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In this lecture, we will learn about resistors and resistor networks, and how to simply them.

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Aresistorischaracterisedbyanumberofparameters:1.  Itsnominalvalue;

2.  Itstoleranceoraccuracy(e.g.±5%);3.  Itspowerrating(i.e.maximumpowerthatitcandissipate);

4.  Itstemperaturecoefficient(howmuchtheresistancevarywithtemperature);

5.  Itsstability(i.e.howmuchitchangesovertime);

6.  Itsselfinductance(somethingwedon’tworryaboutunlessyouareusingresistorsatveryveryhighfrequencies).

Thesecharacteristicsareoftenshownontheresistoritselfasacolourcode.

Thecolourcodeisasshownabove.(Theprintednotesarenotincolour.YoucandownloadthePDFfilefromthecoursewebpage,whichwillbeshowninfullgloriouscolours.)Considerthetopresistor.Ithasfourbands,andthebandcoloursare:

RED,GREEN,ORANGE,agap,BROWN

Thefirsttwocolourbandsarethefirsttwodigitsoftheresistance,i.e.RED=2,GREEN=5.Thethirdbandinthiscaseisthemultiplier.ORANGE=103or1k.Thegapisalwaystheretoseparatevaluebandsfromtoleranceband.BROWN=±5%.

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Sinceresistorshavetolerances,itisnotnecessarynormalevensensibletoprovideresistorsofALLvalues.Letussupposeyouhavea1kΩresistorwithatoleranceof10%.Thisresistorcouldvaryfrom900Ω to1.1kΩinvalue.Youwanttoguaranteethatanotherresistorwithlowernominalvalueisalwayslowerinresistance.Thereforeitdoesnotmakesensetoprovideanyresistancewithavalueabove820Ω,say850Ω. Thisisbecause850Ωat10%wouldgiveyouarangeof765Ωto935Ω, whichwouldbehigherthanthelowestvalueofthe1kresistor!

Thereforeinindustry,onlyselectedvalues(knownasPreferredValues)ofresistorsaremade,dependentonthetolerance.Shownherearethe±20%,±10%and±5%resistorsvaluesinadecaderange.TheyarecalledE6,E12andE24respectivelybecausethereare6,12and24valuesineachdecade(similartomusicalnodes).GOODENGINNERINGPRACTICE:youcanseethatsinceinengineeringdesign,wealwayshavetoconsidertolerance,andeventhehumbleresistoronlyexistsindefinedvalues,itdoesnotmakesensetouseprecisioninyoursolutionshavingmanydigits.Inourlaboratory,wewillbemostlyusingtheE24seriesofresistorsat±5%tolerance.

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Herearethecommonquantitiesusedinelectricalengineering,theirunitsandsymbolicrepresentationsfortheunits.Furthermore,wedonotgenerallyusealldecadesformultipliers(sayofresistors),butthemultipliersareinstepsofTHREEdecade.

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Circuitscanbesimplifiedbycombiningcomponents.Inthetopcircuit,resistersR1,R2andR3areconnectedinseries.ItcanbereplacedbyONEresistorRT=R1+R2+R3.withseriesresistors,theSAMECURRENTflowsthroughallofthem.Notethatinthiscircuit,R3andR4areNOTinseriesbecausetheymaynothavethesamecurrent.Resistorscanbeconnectedinparallel.R1,R2andR3sharethesametwonodes,andthereforetheyhavetheSAMEVOLTAGEacrossthem.R4andR5arealsoinparallel,butonlytoeachother.

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Seriesconnectedresistorshavethesamecurrentthroughthem.SinceV=IR(Ohm’slaw),theresistorsDIVIDEthevoltageinthesameratioastheresistances.ConsiderthevoltageV1,thesimplemathematicsshownhereshowsthatV1/Vx=theratioofR1tothetotalresistanceRT.ThereforeVxisdividedintoV1:V2:V3,byR1:R2:R3.ThehighertheresistanceRx,thehigherthevoltageacrossit.Thisiscalledavoltagedivider.

ItisverycommontouseTWOresistorsshowinthelowerdiagramtoproduceasmallervoltageVYbydividingVXasshown.IftheoutputcurrentIYiszero(thereforenoloadconnecttoVY),thenthevoltagedividerisexact.IfIYisnotzero,aslongasIYismuchsmallertheI,thecurrentthroughtheseriesresistordividercircuit,thevoltagedividerequationisanapproximation.

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Whileseriesconnectedresistorsdivideavoltage,parallelconnectedresistorsdividecurrent.ThecurrentIXisdividedamountthethreeresistorsR1,R2andR3intheratiooftheirCONDUCTANCE(i.e.1/resistance).Thehighertheconductance,thehigherthecurrentthroughthatresistor.Notethatseriesconnectedresistorshaveatotalresistance=sumofallresistancesthatareinseries.Similarforparallelresistors,totalconductance=sumofallconductancesthatareinparallel.

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Wecanreplaceseriesconnectedresistorswithanequivalentresistancetosimplifythecircuit.ThetopcircuithasthreeresistorsR1,R2andR3.ThiscanbereplacedbyoneresistorwithRT=sumofallthreeresistances.TheV-Irelationshipforthetwoversionsofthecircuitareidentical.However,theindividualvoltagesV1,V2andV3acrosseachoftheresistorsarenolongeraccessible.Thecircuitatthetoprightofslide5canthereforebesimplifiedtotheoneshownhere.

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Similarlywecanperformthesamesimplificationwithparallelconnectedresistors.Inthecircuitshownbelow,R1,R2andR2arecombinedtoRP.WecanfurthercombineR4andR5becausetheytooareconnectedinparallel.

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Weoftenwriteparallelconnectedresistorsusingthedoubleverticalbarsymbol.ForthespecialcaseofTWOparallelresistors,theequivalentresistanceisR1*R2/(R1+R2),orproductdividedbythesum.Thisformulaeiswellworthmemorizing–youwillusethisequationoften.Alwaysremember:ifyouhavearesistorR1,andthenconnectaresistorR2inparallelwithR1,theequivalentresistanceisALWAYSSMALLERthanbothR1andR2.

Inotherwords,addingaresistorinseriestoanexistingresistorwillINCREASEthetotalresistance;addingaresistorinparalleltoanexistingresistorwillREDUCEthetotalresistance.

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Hereareafewexamplesofsimplifyingresistornetworks.

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Hereisanotherexampleofavoltagedivider.Itdividesthe9Vbatteryvoltagesourceintotwoparts:lowervoltageV2=6V,andtheuppervoltageV1=3V.

Notthatinthiscase,thediagramalsoshowsthepowerdissipatedbyeachresistors.Ascanbeseen,thepowerdissipatedbythesetworesistorsaremuchlowerthanratedvalueof250mW(thesametypeofresistorsthatyouwillbeusingintheHomeLabKit).

Alsoremember:ifyoukeepresistorsinkW,currentinmA,voltageinV,theneverythingworksoutfinewithOhm’sLaw.Inthiscase,IxR=3x2=6V.

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InTopic2,weconsideredideavoltageDC(directcurrent)sourcessuchasthatfoundinanidealbattery.TheI-Vcharacteristicisaverticallinebecausenomatterwhatcurrentyoudrawfromthebattery,thevoltageremainsconstant.Inpractice,thisdoesnothappen.Inarealbattery,ifyouincreasethecurrentyoudrawfromit,thebatteryvoltagedecreases.ThisisthesameashavingaresistorRBinserieswithanidealvoltagesource.TheI-Vplotnowisalinewithaslightpositiveslope.Asyoudrawsmorecurrentfromthebattery,thecurrentisincreasinglynegative(becausecurrentisflowingOUTofthebatteryandisintheoppositedirectionofthearrowshownhere),thevoltageVdecreases.RBisoftencalledinternalresistanceofthebattery.Astemperaturedrops,RBincreases.That’swhywethatcarsarehardertostartinthewinterbecausethecarbatteryhasincreasedinternalresistance.

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