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13 Magnetically CoupledCircuits Ifyouwouldincreaseyourhappinessandprolongyourlife,forgetyourneighbor’sfaults.. . Forget the peculiarities of your friends, and only remember the good points which mak youfondofthem....Obliterateeverythingdisagreeablefromyesterday;writeupontoday cleansheetthosethingslovelyandlovable. —Anonymous

EnhancingYourCareer

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CareerinElectromagnetics Electromagnetics(EM)isthebranchofelectricalengineering(orphysics)thatdealswiththe analysis and application of electric and magnetic fields. In electromagnetics, electric circui analysisisappliedatlowfrequencies.

Telemetryreceivingstationforspacesatellites.©DV169/GettyImagesRF

TheprinciplesofEMareappliedinvariousallieddisciplines,suchaselectricmachines electromechanical energy conversion, radar meteorology, remote sensing, satellit Alexander, Charles K., and Matthew N. O. Sadiku. Fundamentals of Electric Circuits, McGraw-Hill Higher Education, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unsw/detail.action?docID=5471298. Created from unsw on 2020-04-05 16:43:44.

communications, bioelectromagnetics, electromagnetic interference and compatibility plasmas, and fiber optics. EM devices include electric motors and generators, transformers electromagnets, magnetic levitation, antennas, radars, microwave ovens, microwave dishes superconductors, and electrocardiograms. The design of these devices requires a thoroug knowledgeofthelawsandprinciplesofEM. EM is regarded as one of the more difficult disciplines in electrical engineering. On reasonisthatEMphenomenaareratherabstract.Butifoneenjoysworkingwithmathematic andcanvisualizetheinvisible,oneshouldconsiderbeingaspecialistinEM,inasmuchasfew electrical engineers specialize in this area. Electrical engineers who specialize in EM ar needed in microwave industries, radio/TV broadcasting stations, electromagnetic researc laboratories,andseveralcommunicationsindustries. 554

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Historical

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James Clerk Maxwell (1831–1879), a graduate in mathematics from Cambridge University,in1865wroteamostremarkable paperinwhich hemathematically unified the lawsofFaradayandAmpere.Thisrelationshipbetweentheelectricfieldandmagneticfield served as the basis for what was later called electromagnetic fields and waves, a major fieldofstudyinelectricalengineering.TheInstituteofElectricalandElectronicsEngineers (IEEE)usesagraphicalrepresentationofthisprincipleinitslogo,inwhichastraightarrow representscurrentandacurvedarrowrepresentstheelectromagneticfield.Thisrelationship is commonly known as the right-hand rule. Maxwell was a very active theoretician and scientist. He is best known for the “Maxwell equations.” The maxwell, a unit of magnetic flux,wasnamedafterhim.

Alexander, Charles K., and Matthew N. O. Sadiku. Fundamentals of Electric Circuits, McGraw-Hill Higher Education, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unsw/detail.action?docID=5471298. Created from unsw on 2020-04-05 16:43:44.

LearningObjectives Byusingtheinformationandexercisesinthischapteryouwillbeableto: 1. Understand the physics behind mutually coupled circuits and how to analyze circuits containingmutuallycoupledinductors. 2. Understandhowenergyisstoredinmutuallycoupledcircuits. 3. Understandhowlineartransformersworkandhowtoanalyzecircuitscontainingthem. 4. Understandhowidealtransformersworkandhowtoanalyzecircuitscontainingthem. 5. Understand how ideal auto transformers work and know how to analyze them when usedinavarietyofcircuits.

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13.1

Introduction

Thecircuitswehaveconsideredsofarmayberegardedas conductivelycoupled,becauseon loopaffectstheneighboringloopthroughcurrentconduction.Whentwoloopswithorwithou contactsbetweenthem affecteachotherthroughthe magneticfieldgeneratedby oneofthem theyaresaidtobemagneticallycoupled. Thetransformerisan electricaldevicedesigned onthebasis ofthe conceptof magnetic coupling. It uses magnetically coupled coils to transfer energy from one circuit to another Transformers are key circuit elements. They are used in power systems for stepping up o steppingdownacvoltagesorcurrents.Theyareusedinelectroniccircuitssuchasradioand television receiversforsuch purposes asimpedance matching,isolatingone part of a circui fromanother,andagainforsteppingupordownacvoltagesandcurrents. Wewillbeginwiththeconceptofmutualinductanceandintroducethedotconventionused fordeterminingthevoltagepolaritiesofinductivelycoupledcomponents.Basedonthenotion ofmutualinductance, 555 we thenintroduce the circuit elementknownasthe transformer. We will consider the linea transformer, the ideal transformer, the idealautotransformer,andthe three-phase transformer Finally,amongtheirimportantapplications,welookattransformersasisolatingandmatching devicesandtheiruseinpowerdistribution.

13.2

MutualInductance

Whentwoinductors(orcoils)areinacloseproximitytoeachother,themagneticfluxcaused by current in one coil links with the other coil, thereby inducing voltage in the latter. Thi phenomenonisknownasmutualinductance. Alexander, Charles K., and Matthew N. O. Sadiku. Fundamentals of Electric Circuits, McGraw-Hill Higher Education, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unsw/detail.action?docID=5471298. Created from unsw on 2020-04-05 16:43:44.

Figure13.1 MagneticfluxproducedbyasinglecoilwithNturns.

Letusfirstconsiderasingleinductor,acoilwithNturns.Whencurrentiflowsthroughthe coil, a magnetic flux ϕ is produced around it (Fig. 13.1). According to Faraday’s law, th voltage v induced in the coil is proportional to the number of turns N and the time rate o changeofthemagneticfluxϕ;thatis, (13.1)

But the flux ϕ is produced by current i so that any change in ϕ is caused by a change in th current.Hence,Eq.(13.1)canbewrittenas (13.2)

or (13.3)

which is the voltage-current relationship for the inductor. From Eqs. (13.2) and (13.3), th inductanceLoftheinductoristhusgivenby (13.4)

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Thisinductanceiscommonlycalled self-inductance,becauseitrelatesthevoltageinducedi acoilbyatime-varyingcurrentinthesamecoil.

Figure13.2 MutualinductanceM 21ofcoil2withrespecttocoil1.

Nowconsidertwocoilswithself-inductancesL1andL2thatare in closeproximitywith eachother(Fig.13.2).Coil1hasN1turns,whilecoil2hasN2turns.Forthesakeofsimplicity assumethatthesecondinductorcarriesnocurrent.Themagneticfluxϕ1emanatingfromcoil has two components: One component ϕ11 links only coil 1, and another component ϕ12 link bothcoils.Hence,

Alexander, Charles K., and Matthew N. O. Sadiku. Fundamentals of Electric Circuits, McGraw-Hill Higher Education, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unsw/detail.action?docID=5471298. Created from unsw on 2020-04-05 16:43:44.

ϕ1=ϕ11+ϕ12

(13.5)

Although the two coils are physically separated, they are said to be magnetically coupled Sincetheentirefluxϕ1linkscoil1,thevoltageinducedincoil1is (13.6)

Onlyfluxϕ12linkscoil2,sothevoltageinducedincoil2is (13.7)

556 Again,asthefluxesarecausedbythecurrenti1flowingincoil1,Eq.(13.6)canbewrittenas (13.8)

whereL1=N1dϕ1/di1istheself-inductanceofcoil1.Similarly,Eq.(13.7)canbewrittenas (13.9)

where

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(13.10)

M21isknownasthe mutualinductance ofcoil2withrespecttocoil1.Subscript21indicate thattheinductanceM21relatesthevoltageinducedincoil2tothecurrentincoil1.Thus,th open-circuitmutualvoltage(orinducedvoltage)acrosscoil2is

(13.11)

Supposewenowletcurrent i2flowincoil2,whilecoil1carriesnocurrent(Fig. 13.3) Themagneticfluxϕ2emanatingfromcoil2comprisesfluxϕ22that linksonly coil2andflux ϕ21thatlinksbothcoils.Hence, ϕ2=ϕ21+ϕ22

(13.12)

Theentirefluxϕ2linkscoil2,sothevoltageinducedincoil2is (13.13)

Alexander, Charles K., and Matthew N. O. Sadiku. Fundamentals of Electric Circuits, McGraw-Hill Higher Education, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unsw/detail.action?docID=5471298. Created from unsw on 2020-04-05 16:43:44.

where L2 = N2 dϕ2/di2 is the self-inductance of coil 2. Since only flux ϕ21 links coil 1, th voltageinducedincoil1is (13.14)

where (13.15)

whichisthemutualinductanceofcoil1withrespecttocoil2.Thus,theopen-circuitmutua voltageacrosscoil1is

(13.16)

WewillseeinthenextsectionthatM12andM21areequal;thatis, M12=M21=M

(13.17)

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and we refer to M as the mutual inductance between the two coils. Like self-inductance L mutualinductance Mismeasuredinhenrys(H).Keepinmindthatmutualcouplingonlyexist whentheinductorsorcoilsareincloseproximity,andthecircuitsaredrivenbytime-varyin sources.Werecallthatinductorsactlikeshortcircuitstodc.

Figure13.3 MutualinductanceM 12ofcoil1withrespecttocoil2.

FromthetwocasesinFigs.13.2and 13.3,weconcludethatmutualinductanceresultsif voltageisinducedbyatime-varyingcurrentinanothercircuit.Itisthepropertyofaninducto toproduceavoltageinreactiontoatime-varyingcurrentinanotherinductornearit.Thus, 557 Mutualinductanceistheabilityofoneinductortoinduceavoltageacrossaneighboringinductor,measuredin henrys(H).

Although mutual inductance M is always a positive quantity, the mutual voltage M di/d may be negative or positive, just like the self-induced voltage Ldi/dt. However, unlike th Alexander, Charles K., and Matthew N. O. Sadiku. Fundamentals of Electric Circuits, McGraw-Hill Higher Education, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unsw/detail.action?docID=5471298. Created from unsw on 2020-04-05 16:43:44.

self-inducedLdi/dt,whosepolarityisdeterminedbythereferencedirectionofthecurrentand thereferencepolarityofthevoltage(accordingtothepassivesignconvention),thepolarityo mutual voltage M di/dt is not easy to determine, because four terminals are involved. Th choice of the correct polarity for M di/dt is made by examining the orientation or particula wayinwhichbothcoilsarephysicallywoundandapplyingLenz’slawinconjunctionwithth right-hand rule.Since it is inconvenienttoshowthe construction detailsofcoilson a circui schematic,weapplythe dotconventionincircuitanalysis.Bythisconvention,adotisplace inthecircuitatoneendofeachofthetwomagneticallycoupledcoilstoindicatethedirectio ofthemagneticfluxifcurrententersthatdottedterminalofthecoil.ThisisillustratedinFig 13.4.Givenacircuit,thedotsarealreadyplacedbesidethecoilssothatweneednotbothe about how to place them. The dots are used along with the dot convention to determine th polarityofthemutualvoltage.Thedotconventionisstatedasfollows: Ifacurrententersthedottedterminalofonecoil,thereferencepolarityofthemutualvoltageinthesecondcoilis positiveatthedottedterminalofthesecondcoil.

Alternatively, Ifacurrentleavesthedottedterminalofonecoil,thereferencepolarityofthemutualvoltageinthesecondcoilis negativeatthedottedterminalofthesecondcoil.

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Thus, the reference polarity of the mutual voltage depends on the reference direction of th inducing current and the dots on the coupled coils. Application of the dot convention i illustratedinthefourpairsofmutuallycoupledcoilsinFig.13.5.ForthecoupledcoilsinFig 13.5(a),thesignofthemutualvoltagev2isdeterminedbythereferencepolarityfor v2andth directionofi1.Sincei1enters

Figure13.4 Illustrationofthedotconvention.

Alexander, Charles K., and Matthew N. O. Sadiku. Fundamentals of Electric Circuits, McGraw-Hill Higher Education, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unsw/detail.action?docID=5471298. Created from unsw on 2020-04-05 16:43:44.

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Figure13.5 Examplesillustratinghowtoapplythedotconvention.

558 the dotted terminal of coil 1 and v2 is positive at the dotted terminal of coil 2, the mutua voltageis+Mdi1/dt.ForthecoilsinFig.13.5(b),thecurrenti1entersthe dotted terminal o coil 1 and v2 is negative at the dotted terminal of coil 2. Hence, the mutual voltage is −M di1/dt.ThesamereasoningappliestothecoilsinFigs.13.5(c)and13.5(d).

Alexander, Charles K., and Matthew N. O. Sadiku. Fundamentals of Electric Circuits, McGraw-Hill Higher Education, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unsw/detail.action?docID=5471298. Created from unsw on 2020-04-05 16:43:44.

Figure13.6 Dotconventionforcoilsinseries;thesignindicatesthepolarityofthemutualvoltage:(a)series-aidingconnection,(b)seriesopposingconnection.

Figure 13.6 shows the dot convention for coupled coils in series. For the coils in Fig 13.6(a),thetotalinductanceis

L=L1+L2+2M

  (Series-aidingconnection)

ForthecoilsinFig.13.6(b),

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L=L1+L2−2M

  (Series-opposingconnection)

Now that we know how to determine the polarity of the mutual voltage, we are prepared to analyzecircuitsinvolvingmutualinductance.Asthefirstexample,considerthecircuitinFig 13.7(a).ApplyingKVLtocoil1gives (13.20a)

Forcoil2,KVLgives (13.20b)

WecanwriteEq.(13.20)inthefrequencydomainas Alexander, Charles K., and Matthew N. O. Sadiku. Fundamentals of Electric Circuits, McGraw-Hill Higher Education, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unsw/detail.action?docID=5471298. Created from unsw on 2020-04-05 16:43:44.

V1=(R1+jωL1)I1+jωMI2

(13.21a)

V2=jωMI1+(R2+jωL2)I2

(13.21b)

As a second example, consider the circuit in Fig. 13.7(b). We analyze this in the frequenc domain.ApplyingKVLtocoil1,weget V=(Z1+jωL1)I1−jωMI2

(13.22a)

0=−jωMI1+(ZL+jωL2)I2

(13.22b)

Forcoil2,KVLyields

Equations(13.21)and(13.22)aresolvedintheusualmannertodeterminethecurrents. One of themostimportant things inmakingsureone solvesproblemsaccuratelyistobe able to check each step during the solution process and to make sure assumptions can b verified.Toooften,solvingmutuallycoupledcircuitsrequirestheproblemsolvertotracktwo ormorestepsmadeatonceregardingthesignandvaluesofthemutuallyinducedvoltages.

Figure13.7 Time-domainanalysisofacircuitcontainingcoupledcoils(a)andfrequency-domainanalysisofacircuitcontainingcoupledcoils (b).

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559

Figure13.8 Modelthatmakesanalysisofmutuallycoupledeasiertosolve.

Experiencehasshownthatifwebreaktheproblemintostepsofsolvingforthevalueand the sign into separate steps, the decisions made are easier to track. We suggest that mode (Figure13.8(b))beusedwhenanalyzingcircuitscontainingamutuallycoupledcircuitshown inFigure13.8(a): Noticethatwehavenotincludedthesignsinthemodel.Thereasonforthatisthatwefirs

Alexander, Charles K., and Matthew N. O. Sadiku. Fundamentals of Electric Circuits, McGraw-Hill Higher Education, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unsw/detail.action?docID=5471298. Created from unsw on 2020-04-05 16:43:44.

determine the value of the induced voltages and then we determine the appropriate signs Clearly, I1 induces a voltage within the second coil represented by the value jωI1 and I inducesavoltageof jωI2inthefirstcoil.Oncewehavethevalueswenextusebothcircuitst findthecorrectsignsforthedependentsourcesasshowninFigure13.8(c). SinceI1entersL1atthedottedend,itinducesavoltageinL2thattriestoforceacurren outofthedottedendofL2whichmeansthatthesourcemusthaveaplusontopandaminuso the bottom as shown in Figure 13.8(c). I2 leaves the dotted end of L2 which means that induces a voltage in L1 which tries to force a current into the dotted end of L1 requiring dependentsourcethathasaplusonthebottomandaminusontopasshowninFigure13.8(c) Nowallwehavetodoistoanalyzeacircuitwithtwodependentsources.Thisprocessallow youtocheckeachofyourassumptions. At this introductory level we are not concerned with the determination of the mutua inductances of the coils and their dotplacements. Like R, L,and C, calculation of M would involveapplyingthetheoryofelectromagneticstotheactualphysicalpropertiesofthecoils.In thistext,weassumethatthemutualinductanceandtheplacementofthedotsarethe“givens’ ofthecircuitproblem,likethecircuitcomponentsR,L,andC. 

Example13.1

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CalculatethephasorcurrentsI1andI2inthecircuitofFig.13.9.

Figure13.9 ForExample13.1.

Solution: Forloop1,KVLgives −12+(−j4+j5)I1−j3I2=0 560 or jI1−j3I2=12 Forloop2,KVLgives −j3I1+(12+j6)I2=0

Alexander, Charles K., and Matthew N. O. Sadiku. Fundamentals of Electric Circuits, McGraw-Hill Higher Education, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/unsw/detail.action?docID=5471298. Created from unsw on 2020-04-05 16:43:44.

(13.1.1)

or (13.1.2)

SubstitutingthisinEq.(13.1.1),weget (j2+4−j3)I2=(4−j)I2=12 or  

(13.1.3)

FromEqs.(1...