthebasicbehavioroftheMOS transistor. Notethatforsmallvaluesof V ds ,apreviousequation.37and.42willgiveusthesame I d )]TJ/F11 9.9626 Tf 10.292 0 Td [(V ds behavior,becausewecanignorethe V ds 2 termin.42.Thisiscalledthe linearregime becausewehavea straight-linerelationshipbetweenthedraincurrentandthedrain-sourcevoltage.As V ds startstogetlarger however,thesquaredtermwillbegintokickinandtheplotwillstarttocurveover.Obviously,something iscausingthecurrenttodropoas V ds getslarger.Thisisbecausethevoltagedierencebetweenthegate andthechannelisbecomingless,whichmeansthereislesschargeinthechanneltoprovideconduction. Wecangraphicallyshowthisbymakingthechannellayerlookthinneraswemovefromthesourcetothe drain..42,andinfact,Figure3.18wouldmakeusthinkthatif V ds getslargeenough,thatthedrain current I d shouldactuallystartdecreasingagain,andmaybeevenbecomenegative!.Thisdoesnotseem veryintuitive,soletstakealookinmoredetailattheplacewhere I d becomesamaximum.Wecandene V dsat asthesource-drainvoltagewhere I d becomesamaximum.Wecanndthisbytakingthederivative of I d withrespectto V ds andsettingthederivativeto0. d dV ds I d =0 = s c ox W L V gs )]TJ/F11 9.9626 Tf 9.963 0 Td [(V T )]TJ/F11 9.9626 Tf 9.963 0 Td [(V dsat .43 Ondroppingconstants: V dsat = V gs )]TJ/F11 9.9626 Tf 9.963 0 Td [(V T .44 Rearrangingthisequationgivesusalittlemoreinsightintowhatisgoingon. V gs )]TJ/F11 9.9626 Tf 9.962 0 Td [(V dsat = V T = V gc L .45 Figure3.18: I-Vcharacteristicsshowingturn-over

PAGE 93

87 Figure3.19: Eectof V ds onthechannel Atthedrainendofthechannel,when V ds justequals V dsat ,thedierencebetweenthegatevoltageand thechannelvoltage, V gc L isjustequalto V T ,thethresholdvoltage.Anyfurtherincreasein V ds andthe dierencebetweenthegateandthechannel inthechannelregionjustnearthedrain willdropbelowthe thresholdvoltage.Thismeansthatwhen V ds getsbiggerthan V dsat ,thechanneljustnearthedrainregion disappears!Wenolongerhavesucientvoltagebetweenthegateandthechannelregiontomaintainan inversionlayer,sowesimplyreverttoadepletioncondition.Thisiscalled pincho ,asseeninFigure3.20. Figure3.20: Channelinpinch-o Whathappenstothedraincurrentwhenwehitpincho?Itlookslikeitmightgotozero,butthatis nottherightanswer!Althoughthereisnoactivechannelinthepinch-oregion,thereisstillsilicon-itjust happenstobedepletedofallfreecarriers.Thereisanelectriceld,goingfromthedraintothechannel, andanyelectronswhichmovealongthechanneltothepinch-oregionaresuckedacrossbytheeld,and enterthedrain.Thisisjustlikethecurrentthatowsinthereversesaturationconditionofadiode.There arenofreecarriersinthedepletionregionofthediode,yet I sat does owacrossthejunctionregion. Underpinch-oconditions,furtherincreasesin V ds ,doesnotresultinmoredraincurrent.Youcanthink ofthepinched-ochannelasaresistor,withavoltageof V dsat acrossit.When V ds getsbiggerthan V dsat theexcessvoltageappearsacrossthepinch-oregion,andthevoltageacrossthechannelremainsxedat V dsat .Ifthechannelkeepsthesamecharge,andhasthesamevoltageacrossit,thenthecurrentthrough thechannelandintothedrainwillremainxed,atavaluewewillcall I dsat Thereisoneothergurewhichsometimeshelpsinseeingwhatisgoingon.Wewillplotpotentialenergy foranelectron,asittraversesacrossthechannel.Sincethesourceisatzeropotentialandthedrainisat + V ds ,anelectronwillloosepotentialenergyasitowsfromthesourcetothedrain.Figure3.21showssome examplesforvariousvaluesof V ds :

PAGE 94

88 CHAPTER3.FETS Figure3.21: Electronpotentialenergydropgoingfromsourcetodrain. Forthersttwodrainvoltages, V ds 1 and V ds 2 ,wearebelowpinch-o,andsothevoltagedropacross R channel increasesas V ds increases,andhence,sodoes I d .At V dsat ,wehavejustreachedpinch-o,andwe arestartingtoseethe"higheld"depletionregionbegintodevelop.Sinceelectriceldisjustthederivative ofthepotential,theslopeofcurvesinFigure3.21givesyouanideaofhowbigtheelectriceldwillbe.For furtherincreasesin V ds V ds 4 and V ds 5 alloftheadditionalvoltagejustshowsupasahighelddropatthe endofthechannel.Thevoltagedropacrosstheconductingpartofthechannelstaysxedmoreorlessat V dsat andsothedraincurrentstaysmoreorlessxedat I dsat Substitutingtheexpressionfor V dsat intotheexpressionfor I d wecangetanexpressionfor I dsat I dsat = s c ox W 2 L V gs )]TJ/F11 9.9626 Tf 9.962 0 Td [(V T 2 .46 Wecandeneanewconstant, k ,where k = s c ox W L .47 Sothat I dsat = k 2 V gs )]TJ/F11 9.9626 Tf 9.963 0 Td [(V T 2 .48 WhatthismeansforFigure3.18isthatwhen V ds getsto V dsat ,wesimplyhold I d xedfromthenon, withavalueof I dsat .Fordierentvaluesof V g ,thegatevoltage,wearegoingtohaveadierent I d )]TJ/F11 9.9626 Tf 10.133 0 Td [(V ds curve,andsoonceagain,weendupwithafamilyof"characteristiccurves"fortheMOSFET.Theseare showninFigure3.23.

PAGE 95

89 Figure3.22: CompleteI-VcurvefortheMOSFET Figure3.23: CharacteristiccurvesforaMOSFET Thisalsogivesusafairlyeasywayinwhichto"sketch"asetofcharacteristiccurvesforagivendevice. SupposewehaveaMOSeldeecttransistorwhichhasathresholdvoltageof2volts,awidthof10microns, andachannellengthof1micron,anoxidethicknessof150angstroms,andasurfacemobilityof 400 c Vsec using ox =3 : 3 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(13 F cm ,wegetavalueof 2 : 2 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 F c for c ox .Thisthenmakeskhaveavalueof k = s c ox W L = 400 2 : 2 10 )]TJ/F6 4.9813 Tf 5.396 0 Td [(7 10 1 =8 : 8 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(4 amp volt 2 .49 3.6PlottingMOSI-V 7 Nowweusetwooftheequations.44and.48thatwefoundinthediscussionaboutMOSRegimes Section3.5tocalculateasetof V dsat and I dsat valuesforvariousvalueof V gs .Notethat V gs mustbe greaterthan V T forthetwoequationstobevalid.Whenwegetthenumbers,webuildalittletable. OncewehavethenumbersFigure3.24,thenwesketchapieceofgraphpaperwiththeproperscale, andplotthepointsFigure3.25onit.Oncethe I dsat V dsat pointshavebeendetermined,itiseasyto 7 Thiscontentisavailableonlineat.

PAGE 96

90 CHAPTER3.FETS sketchintheI-Vbehavior.Youjustdrawacurvefromtheoriginuptoanygivenpoint,havingit"peak out"justatthedot,andthendrawastraightlineat I dsat tonishthingso.Onesuchcurveisshownin Figure3.26.AndthennallyinFigure3.27theyareallsketchedin.Yourcurvesprobablywontbeexactly rightbuttheywillbegoodenoughforalotofapplications. V gs V dsat V V dsat mA 3 1 0.44 4 2 1.76 5 3 3.96 6 4 7.04 7 5 11 Figure3.24: Resultsofcalculating V dsat and I dsat Figure3.25: Plotting I dsat and V dsat Figure3.26: SketchinginoneoftheI-Vcurves.

PAGE 97

91 Figure3.27: Thecompletesetofcurves. Thereisaparticularlyeasywaytomeasureby k and V T foraMOSFET.Let'srstintroducethe schematicsymbolfortheMOSFET,itlookslikeFigure3.28.Let'stakeaMOSFETandhookitupas showninFigure3.29. Figure3.28: SchematicsymbolforaMOSFET

PAGE 98

92 CHAPTER3.FETS Figure3.29: Circuitfornding V T and k Sincethegateofthistransistorisconnectedtothedrain,thereisnodoubtthat V gs )]TJ/F11 9.9626 Tf 10.158 0 Td [(V ds islessthan V T .Infact,since V gs = V ds ,theirdierence,iszero.Thus,foranyvalueof V ds ,thistransistorisoperating initssaturatedcondition.Since V gs = V ds ,wecanrewriteapreviousequation.35derivedequationfrom thesectiononMOStransistorSection3.4as I d = k 2 V ds )]TJ/F11 9.9626 Tf 9.963 0 Td [(V T 2 .50 Nowlet'stakethesquarerootofbothsides: p I d = r k 2 V ds )]TJ/F11 9.9626 Tf 9.963 0 Td [(V T .51 SoifwemakeaplotFigure3.30of p I d asafunctionof V ds ,weshouldgetastraightline,withaslope of q k 2 andanx-interceptof V T .

PAGE 99

93 Figure3.30: Obtaining V T and k Becauseoftheexpectednon-ideality,thecurvedoesnotgoallthewayto V T ,butdeviatesabitnear thebottom.Asimplelinearextrapolationofthestraightpartoftheplothowever,yieldsanunambiguous valueforthethresholdvoltage V T 3.7Models 8 Asecond,andsomepeoplethinkmoreaccurate,waytond V T istolookatthecharacteristicsoftheMOS transistorinis linearregime .ThetestcircuitlookslikewhatyouseeinFigure3.31.Inthiscase, V ds is keptquitesmall.2Voltsorsoandthegatevoltage V gs issweptoversomerange.Ifyoulookbackat equationinanothermodule.37,whichwecanslightlyre-writeweseethat I d = s c ox WV ds L V gs )]TJ/F11 9.9626 Tf 9.962 0 Td [(V T .52 Thisequationwillobviouslygiveusalinearplotof I d asafunctionof V gs ,whichwilllooksomethinglike Figure3.32.Obviously,thisisadevicewithathresholdvoltageofabout2volts.Canyougureoutwhat k isforthistransistor?Ifnot,gobackare-readsomestu. 8 Thiscontentisavailableonlineat.

PAGE 100

94 CHAPTER3.FETS Figure3.31: Circuitfornding V T Figure3.32: I d asafunctionof V gs foraMOStransistorinitslinearrange. Nowlet'saddressafundamentalquestionconcerningallofthis:SoWhat?Whatdowehavehere? Oneansweristhatwehaveanotherdevicewhichinsomewaylookslikethebipolartransistorwestudied inthelastchapter.Inthesaturationregime,thedevicelooksandactslikeacurrentsource,andcould probablybeusedasanamplier.Itisprettyeasytomakeasmallsignalmodel.Thedrainactslikea currentsource,whichiscontrolledby V gs .Whatshouldwedoaboutthegateterminal?Thegatereally isnotconnectedtoanythinginsidethetransistor,soitlooksjustlikeanopencircuit.Infact,thereisa capacitance C gate = c ox A gate ,where A gate = WL ,theareaofthegate,butinmostlowfrequencylinear applications,thiscapacitanceisnotsignicant.ThusoursmallsignalmodelfortheMOSFET,ifitis operatinginitsaturationmode,isasseeninFigure3.33.

PAGE 101

95 Figure3.33: SmallsignalMOSFETmodel Thisseemstobeaprettygoodamplier.Ithasinniteinputimpedanceandhencewillnotloaddown thepreviousstageoftheamplierandithasanicebutnon-linearvoltagecontrolledcurrentsourcefor itsoutput.AgureFigure3.21inthesectiononMOSregimesshowsthatas V ds isincreased,thechannel length does ,infact,getabitshorter.Theincreased V ds makesthe pincho regionexpandabit,which, ofcourse,robsfromthechannelregion.Ashorterchannelmeansslightlylesschannelresistance,andso I d actually increasesabitwithincreasing V ds insteadofstayingconstant.Wesawfromthebipolartransistor, thatwhenthisoccurs,wemustaddaresistorinparallelwithourcurrentsource.Thus,let'scompletethe modelwithanadditional r o butinfact,wewillputitinwithadashedline,becauseexceptforveryshort channeldevices,ithasverylittleeectondeviceperformanceFigure3.34. Figure3.34: Addingan r o TheMOSFEThasseveraladvantagesoverthebipolartransistor.Oneofthemainones,asweshallsee, isthatitismucheasiertomake.Youonlyneedtwon-regionsinasinglep-typesubstrate.Itisbasicallya surfacedevice.Thismeansyoudonothavetopileupdierentlayersofnandptypematerialasyoudo withthebipolartransistor.Finally,weshallseethatavariationontheMOSFETtechnologyoersa huge advantageoverbipolardeviceswhenitcomestobuildinglogiccircuitswithalargenumberofgatesVLSI andULSIcircuits. Toseewhythisisso,wehavetodigressforjustalittlebit,anddiscusslogiccircuits.Section3.8

PAGE 102

96 CHAPTER3.FETS 3.8InvertersandLogic 9 Asyoualreadyknow,orwillndoutshortly,fromtakingaclassindigitallogic,logiccircuitsareprimarily baseduponacircuitcalledaninverter.Aninvertersimplytakesasignalandgivesyoutheoppositeone. Forinstance,ifahighvoltagea"one"isplacedontheinputofaninverter,itreturnsalowvoltagea "zero".Figure3.35isasimpleinverterbasedonaMOSFETtransistor: Figure3.35: Invertercircuit If V in iszero,theMOSFETisturnedo V gs is .

PAGE 103

97 theloadresistor, R d ,andthroughthetransistor.Thus,thecorrectvalueofcurrentandvoltageforthe circuitforanygivengatevoltageisthesimultaneoussolutionoftheloadlineequationandthetransistor behavior,which,ofcourse,isjusttheintersectionoftheloadlinewiththeappropriatecharacteristiccurve. Thusitisasimplematterofdrawingverticallinesdownfromeach V in curveor V gs valuedowntothe horizontalaxistondoutwhattheappropriate V dd oroutputvoltagewillbefortheinverter.Assuming that V in onlygoesupto5Volts,theresultingcurvethatwegetlooklikeFigure3.37.Thisisnotagreat transfercharacteristic. V in hastogetfairlylargebefore V out startstofall,andevenwiththefull5Volt input, V out isstillgreaterthan1Volt.Pickingatransistorwithasmall V T andabiggerloadresistorwould giveusabetterresponse,butatleastwiththisexampleyoucanseewhatisgoingon. Figure3.36: Characteristiccurveswithloadline

PAGE 104

98 CHAPTER3.FETS Figure3.37: Transfercharacteristicsfortheinvertercircuit. Basedonthissimpleinvertercircuit,wecanbuildcircuitswhichperformtheNORandNANDfunction. C out = : A + B .55 and C out = : AB .56 Itshould,bynow,beobvioustoyouhowthetwocircuitsinFigure3.38canperformtheNANDandNOR function.ItturnsoutthatwiththecapabilitytodoNANDandNOR,wecanbuildupanykindoflogic functionwedesire. Figure3.38: NANDandNORcircuits

PAGE 105

99 Let'slookattheinverteralittlemorecloselyFigure3.39.Usually,theloadfortheinverterwillbethe nextstageoflogicwhich,alongwiththeassociatedinterconnectwiring,wecanmodelasasimplecapacitor. Thevalueofthecapacitancewillvary,butitwillbeontheorderof 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(12 F. Figure3.39: Drivingacapacitiveload Whentheinputtotheinverterswitchesinstantaneouslytoalowvalue,currentwillstopowingthrough thetransistor,andinsteadwillstarttochargeuptheloadcapacitance.Theoutputvoltagewillfollowthe usual RC chargingcurvewithatimeconstantgivenjustbytheproductof R times C .If C is 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(13 F,then togetarisetimeof1nswewouldhavetomake R about 10 4 Asweshallseelater,itisvirtuallyimpossibletomakea10k resistorusingintegratedcircuittechniques. Remember: R = L A .57 Andthus,togetareallybigresistanceweneedeitheraverytinyAToohardtoachieveandcontrol., areallyBIGLTakesuptoomuchroomonthechiporahuge Again,veryhardtocontrolwhenyou gettotheverylowdopingdensitiesthatwouldberequired. Evenifwecouldndawaytobuildsuchbigintegratedcircuitresistors,therewouldstillbeaproblem. ThecurrentowingthroughtheresistorwhentheMOSFETisonwouldbeapproximately I = V R = 5 V 10 4 =5 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(4 A .58 Whichdoesn'tseemlikemuchcurrentuntilyouconsiderthataPentium microprocessorhasabout6 milliongatesinit.Thiswouldmeananetcurrentof )]TJ/F8 9.9626 Tf 7.749 0 Td [(300 Amps owingintotheCPUchip!We'vegotto comeupwithabettersolutionSection3.9. 3.9TransistorLoadsforInverters 10 ThereareotherkindsofMOSFET'sbesidestheonewehavestudiedsofar.Strictlyspeaking,whatwehave seenuptonowiscalledan n-channelenhancementmodeMOSFET .Itturnsoutthatyoucanbuild 10 Thiscontentisavailableonlineat.

PAGE 106

100 CHAPTER3.FETS aMOSFETwhichlooksjustlikeapreviousgureFigure3.14,exceptthatbyputtingsomeadditional impuritiesunderthegateregion,wecanarrangeitsothatthereisachannelformed,evenwith V g =0 .The transistornowhasa negative V T .Theprocessbywhichtheadditionalimpuritiesareaddediscalleda V T adjust AMOSFETwithanegative V T canbeexpectedtohave I d )]TJ/F11 9.9626 Tf 9.025 0 Td [(V ds curvessimilartothoseforapositive V T device,exceptforonething.For V gs =0 ,thedeviceisalreadyturnedon,andsowegetausualMOSFETtypecurve. Positive gatevoltageturnsitonevenmore,whilenegative V gs tendstoreducethedraincurrent. Ittakesa negative gatevoltagetoturnthethingo.Figure3.40showscomparativecharacteristiccurves foranenhancementanddepletionmodedevices. Figure3.40: Enhancementanddepletioncharacteristiccurves Foranenhancementmodetransistor,youhavetoget V g >V T -1Voltinthisexampleto enhance the conductivityorchanneltomakeitconduct.Foradepletionmodedevice,agatevoltageVgsof0,stillnds thedeviceconducting.Youhavetoputsomenegativevoltageonthegateto deplete thechannel,inorder toturnito.Wenowhavea depletionmoden-channelMOSFET Howwouldweuseadepletionmodedeviceinaninvertergate?Theanswerisfairlystraight-forward.In theschematicinFigure3.41,weindicateadepletionmodeMOSFETbyaddingasecondline,underthegate, tosuggestthatachannelalreadyexistsinthedevice,evenwithno V g .Notethatthegateofthedepletion modetransistoralsosometimescalledthe pullup transistorisconnectedtoitssource,so,infact, V gs doesequal0forthisdevice.Theinputtransistororthe pulldown transistorisjustanenhancement modeMOSFETlikewehadbefore.Itisnothardtochooseappropriate W and L sothat I dsat forthepull uptransistorisontheorderofthe500 Athatweneedtogetour1nsrisetimeonthecapacitiveload.

PAGE 107

101 Figure3.41: Depletionmodeload Inordertogetthetransfercharacteristicforthiscircuit,werstnotethat V sdd = V dd )]TJ/F11 9.9626 Tf 9.963 0 Td [(V sde .59 where V sde isthesource-drainvoltageforthepull-down,orenhancementtransistor,and V sdd ,isthesourcedrainvoltageforthedepletion-modetransistor.Ifwewanttoplotthe load-line forthepull-downtransistor thatiscreatedbythepull-upordepletionmodetransistor,weshouldtakeits V gs =0 characteristiccurve, shiftitoverbyanamount V dd ,andthenreverseitspolarity.Whenwedothiswegetthefollowingshown inFigure3.42.Notingtheintersectionpointsofthe loadline andthecharacteristiccurvesallowsus theopportunityfordrawingthetransfercharacteristicFigure3.43.Thisisabetterlookingcurve.It issymmetricaroundthemid-voltagepoint,andgetsclosertozeroforitsoutput"low"condition.The transitionfrom"high"to"low"isalsosomewhatmoreabrupt,whichisadvantageous.Canyougureout why? Figure3.42: CharacteristiccurveandloadlineforadepletionMOSload

PAGE 108

102 CHAPTER3.FETS Figure3.43: Transfercharacteristicsforadepletionloadinverter Well,wesolvedoneproblem.Atleastwehaveapullupstructurethatwecanmanufacture.Itturns outnottobetoohardtobuildanenhancementMOSFETthathasanequivalentresistanceintherangewe needwithouttakinguptoomuchchiparea.Wehavenotsolvedtheotherproblemhowever.Wearestill lookingata huge currentdrawforlargecircuits.Sinceonaverage,halfoftheinvertergateswillbe"on" inalogiccircuit,westillhavealargecurrentsinktoground.Thisissomethingthatwouldbecompletely prohibitiveinamodern-dayVLSIintegratedcircuit. Fortunately,wehavenotrunoutofoptionsforMOSstructuresSection3.10yet. 3.10CMOSLogic 11 Considerthefollowing,showninFigure3.44. Figure3.44: APMOStransistor ThislooksalotlikeourpreviousMOSFETexceptthatnowwehaveann-typesubstrateandthesource anddrainregionsarep-type.Ifweapplya negative V gs withthesourceconnectedtothen-typesubstrate 11 Thiscontentisavailableonlineat.

PAGE 109

103 thentheinducednegativechargeonthegatewilldriveawaytheelectrons,andifthebandsunderthegate arebentupsuciently,formaninversionlayerFigure3.45of holes thusmakinganenhancementmode p-channelMOSFET ,oraPMOStransistor.AsopposedtoanNMOStransistorwhichwestudiedrst.. NotethataPMOStransistorwillhaveanegative V T .Thatis,thegatevoltagehastobe lessthanthe source/substratevoltage inordertoturnthedeviceon.Themorenegative V gs ,themorecurrentwewill haveowingthroughthedevice. Figure3.45: Inversionofann-typelayer Itturnsoutthatacombinationofbothann-channelandap-channeldeviceonthesamecircuitcanbe veryadvantageous.Suchtechnologyiscalled CMOS ,for"complementaryMOS".Hereishowweusea p-channeltransistorintheinvertercircuit. Firstofall,however,wehavetoseehowwewouldmakeone.Thereisafundamentalproblemintrying tousebothn-channelandp-channeldevicesinthesamecircuit.Whatisit?Itwouldseemweneedtwo dierentkindsofsubstrates,bothap-typesubstrateforthen-channeltransistor,andann-typesubstrate forthep-channeldevice.Thereisawayaroundthisproblembymakingwhatiscalleda tank ora moat Amoatisarelativelydeepregionofonetypeofmaterialplacedintoahostsubstrateoftheoppositetype Figure3.46.Wecanputn-typesource/drainregionsintothep-substrateandp-typesource/drainregions intothen-moat.InFigure3.47,wewillalsoshowthegates,andhowthewholeinverterisconnected together. Figure3.46: PreparingforaCMOSinverter

PAGE 110

104 CHAPTER3.FETS Figure3.47: ACMOSinverter Nowlet'sdrawtheschematicFigure3.48:Ap-channeldeviceisdrawnjustlikeann-channeldevice, exceptweputalittle"bubble"onthegatetosignifythatitisaMOSFETofadierentcolor.Although weusuallydon'tdothisallthetime,wehavealsoshownthesubstrateconnectionsinthisdiagram.These connectionsshowthataMOSFETisatleastafourterminaldevice,notathreeterminaloneaspeople oftenassume.Since,inap-channeldevice,thesubstrateisn-type,weshowthesubstrateconnectionas anoutwardpointingarrow.Thep-typesubstrateforthen-channeldeviceisshownasaninwardpointing arrow.Then-channelsubstrateisconnectedtoground,thep-channelsubstrateisconnectedto V dd .Note thatsincethen-moatisat V dd andthep-substrateisatground,themoat-substratep-njunctionisreverse biased,andsonocurrentshouldowbetweenthem. Figure3.48: SchematicofaCMOSinverter Weusuallydonotlabelthesourceanddraineither,butwedohere,justforcompleteness.Notethat unlikethebipolartransistor,theFETistrulyasymmetricdevice.Thereisreallynowaytotellthesource fromthedrain.Byconvention,wecalltheelementwhichisconnectedtothesubstrateormoatthesource, andtheotherthedrain.Youwillsometimesheartheregionunderthegateeithersubstrateormoat referredtoasthe backbody Nowlet'sseehowthiscircuitworks.If V in ishighatornear V dd theNMOStransistorwillbeturned on.Thevoltagebetweenthegateandsubstrateofthep-channeldeviceisatornearzero.Thegateisat

PAGE 111

105 V dd andsoisthemoat!Hencetheuppertransistorwillbeturnedo.Theoutputwillthusbe low Iftheinputvoltageisatorneargrounda"low"thenthen-channeldeviceisturnedo.Thevoltage betweenthegateandsubstrateofthep-channeldeviceisnow h )]TJ/F11 9.9626 Tf 7.749 0 Td [(V dd .Thegateis h 0 andthesubstrate isat + V dd .IfthePMOStransistorhasathresholdvoltage V T of,say,-2V,thenitwillbeturned on and theoutputwillbe high .Notehowever,thatineitherstate,highorlow, thereisnostaticcurrentowing throughtheinverter Thetransfercharacteristicsforthiscircuit.Arealittlemorecomplicated.First,let'smakesurewehave ourvoltagesandcurrentsdenedFigure3.49.Fromthegure, V gs )]TJ/F10 6.9738 Tf 6.227 0 Td [(n then-channelgate-sourcevoltageis justVin. V gs )]TJ/F10 6.9738 Tf 6.227 0 Td [(p thegate-sourcevoltageforthep-channeldeviceis V in )]TJ/F11 9.9626 Tf 9.05 0 Td [(V dd I d )]TJ/F10 6.9738 Tf 6.227 0 Td [(n = I d )]TJ/F10 6.9738 Tf 6.227 0 Td [(p = I d V ds )]TJ/F10 6.9738 Tf 6.227 0 Td [(p thedrain sourcevoltageforthep-channeltransistorcanbewrittenas V ds )]TJ/F10 6.9738 Tf 6.227 0 Td [(n )]TJ/F11 9.9626 Tf 9.417 0 Td [(V dd .Wehavetwosetsofcharacteristic curvesFigure3.50:Notethatsince V gs )]TJ/F10 6.9738 Tf 6.226 0 Td [(p = V in )]TJ/F11 9.9626 Tf 9.345 0 Td [(V dd ,when V in =0 V, V gs )]TJ/F10 6.9738 Tf 6.227 0 Td [(p = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 Vandsothetransistor isstronglyturnedon. Figure3.49: Deningvoltages

PAGE 112

106 CHAPTER3.FETS Figure3.50: Draincurrentsforthetwotransistorasafunctionofinputvoltageand V ds Wehaveanumberofdierent"loadlines"inthiscase,becauseforeach V in wehaveadierentcurve forboththenandpchanneltransistors.ThisisshowninFigure3.51.Theblackspotsshowthepointof intersection.Followafewofthecurvesalongtoseeifyouagreewithwherethespotshavebeenplaced. Wehavealsoaddedapairofdottedcurvesfor V in =2 : 5 Vsowecangetthe"turn-over"point.Projecting thelocationoftheblackdotstothe V ds )]TJ/F10 6.9738 Tf 6.227 0 Td [(n or V out axiswillgivesusavaluefor V out foreachoftheinput voltages, V in .TheresultingcurveisshowninFigure3.52.Thisgivesusagood"feel"forhowtheinverter works,andhowtheoutputvarieswiththeinput.Notethatthistransfercurveisquitesymmetricabout2.5 volts,andgoesallthewayfrom+5to0voltsontheoutput. Figure3.51: Gettingthetransferfunction

PAGE 113

107 Figure3.52: CMOSinvertertransfercharacteristics 3.11JFET 12 Thereisalotmorethatwecoulddowitheldeectdevices,butitisprobablytimetomoveontonew topics.Foronenalpointhowever,wemightjustlookatsomethingcalledtheJFET,orjunctioneldeect transistor.TheJFETstructurelookslikeFigure3.53.Itconsistsofapieceofp-typesilicon,intowhichtwo n-typeregionshavebeendiused.However,insteadofbeingbothonthesamesurface,aswithaMOSFET, thetworegionsareoppositeoneanotheroneithersideofthecrystal.Incross-section,theJFETlookslike Figure3.54.Wealsoshowthebiasinghere. Figure3.53: JFET 12 Thiscontentisavailableonlineat.

PAGE 114

108 CHAPTER3.FETS Figure3.54: BiasingaJFET Thetwon-regionsareconnectedtogether,andarereversebiasedwithrespecttothep-typesubstrate. Asecondbattery, V ds isusedtopullcurrentoutofthesource,byapplyinganegativevoltagebetweenthe drainandthesource.Thereversebiasedn-pjunctionscreatesadepletionregionwhichextendsintothe p-typematerialthroughwhichtheholestravelastheygofromsourcetodrainachannel?.Byadjusting thevalueof V gs ,onecanmakethedepletionregionsmallerorlarger,thusincreasingordecreasingthedrain current. Theobservantstudentwillalsonotethatthepolarityofthe V ds batterymakesitsothatthereismore reversebiasacrossthep-njunctionsatthedrainendofthechannelthanatthesourceend.Thus,amore accuratedepictionoftheJFETwouldbewhatisshowninFigure3.55.Whenthedrain/sourcevoltage getslargeenough,thetwodepletionregionswilljointogether,and,justaswiththeMOSFET,thechannel pincheso,asshowninFigure3.56. Figure3.55: Depletionregioncontrolscurrent

PAGE 115

109 Figure3.56: Pinch-O Surprisingasitmayseem,whenyouworkouttheequationswhichdescribehowthedepletionregion extendswith V gs andhowthepinch-omechanismchanges I D ,youendupwithbehavior,andequations, whicharequitesimilartothoseofadepletion-modeMOSFET. UsingJFETsisalittlemorecumbersomethananormalMOSFET.Youmustmakesurethatthegatesubstratejunctionalwaysremainsreversebiased,andsincetheJFETcanonlybeadepletion-modedevice, youhavetohaveavoltageonthegateifyouwanttoturnthetransistoro.TheJFET does haveone advantageovertheMOSFEThowever.Awhilebackwecalculatedthevaluefor C ox theoxidecapacitance andfoundthatitwasontheorderof 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 F cm 2 .AtypicalMOSFETgatemightbe1 mlongby20 m wide,andsoitwouldhaveagateareaof20 m 2 or 2 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 cm 2 .Thus,thetotalgatecapacitanceisonly about 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(14 F 3.12ElectrostaticDischargeandLatch-Up 13 Asyouareprobablyaware,youhavetobeverycarefulwhenhandlingMOScircuits,tobesurethatyouare properlygrounded,andthatyoudonottransferanystaticelectricitytothechip.The standardhumanbodymodel assumesastaticchargetransferofabout0.1micro-Coulombs 10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(7 C uponstaticelectricity dischargebetweenahumanandachip.Thisdoesnotseemlikeenoughchargetodoanyharmuntilwe remembertheoldformula: Q = CV .60 or V = Q C .61 LasttimeIlooked 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 dividedby 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(14 isabout 10 7 volts!Addtothisthefactthatthegateoxide thicknessisonlyabout 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(6 cm ,sothatwehaveelectriceldsinthegateoxidewhichareontheorderof 10 13 V cm !Nowonderthethingsbreak.Thisproblemiscalled electrostaticdischarge ,orESD,andisone ofthemajorconcernsofICmanufacturers.ProtectingagainstESDisstillverymucha"blackart"andis somethingthatpeoplearestillstudyingquiteabit.JFET'saremuchmoreruggedstructures,andhave muchhighergatecapacitances,andarenotnearlysopronetoESDfailure. Sinceweareonthesubjectofproblems,letstakealookatonemore"glitch"thatplaguesICdesigners. WehavetogobacktotheCMOScircuit.Remember,themoat/substratejunctionisreversebiased,so wewillhaveanelectriceldinthedepletionregionofthatjunction,pointingasshowninFigure3.57. Suppose,somehow,wehaveoneormorestrayelectronsinthep-typesubstrate.Theywillbesweptacross thesubstrate/moatjunctionbytheelectriceld,andbeattractedtothemoatcontactby V dd .Let'sfocus 13 Thiscontentisavailableonlineat.

PAGE 116

110 CHAPTER3.FETS onwhathappensastheelectronowsoutthe V DD contactFigure3.58.Astheelectronmovesthrough theresistiven-typemoatmaterial,itdevelopsavoltagedropbetweenthen-typematerialunderthe source,andthe V DD contactWhichisalsoatthesourcepotentialsincetheyareconnectedtogetherbythe interconnectonthesurfaceofthewafer.Electronowinonedirectionmeanscurrentowintheotherand sothismakestheregionunderthesourceslightlynegativewithrespecttothesourceregionitself.This, ofcourse,forwardbiasesthesource/moatjunctionslightly,whichcausesaholeortwotobeinjectedinto themoatfromthep-sourceFigure3.59.Theholeswillbeattractedbytheeldacrossthemoat-substrate depletionlayer,and,oncetheygetthere,theywillbesweptintothep-substrateFigure3.60.Oncethe holesgetintothep-substrate,theywillbeattractedtothegroundconnectionsothattheycanleavethe semiconductor.Astheseholesowpastthen-source,andthroughtheresistivep-substrate,theybuildupa potentialbetweenthegroundcontactFigure3.61,andthematerialunderthesourcewithapolaritywhich tendstoforwardbiasthesource-substratejunction,andcauseelectronstobeinjectedintothesubstrate. Theelectrons,inturn,areattractedtotheeldacrossthesubstrate-moatjunctionFigure3.62.Some oftheelectronsmayrecombineinthep-region,butintoday'shigh-qualitysubstrates,thereareveryfew activerecombinationcenters,andsoeventhoughtheelectronsareminoritycarriers,theyhavequitealong minoritycarrierlifetime,andmostofthemmakeittothesubstrate-moatjunction.andaresweptintothe moat.Onceinsidethen-moat,theelectronsarethenattractedtothe+ V dd contact,where,ofcourse,they buildupabiggerforwardbiasacrossthesource-moatjunction,causingmoreholestobeemittedfromthe sourceintothemoatFigure3.63.Theseholesaresweptacrossthemoat-substratejunction,owtothe groundcontactand,well...yougettheidea!Itdoesnottakelongbeforewehaveadeadshortcircuit betweenVddandground.Thisisnothealthyforintegratedcircuitchipsintheleast,andisaprocesscalled latchup Figure3.64. Figure3.57: Thestartoftrouble!

PAGE 117

111 Figure3.58: Electronowbuildsupvoltage Figure3.59: Theforwardbiasedsourceinjectssomeholes Figure3.60: Theholesaresweptintothesubstrate

PAGE 118

112 CHAPTER3.FETS Figure3.61: Voltagedropatthen-channelsourceend. Figure3.62: Theelectronsaresweptintothemoat

PAGE 119

113 Figure3.63: Morecurrentmeansabiggervoltageandmoreholesinjected. Figure3.64: LatchUp! Thereisaninterestingcircuityoucandrawwhichshowswhatishappeningfromasomewhatdierent pointofview.Notethatwecanconsiderthep-source,n-moat,andp-substrateasapnpbipolartransistor. Alsothen-source,p-substrateandn-moatalsomakeanenpnbipolartransistor.Thetwotransistorsare intermingledhowever,withthebaseofthepnpandthecollectorsofthenpnsharingthesamen-moat,and thecollectorofthepnpandthebaseofthenpnsharingthep-substrate.Then-moatandp-substratesare bothcollectors and basesatthesametime.AlittlecarefulinspectionofthecrosssectionoftheCMOS inverterwillleadyoutothefollowingschematicshowninFigure3.65.Weneedsomethingtogetthiscircuit started,sosaywehavealittlecollectorcurrentcomingoutofthetoppnptransistor.Thiscurrentows down,throughtheresistortoground.Asitowsthroughtheresistoritbuildsupalittlevoltagewhich forwardbiasesthebase-emitterjunctionofthelower,npn,transistor,andcausessomecollectorcurrentto owintoit.Thiscurrentcomesfrom V dd throughtheupperresistor,andbuildsupavoltageacrossthat resistorwhichwillforwardbiasthebase-emitterjunctionofthetop,pnp,transistor.This,inturn,causes someadditionalcollectorcurrenttoowoutofthepnptransistor,andawaywego!Latch-upisbad,andis somethingwhichICdesignersworkveryhardtoavoid.

PAGE 120

114 CHAPTER3.FETS Figure3.65: Schematicoflatchupcircuit Youmightwonderwhat actually startsacircuitgoingintolatch-up.ReferbacktotheCMOSinverter Figure3.47,andnotethatthen-drainontheNMOSisconnectedtotheoutput.Theoutput could bea realoutput,goingbeyondthechipintothe"realworld".Ifthe"customer"whoisusingthechipiscareless, andsomehowdragstheoutputdownbelowground,thedrain/p-substratejunctionwillbeforwardbiased, electronswillbeinjectedintothep-substrate,andwearebackatFigure3.57.ICdesignerstrytokeep then-moat/ V dd contactasclosetothePMOSsource,andthep-substrate/groundcontactasclosetothe NMOSsourceastheycantoreducetheresistancebetweenthecontactandthesourceregions,andhence lowerthechanceofthecircuitgoingintolatch-up.

PAGE 121

Chapter4 ICManufacturing 4.1IntroductiontoICManufacturingTechnology 1 Itwouldprobablybeinterestingtospendalittletimeseeinghowintegratedcircuitsaremade.Thischapter willbelongondescription,andrathershortonequationsyay!.Thisisnottosaythatthereisnotalotof analyticalworkintheICfabricationprocess.It'sjustthatthingsget very complicatedinahurry,andso weprobablyarebetterojustlookingatmostprocessesfromaqualitativepointofview. Let'sstartoutbytakingalookatthestateoftheindustry,andremarkonafewtrends.Figure4.1isa plotofICsalesintheUnitedStatesoverthepast30years.Thismightnotbeabadeldtogetinto!Maybe therewillbeajobortwoouttherewhenyouarereadytograduate. Figure4.1: GrowthofICBusiness TherehasbeenasteadyshiftawayfrombipolartechnologytoMOSasisshowninFigure4.2.Currently, about90%ofthemarketiscomposedofMOSdevices,andonlyabout10%ofbipolar.Thisislikelyto bethecaseforsometimetocome.Thechangeinslope,whereMOSstartstakingoverfrombipolarata morerapidrateabout1987iswhenCMOStechnologyreallystartedtocomeintoheavyuse.Atthatpoint, bipolarTTLlogicessentiallyfadedtozero. 1 Thiscontentisavailableonlineat. 115

PAGE 122

116 CHAPTER4.ICMANUFACTURING Figure4.2: PercentageofBusiness Asyouprobablyareaware,deviceshavebeengettingsmallerandsmaller,andchipshavebeengettingbiggerandbiggerwithtime.AmostimpressiveplotFigure4.3isonewhichshowsthenumberof components/chipasafunctionoftime. Figure4.3: Numberoftransistors/chip Oneofthemaindriversforthishasbeenfeaturesize,whichshowsthesamenearlyexponentialbehavior ascomponents/chip.ThisisplottedinFigure4.4foryoureducation.Whatisinterestingtonoteabout thisisthatcertain"doomsayers"havebeenpredictinganabrupthalttothiscurveforsometimenow.It standstoreasonthatyoucannotimagesomethingwhichisnerthan ,thewavelengthofthelightyouuse toprojectitwith.However,bygoingtotheultraviolet,andusingavarietyofimageenhancingtechniques, lithographicengineerscontinuetobeabletomakenerandnerstructures.

PAGE 123

117 Figure4.4: Featuresizewithtime 4.2SiliconGrowth 2 HowisitpossiblefortheICindustrytocontinuetomakesuchgains,andhowdotheybuildsomanycircuits ononechipanyway?Inorderforustobeabletounderstandthis,wehavetotakealookatthe monolithic fabricationprocess Lith comesfromtheGreekwordforstone,and mono meansone,ofcourse.Thus, monolithicconstructionreferstobuildingthecircuitin"onestone"orinonesinglecrystalsubstrate. Inorderforustodothishowever,werstofallneedthe"stone",solet'sseewherethatcomesfrom. Westartoutwithanaturalformofsiliconwhichisveryabundantandrelativelypure;quartziteor SiO 2 sand.Infact,siliconisoneofthemostabundantelementsontheearth.Thisisreactedinafurnace withcarbonfromcokeand/orcoaltomakewhatisknownas metallurgicalgradesilicon MGSwhich isabout98%pure,viathereaction SiO 2 +2 C Si +2 CO .1 Wehaveseenthatontheorderof 10 14 impuritieswillmakemajorchangesintheelectricalbehaviorofa pieceofsilicon.Sincethereareabout 5 10 22 atoms/ cm 3 inasiliconcrystal,thismeansweneedapurity ofbetterthan1partin 10 8 or99.999999%purematerial.Thuswehavealongwaytogofromthepurity oftheMGSifwewanttomakeelectronicdevicesthatwecanuseinsilicon. Thesiliconiscrushedandreactedwith HCl gastomaketrichlorosilane,ahighvaporpressureliquid thatboilsat32 Casin: Si +3 HCl gas SiHCl 3 + H 2 .2 Manyoftheimpuritiesinthesiliconaluminum,iron,phosphorus,chromium,manganese,titanium, vanadiumandcarbonalsoreactwiththe HCl ,formingvariouschlorides.Oneofthenicethingsaboutthe halogensisthattheywillreactwithalmostanything.Eachofthesechlorideshavedierentboilingpoints, andso,byfractionaldistillation,itispossibletoseparateoutthe SiHCl 3 frommostoftheimpurities.The puretrichlorosilaneisthenreactedwithhydrogengasagainatanelevatedtemperaturetoformpure electronicgradesilicon EGS. SiHCl 3 + H 2 2 Si +3 HCl .3 AlthoughtheEGSisrelativelypure,itisinapolycrystallineformwhichisnotsuitablefordevicemanufacture.Thenextstepintheprocessistogrowsinglecrystalsiliconwhichisusuallydoneviathe Czochralski pronounced"cha-krawl-ski"methodtomakewhatissometimescalledCZsilicon.TheCzochralski 2 Thiscontentisavailableonlineat.

PAGE 124

118 CHAPTER4.ICMANUFACTURING processinvolvesmeltingtheEGSinacrucible,andtheninsertingaseedcrystalonarodcalledapuller whichisthenslowlyremovedfromthemelt.Ifthetemperaturegradientofthemeltisadjustedsothat themelting/freezingtemperatureisjustattheseed-meltinterface,acontinuoussinglecrystalrodofsilicon, calleda boule ,willgrowasthepulleriswithdrawn. Figure4.5isadiagramofhowtheCzochralskiprocessworks.Theentireapparatusmustbeenclosedin anargonatmospheretopreventoxygenfromgettingintothesilicon.Therodandthecruciblearerotated inoppositedirectionstominimizetheeectsofconvectioninthemelt.Thepull-rate,therotationrate andthetemperaturegradientmustallbecarefullyoptimizedforaparticularwaferdiameterandgrowth direction.The < 111 > directionalongadiagonalofthecubiclatticestructureisusuallychosenforwafers tobeusedforbipolardevices,whilethe < 100 > directionalongoneofthesidesofthecubeisfavoredfor MOSapplications.Currently,wafersaretypically6"or8"indiameter,although12"diameterwafers mmareloomingonthehorizon. Figure4.5: Czochralskicrystalgrowth Oncethebouleisgrown,itisgrounddowntoastandarddiametersothewaferscanbeusedinautomatic processingmachinesandslicedintowafers,muchlikeasalami.Thewafersareetchedandpolished,and moveontotheprocessline.Apointtonotehowever,isthatdueto"kerf"lossesthewidthofthesaw bladeaswellaspolishinglosses,morethanhalfofthecarefullygrown,verypure,singlecrystalsiliconis thrownawaybeforethecircuitfabricationprocessevenbegins! 4.3Doping 3 Startingwithaprepared,polishedwaferthenhowdowegetanintegratedcircuit?Wewillfocusonthe CMOSprocess,describedinthelastchapter.Let'sassumewehavewaferwhichwasdopedduringgrowthso thatithasabackgroundconcentrationofacceptorsiniti.e.itisp-type.ReferringbacktoCMOSLogic Figure3.46,youcanseethattherstthingweneedtobuildisan-tankormoat.Inordertodothis, weneedsomewayinwhichtointroduceadditionalimpuritiesintothesemiconductor.Thereareseveral waystodothis,butcurrenttechnologyreliesalmostexclusivelyonatechniquecalled ionimplantation Adiagramofanion-implanterisshowninthegureintheprevioussectionFigure4.5.Anionimplanter usesadopantsourcegas,ionizesit,anddrivestheionsintothewafer.Thedopantgasisionizedandthe resultantchargedionsareacceleratedthroughamagneticeld,wheretheyaremass-analyzed.Thevertical magneticeldcausesthebeamofionstospreadout,accordingtotheirmass.Athinapertureselectsthe 3 Thiscontentisavailableonlineat.

PAGE 125

119 ionsofinterest,andletsthempass,blockingalltheothers.Thismakessureweareonlyimplantingtheion wewant,andinfact,evenselectsfortheproperisotope!Theionizedatomsarethenacceleratedthrough severaltenstohundredsofkV,andthendeectedbyanelectriceld,muchlikeinanoscilloscopeCRT.In fact,mostofthetimetheionbeamis"rastered"acrossthesurfaceofthesiliconwafer.Theionsstrikethe siliconwaferandpassintoitsinterior.Ameasurementofthecurrentowinthesystemanditsintegral,is ameasureofhowmuchdopantwasdepositedintothewafer.Thisisusuallygivenintermsofthenumber ofdopant atoms cm 2 towhichthewaferhasbeenexposed. Aftertheatomsenterthesilicon,theyinteractwiththelattice,creatingdefects,andslowingdownuntil nallytheystop.Typicalatomicdistributions,asafunctionofimplantvoltageareshowinFigure4.6 forimplantationintoamorphoussilicon.Whenimplantationisdoneonsinglecrystalmaterial,channeling, theimprovedmobilityofaniondownthe"hallway"ofagivenlatticedirection,canskewtheimpurity distributionsignicantly.Justslightchangesoflessthanadegreecanmakebigdierencesinhowthe impurityatomsarenallydistributedinthewafer.Usually,theoperatoroftheimplantmachinepurposely tiltsthewaferafewdegreesonormaltothebeaminordertoarriveatmorereproducibleresults. Figure4.6: Implantdistributionwithaccelerationenergy Asyoumightexpect,shooting100kVionsatasiliconwaferprobablydoesquiteabitofdamagetothe crystalstructure.Notonlythat,butjusthaving,sayboron,inyourwaferdoesnotmeanyouaregoingto haveholes.Fortheborontobecome"electricallyactive"-thatistoactasanacceptor-ithastoresideon asiliconlatticesite.Eveniftheboronatomdoes,somehow,enduponanactuallatticesitewhenitstops crashingaroundinthewafer,themanydefectswhichhavebeencreatedwillactasdeeptraps.Thus,the holewhichisformedwillprobablybecaughtatatrapsiteandwillnotbeabletocontributetoelectrical conductivityinthewaferanyway.Howcanwexthissituation?Ifwecarefullyheatupthewafer,wecan causetheatomsinthecrystaltoshakearound,andifwedoitright,theyallgetbackwheretheybelong. Notonlythat,butthenewlyaddedimpuritiesenduponlatticesitesaswell!Thisstepiscalled annealing anditdoesjustwhatitissupposedto.Typicaltemperaturesandtimesforsuchanannealare500to1000 C for10to30minutes. Somethingelseoccursduringtheannealstephowever.Wehavejustadded,byourimplantationstep, impuritieswithafairlytightdistributionasshowninFigure4.6.Thereisanobviousgradientinimpurity distribution,andifthereisagradient,thanthingsmaystartmovingaroundbydiusion,especiallyat elevatedtemperatures.

PAGE 126

120 CHAPTER4.ICMANUFACTURING 4.4Fick'sFirstLaw 4 Wetalkedaboutdiusioninthecontextofdiodes,anddescribedFick'sFirstLawofDiusionforsome particleconcentration N x;t : Law4.1: Fick'sFirstLawofDiusion Flux = )]TJ/F11 9.9626 Tf 7.749 0 Td [(D d dx N x;t D isthe diusioncoecient andhasunitsofcm/sec. Inasemiconductor,impuritiesmoveabouteither interstitially ,whichmeanstheytravelaroundinbetweenthelatticesitesFigure4.7.Or,theymoveby substitutionaldiusion ,whichmeanstheyhop fromlatticesitetolatticesiteFigure4.8.Substitutionaldiusionisonlypossibleifthelatticehasa numberof vacancies ,oremptylatticesites,scatteredthroughoutthecrystal,sothatthereareplacesinto whichtheimpuritycanmove.Movinginterstitiallyrequiresenergytogetoverthepotentialbarrierofthe regionsbetweenthelatticesites.Energyisrequiredtoformthevacanciesforsubstitutionaldiusion.Thus, foreitherformofdiusion,thediusioncoecient D ,isastrongfunctionoftemperature. Figure4.7: Interstitialdiusion Figure4.8: Substitutionaldiusion Toaverygooddegreeofaccuracy,onecandescribethetemperaturedependenceofthediusioncoecient withan activationenergy E A ,suchthat: D T = D o e )]TJ/F8 9.9626 Tf 6.227 -0.747 Td [( E a kT .4 4 Thiscontentisavailableonlineat.

PAGE 127

121 Theactivationenergy E A andcoecient D o areobtainedfromaplotofthenaturallogof D vs. 1 kT calledan Arrheniusplot Figure4.9.Theslopegives E A andtheprojectiontoinnite T 1 T 0 gives lnD o Figure4.9: Arrheniusplotofdiusionconstant Thecontinuityequationholdsformotionofimpuritiesjustlikeitdoesforanythingelse,sothedivergence oftheux, divF mustequalthenegativeofthetimerateofchangeoftheconcentrationoftheimpurities, or,inonedimension: d dx Flux = )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( d dx N x;t .5 4.5Fick'sSecondLaw 5 Takingthederivativewithrespectto x ofFick'srstlaw d dx Flux = )]TJ/F1 9.9626 Tf 9.409 14.048 Td [( D @ 2 @x 2 N x;t .6 andthensubstitutingthecontinuityequationintoit,wehave Fick'ssecondlawofdiusion : @ @t N x;t = D @ 2 @x 2 N x;t .7 Thisisastandarddiusionequation,andonewhichshowsupoverandoveragainwhenoneisdealingwith suchphenomena. Inordertogetasolutiontothediusionequation,wemustrstassumesomeboundaryconditions.We willdealwithasemi-innitewafer,andassumethat lim x !1 N x;t =0 .8 Thisisareasonableassumption,sinceatmostourdiusionwillonlypenetrateamicronorsointothe wafer,andthewholewaferitselfisseveralhundredmicronsthick. Wealsohavetodecidesomethingaboutinitialconditions.Wewillmaketheassumptionthatwehave attime t =0 and x =0 somesurfaceconcentrationofimpuritieswhichwewillcall Q 0 impurities cm 2 .This 5 Thiscontentisavailableonlineat.

PAGE 128

122 CHAPTER4.ICMANUFACTURING isthesituationwewouldhaveifweintroducetheimpuritiesusingarelativelyshallowimplantstep.An alternativesurfaceboundaryconditionwouldbeonewheretheconcentrationofimpuritiesremainsatsome xedvalue.Thisiswhathappenswhenthereareimpuritiesinthegasowoverthewaferduringthetime thattheyareinthediusionoven.Thisiscalledan innitesourcediusion Therstconditioniscalleda limitedsourcediusion andthatiswhatweshallconsiderfurtherhere. Itisnottoohardtoshowthatwiththisinitialcondition,thesolutiontothediusionequationis: N x;t = Q 0 p Dt e )]TJ/F27 6.9738 Tf 6.226 7.681 Td [( x 2 4 Dt .9 Notethat N x;t isafunctionofdistanceintothewafer,andtime t .Thetimeis,ofcourse,thetime ofthediusionprocess. D ,thediusionconstant,dependsonthetemperatureatwhichthediusiontakes place.Figure4.10isaplotof D forthreeofthemostcommonlyuseddopantsinsilicon.Phosphorus andboronarethemostcommonacceptoranddonorrespectively.Arsenicissometimesusedbecauseitis signicantlybiggerindiameterthaneitherPorBandthus,movesaroundlessafteranimplant. Figure4.10: Diusionconstantasafunctionof1000/T Supposewedoarelativelyshallowimplantofboronintoourp-typewafer,anddeposita Q 0 of 5 10 13 phosphorus atoms cm 2 .Wethenperformanannealdiusionat1100 Cfor60minutes.At1100 C, D for phosphorusseemstobeabout 2 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(13 cm 2 sec .Wewillmakeaplotof N x forvarioustimes.Ifyoudo thisathome,besuretoputtimeinseconds,notminutes,hours,orfortnights.LookingatFigure4.10,is prettyeasytoseehowtheimpuritiesmoveintothesemiconductor,andhowtheconcentrationatthesurface, N ;t ,decreasesasmoreandmoreoftheimpuritiesmovesdeeperintothewafer. Exercise4.1 Solutiononp.141. Ifthesubstratehadbeendopedat 10 16 acceptors cm 3 wherewouldbethelocationofthep-njunction betweentheimplantedphosphoruslayer,andthebackgroundboron?

PAGE 129

123 Figure4.11: Diusiondistributionatdierenttimes 4.6Photolithography 6 Actually,implantsespeciallyformoatsareusuallydoneatasucientlyhighenergysothatthedopant phosphorusisalreadyprettyfarintothesubstrateoftenseveralmicronsorso,evenbeforethediusion starts.Theanneal/diusionmovestheimpuritiesintothewaferabitmore,andasweshallseealsomakes then-regiongrowlarger. "Then-region"!Wehavenotsaidathingabouthowwemakeourmoatinonlycertainareasofthe wafer.Fromthedescriptionwehavesofar,isseemswehavesimplybuiltann-typelayeroverthewhole surfaceofthewafer.Thiswouldbebad!Weneedtocomeupwithsomekindof"window"toonlypermit theimplantingimpuritiestoenterthesiliconwaferwherewewantthemandnotelsewhere.Wewilldothis byconstructinganimplantation"barrier". Todothis,therstthingwedoisgrowalayerofsilicondioxideovertheentiresurfaceofthewafer.We talkedaboutoxidegrowthwhenwewerediscussingMOSFETsbutlet'sgointoalittlemoredetail.Youcan growoxideineitheradryoxygenatmosphere,orinaanatmospherewhichcontainswatervapor,orsteam. InFigure4.12OxideThicknessasaFunctionofTime,weshowoxidethicknessasafunctionoftimefor growthwithsteam.Dry O 2 doesnotbehavetoomuchdierently,therateisjustsomewhatslower. OxideThicknessasaFunctionofTime Figure4.12 Ontopoftheoxide,wearenowgoingtodeposityetanothermaterial.Thisissiliconnitride, Si 3 N 4 orjustplain" nitride "asitisusuallycalled.Siliconnitrideisdepositedthroughamethodcalledchemical 6 Thiscontentisavailableonlineat.

PAGE 130

124 CHAPTER4.ICMANUFACTURING vapordepositionor"CVD".Theusualtechniqueistoreactdichlorosilaneandammoniainahotwalledlow pressurechemicalvapordepositionsystemLPCVD.Thereactionis: SiH 2 Cl 2 +10 NH 3 Si 3 N 4 +6 NH 4 Cl +6 H 2 .10 Siliconnitrideisagoodbarrierforimpurities,oxygenandotherthingswhichdonotwanttogetintothe wafer.TakealookatFigure4.13InitialWaferCongurationandseewhatwehavesofar.Awordabout scaleanddimensions.Thesiliconwaferisabout250 mthickabout0.01"sinceithastobestrongenough nottobreakasitisbeinghandled.Thetwodepositedlayersareeachabout1 mthick,sotheyshould actuallybedrawnaslinesthinnerthantheotherlinesinthegure.Thiswouldobviouslymakethewhole ideaofasketchridiculous,sowewillleavethingsdistortedastheyare,keepinginmindthatthedeposited anddiusedlayersareactually much thinnerthantherestofwafer,whichreallydoesnotdoanything exceptsupporttheactivecircuitsupontop.Therewegoagain,wastingsilicon.Goodthingit'scheapand plentiful! InitialWaferConguration Figure4.13 Nowwhatwewanttodoisremove part ofthenitride,sowecanmakeourn-well,butnotputin phosphorouswheredonotwantit.Wedothiswithaprocessescalled photolithography and etching respectively.Firstthingwedoiscoatthewaferwithyetanotherlayerofmaterial.Thisisaliquidcalled photoresist anditisappliedthroughaprocesscalled spin-coating .Thewaferisputonavacuumchuck, andalayerofliquidphotoresistissprayeduncapofthewafer.Thechuckisthenspunrapidly,gettingto severalthousandRPMinasmallfractionofasecond.Centrifugalforcecausestheresisttospreadout uniformlyacrossthewafersurfacemostofitinfactieso!.Thesolventforthephotoresistisquite volatileandsothelayerofphotoresistdrieswhilethewaferisstillspinning,resultinginathin,uniform coatingacrossthewaferFigure4.14PhotoresistisSpunOn. PhotoresistisSpunOn Figure4.14 Thename"photoresist"givessomeclueastowhatthisstuis.Basically,photoresistisapolymer mixedwithsomekindoflightsensitizingcompound.In positive photoresist,whereverlightstrikesit,the

PAGE 131

125 polymerisweakened,anditcanbemoreeasilyremovedwithasolventduringthe development process. Conversely, negative photoresistisstrengthenedwhenitisilluminatedwithlight,andismoreresistanttothe solventthanistheunilluminatedphotoresist.Positiveresistisso-calledbecausetheimageofthedeveloped photoresistonthewaferlooksjustlikethemaskthatwasusedtocreateit.Negativephotoresistmakesan imagewhichistheoppositeofwhatthemasklookslike. Wehavetocomeupwithsomewayofselectivelyilluminatingcertainportionsofthephotoresist.Anyone whohaseverseenaprojectorknowhowwecandothis.But,sincewewanttomake small things,notbig ones,wewillchangearoundourprojectorsothatitmakesasmallerimage,insteadofabiggerone.The instrumentthatprojectsthelightontothephotoresistonthewaferiscalleda projectionprinter ora stepper Figure4.15AStepperConguration. AStepperConguration Figure4.15 AsshowninFigure4.15AStepperConguration,thestepperconsistsofseveralparts.Thereisalight sourceusuallyamercuryvaporlamp,althoughultra-violetexcimerlasersarealsostartingtocomeinto use,acondenserlenstoimagethelightsourceonthe mask or reticle .Themaskcontainsanimageofthe pattern wearetryingtheplaceonthewafer.Theprojectionlensthenmakesareducedusually5ximage ofthemaskonthewafer.Becauseitwouldbefartoocostly,ifnotjustplainimpossible,toprojectonto thewholewaferallatonce,onlyasmallselectedareaisprintedatonetime.Thenthewaferis scanned or stepped intoanewposition,andtheimageisprintedagain.Ifpreviouspatternshavealreadybeenformed onthewafer,TVcameras,witharticialintelligencealgorithmsareusedto align thecurrentimagewith thepreviouslyformedfeatures.Thesteppermovesthewholesurfaceofthewaferunderthelens,untilthe waferiscompletelycoveredwiththedesiredpattern.Astepperisnotcheap.Usually,TIorIntelwillfork overseveralmilliondollarsforeachone!Itisoneofthemostimportantpiecesofequipmentinthewhole ICfabhowever,sinceitdetermineswhattheminimumfeaturesizeonthecircuitwillbe. Afterexposure,thephotoresistisplacedinasuitablesolvent,and"developed".Supposeforourexample thestructureshowninFigure4.16AfterPRExpose/Developiswhatresultsfromthephotolithographic step.

PAGE 132

126 CHAPTER4.ICMANUFACTURING AfterPRExpose/Develop Figure4.16 ThepatternthatwasusedinthephotolithographicPLstepexposedhalfofourareatolight,and sothephotoresistPRinthatregionwasremovedupondevelopment.Thewaferisnowimmersedina hydrouoricacidHFsolution.HFacidetchessiliconnitridequiterapidly,butdoesnotetchsilicondioxide nearlyasfast,soaftertheetchwehavewhatweseeinFigure4.17AfterNitrideEtch. AfterNitrideEtch Figure4.17 We now takeourwafer,putitintheionimplanterandsubjectittoa"blast"ofphosphorusions Figure4.18ImplantingPhospohrus. ImplantingPhospohrus Figure4.18 TheionsgorightthroughtheoxidelayerontheRHS,butstickintheresist/nitridelayerontheLHSof ourstructure.

PAGE 133

127 4.7IntegratedCircuitWellandGateCreation 7 Wethenremovetheremainingresist,andperformanactivation/anneal/diusionstep,alsosometimescalled the"drive-in".Thepurposeofthisstepistwofold.Wewanttomakethen-tankdeepenoughsothatwe canuseitforourp-channelMOS,andwewanttobuildupanimplantbarriersothatwecanimplantinto thep-substrateregiononly.Weintroduceoxygenintothereactorduringtheactivation,sothatwegrowa thickeroxideovertheregionwhereweimplantedthephosphorus.Thenitridelayeroverthep-substrateon theLHSprotectsthatareafromanyoxidegrowth.WethenendupwiththestructureshowninFigure4.19 AftertheAnneal/Drive-In. AftertheAnneal/Drive-In Figure4.19 Nowwestriptheremainingnitride.Sincetheonlywaywecanconvertfromptonistoaddadonor concentrationwhichis greater thanthebackgroundacceptorconcentration,wehadtokeepthedopingin thesubstratefairlylightinordertobeabletomakethen-tank.Thelightlydopedp-substratewouldhave toolowathresholdvoltageforgoodn-MOStransistoroperation,sowewilldoa V T adjustimplantthrough thethinoxideontheLHS,withthethickoxideontheRHSblockingtheboronfromgettingintothen-tank. ThisisshowninFigure4.20AdjustImplant,whereboronisimplantedintothep-typesubstrateonthe LHS,butisblockedbythethickoxideintheregionoverthen-well. AdjustImplant Figure4.20: V T adjustimplant Next,westripoalltheoxide,growanewthinlayerofoxide,andthenalayerofnitrideFigure4.21 StripOxide,NewNitride.Theoxidelayerisgrownonlybecauseitisbadtogrow Si 3 N 4 directlyontop ofsilicon,asthedierentcoecientsofthermalexpansionbetweenthetwomaterialscausesdamagetothe 7 Thiscontentisavailableonlineat.

PAGE 134

128 CHAPTER4.ICMANUFACTURING siliconcrystalstructure.Also,itturnsouttobenearlyimpossibletoremovenitrideifitisdepositeddirectly ontosilicon. StripOxide,NewNitride Figure4.21 Thenitrideispatternedcoveredwithphotoresist,exposed,developed,etched,andremovalofphotoresisttomaketwoareaswhicharecalled"active"Figure4.22NitrideAfterEtching.Thisiswherewe willbuildourtransistors.Thewaferisthensubjectedtoahigh-pressureoxidationstepwhichgrowsa thickoxidewhereverthenitridewasremoved.Thenitrideisagoodbarrierforoxygen,soessentiallyno oxidegrowsunderneathit.Thethickoxideisusedtoisolateindividualtransistors,andalsotomakeforan insulatinglayeroverwhichconductingpatternscanberun.Thethickoxideiscalled eldoxide orFOX forshortFigure4.23GrowingFieldOxide. NitrideAfterEtching Figure4.22 GrowingFieldOxide Figure4.23

PAGE 135

129 Then,thenitride,andsomeoftheoxideareetchedo.Theoxideisetchedenoughsothatallofthe oxideunderthenitrideregionsisremoved,whichwilltakealittleotheeldoxideaswell.Thisisbecause wenowwanttogrowthegateoxide,whichmustbeverycleanandpureFigure4.24ReadytoGrowGate Oxide.Theoxideunderthenitrideissometimescalled sacricial oxide,becauseitissacricedinthename ofultraperformance. ReadytoGrowGateOxide Figure4.24 Thenthegateoxideisgrown,andimmediatelythereafter,thewholewaferiscoveredwithpolysilicon Figure4.25PolyDepositionOverGateOxide. PolyDepositionOverGateOxide Figure4.25 Thepolysiliconisthenpatternedtoformthetworegionswhichwillbeourgates.Thewaferiscovered onceagainwithphotoresist.Theresistisremovedovertheregionthatwillbethen-channeldevice,butis leftcoveringthep-channeldevice.Alittleareaneartheedgeofthen-tankisalsouncoveredFigure4.26 PreparingforNMOSChannel/DrainImplant.Thiswillallowustoaddsomeadditionalphosphorusinto then-well,sothatwecanmakeacontactthere,sothatthen-wellcanbeconnectedto V dd .

PAGE 136

130 CHAPTER4.ICMANUFACTURING PreparingforNMOSChannel/DrainImplant Figure4.26 Backintotheimplanterwego,thistimeexposingthewafertophosphorus.Thepolygate,theFOX andthephotoresistallblockphosphorusfromgettingintothewafer,sowemaketwon-typeregionsinthe p-typesubstrate,andwehavemadeourn-channelMOSsource/drainregions.Wealsoaddphosphorousto the V dd contactregioninthen-wellsoasthemakesurewegetgoodcontactperformancethereFigure4.27 PhosphorusS/DImplant. PhosphorusS/DImplant Figure4.27 Notethattheformationofthesourceanddrainwereperformedwitha self-aligningtechnology .This meansthatweusedthegatestructureitselftodenewherethetwoinsideedgesofthesourceanddrain wouldbefortheMOSFET.Ifwehadmadethesource/drainregions before wedenedthegate,andthen triedtolinethegateuprightoverthespacebetweenthem,wemighthavegottensomethingthatlooks likewhatisshowninFigure4.28MisalignedGate.What'sgoingtobetheproblemwiththistransistor? Obviously,ifthegatedoesnotextendallthewaytoboththesource and thedrain,thenthechannelwill noteither,andthetransistorwillneverturnon!Wecouldtrymakingthegatewider,toensurethatit willoverlapbothactiveareas,evenifitisslightlymisaligned,butthenyougetalotofextraneousfringing capacitancewhichwillsignicantlyslowdownthespeedofoperationofthetransistorFigure4.29Wide Gate.Thisisbad!Thedevelopmentoftheself-alignedgatetechniquewasoneofthebigbreakthroughs whichhaspropelledusintotheVLSIandULSIera.

PAGE 137

131 MisalignedGate Figure4.28 WideGate Figure4.29 Wepullthewaferoutoftheimplanter,andstripothephotoresist.Thisissometimesdicult,because theactofionimplantationcan"bake"thephotoresistintoaverytoughlm.Sometimesanrfdischargein an O 2 atmosphereisusedto"ash"thephotoresist,andliterallyburnitothewafer!Wenowapplysome morePR,andthistimepatterntohavethemoatarea,andasubstratecontactexposed,foraboronp+ implant.ThisisshowninFigure4.30Boronp-ChannelS/DImplant. Boronp-ChannelS/DImplant Figure4.30 Wearealmostdone.Thenextthingwedoisremoveallthephotoresist,andgrowonemorelayerof oxide,whichcoverseverything,asshowninFigure4.31FinalOxideGrowth.Weputphotoresistoverthe wholewaferagain,andpatternforcontactholestogothroughtheoxide.Wewillputcontactsforthetwo drains,andforeachofthesources,makesurethattheholesarebigenoughtoalsoallowustoconnectthe sourcecontacttoeitherthep-substrateorthen-moatasisappropriateFigure4.32ContactHolesEtched.

PAGE 138

132 CHAPTER4.ICMANUFACTURING FinalOxideGrowth Figure4.31 ContactHolesEtched Figure4.32 4.8ApplyingMetal/Sputtering 8 Wenowputthewaferina sputterdepositionsystem .Inthesputtersystem,wecoattheentiresurfaceof thewaferwithaconductor.Analuminum-siliconalloyisusuallyused,althoughothermetalsareemployed aswell. AsputteringsystemisshownschematicallyinFigure4.33SputteringApparatus.Asputteringsystem isavacuumchamber,whichafteritispumpedout,isre-lledwithalow-pressureargongas.Ahighvoltage ionizesthegas,andcreateswhatisknownasthe Crookesdarkspace nearthecathode,whichinour case,consistsofametaltargetmadeoutofthemetalwewanttodeposit.Almostallofthepotentialofthe high-voltagesupplyappearsacrossthedarkspace.Theglowdischargeconsistsofargonionsandelectrons whichhavebeenstrippedoofthem.Sincethereareaboutequalnumberofionsandelectrons,thenet chargedensityisaboutzero,andhencebyGauss'law,soistheeld. 8 Thiscontentisavailableonlineat.

PAGE 139

133 SputteringApparatus Figure4.33 Theelectriceldacceleratestheargonatomswhichslamintothealuminumtarget.Thereisanexchange ofmomentum,andanaluminumatomisejectedfromthetargetFigure4.34SputteringMechanismand headstothesiliconwafer,whereitsticks,andbuildsupametallmFigure4.35WaferCoatedwithMetal. SputteringMechanism Figure4.34

PAGE 140

134 CHAPTER4.ICMANUFACTURING WaferCoatedwithMetal Figure4.35 IfyoulookatFigure4.35WaferCoatedwithMetal,youwillnotethatwehaveseeminglydone somethingprettystupid.WehavewiredalloftheelementsofourCMOSinvertertogether!Ah,butallis notlost.Wecandoonemorephotolithographicstep,andpatternandetchthealuminum,soweonlyhave itwhereweneedit.ThisisshowninFigure4.36AfterInterconnectPatterning. AfterInterconnectPatterning Figure4.36 4.9IntegratedCircuitManufacturing:Bird'sEyeView 9 Itwillnodoubtbehelpfulifwealsotakeaplaneor"bird'seye"viewofwhatthiscircuitlookslikeaswell. Thereare,infact,someinterestingthingswecangainbylookingatsomeofthem. Wehavebeenlookingatthedevelopmentofthecircuitfromacross-sectionalpointofview,watching theformationofthevariouslevelswhichmakeupthenishedCMOSinverter.Thisis,infact,nottheway acircuitdesignerlooksatthings.Acircuitdesignerseesthingsfromabove,andonlyworriesaboutthe placementoftransistors,andhowtheywillbeconnectedtogether.Infact,theonlyfactorintheactual designofthelayoutengineerhasanychoiceonisthetransistorwidth,W.Allotherparametersaredecided uponbeforehandbytheprocessengineer.Sowhatdoesthelayoutengineersee?Westartwiththenimplanttomakethen-tank,asshowninFigure4.37Implantedn-Tank.Youshouldgobackandfollow alongwiththecross-sectionalviewsoftheprocess,aswereviewlookingatthingsfromthetop. 9 Thiscontentisavailableonlineat.

PAGE 141

135 Implantedn-Tank Figure4.37 Amaskoppositetothatofthen-tankallowsustoann-channel V T adjust.Wenextdepositandpattern thenitridefortheactiveregions,andgrowtheeldoxideFOXFigure4.38GrowingFOX. GrowingFOX Figure4.38 Weremovethenitride,anddepositandpatternthepoly.,asseeninFigure4.39GatePolyPattern GatePolyPattern Figure4.39 Figure4.40S/DImplantsshowswhatthetwomaskslooklikeforthen+andp+source/drainimplants:

PAGE 142

136 CHAPTER4.ICMANUFACTURING S/DImplants Figure4.40 Notethatthegatepolyextendsbeyondwheretheimplantisbeingperformedinsidethedottedline. Thisisa designrule whichisthewaythecircuitdesignertakesintoaccountthefactthatthemanufacturing processmusthavesometolerancebuiltin,becausethingswillnotalwaysbelinedupjustperfectly.Now wemakesomecontactholes,seeninFigure4.41EtchingContactHoles: EtchingContactHoles Figure4.41 Andnally,wesputterandpatternthemetallization,whichisdepictedinFigure4.42Metallization Patterning.YoushouldgobacktoMOSFETsSection3.1,andconvinceyourselfthatthecircuitshownin Figure4.37Implantedn-TankisindeedwhathasbeenconstructedinFigure4.42MetallizationPatterning.Seeifyoucanidentifyallofthecorrectparts.Notethatthereisaconnectionbetween V ss ground andthep-substrate very closetothen-channelsource.Thereisalsoacontactbetweenthen-moatand V dd whichis very closetothep-channelsource.Whatadvantagewouldthishave?Hint:reviewthediscussion oflatch-upSection3.12.

PAGE 143

137 MetallizationPatterning Figure4.42 4.10DiusedResistor 10 Sometimes,inacircuitdesign,wewillneedaresistor.Thisisusuallymadeeitherwithpolyorwitha diusionFigure4.43:ADiusedResistor.Ifwetookourn-tankorsimilarn-typediusion,wecould makealongnarrowstripofit,anduseitasaresistor.Aslongaswekeepthesubstrateatground,andany voltagesontheresistorgreaterthanground,then-pjunctionwillbereversebiasedandtheresistorwillbe isolatedfromthesubstrate.Nowweallknow R = L A = L nqtW .11 ADiusedResistor Figure4.43 Theonlytroubleis,whatisnforadiusedresistor?AquicklookatthechartFigure4.11showing carrierconcentrationasafunctionofdepthafteradiusionshowsthatwhenwedoadiusion, n isnota constant,butvariesaswegodownintothewafer.Wewillhavetodosomekindofintegral,assuminglots ofparallel,thinresistors,eachwithadierentcarrierconcentration!Thisisnotverysatisfactory. Infact,itissounsatisfactorythatICengineershavecomeupwithabetterdescriptionresistancethan oneinvolving n and .Notethatwecouldwrite.11as R = 1 nqt L W .12 10 Thiscontentisavailableonlineat.

PAGE 144

138 CHAPTER4.ICMANUFACTURING Wedenetherstfractionwhichcontainsthecarrierconcentration,thicknessetc.asthe sheetresistance R s ofthediusion.Whilethiscanbemore-or-lesspredicted,itisusuallyalsoapost-fabricationmeasured value. R s 1 nqt .13 R s hasunitsof"ohms/square",andyouareprobablytemptedtoask"persquarewhat?".Wellitcanbe anysquareatall,cm, m,km,sinceallwereallyneedtoknowis R s andthelengthtowidthratioofthe resistorstructuretondtheresistanceofaresistor.Wedonotneedtoknowwhatunitsareusedtomeasure thelengthandthewidth,solongastheyarethesameforboth.ForinstanceiftheresistorinFigure4.43 ADiusedResistorhasasheetresistivityof50 /square,thenbyblockingtheresistorointosquares W x W indimension,weseethattheresistoris7squareslongFigure4.44CountingtheSquaresandso itsresistanceisgivenas: R =50 square 7 squares =350 .14 CountingtheSquares Figure4.44 4.11Yield 11 Perhapsawordaboutfeaturesize,chipsizeandyieldwouldbeinorder.Wesawearlierthatcircuits arerepeatedmanytimesacrossawafer'ssurfaceduringthephotolithographicstage.Althoughgreatcare isexercisedintryingtopreventdefectsfrombecomingpartofawafersurfacecleanrooms,"bunny" suits,ultra-purechemicalsetc.eachwaferthatgoesthroughafabwillendupwith some "killer"defects distributedacrossthewafersurfaceFigure4.45AWaferwithDefects. 11 Thiscontentisavailableonlineat.

PAGE 145

139 AWaferwithDefects Figure4.45 Imaginethatwetrytomanufacturesomechipsofacertainsize.AglanceatFigure4.46SixKilled Circuitsshowsthatwewouldhave15of21goodchips,forayieldofabout71%.Supposewecould,through improvedtechnology,performa30%"shrink"onthecircuiti.e. makeitsdimensions30%smaller.Now, asFigure4.47LotsMoreGoodOnesshows,weget40goodchips/waferinsteadof15andtheycostno moretoproduceandouryieldhasgoneto40outof46or87%.Wewillberich!Oratleastwewon'tgo outofbusiness! SixKilledCircuits Figure4.46 LotsMoreGoodOnes Figure4.47 Yield,reliabilityandmanufacturabilityareallcriticalissuesinthesemiconductorindustry.Thebusiness

PAGE 146

140 CHAPTER4.ICMANUFACTURING ishighlycompetitive,andthetechnologykeepsmovingrapidly.Itisanexcitingandchallengingeld,one whichdemandstheverybest,butwhichrewardssomeonewhoiswillingtoneverstopthinkingandtobring forththeverybestcreativesolutionstohardproblems.

PAGE 147

141 SolutionstoExercisesinChapter4 SolutiontoExercise4.1p.122 About1.2 mafter1hourofdiusiontime.Youknowthisbecausefor x< 1 : 2 m thephosphorus concentrationisgreaterthanthatofboron,andsothematerialisn-type.For x> 1 : 2 m ,theboron concentrationexceedsthatofthephosphorous,andsothematerialisnowp-type.

PAGE 148

142 CHAPTER4.ICMANUFACTURING

PAGE 149

Chapter5 IntroductiontoTransmissionLines 5.1IntroductiontoTransmissionLines:DistributedParameters 1 Havinglearnedsomethingabouthowwegeneratesignalswithbipolarandeldeecttransistors,wenow turnourattentiontotheproblemofgettingthosesignalsfromoneplacetothenext.EversinceSamuel Morseandthefounderof myalmamater ,EzraCornelldemonstratedtherstworkingtelegraph,engineers andscientistshavebeenworkingontheproblemofdescribingandpredictinghowelectricalsignalsbehave astheytraveldownspecicstructurescalled transmissionlines Anyelectricalstructurewhichcarriesasignalfromonepointtoanothercanbeconsideredatransmission line.Beitalong-haulcoaxialcableusedintheInternet,atwistedpairinabuildingaspartofalocal-area network,acableconnectingaPCtoaprinter,abuslayoutonamotherboard,orametallizationlayeron aintegratedcircuit,thefundamentalbehaviorofallofthesestructuresaredescribedbythesamebasic equations.Ascomputerswitchingspeedsrunintothe100sofMHz,intotheGHzrange,considerationsof transmissionlinebehaviorareevermorecritical,andbecomeamoredominantforceintheperformance limitationsofanysystem. Forourinitialpurposes,wewillintroducea"generic"transmissionlineFigure5.1"Generic"TransmissionLine,whichwillincorporatemostbutnotallfeaturesofrealtransmissionlines.Wewillthenmake someratherbroadsimplications,which,whilerenderingourresultslessapplicabletoreal-lifesituations, nevertheless greatly simplifythesolutions,andleadustoinsightsthatwecanindeedapplytoabroadrange ofsituations. "Generic"TransmissionLine Figure5.1 Thegenericlineconsistsoftwoconductors.Wewillsupposeapotentialdierence V x existsbetween thetwoconductors,andthatacurrent I x owsdownoneconductor,andreturnsviatheother.Forthe 1 Thiscontentisavailableonlineat. 143

PAGE 150

144 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES timebeing,wewillletthetransmissionlinebe"semi-innite",whichmeanswehaveaccesstothelineat somepoint x ,butthelinethenextendsoutinthe + x directiontoinnity.Suchlinesareabitdicultto handleinthelab! Inordertobeabletodescribehow V x and I x behaveonthisline,wehavetomakesomekindof model oftheelectricalcharacteristicsofthelineitself.Wecannotjustmakeupanymodelwewanthowever; wehavetobasethemodelonphysicalrealities. Let'sstartoutbyjustconsideringoneoftheconductorsandthephysicaleectsofcurrentowingthough thatconductor.Weknowfromfreshmanphysicsthatacurrentowinginawiregivesrisetoamagnetic eld, H Figure5.2BuildUpofMagneticField.Multiply H by andyouget B ,themagneticux density,andthenintegrate B overaplaneparalleltothewiresandyouget ,themagneticux"linking" thecircuit.ThisisshowninFigure5.3FindtheFluxLinkageforatleastpartofthesurface.Thedenition of L ,theinductanceofacircuitelement,isjust L I .1 where istheuxlinkingthecircuitelement,and I isthecurrentowingthroughit.Ouronlyproblem innding isthatthelongerasectionofwirewetake,themore wehaveforthesame I .Thus,wewill introducetheconceptofadistributedparameter. Denition1:distributedparameter Adistributedparameterisaparameterwhichisspreadthroughoutastructureandisnotconned toalumpedelementsuchasacoilofwire. Example Forinstance,wewillherebydene L asthe distributedinductance forthetransmissionline. IthasunitsofHenrys/meter.Ifwehavealengthoftransmissionline x 0 meterslong,andifthat linehasadistributedinductanceof L H/m,thentheinductance L ofthatlengthoflineisjust L = L x 0 BuildUpofMagneticField Figure5.2 Likewise,ifwehavetwoconductorsseparatedbysomedistance,andifthereisapotentialdierence V betweentheconductors,thentheremustbesomecharge Q onthetwoconductorswhichgivesrisetothat potentialdierence.Wecanimaginealinearchargedistributiononthetransmissionline, C/m,wherewe have + Coulombs/mononeconductor,and )]TJ/F11 9.9626 Tf 7.749 0 Td [( Coulombs/montheotherconductor.Foralineoflength x 0 ,wewouldhave Q = x 0 oneachsectionofwire.Wheneveryouhavetwochargedconductorswith avoltagedierencebetweenthem,youcandescribetheratioofthechargetothevoltageasacapacitance.

PAGE 151

145 Thetwoconductorswouldhaveacapacitance C = Q V = x 0 V .2 andadistributedcapacitance C F/mwhichisjust V .Alengthofline x 0 longwouldhaveacapacitance C = C x 0 FaradsassociatedwithitFigure5.4LineCapacitance. FindtheFluxLinkage Figure5.3 LineCapacitance Figure5.4 Thus,weseethatthetransmissionlinehasbothadistributedinductance L andadistributedcapacitance C whicharetiedupwitheachother.Thereisreallynowayinwhichwecanseparateonefromtheother. Inotherwords,wecannothaveonlythecapacitance,oronlytheinductance,therewillalwaysbesomeof eachassociatedwitheachsectionoflinenowmatterhowsmallorhowbigwemakeit. Wearenowreadytobuildourmodel.Whatwewanttodoistocomeupwithsomearrangementof inductorsandcapacitorswhichwillrepresentelectrically,thepropertiesofthedistributedcapacitanceand inductancewediscussedabove.Asalengthoflinegetslonger,itscapacitanceincreases,sowehadbetter putthedistributedcapacitancesinparallelwithoneanother,sincethatisthewaycapacitorsaddup.Also, asthelinegetslonger,itstotalinductanceincreases,sowehadbetterputthedistributedinductancesin serieswithoneanother,forthatisthewayinductancesaddup.Figure5.5DistributedParameterModel isarepresentationofthedistributedinductanceandcapacitanceofthegenerictransmissionline.

PAGE 152

146 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES DistributedParameterModel Figure5.5 Webreakthelineupintosections x long,eachonewithaninductance L x andacapacitance C x .If wehalve x ,wewouldhalvetheinductanceandcapacitanceofeachsection,butwe'dhavetwiceasmanyof themperunitlength.Duh!Thepointisnomatterhownewemake C x ,westillhaveLsandCsarranged likeweseeinFigure5.5DistributedParameterModel,withthetwokindsofcomponentsintermixed. We could makeamorerealisticmodelandrealizethatallrealwireshaveseriesresistanceassociatedwith themandthatwhateverweusetokeepthetwoconductorsseparatedwillhavesomeleakageconductance associatedit.Toaccountforthiswewouldintroduceaseriesresistance R ohms/unitlengthandaseries conductance G ohms/unitlength.OnesectionofourlinemodelthenlookslikeFigure5.6Complete DistributedModel. CompleteDistributedModel Figure5.6 Althoughthis is amorerealisticmodel,itleadstomuchmorecomplicatedmath.Wewillstartout anyway,ignoringtheseriesresistance R andtheshuntconductance G .This"approximation"turnsoutto beprettygoodaslongaseitherthelineisnottoolong,orthefrequenciesofthesignalswearesendingdown thelinedonotgettoohigh.Withouttheseriesresistanceorparallelconductancewehavewhatiscalledan ideal losslesstransmissionline 5.2Telegrapher'sEquations 2 Let'slookatjustonelittlesectionoftheline,anddenesomevoltagesandcurrentsFigure5.7Applying Kircho'sLaws. 2 Thiscontentisavailableonlineat.

PAGE 153

147 ApplyingKircho'sLaws Figure5.7 Forthesectionofline x long,thevoltageatitsinputisjust V x;t andthevoltageattheoutput is V x + x;t .Likewise,wehaveacurrent I x;t enteringthesection,andanothercurrent I x + x;t leavingthesectionofline.Notethatboththevoltageandthecurrentarefunctionsof time aswellas position. Thevoltagedropacrosstheinductorisjust: V L = L x @ @t I x;t .3 Likewise,thecurrentowingdownthroughthecapacitoris I C = C x @ @t V x + x;t .4 NowwedoaKVL 3 aroundtheoutsideofthesectionoflineandweget V x;t )]TJ/F11 9.9626 Tf 9.963 0 Td [(V L )]TJ/F11 9.9626 Tf 9.962 0 Td [(V x + x;t =0 .5 Substituting.3for V L andtakingitovertotheRHSwehave V x;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(V x + x;t = L x @ @t I x;t .6 Let'smultiplyby-1,andbringthe x overtothelefthandside. V x + x;t )]TJ/F11 9.9626 Tf 9.963 0 Td [(V x;t x = )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( L @ @t I x;t .7 Wetakethelimitas x 0 andtheLHSbecomesaderivative: @ @x V x;t = )]TJ/F1 9.9626 Tf 9.41 14.047 Td [( L @ @t I x;t .8 NowwecandoaKCL 4 atthenodewheretheinductorandcapacitorcometogether. I x;t )]TJ/F81 9.9626 Tf 9.963 0 Td [(C x @ @t V x + x;t )]TJ/F11 9.9626 Tf 9.963 0 Td [(I x + x;t =0 .9 3 "ElectricCircuitsandInterconnectionLaws":SectionKircho'sVoltageLawKVL 4 "Kircho'sLaws":SectionKircho'sCurrentLaw

PAGE 154

148 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES Anduponrearrangement: I x + x;t )]TJ/F11 9.9626 Tf 9.962 0 Td [(I x;t x = )]TJ/F1 9.9626 Tf 9.409 14.048 Td [( C @ @t V x + x;t .10 Nowwhenwelet x 0 ,thelefthandsideagainbecomesaderivative,andontherighthandside, V x + x;t V x;t ,sowehave: @ @x I x;t = )]TJ/F1 9.9626 Tf 9.409 14.048 Td [( C @ @t V x;t .11 .8and5.11aresoimportantwewillwritethemoutagaintogether: @ @x V x;t = )]TJ/F1 9.9626 Tf 9.41 14.048 Td [( L @ @t I x;t .12 @ @x I x;t = )]TJ/F1 9.9626 Tf 9.409 14.048 Td [( C @ @t V x;t .13 Thesearecalledthe telegrapher'sequations andtheyareallwereallyneedtoderivehowelectrical signalsbehaveastheymovealongontransmissionlines.Notewhattheysay.Therstonesaysthatat somepoint x alongtheline,theincrementalvoltagedropthatweexperienceaswemovedownthelineis justthedistributedinductance L timesthetimederivativeofthecurrentowinginthelineatthatpoint. Thesecondequationsimplytellsusthatthelossofcurrentaswegodownthelineisproportionaltothe distributedcapacitance C timesthetimerateofchangeofthevoltageontheline.Asyoushouldbeeasily aware,whatwehavehereareapairof coupledlineardierentialequationsintimeandposition for V x;t and I x;t 5.3TransmissionLineEquation 5 Weneedtosolvethe telegrapher'sequations @ @x V x;t = )]TJ/F1 9.9626 Tf 9.41 14.048 Td [( L @ @t I x;t .14 @ @x I x;t = )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( C @ @t V x;t .15 Thewaywewillproceedtoasolution,andthewayyoualwaysproceedwhenconfrontedwithapair ofequationssuchasthese,istotakeaspatialderivativeofoneequation,andthensubstitutethesecond equationinforthespatialderivativeintherstandyouendupwith...well,let'stryitandsee. Takingaderivativewithrespectto x of.14 @ 2 @x 2 V x;t = )]TJ/F1 9.9626 Tf 9.409 14.048 Td [( L @ 2 @t@x I x;t .16 Nowwesubstituteinfor @ @x I x;t from.15 @ 2 @x 2 V x;t = LC @ 2 @t 2 V x;t .17 Itshouldbe very easyforyoutoderive @ 2 @x 2 I x;t = LC @ 2 @t 2 I x;t .18 5 Thiscontentisavailableonlineat.

PAGE 155

149 Oh,Iknowyouall love dierentialequations!Well,let'stakealookattheseandjust think foraminute. For either V x;t or I x;t ,weneedtondafunctionthathassomeratherstringentrequirements.First ofall,thefunctionmustbeoftheformsuchthatnomatterwhetherwetakeitssecondderivativeinspace x orintime t ,itmustendupdieringinthewayitbehavesin x or t bynomorethanjustaconstant LC Infact,wecanbemorespecicthanthat.First V x;t musthavethesamefunctionalformfor both its x and t variation.Atmost,thetwoderivativesmustdieronlybyaconstant.Let'strya"lucky"guess andlet: V x;t = V 0 f x )]TJ/F11 9.9626 Tf 9.962 0 Td [(vt .19 where V 0 istheamplitudeofthevoltage,and f issomefunction,ofaformyetundetermined.Well @ @t f x )]TJ/F11 9.9626 Tf 9.963 0 Td [(vt = )]TJ/F8 9.9626 Tf 9.41 0 Td [( vf 0 .20 and @ 2 @t 2 f x )]TJ/F11 9.9626 Tf 9.963 0 Td [(vt = v 2 f 00 .21 Notealso,that @ 2 @x 2 f x )]TJ/F11 9.9626 Tf 9.963 0 Td [(vt = f 00 .22 Now,let'stake.19,.21,and.22andsubstitutetheminto.17: V 0 f 00 = LC V 0 v 2 f 00 .23 Our"lucky"guessworksasasolutionaslongas v = 1 p LC .24 So,whatisthis f x )]TJ/F11 9.9626 Tf 9.963 0 Td [(vt ?Wedon'tknowyetwhatitsactualfunctionalformwillbe,butsupposeatsome pointintime, t 1 ,thefunctionlookslikeFigure5.8fxAtSomePointInTime. fxAtSomePointInTime Figure5.8: f x attime t 1 Atpoint x 1 ,thefunctiontakesonthevalue V 1 .Now,let'sadvancetotime t 2 .Welookatthefunction andweseeFigure5.9fxAtaLaterPointInTime.

PAGE 156

150 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES fxAtaLaterPointInTime Figure5.9: f x atalater t 2 If t increasesfrom t 1 to t 2 then x willhavetoincreasefrom x 1 to x 2 inorderfortheargumentof f to havethesamevalue, V 1 .Thuswend x 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(vt 1 = x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(vt 2 .25 whichcanbere-writtenas x 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 t 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(t 1 = x t v p = 1 p LC .26 where v p isthevelocitywithwhichthefunctionismovingalongthex-axis!Weusethesubscript"p"to indicatedthatwhatwehavehereiswhatiscalledthe phasevelocity .Wewillencounteranothervelocity calledthe groupvelocity alittlelaterinthecourse. Ifwehad"guessed"an f x + vt forourfunction,itshouldbeprettyeasytoseethatthiswouldhave givenusasignalmovinginthe minus x direction,insteadoftheplus x direction.Thusweshalldenote V plus = V + f x )]TJ/F8 9.9626 Tf 20.402 6.74 Td [(1 p LC t .27 the positive goingvoltagefunctionand V minus = V )]TJ/F11 9.9626 Tf 6.725 -4.113 Td [(f x + 1 p LC t .28 whichisthenegativegoingvoltagefunction.Noticethatsincewearetakingthe second derivativeof f withrespectto t ,wearefreetochooseeithera + 1 p LC ora )]TJ/F1 9.9626 Tf 9.41 11.059 Td [( 1 p LC infrontofthetimeargumentinside f Alsonotethattheseareour only choicesforasolution.AsweknowfromDierentialEquations,asecond orderequationhas,atmost,twoindependentsolutions. Since I x;t hasthe same dierentialequationdescribingitsbehavior,thesolutionsfor I mustalsobe oftheexactsameform.Thuswecanlet I plus = I + f x )]TJ/F8 9.9626 Tf 20.401 6.74 Td [(1 p LC t .29 representthecurrentfunctionwhichgoesinthepositive x direction,and I minus = I )]TJ/F11 9.9626 Tf 6.725 -4.113 Td [(f x + 1 p LC t .30 representthenegativegoingcurrentfunction.

PAGE 157

151 Now,let'stake.29and5.27andsubstitutetheminto.14: V + p LC f x )]TJ/F8 9.9626 Tf 20.401 6.74 Td [(1 p LC t = L I + f x )]TJ/F8 9.9626 Tf 20.401 6.74 Td [(1 p LC t .31 Thiscanbesolvedfor V + intermsof I + V + = r L C I + Z 0 I + .32 where Z 0 = q L C iscalledthe characteristicimpedance ofthetransmissionline.Wewillleaveitasan exercisetothereadertoensurethatindeed q L C hasunitsofOhms.Forpractice,andunderstandingabout justhowtheseequationswork,thereadershouldensurehim/herselfthat V )]TJ/F8 9.9626 Tf 9.492 -4.113 Td [(= )]TJ/F1 9.9626 Tf 9.41 17.036 Td [( r L C I )]TJ/F1 9.9626 Tf 6.725 12.923 Td [(! )]TJ/F1 9.9626 Tf 19.926 8.07 Td [()]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(Z 0 I )]TJ/F1 9.9626 Tf 6.725 3.957 Td [( .33 Notethe"subtle"dierencehere,witha"-"signinfrontoftheRHSoftheequation! We'vebeenthroughlotsofequationsrecently,soitisprobablyworthourwhiletosummarizewhatwe knowsofar. 1.Thetelegrapher'sequationsallowtwosolutionsforthevoltageandcurrentonatransmissionline.One movesinthe + x directionandtheothermovesinthe )]TJ/F11 9.9626 Tf 7.749 0 Td [(x direction. 2.Bothsignalsmoveataconstantvelocity v p givenby.34. 3.Thevoltageandcurrentsignalsarerelatedtooneanotherbythecharacteristicimpedance Z 0 ,with .35 v p = 1 p LC .34 Z 0 = r L C .35 V + I + = Z 0 V )]TJETq1 0 0 1 282.474 244.057 cm[]0 d 0 J 0.398 w 0 0 m 14.75 0 l SQBT/F11 9.9626 Tf 283.906 234.732 Td [(I )]TJ/F8 9.9626 Tf 12.12 3.956 Td [(= )]TJ/F11 9.9626 Tf 7.749 0 Td [(Z 0 5.4TransmissionLineExamples 6 Asanexample,andalsobecauseitevenhassomepracticalimportance,let'slookatonekindoftransmission line.Itiscalleda stripline anditlookslikeFigure5.10AStripline.Itconsistsofaatconductor,located betweentwogroundplanes.Itissupportedbyaninsulatingdielectricwithdielectricconstant .Thisis kindoflikethesituationyouwouldndonamulti-levelPCboard,whereperhapsthebuslineswouldbe runningonaninnerlayerwithgroundplanesaboveandbelowthem. 6 Thiscontentisavailableonlineat.

PAGE 158

152 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES AStripline Figure5.10 Betweenthecenterconductorandthegroundplane,therewillbesomecapacitance, C .Ifwecanassume thattheelectriceldismoreorlessconnedtotheregionsbetweenthestripconductorandtheground planewhichoccurswhentheratioof W B isnottoosmallthenforeithercapacitorassumingunitlength intothepicturewewillgetavalue C = W B 2 .36 sincethevalueofacapacitorisjustthedielectricconstanttimestheareaoftheplates,dividedbythe spacingoftheplates. LookingquicklyatFigure5.10AStriplineyoumightthinkthetwocapacitorsareinseries,butyou wouldbewrong!Notethateachcapacitorhasoneleadconnectedtothecenterconductorandtheother leadconnectedtoground,andsothetwocapacitorsareinfact,inparallel,andhencetheircapacitances add.Thus,forthecapacitanceperunitlengthforthisline,wecanwrite: C = 4 W B .37 Itcanbeshownalthoughwewon'tdoitherethatfor any transmissionlinewheretheelectricand magneticeldsareperpendiculartooneanothercalled TEM or transverseelectromagnetic thespeed ofpropagationofthewavedownthelineisjust v p = c p 0 = 3 10 8 m s p r .38 Where r iscalledthe relativedielectricconstant forthematerial.Well,wealsoknowthat v p = 1 p LC .39 Fromwhichwecanwrite L = 1 v p 2 C = B v p 2 4 W .40

PAGE 159

153 Wecannowinsertthisvaluefor L intotheexpressionfor Z 0 ,theimpedanceoftheline. Z 0 = q L C = r B v p 2 4 W 4 W B = B 4 Wv p = B 4 W c p r .41 Andso,wehavederivedanequationfortheimpedance Z 0 ofthelineintermsofthedimensions W and B thedielectricconstantoftheinsulatingmaterial, ,and c ,thespeedoflight.Howgoodisthisexpression, andinparticularhowgoodisourassumptionthattheelectriceldisallconnedtotheregionunderthe conductor?NotsogreatactuallyFigure5.11ExactandApproximateImpedanceForaStripline. ExactandApproximateImpedanceForaStripline Figure5.11: Exactandapproximate Z 0 forastripline Figure5.11ExactandApproximateImpedanceForaStriplineshowstheresultsfromusing.41and amoreexactcalculation,whichtakesintoaccountthefringingelds.Asyoucanseewehavetogetthe ratio W B uptoabout4orsobeforethetwomatch.Butatleastwegettherightbehaviorandwe'renot totallyoutoftheballpark. 5.5ExcitingaLine 7 Wewillnowgoonandlookatwhathappenswhenweexcitetheline.Let'stakeaDCvoltagesource withasourceinternalimpedance R s andconnectittooursemi-inniteline.ThesketchinFigure5.12 ExcitingaTransmissionLineissortofawkwardlooking,andwillbehardtoanalyze,solet'smakeamore "schematiclike"drawingFigure5.13SchematicRepresentation,keepinginmindthatitisasituationsuch asFigure5.12ExcitingaTransmissionLinewhichwetryingtorepresent. 7 Thiscontentisavailableonlineat.

PAGE 160

154 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES ExcitingaTransmissionLine Figure5.12 SchematicRepresentation Figure5.13 Whyhaveweshownan I + anda V + butnot V )]TJ/F15 9.9626 Tf 10.063 -3.616 Td [(or I )]TJ/F15 9.9626 Tf 6.724 -3.616 Td [(?Theansweris,thatifthelineissemi-innite, thenthe"other"endisatinnity,andweknowtherearenosourcesatinnity.Thecurrentowingthrough thesourceresistorisjust I + 1 ,sowecandoaKVLaroundtheloop V s )]TJ/F11 9.9626 Tf 9.963 0 Td [(I + 1 ;t R s )]TJ/F11 9.9626 Tf 9.963 0 Td [(V + 1 ;t =0 .42 Substitutingfor I + 1 intermsof V + 1 usingthisequation.32: V s )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(V + 1 ;t Z 0 R s )]TJ/F11 9.9626 Tf 9.963 0 Td [(V + 1 ;t =0 .43 Whichwere-writeas V + 1 ;t 1+ R s Z 0 = V s .44 Or,onsolvingfor V + 1 ;t : V + 1 ;t = Z 0 Z 0 + R s V s .45 This should lookbothreasonableandfamiliartoyou.Thelineandthesourceresistanceareactingasa voltagedivider.Infact,.45isjusttheusualvoltagedividerequationfortworesistorsinseries.Thus,the generatorcannottellthedierencebetweenasemi-innitetransmissionlineofcharacteristicimpedance Z 0 andaresistorwitharesistanceofthesamevalueFigure5.14LineisInitiallyaVoltageDivider!.

PAGE 161

155 LineisInitiallyaVoltageDivider! Figure5.14 Haveyoueverheardof" 300 twin-lead"ormaybe" 75 co-ax"andwonderedwhypeoplewouldwant tousewireswithsuchahighresistancevaluetobringaTVsignaltotheirset?Nowyouknow.The 300 characterizationisnotameasureoftheresistanceofthewire,ratheritisaspecicationofthetransmission line'simpedance.Thus,ifaTVsignalcomingfromyourantennahasavalueof,say, 30 V ,anditisbeing broughtdownfromtheroofwith 300 twin-lead,thenthecurrentowinginthewiresis I = 30 V 300 =100 nA whichisaverysmallcurrentindeed! Whythen,didpeopledecideon 300 ?Anantennawhichisjustahalf-wavelengthlongWhichturnsout tobebothaconvenientandecientchoiceforsignalsinthe100MHz 3 m rangeactslikeavoltage sourcewithasourceresistanceofabout 300 .IfyourememberfromELEC242,whenwehaveasource withasourceresistance R s andaloadresistorwithloadresistancevalue R L Figure5.15PowerTransfer ToaLoad,youcalculatethepowerdeliveredtotheloadusingthefollowingmethod. PowerTransferToaLoad Figure5.15 P L ,thepowerintheload,isjustproductofthevoltageacrosstheloadtimesthecurrentthroughthe load.Wecanusethevoltagedividerlawtondthevoltageacross R L andtheresistorsumlawtondthe currentthroughit. P L = V L I L = R L R L + R s V s V s R L + R s = R L R L + R s 2 V s 2 .46 Ifwetakethederivativeof.46withrespectto R L ,theloadresistorwhichweassumewecanpick,given

PAGE 162

156 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES somepredetermined R s wehaveignoringthe V s 2 d dR L P L = 1 R L + R s 2 2 R L R L + R s 3 =0 .47 Puttingeverythingon R L + R s 3 andthenjustlookingatthenumerator: R L + R s )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 R L =0 .48 Whichobviouslysaysthatformaximumpowertransfer,youwantyourloadresistor R L tohavethesame valueasyoursourceresistor R s !Thus,peoplecameupwith 300 twinleadsothattheycouldmaximize theenergytransferbetweentheTVantennaandthetransmissionlinebringingthesignaltotheTVreceiver set.Itturnsoutthatforaco-axialtransmissionlinesuchasyourTVcable 75 minimizesthesignalloss, whichiswhythatvaluewaschosenforCATV. 5.6TerminatedLines 8 If,ontheotherhand,wehaveaniteline,terminatedwithsomeloadimpedance,wehaveasomewhatmore complicatedproblemtodealwithFigure5.16AFiniteTerminatedTransmissionLine. AFiniteTerminatedTransmissionLine Figure5.16 Thereareseveralthingsweshouldnote before weheadointoequation-landagain.Firstofall,unlike thetransientproblemswelookedatinapreviouschapterSection5.6,therecanbenomorethan two voltageandcurrentsignalsontheline,just V + and V )]TJ/F15 9.9626 Tf 6.725 -3.616 Td [(,and I + and I )]TJ/F15 9.9626 Tf 6.725 -3.616 Td [(.Wenolongerhavetheluxuryof having V + 1 V + 2 etc. ,becausewearetalkingnowabouta steadystatesystem .Allofthetransientsolutions whichbuiltupwhenthegeneratorwasrstconnectedtothelinehavebeensummedtogetherintojusttwo waves. Thus,onthelinewehaveasingle totalvoltagefunction ,whichisjustthesumofthepositiveand negativegoingvoltagewaves V x = V + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jx + V )]TJ/F11 9.9626 Tf 6.725 -4.114 Td [(e + jx .49 andatotalcurrentfunction I x = I + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jx + I )]TJ/F11 9.9626 Tf 6.724 -4.114 Td [(e + jx .50 Notealsothatuntilwehavesolvedfor V + and V )]TJ/F15 9.9626 Tf 6.725 -3.615 Td [(,wedonotknow V x or I x anywhereontheline. Inparticular,wedonotknow V and I whichwouldtelluswhattheapparentimpedanceislooking 8 Thiscontentisavailableonlineat.

PAGE 163

157 intotheline. Z in = Z = V + + V )]TJETq1 0 0 1 312.124 653.124 cm[]0 d 0 J 0.398 w 0 0 m 30.469 0 l SQBT/F10 6.9738 Tf 314.471 647.197 Td [(I + + I )]TJ/F15 9.9626 Tf 180.195 10.668 Td [(.51 Untilweknowwhatkindofimpedancethegeneratorisseeing,wecannotgureouthowmuchofthe generator'svoltagewillbecoupledtotheline!Theinputimpedancelookingintothelineisnowafunction oftheloadimpedance,thelengthoftheline,andthephasevelocityontheline.Wehavetosolvethis before wecangureouthowthelineandgeneratorwillinteract. Theapproachweshallhavetotakeisthefollowing.Wewillstartatthe load endoftheline,andina mannersimilartotheoneweusedpreviously,ndarelationshipbetween V + and V )]TJ/F15 9.9626 Tf 6.725 -3.616 Td [(,leavingtheiractual magnitudeandphaseassomethingtobedeterminedlater.Wecanthenpropagatethetwovoltagesand currentsbackdowntotheinput,determinewhattheinputimpedanceisbyndingtheratioof V + + V )]TJ/F15 9.9626 Tf 6.725 -3.615 Td [( to I + + I )]TJ/F15 9.9626 Tf 6.725 -3.615 Td [(,andfromthis,andknowledgeofpropertiesofthegeneratoranditsimpedance,determine whattheactualvoltagesandcurrentsare. Let'stakealookattheload.WeagainuseKVLandKCLFigure5.17DoingKirchoattheEndof theLinetomatchvoltagesandcurrentsinthelineandvoltagesandcurrentsintheload: V + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jL + V )]TJ/F11 9.9626 Tf 6.725 -4.113 Td [(e + jL = V L .52 and I + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jL + I )]TJ/F11 9.9626 Tf 6.725 -4.113 Td [(e + jL = I L .53 DoingKirchoattheEndoftheLine Figure5.17: Changevariables! Now,we could substitute V Z 0 forthetwocurrentsonthelineand V L Z L for I L ,andthentrytosolvefor V )]TJ/F15 9.9626 Tf -461.275 -16.981 Td [(intermsof V + using.52and.53butwecanbealittlecleverattheoutset,andmakeourcomplex algebraagoodbitcleanerFigure5.18s=0attheLoadandSotheExponentialsGoAway!.Let'smakea changeofvariableandlet s L )]TJ/F11 9.9626 Tf 9.963 0 Td [(x .54

PAGE 164

158 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES s=0attheLoadandSotheExponentialsGoAway! Figure5.18 Thisthengivesusforthevoltageonthelineusing x = L )]TJ/F11 9.9626 Tf 9.963 0 Td [(s V s = V + e )]TJ/F7 6.9738 Tf 6.226 0 Td [( jL e jL + V )]TJ/F11 9.9626 Tf 6.725 -4.113 Td [(e jL e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jL .55 Usually,wejustfoldtheconstantphaseterms e jL termsinwiththe V + and V )]TJ/F15 9.9626 Tf 10.044 -3.615 Td [(andsowehave: V s = V + e js + V )]TJ/F11 9.9626 Tf 6.724 -4.114 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js .56 Notethatwhenwedothis,wenowhavea positive exponentialinthersttermassociatedwith V + anda negative exponentialassociatedwiththe V )]TJ/F15 9.9626 Tf 10.045 -3.615 Td [(term.Ofcourse,wealsogetfor I s : I s = I + e js + I )]TJ/F11 9.9626 Tf 6.725 -4.114 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js .57 Thischangenowmovesourorigintothe load endoftheline,andreversesthedirectionofpositivemotion. But ,nowwhenwepluginto e js thevaluefor s attheload s =0 ,theequationssimplifyto: V + + V )]TJ/F8 9.9626 Tf 9.492 -4.113 Td [(= V L .58 and I + + I )]TJ/F8 9.9626 Tf 9.492 -4.113 Td [(= I L .59 whichwethenre-writeas V + Z 0 )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(V )]TJETq1 0 0 1 299.009 214.06 cm[]0 d 0 J 0.398 w 0 0 m 14.75 0 l SQBT/F11 9.9626 Tf 300.749 204.735 Td [(Z 0 = V L Z L .60 ThisisbeginningtolookalmostexactlylikeapreviouschapterSection5.1.Asareminder,wesolve.60 for V L V L = Z L Z 0 )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(V + )]TJ/F11 9.9626 Tf 9.962 0 Td [(V )]TJ/F1 9.9626 Tf 6.725 3.956 Td [( .61 andsubstitutefor V L in.58 V + + V )]TJ/F8 9.9626 Tf 9.492 -4.114 Td [(= Z L Z 0 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(V + )]TJ/F11 9.9626 Tf 9.963 0 Td [(V )]TJ/F1 9.9626 Tf 6.725 3.956 Td [( .62 Fromwhichwethensolveforthereectioncoecient )]TJ/F10 6.9738 Tf 6.226 -1.495 Td [( ,theratioof V )]TJ/F15 9.9626 Tf 10.045 -3.615 Td [(to V + V )]TJETq1 0 0 1 260.448 77.838 cm[]0 d 0 J 0.398 w 0 0 m 14.75 0 l SQBT/F11 9.9626 Tf 260.504 68.513 Td [(V + )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [( = Z L )]TJ/F11 9.9626 Tf 9.962 0 Td [(Z 0 Z L + Z 0 .63

PAGE 165

159 Notethatsince,ingeneral, Z L willbecomplex,wecanexpectthat )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( willalsobeacomplexnumberwith bothamagnitude j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j andaphaseangle )]TJ/F15 9.9626 Tf 5.424 1.494 Td [(.Also,aswiththecasewhenwewerelookingattransients, j )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [( j < 1 Sincewenowknow V )]TJ/F15 9.9626 Tf 10.15 -3.615 Td [(intermsof V + ,wecannowwriteanexpressionfor V s thevoltageanywhere ontheline. V s = V + e js + V )]TJ/F11 9.9626 Tf 6.724 -4.114 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js .64 Noteagainthechangeinsignsinthetwoexponentials.Sinceourspatialvariable s isgoingintheopposite directionfrom x ,the V + phasornowgoesas + js andthe V )]TJ/F15 9.9626 Tf 10.045 -3.616 Td [(phasornowgoesas )]TJ/F8 9.9626 Tf 9.409 0 Td [( js Wenowsubstitutein )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( V + for V )]TJ/F15 9.9626 Tf 9.767 -3.616 Td [(in.64,andforreasonsthatwillbecomeapparentsoon,factorout an e js V s = V + e js +)]TJ/F10 6.9738 Tf 16.19 -1.495 Td [( V + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js = V + )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e js +)]TJ/F10 6.9738 Tf 16.189 -1.495 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js = V + e js )]TJ/F8 9.9626 Tf 4.566 -8.069 Td [(1+)]TJ/F10 6.9738 Tf 23.385 -1.495 Td [( e )]TJ/F7 6.9738 Tf 6.226 0 Td [( js .65 Wecouldhavealsowrittendownanequationfor I s ,thecurrentalongtheline.Itwillbeagoodtestof yourunderstandingofthebasicequationswearedevelopingheretoshowyourselfthatindeed I s = V + e js Z 0 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [()]TJ/F10 6.9738 Tf 6.227 -1.495 Td [( e )]TJ/F7 6.9738 Tf 6.226 0 Td [( js .66 5.7BounceDiagrams 9 Nowthis new V + 2 willheadbacktowardstheloadand...Hmmm...thingsaregoingtogetkindofmessy andcomplicated.Fortunatelyforus,transmissionlineengineerscameupwithaschemeforkeepingtrackof allofthewavesbouncingbackandforthontheline.Theschemeiscalleda bouncediagram .Abounce diagramconsistsofahorizontaldistanceline,whichrepresentsdistancealongthetransmissionline,anda verticaltimeaxis,whichrepresentstimesincethebatterywasrstconnectedtotheline.Justtokeepthings conceptuallyclear,weusuallyrststartoutbyshowingtheline,thebattery,theloadandaswitch,S,which isusedtoconnectthesourcetotheline.ItdoesnthurttomakealittlesketchlikeFigure5.19Transient Problem,andwritedownthelengthoftheline, Z 0 and v p ,alongwiththesourceandloadresistances. Nowwedrawthebouncediagram,whichisshowninFigure5.20A"BounceDiagram" TransientProblem Figure5.19 9 Thiscontentisavailableonlineat.

PAGE 166

160 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES A"BounceDiagram" Figure5.20 Normally,youwouldnotputtheformulafor )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [(vS and )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [(vL by0and L inthediagram,butrathertheir values.Thiswillbecomeclearwhenwedoanexample.Thenextthingwedoiscalculate V + 1 anddrawa straightlineonthebouncediagramnominallyataslopeof 1 v p whichwillrepresenttheinitialsignalgoing downtheline.Wemarka = L v p ontheverticalaxistoshowhowlongittakesforthewavetoreachthe endofthelineFigure5.21DiagramWithFirstWave. DiagramWithFirstWave Figure5.21

PAGE 167

161 Oncetheinitialwavehitstheload,asecond,reectedwave V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 =)]TJ/F10 6.9738 Tf 16.743 -1.494 Td [(vL V + 1 issentbacktheotherway.So weaddittothebouncediagram.ThisisshowninFigure5.22AddingtheFirstReectedWave.Sinceall ofthewavesmovewiththesamephasevelocity,weshouldbecarefultodrawallofthelineswiththesame slope.Notethatthetimewhenthereectedwavehitsthegeneratorendisatotalroundtriptimeof 2 Thissimpleconceptisonewhichstudentsoftenforgetcometesttime,sobeforewarned! AddingtheFirstReectedWave Figure5.22 Wesawthatthenextthingthathappensisthatanotherwaveisreectedfromthegenerator,soweadd thattothebouncediagramaswell.ThisisshowninFigure5.23TheThirdWave.

PAGE 168

162 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES TheThirdWave Figure5.23 Finally,onelastwave,aswearealmostbouncedrightothediagram,asshowninFigure5.24Andthe Fourth! AndtheFourth Figure5.24 OK,sowe'vegotabouncediagram,sowhat?Havingthediagramisonlypartofthesolution.Westill havetoseewhatgoodtheyare.Let'sdoanumericalexample,asitismaybealittlemoreillustrative,and certainlywillbeeasiertowriteoutthanalltheseratiosallthetime.Wewilljustpicksometypicalnumbers,

PAGE 169

163 andthenworkouttheanswers.Let'slet V S =40 V R S =150 Z 0 =50 and R L =16 : 7 .Thelinewill be100mlong,and v p =2 10 8 m s Figure5.25ANumericalExample. ANumericalExample Figure5.25 Firstwecalculatethereectioncoecients )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [(vL = R L )]TJ/F10 6.9738 Tf 6.227 0 Td [(Z 0 R L + Z 0 = 16 : 7 )]TJ/F7 6.9738 Tf 6.227 0 Td [(50 16 : 7+50 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(0 : 50 .67 and )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [(vS = R S )]TJ/F10 6.9738 Tf 6.227 0 Td [(Z 0 R S + Z 0 = 150 )]TJ/F7 6.9738 Tf 6.227 0 Td [(50 150+50 =0 : 50 .68 Theinitialvoltagesignal V + 1 is V + 1 = 50 50+150 40 =10 V .69 andthepropagationtimeis = L v p = 100 m 10 8 m s =0 : 5 s .70 SowedrawthebouncediagramsseeninFigure5.26TheBounceDiagram.

PAGE 170

164 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES TheBounceDiagram Figure5.26 Now,here'showweuseabouncediagram,oncewehaveit.Supposewewanttoknowwhat V t ,the voltageasafunctionoftime,wouldlooklikehalf-waydowntheline.Wedrawaverticallineattheplace weareinterestedinthedottedlineinFigure5.26TheBounceDiagramandthenjustgoupalongthe line,addingvoltagetowhateverwehadbeforewheneverwecrossoneofthe"bouncing"signallines.Thus forthelineasshownwewouldhavefor V t whatweseeinFigure5.27Vtat50mDowntheLine. Vtat50mDowntheLine Figure5.27 Fortherst0.25 swehavenovoltage,because V + 1 hasnotreachedthehalf-waypointyet.Thevoltage thenjumpsto+10Vwhen V + 1 comesby.Itstayslikethatuntilthe-5V V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 comesby0.5 slater.The voltagethenremainsconstantat5Vuntilthe-2.5V V + 2 comesalongtodropthetotalvoltagedowntoonly 2.5volts.When V )]TJ/F7 6.9738 Tf -2.214 -6.919 Td [(2 comesalong,ithasbeenswitchedbacktoapositivevoltagewavebythenegativeload reectioncoecient,andsonowthevoltagejumpsbackupto3.75V.Itwillkeeposcillatingbackandforth untilitnallysettlesdowntosomeasymptoticvalue. Whatwillthatasymptoticvaluebe?Oneapproachistowritedownthefollowingequation. V x; 1 = V + 1 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+)]TJ/F10 6.9738 Tf 23.385 -1.494 Td [(L +)]TJ/F10 6.9738 Tf 16.189 -1.494 Td [(L )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [(S +)]TJ/F10 6.9738 Tf 16.189 -1.494 Td [(L 2 )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [(S + ::: .71

PAGE 171

165 Whichwecanre-writeas V + 1 1+)]TJ/F10 6.9738 Tf 23.385 -1.494 Td [(L )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [(S +)]TJ/F10 6.9738 Tf 20.064 -1.494 Td [(L )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [(S 2 + ::: +)]TJ/F10 6.9738 Tf 16.19 -1.494 Td [(L V + 1 1+)]TJ/F10 6.9738 Tf 23.385 -1.494 Td [(L )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [(S +)]TJ/F10 6.9738 Tf 20.063 -1.494 Td [(L )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [(S 2 + ::: .72 Now,rememberingtheinnitesumrelationship: 1 X n =0 x n = 1 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x .73 for j x j < 1 whichis always thecaseforareectioncoecient.Wecansubstitute.73fortheterms insidetheparenthesesin.72andweget V x; 1 = V + 1 1 1 )]TJ/F7 6.9738 Tf 6.227 0 Td [()]TJ/F9 4.9813 Tf 4.926 -1.001 Td [(L )]TJ/F9 4.9813 Tf 4.926 -1.001 Td [(S + )]TJ/F9 4.9813 Tf 4.926 -1.002 Td [(L 1 )]TJ/F7 6.9738 Tf 6.227 0 Td [()]TJ/F9 4.9813 Tf 4.926 -1.001 Td [(L )]TJ/F9 4.9813 Tf 4.926 -1.001 Td [(S = V + 1 1+)]TJ/F9 4.9813 Tf 15.013 -1.001 Td [(L 1 )]TJ/F7 6.9738 Tf 6.226 0 Td [()]TJ/F9 4.9813 Tf 4.926 -1.002 Td [(L )]TJ/F9 4.9813 Tf 4.925 -1.002 Td [(S .74 Wewillleaveitasanexercisetothereadertoshowthatifwesubstitute.57,.62andnally.65 into.74wewilleventuallyget: V x; 1 = R L R L + R S V S .75 LookbackatFigure5.19TransientProblemandseeif.75makesanysense.Itshould.Ifwewaitlong enough,itisreasonabletoexpectthatany"transmissionline"eectsshouldgoaway,andwewouldbeback tothesamesituationwewouldhaveifthelinewasjustsomewireconnectingthesourcetotheload.Inthis case,theloadresistorandthesourceresistorwouldformavoltagedivider,andwewouldexpectthevoltage acrosstheloadtobedeterminedbythevoltagedividerequation.That'sall.75issaying! Whatdowedoifwewant,say,thevoltageacrosstheloadwithtime?TodothiswemoveuptheRHSof thebouncediagram,andcountvoltagewavesaswemoveacrossthem.Westartoutatzero,ofcourse,and donotseeanythinguntilwegetto0.5ms.Thenwecrossthe10V V + 1 wave and wecrossthe-5V V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 wave atthesametime,sothevoltageonlygoesupto+5V.Likewise,another1mslater,wecrossboththe-2.5V V + 2 and the+1.25V V )]TJ/F7 6.9738 Tf -2.214 -6.919 Td [(2 wave,andsothevoltageendsupatthe3.75VpositionFigure5.28VtAcross theLoad. VtAcrosstheLoad Figure5.28 Wecanalsousethebouncediagramtondthevoltageasafunctionofposition,forsomexedtime, t 0 Figure5.29FindingVxatt=0.75 s.

PAGE 172

166 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES FindingVxatt=0.75 s Figure5.29 Todothis,wedrawahorizontallineatthetimeweareinterestedin,say0.75 s.Now,foreachposition x ,wegofromthebottomofthediagram,uptothehorizontalline,addingupvoltageaswego.Thusfor theexample:wegetwhatweseeinFigure5.30Vxatt=0.75 s.Forthersthalfoftheline,wecross the+10V V + 1 ,butthat'sit.Forthesecondhalfofthelinewecross both the+10Vlineaswellas-5V V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 wave,andsothevoltagedropsdownto5V. Vxatt=0.75 s Figure5.30 Ofparticularinteresttomanyofyouwillbethewayinwhicha pulse movesdownalineandisreected etc .Thisisalsoquiteeasytodowithareectiondiagram,ifwesimplybreakthepulseintotwowaves, onewhichhasa positive swingat t =0 andanotherwhichisa negative goingwaveat t = p ,where p is thepulsewidthofthepulsebeinggenerated.ThewaywedothisissuggestedinFigure5.31Simulating aPulseWithTwoBatteriesandTwoSwitches.Wereplacethepulsegeneratorwithtwobattery/switch combinations.Therstcircuitisjustlikewehaveseensofar,withabatteryequaltotheopencircuit pulseheightofthegenerator,andaswitchwhichclosesat t =0 .Thesecondcircuithasabatterywithan amplitudeof minus thepulseheight,andaswitchwhichclosesat t = p ,thepulsewidthofthepulseitself.

PAGE 173

167 SimulatingaPulseWithTwoBatteriesandTwoSwitches Figure5.31 Bysuperposition,wecanjustaddthesetwogenerators,oneaftertheother,andseehowthepulsegoes downtheline.Suppose V p is10volts, p =0 : 25 s R S =50 Z 0 =50 and R L =25 .Withthenumbers, wendthat V + 1 =25 V )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [(vL = )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 and )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [(vS =0 .Let'sassumethatthepropagationtimeonthelineisstill 0.5 stogetfromoneendofthelinetotheother. WedrawthebouncediagramFigure5.32:PulseBounceDiagram,andlaunch twowaves ,onewhich leavesat t =0 hasanamplitudeof V + 1 =5 V .Thesecondwaveleavesatatime p ,later,andhasan amplitudeof-5V. PulseBounceDiagram Figure5.32

PAGE 174

168 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES Nowwhenwewanttoseewhatthevoltageasafunctionoftimelookslike,weagaindrawalineupthe middle,andaddvoltagesaswecrossthem.Herewesee,again,novoltageuntilwecrosstherstwaveat 0.25 s,whichpopsusupto+5V.Atatime0.25 slaterhowever,the-5Vwavecomesalong,andwego backdowntozero.At t =0 : 75 s ,thereected-1.67Vpulsecomesalong,andsoweseethat.Sincethe sourceismatchedtotheline, )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [(vS =0 andsothisistheendofthestoryFigure5.33VtHalf-wayDown theLine. VtHalf-wayDowntheLine Figure5.33 Youcangetsomewhatmoreinterestingwaveformsifyougosomeplacewherethetwopulsesatleast partiallyoverlap.Let'slookatsay, x =87 : 5 m .HereFigure5.34:FindingVtNeartheLoadisthe bouncediagram. FindingVtNeartheLoad Figure5.34 AndhereFigure5.35:VtNeartheLoadisthevoltagewaveformweget.

PAGE 175

169 VtNeartheLoad Figure5.35 Thistimethe1.67Vpulsegetstousbeforethe+5Vpulsehascompletelypassed,andsowedropfrom 5Vto3.33V.Then,whenthe-5Vwavegoesby,wedropdownto-1.67Vforalittlewhile,untilthe+1.67V wavecomesalongtobringusbacktozero. 5.8CascadedLines 10 Wecanusebouncediagramstohandlesomewhatmorecomplicatedproblemsaswell. ArnoldAggiedecidestoaddanadditionalethernetinterfacetotheonealreadyconnectedtohiscomputer. Hedecidesjusttoadda"T"totheterminalwherethecableisconnectedtohis"thin-net"interface,and addonsomemorewire.Unfortunately,heisnotcarefulaboutthecoaxialcableheuses,andsohehassome 75 TVco-axinsteadofthe 50 ethernetcable.HeendsupwiththesituationshownhereFigure5.36: CascadedLineProblem.Thiskindofproblemiscalleda cascadedlineproblem becausewehavetwo dierentlines,onehookedupaftertheother.Theanalysisissimilartowhatwehavedonebefore,justa littlemorecomplicatedisall. CascadedLineProblem Figure5.36 Wewillhavetodoalittlemorethinkingbeforewecandrawoutthebouncediagramforthisproblem. ThedriverforethernetcablecomingtoArnold'scomputercanbemodeledasa10Vopencircuitsource witha 50 internalimpedance.Sincethesourcedoesnotinitiallyknowanythingabouthowthelineitis drivingisterminated,therstsignal V + 1 willbethesameasinourinitialproblem,inthiscasejusta+5V signalheadeddowntheline. 10 Thiscontentisavailableonlineat.

PAGE 176

170 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES Let'sfocusonthe"T"foraminuteFigure5.37AttheJunction. AttheJunction Figure5.37 V + 1 isincidentonthejunction.Whenithitsthejunction,therewillbeareectedwave V )]TJ/F7 6.9738 Tf -2.214 -6.919 Td [(1 andalso now,atransmittedwave V + T 1 .Sincetheincidentwavecannottellthedierencebetweena 75 resistorand a 75 transmissionline,it thinks itisseeingaterminationresistorequaltoa 50 resistor R L 1 inparallel witha 75 resistorthesecondline. 50 inparallelwith 75 is 30 .Let'scallthis"apparent"loadresistor R L ,sothatwecanthencalculate )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [(V 12 ,therstvoltagereectioncoecientingoingfromline1toline2 as: )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [(V 12 = R L )]TJ/F10 6.9738 Tf 6.227 0 Td [(Z 01 R L + Z 01 = 30 )]TJ/F7 6.9738 Tf 6.226 0 Td [(50 30+50 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(0 : 25 .76 Notethatwe could havestartedfromscratchandwrittendownKVLsandKCLsforthejunction V + 1 + V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 = V + T 1 .77 and I + 1 + I )]TJ/F7 6.9738 Tf -0.781 -6.918 Td [(1 = I RL + I + T 1 .78 Then,byre-writing.78intermsofvoltageandimpedanceswehave: V + 1 Z 01 )]TJ/F11 9.9626 Tf 11.404 6.739 Td [(V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 Z 01 = V + T 1 Z 02 + V + T 1 R L .79 Wenowhavetwoequationswithtwounknowns V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 and V + T 1 .Bysolving.79for V + T 1 andthenplugging thatinto.77,wecouldgettheratioof V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 to V + 1 ,orthevoltagereectioncoecient.Theinterested readercanconrmthatindeed,yougettheverysameresultthisway. Inordertocompletelysolvethisproblem,wealsoneedtoknow V + T 1 ,thetransmittedwaveaswell.Since .77says V + T 1 isjustthesumoftheincidentandreectedwavesontherstline V + T 1 = V + 1 +)]TJ/F10 6.9738 Tf 16.189 -1.494 Td [(V 12 V + 1 .80 Wecanthuswrite V + T 1 V + L =1+)]TJ/F10 6.9738 Tf 33.901 -1.494 Td [(V 12 = R L + Z 01 R L + Z 01 + R L )]TJ/F11 9.9626 Tf 9.963 0 Td [(Z 01 R L + Z 01 = 2 R L R L + Z 01 = 60 30+50 =0 : 75 T V 12 .81 Animportantthingtonoteisthat T V =1+)]TJ/F10 6.9738 Tf 33.901 -1.494 Td [(V .82

PAGE 177

171 NOT T V +)]TJ/F10 6.9738 Tf 16.189 -1.494 Td [(V =1 .83 Wedonot"conserve"voltageatatermination,inthesensethatthereectedandtransmittedvoltagehave toadduptobetheincidentvoltage.Rather,thetransmittedvoltageisthe sum oftheincidentvoltageand thereectedvoltage,sothatwecanobeyKircho'svoltagelaw. Wecannowstarttomakeourbouncediagram.Wepropagatea+5Vwaveanda-5Vwaveseparated by100nsdowntowardsthejunction.Sincethelineis40mlong,andthewavesmoveat 2 10 8 m s ,ittakes 200nsforthemtogettothejunction.There,a-1.25Vwaveisreectedbacktowardsthesource,anda +3.75VwaveistransmittedintothesecondtransmissionlineFigure5.38ReectionandTransmissionAt the"T". ReectionandTransmissionAtthe"T" Figure5.38 Sincetheloadforthesecondlineis 50 ,andthecharacteristicimpedance, Z 02 forthesecondlineis 75 ,wewillhaveareectioncoecient, )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [(V 2 = R L 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [(Z 02 R L 2 + Z 02 = 50 )]TJ/F7 6.9738 Tf 6.226 0 Td [(75 50+75 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(0 : 2 .84 Thusa-0.75VsignalisreectedoofthesecondloadFigure5.39ReectionofTransmittedPulse. Exercise5.1 Solutiononp.179. Whatisthemagnitudeofthevoltagewhichisdevelopedacrossthesecondload?

PAGE 178

172 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES ReectionofTransmittedPulse Figure5.39 Whathappenstothe0.75Vpulsewhenitgetstothe"T"?Wellthereisanothermismatchhere,witha reectioncoecient )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [(V 21 givenby )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [(V 21 = 25 )]TJ/F7 6.9738 Tf 6.226 0 Td [(75 25+75 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(0 : 5 .85 The 50 resistorandthe 50 transmissionlinelooklikea 25 terminationtothe 75 lineanda transmissioncoecient T V 21 =1+)]TJ/F10 6.9738 Tf 41.096 -1.494 Td [(V 21 =0 : 5 .86 andsoweaddtothebouncediagramFigure5.40WhentheReectedLoadPulseHitstheJunction. WhentheReectedLoadPulseHitstheJunction Figure5.40 We could keepgoing,butthevoltagereectedoofthesecondloadwillonlybe75mVnow,andsolet's callitaday.

PAGE 179

173 Thereareacoupleofotherinterestingapplicationsofbouncediagramsandthetransientbehaviorof transmissionlinesthatwemightlookatbeforewemoveontootherthings.Therstiscalledthe Charged LineProblem .HereFigure5.41:The"ChargedLine"Problemitis: The"ChargedLine"Problem Figure5.41 Wehaveatransmissionlinewithcharacteristicimpedance Z 0 andphasevelocity v p .Itis L long,andfor sometimehasbeenconnectedtoabatteryofpotential V g Figure5.42InitialConditions.Attime t =0 theswitchS,isthrown,whichremovesthebatteryfromthecircuit,andconnectsthelinetoaloadresistor R L .Thequestionis:whatdoesthevoltageacrosstheloadresistor, V L ,looklikeasafunctionoftime?This is almost likewhatwehavedonebefore,butnotquite. InitialConditions Figure5.42 Intherstplace,wenowhavenon-zeroinitialconditions.For t< 0 wewillhavebothvoltagesand currentontheline.Inordertomatchboundaryconditions,wemustdomorethanhaveonevoltageandone current,becausethevoltageonthelinemustbe V g ,whilethecurrentowingdownthelinemustbe0.So, wewillputinbotha V + anda V )]TJ/F15 9.9626 Tf 9.934 -3.615 Td [(andtheircorrespondingcurrents.Notethat + x isgoingtotheleftthis time.Let'sforgetabouttheswitchandtheloadresistorforaminuteandjustlookatthelineandbattery. Wehavetwoequationswemustsatisfy V + 0 + V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(0 = V g .87 and I + 0 + I )]TJ/F7 6.9738 Tf -0.782 -6.918 Td [(0 =0 .88 Wecanusetheimpedancerelationshiptochange.88to: V + 0 Z 0 )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(V )]TJ/F7 6.9738 Tf -2.214 -6.919 Td [(0 Z 0 =0 .89

PAGE 180

174 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES Ihope most ofyoucanthenseebyinspectionthatwemusthave V + 0 = V )]TJ/F7 6.9738 Tf -2.214 -6.919 Td [(0 = V g 2 .90 OK,theswitchSisthrownat t =0 .NowtheendofthelinelookslikethisFigure5.43:AftertheResistor isConnected. AftertheResistorisConnected Figure5.43 Wehaveanticipatedthefactthatwearegoingtoneedanothervoltageandcurrentwaveifwearegoing tobeabletomatchboundaryconditionswhentheloadresistorisconnected,andhaveaddeda V + 1 anda V )]TJ/F7 6.9738 Tf -2.213 -6.918 Td [(1 totheline.Thesearenewvoltageandcurrentwaveswhichoriginateattheloadresistorpositionin ordertosatisfythenewboundaryconditionsthere.NowwedoKVLandKCLagain. V + 0 + V )]TJ/F7 6.9738 Tf -2.214 -6.919 Td [(0 + V + 1 = V L .91 and V + 0 Z 0 )]TJ/F11 9.9626 Tf 11.158 6.739 Td [(V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(0 Z 0 + V + 1 Z 0 = )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( V L R L .92 Wehavealreadymadetheimpedancesubstitutionforthecurrentequationin.92.Weknowwhatthe sumanddierenceof V + 0 and V )]TJ/F7 6.9738 Tf -2.214 -6.919 Td [(0 are,solet'ssubstitutein. V g + V + 1 = V L .93 and V + 1 Z 0 = )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( V L R L .94 Fromthisweget V L = )]TJ/F1 9.9626 Tf 9.409 14.048 Td [( R L Z 0 V + 1 .95 whichwesubstitutebackinto.93 V g + V + 1 = )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( R L Z 0 V + 1 .96 whichwecansolvefor V + 1 V + 1 = )]TJ/F1 9.9626 Tf 9.409 14.047 Td [( V g 1+ R L Z 0 = )]TJ/F1 9.9626 Tf 9.409 11.059 Td [( Z 0 R L + Z 0 V g .97

PAGE 181

175 Thevoltageontheloadisgivenby.93andisclearlyjust: V L = V g )]TJ/F11 9.9626 Tf 24.01 6.74 Td [(Z 0 R L + Z 0 V g .98 andinparticular,when R L ischosentobe Z 0 whichisusuallydonewhenthiscircuitisused,wehave V L = V g 2 .99 Nowwhatdowedo?Webuildabouncediagram!Letusstaywiththeassumptionthat R L = Z 0 ,inwhich casethereectioncoecientattheresistorendis0.Attheopencircuitendofthetransmissionline )]TJ/F15 9.9626 Tf 9.911 0 Td [(is +1.SowehavethisFigure5.44:BounceDiagramfortheChargedLineProblem. BounceDiagramfortheChargedLineProblem Figure5.44 Notethatfor this bouncediagram,wehaveaddedanadditionalvoltage, V g ,onthebaseline,toindicate thatthereisaninitialvoltageontheline,beforetheswitchisthrown,and t startsonthebouncediagram. Ifweconcentrateonthevoltageacrosstheload,weadd + V g and )]TJ/F1 9.9626 Tf 9.409 11.058 Td [( V g 2 andndthatthevoltage acrosstheloadresistorrisesto V g 2 attime t =0 Figure5.45VoltageAcrosstheLoadResistor.The )]TJ/F1 9.9626 Tf 9.409 11.059 Td [( V g 2 voltagewavetravelsdowntheline,hitstheopencircuit,reectsback,andwhenitgetstotheload resistor,bringsthevoltageacrosstheloadresistorbackdowntozero.Wehavemadeapulsegenerator! VoltageAcrosstheLoadResistor Figure5.45: V t across R L

PAGE 182

176 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES Intoday'sdigitalage,thismightseemlikeastrangewaytogoaboutcreatingapulse.Imaginehowever, ifyouneededapulsewithaverylargepotentialsofthousandsorevenmillionsofvoltsforsay,a particleaccelerator.ItisunlikelythataMOSFETwilleverbebuiltwhichisuptothetask!Infact,ina eldofstudycalled pulsedpowerelectronics justsuchcircuitsareusedallthetime.Sometimestheyare builtwithrealtransmissionlines,sometimestheyarebuiltfromdiscreteinductorsandcapacitors,hooked togetherjustasinthedistributedparametermodelFigure5.5:DistributedParameterModel.Suchcircuits arecalled pulseformingnetworks orPFNsforshort. Finally,justbecauseitaordsusagoodopportunitytoreviewhowwegottowherewearerightnow, let'sconsidertheproblemofanon-resistiveloadontheendofaline.Supposethelineisterminatedwith acapacitor!Forsimplicity,let'slet R s = Z 0 ,sowhenSisclosedawave V + 1 = V g 2 headsdowntheline Figure5.46TransientProblemwithCapacitiveLoad.Let'sthinkaboutwhathappenswhenithitsthe capacitor.Weknowweneedtogenerateareectedsignal V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 ,solet'sgoaheadandputthisinthegure Figure5.47:InitialPulseHitstheLoad,alongwithitscompanioncurrentwave. TransientProblemwithCapacitiveLoad Figure5.46 InitialPulseHitstheLoad Figure5.47 Thecapacitorisinitiallyuncharged,andweknowwecannotinstantaneouslychangethevoltageacross acapacitoratleastwithoutaninnitecurrent!andsothe initial voltageacrossthecapacitorshouldbe zero,making V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 = )]TJ/F11 9.9626 Tf 7.749 0 Td [(V + 1 ,ifwemaketime t =0 bewhentheinitialwavejustgetstothecapacitor.So, at t =0 )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [(V = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 .Notethatwearemaking )]TJ/F15 9.9626 Tf 9.38 0 Td [(afunctionoftimenow,asitwillchangedependingupon thechargestateofthecapacitor.

PAGE 183

177 Thecurrent into thecapacitor, I C isjust I + 1 + I )]TJ/F7 6.9738 Tf -0.781 -6.918 Td [(1 t I C = I + 1 + I )]TJ/F7 6.9738 Tf -0.781 -6.919 Td [(1 = V g Z 0 .100 since I + 1 = V + 1 Z 0 = V g 2 Z 0 .101 and I )]TJ/F7 6.9738 Tf -0.782 -6.918 Td [(1 = )]TJ/F1 9.9626 Tf 9.409 11.059 Td [( V 1 Z 0 = V g 2 Z 0 .102 Howwillthecurrentintothecapacitor I C t behave?Wehavetorememberthecapacitorequation: I C t = C d dt V C t = C )]TJ/F10 6.9738 Tf 7.018 -4.147 Td [(@ @t )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(V + 1 + V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 t = C d dt V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 t .103 since V + 1 isaconstantandhencehasazerotimederivative.Well,wealsoknowthat I C t = I + 1 + I )]TJ/F7 6.9738 Tf -0.781 -6.918 Td [(1 t = V + 1 Z 0 )]TJ/F10 6.9738 Tf 11.158 5.252 Td [(V )]TJ/F6 4.9813 Tf -1.716 -5.395 Td [(1 t Z 0 .104 Soweequate.103and.104andweget C d dt V )]TJ/F7 6.9738 Tf -2.214 -6.919 Td [(1 t = V + 1 Z 0 )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 t Z 0 .105 or d dt V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 t + 1 Z 0 C V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 t = 1 C V + 1 Z 0 .106 whichgetsusbackto another Di-E-Q! Thehomogeneoussolutioniseasy.Wehave d dt V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 t + 1 Z 0 C V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 t =0 .107 forwhichthesolutionisobviously V )]TJ/F7 6.9738 Tf -2.214 -7.268 Td [(1 homo t = V 0 e )]TJ/F27 6.9738 Tf 6.227 7.681 Td [( t Z 0 C .108 Afteralongtime,thederivativeofthehomogeneoussolutioniszero,andsotheparticularsolutionthe constantpartisthesolutionto 1 Z 0 C V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 part = 1 C V + 1 Z 0 .109 or V )]TJ/F7 6.9738 Tf -2.214 -6.919 Td [(1 part = V + 1 .110 Thecompletesolutionisthesumofthetwo: V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 t = V )]TJ/F7 6.9738 Tf -2.214 -7.267 Td [(1 homo t + V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 part = V 0 e )]TJ/F27 6.9738 Tf 6.226 7.682 Td [( t Z 0 C + V + 1 .111

PAGE 184

178 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES Nowallweneedtodoisnd V 0 ,theinitialcondition.Weknow,however,that V )]TJ/F7 6.9738 Tf -2.214 -6.918 Td [(1 = )]TJ/F11 9.9626 Tf 7.748 0 Td [(V + 1 ,sothat makes V 0 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 V + 1 !Sowehave: V )]TJ/F7 6.9738 Tf -2.214 -6.919 Td [(1 t = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 V + 1 e )]TJ/F27 6.9738 Tf 6.227 7.682 Td [( t Z 0 C + V + 1 = V + 1 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 e )]TJ/F27 6.9738 Tf 6.227 7.682 Td [( t Z 0 C .112 Since V + 1 = V g 2 wecanplot V )]TJ/F7 6.9738 Tf -2.214 -6.919 Td [(1 t asafunctionoftimefromwhichwecanmakeaplotFigure5.48: ReectedVoltageasaFunctionofTimeof )]TJ/F10 6.9738 Tf 6.226 -1.495 Td [(V t ReectedVoltageasaFunctionofTime Figure5.48 Thecapacitorstartsolookinglikeashortcircuit,andchargesuptolooklikeanopencircuit,which makesperfectsense.Canyougureoutwhattheshapewouldbeofapulsereectedoofthecapacitor, giventhatthetimeconstant Z 0 C wasshortcomparedtothewidthofthepulse?

PAGE 185

179 SolutionstoExercisesinChapter5 SolutiontoExercise5.1p.171 3Volts!

PAGE 186

180 CHAPTER5.INTRODUCTIONTOTRANSMISSIONLINES

PAGE 187

Chapter6 ACSteady-StateTransmission 6.1IntroductiontoPhasors 1 Wewillnotalwaysbedealingwithtransmissionlinesexcitedwithapulse.Althoughthisisagoodmodel fordigitalcircuitry,itwillnotalwaysapply.Whenwegotoanalogsignalsrf,highfrequencyanalog,etc. wewillneedmoretoolsthanareavailabletousatthispoint.Inthenot-too-distant-past,thematerial wewillnextconsiderwasstartingtobeconsideredpass.Therfspectrumwasmoreorlesslledup,and thewatchwordwas"digital".Now,inthenewageofwirelesscommunication,cellphones,andrfLocal AreaNetworks,demandforengineerswhounderstandacbehaviorontransmissionlinesandwhocandesign systemswhichworkwellwithrfsignalsareverymuchindemand.Payheedtowhatwesayhere,andyou mightwellndyourselfwithmanylucrativejoboersinthefuture. Tobegin,wewanttoconsideratransmissionlinewhichisbeingexcitedwithanoscillatingsource Figure6.1SinusoidalExcitationofaLoadedTransmissionLine. SinusoidalExcitationofaLoadedTransmissionLine Figure6.1 Theusualset-upincludesasource,withasinusoidaloutput,asourceimpedance Z g atransmissionline withimpedance Z 0 L meterslong,andaloadofimpedance Z L attheend. Let'slookatthesourcerst.Wecandescribetheoutputwaveformfromthegeneratoras V t = V g cos !t + .1 WhichwhenplottedlookeslikeFigure6.2ExcitationWaveform. 1 Thiscontentisavailableonlineat. 181

PAGE 188

182 CHAPTER6.ACSTEADY-STATETRANSMISSION ExcitationWaveform Figure6.2 Theoscillatingwaveformhasaperiod T anditsangularfrequency isgivenas = 2 T =2 f .2 Theangle, ,whichspecieshowmuchthewaveisleadingacosinefunctionwithzeroo-setisgivenby =2 T .3 Whatwe donot wanttodo,iscarryabunchofsineandcosinefunctionsaroundwithuseverywhere.Once westartmultiplyinganddividing,thetrigturnsintoabigmess,andgetsinthewayofourunderstanding ofwhatisgoingon.Thewaywedealwiththis,aseverygood242studentknows,istointroduce phasors Sinceweknowfrom Euler'sIdentity V g e j !t + = V g cos !t + + jsin !t + .4 Ifwetakearealpartof V g e j !t + wewillextractthevoltagewaveformwedesire.Wewillre-writethis functionas V g e j !t + = V g e j e j!t .5 andthen dene V g asthe phasorvoltage where V g = V g e j .6 Notethat V g isacomplexquantity,withbothamagnitude j V g j andaphaseangle .Inordertoretrievea realvoltagesignalfromaphasor,wehavetomultiplythephasorby e j!t andthentaketherealpart.Note thatthisisthesamethingasplottingthephasoronthecomplexplane,andthenobservingtheprojectionof thephasorontherealaxis,asthephasorrotatesaroundatarate !t Figure6.3PhasorRepresentation.

PAGE 189

183 PhasorRepresentation Figure6.3 Thismethodofvisualizationwillsometimeshelpmakeresultsseemalittleeasiertounderstand,orat leastcheckforreasonableness. 6.2A/CLineBehavior 2 Ifwearegoingtotrytousephasorsonatransmissionline,thenwehavetoallowforspatialvariationas well.Thisissimpletodo,ifwejustletthephasorbeafunctionof x ,sowehave V x .Howthephasor variesin x isoneofthethingswenowhavetondout. Let'sstartwiththe Telegrapher'sEquations again. @ @x V x;t = )]TJ/F81 9.9626 Tf 7.749 0 Td [(L @ @t I x;t .7 @ @x I x;t = )]TJ/F81 9.9626 Tf 7.749 0 Td [(C @ @t V x;t .8 For V x;t wecannowsubstitute V x e j!t andfor I x;t weplugin I x e j!t .Soweget: @ @x V x e j!t = )]TJ/F81 9.9626 Tf 7.749 0 Td [(L @ @t I x e j!t .9 and @ @x I x e j!t = )]TJ/F81 9.9626 Tf 7.748 0 Td [(C @ @t V x e j!t .10 Wetakethederivativewithrespecttotime,whichbringsdowna j! andthenwecancelthe e j!t fromboth sidesofeachequation: @ @x V x = )]TJ/F1 9.9626 Tf 9.409 11.059 Td [( j! L I x .11 and @ @x I x = )]TJ/F1 9.9626 Tf 9.409 11.058 Td [( j! C V x .12 Viola !Inonesimplemotion,wehavecompletelyeliminatedthetimevariable, t ,fromourequations!Itis notreallygone,ofcourse,foroncewegureoutwhat V x is,wehavetomultiplyitby e j!t andthentake 2 Thiscontentisavailableonlineat.

PAGE 190

184 CHAPTER6.ACSTEADY-STATETRANSMISSION therealpartbeforewecanextractonceagain,theactual V x;t thatwewant.Nonetheless,insofarasthe telegrapher'sequationsareconcerned, t hasdisappearedfromtheradarscreen. Tosolvethesewedojustaswedidwiththetransientproblem.Wetakeaderivativewithrespectto x of.11,whichgivesusa @ @x I x ontherighthandside,forwhichwecansubstitute.12,whichleaves uswith @ 2 @x 2 V x = )]TJ/F1 9.9626 Tf 9.409 11.059 Td [( 2 LC V x .13 -times-is+,but jj = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 andsowehavea-infrontofthe 2 .Wethenre-write.13as @ 2 @x 2 V x + 2 LC V x =0 .14 Thesimplestsolutiontothisequationis V x = V 0 e j! p LC x .15 fromwhichwecanthengettheactualvoltagesignal V x;t = V x e j!t = V 0 e j !t p LC x .16 Notethatwecouldfactoroutan e j! p LC ,fromtheexponent,which,sinceitisjustaconstant,wecould includein V 0 andcallit V 0 ,switchtheorderof x and t ,andwrite.16as V x;t = V 0 e j x 1 p LC t .17 whichlooksalotlikethe"general" f x vt solutionweweretalkingaboutearlier.19! Thenumber p LC isspecial.ItisusuallyrepresentedwithaGreekletter andiscalledthe propagationcoecient .Thuswehave V x;t = V 0 e j !t x .18 Aspreviously,apointonthewaveofconstantphaserequiresthattheargumentinsidetheparenthesis remainsconstant.Thusif V x 1 ;t 1 isgoingtoequal V x 2 ;t 2 i.e. whatwasatpoint x 1 at t 1 isnowat x 2 attime t 2 itmustbethat !t 1 x 1 = !t 2 x 2 .19 or x 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 t 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(t 1 = x t = = ! p LC = 1 p LC v p .20 Whichoneagain,denesthe phasevelocity ofthewave.Otherrelationshipstokeepinmindare = 2 .21 = v p f = ! 2 = 2 .22 Therstcomesfromthefactthatthewavevariesin x as e jx .Thuswhen x = ,thewavelength, just increasesby 2 ,togetthephasortogothroughonefullrotation.Notealso,asbefore,thechoiceofthe minussigninthe in.18representsawavegoinginthe + x direction,whilethechoiceofthe+signwill

PAGE 191

185 giveawavegoinginthe )]TJ/F11 9.9626 Tf 7.749 0 Td [(x direction.Clearly,bystartingouttakingthex-derivativeoftheequationfor I x;t wewouldendupwith I x;t = I 0 e j !t x .23 Let'sconsiderthetwophasorsthen,anddenethevoltagephasorassociatedwiththepositivegoingvoltage waveas V plus x = V + e )]TJ/F7 6.9738 Tf 6.226 0 Td [( jx .24 andthenegativevoltagephasoras V minus x = V )]TJ/F11 9.9626 Tf 6.725 -4.114 Td [(e jx .25 Weshouldkeepinmindthatboth V + and V )]TJ/F15 9.9626 Tf 10.508 -3.616 Td [(canbe,andprobablyare,complexnumbers.Fromnow onwewilldropthelittle overthevariablesbecauseitsverytedioustogetittoshowupwiththisword processor.Youwilljusthavetokeepinmindthatanyvariablewedonotexplicitlyputinsideabsolute valuemarkers i.e. j V + j isgoingtobe,ingeneral,acomplexnumber.Wewill,ofcourse,havesimilar expressionsforthepositiveandnegativegoingcurrentwaves. Let'sconsiderthepositivegoingcurrentandvoltagewaves,andplugtheminto.11. @ @x V + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jx = )]TJ/F1 9.9626 Tf 9.409 11.059 Td [( j! L I + e )]TJ/F7 6.9738 Tf 6.226 0 Td [( jx .26 Thex-derivativebringsdowna )]TJ/F8 9.9626 Tf 9.409 0 Td [( j ,the e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jx 'scancel,andwehave V + = j! L j I + .27 But,since = p LC wehave V + = r L C I + Z 0 I + .28 aswehadbefore. So,whathaschanged?Notmuchfromthecaseoftransientsonaline.Wewillnowassumewehave a steadystate problem.Thismeansweturnedonthegeneratoralongtimeago.Weassumethatithas beenconnectedtothelinelongenoughsothatalltransientbehaviorhasdiedaway,andthatvoltagesand currentsarenotchanginganymoreexceptoscillatingatfrequency ,ofcourse. Ifthelineissemi-inniteormatchedwithaloadequalto Z 0 Figure6.4AWaveOnaSemi-Innite Linethenitisprettyobviousthat V + = Z 0 Z 0 + Z g V g .29 where Z g isthesourceimpedance,and V g isthesourcevoltagephasor. AWaveOnaSemi-InniteLine Figure6.4

PAGE 192

186 CHAPTER6.ACSTEADY-STATETRANSMISSION 6.3TerminatedLines 3 If,ontheotherhand,wehaveaniteline,terminatedwithsomeloadimpedance,wehaveasomewhatmore complicatedproblemtodealwithFigure6.5AFiniteTerminatedTransmissionLine. AFiniteTerminatedTransmissionLine Figure6.5 Thereareseveralthingsweshouldnote before weheadointoequation-landagain.Firstofall,unlike thetransientproblemswelookedatinapreviouschapterSection5.6,therecanbenomorethan two voltageandcurrentsignalsontheline,just V + and V )]TJ/F15 9.9626 Tf 6.725 -3.615 Td [(,and I + and I )]TJ/F15 9.9626 Tf 6.725 -3.615 Td [(.Wenolongerhavetheluxuryof having V + 1 V + 2 ,etc.,becausewearetalkingnowabouta steadystatesystem .Allofthetransientsolutions whichbuiltupwhenthegeneratorwasrstconnectedtothelinehavebeensummedtogetherintojusttwo waves. Thus,onthelinewehaveasingle totalvoltagefunction ,whichisjustthesumofthepositiveand negativegoingvoltagewaves V x = V + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jx + V )]TJ/F11 9.9626 Tf 6.725 -4.113 Td [(e + jx .30 andatotalcurrentfunction I x = I + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jx + I )]TJ/F11 9.9626 Tf 6.724 -4.114 Td [(e + jx .31 Notealsothatuntilwehavesolvedfor V + and V )]TJ/F15 9.9626 Tf 6.725 -3.615 Td [(,wedonotknow V x or I x anywhereontheline. Inparticular,wedonotknow V and I whichwouldtelluswhattheapparentimpedanceislooking intotheline. Z in = Z = V + + V )]TJETq1 0 0 1 312.124 256.352 cm[]0 d 0 J 0.398 w 0 0 m 30.469 0 l SQBT/F10 6.9738 Tf 314.471 250.426 Td [(I + + I )]TJ/F15 9.9626 Tf 180.195 10.668 Td [(.32 Untilweknowwhatkindofimpedancethegeneratorisseeing,wecannotgureouthowmuchofthe generator'svoltagewillbecoupledtotheline!Theinputimpedancelookingintothelineisnowafunction oftheloadimpedance,thelengthoftheline,andthephasevelocityontheline.Wehavetosolvethis before wecangureouthowthelineandgeneratorwillinteract. Theapproachweshallhavetotakeisthefollowing.Wewillstartatthe load endoftheline,andina mannersimilartotheoneweusedpreviously,ndarelationshipbetween V + and V )]TJ/F15 9.9626 Tf 6.725 -3.615 Td [(,leavingtheiractual magnitudeandphaseassomethingtobedeterminedlater.Wecanthenpropagatethetwovoltagesand currentsbackdowntotheinput,determinewhattheinputimpedanceisbyndingtheratioof V + + V )]TJ/F15 9.9626 Tf 6.725 -3.615 Td [( to I + + I )]TJ/F15 9.9626 Tf 6.725 -3.616 Td [(,andfromthis,andknowledgeofpropertiesofthegeneratoranditsimpedance,determine whattheactualvoltagesandcurrentsare. Let'stakealookattheload.WeagainuseKVLandKCLFigure6.6DoingKirchoattheEndofthe Linetomatchvoltagesandcurrentsinthelineandvoltagesandcurrentsintheload: V + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jL + V )]TJ/F11 9.9626 Tf 6.725 -4.114 Td [(e + jL = V L .33 3 Thiscontentisavailableonlineat.

PAGE 193

187 and I + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jL + I )]TJ/F11 9.9626 Tf 6.725 -4.114 Td [(e + jL = I L .34 DoingKirchoattheEndoftheLine Figure6.6: Changevariables! Now,we could substitute V Z 0 forthetwocurrentsonthelineand V L Z L for I L ,andthentrytosolvefor V )]TJ/F15 9.9626 Tf -461.275 -16.982 Td [(intermsof V + using.33and.34butwecanbealittlecleverattheoutset,andmakeourcomplex algebraagoodbitcleanerFigure6.7s=0attheLoadandSotheExponentialsGoAway!.Let'smakea changeofvariableandlet s L )]TJ/F11 9.9626 Tf 9.963 0 Td [(x .35 s=0attheLoadandSotheExponentialsGoAway! Figure6.7 Thisthengivesusforthevoltageonthelineusing x = L )]TJ/F11 9.9626 Tf 9.963 0 Td [(s V s = V + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( jL e js + V )]TJ/F11 9.9626 Tf 6.725 -4.114 Td [(e jL e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js .36 Usually,wejustfoldtheconstantphaseterms e jL termsinwiththe V + and V )]TJ/F15 9.9626 Tf 10.044 -3.615 Td [(andsowehave: V s = V + e js + V )]TJ/F11 9.9626 Tf 6.724 -4.113 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js .37

PAGE 194

188 CHAPTER6.ACSTEADY-STATETRANSMISSION Notethatwhenwedothis,wenowhavea positive exponentialinthersttermassociatedwith V + anda negative exponentialassociatedwiththe V )]TJ/F15 9.9626 Tf 10.045 -3.615 Td [(term.Ofcourse,wealsogetfor I s : I s = I + e js + I )]TJ/F11 9.9626 Tf 6.725 -4.114 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js .38 Thischangenowmovesourorigintothe load endoftheline,andreversesthedirectionofpositivemotion. But ,nowwhenwepluginto e js thevaluefor s attheload s =0 ,theequationssimplifyto: V + + V )]TJ/F8 9.9626 Tf 9.492 -4.114 Td [(= V L .39 and I + + I )]TJ/F8 9.9626 Tf 9.492 -4.114 Td [(= I L .40 whichwethenre-writeas V + Z 0 )]TJ/F11 9.9626 Tf 11.158 6.74 Td [(V )]TJETq1 0 0 1 299.009 525.435 cm[]0 d 0 J 0.398 w 0 0 m 14.75 0 l SQBT/F11 9.9626 Tf 300.749 516.11 Td [(Z 0 = V L Z L .41 ThisisbeginningtolookalmostexactlylikeapreviouschapterSection5.1.Asareminder,wesolve.41 for V L V L = Z L Z 0 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(V + )]TJ/F11 9.9626 Tf 9.962 0 Td [(V )]TJ/F1 9.9626 Tf 6.725 3.956 Td [( .42 andsubstitutefor V L in.39 V + + V )]TJ/F8 9.9626 Tf 9.492 -4.114 Td [(= Z L Z 0 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(V + )]TJ/F11 9.9626 Tf 9.963 0 Td [(V )]TJ/F1 9.9626 Tf 6.725 3.956 Td [( .43 Fromwhichwethensolveforthereectioncoecient )]TJ/F10 6.9738 Tf 6.226 -1.495 Td [( ,theratioof V )]TJ/F15 9.9626 Tf 10.045 -3.615 Td [(to V + V )]TJETq1 0 0 1 260.448 384.06 cm[]0 d 0 J 0.398 w 0 0 m 14.75 0 l SQBT/F11 9.9626 Tf 260.504 374.736 Td [(V + )]TJ/F10 6.9738 Tf 6.226 -1.495 Td [( = Z L )]TJ/F11 9.9626 Tf 9.962 0 Td [(Z 0 Z L + Z 0 .44 Notethatsince,ingeneral, Z L willbecomplex,wecanexpectthat )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( willalsobeacomplexnumberwith bothamagnitude j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j andaphaseangle )]TJ/F15 9.9626 Tf 5.424 1.494 Td [(.Also,aswiththecasewhenwewerelookingattransients, j )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [( j < 1 Sincewenowknow V )]TJ/F15 9.9626 Tf 10.15 -3.615 Td [(intermsof V + ,wecannowwriteanexpressionfor V s thevoltageanywhere ontheline. V s = V + e js + V )]TJ/F11 9.9626 Tf 6.724 -4.114 Td [(e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js .45 Noteagainthechangeinsignsinthetwoexponentials.Sinceourspatialvariable s isgoingintheopposite directionfrom x ,the V + phasornowgoesas + js andthe V )]TJ/F15 9.9626 Tf 10.045 -3.616 Td [(phasornowgoesas )]TJ/F8 9.9626 Tf 9.409 0 Td [( js Wenowsubstitutein )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( V + for V )]TJ/F15 9.9626 Tf 9.767 -3.616 Td [(in.45,andforreasonsthatwillbecomeapparentsoon,factorout an e js V s = V + e js +)]TJ/F10 6.9738 Tf 16.19 -1.495 Td [( V + e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js = V + )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(e js +)]TJ/F10 6.9738 Tf 16.189 -1.495 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js = V + e js )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+)]TJ/F10 6.9738 Tf 23.385 -1.494 Td [( e )]TJ/F7 6.9738 Tf 6.226 0 Td [( js .46 Wecouldhavealsowrittendownanequationfor I s ,thecurrentalongtheline.Itwillbeagoodtestof yourunderstandingofthebasicequationswearedevelopingheretoshowyourselfthatindeed I s = V + e js Z 0 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [()]TJ/F10 6.9738 Tf 6.227 -1.495 Td [( e )]TJ/F7 6.9738 Tf 6.226 0 Td [( js .47

PAGE 195

189 6.4LineImpedance 4 Unfortunately,sincewedon'tknowwhatvaluethephasor V + has,theseequationsdonotdousawhole lotofgood!Onewaytodealwiththisistosimplydividethisequation.47intothisequation.46. Thatgetsridof V + andthe e js andsowenowcomeupwitha new variable,whichweshallcall line impedance Z s Z s V s I s = Z 0 1+)]TJ/F10 6.9738 Tf 23.384 -1.494 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 js 1 )]TJ/F8 9.9626 Tf 9.962 0 Td [()]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 js .48 Z s representstheratioofthetotalvoltagetothetotalcurrentanywhereontheline.Thus,ifwehave aline L long,terminatedwithaloadimpedance Z L ,whichgivesrisetoaterminalreectioncoecient )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [( thenifwesubstitute )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [( and L into.48,the Z L whichwecalculatewillbethe"apparent"impedance whichwewouldseelookingintotheinputterminalstotheline! Thereareseveralwaysinwhichwecanlookat.48.Oneistotrytoputitintoamoretractable form,thatwemightbeabletousetond Z s ,givensomelineimpedance Z 0 ,aloadimpedance Z L and adistance, s awayfromtheload.Wecanstartoutbymultiplyingtopandbottomby e js ,substitutingin for )]TJ/F10 6.9738 Tf 6.226 -1.495 Td [( ,andthenmultiplyingtopandbottomby Z L + Z 0 Z s = Z 0 Z L + Z 0 e js + Z L )]TJ/F11 9.9626 Tf 9.962 0 Td [(Z 0 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js Z L + Z 0 e js )]TJ/F8 9.9626 Tf 9.963 0 Td [( Z L )]TJ/F11 9.9626 Tf 9.962 0 Td [(Z 0 e )]TJ/F7 6.9738 Tf 6.227 0 Td [( js .49 Next,weuseEuler'srelation,andsubstitute cos s jsin s fortheexponential.Lotsofthingswill cancelout,andifwedothemathcarefully,weendupwith Z s = Z 0 Z L + jZ 0 tan s Z 0 + jZ L tan s .50 Forsomepeople,thisequationismoresatisfyingthan.48,butforme,bothareaboutequallyopaque intermsifseeinghow Z s isgoingtobehavewithvariousloads,aswemovedownthelinetowardsthe generator..50 does havethenicepropertythatitiseasytocalculate,andhencecouldbeputinto MATLABoraprogrammablecalculator.Infactyoucouldprogram.48justaswellforthatmatter. Youcouldspecifyacertainsetofconditionsandeasilynd Z s ,butyouwouldnotgetmuchinsightinto howatransmissionlineactuallybehaves. 6.5CrankDiagram 5 Weactuallystillhavesomeoptionsopentous.Oneofthenicest,atleastintermsofgettingsomeinsight, iscalla crankdiagram .Notethatthisequation 6 isacomplexequation,whichrequiresustotakeathe ratiooftwocomplexnumbers; 1+)]TJ/F10 6.9738 Tf 23.384 -1.494 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 s and 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [()]TJ/F10 6.9738 Tf 6.226 -1.494 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 2 s Let'splotthesetwoquantitiesonthecomplexplane,startingat s =0 theloadendoftheline.Wecan represent )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( ,thereectioncoecient,byitsmagnitudeanditsphase, j )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [( j and )]TJ/F15 9.9626 Tf 5.424 1.494 Td [(.Forthenumeratorwe plota1,andthenaddthecomplexvector )]TJ/F15 9.9626 Tf 9.612 0 Td [(whichhasalength j )]TJ/F14 9.9626 Tf 6.226 0 Td [(j andsitsatanangle )]TJ/F15 9.9626 Tf 8.809 1.495 Td [(withrespectto therealaxisFigure6.8Plot.Thedenominatorisjustthesamething,exceptthe )]TJ/F15 9.9626 Tf 9.92 0 Td [(vectorpointsinthe oppositedirectionFigure6.9AnotherPlot. 4 Thiscontentisavailableonlineat. 5 Thiscontentisavailableonlineat. 6 "LineImpedance",

PAGE 196

190 CHAPTER6.ACSTEADY-STATETRANSMISSION Plot Figure6.8: Plotting 1+)]TJ/F59 5.9776 Tf 21.63 -0.996 Td [( AnotherPlot Figure6.9: Plotting 1 )]TJ/F56 8.9664 Tf 9.216 0 Td [()]TJ/F59 5.9776 Tf 5.759 -0.997 Td [( Thetopvectorisproportionalto V s andthebottomvectorisproportionalto I s Figure6.10Another CrankDiagram.Ofcourse,for s =0 weareattheloadso V s =0= V L and I s =0= I L .

PAGE 197

191 AnotherCrankDiagram Figure6.10: Showingthat 1+)]TJ/F59 5.9776 Tf 21.63 -0.997 Td [( = V L V + and 1 )]TJ/F56 8.9664 Tf 9.215 0 Td [()]TJ/F59 5.9776 Tf 5.759 -0.997 Td [( = Z 0 I L V + Aswemovedowntheline,thetwo" )]TJ/F15 9.9626 Tf 6.227 0 Td [("vectorsrotatearoundatarateof )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 s Figure6.11Rotatingthe PhasorsOntheCrankDiagram.Astheyrotate,onevectorgetslongerandtheothergetsshorter,andthen theoppositeoccurs.Inanyevent,toget Z s wehavetodividetherstvectorbythesecond.Ingeneral, thisisnoteasytodo,butthereare some placeswhereitisnottoobad.Oneoftheseiswhen 2 s = )]TJ/F11 9.9626 Tf 7.749 0 Td [( )]TJ/F15 9.9626 Tf -462.576 -10.461 Td [(Figure6.12RotatingaCrankDiagram. RotatingthePhasorsOntheCrankDiagram Figure6.11

PAGE 198

192 CHAPTER6.ACSTEADY-STATETRANSMISSION RotatingaCrankDiagram Figure6.12: Rotatingtoa V max Atthispoint,thevoltagevectorhasrotatedaroundsothatitisjustlyingontherealaxis.Obviouslyits lengthisnow 1+ j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j .Bythesametoken,thecurrentvectorisalsolyingontherealaxis,andhasalength 1 )-222(j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j .Dividingonebytheother,andmultiplyingby Z 0 givesus Z s atthispoint. Z s = Z 0 1+ j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j 1 )-222(j )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [( j .51 Whereisthispoint,anddoesithaveanyspecialmeaning?Forthis,weneedtogobacktoourexpression for V s inthisequation. V s = V + e js )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+)]TJ/F10 6.9738 Tf 23.384 -1.495 Td [( e )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 js = V + e js )]TJ/F8 9.9626 Tf 4.567 -8.069 Td [(1+ j )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [( j e j )]TJ/F13 6.9738 Tf 4.635 0.996 Td [()]TJ/F7 6.9738 Tf 6.227 0 Td [(2 s = V + e js )]TJ/F8 9.9626 Tf 4.567 -8.07 Td [(1+ j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j e j s .52 wherewehavesubstituted j )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [( j e j forthephasor )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( andthendenedanewangle s = )]TJ/F14 9.9626 Tf 7.638 1.494 Td [()]TJ/F8 9.9626 Tf 9.963 0 Td [(2 s Nowlet'sndthemagnitudeof V s .Todothisweneedtosquaretherealandimaginaryparts,add them,andthentakethesquareroot. j V s j = j V + j )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(1+ j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j e j s = j V + j q + j )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [( j cos s 2 + j )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [( j 2 sin 2 s .53 so, j V s j = j V + j q 1+2 j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j cos s + j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j 2 cos 2 s + j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j 2 sin 2 s .54 which,since sin 2 + cos 2 =1 j V s j = j V + j q 1+ j )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [( j 2 +2 j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j cos s .55 Remember, s isananglewhichchangeswith s .Inparticular, s = )]TJ/F14 9.9626 Tf 7.165 1.494 Td [()]TJ/F8 9.9626 Tf 9.489 0 Td [(2 s .Thus,aswemovedown theline j V s j willoscillateas cos s oscillates.Atypicalplotfor V s for j )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [( j =0 : 5 and )]TJ/F8 9.9626 Tf 8.541 1.495 Td [(=45 isshownhereFigure6.13:StandingWavePattern.

PAGE 199

193 StandingWavePattern Figure6.13 6.6StandingWaves/VSWR 7 AStandingWavePattern Figure6.14 InmakingthisplotFigure6.14:AStandingWavePattern,wehavemadeuseofthefactthatthepropagationconstant canalsobeexpressedas 2 ,andsofortheindependentvariable,insteadofshowing s in metersorwhatever,wenormalizethedistanceawayfromtheloadtothewavelengthoftheexcitationsignal, andhenceshowdistanceinwavelengths.Whatweareshowinghereiscalleda standingwave .Thereare placesalongthelinewherethemagnitudeofthevoltage j V s j hasamaximumvalue.Thisiswhere V + and V )]TJ/F15 9.9626 Tf 10.09 -3.615 Td [(areaddingupinphasewithoneanother,andplaceswherethereisavoltageminimum,where V + and V )]TJ/F15 9.9626 Tf 10.435 -3.616 Td [(addupoutofphase.Since j V )]TJ/F14 9.9626 Tf 6.725 -3.616 Td [(j = j )]TJ/F10 6.9738 Tf 6.226 -1.494 Td [( jj V + j ,themaximumvalueofthestandingwavepatternis 1+ j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j times j V + j andtheminimumis 1 )-246(j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j times j V + j .Notethatanywhereontheline,thevoltage is still oscillatingat e j!t ,andsoitisnotaconstant,itisjustthatthe magnitude oftheoscillatingsignal changesaswemovedowntheline.Ifweweretoputanoscilloscopeacrosstheline,wewouldseeanAC signal,oscillatingatafrequency Anumberofconsiderableinterestistheratioofthemaximumvoltageamplitudetotheminimumvoltage amplitude,calledthe voltagestandingwaveratio ,orVSWRforshort.Itiseasytoseethat: VSWR = 1+ j )]TJ/F14 9.9626 Tf 6.226 0 Td [(j 1 )-222(j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j .56 7 Thiscontentisavailableonlineat.

PAGE 200

194 CHAPTER6.ACSTEADY-STATETRANSMISSION Notethatbecause j )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [( j2 [0 ; 1] VSWR 2 [1 ; 1 ] AlthoughFigure6.14AStandingWavePatternlookslikethestandingwavepatternismoreorless sinusoidal,ifweincrease j )]TJ/F14 9.9626 Tf 6.226 0 Td [(j to0.8,weseethatitmostdenitelyisnot.Thereisalsoatemptationtosay thatthespacingbetweenminimaormaximaofthestandingwavepatternis ,thewavelengthofthe signal,butacloserinspectionofeitherFigure6.14AStandingWavePatternorFigure6.15Standing WavePatternwithaLargerReectionCoecient,showsthatinfactthespacingbetweenfeaturesisonly half awavelength,or 2 .Whyisthis?Well, s goesas )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 s and = 2 ,andsoeverytime s increases by 2 s decreasesby 2 andwehavecomeonefullcycleontheway j V s j behaves. StandingWavePatternwithaLargerReectionCoecient Figure6.15 Nowlet'sgobacktotheCrankDiagramFigure6.8:Plot.Atthepositionshown,weareatavoltage maximum,and Z s Z 0 justequalstheVSWR. Z s V max Z 0 = VSWR = 1+ j )]TJ/F9 4.9813 Tf 4.926 -0.996 Td [( j 1 j )]TJ/F9 4.9813 Tf 4.926 -0.996 Td [( j .57 Notealsothatatthisparticularpoint,thatthevoltageandcurrentphasorsareinphasewithoneanother linedupinthesamedirectionandhencetheimpedancemustbe real orresistive. Wecanmovefurtherdowntheline,andnowthe V s phasorstartsshrinking,andthe I s phasor startstogetbiggerFigure6.16MovingFurtherDowntheLine.

PAGE 201

195 MovingFurtherDowntheLine Figure6.16: Movingfurtherdownthelinefroma V max Ifwemoveevenfurtherdowntheline,wegettoapointwherethecurrentphasorisnowatamaximum value,andthevoltagephasorisataminimumvalueFigure6.17MovingEvenFurtherDowntheLine.We arenowatavoltageminimum,theimpedanceisagainrealthevoltageandcurrentphasorsarelinedup withoneanother,sotheymustbeinphaseand Z s V min = 1 VSWR = 1 j )]TJ/F9 4.9813 Tf 4.926 -0.996 Td [( j 1+ j )]TJ/F9 4.9813 Tf 4.926 -0.996 Td [( j .58 MovingEvenFurtherDowntheLine Figure6.17: Crankdiagramata V min Theonlyproblemwehavehereisthatexceptatavoltageminimumormaximum,nding Z s fromthe crankdiagramisnotverystraightforward,sincethevoltageandcurrentareoutofphase,anddividingthe twovectorsbecomessomewhattedious.

PAGE 202

196 CHAPTER6.ACSTEADY-STATETRANSMISSION 6.7BilinearTransform 8 There is awaythatwecanmakethingsagoodbiteasierforourselveshowever.Theonlydrawbackisthatwe havetodosomecomplexanalysisrst,andlookata bilineartransform !Let'sdoonemoresubstitution, anddeneanothercomplexvector,whichwecancall r s : r s j )]TJ/F10 6.9738 Tf 6.227 -1.495 Td [( j e j r )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 s .59 Thevector r s isjusttherotatingpartofthecrankdiagramwhichwehavebeenlookingatFigure6.18 TheVectorrs.Ithasamagnitudeequaltothatofthereectioncoecient,anditrotatesaroundata rate 2 s aswemovedowntheline.Forevery r s thereisacorresponding Z s whichisgivenby: Z s = Z 0 1+ r s 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(r s .60 TheVectorrs Figure6.18 Now,itturnsouttobeeasierifwetalkabouta normalizedimpedance ,whichwegetbydividing Z s by Z 0 Z s Z 0 = 1+ r s 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(r s .61 whichwecansolvefor r s r s = Z s Z 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 Z s Z 0 +1 .62 Thisrelationshipiscalleda bilineartransform .Forevery r s thatwecanimagine,thereisoneand onlyone Z s Z 0 andforevery Z s Z 0 thereisoneandonlyone r s .Whatwewouldliketobeabletodo,is nd Z s Z 0 ,givenan r s .Thereasonforthisshouldbereadilyapparent.Whereas,aswemovealongin s Z s Z 0 behavesinamostdicultmannerdividingonephasorbyanother, r s simplyrotatesaroundonthe complexplane.Givenone r s 0 itis easy tondanother r s .Wejustrotatearound! Weshallndtherequiredrelationshipinagraphicalmanner.SupposeIhaveacomplexplane,representing Z s Z 0 .AndthensupposeIhavesomepoint"A"onthatplaneandIwanttoknowwhatimpedance 8 Thiscontentisavailableonlineat.

PAGE 203

197 itrepresents.Ijustreadalongthetwoaxes,andndthat,fortheexampleinFigure6.19TheComplex ImpedancePlane,"A"representsanimpedanceof Z s Z 0 =4+2 j .WhatIwouldliketodowouldbetoget agridsimilartothatonthe Z s Z 0 plane,butonthe r s planeinstead.Thatway,ifIknewoneimpedence say Z Z 0 = Z L Z 0 thenIcouldndanyotherimpedance,atanyother s ,bysimplyrotating r s aroundby 2 s ,andthenreadingothenew Z s Z 0 fromthegridIhaddeveloped.Thisiswhatweshallattempttodo. TheComplexImpedancePlane Figure6.19 Let'sstartwith.62andre-writeitas: r s = Z s Z 0 +1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Z s Z 0 +1 =1+ )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Z s Z 0 +1 .63 Inordertouse.63,wearegoingtohavetointerpretitinawaywhichmightseemalittleoddtoyou. Thewaywewillreadtheequationistosay:"Take Z s Z 0 andadd1toit.Invertwhatyouget,andmultiply by-2.Thenadd1totheresult."Simpleisn'tit?Theonlyhardpartwehaveindoingthisisinverting Z s Z 0 +1 .This,itturnsout,isprettyeasyoncewelearnoneveryimportantfact. The one factaboutalgebraonthecomplexplanethatweneedisasfollows.Consideraverticalline, s onthecomplexplane,locatedadistance d awayfromtheimaginaryaxisFigure6.20AVerticalLine,s,a Distance,d,AwayFromtheImaginaryAxis.Therearealotofwayswecouldexpresstheline s ,butwe willchooseonewhichwillturnouttobeconvenientforus.Let'slet: s = d )]TJ/F11 9.9626 Tf 9.962 0 Td [(jtan 2 h )]TJ/F1 9.9626 Tf 9.409 11.059 Td [( 2 ; 2 i ; .64

PAGE 204

198 CHAPTER6.ACSTEADY-STATETRANSMISSION AVerticalLine,s,aDistance,d,AwayFromtheImaginaryAxis Figure6.20 Nowweaskourselvesthequestion:whatistheinverseofs? 1 s = 1 d 1 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(jtan .65 Wecansubstitutefor tan : 1 s = 1 d 1 1 )]TJ/F10 6.9738 Tf 6.227 0 Td [(j sin cos = 1 d cos cos )]TJ/F10 6.9738 Tf 6.226 0 Td [(jsin .66 Andthen,since cos )]TJ/F11 9.9626 Tf 9.963 0 Td [(jsin = e )]TJ/F7 6.9738 Tf 6.227 0 Td [( j 1 s = 1 d cos e )]TJ/F6 4.9813 Tf 5.397 0 Td [( j = 1 d cos e j .67

PAGE 205

199 APlotof1/s Figure6.21 AcarefullookatFigure6.21APlotof1/sshouldallowyoutoconvinceyourselfthat.67isan equationforacircleonthecomplexplane,withadiameter = 1 d .If s isnotparalleltotheimaginaryaxis, butratherhasitsperpendiculartotheoriginatsomeangle ,tomakealine s 0 Figure6.22TheLines'. Since s 0 = se j ,taking 1 s simplywillgiveusacirclewithadiameterof 1 d ,whichhasbeenrotatedbyan angle fromtherealaxisFigure6.23InverseofaRotatedLine.Andsowecometothe one factwe havetokeepinmind: "Theinverseofastraightlineonthecomplexplaneisacircle,whosediameteristhe inverseofthedistancebetweenthelineandtheorigin." TheLines' Figure6.22: Theline s multipliedby e j

PAGE 206

200 CHAPTER6.ACSTEADY-STATETRANSMISSION InverseofaRotatedLine Figure6.23 6.8TheSmithChart 9 Nowlet'sseehowwecanuseTheBilinearTransform.63togettheco-ordinatesonthe Z s Z 0 plane transferredoverontothe r s plane.TheBilinearTransform.63tellsushowtotake any Z s Z 0 and generatean r s fromit.Let'sstartwithaneasyone.Wewillassumethat Z s Z 0 =1+ jX ,whichisa verticalline,whichpassesthrough1,andcantakeonwhateverimaginarypartitwantsFigure6.24Complex ImpedenceWithRealPart=+1. ComplexImpedenceWithRealPart=+1 Figure6.24 9 Thiscontentisavailableonlineat.

PAGE 207

201 AccordingtoTheBilinearTransform.63,therstthingweshoulddoisadd1to Z s Z 0 .Thisgivesus theline 2+ jX Figure6.25Adding1. Adding1 Figure6.25 Now,wetaketheinverseofthis,whichwillgiveusacircle,ofdiameter1/2Figure6.26Inverting.Now, accordingtoTheBilinearTransform.63wetakethiscircleandmultiplyby-2Figure6.27Multiplying by-2. Inverting Figure6.26

PAGE 208

202 CHAPTER6.ACSTEADY-STATETRANSMISSION Multiplyingby-2 Figure6.27 Andnally,wetakethecircleandadd+1toit:asshownhereFigure6.28:Adding1OnceAgain. There,wearedonewiththetransform.Theverticallineonthe Z s Z 0 planethatrepresentsanimpedance witharealpartof+1andanimaginarypartwithanyvaluefrom )]TJ/F8 9.9626 Tf 9.409 0 Td [( j 1 to + j 1 hasbeenreducedtoa circlewithdiameter1,passingthrough0and1onthecomplex r s plane. Adding1OnceAgain Figure6.28 Let'sdothesamethingfor Z s Z 0 =0 : 5+ jX and Z s Z 0 =2+ jX .We'llcalltheselinesAandBrespectively, andjustaddthesetothesketcheswealreadyhaveFigure6.29TwoMoreExamples.Followalongwith TheBilinearTransform.63,andseeifyoucangureoutwhereeachofthesesketchescomesfrom.We willsimplybedoingthesamethingsagain:add1;invert;multiplyby-2;add1onceagain.Asyoucan seeinFigure6.30Add+1toEach,Figure6.31Inverting,Figure6.32MultiplyBy-2,andFigure6.33 TheFinalResultwegetmorecircles.Forlinesinsidethe+1realpart,weendupwithacirclethatis larger thanthe+1circle,andforlineswhichhavearealpartgreaterthan+1,weendupwithcircleswhich

PAGE 209

203 aresmallerindiameterthanthe+1circle.Allcirclespassthroughthe+1pointonthe r s planeandare tangenttooneanother. TwoMoreExamples Figure6.29

PAGE 210

204 CHAPTER6.ACSTEADY-STATETRANSMISSION Add+1toEach Figure6.30 Inverting Figure6.31

PAGE 211

205 MultiplyBy-2 Figure6.32 TheFinalResult Figure6.33 Therearetwospeciallinesweshouldworryabout.Oneis Z s Z 0 = jX ,theimaginaryaxis.Wewillput allofthetransformstepstogetheronFigure6.34AnotherTransform.Westartontheaxis,shiftoverone, getacirclewithunitydiameterwhenweinvert,growbytwoandiparoundtheimaginaryaxiswhenwe multiplyby-2,andthenhoponetotherightwhen+1isadded.Onceagain,youshouldworkyourway throughthevariousstepstomakesureyouhaveagoodunderstandingastohowthisprocedureissupposed tohappen.Notethateventheimaginaryaxisonthe Z s Z 0 planegetstransformedintoacirclewhenwego overontothe r s plane.

PAGE 212

206 CHAPTER6.ACSTEADY-STATETRANSMISSION AnotherTransform Figure6.34: Transforming jX tothe r s plane. Theotherlineweshouldworryaboutis Z s Z 0 = 1 + jX .Now 1 +1= 1 ,and )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 =0 : 0+1=1 ,and sotheline 1+ jX getsmappedintoapointat1whenwedoourtransformationontothe r s plane.Even pointsat 1 onthe Z s Z 0 planeenduponthe r s plane,andareeasilyaccessible! OK,Figure6.35OtherConstantRealPartLinesisaplotofthe Z s Z 0 plane.Thelinesshownrepresent therealpartof Z s Z 0 thatwewanttotransform.WerunthemallthroughTheBilinearTransform.63,to getthemontothe r s plane.Nowwehaveawholefamilyofcircles,thebiggestofwhichhasadiameterof2 whichcorrespondstotheimaginaryaxisandthesmallestofwhichhasadiameterof0whichcorresponds topointsat 1 Figure6.36FamilyofCircles.Thecirclesalltwithinoneanother,andsincea+1was addedtoeverytransformasthenalbitofmanipulation,allofthecirclespassthroughthepoint+1, 0 j Circleswithsmallerdiameterscorrespondtolargervaluesofreal Z s Z 0 ,whilethelargercirclescorrespondto thelesservaluesof Z s Z 0 OtherConstantRealPartLines Figure6.35: Addingotherconstantrealpartlinetothe Z s Z 0 plane.

PAGE 213

207 FamilyofCircles Figure6.36: Familyof Re Z s Z 0 Well,we'rehalfwaythere.Nowallwehavetodoisndthetransformfortheco-ordinatelineswhich correspondtotheimaginarypartof Z s Z 0 .Let'slookat Z s Z 0 = R + j 1 .Whenweadd+1tothis,nothing happens!Thelinejustslidesover1unit,andlooksjustthesameFigure6.37ALineofConstantImaginary Part.Nowwetakeitsinverse.Thiswillgivesusacircle,butsincethelineweareinvertingliesatanangle of +90 withrespecttotherealaxis,themajordiameterofthecirclewilllieatanangleof )]TJ/F8 9.9626 Tf 7.749 0 Td [(90 whenwe gothroughtheinversionprocess.Thisgivesusacirclewhichislyinginthe )]TJ/F11 9.9626 Tf 7.748 0 Td [(j regionofthecomplexplane Figure6.38AfterInverting. ALineofConstantImaginaryPart Figure6.37

PAGE 214

208 CHAPTER6.ACSTEADY-STATETRANSMISSION AfterInverting Figure6.38 Thenextthingwedoistotakethiscircleandmultiplyby-2.Thiswillmakethecircletwiceaslarge, butwillalsoreectitbackupintothe + j regionofthecomplexplaneFigure6.39MulitplyBy-2. MulitplyBy-2 Figure6.39 And,nally,weadd1toit,whichcausesthecircletohoponeovertotherightFigure6.40AndAdd 1.

PAGE 215

209 AndAdd1 Figure6.40 Wecandothesamethingtootherlinesofconstantimaginarypartandwecanthenaddmorecircles. Orpartialcircles,foritmakesnosensetogobeyondthe Re Z s Z 0 =0 circles,asbeyondthatistheregion correspondingtonegativerealpart,whichwewouldnotexpecttoencounterinmosttransmissionlines. TakeatleastoneoftheothercirclesdrawnhereFigure6.41:TheCompleteTransformationandseeifyou cangetittoendupinabouttherightplace. TheCompleteTransformation Figure6.41 Thereisonelineofinterestwhichwehaveatakealittlecarewith.Thatistherealaxis, Z s Z 0 =0+ jX Thislineisadistance0awayfromtheorigin,andsowhenweinvertit,wegetacirclewith 1 diameter. That'sOKthough,becausethatisjustastraightline.So,therealaxisofthe Z s Z 0 planetransformsinto therealaxisonthe r s plane. Wehavedoneamostwondrousthing!Althoughyoumaynotrealizeityet.Wehavetakenthe entire halfplaneofcompleximpedance Z s Z 0 andmappedthewholethingintoacirclewithdiameter1!Let'sput thetwoofthemsidebyside.Althoughwecan'tshowthewhole Z s Z 0 planeofcourse.Theseareshown hereFigure6.42:TheMapping,whereweshowhoweachlineon Z s Z 0 mapsintoacurvedlineonthe r s plane.Notealso,thatforeverypointonthe Z s Z 0 plane"A"and"B"thereisacorrespondingpoint onthe r s plane.Pickacouplemorepoints,"C"and"D"andlocatethemeitheronthe Z s Z 0 plane,orthe r s plane,andthenndthecorrespondingpointontheotherplane.

PAGE 216

210 CHAPTER6.ACSTEADY-STATETRANSMISSION TheMapping Figure6.42 Notethatthemappingisnotveryuniform.Alloftheregionwhereeithertherealorimaginarypart of Z s Z 0 is < 1 asmallsquareon Z s Z 0 mapsintoamajorfractionofthe r s planeFigure6.43Mapping whereasalltherestofthe Z s Z 0 plane,allthewayouttoinnityinthreedirections + 1 + j 1 ,and )]TJ/F8 9.9626 Tf 9.409 0 Td [( j 1 mapintotherestofthe r s circleFigure6.44MappingtheRest. Mapping Figure6.43: Mapping1, 1 j

PAGE 217

211 MappingtheRest Figure6.44 Thisgraphortransformationiscalleda SmithChart ,aftertheBellLabsworkerwhorstthoughtit up.Itisamostusefulandpowerfulgraphicalsolutiontothetransmissionlineproblem.InIntroductionto UsingtheSmithChartSection6.9wewillspendalittletimeseeinghowandwhyitcanbesouseful. 6.9IntroductiontoUsingtheSmithChart 10 UsingtheSmithChartFigure6.45:TheSmithChart,wewillinvestigatesomeoftheapplicationanduses ofthe SmithChart .Forthetext,wewillusemynew"miniSmithChart"whichisreproducedbelow. Clearly,thereisnotmuchdetailhere,andouranswerswillnotbeasaccurateastheywouldbeifweuseda fullsizechart,butwewanttogetideasacrosshere,notthebestnumberpossible,andwiththesmallsize, wewon'trunoutofpaperbeforeeverythingisdone. TheSmithChart Figure6.45 Notethatwehaveacoupleof"extras"onthechart.Thetwoscalesatthebottomofthechartcanbe usedtoeithersetormeasureradialvariablessuchasthemagnitudeofthereectioncoecient j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j ,orthe VSWR,asitturnsoutthatinpractice,whatonecanactuallymeasureonalineistheVSWR.Remember, 10 Thiscontentisavailableonlineat.

PAGE 218

212 CHAPTER6.ACSTEADY-STATETRANSMISSION thereisadirectrelationshipbetweentheVSWRandthemagnitudeofthereectioncoecient. VSWR = 1+ j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j 1 )-222(j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j .68 j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j = VSWR )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 VSWR +1 .69 Since j r s j = j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j ,oncewehavetheVSWR,wehave j r s j andsoweknowhowbigacircleweneedonthe SmithChartinordertogofromoneplacetothenext.Notealsothatthereisascalearoundtheoutsideof thechartwhichisgiveninfractionsofawavelength.Since r s rotatesaroundatarate 2 s and = 2 ,we couldeithershowdistanceincmorsomething,andthenchangethescalewheneverwechangewavelength. Or,wecouldjustuseadistancescalein ,andmeasurealldistancesinunitsofthewavelength.Thisis whatweshalldo.Sincetherateofrotationis 2 s ,onetriparoundtheSmithChartisthesameasgoing onehalfofawavelengthdowntheline.Rotationinaclockwisedirectionisthesameasmovingawayfrom theloadtowardsthegenerator,whilemotionalongthelineintheotherdirectiontowardstheloadcalls forcounterclockwiserotation.Thescaleis,ofcourse,arelativeone,andsowewillhavetore-setourzero, dependinguponwheretheloadetc.arereallylocated.Thiswillallbecomecleareraswedoanexample. Let'sstartoutwiththesimplestthingwecan,withjustagenerator,lineandloadFigure6.46TransmissionLineProblem.Ourtaskwillbetondtheinputimpedance, Z in ,fortheline,sothatwecangure theinputvoltage. TransmissionLineProblem Figure6.46 Forthisrstproblem,wearegoingtostartoutwithallthebasics.Inlaterexamples,weprobablywill onlygivelengthsinwavelengths,andimpedancesintermsof Z 0 ,butlet'sdothisthewholewaythrough. 6.10SimpleCalculationswiththeSmithChart 11 So,whatdowedofor Z L ?AquickglanceatatransmissionlineproblemFigure6.46:TransmissionLine Problemshowsthatattheloadwehavearesistorandaninductorinparallel.Thiswasdoneonpurpose, toshowyouoneofthepowerfulaspectsoftheSmithChart.Basedonwhatyouknowfromcircuittheory youwouldcalculatetheloadimpedancebyusingtheformulafortwoimpedancesinparallel Z L = j!LR j!L + R whichwillbesomewhatmessytocalculate. Let'sremembertheformulaforwhattheSmithChartrepresentsintermsofthephasor r s Z L Z 0 = 1+ r s 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(r s .70 11 Thiscontentisavailableonlineat.

PAGE 219

213 Let'sinvertthisexpression Z L Z 0 = 1 Z L 1 Z 0 = Y L Y 0 = 1 )]TJ/F10 6.9738 Tf 6.226 0 Td [(r s 1+ r s .71 .72saysthatiswewanttogetan admittance insteadofanimpedance,allwehavetodoissubstitute )]TJ/F8 9.9626 Tf 9.409 0 Td [( r s for r s ontheSmithChartplane! Y 0 = 1 Z 0 = 1 50 =0 : 02 .72 inourcase.Wehavetwoelementsinparallelfortheload Y L = Y + jB ,sowecaneasilyaddtheir admittances,normalizethemto Y 0 ,putthemontheSmithChart,go 180 aroundsamethingasletting )]TJ/F8 9.9626 Tf 9.409 0 Td [( r s = r s andreado Z L Z 0 .Fora 200 resistor, G ,thecondunctanceequals 1 200 =0 : 005 Y 0 =0 : 02 so G Y 0 =0 : 25 .Thegeneratorisoperatingatafrequencyof 200 MHz ,so =2 f =1 : 25 10 9 s )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 andthe inductorhasavalueof 160 nH ,so j!L =200 j and B = 1 j!L = )]TJ/F8 9.9626 Tf 7.749 0 Td [(0 : 005 j and B Y 0 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(0 : 25 j WeplotthisontheSmithChartFigure6.47:MovingDowntheTransmissionLinebyrstndingthe realpart=0.25circle,andthenwegodownontothelowerhalfofthechartsincethatiswhereallthe negativereactivepartsare,andwendthecurvewhichrepresents )]TJ/F8 9.9626 Tf 7.749 0 Td [(0 : 25 j andwheretheyintersect,weput adot,andmarkthelocationas Y L Y 0 .Nowtond Z L Z 0 ,wesimplyreecthalfwayaroundtotheoppositeside ofthechart,whichhappenstobeabout Y L Y 0 =2+2 j ,andwemarkthataswell.Notethatwecantakethe lengthofthelinefromthecenteroftheSmithCharttoour Z L Z 0 andmoveitdowntothe j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j scaleandnd thatthereectioncoecienthasamagnitudeofabout0.6.OnarealSmithChart,thereisalsoaphase anglescaleontheoutsideofthecirclewhereourdistancescaleiswhichyoucanusetoreadothephase angleofthereectioncoecientaswell.Puttingthatscaleonthe"miniSmithChart"wouldclogthings uptoomuch,butthephaseangleof )]TJ/F15 9.9626 Tf 9.547 0 Td [(isabout 3 : 0 .

PAGE 220

214 CHAPTER6.ACSTEADY-STATETRANSMISSION MovingDowntheTransmissionLine Figure6.47 Nowthewavelengthofthesignalonthelineisgivenas = p f = 2 : 8 10 8 200 10 6 =1 m .73 Theinputtothelineislocated 21 : 5 cm or 0 : 215 awayfromtheload.Thus,westartat Z L Z 0 ,androtate aroundonacircleofconstantradiusadistance 0 : 215 towardsthegenerator.Todothis,weextendaline outfromour Z L Z 0 pointtothescaleandreadarelativedistanceof 0 : 208 .Weadd 0 : 215 tothis,andget 0 : 423 Thus,ifwerotatearoundtheSmithChart,onourcircleofconstantradiusSince,afterall,allweare doingisfollowing r s asitrotatesaroundfromtheloadtotheinputtotheline.Whenwegetto 0 : 423 westop,drawalineoutfromthecenter,andwhereitinterceptsthecircle,wereado Z L Z 0 fromthegrid linesontheSmithChartFigure6.48:UsingaSmithCharttoConvertFromAdmittancetoImpedance. Wendthat Z in Z 0 =0 : 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(0 : 5 j .74

PAGE 221

215 UsingaSmithCharttoConvertFromAdmittancetoImpedance Figure6.48 Thus, Z in =15 )]TJ/F8 9.9626 Tf 9.446 0 Td [(25 j ohmsFigure6.49FindVin.Or,theimpedanceattheinputtothelinelookslike a 15 resistorinserieswithacapacitorwhosereactance jX = )]TJ/F8 9.9626 Tf 7.748 0 Td [(25 j ,or,since X cap = 1 j!C ,wendthat, C = 1 2 200 200 10 6 =31 : 8 pF .75 Tond V in ,thereisnoavoidingdoingsomecomplexmath: V in = 15 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 j 50+15 )]TJ/F8 9.9626 Tf 9.962 0 Td [(25 j 10 .76 Which,wewriteinpolarnotation,divide,gurethevoltageandthenreturntorectangularnotation. V in = 29 : 1 59 69 : 6 )]TJ/F8 9.9626 Tf 9.962 0 Td [(21 10 .77 V in =0 : 418 )]TJ/F8 9.9626 Tf 9.963 0 Td [(38 10 =4 : 18 )]TJ/F8 9.9626 Tf 9.963 0 Td [(38 =3 : 30 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 : 58 j .78

PAGE 222

216 CHAPTER6.ACSTEADY-STATETRANSMISSION FindVin Figure6.49 Ifatthispointweneededtondtheactualvoltagephasor V + wewouldhavetousetheequation V in = V + e jL +)]TJ/F11 9.9626 Tf 16.189 0 Td [(V + e )]TJ/F7 6.9738 Tf 6.226 0 Td [( jL = V + e jL + j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j V + e j r )]TJ/F10 6.9738 Tf 6.227 0 Td [(L .79 Where = 2 isthepropagationconstantforthelineasmentionedinthelastchapter.21,and L isthe lengthoftheline. Forthisexample, L = 2 0 : 215 =1 : 35 radiansand )]TJ/F8 9.9626 Tf 8.191 1.494 Td [(=)-278(=0 : 52 radians.Thuswehave: V in = V + e j 1 : 35 +0 : 52 V + e j : 52 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : 35 .80 Whichthengivesus: V + = V in e j 1 : 35 +0 : 52 e j : 52 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 : 35 .81 Whenyouexpandtheexponentials,addandcombineinrectangularcoordinates,changetopolar,and divide,youwillgetaphasorvaluefor V + .Ifyoudoitcorrectly,youwillndthat V + =5 : 04 )]TJ/F8 9.9626 Tf 9.962 0 Td [(71 : 59 Manytimeswedon'tcareabout V + itself,butaremoreinterestedinhowmuchpowerisbeingdelivered totheload.Notethatpowerdeliveredtotheinputofthelineisalsotheamountofpowerwhichisdelivered totheload!Finding I in iseasy,it'sjust V in Z in .Allwehavetodoischange Z in topolarform. Z in =15 )]TJ/F8 9.9626 Tf 9.963 0 Td [(25 j =29 : 1 59 .82 I in = V in Z in = 4 : 18 38 29 : 1 59 =0 : 144 21 .83 6.11Power 12 Youmightbetemptedtonowsaythat P in = V in I in ,butthatisincorrectforsinusoidalexcitation. V in and I in are phasors !Solet'sdigressforasecondtoseeorreview,Ihopehowtondpowerwhenthevoltage 12 Thiscontentisavailableonlineat.

PAGE 223

217 andcurrentarephasorquantities.Whatreallymattersisnottheabsolutephaseangleofthetwoquantities, butratherthephaseanglebetweenthem.Supposewehaveavoltagephasor, V whichhaszerophaseangle andacompleximpedance Z = j Z j e j z .Obviously,thecurrentisgivenby ~ I = ~ V ~ Z = j V j j Z j e )]TJ/F10 6.9738 Tf 6.226 0 Td [( z .84 Tondpower,wecannotworkjustwithphasors,wehavetogobacktothecompletefunctionoftimeas wellsowewrite: V t = j V j cos !t .85 I t = j V j j Z j cos !t )]TJ/F11 9.9626 Tf 9.963 0 Td [( z .86 I t = j I j cos !t )]TJ/F11 9.9626 Tf 9.963 0 Td [( z .87 Thepowerasafunctionoftimeisgivenas P t = I t V t = j V jj V j cos !t cos !t )]TJ/F11 9.9626 Tf 9.963 0 Td [( z .88 Werememberausefultrigidentity: cos A )]TJ/F11 9.9626 Tf 9.963 0 Td [(B = cos A cos B + sin A sin B .89 Hence: cos !t )]TJ/F11 9.9626 Tf 9.962 0 Td [( z = cos !t cos z + sin !t sin z .90 whichmakes P t P t = cos 2 !t cos z + cos !t sin !t sin z .91 Wearereallyinterestedinnding averagepower sinceenergywhichowsintoandthenbackoutofthe linedoesnoworkforus.Clearlythesecondtermin.91goingas cos !t sin !t hasanaveragevalue ofzero,andsowecanforgetaboutit.Timeforonemoretrigidentity: cos 2 A = 1 2 + 1 2 cos A .92 cos !t haszeroaveragevalueaswell,soweareleftwiththefollowingfortheaveragevalueofthepower P t P t = j V jj I j 2 cos z = j V j 2 2 j Z j cos z .93 Notethatoneusefulwaythatpeoplesometimesusetoexpressthisistosay P t = 1 2 ~ V ~ V .94 Backtoourexample: V in =4 : 18 38 and I in =0 : 144 21 Thus P in t = 1 2 : 18 0 : 144 cos Watts =0 : 155 .95 Asanalternativewayofcalculatingthepowerintothelinenotethatwe know themagnitudeofthecurrent throughboththecapacitorandtheresistoroftheapparent Z in .Theyarejusttwoelementsinseries,andso

PAGE 224

218 CHAPTER6.ACSTEADY-STATETRANSMISSION theybothhavethesamecurrentowingthroughthem,namely, I in .Nopowerisdissipatedinthecapacitor, sowecouldjustaswellhavesaid P in t = 1 2 j I j 2 R = )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(1 2 0 : 144 2 15 =0 : 155 .96 andgottentheanswerinaneveneasierfashion!Notethatwestillhavetokeepthefactorof"1/2"to accountforthetimeaverageofasinusoidalproduct.ForreasonsIdonotunderstand,studentshavealways hadanaversiontondingpower.Itisnotthathard,andintheend,isusuallythe"bottomline"with regardtohowasystemwillperform.Gobackoverthissectionuntilitmakessense,asyoumayseepower cropupsomeplaceelseoneofthesedays! 6.12FindingZL 13 Let'smoveontosomeotherSmithChartapplications.Suppose,somehow,wecanobtainaplotof V s onalinewithsomeunknownloadonit.ThedatamightlooklikeFigure6.50AStandingWavePattern. Whatcanwetellfromthisplot?Well, V max =1 : 7 and V min =0 : 3 whichmeans VSWR = 1 : 7 0 : 3 =5 : 667 .97 andhence j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j = VSWR )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 VSWR +1 = 4 : 667 6 : 667 =0 : 7 .98 AStandingWavePattern Figure6.50 Since j r s j = j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j ,wecanplot r s ontheSmithChart,asshownhereFigure6.51:TheVSWRCircle. Wedothisbysettingthecompassataradiusof0.7anddrawingacircle!Now, Z L Z 0 is somewhere onthis circle.Wejustdonotknowwhereyet!ThereismoreinformationtobegleanedfromtheVSWRplot however. 13 Thiscontentisavailableonlineat.

PAGE 225

219 TheVSWRCircle Figure6.51 Firstly,wenotethattheplothasaperiodicityofabout10cm.Thismeansthat thewavelengthof thesignalonthelineis20cm.Why?Accordingtothis.55equation, j V s j goesas cos s and s = )]TJ/F14 9.9626 Tf 7.368 1.495 Td [()]TJ/F8 9.9626 Tf 9.694 0 Td [(2 s and = 2 ,thus j V s j goesas cos )]TJ/F7 6.9738 Tf 5.762 -4.147 Td [(4 s .Thuseach 2 ,wearebacktowherewestarted. Secondly,wenotethatthereisavoltageminimaatabout2.5cmawayfromtheload.Whereon Figure6.51TheVSWRCirclewouldweexpecttondavoltageminima?Itwouldbewhere r s hasa phaseangleof 180 orpoint"A"showninhereFigure6.52:LocationofaVmin.Thevoltageminima is always wheretheVSWRcirclepassesthroughtherealaxisonthelefthandside.Converselyavoltage maximaiswherethecirclegoesthroughtherealaxisontherighthandside.Wedon'treallycareabout Z s Z 0 atavoltageminima,whatwewantis Z s =0 Z 0 ,thenormalizedloadimpedance.Thisshouldbeeasy!If westartat"A"andgo 2 : 5 20 =0 : 125 towardstheload weshouldendupatthepointcorrespondingto Z L Z 0 Thearrowonthemini-SmithChartsays"Wavelengthstowardsgenerator"IfwestartatA,andwanttogo towardsthe load ,wehadbettergoaroundtheoppositedirectionfromthearrow.Actually,asyoucansee ona real SmithChart,therearearrowspointinginbothdirections,andtheyareappropriatelymarkedfor yourconvenience.

PAGE 226

220 CHAPTER6.ACSTEADY-STATETRANSMISSION LocationofaVmin Figure6.52 Sowestartat"A"go 0 : 125 inacounter-clockwisedirection,andmarkanewpoint"B"whichrepresents our Z L Z 0 whichappearstobeabout 0 : 35 )]TJ/F8 9.9626 Tf 8.894 0 Td [(0 : 95 j orsoFigure6.53MovingfromVmintotheLoad.Thus,the loadinthiscaseassuminga 50 lineimpedanceisaresistor,againbyco-incidenceofabout 50 ,inseries withacapacitorwithanegativereactanceofabout 47 : 5 .Notethatwecouldhavestartedattheminima at12.5cmoreven22.5cm,andthenhaverotated 12 : 5 20 =0 : 625 or 22 : 5 20 =1 : 125 towardstheload.Since 2 =0 : 5 meansonecompleterotationaroundtheSmithChart,wewouldhaveendedupatthesamespot, withthesame Z L Z 0 thatwealreadyhave!Wecouldalsohavestartedatamaxima,atsay7.5cm,marked ourstartingpointontherighthandsideoftheSmithchart,andthenwewouldgo 0 : 375 counterclockwise andagain,we'dendupat"B".

PAGE 227

221 MovingfromVmintotheLoad Figure6.53 Now,hereFigure6.54:AnotherStandingWavePatternisanotherexample.Inthiscasethe VSWR = 1 : 5 0 : 5 =3 ,whichmeans j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j =0 : 5 andwegetacircleasshowninFigure6.55TheVSWRCircle.The wavelength =2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(10=30 cm .Therstminimaisthusadistanceof 10 30 =0 : 333 fromtheload.So weagainstartattheminima,"A"andnowrotateasdistance 0 : 333 towardstheload AnotherStandingWavePattern Figure6.54

PAGE 228

222 CHAPTER6.ACSTEADY-STATETRANSMISSION TheVSWRCircle Figure6.55 6.13Matching 14 Thisgetsusto"B",andwendthat Z L Z 0 =1+1 : 2 j .Nowthisisaveryinteresting Figure6.56: Theloadimpedance result.Supposewetaketheloadotheline,andadd,inseries,anadditionalcapacitor,whosereactance is 1 j C = )]TJ/F8 9.9626 Tf 9.409 0 Td [( j 1 : 2 Z 0 14 Thiscontentisavailableonlineat.

PAGE 229

223 Matchingtheloadwithacapacitor Figure6.57 Thecapacitorandtheinductorjustcanceleachotheroutseriesresonanceandsotheapparentload forthelineisjust Z 0 ,themagnitudeofthereectioncoecient )]TJ/F15 9.9626 Tf 6.227 0 Td [(=0andthe VSWR =1 : 0 !Allofthe energyowingdownthelineiscoupledtotheloadresistor,andnothingisreectedbacktowardstheload. Wewereluckythattherealpartof Z L Z 0 =1 .Iftherewerenotthatcase,wewouldnotbeableto"match" theloadtotheline,right?Notcompletely.Let'sconsideranotherexample.ThenextgureFigure6.58 showsalinewitha Z 0 =50 ,terminatedwitha 25 resistor. )]TJ/F10 6.9738 Tf 6.227 -1.494 Td [(L = )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 3 ,andweendupwiththeVSWR circleshowninthesubsequentgureFigure6.59. Figure6.58: Matchingwithaseriescapacitor

PAGE 230

224 CHAPTER6.ACSTEADY-STATETRANSMISSION Figure6.59: Plotting Z L Z 0 Howcouldwematchthisload?Wecouldaddanother25 inserieswiththerstresistor,butifwewant tomaximizethepowerwedelivertotherstone,thiswouldnotbeaverysatisfactoryapproach.Let'smove downthelineaways.Ifwegotopoint"B",wendthat Figure6.60: Movingtothe"rightspot" atthisspot, Z s Z 0 =1+0 : 8 j .Onceagainwehaveanimpedancewithanormalizedrealpartequals1!How fardowego?Itlookslikeit'salittlemorethan 0 : 15 .Ifweaddanegativereactanceinserieswiththeline atthispoint,withanormalizedvalueof )]TJ/F8 9.9626 Tf 9.409 0 Td [( : 8 j ,thenfromthatpointonbacktothegenerator,theline would"look"likeitwasterminatedwithamatchedload. There'soneawkwardfeaturetothissolution,andthatiswehavetocutthelinetoinsertthecapacitor. Itwouldbealoteasierifwecouldsimplyaddsomethingacrosstheline,insteadofhavingtocutit.This iseasilydone,ifwegooverintotheadmittanceworld.

PAGE 231

225 6.14IntroductiontoParallelMatching 15 Let'sstartwiththeload.Withthesame 25 resistorfortheload,andplotits admittance Y L Y 0 =2 .Ifwe startmovingawayfromtheloadtowardsthegenerator,inabout 0 : 10 weagainrunintothecirclewhich represents Re Y s Y 0 =1 .Thisissuchanimportantcircleishasgaineditsownname,anditisfrequently calledthe matchingcircle Figure6.61GettingtotheMatchingCircle. GettingtotheMatchingCircle Figure6.61 Notethattondouthowfarwehadtomove,wehadtostartatrelativeposition 0 : 25 asourzero,or referencelocation.Point"B"seemstobeatabout 0 : 35 onthescale,andsincewestartedat 0 : 25 ,the distanceis 0 : 35 )]TJ/F8 9.9626 Tf 10.221 0 Td [(0 : 25=0 : 10 .At"B", Y s Y 0 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : 0+0 : 7 j .Thus,ifweaddasusceptance jB withavalue of + j 0 : 014 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 wewouldagainmatchtheline.Positivesusceptancecomesfromacapacitoraswell,andso Figure6.62MatchingWithaShuntCapacitorshowshowwematch. MatchingWithaShuntCapacitor Figure6.62 Notethatwearenot required togotopoint"B".Anypointonthematchingcirclethatwecanget toisfairgame.Anothersuchpointis"C"inFigure6.61GettingtotheMatchingCircle.Thisisata distanceofabout 0 : 40 fromtheload.At"C", Y s Y 0 =1 : 0+0 : 7 j andsowewouldputinaninductor,witha susceptance 1 j!L = )]TJ/F1 9.9626 Tf 9.41 8.069 Td [()]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(j 0 : 014 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 Figure6.63MatchingWithaShuntInductor. 15 Thiscontentisavailableonlineat.

PAGE 232

226 CHAPTER6.ACSTEADY-STATETRANSMISSION MatchingWithaShuntInductor Figure6.63 6.15SingleStubMatching 16 Often,therearereasonswhyusingadiscreteinductororcapacitorformatchingisnotsuchagoodidea. Atthehighfrequencieswherematchingisimportant,lossesinbothLorCmeanthatyoudon'tgetagood match,andmostofthetimeexceptforsomeair-dielectricadjustablecapacitorsitishardtoget just the valueyouwant. Thereisanotherapproachthough.Ashortedoropentransmissionline,whenviewedatitsinputlookslike apurereactanceorpuresusceptance.Withashortasaload,thereectioncoecienthasunitymagnitude j )]TJ/F14 9.9626 Tf 6.227 0 Td [(j =1 : 0 andsowemovearoundtheveryoutsideoftheSmithChartFigure6.64:InputImpedanceofa ShortedLineasthelengthofthelineincreasesordecreases,and Z in Z 0 ispurelyimaginary.Whenwedid thebilineartransformationfromthe Z s Z 0 planetothe r s plane,theimaginaryaxistransformedintothe circleofdiameter2,whichendedupbeingtheoutsidecirclewhichdenedtheSmithChart. InputImpedanceofaShortedLine Figure6.64 Anotherwaytoseethisistogobacktothisequation.50.Therewefound: Z s = Z 0 Z L + jZ 0 tan s Z 0 + jZ L tan s .99 With Z L =0 thisreducesto Z s = jZ 0 tan s .100 16 Thiscontentisavailableonlineat.

PAGE 233

227 Which,ofcourseforvariousvaluesof s ,cantakeonanyvaluefrom + j 1 to )]TJ/F8 9.9626 Tf 9.409 0 Td [( j 1 .Wedon'thavetogo toRadioShack andbuyabunchofdierentinductorandcapacitors.Wecanjustgetsometransmission lineandshortitatvariousplaces! Thus,insteadofadiscretecomponent,wecanuseasectionofshortedoropentransmissionline insteadFigure6.65AShortenedStub.Thesematchinglinesarecalled matchingstubs .Oneofthe majoradvantageshereisthatwithalinewhichhasanadjustableshortontheendofit,wecangetany reactanceweneed,simplybyadjustingthelengthofthestub.Howthisallworkswillbecomeobviousafter wetakealookatanexample. AShortenedStub Figure6.65 Let'sdoone.InFigure6.66AnotherLoadwecanseethat, Z L Z 0 =0 : 2+0 : 5 j ,sowemarkapoint"A" ontheSmithChart.Sincewewillwanttoputthetuningormatchingstubinshuntacrosstheline,the rstthingwewilldoisconvert Z L Z 0 intoanormalizedadmittance Y L Y 0 bygoing 180 aroundtheSmithChart Figure6.67:ConvertingtoNormalizedAdmittancetopoint"B",where Y L Y 0 h 0 : 7 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 : 7 j .Nowwerotate aroundontheconstantradius, r s circleuntilwehitthematchingcircleatpoint"C".Thisisshownin Figure6.68MovingtotheMatchingCircle.At"C", Y S Y 0 =1 : 0+2 : 0 j .Usinga"real"SmithChart,Iget thatthedistanceofrotationisabout 0 : 36 .Remember,allthewayaroundis 2 ,soyoucanveryoften "eyeball"abouthowfaryouhavetogo,anddoingsoisagoodcheckonmakingastupidmatherror.Ifthe distancedoesn'tlookrightontheSmithChart,youprobablymadeamistake! AnotherLoad Figure6.66

PAGE 234

228 CHAPTER6.ACSTEADY-STATETRANSMISSION ConvertingtoNormalizedAdmittance Figure6.67: Convertingto Y L Y 0 MovingtotheMatchingCircle Figure6.68 OK,atthispoint,therealpartoftheadmittanceisunity,soallwehavetodoisaddastubtocancel outtheimaginarypart.Asmentionedabove,thestubsoftencomewithadjustable,or"slidingshort"sowe canmakethemwhateverlengthwewantFigure6.69MatchingwithaShortenedStub.

PAGE 235

229 MatchingwithaShortenedStub Figure6.69 Ourtasknow,istodecidehowmuchtopushorpullontheslidinghandleonthestub,togetthereactance wewant.ThehintonwhatweshoulddoisinFigure6.64InputImpedanceofaShortedLine.Theendof thestubisashortcircuit.Whatistheadmittanceofashortcircuit?Answer: 1 j 1 !Whereisthisonthe SmithChart?Answer:ontheoutside,ontherighthandsideontherealaxis.Now,ifwestartatashort, andstarttomakethelinelongerthan s =0 ,whathappensto Y s Y 0 ?Itmovesaroundontheoutsideofthe SmithChart.Whatweneedtodoismoveawayfromtheshortuntilweget Y s Y 0 = )]TJ/F8 9.9626 Tf 9.409 0 Td [( j 2 : 0 andwewill knowhowlongtheshortedtuningstubshouldbeFigure6.70FindingtheStublength.Ingoingfrom"A" to"B"wetraverseadistanceofabout 0 : 07 andsothatiswhereweshouldsetthepositionofthesliding shortonthestubFigure6.71TheMatchedLine. FindingtheStublength Figure6.70

PAGE 236

230 CHAPTER6.ACSTEADY-STATETRANSMISSION TheMatchedLine Figure6.71 Wesometimesthinkoftheactionofthetuningstubasallowingustomoveinalongthe Re Y s Y 0 to gettothecenteroftheSmithChart,ortoamatchFigure6.72MovingWithaStub.Wearenotinthis case,physicallymovingdowntheline.Ratherwearemovingalonga contourofconstantrealpart because allthestubcandoischangetheimaginarypartoftheadmittance,itcandonothingtotherealpart! MovingWithaStub Figure6.72: Movingalongthe Re Y s Y 0 =1 circlewithastub. 6.16DoubleStubMatching 17 Thereisonelasttechniquewecanlookatwhichissomewhatmoreexiblethanthesinglestubmatching whichwejustlookedat.Thisiscalleddoublestubmatching!Supposewehavethefollowingsituation,as depictedinthegureFigure6.73:DoubleStubMatchingProblem.Thereisaloadof Z L Z 0 =0 : 2+1 : 3 j locatedattheendoftheline,andthensomearbitrarydistanceaway 0 : 11 anadjustablestub.Another 17 Thiscontentisavailableonlineat.

PAGE 237

231 arbitrary 0 : 11 fromtherststub,thereisa second one.Let'splot Y L Y 0 ontheSmithChartFigure6.74: ChangingtheLoadtoanAdmittance,andthen,sincethestubsareinshuntacrosstheline,switchto admittance,andnd Y L Y 0 .Itiseasytoseethat Y L Y 0 =1 : 5+2 : 3 j DoubleStubMatchingProblem Figure6.73 ChangingtheLoadtoanAdmittance Figure6.74 Therstthingwemightaswelldoismovedowntotherststub,andseewhatadmittancewehavethere Figure6.75MovingFromtheLoadtotheFirstStub.Wegofromtheload,totherststubbyrotating ona circleofconstantradius constant j r s j sinceallwearedoingisgoingfromoneplaceonthelineto another.Ifwecallthelocationonthelineoftherststub"A",thenwecanseethat Y A Y 0 =0 : 25+0 : 6 j .

PAGE 238

232 CHAPTER6.ACSTEADY-STATETRANSMISSION MovingFromtheLoadtotheFirstStub Figure6.75 Now,whatcantherststubaccomplish?Ashortedstubcancreateany imaginary admittancewewant, butcannotchangetherealpartoftheadmittance.Thus,byadjustingtherststub,wecanmovearound onacircleof constantrealpart =0 : 25 Y 0 ,andhaveanyimaginarypartwewant.Thisisshownschematically hereFigure6.76:PossibleEectsoftheFirstStub. PossibleEectsoftheFirstStub Figure6.76 Now,wheredowewanttogo?Well,wewouldliketoendupsomeplacesothat,afterwehavemoved fromAtoBonthelinegonefromtherststubtothesecond,weareonthematchingcircle.Ifthiswere so,then,sinceweareonthematchingcircle,wecouldusethesecondstubtomatchthewholelineandwe wouldbedone. Thisistrickynow,soyouhavetopayattentionandthink.IfIwanttondaplacewhich,whenmoved fromAtoB,endsuponthematchingcircle,thenwhatIshoulddoistakethematchingcircleandmove itfromBtoA.Thatis,ifIrotatethematchingcirclearound 0 : 175 towardstheload ,thenanyplaceon thatrotatedmatchingcircleisguaranteedtoenduponthe real matchingcircle,whenwego 0 : 175 back towardsthegenerator. OK,sohere'swhatwedo.First,werotatethematchingcircle0.175aroundtowardstheloadgo counterclockwiseFigure6.77RotatingtheMatchingCircle.Nowwhatwehavetodoissomehowgetfrom

PAGE 239

233 Y A Y 0 withoutstubtosomeplaceontherotatedmatchingcircle.Theonlywaywecandothisistochange theimaginarypartof Y A withthestub.SupposewemoveasshowninFigure6.78MovingtoRotated MatchingCircle.Ingoingfrom Y A Y 0 withoutstubto Y A Y 0 withstubwehavechangedtheimaginarypartfrom )]TJ/F8 9.9626 Tf 9.409 0 Td [( j 0 : 6 to + j 0 : 05 ,thuswehave added j 0 : 65 totheimaginarypartof Y A Y 0 .Thususingourstandardmethod forndingthelengthoftherststub,westartat 1 theshortattheendofthestubandgoaround theoutsideoftheSmithChartFigure6.79:FindingLengthoftheFirstStubuntilwend + j 0 : 05 .Toget fromoneplacetothenextwewent : 25+0 : 09 =0 : 34 andsothelengthoftherststub, L 1 should be 0 : 09 .Nowweareat Y A Y 0 withstub.Thenextthingwehavetodoistorotateanother 0 : 175 towards thegenerator sothatwecangettostubB.Aswedothisrotation,weagainstayonacircleofconstant radius ,becausenowwearemovingdownthetransmissionline not addingreactancebyusingastub!This rotationis guaranteed toendusuponthematchingcirclebecause every pointontherotatedcirclethe onewestartfromisexactly 0 : 175 towardstheload from thematchingcircle.AsshownhereFigure6.80: MovingDowntheSecondStub,wearenowatthepoint Y B Y 0 withoutstub =1 : 0+1 : 6 j .Thusweneedto adjustthelength L 2 ofthesecondstubtogiveus )]TJ/F8 9.9626 Tf 9.409 0 Td [( j 1 : 6 ofreactance,sowecanmovealongacircleof constantrealpart =1.0intothecenteroftheSmithChartFigure6.81:MakingtheMatch.Wehaveto ndthelength L 2 forthesecondstub,butthatisnoweasy!Figure6.82FindingtheLengthoftheSecond Stub RotatingtheMatchingCircle Figure6.77

PAGE 240

234 CHAPTER6.ACSTEADY-STATETRANSMISSION MovingtoRotatedMatchingCircle Figure6.78 FindingLengthoftheFirstStub Figure6.79

PAGE 241

235 MovingDowntheSecondStub Figure6.80 MakingtheMatch Figure6.81

PAGE 242

236 CHAPTER6.ACSTEADY-STATETRANSMISSION FindingtheLengthoftheSecondStub Figure6.82 Thus,bydoing doublestubmatching ,weareable,byaddingtheadditionaldegreeoffreedomoftwo adjustablestubs,nottohavetospecifyexactlywherethestubshavetobeplaced,sotheycanbeinthe linebeforethematchingisattempted.Here'sFigure6.83:DoubleStubMatchingAllPutTogether!the wholesequenceofchangesthatwemade.Seeifyoucanbeginat"Start"andgothroughthenumbers 0 5 andgetfrom Z L Z 0 tothematchingpointatthecenteroftheSmithChart.Remember,whenwemovefrom oneplacetoanotherontheline,wemuststayonacircleofconstantradius.Whenwechangereactanceby adjustingastub,wemustmovealongcirclesofconstantrealpart.Ifyoudothat,it'seasy! DoubleStubMatchingAllPutTogether! Figure6.83 There'sjustonelittleproblem.Whatif Y A Y 0 withoutstubhadendedupasshowninhereFigure6.84: ASituationThatDoesn'tWork.Weareonthe Re Y A Y 0 j withoutstub =2 : 0 circle.NomatterhowhardI try,andnomatterwhereIset L 1 allIcandoisspinaroundonthelittlecircleasshownFigure6.84:A SituationThatDoesn'tWork,andIwillneverendupontherotatedmatchingcircle,andIwon'tbeable tomakeamatch!Well,ifIadda thirdstub ...I'llletyouworkitout!

PAGE 243

237 ASituationThatDoesn'tWork Figure6.84 6.17OddsandEnds 18 Justafewoddsandends.ConsiderthefollowingFigure6.85:CascadedLinewhichiscalleda"cascaded line"problem.Theseareproblemswherewehavetwodierenttransmissionlines,withdierentcharacteristic impedances.Sincewewillgiveallofthedistancesinwavelengths, ,wewillassumethatthe wearetalking aboutistheappropriateoneforthelineinvolved.Ifthephasevelocitiesonthetwolinesisthesame,then thephysicallengthswouldcorrespondaswell.Theapproachisrelativelystraight-forward.Firstlet'splot Z L Z 0 ontheSmithChartFigure6.86:SmithDiagram.Thenwehavetorotate 0 : 2 sothatwecannd Z A Z 01 ,thenormalizedimpedanceatpointA,thejunctionbetweenthetwolinesFigure6.87Towardsthe Generator. CascadedLine Figure6.85 18 Thiscontentisavailableonlineat.

PAGE 244

238 CHAPTER6.ACSTEADY-STATETRANSMISSION SmithDiagram Figure6.86 Thus,wend Z A Z 01 =0 : 32+0 : 6 j .Nowwehaveto renormalize theimpedancesowecanmovetothe linewiththenewimpedance Z 02 .Since Z 01 =300 Z A =96 )]TJ/F8 9.9626 Tf 9.473 0 Td [(180 j .Thisistheloadforthesecondlength ofline,solet'snd Z A Z 02 ,whichiseasilyfoundtobe 1 : 9 )]TJ/F8 9.9626 Tf 10.224 0 Td [(3 : 6 j ,sothiscanbeplottedontheSmithChart Figure6.88:MoreSmithCharts.Nowwehavetorotatearoundanother 0 : 15 sothatwecannd Z in Z 02 Thisappeartohaveavalueofabout 0 : 15 )]TJ/F8 9.9626 Tf 10.385 0 Td [(0 : 45 j ,so Z in =7 : 5 )]TJ/F8 9.9626 Tf 10.384 0 Td [(22 : 5 j Figure6.89EvenMoreSmith Charts. TowardstheGenerator Figure6.87

PAGE 245

239 MoreSmithCharts Figure6.88 EvenMoreSmithCharts Figure6.89 Thereisoneapplicationofthecascadedlineproblemthatisusedquiteabitinpractice.Considerthe following:Weassumethatwehaveamatchedlinewithimpedance Z 02 andweconnectittoanotherline whoseimpedanceis Z 01 Figure6.90SimpliedCascadedLine.Ifweconnectthetwoofthemtogether directly,wewillhaveareectioncoecientatthejunctiongivenby )-278(= Z 02 )]TJ/F11 9.9626 Tf 9.962 0 Td [(Z 01 Z 02 + Z 01 .101

PAGE 246

240 CHAPTER6.ACSTEADY-STATETRANSMISSION SimpliedCascadedLine Figure6.90 Nowlet'simaginethatwehaveinsertedasectionoflinewithlength l = 4 andimpedance Z m Figure6.91 AnotherCascadedLine.AtpointA,thejunctionbetweentherstlineandthematchngsection,wecan ndthenormalizedimpedanceas Z A Z M = Z 02 Z m .102 AnotherCascadedLine Figure6.91 WetakethisimpedenceandrotatearoundontheSmithChart 4 tond Z B Z M Z B Z M = Z m Z 02 .103 wherewehavetakenadvantageofthefactthatwhenwegohalfwayaroundtheSmithChart,theimpedance wegetisjusttheinverseofwhatwehadoriginallyhalfwayaroundturns r s into )]TJ/F8 9.9626 Tf 9.409 0 Td [( r s Thus Z B = Z m 2 Z 02 .104

PAGE 247

241 Ifwewanttohaveamatchforlinewithimpedence Z 01 ,then Z B shouldequal Z 01 andhence: Z B = Z 01 = Z m 2 Z 02 .105 or Z m = p Z 01 Z 02 .106 Thispieceoflineiscalleda quarterwavematchingsection andisaconvenientwaytoconnecttwo linesofdierentimpedance.

PAGE 248

242 GLOSSARY Glossary D distributedparameter Adistributedparameterisaparameterwhichisspreadthroughoutastructureandisnot connedtoalumpedelementsuchasacoilofwire. Example: Forinstance,wewillherebydene L asthe distributedinductance forthe transmissionline.IthasunitsofHenrys/meter.Ifwehavealengthoftransmissionline x 0 meterslong,andifthatlinehasadistributedinductanceof L H/m,thentheinductance L of thatlengthoflineisjust L = L x 0 .

PAGE 249

INDEX 243 IndexofKeywordsandTerms Keywords arelistedbythesectionwiththatkeywordpagenumbersareinparentheses.Keywords donotnecessarilyappearinthetextofthepage.Theyaremerelyassociatedwiththatsection. Ex. apples,1.1 Terms arereferencedbythepagetheyappearon. Ex. apples,1 A acceptors,16,17 accumulationlayer,75 activationenergy,120 admittance,213,225 align,125 allowedstates,7 ambipolar,1.11 ambipolardiusionequation,44 annealing,119 ApplyingMetal,4.8 Arrheniusplot,4.40,121 avalanchebreakdown,39 avalanchemultiplication,39 B backbody,104 band,6 banddiagram,17 bandgap,8 basewidthmodulation,67 bias,62 biascurrent,62 biasvoltage,62 bilineartransform,6.7,196,196 bipolar,2.1 bipolartransistor,55,2.666 Boltzman'sconstant,18 boule,118 bouncediagram,5.7,159 built-inpotential,22 C cableTV,5.5 cascadedlineproblem,169 cascadedlines,5.8 CATV,5.5 cells,1.14 channel,82 characteristicimpedance,5.3,151 characteristics,2.3 chargedensity,1 chargeofanelectron,1 ChargedLineProblem,173 circuit,4.9 circuitwell,4.7127 CMOS,103 CMOSLogic,3.10 coherentlight,48 compensation,17 conduction,1.1 conductionband,10 conductivity,5 contacts,29 continuityequation,42 controlledsource,59 coupledlineardierentialequationsintime andposition,148 crankdiagram,6.5189,189 Crookesdarkspace,132 currentdensity,5 currentreectioncoecient,5.6 curvetracer,35 Czochralski,117 D depletion,1.7 depletionmoden-channelMOSFET,100 depletionregion,1.5,21,1.7 diamond,6 dielectricconstant,24 diusedresistor,4.10 diusion,1.11 diusioncoecient,1.11,42,120 diusiondominatedsituation,37 diode,1.4,19,1.9,1.12 diodeequation,34,1.9 directband-gapsemiconductors,45 distributedinductance,5.1,144 distributedparameter,5.1143,144 donor,12 donors,16 dopedsemiconductors,1.3 doping,13,4.3 doublestubmatching,236 drain,82 Drudetheory,2

PAGE 250

244 INDEX E EarlyVoltage,67,68 electriccurrent,2 electricdisplacementvector,24 electriceldvector,24 Electricpotential,3 electrondiusionlength,44 electrontransport,1.11 electronvolts,22 electron-volts,31 electronicgradesilicon,117 electrons,1.11 electrostaticdischarge,3.12,109 emission,1.13 emitter,2.4 emitterinjectioneciency,57 energybands,8 epitaxiallayer,72 Epitaxy,72 equations,2.2 equilibrium,19 etching,124 Euler'sIdentity,182 excessminoritycarriers,43 excitedline,5.5 F Fermilevel,17 Fermions,1.8,31 Fick'sFirstLaw,1.11 Fick'sFirstLawofDiusion,42,4.4 Fick'sSecondLaw,4.5 Fick'ssecondlawofdiusion,121 eld,4.9 FieldEectTransistor,71 eldoxide,128 forwardactivebiasing,55 ForwardBiased,1.8 FOX,4.9 G gatecreation,4.7 Gauss'Law,1.6,24 groupvelocity,5.3,150 H heterostructure,48 hole,14 holes,16,1.11,103 I I-Vcharacteristiccurve,35 IC,4.1 idealityfactor,38 III-Vcompoundsemiconductors,46 impactionization,39 impedance,5.5 incoherentlight,48 indirectband-gapmaterial,44 innitesourcediusion,122 integrated,4.9 integratedcircuits,4.1,4.11 interstitialdiusion,4.4 interstitially,120 inversionlayer,77 Inverters,3.8,3.9 ionimplantation,118 J JFET,3.11 K kinetic,1 L largesignal,62 latchup,110 lightamplication,1.13 lightemittingdiode,1.12 limitedsourcediusion,122 linebehavior,6.2 lineimpedance,189 linearregime,86,93 load,219 loadline,96,101 load-line,96,101 Logic,3.8 losslesstransmissionline,5.1,146 M magnitude,193 majoritycarriers,17 manufacturing,4.9134 mask,125 masks,4.6 matchedtransmissionline,5.6 Matching,6.13 matchingcircle,225 matchingstubs,227 meanfreepath,8 Metal-Oxide-SemiconductorFieldEect Transistor,71 metallization,4.9 metallurgicalgradesilicon,117 minoritycarrierrecombinationlifetime,43 minoritycarriers,17,1.11 moat,103 mobility,5 Model,3.7 models,2.4,2.5 monolithicfabricationprocess,117 MOS,3.2 MOSRegimes,3.5 MOSTransistor,3.4 MOSFET,3.1,71

PAGE 251

INDEX 245 N n-channelenhancementmodeMOSFET,99 n-tank,4.9 n-type,16 negative,1 normalizedimpedance,196 npn,4.3 O Ohm'slaw,5 onesidedjunction,23 oxide,4.9 oxidecapacitance,75 P p-channelMOSFET,103 P-Njunction,1.4,1.11 p-type,16,17 pattern,125 Pauliexclusionprinciple,8,31 phasevelocity,5.3,150,184 phasor,6.1 phasorvoltage,182 phasors,182 phonons,45 photolithography,4.6,124 photoresist,124 photovoltaicdiode,51 pincho,87,95 PlottingMOSI-V,3.6 pn-junction,19,1.5 pnp,4.3 positive,1 positiveappliedvoltage,30 potential,1 projectionprinter,125 propagationcoecient,184 pulldown,100 pullup,100 pulse,166 pulseformingnetworks,176 pulsedpowerelectronics,176 Q quarterwavematchingsection,241 R radiationdiode,1.13 radiativetransition,45 recombination,19 recombinationcenter,46 recombinationdominated,37 reectedwaves,5.6,5.7 relativedielectricconstant,5.4,152 reliability,4.11 renormalize,238 resistivity,5 reticle,125 ReverseBiased,1.10 ReverseBreakdown,1.10 S SahEquation,86 scanned,125 self-aligningtechnology,130 semiconductor,1.2,1.312 semiconductors,4.2,4.3118, 4.6 sheetresistance,4.10,138 shock,3.12 signal,2.5 signalcurrent,62 signalmodel,2.6 signalvoltage,62 silicon,4.2 smallsignaldevice,62 smallsignalmodel,66 SmithChart,6.8,211,6.9,211, 6.10,6.15 solar,1.14 solarcell,51 source,82 spin-coating,124 spontaneousemission,48 sputterdepositionsystem,132 Sputtering,4.8 standardhuman-bodymodel,109 standingwave,193 state,31 staticelectricity,3.12 steadystatesolutions,44 stepped,125 stepper,125 stimulatedemission,1.13,48 stripline,5.4,151 substitutionaldiusion,4.4,120 superposition,5.6 surfacecharge,83 T tank,103 telegrapher'sequations,5.2,148,148, 183 TEM,152 terminatedlines,6.3 terminatedtransmissionlines,5.6 thermalvoltage,31 threshold,76 thresholdvoltage,76,3.3 totalvoltagefunction,156,6.3,186 transferfunction,96 transistor,2.2,2.3,71

PAGE 252

246 INDEX transistorloads,3.9 transistors,2.1,3.11 transmissionline,5.1,5.4 transmissionlines,143,5.2,5.7 transverseelectromagnetic,152 V vacancies,120 valence,12 valenceband,13 voltagedivider,5.5 voltagereectioncoecient,5.6 voltagestandingwaveratio,6.6,193 VSWR,6.6 VTadjust,100 W wafer,4.7 wafers,4.2,4.3,4.6 wavefunctions,7 wavepattern,6.6 Y yield,4.11 Z ZenerDiodes,40 ZenerEect,40

PAGE 253

ATTRIBUTIONS 247 Attributions Collection: IntroductiontoPhysicalElectronics Editedby:BillWilson URL:http://cnx.org/content/col10114/1.4/ License:http://creativecommons.org/licenses/by/1.0 Module:"SimpleConduction" By:BillWilson URL:http://cnx.org/content/m1000/2.21/ Pages:1-5 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"IntroductiontoSemiconductors" By:BillWilson URL:http://cnx.org/content/m1001/2.13/ Pages:6-11 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"DopedSemiconductors" By:BillWilson URL:http://cnx.org/content/m1002/2.15/ Pages:12-18 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"P-NJunction:PartI" By:BillWilson URL:http://cnx.org/content/m1003/2.13/ Pages:19-21 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"P-NJunction:PartII" Usedhereas:"PN-Junction:PartII" By:BillWilson URL:http://cnx.org/content/m1004/2.14/ Pages:21-24 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"Gauss'Law" By:BillWilson URL:http://cnx.org/content/m1005/2.15/ Pages:24-26 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0

PAGE 254

248 ATTRIBUTIONS Module:"DepletionWidth" By:BillWilson URL:http://cnx.org/content/m1006/2.16/ Pages:26-29 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"ForwardBiasedPNJunctions" Usedhereas:"ForwardBiased" By:BillWilson URL:http://cnx.org/content/m1007/2.19/ Pages:29-34 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"TheDiodeEquation" By:BillWilson URL:http://cnx.org/content/m1008/2.17/ Pages:34-38 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"ReverseBiased/Breakdown" By:BillWilson URL:http://cnx.org/content/m1009/2.11/ Pages:38-41 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"Diusion" By:BillWilson URL:http://cnx.org/content/m1010/2.14/ Pages:41-44 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"LightEmittingDiode" By:BillWilson URL:http://cnx.org/content/m1011/2.23/ Pages:44-48 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"LASER" By:BillWilson URL:http://cnx.org/content/m1012/2.16/ Pages:48-50 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0

PAGE 255

ATTRIBUTIONS 249 Module:"SolarCells" By:BillWilson URL:http://cnx.org/content/m1013/2.13/ Pages:51-54 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"IntroductiontoBipolarTransistors" Usedhereas:"IntrotoBipolarTransistors" By:BillWilson URL:http://cnx.org/content/m1014/2.15/ Pages:55-57 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"TransistorEquations" By:BillWilson URL:http://cnx.org/content/m1015/2.14/ Pages:57-58 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"TransistorI-VCharacteristics" By:BillWilson URL:http://cnx.org/content/m1016/2.15/ Pages:58-61 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"CommonEmitterModels" By:BillWilson URL:http://cnx.org/content/m1017/2.10/ Pages:61-62 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"SmallSignalModels" By:BillWilson URL:http://cnx.org/content/m1018/2.13/ Pages:62-66 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"SmallSignalModelforBipolarTransistor" By:BillWilson URL:http://cnx.org/content/m1019/2.12/ Pages:66-70 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0

PAGE 256

250 ATTRIBUTIONS Module:"IntroductiontoMOSFETs" By:BillWilson URL:http://cnx.org/content/m1020/2.12/ Pages:71-72 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"BasicMOSStructure" By:BillWilson URL:http://cnx.org/content/m1021/2.12/ Pages:72-77 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"ThresholdVoltage" By:BillWilson URL:http://cnx.org/content/m1022/2.15/ Pages:77-82 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"MOSTransistor" By:BillWilson URL:http://cnx.org/content/m1023/2.15/ Pages:82-84 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"MOSRegimes" By:BillWilson URL:http://cnx.org/content/m1024/2.11/ Pages:84-89 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"PlottingMOSI-V" By:BillWilson URL:http://cnx.org/content/m1025/2.13/ Pages:89-93 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"Models" By:BillWilson URL:http://cnx.org/content/m1026/2.11/ Pages:93-95 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"InvertersandLogic" By:BillWilson URL:http://cnx.org/content/m1027/2.12/ Pages:96-99 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0

PAGE 257

ATTRIBUTIONS 251 Module:"TransistorLoadsforInverters" By:BillWilson URL:http://cnx.org/content/m1028/2.10/ Pages:99-102 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"CMOSLogic" By:BillWilson URL:http://cnx.org/content/m1029/2.12/ Pages:102-107 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"JFET" By:BillWilson URL:http://cnx.org/content/m1030/2.12/ Pages:107-109 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"ElectrostaticDischargeandLatch-Up" By:BillWilson URL:http://cnx.org/content/m1031/2.15/ Pages:109-114 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"IntroductiontoICManufacturingTechnology" By:BillWilson URL:http://cnx.org/content/m1032/2.9/ Pages:115-117 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"SiliconGrowth" By:BillWilson URL:http://cnx.org/content/m1033/2.15/ Pages:117-118 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"Doping" By:BillWilson URL:http://cnx.org/content/m1034/2.11/ Pages:118-119 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"Fick'sFirstLaw" By:BillWilson URL:http://cnx.org/content/m1035/2.8/ Pages:120-121 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0

PAGE 258

252 ATTRIBUTIONS Module:"Fick'sSecondLaw" By:BillWilson URL:http://cnx.org/content/m1036/2.11/ Pages:121-123 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"Photolithography" By:BillWilson URL:http://cnx.org/content/m1037/2.10/ Pages:123-126 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"IntegratedCircuitWellandGateCreation" By:BillWilson URL:http://cnx.org/content/m1038/2.12/ Pages:127-132 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"ApplyingMetal/Sputtering" By:BillWilson URL:http://cnx.org/content/m1039/2.11/ Pages:132-134 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"IntegratedCircuitManufacturing:Bird'sEyeView" By:BillWilson URL:http://cnx.org/content/m1040/2.7/ Pages:134-137 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"DiusedResistor" By:BillWilson URL:http://cnx.org/content/m1041/2.9/ Pages:137-138 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"Yield" By:BillWilson URL:http://cnx.org/content/m1042/2.7/ Pages:138-140 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0

PAGE 259

ATTRIBUTIONS 253 Module:"DistributedParameters" Usedhereas:"IntroductiontoTransmissionLines:DistributedParameters" By:BillWilson URL:http://cnx.org/content/m1043/2.9/ Pages:143-146 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"Telegrapher'sEquations" By:BillWilson URL:http://cnx.org/content/m1044/2.12/ Pages:146-148 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"TransmissionLineEquation" By:BillWilson URL:http://cnx.org/content/m1045/2.11/ Pages:148-151 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"TransmissionLineExamples" By:BillWilson URL:http://cnx.org/content/m1046/2.10/ Pages:151-153 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"ExcitingaLine" By:BillWilson URL:http://cnx.org/content/m1047/2.9/ Pages:153-156 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"TerminatedLines" By:BillWilson URL:http://cnx.org/content/m1048/2.12/ Pages:156-159 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"BounceDiagrams" By:BillWilson URL:http://cnx.org/content/m1049/2.11/ Pages:159-169 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0

PAGE 260

254 ATTRIBUTIONS Module:"CascadedLines" By:BillWilson URL:http://cnx.org/content/m1050/2.11/ Pages:169-178 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"ReviewofPhasors" Usedhereas:"IntroductiontoPhasors" By:BillWilson URL:http://cnx.org/content/m1051/2.11/ Pages:181-183 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"A/CLineBehavior" By:BillWilson URL:http://cnx.org/content/m1052/2.14/ Pages:183-185 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"TerminatedLines" By:BillWilson URL:http://cnx.org/content/m1053/2.10/ Pages:186-188 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"LineImpedance" By:BillWilson URL:http://cnx.org/content/m1054/2.5/ Pages:189-189 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"CrankDiagram" By:BillWilson URL:http://cnx.org/content/m1055/2.11/ Pages:189-193 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"StandingWaves/VSWR" By:BillWilson URL:http://cnx.org/content/m1056/2.9/ Pages:193-195 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0

PAGE 261

ATTRIBUTIONS 255 Module:"BilinearTransform" By:BillWilson URL:http://cnx.org/content/m1057/2.13/ Pages:196-200 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"TheSmithChart" By:BillWilson URL:http://cnx.org/content/m1058/2.11/ Pages:200-211 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"IntroductiontoUsingtheSmithChart" By:BillWilson URL:http://cnx.org/content/m1059/2.11/ Pages:211-212 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"SimpleCalculationswiththeSmithChart" By:BillWilson URL:http://cnx.org/content/m1060/2.17/ Pages:212-216 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"Power" By:BillWilson URL:http://cnx.org/content/m1061/2.10/ Pages:216-218 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"FindingtheLoadImpedance" Usedhereas:"FindingZL" By:BillWilson URL:http://cnx.org/content/m1062/2.15/ Pages:218-222 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"Matching" By:BillWilson URL:http://cnx.org/content/m1063/2.12/ Pages:222-224 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0

PAGE 262

256 ATTRIBUTIONS Module:"IntroductiontoParallelMatching" By:BillWilson URL:http://cnx.org/content/m1064/2.14/ Pages:225-226 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"SingleStubMatching" By:BillWilson URL:http://cnx.org/content/m1065/2.13/ Pages:226-230 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"DoubleStubMatching" By:BillWilson URL:http://cnx.org/content/m1066/2.14/ Pages:230-237 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0 Module:"OddsandEnds" By:BillWilson URL:http://cnx.org/content/m1067/2.11/ Pages:237-241 Copyright:BillWilson License:http://creativecommons.org/licenses/by/1.0

PAGE 263

IntroductiontoPhysicalElectronics Anintroductiontosolidstatedeviceincludingeldeectandbipolartransistors.Propertiesoftransmission linesandpropagatingE&Mwaves. AboutConnexions Since1999,Connexionshasbeenpioneeringaglobalsystemwhereanyonecancreatecoursematerialsand makethemfullyaccessibleandeasilyreusablefreeofcharge.WeareaWeb-basedauthoring,teachingand learningenvironmentopentoanyoneinterestedineducation,includingstudents,teachers,professorsand lifelonglearners.Weconnectideasandfacilitateeducationalcommunities. Connexions'smodular,interactivecoursesareinuseworldwidebyuniversities,communitycolleges,K-12 schools,distancelearners,andlifelonglearners.Connexionsmaterialsareinmanylanguages,including English,Spanish,Chinese,Japanese,Italian,Vietnamese,French,Portuguese,andThai.Connexionsispart ofanexcitingnewinformationdistributionsystemthatallowsfor PrintonDemandBooks .Connexions haspartneredwithinnovativeon-demandpublisherQOOPtoacceleratethedeliveryofprintedcourse materialsandtextbooksintoclassroomsworldwideatlowerpricesthantraditionalacademicpublishers.